Encyclopedia of Condensed Matter Physics [1] 0323908004, 9780323908009

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ENCYCLOPEDIA OF CONDENSED MATTER PHYSICS SECOND EDITION

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ENCYCLOPEDIA OF CONDENSED MATTER PHYSICS SECOND EDITION

EDITOR-IN-CHIEF Tapash Chakraborty⁎ Professor, Tier-I Canada Research Chair, University of Manitoba, Winnipeg, MB, Canada VOLUME 1

⁎

Retired.

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge MA 02139, United States Copyright © 2024 Elsevier Ltd. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers may always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

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EDITOR-IN-CHIEF BIOGRAPHY Tapash Chakraborty is a retired professor of physics from the University of Manitoba, Canada, where he was also the Canada Research Chair in Nanoscale Physics (2003–17). His research focus has been on many-body theories of various quantum materials, such as nuclear matter, helium liquids, and electron-hole liquids, that he studied under the tutelage of late Professor Manfred Ristig, at the University of Köln (1979–81). Just a year after the discovery of the fractional quantum Hall effect, he (working with his coworkers) demonstrated that electron–electron interactions could actually lead to quantum Hall states with nontrivial spin configurations, and that the low-lying excitations could be described as “spin-reversed quasiparticles.” This prediction motivated much experimental work from laboratories around the world, as a result of which the concept of spin-reversed quasi-particles was established. Similarly, his work on the quantum Hall states in double layer systems has received much attention from the community. Furthermore, Dr. Chakraborty (working with Dr. P. Maksym and others) made pioneering contributions to the many-electron theory of quantum dots. Dr. Chakraborty has also contributed to the theory of electronic properties of various nanoscale systems, such as spin-orbit coupled quantum dots, quantum rings, and single- and double-layer graphene (primarily with Dr. V. Apalkov). He has authored many books and reviews: the book with Dr. Pietiläinen on the quantum Hall effect and the single-authored book on quantum dots have become standard references in their respective fields.

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EDITORIAL ADVISORY BOARD Ana M. Sanchez Department of Physics, University of Warwick, Coventry, United Kingdom Hideo Aoki Department of Physics, University of Tokyo, Tokyo, Japan Robert H. Blick Center for Hybrid Nanostructures, University of Hamburg & DESY, Hamburg, Germany Roberto Raimondi Mathematics and Physics Department, Roma Tre University, Rome, Italy Rudolf A. Roemer Department of Physics, University of Warwick, Coventry, United Kingdom Vladimir Mihajlovic Fomin Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany

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INTRODUCTION Electrons in orbits contained by the quantum, And atoms conjoined by the dipolar force, Gravity balanced by pressure And flap of a wing. Here, I remember the angles and curves, Calculus, I learned it all – Each bird bit and moon fracture Fixed by the symmetries, Airborne and airless, aloft. – Alan Lightman: Song of Two Worlds

Condensed matter physics (CMP) is the largest discipline in physics, involving a wide variety of interesting concepts and challenges. To quote Leggett (1992), CMP embraces topics “as diverse as traditional solid-state physics, neutron stars and the physics of biological matter.” He further added that, by a liberal definition “just about all of physics outside atomic, nuclear and particle physics and cosmology” fall in this vast enterprise1. One of the primary goals of CMP is to explore the fundamental properties of matter—solids, liquids or gasses—comprising a large number of interacting constituent particles (Laughlin and Pines, 2000). The quantum-mechanical nature of these many-body systems makes condensed-matter highly nontrivial, sometimes counterintuitive and even astonishing. Superconductivity and the fractional quantum Hall effect (FQHE) naturally fall within that category. In fact, for the past four decades, the quantum Hall effects (QHEs) have been the epitome of elegant CMP (von Klitzing et al., 2020). Our fundamental system of units has been redefined following the discovery of the QHE when the “von Klitzing constant” RK ¼ 25, 812.8074593045 O came into being (von Klitzing, 2019; Nawrocki, 2019). Observation of the QHE in graphene2 was crucial in determining the Dirac nature of charge carriers there (see for example, Jiang et al., 2007). Another effect that outwardly resembles the QHE but is conceptually quite different is the FQHE, which is a quintessential manifestation of electron correlations governed by the Coulomb interaction (Störmer, 1992; Eisenstein and Stormer, 1990). To explain this fascinating effect, Laughlin (1983) introduced his eponymous wave function that not only provided the foundation for understanding the effect but also triggered a new wave of ideas with impact far beyond the realm of CMP3. Who would have imagined that a large collection of interacting electrons confined on a plane in a perpendicular magnetic field would result in elementary excitations (quasiholes and quasiparticles) (Chakraborty and Pietiläinen, 1988, 1995; Halperin, 1983; Clark and Maksym, 1989) carrying fractional electron charge and obeying the exotic exchange statistics of anyons (Halperin, 1984; Arovas et al., 1984)? These quasiparticles and quasiholes of the Laughlin state still remain an enigma (Laughlin, 1984; Chakraborty, 1985; Morf and Halperin, 1986). Fractional statistics is a truly pathbreaking concept in CMP that was introduced by Leinaas and Myrheim (1977), and has been recently confirmed experimentally (Bartolomei et al., 2020; Nakamura et al., 2020). Much of the history of CMP can be viewed as a series of paradigm shifts4. The field evolved rapidly through frequent ground-breaking discoveries that challenged existing paradigms and opened up new applications, and this evolution is expected to continue (Leggett, 2018; Cohen, 2008). For example, our current understanding of 1 Interestingly, some similarities do exist between the mathematical frameworks that describe low-temperature CMP and early-universe cosmology. See Kibble and Srivastava (2013). 2

An isolated single layer of graphite consisting of carbon atoms bonded together in a hexagonal structure. See, e.g., Abergel et al. (2010).

3

For example, an important aspect of Laughlin's theory is its seamless concinnity of classical plasma physics and the planar electron gas that has inspired others to use a similar approach to quark-gluon plasma (Lu, 2022). 4

A detailed exposition of the meaning and examples of “paradigm shifts” (Kuhn-type) can be found in Leggett (2018).

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the FQHE required deep mathematical analysis and a wide range of experimental exploration, which revealed a treasure trove of unique phenomena that are yet to be fully understood or explained. Other studies of electron correlations in low spatial dimensions have ushered in many groundbreaking notions, such as Pfaffian states, Majorana fermions in condensed matter, Chern insulators, and topological quantum materials that promise a fault tolerant means to perform quantum computation. The presence of incompressible FQHE states in monolayer graphene (Apalkov and Chakraborty, 2006), bilayer graphene (Apalkov and Chakraborty, 2010, 2011), and double-layer graphene (Liu et al., 2019) has demonstrated the robustness of the effect, and provides invaluable insight into the nature of this effect. The dynamics of an electron in a periodic potential subjected to a perpendicular magnetic field has been an intriguing problem for more than half a century (Obermair and Wannier, 1976; Osadchy and Avron, 2001)5. Within the nearest neighbor tight-binding description of the periodic potential the electron energy spectrum is described by the Harper equation (Harper, 1955). Numerical solution of this equation (Hofstadter, 1976) has revealed that the applied field splits the Bloch bands into subbands and gaps. As the field is varied, the splitting of band continues indefinitely, leading to bands within bands within bands. When plotted as a function of the magnetic flux per lattice cell, the energy spectrum depicts a fractal pattern (a self-similar pattern that repeats at every scale) and is known in the literature as Hofstadter’s butterfly because of the resemblance of the pattern to butterflies6. Observation of the butterfly spectra in real systems is an arduous task. For example, in ordinary crystalline lattices, the required magnetic field to see the butterflies exceeds 1000 Tesla, which is far beyond the reach of present-day technology. Graphene seems to be the ideal system in the quest of those fractal butterflies (Weiss, 2013). It is interesting to note that a mathematical solution of the Harper equation, in contrast to the numerical solution mentioned above, remained a challenging problem (the so-called Ten Martini Problem) for mathematicians until it was solved (Avila and Jitomirskaya, 2006) in 2006. Magnetism has been known to mankind since ancient times, and it remains a major topic in CMP. The road to understanding magnetism has been long and tortuous. At various times in history, magnets were claimed to have the healing power to cure a myriad of ailments that included epilepsy, arthritis, gout, and baldness. Magnets were the prime ingredients for the “elixir of life” by Paracelsus (1493–1541). Then there was the healing process of “animal magnetism” proposed by Franz Anton Mesmer (1734–1815) (mesmerism)7, which was famously mocked in Mozart’s Così fan tutte (Steptoe, 1986). Our current understanding of magnetism is deeply rooted in quantum mechanics as Van Vleck explained (Van Vleck, 1978): “Quantum mechanics is the key to understand magnetism. When one enters the first room with this key there are unexpected rooms beyond, but it is always the master key that unlocks each door.” For centuries, the magnetism of certain rocks and metallic iron was only used for the practical purposes of direction finding, particularly navigation with a compass. Motors, generators, and electrical distribution networks appeared only in the 19th century after the discovery of the deep connection between electricity and magnetism. Magnetic order is a collective phenomenon akin to superconductivity and superfluidity that involves a very large number of particles. The rich variety of magnetically ordered states in solids, not only ferromagnetism but also antiferromagnetism, ferrimagnetism, and a range of noncollinear structures, was confirmed only in the middle of the 20th century, with the availability of neutron beams in research reactors. A range of new magnetic materials are taking their place as active layers in thin film sensors and memory devices (Baltz et al., 2018). Besides information storage (Blundell, 2001), important technical applications include magnetic resonance imaging (MRI) and the use of magnetic nanoparticles in medicine. Many outstanding discoveries in magnetism and magnetic materials have resulted in rapid advances in information storage technology. Over the last two decades spintronics has emerged as an important field of research both from the point of view of fundamental science and advanced technological applications (Yamaguchi et al., 2021; Pinarbasi and Kent, 2022). The giant magnetoresistance (GMR) effect, the spin Hall effect, and the spin galvanic effects are typical examples of developments in the field. Spintronics aims at understanding how the spin degree of freedom of the electrical carriers can be controlled in order to obtain devices with new functionalities and better performance. This endeavor often combines concepts from different aspects of condensed matter systems, giving the topic a fascinating interdisciplinary flavor. Spintronic phenomena are now being studied in an impressive range of different materials ranging from metals to half-metals to semiconductors, graphene and other two-dimensional systems such as transition metal dichalcogenides. 5

For a recent review on butterflies in graphene, see for example Chakraborty and Apalkov (2015).

6

However, as one of the authors of this Encyclopedia has (somewhat humorously) pointed out, this butterfly has Lebesgue measure zero, that is, it cannot fly!.

7

A wonderful account of the use of magnetism in medicine since ancient times can be found in Mourino (1991).

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The GMR effect and the related tunnel magnetoresistance effect (TMR) found their first commercial use in magnetic field sensors for hard-disk read-heads, before broadening their field of application to include magnetic sensors for the automobile industry, solid-state compasses, biosensors, and nonvolatile magnetic memory (Hartmann, 2000). The ability to manipulate the magnetic state of ferromagnetic structures led to initial developments in spintronics, where the spin direction was controlled by externally applied magnetic fields. However, the push for downscaling and faster timescales has prompted research advances in the past decade where the manipulation of magnetic configurations is achieved by local electric currents (the spin torque effect) (Brataas et al., 2012; Locatelli et al., 2014). In recent years, various novel concepts, such as chiral spintronics, skyrmionic spin textures, and magnetic domain walls in spintronics, have been actively pursued. The rate of progress in spintronics is truly astounding (Editorial, 2022). In the past century, superconductivity has raised many theoretical challenges and triggered numerous technical developments. Discovered in mercury in 1911, this unique phenomenon occurs when the electrical resistance of many metals and alloys drops to zero when cooled below about 4 K, the temperature of liquid helium. More significantly, the material expels magnetic flux and becomes perfectly diamagnetic. Superconductivity has been adopted or evaluated for a wide range of technical applications (Rogalla and Kes, 2012). These include electric power generation, electric power transmission, high-speed maglev transportation, and ultra-strong magnetic field generation in high-resolution MRI and other nuclear magnetic resonance (NMR) systems. Interestingly, it took almost 50 years after the original discovery for the phenomena to be explained by the BCS theory (Tinkham, 2004) of reciprocal-space electron pairing, via the electron–phonon interaction. This electron–phonon mediated superconductivity became known as “conventional superconductivity” after the discovery of “high-temperature superconductivity” (Müller and Bednorz, 1987) at liquid nitrogen temperatures in tetragonal copper oxides in 1986. This threw open a new challenge to explain the properties of strongly correlated anisotropic electronic systems. Further new classes of superconductors that remain to be properly understood include the iron-based pnictides discovered in 2008 (Hosono et al., 2018). Recently, a two-dimensional superlattice made from a twisted bilayer of “magic angle” graphene was found to exhibit unconventional superconductivity (Cao et al., 2018), driven by electron–electron interactions. The phenomenon of spontaneous symmetry breaking (SSB) is ubiquitous in physics, and plays a fundamental role in CMP, particle physics, and even in cosmology. A classic illustration of SSB being Heisenberg’s 1928 paper on ferromagnetism (Heisenberg, 1928). It is responsible for various collective modes, such as the famous Nambu-Goldstone (NG) and Higgs modes. Named by Baker and Glashow (Baker and Glashow, 1962), SSB corresponds to the case where the ground or lowest-energy state does not have the symmetry of the underlying dynamics. Instead, multiple degenerate ground states are available and the symmetry breaking is spontaneous because it is not clear, a priori, which one of those ground states will be chosen. In this well-trodden territory a simple analogy of this abstruse concept given by A. Salam is illuminating (CERN, 1977): “Imagine a banquet where guests sit at round tables. A bird’s eye view of the scene presents total symmetry, with serviettes alternating with people around each table. A person could equally well take a serviette from his right or from his left. The symmetry is spontaneously broken when one guest decides to pick up from his left and everyone else follows suit.” Over the years, development of the concept of SSB has taken a circuitous route (Gaudenzi, 2022): “from the land of particle physics to the physics of many bodies (solids and nuclei), from there to the province of superconductivity, and from the latter back to particle physics ... .” SSB’s significance extends beyond the physics of the Higgs boson to the study of low-energy phenomena such as superconductivity, superfluidity, magnetism, and others. Historically, the particle physics community has been guided by this exploration in their quest to understand and discover particles like the Higgs boson. More to the point, when a continuous symmetry is spontaneously broken, a gapless mode, the NG mode, appears which governs the low-energy behavior of the system. As an example, in a solid, the acoustic phonon is the NG mode associated with translational symmetry breaking and determines the behavior of the specific heat at low temperature (the Debye T3 law). Recently, signatures of Higgs and Goldstone modes have been proposed to exist in a perovskite oxide (Marthinsen et al., 2018), in other crystalline structures (Vallone, 2019), and also in various other quantum systems (Léonard et al., 2017). On a lighter note, it is the symmetry breaking that is claimed to have prevented the fabled donkey (Weintraub, 2012) in the parable by Jean Buridan (1300–50) from dying of starvation (Fubini, 1974). Common to the theoretical descriptions of these phenomena across CMP and high-energy physics is Quantum field theory (QFT). QFT was originally developed to describe the fundamental processes in high-energy physics, but has now become a valuable tool to address problems across physics, in particular, in CMP (Fradkin, 2013; see also, Mustafa, 2023), statistical physics, modern quantum chemistry, and even in

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pure mathematics (Folland, 2008). Advances in our understanding of strongly correlated electron systems in CMP such as high-temperature superconductivity and the FQHE have been aided by the QFT. It has facilitated the treatment of spontaneous symmetry broken states, such as superfluids. Interestingly, many of the conceptual developments in CMP have, in return, had a reciprocal impact in QFT. A spectacular example being Nambu’s work in dynamical symmetry breaking (Weinberg, 1986; Nambu, 2009) in particle physics, based on the BCS theory of superconductivity. A glass is a noncrystalline solid obtained by quenching a liquid. It has a liquid-like structure, but a transition from liquid to solid behavior occurs on cooling. It may have one of a continuum of possible ground states that exhibit frozen-in liquid-like disorder. The term “spin glass” was coined by B.R. Coles (Mydosh, 1993) in 1970 for a diluted magnetic material with frozen-in paramagnetic-like spin disorder where the magnetic moments interact with randomly varying positive or negative exchange interactions leading to a multitude of possible ground states with different, random orientations of the spins. As in a normal glass, the energy barriers between the continuum of metastable states prevent it from ever reaching equilibrium. This phase of condensed matter has been an important and productive area of research. Spin-glass models are conceptually simple, but embody sophisticated physics. These models have become a paradigm for the solution of complex optimization problems (Mézard, 2022): much like the undulating flight patterns of starlings around the skies over Rome (Parisi, 2023). After cooling from the paramagnetic phase, the spin glass remains out of equilibrium, and it slowly evolves. This aging phenomenon corresponds to the growth of a “spin-glass order” parameter, whose correlation length can be measured (Joh et al., 1999). A cooling temperature step during aging causes a partial “rejuvenation,” while the “memory” of previous aging is stored and can be retrieved (Refregier et al., 1987; Bouchaud et al., 2001). Many glassy materials present aging, and rejuvenation and memory effects can be found in other cases, but they are usually less pronounced. Numerical simulations of these phenomena are under active investigation using custom-built computers (Baity-Jesi et al., 2023; Vincent, 2023). A general understanding of glassy systems, to which spin glasses bring some special insight, is being pursued. It was realized decades ago that the physics of spin glasses has deep connections with the behavior of complex biological systems. In fact, spin-glass type states in neural network models offer interesting insights into states of high and low brain activity, as observed in mammals (Recio and Torres, 2016). These analogies are potentially useful for applications in fields such as robotics and artificial intelligence (AI) (Andriuschenko et al., 2022). One of the most spectacular phenomena in CMP, discovered in the last century, was Bose-Einstein condensation (BEC). This macroscopic quantum phenomenon had its genesis in the work of Satyendra Nath Bose (Bose, 1924; Blanpied, 1972; Tomczyk, 2022) (whose name is forever associated with bosons), and has fascinated physicists for decades, ever since the first observation of this exotic state of matter in an ultracold gas of rubidium atoms (Georgescu, 2020). Among the advances made in this field, we highlight only a few: simulation of correlated electron states, such as the FQHE state and a bosonic analog of the Laughlin state, and the famous BEC-BCS crossover from a BEC to a BCS superfluid of weakly bound Cooper pairs, as the interparticle interaction strength is varied. Noteworthy is the recent measurement of the dynamic structure factor in an ultracold Fermi gas of lithium-6 atoms in the “unitary” limit, where a linear phonon mode for low momentum transfer can be clearly discerned (Biss et al., 2022). Perhaps such a method might one day shed light on the Laughlin quasiparticle gap and the collective modes that are still largely unresolved (Chakraborty and Pietiläinen, 1988, 1995; Halperin, 1983; Clark and Maksym, 1989). Liquid helium, 4He, and the less common isotope 3He are unique quantum systems (Volovik, 2009). They remain liquid even at absolute zero temperature, a direct manifestation of Heisenberg’s uncertainty principle and zero-point energy. The light mass of helium atoms and their weak interatomic attraction leads to a substantial zero-point kinetic energy and the system remains in the liquid state at ambient pressure. Solidification occurs when a pressure of about 25 MPa is applied. The different quantum statistics of the two helium isotopes (4He is a boson while 3He is a fermion) are responsible for the different physical characteristics of the two systems. Microscopic approaches to understand the ground state and low-energy excitations of the liquid state have occupied researchers for many decades (see for example, Ristig and Clark, 1976; Lam et al., 1977). These highly correlated quantum fluids have been established as systems where the efficacies of different many-body theories can be tested (Kallio and Smith, 1977; Lantto et al., 1977). These systems have been described as Ariadne’s thread in this Encyclopedia, where the strongly interacting constituent particles are treated by a variety of ingenious theoretical and experimental approaches. Interestingly, liquid helium has been proposed to be a medium to design multiexcitation detectors to probe dark matter (Schutz and Zurek, 2016) in the keV mass limit. Although this looks far afield from CMP, the interface between CMP and high-energy particle physics could inspire new horizons in CMP, as discussed earlier.

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CMP also has an enormous impact on our daily lives with unceasing technological innovation stemming directly from the advances in CMP research. It is also closely associated with the progress of our understanding of materials science and mechanical, chemical, and electrical engineering that deals with properties of novel materials and devices. As some authors have very aptly pointed out (Chaikin and Lubensky, 1995): The use and understanding of matter in its condensed (liquid or solid) state have gone hand in hand with the advances of civilization and technology since the first use of primitive tools. So important has the control of condensed matter been to man that prehistoric ages—the Stone Age, the Bronze Age, the Iron Age—are named after the material dominating the technology of the time. In fact, modern civilization is entirely dependent on our deep understanding and practical use of materials, old and new: Steel, glass, and concrete have shaped our modern living with comfort, health, longevity, and material wealth. All around us, from a tiny paper clip to giant skyscrapers, from jet planes to modern medicines, and the ubiquitous plastics that are integrated into nearly all aspects of human life, are testaments to our control over the myriad behavior of materials (Miodownik, 2014). Today, a large number of materials are designed by using principles of quantum mechanics (Freeman, 2002). The Silicon chip that launched the information age is a great example of the effect of CMP on our daily lives. The scaling of transistors down to the nanometer range has enabled them to be embedded in greater numbers in the all-pervasive electronic devices that have transformed every facet of life all around the world. The transition from room-size to pocket-size computers with ever increasing computational power and memory, that occurred in only a few decades, epitomizes the unrelenting progress made in this direction. Advances in nanoscale research have opened the door to exploration of the natural world at the ultra-small scale (Chakraborty, 2022), with the potential to shape our future world. Quantum dots (QDs), the quasi-zero-dimensional electron systems, are one of the most extensively studied nanostructures in the past three decades. They represent the ultimate reduction in dimensionality of a semiconductor device. Here the electrons are confined (either electrostatically or structurally) in all three directions and occupy spectrally sharp energy levels similar to those found in atoms (Chakraborty, 1999), hence the popular moniker artificial atoms (Maksym and Chakraborty, 1990). QDs and other similar nanostructures (Chakraborty, 2003) offer a rich variety of fundamental phenomena that result from manipulation of single electrons (single electronics) or single spins at the nanoscale. The study of interacting electrons in a QD requires large-scale numerical computations involving matrices of dimensions in the millions (“monster matrices”) (Chakraborty and Pietiläinen, 2005; Pietiläinen and Chakraborty, 2006). QDs are also thought to have vast potential for future technological applications. One prominent example being quantum cryptography where the QDs are used in single photon emitters (Nowak et al., 2014), and in single photon detectors (Shields et al., 2000). Quantum cryptography, in turn, forms the basis of the quantum Internet (Liao et al., 2018). Other applications include lasers, and spintronics in quantum computing, entertainment industry (Bourzac, 2013), and even in biology (Bruchez and Holz, 2007). Research in recent years has increasingly revealed that quantum phenomena are also ubiquitous in the natural world (Ball, 2011). They govern how birds are able to navigate around the globe using Earth’s magnetic field or how plants use photosynthesis to harvest sunlight for conversion to chemical energy. All depend on subtle quantum effects that are now coming to light, thereby ushering the topic of quantum biology into CMP. Deoxyribonucleic acid (DNA), the molecule responsible for storage of genetic information in the cells of all living organism, is entrusted with the task of preserving, copying, and transmitting information that are necessary for living beings to grow and function. It is often referred to as the body’s hereditary material as DNA is replicated and transmitted from parents to their offspring. This “molecule of life” was discovered in 1869 by Friedrich Miescher in Tübingen, Germany, who called it “nuclein,” as it was isolated from the nucleus of white blood cells (Dahm, 2005; Thess et al., 2021). It took more than eighty years from that momentous event, to discover that DNA has a right-handed double-helix structure with appropriate base pairing (Watson and Crick, 1953; Klug, 2004). That discovery was so important that it has been compared with the discovery of the atomic nucleus in physics (Frank-Khamenetskii, 1997). Understanding the structure of the atom heralded quantum physics, while understanding of the structure of DNA brought forth the field of molecular biology. DNA has several remarkable properties that make this molecule a prime candidate for nano-electronic materials (Chakraborty, 2007). These include molecular recognition and the ability to self-assemble via the complementarity of the base sequences on the two strands, which means that DNA can be integrated error-free in current semiconductor technology. DNA can also detect information about its own integrity that can trigger its eventual repair mechanism. All these properties are perfectly suitable for developing nanoscale electronics involving DNA

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(Taniguchi and Kawai, 2006). Despite the promise of harnessing biology to realize integrated molecular electronics remains fascinating, much work remains to be done to make it a reality (Braun and Keren, 2004). Protein molecules play a vital role in all living organisms. They are described as Nature’s robots (Tanford and Reynolds, 2001) because for every imaginable task required in a living organism, and for every step of that task, there is a protein designed to carry it out. These proteins are even programmed to know when to turn on or off. Just as the twenty-six letters in the alphabet can spell out hundreds of thousands of words, there are twenty different naturally occurring amino acids that can be combined into many more different proteins than there are atoms in the known universe! Protein’s biological function is determined by its unique and native three-dimensional structures that are characteristic of individual proteins (Gomes and Faísca, 2019; Echenique, 2007). The question of how proteins fold to their unique, compact, and highly organized functional states has remained an enduring challenge (“The protein-folding problem”) (Chan and Dill, 1993; Dill and MacCallum, 2012; Wolynes and Eaton, 1999). Recently, there has been a tremendous development in this area of research. In November 2020, an AI program called AlphaFold, designed to tackle the problem of protein folding, predicted the three-dimensional structures of almost every protein known to science—about 200 millions in all (Jumper et al., 2021). AlphaFold uses a machine learning methodology to bring together information from preexisting knowledge of protein structures, and other geometric and physical constraints to predict protein structures. While this has been heralded a watershed moment in structural biology (Perrakis and Sixma, 2021), there are many key questions that AlphaFold cannot address yet. These include aspects of protein behavior that cannot be understood from a static structure, such as how proteins respond to their environment or how intrinsically disordered proteins actually are organized. Some authors (Nasser et al., 2021) have likened it to solving a crime story: while AlphaFold presents a snapshot in time, the question about how we get there remains unanswered. Biological systems have developed elaborate mechanisms to ensure that proteins fold correctly, or otherwise, they are detected and degraded before any serious harm can result to the host organism. However, protein misfolding does happen and that misfolding in the cell is linked to an array of diseases, including cancers, cardiovascular disease, type II diabetes, and numerous neurodegenerative diseases, such as Alzheimer’s, Parkinson’s, and Huntington’s disease. NMR spectroscopy has been an important technique to study protein folding where valuable insights are gained into many aspects of the folding process (Jane Dyson and Wright, 1996). In this Encyclopedia, we strive to present a taste of all the developments in CMP described above (and much more), written by experts in the field. We are keenly aware that many important and interesting topics are not covered here. However, that is not for want of trying. As some authors had rightly pointed out (Hoddeson, 1992): The field is huge and varied and lacks the unifying features beloved of historians, neither a single hypothesis or set of basic equations underpin the field, as in quantum mechanics or relativity theory, nor a single spectacular and fundamental discovery such as uranium fission for nuclear technology or the structure of DNA for molecular biology. CMP owes what unity it has simply to a common concern with solid and liquid materials. This encompasses such a large range of theories and methods that the field resembles a confederation of interest groups rather than a single entity. Clearly, the task of collecting everything in one place is a monumental endeavor, but we do hope that our compilation of a large number of chapters in this five-volume work, covering a broad range of in-depth descriptions of varied phenomena in CMP will be a treasured source of facts and ideas for the community. It is undeniably true that there would have been no Encyclopedia without the extraordinary contributions of the large number of authors whose valuable time and efforts brought this project to fruition. My sincere thanks to all of them. I also wish to extend my deep appreciation to the staff at Elsevier for their skillful handling of this huge project. The cover image was created by Hong-Yi Chen from National Taiwan Normal University. The image below was created by Elisabetta Collini, one of the authors. My deepest appreciation to them for taking the time to do this work. Finally, my sincere thanks to the Section editors, Vladimir Fomin, Ana Sanchez, Roberto Raimondi, Hideo Aoki, Rudolf Römer, and Robert Blick for the effort they all expended, that has contributed to the success of this venture. Special thanks go to Vladimir Fomin and Ana Sanchez for securing a large share of the chapters and for their enduring help in providing valuable advice and support to bring the project to fruition. Thanks are also due to many of my friends and colleagues, in particular, Michael Coey, Jürg Fröhlich, Sinéad Griffin, Peter Maksym, Eric Vincent, and Jakob Yngvason for their careful and critical reading of the introduction. Tapash Chakraborty St. Catharines Ontario, Canada

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References Abergel DSL, Apalkov V, Berashevich J, Ziegler K, and Chakraborty T (2010) Properties of graphene: A theoretical perspective. Advances in Physics 59: 261. https://doi.org/10.1080/ 00018732.2010.487978. Andriuschenko P, Kapitan D, and Kapitan V (2022) A new look at the spin glass problem from a deep learning perspective. Entropy 24: 697. https://doi.org/10.3390/e24050697. Apalkov VM and Chakraborty T (2006) Fractional quantum Hall states of Dirac electrons in graphene. Physical Review Letters 97: 126801. https://doi.org/10.1103/ PhysRevLett.97.126801. Apalkov VM and Chakraborty T (2010) Controllable driven phase transitions in fractional quantum Hall states in bilayer graphene. Physical Review Letters 105: 036801. https://doi.org/ 10.1103/PhysRevLett.105.036801. Apalkov VM and Chakraborty T (2011) Stable pfaffian state in bilayer graphene. Physical Review Letters 107: 186803. https://doi.org/10.1103/PhysRevLett.107.186803. Arovas D, Schrieffer JR, and Wilczek F (1984) Fractional statistics and the quantum Hall effect. Physical Review Letters 53: 722. https://doi.org/10.1103/PhysRevLett.53.722. Avila A and Jitomirskaya S (2006) Solving the ten martini problem. Lecture Notes in Physics 690: 5. https://doi.org/10.1007/3-540-34273-7_2. Baity-Jesi M, et al. (2023) Memory and rejuvenation effects in spin glasses are governed by more than one length scale. Nature Physics 19: 978–985. https://doi.org/10.1038/ s41567-023-02014-6. Baker M and Glashow SL (1962) Spontaneous breakdown of elementary particle symmetries. Physics Review 128: 2462. https://doi.org/10.1103/PhysRev.128.2462. Ball P (2011) Physics of life: The dawn of quantum biology. Nature 474: 272. https://doi.org/10.1038/474272a. Baltz V, et al. (2018) Antiferromagnetic spintronics. Reviews of Modern Physics 90: 015005. https://doi.org/10.1103/RevModPhys.90.015005. Bartolomei H, et al. (2020) Fractional statistics in anyon collisions. Science 368: 173. https://doi.org/10.1126/science.aaz5601. Biss H, et al. (2022) Excitation spectrum and superfluid gap of an ultracold Fermi gas. Physical Review Letters 128: 100401. https://doi.org/10.1103/PhysRevLett.128.100401. Blanpied WA (1972) Satyendranath Bose: Co-founder of quantum statistics. American Journal of Physics 40: 1212. https://doi.org/10.1119/1.1986805. Blundell S (2001) Magnetism in Condensed Matter. New York: Oxford U.P. Bose SN (1924) Plancks gesetz und lichtquantenhypothese. Zeitschrift für Physik 26: 178. https://doi.org/10.1007/BF01327326. Bouchaud J-P, Dupuis V, Hammann J, and Vincent E (2001) Separation of time and length scales in spin glasses: Temperature as a microscope. Physical Review B 65: 024439. https://doi.org/10.1103/PhysRevB.65.024439. Bourzac K (2013) Quantum dots go on display. Nature 493: 283. https://doi.org/10.1038/493283a. Brataas A, Kent AD, and Ono H (2012) Current-induced torques in magnetic materials. Nature Materials 11: 372. https://doi.org/10.1038/nmat3311. Braun E and Keren K (2004) From DNA to transistors. Advances in Physics 53: 441. https://doi.org/10.1080/00018730412331294688. Bruchez MP and Holz CZ (2007) Quantum Dots: Applications in Biology. New Jersey: Humana Press. https://doi.org/10.1385/1597453692. Cao Y, et al. (2018) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556: 43. https://doi.org/10.1038/nature26160. CERN. (1977) Gauge theories with the lid, partially, off. CERN Courier 17: 271. https://cds.cern.ch/record/1730245/files/vol17-issue9-p271-e.pdf. Chaikin PM and Lubensky TC (1995) Principles of Condensed Matter Physics. Cambridge University Press. https://doi.org/10.1017/CBO9780511813467. Chakraborty T (1985) Elementary excitations in the fractional quantum Hall effect. Physical Review B 31: 4026. https://doi.org/10.1103/PhysRevB.31.4026. Chakraborty T (1999) Quantum Dots: A Survey of the Properties of Artificial Atoms. Amsterdam: North-Holland. https://doi.org/10.1016/B978-0-444-50258-2.X5000-7. Chakraborty T (2003) Nanoscopic quantum rings: A new perspective. Advances in Solid State Physics 43: 79. https://doi.org/10.1007/978-3-540-44838-9_6. Chakraborty T (ed.) (2007) Charge Migration in DNA. Springer. https://doi.org/10.1007/978-3-540-72494-0. Chakraborty T (2022) Nanoscale Quantum Materials: Musings on the Ultra-Small World. CRC Press. https://doi.org/10.1201/9781003090908. Chakraborty T and Apalkov VM (2015) Fractal butterflies of Dirac fermions in monolayer and bilayer graphene. IET Circuits, Devices and Systems 9: 19. https://doi.org/10.1049/ietcds.2014.0275. Chakraborty T and Pietiläinen P (1988) The Fractional Quantum Hall Effect. Springer Verlag. https://doi.org/10.1007/978-3-642-97101-3. Chakraborty T and Pietiläinen P (1995) The Quantum Hall Effects, 2nd. Springer. https://doi.org/10.1007/978-3-642-79319-6. Chakraborty T and Pietiläinen P (2005) Optical signatures of spin-orbit interaction effects in a parabolic quantum dot. Physical Review Letters 95: 136603. https://doi.org/10.1103/ PhysRevLett.95.136603. Chan HS and Dill KA (1993) The protein folding problem. Physics Today 46: 24. https://doi.org/10.1063/1.881371. Clark R and Maksym P (1989) Fractional quantum Hall effect in a spin. Physics World 2: 39. https://doi.org/10.1088/2058-7058/2/9/23.

xvi

Introduction

Cohen ML (2008) Fifty years of condensed matter physics. Physical Review Letters 101: 250001. https://doi.org/10.1103/PhysRevLett.101.250001. Dahm R (2005) Friedrich miescher and the discovery of DNA. Developmental Biology 278: 274. https://doi.org/10.1016/j.ydbio.2004.11.028. Dill KA and MacCallum JL (2012) The protein-folding problem, 50 years on. Science 338: 1042. https://doi.org/10.1126/science.1219021. Echenique P (2007) Introduction to protein folding for physicists. Contemporary Physics 48: 81. https://doi.org/10.1080/00107510701520843. Editorial (2022) New horizons in spintronics. Nature Materials 21: 1. https://doi.org/10.1038/s41563-021-01184-z. Eisenstein JP and Stormer HL (1990) The fractional quantum Hall effect. Science 248: 1510. https://doi.org/10.1126/science.248.4962.1510. Folland GB (2008) Quantum Field Theory: A tourist guide for Mathematicians. American Mathematical Society. https://doi.org/10.1090/surv/149. Fradkin E (2013) Field Theories of Condensed Matter Physics. Cambridge University Press. https://doi.org/10.1017/CBO9781139015509. Frank-Khamenetskii MD (1997) Unraveling DNA: The Most Important Molecule of life. Basic Books. Freeman AJ (2002) Materials by design and the exciting role of quantum computation/simulation. Journal of Computational and Applied Mathematics 149: 27. https://doi.org/ 10.1016/S0377-0427(02)00519-8. Fubini S (1974) Present trends in particle physics. In: Proc. Intl. Conf. on High Energy Physics. Didcot, England: Rutherford Laboratory. https://cds.cern.ch/record/417271/files/ CM-P00060491.pdf. Gaudenzi R (2022) Historical Roots of Spontaneous Symmetry Breaking: Steps Towards an Analogy. Springer. https://doi.org/10.1007/978-3-030-99895-0. Georgescu I (2020) 25 Years of BEC. Nature Reviews Physics 2: 396. https://doi.org/10.1038/s42254-020-0211-7. Gomes CM and Faísca PFN (2019) Protein Folding: An Introduction. Springer. https://doi.org/10.1007/978-3-319-00882-0_1. Halperin BI (1983) Theory of the quantized Hall conductance. Helvetica Physica Acta 56: 75. https://doi.org/10.5169/seals-115362. Halperin BI (1984) Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Physical Review Letters 52: 1583. https://doi.org/10.1103/PhysRevLett.52.1583. Harper PG (1955) Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A 68: 874. https://doi.org/10.1088/03701298/68/10/304. Hartmann U (2000) Magnetic Multilayers and Giant Magnetoresistance: Fundamentals and Industrial Applications. Berlin: Springer Verlag. https://doi.org/10.1007/978-3-66204121-5. Heisenberg W (1928) Zur theorie des ferromagnetismus. Zeitschrift für Physik 49: 619. https://doi.org/10.1007/BF01328601. Hoddeson L (1992) Out of the Crystal Maze: Chapters From the History of Solid-State Physics. N.Y.: Oxford University Press. Hofstadter D (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14: 2239. https://doi.org/10.1103/ PhysRevB.14.2239. Hosono H, et al. (2018) Recent advances in iron-based superconductors toward applications. Materials Today 21: 278. https://doi.org/10.1016/j.mattod.2017.09.006. Jane Dyson H and Wright PE (1996) Insights into protein folding from NMR. Annual Review of Physical Chemistry 47: 369. https://doi.org/10.1146/annurev.physchem.47.1.369. Jiang Z, et al. (2007) Quantum Hall effect in graphene. Solid State Communications 143: 14. https://doi.org/10.1016/j.ssc.2007.02.046. Joh YG, Orbach R, Wood JJ, Hammann J, and Vincent E (1999) Extraction of the spin glass correlation length. Physical Review Letters 82: 438. https://doi.org/10.1103/ PhysRevLett.82.438. Jumper J, et al. (2021) Highly accurate protein structure prediction with alphafold. Nature 596: 583. https://doi.org/10.1038/s41586-021-03819-2. Kallio A and Smith RA (1977) How simple can HNC be and still be right? Physics Letters B 68: 315. https://doi.org/10.1016/0370-2693(77)90483-X. Kibble T and Srivastava A (2013) Condensed matter analogues of cosmology. Journal of Physics. Condensed Matter 25: 400301. https://doi.org/10.1088/0953-8984/25/ 40/400301. Klug A (2004) The discovery of the DNA double helix. Journal of Molecular Biology 335: 3. https://doi.org/10.1016/j.jmb.2003.11.015. Lam PM, Clark JW, and Ristig ML (1977) Density matrix and momentum distribution of helium liquids and nuclear matter. Physical Review B 16: 222. https://doi.org/10.1103/ PhysRevB.16.222. Lantto LJ, Jackson AD, and Siemens PJ (1977) HNC variational calculations of boson matter. Physics Letters B 68: 311. https://doi.org/10.1016/0370-2693(77)90482-8. Laughlin RB (1983) Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50: 1395. https://doi.org/10.1103/ PhysRevLett.50.1395. Laughlin RB (1984) Primitive and composite ground states in the fractional quantum Hall effect. Surface Science 142: 163. https://doi.org/10.1016/0039-6028(84)90301-7. Laughlin RB and Pines D (2000) The theory of everything. Proceedings of the National Academy of Sciences of the United States of America 97: 28. https://doi.org/10.1073/ pnas.97.1.28. Leggett A (1992) On the nature of research in condensed-state physics. Foundations of Physics 22: 221. https://doi.org/10.1007/BF01893613. Leggett A (2018) Reflections on the past, present and future of condensed matter physics. Science Bulletin 63: 1019. https://doi.org/10.1016/j.scib.2018.07.004. Leinaas JM and Myrheim J (1977) On the theory of identical particles. Il Nuovo Cimento B 37: 1. https://doi.org/10.1007/BF02727953. Léonard J, et al. (2017) Monitoring and manipulating higgs and goldstone modes in a supersolid quantum gas. Science 358: 1415. https://doi.org/10.1126/science.aan2608. Liao S-K, et al. (2018) Satellite-relayed intercontinental quantum network. Physical Review Letters 120: 030501. https://doi.org/10.1103/PhysRevLett.120.030501. Liu X, Hao Z, Watanabe K, Taniguchi T, Halperin BI, and Kim P (2019) Interlayer fractional quantum Hall effect in a coupled graphene double layer. Nature Physics 15: 893. https://doi. org/10.1038/s41567-019-0546-0. Locatelli N, Cros V, and Grollier J (2014) Spin-torque building blocks. Nature Materials 13: 11. https://doi.org/10.1038/nmat3823. Lu W (2022) Quark-gluon plasma and nucleons à la Laughlin. International Journal of Geometric Methods in Modern Physics 19: 2250061. https://doi.org/10.1142/ S021988782250061X. Maksym P and Chakraborty T (1990) Quantum dots in a magnetic field: Role of electron-electron interactions. Physical Review Letters 65: 108. https://doi.org/10.1103/ PhysRevLett.65.108. Marthinsen A, et al. (2018) Goldstone-like phonon modes in a (111)-strained perovskite. Physical Review Materials 2: 014404. https://doi.org/10.1103/PhysRevMaterials.2.014404. Mézard M (2022) Spin glasses and optimization in complex systems. Europhysics News 53: 15. https://doi.org/10.1051/epn/2022105. Miodownik M (2014) Stuff Matters: Exploring the Marvelous Materials That Shape Our Man-Made World. N.Y.: Houghton Mifflin Harcourt. Morf R and Halperin BI (1986) Monte Carlo evaluation of trial wave functions for the fractional quantized Hall effect: Disk geometry. Physical Review B 33: 2221. https://doi.org/ 10.1103/PhysRevB.33.2221. Mourino MR (1991) From thales to lauterbur, or from the loadstone to MR imaging: Magnetism and medicine. Radiology 180: 593. https://doi.org/10.1148/radiology.180.3.1871268. Müller KA and Bednorz JG (1987) The discovery of a class of high-temperature superconductors. Science 237: 1133. https://doi.org/10.1126/science.237.4819.1133 Mustafa MG (2023) An introduction to thermal field theory and some of its application. The European Physical Journal Special Topics 232: 1369–1457. https://doi.org/10.1140/epjs/ s11734-023-00868-8. Mydosh JA (1993) Spin Glasses: An Experimental Introduction. CRC Press. Nakamura J, et al. (2020) Direct observation of anyonic braiding statistics. Nature Physics 16: 931. https://doi.org/10.1038/s41567-020-1019-1. Nambu Y (2009) Nobel lecture: Spontaneous symmetry breaking in particle physics: A case of cross fertilization. Reviews of Modern Physics 81: 1015. https://doi.org/10.1103/ RevModPhys.81.1015. Nasser R, Dignon GL, Razban RM, and Dill KA (2021) The protein folding problem: The role of theory. Journal of Molecular Biology 433: 167126. https://doi.org/10.1016/j. jmb.2021.167126. Nawrocki W (2019) Introduction to Quantum Metrology, 2nd. Springer. https://doi.org/10.1007/978-3-030-19677-6. Nowak AL, et al. (2014) Deterministic and electrically tunable bright single-photon source. Nature Communications 5: 3240. https://doi.org/10.1038/ncomms4240.

Introduction

xvii

Obermair GM and Wannier GH (1976) Bloch electrons in magnetic fields: Rationality, irrationality, degeneracy. Physica Status Solidi (B) 76: 217. https://doi.org/10.1002/ pssb.2220760122. Osadchy D and Avron JE (2001) Hofstadter butterfly as quantum phase diagram. Journal of Mathematical Physics 42: 5665. https://doi.org/10.1063/1.1412464. Parisi G (2023) In a Flight of Starlings: The Wonders of Complex Systems. Penguin Press. Perrakis A and Sixma TK (2021) Ai revolutions in biology. EMBO Reports 22: e54046. https://doi.org/10.15252/embr.202154046. Pietiläinen P and Chakraborty T (2006) Energy levels and magneto-optical transitions in parabolic quantum dots with spin-orbit coupling. Physical Review B 73: 155315. https://doi. org/10.1103/PhysRevB.73.155315. Pinarbasi M and Kent AD (2022) Perspectives on spintronics technology: Giant magnetoresistance to spin transfer torque magnetic random access memory. APL Materials 10: 020901. https://doi.org/10.1063/5.0075945. Recio I and Torres JJ (2016) Emergence of low noise frustrated states in E/I balanced neural networks. Neural Networks 84: 91. https://doi.org/10.1016/j.neunet.2016.08.010. Refregier P, Vincent E, Hammann J, and Ocio M (1987) Ageing phenomena in a spin-glass: Effect of temperature changes below Tg. Journal of Physics (France) 48: 1533. https://doi. org/10.1051/jphys:019870048090153300. Ristig ML and Clark JW (1976) Density matrix of quantum fluids. Physical Review B 14: 2875. https://doi.org/10.1103/PhysRevB.14.2875. Rogalla H and Kes PH (2012) 100 Years of Superconductivity. Boca Raton: Taylor & Francis Group. https://doi.org/10.1201/b11312. Schutz K and Zurek KM (2016) Detectability of light dark matter with superfluid helium. Physical Review Letters 117: 121302. https://doi.org/10.1103/PhysRevLett.117.121302. Shields AJ, et al. (2000) Detection of single photons using a field-effect transistor gated by a layer of quantum dots. Applied Physics Letters 103: 3673. https://doi.org/ 10.1063/1.126745. Steptoe A (1986) Mozart, mesmer and ‘cosi fan tutte’. Musics and Letters 67: 248. https://doi.org/10.1093/ml/67.3.248. Störmer HL (1992) Two-dimensional electron correlation in high magnetic fields. Physica B: Condensed Matter 177: 401. https://doi.org/10.1016/0921-4526(92)90138-I. Tanford C and Reynolds J (2001) Nature’s Robots: A History of Proteins. Oxford University Press. Taniguchi M and Kawai T (2006) Dna electronics. Physica E: Low-dimensional Systems and Nanostructures 33: 1. https://doi.org/10.1016/j.physe.2006.01.005. Thess A, et al. (2021) Historic nucleic acids isolated by Friedrich Miescher contain RNA beside DNA. Biological Chemistry 402: 1179. https://doi.org/10.1515/hsz-2021-0226. Tinkham M (2004) Introduction to Superconductivity. N.Y.: Dover. Tomczyk H (2022) Did Einstein predict Bose-Einstein condensation? Studies in History and Philosophy of Science 93: 30. https://doi.org/10.1016/j.shpsa.2022.02.014. Vallone M (2019) Higgs and goldstone modes in crystalline solids. Physica Status Solidi 257: 1900443. https://doi.org/10.1002/pssb.201900443. Van Vleck JH (1978) Quantum mechanics—The key to understanding magnetism. Reviews of Modern Physics 50: 181. https://doi.org/10.1103/RevModPhys.50.181. Vincent E (2023) Rejuvenated but remembering. Nature Physics 19: 926–927. https://doi.org/10.1038/s41567-023-02097-1. Volovik GE (2009) The Universe in a Helium Droplet. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199564842.001.0001. von Klitzing K (2019) Essay: Quantum Hall effect and the new international system of units. Physical Review Letters 122: 200001. https://doi.org/10.1103/PhysRevLett.122.200001. von Klitzing K, Chakraborty T, Kim P, Madhavan V, Dai X, McIver J, Tokura Y, Savary L, Smirnova D, Rey AM, Felser C, Gooth J, and Qi X (2020) 40 years of the quantum Hall effect. Nature Reviews Physics 2: 397. https://doi.org/10.1038/s42254-020-0209-1. Watson JD and Crick FHC (1953) Molecular structure of nucleic acids. Nature 171: 737. https://doi.org/10.1038/171737a0. Weinberg S (1986) Superconductivity for particular theorists. Progress of Theoretical Physics Supplement 86: 43. https://doi.org/10.1143/PTPS.86.43. Weintraub R (2012) What can we learn from Buridan’s ass? Canadian Journal of Philosophy 42: 281. https://doi.org/10.1353/cjp.2012.0013. Weiss D (2013) The butterfly emerges. Nature Physics 9: 395. https://doi.org/10.1038/nphys2680. Wolynes PG and Eaton WA (1999) The physics of protein folding. Physics World 12: 39. https://doi.org/10.1088/2058-7058/12/9/24. Yamaguchi A, Hirohata A, and Stadler BJH (2021) Nanomagnetic Materials: Fabrication, Characterization and Application. Elsevier. (Chapter 5). https://doi.org/10.1016/C2018-005445-6.

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LIST OF CONTRIBUTORS FOR VOLUME 1 Ryosuke Akashi Department of Physics, University of Tokyo, Bunkyo, Japan Yoichi Ando Physics Institute II, University of Cologne, Cologne, Germany Vadym Apalkov Department of Physics and Astronomy, Georgia State University, Atlanta, GA, United States Y Avishai Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel; Yukawa Institute for Theoretical Physics, Kyoto, Japan Ajit C Balram Institute of Mathematical Sciences, CIT Campus, Chennai, India; Homi Bhabha National Institute, Mumbai, India YB Band Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel; Department of Chemistry, BenGurion University of the Negev, Beer-Sheva, Israel; The Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel Jean Bellissard School of Mathematics, Georgia Institute of Technology, Atlanta, GA, the United States Giuliano Benenti Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Como, Italy; Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano, Italy Emil J Bergholtz Department of Physics, Stockholm University, AlbaNova University Center, Stockholm, Sweden

AA Burkov Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada; Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada Massimo Capone International School for Advanced Studies (SISSA) and CNR-IOM, Trieste, Italy Iacopo Carusotto Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, Trento, Italy Giulio Casati Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Como, Italy Tapash Chakraborty Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada; Department of Physics, Brock University, St. Catharines, ON, Canada JT Chalker Theoretical Physics, University of Oxford, Oxford, United Kingdom James R Chelikowsky Center for Computational Materials, Oden Institute of Computational Engineering and Sciences, Departments of Physics and Chemical Engineering, University of Texas at Austin, Austin, TX, United States AG Chronis Department of Materials Science, University of Patras, Patras, Greece ML Cohen University of California at Berkeley, Berkeley, CA, USA

IB Bersuker The University of Texas at Austin, Austin, TX, United States

Dimitrie Culcer School of Physics, The University of New South Wales, Sydney, NSW, Australia

Jordi Boronat Department of Physics, Technical University of Catalonia, Barcelona, Spain

Ippei Danshita Department of Physics, Kindai University, Higashi-Osaka, Osaka, Japan xix

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List of Contributors for Volume 1

Jeanlex Soares de Sousa Departamento de Fí sica, Universidade Federal do Ceará, Fortaleza, Brazil

Yasuhiro Hatsugai Department of Physics, University of Tsukuba, Tsukuba, Japan

SB Dugdale H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom

Philipp Hauke INO-CNR BEC Center and Department of Physics, University of Trento, Italy and INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy

Giuseppe Falci Physics and Astronomy Department "Ettore Majorana", University of Catania, Catania, Italy; INFN Catania Section, Catania, Italy; CNR-IMM, Catania (University) Unit, Catania, Italy Dario Ferraro Dipartimento di Fisica dell'Università degli Studi di Genova & CNR-SPIN, Genova, Italy G Fève Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, Paris, France Christopher Ford Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom Eduardo Fradkin Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, United States Jürg Fröhlich Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland T Giamarchi Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland Henri Godfrin Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, Grenoble, France Mark O Goerbig Laboratoire de Physique des Solides, CNRS - Université Paris-Saclay, Orsay, France Shoushu Gong Department of Physics, Beihang University, Beijing, China Sinéad M Griffin Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, the United States; Molecular Foundry Division, Lawrence Berkeley National Laboratory, Berkeley, CA, the United States Pertti J Hakonen Low Temperature Laboratory, Department of Applied Physics, Aalto University, Otaniemi, Finland; InstituteQ— The Finnish Quantum Institute, Espoo, Finland

O Kieler Quantum Electronics, PTB Braunschweig, Braunschweig, Germany Eckhard Krotscheck Department of Physics, The State University of New York at Buffalo, Buffalo, NY, United States AJ Leggett Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, United States Jon Magne Leinaas Department of Physics, University of Oslo, Oslo, Norway Daniel Leykam Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore Zhao Liu Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, China Douglas Lundholm Department of Mathematics, Uppsala University, Uppsala, Sweden Ken KW Ma National High Magnetic Field Laboratory, Tallahassee, FL, United States Ryusuke Matsunaga The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan; JST-PRESTO, Saitama, Japan Edward McCann Physics Department, Lancaster University, Lancaster, United Kingdom MJ Mehl Department of Mechanical Engineering and Materials Sciences, Duke University, Durham, NC, United States Zi Yang Meng Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam, Hong Kong, China Yoshifumi Morita Faculty of Engineering, Gunma University, Kiryu, Japan Satoshi Moriyama Department of Electrical and Electronic Engineering, School of Engineering, Tokyo Denki University, Tokyo, Japan

List of Contributors for Volume 1

Nathan M Myers Department of Physics, Virginia Tech, Blacksburg, VA, United States

Maura Sassetti Dipartimento di Fisica dell'Università degli Studi di Genova & CNR-SPIN, Genova, Italy

Jan Myrheim Department of Physics, The Norwegian University of Science and Technology, Trondheim, Norway

VW Scarola Department of Physics, Virginia Tech, Blacksburg, VA, United States

Joji Nasu Department of Physics, Tohoku University, Sendai, Japan

DN Sheng Department of Physics and Astronomy, California State University Northridge, Los Angeles, CA, United States

Yusuke Nomura Department of Applied Physics and Physico-Informatics, Keio University, Yokohama, Japan Kozo Okazaki The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan Elisabetta Paladino Physics and Astronomy Department "Ettore Majorana", University of Catania, Catania, Italy; INFN Catania Section, Catania, Italy; CNR-IMM, Catania (University) Unit, Catania, Italy Gaopei Pan Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China; School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China DA Papaconstantopoulos Department of Computational and Data Sciences, George Mason University, Fairfax, VA, United States Zlatko Papic School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom Michael R Peterson Department of Physics & Astronomy, California State University Long Beach, Long Beach, CA, United States Sumathi Rao International Centre for Theoretical Studies, Tata Institute of Fundamental Research, Bangalore, India Raffaele Resta Istituto Officina dei Materiali IOM-CNR, Trieste, Italy; Donostia International Physics Center, Donostia-San Sebastián, Spain

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AL Shluger Department of Physics and Astronomy, University College London, London, United Kingdom MM Sigalas Department of Materials Science, University of Patras, Patras, Greece Tapio Simula Optical Sciences Centre, Swinburne University of Technology, Melbourne, VIC, Australia Daria Smirnova Nonlinear Physics Centre, Research School of Physics, Australian National University, Canberra, ACT, Australia; AM Stoneham Department of Physics and Astronomy, University College London, London, United Kingdom J Strand Department of Physics and Astronomy, University College London, London, United Kingdom Jacques Tempere Departement Fysica, TQC, Universiteit Antwerpen, Antwerp, Belgium MP Tosi Scuola Normale Superiore, Pisa, Italy Shunji Tsuchiya Department of Physics, Chuo University, Kasuga, Tokyo, Japan Naoto Tsuji Department of Physics, University of Tokyo, Hongo, Tokyo, Japan; RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, Japan

Nicolas Rougerie Ecole Normale Supérieure de Lyon & CNRS, UMPA (UMR 5669), Lyon, France

Marco Vallone Department of Electronics and Telecommunications. Politecnico of Turin, Turin, Italy

Eric C Rowell Department of Mathematics, Texas A&M University, College Station, TX, United States

Pedro Vianez Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom

xxii

List of Contributors for Volume 1

G Vignale Department of Physics and Astronomy, University of Missouri, Columbia, MO, United States

Philipp Werner Department of Physics, University of Fribourg, Fribourg, Switzerland

Klaus von Klitzing Max Planck Institute for Solid State Research, Stuttgart, Germany

Kun Yang Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL, United States

Xue-Feng Wang School of physical science and technology, Soochow University, Suzhou, China Jürgen Weis Max Planck Institute for Solid State Research, Stuttgart, Germany G Wendin Department of Microtechnology and Nanoscience— MC2, Chalmers University of Technology, Gothenburg, Sweden

Wei Yang Institute of Physics, Chinese Academy of Sciences, Beijing, China Jakob Yngvason Fakultät für Physik, Universität Wien, Vienna, Austria; Erwin Schrödinger Institute for Mathematics and Physics, Vienna, Austria Guangyu Zhang Institute of Physics, Chinese Academy of Sciences, Beijing, China

CONTENTS OF VOLUME 1 Quantum Hall effect and modern-day metrology

1

Klaus von Klitzing

The pulse-driven AC Josephson voltage standard of Physikalisch-Technische Bundesanstalt (PTB)

9

O Kieler

Field theoretic aspects of condensed matter physics: An overview

27

Eduardo Fradkin

Majorana fermions in condensed-matter physics

132

AJ Leggett

Majorana fermions in Kitaev spin liquids

139

Joji Nasu

Fundamental and emergent particles in condensed matter and high-energy physics

147

Sinéad M Griffin

Phase transitions, spontaneous symmetry breaking, and Goldstone's theorem

158

Jürg Fröhlich

Higgs and Nambu–Goldstone modes in condensed matter physics

174

Naoto Tsuji, Ippei Danshita, and Shunji Tsuchiya

Higgs and Goldstone modes in cold atom systems

187

Jacques Tempere

Higgs and Goldstone modes in crystalline solids

197

Marco Vallone

Quantum mechanics: Foundations

212

AJ Leggett

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

218

Y Avishai and YB Band

Quantum computation of complex systems

237

Giuliano Benenti and Giulio Casati

Quantum information processing with superconducting circuits: A perspective

246

G Wendin

Braids, motions and topological quantum computing

268

Eric C Rowell

Fibonacci anyon based topological quantum computer

279

Tapio Simula

Fractional quantum Hall effect in semiconductor systems

285

Zlatko Papic and Ajit C Balram

From the integer to the fractional quantum hall effect in graphene

308

Mark O Goerbig

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Contents of Volume 1

Fractional quantum Hall effect at the filling factor n = 5/2

324

Ken KW Ma, Michael R Peterson, VW Scarola, and Kun Yang

Interacting Dirac fermions and the rise of Pfaffians in graphene

366

Vadym Apalkov and Tapash Chakraborty

On the stability of Laughlin's fractional quantum hall phase

383

Nicolas Rougerie

Fractional statistics in low-dimensional systems

394

Jon Magne Leinaas and Jan Myrheim

Anyon collisions and fractional statistics

402

G Fève

Fractional statistics, gauge invariance and anomalies in condensed matter physics

417

Jürg Fröhlich

Properties of 2D anyon gas

450

Douglas Lundholm

Abelian and non-abelian anyons

485

Sumathi Rao

Statistical anyons

500

Nathan M Myers

Recent developments in fractional Chern insulators

515

Zhao Liu and Emil J Bergholtz

Quantum Hall states in higher Landau levels

539

Jakob Yngvason

Quantum Hall effect

553

Jürgen Weis

The network model and the integer quantum Hall effect

567

JT Chalker

Photonic quantum Hall effects

575

Daniel Leykam and Daria Smirnova

The Anomalous Hall Effect

587

Dimitrie Culcer

Monolayer and bilayer graphene

602

Edward McCann

Quantum spin Hall effect

623

Shoushu Gong and DN Sheng

Quantum hall and synthetic magnetic-field effects in ultra-cold atomic systems

629

Philipp Hauke and Iacopo Carusotto

Quantum Hall phases of cold Bose gases

640

Nicolas Rougerie and Jakob Yngvason

Valley Currents in Graphene

652

Satoshi Moriyama and Yoshifumi Morita

Bulk-edge correspondence

659

Yasuhiro Hatsugai

Berry phase and geometrical observables

670

Raffaele Resta

Topological properties of Dirac and Weyl semimetals AA Burkov

681

Contents of Volume 1

Topological insulators

xxv 690

Yoichi Ando

Bloch electrons in a magnetic field

700

Jean Bellissard

The Ten Martini problem

712

Jean Bellissard

Hofstadter butterfly in graphene

724

Wei Yang and Guangyu Zhang

Tight-binding method in electronic structure

732

DA Papaconstantopoulos, MJ Mehl, AG Chronis, and MM Sigalas

Electronic Structure Calculations: Plane-Wave Methods

756

ML Cohen

Plasmons in monolayer and bilayer graphene

765

Xue-Feng Wang and Tapash Chakraborty

Electron gas (theory)

778

G Vignale and MP Tosi

The Jahn-Teller effects

797

IB Bersuker

Fermi surface measurements

815

SB Dugdale

Pseudopotential methods

833

James R Chelikowsky

Effective masses

849

Jeanlex Soares de Sousa

Electronic structure: Impurity and defect states in insulators

856

AM Stoneham, J Strand, and AL Shluger

Density functional theory

867

Yusuke Nomura and Ryosuke Akashi

The sign problem in quantum Monte Carlo simulations

879

Gaopei Pan and Zi Yang Meng

Quantum transport and electron-electron interactions in one dimension

894

Pedro Vianez and Christopher Ford

Quantum physics in one dimension

905

T Giamarchi

The Hubbard model and the Mott–Hubbard transition

914

Massimo Capone

Electron quantum optics: A testbed for the Luttinger paradigma

924

Dario Ferraro and Maura Sassetti

Theoretical approaches to liquid helium

934

Jordi Boronat

The dynamics of quantum fluids

946

Henri Godfrin and Eckhard Krotscheck

Quantum fluids of light

959

Iacopo Carusotto

Floquet states Naoto Tsuji

967

xxvi

Contents of Volume 1

Pump-probe spectroscopy for non-equilibrium condensed matter

981

Ryusuke Matsunaga and Kozo Okazaki

Nonequilibirum physics, numerical methods

990

Philipp Werner

1/f noise in quantum nanoscience Giuseppe Falci, Pertti J Hakonen, and Elisabetta Paladino

1003

Quantum Hall effect and modern-day metrology Klaus von Klitzing, Max Planck Institute for Solid State Research, Stuttgart, Germany © 2024 Elsevier Ltd. All rights reserved.

Introduction International system of units (SI units) Universality of quantized Hall resistance Is the quantized Hall resistance identical to h/e 2? Integration of electrical quantum units into the official SI system Conclusion References

1 2 2 4 5 7 7

Abstract The discovery of the quantum Hall effect in 1980 triggered a revolution in metrology. As a result, a modified system of units based on natural constants with a fixed value for the von Klitzing constant h/e2, where h is the Planck constant and e is the elementary charge, was established worldwide on May 20, 2019. Combined with the Josephson effect, all electrical units today are based on natural constants. Moreover, the quantum Hall effect enabled the kilogram to be defined as an internationally standardized unit of mass based on a fixed value for the Planck constant via the Kibble balance.

Key points

• • • • • • • • •

The discovery of the quantum Hall effect in 1980 was based on experiments that showed the presence of plateaus in the Hall resistance, insensitive to the amount of localized electron states due to disorder. High-precision measurements of the quantized Hall resistance performed at various metrology institutes demonstrated the universality of the quantized value. A conventional value for the quantized Hall resistance was introduced for metrological applications between 1990 and 2019 (conventional von Klitzing constant). The quantized resistance is not affected by the gravitational potential but may have corrections at the 10−20 level due to vacuum polarization. The CODATA least-squares fit of fundamental constants failed to detect an inexactness of the quantized Hall resistivity with the fundamental constant h/e2. The revised International System of Units introduced in 2019 with fixed values for the fundamental constants h and e integrates the quantized Hall resistance as a resistance standard into the SI system. The electric quantum standard for the resistance (QHE) and voltage (Josephson effect) allows the mass unit kilogram to be defined in terms of the Kibble balance based on a fixed value for the Planck constant h. The electrical units farad and henry can be traced back to the von Klitzing constant h/e2 via the ac quantum Hall effect. The parallel and series connection of quantum Hall devices allows resistance standards to be defined with arbitrary values.

Introduction Modern metrology, the science of measurements, originated from the French Revolution, when the metric system with its prototypes for the meter and kilogram was introduced. The rotation frequency and circumference of the Earth as well as one cubic decimeter of water originally served as references for global units of time, length and mass, respectively. This system was internationally adopted by the Metre Convention in 1875. However, Scottish physicist J.C. Maxwell criticized this development, stating that, “The properties of the earth are not stable enough for a precise definition of the units length, time and mass. The wavelength, frequency and mass of atoms are much more suitable for time independent and stable basic units” (Maxwell, 1870). Maxwell’s theoretical objections could not be implemented experimentally with high precision until the 1970s, when both the second and the meter were redefined in terms of atomic properties as follows: The standard ephemeris second based on the Earth’s orbit around the Sun was replaced by a fixed value of DnCs, which denotes the transition between the two hyperfine ground states of cesium-133, and the meter was redefined by a fixed value for the wavelength of a special spectral line of krypton-86. However, the finite linewidth and asymmetry of this spectral line were subsequently found to limit the accuracy of this definition of the meter. Consequently, the meter was redefined yet again in 1983 by something that is even more reliable than atomic properties: The value of a fundamental constant. This radical new concept of defining a unit of measure by fixing the value of a fundamental constant became the blueprint for today’s International System of Units and is the most important application of the quantum Hall effect (QHE) (von Klitzing et al., 1980).

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00197-9

1

2

Quantum Hall effect and modern-day metrology

International system of units (SI units) The profound industrial importance of electricity, thermodynamics, optics and chemistry requires globally recognized basic units of measure, in addition to the historically most important units of kilogram, meter and second. The International System of Units (SI system) was introduced in 1960 and consists of the seven base units of second, meter, kilogram, ampere, kelvin, mole, and candela (BIPM, SI Brochure, 2019). This system can accommodate an unlimited number of derived units, all of which can be expressed in terms of these seven base units. The 22 most common coherent derived SI units are generally referred to by their own nomenclature, such as ohm for electrical resistance, which is defined in terms of the base units as 1 O ¼ 1 kg m2 s−3 A−2. The practical importance of standardized units for global commerce requires that the base units can be reproduced worldwide with high accuracy and that these units remain stable over time. Fundamental constants are best suited to fulfill these requirements. For example, in the late 1800s, Max Planck, who discovered the two fundamental constants known today as the Planck and Boltzmann constants h and k, respectively, also developed units for length, mass, time and temperature based on a combination of fundamental constants. Planck determined that “. . .with the help of fundamental constants, we have the capability to establish units of length, time, mass, and temperature, which necessarily retain their significance for all cultures, even unearthly and nonhuman ones” (Planck, 1900). Unfortunately, these Planck units with length scales of 10−35 m or mass units of the order of 10−10 kg could not be obtained with high precision and were therefore not useful for practical applications. Another 80 years would elapse before the velocity of light was chosen as the fundamental constant in terms of which all base units could be defined in today’s International System of Units. With the development of lasers, it became possible to measure the velocity of light with such high precision that this replaced the previous definition of the velocity of light in terms of the meter, which had formerly been defined based on the wavelength of 86Kr. Consequently, the official definition of the meter as the base unit of length was redefined in 1983 “by taking the fixed numerical value of the speed of light in vacuum, c, to be 299,792,458 when expressed in the unit m s–1, where the second is defined in terms of the cesium frequency DnCs” (BIPM, SI Brochure, 2019). This definition is identical to a fixed value for the velocity of light and launched the incorporation of fundamental constants into today’s SI system, which was established worldwide as the new International System of Units on May 20, 2019. This landmark accomplishment would not have been possible without the discovery of new quantum phenomena. The theoretical prediction of the Josephson effect (Josephson, 1962) and the experimental demonstration by Tsai et al. (1983) of a voltage based on fundamental constants with a reproducibility of 2 parts in 1016 paved the way for further applications of fundamental constants in metrology. The US National Bureau of Standards was the first national metrology institute to introduce a Josephson voltage standard (Field et al., 1973). However, it was not possible simply to replace a base unit with this new “quantum volt”, meaning that the volt had to be defined as a unit of electricity outside the official International System of Units. Different countries used different values for the Josephson constant KJ ¼ 2e/h in order to replace electrochemical Weston cells with the Josephson effect as their voltage standard. Four different voltage standards with differences of up to 6 ppm were adopted by the US, France, Russia and the rest of the world. Ultimately, however, this changed with the discovery of the QHE and the worldwide introduction of conventional standards for electrical quantum units in 1990 (Quinn, 1989).

Universality of quantized Hall resistance From the very beginning of the “quantum Hall age”, the high precision of the quantized Hall plateaus was the most surprising finding. Experiments performed in 1980 at the High Magnetic Field Laboratory in Grenoble, France, (see Fig. 1) showed for the first time that the Hall resistance under the condition of a fully occupied Landau level agrees with the fundamental value of h/e2, where h is the Planck constant and e is the elementary charge, and is insensitive to device properties. This quantization on a macroscopic level was unexpected. The article by Jürgen Weis (2022) in this encyclopedia provides an introduction to this quantum effect within a model of independent charge carriers. More general explanations of the QHE are based on gage arguments by Laughlin (1981) and topological numbers (Niu et al., 1985; Hastings and Michalakis, 2015). However, at the time the QHE was discovered, it was assumed that disorder in a device influences the Hall effect and that localized states present in the tails of a Landau level are missing in Hall effect measurements (Aoki and Kamimura, 1977). On the other hand, extensive calculations of the Hall effect in two-dimensional systems by Ando et al. (1975) indicated that impurity bands, separated from a Landau level, do not influence the ideal value of the Hall effect for a fully occupied Landau level. Experimental Hall effect data had been published as early as 1978 (Englert and von Klitzing, 1978; Kawaji, 1978). However, the carrier density for these measurements had always been calibrated on the basis of the gate capacitance and the threshold voltage of the device, but not related to the degeneracy of a Landau level. In addition, the discovery of the QHE in 1980 highlighted that the Hall resistance (calculated directly from the measured Hall voltage and the current through the device) is identical to the resistivity tensor component rxy used in theoretical calculations. The first manuscript to describe the discovery of the QHE was submitted to Physical Review Letters and originally entitled “Realization of a resistance standard based on natural constant.” It remains today the most important application of the QHE in metrology. Peer reviewer Barry Taylor at the National Bureau of Standards requested that the title be changed to “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance” (von Klitzing et al., 1980). This change was justified by the fact that, at the time the QHE was discovered, the uncertainty in demonstrating a resistance in terms of SI ohms via the calculable Thompson and Lampard (1956) capacitor was less than the uncertainty regarding the value of

Quantum Hall effect and modern-day metrology

3

Fig. 1 “Birth” of the quantum Hall effect. Original data (x-y recorder) of resistance measurements on the silicon field effect transistor shown above.

von Klitzing, K. et al. (1980)

25812.68 (8) Ω

BIPM (France)

25812.809 (2) Ω

PTB (Germany)

25812.802 (3) Ω

NBS (USA)

25812.807 (1) Ω

LCIE (France)

25812.802 (6) Ω

NRC (Canada)

25812.814 (6) Ω

NPL (UK)

25812.809 (1) Ω

ETL (Japan)

25812.806 (7) Ω

IMS (Russia)

25812.807 (8) Ω

VSL (The Netherlands)

25812.802 (5) Ω

NIM (China)

25812.806 (16) Ω

EAM (Switzerland)

25812.809 (4) Ω

conven onal von Klitzing constant

25812.807 (0) Ω RK-90

Fig. 2 Summary of high-precision measurements of fundamental quantized Hall resistance up to 1988, starting with the value of the original publication (von Klitzing et al., 1980). The value for the conventional von Klitzing constant was adopted internationally in 1990 (Quinn, 1989).

the fine-structure constant a. With the introduction of a fixed value for the velocity of light in 1983, however, the inverse fine-structure constant was in fact found to be identical to h/e2, apart from the exactly known value for the proportionality constant 2 e0 c  (2p  29.9792458)−1 O−1. (Note that the permittivity constant e0 is no longer a fixed number in the current SI). Ironically, the original title of the manuscript “Realization of a resistance standard. . .” would have proved more appropriate today. As shown in Fig. 2, many National Metrology Institutes (NMI) have since confirmed the reproducibility of the quantized Hall resistance measured with various devices (Taylor and Witt, 1989). The stability of this quantum resistor was so high that a time-dependent variation of the ohm (represented by a wire resistor) officially established by the U.S. National Bureau of Standards was measured to have a drift of −0.055 ppm/year (Fig. 3). Different NMIs observed similar variations of the national resistance

4

Quantum Hall effect and modern-day metrology

Relave change in ΩNBS (ppm)

0.10 DRIFT = −0.055 ppm/year

0.05 0.00 −0.05 −0.10 −0.15 −0.20 1983

1984

1985

1986

1987

Date of quantum Hall effect measurement Fig. 3 National Bureau of Standards (NBS) ohm drift based on QHE measurements (https://www.nist.gov/system/files/documents/calibrations/87mscohm.pdf).

standards, originating mainly from strain in the wire. The stability and reproducibility of the quantized Hall resistance was so convincing that, on January 1, 1990, a fixed value for the conventional von Klitzing constant of RK-90 ¼ 25,812.807 O was adopted worldwide for all resistance calibrations based on the QHE (Quinn, 1989). At the same time, a global value for the conventional Josephson constant was fixed, which resulted in very reproducible electrical units outside the official SI units. Delahaye and Jeckelmann (2003) published a guideline that describes the tests and precautions necessary for reliable precision measurements of the quantized Hall resistance with an uncertainty of only a few parts in 109, according to the official recommendation of the Consultative Committee for Electricity and Magnetism (CCEM).

Is the quantized Hall resistance identical to h /e 2? High-precision comparisons obtained using different quantum Hall devices, the results of which included not only the integer QHE but also the fractional quantum Hall effect (FQHE) and the quantum anomalous Hall effect (QAHE), demonstrated the universality of quantized Hall resistance up to an experimental uncertainty of only a few parts in 1011 as shown in Fig. 4. However, this does not prove that the quantized resistance is identical to h/e2 without corrections. Taylor and Cohen (1991) applied a least-squares adjustment of all experimental data to determine the best value for the fine-structure constant (excluding QHE data) to establish that better consistency is achieved if the measured quantized Hall resistance is about 0.06 ppm greater than the corresponding h/e2 value. However, the 2006 CODATA least-squares adjustment of fundamental constants could not identify a correction for the relation RK ¼ h/e2 within the uncertainty margin of 2 parts in 108. The metrology community accepted the conclusion that there is no experimental evidence for the inaccuracy of KJ ¼ 2e/h nor that of RK ¼ h/e2 (Mohr et al., 2008). Another test to verify the correct description of electric quantum units by fundamental constants resulted from the development of electron pumps (Pekola et al., 2013). Such pumps allow the controlled charging and discharging of a quantum dot with one electron leading to a quantum current I ¼ ef, where f is the pumping frequency, which can be calibrated from the atomic clock. The combination of an electron pump with a quantum Hall resistor and a Josephson voltmeter (quantum metrology triangle QMT)

Fig. 4 Universality of quantized Hall resistance. Comparison between different two-dimensional electron systems including fractional quantum Hall effect (FQHE) and quantum anomalous Hall effect (QAHE) (Hartland et al., 1991; Schopfer and Poirier, 2007; Janssen et al., 2012; Woszczyna et al., 2012; Ahlers et al., 2017; Götz et al., 2018; Okazaki et al., 2022).

Quantum Hall effect and modern-day metrology

5

showed no inconsistency at 1 ppm (Scherer and Camarota, 2012). The accuracy of these experiments is limited by the electron pumps, which produce relatively small currents of approximately 100 pA. Advanced QMT experiments will reduce the uncertainty by about one order of magnitude (Scherer and Schumacher, 2019) and will enhance confidence in our theoretical understanding and the consistency of the quantum electrical effects. Some theoretical publications have discussed various possible corrections of the relation RK ¼ h/e2. Even the finite width W of a Hall device in relation to the diameter of the classical cyclotron orbit could contribute to a correction (Shapiro, 1986). However, experiments using devices with different widths ranging from 1 mm to as small as 10 mm showed no width dependence of the quantized Hall resistance within the margin of experimental uncertainty of 1 part in 109 (Jeanneret et al., 1995). Other theoretical publications have discussed the influence of gravity on the QHE. For example, Hehl et al. (2004) found no influence of the gravitational field if it is perpendicular to two-dimensional electron gas (2DEG), but that an additional electric field due to gravity does exist if the gravitational field is parallel to the Hall field. This was analyzed in detail by Hammad et al. (2020). The mere force of the Earth’s gravitational field on the electron mass should lead to an electric field of 0.55  10−10 V m−1, which is about 13 orders of magnitude less than the Hall field used for high-precision QHE measurements. So far, no experimental data has been published that evaluates the influence of gravity on the quantized Hall resistance. Nevertheless, the gravitational field is not expected to have an effect as long as the gravitational potential does not change within the plane of the two-dimensional electronic system. Penin (2009) took a quantum electrodynamic (QED) approach to predict a small correction to the von Klitzing constant due to vacuum polarization in strong magnetic fields with the following result:   RK  h=e2 1  10 −20 ðB=10 TÞ2 : Such a correction of approximately 1 part in 1020 under a typical magnetic field of 10 T in QHE experiments is negligible for metrological applications of the QHE, meaning that all measurements of the quantized Hall resistance are consistent with the equation RK ¼ h/e2.

Integration of electrical quantum units into the official SI system

Recommended value RK = h/e² (CODATA)

In 1988, the least-squares adjustment of all measurements of the quantized Hall resistance yielded a value of RK ¼ 25,812.807 (5) O (Taylor and Witt, 1989). The uncertainty margin of some 2 parts in 107 was dominated by the uncertainty inherent in expressing ohm in terms of SI units. As the reproducibility of the QHE resistance was significantly greater, a fixed conventional value of RK-90  25,812.807 O was introduced in 1990 for metrological applications. This led to international uniformity of all resistance calibrations with uncertainties of less than 10−8. However, such calibrations were outside the official SI system, and more accurate measurements of h/e2 showed an increasing discrepancy between the von Klitzing constant RK ¼ h/e2 and the conventional value of RK-90 as shown in Fig. 5. Similar discrepancies appeared between the Josephson constant KJ ¼ 2e/h and the conventional value of KJ-90  483,597.9  109 Hz V−1. To prevent quantum units from diverging ever further from SI units, the idea was proposed to integrate the von Klitzing constant and the Josephson constant into the International System of Units. However, this would mean abandoning the definitions of two established base units. The breakthrough solution came from Steiner et al. (2005), who presented high-precision measurements with a watt balance (known since 2017 as the Kibble balance). This created the capability to trace the mass unit of the kilogram back to the Planck constant h with sufficient accuracy. In that experiment, the mechanical force of a mass is compensated by the electrical force of a current in a magnetic field, calibrated in accordance with electrical quantum standards. Hence, electrical quantum units gave way to a new definition of the kilogram based on the fundamental constant h. 25812.807 7 Ω

25812.807 6 Ω

10−9

25812.807 5 Ω

25812.807 4 Ω

25812.807 3 Ω

conven onal value (1990): 25,812.807000 Ω 1998

2002

2006 2

Fig. 5 Recommended CODATA values for the von Klitzing constant RK ¼ h/e .

2010

2014

6

Quantum Hall effect and modern-day metrology

Evidence that the kilogram artifact at the BIPM was not stable in time (Girard, 1994) supported the idea of replacing the kilogram and introducing a new system of units (Mills et al., 2005; Schlamminger and Haddad, 2019). Establishing this revised system was the focus of a historic discussion at the Royal Society in London in 2005 (Quinn and Burnett, 2005). With fixed values for e and h, not only could the base units of ampere and kilogram be related to fundamental constants, but the Josephson and von Klitzing constants could also be integrated into the revised International System of Units. The contribution of C. Borde (2005) at that discussion made it clear that electric metrology is in the midst of a paradigm shift that will play a key role for the entire field of metrology. A revised international system of units was proposed by Mills et al. (2006), but 12 years would elapse before participants at the 26th General Conference on Weights and Measures voted unanimously to implement a new International System of Units as of May 20, 2019. The fixed values of the constants of nature, which form the basis of today’s modified International System of Units, are summarized in Fig. 6. Practical realizations of the SI units have since been published (BIPM, SI Brochure, 2019), which, for example, are defined for the electrical unit ohm as follows: The ohm is the electric resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere, the conductor not being the seat of any electromotive force. The ohm can be defined by using the QHE in a manner consistent with the CCEM Guidelines (Delahaye and Jeckelmann, 2003) and using the value of the von Klitzing constant RK ¼ 25,812.8074593045 O.

With this recommendation, issued some 40 years after the discovery of the QHE, the vision of the original manuscript describing the application of this quantum effect, which, ironically enough, was rejected, has come to fruition. With the discovery of graphene as an ideal two-dimensional electron system, it has become simpler to apply the QHE as a resistance standard. In particular, graphene on SiC (epigraphene) exhibits many advantageous features compared to those of GaAs heterostructures, a well-established material. Ribeiro-Palau et al. (2015) demonstrated that the Hall resistance can be accurately quantized to within 1  10−9 over a 10-T-wide range of magnetic field starting as low as 3.5 T. Moreover, the same precision can be achieved at temperatures as high as 10 K and device currents up to 500 mA. Further research is needed to optimize epitaxial graphene for quantum Hall applications (Meng et al., 2022). A special feature of the QHE is that not only the four-terminal Hall resistance but also the two-terminal resistance between any two contacts at the boundary of a QHE device has the quantized resistance. Errors due to finite contact resistance can be reduced to a negligible value by the multiple-series connection scheme (Delahaye, 1993). This created the possibility to establish quantum Hall array standards (QHARS) by combining QHE devices in parallel and in series with quantized resistances at arbitrary levels (Poirier et al., 2004). A scheme for decade-value standards in the range from 100 O to 1 MO has been presented by Oe et al. (2013). Comparisons of two epigraphene QHARS with 138 devices in parallel (array resistance R  109 O) showed a relative deviation of only 3 parts in 1011 and, compared to a single quantum Hall device, was in agreement with the target value of RK/276 within the uncertainty of 2 parts in 1010 (He et al., 2022). To eliminate lead resistances, the contacts and interconnects were made of NbN superconducting films, which can support currents on the order of 10 mA at 5 T and 2 K. Extending the application of the quantized Hall resistance as the primary resistance standard results from the ac QHE in the frequency range from 50 Hz to 20 kHz. This allows direct measurements of impedances (Schurr et al., 2007; Ahlers et al., 2009) with uncertainties of less than 10−8. Calibration of capacitances (Schurr et al., 2009; Schurr and Lee, 2022) in the range from 10 nF to 10 pF and inductances (Overney and Jeanneret, 2010) in the range from 1 mH to 10H were traced back to the von Klitzing constant. By integrating the conventional electrical quantum units used since 1990 into the International System of Units, the electrical units ohm, farad and henry changed discontinuously on May 20, 2019, by a fractional amount of 17.793  10−9. The corresponding jump for the Josephson volt was nearly four times greater. Such discontinuities for units of measure will never happen again as long as fundamental constants remain constant. Having said that, discussions of possible changes in fundamental constants with time are very important and will focus on the fine-structure constant, which depends on the most important fundamental constants h, e, and c. This has the advantage of being a dimensionless constant independent of units. The most accurate measurements of this constant show that the change is less than 1 part in 1017 per year (Safronova, 2019). Possible variations of the fundamental constants at this level are insignificant in metrology for the foreseeable future.

Quan ty 133Cs hyperfine transi on frequency Speed of light in vacuum Planck constant Elementary charge Avogadro constant Boltzmann constant Luminous efficacy Josephson constant 2e/h von Klitzing constant Fig. 6 Defining constants of the International System of Units (SI).

Symbol Numerical value 'Q Cs 9,192,631,770 c 299,792,458 h 6.62607015 u10–34 e 1.602176634 u10–19 6.02214076 u1023 NA k 1.380649 u10–23 Kcd 683 KJ 483597.848 4 … u109 RK 25812.80745 …

Unit Hz m s–1 J Hz–1 C mol–1 J K–1 lm W–1 Hz V–1 Ω

Quantum Hall effect and modern-day metrology

7

Conclusion The QHE plays a central role in metrology and was crucial for updating the International System of Units based on constants of nature. Even if a complete theory of the QHE for real devices with finite dimensions, electrical contacts, uncontrolled disorder and in the presence of relatively high bias currents has not yet been developed, experiments have demonstrated the reliability of this quantum effect for practical applications. The universality of the quantized Hall resistance has been demonstrated to at least 10 digits, and all high-precision experiments agree with the theoretically predicted value of RK ¼ h/e2 for the quantum Hall plateaus within the experimental uncertainty of 2 parts in 108. The quantized Hall resistance has been officially adopted as a primary realization of the resistance unit ohm. Arrays of quantum Hall devices are used for quantized resistances with arbitrary values in the range from 100 O to 10 MO. Primary realizations of the electrical units farad and henry are possible with the QHE under ac conditions. The development of an electronic kilogram, where the QHE together with the Josephson effect allowed the mass unit to be defined based on a fixed value for the Planck constant h, accelerated the adoption of today’s International System of Units. Fig. 7 shows the official logo of this new system that illustrates the relationship to historically introduced base units.

Fig. 7 Official logo for the base units of the International System of Units. The inner ring contains the constants of nature with fixed values connected to the base units introduced historically (outer ring). © BIPM.

References Ahlers FJ, Jeanneret B, Overney F, Schurr J, and Wood BM (2009) Compendium for precise ac measurements of the quantum Hall resistance. Metrologia 46: R1–R11. Ahlers FJ, Götz M, and Pierz K (2017) Direct comparison of fractional and integer quantized Hall resistance. Metrologia 54: 516. Ando T, Matsumoto Y, and Uemura Y (1975) Theory of Hall effect in a two-dimensional electron system. Journal of the Physical Society of Japan 39: 279–288. Aoki H and Kamimura H (1977) Anderson localization in a two dimensional electron system under strong magnetic fields. Solid State Communications 21: 45–47. BIPM (2019) The International System of Units (SI), 9th edn, updated in 2022. https://www.bipm.org/en/publications/si-brochure. Borde C (2005) Base units of the SI, fundamental constants and modern quantum physics. Philosophical Transactions of the Royal Society A 363: 2177–2201. Delahaye F (1993) Series and parallel connection of multiterminal quantum Hall-effect devices. Journal of Applied Physics 73: 7914. Delahaye F and Jeckelmann B (2003) Revised technical guidelines for reliable dc measurements of the quantized Hall resistance. Metrologia 40: 217–233. Englert T and von Klitzing K (1978) Analysis of rxx minima in surface quantum oscillations on (100) n-type silicon inversion layers. Surface Science 73: 70–80. Field BF, Finnegan TF, and Toots J (1973) Volt maintenance at NBS via 2e/h: A new definition of the NBS volt. Metrologia 9(4): 155. Girard G (1994) The third periodic verification of national prototypes of the kilogram (1988-1992). Metrologia 31: 317–336. Götz M, Fijalkowski KM, Pesel E, Hartl M, Schreyeck S, Winnerlein M, Grauer S, Scherer H, Brunner K, Gould C, Ahlers FJ, and Molenkamp LW (2018) Precision measurement of the quantized anomalous Hall resistance at zero magnetic field. Applied Physics Letters 112: 072102. Hammad F, Landry A, and Mathieu K (2020) A fresh look at the influence of gravity on the quantum Hall effect. European Physical Journal - Plus 135: 449. Hartland A, Jones K, Williams JM, Gallagher BL, and Galloway T (1991) Direct comparison of the quantized Hall resistance in gallium arsenide and silicon. Physical Review Letters 66: 969. Hastings MB and Michalakis S (2015) Quantization of hall conductance for interacting electrons on a torus. Communications in Mathematical Physics 334(1): 433–471. He H, Cedergren K, Shetty N, Lara-Avila S, Kubatkin S, Bergsten T, and Eklund G (2022) Accurate graphene quantum Hall arrays for the new International System of Units. Nature Communications 13: 6933. Hehl FW, Obukhov YN, and Rosenow B (2004) Is the quantum Hall effect influenced by the gravitational field? Physical Review Letters 93: 096804.

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Quantum Hall effect and modern-day metrology

Janssen TJBM, Williams JM, Fletcher NE, Goebel R, Tzalenchuk A, Yakimova R, Lara-Avila S, Kubatkin S, and Falko VI (2012) Precision comparison of the quantum Hall effect in graphene and gallium arsenide. Metrologia 49: 294. Jeanneret B, Jeckelmann B, Buhlmann H-J, Houdre R, and Ilegems M (1995) Influence of the device-width on the accuracy of quantization in the integer quantum Hall effect. IEEE Transactions on Instrumentation and Measurement 44(2): 254–257. Josephson B (1962) Possible new effects in superconductive tunneling. Physics Letters 1: 251. Kawaji S (1978) Quantum galvanomagnetic experiments in silicon inversion layers under strong magnetic fields. Surface Science 73: 46–69. Laughlin RB (1981) Quantized Hall conductivity in two dimensions. Physical Review B 23: 5632. Maxwell JC (1870) Address to the mathematical and physical sections of the British Association, Liverpool, reproduced in: Niven WD (Ed.), The Scientific Papers of James Clerk Maxwell, Vol. 22. Cambridge University Press, Cambridge, 1890, p. 225. Meng L, Panna A, Mhatre S, Rigosi A, Payagala S, Saha D, Tran N, Yeh C, Elmquist R, Hight Walker A, Jarrett D, Newell D, and Yang Y (2022) Quantitative characterization of epitaxial graphene for the application of quantum Hall resistance standard. In: IEEE Proceedings for the Conference on Precision Electromagnetic Measurements 2022, Wellington, NZ. Mills IM, Mohr PJ, Quinn TJ, Taylor BN, and Williams ER (2005) Redefinition of the kilogram: A decision whose time has come. Metrologia 42: 71–80. Mills IM, Mohr PJ, Quinn TJ, Taylor BN, and Williams ER (2006) Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1. Metrologia 43: 227–246. Mohr P, Taylor BN, and Newel DB (2008) CODATA recommended values of the fundamental physical constants: 2006. Journal of Physical and Chemical Reference Data 37: 1258. Niu Q, Thouless DJ, and Wu Y (1985) Quantized Hall conductance as a topological invariant. Physical Review B 31: 3372. Oe T, Matsuhiro K, Itatani T, Gorwadkar S, Kiryu S, and Kaneko N-H (2013) New design of quantized Hall resistance array device. IEEE Transactions on Instrumentation and Measurement 62(6): 1755. Okazaki Y, Kawamura M, Nakamura S, Takada S, Mogi M, Takahashi KS, Tsukasaki A, Kawasaki M, Tokura Y, and Kaneko N-H (2022) Quantum anomalous Hall effect with a permanent magnet defines a quantum resistance standard. Nature Physics 18: 25–29. Overney F and Jeanneret B (2010) Realization of an inductance scale traceable to the quantum Hall effect using an automated synchronous sampling system. Metrologia 47: 690–698. Pekola JP, Saira OP, Maisi VF, Kemppinen A, Möttönen M, Pashkin YA, and Averin DV (2013) Single-electron current sources: Toward a refined definition of the ampere. Reviews of Modern Physics 85(4): 1421. Penin AA (2009) Quantum Hall effect in quantum electrodynamics. Physical Review B 79: 113303. Erratum: Phys. Rev. B 81: 089902. Planck M (1900) Über irreversible Strahlungsvorgänge. Annalen der Physik 306(1): 69–122. Poirier W, Bounouh A, Piquemal F, and André JP (2004) A new generation of QHARS: discussion about the technical criteria for quantization. Metrologia 41: 285–294. Quinn TJ (1989) News from the BIPM. Metrologia 26: 69. Quinn T and Burnett K (2005) The fundamental constants of physics, precision measurements and the base units of the SI. Philosophical Transactions of the Royal Society A 363: 2101–2104. Ribeiro-Palau R, Lafont F, Brun-Picard J, Kazazis D, Michon A, Cheynis F, Couturaud O, Consejo C, Jouault B, Poirier F, and Schopfer F (2015) Quantum Hall resistance standard in graphene devices under relaxed experimental conditions. Nature Nanotechnology 10: 965–971. Safronova MS (2019) The search for variation of fundamental constants with clocks. Annals of Physics (Berlin) 531: 1800364. Scherer H and Camarota B (2012) Quantum metrology triangle experiments: A status review. Measurement Science and Technology 23: 124010. Scherer H and Schumacher HW (2019) Single-electron pumps and quantum current metrology in the revised SI. Annals of Physics 531: 1800371. Schlamminger S and Haddad D (2019) The new International System of Units. The Kibble balance and the kilogram. Comptes Rendus Physique 20: 55–63. Schopfer F and Poirier W (2007) Testing universality of the quantum Hall effect by means of the Wheatstonebridge. Journal of Applied Physics 102: 054903. Schurr J and Lee J (2022) Realization of the farad from the AC quantum Hall resistance at PTB—14 years of experience. IEEE Transactions on Instrumentation and Measurement 71: 1502008. Schurr J, Ahlers FJ, Hein G, and Pierz K (2007) The ac quantum Hall effect as a primary standard of impedance. Metrologia 44: 15–23. Schurr J, Bürkel V, and Kibble BP (2009) Realizing the farad from two ac quantum Hall resistances. Metrologia 46: 619–628. Shapiro B (1986) Finite-size corrections in the quantum Hall effect. Journal of Physics C: Solid State Physics 19: 4709. Steiner RL, Williams ER, Newell DB, and Liu R (2005) Towards an electronic kilogram: An improved measurement of the Planck constant and electron mass. Metrologia 42: 431–441. Taylor BN and Cohen ER (1991) How accurate are the Josephson and quantum Hall effects and QED? Physics Letters A 153: 308–312. Taylor BN and Witt TJ (1989) New international electrical reference standards based on the Josephson and quantum Hall effects. Metrologia 26: 47–62. Thompson AM and Lampard DG (1956) A new theorem in electrostatics with applications to calculable standards of capacitance. Nature 177: 888–890. Tsai JS, Jain AK, and Lukens JE (1983) High-precision test of the universality of the Josephson voltage-frequency relation. Physical Review Letters 51: 316. von Klitzing K, Dorda G, and Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45: 494–497. Weis J (2022) Quantum Hall Effect. this encyclopedia. Woszczyna M, Friedemann M, Götz M, Pesel E, Pierz K, Weimann T, and Ahlers FJ (2012) Precision quantization of Hall resistance in transferred graphene. Applied Physics Letters 100: 164106.

The pulse-driven AC Josephson voltage standard of Physikalisch-Technische Bundesanstalt (PTB) O Kieler, Quantum Electronics, PTB Braunschweig, Braunschweig, Germany © 2024 Elsevier Ltd. All rights reserved. Disclaimer: The commercial instruments and software packages mentioned in this paper merely serve the purpose of adequately specifying the experimental procedure and do not amount to any recommendation or endorsement by PTB whatsoever.

Introduction Fundamental elements of the JAWS JAWS technology JAWS Setup Optical pulse drive and on-chip power divider Applications Conclusion Acknowledgments References

9 10 13 15 16 21 25 25 25

Abstract We present the basic principle of the pulse-driven AC Josephson Voltage Standard, also called “Josephson arbitrary waveform synthesizer” (JAWS). The JAWS is suitable for generating quantized high-precision and arbitrary waveforms in the time and frequency domain with extremely high spectral purity. A brief overview of the fabrication technology for highly integrated JAWS circuits with more than 50,000 junctions per chip will be described. This includes the stacked junction technology with up to 5-stacked junctions. New approaches designed to further increase the output voltage and to simplify the experimental setup are the on-chip power dividers and the optical pulse drive for the JAWS. Some applications of the JAWS that have significant benefits for quantum AC voltage metrology will be presented. Within international research projects, PTB’s JAWS chips have been distributed to NMIs in the Netherlands, the United Kingdom, Norway, Italy, Finland, Turkey, Russia, China and Argentina. The technology transfer from PTB to the Supracon company for commercialization purposes has also been initiated to make JAWS available to other NMIs, to calibration laboratories and to industry in the near future.

Key points

• • • • • • • •

Basic principle of the pulse-driven AC Josephson Voltage Standard (JAWS) Design of integrated circuits for JAWS Key parameter of JAWS: calculable and arbitrary waveforms, quantum precision Circuit fabrication: Nb based thin film and stacked junction technology, fabrication yield Experimental setup Optical pulse-drive and on-chip power divider Applications in Voltage Metrology Conclusion

Introduction This chapter provides an overview of the fundamental principles, practical realization, features and chip design of the AC Josephson voltage standard. To define the electrical units via electrical quantum metrology; the Ohm (quantum Hall effect), the Volt (Josephson effect) and the Ampere (single-electron transport) is one of main tasks of Physikalisch-Technische Bundesanstalt (PTB). The PTB (PTB, 2022) is the national metrology institute of the Federal Republic of Germany. The first such institute in the world, PTB was founded in Berlin as the “Physikalisch-Technische Reichsanstalt “(PTR) in 1887 by Hermann von Helmholtz (its first president) and Werner von Siemens to serve as an interface between science, industry and legal metrology. After World War II, the institute was re-built in Berlin and gained a new, second site in Braunschweig, which was housed in a former research facility. In accordance with the new form of government in Germany established after World War II, the former “Reichsanstalt” was designated “Bundesanstalt” in 1950. PTB is the sole (and thus the highest) authority on the correct and reliable measurements of physical units and forms part of the Federal Ministry for Economic Affairs and Climate Action in Germany. PTB has around 1900 employees and an annual budget of some 200 million euros. Over the years, its advisory board members have included 10 Nobel Prize winners – among them Albert Einstein, Max Planck, Gustav Hertz, Max von Laue and Klaus von Klitzing.

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The so called “Weston cell,” which was previously the electrical voltage standard beginning in 1911, achieved an DC output voltage of about 1 V and precision of about 10−6. This standard was replaced in the 1980s with DC Josephson voltage standards of 1 V/10 V, whose precision was better than 10−9. An AC quantum voltmeter (AC-QVM) based on programmable Josephson voltage standard (PJVS) chips (up to 10 V) was also developed at PTB. Via technology transfer between PTB and the Supracon company (Supracon, 2022), this voltage standard was made commercially available to national metrology institutes (NMIs) and industry worldwide. The AC-QVM only allows waveforms approximated stepwise (i.e., it is only nominally an AC voltage standard). To overcome this issue, a pulse-driven Josephson voltage standard was proposed by NIST (Benz and Hamilton, 1996) and developed at NIST and PTB. This AC Josephson voltage standard is often referred to as the Josephson arbitrary waveform synthesizer (JAWS). In the following section, we will introduce the fabrication of JAWS circuits in PTB’s Clean Room Center via a sophisticated thin-film technology based on Josephson junction properties and stacked-junction technology. The experimental setup used and the most recent developments concerning further improvements to the JAWS will also be discussed, including on-chip power dividers and optical pulse drives. Finally, examples will be given of JAWS applications in electrical quantum metrology where these applications were developed and tested at PTB in close cooperation with our research partners from several NMIs, as well as their impact within the international community.

Fundamental elements of the JAWS The basic principle of JAWS is as follows: a current pulse with a time-dependent pulse-repetition frequency fp(t) transfers N flux quanta F0 through M Josephson junctions (Benz and Hamilton, 1996). The signal voltage V(t) in pulse-mode operation can be calculated via the Josephson equation, i.e., V ðtÞ ¼ N  M  F0  fp ðtÞ

(1)

According to this equation, a time-dependent voltage, which is quantized at all times, is generated at the junction series array output leads, because only flux quanta F0 ¼ h/2e (h is Planck’s constant and e the elementary charge) are transferred and N and M are integer numbers. The frequency fp(t) is very well defined by precise knowledge of the time t. The JAWS delivers quantized arbitrary waveforms, when properly biased. The signal output voltage (RMS) can be obtained from a slight modification of Eq. (1): V signal ðRMSÞ ¼ ASD  N  M  F0  f clock ,

(2)

where ASD is the sigma-delta (SD) code amplitude (details on which are provided below), i.e., a pre-factor in the range 0 < ASD < 1 (Schreier and Temes, 2005). The signal frequency fsignal can easily be calculated from f signal ¼ T SD  f clock =LSD ,

(3)

where TSD is the number of signal periods in the sigma-delta coded waveform, LSD is the length of the sigma-delta code, and fclock is the clock frequency of the pulse pattern generator (PPG), which delivers high-frequency pulses to the Josephson array. For JAWS applications, SNS-type Josephson junctions are used (S ¼ superconductor, N ¼ normal conductor). SNS junctions provide a non-hysteric current-voltage characteristic under microwave irradiation. In pulse-mode operation, Shapiro steps are selected by adjusting the pulse amplitude (see Fig. 1). Large Shapiro steps are generated for all pulse repetition frequencies below the characteristic frequency fc of the SNS junctions: fp  fc (Benz and Hamilton, 1996). The maximum pulse-repetition frequencies most frequently used for the JAWS are the 15 GHz return-to-zero pulse, which is typically limited by the maximum clock frequency of commercially available PPGs. This corresponds to a maximum data rate of 30 Gbit/s. Because a flux-quanta provides only a very small voltage (F 0  2 mV/GHz), many Josephson junctions must be implemented in a series array in order to achieve practicable large output voltages (e.g. 5000 junctions to achieve 100 mV RMS output voltage at

Fig. 1 Current-voltage characteristics of SNS junctions without and with pulse irradiation.

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15 GHz clock frequency). Since the pulse repetition frequency is not constant in time (fp(t) 6¼ const.), the availability of a broadband circuit design is an important precondition for operating the JAWS properly. Fig. 2 shows a typical JAWS circuit layout. Two independent JAWS arrays are integrated on a 10 mm  10 mm silicon chip. The enlarged picture on the right illustrates that the Josephson junctions are embedded in the center line of a coplanar waveguide (CPW). The CPW is matched to a 50 O impedance that corresponds to the impedance of the complete JAWS setup (details on which are provided below). A 50 O termination (load) is realized at the end of the array to prevent pulse reflections. A CPW taper at the HF pulse input side of the chip is implemented to enable a proper Al wire bond of the chip to a printed circuit board (PCB) carrier. At the current input and voltage output leads, LCR filters (Watanabe et al., 2006) are integrated to prevent the propagation of the input pulses along these lines, as this would lead to distortions in the frequency spectrum of the synthesized waveforms. The practical realization of the JAWS, i.e., the way in which an arbitrary waveform is quantized with the JAWS chip, is sketched in Fig. 3. Here, we use a sine waveform for the sake of simplicity. Such a waveform is advantageous because, in an ideal case, there will only be one fundamental tone (the signal frequency) in the frequency spectrum, and no higher harmonics will appear. By means of a higher-order sigma-delta modulation (Kieler et al., 2009) (SD modulation), this analog waveform is digitized into a bit pattern

Fig. 2 Example of JAWS chip circuit layout.

Fig. 3 Sketch of the practical realization of quantized waveforms with JAWS.

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(SD code) which has the restricted values +1, 0 and −1. The aim of a high-quality Sigma Delta modulation is to ensure that the fast Fourier transform (FFT) of this Sigma Delta code is highly pure on the spectral side, that it has a very low noise floor and a quantization noise shifted to very high frequencies (>1 MHz) far beyond the desired signal frequency. At PTB, a second-order modulator with integrated feedback is typically used, i.e., a CIFB modulator topology (CIFB: cascade of integrators with integrated feedback) (Schreier and Temes, 2005). A stable SD modulation is realized if ASD < 1. This bit pattern is stored in the memory of the fast PPG. Typical SD code lengths range from around 2 Mbit to 512 Mbit depending on the desired signal frequency and other properties of the waveform. The ternary pulses (+1, 0, −1) of the PPG are transferred via suitable HF cables to the JAWS chip operating at 4 K. In the Josephson junction array, each input pulse generates a flux pulse. At the array output leads, the quantized waveform can be detected and fed out to a room temperature (300K) environment to be used for applications or further spectral analysis. Ultimately, the JAWS can be considered a quantum standard characterized by these key features:

• • • • • •

high-precision quantized AC voltage source, synthesis of arbitrary waveforms (frequency or time domain), excellent spectral purity (signal-to-noise ratio SNR > 130 dBc), no drift, low noise, wide frequency bandwidth (a few Hz, including 0 Hz, up to the MHz range).

These considerable advantages over semiconductor-based waveform synthesis make the JAWS highly suitable for metrological applications. JAWS signals can be synthesized in a very wide frequency range from a few Hertz up to the Megahertz range and DC voltages are possible too. We will briefly explain the relevance of these features by presenting some waveform examples and data. Fig. 4 shows the frequency spectra of two sine waveforms to demonstrate the concept of “calculable” waveforms. The first waveforms have a signal frequency of 1111.11 Hz and a signal voltage of 111.11 mV RMS, yielding an “11  1” sine. This figure also demonstrates spectral purity: no higher harmonics are above the noise floor and the signal-to-noise ratio is about 123 dBc, which corresponds to the dynamic limit of the spectrum analyzer used. The small distortion peak at about 3.3 kHz is related only to the spectrum analyzer (Benz et al., 2015), whereas the JAWS chip is well quantized, while operating in the quantum-locked state (Burroughs et al., 2005) with sufficient operation margins. The second spectrum shows a similar waveform, i.e., a “1 to 9” sine. Here, the signal frequency is 1234.5 Hz and the signal amplitude is 67.89 mV RMS. Other frequency spectra (Fig. 5) show the wide frequency range of the JAWS. Using an array of 12,000 junctions, different signal frequencies of spectrally pure sine waveforms from 10 Hz to 1 MHz were synthesized at a signal amplitude of 100 mV RMS. Finally, two arbitrary waveforms are presented in Fig. 6. On the left, an arbitrary signal in the frequency domain is shown: a multitone waveform (132 tones) with different signal amplitudes that resembles the word “JAWS” (Behr et al., 2012). On the right,

Fig. 4 Frequency spectra of two example sine waveforms.

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Fig. 5 Frequency spectra of different sine waveforms from 10 Hz to 1 MHz.

Fig. 6 Left: arbitrary frequency domain waveform, right: arbitrary time domain waveform.

an arbitrary time-domain waveform illustrates an ideal heart-beat signal. Both examples demonstrate that virtually any arbitrary waveform, and even complex waveforms, can be created. To confirm the quantum precision of the JAWS, we performed a direct comparison between the JAWS and the PTB AC-QVM for voltages at 1 V RMS and 250 Hz. Excellent agreement was achieved: VJAWS - VQVM ¼ (3.5  12) nV (Behr et al., 2015). This comparison was a highlight paper of the journal “Metrologia” in 2015. This direct comparison achieved even better agreement for DC voltages at a 1 V peak: VJAWS - VQVM ¼ (0.3  04) nV (Behr and Kieler, 2020). This quantum precision is a precondition for any application in voltage metrology. In (Kieler et al., 2017), we were able to make 16 arrays operate in parallel with a total number of 162,000 junctions, achieving an output voltage of 2.25 V RMS. This represented an important milestone toward achieving 7 V RMS with the JAWS in order to increase the number of metrological applications possible. Other current JAWS developments include simplifying the experimental setup in order to lower the overall costs of a complete JAWS system, increasing the number of applications to provide greater benefits to NMIs and industry and initiating the technology transfer necessary to make the JAWS commercially available.

JAWS technology JAWS circuits are fabricated at PTB’s Clean Room Center (class ISO 5). This clean room is fully equipped for the high-quality fabrication of superconducting circuits based on Nb as a superconductor. The main fabrication tools used in this process are a cluster sputtering system for highly reproducible deposition conditions and an electron-beam lithography system for high alignment precision and fast prototyping (thus rendering the use of masks unnecessary, in contrast to UV lithography). Previously, SNS Josephson junctions with HfTi as a normal-metal barrier were used at PTB for JAWS applications (Kieler et al., 2007a; Kieler et al., 2007b; Kieler et al., 2009). These junctions were characterized by a high critical current density in the range of >50 kA/cm2. Consequently, the junction size was limited to the sub-mm range. For large arrays, we observed that these junctions were not suitable because heating effects caused by the high current density degraded the properties of the junctions. This issue prompted the use of SNS Josephson junctions with NbxSi1-x as the normal-metal barrier material (Baek et al., 2006; Kieler et al., 2013a). The SNS

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multilayers are deposited in the cluster sputtering system, which consists of six chambers in total and is fully automatic. The NbxSi1-x barrier is deposited in a co-sputter process from two targets (Nb and Si) onto the rotating 3-in. Si wafer. The homogeneous and reproducible deposition conditions are very important to achieve a low parameter spread of the junction’s characteristic properties. These junction properties can be adjusted nearly independently over a wide range by choosing the composition x and the thickness of the barrier d as shown in Fig. 7, where the critical current density is plotted against the characteristic voltage. The Josephson junction operation frequency can be tuned from about 1 GHz to 150 GHz (and beyond). An x of about 0.2 and a thickness d of about 30–40 nm are typical parameters in JAWS applications. Such SNS junctions are advantageous in that they can be deposited and patterned in junction stacks, which increases the integration density of the JAWS circuits. To manufacture 5-stacked junctions, a 12-layer multilayer is first deposited in-situ in the cluster system, cf. Fig. 8. On top of a thermally oxidized wafer, a thin etch stop layer made of Al2O3 is deposited first to prevent the plasma from etching into the SiO2. The 5-stacked junctions consist of 6 Nb and 5 NbxSi1-x layers. The resulting total layer structure is as follows: Si/SiO2/Al2O3/Nb/NbxSi1-x/Nb/NbxSi1-x/Nb/ NbxSi1-x/Nb/ NbxSi1-x/Nb/NbxSi1-x/Nb. A modified “window” process was developed at PTB (Kieler et al., 2021) to fabricate up to 5-stacked Josephson junction series arrays with a high yield. To achieve this, the thicknesses of the Nb and SiO2 resists were increased. Additionally, the SiO2 was deposited not only by means of PECVD, but also in a thin ALD layer (before PECVD) to ensure perfect edge coverage for the tall junction stacks. This was necessary to avoid shorts at the uppermost junctions in the stack, which had been problematic without ALD. A soft CMP step was also introduced for the SiO2 layer to ensure smoothness of the surface before deposition of the Nb-wiring layers. This was done to prevent this superconducting layer from having a weak current capability due to poor edge coverage. The complete fabrication process consists of six main fabrication steps. First, the Josephson junctions are defined by high-rate RIE-ICP with SF6 (reactive ion etching with inductive coupled plasma); here, the junction sidewall angle is nearly 90 degrees. This ensures a small parameter spread from one junction floor to the next. Second, the Nb base electrode, which is the lower wiring electrode, is patterned. Third, electrical isolation takes place via atomic layer deposition (ALD) and plasma enhanced chemical vapor deposition (PECVD) SiO2 and the contact windows on top of each junction stack are opened by via RIE-ICP with CHF3. Fourth, additional deposition and patterning of the Nb wiring electrode take place via RIE-ICP. Fifth, deposition takes place for AuPd on-chip resistors for impedance matching of the CPW and the on-chip filters by means of a “lift-off” process. Finally, the wire-bond pads are opened by via RIE-ICP to establish contact between the chip and an appropriate PCB carrier. The total process involves 16 layers and about 40 fabrication steps. The clean room cycle time for two wafers is about 3 weeks. More details about this process can be found in (Kieler et al., 2021) for 3-stacked junctions; this process is similar to the modified 5-stacked junction process

Fig. 7 Critical current density vs. characteristic voltage of SNS junctions with a NbxSi1-x barrier.

Fig. 8 Multilayer sequence of 5-stacked SNS junctions with NbxSi1-x barriers deposited in situ.

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Fig. 9 Cross-section SEM image of a JAWS array with 5-stacked Josephson junctions.

presented in (Kieler et al., 2015). Fig. 9 shows a cross-section SEM image of a JAWS array with 5-stacked Josephson junctions. In this picture, we can clearly recognize the steep junction stacks as required. For the stack on the right, a small under-etch into the stack can be seen. This may be the reason for the slight increase in the parameter spread of the critical current. Compared to the total junction area of 4.4 mm  13.6 mm, this effect is small enough to ensure a high-quality current-voltage characteristic (IVC). Due to the relatively thick layers combined with long etching times and resist problems, we observed large Nb fence structures (also visible in Fig. 9) that mainly consist of residues of the photo resist and nonvolatile etch products; however, these fence structures did not negatively impact the operation of the JAWS circuits. Fabricating the highly integrated circuits of JAWS chips, which have many thousands of Josephson junctions, requires a high yield of the functional junctions and a low junction parameter spread. Some information on 3-stacked junctions taken from the wide range of available data will be presented below. The critical current density distribution on a 3-in. wafer shows a deviation of 4.6% from the mean value, which is acceptable for functional JAWS arrays with sufficiently wide operation margins in pulse operation mode. We analyzed the data for 200 arrays that had been fabricated on 20 wafers with a minimum junction count of N ¼ 5000 in each array. This corresponded to a total junction count of N ¼ 1,600,000. Only 48 junctions were defective in the sense that the junction were shorted. This is equivalent to a junction yield of 99.997%, which is more than sufficient for our needs. Clearly, the shorted junctions fail to contribute to the IVC as they are “missing” in Shapiro steps. However, the circuit itself is still functional and can be used for JAWS applications, as the signal voltage levels can be adjusted by choosing the SD code amplitude and the clock frequency. Because these two parameters can be selected with arbitrary precision, there are no limits to the exactness with which the desired signal amplitudes can be adjusted. In contrast to PJVS, the characteristic voltage of a junction does not restrict the voltage resolution of JAWS. Voltage values far below the characteristic voltage can be tuned (see “calibration of nanovoltmeters” in the “Applications” section). For the 5-stacked junction arrays, the database available is smaller but allows spreads and yields to be estimated. Measuring 28 arrays from two wafers yielded the following data for the spread of the critical current: 1. intra-array: 2.4% 2. intra-wafer: 3.6% 3. inter-wafer: 6.9% This data is acceptable for our purposes. The intra-wafer uniformity of the critical current density was determined for one of the 5-stacked junction wafers (3-in.): (6.66  0.07) kA/cm2. Before we implemented ALD into the process, this yield was very poor. By analyzing the data from 58 arrays located on eight wafers, we realized that, on seven wafers, the upper floor of the stack was systematically shorted completely. The reason for this was insufficient edge coverage of the PECVD SiO2 at the top edge of the high pillar, where notches occurred. Only on one wafer were the junctions not shorted. Here, we measured a junction yield of 99.88%. Ever since the implementation of ALD, we have analyzed the data from 28 arrays on two wafers. No stacks were shorted and only 53 out of 440,000 junctions were “missing,” corresponding to a junction yield of 99.99%; this result is slightly lower than that of 3-stacked junctions but still very good. Realizing such 5-stacked junction technology entails significant benefits, as we can synthesize an output voltage of 1 V RMS (2.83 V peak-peak) with four arrays integrated on only two chips instead of eight arrays on four chips (Kieler et al., 2015). The frequency and time domain of a sine waveform generated using 60,000 junctions at 500 Hz is shown in Fig. 10.

JAWS Setup At PTB, eight JAWS systems are used for different applications. Fig. 11 shows the eight-channel JAWS setup. Using this setup, 1.5 V RMS can be synthesized with 93,000 junctions arranged on 8 arrays (i.e., four chips). These four chips are mounted on four specially designed PCBs made of Rogers3006 (Kieler et al., 2013b). These PCBs are arranged in a cryoprobe, as shown in the upper inset, for operation in a liquid helium dewar at 4 K. An eight-channel PPG (Sympuls) that provides ternary pulses, an eight-channel bias device and a spectrum analyzer (PXI 5922, National Instruments) are the main components of the supply electronics for operation

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Fig. 10 Frequency and time domain of a sine waveform at 1 V RMS and 500 Hz synthesized with two chips and 60,000 junctions.

Fig. 11 Photograph of the eight-channel PTB JAWS setup. The inset shows details of the cryoprobe with mounted JAWS chips.

of JAWS circuits. To control the whole system and perform the measurements, customized software (made with Labview) was programmed. In 2013, we also demonstrated how JAWS circuits can be operated in a pulse-tube cryocooler (Kieler et al., 2013b), cf. Fig. 12 (left). For potential users of JAWS, this is a significant advantage, as it allows the liquid helium infrastructure to be circumvented completely. Modern cryocoolers require only a few hours to cool down, are easy to use and require no or only minimal maintenance work. A second cryocooler is designed for daily use for JAWS applications in the impedance bridge (see Section “Applications”), cf. Fig. 12 (right) (Bauer et al., 2017). A third cryocooler will be installed in the near future to shorten the distance between the JAWS chip at 4 K and 300 K from about 1.2 m to about 0.3 m. This will reduce the uncertainty at high signal frequencies in the range of 100 kHz up to 1 MHz (Brom et al., 2016).

Optical pulse drive and on-chip power divider Very recently, an optical pulse drive for the JAWS was developed in a collaborative effort between PTB and NPL, JV and the University of South-Eastern Norway (USN) (Kieler et al., 2019; Ireland et al., 2017; Bardalen et al., 2017; Bardalen et al., 2018; Ireland et al., 2019; Karlsen et al., 2019; Bardalen et al., 2020; Karlsen et al., 2020; Herick et al., 2020). Considerable effort was

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Fig. 12 Photographs of two JAWS setups operated with pulse-tube cryocoolers from Oxford Instruments (left) and TransMIT (right).

Fig. 13 Schematics of “electrical” vs. “optical” JAWS setup.

undertaken by our partner institutions to establish a reliable and robust PD packaging technology with a high yield. The aim of these developments was to create a low-cost solution for the parallel operation of many JAWS arrays by operating fast photodiodes (PD) at 4 K close to the JAWS arrays. Another advantage is that, by using optical fibers (instead of semi-rigid cables), less noise will be introduced – specifically, noise at high signal frequencies in the MHz range. Fig. 13 shows a simplified scheme of the “electrical” JAWS setup vs. the “optical” JAWS setup (Kieler et al., 2019). For this application, we used high-performance InGaAs PD from Albis:

• • • •

dimensions: 350 mm  350 mm integrated backside lens ➔ suitable for flip-chip bonding speed 28 Gbit/s wavelength 1310 nm

In a typical JAWS setup at PTB, the electrical pulses are delivered by a PPG and transmitted by an HF coaxial cable about 1.5 m long from the PPG output to the cryoprobe head. The cryoprobe contains a 1.2 m semi-rigid HF coaxial cable to further transfer the pulses to the K-type connectors at the PCB carrier, where the JAWS chip is mounted at 4 K. This electrical pulse transmission is replaced by an optical pulse drive. The pulse pattern is again provided by the PPG and electrically transferred to the RF modulator driver (RF-MD), which serves as an amplifier for the electrical pulses. The amplified pulses are transferred electrically to the Mach-Zehnder Modulator (MZM). The MZM is a LiNbO3-based intensity modulator designed for 1310 nm lasers. We use an

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Fabry-Perot steady laser. The optical fiber transfers the pulses to the PD at 4 K. The fiber is attached via a glass tube guide to the PD mounted by a flip-chip process to a customized Si-carrier chip. This carrier chip was designed to provide bias lines and CPW for transfer of the electrical pulses. Those carrier chips were fabricated at PTB using Nb for the DC/HF lines and AuPd for the alignment markers and contact pads (PD and wire bond pads). An additional SiO2 layer is deposited by PECVD to effectively protect the sensitive on-chip DC block capacitors against electrical damage during handling and operation. A significant advantage for practical use is that the optical fiber, which is connected via the glass tube, is removable. This tube is precisely aligned on-chip to the PD to ensure high efficiency of the illumination. Because the PD has a relatively large lens diameter of 100 mm, it is easy to implement it in the setup to achieve robust and reliable operation. The flip-chip process of the PD is carried out at USN/JV in cooperation with NPL and PTB by using Au stud bumps and thermo-compression. The Si-carrier chip is glued and Al wire-bonded to a PCB. The JAWS-chip PCB and PD-chip PCB are arranged close together (4 cm) and are connected via a K-type edge launcher. Detailed optimizations and investigations of high-speed performance at room temperature and at 4 K were performed (Bardalen et al., 2017; Bardalen et al., 2018); the optical pulse setup is described in (Karlsen et al., 2019). Fig. 14 shows two PDs mounted on a Si-carrier chip made for the unipolar pulse drive of two JAWS arrays (left), a next-generation Si-carrier chip made for a bipolar pulse-drive of one JAWS array, with two PDs and two glass tubes mounted (center), and a glass tube with the removable optical fiber attached (right). Using a one-channel optical pulse drive, unipolar sine waveforms were synthesized with stable operation margins using a JAWS array of 3000 junctions in a signal frequency range from 60 Hz up to 10 kHz. Clock frequencies up to 15 GHz, which corresponded to the maximum speed of the PPG and the PD, were used to generate output voltages of 6.6 mV RMS resp. 18.6 mV PP with a SD code amplitude of 0.2 (Kieler et al., 2019). Fig. 15 shows the frequency spectrum and time domain of a sine waveform synthesized at 12 GHz with a SD code amplitude of 0.2 and a SD code length of 8 Mbit. This waveform has a signal frequency of 1500 Hz and signal amplitude of 5.26 mV RMS resp. 14.9 mV PP. The signal-to-noise ratio is 90 dBc and the laser-bias operation margins are 1 mA. The time domain clearly shows the DC shift of the waveform generated due to the unipolar pulse-drive with only one PD.

Fig. 14 Photographs of two PDs mounted on a Si-carrier chip (left), a different Si-carrier chip with two PDs and two glass tubes mounted (center), and a glass tube with a removable optical fiber attached (right).

Fig. 15 Frequency spectrum and time domain of a unipolar sine waveform generated with the one-channel optical pulse drive.

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Therefore, only one Shapiro step is used, and the output voltage is reduced by 50%. A two-channel optical pulse-drive setup for generating bipolar waveforms is under development at PTB. The first spectrally pure bipolar waveforms were synthesized with a signal amplitude of 12.5 mV RMS resp. 35.4 mV PP at a signal frequency of 10 kHz (Herick et al., 2020). Another approach to increasing the output voltage of JAWS by parallel operation of several JAWS arrays is to use an on-chip power divider. This approach was successfully implemented by using Wilkinson power dividers for JAWS at NIST (USA) (Flower-Jacobs et al., 2016) and by using CPW-CPS dividers for the programmable voltage standard of AIST (Japan) (Yamamori and Kohjiro, 2016). At PTB, modified versions of both types of dividers were integrated into our JAWS circuits (Tian et al., 2020; Tian et al., 2021; Tian, 2022). Different versions of these dividers were developed to optimize the insertion and return loss and achieve an optimal amplitude balance. Fig. 16 shows an example of the layout (left) and the realized chip (right) of a one-stage three-section Wilkinson divider, wherein two dividers were integrated onto a 10 mm  15 mm chip. The one-stage divider splits the HF chain into two chains to supply two independent JAWS arrays. Because Wilkinson dividers are known to have a small bandwidth, a three-section divider was integrated. Each section had a different center frequency. Here, the intention was to enlarge the overall bandwidth of this divider and consequently improve the performance of the pulse-mode operation. The IVC of both arrays of a one-stage Wilkinson divider is shown in Fig. 17. The circuit contains N ¼ 9000 per array with 3-stacked junctions, i.e., 18,000 junctions in total. Basic Josephson junction parameters can be derived from this: critical current Ic ¼ 3.3 mA, normal resistance Rn ¼ 3 mO. Thus, the following parameters can be calculated: the critical current density jc ¼ 5.5 kA/cm2 (junction area A ¼ 59.84 mm2), the characteristic voltage Vc ¼ 10.3 mV and the characteristic frequency fc ¼ 5.0 GHz. Both IVCs agree very well and an Ic-spread of about 5% was estimated from these IVCs. To evaluate the large bandwidth of these dividers, the IVC was measured for different pulse repetition frequencies in the range from 3 GHz to 15 GHz for one array (Fig. 18). For each IVC, the pulse amplitude was adjusted to 0.7 x Ic. For the sake of clarity, the IVCs were derived and plotted using a color-coding scheme: the white area shows the Shapiro steps, while the multicolored areas are the non-quantized parts of the IVC between those steps. First, the +/− first Shapiro steps (SS) are created over the whole frequency

Fig. 16 Layout (left) and photograph (right) of a JAWS chip with two arrays with a one-stage three-section Wilkinson divider.

Fig. 17 Current-voltage characteristic of both arrays of a one-stage Wilkinson divider containing 9000 junctions per array.

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Fig. 18 Derivation of current-voltage characteristic vs. pulse repetition frequency for one array of a one-stage Wilkinson divider (9000 junctions).

Fig. 19 Frequency spectrum and time domain of a sine waveform generated with the one-section Wilkinson divider.

range with a minimum step width of about 0.8 mA. Second, the Shapiro step generation is (as expected) fairly symmetrical for positive and negative pulses. Both features are the precondition of the synthesis of spectrally pure waveforms in pulse-mode operation. Below, Fig. 19 illustrates the time and frequency domain of a sine waveform synthesized with both arrays of this Wilkinson divider, which has 18,000 junctions in total. At a clock frequency of fclock ¼ 14 GHz and a SD code amplitude of ASD ¼ 0.814, it was possible to synthesize a waveform at fsignal ¼ 100 Hz and Vsignal ¼ 300 mV RMS (848 mV PP), providing small but stable operation margins of 180 mA. An excellent signal-to-noise ratio of 130 dBc was achieved.

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Applications The outstanding features of JAWS have already allowed several applications in AC voltage metrology to be demonstrated at PTB. Several international research projects funded in large part by the European Union have helped establish a strong and fruitful collaboration between several national metrology institutes. The following projects were funded from 2001 until 2019: “JAWS” (EU project, 2001), “JoSy” (EU project, 2007), “Q-Wave” (EU project, 2013), “ACQ-Pro” (EU project, 2015), “Dig-AC” (EU project, 2018) and “QuADC” (EU project, 2016). In this way, we were able to extensively explore the limitations of JAWS in the field of voltage metrology, and to set new milestones in the quality of high-precision measurements. The features of the JAWS investigated included the following (the NMI partners of PTB are denoted in parentheses): 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

AC-DC transfer measurements with a trans-conductance amplifier (NMIA [Australia]) Characterization of Keysight 3458A (SIQ [Slovenia], CEM [Spain]) AC-DC transfer (up to 1 MHz) with Fluke792A in a mini-cryostat (VSL [Netherlands]) Characterization of analog-to-digital converters (ADCs) (CMI [Czech Republic) Systematic error analysis (Tübitak [Turkey]) Characterization of voltage dividers (SP [Sweden]) JAWS based mV-synthesizer JAWS in a cryocooler (INRIM [Italy]) Impedance/quadrature bridge (METAS [Switzerland]) JAWS Johnson-noise thermometry (JNT) (PTB Berlin)

Some of these activities will be described in the following section. A novel method for the calibration of fast ADCs using JAWS with a sophisticated constructed waveform was suggested in (Sira et al., 2019). Here, the aim was to determine the gain and time stability of an ADC for different frequencies and amplitudes almost simultaneously, or even in a single measurement if possible. By reordering the sampled data and the numerous steps involved in the data processing, other basic properties of an ADC can also be estimated such as phase, integral nonlinearity (INL), differential nonlinearity (DNL), effective number of bits (ENOB), and signal-to-noise and distortion ratio (SINAD). The waveform selected is shown in Fig. 20. It consists of 4 sections and 16 subsections, as clearly indicated in the time domain of the signal. The frequency of the sine wave is different for each of the four sections. Each subsection contains four different signal amplitudes, and each amplitude repeats 10 periods of this single-tone sine wave to evaluate the time stability. Frequencies, amplitudes, and periods define the total number of 160 calibration points of this signal. By continuously repeating this waveform, more statistical data can be derived. We selected a National Instruments PXI 5922 as the device under test (DUT). The PXI 5922 is used in many calibration laboratories because of its flexibility. Its maximum sampling frequency is 15 MSa/s. Because the device has internal SD modulation

Fig. 20 Frequency spectrum and time domain of a complex multitone sine waveform for calibration of an ADC.

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electronics, the resolution increases with decreasing sampling frequency. For sampling frequencies below 500 kSa/s, the resolution is nominally 24 bit. Two full-scale ranges, 2 and 10 V, are available. Because of the small maximum output voltage of the JAWS, only the lower range of the digitizer was used. The waveform was constructed to accommodate some restrictions of the devices used – for example, limited values for the sampling rate of the PXI or the minimum bit resolution of 128 bit for the PPG. These restrictions explain the special values of the PPG clock frequency and code length. The main waveform properties and settings of the PPG and PXI5922 were the following: 1. 2. 3. 4. 5. 6. 7. 8.

Number of junctions N ¼ 54,000 (six arrays in series) PPG clock frequency fclock ¼ 13.762 56 GHz PPG code length LSD ¼ 215,040,000 PPG code amplitude ASD ¼ 0.06/ 0.18/ 0.30/ 0.42 (Amain-SD ¼ 0.6 x pre-factors AF ¼ 0.1/ 0.3/ 0.5/0.7) PPG code repetition rate: 32 PXI sampling rate: 480 kSa/s Signal frequencies fsignal ¼ 150 /300/600 Hz/1200 Hz Signal amplitudes Vsignal  0.18/ 0.55/0.92/1.3 V PP

The main output of the calibration (gain errors) can be obtained at different amplitudes and frequencies. By repeating the waveform, the gain stability can also be measured. The advantage of this method is that all gains are measured in a very short period of time. Thus, the stability is measured at multiple calibration points simultaneously. Another application is the characterization of a sampling AC voltmeter. The aim of this approach was to evaluate the noise performance of such a device in sampling mode as a function of different device settings, e.g., aperture time. For this application, a special waveform was again synthesized using the JAWS: a band-limited square waveform (BLSW), which was suggested in (Lapuh et al., 2017) (Fig. 21). This waveform corresponds to a square wave at 52 Hz, where only a certain number of higher harmonics of the frequency spectrum are generated and all other higher harmonics above 20 kHz are cancelled by the signal. As the DUT, we selected an Keysight 3458A digital multimeter, as it provided high accuracy for sampling low-frequency signals and it is widely used in NMIs and calibration laboratories. Detailed knowledge of the noise performance (among other things) is a precondition for high-accuracy measurements with low uncertainties. The main waveform properties and settings of the PPG and PXI5922 were the following: 1. Number of junctions N ¼ 24,000 (two arrays in series) 2. PPG clock frequency fclock ¼ 13 GHz 3. PPG code length LSD ¼ 250,000,000

Fig. 21 Frequency spectrum and time domain of a band-limited square waveform for calibration of an AC voltmeter.

The pulse-driven AC Josephson voltage standard of Physikalisch-Technische Bundesanstalt (PTB) 4. 5. 6. 7. 8. 9. 10.

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PPG code amplitude ASD ¼ 0.32 PPG code repetition rate: 1 JAWS bias current operation margins: 0.5 mA Signal frequencies fsignal ¼ 52 Hz Signal amplitude Vsignal ¼ 150 mV RMS Number of higher harmonics: 192 Only odd harmonics are synthesized: 1, 3, 5, . . ., 383

“Simple” sine waveforms at different frequencies and amplitudes are also suitable for this application, as demonstrated in (de Aguilar et al., 2019). Another application of JAWS is to use it as a precision microvolt synthesizer for the generation of small DC or AC signals (Behr et al., 2017). These signals can be used, for instance, to calibrate nanovoltmeters (DC) or lock-in amplifiers (AC). The JAWS is capable of generating DC/AC signals in the nanovolt range easily if the pulse repetition frequency chosen for the operation of the chip is far below the characteristic voltage of the Josephson junction. In contrast to continuous wave operation, broad Shapiro steps will be generated in this pulse-mode operation regime too (Benz and Hamilton, 1996). We demonstrated this for the first time experimentally for this application. In order to generate DC voltages by means of the JAWS, two methods can be applied. In the first, more obvious method, simple pulse codes with a constant pulse repetition frequency will be generated. By choosing the pulse code length and the clock frequency, the desired signal voltage can be tuned. The more advanced method invokes higher-order SD modulation. Here, the pulse repetition frequency in the pulse code is not constant. The advantage of this method is that any desired output voltage can be adjusted easily and precisely by selecting the corresponding SD code amplitude and (if necessary) the clock frequency. The typical signal voltage range for the calibration of a nanovoltmeter is about V ¼ 10 nV to V ¼ 100 mV. For this calibration, we used a JAWS array with N ¼ 9000 junctions. For each DC signal voltage, a different SD code (−1/0/+1) with a constant length of LSD ¼ 128 Mbit was calculated in advance. The clock frequency was also kept constant: fclock ¼ 5 GHz. To synthesize, for example, a DC voltage of 10 nV with these parameters, the resulting SD code amplitude is very small (ASD  1.5  10−7) and only 1.1  10−5% of the code contains “1”s. To synthesize a “zero” voltage, on the other hand, the SD code contains only “0”s. When this “zero” code is applied to the JAWS array, no flux quanta are transferred, and the Josephson junctions are kept in the superconducting state below the critical current, giving a “perfect” zero signal output as required for the calibration. The significant advantage of a perfect “zero” voltage being set by choosing the “zero” code automatically by means of a program is that no switches are needed (including those inside an instrument) to set the voltage to zero or to reverse its polarity. Due to the small voltages, it was not necessary to apply a compensation signal (Benz et al., 2001), i.e., no additional DC bias currents were needed. The Shapiro steps can be centered around the zero bias current simply by tuning the positive or negative pulse amplitude adjusted at the PPG. A positive pulse amplitude is used to select the first positive Shapiro step and the negative pulse amplitude is adjusted for the first negative Shapiro step. Both pulse amplitudes of the PPG were constant for all pulse repetition frequencies in the SD code (and consequently for all DC voltages) to be generated. Therefore, the entire JAWS setup is comparatively simple. Additionally, this demonstrates that the Shapiro step width is fully independent of the pulse repetition frequency. To avoid high thermal voltages in the setup, twisted-pair copper wires (in contrast to coaxial wires) were used to connect the array to the DUT (Magnicon NVM1, 10 mV-range), keeping the thermal voltages stable and below 500 nV. Stable and low thermal voltages are important for precise DC voltage measurements. The minimum pulse repetition frequency applied was only 540 Hz, which is far below the characteristic frequency of fc ¼ 7.6 GHz, which produces a minimum synthesized voltage of only VJJ ¼ 1.1 pV per Josephson junction: 1. Minimum pulse repetition frequency: fp ¼ 540 Hz  0.00000007  fc 2. Minimum voltage per single junction: V ¼ 1.1 pV  0.00000007  Vc The minimum possible voltage has not yet been explored, as it was possible to use the full available PPG memory of 512 MByte, tune down the clock frequency of the PPG, and also reduce the SD code amplitude. However, from a practical point of view, a minimum voltage of 10 nV is sufficiently small for a typical nanovoltmeter calibration. The complete measurements were taken fully automatically, and we determined 24 voltage calibration points in 60 min, thus providing a low uncertainty. The results are shown in Fig. 22. The deviation from the linearity is plotted in a measurement range from −4 mV to +4 mV. The error bars indicate Type A uncertainties (k ¼ 1). The measurements agree well with the conventional data achieved by means of a potentiometer method, which shows noticeably higher uncertainties. By using a similar procedure, the generation of AC voltages is also possible for the calibration of AC devices that are sensitive to small input signals. In (Behr et al., 2017), a lock-in amplifier in the range from 80 Hz to 100 kHz in a signal voltage range from 1 mV to 50 mV was also calibrated. The following features of this device were characterized: 1. 2. 3. 4. 5.

Linearity Gain vs. frequency Gain vs. voltage range AC gain selection Harmonic content influence

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Fig. 22 Deviation from linearity for DC voltages in the mV-range measured for a digital nanovoltmeter with the JAWS and the conventional potentiometer method.

This shows the excellent suitability of the JAWS for this purpose: uncertainties on the nanovolt level can be achieved within short measurement times and the fully automated calibration procedure makes it a useful tool for metrological applications. Another important application is to integrate two JAWS systems into a quantum-based impedance bridge system for the precise determination of resistor and/or capacitor ratios (Bauer et al., 2017; Bauer et al., 2018; Pimsut et al., 2020; Bauer et al., 2020). In contrast to conventional transformer-based impedance bridges, the JAWS-based setup is very flexible concerning the voltage ratios and phase angles between both voltages. At PTB, a four-terminal-pair impedance bridge was established that can also be directly connected to a graphene quantum Hall resistance (QHR) setup (see schematics in Fig. 23). The noise level of such a combined setup is extremely low. The QHR device showed a deviation from the quantized value of 5  10−8 in a comparison with calibrated 10 nF capacitance standards at 1233.15 Hz. Initial AC measurements with graphene-based quantum Hall resistances show very promising results for measurements with capacitance and resistance standards. Performing a quadrature measurement of the QHR against a capacitance standard allows magnetic field sweeps with lower noise

Fig. 23 Schematics of the four-terminal-pair impedance bridge of PTB combining two JAWS setups and one QHR setup. Courtesy of Stephan Bauer (PTB).

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and therefore reduces the measurement time compared to conventional ratio bridges using room-temperature resistance standards. This can be used to investigate graphene QHR samples and give important feedback on the manufacturing process. Within ratio measurements, the time-dependent drift of a 12.9 k resistance standard was determined in a substitutional measurement and agreed within the fit uncertainty of 0.23 nO/O with the measurements of a 14-bit cryogenic current comparator. This gives an initial impression of the high reproducibility and flexibility of the whole setup. The excellent features of the four-terminal-pair JAWS-based impedance bridge makes it an ideal tool for future metrological applications. The PTB JAWS was recently used for the realization of the quantum-based electronic Kelvin. Johnson-noise thermometry (JNT) will enable the development of a primary thermometer up to 1000 K in the future (Kraus et al., 2018; Drung and Krause, 2021; Drung et al., 2022). The JAWS will be implemented in such a system to provide a pseudo-random quantum noise source such as the one suggested by NIST. These activities were prompted by the 2020 re-definition of the basis unit of the Kelvin by setting a fixed value for the Boltzmann constant; this re-definition has enabled the direct definition of the Kelvin in the whole temperature range via primary thermometry. This is important for future objectives in thermodynamical temperature measurements and future development of the international temperature scale. The work on the JAWS-based JNT is still in progress in a collaborative effort between different departments of PTB.

Conclusion We have presented the basic principle of JAWS and demonstrated the capability of generating quantized high-precision and arbitrary waveforms in the time and frequency domain with extremely high spectral purity. A brief overview of the fabrication technology for highly integrated JAWS circuits with more than 50,000 junctions per chip has been given. We have also introduced stacked junction technology with up to 5-stacked junctions. New approaches designed to further increase the output voltage and to simplify the JAWS setup have been presented, including on-chip power dividers and optical pulse drives for the JAWS. Certain applications of the JAWS that have significant benefits for quantum AC voltage metrology have also been described. These new applications have resulted in numerous publications by PTB scientists in cooperation with our NMI partners over the last several years. For example, from 2018 to 2021, more than 30 papers were published, including conference papers. Within international research projects, PTB’s JAWS chips have been distributed to NMIs in the Netherlands, the United Kingdom, Norway, Italy, Finland, Turkey, Russia, China and Argentina. The technology transfer between PTB and the Supracon company for commercialization purposes has also been initiated to make JAWS available to other NMIs, to calibration laboratories and to industry in the near future.

Acknowledgments R. Behr, L. Palafox, H. Tian, S. Bauer, J. Herick, M. Kraus, Y. Pimsut, J. Kohlmann, T. Weimann, R. Wendisch, R. Gerdau, K. Peiselt, P. Hinze, K. Störr, B. Egeling, J. Felgner, M. Petrich, G. Muchow, S. Gruber, T. Ahbe, A. Zorin, M. Bieler and many international partners from various NMIs.

References Baek B, Dresselhaus PD, and Benz SP (2006) Co-sputtered amorphous NbxSi1-x barriers for josephson-junction circuits. IEEE Transactions on Applied Superconductivity 16(4): 1966–1970. Bardalen E, et al. (2017) Packaging and demonstration of optical-fiber-coupled photodiode array for operation at 4 K. IEEE Transactions on Components, Packaging and Manufacturing Technology 7(9): 1395–1401. Bardalen E, et al. (2018) Reliability study of fiber-coupled photodiode module for operation at 4 K. Microelectronics and Reliability 81: 362–367. Bardalen E, et al. (2020) Bipolar photodiode module operated at 4 K. In: IEEE 8th Electronics System-Integration Technology Conference (ESTC), 15–18 Sept. 2020, Tönsberg, Norway. Bauer S, et al. (2017) A novel two-terminal-pair pulse-driven Josephson impedance bridge linking a 10 nF capacitance standard to the quantized Hall resistance. Metrologia 54(2): 152–160. Bauer S, et al. (2018) Progress on PTB’s Pulse-Driven Josephson Impedance Bridge combined with an AC Quantum Hall Resistance. In: CPEM2018, in Proc. Conference on Precision Electromagnetic Measurements, Paris, France, 8–13 July 2018. Bauer S, et al. (2020) AC quantum hall resistance combined with a four-terminal pair pulse-driven josephson impedance bridge. In: CPEM2020, in Proc. Conference on Precision Electromagnetic Measurements, Denver, USA, 24–28 Aug. 2020. Behr R and Kieler O (2020) unpublished. Behr R, et al. (2012) Development and metrological applications of Josephson arrays at PTB. Measurement Science and Technology 23(12), 124002. Behr R, et al. (2015) Direct comparison of a 1 V Josephson arbitrary waveform synthesizer and an ac quantum voltmeter. Metrologia 52(4): 528–537. Behr R, Kieler O, and Schumacher B (2017) A precision microvolt-synthesizer based on a pulse-driven josephson voltage standard. IEEE Transactions on Instrumentation and Measurement 66(6): 1385. https://doi.org/10.1109/TIM.2016.2619998. (Accessed: 03 March 2022). Benz SP and Hamilton CA (1996) A pulse-driven programmable Josephson voltage standard. Applied Physics Letters 68: 3171–3173. Benz SP, Burroughs CJ, and Dresselhaus PD (2001) AC coupling technique for Josephson waveform synthesis. IEEE Transactions on Applied Superconductivity 11(1): 612–616. Benz SP, et al. (2015) 1 V Josephson arbitrary waveform synthesizer. IEEE Transactions on Applied Superconductivity 25(1). Brom HVD, et al. (2016) AC-DC calibrations with a pulse-driven AC Josephson voltage standard operated in a small cryostat. In: CPEM2016, in Proc. Conference on Precision Electromagnetic Measurements, Ottawa, Canada 1–2. 10–15 July 2016. https://doi.org/10.1109/CPEM.2016.7540605. (Accessed: 03 March 2022). Burroughs CJ, et al. (2005) Precision measurements of AC Josephson voltage standard operating margins. IEEE Transactions on Instrumentation and Measurement 54(2): 624–627.

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The pulse-driven AC Josephson voltage standard of Physikalisch-Technische Bundesanstalt (PTB)

de Aguilar J, et al. (2019) Characterization of an analog-to-digital converter frequency response by a Josephson arbitrary waveform synthesizer. Measurement Science and Technology 30: 035006. Drung D and Krause C (2021) Dual-mode auto-calibrating resistance thermometer: A novel approach with Johnson noise thermometry. The Review of Scientific Instruments 92: 034901. https://doi.org/10.1063/5.0035673. (Accessed: 03 March 2022). Drung D, Kraus M, and Krause C (2022) Calibration of the dual-mode auto-calibrating resistance thermometer with few-parts-per-million uncertainty. Measurement Science and Technology 33. https://doi.org/10.1088/1361-6501/ac2c48. (Accessed: 03 March 2022). EU-project ‘ACQ-Pro’ (2015) 14RPT01. Available at: https://www.euramet.org/measurement-research-innovation-issue-14/#c3965. (Accessed: 03 March 2022). EU-project ‘Dig-Ac’ (2018) 17RPT03. Available at: https://www.euramet.org/research-innovation/search-research-projects/details/project/a-digital-traceability-chain-for-ac-voltageand-current/. (Accessed: 03 March 2022). EU-project ‘JAWS’ (2001) G6RD-CT-2001-00599. Available at: https://cordis.europa.eu/project/id/G6RD-CT-2001-00599. (Accessed: 03 March 2022). EU-project ‘JoSy’ (2007) ERA-NET + no. 217257. Available at: https://cordis.europa.eu/project/id/217257/de. (Accessed: 03 March 2022). EU-project ‘Q-Wave’ (2013) JRP SIB59. Available at: https://www.euramet.org/research-innovation/search-research-projects/details/project/a-quantum-standard-for-sampledelectrical-measurements/. (Accessed: 03 March 2022). EU-project‚QuADC (2016) JRP 15SIB04. Available at: https://www.euramet.org/research-innovation/search-research-projects/details/project/waveform-metrology-based-onspectrally-pure-josephson-voltages/. (Accessed: 03 March 2022). Flower-Jacobs NE, et al. (2016) Two-volt Josephson arbitrary waveform synthesizer using Wilkinson dividers. IEEE Transactions on Applied Superconductivity 26(6): 1400207. https:// doi.org/10.1109/TASC.2016.2532798. (Accessed: 03 March 2022). Herick J, et al. (2020) Realization of an opto-electronic bias for pulse-driven josephson voltage standards at PTB. In: CPEM2020, in Proc. Conference on Precision Electromagnetic Measurements, Denver, USA, 24–28 Aug. 2020. Ireland J, et al. (2017) Josephson arbitrary waveform system with optoelectronic drive. In: Ext. Abstract, ISEC-2017: 16th Int. Supercond. Electron. Conf., Sorrento, Italy, Jun. 2017 12–16. Ireland J, et al. (2019) An optoelectronic pulse drive for quantum voltage synthesizer. IEEE Transactions on Instrumentation and Measurement 68(6). Karlsen B, et al. (2019) Pulsation of InGaAs photodiodes in liquid helium for driving josephson arrays in AC voltage realization. IEEE Transactions on Applied Superconductivity 29(7): 1200308. https://doi.org/10.1109/TASC.2019.2901573. (Accessed: 03 March 2022). Karlsen B, et al. (2020) High-speed pulsation of a cryogenically operable bipolar photodiode module for the Josephson arbitrary waveform synthesizer. In: CPEM2020, in Proc. Conference on Precision Electromagnetic Measurements, Denver, USA, 24–28 Aug. 2020. Kieler O, et al. (2007a) SNS josephson junction series arrays for the Josephson arbitrary waveform synthesizer. IEEE Transactions on Applied Superconductivity 17: 187–189. Kieler O, Kohlmann J, and Müller F (2007b) Improved design of superconductor/normal conductor/superconductor Josephson junction series arrays for an ac Josephson voltage standard. Superconductor Science and Technology 20: 318–322. Kieler O, Iuzzolino R, and Kohlmann J (2009) Sub-mm SNS Josephson junction arrays for the Josephson arbitrary waveform synthesizer. IEEE Transactions on Applied Superconductivity 19(3): 230–233. Kieler O, Behr R, and Kohlmann J (2013a) Development of a pulse-driven AC josephson voltage standard at PTB. In: ISEC 2013, in Proc. 14th International Superconductive Electronics Conference, Cambridge, USA 59–61. 7–11 July 2013. Kieler O, Scheller T, and Kohlmann J (2013b) Cryocooler operation of a pulse-driven AC josephson voltage standard at PTB. World Journal of Condensed Matter Physics 3: 193–198. Kieler O, et al. (2015) Towards a 1 V Josephson arbitrary waveform synthesizer. IEEE Transactions on Applied Superconductivity 25(3): 1400305. https://doi.org/10.1109/ TASC.2014.2366916. (Accessed: 03 March 2022). Kieler O, et al. (2017) Arrays of stacked SNS Josephson junctions for pulse-driven Josephson voltage standards. In: EUCAS 2017 - 13th European Conference on Applied Superconductivity, Geneva, Switzerland, 17–21 September 2017. Kieler O, et al. (2019) Optical pulse-drive for the pulse-driven AC Josephson voltage standard. IEEE Transactions on Applied Superconductivity 29(5): 1200205. Kieler O, et al. (2021) Stacked Josephson junction arrays for the pulse-driven AC josephson voltage standard. IEEE Transactions on Applied Superconductivity 31(5): 1100705. Kraus M, et al. (2018) Frequency-dependent verification of the quantum accuracy of a Quantum Voltage Noise Source. In: CPEM2018, in Proc. Conference on Precision Electromagnetic Measurements, Paris, France, 8–13 July 2018. Lapuh R, et al. (2017) Keysight 3458A noise performance in DCV sampling mode. IEEE Transactions on Instrumentation and Measurement 66(6): 1089. https://doi.org/10.1109/ TIM.2017.2681238. (Accessed: 03 March 2022). Physikalisch-Technische Bundesanstalt (PTB) (2022) Available at: http://www.ptb.de (Accessed: 03 March 2022). Pimsut Y, et al. (2020) A four-terminal-pair pulse-driven Josephson impedance bridge for 10 nF:10 nF capacitance ratio measurements. In: CPEM2020, in Proc. Conference on Precision Electromagnetic Measurements, Denver, USA, 24–28 Aug. 2020. Schreier R and Temes T (2005) Understanding delta-sigma data converters. Pisctaway, New Jersey: IEEE Press. Sira M, Kieler O, and Behr R (2019) A novel method for calibration of ADC using JAWS. IEEE Transactions on Instrumentation and Measurement 68(6): 2091. Supracon AG (2022) Jena. Available at: http://www.supracon.de. (Accessed: 03 March 2022). Tian H (2022) Development of RF power dividers for high integrated circuits of ACJosephson voltage standards. to be published in PhD thesis. Tian H, et al. (2020) Development of RF-power dividers for the Josephson arbitrary waveform synthesizer. IEEE Transactions on Applied Superconductivity 30(5): 1100105. Tian H, et al. (2021) Investigation of broadband wilkinson power dividers for pulse-driven Josephson voltage standards. IEEE Transactions on Applied Superconductivity 31(5): 1100305. Watanabe M, et al. (2006) Resonance-free lowpass filters for the AC Josephson voltage standard. IEEE Transactions on Applied Superconductivity 16(1): 49–53. Yamamori H and Kohjiro S (2016) Fabrication of voltage standard circuits utilizing a serial-parallel power divider. IEEE Transactions on Applied Superconductivity 26(8): 1400404.

Field theoretic aspects of condensed matter physics: An overview Eduardo Fradkin, Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, United States © 2024 Elsevier Ltd. All rights reserved.

Introduction Early years: Feynman diagrams and correlation functions Critical phenomena Classical critical phenomena Landau–Ginzburg theory The renormalization group Scaling The operator product expansion Fixed points Universality RG flows Asymptotic freedom Quantum criticality Dynamic scaling The Ising model in a transverse field Quantum antiferromagnets and nonlinear sigma models Spin coherent states Path integral for a spin-S degree of freedom Quantum ferromagnet Quantum antiferromagnet Topological excitations Topological excitations: Vortices and magnetic monopoles Vortices in two dimensions Magnetic monopoles in compact electrodynamics Nonlinear sigma models and antiferromagnetic quantum spin chains Topology and open integer-spin chains Duality in Ising models Duality in the 2D Ising model The 3D duality: 2 gauge theory Bosonization Dirac fermions in one space dimensions Chiral symmetry and chiral symmetry breaking The chiral anomaly Bosonization, anomalies, and duality Fractional charge Solitons in one dimensions Polyacetylene Fractionally charged solitons Fractional statistics Basics of fractional statistics What is a topological field theory Chern–Simons gauge theory BF gauge theory Quantization of Abelian Chern–Simons gauge theory Vacuum degeneracy a torus Fractional statistics and braids Topological phases of matter Topological insulators Dirac fermions in 2+1 dimensions Dirac fermions and topological insulators Chern invariants The quantum Hall effect on a lattice The anomalous quantum Hall effect

29 29 30 30 30 31 31 31 32 33 33 34 36 36 36 38 38 39 40 40 41 41 42 45 47 48 49 49 50 52 53 55 57 59 62 62 62 63 65 65 66 66 67 67 69 70 72 72 72 73 74 76 79

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https://doi.org/10.1016/B978-0-323-90800-9.00269-9

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Field theoretic aspects of condensed matter physics: An overview

The parity anomaly Three-dimensional 2 topological insulators 2 Topological invariants The axial anomaly and the effective action Theta terms, and domain walls: Anomaly and the Callan–Harvey effect Chern–Simons gauge theory and the fractional quantum Hall effect Landau levels and the integer Hall effect The Laughlin wave function Quasiholes have fractional charge The Jain states Quasiholes have fractional statistics Hydrodynamic effective field theory Composite Boson field theory Composite fermion field theory The compressible states Fractional quantum hall wave functions and conformal field theory Edge states and chiral conformal field theory Point contact tunneling and QH chiral edge states Experimental evidence of fractional charge Quantum interferometers and fractional statistics Particle-vortex dualities in 2+1 dimensions Electromagnetic duality Particle-vortex duality in 2+1 dimensions The 3D XY model Scalar QED in 3D The duality mapping Bosonization in 2+1 dimensions Bosonization of the Dirac theory in 2+1 dimensions Bosonization of the fermi surface Conclusion Acknowledgments References

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Abstract In this chapter, I discuss the impact of concepts of Quantum Field Theory in modern Condensed matter physics. Although the interplay between these two areas is certainly not new, the impact and mutual cross-fertilization has certainly grown enormously with time, and Quantum Field Theory has become a central conceptual tool in condensed matter physics. In this chapter, I cover how these ideas and tools have influenced our understanding of phase transitions, both classical and quantum, as well as topological phases of matter, and dualities.

Key points

• • • • • • • •

Quantum Field Theory has a deep and wide impact on our understanding of modern condensed matter physics. The conceptual framework of the Renormalization Group and of the UV fixed point is the central concept in the theory and classical and quantum phase transitions. The renormalization group is the foundation for universality, scale, and conformal invariance at these fixed points. Here, I discuss quantum criticality in the 1D transverse field Ising model and in quantum antiferromagnets. Topology is also central to our understanding of condensed matter ranging from the role of vortices and magnetic monopoles to topological terms in effective actions. Duality and duality transformations and bosonization and fermionization transformations in condensed matter and gauge theories provide a nonperturbative framework to our understanding of these systems. Topology has provided a sharp definition of fractional charge, fractional statistics, braiding, and fusion of quasiparticles. Topological phases of matter are physical systems with long-range quantum entanglement characterized by topological invariants.

Field theoretic aspects of condensed matter physics: An overview

29

Introduction Many (if not most) puzzling problems in condensed matter physics involve systems with a macroscopically large number of degrees of freedom often in regimes of large fluctuations, thermal and/or quantum mechanical. The description of the physics of systems of this type requires the framework provided by Quantum Field Theory. Although quantum field theory has its origins in high-energy physics, notably in the development of Quantum Electrodynamics, it has found a nurturing home in condensed matter physics. There is a long history of cross-fertilization between both fields. Since the 1950s in many of the most significant developments in Condensed Matter Physics, Quantum Field Theory has played a key role if not in the original development but certainly in the eventual understanding the meaning and the further development of the discoveries. As a result, many of the discoveries and concepts developed in Condensed Matter have had a reciprocal impact in Quantum Field theory. One can already see this interplay in the development of the Bardeen–Cooper–Schrieffer theory of superconductivity (Schrieffer, 1964) and its implications in the theory of dynamical symmetry breaking in particle physics by Nambu (Nambu and Jona-Lasinio, 1961). The close and vibrant relationship between both fields has continued to these days, and it is even stronger today than before. Many textbooks have been devoted to teaching these ideas and concepts to new generations of condensed matter physicists (and field theorists as well). The earlier texts focused on Green functions which are computed in perturbation theory using Feynman diagrams (Abrikosov et al., 1963; Fetter and Walecka, 1971; Doniach and Sondheimer, 1974), while the more modern ones have a broader scope, use path integrals and attack nonperturbative problems (Feynman and Hibbs, 1965; Feynman, 1972; Faddeev, 1976; Fradkin, 2013, 2021; Altland and Simons, 2010). In two recent books I have discussed many aspects of the interrelation between condensed matter physics and quantum field theory in more depth than I can do in this chapter (Fradkin, 2013, 2021).

Early years: Feynman diagrams and correlation functions Quantum field theory played a key role in the development of the Theory of the Fermi Liquid (Pines and Nozières, 1966; Baym and Pethick, 1991). The theory of the Fermi liquid was first formulated by Landau using the framework of hydrodynamics and the quantum Boltzmann equation. Landau’s ideas were later given a microscopic basis using Green functions and Feynman diagrams (Abrikosov et al., 1963), including the effects of quantum fluctuations at finite temperature and nonequilibrium behavior (Kadanoff and Baym, 1962; Kamenev, 2011). Linear response theory was developed, which allowed the computation of response functions (such as electrical conductivities and magnetic susceptibilities) from the computation of correlation functions for a given microscopic theory (Martin, 1968). In turn, correlation functions can be computed in terms of a set of Feynman diagrams. These concepts and tools borrowed many concepts from field theory including the study of the analytic structure of the generalized susceptibilities and the associated spectral functions (together with the use of dispersion relations). These developments led to the derivation of the fluctuation-dissipation theorem. These ideas were widely applied to metals (Baym and Pethick, 1991) and superconductors (Schrieffer, 1964), as well as to quantum magnets (Mattis, 1965). The spectrum of an interacting system has low energy excitations characterized by a set of quantum numbers associated with the symmetries of the theory. These low energy excitations are known as quasiparticles. In the case of the Landau theory of the Fermi liquid (FL), the quasiparticle is a “dressed” electron: it is a low energy excitation with the same quantum numbers (charge and spin) and an electron but with a renormalized effective mass. There are many such quasiparticles in condensed matter physics. The correlation functions (the propagators) of a physical system have a specific analytic structure. In momentum (and frequency) space, the quasiparticle spectrum is given by the poles of the correlators. The role of symmetries and, in particular of gauge invariance, in the structure of correlation functions was investigated extensively. A direct consequence of symmetries is the existence of Ward–Takahashi identities, which must be satisfied by all the correlation functions of the theory. Ward–Takahashi identities are exact relations that relate different correlation functions. Such identities contain a host of important results. For example, in a theory with a globally conserved charge, the Hamiltonian (and the action) have a global U(1) symmetry associated with the transformation of the local field operator f(x) (which can be fermionic or bosonic) to a new field f0 (x) ¼ eiyf(x) (where y is a constant phase). Theories with a global continuous symmetry have a locally conserved current (and satisfy a continuity equation). The Ward–Takahashi identity requires the correlators of these currents (and densities) to be transverse (i.e., they should have vanishing divergence). In the absence of so-called quantum anomalies (which we will discuss below), global symmetries can be made local and become gauge symmetries. In many circumstances, a global symmetry can be spontaneously broken. If the global symmetry is continuous, then the Ward–Takahashi identities imply the existence of gapless excitations known as Goldstone bosons. For example in the case of a superfluid, which has a spontaneously broken U(1) symmetry the Goldstone boson is the gapless phase mode. Instead, the Néel phase of a quantum antiferromagnet has two gapless Goldstone bosons, the magnons of the spontaneously broken SO(3) global symmetry of this state of matter. Another example is the Ward–Takahashi identity of quantum electrodynamics (QED), which

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Field theoretic aspects of condensed matter physics: An overview

relates the electron self-energy to the electron–photon vertex function, which also holds in nonrelativistic electron fluids. In addition, these identities implied the existence of sum rules that the spectral functions must satisfy. All of these results became part of the standard toolkit of condensed matter experimentalists in analyzing their data and for theorists to make predictions. Ward–Takahashi identities and sum rules also imply restrictions on the allowed approximations which are often needed to obtain predictions from a microscopic model.

Critical phenomena Classical critical phenomena The late 1960s and particularly 1970s brought about an intense back and forth between condensed matter physics and field theory in the context of the problem of classical critical phenomena and phase transitions. This was going to become a profound revolution on the description of macroscopic physical systems with large-scale fluctuations. The problem of continuous (“second order”) phase transitions has a long history going back to the work of Landau (Landau, 1937; Landau and Lifshitz, 1959) who introduced the concept of an order parameter field. This turned out to be a powerful concept of broad applicability in many physical systems sometimes quite different from each other at the microscopic level. A simple example is that of a ferromagnet with uniaxial anisotropy in which the spins of the atoms in a crystal are strongly favored to be aligned (or antialigned) along certain directions of the crystal. The simplest microscopic model for this problem is the Ising model, a spin system in which the individual spins are allowed to take only two values, s ¼ 1. The partition function of the Ising model (in any dimension) is 0 1 X X J (1) Z¼ exp@− sðrÞsðr 0 ÞA T 0 ½s hr , r i where J is the exchange coupling constant and T is the temperature (measured in energy units); here [s] denotes the sum over the 2N spin configurations (for a lattice with N sites), and hr, r0 i are nearest neighbor sites of the lattice. The order parameter of the Ising model is the local magnetization hs(r)i which, in the case of a ferromagnet, is uniform. The partition function of the Ising model can be computed trivially in one dimension. The solution of the two-dimensional Ising model by Onsager constituted a tour-de-force in theoretical physics (Onsager, 1944). Its actual meaning remained obscure for some time. The work of Schultz et al. (1964) evinced a deep connection between Onsager’s solution and the problem if the spectrum of one-dimensional quantum spin chains (Lieb et al., 1961) [specifically the one-dimensional Ising model in a transverse field (Pfeuty, 1970)]. One important result was that the Ising model was in fact a theory of (free) fermions which, crudely speaking, represented the configurations of domain walls of the magnet. However, even this simple model cannot be solved exactly in general dimension, and approximate mean field theories of various sorts were devised over time to understand its physics.

Landau–Ginzburg theory Landau’s approach assumed that close enough to a phase transition, the important spin configurations are those for which the local magnetization varies slowly on lattice scales. In this picture, the local magnetization, on long enough length scales, becomes an order parameter field that takes values on the real numbers, and can be positive of negative. Thus, the order parameter field is effectively the average of local magnetizations on some scale large compared to the lattice scale, which we will denote by a real field f(x). The thermodynamics properties of a system of this type in d dimensions can be described in terms of a free energy Z h i k F½f ¼ dd x ðrfðxÞÞ2 + aðT − T c Þf2 ðxÞ + uf4 ðxÞ + ⋯ (2) 2 which is known as the Ginzburg–Landau free energy. Here, k is the stiffness of the order parameter field; Tc is the (mean-field) critical temperature; and a and u are two (positive) constants. This expression makes sense if the transition is continuous and hence that the order parameter is small near the transition. The energy of the Ising model is invariant under the global symmetry [s] 7! [−s]. This is the symmetry of the group 2 . Likewise, the Ginzburg–Landau free energy has the global (discrete) symmetry [f(x)] 7! [−f(x)] and also has a 2 global symmetry. In Landau’s approach, which was a mean field theory, the equilibrium state is the global minimum of this free energy. The nature
¼0 of the equilibrium state depends on whether T > Tc or T < Tc: for T > Tc the global minimum is the trivial configuration, fðxÞ
¼ ðaðT c − TÞ=2Þb , with the (this is the paramagnetic state), whereas for T < Tc, the equilibrium state is two fold degenerate, fðxÞ two degenerate states being related by the 2 symmetry (this is the ferromagnetic state). In the Landau theory, the critical exponent of the magnetization is b ¼ 1/2 and the critical exponent of the correlation length is n ¼ 1/2. However, in the case of the 2D Ising model, the order parameter exponent is b ¼ 1/8 (Yang, 1952) and the correlation length exponent is n ¼ 1. These (and other) apparent discrepancies led many theorists for much of the 1960s believe that each model was different and that these behaviors reflected microscopic differences. In addition, Landau’s theory was regarded as phenomenological and believed to be of questionable validity.

Field theoretic aspects of condensed matter physics: An overview

31

The renormalization group This situation was to change with the development of the Renormalization Group, due primarily to the work of Kadanoff (Kadanoff, 1966a, b; Efrati et al., 2014) and Wilson (Wilson, 1971a, b; Wilson and Fisher, 1972; Wilson and Kogut, 1974; Wilson, 1975, 1983). The renormalization group was going to have (and still has) a profound effect both in condensed matter physics and in Quantum Field Theory (and beyond).

Scaling Several phenomenological theories were proposed in the 1960s to describe the singular behavior of physical observables near a continuous phase transition (Patashinskii and Pokrovskii, 1966; Fisher, 1967, 1974). These early works argued that in order to explain the singular behavior of the observables, the free energy density had to have a singular part which should be a homogeneous function of the temperature, magnetic field, etc. A function f(x) is homogeneous if it satisfies the property that it transforms irreducibly under dilations, that is, f(lx) ¼ lkf(x), there l is a real positive number (a scale) and k is called the degree. These heuristic ideas then implied that the critical exponents should obey several identities. In 1966, Kadanoff wrote an insightful paper in which he showed that the homogeneity hypothesis implied that in that regime these systems should obey scaling. He showed that this can be justified by performing a sequence of block-spin transformations in which the configurations that vary rapidly at the lattice scale a become averaged at the scale of a larger sized block of length scale ba > a, which resulted in a renormalization of the coupling constants from {K} at scale a to {K0 } at the new scale ba (Kadanoff, 1966a; Efrati et al., 2014). In other words, the block spin transformation amounts to a scale transformation and a renormalization of the couplings (and operators). From this condensed matter/statistical physics perspective, the important physics is in the long-distance (“infrared”) behavior. A significant consequence of these ideas was that close enough to a critical point, if the distance |x − y| between two local observables O(x) and O(y) is large compared to the lattice spacing a but small compared to the correlation length x, their correlation function takes the form of a power law hOðxÞOðyÞi 

const: jx − yj2DO

(3)

where DO is a positive real number known as the scaling dimension of the operator O (Patashinskii and Pokrovskii, 1966; Kadanoff, 1966a; Kadanoff et al., 1967). These conjectures were known to be satisfied in the nontrivial case of the 2D Ising model (Kadanoff, 1966b), as well as in the Landau–Ginzburg theory once the effects of Gaussian fluctuations were included (Kadanoff et al., 1967). For example, in the 2D Ising model, the scaling dimension of the local magnetization s is Ds ¼ 1/8 and of the energy density e is De ¼ 1, which were sufficient to explain all the singular behaviors known at that time. The concept of renormalization actually originated earlier in quantum field theory as part of the development of QED. In QED, the notion of renormalization was used to “hide” the short distance (“ultraviolet”) divergencies of the Feynman diagrams needed to compute physical processes involving electrons (and positrons) and photons, that is, their strong, divergent, dependence of an artificially introduced short-distance cutoff or regulator. In particular, the sum of the leading diagrams that enter in the electron–photon vertex amounted to a redefinition (renormalization) of the coupling constant. It was observed by Murray Gell-Mann and Francis Low that this renormalization was equivalent to the solution of a first-order differential equation that governed the infinitesimal change of the coupling, the fine structure constant a ¼ e2/4p, under an infinitesimal change of the UV cutoff L (Gell-Mann and Low, 1954) L

da 2 2  bðaÞ ¼ a + Oða3 Þ 3p dL

(4)

where b(a) is the Gell-Mann–Low beta function. Except for the work by Bogoliubov and Shirkov (1959), this reinterpretation by Gell-Mann and Low was not actively pursued, partly because it predicted that the renormalized coupling became very large at short distances, a ! 1, and, conversely, it vanished in the deep long-distance regime, a ! 0 (if the electron bare mass is zero). In other words, QED is strongly coupled in the UV and trivial in the IR. The same behavior was found in the case of the theory of a scalar field f(x) with an f4 interaction, which is relevant in the theory of phase transitions. In addition to these puzzles, the1960s saw the experimental development of the physics of hadrons which involve strong interactions. For these reasons, for much of that decade most high-energy theorists had largely abandoned the use of quantum field theory, and explored other, phenomenologically motivated, approaches (which led to an early version of string theory.) At any rate the notion that the physics may depend on the scale was present as was the notion that in some regimes field theories may exhibit scale invariance at least in an approximate form.

The operator product expansion The next stage of the development of these ideas was the concept of the operator product expansion (OPE). If we denote by {Oj(x)} the set of all possible local operators in a theory (a field theory or a statistical mechanical system near criticality), then the product of two observables on this list closer to each other than to any other observable (and to the correlation length x) obeys the expansion lim Oj ðxÞOk ðyÞ ¼ lim x!y

x!y

X l

Cjkl jx − yj

Dj +Dk −Dl

Ol

x + y  2

(5)

where this equation should be understood as a weak identity, valid inside an expectation value. Remarkably, this concept was derived independently and simultaneously by Kadanoff (1969) (who was working in critical phenomena), by Wilson (1969)

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Field theoretic aspects of condensed matter physics: An overview

(who was interested in the short distance singularities arising in Feynman diagrams), and by Polyakov (1970, 1974) (also working in critical phenomena). In Eq. (5), {Dj} are the scaling dimensions of the operators {Oj}. The coefficients Cjkl are (like the dimensions) universal numbers. In a follow-up paper, Polyakov showed that if the theory has conformal invariance, that is, scale invariance augmented by conformal transformations which preserve angles, then, provided the operators Oj are suitably normalized, the coefficients Cjkl of the OPE are determined by a three-point correlator hOj ðxÞOk ðyÞOl ðzÞi ¼

Cjkl Djk

jx − yj jy − zjDkl jz − xjDlj

(6)

where Djk ¼ Dj + Dk − Dl. These results constitute the beginnings of conformal field theory (CFT). In a nontrivial check, Kadanoff and Ceva showed that the OPE holds for the local observables of the 2D Ising model (Kadanoff and Ceva, 1971).

Fixed points The next and crucial step in the development of the Renormalization Group was made by Wilson. Wilson was a high-energy theorist who wanted to know how to properly define a quantum field theory and the physical meaning of renormalization. In a Lorentz invariant quantum field theory, one is interested in the computation of the expectation value of time-ordered operators. In the case of a self-interacting scalar field f(x) in D-dimensional Euclidean space-time, obtained by analytic continuation from Minkowski spacetime to imaginary time, the observables are computed from the functional (or path) integral by functional differentiation of the partition function   Z Z (7) Z ¼ Df exp −Sðf, @ m fÞ + dD x JðxÞfðxÞ with respect to the local sources J(x). For a scalar field, the Euclidean action is   Z 1 m2 2 l S ¼ dD x ð@ m fðxÞÞ2 + f ðxÞ + f4 ðxÞ 2 2 4!

(8)

which has the same form as the free energy of the Landau–Ginzburg theory of phase transitions shown in Eq. (2). It is apparent that the Landau–Ginzburg theory is the classical limit of the theory of the scalar field whose partition function is a sum over all histories of the field. It is easy to see that an expansion of the partition function (or of a correlator) in powers of the coupling constant l can be cast in the form of a sum of Feynman diagrams. To lowest order in l a typical Feynman diagram involves a one-loop integral in momentum space of the form Z dD q 1 (9) IðpÞ ¼ ð2pÞD ðq2 + m2 Þððq − pÞ2 + m2 Þ As noted by Wilson in his Nobel Lecture (Wilson, 1983), this integral has large contributions from the IR region of small momenta q  0, but for any dimension D  4 has a much larger contribution form large the UV region of large momenta, which requires the introduction of a UV cutoff L in momentum space (or a lattice spacing a in real space by defining the theory on a hypercubic lattice). In quantum field theory, one then has to require that somehow one takes the limit a ! 0 (or L ! 1). To take this limit in the field theory is very much analogous to the definition of a conventional integral in terms of a limit of a Riemann sum. The difference is that this is a functional integral. While for a function of bounded variation in a finite interval (a, b), the limit of a partition of the interval into N steps each of length Dx such that NDx ¼ b − a exists and defines the integral of the function lim lim

Dx!0 N!1

N X j¼1

Z f ðxj ÞDxj ¼

a

b

dx f ðxÞ,

(10)

the analogous statement does not obviously exists in general for a functional integral, that is, an integral over a space of functions, which is what is required. In fact, although thousands of integrals of a function are known to exist, there are extremely few examples for a functional integral. Moreover, in order to take the continuum limit, the lattice spacing must approach zero, a ! 0. This means that physical scales, such as the correlation length x, must diverge in lattice units so that they can be fixed in physical units. But to do that, one has to be asymptotically close to a continuous phase transition! Hence, the problem of defining a quantum field theory is equivalent to the problem of critical phenomena at a continuous phase transition! Wilson gave a systematic formulation to the Renormalization Group by generalizing the earlier ideas introduced by Kadanoff and the earlier work by Gell-Mann and Low. Wilson’s key contribution was the introduction of the concept of a fixed point of the Renormalization Group transformation (Wilson, 1971a, b; Wilson and Fisher, 1972; Wilson and Kogut, 1974). As we saw, the block-spin transformation is a procedure for coarse graining the degrees of freedom of a physical system resulting in a renormalization of the coupling constants. Upon the repeated action of the RG transformation, its effect can be pictured as a flow in the space of coupling constants. However, in addition to integrating short distance degrees of freedom, one needs to restore the units of length, which have changed under that process. This requires a rescaling of lengths. Once this is done, Wilson showed that the resulting RG flows necessarily have fixed points, special values of the couplings which are invariant (fixed) under the action of the RG

Field theoretic aspects of condensed matter physics: An overview

33

transformation. He then deduced that at a fixed point, the theory has no scales, aside from the linear size L of the system and the microscopic UV cutoff (the lattice spacing a in a spin system). This analysis means that for length scales long compared to a ! 0 but short compared to L !1 the theory acquired a new, emergent, symmetry: scale invariance. Therefore, at a fixed point, the correlators of all local observables must be homogeneous functions (hence, must scale).

Universality A crucial consequence of the concept if the fixed point is that phase transitions can be classified into universality classes. Universality means that a large class of physical systems with different microscopic properties have fixed points with the same properties, that is, the same scaling dimensions, OPEs, and correlation functions at long distances independent on how they are defined microscopically. Although the renormalization group transformation is a transformation scheme that we define and, because of that the location in coupling constant space of the fixed point itself does depends on the scheme we choose, its universal properties are the same. Thus, universality classes depend only on features such as the space (and spacetime) dimension and the global symmetries of the system. But the systems themselves may be quite different. This we speak if the Ising universality class in 2D, on the superfluid (or XY) transition class in 3D, etc. This concept, which originated in the theory of phase transition, has been adopted and generalized in the development of CFT.

RG flows Combined with the condition that the correlators decay at long separations, homogeneity implies that the correlators must have the form of Eq. (3). In addition, this equation also implies that at a fixed point the operators (the local observables) have certain scaling dimensions. Let us consider a theory close to a fixed point whose action we will denote by S . Let {Oj(x)} be a complete set of local observables whose scaling dimensions are {Dj}. The total action of the theory close to the fixed point then can be expanded as a linear combination of the operators with dimensionless coupling constants {gj} Z X S ¼ S + dD x gj aDj −D Oj ðxÞ (11) j

0

Under a change of length scale x ! x ¼ bx, with b > 1, the operators (which must transform homogeneously) change as Oj ðbxÞ ¼ b −Dj Oj ðxÞ. Since the phase space changes as dDx0 ¼ bDdDx, we can keep the form of the action provided the coupling constants also change to compensate for these changes as g0j ¼ bD−Dj gj. Let b ¼ |x0 |/|x| ¼ 1 + da/a, where da is an infinitesimal change of the UV cutoff a. Then, if we integrate out the degrees of freedom in the range a < |x| < a + da, the rate of change of the coupling constants {gj} under this rescaling is a

dgj  bðgj Þ ¼ ðD − Dj Þgj + ⋯ da

(12)

which we recognize as a Gell-Mann–Low beta function for each coupling constant. This result says that if the scaling dimension Dj < D, then the renormalized coupling will increase as we increase the length scale, g0j > gj , and along this direction in coupling constant space, the RG flows away from the fixed point. Conversely, if Dj > D the renormalized coupling flows to smaller values, gj0 < gj and the RG flows into the fixed point. We then say that an operator is relevant if its scaling dimension satisfies Dj < D, and that it is irrelevant if Dj > D. If Dj ¼ D, then we say that operator is marginal. To go beyond this simple dimensional analysis, one has to include the effects of fluctuations. To lowest orders in the couplings one finds (Cardy, 1996) a

X dgj Cjkl gk gl + ⋯ ¼ ðD − Dj Þgj + da k,l

(13)

where {Cjkl} are the coefficients of the OPE shown in Eq. (6). This expression is the general form of a perturbative renormalization group, and it is valid close enough to a fixed point. Wilson and Fisher (1972) used a similar approach to analyze how fluctuations alter the results of the Landau–Ginzburg theory. They considered the partition function of Eq. (7) with the action of Eq. (8). Instead of working in real space, they considered the problem in momentum space and partitioned the field configurations into slow and fast modes fðxÞ ¼ f< ðxÞ + f> ðxÞ

(14)

where f>(x) are configurations whose Fourier components have momenta in the range bL < |p| < L, where L is a UV momentum cutoff and b < 1. Hence, if we choose b ! 1, the fast modes f> (x) have components in a thin momentum shell near the UV cutoff L. Conversely, the slow modes f 4 and the asymptotic IR behavior is the same as predicted by the Landau–Ginzburg theory. However, for D < 4, the free-field fixed point becomes unstable and a new fixed point arises at g ¼ 23 E + OðE2 Þ, where we have set E ¼ 4 − D. This is the Wilson–Fisher fixed point. At this fixed point, E the correlation length diverges with an exponent n ¼ 12 + 12 + OðE2 Þ, which deviates from the predictions of the Landau–Ginzburg theory. The small parameter of this expansion is E, and this is known as the E expansion. The Wilson–Fisher (WF) fixed point is an example of a nontrivial fixed point at which the correlation length is divergent. It has only one relevant operator, the mass term, which in the IR flows into the symmetric phase for t > 0 and flows to the broken symmetry for t < 0. Conversely, in the UV, it flows into the WF fixed point. For these reasons, condensed matter physicists say that this is an IR unstable (or critical) fixed point while high-energy physicists say that it is the UV fixed point. At this fixed point, a nontrivial field theory can be defined with nontrivial interactions. UV fixed points also define examples of what in high-energy physics are called renormalizable field theories and can be used to define a continuum field theory. The D ¼ 4-dimensional theory is special in that the f4 operator is marginal. As can be seen in Eq. (16), at D ¼ 4 the beta function for the dimensionless coupling constant g does not have a linear term and is quadratic in g. In this case the operator is marginally irrelevant, and its beta function has the same behavior as the beta function of Gell-Mann and Low for QED. Such theories are said to have a “triviality problem” since, up to logarithmic corrections to scaling, there are no interactions in the IR and, conversely, become large in the UV. There are also fixed points at which the correlation length x ! 0. These fixed points are IR stable (and in a sense trivial). These stable fixed points are sinks of the IR RG flows. Such fixed points define stable phases of matter, for example the broken symmetry state, and the symmetric (or unbroken state). However in the UV, they are unstable and in high-energy physics such fixed points correspond to nonrenormalizable field theories. More sophisticated methods are needed to go beyond the lowest order beta functions of Eq. (13), and the computation of critical exponents beyond the leading nontrivial order. Possibly, the record high-precision calculations have been done for f4 theory for which the beta function is known to be O(E5). This has been achieved using the method of dimensional regularization (’t Hooft and Veltman, 1972; Bollini and Giambiagi, 1972; Amit, 1980) (with minimal subtraction). Special resummation methods (Borel-Padé) have been used to do these calculations in D ¼ 3 dimensions (Zinn-Justin, 2002). Remarkably, these results are so precise that in the case of the superfluid transition, which is well described by a f4 theory with a complex field, the results could only be tested in the microgravity environment of the International Space Station!

Asymptotic freedom There are several physical systems of great interest whose beta function has the form bðgÞ ¼ Ag2 + ⋯

(17)

The coupling constant has a different interpretation in each theory and the constant A > 0, opposite to the sign of the beta function found in QED and f4 theory in D ¼ 4 dimensions. This beta function means that while the associated operator is marginal, with this sign, it is actually marginally relevant. This also means that the fixed point is unstable in the IR but the departure from the fixed point is logarithmically small. Conversely, in the UV, the RG flows into the fixed point and the effective constant is weak at short distances. This is the origin of the term asymptotic freedom (Gross and Wilczek, 1973). The paradigmatic examples of theories with a beta function of this form are the Kondo problem, the 2D nonlinear sigma model, and the D ¼ 4 dimensional Yang–Mills non-Abelian gauge theory. The Kondo problem is the theory of a localized spin-1/2 degree of freedom in a metal. The electrons of the metal couple to this quantum impurity through an exchange interaction of the impurity and the magnetic moment density of the mobile electrons in the metal with coupling constant J. This problem is actually one-dimensional since only the s wave channel of the mobile (conduction band) electrons actually couple to the localized impurity. In 1970, Philip Anderson developed a theory of the Kondo problem in terms of the renormalization of the Kondo coupling constant g as a function of the energy scale (Anderson, 1970). Anderson used perturbation theory in J to progressively integrate the modes of the conduction electrons close to an effective bandwidth Ec and found that the beta function has the form of Eq. (17) Ec

dJ ¼ −rJ2 + ⋯ dEc

(18)

where r is the density of states at the Fermi energy of the conduction electrons. This work implied that the free-impurity fixed point is IR unstable and that the effective coupling constant J increases as the energy cutoff Ec is lowered. He argued that at some energy scale, the Kondo scale, perturbation theory breaks down and that there is a crossover to a strong coupling regime, which is not accessible in perturbation theory.

Field theoretic aspects of condensed matter physics: An overview

35

Shortly thereafter, in 1973, Wilson developed a numerical renormalization group approach, which showed that the Kondo problem is indeed a crossover from the free impurity fixed point to the “renormalized” FL (Wilson, 1975). In addition, Wilson used the numerical renormalization group to examine the approach to the strong coupling fixed point and showed that it is characterized by a Wilson ratio, a universal number obtained from the low temperature specific heat and the impurity magnetic susceptibility (in suitable units). Wilson’s numerical RG predicted a number close to 2p for this ratio. In 1980, N. Andrei and P. Wiegmann showed (independently) that the Kondo problem is an example of an integrable field theory that can be solved by the Bethe ansatz (Andrei, 1980; Wiegmann, 1980). Their exact result was consistent with Wilson’s RG, including the numerical value of the Wilson ratio. In 1972, Gerard ’t Hooft and Martinus Veltman showed that Yang–Mills gauge theory is renormalizable (’t Hooft and Veltman, 1972). This groundbreaking result opened the door to use quantum field theory to develop the theory of strong interactions in particle physics known as Quantum Chromodynamics (QCD). In 1973, Gross and Wilczek (1973) and, independently, Politzer (1973) computed the renormalization group beta function of Yang–Mills theory with gauge group G and found it to be of the same form as Eq. (17), L

dg g3 11 ¼− C ðGÞ + ⋯ dL 16p2 3 2

(19)

where g is the Yang–Mills coupling constant and L is a UV momentum scale. Here, C2(G) is the quadratic Casimir for a gauge group G. For SU(3), the case of physical interest, C2(SU(3)) ¼ 3. This result implies that under the RG at large momenta (short distances), the Yang–Mills coupling constant flows to zero (up to logarithmic corrections). This result holds in the presence of quarks, provided the number of quark flavors is less than a critical value. Hence at short distances, the effective coupling is weak. Gross and Wilczek called this phenomenon asymptotic freedom. This behavior was consistent with the observation of weakly coupled quarks in deep inelastic scattering experiments. However, the flip side of asymptotic freedom is that at low energies (long distances), the coupling constant grows without limit, which implies that at low energies perturbation theory is not applicable. This strong infrared behavior suggested that in QCD quarks are permanently confined in color neutral bound states (hadrons). However, unlike the Kondo problem we just discussed, QCD is not an integrable theory (so far as we know) and to show that it confines has required the development of lattice gauge theory (Wilson, 1974; Kogut and Susskind, 1975). To this date, the best evidence for quark confinement has been obtained using large-scale Monte Carlo simulations in lattice gauge theory (Creutz et al., 1983). We close this section with a discussion of an important case: the nonlinear sigma models. The O(N) nonlinear sigma model is the continuum limit of the classical Heisenberg model for a spin with N components. Historically, the nonlinear sigma model is the effective field theory for pions in particle physics. We will discuss its role in the theory of quantum antiferromagnets in Section “Quantum antiferromagnets and nonlinear sigma models” and especially in the case of quantum antiferromagnetic spin chains in the Section “Nonlinear sigma models and antiferromagnetic quantum spin chains.” The simplest nonlinear sigma model is a theory of an N-component scalar field n(x) which satisfies the unit length local constraint, n2(x) ¼ 1. The Euclidean Lagrangian is L¼

1 ð@ nðxÞÞ2 2g m

(20)

where g is the coupling constant (the temperature in the classical Heisenberg model). At the classical level, that is, in the broken symmetry phase, where hni 6¼ 0, this model describes the N − 1 massless modes (Goldstone bosons) of the spontaneously broken O(N) symmetry. Dimensional analysis shows that the coupling constant has units of ℓD−2. Hence, we expect to find marginal behavior at D ¼ 2. In 1975, Alexander Polyakov used a momentum shell renormalization group in D ¼ 2 dimensions and showed that the beta function of this model is (here a is the short-distance cutoff ) (Polyakov, 1975b) bðgÞ ¼ a

dg N − 2 2 ¼ g + Oðg3 Þ 2p da

(21)

Hence, in D ¼ 2 dimensions also this theory is asymptotically free. As in the other examples we just discussed, asymptotic freedom here also implies that the coupling constant g grows to large values in the low-energy (long-distance) regime. In close analogy with Yang–Mills theory in D ¼ 4 dimensions, Polyakov conjectured that the O(N) nonlinear sigma model also undergoes dynamical dimensional transmutation (Gross and Wilczek, 1973), that the global O(N) symmetry is restored and that for all values of the coupling constant g the theory is in a massive with a finite correlation length x  expððN − 2Þ=2pgÞ . Extensive numerical simulations, used to construct a renormalization group using Monte Carlo simulations (Shenker and Tobochnik, 1980), showed that there is indeed a smooth crossover from the weak coupling (low temperature) regime to the high temperature regime where the correlation length is finite. The nonlinear sigma model is a renormalizable field theory in D ¼ 2 dimensions (Brézin and Zinn-Justin, 1976). For D > 2 dimensions, it can be studied using the 2 + E expansion (Brézin and Zinn-Justin, 1976; Zinn-Justin, 2002), which predicts the existence of a nontrivial UV fixed point and a phase transition from a Goldstone phase to a symmetric phase. It turns out that there is a significant number of asymptotically free nonlinear sigma models in D ¼ 2 dimensions, many of physical interest (Friedan, 1985), in particular nonlinear sigma models whose target manifold is a coset space, a quotient of a group G and a subgroup H. The O(N) nonlinear sigma model is an example since the broken symmetry space leaves the O(N − 1) subgroup unbroken (the manifold of the Goldstone bosons). In that case, the quotient is O(N)/O(N − 1), which is isomorphic to the N − 1 dimensional sphere SN−1. In later sections, we will discuss other examples in which more general nonlinear sigma models play an important role.

36

Field theoretic aspects of condensed matter physics: An overview

Models on coset spaces arise in the theory of Anderson localization in D ¼ 2 dimensions. Anderson localization is the problem of a fermion (an electron) in a disordered system in which the electron experiences a random electrostatic potential. In the limit of strong disorder Philip Anderson showed that all one-particle states are exponentially localized and the diffusion constant (and the conductivity) vanishes (Anderson, 1958). There was still the question of when it is possible for the electron to have a finite diffusion constant (and conductivity). In D ¼ 2 dimensions, the conductivity is a dimensionless number which suggests that this may be the critical dimension for diffusion. Abrahams, Anderson, Licciardello, and Ramakrishnan used a weak disorder calculation to construct a scaling theory that implied that in D ¼ 2 dimensions, the RG flow of the conductivity at long distances (large samples) flows to zero and all states are localized (Abrahams et al., 1979). Shortly thereafter, Wegner gave strong arguments that showed that the existence of diffusion implied that there are low-energy “diffussion” modes which behaved as Goldstone modes of a nonlinear sigma model on the quotient manifold O(N+ + N−)/O(N+) O(N−) in the “replica limit” N ! 0 (Wegner, 1979). A field theory approach to this nonlinear sigma model was developed by McKane and Stone (1981) and by Hikami (1981).

Quantum criticality Quantum criticality is the theory of a phase transition of a system (e.g., a magnetic system) at zero temperature that occurs as a coupling constant (or parameter) is varied continuously. Although not necessarily under that name, this question has existed as a conceptual problem for a long time. In particular, already in 1973, Wilson considered the problem of the behavior of quantum filed theories below four spacetime dimensions and their phase transitions (Wilson, 1973). The modern interest in condensed matter physics stems from discoveries made since the late 1980s. Since that time the behavior of condensed matter systems at a quantum critical point has emerged as a major focus in the field. There were several motivations for this problem. One was (and still is) to understand the behavior of quantum antiferromagnets in the presence of frustrating interactions. Frustrating interactions are interactions that favor incompatible types of antiferromagnetic orders. The result is the presence of intermediate nonmagnetic “valence bond” phases that favor the formation of spin singlets between nearby spins. These phases typically either break the point group symmetry of the lattice or are spin liquids (which will be discussed below). Another motivation is that doped quantum antiferromagnets typically harbor superconducting phases (among others) whose hightemperature behavior is a “strange” metal that violates the basic assumptions (and behaviors) of FLs. The most studied version of this problem is the case of the copper oxide high temperature superconductors. It was conjectured that there is a quantum critical point inside the superconducting phase at which the antiferromagnetic order (or other orders) disappears and which may be the reason for the strange metal behavior above the superconducting critical temperature. Many of these questions are discussed in depth by Sondhi et al. (1997) and in the textbook by Sachdev (2011).

Dynamic scaling We will consider a general quantum phase transition and assume that it is scale invariance. However, except for the case of relativistic quantum field theories, in condensed matter systems, space and time do not need to scale in the same way. Let us assume that the system of interest has just one coupling constant g and that the system of interest has a quantum phase transition (at zero temperature) at some critical value gc between two phases, for instance one with a spontaneously broken symmetry and a symmetric phase. If the quantum phase transition is continuous, then the correlation length x will diverge at gc and so will the correlation time xt. However, these two scales are in general different and do not necessarily diverge at the same rate. So, in general, if some physical quantity is measured at the quantum critical point at some momentum p and frequency o, the length scale of the measurement is 2p/|p| and at a frequency is o  |p|z, where z is the dynamic critical exponent. Let us say that we measure the observable O at momentum p and frequency o at gc. Scale invariance in both space and time means that at gc, the observable Oðp, oÞ at momentum p and frequency o must scale as ~ z =oÞ Oðp, oÞ ¼ jpj−DO Oðjpj

(22)

where DO is the scaling dimension of the observable O. The situation changes at finite temperature T. A quantum field theory at temperature T is described by a path integral on a manifold, which, along the imaginary time direction t, is finite of length 2p/T and periodic for a theory bosonic fields and antiperiodic for fermionic fields (Fradkin, 2021). Since the imaginary time direction is finite, the behavior for correlation times xt < 2p/T and xt > 2p/T must be different. Indeed, in the first regime, the behavior is essentially the same as at T ¼ 0 while in the second, it should be given by the classical theory in the same space dimension. At the quantum critical point gc, there is only one time scale xt  2p/T and only one length scale x  (2p/T )1/z.

The Ising model in a transverse field The prototype of the quantum phase transition is the Ising model in a transverse field. This model describes a system of spin-1/2 degrees of freedom with ferromagnetic interactions with uniaxial anisotropy in the presence of a transverse uniform magnetic field. The Hamiltonian is

Field theoretic aspects of condensed matter physics: An overview

H ¼ −J

X

s3 ðrÞs3 ðr 0 Þ − h

hr , r 0 i

X r

s1 ðrÞ

37

(23)

where J and h are the exchange coupling constant and the strength of the transverse field, respectively. Here, s1 and s3 are the two Pauli matrices defined on the sites {r} of a lattice with ferromagnetic interactions between spins on nearest neighboring sites. The Hilbert space is the tensor product of the states of the spins at each site of the lattice. At each site, there are two natural bases of states: the eigenstates of s3, which we denote by |"i and |#i (whose eigenvalues are  1), and the eigenstates of s1, which we denote by |i (whose eigenvalues are also  1). It is well known that the Ising model in a transverse field on a hypercubic lattice in D dimensions is equivalent to a classical Ising model in D + 1 dimensions (Fradkin and Susskind, 1978; Kogut, 1979). These two models are related through the transfer matrix. Indeed, a classical Ising model can be regarded as a path integral representation of the quantum model in one dimension less. For simplicity we will see how this works for the 2D classical ferromagnetic Ising model of Eq. (1), but the construction is general. We will regard the configuration of spins on a row of the 2D lattice as a state of a quantum system, and the set of states on all rows as the evolution of the state along the perpendicular direction that we will regard as a discretized imaginary time. The contribution from two adjacent rows to the partition function defines the matrix element of a matrix between two arbitrary configurations. In Statistical Physics, this matrix is known as the transfer matrix T^ and the full partition function (with periodic boundary conditions) is Nt

Z ¼ trT^

(24)

where Nt is the number of rows. For the case of the ferromagnetic Ising model (actually, for any unfrustrated model), the transfer matrix can always be constructed to be Hermitian. This property holds in fact for any theory that satisfies a property known as reflection positivity which requires that all (suitably defined) correlation functions be positive. For theories of this type, and the Ising model is an example, the matrix elements of the transfer matrix can be identified with the matrix element of the evolution operator of a quantum theory for a small imaginary time step (Fradkin and Susskind, 1978). Also, the positivity of the correlators is equivalent to the condition of positivity of the norm of states in the quantum theory. For the classical models that satisfy these properties, all directions of the lattice are equivalent. Moreover, asymptotically close to the critical point, the behavior of all the correlators becomes isotropic, that is, invariant under the symmetries of Euclidean space. This means that the arbitrary choice of the direction for the transfer matrix is irrelevant. Consequently, in the quantum model, its equal-time correlators behave the same way as its correlation functions in imaginary time. In other words, space and time behave in the same way and the quantum theory is relativistically invariant. This implies at the quantum critical point the energy e(p) of its massless excitations should behave as e(p) ¼ v |p|. In a relativistic theory the dynamical critical exponent must be z ¼ 1 and the coefficient v is the speed of the excitations (the “speed of light”). We should note that this is not necessarily always the case. There are in fact classical systems, for example, liquid crystals (Chaikin and Lubensky, 1995), which are spatially anisotropic and map onto quantum mechanical theories in one less dimension for which the dynamical critical exponent z 6¼ 1. One such example are the Lifshitz transitions of nematic liquid crystals in three dimensions and the associated quantum Lifshitz model in D ¼ 2 dimensions, for which the dynamical exponent is z ¼ 2 (Ardonne et al., 2004). Just as in the classical counterpart in D + 1 dimensions, the quantum model in D  1 has two phases: a broken symmetry ferromagnetic phase for J h and a symmetric paramagnetic phase for h J. In the symmetric phase, the ground state is unique (asymptotically is the eigenstate of s1 with eigenvalue + 1) while in the broken symmetry phase, the ground state is doubly degenerate (and asymptotically is an eigenstate of s3) and there is a nonvanishing expectation value of the local order parameter hs3(r)i 6¼ 0. In the symmetric phase, the correlation function of the local order parameter decays exponentially with distance with a correlation length x, as does the connected correlation function in the broken symmetry phase. The model has a continuous quantum phase transition at a critical value of the ratio h/J. For general space dimensions D > 1, this model is not exactly solvable and much of what we know about it is due to large-scale numerical simulations. This problem was solved exactly in one-dimension (Pfeuty, 1970) using the Jordan–Wigner transformation that maps a one-dimensional quantum spin system to a theory of free fermions (Lieb et al., 1961). The fermion operators at site j are ^ w2 ðjÞ ¼ iKðjÞs 3 ðjÞ

w1 ðjÞ ¼ Kðj − 1Þs3 ðjÞ,

(25)

where K( j) is the kink creation operator (i.e., the operator that creates a domain wall between sites j and j + 1 (Fradkin and Susskind, 1978), and is given by Y KðjÞ ¼ s1 ðnÞ (26) nj

The operators w1( j) and w2( j) are Hermitian, w{j ðnÞ ¼ wj ðnÞ and obey the anticommutation algebra fwj ðnÞ, wj0 ðn0 Þg ¼ 2djj0 dnn0 :

(27)

Hence, they are fermionic operators and are Hermitian, anticommute with each other and square to the identity. Operators of this type are called Majorana fermions.

38

Field theoretic aspects of condensed matter physics: An overview

Alternatively, we can use the more conventional (Dirac) fermion operators c(n) and its adjoint c{(n) which are related to the Majorana fermions as c{ ðnÞ ¼ w1 ðnÞ − iw2 ðnÞ

cðnÞ ¼ w1 ðnÞ + iw2 ðnÞ,

(28)

which obey the standard anticommutation algebra fcðnÞ, cðn0 Þg ¼ fc{ ðnÞ, c{ ðn0 Þg ¼ 0,

fcðnÞ, c{ ðn0 Þg ¼ d − nn0 :

(29)

In this sense, a Majorana fermion is half of a Dirac fermion. In terms of the Majorana operators, the Hamiltonian of Eq. (23) becomes X X H ¼ i w1 ðjÞw2 ðjÞ + ig w2 ðjÞw1 ðj + 1Þ j

(30)

j

where we have rescaled the Hamiltonian by a factor of h and the coupling constant is g ¼ J/h. Here, we have not specified the boundary conditions (which depend on the fermion parity). Qualitatively, the Majorana fermions can be identified with the domain walls of the classical models. In the Ising model, the number of domain walls on each row is not conserved but their parity is. Likewise, the number of Majorana fermions NF is not conserved either but the fermion parity ð−1ÞNF is conserved. It is an elementary exercise to show that the spectrum of this theory has a gap G( g) that vanishes at gc ¼ 1 as G( g) |g − gc|n, with an exponent n ¼ 1. Since the Hamiltonian of Eq. (30) is quadratic in the Majorana operators, these operators obey linear equations of motion. In the scaling regime, we take the limit of the lattice spacing a ! 0 and the coupling constant g ! gc ¼ 1 while keeping the quantity m ¼ (g − gc)/a fixed. In this regime, the two-component Hermitian spinor field w ¼ (w1, w1), and w{ ¼ w, satisfies a Dirac equation ði@= − mÞw ¼ 0

(31) T

where we set the speed v ¼ 1, and defined the 2 2 Dirac gamma-matrices g0 ¼ s2, g1 ¼ is3, and g5 ¼ s1. Upon defining w
¼ w g0, we find that the Lagrangian of this field theory is = − L ¼ w
i@w

1 m w
w 2

(32)

which indeed becomes massless at the quantum phase transition of the Ising spin chain. For these considerations, we say that the phase transition of the Ising model (2D classical or 1D quantum) is in the universality class of massless Majorana fermions where m ! 0. In Eq. (32), we have used the standard Feynman slash notation, a/ ¼ gmam, where am is a vector.

Quantum antiferromagnets and nonlinear sigma models As we noted above, the discovery of high temperature superconductors in the copper oxide compounds prompted the study of the behavior of these strongly correlated materials at low temperatures and of possible quantum phase transitions which they may host. The prototypical cuprate material La2CuO4 is a quasi two-dimensional Mott insulator that exhibits long-range antiferromagnetic order below a critical temperature Tc. A simple microscopic model is a spin-S quantum Heisenberg antiferromagnet on the 2D square lattice of the Cu atoms, whose Hamiltonian is H¼

1X Jðjr − r 0 jÞ SðrÞ  Sðr 0 Þ 2 r, r0

(33)

where S are the spin-S angular momentum operators. We will consider the case where the exchange interaction for nearest neighbors J is dominant and a weaker J0 J for next nearest neighbors. In this section we do not consider the regime J0 ’ J in which the interactions compete for incompatible ground states due to frustration effects.

Spin coherent states The simplest way to see the physics of this antiferromagnet is to construct a path-integral representation for a spin-S system using spin coherent states (Perelomov, 1986; Fradkin and Stone, 1988; Wiegmann, 1989). For details, see Fradkin (2013) which we follow here. A coherent state of the (2S + 1-dimensional) spin-S representation of SU(2) is the state |ni, labeled by the spin polarization unit vector n jni ¼ eiyðn0 nÞ  S jS, Si

(34)

where n ¼ 1. The states of the spin-S representation are spanned by the eigenstates of S3 and S , 2

2

S3 jS, Mi ¼ MjS, Mi,

S2 jS, Mi ¼ SðS + 1ÞjS, Mi

(35)

and |S, Si is the highest weight state with eigenvalues S and S(S + 1). In Eq. (34), n0 is a unit vector along the axis of quantization (the direction e3), and y is the colatitude, such that n  n0 ¼ cos y. Two spin coherent states, |n1i and |n2i, are not orthonormal,

Field theoretic aspects of condensed matter physics: An overview

hn1 jn2 i ¼ eiFðn1 ,n2 ,n0 Þ S

 S 1 + n1  n2 Þ 2

39

(36)

where F(n1, n2, n0) is the area of the spherical triangle of the unit sphere spanned by the unit vectors n1, n2, and n0. However, there is an ambiguity in the definition of the area of the spherical triangle since the sphere is a 2-manifold without boundaries: if the “inside” triangle has spherical area F, the complement (“outside”) triangle has area 4p − F. Thus, the ambiguity of the phase prefactor of Eq. (36) is ei4pS ¼ 1

(37)

since S is an integer or a half-integer. So, the quantization of the representations of SU(2) makes the ambiguity unobservable. In addition, the spin coherent states |ni satisfy the resolution of the identity Z   2S + 1 I ¼ jnihnj dðn2 − 1Þ d3 n (38) 4p and hnjSjni ¼ Sn

(39)

Path integral for a spin-S degree of freedom

As an example, consider problem of a spin-S degree of freedom coupled to an external magnetic field B(t) that varies slowly in time. The (time-dependent) Hamiltonian is given by the Zeeman coupling HðtÞ ¼ BðtÞ  S

(40)

As usual, the path integral is obtained by inserting the (over-complete) set of coherent states at a large number of intermediate times. The resulting path integral is a sum of the histories of the spin polarization vector n(t) Z T  Z Y Z ¼ tr exp i dt HðtÞ ¼ Dn expðiS½nÞ dðn2 ðtÞ −1Þ (41) 0

t

where the action is Z T S ¼ S S WZ ½n − S dt BðtÞ  nðtÞ 0

(42)

where S WZ ½n is the Wess–Zumino action Z S WZ ½n ¼ Z ¼

T

0

0

1

dt A½n  @ t n Z

dt

0

T

dt nðt, tÞ  @ t nðt, tÞ @ t nðt, tÞ

(43) (44)

where A[n] is the vector potential of a Dirac magnetic monopole (of unit magnetic charge) at the center of the unit sphere. The vector potential A[n] has a singularity associated with the Dirac string of the monopole. We can write an equivalent expression that is singularity-free using Stokes Theorem. We did this in the second line of Eq. (44) which required to extend the circulation of A on the closed path described by n(t) to the flux of the vector potential through the submanifold S of the unit sphere S2 whose boundary is the history n(t), that is, the area of S. The smooth (and arbitrary) extension of configuration n(t) to the interior of S is done by defining n(t, t) such that n(t, 1) ¼ n0, n(t, 0) ¼ n(t), and n(0, t) ¼ n(T, t). Since S WZ is the area of the submanifold S of the unit sphere S2, just as in Eq. (37), here too there is an ambiguity of 4p in the definition of the area. Here too, this ambiguity is invisible since the spin S is an integer or a half-integer. The path integral of Eq. (41) was derived first by Berry (1984) (and extended by Simon (1983)). The first term (which we called Wess–Zumino by analogy with its field theoretic versions) is called the Berry Phase. The role of this term, which is first order in time derivatives, is to govern the quantum dynamics of the spin which, in the presence of a uniform magnetic field, executes a precessional motion of the (Bloch) sphere. It is also apparent from this expression that in the large-S limit, the path integral can be evaluated by means of a semiclassical approximation. The coherent-state construction shows that this problem is equivalent to the path integral of a formally massless nonrelativistic particle of unit electric charge on the surface of the unit sphere with a magnetic monopole of magnetic charge S in its interior! This is not surprising since the Hilbert space of a nonrelativistic particle moving on the surface of a sphere with and radial magnetic field (the field of a magnetic monopole) has a Landau level-type spectrum with a degeneracy given by the flux. The condition of a massless particle means that only the lowest Landau level survives and all other levels have an infinite energy gap. The coherent state approach has been used to derive a path integral formulation for ferromagnets and antiferromagnets. A detailed derivation can be found in Fradkin (2013).

40

Field theoretic aspects of condensed matter physics: An overview

Quantum ferromagnet

We will consider first the simpler case of a quantum ferromagnet and in Eq. (33) we will set J ¼ −|J| < 0 for nearest neighbors and zero otherwise. The action for the path integral for the spin-S quantum Heisenberg ferromagnet on a hypercubic lattice is Z X jJjS2 X T 2 dt ðnðr, tÞ − nðr 0 , tÞÞ (45) S ¼ S S WZ ½nðr, tÞ − 2 r hr , r 0 i 0 where we have subtracted the classical ground state energy. The order parameter for this theory is the expectation value of the local magnetization, n ¼ hn(r)i, which is constant in space but points in an arbitrary direction in spin space. In the low-energy regime, the important configurations are slowly varying in space and we can simply approximate the action of Eq. (45) by its continuum version in d space dimensions Z Z Z T jJjS2 S d d x dt ðrnðx, tÞÞ2 (46) S ¼ d dd x S WZ ½nðx, tÞ − a0 2ad0 0 where a0 is the lattice spacing. As before, the path integral is done for a field, which satisfies everywhere in space-time the constraint n2(x, t) ¼ 1. This action can be regarded as nonrelativistic nonlinear sigma model. It is straightforward to show that the classical equations of motion for this theory are the Landau–Lifshitz equations @ t n ¼ jJjSa20 n r2 n

(47)

subject to the constraint n ¼ 1. Due to the constraint, the Landau–Lifshitz equation is nonlinear. We will decompose the field into a longitudinal and two transverse components, s and p, respectively   s n¼ (48) p 2

subject to the constraint s2 + p2 ¼ 1. The linearized Landau–Lifshitz equations become (to linear order in p) @ t p1 ’ −jJjSa20 r2 p2 ,

@ t p2 ’ +jJjSa20 r2 p1

(49)

The solution to these equations are ferromagnetic spin waves (magnons or Bloch waves) which satisfy the dispersion relation oðpÞ ’ jJjSa20 p2 + Oðp4 Þ

(50)

which shows that the dynamic exponent for a ferromagnet is z ¼ 2. Notice that in this case the two transverse components are not independent (they are effectively a dynamical pair). These are the Goldstone bosons of a ferromagnet.

Quantum antiferromagnet Formally, the quantum antiferromagnet has a coherent state path integral whose action is Z X JS2 X T S ¼ S S WZ ½nðr, tÞ − dt nðr, tÞ  nðr 0 , tÞ 2 r hr , r 0 i 0

(51)

with J > 0. For a bipartite lattice, for example, the 1D chain, and the square and cubic lattices, the classical ground state is an antiferromagnet with a Néel order parameter, the staggered magnetization. Let m(r) be the expectation value of the local magnetization. A bipartite lattice is the union of two interpenetrating sublattices, and the local magnetization is staggered, that is, it takes values with opposite signs (with equal values) on the two sublattices. Thus, we make the change of variables, n(r, t) ! (−1)rn(r, t) in Eq. (51) and find Z X JS2 X T 2 S ¼ S ð −1Þr S WZ ½nðr, tÞ − dt ðnðr, tÞ − nðr 0 , tÞÞ (52) 2 0 0 r hr , r i We want to obtain the low energy effective action for the field n(r, t). To this end, we decompose this field into a slowly varying part, that we will call m(r, t), and a small rapidly varying part l(r, t) (which represents ferromagnetic fluctuations) nðr, tÞ ¼ mðr, tÞ+ð −1Þr a0 lðr, tÞ

(53)

Since n (r, t) ¼ 1, we will demand that the slowly varying part also obeys the constraint, m (r, t) ¼ 1, and require that the two components be orthogonal to each other, m  l ¼ 0. Due to the behavior of the staggered Wess–Zumino terms of Eq. (52), the resulting continuum field theory turns out to have subtle but important differences between one dimension and higher dimensions. Here we will state the results for two and higher dimensions. We will discuss in detail the one-dimensional below when we discuss the role of topology. It turns out that if the dimension d > 1, the contribution of the staggered Wess–Zumino terms for smooth field configurations is (Fradkin and Stone, 1988; Dombre and Read, 1988; Haldane, 1988b) 2

2

Field theoretic aspects of condensed matter physics: An overview

lim S

a0 !0

X r

ð −1Þr S WZ ½nðr, tÞ ¼ S

Z

d3 x lðx, tÞ  mðx, tÞ @ t mðx, tÞ

The continuum limit of the second term of Eq. (52) in the case of a two-dimensional system is Z Z  JS2 X T JS2 2 lim d3 x ðrmðx, tÞÞ2 + 4l2 ðx, tÞ : dt ðnðr, tÞ − nðr 0 , tÞÞ ¼ a0 2 a0 !0 2 0 0 hr , r i

41

(54)

(55)

The massive field l[x, t] represents ferromagnetic fluctuations. Since this is a massive field, it can be integrated out leading to an effective field theory for the antiferromagnetic fluctuations m(x, t) whose Lagrangian is that of a nonlinear sigma model   1 1 L¼ (56) ð@ t mðx, tÞÞ − vs ðrmðx, tÞÞ2 2g vs where the coupling constant is g ¼ 2/S and the spin-wave velocity is vs ¼ 4a0JS. If we allow for a weak next-nearest-neighbor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi interaction J0 > 0, the coupling constant g and the spin wave velocity vs become renormalized to g0 ’ g= 1 − 2J0 =J and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0s ’ vs 1 − 2J0 =J. We conclude that the quantum fluctuations about a Néel state are well described by a nonlinear sigma model. Provided the frustration effects of the next-nearest-neighbor interactions are weak enough, the long-range antiferromagnetic Néel order should extend up to a critical value of the coupling constant gc where the RG beta function has a nontrivial zero, which signals a quantum phase transition to a strong coupling phase without long-range antiferromagnetic order. Motivated by the discovery of high temperature superconductivity in the strongly correlated quantum antiferromagnet La2CuO4 (at finite hole doping) in 1988, Chakravarty et al. (1988) utilized a quantum nonlinear sigma model to analyze this system and its quantum phase transition. La2CuO4 is a quasi-two-dimensional material and so it exhibits strong quantum and thermal fluctuations. The upshot of this analysis is that while at T ¼ 0 the nonlinear sigma model has a quantum phase transition, at T > 0, the long-range order is absent in a strictly 2D system but present in the actual material due to the weak-three-dimensional interaction. So, in the strict 2D case, there is no phase transition but two different crossover regimes: a renormalized classical regime (without long-range order), a quantum disordered regime and a quantum critical regime. La2CuO4 has long-range Néel (antiferromagnetic) order at T ¼ 0 and is in the renormalized classical regime (with long-range order due to the weak 3D interaction). The nonlinear sigma model does not describe the nature of the ground state for g > gc beyond saying that there is no long-range order. The problem is that, unlike the Ising model in a transverse field, the microscopic tuning parameter is the next nearest neighbor antiferromagnetic coupling J0 , and to reach the regime g ’ gc, one has to make J0 ’ J. This is the regime in which frustration effects become strong. In this regime, the assumption that the important configurations are smooth and close to the classical Néel state is incorrect. The nature of the ground state turns out to depend on the value of S.

Topological excitations Topology has come to play a crucial role both in condensed matter physics and in Quantum Field Theory. Topological concepts have been used to classify topological excitations such as vortices and dislocations and to provide a mechanism for phase transitions, quantum number fractionalization, tunneling processes in field theories, and nonperturbative construction of vacuum states. Here, we will discuss a few representative cases of what has become a very vast subject.

Topological excitations: Vortices and magnetic monopoles In condensed matter physics, topological excitations play a central role in the description of topological defects and on their role in phase transitions. Here, topology integers in the classification of the configuration space into equivalence classes characterized by topological invariants (Mermin, 1979). The most studied example is vortices. Vortices play a key role in the mixed phase of type II superconductors in a uniform magnetic field (Abrikosov, 1957). Vortices also play a key role in the Statistical Mechanics of 2D superfluids and the 2D classical XY model (Kosterlitz and Thouless, 1973; Kosterlitz, 1974; Berezinskii, 1971, 1972; José et al., 1977). Dislocations and disclinations play an analogous role in the theory of classical melting (Kosterlitz and Thouless, 1973; Halperin and Nelson, 1978; Young, 1979), and 2D and 3D classical liquid crystals (Toner and Nelson, 1981; Nelson and Toner, 1981; Chaikin and Lubensky, 1995). A similar problem occurs in Quantum Field Theory. Theories with global symmetries, such as the two-dimensional O(3) nonlinear sigma model discussed above, when formulated in Euclidean space-time have instantons. Typically, instantons are finite Euclidean action configurations, which are also classified into equivalence classes (associated with homotopy groups) labeled by topological invariants (Coleman, 1985; Rajaraman, 1985). Instantons play a central role in understanding the nonperturbative structure of gauge theories. Gauge theories with a compact gauge group coupled to matter fields have nontrivial vortex (Nielsen and Olesen, 1973) and monopole (’t Hooft, 1976b; Polyakov, 1977, 1987) configurations, as do non-Abelian Yang–Mills gauge theories (Callan et al., 1976). Instantons have also played a central role in condensed matter physics as well, notably in Haldane’s

42

Field theoretic aspects of condensed matter physics: An overview

work on 1D quantum antiferromagnets (discussed below), and in the problem of macroscopic quantum tunneling and coherence (Caldeira and Leggett, 1983; Leggett et al., 1987).

Vortices in two dimensions In this section, I will focus on the problem of the superfluid transition in 2D and the closely related problem of the phase transition of a magnet with an easy-plane anisotropy, the classical XY model. A superfluid is described by an order parameter that is a one-component complex field f(x). If electromagnetic fluctuations are ignored, this description also applies to a superconductor. The complex field can be written in terms of an amplitude |f(x)|, whose square represents the local superfluid density, and a phase y(x) ¼ arg(f(x)). Deep in the superfluid phase, the amplitude is essentially constant, that we will set to be a real positive number f0, while the phase field y(x) is periodic with period 2p and can fluctuate. Similarly, an easy-plane ferromagnet is described by a two-component real order parameter field MðxÞ ¼ ðM1 ðxÞ, M2 ðxÞÞ ¼ jMðxÞjðcos yðxÞ, sin yðxÞÞ. Deep in the ferromagnetic phase, the amplitude |M| is essentially constant but the phase field y(x) can fluctuate. We will assume that we are in a regime where the local superfluid density |f0|2 is well formed (or, equivalently that |M| is locally well formed) but that the phase field is fluctuating. In this regime, the problem at hand is an O(2) ’ U(1) nonlinear sigma model, and its partition function takes the form  Z  Z 2 1  (57) Z ¼ Dy exp − d2 x @ m yðxÞ 2g where we defined the coupling constant g ¼ T/J|f0|2, where T is the temperature, J is an interaction strength, and |f0|2 is the magnitude (squared) of the amplitude of the order parameter, which we will take to be constant; k ¼ J|f0|2 is the phase stiffness. Except for the requirement that the phase field be locally periodic, y ’ y + 2p, superficially this seems to be a trivial free (Gaussian) field theory. We will see that the periodicity (or, compactification) condition makes this theory nontrivial. Indeed, configurations of the phase field that are weak enough and do not see the periodicity condition, can for all practical purposes, be regarded as being noncompact and ranging from −1 to + 1. However there are many configurations for which the periodicity condition is essential. Such configurations are called vortices. Even in the absence of vortices, the periodic (compact) nature of the phase field is essential to the physics of this problem. In fact the only allowed observables must be invariant under local periodic shifts of the phase field. This implies that the phase field y itself is not a physical observable but that exponentials of the phase of the form expðinyðxÞÞ are physical. This operator is just the order parameter field of the XY model. In CFT, operators of this type are called vertex operators (Di Francesco et al., 1997; Ginsparg, 1989). We will see below that this theory has a dual field #, associated with vortices, and that there are vertex operators of the dual field. In String Theory, the model of a compactified scalar is known as the compactified boson and represents the coordinate of a string on a compactified space, in this case a circle S1 (Polchinski, 1998). To picture a vortex, consider a large closed curve C on the 2D plane. Hence, topologically a closed curve is isomorphic to a circle, C ’ S1. The phase field y(x) is equivalent to a unit circle S1. Therefore the configuration space are maps of S1 (the large circle) onto S1 (the unit circle of the order parameter space). The configurations can be classified by the number of times the phase winds on the large circle C. The winding number is an integer called the topological charge of the configuration, the vorticity. Thus, a vortex is a configuration of the phase field y(x) that winds by 2pm (where m is an integer): I I 2p ðDyÞC 1 d’ iyð’Þ dx  ryðxÞ  i @ ’ e−iyð’Þ ¼ m (58) ¼ e 2p 2p C 2p 0 where ’ 2 (0, 2p] is the azimuthal angle for a vector at the center of the large circle C. Here n is the vorticity or winding number of the configuration; m > 0 is a vortex and m < 0 is an antivortex. The vorticity is a topological invariant of the field the configuration y(x) which does not change under smooth changes. The winding number of a vortex is a topological invariant that classifies the configurations of the phase field as continuous maps of a large circle S1 onto the unit circle S1 defined by the phase field. In topology such continuous maps are called homotopies. The winding number classifies these maps into a discrete set of equivalence classes, which form a homotopy group under the composition of two configurations. In this case, the homotopy group is called P1(S1). Since the equivalence classes are classified by a topological invariant that takes integer values, the homotopy group P1(S1) is isomorphic to the group of integers,  (Mermin, 1979). The field jm(x) ¼ @ my(x) is the superfluid current, and the vorticity o(x) is the curl of the current, that is, oðxÞ ¼ Emn @ m jn ðxÞ ¼ Emn @ m @ n yðxÞ

(59)

which vanishes unless y(x) has a branch-cut singularity across which the phase field jumps by 2pn. Let o(x) be the vorticity field with singularities at the locations {xj} of vortices with topological charge mj X oðxÞ ¼ 2p mj d2 ðx − xj Þ (60) j

which is satisfied by the phase field configuration

Field theoretic aspects of condensed matter physics: An overview yðxÞ ¼

X 2pmj Im lnðz − zj Þ

43 (61)

j

where we have used the complex coordinates z ¼ x1 + ix2. Away from the singularities {xj}, this configuration obeys the Laplace equation. Hence, it has a Cauchy–Riemann dual field #(x), which satisfies the Cauchy–Riemann equation @ m # ¼ emn @ n y

(62)

−r2 #ðxÞ ¼ oðxÞ

(63)

which satisfies the Poisson equation

whose solution is Z #ðxÞ ¼

d2 y Gðjx − yjÞ oðyÞ

(64)

where G(|x − y|) is the Green function of the 2D Laplacian −r2 Gðjx − yjÞ ¼ d2 ðx − yÞ:

(65)

  1 a ln 2p jx − yj

(66)

In 2D, this Green function is Gðjx − yjÞ ¼

where a is a short distance cutoff (a lattice spacing). In what follows we will assume that the Green function of Eq. (66) has been cutoff so that G(|x − y|) ¼ 0 for |x − y| a. P The energy of a configuration of vortices {nj} with vanishing total vorticity, jmj ¼ 0, is Z  2 Jf2 E½y ¼ 0 d2 x @ m y 2   Z Z (67) X Jf2 a ¼ 0 d2 x d2 y oðxÞ Gðjx − yjÞ oðyÞ ¼ 2pJf20 mj mk ln 2 jxj − xk j>k where we used that configurations with nonvanishing vorticity do not contribute to the partition function since they have infinite energy in the thermodynamic limit. We conclude that, up to an unimportant prefactor, the partition function of Eq. (57) is the same P as the partition function a gas of charges {mj} (the vortices) with total vanishing vorticity, jmj ¼ 0,  ! X jxj − xk j Jf2 X Z2DCG ¼ exp −2p 0 mj mk ln (68) T a j 2, relevant for D < 2 and marginal for D ¼ 2. Hence, vortices are marginal if Dvortex ¼ 2, which happens if g ¼ p/2. This is the same as to say that the system is at the Kosterlitz–Thouless critical temperature T ¼ TKT. Hence, vortices are irrelevant if T < TKT and relevant for T > TKT. This RG analysis tells us that T > TKT, when vortices proliferate, the coupling constant n flows to strong coupling to a regime where # is pinned to an integer value and the theory is in a massive phase. In this phase the connected vortex correlator decays exponentially with distance, which is the same as to say that the vortex charge is screened. This is why in this phase the vortices proliferate. It can be shown that in this phase the correlator of the spins of the XY model, that is, the correlator of the vertex operator expðiyðxÞÞ, decays exponentially with distance and the system is in its disordered phase. There is still the question of the nature of the phase with T < TKT. The Mermin–Wagner Theorem (Mermin and Wagner, 1966) [and its generalizations by Hohenberg (1967) and Coleman (1973)] states that classical statistical mechanical systems with a global continuous symmetry group cannot undergo spontaneous symmetry breaking in space dimensions D  2 (and quantum systems with space-time dimensions D  2). Does this theory violate this the Mermin–Wagner Theorem? The answer is no. It is easy to see that in the phase in which the vortices are irrelevant, that is, for T < TKT, the correlator of the order parameter operator is always a power law of the distance |x − y|, h expðiyðxÞÞ expð−iyðyÞÞi ¼

const: jx − yjg=2p

(81)

with an exponent that depends on temperature and satisfies g T 1  : ¼ 2p 2pJf20 4

(82)

In other words, the entire low temperature phase is not an ordered phase of matter since the correlator is not constant at longdistance. On the other hand, the correlators of the order parameter expðiyÞ and of the vortices expði#Þ have a power-law behavior for T < TKT, we conclude that in this temperature range, the system is scale invariant and that it has a line of critical points. In summary, we succeeded in expressing the partition function as a sum over configurations of the topological excitations, the vortices. We succeeded in doing that because vortices are labeled by their coordinates on the plane. In addition, we found a nontrivial phase transition since the entropy and the energy both scale logarithmically with the linear size of the system or, equivalently, that vortices became marginally relevant at a critical temperature. This is the mechanism behind the Kosterlitz–Thouless transition. One may ask if this construction is generic and the answer is no. As an example, consider the Abelian Higgs model in D ¼ 2. This model has a complex scalar field minimally coupled to a Maxwell gauge field and, hence, its gauge group is U(1). In the classical spontaneously broken phase, the gauge field becomes massive. In this phase the long-range coherence of the phase field of the vortices of the scalar field is screened at the scale of the penetration depth. As a result, the Euclidean action of the vortices is now finite. Furthermore, on longer scales the interaction energy between vortices becomes short ranged. Hence, instead of a 2D Coulomb gas, now one has a gas of particles (vortices and antivortices) with short-range interactions. In this case the entropy always dominates and the vortices proliferate (Schaposnik, 1978). This behavior is quite analogous to the restoration of symmetry by proliferation of domain walls in one-dimensional classical spin chains (Landau and Lifshitz, 1959) and to the analogous problem of tunneling in the path integral formulation of quantum mechanical double-well potentials (Coleman, 1985). As a caveat, we should note that, in spite of the obvious similarities, the 2D Abelian Higgs model does not describe correctly a 2D superconductor (i.e., a superconducting film) since the electromagnetic field is not confined to the film, and this renders the electromagnetic action nonlocal.

Magnetic monopoles in compact electrodynamics Instantons are of great interest in Quantum Field Theory since they provide for a mechanism to understand the nonperturbative structure of these theories. For this reason, they have been used to understand the mechanisms of quark confinement and the role of quantum anomalies in non-Abelian gauge theories (Callan et al., 1976; Polyakov, 1977, 1987; ’t Hooft, 1976a, b). Instantons in non-Abelian gauge theories are magnetic monopoles and the condensation of monopoles has long been argued to be the mechanism behind quark confinement. The simplest non-Abelian gauge theory that has magnetic monopoles is the Georgi–Glashow (Georgi and Glashow, 1974). This model has a three-component real field f and an SU(2) Yang–Mills gauge field Am. In its Higgs phase, the scalar field f acquires an expectation value which breaks the gauge symmetry group SU(2) down to its diagonal U(1) subgroup. Since Uð1Þ SUð2Þ, this Abelian gauge group is compact, meaning that its magnetic fluxes are quantized. Polyakov (1975a) and ’t Hooft (1976a) showed that in 2+1, Euclidean dimensions have nonsingular instanton solutions which at long distances resemble the magnetic monopole originally proposed by Dirac (1931) Bi ðxÞ ¼

q xi − 2pqdi,3 dðx1 Þdðx2 Þyð−x3 Þ 2 jxj2

(83)

The first term in Eq. (83) is the magnetic radial field of a monopole of magnetic charge q. The second term represents an infinitely long infinitesimally thin solenoid ending at the location of the monopole, x ¼ 0, that supplies the quantized magnetic flux 2pq. This singular term is known as the Dirac string. The string itself (and its orientation) is physically unobservable to any electrically charged particle that obeys the Dirac quantization condition, qe ¼ 2p (in units where ℏ ¼ c ¼ 1).

46

Field theoretic aspects of condensed matter physics: An overview

In the language of a lattice gauge theory (Wilson, 1975), a theory with a compact (i.e., periodic) U(1) gauge fields on a D ¼ 3 cubic lattice, describing a compact gauge field in 2 + 1 dimensions. This theory should have instantons that resemble magnetic monopoles much in the same way as a theory with a compact global U(1) symmetry has vortices. The simplest example is Polyakov’s compact electrodynamics (Polyakov, 1977) whose partition function is ! YZ 2p dAm ðxÞ 1 X Z¼ cosðF mn ðxÞÞ (84) exp 2p 4e2 x, m, n x,m 0 P where Fmn(x) ¼Dm An(x) − DnAm(x)  mAm is the magnetic flux through the elementary plaquette labeled by a site x and a pair of directions, m and n, with m ¼ 1, 2, 3. This theory is invariant under local gauge transformations Am(x) ! Am(x) + DmF(x) and it is also invariant under local periodic shifts of the gauge fields Am(x) ! Am(x) + 2pℓm(x), where ℓm ðxÞ 2 . The plaquette flux operator satisfies the lattice version of the Bianchi identity that the product of exponentials of the flux on the faces of every elementary cube of the lattice is Y eiFmn ðxÞ ¼ 1 (85) cube faces

which says that the theory can have magnetic monopoles of integer magnetic charge. We will analyze this theory following an approach analogous to what we used for vortices in Section “Vortices in two dimensions.” To this end, we will consider the partition function   Z Z  2 1 (86) Z½Bmn  ¼ DAm exp − 2 d3 x Fmn ðxÞ − Bmn ðxÞ 4e where Fmn(x) ¼ @ mAn − @ nAm is the field strength of the Abelian U(1) Maxwell gauge field Am. Here Bmn(x) is an (antisymmetric) two-form background gauge field. The coupling constant of this theory is e2. Since Am is a connection, it has units of length−1, and F 2mn is a dimension 4 field. Then, in D ¼ 3 dimensions, e2 has units of length−1. The theory is invariant under two local transformations, namely the usual invariance under gauge transformations Am ðxÞ ! Am ðxÞ + @ m FðxÞ,

Bmn ðxÞ ! Bmn ðxÞ

(87)

where F(x) is an arbitrary smooth function of x. The presence of the background two-form field Bmn now requires invariance under one-form gauge transformations Am ðxÞ ! Am ðxÞ + am ðxÞ,

Bmn ðxÞ ! Bmn ðxÞ + @ m an − @ n am

(88)

The two-form gauge field Bmn essentially represents the magnetic monopoles. Let {mj} be a configuration of monopoles of charges P mj with coordinates {xj}, with total vanishing monopole charge, jmj ¼ 0. Let MðxÞ be the magnetic monopole density at x, X mj d3 ðx − xj Þ (89) MðxÞ ¼ 2p j

which can be expressed as the curl of the two-form gauge field Bmn, 1 MðxÞ ¼ Emnl @ m Bnl ðxÞ 2

(90)

We will proceed next much in the same way as in Eq. (72) and rewrite the partition function of Eq. (86) in terms of a two-form Hubbard–Stratonovich field bmn(x) such that  2Z  Z Z Z  e 1 d3 x b2mn ðxÞ + i d3 x bmn ðxÞ Fmn ðxÞ − Bmn ðxÞ Z½B ¼ DAm Dbmn exp − 4 2  2Z (91) Z Z Z h i e 1 d3 x b2mn ðxÞ + i d3 x Am ðxÞ@ n bmn ðxÞ − bmn ðxÞBmn ðxÞ ¼ DAm Dbmn exp − 4 2 Thus, the gauge field Am plays the role of a Lagrange multiplier field the enforces the constraint @ n bmn ðxÞ ¼ 0

(92)

bmn ðxÞ ¼ Emnl @ l #ðxÞ

(93)

which is solved in terms of a compact scalar field #(x)

Using this identity and the definition of the monopole density MðxÞ we find that the partition function Z[Bmn] of Eq. (86) becomes  Z  2  Z e Z½B ¼ D# exp − d3 x ð@ m #ðxÞÞ2 + iMðxÞ#ðxÞ 2 " # Z Z (94) 2 X e 2 3 d xð@ m #ðxÞÞ + 2pi mj #ðxj Þ ¼ D# exp − 2 j

Field theoretic aspects of condensed matter physics: An overview

47

which requires that the field # obeys the compactification condition # ! # + n, where n is an arbitrary integer. Eq. (94) says that the magnetic monopole instantons of the compact U(1) gauge theory are dual to charges of the dual phase field #, which has a compact U(1) global symmetry. The full partition function is obtained by summing over all monopole configurations satisfying the total neutrality condition, P j mj ¼ 0. As in section “Vortices in two dimensions,” we will weigh the configurations with a coupling u and find ! Z Z X X X e2 d3 x ð@ m #Þ2 + Z¼ Zfmj g D# exp − 2pimj #ðxj Þ − u m2j (95) 2 j j fmj g

which is the same theory we found in Eq. (77) except that now we are in 3D. Moreover, summing only over dilute configurations of monopoles and antimonopoles we find, once again the sine-Gordon theory but now in D ¼ 3 dimensions:  Z  2  Z e (96) ð@ m #Þ2 − v cosð2p#Þ Z ¼ D# exp − d3 x 2 with v ¼ 2 expð−uÞ=a3 . In spite of the similarities between Eq. (96) and the sine-Gordon theory in 2D, Eq. (78), the physics is very different. It is straightforward to see that, in the limit v ¼ 0, the monopole operator correlator is  2   h i 4p p 1 1 (97) h expð2pi#ðxÞÞ expð−2pi#ðyÞÞ ¼ exp 2 ½Gðjx − yjÞ − Gð0Þ ’ exp − 2 2e R a e where a is the short-distance cutoff. Unlike the behavior of the correlator of the vortex operators in 2D found in Eq. (79), Eq. (97) does not show a power-law behavior. The reason is that at v ¼ 0 the compactified field # should be regarded as the Goldstone boson of a spontaneously broken U(1) symmetry. However, the cosine operator is now always relevant and the field # is pinned and its fluctuations are actually massive. Looking back at the partition function of Eq. (95), we could integrate out the field # and obtain an expression with the same form as the Coulomb gas of Eq. (68) except that now this is the three-dimensional neutral Coulomb gas (Polyakov, 1977)  ! X p X 1 1 Z3DCG ¼ exp − 2 mj mk − (98) 2e jxj − xk j a j 0, one can always rescale the time and space coordinates without affecting the form of the Lagrangian, including the coupling constant or, as we will see, the value of the y angle. In what follows we will assume that we have done the rescaling in such a way that we set vs ¼ 1 and, that time and space scale as lengths. Thus we will use a relativistic notation and, after an analytic continuation to imaginary time, we label the time coordinate by x2 and the space direction by x1. The partition function now takes the form   Z Z  2 1 d2 x @ m n + iy Q½m (102) Z ¼ Dm exp − 2g O where O is the spacetime manifold. In this notation, the first term of the exponent is called the Euclidean action of the nonlinear sigma model. In Eq. (102), we denoted by Q½m the quantity Z 1 Q½m ¼ d2 x Emn m  @ m m @ n m (103) 8p O Here we will consider the case in which the spacetime manifold O is closed. In particular we will assume that it is a two-sphere S2. The quantity Q½m is the integral of a total derivative which counts the number of times the field configuration m(x1, x2) wraps around the sphere S2. In other words, it yields a nonvanishing result only for “large” configurations which wind (or wrap) around the sphere S2. Since Q½n is an integer, it has the same for all smooth field configuration m that can be smoothly deformed into each other and are homotopically equivalent. If we demand that the field configurations m have finite Euclidean action, which requires that at infinity the configurations take the same (but arbitrary) value of m, we have effectively compactified the x1 − x2 plane into a two sphere S2. On the other hand the field m is restricted by the constraint m2 ¼ 1 to take values on a two-sphere S2. Therefore, the field configurations m(x1, x2) are smooth maps of the S2 of the coordinate space to the S2 of the target space of the field m. Hence, the integer Q½m classifies the smooth field configurations into a set of equivalence classes each labeled by the integer Q. Under composition homotopies form groups, and the equivalence classes themselves also form a group which, in this case, is isomorphic to the group of integers, . In topology, these statements are summarized by the notation P2 ðS2 Þ ’ . The configurations with nonzero values of Q are called instantons which, in the quantum problem, represent tunneling processes of the nonlinear sigma model. Another consequence of Q being an integer is that the contribution of its term to the weight of the path integral of Eq. (103) is a periodic function of y angle. On the other hand, since the only allowed values of the y are y ¼ 0 (mod 2p) for S integer, or y ¼ p (mod 2p) for S a half-integer, the contribution of the topological invariant Q to the weight of the path integral is expðiy Q½mÞ ¼ ð−1Þ2SQ :

(104)

Therefore, for spin chains with S integer, the weight is 1 and the topological invariant does not contribute to the path integral. But, if S is a half-integer, the weight is ð−1ÞQ½m, and it does contribute. Moreover, its contribution is the same for all half-integer values of S. These results have important consequences for the physics of spin chains which led Haldane to some startling conclusions (Haldane, 1985). In the weak coupling regime, g 1 (equivalently, for large S), we can use the perturbative renormalization group and derive the beta function for all these O(3) nonlinear sigma models (with or without topological terms) and find that their beta functions are the same as in Eq. (21) with N ¼ 3. Hence, for all S, the effective coupling flows to large values. We can make this inference since the topological term yields no contribution for all configurations which are related by smooth deformations. Thus, we infer that all spin chains with S integer are in a massive phase with an exponentially small energy gap  expð−2pSÞ. This result is nowadays known as the Haldane gap. On the other hand, these results also imply that all spin chain with half-integer spin S are also the same and, in particular, the same as spin-1/2 chains. However, the Hamiltonian of the quantum spin chain with spin-1/2 degrees of freedom is an example of an integrable system and its spectrum is known to be gapless from its Bethe Ansatz solution (Bethe, 1931; Yang and Yang, 1969). However, the spin-1/2 chain is not only gapless but its low energy states are gapless solitons with a relativistic spectrum. The low energy description of a theory of this type must be described by a conformal field theory. Haldane concluded that the RG flow for spin-1/2 chains must have an IR stable fixed point at some finite (and large) value of the coupling constant. This conjecture was confirmed by Affleck and Haldane (1987) who showed that the spin-1/2 Heisenberg chains are in the universality class of the SU(2)1 Wess–Zumino–Witten model (Witten, 1984) whose CFT was solved by Knizhnik and Zamolodchikov (1984).

Topology and open integer-spin chains In the preceding section, we showed that integer spin chains have a Haldane gap. We did that by showing that in that case the topological term is absent. However, the derivation is correct provided the spacetime manifold is closed, for example, a sphere and a torus. What happens if the system has a boundary? Let us denote the boundary of O by G ¼ @O. For example we will take G to be along the imaginary time direction and hence that it is a circle of circumference 1/T, where T is the temperature. The topological term has the same form as the Berry phase term of the path integral for spin, the “Wess–Zumino” term of Eq. (44), but its prefactor is 1/2 as big. Thus, for a system with an open boundary the topological term yields a net contribution equal to a Berry phase with a net prefactor of S/2.

Field theoretic aspects of condensed matter physics: An overview

49

In other words, the boundary of the integer spin chain behaves as a localized degree of freedom whose spin is 1/2 (mod an integer). Form the periodicity requirement, we also see that if the spin chain is made of odd-integer degrees of freedom, there should be a spin 1/2 degree of freedom localized at the open boundary!. Conversely, if the chain is made of even-integer spins, there is no boundary degree of freedom! This line of argument implies that an antiferromagnetic chain with odd-integer spins must have a spin-1/2 degree of freedom at the boundary whereas a chain of even-integer spins should not. Notice that the “bulk” behavior is the same for both odd and even integer spin chains. The difference is whether or not they have a nontrivial “zero-mode” state at the boundary. In particular, the existence of this state is robust, that is, it cannot be removed by making smooth changes to the quantum Hamiltonian or, what is the same, the boundary state is topologically protected. A simple system that displays a protected spin-1/2 zero mode at the boundary is a generalized S ¼ 1 spin chain with Hamiltonian H¼a

N X

SðjÞ  Sðj + 1Þ + b

j¼1

N X

ðSðjÞ  Sðj + 1ÞÞ2

(105)

j¼1

where S( j) ¼ (Sx( j), Sy( j), Sz( j) are the spin 1 matrices at each lattice site j. This problem was examined in great detail by Affleck et al. (1988) who showed that at the special value of the parameters a ¼ 1/2 and b ¼ 1/6 this Hamiltonian takes the form of a sum of projection operators X P 2 ðSðjÞ + Sðj + 1ÞÞ (106) H¼ j

where P2 is an operator that projects out the spin 2 states. These authors constructed the exact ground state, known as the AKLT state, of this Hamiltonian by writing each spin 1 degree of freedom of two spin-1/2 degrees of freedom at each site. They showed that the ground state is a projected product state in which the “constituent” spin-1/2 degrees of freedom (each labeled by + and −, respectively) on nearby sites j and j + 1 are in a valence bond singlet state of the form p1ffiffi2 ð|"j,+, #j+1,−i − #j,+, "j+1,−i (and symmetrizing at each site to project onto a spin 1 state). This state of the spin-1 chain is translation invariant and gapped and hence agrees with Haldane’s result. Moreover, it has a spin-1/2 degree of freedom at each open boundary (Hagiwara et al., 1990). The arguments discussed above show that the AKLT state is a gapped topological state. The gapless spin-1/2 boundary states are an example of spin fractionalization. The spin-1/2 boundary degrees of freedom are an example of an edge state which are present in many, thought not all, topological phases of matter. One may ask what symmetry protects the gaplessness of these edge degrees of freedom. The only perturbation that would give a finite energy gap to the spin-1/2 edge states is an external magnetic field. However, this perturbation would break the global SU(2) symmetry of the Hamiltonian as well as time-reversal invariance.

Duality in Ising models Duality plays a significant role of our understanding of statistical physics and of quantum field theory. Many seemingly unrelated correspondences between different theories have come to be called dualities.

Duality in the 2D Ising model One of the earliest versions of duality transformations was used to relate the high-temperature expansion of the 2D classical Ising model and its low-temperature expansion (Kramers and Wannier, 1941). In all dimensions, for concreteness, we will think of a hypercubic lattice, the high temperature expansion is a representation of the partition function as a sum of contributions of closed loops on the lattice. At temperature T, a configuration of loops g contributes with a weight which in the Ising model has the form CðgÞ tanh LðgÞ ðbÞ, where b ¼ 1/T and L(g) is the length of the loop g, that is, the number of links on the loop, and C(g) is an entropic factor that counts the number of allowed loops with fixed perimeter L(g). However not all loop configurations are allowed as in the Ising model they satisfy constraints such as being nonoverlapping, etc. In Section “The Ising model in a transverse field,” we noted that the partition function of the classical Ising model (in any dimension) can be interpreted as the path integral of a quantum spin model in one dimension less on a lattice with a discretized imaginary time. In this picture, the loops g can be regarded as processes in which pairs of particles are created at some initial (imaginary) time, evolve and eventually are annihilated at a later (imaginary) time. In other words, the high-temperature phase is a theory of a massive scalar field. The restrictions on the allowed loop configurations represent interactions among these particles. In the temperature range in which the expansion in loops is convergent, the particles are massive as the loops are small. As the radius of convergence of the expansion is approached, longer and increasingly fractal-like loops begin to dominate the partition function, and concomitantly the mass of the particles decreases. This process is the signal of the approach to a continuous phase transition where the particles become massless. In fact, right at the critical point the particle interpretation is lost as the associated fields acquire anomalous dimensions. Returning to the 2D classical Ising model, Kramers and Wannier also considered the low temperature expansion. This is an expansion around one of the broken symmetry state, for example, the state with all spins up. In this low temperature, regime the partition function is a sum of configurations of flipped spins. A typical configuration is a set of clusters of flipped spins.

50

Field theoretic aspects of condensed matter physics: An overview

In the absence of a uniform field, a configuration of flipped spins has a energy cost only on links of the lattice with oppositely aligned spins (“broken bonds”). Thus a cluster of flipped spins costs an energy equal to 2 (I assumed that I set J ¼ 1) for each broken bond at the boundary of the cluster. This boundary is domain wall, which is a closed loop on the dual lattice. In 2D, the dual of the square lattice is the square lattice of the dual sites (the centers of the elementary plaquettes of the 2D lattice). In 2D, the links of the direct lattice pierce the links of the dual lattice. We can see that this analogous to the geometric duality of forms: sites (“0-forms”) are dual to plaquettes (“2-forms”) and links (“1-forms”) are dual to links (also “1-forms”). Thus, in 2D the low temperature expansion is an expansion in the loops g of the dual lattice that represent the domain walls. We can also regard the domain walls (the dual loops) as the histories the pairs of particles on the dual lattice. However, the weight of each dual loop is expð−2bÞ per link of g . Except for that, the counting and restrictions on the dual loops g are the same as those of g. This means that there is a one-to-one correspondence between the two expansions with the replacement tanh b $ expð−2b Þ. This mapping means that the dual of the 2D classical Ising model (with global 2 symmetry) at inverse temperature b is a dual Ising model (also with global 2 symmetry) on the dual lattice at inverse temperature b . This correspondence is in close analogy with what we discussed in Section “Vortices in two dimensions.” In particular if one assumes that there is a transition at bc ¼ 1/Tc then there should also be a transition at bc ¼ −1=2 ln tanh bc . Moreover, if one further assumes [as Kramers and Wannier did (Kramers and Wannier, 1941)] that there is a unique transition (correct in the Ising pffiffiffimodel but not in other cases), then the critical point must be such that expð−2bc Þ ¼ tanh bc , which yields the value T c ¼ 2= lnð 2 + 1Þ, which agrees with the Onsager’s result (Onsager, 1944). In Section “The Ising model in a transverse field,” we showed that the classical 2D Ising model is equivalent to the one-dimensional Ising model in a transverse field whose Hamiltonian is given in Eq. (23). The 1D Ising model on a transverse field has spin degrees of freedom defined on the sites of a one-dimensional chain labeled by an integer-valued variable n. The 1D Hamiltonian is expressed in terms of local operators, the Pauli matrices s3(n) and s1(n). The 1D chain has a dual lattice whose sites are the midpoints of the chain. Thus in 1D, sites (“0-forms”) are dual to links (“1-forms”) and vice versa. We will now see that there is a Hamiltonian version of the Kramers–Wannier duality (Fradkin and Susskind, 1978). In Eq. (26), we introduced the kink creation operator which flips are s3 operators to the left and including site j. For clarity we will denote the nÞ, where n~ is the site of the dual lattice between the sites n and n + 1 of the original lattice. We will now kink creation operator as t3 ð~ nÞ ¼ s3 ðnÞs3 ðn + 1Þ. The operators t1 ð~ nÞ and t3 ð~ nÞ satisfy the same algebra of the Pauli operators s1(n) and define an operator t1 ð~ s3(n). Furthermore, we readily find the Hamiltonian of the dual theory to be the same (up to boundary conditions) as the original Hamiltonian of Eq. (23) except that the dual coupling constant is l ¼ 1/l. So, once again, if we assume that there is a single (quantum) phase transition we require lc ¼ lc , which is only satisfied by lc ¼ 1. In this language, the kink creation operator plays the role of a disorder operator (Fradkin and Susskind, 1978).

The 3D duality: 2 gauge theory We will now discuss the role of duality in the 3D Ising model on a cubic lattice. This case, and its generalizations to higher dimensions, was considered first by Wegner (1971). In Section “Duality in the 2D Ising model,” we discussed the loop representation of the high temperature expansion and showed that it has the form in all dimensions. Hence, in 3D the high temperature expansion is an expansion in closed loops g with weight of tanh b per unit link of loop. Hence, in 3D as well, the high temperature phase can be regarded as field theory of a massive scalar field. As in the 2D case, the weight of a loop configuration is tanh b for each link of the loop g. However, the low temperature expansion has a radically different form and physical interpretation. Much as in 2D, the low temperature expansion is an expansion in clusters of flipped spins. Here too, in the absence of an external field, the only energy cost resides at the boundary of the clusters of overturned spins, the domain walls. But in 3D, the clusters occupy volumes whose boundaries are closed surfaces S . In 3D a link (bond) is dual to a plaquette (expressing the fact that in 3D a 1-form is dual to a 2-form). Hence the closed domain walls of overturned spins are dual to a closed surface S on the dual lattice. The weight of each configuration of closed surfaces is expð−2bÞ for each plaquette of the surface S . These facts mean that the dual theory of the 3D Ising model is a theory on the dual lattice with coupling constant b , with expð−2bÞ ¼ tanh b, such that its expansion for small b is a sum over configurations of closed surfaces s , and for large b is a sum of configurations of closed loops g on the dual lattice. The dual theory is the naturally defined on (dual) plaquettes, not on links. To this end, let us define a set of Ising-like degrees of freedom sm(x) ¼ 1 located on the links (x, m) of the dual lattice. These degrees of freedom are coupled on each plaquette of the lattice. Since each plaquette has four links, the coupling involves the degrees of freedom on all four links of each plaquette. The partition function of the dual theory is ! X X  Z¼ exp b sm ðxÞsn ðx + em Þsm ðx + em + en Þsn ðxÞ (107) fsm ðxÞg

plaquettes

where the sum in the exponent (the negative of the “action”) runs on all the plaquettes of the dual lattice. The action of the theory of Eq. (107) is invariant under the reversal of all six Ising degrees of freedom on links sharing a given site x. This is a local symmetry. Unlike the 3D Ising model, which has a global 2 symmetry of flipping all spins simultaneously, this theory has local (or gauge) 2 symmetry. This is the simplest example of a lattice gauge theory in which the degrees of freedom are gauge fields that take values on the 2 gauge group (Wegner, 1971; Wilson, 1975; Kogut, 1979).

Field theoretic aspects of condensed matter physics: An overview

51

We saw that in the spin model the high temperature expansion (i.e., the expansion in powers of tanh b ) is a sum over loop configurations which can be interpreted in terms of processes in which pairs of particles are created, evolve (in imaginary time), and then are destroyed (also in pairs). The analogous interpretation of the expansion of the 2 gauge theory in powers of tanh b ¼ expð−2bÞ as a sum over the configurations of closed surfaces is not in terms of the histories of particles but in terms of the histories of closed strings which, as they evolve, sweep the closed surfaces. The physics is, however, more complex as the sum over surfaces runs over all surfaces of arbitrary topology with arbitrary number of handles (or genus). Thus, over (imaginary) time a closed strong is created, evolves, splits into two closed strings, etc. Therefore, the 3D Ising model, which has a 2 global symmetry, is dual to a 2 gauge theory, which has a 2 local symmetry. The duality maps the high temperature (disordered) phase of the Ising model to the strong coupling (small b ) phase of the gauge theory. In the gauge theory language, this is the confining phase. This can be seen by computing the expectation value of the Wilson Q loop operator on the closed loop G, which here reads h (x,m)2G sm(x)i. In the small b phase, this expectation value decays exponentially with the size of the minimal surface bounded by the loop G. This behavior of the area law of the Wilson loop. Under duality, the insertion of this operator is equivalent to an Ising model with a domain wall terminating on the loop G, and the area law of the Wilson loop is the consequence of the fact that if the Ising model has long-range order, the free energy cost of the domain wall scales with its area. Moreover, in this phase of the gauge theory, the closed strings are small meaning that the string tension (the energy per unit length of string) is finite. In the Ising model language, the string tension becomes surface tension of the domain wall. In the preceding section, we discussed a Hamiltonian version of the duality. We will briefly do the same in the case of the 2+1 dimensional Ising model. The Hamiltonian of the Ising model in a transverse field on a square lattice is X X s3 ðrÞs3 ðr + ej Þ (108) H2DTFIM ¼ − s1 ðrÞ − l r r , j¼1, 2 where r labels the sites of the square lattice and ej (with j ¼ 1, 2) are the two (orthonormal) primitive unit vectors of the lattice. Here too, at each site we have a two-level system (the spins), s1 and s3 are Pauli matrices acting on these states at each site and l is the coupling constant. Just as in the 1D case, this system has two phases: an ordered phase for l > lc and a disordered phase for l < lc, where lc is a critical coupling. This Hamiltonian is invariant under the global 2 symmetry generated by the global spin flip operator Q Q ¼ rs1(r) which commutes with the Hamiltonian, [Q, H2D-TFIM] ¼ 0. Let us consider now a 2 gauge theory on the dual of the square lattice. We will now define a two-dimensional Hilbert space on each link, denoted by ð~ r , jÞ (with j ¼ 1, 2) of the dual lattice with sites labeled by r~. We will further define at each link ð~ r , jÞ the Pauli r Þ and s3j ð~ r Þ. The Hamiltonian of the 2 gauge theory is operators s1j ð~ X X r Þ − g s31 ð~ r Þs32 ð~ r + e~1 Þs31 ð~ r + e~1 + e~2 Þs32 ð~ rÞ (109) H2 gauge ¼ − s1j ð~ r~, j r~ where e~j ¼ Ejk ek (with j, k ¼ 1, 2). ~ r Þ be the operator that flips the Unlike the Hamiltonian of Eq. (108), the Hamiltonian of Eq. (109) has a local symmetry. Let Qð~ 2 gauge degrees of freedom on the four links that share the site r~, Y ~ rÞ ¼ Qð~ s1j ð~ r Þs1j ð~ r − e~j Þ (110) j¼1, 2 ~ r Þ ¼ 0. The Hilbert ~ r Þ, Qð~ ~ r 0 Þ ¼ 0, and commute with the Hamiltonian, ½H2 gauge , Qð~ These operators commute with each other, ½Qð~ ~ space of gauge-invariant states is eigenstates of QðrÞ with unit eigenvalue, ~ QðrÞjPhysi ¼ jPhysi

(111)

(for all r). This constraint is the 2 Gauss Law. The Hamiltonian H2 gauge has two phases: a confining phase for g < gc and a deconfined phase for g > gc (where gc is a critical coupling). The duality transformation is defined so that Y s3j ð~ rÞ (112) s1 ðrÞ ¼ plaquetteðrÞ

is the product of the s3j operators of the gauge theory on a dual plaquette centered at the site r, and s11 ð~ r Þ ¼ s3 ðrÞs3 ðr + e2 Þ,

s12 ð~ r Þ ¼ s3 ðrÞs3 ðr + e1 Þ

(113)

These identities imply that the gauge theory constraint of the 2 Gauss Law, Eq. (111), is satisfied. This also means that duality is a mapping of the gauge-invariant sector of the 2 gauge theory to the Hilbert space of the Ising model in a transverse field. Eqs. (112) and (113) imply that the Hamiltonians H2D-TFIM and H2 gauge are equal to each other with the identification of the coupling constants g ¼ 1/l. Hence, the ordered phase of the Ising model, l > lc, maps onto the confining phase of the 2 gauge theory and, conversely, the disordered phase of the Ising model maps onto the deconfined phase of the gauge theory. Eq. (113) also implies that the operator s3(r) of the Ising model can be identified with the operator

52

Field theoretic aspects of condensed matter physics: An overview s3 ðrÞ ¼

Y gðrÞ

s1j ð~ rÞ

(114)

where the product is on the links of the dual lattice pierced by the path g(r) on the direct lattice ending at the site r. While s3(r) is of course just the order parameter of the Ising model, the dual operator defined by Eq. (114) anticommutes with the plaquette term of the Hamiltonian of Eq. (109) and creates an 2 flux excitation at the plaquette. With some abuse of language, this operator can be regarded as creating a 2 “magnetic monopole.” Since in the confining phase this operator has an expectation value, we can regard this phase as a magnetic condensate. We will now discuss the role of boundary conditions, which is different in both theories. Let us examine the behavior of the 2 gauge theory with periodic boundary conditions. Periodic boundary conditions means that the 2D space is topologically a 2-torus. It is straightforward to show that the ground state in the confining phase is unique and insensitive to boundary conditions. The physics of the deconfined phase is more subtle. We will now see that on a 2-torus it has a fourfold degenerate ground state and that the degeneracy is not due to the spontaneous breaking of a global symmetry. Let g1 and g2 be two noncontractible loops on the torus along the directions 1 and 2, respectively. Let us consider the (“electric”) Q Wilson loop operators along g1 and g2, W½g1  ¼ ðr,jÞ2g1 s3j ðrÞ and similarly for g2. Similarly let us consider the (“magnetic”) ’t Hooft Q ~ g1  and W½g ~ 2  defined on the noncontractible closed paths of the dual lattice g~1 and g~2, such that W½~ ~ g1  ¼ ~ s1 ð~ operators W½~ g1 2 r Þ and Q 2 2 ~ ~ g2  ¼ ~ s1 ð~ r Þ. It is easy to see that the Wilson and’t Hooft operators satisfy that W½g  ¼ W½~ g  ¼ I, and that W½~ i j g2 1 ~ gi , W½~ ~ gj  ¼ 0, ½W½gi , W½gj  ¼ ½W½~

~ g2 g ¼ fW½g2 , W½~ ~ g1 g ¼ 0 fW½g1 , W½~

(115)

Let us consider the special limit of g ! 1. In this limit the Wilson loop operators W[gi] on the two noncontractible loops of the 2-torus commute with the Hamiltonian H2 gauge of Eq. (109). Therefore the eigenstates of the Hamiltonian can be chosen to be the eigenstates of these two Wilson loops. Since the Wilson loop operators are Hermitian and obey W[gi]2 ¼ I, the spectrum of each loop is two-dimensional | 1ii (i ¼ 1, 2) with eigenvalues  1, respectively. At g ! 1, these states are also eigenstates of H2 gauge . ~ gi  act as ladder operators in this restricted Hilbert space. Thus, the ground state of H2 gauge is four dimensional. The ’t Hooft loops W½~ The conclusion is that, on a 2-torus, at least in the limit g ! 1, the ground state of H2 gauge is degenerate. However, this degeneracy is not due to the spontaneous breaking of a global symmetry. Rather, it reflects the topological character of the theory. To see this, one can extend this analysis to a theory to be on a more general two-dimensional and instead of a 2-torus we can consider a surface with g handles (not to be confused with the coupling constant!), for example, g ¼ 0 for a sphere (or disk), g ¼ 1 for a 2-torus, g ¼ 2 for a pretzel, etc. For each of these two-dimensional surfaces, the different number of noncontractible loops is g, and the Wilson loops defined on them commute with each other (and with H2 gauge). Hence, in the limit g ! 1, H2 gauge has a ground state degeneracy of 2g which, clearly, depends only on the topology of the surface. We conclude that, at least as g ! 1, the Hamiltonian H2 gauge is in a topological phase. However, is the exact degeneracy found in the limit g ! 1 a property of the entire deconfined phase? For a finite system, the expansion in powers of 1/g is convergent. On the other hand, in the thermodynamic limit, L ! 1, the expansion has a finite radius of convergence with gc being an upper bound. Let us consider the ground state at g ¼ 1 with eigenvalue + 1 for the Wilson loop W ~ g2 , that is, j−1i ¼ W½~ ~ g2  j+1i . The ’t Hooft operator [g1]. The degenerate state with eigenvalue − 1 is created by the ’t Hooft loop W½~ 1 1 is a product of s1 operators on links along the direction 1 crossed by the path g~2 of the dual lattice, which involves (at least) L links. Then, it takes L orders in the expansion in powers of 1/g to mix the state |+ 1i1 with the state |− 1i1, and this amplitude is of order 1/gL, which is exponentially small. The same argument applies to the mixing between all four states. For a finite but large system of linear size L, the degeneracy is lifted but the energy splitting is exponentially small in the system size. Hence, the topological protection is a feature of the deconfined phase in the thermodynamic limit L ! 1. We conclude that the deconfined phase of the 2 , lattice gauge theory is a topological phase. This result extends to the case of a gauge theory with discrete gauge group , the cyclic group of k elements. In general spacetime dimension D > 2, the k gauge theory also has confined and deconfined phases and if k ≳ 4, it also has an intermediate “Coulomb phase” (Elitzur et al., 1979; Ukawa et al., 1980). In Section “BF gauge theory,” we will see that the low-energy (IR) regime of the deconfined phase is described by a topological quantum field theory known as the level k BF theory.

Bosonization The behavior of one-dimensional electronic systems is of great interest in condensed matter physics for many reasons. One is that, even for infinitesimally weak interactions, these one-dimensional metals violate the basic principles of the Landau Theory of the FL (Baym and Pethick, 1991). A central assumption of Femi Liquid theory is that at low energies the excitation energy of the fermion (electron) quasiparticle is always much larger than its width. Hence, at asymptotically low energies, the quasiparticle excitations become increasingly sharp. A manifestation of this feature is that the quasiparticle propagator (the “Green function”) has a pole on the real frequency axis with a finite residue Z. In one-dimensional metals, this assumption always fails since the residue Z vanishes, the Fermi field acquires a nontrivial anomalous dimension, and the pole is replaced by a branch cut. To this date this is the best example of what is called a

Field theoretic aspects of condensed matter physics: An overview

53

“non-Fermi liquid.” In one-dimensional metals, this non-Fermi liquid is often called a “Luttinger Liquid” (Haldane, 1981). A detailed analysis can be found in Chapter 6 of Fradkin (2013), and in other books. Bosonization provides for a powerful tool to understand the physics of these nontrivial systems. Bosonization is a duality between a massless Dirac field in 1+1 dimensions and a (also massless) relativistic Bose (scalar) field (Mattis and Lieb, 1965; Luther and Emery, 1974; Mandelstam, 1975; Coleman, 1975). In this context, bosonization is a set of operator identities relating observables between two different (dual) continuum field theories. These identities have a close resemblance to the Jordan–Wigner transformation, which relates operators of a theory of spinless (Dirac) fermions on a one-dimensional lattice to a theory of bosons with hard cores (i.e., spins) on the same lattice (Lieb et al., 1961).

Dirac fermions in one space dimensions To understand how these operator identities come about and what these fields mean in the Condensed Matter, context we will consider the simple problem of a system of noninteracting spinless fermions c(n) on a one-dimensional chain of length L (with L ! 1) for simplicity with periodic boundary conditions. The Hamiltonian is H0 ¼ −t

L X

cðnÞ{ cðn + 1Þ + h:c:

(116)

n¼1

In momentum space, and in the thermodynamic limit L ! 1, the Hamiltonian becomes Z p dp H0 ¼ ðe0 ðpÞ − mÞ c{ ðpÞcðpÞ −p 2p

(117)

where − p  p  p, m is the chemical potential (which fixes the fermion number) and e0( p) is the (free) quasiparticle energy which, in this case, is e0 ðpÞ ¼ −2t cos p:

(118)

This simple system has a global internal symmetry c0 ðnÞ{ ¼ e−ia cðcÞ

c0 ðnÞ ¼ eia cðnÞ,

where a is a constant phase (with period 2p). This global symmetry reflects the conservation of fermion number X NF ¼ c{ ðnÞcðnÞ: n

(119)

(120)

The free fermion Hamiltonian of Eq. (116) is also invariant under lattice translations. The ground state of this system is obtained by occupying all single particle states with energy below the chemical potential, E  m  EF, which defines the Fermi energy EF. In what follows we will redefine the zero of the energy at the Fermi energy. In the thermodynamic limit, the one-particle states are labeled by momenta defined in the first Brillouin zone [−p, p). The ground state |Gi of this system is obtained by filling up all single-particle states with momentum p in the range [−pF, pF), where pF is the Fermi momentum. Hence, the occupied states have energy e0( p)  EF. The ground state is Y c{ ðpÞj0i (121) jGi ¼ jpjpF

and is called the filled Fermi sea. We will further assume that the fermionic system is dense in the sense that the occupied states are a finite fraction of the available states, that is, we assume that |EF| is a finite fraction of the bandwidth W ¼ 4t. The single-particle excitations have low energy if |e0( p) − EF| |EF|. This range can only be accessed by quasiparticles with momenta p close to  pF. For states with p close to pF, we can a linearized approximation and write e0 ðpÞ ’ vF ðp − pF Þ + ⋯ (122)

0 and similarly for the states near −pF. Here vF ¼ @E @p ¼ 2t sin pF is the Fermi velocity. Let q being the momentum measured from pF

pF (or − pF in the other case), this approximation is correct in a range of momenta −L q  L, where 2p L L p. In this regime we can approximate the lattice fields c(n) with two continuum right moving cR(x) and left moving cL(x) Fermi fields such that (with x ¼ na0) cðnÞ  eipF x cR ðxÞ + e−ipF x cL ðxÞ in terms of which the Hamiltonian takes the continuum form Z   Hcontinuum ¼ dx c{R ðxÞð−ivF Þ@ x cR ðxÞ − c{L ðxÞð−ivF Þ@ x cL ðxÞ Z   dq qv c{ ðqÞcR ðqÞ + c{R ðqÞcR ðqÞ : ¼ 2p F R

(123)

(124)

54

Field theoretic aspects of condensed matter physics: An overview

In what follows I will rescale time and space in such a way as to set vF ¼ 1. Eq. (124) is the Hamiltonian of a massless Dirac field in 1+1 dimensions. It describes the effective low energy behavior of the excitations of the fermionic system defined on a lattice by the Hamiltonian of Eq. (116). Here low energy means asymptotically close to the Fermi energy (which we have set to zero) and momenta close to  pF. We will see that this effective relativistic field theory gives a complete description of the universal low energy physics encoded in the microscopic model of Eq. (116) up to some subtle issues associated with what are called quantum anomalies. Let us denote by c(x) the bi-spinor field c(x) ¼ (cR(x), cL(x)) and the 2 2 Dirac gamma-matrices in terms of the three Pauli matrices are g0 ¼ s1 ,

g1 ¼ −is2 ,

g5 ¼ s3

(125)

which obey the Dirac algebra fgm , gn g ¼ 2gmn where gmn is the metric tensor in 1+1-dimensional Minkowski spacetime   1 0 gmn ¼ : 0 −1

(126)

(127)


Using the standard notation cðxÞ ¼ c{ ðxÞg0 (where I left the spinor index a ¼ 1, 2 implicit), the Lagrangian density of the free massless Dirac fermion is
@c = L ¼ ci

(128)

where we have used the Feynman slash notation, which denotes the contraction of a vector field, say Am with the Dirac gamma / matrices with a slash, Amgm  A. Formally, the Dirac Lagrangian of Eq. (128) (and the Dirac Hamiltonian of Eq. (124)) is invariant under two separate global transformations. It has a global U(1) (gauge) symmetry transformation c0 ðxÞ ¼ eiy cðxÞ

(129)

(where y is constant) under which the two components of the bi-spinor c transform in the same way c0R ðxÞ ¼ eiy cR ðxÞ,

c0L ðxÞ ¼ eiy cL ðxÞ:

(130)

The massless Dirac theory is also invariant under a global gauge U(1) chiral transformation c0 ðxÞ ¼ eiyg5 cðxÞ

(131)

which in components reads c0R ðxÞ ¼ eiy cR ðxÞ,

c0L ðxÞ ¼ e−iy cL ðxÞ:

(132)

In general, if a system has a global continuous symmetry, it should have a locally conserved current and a globally conserved charge. This is the content of Noether’s Theorem. Since the massless Dirac theory has these two global symmetries, one would expect that it should have two separately conserved currents. The global U(1) symmetry has an associated current jm ¼ (j0, j1) given by
c jm ¼ cg m

(133)

which is invariant under the global U(1) symmetry. In terms of the right and left moving Dirac fields, cR and cL, the components of the current are j0 ¼ c{R cR + c{L cL ,

j1 ¼ c{R cR − c{L cL :

(134)

This current is locally conserved and satisfies the continuity equation @ m jm ¼ 0: It has an associated conserved global charge, which we will call fermion number Z 1 Z 1   Q¼ dxj0 ðxÞ ¼ dx c{R cR + c{L cL : −1

−1

(135)

(136)

This is the continuum version of the conservation of fermion number NF of the free fermion lattice model discussed above. Since this current is conserved, it can be coupled to an electromagnetic field through a term in the Lagrangian Lint ¼ −ejm Am

(137)

Field theoretic aspects of condensed matter physics: An overview

55

where Am is the electromagnetic vector potential, which in 1+1 dimensions has only two components, Am ¼ (A0, A1). Here, e is a coupling constant, which is interpreted as the electric charge. This coupling amounts to making the U(1) symmetry a local gauge symmetry under which the fields transform as c0 ðxÞ ¼ eiyðxÞ cðxÞ,

A0m ðxÞ ¼ Am ðxÞ +

1 @ yðxÞ: e m

(138)

Now the conservation of fermion number becomes the conservation of the total electric charge Qe ¼ −eQ:

(139)

In a continuum nonrelativistic model, the Fermi field is c(x) (ignoring spin), for example, an electron gas in one dimension such as a quantum wire, the analog of decomposition shown in Eq. (123) of the electron field c(x) becomes cðxÞ ¼ eipF x cR ðxÞ + e−ipF x cL ðxÞ: 0

(140)

The electron field c(x) transforms as c (x) ¼ e c(x) under the global U(1) gauge symmetry of Eq. (130). In particular, the local density operator r(x) ¼ c{(x)c(x) is invariant under this symmetry. Under both decompositions of Eqs. (123) and (140), the local density operator becomes ia

rðxÞ ¼ c{R ðxÞcR ðxÞ + c{L ðxÞcL ðxÞ + ei2pF x c{R ðxÞcL ðxÞ + e −i2pF x c{L ðxÞcR ðxÞ

(141)

Clearly the nonrelativistic density operator r(x) is invariant under the global U(1) gauge symmetry. The same considerations apply to the lattice version of the density operator (the local occupation number). Thus, the density operator r(x) can be decomposed into a slowly varying part (the first two terms in Eq. (141)) and the last two terms which oscillate with wave vectors Q ¼ 2pF. These observations imply that we can express r(x) in the form rðxÞ ¼ r
+ j0 ðxÞ + eiQx cQ ðxÞ + e−iQx c−Q ðxÞ:

(142)

This decomposition can be interpreted as a Fourier expansion of the density operator in term of newly defined slowly varying fields. In Eq. (142), Q ¼ 2pF and r
is the average density, where we assumed that the Dirac density operator j0(x) has vanishing expectation value (i.e., it is normal-ordered). In Eq. (142), we defined the (bosonic) operators cQ ðxÞ ¼ c{R ðxÞcL ðxÞ,

c−Q ðxÞ ¼ c{L ðxÞcR ðxÞ

(143)

which characterize the oscillatory component of the density. The operators cQ(x) are the order parameters of a charge density wave state in one dimension and Q is the ordering wavevector. Repulsive interactions between the electrons can cause scattering process between the right and left moving components to become relevant (in the Renormalization Group sense) leading to the spontaneous breaking of translation invariance. The resulting state is known as a charge-density-wave (CDW). In this state the operators cQ(x) (or a linear combination of them) acquire a nonvanishing expectation value, and the expectation value of the density operator r(x) has a (static) modulated component, and the Dirac Hamiltonian density becomes   (144) H ¼ c{R ðxÞð−iÞ@ x cR ðxÞ − c{L ðxÞð−iÞ@ x cL ðxÞ + m c{R ðxÞcL ðxÞ + c{L ðxÞcR ðxÞ where m is the Dirac mass. The operator in the second term of Eq. (144) mixes the right and left moving components of the Dirac spinor. This Hermitian operator is known as the Dirac mass term. In relativistic notation this operator is written as
¼ c{ ðxÞc ðxÞ + c{ ðxÞc ðxÞ: cc L R R L

(145)


6¼ 0, the fermionic spectrum has a mass gap, and the electronic states with momenta  pF have When m 6¼ 0, that is, when hcci an energy gap. Alternatively we could have considered a state with the in which the (Hermitian) operator that has an expectation value is  
c ¼ i c{ ðxÞc ðxÞ − c{ ðxÞc ðxÞ icg (146) 5 L R R L which is known as the g5 mass term. In a more general CDW state both mass terms can be present.

Chiral symmetry and chiral symmetry breaking The CDW states we introduced have different symmetries. To see this let us observe that the operators cQ(x) transform nontrivially under a global U(1) chiral transformation c0Q ðxÞ ¼ e2iy cQ ðxÞ:

(147)

56

Field theoretic aspects of condensed matter physics: An overview

Upon substituting this transformation in the expansion of the density r(x) of Eq. (142), we see that a chiral transformation is equivalent to a uniform displacement of the density operator by y/pF, rðxÞ ! rðx + y=pF Þ:

(148)

Under a chiral transformation with arbitrary angle y, the Dirac and g5 mass term operators transform as an orthogonal transformation of a two-component vector. In particular for a chiral transformation with y ¼ p/4,
7! icg
c, cc 5


c 7! −cc
icg 5

(149)

which is equivalent to a displacement of the density by 1/4 of the period of the CDW. In this sense these two states are equivalent, as is the state with a more general linear combination. The microscopic lattice model (and the continuum nonrelativistic model) is invariant under the spatial inversion symmetry x $ −x. This also implies a symmetry under the exchange of right and left moving components of the Dirac spinor, cR $ cL. In the massless Dirac theory this operation is equivalent to the multiplication of the spinor by the Pauli matrix s1. In the massless theory, the multiplication by s2 (followed by a chiral transformation with y ¼ p/2) has the same effect. These symmetries have an important effect on the fermionic spectrum. To understand what they do let us consider the one-particle Dirac Hamiltonian with a Dirac mass m (jn momentum space)   p m h¼ : (150) m −p This operator anticommutes with the Pauli matrix s2, {h, s2} ¼ 0. Let |Ei be an eigenstate of the Hamiltonian h with energy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eigenvalue E ¼ p2 + m2 . Let us consider the state s2|Ei. It is also an eigenstate of h but with energy −E, that is, hs2 jEi ¼ −s2 hjEi ¼ −EjEi,

) s2 jEi ¼ j−Ei:

(151)

Hence if |Ei is an eigenstate of energy E, then s2|Ei is an eigenstate of energy − E. This means that the spectrum is invariant under charge conjugation symmetry. Notice that under this operation the spinor transforms as       cR cL −icL s2 ¼ ¼ e−ig5 p=2 : (152) cL icR cR In other words, this theory is invariant under charge conjugation C and parity P which, combined, it implies that it is invariant under time-reversal T . The same consideration applies in the case of a g5 mass term in which case the one-particle Dirac Hamiltonian now is   p −im5 h¼ : (153) im5 −p This Hamiltonian now anticommutes with the Pauli matrix s1 which also implies that the same charge conjugation symmetry C, |Ei $ |− Ei is present in the spectrum of the case of a g5 mass. We can also repeat the argument on parity invariance P, which is now multiplication by s1, Thus the theory is invariant under time-reversal T . However, if the theory has both a Dirac mass m and a g5 mass m5, these symmetries are broken. Indeed, the one-particle Dirac Hamiltonian now is   p m − im5 h¼ (154) ¼ ps3 + ms1 + m5 s2 m + im5 −p which no longer has a spectral symmetry. In this case CP is broken and, hence, so it T since CPT remains unbroken (as it should). In a lattice system, this is a symmetry transformation only if y ¼ pn/pF is a lattice displacement, which restricts the allowed values of the chiral angle to be discrete. Although this is true, interactions play a significant role in the actual behavior. In fact, there are physical situations in which an effective continuous symmetry actually emerges in this he infrared (long-distance) limit. This is what happens when the CDW is incommensurate and, as we will see in the next section, it slides under the action of an electric field. In the case of a half-filled system (with only nearest-neighbor hopping matrix elements) the Fermi wave vectors are  p/2. In this
7! −cc
(and similarly for ig c) correspondcase the allowed discrete chiral transformation has a chiral angle p/2 under which cc 5 2
ing by a translation by one lattice spacing. The allowed four fermion operator is ðccÞ . If the lattice fermions are spinless this operator reduces to 2


¼ −2jR ðxÞjL ðxÞ + lim c{ ðxÞc{L ðxÞcL ðyÞcR ðyÞ + h:c: ðccÞ y!x R

(155)

where introduced the right and left moving (chiral) components of the current operator 1 jR ðxÞ ¼ ðj0 ðxÞ + j1 ðxÞÞ ¼ c{R ðxÞcR ðxÞ, 2

1 jL ðxÞ ¼ ðj0 ðxÞ − j1 ðxÞÞ ¼ c{L ðxÞcL ðxÞ 2

(156)

Field theoretic aspects of condensed matter physics: An overview

57

The first term in Eq. (155) is known as the backscattering interaction and has scaling dimension 2. Hence, it is a marginal operator. The theory with only the first term is known in condensed matter physics as the Luttinger model and in High-Energy Physics as the (massless) Thirring model. The second operator in Eq. (155) formally violates momentum conservation as its total momentum is 4pF ¼ 2p, which is a reciprocal lattice vector and, as such, it is equivalent to zero (mod 2p). Such an operator is not formally allowed in a (naive) continuum theory. This operator is due to a lattice Umklapp process and breaks the formal continuous chiral symmetry to a discrete 2 subgroup. Although the naive scaling dimension of this operator is 2 (and hence it is formally marginal). If the fermions are spinless, the leading operator actually vanishes and the leading nonvanishing operator actually has dimension 4, which is irrelevant. However, as shown above, backscattering processes of the form jR(x) jL(x) are part of this operator and are exactly marginal. If the interaction is strong enough, the backscattering interaction can make the Umklapp operator relevant. When this happens, the fermionic system has a quantum phase transition to an insulating state with a period 2 (commensurate) CDW state. On the
2 is allowed and is marginally relevant. If the interactions are repulsive, the other hand, for spin 1/2 fermions, the operator ðccÞ resulting state is an antiferromagnetic Néel state at quantum criticality, while for attractive interactions it is a period 2 CDW. This is what happens in the 1D Hubbard model (see, e.g., Fradkin, 2013 for a detailed analysis). There are two theories in relativistic systems, which are closely related to this problem. One if the Gross–Neveu model (Gross and Neveu, 1974) which is a theory of N species of massless Dirac spinors with Lagrangian density
cÞ = a + gðc LGN ¼ c
a i@c a a

2

(157)

with a ¼ 1, . . . , N (summation of repeated induces is implies). The spin-1/2 Hubbard model corresponds to the case N ¼ 2. This Lagrangian is invariant only under the discrete chiral symmetry c
a ca 7! −c
a ca. This is a discrete, 2, symmetry and as such it can be spontaneously broken in 1+1 dimensions. For N  2, the resulting state has a (dynamically) broken 2 chiral symmetry and that
c i 6¼ 0 corresponding to a period 2 CDW. there is a chiral condensate hc a a The other theory is known as the chiral Gross–Neveu model whose Lagrangian density is  
c Þ2 − ðc
g c Þ2 = a + g ðc (158) LcGN ¼ c
a i@c a a a 5 a which has the full continuous chiral symmetry. For N ¼ 1, this theory is equivalent to the Luttinger model (and to the Gross–Neveu model if we ignore the Umklapp term). For N  2 the chiral symmetry is formally broken. If this were true, this theory would violate the Mermin–Wagner theorem. However a detailed study (most easily done using bosonization methods) shows that instead of long-range order the correlator of both mass terms decay as a power law as a function of distance, consistent with the requirements by this theorem. We close this section by noting that if the lattice model is not half filled but its density is either incommensurate or has a higher degree of commensurability, say p/q, the chiral symmetry is actually continuous (in the incommensurate case) or effectively continuous since the requisite Umklapp terms are strongly irrelevant. However, if the lattice model is not at half filling charge conjugation symmetry C is broken at the lattice scale (in the UV), where it is equivalent to particle–hole symmetry, but it is recovered in the low-energy, IR, regime (up to irrelevant operators). In this sense, both the continuous chiral symmetry and charge conjugation symmetry can be regarded as emergent IR symmetries

The chiral anomaly In Section “Chiral symmetry and chiral symmetry breaking” we showed that the theory of massless Dirac fermions, in addition to a global U(1) gauge symmetry, has a second conservation law which we called a global U(1) chiral symmetry, shown in Eq. (131). This symmetry implies that there is a locally associated chiral current j5m, given by
j5m ðxÞ ¼ cðxÞg m g5 cðxÞ

(159)

@ m j5m ¼ 0

(160)

which also satisfies a continuity equation

and there is a globally conserved chiral charge Q5 Z Q5 ¼

1

−1

dx j50 ðxÞ ¼

Z

1

−1

  dx c{R cR − c{L cL :

(161)

It is easy to check that the Dirac current jm and the chiral current j5m are related by j5m ¼ Emn jm

(162)

where emn is the second rank Levi-Civita tensor. The simultaneous conservation of both currents jm and j5m in the massless Dirac theory implies that the right and left moving densities jR and jL, defined in Eq. (156), should be separately conserved. In fact, if the Dirac theory has a mass term

58

Field theoretic aspects of condensed matter physics: An overview
@c
= − mcc L ¼ ci

(163)


5c @ m j5m ¼ 2micg

(164)

it is straightforward to show that

which means that in the massive theory, the axial current is not conserved. This is easy to understand since the mass term mixes the right and left moving components of the Dirac spinor and, hence, the right and left moving densities are not conserved. What happens in the massless limit, m ! 0, is more subtle. This problem was investigated in the late 1960s in 3+1 dimensions by Adler (1969) and by Bell and Jackiw (1969), who were interested in the anomalous decay of a neutral pion into two photons, p0 ! 2g. This process appears at third order of perturbation theory and it involves the computation of a triangle diagram (a fermion loop). In 3+1 dimensions, this process has a UV divergence which needed to be regulated. These authors showed that it is not
c and the axial current j5 ¼ cg
g5 c are conserved. possible to find a regularization in which both the Dirac (gauge) current jm ¼ cg m m m In other words, if gauge invariance is preserved then the axial current is not and has an anomaly and results in a nonconservation of the axial current, @ mj5m 6¼ 0. On the other hand, at least in the case of the physical gauge-invariant regularization, the obtained expression for the anomaly in the axial current is universal, independent of the value of the UV regulator (the cutoff ). Sometime later G. ’t Hooft showed that in non-Abelian gauge theories instant processes also lead to anomalies and, furthermore, the result was also universal (’t Hooft, 1976a, b). Since the result is universal and, hence, independent of the UV scale, this led to the concept of anomaly matching conditions. We will examine this problem in 1+1 dimensions (although it plays a key role in the theory of topological insulators in three space dimensions, Qi et al., 2008). As we noted above, the lattice model is gauge-invariant and has only one conserved current. The conserved axial current appeared only in the low-energy regime in which the lattice model is described by a theory of massless Dirac fermions. To understand this problem, we will consider the theory of fermions in 1+1 dimensions coupled to a U(1) gauge field. In the presence of a background (i.e., not quantized) electromagnetic field in the A0 ¼ 0 gauge, the free fermion Hamiltonian of Eq. (116) becomes H½A ¼ −t

L X

c{ ðnÞeieAðn,tÞ=ℏ cðnÞ + h:c:

(165)

n¼1

An uniform and constant electric field is E, which is represented by a vector potential A ¼ −cEt (with c being the speed of light). The net effect of this gauge field is to shift of the momentum of the fermion quasiparticles p ! p + ecEt=ℏ or, what is the same, to displace the fermion dispersion relation in momentum by the ecEt=ℏ. This means that the Fermi points are also shifted by that amount, pF ! pF + ecEt=ℏ and −pF ! −pF + ecEt=ℏ. This means that the single particle states between pF and pF + ecEt=ℏ that were empty for E ¼ 0 are now occupied and the states between − pF and −pF + ecEt=ℏ that were occupied for E ¼ 0 are now empty. This means that number of right-moving fermions is increasing at a rate of ecE=ℏ and that the number of left-moving fermions decreases at the same rate. This results in a net current. Throughout this process the total number of fermions is not changed, gauge invariance is satisfied, but the number of right and left moving fermions is not separately concerned. Notice that this is an effect that involved the entire Femi sea but the net effect is at low energies. Let us see now how this plays out in the effective Dirac theory. The massless Dirac Lagrangian density in the background of an unquantized electromagnetic field Am is
@= − e= L ¼ cði AÞc:

(166)

Since there is no mass term, the Dirac equation still decouples into two equations, for the right and left moving components of the Dirac spinor. In the A0 ¼ 0, gauge they are i@ 0 cR ¼ ð−i@ 1 − A1 ÞcR ,

i@ 0 cL ¼ ði@ 1 − A1 ÞcL :

(167)

In the temporal gauge, A0 ¼ 0, a uniform electric field E ¼ @ 0A , and A increases linearly with time. As A increases, the Fermi momentum pF (which is equal to the Fermi energy EF) also increases at the rate eE. The density of states of a system of length L is L/(2p). So, the rate of change of the number of right-moving fermions is 1

dN R e ¼ E 2p dx0

1

1

(168)

RL where we defined NR ¼ 0 dx jR and similarly for NL. If the UV regulator of the theory is compatible with gauge invariance, then the total fermion number must be conserved and the total vacuum charge must remain equal to zero, Z L dx j0 ðxÞ ¼ NR + N L ¼ 0: (169) Q¼ 0

Thus, if NR increases, then NL must decrease by the same amount. Or, equivalently, the electric field E creates an equal number of particles NR and of antiparticles NL ¼ −NL .

Field theoretic aspects of condensed matter physics: An overview

59

On the other hand, the chiral charge Q5 ¼ NR − NL must increase at the rate
L e dQ5 dN R dN ¼ + ¼ E: p dx0 dx0 dx0

(170)

Again, the details of the UV regularization do not matter, only that it is gauge-invariant. We can also interpret Eq. (170) as the rate of particle–antiparticle pair creation by an electric field. In a relativistic notation, these results are expressed as @ m j5m ¼

e E F mn : 2p mn

(171)

Hence, the formally conserved current j5m has an anomaly and is not conserved due to quantum effects. Since it is not conserved, we cannot gauge the chiral symmetry. In the next section, we will see that the anomaly is closely related with bosonization.

Bosonization, anomalies, and duality We will reexamine the problem at hand from the point of view of the fermionic currents of the Dirac theory jm as operators. Since the currents obey the continuity equation, @ mjm ¼ 0 one expects that this would imply that it may be possible to write them as a curl of a scalar field, that is, jm(x) ¼ Emn@ nf(x) where f(x) should be a scalar field. Since this should be an operator identity, we will need to understand how the currents act on the physical Hilbert space. The Fermi-Bose mapping in 1D systems is closely related to the chiral anomaly we just discussed. Bosonization of a system of 1+1 dimensional massless Dirac fermions is a set of operator identities understood as matrix elements of the observables in the physical Hilbert space. These identities were first derived by Mattis and Lieb (1965), based on earlier work by Schwinger (1959). These identities were rediscovered (and their scope greatly expanded) by Luther and Emery (1974), by Coleman (1975), by Mandelstam (1975), and by Witten (1978). A non-Abelian version of bosonization was subsequently derived by Witten (1984). The physical Hilbert space is defined as follows: Let |FSi  |0i denote the filled Fermi sea. In what follows we will assume that the physical system is macroscopically large and that local operators create the physical states by acting finitely on the filled Fermi sea. Physical observables, such as the right and left moving densities jR(x) and jL(x), need to be normal-ordered with respect to the physical vacuum state, the filled Fermi (Dirac) sea |0i. The normal ordered densities are jR(x) :  jR(x) − h0|jR(x)|0i and : jL(x) :  jL(x) − h0|jR(x)|0i. Since the densities are products of fermion operators, they need to be defined as a limit in which the operators are separated by a short distance . Crucial to this construction is that the computation of the expectation values be regularized in such a way that the charge (gauge) current jm(x) is locally conserved and satisfies the continuity equation @ mjm ¼ 0 as an operator identity. In what follows, all expectation values will refer to the filled Fermi sea state |0i. The propagators of the right and left moving Fermi fields are given by hc{R ðx0 , x1 ÞcR ð0, 0Þi ¼

−i , 2pðx0 − x1 + iEÞ

hc{L ðx0 , x1 ÞcL ð0, 0Þi ¼

i 2pðx0 + x1 + iEÞ

(172)

The expectation values of the currents at a space location x1 are hjR ðx1 Þi ¼ lim hc{R ðx1 + ÞcR ðx1 − Þi ¼ !0

i , 4p

hjL ðx1 Þi ¼ lim hc{L ðx1 + ÞcL ðx1 − Þi ¼ !0

−i 4p

(173)

which are divergent at short distances. It follows that the normal-ordered right and left moving current densities satisfy the equal-time commutation relations ½jR ðx1 Þ, jR ðx01 Þ ¼ −

i @1 dðx1 − x01 Þ, 2p

½jL ðx1 Þ, jL ðx01 Þ ¼ +

i @1 dðx1 − x01 Þ 2p

(174)

These identities imply that the normal-ordered space-time components of the current jm ¼ (j0, j1) satisfy the equal-time commutation relations i ½j0 ðx1 Þ, j1 ðx01 Þ ¼ − @ 1 dðx1 − x01 Þ, p

½j0 ðx1 Þ, j0 ðx01 Þ ¼ ½ji ðx1 Þ, j1 ðx01 Þ ¼ 0

(175)

The nonvanishing right-hand sides of these commutators are known as Schwinger terms. These identities define the U(1) (Kac-Moody) current algebra. We should note that in a theory of nonrelativistic Fermi fields c(x, t) in all dimensions, the charge density r(x) and the current operators jðxÞ ¼ 2i1 c{ ðxÞrcðxÞ − rc{ ðxÞrcðxÞ satisfy a similar expression (also at equal times) ½rðxÞ, jk ðx0 Þ ¼ −i

e2 @ ðdðx − x0 ÞrðxÞÞ, mc2 k

½rðxÞ, rðx0 Þ ¼ 0,

½jk ðxÞ, jl ðx0 Þ ¼ 0

(176)

In one dimension, and in the regime in which the fermions have a macroscopic density so that r(x) ’hr(x)i, after a multiplicative rescaling of the operators, the nonrelativistic identities of Eq. (176) are equivalent to the U(1) current algebra of Eq. (175).

60

Field theoretic aspects of condensed matter physics: An overview

The U(1) current algebra of Eq,(175) is reminiscent of the equal-time canonical commutation relations of a scalar field. Indeed, if f(x) is a scalar field and P(x) ¼ @ 0f(x) is its canonically conjugate momentum, then they obey the equal-time canonical commutation relations ½fðx1 Þ, Pðx01 Þ ¼ idðx1 − x01 Þ:

(177)

We can then identify the charge density j0 and the current density j1 with the scalar field operators 1 j0 ðxÞ ¼ pffiffiffi @ 1 fðxÞ, p

1 1 j1 ðxÞ ¼ − pffiffiffi PðxÞ ¼ − pffiffiffi @ 0 fðxÞ p p

(178)

which obey the U(1) current algebra of Eq. (175) as a consequence of the canonical commutation relations, Eq. (177). Furthermore, we can rewrite Eq. (177) in the Lorentz covariant form 1 jm ðxÞ ¼ pffiffiffi Emn @ n fðxÞ p

(179)

which is clearly consistent with the local conservation of the current jm, @ m jm ðxÞ ¼ 0:

(180)

Let us examine now the question of the conservation of the chiral current j5m. In Eq. (162), we showed that the gauge current and the chiral current are related by j5m ¼ Emnjn. Therefore the divergence of the chiral current is 1 1 @ m j5m ¼ Emn @ m jn ¼ pffiffiffi Emn Enl @ m @ l f ¼ − pffiffiffi @ 2 f p p

(181)

where we used the identification of the gauge current in terms of the scalar field f, Eq. (179). Therefore we conclude that @ m j5m ¼ 0 , @ 2 f ¼ 0:

(182)

This equation states that the chiral current as an operator identity is conserved if and only if the field f is a free massless scalar field, whose Lagrangian density is 1 LB ¼ ð@ m fÞ2 : 2

(183)

In Eq. (166), we considered the free massless Dirac Lagrangian coupled to a background (not quantized) gauge field Am through the usual minimal coupling which here is Lint ¼ −ejm Am. using the bosonization identity for the gauge current, Eq. (179) we see that the Lagrangian density of the bosonized theory now becomes 1 e LB ½A ¼ ð@ m fÞ2 − pffiffiffi Emn @ n fðxÞ Am ðxÞ 2 p 1 2  ð@ m fÞ + JðxÞfðxÞ 2

(184)

e e JðxÞ ¼ pffiffiffi Emn @ n Am ðxÞ ¼ pffiffiffiffiffiffi F ðxÞ p 4p

(185)

where the source J(x) is

where F (x) ¼ EmnFnm(x) is the (Hodge) dual of the field strength Fmn. Hence, F is (essentially) the source for the scalar field f(x). This implies that the equation of motion of the scalar field must be e −@ 2 fðxÞ ¼ JðxÞ ¼ pffiffiffi Emn @ n Am : p

(186)

Retracing our steps, we find that the chiral current j5m obeys 1 e @ m j5m ¼ − pffiffiffi @ 2 f ¼ E F mn 2p mn p

(187)

which reproduces the chiral anomaly given in Eq. (170). These results suggest that the theory of a free massless Dirac spinor must be equivalent to the theory of the free massless scalar field. This statement is known as bosonization. However for this identification to be correct, there must be an identification of the Hilbert spaces and of all the operators of each theory. We will not do this detailed analysis here but will highlight the most significant statements. Let us begin with the fermion number of the Dirac theory. Consider a system of fermions of total length L. Using the bosonization identity of Eq. (179), we find that the fermion number NF  Q is given by

61

Field theoretic aspects of condensed matter physics: An overview Z NF ¼

0

L

1 dx1 j0 ðx0 , x1 Þ ¼ pffiffiffi p

Z

L

0

1 dx1 @ 1 fðx0 , x1 Þ ¼ pffiffiffi ðfðx0 , LÞ − fðx0 , 0ÞÞ: p

(188)

Thus, the vacuum sector of the Dirac theory, with NF ¼ 0, corresponds to the theory of the scalar field with periodic boundary conditions, f(x1 ¼ 0) ¼ f(x1 ¼ L). Furthermore, since the fermion number is quantized, NF 2 , changing the fermion number is the same as twisting the boundary conditions of the scalar so that pffiffiffi fðx1 + LÞ ¼ fðx1 Þ + pN F : (189) In String Theory (Polchinski, 1998) the scalar field is interpreted as the coordinate of a string. Compactifying the space where the string lives to be a circle of radius R means that the string coordinate is defined modulo 2pRn, where n is an integer. We see that the condition by Eq. (189) is equivalent to say that the scalar field is compactified and that the compactification radius is pffiffiffiffiffiimposed ffi R ¼ 1= 4p . This identification also imposes the restriction that the allowed operators of the scalar field must obey the identification fðxÞ  fðxÞ + 2pRn

(190)

as an equivalency condition. The simplest bosonic operators that obey the compactification condition are the vertex operators Va(x), V a ðxÞ ¼ expðiafðxÞÞ:

(191) pffiffiffiffiffiffi The compactification condition then requires that the allowed vertex operators should have a ¼ n=R ¼ 4p n, where n is an integer. Since the propagator of the scalar field in 1+1-dimensional (Euclidean) spacetime is   jx − x0 j 1 Gðx − x0 Þ ¼ − (192) ln a 2p where a is a short-distance cutoff, p weffiffiffiffiffifind that the scaling dimension of the vertex operator is Da ¼ a2/(4p) ¼ n2. We will see shortly ffi that the vertex operator with a ¼ 4p is essentially the Dirac mass operator (which has scaling dimension 1). The free massless scalar field can be decomposed into right and left moving components, fR and fL respectively, f ¼ fR + fL ,

# ¼ −fR + fL

(193)

dx01 Pðx0 , x01 Þ

(194)

where Z #ðx0 , x1 Þ ¼

x1

−1

is the Cauchy–Riemann dual of the field f(x) since they satisfy the Cauchy–Riemann equation @ m f ¼ Emn @ n #:

(195)

The right and left moving components of the Dirac spinor are found to have the bosonized expression (Mandelstam, 1975) pffiffiffiffiffiffi 1 cR ðxÞ ¼ pffiffiffiffiffiffiffiffi : expði 4pfR ðxÞÞ : , 2pa

pffiffiffiffiffiffi 1 cL ðxÞ ¼ pffiffiffiffiffiffiffiffi : expð−i 4pfL ðxÞÞ : 2pa

(196)

It is easy to check that the propagators of these operators agree with the expressions given in Eq. (172), and that they have scaling dimension 1/2 and spin 1/2. How does a chiral transformation act on the scalar field? A chiral transformation by an angle y, c.f. Eq. (132), acts on the right c0Lffi ¼ expð−iyÞcL . From and moving fermions as c0R ¼ expðiyÞcR and pffiffiffiffiffi pffiffiffiffiffiEq. ffi (196), we see that the right and left moving components of the scalar field transform as f0R ¼ fR + y= 4p and f0L ¼ fL + y= 4p. This means that pffiffiffiffiffiaffi chiral transformation by an angle y of the Dirac fermion is equivalent to a translation (a shift) of the scalar field f0 ¼ f + 2y= 4p. We can use the operator product expansion discussed in Section “The operator product expansion” to show that the fermion
and icg
5 c are given by the following identifications mass terms cc pffiffiffiffiffiffi
¼ 1 : cosð 4pfÞ : , cc 2pa

pffiffiffiffiffiffi
5 c ¼: sinð 4pfÞ : : icg

(197)

These operators have scaling dimension 1 and transform properly under chiral transformations. These identifications imply that a theory of free massive Dirac fermions
@= c − mcc
LD ¼ ci

(198)

is equivalent to the sine-Gordon field theory (Coleman, 1975) whose Lagrangian is pffiffiffiffiffiffi 1 LSG ¼ ð@ m fÞ2 − g : cosð 4pfÞ : 2 where g ¼ m/(2pa).

(199)

62

Field theoretic aspects of condensed matter physics: An overview

Given the central role played by the current algebra identities of Eq. (175), one may wonder if a similar approach might apply in higher dimensions. Schwinger terms in current algebra play an important role in relativistic field theories. However in higher dimensions, their structure is more complex and does not lead to identities of the type we have discussed. The reason at the root of this problem is largely kinematical. The Bose (scalar) field f is qualitatively a bound state, a collective mode in the language of condensed matter physics. In 1+1 dimensions, this collective mode exhausts the spectrum at low energies due to the strong kinematical restriction on one spatial dimension. The equivalency between the theory of free massive Dirac fermions and the sine-Gordon theory is an example of the power of bosonization. On the Dirac side, mapping the theory is free and its spectrum is well understood. But on the sine-Gordon side, the theory is nonlinear. In fact in the sine-Gordon theory, the fermions are essentially solitons, domain walls of the scalar field. For these and many other reasons that we do not have space here bosonization plays a huge role in understanding the nonperturbative behavior of systems both in condensed matter physics and in Quantum Field Theory in 1+1 dimensions. We will see in Section “Bosonization of the Dirac theory in 2+1 dimensions” that to an extent some of these ideas can and have been extended to relativistic systems and classical statistical mechanical systems in 2+1 dimensions.

Fractional charge Solitons in one dimensions We begin by returning to the equivalency between the theory of free massive Dirac fermions and sine-Gordon theory, in 1+1 dimensions. In Eq. (188), we showed that the boundary conditions of the compactified scalar field f(x) are determined by the fermion number NF of the dual Dirac theory, and that the vacuum sector of the Dirac theory maps onto the sine-Gordon theory with periodic boundary conditions. We will now examine the sector with one fermion, NF ¼ 1. This sector of the Dirac theory maps onto the sine-Gordon theory with twisted boundary conditions, pffiffiffi fðLÞ − fð0Þ ¼ p: (200) The Hamiltonian of the sine-Gordon theory is Z 1 h pffiffiffiffiffiffi i 1 1 HSG ¼ dx P2 ðxÞ + ð@ x fðxÞÞ + g cos 4pfðxÞ 2 2 −1

(201)

In the sector with periodic boundary conditions, the classical ground states are static and uniform configurations that minimize the pffiffiffi potential energy. Since the potential energy is a periodic functional of f(x), the classical minima are at fn ðxÞ ¼ ðn + 1=2Þ p, where n is an arbitrary integer. The classical energy of these ground states is extensive and is given by Egnd ¼ −gL where L is the linear size of the system. The classical ground state in the twisted sector is a domain wall (or soliton) that interpolates between the static and uniform pffiffiffi ground states fðxÞ ¼  p=2. The classical ground state in this sector is the static solution of the Euler-Lagrange equation pffiffiffi  pffiffiffi d2 f ¼ −2g p sin 2 pfðxÞ dx2 pffiffiffi such that asymptotically satisfies lim x!1 ¼  p=2. The solution is the classical soliton configuration pffiffiffi  pffiffiffiffiffi p 2 fðxÞ ¼ pffiffiffi tan−1 expð2 pgðx − x0 ÞÞ − 2 p

pffiffiffi The soliton solution represents a domain wall between two symmetry-related classical ground states with f ¼  p=2. The energy of the soliton (measured from the energy of the ground state in the trivial sector) is finite and is given by rffiffiffi g Esoliton ¼ 4 p

(202)

(203)

(204)

where x0 is a zero mode of the soliton solution and represents its coordinate. By coupling the bosonized theory to a weak electromagnetic field Am, as given in Eq. (184), it is easy to check that it has electric charge −e and, in this sense represents the electron. Then the identities of Eq. (196) can be used that as a quantum state it is indeed a fermion.

Polyacetylene In Section “Bosonization, anomalies, and duality,” we saw that solitons of a scalar field can be regarded as being equivalent to electrons, fermions with charge − e. We will now see that in a theory of fermions coupled to a domain wall of a scalar field, the soliton carries fractional charge. This problem has been extensively studied in one-dimensional conductors such as polyacetylene, in

Field theoretic aspects of condensed matter physics: An overview

63

particular by the work of Su et al. (1979) and by Jackiw and Schrieffer (1981). In quantum field theory, this problem was first discussed by Jackiw and Rebbi (1976) and by Goldstone and Wilczek (1981). In Section “Dirac fermions in one space dimensions,” we showed that the physics of lattice fermions in one dimension at low energies is well described by a theory of massless Dirac fermions. In a one-dimensional conductor, such as polyacetylene, the fermions couple to the lattice vibrations (phonons). Su, Schrieffer, and Heeger (SSH) (Su et al., 1979) proposed a simple model in which the electrons couple to the lattice vibrations through a modulation of the hopping amplitude between two consecutive sites n and n + 1, instead of being a constant t, becomes tn,n+1 ¼ t − g(un+1 − un), where un is the displacement of the ion (a CH group in polyacetylene) at site n from its classical equilibrium position and g is the electron–phonon coupling constant. In polyacetylene, the number of electrons (which are spin-1/2 fermions) is equal to the number of sites of the lettuce and the electronic band is half-filled. At half filling, this simple band structure is invariant under a particle–hole transformation. If the coupling to the lattice vibrations is included, this symmetry remains respected provided the displacements change sign un ! −un for all lattice sites. In polyacetylene, the lattice dimerizes (a process known as a Peierls distortion) and the discrete translation symmetry by one lattice spacing is spontaneously broken: the system becomes a period 2 CDW on the bonds of the lattice. The broken symmetry state is still invariant under a particle–hole transformation. The effective field theory of this system is a theory of two Dirac spinors ca,s(x), where a ¼ 1, 2 denotes right and left-moving fermions, and s ¼", # are the two spin polarizations. The Lagrangian density of this system is
ðxÞc ðxÞ − 1 fðxÞ2 : = s − gfðxÞc L ¼ c
s i@c s s 2

(205)

The real scalar field f(x) represents the distortion field of the polyacetylene chain. Here we will assume that the chains have spontaneously distorted and we will regard the scalar field as static and classical. This is a good approximation since the masses of the CH complexes are much bigger than the electron mass. This continuum model is due to Takayama et al. (1980) and further developed by Campbell and Bishop (1981, 1982). Many of the results fund in this (adiabatic) approximation remain qualitatively correct upon taking into account the quantum dynamics of the chain, even in the limit in which the ions are treated as being “light” (provided the spin of the fermions is taken into account) (Hirsch and Fradkin, 1982; Fradkin and Hirsch, 1983). In the field theory, the discrete symmetry of displacements by one lattice spacing becomes the 2 symmetry f ! −f. This is a
! −cc.
symmetry of the electron–phonon system once combined with the discrete chiral transformation c ! g5c under which cc The ground state is twofold degenerate  f0 with   2LvF pv (206) f0 ¼ exp − 2F g g and the Dirac fermion (the electron) has a exponentially small mass, m ¼ gf0.

Fractionally charged solitons Jackiw and Rebbi showed that the 1+1-dimensional classical f4 theory has the following soliton solution that interpolates between the two classically ordered states at  f0 (Jackiw and Rebbi, 1976)   x − x0 fðxÞ ¼ f0 tanh (207) x where x is the correlation length of f4 theory, and x0 is the (arbitrary) location of the soliton. They further showed that, when coupled to a theory of massless relativistic fermions through a Yukawa coupling, as in Eq. (205), this soliton carries fractional charge. The argument goes as follows. The one-particle Dirac Hamiltonian for a Dirac fermion with a position-dependent mass m(x) is   mðxÞ −i@ x H ¼ −is1 @ x + mðxÞs3 ¼ (208) −i@ x −mðxÞ where m(x) ¼ gf(x), with f(x) being the soliton solution of Eq. (207). This Hamiltonian is Hermitian and real. Furthermore, this Hamiltonian anticommutes with the Pauli matrix s2. This implies that for every positive-energy state |Ei with energy + E, there is a negative-energy eigenstate with energy − E given by s2|Ei. Hence, the spectrum is particle–hole symmetric (or, what is the same, charge-conjugation invariant). In addition, and consistent with charge-conjugation symmetry, the Hamiltonian of Eq. (208) has state with E ¼ 0, a zero mode, with a normalizable spinor wave function     Z x −i 1 c0 ðxÞ ¼ pffiffiffi (209) exp −sgnðmÞ dx0 mðx0 Þ 2 1 0 which exists for an arbitrary function m(x), which changes sign once at some location (which we took to be x0 ¼ 0). Jackiw and Rebbi further showed that the soliton (antisoliton) carries fractional charge

64

Field theoretic aspects of condensed matter physics: An overview e Q¼ : 2

(210)

This result follows from the spectral asymmetry identity of the density of states rS(E) in the presence of the soliton Z e 1 e Q¼− ðrS ðEÞ − rS ð−EÞÞ ¼ −  2 0 2

(211)

where  is the spectral asymmetry of the Dirac operator in the soliton background, and it is known as the APS -invariant of Atiyah et al. (1975). Given the one-to-one correspondence that exists between positive and negative energy states in the spectrum, the spectral asymmetry follows from the condition that the zero mode be half-filled, which is required by normal ordering or, what is the same, by charge neutrality. Another way to understand this result is that adding (or removing) a fermion of charge − e results in the creation of a soliton–antisoliton pair, with each topological excitation carrying half of the charge of the electron. In other words, in the dimerized phase, the electron is fractionalized. In this analysis, we ignored the spin of the electron. If we take it into account the spin degree of freedom, the soliton is instead a boson with charge  e. There is an alternative, complementary, way to think about the charge of the soliton. Goldstone and Wilczek (1981) considered a theory in which the (massless) Dirac fermion is coupled to two real scalar fields, ’1 and ’2, with Lagrangian
@c
@c
expðiyg Þc
− ig’ cg
c  ci = − g’1 cc = − gj’jc L ¼ ci 5 5 2

(212)

where j’j2 ¼ ’21 + ’22 and y ¼ tan−1 ð’2 =’1 Þ. They considered a soliton in which g’1 ¼ m is the constant (in space) Dirac mass and ’2 winds slowly between two values  ’0 for x ! 1. In this theory, the one-particle Dirac Hamiltonian is H ¼ −is1 @ x + g’1 s3 + g’2 s2

(213)

which is Hermitian and complex and, hence, it violates CP invariance. A perturbative calculation of the induced (gauge-invariant) current jm, which is given by the triangle diagram of a fermion loop with two gauge field insertions and a coupling of the scalar fields, yields the result (with a ¼ 1, 2) hjm ðxÞi ¼

’a @ n ’b 1 1 ¼ E E E @ny 2p mn ab j’j2 2p mn

(214)

1 2 which is locally conserved. Notice, however, that the induced axial current j5m is not conserved, @ m hj5m i ¼ − 2p @ y 6¼ 0 and, hence, this current is anomalous. We can now compute the total charge accumulated as the soliton is created adiabatically to be given by the Goldstone–Wilczek formula

Q ¼ −e

Dy 2p

where Dy ¼ y(+1) − y(−1). Since limx!1’2(x) ¼ ’0, we obtain the result g’  e 0 : Q ¼ − tan−1 m p

(215)

(216)

In the limit m ¼ g’ ! 0, where CP (or T) invariance is recovered, we get lim Q ¼ −

m!0

e 2

(217)

which is the Jackiw–Rebbi result for the fractional charge of a soliton of Eq. (210). The results for the fractional charge of the soliton can also be derived using the bosonization identities of Section “Bosonization, anomalies, and duality.” Indeed, the bosonized expression for the Lagrangian of Eq. (212) is L¼

pffiffiffiffiffiffi  2 gj’j 1 @mf − cos 4pf − y : 2 2pa

(218)

Deep in the phase in which the Bose field f is massive, the nonlinear term in Eq. (218) locks this field to the chiral angle y, that is, 1 f ¼ pffiffiffiffiffiffi y: 4p

(219)

However, the bosonization identities also tell us that the gauge current jm is given by the curl of the scalar field f. Therefore, in this state the current jm is 1 1 jm ¼ pffiffiffi Emn @ n f ¼ E @ny 2p mn p

(220)

which is the same as the Goldstone–Wilczek result of Eq. (214). That these two seemingly different approaches yield the same result is not accidental as they both follow from the axial anomaly.

Field theoretic aspects of condensed matter physics: An overview

65

Fractional statistics A fundamental axiom of Quantum Mechanics is that identical particles are indistinguishable (Dirac, 1930; Landau and Lifshitz, 1957). In nonrelativistic Quantum Mechanics, this leads to the requirement that the quantum states of a system of identical particles must be eigenstates of the pairwise particle exchange operator. Since two exchanges are equivalent to the identity operation, this implies that the states must be even or odd under pairwise exchanges. This result, in turn, implies that particles can be classified as being either bosons (whose states are invariant under pairwise exchanges) or fermions (whose states change sign under pairwise particle exchanges). A consequence is that bosons obey the Bose-Einstein (and can condense into a single particle state) whereas fermions obey the Fermi–Dirac distribution and must obey the Pauli exclusion principle. It is an implicit assumption of this line of reasoning that all relevant states of a system of identical particles can be efficiently represented by a (suitably symmetrized or antisymmetrized) product state. This classification is present at an even deeper level in (relativistic) Quantum Field Theory where locality, unitarity, and Lorentz invariance require that the fields be classified as representations of the Lorentz group and obey the Spin-Statistics Theorem (Weinberg, 2005). The Spin-Statistics Theorem is actually an axiom of local relativistic Quantum Field theory that requires that fields that transform with an integer spin representation of the Poincaré group (i.e., scalars, gauge fields, gravitational fields) must be bosons while fields that transform with a half-integer spin representation (i.e., Dirac spinors) must be fermions. This spin-statistics connection is intrinsic to the construction of String Theory Polchinski (1998). Given these considerations, there was a general consensus that fermions and bosons were the only possible types of statistics. Nevertheless several exceptions to this rule were known to exist. One is the construction of the magnetic monopole in 3+1 dimensional gauge theory by Tai-Tsun Wu and Chen-Ning Yang who showed that a scalar coupled to a Dirac magnetic monopole behaves as a Dirac spinor (Wu and Yang, 1976). This was an early example of statistical transmutation by coupling a matter field to a nontrivial configuration of a gauge field. As we will note below, the construction of anyons (particles with fractional statistics) in 2+1 dimensions has a close parentage to the Wu-Yang example. Examples of statistical transmutation were known to exist in 1+1 dimensional theories where a system of hard-core bosons was shown to be equivalent to a theory of free fermions using the Jordan–Wigner transformation (Lieb et al., 1961), which represents a fermion as a composite operator of a hard-core boson and an operator that creates a kink (or soliton). This construction also underlies the fermion-boson mapping in 1+1 dimensional field theories (Mattis and Lieb, 1965; Luther and Emery, 1974; Coleman, 1975; Mandelstam, 1975) that we discussed in Section “Bosonization, anomalies, and duality.” Finally, in the late 1970s, it was found that 1+1 dimensional N spin systems harbor operators known as parafermions, which obey the same algebra shortly afterwards found to be obeyed by anyons in 2+1 dimensions (Fradkin and Kadanoff, 1980).

Basics of fractional statistics Jon Magne Leinaas and Jan Myrheim wrote an insightful paper in 1977 in which they examined the structure of the configuration space of the histories of a system of N identical particles (Leinaas and Myrheim, 1977). Using the Feynman path-integral approach, they showed that if the worldlines of the identical particles are not allowed to cross, then the configuration space is topologically nontrivial. Through a detailed analysis they showed that the three and higher dimensions under a pairwise particle exchange the states must be either even or odd and hence the particles are either bosons or fermions. Leinaas and Myrheim also showed that in one and two space dimensions, the wave functions can change by a phase, nowadays known as the statistical angle. In retrospect this result could have been anticipated (but was not) in an earlier paper by Laidlaw and DeWitt (1971), who did a similar analysis of the configuration space of identical particles in the Feynman path integral. Frank Wilczek generalized the Aharonov–Bohm effect (Aharonov and Bohm, 1959) to describe the quantum mechanics of composite objects made of electric charge and magnetic flux in two space dimensions (Wilczek, 1982a). Wilczek showed that composite objects made of a nonrelativistic particle of charge q bound to a magnetic flux of (magnetic) charge F behaves as an object with fractional angular momentum qF/2p. Here F is measured in units of the flux quantum 2p (in units in which ℏ ¼ c ¼ e ¼ 1). Furthermore, in a subsequent paper, Wilczek (1982b) showed that, upon an adiabatic process in which the two composites exchange positions without their worldlines coinciding, the wave function of two identical flux-charge composites changes by a phase factor expðiqFÞ. For instance if F ¼ p (i.e., a half-flux quantum), the acquired phase is equal to expðipÞ ¼ −1. Thus if the particle was a boson, the composite becomes a fermion and vice versa. Wilczek coined the term anyon to describe the behavior of an arbitrary charge-flux composite. Furthermore, this construction also implies that not only fractional statistics but also fractional spin, consistent with a generalization of the Spin-Statistics Theorem. In other terms, “flux-attachment” implies fractional statistics (and fractional spin). Clearly Wilczek’s construction gave an explicit physical grounding to the general arguments of the 1977 paper by Leinaas and Myrheim. It is worth to note the close analogy between this construction in 2+1 dimensions and the Wu-Yang construction in 3+1 dimensions [whose consistency with the Spin-Statistics Theorem had been shown earlier on by Goldhaber (1976)]. In this description, the statistics of the composites (the anyons) enters in the form of complex weights (phases) given in terms of the linking numbers of the worldlines. Hence, the concept of fractional statistics is intimately related to the theory of knots and of the representations of the braid group. These concepts were originally introduced in physics to describe statistical transmutation in the theory of solitons (Finkelstein and Rubinstein, 1968) in the context of the Skyrme model (Skyrme, 1962). Wu (1984) developed an

66

Field theoretic aspects of condensed matter physics: An overview

explicit connection between the work of Leinaas and Myrheim and Wilczek’s work on anyons in terms of operations acting on the worldlines of the anyons and described by the Braid Group. Wu’s work and the somewhat early paper by Wilczek and Zee on the statistics of solitons (Wilczek and Zee, 1983) marked the definite entry of the theory of knots (and of the braid group) in physics in general and in condensed matter physics in particular. The classification of anyons in terms of representations of the braid group labeled by the fractional statistics (determined by linking numbers) as well as of fractional (or topological) spin (determined by the writhing number of the worldlines) leads to a rich set of physical consequences. As it will turn out, Wilczek’s anyons are described one-dimensional representations of the braid group. As we will see below, additional and intriguing (non-Abelian) representations will also play a role.

What is a topological field theory We will now consider a special class of gauge theories known as topological field theories. These theories often (but not always) arise as the low energy limit of more complex gauge theories. In general, one expects that at low energies the phase of a gauge theory be either confining or deconfined. While confining phases have (from really good reasons!) attracted much attention, deconfined phases are often regarded as trivial, in the sense that the general expectation is that their vacuum states be unique and the spectrum of low lying states is either massive or massless. Let us consider a gauge theory whose action on a manifold M with metric tensor gmn(x) is Z pffiffiffi S¼ dD x g Lðg, Am Þ: (221) M

At the classical level, the energy–momentum tensor Tmn(x) is the linear response of the action to an infinitesimal change of the local metric, T mn ðxÞ 

dS : dgmn ðxÞ

(222)

That a theory is topological means that depends only on the topology of the space in which is defined and, consequently, it is independent of the local properties that depend on the metric, for example, distances, angles. Therefore, at least at the classical level, the energy–momentum tensor of a topological field theory must vanish identically, T mn ¼ 0:

(223)

In particular, if the theory is topological, the energy (or Hamiltonian) is also zero. Furthermore, if the theory is independent of the metric, it is invariant under arbitrary coordinate transformations. Thus, if the theory is a gauge theory, the expectation values of Wilson loops will be independent of the size and shape of the loops. Whether or not a theory of this type can be consistently defined at the quantum level is a subtle problem which we will briefly touch on below. It turns out that, due to the nonlocal nature of the observables of a gauge theory, the low-energy regime of a theory in its deconfined phase can have nontrivial global properties. In what follows, we will say that a gauge theory is topological if all local excitations are massive (and in fact we will send their mass gaps to infinity). The remaining Hilbert space of states is determined by global properties of the theory, including the topology of the manifold of their space-time. In several cases, the effective action of a topological field theory does not depend on the metric of the space-time, at least at the classical level. In all cases, the observables are nonlocal objects, Wilson loops and their generalization.

Chern–Simons gauge theory Gauge theories play a key role in physics. In 2+1 dimensions, it is possible to define a special gauge theory which is odd under time-reversal invariance and parity: Chern–Simons gauge theory. Originally introduced in Quantum Field Theory in 1982 by Deser et al. (1982a,b), Chern–Simons gauge theory can be defined for any compact Lie group, as well as an extension of Einstein’s gravity in 2+1 dimensions. In 1989, Witten (1989) showed that Chern–Simons gauge theory computes the expectation values of configurations of Wilson loops, regarded as the worldlines of heavy particles, in terms of a set of topological invariants known as the Jones polynomial that classify knots in three dimensions. We will consider the simplest case, the U(1) Chern–Simons gauge theory. The Chern–Simons action for a U(1) gauge field Am in 2+1 dimension is Z Z k d3 x Emnl Am @ n Al + d3 x J m Am (224) S½A ¼ 4p O O where Jm is a set of conserved currents (representing the worldlines of a set of heavy particles). On a closed 3-manifold O (e.g., a sphere, a torus) the Chern–Simons action is gauge invariant provided the parameter k (known as the level) is an integer. If the manifold O has a boundary, the action is not gauge invariant at the boundary. Gauge invariance is restored by additional boundary degrees of freedom. This structure is general, and not just a feature of the U(1) theory.

Field theoretic aspects of condensed matter physics: An overview

67

Since the Chern–Simons action is first order in derivatives, it is not invariant under time reversal and under parity (which in 2+1 dimensions is a reflection). These symmetries make this theory relevant to the description of the fractional quantum Hall effect. In the absence of external sources, Jm ¼ 0, at the classical level the Chern–Simons action is invariant under arbitrary changes of coordinates. This means that the theory is, at least classically, a topological field theory. For a general non-Abelian gauge group G, the Chern–Simons action becomes Z   k 2 S¼ d3 x tr AdA + A ^ A ^ A : (225) 4p M 3 Here, the cubic term is shorthand for  trðA ^ A ^ AÞ  tr Emnl Am An Al

(226)

m

for a gauge field A that takes values on the algebra of the gauge group G.

BF gauge theory A closely related (abelian) gauge theory is the so-called BF theory (Horowitz, 1989) which, in a general spacetime dimension D (even or odd), is a theory of a vector field Am (a one-form) and an antisymmetric tensor field B with D − 2 Lorentz indices (a D2 form), known as a Kalb–Ramond field. In 2+1 dimensions, the action of the BF theory is Z k S¼ d3 x Emnl Bl @ m An (227) 2p M where, once again, k is an integer. The BF gauge theory has the same content as the topological sector of a discrete k gauge theory. To see this, we will consider the theory of compact electrodynamics which is a U(1) gauge theory minimally coupled to charged scalar field f. The will assume that the gauge theory is defined for a compact U(1) gauge group (meaning that the gauge flux is quantized) and that the complex scalar field has charge integer k 2 . As usual minimal coupling is implemented by introducing the covariant derivative Dm − @ m + ikAm, where Am is the U(1) gauge field. Deep in the phase in which the global U(1) symmetry is spontaneously broken, usually called the Higgs regime, the amplitude of the scalar field is frozen at its (real) vacuum expectation value f0 but its phase o, representing the Goldstone mode, is unconstrained. In this limit, the Lagrangian of this theory becomes  2 1 L ¼ jf0 j2 @ m o − kAm − 2 F2mn 4e

(228)

where e is the coupling constant of the gauge field (the electric charge) and Fmn is the field strength of the gauge field Am. In general spacetime dimension D the charge e has units of length(D−4)/2. In this limit the gauge field becomes massive (this is the Higgs mechanism). This theory has fluxes quantized in units of 2p/k and has only k distinct fluxes. This is the k gauge theory. Here we will consider this theory in 2+1 dimensions. We will use a Gaussian (Hubbard–Stratonovich) decoupling of the first term of Eq. (228) in terms of a gauge field Cm to write the Lagrangian in the equivalent form L¼−

1 1 C2m + Cm ð@ m o − kAm Þ − 2 F 2mn : 4e 4jf0 j2

(229)

Up to an integration by parts, we see that the phase field o plays the role of a Lagrange multiplier field, which forces the vector field Bm to obey the constraint @ mCm ¼ 0. This constraint is solved by writing Cm as Cm ¼

1 E @ n Bl 2p mnl

(230)

of a 1-form gauge field Bm. Upon solving the constraint, the Lagrangian of Eq. (229) becomes L¼

k 1 1 Fmn ðBÞ2 : E Am @ n Bl − 2 Fmn ðAÞ2 − 2p mnl 4e 32p2 jf0 j2

(231)

For spacetime dimensions D < 4 the IR the Maxwell terms for the fields Am and Bm are irrelevant in the IR and, in this limit, this theory reduces to the BF theory at level k of Eq. (227). Therefore, k gauge theory is equivalent to the BF theory at level k. In fact, this result is essentially valid in all dimensions with the main difference being that the field Bm is, in general, a rank D − 2 Kalb–Ramond antisymmetric field.

Quantization of Abelian Chern–Simons gauge theory By expanding the action of Eq. (224), the Lagrangian density becomes   k k L¼ Eij Ai @ 0 Aj + A0 B − J 0 − J i Ai 4p 2p

(232)

68

Field theoretic aspects of condensed matter physics: An overview

where B ¼ Eij@ iAj is the local flux, J0 is a local classical density, and Ji is a local classical current. Then, the first term of Eq. (232) implies that the spatial components of the gauge field obey equal-time canonical commutation relations ½A1 ðxÞ, A2 ðx0 Þ ¼ i

2p dðx − x0 Þ: k

(233)

The second term of the Lagrangian enforces the Gauss Law which for this theory simply implies that the states in the physical Hilbert space obey the constraint BðxÞ ¼

2p J ðxÞ: k 0

(234)

Thus the Gauss’s Law requires that a charge density necessarily has a magnetic flux attached to it. In other terms, the physical states are charge-flux composites as postulated in Wilczek’s theory (Wilczek, 1982b). This is the theoretical basis to the concept of flux attachment which, as we will see in Section “Chern–Simons gauge theory and the fractional quantum Hall effect,” is widely used in the theory of the fractional quantum Hall effect. The third term in Eq. (232) simply states that the Hamiltonian density is just H ¼ J i Ai

(235)

Hence, in the absence of sources, the Hamiltonian vanishes, H ¼ 0. The Chern–Simons action is locally gauge-invariant, up to boundary terms. To see this, let us perform a gauge transformation, Am ! Am + @ mF, where F(x) is a smooth, twice differentiable function. Then, Z S½Am + @ m F ¼ ðAm + @ m FÞEmnl @ n ðAl + @ l FÞ ZM Z (236) ¼ d3 x Emnl Am @ n Al + d3 x Emnl @ m F@ n Al M

M

Therefore, the change is S½Am + @ m F − S½Am  ¼

Z M

d3 x @ m FF m ¼

Z M

d3 x @ m ðFF m Þ −

Z M

d3 x F@ m Fm

(237)

where Fm ¼ Emnl@ nAl is the dual field strength. However, in the absence of magnetic monopoles, this field satisfies the Bianchi identity, @ mF m ¼ @ m(Emnl@ nAl) ¼ 0. Therefore, using the Gauss Theorem, we find that the change of the action is a total derivative and integrates to the boundary Z Z dS ¼ d3 x @ m ðFF m Þ ¼ dSm FF m (238) M

S

where S ¼ @M is the boundary of M. In particular, if F is a nonzero constant function on M, then the change of the action under such a gauge transformation is dS ¼ F fluxðSÞ:

(239)

Hence, the action is not invariant if the manifold has a boundary, and the theory must be supplied with additional degrees of freedom at the boundary. Indeed, the flat connections, that is, the solution of the equations of motion, Fmn ¼ 0, are pure gauge transformations, Am ¼ @ mf, and have an action that integrates to the boundary. Let M ¼ D  where D is a disk in space and  is time. The boundary manifold is S ¼ S1 , where S1 is a circle. Thus, in this case, the boundary manifold S is isomorphic to a cylinder. The action of the flat configurations reduces to Z k S¼ d2 x @ 0 f@ 1 f (240) 4p S1  This implies that the dynamics on the boundary is that of a scalar field on a circle S1, and obeys periodic boundary conditions. Although classically the theory does not depend on the metric, it is invariant under arbitrary transformations of the coordinates. However, any gauge fixing condition will automatically break this large symmetry. For instance, we can specify a gauge condition at the boundary in the form of a boundary term of the form Lgauge fixing ¼ A21. In this case, the boundary action of the field ’ becomes Z k  d2 x (241) S½’ ¼ @ 0 f@ 1 f − ð@ 1 f2 Þ : 4p 1 S  The solutions of the equations of motion of this compactified scalar field have the form f(x1  x0), and are right (left) moving chiral fields depending of the sign of k. This boundary theory is not topological but is conformally invariant. A similar result is found in non-Abelian Chern–Simons gauge theory. In the case of the SU(N)k Chern–Simons theory on a manifold D , where D is a disk whose boundary is G, and  is time, the action is

Field theoretic aspects of condensed matter physics: An overview Z SCS ½A ¼

D 

d3 x



  k 2 tr Emnl Am @ n Al + Emnl Am An Al : 8p 3

69

(242)

This theory integrates to the boundary, G  where it becomes the chiral (right-moving) SU(N)k Wess–Zumino–Witten model (at level k) at its IR fixed point, l2c ¼ 4p=k Z Z   1 k SWZW ½g ¼ 2 d2 x tr @ m g@ m g−1 + Emnl tr g−1 @ m g g−1 @ n g g−1 @ l g : (243) 12p B 4lc G  Here, g 2 SU(N) parametrizes the flat configurations of the Chern–Simons gauge theory. The boundary theory is a nontrivial CFT, the chiral Wess–Zumino–Witten CFT (Witten, 1989).

Vacuum degeneracy a torus We will now construct the quantum version of the U(1) Chern–Simons gauge theory on a manifold M ¼ T 2 , where T2 is a spatial torus, of linear size L1 and L2. Since this manifold does not have boundaries, the flat connections, Eij@ iAj ¼ 0, do not reduce to local gauge transformations of the form Ai ¼ @ iF. Indeed, the holonomies of the torus T2, that is, the Wilson loops on the two noncontractible cycles of the torus G1 and G2, are gauge-invariant observables: Z L1 Z L2 dx1 A1  a
1 , dx1 A2  a
2 (244) 0

0

where a
1 and a
2 are time-dependent. Thus, the flat connections now are A1 ¼ @ 1 F +

a
1 , L1

A2 ¼ @ 2 F +

a
2 L2

(245)

whose action is S¼

k 4p

Z dx0 Eij a
i @ 0 a
j

(246)

Therefore, the global degrees of freedom a
1 and a
2 at the quantum level become operators that satisfy the commutation relations ½a
1 , a
2  ¼ i

2p : k

(247)

We find that the flat connections are described by the quantum mechanics of a
1 and a
2 . A representation of this algebra is a
2  −i Furthermore, the Wilson loops on the two cycles become  Z L1  A1  ei
a1 , W½G1  ¼ exp i 0

2p @ : a1 k @


(248)

Z W½G2  ¼ exp i

L2

0

 A2

 ei
a2

(249)

and satisfy the algebra W½G1 W½G2  ¼ expð−i2p=kÞW½G2 W½G1 :

(250)

Under large gauge transformations a
1 ! a
1 + 2p,

a
2 ! a
2 + 2p:

(251)

Therefore, invariance under large gauge transformations on the torus implies that a
1 and a
2 define a two-torus target space. Let us define the unitary operators U 1 ¼ expðik
a2 Þ,

U 2 ¼ expð−ik
a1 Þ

(252)

which satisfy the algebra U 1 U 2 ¼ expði2pkÞU 2 U 1 :

(253)

The unitary transformations U1 and U2 act as shift operators on a
1 and a
2 by 2p, and hence generate the large gauge transformations. Moreover, the unitary operators U1 and U2 leave the Wilson loop operators on noncontractible cycles invariant, U 1−1 W½G1 U 1 ¼ W½G1 ,

U 2−1 W½G2 U 2 ¼ W½G2 :

(254)

Let |0i be the eigenstate of W[G1] with eigenvalue 1, that is, W[G1]k0i ¼ |0i. The state W[G2]|0i is also an eigenstate of W[G1] with eigenvalue expð−i2p=kÞ, since

70

Field theoretic aspects of condensed matter physics: An overview W½G1 W½G2 j0i ¼ ei2p=k W½G2 W½G1 j0i ¼ e −i2p=k W½g2 j0i:

(255)

W½G1 W p ½G2  k 0i ¼ e −i2pp=k W p ½G2 j0i

(256)

More generally, since

we find that, provided k 2 , there are k linearly independent vacuum states |pi ¼ Wp[G2]|0i, for the U(1) Chern–Simons gauge theory at level k. It is denoted as the U(1)k Chern–Simons theory. Therefore the finite-dimensional topological space on a two-torus is k-dimensional. It is trivial to show that, on a surface of genus g, the degeneracy is kg. We see that in the Abelian U(1)k Chern–Simons theory, the Wilson loops must carry k possible values of the unit charge. This property generalizes to the non-Abelian theories, which are technically more subtle. We will only state some important results. For example, if the gauge group is SU(2), we expect that the Wilson loops will carry the representation labels of the group SU(2), that is, they will be labeled by (j, m), where j ¼ 0, 12 , 1, . . . and the 2j + 1 values of m satisfy |m|  j. However, it turns that out SU(2)k Chern–Simons theory has fewer states, and that the values of j are restricted to the range j ¼ 0, 12 , . . . , 2k .

Fractional statistics and braids Another aspect of the topological nature of Chern–Simons theory is the behavior of expectation values of products of Wilson loop operators. Let us compute the expectation value of a product of two Wilson loop operators on two positively oriented closed contours g1 and g2. We will do this computation in the Abelian Chern–Simons theory U(1)k in 2+1-dimensional Euclidean space. Note that the Euclidean Chern–Simons action is pure imaginary since the action is first order in derivatives. The expectation value to be computed is * !+ I W½g1 [ g2  ¼

exp i

g1 [g2

dxm Am

:

(257)

CS

The result changes depending on whether the loops g1 and g2 are linked or unlinked. In this section, we will compute the contribution to this expression for a pair of contours g1 and g2. Here we will not include the contribution to this expectation value for each contours. We will return to this problem in Section “Bosonization of the Dirac theory in 2+1 dimensions” where we discuss the problem of fractional spin. The expectation value of a Wilson loop on the union of two contours, as in the present case, g can be written as  I    Z  ¼ exp i d3 xJ m Am (258) exp i dxm Am g

CS

CS

where the current Jm is J m ðxÞ ¼ dðxm − zm ðtÞÞ

dzm dt

Here, zm(t) is a parametrization of the contour g. Therefore, the expectation value of the Wilson loop is (Witten, 1989)  I   Z  Z i d3 x  expðiI½gCS Þ ¼ exp − d3 y J m ðxÞGmn ðx − yÞJ n ðyÞ exp i dxm Am 2 g CS

(259)

(260)

where Gmn(x − y) ¼ hAm(x)An(y)iCS is the propagator of the Chern–Simons gauge field. Since the loops are closed, the current Jm is conserved, that is, @ mJm ¼ 0, and the effective action I[g]CS of the loop g is gauge-invariant. The Euclidean propagator of Chern–Simons gauge theory (in the Feynman gauge) is Gmn ðx − yÞ ¼

2p G ðx − yÞEmnl @ l dðx − yÞ k 0

(261)

where G0(x − y) is the propagator of the massless Euclidean scalar field, which satisfies −@ 2 G0 ðx − yÞ ¼ d3 ðx − yÞ: Using these results, we find the following expression for the effective action Z Z p d3 x I½gCS ¼ d3 y Jm ðxÞJn ðyÞG0 ðx − yÞEmnl @ l dðx − yÞ k I I p dxm dyn Emnl @ l G0 ðx − yÞ: ¼ k g g

(262)

(263)

Since the current Jm is conserved, it can be written as the curl of a vector field, Bm, as Jm ¼ Emnl @ n Bl :

(264)

Field theoretic aspects of condensed matter physics: An overview

71

In the Lorentz gauge, @ mBm ¼ 0, we can write Bm ¼ Emnl @ n fl :

(265)

Jm ¼ −@ 2 fm

(266)

Hence,

where Z fm ðxÞ ¼

d3 y G0 ðx − yÞJm ðyÞ

(267)

Upon substituting this result into the expression for Bm, we find Z I Bm ¼ d3 yEmnl @ n G0 ðx − yÞJl ðyÞ ¼ Emnl @ n G0 ðx − yÞdyl : g

Therefore, the effective action I[g]CS becomes I½gCS ¼

p k

I

I g

dxm

g

dyn Emnl @ l G0 ðx − yÞ ¼

p k

(268)

I g

dxm Bm ðxÞ:

(269)

Let S be an oriented open surface of the Euclidean three-dimensional space whose boundary is the oriented loop (or union of loops) g, that is, @S ¼ g. Then, using Stokes Theorem we write in the last line of Eq. (269) as Z Z p p I½gCS ¼ dSm Emnl @ n Bl ¼ dS J : (270) k S k S m m The integral in the last line of this equation is the flux of the current Jm through the surface S. Therefore, this integral counts the number of times ng the Wilson loop on g pierces the surface S (whose boundary is g), and therefore it is an integer, ng 2 . We will call this integer the linking number (or Gauss invariant) of the configuration of loops. In other words, the expectation value of the Wilson loop operator is W½gCS ¼ eipng =k :

(271)

The linking number is a topological invariant since, being an integer, its value cannot be changed by smooth deformations of the loops, provided they are not allowed to cross. We will now see that this property of Wilson loops in Chern–Simons gauge theory leads to the concept of fractional statistics. Let us consider a scalar matter field that is massive and charged under the Chern–Simons gauge field. The excitations of this matter field are particles that couple minimally to the gauge field. Here we will be interested in the case in which these particles are very heavy. In that limit, we can focus on states that have a few of this particles which will be in their nonrelativistic regime. Consider, for example, a state with two particles which in the remote past, at time t ¼ −T ! −1, are located at two points A and B. This initial state will evolve to a final state at time t ¼ T ! 1, in which the particles go back either to their initial locations (the direct process) or to another one in which they exchange places, A $ B. At intermediate times, the particles follow smooth worldlines. These are two processes, direct and exchange. There we see that the direct process is equivalent to a history with two unlinked loops (the worldlines of the particles), whereas in the exchange process, the two loops form a link. It follows from the preceding discussion that the two amplitudes differ by the result of the computation of the Wilson loop expectation value for the loops g1 and g2. Let us call the first amplitude Wdirect and the second Wexchange. The result is W exchange ¼ W direct eip=k

(272)

where the sign depends on how the two worldlines wind around each other. An equivalent interpretation of this result is that if C[A, B] is the wave function with the two particles at locations A and B, the wavefunction where their locations are exchanged is C½B, A ¼ eip=k C½A, B

(273)

where the sign depends on whether the exchange is done counterclockwise or clockwise. Clearly, for k ¼ 1, the wave function is antisymmetric and the particles are fermions, while for k ! 1, they are bosons. At other values of k, the particles obey fractional statistics and are called anyons (Wilczek, 1982a; Leinaas and Myrheim, 1977). The phase factor f ¼ p/k is called the statistical phase. Notice that while for fermions and bosons the statistical phase ’ ¼ 0, p is uniquely defined (mod 2p), for other values of k, the statistical angle is specified up to a sign that specifies how the worldlines wind around each other. Indeed, mathematically the exchange process is known as a braid. Processes in which the worldlines wind clock and counterclockwise are braids that are inverse of each other. Braids can also be stack sequentially yielding multiples of the phase ’. In addition to stacking braids, Wilson loops can be fused: seen from some distance, a pair of particles will behave as a new particle with a well-defined behavior under braiding. This process of fusion is closely related to the concept of fusion of primary fields in CFT.

72

Field theoretic aspects of condensed matter physics: An overview

Furthermore, up to regularization subtleties (Grundberg et al., 1990), the self-linking terms (those with a ¼ b) yield a topological spin 1/2k, consistent with the spin-statistics connection (Polyakov, 1988). For k ¼ 1, this means that the flux-charge composites have spin 1/2. What we have just described is a mathematical structure called the Braid group. The example that we worked out using Abelian Chern–Simons theory yields one-dimensional representations of the Braid group with the phase ’ being the label of the representations. For U(1)k, there are k types of particles (anyons). That these representations are Abelian means that, in the general case of U(1)k, acting on a one-dimensional representation p (defined mod k) with a one-dimensional representation q (also defined mod k) yields the representation one-dimensional p + q (mod k). We will denote the operation of fusing these representations (particles!) as [q]mod k [p]mod k ¼ [q+p]mod k. These representations are in one-to-one correspondence with the inequivalent charges of the Wilson loops, and with the vacuum degeneracy of the U(1)k Chern–Simons theory on a torus. A richer structure arises in the case of the non-Abelian Chern–Simons theory at level k (Witten, 1989), such as SU(2)k. For example, for SU(2)1, the theory has only two representations, both are one-dimensional, and have statistical angles ’ ¼ 0, p/2. However, for SU(2)k, the content is more complex. In the case of SU(2)2, the theory has (a) a trivial representation [0] (the identity, (j, m) ¼ (0, 0)), (b) a (spinor) representation [1/2] ((j, m) ¼ (1/2, 1/2)), and (c) a the representation [1] ((j, m) ¼ (1, m), with m ¼ 0, 1). These states will fuse obeying the following rules: [0] [0] ¼ [0], [0] [1/2] ¼ [1/2], [0] [1] ¼ [1], [1/2] [1/2] ¼ [0] + [1], [1/2] [1] ¼ [1/2], and [1] [1] ¼ [0] (note the truncation of the fusion process!). Of particular interest is the case [1/2] [1/2] ¼ [0] + [1]. In this case, we have two fusion channels, labeled by [0] and [1]. The braiding operations now will act on a two-dimensional Hilbert space and are represented by 2 2 matrices. This is an example of a non-Abelian representation of the braid group. These rather abstract concepts have found a physical manifestation in the physics of the fractional quantum Hall fluids, whose excitations are vortices that carry fractional charge and anyon (braid) fractional statistics. Why this is interesting can be seen by considering a Chern–Simons gauge theory with four quasi-static Wilson loops. For instance in the case of the SU(2)2 Chern–Simons theory the Wilson loops carry the spinor representation, [1/2]. If we call the four particles A, B, C, and D, we would expect that their quantum state would be completely determined by the coordinates of the particles. This, however, is not the case since, if we fuse A with B, the result is either a state [0] or a state [1]. Thus, if the particles were prepared originally in some state, braiding (and fusion) will lead to a linear superposition of the two states. This braiding process defines a unitary matrix, a representation of the Braid Group. The same is true with the other particles. However, it turns out that for four particles, there are only two linearly independent states. This twofold degenerate Hilbert space of topological origin is called a topological qubit. Moreover, if we consider a system with N (even) number of such particles, p the ffiffiffi dimension of the topologically protected Hilbert N space is 2 2 −1. Hence, for large N, the entropy per particle grows as 12 ln 2 ¼ ln 2. Therefore the qubit is not an “internal” degree of freedom of the particles but a collective state of topological origin. Interestingly, there are physical systems, known as non-Abelian fractional quantum Hall fluids that embody this physics and are accessible to experiments! For these reasons, the non-Abelian case has been proposed as a realization of a topological qubit (Kitaev, 2003; Das Sarma et al., 2008).

Topological phases of matter Topological insulators We will now give a brief discussion of the physics of Topological Insulators from a field-theoretic perspective. Topological insulators are systems whose electronic states (band structures) have special topological properties that manifest in the existence of symmetry-protected edge states. For this reason these systems are known as symmetry-protected topological states (or SPTs). The simplest example is found in one space dimension where it is related to the fascinating problem of fractionally charged solitons and electron fractionalization. In several ways many of the concepts involved in these 1+1-dimensional systems can and have been extended to higher dimensions.

Dirac fermions in 2+1 dimensions We will now see that, in spite of the formal similarities with the 1+1 dimensional case, this theory has different symmetries, particularly concerning parity and time-reversal invariance. In addition, in 2+1 dimensions, it is not possible to define a g5 Dirac matrix, which implies that there is no chiral symmetry and no chiral anomaly. We will consider first a theory of a Dirac field which in 2+1 dimensions is a bi-spinor (as is in 1+1 dimensions). The Lagrangian of the Dirac theory coupled to a background gauge field Am is
m @ m c − mcc
− eAm cg
m Dm ðAÞc − mcc

c  cig L ¼ cig m

(274)


c  j is the gauge-invariant (and conserved) Dirac current, and Dm(A) ¼ @ m + ieAm is the covariant derivative. where c
¼ c g0 , cg m m In Eq. (274), gm are the three 2 2 Dirac matrices which obey the algebra, {gm, gn} ¼ 2gmn, where gmn ¼ diag(1, −1, −1) is the metric of 2+1 dimensional Minkowski spacetime. The Dirac matrices gm can be written in terms of the three 2 2 Pauli matrices. For instance we can choose g0 ¼ s3, g1 ¼ is2 and g2 ¼ −is1, which satisfy the Dirac algebra. Since the three gamma matrices involve all tree Pauli matrices, it is not possible to define a g5 matrix, which would anticommute with the gamma matrices. In this sense, it is not possible to define a chirality for bi-spinors in 2+1 dimensions {

Field theoretic aspects of condensed matter physics: An overview

73

We could have also chosen a different set of gamma matrices. For instance, we could have chosen g0 ¼ s3, g1 ¼ is2, and g2 ¼ +is1, which also satisfy the Dirac algebra. These two choices are equivalent to the 2D parity transformation x1 ! x1 and x2 ! −x2. In other words, we can choose the three gamma matrices to be defined in terms of a right-handed or a left-handed frame (or triad). Thus, the choice of handedness of the frame used to define the gamma matrices can be regarded as a chirality. The parity transformation is also equivalent to a unitary transformation (a change of basis) U ¼ g1 ¼ is1 which flips the sign of both g2 and g0. Thus, a parity transformation is equivalent to a change of the sign of g0 or, what is the same, as changing the sign of the Dirac mass m ! −m. Since the Dirac theory is charge conjugation invariant, the parity transformation is equivalent to time reversal. It is easy to see that in 2+1 dimensions, the massive Dirac theory is not invariant under time reversal since the single particle Dirac Hamiltonian H(p) for momentum p involves all three Pauli matrices, HðpÞ ¼ a  p + bm

(275)

where, as usual, a ¼ g0g and b ¼ g0. In fact, time reversal is equivalent to the change of the sign of the Dirac mass. These considerations also apply to the massless Dirac theory coupled to a background gauge field.

Dirac fermions and topological insulators

In addition systems in one space dimension, discussed in Section “Dirac fermions in one space dimensions,” in which Dirac fermions naturally describe the low energy electronic degrees of freedom, Dirac fermions also play a role in many other systems in condensed matter physics. Such systems range from the nodal Bogoliubov fermionic excitations of d-wave superconductors in 2D and p-wave superfluids in 3D, to Chern and 2 topological insulators in 2D and 2 topological insulators and Weyl semimetals in 3D. They also play a role in spin liquid phases of frustrated spin-1/2 quantum antiferromagnets, such as Dirac and chiral spin liquids. Dirac fermions also play a significant role in our understanding of the compressible limit of 2D electron gases (2DEG) in large magnetic fields (see Section “Chern–Simons gauge theory and the fractional quantum Hall effect”). We will not cover here all of these examples. Instead we will focus on the 2D Chern topological insulators (which exhibit the anomalous quantum Hell effect) and on the 3D 2 topological insulators. In condensed matter physics the important low energy electronic degrees of freedom belong to the electronic states close to the Fermi energy. In such systems, the lattice periodic potential determines the properties of their band structures. The only significant exception to this general rule is the case of the 2DEGs in GaAs-AlAs heterostructures (the most commonly used platform for the study of quantum hall effects) whose electronic densities are low enough so that the lattice periodic potential can for practical purposes almost always be ignored. In general, Dirac (and Weyl) fermions arise when the conduction and valence bands cross at isolated points of the Brillouin zone. In such situations, the near the crossing points the low energy electronic states locally (in momentum space) look like cones and, ignoring possible anisotropies, look like the states of a Dirac-Weyl theory. For these reasons, Dirac fermions play a central role in the theory of graphene type materials (Castro Neto et al., 2009) and, more significantly, in the theory of topological insulators (Qi and Zhang, 2011). Dirac fermions play a central role in Quantum Electrodynamics and in the Standard Model of Particle Physics. The problem of quark confinement requires an understanding of these theories in a regime inaccessible to Feynman-diagrammatic perturbation theory and an intrinsically nonperturbative formulation, known as Lattice Gauge Theory (Wilson, 1974; Kogut and Susskind, 1975; Susskind, 1977), needed to be developed. For this reason, in the 1970s, fermionic Hamiltonians with crossings at points became of great interest in High Energy Physics as a way to describing the short-distance dynamics of quarks in lattice gauge theory. This program runs into difficulties when extended to the theory of Weak Interactions, which have an odd number of species of chiral (Weyl) Dirac fermions. All local discretizations of the Dirac equation yield an even number of species. This fact came to be known as the fermion doubling problem. These results are special cases of a general theorem, due to Nielsen and Ninomiya (1981a, b, 1983) [and extended by Friedan (1982)], which proves that for systems whose kinetic energy is local in space (as it must be) there must always be an even number of crossings. Therefore, it is impossible to write a local theory with an odd number of chiral fermionic species. In the context of condensed matter physics, the simplest example of Dirac fermions is found in the electronic structure of graphene (a single layer of graphite) discussed by Castro Neto et al. (2009). Graphene is an allotrope of crystalline carbon in which the carbon atoms are arranged in a 2D hexagonal lattice, which is a lattice with two inequivalent sites in its unit cell, labeled by A and B. Let rA and rB be the sites of the two sublattices. Each site rA has three nearest neighbors on the B sublattice located at r iB ¼ r A + di , where i ¼ 1, 2, 3 and       1 1 1 1 1 pffiffiffi , pffiffiffi , − , d1 ¼ , d2 ¼ d3 ¼ − pffiffiffi , 0 (276) 2 3 2 2 3 2 3 (in units in which the lattice spacing is a ¼ 1). The nearest neighbor A sites are related by the vectors pffiffiffi   pffiffiffi  3 1 3 1 a3 ¼ − ,− , ,− a1 ¼ a2 ¼ ð0, 1Þ, 2 2 2 2

(277)

where a1 and a2 are the primitive lattice vector of the hexagonal lattice. The nearest neighbor sites of the B sublattice are also related by these same three vectors.

74

Field theoretic aspects of condensed matter physics: An overview

The low energy electronic degrees of freedom of graphene can be described by a single fermionic state (ignoring spin) at each carbon atom. Let c{(rA) and c{(rB) be the fermion operators that create an electron at the site rA and at the site nearest neighbor sites rB, respectively. The Hamiltonian is X c{ ðr A Þcðr A + di Þ + h:c: (278) H ¼ t1 r A ,i¼1,2,3 where t1 is the hopping matrix element between nearest neighbor A and B sites. In Fourier space Z Z d2 k d2 k cðr A Þ ¼ cðr B Þ ¼ 2 cA ðkÞ expðik  r A Þ, 2 cB ðkÞ expðik  r B Þ BZ ð2pÞ BZ ð2pÞ

(279)

pffiffiffi pffiffiffi where BZ is the first Brillouin zone of the hexagonal lattice, a hexagonal region of reciprocal space with vertices at 2p= 3ð1, 1= 3Þ 0 (which are denoted by K and K , respectively) and their two images under 2p/3 rotations. In momentum space the Hamiltonian is (Semenoff, 1984) ! P  Z 0 cA ðkÞ d2 k j¼1,2,3 expðik  dj Þ P H ¼ t1 ð c ðkÞ c ðkÞ Þ (280) A B 2 0 cB ðkÞ BZ ð2pÞ j¼1,2,3 expð−ik  dj Þ P The one-particle states of this Hamiltonian have eigenvalues EðkÞ ¼ j j¼1,2,3 expðik  dj Þj. Clearly E(k) vanishes at the K and K0 points, the corners of the Brillouin zone. Hence, there is a crossing of the two bands at the K and K0 points near which the energy eigenvalues are a two cones at K and K0 , respectively. Thus, the low energy states of graphene consist of two massless (gapless) Dirac bi-spinors, with opposite chirality/parity. Other examples with similar fermion content are a theory of fermions on a 2D square lattice with a p magnetic flux (1/2 of the flux quantum) per plaquette which arises, for instance, in the theory of the integer quantum Hall effect on lattices by Thouless et al. (1982) (albeit for more general flux per plaquette), and in the theory of flux phases of 2D spin liquids of Affleck and Marston (1988). It also arises in the theory of the chiral spin liquid of Wen et al. (1989). We will discuss these theories in Section “Topological phases of matter.”

Chern invariants The single-particle quantum states of free fermions (electrons) in the periodic potential of a crystal are Bloch wave functions labeled by the lattice momentum k and band indices m. The Bloch states are periodic functions of the lattice momenta whose periods are the Brillouin zones. Topologically each Brillouin zone is a torus: in 1D is a 1-torus, in 2D a 2-torus, etc. The symmetries (and shapes) of the Brillouin zone are dictated by the symmetries of the host crystal. In this section we will consider the quantum states of systems of fermions on a 2D lattice with M bands with eigenvalues {Em(k)} with m ¼ 1, . . . , M. The eigenstates are Bloch states {|um(k)i} such that the wave functions are cm ðxÞ ¼ um ðkÞ expðik  xÞ, where k is a (quasi) momentum in the first Brillouin zone (Bloch, 1929). We will assume that the band spectrum is such that all the bands are separated from each other by a finite gap for all momenta in the first Brillouin zone. Hence, the eigenvalues obey the inequality |Em(k) − En(k)| > 0. We will assume that the system of interest has N < M filled bands and, consequently, that there is a gap separating the top-most occupied band N (the “valence band”) and the lowest unoccupied band N + 1 (the “conduction band”) that does not close everywhere in the first Brillouin zone. Let us consider specifically a 2D system and a Bloch state |um(k)i at a momentum k, and let @ k|um(k)i be an infinitesimally close Bloch state with momentum k0 ¼ k + dk. The inner product of these two infinitesimally close Bloch states is the vector field (defined on the first Brillouin zone for each band m) ðmÞ

Aj ðkÞ ¼ ihum ðkÞj@ j jum ðkÞi

(281)

where j ¼ 1, 2 are the orthogonal directions in momentum space, and we have used the notation @ i ¼ @/@ki. This vector field is known as the Berry Connection of the electronic states in the mth band. The theory of the quantum states of noninteracting electrons in periodic potentials as developed by Bloch (1929) is the foundation of much of what we know about the physics of many semiconductors and simple metals. This theory is the core subject of many textbooks (Kittel, 1996; Ashcroft and Mermin, 1976). This theory is based on the classification of the electronic states as representations of the group of lattice translations and point (and spatial) group symmetry transformations. A key unstated assumption in much of this body of work is that the Bloch states are globally well-defined functions on the Brillouin zone of a given crystal. It is rather remarkable that this assumption was not stated explicitly since the original work by Bloch in 1929 until the work by Thouless, Kohmoto, Nightingale and den Nijs on the quantum states of electrons on 2D lattices in the presence of an uniform magnetic field (Thouless et al., 1982). The realization that there is a topological obstruction to define the electronic states globally in the Brillouin zone is the key to the development of the theory of topological insulators and, more generally, of the modern theory of band structures (Bradlyn et al., 2017; Cano and Bradlyn, 2021). In quantum mechanics, states are defined up to a phase. This means that Bloch states that differ by a phase describe the same quantum state. In other words, each quantum state |um(k)i is a member of a ray (or fiber) of states each labeled by a phase

Field theoretic aspects of condensed matter physics: An overview jðum ÞðkÞi 7! expðif m ðkÞÞ jum ðkÞi:

75 (282)

M

Changing the states by phase factors defines a U(1) unitary transformation of physically equivalent states. Since we have a fiber at ðmÞ each point k, this amount at defining a fiber bundle. However, under this transformation, the Berry connection Aj ðkÞ changes by a gradient of the phases ðmÞ

ðmÞ

Aj ðkÞ 7! Aj ðkÞ + @ j f m ðkÞ

(283)

where we assumed that the phases fm(k) are continuous and differentiable functions on the entire first Brillouin zone. Since the physical states cannot change by redefinitions of the phases of the basis states, we are led to the condition that only the data that is invariant under the gauge transformations defined by Eqs. (282) and (283) is physically meaningful, which is encoded in the gauge-invariant pseudo-scalar quantity F ðmÞ ðkÞ ðmÞ

F ðmÞ ðkÞ ¼ Eij @ i Aj ðkÞ

(284)

which is known as the Berry curvature. In what follows we will be interested in the quantity I Z 1 1 ðmÞ ðmÞ dkj Aj ðkÞ ¼ d2 k F ðmÞ ðkÞ C1 ¼ 2p G 2p BZ

(285) ðmÞ

where G is the boundary of the 1st Brillouin zone (which we denote by BZ). We will now show that the quantity C1 , which ðmÞ measures the flux of the Berry connection Aj ðkÞ through the first Brillouin zone (in units of 2p), is an integer independent of the ðmÞ particular connection Aj ðkÞ that we have chosen. This integer-valued quantity is a topological invariant known as the first Chern number. However, since the first Brillouin zone is a 2-torus,which is a closed manifold, a Berry connection with a net flux cannot obey periodic boundary conditions. Instead, we must allow for generalized (large) gauge transformations that wrap around the 2-torus of the first Brillouin zone. This problem is similar (and closely related) to the problem of the wave functions of a charged particle moving in the presence of a magnetic monopole (Wu and Yang, 1976). We will consider a 2D system whose first Brillouin zone is spanned by two reciprocal lattice vectors b1 and b2, related by the two primitive lattice vectors a1 and a2 by the relation bi aj ¼ 2pdij (with i, j ¼ 1, 2). The generalized gauge transformations now are jum ðk + b1 Þi ¼ expðifmð1Þ ðkÞÞ jum ðkÞi, ðmÞ

ðmÞ

Aj ðk + b1 Þ ¼ Aj ðkÞ + @ j fmð1Þ ðkÞ,

jum ðk + b2 Þi ¼ expðifmð2Þ ðkÞÞ jum ðkÞi ðmÞ

ðmÞ

Aj ðk + b2 Þ ¼ Aj ðkÞ + @ j fmð2Þ ðkÞ

(286)

where fmð1Þ ðkÞ and fmð2Þ ðkÞ are smooth functions of k. We will now show that the Bloch states cannot be defined globally over the Brillouin zone if the flux of the Berry connection does not vanish. More specifically let us assume that at some point k0 2 T, we know the Bloch state |um(k0)i which satisfies the generalized periodic boundary conditions of Eq. (286). The question is, can we determine the Bloch state at some other also arbitrary point k00 ? We will know that there is a topological obstruction that does not allow it if the flux of the Berry phase is not zero. The reason is that the phase of the Bloch state cannot be defined since, in general, the Bloch state will vanish at some point of the Brillouin zone where the phase is undefined. We will prove that this is true by following an elegant construction due to Mahito Kohmoto which goes as follows (Kohmoto, 1985): The Brillouin zone is a torus that, we will denote by T, can be regarded as the tensor product of two circles, T  S1 S1. Let us S split the torus T into two disjoint subsets (or patches) HI and HII such that T ¼ HI HII. We will assume that in region HI, the Bloch state vanishes at some point k0 2 TI and that the Bloch state does not vanish for all points k 2 HII. This means that we can choose the Bloch state |um(k)i to be real for all k 2 HII. On the other hand, we can always assign some arbitrary phase to the Bloch state at k0 2 TI. Once we have done that, we can extend the definition of the phase to a neighborhood of k0 wholly included in HI. If we assume that there is only one zero, then the phase can be defined over all of HI. The result is that we now have two different definitions of the phase of the Bloch state on HI and on HII. Let |um(k)iI and |um(k)iII be the two resulting definitions of the Bloch state. Let the closed curve g be the common boundary of regions HI and HII, g ¼ @HI ¼ @HII. However on the common boundary g, these two definitions of the Bloch state must be a gauge transformation, jum ðkÞiI ¼ expðif m ðkÞÞ jum ðkÞiII

(287)

on all points k 2 g. The gauge transformation fm(k), known as the transition function, is a smooth periodic function of the points k ðmÞ on the closed curve g. Likewise, the Berry connection Aj ðkÞ has two definitions on the regions HI and HII, which differ by a gauge transformation on all the points k of the common boundary g,



ðmÞ ðmÞ Aj ðkÞ − Aj ðkÞ ¼ @ j f m ðkÞ: (288) I

II

The transition function defines a mapping of the closed curve g, which is topologically equivalent to a circle S1, to the phase of the Bloch state, which is defined mod 2p and hence is also a circle S1. Hence the transition functions fm(k) are homotopies that can be

76

Field theoretic aspects of condensed matter physics: An overview

classified by the homotopy group P1 ðS1 Þ ’ . We recognize that the classification of the transition functions is the same that we used for the vortices of Section “Vortices in two dimensions.” Hence, the change of the transition function on a full revolution of the closed curve g must be an integer multiple of 2p. ðmÞ We will now compute the quantity C1 , given in Eq. (285), which measures the flux of the Berry connection through the 2-torus S T that defines the first Brillouin zone. Using the partition of the torus T ¼ T I T II , we can write Z  Z Z 1 1 ðmÞ d2 k F ðmÞ ðkÞ ¼ d2 k F ðmÞ ðkÞ + d2 k F ðmÞ ðkÞ (289) C1 ¼ 2p T 2p HI HII Using Stokes Theorem on the two regions HI and HII, we get I 

 1 ðmÞ C1 ¼ dkj Aj ðkÞ I − Aj ðkÞ II : 2p g

(290) ðmÞ

Since the two definitions of the Berry connection differ by a gauge transformation, Eq. (288), we can express C1 transition function fm(k) on the curve g I 1 1 ðmÞ C1 ¼ dk  @f m ðkÞ ¼ ðDf m ðkÞÞg ¼ n 2p g 2p

in terms of the

(291)

where we used that the transition functions are classified by the integer-valued quantity n 2 . ðmÞ ðmÞ We conclude that the flux of the Berry curvature C1 takes only integer values. It is known of the first Chern number C1 , which ðmÞ is a topological invariant since it cannot change by a smooth redefinition of the curve g. Since C1 6¼ 0 implies that the Bloch states must vanish at least at some point (or points) k 2 HI, then the integer-valued Chern number can only change if a redefinition of the curve g crosses at least one point k at which the Bloch state vanishes. The conclusion of this analysis is that whenever the flux of the Berry curvature through the Brillouin zone does not vanish, the Bloch states cannot be defined globally on the Brillouin zone. A direct consequence is that in this case the band is characterized by a topological invariant called the first Chern number. This number is a property of a given band and, in general, it is different for each band. This is a particular case of a more general topological classification of the states of free fermions on a lattice which depends on the dimensionality of the system (Schnyder et al., 2008; Kitaev, 2009; Moore and Balents, 2007).

The quantum Hall effect on a lattice We will now show that 2D free fermion systems with Chern bands, that is, bands characterized by a nonvanishing Chern number, are insulators that have a quantized Hall conductivity. We will do this for the theory of the integer quantum Hall effect on a 2D lattice of Thouless, Kohmoto, Nightingale and den Nijs (TKNN) (Thouless et al., 1982; Kohmoto, 1985) [see also, Fradkin and Kohmoto (1987)]. Although this problem played a key role in the development of the theory of topological phases of matter, for many years it was viewed as an academic problem since it would require gigantic magnetic fields to be in the regime of interest for typical solids. The situation changed with the development of twisted bilayer graphene and similar systems which have unit cells large enough for this physics to be observable. These new materials have allowed to study this problem experimentally. TKNN considered a system of free fermions on a planar (actually square) lattice in a uniform magnetic field B with flux 2pp/q per plaquette (in units in which the flux quantum f0 ¼ eℏ=c ¼ 1) with p and q two co-prime integers. The tight-binding Hamiltonian for this simple problem is X t j c{ ðrÞ eiAj ðrÞ cðr + ej Þ + h:c: (292) H¼ r , j¼1, 2 where c(r) and c{(r) are fermion creation and annihilation operators on the lattices sites {r ¼ (m, n)}, ej are lattice unit vectors, and tj, with j ¼ x, y, is a hopping matrix element between nearest neighbor sites of the square lattice along the x and y directions. On each link of the lattice, we defined a vector potential Aj(r), where the oriented sum of the lattice vector potentials on each plaquette 2pf, where f ¼ qp , is the flux on each plaquette of the square lattice. In the axial (Landau) gauge A1(m, n) ¼ 0, A2(m, n) ¼ 2pmf is a periodic function of m with period q. In this gauge, the Hamiltonian becomes P { H¼t c ðm + 1, nÞcðm, nÞ + c{ ðm, nÞcðm + 1, nÞ m,n (293) P { +t c ðm, n + 1Þ ei2pfm cðm, nÞ + c{ ðm, nÞ e−i2pfm cðm, n + 1Þ m,n In this gauge, the problem reduces to a system with a theory with a q 1 unit cell with q inequivalent sites. In momentum space, the (magnetic) Brillouin zone of this system is − pq  k1  pq and −p  k2  p. The Fourier transform of the operators c{(r) and c(r) is, respectively, c{(k) and c(k), which obey the standard anticommutation relations, {c(k), c(q)} ¼ {c{(k), c{(q)} ¼ 0, and {c{(k), c(q)} ¼ d(k − q). In momentum space, the Hamiltonian becomes Z H¼q

p=q

−p=q

dkx 2p

Z

p

−p

dky ~ x , ky Þ Hðk 2p

(294)

Field theoretic aspects of condensed matter physics: An overview

77

where ~ x , ky Þ ¼ 1 Hðk q

q−1 X 

2t x cosðkx + 2pfnÞ c{ ðkx + 2pfn, ky Þ cðkx + 2pfn, ky Þ

n¼0



−iky

+ ty e

{

c ðkx + 2pðn + 1Þf, ky Þ cðkx + 2pnf, ky Þ + e

iky

{

c ðkx + 2pðn −1Þf, ky Þ cðkx + 2pnf, ky Þ



(295)

The spectrum of this system was first investigated by Hofstadter (1876) consists of q bands which for q an odd integer are separated by finite energy gaps. ~ x , ky Þ is the tight-binding model of a For fixed values of kx and ky (in the magnetic Brillouin zone), the Hamiltonian Hðk one-dimensional chain on the 1D lattice of q sites located at kx + 2pnf with nearest neighbor hopping. The single-particle (Bloch) states un(k) of this 1D model obeys the Schrödinger Equation  2t x cosðkx + 2pnfÞun ðkÞ + t y e−iky un−1 ðkÞ + eiky un+1 ðkÞ ¼ En un (296) which is known as Harper’s equation. In general, this equation does not admit an analytic solution. However, the nature qualitative features of the spectrum can be obtained by an expansion either on tx/ty or on ty/tx. TKNN used degenerate perturbation theory to show that the spectrum has q bands and that the r-th band is characterized by two integers sr and ℓr, which are the solution of the Diophantine equation r ¼ q s r + p ℓr

(297)

with ℓ0 ¼ s0 ¼ 0. It is also obvious that for r ¼ q sq ¼ 1 and ℓq ¼ 0 (for all p). Furthermore, TKNN showed that there was an, until then unsuspected, relation between the Hall conductivity sxy of a gapped system of electrons in periodic potentials in a uniform magnetic field and the Berry connection of the filled bands. This relation implies the quantization of the Hall conductivity and its computation in terms of a topological invariant, the Chern number of the ðrÞ occupied bands. They showed that in this context, each band has a nontrivial Berry connection Aj ¼ ihur ðkÞj@ j jur ðki (with @ j ¼ @ kj ) ðrÞ of the form of Eq. (281) with a nonvanishing (first) Chern number C1 , that is, the flux through the magnetic Brillouin zone. The conductivity tensor characterizes the electrical properties of a physical system. Linear Response Theory provides a framework for computing the conductivity tensor by perturbing the system with a weak external electromagnetic field Am and computing the currents that they induce (Martin, 1967; Fradkin, 2021). The expectation value of the gauge-invariant current operator Jm(x) is hJ m ðxÞi ¼ −

i d ln Z½Am  ℏ dAm ðxÞ

(298)

where Z[Am] is the partition function in the presence of a background (i.e., classical) electromagnetic field Am(x). In general spacetime dimension D ¼ d + 1, the lowest order in the vector potential Am, the induced current is given in terms of the polarization tensor Pmn(x, y) Z Z i ln Z½Am  ¼ dD x dD y Am ðxÞ Pmn ðx, yÞ An ðyÞ + OðA3 Þ: (299) 2 Therefore, the induced current is related to the external field by Z hJm ðxÞi  J m ðxÞ ¼ dD y Pmn ðx, yÞ An ðyÞ:

(300)

Expressions of this type are known as a Kubo formula. The polarization tensor can be regarded as a generalized susceptibility. Although we are using a continuum relativistic notation, these expressions are generally valid, even in lattice systems. Gauge invariance requires that the polarization tensor be conserved @ m Pmn ðx, yÞ ¼ 0:

(301)

In general, the (retarded) polarization tensor Pmn(x, y) is related to the (retarded) current–current correlation function DRmn ðx, yÞ, i Dmn ðx, yÞ ¼ − Yðx0 − y0 Þh½Jm ðxÞ, Jn ðyÞi ℏ

(302)

by the identity PRmn ðx, yÞ ¼ DRmn ðx, yÞ − iℏ



 dJm ðxÞ : dAn ðyÞ

(303)

The last term in Eq. (303) is usually called a “contact term.” This term vanishes only for a theory of relativistic fermions (in the continuum). In all other cases, lattice or continuum, relativistic or not, the contact term does not vanish and its form depends on the specific theory. In nonrelativistic systems, for example, in a FL, this term is the origin of the f-sum rule (Martin, 1967; Nozières and Pines, 1966).

78

Field theoretic aspects of condensed matter physics: An overview

When the external field Am represents is a (locally) uniform electric field E, the induced current is Ji ¼ sijEj, where sij is the conductivity tensor, which can be obtained from the polarization tensor as the limit sij ¼ lim

o!0

1 lim P ðo, qÞ io q!0 ij

(304)

In a metal, which has a Fermi surface, the order of limits in which o and q vanish matters and only the order of limits specified above is the correct one to take. In Dirac systems, that we will discuss below, the order does not matter due to relativistic invariance. The other case in which the order does not matter is the Chern insulator that we are interested in. In general, in an isotropic system, the conductivity tensor has a symmetric part and an antisymmetric part. The symmetric part of the conductivity tensor yields the longitudinal conductivity, which has all the effects of dissipation. The antisymmetric part does not vanish in the presence of a magnetic field or, more generally, if time reversal symmetry is broken, and yields the Hall conductivity. A Chern insulator is an insulator and as such is a state with an energy gap. In such a state, the longitudinal conductivity vanishes since there is no dissipation in a gapped state. But in a Chern insulator time-reversal invariance is broken. In the system that we are discussing is broken by the magnetic field. The Hall conductivity can be calculated from the antisymmetric part of the polarization tensor as the limit sxy ¼ lim

o!0

i P ðo, 0Þ: o xy

(305)

In addition, the contact term does not contribute to the Hall conductivity. As a result, the hall conductivity can be computed in terms of the antisymmetric component of the current-correlation function. Let us now return to the theory of free fermions on a lattice in a (commensurate) magnetic field takes. As we saw, the electronic states are split into q bands with single particle states |cn(k)i where k takes values on the first magnetic Brillouin zone. We will compute the Hall conductivity for this system assuming that the Fermi energy EF lies in the gap between the n-th and the n − 1-th bands. Let us label by a the occupied bands and by b the unoccupied bands. Hence Ea(k) < EF < Eb(k). In this case, the Kubo formula for the Hall conductivity becomes Z e2 X d2 k hua ðkÞjJ y jub ðkÞihub ðkÞjJx jua ðkÞi − hua ðkÞjJ x jub ðkÞihub ðkÞjJy jua ðkÞi sxy ¼ i : (306) ℏ E 0

(334)

(335)

there is a nonvanishing gap on the entire Brillouin zone between the occupied valence band, with n−(k) ¼ 1, and the unoccupied conduction band, with n+(k) ¼ 0. For the insulating state, the Fermi energy lies inside this gap and the expression for the Hall conductivity reduces to the following sxy ¼ −

e2 2ℏ

Z

@ h^a ðkÞ @ h^b ðkÞ ^ d2 k h ðkÞ: 2 Eabc @k @ky c x BZ ð2pÞ

(336)

In our discussion of quantum antiferromagnets in one space dimensions in Section “Nonlinear sigma models and antiferromagnetic quantum spin chains,” we found that their effective low-energy action contained a crucial topological term proportional to the integer-valued topological invariant Q (the winding number) of Eq. (103) which classifies the smooths maps (homotopies) of the 2D surface (say a sphere S2) onto the target space of a three-component unit vector field which also a sphere S2. These equivalence classes are represented by the notation P2 ðS2 Þ ’ . In the case at hand, the unit vector ^hðkÞ are points on a 2-sphere. Hence, ^hðkÞ is a map of the first Brillouin zone (which is a 2-torus) to the sphere S2. Such maps are also classified by the same integer-valued topological invariant defined in Eq. (103). These results imply that the Hall conductivity of the two-band system is sxy ¼

e2 Q½^h: 2pℏ

(337)

In other words, we have shown that the Hall conductivity is given in terms of a topological invariant of the occupied band (in units of e2/h), the winding number Q. In the two-band model the topological invariant Q plays the same role as the Chern number does in the work of TKNN (Thouless et al., 1982; Kohmoto, 1985). When Q 6¼ 0, the two-band system exhibits the quantized anomalous quantum Hall effect. This is a property of the entire band of occupied states, and not just a consequence of the low energy approximation. This result implies that the low energy approximation captures all of the topology of the band. It also implies that in these lattice models, the Berry curvature is highly concentrated near the points in momentum space where the two bands are close in energy. We will now consider the problem of the quantum phase transition between the trivial and the Chern insulator. To address this problem, we will tune the parameters of the lattice model discussed in Section “Dirac fermions and topological insulators,” the phase f and the ratio of hopping amplitudes t2/t1, to the point at which the mass of one of the two species of Dirac fermions, say the fermion c1, is zero, m1 ¼ 0, while keeping the fermion c2 massive, m2 6¼ 0. In the low-energy regime we have a massless fermion and a massive fermion. This point in parameter space is a quantum phase transition between a trivial insulator and a Chern insulator. In the low-energy regime, the fermion c1 is massless. A massless fermion is a scale-invariant system in the sense that the correlators of all its observables exhibit power law behavior (free field in this case). We will now discuss briefly the electromagnetic response of this system at the quantum phase transition where one fermion becomes massless. In particular, it is natural to ask if the coefficient of the parity-odd (Chern–Simons) term nonvanishing at the quantum phase transition. To answer this question, we will look at the behavior of the parity-even kernel P0(p2) and the parity-odd kernel PA(p2) in the massless limit for the light fermion, m1 ! 0. We find that the total contribution of both the light fermion and the heavy fermion to the polarization kernels is (assuming m2/m1 ! 1) lim P0 ðp2 Þ ¼

m1 !0

i pffiffiffiffiffi , 16 p2

lim PA ðp2 Þ ¼ −

jm2 j!1

1 sgnðm2 Þ: 4p

(338)

Field theoretic aspects of condensed matter physics: An overview

83

The important conclusion is that at this quantum critical point, the parity-even kernel P0(p2) is nonlocal and that the heavy regulator fermion (the “doubler”) yields the leading finite nonvanishing and local contribution to the parity-odd kernel PA(p2). We can now use these results to compute the conductivity tensor sij at the quantum critical point where m1 ! 0. Since the system is spatially isotropic, the conductivity tensor has the form   sxx sxy sij ¼ (339) −sxy sxx where we used that syy ¼ sxx since the system is isotropic. The longitudinal conductivity sxx and the Hall conductivity sxy are sxx ¼

p e2 , 8h

sxy ¼ 

1 e2 : 2h

(340)

In other words, at the quantum critical point the system has a finite (and universal) longitudinal conductivity. This result may seem surprising as there is no disorder in this model. A finite universal longitudinal conductivity is a standard occurrence in 2D systems at a quantum critical point. For example, at the superconductor–insulator transition, the conductivity is (conjectured to be) sxx ¼ e2/2h (with sxy ¼ 0). In addition, it also has finite and universal Hall conductivity. The Hall conductivity at the quantum critical point is due to the heavy fermionic “doubler” and is equal to 1/2 (in units of e2/h). So, the quantum critical point is not time-reversal invariant since this symmetry is broken at the UV (lattice). We can examine the massless theory using a more formal approach (Witten, 2016; Seiberg et al., 2016). In a gauge-invariant regularization of the massless theory the partition function of a Dirac fermion coupled to a background (unquantized) U(1) gauge field, the partition function is not time-reversal invariant. It is given by  Z  Z  

p

D½A = m c ¼ detðiD½A = m Þ ¼ detðiD½A = m Þ exp i ½Am  exp i d3 x ci (341) Z½Am  ¼ DcDc 2 where the sign depends on how time-reversal invariance is broken by the choice of regularization. The quantity [Am] that appears in the phase factor is the Atiyah-Patodi-Singer -invariant (Atiyah et al., 1975) which we already encountered in the discussion of the fractionally charged solitons in Section “Fractionally charged solitons.” With some caveats (Witten, 2016; Seiberg et al., 2016), the phase factor of the partition function of Eq. (341) is commonly written in the form of a 1/2-quantized Chern–Simons term Z p 1 1 d3 x (342) ½Am    E Am @ n Al 2 2 4p mnl often denoted as a U(1)1/2 Chern–Simons term. This term plays the same role as the contribution of the heavy fermion doubler in the lattice theory. However, we can also wonder if there is a way to have a time-reversal invariant theory of a single massless Dirac fermion, m ! 0. This case cannot be realized in a y 2D lattice model but, as will see, it can be realized on the surface states of a 3D time-reversal invariant 2 topological insulator, which we will discuss in Section “Topological insulators.”

Three-dimensional 2 topological insulators The classification of topological insulators in terms of a Chern number is only possible in even space dimensions in systems with broken time reversal symmetry: in d ¼ 2, the Berry connection is Abelian and the topological invariant is the first Chern number while in d ¼ 4, the Berry connection in a non-Abelian SU(2) gauge field and the topological invariant is the second Chern number (Qi et al., 2008), etc. We will now discuss the time-reversal invariant topological insulators (Fu and Kane, 2007; Moore and Balents, 2007; Qi et al., 2008) and, in particular, those that are invariant under inversion symmetry. Such states exist in both two and space dimensional insulator with strong spin-orbit coupling.

2 Topological invariants

Let {|un(k)i} be the Bloch states. We will represent time reversal by the antiunitary operator Y that acts on the single particle (Bloch) states by complex conjugating the state and reversing the spin. For spin-1/2 fermions Y ¼ expðips2 ÞK , where K is the complex conjugation operator. In this case, Y2 ¼ −1. Let us assume that we have two occupied Bloch bands for each point k of the Brillouin zone. In this case, the states form a rank-2 vector bundle over the torus of the Brillouin zone. In time-reversal invariant systems, the antiunitary time reversal transformation T induces an involution in the Brillouin zone that identifies the points k and −k. Time reversal acts on the one-particle (Bloch) Hamiltonian as YH(k)Y−1 ¼ H(−k). The states |un(k)i are related by time reversal as |un(−k)i ¼ Y|un(k)i, which implies that the bundle is real. The condition Y2 ¼ −1 implies that the bundle is real. In algebraic topology, these bundles are classified by an integer (here the number of occupied bands) and a 2 index that will allow us to classify these states. In a periodic lattice, there exists a set of points Qi of the Brillouin zone with the property that they differ by their images under the action of time reversal by a reciprocal lattice vector, −Qi ¼ Qi + G. In d ¼ 2, there are four such points and d ¼ 3, there are eight P points, and are given by Qi ¼ 12 j nj bj , where nj ¼ 0, 1, j ¼ 1, 2 in d ¼ 2 and j ¼ 1, 2, 3 in d ¼ 3. Here, bj are the primitive lattice

84

Field theoretic aspects of condensed matter physics: An overview

vectors. Kane and Mele (2005) defined the 2N 2N antisymmetric matrix wm,n(k) ¼ hum(−k)|Y|un(k)i. They showed that at each time-reversal invariant point Qi, one can define an index di pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det½wðQi Þ ¼ 1 (343) di ¼ Pf ½wðQi Þ where det[w] and Pf[w] are, respectively, the determinant and the Pfaffian of the matrix w, and det[w] ¼ Pf[w]2. The sign of the quantities di can be made unambiguous by requiring that the Bloch states be continuous. In addition, the quantities di are gaugedependent. However the products ð−1Þn ¼

4 Y

di

(344)

i¼1

in d ¼ 2, and ð−1Þn0 ¼

8 Y i¼1

di ,

ð−1Þnk ¼

Y nk ¼1, nj6¼k ¼0,1

di ðn1 , n2 , n3 Þ

(345)

are gauge and are also topological invariant. The 2-valued indices n and n0 are robust to disorder and are called strong topological indices. Furthermore, Fu and Kane showed that if xn(Qi) ¼ 1 are the parity eigenvalues of the occupied parity eigenstates, the Q quantities di are given by di ¼ N m¼1 x2m ðQi Þ. In d ¼ 3, the index n0 does not rely on the existence of inversion symmetry. In the case of a two-band model in d ¼ 2 and in d ¼ 3, the states are four-component spinors reflecting the two bands and the two spin components. In the context of these systems with strong spin-orbit coupling spin is actually the z-component of the atomic total angular momentum J of the electrons with energies close to the Fermi energy. In these systems, the one-particle Hamiltonian H(k) is a 4 4 Hermitian matrix, which can be expanded as a linear combination of Dirac matrices. A simple and very useful model of systems of this type is the Wilson fermion model (with continuous time) (Wilson, 1974; Bernevig et al., 2006; Qi et al., 2008) of a square (cubic) lattice with 4 states per site (parity and spin) whose Hamiltonian in three dimensions is HðkÞ ¼ sin k  a + MðkÞ b

(346)

where a ¼ g0g and b ¼ g0 are the conventional 4 4 Dirac matrices. The g matrices satisfy the Clifford algebra {gm, gn} ¼ 2gmn1, where gmn ¼ diag(1, −1, −1, −1) is the metric tensor of four-dimensional Minkowski space time. An additional g matrix of interest is g5 ¼ ig0g1g2g3. The (Wilson) mass term M(k) in two dimensions is MðkÞ ¼ M + cos k1 + cos p2 − 2 , and in three dimensions is MðkÞ ¼ M + cos k1 + cos k2 + cos k3 − 3. In Eq. (346), we see that, consistent with the requirements of the Nielsen–Ninomiya theorem (Nielsen and Ninomiya, 1981a,b), in addition to a possible low energy Dirac fermion (if M is small), there are three more massive Dirac fermions (in d ¼ 2) and seven other Dirac fermions in d ¼ 3 so that the total number of Dirac fermions is always even, 8 in this case. With Wilson’s mass term, the additional Dirac fermions (the “doublers”) are always heavy even if the Dirac fermion near the G point Q ¼ 0 is light. In the Dirac basis time reversal is the operation Y ¼ ðis2 IÞK, where K is complex conjugation and parity is P ¼ b. The matrices a and b commute with PY. At the time-reversal and parity invariant points of the Brillouin zone {Qi}, the Hamiltonian only depends on the matrix b, H(Qi) ¼ M(Qi)b. Since the parities of the spinors are the eigenvalues of the matrix b, we conclude that in the two-band models, the quantities di are simply equal to the sign of the mass of the fermions defined at the time-reversal-invariant points Qi of the BZ (Fu and Kane, 2007; Qi et al., 2008), di ¼ d(Qi) ¼ −sgn M(Qi). Using this result, it follows that in two dimensions the system is a 2 topological insulator with index n ¼ 1 (mod 2) if 0 < M < 4 while it is trivial for other values of M, that is, n ¼ 0 mod 2. Similarly, in three dimensions, a 2 (strong) topological insulator exists only if 0 < M < 2 (in the other regimes, this system is in a weak topological insulator state or in a trivial one). We conclude that both in two and three dimensions, the low energy theory of the 2 topological insulators consists of a single Dirac fermion whose mass is small compared to that of the fermion doublers and has the opposite sign. In both cases, there is a quantum phase transition between a trivial insulator and the time-reversal invariant topological insulator at the point where the mass of the light fermion vanishes.

The axial anomaly and the effective action

We will now look at the electromagnetic response of a 2 topological insulator in 3+1 dimensions. This problem can be addressed more easily using the continuum field theory description, which is valid in the regime where the mass M is weak. In this regime, one species of Dirac fermions is light (i.e., its mass is small) while the Dirac doublers remain heavy. Much as we did in our discussion of the Chern insulators and the parity anomaly in Section “The parity anomaly,” we will keep in mind that the fermion doublers play the role of heavy regulators, such as in the Pauli–Villars scheme, in Quantum Field Theory. Here too, we will see that a field-theoretic anomaly, known as the axial anomaly, plays a central role in the physics. The analysis is very similar to what we did with the chiral anomaly in 1+1 dimensions in Section “The chiral anomaly.” Let us begin with a theory of a single massive Dirac fermion in 3+1 dimensions. The Lagrangian of the free massive Dirac theory is

Field theoretic aspects of condensed matter physics: An overview L ¼ c
ð i@= − mÞc:

85 (347)

The equation of motion of the spinor field operator c(x) (I omit the spinor indices here) is the Dirac equation ði@= − mÞc ¼ 0:

(348)


c is locally conserved, The Dirac Lagrangian has a global gauge symmetry c ! e c, which requires that the Dirac current jm ¼ cg m @ mjm ¼ 0. The massless Dirac theory can be decomposed into a theory of two Weyl bi-spinor fields that obey separate Dirac Lagrangians. Furthermore, the massless theory has the additional global symmetry under the transformation c ! eiyg5 c and the
g5 c. However the conservation law of the axial current is violated in the additional formally locally conserved axial current j5m ¼ icg m massive Dirac theory iy


5c @ m j5m ¼ −2micg

(349)

since the two Weyl bi-spinors transmute into each other in the presence of a mass term. This is the origin of the phenomenon of neutrino oscillations. In the condensed matter physics, context there is a similar phenomenon in systems of Weyl semimetals which have crossings between the valence and conducting bands at two locations Q of the BZ, with each crossing associated with each Weyl bi-spinor. A charge density wave with ordering wavevector 2Q mixes the two Weyl fermions that become gapped, becoming effectively a single massive Dirac fermion (Gooth et al., 2019). This state is often called an “axionic”-charge-density-wave. We will now show that the axial symmetry has an anomaly and cannot be gauged. Thus, we will consider the problem of a Dirac theory coupled to a background U(1) gauge field Am and reexamine the putative conservation of the axial current j5m. This question can be addressed in different ways. Quite early on Adler (1969), and Bell and Jackiw (1969) examined this problem by computing a Dirac fermion triangle diagram for the process of a neutral pion decaying into two photons, p0 ! 2g. In particle physics, the pion is the Goldstone boson of the spontaneously broken chiral symmetry. The analog of this problem in condensed matter physics is the phase mode of an incommensurate charge density wave. The relativistic Lagrangian for this problem is a theory of Dirac fermions coupled to a complex scalar field f ¼ f1 + if2 through two Yukawa couplings
Dc
ig5 y c − Vðjfj2 Þ
+ igf cg
5 c − Vðf , f Þ ¼ ci
Dc = + gf1 cc = + gjfjce L ¼ ci 2 1 2 2

f21

(350)

f22 ,

where jfj ¼ + tan y ¼ f2 =f1 , and Dm ¼ @ m + ieAm is the covariant derivative. Here we are regarding the gauge field Am as a background probe field. The triangle Feynman diagram computes the polarization tensor for the electromagnetic field Am with an insertion of the coupling to the complex scalar field in an otherwise massless theory. Assuming a gauge-invariant regularization of the diagram, this computation finds that the axial current j5m is anomalous and is not conserved even in the massless theory (Adler, 1969; Bell and Jackiw, 1969) @ m j5m ¼ −

e2 mn  F Fmn 16p2

(351)

where Fmn ¼ 12 Emnlr F lr is the dual of the electromagnetic field tensor. This is the axial anomaly. To see how the axial anomaly arises, we will follow the physically transparent approach of Nielsen and Ninomiya (1983), which we also employed in Section “The chiral anomaly” in 1+1 dimensions. We will consider a theory of free massless dirac fermions coupled to a background electromagnetic field. Since the theory is massless, the Dirac equation decouples into an equation to the right-handed Weyl fermion cR (with positive chirality g5cR ¼ +cR) and a left-handed Weyl fermion cL (with negative chirality, g5cL ¼ −cL). In the gauge A0 ¼ 0, the Dirac equations become ½i@ 0 − ð−i@ − eAÞ  scR ¼ 0,

½i@ 0 − ði@ − eAÞ  scL ¼ 0

(352)

Let us now consider looking at the solutions of the Weyl equation for right-handed fermions cR, Eq. (352). The left-handed fermions cL are analyzed similarly. Wd will consider a gauge field A1 ¼ 0 and A2 ¼ Bx1 representing a uniform static magnetic field of strength B pointing along the x3 direction. In this gauge, the eigenstates are plane waves along the directions x2 and x3 and harmonic oscillator states along the direction x1. The eigenvalue spectrum consists of Landau levels with energies   h i1=2 1 Eðn, p3 , s3 Þ ¼  2eB n + + p23 + eBs3 2

(353)

for n ¼ 0, 1, 2, . . ., except for the zero mode with n ¼ 0 and s3 ¼ −1, for which Eðn ¼ 0, p3 , s3 ¼ −1Þ ¼ p3

(354)

where the + sign holds for cR and the − sign for cL. Just as in the case of nonrelativistic fermions, the relativistic Landau levels are degenerate. We will consider a system of Dirac fermions at charge neutrality and, hence, EF ¼ 0. The positive and negative energy states are charge conjugate of each other and the negative energy states are filled. For right-handed fermions, the zero mode states with p3 < 0 are filled while for right-handed states the zero modes with p3 > 0 are filled. The density of states of the zero modes is LeB/(4p2) (where L is the linear size of the system).

86

Field theoretic aspects of condensed matter physics: An overview

Let us consider now turning on an external electric field E parallel to the magnetic field B. Just as we saw in 1+1 dimensions in Section “The chiral anomaly,” the electric field leads to pair creation by shifting the Fermi momentum to pF for the zero modes. There is no particle creation for the states with n 6¼ 0 and they do not contribute to the anomaly. The rate of creation of right-handed fermions, NR is dN R 1 LeB pF e2 ¼ ¼ EB: L 4p2 dx0 4p2 dx0

(355)

dN L 1 LeB pF e2 ¼− ¼ − 2 EB L 4p2 dx0 dx0 4p

(356)

The annihilation rate of left-handed particles is

and the creation rate of left-handed antiparticles is
L 1 LeB pF dN e2 ¼ ¼ EB: L 4p2 dx0 4p2 dx0

(357)

The axial anomaly is the total rate of creation of right-handed particles and of left-handed antiparticles:
L dQ5 dN R dN e2 ¼ + ¼ 2 EB dx0 dx0 dx0 2p

(358)

which agrees with the expression of Eq. (351). We now turn to the 2 topological insulator. As we saw, this is a system with two species of (4 component) Dirac spinors. In the topological phase, the sign of the Dirac mass term one of the Dirac fermions (the one near the G point in the lattice model) is opposite (negative) to the sign of the mass term of the of the other Dirac fermion (the fermion doubler). Explicit calculations (Qi et al., 2008; Essin et al., 2009; Hosur et al., 2010) on the lattice model obtain the result that the effective low-energy action for the electromagnetic gauge field Am in the topological phase is   Z 1 y 2 mn lr +⋯ (359) e E F F Seff ½Am  ¼ d4 x − 2 Fmn Fmn + mnlr 4e 32p2 In a time-reversal invariant system, the allowed values of the y angle of Eq. (359) are restricted to be y ¼ np, with n 2 . The case y ¼ 0 (mod 2p) represents a trivial insulator, whereas y ¼ p (mod 2p) holds for a 2 time-reversal invariant topological insulator. The second term in the effective action of the electromagnetic gauge field of Eq. (359), known as the y term, has been extensively discussed in the high-energy physics literature (Fujikawa, 1979; Wilczek, 1987; Peskin and Schroeder, 1995). The derivation of this term is subtle. As it stands, unless y is varying in space-time (in which case this is known a the axion field), this term is a total derivative. In non-Abelian Yang–Mills gauge theories, this term is proportional to a topological invariant known as the Pontryagin index which counts the instanton number of the gauge field configurations (Callan et al., 1976; Coleman, 1985). In the context of the Lagrangian of Eq. (350), this term is induced by the coupling of the Dirac fermion to the Yukawa coupling of the complex scalar field f to the Dirac and g5 mass terms. In this case, the lowest order contribution is given by the triangle diagram. The fact that this term is exact at lowest order reflects the fact that the axial anomaly is in fact a nonperturbative effect, which is the same in both the weak coupling and the strong coupling regimes (’t Hooft, 1976b). In the phase where the potential V (|f|2) in the Lagrangian of Eq. (171) has a minimum at jf0 j expðiyÞ, the chiral symmetry is spontaneously broken and the phase field y(x) is the Goldstone boson of the spontaneously broken chiral symmetry. In this phase, the Dirac fermions become massive through the Yukawa couplings to the complex scalar field, and the phase of the field f enters in the effective action as an axion field which couples to the gauge field Am through the y term of the effective action. We should note that in a recently studied axionic CDW state of a Weyl semimetal (Gooth et al., 2019), the phase of the CDW plays the role of the axion field.

Theta terms, and domain walls: Anomaly and the Callan–Harvey effect

We will now discuss some remarkable behaviors of three-dimensional 2 topological insulators. We will begin with the electromagnetic response encoded in the effective action of Eq. (359). We will assume that the y angle is effectively a slowly varying Goldstone mode of the spontaneously broken U(1) chiral symmetry (i.e., the axion field) present in the Lagrangian of Eq. (350). If the field y is constant, the y term will be a total derivative and it will not contribute to the local equations of motion. We will see below that this term plays a key role in the physics of a domain wall, which we will regard as the interface of a 2 topological insulator and a trivial insulator. In the general case in which y varies slowly (as Goldstone modes do), its presence leads to interesting modification of Maxwell’s equations, known as axion electrodynamics (Wilczek, 1987): r  E ¼ r~ − e2 ry  B, r B ¼ @ t E + ~j + e2 ð@ t y + ry EÞ,

r E ¼ − @t B rB ¼ 0

(360)

where r~ and ~j are external probe electric charge and current densities. The equations of axion electrodynamics have many remarkable properties. Here we will focus on effects: the topological magnetoelectric effect (Qi et al., 2008) and the Witten effect (Witten, 1979).

Field theoretic aspects of condensed matter physics: An overview

87

Let us consider a 2 topological insulator with a flat open boundary (the x1 − x2 plane) perpendicular to the direction x3. We will assume that the topological insulator lies at x3 < 0. This means that for x3 < 0, and far from the surface, y(x3) ! p, while in the trivial vacuum, and also far from the surface, y(x3) ! 0. We will assume that the change of y from 0 to p occurs on a short distance x. We will call this configuration an axion domain wall. In the region where y(x3) is changing, jx3 j≲x, an applied uniform magnetic field B induces a uniform electric field E parallel to B whose magnitude is proportional to the change in y. This is the topological magnetoelectric effect (Qi et al., 2008). A similar striking effect is obtained by considering the case of a magnetic monopole of magnetic charge 2p/e (as required by Dirac quantization) inside a sphere of the trivial region of radius R, with y ¼ 0, surrounded by a region with y 6¼ 0. The two regions are separated by a thin (axion) wall in which y changes for 0 to p in a narrow shell of thickness x R. The equations of axion electrodynamics imply that the magnetic monopole induces an electric charge Qe on the surface of the sphere Qe ¼ e

Dy : 2p

(361)

Thus, a magnetic monopole acquires an electric charge and becomes a dyon. This is the Witten effect (Witten, 1979). In the particular case of a 2 topological insulator, time-reversal invariance requires that Dy ¼ p, which implies that a monopole with unit magnetic charge has an electric charge e/2. A similar argument implies that an external magnetic field perpendicular to the open surface of the 2 topological insulator (which is essentially an axion domain wall) induces an electric charge polarization on the surface proportional to the total magnetic flux, which is another manifestation of the topological magnetoelectric effect. The reader should readily recognize that the result of Eq. (361) for the electric charge induced by the magnetic monopole is the same as the Goldstone–Wilczek equation for the fractional charge for a one-dimensional soliton of Eq. (215). We will now see that this is not just an analogy. We will follow here the general approach of Callan and Harvey (1985) who extended the earlier work of Goldstone and Wilczek (1981). Let us consider the bulk of a 2 topological insulator and assume that y(x) is slowly varying. We can use the triangle diagram calculation to find that an electromagnetic gauge field Am induces a current hJm(x) given by hJm ðxÞi ¼ −i

f ðxÞ@ n fðxÞ − fðxÞ@ n f ðxÞ lr e e Emnlr F ðxÞ ¼ 2 Emnlr @ n yðxÞF lr ðxÞ 2 16p 8p jfðxÞj2

(362)

This result implies that, in the case of a domain wall in the x1 − x2 plane, a magnetic field perpendicular to the wall induces a current toward the wall and, hence, a charge accumulation on the wall. Where does this charge come from? To understand this problem, we will consider a 2 topological insulator occupying a slab of macroscopic size L between a wall at x3 ¼ 0 and a far way “anti-wall” at x3 ¼ L. In this configuration, a magnetic field normal to the wall(s) induces a transfer of charge from one wall to the other. Similarly, an electric field parallel to the wall induces a current also parallel to the wall and perpendicular to the electric field, that is, a Hall current. In 1+1 dimensions, we saw that soliton configurations acquire a fractional charge associated to states bound to the soliton (which are zero modes when Dy ¼ p). We will see that the surfaces of the 3D 2 topological insulators also have zero modes which are Weyl fermions propagating on the wall. To see how this works, we will consider a 3+1 dimensional Dirac fermion with a Dirac mass that changes sign at x3 ¼ 0. The Lagrangian now ill be
@c
= + gfðxÞcc L ¼ ci

(363)

where f(x) is now a real scalar field that has the asymptotic behaviors f(x3) ¼ f0 f(x3) such that lim x3 !1 f ðx3 Þ ¼ 1. We will assume that f(x3) is a monotonous function of x3, but its actual dependence on x3 is immaterial aside from the requirement that it should change sign at some point which we will take to be x3 ¼ 0. This is a special case of Eq. (350) with f1 ¼ f and f2 ¼ 0. We will now recognize that this is just a Dirac fermion with a position-dependent Dirac mass m(x3) ¼ gf(x3). The one-particle Dirac Hamiltonian for this system is H ¼ −ia  r + mðx3 Þb

(364)

where a and b are the four 4 4 Dirac matrices. By symmetry, this Hamiltonian can be split into two Hamiltonians, H ¼ Hwall + H?, Hwall ¼ −ia1 @ 1 − ia2 @ 2 ,

H? ¼ −ia3 @ 3 + mðx3 Þb:

(365)

Let the spinor c be an eigenstate of the anti-Hermitian Dirac matrix g3 ¼ ba3 with eigenvalues  i, respectively g3 c ¼ ic :

(366)

We seek a spinor solution c of the Dirac equation (Eq. 364) such that H?c ¼ 0  @ 3 c + mðx3 Þc ¼ 0

(367)

ðig0 − ig1 @ 1 − ig2 @ 2 Þc ¼ 0:

(368)

which is also a solution of

In other words, it is a solution of the massless Dirac equation in 2+1 dimensions. The requirement that c is an eigenstate of g3 reduces the number of spinor components from four to two. The full solution has the form

88

Field theoretic aspects of condensed matter physics: An overview c ¼  ðx0 , x1 , x2 ÞF ðx3 Þ,

 @ 3 F  ðx3 Þ ¼ −mðx3 ÞF ðx3 Þ:

For a domain wall with lim x3 !1 mðx3 Þ ¼ +m, the normalizable solution of F+(x3) is  Z 1  dx03 mðx03 Þ F+ ðx3 Þ ¼ Fð0Þ exp − 0

(369)

(370)

where F(0) is a constant. For the antidomain wall, for which lim x3 !1 mðx3 Þ ¼ −m, the normalizable solution is F−(x3). We conclude that there is a 2+1-dimensional massless Dirac theory that describes the quantum states bound to the wall, which propagate along the wall. These states are the generalization of the fractionally charged midgap states of solitons in one dimension discussed in Section “Fractionally charged solitons.” The energy of these states is E(p) ¼ |p|, where p ¼ (p1, p2) (where I set the Fermi velocity to unity). Experimental evidence for 2+1 dimensional massless Dirac fermions on the surface of the 3D 2 topological insulator Bi2Te3 has been found in spin-polarized angle-resolved photoemission studies of the surface states, which showed that they have a linear energy–momentum relation (expected of massless Dirac fermions) as well as the spin-momentum locking characteristic of these spinor states (Hsieh et al., 2009). Having succeeded in showing that the surface of the 2 time-reversal invariant 3D topological insulator has a two-component massless Dirac spinor, we now want to determine its electromagnetic response. In fact we have already discussed this problem in our discussion of the parity anomaly in Section “The parity anomaly” where we showed that the effective action for the electromagnetic field Am of Eq. (342) of a single massless Dirac bi-spinor is a Chern–Simons term with a prefactor, which is 1/2 of the allowed value. In that purely 2+1-dimensional context we saw that time-reversal invariance is actually broken. However, the 3D problem is time-reversal invariant so there must be a contribution that cancels this time-reversal anomaly. The answer is that the requisite cancellation is supplied by the bulk. To see how this can happen, we return to the bulk effective action of the 3D topological insulator of Eq. (359) where we observed that the y term is a total derivative. Suppose that the system has a boundary at x3 ¼ 0 and that the topological insulator exists for x3 > 0 (where the fermion mass is negative, m < 0) and trivial for x3 < 0 (where m > 0). Only the region with m < 0 contributes to the y term. The region of four-dimensional space time occupied by the topological insulator is M and in this region the y term becomes Z Z Z y y y d4 x Emnlr Fmn F rl ¼ 2 d4 x Emnlr @ m An @ l Al ¼ 2 d3 x Emnl Am @ n Al (371) Sy ½A ¼ 2 32p M 8p M 8p @M where @M  S  is the boundary of the region M, S is the surface of the 3D topological insulator, and  is time. Thus we see that the y term of the effective action Sy[A] integrates to the boundary where it has the form of a 2+1-dimensional Chern–Simons term. Since for a time-reversal-invariant 3D topological insulator y ¼ p, we see that in this case the boundary Chern–Simons term becomes Z 1 Sy¼p ½A ¼ d3 x Emnl Am @ n Al (372) 8p @M with a coefficient which is 1/2 of the allowed value. This bulk contribution cancels the parity anomaly of the boundary state, rendering the full system time-reversal invariant. In other words, in this system, time-reversal symmetry is realized by cancellation of the anomaly between the bulk of the topological insulator and the boundary or, equivalently, by an inflow of the parity anomaly between the boundary and the bulk of the system. Another way to phrase this result is the statement that a single Dirac fermion cannot exist on its own in 2+1 dimensions but it can as the boundary state of a 3+1-dimensional system whose anomaly cancels the anomaly of the boundary.

Chern–Simons gauge theory and the fractional quantum Hall effect We will now turn to the problem of the quantum Hall effects. This is a problem that revealed the existence of profound and far-reaching connections between condensed matter physics, quantum field theory, CFT, and topology. In particular, the fractional quantum hall effect is the best studied and best understood topological phase of matter. As such, it has become the conceptual springboard for its manifold generalizations. The integer (IQHE) and fractional (FQHE) quantum Hall effects are fascinating phenomena observed in fluids of electrons in two dimensions in strong perpendicular magnetic fields. The integer quantum Hall effect was discovered by von Klitzing et al. (1980) in transport measurements of the longitudinal and Hall resistivity of the surface states of metal oxide field-effect transistors (MOSFET) in magnetic field of up to 15 Tesla. The effect that von Klitzing discovered (for which he was awarded the 1985 Nobel Prize in Physics) was that in the high-field regime the measured Hall conductivity showed a series of sharply defined plateaus at which took the values sxy ¼ ne2/h, where n is an integer, and the longitudinal conductivity appeared to vanish sxx ! 0 as the temperature was lowered down to T ’ 1.5 K. Remarkably the measured value of the Hall conductivity was obtained with a precision of 10−9. To this date, the measurement of the Hall conductivity in the IQHE yields the most precise definition of the fine structure constant. Subsequent transport experiments in ultra-high-purity GaAs-AlAs heterostructures by Dan Tsui, Horst Störmer, and Art Gossard found that, in addition to the IQHE, two-dimensional electron fluids in high magnetic fields exhibit the fractional quantum Hall 2 effect meaning that there are equally sharply defined plateaus of the Hall conductivity at the values sxy ¼ qp eh , where p and q are

Field theoretic aspects of condensed matter physics: An overview

89

co-prime integers (Tsui et al., 1982). Much as in the IQHE case, in the FQHE, the longitudinal conductivity vanishes at low temperatures. It is important to note that both in the IQHE and in the FQHE, the observed temperature dependence in the highest purity samples of the longitudinal conductivity is activated, sxx  expð−W=TÞ. The observed value of the energy scale W is the experimental estimate of an energy gap in the electron fluid in the quantum Hall states. In addition to MOSFETS and GaAs-AlAs, heterostructures, both the IQHE and the FQHE, have been seen in several other experimental platforms, particularly in graphene and other 2D materials (Du et al., 2009; Zibrov et al., 2017). The explanation of these effects and of a panoply of startling consequences that were uncovered in the course of understanding this phenomenon is the focus of this section.

Landau levels and the integer Hall effect At some level, the integer quantum Hall effect can be explained by the Landau quantization of the energy levels of a free charged moving in two dimensions in a perpendicular magnetic field (Landau, 1930). The Hamiltonian for a nonrelativistic particle of charge − e and mass M in a perpendicular magnetic field B is  2 1 e (373) H¼ −iℏr + i AðxÞ 2M c for a uniform perpendicular magnetic field B ¼ B ebz ¼ r AðxÞ. In the circular gauge, the vector potential is Ai ¼ − 12 BEij xj. We will assume that the 2D plane has linear size L. The total magnetic flux is F ¼ BL2 and we will assume that there is an integer number Nf of magnetic flux quanta piercing the plane, f ¼ Nff0, where f0 ¼ hc/e is the flux quantum. In units such that ℏ ¼ e ¼ c ¼ 1, the flux quantum is f0 ¼ 2p and F ¼ 2pNf. In the presence of a magnetic field, the components of the canonical momentum operator p ¼ −iℏr − ec A do not commute with each other ½pi , pj  − i

eℏ BE : c ij

(374)

This means that translations in two directions do not commute with each other. However, the components of the operator k ¼ p (−B) commute with p (and hence with the one-particle Hamiltonian) but do not commute with each other: the commutator is the same as in Eq. (374). Since [k, H] ¼ 0, they act as symmetry generators of the group of magnetic translations. For arbitrary displacements a and b, the translation operators tðaÞ ¼ expðia  k=ℏÞ (and similarly with b) satisfy tðaÞtðbÞ ¼ expðia b  ez =ℓ20 ÞtðbÞtðaÞ:

(375)

Magnetic translations only commute with each other is the area subtended by a and b contains an integer number of magnetic flux quanta. Given the rotational symmetry of the circular gauge, it is natural to work in complex coordinates z ¼ x1 + ix2. We will also use the notation @ z ¼ (@ 1 − i@ 2)/2 and @ z ¼ ð@ 1 + i@ 2 Þ=2. In this gauge, up to a normalization, the eigenstate wave functions have the form   jzj2 (376) cðz, zÞ ¼ f ðz, zÞ exp − 2 4ℓ0 qffiffiffiffiffiffi ℏc where ℓ0 ¼ ejBj is the magnetic length. In this gauge (and in complex coordinates), the angular momentum operator Lz ¼ −iℏðx1 @ 2 − x2 @ 1 Þ ¼ ℏðz@ z − z@ z Þ. Any analytic function f(z) is an eigenstate with energy E0 ¼ 12 ℏoc . A complete basis of analytic functions are the monomials are fn(z) ¼ zn and have energy E0 and angular momentum Lz ¼ nℏ . This is the lowest Landau level whose wave functions cn ðzÞ ¼ zn expð−jzj2 =4ℓ20 Þ. On the other hand, an antianalytic function f N ¼ zN is an eigenstate of energy EN ¼ ℏoc N + 12 , where oc ¼ ejBj Mc is the cyclotron frequency, and angular momentum Lz ¼ −Nℏ. States with angular momentum nℏ have the same energy, and the degeneracy is equal to the number of flux quanta Nf. For the most part, we will be interested in the states in the lowest Landau level. In the absence of disorder, the Landau levels have an extensive degeneracy equal to the number of flux quantum Nf. If we consider a system of N electrons in a Landau level the natural measure of density is not the areal density r ¼ NL2e but the fraction n ¼ NNfe the states in the Landau level which are occupied by electrons. The many-body state in which all Nf states of the lowest Landau level has filling fraction n ¼ 1. The wave function for this state is the Slater determinant of the Landau states in the m ¼ 0 level. After some simple algebra, the wave function of this state is found to be ! Ne Y X jzi j2 Cn¼1 ðz1 , . . . , zNe Þ ¼ ðzi − zj Þ exp − : (377) 4ℓ20 i 0, for R2 big enough, J that is, there is spontaneous magnetization and symmetry breaking if R2 or J are big enough.

Phase transitions, spontaneous symmetry breaking, and Goldstone’s theorem

169

For the Ising model (N ¼ 1) and the classical XY model (N ¼ 2), these results on the existence of phase transitions accompanied by symmetry breaking can be extended to a large class of not necessarily translation-invariant and not necessarily reflection-positive exchange couplings by using what one calls “correlation inequalities.” The relevant inequalities can be found in Messager et al. (1978). One expects that if Jk > J?, then the spontaneous magnetization is parallel to the N-axis (with N ¼ 3 for the Hamiltonian in (46)). In this case, the discrete symmetry SN ! −SN is broken; for results in this direction, see Kennedy (1985) and references given ! there. But if J? > Jk, then the spontaneous magnetization is expected to lie in the plane fS j SN ¼ 0g, and the continuous symmetry O(N − 1) of the Hamiltonian is broken, which, for N  3, is accompanied by the emergence of N − 2 Goldstone bosons. If Jk ¼ J? J, then the model exhibits an O(N)-symmetry, and the spontaneous magnetization can have an arbitrary direction in N , which can ! P ! ! be chosen by introducing a small symmetry breaking term, −h  x S x , in the Hamiltonian and letting jh j tend to 0 eventually. All these predictions can be shown to hold rigorously for N ¼ 1, 2, 3, under suitable assumptions on the parameters. It may be appropriate to sketch a heuristic idea that originally inspired a stronger (exponential) version of the Infrared Bounds in (50), and which could then be extended to quantum lattice systems. We consider a canonical quantum field theory of a real scalar field ’ on d space-time dimensions with a Hamiltonian, denoted by H, whose energy spectrum is bounded from below by 0. Such a theory (if it existed) would have the following properties: h i ! ! ! _ x , tÞ ¼ i H, ’ð x , tÞ ¼: pð x , tÞ, with ’ð h i (55) ! ! ! ! ’ð x , tÞ, pð y , tÞ ¼ i dð x − y Þ, !

where t denotes time and x is a point in space. Let f and g be real-valued test functions on d−1 , and let Z ! ! ! ’ðf , tÞ :¼ d x ’ð x , tÞ f ð x Þ, d−1

Z pðg, tÞ :¼

!

d−1



½’ðf , tÞ, pðg, tÞ ¼ i

!

!

d x pð x , tÞ gð x Þ,

Z

!

d−1

!

with

!

d x f ð x Þ gð x Þ  i hf , gi:

(56) (57) (58)

Since ’ is a real field, we may expect that, at all times t, ’(f, t) is a self-adjoint operator on the Hilbert space, H, of the theory. From our assumption that H  0 it follows that U H U  0, for an arbitrary unitary operator U on H, and we then find that 1 0  ei’ðf ,tÞ H e −i’ðf ,tÞ ¼ H pðf , tÞ+ k f k22 1, 2

(59)

where k f k22 :¼ hf , f i. Using a Feynman–Kac formula, one can translate this bound to one that holds for a corresponding EFT at imaginary time. We denote the Euclidean field corresponding to the quantum field ’ by f, as before, and expectations with respect to the functional measure describing the EFT corresponding to the QFT are denoted by hðÞi. For a real-valued test function h on d, R we define fðhÞ :¼ dd x fðxÞ hðxÞ. Then (59) implies that D ∂f E 2 1 e ∂t ðhÞ  e2khk2 : By Euclidean invariance, it then follows that 1

2

hexp½rfðhÞi  e2khk2 ,

(60)

where h ¼ ðh1 , . . . , hd Þ is a d-tuple of test functions. This bound readily implies that, in momentum space and for k 6¼ 0, D E 1 ^ ^ fðkÞ fð−kÞ  2, jkj which is an Infrared Bound similar to (50) in the special case where N ¼ 1 and where Jk is a nearest-neighbor coupling of strength 1. ! Obviously, one can extend these arguments to canonical theories of N-component real fields, ’ ¼ ð’1 , . . . , ’N Þ, N  2. (Of course, ^ ^ the bound on hfðkÞ fð−kÞi also follows from the Källen–Lehmann representation (15).) For classical lattice spin systems with nearest-neighbor exchange couplings, bounds of the form of (60) have first been proven in Fröhlich et al. (1976). They were subsequently extended, mutatis mutandis, to classical models with general reflection-positive exchange coupling, including long-range couplings, in Fröhlich et al. (1978) and to quantum spin systems in Dyson et al. (1978). A much more ambitious, more robust analysis of classical lattice spin systems with phase transitions accompanied by the spontaneous breaking of continuous O(N)-symmetries has been undertaken in Balaban (1998) and Balaban and O’Carroll (1999), using mathematically controlled renormalization group methods. (See also Bauerschmidt et al. (2021) for a somewhat related, but simpler analysis of a different model.) A new, quite surprising and original approach to proving the existence of phase transitions accompanied by spontaneous symmetry breaking has been proposed in Garban and Spencer (2022). The advantage of their method is that it also applies to

170

Phase transitions, spontaneous symmetry breaking, and Goldstone’s theorem

classical models that are neither translation-invariant nor do they satisfy Reflection Positivity. The price to pay is that the models describe disordered systems and have random exchange couplings. (For the Ising- and the classical XY model, one can, however, use the correlation inequalities in Messager et al. (1978) to extend their results to the usual choice of exchange couplings; see Garban and Spencer, 2022.) It turns out that Infrared Bounds analogous to those in (50)—but for so-called Duhamel two-point correlations—can also be proven for quantum lattice spin systems satisfying Reflection Positivity, as discovered in Dyson et al. (1978). It turns out that, among the quantum Heisenberg models, only the antiferromagnets satisfy Reflection Positivity, but not the ferromagnets. For antiferromagnets, the Infrared Bounds can be used to establish phase transitions accompanied by spontaneous symmetry breaking and the emergence of Goldstone modes (magnons); see Dyson et al. (1978). More general results on phase transitions and spontaneous symmetry breaking for a large family of quantum lattice systems satisfying Reflection Positivity have been established in Fröhlich et al. (1978) and Albert et al. (2006); see also Fröhlich (1978, 2011) for reviews. I cannot go into any details about these results. For the isotropic quantum Heisenberg ferromagnet on the complete graph, the existence of a phase transition accompanied by the spontaneous breaking of the SU(2)-symmetry has been proven in Björnberg et al. (2020), using exact calculations that do not rely on Reflection Positivity; but rigorous results concerning phase transitions in isotropic quantum Heisenberg ferromagnets with continuous internal symmetries on finite-dimensional lattices are not known so far. I conclude this review with a few comments on the Ising model and the classical XY model. By Kramers–Wannier duality, these models are equivalent to gases of “topological defects.” For the Ising model, these defects are domain walls (also called Peierls contours) separating domains with opposite spin orientation. For the XY model the defects are (codimension-2) vortices. It turns out that, in dimension d  2 and at low temperatures, these defect gases are dilute. This feature enables one to analyze them by using mathematically controlled versions of energy-entropy arguments underpinned by “multiscale analysis.” For the Ising model, this strategy is implemented in what is commonly called a “Peierls argument,” in honor of its inventor. For the XY model, the analysis is considerably more subtle, due to the presence of massless Goldstone modes at low temperatures. In particular, the analysis of the low-temperature phase of the two-dimensional classical XY model, viewed as a dilute gas of point vortices with long-range (Coulomb) interactions, is really rather involved. An interesting, fairly precise upper bound on spin–spin correlations in the twodimensional XY model at low temperatures has been proven in McBryan and Spencer (1977). Techniques (involving complex translations in the space of spin configurations) developed in this paper have provided some of the tools used in Fröhlich and Spencer (1981), where, using rather heavy multiscale analysis, the existence of the Kosterlitz–Thouless transition for the twodimensional classical XY model and other related models has been established rigorously; see also Falco (2012). New, simpler proofs have recently appeared in Lammers (2022, 2023) (and references given there). One of the key results established in McBryan and Spencer (1977) and Fröhlich and Spencer (1981) is the following bound on the spin–spin correlation. For ferromagnetic exchange couplings J > Jc, where Jc is the coupling at which the Kosterlitz–Thouless transition occurs, D! ! E   ’ O ½jxj+1−f1=2peðJÞg , with S0  Sx J,h¼0 (61) 0 < eðJÞ ! J, as J ! 1: !

Here, S x ¼ ðcos yx , sin yx Þ, yx 2 ½0, 2pÞ, is the planar spin variable of the classical XY model at site x 2 2, and the external magnetic field h vanishes, h ¼ 0. The Lee–Yang theorem, which holds for the classical XY model, implies that the susceptibility X D! ! E S0  Sx wðJ, hÞ ¼ x22

J,h

is finite if h > 0. By (61), w(J, h) diverges, as h ↘ 0, provided J is large enough. The magnetization, M(J, h), is nonzero, for h > 0, but approaches 0, as h ↘ 0, by the Mermin–Wagner–Hohenberg theorem (Hohenberg, 1967; Mermin and Wagner, 1966; Mermin, 1967). Using that wðJ, hÞ ¼

MðJ, hÞ , h

see (20), we conclude that, for J large enough, MðJ, hÞ ! 0,

MðJ, hÞ diverges; as h h −1 wðJ, hÞ ¼ oðh Þ, as h ↘ 0, but

↘ 0,

and (62)

while, for J < Jc, where the spin–spin correlation decays exponentially, as |x| ! 1, and the susceptibility remains finite, as h ↘ 0, we have that M(J, h) ∝ h, as h ↘ 0. For the three-dimensional classical XY model, the analysis is simpler, because, in d ¼ 3, the vortices are one-dimensional objects and form loops. At high temperatures, these vortex loops condense; but, at low temperatures, they form a dilute gas. They have longrange interactions mediated by massless Goldstone modes. Somewhat surprisingly, these long-range interactions do not cause any major complications in the mathematical analysis of the low-temperature phase of the model. In fact, the vortex-loop

Phase transitions, spontaneous symmetry breaking, and Goldstone’s theorem

171

representation has been used to rigorously prove that, at low enough temperatures, the three-dimensional classical XY model has spontaneous magnetization and that its connected spin–spin correlations decay very slowly, namely like r−1, where r is the distance between the spin insertion points. It is easy to show that, at high temperatures, the spin–spin correlations decay exponentially fast. These findings establish again the existence of a phase transition accompanied by the spontaneous breaking of a U(1)-symmetry and the emergence of Goldstone bosons; see Fröhlich and Spencer (1983). Translation invariance of the Hamiltonian is not essential in this method. Ideas somewhat related to those in Fröhlich and Spencer (1983) have earlier been used in Guth (1980) in an analysis of the deconfining transition in the four-dimensional U(1)-lattice gauge theory; see also Fröhlich and Spencer (1982).

Conclusion I hope this short review convinces the reader that many nontrivial facts about phase transitions accompanied by spontaneous symmetry breaking and about Goldstone bosons of considerable interest in condensed matter and particle physics are now known in the form of mathematically precise results. Work on this subject has given rise to many important insights into properties of physical systems and to the development of numerous interesting techniques of analysis that can also be used in other contexts. Ideas from statistical and condensed matter physics have been fruitfully imported into particle physics and quantum field theory, and conversely. Without this cross-fertilization, many significant results would presumably not have been found, yet. I expect that I have quoted sufficiently much of the relevant literature to enable the reader to access without effort the nitty-gritty of the topics discussed in this review. (I offer my apologies to all colleagues whose work I should have quoted, but did not.) Despite all the progress that has been made during more than half a century, many important questions remain open, at least if one insists on adhering to the standards of mathematical physics. Among the most important such questions are the following ones. 1. Find a continuum model with a translation-invariant Hamiltonian for which one can prove that the ground-state and the extremal low-temperature equilibrium states break translation invariance spontaneously and exhibit crystalline ordering. 2. Show that translation-invariant interacting Bose gases with physically realistic repulsive two-body interactions can exhibit Bose–Einstein condensation as the temperature is lowered at a fixed, positive density. 3. Show that the isotropic quantum Heisenberg ferromagnet in d  3 dimensions undergoes a phase transition accompanied by the spontaneous breaking of its SU(2)-symmetry. 4. Show that the quantum-mechanical XY model and the l|f|4-theory of a complex scalar field in two dimensions undergo a Kosterlitz–Thouless transition. 5. Prove rigorously that the classical Heisenberg model and the classical N-vector models, with N > 3, in two dimensions have a strictly finite correlation length at all positive temperatures. 6. Extend the results in Aizenman et al. (2015) and Aizenman and Duminil-Copin (2021) concerning the magnetization in the three-dimensional Ising model, and the Gaussian nature of the scaling limit of the four-dimensional Ising- and lf4-model (N ¼ 1), respectively, to two-component models, in particular to the classical XY model.

Acknowledgments This review could not have been written without my collaborations, mostly many years ago, with E. H. Lieb, B. Simon, and T. Spencer. They have taught me very many important things about statistical mechanics and quantum field theory. My collaboration and my countless discussions, also in more recent times, with T. Spencer count among the highlights in my scientific life, and I am deeply grateful to him for his generosity and his friendship. I have profited from interactions and collaboration with numerous other colleagues, postdocs and PhD students, too many to list them all. I thank T. Chakraborty for encouraging me to write this review and for his patience.

AIPP-Data availability statement Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References Aizenman M (1982) Geometric analysis of ’4 fields and Ising models. I, II. Communications in Mathematical Physics 86(1): 1–48. Aizenman M and Duminil-Copin H (2021) Marginal triviality of the scaling limits of critical 4D Ising and f44 models. Annals of Mathematics 194(1): 163–235. Aizenman M, Duminil-Copin H, and Sidoravicius V (2015) Random currents and continuity of Ising model’s spontaneous magnetization. Communications in Mathematical Physics 334(2): 719–742. Albert C, Ferrari L, Fröhlich J, and Schlein B (2006) Magnetism and the Weiss exchange field—A theoretical analysis motivated by recent experiments. Journal of Statistical Physics 125(1): 77–124.

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Phase transitions, spontaneous symmetry breaking, and Goldstone’s theorem

Anderson PW (1962) Plasmons, gauge invariance, and mass. Physical Review 130(1): 439–442. Balaban T (1998) The large field renormalization operation for classical N-vector models. Communications in Mathematical Physics 198(3): 493–534. Balaban T and O’Carroll M (1999) Low temperature properties for correlation functions in classical n-vector spin models. Communications in Mathematical Physics 199(3): 493–520. Bauerschmidt R, Crawford N, and Helmuth T (2021) Percolation transition for random forests in d  3. arXiv:2107.01878v2 [math.PR]. Björnberg JE, Fröhlich J, and Ueltschi D (2020) Quantum spins and random loops on the complete graph. Communications in Mathematical Physics 375: 1629–1663. Bratteli O and Robinson DW (1979) In: Balian R, et al. (eds.) Operator Algebras and Quantum Statistical Mechanics. Texts and Monographs in Physics, vol. 1. New York, Berlin, Heidelberg: Springer-Verlag. Bratteli O and Robinson DW (1981) Operator Algebras and Quantum Statistical Mechanics. Texts and Monographs in Physics, vol. 2. New York, Berlin, Heidelberg: Springer-Verlag. Brydges D, Fröhlich J, and Sokal A (1983) A new proof of the existence and non-triviality of the continuum f42 and f43 quantum field theories. Communications in Mathematical Physics 91: 141–186. Buchholz D, Doplicher S, and Longo R (1986) On Noether’s theorem in quantum field theory. Annals of Physics 170: 1–17. Coleman S (1973) There are no Goldstone Bosons in two dimensions. Communications in Mathematical Physics 31: 259–264. Domb C and Green MS (1972–1976) Phase Transition and Critical Phenomena, vols. 1–6. San Diego, New York: Harcourt, Brace & World. Domb C and Lebowitz JL (1983–2001) Phase Transition and Critical Phenomena, vols. 7–20. San Diego, New York: Harcourt, Brace & World. Dunlop F and Newman C (1975) Multicomponent field theories and classical rotators. Communications in Mathematical Physics 44: 223–235. Dyson F, Lieb EH, and Simon B (1978) Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. Journal of Statistical Physics 18: 335–383. Englert F and Brout R (1964) Broken symmetry and the mass of gauge vector mesons. Physical Review Letters 13(9): 321–323. Falco P-L (2012) Kosterlitz-Thouless transition line for the two dimensional Coulomb Gas. Communications in Mathematical Physics 312: 559–609. Feldman J and Osterwalder K (1975) The Wightman axioms and the mass gap for weakly coupled f43 quantum field theories. In: Araki H (ed.) Mathematical Problems in Theoretical Physics. Berlin, Heidelberg, New York: Springer-Verlag. Feldman J and Osterwalder K (1977) The Wightman axioms and the mass gap for weakly coupled f43 quantum field theories. Annals of Physics (NY) 97: 80–135. Fröhlich J (1978) The pure phases (harmonic functions) of generalized processes—Or: Mathematical physics of phase transitions and symmetry breaking. Bulletin of the AMS 84(2): 165–193. Fröhlich J (1982) On the triviality of l’4d theories and the approach to the critical point in d > 4 dimensions. Nuclear Physics B 200(2): 281–296. Fröhlich J (2011) Phase Transitions and Continuous Symmetry Breaking. Vienna: Erwin-Schrödinger Institute (available from the author on request). Fröhlich J and Rodriguez P-F (2012) Some applications of the Lee-Yang theorem. Journal of Mathematical Physics 53. 095218-1–095218-15. Fröhlich J and Rodriguez P-F (2017) On cluster properties of classical ferromagnets in an external magnetic field. Journal of Statistical Physics 166: 828–840. Fröhlich J and Spencer T (1981) The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Communications in Mathematical Physics 81: 527–602. Fröhlich J and Spencer T (1982) Massless phases and symmetry restoration in Abelian Gauge theories and spin system. Communications in Mathematical Physics 83: 411–454. Fröhlich J and Spencer T (1983) The Berezinskii-Kosterlitz-Thouless transition. In: Fröhlich J (ed.) Scaling and Self- Similarity in Physics. Progress in Physics, vol. 7, pp. 29–138. Basel, Boston: Birkhäuser Verlag. Fröhlich J, Simon B, and Spencer T (1976) Infrared bounds, phase transitions and continuous symmetry breaking. Communications in Mathematical Physics 50: 79–95. Fröhlich J, Israel R, Lieb EH, and Simon B (1978) Phase transitions and reflection positivity. I. General theory and long range lattice models. Communications in Mathematical Physics 62: 1–34. Fröhlich J, Morchio G, and Strocchi F (1981) Higgs phenomenon without symmetry breaking order parameter. Nuclear Physics B 190(3): 553–582. Fröhlich J, Knowles A, Schlein B, and Sohinger V (2020) A path-integral analysis of interacting Bose gases and loop gases. Journal of Statistical Physics 180: 810–831. Fröhlich J, Knowles A, Schlein B, and Sohinger V (2023) The Euclidean f42 theory as a limit of an interacting Bose gas. arXiv:2201.07632v2 [math-ph]. Garban C and Spencer T (2022) Continuous symmetry breaking along the Nishimori line. J. Math. Phys. 63(9): 093302. Ginibre J (1969) Existence of phase transitions for quantum lattice systems. Communications in Mathematical Physics 14: 205–234. Glaser V (1974) On the equivalence of the Euclidean and Wightman formulation of field theory. Communications in Mathematical Physics 37(4): 257–272. Glimm J and Jaffe A (1973) Positivity of the f43 Hamiltonian. Fortschritte der Physik 21: 327–376. Glimm J and Jaffe A (1974) ’42 quantum field model in the single-phase region: Differentiability of the mass and bounds on critical exponents. Physical Review D 10: 536–539. Glimm J and Jaffe A (1981) Quantum Physics—A Functional Integral Point of View. New York, Berlin, Heidelberg: Springer-Verlag. Glimm J, Jaffe A, and Spencer TC (1975) Phase transition for f42 quantum fields. Communications in Mathematical Physics 45: 203–216. Goldstone J (1961) Field Theories with “Superconductor” Solutions. Il Nuov Cimento XIX: 154–164. Goldstone J, Salam A, and Weinberg S (1962) Broken symmetries. Physical Review 127: 965–970. Guralnik GS, Hagen CR, and Kibble TWB (1964) Global conservation laws and massless particles. Physical Review Letters 13(20): 585–587. Guth A (1980) Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory. Physical Review D 21: 2291–2307. Heisenberg W (1928) Zur Theorie des Ferromagnetismus. Zeitschrift für Physik 49: 619–636. Higgs PW (1964) Broken symmetries and the masses of gauge bosons. Physical Review Letters 13(16): 508–509. Hohenberg P (1967) Existence of long-range order in one and two dimensions. Physical Review 158: 383–386. Itzykson C and Drouffe J-M (1989) Statistical Field Theory. Cambridge Monographs on Mathematical Physics. , vol. I & II. Cambridge, New York: Cambridge University Press. Jost R (1965) The General Theory of Quantized Fields. Providence RI: AMS publ. Kennedy T (1985) Long range order in the anisotropic quantum ferromagnetic Heisenberg model. Communications in Mathematical Physics 100: 447–462. Kosterlitz M and Thouless DJ (1973) Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C 6: 1181–1203. Lammers P (2022) Height function delocalisation on cubic planar graphs. Probability Theory and Related Fields 182(1): 531–550. Lammers P (2023) Bijecting the BKT transition. arXiv:2301.06905v1 [math.PR]. Lebowitz JL (1974) GHS and other inequalities. Communications in Mathematical Physics 35: 87–92. Lee TD and Yang CN (1952) Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model. Physical Review 87: 410–419. Lee BW and Zinn-Justin J (1972a) Spontaneously broken gauge symmetries. I. Preliminaries. Physical Review D 5: 3121–3137. Lee BW and Zinn-Justin J (1972b) Spontaneously broken gauge symmetries. II. Perturbation theory and renormalization. Physical Review D 5: 3137–3155. Lee BW and Zinn-Justin J (1972c) Spontaneously broken gauge symmetries. III. Equivalence. Physical Review D 5: 3155–3160. Leutwyler H (1994) Non-relativistic effective lagrangians. Physical Review D 49(6): 3033–3043. Leutwyler H (1997) Phonons as Goldstone bosons. Helvetica Physica Acta 70: 275–286. McBryan O and Rosen J (1976) Existence of the critical point in f4 field theory. Communications in Mathematical Physics 51: 91–105. McBryan OA and Spencer TC (1977) On the decay of correlations in SO(n)-symmetric ferromagnets. Communications in Mathematical Physics 53: 299–302. Mermin ND (1967) Absence of ordering in certain classical systems. Journal of Mathematical Physics 6: 1061–1064. Mermin ND and Wagner H (1966) Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Physical Review Letters 17: 1133–1136. Messager A, Miracle-Sole S, and Pfister C (1978) Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model. Communications in Mathematical Physics 58(1): 19–29. Nambu Y and Jona-Lasinio G (1961a) Dynamical model of elementary particles based on an analogy with superconductivity. I. Physical Review 122: 345–358.

Phase transitions, spontaneous symmetry breaking, and Goldstone’s theorem

173

Nambu Y and Jona-Lasinio G (1961b) Dynamical model of elementary particles based on an analogy with superconductivity. II. Physical Review 124: 246–254. Osterwalder K and Schrader R (1975) Axioms for Euclidean Green’s functions II. Communications in Mathematical Physics 42(3): 281–305 (this paper is a sequel to a previous paper by the same authors from the year 1973 that contained a gap in the arguments). Park YM (1977) Convergence of lattice approximation and infinite volume limit in the (lf4 − f2 − f)3 field theory. Journal of Mathematical Physics 18: 354–366. Polyakov AM (1975) Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. Physics Letters 59B: 79–81. Renker, H., 1913. Magnetische Untersuchungen an Legierungen der Eisengruppe oberhalb des Curie-Punktes. 10.3929/ethz-a-000092025. (Promotionsarbeit ETH Zurich (Referent: Herr Prof. Dr. P. Weiss, Korreferent: Herr Prof. Dr. A. Einstein)-and references given there). Ruelle D (1969) Statistical Mechanics: Rigorous Results. Reading MA: W. A. Benjamin, Inc. Simon B (1974) The P(f)2 Euclidean (Quantum) Field Theory. Princeton Series in Physics. Princeton NJ: Princeton University Press. Simon B (1993) The Statistical Mechanics of Lattice Gases, vol. 1. Princeton NJ: Princeton University Press (vol. 2 in preparation). Simon B and Griffiths R (1973) The f42 -field theory as a classical Ising model. Communications in Mathematical Physics 33: 145–164. Spencer T and Fröhlich J (2013–2014) Discussions and Unpublished Notes on the Classical XY Model. Princeton: IAS. Stückelberg ECG (1938a) Die Wechselwirkungs Kräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte (I). Helvetica Physica Acta 11: 225–244. Stückelberg ECG (1938b) Die Wechselwirkungs Kräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte (II). Helvetica Physica Acta 11: 299–312. Stückelberg ECG (1938c) Die Wechselwirkungs Kräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte (III). Helvetica Physica Acta 11: 312–328. Symanzik K (1967) Euclidean proof of the Goldstone theorem. Communications in Mathematical Physics 6: 228–232. ’tHooft G and Veltman M (1972) Regularization and renormalization of gauge fields. Nuclear Physics B 44(1): 189–219. Weinberg S (1995–2000) The Quantum Theory of Fields. Volumes I–III. Cambridge and New York: Cambridge University Press. Weiss P (1907) L’hypothèse du champ moléculaire et la propriété ferromagnétique. Journal de Physique 6: 661–690. Wreszinski W (1976) Goldstone’s theorem for quantum spin systems of finite range. Journal of Mathematical Physics 17: 109–111.

Higgs and Nambu–Goldstone modes in condensed matter physics Naoto Tsujia,b, Ippei Danshitac, and Shunji Tsuchiyad, aDepartment of Physics, University of Tokyo, Hongo, Tokyo, Japan; bRIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, Japan; cDepartment of Physics, Kindai University, Higashi-Osaka, Osaka, Japan; d Department of Physics, Chuo University, Kasuga, Tokyo, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Higgs mode in superconductors Higgs modes in atomic Fermi gases Higgs and Nambu–Goldstone modes in other systems Conclusion References

174 176 179 181 183 184

Abstract Collective dynamics of many particle systems is tightly linked to their underlying symmetry and phase transitions. Higgs and Nambu–Goldstone (NG) modes are, respectively, collective amplitude and phase modes of the order parameter that are widely observed in various physical systems at different energy scales, ranging from magnets, superfluids, and superconductors to our universe. The Higgs mode is a massive excitation, which is a condensed-matter analog of Higgs particle in highenergy physics, while the NG mode is a massless excitation that appears when a continuous symmetry is spontaneously broken. They provide important information on the fundamental aspects of many particle systems, such as symmetry, phases, dynamics, and response to external fields. In this article, we review the physics of Higgs and NG modes in condensed matter physics. Especially, we focus on the development on the study of collective modes in superconductors and cold-atom systems.

Key points

• • •

Introduces collective modes of an order parameter in condensed matter physics, including Higgs and Nambu–Goldstone modes. Describes basic theories for Higgs and Nambu–Goldstone modes in superconductors and cold-atom systems (including fermionic and bosonic superfluids). Reviews experimental observations of Higgs and Nambu–Goldstone modes in condensed matter systems.

Introduction A phenomenon of phase transition is widely observed in various physical systems at different energy scales, ranging from condensed-matter to high-energy physics. For example, when temperature decreases, a paramagnet turns into a ferromagnet, and a metal turns into a superconductor. Those phenomena can be universally understood with the concept of spontaneous symmetry breaking: Physical laws are invariant under a certain symmetry transformation while a physically realized state at low energy does not necessarily exhibit the same symmetry. In the case of magnets having spin rotation symmetry, individual spins are randomly oriented at high temperature, whereas at low temperature, they are aligned to one particular direction, which is spontaneously chosen. In the case of superconductors, a complex phase of a macroscopic wavefunction of electrons is aligned spontaneously. In the conventional Landau paradigm, phase transition is characterized by an order parameter, which grows in ordered phases, such as magnetization or condensate (superfluid) density. When a continuous symmetry is spontaneously broken, there typically appear two types of collective motions of the order parameter: One is a phase fluctuation of the order parameter and the other is an amplitude fluctuation of the order parameter (Fig. 1). The former is often referred to as Nambu–Goldstone (NG) mode while the latter is called Higgs mode, which recently attracts interests in condensed matter physics. As the name suggests, the Higgs mode is a condensed-matter analog of a Higgs particle in high-energy physics, which emerges as a result of quantization of the amplitude fluctuation of a complex scalar field. The Higgs particle plays an important role in generating a mass of gauge fields due to the Higgs mechanism, making a force mediated by gauge bosons short-ranged. Historically, on the other hand, the physics of Higgs particle and Higgs mechanism arose from the study of superconductivity. The macroscopic phenomenological theory of superconductivity had been developed by Ginzburg and Landau (1950), which can be viewed as a nonrelativistic version of the complex scalar field theory adopted by Higgs and others in the study of the Higgs mechanism. The microscopic theory of superconductivity had been presented by Bardeen et al. (1957), known as the BCS theory,

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Fig. 1 Free-energy surface in the plane of a complex order parameter c. The red (blue) arrows represent Higgs (Nambu–Goldstone) mode, corresponding to amplitude (phase) fluctuation of the order parameter.

which had a broad impact on various branches of physics. In 1958, just one year after the BCS theory, Anderson and Bogoliubov had already discussed collective modes in superconductors (Bogoliubov et al., 1958; Anderson, 1958a, b). Especially, Anderson mentioned the existence of what is known as the Higgs mode in superconductors, whose energy was found to be twice as large as the superconducting gap. Physically, the Higgs mode in superconductors corresponds to a collective oscillation of the superfluid density (density of condensed Cooper pairs), which should be distinguished from an individual excitation of Cooper-pair breaking. Stimulated by the development of the BCS theory, Nambu had introduced the concept of spontaneous symmetry breaking in particle physics. Based on an analogy with the BCS theory (Nambu, 1960), a possibility of spontaneous symmetry breaking is postulated for a vacuum ground state in a relativistic field theory (Nambu and Jona-Lasinio, 1961; Goldstone, 1961). It was found that a massless boson appears as a phase fluctuation of the order parameter in a symmetry broken phase, in much the same way as a spin wave appears as a collective excitation in magnets. In parallel, Goldstone and others proved a theorem (Goldstone et al., 1962) that massless particles must appear when a continuous symmetry is spontaneously broken in a relativistic field theory, and the number of those correspond to the number of degrees of broken symmetries. Later, the theorem has been extended to nonrelativistic cases (Watanabe and Murayama, 2012; Hidaka, 2013), where the counting rule of the NG modes has been established. In the application of symmetry breaking in particle physics, one has to avoid Goldstone’s theorem since no such massless particle had been observed in our universe. In 1964, Englert, Brout, Higgs, and others had noticed that there is a way to go around Goldstone’s theorem when the system is coupled to gauge fields (Englert and Brout, 1964; Higgs, 1964; Guralnik et al., 1964): The phase mode is absorbed into the longitudinal component of gauge fields, which results in generating a mass of gauge bosons (Higgs mechanism). What remains after this is the amplitude mode of the scalar field, which corresponds to a massive scalar particle (Higgs boson). In 2012, the Higgs particle has been finally discovered in the LHC experiment at CERN (ATLAS Collaboration, 2012; CMS Collaboration, 2012), after almost 50 years since the theoretical prediction. It should be emphasized that the Higgs mechanism had been discussed earlier by Anderson in the context of superconductivity in condensed matter physics (Anderson, 1958a, b, 1963): In superconductors, an interior magnetic field is completely excluded due to the Meissner—Ochsenfeld effect. This phenomenon can be understood by the fact that electromagnetic fields acquire a mass and cannot propagate freely inside superconductors (Anderson–Higgs mechanism). Anderson’s argument was based on a nonrelativistic theoretical formalism, but essentially the same thing happens in relativistic field theories. While the physics of Higgs phenomena has originated from the study of superconductivity, the Higgs mode in superconductors (Pekker and Varma, 2015; Shimano and Tsuji, 2020), which manifests the existence of an effective scalar potential with a Mexicanhat shape (Fig. 1) that causes the Higgs mechanism, has not been observed for a long time. An exception at an early stage was the Raman scattering experiment reported in 1980 for 2H-NbSe2 (Sooryakumar and Klein, 1980), which is rather a special material in the sense that superconductivity coexists with a charge-density-wave (CDW) order. Only in this particular regime, a resonance peak corresponding to the amplitude mode has been observed in Raman spectra. In 2013, a coherent long-lived oscillation induced by a mono-cycle terahertz laser pulse was observed for a pure superconductor NbN (“pure” means the one without coexistence of any other long-range orders) (Matsunaga et al., 2013). The oscillation frequency agrees with the superconducting gap energy, implying that the Higgs mode is excited in the terahertz laser experiment. In the subsequent experiment, a multicycle terahertz pulse has been applied to NbN, showing a coherent oscillation with the frequency twice as large as the pump laser frequency (Matsunaga et al., 2014). This is consistent with the fact that the Higgs mode can be generated by a two-photon absorption process through the nonlinear light-Higgs coupling (Tsuji and Aoki, 2015). At the same time, the resonant enhancement of third harmonic generation has been observed in NbN when the superconducting gap coincides with twice the pump frequency (Fig. 2) (Matsunaga et al., 2014). Later, it turns out that impurities play an essential role in the resonant enhancement of third harmonic generation mediated by the Higgs mode (Jujo, 2018; Murotani and Shimano, 2019; Silaev, 2019; Tsuji and Nomura, 2020; Seibold et al., 2021).

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THG Intensity (arb. units)

2'(T) (THz)

176

1.5

2Z=1.6 THz

1.0

1.2 THz

0.5 0.0

0.6 THz

(A)

0.8 THz 0.6 THz 0.3 THz

Experiment

1

Tc=15 K

0

(B)

0.4

0.6

0.8

1.0

1.2

T/Tc Fig. 2 Temperature dependence of (A) the superconducting gap compared with twice the pump frequencies and (B) the intensity of third harmonic generation in NbN superconductors. Adapted from Matsunaga R, Tsuji N, Fujita H, Sugioka A, Makise K, Uzawa Y, Terai H, Wang Z, Aoki H, and Shimano R (2014) Light-induced collective pseudospin precession resonating with Higgs mode in a superconductor. Science 345(6201): 1145–1149. ISSN 0036-8075

Table 1

Examples of phase and amplitude modes in various physical systems.

System

Nambu–Goldstone (phase) mode

Higgs (amplitude) mode

Broken symmetry

Magnet (Incommensurate) Charge density wave Superfluid 4He Weakly interacting superfluid Bose gas Strongly interacting superfluid Bose gas in an optical lattice at commensurate filling Superfluid Fermi gas Superfluid 3He (B phase) Superconductor Atomic nuclei Standard model in high-energy physics

Magnon Phason Phase mode Phase mode Phase mode

Amplitude mode Amplitudon – – Amplitude mode

Spin rotation Space translation Phase rotation Phase rotation Phase rotation

Phase mode Phase mode – Pion –

Amplitude mode Squashing mode Amplitude mode s Particle Higgs particle

Phase rotation SO(3)SO(3)U(1) Phase rotation Chiral symmetry SU(2)U(1)

In the case of superconductors and the standard model in high-energy physics, the system is coupled to gauge fields, and the phase mode disappears due to the Anderson–Higgs mechanism.

If one is not limited to charged systems in which the order parameter is coupled with gauge fields, the phase and amplitude modes of order parameters have been widely observed in symmetry broken phases (see Table 1). For example, magnons, or spin waves, correspond to phase modes in magnetically ordered systems. Liquid helium and cold atomic gases are other examples of physical systems in which collective modes of bosonic and fermionic superfluids are well studied. These collective modes provide important information on the underlying system, such as symmetries, phases, dynamics, and response to external fields. Throughout the article, we put ℏ ¼ 1.

Higgs mode in superconductors The emergence of collective modes in symmetry broken phases can be understood with a simple macroscopic phenomenological model. For example, the low-energy effective theory of superconductors corresponds to the Ginzburg–Landau (GL) theory, whose Lagrangian density is given, as a functional of a complex scalar field c(r) (a “macroscopic wavefunction” of electrons), by
 b 1 2 ⁎ + c1 cðrÞ{ ði∂t − e⁎ fÞcðrÞ + c2 jði∂t − e ⁎ fÞcðrÞj2 : jð−ir − e AÞcðrÞj (1) L ¼ − ajcðrÞj2 + jcðrÞj4 + 2 2m⁎ Here a, b, c1, and c2 are constants; m∗ and e∗ are effective mass and charge of condensates, and f and A are scalar and vector potentials for external electromagnetic fields, respectively. The time derivative terms are included phenomenologically. The amplitude of the scalar field c(r) serves as an order parameter, and its squared absolute value has a meaning of the superfluid density (|c(r)|2 ¼ ns). The Lagrangian is invariant under the global phase rotation, c ! eiyc. The coefficient a is assumed to behave as a ¼ a0(T − Tc) (a0 > 0) near the critical point, where T is a temperature of the system, and Tc is a critical temperature at which

Higgs and Nambu–Goldstone modes in condensed matter physics

177

superconductivity emerges. At high temperature, the effective potential (the first two terms in the square bracket in Eq. (1)) has a parabolic cylindrical shape with a single minimum at the origin (c ¼ 0). As the temperature goes down below Tc, the potential turns into a Mexican-hat shape (Fig. 1), where degenerate minima appear along the circle at the bottom of the potential (|c| ¼ c0 > 0). One of the minima is realized as the ground state, which spontaneously breaks the phase rotation symmetry. Without loss of generality, one can assume that the order parameter takes a real positive value in the ground state. One can consider fluctuations of the order parameter around the ground state by expanding as cðrÞ ¼ ðc0 + HðrÞÞeiyðrÞ ,

(2)

where H(r) and y(r) represent amplitude and phase fluctuations, respectively. The Lagrangian is then expanded in the following way: 0 12 1 e ⁎2 @ 1 2 2 L ¼ 2aH − ðrHÞ − A − ⁎ ryA 2m ⁎ e 2m ⁎ 0 1 1  ðc0 + HÞ2 − c1 e ⁎ @f + ⁎ ∂t yAðc0 + HÞ2 (3) e 0 12 1 + c2 ð∂t HÞ2 + c2 e ⁎2 @f + ⁎ ∂t yA ðc0 + HÞ2 e +⋯: The first term indicates that the Higgs mode represented by H has a finite mass, reflecting the fact that the effective potential has a finite curvature along the amplitude direction. The second term in Eq. (3) corresponds to the kinetic term of the Higgs mode. The phase field y appears without a mass term (∝ y2). Hence, y would be thought of as a massless NG mode. However, y always appears in the combination of A − e1⁎ ry or f + e1⁎ ∂t y, due to which one can remove y by performing a gauge transformation, A ! A0 ¼ A − e1⁎ ry and f ! f0 ¼ f + e1⁎ ∂t y (unitary gauge). By rewriting A0 and f0 as A and f, one obtains an expression in terms of H, 1 e ⁎2 2 ðrHÞ2 − A ðc0 + HÞ2 2m ⁎ 2m ⁎ − c1 e ⁎ fðc0 + HÞ2 + c2 ð∂t HÞ2 + c2 e ⁎2 f2 ðc0 + HÞ2 +⋯:

L ¼ 2aH2 −

(4)

The phase mode y disappears from the expression, and a mass term of the gauge field (∝A2) appears in the third term. This is nothing but the Anderson–Higgs mechanism. One can count the number of degrees of freedom before and after the Anderson–Higgs mechanism: 2 ðtransverse components of AÞ+2 ðreal and imaginary parts of cÞ ! 3 ðtransverse and longitudinal components of AÞ+1 ðHiggs modeÞ The electromagnetic field, that mediates the long-range Coulomb interaction, screens phase fluctuations of the superconducting order parameter, pushing them up to high energy (in the scale of plasma frequency). As a result of the Anderson–Higgs mechanism, the electromagnetic field itself becomes short-ranged and decays exponentially within the London penetration depth (Meissner–Ochsenfeld effect). The remaining low-energy excitation mode is the Higgs mode. In the vicinity of Tc, however, the phase mode may have a possibility to revive without damping [known as the Carlson–Goldman mode (Carlson and Goldman, 1975)] due to the presence of a large number of normal electrons which screen the long-range Coulomb interaction. If the system is electrically neutral (i.e., e ¼ 0), the order parameter is not coupled to gauge fields, and the Anderson–Higgs mechanism does not take place. In this case, the NG mode survives as a massless excitation, which dominates the low-energy behavior. In particular, the Higgs mode can hybridize with the NG mode if c1 6¼ 0, which causes a mixing between amplitude and phase modes. In order to decouple the amplitude and phase modes, one often requires an effective particle-hole symmetry, which suppresses the c1 term and makes the Lagrangian relativistic with an emergent Lorentz symmetry. In superconductors, there is an approximate particle-hole symmetry at low energy so that one can practically neglect the c1 term. From Eq. (4), one can see that there is no linear coupling between H and A (or f). This means that the Higgs mode cannot be excited within the linear response regime, which is well expected from the fact that the Higgs mode is a scalar excitation having no electric charge or magnetic moment. On the other hand, if one goes beyond the linear response, the Higgs mode can be excited by electromagnetic fields through the nonlinear coupling (e.g., ∝A2H) (Tsuji and Aoki, 2015), which has been used to observe the Higgs mode in terahertz laser experiments. The GL theory does not predict the precise value of the mass of the Higgs mode. It only tells that the mass is proportional to 1

1

ð− aÞ2 ∝ ðT c − TÞ2 . From a microscopic theory, the mass is found to coincide with the superconducting gap energy 2D. That is, the minimum energy that is required to excite the Higgs mode is the same as the energy to break a single Cooper pair. This causes a problem of using the GL theory in the energy scale of the Higgs mode. In the GL theory, the order parameter is supposed to vary sufficiently slowly in space and time, whereas the order parameter oscillates with the frequency of 2D when the Higgs mode is

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excited. This already reaches the energy scale of pair breaking, and thus one cannot neglect the effect of quasiparticle excitations, invalidating the basic assumption of the GL description. The aforementioned issue motivates one to move to a microscopic description of superconductivity, that is, the BCS theory. In the BCS theory, one adopts the Hamiltonian that includes the kinetic energy of electrons and the pairing interaction (usually mediated by phonons), HBCS ¼

X k, s

VX{ { c c c 0 c0 , N 0 k" −k# −k # k " k, k

ek c{ks cks −

(5)

where cks is the annihilation operator of electrons with momentum k and spin s, ek is the energy dispersion, V (> 0) is the pairing interaction strength, and N is the number of k points. P In the mean-field approximation, the Hamiltonian (5) is replaced by HMF ¼ kC{kh(k)Ck, where Ck ¼ (ck"c{−k#)T is the Nambu spinor and   ek −D hðkÞ ¼ (6) −D −ek is the Bogoliubov–de-Gennes Hamiltonian. The off-diagonal element D is defined by D¼

VX { { hck" c−k# i: N

(7)

VX D : N 2Ek

(8)

k

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We assume that D takes a real positive value. The eigenvalues of h(k) are found to be Ek :¼  e2k + D2, and thus the single-particle spectrum shows an energy gap of 2D. At zero temperature, D satisfies the gap equation, D¼

k

The dynamics of superconductors is determined by the time-dependent Bogoliubov–de-Gennes equation, i

∂ C ¼ hðkÞCk : ∂t k

(9)

To analyze the equation, it is convenient to introduce Anderson’s pseudospins, sak ¼

1 { a hC t Ck i 2 k

ða ¼ x, y, zÞ:

(10)

Here ta denotes the Pauli matrices. Physically, the x and y components of the pseudospins correspond, respectively, to the real and imaginary parts of Cooper pairs’ amplitude hc{k"c{−k#i, and the z component represents the momentum-dependent occupation of electrons. From Eq. (9), one finds that the time evolution of the pseudospins is given by ∂ s ¼ 2bk  sk , ∂t k

(11)

which resembles the Bloch equation that describes precession motion of spins in a magnetic field. Here bk ¼ (−Re D, −Im D, ek) plays a role of a magnetic field for pseudospins. In general, the equation of motion (11) becomes a nonlinear equation since it is supplemented by the self-consistency condition (8). However, one can linearize the equation if the deviation from the initial ground state is sufficiently small. By writing sk(t) ¼sk(0) + dsk(t) and D(t) ¼ D + dD(t), the linearized equation is expressed as ∂ x y ds ¼ −2ek dsk , ∂t k

(12)

∂ y e ds ¼ 2ek dsxk + 2Ddszk − k dD, Ek ∂t k

(13)

∂ z y ds ¼ −2Ddsk : ∂t k

(14)

By using the relation Ddsxk ¼ ekdszk, one can further simplify the equation into 2e2 ∂2 x ds ¼ − 4E2k dsxk + k dD: Ek ∂t 2 k

(15)

2e2k dDðoÞ Ek ð4E2k − o2 Þ

(16)

After Fourier transformation, one obtains dsxk ðoÞ ¼

Higgs and Nambu–Goldstone modes in condensed matter physics V Combining with the linearized gap equation, dDðoÞ ¼ N condition is satisfied:

P

x k dsk ðoÞ,

179

one finds that a nontrivial solution exists if the following

2e2k VX ¼ 1: N Ek ð4E2k − o2 Þ

(17)

k

This is precisely the condition that has been derived by Anderson (1958b). One can quickly verify that o ¼ 2D is a solution of Eq. (17) since it is reduced to the gap equation (8) at this frequency. The solution D(t) consists of a collective amplitude oscillation of the order parameter, and hence is identified to the Higgs mode that has been expected to exist in the GL theory. Subsequently, the equation of motion for a quench problem has been studied by Volkov and Kogan (1974), who showed that the amplitude oscillation of the gap function does not persist but decays in a power law as c DðtÞ  D + pffiffiffiffiffiffiffiffi cosð2Dt + ’Þ 2Dt

(18)

with certain constants c and ’. This slight instability (power-law decay) of the Higgs mode is related to the fact that the energy of the Higgs mode lies right at the bottom edge of the quasiparticle excitation continuum. The decay into quasiparticles occurs without collisions, similar to the case of Landau damping for plasma waves. The energy dispersion of the Higgs mode at finite momentum, oH(q), has the following form (Littlewood and Varma, 1982), rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2 oH ðqÞ ¼ ð2DÞ2 + v2F q2 − i vF q, (19) 24 3 where vF is the Fermi velocity and q ¼ |q|. The presence of the imaginary part indicates that the Higgs mode has a finite lifetime at finite q due to the decay into quasiparticles. Especially, the mode becomes overdamped when vF q=D ≳ 1. In multiband superconductors, the system may possess multiple order parameters (multigap superconductors) such as MgB2. In this case, there can arise a collective oscillation of the relative phase between two order parameters, known as the Leggett mode (Leggett, 1966). Since the relative phase oscillation does not induce the net current and hence does not couple to gauge fields linearly, the Leggett mode can evade the Anderson–Higgs mechanism, remaining to survive at low energy. If there is a hidden competing order that stays close to the primary order, the so-called Bardasis–Schrieffer mode can emerge as a collective oscillation of the amplitude of the secondary order parameter (Bardasis and Schrieffer, 1961). For example, some of ironbased superconductors are expected to have a close competition between the s pairing ground state with the d-wave pairing instability. The Bardasis–Schrieffer mode can be used to probe such hidden instabilities that are difficult to detect by other means. In strongly correlated systems, one may need to go beyond the BCS theory and take into account quantum correction effects to treat Higgs modes, which can be defined by the peak of the superconducting amplitude-amplitude correlation function. It is expected that the lifetime (i.e., inverse of the peak width) of the Higgs mode would become shorter and shorter as one goes into strongly correlated regimes due to the decay to quasiparticles, and the amplitude oscillation of the order parameter would become overdamped at some point. Such kind of behavior has been observed in the ordered phase of the Hubbard model with an interaction quench (Werner et al., 2012; Tsuji et al., 2013).

Higgs modes in atomic Fermi gases The Higgs mode appears not only in superconductors but also in neutral fermionic superfluids. The only traditional condensedmatter system that falls into the latter category had been the system of superfluid 3He. In the last few decades, a new system arose in the field of atomic physics: Superfluidity in a dilute gas of fermionic atoms has been realized in 2004 (Regal et al., 2004; Zwierlein et al., 2004). Since then, this system has been offering a unique opportunity for studying Higgs modes in fermionic superfluids. Atomic gases have several advantages over solid-state systems. One of them is high controllability of system parameters. Using experimental techniques developed in atomic, molecular, and optical physics, various parameters can be tuned with unprecedented precision. In atomic Fermi gases, the interaction strength between fermions can be tuned by external fields via the Feshbach resonance. Employing this technique, one can observe a smooth crossover between the BCS-type state with large overlapping Cooper pairs and Bose–Einstein condensation (BEC) of tightly bound molecules of two fermions, known as the “BEC-BCS crossover,” which has been realized also in 2004 (Regal et al., 2004; Zwierlein et al., 2004). This phenomenon had never been directly observed in condensed-matter systems, though it was theoretically speculated by Leggett (1980) and Eagles (1969). This experimental progress allows one to study the evolution of the Higgs mode in the BEC-BCS crossover. The interaction strength in this system is often described by the dimensionless parameter 1/(kFa), where kF and a are the Fermi wave number and the s-wave scattering length, respectively. The s-wave scattering length can be tuned by an external magnetic field due to the Feshbach resonance. The weak-coupling BCS regime and strong-coupling BEC regime correspond to 1/(kFa) < 0 and 1/(kFa) > 0, respectively. The crossover between the BCS and BEC physics occurs at the unitary limit 1/(kFa) ¼ 0, at which a diverges. The Higgs mode in a superfluid Fermi gas of 6Li atoms has been observed in 2018 (Behrle et al., 2018). A novel scheme to induce the Higgs mode has been developed in the experiment: One of the two hyperfine states involved in pairing is coupled with an initially unoccupied third state by a radiofrequency field. In this manner, a periodic modulation of the amplitude of the gap

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Higgs and Nambu–Goldstone modes in condensed matter physics

Fig. 3 Measured excitation spectra of the Higgs mode in a superfluid Fermi gas for different interaction strength 1/(kFa). Adapted from Behrle A, Harrison T, Kombe J, Gao K, Link M, Bernier J-S, Kollath C, and Köhl M (2018) Higgs mode in a strongly interacting fermionic superfluid. Nature Physics 14(8): 781–785. ISSN 1745-2481

function was induced. The Higgs mode was searched by rapidly sweeping the magnetic field onto the molecular side of the Feshbach resonance and measuring the energy absorption spectrum. The excitation spectra in Fig. 3 show the clear resonance at 2D in the BCS side (1/(kFa) < 0) while the peak broadens significantly signaling the instability of the Higgs mode in the BEC side (1/(kFa) > 0). The time-dependent Bogoliubov–de-Gennes (tdBdG) formalism has been extensively used to study the dynamical properties of superfluid Fermi gases. In this formalism, the gap function D(r, t) obeys the tdBdG equation !    ^h un ðr, tÞ D ∂ un ðr, tÞ i ¼ , (20) ∂t vn ðr, tÞ vn ðr, tÞ D −^h where h^ ¼ −r2 =2m + UðrÞ − m, m is the atomic mass, m is the chemical potential, U(r) is the trapping potential, n is the index for the eigenstates, and un and vn are the time-dependent quasi-particle amplitudes, respectively. The tdBdG equation needs to be solved self-consistently with the gap equation, Dðr, tÞ ¼ and the equation for the total number of atoms N ,

VX u ðr, tÞvn ðr, tÞ, N n n Z

N ¼2

dr

X jun ðr, tÞj2 : n

(21)

(22)

In a uniform system (U(r) ¼ 0), substituting the static solution of Eq. (20) u2k ¼ (1 + ek/Ek)/2 and v2k ¼ (1 − ek/Ek)/2 (ek ¼ k2/2m − m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Ek ¼ e2k + D2 ) into Eqs. (22) and (22), the former reduces to the gap equation (8), while the latter reduces to N ¼

X k

1−

 ek : Ek

(23)

The self-consistent solution of (D, m) for the gap equation (8) and the number equation (23) smoothly connects the weak-coupling pffiffiffiffiffiffiffiffiffiffiffiffiffi 3=4 BCS limit with ðð8eF =e2 Þe −p=2kF jaj , eF Þ and the strong-coupling BEC limit with ð 16=3pjmj1=4 eF , −1=2ma2 Þ, where e is Napier’s constant and eF is the Fermi energy. Note that m coincides with the binding energy of molecules in the BEC limit. The above tdBdG formalism thus correctly captures the dynamical properties of a superfluid Fermi gas including the Higgs mode throughout the entire BEC-BCS crossover.

Higgs and Nambu–Goldstone modes in condensed matter physics

181

One can study small amplitude oscillations of the gap function (Higgs mode) induced by a quench of the interaction strength in the tdBdG formalism. The gap function evolves in time according to Eq. (18) in the BCS regime, where the amplitude oscillates with the frequency oH ¼ 2D and the oscillation decays in a power law ( tg) with the exponent g ¼ −1/2 (Volkov and Kogan, 1974). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the BEC regime, on the other hand, the frequency oH ¼ 2 D2 + m2 and the exponent of the power-law decay g ¼ −3/2 for the amplitude oscillation have been predicted (Gurarie, 2009). The former coincides with twice of the energy threshold for the creation of fermionic excitations by pair-breaking. The evolution of the frequency oH and the power g through the BEC-BCS crossover is of particular interest and worth investigating in future experiments (Tokimoto et al., 2019). The NG mode has also been investigated in atomic fermi superfluids. It manifest itself pffiffiffi as the Anderson–Bogoliubov (AB) phonon (phase) mode as shown in Table 1. Anderson predicted the sound velocity c ¼ vF = 3 for the AB phonon mode in the BCS limit by the generalized random-phase approximation (Anderson, 1958b). The evolution of the excitation spectra throughout the BEC-BCS crossover has been studied using two-photon Bragg spectroscopy, which allows characterization of the dispersion relation of the AB phonon mode (Hoinka et al., 2017).

Higgs and Nambu–Goldstone modes in other systems The NG and Higgs modes exist ubiquitously in quantum many-body systems that possess both a spontaneously broken continuous symmetry and an approximate particle-hole symmetry. In the previous sections, we have taken superconductors and superfluid states of ultracold two-component Fermi gases as specific examples that possess either or both of the NG and Higgs modes. In this section, we explain several other significant examples of the NG and Higgs modes in the contexts of condensed matter and ultracoldatom physics in order to illustrate broad applicability of the concepts. We first consider a superfluid state of ultracold spinless Bose gases in optical lattices. An optical lattice means a periodic potential for atoms that is created by two counter-propagating laser beams. When the optical lattice potential is sufficiently deep, the system is well described by the Bose–Hubbard model,  X { X X { { { ^b ^bl + h:c: + U ^b ^b ^b ^b + ^ BH ¼ −J ðej − mÞ^bj ^bj , (24) H j 2 j j j j j j hj,li where ^bj , U, J, and m denote, respectively, the annihilation operator of bosons at site j, the onsite interaction, the hopping, and the chemical potential. The external field potential ej is present for trapped atoms in standard experiments. In Fig. 4, we show a groundstate phase diagram of the homogeneous Bose-Hubbard model in the (ZJ/U, m/U) plane, where Z is the coordination number. When the filling factor, which is the number of particles per lattice site, is integer and ZJ/U is small, the ground state is a Mott insulating (MI) state. Otherwise, it is a superfluid state. There is a continuous quantum phase transition from the MI to the

2.0

1.5

1.0

0.5

0.0

-0.5 0

0.05

0.10

0.15

0.20

0.25

Fig. 4 Ground-state phase diagram of the homogeneous Bose–Hubbard model obtained by using a mean-field theory. The gray shaded areas near the Mott insulator (MI) regions roughly mark the regions where the superfluid (SF) order parameter is small enough for the fourth-order GL approximation (i.e., the effective action can be written by the expansion up to fourth order in the order parameter) to be valid. On the dashed lines in the SF region, the effective particle-hole symmetry is present. n denotes the filling factor. Adapted from Nakayama T, Danshita I, Nikuni T, and Tsuchiya S (2015) Fano resonance through Higgs bound states in tunneling of Nambu-Goldstone modes. Physical Review A 92(4): 043610

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superfluid state, where the U(1) symmetry is spontaneously broken. In the superfluid region near the quantum phase transitions, which is the gray-shaded area in Fig. 4, low-energy properties of the system can be effectively described by the GL theory (Sachdev, 2011). At the tip of the lobe of the Mott insulating region, where the quantum phase transition to a superfluid with commensurate filling occurs, there emerges particle-hole symmetry. In the nearby superfluid state at the commensurate filling, the emergent particle-hole symmetry survives approximately so that the effective theory coincides with the Lagrangian of Eq. (1) with e ¼ 0. Consequently, the Higgs mode is present as an approximately independent collective mode. Note that there is no Higgs mode in a bosonic superfluid state without particle-hole symmetry as in the cases of superfluid 4He and Bose–Einstein condensates of weaklyinteracting atomic Bose gases. When ZJ/U decreases toward the critical value (ZJ/U)c, the energy gap of the Higgs mode Dg approaches zero as Dg ∝ |zJ/U − (zJ/U)c|n, where n is the critical exponents for the correlation length of the (D + 1)-dimensional classical XY model and D denotes the spatial dimension of the system. In 2012, Endres et al. have investigated dynamics of two-dimensional gas of 87Rb atoms (bosons) in optical lattices in response to the temporal modulation of the lattice depth in order to measure the gap of the Higgs mode (Endres et al., 2012). Fig. 5B shows the temperature of the system after the application of the lattice modulation as a function of the modulation frequency for the superfluid state at J/Jc ¼ 1.2, where Jc is the transition value of the hopping for a given U at unit filling. Above an onset frequency n0, the response exhibits a broad spectrum. Quantitative comparisons with theoretical analyses have revealed that the onset frequency (Fig. 5A) coincides with the gap of the Higgs mode at zero momentum. It is worth noticing that the Higgs mode has not been observed as a resonance peak in the response. This smearing of the resonance can be attributed to combined effects of the spatial inhomogeneity due to the trapping potential and the finite temperatures (Pollet and Prokof’ev, 2012). We next consider antiferromagnetic materials consisting of dimers of S ¼ 1/2 spins, such as TlCuCl3 (Rüegg et al., 2008) and KCuCl3 (Kuroe et al., 2012), which can be effectively described by the following Hamiltonian,  X X ^l,m S ^j,R S ^j,L − Jxx S^x S^x + ^j,m0 : ^ AFD ¼ H JS Jj,l; m,m0 S (25) j,R j,L j j, l; m, m0 ^j,m ¼ ðS^x , S^y , S^z Þ denotes the S ¼ 1/2 spin operator at spin m(¼ L, R) of dimer j. J(> 0), Jxx(> 0) and Jj,l;m,m0 are the Here, S j,m j,m j,m intradimer isotropic interaction, the intradimer anisotropic interaction, and the interdimer isotropic interaction, respectively. In Fig. 6, the spatial configuration of the interdimer interactions is illustrated (Matsumoto et al., 2004). To be concrete, we hereafter focus on the case of TlCuCl3, in which the Higgs mode has been observed via neutron scattering experiments. In this case, the most dominant interdimer interaction is J2 that is antiferromagnetic. Although Jxx is about 1% of J, it is important for discussing the NG and Higgs modes in the sense that it reduces the spin rotation symmetry from SU(2) to U(1). At atmospheric pressure, J is dominant over J2 so that the low-temperature magnetic state is a dimerized state in that two spins on each dimer form a spin singlet and there is no magnetic order. When the hydrostatic pressure p is increased, J2/J increases and a continuous quantum phase transition to an antiferromagnetic ordered state occurs at p ¼ pc ’ 1.08 kbar. Associated with the

j

(A)

1.2

0

0.03

0.06

(B)

0.09

0.12

0.15 0.17

Mott Insulator

1

1

Superfluid

V0 = 8Er j/jc = 2.2

1

0.15

n0

0.13 0.11

0.8

0.18 kBT/U

hn 0/U

2 0.6

V0 = 10Er j/jc = 1.2

0.18 3 0.16 n0 0.14

0.2 0

0.14 0.12

3

0.4

V0 = 9Er j/jc = 1.6

2 n0

0.16

0.12 0

0.5

1

1.5 j/jc

2

2.5

0

400 800 nmod (Hz)

Fig. 5 Experimental observation of the Higgs gap in the system of ultracold Bose gases in optical lattices. (A) Excitation gap versus the hopping parameter j, which is denoted by J in the main text, in unit of its critical value jc. Solid line and dashed line, respectively, stand for the Higgs and Mott gaps calculated by the Gutzwiller mean-field theory for a homogeneous system with unit filling. (B) Temperature response to temporal modulation of the lattice depth for three values of j/jc. Adapted from Endres M, Fukuhara T, Pekker D, Cheneau M, SchaußP, Gross C, Demler E, Kuhr S, and Bloch I (2012) The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition. Nature 487(7408): 454–458. ISSN 1476-4687

Higgs and Nambu–Goldstone modes in condensed matter physics

183

Fig. 6 Schematic diagram of the interdimer interaction between each pair of spins of XCuCl3 in (A) a–c plane and (B) b–c plane, where X is for example Tl or K. One spin with S ¼ 1/2 lives at each vertex. Adapted from Matsumoto M, Normand B, Rice TM, and Sigrist M (2004) Field- and pressure-induced magnetic quantum phase transitions in TlCuCl3. Physical Review B 69(5): 054423

transition, the spin-rotation symmetry is spontaneously broken. The quantum phase transition can be qualitatively captured by a mean-field approximation in which the many-body wave function is assumed to be a product state, jCMF i ¼

j jfij

(26)

where jfij ¼ aj,0 jsij + aj,1 jt 1 ij + aj,2 jt 2 ij + aj,3 jt 3 ij :

(27) 1

1

The local state at dimer j is spanned by the spin-singlet and triplet states, namely, jsi ¼ 2 − 2 ðj", #i − j#, "iÞ, jt 1 i ¼ 2 − 2 ðj", #i+j#, "iÞ, P 1 1 jt 2 i ¼ 2 −2 ðj", "i + j#, #iÞ, and jt 3 i ¼ 2 −2 ðj", "i − j#, #iÞ. The coefficients satisfy the normalization condition 3a¼0 jaj,a j2 ¼ 1. In the limit of Jj,l;m,m0 ! 0, the four states are the eigenstates of each local Hamiltonian, where |si is the ground state, |t1i and |t2i are the degenerate first excited states, and |t3i is the highest energy state. When J2/J is finite but smaller than the critical value, the ground state remains to be |fi ¼ |si for all the dimers and there is the energy gap to create triplet states. The energy cost for creating |t1i is equivalent to that for creating |t2i so that this symmetry can be regarded as a particle-hole symmetry. At the critical value of J2/J, the energy gap closes. When J2/J increases further, the coefficients aj,a for a ¼ 1, 2, 3 become finite and the antiferromagnetic order emerges. The low-energy properties in the ordered phase can be effectively described by the GL theory of Eq. (1), thanks to the spontaneous breaking of the U(1) symmetry and the presence of the particle-hole symmetry. Hence, there are the NG and Higgs modes in the ordered phase. In addition, there is another gaped mode that dominantly includes the |t3i excitation. In 2008, Rüegg et al. have observed the excitation modes explained above by means of inelastic neutron scattering (Rüegg et al., 2008). Fig. 7 shows the energy gaps of the excitation modes at specific momenta, which are detected as resonance peaks in the spectra of inelastic neutron scattering. In the dimerized phase at low pressure, there are three gapped modes corresponding to the creation of a triplet excitation. The gap of the two modes (blue symbols) closes at the quantum critical point whereas that of the other mode (black symbols) remains gappped. In the antiferromagnetic ordered phase at high pressure, the former two modes transform into the Higgs and NG modes, which correspond to longitudinal and transverse fluctuations of the magnetic order parameter. In the last place, we briefly mention another important example, namely, materials with incommensurate CDW (ICDW) order (Grüner, 1988). Specifically, in solid-state materials with quasi-1D lattice structure, such as 2H-NbSe2 (Tsang et al., 1976) and K0.3MoO3 (Yusupov et al., 2010), electron–phonon interactions induce the condensation of particle-hole pairs with momenta 2kF and − 2kF, leading to the formation of ICDW states at low temperatures. In the ICDW state, the electron density is given by rðxÞ ’ r0 + r1 cosð2kF x + ’Þ,

(28)

where kF, r0, r1, and ’ denote, respectively, the Fermi momentum, the average electron density, the amplitude, and the phase of ICDW. The oscillation of the amplitude r1, which is often called amplitudon, corresponds to the Higgs mode whereas that of the phase ’ (phason) does to the NG mode. The Higgs mode has been observed in several ICDW materials via Raman spectroscopy.

Conclusion We have reviewed various order-parameter dynamics that appear in condensed matter systems. Especially, we have focused on Higgs and Nambu–Goldstone (NG) modes, corresponding to amplitude and phase oscillations of order parameters. They provide important information on the fundamental properties of underlying quantum many-body systems, including symmetries, phase transitions, dynamics, and response to external perturbations. Target systems range from superconductors to cold-atom systems (bosonic and fermionic superfluids), quantum magnets, charge density waves, and so on. Here one can see the universality of

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Higgs and Nambu–Goldstone modes in condensed matter physics

p

TN(p) [meV]

c

1.2

Energy [meV]

1

L T2 L

(0 4 0)

L,T1 T2 T1 L

(0 0 1)

0.8 0.6 0.4 0.2 0 0

QD

RC−AFM

QCP 1

2 3 Pressure [kbar]

4

5

Fig. 7 Energy gaps of the magnetic excitations in TlCuCl3 versus pressure, which has been detected via inelastic neutron scattering. The red symbols correspond to the gap of the Higgs mode in the ordered phase, namely, the renormalized classical antiferromagnetic (RC-AFM) phase. QD and QCP stand for the quantum disordered phase and quantum critical point, respectively. pc, L, and T denote the critical pressure, longitudinal magnetic excitation, and transverse magnetic excitation, respectively. (0 4 0) or (0 0 1) means the value of the momentum transfer in the neutron scattering. Adapted from Rüegg C, Normand B, Matsumoto M, Furrer A, McMorrow DF, Krämer KW, Güdel HU, Gvasaliya SN, Mutka H, and Boehm M (2008) Quantum magnets under pressure: Controlling elementary excitations in TlCuCl3. Physical Review Letters 100(20): 205701

order-parameter dynamics, which may not depend so much on details of the system, partly because they are effectively described by a “common language” based on quantum field theories at low energies. There are many open issues that remain to be addressed. In the case of superconductors, it is experimentally important to understand various aspects of Higgs modes in superconductors, especially for those beyond NbN and the coexisting phase of CDW and superconductivity in NbSe2. The difficulty lies in the fact that there are always competitions between the Higgs mode and quasiparticle excitations (Cea et al., 2016), both of which have similar characters with respect to symmetries and excitation energies. To distinguish them, one has to carefully compare experimental results with theoretical calculations, which may depend on details of materials, impurities, electron–phonon couplings, and so on (Tsuji and Nomura, 2020). This reflects the situation that one has not fully understood the universal (i.e., material-independent) condition that the Higgs mode becomes visible in experiments. To see enriched order-parameter dynamics, it will be fascinating to extend the observation of Higgs modes in various multiband superconductors such as MgB2 (Kovalev et al., 2021) and unconventional superconductors (Schwarz et al., 2020) such as cuprates (Barlas and Varma, 2013; Katsumi et al., 2018; Chu et al., 2020) and iron-based superconductors (Vaswani et al., 2021; Grasset et al., 2022). A potential application of the Higgs mode includes detection of light-induced orders (Isoyama et al., 2021; Katsumi et al., 2022) and hidden fluctuations (Katsumi et al., 2020). In the case of cold-atom systems, further investigations, both experimentally and theoretically, are needed to better understand the evolution of the Higgs modes across the BCS-BEC crossover in a superfluid fermi gas and the SF-MI phase transition in a bosonic superfluid in an optical lattice. In the lattice-boson systems, in particular, it is a long-standing problem whether or not the Higgs mode can be detected as a resonance peak in the response to temporal modulation of an external potential. Since recent experiments on optical-lattice systems have realized almost homogeneous samples of ultracold gases (e.g., Mazurenko et al., 2017), it will be interesting to experimentally address such a problem. Last but not least, taking advantage of the high degree of ability to control the system, it may be worthwhile to investigate previously unexplored aspects of Higgs modes, such as tunneling properties (Nakayama et al., 2015; Nakayama and Tsuchiya, 2019), which are typically difficult to study in solid-state systems.

References Anderson PW (1958a) Coherent excited states in the theory of superconductivity: Gauge invariance and the Meissner effect. Physical Review 110(4): 827–835. Anderson PW (1958b) Random-phase approximation in the theory of superconductivity. Physical Review 112(6): 1900–1916. Anderson PW (1963) Plasmons, gauge invariance, and mass. Physical Review 130(1): 439–442. ATLAS Collaboration. (2012) Observation of a new particle in the search for the Standard Model Higgs Boson with the ATLAS detector at the LHC. Physics Letters B 716(1): 1–29. ISSN: 0370-2693. Bardasis A and Schrieffer JR (1961) Excitons and plasmons in superconductors. Physical Review 121(4): 1050–1062. Bardeen J, Cooper LN, and Schrieffer JR (1957) Theory of superconductivity. Physical Review 108(5): 1175–1204. Barlas Y and Varma CM (2013) Amplitude or Higgs modes in d-wave superconductors. Physical Review B 87(5): 054503. Behrle A, Harrison T, Kombe J, Gao K, Link M, Bernier J-S, Kollath C, and Köhl M (2018) Higgs mode in a strongly interacting fermionic superfluid. Nature Physics 14(8): 781–785. ISSN: 1745-2481. Bogoliubov NN, Tolmachev VV, and Shirkov DV (1958) A New Method in the Theory of Superconductivity. Moscow: Academy of Sciences of USSR. Carlson RV and Goldman AM (1975) Propagating order-parameter collective modes in superconducting films. Physical Review Letters 34(1): 11–15.

Higgs and Nambu–Goldstone modes in condensed matter physics

185

Cea T, Castellani C, and Benfatto L (2016) Nonlinear optical effects and third-harmonic generation in superconductors: Cooper pairs versus Higgs mode contribution. Physical Review B 93(18): 180507. Chu H, Kim M-J, Katsumi K, Kovalev S, Dawson RD, Schwarz L, Yoshikawa N, Kim G, Putzky D, Li ZZ, Raffy H, Germanskiy S, Deinert J-C, Awari N, Ilyakov I, Green B, Chen M, Bawatna M, Cristiani G, Logvenov G, Gallais Y, Boris AV, Keimer B, Schnyder AP, Manske D, Gensch M, Wang Z, Shimano R, and Kaiser S (2020) Phase-resolved Higgs response in superconducting cuprates. Nature Communications 11(1): 1793. CMS Collaboration. (2012) Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Physics Letters B 716(1): 30–61. ISSN: 0370-2693. Eagles DM (1969) Physical Review 186(2): 456–463. Endres M, Fukuhara T, Pekker D, Cheneau M, Schauß P, Gross C, Demler E, Kuhr S, and Bloch I (2012) The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition. Nature 487(7408): 454–458. ISSN: 1476-4687. Englert F and Brout R (1964) Broken symmetry and the mass of gauge vector mesons. Physical Review Letters 13(9): 321–323. Ginzburg VL and Landau LD (1950) On the theory of superconductivity. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 20: 1064. Goldstone J (1961) Field theories with superconductor solutions. Il Nuovo Cimento 19(1): 154–164. Goldstone J, Salam A, and Weinberg S (1962) Broken symmetries. Physical Review 127(3): 965–970. Grasset R, Katsumi K, Massat P, Wen H-H, Chen X-H, Gallais Y, and Shimano R (2022) Terahertz pulse-driven collective mode in the nematic superconducting state of Ba1−xKxFe2As2. npj Quantum Materials 7(1): 4. Grüner G (1988) The dynamics of charge-density waves. Reviews of Modern Physics 60(4): 1129–1181. Guralnik GS, Hagen CR, and Kibble TWB (1964) Global conservation laws and massless particles. Physical Review Letters 13(20): 585–587. Gurarie V (2009) Nonequilibrium dynamics of weakly and strongly paired superconductors. Physical Review Letters 103(7): 075301. Hidaka Y (2013) Counting rule for Nambu-Goldstone modes in nonrelativistic systems. Physical Review Letters 110(9): 091601. Higgs PW (1964) Broken symmetries and the masses of gauge Bosons. Physical Review Letters 13(16): 508–509. Hoinka S, Dyke P, Lingham MG, Kinnunen JJ, Bruun GM, and Vale CJ (2017) Goldstone mode and pair-breaking excitations in atomic Fermi superfluids. Nature Physics 13(10): 943–946. ISSN: 1745-2481. Isoyama K, Yoshikawa N, Katsumi K, Wong J, Shikama N, Sakishita Y, Nabeshima F, Maeda A, and Shimano R (2021) Light-induced enhancement of superconductivity in iron-based superconductor FeSe0.5Te0.5. Communications on Physics 4(1): 160. Jujo T (2018) Quasiclassical theory on third-harmonic generation in conventional superconductors with paramagnetic impurities. Journal of the Physical Society of Japan 87(2): 024704. Katsumi K, Tsuji N, Hamada YI, Matsunaga R, Schneeloch J, Zhong RD, Gu GD, Aoki H, Gallais Y, and Shimano R (2018) Higgs mode in the d-wave superconductor Bi2Sr2CaCu2O8+x driven by an intense terahertz pulse. Physical Review Letters 120(11): 117001. Katsumi K, Li ZZ, Raffy H, Gallais Y, and Shimano R (2020) Superconducting fluctuations probed by the Higgs mode in Bi2Sr2CaCu2O8+x thin films. Physical Review B 102(5): 054510. Katsumi K, Nishida M, Kaiser S, Miyasaka S, Tajima S, and Shimano R (2022) Absence of the superconducting collective excitations in the photoexcited nonequilibrium state of underdoped YBa2Cu3Oy. arXiv:2209.01633. Kovalev S, Dong T, Shi L-Y, Reinhoffer C, Xu T-Q, Wang H-Z, Wang Y, Gan Z-Z, Germanskiy S, Deinert J-C, Ilyakov I, van Loosdrecht PHM, Wu D, Wang N-L, Demsar J, and Wang Z (2021) Band-selective third-harmonic generation in superconducting MgB2: Possible evidence for the Higgs amplitude mode in the dirty limit. Physical Review B 104(14): L140505. Kuroe H, Takami N, Niwa N, Sekine T, Matsumoto M, Yamada F, Tanaka H, and Takemura K (2012) Longitudinal magnetic excitation in KCuCl3 studied by Raman scattering under hydrostatic pressures. Journal of Physics Conference Series 400(3): 032042. Leggett AJ (1966) Number-phase fluctuations in two-band superconductors. Progress in Theoretical Physics 36(5): 901–930. ISSN: 0033-068X. Leggett AJ (1980) Diatomic molecules and cooper pairs. In: Pe˛ kalski A and Przystawa JA (eds.) Modern Trends in the Theory of Condensed Matter, vol. 115. Berlin, Heidelberg: Springer. Littlewood PB and Varma CM (1982) Amplitude collective modes in superconductors and their coupling to charge-density waves. Physical Review B 26(9): 4883–4893. Matsumoto M, Normand B, Rice TM, and Sigrist M (2004) Field- and pressure-induced magnetic quantum phase transitions in TlCuCl3. Physical Review B 69(5): 054423. Matsunaga R, Hamada YI, Makise K, Uzawa Y, Terai H, Wang Z, and Shimano R (2013) Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation. Physical Review Letters 111(5): 057002. Matsunaga R, Tsuji N, Fujita H, Sugioka A, Makise K, Uzawa Y, Terai H, Wang Z, Aoki H, and Shimano R (2014) Light-induced collective pseudospin precession resonating with Higgs mode in a superconductor. Science 345(6201): 1145–1149. ISSN: 0036-8075. Mazurenko A, Chiu CS, Ji G, Parsons MF, Kanász-Nagy M, Schmidt R, Grusdt F, Demler E, Greif D, and Greiner M (2017) A cold-atom Fermi-Hubbard antiferromagnet. Nature 545 (7655): 462–466. ISSN: 1476-4687. Murotani Y and Shimano R (2019) Nonlinear optical response of collective modes in multiband superconductors assisted by nonmagnetic impurities. Physical Review B 99(22): 224510. Nakayama T and Tsuchiya S (2019) Perfect transmission of Higgs modes via antibound states. Physical Review A 100(6): 063612. Nakayama T, Danshita I, Nikuni T, and Tsuchiya S (2015) Fano resonance through Higgs bound states in tunneling of Nambu-Goldstone modes. Physical Review A 92(4): 043610. Nambu Y (1960) Quasi-particles and gauge invariance in the theory of superconductivity. Physical Review 117(3): 648–663. Nambu Y and Jona-Lasinio G (1961) Dynamical model of elementary particles based on an analogy with superconductivity. I. Physical Review 122(1): 345–358. Pekker D and Varma CM (2015) Amplitude/Higgs modes in condensed matter physics. Annual Review of Condensed Matter Physics 6(1): 269–297. Pollet L and Prokof’ev N (2012) Higgs mode in a two-dimensional superfluid. Physical Review Letters 109(1): 010401. Regal C, Greiner M, and Jin DS (2004) Observation of resonance condensation of Fermionic atom pairs. Physical Review Letters 92(4): 040403. Rüegg C, Normand B, Matsumoto M, Furrer A, McMorrow DF, Krämer KW, Güdel HU, Gvasaliya SN, Mutka H, and Boehm M (2008) Quantum magnets under pressure: Controlling elementary excitations in TlCuCl3. Physical Review Letters 100(20): 205701. Sachdev S (2011) Quantum Phase Transitions, 2nd edn. Cambridge University Press. Schwarz L, Fauseweh B, Tsuji N, Cheng N, Bittner N, Krull H, Berciu M, Uhrig GS, Schnyder AP, Kaiser S, and Manske D (2020) Classification and characterization of nonequilibrium Higgs modes in unconventional superconductors. Nature Communications 11(1): 287. Seibold G, Udina M, Castellani C, and Benfatto L (2021) Third harmonic generation from collective modes in disordered superconductors. Physical Review B 103(1): 014512. Shimano R and Tsuji N (2020) Higgs mode in superconductors. Annual Review of Condensed Matter Physics 11(1): 103–124. Silaev M (2019) Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with impurities. Physical Review B 99(22): 224511. Sooryakumar R and Klein MV (1980) Raman scattering by superconducting-gap excitations and their coupling to charge-density waves. Physical Review Letters 45(8): 660–662. Tokimoto J, Tsuchiya S, and Nikuni T (2019) Excitation of Higgs mode in superfluid fermi gas in BCS-BEC crossover. Journal of the Physical Society of Japan 88(2): 023601. Tsang JC, Smith JE, and Shafer MW (1976) Raman spectroscopy of soft modes at the charge-density-wave phase transition in 2H −NbSe2. Physical Review Letters 37(21): 1407–1410. Tsuji N and Aoki H (2015) Theory of Anderson pseudospin resonance with Higgs mode in superconductors. Physical Review B 92(6): 064508. Tsuji N and Nomura Y (2020) Higgs-mode resonance in third harmonic generation in NbN superconductors: Multiband electron-phonon coupling, impurity scattering, and polarizationangle dependence. Physical Review Res. 2(4): 043029. Tsuji N, Eckstein M, and Werner P (2013) Nonthermal antiferromagnetic order and nonequilibrium criticality in the Hubbard model. Physical Review Letters 110(13): 136404.

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Vaswani C, Kang JH, Mootz M, Luo L, Yang X, Sundahl C, Cheng D, Huang C, Kim RHJ, Liu Z, Collantes YG, Hellstrom EE, Perakis IE, Eom CB, and Wang J (2021) Light quantum control of persisting Higgs modes in iron-based superconductors. Nature Communications 12(1): 258. Volkov AF and Kogan SM (1974) Collisionless relaxation of the energy gap in superconductors. Soviet Physics–JETP 38: 1018. Watanabe H and Murayama H (2012) Unified description of Nambu-Goldstone Bosons without Lorentz invariance. Physical Review Letters 108(25): 251602. Werner P, Tsuji N, and Eckstein M (2012) Nonthermal symmetry-broken states in the strongly interacting Hubbard model. Physical Review B 86(20): 205101. Yusupov R, Mertelj T, Kabanov VV, Brazovskii S, Kusar P, Chu J-H, Fisher IR, and Mihailovic D (2010) Coherent dynamics of macroscopic electronic order through a symmetry breaking transition. Nature Physics 6(9): 681–684. ISSN: 1745-2481. Zwierlein MW, Stan CA, Schunck CH, Raupach SMF, Kerman AJ, and Ketterle W (2004) Condensation of pairs of Fermionic atoms near a Feshbach resonance. Physical Review Letters 92(12): 120403.

Higgs and Goldstone modes in cold atom systems Jacques Tempere, Departement Fysica, TQC, Universiteit Antwerpen, Antwerp, Belgium © 2024 Elsevier Ltd. All rights reserved.

Introduction Collective modes in Bose condensates Order parameter for the superfluid Fermi gas Gaussian fluctuations of the pair field Anderson–Bogoliubov mode in superfluid Fermi gases Amplitude mode in superfluid Fermi gases Conclusion References

187 188 189 190 192 194 195 195

Abstract Many superfluid and superconducting systems can be well characterized by a complex order parameter that obtains a nonzero expectation value after spontaneous U(1) symmetry breaking. The complex order parameter can exhibit oscillations in phase or in amplitude, corresponding to Goldstone and Higgs excitations, respectively. These modes are also expected to be present in superfluid atomic Bose and Fermi gases. The excellent level of experimental control over interaction strength, atom numbers, and trapping geometry motivated theoretical and experimental efforts to study the Goldstone and Higgs modes in these cold atom systems. In atomic Fermi superfluids in particular, there exists a tunable coupling between these modes, as well as a coupling to a nearby continuum of single-particle excitations, and this leads to a much richer physics than can be initially expected from simple field-theoretical models. In this contribution, the recent theoretical and experimental developments for the collective modes in superfluid atomic systems are reviewed and placed in the context of earlier research both in particle physics and solid-state physics. The focus is on superfluid Fermi gases, as they include the case of the Bose gas in the limit of strong pairing. For this system, the state-of-the-art theoretical understanding of these modes is provided and compared to recent experimental observations of the collective modes.

Key points

• • •

Description of the Anderson–Bogoliubov mode in superfluid quantum gases Description of the Higgs mode in superfluid quantum gases Review of experiment and theory for elementary excitations in superfluid atomic gases

Introduction Collective modes of many-body quantum systems are crucial ingredients in the description of the low-temperature dynamics and thermodynamics of these systems. They allow to construct simplified models for many phenomena, based on fewer and more weakly interacting constituents than those of the original many-body system. Very often, these modes also reveal emergent properties not easily understood from the viewpoint of the original building blocks of the system. Here, we focus on collective modes arising from spontaneously breaking a continuous symmetry. The paradigmatic example that we will explore in more detail is the case of a complex scalar order parameter1, for which the U(1) symmetry can be spontaneously broken. This is up to now also the most relevant case in cold atom systems, both for Bose gases where the order parameter represents the macroscopic wave function of the condensate, and for superfluid Fermi gases where the order parameter corresponds to the wave function characterizing the pair condensate. In both cases, we will denote the order parameter by (r, t). When the U(1) symmetry is spontaneously broken, collective modes of the order parameter appear (for an excellent review, see Pekker and Varma (2015)). One mode is associated to the modulation of the phase of the order parameter. Since any choice of phase is equivalent by the U(1) symmetry, any modulation in the phase in the symmetry-broken phase should have an energy tending to zero as its wavelength tends to infinity. This gapless mode was described by Nambu (1960), and later refined by

1

Superfluid Helium-3 is an example of a system with a more complicated order parameter, and hence more collective modes.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00212-2

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Higgs and Goldstone modes in cold atom systems

Goldstone (1961), in the context of particle physics. From this, it got its name the “Goldstone mode.” In the context of condensed matter physics, more precisely superconductivity, this mode had already been identified earlier by Bogoliubov et al. (1958) and Anderson (1958a), who also realized that for a charged system, the Goldstone mode acquires a gap (Anderson, 1958b). However, in the context of quantum gases, the mode is gapless, since the system consists of neutral atoms. Within the quantum gas community, this mode is most commonly referred to as the “Anderson–Bogoliubov mode.” The complex order parameter is characterized by phase and amplitude, so besides the gapless Goldstone mode, there is also a mode associated with oscillations of the modulus of the order parameter, commonly called the “Higgs” mode. In particle physics, this mode plays an important role, as pointed out by Higgs (1964) and Englert and Brout (1964), who based themselves on an observation by Anderson (1963) on plasma oscillations. In the context of the standard model, the order parameter is associated to the condensate a new quantum field invoked to generate the masses of the gauge bosons. At first, condensed matter physics did not take much note of the standard model Higgs mode. It was not until the detection of the amplitude mode in superconductors by Sooryakumar and Klein (1980) and the subsequent theoretical description (Littlewood and Varma, 1981, 1982) for the case of superconductors that the concept of a Higgs mode came fully to the attention of the condensed matter community. Most recently, the advent of ultracold quantum gases with their exquisite experimental controllability and tunability has spurred a renewed interest in the Higgs/amplitude and Goldstone/Anderson–Bogoliubov modes in superfluid systems. Before we discuss the realizations of these modes in atomic gases in the next sections, there are a few important remarks to be made about the difference with their particle physics counterparts. A well-known graphical depiction of the Higgs field is that of the Mexican hat potential. The Goldstone mode follows the valley, whereas the Higgs mode is a radial oscillation up and down the hat’s rim. An objection to this graph, from the point of view of Fermi superfluids and superconductors, is that it seems to suppose that the (free) energy as a function of the order parameter can be expanded in series around its value in the normal (nonsuperfluid) state, which is not the case, in contrast to the standard model Higgs field where the Mexican hat potential is assumed from the start. Another difference, as noted in Pekker and Varma (2015), is that the original Higgs theory is a locally gauge invariant theory, which need not be the case for realizations in atomic gases. Moreover, in contrast to the field theoretical counterpart, the Higgs mode in Fermi superfluids lies inside the continuum of pair-breaking excitations and is strongly influenced by it. A final point to keep in mind, and that will be discussed in more detail below, is that the phase and amplitude modes are in general coupled in superfluid gases, so the eigenmodes have a mixed nature and may even become degenerate— even in the presence of a Mexican hat potential. To decouple these modes, one needs exact particle–hole symmetry, which is not present in general. It can be realized only in special cases: in the Bardeen–Cooper–Schrieffer (BCS) limit of a fermionic superfluid, or by fine-tuning a system in an optical lattice.

Collective modes in Bose condensates Bose–Einstein condensates (BEC) in cold atomic gases are characterized by a complex scalar order parameter (r, t) that obeys the Gross–Pitaevskii equation (see Pitaevskii and Stringari, 2003), iℏ

∂ ℏ2 2 r + V + gj j2 , ¼ − 2m ∂t

(1)

where m is the mass of the atoms, V is an external potential, and g denotes the strength of the contact potential, related to the s-wave scattering length between the atoms. This equation describes condensates well for temperatures sufficiently below the critical temperature so that effects of thermal excitations are negligible. The description in terms of the order parameter neglects quantum fluctuations of the Bose field, which is justified when the bosons macroscopically occupy a single mode. The order parameter is interpreted straightforwardly: its modulus squared | |2 represents the density n of Bose condensed atoms, and its phase y is related to the velocity field through v ¼ ðℏ=mÞry. This equation does not allow for uncoupled phase and amplitude modes; the only collective mode (in the uniform system, V ¼ 0) is the Bogoliubov mode with dispersion rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 o ¼ ck 1 + ðkxÞ2 , 2

(2)

Higgs and Goldstone modes in cold atom systems

where c ¼

189

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi gn=m is the sound velocity, and x ¼ ℏ 2 =ð2mgnÞ is the healing length of the condensate (Pitaevskii and Stringari,

2003). This can be considered as the Goldstone mode of the condensate, and no separate Higgs mode exists. In the long wavelength limit (k ! 0), the mode turns to a sound mode. As the wave number of the mode grows to become comparable to the healing length kx≳1, the mode no longer represents a collective oscillation. Rather it becomes a single-particle excitation, taking an atom out of the condensate and into the normal-state cloud. The Bogoliubov sound mode was observed experimentally, shortly after the first realization of BEC in cold trapped gases, by modulating the trapping potential to excite these modes (Jin et al., 1996; Andrews et al., 1997). A more refined technique to probe this excitation is through Bragg spectroscopy, where a fixed amount of energy at a fixed wavelength can be imparted on the condensate via two crossed laser beams, as the condensate absorbs a photon from one beam and emits it in the other. This technique reveals the structure factor, in which the Bogoliubov modes were characterized (Stamper-Kurn et al., 1999). As mentioned above, the amplitude mode is not independently present in uniform BECs (nor in the case of a trapping potential that varies slowly with respect to the healing length). However, condensates can be placed in optical lattices, realizing the Bose–Hubbard model, and allowing both the nearest-neighbor hopping amplitude J and the on-site interaction U to be tuned experimentally. Huber et al. (2007) have shown that along certain curves in the J–U phase diagram, particle–hole symmetry is restored. In the superfluid phase, this leads to the existence of gapped amplitude modes. These were subsequently discovered experimentally, again using Bragg scattering, by Bissbort et al. (2011). Following theoretical work (Podolsky et al., 2011; Pollet and Prokofiev, 2012), that showed that the amplitude mode is also present in a two-dimensional optical lattice, the amplitude mode was excited and observed in two-dimensional systems by modulating the lattice depth (Endres et al., 2012), or by Bragg spectroscopy using two crossed optical cavities (Léonard et al., 2017). So, in summary, for cold atomic Bose gases, the Bogoliubov (Goldstone) mode is easily accessible by disturbing and probing the condensate density, but the amplitude (Higgs) mode has to be engineered into the system, by using optical lattices. The study of the Higgs mode and its visibility in quantum simulations of the Bose–Hubbard model has been the subject of a large amount of theoretical work that would require a much longer review paper than this encyclopedia entry. Instead, and for the remainder of this paper, we turn to superfluid Fermi gases. Superfluid Fermi gases are the neutral counterpart of superconductors, and they are most relevant to link the cold atomic systems to solid-state physics in general and superconductors in particular. Moreover, as will be discussed below, the interaction strength between the fermionic atoms can be tuned, and the case of a condensate of point bosons is retrieved in the limit of tightly bound pairs.

Order parameter for the superfluid Fermi gas In atomic Fermi gases, superfluidity requires the presence of pairing between the atoms. The pairing partners are typically different hyperfine states of a given fermionic isotope; for the sake of simplicity we will refer to the two pairing states as “spin-up” and “spindown.” At low temperatures (such that the de Broglie wavelength is much larger than the range of the interatomic potential) scattering between atoms is dominated by the s-wave channel, and fully characterized by the s-wave scattering length as. For fermionic atoms, the requirement of antisymmetry precludes s-wave interactions between same spin atoms. Pairing between ultracold, neutral atoms was first investigated theoretically in the framework of BCS theory by Houbiers and Stoof (1999). In the conventional BCS model, the parameter that controls the strength of the pairing is the inverse product of the BCS interaction strength with the density of states at the Fermi level. In (three-dimensional) cold Fermi gases, this parameter is replaced by 1/(kFas), with kF, the Fermi wave number. Remarkably, the s-wave scattering length as can be tuned experimentally using Feshbach resonances. These are scattering resonances between a bound state in a closed scattering channel, and the scattering state of colliding atoms in the open channel. Here, these channels are characterized by different hyperfine states so that the resonance can be tuned via an external magnetic field through the Zeeman effect. This allows to enhance the strength of the attractive interaction (and raise the critical temperature), by making as more strongly negative. As the bound state of the closed channel dips below the scattering energy (and as becomes positive), the bound molecular state mixes with the scattering state, and allows for a BEC of strongly bound molecules. Experimentally, the BEC–BCS crossover region (1 ≳ 1=ðkF as Þ ≳ −1) is entirely accessible, including the unitary limit (1/(kFas) ¼ 0). This results in a much richer physics than for the limiting case of a BEC discussed in the previous section. The order parameter (r, t) now represents the pair condensate. To see what that means, we turn to the path integral formalism, and start from the imaginary time action functional S for the Fermi gas described by Grassmann variables cs for fermions with spin s:

190

Higgs and Goldstone modes in cold atom systems Z S¼

b 0

"

Z dt

dr

#  X  1 2 c s ∂t − rr − ms cs + gc" c# c# c" , 2m s

(3)

where we have set ℏ ¼ kB ¼ 1, and b ¼ 1/kBT represents the inverse temperature. The chemical potentials for each spin species ms can in principle be set independently. The interaction strength g is related to the scattering length as via the renormalization relation Z 1 m d3 k m − (4) ¼ 3 2: g 4pas k

Fig. 6 Aharonov–Casher Interferometer: Similar to Fig. 3, two wires at different potentials V L V R are connected to a metallic ring. Electrons approaching the ring from the left at polar angle q ¼ 0 are partially reflected back to the left wire and partially transmitted to the right wire at polar angle q ¼ a (reflection matrix r and transmission matrix t ). The ring is subject to a perpendicular electric field so that the Rashba spin–orbit mechanism generates an effective magnetic field along the radial direction ^nðq Þ ¼ ðcos q , sin q , 0Þ (Fig. 5B). The electric field is homogeneous and acts only on the circular area confined by the ring. The local phase factor for any polar angle q is eib ^nðq Þ  s . Taken from Avishai et al. (2019), where an analysis of Aharonov–Casher interferometer subject to some version of Dresselhaus spin–orbit coupling is also analyzed. Taken from Avishai Y, Totsuka K, and Nagaosa N (2019) Aharonov–Casher Phase Factor in Mesoscopic Systems. Journal of the Physical Society of Japan 88(8): 084705.

228

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review 8 2 > < − d CðxÞ ¼ k2 CðxÞ, ðwhen the electron is on left or right wireÞ dx2 2 > : d − b^ nðyÞ  s CðyÞ ¼ k2 CðyÞ ðwhen the electron is on the ringÞ: − i dy

so we have

(32)

We consider an electron with spin projection s ¼", # approaching the ring from the left wire at energy e ¼ k2. It is partly transmitted into the right wire with spin projection m ¼", # and partly reflected back into the left wire with m ¼", #. The corresponding matrix elements of the 2  2 transmission and reflection matrices are tms, rms, explicitly     t "" t "# r "" r "# t¼ r¼ : t #" t ## r #" r ## The technique for calculating t and r is similar to that used in the Aharonov–Bohm interferometer, but is slightly more complicated. 2 The conductance through the interferometer is given by the Landauer formula (Landauer, 1957), g ¼ eh Tr½t { t. 2 Fig. 7 plots the cosine of the Aharonov–Casher phase lAC and the conductance g (in units of eh ) as a function of the spin–orbit strength b for two values of the right splitter angle a.

Conductance and polarization in transmission through a square interferometer ^

It was shown in the discussion of Eq. (31) that the SU(2) phase factor eilAC b  s can be written as a sum of two 2  2 matrices, one has nonvanishing trace and one is traceless. Both depend on the Aharonov–Casher phase lAC, but the role of the traceless term (in ^ was not explained. Using a relatively simple tight-binding model that can be solved analytically, it will particular the unit vector b) be shown that this term is responsible for electron polarization. Note that spin–orbit coupling obeys time reversal invariance but the specification of scattering boundary condition destroys it so that polarization is feasible (Meijer et al., 2005; Entin-Wohlman et al., 2010; Nagasawa et al., 2012, 2013; Avishai and Band, 2017). It can be shown (Avishai et al., 2019) that in two-port systems like the Aharonov–Casher interferometer discussed above, there is no polarization in the outgoing lead (assuming that the incoming beam is not polarized). Here we show that in a multiport system, it is possible to get a polarized outgoing beam due to Rashba spin–orbit coupling. Consider, as in Fig. 8, a tight-binding model of noninteracting electrons moving along two wires and scattered from a region that carries an SU(2) flux where the phase factors in the product do not commute and depend on the spin–orbit strength. The system is laid on the x − z plane and consists of two chains (metallic wires) numbered a ¼ 1, 2 with each chain consisting of sites −1 < n < 1, which are then connected to each other by the two rungs of the square at n ¼ 0 and n ¼ 1 (see Fig. 8). A uniform electric field acts along y^ upon the square, thereby lead to a Rashba spin–orbit coupling along the links of the square. This tight-binding model is treated within the formalism of second quantization.

Fig. 7 Plots of dimensionless conductance g (blue curve) and cosine of the Aharonov–Casher phase cos l AC (orange curve) as function of spin–orbit strength b for the ring interferometer schematically depicted in Fig. 6. The wave number is set to kR ¼ 1.25, and (A) a ¼ p , and (B) a ¼ 1.7. As a property of the closed loop, l AC is independent of a . The conductance, that also encodes the geometric properties of the interferometer, naturally depends on a , but in both cases it is periodic in the Aharonov–Casher phase l AC. Taken from Avishai Y, Totsuka K, and Nagaosa N (2019) Aharonov–Casher Phase Factor in Mesoscopic Systems. Journal of the Physical Society of Japan 88(8): 084705.

0

eiβσx

eiβσz

z

–1

1

2

3

α=1

e–iβσz

n = –2

x e–iβσx

α=2

Fig. 8 Two-channel tight-binding model for electron scattering from a square region in the x-z plane (shown by bold blue lines) subject to a uniform electric field E^y acting only on the square. The corresponding SU(2) hopping matrix elements are given by eibs x along the horizontal links and e ibs z along the vertical links. Taken from Avishai Y, Totsuka K, and Nagaosa N (2019) AharonovCasher Phase Factor in Mesoscopic Systems. Journal of the Physical Society of Japan 88 (8): 084705.

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

229

The creation and annihilation operators for the spin projection s ¼", # are indexed as c{a,n,s and ca,n,s, respectively, and the spin–orbit interaction is active only on the four links forming the square (shown by the bold lines in Fig. 8). The Hamiltonian is written as H¼H0 + H1 with P P { ca, − ðn+1Þ ca, − n H0 ¼ − t a¼1, 2n0 P P { −t ca,ðn+1Þ ca,n + h:c:, (33) a¼1, 2n1 P { − ibsx P { ibsz H1 ¼ c1,n e c2,n + ca,0 e ca,1 + h:c:, n¼0, 1 a¼1, 2 where c{a,n  (c{a,n,", c{a,n,#). The solution of the scattering problem yields the 4  4 (2 for spin and 2 for channel) transmission and reflection matrices t and r whose matrix elements ta0 s0 ;as and r a0 s0 ; as give the transmission and reflection amplitudes for scattering of a spin-s electron impinging from channel a to one in the channel a0 with spin projection s0 . They depend on the SOC strength, and the wave number k that determines the energy e ¼ − 2t cos k. On the other hand, the Aharonov–Casher phase lAC is a property of the closed loop irrespective of the scattering energy. In the present case, it is given by calculating the product of the four matrices shown in Fig. 8, and using Eq. (31), giving cos lAC ðbÞ ¼ 1 − 2 sin 4 b:

(34)

Before presenting results related to the scattering problem, it is worthwhile to check the eigenvalue problem of a system of an electron hopping on an isolated square (without wires; the square shown by bold lines in Fig. 8) as a closed system, especially whether they depend only on the Aharonov–Casher phase. The tight-binding Hamiltonian assumes the following form 1 0 0 eibsz e − ibsx 0 B e − ibsz − ibsx C 0 0 e C B (35) H□ ¼ B ibs C, @ e x 0 0 eibsz A 0

eibsx

e − ibsz

0

where each entry is a 2  2 matrix acting on the spin space at each site of the square. Simple calculations show that there are four different eigenvalues each of which is twofold (Kramers’) degenerate: E□ ¼ 2 cos

lAC ðbÞ l ðbÞ ,  2 sin AC : 4 4

(36)

In any case, the eigenvalues depend on b only through lAC(b) as defined in Eq. (34). The solution of the scattering problem yields the 4  4 (2 for spin and 2 for channel) transmission and reflection matrices t and r whose matrix elements ta0 s0 ;as and r a0 s0 ; as give the transmission and reflection amplitudes for scattering of a spin-s electron impinging from channel a to one in the channel a0 with spin projection s0 . They depend on the spin–orbit coupling strength b and the wave number k that determines the energy e ¼ − 2t cos k. Going back to the scattering problem, we will now inspect the relation between the ACP and a couple of experimental observables. Specifically, we are interested in the conductance g and the transmitted polarization vector P, defined as gðk; bÞ ¼ Tr½t { t, P ¼

Tr½t { St , S ¼ 122 s: g

(37)

The following expressions for the conductance g and the transmitted polarization Py are as follows: 16ð − 5 + 4 cos 2kð1 − cos lAC Þ sin 2 k , − 17 + 8 cos 2k + 8ðcos 4k − 4 cos 2kÞð1 − cos lAC Þ − 16ð1 − cos lAC Þ2 2 sin 2kð1 − cos lAC Þ , P x ¼ P z ¼ 0: Py ¼ 5 − 2 cos 2kð1 − cos lAC Þ

g¼

(38)

When the spin–orbit coupling strength b ! 0 then lAC ! 0. The conductance remains finite but the polarization vanishes (as it must). But when lAC ! 0, the traceless part of the Aharonov–Casher phase factor in Eq. (31) tends to zero as well. Hence, this part of the phase factor is related (albeit in a complex way) to an experimental quantity.

Analogy of Aharonov–Bohm effect in cold atoms physics In the Aharonov–Bohm effect, electrons (charged particles) encircle a magnetic flux in configuration space. There is a beautiful analog pertaining to neutral atoms. When cold atoms are subject to a periodic optical potential the corresponding dynamics is similar to that of Bloch electrons in crystals, and the wave functions are Bloch functions at crystal momentum k and energy en(k). Since the atoms are neutral, and one cannot speak of Aharonov–Bohm phase. However, moving an atom along a closed curve in k space endows its wave function with a phase denoted as Berry phase (Berry 1984; Xiao et al., 2010).

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Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

Thus, in cold atom systems, instead of the Aharonov–Bohm phase, the object of study is the Berry phase, that contains information on the pertinent energy band. In addition, the Berry phase is related to the topological invariant known as the Chern number (Thouless et al., 1982). For a graphene-like optical potential forming a hexagonal lattice, the Berry phase is obtained by moving an atom that encircles a Berry flux along a closed loop C around a Dirac point in momentum space (see Fig. 1 in Duca et al., 2015). For a Bloch function cnk(r) ¼ eikrunk(r) corresponding to an energy band n and lattice momentum k, the analog of the vector potential A is the Berry connection An(k) ¼ ihunk| rk| unki, and the analog of the magnetic field is the Berry curvature On(k) ¼ rk  An(k). The Berry phase is then defined as I Z (39) ’n ¼ An ðkÞdk ¼ On ðkÞd2 k, C

S

where S is the area encircled by the curve C in momentum space. The Berry phase includes information on the geometry of the corresponding Bloch band. It is gauge invariant and hence it can be measured. For a graphene hexagonal lattice, the authors of Duca et al. (2015) built an interferometer that enabled the detection of the Berry flux at each Dirac point. Thereby, they could determine the Berry curvature with high momentum resolution. As it turns out, understanding the physics of multiple bands with degeneracies in topological insulators, requires other concepts beyond the Berry phases. As shown in Li et al. (2016), such systems can be described using the technique of Wilson lines (Yu et al., 2011; Alexandradinata et al., 2014). Chern numbers are topological invariants that classify bands in topological materials (Hasan and Kane, 2010; Asbóth et al., 2016 ; Qi and Zhang, 2011; Goldman et al., 2014; Cooper et al., 2019). They can be calculated using the Berry curvature of the band (Fukui et al., 2005). The Chern numbers determine properties of the edge state of materials. This generalization of Aharonov phase has become a very important part of condensed mater physics.

Magnetic monopoles A magnetic monopole is an elementary particle with a magnetic charge. The existence of a magnetic monopole would violate Gauss’s law for magnetism r  B ¼ 0 and help to explain the law of charge quantization as formulated by Dirac (1931). So far, magnetic monopoles have not been observed (Milton, 2006). The theoretical investigation of Dirac monopoles is extremely rich but will not be considered here. In condensed matter physics (specifically in strongly interacting systems), there are objects that behave as monopole (although they do not exist as elementary particles) are encountered. An example is that of quasi-particles with fractional charge (suggested by Laughlin) in the fractional quantum Hall effect (Willet et al., 2013; Nakamura et al., 2019). In spin ice materials, electron correlations lead to phenomena that are similar to those of magnetic monopoles (e.g., see Castelnovo et al., 2008; Diamentiny et al., 2021, and references therein) but, again, they should not be confused with elementary particles having magnetic charge. An experimental imitation of a magnetic monopole at the tip of a long nanoscopically thin magnetic needle was reported in Béché et al. (2014).

Dirac quantization and phase factors If a monopole does exist, it is interesting to consider the spectrum of an electron of charge −e in the central field of a monopole of magnetic charge g such that the magnetic field is B ¼ g rbr2 , and as shown in Wu and Yang (1975), in this case there does not exist a singularity-free vector potential A(r) in 3D space. To show this, assume the monopole to be fixed at the origin and denote the electron position in spherical coordinates r, y, ’. Denote by F(r, y) the magnetic flux through a cup bounded by the circular loop r, y, 0  ’  2p. From Maxwell equations we have, I Fðr, yÞ ¼ A’ ðr, yÞd’: (40) Evidently, F(r, y) ¼ 2pg(1 − cos y). For y ¼ 0 (the north pole) we indeed get F(r, 0) ¼ 0, which is reasonable, but for y ¼ p (the south pole) we get F(r, p) ¼ 4pg which is not reasonable because the length of the loop is zero. Thus A’ is singular at y ¼ p and the concept of phase factors introduced in the “Gauge invariance” section should be reexamined. This singularity is sometimes termed as the Dirac string. The solution to this contradiction (Wu and Yang, 1975) is to divide the spherical shell for fixed r and ’ into three regions, 0  y  p2 − d, p2 − d  y  p2 + d and p2 + d  y  p where 0 < d < p2 : In the first and third regions, the corresponding vector potentials are, g ð1 − cos yÞ r sin y −g ð1 + cos yÞ A’ ðr, yÞ ¼ r sin y A’ ðr, yÞ ¼

p 0  y  − d, 2 p + d  y  p, 2

(41)

so that there is no singularity, not at y ¼ 0 nor at y ¼ p. In the overlap region the two vector potentials are related by a gauge transformation

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review 2ige

Sð’Þ ¼ e ħc ’ :

231 (42)

Since S(’) must be uniquely defined we must have the Dirac quantization rule, 2ge ¼ n ¼ integer: ħc

(43)

Wu and Yang then show that if Dirac quantization holds, the concept of phase factors is valid for the description of electromagnetism also if monopoles do exist. In a subsequent paper (Wu and Yang, 1976) the authors introduced the concept of Dirac monopole without strings.

Tight-binding spectra on spherical graphs: Avoiding the Dirac string We now analyze a tight-binding model for an electron in the field of a magnetic monopole, in which the Dirac string is avoided (Avishai and Luck, 2008a). It is based on the use of the group U(1) (whose elements are the phase factors) and not on its algebra (whose elements are the path integrals of the vector potentials). The simplest example of this model is that of an electron hopping on the vertices of a tetrahedron at whose center there is a magnetic monopole of magnetic charge g ¼ nħc 2e : The tetrahedron has V ¼ 4 vertices, F ¼ 4 faces, and L ¼ 6 links. According to Euler theorem L ¼ F + V − 2. Denote by a{i and ai the creation and annihilation operators of the electron on vertex i, and by Uij the phase factor between two adjacent vertices i, j by Uij (these are elements of the gauge group U(1) living on these links). The tight-binding Hamiltonian of the system is X { b¼ H ai U ij aj + h:c, (44)

b is complex, i.e., it is not invariant under where the sum runs over the L ¼ 4 oriented links h ij i of a tetrahedron. The Hamiltonian H time reversal. One has U ij ¼ U ji− 1 ¼ U ji∗ :

(45)

The product of the phase factors around each face is related to the outgoing magnetic flux ’ through this face as U ij U jk U ki ¼ eð2pi’=F0 Þ ,

(46)

where F0 ¼ is the quantum flux unit. By Gauss’s theorem the total outgoing magnetic flux is F ¼ 4pg so that ’ ¼ pg. The b is thus represented by a V  V ¼ 4  4 complex Hermitian matrix H such that Hij ¼ Uij. It is easy to check that Hamiltonian H trH ¼ 0 and trH2 ¼ 2L ¼ 12. The main step in the explicit construction of the Hamiltonian matrix consists in finding a configuration of the gauge field, i.e., a set of phase factors Uij, so that Eq. (46) is satisfied. This procedure is carried out in Avishai and Luck (2008a) and results in the spectrum shown in Table 1. Calculations are reported for the five platonic solids tetrahedron, cube, octahedron, dodecahedron, icosahedron (for which the spectrum is periodic in the number of faces F, namely E(n + F) ¼ E(n), n ¼ 0, 1, . . .F − 1), and the fullerene (for which the spectrum is not periodic in n because the ratio between the areas of a hexagon and a pentagon is not a rational number. hc c

Relation to Aharonov–Casher effect In Avishai and Luck (2008b), the authors investigate a tight-binding model for an electron confined to move on a two-dimensional surface with spherical topology and subject to a spin-orbit interaction. The latter is assumed to be generated by the radial electric field produced by a static point charge sitting at the center of the sphere. The tight-binding Hamiltonian considered is a discretization of the familiar form of the spin-orbit Hamiltonian L  S. It involves SU(2) hopping matrices (phase factors) of the form exp(imn  s) living on the oriented links of the graph and n is a unit vector along the link. For a given structure, the dimensionless coupling constant m is the only parameter of the model (see below the expression for m). Table 1 Energy levels E of the tetrahedron and their multiplicities m, for each value of the magnetic charge n over one period. n

E

m

0

3 −1 pffiffiffi 3 pffiffiffi − 3 1 −3

1 3 2 2 3 1

1,3 2

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Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

For definiteness let us consider an electron hoping on the vertices A, B, C, D of a tetrahedron (a spherical graph of sphere radius b The tight-binding model is defined by R), at whose center O there is a charge q ¼ Ze. Denote by Y the angle of the triangle AOB: means of the Hamiltonian  X { b¼ H aA U AB aB + h:c: , (47) ðABÞ

where the sum runs over the six oriented links (AB) of the tetrahedron, whereas     aA" a{A ¼ a{A" , a{A# , a ¼ aA#

(48)

are respectively the electron creation and annihilation operators at site A, and the matrices UAB are elements of the non-Abelian gauge group SU(2), i.e., 2  2 unitary matrices with unit determinant, describing the spin-orbit coupling on an electron hopping from site A to a neighboring site B. In analogy with the Abelian case the phase factor UAB is expressed as a path-ordered integral: ( ) Z U AB ¼ P exp −ig

gðA,BÞ

ðE  sÞ  dr ,

(49)

where g¼

e , 4mc2

E¼q

r : r3

(50)

and g(A, B) is a curve joining A and B. If the path g(A, B). is planar, i.e., entirely contained in the OAB plane, In this case, at every b  B: b As a point of the path the infinitesimal vector r  dr is perpendicular to the latter plane, i.e., aligned with the vector nAB ¼ A consequence, the path-ordering prescription is not needed, and Eq. (49) can be recast as an ordinary integral ( ! ) Z r  dr U AB ¼ exp igq s , r3 (51) gðA,BÞ ¼ expðimðYÞ nAB  sÞ ¼ cos m + i sin m nAB  s: The functional form of m(Y) depends on the form of the planar curve g(A, B). Intuitively, we can choose either a link or an arc of a great circle joining A and B. Consequently, we are left with the following expression for the parameter m, for both choices of the path g(A, B):  2 tanðY=2Þ ðpath along linkÞ, mðYÞ ¼ e  (52) Y ðgreat-circle pathÞ with e¼

gq qe , q ¼ Ze: ¼ R 4mc2 R

(53)

An analysis of the energy spectrum is carried out for the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and the C60 Fullerene. Except for the latter, the m-dependence of all the energy levels is obtained analytically in closed form (Table 2). Special values of m. The following two values of the parameter m: m0 ¼ Y=2,

m1 ¼ Y=2 + p,

(54)

are special in several respects. First, the m $ Y − m symmetry and the semi-periodicity imply that m0 and m1 are symmetry axes of the energy spectrum, if displayed as a function of m. Second, the Kramers degeneracy and the m $ Y − m symmetry imply that all energy levels are at least fourfold degenerate, i.e., that all the multiplicities are integer multiples of 4. Third, there is a striking

Table 2 Energy levels Ea(m) of the tetrahedron and their multiplicities ma. Horizontal lines separate groups of levels related to each other by the m $ Y − m symmetry. a

Ea

ma

1 2 3

3 cos m pffiffiffi −cosm + 2 2 sin m pffiffiffi 3 pffiffiffi − 3 pffiffiffi −cosm − 2 2 sin m

2 2 2 2 4

3

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

233

Fig. 9 Energy levels E of the tetrahedron as a function of the magnetic charge n over one period. Line segments show how individual levels ‘jump’ as n is increased by one unit.

Fig. 10 Plot of the energy spectrum of the tetrahedron against m/(2p ) over one period. Vertical dashed lines: symmetry axes at the special values of m given in Eq. (54). The energies at the intersection point of the first vertical dashed line are identical with the two energies corresponding to the energies of an electron in the field of n ¼ 1 quantized magnetic dipole, shown in Fig. 9.

correspondence between the present problem at the special value m0 and the tight-binding problem in the presence of a magnetic monopole investigated in “Tight-binding spectra on spherical graphs: Avoiding the Dirac string” subsection. For the symmetric point m ¼ Y/2, the problem can be exactly mapped onto a tight-binding model in the presence of the magnetic field generated by a Dirac monopole, studied in “Tight-binding spectra on spherical graphs: Avoiding the Dirac string” subsection. Thus, in Fig. 10, the energies at the intersection point of the first vertical dashed line are identical with the two energies corresponding to the energies of an electron in the field of n ¼ 1 quantized magnetic dipole, shown in Fig. 9. This result, proved analytically in Avishai and Luck (2008b), is a beautiful manifestation of the relation between Aharonov–Bohm and Aharonov–Casher effects.

Conclusion Aharonov–Bohm and Aharonov–Casher effects substantiate the fundamental role of quantum mechanics as a solid theory of Nature (Richter, 2012). In addition, they expose its beauty and peculiarities, and, at the same time, point out the possible occurrence of its puzzles. The Aharonov–Bohm effect constitutes a simple manifestation of Abelian gauge theory, while the Aharonov–Casher effect constitutes a simple manifestation of non-Abelian gauge theory. These two effects are published several decades ago, and remain among the building blocks of quantum mechanics ever since. The question of duality between the Aharonov–Casher and Aharonov–Bohm effects is raised by Hegen (1990), Bogachek and Landman (1994), Borunda et al. (2008), Rohrlich (2010), and Vaidman (2012). A theoretical analysis relating the two effects is suggested in Avishai and Luck (2009). Paradoxes of the Aharonov–Bohm and the Aharonov–Casher effects are discussed by Vaidman (2013). Both effects stress the essential role of phase of the wave functions (Berry, 1984). The Aharonov and Bohm effect involves the motions of charged particles (e.g., electrons) around but away from a region in which the magnetic field is finite. In the Aharonov and Casher effect (Aharonov and Casher, 1984), an electrical field is considered and its effect on particles with nonzero magnetic moment such as electrons (but also neutrons as they need not be charged) is analyzed. Both of these effects have direct relevance to condensed matter systems. Here we described these two effects and reviewed their relevance in condensed matter (mainly in mesoscopic systems). In this short review, some aspects of the relevance of these effects to condensed matter physics are illuminated. In particular, we focus on the physics of mesoscopic systems, for which numerous experiments confirm the two effects. For the Aharonov–Bohm effect, examples of interference around a tube carrying magnetic flux, electrons on the ring (spectrum and persistent current) and an

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Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

interferometer are analyzed. For the Aharonov–Casher effect, examples of an interferometer, and an analysis of transmission and polarization in a two channel system are presented. It is the authors’ hope that the broad list of references will enable the readers to become acquainted with additional aspects of these effects in condensed matter physics.

Acknowledgments The authors thank Tapash Chakraborty for suggesting this project, and Lev Vaidman for valuable comments.

References Aharonov Y and Anandan J (1987) Phase change during a cyclic quantum evolution. Physical Review Letters 58: 1593. Aharonov Y and Bohm D (1959) Significance of electromagnetic potentials in quantum theory. Physics Review 115(3): 485–491. Aharonov Y and Bohm D (1961) Further considerations on electromagnetic potentials in the quantum theory. Physical Review 123(4): 1511–1524. Aharonov Y and Casher A (1984) Topological quantum effects for neutral particles. Physical Review Letters 53(4): 319–321. Aharonov Y, Pearle P, and Vaidman L (1988) Comment on proposed Aharonov-Casher effect: Another example of an Aharonov-Bohm effect arising from a classical lag. Physics Review A 37: 4052–4055. Aharony A, Entin-Wohlman O, Halperin BI, and Imry Y (2002) Phase measurement in the mesoscopic Aharonov-Bohm interferometer. Physical Review B 66: 115311. Aharony A, Entin-Wohlman O, and Imry Y (2003) Measuring the transmission of a quantum dot using Aharonov-Bohm interferometers. Journal of the Physical Society of Japan 72(Supp. A): 112 (cond-mat/0208068). Aharony A, Entin-Wohlman O, and Imry Y (2005) Phase measurements in open and closed Aharonov-Bohm interferometers. Physica E: Low-Dimensional Systems and Nanostructures 28: 283 (cond-mat/0412027). Aharony A, Entin-Wohlman O, Otsuka T, Katsumoto S, Aikawa H, and Kobayashi K (2006) Breakdown of phase rigidity and variations of the Fano effect in closed Aharonov-Bohm interferometers. Physical Review B: 73. Aharony A, Takada S, Entin-Wohlman O, Yamamoto M, and Tarucha S (2014) Aharonov-Bohm interferometry with a tunnel-coupled wire. New Journal of Physics 16. Aharony A, Entin-Wohlman O, Tzarfati L, Hevroni H, Karpovski R, Shelukhina M, Umansky V, and Palevski A (2019) Possible observation of Berry phase in Aharonov-Bohm rings of InGaAs. Solid State Electronics 155: 117. Akkermans E and Montambaux G (2007) Mesoscopic Physics of Electrons and Photons. Cambridge University Press. Alexandradinata A, Dai X, and Bernevig BA (2014) Wilson-loop characterization of inversion-symmetric topological insulators. Physical Review B 89: 155114. Altland A, Gefen Y, and Montambaux G (1996) What is the Thouless energy for ballistic systems? Physical Review Letters 76: 1130. Altshuler BL, Aronov AG, and Spivak BZ (1981) The Aharonov-Bohm effect in disordered conductors. JETP Letters 33: 94. Anandan J (1989) Electromagnetic effects in the quantum interference of dipoles. Physics Letters A 138: 347. Anderson PW (1958) Absence of diffusion in certain random lattices. Physical Review 109: 1492. Aronov AG and Lyanda-Geller Y (1993) Spin-orbit Berry phase in conducting rings. Physical Review Letters 70: 343. Asbóth JK, Oroszlány L, and Pályi A (2016) A Short Course on Topological Insulators. Springer International Publishing. Avishai Y and Band Y (1987) Quantum scattering determination of magneto-conductance for 2D systems. Physical Review Letters 58: 2251. Avishai Y and Band Y (2017) Simple spin-orbit based devices for electron spin polarization. Physical Review B 95: 104429. Avishai Y and Kohmoto M (1993) Quantized persistent currents in quantum dot at strong magnetic field. Physical Review Letters 71: 279. Avishai Y and Luck JM (2008a) Tight-binding electronic spectra on graphs with spherical topology. I. The effect of a magnetic charge. Journal of Statistical Mechanics. P06007, arXiv:0801.1460. Avishai Y and Luck JM (2008b) Tight-binding electronic spectra on graphs with spherical topology. II. The effect of spin orbit interaction. Journal of Statistical Mechanics. P06008, arXiv:0802.0795. Avishai Y and Luck JM (2009) Magnetization of two coupled rings. Journal of Physics A: Mathematical and Theoretical 42: 175301. Avishai Y, Ekstein H, and Moyal JE (1972) Is the Maxwell field local? Journal of Mathematical Physics 13: 1139. Avishai Y, Hatsugai Y, and Kohmoto M (1993) Persistent currents and edge states in a magnetic field. Physical Review B 47: 9501. Avishai Y, Totsuka K, and Nagaosa N (2019) Aharonov-Casher phase factor in mesoscopic systems. Journal of the Physical Society of Japan 88(8): 084705. Bachtold A, Strunk C, Salvetat J-P, Bonard J-M, Forró L, Nussbaumer T, and Schönenberger C (1999) Aharonov-Bohm oscillations in carbon nanotubes. Nature 397(6721): 673. Band Y and Avishai Y (2012) Quantum Mechanics with Application to Nanotechnology and Information Science. Elsevier. Batelaan H and Tonomura A (2009) The Aharonov-Bohm effects: Variations on a subtle theme. Physics Today 62(9): 38–43. Béché A, Van Boxem R, Van Tendeloo G, and Verbeeck J (2014) Magnetic monopole field exposed by electrons. Nature Physics 10: 26. Bergsten T, Kobayashi T, Sekine Y, and Nitta J (2006) Experimental demonstration of the time reversal Aharonov-Casher effect. Physical Review Letters 97: 196803. Berkovits R and Avishai Y (1995) Significant interaction-induced enhancement of persistent currents in 2D disordered cylinders. Europhysics Letters 29: 475. Berkovits R and Avishai Y (1996) Interacting electrons in disordered potentials: Conductance versus persistent currents. Physical Review Letters 76: 261. Berry MV (1984) Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London, Series A 392: 45. Bloch F (1970) Josephson effect in a superconducting ring. Physics Review B 2: 109. Bogachek EN and Landman U (1994) Aharonov-Bohm and Aharonov-Casher tunneling effects and edge states in double-barrier structures. Physical Review B 50(4): 2678–2680. Borunda MF, Liu X, Kovalev AA, Xiong-Jun L, Jungwirth T, and Sinova J (2008) Aharonov-Casher and spin Hall effects in two-dimensional mesoscopic ring structures with strong spinorbit interaction. Physical Review B 78(24): 245315. Büttiker M (1985) Small normal-metal loop coupled to an electron reservoir. Physical Review B 32: 1846. Büttiker M (1986) Four-terminal phase-coherent conductance. Physical Review Letters 57: 1761. Büttiker M (1988) Symmetry of electrical conduction. IBM Journal of Research and Development 32: 317. Büttiker M (1992) Flux-sensitive correlations of mutually incoherent quantum channels. Physical Review Letters 68: 843. Büttiker, M., 2901. Scattering theory of thermal and excess noise in open conductors. Physical Review Letters 65. Büttiker M, Imry Y, and Landauer R (1983) Josephson behavior in small normal one-dimensional rings. Physics Letters 96: 365. Büttiker M, Imry Y, Landauer R, and Pinhas S (1985) Generalized many-channel conductance formula with application to small rings. Physical Review B 31: 6207. Byers N and Yang CN (1961) Theoretical considerations concerning quantized magnetic flux in superconducting cylinders. Physical Review Letters 7: 46. Castelnovo C, Moessner R, and Sondhi SL (2008) Magnetic monopoles in spin ice. Nature 451: 42. Cernicchiaro G, Martin T, Hasselbach K, Mailly D, and Benoit A (1997) Channel interference in a quasi-ballistic Aharonov-Bohm experiment. Physical Review Letters 79: 273.

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

235

Chambers RG (1960) Shift of an electron interference pattern by enclosed magnetic flux. Physical Review Letters 5(1): 3–5. Cheung HF, Riedel EK, and Gefen Y (1989) Persistent currents in mesoscopic rings and cylinders. Physical Review Letters 62(5): 587. Cimmino A, Opat GI, Klein AG, Kaiser H, Werner SA, Arif M, and Clothier R (1989) Observation of the topological Aharonov-Casher phase shift by neutron interferometry. Physical Review Letters 63(4): 380–383. Cooper NR, Dalibard J, and Spielman IB (2019) Topological bands for ultracold atoms. Reviews of Modern Physics 91: 015005. Diamentiny MC, Trugenberger CA, and Vinokur VM (2021) Quantum magnetic monopole condensate. In: Communications Physics, 4. article 25. Dirac PAM (1931) Quantised singularities in the electromagnetic field. Proceedings of the Royal Society A 133: 60–72. Dowker JS (1967) A gravitational Aharonov-Bohm effect. Nuovo Cimento B (1965-1970) 52: 129–135. Dresselhaus G (1955) Spin-orbit coupling effects in zinc blende structures. Physics Review 100(2): 580. Duca L, Reitter T, Bloch M, Schleier-Smith M, and Schneider U (2015) An Aharonov-Bohm interferometer for determining Bloch band topology. Science 347: 288. Edwards JT and Thouless DJ (1972) Numerical studies of localization in disordered systems. Journal of Physics C: Solid State Physics 5: 807. Elion WJ, Wachters JJ, Sohn LL, and Mooij JD (1993) Observation of the Aharonov-Casher effect for vortices in Josephson-junction arrays. Physical Review Letters 71(14): 2311–2314. Engel HA, Rashba EI, and Halperin BI (2006) Theory of spin hall effects in semiconductors. In: Handbook of Magnetism and Advanced Magnetic Materials. John Wiley and Sons, Ltd (cond-mat/0603306). Entin-Wohlman O and Gefen Y (1989) Persistent currents in two-dimensional metallic cylinders: A linear response analysis. Europhysics Letters 8(5): 477. Entin-Wohlman O, Gefen Y, Meir Y, and Oreg Y (1992) Effects of spin-orbit scattering in mesoscopic rings: Canonical-versus grand-canonical-ensemble averaging. Physical Review B 45(20): 11890. Entin-Wohlman O, Imry Y, and Aharony A (2003) Persistent currents in interacting Aharonov-Bohm interferometers, and their enhancement by acoustic radiation. Physical Review Letters 91 (cond-mat/0302146). Entin-Wohlman O, Imry Y, and Aharony A (2004) Effects of external radiation on biased Aharonov-Bohm rings. Entin-Wohlman O, Aharony A, Tokura Y, and Avishai Y (2010) Spin-polarized electric currents in quantum transport through tubular two-dimensional electron gases. Physical Review B 81. 075439–075445. Foldy LL and Wouthuysen SA (1950) On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Physics Review 78(1): 29–36. Fröhlic J and Studer UM (1993) Gauge invariance and current algebra in non-relativistic many-body theory. Journal of Mathematical Physics 65: 733. Frustaglia D and Nitta J (2020) Geometric spin phases in Aharonov-Casher interference. Solid State Communications 311: 113864. Frustaglia D and Richter K (2004) Spin interference effects in ring conductors subject to rashba coupling. Physical Review B 69: 235310. Fukui T, Hatsugai Y, and Suzuki H (2005) Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances. Journal of the Physical Society of Japan 74: 1674. Gefen Y, Imry Y, and Azbel MY (1984) Quantum oscillations and the Aharonov-Bohm effect for parallel resistors. Physical Review Letters 52(2): 129. Goldman N, Juzeliunas G, Öhberg P, and Spielman IB (2014) Light-induced gauge fields for ultracold atoms. Reports on Progress in Physics 77: 126401. Grbic B, Leturcq R, Ihn T, Ensslin K, Reuter D, and Wieck AD (2007) Aharonov-Bohm oscillations in the presence of strong spin-orbit interactions. Physical Review Letters 99: 176803. Hagen CR (1990) Aharonov-Bohm scattering of particles with spin. Physical Review Letters 64(5): 503. Hasan MZ and Kane CL (2010) Colloquium: Topological insulators. Reviews of Modern Physics 82: 3045. Hegen CR (1990) Exact equivalence of spin-1/2 Aharonov-Bohm and Aharonov-Casher effects. Physical Review Letters 64: 2347. Hemanta KK, Biswas S, Ofek N, Umansky V, and Heiblum M (2023) Anyonic interference and braiding phase in a Mach-Zehnder interferometer. Nature Physics 19: 515–521. https:// doi.org/10.1038/s41567-022-01899-z. Heras R (2022) The Aharonov-Bohm effect in a closed flux line. The European Physical Journal Plus 137: 641. Hohensee MA, Estey B, Hamilton P, Zeilinger A, and Muller H (2012) Force-ffee gravitational redshift: Proposed gravitational Aharonov-Bohm experiment. Physical Review Letters 108 (23): 230404. Imry Y (1986) Active transmission channels and universal conductance fluctuations. Europhysics Lett 1: 249. Imry Y (1995) Coherent propagation of two interacting particles in a random potential. Europhysics Letters 30: 405. Imry Y (1997) Introduction to Mesoscopic Physics, p. 110. Oxford University Press (Fig. 5.4). Imry Y and Webb RA (1986) Quantum interference and the Aharonov-Bohm effect. Scientific American 260(4): 56–62. Kane EO (1957) Band structure of indium antimonide. Journal of Physics and Chemistry of Solids 1: 249. Köenig M, Tschetschetkin A, Hankiewicz EM, Sinova J, Hock V, Daumer V, et al. (2006) Direct observation of the Aharonov-Casher phase. Physical Review Letters 96. Landauer R (1957) Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM Journal of Research and Development 1: 223. Lee PA and Stone AD (1985) Universal conductance fluctuations in metals. Physical Review Letters 55: 1622. Levy LP, Dolan G, Dunsmuir J, and Bouchiat H (1990) Magnetization of mesoscopic copper rings: Evidence for persistent currents. Physical Review Letters 64: 2074. Li T, Duca L, Reitter M, Grusdt F, Demler E, Endres M, Schleier-Smith M, Bloch I, and Schneider U (2016) Bloch state tomography using Wilson lines. Science 352: 1094–1097. Li H, Dong Z, Longhi S, Liang Q, Xie D, and Yan B (2022) Aharonov-Bohm caging and inverse Anderson transition in ultra-cold atoms. Physical Review Letters 129: 220403. Loss D, Goldbart P, and Balatsky AV (1990) Berry’s phase and persistent charge and spin currents in textured mesoscopic rings. Physical Review Letters 65: 1655. Mathur H and Stone AD (1992) Quantum transport and the electronic Aharonov-Casher effect. Physical Review Letters 68: 2964. Meijer FE, Morpurgo AF, and Klapwijk TM (2002) One dimensional ring in the presence of rashba spin-orbit interaction: Derivation of the correct hamiltonian. Physical Review B 66. Meijer FE, Morpurgo AF, Klapwijk TM, and Nitta J (2005) Universal spin-induced time reversal symmetry breaking in two-dimensional electron gases with rashba spin- orbit interaction. Physical Review Letters 95: 186805. Meir Y, Gefen Y, and Entin-Wohlman O (1989) Universal effects of spin-orbit scattering in mesoscopic systems. Physical Review Letters 63(7): 798. Meir Y, Entin-Wohlman O, and Gefen Y (1990) Magnetic-field and spin-orbit interaction in restricted geometries: Solvable models. Physical Review B 42(13): 8351. Milton KA (2006) Theoretical and experimental status of magnetic monopoles. Reports on Progress in Physics 69: 1637–1712. Morpurgo AF, Heida JP, Klapwijk TM, van Wees BJ, and Borghs G (1998) Ensemble- average spectrum of Aharonov-Bohm conductance oscillations: Evidence for spin- orbit-induced Berry’s phase. Physical Review Letters 80: 1050. Nagasawa F, Takagi J, Kunihashi Y, Kohda M, and Nitta J (2012) Experimental demonstration of spin geometric phase: Radius dependence of time- reversal Aharonov-Casher oscillations. Physical Review Letters 108. Nagasawa F, Frustaglia D, Saarikoski H, Richter K, and Nitta J (2013) Control of the spin geometric phase in semiconductor quantum rings. Nature Communications 4: 2526. Nakamura J, Fallahi S, Sahasrabudhe H, Rahman R, Liang S, Gardner GC, and Manfra MJ (2019) Aharonov-Bohm interference of fractional quantum Hall edge modes. Nature Physics 15: 563. Nitta J, Akazaki T, Takayanagi H, and Enoki T (1997) Gate control of spin-orbit interaction in an inverted InGaAs/InAlAs heterostructure. Physical Review Letters 78: 1335. Nitta J, Meijer FE, and Takayanagi H (1999) Spin-interference device. Applied Physics Letters 75: 695–697. Niu Q (2006) On a proper definition of spin current. Physical Review Letters 96. Noguchi A, Shikano Y, Toyoda K, and Urabe S (2014) Aharonov-Bohm effect in the tunneling of a quantum rotor in a linear Paul trap. Nature Communications 5: 3868. Olariu S and Popescu S (1985) The quantum effects of electromagnetic fluxes. Reviews of Modern Physics 57(2): 339. Onsager L (1931) Reciprocal relations in irreversible processes. Physics Review 37(4): 405–426. Osakabe N (1986) Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor. Physical Review A 34(2): 815–822. Overstreet C, Asenbaum P, Curti J, Kim M, and Kasevich MA (2022) Observation of a gravitational Aharonov-Bohm effect. Science 375(6577): 226–229.

236

Aharonov–Bohm and Aharonov–Casher effects in condensed matter physics: A brief review

Pauli W (1927) Zur quantenmechanik des magnetischen elektrons. Zeitschrift für Physik 43: 601–623. Pearle P and Rizzi A (2017) Quantum-mechanical inclusion of the source in the Aharonov-Bohm effect. Physical Review A 95(5): 052123. ArXiv:1507.00068. Peshkin M and Tonomura A (1989) The Aharonov-Bohm effect. Berlin Heidelberg New York London Paris Tokyo Hong Kong: Springer-Verlag. Popescu S (2010) Dynamical quantum non-locality. Nature Physics 6(3): 151–153. Qi X-L and Zhang S-C (2011) Topological insulators and superconductors. Reviews of Modern Physics 83: 1057. Qian TZ and Su Z-B (1994) Spin-orbit interaction and Aharonov-Anandan phase in mesoscopic rings. Physical Review Letters 72: 2311. Rashba EI (2006) Spin-orbit coupling and spin transport. Physica E: Low-Dimensional Systems and Nanostructures 34: 31. Recher P, Trauzettel B, Rycerz A, Blanter YM, Beenakker CWJ, and Morpurgo AF (2007) Aharonov-Bohm effect and broken valley degeneracy in graphene rings. Physical Review B 76: 235404. Richter K (2012) Viewpoint: The ABC of Aharonov effects. Physics 5: 22. Rohrlich D (2009) Aharonov-Casher effect. In: Greenberger D, Hentschel K, and Weinert F (eds.) Compendium of Quantum Physics. Berlin, Heidelberg: Springer. Rohrlich D (2010) Duality in the Aharonov-Casher and Aharonov-Bohm effects. Journal of Physics A Mathematical and Theoretical 43(35): 354028. Ronen Y, Werkmeister T, Najafabadi DH, Pierce AT, Anderson LE, Jae Y, Si S, Lee U, Lee YH, Johnson B, Watanabe K, Takashi T, Yacoby A, and Kim P (2021) Aharonov-Bohm effect in graphene-based Fabry-Pérot quantum hall interferometers. Nature Nanotechnology 16: 563. Roy SM (1980) Condition for nonexistence of Aharonov-Bohm effect. Physical Review Letters 44(3): 11–14. Rycerz A and Beenakker CWJ (2007) Aharonov-Bohm effect for a valley-polarized current in graphene. arXiv:0709.3397. Saarikoski H, Vázquez-Lozano JE, Baltanás JP, Nagasawa F, Nitta J, and Frustaglia D (2015) Topological transitions in spin interferometers. Physical Review B 91(R): 241406. Schwarzschild B (1986) Currents in normal-metal rings exhibit Aharonov-Bohm effect. Physics Today 39(1): 17–20. Shapiro B (1983) Quantum conduction on a cayley tree. Physical Review Letters 50: 747. Shech E (2022) Scientific understanding in the Aharonov-Bohm effect. Theoria 88: 943. Shekhter RI, Entin-Wohlman O, Jonson M, and Aharony A (2022) Magnetoconductance anisotropies and Aharonov-Casher phases. Physical Review Letters 129. Sjoqvist E (2014) Locality and topology in the molecular Aharonov-Bohm effect. Physical Review Letters 89(21): 210401. Stern A (1992) Berry’s phase, motive forces, and mesoscopic conductivity. Physical Review Letters 68: 1022. Stern A, Aharonov Y, and Imry Y (1990) Phase uncertainty and loss of interference: A general picture. Physical Review A 41: 3436. Stone AD (1985) Magnetoresistance fluctuations in mesoscopic wires and rings. Physical Review Letters 54: 2692. Sun QF and Xie XC (2005) Definition of the spin current: The angular spin current and its physical consequences. Physical Review B 72: 245305. Thouless DJ, Kohmoto M, Nightingale MP, and den Nijs M (1982) Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters 49: 405–408. Tonomura A (2006) The Aharonov-Bohm effect and its application to electron phase microscopy. Proceedings of the Japan Academy. Series B: 82. Vaidman L (2000) Torque and force on a magnetic dipole. American Journal of Physics 58: 978–983. Vaidman L (2008) Many-worlds interpretation of quantum mechanics. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/qm-manyworlds. Vaidman L (2011) On the role of potentials in the Aharonov-Bohm effect. Physics Review A86. Vaidman L (2012) Role of potentials in the Aharonov-Bohm effect. Physical Review A 86. 040101(R). Vaidman L (2013) Paradoxes of the Aharonov-Bohm and the Aharonov-Casher effects. Arxiv1301/6153, published in Yakir Aharonov 80th birthday Festschrift. Valagiannopoulos C, Marengo EA, Dimakis AG, and Alù A (2018) Aharonov-Bohm-inspired tomographic imaging via compressive sensing. IET Microwaves, Antennas Propagation 12: 1890. van Oudenaarden A, Devoret MH, Nazarov YV, and Mooij JE (1998) Magneto-electric Aharonov-Bohm effect in metal rings. Nature 391(6669): 768. Vidal J, Mosseri R, and Doucot B (1998) Aharonov-Bohm cages in two-dimensional structures. Physical Review Letters 81: 5888. Webb RA, Washburn S, Umbach CP, and Laibowitz RB (1985) Observation of h/e Aharonov-Bohm oscillations in normal-metal rings. Physical Review Letters 54(25): 2696–2699. Willett RL, Nayak C, Shtengel K, Pfeiffer LN, and West KW (2013) Magnetic-field-tuned Aharonov-Bohm oscillations and evidence for non-Abelian anyons at n ¼ 5/2. Physical Review Letters 111: 186401. Wu TT and Yang CN (1975) Concept of non-integrable phase factors and global formulation of gauge fields. Physics Review D 12: 3845–3857. Wu TT and Yang CN (1976) Dirac monopole without strings: Monopole harmonics. Nuclear Physics B 107: 365–380. Xiao D, Chang M-C, and Niu Q (2010) Berry phase effects on electronic properties. Reviews of Modern Physics 82: 1959–2007. Yang CN and Mills RL (1954) Conservation of isotopic spin and isotopic gauge invariance. Physics Review 96: 191. Yau J-B, De Poortere EP, and Shayegan M (2002) AharonovBohm oscillations with spin: Evidence for Berry’s phase. Physical Review Letters 88: 146801. Yerin Y, Gusynin VP, Sharapov SG, and Varlamov AA (2021) Genesis and fading away of persistent currents in a Corbino disk geometry. Physical Review B 104. Yu R, Qi XL, Bernevig A, Fang Z, and Dai X (2011) Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection. Physical Review B 84: 075119. Zvyagin AA (2020) Persistent currents in the two-chain correlated electron model. Low Temperature Physics 46: 507.

Quantum computation of complex systems☆ Giuliano Benentia,b and Giulio Casatia, aCenter for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy; bIstituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano, Italy © 2024 Elsevier Ltd. All rights reserved. This is an update of G. Casati, G. Benenti, Quantum Computation and Chaos, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 9–17, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/01146-3.

Introduction Quantum logic Quantum algorithms Quantum simulation of physical systems Simulating complex dynamics on actual quantum hardware Conclusion Acknowledgments References

237 238 239 240 243 244 244 245

Abstract Miniaturization provides us with an intuitive way of understanding why, in the near future, quantum mechanics will become important for computation. The electronics industry for computers grows hand-in-hand with the decrease in size of integrated circuits. This miniaturization is necessary to increase computational power, that is, the number of floating-point operations per second (flops) a computer can perform. In the 1950s, electronic computers based on vacuum-tube technology were capable of performing approximately 103 floating-point operations per second, while nowadays (2022) there exist supercomputers whose power is greater than 100 petaflops (a 1 petaflops computer is capable of performing 1015 floating-point operations per second). This enormous growth of computational power has been made possible owing to progress in miniaturization, which may be quantified empirically in Moore’s law. This law is the result of a remarkable observation made by Gordon Moore in 1965: the number of transistors on a single integrated-circuit chip doubles approximately every 18–24 months. This exponential growth has not yet saturated and Moore’s law is still valid. At the present time the limit is close to 1010 transistors per chip and the typical size of circuit components is of the order of 5–10 nm. Extrapolating Moore’s law, it is estimated that within a few years, one would reach the atomic size for storing a single bit of information. At that point, quantum effects will become unavoidably dominant.

Key points

• • •

Quantum logic: superposition and entanglement Quantum algorithms for complex dynamical systems Quantum simulations on actual quantum hardware

Introduction Miniaturization provides us with an intuitive way of understanding why, in the near future, quantum mechanics will become important for computation. The electronics industry for computers grows hand-in-hand with the decrease in size of integrated circuits. This miniaturization is necessary to increase computational power, that is, the number of floating-point operations per second (flops) a computer can perform. In the 1950s, electronic computers based on vacuum-tube technology were capable of performing approximately 103 floating-point operations per second, while nowadays (2022) there exist supercomputers whose power is greater than 100 petaflops (a 1 petaflops computer is capable of performing 1015 floating-point operations per second). This enormous growth of computational power has been made possible owing to progress in miniaturization, which may be quantified empirically in Moore’s law. This law is the result of a remarkable observation made by Gordon Moore in 1965: the number of transistors on a single integrated-circuit chip doubles approximately every 18–24 months. This exponential growth has not yet saturated and Moore’s law is still valid. At the present time the limit is close to 1010 transistors per chip and the typical size of circuit components is of the order of 5–10 nm. Extrapolating Moore’s law, it is estimated that within a few years, one would reach the atomic size for storing a single bit of information. At that point, quantum effects will become unavoidably dominant.

☆

Change History: September 2022. G Benenti, G Casati updated the chapter. Section “Simulating complex dynamics on actual quantum hardware” is newly added. Updated outlook.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00071-8

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Quantum computation of complex systems

Quantum physics sets fundamental limitations on the size of the circuit components. The first question under debate is whether it would be more convenient to push the silicon-based transistor to its physical limits or instead to develop alternative devices, such as quantum dots, single-electron transistors or molecular switches. A common feature of all these devices is that they are at the nanometer length scale, and therefore quantum effects play a crucial role. So far, the quantum switches that could substitute silicon-based transistors and possibly be connected together to execute classical algorithms based on Boolean logic were discussed. In this perspective, quantum effects are simply unavoidable corrections that must be taken into account owing to the nanometer size of the switches. A quantum computer represents a radically different challenge: the aim is to build a machine based on quantum logic, that is, it processes the information and performs logic operations in agreement with the laws of quantum mechanics (Benenti et al., 2019).

Quantum logic The elementary unit of quantum information is called a qubit (the quantum counterpart of the classical bit) and a quantum computer may be viewed as a many-qubit system. Physically, a qubit is a two-level system, like the two spin states of a spin-1/2 particle, the vertical and horizontal polarization states of a single photon or two levels of an atom. A classical bit is a system that can exist in two distinct states, which are used to represent 0 and 1, that is, a single binary digit. The only possible operations (gates) in such a system are the identity (0 ! 0, 1 ! 1) and NOT (0 ! 1, 1 ! 0). In contrast, a quantum bit (qubit) is a two-level quantum system, described by a two-dimensional complex Hilbert space. In this space, one may choose a pair of normalized and mutually orthogonal quantum states, called | 0i and |1i (say, the eigenstates of the Pauli operator sz), to represent the values 0 and 1 of a classical bit. These two states form a computational basis. From the superposition principle, any state of the qubit may be written as jCi ¼ aj0i + bj1i,

(1)

where the amplitudes a and b are complex numbers, constrained by the normalization condition | a| + |b| ¼ 1. A quantum computer can be seen as a collection of n qubits and therefore its wave function resides in a 2n- dimensional complex Hilbert space. While the state of an n-bit classical computer is described in binary notation by an integer k e [0,2n − 1], 2

k ¼ kn −1 2n −1 + ⋯ + k1 2 + k0 ,

2

(2)

with k0, k1, . . . kn −1 e [0,1] binary digits, the state of an n-qubit quantum computer is jCi ¼

n 2X −1

ck jki

k¼0

¼

1 X kn −1,⋯ k1 ,k0 ¼0

(3) ckn −1 ,...,k1, k0 jkn −1 ...k1 k0 i,

where |kn−1. . .k1k0i  |kn−1i  ⋯  |k1i  |k0i. Notice that the complex numbers ck are constrained by the normalization condiP n−1 tion 2k¼0 | ck |2 ¼ 1. The superposition principle is clearly visible in Eq. (3): while n classical bits can store only a single integer k, the n-qubit quantum register can be prepared in the corresponding state | ki of the computational basis, but also in a superposition. The number of states of the computational basis in this superposition can be as large as 2n, which grows exponentially with the number of qubits. The superposition principle opens up new possibilities for computation. When one performs a computation on a classical computer, different inputs require separate runs. In contrast, a quantum computer can perform a computation for exponentially many inputs on a single run. This huge parallelism is the basis of the power of quantum computation. The superposition principle is not a uniquely quantum feature. Indeed, classical waves satisfying the superposition principle do exist. For instance, consider the wave equation for a vibrating string with fixed endpoints. Its solutions | ’ki satisfy the superposition principle and one can write the most general state | ’i of a vibrating string as a linear superposition of these solutions, analogously P n to Eq. (3): |’i ¼ 2k¼0−1 ck | ’ki. It is therefore also important to point out the importance of entanglement for the power of quantum computation, as compared to any classical computation. Entanglement is the most spectacular and counter-intuitive manifestation of quantum mechanics, observed in composite quantum systems: it signifies the existence of non-local correlations between measurements performed on well-separated particles. After two classical systems have interacted, they are in well-defined individual states. In contrast, after two quantum particles have interacted, in general, they can no longer be described independently of each other. There will be purely quantum correlations between two such particles, independently of their spatial separation. Examples of     two-qubit entangled state are the four states of the so-called Bell basis, f ¼ 12 ðj00i  j11iÞ and ’ ¼ 12 ðj01i  j10iÞ . The measure of the polarization state of one qubit will instantaneously affect the state of the other qubit, whatever their distance is. There is no entanglement in classical physics. Therefore, in order to represent the superposition of N ¼ 2n levels by means of classical waves, these levels must belong to the same system. Indeed, classical states of separate systems can never be superposed. Thus, any computation based on classical waves requires a number N of levels that grows exponentially with n. If D is the typical energy separation between two consecutive levels, the amount of energy required for this computation is given by D2n. Hence, the amount of physical resources needed for the computation grows exponentially with n. In contrast, due to entanglement, in quantum physics a general superposition of 2n levels may be represented by means of n qubits. Thus, the amount of physical resources (energy) grows only linearly with n. To implement a quantum computation, one must be able to control the evolution in time of the many-qubit state describing the quantum computer. As far as the coupling to the environment may be neglected, this evolution is unitary and governed by the

Quantum computation of complex systems

239

Schrӧdinger equation. It is well known that a small set of elementary logic gates allows the implementation of any complex computation on a classical computer. This is very important: it means that, when one changes the problem, one does not need to modify one’s computer hardware. Fortunately, the same property remains valid for a quantum computer. It turns out that, in the quantum circuit model, any whatever complex unitary transformation acting on a many-qubit system can be decomposed into quantum gates acting on a single qubit and a suitable quantum gate acting on two qubits. Any unitary operation on a single qubit can be constructed using only Hadamard pffiffiffi pffiffiffi and phase-shift gates. The Hadamard gate is defined as follows: it turns |0i into ðj0i + j1iÞ= 2 and |1i into ðj0i −j1iÞ= 2 . The phase-shift gate (of phase d) turns |0i into |0i and | 1i into eid | 1i.. One can decompose a generic unitary transformation acting on a many-qubit state into a sequence of Hadamard, phase-shift and CNOT gates, where CNOT is a two-qubit gate, defined as follows: it turns | 00i into |00i, |01i into |01i, |10i into | 11i, and |11i into |10i As in the classical XOR gate, the CNOT gate flips the state of the second (target) qubit if the first (control) qubit is in the state |1i and does nothing if the first qubit is in the state |0i. Of course, the CNOT gate, in contrast to the classical XOR gate, can also be applied to any superposition of the computational basis states. The decomposition of a generic unitary transformation of a n-qubit system into elementary quantum gates is in general inefficient, that is, it requires a number of gates exponentially large in n (more precisely, O(n24n) quantum gates). However, there are special unitary transformations that can be computed efficiently in the quantum circuit model, namely by means of a number of elementary gates polynomial in n. A very important example is given by the quantum Fourier transform, mapping a P n P n generic n-qubit state 2k¼0−1ak |ki into 2l¼0−1bl |li, where the vector {b0, ⋯, bN−1} is the discrete Fourier transform of the vector P N−1 2pikl/2n ak. It can be shown that this transformation can be efficiently implemented in O(n2) {a0, ⋯ aN−1}, that is, bl ¼ k¼0 e elementary quantum gates, whereas the best known classical algorithm to simulate the Fourier transform, the fast Fourier transform, requires O(n2n) elementary operations. The quantum Fourier transform is an essential subroutine in many quantum algorithms.

Quantum algorithms As shown above, the power of quantum computation is due to the inherent quantum parallelism associated with the superposition principle. In simple terms, a quantum computer can process a large number of classical inputs in a single run. For instance, starting P n from the input state 2k¼0−1ck |ki  |0 ⋯ 0i, one may obtain the output state. n 2 −1 X

ck jkijf ðkÞi:

(4)

k¼0

Therefore, the function f(k) is computed for all k in a single run (note that one needs two quantum registers to compute by means of a reversible unitary transformation f(k); the second register requires enough qubits to load the output f(k) for all input values k ¼ 0, 1, . . ., 2n, with n number of qubits in the first register). However, it is not an easy task to extract useful information from the output state. The problem is that this information is, in a sense, hidden. Any quantum computation ends up with a projective measurement in the computational basis: the qubit polarization is measured along the z-axis for all the qubits. The output of the measurement process is inherently probabilistic and the probabilities of the different possible outputs are set by the basic postulates of quantum  2     mechanics. Given the state (4), one obtains k f k with probability c  , hence, the evaluation of the function f(k) for a single k

k ¼ k, exactly as with a classical computer. However, there exist quantum algorithms that exploit quantum interference to efficiently extract useful information. In 1994, Peter Shor proposed a quantum algorithm that efficiently solves the prime-factorization problem: given a composite odd positive integer N, find its prime factors. This is a central problem in computer science and it is conjectured, though not proven, that for a classical computer it is computationally difficult to find the prime factors. Indeed, the best classical algorithm, the number field sieve, requires exp. (O(n1/3(log n)2/3)) operations. Shor’s algorithm instead efficiently solves the integer factorization problem in O((n2 log n log log n)) elementary quantum gates, where n ¼ log N is the number of bits necessary to code the input N. Therefore it provides an exponential improvement in speed with respect to any known classical algorithm. The integer factoring problem can be reduced to the problem of finding the period of the function f(k) ¼ ak mod N, where N is the number to be factorized and a < N is chosen randomly. The modular exponentiation can be computed efficiently on a quantum computer and, starting from P2n −1 the state p1ffiffiffi k¼0 jkij0 . . . 0i (the equal superposition of all basis states in the first register can be obtained by applying one N P2n −1 Hadamard gate for each qubit), one arrives at p1ffiffiffi k¼0 jkijf ðkÞi: Notice that there are two quantum registers, the first one stores k, N   the second f(k). By measuring the second register, one obtains the outcome f k : Thus, the quantum computer wave function         P collapses onto p1ffiffiffi m−1 k + jr f k , where m is the number of k values such that f ðkÞ ¼ f k , and r is the period of f(k), that is m

j¼0

f(k) ¼ f(k + r). To determine the period r, one has to perform the quantum Fourier transform of the first register. The resulting wave function is peaked around integer multiples of N/r. From the measurement of this state, one can extract the period r. It is worth mentioning that there are cryptographic systems, such as RSA, that are used extensively today and that are based on the conjecture that no efficient algorithms exist for solving the prime factorization problem. Hence Shor’s algorithm, if implemented on a large scale quantum computer, would break the RSA cryptosystem. Other quantum algorithms have been developed. In particular, Grover has shown that quantum computers can also be useful for solving the problem of searching for a marked item in an unstructured database of N ¼ 2n items. The best one can do with a classical computer is to go through the database, until ffiffiffiffi finds the solution. This requires O(N) operations. In contrast, the same problem pone can be solved by a quantum computer in O N operations. In this case, the gain with respect to classical computation is quadratic.

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Quantum computation of complex systems

Quantum simulation of physical systems The simulation of quantum many-body problems on a classical computer is a difficult task as the size of the Hilbert space grows exponentially with the number of particles. For instance, if one wishes to simulate a chain of n spin-1/2 particles, the size of the Hilbert space is 2n. Namely, the state of this system is determined by 2n complex numbers. As observed by Feynman in the 1980s, the growth in memory requirement is only linear on a quantum computer, which is itself a many-body quantum system. For example, to simulate n spin-1/2 particles one only needs n qubits. Therefore, a quantum computer operating with only a few tens of qubits can outperform a classical computer. Of course, this is only true if one can find an efficient quantum algorithm and if one can efficiently extract useful information from the quantum computer. Quite interestingly, a quantum computer can outperform a classical computer not only for the investigation of the properties of many-body quantum systems, but also for the study of the quantum and classical dynamics of complex single-particle systems. For a concrete example, consider the quantum-mechanical motion of a particle in one dimension (the extension to higher dimensions is straightforward). It is governed by the Schrӧdinger equation iℏ

d Cðx, t Þ ¼ HCðx, t Þ, dt

(5)

where the Hamiltonian H is given by H ¼ H0 + V ðx, t Þ ¼ −

ℏ 2 d2 + V ðx, t Þ: 2m dx2

(6)

The Hamiltonian H0 ¼ −(ℏ2/2m) d2/dx2 governs the free motion of the particle, while V(x, t) is a (possibly time-dependent) one-dimensional potential. To solve Eq. (5) on a quantum computer with finite resources (a finite number of qubits and a finite sequence of quantum gates), one must first of all discretize the continuous variables x and t. If the motion essentially takes place inside a finite region, say − d  x  d, decompose this region into 2n intervals of length D ¼ 2d/2n and represent these intervals by means of the Hilbert space of an n-qubit quantum register (this means that the discretization step drops exponentially with the number of qubits). Hence, the wave function | C(t)i is approximated as follows: n 2X −1   C ~ ðt Þ ¼ 1 Cðxi , t Þjii, N i¼0

(7)

  where xi  −d + i + 12 D, |ii ¼ |in−1i  ⋯  |i0i is a state of the computational basis of the n-qubit quantum register a d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   P2n −1 2  ~ provides a N i¼0 jCðxi ,t Þj is a factor that ensures correct normalization of the wave function. It is intuitive that C good approximation to | Ci when the discretization step D is much smaller than the shortest length scale relevant for the motion of the system. The Schrӧdinger Eq. (5) may be integrated by propagating the initial wave function C(x, 0) for each time-step e as follows: i

Cðxt + eÞ ¼ T e −ℏ½H0 + V ðxtÞ2 Cðxt Þ,

(8)

where T is the time-ordering operator. If the time-step e is small enough, it is possible to write the Trotter decomposition i

i

i

e −ℏ½H0 + V ðx, t Þ2  e −ℏH0 2 e −ℏV ðx,t Þ2 ,

(9)

which is exact up to terms of order e2. The operator on the right-hand side of Eq. (9) is still unitary, simpler than that on the left-hand side, and, in many interesting physical problems, can be efficiently implemented on a quantum computer. Advantage is taken of the fact that the Fourier transform can be efficiently preformed by a quantum computer. One can then write the first operator in the right-hand side of (9) as   i ℏ 2 k2 i (10) e −ℏH0 2 ¼ F −1 e+ℏ 2m 2 F, where k is the variable conjugated to x and F the discrete Fourier transform. This represents a transformation from the x-representation to the k-representation, in which this operator is diagonal. Then, using the inverse Fourier transform F−1, one returns to the x-representation, in which the operator exp(− iV(x, t)e/ℏ) is diagonal. The wave function C(x, t) at time t ¼ le is obtained from the initial wave function C(x, 0) by applying l times the unitary operator.   i ℏ 2 k2 i (11) F −1 e+ℏ 2m 2 Fe −ℏV ðx,t Þ2 : Therefore, simulation of the Schrӧdinger equation is now reduced to the implementation of the Fourier transform plus diagonal operators of the form where c is some real constant. Note that an operator of the form jxi ! eicf ðxÞ jxi,

(12)

where c is some real constant. Note that an operator of the form (12) appears both in the computation of exp(− iV (x, t) e/ħ) and of exp(− iH0e/ℏ), when this latter operator is written in the k-representation. The quantum computation of (12) is possible, using an ancillary quantum register | yia, by means of the following steps:

Quantum computation of complex systems   j0ia jx ! jf ðxÞia jx ! eicf ðxÞ jf ðxÞia jxi! eicf ðxÞ j0ia jxi ¼ j0ia eicf ðxÞ jxi:

241 (13)

The first step is a standard function evaluation and may be implemented by means of O(n2n) elementary quantum gates. Of course, more efficient implementations (polynomial in n) are possible when the function f(x) has some structure, as it is the case for the potentials V(x, t) usually considered in quantum-mechanical problems. The second step in (13) is the transformation | yia ! eicy |yia and can be performed in m single-qubit phase-shift gates, m being the number of qubits in the ancillary register. Indeed, one may P j write the binary decomposition of an integer y e [0, 2− 1] as y ¼ m−1 j¼0 yj2 , with yj 2 {0, 1}. Therefore, ! m −1 m −1   X Y exp ðiyÞ ¼ exp (14) icyj 2j ¼ exp icyj 2j , j¼0

j¼0

which is the product of m single-qubit gates, each acting non-trivially (differently from identity) only on a single qubit. The j-th gate operates the transformation | yjia ! exp (icyj2j)| yjia, with |yjia 2 {| 0i, | 1i} vectors of the computational basis for the j-th ancillary qubit. The third step in (13) is just the reverse of the first and may be implemented by the same array of gates as the first but applied in the reverse order. After this step the ancillary qubits are returned to their standard configuration |0ia and it is therefore possible to use the same ancillary qubits for every time-step. Note that the number of ancillary qubits m determines the resolution in the computation of the diagonal operator (12). Indeed, the function f(x) appearing in (12) is discretized and can take 2m different values. An example of an interesting dynamical model that can be simulated efficiently (and without ancillary qubits) on a quantum computer is the so-called quantum sawtooth map. This map represents the dynamics of a periodically driven system and is derived from the Hamiltonian Hðy, I; tÞ ¼

+1 X I2 dðt − jT Þ, + V ðyÞ 2 j¼ −1

(15)

where (I, y) are conjugate action-angle variables (0  y < 2p), with the usual quantization rules, y ! y and I ! I ¼ −i∂/∂ y (set ℏ ¼ 1) and V(y) ¼ −k(y − p)2/2. This Hamiltonian is the sum of two terms, H(y, I; t) ¼ H0(I) + U(y; t), where H0(I) ¼ I2/2 is just the P kinetic energy of a free rotator (a particle moving on a circle parametrized by the coordinate y), while U(y; t) ¼ V(y) jd(t − jT) represents a force acting on the particle that is switched on and off instantaneously at time intervals T. Therefore, its is said that the dynamics described by Hamiltonian (15) is kicked. The (quantum) evolution from time tT− (prior to the t-th kick) to time (t + 1)T− (prior to the (t + 1)-th kick) is described by a unitary operator U acting on the wave function C: Ct + 1 ¼ UCt ¼ U T U k Ct ; U T ¼ e −iTI

2

=2

,U k ¼ eikðy−pÞ

2

=2

:

(16)

This map is called the quantum sawtooth map, since the force F(y) ¼ − dV (y)/dy ¼ k(y− p) has a sawtooth shape, with a discontinuity at y ¼ 0. In the following, an exponentially efficient quantum algorithm for simulation of the map (16) is described. It is based on the forward/backward quantum Fourier transform between action and angle bases. Such an approach is convenient since the operator U is the product of the two operators Uk and UT, which are diagonal in the y and I representations, respectively. This quantum algorithm requires the following steps for one map iteration: 1. Apply Uk to the wave function C(y). In order to decompose the operator Uk into one- and two-qubit gates, we first of all write y in binary notation: y ¼ 2p

n X

aj 2 −j ,

(17)

j¼1

with ai e {0,1}. Here n is the number of qubits, so that the total number of levels in the quantum sawtooth map is N ¼ 2n. One can insert (17) into the unitary operator Uk, obtaining the decomposition 2

eikðy − pÞ

=2

¼

   2 1 1 aj 2 −j − , ei2p k ai 2 −i − 2n 2n i,j¼1 n Y

(18)

which is the product of n2 two-qubit gates, each acting non-trivially only on the 4-dimensional subspace spanned by the qubits i and j. 2. The change from the y to the I representation is obtained by means of the quantum Fourier transform, which requires and 1 2 nðn + 1Þelementary quantum gates. 3. In the I representation, the operator UT has essentially the same form as the operator Uk in the y representation, and therefore it can be decomposed into n2 two-qubit gates, similarly to Eq. (18). 4. Return to the initial y representation by application of the inverse quantum Fourier transform. Thus, overall, this quantum algorithm requires 3n2 + n gates per map iteration. This number is to be compared with the O(n2n) operations required by a classical computer to simulate one map iteration by means of a fast Fourier transform. Thus, the quantum simulation of the quantum sawtooth map dynamics is exponentially faster than any known classical algorithm. Note that the

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Quantum computation of complex systems

resources required to the quantum computer to simulate the evolution of the sawtooth map are only logarithmic in the Hilbert space dimension N. As an example of the efficiency of this quantum algorithm, Fig. 1 shows the Husimi functions, taken after 1000 map iterations. It is noted that n ¼ 9 qubits are sufficient to observe the appearance of integrable islands, while at n ¼ 16 these islands exhibit a complex hierarchical structure in the phase space. However, there is an additional aspect to be taken into account. Any quantum algorithm has to address the problem of efficiently extracting useful information from the quantum computer wave function. Indeed, the result of the simulation of a quantum system is the wave function of this system, encoded in the n qubits of the quantum computer. The problem is that, in order to measure all N ¼ 2n wave function coefficients by means of standard polarization measurements of the n qubits, one has to repeat the quantum simulation a number of times exponential in the number of qubits. This procedure would spoil any quantum algorithm, even in the case, like the present one, in which such algorithm could compute the wave function with an exponential gain with respect to any classical computation. Nevertheless, there are some important physical questions that can be answered in an efficient way. The quantum computation can provide an exponential gain (with respect to any known classical computation) in problems that require the simulation of dynamics up to a time t which is independent of the number of qubits. In this case, provided that one can extract the relevant information in a number of measurements polynomial in the number of qubits, one should compare, in the example of the quantum sawtooth map, O(t(log N)2) elementary gates (quantum computation) with O(tN log N) elementary gates (classical computation). This is, for instance, the case of dynamical correlation functions of the form  t Ct  hC0 jA{t B0 jC0 i ¼ hC0 j U { A{0 U t B0 jC0 i, (19) where U is the time-evolution operator (16) for the quantum sawtooth map. Similarly, one can efficiently compute the fidelity of quantum motion, which is a quantity of central interest in the study of the stability of quantum motion under perturbations. The fidelity f(t) (also called the Loschmidt echo), measures the accuracy with which a quantum state can be recovered by inverting, at time t, the dynamics with a perturbed Hamiltonian. It is defined as  t (20) f ðt Þ ¼ hCj U {2 U t jCi ¼ hCjeiH2 t e −iHt jCi: Here the wave vector | Ci evolves forward in time with Hamiltonian H up to time t and then evolves backward in time with a perturbed Hamiltonian He. If the evolution operators U and Ue can be simulated efficiently on a quantum computer, as is the case in many physically interesting situations, then the fidelity of quantum motion can be evaluated with exponential speed up with respect to known classical computations. As shown in Fig. 2, it is possible to measure the fidelity by means of a Ramsey-type quantum interferometer method. A single ancillary qubit is needed, initially prepared in the state | 0i, while the input state for the other n qubits is a given initial state | C0i for the quantum sawtooth map. Two Hadamard gates are applied to the ancillary qubit, and in between these operations a controlled-W operation is applied (W is a unitary operator), namely W is applied to the other n qubits only if the ancillary qubit is in its |1i state. As a result, one obtains the following final overall state for the n + 1 qubits: 1 ½ðj0i + j1iÞjC0 i + ðj0i −j1iÞW jC0 i: 2

(21)

If W ¼ (U 2 {)tUt, then one can derive the fidelity from polarization measurements of the ancillary qubit. One obtains (A)

(B)

Fig. 1 Husimi function for the sawtooth map for n ¼ 9 (left) and n ¼ 16 (right) qubits, in action angle variables (I, y), with − N/2  I < N/2 (vertical axis, N ¼ 2n) and 0  y < 2p (horizontal axis), averaged in the interval 950  t  1000, for T ¼ 2p/N and kT ¼ −0.1. An action eigenstate, |c0i ¼ |m0i, with m0 ¼ [0.38 N] is considered as initial state at time t ¼ 0. The color is proportional to the density: blue for zero and red for maximal density. Casati G and Benenti G (2005) Quantum computation and chaos, in: Bassani G, Liedl G and Wyder P (eds.) Enciclopedia of Condensed Matter Physics, Oxford, United Kingdom: Elsevier Science, Benenti G and Casati G (2016) Quantum computation of complex systems, in: Hashmi S (eds.) Reference Module in Materials Science and Materials Engineering, United Kingdom: Elsevier Science.

Quantum computation of complex systems

H

H

243

Measurement

W Fig. 2 Schematic drawing of a quantum circuit implementing a Ramsey-type quantum interferometer. The top line denotes a single ancillary qubit, the bottom line a set of n qubits, H the Hadamard gate and W a unitary transformation. Benenti G and Casati G (2016) Quantum computation of complex systems, in: Hashmi, S (ed.) Reference Module in Materials Science and Materials Engineering, United Kingdom: Elsevier Science.

2 f ðt Þ ¼ hsz i2 + sy ,

(22)

where hszi and hsyi are the expectation values of the Pauli operators sz and sy. Provided that the quantum algorithm implementing U is efficient, as it is the case for the quantum sawtooth map, the fidelity can then be computed efficiently.

Simulating complex dynamics on actual quantum hardware Present-day quantum computers, whether they are based on superconducting qubits or on trapped ions, suffer from significant decoherence and the effects of various noise sources. Therefore, achieving the quantum advantage in practically relevant problems such as chemical reactions, new materials design, or biological processes, is an imposing task. Note that quantum advantage is achieved when a quantum computer can solve a problem that no classical computer can solve in a feasible amount of time. The progress of currently available quantum processors can nevertheless be benchmarked by simulating complex dynamics. An illustrative example is again provided by the quantum sawtooth map. The classical limit of such map is chaotic when kT < − 4 or kT > 0. Although the sawtooth map is a deterministic system, in the chaotic regime the motion along the action direction is in practice indistinguishable from a random walk, with diffusion in the action variable. If one considers a classical ensemble of trajectories with fixed initial action m0 and random initial angle y, the second moment of the action distribution grows linearly with the number t of map iterations, h(DI)2i  D(k)t, with a diffusion coefficient D dependent on k. The quantum sawtooth map, in agreement with the correspondence principle, initially exhibits diffusive behavior, with the classical diffusion coefficient D. However, after a break time t⁎, quantum interference leads to suppression of diffusion. For t > t⁎, the quantum distribution reaches a steady state which decays exponentially over the action eigenbasis:

 2jm − m0 j 1 , (23) W m  jhmjCij2  exp − ℓ ℓ where the index m singles out the action eigenstates (I| mi ¼ m |mi), the system is initially prepared in the eigenstate | m0i, and ℓ is known as the localization length of the system. Therefore, for t > t⁎ only q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ðDIÞ2  Dt ⁎  ℓ (24) levels are populated. This phenomenon, known as dynamical localization, is due to quantum interference effects, suppressing the underlying classical diffusion process after a time t⁎  ℓ  D. Fig. 3 shows the results of a dynamical localization experiment with n ¼ 3 qubits on a real and freely available IBM quantum processor, with superconducting qubits, remotely accessed through cloud quantum programming (Pizzamiglio et al., 2021). The initial condition is peaked in action, C0(m) ¼ hm |C0i ¼ dm,m0, with m0 ¼ 0. The quantum algorithm for the sawtooth map allows

Fig. 3 Dynamical localization in the quantum sawtooth map with n ¼ 3 qubits, kT ¼ 1:5, k  0.273. Data from the IBM quantum processors lima (red) are obtained after averaging over 10 repetitions of 8192 experimental runs, and compared with the Qiskit simulator (blue) and the noiseless simulation (green). Pizzamiglio A, Chang SY, Bondani M, Montangero S, Gerace D and Benenti G (2021) Dynamical localization simulated on actual quantum hardware, Entropy 23: 654.

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Quantum computation of complex systems

one to compute the wave vector Ct(m) as a function of the number of map steps, and then the probability distribution Wt(m) ¼ |Ct(m)|2. In the figure k  0:273 < 1, so that the distribution is already localized after a single map step. On the other hand, here kT ¼ 1:5, corresponding to diffusive, chaotic behavior for the underlying classical dynamics. In Fig. 3 the ideal, noiseless distribution after t ¼ 1 map step is compared with the results of the real quantum hardware and with a simulator (Qiskit, provided by IBM), which takes into account a few relevant noise sources, modeling in particular dephasing, relaxation, and readout errors. The results show that the quantum hardware exhibits a localization peak, which emerges from quantum interference. Note that the quantum algorithm performs forward-backward Fourier transform, thus exploring the entire Hilbert space of the quantum register in a complex multiple-path interferometer that leads to wave-function localization. As such, dynamical localization is a very fragile quantum phenomenon, extremely sensitive to noise. The height of the peak, is significantly smaller than the noiseless value and the prediction of the Qiskit simulator. These results show that the Qiskit simulator underestimate some of the relevant noise channels, such as fluctuations of the qubit quality parameters between calibrations of the quantum computer, memory effects, and cross-talks between qubits. The presence of these noise channels also shows the imposing difficulties in scaling quantum algorithms to a large number of qubits and of quantum gates. On the other hand, striking progress has been reported in recent years, quantified for instance by the quantum volume VQ, a single number meant to encapsulate the quantum computer performance, including number of available qubits and number of quantum gates that can be reliably implemented, before errors dominate (Cross et al., 2019). The quantum volume is defined as log 2 V Q ¼ arg max kn f min ½k, dðkÞg,

(25)

where n is the number of qubits in the quantum computer, and d(k) ¼ 1/(keeff (k)), known as circuit depth, is determined by an effective error rate eeff (k) for a subset of k  n qubits, on which sequences of random two-qubit unitaries are implemented. From January 2020 to December 2021, the reported values of VQ have increased from VQ ¼ 32 to VQ ¼ 2048 (for a comparison, data from Fig. 3 have been obtained with a machine with quantum volume VQ ¼ 8).

Conclusion A few significant examples have been discussed showing the capabilities of a quantum computer in the simulation of complex physical systems. A quantum computer with a few tens of qubits and a long enough decoherence time to allow the implementation of a large number of quantum gates, would outperform a classical computer in this kind of problems. Any practical implementation of a quantum computer has to face errors, due to the inevitable coupling of the computer to the surrounding environment or to imperfections in the quantum computer hardware. The first kind of error is known as decoherence and is a threat to the actual implementation of any quantum computation. More generally, decoherence theory has a fundamental interest beyond quantum information science, since it provides explanations for the emergence of classicality in a world governed by the laws of quantum mechanics (Zurek, 2003). The core of the problem is the superposition principle, according to which any superposition of quantum states is an acceptable quantum state. This entails consequences that are absurd according to classical intuition, like the superposition of “cat alive” and “cat dead” that is considered in the Schrӧdinger’s cat paradox. The interaction with the environment can destroy the coherence between the states appearing in a superposition (for instance, the “cat alive” and “cat dead” states). Therefore, decoherence invalidates the quantum superposition principle, which is at the heart of the power of quantum algorithms. The presence of device imperfections, although not leading to any decoherence, also hinders the implementation of any quantum computational task, introducing errors. Therefore, decoherence and imperfection effects appear to be the ultimate obstacle to the realization of a large-scale quantum computer. Note that a quantum computer is not necessarily required for implementing quantum simulation. Simpler quantum devices, called (analog) quantum simulators can mimic the evolution of other quantum systems in an analog manner. Such simulators are problem-specific quantum machines, namely controllable quantum systems used to simulate other quantum systems (Georgescu et al., 2014). At present (2022) it is not clear if and when a useful quantum computer, capable of outperforming existing classical computers in important computational tasks, will be built. In order to perform coherent controlled evolution of a many-qubit system, one needs to take into account the problem of decoherence, and therefore large-scale quantum computers appear unrealistic with present technology. On the other hand, progress in the field has been huge in recent years. Moreover, we should bear in mind that technological breakthroughs (such as the transistor was for the classical computer) are always possible and that no fundamental objections have been found against the possibility of building a quantum computer.

Acknowledgments We acknowledge use of the IBM Quantum Experience for Fig. 3 of this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM company or the IBM-Q team.

Quantum computation of complex systems

References Benenti G, Casati G, Rossini D, and Strini G (2019) Principles of Quantum Computation and Information (A comprehensive textbook). Singapore: World Scientific. Cross AW, Bishop LS, Sheldon S, Nation PD, and Gambetta JM (2019) Validating quantum computers using randomized model circuits. Physical Review A 100: 032328. Georgescu IM, Ashhab A, and Nori F (2014) Quantum simulation. Reviews of Modern Physics 86: 153. Pizzamiglio A, Chang SY, Bondani M, Montangero S, Gerace D, and Benenti G (2021) Dynamical localization simulated on actual quantum hardware. Entropy 23: 654. Zurek WH (2003) Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics 75: 715.

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Quantum information processing with superconducting circuits: A perspective G Wendin, Department of Microtechnology and Nanoscience—MC2, Chalmers University of Technology, Gothenburg, Sweden © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Quantum processor systems: Hardware and software Quantum algorithms Quantum supremacy Performance metrics Cross entropy benchmarking—XEB Quantum volume—QV Relevance of metrics for usefulness Applications Quantum approximate optimization algorithm—QAOA QAOA basics QAOA applied to air transportation—Tail assignment Variational quantum eigensolver—VQE VQE basics VQE applied to chemistry Simulating physical systems on engineered quantum platforms Quantum transport and localization Quantum information scrambling Many-body Hilbert space scarring Key issues Noise and loss of information—A common experience Fighting imperfections and noise in quantum processors Quantum error suppression Quantum error mitigation Quantum error correction Scaling up for practical quantum advantage QPU-centric approach HPC-centric approach Useful NISQ digital quantum advantage—Mission impossible? Future directions Improved and alternative superconducting qubits Hybrid distributed computing Continuous variables—Computing with resonators Biochemistry and life science—Drivers of quantum computing? Conclusion Acknowledgments References

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Abstract The last 5 years have seen a dramatic evolution of platforms for quantum computing, taking the field from physics experiments to quantum hardware and software engineering. Nevertheless, despite this progress of quantum processors, the field is still in the noisy intermediate-scale quantum (NISQ) regime, seriously limiting the performance of software applications. Key issues involve how to achieve quantum advantage in useful applications for quantum optimization and materials science. In this article we will describe recent work to establish relevant benchmarks for quantum supremacy and quantum advantage, present recent work on applications of variational quantum algorithms for optimization and electronic structure determination, discuss how to achieve practical quantum advantage, and finally outline current work and ideas about how to scale up to competitive quantum systems.

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00226-2

Quantum information processing with superconducting circuits: A perspective

247

Key points

• • • • • • • •

Progress of gate-based superconducting quantum processors Status of quantum supremacy and quantum advantage Quantum approximate optimization algorithm (QAOA) applied to flight logistics Variational quantum eigensolver (VQE) applied to molecular ground state energies Methods for quantum error mitigation in noisy quantum processors Prospects of useful quantum advantage in current efforts to scale up superconducting quantum processors Improved and alternative superconducting qubits and architectures Superconducting platforms for quantum simulation of fundamental and applied physics problems

Introduction Since around 1980, quantum computing (QC) and quantum simulation (QS) have gone from fantasy to possibility, from concept to application, from basic science to engineering (Wendin, 2017; Gu et al., 2019; Krantz et al., 2019; Kjaergaard et al., 2020; Blais et al., 2020, 2021; Bruzewicz et al., 2019; Brown et al., 2021; Daley et al., 2022). In 2011, at a workshop at Benasque in the Spanish Pyrenees, at a memorable session we discussed in particular the future of QC. Myself, I predicted 20–30 years for useful applications, while Rainer Blatt emphasized that if we did not have any decisive results in 5 years, QC would soon be dead. It seems we were both right: QC did take off around 2016 at engineering levels, while really useful competitive applications showing practical quantum advantage may still be 10–20 years ahead. The intense discussions in the European quantum community then led to the 2016 Quantum Manifesto, and to the EU Quantum Flagship setting sail in 2018. From 2019, the field has seen a huge development of platforms for QC and simulation with superconducting devices and systems (Arute et al., 2019; Wu et al., 2021; Zhu et al., 2022; Córcoles et al., 2020; Gambetta, 2022a), including experiments indicating quantum supremacy (Arute et al., 2019; Wu et al., 2021; Zhu et al., 2022), a concept introduced by Preskill (2012). However, we are living in the era of noisy intermediate-scale quantum (NISQ) devices (Preskill, 2018), and it is currently impossible to build superconducting quantum processing units (QPUs) where one can entangle more than 10–15 qubits with high probability during the coherence time. Which means there is no time for useful computation with deep quantum circuits, only time to characterize the device and demonstrate physical entanglement—impressive but not necessarily useful. Nevertheless, IBM is now scaling superconducting QPUs to more than 1000 qubits in 2023 and over 4000 qubits in 2025 (Gambetta, 2022a; Bravyi et al., 2022), aiming for seamless integration of high-performance computers (HPCs) and QPU accelerators. Google (Lucero, 2021) seems to focus on modest-size quantum error-corrected QPUs for large-scale quantum computational breakthroughs already by 2029, while at the same time there seems to be some consensus (Sanders et al., 2020; Babbush et al., 2021; Beverland et al., 2022) that practical quantum advantage may take much longer to achieve. To create useful applications showing quantum advantage, it is necessary to scale up QPUs and related classical-quantum hybrid (HPC+QC) infrastructure. This explains a number of current trends: (i) stay at “small” scales (
100 qubits) and try to solve coherence problems and create useful applications before scaling up; (ii) go for large scales (1000 qubits) and try to implement quantum error correction (QEC) for quantum advantage or superiority while scaling up; (iii) scale up and solve large-scale hardware (HW) and software (SW) integration at systems levels, waiting for practical quantum advantage for use cases to emerge. The question of the feasibility of powerful quantum computers beating classical super-HPC hinges on that it will be ultimately possible to perform QEC. When John Martinis’ group announced that their superconducting quantum circuits were at the surface code threshold for fault tolerance (Barends et al., 2014), then the field opened up and went from quantum physics toward QC, scaling up HW and SW (Barends et al., 2015, 2016; Song et al., 2017; Kandala et al., 2017, 2019; Arute et al., 2019; Gong et al., 2019, 2021, 2022), and implementing significant quantum algorithms and quantum physics experiments (Barends et al., 2015, 2016; Kandala et al., 2017, 2019; Arute et al., 2020b, a; Neill et al., 2021; Gong et al., 2019, 2021, 2022; Zhang et al., 2022b; Mi et al., 2022b; Stanisic et al., 2022), including demonstrations of significant steps toward QEC (Andersen et al., 2020; Chen et al., 2021b; Marques et al., 2022; Krinner et al., 2022; Acharya et al., 2023). Much of the current discussion concerns how to get from proofs of concept to useful applications. Consider the way QC is being promoted by Google (Lucero, 2021): “Within the decade, Google aims to build a useful, error-corrected quantum computer. This will accelerate solutions for some of the world’s most pressing problems, like sustainable energy and reduced emissions to feed the world’s growing population, and unlocking new scientific discoveries, like more helpful AI.” This tells us that the world’s most pressing problems, like benchmarking climate models, are already subject to large-scale calculations, pushing super-HPCs to their limits set by NP-hard problems. The intended role of QPUs is to provide quantum superiority and to go far beyond those classsical limits. However, in the short term, at least during this decade, this can only be achieved by experimental quantum co-processors running specific subroutines addressing classically hard problems, omnipresent in industrial use cases (QUTAC, 2021; Kim et al., 2022). In the short term, industry will effectively be co-developing quantum algorithms as subroutines, and benchmark them against competing classical algorithms. If this leads to quantum advantage already in the short term, that will be a great bonus. The important thing to understand is that lack of quantum advantage for now does not

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Quantum information processing with superconducting circuits: A perspective

jeopardize powerful computing—exascale super-HPC platforms will continue addressing the world’s most pressing problems, and eventually QPU accelerators may provide quantum leaps. This perspective article can be looked upon as a self-contained second part of a research and review paper (Wendin, 2017) that stopped short at the beginning of the current engineering era of scaling up devices and building QC ecosystems. The purpose is to focus on addressing the quite dramatic development since then, trying to “predict the future” based on current visions, roadmaps, efforts, and investments that aim for the next 10 years (Alexeev et al., 2021; Altman et al., 2022; Ang et al., 2022), outlining a sustainable quantum evolution that hopefully survives the quantum hype (Ezratty, 2022; Das Sarma, 2022). To be able to do so in this brief article, we will frequently refer to Wendin (2017) and to recent reviews for basic background, technology, and methods. This review will focus on superconducting technology and systems based on circuit quantum electrodynamics (cQED) (Gu et al., 2019; Blais et al., 2021), but will also provide glimpses of the broader development.

Overview Quantum processor systems: Hardware and software With NISQ devices with limited coherence time, the necessary circuits are much too deep to achieve reasonable accuracy. It is therefore necessary to break up the quantum circuits in short, low-depth pieces with a small number of gates that can be run on QPUs during the coherence time. These often make use of variational quantum algorithms (VQAs) where one calculates the expectation value of a cost function, for example, a Hamiltonian, using a parameterized trial function. A classical HPC controls and executes the classical optimization loop: computes averages, searches for improved energies, computes new parameters, and updates the quantum circuit. Fig. 1 illustrates the basics of QC from a user perspective. The program code is typically prepared on a small classical computer and then submitted to the application programming interface (API) of an HPC frontend. The HPC interprets the quantum part and prepares the code defining the quantum circuit. This is finally loaded into the stack of the QPU. In the case of an ideal QPU, the code is executed on the QPU until solution is achieved, and the results finally read out and sent back to the HPC for postprocessing. In the NISQ world of QPUs, the quantum execution has to be limited to low-depth (shallow) quantum circuits that can be executed within the coherence time. This necessitates repeated quantum-classical processing loops for optimization of variational problems. Here the classical computer is a bottleneck. The HPC calculation takes much longer time than the QPU execution time, even if the HPC responds without delay (no latency) and the QPU backend is available immediately on request. This is hybrid HPC+QC computation, and is algorithm dependent. Quantum advantage in the NISQ era depends critically on efficient representation and coding of problems. There is a distinct difference between binary decision and optimization problems that can be mapped directly on qubits, and electron structure problems that need additional extensive transformations from fermionic operators to qubits.

Quantum algorithms An ideal digital quantum computer executes perfect gates on ideal qubits with infinite coherence time. It evolves the time-evolution operator e−iHt corresponding to a given Hamiltonian describing the problem (see e.g., Wendin, 2017). e−iHt is then broken down

Fig. 1 HPC + QC. The user prepares a program and submits it to the classical frontend. The HPC prepares the quantum circuit and sends it to the QC. The HPC/QC registers have N bits/qubits, that is, n ¼ 2N possible configurations/states. The HPC register can only be in one of 2N states: j00 . . . 00i, j00 . . . 01i, j00 . . . 10i, . . . j11 . . . 11i at each instance of time t, while the QC register can be in a superposition of all states: f 1 ðtÞj00 . . . 00i + f 2 ðtÞj00 . . . 01i + f 3 ðtÞj00 . . . 10i, . . . +f n −1 ðtÞj11 . . . 10i + f n ðtÞj11 . . . 11i. This describes time-dependent quantum superposition and entanglement and can, at best, lead to exponential quantum advantage (EQA).

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into a product of factors for the different terms of the Hamiltonian. These factors are finally represented in terms of products of quantum gates constituting a quantum circuit. In this case, the role of a classical computer is exclusively pre- and post-processing: preprocessing to construct the quantum circuit and postprocessing to read out and treat the results. Both of these are in principle NP-hard. To get desired results, the QC must be able to run for sufficiently long times to execute deep quantum circuits, which requires perfect qubits and gates. With NISQ devices, it is not possible to run phase-estimation algorithms (PEAs) to compute the energies of molecules—the needed quantum circuits are far too long with respect to the coherence time. This has led to alternative approaches, calculating the expectation value of the problem Hamiltonian with respect to parametrized trial functions and then optimizing the parameters for lowest energy. VQAs are generally based on constructing parametrized trial functions to compute and minimize the expectation value of a cost function. In the quantum case, the specific quantum computation involves computing the expectation value of a Hamiltonian cost function, while the classical computer prepares the trial function, computes the energy, updates the trial function parameters and minimizes the energy in an optimization loop. Extensive discussions and reviews of quantum methods and algorithms are presented by Wang et al. (2019), Cerezo et al. (2021), Franca and Garcia-Patron (2021), Tilly et al. (2022), Bharti et al. (2022), Abhijith et al. (2022), and Kowalsky et al. (2022).

Quantum supremacy John Preskill was the first one to explicitly introduce the concept of quantum supremacy in a 2012 paper discussing QC and the entanglement frontier (Preskill, 2012). In 2016, Boixo et al. then described how to characterize quantum supremacy in near-term devices (Boixo et al., 2018) preparing for the 2019 Google experiment to demonstrate quantum supremacy (Arute et al., 2019). The idea was to measure the output of a pseudo-random quantum circuit (Fig. 2) to produce a distribution of samples, and to compute the cross-entropy describing the “overlap” between the quantum and classical distributions (see Hangleiter and Eisert (2022) for a review of quantum random sampling). Aaronson and Chen (2017) put this on complexity-theoretic foundations. They noted that for the sampling tasks, not only simulation but also verification might need classical exponential time. This made it advantageous to directly consider the probability of observed bitstrings, rather than the distribution of sampled bitstrings. To this end, Aaronson and Chen (2017) showed that there is a natural average-case hardness assumption (Heavy Output Generation, HOG), which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test passed by the outputs of the quantum sampling procedure. The quantum volume benchmark of IBM (see Section “Quantum volume—QV”) is based on HOG. As mentioned, Google based their quantum supremacy demonstration (Arute et al., 2019) on sampling quantum and classical distributions, calculating the cross-entropy as described by Boixo et al. (2018). Cross-entropy benchmarking (XEB) has the advantage that it provides deeper insight than HOG, including measures of fidelity, and allows tracing of the development from small processors to devices that can only be simulated approximately. The Google paper (Arute et al., 2019) stated that an HPC would take 10,000 years. That statement immediately met with a rebuttal on the IBM Research blog by Pednault et al. (2019), explaining that an ideal simulation of the same task in a conservative, worst-case estimate could be performed on a classical system in 2.5 days and with far greater fidelity. Therefore the quantum supremacy threshold had not been met by Google using 53 qubits. This was of course valid criticism, but effectively just delaying the inevitable. An experiment that more decisively passed the quantum supremacy threshold was soon announced by Chinese researchers (Wu et al., 2021; Zhu et al., 2022) using the Zuchongzhi processor, closely following Google’s recipes, to demonstrate distinct quantum computational advantage. In the most recent experiment (Zhu et al., 2022) using 60-qubit 24-cycle random circuit sampling (RCS), the state-of-the-art HPC classical

Fig. 2 Control operations for generating the pseudo-random quantum circuits for Google’s quantum supremacy benchmarking protocol. Adapted from Arute F, et al. (2019) Quantum supremacy using a programmable superconducting processor. Nature 574: 505–510.

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simulation would have taken tens of thousands of years, while Zuchongzhi 2.1 only took about 4.2 h, thereby significantly enhancing the quantum computational advantage. As emphasized by Pednault et al. (2019), quantum supremacy is a threshold that does not automatically certify the quantum processor to be useful for running useful algorithms. However, it does benchmark the quality of the processor. The original Google experiment used a 53-qubit quantum processor that implements a large two-qubit gate quantum circuit of depth 20, with 430 twoqubit and 1,113 single-qubit gates, and with predicted total fidelity of FXEB ¼ 0.2% > 0. The condition for quantum supremacy, FXEB > 0, is based on statistics of creating an ensemble of a million runs of quantum circuits. The problem is that to solve, for example, a quantum chemistry problem based on 53 qubits, the depth of the quantum circuit would have to be in the range of a million. To solve useful problems challenging HPCs one must be able to run perfect quantum circuits with depth much larger than the circuit width, involving a very large number of 2-qubit gates. The name of the game is how to achieve practical quantum advantage.

Performance metrics Cross entropy benchmarking—XEB

The task is to sample the 2N bitstring output of a pseudo-random quantum circuit (Fig. 2). XEB compares the probability for observing a bitstring experimentally with the corresponding ideal probability computed via simulation on a classical computer. For a given circuit, one collects the measured bitstring sample {xi} and computes the linear XEB fidelity (Arute et al., 2019) FXEB ¼ 2N hPðxi Þii − 1

(1)

where N is the number of qubits, P(xi) is the probability of the experimental bitstring {xi} computed for the ideal quantum circuit, and the average is over the observed bitstrings. FXEB is correlated with how often one samples high-probability bitstrings. If the distribution is uniform, then hPðxi Þii ¼ 1=2N and FXEB ¼ 0. Values of FXEB between 0 and 1 correspond to the probability that no error has occurred while running the circuit. In the Google case (Arute et al., 2019), the computed values are very small, FXEB  10−3. This may represent proof of principle, but hardly provides any useful result. One needs to have FXEB  1 to be able to run algorithms, useful or not. To demonstrate quantum supremacy, one must achieve a high enough FXEB for a circuit with sufficient width and depth such that the classical computing cost of P(xi) for the full circuit is intractable. P(xi) must be calculated classically by simulating the ideal quantum circuit, which is formally intractable in the region of quantum supremacy. Since 2016, or even before, it has been understood that RCS, the task to sample the 2N bitstring output of a pseudo-random quantum circuit, will not scale to arbitrarily many qubits without error correction (Aaronson and Chen, 2017). Bouland et al. (2019) provided strong complexity theoretic evidence of classical hardness of RCS, placing it on par with the best theoretical proposals for supremacy. However, very recently Aharonov et al. (2022) (see also Brubaker, 2023) produced a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit. This gives strong evidence that, in the presence of a constant rate of noise per gate, RCS cannot be the basis of a scalable experimental violation of the extended Church-Turing thesis. Noise kills entanglement and makes RCS classically tractable (provided the HPC has enough memory to do the calculation). However, the algorithm does not directly address finite-size RCS-based quantum supremacy experiments (Aharonov et al., 2022), so the result is not directly applicable to current attempts to invalidate the quantum supremacy results (Arute et al., 2019; Wu et al., 2021; Zhu et al., 2022) using classical HPC. Pan et al. (2022) solved the Google sampling problem classically in about 15 h on a computational cluster with 512 GPUs with state fidelity 0.0037 (Google 0.00224), and claimed that it would only take few dozen seconds on an exascale machine, much faster than Google. Clearly it provides some satisfaction to demonstrate in practice that an HPC can beat the noisy 53q Sycamore QPU. However, a more challenging target for the HPC may now be to beat the 66q Zuchongzhi 2.1 with its 60-qubit 24-cycle RCS (Zhu et al., 2022).

Quantum volume—QV The fundamental challenges in the NISQ era can be illustrated using the concept of quantum volume (QV) introduced by IBM (Cross et al., 2019). QV is linked to system error rates, and quantifies the largest random circuit of equal width and depth that a specific computer can successfully implement given decoherence, gate fidelities, connectivity, and more (Cross et al., 2019; Pelofske et al., 2022). QV is a benchmarking protocol based on the execution of a pseudo-random quantum circuit with a fixed but generic form producing a bitstring {x} (Fig. 3). QV quantifies the largest random circuit U of equal width N (number of qubits) and depth d (number of layers) that the computer successfully implements U ¼ UðdÞ, . . . , Uð2ÞUð1Þ and produces the ideal output distribution pU ðxÞ ¼ jhxjU j0ij2 where {x} is an observable bit string. Benchmarking the QV, one runs circuits with an increasing number of cycles d ¼ 1, . . . , dmax with d ¼ N, and measures the success rate for increasing the depth d until one reaches a prescribed success threshold. To define when a model circuit U has been successfully implemented in practice, Cross et al. (2019) use the HOG problem formulated by Aaronson and Chen (2017): “Given as input a random quantum circuit C (drawn from some suitable ensemble), generate output strings x1 , . . . , xk, at least a 2/3 fraction of which have greater than the median probability in C’s output distribution.” This means that the set of output probabilities pU(x) are sorted in ascending order of probability, and the heavy (high probability) output generation problem is to produce a set of output strings {x} such that more than two-thirds are heavy, that is, greater than the median probability.

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Fig. 3 IBM QV pseudo-random quantum circuit (Cross et al., 2019) consisting of d layers (depth) of random permutations p of the N qubit labels, followed by random SU(4) two-qubit gates. When the circuit width N is odd, one of the qubits is idle in each layer. From Cross AW, Bishop LS, Sheldon, S, Nation, PD, Gambetta, JM, (2019) Validating quantum computers using randomized model circuits. Physical Review A 100: 032328.

Aaronson and Chen (2017) state: “HOG is easy to solve on a quantum computer, with overwhelming success probability, by the obvious strategy of just running C over and over and collecting k of its outputs,” and demonstrate that HOG is exponentially hard for a classical computer. The important thing is that the approach makes no reference to sampling or relation problems. Thus, one can shift focus from sampling algorithms to algorithms that simply estimate amplitudes. Pelofske et al. (2022) recently published a guide to the QV: “Quantum Volume in Practice: What Users Can Expect from NISQ Devices.” QV provides a standard benchmark to quantify the capability of NISQ devices. Interestingly, the QV values achieved in the tests (Pelofske et al., 2022) typically lag behind officially reported results and also depend significantly on the classical compilation effort. This is important to have in mind when popular articles announce QC breakthroughs in terms of higher QV values.

Relevance of metrics for usefulness

The definition of QV: d ¼ N stops short of benchmarking what is needed for useful applications. Useful algorithms often require the quantum circuit depth d to be much larger than the width N (number of qubits): d  N. This is typically the case when describing the ground-state energy of a molecule with reasonable accuracy. For example, a small molecule like HCN can be described (STO-6G basis) using Qiskit with N ¼ 15 and d  33000 ¼ 2200 N (N ¼ 14 and d  3000  200N within an optimized scheme of Tranter et al., 2022). Similarly, HCN (6–31 G basis) can be described using Qiskit with N ¼ 69 and d ¼ 6  106  87000 N (Lolur et al., 2021). These huge circuit depths can most likely be reduced with improved compilation methods (like in Tranter et al. (2022)), but nevertheless indicate the nature of the problem to perform useful calculations. For comparison, instead of using random circuits and XEB or QV/HOG as targets, one can generate specific quantum states showing genuine multipartite entanglement (GME) with sufficient fidelity. Mooney et al. (2021) investigated multiple quantum coherences of Greenberger-Horne-Zeilinger (GHZ) states on 11–27 qubits prepared on the IBM Quantum Montreal (ibmq_montreal) device (27 qubits), applying quantum readout error mitigation and parity verification error detection to the states. In this way, a fidelity of 0.546  0.017 > 0.5 was recorded for a 27-qubit GHZ state, demonstrating rare instances of GME across the full device. Although this experiment may feel more interesting and useful than testing with random circuits, it nevertheless demonstrates that there is a very low probability for creating a 27-qubit GHZ state. For it to be useful, the GHZ state must be created with 100% probability to serve as starting point for useful information processing.

Applications Quantum approximate optimization algorithm—QAOA The quantum approximate optimization algorithm (QAOA) was proposed as a heuristic variational method for solving NP-hard combinatorial optimization problems on near-term quantum computers (Farhi et al., 2014; Farhi and Harrow, 2016), and constitutes one of the most widespread and active current methods for using NISQ computers (Farhi et al., 2022; Zhou et al., 2020; Morales et al., 2020; Zhang et al., 2022a; Harrigan et al., 2021; Herrman et al., 2021; Borle et al., 2021; Dupont et al., 2022; Chen et al., 2022b; Sreedhar et al., 2022; Willsch et al., 2020; Bengtsson et al., 2020; Fitzek et al., 2021; Lacroix et al., 2020; Weggemans et al., 2022; Verdon et al., 2019; Moussa et al., 2022; Patel et al., 2022; Bravyi et al., 2020, 2022; Egger et al., 2021; Guerreschi and Matsuura, 2019; Wurtz and Love, 2022; Chandarana et al., 2022; Hegade et al., 2022).

QAOA basics

The QPU prepares a variational quantum state j ðg, bÞi with N qubits starting from an initial uniform superposition of all possible computational basis states j + i N generated by Hadamard gates from j0i N (Fig. 4). The second step of QAOA is then to apply an alternating sequence U ¼ UðpÞ, . . . , Uð2ÞUð1Þj + i N of two parametrized noncommuting quantum gates ^ ^ UðiÞ ¼ Uði, bi ÞUði, gi Þ ¼ e −ibl B e −igl C , followed by measurement generating an n-qubit bitstring. Many repetitions (shots) of the

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^ An alternating Fig. 4 Quantum circuit for the quantum approximate optimization algorithm (QAOA). The QAOA for a problem specified by the Ising Hamiltonian C. sequence of the Ising Hamiltonian C^ and the transverse mixing Hamiltonian B^ is applied to an equal superposition of N qubits, producing a trial state function Q ^ ^ j ðg, bÞi ¼ pl¼1 e −ibl B e −igl C j + in . Measurement of the qubit state produces a specific N-qubit bitstring, and many repetitions (shots) of the identical quantum circuit (loop not shown) creates a distribution used for estimating the cost function h ðg, bÞjCj ðg, bÞi. A classical optimization loop then minimizes the cost function by updating the variational parameters g, b. The level p represents the depth of the circuit, determining the number of variational parameters and gates used in the trial function j ðg, bÞi. A large circuit width N requires a (very) large depth p for accuracy. Adapted from Bengtsson A, et al. (2020) Quantum approximate optimization of the exact cover problem on a superconducting quantum processor. Physical Review Applied 14: 034010.

same circuit generate a distribution of bitstrings used to evaluate the cost function h ðg, bÞjCj ðg, bÞi. The variational parameters are then updated in a closed loop using a classical optimizer to minimize the cost function.

QAOA applied to air transportation—Tail assignment Industrial optimization has a long history (Ong and Teo, 2021), one of the most famous applications being Toyota’s Just-In-Time production system first implemented in 1973 (Monden, 2012). There is an extensive recent literature on optimization for industrial engineering and logistics (see e.g., de Sousa Junior et al., 2019; Csalódi et al., 2021; Ong and Teo, 2021; Monden, 2012), since around 2015 often referred to as Industry 4.0 (Csalódi et al., 2021; Rai et al., 2021). In the following we will discuss one specific example addressing airline scheduling (Wedelin, 1995): the performance of the QAOA algorithm for optimizing small but realistic instances of logistic scheduling relevant to airlines. The problem addressed is called tail assignment (TAS) (Grönkvist, 2005; Vikstål et al., 2020; Svensson et al., 2021), assigning individual aircraft (identified by the number on its tail fin) to particular routes, deciding which individual aircraft (tail) should operate each flight. A full approach to TAS is discussed in detail by Svensson et al. (2021), separating TAS into a generation problem and a selection problem. In this way, the complex rules only affect the generation problem, whereas the selection problem is often a pure Set Cover or Set Partitioning problem. The TAS generation problem is responsible for generating the complex aircraft routes. A flight is a connection between two airports. A set of flights operated in sequence by the same aircraft (tail) is called a route (Vikstål et al., 2020). To formulate the TAS problem, let F denote the set of flights f, T the set of tails t, and R the set of all legal routes r. In order for a route to be considered legal to operate, it needs to satisfy a number of constraints. In a full problem description one would include various costs, like the cost of flying a route and the cost of leaving a flight unassigned. In the decision version of TAS, the goal is to find any solution satisfying all the constraints, disregarding the costs. Essential aspects of the full TAS selection problem can then be reduced to an Exact Cover decision problem with the constraint X af r xr ¼ 1; xr 2 f0, 1g (2) r2R

The constraint matrix {afr} defines the relationship between F and R and tells whether a flight f is included in route r: afr ¼ 1 if flight f is covered by route r and 0 otherwise. Given the generated constraint matrix {afr}, the solutions for the decision variable xr will follow from the solution of the Exact Cover decision problem: xr ¼ 1 if route r should be used in the solution, and is 0 otherwise. The constraint can be turned into a cost function !2 X X C¼ af r xr − 1 (3) f 2F

r2R

that can be converted to the classical QUBO model—quadratic unrestricted binary optimization, which then maps over onto the quantum Ising model (Glover et al., 2022a, b; Ajagekar et al., 2022). In the final cost function, the Ising Hamiltonian, the constants (external field and spin interactions) are then determined by the constraint matrix {afr} (Eq. 2).

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Vikstål et al. (2020) reduced real-world instances obtained from real flight scheduling to instances with 8, 15, and 25 decision variables, which could be solved using QAOA on a quantum computer laptop simulator with 8, 15, and 25 qubits (routes), respectively. For these small instances, the problem was reduced to an exact cover problem with one solution in each instance. Elementary instances have been studied experimentally by Bengtsson et al. (2020) and Lacroix et al. (2020). The same TAS problem was studied by Willsch et al. (2022) mapped onto a 40-qubit problem and 472 flights. For each of the 40 routes, the constraint matrix defines all flights that are covered by this route. As explained, the exact cover problem is to find a selection of routes (i.e., a subset of rows of the constraint matrix) such that all 472 flights are covered exactly once. The exact cover problem was programmed on D-Wave Advantage and 2000Q. The problem instance has the unique ground state j0000000001010010011001000001000000000110i, where each qubit represents a flight route. The ground state contains nine 1’s meaning that for this particular instance, the solution consists of nine routes. Each route is assigned to an aircraft. All other states represent invalid solutions, in the sense that not all 472 flights are covered exactly once. However, the fact that the D-Wave quantum annealers (simulators) were able to solve this problem does not imply quantum advantage (Kowalsky et al., 2022).

Variational quantum eigensolver—VQE VQE basics The VQE implements the Rayleigh-Ritz variational principle (Fig. 5) (Wendin, 2017; Lee et al., 2019): ^ j ðyÞi  E0 EðyÞ ¼ h ðyÞj H

(4)

The VQE is a classical-quantum hybrid algorithm where the trial function j ðyÞi is created in the qubit register by gate operations. Calculating the expectation value on a QPU, the energy is estimated via repeated generation and readout of the quantum state, ^ In quantum simulation on an HPC, the state vector generating a distribution used to evaluate the Pauli operator products of H. is available classically, and the expectation value of H can be evaluated directly. The VQE scales badly for large molecules due ^ Nevertheless, the VQE is the to repeated measurements or tomography to form the expectation value of the Hamiltonian, hHi. common approach for small molecules with present NISQ QPUs (Cao et al., 2019; McArdle et al., 2020; McClean et al., 2021; Fedorov et al., 2022). The PEA scales better, but involves much deeper circuits, puts much higher demands on the coherence time of the q-register, and needs advanced QEC.

VQE applied to chemistry For an overview of applications to chemistry, see reviews (McArdle et al., 2020; McClean et al., 2021; Fedorov et al., 2022) and specific applications (Lee et al., 2019; Kim et al., 2022; Sokolov et al., 2020; Elfving et al., 2021; Sapova and Fedorov, 2022; Lolur et al., 2021; Tranter et al., 2022; Lolur et al., 2023; Sugisaki et al., 2019, 2022; Chen et al., 2021a; Zhang et al., 2022c; de GraciaTreviño et al., 2023; Grimsley et al., 2019; Tang et al., 2021; Tkachenko et al., 2021; Lan and Liang, 2022).

Fig. 5 The Variational Quantum Eigensolver (VQE) implements the Rayleigh-Ritz variational principle: EðyÞ ¼ h ðyÞjH^ j ðyÞi  E0 . The trial function j ðyÞi is created as a quantum circuit by the HPC and sent to the backend QPU (or simulator). The gates are then executed in the QPU by pulses generated by the classical electronics and software control system in the stack. The quantum register is finally measured, producing a bitstring used to evaluate the Hamiltonian ^ This is repeated many times to evaluate the energy h ðyÞjH^i j ðyÞi with sufficient accuracy. The optimization loop then updates the cost function H. parameters {yi } to minimize the energy.

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In VQE calculations for quantum chemistry (Wendin, 2017; Lee et al., 2019), one typically starts from an ansatz of the quantum state j ðyÞi ¼ UðyÞj ref i with variational parameters y, where U(y) is a unitary operator describing the quantum circuit, and j ref i is the initial state. U(y) could be a heuristic “hardware efficient” quantum circuit (Kandala et al., 2017) or a more elaborate unitary coupled cluster (UCC) expansion, with a Hartree-Fock (Wendin, 2017; Lee et al., 2019; Sokolov et al., 2020) or multiconfiguration (Sugisaki et al., 2019, 2022) initial reference state. The UCC ansatz of the quantum state j ðyÞi: ^ j ðyÞi ¼ UðyÞj

ref i

{

¼ eTðyÞ − TðyÞ j

ref i

(5)

can be expanded: TðyÞ ¼ T 1 + T 2 + T 3 + . . . +T N producing 1, 2, 3, . . . , N electron-hole excitations from the N-electron reference state. The first two terms X X T1 ¼ tðyÞpq c{p cq ; T 2 ¼ tðyÞpqrs c{p c{q cr cs pq

pqrs

(6)

(7)

with fermionic creation (c{i ) and annihilation (ci) operators generate single (S) and double (D) excitations and produce the parametrized UCCSD trial-state approximation. In particular t(y)pq ¼ yi and t(y)pqrs ¼ yj for all combinations of the indices pqrs. The trial-state fermionic operator U(y) must now be mapped onto qubit spin operators. Common transformations (codings) are Jordan-Wigner (JW), Bravyi–Kitaev (BK) and Parity, all designed to impose the anticommutation rules. In the case of the UCC ansatz, the exponential is expanded into exponentials of large numbers of products of Pauli spin operators acting on qubits. The initial trial state is then constructed through entangled quantum circuits: combinations of parametrized 1q-rotation gates and entangling 2q gates. The size of the quantum circuit can finally be reduced by qubit reduction schemes. All this results in a state vector for the trial state. The fermionic operators c{i and ci in the molecular Hamiltonian ^¼ H

X pq

hpq c{p cq +

1X h c{ c{ c c 2 pqrs pqrs p q r s

(8)

must also be expanded in products of Pauli spin operators using codings like JW, BK, or Parity, resulting in the generic interaction form: X X X ^¼ H hia sia + hia,jb sia sjb + hia,jb,kg sia sjb skg + . . . . . . (9) ia ia, jb ia, jb, kg ^ can then be where sia corresponds to the Pauli matrix sa for a 2{0, x, y, z}, acting on the i-th qubit. The expectation value hHi calculated in two ways: (1) State vector approach: direct calculation of by matrix operations; (2) Measurement approach: generating ^ i. an ensemble of identical trial states and evaluating the Pauli operators of the Hamiltonian terms H The original UCC exponential (Eq. 5) is then expanded into exponentials of large numbers of products of Pauli spin operators acting on qubits: e −iys1z s2z ; e −iys1z s2z s3z , etc. The parametrized initial entangled quantum circuit U(y) for the UCCSD trial state is then finally constructed through combinations of parametrized one-qubit rotation gates and entangling two-qubit CNOT gates, resulting in a state vector j ðyÞi ¼ UðyÞj HF i for the trial state. Lolur et al. (2021) have benchmarked the VQE as implemented in the Qiskit software package on laptops and HPCs, applying it to computing the ground state energy of water, H2O, hydrogen cyanide, HCN, and a number of related molecules.a The energies ^ j ðyÞi through matrix multiplihave been determined using the noise-free Qiskit statevector backend to directly calculate h ðyÞj H cation rather than repeated measurement. Clearly, substantial classical computational resources are needed to compute these systems on classical HPC quantum simulators. It is evident that for problems with QChem-inspired ansätze, even small numbers of qubits lead to large numbers of gates. And this is then amplified by the variational procedure with many parameters and iterations. The large number of gates will severely limit the types of molecules that can be used for benchmarking real quantum HW. And it will also limit what can be simulated on HPC quantum simulators. QChem problems will provide serious challenges and benchmarks for testing HPC and quantum HW NISQ implementations. To utilize VQE and achieve near chemical accuracy will be extremely challenging for NISQ processors. It is problematic or impossible to achieve chemical accuracy with conventional HPC VQE simulators already for small molecules such as HCN. But, there is no way around it: one must benchmark and challenge existing quantum HW and SW with available resources. The water molecule is a kind of “gold standard” even for forefront HPC applications, and H2O is an excellent candidate for testing the VQE on a Wendin G., ’The Main Results of Lolur et al. (2021) are Produced with an iMac i9, 8-Cores Workstation with 128 GB RAM (max 32 Qubits). The Practical Limit for the iMac to Achieve an Accurate Ground-State Energy Minimum is so far Connected with H2O (6-31G): 20 Qubits and 73 000 Gates, Taking 66.5 Hours to Converge to 10−6 Ha Precision of the Ground-State Energy for the Equilibrium Geometry. In the Case of N¼C¼C¼N (STO-6G: 29 Qubits, 2815 Variational Parameters, 515611 Gates) it Took 132 CPU-Hours (5.5 days) to Produce the First Energy Value. This Must then be Multiplied by 2815 to Evaluate one Gradient Vector for the First Energy Estimate by the Optimizer. And then this will need to be Iterated at Least 10 Times to Converge to an Accurate Ground State Energy Value. The Time-to-Solution (TTS) is then about 3.7 106 CPU-Hours. For Comparison, SUMMIT at ORNL, USA, with in total 202752 Cores, Would Finish the N¼C¼C¼N Instance in 147 Hours ¼ 6 days. In the QChem Case, the Challenge Lies in the Number of Variational Parameters and the number of Gates Needed, not in the number of Qubits. Smart Compilation can Greatly Reduce the number of Gates Tranter et al. (2022), but does not really change the poor scaling.

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quantum HW. Nevertheless, at the present stage of the NISQ era, one has to start with “easy” applications and simple approximations just to benchmark the quantum HW. The concept of “hardware-efficient” trial functions (Kandala et al., 2017) is an attempt to short-circuit standard UCCSD approaches and still introduce essential electron correlation. The recently developed adaptive VQE (Grimsley et al., 2019; Tang et al., 2021) and related further developments (Tkachenko et al., 2021; Lan and Liang, 2022) provide a more systematic approach to including electron correlation processes in order of monotonically decreasing weight. Nevertheless, the electron-correlation problem is computationally hard (NP-hard), so there is no easy way around it. State-of-the-art HPC computation of accurate molecular energies based on the Schrödinger equation defines the resources needed, and they are indeed huge (Calvin et al., 2021; Jones et al., 2022). HPC quantum simulators cannot be more efficient than systematic HPC brute-force full CI calculations. Quantum advantage will be possible by definition as soon as a quantum register exceeds the available RAM memory of an HPC. But to profit from that potential quantum advantage, the QPU must be able to run the q-algorithm to solution, and that will involve a very large number of gate operations even for the VQE. So, this is the ultimate challenge of the NISQ era.

Simulating physical systems on engineered quantum platforms Feynman’s original idea was to simulate quantum systems with engineered quantum systems. Using analog quantum circuits one tunes the interactions in a controllable quantum substrate to describe the Hamiltonian of the system to be simulated, and then anneals the systems toward their ground states by lowering the temperature. A comprehensive review (Daley et al., 2022) describes how quantum simulation can be performed already today through special-purpose analog HW quantum simulators, arguing that a first practical quantum advantage already exists in the case of specialized applications of analog devices. A particular example is simulation of 2D antiferromagnets with hundreds of Rydberg atoms trapped in an array by optical tweezers (Scholl et al., 2021), and the development of digital-analog QS for superconducting circuits (Lamata et al., 2018; Yu et al., 2022). The recent development of large-scale superconducting arrays makes it possible to design qubit circuits that simulate a specific physical device, and perform experiments on it (Arute et al., 2020b; Neill et al., 2021; Gong et al., 2021; Mi et al., 2022b; Zhang et al., 2022b; Karamlou et al., 2022; Landsman et al., 2019; Touil and Deffner, 2020; Mi et al., 2021; Braumüller et al., 2022; Frey and Rachel, 2022; Mi et al., 2022a). In this way, Arute et al. (2020b) simulated separation of the dynamics of charge and spin in the Fermi-Hubbard model, Neill et al. (2021) simulated the electronic properties of a quantum ring created in the Sycamore substrate, and Mi et al. (2022b) investigated discrete time crystals in an open-ended, linear chain of 20 superconducting transmon qubits that were isolated from the two-dimensional Sycamore grid. Kim et al. (2023) studied the evolution of quench dynamics of a 2D transverse-field Ising model with up to 26 spins on a 27q IBM Falcon processor, and similar experiments with the 127q Eagle processor are in the pipeline. In this case the results may be classically intractable. We will now discuss a few recent experiments addressing transport, information scrambling, and scarring in quantum circuits.

Quantum transport and localization Tight-binding lattice Hamiltonians are canonical models for particle transport and localization phenomena in condensed-matter systems. To study the propagation of entanglement and observe Anderson and Wannier–Stark localization, Karamlou et al. (2022) experimentally investigated quantum transport in one- and two-dimensional tight-binding lattices, emulated by a fully controllable 3 x 3 array of superconducting qubits in the presence of site-tunable disorder strengths and gradients. The dynamics are hard to observe in natural solid-state materials, but they can be directly emulated and experimentally studied using engineered quantum systems. The close agreement between the experimental results, simulations, and theoretical predictions (Karamlou et al., 2022) results from high-fidelity, simultaneous qubit control and readout, accurate calibration, and taking into account the relevant decoherence mechanisms in the system. Karamlou et al. (2022) emphasize that although the experiments are performed on a small lattice that can still be simulated on a classical computer, they demonstrate a platform for exploring larger, interacting systems where numerical simulations become intractable.

Quantum information scrambling Quantum scrambling is the dispersal of local information into many-body quantum entanglements and correlations distributed throughout an entire system, leading to the loss of local recoverability of quantum information (Landsman et al., 2019; Touil and Deffner, 2020; Mi et al., 2021; Braumüller et al., 2022). ^ Mji ^  ðtÞM ^ ^  OðtÞ Following Mi et al. (2021), the approach is based on measuring the out-of-time-order correlator (OTOC) C ¼ hjO ^ and unitary operator M ^ which is a Pauli operator on a different qubit. Scrambling between a unitary local perturbation operator OðtÞ ^ means a local perturbation is rapidly amplified over time. During the time evolution, OðtÞ becomes increasing nonlocal, which leads to decay of correlation function due to the spreading of the excitation all over the system. P ^ ¼ ^ ^ The perturbation operator can be modeled as OðtÞ i wi Bi , where Bi ¼ b1 ðiÞ b2 ðiÞ b3 ðiÞ . . . is a string of single-qubit basis operators acting on different qubits, and wi are the weights of the operator strings. Scrambling involves two different mechanisms: (i) Operator spreading, and (ii) generation of operator entanglement. Operator spreading (i) means that the strings of single-qubit

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basis operators B^i get expanded, spreading over more qubits, while (ii) generation of operator entanglement is reflected in the P ^ ¼ ^ growth in time of the minimum number of terms needed to expand OðtÞ i wi Bi with a broad distribution of coefficients wi. By measuring the OTOC, Mi et al. (2021) experimentally investigated the dynamics of quantum scrambling on a 53-qubit Sycamore quantum processor. Engineering quantum circuits that distinguished between operator spreading and operator entanglement, they showed that operator spreading is captured by an efficient classical model, while operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. However, the quantum-supremacy discussion of the influence of noise, making possible classical simulation of large noisy QPUs, suggests that the noise level needs to be reduced substantially before exponentially scaled computational resources are needed. Recently Braumüller et al. (2022) also probed quantum information propagation with out-of-time-ordered correlators (OTOCs). They implemented a 3  3 two-dimensional hard-core Bose–Hubbard lattice with a superconducting circuit, studied its time reversibility, and measured out-of-time-ordered correlators. The method relies on the application of forward and backward time evolution steps implemented by interleaving blocks of unitary time evolution and single-qubit gates. Extracting OTOCs made it possible to study quantum information propagation in lattices with various numbers of particles, enabling observation of a signature of many-body localization in the 2D hard-core Bose–Hubbard model. Braumüller et al. (2022) propose that applying the technique to larger lattices may improve our understanding of quantum thermodynamics and black-hole dynamics, as well as of using many-body systems for quantum memories. In addition, experimentally accessing OTOCs in large quantum circuits may provide a powerful benchmarking tool to study future quantum processors. But again, here noise will likely become an issue.

Many-body Hilbert space scarring Moudgalya et al. (2022) and Chandran et al. (2023) recently presented reviews of work on quantum many-body scars and Hilbert space fragmentation, describing dynamical process that is weakly nonergodic, not completely filling the entire phase space. Zhang et al. (2022b) and Chen et al. (2022a) studied many-body Hilbert space scarring (QMBS) on superconducting processors. QMBS is a weak form of ergodicity breaking in strongly interacting quantum systems, meaning that the system does not visit all parts of phase space. This presents opportunities for mitigating thermalization-induced decoherence due to scrambling in quantum information processing applications. Utilizing a programmable superconducting processor with 30 qubits and tunable couplings, Zhang et al. (2022b) created Hilbert space scarring in a nonconstrained model in different geometries, including a linear chain and quasi-one-dimensional comb geometry, by approximately decoupling from the qubit substrate. By reconstructing the full quantum state through quantum state tomography on four-qubit subsystems, they present strong evidence for QMBS states by measuring qubit population dynamics, quantum fidelity, and entanglement entropy after a quench from initial unentangled states. The QMBS is found to be robust to various imperfections such as random cross-couplings between qubits, and it persists beyond 1D systems. The experimental findings also broaden the realm of scarring mechanisms and identify correlations in QMBS states for quantum technology applications. Comparing with other qubit platforms, Zhang et al. (2022b) state that the superconducting platform can process the same quantum information in a shorter time, implying advantages of QMBS in a superconducting platform for more practical quantum-sensing and metrology applications. Also, Chen et al. (2022a) recently used pulse-level control and a variety of quantum error-mitigation (QEM) techniques to simulate the dynamics of a spin chain with QMBS on IBM QPUs for chains of up to 19 qubits. They conclude that the results demonstrate the power of error-mitigation techniques and pulse-level control to probe many-body coherence and correlation effects on present-day quantum hardware.

Key issues Noise and loss of information—A common experience A common classical experience might illuminate what QC is facing in the present NISQ era. Onboard an airplane, listening to music using the cheap versions of headphones offered by airlines in economy class can be a less-than-satisfactory experience. To start with, the headphone sound emitters have narrow bandwidth and large distortion, certainly not improving the limited quality of the source itself. Then the high-frequency background noise in the cabin from the air conditioning and engines may swamp the music signal. And finally, the sensitivity and frequency response of passenger’s ears, and the processing in the brain (CPS, 2023; CAPD, 2023; Souffi, 2023), may be less than perfect, making it difficult to discriminate against the noise. For the traveler, high-quality headphones with noise suppression therefore make a big difference. Then the external noise from the environment is processed in real time: recorded, inverted, and subtracted. This is a useful analogy in the case of a single qubit in a noisy environment. However, in order to describe the influence of noise on a multiqubit processor, one might better illustrate the situation in terms of the “Cocktail Party Syndrome” (CPS, 2023), referring to the difficulty to entertain a meaningful conversation within a group of people in a very noisy environment. Here, also simultaneous “two-body interactions” between members of the group add “correlated” noise to the “random” noise from the background. In our classical example, one can informally define error suppression, error mitigation, and error correction: Error suppression means creating high-quality hardware: the signal input is perfect; the classical bits are perfect; the sound generators in the

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headphones are perfect. Error mitigation means eliminating background cocktail party (channel) noise, for example, via noise inverting devices, as well as eliminating noise generated within the group (e.g., noise within the brain from alcohol consumption and tinnitus). Error mitigation would also include recording of the session and providing a clean edited transcript afterwards (postprocessing). Error correction means coding the information such that any errors can be traced and corrected, in real time or at the end. When it comes to real quantum computers, similar concepts and actions apply: one talks about quantum error suppression (QES), quantum error mitigation (QEM), and quantum error correction (QEC).

Fighting imperfections and noise in quantum processors Key issues concern the impact of imperfections and noise on computational capacity. NISQ devices are noisy, which creates decoherence and computational errors due to qubit relaxation and dephasing. One talks about three main types of noise: incoherent, coherent, and correlated. JJ qubits are embedded in a sea of fluctuating defects creating stochastic charge fluctuations—incoherent noise—capable of driving unwanted qubit transitions causing relaxation. Moreover, qubit control via microwave waveguides and magnetic flux lines is subject to stochastic fluctuations influencing the precision of quantum operations. These fluctuating fields may lead to incoherent dephasing of individual qubits, as well as to systematic miscalibrations, drift, and crosstalk—coherent noise—that in principle can be reversed (Krantz et al., 2019; Hashim et al., 2021; Ahsan et al., 2022). Finally, recent work shows that impact of cosmic rays can generate quasiparticles that create correlated charge noise and qubit relaxations on a length scale of hundreds of micrometers (Vepsäläinen et al., 2020; Wilen et al., 2021).

Quantum error suppression QES refers to various efforts to maximize the QV by improving the fabrication and operation of the quantum hardware (HW). (This excludes active feedback, treated as QEC). On the fabrication side, the main issues concern minimizing decoherence (Krantz et al., 2019; Burnett et al., 2019; Vepsäläinen et al., 2020; Wilen et al., 2021; Spring et al., 2022; Müller et al., 2019; Bilmes et al., 2020; Carroll et al., 2022) and cross talk (Zhao et al., 2022; Spring et al., 2022). For a given QPU circuit, the issue is to maximize gate fidelities and speed via optimal control (e.g., pulse shaping) based on advanced characterization of the device (Wittler et al., 2021).

Quantum error mitigation QEM aims to produce accurate expectation values of observables. It refers to various software methods to alleviate the effects of noise on computational results during execution of an algorithm on a QPU (Temme et al., 2017; Li and Benjamin, 2017; Kandala et al., 2017; Endo et al., 2018; Song et al., 2019; van den Berg et al., 2022; Takagi et al., 2022; Lolur et al., 2023). QEM effectively involves creating a noisy distribution of results, and then extracting the desired quantum information via postprocessing. Currently, the main principles are zero-noise extrapolation (ZNE) (Temme et al., 2017; Li and Benjamin, 2017; Kandala et al., 2017; Endo et al., 2018) and probabilistic error cancellation (PEC) (Temme et al., 2017; Li and Benjamin, 2017; Endo et al., 2018; Song et al., 2019; van den Berg et al., 2022; Takagi et al., 2022). The first scheme (ZNE) does not make any assumption about the noise model other than it being weak and constant in time (Temme et al., 2017). The second scheme (PEC) can tolerate stronger noise; however, it requires detailed knowledge of the noise model (Temme et al., 2017). ZNE works by physically increasing the impact of noise, determining a curve describing how the expectation value of some observable varies with noise. Variation of the noise strength can be done or simulated by varying the 1q- and 2q-gate times. Given enough points to determine the variation, the curve is then extrapolated back to zero noise, providing a best estimate of the expectation value. This has been implemented successfully in a number of experimental and theoretical investigation (Kandala et al., 2019; Giurgica-Tiron et al., 2020; Schultz et al., 2022; Pascuzzi et al., 2022, Kim et al., 2023). The ZNE requires sufficient control of the time evolution to implement the rescaled dynamics and hinges on the assumption of a large time-scale separation between the dominant noise and the controlled dynamics (Temme et al., 2017). PEC works by measuring the noise spectrum and applying an inverted quasiprobability distribution to the result of the computation via postprocessing (Temme et al., 2017; Li and Benjamin, 2017; Endo et al., 2018; Song et al., 2019; van den Berg et al., 2022; Takagi et al., 2022). PEC requires a full characterization of the noisy computational operations. To obtain this to sufficient precision is challenging in practice (Temme et al., 2017). Nevertheless, Song et al. (2019) experimentally demonstrated that PEC based on a combination of gate set tomography (GST) and quasiprobability decomposition can substantially reduce the error in quantum computation on a noisy quantum device. Moreover, van den Berg et al. (2022) have presented a practical protocol for learning and inverting a sparse noise model that is able to capture correlated noise and scales to large quantum devices, demonstrating PEC on a superconducting quantum processor with crosstalk errors. In contrast, Leyton-Ortega et al. (2023) present a method to improve the convergence of variational algorithms by replacing the hardware implementation of certain Hermitian gates with their inverses, resulting in noise cancellation and a more resilient quantum circuit. This is demonstrated on superconducting quantum processors running the VQE algorithm to find the H2 ground-state energy. Another QEM method has been developed by Lolur et al. (2023) for quantum chemical computations on NISQ devices— Reference-State Error Mitigation (REM). The method relies on determining the exact error in energy due to hardware and

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environmental noise for a reference wavefunction that can be feasibly evaluated on a classical computer. REM is shown to drastically improve the computational accuracy at which total energies of molecules can be computed using current quantum hardware.

Quantum error correction QEC refers to methods to code quantum information into logical qubits that can be measured, errors detected and identified, and logical qubits restored (Fowler et al., 2012; Kelly et al., 2015; Terhal, 2015; Wendin, 2017; Roffe, 2015; Bravyi et al., 2022; RyanAnderson et al., 2021, 2022; Postler et al., 2022; Andersen et al., 2020; Marques et al., 2022; Chen et al., 2021b; Krinner et al., 2022; Acharya et al., 2023; Piveteau et al., 2021; Ouyang, 2021; Suzuki et al., 2022). The initial application of the surface code to superconducting devices (Kelly et al., 2015) was followed up by several groups (Andersen et al., 2020; Marques et al., 2022; Chen et al., 2021b), and has recently resulted in demonstrations of repeated cycles of QEC. Krinner et al. (2022) implemented a 17-qubit distance-3 surface code, while Acharya et al. (2023) implemented a 49-qubit distance-5 code. Acharya et al. (2023) draw particular attention to the question whether scaling up the error-correcting code size will reduce logical error rates in a real device. They answer it by reporting on a 72-qubit superconducting device supporting a 49-qubit distance-5 (d ¼ 5) surface code that narrowly outperforms its average subset 17-qubit distance-3 (d ¼ 3) surface code, demonstrating a critical step toward scalable QEC. Even though John Martinis’ group at USCB in 2014 announced that superconducting quantum circuits were at the surface code threshold for fault tolerance (Barends et al., 2014), this did not mean that the route to QEC was a straight one—it is all about scaling. When the physical error rate is high, the logical error probability increases with increasing system size while sufficiently low physical error rates will eventually lead to the desired exponential suppression of logical errors. Acharya et al. (2023) show that their experiment lies in a crossover regime where increasing system size initially suppresses the logical error rate before, due to finite-size effects, later increasing it. They estimate that component performance must improve significantly to achieve practical scaling. In any case, the work demonstrates the first step in the process of suppressing logical errors by scaling up a quantum error-correcting code.

Scaling up for practical quantum advantage The concept of quantum advantage (QA) has emerged as a reaction to the more dramatic notion of quantum superiority (QS) (Preskill, 2012). In principle, QS is what we need—exponential advantage of over classical computers. However, this is only possible with functional QEC. QA emphasizes enhanced performance relative to specific classical algorithms for real-world use cases, typically addressing variational problems like QAOA and VQE. Currently there seems to be two opposite uses of practical QA: (i) effectively describing QS in huge QEC machines and (ii) mainly providing some useful speedup relative to classical algorithms. In the present NISQ era, there are essentially two ways to look at the scaling up of QPUs, what we will refer to as QPU-centric and HPC-centric.

QPU-centric approach Here QPUs are scaled up in tune with HW progress to push the limits in experiments testing quantum supremacy (Arute et al., 2019; Wu et al., 2021; Zhu et al., 2022), QEC (Andersen et al., 2020; Chen et al., 2021b; Marques et al., 2022; Krinner et al., 2022; Acharya et al., 2023), and physics (Barends et al., 2015, 2016; Kandala et al., 2017, 2019; Gong et al., 2019; Arute et al., 2020b, a; Neill et al., 2021; Gong et al., 2021, 2022; Zhang et al., 2022b; Mi et al., 2022b; Stanisic et al., 2022). The main role of the classical computer is to serve a single “free-standing” QPU with pre- and post-processing resources for running quantum circuits. The maximum number of qubits in the QPU so far is 72 (Acharya et al., 2023). Further scaling up the number of qubits will make it possible to systematically build larger logical qubits and larger code distances. In a blog post in May 2021, Erik Lucero at Google (Lucero, 2021) began with a bold statement: “Within the decade, Google aims to build a useful, error-corrected quantum computer.” The key issue in 2029 will most likely be “useful to whom?” Researchers or industry?

HPC-centric approach Here the HPC supercomputer seamlessly integrates collections of parallel CPUs, GPUs, and QPUs. The quantum big picture is to boost classical performance by including QPU subroutines approximately solving specific NP-hard problems that form classical bottlenecks. In this sense, the IBM roadmap and philosophy look HPC-centric, even though it is described by IBM rather as quantum-centric supercomputing (Gambetta, 2022a; Bravyi et al., 2022). The maximum number of qubits is currently 433, in the Osprey QPU (IBM, 2022). Osprey will be operated as a system of small parallel QPUs to achieve a computational advantage in the near term by combining multiple QPUs through circuit knitting techniques (Bravyi et al., 2022; Piveteau and Sutter, 2022), improving the quality of solutions through error suppression and mitigation, and focusing on heuristic versions of quantum algorithms with asymptotic speedups. Much of this is controlled by fast electronics and software that must effectively be incorporated as a dedicated HPC in the QPU stack. In this sense, the IBM approach is indeed QPU-centric, and the HPC mainframe becomes more of a server. The work distribution between the HPC and the QC/QPU is basically floating, and the terminology rather reflects HPC/QC attitudes: who serves who? The IBM Q Experience has created an ecosystem based on Qiskit, providing a versatile programming and testbed environment (García-Pérez et al., 2020; Bravyi et al., 2022), beginning to emerge as an industry standard. However, superpolynomial speedup does not belong to the NISQ era, and practical quantum advantage is elusive. Realistically, industrial users will not profit from

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quantum accelerators in the near term, so how can this large quantum effort be justified? The answer seems to be that IBM is going for useful quantum advantage via quantum parallel processing provided by QPU accelerators integrated in an efficient runtime HPC environment. Again the question is: useful to whom? Researchers or industry? And is useful quantum advantage possible without QEC? The quantum volume (QV) effectively represents the size of a qubit register for which one can entangle all of its qubits in a single shot. Currently, for IBM the best published value is QV ¼ 29 ¼ 512, corresponding to a 9-qubit quantum circuit 9 levels deep. This means that there is no point running algorithms requiring more than that. Instead, one can configure the operating system to run a number of 9-qubit mini-QPUs in parallel to speed up the rate for creating measured distributions, thus reducing the time to solution for computing expectation values of physical variables, like energy. Scale, quality, and speed are three key attributes to measure the performance of near-term quantum computers (Gambetta, 2022a; Bravyi et al., 2022). In the NISQ era, the QPU will spend very little time computing compared with the time spent by the CPU on pre- and post-processing before and after every call to the QPU. Calls that will be very frequent when solving variational problems. Circuit Layer Operations per Second (CLOPS) (Wack et al., 2021) is a measure correlated with how many QV circuits (mini-QPUs) a QPU can execute per unit of time, therefore shortening the time to solution. At the IBM Summit 2022, Gambetta (2022b) pledged that in 2024, IBM will offer a system that will generate reliable outcomes running 100 qubits with gate depth of 100: “So creating this 100  100 device will really allow us to set up a path to understand how can we get quantum advantage in these systems and lay a future going forward.” It must be noted, however, that this does not mean executing a 100q quantum circuit with depth 100 coherently “in a single shot,” achieving QV ¼ 2100—that would be a quantum-earth shaking demonstration of quantum supremacy. Microsoft has developed a framework for quantum resource estimation (Beverland et al., 2022) to estimate resources required across the stack layers for large-scale quantum applications, finding (as expected) that hundreds of thousands to many millions of physical qubits are needed to achieve practical quantum advantage. Beverland et al. (2022) maintain that the best solution is a monolithic QPU with, say, 10 million controllable, fast, and small qubits. The stack control system must be able to run millions of parallel high-fidelity gates at high speed, as well as reading out millions of qubits in parallel. Not surprisingly, no qubit technology currently implemented satisfies all of these requirements. However, Microsoft suggests that the recent proposals of electro-acoustic qubits (Chamberland et al., 2022) and the topological qubit approach based on Majorana Zero Modes (MZMs) (Karzig et al., 2017) might do it in future. Practical quantum advantage is on the horizon but needs to be accelerated through a variety of breakthrough techniques. The Microsoft view is that these research directions can be best studied in the context of resource estimation to unlock quantum at scale.

Useful NISQ digital quantum advantage—Mission impossible? The short answer is: yes, unfortunately probably mission impossible in the NISQ era. There are two fundamental questions: (1) Does the physical problem itself provide a quantum advantage? and (2), does a quantum algorithm have any advantage over a corresponding classical algorithm? Both questions were the original drivers of QC: the exponential advantage of Shor’s algorithm for factorization into prime numbers. However, the need for QEC put that problem far in the future. Instead, Matthias Troyer and coworkers (Reiher et al., 2017) promoted quantum chemistry for catalysts as the most useful killer application motivating the quest for scaling up QC. The paper argues that “quantum computers will be able to tackle important problems in chemistry without requiring exorbitant resources,” but at the same time concludes that “The required space and time resources for simulating FeMoco are comparable to that of Shor’s factoring algorithm.” Berry et al. (2019) improved on those results, obtaining circuits requiring less surface code resources, despite using a larger and more accurate active space. Nevertheless, also this needs extensive QEC and is far beyond NISQ computers. Liu et al. (2022) further elaborate on the potential benefits of QC in the molecular sciences, that is, in molecular physics, chemistry, biochemistry, and materials science, emphasizing the competition with classical methods that will ultimately decide on the usefulness of quantum computation for molecular science. Lee et al. (2022) have examined the case for the exponential quantum advantage (EQA) hypothesis for the central task of ground-state determination in quantum chemistry. Key for EQA is for the quantum state preparation to be exponentially easy compared to classical heuristics, which is far from clear and perhaps not even likely (Wang et al., 2021; Bittel and Kliesch, 2021). Identifying relevant quantum chemical systems with strong evidence of EQA remains an open question (Lee et al., 2022). The second question: “does a quantum algorithm have any advantage over a corresponding classical algorithm?” is currently a hot topic. Understanding whether, for example, quantum machine learning (QML) algorithms present a genuine computational advantage over classical approaches is extremely challenging. It seems that quantum-inspired classical algorithms "dequantizing" quantum algorithms (Tang, 2021, 2022; Cotler et al., 2021; Gharibian and Gall, 2022) can compete in polynomial time as long as one is not demanding exponentially accurate results. Tang and coworkers (Tang, 2022) developed a dequantization framework for analyzing QML algorithms to produce formal evidence against EQA. These are fully classical algorithms that, on classical data, perform only polynomially slower than their quantum counterparts. The existence of a dequantized algorithm means that its quantum counterpart cannot give exponential speedups on classical data, suggesting that the quantum exponential speedups are simply an artifact of state preparation assumptions. QML has the best chance of achieving large speedups whenever classical computation cannot get access to this data (which occurs when input states come from quantum circuits and other physical quantum systems). This does not yet rule out the

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possibility of large polynomial speedups on classical data, which could still lead to significant performance improvements in practice with sufficiently good quantum computers (Tang, 2022). Lloyd et al. (2016), however, proposed an algorithm for topological data analysis (TDA) that could not to be directly dequantized using the same techniques, raising the question whether a greater speedup was possible with TDA algorithms. This question has now been analyzed in depth by Berry et al. (2022), proposing a dequantization of the quantum TDA algorithm which shows that having exponentially large dimension and Betti number is necessary for superpolynomial advantage. The speedup is quartic, which will not be killed by QEC overhead (Babbush et al., 2021). Based on that, Berry et al. (2022) estimate that tens of billions of Toffoli gates will be sufficient to estimate a Betti number that should be classically intractable. This number of Toffoli gates is considered to be reasonable for early generations of fully fault-tolerant quantum computers (Berry et al., 2022), falling somewhere in between quantum chemistry applications and Shor’s algorithm in terms of the resources required for quantum advantage. Recently Akhalwaya et al. (2022) presented an NISQ-TDA QML algorithm needing only a short circuit depth with provable asymptotic speedup for certain classes of problems. How this goes together with Berry et al. (2022) remains to be understood in greater depth, as nicely explained by Quanta Magazine (Parshall, 2022), pointing out that the number of high-dimensional holes needs to be unthinkably large for quantum advantage. Reconnecting here to quantum chemistry, Gharibian and Gall (2022) have shown how to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix, implying that the ground state energy in quantum chemistry can be solved efficiently with constant precision on a classical computer. However, Gharibian and Gall (2022) also prove that with inverse-polynomial precision, the same problem becomes BQP-complete, suggesting that the superiority of quantum algorithms for chemistry stems from the improved precision achievable in the quantum setting. Finally, Huang et al. (2022) investigate quantum advantage in learning from experiments that processes quantum data with a quantum computer. That could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. Huang et al. (2022) prove that quantum machines can learn from exponentially fewer experiments than the number required by conventional experiments. They do that by assuming having access to data obtained from quantum-enhanced experiments like quantum sensing systems and stored in quantum memory (QRAM), allowing the QPU to process quantum input data. Exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Huang et al. (2022): “Although for now we lack suitably advanced sensors and transducers, we have conducted proof-of-concept experiments in which quantum data were directly planted in our quantum processor.” In the absence of perfect physical qubits, or QEC, quantum memory is a great challenge. Quantum memory may be far away for QC as needed by, for example, Huang et al. (2022), but it is essential for the development of quantum repeaters for quantum communication networks. O’Sullivan et al. (2022) investigate random-access quantum memory using chirped-pulse phase encoding. The protocol is implemented using donor spins in silicon coupled to a superconducting cavity, offering the potential for microwave random access quantum memories with lifetimes exceeding seconds.

Future directions Improved and alternative superconducting qubits A recent comprehensive review by Calzona and Carrega (2023) describes and analyzes the basic concepts and ideas behind the implementation of novel superconducting circuits with intrinsic protection against decoherence at the hardware level. The review explains the basics and performance of state-of-the-art transmons and other single-mode superconducting quantum circuits, and goes on to describe multimode superconducting qubits, toward the realization of fully protected qubits engineered in systems with more than one degree of freedom and/or characterized by the presence of specific symmetries. Regarding state-of-the-art tantalum-based transmons Place et al. (2021), Wang et al. (2022b) and Tennant et al. (2022) performed low-frequency charge-noise spectroscopy on Ta-based transmons and found distinctly different behavior compared with Al- and Nb-based transmons. They conclude that the temperature-dependent behavior of the neighboring chargeconfiguration transitions is caused by jumps between local charge configurations in the immediate vicinity of the transmon. This is in contrast to Al- and Nb-based transmons which are dominated by a TLS distribution giving rise to 1/f noise, apparently ruling out a TLS collection as the basis of the quasistable charge offsets in Ta-based transmons. Very different types of superconducting devices are semiconductor–superconductor hybrid structures containing Andreev bound states (ABS) (Wendin and Shumeiko, 2022) and topological MZMs (Das Sarma et al., 2015; Pikulin et al., 2021; Aghaee et al., 2022). Recently Pikulin et al. (2021) developed an experimental protocol (Topological Gap Protocol, TGP) to determine the presence and extent of a topological phase with MZMs in a hybrid semiconductor–superconductor three-terminal device with two normal leads and one superconducting lead. Now Aghaee et al. (2022) have presented measurements and simulations of InAs-Al hybrid three-terminal devices that are consistent with the observation of topological superconductivity and MZM, passing the TGP. Passing the protocol indicates a high probability of detection of a topological phase hosting MZMs as determined by large-scale disorder simulations and is a prerequisite for experiments involving fusion and braiding of MZMs.

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Hybrid distributed computing In Section “HPC-centric approach” we talked about distributed quantum processing, running many QPU modules in parallel. It is a matter of debate whether future large-scale algorithms can be run on monolithic or modular QPUs fitting inside a single fridge, or whether the algorithms have to be distributed over several fridges and locations. Eventually one will be able to work with clusters of quantum computers connected via local or global quantum networks, but for now this represents a great challenge waiting for great breakthroughs and a quantum infrastructure. On a smaller scale, recent work has connected superconducting qubit circuits between two separate chips linked by a microwave cable: Wallraff’s group (Magnard et al., 2020) produced multiqubit entanglement between single transmons in separate fridges while Cleland’s group (Zhong et al., 2021; Yan et al., 2022) demonstrated entanglement, purification, and protection between two separately enclosed three-transmon nodes inside the same fridge. The microwave connections are photonic, but limited to local clusters. The needed interfaces for long-distance optical connections are emerging (Chu and Gröblacher, 2020), but will probably remain research endeavors for quite some time (Arnold et al., 2020; Wang et al., 2022a; Kumar et al., 2022; Tsuchimoto et al., 2022). On a larger scale, Ang et al. (2022) have developed architectures for superconducting modular, distributed, or multinode quantum computers (MNQCs), employing a “co-design”-inspired approach to quantify overall MNQC performance in terms of hardware models of internode links, entanglement distillation, and local architecture. In the particular case of superconducting MNQCs with microwave-to-optical interconnects, Ang et al. (2022) describe how compilers and software should optimize the balance between local gates and internode gates, discuss when noisy quantum internode links have an advantage over purely classical links, and introduce a research roadmap for the realization of early MNQCs. This roadmap illustrates potential improvements to the hardware and software of MNQCs and outlines criteria for evaluating the improvement landscape, from progress in entanglement generation to the use of quantum memory in entanglement distillation and dedicated algorithms such as distributed quantum phase estimation. As a concrete example, DiAdamo et al. (2021) consider an approach for distributing the VQE algorithm over distributed quantum computers with arbitrary number of qubits in a systematic approach to generate distributed quantum circuits for QC. This includes a proposal for software-based system for controlling quantum systems at the various classical and quantum hardware levels. DiAdamo et al. (2021) emphasize that much effort has gone into distributed computing in the classical computing domain. And since the overlap between the fields is high, one can use this knowledge to design robust and secure distributed quantum computers, and as quantum technologies improve, this may become a reality.

Continuous variables—Computing with resonators In this field, the resonator modes are the logical qubits, and the transmon qubits provide ancillas for loading and readout. The Yale group is leading the development, and has recently demonstrated some decisive breakthroughs (Sivak et al., 2022b). The name of the game is to construct logical qubits from linear combinations of (already long-lived) resonator states representing the Gottesman-Kitaev-Preskill (GKP) bosonic code, encoding a logical qubit into grid states of an oscillator. Sivak et al. (2022b) demonstrate a fully stabilized and error-corrected logical qubit whose quantum coherence is significantly longer than that of all the imperfect quantum components involved in the QEC process, beating the best of them with a coherence gain of G  2.3. This was achieved by combining innovations in several domains including the fabrication of superconducting quantum circuits and modelfree reinforcement learning (Sivak et al., 2022a). To correct for single-photon loss, Kudra et al. (2022) have implemented two photon transitions that excite the cavity and the qubit at the same time. The additional degree of freedom of the qubit makes it possible to implement a coherent, unidirectional mapping between spaces of opposite photon parity. The successful experimental implementation, when supplemented with qubit reset, is suitable for autonomous QEC in bosonic systems, opening up the possibility to realize a range of nonunitary transformations on a bosonic mode. For full-scale QEC, various groups have recently investigated the concatenation of CV and DV codes, such as concatenating the single-mode GKP code with the surface code. Instead, Guillaud and Mirrahimi (2019) present a 1D repetition code based on a cat code as the base qubit for which the noise structure is modified in such a way that QEC becomes of similar complexity as classical error correction and can be performed using a simple repetition code. According to Guillaud and Mirrahimi (2019), the specific noise structure can be preserved for a set of fundamental operations which at the level of the repetition code lead to a universal set of protected logical gates. Regarding scaling up CV resonator technology, Axline et al. (2018) have experimentally realized on-demand, high-fidelity state transfer and entanglement between two isolated superconducting cavity quantum memories. By transferring states in a multiphoton encoding, they show that the use of cavity memories and state-independent transfer creates the striking opportunity to deterministically mitigate transmission loss with QEC. The results establish a basis for deterministic quantum communication across networks, and will enable modular scaling of CV superconducting quantum circuits. The size of superconducting 3D microwave resonators makes it challenging to scale up large 3D multiqubit CV systems. An alternative may be provided by nanomechanical phononic nanostructures (Chu and Gröblacher, 2020). Chu et al. (2017) experimentally demonstrated strong coupling between a superconducting transmon qubit and the long-lived longitudinal phonon modes of a high-overtone bulk acoustic wave disk resonator (HBAR) formed in thin-film aluminum nitride (AlN). Recently, von

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Lüpke et al. (2022) demonstrated HBAR parity measurement in the strong dispersive regime of circuit quantum acoustodynamics, providing basic building blocks for constructing acoustic quantum memories and processors. Moreover, Schrinski et al. (2022) measured long-lived HBAR Wigner states, monitoring the gradual decay of negativities over tens of microseconds. Wollack et al. (2022) use a superconducting transmon qubit to control and read out the quantum state of a pair of nanomechanical resonators made from thin-film lithium niobate (LN). The device is capable of fast swap operations, used to deterministically manipulate the nonclassical and entangled mechanical quantum states. This creates potential for feedback-based operation of quantum acoustic processors. Finally, Chamberland et al. (2022) have presented a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes applied to a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout.

Biochemistry and life science—Drivers of quantum computing? Biochemistry and life science are, as always, at the focus of high-performance computing, driving the development of exascale and postexascale supercomputers, experiencing the limitations and bottlenecks. These are topics and areas that would profit immensely from quantum advantage. As already mentioned, Reiher et al. (2017) practically created the vision of quantum chemistry as the killer application for quantum computers, needed to design new catalysts for efficient biochemistry, beyond the reach of classical computers. Since then, if not before, quantum chemistry and materials science present a kind of mission for QC to benchmark methods, algorithms, and computers (Santagati et al., 2023; Rubin et al., 2023; Babbush et al., 2023; O’Brien et al., 2022; Liu et al., 2022; Lee et al., 2022; Goings et al., 2022; Cheng et al., 2020). In computational science there is the well-established method of multiscale modeling (Fish et al., 2021) that gave the Nobel Prize in Chemistry in 2014 to Arieh Warshel for modeling biological functions, from enzymes to molecular machines (Warshel, 2014). Multiscale modeling describes methods that simulate continuum-scale behavior using information derived from computational models of finer scales in the system, down to molecular quantum levels, bridging across multiple length and time scales. It is then natural to consider using a quantum computer to address the case of a quantum system embedded in a multiscale environment. Cheng et al. (2020) review these methods and propose the embedding approach as a method for describing complex biochemical systems, with the parts treated at different levels of theory and computed with hybrid classical and quantum algorithms. Considering healthcare, life science, and artificial intelligence in a broad sense, from the huge databases of cell biology and human diseases, one can design network models describing the networks of Life. In particular Barabasi, Loscalzo, and collaborators (Barabasi et al., 2011; Lee and Loscalzo, 2019) developed the science of network medicine, and machine learning is essential for creating models for therapies that can design and control the action of drugs (de Siqueira Santos et al., 2022; Infante et al., 2021). Maniscalco et al. (2022) recently published a forward-looking white paper: “Quantum network medicine: rethinking medicine with network science and quantum algorithms,” and posit that QC may be a key ingredient in enabling the full potential of network medicine, laying the foundations of a new era of disease mechanism, prevention, and treatment. This challenging vision reflects the mission of the entire field of QC—to achieve the elusive Quantum Advantage. The difficulty is amply illustrated by Goings et al. (2022) who discussed the ways to “explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical-quantum advantage boundary.” One conclusion (Goings et al., 2022) is that a large classically intractable CYP model simulation may need 5 million qubits and nearly 10 billion Toffoli gates, and may take 100 QPU hours. Another, less surprising, conclusion is that deep classical chemical insight is essential for guiding quantum algorithms and defining the computational frontier for chemistry.

Conclusion Quantum computers and QC have evolved in ways that seemed almost impossible a few decades ago. When David DiVincenzo published his famous seven points stating the conditions for creating functional quantum computers (DiVincenzo, 2000), the first superconducting solid-state qubit had just been born (Nakamura et al., 1999). And 10 years later, demonstrating elementary operations on two-three superconducting transmon qubits still represented the state of the art in that field. Scaling to larger systems was not meaningful until the quality of qubits and the technology of electronic and optical control systems for driving and reading out qubits had improved. However, during the following 10 years, the telecommunication development has provided many of the tools needed, and some areas of the field have gone from academic science to engineering. Strictly speaking, the current technology for quantum computers is nevertheless not yet scalable. Classical digital computers became scalable when high-quality integrated transistor circuits were combined with error correction in the 1960s—then the road was open toward scaling, as manifested by Moore’s Law (Moore, 1965). Morover, the subsequent invention of microprocessors in the 1970s opened up the landscape for classical computers. Nevertheless it took 60 years to get to where we are now, in continuous mutual adaptation of problem solving and technology development. This has led to exascale supercomputers with currently ten million cores that still uphold Moore’s Law when it comes to computing power. However, as Landauer (May, 1991) pointed out, information is physical, which explains the current explosion of electrical power consumption for computing. That long timescale is very different from today’s visions of what QC will be able to do already in a not-so-distant future, and certainly different from the mission of several academic and industry projects to deliver useful quantum advantage by the end of this

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decade. Fortunately there are very few problems of importance to mankind that rely on the imminent arrival of quantum computers with quantum advantage. This is not to say that we do not need still more powerful computers—the problem is that the complex problems we may urgently want to solve, like the evolution of global climate dynamics, are NP-hard and can only be solved approximately, even by quantum computers. This means that predictions over time are necessarily uncertain. Adding to this the necessarily incomplete knowledge of models and background information, computation of our future unavoidably has huge error bars— uncertain predictions of the effects of global warming around the year 2100 are just one aspect of that. The advantage of digital quantum computers beyond classical computing and algorithms must be established by benchmarking performance in practical applications. A common view in the HPC community is to regard QPUs as a kind of GPUs, closely integrated with the HPC and expected to accelerate the CPUs when handling hard problems that present classical bottlenecks. The development of HPC+QC integration is happening right now, and during the next 5 years, it will be taken to high levels—IBM is one prominent example. For sure this will challenge and boost the development of competitive classical algorithms and dedicated hardware—but useful quantum advantage for digital quantum computers may remain problematic in the NISQ era. It is therefore desirable to dampen some of the hype and promote realistic views of the near-future power of quantum computers. Industry currently involved in testing QC is certainly aware of what to expect, but that is not obvious when it comes to journalists and the public. In the near term, the most important use of quantum processors may actually be to follow Feynman’s original idea and create quantum hardware platforms for emulating physical systems with properties that are hard to describe by classical computers. An important use of HPC is to make virtual rather than real-life experiments. One example is to describe the liquid environment of molecules via molecular dynamics simulations (Warshel, 2014). Another example is simulation of the behavior of aircraft in wind tunnels through extensive hydrodynamics calculations without need for real wind tunnels. In the same spirit, quantum processors already provide platforms for performing computer experiments, simulating fundamental quantum physics problems that are hard or impossible, in practice, for classical computers to describe. There are already impressive results for ion traps (Kokail et al., 2019, 2021), arrays of atoms (Scholl et al., 2021; Daley et al., 2022), and superconducting transmon processors (Arute et al., 2020b; Neill et al., 2021; Gong et al., 2021; Mi et al., 2022b; Zhang et al., 2022b; Karamlou et al., 2022; Landsman et al., 2019; Touil and Deffner, 2020; Mi et al., 2021; Braumüller et al., 2022; Mi et al., 2022a; Frey and Rachel, 2022; Kim et al., 2023). The near-term mission for superconducting processors might therefore be to emulate basic and applied physics experiments by mapping them on large superconducting controllable multiqubit networks that are impossible for classical digital supercomputers to simulate. The next assessment of the future of quantum information processing is planned for 2028 (Wendin, 2028) and then we can perhaps compare notes.

Acknowledgments This work was supported from the Knut and Alice Wallenberg Foundation through the Wallenberg Center for Quantum Technology (WACQT).

References Aaronson S and Chen L (2017) Complexity-theoretic foundations of quantum supremacy experiments. 32nd Computational Complexity Conference CCC’17; arXiv: 1612.05903, pp. 1–67. Abhijith J, et al. (2022) Quantum algorithm implementations for beginners. ACM Transactions on Quantum Computings 3(4): 1–92. Acharya R, et al. (2023) Suppressing quantum errors by scaling a surface code logical qubit. Nature 614(7949): 676–681. Aghaee M, et al. (2022) InAs-Al hybrid devices passing the topological gap protocol. arXiv:2207.02472v3. Aharonov D, Gao X, Landau Z, Liu Y, and Vazirani U (2022) A polynomial-time classical algorithm for noisy random circuit sampling. arXiv:2211.03999v1. Ahsan M, Abbas S, Naqvi Z, and Anwer H (2022) Quantum circuit engineering for correcting coherent noise. Physical Review A 15: 022428. Ajagekar A, Hamoud KA, and You F (2022) Hybrid classical-quantum optimization techniques for solving mixed-integer programming problems in production scheduling. IEEE Transactions on Quantum Engineering 3: 3102216. Akhalwaya IY, et al. (2022) Towards quantum advantage on noisy quantum computers. arXiv:2209.09371v2. Alexeev Y, et al. (2021) Quantum computer systems for scientific discovery. PRX Quantum 2: 017001. Altman E, et al. (2022) Quantum simulators: Architectures and opportunities. PRX Quantum 2: 017003. Andersen CK, Remm A, Lazar S, Krinner S, Lacroix N, Norris GJ, Gabureac M, Eichler C, and Wallraff A (2020) Repeated quantum error detection in a surface code. Nature Physics 16(8): 875. Ang J, et al. (2022) Architectures for multinode superconducting quantum computers. arXiv:2212.06167v1. Arnold G, Wulf M, Barzanjeh S, Redchenko ES, Rueda A, Hease WJ, Hassani F, and Fink JM (2020) Converting microwave and telecom photons with a silicon photonic nanomechanical interface. Nature Communications 11: 4460. Arute F, et al. (2019) Quantum supremacy using a programmable superconducting processor. Nature 574: 505–510. Arute F, et al. (2020a) Hartree-Fock on a superconducting qubit quantum computer. Science 369: 1084–1089. Arute F, et al. (2020b) Observation of separated dynamics of charge and spin in the Fermi-Hubbard model. arXiv:2010.07965v1. Axline CJ, Burkhart LD, Pfaff W, Zhang M, Chou K, Campagne-Ibarcq P, Reinhold P, Frunzio L, Girvin SM, Jiang L, Devoret MH, and Schoelkopf RJ (2018) On-demand quantum state transfer and entanglement between remote microwave cavity memories. Nature Physics 14: 705–710. Babbush R, McClean JR, Newman M, Gidney C, Boixo S, and Neven H (2021) Focus beyond quadratic speedups for error-corrected quantum advantage. PRX Quantum 2: 010103. Babbush R, Huggins WJ, Berry DW, Ung SF, Zhao A, Reichman DR, Neven H, Baczewski AD, and Lee J (2023) Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods. arXiv:2301.01203v1.

264

Quantum information processing with superconducting circuits: A perspective

Barabasi AL, Gulbahce N, and Loscalzo J (2011) Network medicine: A network-based approach to human disease. Nature Reviews. Genetics 12: 56e68. Barends R, et al. (2014) Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508: 500–503. Barends R, et al. (2015) Digital quantum simulation of fermionic models with a superconducting circuit. Nature Communications 6: 7654. Barends R, et al. (2016) Digitized adiabatic quantum computing with a superconducting circuit. Nature 534(7606): 222–226. Bengtsson A, et al. (2020) Quantum approximate optimization of the exact-cover problem on a superconducting quantum processor. Physical Review Applied 14: 034010. Berry DW, Gidney C, Motta M, McClean JR, and Babbush R (2019) Qubitization of arbitrary basis quantum chemistry by low rank factorization. Quantum 3: 208. Berry DW, et al. (2022) Quantifying quantum advantage in topological data analysis. arXiv:2209.13581v1. Beverland ME, et al. (2022) Assessing requirements to scale to practical quantum advantages. arXiv:2211.07629v1. Bharti K, et al. (2022) Noisy intermediate-scale quantum algorithms. Reviews of Modern Physics 94: 015004. Bilmes A, Megrant A, Klimov P, Weiss G, Martinis JM, Ustinov AV, and Lisenfeld J (2020) Resolving the positions of defects in superconducting quantum bit. Science Reports 10: 3090. Bittel L and Kliesch M (2021) Training variational quantum algorithms is NP-hard. Physical Review Letters 127: 120502. Blais A, Girvin SM, and Oliver WD (2020) Quantum information processing and quantum optics with circuit quantum electrodynamics. Nature Physics 16: 247. Blais A, Grimsmo AL, Girvin SM, and Wallraff A (2021) Circuit quantum electrodynamics. Reviews of Modern Physics 93: 025005. Boixo S, Isakov SV, Smelyanskiy VN, Babbush R, Ding N, Jiang Z, Bremner MJ, Martinis JM, and Neven H (2018) Characterizing quantum supremacy in near-term devices. Nature Physics 14: 595–600. Borle A, Elfving VE, and Lomonaco SJ (2021) Quantum approximate optimization for hard problems in linear algebra. SciPost Physics Core 4: 031. Bouland A, Fefferman B, Nirkhe C, and Vazirani U (2019) On the complexity and verification of quantum random circuit sampling. Nature Physics 15: 159–163. Braumüller J, et al. (2022) Probing quantum information propagation with out-of-time-ordered correlators. Nature Physics 18: 172–178. Bravyi S, Kliesch A, Koenig R, and Tan E (2020) Obstacles to variational quantum optimization from symmetry protection. Physical Review Letters 125: 260505. Bravyi S, Dial O, Gambetta JM, Gil D, and Nazario Z (2022) The future of quantum computing with superconducting qubits. Journal of Applied Physics 132(16): 160902. Brown KR, Chiaverini J, Sage JM, and Häffner H (2021) Materials challenges for trapped-ion quantum computers. Nature Reviews Materials 6: 893–905. Brubaker B (2023) New algorithm closes quantum supremacy window. Quanta Magazine January 9, 2023. Bruzewicz CD, Chiaverini J, McConnell R, and Sage JM (2019) Trapped-ion quantum computing: Progress and challenges. Applied Physics Reviews 6: 021314. Burnett JJ, Bengtsson A, Scigliuzzo M, Niepce D, Kudra M, Delsing P, and Bylander J (2019) Decoherence benchmarking of superconducting qubits. npj Quantum Information 5: 54. Calvin JA, Peng C, Rishi V, Kumar A, and Valeev EF (2021) Many-body quantum chemistry on massively parallel computers. Chemical Reviews 121: 1203–1231. Calzona A and Carrega M (2023) Muti-mode architectures for noise-resilient superconducting qubits. Superconductor Science and Technology 36: 023001. Cao Y, et al. (2019) Quantum chemistry in the age of quantum computing. Chemical Reviews 119: 10856. CAPD. (2023) Central Auditory Processing Disorder (CAPD) is a Deficiency in the Central Auditory Nervous System (CANS). https://www.asha.org/practice-portal/clinical-topics/centralauditory-processing-disorder/; https://www.additudemag.com/central-auditory-processing-disorder/. Carroll M, Rosenblatt S, Jurcevic P, Lauer I, and Kandala A (2022) Dynamics of superconducting qubit relaxation times. npj Quantum Information 8: 132. Cerezo M, et al. (2021) Variational quantum algorithms. Nature Reviews Physics 3: 625–644. Chamberland C, et al. (2022) Building a fault-tolerant quantum computer using concatenated cat codes. PRX Quantum 3: 010329. Chandarana P, Hegade NN, Paul K, Albarrán-Arriagada F, Solano E, del Campo A, and Chen X (2022) Digitized-counterdiabatic quantum approximate optimization algorithm. Physical Review Research 4: 013141. Chandran A, Iadecola T, Khemani V, and Moessner R (2023) Quantum many-body scars: A quasiparticle perspective. Annual Review of Condensed Matter Physics 14: 443–469. Chen H, Nusspickel M, Tilly J, and Booth GH (2021a) Variational quantum eigensolver for dynamic correlation functions. Physical Review A 104: 032405. Chen Z, et al. (2021b) Exponential suppression of bit or phase errors with cyclic error correction. Nature 595(7867): 383–387. Chen I-C, et al. (2022a) Error-mitigated simulation of quantum many-body scars on quantum computers with pulse-level control. Physical Review Research 4: 043027. Chen Y, Zhu L, Liu C, Mayhall NJ, Barnes E, and Economou SE (2022b) How much entanglement do quantum optimization algorithms require?. Quantum 2.0 Conference 2022; arXiv:2205.12283. Cheng H-P, Deumens E, Freericks JK, Li C, and Sanders BA (2020) Application of quantum computing to biochemical systems: A look to the future. Frontiers in Chemistry 8: 587143. Chu Y and Gröblacher S (2020) A perspective on hybrid quantum opto- and electromechanical systems. Applied Physics Letters 117: 150503. Chu Y, Kharel P, Renninger WH, Burkhart LD, Frunzio L, Rakich PT, and Schoelkopf RJ (2017) Quantum acoustics with superconducting qubit. Science 358(6360): 199–202. Córcoles AD, Kandala A, Javadi-Abhari A, McClure DT, Cross AW, Temme K, Nation PD, Steffen M, and Gambetta JM (2020) Challenges and opportunities of near-term quantum computing systems. Proceedings of the IEEE 108(8): 1338. Cotler J, Huang H-Y, and McClean JR (2021) Revisiting dequantization and quantum advantage in learning tasks. arXiv:2112.00811v2. CPS. (2023) The “Cocktail Party Syndrome” is the name given to the inability to differentiate sound from background noise. https://en.wikipedia.org/wiki/Cocktail_party_effect. Cross AW, Bishop LS, Sheldon S, Nation PD, and Gambetta JM (2019) Validating quantum computers using randomized model circuits. Physical Review A 100: 032328. Csalódi R, Süle Z, Jaskó S, Holczinger T, and Abonyi J (2021) Industry 4.0-driven development of optimization algorithms: A systematic overview. Complexity 2021: 1–22. Daley AJ, Bloch I, Kokail C, Flannigan S, Pearson N, Troyer M, and Zoller P (2022) Practical quantum advantage in quantum simulation. Nature 607: 667–676. Das Sarma S, Freedman MH, and Nayak C (2015) Majorana zero modes and topological quantum computation. npj Quantum Information 1: 15001. Das Sarma S (2022) Quantum computing has a hype problem. MIT Technology Review (March 28, 2022). de Gracia-Treviño JA, Delcey MG, and Wendin G (2023) Complete active space methods for NISQ devices: The importance of canonical orbital optimization for accuracy and noise resilience. ChemRxiv. de Siqueira Santos S, Torres M, Galeano D, del Mar Sánchez M, Cernuzzi L, and Paccanaro A (2022) Machine learning and network medicine approaches for drug repositioning for COVID-19. Patterns (New York) 3(1): 100396. de Sousa Junior WT, Montevechi JAB, de Carvalho Miranda R, and Campos AT (2019) Discrete simulation-based optimization methods for industrial engineering problems: A systematic literature review. Computers & Industrial Engineering 128: 526–540. DiAdamo S, Ghibaudi M, and Cruise J (2021) Distributed quantum computing and network control for accelerated VQE. IEEE Transactions on Quantum Engineering 2: 3100921. DiVincenzo D (2000) The physical implementation of quantum computation. Fortschritte der Physik 9–11: 771–783. Dupont M, Didier N, Hodson MJ, Moore JE, and Reagor MJ (2022) An entanglement perspective on the quantum approximate optimization algorithm. Physical Review A 106: 022423. Egger DJ, Marecek J, and Woerner S (2021) Warm-starting quantum optimization. Quantum 5: 479. Elfving VE, Millaruelo M, Gámez JA, and Gogolin C (2021) Simulating quantum chemistry in the seniority-zero space on qubit-based quantum computers. Physical Review A 103: 032605. Endo S, Benjamin SC, and Li Y (2018) Practical quantum error mitigation for near-future applications. Physical Review X 8: 031027. Ezratty O (2022) Mitigating the quantum hype. arXiv:2202.01925. Farhi E and Harrow AW (2016) Quantum supremacy through the quantum approximate optimization algorithm. arXiv:1602.07674v1. Farhi E, Goldstone J, and Gutmann S (2014) A quantum approximate optimization algorithm. arXiv:1411.4028. Farhi E, Goldstone J, Gutmann S, and Zhou L (2022) The quantum approximate optimization algorithm and the Sherrington-Kirkpatrick model at infinite size. Quantum 6: 759. Fedorov DA, Peng B, Govind N, and Alexeev Y (2022) VQE method: A short survey and recent developments. Materials Theory 6: 2. Fish J, Wagner GJ, and Keten S (2021) Mesoscopic and multiscale modelling in materials. Nature Materials 20: 774. Fitzek D, Ghandriz T, Laine L, Granath M, and Kockum AF (2021) Applying quantum approximate optimization to the heterogeneous vehicle routing problem. arXiv:2110.06799.

Quantum information processing with superconducting circuits: A perspective

265

Fowler AG, Mariantoni M, Martinis JM, and Cleland AN (2012) Surface codes: Towards practical large-scale quantum computation. Physical Review A 86: 032324. Franca DS and Garcia-Patron R (2021) Limitations of optimization algorithms on noisy quantum devices. Nature Physics 17: 1221. Frey P and Rachel S (2022) Realization of a discrete time crystal on 57 qubits of a quantum computer. Science Advances 8: eabm7652. Gambetta J (2022a) Expanding the IBM Quantum Roadmap to Anticipate the Future of Quantum-Centric Supercomputing (10 May 2022). https://research.ibm.com/blog/ibmquantum-roadmap-2025. Gambetta J (2022b) IBM Quantum Summit 2022, IBM Quantum State of the Union 2022. https://www.ibm.com/quantum/summi. García-Pérez G, Rossi MAC, and Maniscalco S (2020) IBM Q Experience as a versatile experimental testbed for simulating open quantum systems. npj Quantum Information 6: 1. Gharibian S and Gall FL (2022) Dequantizing the quantum singular value transformation: Hardness and applications to quantum chemistry and the quantum PCP conjecture. STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingJune (STOC 2022), pp. 19–32. Giurgica-Tiron T, Hindy Y, LaRose R, Mari A, and Zeng WJ (2020) Digital zero noise extrapolation for quantum error mitigation. 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). Glover F, Kochenberger G, and Du Y (2022a) Quantum bridge analytics I: A tutorial on formulating and using QUBO models. Annuals of Operations Research 17: 335–337. Glover F, Kochenberger G, Ma M, and Du Y (2022b) Quantum bridge analytics II: QUBO-Plus, network optimization and combinatorial chaining for asset exchange. Annals of Operations Research 314: 185. Goings JJ, White A, Lee J, Tautermann CS, Degroote M, Gidney C, Shiozaki T, Babbush R, and Rubin NC (2022) Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers. PNAS 119. e2203533119. Gong M, et al. (2019) Genuine 12-qubit entanglement on a superconducting quantum processor. Physical Review Letters 122: 110501. Gong M, et al. (2021) Quantum walks on a programmable two-dimensional 62-qubit superconducting processor. Science 372: 948–952. Gong M, et al. (2022) Quantum neuronal sensing of quantum many-body states on a 61-qubit programmable superconducting processors. arXiv:2201.05957v1. Grimsley HR, Economou SE, Barnes E, and Mayhall NJ (2019) An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications 10: 3007. Grönkvist, M., 2005. The Tail Assignment Problem. (Ph.D. thesis), Chalmers University of Technology. Gu X, Kockum AF, Miranowicz A, Liu Y-X, and Nori F (2019) Microwave photonics with superconducting quantum circuits. Physics Reports 718–719: 1. Guerreschi GG and Matsuura AY (2019) QAOA for Max-Cut requires hundreds of qubits for quantum speed-up. Scientific Reports 9: 6903. Guillaud J and Mirrahimi M (2019) Repetition cat qubits for fault-tolerant quantum computation. Physical Review X 9: 041053. Hangleiter D and Eisert J (2022) Computational advantage of quantum random sampling. aXiv:2206.04079v3. Harrigan MP, et al. (2021) Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nature Physics 17: 332–336. Hashim A, et al. (2021) Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor. Physical Review X 11: 041039. Hegade NN, Chen X, and Solano E (2022) Digitized counterdiabatic quantum optimization. Physical Review Research 4: L042030. Herrman R, Ostrowski J, Humble TS, and Siopsis G (2021) Lower bounds on circuit depth of the quantum approximate optimization algorithm. Quantum Information Processing 20: 59. Huang H-Y, Broughton M, Cotler J, Chen S, Li J, Mohseni M, Neven H, Babbush R, Kueng R, Preskill J, and McClean JR (2022) Quantum advantage in learning from experiments. Science 376: 1182. IBM. (2022) Osprey. https://newsroom.ibm.com/2022-11-09-IBM-Unveils-400-Qubit-Plus-Quantum-Processor-and-Next-Generation-IBM-Quantum-System-Two. Infante T, et al. (2021) Machine learning and network medicine: A novel approach for precision medicine and personalized therapy in cardiomyopathies. Journal of Cardiovascular Medicine 22(6): 429. Jones GM, Pathirage PDVS, and Vogiatzis KD (2022) Data-driven acceleration of coupled-cluster theory and perturbation theory methods. In: Pavlo Dral (ed.) Quantum Chemistry in the Age of Machine Learning, pp. 509–529. Elsevier. Kandala A, Mezzacapo A, Temme K, Takita M, Brink M, Chow JM, and Gambetta JM (2017) Hardware-efficient quantum optimizer for small molecules and quantum magnets. Nature 549(7671): 242–246. Kandala A, Temme K, Córcoles AD, Mezzacapo A, Chow JM, and Gambetta JM (2019) Error mitigation extends the computational reach of a noisy quantum processor. Nature 567 (7749): 491. Karamlou AH, et al. (2022) Quantum transport and localization in 1d and 2d tight-binding lattices. npj Quantum Information 8: 35. Karzig T, et al. (2017) Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majorana zero modes. Physical Review B 95: 235305. Kelly J, et al. (2015) State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519: 66–69. Kim IH, Liu Y-H, Pallister S, Pol W, and Roberts S (2022) Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules. Physical Review Research 4: 023019. Kim Y, et al. (2023) Scalable error mitigation for noisy quantum circuits produces competitive expectation values. Nature Physics. https://doi.org/10.1038/s41567-022-01914-3. Kjaergaard M, Yan F, Orlando TP, Gustavsson S, and Oliver WD (2020) Superconducting qubits: Current state of play. Annual Review of Condensed Matter Physics 11: 369. Kokail C, et al. (2019) Self-verifying variational quantum simulation of lattice models. Nature 569: 355–360. Kokail C, van Bijnen R, Elben A, Vermisch B, and Zoller P (2021) Entanglement Hamiltonian tomography in quantum simulation. Nature Physics 17: 936–942. Kowalsky M, Albash T, Hen I, and Lidar DA (2022) 3-regular three-XORSAT planted solutions benchmark of classical and quantum heuristic optimizers. Quantum Science and Technology 7: 025008. Krantz P, Kjaergaard M, Yan F, Orlando TP, Gustavsson S, and Oliver WD (2019) A quantum engineer’s guide to superconducting qubits. Applied Physics Reviews 6: 021318. Krinner S, et al. (2022) Realizing repeated quantum error correction in a distance-three surface code. Nature 605: 669. Kudra M, Abad T, Kervinen M, Eriksson AM, Quijandrá F, Delsing P, and Gasparinetti S (2022) Experimental realization of deterministic and selective photon addition in a bosonic mode assisted by an ancillary qubit. arXiv:2212.12079v1. Kumar A, Suleymanzade A, Stone M, Taneja L, Anferov A, Schuster DI, and Simon J (2022) Quantum-limited millimeter wave to optical transduction. arXiv:2207.10121v1. Lacroix N, et al. (2020) Improving the performance of deep quantum optimization algorithms with continuous gate sets. PRX Quantum 1: 020304. Lamata L, Parra-Rodriguez A, Sanz M, and Solano E (2018) Digital-analog quantum simulations with superconducting circuits. Advances in Physics: X 3: 1457981. Lan Z and Liang WZ (2022) Amplitude reordering accelerates the adaptive variational quantum eigensolver algorithms. Journal of Chemical Theory and Computation 18: 5267. Landauer R (1991) Information is physical. Physics Today, May 1991. Landsman KA, Figgatt C, Schuster T, Linke NM, Yoshida B, Yao NY, and Monroe C (2019) Verified quantum information scrambling. Nature 567: 61–65. Lee LY-H and Loscalzo J (2019) Network medicine in pathobiology. The American Journal of Pathology 189: 1311. Lee J, Huggins WJ, Head-Gordon M, and Whaley KB (2019) Generalized unitary coupled cluster wave functions for quantum computation. Journal of Chemical Theory and Computation 15: 311. Lee S, et al. (2022) Is there evidence for exponential quantum advantage in quantum chemistry? arXiv:2208.02199v3. Leyton-Ortega V, Majumder S, and Pooser RC (2023) Quantum error mitigation by hidden inverses protocol in superconducting quantum devices. Quantum Science and Technology 8: 014008. Li Y and Benjamin SC (2017) Efficient variational quantum simulator incorporating active error minimization. Physical Review X 7: 021050. Liu H, Low GH, Steiger DS, Häner T, Reiher M, and Troyer M (2022) Prospects of quantum computing for molecular sciences. Materials Theory 6: 11. Lloyd S, Garnerone S, and Zanardi P (2016) Quantum algorithms for topological and geometric analysis of data. Nature Communications 6: 10138. Lolur P, Rahm M, Skogh M, García-Álvarez L, and Wendin G (2021) Benchmarking the variational quantum eigensolver through simulation of the ground state energy of prebiotic molecules on high-performance computers. AIP Conference Proceedings 2362: 030005.

266

Quantum information processing with superconducting circuits: A perspective

Lolur P, Skogh M, Dobrautz W, Warren C, Biznárová J, Osman A, Tancredi G, Wendin G, Bylander J, and Rahm M (2023) Reference-state error mitigation: A strategy for high accuracy quantum computation of chemistry. Journal of Chemical Theory and Computation 19: 783–789. Lucero E (2021) Google: Unveiling our new Quantum AI campus, May 18, 2021. https://qt.eu/app/uploads/2018/04/93056_Quantum-Manifesto_WEB.pdf. Magnard P, et al. (2020) Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems. Physical Review Letters 125: 260502. Maniscalco S, et al. (2022) Quantum network medicine: Rethinking medicine with network science and quantum algorithms. arXiv:2206.12405v1. Marques JF, et al. (2022) Logical-qubit operations in an error-detecting surface code. Nature Physics 18(1): 80–86. McArdle S, Endo S, Aspuru-Guzik A, Benjamin SC, and Yuan X (2020) Quantum computational chemistry. Reviews of Modern Physics 92: 015003. McClean JR, et al. (2021) What the foundations of quantum computer science teach us about chemistry. The Journal of Chemical Physics 155: 150901. Mi X, et al. (2021) Information scrambling in quantum circuits. Science 374: 1479–1483. Mi X, et al. (2022a) Noise-resilient edge modes on a chain of superconducting qubits. Science 378(6621): 785–790. Mi X, et al. (2022b) Time-crystalline eigenstate order on a quantum processor. Nature 601(7894): 531–536. Monden Y (2012) TOYOTA Production System an Integrated Approach to Just-In-Time. CRC Press, Taylor & Francis Group. Mooney GJ, White GAL, Hill CD, and Hollenberg LCL (2021) Generation and verification of 27-qubit Greenberger-Horne-Zeilinger states in a superconducting quantum computer. Journal of Physics Communications 5: 095004. Moore GE (1965) Cramming more components onto integrated circuits. Electronics. April 19, 1965. Morales MES, Biamonte JD, and Zimborás Z (2020) On the universality of the quantum approximate optimization algorithm. Quantum Information Processing 19: 291. Moudgalya S, Bernevig BA, and Regnault N (2022) Quantum many-body scars and Hilbert space fragmentation: A review of exact results. Reports on Progress in Physics 85: 086501. Moussa C, Wang H, Bäck T, and Dunjko V (2022) Unsupervised strategies for identifying optimal parameters in quantum approximate optimization algorithm. EPJ Quantum Technology 9: 11. Müller C, Cole JH, and Lisenfeld J (2019) Towards understanding two-level-systems in amorphous solids: Insights from quantum circuits. Reports on Progress in Physics 82: 124501. Nakamura Y, Pashkin YA, and Tsai J-S (1999) Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature 398: 786–788. Neill C, et al. (2021) Accurately computing the electronic properties of a quantum ring. Nature 594(7864): 508–551. O’Brien TE, et al. (2022) Efficient quantum computation of molecular forces and other energy gradients. Physical Review Research 4: 043210. Ong JH and Teo J (2021) Systematic review and open challenges in hyper-heuristics usage on expensive optimization problems with limited number of evaluations. 2021 IEEE Symposium on Industrial Electronics & Applications (ISIEA). O’Sullivan J, et al. (2022) Random-access quantum memory using chirped pulse phase encoding. Physical Review X 12: 041014. Ouyang Y (2021) Avoiding coherent errors with rotated concatenated stabilizer codes. npj Quantum Information 7: 87. Pan F, Chen K, and Zhang P (2022) Solving the sampling problem of the Sycamore quantum circuits. Physical Review Letters 129: 090502. Parshall A (2022) After a quantum clobbering, one approach survives unscathed. Quanta Magazine Dec 7, 2022. Pascuzzi VR, He A, Bauer CW, de Jong WA, and Nachman B (2022) Computationally efficient zero-noise extrapolation for quantum-gate-error mitigation. Physical Review A 105: 042406. Patel YJ, Jerb S, Bäck T, and Dunjko V (2022) Reinforcement learning assisted recursive QAOA. arXiv:2207.06294v1. Pednault E, Gunnels J, Maslov D, and Gambetta J (2019) On “Quantum Supremacy”. https://qt.eu/app/uploads/2018/04/93056_Quantum-Manifesto_WEB.pdf. Pelofske E, Bärtschi A, and Eidenbenz S (2022) Quantum Volume in practice: What users can expect from NISQ devices. arXiv:2203.03816v4. Pikulin DI, et al. (2021) Protocol to identify a topological superconducting phase in a three-terminal device. arXiv:2103.12217v1. Piveteau C and Sutter D (2022) Circuit knitting with classical communication. arXiv:2205.00016. Piveteau C, Sutter D, Bravyi S, Gambetta JM, and Temme K (2021) Error mitigation for universal gates on encoded qubits. Physical Review Letters 127: 200505. Place APM, et al. (2021) New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nature Communications 12: 1779. Postler L, et al. (2022) Demonstration of fault-tolerant universal quantum gate operations. Nature 605(7911): 675–680. Preskill J (2012) Quantum computing and the entanglement frontier. arXiv:1203.5813. Preskill J (2018) Quantum computing in the NISQ era and beyond. Quantum 2: 79. QUTAC. (2021) QUTAC, industry quantum computing applications. EPJ Quantum Technology 8: 25. Rai R, Tiwari MK, Ivanov D, and Dolgui A (2021) Machine learning in manufacturing and industry 4.0 applications. International Journal of Production Research 59: 4773. Reiher M, Wiebe N, Svore KM, Wecker D, and Troyer M (2017) Elucidating reaction mechanisms on quantum computers. PNAS 114: 7555. Roffe J (2015) Quantum error correction: An introductory guide. Contemporary Physics 60(3): 226. Rubin NC, et al. (2023) Fault-tolerant quantum simulation of materials using Bloch orbitals. arXiv:2302.05531v1. Ryan-Anderson C, et al. (2021) Realization of real-time fault-tolerant quantum error correction. Physical Review X 11: 041058. Ryan-Anderson C, et al. (2022) Implementing fault-tolerant entangling gates on the five-qubit code and the color code. arXiv: 2208.01863v1. Sanders YR, Berry DW, Costa PCS, Tessler LW, Wiebe N, Gidney C, Neven H, and Babbush R (2020) Compilation of fault-tolerant quantum heuristics for combinatorial optimization. PRX Quantum 1: 020312. Santagati R, et al. (2023) Drug design on quantum computers. arXiv:2301.04114v1. Sapova MD and Fedorov AK (2022) Variational quantum eigensolver techniques for simulating carbon monoxide oxidation. Communications Physics 5: 19. Scholl P, et al. (2021) Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature 595: 233–238. Schrinski B, et al. (2022) Macroscopic quantum test with bulk acoustic wave resonators. arXiv:2209.06635v1. Schultz K, et al. (2022) Impact of time-correlated noise on zero-noise extrapolation. Physical Review A 106: 052406. Sivak VV, et al. (2022a) Model-free quantum control with reinforcement learning. Physical Review X 12: 011059. Sivak VV, et al. (2022b) Real-time quantum error correction beyond break-even. arXiv:2211.09116v1. Sokolov IO, Barkoutsos PK, Ollitrault PJ, Greenberg D, Rice J, Pistoia M, and Tavernelli I (2020) Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equivalents? The Journal of Chemical Physics 152: 124107. Song C, et al. (2017) 10-qubit entanglement and parallel logic operations with a superconducting circuit. Physical Review Letters 119: 180511. Song C, Cui J, Wang H, Hao J, Feng H, and Li Y (2019) Quantum computation with universal error mitigation on a superconducting quantum processor. Science Advances 5: eaaw5686. Souffi S (2023) Reduction in sound discrimination in noise is related to envelope similarity and not to a decrease in envelope tracking abilities. The Journal of Physiology 601(1): 123. Spring PA, et al. (2022) High coherence and low cross-talk in a tileable 3D integrated superconducting circuit architecture. Science Advances 8: eabl6698. Sreedhar R, Vikstål P, Svensson M, Ask A, Johansson G, and García-Álvarez L (2022) The quantum approximate optimization algorithm performance with low entanglement and high circuit depth. arXiv:2207.03404arXiv1. Stanisic S, Bosse JL, Gambetta FM, Santos RA, Mruczkiewicz W, O’Brien TE, Ostby E, and Montanaro A (2022) Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer. Nature Communications 13(1): 5743. Sugisaki K, Nakazawa S, Toyota K, Sato K, Shiomi D, and Takui T (2019) Quantum chemistry on quantum computers: A method for preparation of multiconfigurational wave functions on quantum computers without performing post-Hartree-Fock calculations. ACS Central Science 5: 167. Sugisaki K, Kato T, Minato Y, Okuwaki K, and Mochizuki Y (2022) Variational quantum eigensolver simulations with the multireference unitary coupled cluster ansatz: A case study of the C2v quasi-reaction pathway of beryllium insertion into a H2 molecule. Physical Chemistry Chemical Physics 24: 8439–8452.

Quantum information processing with superconducting circuits: A perspective

267

Suzuki Y, Endo S, Fujii K, and Tokunaga Y (2022) Quantum error mitigation as a universal error reduction technique: Applications from the NISQ to the fault-tolerant quantum computing eras. PRX Quantum 3: 010345. Svensson M, Andersson M, Grönkvist M, Vikstål P, Dubhashi D, Ferrini G, and Johansson G (2021) A hybrid quantum-classical heuristic to solve large-scale integer linear programs. arXiv:2103.15433v2. Takagi R, Endo S, Minagawa S, and Gu M (2022) Fundamental limits of quantum error mitigation. npj Quantum Information 8: 114. Tang E (2021) Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions. Physical Review Letters 127: 060503. Tang E (2022) Dequantizing algorithms to understand quantum advantage in machine learning. Nature Reviews Physics 4: 693. Tang HL, Barnes E, Grimsley HR, Mayhall NJ, and Economou SE (2021) Qubit-ADAPT-VQE: An adaptive algorithm for constructing hardware-efficient ansätze on a quantum processor. PRX Quantum 2: 020310. Temme K, Bravyi S, and Gambetta JM (2017) Error mitigation for short-depth quantum circuits. Physical Review Letters 119: 180509. Tennant DM, et al. (2022) Low-frequency correlated charge-noise measurements across multiple energy transitions in a tantalum transmon. PRX Quantum 3: 030307. Terhal BM (2015) Quantum error correction for quantum memories. Reviews of Modern Physics 87: 307. Tilly J, et al. (2022) The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986: 1–128. Tkachenko NV, et al. (2021) Correlation-informed permutation of qubits for reducing ansatz depth in the variational quantum eigensolver. PRX Quantum 2: 020337. Touil A and Deffner S (2020) Quantum scrambling and the growth of mutual information. Quantum Science and Technology 5: 035005. Tranter A, et al. (2022) Benchmarking the performance of variational quantum eigensolvers (VQE) applied to the HCN molecule. APS March Meeting 67. abstract id.B01.010. Tsuchimoto Y, et al. (2022) Large-bandwidth transduction between an optical single quantum dot molecule and a superconducting resonator. PRX Quantum 3: 030336. van den Berg E, Minev ZK, Kandala A, and Temme K (2022) Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors. arXiv:2201.09866v2. Vepsäläinen AP, et al. (2020) Impact of ionizing radiation on superconducting qubit coherence. Nature 584(7822): 551–556. Verdon G, Broughton M, McClean JR, Sung KJ, Babbush R, Jiang Z, Neven H, and Mohseni M (2019) Learning to learn with quantum neural networks via classical neural networks. arXiv:1907.05415v1. Vikstål P, Grönkvist M, Svensson M, Andersson M, Johansson G, and Ferrini G (2020) Applying the quantum approximate optimization algorithm to the tail-assignment problem. Physical Review Applied 14: 034009. von Lüpke U, Yang Y, Bild M, Michaud L, Fadel M, and Chu Y (2022) Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics. Nature Physics 18: 794. Wack A, Paik H, Javadi-Abhari A, Jurcevic P, Faro I, Gambetta JM, and Johnson BR (2021) Scale, quality, and speed: Three key attributes to measure the performance of near-term quantum computers. arXiv:2110.14108v2. Wang D, Higgott O, and Brierley S (2019) A generalised variational quantum eigensolver. Physical Review Letters 122: 140504. Wang S, Fontana E, Cerezo M, Sharma K, Sone A, Cincio L, and Coles PJ (2021) Noise-induced barren plateaus in variational quantum algorithms. Nature Communications 12: 6961. Wang C, Gonin I, Grassellino A, Kazakov S, Romanenko A, Yakovlev VP, and Zorzetti S (2022a) High-efficiency microwave-optical quantum transduction based on a cavity electro-optic superconducting system with long coherence time. npj Quantum Information 8: 149. Wang C, et al. (2022b) Towards practical quantum computers: Transmon qubit with a lifetime approaching 0.5 milliseconds. npj Quantum Information 8: 3. Warshel A (2014) Multiscale modeling of biological functions: From enzymes to molecular machines (nobel lecture). Angewandte Chemie, International Edition 53: 10020. Wedelin D (1995) An algorithm for large scale 0–1 integer programming with application to airline crew scheduling. Annals of Operations Research 57: 283–301. Weggemans JR, et al. (2022) Solving correlation clustering with QAOA and a Rydberg qudit system: A full-stack approach. Quantum 6: 687. Wendin G (2017) Quantum information processing with superconducting circuits: A review. Reports on Progress in Physics 80: 106001. Wendin, G., 2028. Five Years Later, What Happened? (In preparation). Wendin G and Shumeiko VS (2022) Coherent manipulation of a spin qubit. Science 373: 390. Wilen CD, et al. (2021) Correlated charge noise and relaxation errors in superconducting qubits. Nature 594(7863): 369–373. Willsch M, Willsch D, Jin F, Raedt HD, and Michielsen K (2020) Benchmarking the Quantum Approximate Optimization Algorithm. Quantum Information Processing 19: 197. Willsch D, Willsch M, Calaza CDG, Jin F, Raedt HD, Svensson M, and Michielsen K (2022) Benchmarking advantage and D-Wave 2000Q quantum annealers with exact cover problems. Quantum Information Processing 21: 141. Wittler N, Roy F, Pack K, Werninghaus M, Roy AS, Egger DJ, Filipp S, Wilhelm FK, and Machnes S (2021) Integrated tool set for control, calibration, and characterization of quantum devices applied to superconducting qubits. Physical Review Applied 15: 034080. Wollack EA, Cleland AY, Gruenke RG, Wang Z, Arrangoiz-Arriola P, and Safavi-Naeini AH (2022) Quantum state preparation and tomography of entangled mechanical resonators. Nature 604: 463. Wu Y, et al. (2021) Strong quantum computational advantage using a superconducting quantum processor. Physical Review Letters 127: 180501. Wurtz J and Love PJ (2022) Counterdiabaticity and the quantum approximate optimization algorithm. Quantum 6: 635. Yan H, et al. (2022) Purification and protection in a superconducting quantum network. Physical Review Letters 128: 080504. Yu J, Retamal JC, Sanz M, Solano E, and Albarrán-Arriagada F (2022) Superconducting circuit architecture for digital-analog quantum computing. EPJ Quantum Technology 9: 90. Zhang B, Sone A, and Zhuang Q (2022a) Quantum computational phase transition in combinatorial problems. npj Quantum Information 8: 87. Zhang P, et al. (2022b) Many-body Hilbert space scarring on a superconducting processor. Nature Physics 19(1): 120–125. Zhang Y, et al. (2022c) Variational quantum eigensolver with reduced circuit complexity. npj Quantum Information 8: 96. Zhao P, Linghu K, Li Z, Xu P, Wang R, Xue G, Jin Y, and Yu H (2022) Quantum crosstalk analysis for simultaneous gate operations on superconducting qubits. PRX Quantum 3: 020301. Zhong Y, et al. (2021) Deterministic multi-qubit entanglement in a quantum network. Nature 590(7847): 571–575. Zhou L, Wang S-T, Choi S, Pichler H, and Lukin MD (2020) Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Physical Review X 10: 021067. Zhu Q, et al. (2022) Quantum computational advantage via 60-qubit 24-cycle random circuit sampling. Science Bulletin 1667(3): 240–245.

Braids, motions and topological quantum computing Eric C Rowell, Department of Mathematics, Texas A&M University, College Station, TX, United States © 2024 Elsevier Ltd. All rights reserved.

Introduction Objectives The braid group Braids and knots in physics: An incomplete history Topological quantum computation The origins of topological quantum computation Modeling topological phases with categories The role of the braid group in TQC Detecting non-abelian anyons via braiding Braiding universality Measurement assisted universality Distinguishing anyons and anyon systems via braiding Beyond braids Conclusion Appendix on categorical notions Acknowledgments References

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Abstract The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the mathematical study of braids is crucial for the theory. We provide some brief historical context as well, emphasizing ways that braiding appears in physical contexts. We also briefly discuss the 3-dimensional generalization of braiding: motions of knots.

Introduction Quantum computation is predicated upon the ability to create, manipulate and measure the states in quantum systems (Freedman et al., 2003). In the topological model for quantum computation (Freedman et al., 2003; Nayak et al., 2008) the quantum systems of interest are topological phases of matter. Typically, these are 2 dimensional systems harboring point-like excitations–anyons (Wilczek, 1990). This alternative to the more traditional quantum circuit model is theoretically robust against decoherence due to the topological nature of the states: the information is stable due to invariance of the states under small, local perturbations. On the other hand, in order to implement quantum gates more drastic evolution of the states must be employed. The most natural choice is particle exchange of the anyons, so that their trajectories in 2 + 1 dimensional space-time form braids. The corresponding transformations are the quantum gates, which are mathematically encoded in unitary representations of the braid group. The process of creating anyons, braiding them and then measuring form knots or links in (2 +1)-dimensional spacetime, which is interpreted as a (single run) of a topological quantum computation as illustrated in Fig. 1. Notice that the motions of the point-like excitations in the effectively two dimensional medium have a natural mathematical interpretation as the group of braids. Thus it comes as no surprise that the braid group ℬn plays an outsized role in the development of topological quantum computation. In this article we will describe a selection of topics illustrating the connection between braids and topological quantum computation as well as some generalizations. We make no attempt to be exhaustive on the subject, nor will we provide a completely self-contained background. There are a number of much more complete treatments of topological quantum computation from various perspectives and for various tastes, a few of which we list here: (Wang, 2006, 2013, 2010; Delaney et al., 2016; Nayak et al., 2008; Pachos, 2012; Ogburn and Preskill, 1999; Freedman et al., 2003; Rowell, 2016; Rowell and Wang, 2018). For deeper and more complete mathematical details on the braid group and invariants of knots and links we suggest (Kassel and Turaev, 2008), while for the categorical background the text (Etingof et al., 2015) is an excellent resource. We provide a very brief tour of categorical notions in Appendix A.

Objectives In the article we will: • Introduce braids and the braid group, briefly discussing the history of braids in physics. • Introduce topological quantum computation through a historical perspective.

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Braids, motions and topological quantum computing Computation

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Measure (fusion)

Compute (applygates)

Braid anyons

Initialize

Create anyons

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Vacuum Fig. 1 A single iteration of a topological quantum computation via braiding.

• • •

Discuss the categorical model for the relevant topological phases of matter. Elucidate the role of the braid group in topological quantum computation through a series of examples. Give a higher dimensional generalization of the braid group and speculate on its relevance to topological quantum computation.

The braid group The braid group Bn on n strands is most concisely described using the abstract group presentation by generators and relations due to Artin (Artin, 1925): Bn ¼ hs1 , s2 , . . . , sn −1 j si sj ¼ sj si for |i − j|  2, si si+1 si ¼ si+1 si si+1 , for 1  i, j  n −1i The first relation is referred to as far commutativity and the second is the braid relation. For visualization purposes we display the generator s1 and its inverse s−1 1 geometrically as members of B4:

σ1 =

σ1−1 =

The composition in the group then corresponds to stacking, whereas the identity is clearly represented by unbraided strands. Predating Artin’s work on braids by 30 years is the work of Hurwitz (1891) on motions of points on the 2-dimensional sphere. This is much closer to their use in condensed matter physics: the world-lines of anyons under particle exchange can be viewed as trajectories of motions of points in the disk D2. Connecting braids to knots and links are two key mathematical results: Alexander’s theorem (Alexander, 1923) and Markov’s theorem (Markov, 1936). The key consequence of Alexander’s theorem is that any knot/ link may be obtained the plat closure: capping off the strands of a braid in B2n pairwise, on both the top and bottom. Birman’s (1976) analogue of Markov’s theorem establishes a way to determine if two distinct braids have the same plat closure.1

Braids and knots in physics: An incomplete history Around the 7th century B.C.E. there was a debate between the natural philosopher Gargı Vachaknavı (daughter of Vachaknu) and the sage Yajnavalkya (Olivelle, 1998). She asks him: Since this whole world is woven back and forth on water, on what then is it woven back and forth?

To which he replies “On air, Gargı.” The exchange continues with Gargı asking what air and the intermediate regions are woven upon, and eventually what the universe is woven upon. This could be the most ancient reference to braids in physics. That the universe consists of a sequence of braidings may seem quaint by today’s scientific standards, but the similitude with topological quantum circuits is nonetheless intriguing.2 Many point to Gauss as the first person to study knots mathematically and suggest he was inspired by physical considerations such as magnetic potential, see, e.g., (Epple, 1998; Przytycki, 1998; Ricca and Nipoti, 2011). Maxwell himself (Maxwell, 1873) 1 Both Alexander’s and Markov’s theorems actually deal with a different closure in which one glues the ends of top strands to the bottom strands. But the plat closure is more physically relevant. 2

We thank Paul Martin for informing us of this ancient reference.

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mentions Gauss’ integral formula for the linking number of a two component link and its physical interpretation in electromagnetic induction.3 Gauss’ interest in braids goes back even earlier according to Epple (1998), as he drew the following braid sometime between 1815 and 1830:

One might naturally surmise that Gauss’ early interest in braids and knots were influential in his discovery of the linking number. Yet another connection between physics and knots came some years later (see (Silver, 2014) for a full account) when Thompson (Lord Kelvin) learned of some experiments of Helmholtz in the 1850s producing knotted vortices of smoke. This led to the speculation by Thompson that matter was made up of knotted loops of luminiferous æther. This short-lived “vortex-atoms” theory suggested that different elements were distinguished by their knot-type. Yet again, we find an archaic theory resonating with a modern idea: in topological quantum computation information is stored in knotted world-lines of anyons. Mathematics reaped the benefit, as Tait produced the first reasonably comprehensive table of knots with few crossings, inspired by his friend Thompson’s vortex-atom theory.

Topological quantum computation Here we provide a historical perspective on topological quantum computation and set up the mathematical framework in terms of categories.

The origins of topological quantum computation Topological quantum computation emerged as a confluence of ideas of Freedman, in topology, and Kitaev, in physics, (Freedman, 1998; Yu Kitaev, 2003) in the late 1990s. The framework was then laid out quickly in a series of papers of Freedman, Kitaev, Wang and Larsen (Freedman et al., 2002b,a, 2003). Freedman postulated that physical systems described by topological quantum field theories (TQFTs) (Witten, 1989) could potentially be employed to perform certain Jones polynomial (Jones, 1985) evaluations. The Jones polynomial is a remarkably powerful invariant of knots and links. Its discovery in the early 1980s (Jones, 1985) precipitated the field of quantum topology. Although it is straightforward to compute for a given knot, the complexity of the competition grows quickly with the number of crossings. Indeed, evaluating the Jones polynomial at roots of unity is classically a #P-hard computation at most values (Jaeger et al., 1990). A clue to how physics could be useful for such computations is found in the original formulation of the Jones polynomial as the trace of the braid group representation associated with the SU(2) Chern-Simons/Reshetikhin-Turaev TQFTs. Meanwhile, Kitaev was interested in topological phases of matter harboring anyons for their potential use in fault-tolerant quantum computation. Topological phases of matter were studied going back to the 1970s and 1980s by the recipients of the 2016 Nobel Prize in Physics (Nobel Prize Committee, 2016): Kosterlitz, Thouless, and Haldane. Well-studied examples of include the fractional quantum Hall liquids (Moore and Read, 1991; Wen, 1991) that are expected to yield non-abelian anyons. Further examples include topological insulators (Nayak et al., 2008; das Sarma et al., 2015). More recently nanowires have been studied for their potential for harboring anyons (Mourik et al., 2012; Albrecht et al., 2016). The need for expensive error-correction due to decoherence continues to be a major hurdle for scaled up quantum computation. Kitaev’s idea was that quantum gates could be implemented by braiding anyons, so that error-correction comes from the topological nature of the systems. Remarkably, Freedman’s and Kitaev’s proposals are two sides of the same coin: 2D topological phases of matter are modeled by TQFTs! It is a rare instance when two essentially independent ideas converge yielding a powerful new paradigm. To breathe life into this new paradigm some deep mathematics was needed from both quantum topology and group theory. The resulting proof-of-principle of topological quantum computation is found in Freedman et al. (2002b), Freedman et al. (2002a). There they show that (1) the topological model for quantum computation could be as powerful as the more standard quantum circuit model based on qubits and (2) it is not more powerful than the quantum circuit model. The second result is proved by showing that any topological quantum computation can be simulated on a universal qubit machine with polynomial overhead. The first result show that universal topological quantum computers exist, in principle. We will return to this below.

3

There is some historical controversy here, which the reader may find in the references.

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Modeling topological phases with categories Anyons are topological quantum fields emerging as finite energy particle-like excitations in topological phases of matter. Like particles, they can be moved, but cannot be created or destroyed locally. To model them we consider the fusion and braiding structures of these elementary excitations in the plane. The anyon system is then modeled by a unitary modular category (UMC). For this reason we will use anyon model and unitary modular category synonymously. There is a philosophical explanation for why category theory is suitable for describing some quantum physics. In quantum physics we appeal to measurements to “see” the elementary particles by analyzing their responses to measuring devices. Anyons are defined by how they interact with other particles, and their responses to measuring devices. According to Kapranov and Voevodsky (1994), the main principle in category theory is: “In any category it is unnatural and undesirable to speak about equality of two objects.” In category theory an object X is determined by the vector spaces of morphisms Hom(X, Y) for all Y in the tensor category. Therefore, it is natural to treat objects as anyons, and the morphisms as models of the quantum processes between them. For a more complete treatment see the survey (Rowell and Wang, 2018).4 The notion of a modular tensor category was invented by Moore and Seiberg using tensors (Moore and Seiberg, 1989), and its coordinate-free version: modular category was defined by Turaev (1992). In this model, an anyon X is a simple object in the category, while the vacuum is regarded as the monoidal unit 1. The Hilbert space of states of a system of anyons on a disk with prescribed boundary condition is modeled by the space of morphisms in the category so that the braiding of anyons provides unitary operators as the system evolves: the quantum gates in topological quantum computation. For example, the Hilbert space associated with n indistinguishable X anyons in the disk with boundary condition (total charge) Y is the space Hom(Y, Xn). One assumption is that there are finitely many distinguishable indecomposable anyon types (including the vacuum type): {X0 ¼ 1, X1, . . ., Xr}, so that that every anyon corresponds to some fixed type a 2 {0, . . ., r}. A dictionary of terminologies between categories and anyons systems is given in Table 1, adapted from Wang (2010). The Hilbert space Vcab ¼ Hom(a  b, c) represents the span of the linearly independent ways of fusing anyons of type a and b to obtain the anyon of type c. In topological phases of matter the ground state degeneracy is typically encoded as the dimension of the spaces c Vcab where the sum is over all anyon types c. Most often this is consider for a ¼ b so that we can speak of the ground state degeneracy of a given anyon. In the vernacular one says a has non-trivial ground-state degeneracy if dim(c Vcaa) > 1. There are two basic local events that can occur during a topological quantum computation besides braiding, namely splitting and fusion. Fusion is when two anyons combine to produce another anyon, while splitting it the reverse. These events are conveniently represented using the diagrams in Fig. 2. A special case of fusion is annihilation when an anyon particle and its anti-particle fuse to the vacuum. Similarly creation is a special case of splitting in which a particle/anti-particle pair are drawn from the vacuum. We often depict the splitting/fusion involving the vacuum anyon type 1 by a dotted line, or sometimes we omit it entirely. This is justified as the vacuum anyon may be freely inserted/deleted in the system. Braiding is the isomorphism denoted cX,Y : X  Y 7! Y  X. c As there are finitely many anyon types one may encode the fusion rules into a matrix: for a fixed type a, define [Na]c,b : ] dim(Vab ). Running over all pairs (b, c) we obtain the fusion matrix Na. An important invariant of an anyon is the dimension da of the corresponding anyon type: this is simply the largest eigenvalue of Na, which is guaranteed to be real and positive by the Perron-Frobenius theorem. However, this dimension is not necessarily an integer–in general it is a real cyclotomic integer: a real number that can be expressed as a finite sum of roots of unity. We shall see later that the computational utility of anyons of type a in a given system is intimately intertwined with the value of da. Table 1

A dictionary of categorical terms and their interpretations in anyonic systems.

Anyon model (UMC)

Anyonic system

Simple object Label Tensor product c or Vcab Triangular space Vab Dual Birth/death Braid representation Morphism Braids

Anyon Anyon type or topological charge Fusion Fusion/splitting space Antiparticle Creation/annihilation Anyon statistics Physical process or operator Anyon trajectories

Y

Z

Y

X

X

Z

X

X

X

X

Fig. 2 Basic local events: fusion, splitting and braiding. 4

We shall not need a full treatment here, we will introduce the key ideas as they become necessary.

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We are now ready to describe the braid group representations associated with anyons of type a in a topological phase of matter. Suppose we have n type a anyons localized at positions 1, . . ., n in the system, whose total charge is b. Interchanging the anyons at positions i and i + 1 induces a unitary operator Ui(b) on the state space H(b; a, . . ., a) ¼ Hom(b, an), which represents the braid group generator si. Extending to all such interchanges and summing over all possible boundary conditions we obtain a representation
 ra : Bn ! U Hðb; a; . . . ; aÞ via si 7! Ui ðbÞ:





b

b

With this formulation in hand, we can mathematically analyze topological gates. Note that dimension of the representation bH(b; a, . . ., a) is asymptotic to dna , so that the ground state degeneracy of n type a anyons in the disk can be approximated using the quantum dimension. It is p useful to keep in mind three examples: (1) the Fibonacci anyon t with fusion rule t2 ¼ ffiffi pffiffiffi1 + t and quantum dimension dt ¼ 1 +2 5 (2) the Ising anyon s with fusion rules s2 ¼ 1 + c, c  s ¼ s, and f2 ¼ 1 with ds ¼ 2 and (3) the Z/3 anyon o with fusion rules o2 ¼ o∗, o  o∗ ¼ 1 and do ¼ 1. The ground state degeneracy of Fibonacci anyons in a disk is a Fibonacci number, while the Z/3 has no disk ground state degeneracy and the Ising theory has disk ground state degeneracy of the form 2n.

The role of the braid group in TQC In this section we will review a number of ways that the braid group plays an essential role in topological quantum computation. The examples presented here are representative of the author’s tastes and are by no means exhaustive.

Detecting non-abelian anyons via braiding If the braid group representation ra associated with an anyon of type a has abelian image, we say that a is an abelian anyon. Abelian anyons have been convincingly demonstrated in the laboratory, and can even be directly shown to admit braidings (Nakamura et al., 2020). Notice that anyons that have no ground state degeneracy are automatically abelian: if dim(Vcaa) ¼ 1 for a unique anyon type c then it can be shown that Hom(b, an) is 1-dimensional for a unique type b for each n, and 0-dimensional for all other types c 6¼ b. Thus the braid group representation ra described above is 1 dimensional, hence has abelian image. In particular if da ¼ 1, then a is an abelian anyon. On the other hand, non-abelian anyons are essential if one is to have any interesting (i.e., not phases) braiding gates. How can one detect if an anyon is non-abelian? In Rowell and Wang (2016) it is shown that if a is abelian then necessarily da ¼ 1. The idea is the following where we assume that a is its own anti-particle type for convenience5: Suppose that an anyon X of type a has da > 1, yet is abelian. This implies that there is some anyon Y of non-vacuum type b 6¼ 1 so that dim(Vbaa)  1 by a simple eigenvector argument. We construct a non-zero state in Hom(a, a3) by drawing a pair of type a anyons out of the vacuum (we have assumed that the splitting space Hom(1, a2) is non-trivial). Next we perform the braiding 2 −1 decomposes as a sum of simple objects that includes b, so that there is a non-zero vector in s1s2s−1 1 s2 2 B3 on this state. Now a Hom(a, b  a), which can be achieved as a consequence of a fusion of the left two type a anyons to a type b anyon Y. So the process −1 yields the state in Fig. 3. Now if the X anyon were abelian, then the image of the braid s1s2s−1 1 s2 must be the identity, as illustrated in Fig. 4. Thus, since we cannot create a single anyon out of the vacuum, we obtain the zero state as illustrated in Fig. 5. This contradicts the assumption that da > 1. pffiffi pffiffiffi Computing the quantum dimensions of the Fibonacci, Ising and Z/3 anyons we get 1 +2 5 , 2 and 1, respectively. Thus the Fibonacci and Ising anyons are non-abelian, while the Z/3 anyons are abelian.

Y

X

1

X

Fig. 3 A state achieved by creating a pair of X anyons from the vacuum, braiding, and then fusing two X anyons to an anyon Y. This is a non-zero state.

5 This is not a crucial restriction: a slightly more complicated argument works for all types and in most examples one hopes to construct in the laboratory this is satisfied anyway.

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=

Fig. 4 If X is abelian, the sequence of braids has the same effect as the identity braid.

Y

X

= 0 X

X 1

X

Fig. 5 Since Y 6¼ 1, the state on the left must be zero, and therefore the whole state must be 0.

Braiding universality The foundational paper (Freedman et al., 2002b) exhibited the theoretical viability of topological quantum computers by showing that the so-called Fibonacci anyons are universal for quantum computation. This was achieved by demonstrating that one could efficiently approximately simulate a universal qubit computer on the Fibonacci topological model. Let us briefly recall the qubit model (Nielsen and Chuang, 2000). A qubit is a 2-dimensional vector space V ¼ ℂ2, typically modeled by some 2-level quantum system, e.g., a spin-system. One fixes a gate set G consisting of unitary operators Ui 2 U(V ni). N-qubit quantum circuits are built as b compositions of promotions of the Ui to V N, i.e., Ia V  Ui  IV where N ¼ a + ni + b. Each application of a gate is considered a single step, hence the length of the quantum circuit in an algorithm represents consumed time, and is a complexity measure. A gate set should be physically realizable and complicated enough to perform any computation given enough time. Definition 3.1: The gate set G is said to be universal if any unitary operator X 2 U(V N) can be efficiently approximated (up to a phase) by a quantum circuit built from G: A typical choice of a universal gate set is n o S ¼ H, s1=4 z , CNOT consisting of the Hadamard, p/8 and CNOT gates: 0

1 H ¼ pffiffiffi 2



1 1



1 , −1

sz1=4 ¼



1 0



0 , epi=4

1 B0 B CNOT ¼ B @0 0

0 1 0 0

0 0 0 1

1 0 0C C C 1A 0

Here the efficiency should be that the length of the quantum circuit is a polynomial in the desired accuracy. The Kitaev-Solovay theorem (see Freedman et al., 2002b for the most appropriate formulation) states that it is enough to show that the closure of the N-qubit circuits in the operator norm topology contains all unitaries in U(V N) up to phases. For the topological model with gates obtained from braiding as above, the following is the equivalent formulation of universality: Definition 3.2: An anyon X is called braiding universal if, for some n0, the images of B n on the irreducible sub-representations W  End(Xn) are dense in SU(W) for all n  n0. This is a mathematically precise definition, but in practice pffiffiit can be quite difficult to verify universality (Freedman et al., 2002b). 1+ 5 is universal, while the Ising anyon s which has quantum dimension The Fibonacci anyon t, which has quantum dimension 2 pffiffiffi 2 is not. In fact the image of the braid group representations associated with the Ising anyon s have finite groups as their images–very far from universal! In practice this appears to be the dichotomy: either an anyon X is braiding universal, or the representations rX have image a finite group, in which case we say X is a property F anyon. In principle, to determine if the image is finite or infinite requires fairly explicit descriptions of the representations, which are not always available. It would be useful to have an indirect way of detecting when an anyon X has associated braid group representations rX with infinite image. The following gives a very concise conjectural condition: Conjecture 3.1: (Naidu and Rowell, 2011) An anyon X is braiding universal if and only d2x 2 = Z.

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For simplicity we are conflating “infinite image” with “universality.” But we do not lose too much by doing so: if an irreducible unitary braid group representation rX : Bn ! U(Vn) has infinite image in U(Vn) then its closure rX ðBn Þ is some infinite compact Lie group G. By analyzing the eigenvalues of the braid group generators’ images one finds that G contains, with few exceptions, SU(Vn), SO(Vn) or Sp(Vn) (see Larsen et al., 2005 for the general method).6 To obtain universality on the nose one hopes for the first case, but one can typically lift the second two cases to special unitaries by taking n slightly larger. This conjecture has been verified in numerous cases, see, e.g., (Etingof et al., 2008; Rowell, 2010; Rowell and Wenzl, 2017; Green and Nikshych, 2021; Larsen et al., 2005). In particular it is known to be true for anyon models obtained from quantum groups at roots of unity (equivalently, Kac-Moody algebras) as well as all so-called weakly group-theoretical braided fusion categories. This latter class includes the metaplectic anyons of Hastings et al. (2014).

Measurement assisted universality Convincing laboratory confirmation of the existence of non-abelian anyons is still a major hurdle. The simplest model for non-abelian anyons correspond to the Ising categories, known as the Majorana zero modes (or more colloquially, Majorana fermions) (das Sarma et al., 2015). It would be a breakthrough to have such a confirmation for the Majorana zero modes. On the other hand, this could not be used for braiding universal topological quantum computation. The next simplest non-abelian anyons correspond to the SU(2)4 model (certain metaplectic anyons (Hastings et al., 2014)), with some numerical evidence that this is realized by the fractional quantum Hall liquids at filling fraction n ¼ 8/3[PWC+15]. But again, these are not braiding universal. If nature conspires against us (which she so often does) it might be the case that braiding universal anyons are out of reach. While disappointing, this does not mean that one cannot have topological quantum computers. In Cui et al. (2015), Cui and Wang (2015) protocols for recovering universality from metaplectic anyons are presented. The idea is to produce a universal gates set from braiding gates and one non-braiding gate achieved through a projective measurement of topological charge. Specifically, the system modeled by the SU(2)4 includes anyons 1, Z, X, and Y with the fusion rules X2 ¼ 1 + Y and Y2 ¼ 1 + Z +Y, where Z is an abelian anyon satisfying Z2 ¼ 1 and Z  Y ¼ Y. A single qutrit is encoded as in Fig. 6. The space is 3 dimensional, corresponding to the pairs (a, b) which can be (1, Y), (Y, 1), or (Y, Y). While braiding the four X-type anyons provides many gates, braiding alone is not universal. The additional measurement that must be performed is a projection of the left two X-type anyons onto the vacuum 1-type, and the orthogonal complement. If the left pair total charge is not 1 then the right pair total charge is in a coherent superposition of (Y, 1) and (Y, Y). This operation, together with the braiding gates is universal.

Distinguishing anyons and anyon systems via braiding One of the key axioms of a modular category is the non-degeneracy of the braiding: this says that the only anyon type X so that the full exchange (double braid, see Fig. 7) with all anyon types Y is trivial is the vacuum type 1. This turns out to be equivalent to the condition that the matrix with entries Sab :] tr(ca,b ∘ cb,a) for all pairs of anyon types (a, b) has det(S) 6¼ 0 (Bruguiéres, 2000). This implies that, in principle, one may distinguish anyons X and Y by performing all exchanges of X and Y with an anyon of each other type. One would of course need to perform measurements–in this case the right computation is essentially the one illustrated

Fig. 6 One qutrit from SU(2)4.

Y

X

Y

X

Y

X

= Y

X

Fig. 7 If the full exchange is trivial for all Y, anyon X is transparent.

6

These are the special unitary, special orthogonal and symplectic Lie groups.

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in Fig. 1, creating two particle/anti-particle pairs and then projecting onto the vacuum-vacuum state. These are precisely the entries of the matrix S, in diagrams:

a

b

The S-matrix entries are the invariants of the Hopf link, with components “colored” by pairs of anyon types. Thus the non-degeneracy condition implies we may distinguish anyons by the set of double braidings. The S-matrix records the traces of all double braidings, and hence gives an invariant of the anyon model. Suppose we wish to pin down precisely which modular category is the model for our anyon system. What computations/measurements must one perform? The quantum dimension da of each anyon type appears in the S-matrix: it corresponds to setting b ¼ 1. Another invariant is obtained by taking the trace of a single braiding of X with itself, see Fig. 8: this yields the quantity dXyX for a root of unity yX, known as the topological spin of X. The S-matrix together with the twists ya for all anyon types a form the modular data of the theory. Is the modular data enough to determine the anyon model? The answer turns out to be “no” (Mignard and Schauenburg, 2017). It is possible for two inequivalent anyon models to have exactly the same modular data: known as modular isotopes (Delaney et al., 2021). By the philosophy that the anyon model should be determined by the topological measurements (mathematically, invariants) one might expect that there is a finite set of measurements that would determine the theory. That is, one seeks a set of knots and links, including the modular data, so that the invariants one obtains by “coloring” the components by anyon types completely determines the anyon model. The two component Whitehead link is one candidate (see Fig. 9, left) which together with the modular data can be used to distinguish some modular isotopes Bonderson et al., 2019. The Whitehead link is named after the topologist J. C. Whitehead but predates him by about one millennium–it is found on Viking artifacts from around 1000C.E (Fig. 9, right). Interestingly, the Whitehead link has Gauss linking number 0.

Beyond braids In the Gargı -Yajnavalkya debate one finds the suggestion that the cosmos are braids upon braids upon braids. Lord Kelvin, in his the vortex-atom theory, proposed that all matter is built from knots. From our modern perspective these ideas are easily dismissed, but we see glimmers of them in topological phases of matter. Is there more to this? So far we have focused on braids as trajectories of point-like excitations with 2-dimensional ambient space, and have seen that the braid group plays a key role. Many interesting computational and physical questions can be answered by studying braids. Can we push this theory to higher dimensions? Point-like excitations in 3 dimensions are not interesting: they must be bosons or fermions, and are thus abelian in the sense that braiding them has abelian image. But what about other kinds of excitations, such as loop-like vortices? Or, for that matter, knot-like vortices? At least mathematically, we can describe the dynamics of system in which knots are braided! The simplest example is known as the loop braid group: the motions of n oriented circles in S3. It has two kinds of generators: the first by “leapfrogging” the ith circle through the (i + 1)st si and the second by loop interchanges si for 1  i  n − 1. This group is denoted LBn and has only been studied relatively recently (see Damiani, 2017 for a survey). Abstractly, LBn is the

X

X

cX,X = Fig. 8 The braiding of X with itself: the trace of this operator has value dXyX.

Fig. 9 Left: A braid with closure the Whitehead link colored to emphasize components. Right: Thor’s hammer (Mjölnir) artifact featuring the Whitehead link, dating from around 1000 C.E.

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3

n-2 2

n-1 n

1

Fig. 10 The Necklace: motions include leapfrogging and moving each small circle by a rotation of 2p/n around the center point of the large circle.

Fig. 11 Three trefoils: distinct motions could include passing one such knot through any of the four holes in a neighboring knot.

group generated by one copy of the n-strand braid group (generated by the si) and one copy of the symmetric group (generated by the si) with the additional (mixed) relations: si si+1 si ¼ si+1 si si+1 ,

si si+1 si ¼ si+1 si si+1 ,

1  i  n − 2, si sj ¼ sj si

if

|i −j| > 1:

Other configurations are possible, such as “the necklace” as in Fig. 10. Representations of the motion group of the necklace, the necklace braid group have been studied, for example, in Bullivant et al. (2020). A more complicated, but nonetheless intriguing example is the motions of three or more trefoil knots as in Fig. 11. One can imagine many different “leapfrog” motions between a pair of such knots. Equipped with the mathematical description of these motions, one should be able to approach the theoretical analogues of those described above for 2-dimensional systems in topological phase, such as: How do we model these systems? and What is the computational power of such a system?

Conclusion The landscape of topological quantum computation is still in flux as the goals of computer science and the realities of physics are reconciled within the bounds of engineering. It seems clear that on the mathematics side the braid group will continue to play a key role, with adjustments as dictated by the relevant models, as in the example of measurement-assisted universality. As new sources of quantum systems with topological features are found and our understanding of the extant systems is deepened, new ideas will be needed to find mathematical models and explore their utility in quantum technologies.

Appendix on categorical notions We have used a number of standard notions from category theory that the reader may not be familiar with. Here we lay down some of the key ideas for the reader’s convenience, always deferring to Etingof et al. (2015) for a comprehensive treatment. A category consists of a collection of objects X, Y, . . . (not necessarily a set, and the objects have no assumed structure), together with a collection of morphisms Hom(X, Y) for each pair of objects, with a distinguished element idX 2 Hom(X, X) ¼ End(X) for each object X. These morphisms can be composed when it makes sense to, and the composition is assumed to be associative. A basic idea in category theory is to axiomatize the notion of “sameness.” Two objects X, Y are said to be isomorphic if there are morphisms f 2 Hom(X, Y) and g 2 Hom(Y, X) so that f ∘ g ¼ idY and g ∘ f ¼ idX. It is worth pointing out that while the objects have no structure (nor do the collection of objects), the morphisms do have structure. In modern category theory one adds additional structures and requires additional properties. It can often be confusing to distinguish between structures and properties, but typically there are choices for a structure, while a property is either satisfied or isn’t. For an instructive example from algebra, consider a set X with a binary operation (x, y) 7! x ∘ y 2 X. This is called a magma and it depends a structure–there are many choices for ∘. On the other hand, if the property that (x ∘ y) ∘ z ¼ x ∘ (y ∘ z) holds for all x, y, z then the magma (X, ∘) is a called a semigroup. If there is an element e 2 X so that e ∘ x ¼ x ∘ e ¼ x for all x, this is an additional property and the semigroup is called a monoid. If the property that x ∘ y ¼ y ∘ x for all x, y the monoid is called commutative, or abelian. To be a group the monoid must satisfy another property: every element should have an inverse. We could continue to build structures and insist on properties: a commutative group with an additional structure of scalar multiplication by the real numbers ℝ satisfying several more properties defines a vector space, and so on. We are often interested in classifying categories with certain

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structures and properties. Moreover, many of the structures and properties are categorical generalizations of algebraic constructions we are already familiar with. The categories we are concerned with are modular categories: concisely (albeit obfuscating), these are non-degenerate spherical braided fusion categories. Each of these adjectives carries with it some structure and/or assumes some properties hold. We will first unravel this definition in somewhat broad strokes before providing some examples. (1) Fusion categories are modeled after the category of finite-dimensional complex representations Rep(G) for a finite group G. This axiomatizes notions like direct sum (additive structure), tensor product (monoidal structure ) with proscribed associativity morphisms aX,Y,Z, trivial representation (monoidal unit 1), linearity (Hom(X, Y) is a finite dimensional vector space), duality (objects V ∗ for each V), irreducibility (simple objects), finiteness (finitely many simple objects up to isomorphism, among which is 1), and semisimplicity (all objects are sums of simples). Sphericity is a somewhat technical property having to do with choices (“pivotal” structure) of isomorphisms V ffi V ∗∗ that lead to a notion of left and right traces for morphisms. Essentially, this structure is spherical if the ensuing left and right traces coincide. This trace is used to define the S-matrix in Section 3.4 for example. (2) A braiding structure on a fusion category is a collection of morphisms cX,Y 2 Hom(X  Y, Y  X) that satisfy a version of the braid equation. This is of course crucial for our exposition above: the key fact is that −1 si 7! idi  cX,X  idXn −i −1 X

can be modified to a braid group representation after employing appropriate re-coupling of the tensor factors. In practice one often assumes the category is strict so that the associativity constraints are identities. In the presence of a braiding the spherical structure is encoded in the topological twists YX 2 Hom(X, X) for each object X. For simple objects we have YX ¼ yX  idX where yX is a root of unity. They satisfy the balancing equation: YXY ¼ cY,XcX,YYX  YY which allows one to represent morphisms as labeled ribbons and manipulate them in a topologically consistent way (graphical calculus). (3) The braiding has the property of being non-degenerate if the only simple object X such that cY,X ∘ cX,Y ¼ idXY for all Y is the monoidal unit 1. For a fixed spherical structure non-degeneracy can be encoded in terms of the invertibility of the S-matrix. Three examples to keep in mind are the categories Fib, Ising and CðZ=3Þ alluded to above. You may find further details on these categories in Rowell et al. (2009), Section 5.3, in which modular categories with at most four simple objects are classified. Example A.1: The category Fib has 2 simple objects 1 and t with fusion rules t  t ¼ t  1 as mentioned in the main text. The   pffiffi 1 ’ S-matrix is where ’ ¼ 1 +2 5 : The twist of t is yt ¼ e4pi/5, while y1 ¼ 1 (for any category, in fact). The anyon t is universal. ’ −1 We have t∗ ffi t–the Fibonacci anyon is self-dual. Example A.2: Ising has 3 simple objects 1, c and s with the fusion rules described in the main text. The S-matrix is 1 0 pffiffiffi 1 2 1 p ffiffiffi p ffiffiffi C B pi/8 @ 2 0 − 2 A while the non-trivial twists are yc ¼ − 1 and ys ¼ e . The object c is a mathematical fermion: it is an pffiffiffi 1 − 2 1 abelian anyon with twist −1. The anyon s is non-abelian, but non-universal. All of the objects in the Ising category are self-dual.



Example A.3: The category CðZ=3Þ consists of abelian anyons 0 1 1 B q object o is not self-dual as o ffi o∗: The S-matrix is @ 1 1 1=q

1, o, o∗ with the fusion rules like the group Z/3. In this case the 1 1 C 1=q A where q ¼ e2pi/3. and the twists are yo ¼ yo∗ ¼ q. q

We have introduced modular categories as a kind of axiomatization of various algebraic structures that some familiar categories have. Another perspective on modular categories is that it is the categorical structure underlying (2 + 1)-dimensional topological quantum field theory. As a fundamental level then modular categories may be thought of as the categorification of ribbon diagrams: essentially the various diagrams one may draw using the basic events in Fig. 2, interpreted as morphisms in some category, so that composition is stacking and the labels represent a finite set of colors for the ribbons. While we have drawn the ribbons as curves, it should be emphasized that these really have some width and a “framing” which is essentially the number of times the ribbon is twisted along its axis. The modularity condition is natural in the sense that it says we may distinguish the colors by means of braiding and traces.

Acknowledgments Rowell is partially supported by NSF grants DMS-1664359 and DMS-2000331, and thanks Zhenghan Wang, Paul Martin and Xingshan Cui for helpful conversations.

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References Albrecht SM, Higginbotham AP, Madsen M, Kuemmeth F, Jespersen TS, Nygård J, Krogstrup P, and Marcus CM (2016) Exponential protection of zero modes in Majorana islands. Nature 531(7593): 206–209. Alexander JW (1923) A lemma on systems of knotted curves. Proceedings of the National Academy of Sciences of the United States of America 9(3): 93–95. Artin E (1925) Theorie der Zöpfe. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 4(1): 47–72. MR 3069440. Birman JS (1976) On the stable equivalence of plat representations of knots and links. Canadian Journal of Mathematics 28(2): 264–290. MR 402715. Bonderson P, Delaney C, Galindo C, Rowell EC, Tran A, and Wang Z (2019) On invariants of modular categories beyond modular data. Journal of Pure and Applied Algebra 223(9): 4065–4088. MR 3944465. Bruguiéres A (2000) Catégories prémodulaires, modularisations et invariants des variétés de dimension 3, (French). Mathematische Annalen 316(2): 215–236. Bullivant A, Kimball A, Martin P, and Rowell EC (2020) Representations of the necklace braid group: topological and combinatorial approaches. Communications in Mathematical Physics 375(2): 1223–1247. MR 4083878. Cui SX and Wang Z (2015) Universal quantum computation with metaplectic anyons. Journal of Mathematical Physics 56(3), 032202. 18. MR 3390917. Cui SX, Hong S-M, and Wang Z (2015) Universal quantum computation with weakly integral anyons. Quantum Information Processing 14(8): 2687–2727. MR 3370675. Damiani C (2017) A journey through loop braid groups. Expositiones Mathematicae 35(3): 252–285. MR 3689901. Delaney C, Rowell EC, and Wang Z (2016) Local unitary representations of the braid group and their applications to quantum computing. Revista Colombiana de Matemáticas 50(2): 207–272. MR 3605648. das Sarma S, Freedman M, and Nayak C (2015) Majorana zero modes and topological quantum computation. Physical Review Letters 94: 166802. Delaney C, Kim S, Plavnik J (2021) Zesting produces modular isotopes and explains their topological invariants, Preprint. Epple M (1998) Orbits of asteroids, a braid, and the first link invariant. The Mathematical Intelligencer 20: 45–52. Etingof P, Rowell E, and Witherspoon S (2008) Braid group representations from twisted quantum doubles of finite groups. Pacific Journal of Mathematics 234(1): 33–41. MR 2375313. Etingof P, Gelaki S, Nikshych D, and Ostrik V (2015) Tensor categories, Mathematical Surveys and Monographs. vol. 205. Providence, RI: American Mathematical Society. Freedman MH (1998) P/NP, and the quantum field computer. Proceedings. National Academy of Sciences. United States of America 95(1): 98–101. MR 1612425. Freedman MH, Kitaev A, and Wang Z (2002a) Simulation of topological field theories by quantum computers. Communications in Mathematical Physics 227(3): 587–603. Freedman MH, Larsen M, and Wang Z (2002b) A modular functor which is universal for quantum computation. Communications in Mathematical Physics 227(3): 605–622. Freedman MH, Kitaev A, Larsen MJ, and Wang Z (2003) Topological quantum computation. American Mathematical Society 40(1): 31–38. Mathematical challenges of the 21st century (Los Angeles, CA, 2000). MR 1943131. Green J and Nikshych D (2021) On the braid group representations coming from weakly group-theoretical fusion categories. Journal of Algebra and Its Applications 20(1). Paper No. 2150210, 20. MR 4209972. Hastings MB, Nayak C, and Wang Z (2014) On metaplectic modular categories and their applications. Communications in Mathematical Physics 330(1): 45–68. MR 3215577. Hurwitz A (1891) Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Mathematische Annalen 39(1): 1–60. MR 1510692. Jaeger F, Vertigan DL, and Welsh DJA (1990) On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108: 35–53. MR 1049758. Jones VFR (1985) A polynomial invariant for knots via von Neumann algebras. American Mathematical Society 12(1): 103–111. Kapranov MM and Voevodsky VA (1994) 2-categories and Zamolodchikov tetrahedra equations. Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math. vol. 56, pp. 177–259. Providence, RI: Amer. Math. Soc. MR 1278735. Kassel C and Turaev V (2008) Braid groups, Graduate Texts in Mathematics. vol. 247. New York: Springer. With the graphical assistance of Olivier Dodane. MR 2435235. Larsen MJ, Rowell EC, and Wang Z (2005) The N-eigenvalue problem and two applications. International Mathematics Research Notices (64): 3987–4018. MR 2206918. Markov AA (1936) Uber die freie Aquivalenz der geschlossenen Zopfe. Recueil Mathematique Moscou 43(1): 73–78. Maxwell JC (1873) A Treatise on Electricity and Magnetism. vol. 1. Oxford. Mignard M and Schauenburg P (2017) Modular categories are not determined by their modular data. Letters in Mathematical Physics 111: preprint arXiv:1708.02796. Moore G and Read N (1991) Nonabelions in the fractional quantum Hall effect. Nuclear Physics B 360(2): 362–396. Moore G and Seiberg N (1989) Classical and quantum conformal field theory. Communications in Mathematical Physics 123(2): 177–254. MR 1002038. Mourik V, Zuo K, Frolov SM, Plissard SR, Bakkers EPAM, and Kouwenhoven LP (2012) Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336(6084): 1003–1007. Naidu D and Rowell EC (2011) A finiteness property for braided fusion categories. Algebr. Represent. Theory 14(5): 837–855. Nakamura J, Liang S, Gardner G, and Manfra MJ (2020) Direct observation of anyonic braiding statistics. Nature Physics 16(9): 931–936. Nayak C, Simon SH, Stern A, Freedman M, and Sarma SD (2008) Non-abelian anyons and topological quantum computation. Reviews of Modern Physics 80(3): 1083. Nielsen MA and Chuang IL (2000) Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. MR 1796805. Nobel Prize Committee (2016) Topological phase transitions and topological phases of matter. Tech. report . Ogburn RW and Preskill J (1999) Topological quantum computation, Quantum computing and quantum communications (Palm Springs, CA, 1998). Lecture Notes in Computer Science 1509: 341–356. Springer, Berlin. MR 1750535. Olivelle P (1998) Upanisads. Oxford University Press. Pachos JK (2012) Introduction to Topological Quantum Computation. Cambridge University Press. Przytycki JH (1998) Classical roots of knot theory. Chaos, Solitons & Fractals 9(4): 531–545. Knot Theory and Its Applications: Expository Articles on Current Research. Ricca RL and Nipoti B (2011) Gauss’ linking number revisited. Journal of Knot Theory and its Ramifications 20(10): 1325–1343. MR 2851711. Rowell EC (2010) Braid representations from quantum groups of exceptional Lie type. Revista de la Unión Matemática Argentina 51(1): 165–175. MR 2681262. Rowell EC (2016) An invitation to the mathematics of topological quantum computation. Journal of Physics: Conference Series 698: 012012. IOP Publishing. Rowell EC and Wang Z (2016) Degeneracy and non-Abelian statistics. Physical Review A 93(3), 030102. Rowell EC and Wang Z (2018) Mathematics of topological quantum computing. Bulletin of the American Mathematical Society 55(2): 183–238. MR 3777017. Rowell E and Wenzl H (2017) SO(N )2 braid group representations are Gaussian. Quantum Topol. 8(1): 1–33. MR 3630280. Rowell E, Stong R, and Wang Z (2009) On classification of modular tensor categories. Communications in Mathematical Physics 292(2): 343–389. Silver DS (2014) Knots in the nursery: (Cats) Cradle Song of James Clerk Maxwell. Notices of the American Mathematical Society 61(10): 1186–1194. MR 3241299. Turaev VG (1992) Modular categories and 3-manifold invariants. International Journal of Modern Physics B 6(11–12): 1807–1824. Wang Z (2006) Topologization of electron liquids with Chern-Simons theory and quantum computation. Differential geometry and physics, Nankai Tracts Math. vol. 10, pp. 106–120. Hackensack, NJ: World Scientific Publishing. MR 2322390. Wang Z (2010) Topological quantum computation. no. 112. American Mathematical Soc. Wang Z (2013) Quantum computing: A quantum group approach. Symmetries and groups in contemporary physics, Nankai Ser. Pure Appl. Math. Theoret. Phys. vol. 11, pp. 41–50. Hackensack, NJ: World Sci. Publ. MR 3221449. Wen X-G (1991) Non-abelian statistics in the fractional quantum Hall states. Physical Review Letters 66(6): 802. Wilczek F (1990) Fractional statistics and anyon superconductivity. vol. 5. World Scientific. Witten E (1989) Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121(3): 351–399. Yu Kitaev A (2003) Fault-tolerant quantum computation by anyons. Annals of Physics 303(1): 2–30.

Fibonacci anyon based topological quantum computer Tapio Simula, Optical Sciences Centre, Swinburne University of Technology, Melbourne, VIC, Australia © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Conventional two-level quantum computer Fibonacci topological quantum computer Fusion rules Topological qubit Braiding rules Universality Key issues Leakage error Quantum compilation error Hardware error sources Conclusion Acknowledgments References

279 279 280 281 281 281 282 283 283 283 283 283 284 284 284

Abstract The idea of topological quantum computation with non-abelian anyons is presented using Fibonacci anyons as the building blocks of topological qubits. Using a single qubit conventional quantum computer as a reference, we outline how braiding Fibonacci anyons may be used for realizing fault tolerant topological qubit operations. We mention some key issues affecting topological quantum computers, and conclude with a brief discussion on the future prospects of topological quantum computing.

Objectives

• • •

To introduce the idea of topological quantum computation by using the Fibonacci anyon model as an exemplar. To mention certain technicalities that affect the efficacy of braid-based topological quantum computers. To embed the Fibonacci anyon model within the broader context of nonabelian anyons and topological quantum computers.

Introduction The material world around us is composed of two kinds of elementary particles, the bosons and the fermions (Leggett, 2005). This observation is formalized by the spin-statistics theorem, which essentially states that upon transporting a particle around another indistinguishable particle, the quantum state describing these particles must either remain unchanged (bosons) or change its sign (fermions). This is related to the fact that such a world line path can be contracted to a point. However, in systems where the (quasi) particles are constrained to move in two-dimensional plane, the mutual quasiparticle exchange statistics does not need to be bosonic or fermionic but can in principle be anything in between (Leinaas and Myrheim, 1977). Such quasiparticles are called anyons (Wilczek, 1982). The essence of the topological difference between the three and two dimensional systems is illustrated in Fig. 1. The topological richness afforded by two dimensional systems allows for an even more exotic possibility where an encirclement of one quasiparticle by another one may cause such a quasiparticle to become transmutated into an entirely different kind of quasiparticle! Such quasiparticles are called non-abelian anyons and they could be used as building blocks for constructing a topological quantum computer. For a comprehensive technical introduction to the topic of this encyclopedia entry on Fibonacci anyon based topological quantum computation, we refer the reader to Nayak et al. (2008), Trebst et al. (2008), Lahtinen and Pachos (2017), Field and Simula (2018).

Overview It is useful to briefly outline some key concepts of conventional quantum computation before discussing the basics of a topological quantum computer. Note that, in addition to conventional and braid based topological quantum computers, several other kinds of quantum computer models also exist, such as a measurement-based one-way quantum computer, that will not be discussed here. Encyclopedia of Condensed Matter Physics, Second Edition

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particle particle

3D world contractible path

quasiparticle quasiparticle

2D world non-contractible path

Fig. 1 Illustration of an exchange of two bosonic or fermionic particles along a contractible path (left) and two anyon quasiparticles along a non-contractible path (right). In the former case the particles can be “lifted off the world line,” where as in the latter case this is not topologically allowed.

Fig. 2 Bloch sphere representation of single-qubit quantum computation. (a) the qubit is initialized in state | 0⟩. It is then rotated (dotted trajectory) using unitary quantum gates to the final superposition state |c⟩. A projective measurement of the final state will yield the results | 0⟩ and | 1⟩ with probabilities | a|2 and |b |2, respectively. (b) the initial state | 0⟩ is rotated using the Hadamard gate to a final state H |0⟩. A noisy Hadamard gate H0 leads to a deviation in the state vector and such errors rapidly accumulate during quantum computation.

Conventional two-level quantum computer The “transistor” of a conventional quantum computer is called a qubit or an artificial two-level atom. The qubit has two possible states

 1 0 and j1i ¼ j#i ¼ (1) j0i ¼ j"i ¼ 0 1 referred to as the spin-up and spin-down, respectively.1 An arbitrary single qubit state may be expressed as a superposition   pffiffiffi jci ¼ j0i + eiy j1i = 2

(2)

of the two possible qubit eigenstates, where y determines the azimuthal phase angle of the state vector, and every one of such states can be visualized by a state vector on the Bloch sphere, shown in Fig. 2.2 The state vectors can be rotated using unitary quantum gate operators U such as the Hadamard gate
 1 1 1 H ¼ pffiffiffi (3) 2 1 −1 to yield new state vectors | c⟩ ¼ U|c0⟩ from the initial states | c0⟩. Quantum computation corresponds to sequential application of a predetermined set of unitary logic gates acting on a specified initial state | c0⟩, after which a probabilistic measurement of the final quantum state is made. In a single qubit quantum computer every logic gate operation rotates the quantum state vector on the Bloch sphere to a new location such that during the computation the state vector traces out a specific path on the Bloch sphere. After that a measurement yields either spin-up or spin-down outcome and to find out the respective probabilities the computation must be repeated very many times. Gate fidelity is a major issue of conventional quantum computation. That is, how accurately the action of the quantum gates can be realized by the quantum hardware as any such errors rapidly accumulate in the course of the computation. In a single qubit quantum computer every imperfect unitary gate operation will rotate the state vector slightly off the target, see Fig. 2(b), and these errors add up such that after all gates have been applied the final state vector may be far away from its correct location such that the probability of repeated measurements will ultimately converge to the completely random classical 50/50 measurement outcome. To certain extent the adverse effects of a noisy environment may be mitigated by implementing sophisticated error correcting protocols. To go beyond single qubit quantum computation, generic many qubit states can be constructed by summing up tensor products of single qubit states. The many-qubit state (Hilbert) space whose dimension is 2N ¼ eN ln(2) grows exponentially rapidly with the number N of added qubits.3 An important resource in many-qubit quantum systems is entangled states, which cannot be expressed 1

Eq. (1) uses the non-unique yet conventional representation of the two qubit basis states.

On each point on the Bloch sphere there is also an attached fiber that accounts for the quantum mechanically unmeasurable global phase degree of freedom f that amounts to transformation | c⟩ ¼ eif | f0⟩. 2

3 Note that also the number of possible configurations of classical two-state on-off switches is equal to 2N, however, where as after the computation (pre-determined flicking of the switches) there is one and only one observable outcome, the state of a quantum computer may be in a state where all of the 2N states can have a non-zero probability of being measured.

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as a tensor product of any two quantum states. In addition to these rudiments, a practically useful quantum computer will need to satisfy a range of additional criteria such as scalability to very large numbers of qubits (Divincenzo, 2000).

Fibonacci topological quantum computer At the software level the operation of a topological quantum computer can be made to appear identical to that of a conventional quantum computer. The crucial difference occurs at the hardware level that enables the realization of fault tolerant qubits based on topology. Simplifying, the state vector rotations in a topological quantum computer can, in principle, be realized exactly even in the presence of noise sources that would cause the qubit rotations in a conventional quantum computer to be imprecise. In what follows, we will focus on the theoretical underpinnings of a braid group based topological quantum computer that uses Fibonacci anyons for its quantum logic operations.

Fusion rules An anyon model that facilitates topological quantum computation is defined by its particle content and their fusion rules. The Fibonacci anyon model contains only one type of anyon, the non-abelian Fibonacci anyon t in addition to the trivial vacuum state denoted by 1. The fusion rules of the Fibonacci anyon model are tt ¼ 1t;

1t ¼ t1 ¼ t;

and

11 ¼ 1:

(4)

The first one of these is the most important one and states that when two Fibonacci anyons are brought together (fused) they may either annihilate each other yielding the vacuum 1 or combine to form a single Fibonacci anyon t. This double valuedness of the fusion outcome can be used for defining a twolevel system—a topological qubit. It also means that the Fibonacci anyon may act as its own antiparticle. The latter two fusion rules correspond to “trivial” fusion outcomes. When multiple Fibonacci anyons are fused sequentially, the possible fusion paths can be expressed using a fusion tree as illustrated in Fig. 3. The number of distinct fusion paths PN as a function of the number N of anyons form a Fibonacci pffiffiffi  sequence, which gives the Fibonacci anyon its name. In the limit N ! 1 the ratio PN/PN−1 converges to the golden ratiof ¼ 1+ 5 =2, which is also the so-called quantum dimension of the Fibonacci anyon model.

Topological qubit

A well defined topological qubit may be conveniently constructed out of an array of four Fibonacci anyons (t1, t2, t3, t4), where the subscript denotes their spatial position in the array, by creating them pairwise from the vacuum, which guarantees that the initial state satisfies t1  t2 ¼ 1 and t3  t4 ¼ 1. The conservation of the topological charge t1  t2  t3  t4 ¼ 1  1 ¼ 1 thus means that there remains two possible fusion paths for these four anyons tttt ! 1tt ! tt ! 1 ffi j0i and tttt ! ttt ! tt ! 1 ffi j1i that can be used for the definition of a topological qubit. Measuring the state of the topological qubit may thus be determined solely by the fusion outcome of the first two anyons t1 and t2. In words, if t1  t2 ¼ 1 the qubit state is | 0⟩ and if t1  t2 ¼ t the qubit state is |1⟩. The state of the topological qubit can be changed by braiding the world lines of the anyons, which physically corresponds to moving the anyons around in the two-dimensional plane. Shuffling around the anyons in the two-dimensional physical space thus realizes unitary qubit rotations of the quantum mechanical state vector in the abstract state space, which is equivalent to the Bloch sphere in the case of a single qubit.

fusion tree

τ ⊗τ ⊗τ ⊗τ a b

c

fusion paths a b c

1→τ → 1 1→τ →τ τ→ 1 →τ τ→τ→ 1

τ→τ→τ

state |0 not used not used |1 not used

Fig. 3 Fusion of four Fibonacci anyons. Fusion of the four anyons sequentially from left to right leads to a fusion tree where the fusion outcome of the first two anyons is a, fusion of a with the third anyon yields b and fusion of b with the fourth anyon yields c. For four Fibonacci anyons there are five possible fusion paths that span the Hilbert space of possible state vectors. Topological charge conservation reduces the accessible state space from five to two, and these two fusion paths define a two level system (a topological qubit).

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Fibonacci anyon based topological quantum computer

Braiding rules When the anyons 1 and 2 are braided by moving them around each other counter-clockwise in the physical space to swap their positions, their world lines (string-like trajectories through space and time) become “knotted.” This operation is denoted by s1. The inverse operation where the anyons are moved clockwise is denoted by s−1 1 . If these two operations are performed one after another, −1 the second operation undoes the first one such that s−1 1 s1 ¼ s1s1 ¼ 1. Generally, braiding the anyons n and n + 1 is denoted by braid operators sn and s−1 n . The braid operators acting on n strands (anyons) generate a braid group Bn. For a single Fibonacci anyon topological qubit it is sufficient to braid anyons 1, 2 and 3, while the fourth anyon does not need to participate in braiding so that its sole purpose is to ensure topological charge conservation of the four anyons. The qubit transformations may therefore be mapped onto the braid group B3. The two elementary braid operators of B3 have explicit (truncated to a separable two-dimensional qubit subspace) matrix representations ! e −i4p=5 0 s1 ¼ (5) 0 ei3p=5 and 0

ei4p=5 B f B s2 ¼ B −i3p=5 @e pffiffiffiffi f

1 e −i3p=5 pffiffiffiffi C f C C 1 A − f

(6)

−1 and the operators s−1 1 ¼ and s2 are the matrix inverses of s1 and s2, respectively. These representations may be derived by solving the so-called pentagon and heptagon equations. An elementary topological quantum computation operation with Fibonacci anyons may thus be compactly expressed using standard notation 1 0 i4p=5 e e −i3p=5 pffiffiffiffi C
 B f f C 1 B ¼ aj0i + bj1i (7) jci ¼ s2 j0i ¼ B −i3p=5 C @e 1 A 0 pffiffiffiffi − f f

and it physically corresponds to braiding the second and third anyon by exchanging their positions in real space, see Fig. 4, which in the abstract two-dimensional Hilbert space corresponds to creating a superposition state | c⟩ in Eq. (7). By fusing the anyons after this braiding, the topological qubit would be found to be in the state |0⟩ with a probability P0 ¼ |h0| ci|2 ¼ |a|2 ¼ f−2  0.4 and in the state |1⟩ with a probability P1 ¼ 1 − P0 ¼ |b|2 ¼ f−1  0.6.. Any single-qubit operation can be realized by concatenating L elementary braids to form a braid word WL ¼ siL. . .si2 si1 of length L and then acting with this braid word quantum gate on the initial state jci ¼ W L j0i:

(8)

state preparation = pair creation

1

1

1 arrow of time

computation

1 100% 1 state measurement = fusion

1 σ2

1 40% 1

τ

60%

τ

Fig. 4 Topological quantum computation is realized by braiding Fibonacci anyons. Four Fibonacci anyons are first spawned pairwise from the vacuum to create a topological qubit. Identity operator (left diagram) leaves the qubit intact and a fusion at the end of the computation will yield a definite outcome with a probability 1. If the anyons two and three are braided (acting on the qubit with a s2 operator) the qubit is rotated into a superposition state and final fusion will then have two possible outcomes with respective probabilities of occurrence that add up to one.

Fibonacci anyon based topological quantum computer

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Universality An important enabling feature of the Fibonacci anyon model is that it is universal for the purposes of quantum computation. In simple terms this means that every quantum state that may be expressed in terms of the available qubits may also be transformed to any other one by braiding the anyons and without requiring any other operations. In the single-qubit case this means that it is possible to construct all possible quantum gates as products of the elementary braid matrices, such that it is possible to connect any two state vectors (points on the Bloch sphere) by braiding the anyons. This property holds true also in the case of many qubits that may be constructed by sequencing the available anyons into groups of four. For instance, using eight anyons one may construct two logical topological qubits each of which is comprised of four Fibonacci anyons. Two-qubit gates may then be realized by braiding anyons belonging to the different logical qubits.

Key issues Presently, by far the most serious issue regarding Fibonacci anyons is their (non-)existence. Although it has been suggested that certain quasiparticles in exotic fractional quantum-Hall states could potentially serve for realizations of Fibonacci anyons, proving or refuting this is beyond current experimental capabilities. Even if this most severe limitation would be overcome in the future, several other technical obstacles remain that would need to be resolved. While topological quantum computers are often stated to be immune to the kinds of environmental noise that affects the conventional quantum computers, they nevertheless are susceptible to error sources and challenges of their own, both at the software and the hardware levels.

Leakage error Even in the single-qubit case where only the first three anyons are being braided, the full Hilbert space representable by the anyons is five dimensional. However, the braiding only acts on the two dimensional qubit subspace. For two or more qubits, this is no longer the case. For instance, the braid operators of a two-qubit system involving eight anyons act on a 13 dimensional Hilbert space, only 4 of which are used for defining the two-qubit states, such that operators s1(1,2) and s2(2,3) are intrinsic to the first qubit and the operators s4(5, 6) and s5(6, 7) are intrinsic to the second qubit, where the numbers in the parenthesis enumerate the anyons. However, the inter-qubit braid operator s3(3, 5) that braids the anyon t3 (in the first qubit) and the anyon t5 (in the second qubit) transfers population to some of the 9 so-called non-computational basis states. This is referred to as a leakage error as the probability leaks to quantum states not used for computational logic and this breaks the unitarity of the logic gates acting on the computational subspace. Although the amount of leakage can be mitigated by clever braiding protocols, this typically involves an additional computational overhead. This type of error could be completely avoided by making use of the full accessible Hilbert space and using multi-level qudits for computation instead of the more restricted two-level qubit basis and in this sense the leakage error is not a fundamental limitation of topological quantum computation.

Quantum compilation error Another kind of error that affects topological quantum computers is the quantum compilation error. For instance, if one wishes to implement the Hadamard gate using a Fibonacci anyon based topological quantum computer, one must first compile the Hadamard gate. This means finding a braid word representation of H such that WL  H. However, finding the optimal braid word of a fixed length L out of the exponentially, as a function of L, growing search space is a formidable task in its own right. Moreover, although universality of the Fibonacci anyon model guarantees that for every unitary U an arbitrarily good braidword approximation WL may be obtained by increasing L, in practice for finite L, even the optimal WL will only realize the target gate approximately. The compilation error thus has two parts, one due to the difficulty in finding the optimal braid word of given length L and another one due to the fact that even the optimal braid word will not in general be identical to the target gate. As for the leakage error, the compilation error too could be mitigated by designing quantum computation algorithms that make direct use of the natural braid gates of the Fibonacci anyon model, rather than having to first compile a standard gate set from the elementary braids. The trade-off here is between computational efficiency and end-user convenience.

Hardware error sources In addition to the error mitigation at the software level, topological quantum computers would also need to consider sources of errors originating from the hardware implementation of the anyons themselves such as the so-called quasiparticle poisoning. For instance, a pair of Fibonacci anyons could spontaneously form out of the vacuum and interfere with the braiding of the computational anyons. Material impurities or external causes could result in the appearance of patches of non-topological phases of matter, and in any physical realization there could exist so-far unidentified effects that could mimic braiding depending on how faithfully the physical system could be represented by the anyon model. These kinds of hardware level issues become important engineering considerations if non-abelian anyons will be proven to exist by experiments.

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Conclusion In summary, topological quantum computation is a form of quantum computation which is conducted using special kind of quantum hardware called non-abelian anyons. In a conventional quantum computer a rotation of a quantum state by a quantum gate is a continuous error prone operation in the sense that environmental noise may change the action of the quantum logic gate proportional to the strength of the noise leading to imprecise qubit rotations whose effects accumulate over the entire computation. By contrast, a topological quantum gate acts on the state vector either exactly or not at all, depending on the topology encoded in the world lines of the braided anyons. This special property could be harnessed to facilitate fault tolerant computations. Here we have considered only one special kind of topological quantum computation model based on non-abelian Fibonacci anyons to illustrate some generic principles of topological quantum computation. Nevertheless, there exists a large variety of other types of anyon models. Perhaps the most promising one from the present day experimental physics perspective, the Ising anyon model, is based on Majorana fermion zero modes. Such non-abelian anyons are thought to be realizable by quantized vortices (Simula, 2019) in topological superfluids and at the end points of superconducting nano-wire structures. Both the Fibonacci anyons and the Ising anyons belong to a broader (countably infinite) class of the so-called SU(2)k anyon models where the integer level k ¼ 3 for the Fibonacci anyons and k ¼ 2 for the Ising anyons. A major distinction between the k ¼ 2 and k ¼ 3 cases is that while the Fibonacci anyon model is universal for quantum computation, the Ising anyon model is not. This means that it is not possible to achieve full topological protection for the quantum computation in the k ¼ 2 case, and that at least one quantum gate in the gate set of such a general purpose quantum computer must be implemented via non-topological means. Another generic class of topological quantum computation models are the so called D(H) quantum double models (de Wild Propitius and Bais, 1999). It has been suggested that such quantum double anyon models could potentially be realized by quantized vortices in spinor Bose–Einstein condensates (Mawson et al., 2019; Génetay Johansen and Simula, 2021). Such anyon models are based on representations of finite quantum groups and therefore, similarly to the Ising anyon models, they are not capable of universal quantum computation by braiding alone and need to be supplemented with non-topological gate operations. At the time of writing, evolution of conventional quantum computers is still in its infancy and topological quantum computers do not yet exist. It will be exciting to see whether the future of scalable quantum computation will be based on topological or conventional qubits, or perhaps a combination of both?

Acknowledgments This work was funded by the Australian Government through the Australian Research Council (ARC) Future Fellowship FT180100020.

References de Wild Propitius M and Bais FA (1999) Discrete gauge theories. In: Semenoff G and Vinet L (eds.) Particles and Fields, p. 353. Springer-Verlag New York, Inc. Divincenzo DP (2000) The physical implementation of quantum computation. Fortschritte der Physik 48: 771–783. https://doi.org/10.1002/1521-3978(200009)48:9/113.0.CO;2-E. Field B and Simula T (2018) Introduction to topological quantum computation with non-Abelian anyons. Quantum Science and Technology 3: 045004. https://doi.org/ 10.1088/2058-9565/aacad2. Génetay Johansen E and Simula T (2021) Fibonacci anyons versus Majorana fermions: A Monte Carlo approach to the compilation of braid circuits in SU(2)k anyon models. PRX Quantum 2: 010334. https://doi.org/10.1103/PRXQuantum.2.010334. Lahtinen V and Pachos JK (2017) A Short introduction to topological quantum computation. SciPost Physics 3: 021. https://doi.org/10.21468/SciPostPhys.3.3.021. Leggett A (2005) Quantum mechanics: Foundations. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 51–56. Oxford: Elsevier. https://doi. org/10.1016/B0-12-369401-9/00616-1. Leinaas JM and Myrheim J (1977) On the theory of identical particles. Nuovo Cimento B Series 37: 1–23. https://doi.org/10.1007/BF02727953. Mawson T, Petersen TC, Slingerland JK, and Simula TP (2019) Braiding and fusion of non-abelian vortex anyons. Physical Review Letters 123: 140404. https://doi.org/10.1103/ PhysRevLett.123.140404. Nayak C, Simon SH, Stern A, Freedman M, and Das Sarma S (2008) Non-abelian anyons and topological quantum computation. Reviews of Modern Physics 80: 1083–1159. https:// doi.org/10.1103/RevModPhys.80.1083. Simula T (2019) Quantised Vortices. Morgan Claypool Publishers2053–2571. https://doi.org/10.1088/2053-2571/aafb9d. Trebst S, Troyer M, Wang Z, and Ludwig AWW (2008) A short introduction to fibonacci anyon models. Progress of Theoretical Physics Supplement 176: 384–407. https://doi.org/ 10.1143/PTPS.176.384. Wilczek F (1982) Quantum mechanics of fractionalspin particles. Physical Review Letters 49: 957–959. https://doi.org/10.1103/PhysRevLett.49.957.

Fractional quantum Hall effect in semiconductor systems Zlatko Papi
ca and Ajit C Balramb,c , aSchool of Physics and Astronomy, University of Leeds, Leeds, United Kingdom; bInstitute of Mathematical Sciences, CIT Campus, Chennai, India; cHomi Bhabha National Institute, Mumbai, India © 2024 Elsevier Ltd. All rights reserved.

Introduction Phenomenology of the FQH effect Incompressible fluids Broken symmetry phases Wigner crystals in the lowest Landau level Stripes, nematics and bubble crystals in higher LLs Laughlin states Laughlin’s wave function Charged excitations Magnetoroton excitation Edge excitations Parent Hamiltonian and Haldane pseudopotentials Hierarchy and composite fermions Haldane-Halperin hierarchy Composite fermions Half-filled Landau level Gapless composite fermion Fermi liquid in the half-filled Landau level n ¼ 5/2 state Anti-Pfaffian Non-Abelian states Non-Abelian anyons and topological quantum computation n ¼ 12/5 state Other non-Abelian states Multicomponent fractional quantum Hall effect Spinful systems Bilayer systems Recent developments Parton theory Effective field theory Entanglement-based approaches Conclusion Acknowledgments References

286 287 289 290 290 291 291 291 292 292 292 293 293 293 294 295 295 295 296 296 296 297 297 297 298 298 300 300 300 301 302 302 302

Abstract The fractional quantum Hall (FQH) effect refers to the strongly-correlated phenomena and the associated quantum phases of matter realized in a two-dimensional gas of electrons placed in a large perpendicular magnetic field. In such systems, topology and quantum mechanics conspire to give rise to exotic physics that manifests via robust quantization of the Hall resistance. In this chapter, we provide an overview of the experimental phenomenology of the FQH effect in GaAs-based semiconductor materials and present its theoretical interpretations in terms of trial wave functions, composite fermion quasiparticles, and enigmatic non-Abelian states. We also highlight some recent developments, including the Parton theory and the Dirac composite fermion field theory of FQH states, the role of anisotropy and geometrical degrees of freedom, and quantum entanglement in FQH fluids.

Key points

• • • •

describe the rich phenomenology exhibited by two-dimensional electrons hosted in a semiconductor system placed in a perpendicular magnetic field; overview of different theoretical approaches for describing fractional quantum Hall (FQH) phases of matter; introduce the concept of non-Abelian FQH states and their potential applications in fault-tolerant topological quantum computation; summarize recent developments in the field, including parton theory of the FQH effect, geometric properties of FQH states, entanglement-based and field-theoretic approaches to understanding FQH systems.

Encyclopedia of Condensed Matter Physics, Second Edition

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Notations and acronyms z ¼ x – iy n Q F1  i 1: In general, one can add longer-range hopping in a jCj ¼ 1 model or appropriately stack multilayers of jCj ¼ 1 models to get jCj > 1 bands (Wang and Ran, 2011; Zhang et al., 2011; Trescher and Bergholtz, 2012; Yang et al., 2012; Sticlet and Piéchon, 2013; Wu et al., 2015; Behrmann et al., 2016; Zhang et al., 2019a; Chittari et al., 2019; Lee et al., 2019). This strategy also works for moiré materials. One can systematically construct jCj ¼ n bands near the CNP by twisting two sheets of Bernal-stacked n graphene layers (Wang and Liu, 2022; Ledwith et al., 2022). TBG corresponds to the n ¼ 1 case. n ¼ 2 gives the twisted double bilayer graphene (TDBG), for which jCj ¼ 2 flat bands near the CNP have been confirmed (Zhang et al., 2019a; Lee et al., 2019; Chebrolu et al., 2019). Strikingly, the model constructed in this way gains chiral symmetry for any n  1 if only intersublattice couplings are kept. At magic angles, these chiral models all possess perfectly flat valence and conduction bands with ideal band geometry. FCIs can be diagnosed in high-Chern number bands by the same identifications as in jCj ¼ 1 models. Excitingly, series of high-C FCI states have been numerically discovered in realistic moiré bands (Liu et al., 2021a; Wang and Liu, 2022) inspired by earlier numerical findings in tight-binding toy models (Yang et al., 2012; Wang et al., 2012a; Liu et al., 2012; Wang et al., 2013; Sterdyniak et al., 2013; Wu et al., 2013; Bergholtz et al., 2015; Wu et al., 2015; Behrmann et al., 2016; Möller and Cooper, 2015; Andrews and Moller, 2018). These FCIs appear at band fillings, for instance, n¼

k jCjðm −1Þ + 1

(25)

with integer k  1 and m  2. Here k ¼ 1 and k > 1 correspond to Abelian and non-Abelian states, respectively, and m is even (odd) for bosons (fermions). Further fillings of high-CFCIs can be predicted by applying the composite-fermion theory to Hofstadter bands (Möller and Cooper, 2015). Some of the observed high-CFCIs are even model states occurring as the zero-energy ground states of proper short-ranged interactions (Behrmann et al., 2016; Wang and Liu, 2022). Ample attention has been directed toward understanding these high-CFCIs. Intuitively, they may have a relation with the conventional FQH states in jCj copies of the LLL, which also carry Chern number C as a whole. However, an obvious discrepancy exists between a single Chern number C band and jCj copies of the LLL: the number of single-particle orbitals in the latter must be divisible by jCj as each LLL contains an integer number of orbitals, while the number of single-particle states in the former does not necessarily to be a multiple of jCj: To resolve this discrepancy, a new type of jCj-component LLL was proposed. Here component refers to internal degrees of freedom of particles, like spin or physical layer. In contrast to the usual multicomponent LLL where the jCj components are decoupled, a new set of boundary conditions, called “color-entangled” boundary conditions, is adopted to connect different components, i.e., a particle can change its component index when crossing the boundary of the system (Wu et al., 2013, 2014). Under this scenario, one can construct a single manifold of Bloch-like states with Chern number C, rather than jCj separate Chern-number-one manifolds, such that the number of orbitals per LLL does not need to be an integer. Diagonalizing Haldane’s pseudopotentials in the “color-entangled” LLL basis produces unconventional FQH states, which were found to have high overlaps and the same PES countings with high-CFCIs in lattice models (Wu et al., 2013). However, the analytical ansatz wavefunctions of the high-CFCIs are still unknown, and it remains unclear whether the color-entangled boundary conditions lead to new topological orders compared with conventional jCj -component FQH states.

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Experimental observation of FCIs We have seen that FCIs arise from the interplay between interactions and the dispersion, topology, and geometry of the partially filled band. Therefore, their experimental realizations likely require highly controllable platforms in which band properties and interactions can be easily tuned to the suitable regime. Given this, van-der-Waals heterostructures with moiré patterns (Geim and Grigorieva, 2013; Novoselov et al., 2016) naturally emerge as a promising candidate to host FCIs. There are many experimental knobs in this kind of setups that can affect band properties and interactions, including the twist angle, the external fields, the dielectric environment, and even the material itself. In the past few years, rapid progress has been made on manufacturing van-der-Waals heterostructures and investigating the exotic correlated phases thereof.

Experiment in Bernal-stacked bilayer graphene The first observation of FCI states was reported in a van-der-Waals heterostructure made of Bernal-stacked bilayer graphene aligned with hBN (BLG-hBN), penetrated by a perpendicular magnetic field B  30 T (Spanton et al., 2018). In this system, electrons experience a moiré superlattice potential from the mismatched graphene and hBN lattices. The interplay between the magnetic field and the moiré potential creates topological Harper-Hofstadter bands that act as parents of potential FCI states. By determining the compressibility of the system via measuring the penetration field capacitance for variable electron density ne and magnetic flux density nf per moiré superlattice cell, multiple incompressible states are identified along trajectories obeying Streda formula ne ¼ tnf + s,

(26)

where t ¼ sxyh/e is the dimensionless Hall conductance and s is the zero-field filling factor Streda (1982); MacDonald (1983). The observed incompressible states can be distinguished by their values of t and s (see Table 2): non-integer t signals intrinsic topological order whereas fractional s may originate from either topological order or translational symmetry (TS) breaking. Remarkably, apart from the trivial states, QHE states in LLs, and Chern insulators in Hofstadter bands, clear evidence of states with fractional s and fractional t was observed in the experiment, which correspond to FCIs. These FCI states develop from electron/ hole doping the nearby parent Chern insulating states with an effective filling v in the partially filled band. Some of them exist in a C ¼ −1 band at v ¼ 1=3,2=3,2=5,3=5, falling into the odd-denominator Laughlin and composite fermion sequence. Notably, several other states exist in a C ¼ 2 band, for instance, at v ¼ 1=3 and v ¼ 1=6: The C ¼ 2,v ¼ 1=3 state, with (t, s) ¼ (8/3, −1/3), preserves TS of the moiré superlattice and belongs to the sequence predicted in Ref. Möller and Cooper (2015). However, the C ¼ 2,v ¼ 1=6 state carries (t, s) ¼ (7/3, −1/6), which means that only half fundamental charge e/3 is bound to each moiré unit cell. This feature suggests doubling of unit cell in the electron density distribution which spontaneously breaks TS of the moiré superlattice. The TS-broken FCI, as well as the observed TS-broken Chern insulators, imply an intriguing hybridization of conventional CDW and topological phase. Such combination of CDW and topological features is also numerically discovered in C ¼ 0,  1, 2 bands of several toy tight-binding models (Kumar et al., 2014; Kourtis and Daghofer, 2014; Kourtis, 2018) and experimentally observed in other moiré materials (Pierce et al., 2021; Polshyn et al., 2022). The hybrid states in jCj ¼ 2 bands may originate from ferromagnetism of internal degree of freedoms and their entanglement with real-space translation in high Chern number bands (Kumar et al., 2014; Polshyn et al., 2022). 2

Table 2 The observed incompressible states in Refs. Spanton et al. (2018) and Xie et al. (2021b) classified by their values of t and s. While some of the observed Chern insulators and FCIs preserve translational symmetry (TS) of the moiré superlattice, others spontaneously break this symmetry with an expanded unit cell of electron density distribution, implying an intriguing combination of conventional charge density wave and topological features. Those exotic fractional Chern insulators in the last row are neither simple TS preserving nor TS broken states, and may originate from the interplay between the multicomponent nature of the material and the spatial symmetry. Incompressible states

Values of t and s

Trivial correlated insulator Charge density wave Integer quantum Hall states in LLs TS-preserving Chern insulators TS-broken Chern insulators (also observed in Refs. Pierce et al. (2021) and Polshyn et al. (2022)) Fractional quantum Hall states in LLs TS-preserving fractional Chern insulators TS-broken fractional Chern insulators Exotic fractional Chern insulators

t ¼ 0 and integer s 6¼ 0 t ¼ 0 and fractional s Integer t 6¼ 0 and s ¼ 0 Integer t 6¼ 0 and integer s 6¼ 0 Integer t 6¼ 0 and fractional s Fractional t and s ¼ 0 Fractional t and fractional s, t and s have the same denominator Fractional t and fractional s, denominator of s is a multiple of that of t Fractional t and fractional s, t and s have coprime denominators

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Experiment in twisted bilayer graphene While the experiment above realizes the FCI states in the sense that they have no direct LL counterparts, a quite high magnetic field is still needed to produce the topological band structure. Therefore, to turn the dream of zero-field FCI into reality, we must find systems hosting intrinsic topological flat bands in the absence of (or at least with weak) external magnetic fields. As predicted by theoretical works (Zhang et al., 2019a; Song et al., 2019; Liu et al., 2019; Po et al., 2019; Zhang et al., 2019b), TBG appears as a promising candidate to satisfy this requirement. A plethora of recent experiments down to zero field (Zondiner et al., 2020; Nuckolls et al., 2020; Wu et al., 2021; Saito et al., 2021; Das et al., 2021; Choi et al., 2021; Park et al., 2021; Stepanov et al., 2021; Serlin et al., 2020; Pierce et al., 2021) have confirmed the existence of Chern insulating states at integer band fillings, thus indeed raising the possibility of realizing zero-field FCIs in TBG systems. A significant progress was made in this direction in which FCI states were observed in TBG aligned with hBN at weak magnetic fields as low as 5 T (Xie et al., 2021b). In this experiment, TBG devices were prepared near the magic twist angle y  1.06 , then a perpendicular magnetic field is applied. Local electronic compressibility measurements were preformed using a scanning single-electron transistor. Similar to the BLG-hBN experiment (Spanton et al., 2018), the dependence of this compressibility on the electron density and magnetic field, as shown in Fig. 5(a), reveals a large number of incompressible states (Table 2) satisfying Eq. (26), in which multiple FCI states with fractional t and s are identified. Remarkably, the first two FCI states appear in the range of 3 < n < 4 at a weak magnetic field of only 5 T [state 1 with (t, s) ¼ (2/3, 10/3) and state 2 with (t, s) ¼ (1/3, 11/3) in Fig. 5(b)]. According to their (t, s) values, these two states are formed at v ¼ 1=3 and v ¼ 2=3 filling of a C ¼ −1 band and preserve the TS of the moiré superlattice, thus they can be interpreted as lattice analogs of the n ¼ 1/3 Laughlin FQH state and its particle-hole conjugate. The energy gaps for both states are estimated as 0.6 K, which is comparable to the typical temperature required to realize Laughlin states in 2DEGs. Notably, these two FCI states are suddenly replaced by CDWs with the decreasing of the magnetic field. Numerical simulations in Refs. Xie et al. (2021b) and Parker et al. (2021) interpreted this phenomenon in terms of quantum geometry of the partially filled band: the applied magnetic field flattens the band Berry curvature, such that favorable band geometry conditions for the emergence of FCIs are created. However, unlike in the BLG-hBN experiment where the whole Chern band is created by the

(a)

(b) 6 8

5 7 4

3 1 2

Fig. 5 Incompressible states observed in TBG-hBN. (a) Local inverse compressibility dm/dn as a function of electron filling n and the magnetic field B. Those red trajectories correspond to incompressible states. (b) Wannier diagram extracted from the experimental data in (a). Black lines correspond to integer quantum Hall states/TS-preserving Chern insulators (IQH/ChI); green lines correspond to trivial correlated insulators (CI); blue lines correspond to charge density waves (CDW); yellow lines correspond to TS-broken Chern insulators (SBCI); and orange lines correspond to (both TS-preserving and TS-broken) fractional Chern insulators (FCI). The detailed definitions of these states can be found in Table 2. The red numbers 1–8 indicate the eight observed FCI states. Modified from Ref. Xie Y., Pierce A.T., Park J.M., Parker D.E., Khalaf E., Ledwith P., Cao Y., Lee S.H., Chen S., Forrester P.R., Watanabe K., Taniguchi T., Vishwanath A., Jarillo-Herrero P., Yacoby A. (2021b). Fractional chern insulators in magic-angle twisted bilayer graphene. Nature 600(7889):439–443. doi: 10.1038/s41586-021-04002-3.

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magnetic field, here the Berry curvature stems from the zero-field topological band of TBG-hBN and the role of the magnetic field is merely to redistribute Berry curvature. Once the Berry curvature can be flattened by other effects, the magnetic field will probably become unnecessary and these v ¼ 1=3 and v ¼ 2=3 FCIs are expected to survive at zero field. Indeed, zero-field v ¼ 1=3 FCIs in TBG-hBN were first numerically predicted at weaker w1 ¼ 90 meV (Abouelkomsan et al., 2020), where the Berry curvature is flatter. Alternatively, a reduction of w0/w1 or slightly increasing the twist angle can also improve the band geometry (Parker et al., 2021). Apart from the weak-field FCI states 1 and 2, additional FCIs were also observed in TBG-hBN away from 3 < n < 4 at slightly stronger magnetic fields, including TS-preserving states [state 6 with (t, s) ¼ (−13/5, −2/5) and state 7 with (t, s) ¼ (−4/3, −5/3)] and TS-broken states [state 3 with (t, s) ¼ (−8/5, 11/10) and state 4 with (t, s) ¼ (−7/3, 2/9)]. More intriguingly, there are also FCI states with coprime denominators of t and s, like state 5 with (t, s) ¼ (−5/2, −1/5) and state 8 with (t, s) ¼ (−1/2, −8/3), which are neither simple TS-preserving nor TS-broken states. These exotic FCIs may originate from complex interplay between the multicomponent nature of TBG and spatial symmetry.

FCIs in cold-atom systems Beside solid-state moiré systems, the high microscopic control and precision that are achievable in ultracold atoms make them also among the most promising candidates to realize rich correlated physics. In fact, there is a long-standing interest to realize the FQHE via cold atoms. The key task is to achieve a tight-binding model with complex hopping. Theoretical works along this direction have proposed several schemes, such as dipole systems (Weber et al., 2022; Yao et al., 2012) and bosons in optical flux lattice (Cooper, 2011; Cooper and Moessner, 2012; Cooper and Dalibard, 2013; Sterdyniak et al., 2015), and predicted FCIs therein. In particular, tremendous effort is made to study onsite-interacting bosons confined in a square optical lattice penetrated by a uniform effective magnetic field. This system is described by the 2D Harper-Hofstadter-Hubbard (HHH) Hamiltonian

X { UX b ðn b −1Þ, am,n+1 am,n + ei2pfn a{m+1,n am,n + H:c: + (27) H ¼ −J n 2 m,n m,n m,n m,n where J is the nearest-neighboring hopping strength, a{m,n(am,n) creates (annihilates) a boson on the lattice site (m, n), f is the bm,n ¼ a{m,n am,n is effective magnetic flux through each elementary square of the lattice, U is the magnitude of onsite repulsion, and n the occupation number operator on site (m, n). A plethora of numerical works find bosonic Abelian and non-Abelian FCIs as ground states of the HHH model at rational flux density f  1/3 when the interaction is sufficiently strong and the lowest band is partially filled (Palmer and Jaksch, 2006; Sørensen et al., 2005; Hafezi et al., 2007; Hormozi et al., 2012; Gerster et al., 2017; Möller and Cooper, 2015; Hudomal et al., 2019; Palm et al., 2021; Boesl et al., 2022). The introduction of longer-range hopping and reduction of flux density may further stabilize these states. Fine-tuned exponentially decaying hopping can even perfectly flatten the lowest band and support model FCIs as the zero-energy ground states (Kapit and Mueller, 2010; Liu et al., 2013b). The FCI states found in the HHH model can be prepared quasi-adiabatically, that is, one begins with a trivial state with low entropy, then slowly ramps the system to the desired final state (Popp et al., 2004; Grusdt et al., 2014; Barkeshli et al., 2015; He et al., 2017; Motruk and Pollmann, 2017; Palm et al., 2021). This protocol replies on continuous phase transitions between FCIs and other competing phases like CDW and superfluid. While dissipation is often thought of being harmful to the success of state preparation, some protocols suggest using special dissipative mechanism to pumping bosons from higher bands to the lowest band to make the state closer to FCIs (Liu et al., 2021b). Theoretical works are further encouraged by the rapid development of experimental techniques on artificial gauge field and quantum gas microscope, which make the HHH Hamiltonian a paradigmatic model that has been realized in cold-atom experiments. Once bosons are trapped in an optical lattice, the effective magnetic field can be achieved by generating an artificial gauge field through lattice shaking techniques (Aidelsburger et al., 2013; Miyake et al., 2013). The nontrivial band topology in the HHH model has been confirmed by measuring the transverse deflection of the atomic cloud in response to an optical gradient (Aidelsburger et al., 2015). Further, the model has been pushed into the strong-interaction regime (Eric Tai et al., 2017), such that FCIs of ultracold atomic gases may potentially be within reach in the not too distant future. Motivated by the encouraging theoretical and experimental progress, recent research focuses on how to detect FCI states once they are ready in cold-atom systems. In particular, such detection should be experimentally feasible and valid for those few-particle settings limited by the current technology. One way of detection is to probe the Hall conductance. However, because the transport measurements akin to those in solid state systems cannot be straightforwardly adopted into the cold-atom setting, alternative schemes have to be designed to extract Hall conductance. There have been several protocols on this aspect. One can apply a weak external force to the atomic cloud and track its center of mass drift (Repellin et al., 2020a; Motruk and Na, 2020). It is also possible to evaluate Hall conductance using interferometric protocols (Grusdt et al., 2016), randomized measurement scheme (Cian et al., 2021), or through circular-dichroic measurement (Tran et al., 2017; Repellin and Goldman, 2019). Apart from Hall conductance measurements, other promising schemes include probing fractionalized quasiholes through density measurement (Mantas Raci
unas et al., 2018; Umucalılar, 2018; Macaluso et al., 2020; Wang et al., 2022) and detecting chiral edge excitation (Kjäll and Moore, 2012; Dong et al., 2018). All of these proposals have been examined by numerical simulations in small cold-atom systems. Yet, it remains unclear which protocol is the most experimentally feasible before FCIs of cold atoms really turn to reality.

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Conclusion In this chapter, we have presented an introduction to fractional Chern insulators in topological flat bands. While FCIs share key features with conventional FQH states, the significant discrepancies between a Chern band and a Landau level make FCIs an ideal stage to realize the FQH physics under more general experimental conditions, to explore in the pursuit for qualitatively new topologically ordered states, and for unveiling the profound interplay between interaction, topology and geometry. Although it is impossible to provide a fully comprehensive account of the entire field here, we hope that our introduction has demonstrated the major methodology that has been used to understand FCIs and the new research trends which are emerging recently in this field. The enthusiasm of studying FCIs has lasted over the past decade, and is now gaining renewed and extended interest due to their direct relevance in moiré materials. We would like to close this chapter by listing several directions which in our opinion are important for the development of this field in near future.

•

Exploring new platforms. The recent trend of studying FCIs in moiré materials has clearly demonstrated the intimate relation between FCI and material science. Searching for new solid-state platforms that can host FCIs will significantly stimulate further development of this field. The research in this direction will need contributions from material calculations via, for instance, density functional theory. In fact, progress is already being made in this direction. In addition to more graphene based materials (Park et al., 2020; Rademaker et al., 2020; Chen et al., 2021), moiré systems built on transition metal dichalcogenides (TMD) are found to support topological flat bands (Wu et al., 2019; Pan et al., 2020; Li et al., 2021b; Zhang et al., 2021; Devakul et al., 2021; Pan et al., 2021; Zhou et al., 2022). Excitingly, a few numerical works have identified FCIs in twisted TMD homobilayers, like twisted bilayer MoTe2 (Li et al., 2021a) and WSe2 (Crépel and Fu, 2022). It requires more effort to thoroughly investigate FCI states and their competing phases in these new materials. These theoretical explorations will be greatly helpful to accelerate the experimental realization of zero-field FCIs. • Experimental realization of high-temperature zero-field FCIs. Despite of the encouraging experimental observation of FCI states in TBG-hBN at weak magnetic fields down to 5 T (Xie et al., 2021b), how to stabilize these states at further reduced fields remains a crucial challenge for really achieving zero-field FCIs. Theoretical works have suggested to probe TBG-hBN at twist angles above the magic angle, which could improve the band geometry such that the magnetic field might become unnecessary (Parker et al., 2021). Alternatively, one can also try other materials like twisted bilayer TMD. These proposals should be examined by experiments, then the optimal setups which facilitate the realization of zero-field FCIs the most can be selected. Moreover, apart from the compressibility measurement used in current experiments, transport features, which directly show the characterizing Hall conductance plateaus, is in an ideal situation a preferable method to identify FCIs in the future. In addition to the static realization of zero-field FCIs, one may also consider the dynamic preparation of these states in periodically driven systems. Recently, there have been a series of works studying the photon-dressed band structure of moiré materials driven by an external laser light field (Topp et al., 2019; Katz et al., 2020; Vogl et al., 2020; Li et al., 2020; Rodriguez-Vega et al., 2020; Assi et al., 2021; Topp et al., 2021; Vogl et al., 2021; Benlakhouy et al., 2022). Based on these Floquet band structures, numerical simulations have demonstrated the possibility of Floquet FCIs (Grushin et al., 2014; Anisimovas et al., 2015) in light-driven TBG (Hu and Liu, 2022). The energy gaps of the observed v ¼ 1=3 FCIs in TBG-hBN are about 0.6 K (Xie et al., 2021b), which are still close to that of the conventional Laughlin state in 2DEGs. Considering the expectation that FCIs can in principle exist at much higher temperature than conventional FQH states, it is definitely worthy of seeking a way to further increase the gap. • Searching and understanding new states. With more theoretical and experimental explorations in various platforms, we should pay special attention on searching and understanding new states. Clearly, non-Abelian FCI states are still absent in experimental observation. In fact, it is difficult to get them even in numerical simulations, where multi-body interactions are often required (Wang et al., 2012b; Andrei Bernevig and Regnault, 2012; Wu et al., 2012a; Sterdyniak et al., 2013; Behrmann et al., 2016). Considering the potential application of non-Abelian anyons in topological quantum computation, it is a challenging albeit urgent task to find schemes that can provide non-Abelian FCIs by realistic two-body interactions. The valley and spin degrees of freedom in moiré systems make it interesting to probe novel multicomponent FCI states. On the other hand, the symmetry-broken and other exotic FCI states (Table 2) found in the recent TBG-hBN experiment (Xie et al., 2021b) fall beyond the scope of conventional translational-symmetry-preserving FCIs. Apparently they have no analog in continuum FQH states. More theoretical and experimental effort is needed to understand these enigmatic states. Crucial open questions include: (i) Can these states survive at zero magnetic field? (ii) Can numerical simulations probe these states and extract their topological orders? (iii) Can experiments further identify the nature of these states, for example, their quasihole charges? Motivated by seeking solutions of the questions above, we are convinced that the study of FCIs will remain an active field providing exciting and beautiful new physics.

Acknowledgments Z. L. is supported by the National Key Research and Development Program of China through Grant No. 2020YFA0309200.

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References Abouelkomsan A, Liu Z, and Bergholtz EJ (2020) Particle-hole duality, emergent fermi liquids, and fractional chern insulators in moiré flatbands. Physical Review Letters 124: 106803. https://doi.org/10.1103/PhysRevLett.124.106803. Abouelkomsan A, Yang K, and Bergholtz EJ (2022) Quantum Metric Induced Phases in Moiré Materials. https://doi.org/10.48550/ARXIV.2202.10467. Aidelsburger M, Atala M, Lohse M, Barreiro JT, Paredes B, and Bloch I (2013) Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices. Physical Review Letters 111: 185301. https://doi.org/10.1103/PhysRevLett.111.185301. Aidelsburger M, Lohse M, Schweizer C, Atala M, Barreiro JT, Nascimbène S, Cooper NR, Bloch I, and Goldman N (2015) Measuring the chern number of hofstadter bands with ultracold bosonic atoms. Nature Physics 11(2): 162–166. https://doi.org/10.1038/nphys3171. Andrei Bernevig B and Regnault N (2012) Emergent many-body translational symmetries of abelian and non-abelian fractionally filled topological insulators. Physical Review B 85: 075128. https://doi.org/10.1103/PhysRevB.85.075128. Andrei Bernevig B, Song Z-D, Regnault N, and Lian B (2021) Twisted bilayer graphene. iii. Interacting hamiltonian and exact symmetries. Physical Review B 103: 205413. https://doi. org/10.1103/PhysRevB.103.205413. Andrei EY and MacDonald AH (2020) Graphene bilayers with a twist. Nature Materials 19(12): 1265–1275. https://doi.org/10.1038/s41563-020-00840-0. Andrews B and Soluyanov A (2020) Fractional quantum hall states for moiré superstructures in the hofstadter regime. Physical Review B 101: 235312. https://doi.org/10.1103/ PhysRevB.101.235312. Andrews B, Mohan M, and Neupert T (2021a) Abelian topological order of n ¼ 2/5 and 3/7 fractional quantum hall states in lattice models. Physical Review B 103: 075132. https:// doi.org/10.1103/PhysRevB.103.075132. Andrews B and Möller G (2018) Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1. Physical Review B 97(3): 035159. https://doi.org/10.1103/PhysRevB.97.035159. Andrews B, Neupert T, and Möller G (2021b) Stability, phase transitions, and numerical breakdown offractional chern insulators in higher chern bands of the hofstadter model. Physical Review B 104: 125107. https://doi.org/10.1103/PhysRevB.104.125107. Anisimovas E, Žlabys G, Anderson BM, Juzeli
unas G, and Eckardt A (2015) Role of real-space micromotion for bosonic and fermionic floquet fractional chern insulators. Physical Review B 91: 245135. https://doi.org/10.1103/PhysRevB.91.245135. Assi IA, LeBlanc JPF, Rodriguez-Vega M, Bahlouli H, and Vogl M (2021) Floquet engineering and nonequilibrium topological maps in twisted trilayer graphene. Physical Review B 104: 195429. https://doi.org/10.1103/PhysRevB.104.195429. Avron JE, Seiler R, and Simon B (1983) Homotopy and quantization in condensed matter physics. Physical Review Letters 51: 51–53. https://doi.org/10.1103/PhysRevLett.51.51. Barkeshli M and Qi X-L (2012) Topological nematic states and non-abelian lattice dislocations. Physical Review X 2: 031013. https://doi.org/10.1103/PhysRevX.2.031013. Barkeshli M, Jian C-M, and Qi X-L (2013) Twist defects and projective non-abelian braiding statistics. Physical Review B 87: 045130. https://doi.org/10.1103/PhysRevB.87.045130. Barkeshli M, Yao NY, and Laumann CR (2015) Continuous preparation of a fractional chern insulator. Physical Review Letters 115: 026802. https://doi.org/10.1103/ PhysRevLett.115.026802. Bauer D, Talkington S, Harper F, Andrews B, and Roy R (2022) Fractional chern insulators with a non-landau level continuum limit. Physical Review B 105: 045144. https://doi.org/ 10.1103/PhysRevB.105.045144. Behrmann J, Liu Z, and Bergholtz EJ (2016) Model fractional chern insulators. Physical Review Letters 116: 216802. https://doi.org/10.1103/PhysRevLett.116.216802. Benlakhouy N, Jellal A, Bahlouli H, and Vogl M (2022) Chiral limits and effect of light on the hofstadter butterfly in twisted bilayer graphene. Physical Review B 105: 125423. https:// doi.org/10.1103/PhysRevB.105.125423. Bergholtz EJ and Liu Z (2013) Topological flat band models and fractional chern insulators. International Journal of Modern Physics B 27(24): 1330017. https://doi.org/10.1142/ S021797921330017X. Bergholtz EJ, Zhao Liu M, Trescher RM, and Udagawa M (2015) Topology and interactions in a frustrated slab: Tuning from weyl semimetals to C > 1 fractional chern insulators. Physical Review Letters 114: 016806. https://doi.org/10.1103/PhysRevLett.114.016806. Bessler R, Duerig U, and Koren E (2019) The dielectric constant of a bilayer graphene interface. Nanoscale Advances 1: 1702–1706. https://doi.org/10.1039/C8NA00350E. Bistritzer R and MacDonald AH (2011) Moiré bands in twisted double-layer graphene. Proceedings of the National Academy of Sciences of the United States of America 00278424108(30): 12233–12237. https://doi.org/10.1073/pnas.1108174108. Boesl J, Dilip R, Pollmann F, and Knap M (2022) Characterizing fractional topological phases of lattice bosons near the first mott lobe. Physical Review B 105: 075135. https://doi.org/ 10.1103/PhysRevB.105.075135. Bultinck N, Chatterjee S, and Zaletel MP (2020) Mechanism for anomalous hall ferromagnetism in twisted bilayer graphene. Physical Review Letters 124: 166601. https://doi.org/ 10.1103/PhysRevLett.124.166601. Chang C-Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Yunbo O, Wei P, Wang L-L, Ji Z-Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S-C, He K, Wang Y, Li L, Ma X-C, and Xue Q-K (2013) Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator. Science 340(6129): 167–170. https://doi.org/10.1126/ science.1234414. Chebrolu NR, Chittari BL, and Jung J (2019) Flat bands in twisted double bilayer graphene. Physical Review B 99: 235417. https://doi.org/10.1103/PhysRevB.99.235417. Chen L, Mazaheri T, Seidel A, and Tang X (2014) The impossibility of exactly flat non-trivial chern bands in strictly local periodic tight binding models. Journal of Physics A: Mathematical and Theoretical 47(15): 152001. https://doi.org/10.1088/1751-8113/47/15/152001. Chen S, He M, Zhang Y-H, Hsieh V, Zaiyao Fei K, Watanabe TT, Cobden DH, Xiaodong X, Dean CR, and Yankowitz M (2021) Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nature Physics 17(3): 374–380. https://doi.org/10.1038/s41567-020-01062-6. Cheng R (2010) Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A Pedagogical Introduction. https://doi.org/10.48550/ARXIV.1012.1337. Chittari BL, Chen G, Zhang Y, Wang F, and Jung J (2019) Gate-tunable topological flat bands in trilayer graphene boronnitride moiré superlattices. Physical Review Letters 122: 016401. https://doi.org/10.1103/PhysRevLett.122.016401. Choi Y, Kim H, Peng Y, Thomson A, Lewandowski C, Polski R, Zhang Y, Arora HS, Watanabe K, Taniguchi T, Alicea J, and Nadj-Perge S (2021) Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589(7843): 536–541. https://doi.org/10.1038/s41586-020-03159-7. Cian Z-P, Dehghani H, Elben A, Vermersch B, Zhu G, Barkeshli M, Zoller P, and Hafezi M (2021) Many-body chern number from statistical correlations of randomized measurements. Physical Review Letters 126: 050501. https://doi.org/10.1103/PhysRevLett.126.050501. Claassen M, Lee CH, Thomale R, Qi X-L, and Devereaux TP (2015) Position-momentum duality and fractional quantum hall effect in chern insulators. Physical Review Letters 114: 236802. https://doi.org/10.1103/PhysRevLett.114.236802. Cooper NR (2011) Optical flux lattices for ultracold atomic gases. Physical Review Letters 106: 175301. https://doi.org/10.1103/PhysRevLett.106.175301. Cooper NR and Dalibard J (2013) Reaching fractional quantum hall states with optical flux lattices. Physical Review Letters 110: 185301. https://doi.org/10.1103/ PhysRevLett.110.185301. Cooper NR and Moessner R (2012) Designing topological bands in reciprocal space. Physical Review Letters 109: 215302. https://doi.org/10.1103/PhysRevLett. 109.215302. Crépel V and Fu L (2022) Anomalous Hall Metal and Fractional Chern Insulator in Twisted Transition Metal Dichalcogenides. https://doi.org/10.48550/ARXIV.2207.08895. Das I, Xiaobo L, Herzog-Arbeitman J, Song Z-D, Watanabe K, Takashi Taniguchi B, Bernevig A, and Efetov DK (2021) Symmetry-broken chern insulators and rashba-like landau-level crossings in magic-angle bilayer graphene. Nature Physics 17(6): 710–714. https://doi.org/10.1038/s41567-021-01186-3.

Recent developments in fractional Chern insulators

533

Devakul T, Crépel V, Zhang Y, and Liang F (2021) Magic in twisted transition metal dichalcogenide bilayers. Nature Communications 12(1): 6730. https://doi.org/10.1038/s41467021-27042-9. Dong X-Y, Grushin AG, Motruk J, and Pollmann F (2018) Charge excitation dynamics in bosonic fractional chern insulators. Physical Review Letters 121: 086401. https://doi.org/ 10.1103/PhysRevLett.121.086401. Eric Tai M, Lukin A, Rispoli M, Schittko R, Menke T, Borgnia D, Preiss PM, Grusdt F, Kaufman AM, and Greiner M (2017) Microscopy of the interacting harper–hofstadter model in the two-body limit. Nature 546(7659): 519–523. https://doi.org/10.1038/nature22811. Feldman DE and Halperin BI (2021) Fractional charge and fractional statistics in the quantum hall effects. Reports on Progress in Physics 84(7)076501. https://doi.org/10.1088/13616633/ac03aa. Fukuyama H, Platzman PM, and Anderson PW (1979) Two-dimensional electron gas in a strong magnetic field. Physical Review B 19: 5211–5217. https://doi.org/10.1103/ PhysRevB.19.5211. Geim AK and Grigorieva IV (2013) Van der Waals heterostructures. Nature 00280836499(7459): 419–425. https://doi.org/10.1038/nature12385. Gerster M, Rizzi M, Silvi P, Dalmonte M, and Montangero S (2017) Fractional quantum hall effect in the interacting hofstadter model via tensor networks. Physical Review B 96: 195123. https://doi.org/10.1103/PhysRevB.96.195123. Grusdt F, Letscher F, Hafezi M, and Fleischhauer M (2014) Topological growing of laughlin states in synthetic gauge fields. Physical Review Letters 113: 155301. https://doi.org/ 10.1103/PhysRevLett.113.155301. Grusdt F, Yao NY, Abanin D, Fleischhauer M, and Demler E (2016) Interferometric measurements of many-body topological invariants using mobile impurities. Nature Communications 7(1): 11994. https://doi.org/10.1038/ncomms11994. Grushin AG, TitusNeupert CC, and Mudry C (2012) Enhancing the stability of a fractional chern insulator against competing phases. Physical Review B 86: 205125. https://doi.org/ 10.1103/PhysRevB.86.205125. Grushin AG, Gómez-León Á, and Neupert T (2014) Floquet fractional chern insulators. Physical Review Letters 10797114112(15): 1–5. https://doi.org/10.1103/ PhysRevLett.112.156801. Grushin AG, Motruk J, Zaletel MP, and Pollmann F (2015) Characterization and stability of a fermionic n ¼ 1/3 fractional chern insulator. Physical Review B 91: 035136. https://doi. org/10.1103/PhysRevB.91.035136. Guinea F and Walet NR (2018) Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers. Proceedings of the National Academy of Sciences of the United States of America 115(52): 13174–13179. https://doi.org/10.1073/pnas.1810947115. Guinea F and Walet NR (2019) Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters. Physical Review B 99: 205134. https://doi.org/ 10.1103/PhysRevB.99.205134. Haddadi F, QuanSheng W, Kruchkov AJ, and Yazyev OV (2020) Moiréflat bands in twisted double bilayer graphene. Nano Letters 20(4): 2410–2415. https://doi.org/10.1021/acs. nanolett.9b05117. Hafezi M, Sørensen AS, Demler E, and Lukin MD (2007) Fractional quantum hall effect in optical lattices. Physical Review A 76: 023613. https://doi.org/10.1103/ PhysRevA.76.023613. Haldane FDM (1983) Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states. Physical Review Letters 51: 605–608. https://doi.org/10.1103/ PhysRevLett.51.605. Haldane FDM (1988) Model for a quantum hall effect without landau levels: Condensed-matter realization of the “parity anomaly” Physical Review Letters 61: 2015–2018. https://doi. org/10.1103/PhysRevLett.61.2015. Haldane FDM, Rezayi EH, and Yang K (2000) Spontaneous breakdown of translational symmetry in quantum hall systems: Crystalline order in high landau levels. Physical Review Letters 85: 5396–5399. https://doi.org/10.1103/PhysRevLett.85.5396. He Y-C, Grusdt F, Kaufman A, Greiner M, and Vishwanath A (2017) Realizing and adiabatically preparing bosonic integer and fractional quantum hall states in optical lattices. Physical Review B 96: 201103. https://doi.org/10.1103/PhysRevB.96.201103. He A-L, Luo W-W, Yao H, and Wang Y-F (2020) Non-abelian fractional chern insulator in disk geometry. Physical Review B 101: 165127. https://doi.org/10.1103/ PhysRevB.101.165127. Hormozi L, Möller G, and Simon SH (2012) Fractional quantum hall effect of lattice bosons near commensurate flux. Physical Review Letters 108: 256809. https://doi.org/10.1103/ PhysRevLett.108.256809. Hu P-S and Liu Z (2022) Floquet Fractional Chern Insulators and Competing Phases in Twisted Bilayer Graphene. https://doi.org/10.48550/ARXIV.2207.07314. Hudomal A, Regnault N, and Vasic I (2019) Bosonic fractional quantum hall states in driven optical lattices. Physical Review A 100: 053624. https://doi.org/10.1103/ PhysRevA.100.053624. Hunt B, Sanchez-Yamagishi JD, Young AF, Yankowitz M, LeRoy BJ, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, and Ashoori RC (2013) Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure. Science 340(6139): 1427–1430. https://doi.org/10.1126/science.1237240. Jackson TS, Möller G, and Roy R (2015) Geometric stability of topological lattice phases. Nature Communications 6(1): 8629. https://doi.org/10.1038/ncomms9629. Jaworowski B, Güçlü AD, Kaczmarkiewicz P, Kupczynski M, Potasz P, and Wójs A (2018) Wigner crystallization in topological flat bands. New Journal of Physics 20(6): 063023. https://doi.org/10.1088/1367-2630/aac690. Jaworowski B, Regnault N, and Liu Z (2019) Characterization of quasiholes in two-component fractional quantum hall states and fractional chern insulators in |c| ¼ 2 flat bands. Physical Review B 99: 045136. https://doi.org/10.1103/PhysRevB.99.045136. Jian C-M and Qi X-L (2013) Crystal-symmetry preserving wannier states for fractional chern insulators. Physical Review B 88: 165134. https://doi.org/10.1103/ PhysRevB.88.165134. Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, and Esslinger T (2014) Experimental realization of the topological haldane model with ultracold fermions. Nature 515(7526): 237–240. https://doi.org/10.1038/nature13915. Jung J, Raoux A, Qiao Z, and MacDonald AH (2014) Ab initio theory of moiré superlattice bands in layered two-dimensional materials. Physical Review B 89: 205414. https://doi.org/ 10.1103/PhysRevB.89.205414. Jung J, DaSilva AM, MacDonald AH, and Adam S (2015) Origin of band gaps in graphene on hexagonal boron nitride. Nature Communications 6: 6308. https://doi.org/10.1038/ ncomms7308. Kang J and Vafek O (2019) Strong coupling phases of partially filled twisted bilayer graphene narrow bands. Physical Review Letters 122: 246401. https://doi.org/10.1103/ PhysRevLett.122.246401. Kapit E and Mueller E (2010) Exact parent hamiltonian for the quantum hall states in a lattice. Physical Review Letters 105: 215303. https://doi.org/10.1103/ PhysRevLett.105.215303. Katz O, Refael G, and Lindner NH (2020) Optically induced flat bands in twisted bilayer graphene. Physical Review B 24699969102(15): 155123. https://doi.org/10.1103/ PhysRevB.102.155123. Kitaev A and Preskill J (2006) Topological entanglement entropy. Physical Review Letters 96: 110404. https://doi.org/10.1103/PhysRevLett.96.110404. Kjäll JA and Moore JE (2012) Edge excitations of bosonic fractional quantum hall phases in optical lattices. Physical Review B 85: 235137. https://doi.org/10.1103/ PhysRevB.85.235137. Klitzing KV, Dorda G, and Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Physical Review Letters 45: 494–497. https://doi.org/10.1103/PhysRevLett.45.494.

534

Recent developments in fractional Chern insulators

Knapp C, Spanton EM, Young AF, Nayak C, and Zaletel MP (2019) Fractional Chern insulator edges and layer-resolved lattice contacts. Physical Review B 99(8): 081114. https://doi. org/10.1103/PhysRevB.99.081114. Koshino M, Yuan NFQ, Koretsune T, Ochi M, Kuroki K, and Fu L (2018) Maximally localizedwannier orbitals and the extended hubbard model for twisted bilayer graphene. Physical Review X 216033088(3): 31087. https://doi.org/10.1103/PhysRevX.8.031087. Kourtis S (2018) Symmetry breaking and the fermionic fractional chern insulator in topologically trivial bands. Physical Review B 97: 085108. https://doi.org/10.1103/ PhysRevB.97.085108. Kourtis S and Daghofer M (2014) Combined topological and landau order from strong correlations in chern bands. Physical Review Letters 113: 216404. https://doi.org/10.1103/ PhysRevLett.113.216404. Kourtis S, Venderbos JWF, and Daghofer M (2012) Fractional chern insulator on a triangular lattice of strongly correlated t2g electrons. Physical Review B 86: 235118. https://doi.org/ 10.1103/PhysRevB.86.235118. Kourtis S, Neupert T, Chamon C, and Mudry C (2014) Fractional chern insulators with strong interactions that far exceed band gaps. Physical Review Letters 112: 126806. https://doi. org/10.1103/PhysRevLett.112.126806. Kourtis S, Neupert T, Mudry C, Sigrist M, and Chen W (2017) Weyl-type topological phase transitions in fractional quantum hall like systems. Physical Review B 96: 205117. https:// doi.org/10.1103/PhysRevB.96.205117. Kumar A, Roy R, and Sondhi SL (2014) Generalizing quantum hall ferromagnetism to fractional chern bands. Physical Review B 90: 245106. https://doi.org/10.1103/ PhysRevB.90.245106. Kupczynski M, Jaworowski BZ˙ , and Wójs A (2021) Interaction-driven transition between the wigner crystal and the fractional chern insulator in topological flat bands. Physical Review B 104: 085107. https://doi.org/10.1103/PhysRevB.104.085107. Läuchli AM, Liu Z, Bergholtz EJ, and Moessner R (2013) Hierarchy of fractional chern insulators and competing compressible states. Physical Review Letters 111: 126802. https://doi. org/10.1103/PhysRevLett.111.126802. Laughlin RB (1983) Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50: 1395–1398. https://doi.org/ 10.1103/PhysRevLett.50.1395. Ledwith PJ, Tarnopolsky G, Khalaf E, and Vishwanath A (2020) Fractional chern insulator states in twisted bilayer graphene: An analytical approach. Physical Review Research 2: 023237. https://doi.org/10.1103/PhysRevResearch.2.023237. Ledwith PJ, Vishwanath A, and Khalaf E (2022) Family of ideal chern flatbands with arbitrary chern number in chiral twisted graphene multilayers. Physical Review Letters 128: 176404. https://doi.org/10.1103/PhysRevLett.128.176404. Lee CH, Thomale R, and Qi X-L (2013) Pseudopotential formalism for fractional chern insulators. Physical Review B 88: 035101. https://doi.org/10.1103/PhysRevB.88.035101. Lee JY, Khalaf E, Liu S, Liu X, Hao Z, Kim P, and Vishwanath A (2019) Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene. Nature Communications 10(1): 5333. https://doi.org/10.1038/s41467-019-12981-1. Levin M and Wen X-G (2006) Detecting topological order in a ground state wave function. Physical Review Letters 96: 110405. https://doi.org/10.1103/PhysRevLett.96.110405. Li H and Haldane FDM (2008) Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states. Physical Review Letters 101: 010504. https://doi.org/10.1103/PhysRevLett.101.010504. Li W, Liu Z, Yong-Shi W, and Chen Y (2014) Exotic fractional topological states in a two-dimensional organometallic material. Physical Review B 89: 125411. https://doi.org/10.1103/ PhysRevB.89.125411. Li Y, Fertig HA, and Seradjeh B (2020) Floquet-engineered topological flat bands in irradiated twisted bilayer graphene. Physical Review Research 2: 043275. https://doi.org/10.1103/ PhysRevResearch.2.043275. Li H, Kumar U, Sun K, and Lin S-Z (2021a) Spontaneous fractional chern insulators in transition metal dichalcogenide moiré superlattices. Physical Review Research 3: L032070. https://doi.org/10.1103/PhysRevResearch.3.L032070. Li T, Jiang S, Shen B, Zhang Y, Li L, Tao Z, Devakul T, Watanabe K, Taniguchi T, Liang F, Shan J, and Mak KF (2021b) Quantum anomalous hall effect from intertwined moirébands. Nature 600(7890): 641–646. https://doi.org/10.1038/s41586-021-04171-1. Liu Z and Bergholtz EJ (2013) From fractional chern insulators to abelian and non-abelian fractional quantum hall states: Adiabatic continuity and orbital entanglement spectrum. Physical Review B 87: 035306. https://doi.org/10.1103/PhysRevB.87.035306. Liu Z and Bergholtz EJ (2019) Fractional quantum hall states with gapped boundaries in an extreme lattice limit. Physical Review B 99: 195122. https://doi.org/10.1103/ PhysRevB.99.195122. Liu J and Dai X (2021) Theories for the correlated insulating states and quantum anomalous hall effect phenomena in twisted bilayer graphene. Physical Review B 103: 035427. https://doi.org/10.1103/PhysRevB.103.035427. Liu Z, Bergholtz EJ, Fan H, and Läuchli AM (2012) Fractional chern insulators in topological flat bands with higher chern number. Physical Review Letters 109: 186805. https://doi.org/ 10.1103/PhysRevLett.109.186805. Liu T, Repellin C, Bernevig BA, and Regnault N (2013a) Fractional chern insulators beyond laughlin states. Physical Review B 87: 205136. https://doi.org/10.1103/ PhysRevB.87.205136. Liu Z, Bergholtz EJ, and Kapit E (2013b) Non-abelian fractional chern insulators from long-range interactions. Physical Review B 88: 205101. https://doi.org/10.1103/ PhysRevB.88.205101. Liu Z, Kovrizhin DL, and Bergholtz EJ (2013c) Bulk-edge correspondence in fractional chern insulators. Physical Review B 88: 081106. https://doi.org/10.1103/ PhysRevB.88.081106. Liu Z, Bhatt RN, and Regnault N (2015) Characterization of quasiholes in fractional chern insulators. Physical Review B 91: 045126. https://doi.org/10.1103/PhysRevB.91.045126. Liu Z, Möller G, and Bergholtz EJ (2017) Exotic non-abelian topological defects in lattice fractional quantum hall states. Physical Review Letters 119: 106801. https://doi.org/10.1103/ PhysRevLett.119.106801. Liu J, Liu J, and Dai X (2019) Pseudo landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase. Physical Review B 99: 155415. https://doi.org/10.1103/PhysRevB.99.155415. Liu Z, Abouelkomsan A, and Bergholtz EJ (2021a) Gate-tunable fractional chern insulators in twisted double bilayer graphene. Physical Review Letters 126: 026801. https://doi.org/ 10.1103/PhysRevLett.126.026801. Liu Z, Bergholtz EJ, and Budich JC (2021b) Dissipative preparation of fractional chern insulators. Physical Review Research 3: 043119. https://doi.org/10.1103/ PhysRevResearch.3.043119. Lopes dos Santos JMB, Peres NMR, and Castro Neto AH (2007) Graphene bilayer with a twist: Electronic structure. Physical Review Letters 99: 256802. https://doi.org/10.1103/ PhysRevLett.99.256802. Lopes dos Santos JMB, Peres NMR, and Castro Neto AH (2012) Continuum model of the twisted graphene bilayer. Physical Review B 86: 155449. https://doi.org/10.1103/ PhysRevB.86.155449. Luo W-W, Chen W-C, Wang Y-F, and Gong C-D (2013) Edge excitations in fractional chern insulators. Physical Review B 88: 161109. https://doi.org/10.1103/PhysRevB.88.161109. Luo W-W, He A-L, Zhou Y, Wang Y-F, and Gong C-D (2020) Quantum phase transitions in a v¼1/2 bosonic fractional chern insulator. Physical Review B 102: 155120. https://doi.org/ 10.1103/PhysRevB.102.155120. Luo W-W, He A-L, Wang Y-F, Zhou Y, and Gong C-D (2021) Bosonic fractional chern insulating state at integer fillings in a multiband system. Physical Review B 104: 115126. https:// doi.org/10.1103/PhysRevB.104.115126. Macaluso E, Comparin T, Umucalılar RO, Gerster M, Montangero S, Rizzi M, and Carusotto I (2020) Charge and statistics of lattice quasiholes from density measurements: A tree tensor network study. Physical Review Research 2: 013145. https://doi.org/10.1103/PhysRevResearch.2.013145.

Recent developments in fractional Chern insulators

535

MacDonald AH (1983) Landau-level subband structure of electrons on a square lattice. Physical Review B 28: 6713–6717. https://doi.org/10.1103/PhysRevB. 28.6713. Manna S, Wildeboer J, Sierra G, and Nielsen AEB (2018) Non-abelian quasiholes in lattice moore-read states and parent hamiltonians. Physical Review B 98: 165147. https://doi.org/ 10.1103/PhysRevB.98.165147. Mantas Raci
unas F, Ünal N, Anisimovas E, and Eckardt A (2018) Creating, probing, and manipulating fractionally charged excitations of fractional chern insulators in optical lattices. Physical Review A 98: 063621. https://doi.org/10.1103/PhysRevA.98.063621. Mera B and Ozawa T (2021) Engineering geometrically flat Chern bands with Fubini-Study Kähler structure. Physical Review B 104(11): 115160. https://doi.org/10.1103/ PhysRevB.104.115160. Miyake H, Siviloglou GA, Kennedy CJ, Burton WC, and Ketterle W (2013) Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices. Physical Review Letters 111: 185302. https://doi.org/10.1103/PhysRevLett.111.185302. Moessner R and Chalker JT (1996) Exact results for interacting electrons in high landau levels. Physical Review B 54: 5006–5015. https://doi.org/10.1103/PhysRevB.54.5006. Möller G and Cooper NR (2015) Fractional chern insulators in harper-hofstadter bands with higher chern number. Physical Review Letters 115: 126401. https://doi.org/10.1103/ PhysRevLett.115.126401. Möller G and Simon SH (2005) Composite fermions in a negative effective magnetic field: A monte carlo study. Physical Review B 72: 045344. https://doi.org/10.1103/ PhysRevB.72.045344. Moore G and Read N (1991) Nonabelions in the fractional quantum hall effect. Nuclear Physics B 0550-3213360(2): 362–396. https://doi.org/10.1016/0550-3213(91)90407-O. Motruk J and Na I (2020) Detecting fractional chern insulators in optical lattices through quantized displacement. Physical Review Letters 125: 236401. https://doi.org/10.1103/ PhysRevLett.125.236401. Motruk J and Pollmann F (2017) Phase transitions and adiabatic preparation of a fractional chern insulator in a boson cold-atom model. Physical Review B 96: 165107. https://doi.org/ 10.1103/PhysRevB.96.165107. Motruk J, Zaletel MP, Mong RSK, and Pollmann F (2016) Density matrix renormalization group on a cylinder in mixed real and momentum space. Physical Review B 93: 155139. https://doi.org/10.1103/PhysRevB.93.155139. Nam NNT and Koshino M (2017) Lattice relaxation and energy band modulation in twisted bilayer graphene. Physical Review B 2469996996(7): 1–12. https://doi.org/10.1103/ PhysRevB.96.075311. Neupert T, Santos L, Chamon C, and Mudry C (2011) Fractional quantum hall states at zero magnetic field. Physical Review Letters 106: 236804. https://doi.org/10.1103/ PhysRevLett.106.236804. Neupert T, Chamon C, Iadecola T, Santos LH, and Mudry C (2015) Fractional (chern and topological) insulators. Physica Scripta T164: 014005. https://doi.org/10.1088/00318949/2015/t164/014005. Niu Q, Thouless DJ, and Wu Y-S (1985) Quantized hall conductance as a topological invariant. Physical Review B 31: 3372–3377. https://doi.org/10.1103/PhysRevB.31.3372. Novoselov KS, Mishchenko A, Carvalho A, and Castro Neto AH (2016) 2D materials and van der Waals heterostructures. Science 10959203353(6298). https://doi.org/10.1126/ science.aac9439. Nuckolls KP, Myungchul O, Wong D, Lian B, Watanabe K, Takashi Taniguchi B, Bernevig A, and Yazdani A (2020) Strongly correlated chern insulators in magic-angle twisted bilayer graphene. Nature 588(7839): 610–615. https://doi.org/10.1038/s41586-020-3028-8. Okamoto S, Mohanta N, Dagotto E, and Sheng DN (2022) Topological flat bands in a kagome lattice multiorbital system. Communications Physics 5(1): 198. https://doi.org/10.1038/ s42005-022-00969-1. Ozawa T and Mera B (2021) Relations between topology and the quantum metric for Chern insulators. Physical Review B 104(4): 045103. https://doi.org/10.1103/ PhysRevB.104.045103. Palm FA, Buser M, Léonard J, Aidelsburger M, Schollwöck U, and Grusdt F (2021) Bosonic pfaffian state in the hofstadter-bose-hubbard model. Physical Review B 103: L161101. https://doi.org/10.1103/PhysRevB.103.L161101. Palmer RN and Jaksch D (2006) High-field fractional quantum hall effect in optical lattices. Physical Review Letters 96: 180407. https://doi.org/10.1103/PhysRevLett.96.180407. Pan H, Wu F, and Sarma SD (2020) Band topology, hubbard model, heisenberg model, and dzyaloshinskii-moriya interaction in twisted bilayer wse2. Physical Review Research 2: 033087. https://doi.org/10.1103/PhysRevResearch.2.033087. Pan H, Xie M, Wu F, and Sarma SD (2021) Topological Phases in ab-Stacked mote2/wse2: Z2 Topological Insulators, Chern Insulators, and Topological Charge Density Waves. https:// doi.org/10.48550/ARXIV.2111.01152. Parameswaran SA, Roy R, and Sondhi SL (2012) Fractional chern insulators and the W1 algebra. Physical Review B 85: 241308. https://doi.org/10.1103/PhysRevB.85.241308. Parameswaran SA, Roy R, and Sondhi SL (2013) Fractional quantum hall physics in topological flat bands. Comptes Rendus Physique 1631-070514(9): 816–839. https://doi.org/ 10.1016/j.crhy.2013.04.003. Topological insulators/Isolants topologiques. Park Y, Chittari BL, and Jung J (2020) Gate-tunable topological flat bands in twisted monolayer-bilayer graphene. Physical Review B 102: 035411. https://doi.org/10.1103/ PhysRevB.102.035411. Park JM, Cao Y, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2021) Flavour hund’s coupling, chern gaps and charge diffusivity in moirégraphene. Nature 592(7852): 43–48. https://doi.org/10.1038/s41586-021-03366-w. Parker D, Ledwith P, Khalaf E, Soejima T, Hauschild J, Xie Y, Pierce A, Zaletel MP, Yacoby A, and Vishwanath A (2021) Field-Tuned and Zero-Field Fractional Chern Insulators in Magic Angle Graphene. https://doi.org/10.48550/ARXIV. 2112.13837. Pierce AT, Xie Y, Park JM, Khalaf E, Lee SH, Cao Y, Parker DE, Forrester PR, Chen S, Watanabe K, Taniguchi T, Vishwanath A, Jarillo-Herrero P, and Yacoby A (2021) Unconventional sequence of correlated chern insulators in magic-angle twisted bilayer graphene. Nature Physics 17(11): 1210–1215. https://doi.org/10.1038/s41567-021-01347-4. Po HC, Zou L, Senthil T, and Vishwanath A (2019) Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Physical Review B 99: 195455. https://doi.org/ 10.1103/PhysRevB.99.195455. Polshyn H, Zhang Y, Kumar MA, Soejima T, Ledwith P, Watanabe K, Taniguchi T, Vishwanath A, Zaletel MP, and Young AF (2022) Topological charge density waves at half-integer filling of a moirésuperlattice. Nature Physics 18(1): 42–47. https://doi.org/10.1038/s41567-021-01418-6. Popov AM, Lebedeva IV, Knizhnik AA, Lozovik YE, and Potapkin BV (2011) Commensurate-incommensurate phase transition in bilayer graphene. Physical Review B 1098012184(4): 1–6. https://doi.org/10.1103/PhysRevB.84.045404. Popp M, Paredes B, and Cirac JI (2004) Adiabatic path to fractional quantum hall states of a few bosonic atoms. Physical Review A 70: 053612. https://doi.org/10.1103/ PhysRevA.70.053612. Qi X-L (2011) Generic wave-function description of fractional quantum anomalous hall states and fractional topological insulators. Physical Review Letters 107: 126803. https://doi. org/10.1103/PhysRevLett.107.126803. Rademaker L, Protopopov IV, and Abanin DA (2020) Topological flat bands and correlated states in twisted monolayer-bilayer graphene. Physical Review Research 2: 033150. https:// doi.org/10.1103/PhysRevResearch.2.033150. Regnault N and Andrei Bernevig B (2011) Fractional chern insulator. Physical Review X 1: 021014. https://doi.org/10.1103/PhysRevX.1.021014. Repellin C and Goldman N (2019) Detecting fractional chern insulators through circular dichroism. Physical Review Letters 122: 166801. https://doi.org/10.1103/ PhysRevLett.122.166801. Repellin C and Senthil T (2020) Chern bands of twisted bilayer graphene: Fractional chern insulators and spin phase transition. Physical Review Research 2: 023238. https://doi.org/ 10.1103/PhysRevResearch.2.023238. Repellin C, Andrei Bernevig B, and Regnault N (2014) Z2 fractional topological insulators in two dimensions. Physical Review B 90: 245401. https://doi.org/10.1103/ PhysRevB.90.245401.

536

Recent developments in fractional Chern insulators

Repellin C, Léonard J, and Goldman N (2020a) Fractional chern insulators of few bosons in a box: Hall plateaus from center-of-mass drifts and density profiles. Physical Review A 102: 063316. https://doi.org/10.1103/PhysRevA.102.063316. Repellin C, Dong Z, Zhang Y-H, and Senthil T (2020b) Ferromagnetism in narrow bands of moiré superlattices. Physical Review Letters 124: 187601. https://doi.org/10.1103/ PhysRevLett.124.187601. Rezayi EH, Haldane FDM, and Yang K (1999) Charge-density-wave ordering in half-filled high landau levels. Physical Review Letters 83: 1219–1222. https://doi.org/10.1103/ PhysRevLett.83.1219. Rodríguez ID and Nielsen AEB (2015) Continuum limit of lattice models with laughlin-like ground states containing quasiholes. Physical Review B 92: 125105. https://doi.org/10.1103/ PhysRevB.92.125105. Rodriguez-Vega M, Vogl M, and Fiete GA (2020) Floquet engineering of twisted double bilayer graphene. Physical Review Research 2: 033494. https://doi.org/10.1103/ PhysRevResearch.2.033494. Roy R (2014) Band geometry of fractional topological insulators. Physical Review B 90: 165139. https://doi.org/10.1103/PhysRevB.90.165139. Saito Y, Ge J, Watanabe K, Taniguchi T, and Young AF (2020) Independent superconductors and correlated insulators in twisted bilayer graphene. Nature Physics 16(9): 926–930. https://doi.org/10.1038/s41567-020-0928-3. Saito Y, Ge J, Rademaker L, Watanabe K, Taniguchi T, Abanin DA, and Young AF (2021) Hofstadter subband ferromagnetism and symmetry-broken chern insulators in twisted bilayer graphene. Nature Physics 17(4): 478–481. https://doi.org/10.1038/s41567-020-01129-4. Scaffidi T and Möller G (2012) Adiabatic continuation of fractional chern insulators to fractional quantum hall states. Physical Review Letters 109: 246805. https://doi.org/10.1103/ PhysRevLett.109.246805. Serlin M, Tschirhart CL, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, and Young AF (2020) Intrinsic quantized anomalous hall effect in a moiré heterostructure. Science 367(6480): 900–903. https://doi.org/10.1126/science.aay5533. Sheffer Y and Stern A (2021) Chiral magic-angle twisted bilayer graphene in a magnetic field: Landau level correspondence, exact wave functions, and fractional chern insulators. Physical Review B 104: L121405. https://doi.org/10.1103/PhysRevB.104.L121405. Sheng DN, Zheng-Cheng G, Sun K, and Sheng L (2011) Fractional quantum hall effect in the absence of landau levels. Nature Communications 2(1): 389. https://doi.org/10.1038/ ncomms1380. Simon SH, Harper F, and Read N (2015) Fractional chern insulators in bands with zero berry curvature. Physical Review B 92: 195104. https://doi.org/10.1103/PhysRevB.92.195104. Song Z, Wang Z, Shi W, Li G, Fang C, and Bernevig BA (2019) All magic angles in twisted bilayer graphene are topological. Physical Review Letters 123: 036401. https://doi.org/ 10.1103/PhysRevLett.123.036401. Sørensen AS, Demler E, and Lukin MD (2005) Fractional quantum hall states of atoms in optical lattices. Physical Review Letters 94: 086803. https://doi.org/10.1103/ PhysRevLett.94.086803. Spanton EM, Zibrov AA, Zhou H, Taniguchi T, Watanabe K, Zaletel MP, and Young AF (2018) Observation of fractional chern insulators in a van der waals heterostructure. Science 360(6384): 62–66. https://doi.org/10.1126/science.aan8458. Srivatsa NS, Li X, and Nielsen AEB (2021) Squeezing anyons for braiding on small lattices. Physical Review Research 3: 033044. https://doi.org/10.1103/ PhysRevResearch.3.033044. Stepanov P, Das I, Xiaobo L, Fahimniya A, KenjiWatanabe TT, Koppens FHL, Lischner J, Levitov L, and Efetov DK (2020) Untying the insulating and superconducting orders in magic-angle graphene. Nature 583(7816): 375–378. https://doi.org/10.1038/s41586-020-2459-6. Stepanov P, Xie M, Taniguchi T, Watanabe K, Xiaobo L, MacDonald AH, Andrei Bernevig B, and Efetov DK (2021) Competing zero-field chern insulators in superconducting twisted bilayer graphene. Physical Review Letters 127: 197701. https://doi.org/10.1103/PhysRevLett.127.197701. Sterdyniak A, Regnault N, and Bernevig BA (2011) Extracting excitations from model state entanglement. Physical Review Letters 106: 100405. https://doi.org/10.1103/ PhysRevLett.106.100405. Sterdyniak A, Regnault N, and Möller G (2012) Particle entanglement spectra for quantum hall states on lattices. Physical Review B 86: 165314. https://doi.org/10.1103/ PhysRevB.86.165314. Sterdyniak A, Repellin C, Andrei Bernevig B, and Regnault N (2013) Series of abelian and non-abelian states in c > 1 fractional chern insulators. Physical Review B 87: 205137. https://doi.org/10.1103/PhysRevB.87.205137. Sterdyniak A, Andrei Bernevig B, Cooper NR, and Regnault N (2015) Interacting bosons in topological optical flux lattices. Physical Review B 91: 035115. https://doi.org/10.1103/ PhysRevB.91.035115. Sticlet D and Piéchon F (2013) Distant-neighbor hopping in graphene and haldane models. Physical Review B 87: 115402. https://doi.org/10.1103/PhysRevB.87.115402. Streda P (1982) Theory of quantised hall conductivity in two dimensions. Journal of Physics C: Solid State Physics 15(22): L717–L721. https://doi.org/10.1088/0022-3719/15/ 22/005. Sun K, Zhengcheng G, Katsura H, and Sarma SD (2011) Nearly flatbands with nontrivial topology. Physical Review Letters 106: 236803. https://doi.org/10.1103/ PhysRevLett.106.236803. Tang E, Mei J-W, and Wen X-G (2011) High-temperature fractional quantum hall states. Physical Review Letters 106: 236802. https://doi.org/10.1103/PhysRevLett.106.236802. Tarnopolsky G, Kruchkov AJ, and Vishwanath A (2019) Origin of magic angles in twisted bilayer graphene. Physical Review Letters 122: 106405. https://doi.org/10.1103/ PhysRevLett.122.106405. Thouless DJ, Kohmoto M, Nightingale MP, and den Nijs M (1982) Quantized hall conductance in a two-dimensional periodic potential. Physical Review Letters 49: 405–408. https:// doi.org/10.1103/PhysRevLett.49.405. Topp GE, Jotzu G, McIver JW, Xian L, Rubio A, and Sentef MA (2019) Topological floquet engineering of twisted bilayer graphene. Physical Review Research 1: 023031. https://doi. org/10.1103/PhysRevResearch.1.023031. Topp GE, Eckhardt CJ, Kennes DM, Sentef MA, and Törmä P (2021) Light-matter coupling and quantum geometry in moiré materials. Physical Review B 104: 064306. https://doi.org/ 10.1103/PhysRevB.104.064306. Tran DT, Dauphin A, Grushin AG, Zoller P, and Goldman N (2017) Probing topology by heating: Quantized circular dichroism in ultracold atoms. Science Advances 3(8)e1701207. https://doi.org/10.1126/sciadv.1701207. Trescher M and Bergholtz EJ (2012) Flat bands with higher chern number in pyrochlore slabs. Physical Review B 86: 241111. https://doi.org/10.1103/PhysRevB.86.241111. Tsui DC, Stormer HL, and Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters 48: 1559–1562. https://doi.org/10.1103/ PhysRevLett.48.1559. Uchida K, Furuya S, Iwata J-I, and Oshiyama A (2014) Atomic corrugation and electron localization due to moiré patterns in twisted bilayer graphenes. Physical Review B 90: 155451. https://doi.org/10.1103/PhysRevB.90.155451. Umucalılar RO (2018) Real-space probe for lattice quasiholes. Physical Review A 98: 063629. https://doi.org/10.1103/PhysRevA.98.063629. van Wijk MM, Schuring A, Katsnelson MI, and Fasolino A (2015) Relaxation of moiré patterns for slightly misaligned identical lattices: Graphene on graphite. 2D Materials 2(3): 034010. https://doi.org/10.1088/2053-1583/2/3/034010. Varjas D, Abouelkomsan A, Yang K, and Bergholtz EJ (2022) Topological lattice models with constant berry curvature. SciPost Physics 12: 118. https://doi.org/10.21468/ SciPostPhys.12.4.118. Varney CN, Sun K, Rigol M, and Galitski V (2010) Interaction effects and quantum phase transitions in topological insulators. Physical Review B 82: 115125. https://doi.org/10.1103/ PhysRevB.82.115125.

Recent developments in fractional Chern insulators

537

Verlinde E (1988) Fusion rules and modular transformations in 2d conformal field theory. Nuclear Physics B 0550-3213300: 360–376. https://doi.org/10.1016/0550-3213(88) 90603-7. Vogl M, Rodriguez-Vega M, and Fiete GA (2020) Effective floquet hamiltonians for periodically driven twisted bilayer graphene. Physical Review B 101: 235411. https://doi.org/ 10.1103/PhysRevB.101.235411. Vogl M, Rodriguez-Vega M, Flebus B, MacDonald AH, and Fiete GA (2021) Floquet engineering of topological transitions in a twisted transition metal dichalcogenide homobilayer. Physical Review B 103: 014310. https://doi.org/10.1103/PhysRevB.103.014310. Wang J and Liu Z (2022) Hierarchy of ideal flatbands in chiral twisted multilayer graphene models. Physical Review Letters 128: 176403. https://doi.org/10.1103/ PhysRevLett.128.176403. Wang F and Ran Y (2011) Nearly flat band with chern number c ¼ 2 on the dice lattice. Physical Review B 84: 241103. https://doi.org/10.1103/PhysRevB.84.241103. Wang Y-F, Gu Z-C, Gong C-D, and Sheng DN (2011) Fractional quantum hall effect of hard-core bosons in topological flat bands. Physical Review Letters 107: 146803. https://doi.org/ 10.1103/PhysRevLett.107.146803. Wang Y-F, Yao H, Gong C-D, and Sheng DN (2012a) Fractional quantum hall effect in topological flat bands with chern number two. Physical Review B 86: 201101. https://doi.org/ 10.1103/PhysRevB.86.201101. Wang Y-F, Yao H, Zheng-Cheng G, Gong C-D, and Sheng DN (2012b) Non-abelian quantum hall effect in topological flat bands. Physical Review Letters 108: 126805. https://doi.org/ 10.1103/PhysRevLett.108.126805. Wang D, Liu Z, Cao J, and Fan H (2013) Tunable band topology reflected by fractional quantum hall states in two-dimensional lattices. Physical Review Letters 111: 186804. https:// doi.org/10.1103/PhysRevLett.111.186804. Wang J, Cano J, Millis AJ, Liu Z, and Yang B (2021) Exact landau level description of geometry and interaction in a flatband. Physical Review Letters 127: 246403. https://doi.org/ 10.1103/PhysRevLett.127.246403. Wang B, Dong X-Y, and Eckardt A (2022) Measurable signatures of bosonic fractional Chern insulator states and their fractional excitations in a quantum-gas microscope. SciPost Physics 12: 95. https://doi.org/10.21468/SciPostPhys.12.3.095. Weber S, Bai R, Makki N, Mögerle J, Lahaye T, Browaeys A, Daghofer M, Lang N, and Büchler HP (2022) Experimentally accessible scheme for a fractional Chern insulator in Rydberg atoms. arXiv. 2202.00699. https://doi.org/10.48550/ARXIV.2202.00699. Wen XG (1990) Topological orders in rigid states. International Journal of Modern Physics B 04(02): 239–271. https://doi.org/10.1142/S0217979290000139. Wen XG and Niu Q (1990) Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus Riemann surfaces. Physical Review B 41: 9377–9396. https://doi.org/10.1103/PhysRevB.41.9377. Wilhelm P, Lang TC, and Läuchli AM (2021) Interplay of fractional chern insulator and charge density wave phases in twisted bilayer graphene. Physical Review B 103: 125406. https://doi.org/10.1103/PhysRevB.103.125406. Wu Y-L, Bernevig BA, and Regnault N (2012a) Zoology of fractional chern insulators. Physical Review B 85: 075116. https://doi.org/10.1103/PhysRevB.85.075116. Wu Y-L, Regnault N, and Andrei Bernevig B (2012b) Gauge-fixed wannier wave functions for fractional topological insulators. Physical Review B 86: 085129. https://doi.org/10.1103/ PhysRevB.86.085129. Wu Y-L, Regnault N, and Bernevig BA (2013) Bloch model wave functions and pseudopotentials for all fractional chern insulators. Physical Review Letters 110: 106802. https://doi. org/10.1103/PhysRevLett.110.106802. Wu Y-L, Regnault N, and Bernevig BA (2014) Haldane statistics for fractional chern insulators with an arbitrary chern number. Physical Review B 89: 155113. https://doi.org/10.1103/ PhysRevB.89.155113. Wu Y-H, Jain JK, and Sun K (2015) Fractional topological phases in generalized hofstadter bands with arbitrary chern numbers. Physical Review B 91: 041119. https://doi.org/ 10.1103/PhysRevB.91.041119. Wu F, Lovorn T, Tutuc E, Martin I, and MacDonald AH (2019) Topological insulators in twisted transition metal dichalcogenide homobilayers. Physical Review Letters 122: 086402. https://doi.org/10.1103/PhysRevLett.122.086402. Wu S, Zhenyuan Zhang K, Watanabe TT, and Andrei EY (2021) Chern insulators, van hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nature Materials 20(4): 488–494. https://doi.org/10.1038/s41563-020-00911-2. Xie Y, Lian B, Jäck B, Liu X, Chiu C-L, Watanabe K, Takashi Taniguchi B, Bernevig A, and Yazdani A (2019) Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene. Nature 572(7767): 101–105. https://doi.org/10.1038/s41586-019-1422-x. Xie F, Cowsik A, Song Z-D, Lian B, Bernevig BA, and Regnault N (2021a) Twisted bilayer graphene. vi. an exact diagonalization study at nonzero integer filling. Physical Review B 103: 205416. https://doi.org/10.1103/PhysRevB.103.205416. Xie Y, Pierce AT, Park JM, Parker DE, Khalaf E, Ledwith P, Cao Y, Lee SH, Chen S, Forrester PR, Watanabe K, Taniguchi T, Vishwanath A, Jarillo-Herrero P, and Yacoby A (2021b) Fractional chern insulators in magic-angle twisted bilayer graphene. Nature 600(7889): 439–443. https://doi.org/10.1038/s41586-021-04002-3. Yang K, Haldane FDM, and Rezayi EH (2001) Wigner crystals in the lowest landau level at low-filling factors. Physical Review B 64: 081301. https://doi.org/10.1103/ PhysRevB.64.081301. Yang S, Zheng-Cheng G, Sun K, and Sarma SD (2012) Topological flat band models with arbitrary chern numbers. Physical Review B 86: 241112. https://doi.org/10.1103/ PhysRevB.86.241112. Yao NY, Laumann CR, Gorshkov AV, Bennett SD, Demler E, Zoller P, and Lukin MD (2012) Topological flat bands from dipolar spin systems. Physical Review Letters 109: 266804. https://doi.org/10.1103/PhysRevLett.109.266804. Yoshioka D and Lee PA (1983) Ground-state energy of a two-dimensional charge-density-wave state in a strong magnetic field. Physical Review B 27: 4986–4996. https://doi.org/ 10.1103/PhysRevB.27.4986. Zaletel MP, Mong RSK, and Pollmann F (2014) Flux insertion, entanglement, and quantized responses. Journal of Statistical Mechanics: Theory and Experiment 2014(10): P10007. https://doi.org/10.1088/1742-5468/2014/10/p10007. Zeng T-S (2021) Fractional quantum hall effect of bose-fermi mixtures. Physical Review B 103: L201118. https://doi.org/10.1103/PhysRevB.103.L201118. Zeng T-S and Sheng DN (2018) SU(n) fractional quantum hall effect in topological flat bands. Physical Review B 97: 035151. https://doi.org/10.1103/PhysRevB.97.035151. Zeng T-S and Zhu W (2022) Chern-number matrix of the non-abelian spin-singlet fractional quantum hall effect. Physical Review B 105: 125128. https://doi.org/10.1103/ PhysRevB.105.125128. Zeng T-S, Sheng DN, and Zhu W (2019) Topological characterization of hierarchical fractional quantum hall effects in topological flat bands with su(n) symmetry. Physical Review B 100: 075106. https://doi.org/10.1103/PhysRevB.100.075106. Zeng T-S, Liangdong H, and Zhu W (2022) Bosonic halperin (441) fractional quantum hall effect at filling factor n ¼ 2/5. Chinese Physics Letters 39(1)017301. https://doi.org/ 10.1088/0256-307x/39/1/017301. Zhang F, Jung J, Fiete GA, Niu Q, and MacDonald AH (2011) Spontaneous quantum hall states in chirally stacked few-layer graphene systems. Physical Review Letters 106: 156801. https://doi.org/10.1103/PhysRevLett.106.156801. Zhang Y, Grover T, Turner A, Oshikawa M, and Vishwanath A (2012) Quasiparticle statistics and braiding from ground-state entanglement. Physical Review B 85: 235151. https://doi. org/10.1103/PhysRevB.85.235151. Zhang Y-H, Mao D, Cao Y, Jarillo-Herrero P, and Senthil T (2019a) Nearly flat chern bands in moiré superlattices. Physical Review B 99: 075127. https://doi.org/10.1103/ PhysRevB.99.075127. Zhang Y-H, Mao D, and Senthil T (2019b) Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous hall effect and a lattice model. Physical Review Research 1: 033126. https://doi.org/10.1103/PhysRevResearch.1.033126.

538

Recent developments in fractional Chern insulators

Zhang Y, Devakul T, and Liang F (2021) Spin-textured chern bands in ab-stacked transition metal dichalcogenide bilayers. Proceedings of the National Academy of Sciences 118(36): e2112673118. https://doi.org/10.1073/pnas.2112673118. Zhou BT, Egan S, and Franz M (2022) Moiré flat chern bands and correlated quantum anomalous hall states generated by spinorbit couplings in twisted homobilayer mos2. Physical Review Research 4: L012032. https://doi.org/10.1103/PhysRevResearch.4.L012032. Zhu W, Sheng DN, and Haldane FDM (2013) Minimal entangled states and modular matrix for fractional quantum hall effect in topological flat bands. Physical Review B 88: 035122. https://doi.org/10.1103/PhysRevB.88.035122. Zhu W, Gong SS, Haldane FDM, and Sheng DN (2014) Identifying non-abelian topological order through minimal entangled states. Physical Review Letters 112: 096803. https://doi. org/10.1103/PhysRevLett.112.096803. Zhu W, Gong SS, and Sheng DN (2016) Interaction-driven fractional quantum hall state of hard-core bosons on kagome lattice at one-third filling. Physical Review B 94: 035129. https://doi.org/10.1103/PhysRevB.94.035129. Zondiner U, Rozen A, Rodan-Legrain D, Cao Y, Queiroz R, Taniguchi T, Watanabe K, Oreg Y, von Oppen F, Ady Stern E, Berg PJ-H, and Ilani S (2020) Cascade of phase transitions and dirac revivals in magic-angle graphene. Nature 582(7811): 203–208. https://doi.org/10.1038/s41586-020-2373-y.

Quantum Hall states in higher Landau levels Jakob Yngvason, Fakultät für Physik, Universität Wien, Vienna, Austria; Erwin Schrödinger Institute for Mathematics and Physics, Vienna, Austria © 2024 Elsevier Ltd. All rights reserved.

Introduction The magnetic Hamiltonian and its Eigenfunctions The Landau levels The two oscillators Complex notation Eigenfunctions Factorization The lowest Landau level as a Bargmann space of holomorphic functions Unitary maps between Landau levels Many body states and ℓ-particle densities Mapping Hamiltonians to the LLL Another proof of Coherent state representations Coherent states Integral kernels Recap of the different expressions for the unitary maps Some special states Laughlin states Filling factor n ¼ 5/2 Conclusion Acknowledgments References

540 540 540 541 542 543 543 543 544 545 546 546 547 547 548 548 549 549 550 551 551 551

Abstract The unitary correspondence between Quantum Hall states in higher Landau levels and states in the lowest Landau level is discussed together with the resulting transformation formulas for particle densities and interaction potentials. This correspondence leads in particular to a representation of states in arbitrary Landau levels in terms of holomorphic functions. Some important special Quantum Hall states in higher Landau levels are also briefly discussed.

Key points

• • • • •

Describe the emergence of Landau levels from for the quantized two-dimensional motion of charged particles in a perpendicular magnetic field. Introduce non-commutative guiding center and cyclotron variables and the associated two commuting harmonic oscillators. Discuss the description of states in higher Landau levels in terms of holomorphic wave functions in the lowest Landau level. Derive formulas that express particle densities and interaction potentials in an arbitrary Landau level in terms of corresponding quantities in the lowest Landau level. Discuss briefly Laughlin states in higher Landau levels and recent investigations of states with filling factor 5/2.

Notations

ℓB Magnetic length LLL Lowest Landau level (single particle) LLLN Lowest Landau level (N particles) nLL n-th Landau level (single particle) nLLN n-th Landau level (N particles) P0 Projector on LLL Pn Projector on nLL

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00062-7

539

540

Quantum Hall states in higher Landau levels

Introduction The quantum states of charged particles moving in a plane orthogonal to a homogeneous magnetic field are naturally grouped into Landau levels which correspond to quantization of the cyclotron motion of the particles in the magnetic field. The interplay of this motion with external electric fields and impurities as well as Coulomb interactions between the particles is the subject of Quantum Hall Physics which has developed into a major subfield of condensed matter physics since the discovery of the integer and fractional Quantum Hall Effects in Klitzing et al. (1980) and Störmer et al. (1982) respectively. These developments are extensively discussed in many monographs and review articles, see, e.g., (Chakraborty and Pietiläinen, 1988; Chakraborty and Pietiläinen, 1995; Goerbig, 2009; Fradkin, 2013; Laughlin, 1987, 1999; Prange and Girvin, 1987; Störmer et al., 1999; Goerbig and Lederer, 2006; Greiter, 2011; Jain, 2007; Tong, 2016). The present chapter focuses on one particular theoretical aspect, namely the relations between some basic properties of Quantum Hall states in different Landau levels. In strong magnetic fields the lowest Landau level, corresponding to the smallest of the discrete values for the cyclotron radius, plays a special role. The fractional quantum Hall effect was first discovered in strongly correlated states within this level and the celebrated Laughlin wave function (Laughlin, 1983), describing a quantum liquid at filling factor 1/3 (more generally 1/q with small odd values of q), turned out to be very successful in describing ground state properties as well as excitations. An important mathematical feature of wave functions in the lowest Landau level is the fact that they form a Bargmann space of holomorphic functions when the position coordinates are expressed as complex numbers in the symmetric gauge. The holomorphy has been used for derivations of some basic properties of the states, for examples bounds on the local particle density (Lieb et al., 2018; Lieb et al., 2019). Wave functions in higher Landau levels, on the other hand, involve also powers of the complex conjugate position variables in the standard representation and at first sight it might appear that the advantage of holomorphic representations gets lost. It was, however, noted early (MacDonald, 1984) that, due to a unitary correspondence between states in different Landau levels, wave functions in higher Landau levels can also be represented by holomorphic functions. In particular, there are Laughlin states in any Landau level. The physical picture behind this correspondence is based on the fact that the position variable has a natural decomposition into two sets of non-commutative variables, one associated with the cyclotron motion and the other with the guiding centers around which the cyclotron motion takes place. Mathematically, these correspond to two sets of mutually commuting harmonic oscillators. For a given Landau level, the cyclotron motion can be thought of as fixed, while the physics takes place in the guiding center degrees of freedom where holomorphic wave functions appear naturally. In this chapter the mathematical implementation of this idea is presented in two versions, one employing ladder operators and another one based on coherent states. Each brings different aspects of the unitary correspondence between states in higher Landau levels with those in the lowest Landau level into focus. Furthermore we derive formulas that relate effective potentials in higher Landau levels to their counterparts in the lowest Landau level. In the last section we discuss some important special states in higher Landau levels and their relation to states in the lowest Landau level.

The magnetic Hamiltonian and its Eigenfunctions The Landau levels The magnetic Hamiltonian of a single spinless (or spin-polarized) particle of charge q and effective mass m∗, moving in a plane with position variables r ¼ (x, y), is
 1 (1) H¼ p2x + p2y  2m where   p ¼ px , py ¼ p − qA

(2)

  p ¼ −iℏ ∂x , ∂y

(3)

is the gauge invariant kinetic momentum with

the canonical momentum and A the magnetic vector potential. We assume a homogeneous magnetic field of strength B perpendicular to the plane and choose the symmetric gauge A¼

B ð −y, xÞ: 2

(4)

Moreover, we choose units and signs such that |q| ¼ 1, qB  B > 0, ℏ ¼ 1 and m ¼ 1. Then px ¼ −i∂x +

1 By, 2

1 py ¼ −i∂y − Bx 2

(5)

Quantum Hall states in higher Landau levels and the kinetic momentum components satisfy the canonical commutation relations (CCR)   px , py ¼ iℓB−2

541

(6)

with ℓB ¼ B −1=2

(7)

the magnetic length. In terms of the creation and annihilation operators  ℓ  a{ ¼ pBffiffiffi −py −ipx , 2

 ℓ  a ¼ pBffiffiffi −py + ipx 2

(8)

with [a, a{] ¼ 1 one can write (1) as
 1 H ¼ 2B a{ a + 2 which is the Hamiltonian of a harmonic oscillator with eigenvalues
 1 en ¼ n + 2B, n ¼ 0, 1, . . . : 2

(9)

(10)

The Gaussian 2 2 2 1 ’0 ðrÞ ¼ pffiffiffi e−ðx +y Þ=4ℓB p

(11)

with a’0 ¼ 0 is a ground state for H with energy e0 ¼ B. The discrete energy values (10) result from the quantization of the classical cyclotron motion of a charged particle around guiding centers as discussed below. Every energy eigenvalue is infinitely degenerate due to the different possible positions of the guiding centers. The degeneracy per unit area is (2pℓ2B)−1. The eigenspace in L2(ℝ2) to the eigenvalue en is called the n-th Landau level and denoted by nLL. The lowest Landau level, corresponding to n ¼ 0 will be denoted by LLL. The corresponding fermionic spaces for N electrons, denoted by nLLN and LLLN respectively, are the antisymmetric tensor powers LLLN ¼ LLLa N ,

nLLN ¼ nLLa N :

(12)

For bosons the antisymmetric tensor product is replaced by the symmetric one; this is e.g. relevant for cold atomic gases in rapid rotation, where the rotational velocity takes over the role of the magnetic vector potential (Cooper, 2008). The filling factor v in a given Landau level is by definition the ratio of the particle density in that level to the degeneracy (2pℓ2B)−1 per unit area. In a sample with area A a completely filled Landau level of fermions (filling factor n ¼ 1) thus contains N ¼ (2pℓ2B)−1A particles. Remark. In the whole chapter, with the exception of the last Section 8.2, we focus on the orbital motion of the charged particles, P ignoring spin and the accompanying Zeeman energy i g∗mBB  Si (with Si the spin operator for particle i, B the magnetic field, mB the Bohr magneton and g∗ the effective Landé g-factor). This is justified if the particles are either spinless, or their spin is fully polarized due to a strong magnetic field. Moreover, in semiconductors the Zeeman energy is typically much smaller than the gap between Landau levels because the effective g-factor is small. See, e.g., (Tong, 2016, p. 16).

The two oscillators Quantum mechanically the dynamics of the guiding centers is described by another harmonic oscillator commuting with the cyclotron oscillator (1). While the latter determines the Landau energy spectrum (10), the quantization of the guiding center oscillator parametrizes the degeneracy creating a natural basis of states in each Landau level. One arrives at this picture by splitting the (gauge invariant) position operator r into a guiding center part R and the cyclotron part ~ ¼ ℓ2 n  p, R B

(13)

~ are gauge invariant and they commute with each other. On the other hand with n the unit normal vector to the plane. Both R and R ~ More precisely, we have the two components (Rx, Ry) of R do not commute and likewise for the components of R. ~ r¼R+R

(14)

with 1 Rx ¼ x + ℓ2B py ¼ x − iℓ2B ∂y , 2

1 Ry ¼ y − ℓ2B px ¼ y + iℓ2B ∂x , 2

(15)

542

Quantum Hall states in higher Landau levels 1 R~x ¼ −ℓ2B py ¼ x + iℓ2B ∂y , 2

1 R~y ¼ ℓ2B px ¼ y − iℓ2B ∂x 2

(16)

and the commutation relations   ~ ¼ 0, R, R

  Rx , Ry ¼ −iℓ2B ,



 R~x , R~y ¼ iℓ2B :

(17)

~ are the same as (8), namely The creation and annihilation operators for R  1  a{ ¼ pffiffiffi R~x − iR~y , 2ℓB

 1  a ¼ pffiffiffi R~x + iR~y : 2 ℓB

(18)

 1  b ¼ pffiffiffi Rx − iRy : 2ℓB

(19)

Those for the guiding centers, on the other hand, are  1  b{ ¼ pffiffiffi Rx + iRy , 2ℓB

with [b, b{] ¼ 1 and [a#, b#] ¼ 0. Note the different signs compared to (18) due to the different signs in (17).

Complex notation The two-dimensional configuration space ℝ2 can be identified with the complex plane ℂ. Defining complex coordinates and derivatives by z ¼ x + iy,

z ¼ x − iy,

∂z ¼

 1 ∂ − i∂y , 2 x

∂z ¼

 1 ∂ + i∂y 2 x

(20)

we can write
 1 1 z − 2ℓ2B ∂z , a{ ¼ pffiffiffi 2 2ℓB


 1 1 a ¼ pffiffiffi z + 2ℓ2B ∂z : 2 2ℓB

(21)

Choosing units so that B ¼ 2, or equivalently, defining z ¼ p1ffiffi2ℓ ðx + iyÞ, this simplifies to B

1 a ¼ z − ∂z , 2 {

1 a ¼ z + ∂z : 2

(22)

Also, the gaussian factor e−(|x| +|y| )/4ℓB becomes e−|z| /2. We note that besides the standard definition z ¼ x + iy, other complexifications of ℝ2 are possible and can be useful, as stressed in Haldane (2018). { a which act only on the pre-factors to the gaussian For computations it is often convenient to use instead of (22) the operators ab , b and are defined by h i h i 2 2 # a f ðz, zÞ e−jzj =2 : a# f ðz, zÞe−jzj =2 ¼ b (23) 2

2

2

2

These are { ab ¼ z − ∂z ,

ab ¼ ∂z :

(24)

In the sequel we shall generally use the hat ^on operators and functions to indicate that gaussian normalization factors are dropped from the notation. The creation and annihilation operators for the guiding center oscillator are in complex notation

 1 1 1 1 (25) b{ ¼ pffiffiffi z + 2ℓ2B ∂z : z − 2ℓ2B ∂z , b ¼ pffiffiffi 2 2 2ℓB 2ℓB For B ¼ 2 1 b { ¼ z − ∂z , 2

1 b ¼ z + ∂z 2

(26)

and { bb ¼ z − ∂z ,

bb ¼ ∂z :

(27)

The splitting (14) corresponds to     ðz, zÞ ¼ b{ , b + a, a{ : {

(28) {

While the operators a , a increases or decrease the Landau level index, the operators b , b leave each Landau level invariant. Pictorially speaking we can say that operators a# associated with the cyclotron oscillator are ladder operators moving states “vertically” on the energy scale, while the ladder operators b# associated with the guiding center oscillator move states “horizontally”, keeping the energy fixed.

Quantum Hall states in higher Landau levels

543

Eigenfunctions With the gaussian ’0,0 ≔ ’0 (11) as the common, normalized lowest energy state for the two oscillators the states 1  n  m 1  m  { n ’n,m ¼ pffiffiffiffiffiffiffiffiffi a{ b{ ’0,0 ¼ pffiffiffiffiffiffiffiffiffi b{ a ’0,0 , n!m! n!m!

n, m ¼ 0, 1, . . .

(29)

form a basis of eigenstates of the Hamiltonian (1). For fixed n the states ’n,m, m ¼ 0, 1, . . . generate the Hilbert space nLL of the n’th Landau level with n ¼ 0 corresponding to the lowest Landau level LLL. In complex coordinates the wave functions with n ¼ 0 respectively m ¼ 0, are 2 1 ’0,m ðrÞ ¼ pffiffiffiffiffiffiffiffi zm e−jzj =2 , pm!

2 1 ’n,0 ðrÞ ¼ pffiffiffiffiffiffiffi zn e−jzj =2 : pn!

(30)

More generally, one can write 2 2 1 1 ’n,m ðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ½ðz − ∂z Þm zn e−jzj =2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ½ðz − ∂z Þn zm e−jzj =2 : pn!m! pn!m!

(31)

These functions can be written in polar coordinates in terms of associated Laguerre polynomials (Landau and Lifschitz, 1977; Gradshteyn and Ryzhik, 1994). They are simultaneous eigenfunctions of the Hamiltonian (1) and the angular momentum operator L ¼ r  p in the symmetric gauge. Its action on the pre-factor of the gaussian is Lb ¼ z∂z − z∂z

(32)

{

with eigenvalues M ¼ m − n, m ¼ 0, 1, . . . in the nLL. The operators b and b shift the angular momentum within each Landau level.

Factorization The mutual commutativity of the two oscillators has an important consequence: Lemma 2.1: (Factorization). If A is a function of a{, a and B of b{, b, then





’n0 ,m0 jABj’n,m ¼ ’n0 ,0 jAj’n,0 ’0,m0 jBj’0,m :

(33)

The proof follows from the fact that the two commuting harmonic oscillators (8) and (19) can be represented, in a unitarily equivalent way, in the tensor product of two spaces with basis vectors ’n,0 and ’0,m respectively. In this representation ’n,m ’ ’n,0  ’0,m and the operators A and B act independently on each of the tensor factors. (Alternatively, one can choose A, B to be polynomials in the creation and annihilation operators and use the CCR to prove (33) directly.) Note, however, that in the representation (31) the functions ’n,m ðz, zÞ are not simply products of the functions ’n,0(z) and ’0,m ðzÞ. Indeed, the variables z and z do not act independently in the tensor factors because they involve sums of a#‘s and b#’‘s, cf. Eq. (28). The lemma, however, is in accord with the intuitive picture stressed in Haldane (2013) that in a given Landau level all the non-trivial structure is in the guiding center degrees of freedom while the cyclotron motion just produces a decoration by a fixed factor in matrix elements.

The lowest Landau level as a Bargmann space of holomorphic functions Wave functions c in the lowest Landau level LLL are characterized by the equation ac ¼ 0

(34)

which by (24) means that cðrÞ ¼ f ðzÞe−jzj

2

=2

(35)

with f satisfying the Cauchy-Riemann equation ∂z f ¼ 0 , i.e., f is a holomorphic function of z. Holomorphic functions that are 2 square integrable on ℝ2 with respect to the Lebesgue measure with the gaussian weight factor e−|z| form a Bargmann space (Bargmann, 1961). In the same way, N-particle fermionic wave functions in LLLN have the form Cðr1 . . ., rN Þ ¼ F ðz1 , . . . , zN Þe−ðjz1 j

2

+⋯+jzN j2Þ=2

(36)

with a holomorphic, antisymmetric function F of the N variables z1, . . ., zN. In the sequel we shall show that representations of states by holomorphic functions are not limited to the lowest Landau level.

544

Quantum Hall states in higher Landau levels

Unitary maps between Landau levels The structure of the eigenfunctions (29) with the mutually commuting sets of creation and annihilation operators a# and b# for the two oscillators leads immediately to a unitary correspondence nLL $ LLL. In fact, the operator U n ¼ ðn!Þ−1=2 an

restricted to nLL

(37)

maps ’n,m ↦ ’0,m because a commutes with b{, and the inverse  n U n−1 ¼ ðn!Þ−1=2 a{

restricted to LLL

(38)

takes ’0,m ↦ ’n,m. Hence Un is a unitary map nLL ! LL. The wave functions in nLL are by (31) polynomials in z of order n with coefficients that are holomorphic functions of z. Using the representations (24) for the creation and annihilation operators we conclude that the following holds: Proposition 3.1: (Unitary map between nLL and LLL) Let cn 2 nLL have wave function cn ðr Þ ¼

n X k¼0

2

zk f k ðzÞe−jzj

=2

,

(39)

with fk holomorphic for k ¼ 0, . . ., n. Then c0 ¼ Uncn 2 LLL has the wave-function c0 ðr Þ ¼

pffiffiffiffi 2 n!f n ðzÞe−jzj =2 :

(40)

Conversely, the wave function of cn ¼ U−1 n c0 2 nLL is 2

cn ðrÞ ¼ ½ðz − ∂zÞn f n ðzÞe−jzj =2 " n X ¼ zn f n ð zÞ + ð −1Þk k¼1

n k

!

# n−k k

∂ f n ðzÞ e−jzj

z

2

=2

:

(41)

Note that Eq. (41) implies in particular that the holomorphic factor fn(z) to the highest power n of z determines uniquely the factors to the lower powers zk :  n ðn−kÞ f k ðzÞ ¼ ð −1Þn−k f ðzÞ: (42) k n A wave function in nLL is thus completely fixed by the holomorphic function fn and the Landau level index n. These considerations lead also to a formula for projecting functions into the lowest Landau level: Proposition 3.2: (LLL projection). Let f be a state in nk¼0kLL with wave function fðrÞ ¼

n X k¼0

2

zk gk ðzÞe−jzj

=2

(43)

with holomorphic functions gk. It’s orthogonal projection into LLL is P0 fðzÞ ¼

n X k¼0

∂k gk ðzÞe−jzj

2

=2

:

(44)

This is known as the recipe “move all z factors to the left and replace them by derivatives in z”, see e.g. (Jain, 2007). Proof. The previous considerations lead by induction to a splitting of the state (43) into its components in the different kLL, k ¼ 0, . . ., n: Start with a wave function as in (43). It’s component cn in the nLL is given by (41) with fn ≔ gn and fk for ~ ¼ f −c is now in n−1 0 k n − 1 defined by (42). The difference f k¼0kLL and we can repeat the procedure with n replaced by n −1, n  n P n ~ etc. until we obtain the splitting f ¼ k¼0ck with ck 2 kLL. By induction over n, using that Pn ð −1Þk ¼ ð1 − 1Þn ¼ 0, f by f k¼0 k this procedure implies (44).

Quantum Hall states in higher Landau levels

545

Many body states and ℓ-particle densities The considerations of the previous sections carry in a straightforward manner over to many-body states in symmetric or anti-symmetric tensor powers nLLN  nLLs,aN of single-particle states by applying the single-particle formulas to each tensor factor. Let Cn be a state in nLLN with wave function b n ðr1 , . . . , rN Þe−ðjz1 j Cn ð r 1 , . . . , r N Þ ¼ C

2

+⋯+jzN j2 Þ=2

:

(45)

zki i f k1 ,...,kN ðz1 , . . . , zN Þ:

(46)

Expanding in powers of zi we can write b n ðr1 , . . . , rN Þ ¼ C

N Y i¼1

zni f n ðz1 , . . . , zN Þ +

N XY i¼1

The sum is here over N-tuples (k1, . . .kN) such that ki < n for at least one i. The functions fn and fk1, . . ., kN are holomorphic and the latter ones are, in fact, derivatives of fn, cf. (42). Consider now the state C0 ¼ U N,n Cn in LLLN where U N,n ¼ U n ⋯U n

with U n defined by ð37Þ:

(47)

Its wave function is b 0 ðz1 , . . . , zN Þe−ðjz1 j C0 ð r 1 , . . . , r N Þ ¼ C

2

+⋯+jzN j2 Þ=2

(48)

with the holomorphic function b 0 ðz1 , . . . , zN Þ ¼ ðn!Þ−N=2 C

N Y i¼1

b n ðz1 , z1 ; . . . ; zN , zN Þ ¼ ðn!ÞN=2 f ðz1 , . . . , zN Þ: ∂nzi C n

(49)

−1 b n can by (38) and (24) be expressed in terms of the holomorphic function fn: C0 so the wave function C Now Cn ¼ U N,n

b n ðz1 , z1 ; . . . ; zN , zN Þ ¼ ðn!Þ−N=2 C

N Y i¼1

b 0 ðz1 , . . . , zN Þ ¼ ðzi − ∂zi Þn C

N Y i¼1

ðzi − ∂zi Þn f n ðz1 , . . . , zN Þ:

Next we consider ℓ-particle densities, ℓ ¼ 1, 2, . . ., which for a general N-particle wave function C are defined by  Z   N ðℓÞ rC ðr1 , . . . , rℓ Þ ¼ jC r1 ⋯rℓ ; r0ℓ+1 ⋯r0N j2 dr0ℓ+1 ⋯dr0N : 2 ð N −ℓ Þ ℓ ℝ

(50)

(51)

Theorem 4.1: (Particle densities in the n-th Landau level) The ℓ-particle densities of Cn 2 nLLN and C0 ¼ U N,n Cn 2 LLLN are connected by ðℓÞ

rCn ðr1 , . . . , rℓ Þ ¼


 1 ðℓÞ Ln − Dri rC0 ðr1 , . . . , rℓ Þ 4 i¼1

ℓ Y

(52)

where Ln is the n-th Laguerre polynomial Ln ð t Þ ¼

n  X n ð −t Þl l¼0

l

l!

:

(53)

The proof is a simple consequence of the following Lemma: Lemma 4.2: (Reshuffling differentiations) Let f be a holomorphic function. Then h i ðn!Þ−1 ½ðz − ∂z Þn f ðzÞ½ðz − ∂z Þn f ðzÞe−zz ¼ Ln ð−∂z ∂z Þ f ðzÞf ðzÞe−zz :

(54)

Proof. This is a straightforward computation by induction over n, using the recursion relation for the Laguerre polynomials, ðn + 1ÞLn + 1 ðuÞ ¼ ð2n + 1ÞLn ðuÞ − nLn − 1 ðuÞ − uLn ðuÞ:

(55)

h i ∂z ∂z ½ðz − ∂z Þn f ðzÞ ½ðz − ∂z Þn f ðzÞe −zz ,

(56)

To compute

starting with n ¼ 0 and L0 ¼ 1, one uses the commutation relations

546

Quantum Hall states in higher Landau levels ∂z ðz − ∂z Þn ¼ ðz − ∂Þn ∂z + nðz − ∂z Þn − 1 ,

∂z ðz − ∂z Þn ¼ ðz − ∂Þn ∂z ,

(57)

and the fact that ∂z f ðzÞ ¼ ∂z f ðzÞ ¼ 0 for holomorphic f. Theorem 4.1 follows by applying the Lemma to the first ℓ variables of C0, noting that ∂z ∂z ¼ 14 D:

Mapping Hamiltonians to the LLL Consider an N-body Hamiltonian of the form HV,w ¼

N h X

i X   HðiÞ + V ðri Þ + w r i − rj

i¼1

(58)

i ln l and the other regime, called Diophantine, looks at numbers with b < ln l . Actually there is a special treatment to be done for b  ln l  2b. The treatment of this delicate regime will be skipped. This article is devoted to make the main ideas accessible to physicists.

The Diophantine regime While the Diophantine Regime is most technical and requires fine analysis, it has been considered since the earliest work by Dinaburg and Sinai (1975) on the Almost Mathieu equation. The strategy they used was inspired by the method of the KAM Theorem, proposed first by Kolmogorov in 1952 and then by Arnold and Moser in 1962–1963 to treat perturbation theory in Classical Mechanics. The main difficulty comes from what astronomers of the 19th century called secular terms and called today the small divisor problem. Such a situation is already contained in the solution of the cocycle equation (see Eq. 24). Indeed, to solve this equation, let g(x) be expanded in Fourier series Z 1 mX ¼ +1 mX ¼ +1  gðxÞ ¼ e2ıpmx g^m , bðxÞ ¼ e2ıpmx b^m , b^0 ¼ bðxÞdx ¼ b: m ¼ −1

0

m ¼ −1

Thus g^m ¼

b^m , e2ıpma −1

8m 6¼ 0:

If a is irrational, the Fourier coefficients of g are all well defined apart from g^0, which can be chosen arbitrarily. The problem comes from numbers m’s such that ma is very close to a rational number. For instance if m ¼ qn, one of the continued fraction rational approximations, then |qna − pn| 1/qn+1. Therefore, using the definition of b this gives je2ıpqn a − 1j  2pe −bqn , which can be very small, leading the Fourier series for g to diverge. To solve this problem, the trick is to use functions b(x) that can be analytically continued in the complex plane replacing x by z ¼ x + ıy with |y| < d for a suitable d > 0. Then this is possible provided jb^m j  const:e −djmj for all m. In this case as long as b < d, the cocycle equation can be solved with g analytic in |y| ¼ |Iz| < d − b. Why is this relevant? Because, as guessed by Aubry and André (1980), the decay of the wave functions solution of the dual equation (21) is given by the Lyapunov exponent g, namely, the behavior at infinity of the transfer matrix between the sites N and N + n as n ! 1. A beautiful argument from Herman (1983) using the properties of subharmonic function leads to  0 if 0  l  1 g¼ : ln l if l  1 Consequently, as proved by Sinai (see Puig, 2004) Theorem 5. If 0 < l < 1, the dual equation (21) admits a unique solution ’^ such that, for some C > 0 ^ + 1Þ2 + ’ðlÞ ^ 2 ’ðl  C e −2j ln ljjlj , 2 ^ 2 ^ + ’ð0Þ ’ð1Þ

jlj ! 1:

Equivalently the Almost Mathieu equation (20) admits a unique Floquet–Bloch solution that can be extended by analyticity to z ¼ x + ıy in the complex plane, in the strip defined by jIzj < j ln lj. Corollary 1. If 0 < l < 1, the cocycle equation (24) admits a solution if a satisfies bðaÞ < j ln lj, analytic in the strip jIzj < j ln lj −bðaÞ. In particular, all spectral gaps of the Almost Mathieu operator, but at E ¼ 0, are open. Proof. Indeed the function b(x) obtained in Eq. (23) is a function computable in terms of the Floquet–Bloch solution ’ and is analytic in the same trip as ’. From Theorem 5, this domain is the strip jIzj < j ln lj. From solving the cocycle equation, the function g is analytic in jIzj < j ln lj − bðaÞ. At last if E 6¼ 0 due to the uniqueness of the solution of the dual equation, the constant b cannot vanish, proving that all gaps, but E ¼ 0 are open for such as.

The Liouville regime If b(a) is large enough though, a different strategy can be used, which was initiated in the seminal paper of Choi et al. (1990). Indeed, the Hölder continuity of the gap edges can be used to estimate the gap size near a rational approximation of a to prove that the gap stay open. The main difference brought by Avila and Jitomirskaya is the use of the improved Hölder estimate proved at about the same time by Avron et al. (1991). This improved estimate permits to extend the arguments to irrational numbers with bðaÞ > 2j ln lj.

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The argument (with many details skipped) goes as follows: let ga ¼ ðE − ðaÞ, E+(a)) be a spectral gap for the Almost Mathieu pffiffiffiffiffiffiffiffiffiffiffiffiffiffi operator. Thanks to Avron et al. (1991), there is a constant CH > 0 such that jEs ðaÞ −Es ða0 Þj  CH ja − a0 j for s ¼ 1. If pn/qn is a rational approximation of a obtained from the continued fraction expansion, it follows that ja −pn =qn j  c1 e −bqn if b > b(a) and if n large enough while c1 is some positive constant. Therefore 1=2        Es ðaÞ − Es pn   CH a − pn   c1 ebqn =2 :    qn qn  It follows that for the gap width this leads to the following inequality 

  p p  jE − ðaÞ − E+ ðaÞj  E − n − E+ n  − 2c1 e−bqn =2 : qn qn Thanks to Choi et al. (1990), the gap width for pn/qn is bounded from below by 

   E − pn − E+ pn   c0 ðlÞðc1 lÞqn =2 , 0 < c1 < 1:  qn qn  where c1 > 0 is some numerical constant (see Eq. 17 valid for qn odd, for instance). As n ! 1, one can choose b sufficiently close to b(a). From this it follows that if bðaÞ > j ln lj + j ln Dj, the gap ga is open. So, there might still be a gap in the range of values of b(a) for which the sketch of the proof presented above might fail to show that the gaps stay open. The paper by Avila and Jitomirskaya fills the gap in between.

Conclusion This article is aimed at explaining to an audience of nonexpert scientists the strategy required to solve the Ten Martini Problem. Nevertheless entering into some technical details is unavoidable. Many contributions were necessary, which spread over a quarter of a century to clarify each point until a synthesis of all the arguments could lead to the solution. Several contributions were appreciated as major steps, so much so as to be published by the highest rated Mathematical or Mathematical Physics journals.

References Aubry S (1978) The new concept of transitions by breaking of analyticity in a crystallographic model. In: Solitons and Condensed Matter Physics. Springer Series Solid-State Science, vol. 8. Berlin-New York: Springer-Verlag. xi+341 pp. ISBN: 3-540-09138-6. Aubry S and André G (1980) Analyticity breaking and Anderson localization in incommensurate lattices. Annals of the Israel Physical Society 3(133): 18. Avila A and Jitomirskaya S (2006) Solving the Ten Martini problem. In: Asch J and Joye A (eds.) Mathematical Physics of Quantum Mechanics. Lecture Notes in Physics, vol. 690, pp. 5–16. Springer, Berlin, Heidelberg. Avila A and Jitomirskaya S (2009) The ten Martini problem. Annals of Mathematics 170(1): 303–342. ISSN: 0003486X. https://doi.org/10.4007/annals.2009.170.303. Avron J and Simon B (1983) Almost periodic Schrödinger operators, II. The density of states. Duke Mathematical Journal 50: 369–391. Avron JE, van Mouche P, and Simon B (1991) On the measure of the spectrum for the Almost Mathieu operator. Communications in Mathematical Physics 139(1): 215. ISSN: 0010-3616. https://doi.org/10.1007/BF02102736. 215. Beckus S and Bellissard J (2016) Continuity of the spectrum of a field of self-adjoint operators. Annales Henri Poincaré 17(12): 3425–3442. ISSN: 1424-0637. https://doi.org/ 10.1007/s00023-016-0496-3. Bellissard J (1982) Schrödinger operators with almost periodic potential: An overview. In: Mathematical Problems in Theoretical Physics, pp. 356–363. Springer Berlin Heidelberg. Bellissard J (1994) Non commutative methods in semiclassical analysis. In: Transition to Chaos in Classical and Quantum Mechanics, pp. 1–64. Berlin: Springer. Bellissard J and Simon B (1982) Cantor spectrum for the almost Mathieu equation. Journal of Functional Analysis 48(3): 408–419. ISSN: 0022-1236. https://doi.org/10.1016/00221236(82)90094-5. Chambers WG (1965) Linear-network model for magnetic breakdown in two dimensions. Physical Review 140(1A): A135–A143. ISSN: 0031899X. https://doi.org/10.1103/ PhysRev.140.A135. Choi MD, Elliott GA, and Yui N (1990) Gauss polynomials and the rotation algebra. Inventiones Mathematicae 99(1): 225–246. ISSN: 00209910. https://doi.org/10.1007/ BF01234419. Dinaburg EI and Sinai YG (1975) The one-dimensional Schrodinger equation with quasi-periodic potential. Functional Analysis and Its Applications 9(4): 8–21. Eliasson LH (1992) Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Communications in Mathematical Physics 146(3): 447–482. ISSN: 0010-3616. https://doi.org/10.1007/BF02097013. Eliasson LH (1999) On the discrete one-dimensional quasi-periodic Schrödinger equation and other smooth quasi-periodic skew products. Hamiltonian Systems With Three or More Degrees of Freedom 1995: 55–61. https://doi.org/10.1007/978-94-011-4673-9_6. Elliot GA (1982) Gaps in the spectrum of an almost periodic Schrodinger operator. Comptes Rendus des Seances de l’Academie des Sciences, Serie A: Sciences Mathematiques Canada IV(5): 255–259. Hardy GH and Wright EM (2008) An Introduction to the Theory of Numbers, 6th. Clarendon: Oxford. https://www.ams.org/bull/1929-35-06/S0002-9904-1929-04793-1/. 1938. Harper PG (1955a) The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals. Proceedings of the Physical Society. Section A 68(10): 879. ISSN: 0370-1298. https://doi.org/10.1088/0370-1298/68/10/305. Harper PG (1955b) Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A 68(10): 874–878. ISSN: 0370-1298. https://doi.org/10.1088/0370-1298/68/10/304. Herman MR (1983) Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Commentarii Mathematici Helvetici 58(1): 453–502. ISSN: 0010-2571. https://doi.org/10.1007/BF02564647.

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Hofstadter DR (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14(6): 2239–2249. ISSN: 01631829. https:// doi.org/10.1103/PhysRevB.14.2239. Johnson R and Moser J (1983) The rotation number for almost periodic potentials. Communications in Mathematical Physics 90(2): 317–318. ISSN: 00103616. https://doi.org/ 10.1007/BF01205510. Last Y (2005) Spectral theory of Sturm-Liouville operators on infinite intervals: A review of recent developments. In: Sturm-Liouville Theory, pp. 99–120. Basel: Birkhäuser-Verlag. Puig J (2004) Cantor spectrum for the Almost Mathieu operator. Communications in Mathematical Physics 244(2): 297–309. ISSN: 00103616. https://doi.org/10.1007/s00220003-0977-3. Rammal R and Bellissard J (1990) An algebraic semi-classical approach to Bloch electrons in a magnetic field. Journal de Physique 51(17): 1803–1830. ISSN: 0302-0738. https:// doi.org/10.1051/jphys:0199000510170180300. Simon B (2000) Schrödinger operators in the twenty-first century. In: Mathematical Physics 2000, pp. 283–288. Published by Imperial College Press and Distributed by World Scientific Publishing Co. Zak J (1964a) Magnetic translation group. Physical Review 134(6A): A1602. ISSN: 0031899X. https://doi.org/10.1103/PhysRev.134.A1602. Zak J (1964b) Magnetic translation group. II. Irreducible representations. Physical Review 134(6A): A1607. ISSN: 0031899X. https://doi.org/10.1103/PhysRev.134.A1607.

Hofstadter butterfly in graphene Wei Yang and Guangyu Zhang, Institute of Physics, Chinese Academy of Sciences, Beijing, China © 2024 Elsevier Ltd. All rights reserved.

Introduction Graphene superlattices Hofstadter butterfly in graphene/hBN superlattice Hofstadter butterfly in twisted bilayer graphene Conclusion References

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Abstract This chapter reports recent progresses on the observation of Hofstadter butterfly in graphene, and organized as follows. We first briefly introduce the concept of Hofstadter butterfly, including the theoretical model of Bloch electrons in the magnetic field, the resulting Harper’s equation and its solution, and its realizations in non-graphene systems. Then, we discuss the ways to realize the fractal Hofstadter spectra in graphene, and introduce three types of graphene superlattice structure, including graphene/hBN, twisted graphene layers, nanofabricated graphene superlattice. In particular, details about the fractal Hofstadter spectrum in graphene will be focused and discussed in the graphene/hBN superlattice and the twisted bilayer graphene.

Key points

• • • •

Introduction of Hofstadter Butterfly and the realization in non-graphene systems Graphene superlattices: graphene/hBN, twisted graphene layers, nanofabricated graphene superlattice The realization of fractal Hofstadter butterfly in Graphene/hBN superlattice Topological Chern insulators and Hofstadter butterfly in Twisted bilayer graphene

Introduction Hofstadter butterfly refers to the energy spectrum of two-dimensional (2D) electrons being subjected to both the periodic electrostatic potential and a perpendicular magnetic field (B), which leads to a self-similar recursive Landau level spectrum resembling a butterfly. It is named after D. Hofstadter who first obtained the fractal energy spectrum mathematically for a square lattice in 1976 (Hofstadter, 1976). He assumed that the tight-binding Bloch energy has the form of 2E0(coskxl + cos kyl), where E0 is an empirical parameter, l is the electrostatic potential period, and k is the wave vector. He then replaced the momentum by doing a Peierls substitution, i.e., ℏk ! ℏk − eA, where e is the electron charge and A is the vector potential with B ¼ r  A. By solving the time-independent Schrodinger equation with the ansatz of c(x, y) ¼ gneiam, where x ¼ nl and y ¼ ml (n and m the integers) with a depends on energy, one could obtain Harper’s equation: gn+1 + gn−1 + 2 cosð2pnf=f0 − aÞgn ¼ egn , with f ¼ Bl is the magnetic flux through the unit cell, f0 ¼ h/e is the magnetic quantum flux with h the Planck constant, and e ¼ 2E/E0 with E the energy. At rational fillings with f/f0 ¼ p/q, where p and q are co-prime integers, the solution of the Harper’s equation reveals that the Bloch band would split into q distinct energy subbands, resulting in a recursive fractal energy diagram as shown Fig. 1, the so called Hofstadter butterfly. In 1978, Wannier (1978) took the density of state into account and transformed the fractal energy spectrum into a linear trajectory of density-field diagram, so called Wannier diagram. In this representation, the Hofstadter butterfly is described by the Diophantine relation: 2

n=n0 ¼ t ðf=f0 Þ + s Here n is the carrier density and n0 is the carrier density of a unit cell. The first quantum number t corresponds to the topological properties of Landau level flat band and the second quantum number s is the Bloch band filling index. Later on, the fractal spectrum was also calculated for triangular lattices (Claro and Wannier, 1979) and honeycomb lattices (Rammal, 1985). To experimentally observe the Hofstadter butterfly is rather challenging. The minigaps of the p fractal ffiffiffiffiffiffiffiffiffiffi spectrum is considerable when magnetic flux is comparable to the quantum flux, in other words, the magnetic length lB ¼ ℏ=eB is of the same order as l, the period of electrostatic potential of the Bloch band. For a real crystal, the period is the lattice constant, which is of the order of

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Fig. 1 Calculated Fractal spectrum in a unit cell by D. Hofstadter. Reproduced from Hofstadter D.R. (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14, 2239–2249.

sub-nanometer, and thus it usually requires an extremely large magnetic field over 1000T that is experimentally inaccessible in the lab. Back in the 1990s, electrostatic periods much larger than its lattice constant were achieved by fabricating lateral superlattices for 2D electron gas in GaAs/AlGaAs heterostructures (Schlösser et al., 1996; Albrecht et al., 1999, 2001, 2003; Geisler et al., 2004). Although signatures of Hofstadter spectrum were observed in these artificial superlattices, the device quality degradation due to nanofabrication as well as the difficulty in tuning carrier density hindered the revelation of the fractal structure. The fractal Hofstadter spectrum was vividly realized in a microwave waveguide with a periodic arrangement of scatterers (Kuhl and Stöckmann, 1998). In this photonic system, the transmission matrix mimics the Hamiltonian of Bloch electrons in the magnetic field, and equivalently leads to the Harper equation and a fractal Hofstadter butterfly. Similar realizations of the Hofstadter Hamiltonian were also reported in optical lattices with ultracold atoms (Aidelsburger et al., 2013) and in superconducting qubits (Roushan et al., 2017).

Graphene superlattices Being atomically thin, graphene is an ideal 2D electron system. Soon after the discovery of quantum Hall effect in graphene (Novoselov et al., 2005; Zhang et al., 2005), calculations of the fractal Hofstadter spectrum were carried out in monolayer graphene (Hasegawa and Kohmoto, 2006) and bilayer graphene (Nemec and Cuniberti, 2007). Yet, the problem is obvious as the graphene lattice constant is small, a ¼ 0.246 nm, thus requiring a superlattice structure. One solution is to realize graphene superlattice structures with external periodic potentials, whose band structure was calculated by Park et al. (2008), revealing an emergence of secondary Dirac point at the superlattice Brillouin zone boundary. Similar calculations were also done by Brey and Fertig, 2009, and Barbier et al., 2010. The first and also the best experimental demonstration is the moiré superlattice structure on graphene/hexagonal boron nitride (hBN) heterostructure, formed due to their lattice mismatch as illustrated in Fig. 2A. The graphene/hBN superlattice was initially observed in a scanning tunneling microscope/spectrum (STM/STS) study by Yankowitz et al. (2012). Perfect twist angle control with the y ¼ 0o was realized via direct epitaxy of graphene on hBN with a superlattice period of  15.6 nm (Yang et al., 2013). Note that the twist angle between graphene

(A)

(B)

(C)

Fig. 2 Sketches of graphene superlattice structures formed by lattice constant mismatch (A), nanofabrication (B), and twisting between graphene layers (C).

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and hBN could also be tuned by thermal annealing (Wang et al., 2016) and AFM tip manipulation (Koren et al., 2016; Wang et al., 2016; Ribeiro-Palau et al., 2018). The fractal Hofstadter spectrum was realized in transferred graphene/hBN superlattices (Dean et al., 2013; Hunt et al., 2013; Ponomarenko et al., 2013; Yu et al., 2014; Wang et al., 2015; Krishna Kumar et al., 2017, 2018; Spanton et al., 2018), in epitaxial graphene/hBN superlattices (Yang et al., 2013, 2016; Chen et al., 2017; Lu et al., 2020), and in ABC-trilayer graphene/hBN superlattice (Chen et al., 2020a). Soon after the experimental observations, the fractal Hofstadter spectrum were calculated in graphene/hBN superlattice (Chen et al., 2014). Besides, there are also other graphene superlattices based on nanofabrication, such as one-dimensional (1D) superlattices with a comb-like gate (Dubey et al., 2013; Drienovsky et al., 2014), etched graphene antidots (Sandner et al., 2015; Yagi et al., 2015), and patterned dielectric superlattice (Forsythe et al., 2018), as sketched in Fig. 2B. However, the results are not promising. The 1D gated superlattices suffered from serious p-n junction effects, and the graphene antidots were limited by its big superlattice size and thus dominated by classic commensurate effect between the superlattice period and the cyclotron orbits. The patterned dielectric superlattices with a reduced period of  40 nm and a stronger tunability of the periodic potential revealed signatures of the fractal spectrum; however, the quality was still limited as it suffered from similar problems as those of lateral superlattices of the 2D electron gas in GaAs/AlGaAs heterostructures. Another strategy in this effort is a moiré butterfly by rotating two graphene layers with the period of l ¼ a/(2 sin (y/2)) at a twisted angle (y) (Fig. 2C). This was firstly proposed in (Bistritzer and MacDonald, 2011b), where fractal spectrum at a strong field limit (f/f0 > 1) was investigated. Later, calculations of the fractal spectrum at a weak field limit (1 > f/f0 > 0) were carried out in twisted bilayer graphene (TBG) for different twist angles by Moon and Koshino (2012) and Wang et al. (2012). In fact, the moiré superlattice structure of the TBG were observed in a series of the STM experiments (Li et al., 2010; Luican et al., 2011; Brihuega et al., 2012). In the early works of quantum transport on TBG prepared either by CVD growth on SiC (Lee et al., 2011) or by folded graphene (Schmidt et al., 2014), the device quality was much limited and no signature of the fractal band was observed. Thanks to the development of “pick-up” transfer technique (Wang et al., 2013) and subsequent “tear and stack” technique using the same graphene flake (Kim et al., 2016a), accurate control of twist angle in TBG and high-quality devices were possible. In 2016, realizations of TBG with y  2o (corresponding to l 7nm) were reported by Kim et al. (2016b) and Cao et al. (2016), where two sets of LLs were observed; one stemming from the main charge neutral point (CNP) and the other from the superlattice gaps (or called full filling n ¼ 4n0). However, the two LLs intersect each other without revealing the recurring fractal structures. Then, the magic angle TBG with y 1.1o (corresponding to l  12.8 nm) was firstly realized by Cao et al. (2018a,b), where the LLs are observed to fan out of the main CNP and superlattice gaps, and later by (Lu et al., 2019; Yankowitz et al., 2019) with the LLs developed from half filling and quarter fillings. The fractal Hofstadter spectrum was also observed in other twisted graphene multilayer moiré systems including but not limited to the twisted double bilayer graphene (TDBG) (Burg et al., 2019; Cao et al., 2020; Liu et al., 2020; Shen et al., 2020), twisted monolayer bilayer graphene (TMBG) (Chen et al., 2020b; Polshyn et al., 2020, 2021; He et al., 2021; Xu et al., 2021), and twisted trilayer graphene (TTG) (Hao et al., 2021; Park et al., 2021b; Siriviboon et al., 2021).

Hofstadter butterfly in graphene/hBN superlattice In graphene/hBN heterostructure, graphene and hBN share a similar triangular lattice with a mismatch of d¼  1.7%, which gives a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi triangular moiré superlattice with a period of l ¼ ð1 + dÞa= 2ð1 + dÞð1 − cosyÞ + d2 at a given twist angle (y). The area of the pffiffiffi superlattice unit cell A ¼ 3l2 =2 defines the number of electron states per area of a fully filled Bloch band n0 ¼ 1/A. Here, the carrier density n can be largely tuned by gating n ¼ CGVG/e, where CG (VG) is the geometrical capacitance (gate voltage), and the magnetic flux is defined by f ¼ BA. Then, the measured Landau fan diagram, a mapping of longitudinal resistance (Rxx) or Hall resistance (Rxy) to the change of VG and B, could be transformed into the Wannier diagram of the fractal Hofstadter butterfly spectrum, as shown in Fig. 3, following the Diophantine relation: n/n0 ¼ t(f/f0) + s, where both t and s are integers in the single-particle picture. At zero magnetic field, i.e., f/f0 ¼ 0, the Rxx peak at s ¼ 0 corresponds to the charge neutral point (CNP) of graphene, while two satellite peaks developed at s ¼  4 are referred to the gap openings at the moiré mini-Brillouin zone boundary (or called secondary Dirac points in some literatures), and the number 4 also means the fourfold degeneracy (spin and valley) of the Bloch band, i.e., 4 electrons/holes to fully filled one moiré unit cell. There is electron-hole asymmetry, and usually the superlattice effect is stronger for holes than that for electrons. At s ¼ 0, the Diophantine relation is simplified to n/n0 ¼ t(f/f0). This is exactly the case of 2D free electrons/holes subjected to a perpendicular magnetic field, and the quantum number t has the same physical meaning as the filling factors of Landau levels in graphene where the Hall conductance is quantized to te2/h. Let us take the monolayer (bilayer) graphene/hBN superlattice as an example. The states developed at t ¼ 4N+ 2 (4N for bilayer graphene) means that the Landau quantization gaps with fourfold symmetry, while those at other integers are corresponding to the gaps with broken symmetry, with N ¼ 0, 1, 2. . .Landau level index (N ¼ 1, 2, 3. . .for bilayer graphene). At s ¼  4, the Diophantine relation can be modified as (n/n0  4) ¼ t(f/f0). At a small magnetic field before the Landau quantization, the sign of Rxy changes at s ¼  4, which is exactly the same as sign change of Rxy at CNP. As the magnetic field increases further, features of the Landau level states are found fanning out from s ¼  4 with a slop of t ¼(f/f0)/(n/n0  4), similar

Hofstadter butterfly in graphene

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Fig. 3 Fractal Hofstadter butterfly revealed in a color mapping of conductance to magnetic field B and gate voltage Vg (A) and corresponding Wannier diagram (B) in graphene/hBN superlattice. Reproduced from Hunt B., Sanchez-Yamagishi J.D., Young A.F., Yankowitz M., LeRoy B.J., Watanabe K., Taniguchi T., Moon P., Koshino M., Jarillo-Herrero P., et al. (2013) Massive Dirac Fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430.

to those fanning out from CNP with a slope of t ¼(f/f0)/(n/n0). The whole spectra stemming at s ¼  4 is like a replica of those at CNP, a signature of fractal Hofstadter butterfly. Note that the LLs stemmed from s ¼ −4 show up at almost the same magnetic field as those at the CNP. The most intriguing evidence of the fractal Hofstadter spectrum is revealed at the fractional filling f/f0 ¼ p/q. If p ¼ 1 as in most cases, the LLs from s ¼ −4 and those from s ¼ 0 intersect at f/f0 ¼ 1/q. For instance, the LLs with t ¼ 2 from s ¼ −4 will meet those with t ¼ −2, −6, and −10 from s ¼ 0 at q ¼ 1, 2, and 3, respectively, as shown in Fig. 3B (Hunt et al., 2013). At these crossing points, mini Hofstadter gaps open at f/f0 ¼ 1/q, and moreover LLs with different t are developed at an effective magnetic field Beff ¼  |B − B1/q |, whose traces at B ¼ 0 could be any integer s between 0 and −4. In principle, similar recursive fractal gaps should be found at other crossing points in the Wannier diagram, and the whole spectrum constructs the fractal Hofstadter butterfly. Note that the fractional s with an integer t (also named symmetry broken Chern insulator) as well as a fractional t with s ¼ 0 (fractional quantum Hall insulator) are observed in graphene/hBN superlattice (Wang et al., 2015). Moreover, gaps with both fractional s and fractional t (also named fractional Chern insulator) are revealed in bilayer graphene/hBN superlattices (Spanton et al., 2018), indicating fractional Hofstadter butterfly spectra in the many-body picture. The fractal Hofstadter spectrum can be probed in conductance measurement (Dean et al., 2013; Hunt et al., 2013; Ponomarenko et al., 2013; Wang et al., 2015; Yang et al., 2016; Chen et al., 2017; Krishna Kumar et al., 2017) and also in capacitance measurements (Hunt et al., 2013; Yu et al., 2014; Spanton et al., 2018). Note that the fractal Hofstadter butterfly spectra can survive at elevated temperature up to 100 K (Krishna Kumar et al., 2017, 2018), even though the fractal Hofstadter butterfly spectra are thermally smeared into Brown-Zak oscillations, i.e., the conductance oscillations in f0/f.

Hofstadter butterfly in twisted bilayer graphene The situation in TBG are more complicated and interesting, because now the twisted angle (y) determines not only the size of the moiré superlattice l ¼ a/(2 sin (y/2)), but also the interlayer coupling. At a magic angle of  1.05o, the resulting moiré bands are extremely flat with a bandwidth of  10 meV, implying a reduced Fermi velocity (Bistritzer and MacDonald, 2011a). The moiré flat band in the magic angle TBG gives rise to the strong electron correlation effects, including the correlated insulators at half fillings and superconductivity nearby (Cao et al., 2018a,b), as well as correlator insulators at quarter fillings (Lu et al., 2019; Yankowitz et al., 2019). In TBG, it is convenient to define the moiré band filling factor (v ¼ n/n0), corresponding to the number of carriers filled

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pffiffiffi in the moiré conduction/valence band, with n0 ¼ 1/A and A ¼ 3l2 =2. As a result, the resistance peaks at v ¼  1 (3), 2, 4 in the transfer curve are corresponding to the correlated insulators at quarter (three quarters) fillings, half fillings, and moiré band insulators, respectively. The fractal Hofstadter butterfly spectrum in TBG is also revealed in the Wannier diagram, following the Diophantine relation: n/n0 ¼ t(f/f0) + s, where both t and s are integers in the single-particle picture. The strong electron-electron interactions in TBG could substantially reshape the fractal Hofstadter spectrum. Usually, the correlated insulators are quite robust against the magnetic field, and the replica of the LLs emerges at these integer fillings v, unlike the dominating LLs at v ¼ 0 and 4 in graphene/hBN superlattice. Note that the LL filling factor vLL in the replica spectrum at non-zero integer v is usually twofold or even onefold degenerate, and the filling factors vLL have the same sign as those of v. In addition, the quantum oscillations in TBG are usually much weaker than those in the graphene/hBN superlattice. This is due to the reduced Fermi velocity of the flat band, especially for MATBG, and therefore the Landau quantization gaps as well as the fractal Hofstadter gaps are smaller compared to the dispersive moiré bands in the graphene/hBN superlattice. Additionally, these moiré flat bands in TBG are topological with non-trivial Berry curvatures (Liu et al., 2019a,b) at zero magnetic fields. The zero-field topological bands are mathematically the same as those LLs or fractal LLs at finite magnetic fields. Note that the quantum Hall insulator is in principle a Chern insulator where the LL filling factor corresponds to the Chern number, and here we use “Chern” specifically for the moiré Chern bands to avoid confusion. Considering spin, valley, and the moiré valley, the moiré bands are eightfold degenerate at the Dirac point, protected by the C2T symmetry where C2 is the inversion symmetry and T is time reversal symmetry. If C2T symmetry is unbroken, then the total Chern number is zero since the moiré bands with opposite Chern numbers touches at the Dirac point, even though each of the moiré bands are topological with C ¼  1. This can be related to the fragile topology (Ahn et al., 2019; Po et al., 2019; Song et al., 2019; Lian et al., 2020), whose signature is revealed in the observation of the C ¼ 0 band gaps being intersected by the non-zero LLs at f/f0 ¼ 2 (B  9T) in the TBG with y 0.45o (Lu et al., 2021). If the inversion symmetry is broken by a staggered potential mass while the T symmetry is preserved, it leads to C ¼ + 1 at the K valley and C ¼ −1 at the K0 valley in the moiré conduction band, and C ¼ −1 at the K valley and C ¼ +1 at the K0 valley in the moiré valence band. As a result, the Chern insulator with C ¼ + 1 or −1 emerge when the odd number of carriers is filled in the moiré band. The breaking of the C2 symmetry happens when the TBG is aligned with the hBN lattice, first realized by A. Sharpe (Sharpe et al., 2019) with the observations of ferromagnetism and Chern insulator with C ¼ 1 at v ¼ 3, and then by Serlin et al. (2020) with the observation of quantized anomalous Hall effect (QAH) with C ¼ 1 at v ¼ 3, as shown Fig. 4A. Similar observations have also been observed in ABC-trilayer graphene/hBN superlattice (Chen et al., 2020a), twisted monolayer bilayer graphene (Chen et al., 2020b; Polshyn et al., 2020). These observations are in stark contrast to the conventional fractal Hofstadter butterfly that strongly depends on the magnetic flux in the moiré superlattice. If the T symmetry is broken by the electron-electron interaction induced Haldane mass (Haldane, 1988), it leads to C ¼ +1 at both the K and K0 valley in the moiré conduction band, and C ¼ − 1 at the K and K0 valley in the moiré valence band. As a result, it gives a Chern insulator with Chern number depending on a sequential filling of carriers, i.e., C ¼ −4 − v for holes and C ¼ 4 − v for electrons (as shown Fig. 4B). The Chern insulators with the T symmetry-broken are observed in various experiments, for instance in

Fig. 4 (A) Observation of quantized anomalous Hall effect in hBN aligned TBG, i.e., Rxy ¼ h/(Ce2) and Rxx ¼ 0 h/e2 with Chern number C ¼ 1. (B) Observation of sequential filled Chern insulator with C ¼ 4 − v. Panel (a): Reproduced from Serlin, M., Tschirhart C.L., Polshyn H., Zhang Y., Zhu J., Watanabe K., Taniguchi T., Balents L., Young A.F. (2020) Intrinsic quantized anomalous Hall effect in a moire heterostructure. Science 367, 900–903. Panel (b): Reproduced from Das I., Lu X., Herzog-Arbeitman J., Song Z.-D., Watanabe K., Taniguchi T., Bernevig B.A., Efetov D.K. (2021) Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nature Physics 17, 710–714.

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transport measurements (Das et al., 2021; Park et al., 2021a; Saito et al., 2021; Shen et al., 2021; Wu et al., 2021b) and scanning spectroscopic measurements (Nuckolls et al., 2020; Choi et al., 2021; Pierce et al., 2021). These Chern insulators are found to compete with the zero-field ground states in MATBG (Stepanov et al., 2021). These Chern insulators are also observed for the non-magic angle TBG fractal Hofstadter spectra at finite magnetic fields (Choi et al., 2021; Shen et al., 2021). Moreover, fractional Chern insulators with fractional C and v (referred to the fractional t and fractional s in the Diophantine language), symmetry-broken Chern insulators (non-zero integers t and fractional s), and charge density waves (t ¼ 0 and fractional s) are observed in local compressibility measurements (Xie et al., 2021), and are also studied in the numerical calculations (Andrews and Soluyanov, 2020; Wang and Santos, 2020). Similar topological phases with a fractional v are also observed in the twisted monolayer-bilayer graphene (Polshyn et al., 2021) and twisted trilayer graphene (Siriviboon et al., 2021). All these Chern insulators as well as CDW phases are weakly dependent on the filling of the magnetic flux, unlike the fractal and fractional fractal Hofstadter spectra in graphene/hBN superlattices. In fact, it is also possible to suppress the Chern insulators and recover the conventional fractal Hofstadter spectrum by twisting away from the magic angle (Shen et al., 2021). Because the moiré bands are more dispersive for the non-magic angle, which leads to stronger Landau quantization (bigger fractal Hofstadter gaps) and weaker electron interactions (smaller Chern gaps). Also note that the moiré bands in TBG at a non-magical angle might be topologically trivial. Besides, it is also possible to reveal the fractal Hofstadter spectra in the MATBG by tuning the Fermi level from the flat moiré bands to the dispersive remote bands (Das et al., 2021). Last but not least, aside from TBG, other twisted graphene multilayer system offers a good opportunity to study the electrical field tunable Hofstadter spectrum. Take the TDBG for example, the Landau fan diagram of the Hofstadter butterfly spectrum is strongly displacement field tunable (Liu et al., 2022a) and stacking order dependent (Crosse et al., 2020; Wu et al., 2021a), and the valley polarized correlated insulators at half fillings in TDBG are observed topological (Liu et al., 2022b) and could host insulating quantum oscillations (Liu et al., 2022a). These observations suggest a rich interplay among the moiré flat band topology, symmetry, Landau quantization and others.

Conclusion Graphene is an ideal platform to experimentally realize the fractal Hofstadter butterfly. The development of van der Waals stacking techniques allows the realizations of high-quality graphene superlattice structures, including the graphene/hBN superlattice and the TBG and other twisted graphene multilayers. In a graphene/hBN superlattice, the recursive nature of the fractal Hofstadter spectra has been vividly revealed at rational fillings of f/f0. In TBG, the fractal Hofstadter spectra is strongly twist-angle dependent. In particular, the magic angle TBG reveals abundant topological Chern insulators including the quantized anomalous Hall effect at zero magnetic field, which is distinct from the conventional fractal Hofstadter spectrum.

References Ahn J, Park S, and Yang B-J (2019) Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle. Physical Review X 9: 021013. Aidelsburger M, Atala M, Lohse M, Barreiro JT, Paredes B, and Bloch I (2013) Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Physical Review Letters 111: 185301. Albrecht C, Smet J, Weiss D, Von Klitzing K, Hennig R, Langenbuch M, Suhrke M, Rössler U, Umansky V, and Schweizer H (1999) Fermiology of two-dimensional lateral superlattices. Physical Review Letters 83: 2234. Albrecht C, Smet JH, von Klitzing K, Weiss D, Umansky VV, and Schweizer H (2001) Evidence of Hofstadter’s fractal energy spectrum in the quantized hall conductance. Physical Review Letters 86: 147–150. Albrecht C, Smet JH, von Klitzing K, Weiss D, Umansky V, and Schweizer H (2003) Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance. Physica E: Low-dimensional Systems and Nanostructures 20: 143–148. Andrews B and Soluyanov A (2020) Fractional quantum Hall states for moir\’e superstructures in the Hofstadter regime. Physical Review B 101: 235312. Barbier M, Vasilopoulos P, and Peeters FM (2010) Extra Dirac points in the energy spectrum for superlattices on single-layer graphene. Physical Review B 81: 075438. Bistritzer R and MacDonald AH (2011a) Moire bands in twisted double-layer graphene. Proceedings of the National Academy of Sciences of the United States of America 108: 12233–12237. Bistritzer R and MacDonald AH (2011b) Moiré butterflies in twisted bilayer graphene. Physical Review B 84: 035440. Brey L and Fertig HA (2009) Emerging zero modes for graphene in a periodic potential. Physical Review Letters 103: 046809. Brihuega I, Mallet P, González-Herrero H, Trambly de Laissardière G, Ugeda MM, Magaud L, Gómez-Rodríguez JM, Ynduráin F, and Veuillen JY (2012) Unraveling the intrinsic and robust nature of Van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis. Physical Review Letters 109: 196802. Burg GW, Zhu J, Taniguchi T, Watanabe K, MacDonald AH, and Tutuc E (2019) Correlated insulating states in twisted double bilayer graphene. Physical Review Letters 123: 197702. Cao Y, Luo JY, Fatemi V, Fang S, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Kaxiras E, and Jarillo-Herrero P (2016) Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Physical Review Letters 117: 116804. Cao Y, Fatemi V, Demir A, Fang S, Tomarken SL, Luo JY, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Kaxiras E, et al. (2018a) Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556: 80–84. Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, and Jarillo-Herrero P (2018b) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556: 43–50. Cao Y, Rodan-Legrain D, Rubies-Bigorda O, Park JM, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2020) Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature 583: 215–220. Chen X, Wallbank JR, Patel AA, Mucha-Kruczynski M, McCann E, and Fal’ko VI (2014) Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices. Physical Review B 89: 075401.

730

Hofstadter butterfly in graphene

Chen G, Sui M, Wang D, Wang S, Jung J, Moon P, Adam S, Watanabe K, Taniguchi T, Zhou S, et al. (2017) Emergence of tertiary dirac points in graphene moire superlattices. Nano Letters 17: 3576–3581. Chen G, Sharpe AL, Fox EJ, Zhang Y-H, Wang S, Jiang L, Lyu B, Li H, Watanabe K, Taniguchi T, et al. (2020a) Tunable correlated Chern insulator and ferromagnetism in a moire superlattice. Nature 579: 56–61. Chen S, He M, Zhang Y-H, Hsieh V, Fei Z, Watanabe K, Taniguchi T, Cobden DH, Xu X, Dean CR, et al. (2020b) Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nature Physics 17: 374–380. Choi Y, Kim H, Peng Y, Thomson A, Lewandowski C, Polski R, Zhang Y, Arora HS, Watanabe K, Taniguchi T, et al. (2021) Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589: 536–541. Claro FH and Wannier GH (1979) Magnetic subband structure of electrons in hexagonal lattices. Physical Review B 19: 6068–6074. Crosse JA, Nakatsuji N, Koshino M, and Moon P (2020) Hofstadter butterfly and the quantum Hall effect in twisted double bilayer graphene. Physical Review B 102: 035421. Das I, Lu X, Herzog-Arbeitman J, Song Z-D, Watanabe K, Taniguchi T, Bernevig BA, and Efetov DK (2021) Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nature Physics 17: 710–714. Dean CR, Wang L, Maher P, Forsythe C, Ghahari F, Gao Y, Katoch J, Ishigami M, Moon P, Koshino M, et al. (2013) Hofstadter’s butterfly and the fractal quantum Hall effect in moire superlattices. Nature 497: 598–602. Drienovsky M, Schrettenbrunner F-X, Sandner A, Weiss D, Eroms J, Liu M-H, Tkatschenko F, and Richter K (2014) Towards superlattices: Lateral bipolar multibarriers in graphene. Physical Review B 89: 115421. Dubey S, Singh V, Bhat AK, Parikh P, Grover S, Sensarma R, Tripathi V, Sengupta K, and Deshmukh MM (2013) Tunable superlattice in graphene to control the number of Dirac points. Nano Letters 13: 3990–3995. Forsythe C, Zhou X, Watanabe K, Taniguchi T, Pasupathy A, Moon P, Koshino M, Kim P, and Dean CR (2018) Band structure engineering of 2D materials using patterned dielectric superlattices. Nature Nanotechnology 13: 566–571. Geisler MC, Smet JH, Umansky V, von Klitzing K, Naundorf B, Ketzmerick R, and Schweizer H (2004) Detection of a Landau band-coupling-induced rearrangement of the Hofstadter butterfly. Physical Review Letters 92: 256801. Haldane FDM (1988) Model for a quantum hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Physical Review Letters 61: 2015–2018. Hao Z, Zimmerman A, Ledwith P, Khalaf E, Najafabadi DH, Watanabe K, Taniguchi T, Vishwanath A, and Kim P (2021) Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371: 1133–1138. Hasegawa Y and Kohmoto M (2006) Quantum Hall effect and the topological number in graphene. Physical Review B 74: 155415. He M, Zhang YH, Li Y, Fei Z, Watanabe K, Taniguchi T, Xu X, and Yankowitz M (2021) Competing correlated states and abundant orbital magnetism in twisted monolayer-bilayer graphene. Nature Communications 12: 4727. Hofstadter DR (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14: 2239–2249. Hunt B, Sanchez-Yamagishi JD, Young AF, Yankowitz M, LeRoy BJ, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, et al. (2013) Massive Dirac Fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340: 1427–1430. Kim K, Yankowitz M, Fallahazad B, Kang S, Movva HCP, Huang S, Larentis S, Corbet CM, Taniguchi T, Watanabe K, et al. (2016a) van der Waals heterostructures with high accuracy rotational alignment. Nano Letters 16: 1989–1995. Kim Y, Herlinger P, Moon P, Koshino M, Taniguchi T, Watanabe K, and Smet JH (2016b) Charge Inversion and topological phase transition at a twist angle induced van hove singularity of bilayer graphene. Nano Letters 16: 5053–5059. Koren E, Leven I, Lortscher E, Knoll A, Hod O, and Duerig U (2016) Coherent commensurate electronic states at the interface between misoriented graphene layers. Nature Nanotechnology 11: 752–757. Krishna Kumar R, Chen X, Auton G, Mishchenko A, Bandurin DA, Morozov SV, Cao Y, Khestanova E, Ben Shalom M, Kretinin A, et al. (2017) High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices. Science 357: 181–184. Krishna Kumar R, Mishchenko A, Chen X, Pezzini S, Auton GH, Ponomarenko LA, Zeitler U, Eaves L, Fal’ko VI, and Geim AK (2018) High-order fractal states in graphene superlattices. Proceedings of the National Academy of Sciences of the United States of America 115: 5135–5139. Kuhl U and Stöckmann H-J (1998) Microwave realization of the Hofstadter butterfly. Physical Review Letters 80: 3232. Lee DS, Riedl C, Beringer T, Castro Neto AH, von Klitzing K, Starke U, and Smet JH (2011) Quantum Hall effect in twisted bilayer graphene. Physical Review Letters 107: 216602. Li G, Luican A, Lopes dos Santos JMB, Castro Neto AH, Reina A, Kong J, and Andrei EY (2010) Observation of Van Hove singularities in twisted graphene layers. Nature Physics 6: 109–113. Lian B, Xie F, and Bernevig BA (2020) Landau level of fragile topology. Physical Review B 102: 041402. Liu J, Liu J, and Dai X (2019a) Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase. Physical Review B 99: 155415. Liu J, Ma Z, Gao J, and Dai X (2019b) Quantum valley Hall effect, orbital magnetism, and anomalous hall effect in twisted multilayer graphene systems. Physical Review X 9: 031021. Liu X, Hao Z, Khalaf E, Lee JY, Ronen Y, Yoo H, Haei Najafabadi D, Watanabe K, Taniguchi T, Vishwanath A, et al. (2020) Tunable spin-polarized correlated states in twisted double bilayer graphene. Nature 583: 221–225. Liu L, Chu Y, Yang G, Yuan Y, Wu F, Ji Y, Tian J, Yang R, Watanabe K, and Taniguchi T (2022a) Quantum oscillations in correlated insulators of a moir\’e superlattice. arXiv. preprint arXiv:2205.10025. Liu L, Zhang S, Chu Y, Shen C, Huang Y, Yuan Y, Tian J, Tang J, Ji Y, Yang R, et al. (2022b) Isospin competitions and valley polarized correlated insulators in twisted double bilayer graphene. Nature Communications 13: 3292. Lu X, Stepanov P, Yang W, Xie M, Aamir MA, Das I, Urgell C, Watanabe K, Taniguchi T, Zhang G, et al. (2019) Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574: 653–657. Lu X, Tang J, Wallbank JR, Wang S, Shen C, Wu S, Chen P, Yang W, Zhang J, Watanabe K, et al. (2020) High-order minibands and interband Landau level reconstruction in graphene moiré superlattices. Physical Review B 102: 045409. Lu X, Lian B, Chaudhary G, Piot BA, Romagnoli G, Watanabe K, Taniguchi T, Poggio M, MacDonald AH, Bernevig BA, et al. (2021) Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle. Proceedings of the National Academy of Sciences of the United States of America 118: e2100006118. Luican A, Li G, Reina A, Kong J, Nair RR, Novoselov KS, Geim AK, and Andrei EY (2011) Single-layer behavior and its breakdown in twisted graphene layers. Physical Review Letters 106: 126802. Moon P and Koshino M (2012) Energy spectrum and quantum Hall effect in twisted bilayer graphene. Physical Review B 85: 195458. Nemec N and Cuniberti G (2007) Hofstadter butterflies of bilayer graphene. Physical Review B 75: 201404. Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV, and Firsov AA (2005) Two-dimensional gas of massless Dirac fermions in graphene. Nature 438: 197–200. Nuckolls KP, Oh M, Wong D, Lian B, Watanabe K, Taniguchi T, Bernevig BA, and Yazdani A (2020) Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588: 610–615. Park CH, Yang L, Son YW, Cohen ML, and Louie SG (2008) New generation of massless Dirac fermions in graphene under external periodic potentials. Physical Review Letters 101: 126804. Park JM, Cao Y, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2021a) Flavour Hund’s coupling, Chern gaps and charge diffusivity in moire graphene. Nature 592: 43–48. Park JM, Cao Y, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2021b) Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590: 249–255.

Hofstadter butterfly in graphene

731

Pierce AT, Xie Y, Park JM, Khalaf E, Lee SH, Cao Y, Parker DE, Forrester PR, Chen S, Watanabe K, et al. (2021) Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene. Nature Physics 17: 1210–1215. Po HC, Zou L, Senthil T, and Vishwanath A (2019) Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Physical Review B 99: 195455. Polshyn H, Zhu J, Kumar MA, Zhang Y, Yang F, Tschirhart CL, Serlin M, Watanabe K, Taniguchi T, MacDonald AH, et al. (2020) Electrical switching of magnetic order in an orbital Chern insulator. Nature 588: 66–70. Polshyn H, Zhang Y, Kumar MA, Soejima T, Ledwith P, Watanabe K, Taniguchi T, Vishwanath A, Zaletel MP, and Young AF (2021) Topological charge density waves at half-integer filling of a moiré superlattice. Nature Physics 18: 42–47. Ponomarenko LA, Gorbachev RV, Yu GL, Elias DC, Jalil R, Patel AA, Mishchenko A, Mayorov AS, Woods CR, Wallbank JR, et al. (2013) Cloning of Dirac fermions in graphene superlattices. Nature 497: 594–597. Rammal R (1985) Landau level spectrum of Bloch electrons in a honeycomb lattice. Journal de Physique 46: 1345–1354. Ribeiro-Palau R, Zhang C, Watanabe K, Taniguchi T, Hone J, and Dean CR (2018) Twistable electronics with dynamically rotatable heterostructures. Science 361: 690–693. Roushan P, Neill C, Tangpanitanon J, Bastidas VM, Megrant A, Barends R, Chen Y, Chen Z, Chiaro B, Dunsworth A, et al. (2017) Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358: 1175–1179. Saito Y, Ge J, Rademaker L, Watanabe K, Taniguchi T, Abanin DA, and Young AF (2021) Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nature Physics 17: 478–481. Sandner A, Preis T, Schell C, Giudici P, Watanabe K, Taniguchi T, Weiss D, and Eroms J (2015) Ballistic transport in graphene antidot lattices. Nano Letters 15: 8402–8406. Schlösser T, Ensslin K, Kotthaus JP, and Holland M (1996) Landau subbands generated by a lateral electrostatic superlattice-chasing the Hofstadter butterfly. Semiconductor Science and Technology 11: 1582. Schmidt H, Rode JC, Smirnov D, and Haug RJ (2014) Superlattice structures in twisted bilayers of folded graphene. Nature Communications 5: 5742. Serlin M, Tschirhart CL, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, and Young AF (2020) Intrinsic quantized anomalous Hall effect in a moire heterostructure. Science 367: 900–903. Sharpe AL, Fox EJ, Barnard AW, Finney J, Watanabe K, Taniguchi T, Kastner MA, and Goldhaber-Gordon D (2019) Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365: 605–608. Shen C, Chu Y, Wu Q, Li N, Wang S, Zhao Y, Tang J, Liu J, Tian J, Watanabe K, et al. (2020) Correlated states in twisted double bilayer graphene. Nature Physics 16: 520–525. Shen C, Ying J, Liu L, Liu J, Li N, Wang S, Tang J, Zhao Y, Chu Y, Watanabe K, et al. (2021) Emergence of chern insulating states in non-magic angle twisted bilayer graphene. Chinese Physics Letters 38: 047301. Siriviboon P, Lin J-X, Scammell HD, Liu S, Rhodes D, Watanabe K, Taniguchi T, Hone J, Scheurer MS, and Li J (2021) Abundance of density wave phases in twisted trilayer graphene on WSe $ _2$. arXiv. preprint arXiv:2112.07127. Song Z, Wang Z, Shi W, Li G, Fang C, and Bernevig BA (2019) All magic angles in twisted bilayer graphene are topological. Physical Review Letters 123: 036401. Spanton EM, Zibrov AA, Zhou H, Taniguchi T, Watanabe K, Zaletel MP, and Young AF (2018) Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360: 62–66. Stepanov P, Xie M, Taniguchi T, Watanabe K, Lu X, MacDonald AH, Bernevig BA, and Efetov DK (2021) Competing zero-field chern insulators in superconducting twisted bilayer graphene. Physical Review Letters 127: 197701. Wang J and Santos LH (2020) Classification of topological phase transitions and van hove singularity steering mechanism in graphene superlattices. Physical Review Letters 125: 236805. Wang ZF, Liu F, and Chou MY (2012) Fractal Landau-level spectra in twisted bilayer graphene. Nano Letters 12: 3833–3838. Wang L, Meric I, Huang P, Gao Q, Gao Y, Tran H, Taniguchi T, Watanabe K, Campos L, Muller D, et al. (2013) One-dimensional electrical contact to a two-dimensional material. Science 342: 614–617. Wang L, Gao Y, Wen B, Han Z, Taniguchi T, Watanabe K, Koshino M, Hone J, and Dean CR (2015) Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350: 1231–1234. Wang D, Chen G, Li C, Cheng M, Yang W, Wu S, Xie G, Zhang J, Zhao J, Lu X, et al. (2016) Thermally induced graphene rotation on hexagonal boron nitride. Physical Review Letters 116: 126101. Wannier G (1978) A result not dependent on rationality for Bloch electrons in a magnetic field. Physica Status Solidi 88: 757–765. Wu Q, Liu J, Guan Y, and Yazyev OV (2021a) Landau levels as a probe for band topology in graphene Moir\’e superlattices. Physical Review Letters 126: 056401. Wu S, Zhang Z, Watanabe K, Taniguchi T, and Andrei EY (2021b) Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nature Materials 20: 488–494. Xie Y, Pierce AT, Park JM, Parker DE, Khalaf E, Ledwith P, Cao Y, Lee SH, Chen S, Forrester PR, et al. (2021) Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600: 439–443. Xu S, Al Ezzi MM, Balakrishnan N, Garcia-Ruiz A, Tsim B, Mullan C, Barrier J, Xin N, Piot BA, Taniguchi T, et al. (2021) Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene. Nature Physics 17: 619–626. Yagi R, Sakakibara R, Ebisuoka R, Onishi J, Watanabe K, Taniguchi T, and Iye Y (2015) Ballistic transport in graphene antidot lattices. Physical Review B 92: 195406. Yang W, Chen G, Shi Z, Liu C-C, Zhang L, Xie G, Cheng M, Wang D, Yang R, Shi D, et al. (2013) Epitaxial growth of single-domain graphene on hexagonal boron nitride. Nature Materials 12: 792–797. Yang W, Lu X, Chen G, Wu S, Xie G, Cheng M, Wang D, Yang R, Shi D, and Watanabe K (2016) Hofstadter butterfly and many-body effects in epitaxial graphene superlattice. Nano Letters 16: 2387–2392. Yankowitz M, Xue J, Cormode D, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Jarillo-Herrero P, Jacquod P, and LeRoy BJ (2012) Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nature Physics 8: 382–386. Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young AF, and Dean CR (2019) Tuning superconductivity in twisted bilayer graphene. Science 363: 1059–1064. Yu GL, Gorbachev RV, Tu JS, Kretinin AV, Cao Y, Jalil R, Withers F, Ponomarenko LA, Piot BA, Potemski M, et al. (2014) Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nature Physics 10: 525–529. Zhang Y, Tan Y-W, Stormer HL, and Kim P (2005) Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438: 201–204.

Tight-binding method in electronic structure☆ DA Papaconstantopoulosa, MJ Mehlb, AG Chronisc, and MM Sigalasc, aDepartment of Computational and Data Sciences, George Mason University, Fairfax, VA, United States; bDepartment of Mechanical Engineering and Materials Sciences, Duke University, Durham, NC, United States; cDepartment of Materials Science, University of Patras, Patras, Greece © 2024 Elsevier Ltd. All rights reserved. This is an update of D.A. Papaconstantopoulos, M.J. Mehl, Tight-Binding Method in Electronic Structure, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 194–206, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00452-6.

Introduction NRL tight-binding method Technical procedure Ground-state behavior and phase stability Elastic constants Vacancies Surfaces Stacking faults Phonons Point defects Finite temperature properties from molecular dynamics Multicomponent systems One- and two-dimensional structures Conclusion Acknowledgments References

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Abstract The Slater-Koster (SK) method was originally developed as a technique for interpolating first-principles energy bands from a limited number of k-points to a large one, for the purpose of accurately determining Fermi surfaces and densities of states. Numerous applications of the SK method have been made and extensions to include total energy determinations. One extension of the SK approach is the Naval Research Laboratory Tight-Binding (NRL-TB) method which has been very successful in fitting a set of first-principles electronic structure results and in addition being capable to predict properties that were not fitted and to also perform molecular dynamics simulations. This review presents a summary of applications using the NRL-TB scheme which covers most of the elements in the periodic Table and examples for multi-component systems.

Key points

• • • • • •

The NRL tight-binding method uses the Slater-Koster method to reproduce first-principles results. The resulting parameters can predict results not included in the database. The reduced size of the Hamiltonian allows rapid evaluation of systems containing hundreds of atoms. Phonon spectra as well as defect, stacking fault, and surface energies can be computed. Finite-temperature molecular dynamics simulations can compute thermodynamic properties. One- and two-dimensional systems, as well as nanoparticles, can be modeled.

Introduction In the linear combination of atomic orbitals (LCAO) method, the one-electron wave function is expressed as a linear combination of Bloch sums. When such an expansion is made to the wave function in the Schrödinger equation, one obtains a set of simultaneous linear equations that has a nonzero solution if the determinant of the coefficients vanishes, that is, ~ − eS~ ¼ 0 H

(1)

☆

Change History: September 2022. DA Papaconstantopoulos and MJ Mehl updated all sections with minor changes. New figures have been added and changes made in tables.

732

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00103-7

Tight-binding method in electronic structure

733

The matrix elements in this equation have the form X

Hnm ¼

Z

   exp ik  Rj −Ri

Rj

  ~ m r−Rj d3 r fn∗ ðr−Ri ÞHf



(2)

and Snm ¼

X

   exp ik Rj −Ri

Rj

Z 

  fn∗ ðr−Ri Þfm r−Rj d3 r

(3)

where Ri and Rj denote the positions of atoms located on orbitals fn and fm, respectively. The integrals in the above equations can, in principle, be calculated directly, however the most common practice has been to follow Slater and Koster (SK) (Slater and Koster, 1954). They suggested replacing these integrals by adjustable parameters that could be estimated from experiment, but which are generally determined by fitting to more elaborate electronic structure calculations. The size of the matrices  H and  S is determined by the number of atoms in the unit cell and the number of atomic orbitals on each site. So, for face-centered cubic (f.c.c.), body-centered cubic (b.c.c.), and simple cubic (s.c.) lattices with one atom per unit cell,  H and  S are 9  9 matrices representing one s-function, three p-functions, and five d-functions. The f-states have been omitted in most works, although there have been papers that provide extensions of the SK scheme that include f-orbitals (Durgavich et al., 2016). For diatomic lattices, such as hexagonal close-packed (h.c.p.) and diamond and in binary compounds the spd model will result in an 18  18 matrix. Often in semiconductors such as Si and Ge the d-states are not included and hence the size of the Hamiltonian is 8  8. The full SK Hamiltonian for a monatomic material could require the determination of 81 parameters between each pair of atoms; however, SK showed that these parameters are related by symmetry operations that considerably reduce their actual number. The resulting independent matrix elements may be found in the tables given by SK. The reader is cautioned that some obvious typographical errors appear in these tables which have been corrected by users. Since the above summations are over the interatomic distances Rj, the number of parameters increases for calculations that include more than first nearest neighbors. In practice, no more than the third nearest neighbors are included. The integrals in [2] are three-centered since they are the product of an atomic wave function f⁎(r −Ri) centered on the atom at position Ri, an atomic wave function f(r−Rj) located on the atom at position Rj, and a potential function inside the Hamiltonian H, centered on a third atom. These three-center integrals can be reduced to two-center integrals by approximating the potential energy as a sum of spherical potentials located on the same two atoms where the atomic orbitals are located. For the relationships between three-and two-center integrals, see Papaconstantopoulos (2015). Although working in three-center formalism has the advantage of fitting band structures more accurately, the two-center approximation has the distinct advantage that its parameters are transferable from one structure to another. This property of the two-center scheme makes it suitable to introduce bond-length dependence in the parameters which leads one to implementing a total energy capability in the SK method. Since the 1960s there have been numerous applications of tight-binding (TB) theories, in particular the SK approach, to calculate electronic energy bands, densities of states, and, more recently, total energies, forces, and molecular dynamics simulations (Cleri and Rosato, 1993; Cohen et al., 1994; Gross et al., 1999; Groß et al., 2003; Haftel et al., 2004; Harrison, 1989; Mehl et al., 2000; Mehl and Papaconstantopoulos, 2002; Papaconstantopoulos and Mehl, 2003; Silayi et al., 2018; Shi and Papaconstantopoulos, 2004; Yang et al., 1998). There are two main uses of the TB theory. One is to determine a small number of parameters with the aim of providing a physical insight into the electronic structure without the necessity of going beyond a qualitative description. A classic example of this approach is the universal TB parameters of Harrison (1989). Harrison’s TB theory has been of great educational value for a whole generation of scientists. More recently, extensions to this theory have been provided that greatly improve its accuracy. The other approach is to place emphasis on accurately reproducing the results of first-principles calculations by applying a least-squares procedure using a large number of parameters. The authors of this article have generally followed the second approach. A handbook by Papaconstantopoulos (2015) provides TB parameters for all elements in the periodic table, except for those with f-states. The handbook provides TB parameters in both two-center and three-center bases as well as in orthogonal and nonorthogonal representations, however these parameters are fitted to the ground state of each element and have very limited transferability to other volumes and structures. In addition, these TB Hamiltonians are not designed to compute total energies. The NRL-TB scheme (Cohen et al., 1994; Mehl and Papaconstantopoulos, 1996), on the other hand, which is discussed in the next section, is a two-center nonorthogonal TB method that uses environment-dependent parameters that capture the volume dependence of both the energy bands and the total energy. To complete this introduction, the attention of the reader is drawn to a simple level of the TB theory, known as the second moment approximation (SMA), which originates from the Friedel model of rectangular density of states for the d-bands. The SMA method does not account for the band structure but, at least for the f.c.c. metals, reproduces well the total energy which has led to its wide use in molecular dynamics simulations. This method contains four adjustable parameters fitted to reproduce either experimental quantities or the total energies from first-principles calculations.

734

Tight-binding method in electronic structure

NRL tight-binding method In the (Naval Research Laboratory) NRL-TB scheme, the on-site terms hil are written as a polynomial 2=3

hil ¼ al + bl ri

4=3

+ cl ri

+ di r2i ,

where i labels the atom and l the angular momentum of s, p, and d characters in the present form of the method. The quantity ri that appears in [4] is an embedded-atom-like “density” per atom given by the expression X     exp −l2 Rij F Rij , ri ¼

(4)

(5)

where Rij denotes the position of neighboring atoms from a central atom i, and F(Rij) is the cutoff function. The quantities al, bl, cl, dl, and l are 13 parameters to be determined by a least-squares fit to the first-principles results. It is possible, within this scheme, for example, to split the d-onsite parameters to t2g and eg, or going even further, to have different parameters for the three p-orbitals and the five d-orbitals. It is also to be noted that this scheme has the flexibility of adding more terms in [4] to describe the density contributions from atom A to atom B (and vice versa) in the case of a binary material. It was shown by SK that the two-center (spd) hopping integrals can be constructed from ten independent parameters Hll’m, where  0  ll m ¼ sss,sps,pps,ppp,sds, pds,pdp,dds,ddp,and ddd: One notes that in the case of a binary material AB, there will be a set of four additional parameters  0  ll m ¼ pss,dss,dps,and dpp: In the NRL-TB, each of these parameters is expressed in the form of a second-order polynomial-times an exponential function   Hll0 m ðr Þ ¼ ell0 m + f ll0 m r + gll0 m r 2   , (6)  exp −t 2ll0 m r Fðr Þ where r is the distance between atoms, F(r) the cutoff function as in [5], and ell’m, fll’m, gll’m, and tll’m are an additional set of 40 parameters (or 56 for the, A-B interactions) to be determined by fitting to the first-principles results. Since one usually fits to a nonorthogonal Hamiltonian, an additional set of 40 parameters is used (56 for A-B) for the overlap matrix following [6]. In cases where the validity of the parameters for interatomic distances much smaller than those in the original databases needs to be extended, one finds that the following form of overlap parameters is more successful   Sll0 m ðr Þ ¼ dll0 + pll0 m r + qll0 m r 2 + r ll0 m r 3  2  , (7)  exp −sll0 m r Fðr Þ where pll’m, qll’m, rll’m, and Sll’m are the corresponding parameters of the overlap matrix and dll’ is the Kronecker delta. In most TB approaches (Cleri and Rosato, 1993; Harrison, 1989), as well as in all the so-called “glue” potential atomistic methods (Finnis and Sinclair, 1984), one writes the total energy as a sum of a band energy term (sum of eigenvalues) and a repulsive potential G[n(r)] that can be viewed as replacing all the charge density-dependent terms appearing in the total energy expression of the density-functional theory. The NRL-TB method has the unique feature that eliminates G by the following ansatz: a quantity V0 is defined as V 0 ¼ G½nðr Þ=Ne ,

(8)

where Ne is the number of valence electrons and n(r) is the charge density. All the first-principles eigenvalues ei(k) are then shifted by the constant V0, and the shifted eigenvalue is defined as e0i ðkÞ ¼ ei ðkÞ + V 0 The result of this manipulation is that the first-principles total energy E is given by the expression X E¼ e0i ðkÞ

(9)

(10)

where the sum is over all occupied bands and all k-points in the Brillouin zone. It is noted that the constant V0 is different for each volume and structure of the first-principles database. The reader should recognize that each band structure has been shifted by a constant, retaining the exact shape of the first-principles bands. It should also be stressed that all this is done to the first-principles database before one proceeds with the fit that will generate the TB Hamiltonian. The next step is to use a least-squares procedure to fit this database with the shifted eigenvalues ei to the TB Hamiltonian. A typical database in the NRL-TB method contains, for example, in a transition metal, five volumes each for the f.c.c. and b.c.c. structures and often the s.c. lattice to achieve better transferability to other periodic structures or defect structures. The TB method can also be extended to cover magnetism. In this case, a set of spin-polarized parameters are constructed by fitting to spin-polarized linearized augmented plane wave (LAPW) calculations. Common hopping [6] and overlap [7] parameters

Tight-binding method in electronic structure

735

are used for the majority and minority spin cases, but separate onsite [4] parameters for each channel are developed. Each channel is then diagonalized separately. A set of paramagnetic TB parameters is constructed by averaging the majority and minority spin onsite parameters. Next the pressure-induced ferromagnetic-to-paramagnetic transitions are studied by noting whether the spin-polarized or paramagnetic parameters produce lower energies. Currently well-tested parameters for Fe, Co, and Ni are available (Bacalis et al., 2001; Mehl and Papaconstantopoulos, 2005). The parameters are fit to reproduce the first-principles total energies and eigenvalues by using the Levenberg-Marquardt algorithm to minimize the mean square error of the expression (McGrady and Papaconstantopoulos, 2021) M¼

j X

wE ðiÞj ELAPW ðiÞ−ETB ðiÞj2 +

i

X

wB ði, k, nÞjeLAPW ði, k, nÞ−eTB ði, k, nÞj2

(11)

i,j,n

In Eq. 11, wE is the weighting factor for the total energy for structure i, wB the weight for the bands, ELAPW and ETB are the respective first-principles and TB total energies for the ith structure and eLAPW and eTB are the eigenvalues for the nth band and the kth k-point of the ith structure.

Technical procedure The first-principles calculations were done either with the muffin-tin potential augmented plane wave (APW) method or with the full-potential LAPW method. The small differences between these two approaches, especially for closed packed structures, are within the error one makes after fitting to the TB Hamiltonian. For the f.c.c., b.c.c., and s.c. structures uniform k-point meshes that include the origin and contain 89, 55, and 35k-points in the irreducible part of the Brillouin zone, respectively, are used. In most cases, the Hedin-Lundqvist parametrization of the local density approximation (LDA) to the density-functional theory (DFT) is used. In some cases, such as for spin-polarized iron (Bacalis et al., 2001), the generalized gradient approximation (GGA) has been used. Spin-polarized calculations were performed for the appropriate metals in the 3d transition series. For a typical transition metal, there are  4000 eigenvalues and energies in the database. An IMSL package is used, based on a finite-difference Levenberg-Marquardt algorithm, to adjust the 93 parameters involved to reproduce this database by means of a nonlinear least-squares fit, with the total energies typically weighted  200 times larger than the eigenvalues in a single band. Starting parameters are selected guided by those found in the previous work and it is ensured that the symmetry of the eigenstates is taken into account. This is an important issue because a block-diagonalization of the Hamiltonian avoids the possibility of incorrectly aligning the bands and preserves the angular momentum character of the states. For details of this symmetrization procedure, refer to (Mehl and Papaconstantopoulos, 1996). Another issue is the number of bands per k-point used in the fit. Both occupied and empty states are fitted. For example, in the transition metals, the lowest six bands for all k-points are fitted but the bands 7–9 are also included for the four high-symmetry points. With this choice, one obtains reliable values for the SK parameters that correspond to the p-orbitals. The fitting RMS errors are in the range 1–5 mRy for six bands and  0.5 mRy for the total energies. For the semiconductors, eight bands, divided into four valence and four conduction bands in an s-p basis are fitted. Here, the RMS error is  5 mRy for the valence bands but much higher for the conduction bands. There is a very significant improvement in the conduction bands upon inclusion of the d-orbitals. However, the s-p basis gives a very good fit to the total energy with an RMS error not exceeding 0.5 mRy. Fig. 1 shows an example of the accuracy of the TB fit: we show a comparison between the TB and LAPW energy bands for hcp beryllium (McGrady and Papaconstantopoulos, 2021). Clearly the TB Hamiltonian accurately reproduces the first-principles results.

Ground-state behavior and phase stability The NRL-TB Hamiltonians have been tested to give the correct ground state for all materials for which TB parameters have been generated. The results are summarized in Tables 1 and 2. Table 1 shows the equilibrium lattice constants and bulk moduli for all the materials studied, comparing them to first-principles LDA calculations and to experiment. The equilibrium lattice constant is typically within 1–2% of the LDA value and correspondingly very close to the measured values. In this table, the magnetic elements are spin-polarized in a manner consistent with the observed magnetic structure. Thus, one can find that the ground state of chromium is an anti-ferromagnetic CsCl phase, which is a good approximation to the experimentally observed spin density wave. Similarly, the equilibrium bulk moduli are in good agreement with the first-principles results, within 10% for most elements. Except for Ti, the h.c.p. phases were not fit to first-principles results, but instead predicted as an output of the TB scheme. As a result, the equations of state predicted by the TB model are not as accurate as for the cubic lattices. The largest error is for zirconium, where a is 7% smaller than experiment and c is 8% larger. Yttrium and hafnium also show large discrepancies between the TB model and experiment. The lattice constants for the other elements are within 2% of experiment, consistent with the errors one would find in first-principles calculations. The discrepancies from experiment found in Zr, Y, and Hf can be removed by including h.c.p. and, if necessary, the simple cubic lattice. For Ti, the GGA calculations were fitted with the h.c.p. and s.c. lattices (Mehl and Papaconstantopoulos, 2005). In this more elaborate fit, it was possible to improve on the lattice constants and differentiate between the h.c.p. and o phases. This TB method

736

Tight-binding method in electronic structure

Band structure of Be c/a=1.6 V=106 Bohr3 1.5 Be-TB Be-LAPW

Energy (Ry)

1

0.5

0

-0.5

-1

Γ

M

L

A

Γ

K

H

A

Fig. 1 Comparison of the bands from the NRL-TB and LAPW methods for the HCP Be structure with c/a ¼ 1.6 and a volume of 106 Bohr3 (McGrady and Papaconstantopoulos, 2021). The LAPW bands are shown with a dashed line and the NRL-TB results are shown with a solid line. The Fermi energy is at 0 Ry.

Table 1 Equilibrium lattice constants and bulk moduli for the experimentally observed ground-state structures of the elements, comparing the results of the TB parametrization (Bernstein et al., 2000, 2002; Gotsis et al., 2002; McGrady et al., 2014; McGrady and Papaconstantopoulos, 2021; Mehl and Papaconstantopoulos, 1996; Wang et al., 2003; Yang et al., 1998), first-principles DFT, where available, and experiment Element

Structure

Li Be B C Mg Al Si Ca Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga

b.c.c. h.c.p. rhomb dia h.c.p. f.c.c. dia f.c.c. f.c.c. h.c.p. h.c.p. b.c.c. CsCl aMn b.c.c. h.c.p. f.c.c. f.c.c. h.c.p aGa

Ge Sr Sr Y Zr Nb Mo

dia f.c.c. f.c.c. h.c.p. h.c.p. b.c.c. b.c.c.

˚) a (A

˚) c (A

B0 (GPa)

TB

DFT

Expt.

TB

DFT

Expt.

TB

DFT

Expt.

3.40 4.25 4.90 3.52 3.22 4.00 5.43 5.35 5.51 3.26 2.94 2.94 2.79 8.64 2.84 2.54 3.43 3.54 4.87 4.63 7.63 5.60 5.73 6.09 3.59 2.99 3.25 3.12

3.39 4.28

3.49 4.32 12.43 3.57 3.21 4.05 5.43 5.58 5.58 3.31 2.95 3.03 2.88 8.91 2.87 2.51 3.52 3.61 5.03 4.51 7.64 5.66 6.08 6.08 3.65 3.23 3.30 3.15

a 6.80 4.91 a 5.26 a a a a 4.91 4.56 a a a a 4.01 a a 8.89 4.52

a 6.72

26 115 250 480 34 80 108 16 15 63 112 211 305 248 180 237 264 106 41 651

30 124 226 468 39 70 96 19 16 65 112 196

12 127

4.35

a 6.78 12.57 a 5.21 a a a a 5.27 4.68 a a a a 4.07 a a 9.35 4.52

669

443 35 72 99 15 15 44 107 162 190 60 168 191 186 137 66 613

a a a 5.61 5.14 a a

a a a 5.73 5.15 a a

67 15 11.5 46 108 185 283

78 16 11 48 119 193 291

77 12 12 37 83 170 272

3.53 3.16 3.99 5.40 5.28 5.55 3.21 2.94 2.93 2.83 3.52 3.56 4.38 7.39 5.61 5.74 6.05 3.52 3.17 3.25 3.12

a a a 5.35 5.57 a a

a 5.02 a a a a 5.01 4.65 a a a a a a

197 200 195

737

Tight-binding method in electronic structure Table 1 Element

(Continued) Structure

˚) a (A TB

Tc Ru Rh Pd Ag Cd In Sn Ba La Yb Hf Ta W Re Os Ir Pt Au Pb Po Ac Th

h.c.p. h.c.p. f.c.c. f.c.c. f.c.c. h.c.p. b.c.t. dia b.c.c. d.h.c.p. f.c.c. h.c.p. b.c.c. b.c.c. h.c.p. h.c.p. f.c.c. f.c.c. f.c.c. f.c.c. s.c. f.c.c. f.c.c.

2.72 2.68 3.77 3.85 4.02 5.51 4.29 6.48 4.82 3.62 5.40 3.07 3.30 3.14 2.78 2.75 3.86 3.90 4.06 4.88 3.26 5.52 5.08

˚) c (A DFT

3.76 3.85 4.00 4.60 6.47 4.80 3.62 5.40 3.18 3.24 3.14 2.77 2.73 3.82 3.90 4.06 4.88 3.34 5.50 5.08

Expt.

TB

2.74 2.71 3.80 3.89 4.09 5.63 5.10 6.49 5.02 3.77 5.48 3.19 3.30 3.16 2.76 2.74 3.84 3.92 4.08 4.95 3.35 5.31 5.08

4.34 4.26 a a a 10.37 a a a 11.68 a 5.08 a a 4.39 4.31 a a a a a a a

B0 (GPa) DFT

a a a 10.62 a a a 11.68 a 5.15 a a 4.50 4.39 a a a a a a a

Expt.

TB

4.40 4.28 a a a

304 360 306 212 109 53

a a a 12.1 a 5.05 a a 4.46 4.32 a a a a a a a

45 10 28.0 26.4 111 185 319 371 441 389 318 181 50 59 25.9 47.3

DFT

309 220 154 45 11 30.0 16 110 224 333 388 440 401 305 198 54 44 27.6 57.7

Expt. 297 321 270 181 132 52 53 10 24.3 13.3 109 200 323 372 418 355 278 169 45 26 54.0

First-principles calculations were done within the LDA except for those rows marked with an asterisk, which have GGA calculations. The “CsCl” structure given for Cr is described in the text. Note that gallium has an orthorhombic structure, so values for lattice constants a and b are both given in the a columns.

Table 2

TB energies discussed in the text.

Struct. Struk.

f.c.c. A1

b.c.c. A2

h.c.p. A3

dia A4

bSn A5

b.c.t. A6

gra. A9

aMn A12

s.c. Ah

Li Be B C Mg Al Si Ca Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ge Sr Sr Y Nb Mo

−0.7 6.2 108.7 306.6 0.8 0.0 36.4 0.0 0.0 4.7 5.1 19.7 28.7 7.4 9.8 2.7 0.0 0.0 3.2 17.3 0.0 0.0 2.3 29.6 29.7

0.0 6.6 135.9 302.3 3.6 8.1 34.8 2.2 1.4 8.6 7.5 0.0 0.0 14.3 0.0 13.1 8.1 2.6 8.4 20.7 16.4 0.2 8.8 0.0 0.0

−0.4 0.0

– 118.

–

–

–

–

307.4 0.0 2.0 36.6 0.8 2.8 0.0 0.0 20.9 30.6 2.8 11.8 0.0 2.3 0.4 0.0 19.6 15.3 0.2 0.0 28.8 30.6

1.1 74.5 61.0 0.0 144.1 70.1 101.1 171.6 180.3 228.2 171.3 76.7 92.4 89.7 81.3 40.4 0.0 87.2 71.8 97.9 187.7 147.0

44.9 1.1 19.9 22.5 26.5 28.2 – 13.5 26.9 48.2 78.6 60.4 34.0 28.0 33.7 27.2 14.0 15.5 30.6 – 16.7 42.8 48.4

107.6 120.2 2.7 5.2 34.8 2.3 – 7.8 7.0 13.5 26.7 7.4 8.4 11.1 5.4 2.6 – 19.4 14.9 – 8.0 14.0 24.9

0.0 68.0 42.2 58.6 85.9 – 49.8 73.8 112.3 134.4 80.9 61.4 48.4 53.9 59.0 30.8 48.1 59.1 – 46.9 111.6 90.8

41.7 5.0 4.2 32.2 6.0 – 10.3 14.1 11.8 20.4 0.0 8.0 3.4 3.8 6.1 5.2 19.8 24.4 – 8.6 15.4 17.2

9.5 73.5 93.1 206.0 28.3 29.4 20.4 39.9 29 35.5 57.9 76.7 119.3 90.0 47.6 57.8 55.7 39.3 15.9 12.4 36.2 26 34.3 74.1 68.8 (Continued )

738 Table 2

Tight-binding method in electronic structure (Continued)

Struct. Struk.

f.c.c. A1

b.c.c. A2

h.c.p. A3

dia A4

bSn A5

b.c.t. A6

gra. A9

aMn A12

s.c. Ah

Tc Ru Rh Pd Ag Cd Ba Hf Ta W Re Os Ir Pt Au Ac Th Pb Po

6.1 7.5 0.0 0.0 0.0 2.5 0.8 0.9 24.6 35.8 9.1 7.9 0.0 0.0 0.0 0.0 0.0 0.0 21.3

23.4 50.9 31.6 10.2 2.8 8.5 0.0 7.1 0.0 0.0 28.1 66.0 49.9 10.0 1.0 10.0 15. 4.6 14.4

0.0 0.0 5.0 2.6 0.6 0.0 0.0 0.0 25.4 38.1 0.0 0.0 8.3 4.8 0.7 2.0 7. 2.2 5.7

72.9 134.5 159.7 148.8 66.1 45.5 12.0 380.3 188.7 159.3 38.8 11.8 83.7 169.0 70.6 125. 149. 14.6 23.9

43.4 80.8 68.0 60.9 18.2 17.0 14.1 73.8 39.8 76.5 35.1 44.4 57.7 54.8 15.4

21.8 42.0 17.2 8.6 2.5 – 0.8 6.9 11.8 31.7 26.2 47.5 20.2 5.0 0.5

53.3 124.8 120.5 109.1 50.2 35.1 52.4 203.9 107.4 124.8 35.1 65.5 96.6 131.9 55.4

0.2 16.4 16.2 9.3 6.0 4.2 2.8 22.9 9.0 18.2 0.2 17.8 23.9 14.5 8.1

13.3 7.4

2.5 22.0

24.2 14.5

1.2 1.3

56.8 106.2 96.2 85.0 24.6 21.8 23.9 115.9 64.6 115. 0 55.3 59.0 83.4 79.1 20.4 49 66 19.1 0.0

The energy of the experimental ground-state structure is arbitrarily set to zero. Energies for boron are relative to the R105 ground state (McGrady et al., 2014). All energies are calculated at the equilibrium volume found by the TB fit and are expressed in mRy. Below the common name of each phase is its Strukturbericht designation. The fit was made using LDA first-principles results except for those rows marked with an asterisk, which were fit to GGA results (Mehl and Papaconstantopoulos, 1996; Bernstein et al., 2000, 2002; Gotsis et al., 2002; McGrady et al., 2014, McGrady and Papaconstantopoulos, 2021; Sha et al., 2011; Durgavich et al., 2016)

succeeds in predicting the correct ground state for all materials presented here. Fig. 2 shows the energy volume curves for a representative sample of materials that crystallize in different structures. The results for all the elements are summarized in Table 2, which shows the equilibrium energy of each of these phases expressed as the difference in energy between that phase and the equilibrium energy of the experimental ground state. The energy of each crystal structure is computed by doing a conjugate-gradient minimization of the total energy with respect to all the parameters in the lattice, internal (atomic positions) as well as external (a, c, etc.). The TB method correctly predicts the ground-state structure for all metals including the ferromagnetic metals iron, cobalt, and nickel. The results are also consistent with the accepted crystal structure for the semiconductors Si and Ge, and with the more complex structures of Mn, Ga, and In. As a further example, Fig. 3 shows the energy/volume behavior of boron (McGrady et al., 2014) in a variety of structures. The low energy states of boron are all consist of differing arrangements of regular icosahedra, a structure not seen in other elements. Tight-binding parameters were fit to DFT results for the f.c.c., b.c.c., s.c., and dhcp and boron R12 structure, the simplest of the boron structures. These parameters correctly predict the R105 structure to be lower in energy than the other known boron structures, however the R105 + vacancy structure is found to be the ground state. There is some uncertainty in the R105 structure, with some models only partially filling some sites. The prediction of a vacancy structure may be a reflection of these models. The fact that this method finds the correct ground states of the h.c.p. metals and the other complex structures that were not included in the fit of the TB parameters should be emphasized. The robustness of the TB parameters has also been established by studying crystal structures of lower symmetry such as the volume-conserving tetragonal strain, the so-called Bain path. The Bain path of Mo at the experimentally observed equilibrium volume of the b.c.c. phase is shown in Fig. 4 (Mehl and Papaconstantopoulos, 1996). The energy is properly a minimum for the b.c. c. lattice, where c/a ¼ 1, and attains a local maximum at the f.c.c. structure (c/a ¼ √2). This shows that the f.c.c. elastic constant associated with tetragonal shear, C11 −C12, is negative; hence, the f.c.c. structure is unstable.

Elastic constants Elastic constants measure the proportionality between strain and stress in a crystal, provided that the strain is not so large as to violate Hook’s law. Computationally, the elastic constant is determined by applying a strain to a crystal, measuring the energy versus strain, and determining the elastic constant from the curvature of this function at zero strain (Mehl et al., 1994). A given strain is associated with a certain linear combination of elastic constants. For cubic systems, including diamond, there are three independent elastic constants, C11, C12, and C44. One linear combination of the elastic constants is obtained from the bulk modulus BðV Þ ¼ VE00 ðV Þ ¼ 1=3ðC11 + 2C12 Þ

(11)

Tight-binding method in electronic structure

739

Fig. 2 Energy vs. volume curves for various phases of (clockwise from the top left) copper, vanadium, germanium, and titanium, using the NRL-TB parameters described in the text. Note that the method correctly predicts the equilibrium structure (Bernstein et al., 2002; Mehl and Papaconstantopoulos, 1996, 2002).

Fig. 3 Energy per atom versus volume per atom for possible structures of boron using the NRL tight-binding fit (McGrady et al., 2014). The R105hexvac structure is the minimum energy of the R105 structure with one vacancy. The R12 structure was included in the tight-binding fit along with the standard structures. The energies of the remaining boron structures (R105, R105hexvac, T190, T50, g-28) are predicted from this fit.

where V is the unit cell volume. A second combination is most easily obtained by straining the crystal in the (1 0 0) direction while simultaneously compressing it in the (0 1 0) direction to conserve the volume, with lengths in the (0 0 1) direction remaining fixed. If x is the fractional change in the (10 0) direction, then, EðxÞ ¼ E0 + V ðC11  C12 Þ x2 + Oðx4 Þ

(12)

where the function, E(x) measures the energy as a function  of strain. Finally, one finds C44 by straining the crystal in the (1 1 0) direction and fixing the volume by compressing in the 110 direction. In this case,

740

Tight-binding method in electronic structure

Fig. 4 TB calculation of the energy of molybdenum, at the experimental equilibrium volume, under a tetragonal strain, as a function of c/a. The vertical lines denote the positions of the b.c.c. and f.c.c. lattices (Mehl and Papaconstantopoulos, 1996).

EðxÞ ¼ E0 +

1 V C44 x2 + Oðx4 Þ 2

(13)

Note that for diamond the calculation of C44 requires the minimization of the total energy with respect to an internal parameter at each strain. The calculation of elastic constants assesses the ability of the method to determine properties not included in the fitting database. The elastic constants of the cubic materials calculated by the NRL-TB approach are compared to the experimental values in Table 3. These calculations were performed at the experimental volume and therefore they are not consistent with the equilibrium value of B found in Table 1. The deviation of the calculated C11 and C12 from the measured values is within 10–15% which is very close to the error one finds when comparing with a direct evaluation from first-principles calculations. An inspection of Table 3 reveals somewhat larger errors for C44. A solution to this problem is to fit C44 either to LAPW calculations or to experiment. It should be noted here that calculations of Cij in the alkaline-earth metals is problematic because the lattice is extremely soft. One can also note that the method correctly reproduces the sign of the elastic constant difference C12 −C44, even in the metals rhodium and iridium, and the semiconductors Si and Ge where it is negative. It is worth mentioning that this negative sign cannot be obtained from the standard embedded-atom method, and that a model with only central forces would have C12 ¼ C44, the Cauchy relationship. The elastic constants for the h.c.p. structure are C11, C12, C13, C33, and C44 (Mehl et al., 1994). The elastic constants found by the NRL-TB method show larger relative deviations from experiment than found in the cubic crystals. These were found from TB parameters fitted only to cubic lattices. It is shown, in the case of Ti, that if one extends the fitting database to include the h.c.p. lattice as well, there is a dramatic improvement in the comparison of TB and measured elastic constants.

Vacancies Vacancy formation energies have been calculated by a supercell method. One atom in the supercell is removed and neighboring atoms are allowed to relax around this vacancy while preserving the symmetry of the lattice. The great advantage of the NRL-TB method over first-principles approaches is that one can do the calculation in a very large supercell, in a computationally efficient manner, including relaxation. It is found that a supercell containing 128 atoms is sufficient to eliminate the vacancy-vacancy interaction. The vacancy formation energy is given by Evac ðN Þ ¼ EðN−1Þ−ðN−1Þ EðN, 0Þ=N

(14)

where E(M, Q) is the total energy of a supercell containing M + Q sites, where M are occupied by atoms and Q are vacant, whence E(N,0) is N times the energy of one atom in its bulk crystal structure. The experimental lattice constant is used to set the volume of the system, since under experimental conditions the lattice constant of a metal containing isolated vacancies will be the lattice constant of the bulk metal. Calculated values of the vacancy formation energy of the metals (Mehl and Papaconstantopoulos, 1996) (in eV) are shown in Table 4. (Note that vacancy formation energies for Si and Ge are shown in Table 6). Relaxation around the vacancy was included via a conjugate gradient procedure but it introduced, not unexpectedly, a small effect. Where available, it is compared to both first-principles calculations and experiment (Schaefer, 1987). The results for niobium, silver, tantalum, and iridium are in excellent agreement with experiment. The vacancy formation energies for rhodium, tungsten, and platinum do not compare well with

Table 3

Elastic constants for cubic elements (in GPa). TB

Exp.

Element

Structure

C11

C12

C44

Li C Al Si Ca Ca V Fe Cu Ge Sr Sr Nb Mo Rh Pd Ag Sn Yb Ta W Ir Pt Au Ac Th Pb Po

b.c.c. dia f.c.c. dia f.c.c. f.c.c. b.c.c. b.c.c. f.c.c. dia f.c.c. f.c.c. b.c.c. b.c.c. f.c.c. f.c.c. f.c.c. dia f.c.c. b.c.c. b.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. s.c.

36.7 1036 125 179 15 18. 224 223 146 133 8 13.8 277 453 491 233 150 68 36.3 275 529 694 380 195 41.21 91.50 53 128

20.7 48 58 73 10 12.7 106 95 86 20 3 10.3 139 147 171 163 87 30 21.42 140 170 260 257 174 18.2 25.20 35 13

8.2 601 26 95 14 17. 92 78 76 107 −3 11.3 37 120 260 63 48 38 36.17 78 198 348 71 40 22.68 33.82 19 6

C11

C12

C44

1076 103 166 16 16 228 237 168 131 15 15 246 450 433 227 130 67

125 53 64 12 12 119 141 122 49 6 6 139 173 185 176 90 36

576 28 80 8 8 43 69 75 68 10 10 29 125 206 72 46 30

261 523 590 347 189

157 203 249 251 159

82 160 262 76 42

95.93 47

33.04 39

31.0 14

All calculations were fit to LDA first-principles results except for those rows marked with an asterisk, which were fit to GGA calculations (Bernstein et al., 2000, 2002; McGrady et al., 2014; McGrady and Papaconstantopoulos, 2021; Mehl and Papaconstantopoulos, 1996, 2002; Papaconstantopoulos et al., 1998; Wang et al., 2003; Durgavich et al., 2016). All elements for which a set of TB parameters have been constructed and for which one has a cubic ground state (except manganese) are presented here. A comparison is made between the results of the TB parametrization and experiment. Calculations were performed at the experimental volume.

Table 4

TB vacancy formation energies compared to first-principles calculations and experiment. TB

Element

Fixed

Relaxed

DFT

Exp.

Al Ca Cu Sr Nb Mo Rh Pd Ag Ta W Ir Pt Au Ac Th Pb

0.49 – 1.29

0.40 0.66 1.18 1.03 2.82 2.46 3.35 2.45 1.24 2.95 6.43 2.17 2.79 1.12 1.68 1.71 0.64

0.56, 0.84 EAM 0.95 1.41, 1.29 EAM 0.97

0.66

2.84 2.63 3.39 2.46 1.31 3.17 6.86 2.19 2.79 1.24 0.76

2.26 1.57 1.20, 1.06

1.28–1.42 2.62  0.03 3.0–3.6 1.71 1.85  0.25 1.11–1.31 2.9  0.4 4.6  0.8 1.97 1.35  0.09 0.89  0.04 0.95 0.54

The LDA density functional was used for fitting and comparisons, except for those rows marked with an asterisk, which use GGA. All energies are in eV (Mehl and Papaconstantopoulos, 1996). Energies were computed using a 128-atom supercell. Calculations with the atoms at the primitive lattice sites in the crystal (fixed) and allowing relaxation around the vacancy (relaxed) are shown. First-principles calculations of the vacancy formation energy are given in the column labeled “LDA.” The experimental column shows a range of energies if several experiments have been tabulated. Otherwise, the estimated error in the experiment is given. TB results are from Mehl and Papaconstantopoulos (1996). Experimental data are from Schaefer (1987).

742

Tight-binding method in electronic structure

experiment. Aluminum is an intermediate case. It is possible that the measured values are not reliable. Otherwise, there may be a need to check these results against first-principles data and perhaps fit to DFT calculations.

Surfaces Surface energies are also calculated by a supercell technique (Haftel et al., 2004; Lekka et al., 2003a, 2003b; Mehl and Papaconstantopoulos, 1996). A slab of metal is formed by cleaving the crystal along the desired plane, creating two identical free surfaces. The distance between the two surfaces is increased, creating a set of slabs that repeat periodically in the direction perpendicular to the surfaces. The slabs are separated by a large region of vacuum so that the electrons on one slab cannot hop to a neighboring slab. In addition, the slabs must be thick enough so that the atoms at the center of the slab have the electronic properties of atoms in the bulk material and the two surfaces on the same slab cannot interact with each other. It is found that these criteria are met if slabs containing 25 atomic layers and 7–13 layers between slabs depending on the direction of the surface are used. Care should be taken with respect to the k-point mesh convergence. Depending on the surface and underlying bulk structure, this requires  200k-points in the two-dimensional Brillouin zone. The surface energy, expressed as the energy required to create a unit area of new surface, is then given by the formula Esurf ¼ ðEslab  N Ebulk Þ=ð2 AÞ,

(15)

where A is the area occupied by one unit cell on the surface of the slab, Eslab is the total energy of the slab, N is the number of atoms in the unit cell, and Ebulk is the energy of one atom in the bulk at the lattice constant of the atoms in the interior of the slab. In the first surface calculations, the bulk equilibrium lattice parameters with no relaxation or reconstruction at the surface were used. These calculations showed reasonable agreement with experiment indicating that relaxation does not seriously affect the surface energy. However, recently calculations with all atoms completely relaxed were performed, including charge self-consistency. The new results for Nb (Lekka et al., 2003b) and several f.c.c. metals (Haftel et al., 2004) show significant differences in the interlayer separations and reconstruction. The results for the f.c.c. metals are particularly gratifying. For the f.c.c. metals, it is found that E(1 1 1) < E (1 0 0) < E (1 1 0), which means that close-packed surfaces are the most stable for the f.c.c. metals. For the b.c.c. metal surface energies, the opposite inequality is satisfied, E(1 1 0) < E(1 0 0) < E (1 1 1).

Stacking faults Stacking faults are introduced in a crystal by cutting a perfect crystal block along a plane and shifting the upper part with respect to the lower part by a vector f, defining the generalized stacking fault energy surface (Mehl et al., 2000). Local energy minima on this surface are called pffiffiffi stable intrinsic stacking faults. In f.c.c. systems and for the (111) slip plane, the stable stacking fault corresponds to a slip of a= 6 in the direction. The < 1 2 1> slip is modeled on a (1 1 1) slip plane by a supercell consisting of nine close-packed (1 1 1) planes of atoms. The primitive vectors of the supercell are 1 1 a1 ¼ ab y + ab z 2 2 1 1 x + ab z a2 ¼ ab 2 2   q q  q ab x+ 3+ ab y− 3− ab z a3 ¼ 3 + 6 6 3

(16)

where q is the stacking fault displacement of the atoms in the boundary layer along the vector f in the < 1 1 2> direction. At q ¼ 0, the atoms are in f.c.c. positions and at q ¼ 1, at the h.c.p. positions. Since the calculations are computationally intensive because of the nine planes of atoms and the 4730k-points used in the irreducible part of the Brillouin zone, the NRL-TB method proves to be very efficient. The results of the stacking fault energy as a function of the displacement q are shown in Fig. 5. The results correctly predict the qualitative effect that the energies for Ir are much larger than that for the noble metals and they also agree with full-potential linearized muffin-tin orbital (FLMTO) results for Al, Ag and Ir. Finally, the value at the maximum energy in Fig. 5 has been used to extract the unstable stacking fault energy gus, which is used together with the surface energy gsurf to establish the Rice criterion for ductility, D ¼ 0:3 gus =gsurf :

(17)

One finds, in agreement with experiment, that the noble metals have the largest values of D.

Phonons Phonon frequencies at high-symmetry points in the Brillouin zone computed with the NRL-TB model using the frozen phonon approximation are compared with experimentally measured values in Table 5. In this table, results from metals and semiconductors

Tight-binding method in electronic structure

743

Fig. 5 Stacking fault energy as a function of the parameter q in [16] for the elemental f.c.c. metals, as determined by the NRL-TB method. The labels gus and gis indicate the positions of the unstable and intrinsic stacking faults, respectively, while the f.c.c. label on the left vertical axis indicates the position of the ground-state bulk structure (Mehl et al., 2000).

Table 5 Phonon frequencies (in THz) at high symmetry points for various elements (Gotsis et al., 2002; Mehl and Papaconstantopoulos, 2002; Papaconstantopoulos et al., 1998).

C (dia)

Mg (h.c.p.) Si (dia)

Ca (f.c.c)

Ti (h.c.p.) Cu (f.c.c.)

Sym.

TB

Exp.

G250

39.3 34.2 29.8 24.7 39.3 36.5 17.6 35.3 12.1 4.3 7.1 3.9 15.9 12.1 15.2 4.8 3.8 16.0 3.45 3.12 4.26 2.17 2.62 3.20 3.17 2.55 5.7 4.2 5.3 2.9 7.2 4.4

39.9 35.5 32.1 24.2 36.6 31.0 16.9 36.2 7.3 3.7 2.9 5.2 15.5 12.3 13.9 4.5 3.4 14.7 4.52 3.63 4.61 2.36 3.66 4.61 3.66 2.36 5.5 4.1 5.7 3.0 7.3 5.1

X1 X3 X4 L1 L20 L+3 L−3 G+3 G+5 A1 A3 G250 X1 X3 X4 L+3 L−3 X3 X5 L2’ L3 W2’ W3 D1 D5 G+3 G+5 A1 A3 X3 X5

(Continued )

744

Tight-binding method in electronic structure Table 5

Ge (dia)

Sr (f.c.c.)

Nb (b.c.c.)

Mo (b.c.c.) Au (f.c.c.) Ac (f.c.c.)

Th (f.c.c.)

Pb (f.c.c.)

(Continued) Sym.

TB

Exp.

L2 L3 G250 X1 X3 X4 L1 L20 L+3 L−3 X3 X5 L2’ L3 W2’ W3 D1 D5 S1 S2 S3 H P N2 N3 N4 H

7.9 3.5 9.7 6.4 8.9 3.2 9.3 5.2 2.5 7.3 3.56 2.32 3.52 1.49 2.31 3.04 2.32 1.62 2.92 1.46 2.49 6.3 5.7 4.9 5.9 4.4 4.1

7.3 3.4 9.1 7.2 8.3 2.4 8.7 6.7 1.9 7.3 3.17 2.48 3.08 1.74 2.57 – 1.90 1.50 2.70 1.40 2.40 6.5 5.0 5.1 5.7 3.9 5.5

X3 X5 L2 L3 X3 X5 L2’ L3 W2’ W3 D1 D5 S1 S2 S3 X3 X5 L2’ L3 W2’ W3 D1 D5 S1 S2 S3 X3 X5 L2 L3

5.3 2.9 5.5 1.9 2.35 1.84 1.35 0.53 1.83 1.58 1.56 1.04 1.71 1.12 1.35 2.03 3.34 2.78 1.94 2.34 3.35 2.95 1.97 3.39 1.94 2.65 1.9 1.0 2.2 1.0

4.6 2.7 4.7 2.0

3.47 2.26 3.24 1.28 2.88 2.22 2.42 1.59 3.06 2.24 1.50 1.8 0.9 2.2 0.9

Tight-binding method in electronic structure

745

Fig. 6 Phonon dispersion curves for diamond-structured Si computed from the velocity-velocity correlation function found during a molecular-dynamics simulation at 300 K (solid line) and 1500 K (dashed line) (Bernstein et al., 2000). Dots are taken from experimental data (Tubino et al., 1972).

Fig. 7 Phonon frequencies along high symmetry lines in Ge using the sp3 basis TB parameters (Bernstein et al., 2002) (solid symbols) compared to DFT/LDA results (Giannozzi et al., 1991) (open symbols) and experiment (crosses) at 80 K (Nelin and Nilsson, 1972).

have been included. The agreement with experiment is similar to that with elastic constants. For most modes the TB results are within 15–20% of experiment, which is also comparable to phonon frequencies obtained directly from first-principles calculations. For some materials, larger errors as for the L1, and W2 modes in silicon, and the P and N modes in niobium have been found. This issue has been addressed for Nb (Lekka et al., 2003b) whereby augmenting the first-principles database by including the energies at displacement of 0.005 lattice coordinates for the high symmetry points P and N, the overall fit of the phonon spectrum has been drastically improved. Figs. 6 and 7 show the phonon dispersion curves of Si and Ge that indicate a good agreement with the measured spectra.

Point defects Point defects in the semiconductors Si (Bernstein et al., 2000) and Ge (Bernstein et al., 2002) have been calculated. Point defects are a major source of disorder in semiconductors since they are thermodynamically favored to occur at finite concentrations and temperatures. Because they are far more mobile than perfectly bonded atoms, they dominate diffusion. The formation energy of such defects strongly influences their concentrations and is, therefore, an important material property. Table 6 lists the formation energies of three interstitial configurations: the tetrahedral, hexagonal, and 110 split, for Si and Ge. These defect energies were computed using a cubic supercell at the equilibrium lattice constant with 216 atoms, sampling the Brillouin zone at the G point. The relaxed configurations were obtained by a conjugate-gradient algorithm using as a criterion the force on each atom to be less than 3 meVA−1. The formation and relaxation energies of all three interstitial configurations are within 10% of the range of LDA calculations for Si but not as good for Ge. On the other hand, the relaxed geometries show more serious discrepancies from ab initio results.

746

Tight-binding method in electronic structure

Table 6 Formation energies Ef (ideal) and relaxation energies DEf (relaxed), in eV, for point defects computed using the TB model and comparison to DFT/LDA results (Bernstein et al., 2000, 2002). TB

V It Ih h110i

LDA

TB

LDA

E0f

Ef

E0f

Ef

E0f

Ef

Ef

Si 4.2 4.8 5.6 3.7

3.2 4.5 5.1

3.3–4.3 3.7–4.8 4.3–5.0 3.3

2.9–3.7 3.6–4.6 3.7–3.9

Ge 4.6 3.1 5.6 4.3

3.6 2.5 4.4 3.3

1.9 3.2 2.9 2.3

Since the structure of the ideal split interstitial is not uniquely defined, the energy listed under ideal is actually the relaxed formation energy.

Fig. 8 Mean-squared displacement of atoms in diamond-structured silicon as a function of temperature, as determined by TBMD (Bernstein et al., 2000). The dots represent the TBMD data, and the lines are linear (solid) and higher-order representations of the experimental data. From Batterman BW and Chipman DR (1962) Vibrational amplitudes in germanium and silicon. Physical Review 127: 690.

Finite temperature properties from molecular dynamics The TB method described above can be used with the NRL-TB molecular dynamics (TBMD) package developed by Kirchhoff et al. (2001) to determine thermodynamic properties. For example, the TBMD package can be used to describe computed mean square displacements, and pair correlation functions as a function of temperature, as well as thermal expansion, in silicon, germanium, and gold. It is also possible to extract phonon-dispersion curves from the velocity autocorrelation function determined in an MD simulation. As an example of the power of the method, one can describe the calculation of the mean-squared displacement of diamond structure silicon as a function of temperature (Bernstein et al., 2000). The calculation used a 512-atom unit cell, evolving at constant energy with fixed initial kinetic energy for 2000 timesteps of length 2.0 fs. As seen in Fig. 8, results are compared with the experimentally determined values, computed from the experimental measurements of the temperature dependence of broadening of the X-ray diffraction peak. The agreement is excellent. The TBMD code can also determine the change in pressure versus temperature at constant volume. Coupled with the bulk modulus found in Table 1, we can determine the linear coefficient of thermal expansion,     1 ∂V 1 ∂P ¼ (18) a ¼ 3V ∂T P 3B ∂T V Table 7 compares the results for selected elements to experiment. da Silva et al. (2001) used the NRL-TB parameters for Au to present a very interesting simulation of the formation, evolution, and breaking of Au nanowires.

Multicomponent systems The above work was confined to single-component systems. The method is extendable to systems with more than one atomic species by making two modifications. First, one adds additional terms to the onsite energies which couple the density from atoms of type i to atoms of type j. Second, one notes that the SK parameters pss, dss, dps, and dpp, which in the single-atom case are

Tight-binding method in electronic structure Table 7 elements.

747

Linear thermal expansion coefficient a (in K−1) calculated using the TBMD with the NRL-TB parameters for selected

Element

Calculated

Experiment

Ca Sr Be Cu Ag Au

3.07  10−5 2.45  10−5 1.52  10−5 2.50  10−5 1.07  10−5 1.43  10−5

2.23  10−5 2.25  10−5 1.20  10−5 1.80  10−5 2.00  10−5 1.40  10−5

The parameters were fit to first-principles LDA calculations except for those elements marked with an asterisk, which are GGA (Chellathurai et al., 2020; McGrady and Papaconstantopoulos, 2021; Silayi et al., 2018).

equivalent (within a sign) to sps, sds, pds, and pdp, respectively, are now independent parameters. These complications raise the total number of parameters from 93 in the single-atom case, to 330 for two-atom type systems. This method has been applied to several compounds, including NbdC, NbdMo, PddH, FedAl, PtdO, CuZr, CudAg, and CudAu (Mehl and Papaconstantopoulos, 1996; Lekka et al., 2009; Silayi et al., 2020). For CudAu, the energy of a variety of structures was computed, and it was found that the expected ordered phases (the L12, Cu3Au, and L10 CuAu) were stable, while other structures were not. However, the advantage of the TB model is that one can compared to structures which are not found experimentally, allowing one to explore configurations that might occur locally in an alloy, but which do not form long-range ordered states. To check the correctness of the CudAu parameters in this case, it is compared to the cluster-expansion and first-principles calculations of Ozolin¸š et al. Fig. 9 shows formation energies computed for several ordered phases in the CuxAu1− x system and compares them to the energies obtained by Ozolin¸š et al. (1998). It is found that the agreement is excellent. We have also applied the NRL-TB method to understanding the electronic structure of the high-temperature superconductor SH3 (Drozdov et al., 2015; Papaconstantopoulos et al., 2015). The tight-binding Hamiltonian is a 12  12 matrix including the s, p, and d orbitals of sulfur and three hydrogen s-orbitals. We have simultaneously fitted the energy bands and total energies as a function of volume for the Im3m (superconducting), A15 and L12 structures. The total energy (or formation energy) for these three structures is shown in Fig. 10 as a function of volume. The agreement between TB and LAPW is excellent. The energy bands and DOS are shown in Fig. 11 for the bcc-like Im3m structure at the lattice constant a ¼ 5.6 Bohr, corresponding to a pressure of 200 GPa where high temperature superconductivity of Tc ¼ 190 K has been computationally predicted and observed experimentally. The excellent agreement between TB and LAPW for both bands and DOS is also noted. Table 8 summarizes the DOS at the Fermi level at its decomposition per site and angular momentum the superconducting Im3m structure at a lattice constant of 5.6 Bohr. The strongest contributions to N(Ef) are from the sulfur p- and hydrogen s-states, with a significant contribution from the sulfur d-states.

Fig. 9 Formation energy of several ordered phases in the CuxAu1− x system, calculated using the TB parameters (blue bars), and compared to first-principles calculations performed by Ozolin¸š et al. (1998) (red bars). On this scale, the cluster-expansion energies are indistinguishable from the corresponding LAPW results. For comparison, the first-principles LAPW results (green bars), which were used in the Cu-Au TB fitting process are also plotted.

748

Tight-binding method in electronic structure

0.20

L12

Energy (Ry)

0.15

0.10 A15 0.05 Im3m 0.00 100

110

120

130

140

150

160

170

180

3

Volume (Bohr ) Fig. 10 Comparison of tight-binding (dashed lines) and LAPW (solid lines) total energies for three phases of SH3. Energies and volumes are for one formula unit of each compound.

One- and two-dimensional structures The NRL-TB method was applied to one- and two-dimensional structures of Cu and Au in conjunction with the conjugate gradient method to study nanoparticles. The TB parameters used (Silayi et al., 2018) were derived by fitting to LAPW calculations of periodic f.c.c., b.c.c., and simple cubic structures. It is impressive that this TB Hamiltonian works well for one- and two-dimensional structures. These TB results are in good agreement with binding energies and bond lengths found by independent first-principles DFT calculations which were not included in the fit. The conjugate gradient method is used to relax the clusters. Nanoparticles with elongated geometries were made by cutting atoms from the f.c.c. crystalline structure. There are four categories of structures which we labeled A, B, C and D. In category (A) the elongation grows along the [100] direction. In category (B) the elongation is in the [011] direction. In category C the elongation starts with the atoms placed as in (B) and then grows in the [0–11] direction. In category (D) the structure is similar to (A) but it continues along the [1–10] axis. The TB results for binding energy and bond lengths were compared with DFT results for Cu and Au nanoparticles of small number of atoms (between 40 and 60 atoms) where DFT calculations can be done within reasonable computer times. The TB v. DFT comparison was good, motivating us to use the NRL-TB method to perform calculations in larger structures where, due to computational time and memory issues, DFT calculations are problematic. Our TB calculations were able to simulate structures with up to 800 atoms efficiently. The calculation times show that TB is at least three orders of magnitude faster than DFT. For each category we started with a small structure (for example in category (A), we started with a structure with 40 atoms). Next, we cut the structure in half, moving the two parts in opposite directions. Finally, we add the corresponding atoms to create the largest structure (in this case the structure with 50 atoms, next the structure with 60 atoms, etc.). An important issue in this process was that the nearest neighbor distance of the initial structure must be reasonable (2.6 A˚ ngstrom or smaller) to prevent structure breaking. The binding energies (BE) per atom for Cu and Au NPs, as calculated with the TB method, are shown in Fig. 12. By comparing with the DFT results for both Cu and Au we find that the TB results follow the same trend as the DFT results. That is, the BE of category (B) has the lowest values, the category (D) follows, and the category (A) has the highest values of all categories. Specifically, the BE energies for Cu with 200 atoms we have: category (A) BE 3.94 eV, category (D) 3.90 eV and category (B) 3.72 eV. The corresponding values for Au are: category (A) 3.29 eV, category (D) 3.27 eV and category (B) 3.22 eV. Furthermore, we observe that the TB BE difference between smaller and larger 0D structures are 1.50 eV for Cu and 1.03 eV for Au. The corresponding results for DFT are 1.20 eV and 0.88 eV for Cu and Au, respectively; a quantitative difference but with the same trend. After establishing that the TB results for small systems are in good agreement with the DFT ones, larger structures were considered. Fig. 13 shows TB results for Cu NPs of the (D) category. The lengths along the longest direction are 5.18, 7.73, 10.27, 14.39 and 19.26 nm respectively for Cu205, Cu304, Cu403 Cu601 and Cu808. Similarly, Au NPs have lengths of 4.79 and 9.78 nm for Au205 and Au403 respectively. The BE per atom of Cu NPs (with 300 atoms), converge to values of 3.97, 3.92 and 3.84 eV for the (A), (D) and (B) categories respectively. The corresponding saturated BE values for Au NPs (with 300 atoms), are 3.284, 3.287 and 3.220 eV for the (A), (D) and (B) categories.

Tight-binding method in electronic structure

(A)

1

LAPW TB

0.5

ε − εF (Ry)

0

−0.5

−1

−1.5

−2

Γ

Δ

H

G

N

Σ

Γ

Λ

P D N D P

F

H

(B)

14

LAPW TB

ρ(ε) (States/eV/cell)

12 10 8 6 4 2 0 −2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

ε − εF (Ry) Fig. 11 Comparison of the LAPW (solid lines) and tight-binding (dashed lines) electronic structure (a) and density of states (b) for the high-temperature superconductor SH3 at the cubic lattice constant 5.60 Bohr.

Table 8

Total electronic density of states and the orbital contributions to the DOS at the Fermi level of the Im3m superconducting phase of SH3.

N(Ef)

S-s

S-p

S-d

H-s

6.05

0.30

2.07

1.62

2.06

This data was extracted from the tight-binding fit at the lattice constant of 5.6 Bohr. The DOS is given in units of states/Rydberg/cell for both spins. The hydrogen s value includes the contributions from all three atoms.

749

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Tight-binding method in electronic structure

Fig. 12 The Binding Energy per atom of (a) Cu and (b) Au structures, using Tight Binding.

We have also calculated the HOMO-LUMO (HL) Gap and the Density of States (DOS). The TB results are compared with the corresponding DFT results in Fig. 14 for the HL gap and Fig. 13 for the DOS showing good agreement. We note that the HL gap for the smaller structures have higher values compared to the HL gap for the larger structures. This happens due to the transition from semiconductor to metallic behavior. This transition occurs at about 30–50 atoms systems. Generally, the HL gap of DFT has somewhat lower values compared to TB. For DFT, the values of the HL gap for Cu25 (category D) and Cu80 (category B) structures are 0.69 eV and 0.10 eV, respectively. The corresponding values for TB are 0.75 eV and 0.18 eV. This discrepancy is probably due to the following two factors: (a) the TB parameters used result from a fit to LAPW results from periodic cubic structures where the exchange and correlation functional (Hedin-Lundqvist) is different than that in the DFT code, and (b) different temperature broadening parameters used in the respective methods for calculating the DOS. In Fig. 15 we show the per one atom DOS for different sizes of Cu and Au systems (24 and 108 atoms). The per one atom DOS shown are similar, characterized by the dominant d- states of Cu and Au, below the Fermi level and very small s-like DOS values at Ef reminiscent of the DOS of bulk periodic Cu. The DOS for small number of atoms shows a gap at the Fermi level (see Cu24 and Au24 in Fig. 14). For larger systems like Cu108 and Au108, this gap is filled by s-like states. Furthermore, we calculate the per one atom DOS for the Cu601 and Cu808 cases. The graphs for those cases are similar to the corresponding Cu108 graph.

Tight-binding method in electronic structure

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Fig. 13 The Copper structures over 200 atoms, that grow along the [110] direction (category D), using Tight Binding. More specifically, (a) Cu205, (b) Cu304, (c) Cu403, (d) Cu601 and Cu808.

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Tight-binding method in electronic structure

Fig. 14 The HOMO-LUMO Gap of the Cu (solid lines) and Au (dashed lines) structures (A, B and D categories), calculated with (a) DFT and (b) Tight Binding method.

Tight-binding method in electronic structure

Fig. 15 The DOS per one atom for (a) Cu24—category B, (b) Cu108—category B, (c) Cu808—category D, (d) Au24—category B and (e) Au108—category B.

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Tight-binding method in electronic structure

Conclusion We have shown that the NRL-TB method generates TB Hamiltonians that fit very well first-principles electronic structure results such as LAPW energy bands, density of states and total energies for most elements in the periodic Table and several compounds. This scheme accurately determines the ground state and the equilibrium lattice parameter of the system and positions well the total energies of other structures that were not included in the fit. Also, mechanical properties like elastic constants and bulk moduli as well as phonon frequencies are found in good agreement with experiment without being included in the database of the original fit. Large scale molecular dynamics simulations give a good account of mean-squared displacements, vacancy formation energies and the coefficient of thermal expansion, with a much smaller computational cost than most first-principles calculations. It should be emphasized that the NRL-TB approach is successful in obtaining a wide spectrum of properties unlike other TB or empirical potential schemes which usually have a restricted range of applications.

Acknowledgments The authors thank the collaborators in this work. This work was supported by the US Office of Naval Research and the US Department of Energy.

References Bacalis NC, Papaconstantopoulos DA, Mehl MJ, and Lach-hab M (2001) Transferable tight-binding parameters for ferromagnetic and paramagnetic iron. Physica B: Condensed Matter 296: 125. Bernstein N, Mehl MJ, Papaconstantopoulos DA, Papanicolaou NI, Bazant MZ, et al. (2000) Energetic, vibrational, and electronic properties of silicon using a nonorthogonal tight-binding model. Physical Review B 62: 4477. Erratum: Physical Review B 65: 249002(E). Bernstein N, Mehl MJ, and Papaconstantopoulos DA (2002) Nonorthogonal tight-binding model for germanium. Physical Review B 66: 075212. Chellathurai M, Gogovi GK, and Papaconstantopoulos DA (2020) Electronic structure and tight-binding molecular dynamics simulations for calcium and strontium. Materialia 14: 100915. Cleri F and Rosato V (1993) Tight-binding potentials for transition metals and alloys. Physical Review B 48: 22. Cohen RE, Mehl MJ, and Papaconstantopoulos DA (1994) Tight-binding total-energy method for transition and noble metals. Physical Review B 50: 14694. da Silva EZ, da Silva JR, and Fazzio A (2001) How do gold nanowires break? Physical Review Letters 87: 256102. Drozdov AP, Eremets MI, Troyan IA, Ksenofontov V, and Shylin SI (2015) Conventional superconductivity at 203 Kelvin at hi pressures in the sulfur hydride system. Nature 525: 73. Durgavich J, Sayed S, and Papaconstantopoulos D (2016) Extension of the NRL tight-binding method to include f orbitals and applications in Th, Ac, La and Yb. Computational Materials Science 112: 395. Finnis MW and Sinclair JE (1984) A simple empirical N-body potential for transition metals. Philosophical Magazine A 50: 45. Giannozzi P, de Gironcoli S, Pavone P, and Baroni S (1991) Ab Initio calculation of phonon dispersions in semiconductors. Physical Review B 43: 7231. Gotsis HJ, Papaconstantopoulos DA, and Mehl MJ (2002) Tight-binding calculations of the band structure and total energies of the various phases of magnesium. Physical Review B 65: 134101. Gross A, Scheffler M, Mehl MJ, and Papaconstantopoulos DA (1999) Ab initio based tight-binding Hamiltonian for the dissociation of molecules at surfaces. Physical Review Letters 82: 1209. Groß A, Eichler A, Hafner J, Mehl MJ, and Papaconstantopoulos DA (2003) Unified picture of the molecular adsorption process: O2/Pt (111). Surface Science 539: L542. Haftel MI, Bernstein N, Mehl MJ, and Papaconstantopoulos DA (2004) Interlayer surface relaxations and energies of fcc metal surfaces by a tight-binding method. Physical Review B 70: 125419. Harrison WA (1989) Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond. New York: Dover. Kirchhoff F, Mehl MJ, Papanicolaou NI, Papaconstantopoulos DA, and Khan FS (2001) Dynamical properties of Au from tight-binding molecular-dynamics simulations. Physical Review B 63: 195101. Lekka CE, Bernstein N, Mehl MJ, and Papaconstantopoulos DA (2003a) Electronic structure of the Cu3Au (111) surface. Applied Surface Science 219: 158. Lekka CE, Mehl MJ, Bernstein N, and Papaconstantopoulos DA (2003b) Tight-binding simulations of Nb surfaces and surface defects. Physical Review B 68: 035422. Lekka CE, Papaconstantopoulos DA, Mehl MJ, Finkenstadt D, and Evangelakis G (2009) Static and dynamic tight-binding simulations of the binary NbMo and CuZr alloys. Journal of Alloys and Compounds 483: 627. McGrady JW and Papaconstantopoulos DA (2021) Tight-binding study of beryllium. Computational Materials Science 188: 110142. McGrady JW, Papaconstantopoulos DA, and Mehl MJ (2014) Tight-binding study of boron structures. Journal of Physics and Chemistry of Solids 75: 1106. Mehl MJ and Papaconstantopoulos DA (1996) Applications of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals. Physical Review B 54: 4519. Mehl MJ and Papaconstantopoulos DA (2002) Tight-binding study of high-pressure phase transitions in titanium: Alpha to omega and beyond. Europhysics Letters 60: 248. Mehl MJ and Papaconstantopoulos DA (2005) Tight-binding total energy methods for magnetic materials and multi-element systems. In: Yip S (ed.) Handbook of Materials Modeling, pp. 275–305. The Netherlands: Springer. ch. 1.14. Mehl MJ, Klein BM, and Papaconstantopoulos DA (1994) First-principals calculation of elastic properties. In: Westbrook JH and Fleischer RL (eds.) Intermetallic Compounds—Principles and Practice. vol. 1, pp. 195–210. London: Wiley. ch. 9. Mehl MJ, Papaconstantopoulos DA, Kioussis N, and Herbranson M (2000) Tight-binding study of stacking fault energies and the Rice criterion of ductility in the fcc metal. Physical Review B 61: 4894. Nelin G and Nilsson G (1972) Phonon density of states in germanium at 80 K measured by neutron spectrometry. Physical Review B 5: 3151. Ozolin¸š V, Wolverton C, and Zunger A (1998) Cu-Au, Ag-Au, Cu-Ag, and Ni-Au intermetallics: First-principles study of temperature-composition phase diagrams and structures. Physical Review B 57: 6427. Papaconstantopoulos DA (2015) Handbook of the Band Structure of Elemental Solids. New York: Springer. Papaconstantopoulos DA and Mehl MJ (2003) The Slater-Koster tight-binding method: A computationally efficient and accurate approach. Journal of Physics. Condensed Matter 15: R413. Papaconstantopoulos DA, Mehl MJ, Erwin SE, and Pederson MR (1998) Tight-binding Hamiltonians for carbon and silicon. Materials Research Bulletin 491: 221.

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Papaconstantopoulos DA, Klein BM, Mehl MJ, and Pickett WE (2015) Cubic H3S around 200 GPa: An atomic hydrogen superconductor stabilized by sulfur. Physical Review B 91: 184511. Schaefer H-E (1987) Investigation of thermal equilibrium vacancies in metals by positron annihilation. Physica Status Solidi 102: 47. Sha XW, Papaconstantopoulos DA, Mehl MJ, and Bernstein N (2011) Tight-binding study of hcp Zn and Cd. Physical Review B 84(18): 184109. https://doi.org/10.1103/ PhysRevB.84.184109. Shi L and Papaconstantopoulos DA (2004) Modifications and extensions to Harrison’s tight-binding theory. Physical Review B 70: 205101. Silayi S, Papaconstantopoulos DA, and Mehl MJ (2018) A tight-binding molecular dynamics study of the noble metals Cu, Ag, and Au. Computational Materials Science 146: 278. Silayi S, Chang P-H, and Papaconstantopoulos DA (2020) Electronic structure and dynamics analysis of the Cu-Ag binary alloy by a tight-binding parameterization. Computational Materials Science 173: 109454. Slater JC and Koster GF (1954) Simplified LCAO method for the periodic potential problem. Physical Review 94: 1498. Tubino R, Piseri L, and Zerbi G (1972) Lattice dynamics and spectroscopic properties by a valence force potential of diamondlike crystals: C, Si, Ge, and Sn. Journal of Chemical Physics 56: 1022. Wang GM, Papaconstantopoulos DA, and Blaisten-Barojs E (2003) Pressure induced transitions in calcium: A tight-binding approach. Journal of Physics and Chemistry of Solids 64: 185. Yang SH, Mehl MJ, and Papaconstantopoulos DA (1998) Application of a tight-binding total-energy method for Al, Ga, and In. Physical Review B 57: R2013.

Electronic Structure Calculations: Plane-Wave Methods ML Cohen, University of California at Berkeley, Berkeley, CA, USA © 2005 Elsevier Ltd. All rights reserved. This is an update of M.L. Cohen, Electronic Structure Calculations: Plane-Wave Methods, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 85–93, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00449-6.

Introduction The Standard Model of Solids Pseudopotentials Total Energy Calculations and Structural Properties Superconductors, Optical Properties, and Novel Materials Conclusion Acknowledgments Further Reading

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Introduction Understanding the electronic structure of solids is basic to explaining and predicting solid-state properties, but what is an appropriate description of electronic states in solids? Although it is clear that solids are made of atoms and that one can think of the gas-to-liquid-to-solid transitions as a process in which atoms just get closer together, the electronic properties of solids often differ significantly from what one would expect from a collection of isolated atoms. In cases where the interactions between the atoms are weak, as in the case of rare gas solids, it is reasonable to consider these interactions as perturbations of atomic states. However, when atoms are close to each other in a solid, sometimes they can give up their outermost electrons easily, and it is even possible to end up with a sea of nearly free electrons. In fact, a nearly free-electron model is appropriate for metals such as the alkali metals, aluminum, and many others. It is also fortunate and useful that this nearly free-electron model applies to a wider class of solids when a few modifications are added. A totally free electron is described by a plane-wave wave function. Hence, if one assumes that the electrons describing a metal behave as if they are nearly free, then a plane-wave position-dependent wave function C(r) confined to a box of volume O, having a wave vector (k) which can be written as 1 cðr Þ ¼ pffiffiffiffi eikr O may be a good starting point for describing the electrons in metals such as sodium. If one examines the resulting electronic density corresponding to a plane-wave wave function, it is constant since it is proportional to c ðr Þcðr Þ. Therefore, the collection of atoms with localized wave function on each atomic site and peaked electronic density yields a valence electron density which is vastly different. What about bonds? For example, covalent bonds arise because electrons tend to concentrate in certain regions between atoms. Can the bonds and charge modulation known to exist in metals and especially in covalent semiconductors be described using plane waves as a starting point? The answer is yes for a broad class of solids. A plane wave is a solution to Schrödinger’s equation when the electron–ion and electron–electron potentials are zero. However, when these potentials are included (and if they are not too strong and if the electrons are not strongly correlated), a perturbative scheme can be used to modify the free-electron picture, and this is the “nearly free-electron model” which is described in many textbooks. However, more generally, Schrödinger’s equation is solved by diagonalizing a Hamiltonian matrix with appropriate potentials and a basis set composed of plane waves to describe the wave function. What about alternatives to the nearly free-electron model? The opposite approach to using plane waves is the utilization of atomic wave functions for the basis set. There are tight-binding methods where linearized combinations of atomic orbitals form the basis. This description is also very useful especially when the solid-state wave functions are not too different from the atomic wave functions forming them. If done correctly with sufficient orbitals in the basis set, both the above methods work and are useful. In the following discussion, the plane-wave basis is featured. In particular, a pseudopotential-density functional approach is described which has been applied successfully to a large number of solids, clusters, nanostructures, and molecules.

The Standard Model of Solids In Figure 1, a schematic picture of a solid is presented where the nuclei and core electrons for the atoms are represented as positive cores and the stripped-off valence electrons are contributed to the electron sea which flows through the periodic array of cores. In this model, the solid system is viewed as having two types of particles: cores and valence electrons. The interaction between the positive cores is taken to be a standard Coulomb interaction describable by Madelung sums. The electron–core and electron–electron interactions are essential inputs to this model. As mentioned above, the form of the standard model of solids, represented by Figure 1, which is focused here, will model the electron–core interaction using pseudopotentials and the electron–electron interaction using the density-functional theory – in particular, the local density approximation (LDA).

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Nucleus Core electrons Valence electrons Figure 1 Schematic view of a solid with positive cores at fixed lattice positions in a periodic array and a sea of relatively free valence electrons moving through the lattice.

Pseudopotentials Arguments for the use and/or appropriateness of a weak electron–core potential have been made since the 1930s. In 1934, Fermi constructed a pseudopotential to describe highly excited electronic states of alkali atoms. His weak potential was designed to yield an accurate description of the outer portions of the wave function and to avoid the more difficult calculation of the properties of the oscillations of the wave function near the core. This would require a strong potential. Since solid-state effects are dominated by the outer portions of atomic wave functions, this is a viable approach for constructing pseudopotentials for solids. As shown in Figure 2,

1.0 3s Radial wave function of Si

0.5

Pseudoatom All-electron 0

0.5

0

1

2 3 Radial distance (a.u.)

4

5

Figure 2 The solid line represents the Si 3s pseudo-wave-function resulting from the use of a pseudopotential. The dashed line results from an “all electron” calculation.

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Electronic Structure Calculations: Plane-Wave Methods

a “pseudoatom” wave function is generated which is identical to an “all-electron” 3s radial wave function for Si away from the core and smooth near the core. The weak pseudopotential which produced the pseudoatom wave function can be computed from first principles using only the atomic number as input. There are many schemes available for producing these ab initio pseudopotentials, which in principle can be constructed for all atoms. The electron–electron interactions can be approximated using the electronic density. This density-functional approach represents the Hartree and exchange-correlation potentials as functionals of the electronic density. Hence, a self-consistent calculation can be done with a plane-wave basis set, pseudopotentials, and the LDA to density-functional theory. The calculated electronic density can be used to produce LDA potentials which in turn can produce updated wave functions and subsequently, new LDA potentials. This scheme and variants using pseudopotentials and plane waves are often referred to as the “plane-wave pseudopotential method” (PWPM). In general, the method has been extremely successful for determining the electronic structure and many other properties of solids. Although most modern pseudopotential calculations use ab initio pseudopotentials of the type described above, the pseudopotential method went through a semi-empirical stage of development where experimental input was used to generate a potential representing the total valence electron potential (electron–core plus electron–electron). This approach, called the empirical pseudopotential method (EPM), used plane waves and a few Fourier coefficients of the potential which were fit to experimental data. In addition to explaining the origin of the optical structure in the visible and UV in terms of interband electronic transitions and many other properties of solids, the EPM set the stage for the first-principles methods and the standard model. A schematic real space potential is illustrated in Figure 3. The strong ionic Coulomb potential is “cut off” in the core region. The physical origin of the cancelation of the ionic potential in the core region was attributed by Phillips and Kleinman to a repulsive potential arising because of the Pauli exclusion principle preventing the valence electrons from occupying core states. The result is again a weak net potential describable by a few Fourier form factors and a wave function which can be expanded in a plane-wave basis set. Usually, three pseudopotential form factors per atom gave highly accurate electronic band structures. The EPM was the first successful plane-wave method for real materials with applications to metals, semiconductors, and insulators. Dozens of band structures and optical response functions were calculated with high precision for the first time. The optical spectra of semiconductors were interpreted in terms of critical points in the electronic band structure, and when the wave functions were used to compute the electronic density, the covalent and ionic bonds were pictorially displayed – again for the first time. In Figure 4, a plot of the predicted electronic density for Si is displayed and compared with results based on subsequent X-ray studies. In addition to the striking agreement between theory and experiment, these plots reveal the nature of the covalent bond. It is particularly satisfying to see the local build-up of charge in the Si–Si bond region even though the basis set is composed of plane waves which yield a constant density before the potential is “turned on.” There is no prior prejudice for this pile-up of charge in the bond region built into the basis set.

V (r ) a1/2 Bond length

Core region

Ion potential

Figure 3 Schematic drawing of a pseudopotential (solid curve) compared to a Coulomb ion potential (dashed line).

Electronic Structure Calculations: Plane-Wave Methods

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Experiment (silicon)

24 16 12 8 4

Bond Theory

24 20 16 12

12 8 4

Figure 4 Valence electron density for silicon. The contour spacings are in units of electrons per unit cell volume. The shaded disks represent atomic cores.

Total Energy Calculations and Structural Properties The EPM calculations were constructed to deal with bulk crystals which were not terminated. Standard techniques, described in textbooks, use periodic boundary conditions to avoid having surfaces or interfaces. The EPM pseudopotential V(r) has the periodicity of the lattice so it could be expressed as an expansion using reciprocal lattice vectors G: X SðGÞV ðGÞeiG:r V ðr Þ ¼ G

where S(G) is the structure factor that locates the atoms relative to a lattice point. It is the form factors V(G) which are fit to experiment in the EPM scheme. In order to deal with surfaces and interfaces, one has to allow charge to redistribute at a surface or interface and break translational invariance. To do this, the EPM evolved into a scheme where the electron–core and the electron–electron potentials were separated. As in modern calculations, the electron–core potential is referred to as the “pseudopotential” and the electron–electron potential was expressed in terms of the density. In early calculations, the Slater and Wigner approximations were used. This allowed the charge density to readjust to a surface or interface and because of self-consistency, so did the potentials. The surface or interface was modeled by introducing a supercell which contained the geometric configuration of interest. For example, a (1 1 1) Si surface was modeled using a slab containing 12 atomic layers separated from the next slab by an equivalent space with no atoms. The surface of the slab represented the surface of the Si crystal and self-consistency required the rearrangement of the electronic density to correspond to the new geometry. Figure 5 shows this rearrangement of electron density on an ideal Si (1 1 1) surface in one of the first calculations done with this method. Later calculations used supercells to examine Schottky barriers, heterojunctions, and reconstructions at surfaces and interfaces. The evolution of the use of the LDA was straightforward since in this approach the electron–electron potentials are expressed as functionals of the electron density. Hence, surfaces, interfaces, and localized configurations in general could be dealt with using supercells and by letting the charge density adjust to the geometric constraints. The use of the PWPM and the LDA also allowed the exploration of other solid-state properties. Important applications became possible once a scheme was developed to calculate the total structural energyESof the solids for given configurations of the atomic cores. OnceESis evaluated for a series of candidate structures, a comparison of their structural energies immediately reveals the lowest energy structure for the group. At a given volume, this lowest energy structure would be the

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Surface "heals" near dangling bond region Channel

9

9 12

12

9

20

20

12

9

20 9

12

20 Figure 5 Valence electron density for the ideal Si (1 1 1) surface. Contours are plotted in the perpendicular (1 1 0) plane with units corresponding to one electron per primitive cell. The shaded disks represent atomic cores.

most stable structure if all possible structures are considered. Since it is not possible to test all structures, a limitation on this method is the possible omission of the most stable structure. However, in practice, it has often been possible to choose the most likely structures, and the success rate of the PWPM is high for predicting new structures and structural properties. BecauseESdepends on volume (or lattice separations), changes induced by pressure can cause solid–solid phase transitions where crystal structures can change. Therefore, calculations of the dependence of on volume can predict transition pressures and volumes for the structural phase changes. Similar calculations for lattice constant changes caused by strains or alloying can yield elastic constants and even vibrational spectra. Using the standard model with cores and valence electrons as illustrated in Figure 1, the total structural energy of the solid can be expressed as ee ES ¼ Ecc +Eec +Eek +Eee c +Ex

and all the energy components can be evaluated using the PWPM. The core–core Coulomb energy Ecc can be calculated from point–ion sums as discussed earlier, Eec is evaluated using the pseudopotential, and both the Hartree (Coulomb) electron–electron ee c energy Eee c and the exchange-correlation energy Ex can be approximated using the electron density. The electron kinetic energy Ek is obtained once the wave function is computed. Therefore, with pseudopotentials and the LDA approximation, the total energy ES and many ground-state properties can be determined for different structural configurations. It is convenient to evaluate the above energy contributions to ES using expressions for the potentials and energies in reciprocal space. An early application was the examination of the ground-state properties of Si. Figure 6 contains the results for calculations of ES as a function of volume for seven candidate structures. The volume is normalized by the experimental volume of the diamond phase at atmospheric pressure. Therefore, when the volume is unity, this corresponds to ambient pressure and the diamond structure has the lowest energy. As the pressure increases and the volume decreases, it is noted that the hexagonal diamond-structured material is always at a higher energy than the diamond (cubic) structure; therefore, diamond is the more stable phase. At smaller volumes (higher pressures), the b-tin phase has a lower energy than diamond-structured Si, and b-tin is therefore the stable phase. If one views the transition as occurring along the dashed line (Figure 6) which is the common tangent between the diamond and b-tin curves, the transition volumes (points 2 and 3) can be evaluated along with the transition pressure which is the slope of the common tangent. These values are all found to be in good agreement with the measured data. In addition, the minimum energy volume for the diamond structure yields a calculation of the equilibrium lattice constant while the curvature of ES as a function of volume near the minimum energy point yields the bulk modulus or compressibility for Si. In all the calculations, it is assumed that the temperature is zero. Using the above method, the lattice constants and bulk moduli have been calculated for many solids. Typical accuracies are less than 1% for lattice constants and less than 5% for bulk moduli. Discrepancies between theory and experiment for transition volumes and pressures are normally in the few percent range. In addition, modifications of structures at interfaces and surfaces can

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7.84 Si

7.86 E structure (Ry atom 1)

f.c.c.

h.c.p.

b.c.c. 7.88

sc

4

-Tin 3

7.90 2

Hexagonal diamond

Diamond

1

7.92 0.6

0.7

0.8 0.9 Volume

1.0

1.1

Figure 6 Structural energies for seven structures of Si as a function of volume normalized to the experimental volume at ambient pressure (solid lines). The common tangent represented by the dashed line illustrates the structural transition between the diamond and b-tin structures.

also be addressed with this approach. By calculating the changes in energy with movements of specific atoms or by computing Hellmann–Feynman forces, it is possible to determine reconstructions of surfaces and interfaces. In the calculations described above, the only input is the atomic number of the constituent atoms to calculate the pseudopotential and the candidate structure. Another important application of this method is the determination of vibrational spectra. The method often used within the PWPM is called the “frozen phonon” approach, where the crystal structure is distorted to represent the displaced atoms associated with a particular phonon mode. By comparing the energies of the undistorted and distorted structures or by directly calculating the forces on the atoms, the phonon spectrum can be determined. Here, one needs to input the atomic masses of the constituent atoms as well as their atomic numbers. Again, agreement with experiment is excellent.

Superconductors, Optical Properties, and Novel Materials A particularly striking success of this approach is its application to superconductivity. The systems considered are expected to be “BCS electron–phonon superconductors.” A variation on the frozen phonon method allows a determination of the electron–phonon coupling constants. As described above, the PWPM within the LDA yields ground-state properties with high precision. Since the superconducting transition temperature in the BCS description is extremely sensitive to the strength of the electron–phonon parameter l and, to a lesser extent, the Coulomb parameter m , it is imperative that the normal-state properties of the material studied be known with high precision. Only a few first-principles calculations of m are available; however, this parameter does not vary a great deal for different materials, and it scales reasonably well with the density of states at the Fermi energy. In contrast, l can have a much larger variation and therefore the focus for first-principles calculations for superconducting materials has been on the determination of l. The enhancement of the effective mass is also related to l: m ¼ mb ð1 +lÞ where mb is the band mass calculated without including the electron–phonon interactions. The strong dependence on l for the superconducting transition temperature, Tc, can be illustrated by equations such as the McMillan equation which have the generic form   1 T c ¼ AT D exp   l −m 

where A is a constant, TD is the Debye temperature, and l ¼ l=ð1 +lÞ.

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The PWPM formulation for the calculation of l had as its first application a study of high-pressure phases of Si. The results were dramatic because the existence of the two high-pressure solid metallic phases, simple (or primitive) hexagonal (s.h.) and hexagonal close packed (h.c.p.) were predicted by PWPM calculations. In addition to predicting the transition pressures needed to obtain these phases, their electronic, structural, vibrational, and superconducting properties were successfully predicted. The input information was only the atomic number and atomic mass of Si, the candidate structures, and an estimate of m . Although the PWPM has limited use for highly correlated materials, it is sometimes used as a starting point to obtain the electronic structure before correlation effects, such as a Hubbard U parameter, are considered to estimate the expected changes when correlation is considered. However, when new superconductors are found, it is common to apply the approach described above as a test of whether one is dealing with a conventional superconductor or not. Superconducting doped C60 materials, nanotubes, and MgB2 are good representatives of materials in this category. The case of MgB2 serves as an excellent example of what can be done now with the PWPM and Eliashberg theory, which is an extension of the BCS theory. The method yielded the electronic and vibrational properties of MgB2 and demonstrated that the strongest electron–phonon coupling was associated with a particular phonon mode. By considering phonon anharmonicity and including the anisotropy of the superconducting energy gap which was found to be important when doing the calculation, it was possible to reproduce the experimental data and predict new properties. The results were consistent with the proposal that MgB2 is a BCS superconductor with electron pairing caused by electron–phonon coupling and that the superconducting energy gap was multivalued. The results for MgB2 are particularly striking since this PWPM calculation had no experimental input except for the estimate of m and the known structure of MgB2. The optical properties of solids are of particular interest when one is describing a method for computing an electronic structure. The interplay between the development of quantum mechanics and the study of atomic optical spectra was extremely important to the evolution of both fields. However, identifying peaks in optical response functions for solids appeared to be a formidable task. Unlike atomic spectra that contained very sharp peaks which could be interpreted in terms of electronic transitions between narrow electronic states, solid-state spectra were broad and featureless. The challenge remained in essence until the 1960s, and, for the most part, it was the use of the EPM which provided the tools to deal with these spectra, particularly for semiconductors. An important feature is the fact that peaks in optical response functions, such as reflectivity, arise from interband transitions between states where the bands are parallel. For example, if an electron in a valence band n at point k in the Brillouin zone with energy En ðkÞ is excited to an empty state m with energy Em ðkÞ, this requires a photon having energy ħo ¼ Em ðkÞ −En ðkÞ. This transition will result in a more prominent structure or “signature” in the reflectivity if the initial occupied and final unoccupied bands are parallel at the point k. The condition is rk En ðkÞ ¼ rk Em ðkÞ, and this is the requirement for a “critical point” to occur in the band structure. It can be shown that this critical point is associated with the structure in the joint density of states between the two bands. This structure, referred to as a Van Hove singularity, in turn, appears in the optical response functions such as the reflectivity or frequency-dependent dielectric function. It was the analysis of these critical points in semiconductor spectra that yielded the energy separations used to fit the pseudopotential form factors for the EPM. A typical result is shown in Figure 7 where the modulated or derivative reflectivity spectrum of Ge is displayed. Derivative spectroscopy is used to accentuate the Van Hove structures, and the zeros and peaks arise because of the nature of the critical points. As discussed before, the positions of these are related to the energies of the critical points. In Figure 7, near the 2 eV region, it is noted that although there is reasonable agreement between theory and experiment for the critical point energies, the magnitudes of the curves differ considerably. This arises because of electron–hole attraction effects which are not included in the calculation. These are discussed later. 2 Ge Theory Experiment

I 'R (eV1) R 'E

1

0

1

2

0

1

2

3

4

5

6

7

Energy (eV) Figure 7 The modulated or derivative reflectivity spectrum for Ge. A comparison is made between the experimental spectrum and the theoretical calculation.

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Although the transition from the EPM to ab initio PWPM applications was fairly straightforward, there were particularly interesting problems associated with the calculations of energy-band separations, the minimum bandgap, optical spectra, and the role of electron–hole effects described above. The most dramatic signal that one could not just apply the LDA and the PWPM to obtain experimental band structures was the fact that researchers using this approach were consistently underestimating bandgaps. The specific focus was on Si where the measured and EPM values for the minimum gap were 1.1 eV, while the PWPM–LDA gap was half this value. The EPM was fit to give the correct result, but the ab initio method had as its input the LDA, the pseudopotential, and the plane-wave bases. It was soon found that other methods using the LDA gave similar results. Hence, it was concluded that it was the use of the LDA which resulted in calculations which underestimated the bandgap. Since it was emphasized by the inventors of the LDA that, in principle, this approximation should not be applicable for bandgap determination, this result came as no surprise. However, because of the ease of use of the LDA and its success for determining ground-state properties, it was hoped that modifications could be made to standard LDA methods to allow the computation of bandgaps and optical spectra. A successful practical solution to this dilemma that was often referred to as “the bandgap problem” involved the use of the “GW” method. In this approach, the quasiparticle energies appropriate to excited electron and hole states are calculated starting with LDA calculations. In this scheme, the electron self-energy is computed using the Green’s function G and a screened Coulomb potential W. When the resulting self-energy is included in the LDA calculation, the electronic states are renormalized and yield bandgaps in excellent agreement with experiment for a broad class of materials. An analysis of the results of applying the GW method reveals that two important physical features are mainly responsible for the success of the method. First, by using an appropriate dielectric function which includes local field effects, the variations in the charge density arising because of bonding effects are correctly included. Second, the renormalization of the electronic energies because of their interactions with plasmons and electron–hole excitations is included to produce appropriate quasiparticle states produced because of the excitations. Hence with the inclusion of GW modifications to the LDA, the calculations employing the PWPM can yield bandgaps and band structures in agreement with experiment on a level of accuracy found for the experimentally fit EPM results. In addition, these ab initio calculations have been extended further and have addressed the problems of the effects on the optical response functions arising from electron–hole interactions. This problem was raised earlier when discussing the results shown in Figure 7. Modifications of spectra of this kind are sometimes referred to as exciton enhancements of oscillator strengths near band edges. However, even when excitons, which are bound states of electron–hole pairs, are not formed, the electron–hole interactions in the excited state can still modify the oscillator strengths and in turn the optical response functions. Applications of the Bethe–Salpeter approach for two-particle interactions to this problem with electron and hole wave functions computed using the PWPM have been successful. At this point, applications of the GW approximation including electron–hole effects have been done for several materials. The computational requirements are somewhat heavy, but the results in all cases tested yield excellent spectra. A major and important component of condensed matter physics is the research focused on the development of new and novel materials. In the past two or three decades, new systems have revealed new physical properties, new physics, and important application. Systems such as superlattices, fullerenes, nanotubes, and high-temperature superconductors have broadened the horizons. Often novel materials are complex, structurally or electronically, and they offer new challenges for theoretical methods such as the PWPM. In particular, the number of atoms in a unit cell or supercell is a constraint on this method. Usually, complex materials or configurations of materials can only be modeled with a large number of atoms, and this may require a very large number of plane waves in the basis set. If the electronic structure results in very localized configurations as might be appropriate for complex molecules, again many plane waves may be required. A further and interesting aspect of most novel systems on the nanoscale level is that there are effects of confinement. These effects can be dominant for lower-dimensional or very small structures, and they must be accounted for in the calculational scheme. A typical example is the case of research on nanocrystals where there have been significant advances in recent years. Here, supercells can be employed along with the PWPM. Other techniques which start out with plane waves have had success. One such approach is to use the EPM to generate Bloch functions for a bulk crystal and then convert these itinerant states into more localized states represented by Wannier functions. If the nanocrystal is then modeled using a potential which is bulk-like inside the nanocrystal and zero outside, the Wannier functions can be used to solve the resulting Hamiltonian appropriate to the nanocrystal. One example of the application of this method is a study of the bandgap of InAs as a function of the nanocrystal radius. The results are in excellent agreement with experiment. Many applications of the PWPM to C60 and nanotubes have appeared in the literature. Electronic structure calculations for C60 were used to interpret optical, photoemission, superconducting, and other properties. For nanotubes, the explorations have been even more extensive. For carbon nanotubes, calculations of the properties of semiconductor and metal nanotubes have revealed interesting aspects of their electronic structure for interpreting transport and optical properties. Theoretical predictions of Shottky barrier and heterojunction systems composed of semimetallic and semiconductor nanotubes have been verified. There is even evidence of superconductivity which is consistent with semi-empirical calculations based on PWPM results. No completely ab initio calculations of the superconducting properties of fullerenes or nanotubes as in the case of MgB2 have as yet been done. A dramatic success of applications of the methods described here to this area is the successful prediction of compound nanotubes. In particular, the prediction of BN nanotubes was verified. In this case, unlike the situation for carbon nanotubes, all BN nanotubes without defects or impurities are semiconducting independent of their “chirality.” Theory also suggests that when doped, BN nanotubes will have interesting behavior. An example is the nature of the free electron-like wave function associated with the lowest conduction band for this system. It is predicted that when doped appropriately, the electron density for this state will reside in the center of the BN nanotube. It is expected that this system will resemble a tube with electrons filling its center. Other

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interesting predictions are that, for BC2N nanotubes, doping can result in electrons traveling along a spiral path producing effects resembling a nanocoil. For BC3N, it is expected that each tube will behave as if it were a semiconductor, but a collection of these tubes would be metallic. Although the existence of these predicted compound nanotubes has been established, in general, their transport properties have not been measured, so tests for these unusual predicted properties will have to wait until the appropriate experimental techniques become available.

Conclusion Summarizing, the history of the use of plane-wave methods for electronic structure calculations has many components. As was discussed, the physical justifications came in different forms, and the development of the techniques required physical interpretation. For example, the major changes in the development of the PWPM were motivated by the desire to answer new questions or to broaden the application of the method. Although the advances in computer hardware had a large effect, the largest impact came from the knowledge gained from doing the calculations. It is important to know what is important, or as Linus Pauling said, “The secret to having good ideas is to have many ideas and the judgment to know which ones should be discarded.”

Acknowledgments This work was supported by National Science Foundation Grant No. DMR00-87088 and by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, US Department of Energy under Contract No. DE-AC0376SF00098.

Further Reading Balkanski M and Walis RF (2000) Semiconductor Physics and Applications. Oxford: Oxford University Press. Bassani F and Pastori Parravicin G (1975) Electronic States and Optical Transitions in Solids. Oxford: Pergamon Press. Cohen ML (1982) Pseudopotentials and total energy calculations. Physica Scripta , vol. T1, 5–10. Cohen ML and Chelikowsky JR (1988) Electronic Structure and Optical Properties of Semiconductors. Berlin: Springer-Verlag. Kittel C (1996) Introduction to Solid State Physics, 7th edn. New York: Wiley. Phillips JC (1973) Bonds and Bands in Semiconductors. New York: Academic Press. Yu PY and Cardona M (1996) Fundamentals of Semiconductors. Berlin: Springer.

Plasmons in monolayer and bilayer graphene Xue-Feng Wanga and Tapash Chakrabortyb,c, aSchool of physical science and technology, Soochow University, Suzhou, China; b Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada; cDepartment of Physics, Brock University, St. Catharines, ON, Canada © 2024 Elsevier Ltd. All rights reserved.

Introduction Plasmons in monolayer graphene Plasmons in bilayer graphene Plasmon in biased bilayer graphene Conclusion References

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Abstract The collective oscillations of electrons in monolayer and bilayer graphene show different characteristics from those in ordinary two-dimensional Fermi gases. The chirality of the electrons in graphene sets up extra selection rules to their dynamics. The unconventional energy dispersion relation in graphene modifies the electron movement. The narrow gap encourages the carrier exchange between the conduction and the valence bands. The competition between the intra- and the inter-band interactions may introduce extra plasmon branches. The potential bias between the sublattices in monolayer graphene and the monolayers in bilayer graphene opens gap in the electron-hole single particle continuum. Undamped plasmon modes with almost zero group velocity may appear in the gap.

Key points

• • • •

Coulomb screening and collective excitations of electrons in monolayer and bilayer graphene are studied in the random phrase approximation. Physical mechanism behind the evolution of plasmon spectra in parameter spaces are described. Approaches to manipulating the plasmon spectra in graphene are proposed. Optimized conditions for undamped plasmon modes with low group velocity are discussed.

Introduction Mobile electrons in a charge neutral solid material may be a plasma system with negative charges immersed in a uniform background of equal and opposite positive charges. An external charge therein redistributes the negative charges and result in an electric field different from that in vacuum. This Coulomb screening effect can be described by the dielectric function e(o) which relates the modified electric field E and the electric field in vacuum or the electric displacement D by E ¼ D/e at frequency o. Collective plasma oscillations may be excited spontaneously at some specific plasma frequency op if e(op) ¼ 0. As in the case of a simple harmonic oscillator, the energy of each plasma oscillation is quantized into units of ħop according to the quantum mechanics. The corresponding quasi particles with the energy unit are known as plasmons Yu and Cardona (2003). In an ordinary two-dimensional electronic Fermi gas (2DEG), an effective mass m can be used to describe the energy dispersion of the electrons as 1

2 ⁎ 2 E ¼ ħ2k2/2m⁎, and o2D p ¼ ½ne q=2e0 eb m  in the long wavelength limit q  0 (Ando et al., 1982). Here k is the electronic wave vector, n is the sheet density of the electron gas, q is the wave vector of plasmon, e0 is the permittivity of vacuum, and eb is the background high-frequency dielectric constant. An electron gas in graphene may differ from the normal Fermi gases in at least three aspects, the energy dispersion, the chirality, and the energy gap between the conduction and the valence bands. The random phase approximation (RPA) for many-body Coulomb interaction can be employed to describe approximately the screening effect and understand the behavior of plasmons in graphene as in normal Fermi gas (Castro Neto et al., 2009; Abergel et al., 2010; Das Sarma et al., 2011; Goerbig, 2011; Kotov et al., 2012; Vafek, 2006; Wang and Chakraborty, 2007a, b, 2010; You and Wang, 2012; Hwang and Das Sarma, 2008). The RPA Coulomb interaction in the Fourier space V(q, o) obeys the equation

^ 0 ðq, oÞV ðq, oÞ V ðq, oÞ ¼ v0 + v0 P

(1)

with the electron-hole propagator

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00063-9

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Fig. 1 Diagrammatic illustration of the RPA dressed Coulomb interaction.

h ’ i   2 f Elk+q − f Elk X  ’  ^ 0 ðq, oÞ ¼ 4 , P gl,l ’ k ðqÞ o + Elk+q − Elk + id ’ l,l ,k

(2)

as illustrated by the Feynmann diagram in Fig. 1. Here the factor 4 comes from the degenerate two spins and two nonequivalent valleys K and K’ of honeycomb lattice in graphene, f(E) is the Fermi-Dirac distribution function for electrons of energy E, v0 ¼ e2/(2e0ebq) is the two-dimensional bare Coulomb interaction (in Fourier space) (Peres et al., 2005), the index l ¼  marks the l’ (q) is the Coulomb interaction vertex factor which depends on the electron wave functions. conduction and valence bands, gl, k The plasmon spectrum can be obtained by finding the zeros of the real part of the dielectric function ^ 0 ðq, oÞ: ^eðq, oÞ ¼ 1 −v0 ðqÞP

(3)

The imaginary part of the dielectric function gives the damping rate of the collective excitations to the electronhole single-particle excitations (Mahan, 2000).

Plasmons in monolayer graphene Monolayer graphene (MLG) has a honeycomb lattice of carbon atoms with two sublattices A and B. The two sublattices are displaced from each other along an edge of the hexagons by a distance of a0 ¼ 1.42A˚ . Its energy band can be calculated by the tight-binding model and an intrinsic graphene is a semimetal with the Fermi energy located at the nonequivalent K and K’ points at opposite corners of its hexagonal Brillouin zone (Wallace, 1947; Slonczewski and Weiss, 1958; Zunger, 1978; Ando, 2002). In the effective mass approximation, the Hamiltonian of electrons near the Fermi energy is expressed by a 8  8 matrix. In the case of spin and valley degeneracies, the matrix can be reduced to four independent 2  2 blocks. In the representation of the two sublattices, the Hamiltonian of a spin-up electron near the K point of the reciprocal space reads   D vmg p − Hmg ¼ vmg p  s + Dsz ¼ , (4) vmg p+ −D with s ¼ (sx, sy, sz) the Pauli matrices in the pseudospin space of the two sublattices and p ¼ − iħ(∂/∂ x  i∂/∂ y) (Kane and Mele, 2005;DiVincenzo and Mele, 1984; Sinitsyn et al., 2006). Here vmg ¼ 106 m/s is the ‘light’ velocity of the Dirac electron gas in MLG. 2D is the potential bias between the two sublattices, which is zero in intrinsic graphene but can be nonzero in graphene grown on substrates like boron nitride. The eigenenergy and eigenfunction of the above Hamiltonian read qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Elk ¼ l D2 + ℏ2 v2mg k2 , (5) Clk ¼ eikr



sin ½ak =2 + ð1 + lÞp=4 − cos ½ak =2 + ð1 + lÞp=4eifk

(6)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi l’ with tanfk ¼ ky/kx, tan ak ¼ ħvk/D, and k ¼ k2x + k2y . The interaction vertex factor reads |gl, (q)|2 ¼ [1 + ll’ cos ak+q k ’ cos ak +ll sin ak+q sin ak(k + q cos y)/| k + q|]/2 with y being the angle between k and q. Since the intra-band backward scattering at q ¼ 2 k and the inter-band vertical transition at q ¼ 0 are not allowed under Coulomb interaction in the system, we have 2 2 l, l | gl,−l k (0)| ¼ | gk (2k)| ¼ 0. In intrinsic MLG, the net electron density is equal to zero and EF ¼ 0. There exists no plasmon mode at zero temperature. Fig. 2A and B show the real (er) and imaginary (ei) parts of the dielectric function at temperature T ¼ 0 for D ¼ 0 (solid), 0.08 (dashed), and 0.1 meV (dotted) at a wave vector q ¼ 0.05  105 cm−1. At zero temperature, only inter-band transitions are allowed h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ^ 0 ðq, oÞ ¼ −q2 = 4 ℏ2 v2 q2 −o2 (J. González et al., 1994, 1999, for electrons. The electron-hole propagator for D ¼ 0 is given by P mg

2001; Khveshchenko, 2006). Therefore we have er ¼ 1 (for o > ħvmgq) above the interband electron-hole continuum (EHC) edge and ei ¼ 0 (for o < ħvmgq) below it. With increasing D, the peaks of er and ei, which are located at the edge of the interband EHC, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oH ¼ 2 D2 + ℏ2 v2mg q2 =4 , shift to higher energies. At the same time, for o > oH, er increases continuously with D and remains positive. At a finite temperature, the intra-band transitions are allowed and contribute to the electron-hole propagator of Eq. (2) and a er dip at the intra-band EHC edge oL ¼ ħvmgq is formed as shown in Fig. 2C. For D ¼ 0 where oL ¼ oH, two plasmon modes may

Plasmons in monolayer and bilayer graphene

(A)

(C)

(B)

(D)

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Fig. 2 Real (er) and imaginary (ei) parts of the dielectric function vs o in MLG at T ¼ 0 (left panels) and T ¼ 2 K (right panels) for EF ¼ 0 and D ¼ 0 (solid), 0.08 meV (dashed), and 0.1 meV (dotted). The wave vector is q ¼ 0.05  105 cm−1.

appear, one strongly Landau damped at o ¼ ħvmgq and another weakly damped at a higher energy. The damping rates of the plasmon modes are indicated in Fig. 2D by ei at the corresponding energies. Comparing the curves in Fig. 2D with those in (B), at finite temperatures a finite ei at o < oL is introduced by the intra-band transition and a decreased ei for o > oL due to the weakening of the inter-band transitions, a result of the electronic occupation of the conduction band. With increasing D, the er peak shifts with oH to higher energy while the er dip stays with oL. Since the er peak has a lower energy than the er dip initially at D ¼ 0, the peak and the dip merge at first and split again. As a result, at temperature T ¼ 2 K as shown in Fig. 2C, the er curve for D ¼ 0.08 meV is deformed in such a way that two extra zeros of er or two new plasmon modes emerge. Corresponding to the separation of the er peak from its dip, the ei curve evolves a gap between oL and oH as illustrated in Fig. 2D. One of the plasmon modes is located in this ei gap or the EHC gap and is undamped, and so it can be observed experimentally. For D ¼ 0.1 meV, there is no damped plasmon mode in the inter-band EHC. The appearance of the undamped plasmon mode in the existence of energy gap is a result of the interplay between intra- and inter-band correlations which can be adjusted by varying the temperature of the system in experiments. For an MLG of D ¼ 0.08 meV, with increasing temperature from T ¼ 0, the ratio of the intrato the inter-band correlation increases and er in the EHC gap (oL < o < oH) decreases and crosses zero. There is no plasmon mode when inter-band correlation dominates at T  1.1 K and only two damped modes exist when the intra-band correlation dominates at T 3.3 K. In the temperature regime 1.1 K  T  3.3 K or T 2D when the intra- and inter-band correlations can match with each other, however, er(oL) < 0 while er(oH) > 0 and one undamped plasmon mode exists. Furthermore, in the upper end of this regime at 1.9 K  T  3.3 K, one undamped mode and three damped modes exists. The o versus q spectrum of the plasmon modes at T ¼ 2 K is plotted by thick solid curves for D ¼ 0 in Fig. 3A and for pffiffiffi D ¼ 0.08 meV in Fig. 3B. The weakly damped plasmon mode in the D ¼ 0 case has an approximate dispersion of o∝ q at small q. A finite D separates oH from oL and opens a gap between the intraand inter-band EHC’s. As a result, the formerly weakly damped plasmon mode becomes undamped near q ¼ 0 because it is now located in the gap. When it approaches the inter-band EHC edge, the dispersion curve of this mode is squeezed to a lower energy by the er peak near the inter-band EHC edge as shown in Fig. 2C and then is split into three plasmon modes, viz., two new plasmon modes emerge near the inter-band EHC edge. One of the modes remain undamped in the EHC gap and the other Landau damped two are located inside the inter-band EHC. The latter two modes merge and disappear at q near 0.07  105 cm−1 for D ¼ 0.08 meV. The undamped mode survives in larger wavevectors until it merges with the strongly damped mode to the intra-band EHC edge when the inter-band correlation dominates and er becomes positive in the all frequency regime. The insets of Fig. 3 show the excitation spectral weight or the dynamical structure factor Sðq, oÞ ¼ −ð1=nv0 ðqÞÞIm½1=^eðq, oÞ at q ¼ 0.05  105 cm−1 (Mahan, 2000; Hwang and Das Sarma, 2008). The damped plasmon spectra is hidden in the EHC’s while the undamped plasmon spectra appears as a sharp peak in the EHC spectral gap and should be observable in experiments (Grimes and Adams, 1976). pffiffiffi As discussed above, dominant intra-band correlation results in the appearance of the plasmon mode with dispersion o∝ q at small q in Dirac gases as well as in Fermi gases (Stern, 1967). Different from 2DEGs, Dirac gases of EF ¼ 0 show only inter-band correlation at zero temperature and there is a strong competition between intra- and inter-band correlations as the parameters of the system vary. Controlling the sublattice bias and the carrier density can change significantly the ratio between intra- and inter-band many-body correlations of the Dirac electron gas and may introduce undamped plasmon modes (Novoselov et al., 2005; Zhang et al., 2005).

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(A)

(B)

Fig. 3 Plasmon spectrum (thick curves) of MLG (EF ¼ 0) at temperatures T ¼ 2 K with D ¼ 0 in (A) and 0.08 meV in (B). Intra- (dark shaded) and inter- (light shaded) band single-particle continuums are also shown. oL and oH are the lower and upper borders separating the white (EHC gap) and shaded areas respectively in (B). The vertical dotted lines indicate the q values for which excitation intensity is shown in the insets and the dielectric function is shown in Fig. 2C.

Due to the symmetry of the band structure, MLG having the same density of electrons or holes are similar and the result can be shown by a system with a net electron density and positive Fermi energy EF. For a Dirac gas in a MLG, the extra electrons in the conduction band reduce the inter-band scattering rate but enhances the intra-band one by increasing the length of the Fermi ring. pffiffiffi Similar to the case of finite temperature, a plasmon mode with an approximate dispersion o∝ q at small q appears even at zero temperature as shown in Fig. 4A. For D ¼ 0 and at T ¼ 0, a EHC gap of width 2ħvmg(kF − q) is opened above the intra-band EHC edge pffiffiffi in the range 0 < q < kF. This EHC gap at finite Fermi energy lets the o∝ q plasmon mode undamped near q ¼ 0. Another character of this mode at finite fermi energy is its flat dispersion slope as shown by the dashed curve in Fig. 4A, compared with the case at finite temperature as shown in Fig. 3A, near its entrance into the interband EHC. Furthermore, in contrast to the emergence of new plasmon modes occurred when its dispersion curve enters the inter-band EHC from a EHC gap, the dispersion curve enters the inter-band EHC smoothly here. We can characterize the carrier density by the Fermi energy to emphasize the interesting energy regime in the existence of energy gap. The corresponding carrier density can be estimated with the help of their simple relation at zero temperature n ¼ (E2F − D2)/(pv2mg). For example, the carrier density of a system with EF ¼ 1 meV is n ¼ 7.34  107 cm−2. The presence of the energy gap changes the physical scenario of the excitation spectrum in the D ¼ 0 case described above. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi At T ¼ 0, the intraband EHC edge deviates from o ¼ ħvmgq to a lower energy D2 + ℏ2 v2mg ðkF + qÞ2 − D2 + ℏ2 v2mg k2F ; the inter-band qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EHC edge shifts to a higher energy D2 + ℏ2 v2mg k2F + D2 + ℏ 2 v2mg ðkF −qÞ2 for q < 2kF while remains oH when q > 2kF. (A)

(B)

Fig. 4 The same as in Fig. 3 with EF ¼ 0.25 meV at T ¼ 0 (A) and T ¼ 2 K (B). The solid curves are for D ¼ 0.08 meV and the dashed in (A) for D ¼ 0. The inset pffiffiffi of (B) shows the zoomed area where the q plasmon mode enters the interband EHC and splits into three modes.

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More interestingly, the sublattice bias may shift the plasmon mode in the interband EHC to the EHC gap and makes it undamped. For a system of EF ¼ 0.25 meV, the plasmon mode at q ¼ 0.05  105 cm−1 becomes undamped if D > 0.02 meV. In a MLG of D ¼ 0.08 meV, the plasmon mode remains undamped up to q ¼ 0.1  105 cm−1 where it merges with the mode in the intra-band EHC as illustrated by the solid curves in Fig. 4A. At higher temperature the restriction to single-particle transitions by the Fermi energy is relaxed and the EHC is independent of the Fermi energy as shown by the shades in Fig. 3B and Fig. 4B. However, the collective excitation spectrum changes as the ratio of intra- to interband correlation increases with the net electron density. The relative weakening of inter-band correlation when EF > D lowers the er peak at the inter-band EHC edge, shown in Fig. 2C, and consequently changes the number or shifts the position of the zeros of er for different q. Comparing the resulting plasmon spectrum plotted in Fig. 4B with that in Fig. 3B, we find that for q between 0.006 − 0.04  105 cm−1 the undamped mode disappears because the er peak becomes lower than zero and for q bigger than 0.07  105 cm−1 two damped modes exist in the inter-band EHC.

Plasmons in bilayer graphene The bilayer graphene (BLG) is formed by stacking two graphene monolayers in the same way as the stacking occurs in graphite, i.e., the Bernal stacking (McCann and Fal’ko, 2006; Koshino and Ando, 2006; Nilsson et al., 2006; Ohta et al., 2006). The two graphene monolayers are arranged in such a way that the A sublattice is exactly on top of the sublattice B’ with a vertical separation of b0 ¼ 3.4 A˚ (Trickey et al., 1992). The system can be described by a tight-binding model characterized by three coupling parameters, g0 ¼ 3.16 eV between the atoms A and B or A’ and B’ (intra-layer coupling), g1 ¼ 0.39 eV between A and B’ (the direct inter-layer coupling), and g3 ¼ 0.315 eV between A’ and B, A and A’, or B and B’ (the indirect inter-layer coupling) (Koshino and Ando, 2006). In the k space, the BLG has the same hexagonal Brillouin zone as that of a monolayer. Its physical properties are mainly determined by the energy spectrum and the wavefunction near the two inequivalent corners of the Brillouin zone K and K’, where the p⁎ conduction band and the p valence band meet at the Fermi surface (Wallace, 1947). Due to the strong interlayer coupling (the p orbit overlap) both the conduction band and the valence band in a bilayer are split by an energy of  0.4 eV near the K and K’ valleys (Trickey et al., 1992; Yoshizawa et al., 1996; Ohta et al., 2006). Since this energy splitting is larger than the energy range we are interested in from the bottom of the energy band, we take into account only the upper valence band and the lower conduction band. The BLG cannot be treated as two independent graphenes monolayer with the interlayer coupling as a perturbation because of the strong interlayer overlap of the p orbits. In contrast, the perturbation treatment of the interlayer coupling is valid for a normal double quantum well system or in an intercalated graphite (Wang, 2005; Shung, 1986; Visscher and Falicov, 1971). In the effective-mass approximation (McCann and Fal’ko, 2006; Koshino and Ando, 2006), the electrons in the K valley are described by a Hamiltonian with a mixture of the linear and the quadratic terms of p   0 p2− ℏ2 kbg 0 p+ ℏ2 − : (7) Hbg ¼ 2m ⁎ p2+ 0 2m ⁎ p − 0 The effective mass of the quadratic term is m⁎¼ 2ħ2g1/(3a0g0)2 0.033m p0ffiffiffi with m0 the free electron mass and the ‘light’ velocity of the linear term is vbg ¼ ħkbg/2m⁎ ¼ 3a0g3/2ħ 105 m/s with kbg 108 = 3 m −1 . The eigenenergy and eigenfunction of the above Hamiltonian read qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Elk ¼ lℏ 2 k k2 − 2kbg k cos 3f + k2bg =2m ⁎ , (8) eikr Clk ðr Þ ¼ pffiffiffi 2



ei’k l

:

(9)

with the pseudospin angle l’k ¼ l arg (ke−2if − kbgeif). Here arg(z) is the argument y of a complex z ¼ | z | eiy. For k  kbg the electron states are chiral with ’ ¼ − 2f and have an approximately isotropic parabolic energy dispersion Elk ¼ lħ2k2/2m⁎. Near k ¼ kbg, the energy dispersion becomes highly anisotropic as shown in Fig. 5A and B. The corresponding characteristic energy is E0 ¼ ħ2k2bg/2m⁎ ¼ 3.9 meV. At E ¼ 0, where the Fermi energy is located in the undoped BLG, there are four contact points between the conduction and the valence bands: One at k ¼ 0, the center of the valley and the three satellites at k ¼ kbg in the directions of f ¼ 0, 2p/3, and 4p/3. They can be treated as four Dirac points because the electronic states near each point have a linear energy dispersion and have the same chirality as those near a Dirac point in the monolayer graphene. However, compared to the monolayer graphene, the energy dispersion here is anisotropic and the ‘light’ velocity is about 10 times slower. As illustrated in Fig. 5A and B, there is an energy pocket with a depth of about E0/4 at each Dirac point. The above peculiar characteristics of the BLG makes it quite different from the MLG in their Coulomb screening properties as described in the following. 2 ’ l’ The Coulomb interaction vertex factor reads | gl, k (q)| ¼ [1 + ll cos (’k − ’k + q)]/2. Near the central Dirac point at k ¼ 0, the l, l 2 2 intraband backward scattering and interband vertical Coulomb scattering are forbidden and | gl,−l k (0)| ¼ |gk (−2k)| ¼ 0. The 2 2 l,−l l,−l l, l same rules also hold for the three satellite Dirac points. For a large k(k  kbg), |gk (0)| ¼ |gk (−2k)| ¼ 0 but | gk (−2k)|2 ¼ 1, i.e. the intraband backward transition is allowed but both the interband backward and vertical transitions are forbidden. The above selection rules together with the energy dispersion of the carriers in the BLG indicate that the electrons (holes) close to the bottom (top) of the conduction (valence) band have very different behaviors from those away from the bottom (top).

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(A)

(B)

Fig. 5 (A) The contour lines of the energy in the kx − ky plane near the K point in BLG. The corresponding energies, starting with the innermost curve are 0.1E0 to E0 with an increment of 0.1E0. (B) The energy spectrum for equally separated f from 0 to p/6 (curves with increasing energy).

Note that in a MLG the dielectric function ^e is invariant if all the parameters with the energy unit, o, EF, and kBT, and with the wavevector unit, k and q, vary proportionally because of the linear energy dispersion of the Dirac gas (Wang and Chakraborty, 2007a). As a result, the dielectric function and the plasmon dispersion is uniform for systems with proportional parameters. In a BLG, however, this is not true anymore because of the nonlinear energy dispersion. The static dielectric function at zero temperature versus q is plotted in Fig. 6A. Its long wavelength limit is given by the properties of the four Dirac points. The central point has an isotropic ‘light’ velocity vbg while the satellite ones have the elliptic form of equi-energy lines with a minimum ‘light’ velocity equal to vbg along their radical direction and a maximum of 3vbg along the azimuthal direction. The static dielectric constant at q ¼ 0 is estimated to be es ¼ 1 + 3e2/(8e0ħvbg) 105. This value is much bigger than that of the MLG (4.5) (Wang and Chakraborty, 2007a). This means that the long-range Coulomb interaction is much more strongly screened for the bilayer system, thanks to a much bigger density of states near the Fermi energy in a BLG. Another characteristic of the BLG is its screening anisotropy, especially for scattering at a distance range of about 10 nm. This is shown by the difference between the solid and the dotted curves in Fig. 6, corresponding to the directions of q pointing to any satellite from the central pffiffiffiDirac points (a ¼ 0) or connecting any two satellites (a ¼ p/6) respectively. Here a is the angle between q and the x-axis. At q ¼ 3kbg ¼ 108 m −1 , the wavevector distance between any two satellite Dirac points, the anisotropy of es reaches its maximum with a mismatch of 20% along the different directions. The shoulder near q ¼ kbg ¼ 0.58  108 m−1 in the solid curve reflects the strong scattering between the carriers in the central and the f ¼ 0 setellite Dirac points. At a finite temperature, the energy pockets near the Dirac points are partially occupied and the intraband scattering strength is greatly enhanced. As a result, the static dielectric function near q ¼ 0 increases rapidly, as shown in Fig. 6B at T ¼ 4.2 K. The real and imaginary parts of the dynamic dielectric function versus the energy for a small wavevector q ¼ 0.005 along a ¼ 0 at T ¼ 0 and at T ¼ 4.2 K are shown in Fig. 6C and Fig. 5D.

(A)

(C)

(B)

(D)

0.0 Fig. 6 (A) The static dielectric function es vs the wavevector q in BLG along the direction a ¼ 0 (solid) and p/6 (dotted) at T ¼ 0. (B) The same as (A) but at T ¼ 4.2 K. (C) The real part of the dielectric function er vs frequency o at T ¼ 0 (dotted) and at T ¼ 4.2 K (solid). (D) The imaginary part of the dielectric function ei vs o at T ¼ 0 (dotted) and at T ¼ 4.2 K (solid). In (C) and (D), q ¼ 0.005  108 m−1 and a ¼ 0. In the limit o ! 1 , er gradually approaches to one while ei approaches to zero.

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771

For o > E0/2, the dielectric function of a BLG is similar to that of a normal Fermi gas and its temperature dependence is weak. The step of er and the peak of ei near o ¼ E0/2 ¼ 2 meV correspond to the single-particle excitation coupling states with a vanishing group velocity and having wavevectors located near the middle between the central and the satellite Dirac points. For small o, however, the dielectric function becomes more sensitive to the temperature and shows characteristics of the Dirac gas. One sign of the Dirac gas is the lack of Coulomb screening (er 1) in the energy window between 1 and 2 meV. Another sign is that a low-energy plasmon mode appears only at a finite temperature. As shown in Fig. 6C, the er has no negative value for the energy o < E0/2 at T ¼ 0 but evolves into a deep negative dip at a finite temperature T ¼ 4.2 K, when the energy pockets near the Dirac pffiffiffi points is partially occupied. As a result, one observes a weakly Landau damped plasmon mode of dispersion o  q at T ¼ 0 and may observe two such weakly damped modes at finite temperatures. Fig. 7A presents the plasmon spectrum of an intrinsic BLG (EF ¼ 0). The dispersion of the weakly Landau damped mode is pffiffiffi indicated by the thick curve and has a q dependence. Interestingly, the plasmon mode exists only at the energy higher than E0/2, i.e., double the depth of the energy pockets in the Dirac points. At a finite temperature T ¼ 4.2 K, another weakly damped plasmon pffiffiffi mode appears at the energy lower than E0/2 and also has a dispersion of q near q ¼ 0, as illustrated in Fig. 7B. The plasmon mode of higher energy that exists at T ¼ 0 is not sensitive to the temperature. This temperature dependence of the low and high energy plasmon spectra represents the marked difference between the electron gases having linear (without the collective excitation) and quadratic (with the collective excitation) energy dispersion at T ¼ 0. We have established earlier that the collective excitations appear only at a finite temperature for the Dirac gas (Wang and Chakraborty, 2007a). The electronic states in a BLG are similar to the Fermi type at the high energies but reverts to a Dirac type at the low energy. The carrier density of the system can be changed by doping. For a typical doping density of 1012 cm−2, the Fermi energy is high enough from the conduction bottom and the linear k term in the Hamiltonian can be neglected (Koshino and Ando, 2006; Ohta et al., 2006). The electrons then have a quadratic dispersion but with chirality and ’ ¼ − 2f. Near q ¼ 0, the plasmon dispersion in pffiffiffi a doped BLG has again a q dispersion as shown by the solid curve in Fig. 8 and shares the same dispersion with a normal 2DEG. The effect of the chirality is shown by the dotted curve for the plasmon dispersion of a normal two-dimensional Fermi gas with two valleys, for comparison. pffiffiffi The two curves overlap for the small q but separate as q increases. The maximum difference in the dispersion appears near q ¼ 2kF when k and k + q form a right angle in the Fermi plane and the corresponding transition is forbidden in the BLG due to its chirality.

Plasmon in biased bilayer graphene Under a perpendicular electric field, an energy gap opens between the conduction and the valence bands in BLG (McCann and Fal’ko, 2006), when an electrostatic potential bias U exists between the two graphene layers. In the effective-mass approximation, the Hamiltonian describing electrons of moderate energies in the K valley of a biased BLG reads   0 p2− ℏ2 U 1 0 + Hbbg ¼ : (10) 2m ⁎ p2+ 0 2 0 −1

(A)

(B)

Fig. 7 The plasmon spectrum of intrinsic BLG at T ¼ 0 (A) and at T ¼ 4.2 K (B). The thick curves indicate the weakly Landau damped modes while the thin curves represent the strongly damped modes.

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Plasmons in monolayer and bilayer graphene

Fig. 8 The plasmon spectrum of a doped BLG (solid curve) with a typical carrier density of 1012 cm−2, corresponding to EF ¼ 36.3 meV and kF ¼ 1.77  108 m−1. The plasmon spectrum in the same system but without chirality is plotted as a dotted curve for comparison. Intra- (dark shaded) and inter- (light shaded) band single-particle continuums are also shown.

The indirect inter-layer coupling is neglected since it affects only the energy band in the range of less than 2 meV from the middle of the conduction-valence band gap. The eigenenergy and eigenfunction of the above Hamiltonian read v0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 12 ffi 1 u u 2 2 2 u ℏ k U (11) Elk ¼ lt@ ⁎ A + @ A , 2 2m Clk ¼ eikr



sin ½bk =2 + ð1 + lÞp=4 − cos ½bk =2 + ð1 + lÞp=4ei2fk

:

(12)

Here bk indicates the ratio of the kinetic energy to the potential bias with tanbk ¼ ħ2k2/(m⁎ U). For such an energy dispersion the DOS of the system is Z 1 dS m⁎ E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðEÞ ¼ ¼ (13) 2

2 , jrk Ej pℏ2 ðpÞ 2 E − U2 Under finite bias U, the DOS of the BLG diverges on the edge of the energy gap E ¼ | U/2 |. The vertex factor reads D  ’ 2 h

i 1    l,l 2 gk ðqÞ ¼  k + q, l’ jeiqr jk, lij ¼ 1 + ll’ cos bk cos bk+q + ll’ sin bk sin bk+q cos 2fk −2fk+q : 2  ’ 2

  ’ 1 When q ¼ 0 and q ¼ −2k,gl,l . Similar to unbiased BLG, the interband vertical and back scatterings are both ð q Þ ¼ 1 + ll  k 2 forbidden but the intraband back scattering is allowed in biased BLG. For intraband scattering with k ? k + q, we have  ’ 2 h

i   l,l gk ðqÞ ¼ 12 1 + cos bk + bk+q , which becomes zero in unbiased BLG. The intraband component (l ¼ l’) of the electronhole propagator is expected to be similar to that in normal 2DEG except the effect of chirality and deformation of energy band. The interband one (l ¼ − l’), which can be manipulated by the bias voltage, modifies qualitatively the screening properties of the system. Since the approximate energy spectrum we use in this study is isotropic, the obtained properties are also isotropic and we use q | q| and k | k| in the following discussion. Fig. 9 shows the negative intraband propagator in unit of N0 ¼ 2m⁎/p (the DOS of intrinsic BLG) versus frequency o for three typical wavevector q values. Similar to the 2DEG result at zero temperature, the imaginary part is nonzero in the single-particle continuum 0 < o < ou for 0 < q < 2kF and ol < o < ou for q > 2kF, with ol ¼ |EkF1−q − EkF1 | and ou ¼ EkF1+q − EkF1. The derivative of the imaginary part is not continuous at the continuum edges and for 0 < q < 2kF is also not continuous at o ¼ ol besides o ¼ 0 and ou. When the bias U increases, the structures of the curves shift to lower energy because the energy band is narrowed and the group velocity of electrons at the bottom of the conduction band decreases. When the temperature increases, the sharp edges become smoother and the nonzero range gets wider as expected. Different from the 2DEG result, as illustrated in Fig.9C for q > kF at U ¼ 0 and T ¼ 0, the imaginary part has a sharp dip with a derivative discontinuity at om ¼ q2/2m⁎ between the edges due to the chiral nature of the wave functions (Sensarma et al., 2010). The dip and discontinuity persist at finite temperature but become softened and disappear under finite bias voltage. The real part, presented in the lower panels of Fig.9, shows a sharp peak at ol and a sharp dip at ou with an extra peak near om for q > kF and moves in a similar way as the imaginary part with bias and temperature. However, the structure near om for both the imaginary and real part develops in a different way from those at ol and ou when U and T increase. It remains at high temperature and removes off with the bias voltage while those at ol and ou decay with temperature. On average, the variation range of the propagator decreases with q. Fig.11 illustrates the negative interband propagator in unit of N0 versus o for the same three q values as in Fig.10. Its imaginary 1 part and the single-particle continuum have a minimal energy limit o1 ¼ EkF1 − EkF−1 −q at zero temperature or o2 ¼ 2Eq/2 at finite

(A)

(B)

(C)

(D)

(E)

(F)

Fig. 9 The negative intraband electron-hole propagator in unit of N0 versus frequency o in unbiased/biased BLG at zero/room temperature. The imaginary part for q ¼ 0.06, 0.8, and 2.5kF is plotted in panel (A), (B), and (C), respectively, and the corresponding real part in (D), (E), and (F). The electron density is n ¼ 1012 cm−2 with the Fermi wavevector kF ¼ 1.77  108 m−1. The corresponding Fermi energy is EF ¼ 34 meV at U ¼ 0 and EF ¼ 45.4 meV at U ¼ 60 meV. Solid curves are for T ¼ 0, U ¼ 0; dashed T ¼ 0, U ¼ 60 meV; dotted T ¼ 300 K, U ¼ 0; dash-dotted T ¼ 300 K, U ¼ 60 meV.

(A)

(B)

(C)

(D)

(E)

(F)

Fig. 10 The negative interband electron-hole propagator of BLG in unit of N0 versus frequency o is plotted for the same parameters set and arrangement as in Fig.9.

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temperature. For small q [Fig.10A] the imaginary part increases sharply and reaches a peak before decreases in a way  1/o. For mediate q ¼ 0.8kF [Fig.10B] the main peak becomes round and smooth while a sharp peak near o3 ¼ EkF1 − EkF−1 +q appears under bias as shown in the inset of Fig.10B. This latter peak grows and becomes more visible as U increases. For q > kF a peak with discontinuity appears near om ¼ q2/2m⁎ at zero temperature in both biased and unbiased BLG as shown in Fig.10C (Sensarma et al., 2010). Corresponding to the continuum edges of the imaginary part, on the curve of the real part as shown in the lower panels of Fig.10, there is a peak near o1(o2) at low (high) temperature when the electron system is degenerate (nondegenerate). The peak near o2 may become very sharp for small q in biased BLG because the bottom (top) of the conduction (valence) band becomes flat and the DOS diverges on the edge of energy gap. The overal1 contribution of interband excitation to the propagator increases with q and in a way  q2 at small q. The electron-hole propagator reflects the electric polarizability of a many-body system screening a Coulomb potential. After being reduced by the background dielectric constant eb, it determines the dielectric function e(q, o) ¼ er + iei. The zero of the real part er gives the collective excitation of the system in the absence of external electromagnetic field. The imaginary part ei gives the spectrum of the single-particle excitation. In the presence of only intraband single particle excitation as in conventional 2DEG, there exist maximally two plasmon modes, one acoustic mode of frequency oA within the single-particle excitation and another optical mode oO, because −Re [P(q, o)] has only one dip below zero as shown in Fig.9. The acoustic mode is thus always overdamped with little spectral weight and not experimentally relevant. The contribution from interband excitation introduces fine structures to er near zero and at least two extra modes, op3 and op4 may emerge. Fig. 10 displays er versus o for various temperature T (A), bias voltage U (B), electron density n (C), and wavevector q (D). In the high frequency limit, the effect of polarization vanishes and eb ¼ 1. For intrinsic BLG where n ¼ 0, the intraband polarization is negligible and er > 0 at T ¼ 0. There is no collective mode. In other cases, the er versus o curve has a deep dip at ou and a peak at o1 or/and o2. The competition among value one, the intra-, and the inter-band contributions to the polarizability determine its fine features. For typical parameters, as illustrated in Figs. 9 and 10, the effect of the intra-(inter-) band polarization decreases (increases) with q so we expect that er mainly shows intra- (inter-) band characteristics for long (short) wavelength. In this study we are mostly interested in the screening and collective excitation properties of long wavelength in the system, and in Fig.11 we present the details only for q kF near er ¼ 0. The result for a system of U ¼ 30 meV, n ¼ 1012cm−2, and q ¼ 0.1  108 m−1 is exhibited in Fig.11A. At low temperature T ¼ 4 K (solid) when the system is degenerate, the intraband contribution dominates and a peak appear near 1 o1 ¼ EkF1 − EkF−1 −q ’ 2EkF ¼ 74.4 meV which is out of the panel. When the temperature increases to 200 K (dashed) and the system becomes nondegenerate, a sharp peak emerge near o2 ¼ 2E1q/2 ’ 30 meV. Since this interband peak is located just below the optical

(A)

(C)

(B)

(D)

Fig. 11 The real part of the dielectric function er vs frequency o when eb ¼ 1 in variety of the BLG bias U, temperature T, electron density n(1012 m−2), and wavevector q(108 m−1) is plotted in (A)–(D), respectively. Parameters U ¼ 30 meV n ¼ 1012 m−2, q ¼ 0.1  108 m−1, T ¼ 300 K are used if not specified in the panels.

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plasmon energy oO, it can introduce two extra roots op3 and op4 to the equation er ¼ 0, i.e., two extra plasmon modes in the system. The extra plasmon mode of lower energy op3 is out of the interband single-particle continuum and can be only weakly damped. When the temperature increases further (dotted and dash-dotted) the enhanced intraband contribution shifts the peak below zero and the extra plasmon modes disappear. In the same time, oO increases with the temperature as more electrons (holes) exist in the conduction (valence) band. As shown in Fig.11B, the bias voltage can sensitively shift the interband peak and control the emergence of the extra plasmon modes. These modes have energies (frequencies) proportional to the bias voltage and group velocities close to zero. The increase of the electron density enhances the degeneracy of the system and reduces the amplitude of the interband peak at o2 ¼ 2E1q/2 as plotted in Fig.11C. The enhanced intraband contribution at higher density also increases the frequency of the optical plasmon mode oO. Fig.1D illustrates how the er − o curve develops with q at room temperature T ¼ 300 K. Overall the intra-(inter-) band contribution to e simply decreases (increases) with q as shown previously in Figs.10 and 11, but their effect on the plasmon spectrum is more complicated due to their competition with each other. When q increases, the intraband introduced er dip becomes wide which usually results in the increase of the plasmon frequency. The interband peak of er is located at the fixed energy o2 ’ 30meV and its amplitude increases with q. As a result, only when o2 is close to oO, the interband contribution can affect significantly the plasmon spectrum. The plasmon spectrum in intrinsic BLG (unbiased and undoped) depicted in Fig.12 at various temperatures T (A) and for various background dielectric constant eb (B). At zero temperature, there is no carrier and no plasmon mode in the system. At finite temperature, electrons (holes) are excited in the conduction (valence) band and two plasmon modes emerge. In the pffiffiffi long-wavelength limit, their frequencies are proportional to T at high temperature. Similar to the 2DEG result, we also observe a pffiffiffi dispersion oO ∝ q and oA ∝ q. The background dielectric constant can also be employed to modify the plasmon frequency as shown in Fig.12B. The acoustic mode is not sensitive with eb but the frequency of optical mode decreases quickly with eb. In Fig. 13, the plasmon spectrum for various electron densities at zero temperature is illustrated in unbiased BLG (A) and in biased one with U ¼ 60 meV (B). In the long-wave limit, q  0, the dielectric function is dominated by the intraband contribution. The properties of the system at zero temperature is mainly determined by the group velocity of electrons at the Fermi energy. The plasmon spectrum of unbiased system is similar to that of 2DEG. If the electron density is not high, e.g. n ≲ 1012cm−2 for U ¼ 60 meV as shown in Fig.13B, the Fermi group velocity of electrons decreases when the external electric field is turned on, due to the band deformation, and the plasmon mode is softened. With increasing q the interband contribution becomes more important, which reduces the group velocity of the optical plasmon mode. In some cases, the group velocity of plasmon can be close (A)

(B)

Fig. 12 Plasmon spectrum of intrinsic BLG (A) in vacuum with eb ¼ 1 at various temperature and (B) with various dielectric constant at room temperature T ¼ 300 K. The shadow shows the electron-hole single-particle continuum at zero temperature.

(A)

(B)

Fig. 13 Plasmon spectrum at zero temperature of (A) unbiased (B) U ¼ 60 meV biased BLG in vacuum for density n ¼ 0.1 (solid), 0.5 (dashed), 1.0 (dotted), and 2.0 (dash-dotted) 1012 cm−2. The single-particle continuum at n ¼ 1012 cm−2 is also present (light/dark shadow for the inter−/intra-band part).

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Fig. 14 Plasmon spectrum of biased and doped BLG with U ¼ 30 meV and n ¼ 1012 cm−2 at room temperature T ¼ 300 K in environment of different background dielectric constants eb. The inset shows the zoomed spectrum of the plasmon branches oO, op3, op4, and oA for eb ¼ 1 at small q. The light (dark) shadow shows the interband (intraband) single-particle continuum at zero temperature.

to zero, a favorite situation for the stimulated plasmon emission (Rana, 2008). For large q, when the optical plasmon branch enters the interband single-particle continuum, i.e. oO > o1, the effect of interband contribution decreases and the plasmon spectrum has a long tail in unbiased system or when the carrier density is high. However, in biased systems with low carrier density, the effect of the interband contribution can be significant due to the flat band and the plasmon spectrum ends near where the intra- and inter-band single-particle continua meet at ou ¼ o1. The competition between the intra- and inter-band contributions may result in two extra plasmon modes for proper U at finite temperature when thermal excitation becomes important as previously discussed in Fig.11. This is an interesting phenomenon because those modes have unique properties and might be used in nanotechnology. In Fig.14, we plot the plasmon spectra of a biased system with U ¼ 30 meV at T ¼ 300 K and n ¼ 1012cm−2 for background dielectric constant eb ¼ 1 (solid), 5 (dash-dotted), and 15 (dotted). At room temperature T ¼ 300 K, the electronic system is not degenerate and the plasmon spectrum is in general similar to the one of intrinsic BLG as shown in Fig.12B. However, as shown in Figs.10 and 11, the interband contribution adds a sharp peak at o2 which is around U for small q. As a result, the plasmon spectra are deformed at o2 and bifurcate in some cases where two extra plasmon modes op3 and op4 emerge as shown in the inset of Fig.14. The emerged two plasmon modes have almost zero group velocities and their frequencies are proportional to the bias voltage. In addition, their energies are in the gap of the zero-temperature single particle continuum and the lower one is below the lower limit of single-particle excitation, o2, at high temperature. This suggests that the modes are undamped or weakly damped and have long lifetime. These undamped plasmon modes are similar to those in MLG but have lower group velocity. Their energies can be easily manipulated by the bias voltage. These undamped modes with almost zero group velocities.

Conclusion The Coulomb screening properties and the collective excitation modes in monolayer and bilayer graphene are discussed in the random phase approximation. In intrinsic monolayer graphene there is no collective excitation at zero temperature. A potential bias between the sublattices transfers the semimetal to a narrow gap semiconductor system and opens a gap between the intra- and the inter-band electron-hole continuum (EHC). At finite temperature or under the gate voltage, the switch-on of intra-band correlation introduces plasmon modes to the system and the interplay between the intra- and interband correlations play an important role in determining the plasmon spectrum. Undamped plasmon modes exist in the EHC gap. In addition, in a gated graphene with net pffiffiffi electrons or holes, a EHC gap is formed for q < kF and an undamped plasmon mode of frequency o∝ q exist in monolayer graphene gas even at zero temperature, similar to the situation in a normal Fermi gas. In semimetal intrinsic bilayer graphene, the static dielectric constant is much bigger than that in monolayer graphene, due to the existence of four Dirac points in each energy valley and the much lower ‘light’ velocity of the Dirac points than the one in monolayer graphene. The Coulomb screening also shows a strong anisotropy. The dynamic Coulomb screening has the characteristics of a Dirac gas on the low energy side and those of a normal Fermi gas on the high energy side. This transition from the Dirac to a Fermi

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gas is also reflected in the plasmon dispersion as the wavevector or the energy increases from zero. In doped bilayer graphene, the long wavevector limit of plasmon spectrum is the same as in normal two-dimensional Fermi gas with two valleys. A potential bias between the two monolayer in bilayer graphene opens a gap between the conduction and valence bands and increases the DOSs at the band edges. As a result, the bias modifies the dielectric function greatly. In the long-wave limit, a sharp and controllable dielectric peak might appear at the energy equal to the band gap at high temperature when the system is nondegenerate or at the energy of double the Fermi energy at low temperature when the system is degenerate. In some cases, two extra undamped plasmon modes appear at energies close to the band gap energy and have almost zero group velocities.

References Abergel DSL, et al. (2010) Properties of graphene: A theoretical perspective. Advances in Physics 59: 261–482. Ando T (2002) Electronic states and transport in carbon nanotubes. In: Chakraborty T, et al. (ed.) Nanophysics & Bio-electronics: A new Odyssey, pp. 2–59. Amsterdam, New York: Elsevier. Ando T, et al. (1982) Electronic properties of twodimensional systems. Reviews of Modern Physics 54: 437–672. Castro Neto AH, et al. (2009) The electronic properties of graphene. Reviews of Modern Physics 81: 109–162. Das Sarma S, et al. (2011) Electronic transport in twodimensional graphene. Reviews of Modern Physics 83: 407–470. DiVincenzo DP and Mele EJ (1984) Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds. Physical Review B 29: 1685–1694. Goerbig MO (2011) Electronic properties of graphene in a strong magnetic field. Reviews of Modern Physics 83: 1193–1243. González J, et al. (1994) Non-Fermi liquid behaviour of electrons in the half-filled honeycomb lattice (a renormalization group approach). Nuclear Physics B 424: 595–618. González J, et al. (1999) Electronic structure: Wideband, narrow-band, and strongly correlated systems-Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction. Physical Review B 59: R2474. González J, et al. (2001) Electron-electron interactions in graphene sheets. Physical Review B 63: 134421. Grimes CC and Adams G (1976) Observation of two-dimensional plasmons and electron-ripplon scattering in a sheet of electrons on liquid helium. Physical Review Letters 36: 145–148. Hwang EH and Das Sarma S (2008) Screening, kohn anomaly, friedel oscillation, and RKKY interaction in bilayer graphene. Physical Review Letters 101: 156802. Kane CL and Mele EJ (2005) Quantum spin hall effect in graphene. Physical Review Letters 95: 226801. Khveshchenko DV (2006) Coulomb-interacting Dirac fermions in disordered graphene. Physical Review B 74: 161402(R). Koshino M and Ando T (2006) Transport in bilayer graphene: Calculations within a self-consistent Born approximation. Physical Review B 73: 245403. Kotov VN, et al. (2012) Electron-electron interactions in graphene: Current status and perspectives. Reviews of Modern Physics 84: 1067–1125. Mahan GD (2000) Many-Particle Physics, 3rd edn. Kluwer Academic/Plenum Publishers. McCann E and Fal’ko VI (2006) Landau-level degeneracy and quantum hall effect in a graphite bilayer. Physical Review Letters 96: 086805. Nilsson J, et al. (2006) Electron-electron interactions and the phase diagram of a graphene bilayer. Physical Review B 73: 214418. Novoselov KS, et al. (2005) Two-dimensional gas of massless Dirac fermions in graphene. Nature 438: 197–200. Ohta T, et al. (2006) Controlling the electronic structure of bilayer graphene. Science 313: 951–954. Peres NMR, et al. (2005) Coulomb interactions and ferromagnetism in pure and doped graphene. Physics Review 72: 174406. Rana F (2008) Graphene terahertz plasmon oscillators. IEEE Transactions on Nanotechnology 7: 91–99. Sensarma R, et al. (2010) Physical Review B 82: 195428. Shung KWK (1986) Dielectric function and plasmon structure of stage-1 intercalated graphite. Physical Review B 34: 979–993. Sinitsyn NA, et al. (2006) Charge and spin hall conductivity in metallic graphene. Physical Review Letters 97: 106804. Slonczewski JC and Weiss PR (1958) The band structure of graphite. Physics Review 109: 272–279. Stern F (1967) Polarizability of a two-dimensional electron gas. Physical Review Letters 18: 546–548. Trickey SB, et al. (1992) Interplanar binding and lattice relaxation in a graphite delayer. Physical Review B 45: 4460–4468. Vafek O (2006) Thermoplasma polariton within scaling theory of single-layer graphene. Physical Review Letters 97: 266406. Visscher PB and Falicov LM (1971) Dielectric screening in a layered electron gas. Physics Review 3: 2541–2547. Wallace PR (1947) The band structure of graphite. Physics Review 71: 622–634. Wang XF (2005) Semiconductors II: Surfaces, interfaces, microstructures, and related topics-Plasmon spectrum of two-dimensional electron systems with Rashba spin-orbit interaction. Physical Review B 72: 85317. Wang XF and Chakraborty T (2007a) Collective excitations of Dirac electrons in a graphene layer with spinorbit interactions. Physical Review B 75: 033408. Wang XF and Chakraborty T (2007b) Coulomb screening and collective excitations in a graphene bilayer. Physical Review B 75: 041404(R). Wang XF and Chakraborty T (2010) Coulomb screening and collective excitations in biased bilayer graphene. Physical Review B 81: 081402(R). Yoshizawa K, et al. (1996) Interlayer interactions in two-dimensional systems: Second-order effects causing ABAB stacking of layers in graphite. The Journal of Chemical Physics 105: 2099–2105. You WL and Wang XF (2012) Dynamic screening and plasmon spectrum in bilayer graphene. Nanotechnology 23: 505204. Yu PY and Cardona M (2003) Fundamentals of Semiconductors: Physics and Materials Properties, 3rd edn. Springer. Zhang Y, et al. (2005) Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438: 201–204. Zunger A (1978) Self-consistent LCAO calculation of the electronic properties of graphite. I. The regular graphite lattice. Physical Review B 17: 626–641.

Electron gas (theory) G Vignalea and MP Tosib,⁎, aDepartment of Physics and Astronomy, University of Missouri, Columbia, MO, United States; bScuola Normale Superiore, Pisa, Italy © 2024 Elsevier Ltd. All rights reserved.

Introduction Ideal Fermi gas and Fermi liquid theory Weakly interacting electron gas: Exchange effects Electron correlations: Calculation of the ground state energy Electron gas and density functional theory Static screening Dynamical screening and plasmons Wigner crystallization Electron gas in reduced dimensionality Two-dimensional electron gas in graphene Two-dimensional electron gas at high magnetic field Transport coefficients Spin-orbit interaction Conclusion References

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Abstract This article provides an overview of the uniform electron gas model (also known as jellium model) for electrons in solid matter. Historically, this model has provided the basis for the understanding of electron-electron interaction effects in metals and semiconductors and it currently plays a crucial role at the foundations of Density Functional Theory. We review the basic properties of the model, such as the existence of the Fermi surface, the exchange and correlation energy, static and dynamic screening, quasiparticles and plasmons and the possibility of electron crystallization and magnetic phases. The two-dimensional electron gas in systems such as quantum wells and graphene and in the presence of spin-orbit interactions and magnetic fields is also discussed.

Key points

• • • • • • •

An electron gas pervades all known materials and is responsible for many of their observed properties Electrons in metals are described by Fermi liquid theory, whereby each electron behaves essentially like a free particle, dressed by interactions. The energy of the electron gas is controlled by the so-called “exchange-correlation” hole, arising in part from the Pauli exclusion principle, in part from the Coulomb interaction, whereby the average electrons density is reduced in the vicinity of an electron pinned at a given position. Electrons in metals rearrange around an external charged impurity in such a way that the impurity is “screened,” i.e., it becomes essentially invisible at large distance from the impurity. This screening effect is very effective in three dimensions, less so in lower dimension. The electron gas supports collective excitations of the charge density known as plasmons. Collective oscillation dominate the high-frequency response of the electron gas and give a large contribution to the correlation energy. Besides the liquid phases, other phases of the electron gas are possible, including crystalline and magnetic ones, and have been investigated by numerical Quantum Monte Carlo methods. The electron gas in two and one-dimensional structures may present non Fermi liquid behavior, such as the Luttinger liquid in one dimension and the quantum Hall liquid in two dimensions.

⁎

Deceased.

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00167-0

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Notations and Acronyms

e(q) eF aB B e g(r) H h ħ K kB kF ℓ m mc n pF q QMC r rs S T aee l lTF nF x(q) x0(q) xS vc vP

Dielectric function Fermi energy Bohr radius Bulk modulus Electron charge Pair distribution function Magnetic field Planck constant Planck constant h-crossed (h divided by 2p) Compressibility Boltzmann constant Fermi wave number Magnetic length Electron mass Cyclotron mass Electron density Fermi momentum Wave number Quantum Monte Carlo Distance Dimensionless electron-electron coupling strength Shear modulus Temperature Electron-electron interaction coupling constant in graphene de Broglie wavelength Thomas-Fermi screening length Density of states at the Fermi surface Density response function Static density susceptibility Spin susceptibility Cyclotron frequency Plasma frequency v(q): Fourier transform of the Coulomb interaction

Introduction The homogeneous gas of electrons (also known as jellium) is one of the most important basic models in the physics of condensed matter. It was introduced by Drude in 1900 to account for the electromagnetic properties of plasmas and was used by the pioneers of quantum mechanics in the early 1930s to model the sea of interacting valence electrons in a metal. The valence electrons are the outer electrons of an atom and determine how it interacts with its surroundings, but in a metal they are released by the atoms into a collective state. In the early studies, the ions of the crystalline lattice in the metal were smeared into a rigid uniform background of positive charge, which exactly balances the negative charge associated with the distribution of outer electrons. More generally, the background charge can be endowed with an ability to deform under pressure leading to a less widely known, but very useful “deformable jellium model.” Following the explosive growth of density-functional theory in the 1980s, the electron gas has become the basic reference system in most calculations of electronic structure not only for metallic or insulating solids and liquids, but also for atomic-scale systems and biomolecules. In metals the density of valence (or conduction) electrons is high enough that many of their properties can be understood in terms of a picture in which they move almost independently of each other, forming a paramagnetic fluid made of equal numbers of up and down spins. The success of this counterintuitive single-electron picture of metals rests on the work carried out in the 1950s by Landau in developing the theory of Fermi liquids, as will be discussed in the next section. With decreasing density the potential energy associated with the Coulomb interactions between the electrons grows to dominate over their kinetic energy and ultimately at very low density the electron gas freezes into the “Wigner crystal,” in which the electrons become localized on lattice sites. This behavior is just the opposite of what happens with classical systems, where it is high density that favors a lattice structure. It has long been believed that crystallization of the electron gas in the jellium model is preceded, in an intermediate range of density, by the stabilization of a spin-polarized state in which the spins spontaneously align into a ferromagnetic state of order. This possibility was foreseen by Bloch in 1929 in the context of the Hartree-Fock approximation, and was initially confirmed by accurate calculations of the ground state energy based on quantum Monte Carlo (QMC) methods implemented on modern computers. The traditional

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Fig. 1 The traditional QMC phase diagram for a three dimensional electron gas (Ceperley, 2004) Recent studies suggest that the narrow “polarized fluid” region is a computational artifact and does not exist in reality. Ceperley DM (2004) Introduction to quantum Monte Carlo methods applied to the electron gas. Proceedings of the International School of Physics Enrico Fermi, Course CLVII, edited by Giuliani GF and Vignale G. Amsterdam: IOS Press used with permission.

phase diagram of the electron gas in three dimensions is shown in Fig. 1 (Ceperley, 2004). However, more recent studies, also based on quantum Monte Carlo, have cast serious doubts on the existence of the ferromagnetic phase both in two-dimensions (Drummond and Needs, 2009) and in three dimensions (Holzmann and Moroni, 2020). The electron gas framework, when widened to include artificial semiconductor heterostructures in which the electronic carriers are constrained to move in a plane (as in a quantum well) or on a line (as in a quantum wire) or under three-dimensional confinement (as in a quantum dot), provides much of the basic understanding that is necessary for modern condensed matter physics and nanotechnology. However, it is appropriate to note at this point that some more exotic properties of the valence-electron gas have come to the fore during the last few decades. These developments started with the idea of Cooper pairs in the BCS theory of superconductivity in metals and have been stimulated by discoveries of novel materials, such as the cuprate high-Tc superconductors and other oxides, or of novel physical effects such as the Kondo scattering against magnetic impurities and the fractional quantum Hall effect. The quantum theory of solids is now no longer confined to the electron as the quasiparticle of a Fermi liquid carrying an elementary charge and a spin, but in special situations the system of valence electrons reconstructs to produce quasiparticles that may behave like fragments of an electron, or more complicated objects such as composite fermions, composite bosons, holons and spinons. Also spin-orbit interaction effects, and linearly dispersing electrons in novel two-dimensional materials consisting of a single atomic layer (e.g., graphene) have been intensively studied. A complete discussion of electron gas theory up to 2005 can be found in the monograph by G. F. Giuliani and G. Vignale (Giuliani and Vignale, 2005). More recent topics will be briefly covered later in this article.

Ideal Fermi gas and Fermi liquid theory Conduction electrons in normal metals under ordinary laboratory conditions form a highly degenerate Fermi fluid, in which the mean interparticle spacing a is a small fraction of the characteristic de Broglie wavelength l ¼ h/pth (here h is the Planck constant and pth is the momentum of a particle of mass m having kinetic energy equal to the thermal energy kBT, with kB the Boltzmann constant and T the temperature). If for the moment one neglects all interactions, one can write the energy levels E of the gas in terms of the P single-particle kinetic energies ek and of the number nk of electrons in a single-particle state at energy ek : E ¼ k nkek. Here k is a 2 2 h ) is the kinetic energy of each such electron. label specifying the momentum and the spin of each electron and ek ¼ ħ2mk (with ħ ¼ 2p The Pauli exclusion principle restricts the values that nk may take for particles of half-integer spin to nk ¼ 0 or 1. A cell of volume h3 in phase space can thus contain at most two electrons with opposite spins, so that in the ground state at T ¼ 0 the electrons must have a spread of momenta in a range up to a maximum momentum pF (the “Fermi momentum”). For N electrons with up and down spins inside a volume V and defining kF ¼ pF/ħ (the Fermi wave number), one can write the density n ¼ N/V as n ¼ k3F /3p2 (The expression for the Fermi wave number becomes (2pn)1/2 in two dimensions and pn/2 in one dimension, where n is given by N/A and N/L respectively). The Fermi wave number kF is the radius of a spherical surface in wave number space

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Fig. 2 Schematic representation of a section of the Fermi sphere for a paramagnetic electron gas, showing also a process of excitation of an electron-hole pair.

(the “Fermi surface”) and the energy of an electron at the Fermi surface is, by definition, the Fermi energy eF ¼ ħ2k2F /2m. This also is the lowest energy cost at which one may add a further electron to the gas (i.e., eF is the chemical potential of the gas at T ¼ 0) and evidently increases with the electron density, since kF ¼ (3p2n)1/3. The notion of a spherical Fermi surface allows a vivid picture of ground state and excitations in a normal Fermi fluid. At T ¼ 0, the single-particle states inside the Fermi sphere are fully occupied by the fermions and the states outside of it are empty. The momentum distribution (i.e., the occupation number of an electron state of given momentum and spin) thus takes a discontinuous change from 1 to 0 as the momentum crosses the Fermi surface. Thermal excitations at finite temperature bring fermions from states inside to states outside the Fermi surface, leaving “holes” behind and lowering the chemical potential of the gas (see the schematic picture in Fig. 2). It is somewhat surprising that a number of properties of normal metals are observed to behave as predicted by the ideal Fermi-gas model, in spite of the electron-electron and electron-ion interactions that may be expected to lead to substantial departures from ideality. In particular, the electronic contribution to the specific heat of a metal increases linearly with temperature and the paramagnetic spin susceptibility is independent of temperature, both quantities being proportional to the density of electronic states at the Fermi surface, denoted by nF. These results are in agreement with the noninteracting electron gas model. The explanation of these behaviors was first given by Landau in what is now known as the “Landau theory of Fermi liquids.” Landau realized that at low temperature electrons with momenta close to the Fermi surface have limited opportunity to scatter against each other and thus change their state of motion. This is due to the combined constraints of energy conservation and the Pauli exclusion principle. These electrons therefore behave as if they were essentially free and their individual momentum is almost exactly conserved: such “almost free” particles are appropriately termed “quasiparticles.” Once one deals with quasiparticles rather than with “bare” electrons, the excitations of the fluid can be described as an almost ideal gas of quasiparticles. A crucial point in Landau’s theory is that the quasiparticles obey Fermi statistics, so that at low temperature the low-lying excitations of the interacting system are in one-to-one correspondence with those of the ideal noninteracting system, i.e., they are quasi-electron/quasi-hole pairs across the sharply defined Fermi surface. The correct behavior of the electronic specific heat and spin susceptibility follows at once. There are, however, some major differences to be emphasized between quasiparticles and bare particles. First of all, a quasiparticle can be visualized as a bare particle “dressed” by a cloud of surrounding particles: the effect of this “dressing” is to rescale the relation between energy and momentum, introducing an effective mass m that depends on the strength of the electron-electron interaction. Thus, while the energy of a bare particle, relative to the chemical potential, is ek − m ’ ħkF(k − kF)/ m for k ’ kF (with m the bare mass of the electron), the energy of a quasiparticle is e∗k − m ’ ħkF(k − kF)/m∗ where m is the effective mass. m /m is usually of order 1 in the electron gas at metallic densities, but it can rise into the hundreds for electrons in strongly correlated materials (heavy-fermion materials). Second, there are residual interactions between the quasiparticles, which cause the energy of a quasiparticle to change in proportion to the density of quasiparticle excitations already present in the system as a result of finite temperature and/or external excitation. These interactions are described by an (in principle, infinite) set of numbers known as “Landau parameters.” Third, the lifetime t of a quasiparticle is not really infinite: its inverse goes to zero in the low-energy limit as ħ/t ∝ (ek − eF)2/eF (for T ¼ 0) or in the low-temperature limit as ħ/t ∝ (kBT )2/eF (for ek ¼ eF). While this is much smaller than either ek − eF or kBT, justifying a posteriori the validity of the quasiparticle concept, it cannot be entirely neglected because it plays an important role in the calculation of the transport coefficients, when the system is driven out of equilibrium. The effective mass of quasiparticles causes a shift in the density of states at the Fermi surface, n∗F/nF ¼ m∗/m, which affects the values of the thermodynamic properties of the interacting electron gas relative to the ideal Fermi gas. For example, the low-temperature heat capacity of the interacting electron gas in three dimensions is given by Cv ¼ (p2/3)k2BTn∗F, which is still linear in T but is modified by a factor m /m relative to its noninteracting counterpart. Remarkably, the interaction between quasiparticles

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(Landau parameters) does not enter this result. In contrast, the bulk modulus n2d m/d n and the spin susceptibility (defined as the derivative of the spin magnetization with respect to an applied magnetic field) are affected both by the effective mass and by the quasiparticle interaction. In the electron gas these interactions reduce the bulk modulus and increase the spin susceptibility: the second effect can be seen as a consequence of interactions favoring the alignment of neighboring electron spins in a parallel configuration (see below). Theoretical calculations and experimental measurements of the effective mass and the quasiparticle interaction parameters in the electron gas are reviewed in Chapter 8 of Ref. (Giuliani and Vignale, 2005), to which we refer the reader.

Weakly interacting electron gas: Exchange effects As a start to a calculation of the ground-state energy of the interacting electron gas, one might ask what is the distribution of electron pairs in the ideal Fermi gas as a function of their relative distance r. The pair distribution function, which is written in general as g(r), is defined by computing the average number of electrons that lie within a spherical shell of radius r and thickness dr centered on any given electron and equating this to the quantity ng(r)(4pr2dr). For the electron gas with equal numbers of up and down spins it can be written as g(r) ¼ [g""(r) + g"#(r)]/2, the average of the values for pairs of electrons with parallel spins and antiparallel spins. Whereas g(r) ¼ 1 in a classical ideal gas, the probability of finding two electrons with the same spin at a distance r vanishes exactly as r ! 0. This property derives from the symmetry imposed by the Pauli exclusion principle, according to which the wave function of a Fermi fluid changes sign under the operation of exchange of two identical fermions. That is, the many-body wave function of the electron gas has a zero whenever two parallel-spin electrons overlap in space. The property g""(0) ¼ 0 is an exact result and implies, by continuity, that each electron of given spin induces a local depletion of the density of electrons with the same spin in its surroundings. If we bear in mind that the Coulomb interactions between electrons are repulsive, it is seen that the presence of the “Pauli hole” that was introduced above is accompanied by a gain in potential energy of the electron gas. Exchange effects keep apart electrons with parallel spins and therefore reduce on average the Coulomb repulsion energy of parallel-spin pairs. In the ideal Fermi gas g(r) is given by the expression gideal ðr Þ ¼ 1 −

9 ½j ðk r Þ=ðkF r Þ2 , 2 1 F

(1)

where j1(x) ¼ (sin x − x cos x)/x2 is a spherical Bessel function. The shape of this function, which describes the Pauli hole in the ideal Fermi gas, is shown in Fig. 3. Of course, no correlations exist at this level between pairs of electrons with antiparallel spins, therefore g"#(r) ¼ 1. The expression for gideal(r) can be used to evaluate the ground-state energy in first-order perturbation theory. This calculation corresponds to the energy of the electron gas in the Hartree-Fock approximation, since by translational symmetry the self-consistent Hartree-Fock single-particle orbitals must be the plane-wave orbitals of the ideal gas. The result (in three dimensions) is   3 3 F Ryd − EH g ðr s Þ ¼ 2 2 2par s 5a r s   2:21 0:916 ’ Ryd − rs r 2s per electron, with a ¼ (9p/4)−1/3 and in units of the Rydberg (1 Ryd ¼ e2/(2aB), with aB ¼ ħ2/(me2) being the Bohr radius). Here, we have introduced the crucial density parameter rs by the definition 4p(rsaB)3 ¼ n−1, that is to say, rs represents the mean distance between electrons in units of the Bohr radius. Electronic densities that are usually found in metals (metallic densities) have values of rs ranging from ’2 for Al to ’6 for Cs. Much smaller as well as much higher values of rs, corresponding to higher and lower densities respectively, can be found in semiconductors and semiconductor heterostructures.

r/rsaB

3D 2D g(0)(r)

1D

Fig. 3 The Pauli hole surrounding an electron of given spin in the ideal Fermi gas in three, two, and one dimension.

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The two terms in EHF g (rs) are the average kinetic energy and the average potential energy per electron, respectively, and their ratio is evidently proportional to 1/rs. This shows that the role of the electron-electron interactions becomes progressively more important as rs increases, that is, as the density of the electron gas decreases. This counterintuitive fact, already noted in the Introduction, is now clearly seen to be a consequence of the Pauli exclusion principle which forces the electrons to occupy states of higher and higher momentum, and therefore higher kinetic energy, as their density increases. This is reflected in the behavior of the kinetic energy, which increases as 1/r2s . The Coulomb interaction energy, on the other hand, scales as 1/rs, as expected from the 1/r behavior of the Coulomb potential, and therefore becomes negligible in comparison with the kinetic energy for rs ! 0. Thus, the electron gas becomes effectively noninteracting in the limit of very high density.

Electron correlations: Calculation of the ground state energy Naturally, the expression for the energy presented above (Eq. 2) is valid only when the effect of interactions is weak, i.e., in the high-density limit. Going beyond the Hartree-Fock expression requires the full machinery of many-body theory. This has been done in several steps, starting with the celebrated Random Phase Approximation (RPA) in the 1950s (Bohm and Pines, 1953; Gell-Mann and Brueckner, 1957) and continuing with semi-analytical theories such as the STLS theory (Singwi et al., 1968) (an extension of the RPA based on the Hubbard concept of “local field factor,” see below) and culminating with the essentially exact quantum Monte Carlo calculations by Ceperley et al. in 1980 (Ceperley and Alder, 1980). All these calculations take into account, with varying degrees of accuracy, the Coulomb correlations between electrons, i.e., the correlations that are caused by the Coulomb interaction in addition to the ones that are required by the Pauli principle. It has been already remarked that no correlations between antiparallel spins are present in the ideal Fermi gas. This is no longer true in the presence of Coulomb interaction. The main effect is that the motions of opposite-spin electrons, which were completely uncorrelated in the ideal case, start to be correlated, and the depletion of the local electron density around each electron deepens since pairs of electrons with antiparallel spins are now kept apart by the Coulomb repulsion (The depletion of parallel-spin electrons is also somewhat deepened, but this is a relatively small effect since those electrons were already kept apart by the Pauli exclusion principle). Fig. 4 shows the “Pauli-Coulomb hole,” calculated from QMC for increasing values of the coupling strength rs (Ortiz and Ballone, 1994). An exact property of this function (also shared by the ideal pair correlation function of Eq. (1)) is that the total local depletion of electron density around the electron at the origin corresponds to taking away exactly one electron from its neighborhood. This ensures that the central electron plus the “exchange-correlation” (xc) hole surrounding it is an overall charge-neutral object. Within this constraint strong Coulomb interactions at low density cause a deepening of the xc hole and also a diminished contrast between parallel- and antiparallel-spin correlations (i.e., the Pauli exclusion principle becomes less important in the

1

g (r/r s )

g (r)

0 1

g (r)

0

1

r/rsaB

2

3

Fig. 4 The Pauli-Coulomb hole in the interacting spin-unpolarized electron gas for various values of the coupling strength rs: rs ¼ 1 (dotted line), rs ¼ 3 (dash-dotted line), rs ¼ 5 (dashed line), and rs ¼ 10 (full line). Adapted from Ref. Ortiz G and Ballone P (1994) Physical Review B 50: 1391.

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presence of strong Coulomb repulsion). The diminished contrast between parallel and antiparallel spins plays a crucial role in keeping the electron gas paramagnetic, rather than ferromagnetic, at metallic densities. Besides deepening the exchange-correlation (xc) hole, correlations also affect the kinetic energy of the electron gas by modifying the occupation numbers nk of momentum states. In the ideal gas these occupation numbers are distributed according to a step function: nk ¼ 1 for k < kF and nk ¼ 0 for k > kF. In the interacting electron gas the occupation numbers are reduced below 1 for k > kF and do not vanish for k > kF resulting in an increased kinetic energy (see Fig. 4). However a discontinuity in nk persists at k ¼ kF, being a key signature of the existence of Landau quasiparticle. The discontinuity Z, known as quasiparticle renormalization constant, is loosely speaking, a measure of the fraction of the spectral weight of single-particle excitations that fall under the description of quasiparticles. The calculation of the ground state energy of the electron gas requires both the pair correlation function and the momentum space occupation numbers. Alternatively, the energy can be calculated from the pair correlation function alone if the latter is known as a function of the coupling constant rs0 in the range 0 < rs0 < rs (this is denoted by grs0 (r)). Then the formula for the total ground state energy is Z rs Z h i Eg 1 e2 d r n gr 0s ðr Þ −1 dr 0s (2) ¼ ekin ðr s Þ + N r rs 0 where ekin(rs) ’ 2.21/r2s Ryd in 3D is the noninteracting energy per particle. In the early stages of the electron gas theory the pair correlation functions was obtained either perturbatively, or from approximate theories of the density response function which were developed to describe the phenomenon of screening (described in the next section). Modern approaches, like QMC, extract g(r) directly from the many-body wave function. The results of these calculations have been cast in convenient parametrized form, which can be found, for example, in Chapter 1 of Ref. (Giuliani and Vignale, 2005) (Fig. 5). Fig. 6 shows the energy per electron of the electron gas in 3D and 2D obtained from analytic fits to the QMC results. Qualitatively, its behavior is not very different from that of the Hartree-Fock approximation (Eq. 2), but of course it lies considerably below Hartree-Fock in the low density limit, going in 3D as ’ − 1.8/rs Ryd rather than −0.916/rs Ryd. Both energy curves shows a minimum, followed by and an inflection point (change in the sign of the curvature) at a slightly higher value of rs. These two special points are marked by arrows in Fig. 6. The minimum indicates vanishing pressure P, and the inflection point is associated with a vanishing bulk modulus B. One may wonder why negative values of P and B, occurring for values of rs larger than the minimum and the inflection point respectively, are allowed at all. Normally, such negative values would indicate a thermodynamic instabilities of the system. The reason why these apparently unphysical values of P and B are allowed is that the jellium model includes a rigid background of positive charge. This rigid background fixes the density of the electron gas, which is therefore not allowed nor required to adjust its density to the minimum of the energy vs density curve. Likewise, the rigid background neutralizes any tendency of the electron gas to spontaneously contract or expand. Fig. 7 shows the inverse bulk modulus K ¼ 1/B (also known as compressibility) and the spin susceptibility of the three-dimensional electron gas as calculated from QMC. Both quantities are plotted in units of their ideal-gas (i.e., noninteracting) counterparts. We also plot the same quantities obtained from the Hartree-Fock approximation. It is evident from these plots that while interactions generally enhance both the compressibility and the spin susceptibility, the specific effect of correlations (primarily correlations between antiparallel spin electrons) as revealed by the difference between Hartree-Fock and fully interacting results, is to further enhance the effect of exchange on the compressibility but at the same time to suppress the effect of exchange on the spin susceptibility. The second effect is the reason why ferromagnetism does not occur at metallic densities, and most likely at no density.

nk 1 0.8 0.6

Z 0.4 0.2

0.5

1

1.5

2

2.5

3

k/kF Fig. 5 Occupation numbers in momentum space for the three dimensional electron gas from Ref. (Senatore et al., 1996) for rs ¼ 2 (solid line) and rs ¼ 5 (dashed line). The occupation number discontinuity Z is shown.

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Fig. 6 Energy vs rs curves of a three- and two-dimensional electron gas. The arrows at the lower value of rs indicate the points where the energy has a minimum (zero pressure), while the arrows at the higher values of rs indicate the inflection points in the energy vs density relation (zero bulk modulus).

χP χS

1 0.5

2

4

6

8

10

rs

K0 K

-0.5 -1

Fig. 7 Plot of the dimensionless compressibility K and spin susceptibility w of a three-dimensional electron gas as a function of rs. K0 is the non-interacting compressibility and wP is the non-interacting spin susceptibility. In the Hartree-Fock approximation, which includes only exchange, both K and w are enhanced relative to their non-interacting values. Correlations between opposite spin electron cause K to be even more enhanced, while w is reduced.

Electron gas and density functional theory The ground state energy of the uniform electron gas can be written as Eg =N ¼ ekin ðnÞ + exc ðnÞ,

(3)

where ekin ’ 2.21/r2s Ryd is the noninteracting energy per electron in 3D. The remainder exc(n) is known as the exchange-correlation (xc) energy (per particle): it includes the interaction contribution to the kinetic energy. The xc energy has found widespread use in density functional theory. According to this theory, the effect of the electron-electron interaction on the ground state of an electronic system can be mimicked by a local exchange-correlation potential, vxc(r), which is added to the external potential and to the classical electrostatic potential of the charge distribution, so that interactions do not need to be taken into account explicitly. In the local density approximation the xc potential is given by vxc ðr Þ ¼

 d ½nexc ðnÞ n¼nðrÞ dn

(4)

i.e., it is the derivative of the uniform xc energy density with respect to density, evaluated at the local density of the system. Several parametrized expression of exc(rs) (where rs is a proxy for the density) are available in the literature (see Chapter 7 in Ref. (Giuliani and Vignale, 2005)).

Static screening When a foreign charged particle (impurity) is inserted in the electron gas, the electrons respond by rearranging their density around the impurity in such a way that, at large distance from the impurity, the electric field created by the impurity is largely cancelled by the electric field created by the displaced electrons. This phenomenon is known as screening. To be more precise, let us denote by Vext(q) the Fourier transform of the potential created by the impurity. Then in the linear response approximation (valid at large distance from the impurity, where the potential is weak) the Fourier transform of the induced electronic density is dnðqÞ ¼ wðqÞV ext ðqÞ:

(5)

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Electron gas (theory)

The static density response function w(q) is a fundamental property of the electron gas. It allows us to calculate the net electric potential from the impurity and the displaced electrons as V sc ðqÞ ¼ V ext ðqÞ + vðqÞdnðqÞ ¼ ½1 + vðqÞwðqÞV ext ðqÞ V ðqÞ ¼ ext eðqÞ

(6)

where v(q) is the Fourier transform of the Coulomb interaction (v(q) ¼ 4pe2/q2 in three dimensions), and e(q)  [1 + v(q)w(q)]−1 is known as the static dielectric function of the electron gas. The calculation of w(q) and e(q) is a classic problems in electron gas theory. For the ideal noninteracting gas one has    2 q − 2kF  1 q2 − 4kF  , (7) ln  w0 ðqÞ ¼ − nF − 2 8kF q + 2kF  in three dimensions (similar analytical results are obtained in two and one dimension, see Fig. 8). Here nF is the density of states at the Fermi surface which for a three-dimensional ideal gas is given by nF ¼ 3n/(2eF). For the interacting electron gas one has ws ðqÞ , (8) wðqÞ ¼ 1 − vðqÞ ws ðqÞ where ws(q), is known as the proper density response function and describes the response of the electronic density to the screened potential, i.e., dn(q) ¼ ws(q)Vsc(q). The exact form of ws(q) is not known. The most popular approximation for ws(q) is the random phase approximation, which simply posits ws(q) ¼ w0(q). That is to say, the random phase approximation assumes that the electron gas responds as an ideal gas to the self-consistently screened potential. This leads to the widely used expression w0 ð q Þ : 1 − vðqÞw0 ðqÞ

(9)

eR PA ðqÞ ¼ 1 − vðqÞw0 ðqÞ

(10)

wR PA ðqÞ ¼ The corresponding dielectric function

is easily seen to go to infinity as 1/q for q ! 0 (since w0(q) ! –nF in this limit). This demonstrates the screening property described at the beginning of this section. The screened potential is suppressed by a diverging dielectric function in the limit of q ! 0, which physically corresponds to large distances from the impurity. Indeed, in this limit the RPA dielectric function reduces to eTF(q) ¼ 1 + (qlTF)−2 where lTF ¼ (4pe2nF)−1/2 is the Thomas-Fermi screening length. The Fourier transform of the screened potential v(q)/eTF(q) is easily seen to decay exponentially as e−r/lTF, where r is the distance from the impurity. However, this result is not quite accurate. A careful consideration of the logarithmic singularity of w0(q) at q ¼ 2kF (see Eq. 7) reveals that the electronic density at a distance r from the impurity is asymptotically described by the expression 2

nðr Þ − n∝

cosð2kF r + fÞ : r3

Fig. 8 Static susceptibility of the Fermi gas in dimensions 1,2, and 3, as a function of wave number q; the vertical dashes show the position q ¼ 2kF.

(11)

Electron gas (theory)

787

The presence of the Fermi surface induces microscopic “Friedel oscillations” of wavelength p/kF in the electron density. This result should be contrasted with the monotonic decay of the screened potential in a classical plasma, as found for instance in the DebyeHückel theory of electrolytic solutions. Instead the effective potential created by an ion inserted in the electron gas, as well as the effective potential with which each electron interacts with the other electrons, are oscillating functions of distance showing both repulsive and attractive regions. These results of the random phase approximation are essentially confirmed by more accurate many-body theories of the density response function. In these theories one often assumes that the electron gas responds as an ideal gas not to the self-consistently screened electrostatic potential, but to an effective self-consistent potential that is modified to take into account short-range exchange-correlation effects. In other words, one posits dnðqÞ ¼ w0 ðqÞ½V ext ðqÞ + vðqÞdnðqÞ − vðqÞGðqÞdnðqÞ

(12)

where the second term in the brackets is the electrostatic potential created by the charge density distribution and the last term, containing the so-called local-field factor G(q), generates the additional exchange-correlation potential (this is the term that is neglected in the RPA). Different many-body theories propose different approximate forms for G(q), beginning with the first and perhaps most famous one, due to Hubbard, which goes as G(q) ’ (q2/2)/(q2 + k2F ). In all cases G(q) tends to zero as q2 for q ! 0 reflecting the short-range nature of the xc effects. In these theories the density response takes the form wðqÞ ¼

w0 ðqÞ , 1 − vðqÞ½1 − G ðqÞ w0 ðqÞ

(13)

and the finite q ! 0 limit of v(q)G(q) amounts to a many-body renormalization of the density of states at the Fermi surface, hence of the screening length. Fast-forwarding over many years of theoretical development (the interested reader can find a review in Ref. Giuliani and Vignale, 2005). Fig. 9 shows the plot of an analytical fit (Corradini et al., 1998) to the local field factor calculated directly from QMC by applying a periodic potential to electrons in the simulation box. More recent calculations and parametrization of G(q) are also available (Chen and Haule, 2019; Ruzsinszky et al., 2020). The local field factor (also known in the literature through its alias, the exchange correlation kernel fxc(q) ¼ −v(q)G(q)), and its spin-dependent and frequency-dependent generalizations Gss0 (q) and Gss0 (q,w) respectively (where s and s0 are spin indices), plays a crucial role in density functional theory and in many real-world applications of the electron gas model. In particular, the effective interaction between two electrons in the electron gas (Kukkonen and Overhauser, 1979; Kukkonen and Chen, 2021; Dornheim et al., 2022), which is constructed in terms of local field factors and differs from the effective interaction between two impurities embedded in the electron gas can be used to analyze the possibility of superconductivity in electronic systems (Kukkonen and Chen, 2021).

Dynamical screening and plasmons While the low energy excitations of the electron gas are fermionic quasi-particles, the spectrum of long-wavelength density fluctuations is overwhelmingly dominated by high-frequency collective oscillations known as “plasmons.” The physical origin of plasmons is very simple. When the density of the electron gas is locally perturbed (say increased) electrons from neighboring regions move to screen the charge inhomogeneity, but in doing so they tend to overshoot the mark. Then they are pulled back toward the original disturbance and overshoot again, setting up an oscillation that decays slowly in time. The process is very similar to the one giving rise to sound waves in an ordinary gas, but there is an important difference: the restoring force responsible for the oscillation is the average long-range field created by all the electrons as opposed to, in the case of sound, the force arising from frequent

Fig. 9 Analytical fit to the local field factor calculated from QMC at rs ¼ 2 (Corradini et al., 1998). Notice that, unlike earlier approximations, this local field factor appears to grow indefinitely for large wave vector q.

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Electron gas (theory)

short-range collisions between particles. Because of the long-range nature of the Coulomb interaction, the frequency of oscillations at wave vector q, denoted by op(q), tends to be high and is given in the long-wavelength limit by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi op ðq ! 0Þ ¼ nq2 vðqÞ=m ¼ 4pne2 =m (14) for the three-dimensional electron gas. The possibility of such oscillations was anticipated by Langmuir in 1928, long before the central importance of plasmons in the theory of the electron gas was recognized in the seminal work of Bohm and Pines in the 1950s. Notice that in Eq. (14) m is the bare electron mass in vacuum, not the quasiparticle effective mass. This simple long-wavelength result is exact, regardless of the relative strength of the electron-electron interaction, because a long-wavelength plasmon can be viewed as a rigid motion of the center of mass of the entire system, which does not involve the complex exchange and correlation effects that dress the motion of individual electron quasiparticles. Plasmon excitations are observed in inelastic scattering experiments on metals, using either a beam of fast electrons in electron energy loss spectroscopy or a beam of X-rays. The simplest way to calculate the plasmon frequency is to look for the poles of the density response function, properly generalized to accommodate periodic potentials oscillating at a frequency o, as a function of o. This is easily understood by considering the definition of the dynamical density response function w(q, o): dnðq, oÞ ¼ wðq, oÞV ext ðq, oÞ

(15)

where dn(q, o), Vext(q, o)are Fourier transforms of dn and Vext in both space and time. The existence of a self-sustained plasma oscillation implies the possibility of a finite density fluctuation dn in the absence of an external field Vext. This can only happen if w(q, o) is infinite at the plasmon frequency. The general theory of the linear response guarantees that poles of w(q, o) can only occur in the lower half of the complex frequency plane, but nothing prevents these poles from coming arbitrarily close to the real frequency axis. When this happens, the real part of the frequency is the plasmon frequency, while the imaginary part (small and negative) is related to the damping rate of the oscillation. These general features are clearly brought out by the simplest approximation, the RPA, in which one has (with obvious generalization of Eq. (16)) wR PA ðq; oÞ ¼

w0 ðq; oÞ , 1 − vðqÞw0 ðq; oÞ

(16)

where w0(q, o) is the frequency-dependent response function of the ideal electron gas, which is known analytically and usually referred to as the Lindhard function (Giuliani and Vignale, 2005). The plasmon frequency is then obtained by the solution of the equation 1 − vðqÞw0 ðq, oÞ ¼ 0,

(17)

which, in view of the exact result w0(q, o) ! nq /(mo ) for q ! 0 and finite o leads directly to Eq. (14). It is worth noting that the solution is purely real, i.e. the plasmon is not damped in the RPA, The situation is illustrated in Fig. 10A, where we see that plasmons 2

(A)

2

(B)

Fig. 10 (A) Plasmon dispersion in three and two dimensional electron gas (full lines) for rs ¼ 2. The dashed line represents the approximate two dimensional dispersion given by Eq. (17). Notice that both in three and two dimensions the long wavelength plasmons lie well above the edge of the particle hole continuum (shaded region), which protects them from decay into single electron-hole pairs. (B) Plasmon dispersion normalized to the plasmon frequency at q ¼ 0. The symbols represent the experimental data for Na (rs ¼ 3.93, downward triangles), K (rs ¼ 4.86, upward triangles), Rb (rs ¼ 5.20, dots) and Cs (rs ¼ 5.62, squares). The solid lines give the theoretical plasmon dispersions obtained with the use of the STLS static local field factor.

Electron gas (theory)

789

oscillations being much higher in frequency than electron-hole pair excitations at the same wave vector, have no viable mechanism for decaying into single electron-hole pairs while obeying the conservation of momentum and energy. The situation changes, however, when one consider more sophisticated many-body theories beyond the RPA. In these theories the plasmon can decay by emitting two electron-hole pairs with large but nearly opposite momenta, so that the total energy and momentum match those of the plasmons. But even these more sophisticated theories predict plasmon damping to vanish as q2 for q ! 0 in agreement with the general physical argument outlined above, namely the decoupling of the center of mass motion from the internal degrees of freedom of the electron gas (Kohn’s theorem). Fig. 10B shows the plasmon dispersion op(q) for the three dimensional electron gas at various densities. These results are obtained using the STLS theory, which includes xc local field factors beyond the RPA. At high density (small rs) the frequency increases with increasing q until a critical q is reached where the plasmon merges with the continuum of electron-hole excitations and is no longer a well-defined excitation. Here the RPA is qualitatively correct. But, as the density decreases, the plasmon dispersion flattens and even bends downward. This effect cannot be reproduced within the RPA and reflects the increasing importance of correlations at low density. A vanishing plasmon frequency at finite q would in principle indicate an instability of the uniform electron gas and the emergence of a highly correlated phase such as the Wigner crystal (see below).

Wigner crystallization It has already been remarked that as the dimensionless electron spacing rs increases (i.e., as the electron density decreases) the kinetic energy becomes less and less important relative to the potential energy associated with the Coulomb repulsions. In the 1930s Wigner had noticed that an optimal value is obtained for the potential energy of jellium if the electrons are placed on the sites of a body-centered cubic lattice structure. Localization of the electrons raises their kinetic energy and an electron crystal may become favored only at very low density, where the potential energy becomes dominant. Of course, in the crystal the electrons are not strictly localized on the lattice sites but execute vibrational motions around them. But the kinetic energy associated with this zero-point and becomes negligible compared to the interaction energy r−1 motion scales as r–3/2 s s in the low density limit. At the same time, the which is much smaller than rs when rs is large, justifying the picture of amplitude of the zero-point oscillations scales as r3/4 s well-localized electrons in the limit under consideration. Of course, strong anharmonicity and high concentrations of lattice defects will be present in this quantum crystal near the transition density, or even at lower densities as the temperature is raised and the melting point is approached. The problem of “Wigner crystallization” has become a classic in many-body physics, from both the theoretical and the experimental point of view. Quantum Monte Carlo studies have indicated that the ground state of three-dimensional (3D) jellium undergoes a first-order transition to a body-centered-cubic crystal around rs ’ 100, which is probably beyond the reach of experiments. In two dimensions, however, a first-order transition between the paramagnetic liquid and a triangular Wigner crystal is predicted to occur at a much lower value of rs ’ 35. The search for Wigner crystallization in the laboratory has therefore focused on quasi-2D assemblies of electronic carriers, mostly in artificial semiconductor heterostructures. The first unambiguous observation of Wigner crystallization was reported by Grimes and Adams in 1979 for electrons floating on top of a substrate of liquid 4He in a quasi-classical regime. Metal-insulator transitions have subsequently been reported in quasi-2D systems of carriers subject to very strong magnetic fields in the fractional quantum Hall effect regime or in the presence of disorder and also in high-purity quasi-2D samples having exceptionally high carrier mobility. The observation of shear phonon modes, pinning-depinning transitions, and, more recently, geometric resonances occurring when the cyclotron radius of electrons becomes commensurate with the lattice constant, have been considered “smoking gun” evidence for Wigner crystallization.

Electron gas in reduced dimensionality Electron gases in reduced dimensionalities present strong fundamental interest and high technological relevance. A general consequence of reducing the dimensionality is an increasing role of the electron-electron correlations at each given value of the dimensionless coupling strength rs. This is ultimately a consequence of the modified topology of the electronic motions as the electrons are constrained to move in a plane or along a line or inside a quantum dot. This profoundly modifies the excitation spectrum of the ideal Fermi gas and often enhances the susceptibility to external perturbations. Two-dimensional electron gases (2DEGs) are typically realized in semiconductor heterostructures at the interface between two different semiconductors. Typical examples are the inversion layer that forms at the surface of p-type Si in a metal-oxidesemiconductor (MOS) heterostructure with M ¼ polycrystalline highly doped Si, I]SiO2, S ¼ p-type Si, and GaAs-AlGaAs-GaAs quantum wells, where the electrons reside in the AlGaAs region. These system are two-dimensional in the sense that the motion of electrons in one direction (perpendicular to the heterostructure) is strongly quantized (i.e., all the electrons reside in the lowest quantized state of motion, which is separated from other states by a large energy gap) but the electrons move freely in the remaining two directions. What makes the systems particularly attractive is (i) the possibility of changing the electronic density by orders of magnitude by electrostatic gating (whereas the electron density in traditional metallic systems is essentially fixed by the ionic background) and (ii) the possibility of spatially separating the electrons from sources of disorder such as donor impurities, resulting in much higher mobilities. In recent years a new class of two-dimensional materials has emerged, which are based on single atomic

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Electron gas (theory)

layers, such as a single layer of C atom (known as graphene) where the electronic density can be controlled by doping and by electrostatic gating. Within this family of systems many variations are possible, such as double electronic layers, multiple layers, and periodic superlattices. Recently, it has also become possible to twist two adjacent atomic layers relative to each other, thus generating incommensurate patterns known as moiré superlattices. The Landau theory of Fermi liquids remains generally applicable to 2DEGs, in spite of some subtle differences. For example, the inverse lifetime of Landau quasiparticle in 2D varies as (ek − eF)2 ln | ek − eF | /eF as a function of energy and as (kBT)2 ln |kBT/EF|/EF as a function of temperature. The logarithmic terms, which are absent in three dimensions, tend to decrease the lifetime of quasiparticles in 2D. 2DEGs are also much more sensitive to impurity scattering than their 3D counterparts. For example in the non-interacting 2DEG it is widely believed that all electronic states are localized in the presence of disorder. Electron-electron interactions restore the metallic state, but the inverse lifetime of quasiparticles now scales as kBT, which is much larger that kBT)2 ln T, making these systems marginal Fermi liquids. Static screening is greatly reduced in two dimensions as the electric field created by, say, an impurity can escape out of the plane and reach far away electrons in spite of the presence of compensating charge in the plane. Plasmon excitation are similarly reduced in frequency, their long-wavelength dispersion going as  1=2 2 o2D , p ðq ! 0Þ ¼ 2pne q=m

(18)

which goes to zero for q ! 0 but still lies much above the energies of electron-hole pairs (linear in q). The above formula, incidentally, is exactly what one gets from Eq. (14), provided one takes care to use the correct form of the Fourier transform of the Coulomb potential in two dimensions, namely v2D(q) ¼ 2pe2/q. Thus, Eq. (14) is valid for the Galilean-invariant electron gas in any number of dimensions. Similarly, the formula for the Friedel oscillations in the static screening density is modified due to the fact that the static density response function has a cusp at q ¼ 2kF (see Fig. 6): this causes the amplitude of the oscillations to decay more slowly as a function of distance, v.i.z., as 1/r2 rather than as 1/r3. Some of the most impressive examples of 2DEG physics have emerged in the presence of high magnetic field (quantum Hall effect) and in graphene where a peculiar band structure causes the free particle dispersion to be linear in k (mass less Dirac fermions). We will say more about these in the next two sections. Consider now a system in which the conduction electrons are restricted to move along a line. The “Fermi surface” reduces to two points located at k ¼ kF and only forward or backward scattering processes are allowed at the “Fermi surface”. Here the Landau theory of Fermi liquid breaks down completely even in the absence of disorder. In fact, one can show that the Landau quasiparticles are no longer well-defined, as their inverse lifetime becomes comparable to or larger than their energy. Fermionic quasiparticles are superseded by linearly dispersing collective excitation (similar to plasmons) with bosonic character. This leads to a new paradigm of interacting electrons known as the “Luttinger liquid”. Characteristics of the Luttinger liquid are (i) There are no single-particle excitations: sharp peaks in the one-particle spectral function at k ¼ kF are replaced by power-law singularities (ii) there is no Fermi surface: the discontinuity in the momentum distribution across the Fermi surface is replaced by a different type of nonanalytic behavior, (iii) Correlation functions between observables at different points in space decay with power laws with fractional exponents that depend on the strength of the interaction (iv) spin-charge separation occurs, i.e., spin-carrying collective excitations and charge-carrying collective excitations propagate with different velocities (in a normal Fermi liquid both spin and charge reside in the Landau quasiparticle and travel with the same velocity). Even a simple look at the density response function of the noninteracting 1D Fermi gas shows how unusual this system can be. We have   q + 2kF  2m :  (19) ð q Þ ¼ − ln w1D 0 pħ2 q  q −2kF  In Fig. 6 this density response function is plotted visa-vis its counterparts in 2 and 3 dimensions. In all cases there is a singularity at |q| ¼ 2kF, but the singularity gains strength as dimensionality is reduced, going from a logarithmic divergence in the derivative of w0(q) in 3D, to a discontinuity in the derivative of w0(q) in 2D, and finally to a logarithmic divergence in w0(q) itself in 1D. The divergence of the response means that the system is on the verge of an instability which will cause a massive reconstruction of the ground state as soon as additional interactions are turned on. For example, a monoatomic 1D conductor with a half-filled conduction band (one electron per atom) will gain electronic energy as soon as the electron-lattice coupling is turned on. The system undergoes a lattice distortion which opens a gap at the Fermi level through the formation of dimers. This is known as “Peierls instability”. Similarly, electron-electron interactions, even in the absence of electron-lattice coupling, lead to the formation of a Luttinger liquid.

Two-dimensional electron gas in graphene In its pristine state graphene – a single layer of C atoms arranged in a honeycomb lattice – is a semimetal. Its valence and conduction bands touch at two points in the Brillouin zone (Dirac points) and the band dispersion is near these points with the two bands being mirror images of each other (see Fig. 11). This unusual form of the energy-momentum relation arises from the fact that the

Electron gas (theory)

791

Fig. 11 Band dispersion for massless Dirac fermions in the vicinity of the crossing point (Dirac point). In pristine graphene the Fermi level cuts the bands at the crossing point, so that there are no free carriers at zero temperature. Doping and/or electrostatic doping can shift the Fermi level away from the crossing point, as shown in this figure, creating a two-dimensional gas of massless Dirac electrons (case shown) or holes.

honeycomb lattice has two inequivalent sites in each unit cell, thus endowing the electron with an additional degree of freedom, formally equivalent to a spin, which has eigenvalues 1 or −1 depending on whether the electron is on or the other site. Under these conditions the electron wave function can no longer be described by a scalar plane wave: it becomes instead a two-component spinor, similar to the spinors that occur in the theory of massless relativistic Dirac fermions, but, of course, in a completely non-relativistic context. The pseudospin degree of freedom plays a crucial role in the calculation of the electronic properties of graphene. For a detailed review see Ref. (Castro Neto et al., 2009). This system can host a 2DEG if we add electrons to the conduction band by doping or by electrostatic gating. Or it can host a two-dimensional hole gas if we remove electrons from the valence band. We will focus on the former case. Let us set the zero of the energy at one of the Dirac points, and let us also choose that point as the origin of momentum space. The main differences between this and the ordinary 2DEG stem from (i) the linearity of the conduction band dispersion ek ¼ ħvk (notice the energy vanishing at the Dirac point k ¼ 0) and (ii) the presence of occupied valence band states which are separated from empty conduction band states by energies of the order of the Fermi energy. Concerning point (i) we notice that the linear dispersion breaks Galilean invariance, which mandates a quadratic relation between energy and momentum. On the contrary, this is the dispersion of massless relativistic particles (e.g., photons) which move with a preassigned speed v (v ¼ speed of light in the case of photons) independent of the momentum. For electrons in graphene, v is about 1/300 the speed of light. As usual, we introduce the Fermi wave number kF, which separates occupied from empty states within the conduction band (see Fig. 4): it is related to the two-dimensional electron density by kF ¼ (pn)1/2, where we have taken into account not only the spin degeneracy but also the existence of two degenerate states associated with the same k in the vicinity of two different Dirac points. The Fermi energy is then eF ¼ ħvkF ¼ ħv(pn)1/2.. An important difference between the 2DEG in graphene and the ordinary 2DEG emerges immediately from the above. Namely, the kinetic energy per particle eF ∝ ħvn1/2 and the potential energy per particle e2/r ∝ e2n1/2 scale with the same power of density and the ratio of the second to the first is aee ¼ e2/(ħv)  2 in graphene. Thus, the electron gas in graphene does not seem to become “more ideal” as the density is increased, or “less ideal” as it is decreased – rather it presents us with an intermediate-coupling scenario at all densities (the actual value of aee is strongly affected by the dielectric environment of the graphene sheet, and can be larger or smaller than 2. The value quoted above is for a graphene sheet in vacuum). However, there is now another measure of the importance of interaction effects, and that is L/kF where L is an cutoff wave vector, which determines the largest momentum of the occupied states for which the linear band model is still valid. The precise value of this cutoff wave vector is somewhat fuzzy, but is expected to be of order 1/a, where a is the lattice constant. So even though aee is constant, interaction effects become stronger as kF tends to zero, which is similar to the familiar situation, but leads to very different phenomenology in this case. For example it can be shown that the bulk modulus, proportional to ∂m/∂n, is increased by interactions rather than decreased (Polini et al., 2007). This happens because when the electronic density is increased the Fermi level in the upper band moves farther away from the lower band: the negative exchange energy that is lost due to this effect outweighs the negative exchange energy that is gained by having more electrons in the upper band. The same phenomenon is observed for the spin susceptibility, which is now reduced, rather than enhanced, by interactions. The existence of a finite kF allows one to define an effective mass, even though the underlying particles are massless. To do this notice that in the ordinary electron gas the energy of a particle near the Fermi level can be approximated as ek − eF ¼ ħkF(k − kF)/m, where m is the mass of the particles. Comparing this with the corresponding expression for massless fermions ek − eF ¼ ħv(k − kF) one is led to the identification of an effective mass mc ¼ ħkF/v, which is also known as the cyclotron mass. Thus, massless electrons in the vicinity of the Fermi surface behave like ordinary massive electrons with an effective mass ħkF/v. This would seem a rather uninteresting story until we realize that this effective mass is density-dependent, because kF is, and does indeed tend to zero in the limit of low carrier density.

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Electron gas (theory)

These differences are duly reflected in the calculation of the ideal-gas linear response functions w0 (q) and w0 (q, o). For example in the ordinary 2DEG one has w0 (q, o) ! nq2/(mo2) for q ! 0 and finite o. For electrons in graphene this relation is replaced by w0(q, o) ! eFq2/(pħ2o2), which follows directly from the replacement of m by the cyclotron mass. Making use of this limiting form the plasmon dispersion for electrons in graphene is found to be  1=2 op ðq ! 0Þ ’ 2e2 eF q =ħ,

(20)

which is qualitatively different from Eq. (18) because the coefficient of q1/2 scales with density as n1/4 rather than n1/2. More importantly, the coefficient of q1/2 is no longer “universal”, in the sense of being independent of the electron-electron interaction strength. In the Galilean-invariant electron gas, that universality stems from the fact that long-wavelength plasmons correspond to a rigid motion of the center of mass of the electron gas, and the center of mass is decoupled from the internal degrees of freedom. This is no longer the case for the electron gas in graphene. The center of mass motion does not decouple from internal degrees of freedom and the coefficient of q1/2 in Eq. (18) (also known as Drude weight) is found to depend on the strength of the electron-electron interaction (i.e., the value of the coupling constant aee). Additional differences between ordinary 2DEG and graphene 2DEG emerge when one consider the structure of the electron-hole excitation spectrum (see Fig. 12A), which now contains an entirely new branch in which electrons are promoted from a valence band state to a conduction band state with excitation energies comparable to the Fermi energy. Also plotted in Fig. 12B is the static density response function w0 (q), which is distinctly different from the one plotted in Fig. 6, rising linearly, rather than decreasing, as a function of q, for q > 2kF. The different form of the singularity at q ¼ 2kF is reflected in a different form of the Friedel oscillations induced by an impurity. The amplitude of these oscillations decays as cos(2kFr)/r3, at variance with the 2DEG result cos(2kFr)/r2. Coming to electron-electron interaction effects, we point out that, as the Fermi level approaches the crossing point with decreasing density, the Fermi liquid concept remains in force as long as kF > 0, but the effective mass of quasi-particles is found to be logarithmically suppressed relative to the noninteracting cyclotron mass (Kotov et al., 2012). Another way of saying this is that the Fermi velocity v is renormalized to v (kF) >v where v (kF) diverges logarithmically in first order perturbation theory as kF ! 0:   v ∗ ðkF Þ a L : (21) ¼ 1 + ee ln v 4 kF While an increase of the velocity, leading to a reshaping of the bands near the Dirac point, has been experimentally observed, the divergence of v (kF) poses a problem of legitimacy for the microscopic perturbation theory on which this prediction is based. A non-perturbative approach is needed to analyze the kF ! 0 limit. This approach is provided by the renormalization group, which is reviewed in Ref. (Kotov et al., 2012), and generally confirms the predictions of the weak coupling theory. What is the effect of the interaction of the compressibility and the spin susceptibility of a the electron gas in graphene? In an ordinary 2DEG both are strongly enhanced by the interaction because the exchange energy becomes larger in absolute value (i.e., actually more negative) when the electron gas is spin polarized (spin susceptibility) or when the Fermi energy is pushed toward higher values. Surprisingly, the situation is reversed in the case of graphene 2DEG: detailed microscopic calculations (Polini et al., 2007) reveal that both the spin susceptibility and the compressibility are reduced by electron interaction. The explanation of this counterintuitive result lies in the fact that the exchange energy has a large contribution from the “spectator”

(A)

(B)

Fig. 12 (A) Schematic diagram of electron-hole pairs and collective excitations in graphene. Notice that the electron-hole pair excitation spectrum now contains transitions from the valence to the conduction band, as well as the usual transitions within the conduction band (B) The static density response function of the 2DEG in graphene. Notice the rapid linear increase as a function of q for q > 2kF. This unusual behavior is a consequence of interband transitions which contribute to the polarizability of the electron gas in graphene.

Electron gas (theory)

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electrons in the valence band. At first sight, it may seem that the distribution of electrons in the valence band should not be affected by the position of an electron in the conduction band for the same reason that the distribution of spin-down electrons is not affected by the presence of a spin-up electron in an ideal electron gas. But this is not true because valence and conduction band states of the same spin orientation (unlike states of opposite spin orientations) are not completely distinguishable at a given point in space: this gives rise to an inter-band exchange energy. When the chemical potential in the conduction band is raised, it turns out that the familiar gain in intra-band exchange is more than compensated by the loss of inter-band exchange (due to decreased overlap of the electron wave function at the Fermi level with valence band wave functions). Hence the sign of the effect is reversed.

Two-dimensional electron gas at high magnetic field One of the most spectacular effects in the two-dimensional electron gas is the Quantum Hall Effect, which occurs when the gas is exposed to a strong perpendicular magnetic field H of the order of several Teslas and sufficiently low temperatures (von Klitzing et al., 1980; Tsui et al., 1982). The effect can be described as the quantization of the Hall conductance (i.e., the ratio IH/VL, where IH is the transverse current and VL is the longitudinal voltage) in multiples of the fundamental quantum of conductance e2/h ’ 10−4S. Thus one has IH ¼ v(e2/h)VL where v is an integer in the integer quantum Hall effect or an odd-denominator fraction in the fractional quantum Hall effect. (The two versions of the effect, integral and fractional, have different physical origins, as we will see). The quantization is spectacularly accurate (one part in 1010), in stark contrast with ordinary conductances, with or without magnetic fields, which depend strongly on the preparation of the sample, the level of purity, the temperature, and so on. To explain this effect, we recall that the energy levels of an electron in 2D in the presence of a perpendicular magnetic field are the so-called Landau levels

 1 ħoc ej ¼ j + (22) 2 where oc ¼ eH/m is the cyclotron frequency and j is a nonnegative integer. These discrete levels replace the continuous distribution of energies ek ¼ ħ2k2/(2m) of a free particle. However, since the number of states must be conserved, each of the Landau levels is macroscopically degenerate, i.e., for a particle confined to a 2D box of area A the number of degenerate states in each Landau level is D¼

HA , F0

(23)

where F0 ¼ h/e is the quantum of magnetic flux. Notice that this degeneracy is the same for all Landau levels, regardless of j. We note in passing that the situation is qualitatively similar in graphene, except for the fact that the “mass” m is no longer a constant but scales with the square root of the electronic density. And since the electronic density, for a fixed number of occupied Landau levels, is proportional to H, as follows from the degeneracy of Landau levels, Eq. (23), it is quite plausible that pffiffiffiffithe effective cyclotron frequency, which determines the energy of the Landau levels in Eq. (22) should become proportional to H rather than H. Indeed, the correct formula for the cyclotron frequency in graphene is v/ℓ, where ℓ ¼ ħ/(eH) is the magnetic length, related to the radius of a semiclassical orbit and described more precisely below. The integer quantum Hall effect occurs whenever the Fermi level falls in the gap between two Landau levels, so that one has an integer number of fully occupied Landau levels. In a perfectly clean electron gas this situation would require an extremely accurate tuning of the electronic density, impossible to achieve in practice. Fortunately, the 2DEG in semiconductor heterostructures is sufficiently dirty to allow a significant density of localized electronic states to exist in the gap between Landau levels. These localized states do not contribute to the Hall effect, but play an essential role in allowing the Fermi level to remain inside the gap over a finite range of densities. According to conventional wisdom, this system should be an insulator, since all the states at the Fermi level are localized and unable to conduct electricity. Nevertheless, it is observed that the Hall conductance is ne2/h where n is the number of occupied Landau levels. The apparent contradiction can be resolved in two ways. The first approach is to recognize that the gap exists only in the bulk of the 2DEG. As the edges of the Hall bar are approached the Landau levels rise under the action of a confinement potential, and the n fully occupied ones eventually cross the Fermi level, thus closing the gap and generating conducting states on each edge. These “edge states” are like a one way street, allowing electrons to propagate only in one direction, with no possibility of backscattering. It is then shown that each pair of perfectly conducting edge channels (one state on each edge, with a potential difference between the two edges) contributes exactly a quantum of a conductance, e2/h, so that the total conductance equals ne2/h. The second approach focuses on the bulk of the sample, which is now bent in the shape of an annulus (the so-called Corbino disk). In this description, one notices that the nominally insulating electron in the occupied Landau levels can still transport a current if they all move together in rough analogy with bags on a conveyor belt. The electromotive force is generated by a varying magnetic flux threading the annulus and is directed tangentially along the annulus. At the same time, the varying magnetic flux causes the centers of the electronic states in each Landau level to slowly drift in the radial direction, thus creating the Hall current. The precise quantization of the current follows from the fact that each additional flux quantum threaded through the annulus causes an integer number of electron to be transported from the inner to the outer edge.

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Electron gas (theory)

The precise relation between the edge and bulk descriptions of the quantum Hall effect is still somewhat unsettled to date, particularly with regard to the spatial distribution of the Hall current. What is certain is that the effect has a topological origin, related to the existence of exactly n bundles of extended states (the Landau levels) below the Fermi level. In contrast to the integral QHE, the fractional QHE occurs when the Fermi level falls in a gap created by electron-electron interactions within the lowest Landau level. The many-body origin of this gap has attracted tremendous attention from many-body theorists. A microscopic theory of the gap responsible for the 1/3 effect was first proposed by Laughlin (Laughlin, 1983), who proposed that the state of the 2DEG in the 1/3 quantum Hall state could be described by the elegant wave function cðz1 , . . . , zN Þ ¼

N  Y

3 PN 2 2 zi − zj e − i¼1 jZi j =4ℓ

(24)

i D, that it penetrates the high-symmetry ground state, producing a global minimum with a distorted configuration; (d) Hidden-PJTE: the same situation as in the hidden-JTE, but the intervening excited state has a strong PJTE with EPJT > D0, instead of the JTE.

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energy surfaces (APES), showing only one distortion coordinate of the system. In all the four cases the APES has no global minimum in the high-symmetry configuration, meaning possible SSB, as in the original Landau formulation for the JTE. This simplified, one-dimensional presentation of the four JTEs is for visual demonstration only, allowing one to get a glimpse on the rich variety of the specific vibronic coupling effects in polyatomic systems, which lead to SSB. We see that while the JTE, nominally, takes place in all the cases of electronic degeneracy, the other three cases are rather limited by the energy gaps and the vibronic coupling strength between the involved electronic states. In particular, the condition of pseudodegeneracy leading to the PJTE is as follows: D < F2 =K 0

(1)

where 2D is the energy gap between the coupled electronic states, F is the vibronic coupling constant, and K0 is primary force constant (the stiffness of the systems with respect to the Q displacements). The analytical conditions for the h-JTE (EJT > D) and h-PJTE (EPJT > D0), graphically seen in Fig. 1, are slightly more complicated (Bersuker, 2009, 2020a). In reality, the APES and the consequent effects are multidimensional with many ramifications and particular cases, including (most important) strong dependence on external perturbations. The picture as a whole for the JTE and PJTE, their modifications and applications, are well presented in books and reviews, cited in the Introduction (see, e.g., Bersuker, 2006, 2021 and references therein). The present brief account of the subject is just to review the latest achievements in the evolution of understanding of these effects and their general applications, allowing the reader to find the necessary details and proofs (most important, mathematical proofs and deduction of these effects based on first principles) in the cited literature. The hidden modifications, h-JTE and h-PJTE, are relatively new, much less studied, and so far not widely applied to relevant local properties in solids. Therefore, we bring here some examples that further illustrate their physical meaning. For illustration of the h-JTE, consider the ozone molecule O3, which is unstable in its high-symmetry D3h configuration. Its full APES has three equivalent minima, in each of which the regular triangular configuration is distorted to an obtuse triangle. The distortion could not be explained, neither by the JTE, as its ground state in the high-symmetry configuration is nondegenerate, nor by the PJTE, because of the very high energy of the relevant excited state, which does not follow the condition of pseudodegeneracy (1). Fig. 2 illustrates the results of ab initio calculations of the energy profiles of the O3 APES in the cross-section along one of the coordinates of the e type displacements from the high-symmetry configuration, in the ground and the lowest excited state (Garcia-Fernandez et al., 2006). It shows convincingly that the instability of the high-symmetry configuration in the ground state is due to the very strong JTE in its excited E state, which penetrates the ground state to produce the three distorted configuration of the JTE in the E  e problem, exactly following the above definition of the h-JTE (Fig. 1c). As for the h-PJTE, a series of demonstrative examples emerged from a class of molecular systems and local centers in crystals with electronic configurations e2 and t3 half-occupied by 2 and 3 electrons with parallel spins, respectively (half-filled closed shells), which take place, for example, in systems with at least one threefold (or higher) symmetry axis (Garcia-Fernandez and Bersuker, 2011). For instance, the CuF3 molecule (and any similar local formation in crystals) falls in this category. The electronic configuration of the Cu3+ ion in this molecule (or cluster) is (t2g)6e2, occupied by 8 electrons, two in the e2 configuration. Its

E

A Fig. 2 Cross-section of the APES of the ozone molecule along the Qy component of the double degenerate e mode (one of the three equivalent distortions in the E  e problem of the JTE), obtained by numerical ab initio calculations including the highly excited E state (at D  9 eV), explicitly demonstrating that the ground state distorted configurations are due to the JTE in the excited state. The global minimum is at Qy ¼ 0.69 A˚ and the E-A avoided crossing takes place at Qy  0.35 A˚ . Reprinted from Bersuker I.B. (2020a) Hidden Jahn-Teller and Pseudo-Jahn-Teller Effects, Encyclopedia, General & Theoretical Physics, Topic Review. https://encyclopedia.pub/4283. Copyright (2020) MDPI.

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Fig. 3 Ab initio calculated energy profiles of the planar CuF3 molecule (cluster) in the ground and lowest excited states as a function of the angle a (e mode distortion of the D3h configuration), showing the formation of two equilibrium geometries with the lower energy one distorted at a  93 , induced by the PJTE on two excited states. The distorted and undistorted configurations have different spin multiplicity (magnetic moments) and different dipole moments. Reproduced from Bersuker I.B. (2020a) Hidden Jahn-Teller and Pseudo-Jahn-Teller Effects, Encyclopedia, General & Theoretical Physics, Topic Review. https://encyclopedia.pub/ 4283. Copyright 2020 MDPI.

Fig. 4 The e2 electronic configuration spans the states 3A2, 1A1, and 1E. While the magnetic 3A2 state is lowest in energy and stable in the high-symmetry configuration (left), the excited two states are coupled via the PJTE leading to a lower-energy non-magnetic and distorted equilibrium configuration (right). Reprinted from: Bersuker I.B. (2020a) Hidden Jahn-Teller and Pseudo-Jahn-Teller Effects, Encyclopedia, General & Theoretical Physics, Topic Review. https://encyclopedia. pub/4283. Copyright 2020 MDPI.

calculated energy level scheme is shown in Fig. 3. We see that the ground state of the undistorted regular triangular configuration of this system is a spin triplet, 3A2, but a strong PJTE mixing of two excited singlet states, 1A1, and 1E, produces an additional lower-energy state with a distorted configuration. The latter becomes the global minimum of the APES when the stabilization energy EPJT is larger than the energy gap D to the ground state. This example is of special interest also because it leads to an interesting molecular magnetic-dielectric dualism (Garcia-Fernandez and Bersuker, 2011; Bersuker, 2020b). Indeed, the distorted configuration has a non-zero dipole moment, but a zero magnetic moment (zero spin), whereas the undistorted configuration has no dipole moment, but a nonzero magnetic moment of the spin-triplet electronic state. As mentioned above, the h-PJTE may take place in any system with half-filled e2 or t3 electronic configurations. Fig. 4 shows the general scheme of the energy level states of systems with e2 configurations (similar schemes are available for t3 systems). The CO3 molecule may serve as an example of h-JTE combined with h-PJTE (Liu et al., 2009). An important feature of the JTEs is that they are defined in a general way by the specifics of their APES. However, APES are not directly observable properties of the system. Ignoring this circumstance played as a difficulty in finding the first experimental confirmation of the original JTE, and continues to be an impediment in correctly understanding of what observable changes in the properties of polyatomic systems, caused by these effects, are expected. The situation is aggravated by its being continuously ignored in many books, reviews, academic lecturing, and even in some scientific publication, leading to erroneous conclusions in prediction of observable properties, induced by the JTEs. Nevertheless, serving as the potential energy in the Schrödinger equation for observables, the specific APES features of the JTEs may allow for qualitative predictions of some of the expected properties. First, we note that for any given high-symmetry configuration, there are several (or an infinite number of ) equivalent low-symmetry distortions that satisfy the JTE requirement of SSB. This means that the APES should be an equivalent-multiminimum surface, or a trough of continuing minima. In other words, the JTEs, in general, predict a multiminimum potential with a complicated picture of nuclear dynamics and observables that depends of the form of this potential. However, again, the multiminimum potentials of the APES, by themselves, are not directly observable experimentally.

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The consequences for observable properties of the system induced by its JTEs are inexhaustible, they are considered in many books, reviews, and original publications. In general, without additional consideration, they cannot be reduced to observable distortion of space configuration of free polyatomic systems, as it is stated in “primitive” definitions in some literature on the JTEs. More important, the form of the APES is sometimes erroneously taken as the direct observable property of the system. For instance, in some recent publications devoted to the influence of spin-orbital coupling (SOC) on the JTE, the authors calculated the changes of the APES by SOC perturbation, assuming that this is the observable effect. Excluding the possibility of exact solution, the correct approximate calculations would get first the energy levels and wave functions with the APES of the JTE system, and then include the SOC as a perturbation (if it is sufficiently small), which would yield the vibronic reduction of the SOC (Ham, 1972). It follows that, strictly speaking, we cannot define the observable properties of the system with JTEs just based on their APES, without further investigation. The direct way to do this is to solve the relevant Schrödinger equation including the nuclear dynamics. In overcoming the difficulties in solving the problem with arbitrary APES of JTEs, we explore, as usual, approximations and limiting cases (see in: Englman, 1972, Bersuker and Polinger, 1989, Bersuker, 2006, Köppel et al., 1912, etc.). In condensed matter problems, the most important way to reveal the observable properties in the JTEs is to involve external perturbations (see below).

JTEs in local properties of solids. Examples As mentioned above, exploration of the observable JTEs in molecular systems and condensed matter is presently well developed and continuing, presented in a series of books, reviews, and original papers, cited in the Introduction (for more citations, see Bersuker, 1984; Perlin and Wagner, 1984; Bersuker, 2021). As it is clearly seen from the definitions in section “Four modification of the JTEs. Adiabatic potentials vs observables,” all JTEs are of local origin, grounded in the specifics of the local interactions between electronic states and nuclear displacements (vibronic coupling); they define the local properties of solids directly. Note that in the overwhelming majority of cases, point defects (local impurity, dopant, and vacancy centers), as well as the bulk local centers in crystals, have degenerate or pseudodegenerate electronic ground and/or excited states (both explicit and hidden), subject to the JTEs, which influence all their properties. Below are several examples of the latest achievements in this field. Obviously, in condensed matter, the local JTEs distortions may interact strongly, resulting in cooperative effects, discussed in the next section. With regard to JTEs, local properties in solids (inherent to neutral molecules, ions, clusters, impurity and vacancy centers in solids, qubits, quantum dots, thin films, etc.) are studied by a variety of spectroscopies, including optical, electron spin resonance (ESR), X-ray, EXAFS, and other related methods, and also with the same methods under influence of external fields and pressure. Cooperative interactions in crystals with JTEs lead to structural, magnetic, ferroelectric, multiferroic, etc., phase transitions. The study of the JTEs shows also how we can manipulate condensed matter properties by employing external perturbations that directly influence the underlying JTEs parameters. Below we show examples from the latest achievements on these topics.

Ultrasonic exploration Recently developed studies based on measurements of ultrasonic attenuation and velocity change induced by JTEs impurities in crystals (Gudkov and Bersuker, 2012; Averkiev et al., 2017a,b; Gudkov et al., 2020), demonstrated high efficiency in obtaining important information about the structure and properties of the system. Distinguished from ESR, optical, and other similar experimental methods, in which the measurement perturbation engages primarily with the electronic subsystem, ultrasound interacts directly with the nuclear arrangement, thus determining the main features of the APES of the JTEs centers. When propagating through the crystal containing the latter, and dependent on direction of propagation and polarization, the ultrasonic wave unequally shifts the minima of the multiminimum APES, making them nonequivalent, thus creating a non-equilibrium distribution of the JT complexes over their energy levels (Fig. 5). This non-equilibrium state relaxes to equilibrium during a characteristic relaxation time t. The relaxation process introduces an additional channel of energy loss, which results in the ultrasonic wave attenuation, velocity change, and dispersion (Gudkov and Bersuker, 2012; Averkiev et al., 2017a). It follows that in relation to ultrasonic experiments the subsystem of JT centers in crystals is very special: due to their well-defined local multiminimum APES and strong anisotropy with respect to the polarized ultrasound wave, it produces specific anomalies in the ultrasound propagation that are absent in other ultrasound absorbers (the latter thus can be easily excluded). Ultrasonic experiments with doped crystals directly reveal the symmetry properties and the local APES parameters, including the JT active modes (trigonal, tetragonal, and orthorhombic JTE distortions, respectively), linear and quadratic vibronic coupling constants, extrema point positions on the APES, including minima positions and energy barriers between them. In other words, ultrasonic investigations provide for a unique possibility to reveal the whole APES of JTE centers in crystals and consequent properties. Below we show an example that illustrates this statement. The dopant Cr2+ center in a fluorite crystal, SrF2:Cr, is subject to the JTE problem T  (e + t2), which (with the quadratic vibronic coupling terms included) yields the predicted by the theory six-dimensional APES in the space of two tetragonal and three trigonal symmetrized atomic displacements around the Cr2+ center (Bersuker and Polinger, 1989; Bersuker, 2006). It includes six equivalent orthorhombic global minima with four trigonal and three tetragonal saddle-points between them. This most complicated APES of a JTE system was fully reconstructed experimentally based on ultrasonic measurements, thus demonstrating their unique possibilities (Averkiev et al., 2017a). The temperature-dependent relaxation attenuation of the longitudinal ultrasonic waves (propagating along

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Fig. 5 Distortions of the APES of a JTE center in crystals (subject to the E  e problem), produced by the e4-type ultrasonic mode deformations, which violate the equivalency of the minima and the equilibrium conditions within the JT subsystem. Panels (a), (b), (c) show the APES distortions in accordance with the wave’s profile shown in panel (d) (Uy is the instant displacement of the atoms perpendicular to the propagation direction z axis at t ¼ t0). Gray arrows indicate the relaxation process. Reproduced from Gudkov V.V., Sarychev M.N., Zherlitsyn S., Zhevstovskikh I.V., Averkiev N.S., Vinnik D.A., Gudkova S.A., Niewa R., Dressel M., Alyabyeva L.N., Gorshunov B.P., Bersuker I.B. (2020) Sub-lattice of Jahn-Teller centers in hexaferrite crystal. Nature Scientific Reports 10: 7076–7091. Copyright 2020 Nature Scientific Reports.

Δαrel/k0 (rel. un.)

0.0004 0.0003 0.0002

2

0.0001

1

0.0000 25

50

75

100

T (K) Fig. 6 Temperature dependence of the relaxation attenuation of the longitudinal ultrasonic wave propagating along the [110] axis of the SrF2:Cr crystal, at the wave frequencies of 52 MHz (curve 1), and 162 MHz (curve 2). Reproduced from Averkiev NS, Bersuker IB, Gudkov V, Zhevstovskikh IV, Sarychev MN, Zherllitsyn S, Yasin S, Shakurov GS, Ulanov VA, Surikov VT (2017) Manifestation of the Jahn-Teller effect in elastic moduli of strontium fluorite crystals doped with chromium ions. Journal of Physics Conference Series 833: 012003. Copyright 2017 IOP Publishing.

the [110] axis of this crystal), induced by the JTE distortions, is shown in Fig. 6. It allows extracting the relaxation time t as a function of temperature (Fig. 7) using well-established relations (Averkiev et al., 2017a). The relaxation attenuation of the shear ultrasonic wave is directly determined by the linear vibronic coupling constants in tetragonal FE and trigonal FT distortions: |FT | ¼ 3.6610−5 dyn and |FE | ¼ 0.6310−5 dyn (Averkiev, 2017a). On the other hand, the temperature dependence of the relaxation time in Fig. 7 shows that there are two barriers of relaxation. At low temperatures, first the lower barrier between the tetragonal distortions (V0 ¼ 45.6 cm−1) is overcome, followed by the barrier between the larger trigonal distortions (V0 ¼ 271 cm−1) at higher temperatures. The two barrier heights give us also the energy positions of the tetragonal and trigonal saddle-points, respectively. Involving the more complicated part of the JTE theory for a T term with the quadratic terms of the vibronic coupling included (Bersuker and Polinger, 1989) and some additional considerations about the elastic properties and local low-frequency vibrations of this crystal, the authors revealed also the stabilization energy of the orthorhombic minima

W (s)

The Jahn-Teller effects

10

-6

10

-7

10

-8

10

-9

803

1

2

0.02

0.03

0.04

0.05

0.06

0.07

-1

1/T (K ) Fig. 7 Temperature dependence of the relaxation time obtained from the attenuation of the c44 normal mode at the frequency of o ¼ 105 MHz. The square symbol corresponds to the condition ot ¼ 1. The lines represent two exponential functions t ¼ t(i0 ) exp (V0(i )/kT ) overcoming tetragonal (Curve 1, −8 t0(1) ¼ 5  10−11 s, V 0(1) ¼ 390 K ¼ 271 cm−1) and trigonal (curve 2, t(2) s, V 0(2) ¼ 65 K ¼ 45.2 cm−1) potential barriers. Reproduced from Averkiev 0 ¼ 0.5  10 N.S., Bersuker I.B., Gudkov V., Zhevstovskikh I.V., Sarychev M.N., Zherllitsyn S., Yasin S., Shakurov G.S., Ulanov V.A., Surikov V.T. (2017a) Manifestation of the Jahn-Teller effect in elastic moduli of strontium fluorite crystals doped with chromium ions. Journal of Physics Conference Series 833: 012003. Copyright 2017 IOP publishing. −1 −4 Eor dyn/cm, the primary force constants K0E and K0T, and JT ¼ 562 cm , the quadratic vibronic coupling constant WE,T ¼ −2.1  10 the symmetrized coordinates of the tetragonal, trigonal, and orthorhombic extrema points of the six-dimensional APES, as well as the splitting of the lowest vibrational states in the minima produced by the tunneling between the six equivalent orthorhombic minima, G  19 cm−1. Thus the first full description of a rather complicated APES created by the JTE in a crystal center was obtained experimentally by means of ultrasonic investigation (Averkiev, 2017a). Another example of ultrasound investigation of JTEs in local properties of crystals illustrates its efficiency in combination with magnetic field influence (Averkiev et al., 2017b), which is important in view of possible applications in electronics and spintronics. It was shown that in some JTE systems, an external magnetic field induces tunneling transitions between local configurations in polyatomic systems with orthogonal electronic states, for which no tunneling is expected, the tunneling levels being produced by the magnetic field. In a variety of impurity and vacancy centers in cubic crystals and similar molecular systems with cubic and icosahedral symmetry in different aggregate states (including thin films and quantum dots) the ground or excited electronic state may be threefold degenerate T state, subject to the T  e problem of the JTE. In these cases (ignoring the SOC reduced by the vibronic coupling), the APES has three equivalent minima along the three tetragonal distortions, in which, distinguished from the E  e problem, the three electronic functions of the three distorted configurations in the minima are orthogonal, so no tunneling or direct relaxation transitions between them are allowed. This is one of the JTE problems where tunneling between the equivalent minima in the ground state is forbidden. It was shown that an external magnetic field removes this prohibition, producing a new channel of tunneling and direct relaxation transitions between the distorted orthogonal configurations (Averkiev, 2017b). The theory of the JTE T  e problem in magnetic fields, and in interaction with ultrasound, and the ultrasonic experimental observations of the novel effect were carried out using the Cr2+ impurities in the crystal ZnSe:Cr2+ as a typical example. In this impurity center the ground state is electronically threefold degenerate with three orbital wave functions of a hole in the closed-shell 3d5 configuration, transforming as vectors. They can be described by an orbital momentum operator with L ¼ 1, hence being subjects to magnetic field influence. Experimental data show that the JTE distortions in the impurity center under consideration are tetragonal, thus evidencing for the T  e problem. Tunneling effects in JTE systems, in general, were predicted in 1961–1962 (Bersuker, 1961, 1962) and observed in optical, ESR, and ultrasonic experiments in a variety of systems with non-orthogonal minima states. In ultrasonic experiments tunneling manifests itself in the temperature dependence of relaxation time trel, where, by lowering the temperature, the function trel(1/T) changes from trel  (n0)−1 exp (V0/kBT) in the activation mechanism with over-the-barrier transitions to the tunneling mechanism with vanishing temperature dependence. The analysis of the temperature dependence of the relaxation time in zero magnetic field shows the absence of tunneling in the JT system of Cr2+ centers in the ZnSe crystal. When a weak magnetic field is applied to the system at low temperatures, a sharp increase in the attenuation of ultrasound is observed, illustrated in Fig. 8 (Averkiev, 2017b). A microscopic theory was developed showing that the magnetic field produces a new relaxation channel associated with tunneling between the orthogonal distorted configurations, made possible due to magnetic-induced coupling between their (otherwise orthogonal) orbital states (Averkiev, 2017b). The rather cumbersome deductions carried out for the ultrasound attenuation in magnetic fields in ZnSe:Cr2+ can be concluded as follows. In zero magnetic field there is a relaxation attenuation of ultrasound driven by the relaxation time, which is determined by one-phonon processes in the absence of tunneling between

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Fig. 8 Magnetic field dependence of ultrasound attenuation Da ¼ a(B) − a(0) at different temperatures, for the ultrasound frequency o/ 2p ¼ 33 MHz and magnetic field induction B‖[110]. Inset: the peak of attenuation in magnetic fields at T ¼ 1.3 K at the same frequency, but different directions of B: curve 1, B‖[110]; curve 2, B‖ [001]. Reproduced with permission from ref. Averkiev N.S., Bersuker I.B., Gudkov V.V., Zhevstovskikh I.V., Baryshnikov K.A., Sarychev M.N., Zherlitsyn S., Yasin S., Korostelin Yu.V. (2017b) Magnetic field induced tunneling and relaxation between orthogonal configurations in solids and molecular systems. Physical Review B 96: 094431. Copyright 2017 American Physical Society.

equivalent JTE configurations. In weak magnetic fields up to 0.1 T (up to 3 mT for B‖ [001]) the ultrasonic attenuation is still determined by the relaxation mechanism, but due to magnetic field induced tunneling there is a sharp increase in relaxation rate, which leads to a sharp increase of attenuation. Finally in magnetic fields above 0.1 T there is a transition to a quasi-resonance mechanism of attenuation with a smooth decrease of attenuation up to the fields of 2 T (Averkiev, 2017b).

Qubits for spintronics in quantum information devices Investigations are continuously developing in application of the JTE and PJTE theory to revealing the structure and properties of specific defect centers in diamond and similar crystals (impurity, dopant and vacancy centers and their combinations), which are promising candidates as quantum bits (qubits) for a wide variety of quantum technologies, including quantum sensors to repeater nodes for long-range quantum networks. The studied qubit centers include a large variety of neutral atoms as impurities plus a vacancy, denoted as XV0 (X ¼ N, Si, Ge, Sn, Pb, etc.), or a charged atom plus vacancy (e.g., NV−), as well as many other combinations. As could be expected, all of these artificial defects in diamond, potential qubits for spintronics, are strongly affected by the JTE, and many of them are also influenced by the PJTE (the possibility of the hidden JTEs was not explored as yet). The JTEs essentially alter the energy level positions and their ordering, including the spin-orbit splitting, as well as optical transitions between them with intersystem crossing and zero-phonon lines, making electronic structure calculations without the JTEs absolutely unreliable. For instance, one of the most important features of such qubits, the presence of an excited spin state with a sufficiently long lifetime, strongly depends on the spin-orbit coupling (SOC). As mentioned above, the dynamic JTE may reduce the SOC by orders of magnitude (Ham, 1972). With regard to qubits, this was confirmed by direct calculations in exploring the active electronic 3E state of the NV(−) center in diamond: the calculation of the SOC without the JTE yields it equal to 15.8 GHz, which is about three times larger than the measured one (Thiering and Gali, 2017). The idea was further developed by more accurate calculations of the JTE and SOC simultaneously (non-perturbatively) for a series of qubit centers in diamond, GeV0, SnV0 and PbV0, which show that the JTE reduces the SOC by an order of magnitude (Ciccarino et al., 2020). These fine structure details reveal the new physics of color center qubits in diamond, and present a pathway to identify them experimentally. To show the JTE implication in these problems in somewhat more technical detail, we bring here a brief demonstration of the JTE in the XV0 (X ¼ Si, Ge, Sn, Pb) centers in diamond with numerical results for SiV0 (Thiering and Gali, 2019). The JTE in these centers is produced by multi-electron states, in which the JTE stabilization energy EJT is larger than (or of the same order of magnitude of ) the interelectron repulsion that leads to the multiplet term formation. In such cases it is more convenient to consider first the vibronic coupling in the separate one-electron molecular-orbital (MO) states, and then their interelectron interaction. This leads to the product-JTE (see in: Bersuker, 2006); in the case under consideration, it yields (in the excited state) two orbital degenerate MOs, eg and eu, coupled via Eg displacements, denoted as the (eg  eu)  Eg product-JTE, which, upon interelectron interaction, results in two triplet terms, 3´2u and 3Ê´u. Fig. 9 illustrates the results of the calculations performed for the SiV0 center in diamond (Thiering and Gali, 2019). Similar in trend (but different in detail) APES with ground and excited energy levels and transitions between them, showing the defining role of the JTE or PJTE (or both), were found in all the other candidates for qubits in quantum information systems, that were explored theoretically; there are no known defects in diamond and other similar systems with no JTE or PJTE implications.

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Fig. 9 (a) Adiabatic potential energy surface of the SiV(0) center in diamond that shows the (eg  eu)  Eg product JTE in the excited state. The separate JTE in the two MOs, eg and eu, produce two “Mexican hat”-type surfaces with stabilization energies EJT(1) and EJT(2); (b) Their resulting electronic triplet terms 3´2u and 3Ê´u, respectively, with an energy gap of 6.7 meV. The vertical absorption, luminescence, and zero-phonon lines are shown by blue, red, and black arrows, respectively; (c) geometry of the undistorted SiV(0) center with D3d symmetry, showing also the carbon-localized X and Y coordinates of the JTE active normal mode Eg (Bersuker, 2006). Reproduced from Thiering G., Gali A. (2019) The (egeu)Eg product Jahn–Teller effect in the neutral group-IV vacancy quantum bits in diamond. Nature Computational Materials 5: 1–6. Copyright 2019 Nature Computational Materials.

Cooperative properties in crystals. Ferroelectricity, multiferroicity, and orientational polarization As mentioned above, cooperative interactions between crystal centers with JTEs lead to important observable properties. The first investigations of cooperative JTE were performed on rare-years orthovanadates and similar systems (Ghering and Ghering, 1975; for more details, see Bersuker and Polinger, 1989; Polinger, 2009; Kaplan and Vekhter, 1995), showing their rich influence on solid state properties, leading to structural phase transitions. Before that, the cooperative PJTE as leading to ferroelectricity was suggested (Bersuker, 1966), its full theory being developed in more recent times. We bring here an outlook of its latest achievements (Bersuker, 2017a, 2018; Bersuker and Polinger, 2020; Polinger and Bersuker, 2018).

Cooperative PJTE, ferroelectricity Actually, the explanation of the origin of ferroelectricity in perovskite type crystals (like BaTiO3) as triggered by the local PJTE was the first application of this effect to solving a major solid state problem (Bersuker, 1966). It was shown that in the centrosymmetric configuration of the perovskite crystals, local dipolar distortions in the metal centers may occur due to the PJTE. Without the latter, the existing theories implied that the spontaneous polarization of such crystals is due to a self-consistent cooperative mechanism, in which the local (repulsive) dipolar displacements are compensated by their long-range dipole-dipole attractions. However, all the experimental data, accumulated in years, show unambiguously that the instant local dipolar distortions are there in all the phases of the crystal, including the paraelectric phase, before the phase transition to the ferroelectric one (see, e.g., Ravel et al., 1998; Stern, 2004). The PJTE theory solves this problem (Bersuker, 1966, 2018, 2019; Bersuker and Polinger, 2020). Fig. 10 shows the local electronic structure of the octahedral [TiO6]8− cluster in the paraelectric phase of BaTiO3. Very briefly, the local PJTE emerges from the vibronic mixing of the ground state A1g, formed by the fully populated highest occupied (HO) six molecular orbitals (HOMO) t1u and t2u (mostly oxygen 2p orbitals), with three lowest unoccupied MO (LUMO) t2g (mostly titanium 3d orbitals), a total of 9 MO’s, by the polar t1u type normal coordinates Qx, Qy, and Qz. This PJTE (A1g + T1u)  t1u problem results in a 9  9 secular equation. Its solution yields the following APES of the TiO68− cluster, obtained already in the first paper on the subject (Bersuker, 1966, 2018):

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4p

e*g

4s 3d

* t2g

2D

t1u t2u t1g

2p 2s t2g eg

Ti

O

Fig. 10 The energy-level correlation diagram (not to scale) for the octahedral cluster [TiO6]. HOMO t1u and t2u (of mostly oxygen origin) and LUMO t2g (mostly titanium) are one-electron energy levels of the orbitals that are coupled in the PJTE. The shown electron populations correspond to the particular case of [TiO6]8−. Reprinted from Bersuker I.B. (2006) The Jahn-Teller Effect. Cambridge, UK: Cambridge University Press. Copyright 2006 I. B. Bersuker.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi 1 2 U ðQÞ ¼ K 0 Q −2 D2 + 2F2 Q2 −Q2x + D2 + 2F2 Q2 −Q2y + D2 + 2F 2 Q2 −Q2z 2

(2)

where Q2 ¼ Q2x + Q2y + Q2z , 2D is the energy gap between the mixing electronic states, K0 is the primary force constant for the Q displacements (stiffness of the crystal without the vibronic coupling), and F is the vibronic coupling constant (H is the Hamiltonian), 



 ∂H  3dxz ðTiÞ  (3) F ¼ 2pz ðOÞ  ∂Qx 0 In a more rigorous treatment by the Green’s functions approach (Polinger et al., 2015), the local PJTE in the cluster unit was appended with the bulk crystal properties by taking into account the interaction of the Ti ion with the whole crystal via its electronic and vibrational bands. It improved the results by yielding appropriately band-averaged parameters instead of the local cluster ones. The three-dimensional APES (2) has a specific form. Under the condition. D < 8F2 =K0

(4)

the surface (2) has a maximum (meaning instability) when the Ti ion is in the center of the octahedron, eight equivalent minima placed along the four trigonal axes, in which the Ti ion is displaced toward three oxygen ions (away from the other three); higherin-energy 12 equivalent saddle points along the six C2v axes, at which the Ti ion is displaced toward two oxygen ions (at the top of the lowest barrier between two near-neighbor minima); and next six higher-in-energy equivalent saddle points, at which the Ti ion is displaced to one of the oxygen ions along the fourfold axes (Bersuker, 1966, 2018). With additional two experimentally determined structural constants, the band-averaged energy gap 2D ¼ 2.8 eV and the vibrational frequency at the bottom of the trigonal minimum ħoE ¼ 193 cm−1, all the main parameters of this APES, shown in Table 1, were estimated, including K0, F, the positions of the minima Qx ¼ Qy ¼ Qz ¼ Q0 and first saddle points Qx ¼ Qy ¼ q0, Qz ¼ 0, their PJTE stabilization energies, and the tunneling splitting G. The latter is a characteristic measure of the energy barrier between the near-neighbor minima of the APES. The four phases of BaTiO3 emerge directly from this APES by gradually populating the states in the minima with temperature and stepwise overcoming its barriers (Bersuker, 1966; Polinger, 2013; Polinger et al., 2015). Of particular interest is also the prediction of disorder in the orthorhombic and tetragonal ferroelectric phases and in the cubic paraelectric phase. As mentioned above, it was confirmed later together with a variety of other predictions of the theory that have no explanation in displacive theories (see in: Bersuker, 2018). Using the numerical estimates of Table 1, the interaction between the PJTE induced local dipole moments in BaTiO3 was taken into account in a mean-field approximation, in which both the local off-center displacement and the mean field of the environment are interdependent in a self-consistent way (Polinger, 2013). It yields the experimentally observed phase transitions in BaTiO3 with reasonable values of Curie temperatures.

Table 1

Numerical values of the PJTE vibronic coupling and APES parameters of the Ti active centers in the BaTiO3 crystal (Polinger et al., 2015).

K0

2D

F

ħoE

Q0

q0

EJT[111]

EJT[110]

G

55 eV/A˚ 2

2.8 eV

3.42 eV/A˚

193 cm−1

0.14 A˚

0.16 A˚

−1250 cm−1

−1130 cm−1

35 cm−1

The Jahn-Teller effects

807

Multiferroicity The PJTE theory was extended to formulate the necessary condition of coexisting magnetic and ferroelectric (multiferroicity) properties in ABO3 perovskite crystals with B as a transition metal ion in a dn configuration, n ¼ 1, 2, . . .10 (Bersuker, 2012, 2013b), that found already confirmation in experiments (Domracheva et al., 2013; Raymond et al., 2014; Weston et al., 2016) and ab initio calculations. Multiferroicity implies that the ferroelectric crystal, which is a dielectric, has also a nonzero magnetic moment, meaning unpaired electrons. In the ferroelectric BaTiO3 with perovskite ABO3 structure the d0 configuration of the Ti4+ ion has no unpaired electrons, and attempts to obtain ferroelectricity in perovskites with dn, n > 0, configurations of the transition metals B ions were unsuccessful for a long time (Barone and Picozzi, 2015). The PJTE origin of ferroelectricity not only explains how the possible ferroelectric properties of perovskite crystal may coexist with unpaired electrons, but it shows also when this coexistence may take place. Moreover, for dn ions with n ¼ 3, 4, 5, 6, and 7, which are directly influenced by the well-known transition metal high-spin/low-spin crossover, one may expect the coexistence of three phenomena: ferroelectricity (FE), magnetism (M), and spin crossover (SCO). This, in turn, leads to a quite novel phenomenon: magnetic-ferroelectric (multiferroics) crossover (MFCO), creating a rich variety of possible magnetoelectric and related effects (Bersuker, 2012, 2013b). The vibronic (PJTE) theory, exclusively, reveals the role of spin in spontaneous polarization of crystals. The above MO energy scheme for an octahedral cluster BO68− in Fig. 10 shows the MO electron population of the d0 transition metal ion, e.g., when B Ti, with the HOMO electron configuration (t1u)6 ¼ (t1u #)3(t1u ")3, where the arrows up and down indicate the two spin states. The energy term of this configuration is 1A1g. The excited state with opposite parity is formed by the one-electron, (t1u ") ! (t2g ") or (t1u ") ! (eg "), excitation, resulting in the lowest excited ungerade term 1T1u at the energy gap 2△. In this case, the PJTE at the B center, under the condition of instability △ < 8F2/K0 (Eq. 4), produces a polar displacements of the B atom along [111]-type directions, which triggers the ferroelectric polarization (see section “Cooperative PJTE, ferroelectricity”). For the d1 configuration of the B ion instead of the d0, the HOMO becomes (t1u #)3(t1u ")3(t2g ")1 with the term 2T2g, and the LUMO, taking into account Hund’s rule, is (t1u #)2(t1u ")3(t2g ")2 with the lowest excited ungerade term 4T1u. Hence the two closest terms of different parity, possess different spin multiplicity, and hence they do not mix by the vibronic coupling; the latter does not contain spin operators. In principle, there may be higher in energy electronic configurations of opposite parity with the same spin as the ground state one, but they are at much larger energy gaps △, and therefore less appropriate to satisfy the condition of instability (4) (numerical estimates show that the condition (4) may be very restricting). A similar picture emerges for d2 configurations for which the two lowest terms of opposite parity are 2T2g and 5T1u. The situation changes for d3. Indeed, in this case the HOMO is (t1u #)3(t1u ")3(t2g ")3 with the ground state term 4A1g, and in the low-spin conditions of the strong ligand fields (large t2g-eg separation in Fig. 10), the LUMO is (t1u #)2(t1u ")3(t2g ")3(t2g #)1 with the lowest ungerade term 4T1u. Therefore, for d3 configurations (e.g., Mn4+) in sufficiently strong ligand fields the situation becomes again favorable for the PJTE and polar distortions, but in this case, distinguished from the d0 case, the system possess also a magnetic moment created by three unpaired electrons. However, if the ligand field is weak and the separation t2g-eg is small, the high-spin arrangement of the excited electronic configuration takes place, and the excitation electron occupies the eg " orbital instead of t2g #; the LUMO configuration under Hund’s rule becomes (t1u #)2(t1u ")3(t2g ")3(eg ")1 with the lowest ungerade state 6T1u. Here again there is no PJTE on dipolar distortions and no ferroelectric instability. This is one of the examples, which shows explicitly how the spin states interfere directly into the possible local polar displacement and ferroelectricity. In this way all the dn configurations with n ¼ 0, 1, 2, . . .10, were explored (Bersuker, 2012) showing that in perovskite ABO3 crystals, only B ions with configurations d0, d3-low-spin, d4-low-spin, d5-low-spin and high-spin, d6-high-spin and intermediate-spin, d7-high-spin, d8, and d9 can, in principle, produce multiferroics, provided the criterion of instability (4) is satisfied. If the (in general, hardly significant) contribution of higher excited states can be ignored, transition metal ions B with configurations d1, d2, d3-high-spin d4-high-spin, d6-low-spin, d7-low-spin, and d10 are not expected to produce multiferroics under this mechanism of proper ferroelectricity. Experimentally observed multiferroics with such B ions, for example, Mn4+(d3), Cr3+(d3), Mn3+(d4), Fe3+(d5), Fe2+(d6), Co2+(d7), etc., fit well with this conclusion; there are no perovskite multiferroics with d0, d1, d2, and d10 configurations. Of special importance is also the fact that some dn ions with n ¼ 3, 4, 5, 6, and 7, dependent on the ligands of the octahedral environment, may produce two types of magnetic centers, high-spin (HS) and low-spin (LS), and in d6 case there maybe also intermediate spin (IS) states (d3 has two spin configurations in the one-electron excitation). According to the analysis (Bersuker, 2012), only d5 systems follow the necessary condition of potential multiferroics in both spin states, but the PJTE condition of instability and the magnetic moments are different in these two cases. For d3, d4, d6, and d7 ions only one of the two spin states may serve as a candidate of potential multiferroics. On the other hand, in many cases the two spin states are close in energy, producing the mentioned above well-known phenomenon of transition metal spin crossover (SCO), in which case the system can be relatively easily transferred from one spin state to another by external perturbations like heat, light, and magnetic fields. Since, as shown above, the change of the spin state also changes the possibility of ferroelectric polarization, the SCO in some perovskite crystals is simultaneously a magnetic-ferroelectric (multiferroic) crossover (MFCO). This coexisting magnetic, ferroelectric, and spin-crossover phenomena opens a variety of new possibilities to manipulate the properties of the system with exciting novel functionalities to electronics and spintronics. In addition to many examples that confirm the (outlined by the PJTE theory) necessary conditions of multiferroicity (see, e.g., Barone and Picozzi, 2015), several papers reported observing also the predicted magnetic-ferroelectric spin-crossover effect (Domracheva et al., 2013; Raymond et al., 2014; Weston et al., 2016), and it was also shown that in BiCoO3 the ferroelectric

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polarization is greatly enhanced when the Co3+ ion is in the high-spin state, as compared to the nonmagnetic state with the Co3+ ion in the low-spin configuration (Weston et al., 2016). The authors demonstrated also the predicted electric magnetization (Bersuker, 2012, 2013b), when by means of induced polarization the spin state changes from low-spin (S ¼ 0) to high-spin (S ¼ 2). They concluded that in contrary to the widespread belief, “unpaired electron spins actually drive ferroelectricity, rather than inhibit it, which represents a shift in the understanding of how ferroelectricity and magnetism interact in perovskite oxides” (Weston et al., 2016). This conclusion follows directly (and exclusively) from the above PJTE theory of ferroelectricity and multiferroicity.

Orientational polarization of solids If the interactions between the JTEs centers in crystals are not sufficiently strong (at temperatures above the structural phase transitions to the cooperative effects), and in the absence of external perturbations, the local distortions at each center are not fully correlated with the others, and they resonate between the different equivalent orientations controlled by the local multiminimum APES. If the barriers between the minima are not very high, the picture as a whole looks like the local dipole moments performing hindered rotations. Under a uniaxial external perturbation (e.g., strain, stress, or electric field E) the equivalency between the minima of the APES is violated, and the system becomes trapped in one of the minima, or in an orientational controlled linear combination of two or several minima that remain equivalent in the external field, in the same direction for all the PJTE centers under the same perturbation. In other words, under external perturbations, the induced by the PJTE local dipole moments rotate to adjust along the external field, resulting in orientational polarization of solids. It is an essentially novel property, which, similar to orientational polarization of liquids, results in drastically enhanced polarization susceptibility in terms of orders of magnitude (Bersuker, 2015a,b; Polinger and Bersuker, 2017; Bersuker and Polinger, 2020). Remarkably, the possible existence of dielectric crystals with randomly oriented dipole moments (similar to ferromagnetics), which may behave like a polar liquid above its freezing point, was first suggested by P. Debye more than a century ago (Debye, 1912), but not discovered till the recent works (Bersuker, 1966, 2015a,b). Below follow several examples of experimentally observable orientational polarization of solids.

Flexoelectricity There are many consequences of this orientational effect induced by the multiminimum nature of the APES of systems with JTEs. We show here several examples beginning with polarization of centrosymmetric dielectric solids under a strain gradient, called flexoelectricity. The physical origin of flexoelectricity is, in general, quite transparent: a strain gradient removes the inversion symmetry of the lattice in at least one direction, thus inducing a polar distortion of the charge distribution (see, e.g., Tagantsev and Yudin, 2017). Without JTEs the polarization under the flexoelectric effect in usual ionic crystals is expected to be small, of the order of nCm−1, but the experimental measurements of the flexoelectricity effect in ferroelectric systems show that it can be much larger than the theoretical predictions by 3–4 orders of magnitude (Ma and Cross, 2006; Ma, 2010; Nguyen et al., 2013; Zubko et al., 2013). The largest effect is obtained in the paraelectric phase of BaTiO3, but it is much smaller in the similar SrTiO3 crystal, while it approximately follows the theoretical predictions in the majority of other non-ferroelectric crystals. For BaTiO3 the flexoelectric coefficient f (the component of the tensor coefficient of proportionality between the polarization and strain gradient in the one-dimensional model) decreases significantly when moving from the paraelectric phase to its ferroelectric phases of consequent lower symmetry and becomes “normal” (of the order of magnitude predicted by the theory) in the rhombohedral phase. In essence, the publications on this subject based on displacive theories give no reasonable explanation of this controversy beyond the unproved assumption of different crystal imperfections. The essential controversies in the origin of flexoelectricity with several orders of magnitude in the difference between its calculated and measured values disappear when the JTE or PJTE leading to orientational polarization, are taken into account. As outlined above, these effects under certain conditions, produce a local dipolar instability and dipolar distortions, which in the free system are of dynamic nature, but become oriented by the gradient of strain. The latter restores a static orientation of the dipoles, in the same direction for all the centers, thus polarizing the centrosymmetric system. This explains the (by-ordersof-magnitude) bigger flexoelectric effect in perovskite crystals as due to the PJTE: restoring a resonating dipole moment (stopping its tunneling between equivalent minima by violating their equivalency) is much easier than separating opposite charges (polarizing) a rigid ionic lattice. For a demonstration of the orientational polarization, we bring here some numerical estimates of the flexoelectric properties of the BaTiO3 crystal, for which numerical values of the PJTE induced APES parameters are given in Table 1 (Bersuker, 2015a,b; for a more rigorous treatment, see Polinger and Bersuker, 2017). In the paraelectric BaTiO3 the Ti ions are (instantly) dipolar displaced in one of the eight equivalent trigonal minima, with tunneling between them (see section “Cooperative PJTE, ferroelectricity”). Any small perturbation in the form of strain gradient makes these PJTE minima nonequivalent, thus violating the conditions of tunneling between them and restoring some of the local dipole moments; the strain gradient acts as a “stop-flow” terminating a part of the tunnelings. Since the strain gradient has the same direction for all the unit cells, this leads to the polarization of the crystal. A rough (qualitative) estimate of the flexoelectric coefficient f in the paraelectric phase of BaTiO3, employing the vibronic coupling parameters of Table 1, yields the following relation: f ¼ ðd0 =aÞð20K0 =GÞ1=2

(5)

The Jahn-Teller effects

809

where we used the expression for the polarization produced by the dipole moments in the minima P ¼ d0/a3. The dipolar minimum position from Table 1 is x0 ¼ √3Q0 and the effective (Born) charge of the Ti ion is Z ¼ 8.7e, hence d0 ¼ −√3Z eQ0. With numerical values of the parameters and K0 from Table 1 we get the following estimate: f ¼ − 0.4310−6 Cm−1. Thus instead of (and in addition to) producing a polar charge redistribution in a rigid cubic crystal of the order of nCm−1 the strain gradient in the paraelectric phase of a PJTE ferroelectric in an orientational polarization recovers the virtual local dipole moments (suppresses their dynamical averaging) inducing the ferroelectric polarization of the order of mCm−1. This removes the main controversy in the theory of flexoelectricity explaining the origin of the 3–4 orders of magnitude larger flexoelectric coefficient f in paraelectric BaTiO3 as compared with nonferroelectric crystals. As the dynamic JTE and the PJTE are of local origin, the enhanced flexoelectric effect takes place everywhere these effects produce local dipolar distortions, and the density of such JTE or PJTE centers is high enough for experimental observation.

Enhanced permittivity Similar to flexoelectricity, permittivity of ferroelectric crystals is strongly influenced by the orientational polarization of the dipolar distortions produced by the JTE or PJTE (Bersuker, 2015b). A more rigorous treatment of permittivity of crystals with these effects is given in (Polinger and Bersuker, 2017), but a rough estimation, yielding orders of magnitude, can be obtained directly from the data in Table 1. In the case of permittivity the equivalency between the minima of the APES is violated by an external electric field E, due to which the dynamic polar distortions become trapped in one of the minima (or in an orientational controlled linear combination of the two or several minima that remain equivalent in the field E), in the same direction for all the JTE or PJTE centers. In case of polar distortions this results in polarization of the system, Po ¼ woE, where wo is the contribution to the dielectric susceptibility, which is thus of orientational nature: the virtual dipole moment that resonates between the equivalent minima becomes real, oriented along the field. In addition to this orientational effect there is also the usual displacive susceptibility wd. The orientational contribution to the polarization is always much larger than the displacive one, provided there are orientational degrees of freedom in the system (for example, in water wo ¼ 72, whereas wd ¼ 5). This is quite understandable: similar to the case of flexoelectricity, discussed above, it is much easier to rotate a ready-made dipole along the electric field, than to displacive-polarize a stable polyatomic formation. Similar to the above rough estimate for the coefficient of flexoelectricity in BaTiO3, we get the following relation for the orientational dielectric susceptibility wo in the relation Po ¼ woE, based on the vibronic PJTE theory (Bersuker, 2015b): wo ¼ 2d20 =kGa3

(6)

where k ¼ 0.1–1.0 is an adaptive coefficient. By using the numerical data for the BaTiO3 crystal in Table 1 we can estimate the numerical value of the susceptibility: wo ¼ 0.45  103–0.45  104. This agrees well by orders of magnitude with experimental measurements of permittivity in this system. On the other hand, ignoring the PJTE, the displacive polarization of the unit cell Pd is limited by its stiffness (force constant) K0 with respect to the polar distortions along the above normal coordinate Q. Hence the electric force 2|e |Z E, producing the dipole

moment |e |Z Q, is compensated by the elastic force K0Q. Thus, 2| e|Z E ¼ K0Q and P d ¼ jejZa3 Q ¼ wd E, which leads us to the expression: wd ¼ 2ðeZ Þ2 =a3 K 0

(7)

Using the same numerical values as above and the K0 value from Table 1 we get the following estimate: wd ¼ 1.22. This yields a typical value expected for permittivity in ionic crystals with just displacive (no orientational) contribution (and in the paraelectric region far from the phase transition). Compared with the value wo ¼ 0.45  103 – 0.45  104, obtained above for the orientational contribution, we see that by taking into account the dynamic PJTE the calculated permittivity of BaTiO3 increases by more than three orders of magnitude. In fact the whole permittivity in the field E up to the orientational saturation at E ¼ Em is of PJTE origin, acquiring a less-significant displacive-only addition in higher fields. The value of Em can be estimated from the above expression Em ffi kG/2d0. For BaTiO3 Em ffi k  1.15  107 V/m. It follows that up to this voltage the permittivity is strongly enhanced by the orientational contribution of the dynamic PJTE, reaching the value of polarization P ¼ d0/a3 ffi 0.47 Cm−2 which coincides approximately with that of the ferroelectric phase. The estimates above demonstrate qualitatively the strong effect of orientational polarization in the properties of crystals with JTEs centers. In a more rigorous consideration (Polinger and Bersuker, 2017) the orientational effect is strongly dependent on the speed of the inter-minima dipoles orientation, characterized by the tunneling splitting parameter G in comparison with the local dipole moment d0 in the minima of the APES, the relative strength of the applied electric field, and even its direction with respect of perovskite crystal axes, illustrated in Fig. 11 (Polinger and Bersuker, 2017). The enhanced polarization may take place in any dielectric polyatomic formation in the presence of dynamic JTE or PJTE, and the number of such systems in solid, liquid, and gas phases is innumerable. If the centers of the system with local trigonal symmetry are in a degenerate electronic E state, the local E  e type JTE results in three dipolar minima or a trough of dipolar distortions of the Mexican-hat type (Bersuker, 2006). As in the BaTiO3 case, their contribution to the polarization in electric fields is expected to strongly enhance the permittivity. Similarly, enhancement is expected when the centers with trigonal symmetry are subject to a PJTE.

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The Jahn-Teller effects

Fig. 11 Temperature dependence of the dielectric susceptibility w of the cluster [BO6] in perovskite crystals at three directions of the external electric field, E || [111], E || [100] and E || [110], and three absolute values, E ¼ 0.01G/d0, G/d0, and 2G/d0, [in units of wsc ¼ d 20/(3Ge0a3) with e0 ss the vacuum constant]. The temperature is given in dimensionless units kT/G. Reprinted from Polinger V.Z., Bersuker I.B. (2017) Pseudo Jahn-Teller effect in permittivity of ferroelectric perovskites. Journal of Physics Conference Series 833: 012012. Copyright 2017 IOP publishing.

Giant electrostriction Since the electric field along the trigonal axis of BaTiO3 suppresses the tunneling between the dipolar minima of the APES, trapping all the units of the system in the same kind of trigonal minimum along the whole crystal, the latter will be not only polarized, but also strained in this direction. Similar to permittivity and flexoelectricity, the change in the dimensions of the unit cell in the electric field E, the electrostriction, can be estimated for the BaTiO3 crystal (Bersuker, 2015b). In fact the electrostriction problem is closely related to the inverse of flexoelectricity: instead of polarization P induced by strain gradient G in flexoelectricity, electrostriction is the strain S (not strain gradient) induced by the polarization. In the one-dimensional model with the electric field and strain along the same trigonal axis of the crystal we have: S ¼ gP2, where g is the coefficient of electrostriction. More convenient is to measure the strain as a function of the electric field intensity E, S ¼ E2. As shown above, at Em  (106–107) Vm−1 in the trigonal direction the full orientational ordering of the local unit cell dipoles is reached resulting in the strain S ¼ Q/a ffi √ 3Q0/4 ffi 0.05. Without the orientational contribution of the dynamic PJTE, from the data above we have: 2 |e|Z E ¼ K0Q which at E ¼ Em yields the strain S  10−5, a typical value for a regular ionic crystal. Again, as for permittivity and flexoelectricity, the orientational polarization induces giant electrostriction of paraelectric BaTiO3, about three orders of magnitude bigger than that in similar systems without the dynamic PJTE, and this effect is in accord with experimental measurements.

Manipulation of solids and 2D systems by influencing their JTEs parameters The knowledge that the structural instability and distortions of high-symmetry configurations of molecular systems and solids are induced by the JTEs allows one to restore the symmetry of the system by suppressing these effects with external perturbations that influence (modify) the JTEs (vibronic coupling) parameters. Conversely, specific external perturbations can induce the JTEs, producing polyatomic structures with distorted configurations and novel properties. Manipulation of atomic systems by influencing their features induced by the vibronic coupling effects is a novel trend (see in: Bersuker, 2017b, 2019, Gorinchoy et al., 2011, Gorinchoy and Bersuker, 2017, Gudkov et al., 2020); below are several examples.

Inducing the JTE in crystal sublattices There are two possibilities to manipulate JTE-induced properties of the system by means of external influence targeting the corresponding JTE parameters: (1) destroying the high-symmetry configuration that produces the degeneracy of the electronic state by means of lower-symmetry perturbations; (2) adding or removing valence electrons in a redox process that removes or installs the degeneracy of the state under consideration (without destroying the reference symmetry of the configuration). Low-symmetry external perturbations (e.g., strain, crystal field environment, coordination to lower symmetry groups, electric or magnetic fields, etc.), at first sight, may produce the same kind of distortions, as expected from the JTE, but actually they are essentially different: external perturbation produce fixed local distortions without the local JTE dynamics. Obviously, the degeneracies can be manipulated also by adding or removing electronic charge in oxidation, reduction, excitation, coordination, and via other fractional charge transfers (Gorinchoy et al., 2011). One of these possibilities was recently realized by inducing the JTE in a crystal sublattice of the hexaferrite BaFe12O19 crystal (Gudkov, 2020). In this crystal structure, all iron ions are in the Fe3+(d5) oxidation state with the high-spin configuration (S ¼ 5/2),

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811

and they occupy five different crystallographic locations with octahedral, tetrahedral, and bi-pyramidal coordination. In all of them the electronic ground state is orbitally non-degenerate, 6A1(t32e2), and hence there is no JTE. The authors (Gudkov, 2020) suggested and realized a doping procedure that targets the nondegenerate Fe3+ centers, reducing them to Fe2+(d6), with the ground state electronic term 5T2(t42e2) in the octahedral, and 5E(t32e3) in the tetrahedral coordination, both becoming subjects of the JTE. This process was achieved by doping the crystal with electron-donor Ti atoms. It cames out that the dopants target first a tetrahedral site (which is in an easier way of reduction). Therefore, by regulating the concentration of dopants, a whole tetrahedral sublattice of the crystal was transformed in a sublattice with the JTE (the E  e problem) on each of its centers. Actually, by inducing the JTE in a whole sublattice of the host lattice, we get a modified crystal with properties different from that of the parent crystal. In particular, if the JT centers in the transformed sublattice are sufficiently close to each other, their interaction may lead to the cooperative JTE (see section “Cooperative PJTE, ferroelectricity”) with thermally controlled structural phase transitions within the JTE-modified sublattice. Such a (quite novel) phenomenon with sublattice phase transitions in a complex crystal remains to be explored in more detail. By comparing the properties of the JT modified crystal with the parental crystal, one gets important information about the crystal structure in both of them (Gudkov, 2020). The formation of the JTE sublattice in the tetrahedral sites of the hexaferrite crystal by means of Ti doping was confirmed by ultrasonic and terahertz-infrared experiments performed at temperatures 2–200 K in external magnetic fields up to 13 T. The vibronic coupling in the tetrahedral E  e type JTE, was analyzed by ultrasound experiments, employing the mentioned above methodology (see section “Ultrasonic exploration”). Using terahertz-infrared spectroscopy made also possible to determine the crystallographic sites of the induced JT centers. The realization of the modified sublattice in a complex crystal seems to be the first case of such manipulation of the JTE by targeted external influence.

Planarization of puckered two-dimensional systems The origin of instability and distortions of high-symmetry configurations of molecular systems in nondegenerate states as due to the PJTE is already well established with many examples well studied (Bersuker, 2006, 2013a), but manipulation of their structural features based on this knowledge is relatively new and so far applied mainly to planar and quasi planar (two-dimensional (2D) and quasi 2D) systems and their extended forms (see, e.g., Bersuker, 2017b, 2021; Gorinchoy and Bersuker, 2020, and references therein). The enhanced attention to these systems is due to their special structural feature, their out-of-plane distortions in the form of puckering or buckling, which influence all their properties, notably the chemical reactivity, charge mobility, and band gaps. It raises the importance of manipulating this structural feature by means of external implications (perturbations). Many attempts to influence the planarity of 2D systems employ the considerations of atomic orbital hybridization, which correctly illustrates the origin of the out-of-plane distortion effect in some particular cases, but it is insufficient to explain the origin of puckering or buckling in more complicated situations, which cannot be reduced to just hybridizations. Based on the general proof that all the deviations from high-symmetry configuration of polyatomic systems in nondegenerate states are of PJTE origin, we can find out the main features of the planar configuration of 2D systems that are responsible for the outof-plan instability and try to influence them by external perturbations. The procedure starts with electronic structure calculations of the ground and low-lying excited states in the planar configuration, their energy, symmetry, MO origin, and the numerical values of the parameters in the condition of instability (1). This gives us the necessary insight on how we can enhance or suppress the distortions, meaning manipulate the puckering or buckling of the system. In case of the PJTE induced distortions the following manipulation possibilities were suggested: (1) increasing the energy gap D between the PJTE active electronic states, e.g., by means of weak coordination with outside systems (van-der-Waals or polarizing external influence); (2) changing the symmetries of the active ground or excited states by removing electrons from, or adding electrons to targeted orbitals, that would nullify the vibronic coupling for the distortion under consideration; (3) changing the numerical value of the constant of the vibronic coupling between the active electronic states by means of partial charge transfer via coordination to outer groups; (4) making targeted chemical substitutions that change the planarity, but do not violate the quasi-2D character of the system; (5) by spectroscopic excitations. All these possibilities were realized (see: Bersuker, 2017b, and references therein). For illustration, we show here the example of the carbon nitride g-C3N4, where the procedure of planarization was realized by influencing the vibronic coupling parameters via charge transfer by coordination. Graphitic carbon nitride g-C3N4 is a 2D system, which is similar to graphene in many aspects, but unlike graphene, g-C3N4 is not regular planar, it is buckled as shown in Fig. 12. The PJTE origin of the buckling in this system was studied in detail with the goal to influence it by external perturbations (Ivanov et al., 2015). It was shown that its planar configuration is unstable along the e00 symmetrized displacements for which the coupled HOMO and LUMO responsible for the PJTE instability were revealed by ab initio calculations. With these data and previous experience, the authors suggested suppressing the PJTE and restoring the planar configuration of this system by adding Be2+ ions below and above the fragment unit in an inverse sandwich arrangement. The calculations show indeed that the electron acceptor ions change the HOMO-LUMO positions thus quenching the PJTE and flattening the system. Fig. 13 shows the optimized structures of both the fragment unit and the extended sheet of g-C3N4 with appropriately positions of the Be2+ ions, which restore the planar configuration, showing also the electronic band structure. Similar planarization procedures involving appropriate external influencing the PJTE parameters were carried out on a series of other 2D systems (see: Bersuker, 2017b, and references therein).

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Fig. 12 Buckled 2D structure of graphitic carbon nitride g-C3N4 shown in top and side view. Here and below the gray, blue, and white spheres represent carbon, nitrogen, and hydrogen atoms, respectively. Reprinted with permission from Ivanov AS, Miller E, Boldyrev AI, Kameoka Y, Sato T, and Tanaka K (2015) Pseudo Jahn − Teller origin of buckling distortions in two-dimensional triazine-based graphitic carbon nitride (g-C3N4) sheets. Journal of Physical Chemistry C 119: 12008–12015. Copyright 2015 American Chemical Society.

Fig. 13 Top and side views of the optimized complex of the graphitic carbon nitride, g-C3N4, local unit with two Be2+ ions in an inverse sandwich arrangement (a), the corresponding extended 2D sheet (b) and its electronic band structure (c). Reprinted with permission from Ivanov AS, Miller E, Boldyrev AI, Kameoka Y, Sato T, and Tanaka K (2015) Pseudo Jahn − Teller origin of buckling distortions in two-dimensional triazine-based graphitic carbon nitride (g-C3N4) sheets. Journal of Physical Chemistry C 119: 12008–12015. Copyright 2015 American Chemical Society.

Conclusion The definition of the JTEs, followed by illustrative examples of some of their more recent applications to the study of structure and properties of polyatomic systems, with examples of applications to local and cooperative properties of solids shows that presently these effects are unalienable features of condensed matter. The theory of JTEs continues developing, and their implication in observable properties is rapidly expanding. All the four JTEs are of local origin, steaming from the specifics of the electron-nuclear (vibronic) interactions. Of them, the best known is the JTE, less known, but already in wider use, is the PJTE, while the exploration of the hidden effects, h-JTE and h-PJTE, just started. From the most novel applications to condensed matter, local and cooperative properties of crystals are most influenced, the latter including ferroelectricity, multiferroicity, and a novel property of solids, orientational polarization. Quite new is also the developing trend of manipulation of polyatomic properties via influencing their JTEs parameters by means of external perturbations.

References Averkiev NS, Bersuker IB, Gudkov V, Zhevstovskikh IV, Sarychev MN, Zherllitsyn S, Yasin S, Shakurov GS, Ulanov VA, and Surikov VT (2017a) Manifestation of the Jahn-Teller effect in elastic moduli of strontium fluorite crystals doped with chromium ions. Journal of Physics Conference Series 833: 012003. Averkiev NS, Bersuker IB, Gudkov VV, Zhevstovskikh IV, Baryshnikov KA, Sarychev MN, Zherlitsyn S, Yasin S, and Korostelin YV (2017b) Magnetic field induced tunneling and relaxation between orthogonal configurations in solids and molecular systems. Physical Review B 96: 094431.

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Barone P and Picozzi S (2015) Mechanisms and origin of multiferroicity. Comptes Rendus Physique 16: 143–152. Bersuker IB (1961) Hindered motions in transition metal complexes. Optics and Spectroscopy 11: 319–324. Bersuker IB (1962) Inversion splitting of energy levels in free transition metal complexes. Zhurnal Éksperimental’noĭ i Teoreticheskoĭ Fiziki 43: 1315–1322. [English translation: 1963. Sov. Phys. JETP 17, 933–938]. Bersuker IB (1966) On the origin of ferroelectricity in perovskite type crystals. Physics Letters 20: 589–591. Bersuker IB (ed.) (1984) The Jahn-Teller Effect. A Bibliographic Review, pp. 1–590. New York: IFI/Plenum. Bersuker IB (2006) The Jahn-Teller Effect. Cambridge (UK): Cambridge University Press. Bersuker IB (2009) Recent developments in the Jahn-Teller effect theory. The hidden Jahn-Teller effect. In: Köppel H, Yarkony DR, and Barentzen H (eds.) The Jahn-Teller Effect. Fundamentals and Implications for Physics and Chemistry, Springer Series of Chemical Physics. vol. 97, pp. 3–23. Heidelberg: Springer. Bersuker IB (2012) Pseudo Jahn-Teller origin of perovskite multiferroics, magnetic-ferroelectric crossover, and magnetoelectric effects. The d 0-d10 problem. Physical Review Letters 108: 137202. Bersuker IB (2013a) The pseudo Jahn-Teller effect—A two-state paradigm in formation, deformation, and transformation of molecular systems and solids. Chemical Reviews 113: 1351–1390. Bersuker IB (2013b) A local approach to solid state problems: Pseudo Jahn-Teller origin of ferroelectricity and multiferroicity. Journal of Physics Conference Series 428: 012028. Bersuker IB (2015a) Pseudo Jahn-Teller effect in the origin of enhanced flexoelectricity. Applied Physics Letters 107: 202904. Bersuker IB (2015b) Giant permittivity and electrostriction induced by dynamic Jahn-Teller and pseudo Jahn-Teller effects. Applied Physics Letters 107: 202904. Bersuker IB (2016) Spontaneous symmetry breaking in matter induced by degeneracy and pseudodegeneracy. In: Rice S and Dinner R (eds.) Advances in Chemical Physics. vol. 160, pp. 159–208. New York: Wiley. Bersuker IB (2017a) The Jahn-Teller and pseudo Jahn-Teller effect in materials science. Journal of Physics Conference Series 833: 01200. Bersuker IB (2017b) Manipulation of structure and properties of two-dimensional systems employing the pseudo Jahn-Teller effect. FlatChem 6: 11–27. Bersuker IB (2018) The vibronic (pseudo-Jahn-Teller) theory of ferroelectricity. Novel aspects and applications. Ferroelectrics 536: 1–59. Bersuker IB (2019) The Jahn-Teller and pseudo Jahn-Teller effect in materials science. In: Lind PR (ed.) Recent Studies in Materials Science, pp. 1–95. Nova Science Publishers. Bersuker IB (2020a) Hidden Jahn-Teller and Pseudo-Jahn-Teller effects, Encyclopedia, General & Theoretical Physics, Topic Review. https://encyclopedia.pub/4283. Bersuker IB (2020b) Spin crossover and magnetic-dielectric bistability induced by hidden pseudo-Jahn-Teller effect. Magnetochemistry 6(4): 64. Bersuker IB (2021) Jahn–Teller and pseudo-Jahn–Teller effects: From particular features to general tools in exploring molecular and solid state properties. Chemical Reviews 121: 1463–1512. Bersuker IB and Polinger VZ (1989) Vibronic Interactions in Molecules and Crystals. Berlin-Heidelberg: Springer-Verlag. Bersuker IB and Polinger V (2020) Perovskite crystals: Unique pseudo-Jahn-Teller origin of ferroelectricity, multiferroicity, permittivity, flexoelectricity, and polar nanoregions. Condensed Matter 5(4): 68. Bersuker IB, Gorinchoy NN, and Polinger VZ (1984) On the origin of dynamic instability of molecular systems. Theoretica Chimica Acta 66: 161–172. Chancey CC and O’Brien MCM (1997) The Jahn-Teller effect in C60 and other icosahedral complexes. Princeton, New Jersey: Princeton University Press. Ciccarino CJ, Flick J, Harris IB, Trusheim ME, Englund DR, and Narang P (2020) Strong spin–orbit quenching via the product Jahn–Teller effect in neutral group IV qubits in diamond. npj Quantum Materials 5: 75. https://doi.org/10.1038/s41535-020-00281-7. Debye P (1912) Eine Resultate einer Kinetischen Theorie der Isolatoren. Physikalische Zeitschrift 13: 97–100. Domracheva NE, Pyataev AV, Vorobeva VE, and Zueva EM (2013) Detailed EPR study of spin-crossover dendrimeric iron(III) complex. The Journal of Physical Chemistry B 117: 7833–7842. Englman R (1972) The Jahn-Teller Effect in Molecules and Crystals. London: Wiley1972. Garcia-Fernandez P and Bersuker IB (2011) A class of molecular and solid state systems with correlated magnetic and dielectric bistabilities induced by the pseudo Jahn-Teller effect. Physical Review Letters 106: 246406. Garcia-Fernandez P, Bersuker IB, and Boggs JE (2006) Lost topological (Berry) phase factor in electronic structure calculations. Example: The ozone molecule. Physical Review Letters 96: 163005. Ghering GA and Ghering KA (1975) Cooperative Jahn-Teller effect. Reports on Progress in Physics 38: 1–89. Gorinchoy NN and Bersuker IB (2017) Pseudo Jahn-Teller effect in control and rationalization of chemical transformations in two-dimensional compounds. Journal of Physics Conference Series 833: 012010. Gorinchoy NN, Balan I, and Bersuker IB (2011) Jahn-Teller, pseudo Jahn-Teller, and Renner-Teller effects in systems with fractional charges. Computational & Theoretical Chemistry 976: 113–119. Gorinchoy N and Bersuker IB (2020) Origin of puckering (buckling) of planar heterocycles and methods of its suppression. In: Danielsen JM (ed.) Heterocycles: Synthesis, Reactions and Applications, pp. 129–188. Nova Science Publishers. Gudkov VV and Bersuker IB (2012) Experimental evaluation of the Jahn-Teller parameters by means of ultrasonic measurements. Application to impurity centers in crystals. In: Atanasov M, Daule C, and Tregenna-Pigot P (eds.) Vibronic Interactions and the Jahn-Teller Effect: Theory and Applications; Progress in Theoretical Chemistry and Physics. vol. 23, pp. 143–161. Heidelberg: Springer. Gudkov VV, Sarychev MN, Zherlitsyn S, Zhevstovskikh IV, Averkiev NS, Vinnik DA, Gudkova SA, Niewa R, Dressel M, Alyabyeva LN, Gorshunov BP, and Bersuker IB (2020) Sub-lattice of Jahn-Teller centers in hexaferrite crystal. Nature Scientific Reports 10: 7076–7091. Ham FS (1972) Jahn-Teller effect in electron paramagnetic resonance spectra. In: Geschwind S (ed.) Electron Paramagnetic Resonance, p. 1972. New York: Plenum Press. Ivanov AS, Miller E, Boldyrev AI, Kameoka Y, Sato T, and Tanaka K (2015) Pseudo Jahn −Teller origin of buckling distortions in two-dimensional triazine-based graphitic carbon nitride (g-C3N4) sheets. Journal of Physical Chemistry C 119: 12008–12015. Jahn HA and Teller E (1937) Instability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proceedings of the Royal Society of London. Series A 161: 220–235. Kaplan MD and Vekhter BG (1995) Cooperative Phenomena in Jahn-Teller Crystals. New York: Plenum Press1995. Köppel H, Domcke W, and Cederbaum LS (1984) Multimode molecular dynamics beyond the Born-Oppenheimer approximation. Advances in Chemical Physics 57: 59–246. Köppel H, Yarkony DR, and Barentzen H (1912) The Jahn-Teller Effect. Fundamentals and Implications for Physics and Chemistry. Springer Series of Chemical Physics 97: Heidelberg: Springer. Liu Y, Bersuker IB, Zou W, and Boggs JE (2009) Combined Jahn-Teller and pseudo Jahn-Teller effects in the CO3 molecule: a seven-state six-mode problem. Journal of Chemical Theory and Computation 5: 2679–2686. Ma W (2010) Flexoelectric charge separation and size dependent piezoelectricity in dielectric solids. Physica Status Solidi B 247: 213–218. Ma W and Cross LE (2006) Flexoelectricity in barium titanate. Applied Physics Letters 88: 232902. Nguyen TH, Mao S, Yeh Y-W, Purohit PK, and McAlpine MC (2013) Nanoscale flexoelectricity. Advanced Materials 25: 946–974. Öpik U and Pryce MHL (1957) Studies of the Jahn-Teller effect I. A survey of the static problem. Proceedings of the Royal Society of London. Series A 238: 425–447. Perlin YE and Wagner M (eds.) (1984) Dynamical Jahn-Teller Effect in Localized Systems. Amsterdam: Elsevier. Polinger VZ (2009) Orbital ordering versus the traditional approach in the cooperative Jah-Teller effect: A comparative study. In: Köppel H, Yarkony DR, and Barentzen H (eds.) The Jahn-Teller Effect. Fundamentals and Implications for Physics and Chemistry, Springer Series of Chemical Physics. vol. 97, pp. 685–725. Heidelberg: Springer. Polinger VZ (2013) Ferroelectric phase transitions in cubic perovskites. Journal of Physics Conference Series 428: 012026. Polinger VZ and Bersuker IB (2017) Pseudo Jahn-Teller effect in permittivity of ferroelectric perovskites. Journal of Physics Conference Series 833: 012012.

814

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Polinger V and Bersuker IB (2018) Origin of polar nanoregions and relaxor properties of ferroelectrics. Physical Review B 98: 214102. Polinger VZ, Garcia-Fernandez P, and Bersuker IB (2015) Pseudo Jahn-Teller origin of ferroelectric instability in BaTiO3 type perovskites. The green’s function approach and beyond. Physica B: Condensed Matter 457: 296–309. Ravel B, Stern EA, Vedrinskii RI, and Kraisman V (1998) Local structure and the phase transitions of BaTiO3. Ferroelectrics 206–207: 407–430. Raymond O, Ostos C, Font R, Curiel M, Bueno-Baques D, Machorro R, Mestres L, Portelles J, and Siqueiros JM (2014) Multiferroic properties and magnetoelectric coupling in highly textured Pb(Fe0.5Nb0.5)O3 thin films obtained by RF sputtering. Acta Materialia 66: 184–191. Stern E (2004) Character of order-disorder and displacive components in barium titanate. Physical Review Letters 93: 037601. Tagantsev AK and Yudin PV (2017) Flexoelectricity in Solids. From Theory to Applications. Singapore: World Scientific. Teller E (1972) An historical note. In: Englman R (ed.) The Jahn-Teller Effect in Molecules and Crystals. London, Foreword: Wiley. Thiering G and Gali A (2017) Ab initio calculation of spin-orbit coupling for an NV center in diamond exhibiting dynamic Jahn-Teller effect. Physical Review B 96: 81115(R). Thiering G and Gali A (2019) The (eg  eu) Eg product Jahn–Teller effect in the neutral group-IV vacancy quantum bits in diamond. Nature Computational Materials 5: 1–6. Weston L, Cui XY, Ringer SP, and Stampfl C (2016) Multiferroic crossover in perovskite oxides. Physical Review B 93: 165210. Zubko P, Catalan G, and Tagantsev AK (2013) Flexoelectric effect in solids. Annual Review of Materials Research 43: 387–421.

Fermi surface measurements☆ SB Dugdale

, H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom

© 2024 Elsevier Ltd. All rights reserved. This is an update of C. Bergemann, Fermi Surface Measurements, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 185–192, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00456-3.

Introduction Quantum oscillations Experimental realization Angle-resolved photoemission Experimental realization Angle-dependent magnetoresistance oscillations Types of oscillation Experimental realization The electron momentum distribution—Compton scattering and positron annihilation Compton scattering Positron annihilation Spin-resolved measurements Strengths and weaknesses of the various techniques Sensitivity to correlations Sample purity Temperature Dimensionality k-space resolution Bulk sensitivity Other techniques Anomalous skin effect Cyclotron resonance Kohn anomalies, RKKY and Friedel oscillations Conclusion Acknowledgments References

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Abstract The Fermi surface separates occupied from unoccupied states in momentum space at zero temperature. For free electrons, the Fermi surface is a sphere, but those of real metals exhibit a spectacular range of topologies. Knowledge of those Fermi surface topologies is the key to understanding the dynamical properties of electrons in metals. An arsenal of ingenious experimental techniques has been developed to reveal the Fermi surfaces of a wide range of materials. This article explores some of these methods, providing a theoretical background and an overview of how the experiments are carried out. The strengths and weaknesses of the different experimental approaches are summarized.

Key points

• • •

Motivate the experimental determination of the Fermi surface. Describe the techniques which are able to measure the Fermi surface, with examples. Discuss the strengths and weaknesses of those techniques.

Introduction An isolated atom can meaningfully be thought of as being composed of a positively charged ion core (comprising the nucleus of neutrons and protons as well as the most tightly bound electrons) surrounded by a small number of more loosely bound valence electrons (Chambers, 1966). The atoms in a crystalline solid are sufficiently close together for the wavefunctions of valence ☆

Change History: October 2022. SB Dugdale uodated the chapter.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00123-2

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electrons, on one ion core, to overlap with those of its neighbors, allowing the electrons in a metal to move from one ion to the next. These mobile electrons are the conduction electrons of the metal, and one can get a surprisingly good description of the motion of these electrons by only considering the motion of a single electron in the average distribution of all the other electrons in the solid. Perhaps even more surprisingly, substantial insight can be gained by solving the simplest quantum mechanical model of electrons confined to a box (with periodic boundary conditions) in which the energy eigenfunctions are plane waves with eigenvalues which depend quadratically on the (crystal) momentum of the electron. Since electrons are fermions, the Pauli exclusion principle permits each discrete momentum state allowed by the periodic boundary conditions to be filled by (at most) two electrons of opposite spin. Once all the electrons in the crystal have been accommodated, the resulting surface in reciprocal space which separates occupied from unoccupied momentum states at zero temperature is called the Fermi surface (Dugdale, 2016). For free electrons the Fermi surface is simply a sphere. When the interaction between the electrons and the lattice of ion cores, and indeed the correlations between electrons are taken into account, the Fermi surface continues to exist, but, in general will not be spherical and the electrons will behave as though they have masses different from the free-electron mass. The significance of the Fermi surface lies in the fact that the electrical and thermal properties of a metal are essentially determined by a tiny fraction of its electrons, namely those that lie at or very close to the Fermi surface. The dynamical properties of those electrons depend on where the electron is on the Fermi surface, which means determining the shape of the Fermi surface is a crucial step in understanding the properties of a metal (Dugdale, 2016). The development of density functional theory (DFT), and its practical implementation in widely available computer codes, has facilitated the calculation of electronic band structure. Theoretical predictions for the topologies of Fermi surfaces have been able to provide great insight. Nevertheless, the experimental determination of Fermi surface remains vitally important as a test of the predictive capability of those calculations, particularly when the presence of strong electron correlations precludes a conventional DFT description.

Quantum oscillations The phenomenon of quantum oscillations is most easily understood within a semi-classical picture. The classical motion of a charged particle in the presence of a magnetic field B, will be helical in real space. Thus in reciprocal (momentum) space the electron will trace out a circular path. The angular frequency of the motion, oc, is called the cyclotron frequency and is given by oc ¼

eB , m∗

(1)

where e is the charge on the electron and m is the cyclotron mass (which for a free electron is the same as the free-electron mass, me). While any cyclotron radius is classically allowed, the necessity of the quantum mechanical wavefunction to be single-valued means that only a discrete set of orbits are permitted. The Bohr-Sommerfeld quantization of the motion of the electron restricts the areas of real space orbits, in the plane perpendicular to the magnetic field, to those which contain an integer multiple of the flux quantum. The area of the nth allowed orbit in real space, An, is given by
 1 h An ¼ n + , (2) 2 eB where e is the charge on the electron and h is Planck’s constant. The circular orbits in reciprocal space will be similarly quantized; they are called ‘Landau levels’ and will have areas A∗n, given by
 1 2pe B An∗ ¼ n + : (3) 2 ℏ Since there is no additional quantization of the orbits parallel to the magnetic field, in three dimensions these semi-classical orbits lie on ‘Landau tubes’, as shown in Fig. 1. The Fermi surface, which was quasi-continuous in zero magnetic field, is now constrained to lie on these Landau tubes. The density of states will be a set of sharp d-function-like peaks, separated in energy by ℏo∗c . If the energy of an electron state on one of these Landau tubes is below the chemical potential, the state will be occupied. As the magnetic field B is changed, the separation of the Landau levels will change, and as a Landau level passes through the chemical potential m, the density of states at the chemical potential will change. The chemical potential is identical to the Fermi energy at zero temperature, but at low temperature in a metal the Fermi energy is a very good approximation to the chemical potential (and the two are often used interchangeably, even if this is not strictly correct). Consider a two-dimensional free-electron gas of non-interacting electrons at low temperature and in zero magnetic field. The Fermi surface will be a circle with a radius equal to the Fermi wavevector, kF, enclosing an area A∗FS(¼pk2F) containing occupied states. In the presence of a magnetic field B, the allowed electron states will coalesce onto Landau levels, the energy separation of which will increase with B such that successive Landau levels will pass through the chemical potential when they have an area A∗FS. As each Landau level is depopulated, electrons will be transferred to and accommodated in lower Landau levels, this being permitted since the degeneracy of each level is proportional to B. Consider a situation in which at some magnetic field, Bn, the nth Landau level is coincident with the Fermi surface i.e., it has an area A∗FS. If the magnetic field is changed so that the separation between Landau levels changes then at some new magnetic field, Bn+1, the next ((n + 1)th) Landau level will also have the same area, A∗FS. For that to be true, then
 1 2pe Bn ∗ ¼ n+ AFS ℏ 2

(4)

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817

B

Fig. 1 A spherical Fermi surface (dashed line) in a magnetic field together with some Landau tubes.

and
 1 2peBn+1 ∗ AFS ¼ n+1+ : ℏ 2

(5)

It follows that ∗ AFS



1

Bn+1

−

1 Bn

 ¼

2pe , ℏ

(6)

and therefore the change in (inverse) magnetic field (D(1/B)) needed to make two consecutive Landau levels have areas which coincide with the Fermi surface is
 1 2pe ¼ : (7) D ∗ B ℏAFS   If measurements of D B1 can be made by determining the period of oscillations as a function of inverse magnetic field as B is changed, then Eq. (7) (the so-called ‘Onsager relation’ (Onsager, 1952)) can be used to infer the area of the Fermi surface, A∗FS. Almost all of the physical properties of a metal are sensitive to the density of states at the chemical potential and therefore it is to be expected that there will be modulation of these physical properties as successive Landau levels pass through the chemical potential. Moving from two to three dimensions, it can be shown using the stationary phase approximation that the contributions of different non-extremal areas will cancel such that the signal is dominated by the extremal areas. The extremal areas are usually identified as distinct frequencies (and their harmonics) in the Fourier transform of the measured oscillations. In a three-dimensional metal, a measurement will provide information about extremal (maximal and minimal) Fermi surface cross-sectional areas in planes perpendicular to the magnetic field. For a spherical Fermi surface, such a measurement immediately provides the equatorial area of the Fermi surface and measurements with the magnetic field aligned along different crystallographic directions would always yield the same extremal cross-sectional area. For a cylindrical Fermi surface (Fig. 2), the extremal area will increase like 1/cos y as the magnetic field is moved away from the cylindrical axis. For more complicated topologies (e.g. Fig. 3), or when more than one sheet of Fermi surface exists (owing to more than one band crossing the Fermi energy), it is necessary to try and solve the inverse problem of extracting the Fermi surface topology from the set of measured extremal areas as a function of the magnetic field direction (Mueller and Priestley, 1966). In practice, the experimentally extracted extremal areas are compared with predictions of DFT calculations (Rourke and Julian, 2012). For example, in their measurement of the Fermi surface of the nodal-line semi-metal CaAgAs, Kwan et al. (2020) were able to match their quantum oscillations measurements as a function of angle to the geometry of a toroid.

Experimental realization In practice, many measurable quantities will exhibit quantum oscillations, including the magnetic susceptibility (de Haas-van Alphen effect) (Shoenberg, 1984), resistivity (Shubnikov-de Haas effect), specific heat and magnetostriction (lattice deformation in

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Fermi surface measurements T B

area

A area=A T

Fig. 2 For a cylindrical Fermi surface with cross-sectional area A, the extremal area increases as cylindrical axis.

A cos y

the magnetic field is rotated by y away from the

ky

kx kz Fig. 3 Extremal orbits on the Fermi surface. The white orbit shows the single extremal orbit when the magnetic field parallel to ky. When the field is parallel to kz there will be both a maximum orbit (dashed black line) and a minimum orbit (solid black line).

the presence of magnetization). The observation of quantum oscillations in metals relies on very low temperatures (typically less than 1 K), very large magnetic fields (typically greater than 10 T) and for the electron mean-free-path to be very long (typically greater than 1000 A˚ , implying a residual resistivity of less than 1 m O cm). The development of several laboratories around the world which are dedicated to providing high magnetic field environments (both DC and pulsed) has meant that the restriction on sample purity can be less stringent. De Haas-van Alphen oscillations are usually measured using a field modulation technique in which a modulation field of a few mT and a frequency of a few Hz is applied on top of a background field (Bergemann et al., 2003). This is combined with torque cantilever magnetometery in which the sample is attached to one end of a cantilever which acts as one plate of a capacitor (Sheikin et al., 2003). The torque in the magnetic field bends the cantilever with respect to the fixed capacitor plate, changing the capacitance. By using a capacitance bridge, such an arrangement can detect changes to less than one part in a million (Hall et al., 2001). Piezo-resistive cantilevers (Rossel et al., 1996), combined with resistance bridges have also been used (Cooper et al., 2003). B , if kBT becomes comparable with that separation Given that the energy separation of two neighboring Landau levels is ℏoc∗ ¼ ℏe m∗ then the amplitude of the oscillations will be strongly suppressed. One term in the Lifshitz-Kosevich formula describes the
 X  2p2 kB Tm ∗ (Lifshitz and Kosevich, 1956; Shoenberg, temperature damping of the oscillations by a factor sinh X , where X ¼ ℏe B 1984). By tracking the amplitude of the oscillations as a function of temperature, very accurate quasiparticle effective masses can be determined. The effective mass m measured in a quantum oscillations experiment will be renormalized by both electron-electron and electron-phonon interactions (Kohn, 1961; Quader et al., 1987). The presence of disorder (for example from impurities) will introduce a finite relaxation time, t, to the quasiparticles which will broaden the Landau levels. This broadening in energy can be estimated from the Heisenberg Uncertainty Principle as being of the order of ℏ/t. A full treatment shows that owing to a finite t, the amplitude of the oscillations will be reduced by a factor proportional to eð−p=oc tÞ, and when its full form is considered in the Lifshitz-Kosevich formula, it is known as the ‘Dingle factor’ (Shoenberg, 1984). This factor enables the relaxation time t to be extracted. There will be a further reduction in the amplitude of the oscillations due to electron spin. In the presence of a magnetic field, the spin-up and spin-down electrons will have separate Landau tubes. This will lead to interference between oscillations coming from electrons of opposite spin which reduces the amplitude of the signal. Further details can be found in Shoenberg (1984). The observation of ‘giant’ orbits in de Haas-van Alphen measurements of Mg which were larger than the cross-sectional area of the Brillouin zone (Priestley and Shoenberg, 1963) were explained by Cohen and Falicov (1961) in terms of electrons tunneling from one orbit on one section of the Fermi surface to another separated from it by a small energy gap, eg. This was referred to as ‘magnetic breakdown’. It was subsequently shown that the criterion for magnetic breakdown to occur is that ℏoc  e2g /EF, where EF is

Fermi surface measurements

819

the Fermi energy (Blount, 1962). In practice, this means that magnetic breakdown could occur in magnetic fields of the order of 1 T (Shoenberg, 1984). Shubnikov-de Haas oscillations of the (magneto)resistivity in metals are often weak (Shoenberg, 1984) and challenging to observe and the theory is quite complicated (Adams and Holstein, 1959). The effect is easier to observe in semiconductors and semimetals. As an example, the observation of Shubnkiov-de Haas oscillations in ‘magic-angle’ twisted bilayer graphene have confirmed the presence of a small Fermi surfaces which emerges from the correlated insulating state (Cao et al., 2018). In sample geometries where there is limited scope for pick-up coils, such as inside pressure cells, measurements of Shubnikov-de Haas oscillations are favored. Putzke et al. (2016) were able to follow the behavior of the effective mass m in YBa2Cu4O8 under hydrostatic pressure (up to about 0.8 GPa) and correlate this with the superconducting critical temperature.

Angle-resolved photoemission Angle-resolved photoemission spectroscopy (ARPES) can measure the single-particle spectral function, A(k, o), which contains information not only about the band structure and Fermi surface, but also about the self-energy S(k, o), since the real part of the self-energy will change the centers of the bands and the imaginary part will broaden them. In the last 30 years, ARPES has developed into a powerful tool for revealing Fermi surfaces and providing significant insight into the impact of electron correlations on electronic structure (Sobota et al., 2021). In the photoelectric effect, the energy of a photon which is incident on the surface of a material is absorbed by an electron which then escapes from the material’s surface. In an ARPES experiment, by measuring properties of the electron outside the material, the kinematics of photoemission can be used to infer the binding energy, EB, and the crystal momentum ℏk of the electron prior to its emission. An ARPES experiment requires a monochromatic source of photons, typically in the UV regime which then impinge on the surface of a single-crystal sample, as shown in Fig. 4. Some fraction of the photoelectrons, which are emitted in all directions, are then detected by an analyzer. The two-dimensional detector of the popular hemispherical analyzer chromatically disperses the electron beam such that the kinetic energy, Ekin can be recorded along one spatial dimension and the angular distribution along the other (Mårtensson et al., 1994). The emission angle (y, f) is defined such that the polar angle y is described with respect to the surface normal, and the azimuthal angle f is usually defined relative to a crystallographic direction. Two important relationships can be derived by considering the conservation of energy and momentum EB ¼ hn −F −EB −Ekin ,

(8)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mEkin sin y,

(9)

where F is the work function, and ℏkk ¼

where ℏkk is the crystal momentum parallel to the surface in the extended zone scheme and where the photon momentum (negligible in the UV regime) has been neglected in Eq. (9). The parallel component, ℏkk is conserved (up to an in-plane reciprocal lattice vector Gk, due to the discrete in-plane translational invariance). Although the crystal momentum perpendicular to the surface, ℏk?, is not conserved, it is usually possible to infer it if one assumes that the final-state dispersion of the photoelectron inside the crystal can be parameterized using a free-electron dispersion shifted by a so-called ‘inner potential’. It should be noted that in truly 2D materials, ℏk? is not relevant. Spin-resolved ARPES measurements are also feasible (Osterwalder, 2006). If a spin detector (e.g., a Mott polarimeter) is coupled to the analyzer, spin- and angle-resolved photoemission spectroscopy is possible. For example, in the 3D topological insulator Bi1−x Sbx, the spin-polarized nature of the surface states has been observed using such a Mott detector (Hsieh et al., 2010). electron analyzer photon source

z T y M x Fig. 4 An ARPES experiment. Photons, typically from a synchrotron source impinge on a sample, ejecting a photoelectron which is emitted at polar and azimuthal angles (y, f), respectively. The electron is detected by the hemispherical analyzer.

820

Fermi surface measurements

Experimental realization Synchrotron sources provide photon energies which are continuously tunable from the vacuum ultraviolet (VUV) through to hard X-rays (although not on the same synchrotron beamline). This is an important advantage of a synchrotron photon source, since it permits measurements to be made at different energies and polarizations, both of which influence the matrix elements for the photoemission process (Bansil and Lindroos, 1999). A tunable photon source also allows the kz dispersion to be investigated. Photoemission is surface sensitive, the escape of the photoelectron being limited by its inelastic mean-free-path. This is strongly dependent on the incident photon energy, but only varies weakly from one material to another, giving rise to the so-called ‘universal curve’ shown in Fig. 5 (Seah and Dench, 1979). At typical photon energies from UV sources, the inelastic mean-free-path is less than 1 nm. In order to obtain bulk information, a necessary, but not sufficient condition, is that the sample surface is atomically flat and clean. The presence of surface states and/or surface reconstruction would still prevent a bulk measurement. Therefore surface preparation is key and treatments such as sputtering and annealing are often employed if the crystal cannot be cleaved in situ to expose a pristine surface. Measurements at temperatures of 10 K are commonplace, and ARPES has even successfully been performed below 1 K (Borisenko et al., 2010). Once the surface has been prepared, it needs to be kept at pressures of the order of 10−9 Pa. An estimate of the time (in seconds) for a monolayer of a highly reactive gas (e.g., oxygen) to form on a clean transition metal surface under a pressure of p Pa is of the order of tm(s)  2  10−4/p; this is only 2 days at p  10−9 Pa (Hofmann, 2013). In practice, higher cross-sections and better resolution for a given resolving power mean that the majority of experiments are conducted in the VUV range. The typical beam spot size is between 10 mm and 100 mm, although by focusing the photon beam onto an area of the sample of the order of a 100 nm wide, recent advances in so-called nano-ARPES offer the possibility of probing the electronic structure of nano-scale samples, single domains within a crystal, and permitting spatially resolved measurements by raster scanning across the surface of a sample (Avila and Asensio, 2014). UV lasers can also be used to produce high photon fluxes with great energy stability and energy/momentum resolutions, with spot sizes smaller than 5 mm having been achieved. Gas discharge lamps combined with a monochromator can also be used as a photon source, with spot sizes typically of about 1 mm, although smaller sizes down to about 200 mm are possible. The ability of ARPES to rapidly deliver great insight into the Fermi surface topologies of new classes of material in the form of beautiful images and multi-dimensional datasets which contain a wealth of information about the nature of their electronic structure is most certainly one of its great strengths. Such has been its success that it has become almost commonplace to expect that ARPES can deliver many important answers to questions about the underlying physics at play in a material. The recent exposition of the Fermi surface of the Kagome-lattice superconductor KV3Sb by Luo et al. (2022) showcases how such experiments can reveal not only the Fermi surface in exquisite detail, but also evidence of charge-density-wave reconstruction including the extraction of the size and anisotropy of the induced gaps, signatures of electron-phonon coupling such as kinks in the band dispersions, and self-energy effects in the electronic structure. In another topological Kagome metal, CsV3Sb5, ARPES measurements have been able to probe the role of van-Hove singularities in electronic symmetry breaking (Kang et al., 2022). Being able to follow electronic structure through phase transitions governed by some control parameter other than temperature is a highly desirable objective. By subjecting a sample of Pr-doped Ca2RuO4 to uniaxial strain, Riccò et al. (2018) were able to suppress the Mott-insulating phase and induce metallicity as evidenced by the appearance of a Fermi surface. Time- and angle-resolved photoemission spectroscopy (trARPES) can provide new insights into the dynamics and out-of-equilibrium behaviors of electrons in solids. This pump-probe approach typically uses a femtosecond infra-red laser pulse to first excite the sample which is subsequently probed by the UV pulse at some time delay with respect to the excitation. An example of this approach is a study of the dynamics associated with the melting and recovery of a charge-density wave (CDW) (Zong et al., 2019). The investigation revealed the re-appearance (upon melting) of the nested parts of the Fermi surface of LaTe3 and their subsequent disappearance as both the CDW coherence and the gap are re-established. In a spin, time and angle-resolved study, the pump-probe measurements of Jozwiak et al. (2016) were able to reveal spin texture in the unoccupied bands above the Fermi energy in Bi2Se3.

Fig. 5 The depth from which a photoelectron can escape is limited by its inelastic mean-free-path. This ‘universal curve’ as a function of electron’s kinetic energy is based on data from Seah and Dench (1979).

Fermi surface measurements

821

ARPES measurements are usually made by rotating the sample with respect to a fixed analyzer. In order to scan the sample surface in real space to locate regions which are of interest (for example, a particular domain) before performing ARPES, either the sample has to be moved with respect to the photons which are confined to a small spot size, or by using an instrument which permits the both real space and k-space to be imaged using strong electric fields to extract the electrons (Cattelan and Fox, 2018). However, there is another category of instrument which is generally referred to as an energy-filtered photoemission electron microscope (EF-PEEM), or more succinctly as a ‘momentum microscope’. Here, the photoelectrons are accelerated into a column containing electron optical elements, as well as apertures in the image plane which can be used to select mm-sized regions of a sample. The optical elements can be adjusted to image either in real-space or reciprocal space. This means that interesting regions of the sample in real space can be selected before switching to reciprocal space where ARPES is carried out by acquiring images of the available portion of k-space at a series of different kinetic energies. The Fermi surface can be mapped by acquiring images of the spectral weight at the Fermi energy (Cattelan and Fox, 2018). One clear advantage of EF-PEEM instruments over conventional ARPES for FS measurements is that all of the available kk ¼ (kx, ky) momentum space is measured simultaneously (this sometimes being referred to as ‘full wavevector ARPES’) i.e., a 2D image of the FS is acquired at once. Many more details about instrumentation and the current capabilities of ARPES, can be found in the review by Iwasawa (2020).

Angle-dependent magnetoresistance oscillations In materials whose electronic structure is quasi one- or two-dimensional, the magnetoresistivity (which describes the change in the resistivity in an external magnetic field) for currents along the high-resistivity axis, is particularly sensitive to the direction of the applied magnetic field. In the previous section, Shubnikov-de Haas quantum oscillations in the resistivity and the connection between the period of those oscillations (in inverse magnetic field) and the extremal area of the Fermi surface has been described. The non-extremal orbits, while not contributing to the oscillations, do generate a non-oscillatory background magnetoresistance which can be quite sensitive to the direction of the magnetic field (Blundell and Singleton, 1996a, 1996b). In some cases, very large angle-dependent magnetoresistance oscillations (AMROs) can be found at fixed magnetic field. AMROs can be measured experimentally by rotating the sample in a fixed magnetic field. They can provide complementary information to quantum oscillations since the signal comes from all electrons on the Fermi surface, not just those associated with the extremal orbits. Since AMROs don’t come from Landau levels passing through the chemical potential, they should be observable at much higher temperatures and much lower magnetic fields than quantum oscillations. Scattering from impurities will, however, suppress the oscillations. AMROs should be visible if oct  1, where t is the relaxation time (inverse scattering rate). In practice, however, the usefulness of this powerful technique is limited to lower dimensional electronic structures (1D and 2D) and to metals with only one or two sheets of Fermi surface since analysis becomes difficult when there are contributions from more than one.

Types of oscillation In general, the different kinds of oscillations that can be found in quasi-1D and quasi-2D materials have been given different names (for example, the ‘Danner-Kang-Chaikin’ oscillations seen in quasi-1D metals (Danner et al., 1994)). A description of the types of oscillations associated with some of these can names can be found in Goddard et al. (2004) and an excellent description of AMROs in quasi-one-dimensional metals can be found in Blundell and Singleton (1996a, 1996b). As an example, considered a quasi-2D Fermi surface such as a warped cylinder, as shown in Fig. 6. On the simplest level, for such a Fermi surface there will be a maximum in the magnetoresistance whenever the magnetic field is oriented such that the Fermi surface cross-sectional areas in the plane perpendicular to the field are all the same. The phenomenon can be understood from a semi-classical perspective within the relaxation time approximation in which oscillations as a function of the angle of the magnetic B TY vF

vF z

Fig. 6 When the magnetic field B is rotated away from the cylindrical c-axis by a Yamaji angle (yY), the c-axis conductivity disappears because the c-axis component of the Fermi velocity vFz averages to zero over the cyclotron orbit. Adapted from Bergemann C, Mackenzie AP, Julian SR, Forsythe D, Ohmichi E (2003) Quasi-two-dimensional fermi liquid properties of the unconventional superconductor Sr2RuO4. Advances in Physics 52: 639–725. https://doi.org/ 10.1080/00018730310001621737.

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Fermi surface measurements

B

B

(a)

(b)

B

(c)

Fig. 7 When the magnetic field B is parallel to the cylindrical c-axis, all the cyclotron orbits are different, as indicated by the extremal black and white orbits shown in (a). When the field is rotated to a Yamaji angle, as shown in (b) and (c), all the Fermi surface cross-sections perpendicular to the magnetic field have the same area.

field originate from the way the components of the quasiparticle velocity is averaged over the cyclotron orbits which occur at a particular orientation of the magnetic field, yY, known as a ‘Yamaji angle’, as indicated in Fig. 6. Consider the warped cylindrical Fermi surface shown repeated over several Brillouin zones in Fig. 7. The applied magnetic field B will make the electrons execute cyclotron orbits on the Fermi surface in planes perpendicular to the field. When the magnetic field is rotated away from the high-resistivity axis then at certain angles yY, determined by the shape of the Fermi surface, the areas of all orbits will be equal. In such cases, the component of the velocity parallel to the high-resistivity axis will average to zero, and there will be a maximum in the resistivity. These Yamaji ‘magic’ angles are periodic in tan y.

Experimental realization The c-axis resistivity is measured as a function of the direction of the magnetic field. The analysis of AMROs involve the fitting of the experimental data to a parameterization of the underlying Fermi surface. A description of how this can be done efficiently can be found in Bergemann et al. (2003). The measurement of a coherent 3D Fermi surface in a cuprate superconductor by Hussey et al. (2003) is an excellent example of how AMROs can provide important insight. The Fermi surface of Tl2Ba2CuO6+d was revealed by fitting the oscillations, measured using a single-axis rotator in a magnetic fields of 45 T (provided by a hybrid magnet) and a temperature of 4.2 K, to a parameterized Fermi surface. These AMRO results for the Fermi surface topology were subsequently confirmed by ARPES (Platé et al., 2005), and quantum oscillations (both Shubnikov-de Haas and de Haas-van Alphen) (Vignolle et al., 2008).

The electron momentum distribution—Compton scattering and positron annihilation Within the one-electron approximation, the electron momentum distribution (EMD) can be expressed in terms of the one-electron wavefunction c(r) as 2 Z
 1 3 X dr: c ð r Þ exp ð −ipr=ℏ Þ rðpÞ ¼ (10) 2pℏ occ The EMD contains information about the occupied states in momentum space and therefore in a metal will possess signatures associated with the presence of the Fermi surface due to the sum being over the occupied momentum states. Experimentally, the EMD can be accessed using Compton scattering and positron annihilation. The main advantage of these experimental techniques is that they are bulk probes which can be used at elevated temperatures and which are not limited by the electron mean-free-path (meaning, for example, that the Fermi surfaces of disordered alloys can be measured (Dugdale et al., 2006)). Compton scattering is a unique probe of the many-body ground state wavefunction. One significant strength of both techniques is that measurements can be made at elevated temperature, including above room temperature which is required to probe the Fermi surface of paramagnetic Cr (Fretwell et al., 1995). Positrons are susceptible to being trapped by open-volume defects such as vacancies (resulting in a partial or complete suppression of any anisotropy in the

Fermi surface measurements

823

measured distributions (Dugdale and Laverock, 2014)). Compton scattering has no such restriction. Although both techniques measure a projection of the momentum distribution (rather than the distribution itself, as described below), it is straightforward (if time consuming because of the number of projections which must be measured) to tomographically recover the full three-dimensional momentum distribution by combining measurements made along different crystallographic directions (Kontrym-Sznajd and Samsel-Czekała, 2000; Kontrym-Sznajd, 1990). Both positron annihilation and Compton scattering experiments measure the real momentum p (as opposed to the crystal momentum k) of the electrons in the material. In the presence of a Fermi surface (such as that of Li whose Fermi surface, calculated from DFT in the non-interacting zero-temperature limit, is shown in Fig. 8), the discontinuities in the momentum distribution arise when the contribution of a band appears or disappears as the band passes through the Fermi energy (at k ¼ kF) (see Fig. 9). These will appear not only at p ¼ kF, but also at all p ¼ kF + G, where G is a reciprocal lattice vector. This is illustrated in Fig. 10 in which the EMD, r(p), is plotted along the [100] direction. At p ¼ kF, the contribution of the 2 s band drops to zero, before reappearing in what would be the second Brillouin zone in the extended zone scheme. The two insets in Fig. 10 show these so-called ‘higher momentum components’. However, it is straightforward to restore the discrete translational invariance of k-space by bringing the measured densities back into the first Brillouin zone (using the reciprocal lattice vectors) by taking the contributions in higher Brillouin zones and adding them to that in the first Brillouin zone (in a process known as ‘LCW folding’ (Lock et al., 1973)). The discontinuities associated with the partially occupied bands will reinforce constructively, and the contributions from fully occupied bands will combine to give a constant background. In this LCW procedure, the momentum density in p-space becomes an occupation number in k-space. Returning to our Li example, the resulting occupation number can be seen in Fig. 11. The Fermi surface will then be visible as a

Fig. 8 The Fermi surface of Li is almost spherical. The X symmetry point on the Brillouin zone is indicated. The program FermiSurfer was used to plot the Fermi surface (Kawamura, 2019).

band 2 (2s)

band 1 (1s)

Fig. 9 The band structure of Li, plotted along the [100] direction in the repeated zone scheme. The tightly-bound lower band is due to the core-like 1 s electrons and as such is almost flat (dispersionless). The upper band is due to the (much larger) overlap of 2s electron wavefunctions, resulting in a much more dispersive band. Note that the energy scale has been split since the 1s is a long way below the Fermi energy.

824

Fermi surface measurements

Fig. 10 The electron momentum distribution of Li resolved along the [100] direction. Note that 1 atomic unit (a.u.) of momentum is equal to 1.99285191410  10−24 kg m s−1. The two insets (with their ordinate axes scaled by 102 and 104, respectively) show how the momentum density for band 2, which decreases to zero at p ¼ kF, reappears to make contributions in the second and third Brillouin zones.

n(k) 4 2 kF

2 a

k

Fig. 11 Occupation number for Li along the [100] direction obtained by LCW folding (Lock et al., 1973) the EMD back into the first Brillouin zone. At the origin (the G point), both bands are occupied giving four electrons. The completely occupied 1s band sums to a constant, but the partially occupied 2s exhibits a discontinuity at   the Fermi wavevector kF as the occupation number for that band drops to zero. Note that the Brillouin zone boundary at the symmetry point X is at 2p a in the body-centered cubic lattice of Li.

change in the occupation number. An example of fitting several Fermi surface breaks to an occupation number reconstruction from several high-resolution Compton profiles measured on the high-entropy alloy NiFeCoCr can be found in Robarts et al. (2020). Since the Fermi surface is highly smeared due to the extreme compositional disorder (the atoms are randomly distributed on an underlying face-centered cubic lattice), the smeared Fermi breaks can be clearly resolved, and the local contributions to the electron mean-free-path determined across the Fermi surface from the degree of smearing.

Compton scattering Following Schülke (2007), the basic kinematics of an inelastic scattering experiment are shown in Fig. 12. A photon of energy ℏo1, wavevector K1, and (unit) polarization vector e1 scatters at an angle y off a target which is characterized by a state vector |ii and energy Ei, resulting in a photon of energy ℏo2, wavevector K2 and polarization vector e2. The target is left in a final state |fi with an energy Ef. Conservation of energy and momentum imply that an energy ℏo, given by ℏo ℏ(o1 − o2) ¼ Ef − Ei, and a momentum ℏq K1 − K2 are transferred to the target. As long as o o1, q  2 K1 sin (y/2) (Schülke, 2007). In a Compton scattering experiment, monoenergetic incident photons are scattered by the electrons in the sample and the energy distribution of the scattered photons is measured to a resolution dℏo2 within some solid angle dO2. This is the double differential scattering cross-section, which can be expressed in the non-relativistic limit as (Schülke, 2007): * + 2   2 o2 X X iqr   d2 s ¼ r 20 e1 e2∗ pi f e j i d Ef −Ei −ℏo , (11) o1 dO2 dℏo2 j i,f where pi is the probability of the initial state |ii and r0 is the classical electron radius. The sum over j is over all the electrons in the system.

Fermi surface measurements

825

K 2, e 2 , Z 2 K 1, e 1, Z 1

T q

Fig. 12 Compton scattering of a photon with wavevector K1, unit polarization vector e1 and frequency o1 by an angle y by a moving electron in an initial state |ii, leaving it in a final state |fi.

Fig. 13 The Compton profile J(pz) of Li, resolved along the [100] direction.

In order to probe single-particle properties (rather than their collective behavior), q−1 must be much smaller than the distance between the electrons. For the properties to be representative of the ground state, the scattering process should be fast such that the energy transfer ℏ(o1 − o2) is sufficiently high to prevent the remaining system of electrons from rearranging itself. This is usually described within the so-called ‘impulse approximation’ which is valid when the energy transferred in the scattering process is much larger than the binding energies of any electrons involved; the other electrons are mere spectators since the scattering occurs on a very short timescale on the order of 10−19 s. If the photons were scattering off stationary electrons then the scattered photons would all have the same energy given by the usual Compton formula Compton (1923). However, the electrons have finite momentum and therefore the photon energies will be Doppler-broadened and the distribution of energies recorded at that fixed angle can be related to the momentum distribution of the electrons along the scattering vector integrated over the two orthogonal momentum components. It can be shown that Z Z d2 s ∝ rðpÞdpx d py , (12) dO2 d ℏo2 where r(p) is the momentum density. The quantity

Z Z Jðpz Þ

rðpÞdpx dpy

(13)

is called the Compton profile; it is the projection of the momentum density along the scattering vector. A detailed description of the theoretical background can be found in Refs. (Cooper et al., 2004; Eisenberger and Platzman, 1970; Eisenberger and Reed, 1974; Ribberfors, 1975a, 1975b). Revisiting the Li example, its Compton profile along the [100] direction is shown in Fig. 13. In this case, the fully occupied first band gives rise to a broad, smooth, Gaussian-like distribution, whereas the almost spherical Fermi surface contributes an inverted

826

Fermi surface measurements

position-sensitive detector

Rowland circle R=3650mm incident x-rays 115keV

slits Compton scattered x-rays 70-90keV

crystal analyser

165°

500m sample

Fig. 14 Schematic of a Cauchois-type high-resolution Compton spectrometer, based on Itou et al. (2001).

parabola when it is integrated over two momentum components. However, the Fermi break kF can still be identified (in Fig. 13 at about p  0.6 (2p/a)) and the higher momentum components are just visible as bumps at momenta outside the first Brillouin zone. If the LCW procedure were applied to the Compton profile, a projected occupation number would be obtained. The two ‘lost’ momentum components can be reconstructed tomographically, typically by measuring 10—30 Compton profiles along different crystallographic directions. With reconstruction techniques based on polynomial expansions, the number of profiles required for a faithful reconstruction can be reduced by carefully choosing the directions. Experiments are usually performed at synchrotron sources. As an example of an experimental geometry, the Cauchois-type high-resolution Compton spectrometer at SPring-8 (shown in Fig. 14) offers a typical momentum resolution of 0.1 a.u. (Itou et al., 2001) and a photon energy of 115 keV. At these kinds of photon energies, it is clear that it is the bulk being probed. The Compton-scattered photons are dispersed by a bent crystal analyzer and registered on a position-sensitive detector. In some spectrometers, a trade-off can be made between count rate and resolution, and ultra-high resolution measurements (with a momentum resolution of about 0.02 a.u.) measurements have been performed (Schülke et al., 2001). An excellent example of the kind of information that Compton scattering can provide can be found in Tanaka et al. (2001). Since bcc Li has a martensitic transformation at 75 K (Smith, 1987), quantum oscillatory probing of the Fermi surface is challenging (Hunt et al., 1989). By reconstructing the full 3D EMD from high-resolution Compton profiles, the small anisotropy (4%) in its Fermi surface has been revealed (Schülke et al., 1996; Tanaka et al., 2001). Moreover, both studies were able to show electron correlation substantially reducing the size of the break Zk in the momentum density (sometimes called the ‘quasiparticle residue’) at the Fermi surface. Compton scattering has also been able to map out the evolution of the Fermi surface of La2−xSrxCuO4 as a function of doping (Sakurai et al., 2011) and expose the possible role of nematicity in the CuO2 planes in deforming the Fermi surface in the underdoped regime (Yamase et al., 2021).

Positron annihilation The electron momentum distribution can also be probed using positrons. In this case, it is the electron momentum distribution as seen by the positron which is measured, and it is often called the electron-positron momentum distribution or the ‘two-photon momentum distribution’ (TPMD, where the ‘two-photon’ refers to the pair of annihilation g-photons which are experimentally measured). Within the independent particle model (in which the electron and positron are treated as non-interacting independent particles), the TPMD can be expressed as 2 X
1 3 Z cðrÞc ∗ ðrÞ exp ð −ipr=ℏÞ dr, r2g ðpÞ ¼ (14) + 2pℏ occ where c + ∗(r) represents the (complex conjugate of the) positron wavefunction. The technique is commonly known as 2D angular correlation of annihilation radiation (2D-ACAR). Positrons, usually obtained from a 22Na source, are implanted into the sample being studied. The positrons have a typical b energy spectrum with a maximum energy of about 0.5 MeV, and thus penetrate into the bulk of the sample. Typical source activity and annihilation rates mean that there will only ever be one positron in the sample at any time. After thermalization, which happens on a timescale of several picoseconds in a metal (Kubica and Stewart, 1975), the positron will propagate in its own Bloch state (at the bottom of its own positronic band) for 100 ps before annihilation with an electron which proceeds predominantly through the creation of two

Fermi surface measurements Δx 2

y

detector 1

x

z

827

detector 2

Δy2

face of detector 1 Δy1 Δx 1 Fig. 15 Geometry of a 2D-ACAR experiment. A pair of g photons resulting from electron-positron annihilation are detected in coincidence on a pair of position-sensitive detectors. The angular deviation from being back-to-back can then be calculated from the positions DX and DY on the two detectors.

g-photons which, owing to conservation of energy and momentum will be emitted back-to-back in the center-of-momentum frame. However, in the laboratory frame, in which the electron and positron have a finite momentum, there will be a small angle (mrad) between the gs. Since the positron has only a thermal momentum, the combined momentum of the pair is completely dominated by that of the electron. Two components of the momentum can be directly inferred by measuring the angular deviation of the two g from being back-to-back by recording where they hit position-sensitive detectors (Dugdale et al., 2013) typically located about 10 m away on either side of the sample, as shown in Fig. 15. If the momentum of the pair has components (px,py,pz), then px  mcyx ,

(15)

py  mcyy ,

(16)

and

where m is the electron rest mass, c is the speed of light and yx and yy can be calculated from the detected g positions if the distance of each detector from the sample is L. yx ¼

D x1 + Dx2 , L

(17)

yy ¼

Dy1 + D y2 : L

(18)

and

The remaining momentum component, which would be encoded in a Doppler shift in the energy of the photons, cannot be measured with the same precision. This means that this third component is integrated over, and the resulting spectrum is a projection of the underlying momentum distribution. Tomographic reconstruction is necessary to regain the third momentum component, and this is achieved by measuring down a small number (typically fewer than six) crystallographic directions. However, in many cases, an excellent understanding of the Fermi surface topology can be gained from a single high-symmetry projection. The momentum resolution is normally dominated (at least at low temperatures) by the precision with which the coordinates of g-photons can be recorded on the two position-sensitive detectors as well as uncertainty in the origin of annihilation due to the source spot on the sample. The latest generation of detectors offer sub-mm position resolutions Dugdale et al. (2013), which at typical distances from the sample means momentum resolutions of a few percent of the Brillouin zone of Cu. Not all electrons are seen by the positron equally, since the positron’s positive charge means that it is strongly repelled by the nucleus. This results in a smaller overlap between the positron’s wavefunction and those of the most tightly bound electrons. Their contribution is therefore strongly suppressed, but that from electrons at the Fermi surface is enhanced since annihilation is more likely to occur with one from among the screening cloud of electrons at the Fermi surface. While many-body electron-positron correlations can modulate the shape of the momentum density, the position of the discontinuity is not affected (Majumdar, 1965) meaning that the Fermi surface can be clearly identified. Although the TPMD is sensitive to electron-positron correlations (for example, see Laverock et al. (2010)), it is still possible to study the effect of electron correlation on the occupation numbers in k-space (Harris-Lee et al., 2021). Positron annihilation experiments have been used to investigate the impact of electron correlation on the Fermi surface of vanadium (Weber et al., 2017). This was achieved by comparing the densities calculated using a combination of DFT and dynamical mean-field theory with experimental measurements.

828

Fermi surface measurements

Spin-resolved measurements Both Compton scattering and positron annihilation can be operated in a spin-resolved mode. In the case of Compton scattering, magnetic measurements exploit the spin-dependent term in the scattering cross-section for circularly polarized photons (Duffy, 2013). Magnetic Compton scattering experiments are typically performed using a solid-state detector which has a relatively poor energy resolution corresponding to a momentum resolution which is comparable to the size of the Brillouin zone, making any sensible study of the Fermi surface impossible. However, high-resolution magnetic Compton profiles with a resolution acceptable for Fermi surface studies have been measured (Sakurai et al., 1994). Spin-resolved positron annihilation relies on the parity violation in beta decay which manifests as a preferential positron spin parallel to its velocity vector. It is possible to extract a spin-resolved Fermi surface (Hanssen et al., 1990; Weber et al., 2015) by making measurements with a magnetic field in different directions, since two-photon annihilation relies on the electron and positron having opposite spins.

Strengths and weaknesses of the various techniques There is a great deal of complementarity between the techniques which have been discussed. All have their advantages and disadvantages, their strengths and weaknesses.

Sensitivity to correlations Through their ability to measure the effective mass, quantum oscillations can probe electron correlation on the low energy ( requires an operation wherein the wave function is projected on Cc and the functional dependence now corresponds to Cc. If we define a pseudopotential as V p ¼ V + V R we now have now transformed the one-electron Schrödinger equation to  2 2  −ℏ r + Vp Fpv ¼ Ev Fpv : Hp Fpv ¼ (8) 2m This description of an atom has a number of advantages. The repulsive part of the potential, V R , largely cancels the attractive part of the all electron potential in the core region. The pseudo wave function is smooth as the core oscillations have been removed and the eigenvalue, Ev, is identical to the all electron potential. While the wave function is now amenable to a simple basis, the pseudopotential within this construction is more complex than the all electron potential. The potential is weak and only binds the valence state, but it is also energy dependent, state dependent and involves a nonlocal, non-Hermitian operator. Owing to these complexities, the Phillips-Kleinman potential is rarely, if ever, used for calculating pseudopotentials. However, the cancellation theorem is useful in demonstrating the essential features of a pseudopotential.

Model potentials For many purposes, a model pseudopotential can be used to capture the essential physics and advance the essential understanding of the system of interest. As a starting point, we consider a simple model for the ion-core pseudopotential. This potential is expected to be transferable since it depends only on the nuclear and core states. In a vein similar to what Hellmann proposed, we write  0 r  rc , V pc ðr Þ ¼ (9) −Zv e2 =r r > r c : Within the core, this potential is defined as an “empty core.” rc is defines the size of the core region of the potential. The electron charge is e, and Zv is the number of valence electrons. In most cases, the number of valence electrons is simply given by the column of the periodic table. Some issues arise in the presence of loosely bound core states. An example is the 3d state in Cu, which can be taken as a valence state or a core state. This issue can be tested by considering both cases and assessing any differences. To form a solution for the electronic structure of a crystal, we need to input the structure of the crystal and account for the interactions of the valence electrons. This is often handled by expressing the wave functions in terms of a plane wave basis, which requires an expression of the potential in Fourier-space characterized by an atomic form factor, Vpc (q) Z
!  1 ! V pc ðqÞ ¼ V pc ðr Þ exp i q  r d3 r: (10) Oa where Oa is the atomic volume. The atomic form factor for empty core pseudopotential is given by V pc ðqÞ ¼

−4pZv e2 cos ðqr c Þ: Oa q2

(11)

The long range nature of the Coulomb tail requires special handling in executing this integral when q ! 0. Also, the form factor “rings”, i.e., the form factor oscillators as a cosine function owing to the discontinuity in the potential at rc. The ion core pseudopotential needs to be screened by the valence electrons. For simplicity, Thomas-Fermi screening is often used for this purpose (Kittel, 2005) V pa ðqÞ ¼

V pc ðqÞ , eT F ðqÞ

(12)

where the screening is characterized by a dielectric function given by eT F ðqÞ ¼ 1 +

K 2s , q2

(13)

where 1/Ks is the Thomas-Fermi screening length K2s ¼ 6pne2/EF, which depends on the valence electron density, (n ¼ Zv/Oa) and EF is the Fermi energy for a free electron gas of density, n. The atomic form factor is now given by V pa ðqÞ ¼

−4pe2 n cos ðqr c Þ: q2 + K 2s

(14)

A common practice is to set the potential to zero outside of the second node to remove the oscillatory or ringing behavior of the cos(qrc) term. This simple potential is characterize by only one parameter, rc and the electron density, n. An interesting consequence of Thomas-Fermi screening is that the potential now has a well defined value when q ¼ 0,

Pseudopotential methods

837

Fig. 3 Model of an atomic pseudopotential. The required form factors can be extracted from this model. Note the limiting case of G ¼ 0 for a metallic system.

Vap ð0Þ ¼

−4pe2 n 2 ¼ − EF : Ks2 3

(15)

The q ¼ 0 term corresponds to the average of the crystalline potential. This limiting case from Thomas-Fermi screening is not a very good approximation for a semiconductor like silicon and is more appropriate for a metal. This is expected as the Thomas-Fermi expression works best for a “free electron” gas, which is likely to be a better model for a simple metal as opposed to a semiconductor. We have considered a general value, q, in Fourier space, but symmetry restricts the value of q to be a reciprocal space vector, determined by the crystal lattice (Kittel, 2005). This is depicted in Fig. 3 where a schematic of a screened atomic potential is illustrated. We can write the total crystalline potential as
 X
 ! ! ! ! (16) Vap r − R − t , Vp r ¼ ! ! R, t !

where the position of the lattice point are given by the lattice vector, R , and the atomic positions associated with the lattice point ! by the basis vector, t : Using the atomic form factors, we can express this pseudopotential as
 X
!
!  ! ! (17) Vp r ¼ V pa ðGÞS G exp i G r , !!


! where S G is the structure factor:

G, t


!
!  1 X ! S G ¼ exp iG t : Na t

(18)

!

G is a reciprocal lattice vector and Na is the number of atoms in the unit cell (Kittel, 2005). This expression assumes an elemental crystal, but generalizing the potential to a crystal with different atomic species present is straightforward. The total pseudopotential is determined by the atomic structure factor, which represents the screened ion core potential, and the crystal structure determined by the reciprocal lattice and the atomic positions. This simple pseudopotential is called a “local” pseudopotential as the ion core is a simple function of position. The potential does not reflect elements present in the Phillips-Kleinman cancellation theorem wherein the potential depends on the nature of the core states. For example, in carbon there is no 1p core state and the valence 2p is not required to be orthogonal to the core. This is not true for silicon in which both s and p states exist in the core, both the s and p valence states experience a repulsive term. Historically, this difference has been used to explain why silicon and carbon chemistry are different, the former representing “geology” and the later “biology” (Kittel, 2005). The empty core model potential can be modified to reflect the state dependence  r  r c,l , Al p V c,l ðr Þ ¼ (19) −Zv e2 =r r > r c,l : This potential is not a simple function of position, but rather acts on the l-component present in the wave function (Cohen and Chelikowsky, 1982, 1989; Chelikowsky and Cohen, 1992). This can be accomplished by using a projection operator as will be discussed later. There are additional parameters to be determined as the well depth and size must be fixed for each l-component. A common practice in the early days of pseudopotential involved estimating the well depths and sizes (Al, rc,l) to replicate optical data of ionized atoms. For example, to construct an ion core pseudopotential for an Al atom, one needs to consider the optical excitations for an Al+2 ion. In some cases, this is relatively easy, e.g., treating the Na atom does not involve an ionized atom.

838

Pseudopotential methods

However, determining such parameters for an O atom requires the examination of an O+5 ion. The experimental task of multiply ionizing atoms and measuring the resulting atomic levels is difficult. Often the potentials were estimated by extrapolation from lighter atoms to heavier ones. Another issue concerns using more accurate dielectric functions screening the potentials. Both issues were the subject of a number of studies. By the mid 1960s an inventory of such model potentials existed (Animalu and Heine, 1965; Cohen and Heine, 1970) with some success.

The empirical pseudopotential method One of the most significant advances in applying pseudopotential theory was the development of the empirical pseudopotential method (EPM). The intuitive picture of Fermi and Hellmann was greatly advanced by Phillips and Kleinman; however, a practical method of predicting accurate energy band structures was not achieved until the EPM. This method developed in the late 1960s was based on fixing an energy band so that a measured response function, such as the reflectivity of a solid, was reproduced (Cohen and Bergstresser, 1965). To accomplish this feat, one had to develop a framework of translating an energy band solution to a response function. Determining the energy band structure for a simple pseudopotential is not difficult. If we are given the crystalline potential, we can solve the corresponding eigenvalue problem (Eq. 1) using a variety of approaches. Since the pseudopotential is weak compared to the all electron potential, we can write the wave function in terms of a plane wave basis, which are expected to converge quickly
 X
! !

! !   ! (20) c ! r ¼ an k , G exp i k + G r : n, k

!

G

n !o ! n is a band index, k is the wave vector. The sum is over all reciprocal lattice vectors, G , but in principle the sum is truncated for !

!

| k + G| > G max . This form of the wave function is consistent with the Bloch form of the wave function for a periodic potential. The Bloch form of the wave function allows one to express the corresponding periodicity of the wave function, save a phase factor determined by the wave vector

! !
 ! ! ! c ! r + R ¼ exp i k R c ! r : (21) n, k

n, k

Inserting the crystalline potential (Eq. 17) and the plane wave basis (Eq. 20) in the one-electron Schrödinger equation, yields the following: 82 3 !

!

!


! ! 0 ! 2 ! ! P < ℏ2
! 0 0 p 4 K + G − En k 5 d! !0 +Va j G − G j S j G − G j (22) an k G ¼ 0, GG 2m ! : 0 G

!

which corresponds to a set of linear equations, one for each G-vector in the set. For a non-trivial solution, we require the following: 2 3 !

!


! ℏ 2
! ! 2 ! ! 0 0 p 4 5 ! (23) d! 0 +Va j G − G j S j G − G j ¼ 0, k + G − En k det GG 2m The diagonal elements of this determinant contain the kinetic energy contribution; the off diagonal elements contain the form factors for the pseudopotential and the structure factor. This is an eigenvalue problem and a standard mathematical problem, which is easily handled, save for large systems, i.e., the matrix is large and dense (Saad et al., 2010). Owing to the rapid convergence of the pseudopotentials in Fourier space for common semiconductors or simple metals, one needs only a few hundred plane waves to obtain a converged result. As a specific example, consider the energy band structure of germanium as computed by pseudopotentials. The band structure is
! ! shown in Fig. 4. The energy at point k and band n, En k , is plotted along different high symmetry directions, e.g., the energy band !

!

along G to X is plotted from k ¼ ð2p=aÞð0, 0, 0Þ to k ¼ ð2p=aÞ ð1, 0, 0Þ along the h100i or D direction. The input required to compute the energy band includes the crystal structure of germanium, which occurs in a diamond structure, and the form factors required to construct the pseudopotential for the crystal. As indicated in Fig. 3, only three form factors are required for this case. The empirical pseudopotential method treats these form factors as adjustable parameters. The form factors are not linearly independent, so the information required to fix the form factors is not great. Three or four band structure features are sufficient. As a starting point, one can use a model pseudopotential based on optical spectra for atoms. The form factors are then refined by using optical or photoemission data for the crystal of interest. Given the energy band of a crystal, one can compute the complex index of refraction and the reflectivity. The first such computations were based on simplified dielectric theories that excluded electron-hole interactions (Cohen and Chelikowsky, 1989). Four ingredients are included: conservation of energy, conservation of crystal momentum, wave function symmetry and the number of states accessible. Energy conservation requires a photon energy sufficient to excite an electron from an occupied

Pseudopotential methods

839

Fig. 4 Band structure of germanium using pseudopotentials (Cohen and Chelikowsky, 1989). The valence band maximum is set to zero.

valence band to an empty conduction band. Such transitions are direct, i.e., the wave vector of the valence band and conduction bands are the same so that crystal momentum is conserved. Just as in an atomic state, the overlap of initial and final states should have a dipole component to couple to the electric field of the photon. Finally, for a given energy difference and wave vector, one can compute the number of possible valence and conduction band states accessible. The larger the number of states, the more probable is an optical excitation. The computed reflectivity of germanium is given in Fig. 5. The agreement between experiment and theory is sufficient to identify features in the reflectivity spectrum that can be associated with energy band features. For example, the reflectivity peak around 2 eV can be identified with transitions occurring between the top of the valence band and lowest conduction bands near the L-high symmetry point. If the theoretical reflectivity for this part of the spectrum does not agree with experiment, we know that the energy band structure is incorrect. If the theoretical reflectivity agrees with experiment, we can be reasonably confident that the energy band structure is correct.

840

Pseudopotential methods

Fig. 5 Experimental (Philipp and Ehrenreich, 1963) and theoretical reflectivity of crystalline germanium (Cohen and Chelikowsky, 1989).

The empirical pseudopotential method provided the first realistic energy band structures for many simple metals and semiconductors (Cohen and Chelikowsky, 1989). Most of our knowledge of the electronic structure of these materials was pioneered by this type of pseudopotential. However, the empirical pseudopotential method is somewhat limited for several reasons. The potential requires experimental input and this input can bias the pseudopotential. For example, suppose we fit the optical properties of GaSb and extract a Ga potential, we must be careful in trying to use this fit Ga potential in a different crystal. This potential might work fine in GaAs, which has a similar bonding configuration when compared to GaSb, but the potential may fail to describe GaN, which is considerably more ionic than GaSb. A related problem concerns how the form factors are fit. For high symmetry systems, only a small subset of reciprocal lattice vectors come into play. If the crystal volume or structure changes, one must extrapolate to a new subset of vectors. The required set of data may notably exceed the available experimental data. This problem can occur at a surface or at a defect. In crystalline GaAs, each Ga atom is surrounded by four As atoms. For the cleavage plane, the (110) surface, the Ga atom is bonded to three As atoms. Given the coordination change, the surface Ga atom cannot be expected to retain a” bulk-like” screening potential. There is no reason to be confident that the Ga pseudopotential extracted from the crystalline environment will be very accurate at the surface. Of course, this problem is made worse if one wants to examine a material where the structure is not known as for an amorphous solid or a structure varying in time such as in a liquid state, where the coordination around an atom may continuously change.

First principles pseudopotentials using density-functional theory A key issue is how to account for the valence charge density screening the ion core pseudopotential. We can consider the form factors to be composed of two interactions: the “ion core-valence electron” interaction and the “valence electron-valence electron” interaction. A fundamental postulate of the pseudopotential method is that the ion core pseudopotential is not dependent on the chemical environment. We assume that this part of the potential can be transferred from one environment to another with no loss in accuracy. A notable problem is the determination of the total pseudopotential by including interactions between the valence electrons. We illustrated the use of Thomas-Fermi dielectric screening in model potentials, which can be used as a first pass for empirical potentials. A better approach constructs a self-consistent potential by using the valence charge density within density functional theory. Let us assume a one-electron Hamiltonian, which can be based on density functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965)  2 2




 −ℏ r ! ! ! ! ! p Cn r ¼ En Cn r : (24) + Vion r + VH r + Vxc r 2m The ion-core pseudopotential, Vpion, can be taken as a linear superposition of ion core atomic potentials. Determining the ionic potential can be accomplished by resort to atomic structure calculations. The potential arising from the valence-electron interactions can be divided into two parts. One part represents the “classical” electrostatic terms, named the Hartree or Coulomb potential

 ! ! r2 VH r ¼ − 4per r ,

 (25) P ! ! 2 r r ¼e Cn r : n, occup

Pseudopotential methods

841

Assume initial density:

VH = 4 e

2

Solve:

Form: VT = Vionp + VH + Vxc

2

Solve:

2

Vionp + VH + Vxc

2m

Form:

n

= En

n

2

=e

n n, occup

Fig. 6 Self-consistent field loop. The loop is repeated until the “input” and “output” charge densities are equal to within some specified tolerance.

where r is the valence electron charge density. The second part of the screening potential, the effective “exchange-correlation” part of the potential, Vxc, is quantum mechanical in nature. A common approximation for this part of the potential is the local density approximation. The potential depends only on the
 h
i ! ! charge density at the point of interest, V xc r ¼ V xc r r : In principle, density functional theory is exact, provided one can obtain an exact functional for Vxc. One productive approach is to assume that the functional extracted for a homogeneous electron gas (Ceperley and Alder, 1980) is “universal” and can be applied to an inhomogeneous gas. The procedure for generating a self-consistent field (SCF) potential is given in Fig. 6. The SCF cycle is initiated with a potential constructed by a superposition of atomic densities. These densities are then used to solve a Poisson equation for the Hartree potential and a density functional is used to obtain the exchange-correlation potential. A screening potential composed of the Hartree and exchange-correlation potentials is then added to the fixed ion-core pseudopotential, after which the one-electron Schrödinger equation, or Kohn-Sham equation, is solved. The resulting wave functions from this solution are then employed to construct a new potential and the cycle is repeated. In practice, the “output” and “input” potentials are mixed using a scheme that accounts for the history of the previous iterations (Broyden, 1965; Chelikowsky and Cohen, 1992). Methods for constructing ion core pseudopotentials can be based on this density functional framework. The first step is to consider an electronic structure calculation for a free atom, including all the electrons. This is an easy numerical calculation as the atomic densities are assumed to possess spherical symmetry and the problem reduces to a one dimensional radial integration. Once we know the solution for an “all electron” potential, we can invert the Kohn-Sham equation and find the total pseudopotential, i.e., one can “unscreen” the total potential and extract the ion core pseudopotential. This ion core potential, which arises from tightly bound core electrons and the nuclear charge, is not expected to change from one environment to another. It should be transferable from the atom to a molecular state to a solid state or liquid state. The issue of this transferability must be addressed according to the system of interest. An immediate issue focuses on the pseudo-wave functions, which can be used to define the corresponding ion core pseudopotential. Suppose we insist that the pseudo-wave function be identical to the all-electron wave function outside of the core region. As a specific example, let us consider the 3s state for a silicon atom. We want the pseudo-wave function to be identical to the all electron state outside the core region p

f3s ðr Þ ¼ c3s ðr Þ fp3s

r > rc ,

(26)

where is a pseudo-wave function and rc defines the core size. This assignment will guarantee that the pseudo-wave function will possess properties identical to the all electron wave function, c3s in the region away from the ion core. For r < rc, we alter the all-electron wave function. We are free to do this as we do not expect the valence wave function within the core region to affect the chemical properties of the system. We can make the pseudo-wave function smooth and nodeless in the core region. This will provide rapid convergence with simple basis functions. One other criterion is mandated. Namely, the integral of the pseudocharge density within the core should be equal to the integral of the all-electron charge density. Without this condition, the normalized pseudo wave function differs by a scaling factor from the all-electron wave function. Pseudopotentials constructed with this constraint are called “norm conserving” (Hamann et al., 1979). Since we expect the bonding in a solid to be highly dependent on the tails of the valence wave functions, it is imperative that the normalized pseudo wave function be identical to the all-electron wave functions. There are many ways of constructing “norm conserving” pseudopotentials as within the core the pseudo wave function is not unique. One of the most straightforward construction procedures places conditions on the pseudo-wave functions and then inverts a density functional solution for the atom (Kerker, 1980; Troullier and Martins, 1991).

842

Pseudopotential methods

Fig. 7 An all electron and a pseudo-wave function for the silicon 3 s radial wave function.

Suppose one assumes the pseudo-wave function, fpl (r) has the following properties: ( r l exp ðpðr ÞÞ r  r c , p fl ðr Þ ¼ cl ðr Þ r > rc :

(27)

p(r) is taken to be a polynomial of the form: pðr Þ ¼ c0 +

6 X

c2n r 2n :

(28)

n¼1

This form assures us that the pseudo-wave function is nodeless and by taking even powers in the polynomial there is no cusp associated with the pseudo-wave function. The parameters, c2n, are fixed by the following criteria: (a) The all-electron and pseudo-wave functions have the same valence eigenvalue. (b) The pseudo-wave function is nodeless and be identical to the all-electron wave function for r > rc. (c) The pseudo-wave function must be continuous as well as the first four derivatives of the wave function at rc. (d) The pseudopotential has zero curvature at the origin. This approach to pseudopotential construction is easy to implement and to include other possible constraints. An example of an atomic pseudo-wave function for Si is given in Fig. 7 where it is compared to an all-electron wave function. Unlike the 3s all electron wave function, the pseudo-wave function has no nodes. Once the pseudo-wave function is constructed, then the Kohn-Sham equation can be inverted to arrive at the ion-core pseudopotential p

p

Vion, l ðr Þ ¼

ℏ2 r2 fl p − En, l − VH ðr Þ − Vxc ½rðr Þ: 2mfl

(29)

The ion core pseudopotential is well behaved as fp has no nodes and does not vanish. However, the potential is state dependent and energy dependent. The energy dependence is usually weak. For example, the 4s state in silicon computed by the pseudopotential constructed from the 3s state is usually accurate. Physically this happens because the 4 s state is extended and experiences the potential in a region where the ion core potential has assumed a simple −Zve2/r behavior. However, the state dependence through l is an issue, the difference between a potential generated via a 3s state and a 3p can be significant. In particular, for first row elements such as C or O, the nonlocality or state dependence is quite large as there are no p states within the core region. For the first row transition elements for such as Fe , this is also an issue as again there are no d-states within the core. An additional advantage of the norm conserving potential concerns the logarithmic derivative of the pseudo-wave function (Bachelet et al., 1982). An identity exists − 2p

ðrfÞ2

ZR d2 ln f ¼ 4p f2 r 2 dr ¼ QðRÞ: dE dr R 0

(30)

843

Pseudopotential methods

Fig. 8 Ion core pseudopotentials for carbon generated by four different methods in real and reciprocal space: (A) Troullier and Martins (1991), (B) Kerker (1980), (C) Hamann et al. (1979) and (D) Vanderbilt (1985). The dotted and solid lines correspond to the s and p pseudopotentials, respectively.

The energy derivative of the logarithmic derivative of the pseudo-wave function is fixed by the amount of charge, Q, within a radius, R. The radial derivative of the wave function f is related to the scattering phase shift from elementary quantum mechanics. For a norm-conserving pseudopotential, the scattering phase shift at R ¼ rc and at the eigenvalue of interest is identical to the all-electron case as Q is the same for the all-electron potential and the pseudopotential. As a consequence, the scattering properties of the pseudopotential and the all-electron potential have the same energy variation to first order when transferred to other systems. The pseudo wave function is not unique. This property was recognized early in its inception. For example, within the Phillips-Kleinman formulation, one can always add a function, f, to the pseudo-wave functions without altering the pseudopotential provided f is orthogonal to the core states. Consider the matrix element in Eq. 5. If one changes Fpv to Fpv + f, then hCc| Fpv + fi ¼ hCc| Fpvi + hCc| fi ¼ hCc| Fpv i. Nothing is changed in the Phillips-Kleinman pseudopotential by this addition. The non-uniqueness of the pseudopotential can be exploited to optimize the convergence of the pseudopotentials for the basis of interest. Much effort has been made to construct “soft” pseudopotentials. By “soft,” one means a “rapidly” convergent calculation using plane waves as a basis. Such potentials are characterized by a “large” core size, i.e., a large value for rc. However, as the core becomes larger, the “goodness” of the pseudo-wave function can be compromised as the transferability of the pseudopotential becomes more limited. In Fig. 8, the ion core pseudopotential for carbon is plotted in real space and in reciprocal space. Although the potentials look quite different in the core region, they all give reliable electronic structure properties for carbon. Pseudopotentials constructed within density functional theory are inherently nonlocal as they depend on the all electron state of the atom. This feature can also be included in empirical pseudopotentials using parameterized potentials (Cohen and Chelikowsky, 1989). In setting up the eigenvalue problem, we need to include an additional matrix element in Eq. (22). One can consider a local potential as a reference potential. For example, we can write DVl ¼ Vlocal − Vl, where Vlocal can be any local potential, but is often chosen to a specific component. As an example, suppose we have a carbon atom. We can find a pseudopotential for the s-state and take this as a local component. The p potential is then referenced as V s + P {p V p −V s P p where P p is a projection operator that only acts on the l ¼ 1 component of the wave function. In general, the nonlocal elements are very short ranged in real space as the size of the ion core is determined by rc. Outside of the core the potential components converge to the same limit. A major strength of density functional pseudopotentials is that they can be employed for systems with many atoms of low symmetry, e.g., clusters, liquids, and surfaces, without having to fit any form factors. However, a plane wave basis will require a large !

!

cutoff for such systems. In this case, the nonlocal matrix elements can be difficult to evaluate as elements depend on both k + G and

!

!0

k + G : An efficient form for the required nonlocal matrix elements (Kleinman and Bylander, 1982) is given by

844

Pseudopotential methods



!d! P ∗ !d! 0 Y Y + G + G k k lm lm ! ! ! !0 m Z DV Kl B k + G, k + G ¼ p p 4pOa Fl ðr Þ DV l ðr Þ Fl ðr Þ dr

Z ! !0 p  Fl ðr Þ DV l ðr Þjl j k + G jr r 2 dr Z
! !  p  Fl ðr Þ DV l ðr Þjl j k + Gjr r 2 dr,



(31)

where Fpl is the atomic reference pseudo-wave function and Y∗lm is a spherical harmonic with the angles determined by the unit !d! ! ! vector: k + G: This matrix element form has a great advantage as the integrals are separable in that they are solely a function of k + G !

!0

or a function of k + G : The individual integrals can be stored and retrieved as required in setting up the matrix. The Kohn-Sham problem can also be solved directly in real space (Chelikowsky et al., 1994). In this case the nonlocal matrix elements can be cast as DV Kl B ðx, y, zÞfp ðx, y, zÞ ¼ Z Glm ¼ Z

X

p

Glm ulm ðx, y, zÞ DV l ðx, y, zÞ

lm p

ulm DV l fp dx dy dz p

p

(32) ,

ulm DV l ulm dx dy dz

where uplm are the reference atomic pseudo-wave functions. The nonlocal nature of the pseudopotential is apparent from the definition of Glm, the value of these coefficients are dependent on the pseudo-wave function, fp, acted on by the operator DVl. While our focus has been on “norm conserving” pseudopotentials, one can generalize the pseudopotential method. It is possible to make the pseudopotentials even weaker by relaxing this condition. (Vanderbilt, 1990) proposed such a method for constructing ultra soft pseudopotentials, which are constructed as a generalized eigenvalue problem. The norm conservation constraint is relaxed, and the charge density within the ion core is not explicitly considered as part of the pseudo-wave function. The relaxation of the norm conservation constraint allows one to consider a much larger core radius and a much softer potential. One issue which is relevant for pseudopotential constructions, regardless of whether the potential is intended for use with a plane wave basis or not, concerns the issue of unbound, or weakly bound, atomic states. If an atom does not bind a state of interest, then the atomic wave function corresponding to this state is clearly not normalizable. Nonetheless, the pseudopotential corresponding to this state might be of some interest, e.g., in a crystal such diverging wave functions are “captured” by the potentials of neighboring atoms. For example, Ba has a strong f-component resonance. However, these f-states are not bound for neutral Ba atom. In order to bind such states, one must consider highly ionized atomic states, which result in very strong pseudopotentials. Sometimes these potentials are so strong as to be useless for a plane wave basis, or so far removed from the chemical environment of interest that their transferability may be suspect. Hamann (1989) has suggested a method for handling such cases by integrating out the Kohn-Sham equation to a large distance and at that point terminating the pseudo-wave function. The corresponding “terminated” wave function is then used to generate a pseudopotential for the component of interest. Empirical pseudopotentials provide effective tools in understanding the optical and dielectric properties of semiconductor crystals (Cohen and Chelikowsky, 1989); however, to understand structural properties, we need to employ a different approach. Since the form factors in the empirical pseudopotential have been fit to optical transitions, there is no reason to believe that they will be very accurate for structural properties such as the phase stability, the equilibrium bond length, and the bulk modulus of a crystal. Pseudopotentials constructed within density functional theory can serve this purpose. Evaluating the total energy of a crystal can be a difficult task. The total energy of the system contains terms that individually diverge, e.g., the repulsive Coulomb terms between the ion cores and the electron-electron interactions. These terms can be handled in Fourier or momentum space using a plane wave basis (Ihm et al., 1979). The total energy of the system can be written as ET ¼ Ekin + Eec + EH + Exc + Ecc :

(33)

ET is the total electronic energy, Ekin represents the electronic kinetic energy, the electron-ion core interaction energy is given by Eec, the electrostatic coulomb energy is given by the Hartree energy, EH, the non-classical exchange-correlation energy is given by Exc and the ion core-ion core classical term is given by Ecc. The momentum-space expressions for the electronic energy terms are as follows:

Ekin

! ! 2 2 !  2 ℏ k + G 1 X p
! , ¼ f k + G N ! ! n 2m n, k , G

(34)

Pseudopotential methods Eec ¼


! !  ! 1 X p ∗
! fn k + G fpn k + G 0 N ! ! 0n, k , G 1 Z
! ! !  1 Ze2 3 A p ! 0 @ d r ,  Vc k + G ; k + G + dG!, G!0 Oc r EH ¼

845

(35)


! Oc X 4pe2 2 2 jr G j , 2 ! G

(36)



! Oc X ∗ ! r G exc G , 2 !

(37)

G,G6¼0

Exc ¼

G

8 2

= P 6 erfc  R − t s + t s0 7 2 ! − pffiffiffi ds, s0 : + 5 4 ! ! > p ! ; R − t s + t s0 R , R6¼0 !

(38)

!

Each term yields the energy per cell; N is the total number of cells, Oc is the cell volume,R is a lattice vector,G is a reciprocal !

!

lattice vector, Z is the total core charge, which is a sum of the individual core charges Zv, t is the basis vector, k is the crystal
!  ! ! ! ! 0


! ! momentum and fpn is wavefunction for state n. r G V pc k + G, k + G , and fpn k + G represent the Fourier components of the charge, ion core potential and the pseudo-wave function. The sums involving the pseudo-wave functions run only over the occupied states.  is parameter that controls the convergence of the Ewald summation in real space versus momentum space (Ihm et al., 1979). In practice, it is easier to make use of the eigenvalue explicitly and subtract off the “double counting terms,” i.e., one can write the total energy as
! 1X ET ¼ E k −EH + DExc + Ecc : (39) N ! n n, k

The sum is over all occupied states. The eigenvalue term contains the exchange-correlation term whereas the total energy should contain the exchange-correlation energy. This term can be written as
! i X
! h
! exc G − Vxc G , (40) r∗ G DExc ¼ Oc !

G

where exc is related to the exchange-correlation via a functional derivative: DV xc ¼

dðr exc ½rÞ , dr

In the local density approximation (Kohn and Sham, 1965) Z
 h
i ! ! d3 r: Exc ½r ¼ r r exc r r

(41)

(42)

In the simplest formalism, exc is extracted from a homogeneous electron gas (Kohn and Sham, 1965). The total energy, ET, for a crystal structure can be calculated using Eq. (39) as a function of the lattice constant and any internal structural parameters. This calculation can be used to find the equation of state for a given phase (Yin and Cohen, 1980). This approach represents a seminal point in condensed matter physics, e.g. it demonstrated a workable scheme for examining structural properties of solids and its extension led to molecular dynamics using quantum forces (Car and Parrinello, 1985). The first application of phase stability using pseudopotential-density functional theory centered on a number of silicon crystal structures: face centered cubic, body centered cubic, hexagonal close packed, hexagonal diamond, cubic diamond, white tin and simple cubic. For each structure, the energy was minimized for a given volume by optimizing the internal coordinates, e.g., for a hexagonal structure the c/a ratio was optimized. Depending on the system studied the range of volumes considered is varied.

846

Pseudopotential methods

Fig. 9 Total electronic energy for seven phases of crystalline silicon as function of the atomic volume (Yin and Cohen, 1980). The volume has been normalized to the measure value of diamond. The dashed line is a common tangent of the energy curves for the diamond and b-tin phases. The total energy reference is the energy per pseudo Si atom.

The results are impressive when one considers that this is a calculation requiring only the atomic number and crystal structure as input. Generally, the pseudopotential-density functional method yields lattice constants and bulk moduli to an accuracy of about 1% and 5%, respectively. The cohesive energy can be calculated by comparing the total energy at the equilibrium lattice constant with the energy of the isolated atoms. Spin polarization effects (von Barth and Hedin, 1972; Gunnarsson et al., 1974) and zero point vibrational energy need to be considered. Early results computed within the local density approximation gave reasonable estimates of the cohesive energies; however, in general the generalized gradient approximation yields more accurate cohesive energies (Becke, 1992). The total energy versus volume curves for seven crystal phases of silicon and are shown in Fig. 9. The lowest energy state corresponds to the diamond structure, which is observed experimentally under ambient conditions. As with any such calculation, there is no guarantee that there is an untried structure with a lower energy state. For small volumes other structural phases are lower in energy than the diamond phase, hence pressure induced solid-solid structural phase transitions are predicted. As hydrostatic pressure is applied the path indicated in the figures, 1 ! 2 ! 3 ! 4, illustrates the change from diamond at 1 to b-tin at 4, with the transformation occurring along the 2 ! 3 segment of the path where both structures can coexist. The predicted phase transition pressure for the diamond ! b-tin phase is 99 kbar; the measured values are 125 and 100 kbar, respectively (Yin and Cohen, 1982). The successful prediction of high pressure structural phases such as hexagonal forms of Si and Ge and fcc Si are one of the impressive results of the pseudopotential-density functional theory calculations (Chang and Cohen, 1984). Moreover, these high pressure phases are metallic and predicted to be superconductors (Chang et al., 1985). Since these calculations, total energy computations using pseudopotentials constructed within density functional theory have become routine for a wide variety of materials. As with any computational approach, an accurate solution is limited by the least accurate approximation. In the case of density functional pseudopotentials, the weakest aspect centers on the nature of the exchange-correlation functional. For example, computations for highly correlated systems are likely to fail. A pseudopotential constructed within a given functional will accurately reproduce a solution of the functional, but if the functional is flawed the pseudopotential will reflect his flaw. In contrast to density functional theory, the accuracy of the pseudopotential can be validated by replacing it with an all electron potential and assessing any differences. What cannot be done is to take an approximate density functional and replace it with an exact functional, as an exact functional does not exist, save in limiting cases like a homogeneous electron gas.

Conclusion The pseudopotential concept has had a profound impact on our understanding of the electronic structure of semiconductors. In this chapter, both empirical and density functional pseudopotential concepts were outlined and some central applications discussed.

Pseudopotential methods

847

Empirical pseudopotentials provided a means for understanding the optical and dielectric properties of semiconductors and the underlying energy band structures. It is sometimes stated that the empirical pseudopotential method (EPM) established the validity of the energy band concept for solids in general. There is truth to this statement. In the 1950s, there were extended discussions about the validity of the one electron picture and whether it could be applied to solids and specifically to semiconductors such as silicon. Some workers thought that the electron-hole interactions for a optical excitation would be so strong as to obscure any attempt at understanding such excitations using energy band theory. Given that the EPM allowed adjustments to the one electron potential, if the EPM had failed to work with the flexibility of choosing a potential, one would have questioned whether it was possible to construct a meaningful one-electron potential and corresponding energy band structure. The consequences of this failure would have been very damaging to prospects for using energy bands to interpret optical, dielectric, photoemission, and transport properties of semiconductors. Fortunately, this was not the case, the EPM provided a simple and effective way to couple band structure theory to experimental work and its impact was immediate. While much of this effort took place in the 1960s and 1970s, the EPM still serves as an effective means to interpret optical data for semiconductors. While more sophisticated methods now exist to examine optical properties, few of the conclusions of the EPM have been overturned by these methods. The coupling of density functional theory with pseudopotentials in the early 1980s resulted in another impressive advance for understanding the electronic structure of materials. The accuracy of density functional theory for predicting electronic and structural energies was problematic at that time. However, the direct numerical application to structural energies to problems involving bond lengths, compressibilities, phonon modes, and phase stabilities showed conclusively the applicability of pseudopotential methods to these problems. Over the last 10 years, more than 10,000 papers have appeared with pseudopotentials in the title or abstract and these papers have been cited over 100,000 times. The future of pseudopotentials constructed within density functional theory clearly depend on advances in creating more accurate functionals. One proposed advance centers on mixing exact exchange with local density functionals (Perdew et al., 1996). These so called hybrid functionals should capture nonlocal contributions and result in a more accurate description of the many electron problem. Of course, such approaches can be computationally complex to implement. As these difficulties are overcome (Boffi et al., 2016), pseudopotentials constructed within hybrid functionals will become more commonplace. The vitality and future of the pseudopotential method is without question. The method is now used to examine new and exciting materials systems. These systems include nanoscale systems, amorphous solids, glasses, and liquids. Moreover, owing to strong advances in both hardware and software (algorithms) it is now possible to address systems with thousands of atoms on an almost routine basis (Liou et al., 2021).

References Animalu AOE and Heine V (1965) The screened model for 25 elements. Philosophical Magazine 12: 1249. Bachelet G, Hamann DR, and Schlüter M (1982) Pseudopotentials that work: From hydrogen to plutonium. Physical Review B 26: 4199. Becke AD (1992) Density-functional thermochemistry 1. The effect of the exchange-only gradient correction. The Journal of Chemical Physics 96:2155. Boffi NM, Jain M, and Natan A (2016) Efficient computation of the Hartree–Fock exchange in real-space with projection operators. Journal of Chemical Theory and Computation 12: 3614. Broyden CG (1965) A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation 19: 577. Car R and Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Physical Review Letters 55: 2471. Ceperley DM and Alder BJ (1980) Ground state of the electron gas by a stochastic method. Physical Review Letters 45: 566. Chang KJ and Cohen ML (1984) Structural and electronic properties of the high-pressure hexagonal phases of silicon. Physical Review B 30: 5376. Chang KJ, Dacorogna MM, Cohen ML, Mignot JM, Chouteau G, and Martinez G (1985) Superconductivity in high-pressure metallic phases of Si. Physical Review Letters 54: 2375. Chelikowsky JR and Cohen ML (1992) Ab initio pseudopotentials for semiconductors. In: Moss TS and Landsberg PT (eds.) Handbook of Semiconductors, p. 59. Amsterdam: Elsevier. Chelikowsky JR, Troullier N, and Saad Y (1994) The finite-difference-pseudopotential method: Electronic structure calculations without a basis. Physical Review Letters 72: 1240. Cohen M and Chelikowsky J (1982) Pseudopotentials for semiconductors. In: Paul W (ed.) Handbook on Semiconductors, vol. 1, p. 219. Amsterdam, North Holland. Cohen ML and Bergstresser TK (1965) Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Physics Review 141: 789. Cohen ML and Chelikowsky JR (1989) Electronic Structure and Optical Properties of Semiconductors, 2nd edn. Berlin: Springer-Verlag. Cohen ML and Heine V (1970) The fitting of pseudopotentials to experimental data and their subsequent application. In: Ehrenreich H, Seitz F, and Turnbull D (eds.) Solid State Physics. vol. 24, p. 37. New York: Academic Press. Dirac PAM (1929) Quantum mechanics of many elecrton systems. Proceed Royal Soc London A 123: 714. Fermi E (1934) Sullo spostamento per pressione dei termini elevati delle serie spettrali (on the pressure displacement of higher terms in spectral series). Nuovo Cimento 11: 157. Gunnarsson O, Lundqvist BI, and Wilkins JW (1974) Cohesive energy of simple metals. Spin-dependent effect. Phys Rev B 10: 1319. Hamann DR (1989) Generalized norm-conserving pseudopotentials. Physical Review B 40: 2980. https://doi.org/10.1103/PhysRevB.40.2980. Hamann DR, Schlüter M, and Chiang C (1979) Norm-conserving pseudopotentials. Physical Review Letters 43: 1494. Hellman H (1935) A new approximation method in the problem of many electrons. Journal of Chemical Physics 3: 61. Herring C (1940) A new method for calculating wave functions in crystals. Physics Review 57: 1169. https://doi.org/10.1103/PhysRev.57.1169. Hohenberg P and Kohn W (1964) Inhomogeneous electron gas. Physics Review 136: B864. https://doi.org/10.1103/PhysRev.136.B864. Ihm J, Zunger A, and Cohen ML (1979) Momentum-space formalism for the total energy of solid. Journal of Physics C 12: 4409. Kerker GP (1980) Nonsingular atomic pseudopotentials for solid state applications. Journal of Physics C 13: L189. Kittel C (2005) Introduction to Solid State Physics, 8th edn. New York: Wiley. Kleinman L and Bylander DM (1982) Efficacious form for model pseudopotentials. Physical Review Letters 48: 1425. Kleinman L and Phillips JC (1960a) Crystal potential and energy bands of semiconductors. II. Self-Consistent calculations for cubic boron nitride. Physics Review 117: 460. https://doi. org/10.1103/PhysRev.117.460.

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Kleinman L and Phillips JC (1960b) Crystal potential and energy bands of semiconductors. III. Self-Consistent calculations for silicon. Physics Review 118: 1153. https://doi.org/ 10.1103/PhysRev.118.1153. Kohn W and Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Physical Review A 140: 1133. https://doi.org/10.1103/PhysRev.140.A1133. Liou KH, Biller A, Kronik L, and Chelikowsky J (2021) Space-filling curves for real-space electronic structure calculations. Journal of Chemical Theory and Computation 17: 4039. Perdew JP, Ernzerhof M, and Burke K (1996) Rationale for mixing exact exchange with density functional approximations. The Journal of Chemical Physics 105: 9982. Philipp HR and Ehrenreich H (1963) Optical properties of semiconductors. Physics Review 129: 1550. Phillips JC and Kleinman L (1959) New method for calculating wave functions in crystals and molecules. Physics Review 116: 287. Phillips JC and Kleinman L (1962) Crystal potential and energy bands of semiconductors. IV. Exchange and correlation. Physics Review 128: 2098. https://doi.org/10.1103/ PhysRev.128.2098. Saad Y, Chelikowsky J, and Shontz S (2010) Numerical methods for electronic structure calculations of materials. SIAM Review 52: 3. Schwarz W, Andraea D, Arnold S, Heidberg J, Hellmann H Jr, Hinze J, Karachalios A, Kovner M, Schmidtand P, Zülicke L, and Hans GA (1999a) Hellmann (1903-1938): Part I. A pioneer of quantum chemistry. Bunsen-Magazin 1: 10. Schwarz W, Andraea D, Arnold S, Heidberg J, Hellmann H Jr, Hinze J, Karachalios A, Kovner M, Schmidtand P, Zülicke L, and Hans GA (1999b) Hellmann (1903-1938): Part II. A German pioneer of quantum chemistry in Moscow. Bunsen-Magazin 2: 60. Troullier N and Martins J (1991) Efficient pseudopotentials for plane-wave calculations. Physical Review B 43: 1993. Vanderbilt D (1985) Optimally smooth norm-conserving pseudopotentials. Physical Review B 32: 8412. Vanderbilt D (1990) Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Physical Review B 41: 7892. von Barth U and Hedin L (1972) Local exchange-correlation potential for the spin-polarized case. I. Journal of Physics C 5: 1629. Yin MT and Cohen ML (1980) Microscopic theory of the phase transformation and lattice dynamics of Si. Physical Review Letters 45: 1004. Yin MT and Cohen ML (1982) Ab initio calculation of the phonon dispersion relation: Application to silicon. Physical Review B 25: 4317.

Effective masses☆ Jeanlex Soares de Sousa, Departamento de Física, Universidade Federal do Ceará, Fortaleza, Brazil © 2024 Elsevier Ltd. All rights reserved. This is an update of A.L. Wasserman, Effective Masses, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 1–5, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00457-5.

Introduction Electron dynamics Band curvature and types of charge carriers Parabolic band approximation Effective mass theory Heterostructures Density of states and carriers concentration Two-dimensional materials Measurement methods Conclusion References

849 850 850 851 852 852 853 854 854 855 855

Abstract The effective mass of a particle in an energy band is its inertia due to the interaction with the lattice during its motion through the crystal. Effective masses are important quantities to parameterize the band structure of materials and their electronic properties like the intrinsic carrier concentrations as function of temperature, electrical conductivity, speed of integrated circuits or the efficiency of solar cells. This chapter aims to review the basic principles and quantum theory behind the concept of effective mass in solid state physics.

Key points

• • • • •

Parabolic band approximation Effective mass theory Envelope function approximation Density-of-states effective mass Effective mass in Dirac cones

Introduction Electrons move very differently in a periodic lattice than when they are free in space. In vacuum, the electron mass is a fundamental constant of nature given by m0 ¼ 9, 66  10−31 kg. In crystalline solids, electrons sense the environment and interact with (i) the lattice atoms, (ii) other electrons moving around the lattice, and (iii) with vibrations of the lattice. As a consequence of all these interactions, electrons move as if their mass was different from m0. This is the most fundamental idea behind the concept of effective masses. The well-known single particle energy of free electrons is
! ℏ2 k2 E k ¼ , 2mo

(1)

!

where k is the electron wave vector (Ashcroft and Mermin, 1976). In a crystal close to an energy band extremum, the single particle energy of electrons including the effect of all interactions with the surroundings can be described as
! ℏ2 k2 E k ¼ , 2m ∗

(2)

☆ Change History: September 2022. JS de Sousa updated sections Abstract, Introduction, Electron dynamics, Band curvature and types of charge carriers, Parabolic band approximation, Effective mass theory, Heterostructures, Density of states and carriers concentration, Two-dimensional materials, Measurement methods, Conclusions and 2 figures.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00089-5

849

850

Effective masses

where m∗ is the effective mass of the electron in the crystal. Such a surprisingly simple result is successful in describing many physical phenomena in crystalline condensed matter. In fact, effective masses provide a very useful and powerful way to parameterize solid state materials like semiconductors and insulators.

Electron dynamics !

In a crystalline solid, electrons occupy electronic states characterized by their wave vector k and a band index n. The energy in any
!
! band is a quasi-continuous function En k , and the ensemble of functions En k constitutes the energy band structure of a given

 !! ! ! crystal. Due to the periodicity of the lattice, a quantum state may be described by a Block function c ! r ¼ u ! r ei k  r , n, k n, k h
i

!

 2 ! ! ! ! that satisfies the Schrödinger equation b p =2m + V r c ! r ¼ En k c ! r , where V r is the periodic potential energy n, k

n, k

created by the crystalline lattice.
 ! As the electronic states c ! r extend over the whole crystal with the same periodicity as the lattice, it may not be very insightful n, k to describe the dynamics of electrons. Instead, one constructs a moving wavepacket localized in space, whose group velocity is given by
! 1 ! (3) v n ¼ rk En k , ℏ
 ! where the index n indicates that the wavepacket was constructed with c ! r states belonging to a single energy band (it will be n, k dropped in the following analysis). The acceleration of this wavepacket is then given by:
! ! ! !
!
!
! d ℏ k dv 1 d 1 dk 1 ! a¼ ¼ r rk E k ¼ 2 HE k , (4) rE k ¼ ℏ dt k ℏ dt k dt dt ℏ
!

! ði, jÞ ! whose elements are given by HE k ¼ ∂2 E k =∂ki ∂kj : An external force acting on the where HE k is the Hessian
matrix, ! ! ! ! wavepacket is given byF ¼ d ℏ k =dt, where p ¼ ℏ k is the electron momentum. Comparing Eq. (4) with Newton’s second law ! ! a ¼ F =m, one obtains that the electron mass is replaced by an effective mass tensor m∗, whose elements are given by: 0
!1 −1 ∂2 E k C B1 ∗ mi,j ¼ @ 2 2 (5) A : ℏ ℏ ∂ki ∂kj Eq. (5) indicates that the effective mass is inversely proportional to the curvature of the band structure. Large curvatures imply small effective masses, and vice-versa. The importance of this result is paramount: It indicates that despite the complexity of the multiple interactions undergone by electrons across the crystal, they all can be accounted for by a single parameter in their dynamics. The tensor nature of the effective mass indicates that materials can be anisotropic, i.e., their properties depend on orientation of the material’s body.

Band curvature and types of charge carriers !

In a typical band structure, shown in Fig. 1, there are many energy bands for which curvatures vary as function of k . Hence, near the bottom of any band the curvature is positive, whereas near the top of any band the curvature is negative. This indicates that the effective masses can be either positive or negative.

EG direct gap

E(k)

indirect gap EG

Conduction band Heavy-hole band Light-hole band Spin-orbit band

K

*

Parabolic approx. region

L

wave vector Fig. 1 Schematics of a semiconductor energy band structure.

W

X

Effective masses

851

In pure dielectric materials at T ¼ 0 K, there are just enough electrons to fill the valence band, and no electrons to occupy the conduction band. The band gap EG provides a primary indication of the metallic, semiconductor, or insulator character of materials. EG  3.5 eV are representative of insulators, EG  0 eV are found in metallic materials and EG ranging between 0.3 eV and 3.5 eV are typical of semiconductor materials. One important feature of the band structure of real materials is that the maximum of the valence ! band (VBM) and the minimum of the conduction band (CBM) may be located at distinct k points in the reciprocal space. If so, they are called indirect gap materials, otherwise they are called direct gap materials. At finite temperatures, some of those electrons occupying the top of the valence band are transferred across the band gap to the bottom of the conduction band, leaving unoccupied states behind in the valence band. Particles occupying conduction band states exhibit positive effective mass, and are called electrons. On the other hand, particle occupying valence band states are called holes, and exhibit negative effective mass. This means that holes act as if they were positively charged and are accelerated by an external electric field in the reverse direction expected for electrons. Usually, the top of the valence band is degenerate exhibiting energy bands with different curvatures, giving rise to the heavy-hole (smaller curvature) and light-hole (larger curvature bands). There is even a third type of hole band called spin-orbit split-off band, which arises from the spin-orbit interaction. In general, the effective masses of electrons and holes are different.

Parabolic band approximation


! Near band edges, like the top of the valence ! band and/or bottom of the conduction band, E k can be expanded into Taylor series. For simplicity, we temporarily assume that k is a scalar     ðk −k0 Þ2 d2 E dE + O ðk −k0 Þ3 , EðkÞ ¼ Eðk0 Þ + ðk −k0 Þ  + (6) 2 dk k0 dk2 k0 where k0 is the position of the band edge in the reciprocal space. We keep in mind that the energy bands at the band edges are extrema ([dE(k)/dk]k0 ¼ 0), and that the energy states above/below few kBT of the band edges are hardly occupied at finite temperatures. We can disregard high order terms O[(k−k0)3] in the above expansion to obtain the well-known parabolic band approximation  ðk −k0 Þ2 d2 E ℏ 2 ðk −k0 Þ2 ¼ , (7) EðkÞ −Eðk0 Þ ¼  2 2 2m ∗ dk k0 where we used the definition of the effective mass in terms of the band curvature near k ¼ k0. One should note that the parabolic band approximation is similar to the energy dispersion of the free electrons replacing m0 by m∗ in Eq. (1). Recovering the vector ! form of k , the parabolic band approximation becomes:
!
! 


! !  ℏ2 ! ! 1 k −k0  ∗  k −k0 , E k ¼ E k0 + 2 m

(8)

where the effective mass is written in the tensor form whose elements are given by Eq. (5). We are now in position to describe some few cases of energy bands in terms of effective
! masses.  The simplest possible case is for isotropic direct gap materials (e.g., GaAs, InAs) whose band edges are located at G point k 0 ¼ 0 in the Brillouin zone. In this case, the conduction, light-hole, heavy hole, and split-off bands are respectively given by:
! ℏ2 k2 Ec k ¼ Ec ð0Þ + , 2mc∗

(9)


! ℏ2 k2 Elh k ¼ Ev ð0Þ + , 2mlh∗

(10)


! ℏ2 k2 : Ehh k ¼ Ev ð0Þ + ∗ 2mhh

(11)


! ℏ 2 k2 : Eso k ¼ Ev ð0Þ + 2mso∗

(12)

In the above equations, m∗e , m∗hh, m∗lh and m∗SO represent the effective masses of electrons, heavy-holes, light-holes and split-off band, respectively. D0 is the spin–orbit split-off energy, and the band gap is given by EG ¼ Ec(0) − Ev(0). For indirect gap materials (e.g., Si, Ge), the VBM is located at the G point, but the CBM is located away from the Brillouin zone center. In this case, the light-hole and heavy-hole bands are considered reasonably isotropic, and can be described in the same way of Eqs. (10) and (11). However, the CBM is degenerated, highly anisotropic and can be written as

!
!  ℏ2 k2 ℏ2 k2y’ + k2z’ x’ Ec k ¼ Ec k 0 + + (13) 2mt∗ 2ml∗ !

where k 0 is located near the X of the Brillouin zone for Si, and at L point for Ge. Those are symmetric points with six degenerated equivalent directions. kx0 is measured from the CBM and it is oriented along the GX direction for Si, and along GL direction for Ge.

852

Effective masses Table 1

Effective masses of some of the most common semiconductors.

Material

m∗e /m0

m∗hh/m0

m∗lh/m0

m∗SO/m0

Si Ge C GaAs InAs InP GaP GaSb InSb

0.91 (0.19) 1.58 (0.082) 1.40 (0.36) 0.063 0.023 0.008 1.12 (0.22) 0.041 0.014

0.49 0.33 2.12 0.51 0.41 0.60 0.79 0.40 0.43

0.16 0.043 0.7 0.082 0.026 0.089 0.14 0.05 0.015

0.24 0.084 1.06 0.15 0.16 0.17 0.83 0.14 0.19

ky0 ,z0 lie along perpendicular directions with respect! to kx0 . m∗l and m∗t refer to longitudinal and transverse effective masses with respect to direction connecting G point and the position k 0 of the CBM. Table 1 lists the effective masses of some of the most common elemental and compound semiconductors.

Effective mass theory Although we have described the concept of effective masses phenomenologically so far, there is a solid theoretical


!
 ! ! ! framework behind it. Consider the general Schrödinger equation for a Bloch state H r C ! r ¼ En k C ! r , where n, k n, k

 !!
 ! ! ! c ! r ¼ u ! r ei k  r : Written in terms ofu ! r , it becomes n, k

n, k

n, k



 
 
!
 ℏ2 ! ! ℏ2 k2 ! ! u ! r : k  p u ! r ¼ En k − n, k n, k m0 2m0

H0 +

(14)




! ! ! p2 where H0 ¼ 2m + V r is the unperturbed crystal Hamiltonian, and H1 ¼ ℏ2 =m0 k  p is a perturbation. The unperturbed 0

 ! ! problem obeys H0 un,0 r ¼ En ð0Þun,0 r : The time-independent perturbation theory gives the energy to 2nd order in perturbation: ! ! 2  
! 2 2 2 X  k  p n,n0  ℏ k ℏ En k ¼ En ð0Þ + , + 2 2m0 m0 n0 6¼n En ð0Þ −En0 ð0Þ

(15)

D E
! !! ! ! where p n,m ¼ un,0 j p jum,0 is the momentum matrix element. The first order correction Eðn1Þ k ¼ ðℏ=m0 Þk  p n,n vanishes because !

p n,n ¼ 0 due to symmetry considerations. This equation can be written as:
!
 ℏ2 X 1 ka kb, En k ¼ En ð0Þ + 2 m ∗ ab

(16)

a,b

where


1 m∗

 ab

¼

pan,n’ pbn’ ,n 1 2 X : dab + 2 m0 m0 n’ 6¼n En ð0Þ −En’ ð0Þ

(17)

!

!

Although the above expression was developed for a band whose edge is located at k ¼ 0, it is valid for any location k 0 : The above theory is based on non-degenerated time-independent perturbation theory, and it needs modification in order to accurately include the light-hole, heavy-hole and spin-orbit bands. Further accuracy can be obtained by noticing that the functions u u

!

!

m, k 0 !

m, k 0

(for a given k 0 ) deliver a complete set of eigenfunctions. The eigenfunctions u , for a proper choice of

! k 0,

!

n, k

can be expressed as linear combination of

and the Eq. (14) can be solved using this basis set. This approach gave rise to widely used models in

the literature as Kane and Luttinger-Kohn models (Chuang, 1995; Kane, 1966; Luttinger and Kohn, 1956).

Heterostructures One of the most important consequences of the effective mass theory is the envelope function approximation to study the electronic properties of heterostructures (e.g., quantum wells, wires and dots). In this approach, the single band effective Schrödinger equation for the whole heterostructure is written as (Bastard, 1988; Burt, 1992):

Effective masses 3 0 1


 2 1 A ! 5 ! 4 − ℏ r@
 r+U r F ! r ¼ EF r ! 2 ∗ m r

853

2

(18)


 ! where U r is the heterostructure potential constructed from the alignment of band gaps at the interfaces of different materials
 ! and/or an external potentials, m ∗ r is the position-dependent effective mass, E is measured from the band edge of the quantum
 ! well (QW) region, and F r is known as envelope wavefunction. In the envelope function approximation, the atomistic description of the different materials in the heterostructure are embedded in their effective masses. The approximate atomistic


 ! ! ! ! wavefunction is linked to the envelope wavefunction as c r ¼ F r u ! r ,where k 0 is the position of the band edge for which n, k 0

the envelope function description is being constructed. The presence of boundaries between materials demands the application of
 h
i −1
 ! ! ! rF r at the interfaces between continuity boundary conditions in the envelope function F r and in the quantity m ∗ r materials. Fig. 2 shows an example of how the alignment of band structures in the assembly of a QW heterostructure gives rise to a position-dependent effective mass. If the QW is thin enough (up to few tens of nanometers), discrete energy states in the conduction and valence bands appear in the QW region in the direction of confinement. Assuming infinite confinement barriers, those energy states measured from the bottom of their respective band edges, are given by: Evn ¼

ℏ 2 p2 2 n : 2mn∗ L2

(19)

where n ¼ e, hh, lh, and L is the QW width.

Density of states and carriers concentration The concentration of mobile carriers in the conduction (n) and valence ( p) bands are given by: Z 1 n¼ gc ðEÞf ðEÞdE,

(20)

Ec

Energy

quantum confined states in the CB of the QW region

CB misalignment

Position quantum confined states in the VB of the QW region

VB misalignment

barrier material

Quantum well material

Fig. 2 Schematics of band alignment and position-dependent effective mass in heterostructures.

barrier material

854

Effective masses Z p¼

Ev −1

gv ðEÞ½1 −f ðEÞdE,

(21)

where gc,v(E) are the density of states in the conduction and valence bands, respectively. f(E) ¼ 1/1 + e−(E−Ef)/eBT is the Fermi-Dirac distribution, Ef is the Fermi energy, kB is the Boltzmann constant, and T the absolute temperature. The density of states gn(E) (n ¼ c, v) represents the number of states per unit volume per unit energy in n band, and it is obtained by counting all states in the reciprocal space for each energy band. Assuming the parabolic band approximation, the respective densities of states (DOS) are given by: g c ðEÞ ¼

1 2p2

gv ðEÞ ¼

1 2p2

 ∗ 3=2 pffiffiffiffiffiffiffiffiffiffiffi 2me E −Ec ðE > Ec Þ, ℏ2 

2mh∗ ℏ2

3=2

pffiffiffiffiffiffiffiffiffiffiffi Ev −E ðE < Ev Þ:

(22) (23)

For anisotropic materials, m∗e and mh may be slightly different from the values obtained from the curvatures of the band structure. For Si, for example, the DOS effective mass becomes m∗e ¼ (mlm2t )1/3. Solving Eqs. (20) and (21) for nondegenerate semiconductors i.e., (Ec − Ef)  3kBT and (Ef − Ev)  3kBT, one obtains n ¼ Nc e −ðEc −Ef Þ=kB T ,

(24)

p ¼ N v e −ðEv −Ef Þ=kB T :

(25)

Nn (n ¼ c, v) is known as the effective density of states in each band, and they are given by:  ∗ 3=2 mv k B T Nv ¼ 2 2pℏ2

(26)

In intrinsic (pure) materials, for every electron in the conduction band there is a hole in the valence band. This equilibrium condition is commonly written as n ¼ p ¼ ni, where ni is the intrinsic carrier concentration. Applying the condition np ¼ n2i (that holds even for doped materials), one obtains: pffiffiffiffiffiffiffiffiffiffiffiffi ni ¼ Nc N v e −Eg =2kB T : (27) The above quantities provide a drastically simplified method to parameterize carriers concentrations of solid state materials in terms of their effective masses, energy gap and temperature.

Two-dimensional materials The recent discovery of graphene and other two dimensional (2D) materials (e.g., phosphorene, transition metal dichalcogenides, boron nitride just to mention a few) motivated huge efforts to investigate their electronic properties and to develop practical applications with those materials. Nowadays, there is a plethora of reported 2D materials with a wide variety of properties, ranging from metallic to semiconductors (Miró et al., 2014). In 2D materials, the definition of effective masses in terms of band curvature (Eq. 5) still holds with a few
! important differences.  The most intriguing difference is found in graphene, whose band structure exhibit linear dispersion, i.e., E k ∝ k near the Fermi energy forming the so called Dirac cones, for which the absence of band curvature clearly results in an infinite effective mass. On the other hand, electrons in graphene are said to be massless in a clear contradiction with the conventional solid state definition of effective mass. This confusion is explained if we consider electrons in graphene as relativistic particles, for which the energy-momentum relationship is given by E2 ¼ (pc)2 + (m0c2)2 ¼ (ħkc)2 + (m0c2)2, where m0 is the rest mass of the particle. For massless particle we simply obtain E ¼  ħkc. Such a linear energy dispersion of massless relativistic is similar to the band structure of graphene at low energies. In practice, cyclotron resonance measurements have shown that the electrons in graphene are not massless (Novoselov et al., 2005), and the electronics properties of graphene are better described by Dirac equation rather than Schrödinger equation. Thus, the alternative definition of effective mass for materials exhibiting linear band structure is (Ariel and Natan, 2012)  −1 dEðkÞ m ∗ ¼ ℏ2 k : (28) dk As for 2D materials with non-diverging effective masses defined by Eq. (5), the most important difference is the absence of heavy-hole, light-hole and spin-orbit split-off bands as compared to three-dimensional materials.

Measurement methods Effective masses have been traditionally measured using cyclotron resonance (Dresselhaus et al., 1955). In this method, the microwave absorption of an electron moving in an orbit perpendicular to a static magnetic field exhibits a peak at oc ¼ meB∗ ,

Effective masses

855

where e is the electron charge, and B
!isthe magnetic field intensity. The effective mass can be also determined by methods that directly measure the band structure E k like low energy angular-resolved photoemission spectroscopy (ARPES) (Damascelli, 2004). Measurements of specific heat Cv in Fermi systems can be also used to determine the effective mass. At very low temperatures, the phonons contribution to Cv becomes negligible, and Cv becomes dominated by the electronic contribution. At this regime, one has Cv ¼ p2k2Bg (Ef)T/3, where g(Ef) is the density of states at the Fermi energy (Ashcroft and Mermin, 1976). Using Eq. (22), one obtains Cv ¼ k2B kfm∗e T/3ħ2, where kf is the radius of the Fermi sphere in the reciprocal space. However, thermodynamic experiments are not able to distinguish directional effective masses, and the values estimated from Cv represent the DOS effective mass. Effective masses are also helpful for the analysis of other experiments associated to electronic properties:

•

The electrical conductivity s is given by s¼

ne2 t , me∗

(29)

where n is the density of mobile electrons, t is a transport relaxation time. An electrical conductivity for holes can also be determined by replacing m∗e by m∗hand n by p, where p is the density of mobile holes. In the most general case where electrons and holes contribute to the charge transport, the total conductivity is s ¼ se + sh.For anisotropic materials, the electrical conductivity assumes a tensor form. • Plasma oscillation is a collective excitation of all available mobile electrons whose oscillation frequency (also known as plasma frequency) is: o2p ¼

ne2 me∗ e

(30)

where e is the electron charge, and e is the dielectric permittivity of the material. • Excitons are fundamental electronic excitations associated to bound states of e-h pairs acting as a hydrogen-like atom. The exciton binding energy is given by: Eb ¼

m e20 Ry ðn ¼ 1, 2, 3, . . .Þ m0 e2 n2

(31)

where Ry ¼ 13.6 eV is the Rydberg energy, and m ¼ m∗e m∗h/(m∗e + m∗h) is the reduced effective mass of the exciton.

Conclusion In conclusion, effective mass is a cornerstone concept of condensed matter physics whose importance and applicability is far greater than what it is possible to cover in such a short chapter. The beauty of the effective mass theory relies on the fact that it provides a rather simple method to describe materials in terms of a reduced number of parameters (e.g., band gap energy and band curvatures at high symmetry points of the Brillouin zone). Perhaps its most important practical consequence is the envelop function approximation that allows to accurately model the electronic and transport properties of quantum heterostructures, which are ubiquitous in modern technologies.

References Ariel V and Natan A (2012) Electron effective mass in graphene. Ashcroft NW and Mermin ND (1976) Solid State Physics. Saunders College. Bastard G (1988) Wave Mechanics Applied to Semiconductor Heterostructures, Monographies de physique. Les Editions de Physique. Burt MG (1992) The justification for applying the effective-mass approximation to microstructures. Journal of Physics: Condensed Matter 4(32): 6651–6690. Chuang SL (1995) Physics of Optoelectronic Devices. New York: Wiley. Damascelli A (2004) Probing the electronic structure of complex systems by ARPES. Physica Scripta T109: 61. Dresselhaus G, Kip AF, and Kittel C (1955) Cyclotron resonance of electrons and holes in silicon and germanium crystals. Physical Review 98(2): 368–384. Kane EO (1966) The k.p method. In: Willardson RK and Beer AC (eds.) Semiconductor and Semimetals. vol. 1, pp. 193–217. New York: Academic Press. Luttinger JM and Kohn W (1956) Quantum theory of cyclotron resonance in semiconductors: General theory. Physics Review 102: 1030–1041. Miró P, Audiffred M, and Heine T (2014) An atlas of two-dimensional materials. Chemical Society Reviews 43(18): 6537–6554. Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV, and Firsov AA (2005) Two-dimensional gas of massless dirac fermions in graphene. Nature 438(7065): 197–200.

Electronic structure: Impurity and defect states in insulators☆ AM Stoneham, J Strand, and AL Shluger, Department of Physics and Astronomy, University College London, London, United Kingdom © 2024 Elsevier Ltd. All rights reserved. This is an update of A.M. Stoneham, A.L. Shluger, Insulators, Impurity and Defect States in, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 380–388, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00471-X.

Introduction Theoretical methods for defect electronic structure Coupling of electrons and lattice deformation Self-trapping and localization Defects in disordered materials Surfaces and interfaces Conclusion References

856 858 860 862 863 863 865 865

Abstract Defects in insulators control properties of materials and devices. We first define the main issues and describe the theoretical methods for calculating defect electronic structure. Then we describe the main point defect types and consider how defect electrons couple to the motion of nearby ions. The concepts of electron and hole localization and self-trapping of polarons and excitons are introduced next. Properties of defects in disordered systems are briefly outlined and key points concerning defects at surfaces and interfaces are discussed including the dependence of the defect charge state on the position of the system Fermi energy. Finally, a short outlook is provided.

Key points

• • • • • • •

Outline the main issues pertaining to defects in materials and defect types Define the main structure parameters of interest Outline the main theoretical models and methods used to study defect properties Introduce the electron-lattice coupling Describe intrinsic and extrinsic polaron and exciton localization and self-trapping Outline properties of defects in disordered materials Describe the main defect-induced properties of surfaces and interfaces

Introduction Serious studies of materials are often serious studies of defects, for control of properties of materials implies control of defects or impurities. Understanding the electronic structure of defects in insulators has helped the development of microelectronic and quantum devices, the photographic process, solid electrolytes, and the applied surface physics of sensors and catalyst substrates (Crawford and Slifkin, 1972; Fleetwood et al., 2009; Hayes and Stoneham, 1985; Lannoo and Bourgoin, 1981; Stoneham, 1975). Here we will focus mainly on point defects, but dislocations and grain boundaries are very important too. The understanding of defect properties is based on a synthesis of theories, modelling, and experiment. Defining the issues. Some of the common point defects are shown schematically in Fig. 1. They can be created in different charge states and undergo transformations during the lifetime of materials and devices. When a new material is created, which defects are important? Will there be vacancy-interstitial pairs (Frenkel defects) or just vacancies of several types (Schottky defects)? Will there be non-standard ionic charge states, like 3+ Fe ions in FeO? Could one species move on to the sites of another, e.g. As antisites on Ga sites in GaAs? Can the material be doped with electrically-active impurities? Are there understandable trends of properties from material to material in some class, like II-VI compounds? Electronic structure is subtle and varied, with the key ideas identified by a mix of theory and experiment. In most insulators Coulomb energies dominate, so antisite defects are not energetically easy, and charge transfer across sublattices is difficult, instead non-stoichiometry is common. Thus vacant lattice sites (vacancies), ions placed at normally unoccupied sites (interstitial ions), foreign ions present as impurity or dopant, and ions with charges different from those of the ☆

Change History: September 2022. J Strand and AL Shluger prepared the update. Figs. 1, 3, 5, 7, 8 are new. All chapters and figures have been updated.

856

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00031-7

Electronic structure: Impurity and defect states in insulators

857

Fig. 1 Schematic of basic defect types in a binary insulator AB. (A) and (C) are vacancies of atoms type A and B, respectively. (B) An impurity atom on the site of atom A. (D) and (E) are two types of interstitial configurations of atom A: (D) – self-interstitial and (E) dumbbell interstitial. A pair of vacancy and interstitial defects, such as (A) + (D) or (A) + (E), is called a Frenkel pair. A pair of vacancies of atoms A and B, such as (A) + (C), is called a Schottky pair. (F) is a pair of anti-site defects, where the atom of type B occupies the site of the atom type A and the atom of type A occupies the site of the atom type B.

host ions are prevalent defect types (Fig. 1). Electronic defects may arise either as ions with charges deviating from those on lattice ions, or from excitation of electrons from filled valence bands to empty conduction bands. When a valence-band electron is missing, one has an electron hole (hole) and its behavior can be compared with that of an electron in a normally empty conduction band. Electron and hole together can form a transient neutral species, the exciton. In the absence of macroscopic electric fields or irradiation, ionic lattices must be electrically neutral overall, implying that one charged defect must be compensated by another charged defect or defects to satisfy the electro-neutrality condition locally or over the whole sample. The charges of defects and of the regular lattice particles, defined with respect to the neutral ideal lattice, are called effective charges. Complexity is the norm for advanced materials. When solid-state lasers degrade, when mechanical properties change under irradiation, defect processes occur in parallel on timescales from femtoseconds upwards. In a composite material, defect phenomena may happen in the matrix, the fiber or at the interface. When a metal is oxidized, interfacial, diffusional and impurity processes co-exist. Separating these processes by experiment alone may not be feasible. The electronic structure theory establishes priorities, and is used to organize and analyze massive amounts of data, to identify what is novel and point to significant issues. Which electronic structure parameters are of interest? Defects control the properties and performance of materials and devices in various ways. For luminescence, whether of rare earths in oxides or sulfides or CsI:Na, the spectroscopic properties of impurities, optical absorption and luminescence energies, are of the utmost importance. Bragg grating formation in silica fibers is associated with the optical absorption of induced defects. Defect species in gate dielectrics, such as oxygen vacancies, impurities and hydrogen, influence device performance. In microelectronic devices these defects may initiate degradation, trap charge, or create fixed charge, which scatters carriers in the Si substrate. Their electron and hole trapping energies and cross-sections are of main interest. Many radiation-induced processes in insulators involve forming self-trapped excitons and holes, so the electron coupling to lattice deformation plays a crucial role. Designing qubit formation in wide band gap insulators requires deep defect states which can be initialized, manipulated, and measured with high fidelity at room temperature. Among other criteria, they should possess optical transitions that do not introduce interference from the electronic states of the host. In photovoltaic materials, except for selected dopants, defects often have detrimental effects and should be eliminated to minimize charge recombination. On the other hand, in photocatalysis defects often play an active role by stabilizing charge separation and mediating rate-limiting catalytic steps. Experimental defect characterization is vital for our understanding which defects influence the optical, magnetic and electrical properties of materials. A broad range of characterization techniques includes vibrational spectroscopy, electron paramagnetic resonance (EPR), optical techniques, e.g. photoluminescence spectroscopy, and electrical techniques, such as deep-level transient spectroscopy (DLTS), as well as optoelectrical techniques, e.g. photocurrent spectroscopy (Bassani and Pastori Parravicini, 1975; Fleetwood et al., 2009; Spaeth and Overhof, 2003; Toyozawa, 2003; Weil and Bolton, 2007). Each method has its advantages and disadvantages. Therefore, the different methods complement each other as there is not a single method to fully characterize any defect. The raw data of defect studies describe spectroscopic observations and their dependence on time and other parameters, such

858

Electronic structure: Impurity and defect states in insulators

(B)

(C)

(D)

(A)

(E)

Fig. 2 Schematics of defect models developed for amorphous silica. (A) A fragment of perfect SiO2 lattice showing two tetrahedral SiO4 units connected via a common oxygen ion; smaller balls are oxygen ions. (B) An oxygen vacancy with an unpaired electron on a dangling bond of a three-coordinated Si atom. (C) A non-bridging oxygen atom trapping a hole; the partially filled p-orbitals of the oxygen atom are shown. (D) A bridging peroxy linkage center formed by incorporation of an oxygen atom at a regular Si – O – Si bond. (E) A peroxy radical center formed by attachment of an oxygen ion to a regular oxygen site.

as dopant concentration, temperature, pressure, irradiation power and dose. Carefully-obtained data provide the basis for formulating defect models. Fig. 2 shows several such models for major paramagnetic defects in irradiated fused silica. The most reliable models knit together the best-available experimental data from many sources with solid-state physics theory, often aided by detailed calculations. Some of the key ideas underpinning modern calculations of the electronic structure of point defects in insulators are reviewed here. We start from outlining some of the theoretical methods used to study defect electronic structure. Then we review some of the concepts necessary for understanding the electronic properties of point defect in crystalline and disordered solids and at interfaces.

Theoretical methods for defect electronic structure Even simple answers can be useful answers. At one level, electrons and holes can be treated implicitly only. Born’s model of ionic solids as polarizable point ions, interacting through transferable inter-atomic potentials, was the basis for describing cohesion, dielectric and vibrational properties of perfect solids. The large Coulomb energies are the reason why accurate defect studies were first practical for ionic crystals. Yet polarization energies are large too (for an ion of charge Z + 2 substituting for a 2+ Mg ion in MgO, Madelung terms are about 24
Z eV and polarization about 6
Z2 eV) and explain why so many charge states of impurities can be stable. Mott and Littleton’s systematic way of calculating the Coulomb and polarization contributions to defect formation energies in ionic solids forms the basis of many current state-of-the art codes (Hayes and Stoneham, 1985; Stoneham, 1975). Modelling simple one-electron systems: For non-polar semiconductors, effective mass theory is superb, often working well beyond its justification using empirical bulk properties. For ionic crystals, point ions are a surprisingly good start, but pseudo-potential methods were necessary to explain why, and to give the “ion-size corrections” needed for accurate trends over the whole series of alkali halides and oxides. The F center (electron at an anion vacancy) was the first deep trap to be predicted well over a very wide range of halide and oxide hosts, and the same techniques were successful in early modelling states of self-trapped excitons. For most systems, however, the many-electron treatment is essential. Here both first-principles and empirical theories are often used. Empirical methods help most in new situations where what is important is still not certain, for more complex systems, where computer power and technique has yet to be sufficient for the best methods, or for extrapolation across a broad range of materials. Many of the problems addressed empirically today will be solved by far better methods tomorrow. By choosing first-principles methods one is not reliant on parametrized theories or fitting possibly inaccurate experiments (Martin, 2004). These theories can be applied more reliably to hypothetical materials, defects for which data are not available, or for extreme temperature, pressure and irradiation conditions. Computational techniques for modelling defects in solids exist in a form of computer codes implementing a model of a solid with a defect and methods for calculation of defect structure and properties (Freysoldt et al., 2014).

Electronic structure: Impurity and defect states in insulators

859

Most first-principles methods are based on Hartree-Fock and Density Functional Theory (DFT) approximations. DFT states that the ground state properties of a system are given by the charge density, and state-of-the-art methods allow us to compute the charge density and energy self-consistently using various effective exchange-correlation potentials that account for the quantum mechanical interactions between electrons. The local density approximation (LDA) takes the exchange-correlation potential from uniform electron gas theory at the density for each point in the material. The generalized gradient approximation (GGA) includes the effects of local gradients in the density. DFT’s success is facilitated by the computational efficiency of these approximations, which made it applicable to studies of complex materials and defects. The continuous development of more sophisticated density functionals as well as Quantum Monte Carlo methods significantly improved the accuracy of predicted defect properties. As computing power has increased, hybrid DFT functionals have also been employed, which combine DFT estimation of exchange with a percentage of Hartree-Fock exchange (called “exact” exchange). Hybrid DFT has been successful in predicting band gaps and describing the behavior of localized states. The main computational models of a point defect in a crystal include a periodic model, an embedded cluster, and molecular cluster models, illustrated in Fig. 3. They differ by the boundary conditions imposed on a system wave-function. The periodic model was first developed for calculating electronic structure and properties of ideal crystals (Kantorovich, 2004; Kittel, 2004), whereas the cluster model evolved from molecular calculations (Grimes et al., 1992). They are compared in Table 1. Within the periodic model a unit cell with a defect is periodically translated. It is ideal if one is interested in the properties of large concentration of neutral defects, which weakly perturb the surrounding lattice. Many molecular cluster models treat a defect with surrounding atoms simply as a molecule or cluster. They disregard the effects of the rest of the atoms outside the cluster, but apply semi-empirical fixes to the atoms at the cluster surface to diminish their adverse effect. This major problem is addressed and partially solved in “embedded

(A)

(B)

Fig. 3 Schematic of two prevalent computational models used for simulations of defects. (A) A periodic model, where each rectangle represents a periodic cell. (B) An embedded cluster model, where a cluster of atoms in the region I is treated using a high level quantum-mechanical theory (DFT, Coupled Cluster, GW); the surrounding region II is treated on a more approximate level of theory (semi-empirical quantum chemistry, atomistic simulations using a polarizable ion model to treat atomic polarizations); the region III is a polarizable continuum representing the rest of the infinite solid. The boundary between regions I and II is represented by link atoms or embedding potential to provide a seamless link between the two regions.

Table 1

Comparison of computational methods used in defect calculations.

Models

Methods

Properties

Molecular cluster: molecule represents local defect environment in a solid

Density Functional Theory and ab initio and semi-empirical Hartree-Fock Theory in localized basis sets Density Functional Theory and ab initio and semi-empirical Hartree-Fock Theory in localized basis sets

Can predict local defect structure, vibrational, optical and EPR properties providing intermediate and long-range order are not important. Can predict local defect structure, ionization energies and electron and hole affinities, vibrational, optical and EPR properties and diffusion parameters. Can treat infinite amorphous structures. Can predict local defect structure, electrical levels, vibrational and EPR properties and diffusion parameters providing the interaction between periodically translated defects is small. Treats amorphous structures as periodic arrangement of disordered cells.

Embedded cluster: local defect environment is treated quantum-mechanically and the rest of the solid structure treated classically Periodic: defects are repeated periodically in a lattice

Density Functional Theory, Hartree-Fock Theory, tight-binding methods in plane wave and localized basis sets.

860

Electronic structure: Impurity and defect states in insulators

cluster” models, where a cluster with a defect is embedded in an infinite solid (Fig. 3B). This model works well in ionic insulators, such as MgO, however, for insulators with ionic-covalent character of bonding, such as SiO2 or Ga2O3, an accurate embedding is challenging to achieve. It is ideal for modelling individual defects and allows one to self-consistently include the effect of polarizable environment into predicting the defect characteristics, which is particularly important for charged defects in insulators. Both static Hartree-Fock and DFT methods are ground state theories, in the form of an effective one-particle Schrödinger equation (the Kohn-Sham equation in the case of DFT). The eigenvalues of these equations are often interpreted as electron energies and their differences as optical excitation energies. This interpretation is linked to an intuitive picture of independent electrons occupying well-defined energy levels. However, this is a gross simplification: the electrons are not independent, and when, for example, an electron is removed to form a hole, all remaining electrons respond by changing their electronic states. This electron relaxation and other quantum-mechanical contributions must be taken into account for energy differences to be calculated correctly. This can be achieved using advanced methods of quantum chemistry, such as Coupled Cluster, as well as other techniques often implemented in periodic boundary conditions, such as GW (Golze et al., 2019). The improved efficiency of GW facilitated accurate predictions of electronic excitations in materials and defects as well as of charged excitations as measured in direct or inverse photoemission spectroscopy. One can retain the picture of one-particle energy levels, but these particles are quasi-electrons and quasi-holes, which embody the effects of interaction with all the other particles. The concept of quasi-particles is often used to represent the electronic states of an insulator with a defect schematically, as shown in Fig. 4. Such energy diagrams usually include the highest valence band and the lowest conduction band of the system (Freysoldt et al., 2014). Point defects and impurities may induce occupied and unoccupied states in the gap and so-called resonant states in the valence and conduction bands of the system. The wavefunctions of defect states are often strongly localized in the defect area (Fig. 5) and the corresponding quasi-electron energies are located in the band gap of the solid. The resonant states differ from the ordinary band states in that they are localized in the defect region with the major wavefunction component decaying exponentially from the defect, but their quasi-electron energies are located within the occupied or unoccupied bands. Quasi-particle energy diagrams similar to that shown in Fig. 4 are used to summarize pictorially the electronic structure and processes involving optical excitation, luminescence, ionization and energy transfer between defects.

Coupling of electrons and lattice deformation One of the issues crucial for the understanding defect properties is how defect electrons couple to the motion of nearby ions or, alternatively, the way defect electrons apply forces to host ions (Hayes and Stoneham, 1985; Itoh and Stoneham, 2001; Song and Williams, 1993; Stoneham, 1975). The static polarization and distortion of the host lattice affects defect formation and transition energies; the dynamics leads to transitions between electronic states and to defect diffusive motion. Optical spectra change qualitatively: the optical line width, the existence of a zero-phonon line, and the Stokes shift, the energy difference between optical absorption and emission energies, give direct measures of this coupling. Although any of a large number of atomic displacements might influence a defect in a solid, often a very specific combination of displacements is especially important. For example, the radial displacement of the nearest neighbors to the defect can be represented by a single configuration coordinate Q, as illustrated in Fig. 6. We note that it is most unlikely that the coordinate Q will be one of the dynamically-independent normal vibrational modes q. It is almost certain that normal modes of many different frequencies will be needed to make up Q. Moreover, the precise mode combinations making up Q will depend on the defect electronic state (it will be different in the excited state of defects shown in E CONDUCTION BAND RC 5 2

3

6

4

7

1 RV

VALENCE BAND

Defect A

Defect B

Fig. 4 The quasi-particle energy diagram representing total energies of a solid with two defects A and B. The defect A produces local single-occupied and unoccupied states within the band gap, and resonant states (denoted RV and RC) in the valence and conduction band, respectively. The defect B is characterized only by two local states in the gap. The arrows 1–4 correspond to optical transitions between states in the defect A. The arrow 5 corresponds to thermal ionization of an electron from the excited state of the defect A into the conduction band. Process 6 is non-radiative trapping of an electron from the conduction band on an empty state of the defect B. Process 7 is the de-excitation of the defect B which may be either radiative or non-radiative.

Electronic structure: Impurity and defect states in insulators

861

Fig. 5 The geometric configurations and electronic states of positively charged oxygen vacancies in oxides. (A) The relaxed configuration of the positively charged oxygen vacancy, E0 1 center, in a-quartz; red are oxygen ions. The envelope of the density of the unpaired electron mostly localized on a p-orbital of one Si ion is shown pointing into the vacancy. The arrow indicates the direction of the displacement of the second Si ion of the vacancy. This is the so-called “puckered” configuration of the E0 1 center. (B) The relaxed configuration of the positively charged oxygen vacancy, F+ center, in MgO; red are oxygen ions. The envelope of the density of the unpaired electron localized on an s-orbital inside the vacancy is shown. Note the electron density of the unpaired electron propagates much further out from the vacancy. Depressions in the envelope show that the unpaired electron avoids the core electrons of surrounding Mg ions. The arrows indicate the symmetric displacements of the six nearest-neighbor Mg ions surrounding the vacancy. This radial displacement of the six Mg ions is often used as a configuration coordinate, Q, in considering the spectroscopic properties of the F+ center.

Fig. 6 Configuration coordinate diagram. The figure shows the main characteristic energies and transitions, both radiative and non-radiative. Accepting coordinate represents one of the configuration coordinates effective in vibrational energy dissipation. The vertical arrow A – B corresponds to a vertical (so called Franck-Condon) transition between the ground and excited states of the system represented by two harmonic energy surfaces. Horizontal lines indicate vibrational energy levels with small arrows showing transitions between these levels (cooling transitions) dissipating the excess vibrational energy (relaxation energy). The transition C – D between the lowest vibrational state in the excited state back to the ground state energy surface corresponds to the Franck-Condon de-excitation with photon emission (luminescence or phosphorescence). It is accompanied by further cooling transitions in ground electronic state. The area near the crossing point X is where the system can cross nonradiatively from the excited to the ground state. In this case the relaxation energy, which is the energy difference between X and A or between C and A, is dissipated into lattice vibrations. The zero-phonon transition energy is the energy difference between C and A.

Fig. 4). Yet this configuration coordinate model is surprisingly effective, giving an invaluable tool for showing what happens after electronic excitation, and a simple language for describing defect processes. An example of such a diagram is given in Fig. 6. A central issue for optical and electronic materials is how they recover after excitation. A good X-ray phosphor emits the most visible light after excitation and degrades least from defect production. A solid-state laser needs both population inversion and a lack of wasteful competing processes; carrier capture in semiconductors determines what defects and materials are acceptable; photographic (not to mention the photosynthetic) processes have their own demands. In such cases, non-radiative transitions are central, a role evident from over half a century of theory. In nonradiative decay, excitation energy is transformed into heat, defect production, or to excitation of other electrons, the so-called Auger processes which are especially important at higher carrier densities. The recombination of carriers in semiconductors and insulators is governed mainly by nonradiative processes often described by the Shockley-Read-Hall model. The probability of this process decreases when the number of the emitted phonons increases. Therefore, recombination is facilitated by the presence of

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deep impurity or other defect states (recombination centers), because this makes it possible to reduce the number of phonons emitted in one event. Hence, the rate of recombination is governed by the nature and concentration of recombination centers in a crystal. For deep centers, the nonradiative carrier capture occurs via multiphonon emission. Another common manifestation of nonradiative decay of an excited state concerns suppression of luminescence. Can one tell from optical absorption data alone whether or not luminescence will occur? Huang and Rhys’s (1950) seminal study showed one possible criterion: the larger the excitation energy, the faster the optical transition and the slower the non-radiative decay, since larger numbers of phonons would be needed (see Fig. 6). Dexter et al. (1956) developed other criteria relating to the final state energy achieved by optical excitation and the energy at which the ground and excited electronic states intersected on the configuration coordinate diagram: the energy surfaces, rather than some matrix element, controlled the result. This led Bartram & Stoneham to show that simple expressions could be given for branching ratios at critical steps, and that these ideas - which could be applied to much more complex cases correctly predicted behavior over a wide range of systems.

Self-trapping and localization Landau made the remarkable suggestion of self-trapping: an electron could be so strongly coupled to the lattice that, even without a defect, it would lead to lattice distortion and be effectively immobilized. The suggestion was the basis of the important ideas of small polarons (incoherent hopping of localized carriers, as in some transition metal oxides, rather than the long mean-free-path motion of carriers in Si or metals), and even of the recently-observed barrier to self-trapping (Itoh and Stoneham, 2001; Song and Williams, 1993). The observed small polarons (e.g. self-trapped holes in halides in Fig. 7) show both the soundness of Landau’s idea and (since naive belief in band theory appears to be violated) the importance of theorists learning from experiment too. Hole polarons are ubiquitous in oxides and halides, but electron polarons are less common. If a polaron can trap two electrons (or holes) with the second bound more strongly than the first, it has negative-U properties (where pairing of two electrons and two holes in a localized region is favored against single polaron formation), as originally suggested by Anderson (1975). Distortion and polarization dominate too in negative-U behavior of defects, where a charge disproportionation is exothermic (so the

(A) (B)

(C)

(D)

Fig. 7 Atomic structure of the trapped hole centers in halides and MgO: (A) AgCl – the hole is localized on a silver ion; (B) alkali halides of NaCl structure – the hole is localized on two halogen ions forming a Cl−2 molecule; (C) CaF2 – the hole is localized on two fluorine ions forming a F−2 molecule. Note that in (B) – (C) the X−2 (X ¼ F, Cl) molecular ion occupies two nearest anion lattice sites. (D) The hole localized on an O− ion neighboring the Mg vacancy (VMg) in MgO (the so-called V− center). In all cases, the hole localization is accompanied by a strong distortion of the surrounding lattice (not shown in the figure).

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positive vacancies in Si decay from 2 V+ to V0 + V2+ as well as H+ and H− are the preferred thermodynamic states of hydrogen in the bulk of many oxides). Localization can correspond to carrier or exciton trapping by an isolated defect (extrinsic trapping). If there is an attractive perturbation DV at a defect site, then, for large enough DV, a bound state emerges for the extra electron, hole or exciton. Both impurity-induced potentials and electron-phonon interaction play crucial roles in this localization. Extrinsic electron trapping near defects is quite rare in insulators because electron polarons are not easily formed in these materials. However, exciton trapping by impurities is very common in most insulators. As well-established examples of extrinsic hole trapping, one can consider holes trapped near cation vacancies in MgO (so called Vo and V− centers) and holes trapped near Ge and Al impurities in a-quartz (Schirmer, 2006). In particular, cation vacancies in MgO have an effective negative charge of –2e with respect to the lattice and therefore attract positive holes. When one hole is trapped, a paramagnetic V− center is formed, well-studied experimentally using Electron Paramagnetic Resonance and optical spectroscopy (Fig. 7D). Similar centers occur in CaO, SrO and BaO, in the fourfold coordinated oxides like BeO and ZnO and in related sulfides, selenides and tellurides. In MgO, at low temperatures the hole is strongly localized on one of the six oxygen ions surrounding the vacancy. As the temperature rises, the hole moves more and more rapidly among the equivalent oxygens until it appears delocalized at room temperature. The nature of optical transitions in this center is characteristic to many other hole centers in oxides: there are two types of optical transitions. One is internal to the O− ion on which the hole is localized. This is a crystal-field transition, since the transition energy is largely determined by the electrical field at the O− ion due to the Mg2+ vacancy. The other, more important, transitions involve charge transfer of the hole to other neighbors of the vacancy. These are analogous to electron excitation from resonant states in the valence band into the unoccupied hole state shown in Fig. 4. Similar charge transfers are important in determining the color of gemstones. Extensive experimental and theoretical studies show that excitons self-trap in both a-quartz (the most stable crystalline polymorph of SiO2) and amorphous SiO2. However, EPR studies show that holes and electrons do not self-trap in a-quartz. Yet, in high-purity Ge-doped quartz, low-temperature irradiation leads to the formation of a hole center localised at an oxygen site next to Ge. This formation of the localized hole centers associated with Ge impurity can be described as “extrinsic self-trapping,” where both impurity-induced potential and electron-phonon interaction play crucial roles in the hole localization. In amorphous SiO2, two types of localized hole centers have been determined that are free from pre-existing lattice defects such as vacancies, interstitials, or impurities, and are trapped by some precursor states induced by disorder of amorphous structure. This example demonstrates that the mechanisms of localization processes in amorphous solids are more complicated than in crystalline materials due to co-existence of electron-phonon interaction, structural defects and static potential fluctuations. It is not straightforward to distinguish experimentally the polaron self-trapping due to strong electron-phonon interaction from charge trapping by structural defects in amorphous structure.

Defects in disordered materials Defect structures and processes in amorphous materials have been studied extensively, partly as fundamental research on model amorphous materials, and partly to understand their extremely important role in technology (Catlow, 1994; Mott and Davis, 1979; Pacchioni et al., 2000). For example, the mechanisms and kinetics of relaxation of electronic excitations in SiO2 are crucial in microand opto-electronics applications (Pacchioni et al., 2000). There are at least four basic fallacies concerning amorphous systems. First, it is wrong to assume there is only a single amorphous structure for a given composition, since the method of preparation can have significant effects. Secondly, it is wrong to think that the mean energies of defect formation or of trap ionization in an amorphous structure are sufficient to understand the behavior: the form and tails of the distribution are equally important. Thirdly, one should not assume that crystals and amorphous systems of the same composition have the same values for defect and trap energies. Finally, amorphous systems may have several accessible metastable states, leading to easy reorganization of the structure when the system is perturbed by defect processes, such as carrier trapping and detrapping. Structural disorder in amorphous materials generally means that all sites are different. In most experimental studies the effect is hidden in broadening of the spectral features and only rarely has disorder been implicated in formation of different structural types of the same defect. Predicting defect properties and relative abundance of different defect configurations generally requires considering not one or several, as in crystals, but a statistical ensemble of structural sites. Additional factors include the sample history and the mechanism of defect formation. This provides new challenges for both theoretical and experimental analysis of defects in amorphous materials.

Surfaces and interfaces No discussion of defects could be complete without mention of interfaces. Their structures are rarely, if ever, the ideal terminations of perfect crystal structures, and the variety of reconstructions is becoming clearer as microscopies (e.g. the Transmission Electron Microscope, Scanning Tunnelling Microscope and Atomic Force Microscope) develop (Carter and Williams, 2016; Hofer et al., 2003). Defects or impurities at surfaces and interfaces are especially important for most applications. It is impurities close to grain boundaries, which are most strongly involved in the operation of solid-state gas sensors, and the framework of Debye-Hückel

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theory is central (Maier, 2004). The space-charge in silver halides relates to their photographic effectiveness; the Mott and Gurney theory provided the basic framework. Space-charge layers play significant role in the overall performance of electrochemical cells and ionic conduction. Defects at the interface between silicon and gate dielectrics control the reliability of microelectronic devices. Ceramists recognize the importance of segregation in oxides, where experiment shows Mg in alumina gives dramatic improvements in sintering. Defects also influence adhesion. What controls the variation from one oxide to another of the adhesion of a non-reactive metal? Why (a related question) does liquid Cu wet some oxides (NiO, urania, chromia) but not other similar oxides (MgO, thoria, alumina)? The answer lies largely in the defect populations and basic electrostatics: charged defects close to the highly-polarizable metal have lower energy; those close to the unpolarizable vacuum have higher energy. The operation of gas sensors relies on electron transfer across interfaces, as does the operation of contacts to semiconductors (whether conventional or organic) and electrodes. Yet another issue, specific for microelectronics applications, is that the dielectric of interest can be in contact with the electrode(s) - semiconducting (e.g. silicon) or metallic - which provides a source of extra electrons. Also, a dielectric film can be placed in the electric field between the two electrodes, which may alter the defect charge states. Charged defects can cause substantial random fields, which may alter breakdown thresholds locally. Typically, for 100 ppm of charged defects, there will be random fields of order 105 V/cm. Defects can be involved in stress-induced leakage currents (SILC) and dielectric breakdown. In such cases, the defect offers a channel for electron transfer across the dielectric: an electron tunnels from one electrode into an oxide defect; there are then relaxation processes or a sequence of defect processes, and the electron can then tunnel to the other electrode (Fleetwood et al., 2009). What can we say about the defects responsible, their electrical energy levels and relaxation energies (differences in energy for electron capture and emission)? Can we give useful estimates of electron and hole trapping cross-sections? Will there be any field-stimulated diffusion of defects? To define which charge state of an intrinsic defect (e.g. vacancy) or incorporated defect species is favored, assuming the common case of oxide film grown on silicon, the source or sink of electrons is the bottom of silicon conduction band at the Si/SiO2 interface. Hence the chemical potential of the electron can be chosen at the bottom of Si conduction band or at the silicon midgap energy. Electrons are assumed to be able to tunnel elastically or inelastically from/to these states to/from defect states in the oxide and create charged species as illustrated in Fig. 8. Most calculations use experimental information to estimate the energy of an electron at the bottom of the conduction band of Si with respect to the defect states.

Fig. 8 Potential energy diagrams of an electronic device with a defect. (A) Illustrates an elastic tunnelling of an electron from the bottom of the conduction band of the left electrode (Si, Ge) into the gate metal electrode; electron scatters on the oxide trap. (B) Inelastic tunnelling of an electron from the bottom of the conduction band of the left electrode (Si, Ge) into the gate metal electrode. The electron first tunnels into the localized state of the oxide trap and, after vibrational relaxation, tunnels into the metal electrode. This type of process is also called multi-phonon trap assisted tunnelling. (C) Diagram showing the formation energies of O vacancies in the monoclinic phase of HfO2 as a function of the Fermi level position (electron chemical potential) in the band gap calculated using DFT in a periodic model. Lines show the dependence of the defect formation energy in different charge states on the electron chemical potential. Crossing points are the charge transition energies. They correspond to chemical potential above/below which the existence of the defect in the different charge state is becoming more energetically favorable. One can see that positively charged vacancy states dominate up to about 4.3 eV.

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The key point about defect states in dielectrics from a device point of view is to control the dependence of the defect charge state on the position of the system Fermi energy (Fleetwood et al., 2009; Freysoldt et al., 2014). It is useful to distinguish two charge transition levels: the thermodynamic level, Eth, and the transient level, Etr. The thermodynamic charge transition level corresponds to the system Fermi energy for which a defect changes its charge in thermal equilibrium. Eth is defined as a difference of total energies of systems of N and N + 1 electrons, each in its fully relaxed ground state: Eth ¼ E(N + 1) - E(N). Thus, if EF is the energy required to remove an electron from some electron reservoir (e.g. Si at the Si/SiO2 interface), Eth – EF is the energy required to add an electron from the reservoir to the lowest unoccupied charge-state level. The position of the defect charge transition levels indicates whether the defect is a deep or shallow donor or an acceptor. Furthermore, if the defect charge transition level is in the same position as the electrode Fermi level (said to be in resonance), then tunnelling rates between the defect and electrode will be very high, provided the defect is close enough to the interface (typically up to 3 nm). An example of defect formation energy diagram showing charge transition levels is shown in Fig. 8C. Taking into account that the electrical measurements and the electronic processes are often fast compared with the time needed for the system to come to equilibrium, another, transient, level is introduced, which is consistent with the Frank-Condon principle. Etr is defined as the energy required for trapping an electron or a hole in the geometry of the system as it was prior to the trapping. Thus Etr are obtained as the differences of the total energies of the system with N and N + 1 electron, but in the local atomic geometries, which correspond to one of these systems. For example, Etr (0/−) ¼ E(−) – E(0) denotes the energy required to add an electron to the neutral defect calculated using the neutral defect geometry. Once the charge-transition levels are calculated, the dependence of the defect charge state on the electrode Fermi energy can be introduced via a correction term q
EF, where EF is varied in the energy range from the maximum of the last valence band of the oxide to the bottom of the first conduction band. The results are presented as plots of Eth vs. EF or Etr vs. EF calculated for different values of EF (Fig. 8C). The crossing points of the corresponding lines determine the regions where one charge state of a defect is more favorable than the other. Although the Fermi energy remains unknown in the experiment, such diagrams allow one to identify some critical cases and help in the analysis of the experimental data. The same approach is used to determine the dependence of the defect formation energies on the position of the Fermi energy.

Conclusion Our understanding of point defects in insulators has progressed enormously since the pioneering works of Frenkel and Schottky. Advances in experimental defect characterization combined with increasing accuracy of theoretical calculations of defect properties and reactions provide defect models of increasingly complex processes in the bulk and at surfaces of insulators. Transitioning from understanding fundamental materials behavior to development of quantitative approaches to explain and predict experimental observations requires further advances in the computational methods that take advantage of the increasingly powerful computing hardware. Material function is, however, multi-scale and is intimately linked not only to point defects considered here but also to microstructural processes (diffusion, electron transfer, chemical reactions) at the internal grain boundaries and interfaces to electrodes or contacts between insulators and other materials. The roles of microstructure of materials (particularly texture) are well established in metallurgy, but much less understood in insulators, e.g. functional ceramics, particularly when it comes to linking the atomic structure of polycrystalline materials to their function. The challenge is to merge this information at various length scales and to develop a fundamental understanding of the evolution of the microstructure and its effect on properties, such as thermal-mechanical response, electrical response, degradation, and failure. A step-change will be achieved when it is possible to predict the effect of changing the microstructure on system function and so guide the materials synthesis and processing.

References Anderson PW (1975) Model for the electronic structure of amorphous semiconductors. Physical Review Letters 34: 953–955. Bassani F and Pastori Parravicini G (1975) Electronic States and Optical Transitions in Solids. New York: Pergamon Press. Carter CB and Williams DB (eds.) (2016) Transmission Electron Microscopy. Berlin: Springer. Catlow CRA (ed.) (1994) Defects and Disorder in Crystalline and Amorphous Solids. Berlin: Springer NATO series. Crawford JH and Slifkin LM (1972) Point Defects in Solids: General and Ionic Crystals. New York: Plenum Press. Dexter D, Klick CC, and Russell GA (1956) Configuration coordinate curves for F-centers in alkali halide crystals. Physics Review 101: 1473–1479. Fleetwood DM, Pantelides ST, and Schrimpf RD (eds.) (2009) Defects in Microelectronic Materials and Devices. New York: CRC Press. Freysoldt C, Grabowski B, Hickel T, Neugebauer J, Kresse G, Janotti A, and Van de Walle CG (2014) First-principles calculations for point defects in solids. Reviews of Modern Physics 86: 253–305. Golze D, Dvorak M, and Rinke P (2019) The GW Compendium: A practical guide to theoretical photoemission spectroscopy. Frontiers in Chemistry 7(paper 377): 1–66. Grimes RW, Catlow CRA, and Shluger AL (eds.) (1992) Quantum Mechanical Cluster Calculations in Solid State Studies. Singapore: World Scientific Publishing. Hayes W and Stoneham AM (1985) Defects and Defect Processes in Non-Metallic Solids. New York: Wiley. Hofer WA, Foster AS, and Shluger AL (2003) Theories of scanning probe microscopes at the atomic scale. Reviews of Modern Physics 75: 1287–1331. Huang K and Rhys A (1950) Theory of light absorption and non-radiative transitions in F-centres. Proceedings of the Royal Society of London A 204: 406–423. Itoh N and Stoneham AM (2001) Materials Modification by Electronic Excitation. Cambridge: Cambridge University Press. Kantorovich L (2004) Quantum Theory of the Solid State: The Introduction. London: Kluver. Kittel C (2004) Introduction to Solid State Physics, 8th edition New Jersey: Wiley. Lannoo M and Bourgoin J (1981) Point Defects in Semiconductors, I Theoretical Aspects. Berlin: Springer Verlag.

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Maier J (2004) Physical Chemistry of Ionic Materials: Ions And Electrons in Solids. New Jersey: Wiley. Martin RM (2004) Electronic Structure: Basic theory and Practical Methods. Cambridge: Cambridge University Press. Mott NF and Davis EA (1979) Non-Crystalline Solids. Oxford: Oxford University Press. Pacchioni G, Skuja L, and Griscom DL (eds.) (2000) Defects in SiO2 and Related Dielectrics: Science and Technology. Berlin: Springer NATO series. Schirmer OF (2006) O− bound small polarons in oxide materials. Journal of Physics: Condensed Matter 18: R667–R704. Song KS and Williams RT (1993) Self-Trapped Excitons. Berlin: Springer Verlag. Spaeth J-M and Overhof H (2003) Point Defects in Semiconductors and Insulators. Berlin: Springer Verlag. Stoneham AM (1975) Theory of Defects in Solids. Oxford: Oxford University Press. Toyozawa Y (2003) Optical Processes in Solids. Cambridge: Cambridge University Press. Weil JA and Bolton JR (2007) Electron Paramagnetic Resonance: Elementary Theory and Practical Applications. New Jersey: Wiley.

Density functional theory☆ Yusuke Nomuraa and Ryosuke Akashib, aDepartment of Applied Physics and Physico-Informatics, Keio University, Yokohama, Japan; b Department of Physics, University of Tokyo, Bunkyo, Japan © 2024 Elsevier Ltd. All rights reserved. This is an update of S. Kurth, M.A.L. Marques, E.K.U. Gross, Density-Functional Theory, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 395–402, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00445-9.

Introduction Basic formalism Born-Oppenheimer approximation Hohenberg-Kohn theorem and Kohn-Sham equation Extensions Approximate exchange-correlation functionals LDA, GGA, and orbital dependent functionals Notes on constructing approximate exchange-correlation functionals Electronic structure Single-particle spectrum Response function Kohn-Sham states in practice Nuclear dynamics Forces acting on nuclei Interatomic force constants Recent topics Practical guide Program packages Reproducibility Conclusion Acknowledgments References

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Abstract Density functional theory (DFT) is an essential building block for modern theoretical physics, chemistry, and engineering, especially those concerning electronic properties. Through decades of development, various program packages for first-principles electronic structure calculation are now available. Their sophisticated interfaces allow users to apply DFT to actual systems, even without knowing the theory. It is hence becoming more and more important to recall the fundamentals of how DFT enables accurate calculations. This article attempts to provide such knowledge with a minimal overview of DFT—its basic foundation, relations to observable electronic and nuclear dynamical properties, and some of its cutting-edge applications.

Key points

• • •

Theoretical foundation of density functional theory is reviewed. Relations to experimentally observable quantities are highlighted. Modern applications of density functional theory are introduced.

Introduction The motion of the electrons in atoms, molecules, and solids is described by the Schrödinger (or Dirac) equation. The computational complexity to obtain the exact solution grows exponentially with the number of electrons N. Calculating exact ground-state wave function is prohibitively expensive when N is more than a few dozen. Therefore, it has been a great challenge to develop a powerful and accurate numerical method to calculate electronic structures.

☆

Change History: November 2022. Y Nomura and R Akashi updated Sections: “Introduction” “Born-Oppenheimer Approximation” “Hohenberg-Kohn theorem and Kohn-Sham equation” “Extensions” “LDA, GGA, and orbital dependent functionals” “Single-particle spectrum” “Kohn-Sham states in practice” “Nuclear dynamics”.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00148-7

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There are mainly two numerical approaches. One is the wave function theory. It directly deals with the many-body wave function itself and attempts to find good approximations to the exact wave function. The other is the density functional theory (DFT), for which we will give a brief review. By employing the electron density r(r) (a function of three coordinate variables) as the fundamental variable instead of the many-body wave function (a function of 3N coordinate variables), DFT has drastically reduced the computational cost. Therefore, DFT is the most widely used method for electronic structure calculations of solids. Here, we briefly review the fundamental and practical aspects of DFT. Section “Basic formalism” reviews the Hohenberg-Kohn theorem and the Kohn-Sham equation, which gives a foundation of DFT. We also describe several extensions of DFT. In DFT, as described below, the exchange-correlation functional is one of the most fundamental quantities. Section “Approximate exchange-correlation functionals” is devoted to a discussion of approximations to the exchange-correlation functional. Section “Electronic structure” presents several topics related to electronic structure calculations. In Section “Nuclear dynamics,” we show that DFT can also be used to calculate the properties of atomic vibrations. Section “Practical guide” provides a practical guide of DFT. Finally, we give a summary in Section “Summary.”

Basic formalism We start from the full non-relativistic Hamiltonian, in which electrons and atomic nuclei interact with each other. Because the masses of nuclei are much heavier than that of electrons, the kinetic energies of nuclei are usually much smaller than that of electrons. Then, to a good approximation, we can treat electron and nuclear dynamics separately. DFT is a theory for dealing with the electronic part under this approximation. In this section, we first briefly discuss the Born-Oppenheimer approximation (Born and Oppenheimer, 1927), which is the most widely used framework to derive separate equations for the electron and nuclear dynamics. Then, we discuss the Hohenberg-Kohn theorem (Hohenberg and Kohn, 1964) and the Kohn-Sham equation (Kohn and Sham, 1965), which form the basis of DFT.

Born-Oppenheimer approximation The Hamiltonian for interacting electrons and nuclei reads X ℏ 2 ∂2 X ℏ 2 ∂2 c¼ − ℋ − + V ðfrg, fRgÞ, 2MI ∂R2I 2m ∂r2i I i

(1)

where X

V ðfrg; fRgÞ ¼

i <  0, detðI + MÞ 0, > : 0,

if M 2 O++ ðn, nÞ, if M 2 O − − ðn, nÞ, otherwise,

(30)

where the split orthogonal group O(n, n) is formed by all 2n  2n real matrices that preserve the metric 0 1  ¼ diag@1, . . . , , −1, . . . , −1A,MT M ¼  . Split orthogonal group O(n, n) means that if we write the matrix M in the block |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} n n  M11 M12 form like: M ¼ , then we have |det (M11)|  1 and |det (M22)|  1. And we use the notation O(n, n) to classify the M21 M22 four components of O(n, n) by considering the signs of det (M11) and det (M22). Since the fermion sign problem is related to the basis of decoupling, it can be found that some models translated to Majorana representation are sign-problem free (Li et al., 2015, 2016; Li and Yao, 2019). Majorana representation is turning fermions into Majorana fermions ^ci ¼

  1 1 1 g + ig2i , ^c{i ¼ g1i + ig2i : 2 i 2

(31)

These findings inspired research on Majorana reflection positivity or the Majorana Kramers positivity (Wei et al., 2016). One can write the H0 and decoupled Hamiltonian under the Majorana representation: Hbl ¼ gTVg. Then. (a) if V is a Majorana reflection positive kernel, it can be represented as 

A iB , −iBT A ∗ where A and B are N  N matrices. A is complex antisymmetric matrix and B is a Hermitian matrix. Or,

(32)

V¼

(b) if there exist two transformation operators S and P such that ST VS ¼ V ⁎ , (33) PVP −1 ¼ V, where S is a real antisymmetric matrix and P is a symmetric or antisymmetric Hermitian matrix, and we need P anticommutes with S. There is no sign problem in the DQMC simulation. In Xu et al. (2019b), there is a similar discussion on the sign problem related to the structure of the fermion determinant. It is proven that if D 2 SU(n, m), then det (In+m + D) 2 ℝ, where SU(n,m) is a pseudo-unitary group. Such condition is used to guarantee the DQMC simulation for U(1) gauge field coupled to Dirac fermions in (2 + 1)d, a fundamental problem for both high-energy physics and condensed matter physics related to quantum electrodynamics and deconfined quantum criticalities, is sign-problem free (Xu et al., 2019b; Wang et al., 2019; Janssen et al., 2020).

Sign bound theory As discussed in section “What is the sign problem?,” for systems with sign problems, we can usually sample them by reweighting and the average sign is usually ⟨s⟩ ¼ Z/Z0  e− bNDf. However, the difference of free energy Df between the reference system Z0 and the system Z is not necessarily a constant with little change, but may be a quantity depending on the system size N ¼ Ld and inverse temperature b ¼ kB1T . In some systems, we find that in the zero-temperature limit, the sign does not decay exponentially with respect to size L but rather decays algebraically (Ouyang and Xu, 2021; Zhang et al., 2021), implying that Df could be a logarithmic function of L. In fact, −bE0 in the zero-temperature limit, the mean of the signs is hsiR ¼ ZZR ¼ geR −bER , as given in Eq. (19). We can choose a good reference system g e 0 R R Z , which has the same ground state energy E0 ¼ E0 with Z. Then the mean of the signs is ⟨s⟩R ¼ g/gR, which is only related to the ratio of ground state degeneracy. And as shown in the cases below, in a few common situations, such as the lattice model for quantum Moiré materials in momentum space (Xu et al., 2021a,b; Zhang et al., 2021; Pan et al., 2022) and extended Hubbard model in real space (Ouyang and Xu, 2021; Da Liao et al., 2022, 2021; Liao et al., 2021), the ground state degeneracy is a polynomial function of the size of the system. Even if most of the time we cannot find such a perfect ZR, we can still consider the upper bound on the value of ZR, which means −bE0

if we know ZR C, then in the zero temperature limit we can get a lower bound of the sign: hsiR ¼ ZZR  ge C . This is the Sign bound pffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 R theory (Zhang et al., 2021). Two cases of the theory will be shown below. In case 1: ZR ¼ gR e −bE0 g0 e −bE0 and E00 /2 ¼ E0 such −bE0 pffiffiffiffi pffiffiffiffi 0 0 R that hsi ¼ ZR  pgeffiffiffiffiffiffiffiffiffiffiffi ¼ g= g0 e −bðE0 − E0 =2Þ ¼ g= g0 , and in case 2: ZR ¼ gRe− bE0 g0 e− bE0 and E 0 ¼ E such that R

hsiR ¼ ZZR 

Z

ge −bE0 −bE0 g0 e 0

g0 e

0

−bE0 0

¼ g=g0 e −bð

E0 −E00

0

Þ ¼ g=g0 . If we succeed in finding such an upper bound on ZR which can cancel out the e− bE0

The sign problem in quantum Monte Carlo simulations

889

term in the denominator, then the lower bound of the sign is just a function of the ground state degeneracy. If the ground state degeneracy is a polynomial function of the size of the system, then we can obtain a lower bound of the average sign decay with the system size algebraically.

Case 1

P P P For example, we choose ZR ¼ {l}P({l})| R(D({l}))|  h|R(D)|i and Z0 ¼ {l}P({l})|D |2({l})  h| D |2i, where Z ¼ {l}P({l}) D({l})  hDi ¼ hR(D)i. Here D({l}) is the determinant term in weight, and P({l}) is a normalized probability distribution. q ffiffiffiffiffiffiffiffiffiffiffiffiffi −bE0 pffiffiffiffi pffiffiffiffi 0 ffi ¼ g= g0 e −bðE0 − E0 =2Þ ¼ g= g0 , if E00 /2 ¼ E0. Because hjRðDÞji jDj2 then hsi ¼ ZZR  pgeffiffiffiffiffiffiffiffiffiffi −bE0 g0 e

0

The twisted-bilayer-graphene (TBG) models, projected to the flat bands, belong to such a case (Xu et al., 2021a; Pan et al., 2022). In the chiral limit, the Hamiltonian only has density-density interaction in momentum space X X H¼ V ðqÞr −q rq ¼ V ðqÞr{q rq , (34) q6¼0

P 

q6¼0

 − 12 mq . where rq ¼ If we set l as random number and mq ¼ 0, and there is no other symmetry, then E0 ¼ E00 /2. For ground state degeneracy, it’s pffiffiffiffiffiffiffiffiffiffiffiffi P easy to know g ¼ 2 and g0 ¼ N + 3, by introducing a raising operator D{ ¼ i0 c{i0 , +ci0 , −. Then at low temperature, hsi  2= N + 3 (as shown in Fig. 5(a)). We consider single valley/spin two band (m, n 2 {1,−1}) TBG model at hall-fing and chiral limit, which means li, j,m,n(q) ¼ m ∙ n ∙ li, j,m,n(q) and mq 6¼ 0. Similarly, we still have E0 ¼ E00 and g ¼ 2, and now g0 ¼ (N + 1)2 + 2. Then at low qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi temperature, hsi  2= ðN + 1Þ2 + 2 (as shown in Fig. 5(b)). c{i cj i,j li,j ðqÞ^

Case 2 If we consider the real space extended Hubbard model (Ouyang and Xu, 2021; Liao et al., 2021; Da Liao et al., 2021), which also offers the description of the TBG systems once the interactions are projected to the flat bands with truncation ð35Þ , n is used to control filling, s, t

,

where

are spin and valley indexes, R + dl represents site l in a single R hexagon and U, a is real number coming from the overlap of Wannier functions on the Moiré scale. P P P Here, we choose Z0 ¼ {l}P({l}) | D | ({l})  h| D | i, still ZR ¼ {l}P({l}) | R(D({l})) |  h| R(D)| i and Z ¼ {l}P({l}) −bE0 0 D({l})  hDi ¼ hR(D)i. Because h| R(D)| i h| D | i then hsi ¼ ZR  ge 0 ¼ g=g0 e −bðE0 − E0 Þ ¼ g=g0 . Similarly, we still have Z

g0 e

−bE 0

E00 ¼ E0. It can be computed from tensor Young tableau method that, at n ¼  2, g ¼ (N + 3)(N + 2)(N + 1)/6, and at n ¼ 0, g0 ¼ (N + 3)(N + 2)2(N + 1)/12 (Zhang et al., 2021). Then at low temperature, ⟨s⟩ ⩾ g/g0 ¼ 2/(N + 2) (as shown in Fig. 5(c)). For the model we mentioned above and its reference system, the exponential decay part cancels out, and we can easily calculate the ground state degeneracy. As shown in Fig. 5, numerically we also see that at very low temperatures the average sign decays polynomially. (A)

(B) 100

(C) 100

100

10-1 10-1 10-2 10-1

100

QMC Bounds Fitting Region

101

102

103

10-2 100

QMC Bounds Fitting Region

101

102

10-3 100

QMC Bounds Fitting Region

101

102

103

Fig. 5 The average sign for three different cases according to the sign bound theory. All measurement are carried out at a average sign converged temperature. The errorbars in QMC data are smaller than the symbol size. (a) momentum space case, now l is randomly set, and the red line gives us the lower bound pffiffiffiffiffiffiffiffiffiffiffi hsi  2= N + 3. Here N ¼ 18, 32, 50, 72, 98, 128, 162. Fitting line ∝ N−0.23. (b) momentum space case, we consider n ¼ 0 single valley/spin TBG model qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in chiral limit, and the red line gives us the lower bound hsi  2= ðN + 1Þ2 + 2. N ¼ 9, 16, 25, 36, 49 and T ¼ 0.91 meV. Fitting line ∝ N−0.70. (c) For real space TBG model, the red line lower bound is ⟨s⟩ ⩾ 2/(N + 2) and N ¼ 18, 72, 162, 288, 450, 648, 882. The parameters of QMC are detailed in (Ouyang and Xu, 2021). Fitting line ∝ N−0.98.

890

The sign problem in quantum Monte Carlo simulations

Lefschetz thimble This is also a method that can be used to cure or reduce the sign problem in lattice fermion QMC simulations (Ulybyshev et al., 2020; Alexandru et al., 2020; Cristoforetti et al., 2013; Fukuma et al., 2019; Mukherjee and Cristoforetti, 2014; Mukherjee et al., 2013; Mooney et al., 2021; Mishchenko et al., 2021; Körner et al., 2020), here we only briefly outline its main idea. For computing the expectation value ⟨O[x]⟩ in Monte Carlo sampling on the configuration space [x], we have Z 1 dxe − S½x O½x hO½xi ¼ Z Z dxe − SR ½x e −iSI ½x O½x ¼ Z dxe − SR ½x e − iSI ½x Z Z (36) dxe − SR ½x e − iSI ½x O½x dxe − SR ½x e −iSI ½x Z ¼ = Z dxe − SR ½x dxe −SR ½x

−iS  e IO R ¼ , he −iSI iR R −S ½x dxe R O where R and I means real and imaginary parts of action S [x]. And ⟨ ⟩R is defined as hOiR ¼ R −SR ½x . dxe

If the action S [x] is analytic for all x 2 ℂn, its saddle points are nondegenerate and Eq. (36) is convergent, then Morse theory (Banyaga and Hurtubise, 2004; Witten, 2010, 2011) told us these integrals can be evaluated using the steepest descent cycles J s . They are called Lefschetz thimbles, which means we can replace the integration over the real domain Rn with that over curved complex n-dimensional manifolds J s Z X Z dn xe −SðxÞ ¼ ns dn ze −SðzÞ , (37) ℝn

s

Js

where the Lefschetz thimbles J s are the union of all solutions of the gradient flow equations
dzðtÞ ∂S ¼ − , ∂z dt

(38)

and it starts from the corresponding saddle point zs, which means limt!1 z(t) ¼ zs. Here s is the index for different saddle point/ Lefschetz thimbles, t is a parameter and the overline represents complex conjugation, and saddle point is the point on the surface of S where the derivatives in orthogonal directions are all zero, but is not a local extremum. While the thimbles always end inside regions of stability, duals thimbles (anti-thimbles) Ks are also such solutions at a given saddle point limt!−1 z(t) ¼ zs and end inside regions of instability. We can make an analogy between the dual thimble and the contour of the steepest ascent, as we did with the thimble and the contour of the steepest descent. And ns are intersection numbers (which decide the contribution of a particular saddle point zs to the partition function) of hypersurface generated by the paths of steepest ascent duals Ks and the original region of integration ℝn ns ¼ hKs , ℝn i:

(39)

The important point of such construction is: Im[S] is constant on Lefschetz thimbles (mod 2p), which means there is no sign problem on each thimble. The ratio of weights in the same thimble must be a positive real number, since the phase of the weight is the same. Let’s take a trivial 1d case for the purpose of illustration (Kanazawa and Tanizaki, 2015; Bharathkumar and Joseph, 2020) Z X Z 1 2 dx e −x =2 ¼ ns dz e −SðzÞ , (40) x ℝ+ie Js s which corresponds to the case S [z] ¼ log z + z2/2. The two critical points of this action are: z1 ¼ i, z2 ¼ −i. Since we know that the imaginary parts Im[S] are constant on Lefschetz thimbles, then we have: Im(S [z])|along ℝ+ie ¼ Im S [zs] ¼ ip/2. Following the procedure described above we can obtain the corresponding thimbles and dual thimbles (as shown in Fig. 6), and then R R 1 −x2 =2 ¼ J 1 dz e −SðzÞ . ℝ+ie dx x e The Lefschetz thimbles method is commonly used to calculate the value of certain integrals and in high-energy physics and real time calculations. Recently, with the application of hybrid Monte Carlo in condensed matter, there has been a lot of work using the Lefschetz thimbles method to study common condensed matter problems which have sign problems. For example, Hubbard model on the hexagonal (honeycomb) lattice at finite chemical potential with linear system size L ¼ 6 and inverse temperature b ¼ 20, has been successfully simulated with Lefschetz thimbles method in DQMC (Ulybyshev et al., 2020; Körner et al., 2020).

The sign problem in quantum Monte Carlo simulations

891

Fig. 6 A schematic plot of thimbles and dual thimbles. Here the black dots are saddle points z1,2, while the solid blue/purple lines correspond to thimbles and the dotted red/yellow lines correspond to dual thimbles. And the arrows show the directions of the flows. See intersection numbers of dual thimbles and ℝ + ie, we know n1 ¼ 1, n2 ¼ 0.

Conclusion In this article, we give a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum space quantum Monte Carlo for interacting fermions. We have elaborated on the definition of the sign problem and its possible origins, including the Pauli exclusion principle, geometric frustration and the lack of symmetry requirements, etc. In addition, we also point out the sign problem is actually basis-dependent and summarize established ways to ensure that some models are free of sign problems such as checking symmetry and Majorana positivity. We explain what to do when there is a sign problem in general: reweighting, and how to reduce the severity of sign problems, in particular that based on the properties of the finite size partition functions, the recent sign bound theory could distinguish when the bounds have the usual exponential scaling, and when they are bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.

Acknowledgments We thank Xu Zhang, Weilun Jiang, Yuan Da Liao and Xiao Yan Xu for insightful discussions and fruitful collaborations on related topics over the years. GPP and ZYM acknowledge the support from the Research Grants Council of Hong Kong SAR of China (Grant Nos. 17303019, 17301420, 17301721 and AoE/P-701/20), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000), the K. C. Wong Education Foundation (Grant No. GJTD-2020-01) and the Seed Funding “QuantumInspired explainable-AI” at the HKU-TCL Joint Research Centre for Artificial Intelligence. We thank the Information Technology Services at the University of Hong Kong and the Tianhe platforms at the National Supercomputer Centers for their technical support and generous allocation of CPU time.

References Alet F, Damle K, and Pujari S (2016) Sign-problem-free monte carlo simulation of certain frustrated quantum magnets. Physical Review Letters 117: 197203. Alexandru A, Basar G, Bedaque PF, and Warrington NC (2020) Complex paths around the sign problem. arXiv. preprint arXiv:2007.05436. Assaad FF and Evertz HG (2008) World-Line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons. Berlin, Heidelberg: Springer Berlin Heidelberg 277–356. Banyaga A and Hurtubise D (2004) Lectures on Morse Homology. vol. 29. Springer Science & Business Media. Barahona F (1982) On the computational complexity of ising spin glass models. Journal of Physics A: Mathematical and General 15(10): 3241–3253. Batrouni GG and Scalettar RT (1992) World-line quantum monte carlo algorithm for a one-dimensional bose model. Physical Review B 46: 9051–9062. Bharathkumar R and Joseph A (2020) Lefschetz thimbles and quantum phases in zero-dimensional bosonic models. The European Physical Journal C 80(10): 1–20. Broecker P, Carrasquilla J, Melko RG, and Trebst S (2017) Machine learning quantum phases of matter beyond the fermion sign problem. Scientific Reports 7(1): 1–10. Carlson J, Gandolfi S, Pederiva F, Pieper SC, Schiavilla R, Schmidt KE, and Wiringa RB (2015) Quantum monte carlo methods for nuclear physics. Reviews of Modern Physics 87: 1067–1118. Chandrasekharan S and Wiese UJ (1999) Meron-cluster solution of fermion sign problems. Physical Review Letters 83(16): 3116–3119. Congjun W and Zhang S-C (2005) Sufficient condition for absence of the sign problem in the fermionic quantum monte carlo algorithm. Physical Review B 71: 155115. Cristoforetti M, Di Renzo F, Mukherjee A, and Scorzato L (2013) Monte carlo simulations on the lefschetz thimble: Taming the sign problem. Physical Review D 88(5), 051501. D’Emidio J, Wessel S, and Mila F (2020) Reduction of the sign problem near t ¼ 0 in quantum monte carlo simulations. Physical Review B 102: 064420. Da Liao Y, Kang J, Breiø CN, Xu XY, Wu H-Q, Andersen BM, Fernandes RM, and Meng ZY (2021) Correlation-induced insulating topological phases at charge neutrality in twisted bilayer graphene. Physical Review X 11: 011014. Da Liao Y, Xu XY, Meng ZY, and Qi Y (2022) Gross-neveu and deconfined quantum criticalities in an extended hubbard model on the square lattice. arXiv . preprint arXiv:2204.04884. Dai X (2022) Quantum monte carlo simulations in momentum space. Chinese Physics Letters 39(5): 050101.

892

The sign problem in quantum Monte Carlo simulations

De Raedt H and Lagendijk A (1985) Monte carlo simulation of quantum statistical lattice models. Physics Reports 127(4): 233–307. Evertz HG, Lana G, and Marcu M (1993) Cluster algorithm for vertex models. Physical Review Letters 70: 875–879. Foulkes WMC, Mitas L, Needs RJ, and Rajagopal G (2001) Quantum monte carlo simulations of solids. Reviews of Modern Physics 73: 33–83. Fukuma M, Matsumoto N, and Umeda N (2019) Applying the tempered lefschetz thimble method to the hubbard model away from half filling. Physical Review D 100(11), 114510. Fye RM (1986) New results on trotter-like approximations. Physical Review B 33(9): 6271. Fye RM and Scalettar RT (1987) Calculation of specific heat and susceptibilities with the use of the trotter approximation. Physical Review B 36(7): 3833. Geyer CJ (2011) Introduction to markov chain monte carlo. In: Handbook of Markov Chain Monte Carlo, 20116022, 45. Golan O, Smith A, and Ringel Z (2020) Intrinsic sign problem in fermionic and bosonic chiral topological matter. Physical Review Research 2: 043032. Hands S, Montvay I, Morrison S, Oevers M, Scorzato L, and Skullerud J (2000) Numerical study of dense adjoint matter in two color qcd. The European Physical Journal C—Particles and Fields 17(2): 285–302. Hangleiter D, Roth I, Nagaj D, and Eisert J (2020) Easing the monte carlo sign problem. Science Advances 6(33): eabb8341. Hastings MB (2016) How quantum are non-negative wavefunctions? Journal of Mathematical Physics 57(1), 015210. Hatano N and Suzuki M (1992) Representation basis in quantum monte carlo calculations and the negative-sign problem. Physics Letters A 163(4): 246–249. Hirsch JE (1985) Two-dimensional hubbard model: Numerical simulation study. Physical Review B 31: 4403–4419. Hirsch JE, Sugar RL, Scalapino DJ, and Blankenbecler R (1982) Monte carlo simulations of one-dimensional fermion systems. Physical Review B 26: 5033–5055. Hohenadler M, Meng ZY, Lang TC, Wessel S, Muramatsu A, and Assaad FF (2012) Quantum phase transitions in the kane-mele-hubbard model. Physical Review B 85: 115132. Honecker A, Wessel S, Kerkdyk R, Pruschke T, Mila F, and Normand B (2016) Thermodynamic properties of highly frustrated quantum spin ladders: Influence of many-particle bound states. Physical Review B 93: 054408. Huffman EF and Chandrasekharan S (2014) Solution to sign problems in half-filled spin-polarized electronic systems. Physical Review B 89: 111101. Iazzi M, Soluyanov AA, and Troyer M (2016) Topological origin of the fermion sign problem. Physical Review B 93: 115102. Ibarra-García-Padilla E, Dasgupta S, Wei H-T, Taie S, Takahashi Y, Scalettar RT, and Hazzard KRA (2021) Universal thermodynamics of an SU(n) fermi-hubbard model. Physical Review A 104: 043316. Iglovikov VI, Khatami E, and Scalettar RT (2015) Geometry dependence of the sign problem in quantum monte carlo simulations. Physical Review B 92: 045110. Janssen L, Wang W, Scherer MM, Meng ZY, and Xu XY (2020) Confinement transition in the qed3-gross-neveu-xy universality class. Physical Review B 101: 235118. Johannes S, Hofmann EK, Vishwanath A, Berg E, and Lee JY (2021) Fermionic monte carlo study of a realistic model of twisted bilayer graphene. arXiv . preprint arXiv:2105.12112. Kanazawa T and Tanizaki Y (2015) Structure of lefschetz thimbles in simple fermionic systems. Journal of High Energy Physics 2015(3): 1–36. Kaul RK, Melko RG, and Sandvik AW (2013) Bridging lattice-scale physics and continuum field theory with quantum monte carlo simulations. Annual Review of Condensed Matter Physics 4(1): 179–215. Kim AJ, Werner P, and Valentí R (2020) Alleviating the sign problem in quantum monte carlo simulations of spin-orbit-coupled multiorbital hubbard models. Physical Review B 101: 045108. Klassen J, Marvian M, Piddock S, Ioannou M, Hen I, and Terhal BM (2020) Hardness and ease of curing the sign problem for two-local qubit hamiltonians. SIAM Journal on Computing 49(6): 1332–1362. Koonin SE, Dean DJ, and Langanke K (1997) Shell model monte carlo methods. Physics Reports 278(1): 1–77. Körner M, Langfeld K, Smith D, and von Smekal L (2020) Density of states approach to the hexagonal hubbard model at finite density. Physical Review D 102: 054502. Lang GH, Johnson CW, Koonin SE, and Ormand WE (1993) Monte carlo evaluation of path integrals for the nuclear shell model. Physical Review C 48: 1518–1545. Lawrence S (2020) Perturbative removal of a sign problem. Physical Review D 102: 094504. Levy R and Clark BK (2021) Mitigating the sign problem through basis rotations. Physical Review Letters 126: 216401. Li Z-X and Yao H (2019) Sign-problem-free fermionic quantum monte carlo: Developments and applications. Annual Review of Condensed Matter Physics 10(1): 337–356. Li Z-X, Jiang Y-F, and Yao H (2015) Solving the fermion sign problem in quantum monte carlo simulations by majorana representation. Physical Review B 91: 241117. Li Z-X, Jiang Y-F, and Yao H (2016) Majorana-time-reversal symmetries: A fundamental principle for sign-problem-free quantum monte carlo simulations. Physical Review Letters 117: 267002. Liao Y-D, Xiao-Yan X, Meng Z-Y, and Kang J (2021) Correlated insulating phases in the twisted bilayer graphene. Chinese Physics B 30(1), 017305. Loh EY, Gubernatis JE, Scalettar RT, White SR, Scalapino DJ, and Sugar RL (1990) Sign problem in the numerical simulation of many-electron systems. Physical Review B 41: 9301–9307. Marvian M, Lidar DA, and Hen I (2019) On the computational complexity of curing non-stoquastic hamiltonians. Nature Communications 10(1): 1–9. Meng ZY, Hung H-H, and Lang TC (2014) The characterization of topological properties in quantum monte carlo simulations of the kane–mele–hubbard model. Modern Physics Letters B 28(01): 1430001. Mishchenko PA, Kato Y, and Motome Y (2021) Quantum monte carlo method on asymptotic lefschetz thimbles for quantum spin systems: An application to the kitaev model in a magnetic field. Physical Review D 104: 074517. Mondaini R, Tarat S, and Scalettar RT (2022) Quantum critical points and the sign problem. Science 375(6579): 418–424. Mooney TC, Bringewatt J, and Brady LT (2021) Lefschetz thimble quantum monte carlo for spin systems. arXiv. preprint arXiv:2110.10699. Mukherjee A and Cristoforetti M (2014) Lefschetz thimble monte carlo for many-body theories: A hubbard model study. Physical Review B 90(3), 035134. Mukherjee A, Cristoforetti M, and Scorzato L (2013) Metropolis monte carlo integration on the lefschetz thimble: Application to a one-plaquette model. Physical Review D 88(5), 051502. Ng K-K and Yang M-F (2017) Field-induced quantum phases in a frustrated spin-dimer model: A sign-problem-free quantum monte carlo study. Physical Review B 95: 064431. Ouyang Y and Xu XY (2021) Projection of infinite-u hubbard model and algebraic sign structure. Physical Review B 104: L241104. Pan G, Zhang X, Li H, Sun K, and Meng ZY (2022) Dynamical properties of collective excitations in twisted bilayer graphene. Physical Review B 105: L121110. Prokof’Ev NV, Svistunov BV, and Tupitsyn IS (1998) Exact, complete, and universal continuous-time worldline monte carlo approach to the statistics of discrete quantum systems. Journal of Experimental and Theoretical Physics 87(2): 310–321. Ringel Z and Kovrizhin DL (2017) Quantized gravitational responses, the sign problem, and quantum complexity. Science Advances 3(9), e1701758. Rossi R (2017) Determinant diagrammatic monte carlo algorithm in the thermodynamic limit. Physical Review Letters 119: 045701. Rossi R, Prokof’ev N, Svistunov B, Van Houcke K, and Werner F (2017) Polynomial complexity despite the fermionic sign. EPL (Europhysics Letters) 118(1): 10004. Sandvik AW (1992) A generalization of handscomb’s quantum monte carlo scheme-application to the 1d hubbard model. Journal of Physics A: Mathematical and General 25(13): 3667. Sandvik AW (2010) Computational studies of quantum spin systems. AIP Conference Proceedings 1297(1): 135–338. Sandvik AW and Kurkijärvi J (1991) Quantum monte carlo simulation method for spin systems. Physical Review B 43(7): 5950. Shinaoka H, Nomura Y, Biermann S, Troyer M, and Werner P (2015) Negative sign problem in continuous-time quantum monte carlo: Optimal choice of single-particle basis for impurity problems. Physical Review B 92: 195126. Smith A, Golan O, and Ringel Z (2020) Intrinsic sign problems in topological quantum field theories. Physical Review Research 2: 033515. Stapmanns J, Corboz P, Mila F, Honecker A, Normand B, and Wessel S (2018) Thermal critical points and quantum critical end point in the frustrated bilayer heisenberg antiferromagnet. Physical Review Letters 121: 127201. StefanWessel BN, Mila F, and Honecker A (2017) Efficient Quantum Monte Carlo simulations of highly frustrated magnets: the frustrated spin-1/2 ladder. SciPost Physics 3: 005.

The sign problem in quantum Monte Carlo simulations

893

Suzuki M (1985) General correction theorems on decomposition formulae of exponential operators and extrapolation methods for quantum monte carlo simulations. Physics Letters A 113(6): 299–300. Suzuki M (1976) Relationship between d-dimensional quantal spin systems and (d+ 1)-dimensional ising systems: equivalence, critical exponents and systematic approximants of the partition function and spin correlations. Progress of Theoretical Physics 56(5): 1454–1469. 11. Takasu M, Miyashita S, and Suzuki M (1986) Monte Carlo simulation of quantum Heisenberg magnets on the triangular lattice. Progress of Theoretical Physics 75(5): 1254–1257. Tarat S, Xiao B, Mondaini R, and Scalettar RT (2022) Deconvolving the components of the sign problem. Physical Review B 105: 045107. Torlai G, Carrasquilla J, Fishman MT, Melko RG, and Fisher MPA (2020) Wave-function positivization via automatic differentiation. Physical Review Research 2: 032060. Trotter HF (1959) On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10(4): 545–551. Troyer M and Wiese U-J (2005) Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations. Physical Review Letters 94: 170201. Ulybyshev M, Winterowd C, and Zafeiropoulos S (2020) Lefschetz thimbles decomposition for the hubbard model on the hexagonal lattice. Physical Review D 101: 014508. Vaezi M-S, Negari A-R, Moharramipour A, and Vaezi A (2021) Amelioration for the sign problem: An adiabatic quantum monte carlo algorithm. Physical Review Letters 127: 217003. Wan Z-Q, Zhang S-X, and Yao H (2020) Mitigating sign problem by automatic differentiation. arXiv. preprint arXiv:2010.01141. Wang L, Liu Y-H, Iazzi M, Troyer M, and Harcos G (2015) Split orthogonal group: A guiding principle for sign-problem-free fermionic simulations. Physical Review Letters 115: 250601. Wang W, Lu D-C, Xu XY, You Y-Z, and Meng ZY (2019) Dynamics of compact quantum electrodynamics at large fermion flavor. Physical Review B 100: 085123. Wei Z-C (2017) Semigroup approach to the sign problem in quantum monte carlo simulations. arXiv. preprint arXiv:1712.09412. Wei ZC, Wu C, Li Y, Zhang S, and Xiang T (2016) Majorana positivity and the fermion sign problem of quantum monte carlo simulations. Physical Review Letters 116: 250601. Wessel S, Niesen I, Stapmanns J, Normand B, Mila F, Corboz P, and Honecker A (2018) Thermodynamic properties of the shastry-sutherland model from quantum monte carlo simulations. Physical Review B 98: 174432. White SR, Scalapino DJ, Sugar RL, Loh EY, Gubernatis JE, and Scalettar RT (1989) Numerical study of the two-dimensional hubbard model. Physical Review B 40: 506–516. Witten E (2010) A new look at the path integral of quantum mechanics. arXiv. preprint arXiv:1009.6032. Witten E (2011) Analytic continuation of chern-simons theory. AMS/IP Studies in Advanced Mathematics 50: 347. Wynen J-L, Berkowitz E, Krieg S, Luu T, and Ostmeyer J (2021) Machine learning to alleviate hubbard-model sign problems. Physical Review B 103: 125153. Xu XY, Law KT, and Lee PA (2018) Kekulé valence bond order in an extended hubbard model on the honeycomb lattice with possible applications to twisted bilayer graphene. Physical Review B 98: 121406. Xu XY, Liu ZH, Pan G, Qi Y, Sun K, and Meng ZY (2019a) Revealing fermionic quantum criticality from new monte carlo techniques. Journal of Physics: Condensed Matter 31(46): 463001. Xu XY, Qi Y, Zhang L, Assaad FF, Xu C, and Meng ZY (2019b) Monte carlo study of lattice compact quantum electrodynamics with fermionic matter: The parent state of quantum phases. Physical Review X 9: 021022. Xu Z, Pan G, Zhang Y, Kang J, and Meng ZY (2021a) Momentum space quantum monte carlo on twisted bilayer graphene. Chinese Physics Letters 38(7): 077305. Xu Z, Sun K, Li H, Pan G, and Meng ZY (2021b) Superconductivity and bosonic fluid emerging from moir\’e flat bands. arXiv. preprint arXiv:2111.10018. Zhang X, Pan G, Xu XY, and Meng ZY (2021) Sign problem finds its bounds. arXiv. preprint arXiv:2112.06139.

Quantum transport and electron-electron interactions in one dimension Pedro Vianez and Christopher Ford, Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom © 2024 Elsevier Ltd. All rights reserved.

Introduction The mesoscopic regime Experimental realization of a (quasi-)1D system and quantization of conductance Interaction effects in 1D Conclusion Further reading References

894 895 896 899 903 903 903

Abstract When confined to lower dimensions, electrons often start behaving in new and unexpected ways. This is particularly striking in one dimension (1D), for here, both individual, as well as collective, properties can start to emerge, making these systems a very rich playground on which single- and many-body physics can be studied. In this chapter, we introduce the field of one-dimensional electronic systems by discussing a series of results in fundamental physics obtained over the past few decades using semiconductor devices at low temperatures. We start by reviewing the transport properties of these systems and, specifically, the observation of conductance quantization in 1D. We then focus on a series of interaction effects that have been observed in more recent years, from the well-known “0.7 structure” to the Tomonaga-Luttinger liquid and its more recent nonlinear counterparts.

Key points This chapter describes the one-dimensional electronic system with particular emphasis on its experimental realization using semiconducting devices. It aims to provide a first introduction to the subject by highlighting some of the early work, as well as more recent developments in the field. The topics covered are:

• • • • •

the experimental realization of a quasi-1D system; its subbands, eigenstates and conductance quantization in units of 2e2/h; the “0.7 structure,” which is a feature below the lowest conductance plateau and requires an explanation in terms of many-body physics; the Tomonaga-Luttinger liquid, which is the exact many-body description of a true 1D system at low energies, with particular emphasis on the observation of spin-charge separation and zero-bias anomaly; the nonlinear Luttinger liquid, which tries to describe interacting 1D systems beyond the low-energy limit, including recent developments on 1D “replica” modes and separate Fermi seas for spin and charge.

Introduction How do electrons move inside solids? Do they flow, like a stream down a hill, or instead remain static, close to their atomic hosts? Furthermore, when in the presence of other electrons, do they behave collectively or are they instead, for the most part, unaware of the presence of each other? The challenge posed by these questions is the bedrock of modern-day condensed-matter physics, and to answer them we must first understand how and where electrons are moving in the first place. The world is three-dimensional, but we mainly live on a two-dimensional flat surface, and sometimes a flow is confined to a single dimension, a line, for example, in rivers and roads. In a conducting solid, electrons flow to carry a current. But a wire is huge on the scale of the distance an electron can move before being scattered, so it is not usually 1D. With the advent of nanoelectronics, we can grow and pattern materials that are so pure that scattering can almost be ignored on the length scales of the microscopic wires in which we can channel electrons. We then start to see extraordinary new phenomena, where electrons act as waves and form standing waves across the wire, showing beautiful, equally spaced steps in the inverse of the resistance, which is called the conductance. However, electrons are charged particles and repel each other, so interactions should be important. Remarkably, these interactions are usually hidden because excitations of a collection of interacting electrons actually behave like single electrons. In 1D, on the other hand, this is not the case—as on a single-track road, cars (or electrons) cannot get past each other. So, instead, waves of charge and spin travel along the queues of electrons, more like sound waves, and these can be detected in special experiments, as we will now discuss.

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The mesoscopic regime The first attempt at describing charge transport inside solids (and specifically, metals) led to what came to be known as the Drude model (Drude, 1900; Ashcroft and Mermin, 1976). The goal was to explain what was empirically already established as Ohm’s law, that is, that the current through a conductor is directly proportional to the voltage applied along it, with the proportionality constant defining the resistance, R, of a given material. Drude managed to explain this observation in terms of scattering of electrons against a static background of ions which, macroscopically, gave rise to an electrical resistance. Drude’s model was essentially an application of the kinetic theory of gases, whereby the electrons inside a conductor were seen as bouncing about like billiard balls and following the laws of classical Newtonian mechanics. However, once the theory of Quantum Mechanics was developed in the 1920s, it became clear that subatomic particles such as electrons should be treated as waves (or wave functions), rather than simple point-like particles. Bloch showed that the periodic potential from the crystal lattice in a solid had the effect of combining all the quantum-mechanical states on each atom into a set of waves that modulate the wave functions on each atom and broaden their quantized energy levels into “bands” (Bloch, 1929). We generally say that these plane “Bloch” waves have wavelength 2p/k, where k is called the wavenumber (or, including the direction, k, the wavevector). Having established the existence of electronic bands, however, one must now question how to populate the available states. For simplicity, let us assume that no interactions are present, that is, everything can be described in terms of plane waves Because electrons are spin-1/2 particles, they are fermions, meaning that they follow Fermi-Dirac statistics. A well-known consequence of this is the Pauli exclusion principle. Put simply, this prevents any two fermions from occupying the same state, unless they differ in spin direction (“up” or “down”). This has profound consequences in the way electrons are distributed energetically inside a crystal, and leads us to the concept of a Fermi surface, see Fig. 1. At zero temperature, all the states are filled with electrons, starting with the lowest energy until all the available electrons have been accommodated. The Fermi energy EF lies between the energy of this highest occupied state and the next, empty, state. In three dimensions (3D), the Fermi surface for a system of free electrons is a simple sphere or, in two dimensions (2D), a circle. Here, the most energetic state defines the Fermi energy, EF ¼ ħ2k2F /2me, where kF is the Fermi wavevector and me the free-electron mass. In one dimension (1D), however, things start to change, as the Fermi “surface” now is simply two degenerate points at kF! As we will see later, this drastically affects the dynamics of electrons when confined to 1D. The principal effect of the presence of band structure is to change me to the “effective” mass m , which is a constant near the extremities of each band. Generally speaking, however, m tends to be a little less than me in semiconductors (e.g., 0.067 me in GaAs). The implicit assumption behind all these pictures is that the electrons can be treated as occupying single-particle states, each with a different wavevector k, in other words neglecting the possibility of many-body effects. Similarly, by treating the electrons as purely classical point-like particles, or quantum-mechanical “wave packets” with well-defined momentum and some idea of their position, collisions are taken to be instantaneous or, at least, much shorter than the average time it takes for them to occur. This means that, when in the presence of some sort of physical gradient, (e.g., temperature, electric or magnetic fields, stress, etc.), transport across a system of characteristic length L  ℓ is typically diffusive, with electrons behaving as if undergoing a random walk of typical step size ℓ. Alternatively, if L  ℓ, then we enter into the ballistic regime, meaning that electron motion can only be altered by collision with a boundary. In the middle between these two limits, we enter the mesoscopic regime, where one cannot average over scattering sites, such as impurities, and each “device” may behave slightly differently. In order for mesoscopic or ballistic effects to be observed, electrons must be confined to sufficiently small structures and/or cooled to sufficiently low temperatures. The field of quantum transport concerns, generally speaking, the study of electron flow inside materials and devices confined in at least one of the dimensions to a nanometer length-scale ( Uc, in the Mott insulating phase). The vertical dotted line marks the Fermi energy. W is the noninteracting bandwidth, ZW the width of the renormalized quasiparticle peak.

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The Hubbard model and the Mott–Hubbard transition

of weak coupling (Gutzwiller) and strong coupling (Hubbard), but neither of them provides a unified picture which is sufficiently accurate in both limits.

Dynamical mean-field theory A turning point in the modern understanding of the Mott–Hubbard transition has been the development of Dynamical Mean-Field Theory (DMFT) (Georges et al., 1996), a powerful approach which treats accurately both limits of weak and strong interactions. While the origin of the method lies in pioneering investigations of the limit of infinite dimensions (Metzner and Vollhardt, 1989), the best way to understand the method is a quantum (or dynamical) version of a mean-field theory, where the effect of the rest of the lattice on a single site—representative of any other site—is described by a frequency (or time)-dependent effective field which is determined self-consistently. The self-consistency is enforced by assuming a local self-energy and that the Green’s function of the effective local theory coincides with the local component of the lattice Green’s function. The dynamical nature of the effective field substantially extends the scope of the method with respect to static mean field, allowing to study intrinsically many-body phenomena, among which the Mott–Hubbard transition has been the epitome. Starting from the Hubbard model Eq. (1), DMFT defines an effective local theory for an arbitrary site “0” associated with the effective action Z Z Z (4) Seff ¼ dt dt0 c{0s ðtÞG0−1 ðt − t0 Þc0s ðt0 Þ + U dtn0" ðtÞn0# ðtÞ, where t and t0 are imaginary time variables, c{0s and c0s are Grassman fields associated with creation and annihilation of fermions with spin s on site 0, n0s ¼ c{0s c0s. The effective action describes the (imaginary) time evolution of site 0 in the presence of the local repulsion, which is controlled by the dynamical Weiss field G0−1 ðt − t0 Þ, which contains the effect of the rest of the lattice on site 0. Similarly to a classical mean-field theory, the dynamical mean field is determined by a self-consistency condition which requires that the Green’s function of the effective theory GðtÞ ¼ −hTc0s ðtÞc{0s ð0ÞiSef f

(5)

coincides with the local component of the lattice Green’s function. Namely, in Matsubara frequencies Z 1X 1 1 Gðion Þ ¼  deDðeÞ , N io io + m − e − Sðio Þ + m − e − Sðion Þ n n n k k

(6)

where S(ion) is the Green’s function of the effective theory, connected to the Weiss field and the Green’s function by Sðion Þ ¼ G0−1 ðion Þ − G −1 ðion Þ:

(7) G0−1 ðion Þ,

−1

and G (ion). The three Eqs. (5), (6), and (7) are a closed set of equations that allow to find a self-consistent set of S(ion), The lattice enters only via the self-consistency condition which depends on the noninteracting density of states. A practical implementation of DMFT amounts to a recursive solution of the effective theory (4) computing the Green’s function (5) starting from a guess for G0−1 ðion Þ . Then the self-consistency is used to compute a new G0−1 ðion Þ and the procedure is iterated until convergence is achieved. The solution of the effective theory usually exploits a mapping onto an impurity Hamiltonian which, for the Hubbard model, is an Anderson impurity model. Different “impurity solvers” can be used to obtain the Green’s function, each with specific advantages and disadvantages. The combined use of different solvers led the community to a complete understanding of the Mott transition in the single-band Hubbard model which represents the basis for extensions including more orbitals or partially relaxing the DMFT approximation. As we mentioned above, DMFT is exact not only in the limit of infinite coordination (or dimensionality) but also in finite dimensionality in the opposite limits of U ¼ 0 (noninteracting limit) and t ¼ 0 (atomic limit). This makes the theory ideally suited to study the Mott transition, as it correctly reproduces both the opposite limits and, being exact in one well-defined limit, it is a conserving approximation. We notice in passing that DMFT remains exact in the same limit also including other local interaction terms, like exchange terms in multiorbital systems (Georges et al., 2013) or electron–phonon coupling (Capone et al., 2010).

Key issues The Mott–Hubbard transition in DMFT The picture of the Mott transition in DMFT is summarized in Figs. 1 and 2. In Fig. 1, we report a schematic sketch of the evolution of the local many-body one-particle spectral function 1 rðoÞ ¼ − ImGðoÞ p

(8)

as the ratio U/t is increased. r(o) can be seen as an interacting local density of states or the integral over momentum of the spectral density A(k, o).

The Hubbard model and the Mott–Hubbard transition

T

919

PM-I

PM-M

TN ∝

t

TN ∝ e−α U

t2 U

U/t Fig. 2 Schematic phase diagram of the Mott transition for the half-filled single-band Hubbard model in the U/t-T plane. The dashed red line is the Néel temperature for anriferromagnetic order. The blue dotted line is the first-order Mott transition. The endpoints marked with a blue dot are second-order transitions. PM-M and PM-I label the paramagnetic metallic and paramagnetic insulating solutions.

For U ¼ 0, the system is a metal with a half-filled band of total width W. When we increase the strength of the interaction, a part of the spectral weight shifts toward higher energy. At first this effect is essentially a broadening of the spectrum, but further increasing the interaction reveals a three-feature spectrum where a low-energy quasiparticle peak is flanked by two high-energy features centered around o ’ U/2. As the interaction grows this feature becomes clearly separated from the quasiparticle peak, which in turn shrinks. The shrinking of the low-energy feature is measured by the quasiparticle weight Z, which is obtained from the self-energy from Z −1 ¼ 1 − ∂SðoÞ ∂o . For U ¼ 0 we have Z ¼ 1, while at a critical value of the interaction Uc, Z vanishes, which means that the quasiparticle feature completely disappears, leaving behind two high-energy features which are usually referred to as Hubbard bands in connection with the original work by Hubbard. The continuous disappearance of the quasiparticle peak is the hallmark of the Mott–Hubbard transition at Uc. For a model with a semicircular density of states with half-bandwidth D (proportional to the hopping t), Uc ’ 2.94D (Bulla et al., 2001). We notice that Uc is also the largest value for which a metallic solution exists, while the insulating solution exists also for Uc1 < U < Uc, with Uc1 ’ 2D, leading to a wide region of parameters where two solutions exist (Georges et al., 1996). In Fig. 2, we show the resulting phase diagram in the plane of interaction (U/t) and temperature (T ). The coexistence of solutions extends to finite temperature so that the second-order transition at zero temperature becomes at finite temperature a first-order line that ends in a finite-temperature critical point at T ’ 0.04D (Zhang et al., 1993; Georges and Krauth, 1992; Georges et al., 1996). At temperatures above the critical point, a continuous crossover connects solution with metallic and insulating character. In Fig. 2, we have also reported a sketch of the Néel temperature for a three-dimensional system. We have to stress that this curve strongly depends on the chosen lattice and on the existence of long-range hopping. For nearest-neighbor hopping the Néel temperature is indeed larger than the critical point of the Mott transition and the whole first-order line is hidden by the region of stability of the antiferromagnetic state. Yet, already for moderate values of next-neighbor hoppings, the magnetic ordering temperature is reduced and we obtain a phase diagram which resembles that of Fig. 1A, where the first-order paramagnetic Mott line emerges from the magnetic dome (Georges et al., 1996). Crucially, this phase diagram mirrors the celebrated pressure-temperature diagram of V2O3 (McWhan et al., 1973), which is often considered the paradigm of Mott transitions. We summarize here some of the main new results of DMFT for which the consensus is general and many experimental confirmations have been obtained

• • •

The Mott–Hubbard transition is of first-order at finite temperature except for the second-order endpoint without invoking the coupling with the phonons or any structural transition. In the strongly correlated metallic solution, metallic features and insulating features are simultaneously present at different energy scales. The competition with ordered phases depends on the details of the lattice, but realistic parameters provide a phase diagram close to those experimentally observed in transition-metal oxides.

The DMFT scenario for the Mott transition has been extremely influential in the field of strongly correlated systems providing the community with a solid, even if approximate, picture which is easy to visualize and compare with experiments. In particular, combining DMFT with density-functional theory (DFT) (Hohenberg and Kohn, 1964) has been extremely successful in providing a method to compute ab initio the properties of strongly correlated materials which are not accessible by standard implementations of DFT (Kotliar et al., 2006). Another fundamental extension of DMFT is given by methods that include nonlocal dynamical correlations besides the local frequency-dependent self-energy. Several approaches have been devised and developed that can be grouped in two main categories: (i) cluster extensions of DMFT, where the single-site theory is generalized into a cluster theory and a similar self-consistent scheme is derived (Maier et al., 2005); (ii) diagrammatic extensions generalizing the DMFT construction in different directions (Rohringer et al., 2018). These approaches have a somewhat complementary character and they allowed to apply the DMFT ideas to the twodimensional Hubbard model. As we already mentioned, this field is however still competitive and we prefer to defer the interested reader to previous review work (Maier et al., 2005; Rohringer et al., 2018).

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The Hubbard model and the Mott–Hubbard transition

Finally, a realistic description of most strongly correlated materials requires to go beyond the single-band Hubbard model including more than one correlated orbital in the theoretical modeling. In the next section we discuss some aspects of this physics in order to emphasize some novel phenomena characteristic of multiorbital (or multiband) Hubbard models that are simply absent in the single-band system.

Multiorbital Hubbard model The most straightforward multiorbital extension of the Hubbard model to include M atomic orbitals (each hosting spin-1/2 fermions) is an SU(2M) invariant model of the form H¼

X

t mn c{ims ^cjns + ij ^

i, j, m, n, s

UX 2 n^ , 2 i i

(9)

where i and j are site indices, m and n are orbital indices ranging from 1 to M, and s is a spin label. Hence ^c{ims creates a fermion with P spin s on site i and orbital m. The Coulomb term involves the total charge on site i n^i ¼ ms ^c{ims ^cims . This term generalizes the Hubbard repulsion to the multiorbital case, simply assuming a density–density repulsion between fermions with different quantum numbers with the same strength for any pair of indices. The first novelty of a multiorbital model is that the Mott transition is no longer limited to the case of half-filling, which here would correspond to M electrons per site, but it can take place for any integer filling N ¼ 1, . . . 2M −1. The physical picture is quite obvious. If U is the largest energy scale, the ground state has N electrons on any site, while any charge excitation has an energy cost U. Thus, for large U, any charge excitation is forbidden and N electrons remain localized on each site. Solving this model with DMFT does not change qualitatively the picture we described in the previous section. The metal is strongly affected by an increased U until the quasiparticle low-energy feature is destroyed and a Mott insulator is realized. There is only a quantitative dependence of the results on the number of orbitals and the filling (Florens et al., 2002). While these result would suggest that increasing the number of orbitals has only a quantitative effect on Mott physics, it must be noted that Eq. (9) is only an approximation of the correct Hamiltonian that one obtains expanding the Coulomb interaction on a basis of localized orbitals. The generic Coulomb integral for a given site is Z   (10) U mnm0 n0 ¼ drdr 0 fm ðrÞfn ðr 0 ÞWðr − r 0 Þfn0 ðr 0 Þfm0 ðrÞ where fm(r) is a localized wave function for an electron in the orbital m (the site index is the same for every function and it is omitted), while W(r − r0 ) is the Coulomb interaction screened by the other electronic bands. In rather general terms we can define an intra-orbital repulsion Z U m ¼ drdr 0 jfm ðrÞj2 Wðr − r 0 Þjfm ðr 0 Þj2 , (11) an interorbital density–density repulsion U 0mn ¼

Z

drdr 0 jfm ðrÞj2 Wðr − r 0 Þjfn ðr 0 Þj2 ,

(12)

an exchange term Z Jmn ¼





(13)





(14)

drdr 0 fm ðrÞfn ðr 0 ÞV c ðr − r 0 Þfm ðr 0 Þfn ðrÞ

and a pair integral J0mn ¼

Z

drdr 0 fm ðrÞfm ðr 0 ÞV c ðr − r 0 Þfn ðr 0 Þfn ðrÞ:

If we assume that the various orbitals are equivalent Um  U, U 0m  U 0 , Jm  J, and J0m  J0 . For real wave functions we also obtain J0 ¼ J. It follows from the definitions that U > U0 > J. Thus the interaction Hamiltonian becomes P P P Hint ¼ U n^im" n^im# + U 0 n^im" n^in# + ðU 0 − JÞ n^ims n^ins + im i , m>n ,s i, m6¼n (15) PP { P P ^cim" ^cim# ^c{in# ^cin" + J ^c{im" hatc{im# ^cin# ^cin" −J i m6¼n

0

i m6¼n

If we finally assume U ¼ U − 2J, the interaction becomes invariant under orbital and spin rotations and it can be written in the form  X 2 n^ ðn^ −1Þ J 2 5 (16) ðU − 3JÞ i i − 2J S^i − L^i + J n^i , Hint ¼ 2 2 2 i

The Hubbard model and the Mott–Hubbard transition

921

where L^i and S^i are the orbital and spin angular momentum of the electrons on site I. The final form of the interaction conveys a clear physical meaning. The first term is a Hubbard-like interaction where the repulsion U becomes U − 3J, while the second and third terms, both proportional to J, give rise to the first two Hund’s rules. For this reason, J is called the Hund’s coupling. The fourth term can be reabsorbed in a redefinition of the chemical potential.

Role of the Hund’s coupling The inclusion of the Hund’s coupling J and the corresponding terms turns out to have remarkable effects on the Mott–Hubbard transition and on the correlation properties of the metallic phase which the object of current research, both in terms of model investigations and of applications to materials, including iron-based superconductors (Haule and Kotliar, 2009; de’ Medici and Capone, 2017) and ruthenates (Kim et al., 2018). In this work we do not attempt a full review of this very active field of research, for which we refer to Georges et al. (2013). We provide however some arguments to provide some intuition about the role of the Hund’s coupling. In order to estimate the effect of J on the Mott transition, we can easily estimate the critical U for the Mott transition via the atomic value of the Mott gap. For a system with a filling of n electrons per site, we can compute the atomic Mott gap as the cost of exciting one electron from one site to another as D(U, J, N) ¼ E(N + 1) + E(N − 1) − 2E(N), where E(N) is the atomic ground state for a system of N electrons. From this quantity we can estimate Uc from the condition D(U, J, N) ¼ W, where W is the bandwidth. For a single-band model, or for J ¼ 0, it is easy to see that D ¼ U. When we include J, we obtain two radically different results for N ¼ M, that is, when the system is globally half filled, and for other integer filling N6¼M (de’ Medici et al., 2011; Georges et al., 2013). For N ¼ M, we obtain D ¼ U + (M − 1)J, that is, the Mott gap is increased by J. Using the above criterion, one finds that Uc(J) ¼ Uc(0) − (M − 1)J. This means that the Hund’s coupling reduces the critical U for the Mott transition. On the other hand, for N 6¼ M, D ¼ U − 3J, that is, a reduction of the Mott gap. The critical U is increased by J according to Uc(J) ¼ Uc(0) + 3J. These estimates are indeed qualitatively confirmed by DMFT calculations for that unveiled a richer scenario with respect to the single-band model. The key point is that for small values of U and J, the latter parameter cooperates with U in localizing the electrons. This is reflected by a quasiparticle weight Z that drops faster as a function of U than in the J ¼ 0 case. For N ¼ M, this trend is compatible with the reduction of the critical U: As a result the Mott transition occurs in a similar way as in the absence of J (or in the single-band model), only for smaller values of U. On the other hand, for N6¼M we have a dichotomy in the effect of J it first favors Mott localization but, for values of the interactions, it leads shifts he critical U to larger values. We have thus a two-step localization where a first rapid reduction of the electronic mobility is followed by a large region where Z is almost flat before a Mott transition takes place (de’ Medici et al., 2011). This interaction-resistant metal has been called a Hund’s metal (Haule and Kotliar, 2009). The investigation of the properties of Hund’s metals is currently a very active field of research encompassing fundamental research and applications to different materials, including iron-based superconductors (de’ Medici et al., 2014; de’ Medici and Capone, 2017), ruthenates (Kim et al., 2018). We notice in passing that the anomalous phonondriven superconductivity of alkalidoped fullerides is understood in the terms of a model of the form (16) with a negative Hund’s coupling J ¼ −|J| arising from electron–phonon coupling (Capone et al., 2009; Nomura et al., 2015). Another quite general phenomenon which is realized in multiorbital correlated systems, especially when the Hund’s coupling is sizable, is orbital-selective Mott physics. An orbital-selective Mott insulator is defined as a system in which the electrons is some orbitals are Mott localized, while those of other orbitals remain itinerant (Vojta, 2010). The metallic region close to an orbitalselective Mott transition is characterized by orbital-selective correlations, that is, the coexistence of carriers with remarkably different correlation properties. A large body of experimental data on iron-based superconductors can indeed be understood within this framework, both in the metallic normal state (de’ Medici et al., 2014; Capone, 2018; Kostin et al., 2018) and in the superconducting state, in which orbital-selective pairing has been observed (Sprau et al., 2017).

Conclusion In this work we have outlined the scenario of the Mott–Hubbard transition obtained within DMFT (Georges et al., 1996). Regardless of the accuracy of the method, the DMFT description of the Mott transition can be considered as a paradigm of “pure” Mott physics because the method treats accurately the competition between the delocalizing effect of the hopping term and the blocking effect of the Hubbard local repulsion and it allows to disentangle this intrinsic Mott localization with the tendency to form spatially ordered phases, like the antiferromagnetic phase of the single-band Hubbard model. In this sense, we consider the DMFT scenario as a paradigm to be used as the basic backbone of calculations including nonlocal dynamical correlations and/or more realistic descriptions of materials. The picture that emerges within DMFT is richer than previously expected. The evolution from a standard metal to a paramagnetic Mott insulator is characterized by the development of a strongly correlated metallic state in which the low-energy properties of the system are those of a strongly renormalized Fermi liquid, while the high-energy behavior is similar to a precursor of an insulator. The Mott–Hubbard transition is associated with the disappearance of the low-energy quasiparticle features, leaving behind a preformed gap (Georges et al., 1996). At finite temperature, the transition becomes a first-order line which ends in a critical point (Zhang et al., 1993; Georges and Krauth, 1992; Georges et al., 1996). The first-order character of the Mott transition is a remarkable prediction of DMFT which is

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actually in agreement with the phenomenology of transition-metal oxides, where a first-order line emerges from a magnetic lowtemperature state. This demonstrates that the first-order character of the transition is of electronic origin, in contrast with previous expectations that the coupling with the lattice was a necessary condition to make the transition discontinuous. The DMFT scenario of the Mott transition and its prediction on several observables have been verified in three-dimensional strongly correlated materials (McWhan et al., 1973; Furukawa et al., 2015), leading to a substantial advance both in our understanding and in our ability to accurately compute the properties of this class of materials. However, a full account of the properties of many materials requires to relax some of the approximations behind the single-band Hubbard model and/or to address the role of low dimensionality. In particular, a realistic description of most correlated materials requires to include multiple low-energy orbitals. This extension is far from trivial, in particular when the Hund’s exchange is not small (Georges et al., 2013). In this case, novel phenomena like a two-step Mott localization and orbital-selective Mott physics have been identified in the recent years. Finally, several methods have been proposed to extend DMFT including nonlocal correlation effects. Different approaches, ranging from cluster extensions (Maier et al., 2005) to diagrammatic methods focusing on two-particle correlations (Rohringer et al., 2018) are currently being developed in order to close the gap that separates us from a complete theory of the Mott transition in two dimensions.

References Anderson PW (1987) The resonating valence bond state in La2CuO4 and superconductivity. Science 235(4793): 1196–1198. https://doi.org/10.1126/science.235.4793.1196. Arovas DP, Berg E, Kivelson SA, and Raghu S (2022) The Hubbard model. Annual Review of Condensed Matter Physics 13(1): 239–274. https://doi.org/10.1146/annurevconmatphys-031620-102024. Bednorz JG and Müller KA (1986) Possible high Tc superconductivity in the Ba-La-Cu-O system. Zeitschrift fur Physik B Condensed Matter 64: 189–193. https://doi.org/10.1007/ BF01303701. Bulla R, Costi TA, and Vollhardt D (2001) Finite-temperature numerical renormalization group study of the Mott transition. Physical Review B 64: 045103. https://doi.org/10.1103/ PhysRevB.64.045103. Capone M (2018) Orbital-selective metals. Nature materials 17(10): 855–856. Capone M, Fabrizio M, Castellani C, and Tosatti E (2009) Colloquium: Modeling the unconventional superconducting properties of expanded A3C60 fullerides. Reviews of Modern Physics 81: 943–958. https://doi.org/10.1103/RevModPhys.81.943. Capone M, Castellani C, and Grilli M (2010) Electron-phonon interaction in strongly correlated systems. Advances in Condensed Matter Physics 2010: 920860. ISSN: 1687-8108. https://doi.org/10.1155/2010/920860. Castellani C, Di Castro C, and Grilli M (1995) Singular quasiparticle scattering in the proximity of charge instabilities. Physical Review Letters 75: 4650–4653. https://doi.org/10.1103/ PhysRevLett.75.4650. de Boer JH and Verwey EJW (1937) Semi-conductors with partially and with completely filled 3d-lattice bands. Proceedings of the Physical Society 49(4S): 59. https://doi.org/ 10.1088/0959-5309/49/4S/307. de’ Medici L and Capone M (2017) Modeling many-body physics with slave-spin mean-field: Mott and Hund’s physics in Fe-superconductors. In: Mancini F and Citro R (eds.) The Iron Pnictide Superconductors: An Introduction and Overview, pp. 115–185. Cham: Springer International. de’ Medici L, Mravlje J, and Georges A (2011) Janus-faced influence of Hund’s rule coupling in strongly correlated materials. Physical Review Letters 107: 256401. https://doi.org/ 10.1103/PhysRevLett.107.256401. de’ Medici L, Giovannetti G, and Capone M (2014) Selective Mott physics as a key to iron superconductors. Physical Review Letters 112: 177001. https://doi.org/10.1103/ PhysRevLett.112.177001. Florens S, Georges A, Kotliar G, and Parcollet O (2002) Mott transition at large orbital degeneracy: Dynamical mean-field theory. Physical Review B 66: 205102. https://doi.org/ 10.1103/PhysRevB.66.205102. Furukawa T, Miyagawa K, Taniguchi H, Kato R, and Kanoda K (2015) Quantum criticality of Mott transition in organic materials. Nature Physics 11(3): 221–224. ISSN: 1745-2481. https://doi.org/10.1038/nphys3235. Georges A and Krauth W (1992) Numerical solution of the d ¼ 1 Hubbard model: Evidence for a Mott transition. Physical Review Letters 69: 1240–1243. https://doi.org/10.1103/ PhysRevLett.69.1240. Georges A, Kotliar G, Krauth W, and Rozenberg MJ (1996) Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics 68: 13–125. https://doi.org/10.1103/RevModPhys.68.13. Georges A, de’ Medici L, and Mravlje J (2013) Strong correlations from hund’s coupling. Annual Review of Condensed Matter Physics 4(1): 137–178. https://doi.org/10.1146/ annurev-conmatphys-020911-125045. Gutzwiller MC (1963) Effect of correlation on the ferromagnetism of transition metals. Physical Review Letters 10: 159–162. https://doi.org/10.1103/PhysRevLett.10.159. Haule K and Kotliar G (2009) Coherence-incoherence crossover in the normal state of iron oxypnictides and importance of hund’s rule coupling. New Journal of Physics 11(2): 025021. https://doi.org/10.1088/1367-2630/11/2/025021. Hohenberg P and Kohn W (1964) Inhomogeneous electron gas. Physical Review 136: B864–B871. https://doi.org/10.1103/PhysRev.136.B864. Hubbard J (1963) Electron correlations in narrow energy bands. Proceedings of the Royal Society of London, Series A 276: 238–257. https://doi.org/10.1098/rspa.1963.0204. https://www.osti.gov/biblio/4151988. Imada M, Fujimori A, and Tokura Y (1998) Metal-insulator transitions. Reviews of Modern Physics 70: 1039–1263. https://doi.org/10.1103/RevModPhys.70.1039. Kanamori J (1963) Electron correlation and ferromagnetism of transition metals. Progress of Theoretical Physics 30(3): 275–289. ISSN: 0033-068X. https://doi.org/10.1143/ PTP.30.275. https://academic.oup.com/ptp/article-pdf/30/3/275/5278869/30-3-275.pdf. Kim M, Mravlje J, Ferrero M, Parcollet O, and Georges A (2018) Spin-orbit coupling and electronic correlations in Sr2RuO4. Physical Review Letters 120: 126401. https://doi.org/ 10.1103/PhysRevLett.120.126401. Kostin A, Sprau PO, Kreisel A, Chong YX, Böhmer AE, Canfield PC, Hirschfeld PJ, Andersen BM, and Davis JCS (2018) Imaging orbital-selective quasiparticles in the Hund’s metal state of FeSe. Nature Materials 17(10): 869–874. ISSN: 1476-4660. https://doi.org/10.1038/s41563-018-0151-0. Kotliar G, Savrasov SY, Haule K, Oudovenko VS, Parcollet O, and Marianetti CA (2006) Electronic structure calculations with dynamical mean-field theory. Reviews of Modern Physics 78: 865–951. https://doi.org/10.1103/RevModPhys.78.865. Kozik E, Burovski E, Scarola VW, and Troyer M (2013) Néel temperature and thermodynamics of the half-filled three-dimensional Hubbard model by diagrammatic determinant Monte Carlo. Physical Review B 87: 205102. https://doi.org/10.1103/PhysRevB.87.205102.

The Hubbard model and the Mott–Hubbard transition

923

LeBlanc JPF, Antipov AE, Becca F, Bulik IW, Chan GK-L, Chung C-M, Deng Y, Ferrero M, Henderson TM, Jiménez-Hoyos CA, Kozik E, Liu X-W, Millis AJ, Prokof’ev NV, Qin M, Scuseria GE, Shi H, Svistunov BV, Tocchio LF, Tupitsyn IS, White SR, Zhang S, Zheng B-X, Zhu Z, and Gull E (2015) Simons Collaboration on the Many-Electron Problem. Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms. Physical Review X 5 5: 041041. https://doi.org/10.1103/ PhysRevX.5.041041. Lee PA, Nagaosa N, and Wen X-G (2006) Doping a Mott insulator: Physics of high-temperature superconductivity. Reviews of Modern Physics 78: 17–85. https://doi.org/10.1103/ RevModPhys.78.17. Lieb EH (1989) Two theorems on the Hubbard model. Physical Review Letters 62: 1201–1204. https://doi.org/10.1103/PhysRevLett.62.1201. Lieb EH and Wu FY (1968) Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension. Physical Review Letters 20: 1445–1448. https://doi. org/10.1103/PhysRevLett.20.1445. Lin HQ and Hirsch JE (1987) Two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping. Physical Review B 35: 3359–3368. https://doi.org/10.1103/ PhysRevB.35.3359. MacDonald AH, Girvin SM, and Yoshioka D (1988) Ut expansion for the Hubbard model. Physical Review B 37: 9753–9756. https://doi.org/10.1103/PhysRevB.37.9753. Maier T, Jarrell M, Pruschke T, and Hettler MH (2005) Quantum cluster theories. Reviews of Modern Physics 77: 1027–1080. https://doi.org/10.1103/RevModPhys.77.1027. Maiti S and Chubukov AV (2013) Superconductivity from repulsive interaction. AIP Conference Proceedings 1550(1): 3–73. https://doi.org/10.1063/1.4818400. McWhan DB, Menth A, Remeika JP, Brinkman WF, and Rice TM (1973) Metal-insulator transitions in pure and doped V2O3. Physical Review B 7: 1920–1931. https://doi.org/ 10.1103/PhysRevB.7.1920. Mermin ND and Wagner H (1966) Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Physical Review Letters 17: 1133–1136. https://doi.org/10.1103/PhysRevLett.17.1133. Metzner W and Vollhardt D (1989) Correlated lattice fermions in d ¼ 1 dimensions. Physical Review Letters 62: 324–327. https://doi.org/10.1103/PhysRevLett.62.324. Mott NF and Peierls R (1937) Discussion of the paper by de Boer and Verwey. Proceedings of the Physical Society 49(4S): 72. https://doi.org/10.1088/0959-5309/49/4S/308. Nomura Y, Sakai S, Capone M, and Arita R (2015) Unified understanding of superconductivity and Mott transition in alkali-doped fullerides from first principles. Science Advances 1(7): e1500568. https://doi.org/10.1126/sciadv.1500568. Qin M, Schäfer T, Andergassen S, Corboz P, and Gull E (2022) The hubbard model: A computational perspective. Annual Review of Condensed Matter Physics 13(1): 275–302. https://doi.org/10.1146/annurev-conmatphys-090921-033948. Rohringer G, Hafermann H, Toschi A, Katanin AA, Antipov AE, Katsnelson MI, Lichtenstein AI, Rubtsov AN, and Held K (2018) Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. Reviews of Modern Physics 90: 025003. https://doi.org/10.1103/RevModPhys.90.025003. Sachdev S (2010) Where is the quantum critical point in the cuprate superconductors? Physica Status Solidi (B) 247(3): 537–543. https://doi.org/10.1002/pssb.200983037. Sangiovanni G, Toschi A, Koch E, Held K, Capone M, Castellani C, Gunnarsson O, Mo S-K, Allen JW, Kim H-D, Sekiyama A, Yamasaki A, Suga S, and Metcalf P (2006) Static versus dynamical mean-field theory of Mott antiferromagnets. Physical Review B 73: 205121. https://doi.org/10.1103/PhysRevB.73.205121. Schäfer T, Wentzell N, Šimkovic F, He Y-Y, Hille C, Klett M, Eckhardt CJ, Arzhang B, Harkov V, Le Régent F-M, Kirsch A, Wang Y, Kim AJ, Kozik E, Stepanov EA, Kauch A, Andergassen S, Hansmann P, Rohe D, Vilk YM, LeBlanc JPF, Zhang S, Tremblay A-MS, Ferrero M, Parcollet O, and Georges A (2021) Tracking the footprints of spin fluctuations: A multimethod, multimessenger study of the two-dimensional Hubbard model. Physical Review X 11: 011058. https://doi.org/10.1103/PhysRevX.11.011058. Slater JC (1951) Magnetic effects and the Hartree-Fock equation. Physical Review 82: 538–541. https://doi.org/10.1103/PhysRev.82.538. Sprau PO, Kostin A, Kreisel A, Böhmer AE, Taufour V, Canfield PC, Mukherjee S, Hirschfeld PJ, Andersen BM, and Davis JCS (2017) Discovery of orbital-selective Cooper pairing in FeSe. Science 357(6346): 75–80. https://doi.org/10.1126/science.aal1575. Vojta M (2010) Orbital-selective Mott transitions: Heavy fermions and beyond. Journal of Low Temperature Physics 161(1): 203–232. ISSN: 1573-7357. https://doi.org/10.1007/ s10909-010-0206-3. Zhang XY, Rozenberg MJ, and Kotliar G (1993) Mott transition in the d ¼ 1 Hubbard model at zero temperature. Physical Review Letters 70: 1666–1669. https://doi.org/10.1103/ PhysRevLett.70.1666.

Electron quantum optics: A testbed for the Luttinger paradigma Dario Ferraro and Maura Sassetti, Dipartimento di Fisica dell’Università degli Studi di Genova & CNR-SPIN, Genova, Italy © 2024 Elsevier Ltd. All rights reserved.

Introduction Wave-guides, beam-splitters and on-demand single-electron sources Hanbury-Brown-Twiss and Hong-Ou-Mandel interferometry with electrons Luttinger liquid description of interacting quantum Hall edge channels Effects of interaction in the Hong-Ou-Mandel profile: The Leviton case Crystallization of Levitons in the fractional quantum Hall regime Conclusion Acknowledgments References

924 925 926 928 930 930 932 933 933

Abstract Electron quantum optics recently emerged as a new branch of condensed matter physics aimed at creating, manipulating and detecting electronic wave-packets ballistically propagating along one-dimensional mesoscopic channels. Here, theoretical tools elaborated in the framework of the conventional quantum optics based on photons have been reconsidered in order to properly describe interferometric experiments showing an interplay between the fermionic nature of the electrons and the ubiquitous many-body interaction among them. This allowed a direct comparison between the predictions derived within the Luttinger liquid picture for interacting electrons in one-dimension and the experimental observations. This Chapter will provide an overview on this subject, with main focus on the case of wave-packets generated by means of time-dependent voltage pulses.

Key points

• • • •

From quantum optics to electron quantum optics: the dictionary. Experimental milestones: electron partitioning and collisional interferometry. Effects of interaction: the Luttinger paradigma in action. Strongly correlated systems: crystallization in the time domain.

Introduction The technological advancement in the generation, control and measurement of individual electronic degrees of freedom in solid-state devices opened the way to the development of Electron Quantum Optics (EQO) (Bocquillon et al., 2013). This branch of condensed matter physics aims at realizing quantum optics-like experiments with electrons ballistically propagating along mesoscopic channels and has been characterized since the beginning by a remarkable synergy between experimental investigations and theoretical analysis. In this direction, the edge channels of an Hall bar in the quantum regime represent the ideal candidate for the realization of one-dimensional chiral wave-guides (Das Sarma and Pinczuk, 2004). Moreover, thank to the applications of electrostatic gates, these states can be put in contact realizing a Quantum Point Contact (QPC), which mimics the functioning of a beam-splitter. The last ingredient to complete this translation from the photonic to the electronic domain is provided by on-demand single-electron sources. Even if various techniques have been proposed to implement this essential building block, two of them emerged as more promising. The first is based on a quantum dot coupled to the Hall bar whose Fermi level is periodically driven above and below the one of the outer channel of the Hall bar in such a way to trigger the controlled injection of electrons and holes (Fève et al., 2007). The second requires the application of a periodic drive directly on the one-dimensional channels. Despite apparently simpler, this approach needs an extremely high level of control on the form of the applied voltage. Indeed, only a properly quantized train of voltage pulses with Lorentzian profile in time is able to guarantee the injection of purely electronic states, with no additional spurious particle-hole pairs, usually called Levitons (Levitov et al., 1996; Glattli and Roulleau, 2017). Thank to the tools discussed above, two seminal electron interferometers have been realized within a condensed matter setting (Bocquillon et al., 2013). When only one train of excitations reaches the QPC, the measurement of the outgoing current fluctuations allows to access a particle partitioning reminiscent of what observed in the Hanbury-Brown-Twiss (HBT) intensity interferometer. Moreover, when two trains are injected on the opposite sides of the QPC, the electronic analogous of the Hong-Ou-Mandel (HOM) collisional interferometer is achieved. It is characterized by the presence of a dip in the current fluctuations for synchronized

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00073-1

Electron quantum optics: A testbed for the Luttinger paradigma

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injections, signature of the fact that identical fermionic excitations cannot emerge on the same outgoing channel due to the Pauli principle. A similar feature, involving the number of coincidences instead of the fluctuations, represents the footprint of its photonic counterpart. Despite this simple free-electron picture provides important hints of the physics at work in these devices, for technical reasons experiments in this domain are usually carried out with edge states at filling factor n ¼ 2 or even higher, where the Coulomb interaction among the edge channels cannot be neglected. This represents a further major difference between the photonic and the electronic cases and needs to be carefully taken into account. The conventional way to do this is based on the Luttinger liquid description, which properly describes interacting one-dimensional systems, replacing the standard Fermi liquid picture valid at higher dimensions (Giamarchi, 2003). While this interaction only marginally affects the functioning of the HBT interferometer, its role dramatically emerges in HOM interferometers as we will see in the next Sections. Even more interesting are the effects of electron-electron interaction in strongly correlated systems such as the fractional quantum Hall edge states (Das Sarma and Pinczuk, 2004). Here, the HOM noise generated by the collision of Levitons carrying a charge multiple of the electron one shows the emergence of various side dips (Ronetti et al., 2018). These can be interpreted in terms of an interaction-induced crystallization of the injected wave-packets, real-time analog of what observed in Luttinger liquids confined in a potential well. The present Chapter aims at providing a review of the role of interaction in electron quantum optics addressing in detail the points sketched above.

Wave-guides, beam-splitters and on-demand single-electron sources In quantum optics experiments individual photonic degrees of freedom are injected into wave-guides, such as optical fibers, and manipulated using beam-splitters, realized for example by means of half-silvered mirrors. Starting from these elementary building blocks various interferometers have been realized and used to investigate the coherence property of light, the wave/particle nature of the photons and the consequences of their bosonic statistics. The possibility to realize analogous experiments using electrons ballistically propagating along one-dimensional mesoscopic channels in solid-state devices has been investigated since the nineties. A particularly suitable platform to implement this translation from the photonic to the electronic domain is represented by the edge states of the quantum Hall effect (Das Sarma and Pinczuk, 2004). Here, a very clean two-dimensional electron gas realized in GaAs/AlGaAs heterostructures, placed in an intense magnetic field ( 10 T) and at low temperature (below  1 K), is characterized by a chiral current flowing only along one-dimensional channels localized at the edge of the Hall bar. Under these conditions, the electron propagation is unaffected by impurity-induced random backscattering, leading to impressive coherence lengths ( 20 mm) at cryogenic temperatures ( 20 mK). Moreover, thanks to a properly designed electrostatic confinement induced by gates, the electron gas can be locally pinched off. This allows to approach the opposite edge of the Hall bar creating a QPC, characterized by a controllable probability of backscattering for the electrons. The experimental set-up described so far offers the ideal conditions for the realization of the electronic counter-part of both wave-guides and beam-splitters (see Fig. 1 below). However, the realization of on-demand sources of controlled electronic wave-packets required additional experimental effort. In this direction, two main technologies emerged during the years. The first, developed by the experimental group of the Laboratoire Pierre Aigrain in Paris (Fève et al., 2007), is based on a quantum dot capacitively coupled to the outer channels of an Hall bar. By periodically driving the Fermi level of this confined system and properly tuning the coupling, it is possible to inject into the edge channel one electron and one hole per period T. The wave-packets generated in this way are characterized by a Lorentzian profile in the energy domain (Bocquillon et al., 2013). A conceptually easier, but experimentally more challenging, way to realize an on-demand electron source has been implemented by the experimental Nanoelectronics group in Saclay (Glattli and Roulleau, 2017). It consists in the direct application of a periodic drive to a mesoscopic channel. According to theoretical analysis (Levitov et al., 1996), the form of this voltage need to be chosen carefully. In particular, a train of Lorentzian voltage pulses in time of the form (from now on we assume ℏ ¼ 1)

!

Fig. 1 Cartoon view of a two-dimensional electron gas (light blue) subject to a perpendicular magnetic field B. In the quantum regime, two edge channels (outer in red and inner in blue) propagate along the boundary of the sample and can be approached by means of a gate voltage (Vg).

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Electron quantum optics: A testbed for the Luttinger paradigma

Fig. 2 Injection in a mesoscopic channel of electronic wave-packets with typical width 2 t0, through the application of a periodic train of Lorentzian voltage pulses in time V(t).

V ðt Þ ¼

+1 q X 2t0 e m¼ −1 ðt −mT Þ2 + t20

(1)

with e charge of the electron and q a real parameter, leads to the periodic injection (again with period T ) of purely electronic Lorentzian wave-packets of width t0 in the time domain in the case of integer values q (see Fig. 2 below). This is a direct consequence of the quantization of edge conductance, given by G ¼ e2/(2p), as can be seen by the simple chain of relations involving the current I(t) flowing along the edge, namely Q¼

1 T

Z

+T2 −T2

dtIðt Þ ¼

1 T

Z

+T2 −T2

dt

e2 V ðt Þ ¼ qe: 2p

(2)

In addition, if one chooses and integer value of q the created excitations, called Levitons, are characterized by the absence of spurious additional particle-hole contributions and less affected by the interaction with respect to other injected pulses. This kind of source is therefore able to generate clean and robust electronic wave-packets. The two on-demand single-electron sources discussed above are complementary, showing a Lorentzian behavior in the energy and time domain respectively, and opened the way to the first seminal interferometric experiments in the field of EQO.

Hanbury-Brown-Twiss and Hong-Ou-Mandel interferometry with electrons We start this discussion with a description of the electronic version of the HBT interferometer. In ideal conditions, a controlled train of electrons (and eventually holes) is generated by means of one of the two sources discussed above and injected into a quantum Hall edge channel. Once arrived at a QPC they have a probability T to be transmitted along the same edge and a probability R ¼ 1 −T to be reflected on the opposite edge, the constraint being imposed by particle number conservation (see Fig. 3 below). To be more precise, the action of the QPC on free electrons is encoded by the unitary scattering matrix pffiffiffiffiffi pffiffiffiffi ! R i T pffiffiffiffi pffiffiffiffiffi s¼ (3) i T R in such a way that the outgoing annihilation fermionic operators (in channels 3 and 4) are connected to the incoming ones (in channels 1 and 2) through the simple relation     c1 c3 ¼s : (4) c4 c2 Notice that great care is devoted by experimentalists to find a proper working point of the QPC where T and R are energy independent. This guarantees that wave-packets with a finite spread in energy are not improperly distorted at this level. According to this we have suppressed all the energy dependence in (3) and (4). The excitations emerging from the QPC are then detected in one of the outgoing edge channels in such a way to investigate the partitioning associated with their corpuscular nature. Assuming, for sake of clarity, a unique excitation injected in the incoming

Fig. 3 Cartoon view of a set-up for partitioning and collisional experiments. Excitations flowing along the incoming channels 1 and 2 (green arrows) reach the central QPC (cyan) and emerge in the outgoing channels 3 and 4 (red arrows).

Electron quantum optics: A testbed for the Luttinger paradigma

927

channel 1 one has obviously an average number of excitations hN1i ¼ hc1 { c1i ¼ 1 and hN2i ¼ hc2 { c2i ¼ 0 in the incoming channels with no fluctuations: h(DN1)2i ¼ h(DN2)2i ¼ 0, with DN1,2 ¼ N1,2 − hN1,2i. Due to the action of the QPC, in the outgoing channels one has hN3 i ¼ R,hN 4 i ¼ T with fluctuations     (5) ðDN3 Þ2 ¼ ðDN4 Þ2 ¼ −hDN 3 DN4 i ¼ RT : The above expression characterizes the random partitioning of corpuscular excitations at the level of the QPC and is usually indicated as partition noise. Even more interesting is the situation for what it concerns the electronic translation of the HOM interferometer. This consists in a collisional experiment in which two identical trains of excitations are injected, with a tunable delay tD, on the opposite sides of a QPC (channels 1 and 2) and detected in the outgoing channels. For a time delay longer than the typical width of the wave-packets in time the partitioning of the two trains of excitations at the QPC is independent and one recovers the HBT phenomenology discussed above. However, for synchronized injections (tD ¼ 0) the colliding fermionic excitations are forced to always emerge on the opposite sides of the QPC. Also in this case we can understand this phenomenology considering the action of the QPC scattering matrix (3). In this case one has hN1i ¼ hN2i ¼ 1 in the incoming channels, again with no fluctuations: h(DN1)2i ¼ h(DN2)2i ¼ 0. In the outgoing channels one has hN3i ¼ hN4i ¼ 1 with     (6) ðDN 3 Þ2 ¼ ðDN4 Þ2 ¼ −hDN3 DN 4 i ¼ 0: This is a direct consequence of the Pauli principle which prevents two identical electrons to emerge in the same outgoing channel, leading to an anti-bunching effect. In order to further strengthen this statement, we observe that in a photonic HOM experiments the excitations always bounch with the emergence of pairs of photons on one of the two outgoing branch of the interferometer, namely (Bocquillon et al., 2013)    

D E ðphÞ 2 ðphÞ 2 ðphÞ ðphÞ ¼ 4RT : (7) DN 3 ¼ DN4 ¼ − DN 3 DN4 In order to go beyond this simplified picture and to better understand the relevant physics at work in actual experiments, we need to properly theoretically address what can be directly measured in electronic interferometers. Indeed, while in quantum optics experiments the measurement of photo-counting and coincidences as a function of time is possible thanks to detectors placed in the outgoing branches of the interferometer, the sensitivity of their solid-state counterparts is not enough to achieve a real-time counting of outgoing excitations. To overcome this issue, experimentalists in the field investigate the current correlations, in particular the cross-correlated zero frequency noise averaged over the period T of the injection, which is defined as Z S¼

+T2 −T2

dt T

Z

+1 −1

D    E D  E D  E t t t t I4 t − − I3 t + : dt I3 t + I4 t − 2 2 R 2 R 2 R

(8)

In the above equation Ia is the operator associated to the current flowing along the outgoing channel a ¼ 3, 4 and the expectations values are taken with respect to the initial density matrix R, which takes into account the injection of additional excitations with respect to the Fermi level of the edge channel. The noise discussed above is directly related to the fluctuations of the charge outgoing form the QPC in a given period. According to what discussed in (3) and (4) it is possible to connect the fluctuations of the outgoing currents with the incoming ones. This leads to the decomposition X SHBT + SHOM ðt D Þ: (9) S ¼ SFS + a a¼1,2

FS

The first terms (S ) only depends on the properties of the many-body Fermi sea of the edge channels when the electron source are switched-off, namely temperature and dc bias, and is typically subtracted in experiments. The second term is the sum of two contributions of the form ¼ RT SHTB a

e2 2p

Z

+T2 −T2

dt T

Z

+1 −1

h i doDW a ðt, oÞ 1 −2f m ðoÞ :

(10)

Here, we have introduced the Fermi distribution fm at fixed temperature and chemical potential m, supposed the same for both edges of the Hall bar for sake of simplicity, and the excess Wigner function (Ferraro et al., 2013) defined as D    E t t dteiot C{a t − Ca t + 2 2 R D −1    E i t t { : − Ca t − Ca t + 2 2 F Z

DW a ðt, oÞ ¼ v

+1

(11)

This latter quantity completely characterizes the time-energy profile of the excitations injected into the channel a ¼ 1, 2. In the above expression, v is the propagation velocity of the excitations along the edge channels,

928

Electron quantum optics: A testbed for the Luttinger paradigma 1 Ca ðt Þ ¼ pffiffiffi v

Z

+1 −1

doca ðoÞe −iot

(12)

is the time representation of the annihilation operator for an electron in the incoming channel a and with F we indicate the average taken with respect to the Fermi sea in absence of injected excitations. Finally, the last term in (9) reads SHOM ðt D Þ ¼ −RT

e2 p

Z

+T2 −T2

dt T

Z

+1 −1

doDW 1 ðt, oÞ

(13)

DW 2 ðt + t D , oÞ and characterizes the overlap between excitations injected in the two incoming channels with a controlled delay tD. Notice that this contribution goes to zero when the delay exceeds the typical time width of the injected wave-packets. According to the above considerations, an HBT experiment is properly described by a noise contribution of the form (10). It characterizes the partition of the excitations at the QPC discussed in (5) and described here by the excess Wigner function, but takes into account also a suppression of the noise due to the unavoidable Pauli repulsion between the incoming excitations and the ones already present in the edge channels described by a Fermi sea at a cryogenic but finite temperature. For what it concerns HOM measurements, the ratio Qðt D Þ ¼

SHTB + SHTB + SHOM ðt D Þ 1 2 HTB S1 + SHTB 2

(14)

is typically reported (Bocquillon et al., 2013; Glattli and Roulleau, 2017). As an example, in the case of a Leviton carrying a charge q ¼ 1 one can find the analytical expression for the above ratio

 sin 2 p tTD



: (15) Qðt D Þ ¼ sinh 2 2p tT0 + sin 2 p tTD It shows a dip reaching zero at tD ¼ 0, which is consistent with the previously discussed idea of total suppression of the current fluctuations in case of synchronized collisions associated to the certainty of finding exactly one excitation in each outgoing channel as a consequence of the Pauli repulsion. Moreover, it asymptotically approaches one, in the limit t0  T, recovering the HBT physics. Notice that, differently from the HBT signal, the HOM one is only marginally affected by thermal effects that will be neglected in the following. The simple analysis given so far, inherently based on a free fermion picture, is only able to provide a qualitative description of the observations. Indeed, despite the HBT behavior is typically well depicted, discrepancies in the form of the HOM dip with respect to the reported theoretical results are observed (Bocquillon et al., 2013). This is related to the fact that measurement are typically carried out at filling factors n ¼ 2, where two channels co-propagates at edge of the Hall bar. Electron-electron interaction between them need to be carefully taken into account in order to accord theory and experiments. In the following, we will discuss how to handle inter-channel interaction in the framework of the chiral Luttinger liquid picture.

Luttinger liquid description of interacting quantum Hall edge channels Non-interacting electrons form a Fermi gas characterized by infinity long-lived electron and hole excitations. In two and three dimensions this picture can be extended to include interacting cases, leading to the notion of Fermi liquid elaborated by L. D. Landau during the fifties. Here, virtual particle-hole pairs dress the excitations with the consequent renormalization of the physical parameters, such as the effective mass, and the emergence of a finite lifetime for the excitations. Despite this, these quasi-particles maintain their individual nature. Radically different is the situations for interacting one-dimensional electrons. Here, due to the geometrical constraint, every individual degree of freedom immediately turns into density fluctuations of the overall system, leading a failure of the Fermi liquid description. These collective excitations, corresponding to electron-hole pairs close to the Fermi energy, are properly described in term of a bosonic theory known as the Luttinger liquid model (Haldane, 1981; Giamarchi, 2003). In the following, we will investigate the dynamics of the edge states of a quantum Hall bar at filling factor n ¼ 2 in this framework. On each edge of the Hall bar, the two co-propagating channels (indicated in the following with a and b respectively) interact capacitively along a region of finite length L via screened Coulomb interaction (see Fig. 4 below).

Fig. 4 Scheme of a quantum Hall edge state at filling factor n ¼ 2. Here, the two channels a and b interact with a short range capacitive coupling u over a finite length L.

Electron quantum optics: A testbed for the Luttinger paradigma

929

Consistently to experimental observations no electron tunneling occurs between the two channels. Due to the presence of the magnetic field, this situation is well represented by a chiral version of the Luttinger liquid theory in which the interacting one-dimensional edge channels are described in terms of bosonic modes usually called edge-magnetoplasmons (Wen, 1995). The Hamiltonian density of the systems can be written as ℋ¼

1 X V i,k ½∂x fi ðxÞ∂x fk ðxÞ 4p

(16)

i,k¼a,b

with  V¼

v u

u v

 (17)

and where the bosonic modes fi are related to the electron particle density operator of each channel through ri ðxÞ ¼

1 ∂ f ð xÞ 2p x i

i ¼ a,b:

(18)

The annihilation operator for an electron at point x of one of the two channels can be written in this picture in the form ℱ i −ifi ðxÞ Ci ðxÞ ¼ pffiffiffiffiffiffiffiffi e 2px

i ¼ a,b

(19)

with x a finite length cut-off and ℱi playing the role of Klein factors. In (16), for sake of simplicity, we have assumed identical propagation velocity v for both the excitations along the two edge channels and we have indicated with u the intensity of their capacitive coupling. Under these conditions the full interacting problem can be easily diagonalized through a rotation in the field space. The full Hamiltonian density then becomes h i2 1 X vb ∂x fb ðxÞ (20) ℋ¼ 4p b¼r,s

in terms of the rotated fields 1 fr ðxÞ ¼ pffiffiffi ½fa ðxÞ + fb ðxÞ 2 1 fs ðxÞ ¼ − pffiffiffi ½fa ðxÞ −fb ðxÞ: 2

(21)

These fields are associated with a fast charged mode (fr(x)) and a slow dipole mode (fs(x)) propagating with velocities vr, s ¼ v  u, respectively. Focusing, for pedagogical reasons, on the case of Levitons injection one can model the electron source as an ohmic contact which couples the outer channel a with a time-dependent voltage source V(t). This leads to an additional term in the Hamiltonian density of the form ℋV ¼ eV ðt Þra ðxÞYð−xÞ,

(22)

where we have assumed a macroscopic contact from −1 to x ¼ 0 in correspondence of the starting of the interacting region. This allows to fix the proper initial conditions for the quantum fields at the beginning of the interacting region. Taking into account the transformation in (21) and solving the complete dynamics of the system, the voltages outgoing form the interacting region and reaching the QPC in the channels a and b respectively are related to the incoming one through   1 V t −tr + V ðt −ts Þ 2   1 V 0b ðt Þ ¼ V t −tr −V ðt −ts Þ , 2 V 0a ðt Þ ¼

(23)

with tr,s ¼ vr,s/L times of flight required by the charged and dipole mode to cross the interacting region. This leads to the fractionalization of the voltage pulse illustrated in Fig. 5 below. The consequences of this phenomenology on the form of the HOM dip will be discussed in the following.

Fig. 5 Cartoon view of the fractionalization of a peaked voltage profile injected into the a channel, crossing an interacting region and reaching a QPC.

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Electron quantum optics: A testbed for the Luttinger paradigma

Effects of interaction in the Hong-Ou-Mandel profile: The Leviton case According to what reported in (23) it is possible to Fourier expand both the HBT and the HOM contributions to the noise. The ratio in (14) can then be written, at zero temperature, as 2 P+1 P ∗ imOt D   |l| l¼ −1  m pl+m ðqÞpm ðqÞe (24) Qðt D Þ ¼  P+1  2 2 l¼ −1 jpl ðqÞj jl + qj where we have made explicit the dependence with respect to the number of the injected excitations q. In the above expression we have introduced O ¼ 2p/T and the quantity pl ðqÞ ¼

+1 X n¼ −1

pl −n

q q p eiOtr ðl −nÞ eiOts n 2 n 2

(25)

with pl ðzÞ ¼ z

+1 X ð −1Þs Gðz + l + sÞe −2pt0 ð2s+lÞ=T Gðz + 1 −sÞGð1 + sÞGð1 + l + sÞ s¼0

(26)

giving the probability amplitude for an electron to absorb (l > 0) or emit (l < 0) l photons in the case of Lorentzian voltage pulse in time (Glattli and Roulleau, 2017). We consider now, for sake of simplicity, the ratio Q evaluated in a symmetrical set-up with respect to the QPC, namely a configuration where the lengths of the two interacting regions in the incoming channels are identical. When interaction is present, by changing the time delay tD between the right and the left moving trains of excitations, we find three characteristic signatures in the noise profile (see Fig. 6 below). At tD ¼ 0 a central dip appears for both the free electron case described by (15) and the interacting case, while two previously absent side-dips emerge at a position tD ¼  |tr − ts |. The shape of these three dips is reminiscent of the Lorentzian profile in time of the wave-packet of the injected excitations while their width depends on its typical timescale (Rebora et al., 2020). This interference pattern can be easily understood in terms of the Luttinger liquid picture discussed in the previous Section. After injection, the electrons are fractionalized into two modes: a slow dipole mode and a fast charged mode (see Fig. 5). The central dip, which corresponds to the symmetric situation of simultaneous injection of electrons, start from zero because these identical excitations reach the QPC at the same time. The destructive interference is responsible for the appearance of side-dip structures. Indeed, when tD ¼  |tr − ts | the fast right moving mode and the slow dipole mode reach the QPC at the same time. Even more dramatic features are predicted to appear in strongly interacting systems like fractional quantum Hall edge channels. These will be discussed in the following Section.

Crystallization of Levitons in the fractional quantum Hall regime A prototypical example of strongly correlated electron systems is represented by fractional quantum Hall states with filling factor belonging to the so called Laughlin sequence, namely with n ¼ 1/(2n + 1) with n 2 ℕ (Das Sarma and Pinczuk, 2004). According to the chiral Luttinger liquid picture discussed above (Wen, 1995), the up u and down d edge states of the Hall bar in

Fig. 6 Ratio Q for Lorentzian voltage pulses, carrying unitary charge q ¼ 1, as a function of tD/T for a symmetric setup shown for the non-interacting (black curve) and the interacting case (red curve). Other parameters are: t0/T ¼ 0.05, vr ¼ 4  105 m/s, vs ¼ 1.8  105 m/s and L ¼ 2 mm.

Electron quantum optics: A testbed for the Luttinger paradigma

931

this regime can be both described in terms of a single chiral bosonic mode for each edge. The low energy effective Hamiltonian density for the system reads h i2 v X ∂x fj ðxÞ (27) ℋ¼ 4p j¼u,d

with particle density operators given by ru,d ðxÞ ¼ 

pffiffiffi n ∂ f ðxÞ: 2p x u,d

(28)

We assume again that a Lorentzian train of voltage pulses V(t) is applied to one (HBT configuration) or both (HOM configuration) the channels by means of a capacitive coupling between the channel and an ohmic contact. Differently from what done before, in this case it is no more possible to assume a non-interacting picture for tunneling of excitations at the level of the QPC and the scattering matrix derived from it. Therefore, a more convenient way to handle the problem requires a perturbative approach. As an additional, despite very interesting, complication in the present case the more relevant tunneling process is represented by emergent fractionally charged excitation with charge e∗ ¼ ne whose annihilation is ~ u,d ðxÞ. Their tunneling from u to d and vice versa is taken into account by the Hamiltonian described in terms of the operator C contribution (Wen, 1995) ~ { ðxÞC ~ d ðxÞdðxÞ + H:c: ℋL ¼ LC u ¼

L e 2px

pffiffi pffi −i nfu ðxÞ i nfd ðxÞ

e

dðxÞ + H:c:

(29) (30)

Here, we have introduced the tunneling amplitude L. Taking into account the chiral propagation of the modes along the edge, which relates the space coordinate x with the time t, the backscattering current operator due to the quasiparticles is given by IB ðt Þ ¼ ie ∗

L −ipffiffinfu ðt Þ ipffinfd ðt Þ e + H:c: e 2px

(31)

Starting from this, it is possible to evaluate the noise perturbatively in L. In the following we will stop at the leading order in this expansion. In case of identical injection on the two incoming channels, the two identical HBT contributions to the noise can be given again in terms of a Fourier series, namely +1

2 jLj2 X ðqÞ ¼ −2 e ∗ SHBT jpl ðqÞj2  a 2 ð2pxÞ l¼ −1

(32)

½P 2n ððq + lÞOÞ + P 2n ð −ðq + lÞOÞ where we have introduced the function P g ðoÞ ¼

2p g −1 YðoÞ gx GðgÞoc

(33)

with G( g) the Euler’s Gamma function of argument g and oc ¼ v/x In the case of a train of Lorentzian voltage pulses, the HBT noise shows a Poissonian behavior each time q is an integer (Rech et al., 2017). This is another signature of the fact that integer charged Levitons are robust excitations also in presence of strong interaction. In the HOM case, considering the injection of two identical train of excitations carrying the same charge q on the two side of the Hall bar and following the same procedure we have used for the HBT noise, we get 2   + 1 X

∗ 2 jLj2 X  ∗ imOt D  HOM ðt D Þ ¼ −2 e S   pl+m ðqÞpm ðqÞe 2 (34)   m ð2pxÞ l¼ −1

½P 2n ðloÞ + P 2n ð −loÞ: The profile of the corresponding ratio Q as a function of the injection delay and for different values of injected charge q is shown in Fig. 7 below. It is strongly different with respect to what shown in both the free fermion case and the interacting case at filling factor n ¼ 2. Indeed, one still has a central dip reaching zero at tD ¼ 0. However, by increasing the delay one observes the emergence of subdips for q > 1. It is worth noticing that the number of these sub-dips is equal to 2q-2. Due to the presence of a unique edge channel, one cannot interpret this emerging structure in terms of the fractionalization discussed in the previous Sections. In contrast, this phenomenology can be explained in terms of a different process, namely a crystallization of Levitons carrying an integer charge q occurring in the time domain. This is a consequence of the fact that the injected wave-packet reorganizes into several sub-peaks at the output of the QPC due to the strong repulsion between the electrons composing it as show by the excess of particle density at the output of the QPC reported in Fig. 8 below.

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Electron quantum optics: A testbed for the Luttinger paradigma

Fig. 7 Ratio Q as a function of tD/T for equal injected charge q ¼ 1 (black curve), q ¼ 4 (green curve), q ¼ 5 (red curve) and q ¼ 6 (blue curve). The free fermion case at n ¼ 1 (dashed lines) is compared to the fractional case at n ¼ 1/3 (solid lines). The other parameters are t0/T ¼ 0.04 and oc ¼ 100O. Reprinted figure with permission from Ronetti F, Vannucci L, Ferraro D, Jonckheere T, Rech J, Martin T, and Sassetti M (2018) Crystallization of levitons in the fractional quantum Hall regime. Physical Review B 98: 075401. Copyright 2018 by the American Physical Society.

Fig. 8 Variation of the charge density Dru(t) due to the injection of a train of Levitons carrying q ¼ 6 electrons, evaluated in correspondence of a detector placed at after the QPC. Two different filling factors showing crystallization are considered: n ¼ 1/3 (solid line) and n ¼ 1/5 (dashed line). The other parameters are t0/ T ¼ 0.04, oc ¼ 100O. Reprinted figure with permission from Ronetti F, Vannucci L, Ferraro D, Jonckheere T, Rech J, Martin T, and Sassetti M (2018) Crystallization of levitons in the fractional quantum Hall regime. Physical Review B 98: 075401. Copyright 2018 by the American Physical Society.

Conclusion This Chapter represents an overview of the subject of Electron Quantum Optics, with particular focus on the role played by electron-electron interaction in collisional interferometric experiments. The analogies between conventional quantum optics experiments, based on the manipulation of individual photonic degrees of freedom, and their fermionic counterparts, involving electronic wave-packets ballistically propagating along mesoscopic channels realized in solid-state devices such as the edge states of the quantum Hall effect have been addressed. The behavior of the electronic versions of Hanbury-Brown-Twiss and Hong-Ou-Mandel interferometers has been discussed in detail, underlying the fact that while the first provide information about the corpuscular nature of the excitations, the second can shed light on the interplay between Fermionic statistics and interaction effects. In particular, the role of statistics is encoded in the so called Pauli dip, a suppression of the current fluctuations associated to the anti-bunching of electrons that reach a quantum point contact at the same time.

Electron quantum optics: A testbed for the Luttinger paradigma

933

To describe the effects of interaction, we have introduced the Luttinger liquid picture, universally accepted as the proper way to treat interacting systems confined in one spatial dimension, where the conventional Fermi liquid paradigma fails. To better address experimentally relevant situations, we have focused on the case of a quantum Hall state at filling factor n ¼ 2, where two capacitively coupled edge channels co-propagate at the edge of the Hall bar. Here, a fractionalization of the injected voltage pulses occurs. At the level of collisional Hong-Ou-Mandel experiments this phenomenology leads to a deformation of the Pauli dip with the emerge of side dips. Even more dramatic departures form the single Pauli dip profile, typical of the free fermion picture, can be observed in strongly interacting electron systems such as the fractional quantum Hall edge channels. Here, collisional experiments are characterized by the emergence of a number of side dips which grows with the charge carried by the injected pulses. This can be interpreted as a crystallization in the time domain, namely a rearrangement of the profile of a wave-packet carrying multiple electrons, as a consequence of the intense Coulomb interaction.

Acknowledgments Authors acknowledge the support of the “Dipartimento di Eccellenza MIUR 2018-22.” We would like also to thank G. Rebora, F. Ronetti and L. Vannucci for useful discussions.

References Bocquillon E, Freulon V, Parmentier FD, Berroir J-M, Plaçais B, Wahl C, Rech J, Jonckheere T, Martin T, Grenier C, Ferraro F, Degiovanni P, and Gwendal F (2013) Electron quantum optics in ballistic chiral conductors. Annalen der Physik 526: 1. Das Sarma S and Pinczuk A (2004) Perspectives in Quantum Hall Effects. Wiley-VCH. Ferraro D, Feller A, Ghibaudo A, Thibierge E, Bocquillon E, Fève G, Grenier C, and Degiovanni P (2013) Wigner function approach to single electron coherence in quantum Hall edge channels. Physical Review B 88: 205303. Fève G, Mahé A, Berroir J-M, Kontos T, Plaçais B, Glattli DC, Cavanna A, Etienne B, and Jin Y (2007) An on-demand coherent single-electron source. Science 316: 1169. Giamarchi T (2003) Quantum Physics in One Dimensions. Oxford University Press. Glattli DC and Roulleau P (2017) Levitons for electron quantum optics. Physica Status Solidi B: Basic Solid State Physics 254: 1600650. Haldane FDM (1981) Luttinger liquid theory of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. Journal of Physics C: Solid State Physics 14: 2585. Levitov LS, Lee H, and Lesovik GB (1996) Electron counting statistics and coherent states of electric current. Journal of Mathematical Physics 37: 4845. Rebora G, Acciai M, Ferraro D, and Sassetti M (2020) Collisional interferometry of levitons in quantum Hall edge channels at n ¼ 2. Physical Review B 101: 245310. Rech J, Ferraro D, Jonckheere T, Vannucci L, Sassetti M, and Martin T (2017) Minimal excitations in the fractional quantum Hall regime. Physical Review Letters 118: 076801. Ronetti F, Vannucci L, Ferraro D, Jonckheere T, Rech J, Martin T, and Sassetti M (2018) Crystallization of levitons in the fractional quantum Hall regime. Physical Review B 98: 075401. Wen X-G (1995) Topological orders and edge excitations in fractional quantum Hall states. Advances in Physics 44: 405.

Theoretical approaches to liquid helium Jordi Boronat

, Department of Physics, Technical University of Catalonia, Barcelona, Spain

© 2024 Elsevier Ltd. All rights reserved.

Introduction Early approaches Microscopic theories Quantum Monte Carlo methods Conclusion Acknowledgments References

934 934 935 940 943 944 944

Abstract In this article, we review the main theoretical methods applied to the study of liquid Helium adopting a microscopic approach, that is, starting from the many-particle Hamiltonian of the system. Following an introduction on the first early approaches, we discuss two main issues. In the first one, we report the main ingredients of a theoretical approach based on the variational method, with the discussion of the high accuracy obtained with them and the related progress in the design of highly accurate trial wave functions. In the second part, the main stochastic methods used in this study are briefly discussed. Altogether reflects the strong links between the study of liquid Helium and the progress in the development of new and extremely powerful approaches to deal with one of the most strongly quantum many-body systems in Nature.

Key points

• • • •

Main physical properties of liquid Helium, both in its superfluid and normal states Early phenomenological theoretical approaches to account for the main experimental data Development of theoretical approaches from a microscopic view: variational and correlated basis function theories Theoretical study based on stochastic methods: quantum Monte Carlo

Introduction Liquid Helium was first liquefied by Kammerlingh Onnes in 1908 (Kamerlingh Onnes, 1908) and, since then, it has been one of the most studied quantum systems. Two of its isotopes are stable, 4He and 3He. In naturally produced Helium, the fraction of 3He is extremely small, 1 part in 106, and is produced from radioactive decay of tritium. Both isotopes remain in liquid state even in the limit of zero temperature, manifesting in this way their intrinsic quantum nature. Contrarily to classical theory, that predicts that any material becomes solid in that limit, quantum mechanics explain this fail of the classical approach by considering the zero-point kinetic energy. The combination of the light mass of Helium atoms and their weak attraction produces the unique opportunity of observing the most paradigmatic quantum liquid. 4He and 3He have different masses but the most crucial difference between both isotopes affects their behavior as quantum many-body systems: 4He is composed by an even number of spin 1/2 particles and thus, it is a boson, whereas 3He has an odd number of constituents and behaves as a fermion. As we will discuss in this article, the different quantum statistics of the two Helium isotopes produces dramatic differences between them. In 4He, the specific heat shows a large excess at a temperature of 2.17 K, signaling the emergence of a phase transition, first observed by Keesom and Clusius (1932). The specific heat shows a shape similar to the l Greek letter and is now universally known as the l phase transition, with a critical temperature Tc ¼ 2.17 K. This second-order transition separates two liquids with different properties and were termed liquid He I and liquid He II for T > Tc and T < Tc, respectively. In 1938, P. Kapitza (Kapitza, 1938), from one side, and Allen and Misener (1938), on the other, published back-to-back papers announcing the discovery of superfluid 4He in the Helium II phase. For an insightful historical analysis of the role played by the two independent teams on that discovery, we recommend the paper by Balibar (2007). Below Tl, 4He flows with vanishingly small viscosity producing a set of surprising effects such as the fountain effect, absence of boiling due to the large thermal conductivity, and second sound (Wilks, 1967).

Early approaches In the early theoretical approaches to understand the extraordinary behavior of liquid Helium around the l point, four names appear above many others: Laszlo Tisza, Lev Landau, Fritz London, and Nikolai Bogoliubov. Tisza and Landau worked in a

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00180-3

Theoretical approaches to liquid helium

935

quantum hydrodynamic approach with the main idea of Helium as a liquid composed by two independent liquids (velocity fields). This is the basis of the two-fluid model first introduced by Tisza (1938) and then completed by Landau (1941). The two-fluid model assumes that the total density is the sum of two partial densities: the normal and the superfluid. Below Tl, the superfluid component dominates, thus explaining the superfluid properties of Helium II. Increasing the temperature, at T  Tl, the superfluid density turns to be zero and the total density equals the normal one. Landau (1941) introduced the concept of quasiparticle to postulate the shape of the excitation spectrum. Instead of single-particle excitations, the system presents collective modes that at low momenta are termed phonons, in resemblance with the excitation spectrum of a crystal. At low momenta, the excitations are linear with the momenta, the slope being the speed of sound. At larger momenta, Landau defined another type of collective excitations which were named rotons, since where initially related to the presence of quantized vortices. Landau predicted that below a certain velocity, termed critical velocity, the system cannot be excited and thus liquid Helium flows without friction. Combining phonons and rotons, Landau (1947) was able to draw the excitation spectrum of Helium before any measure of it was made. The hydrodynamic theory by Tisza and Landau did not take into account the quantum statistics of 4He because they believed that this feature was not relevant to understand the superfluidity and the l phase transition. On the contrary, London (1938) argued that a Bose gas experiences a second-order phase transition that can explain, at least qualitatively, the l transition in 4He. Applying the critical temperature derived in the Bose-Einstein statistics, kB T BEC ¼

2pħ2 m

n g3=2 ð1Þ

!2=3 ,

(1)

with g3/2(1) ’ 2.612 and n the density, to the case of Helium London obtained 3.1 K, not so far from Tl. Landau always ignored London approach to the problem (Balibar, 2007) because with his hydrodynamic theory he was able to account for many experimental facts. However, we know now that the quantum statistics of 4He is crucial to understand superfluid 4He. When samples of 3He were produced, we learned that 3He, which is a fermion, do not show the l phase transition. Bogoliubov, in 1947 (Bogoliubov, 1947), was the first to connect the elementary excitation spectrum with a Bose–Einstein gas with repulsive but small interatomic interaction. He proved that the lowest energy excitations are collective modes (phonons) that at low momenta are proportional to the speed of the sound, in agreement with Landau theory. The Bogoliubov excitation spectrum (Pitaevskii and Stringari, 2016) "

gn 2 eðpÞ ¼ p + m



p2 2m

2 #1=2 ,

(2)

with g as the interaction strength predicts a critical velocity which is always equal to the speed of sound. Bogoliubov theory was again qualitatively correct but Helium is not a rarefied gas but a liquid with strong interparticle interactions.

Microscopic theories We discuss first liquid 4He. 3He was obtained later in time due to its complex experimental production. As we commented in the previous section, Bogoliubov (1947) introduced field theoretical models in the theoretical description of liquid 4He, but his approach was only applicable to dilute Bose gases, and this is far form the real properties of Helium. This is specially evident from the explicit assumption in Bogoliubov analysis that all the particles of the system are in the Bose–Einstein condensate, whereas in 4 He only 7% of them are really in this state due to the unavoidable role played by atomic correlations. In subsequent work, Beliaev (1958) introduced Green functions Ga,b(q, o) in the formalism, deriving a general equation for them at zero temperature. The quantum field theory for a Bose fluid was reanalyzed by Hugenholtz and Pines (1959) by the introduction of the chemical potential m to guarantee the number conservation of the theory. Importantly, they removed the constraint of a full condensate, approaching better the characteristics of liquid Helium. An important result of this approach was the calculation of the energy of a Bose gas incorporating the first corrections to the Hartree–Fock energy pffiffiffi   E 128 pffiffiffiffiffiffiffi3ffi 8ð4p − 3 3Þ 3 pffiffiffi na + (3) na lnðna3 Þ + . . . , ¼ 4pna3 1 + 3 N 15 p the energy per particle E/N being in units of ħ2 =ð2ma2 Þ. Hugenholtz and Pines (1959) proved that, beyond the expansion terms in Eq. (3), additional contributions will depend on the specific shape of the interatomic potential and not only on the s-wave scattering length a. The coefficient of the (na3)3/2 term was first calculated by Lee et al. (1957), while the coefficient of the last term was first obtained by Wu (1959). Both of them were originally derived for hard spheres, but it was shown that the same expansion is valid for any repulsive potential with scattering length a (Hugenholtz and Pines, 1959). Eq. (3) has been very useful in the study of dilute Bose gases, composed by alkali atoms, due to their extreme diluteness (Pitaevskii and Stringari, 2016). However, liquid Helium is a strongly interacting quantum many-body system in which any expansion in terms of the gas parameter na3 is completely useless. The first theoretical approach based on the wave function of the N-body quantum system was made by Feynman (1953). This work opened a new way to deal with the properties of Helium based on a microscopic approach. Starting from general arguments,

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Theoretical approaches to liquid helium

that included the symmetry of the 4He atoms and their hard-core interactions at short distances, Feynman argued that the lowest energy excitations have a collective behavior since single-particle ones have always higher energy. Based on these considerations, he wrote the wave function of the excited state of the N-body fluid as ðr 1 , . . . , r N Þ ¼

N Y

f ðr i Þ Fðr 1 , . . . , r N Þ,

(4)

i¼1

where Fðr 1 , . . . , r N Þ is the ground-state wave function. Eq. (4) is a variational approach to the problem and thus the energy of the excitation is an upper bound to the exact energy. He obtained the lowest excitation energy ħ2 q2 =2m , (5) SðqÞ P where S(q) ¼ 1/Nhr{(q)r(q)i is the static structure factor, and rðqÞ ¼ i expð −iq  r i Þ is the density fluctuation operator. When q ! 0, SðqÞ ¼ ħq=ð2mcÞ, where c is the speed of sound. In this way, Feynman recovers the phonon relation previously found by Landau and Bogoliubov, ħo ¼ cq, that is, a linear behavior. At large momenta, the static structure factor tends to 1 and so one recovers the excitation energy of a free particle ħ2 q2 =2m, quadratic with q. At intermediate q values, S(q) shows a peak related to the mean interparticle distance and thus the excitation energy (5) shows a local minimum that we can identify with the roton. Qualitatively, the Feynman spectrum is correct up to momenta not much larger than the one of the roton but the roton energy is nearly two times larger than the experimental one. The minimum energy (5) is obtained with a function f ðrÞ ¼ expðiq  rÞ, so the total wave function (4) is given by ħoF ðqÞ ¼

ðr 1 , . . . , r N Þ ¼

N Y

eiqr i Fðr 1 , . . . , r N Þ ¼ r{ ðqÞFðr 1 , . . . , r N Þ,

(6)

i¼1

that emerges as the creation of a collective mode of momentum q acting on the ground-state, that is not changed by the excitation. Feynman and Cohen (1956) improved the initial theory by introducing in the wave function two-body correlations. Their idea is that when one particle moves in the fluid its neighbors feel the movement or, in other words, the movement of each particle depends on its local environment. They termed these correlations as backflow correlations. Explicitly, the wave function of the collective excitation turns to " # N Y X ðr 1 , . . . , r N Þ ¼ eiqr i eigðr i −r j Þ Fðr 1 , . . . , r N Þ: (7) i¼1

j6¼i

The function g(r) in Eq. (7) couples the movement of the excited particle i with the neighboring ones. In Feynman and Cohen (1956), it is taken as a dipolar term, g(r) ¼ aq  r/r3 and the imaginary exponential containing g(r) is linearized to make the calculation of the energy simpler. With this correction, the difference between the Feynman roton energy (5) and the experimental one is reduced to a factor of  2 but still there is a significant difference. To reproduce as better as possible the elementary excitation spectrum of liquid 4He is not the only goal of a microscopic theory. It is fundamental to have an accurate description of the ground state of the system. The most fruitful approach to account for the equation of state of liquid 4He has been the variational theory since standard perturbative approaches are not applicable due to the hard core of the He–He interaction at short distances. The Hamiltonian of the N-particle system is H¼ −

N N ħ2 X 2 X ri + Vðr ij Þ, 2m i¼1 i0) is the molecular field coefficient representative of the interactions within a given sublattice, and n0 ( n1/4. When the magnetic moments are confined in the plane perpendicular to the chain by the magnetic anisotropy, a helimagnetic structure is found in which the moments rotate progressively from one to the next, along the chain (Figure 2d). This structure is found in some rare-earth metals, such as terbium or dysprosium. Reciprocally, when the moments are forced to align along the chain axis, a modulated structure occurs in which the amplitude of the moments oscillates periodically (Figure 2e). This occurs in thulium. Helimagnetic and modulated structures belong to the large class of antiferromagnetic structures, which are characterized by the zero value of the spontaneous magnetization. However, in these two cases, the stable magnetic structure is the result of a compromise between the various interactions involved, and is said to be frustrated. The analysis of frustrated structures has recently been the topic of intense research, both theoretically and experimentally.

Spin Glasses and Other Frustrated Systems When the magnetic atoms do not form a periodic arrangement, another type of frustrated magnetic arrangement may exist. This is the case in substitutional alloys in which two or more types of atoms are randomly distributed on a given periodic lattice, or in amorphous alloys, obtained by very fast quenching from the melt. Since the magnitude and sign of interactions depend on the distance, it may be expected that the random distribution of atoms leads to a distribution of J values which is centered at 0 such that there are as many positive and negative interactions. As long as the interactions nij are evenly distributed from positive to negative, with zero mean value, no stable magnetic order may exist. However, if eqn [20] is squared, one obtains !2 X   2 2 2 T hmi i ¼ C nij mj [24] j

   For a fully disordered system of infinite dimensions, the sum over the terms nij nij mj mj 0 is zero and eqn [24] can be expressed as

Magnetic Order X

T 2 hmi i2 ¼ C2

j

33

! n2ij hmi i2

[25]

This expression has a critical temperature given by Tc ¼ C

X j

!1=2 n2ij

[26]

Below Tc, the system is in a spin-glass state where the moments are randomly oriented. Equation [25] does not apply rigorously to finite-size systems. It has been a long debate to decide whether or not a real phase transition occurs in spin glasses. Experimentally, a peak is observed in the thermal variation of the magnetic susceptibility, but the temperature at which it occurs is found to depend on the time taken for the measurement (Figure 3). Initially, this type of measurement suggested that the spin-glass transition is not a real phase transition. Actually, it is now accepted that the spin-glass transition constitutes a new class of phase transitions in which the first derivative of the magnetic susceptibility versus temperature does not show any anomaly, but derivatives of higher orders do. The frequency dependence of the temperature at which the peak occurs in w(T ) is related to the fact that the critical slowing down of the magnetic fluctuations extends in spin glasses much farther from the value of the critical temperature than it does in other ordered magnetic systems.

Experimental Studies of Magnetic Structures The macroscopic magnetic properties of magnetic materials may be derived from the magnitude of the stray field generated by the material magnetization, whether it is spontaneous or induced by the applied field. However, such measurements do not allow the moments of the various constitutive elements to be determined individually. Additionally, in the absence of applied field, antiferromagnetic-like materials do not generate any stray field. For this type of characterization, local probes such as Mössbauer spectroscopy, nuclear magnetic resonance (NMR), or muon-spin rotation (mSR) may be used. These methods measure the local magnetic field at a particular crystallographic site. Uning these methods, the magnetization of individual sublattices in an antiferromagnet can be measured. The mSR spectroscopy is ideal to investigate a material having small ordered state moment 0.1mB or a material having a heterogenous structure, such as ‘magnetic’ and ‘nonmagnetic’ regions. Neutron diffraction constitutes an invaluable tool for the study of magnetic arrangements. The neutron bears a small magnetic moment which probes the magnetism existing within matter. Neutron diffraction is the strict analog for the analysis of magnetic structures as X-ray diffraction is for the studies of crystallographic structures. Neutron diffractograms obtained with MnO are shown in Figure 4. In (a), the peak observed at 12 is due to the (1 1 1) reflection; it is characteristic of the face-centered cubic crystallographic structure of this compound. In (b), an additional peak is observed at around 6  . This peak can be indexed as 1/2, 1/2, 1/2. It is the signature of periodicity doubling. In an antiferromagnetic structure, the alternation of moment orientations leads to periodicity doubling (see scheme in the inset to Figure 4). The diffractograms shown in Figure 4 have a special historical meaning. They constituted, when they were published, the first experimental proof of the existence of antiferromagnetism.

Figure 3 Thermal variation of the susceptibility in a spin glass: a Cu–Mn alloy containing 0.95% of Cu. The measurements of the susceptibility were realized under AC field of different frequencies (□1330 Hz, ○234 Hz, X 10.4 Hz, D 2.6 Hz). The frequency dependence of the temperature at which the peak occurs in w(T ) is related to the fact that the critical slowing down of the magnetic fluctuations extends in spin glasses much farther from the value of the critical temperature than it does in other ordered magnetic systems.

34

Magnetic Order

Figure 4 Neutron diffractograms obtained with MnO. (a) At 300 K, the material is paramagnetic. The observed (1 1 1) peak near y ¼ 12 is characteristic of the cubic crystallographic structure. (b) The additional peak near y ¼ 6 observed at 120 K reveal a doubling of the unit cell. It may be associated with the existence of an antiferromagnetic arrangement of the moments (inset).

Today, neutron diffraction has developed to a very sophisticated level; it allows very complex magnetic arrangements, including those found in spin glasses and other disordered systems, to be examined. X-rays interact with the electrons within matter. Since spin–orbit coupling makes the bridge between electron distribution and magnetism, they probe magnetism as well. Although the magnetic interaction term is very small, magnetic studies using X-rays (X-ray magnetic circular dichroism and X-ray diffraction) have recently developed, thanks to the availability of very intense X-ray sources at synchrotron radiation facilities. A specific feature of X-ray studies is element selectivity, which occurs when the wavelength of the incident radiation corresponds to the characteristic energy of a given absorption edge for some element contained in the material examined. Using magnetic X-ray circular dichroism, the spin and orbital contributions to the magnetization density can be measured separately.

Further Reading Bacon GE (1997) Fifty Years of Neutron Diffraction, The Advent of Neutron Scattering. Boca Raton, FL: CRC Press. Chikazumi S (2009) Physics of Ferromagnetism. Oxford: Oxford Science Publisher. Coey JMD (2010) Magnetism and Magnetic Materials. Cambridge: Cambridge University Press. de Lacheisserie E, Gignoux D, and Schlenker M (2002) Magnetism. vol. 1 and 2. Boston: Kluwer Academic. Levitt MH (2008) Spin Dynamics: Basics of Nuclear Magnetic Resonance. Chichester: Wiley. Mydosh JA (1993) Spin Glasses: An Experimental Introduction. London: Taylor and Francis. Schenck A and Gygax FN (1995) Magnetic Materials Studied by Muon Spin Rotation Spectroscopy. Handbook of Magnetic Materials. vol. 9, p. 57. Amsterdam: Elsevier. Slichter CP (1996) Principles of Magnetic Resonance. Berlin: Springer. Stein DL and Newman CM (2013) Spin Glasses and Complexity (Primers in Complex Systems). Princeton, NJ: Princeton University Press. Willis BTM and Carlile CJ (2013) Experimental Neutron Scattering. Oxford, NY: Oxford University Press.

Paramagnetism

☆

PF de Châtel, New Mexico State University, Las Cruces, NM, USA © 2016 Elsevier Ltd. All rights reserved. This is an update of P.F. de Châtel, Paramagnetism, Reference Module in Materials Science and Materials Engineering, Elsevier, 2016, ISBN 9780128035818, https://doi.org/10.1016/B978-0-12-803581-8.01113-9.

Nomenclature B Hm L mat M S r(e)

magnetic field (T) molecular field (A m–1) orbital angular momentum (kg m2 s−1) atomic magnetic momentum (kg m2 s−1) magnetization (A m−1) Stoner enhancement factor (dimensionless) density of states (J−1 m−3)

Paramagnetic materials do not order magnetically and have a positive magnetic susceptibility, which means the magnetization is pointing in the same direction as the magnetic field. Inevitably, there is also a diamagnetic susceptibility. The material is classified as paramagnetic if the sum of the two contributions is positive. Nevertheless, the paramagnetic susceptibility can be determined in both kinds of materials. In addition, the paramagnetic susceptibility of magnetically ordered materials (ferro-, ferri-, and antiferromagnets) above the transition temperature can be a subject of study. In the most common paramagnetic materials, the response to an external magnetic field is due to the alignment of atomic magnetic moments under the torque exerted by the field. The potential energy of the moment mat in a field B is V ¼ −mat  B

[1]

The magnitude of the atomic moment is of the order of the Bohr magneton, mB ¼ eħ/2me ¼ 9.27  10−24 JT−1, where e is the electron charge, ħ ¼ b/2p the Planck constant, and me the electron mass. Eqn [1] shows that mat tends to align parallel to the magnetic field and the energy gained by alignment in a field of 1T is 10−23 J, less than 1% of the thermal energy per degree of freedom, 1=2 kB T, at room temperature (kB is the Boltzmann constant). At ambient temperature, in a regular electromagnet, one should not expect a substantial paramagnetic magnetization. On the other hand, if the moments are free, that is, no interaction hampers their rotation, lowering the temperature can reduce the thermal fluctuations, whereupon the paramagnetic susceptibility will increase. If none of the constituent atoms of a material has a magnetic moment, there can still be a paramagnetic moment, due to moments induced by the magnetic field. The resulting Van Vleck susceptibility is independent of temperature (see below). The value of the atomic moment depends on the number of electrons and on their angular momentum. The filled shells of the ion cores and of fully ionized species, like O2− or Cu+, have zero angular momentum and hence no magnetic moment. The outermost electron shells are often but partially occupied and can carry a magnetic moment. However, in crystalline solids, these shells provide the delocalized Bloch states and contribute to the cohesive energy of the crystal. This requires a considerable overlap of the atomic wave functions on neighboring sites. The corresponding states are severely perturbed and cannot be labeled by the angular momentum quantum numbers ℓ and mℓ. In the case of transition metals, lanthanides, and actinides, under certain circumstances, there is a distinction between the outermost s and p states and the more localized states in the d and f shells, which are not fully occupied either. The former are responsible for the cohesion and determine the interatomic distances, while the latter remain localized and can form localized moments. This subtle division of electronic states makes transition and rare-earth metals and their compounds the most important ordered magnetic materials and also the strongest paramagnets. A given number of d or f electrons will form a total angular momentum in accordance with Hund’s rules, which require that (1) the total spin angular momentum S be as large as possible without violating the Pauli principle; (2) the total orbital angular momentum L be as large as possible without violating the Pauli principle and requirement (1); and (3) S and L be aligned antiparallel and form the total angular momentum, J ¼ | L − S | , if the shell is less than half filled, and parallel, J ¼ L + S, if it is more than half filled. The hierarchy of the three rules of Hund reflects a hierarchy among the interactions involved. Of these, the exchange interaction, which favors the parallel-spin arrangement, rule (1), is the strongest. The preference for the alignment of orbital moments, rule (2), is dictated by the less important correlation energy, while the spin–orbit coupling, responsible for rule (3), is ineffective in transition metals and quite important in the rare earths. Table 1 shows the ground states following from the three rules for transition-metal ions in the conventional notation (2S + 1)XJ, where X ¼ S, P, D, and F stand for L ¼ 0,1,2, and 3, respectively. Also shown are the ☆

Change History: March 2015. P.F. de Châtel made minor changes to the text and added one reference to the Further Reading section.

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36

Paramagnetism

Table 1

The ground states and effective moments of transition-metal ions

Ion

Configuration

Ground state

meff ¼ g

Ti3+, V4+ V3+ Cr3+, V2+ Mn3+, Cr2+ Fe3+, Mn2+ Co3+, Fe2+ Co2+ Ni2+ Cu2+

3d1 3d2 3d3 3d4 3d5 3d6 3d7 3d8 3d9

2D3/2 3F2 4F3/2 5D0 6S5/2 5D4 4F9/2 3F4 2D5/2

1.55 1.63 0.77 0 5.92 6.70 6.63 5.59 3.55

Table 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JðJ +1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi meff ¼ 2 SðS +1Þ 1.73 2.83 3.87 4.90 5.92 4.90 3.87 2.83 1.73

The ground states and effective moments of tripositive rare-earth ions

Ion

Configuration

Ground state

meff ¼ g

Ce3+ Pr3+ Nd3+ Pm3+ Sm3+ Eu3+ Gd3+ Tb3+ Dy3+ Ho3+ Er3+ Tm3+ Yb3+

4f1 4f2 4f3 4f4 4f5 4f6 4f7 4f8 4f9 4f10 4f11 4f12 4f13

2F5/2 3H4 4I9/2 5I4 6H5/2 7F0 8S7/2 7F6 6H15/2 5I8 4I15/2 3H6 2F7/2

2.54 3.58 3.62 2.68 0.84 0 7.94 9.72 10.63 10.60 9.59 7.57 4.54

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JðJ +1Þ

meff (measured) 2.4 3.5 3.5 – 1.5 3.4 8.0 9.5 10.6 10.4 9.5 7.3 4.5

effective moments, calculated for the L – S coupled as well as spin-only states, which will be discussed below. Table 2 gives the corresponding data for the tripositive rare-earth ions (H and I stand for L ¼ 5 and 6, respectively). If the quantum numbers of the ground state are known, the atomic magnetic moment can be calculated as mat ¼ gJmBJ, where gJ ¼ 1 +

JðJ +1Þ +SðS +1Þ −LðL +1Þ 2J ðJ +1Þ

is the Landé g factor. There is no classical analogy of the above procedure of determining the magnitude of the atomic moments as there should not be one, considering that the Bohr–van Leeuwen theorem precludes orbital magnetism in a classical system. What is often referred to as the classical derivation of the paramagnetic magnetization is the application of classical statistics to the quantum mechanical objects mat. In the context of classical statistics, the partition function involves the potential energy (eqn [1]), where mat  B ¼ matB cos W, where W is a continuous variable and integration is to be performed over ’ and W, the angular coordinates of the vector mat. In the proper quantum mechanical description, a summation over the possible values of the z component of mat is required and the partition function takes the form Z¼

J X MJ ¼ −J

  exp gJ mB BMJ =kB T

!Nat

where Nat is the number of atoms or ions. Summation of the geometrical series leads to  !Nat sinh gJ mB BðJ +1=2Þ=kB T   z¼ sinh gJ mB B=2kB T Deriving the free energy as F ¼ ð −1=kB T Þ and using M ¼ (−1 @F/V@B), the magnetization is found in the form M¼

  N at gm B gm B V J B J J B

[2]

Paramagnetism

37

where BJ ðxÞ ¼

2J +1 2J +1 1 1 coth x − coth x 2J 2J 2J 2J

is the Brillouin function. For large values of J, BJ(x) approaches the Langevin function, L(x) ¼ coth x − 1/x which is the result of the classical treatment outlined above. The susceptibility is found to obey Curie’s law,  2   g mB J ðJ +1Þ C @M N wc ¼ ¼ at m0 J ¼ [3] V 3K B T @H H¼0 T where m0 ¼ 4p10−7TmA−1 is the permeability of free space and the Curie constant, C, is implicitly defined by this equation. The Curie law is well obeyed by various salts and other compounds, where the atoms or ions carrying magnetic moments are far enough from each other,pso that their interaction, the exchange interaction, is negligible. In these cases the effective magnetic moment, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi meff ¼ gJ mB JðJ +1Þ, can be determined from the slope of w−1 versus T plots. In paramagnetic rare-earth materials, the measured meff values agree with the Hund’s rule values listed in Table 2. In most transition-metal oxides and salts, however, the meff values calculated from the Hund’s rule spin quantum numbers alone are closer to the measured values than the ones involving J, indicating that crystal fields are strong enough to suppress (‘quench’) the orbital angular moment. If the exchange interaction between magnetic moments is not negligible, the Curie–Weiss law, wCWa(T − y)−1, often holds. To account for the interaction, whose source was unknown at that time (1907), Weiss has introduced the molecular field Hm, which acts, in addition to the external field H, on each atomic magnetic moment. As Hm is supposed to describe the tendency of atomic moments to point in the same direction, it is assumed to be proportional with the magnetization, Hm ¼ lM. Substituting this expression in the expression for the magnetization in the total field, M ¼ wc ðH +Hm Þ ¼

C ðH +lMÞ T

the magnetization can be expressed as M ¼ wcwH with wcw ¼

wc C ¼ 1− lwc T−y

[4]

where y ¼ Cl is the paramagnetic Curie temperature. It is seen that the slope of w−1 cw versus T plots are determined by the effective moment the same way as in the case of the Curie law, but the straight lines representing the measured data do not go through the origin. Deviations from the straight-line behavior may be due to crystal fields. In most metals, the effect of the crystalline environment is so strong that all couplings imposed by Hund’s rules are broken and all but the ion-core electrons are delocalized. In rare-earth materials, the f electrons are localized and form magnetic moments conforming to Hund’s rules. In insulators, the d electrons of transition-metal cations remain localized, but the spin–orbit coupling is broken and the orbital angular momentum itself is destroyed. The latter process is called the quenching of angular momentum, not a particularly enlightening term. What it stands for is that, the spherical symmetry around a particular ion being broken by its neighbors in the crystal, the stationary states are no more eigenstates of the angular momentum. Due to the symmetry of the crystalline environment, some degeneracy of the eigenstates may remain. However, if all matrix elements of the three components ^z , between the degenerate wave functions vanish, its expectation value vanishes for any of the angular momentum, L^x , L^y , and L linear combination of these functions; the angular momentum is ‘quenched.’ If that is not the case, the angular momentum may be partially quenched and can be described by an effective angular momentum operator working within the multiplet of the lowest-lying crystal-field level. Partial or complete quenching of the orbital angular momentum influences the temperature dependence of the paramagnetic susceptibility if the temperature is high enough for the thermal energy kB T to be comparable to the splitting between the lowest and next-to-lowest crystal-field levels. At temperatures exceeding D=kB , D being the total splitting of the ground-state multiplet, the Curie or Curie–Weiss susceptibility is recovered with the effective moment appropriate to that multiplet. Higher-lying crystal-field states can influence the paramagnetic susceptibility also at very low temperatures, T  D=kB : Even though the thermal occupancy of such states is negligible in this limit, the magnetic field will mix them into the ground state via the perturbing Hamiltonian (eqn [1]). Taking account of this perturbation to second order, the corresponding contribution to the susceptibility, the Van Vleck paramagnetic susceptibility, is found to be R 2 X | c ∗ ðm Nat 0 ^ at Þz ci dt | m0 2 wVV ¼ [5] e V − e i 0 i independent of temperature. Here c0, c1 are the wave functions and e0, ei the energies of the ground state and higher-lying crystal-field states, respectively. The operator representing the atomic magnetic moment is   mat ¼ mB L^ + 2S^

38

Paramagnetism

^ at Þz has nonvanishing matrix elements between the Hund’s rule ground state and evidently not proportional to ^J: Therefore, ðm higher-lying eigenstates of different J values as well. In fact, Van Vleck derived for that case, with ions having small spin–orbit splitting in mind. However, the term ‘Van Vleck susceptibility’ is used with reference to any temperature-independent contribution to the paramagnetic susceptibility due to the quantum mechanical admixture of higher-lying states. As mentioned above, crystal fields in insulators constitute a subtle effect, compared to the full delocalization of conduction electrons in metals. The consequence of the latter for the paramagnetic response is indeed drastic. The conduction electrons, instead of following the Curie law (eqn [3]) with g ¼ 2 and S ¼ 1/2, contribute only the Pauli paramagnetism, a small, temperature-independent susceptibility, wP ¼ m0 m2B rðeF Þ

[6]

where r(eF) is the density of states per unit volume at the Fermi energy. It is indeed the Pauli principle that suppresses the 1/T divergence of w: most of the electrons are unable to flip their spin, because the opposite-spin state of the same energy is occupied. Only the electrons in states with energies within kB T from the Fermi energy are given the choice of ms values implied by eqn [2]. As the density of such electrons is about KBTr(eF), the temperature in the denominator of the Curie susceptibility is canceled out and eqn [6] can be seen to follow from eqn [3]. In the alkali metals, r(eF)  1047 J−1 m−3 whence from eqn [6] wP  10−5, in good agreement with the paramagnetic susceptibility derived from experiment. In transition metals, the density of states is about one order of magnitude larger, but even so, eqn [6] underestimates the paramagnetic susceptibility, which is enhanced by the exchange interaction between the d electrons. Like in the derivation of the Curie–Weiss law, here too the effect of this interaction can be described by the molecular field. The result, x¼

wP 1− lwP

[7]

is formally similar to eqn [4], but it should be noted that it is independent of temperature in this case. Equation [7] gives the exchange-enhanced paramagnetic susceptibility, S ¼ (1 − lwP)−1 is the Stoner enhancement factor. Among the transition metals, the largest value of S is observed in palladium, where S ¼ 10.

Further Reading Ashcroft NW and Mermin ND (1976) Solid State Physics. New York: Holt, Reinhart and Winston. Blundell S (2001) Magnetism in Condensed Matter. Oxford University Press. Chikazumi S (1965) Physics of Magnetism. New York: Wiley. Kittel C (1986) Introduction to Solid State Physics, sixth edn New York: Wiley. O’Handley RC (2000) Modern Magnetic Materials Principles and Applications. New York: Wiley. Van Vleck JH (1932) The Theory of Electric and Magnetic Susceptibilities. Oxford: Clarendon Press.

Diamagnetism PF de Châtel, New Mexico State University, Las Cruces, NM, USA © 2005 Elsevier Ltd. All rights reserved. This is an update of P.F. de Châtel, Diamagnetism, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 414–417, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00518-0.

Nomenclature A B L M mer
v mcv p

vector potential (A) magnetic field (T) canonical angular momentum (kg m2 s–1) magnetization (A m–1) kinetic angular momentum (of electron) (kg m2 s–1) kinetic momentum (of electron) (kg m s–1) canonical momentum (kg m s–1)

Diamagnetic materials do not order magnetically and have a negative susceptibility, which means the magnetization is pointing opposite to the magnetic field. Possibly, there is also a paramagnetic susceptibility. The material is classified as diamagnetic if the sum of the two contributions is negative. Nevertheless, the diamagnetic susceptibility can be determined in both kinds of materials. Diamagnetism is a very small effect, but all the more puzzling. It entails a response of all known materials to an external magnetic field, which manifests itself in a magnetization opposing that field. It appears thus that the magnetic field generates a magnetic dipole, whose orientation is most unfavorable, being at a maximum of the potential energy as a function of the angle between the magnetic field and the magnetization, E ¼ −B  M ¼ BM > 0 There is no simple, intuitive explanation for this paradox. This is not surprising though, considering that our intuition is mostly based on classical physics. The celebrated Bohr–van Leeuwen theorem states that the classical free energy of a system of charged particles moving in a magnetic field is independent of the field. Since the magnetization is proportional to the derivative of the free energy with respect to the field, 1 @F M¼− V @B

[1]

this theorem implies that classical physics, in this case classical mechanics combined with classical statistics, cannot account for any field-induced magnetization, be it paramagnetic (aligned along the field) or diamagnetic (opposing). In the case of diamagnetic metals, whose charge carriers behave like free electrons, this conclusion contradicts what one would expect from electrodynamics. Charged classical free particles exposed to a magnetic field move in circles. Lenz’s law of induction requires that the current loops represented by the orbiting electrons produce a magnetization opposed to the field which induces the currents; this should result in a diamagnetic response. It is no wonder that Niels Bohr was fascinated by this paradox. His main interest was the stability of atomic orbitals, which also contradicts classical expectation. He has formulated the rule of angular-momentum quantization, which, imposed on the classical treatment, accounted for the observed stable orbitals. As it will be shown below, the same rule, applied in the presence of a magnetic field, gives the correct answer for the diamagnetism of free atoms. The diamagnetism of free electrons cannot be understood in this semiclassical way, here a fully quantum mechanical treatment is needed. The formal proof of the Bohr–van Leeuwen theorem is rather simple, but conceals the clue to its counterintuitive implication. Indeed, if one writes the free energy, F ¼ −kB T ln Z, of a system of N identical particles (taking electrons of mass me and charge −e), the partition function has the form Z N Z Y ⋯ dr i dpi
expð−Hðfr i g, fpi gÞ=kB TÞ [2] Z¼ i¼1

where H is the Hamiltonian, a function of all position vectors ri and canonical momenta pi. The latter are written as pi ¼ me vi −eAðr i Þ where A is the vector potential describing the magnetic field, which enters the Hamiltonian via the kinetic energy, 1 1 Ekin ¼ me v2i ¼ ½p +eAðr i Þ2 2 2me i

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40

Diamagnetism

To prove the independence of Z from A, one changes the integration variables pi to p0i ¼ pi +Aðr i Þ and realizes that this will not change the result of integration, because the shift of the limits of integration, 1, has no effect. The vector potential thus being eliminated from eqn [2], the partition function must be independent of the magnetic field. To understand the implications of this freedom to shift parameters, it is instructive to carry out the differentiation in eqn [1] for the simple case of a system of Na identical atoms. If the electron–electron interaction is treated in the mean-field approximation, the N electrons localized on the same atom are described by the Hamiltonian H¼

N X i¼1

Hð1Þ ðr i , pi Þ

with Hð1Þ ðr, pÞ ¼ VðrÞ + ½p + eAðrÞ2 =2me

[3]

where V(r) is the potential consisting of the potential of the nucleus and of the “other” N − 1 electrons, the latter contribution being spherically averaged (central-field approximation). The partition function (eqn [2]) becomes then the product of N identical integrals, Z Z N N Z¼ f dr dp expð−Hð1Þ ðr, pÞ=kB Tg ¼ fZð1Þ g and, consequently, the free energy is a sum of N identical terms, F ¼ −kB TN ln Zð1Þ . Choosing the gauge 1 A ¼ − ðr
BÞ 2

[4]

one obtains M¼ −

Na @F N eN ¼− a V @B V 2me

Z Z

expð−Hð1Þ ðr, pÞ=kB TÞ e dr dp r
½p− ðr
BÞ
2 Zð1Þ

[5]

where Na is the number of atoms. The result is seen to be proportional to the thermal average of the kinetic angular momentum, r
me v . This is not unexpected, because currents, the sources of magnetization, are related to the velocity, not the canonical momentum p, of the charge carriers. The same holds for the kinetic energy in the Hamiltonian [3], which appears in the statistical weight factor expð−Hð1Þ =kB TÞ. Consider now two points in the phase space of a single electron, which are given by the same position vector r and opposite velocities, r1 ¼ r2 ¼ r e e p1 − ðr 1
Bðr 1 ÞÞ ¼ −ðp2 − ðr 2
Bðr 2 ÞÞÞ 2 2 The statistical weight factors associated with these two points are identical, whereas the kinetic angular momenta have opposite signs and the same magnitude. Therefore, the sum of their contributions to the integral will vanish. The cancelation would have been more obvious, if p was shifted to p0 ¼ p +ðe=2Þðr
BðrÞÞ. In that case, the integrand would have become an odd function of p0 resulting obviously in a vanishing M. Again, the freedom to shift the canonical momentum at a given value of r, which amounts to shifting the canonical angular momentum L ¼ ðr
pÞ, turns out to be important in the derivation. As classical mechanics puts no constraints on L and, according to the ergodic theorem underlying classical statistics, a system will visit all possible states of different values of L, it is understandable that M has to vanish on average. These considerations suggest that the classical realm need not be left altogether. It may suffice to impose some constrains on the angular momentum, in the spirit of the Bohr model of atoms, to get sensible results for the magnetization. As a uniform magnetic field along the z-axis, B ¼ (0,0,B), leaves all rotations around the z-axis as symmetry operations, it is postulated that Lz be limited to the values mℏ, m ¼ 1, 2, 3, . . .. To impose this condition, each integral over the entire six-dimensional (r,p) phase space must be replaced by a sum of integrals, each over a subspace Pm containing only (r,p) points, for which ðr
pÞz ¼ mℏ. In particular, X Z Z X dr dp expð−Hð1Þ ðr, pÞ=kB TÞ¼ Zð1Þ Zð1Þ ¼ m m

Pm

m

By the definition of the subspace Pm Z Z Pm

so that eqn [5] can be written as

dr dpðr
pÞz expð−Hð1Þ ðr, pÞ=kB TÞ¼ mℏZð1Þ m

Diamagnetism

Mz ¼ −

41

Na X eN Zð1Þ e2 N 2 th m f ℏm + hr im Bg V m 2me Zð1Þ 4me

ð1Þ where NZ ð1Þ ¼ pm is the thermal occupation probability of the Lz ¼ mℏ orbit. Introducing the Bohr magneton, mB ¼ eℏ=2me, the m =Z first term takes the form pm mmB , which can be recognized as the paramagnetic contribution to the magnetization. In the second term, which makes evidently a negative, diamagnetic contribution to the magnetization, hr2 ith m ¼ R R ð1Þ ð1Þ 2 dr dp r expð−H ðr, pÞ=k TÞ=Z is the thermal average of r for the same orbit m. Here, it is assumed that the r vector is B 2 Pm in the z ¼ 0 plane and jrj ¼ r, constant. It then follows that

Mdia ¼ −

Na e2 X 2 th hr im B V 4me m

In practice, no thermal excitation out of the ground state will occur and Nhr2 ith m can be replaced by the average of the squared radii of occupied orbits. Clearly, at this stage the simple Bohr model fails. It would be a tortuous extension to introduce the principal quantum number n on the basis of the Bohr–Sommerfeld model and the angular momentum quantum number ℓ on an ad hoc basis. Even so, only for T ¼ 0 would the introduction of the Pauli principle suffice to give the correct result. For T>0, Fermi–Dirac statistics should be applied, which is a further step away from classical physics. In a proper quantum mechanical treatment, (2/3) hr2i enters, instead of hr2 ith where h . . . i stands for the expectation value and the factor 2/3 accounts for the difference between the expectation values of r 2 ¼ x2 +y2 +z2 and r2 ¼ x2 +y2 . The diamagnetic susceptibility is thus ðLangevinÞ

wdia

Mdia N m e2 X 2 hr j i ¼− a 0 H V 6me j

¼

[6]

where the summation is over the occupied orbits j on each atom. Langevin’s result for atomic diamagnetism was brought to this form by Pauli. It is remarkable that Planck’s constant does not appear in the expression for the diamagnetic susceptibility, whereas the importance of angular-momentum quantization in its derivation is obvious. Also, eqn [6] is an appealing result, as it enables a seemingly classical interpretation in terms of currents ij, induced on each orbit j. Such currents will generate magnetic moments mj ¼ ij aj ∝hr 2j i, where aj is the area enclosed by the orbit j. The sign of the moments (opposite to B) can be understood in terms of Lenz’s law and their magnitudes can be derived from Faraday’s law of induction. This derivation, however, invokes well-defined orbits, which are modeled by tiny current loops. Bohr has grafted such orbits onto the classical treatment of particles moving in a potential proportional to r−1 and postulated the quantization of angular momentum. Remarkably, this supplement to the classical description was sufficient to understand quantized energy levels and line spectra, as well as diamagnetism. The validity of eqn [6] turns out to extend beyond the case of atomic orbits. Most notably, it is applicable to ring-shaped molecules, whose structure offers extended orbits. The diamagnetic susceptibility of substances containing such molecules is known to be anisotropic, being larger in magnitude if the magnetic field is applied perpendicular to the plane of the ring-shaped molecules, k

jw? dia j > jwdia j The interpretation of this phenomenon in terms of “ring currents” goes beyond the qualitative argument that the induced magnetic moments are maximized in the perpendicular configuration. For benzene, C6H6, the anisotropic susceptibility k

wan ¼ w? dia −wdia is found to be close to what eqn [6] gives. The summation in eqn [6] is to be taken over the six carbon p electrons, which, not being localized in sp2 hybrid states, are free to move around the molecule: wan benzene ¼ −6

A m 0 e2 2 A ir r ¼− a V m 4me V m H hex

[7]

Here, A/Vm is the density of molecules, A being Avogadro’s number and V m ¼ 89
10−6 m3 the molar volume of benzene, r ¼ 0.139 nm is the distance from the axis of the molecule to the carbon nuclei and ir is the total ring current carried by the six delocalized electrons. Pauling has shown in 1936 that the ratio of the anisotropic susceptibility of other, more complicated aromatic hydrocarbons to that of benzene is correctly given by a simple model of conducting networks, assuming that the “resistance” of a carbon–carbon bond is the same in all such molecules. The ultimate success of this model is the correct estimation of the anisotropic susceptibility of graphite. The crystal structure of graphite consists of honeycomb-like layers built of benzene-like units. Assuming that in each hexagonal unit a ring current is induced of the same magnitude as in benzene, one finds that the current of neighboring unit cells cancel and the current density inside the material vanishes. However, the outermost hexagons have two sides with uncompensated currents ir flowing in a zigzag pattern at 30 from the orientation of the surface. The resulting surface current density is jS ¼ ir cos 30 =d, where pffiffiffi d ¼ 0.335 nm is the interlayer distance in graphite. The surface current density jS generates a magnetization Mgraphite ¼ −jS ¼ −ir 3=2d inside the sample. This is to be compared with Mbenzene ¼ −ðA=V m Þir ahex (cf. eqn [7]), leading to

42

Diamagnetism pffiffiffi wan 3V m graphite ¼ ¼ 7:6 an wbenzene 2Aahex d

[8]

which is close enough to the experimental value of 6.9, considering the simplicity of the model. Semiclassical and empirical schemes like the ones used above for free atoms and aromatic compounds, respectively, cannot be applied to the diamagnetic response of conduction electrons in metals. Even the simplest model, the free-electron gas, has been a major challenge, until Landau gave his derivation of what has come to be called Landau diamagnetism. In fact, in most textbooks the Bohr–van Leeuwen theorem is only mentioned in this context, even though its validity is not violated by the presence of atomic or molecular potentials. The Schrödinger equation was four years old, when Landau (who was twenty-one at the time) solved it for free electrons in the uniform magnetic field (eqn [6]). Instead of the symmetric gauge (eqn [4]), he has chosen A ¼ ð0, xB, 0Þ

[9]

which enabled the separation of variables. The resulting differential equation in x was found to be identical with the Schrödinger equation of a harmonic oscillator with a force constant e2B2/me, while the ones in y and z were satisfied by plane-wave solutions. The latter is reasonable for the z dependence, considering that the Lorentz force is perpendicular to B, so that the motion along the z axis is free. The density of the eigenstates described by eikz z is that of a one-dimensional electron gas, rðeÞ ¼

2eB pffiffiffiffiffiffiffiffiffiffiffi 2me e h2

[10]

The motion in the (x,y) plane gives a discrete spectrum, with highly degenerate states at the energies 1 ℏe en ¼ ðn + Þ B 2 me

[11]

the Landau levels, which correspond to the stationary states of the harmonic oscillator. The resulting total density of states has a spiked structure. A singular contribution of the form [10] begins at each level [11]. These rðe−en Þ contributions stand for the motion in the z direction while the motion in the (x,y) plane corresponds to the nth level of the harmonic oscillator. As to the x- and y-dependence of the wave functions, they are not suggestive of the circular motion resulting from the classical equations of motion for a charged particle subjected to a uniform magnetic field. This may seem to be due to the choice of the asymmetric gauge [9]. However, it is hardly surprising that the classical motion is not implied by the calculation of an effect, which does not occur at all in the classical theory. The maxima in the density of states at energies separated by ℏeB/me result in a total energy, which is a periodic function of 1/B. Through eqn [1], a similar periodicity arises in the magnetization, which is the signature of the de Haas–van Alphen effect. In the low-field limit, the same B-dependent total-energy expression yields the diamagnetic susceptibility ðLandauÞ

wdia

¼−

nm2B eF

where n is the electron density and eF is the Fermi energy. This is seen to be −1/3 times the Pauli paramagnetic susceptibility, a relationship, which cannot be expected to be valid for metals in general, but seems to hold for the so-called “simple” metals, for which the nearly free-electron approximation works.

Further reading Ashcroft NW and Mermin ND (1976) Solid State Physics. New York: Holt, Reinhart and Winston. Chikazumi S (1965) Physics of Magnetism. New York: Wiley. O’Handley RC (2000) Modern Magnetic MaterialsPrinciples and Applications. New York: Wiley. Peierls RE (1955) Quantum Theory of Solids. Oxford: Oxford University Press. Van Vleck JH (1932) The Theory of Electric and Magnetic Susceptibilities. Oxford: Clarendon Press.

Ferromagnetism N Magnani, Università di Parma, Parma, Italy © 2005 Elsevier Ltd. All rights reserved. This is an update of N. Magnani, Ferromagnetism, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 201–210, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/01115-3.

Introduction The Molecular Field Approach to Ferromagnetism Magnetostatic Energy and Demagnetizing Field Magnetic Anisotropy Magnetocrystalline Anisotropy Shape Anisotropy Stress Anisotropy Anisotropy Induced by Magnetic Annealing Magnetic Domains The Magnetization Curve The Hysteresis Cycle Magnetization Rotation and Anisotropy Further Reading

43 43 46 46 48 48 49 49 49 51 51 53 54

Nomenclature B H M T m

magnetic induction (T) magnetic field (A m−1) magnetization (A m−1) temperature (K) magnetic moment (mB)

Introduction As far as isolated atoms are concerned, only two different magnetic behaviors exist: diamagnetism and paramagnetism. The latter property is only present if the considered atoms possess a nonzero permanent magnetic dipole, and is responsible for the weak attractive force experienced by a paramagnetic substance in an applied magnetic field. However, the most interesting magnetic properties of condensed matter arise when the interaction between the individual elementary moments of each atom is considered. In particular, ferromagnetism concerns the situation where the moments have the tendency to align, one parallel to another, even in the absence of any external magnetic field, thus leading to the possibility of obtaining very large values of the magnetization with very little values of the applied magnetic field (spontaneous magnetization). Only a few elements in the periodic table (Fe, Co, Ni, Gd, Dy, and Tb) order ferromagnetically, but an extremely large number of compounds and alloys do. From the practical point of view, apart from their high values of the magnetic susceptibility and relative permeability, one of the most interesting properties of ferromagnetic compounds is the possibility of retaining a large amount of the induced magnetization when the external field is completely removed. This makes it possible to develop devices such as permanent magnets (i.e., magnetized bodies which produce a constant magnetic field in a given volume of space without the need to continuously supply electrical or chemical energy), memory storage devices, magnetic circuits, etc.

The Molecular Field Approach to Ferromagnetism At the beginning of the twentieth century, Pierre Weiss interpreted the tendency of ferromagnets to spontaneous magnetization by introducing a “molecular field” responsible for the ordering of the elementary magnetic moments, and he assumed that this field is linearly proportional to the bulk magnetization (total magnetic moment per volume unit). It may be recalled that the magnetization of a set of identical noninteracting ions, each having total angular momentum J, in a magnetic field H can be expressed as
 gmB JH M ¼ ngmB JBJ [1] kB T where g is the Landé factor and

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00280-8

43

44

Ferromagnetism BJ ðxÞ ¼



 2J +1 2J +1 1 x coth x − coth 2J 2J 2J 2J

[2]

is the Brillouin function; the magnetization of a ferromagnet can then be obtained by substituting in eqn [1] the applied magnetic field H with H + lM, leading to the implicit equation
 gmB J gm J M H + B lM [3] ¼ BJ ngmB J kB T kB T In order to investigate the possibility of spontaneous magnetization, the H ¼ 0 case is considered; eqn [3] can be rewritten as m ¼ tx ¼ BJ ðxÞ

[4]

with x¼

gmB JlM ; kB T

t¼

kB T nlðgmB JÞ2

[5]

A graphical solution of eqn [4] is given in Figure 1; if t  tC ¼ (J + 1)/3J, only the trivial solution x ¼ 0 is present, while otherwise there is also a possible solution with x 6¼ 0. This means that there exists a critical temperature TC (Curie temperature), below which a ferromagnetic compound can have a nonzero magnetization in the absence of an applied magnetic field. The Curie temperature and saturation magnetization for several ferromagnetic elements are listed in Table 1. It is worth noting that eqn [4] can be expressed in terms of J, m ¼ M(T )/M(0) and t ¼ (J + 1)/(3J) T/TC only. In the classical limit (J ! 1), BJ(x) is substituted by the Langevin function coth x − 1/x, and the function m(t) is the same for all ferromagnetic materials; the latter statement is often referred to as the law of corresponding states. Figure 2 shows the temperature dependence of the magnetization measured for nickel (Ni), together with the curve derived from eqn [4] considering a pure-spin moment J ¼ S ¼ 1/2. The overall agreement is fairly good, and can be further improved by the use of more accurate models (the low-temperature decrease due to spin waves and the critical behavior near TC are shown as dashed lines for comparison). Above TC, the sample is in a paramagnetic phase (disordered moments), and its magnetic susceptibility can be calculated by noting that BJ(x) ’ (J + 1) x/(3J) for small values of x, leading to 1.5

t tC

t > tC

t < tC

1 y

Spontaneous magnetization 0.5

y BJ (x) y tx

0

1

x

2

3

Figure 1 Graphical solution of eqn [4] (see text for details). Table 1 Curie temperature, saturation magnetization at room-temperature, and magnetic moment per formula unit (at 0 K) of ferromagnetic elements Element

TC (K)

MS (106 A m−1

mB/formula unit

Cobalt Iron Nickel Gadolinium Terbium Dysprosium

1388 1043 627 293 221 85

1.42 1.71 0.48

1.72 2.22 0.61 7.10 9.34 10.0

Data from: (1) Kittel C (1962) Introduction to Solid State Physics, 7th edn. London: Wiley; (2) Jiles D (1991) Introduction to Magnetism and Magnetic Materials. London: Chapman and Hall; (3) Elliott JF, Legvold S, and Spedding FH (1954) Some magnetic properties of dy metal. Physical Review 94: 1143; (4) Hegland DE, Legvold S, and Spedding FH (1963) Magnetization and electrical resistivity of terbium single crystals. Physical Review 131: 158; (5) Heller P (1967) Experimental investigations of critical phenomena. Reports on Progress in Physics 30: 731.

Ferromagnetism

45

1 0.8 m 0.6 0.4

Experimental Calculations

0.2 0

0.2

0.4

t

0.6

0.8

1

Figure 2 Experimental (diamonds) and calculated (lines) value of m ¼ M(T )/M(0) vs. t ¼ T/TC for nickel (J ¼ S ¼ 1/2). Experimental data are taken from Weiss P and Forrer R (1926) Aimantation et phenomene magnetocalorique du nickel. Ann. Phys. 5: 153. The full black line is calculated by means of eqn [4]; dashed lines take into account the spin-wave excitations at low T and the critical behavior near the Curie point.

w

M C ¼ H T−lC

[6]

where C can be identified with the paramagnetic Curie constant (w ¼ C/T if no ferromagnetic order exists, i.e., l ¼ 0) and TC ¼ lC. This very simple model gives a fairly accurate description of the experimental magnetic susceptibilities of ferromagnetic compounds at high temperatures. It must be noted that lM can be as large as several times the maximum magnetic field which can be produced in a standard laboratory, and is about four orders of magnitude larger than the dipolar interaction due to the other magnetic ions in the crystal. The physical origin of this huge molecular field was not clear until the development of quantum mechanics made it possible to point out the role of the exchange interaction (a purely quantum effect and a direct consequence of Pauli’s principle) in determining the magnetic spin alignment. The exchange interactions between a collection of N spins can be described by the Heisenberg–Dirac Hamiltonian, which has the form X [7] HH−D ¼ − Jij Si  Sj i, j where the sign of the exchange constants Jij depends on whether the two spins labeled i and j are coupled ferromagnetically (positive) or antiferromagnetically (negative), and the sum usually involves only the z nearest neighbors since the interaction strength drops very rapidly as the distance increases. In the presence of an applied magnetic field H, the Zeeman term X [8] HZ ¼ −2mB H  Si i

must also be considered (a gyromagnetic factor of 2 was taken, in the hypothesis that pure spin moments are being dealt with). The molecular field approximation consists in reducing the N-body Hamiltonian HH–D+HZ to N one-body Hamiltonians Hi, by replacing all the spin operators except Si with their average values: " # X Jij hSj i +2mB H [9] Hi ¼ −Si  j

The case now considered is that of a ferromagnetic body, with all the relevant exchange constants Jij ¼ J > 0 for simplicity. Since X [10] M ¼ 2mB hSl i l

is the total magnetization of the sample, eqn [9] can be rewritten as Hi ¼ −2mB Si  ½H +lM

[11]

with l¼

zJ 4m2B N

[12]

Equation [11] immediately shows that the mean-field analysis based on the Heisenberg–Dirac exchange Hamiltonian provides an a posteriori justification to Weiss’ molecular-field hypothesis; as one might expect, the accuracy of this approximation is higher when z and J are larger, since in this case the role of the fluctuations Si − h Si i is practically negligible and the substitutions made in eqns [7] and [8] are well-grounded.

46

Ferromagnetism

One word of caution is required before the end of this section. The existing theories of magnetism may roughly be divided into two groups, dealing respectively with localized and itinerant magnetic moments. The Heisenberg approach which was mentioned before belongs to the former; to give a meaning to eqn [7], it must be explicitly assumed that each ion in the crystal carries on it a well-defined magnetic moment Si due to the electrons which are bound to it. Although this approach is very useful when dealing with certain substances and alloys and for studying critical magnetic behaviors, its validity for metals is questionable due to the presence of conduction electrons, not tied to a specific ion but belonging to the whole crystal. On the other hand, band models considering the role of itinerant electrons have managed to solve several flaws of localized theories, such as the noninteger values of the magnetic moment per atom experimentally measured in most ferromagnets (Table 1), which could not be explained within a localized-moment framework.

Magnetostatic Energy and Demagnetizing Field One may consider a sample of a magnetic substance, composed of n magnetic atoms per unit volume, each carrying an elementary moment m, and imagine applying a magnetic field which is strong enough to align all these dipoles along the same directions. As a result of this process, the sample will display a saturation magnetization MS ¼ njmj (it may be recalled that the magnetization is defined as the total magnetic moment per volume unit). The energy required to obtain this configuration is Z E ¼ −m0 HdM

[13]

[14]

where H indicates the magnetic field and M is the magnetization vector. In the absence of dissipative processes, an equivalent amount of magnetic potential energy is stored within the sample. In turn, any magnetized body produces a magnetic field in its surrounding space, as one can immediately witness by means of a compass needle. In close analogy with the charge polarization of a dielectric material, one can describe this field as being generated by “free” magnetic poles (i.e., positive poles not neutralized by the presence of a negative pole in their immediate neighborhood and vice versa) on the surface; in addition to this, the presence of a magnetic field generated by an external source is usually considered. Outside the sample, the magnetic induction B and the magnetic field H are always proportional (as B ¼ m0[H + M] and M ¼ 0) and their flux lines are oriented from the north (N) to the south (S) poles. On the other hand, the total magnetic field Hint inside the sample is found by summing the external field, which is parallel to the magnetization, and the contribution from the free poles, which is antiparallel to it (in fact, the flux lines of H always begin on N poles and end on S poles except when the magnetic field is due to electric currents, in which case they are closed and continuous). The free-poles contribution is also called demagnetizing field, and can be represented as Hd ¼ −N d M

[15]

where Nd is, in general, a second-order tensor whose components depend on the geometry of the studied system; in some particular cases and/or for special directions of the magnetic field, M and H are collinear, and one can write a simpler relation where Nd is a dimensionless number called demagnetizing factor. For example, a homogeneous spherical sample (Figure 3) has Nd ¼ 4p/3, hence Hd ¼ −

4p M 3

[16]

Demagnetizing factors for samples of various geometries and for different field directions can be found in specialized textbooks.

Magnetic Anisotropy Another contribution to the energy balance, that is, magnetic anisotropy, will be dealt with now. While the meaning of such a locution is quite simple (the magnetic behavior of a material depends on the direction), it is not easy to understand the physical reasons behind its very existence. In fact, magnetic anisotropy may be produced by several different mechanisms, which include (but are not limited to) magnetocrystalline (or crystal-field) anisotropy, shape anisotropy, stress anisotropy, annealing in a magnetic field, etc. For the sake of simplicity, the considerations made in the following will be referred to a uniformly magnetized single-crystal sample. From a phenomenological point of view, the effect of magnetic anisotropy may be taken into account by adding to the energy balance an extra term EA (anisotropy energy), which depends on the relative orientation of the magnetization vector with respect to the crystallographic axes. As shown by the Russian physicist Akulov, the free anisotropy energy per unit volume can always be expressed by an infinite series

47

Ferromagnetism

H

N

S

Figure 3 Magnetic field generated inside and outside a uniformly magnetized sphere. The magnetization vector M is directed from the S to the N pole; the resulting demagnetizing field (inside the sphere) is antiparallel to M.

EA ¼

+1 X 3 X 3 X n¼1

¼

3 X

i1 ¼1 i2 ¼1

ci ai +

i¼1

⋯

3 X in ¼1

3 X 3 X

! ci1 ,i2 ,...,in ai1 ai2 . . . ain

3 X 3 X 3 3 X 3 X 3 X 3 X X ci,j ai aj + ci,j,k ai aj ak + ci,j,k,l ai aj ak al +⋯

i¼1 j¼1

i¼1 j¼1 k¼1

[17]

i¼1 j¼1 k¼1 l¼1

where a1, a2, and a3 are the direction cosines of the magnetization vector with respect to the Cartesian axes x, y, and z, and several coefficients may equal zero for symmetry reasons; for example, all the terms of the sum with odd n vanish if eqn [17] is invariant under inversion of the magnetization. In most relevant cases, eqn [17] can be rewritten as EA

+1 X

K n An ða1 , a2 , a3 Þ

[18]

n¼1

for example, a ferromagnetic crystal with cubic lattice symmetry has A0 ða1 , a2 , a3 Þ ¼ 1 ; A1 ða1 , a2 , a3 Þ ¼ a21 a22 +a21 a23 +a22 a23 ; A2 ða1 , a2 , a3 Þ ¼ a21 a22 a23 ; etc. The coefficients Kn, named anisotropy constants, can be determined by magnetization measurements; a knowledge of a reliable set of anisotropy constants for a given sample can lead to a direct phenomenological interpretation of the magnetic processes and is useful to make comparisons between different systems. For lattices with uniaxial symmetry (e.g., cylindrical, hexagonal, and tetragonal), compact expressions for the anisotropy energy can be derived as a function of the polar and azimuthal angles y and ’, which define, respectively, the angle between the magnetization vector M and the z-axis (coincident with the symmetry axis), and the angle between the projection of M on the xy-plane and the x-axis. In this case, EA ¼

+1 X

X

sin2n y

n¼0

0l k). Eex and EA are opposite in sign, and the absolute value of both grows with N; the actual size of the Bloch wall then results from a competition of the two, being smaller when the anisotropy dominates and larger when the exchange interaction dominates. In the large-N limit Eex ’ JS2p2/(Na2), and minimizing the total energy gives an approximate domain wall width of rffiffiffiffiffiffiffiffi 2J [30] ℓ ¼ Na ’ 2pS aK 1 typically a few hundred lattice constants for iron.

The Magnetization Curve From the macroscopic point of view, ferromagnets may be considered at a first glance as magnetic materials with extremely large susceptibility and permeability. In practice, as both of them are strongly dependent not only on the temperature, but also on the applied field and on the past history of the sample, they are not very useful parameters and it is quite simple to describe the physical properties of a ferromagnet by means of M versus H (or B versus H, being B ¼ m0(H +M)) plots, whose main characteristics will be discussed in the following sections.

The Hysteresis Cycle When an external magnetic field is applied to a ferromagnetic sample in a demagnetized state in order to bring its resulting magnetization to saturation, one obtains the magnetization curve shown in Figure 7. Two distinct and coexistent processes must be considered in order to understand this (Figure 8): boundary motion and domain rotation. In the former case, those domains which are favorably oriented with respect to the applied field grow in size at the expense of the others, without changing their overall magnetization direction; this process is dominant at low fields. At intermediate fields, the process of sudden rotation of the magnetization of unfavorably oriented domains to the easy-axis direction(s) nearest to that of the applied field also becomes significant. At a given value of the magnetic field (labeled HS), saturation is finally achieved (M ¼ MS, defined in eqn [13]), mainly by coherent rotation of the domain magnetization toward the applied field direction. For H > HS, the M(H) curve is flat since the applied field is already strong enough to align all the elementary dipoles in the sample along its direction.

H1

H2

Magnetization saturated

Reversible coherent rotation of domains

Domain wall motion

Boundary motion  irreversible domain : rotation

Ferromagnetism

M

52

Hs

H

Figure 7 “Virgin” magnetization curve of a ferromagnet. The main mechanisms by which the magnetization process advances for several applied field values are indicated.

Easy magnetization direction Magnetic field

H 

H H1

H H2

H Hs

Figure 8 Schematic view of the main mechanisms by which the magnetization process advances for several applied field values.

If the value of the applied field is now reduced to zero, it is noticed that the obtained M(H) curve does not coincide with the previous one apart from a small region near HS. In particular, the magnetization at zero applied field does not go back to zero (retentivity), and its value MR is called remanence or remanent magnetization. In order to force the magnetization to zero, it is necessary to apply a magnetic field HC in the opposite direction; this value is called coercivity or coercive field. (Some authors make use of a distinction between remanence/coercivity (indicating the sample behavior after it has reached saturation) and remanent

Ferromagnetism

53

magnetization/coercive field (relative to a magnetization curve reaching an arbitrary value M < MS).) Raising the reverse field strength to HS leads to the saturation of the magnetization in the opposite direction as before. Reversing the field once again, one obtains the so-called hysteresis loop (Figure 9). While “hard” magnetic materials (i.e., those displaying a large coercivity) are generally useful as permanent magnets or for memory-storage purposes (in order to avoid demagnetization due to stray magnetic fields), “soft” ones may be good candidates for the realization of power transformers or electromagnets. In the former case, apart from retentivity, coercivity, and Curie temperature, another figure of merit which is often used in practice is the maximum energy product BHmax, which is the maximum value of the product B  H in the demagnetizing quadrant (M > 0, H < 0) of the hysteresis curve. This corresponds to the energy stored in the considered material within a magnetic circuit operating at an optimized workpoint, and should not be confused with the energy lost due to the irreversible processes during one hysteresis loop, that is, ∮BdH

[31]

Magnetization Rotation and Anisotropy Most of the energy loss during the hysteresis cycle occurs at low fields as a result of irreversible domain boundary motion, due to inhomogeneous microstrains (dislocations) which obstructs the rotation of the magnetic moments. On the other hand, processes involving domain rotation are governed by a competition between the anisotropy and the Zeeman energy EZ ¼ −m0H  M. To study the role of anisotropy in determining the shape of the magnetization curve, assume that the boundary motion of domain walls can be achieved with a negligibly small magnetic field. In the case of a uniaxial sample, M(H) can be found by minimizing X K n sin2n y −m0 H  M [32] EA +EZ ¼ n

Let y be the angle between the magnetization vector and the c-axis, which will be considered also as the EMD, for simplicity. If H is applied along the c-axis, both EA and EZ are minimized M k H and saturation is reached immediately when the external applied field is equal to the demagnetizing field. If H is applied perpendicularly to the easy-axis, differentiating eqn [32] and knowing that M ¼ MS sin y leads to
2n −1 X M HMS ¼ 2nK n [33] M S n an implicit expression for the magnetization curve. By putting M ¼ MS, one can immediately find HS ¼

2 X nK n MS n

[34]

and M2 dM ¼ S j dH H¼0 2K 1

[35]

Similar calculations can be made for any other orientations of the field and of the EMD, leading to the remarkable result that, in principle, complete saturation cannot be achieved if H is not parallel to an “extremal” direction (i.e., one which minimizes or maximizes the anisotropy energy).

M MR

HC

H

Figure 9 Hysteresis cycle of a ferromagnet. Remanence (MR) and coercivity (HC) are indicated on the graph axes.

54

Ferromagnetism

Further Reading Ashcroft NW and Mermin ND (1976) Solid State Physics. Philadelphia: Saunders College. Bozorth RM (1951) Ferromagnetism. Princeton, NJ: Van Nostrand. Brailsford F (1966) Physical Principles of Magnetism. London: Van Nostrand. Cullity BD (1972) Introduction to Magnetic Materials. Reading, MA: Addison-Wesley. Hubert A and Schaefer R (1998) Magnetic Domains. Berlin: Springer. Mattis DC (1965) The Theory of Magnetism. New York: Harper and Row.

Localized and Itinerant Magnetism H Yamadaa and T Gotob, aShinshu University, Matsumoto, Japan; bUniversity of Tokyo, Kashiwa, Japan © 2005 Elsevier Ltd. All rights reserved. This is an update of H. Yamada, T. Goto, Localized and Itinerant Magnetism, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 152–160, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00517-9.

Introduction Localized-Electron Model Ferromagnetism Antiferromagnetism Magnetizing Process of Antiferromagnets Spin Waves Itinerant-Electron Model Band Theory of Ferromagnetism Description of Ferromagnetism by the Landau Theory Band Theory of Antiferromagnetism Spin Fluctuations Further Reading

55 55 55 56 57 58 59 60 60 61 62 63

Introduction The quantum theory of magnetism has been developed from two opposite starting points. One is the localized-electron model, where the electrons remain on the atoms in the crystal and contribute to magnetism. The intra-atomic interaction between these electrons is so strong that the magnetic moment is determined on each individual atom by Hund’s rules. The interatomic exchange interaction is, on the other hand, much smaller than the intra-atomic one, and competes with thermal disorder to determine the long-range magnetic ordering. This model is applied to insulating crystals and rare-earth metals. The other model of magnetism is based on the band structure of electrons. These electrons move around in the crystal and contribute to the electric conduction. The ferromagnetic state may be stabilized for materials with a strong exchange interaction and a high density of states at the Fermi level. This is the band model or itinerant-electron model, which is commonly applied to 3d transition metals, alloys, and compounds. In this section, magnetic properties as well as magnetic ordering are discussed for both localized-electron and itinerant-electron models. A molecular field theory is extremely useful in describing the ground state and finite temperature properties. In this approximation, the focus is on how various magnetic properties can be explained, and then, how spin wave excitations and spin fluctuations at finite temperature can be studied. SI units are used in this article. The magnetic induction, B, is given by m0 ðH+MÞ, where H, M, and m0 are the magnetic field, magnetization, and permeability in vacuum, respectively. The Bohr magneton is denoted by mB ¼ eh=2m in this unit system, where e and m are the charge and mass of an electron, respectively, and where h is the Planck constant divided by 2p. To avoid confusion, mB is taken to be positive, so that the direction of magnetization is the same as that of the spin.

Localized-Electron Model The concept of exchange coupling between the spins of two or more nonsinglet atoms arose with the Heitler–London theory of chemical bonding and was applied in 1928 to the theory of ferromagnetism by Heisenberg. The spin Hamiltonian is given by P H ¼ − i6¼j Jij Si  Sj , where Jij is the interatomic exchange integral between the electrons on atoms at Ri and Rj. Si and Sj are spin operators on these atoms. The exchange integral Jij comes from the antisymmetric nature of wave functions with respect to the spatial and spin coordinates, and is given by Jij ¼

e2 4pe0

Z Z

cj ðrÞci ðr0 Þci ðrÞcj ðr0 Þ drdr0 jr−r0 −Ri +Rj j

  where ci ðrÞ is the wave function on the atomic site Ri. The value of Jij becomes small obviously when Ri −Rj  is large, as ci ðrÞ is localized on the atom at Ri in this model. Here, Jij is assumed to take a constant value, J for the nearest neighbor atomic pair at Ri and P Rj, and to be zero for other pairs. In this case, the Hamiltonian is written by H ¼ −2J hi,ji Si  Sj , where hi, ji on the summation denotes the sum over nearest neighbor pairs at Ri and Rj.

Ferromagnetism Before the concept of quantum mechanics had been established, Weiss proposed in 1907 a very successful description of ferromagnetism by introducing a molecular field acting on atomic magnetic moments, which is assumed to be proportional to

Encyclopedia of Condensed Matter Physics, Second Edition

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55

56

Localized and Itinerant Magnetism

the net magnetization. The origin of the molecular field is nowadays known to be a quantum mechanical exchange interaction. A material, which consists of the same type of atoms, is considered. When the magnetic field is applied to the z direction (quantized axis of spin), the average value (or expectation value) of the spin operator Si has a z component hSz i only. The magnetization is now written as M ¼ NgmB hSz i, where N is the number of atoms in the material, g ¼ 2 (g-factor for spin system), and mB is the Bohr magneton. The term Si  Sj in the Hamiltonian is approximately written by Si  Sj ’ hSz iSj,z +Si,z hSz i−hSz i2 . One gets, together with P P P the Zeeman energy by the magnetic induction B, H ’ −2zJhSz i i Si,z −gmB B i Si,z +NzJhSz i2 ¼ −gmB Beff i Si,z +NzJhSz i2, where z is the number of the nearest neighbor atoms in the crystal. Here, J is taken to be positive as the ferromagnetic state is discussed. The effective magnetic field is given by Beff ¼ 2zJhSz i=gmB +B. The first term 2zJhSz i=gmB corresponds to the molecular field proposed by Weiss. The molecular field coefficient o, defined by Beff ¼ oM+B, is given by o ¼ 2zJ=NðgmB Þ2 . The magnetic quantum number mS on an atom may take S, S−1, . . ., −S. The average value hSz i is evaluated by the standard statistics as hSz i ¼ SBS ðgmB SBeff =kB TÞ at finite temperature T, where BS(x) is the Brillouin function defined by     2S+1 2S+1 1 x B S ð xÞ ¼ coth x − coth 2S 2S 2S 2S where x ¼ gmB SBeff =kB T and kB is the Boltzmann constant. (In the limiting case of S ¼ 1, BS ðxÞ becomes the Langevin function, which appears when Si,z is treated as a classical spin vector.) The average value hSz i is obtained as a function of T and B by solving the equation for hSz i self-consistently, as the argument in the Brillouin function contains hSz i itself. Figure 1 shows the calculated result of the magnetization for S ¼ 5=2 as a function of T and B. TC is the Curie temperature where hSz i becomes zero at B ¼ 0. The T-dependence of M at B ¼ 0 is very weak at low T in this theory, being different from the observed results. At low T, M decreases with increasing T as T3/2 by the thermal excitations of spin waves. This is discussed in the subsection, “Spin waves.” The susceptibility wp ð ¼ m0 M=BÞ above TC is written at high T as wp ¼ m0 C=ðT−T C Þ. This is called the Curie–Weiss law. Here, TC and C are the Curie temperature and the Curie constant respectively, given by T C ¼ 2zJSðS+1Þ=3kB and C ¼ NðgmB Þ2 SðS+1Þ=3kB. This is derived by using the approximation BS ðxÞ  xðS+1Þ=3S at small x. At T ¼ T C , M is proportional to B1=3 . Then, the susceptibility wp diverges as B–2/3 when B tends to zero. The critical index d, defined by M∝B1=d, is 3 in the molecular field theory. However, the observed result of d for conventional ferromagnets is 4. The molecular field theory breaks down at T ’ T C .

Antiferromagnetism The crystal, which consists of two equivalent sublattices A and B, may have an antiferromagnetic spin structure when the exchange integral JAB between the nearest neighbor spins on the different sublattices is negative. Sublattice moments are given by MA ¼ NgmB hSAz i=2 and MB ¼ NgmB hSBz i=2, where the number of atoms on each sublattice is N=2 . Here, SAz and SBz are the z components of S on these sublattices. Magnitudes of MA and MB without magnetic fields are equal to M, that is, MA ¼ −MB ¼ M. Effective fields acting on the A and B sublattice spins are written by BAeff ¼ −o0 MA −oMB +B and BBeff ¼ −oMA +o0 MB +B, where o and o0 are the molecular field coefficients of inter- and intra-sublattice moments given by o ¼ 4zAB jJAB j=NðgmB Þ2 and o0 ¼ 4zAA JAA =NðgmB Þ2 . Here, JAA ð ¼ J BB Þ is the intra-sublattice exchange integral and zAB and zAA ð ¼ zBB Þ are the numbers of the nearest neighbor atoms on the different sublattices and on the same sublattice. By the same procedure as used in the ferromagnetic

1.0

M (T,B)

0.8 0.6 0.4 0.2 1.0

0 0.2

0.4

0.6 ~ T

0.8

1.0

1.2

0

0.5 ~ B u102

   Figure 1 Temperature and field dependences of the magnetization M for S ¼ 5=2. M ¼ M=NgmB S,T ¼ T=T C and B ¼ gmB B=2zJS.

Localized and Itinerant Magnetism

57

case, the thermal averages hSAz i and hSBz i are obtained by the Brillouin function as hSAz i ¼ SBS ðxA Þ and hSBz i ¼ SBS ðxB Þ, where xA ¼ gmB SBAeff =kB T and xB ¼ gmB SBBeff =kB T. The Néel temperature TN, where the sublattice moment at B ¼ 0 vanishes, is given by T N ¼ 2SðS+1ÞðzAB jJAB j+zAAJ AA Þ=3k B . Solving the simultaneous equations for MA and MB, the susceptibility wp ðT Þ above TN is obtained as wp ðT Þ ¼ m0 C= T+Yp , where Yp ¼ 2SðS+1ÞðzAB jJ AB j−zAA J AA Þ=3kB and C ¼ NðgmB Þ2 SðS+1Þ=3kB . At T ¼ T N , wp ðT N Þ is given by wp ðT N Þ ¼ m0 =o. When J AA ¼ 0, T N is equal to Yp . The observed value of TN for conventional antiferromagnets is smaller than that of Yp . Below TN, there are two kinds of susceptibility. One is the susceptibility when the magnetic field is applied along the direction of MA. In this case, MA increases by dMk =2, and MB shrinks by dMk =2. dMk is obtained by solving simultaneous equations of MA and MB given by the Brillouin functions with BAeff and BBeff . Noting that MA ¼ −MB ¼ M at B ¼ 0 , one gets the susceptibility  wk ðT Þ ¼ m0 dMk =B parallel to the sublattice moment as wk ðTÞ ¼ m0 C=fYp +ðS+1ÞT=3SB0S ðxÞg, where x ¼ 6MT N =NgmB ðS+1ÞT and B0S ðxÞ is the derivative of BS ðxÞ with respect to x. At T ¼ 0, wk ð0Þ ¼ 0 as limT!0 T=B0S ðxÞ ¼ 1. At T ¼ T N , wk ðT N Þ ¼ wp ðT N Þ ¼ m0 =o as B0S ð0Þ ¼ ðS+1Þ=3S. The other is the susceptibility when the magnetic field is applied in the perpendicular direction to MA and MB. In this case, the induced moment dM? =2 appears on each sublattice in the field direction. The susceptibility w? ð¼ m0 dM? =BÞ perpendicular to the sublattice moment is obtained by the balance of effective fields shown in Figure 2. The sublattice moments rotate by an angle y so that B−oðMA +MB Þ ¼ 0. One gets w? ¼ m0 =o, which does not depend on T and also on the value of o0 . In Figure 3, wk ðT Þ and w? ðT Þ below TN, and wp ðT Þ above TN are shown as a function of T for S ¼ 5=2 and o0 ¼ 0. For polycrystalline or powder samples, the antiferromagnetic moments in domains or particles will orientate in random directions, so that wðT Þ ¼ wk ðT Þ=3+2w? ðT Þ=3 below TN.

Magnetizing Process of Antiferromagnets For the magnetizing process of antiferromagnets, magnetic anisotropy plays an important role. The magnetic moment prefers to align in a certain crystallographic direction without a magnetic field. For the sake of simplicity, consider the case of uniaxial anisotropy. Taking the easy axis of magnetization along the z-axis, the sublattice moment MA prefers to align in the positive (negative) z direction and MB in the negative (positive) z direction. The anisotropy energy is phenomenologically given by Eani ¼ −ðK u =2Þðcos2 yA +cos2 yB Þ, where yA and yB are angles between the sublattice moments and the easy axis. The coefficient Ku is called the uniaxial anisotropy constant. The anisotropy fields, BAani and BBani acting on the A and B sublattice moments, are defined by BAani ¼ K u MzA =M2 and BBani ¼ K u MzB =M2 . GMA MA

( (

 MB

MB

) )

 MA

B Figure 2 Magnetic induction B of the applied field and molecular fields −oMA and −oMB on the sublattice moments MA and MB, respectively. dM? is the induced moment.

1.5 ~ (T )/ p(T N)

A

1.0

~

p

~ 0.5

0

__

0.5

1.0 ~ T

1.5

2.0

Figure 3 Parallel and perpendicular susceptibilities  wk ðTÞ and  w? ðTÞ below T N , and of  wp ðTÞ above T N for S ¼ 5=2 and o0 ¼ 0.  wk ðTÞ ¼ wk ðTÞ=wp ðT N Þ,    w? ðTÞ ¼ w? ðTÞ=wp ðT N Þ, wp ðTÞ ¼ wp ðTÞ=wp ðT N Þ, and TT=T N .

58

Localized and Itinerant Magnetism

The induced moment dM? by the field applied perpendicular to the z-axis (hard axis of magnetization) is suppressed by the  2 anisotropy field. The susceptibility w is given by w ¼ m = o+K =2M . This is derived in the following way. Eani is rewritten as u ? ? 0  2  2 ðK u =2Þ sin yA +sin yB −K u. The anisotropy field to the hard axis of magnetization is written by −K u dM? =2M2 (negative field), and then one can get the above expression of w? . On the other hand, when the magnetic field is applied along the easy axis of magnetization and when the sublattice moments align in the hard axis, the induced moment dM? in the field direction is enhanced by the anisotropy field. The susceptibility, in this case, is written by w0? ¼ m0 =ðo−K u =2M2 Þ. When the magnetic field is applied along the easy axis of magnetization, the magnetic energy together with Eani is approximately given by DEk ¼ −wk ðT ÞB2 =2m0 −K u at low fields. On the other hand, when the magnetic field is applied along the easy axis of magnetization and when the sublattice moments align in the hard axis, the magnetic energy is given by DE? ¼ −w0? ðTÞB2 =2m0 . At weak magnetic fields applied along the direction of the easy axis, the sublattice moments keep to align in the easy axis by the gain of the anisotropy energy. However, under strong magnetic fields they may align in the direction perpendicular to the magnetic field, as w0? is larger than wk, and then, DE? becomes lower than DEk. That is, a spin flipping takes place from the parallel direction to B to the perpendicular one at a certain magnetic field. The critical value Bsf of the spin flipping is given by the equation DEk ¼ DE? , and one obtains Bsf ðTÞ ¼ ½2m0 K u =fw0? ðTÞ−wk ðTÞg1=2 . At T ¼ 0, Bsf is written by Bsf ¼ fBani ð2Bex −Bani Þg1=2, where Bex ¼ oM and Bani ¼ K u =M . When B > 2Bex −Bani , the sublattice moments become parallel to each other. The critical value is denoted by Bc ð¼ 2Bex −Bani Þ. The spin structures are the antiferromagnetic one at B < Bsf , the canted one at Bsf < B < Bc, and the ferromagnetic one at B > Bc . When T increases, Bc decreases, while Bsf increases slightly, as shown in Figure 4. When Bani is larger than Bex, the canted spin state does not appear. The magnetization curve at T ¼ 0 shows a jump from the antiferromagnetic state to the ferromagnetic one at B ¼ Bex. This is a field-induced metamagnetic transition. However, the transition at finite T becomes gradual at Bex, as shown in Figure 5. When the magnetic field is applied to the hard axis of magnetization, the induced moment dM? increases as w? B=m0 up to B0C ¼ 2Bex +Bani and saturates to 2M.

Spin Waves The low-lying energy states of spin systems coupled by exchange interactions are wave-like, as originally discussed by Bloch. This is called a spin wave or a magnon. Spin waves have been studied for all types of ordered spin arrays. In this subsection, the ferromagnetic spin wave is studied. The energy associated with nonuniform distributions of the spin direction is given by the Heisenberg exchange energy PP W ¼ −J i d Si  Si+d , where d is the nearest neighbor vector of Ri. Here, the Ss are quantum mechanical spin operators. They are treated as if they were classical vectors. Si+d is expanded in a Taylor series as Si +ðd  rÞSi +ð1=2Þðd  rÞ2 Si . For lattices with  inversion  P PP symmetry, the first-order term in d is cancelled out by the summation over d, and one gets W ¼ −J i jSi j2 −ðJ=2Þ i d Si  ðd  rÞ2 Si : exchange energy for the local magnetization It is noted here that r2 ðSi  Si Þ ¼ 0, as jSi jR2 is a constant. Then, one can get the  P P  MðrÞ ¼ gmB i Si dðr−Ri Þ as Eex ¼ ð1=2VÞD jrMðrÞj2 dr , where jrMðrÞj2 ¼ a ð@Ma =@xÞ2 +ð@Ma =@yÞ2 +ð@Ma =@zÞ2 , a ¼ x, y, z, and V the volume of the sample. The coefficient D in Eex is called the Landau–Lifshitz constant. The exchange field is defined by Bex ðrÞ ¼ @Eex =@MðrÞ. Neglecting the boundary effects, one gets Bex ðrÞ ¼ Dr2 MðrÞ. Then, the torque equation is given by

2.0

~ Mind(T,B )

1.5

1.0 0.5

3 2

0 0.2

0.4

0.6 ~ T

1 0.8

1.0

1.2 0

~ B

  Figure 4 Temperature and field dependences of the induced moment Mind for S ¼ 5=2, o0 ¼ 0, and 2K u =Nz AB jJAB jS2 ¼ 0:2. Mind ¼ 2Mind =NgmB S, TT=T N,  and BgmB B=2z AB jJAB jS.

Localized and Itinerant Magnetism

59

2.0

~ Mind(T,B)

1.5 1.0 0.5 1.5 0

1.0 0.2

0.4 ~ T

0.6

0.5 0.8

1.0

1.2

~ B

0

Figure 5 Temperature and field dependences of the induced moment Mind for S ¼ 5=2 and o0 ¼ 0 for the case of the strong anisotropic field ðBani > Bex Þ.    Mind ¼ 2Mind =NgmB S, TT=T N , and BgmB B=2z AB jJAB jS.

@Mðr, t Þ=@t ¼ gDMðr, t Þr2 Mðr, t Þ, where g is the gyromagnetic ratio gmB =h . One looks for travelling wave solutions of the form Mx ðr, tÞ ¼ dmx expfiðq  r−otg, My ðr, tÞ ¼ dmy expfiðq  r−otg and Mz ’ Ms . One gets hoq ¼ gmB DMs q2 for small q. The coefficient of q2 is called the exchange stiffness constant, which is observed by neutron scattering and spin wave resonance measurements. A schematic representation of Mðr, t Þ at a fixed time t is shown in Figure 6. The wave of local magnetization, that is, the spin wave is quantized. By the analogy with phonons, the Hamiltonian for the P quantized spin wave (magnon) is written by H ¼ q nq +1=2 ℏoq, where nq is the occupation quantum number of magnons with P the wave-number vector q. The temperature dependence of the magnetization is given by MðTÞ ¼ M  s −2mB q hnqi,−1where Ms is the . One obtains spontaneous magnetization and the thermal average hnq i is a Bose distribution function exp hoq =kB T −1 DMðT Þ ¼ Ms −MðT Þ∝T 3=2 at low temperature where ℏoq is proportional to q2. The theory of spin waves is also established in the itinerant-electron model of magnetism.

Itinerant-Electron Model A quantum theory of ferromagnetism of conduction electrons in metals began with Bloch’s paper in 1929. A free-electron gas with sufficiently low density was shown to be ferromagnetic. This means that the carriers of the magnetic moment in metals also participate in electric conductions. In 1933, Stoner pointed out that nonintegral values of the observed magnetic moment per atom

q

Figure 6 Schematic representation of the spin wave with wave-number vector q.

60

Localized and Itinerant Magnetism

in 3d transition metals is explained if 3d electrons are itinerant, having a suitable density of states (DOS) and an exchange interaction parameter. Slater discussed ferromagnetism of Ni metal, using calculated results of the electronic structure and estimating the exchange couplings between electrons from spectroscopic data. The subsequent development of itinerant-electron magnetism has proceeded mainly along two lines. One is the study on refinements of the electronic structure. The other is the study on the interaction between electrons. For both studies, electron correlations make the problem difficult. In the former, the electron correlations are taken into account in the potential of the electrons. A local spin-density functional formalism for the band calculation is established. Ferromagnetic, antiferromagnetic, and spin-density wave states are discussed for real materials by this formalism. In the latter, the interaction between electrons is treated as a many-body problem. To discuss the electron correlations, the Hubbard Hamiltonian is often made use of.

Band Theory of Ferromagnetism The band theory (or Stoner theory) is widely used in the study of magnetism for transition metals, alloys, and compounds. In the original paper of Stoner, the DOS of electrons is assumed to be a parabolic one with respect to the energy. However, the DOS for actual transition metals is not so simple, but has a lot of peaks, as many energy levels overlap with one another. Figure 7 denotes the calculated DOS curve for a b.c.c. Fe metal in the nonmagnetic state at the lattice constant 28.5 nm. EF denotes the Fermi level. Electrons occupy energy levels up to EF . 0 The Pauli spin susceptibility is written in terms of the DOS curve, DðEÞ, as w0 ðTÞ ¼ 2m0 m2B ½D0 −ðp2 k2B T 2 =6ÞfD02 0 =D0 −D0 g at low T, 0 00 where D0, D0 and D0 are the values of D(E) at EF and the first and second derivatives of D(E) with respect to E at EF, respectively. The T-dependence comes from the low temperature expansion of the integrals connected with the Fermi distribution function. One takes into account the exchange interaction by the molecular field approximation. The induced magnetic moment, M, by the external field is given by M ¼ w0 ðT ÞðB+IMÞ=m0, where I is the molecular field coefficient. Then, the susceptibility wðT Þð ¼ m0 M=BÞ is obtained as wðT Þ ¼ w0 ðT Þ=f1−Iw0 ðT Þ=m0 g. The factor 1=f1−Iw0 ðT Þ=m0 g is the exchange-enhancement factor of the susceptibility. When the value of I is so large that the denominator in the enhancement factor becomes zero, the susceptibility wðT Þ diverges. This is the magnetic instability in the paramagnetic state, which is called the Stoner condition for the appearance of ferromagnetism. When Iw0 ð0Þ=m0 > 1 , the ferromagnetic state is stable at T ¼ 0 . In this case, the Curie temperature TC is obtained by Iw0 ðT C Þ=m0 ¼ 1. In the native Stoner theory, the value of I is an adjustable parameter to fit the Curie temperature. However, the value of I is nowadays obtained by the local spin-density functional formalism of the band calculation. The value of TC estimated with the calculated value of I is much larger than the observed one, although reasonable values of the magnetic moment for 3d transition metals are obtained. This is because the effect of spin fluctuations at finite temperature is not taken into account in the theory.

Description of Ferromagnetism by the Landau Theory

D (E ) (eV1atom1spin1)

By the fixed-spin-moment method developed recently, the magnetic energy DEðMÞ can be calculated numerically as a function of the magnetic moment M. The open circles in Figure 8 denote the calculated results of DEðMÞ for a b.c.c. Fe metal at the lattice constant 28.5 nm. The equilibrium state is achieved at M  2:2mB per Fe atom, being very close to the observed one. The calculated DEðMÞ can be fitted in the form of DEðMÞ ¼ ða0 =2ÞM2 +ðb0 =4ÞM4, as shown by the dotted curve in the figure. This is just the Landau expansion of the magnetic energy. For some materials, the calculated DEðMÞ is expanded up to much higher order terms of M. The Landau coefficients a0 and b0 are also obtained by the Stoner theory. At T ¼ 0, they are written by a0 ¼ ½1=2D0 −I=m0 m2B, and 2 00 3 4 b0 ¼ ½f3D02 0 =D0 −D0 =D0 g=48D0 =m0 mB. These expressions of a0 and b0 are derived by a rather crude approximation of the rigid band EF

3

2

1

0

6.0

4.0

2.0

0.0

2.0

E (eV) Figure 7 DOS curve for a b.c.c. Fe metal in the nonmagnetic state. EF denotes the Fermi level. The value of D0 is 2.05 eV−1 atom−1 spin−1. The unit of energy 1 eV is 1.60210−19 J.

Localized and Itinerant Magnetism

61

'E(M) (eV atom1)

0.0

0.2

0.4 0

5 (M/

2 B)

10

(atom2)

Figure 8 Magnetic energy DE ðMÞ as a function of M2 for a b.c.c. Fe metal. The unit of energy 1 eV is 1.60210−19 J. The dotted curve denotes the form of 3 3 −3 DE ðMÞ ¼ ða0 =2ÞM2 +ðb0 =4ÞM4 with a0 ¼ −5:59103 Tatomm−1 B and b0 ¼ 1:10  10 Tatom mB .

model, where DOS curves in the up and down spin bands are assumed not to change their shapes from those in the paramagnetic state. Nevertheless, those expressions of a0 and b0 are still relevant. When EF lies near a peak of the DOS, D00  0, and D000 < 0, the value of b0 is positive at 0 K. On the other hand, when EF lies near a minimum of the DOS, D00  0, and D000 > 0, b0 is negative. Even if D0 is so small that the Stoner condition is not satisfied, the ferromagnetic state would be stabilized when the value of b0 is negative and large. In the case of a0 > 0 and b0 < 0, a magnetic field-induced metamagnetic transition from the paramagnetic state to the ferromagnetic one may take place. This is the itinerant-electron metamagnetism. Moreover, the value of I can be estimated from the calculated values of a0 and D0. The evaluated value of I is 0.57 eV atom for a b.c.c. Fe metal at the lattice constant 28.5 nm. The magnetic equation of state is given by B ¼ a0 M+b0 M3 , as B ¼ @DEðMÞ=@M. The spontaneous magnetic moment Ms at B ¼ 0 is given by Ms ¼ ðja0 j=b0 jÞ1=2 if a0 < 0 and b0 > 0. It is noted that the magnetic moment in this theory increases with increasing B even at T ¼ 0. The differential susceptibility whf ð ¼ m0 dM=dBÞ is finite ð2m0 ja0 jÞ at T ¼ 0. This is the high-field susceptibility in the ferromagnetic state. In the localized electron model, whf vanishes at T ¼ 0. The finite value of whf at T ¼ 0 is one of the characteristics of the itinerant-electron model.

Band Theory of Antiferromagnetism An antiferromagnetic state in the itinerant-electron model can also be described by the electronic structure. For the sake of simplicity, consider a 1D lattice, where atoms are arranged along the x direction with a lattice constant a. The wave function fk ðxÞ in the nonmagnetic state is given by a Bloch function in the periodic potential by the ions. The one-electron energy eðkÞ is expressed with the wave number k. The wave numbers are restricted to the first Brillouin zone ð−p=a < k ⩽ p=aÞ . A schematic representation of eðkÞ is shown by the solid curve in Figure 9a. When the antiferromagnetic moment is given by MQ ðxÞ ¼ M0 expðiQx Þ, where Q ¼ p=a, the potential of electrons includes a term with the modulation of the wavelength 2a. This modulated potential is treated by the perturbation theory of quantum mechanics. The wave function ck ðxÞ in the antiferromagnetic state is given by a linear combination of the unperturbed wave and ’k+Q ðxÞ, as ck ðxÞ ¼ ak ’k ðxÞ+bk ’k+Q ðxÞ with jak j2 +jbk j2 ¼ 1. One gets two energy curves functions ’k ðxÞ h i  1=2 , where D is the off-diagonal matrix element of the perturbed potential. E ðkÞ ¼ ð1=2Þ eðkÞ+eðk+QÞ  ðeðkÞ−eðk+QÞÞ2 +D2 The broken curve in Figure 9a denotes eðk+QÞ. The two energy curves cross each other at k ¼ Q=2. A bandgap D appears at the crossing point, as shown in Figure 9b. The lower energy band E− ðkÞ is that for electrons with up spin at the up-spin sites, or for E r(k)

(k)

 /a (a)

0 k

/a

 /a Q/2

0

(b)

k

Q/2

/a

Figure 9 (a): One-electron energy eðk Þ in the nonmagnetic state, and (b): E  ðk Þ in the antiferromagnetic state. A thin horizontal line denotes the Fermi energy.

62

Localized and Itinerant Magnetism

Figure 10 Main parts of the Fermi surfaces of Cr metal. The hole surface (octahedron-like) is located around H in the Brillouin zone. The surface centered at Ɣ (center of the Brillouin zone) is for electrons (K Nakada, unpublished work).

electrons with down spin at the down-spin sites. The higher energy band E+ ðkÞ is that for electrons with opposite spin at the up and down spin sites. This is a simple mechanism for the appearance of the antiferromagnetic state in the band theory. It is called the split band model of antiferromagnetism. In actual materials, the situation becomes more complicated. In Figure 10, the calculated Fermi surfaces of Cr metal in the nonmagnetic state are shown. A Fermi surface denotes the equal energy surface in the reciprocal lattice space at the Fermi level. A hole surface has the octahedron-like shape centered at the H-point in the Brillouin zone and the electron surface is centered at the G-point (center of the Brillouin zone). Here, H and G are conventional symbols labeled to the symmetry points in the Brillouin zone. When the electron surface centered at Ɣ is shifted toward H, some portions of the electron and hole surfaces may overlap with one another. On these overlapping surfaces, the bandgap appears and the antiferromagnetic state is stabilized. Strictly speaking, the shift of the electron surface is not exactly the same as the length between the G- and H-points in this case. The antiferromagnetic wave-number vector Q is slightly different from that at the H-point. Then, the spin-density wave state is stabilized in the Cr metal. This is the nesting model of the Fermi surfaces.

Spin Fluctuations As mentioned in the section, “Band theory of ferromagnetism,” the value of TC, estimated by the calculated value of I, becomes rather higher than the observed one. This is because the excitations of the collective modes of spins are not taken into account in the

Localized and Itinerant Magnetism

63

theory. The excitations of collective modes play an important role in the magnetic properties at finite temperature. In this subsection, it is studied within the phenomenological Ginzburg–Landau theory. R The magnetic free energy DF m is written by DF m ¼ ð1=V Þ drDf m ðrÞ , where the free energy density Df m ðrÞ is given by Df m ðrÞ ¼ ða0 =2ÞjmðrÞj2 +ðb0 =4ÞjmðrÞj4 +ðD=2ÞjrmðrÞj2 . D is the Landau–Lifshitz constant given in the section “Spin waves.” DFm is rewritten by using the Fourier transformation for the ith component of the magnetization density, P mi ðrÞ ¼ Mdi,z +ð1=VÞ 0q mi ðqÞexpðiq  rÞ, where M is the bulk moment in the z direction. The prime on the summation denotes the sum over q except q ¼ 0. The expression of DFm contains terms up to the fourth power of mi ðqÞ. Among them, only the terms with even power of mi ðqÞ are considered. In this case, DF m is given by a functional of only M and jmi ðqÞj2 . The equation of state, defined by B ¼ h@DF m =@Mi, is written as B ¼ aðT ÞM+bðT ÞM3, where h⋯i denotes a thermal average. The Landau coefficients a(T ) and b(T ) are given by aðTÞ ¼ a0 +b0 f3hðdmk Þ2 i+2hðdm? Þ2 ig and bðT Þ ¼ b0 . hðdmk Þ2 i and hðdm? Þ2 i are P mean square amplitudes of the longitudinal and transverse magnetization densities given by hðdmi Þ2 i ¼ ð1=VÞ q hjmi ðqÞj2 i. The T-dependence of the Landau coefficients comes from that of the mean square amplitudes of spin fluctuations xðTÞ2 ¼ hðdmk Þ2 i+2hðdm? Þ2 i . Isotropic spin fluctuations are assumed, that is, hðdmk Þ2 i ¼ hðdm? Þ2 i . In this case, the Landau coefficients are given by aðT Þ ¼ a0 +ð5=3Þb0 xðT Þ2 and bðT Þ ¼ b0 . The present T-dependence of the Landau coefficients is very different from that in the Stoner theory where it comes only from the Fermi distribution function. The degeneracy temperature for electrons estimated from the Fermi level is very high, and then the T-dependence in the Stoner theory is very weak in the conventional ferromagnets. Thus, the spin fluctuations are found to play a dominant role at finite T. In the case of a0 < 0 and b0 > 0, the Curie temperature is given by xðT C Þ2 ¼ 3ja0 j=5b0 as the inverse of the susceptibility should be zero at TC, that is, aðT C Þ ¼ 0. By classical thermodynamics, the mean square amplitude of thermal fluctuations is known to increase monotonically as T increases. The susceptibility (at xðTÞ2 ¼ pT) above TC is written in the form of the Curie–Weiss law as wðT Þ ¼ m0 C=ðT−T C Þ, where C ¼ 3=5b0 p, T C ¼ 3ja0 j=5b0 p, and p a positive constant. Such a Curie–Weiss law cannot be obtained in the Stoner model. By analogy with the localized-electron model, the square of the effective magnetic moment M2p in the paramagnetic state, estimated from the susceptibility, is proportional to the Curie constant On the other hand, the square of the  C. 2 spontaneous magnetization is given by M2s ¼ ja0 j=b0 at T ¼ 0. Then, the value of Mp =Ms is shown to be proportional to ja0 j−1, and subsequently to T −1 C . This type of behavior of Mp =Ms is observed for 3d transition metals, alloys, and compounds (Rhodes–Wohlfarth plot). Spin fluctuations in itinerant-electron antiferromagnets have also been studied.

Further Reading Arrott A (1966) Antiferromagnetism in metals and alloys. In: Rado GT and Shul H (eds.) Magnetism, vol. II B, pp. 295–416. New York: Academic Press. Bozorth RM (1951) Ferromagnetism. New York: Van Nostrand. Chikazumi S (1964) Physics of Magnetism. New York: Wiley. Gautier F (1982) Itinerant magnetism. In: Cyrot M (ed.) Magnetism of Metals and Alloys. Amsterdam: North-Holland. Herring C (1966) Exchange interactions among itinerant electrons. In: Rado GT and Shul H (eds.) Magnetism, vol. IV, pp. 1–396. New York: Academic Press. Kittel C (1962) Magnons. In: Dewitt B, Dreyfus B, and de Gennes P (eds.) Low Temperature Physics, pp. 441–478. New York: Gordon and Breach. Kübler J (2000) Theory of Itinerant Electron Magnetism. Oxford: Clarendon Press. Moriya T (1985) Spin Fluctuations in Itinerant Electron Magnetism. Berlin: Springer. Vonsovskii SV (1974) Magnetism, vols. 1 and 2. New York: Wiley. Wohlfarth EP (ed.) (1980) In: Iron, cobalt and nickel, vol. 1, pp. 1–70. Amsterdam: North-Holland. Ferromagnetic Materials.

Magnetic Domains O Portmann, A Vaterlaus, C Stamm, and D Pescia, Laboratorium für Festkörperphysik der ETH Zürich, Zürich, Switzerland © 2005 Elsevier Ltd. All rights reserved. This is an update of O. Portmann, A. Vaterlaus, C. Stamm, D. Pescia, Magnetic Domains, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 191–197, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00525-8.

Introduction The Physical Basis of Magnetic Domains Domain Walls An Example of Magnetic Domains: Stripes Acknowledgments Appendix A: Magnetostatic Energy of a Localized Distribution M(x) Appendix B: The Equilibrium Stripe Width Further reading

64 64 65 67 68 68 69 70

Introduction The Dirac equation for the relativistic quantum mechanical electron is most famous for having predicted the existence of antiparticles. One further important aspect of this equation is the explanation of the electron spin and the magnetic moment associated with it. When atoms are formed, the electron spins interact by means of the intra-atomic exchange interaction to produce atomic magnetic moments. The correct value of the atomic magnetic moments can be calculated empirically using Hund’s rules and Pauli’s principle. When atoms are brought together to form a body, the atomic wave functions change drastically: the atomic magnetic moments are, in general, reduced, but may remain finite in certain special cases such as Fe, Co, and Ni. Moreover, as soon as neighboring atoms are provided, the interatomic exchange interaction comes into play and may favor the parallel alignment of the neighboring magnetic moments. In this way, macroscopic regions of aligned magnetic moments are formed in the body, and the body is said to be in a ferromagnetic state. This microscopic description of magnetism, based on the quantum mechanical exchange interaction, is not complete. Each individual magnetic moment can be viewed as an atomic current generating a magnetic field, which interacts with other magnetic moments and contributes another term to the total magnetic energy – the energy of the dipole–dipole interaction. This term is also called the magnetostatic energy, as it is described by the Maxwell equations of magnetostatics. In contrast to the exchange interaction, this term is purely classical. It is later shown that, depending for example on the shape and size of the body, this term perturbs the parallel alignment of the magnetic moments favored by the exchange interaction. Thus the parallel alignment does not extend over the whole body, but is restricted to elementary regions called magnetic domains. Within such domains, the magnetic moments are aligned, while the orientation of the magnetic moments changes upon moving to neighboring domains. Another class of magnetic interactions arises because of the spin–orbit coupling also contained in the Dirac equation: magnetic anisotropies. A magnetic anisotropy energy determines one or more spatial directions along which the exchange-coupled magnetic moments within the domains prefer to be aligned, where “preferring” means that the total energy of the system is reduced when the moments pointf along that direction in space. Magnetic anisotropies play a crucial role in domain formation. In fact, the change of the magnetic moment direction between consecutive domains does not occur abruptly: a domain wall with finite width connects domains with different magnetic moment orientation and magnetic anisotropies stabilize this wall. The purpose of this article is to account for the state-of-the-art knowledge on magnetic domains and the walls between them. Before going into details, it is important to estimate the energy scale of the various magnetic interactions, given by their typical strength per atom. The intra-atomic exchange interaction is of the order of eV, on a temperature scale this corresponds to 104 K. This energy is required to produce an excited state with no atomic magnetic moment. The interatomic exchange interaction is some 10 meV (100 K). This means that the excited state with misaligned nearest-neighbor magnetic moments is attained when the temperature exceeds some 100 K (at a temperature which is called the Curie temperature). Magnetic anisotropies range from meV in three-dimensional solids to 0.1 meV in ultrathin films (0.01 to 1 K). The reason for this broad range is that the magnetic anisotropies are strongest when the symmetry of the system is lowered, for example, when the dimensionality is reduced. The magneto-static interaction is
0.1 meV (1 K). Clearly, the interactions relevant for domain formation are much smaller than the exchange interaction. Thus, it is not immediately evident why such small energies should have an impact in determining the total energy of the system and its ground-state magnetic moment configuration. This question is also carefully investigated in the remaining sections.

The Physical Basis of Magnetic Domains The physical basis for the explanation of magnetic domains was laid down in a paper by Landau and Lifshitz in 1935. One can consider a body in a situation where each lattice site carries a magnetic moment vector (for simplicity, a spin vector is used, noticing that, according to Dirac’s theory, the two vectors are proportional to each other). One important caveat: it will turn out that the relevant length scales intervening in domain formation are large with respect to the lattice constant. On these characteristic length

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scales, the number of spins is so large that one can give up the quantum description of the spin vector and consider it a classical vector. The strongest contribution to the total magnetic energy comes from the exchange interaction, which measures the energy required to turn two neighboring spins into an antiparallel configuration. If this energy is positive, the total exchange energy is minimized by a uniform spin configuration. The spin ensemble can also have a magnetic anisotropy energy contribution, which favors certain spatial directions for the spins, the so-called easy direction. In the simplest case of uniaxial anisotropy, two directions in space, say e and its negative, are energetically favored. Thus, the uniform spin configuration will be aligned either along +e or along −e, as either of these directions minimizes the total energy arising from the exchange and the magnetic anisotropy. How stable is the resulting spin configuration with respect to rotations or spin nonuniformities? The magnetic anisotropy per spin is very small but it is not possible to rotate spins individually, as they are bound together by the exchange interaction. Thus, a rotation excitation would involve turning a macroscopically large amount of spins and this requires a large excitation energy which is not available at sufficiently low temperatures. If the body is divided into oppositely magnetized regions, it would not entail any lowering of the total energy; on the contrary, some energy must be introduced to form boundaries between oppositely magnetized regions, causing an increase of the total energy. A more complicated anisotropy term may have a larger symmetry, but in no case will the magnetic anisotropy lead to magnetic domains as there is no energy to be gained upon changing from one easy direction to another. The behavior is quite different when the dipolar interaction between the magnetic moments is considered. Each moment produces a magnetic field, which interacts with all the other magnetic moments by means of the Zeeman energy contribution. The most striking feature of this magnetostatic energy is that the magnetic fields arising from a magnetic moment are very long ranged, they decay only with the third power of the distance. This is in opposition to exchange and magnetic anisotropies, which are local. The energy due to the dipolar field is the most complicated (see Appendix A), but contains a distinct feature which makes it compete directly with the exchange interaction: one part of it favors an antiparallel alignment of the spins. Of course, the interaction is very weak, but its long-range character amplifies its role with respect to the competing exchange interaction. Thus, under certain circumstances, which Landau and Lifshitz showed to depend on the size and the geometry of the body, the dipolar interaction can provide enough energy gain to sustain a nonuniform spin configuration: magnetic domains arise. The complexity of the dipolar field and the long-range character of the dipolar interaction makes it very difficult to find the spin configuration that minimizes the total energy. However, there are some rules of thumb that help in analyzing the physical situation. One of them is provided by eqn [3] in Appendix A. This equation has a simple meaning: the total magnetostatic energy of a body is : the Coulomb energy of an effective magnetic charge distribution rM ¼ −r  MðxÞ, where M(x) is the magnetic moment per unit volume. Its length is determined by the total magnetic moment in a sufficiently small cell centered at x divided by the volume of the cell, and its direction is given by the direction of the total magnetic moments within this cell. The size of this cell has to be chosen in such a way that it is small with respect to the smallest length occurring in domain formation, that is the size of the domain wall (see next section). The effective magnetic charge is provided by the divergence of M(x), thus minimizing the magnetostatic energy means avoiding regions where −r  MðxÞ is too large. A sizeable divergence of Mx exists, for instance, at the boundaries of a uniformly magnetized body. Such surface charges increase the energy of the uniform spin configuration above the value specified by the exchange interaction. To avoid such charges, the body can split into elementary regions of different magnetization. At the boundaries between such regions, the spin vectors make a finite angle so that the exchange energy is increased (at the walls) with respect to the uniform configuration. Despite this, Landau and Lifshitz have explicitly shown that by introducing domains, the total energy might effectively be lowered. In Appendix B, a simple case is worked out where the subtleties of such total-energy minimization calculations are shown. At the end of this section, an example illustrating this “magnetic-charges” avoidance rule is discussed. One can compare a small ferromagnetic plaquette with square geometry and a uniform spin configuration (see Figure 1a) to an identical plaquette with a particular nonuniform spin configuration where the magnetization vector circulates along a closed path within the plaquette (Figure 1b). The spin configuration in Figure 1a contains some effective magnetic charges appearing at the boundaries perpendicular to the magnetization vector. The spin configuration in Figure 1b is a very peculiar one, the magnetization vectors between two consecutive domains are orthogonal and no divergence develops at the boundary between two consecutive domains. The spin configuration in Figure 1b is free of magnetic charges and has no magnetostatic energy. One can show that when the plaquette is sufficiently large, the gain in magnetostatic energy overcomes the cost of introducing domain walls: the configuration in Figure 1b is the ground state. Experimentally, this closed path configuration is indeed observed, see Figure 1c as an illustration. The butterfly shape of the plaquette in Figure 1c is more complicated than the square plaquette just considered. Correspondingly, the experimental domain distribution (indicated by arrows), is more complex. It is also evident from Figure 1c that similar triangular plaquettes – the wings in the figure – have a slightly different domain distribution, showing that unresolved differences play a crucial role in determining the actual domain configuration. However, one essential element dominates the central parts of both wings: the domains are arranged in such a way that the magnetization vector circulates within the plane of the plaquette, avoiding the formation of magnetic charges, in agreement with the rule of thumb.

Domain Walls Although it is the magnetostatic energy contribution that drives the rearrangement of the spin configuration into domains, it is the magnetic anisotropy that stabilizes the walls between them and makes the whole rearrangement process possible. For the sake of simplicity, the creation of a domain wall in a one-dimensional spin chain is illustrated. One can consider a chain extending along

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 (a)

(b) Mx

My

(c) Figure 1 (a) and (b): Schematic domain distribution in a plaquette. In (a) the uniform magnetization produces magnetic surface charges, indicated in the figure. In (b) the magnetization vector follows a closed path, thus avoiding magnetic charges. (c) Image of the domain distribution in a plaquette with butterfly shape. The images were taken using scanning electron microscopy with polarization analysis (SEMPA). In SEMPA, a finely focused electron beam impinges onto the surface of a solid. Secondary electrons are ejected and used to map the topography of the surface while the beam is scanned across the surface. In the present case a
14 nm thick Co film was deposited onto a Cu(1 0 0) surface by molecular beam epitaxy (MBE) through a mask inserted between MBE source and substrate. In this mask, a butterfly-profile was carved in by focused ion beam milling, which gives rise to the laterally patterned Co film visible in the topography image. In SEMPA, not only the secondary electron current is measured, but their spin polarization is also detected. This allows mapping the surface magnetization vector. The in-plane horizontal (Mx) and vertical (My) components of M are reproduced in (c) (the component perpendicular to the film plane vanishes). The black-white constrast is indicative of a strong magnetization in opposite directions. The Cu-substrate is nonmagnetic and appears gray. From the contrast maps of the two in-plane components, the domain distribution can be reconstructed as schematically summarized by the arrows.

the x direction with fixed-length spin vectors S(x) being allowed to rotate in the y–z plane. Thus, S ¼ Sð0, sinyðxÞ, cos yðxÞÞ, S being the (dimensionless) length of the spin vector and y the angle between the vector S and the y direction. The variable x is allowed to assume continuous values which is equivalent to say that, if any deviation from the parallel spin alignment occurs, it occurs over spatial length scales much larger than the lattice constant a. Assuming an exchange coupling between nearest neighbors of the simple form −J  SðxÞ  Sðx + aÞ (J>0 being the exchange constant), anyR deviation from the parallel spin alignment can be worked out to increase the exchange energy by an amount DG ¼ fðG  aÞ=2  dx½dyðxÞ=dx2 g with the coupling constant G ¼ ðJ=2ÞS2 z (z being the number R of nearest neighbors, that is, 2 for a chain). A magnetic anisotropy term which favors the y direction for the spin is Dl ¼ ðl=ð2aÞÞ dxsin2 yðxÞ with the coupling constant l measuring the energy required to turn the spin away from the y direction. Notice that, in accordance with the above discussion, G > l. The equilibrium y(x) is found by setting the functional variation of DG +Dl to zero, which produces the Euler equation of the variational problem: Ga

d2 y l − siny cos y ¼ 0 dx2 a

[1]

One can study the stability of a wall by solving this equation with the following boundary conditions: y¼p (0) for x ¼ −1ð+1Þ, and dy/dx¼0 for x ¼ 1. This establishes a spin profile with domains of opposite spins on the left- and right-hand side of x ¼ 0. The solution to the Euler equation with these boundary conditions exists and was found in the work of Landau and Lifshitz: cos y ¼ tanh

2x w

[2]

pffiffiffiffiffiffiffiffi w ¼ 2a G=l being the width of the region near x¼0 over which the spin rotates from one domain to the domain of opposite magnetization, that is, the domain wall width. This solution is plotted in Figure 2b as a continuous curve. The wall energy (i.e., the energy required to establish per ffi unit wall area can be calculated analytically by inserting the solution into the   a wall) pffiffiffiffiffiffiffiffiffi l  G . The prefactor contains the dimensionality of the system d. For the interaction total energy expression: sw ¼ 1= ad − 1 strengths discussed in the introduction, w is expected to be of the order of some 100 lattice constants, which is a mesoscopic scale. A direct measurement of the spin profile inside the domain wall at the surface of an Fe single crystal indeed gives 200 nm. As an example, one can consider the domain image of a head-on magnetization distribution recorded in a thin epitaxial Fe film on W(1 1 0) (Figure 2a). The spatial resolution of the image is such that the continuous in-plane rotation of the magnetization

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(a)

67

[0 0 1] 2

M (a.u.)

1 0 1 2 10 (b)

5

0 x/w

5

10

Figure 2 (a) Domain wall formation between head-on magnetized domains. The two domains, represented by different colors, are magnetized in opposite directions parallel to the horizontal axis (the [0 0 1]-direction of the cubic lattice, in this case). At the boundary between them, one recognizes that the magnetization vector rotates in the plane from one direction into the opposite one. A color code is used to represent the direction of the magnetization vector. Image size: 4.64.6 mm2 (b) The horizontal component of the magnetization vector (divided by its value well within the domains) is plotted along an axis x, drawn perpendicular to the domain wall (full circles). At the domain boundary, the value of the component makes a transition that can be fitted by the calculated tanh - profile (continuous curve), with a characteristic wall width w140 nm. To improve the statistical accuracy of the experimental profile, all experimental data points within a stripe going across the wall were averaged.

vector from one direction into the opposite direction occurring at the boundary between the two domains can be detected in detail. The rotation is illustrated with a color code explained in the caption. The experimental profile of the magnetization component that changes sign, given by the circles in Figure 2b, follows the calculated profile (continuous curve in Figure 2b), obtained by fitting the experimental data with tanh (2x/w).

An Example of Magnetic Domains: Stripes Having established the physical basis for domain formation, the details can be worked out, which will help to achieve a deeper understanding of the energy balance which leads to domain formation. As anticipated, magnetostatic charges have to be considered, which means that the geometry and size of the body are central to the problem. For one special geometry, the appearance of domains is illustrated (for many other geometries which have been investigated so far, see the “Further reading” section). The special case of ultrathin magnetic film extending to infinity in the x1–x2 plane and having a finite but small thickness d in the x3 direction (Figure 3a) is worked out in detail. The magnetization vector is taken to be perpendicular to the film plane. The perpendicular magnetization in ultrathin films is a very special configuration, requiring a strong uniaxial magnetic anisotropy to overcome the demagnetizing effect of the dipolar interaction. This demagnetizing effect is best appreciated by comparing a uniform in-plane spin configuration with a uniform perpendicular spin configuration. The in-plane configuration has no magnetostatic energy, provided the film extends to infinity. The perpendicular configuration, on the other hand, produces opposite charges at the two boundaries of the film with vacuum. This gives rise to an electrical plane condenser geometry. Using eqn [4] in Appendix A, the magnetostatic energy (per unit area) can easily be calculated to be 2pM20 d and must be overcome by a strong magnetic anisotropy favoring the perpendicular configuration in order for a perpendicular magnetization to appear.

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e

x3

d (a)

L

x2 x1

(b) Figure 3 (a) Schematic view of an ultrathin film (thickness d) with a stripe configuration (stripe width: L) of alternating perpendicular magnetization. The geometrical elements used in Appendix B are also indicated. (b) Room temperature SEMPA image of the stripe phase in an ultrathin (d  0.3 nm) Fe film deposited on Cu(1 0 0) by means of MBE. The image is 92.2 mm long. The component of the magnetization vector perpendicular to the film plane is shown. The black-white contrast is indicative of oppositely magnetized stripes.

The possibility of the existence of a nonuniform perpendicular spin configuration that might lower the energy of the system with respect to the uniform configuration in the given perpendicular magnetic configuration is examined. A possible nonuniform spin configuration with perpendicular magnetization is the one consisting of stripes of opposite perpendicular magnetization running parallel to say x2 and repeating periodically along x1, see Figure 3a. Such a configuration does certainly not introduce any extra magnetic anisotropy energy. Domain walls, however, are created and cause an increase of the exchange energy. The striped phase can only become energetically more favorable if the gain in magnetostatic energy is large enough to overcome the extra domain wall  L energy (this is shown to happen exactly in Appendix B). The stripes have an equilibrium L ∝wep 0 =2d . L0 is the so-called   width : 2 2 3 dipolar length L0 ¼sw = 2pM0 , and is related to the characteristic dipolar energy per atom 2pM0 a . Using the characteristic values as discussed in the introduction, one obtains L0 of the order of 100a. Thus, in the ultrathin limit d  a, the equilibrium stripe width L can reach astronomical values at T ¼ 0: the ground state will appear to be a uniform single-domain spin configuration because the stripe width might be much larger than the size of the sample. However, when the temperature is increased, the various coupling constants renormalize and the stripe phase can penetrate the sample as its width reduces to mesoscopic lengths. This situation is indeed observed to occur in several perpendicularly magnetized ultrathin films where the stripe phase has been observed to develop. An example is given in Figure 3b for Fe films on Cu(1 0 0).

Acknowledgments The authors thank the Swiss National Fund and ETH Zürich for financial help. They are grateful to G M Graf for making his calculation of the equilibrium stripe width (Appendix B) available for this article.

Appendix A: Magnetostatic Energy of a Localized Distribution M(x) A useful starting point for finding the magnetostatic energy of a continuous static distribution MðxÞ, are the Maxwell equations of : magnetostatics: rB ¼ 0 and r  ðB −4pMÞ ¼ 0. The magnetic field H defined as H¼B −4pM can be written in terms of a magnetic : potential, FM : H¼ −rFM . The first Maxwell equation now corresponds to the Poisson equation of an “effective magnetic charge” : rM ¼rM : r2 FM ¼ −4prM. Thus, the magnetostatic problem has been transformed into an electrostatic problem involving charges rM. Their energy amounts to (Coulomb) Z r ðxÞrM ðyÞ 1 EM ¼ d3 xd3 y M [3] 2 jx − y j This expression for EM states that a possible way to minimize the magnetostatic energy is to avoid magnetic charges, that is to : minimize the magnitude of rM ¼ −rM. A further useful expression for the energy involves the magnetic field H: Z Z Z Z r ðxÞrM ðyÞ 1 1 1 1 d3 xd3 y M d3 xrM ðxÞFM ðxÞ ¼ − d3 xðr2 FM ðxÞÞFM ðxÞ ¼ d3 xH 2 ðxÞ EM ¼ [4] ¼ 2 2 8p 8p jx − y j R where the general solution of the Poisson equation FM ðxÞ ¼ d3 yrM ðy Þ=jx − y j is used. According to eqn [4], minimizing EM means searching for a spin configuration which avoids the formation of stray fields H.

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From eqn [3], an often quoted expression for EM can be derived. After a partial integration, the surface integral vanishes provided the magnetization distribution is localized. Using r2 1=jx − y j ¼ −4pdðx − y Þ, one obtains Z Z ry  MðyÞ 1 1 1 d3 xd3 yMðxÞ rx d3 xd3 yMðxÞ  rx ðMðyÞ  ry Þ EM ¼ − ¼ 2 2 jx − y j jx − y j Z Z [5] MðxÞ  Mðy Þ MðxÞ  ðx − y Þ½Mðy Þ  ðx − y Þ 2p 1 2 3 d3 xd3 y dx ¼ −3 + MðxÞ 2 3 jx − y j3 jx − y j5 This equation states that the magnetic “dipole” M(x) interacts with a H field Z MðyÞ ½MðyÞ  ðx − yÞðx − yÞ 4p HðxÞ ¼ dy3 ð − +3 − MðyÞdðx − yÞÞ 3 jx − y j3 jx − y j5

[6]

produced at x by all other dipoles at y (the last term is the own field of the dipole at x). This expression for EM is often called the dipolar energy due to the dipole–dipole interaction. R A third useful equation can be derived from eqn [4] d3 xH  B ¼ 0, holding for a localized distribution M(x): Z Z 1 d3 xB2 x [7] EM ¼ 2p d3 xM2 x − 8p Accordingly, if the length of the magnetization vector is a constant, a spin configuration with a low magnetostatic energy requires maximizing the magnetic field B. Using the same identity, one can immediately derive Z 1 d3 xHM [8] EM ¼ − 2 : Finally, one can transform eqn [7] into the interaction energy of a system of effective currents J M ¼ cr  M. With B ¼ r  A and 2 2 r  ðr  AÞ ¼ −r A (in the Coulomb gauge), one obtains r A ¼ −4p=cJ M . Finally, from this Poisson equation for A, one can obtain Z Z J ðxÞJ M y 1 EM ¼ 2p d3 xM2 x − 2 d3 xd3 y M [9] 2c jx − y j Thus, minimizing the magnetostatic energy means searching for a spin configuration that maximizes the curl of M.

Appendix B: The Equilibrium Stripe Width It is very instructive for the reader to follow in detail the appearance of domains for a very specific case. One can consider perpendicularly magnetized ultrathin films of thickness d extending over a large area L1  L2 in the x1 – x2 plane, see Figure 3a. The magnetization within each stripe points along x3. The stripes of width L are taken to be parallel to x2. They are joined by walls with a finite width o with d < w < L. Consecutive stripes have opposite perpendicular magnetizations. The total wall energy Ew amounts to sw ðL1 =LÞL2 d. This is just the energy per unit wall area multiplied by the length of the wall L2, the film thickness d, and the number of walls within the length L1 . Here, the wall energy per unit film area sw d=L is the main interest. The lowering of the magnetostatic energy upon introducing stripes can immediately be seen when one considers eqn [9]. A uniform perpendicular spin configuration is curl free, so that there is no effective current JM anywhere in space and the total energy  per unit area is simply 2pM20 d . The stripe configuration, instead, develops some curl (and correspondingly some magnetic field B), thus necessarily lowering the magnetostatic energy. The following calculation shows explicitly that the energy gain is enough to overcome the wall energy introduced by the stripes. First, the effective currents are found. The stripe magnetization distribution develops a curl at the boundaries (labeled with the index occur where n) between consecutive stripes. Along these boundaries, effective current densities J n ¼ cjn e jn ¼ ð −1Þn 2M0 =wðn ¼ 0, 1, . . . , ðN − 1ÞÞ and jN ¼ ð−1ÞN M0 =w. One contribution to the Coulomb energy, eqn [9] arises from the interaction of each current with the remaining currents. There are L1 =L of such interaction terms and the calculation of this part of the interaction energy reduces to the computation of the sum of elementary integrals of the type Z

L2 =2 Z L2 =2

−L2 =2

−L2 =2

1 dx2 dy2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 − y2 Þ2 +L2n

[10]

(Ln¼n, L and n6¼0). Notice that the integration along (x3, y3) can be approximated by multiplying the remaining two-dimensional integrals with d2. This is possible because d is considered small with respect to L, so that one can set x3 ¼ y3 ¼ 0 in the integrand. The integration over (x1, y1) can be approximated by a factor w2, because w is considered to be much smaller than L, allowing one to set ðx1 −y1 Þ2 ¼ L2n in the integrand. Thus, eqn [9] reduces to the elementary integral of eqn [10]. After computing these integrals, one is left with a sum that can be solved exactly because it contains in the limit N!1 the product of Wallis. The other contribution arises

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from the self energy of each current. Under the integral of the self energy, (x1 − y1) may be smaller than |x2 − y2|. Thus, the integrations over x1, y1 must be performed explicitly. There are again L1 =L such terms and their self energy per unit area contains the integral Z 0

wZ w 0

Z dx1 dy1

L2 =2 Z L2 =2

1 dx2  dy2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 −L2 =2 ðx2 − y 2 Þ +ðx1 − y1 Þ2

−L2 =2

Summarizing all contributions, the total energy per unit film area is    3 2 2 2 w d 2 d d d − ln  + + ln ln +sw 2pM20 df1 − p p p p d L p L L L

[11]

[12]

: The equilibrium stripe width L is found by minimizing the total energy per unit area with respect to the variable x¼d=L. 

L ¼ wepðb +1Þ=2 epL0 =2d

[13]

where the numerical factor b ¼ −ð3=p +2=pln2=pÞ. Notice that the equilibrium stripe width is a nonanalytic function of d for small d. It can be readily verified that the total energy of the striped phase stripe width is indeed lower than the total  at the equilibrium  energy of the uniform perpendicular state, by an amount −ð2=pÞ 2pM20 d=L .

Further reading Hubert A and Schäfer R (2000) Magnetic Domains. Berlin: Springer. Jackson, J D Classical Electrodynamics, Third Edition, ch 5. New York: Wiley. Kaplan B and Gehring GA (1993) Journal of Magnetism and Magnetic Materials 128: 111–116. Rowlands G (1994) Journal of Magnetism and Magnetic Materials 136: 284–286. Ter Haar D (ed.) (1967) Collected Papers of L.D. Landau, pp. 101–116. New York: Gordon and Breach.

Magnetic Interactions G Hilscher and H Michor, Vienna Institute of Technology, Vienna, Austria © 2005 Elsevier Ltd. All rights reserved. This is an update of G. Hilscher, H. Michor, Magnetic Interactions, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 197–204, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00515-5.

Introduction Crystalline Electric Field Interactions Origin of Long-Range Magnetic Order – Exchange Interactions Origin of Exchange Direct Exchange Indirect Exchange in Insulators – Superexchange Double Exchange Indirect Exchange in Metals – RKKY Interaction Exchange Interactions in Free Electron Systems – Itinerant Exchange Further Reading

71 71 73 73 75 75 75 75 77 77

Introduction Magnetism in matter is a purely quantum mechanical phenomenon according to the Bohr–van Leeuwens theorem: van Leeuwen demonstrated in 1918 that classical Boltzmann statistics applied rigorously to any dynamical system leads neither to a susceptibility nor to a magnetic moment, which means that classical mechanics cannot account for diamagnetism, and paramagnetism or any type of collective magnetic order. However, under the assumption that a molecule or atom has a finite magnetic moment m, the diamagnetic and paramagnetic susceptibility was derived already in 1905 by Langevin using classical statistical thermodynamics and treating these moments without interactions. The assumption of a finite magnetic moment, however, requires quantized values for the spin S and angular momentum L that is a result of quantum mechanics. The coupling of spin and angular momentum to the total angular momentum J is a relativistic interaction, which can be treated as a perturbation for not too heavy elements (Russel Sounders or L–S coupling) and the ground state is described by Hunds rules. Accordingly, magnetism arises from quantized motions of electrons giving rise to the fundamental unit of the magnetic moment, the Bohr magneton, defined by mB ¼ eh=2me that takes the value 9:27410 −24 Am2 orJT −1 . Thus, the magnetic moment of an atom or a molecule is given in units of mB (e.g., the magnetic moment of pure Fe in the metallic state is 2:2mB per atom). The magnetic properties of a system of electrons and nuclei are determined by the interactions of these particles with electromagnetic fields. These fields arise from two types of sources: first, there are fields due to sources external to the system such as applied electric and magnetic fields. Second, there are internal sources, which come from the charges and currents due to the particles of which the system is composed. Interactions with and between nuclear moments mN are not discussed here since mN is by a factor of 1800 smaller than mB. A first step is to consider the magnetic moments associated with itinerant and/or localized electrons on the respective ions of a crystal to behave completely free without interacting with each other or with their surrounding. This leads to diamagnetism and paramagnetism. In a free atom the quantum number J, the total angular momentum, is a good quantum number, which reflects the fact that a free atom has a complete rotational symmetry in the absence of external fields and its ground state is 2J +1 fold degenerate. In a crystal, however, the surrounding ions produce an electric field to which the charge of the central atom adjusts and the 2J +1 degeneracy is removed giving rise to a splitting of the ground multiplet. The object of the crystalline electric field (CEF) theory is to describe how the moments are affected by their local environment in a crystal. Finally, one has to consider the various interactions of magnetic moments with each other leading eventually to a long-range magnetic order.

Crystalline Electric Field Interactions The charge distribution around a central ion produces an electric field with the local symmetry of the environment. This crystal field makes a contribution to the potential energy Z X q erðRÞ i V C ðr Þ ¼ dR  e jr −Rj jr −Ri j i where the charge distribution r(R) in the point charge model is approximated by point charges qi at the position Ri. The denominator may be expanded in spherical harmonics. If VC is small compared to the spin–orbit coupling (as in the rare-earth elements where the incomplete 4f shell is responsible for the magnetic moment), the eigenstates are adequately accounted for by the first-order perturbation theory. Provided that the ground state and the next excited states do not mix, the matrix elements of VC(r)

Encyclopedia of Condensed Matter Physics, Second Edition

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72

Magnetic Interactions

are proportional to those obtained with operator equivalents written in terms of J, which are called the Stevens operator equivalents Om l ðJ i Þ and the CEF Hamiltonian reads XX m ^ CF ¼ H Bm l Ol ðJ i Þ i

lm

Bm l

The CEF parameters can, in principle, be calculated from the charge distribution of the environment, which yields reasonable results for insulators but is of limited success for metals. The electrostatic interactions destroy the rotational symmetry of the ion and the electronic orbitals are linked to the symmetry of the lattice removing the 2J +1 degeneracy of the ground multiplet of the free ion. Schematically illustrated in Figure 1 is the CEF splitting (e.g., Mn in the perovskite LaMnO3) resulting from the interaction between the nonspherical d-orbitals and the electrostatic field of the environment, which is also nonspherical. As an example, the dxy-orbital in an octahedral surrounding (a) is energetically favored in comparison to the dx2 −y2 orbital (b) in the same environment. Those lobes, which point to neighboring charges, have higher overlap and correspond to higher electrostatic energy than those that point between. This gives rise to the CEF splitting displayed in Figure 1c. An improvement on the point charge approximation for d-transition metal ions is the ligand field theory which is an extension of the molecular orbital theory that focuses on the role of the d-orbitals of the central ion and their overlap with orbitals on surrounding ions (ligands). Also for rare-earth compounds, CEF parameters can be calculated from first principles using the density-functional theory which, however, should be checked experimentally as, for example, by inelastic neutron scattering. It is instructive to consider two rather different cases, namely the 3d-transition metal series (Fe group) and the 4f series (rare earth), where the relative importance of the spin–orbit coupling and CEF splitting of the ground multiplet is inverted. The spin–orbit coupling which is proportional to Z2 (atomic number) is much larger in the 4f series than in the 3d-transition metals since rare-earth elements are significantly heavier. Furthermore, the spatial extension of the 3d wave functions is much more delocalized than the 4f wave functions that are additionally screened from CEF by the outer 6s- and 5d-shells. Thus, in rare-earth compounds CEF splitting is 102 K and spin–orbit coupling of 104 K, which means that the Hunds rule ground state is usually observed both in metals and insulators and the effect of CEF interaction can be treated in terms of Stevens operators. In 3d-compounds, VC is typically of the order of 104 K which is larger than the spin–orbit coupling (102 K) and VC couples mainly to the orbital part of the wave function which lifts the orbital degeneracy of the 3d-states and Hunds rule ground state is usually not obeyed. If the CEF perturbation is strong enough and the symmetry low enough, the orbital degeneracy is completely removed. Thus, the orbital ground state is a singlet which is called the quenching of the orbital momentum (L is not anymore a good quantum number and virtually reduced to L ¼ 0) and “spin-only” magnetic properties are observed. In 4d-and 5d-series (Pd and Pt groups),

dx 2  y 2

dxy (b)

(a)

y

eg

z x

dx 2  y 2

x z

dz 2

x y t 2g

z

y x

(c)

dxy

z x

dxz

y dyz

Figure 1 (a) The dxy-orbital is lowered in energy with respect to the d x 2 −y 2 -orbital in an octahedral environment. (b) Schematic crystal field splitting in an octahedral environment (c) The dz 2 -and d x 2 −y 2 -orbitals are grouped together and called eg levels (twofold degenerate). The dxy, dxz and dyz are grouped together and called the t2g levels (threefold degenerate).

Magnetic Interactions

73

the situation is less clear-cut because the heavier ions have a larger spin–orbit splitting, and the CEF and spin–orbit interaction can be of comparable magnitude. The impact of CEF interaction on the magnetic properties is important and in combination with spin–orbit interaction it is the principal source of magnetic anisotropy, which is the dependence of magnetic properties on the direction in the compound. Further, properties influenced by CEF are magnetoelastic phenomena and transport phenomena as well as thermodynamic properties as the specific heat.

Origin of Long-Range Magnetic Order – Exchange Interactions In the previous section, magnetic ions embedded in the CEF environment of the crystal were considered but the interaction between the magnetic moments was neglected. The classical interaction between two magnetic dipoles m1 and m2 at a distance r between each other is known as the dipole–dipole interaction. As the magnetic energy is EM ¼ −mH, the dipolar energy reads ED ¼

m0 3 ½m m − ðm r Þðm2 r Þ 4pr 3 1 2 r 2 1

Thus ED/kB is of the order of 1 K, if one considers two moments of 1 mB which are separated by 1 Å. Accordingly, this interaction cannot account for Curie temperatures of 1000 K, but nevertheless can become of relevance at low and ultra-low temperatures. Dirac and Heisenberg discovered independently that electrons are coupled by exchange interactions among which direct, indirect, itinerant exchange, and superexchange can be distinguished. Direct, indirect, and itinerant exchange occurs in metallic systems. Direct and superexchange is important in nonmetallic systems.

Origin of Exchange In order to show how the Pauli exclusion principle together with electrostatic interactions leads to magnetic effects, consider the simplest possible case – a two-electron system – where the Hamiltonian ^ H

2 X p2i +Vðr 1 , r 2 Þ 2m i¼1

does not depend on the spin. In quantum mechanics one cannot distinguish between identical particles, which means that the particle coordinates q1 and q2 can be permuted (exchanged) in the wave function and the system remains the same physically; the new wave function differs from the original one by a phase factor: cðq1 , q2 Þ ¼ eif cðq2 , q1 Þ. Permuting once more gives the original wave function. Thus e2if ¼ 1 and cðq1 , q2 Þ ¼ cðq2 , q1 Þ , which means the wave function must be either totally symmetric or antisymmetric under permutation. Electrons (Fermions) have totally antisymmetric wave functions; so the spin part must either be an antisymmetric singlet state wS ðS ¼ 0Þ in the case of a symmetric spatial wave function or a symmetric spin triplet state wT ðS ¼ 1Þ in the case of an antisymmetric spatial state. The overall wave function for the singlet cS and the triplet state cT for two electrons where both the orbital and spin part is included reads: 1 1 cS ¼ pffiffiffi wS ½c1 ðr 1 Þc2 ðr 2 Þ +c1 ðr 2 Þc2 ðr 1 Þ, withwS ¼ pffiffiffi ðj "#i −j #"iÞ 2 2 1 1 cT ¼ pffiffiffi wT ½c1 ðr 1 Þc2 ðr 2 Þ −c1 ðr 2 Þc2 ðr 1 Þ, with wT : j ""i; pffiffiffi ðj "#i +j #"iÞ; j ##i 2 2 If the interaction V ¼ e2 =r 12 is absent, then these two states would have the same energy. For V 6¼ 0, the energies of the triplet and the singlet state are different Z  ^ S ¼ U +J C HC ES ¼ Z S  ^ T ¼ U −J CT HC ET ¼ with

Z Z U¼

and dE ¼ ES −ET ¼ 2J

Z Z J¼

jc1 ðr 1 Þj2 Vðr 1 , r 2 Þjc1 ðr 2 Þj2 dr 1 dr 2

c1 ðr 1 Þc1⁎ ðr 2 ÞV ðr 1 , r 2 Þc2 ðr 2 Þc2⁎ ðr 1 Þdr 1 dr 2

where J is the exchange interaction or exchange integral, and U the Coulomb integral. The original Hamiltonian does not depend on ^ may now be replaced by an effective the spin but produces an energetically favorable spin configuration for parallel spins. Thus H ^ spin acting upon the spin components only with the requirement that it should produce the same dE ¼ ES −ET . What will be one H ^ spin ? the form of H

74

Magnetic Interactions For the two-electron system, the spin operators yield the following eigenvalues: 3 2 S^i ¼ ℏ2 Si ðSi + 1Þ ¼ ℏ2 and 4 3 2 2 2 2 2 S^ ¼ ℏ2 SðS + 1Þ ¼ S^1 + S^2 + 2S^1 S^2 ¼ ℏ2 + 2S^1 S^2 2

2 2 Accordingly, the product S^1 S^2 =ℏ 2 ¼ SðS +1Þ=2 −3=4 of both operators gives −3/4 for the singlet S ¼ 0 and 1/4 for triplet state S ¼ 0. Hence the original spin-independent Hamiltonian can be written in the form of an effective spin Hamiltonian yielding the same eigenvalues ES and ET:

^ ^ ^ spin ¼ 1 ðES +3ET Þ − ðES −ET ÞS1 S2 H 4 ℏ2 ^ spin ¼ U −JS^1 S^2 H with U ¼ 1=4ðES +3ET Þ and J ¼ ðES −ET Þ=ℏ2 . If J > 0, ES −ET > 0 and the “parallel” coupling of the spins in the triplet state with S¼1 is favored, while for Jj

where Jij is the exchange interaction between the ith and jth spins. Frequently, Jij is taken as a constant for the nearest neighbors (and next nearest neighbors) and zero otherwise. Jij is of the order of 102–103 K for direct exchange in 3d-compounds, 10–102 K for indirect exchange and superexchange. A reliable calculation of Jij is a rather complex task in general. A simple but straightforward approach to obtain a convenient expression for the exchange interaction is the mean field P approximation where the exchange interaction between the ith spin and its neighbors j Jij S^j is approximated by a phenomenological molecular field Hm acting upon Si. In the Weiss molecular field model, the molecular field is set proportional to the magnetization Hm ¼ lM, where the molecular field constant l is determined by the exchange interaction yielding fields m0Hm of the order of 1000 T for 3d-intermetallics such as Fe–Co alloys to account for their Curie temperatures of 1000 K ðmH ¼ kB T c Þ . Nevertheless, this phenomenological molecular field is a convenient fiction but not experienced by the electrons since exchange origins from Coulomb interaction. Some general features of exchange interactions are noteworthy. If two electrons on an isolated atom are considered, J is usually positive and favors the parallel spin configuration (triplet state) associated with an antisymmetric orbital state which reduces the Coulomb repulsion by keeping them apart. This is consistent with Hund’s first rule yielding maximum total spin under the constraint of the Pauli principle. The situation is different for electrons on neighboring ions in molecules or in a solid. In contrast to the free electron description for a simple metal, the model of tightly bound electrons is used as an approximation not only for ionic solids but also for transition metal compounds. The total wave function is a linear combination of atomic orbitals (LCAO) centered on the respective ions (which has to be in overall antisymmetric realized by a Slater determinant.) The electrons can save kinetic energy by forming bonds because their common wave function is more extended than the original atomic orbital and comprises both ions. These molecular Antibonding E ab

Molecule

E0

10N W

Free ions Eb a0

Bonding (a)

Interatomic distance

(b)

Figure 2 Schematical energy splitting of bonding (spatially symmetric) and antibonding (spatially antisymmetric) molecular orbitals in a molecule (a) yielding a singlet ground state as, e.g., in H2 molecule. (b) Schematical formation of the bandwidth W in a solid by the quasicontinuous distribution of the 10N d-states between the lower and upper bonding states of the originally discrete and degenerated 5d-atomic states for up and down spin of N transition metal atoms condensed to a crystal; Eb is the energy of the bonding state and Eab of the antibonding state.

Magnetic Interactions

75

orbitals can be bonding (spatially symmetric) or antibonding (spatially antisymmetric) as shown schematically in Figure 2a for the hydrogen molecule. The latter are more localized associated with higher kinetic energy. This favors the “antiparallel” singlet state as observed in H2. In a d-transition metal, the atomic 5N d-states for spin up and down give rise to a band of 10N levels distributed quasicontinuously between the lower bonding and the upper antibonding levels yielding together with transfer integrals and hopping integrals, the bandwidth W (Figure 2b). One has to distinguish between interatomic terms of the exchange and Coulomb integrals (Jij and Uij, i 6¼ j) and intra-atomic terms Jii and Uii. According to the more or less localized character of the wave function, the interatomic terms are much weaker than the intra-atomic terms. The U terms tend to avoid two electrons with antiparallel spins in the same orbital and the exchange terms J favor electrons in different orbitals with parallel spins. The latter are about one order of magnitude smaller. Thus, one may define an average exchange energy I per pair of electrons per atom, which represents the interaction favoring “ferromagnetic” alignment. On the other hand, the transfer integrals associated with the bandwidth work in the opposite direction, since the creation of parallel spins implies the occupation of higher energies in the band (i.e., with higher kinetic energy). Following this very simplified approach, the balance between kinetic and exchange energy favors the appearance of a spontaneous magnetic order for the end of the 3d-elements (Cr, Mn, Fe, Co, and Ni). A proper description is performed in terms of the density-functional theory where the exchange correlation potential is added to the Coulomb correlation, which allows a straightforward determination of the exchange correlation. This treatment is called the local spin density approximation (LSDA). To summarize, exchange can be explained in the following way: the Pauli exclusion principle keeps the electrons with parallel spins apart and so reduces their Coulomb repulsion. The difference in energy between parallel and antiparallel spin configuration is the exchange energy. This, however, is favorable for ferromagnetism if the increase in kinetic energy associated with parallel spins does not outweigh the gain in potential energy, as in the familiar examples of the H2 molecule or the He atom. In metallic Fe, Co, and Ni, the number of parallel spins produces ferromagnetism because the cost in kinetic energy is not as great as in some other metals, where the gain in potential energy is significant yielding a paramagnetic ground state.

Direct Exchange If there is sufficient overlap of the relevant wave functions (e.g., 3d-orbitals) between neighboring atoms in the lattice, the exchange interaction is called direct exchange. Although Fe, Co, and Ni are frequently referred to as typical examples for direct exchange, the situation is not that simple because of the metallic state where magnetism of itinerant electrons plays an important role. Accordingly, both the localized and band character of the electrons involved in transition metals and their intermetallics have to be taken into account as is seen in the model of covalent magnetism for transition metal intermetallics.

Indirect Exchange in Insulators – Superexchange Superexchange is typical for insulators or materials on the border of a metal insulator transition where the ions with a magnetic moment are surrounded by ions which do not carry a moment but mediate the exchange. This type of exchange is observed in transition metal oxides. An archetypical example is MnO where two cations (Mn2+ in the 3d5 state with S ¼ 5/2) are coupled via an anion (O2− in the 2p6 state with S ¼ 0); the ground state is shown schematically in Figure 3a. Because of the overlap of the wave functions, one of the p-electrons of the anion (O2−) hops over to one of the cations Mn2+. The remaining unpaired p-electron on the anion enters into a direct exchange with the other cation. In principle, this exchange may be ferromagnetic or antiferromagnetic. The excited state shown in Figure 3b for an antiferromagnetic coupling is in most cases energetically in favor due to the interplay of two effects: the hopping of electrons between ligands, characterized by a hopping matrix element t being proportional to the bandwidth, and an average Coulomb interaction U between electrons on the same orbital. Superexchange involves oxygen p-orbitals and transition metal d-orbitals and is therefore a second-order process. In second-order perturbation theory, the exchange interaction is given by J∝ −t 2 =U, and normally yields antiferromagnetic coupling.

Double Exchange Double exchange is essentially a ferromagnetic type of superexchange in mixed-valent systems. Mixed-valency means that the transition metal ion on crystallographically equivalent lattice sites exhibits different ionic states, for instance, trivalent and tetravalent Mn in perovskites (La,Sr)MnO3 or Fe2+ and Fe3+ in magnetite (Fe2O3). Due to the mixed valency, holes (i.e., unoccupied states) exist in the eg states of the transition metal ion with a higher valency with respect to a cation of the same species with a lower valency. The simultaneous hopping of an electron from Mn3 + ! O2 − and concomitantly from O2 − ! Mn4 + is known as double or Zehner exchange. Hopping is favored if the neighboring ion is ferromagnetically aligned since kinetic energy is saved by delocalization, while it is hindered by antiferromagnetic alignment as shown schematically in Figure 4.

Indirect Exchange in Metals – RKKY Interaction In transition metal alloys, intermetallic compounds, and diluted magnetic alloys where magnetic ions are dissolved in a weak paramagnetic or diamagnetic host (e.g., Fe–Au or Fe–Cu alloys), the magnetic moment carrying ions are separated by other ions and a direct overlap of wave functions responsible for the magnetic moment is hardly possible. Thus, the various types of indirect

76

Magnetic Interactions

3d 5

S

2p6

5/2

Mn 2

3d 5

S = 5/2 Mn 2

O2

(a)

3d 6

S

2p 5

4

Mn1

O1

3d 5

S

5/2

Mn2

(b) Figure 3 (a) Superexchange of a transition metal oxide as, e.g, MnO in the ground state, (b) hopping of electrons in the excited state favors antiferromagnetic coupling.

eg O2 t 2g Mn3

3d 4

Mn4

3d 3

Mn4

3d 3

(a)

eg O2 t 2g Mn3

3d 4

(b) Figure 4 (a) Double exchange interaction favors hopping via an anion (O2−) for ferromagnetic alignment of the transition metal ions by reducing kinetic energy, (b) while hopping is hindered by antiferromagnetic alignment.

exchange interactions between magnetic moments are mediated by conduction electrons polarized by the local moment, which is known as RKKY interaction according to Ruderman, Kittel, Kasuya, and Yoshida. Under the assumption of a spherical Fermi surface with radius kF, the interaction is given by J RKKY ∝

cosð2kF r Þ r3

The RKKY interaction is long ranged and oscillates as a function of the distance r from the polarizing local moment. Hence depending on the distance, the conduction electron polarization is positive or negative, yielding ferromagnetic or antiferromagnetic

Magnetic Interactions

77

Ef n/2

n/2

n

n

2 N (E )

N (E )

(a)

N (E )

BH

N (E )

(b)

Figure 5 (a) Density of states of the free electron gas in the paramagnetic state, (b) splitting of the spin up and down bands by either an external field H0 or by an effective exchange field Heff.

coupling with the next local moment at the position r, and is an important mechanism to describe complex magnetic structures. Archetypical examples are rare earths (R) and their intermetallics, where the localized 4f-electrons are responsible for magnetism without any overlap with their nearest-neighbor 4f wave functions. The 4f-moments in these R-intermetallics compare well with those of their free ion value of R3+ ions according to the Hund’s rules and their Curie temperature are of the order of 10 to 102 K. In particular for systems containing f-electrons, the isotropic exchange interaction has to be extended by anisotropic terms. The anisotropy of the coupling is associated with the orbital moments of the electrons involved and the general exchange operator contains, besides the bilinear term S^i S^j , higher order isotropic interactions and all types of anisotropic exchange as, for example, Dzialoshinsky–Moriya exchange S^i  S^j , which produces magnetic components perpendicular to the principal spin axis of the ferromagnetic or antiferromagnetic alignment. The 3d–4f exchange interactions, together with CEF effects, are the origin of the large magnetic anisotropy in R-3d-intermetallics needed for the high coercitivity of permanent magnets such as Nd2Fe14B or SmCo5 and Sm2Co17.

Exchange Interactions in Free Electron Systems – Itinerant Exchange Paramagnetism of the free noninteracting electron gas is described by the Pauli susceptibility wP ¼ 2m2B N ðEF Þ, which relates the paramagnetic response of the system to an external field with the density of states N(E) at the Fermi energy EF. In the paramagnetic state, the number of up and down spins is equal ðn "¼ n #¼ n=2Þ; thus, the magnetic moment m ¼ mB ðn " −n #Þ given by the number of unpaired spins is zero; here, it is assumed that each electron carries a magnetic moment of 1 mB, since S ¼ 1/2 and gyromagnetic ratio g ¼ 2 which is a good approximation for conduction electrons. If an external field H0 is applied, the energy of an electron with spin up or down is mB m0 H depending upon the orientation of the spin with respect to the field. Thus, the external field splits the up and down spin bands with respect to each other by the energy 2mBm0H (see Figure 5) yielding a small but finite magnetic moment m ¼ mB ðn " −n #Þ and a Pauli susceptibility wP ¼ m=m0 H0 . In the Stoner Wohlfahrth model, the susceptibility of the interacting electron gas, wS, is enhanced due to exchange interactions by a factor S ¼ ð1 −IN ðEF ÞÞ −1 , known as the Stoner enhancement yielding, wS ¼

2m2B NðEF Þ ¼ wP S 1 −IN ðEF Þ

The quantity I is the Stoner exchange factor describing phenomenologically, the exchange interactions of the interacting electron gas that can be interpreted as the above mentioned average exchange per pair of electrons I favoring their ferromagnetic alignment, that is, to increase n" with respect to n#. In a more advanced treating, I corresponds to the Hubbard U which models the Coulomb interaction in the Hubbard Hamiltonian in combination with the hopping integrals t. If IN(EF) approaches 1, wS diverges and a second-order phase transition into a magnetically ordered state takes place. Thus IN ðEF Þ 1 is the Stoner criterion for the onset of magnetism. That means for transition metals: a reasonably high density of states at EF due to narrow enough d-bands together with exchange is needed for magnetic order. By analogy to the band splitting due to an external field in the paramagnetic state, the spontaneous moment in the ordered state arises from the spontaneously spin–split bands caused in a simple mean field model by an effective exchange field Heff ¼ Im.

Further Reading Barbara B, Gignoux D, and Vettier C (eds.) (1988) Lectures on Modern Magnetism. Berlin: Spinger. Blundells S (ed.) (2001) Magnetism in Condensed Matter. Oxford: Oxford University Press. Fazekas P (1999) Lectures on Electron Correlation and Magnetism, Series in modern condensed matter physics, 5th edn. Singapore: World Scientific. Jensen J and Mackintosh AR (eds.) (1991) Rare Earth Magnetism. Oxford: Clarendon. Mattis DC (1981) Theory of Magnetism I. Solid state sciences, 17th edn. Berlin: Spinger. Mohn P (2003) Magnetism in the Solid State, Solid state sciences, 134 edn. Berlin: Springer. Richter MJ (1998) Band structure theory of magnetism in 3d–4f compounds. Journal of Physics D: Applied Physics 31: 1017–1048.

Magnetic Materials and Applications H Szymczak, Polish Academy of Sciences, Warsaw, Poland © 2005 Elsevier Ltd. All rights reserved. This is an update of H. Szymczak, Magnetic Materials and Applications, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 204–211, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00523-4.

Introduction Soft Magnetic Materials Iron and Soft Steels Nickel–Iron Alloys Iron–Cobalt Alloys Soft Ferrites Ferrimagnetic Iron Garnets Amorphous Alloys Nanocrystalline Alloys Hard Magnetic Materials Alnicos Hexagonal Ferrites Cobalt–Samarium Magnets Neodymium–Iron–Boron Alloys Giant Magnetostrictive Materials Manganites Further Reading

78 78 79 80 80 80 80 80 81 82 82 82 82 83 83 84 84

Introduction Magnetic materials play a prominent role in modern technology. They are key components of motors, generators, and transformers. In this article, various magnetic materials of technological importance are described with an emphasis on recent developments. Magnetic materials have a long history of usage. According to Chinese data, the compass was discovered in China more than 4500 years ago. There are also claims that a natural magnet was discovered in Magnesia (Asia Minor) more than 3500 years ago. Magnetic materials have contributed vitally to the history of civilization. Traditionally, only those materials that exhibit ferromagnetic or ferrimagnetic properties are called “magnetic.” Only nine elements are ferromagnets. All are metals, of which three (Fe, Co, Ni) are iron-group metals and the other six (Gd, Tb, Dy, Ho, Er, Tm) are rare-earth metals. The transition metals Fe and Co are the basic elements for preparation of alloys and compounds with large Curie temperature (TC) and a large spontaneous magnetization (Ms) (see Table 1). Some of the intermetallic compounds with rare-earth metals are characterized by a very high value of magnetocrystalline anisotropy and magnetostriction. The primary criterion allowing for classification of magnetic materials is coercivity, which is a measure of stability of the remanent state. Soft magnetic materials are characterized by low values of coercivity (Hc105. The permeability m describes the response of magnetic materials to a weak magnetic field. For small fields, this response is linear and may be characterized by the initial permeability mi. Generally, the permeability is determined by two mechanisms: the growth of some magnetic domains at the expense of others (wall mechanism) and the rotation of magnetization within each domain (rotation mechanism). The same two mechanisms are also responsible for the hysteresis loop in soft magnetic materials. There are two classes of losses in these materials: hysteresis losses (determined by the area of hysteresis loop) and eddy current losses (related to the DC electrical resistivity). After a thermal demagnetization or after rapid changing of magnetization, almost all soft magnetic materials show a slow decrease of their permeability as a function of time (t), according to a law: dm ¼ −D ln t

[2]

This decay of permeability with time is called “disaccommodation.” The disaccommodation is ascribed to an increase in the domain wall stiffness, which originates in the presence of mobile defects interacting with magnetization.

Iron and Soft Steels Iron is a very good and relatively cheap soft magnetic material. It has a very high saturation magnetization at room temperature and a large Curie temperature (see Table 2). Pure iron or soft steels (low-carbon steels, silicon steels) are used in magnetic shielding, in power and distribution transformers, in small motors, in electromagnets, and in various electromechanical devices. In AC applications, low-carbon steels are used because of their high permeability and low core loss. Silicon steels are widely used in transformers, rotating power equipment, and relays. Silicon is added to pure iron to increase resistivity and permeability and to decrease magnetostriction and magnetocrystalline anisotropy. In silicon steels, the basic magnetostriction constants approach zero at a silicon level slightly greater than 6 wt.%. Such high silicon content alloys are usually produced by a fast solidification technique. This melt-spinning technique was initially developed for amorphous alloys. Silicon steels are divided into two main categories: Fe–Si sheets with nonoriented grain texture and grain-oriented Fe–Si sheets. In the latter case, a preferred crystallographic texture leads to a reduction of core loss and increase of permeability. Annealing of iron and steels removes stresses and improves the time stability of magnetic parameters. In order to avoid the a!g phase transition of iron the temperature of annealing should not exceed 1125 K. In the case of silicon steel, this temperature is even higher since silicon stabilizes the a-Fe phase. Table 2

Soft magnetic materials (crystalline)

Material

Saturation induction Bs (T)

Coercivity Hc (A m−1)

Iron (ingot) Carbon steel Silicon iron (3% Si) Unoriented Oriented Permalloy (79% Ni, 5% Mo) Permendur (49%Co, 2% V) Supermendur (49%Co, 2% V) Alfenol (16%Al) MnZn ferrite NiZn ferrite

2.14 2.14

80 100

2.01 2.01 0.80 2.40 2.40 0.80 0.40 0.33

60 7 1 160 16 3.5 7 80

Initial permeability mi 150 200 270 1 400 40000 800 1000 4000 10000 290

80

Magnetic Materials and Applications

Nickel–Iron Alloys Nickel–iron alloys (known as “permalloys”) are used for electronic transformers and inductor applications in the 1–100 kHz range. These are characterized by high initial permeability (see Table 2) and good ductility for forming into very thin strips. Fe–Ni alloys with 80% of Ni are characterized by the almost simultaneous vanishing of anisotropy and magnetostriction. The best alloys have the composition Fe15Ni80Mo5. Alloys with very high permeability are mainly used in safety devices and in magnetic field shielding.

Iron–Cobalt Alloys The most important iron–cobalt soft magnetic alloys are “Permendur” (49%Fe–49%Co–2%V) and “Supermendur” (see Table 2). The technological importance of these alloys is a consequence of their high saturation magnetization and high Curie temperature. They are characterized by high permeabilities and low coercivities. Because of high cobalt content, Fe–Co alloys are relatively costly and they are used only for special purposes where cost of the magnetic material is not of great importance, for example, in airborne medium-frequency electrical engineering and in high-temperature magnetic devices.

Soft Ferrites The term “ferrite” refers to spinel ferrites with the general formula MO Fe2O3, where M is a divalent cation (e.g., Mg2+, Zn2+, Cd2+, Mn2+, Fe2+, Co2+, Cu2+). The spinel structure is cubic, with a face-centered cubic oxygen anion lattice. Metallic ions are distributed between tetrahedral (a) and octahedral (d) sites. In the “normal spinel” structure, tetrahedral sites are occupied by divalent cations and the octahedral sites by trivalent cations. In the “inverse spinel” structure, the tetrahedral sites are occupied by trivalent cations while the octahedral sites are occupied by divalent and trivalent cations. Magnetism of ferrites is usually determined by antiferromagnetic interactions between magnetic cations located at octahedral and tetrahedral sites. It leads to a relatively low value of saturation magnetization. The Curie temperature in most ferrites is lower than that in metals. The anisotropy constant K1 of soft ferrites is usually small and negative with the h1 1 1i axis being the easy magnetization direction. Additions of Fe2+ or Co2+ ions to the lattice provide positive contribution to K1 and a tendency to change the easy magnetization direction (to h1 0 0i axis). It also gives a possibility to achieve a near-zero value of the anisotropy constant. The resistivity of ferrites is in the insulating range and much larger than that of metallic alloys. The dominant mechanism for the conduction in soft ferrites is the transfer of electrons between Fe2+ and Fe3+ ions on the octahedral sites. It means that resistivity of ferrites is dependent on the ferrous ions content. Resistivities of Mn–Zn ferrites range from 10–2 to 10 O m, while Ni–Zn ferrites have resistivities 107 O m. The hopping Fe2+–Fe3+ also determines the frequency dependence of mi. Ferrites have some disadvantages: low saturation magnetization (see Table 2), relatively low Curie temperature, and inferior mechanical properties. Despite these disadvantages, ferrites are widely used in modern technology. They are used in two main fields: power electronics (mainly Mn–Zn and Ni–Zn ferrites) and low-power techniques (Ni–Zn ferrites up to 300 MHz and Mn–Zn high-permeability ferrites up to some MHz).

Ferrimagnetic Iron Garnets 3+ 3+ The basic formula of the ferrimagnetic rare-earth iron garnets is {R3+ 3 }[Fe2 ](Fe3 )O12, where R is either a rare-earth element or yttrium. These ions occupy dodecahedral {c} sites. The iron ions are distributed between tetrahedral (d) sites and octahedral [a] sites. Garnets crystallize in the cubic system. In a ferrimagnetic garnet, the magnetic moments of the octahedral and tetrahedral sublattices are antiparallel to each other. The moments of the Gd3+ and heavier R3+ ions couple antiferromagnetically to the net moment of the Fe3+ sublattices. The resulting magnetization of the garnet is the sum of these three contributions:

MðT Þ ¼ jMa ðT Þ −Md ðT Þ +Mc ðT Þj At low temperatures the rare-earth magnetization dominates. This effect is exactly canceled out by the net magnetization of iron sublattices at the “compensation temperature” Tcomp. Below Tcomp, the magnetization Mc lies parallel to a saturating field, and above Tcomp it is antiparallel. The moments of the light rare-earth ions couple ferromagnetically to the net moment of the Fe3+ sublattices and no compensation points exist. In the compound Y3Fe5O12 (YIG) the most attractive property is the extremely narrow ferrimagnetic resonance line width, of the order of 50 A m–1. Thus, YIG with various partial substitutions for Y finds wide applications in microwave devices. Garnets find major application at lower frequencies ( l100. These alloys are produced either by a directional solidification method, sintering, or by a polymer bonding. The last method has an advantage over the other two because including the polymer binder with a high electrical resistivity, the frequency limit of the polymer-bonded composites can be extended to hundreds of kHz. Favorable magnetostrictive properties are also observed in amorphous R–Fe alloys. The main industrial applications of “Terfenol-D” are linear actuators.

Manganites Doped perovskite manganites La1–xAxMnO3 (where A is Ba, Ca, Sr, or Pb) show a wide variety of magnetic field-induced phenomena, including the gigantic decrease of resistance under the application of a magnetic field. It is termed “colossal magnetoresistance effect (CMR).” By replacement of a trivalent rare earth by a divalent alkaline earth, the nominal valence of Mn can be continuously tuned between 3+ (for x¼0) and 4+ (for x¼1). The magnetic and magneto-transport properties of manganites are governed mainly by the mixed valence Mn3+/Mn4+ ions and less affected by the oxygen ions. The d-electrons are subject to a variety of interactions, of which the most important are the double exchange, superexchange, and Jahn–Teller coupling. The double exchange interaction is responsible for a strong correlation between magnetization and resistivity (or magnetoresistivity). The large resistance is related to the formation of small lattice polarons in the paramagnetic state. In these interactions, an important role is played by the eg electrons on the Mn3+ ions. The orbital degeneracy leads to a variety of instabilities (such as magnetic, structural, and metal–insulator transitions, collapse of charge-ordered states under a magnetic field) and is responsible for a strong magnetoelastic coupling. The eg electrons can be itinerant and hence play the role of conduction electrons, when electron vacancies or holes are created in the eg orbital states of the crystal. The other characteristic feature of manganites is the appearance of unusual collective state of charge order, observed mostly for x>0.3. This insulating state competes with two other basic phases in manganites, namely with the ferromagnetic metallic phase and the nonmetallic paramagnetic phase. A number of experiments have demonstrated that over wide ranges of compositions there can be a coexistence between the ferromagnetic metallic and charge-ordered insulating phases.

Further Reading Baden-Fuller A (1987) Ferrites at Microwave Frequencies. London: Peregrinus. Chen CW (1977) Magnetism and Metallurgy of Soft Magnetic Materials. Amsterdam: North-Holland Publishing Company. Chikazumi S (1997) Physics of Ferromagnetism. Oxford: Clarendon. Coey JMD, Viret M, and von Molnar S (1999) Mixed-valence manganites. Advances in Physics 48: 167. Craik DJ (ed.) (1975) Magnetic Oxides. London: Wiley. Cullity BD (1972) Introduction to Magnetic Materials. Reading Mass. and Magnetic Materials. Reading, MA: Addison-Wesley. Engdahl G (2000) Handbook of Giant Magnetostrictive Materials. London: Academic Press. Gibbs MRJ (ed.) (2001) Modern Trends in Magnetostriction Study and Application. Amsterdam: Kluwer Academic Publishers. Kaplan TA and Mahanti SD (1999) Physics of Manganites. New York: Kluwer Academic Publishers. Moorjani K and Coey JMD (1984) Magnetic Glasses. Amsterdam: Elsevier. O’Handley RC (2000) Modern Magnetic Materials: Principles and Applications. New York: Wiley. Skomski R and Coey JMD (1999) Permanent Magnets. Bristol: Institute of Physics Publishing. Snelling EC (1988) Soft Ferrites: Properties and Applications. London: Butterworth. Tokura Y (ed.) (1997) Colossal Magnetoresistance Oxides. Amsterdam: Gordon and Breach. Tokura Y and Tomioka Y (1999) Colossal magnetoresistive manganites. Journal of Magnetism and Magnetic Materials 200: 1. du Tremolet de Lacheisserie E, Gignoux D, and Schlenker M (2002) Magnetism II – Materials and Application. Dordrecht: Kluwer Academic Publishers.

Magnetocaloric Effect VK Pecharsky and KA Gschneidner Jr., Iowa State University, Ames, IA, USA © 2005 Elsevier Ltd. All rights reserved. This is an update of V.K. Pecharsky, K.A. Gschneidner, Magnetocaloric Effect, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 236–244, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/01127-X.

Introduction Discovery and First Application of the Magnetocaloric Effect Fundamentals of the Magnetocaloric Effect Magnetic Order and the Magnetocaloric Effect Active Magnetic Regenerator Cycle and Near-Room-Temperature Magnetic Refrigeration The Future of the Magnetocaloric Effect Acknowledgments Further Reading

85 85 86 87 90 92 92 92

Introduction The word “magnetocaloric” is derived from Greek Magnes (lithos) (literally, stone of Magnesia, ancient city in Asia Minor), that is, magnesian (stone) or “magnetite” that has been known for thousands of years to attract articles made from iron, and Latin “calor” for heat, which evolved into French “calorique” – the name of the invisible and indestructible fluid that, according to a theory prevailing in 1700s, was thought to be responsible for virtually everything related to heat. This two-part word describes one of the most fundamental physical properties of magnetic materials: that is, when a solid is exposed to a varying magnetic field, its temperature may be measurably increased or decreased with both the sign and the magnitude of the temperature difference between the final and the initial states of the material dependent on numerous intrinsic and extrinsic factors. The chemical composition, crystal structure, and magnetic state of a compound are among the most important intrinsic material parameters. The temperature, surrounding pressure, and sign of the magnetic field change are examples of extrinsic variables that affect these magnetic-field-induced temperature changes and, therefore, play a role in defining the magnetocaloric effect (MCE). Inherent to every magnetic solid, the magnetocaloric effect has exceptional fundamental importance because it covers length, energy, and timescales spanning over many orders of magnitude: from quantum mechanics to micromagnetics, from statistical to macroscopic thermodynamics, and from spin dynamics to bulk heat flow and thermal conductivity. Mapping out this vast landscape is an enormously challenging task, yet even partial successes along the way facilitate a better understanding of how to design novel magnetic solids; in other words, they help in creating an environment in which a material could be tailored to exhibit a certain combination of magnetic and thermal properties. In addition to its basic significance, the magnetocaloric effect is the foundation of near-room-temperature magnetic refrigeration, which is poised for commercialization in the foreseeable future and may soon become an energy-efficient and environmentally friendly alternative to vapor-compression refrigeration technology. Practical applications of the magnetocaloric effect, therefore, have the potential to reduce the global energy consumption, and eliminate or minimize the need for ozone depleting, greenhouse, and hazardous chemicals.

Discovery and First Application of the Magnetocaloric Effect MCE was originally discovered in iron by E Warburg, who reported the phenomenon in 1881. The nature of the MCE was explained and its usefulness to reach temperatures below 1 K by adiabatically demagnetizing a paramagnetic substance was suggested independently in the mid-1920s by P Debye and W F Giauque. Namely, after cooling a paramagnetic material in a large magnetic field to as low a temperature as possible by conventional means, such as pumping on liquid helium, the material was to be thermally isolated from its surroundings, the magnetic field removed, and the sample was predicted to cool well below 1 K due to the magnetocaloric effect. Together with his student D P MacDougall, W F Giauque was the first to verify this prediction experimentally. As Giauque and MacDougall wrote in their 12 April 1933 letter to the editor of the Physical Review, “On March 19, starting at a temperature of about 3.4 K, the material cooled to 0.53 K. On April 8, starting at
2 K, a temperature of 0.34 K was reached. On April 9, starting at
1.5 K, a temperature of 0.25 K was attained.” They achieved these breakthroughs by using 61 g of hydrated gadolinium sulfate, Gd2(SO4)38H2O, and by reducing the magnetic field from 0.8 to 0 T. Giauque and MacDougal conclude their short (about 200 words) letter that was published in 1 May 1933 issue of the Physical Review with the following statement: “It is apparent that it will be possible to obtain much lower temperatures, especially when successive demagnetizations are utilized.” This technique is known as adiabatic demagnetization refrigeration and it is successfully employed today in ultra-low-temperature research – for example, nuclear adiabatic demagnetization has been and is being used to reach micro-kelvin temperatures.

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Fundamentals of the Magnetocaloric Effect The MCE occurs due to the coupling of a magnetic sublattice with an external magnetic field, which affects the magnetic part of the total entropy of a solid. Similar to isothermal compression of a gas during which positional disorder and, therefore, the corresponding component of the entropy of a system are suppressed, isothermal magnetizing of a paramagnet near the absolute zero temperature or a ferromagnet near its spontaneous magnetic ordering temperature – the Curie temperature, TC – greatly reduces disorder of a spin system, thus substantially lowering the magnetic part of the total entropy. In a reversible process, which resembles the expansion of a gas at constant temperature, isothermal demagnetization restores the zero field magnetic entropy of a system. The MCE, therefore, can be measured as an extensive thermodynamic quantity – the isothermal magnetic entropy change, DSM, which is illustrated in Figure 1 as a difference between the two entropy functions taken at the same temperature and is marked by a vertical arrow. When a gas is compressed adiabatically, its total entropy remains constant whereas velocities of the constituent molecules, and therefore, the temperature of the gas both increase. Likewise, the sum of the lattice and electronic entropies of a solid must be changed by –DSM as a result of adiabatically magnetizing (or demagnetizing) the material, thus resulting in an increase (decrease) of the lattice vibrations and the adiabatic temperature change, DTad, which is an intensive thermodynamic quantity also used to measure and express the MCE. In Figure 1, DTad is illustrated as a difference between the two entropy functions taken at the same entropy and is indicated using a horizontal arrow. It is worth noting that in the case of a gas, it is a “change of pressure” that results either in the adiabatic temperature rise or drop, or the isothermal entropy change, while in the case of a magnetic solid it is a “change of the magnetic field” that brings about the MCE. Needless to say, no matter how strong the magnetic field around the sample, the MCE will remain zero as long as the field is kept constant. For a given material at a constant pressure, the two quantitative characteristics of the magnetocaloric effect are functions of the absolute temperature (T ) and the magnetic field change (DB ¼ Bf  Bi ), where Bf and Bi are the final and initial magnetic fields, respectively, experienced by the material. The MCE can be easily computed provided the behavior of the total entropy (S) of a compound is known as a function of temperature in both the initial and final magnetic fields, e.g., see Figure 1: DSM ðT, dBÞDB¼Bf −Bi ¼ SðT, BÞB¼Bf −SðT, BÞB¼Bi

[1]

DT ad ðT, DBÞdB¼Bf

[2]

−Bi

¼ T ðS, BÞB¼Bf −T ðS, BÞB¼Bi

Equation [2], in which the entropy is the independent variable and the temperature is the dependent variable, is straightforwardly employed in direct measurements of DTad. Thus, the temperature of a sample is measured in both Bi and Bf, that is, before and after the magnetic field has been altered. The difference between the two temperatures yields the intensive MCE value, which is usually reported as a function of temperature for Bi ¼ 0. At equilibrium, both DSM and DTad are correlated with the magnetization (M), magnetic flux density (B), heat capacity at constant pressure (C), and absolute temperature by one of the following fundamental Maxwell equations:  Z Bf  @MðT, BÞ DSM ðT, DBÞDB¼Bf −Bi ¼ dB [3] @T Bi B Z DT ad ðT, DBÞDB¼Bf

¼ −

Bf 

Bi

@MðT, BÞ T  @T CðT, BÞ



B

dB

[4]

Gd, B || [0001]

70 69 B=0T

Intensive,

67

'T ad 'SM

68

Extensive,

Total entropy, S (J mol1K)

71

−Bi

66 B = 7.5 T

65 64 280

285

290

295 300 305 310 Temperature, T (K)

315

320

Figure 1 The total entropy of gadolinium in the vicinity of TC¼294 K in 0 and 7.5 T magnetic fields. The data are for a high-purity single crystalline Gd with the magnetic field vector parallel to the [0001] crystallographic direction of the specimen. The vertical arrow illustrates DSM; the horizontal arrow represents DTad.

2.2

16

Gd, B || [0001] 'T ad

2.1 T C = 294 K

2.0

15

14

13

1.9 'S M 1.8

12

1. 1.7

11 'B = 7.5 T

1.6 280

87

Adiabatic temperature change, 'T ad (K)

Isothermal entropy change, 'SM (J g1 at K)

Magnetocaloric Effect

285

290

295 300 305 Temperature, T (K)

310

315

320

Figure 2 The extensive, DSM, and intensive, DTad, MCE of the elemental Gd in the vicinity of its Curie temperature, TC¼294 K, for a magnetic field change from 0 to 7.5 T. The data are for a high-purity single crystalline Gd with the magnetic field vector parallel to the [0001] crystallographic direction of the specimen.

As immediately follows from eqns [1]–[4], materials whose total entropy is strongly influenced by a magnetic field and where the magnetization varies rapidly with temperature, are expected to exhibit an enhanced MCE. The latter peaks when |(@M(T, B)/ @T )B| is the greatest, that is, around the TC in a conventional ferromagnet or near the absolute zero temperature in a paramagnet. Usually, the MCE of a simple ferromagnet is gradually lowered both below and above the TC, as is clearly seen in Figure 2. Equations [3] and [4] are easily derived from general thermodynamics, yet both fail to describe the MCE in the vicinity of a truly discontinuous first-order phase transition when either or both |[@M(T, B)/@T]B| and [T/C(T, B)]B do not exist even if the phase volume change is negligibly small. (By definition, partial first derivatives of Gibbs free energy with respect to intensive thermodynamic variables, for example, T, P, or B, vary discontinuously at the first-order phase transition. As a result, the bulk magnetization is expected to undergo a discontinuous change at constant temperature; and the heat capacity is expected to be infinite during a first-order phase transformation. Thus, in theory, [@M(T, B)/@T]B and [T/C(T, B)]B do not exist at the temperature of the first-order transition. In reality, these changes occur over a few-kelvin-wide temperature range and both functions can be measured experimentally.) Equations [1] and [2], on the other hand, define the MCE regardless of the thermodynamic nature of the phase transformation that occurs, if any, in a material. For a first-order phase transition, it is also possible to employ an approximation, which is based on the Clausius–Clapeyron equation:     dB DS ¼ [5] DM T dT eq In eqn [5], the left-hand-side derivative is taken under equilibrium conditions, that is, when Gibbs free energies of the two phases are identical to one another. For the right-hand side, DS ¼ S2 −S1 and DM ¼ M2 −M1 , where the subscripts 1 and 2 correspond to the states of the material in the initial and final magnetic fields, respectively. Obviously, eqn [5] is only applicable when Bf is strong enough to complete the transformation from a state 1 to state 2 and when the quantity dB/dT at equilibrium is known. In other words, the B–T phase diagram for the system must be well established. By using the Clausius–Clapeyron equation, only an estimate of the extensive MCE, DSM ¼ DS, is possible.

Magnetic Order and the Magnetocaloric Effect It is easy to see (Eqns [3] and [4]) that both DSM and DTad are proportional to the derivative of bulk magnetization with respect to temperature at constant magnetic field. The DTad is also proportional to the absolute temperature and inversely proportional to the heat capacity at constant magnetic field. It is predictable, therefore, that any material should have the largest MCE when its magnetization is changing most rapidly with temperature. This, for example, occurs in the vicinity of a Curie temperature of a ferromagnet. (From eqn [4] it is not obvious that DTad(T )DB is at its maximum at the Curie temperature because heat capacity of a material ordering ferromagnetically also peaks around TC. It can be shown, however, that the maximum of DTad(T )DB does indeed occur in the immediate vicinity of TC, especially when Bi¼0 and DB!0. By directly measuring DTad(T )DB in low magnetic fields, it is, therefore, possible to determine the Curie temperature of a ferromagnet.) The MCE will decrease below TC, where the magnetization

88

Magnetocaloric Effect

approaches saturation and becomes weakly dependent on temperature, and above the Curie temperature when the magnetization shows only a paramagnetic response, that is, it is proportional to T−1, while its derivative maintains a proportionality to T−2. Conventional ferromagnets, therefore, typically display “caret-like MCE.” This kind of behavior is easily recognizable in Figure 2, and it is also shown in Figure 3a for a Gd5Si2.5Ge1.5 compound. The latter orders ferromagnetically via a second-order phase transformation at 312 K. As the DB increases, the magnitude of the MCE of a conventional ferromagnet also increases. The rate of change of the MCE with magnetic field, that is, the instantaneous derivative of the MCE with respect to DB is, however, the largest at the lowest Bf. The intensive measure of the MCE can vary from a fraction of 1 K T−1 to several K T−1. Elemental Gd, for example, exhibits a d[DT(T, DB)]/d(DB) of
3 K T−1 around TC when Bf is on the order of 1 T. The derivative falls to
1.8K T−1 when Bf approaches 10 T. It is worth noting that away from absolute zero, d[DT(T, DB)]/d(DB) in conventional ferromagnetic materials is only weakly dependent on temperature (see Figure 3a). Anomalous behaviors of the MCE are closely related to anomalous changes in the magnetic structures of solids causing unusual behaviors of the two functions, [@M(T, B)/@T]B and [T/C(T, B)]B, which are carried over to both DSM(T )DB and DTad(T )DB (see eqns [3] and [4]). One of the most commonly observed MCE anomalies is when a material undergoes two or more successive magnetic transitions in close proximity to one another. Then, instead of a conventional caret-like shape, a skewed caret, sometimes approaching a flat and an almost constant value as a function of temperature, that is, a “table-like MCE” can be observed. An example of such behavior is shown in Figure 3b for Gd0.54Er0.46AlNi. Both DSM(T )DB (Figure 3b) and DTad(T )DH (not shown) of this material are almost constant between the two successive magnetic phase transition temperatures, which are
20 K apart. Anomalous MCE behaviors may also be due to low-lying crystalline electric fields and strong quadrupolar and/or

Gd5Si2.5Ge1.5

0 =1

0.8

'B

0.8

'B =

0.6

'B =

0T

Gd0.54Er0.46AlNi

T 7.5

T

'B

.5

=7 T

0.6

5T

'B 'B =

0.2

T

=5

0.4

0.4

2T

0.2

0.0 (a)

=1

1.0

'B

'SM /['SM(max) @ 'B = 10 T ]

1.0

'B =

2T

0.0 40 30 20 10 0 10 20 30 40 Temperature, TT C(K)

10 0 10 20 30 40 50 60 70 Temperature, TT C(K) Gd5Si2Ge2

1.0

0.8

0.6 'B =10 T

'B =7.5 T

'B =5 T

0.4

'B =2 T

'SM /['SM(max)]@ 'B = 10 T

(b)

0.2

0.0 (c)

20 10 0 10 20 30 40 50 60 Temperature, TT C(K)

Figure 3 Normalized magnetocaloric effect of (a) Gd5Si2.5Ge1.5 (TC¼312 K), (b) Gd0.54Er0.46AlNi (TC¼14.5 K (the normalization temperature) and 35 K) and (c) Gd5Si2Ge2 (TC¼273 K) shown as a function of the reduced temperature for various magnetic field changes. In all cases, Bi¼0. The DSM(max) values are 1.57 J g−1, 0.63 J g−1, and 2.61 J g−1 at DB¼10 T for Gd5Si2.5Ge1.5, Gd0.54Er0.46AlNi, and Gd5Si2Ge2, respectively.

Magnetocaloric Effect

89

magnetoelastic interactions. Furthermore, the MCE can be adjusted over a broad range of behaviors by mixing two or more individual compounds or phases, each with a simple caret-like MCE, into a composite where the resulting isothermal magnetic entropy change will be a weighted sum of the corresponding values of the individual components. Generally, a more complicated magnetic behavior results in an anomalous and a more complicated behavior of the MCE. For instance, upon cooling in a zero magnetic field, elemental Dy orders antiferromagnetically at
180 K with a helical magnetic structure, and then transforms from the helical antiferromagnetic state to a ferromagnet at
90 K. When the magnetic field is between 0 and
2 T, a few additional magnetic structures emerge in pure Dy between 90 and 180 K. Thus, when the magnetic field is low, the MCE of Dy shows a sharp step-like increase at
90 K due to a first-order ferromagnetic–antiferromagnetic transition, and then goes through a minimum immediately followed by a weak maximum at
180 K for DB¼1 T (see Figure 4). The minimum exists because the application of a magnetic field to an antiferromagnet increases the magnetic entropy, thus inverting the sign of the MCE (in a ferromagnet the increasing field decreases magnetic entropy). When Bf is increased to 2 T, the magnetic field is strong enough to quench the first-order ferromagnetic–antiferromagnetic phase transition and it induces a noncollinear magnetic structure which yields a broad MCE maximum at
127 K. Since a 2 T magnetic field is not strong enough to destroy this noncollinear structure, the slightly negative DTad is still observed at
174 K. The latter is followed by a weak caret-type peak at
181 K (Figure 4). Upon increasing the magnetic field to 5 T, it becomes strong enough to suppress all of the magnetic structures except the ferromagnetic phase and the MCE of Dy for DB¼5 T has a single, somewhat skewed, caret-type peak at
181 K. Most magnetic materials on cooling undergo a second-order phase transition from a paramagnet to a ferromagnet with a conventional MCE behavior (Figures 2 and 3a), or from a paramagnet to an antiferromagnet with a skewed caret-like MCE if the magnetic field is large enough to destroy the antiferromagnetism and cause it to change to a ferromagnetic structure (Figure 4). A few materials, however, form a ferromagnetically ordered phase through a first-order magnetic phase transition, which may also be coupled with a change of the crystal lattice. One such example is illustrated in Figure 3c. Here, since the phase transition in Gd5Si2Ge2 is of the first order, the |[@M(T, B)/@T]B| is larger than usual (but not infinite and undefined as could happen if the transition occurs infinitely fast at constant temperature, pressure, and magnetic field), and therefore, the MCE is also larger than usual (see eqns [3] and [4]). Technically, this behavior is known as the “giant MCE.” If one compares the behavior of the giant MCE in Gd5Si2Ge2 with that of the other two materials shown in Figures 3a and 3b the most obvious difference is that at large magnetic fields the MCE versus temperature is extended considerably toward the high-temperature side of the peak, but the increase in the magnitude of the MCE with increasing field is affected to a much lesser extent than that in second-order phase transition compounds. The MCE in first-order phase transition materials undergoing coupled magnetostructural transformations may be further enhanced by the added difference of the entropies of the two crystallographic modifications of the material, DSst¼(SHF–SLF), where HF and LF designate the high field and the low field phases, respectively:  Z Bf  @MðT, BÞ dB +DSst [6] DSM ðT, DBÞDB¼Bf −Bi ¼ @T Bi B It is worth noting that unlike the first factor of the right-hand side of eqn [6], which represents the conventional contribution to the isothermal magnetic entropy change (see eqn [3]), DSst is magnetic field independent provided the change from Bi to Bf is large enough to complete the structural transformation. Although the last factor in eqn [6] is a hidden parameter in conventional measurements of the MCE, an estimate based on comparing the MCE exhibited by the closely related materials with and without magnetic field-induced structural transformations indicates that DSst may account for more than a half of the total magnetic entropy change in magnetic fields below 5 T. Advanced magnetocaloric materials should, therefore, exist in solid systems where extensive structural changes are coupled with ferromagnetic

Magnetocaloric effect, 'T ad (K)

10

Solid-state electrolysis purified Dy

'B=5 T

8 6 'B=2 T

4

'B=1 T

2 0 40

80

120 160 Temperature, T (K)

Figure 4 The magnetocaloric effect in ultrapure Dy calculated from the heat capacity data.

200

240

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Magnetocaloric Effect

ordering, and therefore, can be triggered by a magnetic field. Considering eqn [6], the strongest MCE should be found in novel compounds engineered in order to maximize the entropy of a structural transformation, DSst. An extensive listing of DSM and DTad values can be found in the reviews by Gschneidner and Pecharsky, and Tishin and Spichkin – see the “Further reading” section.

Active Magnetic Regenerator Cycle and Near-Room-Temperature Magnetic Refrigeration Adiabatic demagnetization refrigeration, described at the beginning of this article, is a discontinuous process for all practical reasons because it may take as much as a few hours to repeat the refrigeration cycle and reach the target temperature. The history of continuous magnetic refrigeration can be traced to the work of S C Collins and F J Zimmerman, and C V Heer, C B Barnes and J C Daunt, who in the early 1950s built and tested two magnetic refrigerators operating between
1 and 0.2 K by periodically magnetizing and demagnetizing iron ammonium alum. The apparatus built by Heer, Barnes, and Daunt operated at 1/120 Hz frequency (2 min per cycle) and on average extracted 12.3 mJ s−1 from the cold reservoir at 0.2 K. It was not, however, until 1976 when G V Brown reported a near-room-temperature continuously operating magnetic refrigerator, that it became clear that magnetic refrigeration may be successfully utilized at significantly higher temperatures and achieve much larger temperature spans than the maximum observed MCE. Brown was able to attain a 47 K no-load temperature difference between the hot end (319 K) and cold end (272 K) of his unit by regenerating a column of fluid using Gd metal and a magnetic field change from 0 to 7 T and from 7 to 0 T. The achieved temperature difference was more than three times the MCE of Gd for DB¼7 T (see Figure 2, where for a slightly greater DB¼7.5 T, the maximum DTad of Gd is around 16 K at TC¼294 K; the MCE is
13 K and
11 K at 319 K and 272 K, respectively). Following the early work of Brown, the concept of active magnetic regenerator (AMR) refrigeration was introduced by W A Steyert in 1978 and developed by J A Barclay and W A Steyert in the early 1980s. It was subsequently brought to life in the later 1990s, when various magnetic refrigeration units were built in the US, Europe, and Japan. In the AMR cycle, shown schematically in Figure 5, a porous bed of a magnetic refrigerant material acts as both the refrigerant that produces refrigeration (i.e., temperature lift) and the regenerator for the heat transfer fluid. Assume that the bed is at a steady state with the hot heat exchanger at 18 C and the cold heat exchanger at 2 C. In Figure 5a, the initial temperature profile for the bed is in its demagnetized state in zero magnetic field (dashed line). When a magnetic field is applied to the refrigerant, each particle in the bed warms because of the MCE to form the final magnetized bed temperature profile (solid line). The amount each particle warms is equal to DTad reduced by the effect of the heat capacity of the heat transfer fluid in the pores between the particles. Next, the 2 C fluid flows through the bed from the cold end to the hot end (Figure 5b). The bed is cooled by the fluid, lowering the temperature profile across the bed (from the dashed line to the solid line), and the fluid in turn is warmed by the bed, emerging at a temperature close to the temperature of the bed at the warm end. This temperature is higher than 18 C, so heat is removed from the fluid at the hot heat sink as the fluid flows through the hot heat exchanger. After the fluid flow is stopped, the magnetic field is removed, cooling the bed by the MCE (from the dashed line to the solid line in Figure 5c). The refrigeration cycle is completed by forcing the 18 C fluid to flow from the hot to the cold end of the bed (Figure 5d). The fluid is cooled by the bed, emerging at a temperature below 2 C and removes heat from the cold sink as the fluid passes through the cold heat exchanger. After completion of this last step, the cycle is repeated beginning from step (a) (Figure 5).

Magnet

Magnet HHEX

C HEX CHEX

Distance

20 15 10 5 0

HHEX HHEX

T(qC)

20 15 10 5 0

T (qC)

CHEX

Distce Distance

Field increase

Magnet

(a)

Magnet

Cold to hot flow

(b)

Magnet

Magnet

(c)

HHEX

CHEX

D Distance

Field decrease

20 15 10

T(qC)

20 15 10 5 0

T(qC)

CHEX

5 0

HHEX

Distance

Hot to cold flow (d)

Figure 5 The four steps of the active magnetic regenerator cycle: magnetizing (a), flow from the cold heat exchanger (CHEX) to the hot heat exchanger (HHEX) (b), demagnetizing (c), and flow from HHEX to CHEX (d). This description follows that introduced by C B Zimm and A J DeGregoria in 1993.

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91

The AMR cycle outlined above has several positive features useful for practical application in magnetic refrigeration devices. First, the temperature span of a single stage can greatly exceed that of the MCE of the magnetic refrigerant because the MCEs of the individual particles collectively change the entire temperature profile across the bed. Second, because the bed acts as its own regenerator, heat need not be transferred between two separate solid assemblies, but rather between the solid particles in a single bed via the action of a fluid. Third, the individual particles in the bed do not encounter the entire temperature span of the stage, and hence the bed may be made into layers, each containing a magnetic material with properties optimized for a particular temperature range. The AMR refrigeration cycle has been successfully realized in a laboratory prototype magnetic refrigerator, which was recently designed and constructed by the Astronautics Corporation of America. The device is shown together with its schematic diagram in Figure 6. The refrigerator operates near room temperature in a magnetic field between 0 and
1.5 T created by a permanent magnet. As the wheel spins, it brings one of its sections containing the magnetic refrigerant (the regenerator bed) into the high magnetic field volume, where the bed is heated due to the MCE (see Figure 6a). As long as this section of the wheel moves between the poles of the permanent magnet, it remains in the magnetized state and the heat exchange fluid (water) flows from the cold heat exchanger through the bed and into the hot heat exchanger. This corresponds to the second step of the AMR cycle shown in Figure 5b. While the wheel continues to spin, the same section exits from the high magnetic field volume. As the bed cools due to the MCE, the heat exchange fluid flow is reversed and it now flows from the hot heat exchanger through the bed toward the cold heat exchanger, thus completing steps (c) and (d) of the AMR cycle illustrated in Figure 5. This permanent magnet-based rotating bed magnetic refrigerator was recently awarded US patent no. 6 526 759. The unit can operate at frequencies exceeding 1 Hz and it has achieved a maximum cooling power of 95 W and a maximum temperature span of 20 C. As tested by engineers at the Astronautics Corporation of America, the Carnot efficiency of this laboratory prototype system

Hot heat exchanger Magnetic field Magnetic material (magnetized) Magnetic material (demagnetized) Load Rotation Cold heat exchanger

Wheel

(a)

HHEX

CHEX

Magnet Wheel (b) Figure 6 (a) The principal schematic diagram of the device and (b) laboratory prototype magnetic refrigerator designed and constructed by the Astronautics Corporation of America in 2001. A
1.5 T magnetic field around the wheel filled with Gd spheres is produced by a permanent magnet. The refrigerator operates at ambient conditions with a maximum temperature span of
20 C and maximum cooling power of 95 W. ((b) Courtesy of the Astronautics Corporation of America, 4115 N. Teutonia Avenue, Milwaukee, WI 53209.)

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compares favorably with conventional vapor-cycle refrigeration devices, thus making magnetic refrigeration the first energy-efficient solid-state refrigeration technology.

The Future of the Magnetocaloric Effect Considering practical applications of the MCE, its extent is one of the most critical parameters defining the performance of a magnetic refrigerator – the stronger the MCE the higher the efficiency of the device, all other things being equal. As is often the case with emerging technologies, numerous applications of magnetic refrigeration may stem from a reliable foundation formed by advanced magnetocaloric materials. For example, the availability of low-cost high-performance solids exhibiting enhanced MCE between
250 and
350 K is an important requirement in order to facilitate commercialization of magnetic refrigeration for a variety of consumer uses – from home appliances to climate control in motor vehicles. In another example, when suitable magnetocaloric compounds supporting continuous magnetic cooling from
300 to
20 K are developed, the energy penalty incurred during hydrogen liquefaction using the conventional gas-compression approach may no longer be a limiting factor preventing the widespread use of liquid hydrogen fuel in transportation. It is conceivable that many other areas where conventional refrigeration is inapplicable because of scaling difficulties and/or where thermoelectric refrigeration demands too much energy, magnetic refrigeration will eventually emerge as the cooling technology of choice. To amplify the MCE, conventional wisdom calls for increasing the magnetic field change in addition to maximizing both the derivative, |(@M(T, B)/@T )B|, and the region of magnetic fields and temperatures where the magnetization remains highly sensitive to temperature (see eqns [1]–[5]). Considering the current state-of-the-art of permanent magnet alloys and magnet design, it is quite unlikely that magnetic fields in excess of about 2 T at a reasonable cost will become common in the foreseeable future. That is why maximizing the MCE by manipulating chemical and phase compositions, atomic, microscopic, and magnetic structures of a material appears to be the most realistic option in order to reach the strongest possible MCEs in readily available magnetic fields. Although it remains a formidable challenge for basic science, a better understanding of the MCE, including the ability to control a variety of chemical, structural, and physical degrees of freedom that define the properties of solids will lead to improved existing materials and should eventually result in novel compounds exhibiting large MCE. An especially promising area of research is the study of the extensive measure of the MCE, that is, the isothermal magneticfield-induced entropy change that can be strongly enhanced by the added entropy of a structural transition (see eqn [6]). Provided the magnetic field completes the polymorphic transformation, this additional entropy is magnetic field independent, and therefore, may account for more than half of the observed MCE, especially in relatively weak magnetic fields. The design of novel materials exhibiting potent MCEs in weak magnetic fields is, therefore, possible by ensuring that (i) a structural transition can be triggered and completed (or nearly completed) by a weak magnetic field; (ii) the transformation has low thermal and magnetic field hystereses, that is, is reversible, and (iii) the difference between the entropies of different polymorphic modifications is maximized.

Acknowledgments This work was supported by the Materials Sciences Division of the Office of Basic Energy Sciences of the US Department of Energy under a contract No. W-7405-ENG-82 with Iowa State University. The authors wish to thank Dr. Steven Russek and Dr. Carl Zimm of Astronautics Corporation of America for permission to use the photograph of their rotary magnetic refrigerator.

Further Reading Gschneidner KA Jr. and Pecharsky VK (2000) Magnetocaloric materials. Annual Review of Materials Science 30: 387–429. Gschneidner KA Jr. and Pecharsky VK (2002) Magnetic refrigeration. In: Westbrook JH and Fleischer RL (eds.) Intermetallic Compounds. Principles and Practice, vol. 3, pp. 519–539. New York: Wiley. Morrish AH (1965) The Physical Principles of Magnetism. New York: Wiley. Pecharsky VK and Gschneidner KA Jr. (1999) Magnetocaloric effect and magnetic refrigeration. Journal of Magnetism and Magnetic Materials 200: 44–56. Tegus O, Brück E, Zhang L, Dagula, Buschow KHJ, and de Boer FR (2002) Magnetic phase transitions and magnetocaloric effects. Physica B 319: 174–192. Tishin AM (1999) Magnetocaloric effect in the vicinity of phase transitions. In: Buschow KHJ (ed.) Handbook of Magnetic Materials, vol. 12, pp. 395–525. Amsterdam: Elsevier. Tishin AM and Spichkin YI (2003) The Magnetocaloric Effect and its Applications. Bristol: Institute of Physics Publishing. www.m-w.com

Manganites VZ Kresin, University of California at Berkeley, Berkeley, CA, USA © 2005 Elsevier Ltd. All rights reserved. This is an update of V.Z. Kresin, Manganites, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 261–265, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/01159-1.

Introduction Structure Doping: Phase Diagram Main Interactions: Hamiltonian Metallic Phase: CMR Phenomenon Insulating State Percolative Transition Further Reading

93 93 93 94 95 96 96 97

Introduction Manganites were discovered in 1950 (Jonner and van Santen). The materials are named after the manganese ion which is a key ingredient of the compounds due to its mixed-valence state. Unlike the usual ferromagnetics, the transition of manganites to ferromagnetic state is accompanied by a drastic increase in conductivity. Such transition from an insulating to a metallic and magnetic state is a remarkable fundamental feature of these materials. Another fundamental property is the colossal magnetoresistance effect (CMR); one can observe a thousandfold (!) change in the resistance in the presence of a moderate applied magnetic field. The chemical composition of the manganites is A1–x Rx MnO3; usually A
La, Pr, Nd and R
Sr, Ca, Ba. The compound La1–xSrxMnO3 is the most studied material.

Structure The parent (undoped) compound LaMnO3 has a perovskite structure which is a distorted cubic lattice. The ideal crystal unit is shown in Figure 1. It is essential that the Mn ions are caged into oxygen octahedron, so that it is surrounded by six oxygen ions (Figure 2). Each oxygen ion is shared by neighboring octahedra. The fivefold orbital degeneracy is split by the crystal field into two well-separated terms, t2g and e2g (Figure 3). The t2g-level contains three electrons forming the so-called “t-core.” The last d-electron (e2g-electron) is in a loosely bound state. The e2g-electron plays a crucial role in conduction and in other properties of manganites as well as in determining its magnetic order.

Doping: Phase Diagram The phase diagram of manganites is very rich and contains many different phases. The parent compound, LaMnO3, is an insulator and its transition to the conducting state is provided by doping. The doping is realized through a chemical substitution, for example La3+!Sr2+, that is, by placing a divalent ion into the local La3+ position. Such substitution leads to the change in the valence of the manganese ion valence: Mn3+ ! Mn4+. That is why manganites are often called “mixed-valence magnetic oxides.” The four-valent state of the Mn means that the ion loses its e2g-electron. The missing electron can be described as a hole which is spread over the unit cell, being shared by eight Mn ions (see Figure 1) As a result, one obtains a crystal La1–xSrxMnO3, where a number of La centers are randomly substituted by the Sr-ions. Even in the presence of some holes, the crystal, at first, continues to behave as an insulator. This is due to the fact that each hole is localized and the localization corresponds to the formation of local polarons. Such insulating state is preserved with an increase in doping up to some critical value x¼xc0.16–0.17. At x¼xc, the material makes a transition into the conducting (metallic) state which persists with further doping up to x0.5–0.6, depending on the chosen composition. It is remarkable that the transition at x¼xc is also accompanied by the appearance of the ferromagnetic state. As was noted above, the correlation between conductivity and magnetism is the most fundamental feature of manganites. The conductivity of the best samples of the Sr-doped films at low T is of order s ¼ 104  105 O −1 cm −1 , and this corresponds to a typical metallic regime. Continuous increase in doping leads to the transition from the metallic to insulating state. For many compounds one can observe a peculiar charge-ordered state. The right end of the phase diagram (e.g., SrMnO3) describes the compound which does not have the e2g-electrons. Naturally, this material is an insulator. If the compound is in the metallic ferromagnetic state (e.g., it corresponds to 0.2 6.3 nm), the nominally insulating regime shows a plateau of residual conductance close to 2e2/h. The residual conductance is independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance is destroyed by a small external magnetic field.

Spin manipulation by Aharonov-Casher spin interference An electron acquires a phase around magnetic flux due to the vector potential leading to the Aharonov-Bohm (AB) effect in an interference loop (Aharonov and Bohm, 1959; Tonomura et al., 1986). From the view point of inherent symmetries between magnetic field and electric field in the Maxwell equations, Aharonov and Casher have predicted that a magnetic moment acquires a phase around a charge flux line (Aharonov and Casher, 1984). It is pointed out that the AC phase shift can be derived from SOI (Balatsky and Al’tshuler, 1993). A. G. Aronov and Y. B. Lyanda-Geller have derived a spin-orbit Berry phase in conducting rings with SOI (Aronov and Lyanda-Geller, 1993). T. Qian and Z. Su have obtained the AC phase, which is the sum of spin-orbit Berry phase and spin dynamical phase in a one-dimensional ring with SOI (Qian and Su, 1994). Mathur and Stone have theoretically shown (Mathur and Stone, 1992) that the effects of SOI in disordered conductors are manifestations of the AC effect in the same sense as the effects of weak magnetic fields are manifestations of AB effect. They have proposed the electronic AC effect in a mesoscopic interference loop made of GaAs 2DEG with the Dresselhaus SOI. The AC spin interference based on this proposal was not reported since the Dresselhaus SOI strength is not controlled by an electrostatic way. Electrostatic manipulation of spins is of crucial for spintronics. An AC spin-interference device has been proposed on the basis of the gate controlled Rashba SOI (Nitta et al., 1999; Meijer et al., 2002). The schematic structure of the spin interference device is shown in Fig. 8. The spin interference can be expected in a ring-shaped loop with the Rashba SOI because the spins of electrons precess in opposite directions between clockwise and counter clockwise traveling directions in the ring. The relative difference in spin precession angle at the interference point causes the phase difference in the spin wave functions. The gate electrode, which covers the whole area of the ring, controls the Rashba SOI, and therefore, the interference. The advantage of this proposed spin-interference device is that conductance modulation is not washed out even in the presence of multiple modes since the spin precession angle does not depend on the wavenumber as shown in Eq. (5). Furthermore, this Rashba spin interferometer works without ferromagnetic electrodes in contrast to spin-FET. The conductance of a one-dimensional ring with the Rashba SOI a and its radius r when electrons travel halfway around the ring is given by Nitta et al. (1999), Frustaglia and Richter (2004), Wang and Vasilopoulos (2005) 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi93 2  2 = 

 < 2 2 ∗ e 2m∗ ar 5 ¼ e 1 − cos 2prm a sin y − pð1 − cos yÞ (9) GR ¼ 41 − cos p 1 + h h : ; ℏ2 ℏ2 The acquired AC spin phase can be written as the sum of two phases; 2prm∗ a/ℏ2 and p(1 − cos y), as is shown in the last expression in Eq. (9). Here, y is defined by tan y ¼ 2m∗ ar/ℏ2, and corresponds to the effective magnetic field direction in the rotational frame of

Gate

InGaAs/InAlAs 2DEG Fig. 8 Schematic illustration of spin interference device based on Rashba SOI. J. Nitta, F. Meijer, H. Takayanagi (1999) Spin-interference device. Applied Physics Letters 75: 695.

Spin-orbit interaction based spintronics

201

reference. The former term is sometimes called the dynamical part of the AC spin phase, because of its dependence on the distance traveled by the electrons and its relation to spin precession angle. The latter term is a geometrical phase since it only depends on the solid angle y, and not on spatial parameters. Such geometric phases were discovered by Berry from the basic laws of quantum mechanics (Berry, 1984), and received considerable attention. Berry shows that the wavefunction obtains a non-trivial phase when a parameter in the Hamiltonian is changed in a cyclic and adiabatic way. The conductance of a mesoscopic ring is affected by several quantum interference effects. The well-known AB effect results in a resistance oscillation with a magnetic flux period of h/e. The AB effect is sample specific and very sensitive to the Fermi wave length, therefore, the interference pattern is rapidly changed by the gate voltage. Complex gate voltage dependence has been reported in an AC experiment in a single ring fabricated from HgTe/HgCdTe QWs (König et al., 2006). Therefore, a detailed analysis is necessary to compare with the AC theory. In order to detect the AC effect clearly, it is better to focus another quantum interference phenomenon, the Al’tshuler-Aronov-Spivak (AAS) effect (Al’tshuler et al., 1981). The AAS effect is the AB effect of time reversal symmetric paths, where the two wave function parts go all around back to the origin on identical paths, but in opposite directions. If there is magnetic flux inside the paths the resistance will oscillate with the period of h/2e. In this time reversal symmetric paths, any other phases which are acquired by path geometry and/or the Fermi wave length (carrier density) differences will be identical and will be canceled out. However, the time reversal AAS effect depends on the spin phase modulated by the Rashba SOI. The time reversal symmetric AC spin interference can be picked up by using an array of rings instead of a single ring. The experiments to detect the AC spin interference have been performed by using gate fitted 40  40 rings array made of an InAlAs/InGaAs 2DEG as shown in inset of Fig. 9. Magnetoresistance curves were measured for various gate voltages to tune the Rashba SOI strengths. The AAS oscillations at the gate voltage of −2.4 V are shown in Fig. 9(a). 2D plot for the gate voltage dependence of AAS oscillations is displayed in Fig. 9(b). At zero magnetic field, a phase contribution from the orbital part of the wave-function is canceled out in the time-reversal paths, therefore, the spin part of the wave-function controls the AAS amplitude through the Rashba SOI modulation by Vg. Fig. 9(c) shows the AAS amplitude plotted against the gate voltage at zero magnetic field. The result shown in Fig. 9(c) is consistent with gate voltage dependence of the Rashba SOI parameters obtained by analysis of SdH oscillations using the same InAlAs/InGaAs 2DEG. Therefore, the AAS amplitude modulation is ascribed to the spin interference in the time-reversal paths, i.e., the time-reversal AC effect (Bergsten et al., 2006; Nagasawa et al., 2012). A geometric phase of the most fundamental spin-1/2 system, the electron spin, had not been observed directly and controlled independently from dynamical phases. The geometrical phase shift and its topological transition by in-plane Zeeman field was theoretically investigated in the AC spin interference device (Saarikoski et al., 2015). Control of the geometric phase appeared in Eq. (9) was demonstrated by gate voltage (Nagasawa et al., 2012) and Zeeman in-plane field (Nagasawa et al., 2013). It was also shown that the Aharonov-Casher oscillations in various radius arrays collapse onto a universal curve if the radius and the strength of Rashba SOI are taken into account. The result is interpreted as the observation of the effective spin dependent flux through a ring.

Fig. 9 (a) AAS oscillation of 40  40 rings array as shown inset SEM picture. Horizontal axis represents magnetic field. (b) Gate voltage dependence of the AAS oscillations. (c) AAS amplitude at B ¼ 0 mT as a function of gate voltage (time-reversal AC effect). Vertical axis represents gate voltage. Red solid line is calculated by theory. F Nagasawa, J Takagi, Y Kunihashi, M Kohda, J Nitta (2012) Experimental demonstration of spin geometric phase: Radius dependence of time-reversal aharonov-casher oscillations. Physical Review Letters 108: 086801.

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Spin manipulations by SOI-based ESR Electron spin resonance (ESR) is of importance for the manipulation of individual electron spins in quantum information technologies. In general, ESR requires two external magnetic fields: a static field (B0) to split the spin states in energy and an oscillating field (B1) with the frequency resonant to the splitting energy. However, spin manipulation methods relying on real magnetic fields—much broader than the size of individual electrons—are energetically inefficient and unsuitable for future device applications. One of SOI-based ESR set-ups for spin manipulation is shown in Fig. 10(a). Quantum dot is formed in an InAs wire between two gate electrodes. In addition to static magnetic field B0ext, SOI induces an oscillating B1eff when an electron continuously moves back and forth by using time-dependent gate voltages Vg1 and Vg2. Spin manipulations based on SOI-induced ESR have been realized by using semiconductor narrow wires, quantum dots and winding channels. A ballistic spin resonance (BSR) was realized by utilizing oscillating field induced by the Rashba and Dresselhaus SOIs driven by the free motion of electrons that bounce at frequencies of tens of GHz in high mobility GaAs narrow channels (Frolov et al., 2009). BSR is manifested as a strong suppression of spin relaxation length when the motion of electrons is in resonance with spin precession. Manipulation of single spins is essential for spin-based quantum information processing. Electrical control instead of magnetic control is particularly important for this purpose, because electric fields are easy to generate locally on-chip. Novak et al. experimentally realized coherent control of a single-electron spin in a GaAs/AlGaAs gate-defined quantum dot using an oscillating electric field generated by a local gate (Nowack et al., 2007). Nadj-Perge et al. (2010) implemented a SO quantum bit (qubit) in an InAs nanowire, and realized fast qubit rotations and universal single-qubit control using electric fields; the qubits are hosted in single-electron quantum dots that are individually addressable. The authors claim that nanowires offer various advantages for quantum computing: they can serve as one-dimensional templates for scalable qubit registers, and it is possible to vary the material even during wire growth. Rabi frequencies exceeding 100 MHz were demonstrated in InSb nanowires (Van den Berg et al., 2013). The demonstration of SO qubits coupled to superconducting resonators paves the way for a scalable quantum computing architecture. Circuit quantum electrodynamics (cQED) allows spatially separated superconducting qubits to interact via a superconducting microwave cavity that acts as a “quantum bus,” making possible two-qubit entanglement and the implementation of simple quantum algorithms. Petersson et al. (2012) combine the cQED architecture with spin qubits by coupling an InAs nanowire double quantum dot to a superconducting cavity. The demonstration of SO qubits coupled to superconducting resonators paves the way for a scalable quantum computing architecture. (B)

(A)

Gate1 Q-dot Gate2 InAs wire y x

=

+

Fig. 10 (a) An SOI-based ESR example. An electron confined in quantum dot(Q-dot) is moving back and forth by time dependent gate voltages Vg1 and Vg2, leading to oscillating effective magnetic field B1eff. Frequency of the oscillation field should be resonant with spin splitting energy induced by static external magnetic field B0ext to manipulate the spin in Q-dot. (b) SOI in a winding channel induces effective static field B0eff and oscillating field B1eff. These fields are created when electron spins are carried by surface acoustic waves along the winding channel. H. Sanada, Y. Kunihashi, H. Gotoh, K. Onomitsu, M. Kohda, J. Nitta, P. V. Santos and T. Sogawa (2013) Manipulation of mobile spin coherence using magnetic-field-free electron spin resonance. Nature Physics 9: 280.

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203

For ESR experiments using the SOIs so far, the spin splitting energies are induced by an external Zeeman field (or local ferromagnet) and oscillating resonant SOI fields driven by ac electric fields. Sanada et al. (2013) have demonstrated an alternative approach where the SOI of trajectory-controlled electrons induces effective static field B0eff and oscillating field B1eff. These fields are created when electron spins are carried by surface acoustic waves along winding semiconductor channels as shown in Fig. 10(b). The resultant spin dynamics—mobile spin resonance—is equivalent to the usual ESR but requires neither static nor time-dependent real magnetic fields to manipulate electron spin coherence.

Conclusion Rashba SOI in semiconductor heterostructures is not materials constant but its strengths can be tuned by the design of QWs and an external gate voltage. Electrical spin generation, manipulation, and detection are prerequisite for future spintronics, and these spin functionalities can be realized solely by utilizing SOI. Spin polarized carriers have been generated by spin Hall effect and Stern-Gerlach spin filter. Both inverse spin Hall effect and spin filter can be also utilized for electrical spin detection. The electrical control of the magnetization direction of small magnets is currently among the most active areas in spintronics due to its interest for memory, logic and data storage applications. Spin-orbit torque (SOT) was first observed in (Ga,Mn)As (Chernyshov et al., 2009) where strained zinc-blende structure is responsible for the emergence of linear Dresselhaus SOI. SOT induced by spin Hall effect and Rashba-Edelstein effect are of importance for magnetization reversal of ferromagnets placed adjacent to SOI materials. Quantum spin Hall effect is discovered in HgTe/CdTe QWs by band inversion due to strong SOI. The quantized conductance is caused by the edge states where opposite spin-polarized currents are counter-propagating. Gate controlled spin precession is confirmed in AC spin interference devices. This electrical control of spin precession is a basis of spintronics devices such as spin-FET. SOI-based ESR techniques are developed by using ac-electric fields for spin manipulations. Furthermore, magnetic-field-free ESR techniques based on SOI are demonstrated and can be applicable for quantum information process. Generation of both static and oscillating SOI effective fields is possible in an optimized winding channel since the SOI induced field is momentum dependent. Although SOI is an origin of spin relaxation, long spin coherence in the PSH state has been realized by making Rashba and Dresselhaus SOI equal strength. It should be emphasized that the concept of SOI demonstrated in semiconductors has inspired metal spintronics devices such as magnetization reversal using SOT, discoveries of new topological materials and innovative concepts such as Majorana fermions (Manchon et al., 2015).

References Aharonov Y and Bohm D (1959) Physics Review 115: 485. Aharonov Y and Casher A (1984) Physical Review Letters 53: 2964. Al’tshuler BL, Aronov AG, and Spivak BZ (1981) JETP Letters 33: 94. Aronov AG and Lyanda-Geller YB (1993) Physical Review Letters 70: 343. Balatsky A and Al’tshuler B (1993) Physical Review Letters 70: 1678. Bergsten T, Kobayashi T, Sekine Y, and Nitta J (2006) Physical Review Letters 97: 196803. Bernevig BA, Orenstein J, and Zhang S-C (2006a) Physical Review Letters 97: 236601. Bernevig BA, Hughes TL, and Zhang SC (2006b) Science 314: 1757. Berry MV (1984) Proceedings of the Royal Society of London A 392: 45. Bychkov YA and Rashba EI (1984) Journal of Physics C: Solid State Physics 17: 6039. Chernyshov A, et al. (2009) Evidence for reversible control of magnetization in a ferromagnetic material by means of spin–orbit magnetic field. Nature Physics 5: 656–659. Datta S and Das B (1990) Applied Physics Letters 56: 665. Dresselhaus G (1955) Physics Review 100: 580. Dyakonov MI and Perel VI (1971) Soviet Physics - JETP 33: 1053. Edelstein VM (1990) Spin Polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Communications 73: 233–235. Engel H-A, Halperin BI, and Rashba EI (2005) Theory of spin hall conductivity in n-doped GaAs. Physical Review Letters 95: 166605. Engels G, Lange J, Schäpers T, and Lüth H (1997) Physical Review B 55: R1958. Frolov SM, et al. (2009) Ballistic spin resonance. Nature 458: 868–871. Frustaglia D and Richter K (2004) Physical Review B 69: 235310. Ganichev SD (2008) International Journal of Modern Physics B 22: 1–26. Ganichev SD, Danilov SN, Schneider P, Bel’kov VV, Golub LE, Wegscheider W, Weiss D, and Prettl W (2006) Electric current-induced spin orientation in quantum well structures. Journal of Magnetism and Magnetic Materials 300: 127–131. Holleitner AW, Sih V, Myers RC, Gossard AC, and Awschalom DD (2006) Physical Review Letters 97: 036805. Inoue J, Bauer GE, and Molenkamp LW (2004) Physical Review B 70: 041303(R). Ishihara J, Ohno Y, and Ohno H (2014) Applied Physics Express 7: 013001. Ivchenko EL and Pikus GE (1978) New photogalvanic effect in gyrotropic crystals. JETP Letters 27: 604. Kato YK, Myers RC, Gossard AC, and Awschalom DD (2004a) Science 306: 1910. Kato YK, Myers R, Gossard A, and Awschalom DD (2004b) Current-induced spin polarization in strained semiconductors. Physical Review Letters 93: 176601. Kettemann S (2007) Physical Review Letters 98: 176808. Koenig M, et al. (2007) Science 318: 766. Koga T, Nitta J, Akazaki T, and Takayanagi H (2002) Physical Review Letters 89: 046801. Kohda M, Lechner V, Kunihashi Y, Dollinger T, Olbrich P, Schönhuber C, Caspers I, Bel’kov VV, Golub LE, Weiss D, Richter K, Nitta J, and Ganichev SD (2012a) Physical Review B 86: 081306(R). Kohda M, Nakamura S, Nishihara Y, Kobayashi K, Ono T, Ohe J, Tokura Y, Mineno T, and Nitta J (2012b) Nature Communications 3: 1038.

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Spin-orbit interaction based spintronics

Kohda M, Okayasu T, and Nitta J (2019) Spin-momentum locked spin manipulation in a two-dimensional Rashba system. Scientific Reports 9: 1909. König M, Tschetschetkin A, Hankiewiccz EM, Sinova J, Hock V, Daumer V, Schaefer M, Becker CR, Buhmann H, and Molenkamp LW (2006) Physical Review Letters 96: 076804. Koralek JD, Weber CP, Orenstein J, Bernevig BA, Zhang S-C, Mack S, and Awschalom DD (2009) Nature 458: 610. Kumihashi Y, Sanada H, Gotoh H, Onomitsu K, Kohda M, Nitta J, and Sogawa T (2016) Nature Communications 7: 10722. Kunihashi Y, Kohda M, and Nitta J (2009) Physical Review Letters 102: 226601. Kunihashi Y, Kohda M, Sanada H, Gotoh H, Sogawa T, and Nitta J (2012) Applied Physics Letters 100: 113502. Li J, Hu L, and Shen S-Q (2005) Physical Review B 71: 241305 (R). Lin Y, Koga T, and Nitta J (2005) Physical Review B 71: 045328-1. Mal’shukov AG and Chao KA (2000) Physical Review B 61: R2413. Manchon A, Koo HC, Nitta J, Frolov SM, and Duine RA (2015) Nature Materials 14: 871–882. Mathur H and Stone AD (1992) Physical Review Letters 68: 2964. Meier L, Salis G, Shorubalko I, Gini E, Schoen S, and Ensslin K (2007) Nature Physics 3: 650. Meijer FE, Morpurgo AF, and Klapwijk TM (2002) Physical Review B 66: 033107. Murakami S, Nagaosa N, and Zhang SC (2003) Science 301: 1348. Nadj-Perge S, Frolov SM, Bakkers EPAM, and Kouwenhoven LP (2010) Spin-orbit qubit in a semiconductor nanowire. Nature 468: 1084–1087. Nagasawa F, Takagi J, Kunihashi Y, Kohda M, and Nitta J (2012) Physical Review Letters 108: 086801. Nagasawa F, Frustaglia D, Saarikoski H, Richter K, and Nitta J (2013) Nature Communications 4: 2526. Nitta J, Akazaki T, Takayanagi H, and Enoki T (1997) Physical Review Letters 78: 1335. Nitta J, Meijer FE, and Takayanagi H (1999) Applied Physics Letters 75: 695. Nowack KC, Koppens FHL, Nazarov YV, and Vandersypen LMK (2007) Coherent control of a single electron spin with electric fields. Science 318: 1430–1433. Ohe J, Yamamoto M, Ohtsuki T, and Nitta J (2005) Physical Review B 72: 041308(R). Petersson KD, et al. (2012) Circuit quantum electrodynamics with a spin qubit. Nature 490: 380–383. Qian T-Z and Su Z-B (1994) Physical Review Letters 72: 2311. Rashba EI (1960) Soviet Physics - Solid State 2: 1109. Rowe ACH, Nehls J, and Stradling RA (2001) Physical Review B 63: 201307. Saarikoski H, Vazquez-Lozano JE, Baltanas JP, Nagasawa F, Nitta J, and Frustaglia D (2015) Physical Review B 91: 241406(R). Sanada H, Kunihashi Y, Gotoh H, Onomitsu K, Kohda M, Nitta J, Santos P, and Sogawa T (2013) Nature Physics 9: 280. Sasaki A, Nonaka S, Kunihashi Y, Kohda M, Bauernfeind T, Dollinger T, Richter K, and Nitta J (2014) Nature Nanotechnology 9: 703–709. Schäpers T, Engels G, Lamge J, Klocke T, Hollfelder M, and Lüth H (1998) Journal of Applied Physics 83: 4324. Schäpers T, Guzenko VA, Pala MG, Zülich U, Governale M, Knobbe J, and Hardtdegen H (2006) Physical Review B 74: 081301(R). Scheid M, Kohda M, Kunihashi Y, Richter K, and Nitta J (2008) Physical Review Letters 101: 266401. Schiemann J and Loss D (2005) Physical Review B 71: 085308. Schliemann J, Carlos Egues J, and Loss D (2003) Physical Review Letters 90: 146801. Schmidt G, Ferrand D, Molenkamp LW, Filip AT, and van Wees BJ (2000) Physical Review B 62: R4790. Sinova J, Culcer D, Niu Q, Sinitsyn NA, Jungwirth T, and MacDonald AH (2004) Physical Review Letters 92: 126603. Tonomura A, Osakabe N, Matsuda T, Kawasaki T, and Endo J (1986) Physical Review Letters 56: 792. Van den Berg J, et al. (2013) Fast spin-orbit qubit in an indium antimonide nanowire. Physical Review Letters 110: 066806. Walser MP, Reichl C, Wegscheider W, and Salis G (2012) Nature Physics 8: 757–762. Wang XF and Vasilopoulos P (2005) Physical Review B 72: 165336. Weigele OJ, Marinescu DC, Dettwiler F, Fu J, Mack S, Egues JC, Awschalom DD, and Zumbühl DM (2020) Physical Review B 101: 035414. Winkler R (2003) Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, p. 191. Berlin: Springer-Verlag. Wunderlich J, Kaestner B, Sinova J, and Jungwirth T (2005) Physical Review Letters 94: 047204. Wunderlich J, Irvine AC, Sinova J, Park BG, Zârbo LP, Xu XL, Kaestner B, Novák V, and Jungwirth T (2009) Spin-injection Hall effect in a planar photovoltaic cell. Nature Physics 5: 675–681. Wunderlich J, Park B-G, Irvine AC, Zârbo LP, Rozkotová E, Nemec P, Novák V, Sinova J, and Jungwirth T (2010) Science 330: 1801. Yoshizumi K, Sasaki A, Kohda M, and Nitta J (2016) Applied Physics Letters 108: 132402.

Spintronics in 2D graphene-based van der Waals heterostructures David TS Perkins and Aires Ferreira, School of Physics, Engineering and Technology and York Centre for Quantum Technologies, University of York, York, United Kingdom © 2024 Elsevier Ltd. All rights reserved.

Introduction Graphene and van der Waals heterostructures Honeycomb monolayers: A tight-binding description Honeycomb monolayers: SOC interactions Honeycomb monolayers with proximity-induced SOC: Continuum theory Relativistic spin–orbit coupled transport phenomena Electronic signatures Spin Hall effect Inverse spin galvanic effect Observing spin–charge interconversion Conclusion Acknowledgments References

206 207 209 210 211 213 213 214 216 218 219 219 219

Abstract Spintronics has become a broad and important research field that intersects with magnetism, nano-electronics, and materials science. Its overarching aim is to provide a fundamental understanding of spin-dependent phenomena in solid-state systems that can enable a new generation of spin-based logic devices. Over the past decade, graphene and related 2D van der Waals crystals have taken center stage in expanding the scope and potential of spintronic materials. Their distinctive electronic properties and atomically thin nature have opened new opportunities to probe and manipulate internal electronic degrees of freedom. Purely electrical control over conduction-electron spins can be attained in graphene-transition metal dichalcogenide heterostructures, due to proximity effects combined with graphene’s high electronic mobility. Specifically, graphene experiences a proximity-induced spin–orbit coupling that enables efficient spin-charge interconversion processes; the two most well-known and at the forefront of current research are the spin Hall and inverse spin galvanic effects, wherein an electrical current yields a spin current and nonequilibrium spin polarization, respectively. This article provides an overview of the basic principles, theory, and experimental methods underpinning the nascent field of 2D material-based spintronics.

Key points This chapter aims to achieve the following:

• • • • • • •

Provide a brief historic overview of how spintronics developed before and after the discovery of graphene, Discuss the various properties of the 2D honeycomb materials at the heart of spintronics in van der Waals heterostructures, Outline how the electronic structure of 2D monolayers, such as graphene, is altered by proximity-induced spin–orbit coupling using a tight-binding formalism informed by a symmetry analysis, Construct a low-energy theory representative of the van der Waals heterostructures used in realistic spintronic devices, Introduce the idea of spin–charge interconversion: how an electrical current can generate a macroscopic spin current or spin polarization, and vice versa, Present the framework of linear response theory used in the mathematical description of the spin Hall and inverse spin galvanic effects, Explain how spin–charge interconversion processes can be detected via electrical means.

Abbreviations 2DEG BR CEE DOF FM

Two-dimensional electron gas Bychkov–Rashba Collinear Edelstein effect Degree of freedom Ferromagnet or ferromagnetic

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hBN ISGE ISHE KM SdH SGE SHE SOC SOT SV TB TMD vdW WAL WL

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Hexagonal boron nitride Inverse spin galvanic effect Inverse spin Hall effect Kane–Mele Shubnikov–de Haas Spin galvanic effect Spin Hall effect Spin–orbit coupling Spin–orbit torque Spin-valley Tight binding Transition metal dichalcogenide van der Waals Weak localization Weak antilocalization

Introduction Spintronics can be considered the magnetic counterpart to electronics whereby the transfer and processing of information can be conducted through the electron’s spin degree of freedom (DOF), rather than or in addition to its electronic property of charge. Specifically, spintronics concerns itself with the electrical manipulation of the electron’s spin and one of its first milestones was the observation of a spin-polarized current by Tedrow and Meservey (1973). Despite this and the first controlled injection of a spinpolarized current into a nonmagnetic material being achieved in 1985 (Johnson and Silsbee, 1985), it was only in 1988 that spintronics began to bloom and flourish into the field we know today. What sparked this scientific boom was the independent observation of a giant magnetoresistance attributed to the relative orientation of the ferromagnetic (FM) layers in Fe-Cr-based systems by Baibich et al. (1988) and Binasch et al. (1989). Since the Nobel prize winning discovery of giant magnetoresistance, initial efforts in the field focused on spin valve setups to study how the electron’s spin would diffuse and relax through a normal metal when injected via a spin-polarized current using an FM contact. However, a more exotic method of spin current generation had already been proposed in 1971 by Dyakonov and Perel (Dyakonov and Perel, 1971a,b; Sinova et al., 2015), the spin Hall effect (SHE), where the application of an electrical current would yield a perpendicular pure spin current without the need for magnetic materials. The origin of this effect can be found in the asymmetric scattering of electrons based upon their spin due to the presence of spin–orbit coupling (SOC). Initially observed in 2004 via optical methods (Kato et al., 2004), the SHE and its inverse (ISHE) have become paradigmatic phenomena in spintronics due to their lack of reliance upon magnetic components. A major focus for the application of the SHE is in low-power magnetic memory devices, where the manifestation of a spin current results in a spin accumulation, and hence a net spin polarization, which can exert a torque on the magnetization of a nearby FM. With a large enough spin accumulation, dramatic effects like magnetization switching can be induced. Clearly, with the use of a simple electrical current, we can manipulate the spin of charge carriers in a material and induce interesting nonequilibrium phenomena at interfaces between materials, thus yielding important applications in modern information technology. In addition to spin current, another major concept in spintronics is spin texture: the momentum dependence of the transport electrons’ spin within a solid. In many low-dimensional systems and at heterostructure interfaces, the enhancement of the relativistic spin–orbit interaction is key to generating nontrivial spin textures that give rise to a plethora of phenomena, ranging from the spin-momentum-locked surface states in topological insulators to the topologically protected real-space spin textures seen in chiral magnets, such as skyrmions and spin spirals. Of particular interest is the emergence of a spin polarization, as in the SHE, but without a resulting spin current. This effect, the sole spontaneous generation of a spin polarization via application of an electric current, is known as the inverse spin galvanic effect (ISGE), though in some literature it may also be referred to as the Rashba–Edelstein or Edelstein effect. Based upon Bychkov–Rashba-type SOC (Bychkov and Rashba, 1984), arising when the mirror symmetry about a given plane is broken, this effect was initially predicted in two-dimensional electron gases (2DEGs) in semiconductor heterostructures in 1989 (Aronov and Lyanda-Geller, 1989; Edelstein, 1990; Aronov et al., 1991) but remained unobserved in experiment until 2002, when its reciprocal effect (i.e., spin-to-charge conversion) was observed (Ganichev et al., 2002). Combining both the SHE and ISGE, we find ourselves in a position with great control over the motion and net orientation of the electron spins in materials. However, to truly construct realistic devices using a spin-focused infrastructure, we must understand what makes an ideal material for providing such intricate control over spin. The two most important factors governing the effectiveness of spintronic devices are disorder and SOC: both mechanisms yield significant consequences on the transport range of electron spins and our control over them. While a larger SOC might ensure more efficient generation of spin currents and polarizations, it comes at the cost of faster spin dephasing and hence spin information is lost over a much shorter distance. In contrast, a weak SOC might allow for long range transport, but results in less efficient charge-to-spin conversion. Similarly, disorder can also change the efficiency of spin–charge interconversion. In a pristine system, the only processes present will be those intrinsic to the system. However, upon the

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inclusion of disorder, some intrinsic mechanisms are completely suppressed in favor of extrinsic mechanisms. Furthermore, spin–orbit active impurities can have similar effects to including a SOC into the system, due to their ability to change an electron’s spin orientation. Clearly, there is a balancing act to be handled in constructing ideal spintronic devices with both disorder and SOC acting as the tuning knobs. Prior to 2004, experiments studying spin transport focused primarily on 2DEGs realized in a multitude of systems including thin metallic films (Jedema et al., 2001), permalloys (Steenwyk et al., 1997; Dubois et al., 1999), and semiconductor heterostructures (Yang et al., 1994; Kikkawa and Awschalom, 1999; Ohno et al., 1999; Malajovich et al., 2000). In most cases, spin information was passed into the system by either a spin-polarized charge current, or via optical excitation of a semiconductor resulting in a spin imbalance of the conduction band. Despite these initial endeavors, the electrical processing of spin-encoded information was hindered by the difficulty of combining effective spin control with large enough spin lifetimes. However, with the discovery of graphene in 2004 as the first truly 2D, that is, atomically thin, solid state system, a new epoch dawned in spintronics. It was quickly established that bare graphene offered the largest spin diffusion lengths of any material to date, with many reports finding ls
1 − 20 mm, courtesy of the weak intrinsic spin–orbit and hyperfine interactions of sp2 hybridized carbon. Naturally, graphene’s extremely small intrinsic SOC makes it difficult to have precise electrical control over the net spin orientation, but does allow for long-distance spin-information transfer. However, with the many advances in nanofabrication over the past two decades and the isolation of other 2D materials, such as transition metal dichalcogenides (TMDs), layer-by-layer assemblies of 2D materials have been imagined and constructed with the ability to alter the net behavior of the composite system based purely on the individual layers used. Thus the concept of bespoke devices combining the desirable properties of different materials has become possible in the form of van der Waals (vdW) heterostructures. What single 2D crystals lack, vdW heterostructures may offer a way to access by enhancing certain properties by matching an appropriate set of materials. Through simple proximity effects, the properties absent or deemed too weak in the original isolated crystal can be enhanced. Semiconducting TMDs, such as MoS2 and WTe2, are a classic example of this; due to their composition involving transition metal atoms, they naturally possess a large SOC. By stacking this with a graphene monolayer, the electrons of the graphene sheet will experience a proximity-induced SOC effect due to their ability to now hop between the two layers. Furthermore, graphene’s transport nature is not jeopardized by the proximity of the TMD since the low-energy states of graphene lie well within the TMD band gap. Consequently, transport is still dominated by the more conductive graphene layer, though the electronic states are now endowed with a substantial SOC. The typical size of these induced SOCs is up to order 10 meV (Wang et al., 2019), which is three orders of magnitude larger than the intrinsic SOC present in regular graphene (Kane–Mele type of order 10 meV (Sichau et al., 2019)). The specific values of the SOCs depend on the choice of TMD partner and can be further tuned by means of external pressure (Fulop et al., 2021). This dramatic change from an isolated crystal’s properties with the introduction of material partners is what makes vdW heterostructures the modern candidates par excellence for many condensed matter experiments.

Graphene and van der Waals heterostructures Formed by stacking atomically thin layers of hexagonally packed atoms with weak vdW forces binding neighboring layers together, Fig. 1a (Geim and Grigorieva, 2013), vdW heterostructures constitute a specific class of 2D materials. Given the breadth of materials with 2D behavior—encompassing semiconductors, insulators, and semimetals—which can be exfoliated down to a monolayer, it is no surprise that the variety of vdW heterostructures is equally diverse. Furthermore, the electronic properties of such 2D compounds are sensitive to the number of layers, stacking sequence, and atomic coordination, while also being tunable “on-demand” through the controlled application of strain and electric fields (Castro Neto et al., 2009; Wang et al., 2012; Yun et al., 2012). Monolayer graphene presents itself as a zero-gap semiconductor with a linear dispersion relation for both electrons and holes, whose unit cell consists of two distinct lattice sites that are individually referred to as A and B sublattices (Fig. 1b). The band structure of graphene is characteristic of massless chiral Dirac fermions, which derives from the sp2 network of carbon atoms forming a honeycomb lattice with preserved inversion symmetry (Fig. 1b), whose presence gives rise to graphene’s notable electronic properties (Castro Neto et al., 2009). Other 2D materials relevant to vdW heterostructures include (i) hexagonal boron nitride “hBN,” an insulator (ii) bilayer graphene, a zero-gap semiconductor with parabolic band dispersion; and (iii) group-VI dichalcogenides, direct band-gap semiconductors with strong SOC. What makes a material useful in vdW heterostructures varies from material to material. The band gap opening in polyatomic compounds (e.g., hBN) is a direct consequence of broken inversion symmetry. The electronic band gap in monolayer hBN is about 7 eV, making it an insulating analog of graphene. The small lattice mismatch between hBN and graphene (about 1.8%) allows for easy integration in graphene-based devices using dry transfer techniques. hBN encapsulation yields major improvements in the electronic mobility of graphene-based devices and is currently the gold standard for the fabrication of high-performance vdW heterostructures. In lateral spin transport experiments, the hBN encapsulation of graphene has enabled a 10-fold increase in roomtemperature spin lifetimes compared to devices using the now obsolete silicon oxide substrates (Drögeler et al., 2014). The hBN induces an orbital gap in graphene (Fig. 2a), which results from virtual interlayer hopping processes between carbon sites in graphene and the distinct chemical species occupying the A and B sublattices of the partnered hBN layer.

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Spintronics in 2D graphene-based van der Waals heterostructures (a)

(b)

B A

Fig. 1 (a) Vertical heterostructure built from 2D crystals. (b) The honeycomb lattice composed of two triangular Bravais lattices (top right corner shows the first Brillouin zone with two inequivalent Dirac points, K). The nearest-neighbor vectors read as di ¼ a ðsin ai , − cos ai ÞT , where a is the bond length and ai ¼ 2p 3 ði − 3Þ and i ¼ 1, 2, 3. (a) Reprinted with permission from Geim AK and Grigorieva IV (2013) Van der Waals heterostructures. Nature 499(7459): 419–425. https://doi.org/10.1038/nature12385.

(a)

(b)

SOC

orbital gap

spin splitting

Fig. 2 (a) Low-energy dispersion of a honeycomb monolayer with a small orbital gap (m 6¼ 0). (b) Lifting of spin degeneracy due to symmetry breaking SOC (arrows indicate different spin states).

Bilayer graphene can exist in Bernal-stacked “AB” form, with atoms on opposite layers stacked on top of each other in a staggered configuration, or, less frequently, in the “AA” form, with sublattices in adjacent layers perfectly aligned (Liu et al., 2009). Similar to monolayer graphene, its conduction and valence bands touch at the Brillouin zone corners (Fig. 2a); however, low-energy excitations around the Fermi points are associated with quadratic energy dispersion (McCann and Fal’ko, 2006; Guinea et al., 2006). The finite density of states near the Fermi level exacerbate the effect of interactions leading to rich broken symmetry states even in the absence of external fields (Zhang et al., 2010; Vafek and Yang, 2010; Nandkishore and Levitov, 2010; Weitz et al., 2010). Individual layers in AB bilayer graphene can be addressed separately allowing important device functionalities, including band gap opening through gating (McCann, 2006; Castro et al., 2007). Of particular interest for spintronics is the ability to fine-tune both the SOC (proximity with a TMD) and exchange interaction (proximity with an FM) experienced by the electrons. Only the layer that is the immediate neighbor to a partner material will experience significant proximity effects. Therefore, by applying an electric field perpendicular to the graphene bilayer, the electron density of each layer can be adjusted, which in turn changes the proximity-

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induced SOC and exchange interaction experienced by the electrons by shifting them toward or away from the partner material (Zollner et al., 2020). Group-VI dichalcogenides [MX2 (M ¼ Mo, W; X ¼ S, Se,Te)] can be either trigonal-prismatically or octahedrally coordinated (socalled 2H and 1T phases, respectively). These polytypic structures have vastly different electronic properties: while TMDs of the 2H-MX2 type are large-gap semiconductors, 1T-MX2 are predominantly metals (Mattheiss, 1973; Eda et al., 2012; Voiry et al., 2015). 2H-TMD monolayers have direct band gaps in the near-infrared to the visible region, which make them well suited for a broad range of applications in optoelectronics and photonics (Mak and Shan, 2016). Owing to ultimate (2D) quantum confinement, electrons and holes are tightly held together which is responsible for enhanced light–matter interactions. Excitons have typical binding energies of 0.5 eV, which strongly impact the spin and optoelectronic properties of semiconducting TMDs (Wang et al., 2018). Moreover, the spin–valley locking of energy states in the vicinity of K points, stemming from lack of inversion symmetry, lead to spin–valley-dependent optical transition rules (Wang et al., 2012), which enable the addressing of individual valleys with circularly polarized light. The spin–valley coupling in TMDs has been explored to convert optically driven valley currents into charge currents via the inverse valley Hall effect (Mak et al., 2014). In free-standing conditions, the 1T phase undergoes a spontaneous lattice distortion to a semiconducting phase dubbed 1T0 , which supports robust nontrivial topological behavior (quantum spin Hall effect) (Qian et al., 2014; Tang et al., 2017; Wu et al., 2018; Shi et al., 2019). The weak vdW forces between planes of 2D crystals offer a practical route for band structure design. A remarkable example is “twisted” bilayer graphene, where the two graphene monolayers forming bilayer graphene are offset from one another by a simple rotation about the out-of-plane axis (Lopes dos Santos et al., 2007), in addition to their stacking arrangement. For “magic angle” twisted bilayer graphene, the superlattice created by the two graphene sheets leads to strong renormalization of the band structure, which can exhibit flat bands at the Fermi level, leading to possible strongly correlated insulating states in the ultimate 2D (atomically thin) limit (Cao et al., 2018). With the above discussion of different 2D systems and their various combinations, it is clear that vdW heterostructures offer a route toward 2D designer materials. The exact combination of the individual layers is dictated by the purpose of the device being constructed. This is particularly relevant when trying to study and apply the SHE and ISGE from a technological perspective. The specifics behind these phenomena are discussed toward the end of this article while the effect of proximity-induced SOC shall be covered shortly.

Honeycomb monolayers: A tight-binding description The distinctive electronic properties of 2D layered materials and the special role played by the sublattice DOF can be best appreciated within a tight-binding (TB) model of electrons hopping on a honeycomb lattice (Fig. 1b). The minimal Hamiltonian (without SOC) reads as   X { X { (1) H2D ¼ −t ai bj + H:c: + m ai ai − b{i bi i hi, ji where the fermionic operator ai (bi) annihilates a quasiparticle on site i belonging to sublattice A(B), t is the nearest neighbor hopping energy (t  2.8 eV in graphene Castro Neto et al., 2009), and hi, ji denotes the sum over nearest neighbors. The second term describes a staggered on-site energy with amplitude m ¼ (eA − eB)/2 and is relevant for noncentrosymmetric 2D crystals (e.g., m  3.5 eV in hBN (Román et al., 2021) and m  0.7 − 0.9 eV in semiconducting TMDs (Xiao et al., 2012)), as well as graphene-based heterostructures displaying Moiré superlattice effects (Dean et al., 2010; Jung et al., 2015; Wallbank et al., 2015). This tight-binding Hamiltonian provides a starting point for understanding the low-energy properties of prototypical 2D P materials. The energy bands are obtained by means of the Fourier transform ai (bi) ¼ N−1 ke−ikxiak (bk), where k ¼ (kx, ky)T is the 2D wavevector and N is the number of sites in each sublattice (N ¼ NA ¼ NB). After straightforward algebra, one finds !   X { {  ak m f ðkÞ H2D ¼ , (2) ak bk bk fðkÞ −m k with the geometric form factor fðkÞ ¼ −t

X a¼1, 2, 3

eikda ,

where the bond vectors, da, are defined in the caption of Fig. 1. The energy dispersion is readily obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðkÞ ¼  t 2 jfðkÞj2 + m2 ,

(3)

(4)

where the sign  selects positive (negative) energy branch of the spectrum. A band structure representative of graphene with a sublattice staggered potential is shown in Fig. 2a. Of particular note is the appearance of local extrema in the spectrum at the corners of the Brillouin zone, K, where E ¼ m, known as Dirac points. In this example, the small orbital gap (m  t) could be due to use of a lattice-matching substrate (e.g., hBN). At half-filling, the negative energy states are completely filled, and hence the low-energy physics is controlled by excitations about the Dirac points (the region around a Dirac point is known as a valley). For

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semiconducting TMDs and hBN, the staggered on-site energy is comparable to (or larger than) t, resulting in sizable orbital gaps at the Dirac points. The generalization of this Hamiltonian to describe the strong intrinsic SOCs inherent in semiconducting TMDs, as well as symmetry breaking SOCs in vdW heterostructures, is presented in the next section. While the extension of the effective TB model to multilayers (e.g., bilayer graphene) is straightforward, the complete details are beyond the scope of this article, though the interested readers can find details in Peres (2010) and McCann and Koshino (2013).

Honeycomb monolayers: SOC interactions The electronic structure of 2D materials containing heavy elements is strongly modified by spin–orbit effects generated by the periodic crystal potential. In the nonrelativistic approximation to the Dirac equation, the intrinsic spin–orbit interaction reads as HSO ¼ −

ħ sðp  rVÞ, 4m2e c2

(5)

where V (x) is the periodic crystal potential, p is the momentum operator, me is the electron mass, and s is the vector of Pauli matrices describing spin-1/2 particles (i.e., acting on the spin DOF). The (spin-dependent) hopping generated by Eq. (5) can be obtained by exploring time-reversal symmetry (T ) and the crystal symmetries. The most general HSO for honeycomb lattices may therefore be written as   X { X { { ^ ^{ ba ^j + ^ ^j + b^i T bb (6) HTSO ¼ a^i T ab a^i T aa ij bj + bi T ij a ij a ij bj , hi, ji hhi, jii { where a^{i /^ ai (b^j /b^j ) are creation/annihilation operators for the A(B) sublattice with a 2-component spinor structure acting on spin ðBÞ

{

space, T ij (with B ¼ aa, ab, bb) are spin-dependent hopping coefficients with a 2  2 complex matrix structure satisfying T Bij ¼ ½T Bji , and hhi, jii denotes the sum over next-nearest neighbors. The neglect of hoppings beyond next-nearest neighbors in Eq. (6) is justified given the exponential decay of matrix elements with the distance. Note that on-site spin-dependent terms, such as a^{i sz a^i , change sign under T and thus are not allowed. ðBÞ The symmetries of the system are contained within the T ij . Expansion of the spin hopping matrices into elements of the SU(2) spin-algebra yields y

T ij ¼ oxij sx + oij sy + ozij sz ,

(7)

where sublattice superscripts have been omitted. The antiunitary operator enacting T is given by Y ¼ isyK, where K denotes complex  conjugation. The requirement, Tij ¼ Y Tij Y−1, leads to the following constraint: oaij ¼ −ðoaij Þ (with a ¼ x, y, z). As such, the coefficients of spin-dependent hopping mediated by HSO are purely imaginary. The full set of spin–orbit interactions up to nextnearest neighbors are shown in Fig. 3, where the allowed hoppings are dictated by the point group symmetry. For example, D6h-invariant Hamiltonians (e.g., flat pristine graphene) only permit spin-conserving next-nearest neighbor hoppings. This is the result of three symmetries. First, mirror inversion about the plane, Shxy , which reverses the sign of x- and y- components in Eq. (7), ðx,yÞ forbids all spin flip processes, oij ¼ 0. Second, the mirror symmetry Sdyz , which sets y !−y and consequently reverses the sign z of sz in Eq. (7), ensures that oij ¼ 0 for all in-plane vertical hoppings, which are naturally nearest neighbor processes. Finally, combining the Sdyz mirror symmetry with 6-fold rotational symmetry about the z-axis, we see that the vertical hoppings must map onto the other nearest neighbor hoppings and hence they too must vanish. More generally, when space inversion or horizontal reflection symmetries are broken, other terms are allowed. Of particular relevance is the C3v point group, given that it is a common subgroup of D3h (e.g., hBN and TMD monolayers), D3d (e.g., rippled graphene and Si- and Ge-based graphene analogs), and C6v (e.g., graphene on a nonlattice-matched substrate) and hence defines the most general class of Hamiltonians compatible with the honeycomb lattice symmetries (Saito et al., 2016; Kochan et al., 2017). Altogether, C3v-invariant models allow for 3 types of SOC   h i   { i X 2i X HCSO3v ¼ pffiffiffi nij lA a^{i sz a^j + lB b^i sz b^j + l a^{i s  dij z b^j + ðb $ aÞ 3 3 3 hhi, jii hi, ji (8) h    i 2i X A {  B ^{ + lnn a^i s  dij z a^j + lnn bi s  dij z b^j , 3 hhi, jii namely, spin-conserving sublattice-resolved SOC, lA(B), Bychkov–Rashba (BR) interaction, l, and spin-flipping sublattice-resolved SOC, lAðBÞ nn . Here, dij is the unit vector along the line segment connecting site i and j while nij ¼ 1 distinguishes between clockwise and anticlockwise electron hopping, respectively. The spin-conserving SOC (first term in Eq. 8) is a fingerprint of pseudospin-spin coupling in 2D layered materials. Two cases are of noteworthy interest: (i) centrosymmmetric crystals, where only a spin conserving SOC is permitted with lA ¼ lB, as is the case in pristine graphene (D6h) discussed above, and graphene-like materials with D3d point group, and (ii) systems with broken inversion symmetry leading to sublattice-resolved SOC (lA 6¼ lB), as seen, for example, in 2H-TMD monolayers (D3h) and graphene-TMD

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Fig. 3 Spin–orbit hopping terms compatible with subgroups of D6h. Horizontal (h), vertical (v), and dihedral (d) reflections are shown (bottom, middle). Black (white) arrows indicate spin-conserving (spin-flip) hoppings. D6h-, D3h-, and D3d-invariant systems (top panel) admit next-nearest hoppings. C6v- and C3v-invariant systems (bottom panel, right and left) allow for nearest neighbor spin-flip hopping.

heterostructures (C3v). The BR interaction generated by the nearest-neighbor spin-flip hoppings (second term in Eq. 8) signals the lack of mirror inversion symmetry, Shxy , associated with reduction of the point group from D6h to C6v, which may occur through interfacial effects in heterostructures or via application of an electric field perpendicular to the 2D plane. Finally, next-nearest(third term in Eq. 8) are also allowed in systems lacking neighbor spin-flip processes, parameterized by the couplings lAðBÞ nn out-of-plane reflection symmetry, Shxy , such as graphene-based vdW heterostructures or graphene placed on a generic substrate. Note that when the in-plane inversion symmetry is also broken, these processes become sublattice-resolved (lAnn 6¼ lBnn). This occurs, for example, in graphene-TMD heterostructures.

Honeycomb monolayers with proximity-induced SOC: Continuum theory The low-energy electronic structure of 2D materials can be conveniently modeled using a (long-wavelength) continuum description. pffiffiffi ¼ 4p=ð3 3 aÞð1, 0ÞT , that is f(k) ’ Expanding the geometric form factor (Eq. 3) to first order around each Dirac point, K    (3ta/2)(kx + iky), yields the effective single-particle Hamiltonian H0k ¼ ħv tz sx kx + t0 sy ky s0 + mt0 sz s0 when written in the conventional basis, jci ¼ ðK + , K − ÞT ðA, BÞT ð", #ÞT . Here k is the 2D wavevector measured with respect to a Dirac point, v ¼ 3at=2ħ is the Fermi velocity of massless Dirac fermions, s0 is the identity matrix acting on the spin DOF, and si and ti are the Pauli matrices supplemented with identity acting on sublattice and valley DOFs, respectively. This low-energy Hamiltonian admits an even simpler and more elegant form when written using the valley basis, jci ¼ ðK + A, K + B, −K − B, K − AÞT ð", #ÞT , that is, H0k ¼ t0 H0k s0 , with H0k ¼ −iħv sk + msz :

(9)

The Dirac–Weyl equation (Eq. 9) governs the low-energy properties of honeycomb monolayers with low SOC and has been extensively used in investigations of opto-electronic and transport phenomena in single-layer graphene (Peres, 2010; McCann and Koshino, 2013). For ease of notation, the writing of the tensor product between matrices acting on different DOFs is omitted from here onward. The spin–orbit interactions in the continuum can be derived by expanding the Fourier transformed TB Hamiltonian (Eq. 8) around the K points. Here, as in the previous section for the full TB Hamiltonian, T and unitary (spatial) symmetries are exploited so as to constrain the allowed spin–orbit terms in the continuum theory. The time-reversal operation reverses spins s ! −s and swaps valleys (as momentum reversal sends K+ $ K−). Since the time reversal operator is antiunitary, the sy pseudospin operator is

212

Spintronics in 2D graphene-based van der Waals heterostructures

also affected, T : sx ! sx , sy ! −sy , sz ! sz. Exploiting the T -symmetry transformations, one finds that there are 4  3 ¼ 12 possible terms  X  ðiÞ ðiÞ HTSO ¼ DðiÞ t0 sz + lðiÞ (10) x t0 sx + ly t0 sy + o tz s0 si : i¼x, y, z Direct inspection of Eq. (10) shows that the majority of SOC terms lead to anisotropic energy dispersion, and hence are incompatible with the C3v point group symmetry. To preserve the continuous rotational symmetry of the low-energy theory about each Dirac point, the Hamiltonian within each valley should commute with the generator of rotations within the 2D plane for that valley (the total angular momentum operator). In the valley basis, the total angular momentum is independent of the valley index and takes the form J z ¼ −i ħ∂f t0 s0 s0 +

ħ ðt s s + t0 s0 sz Þ, 2 0 z 0

(11)

where f is the azimuthal angle. The first term of Jz is simply the orbital angular momentum while the sz piece is the usual spin contribution. The sz term arises from spin-like sublattice DOF and is therefore characteristic of vdW materials. The requirement of ðyÞ Jz’s commutation with the Hamiltonian yields Dðx,yÞ ¼ oðx,yÞ ¼ lzxðyÞ ¼ 0 , lxðyÞ ¼ −lyðxÞ and lðxÞ x ¼ ly . One of such terms, a ðxÞ Dresselhaus-type SOC, lx t0 ðsx sx − sy sy Þ, breaks the mirror reflection symmetry about the yz plane, rendering atomic sites within the same sublattice inequivalent, and is thus forbidden in C3v invariant systems. The full C3v-invariant low-energy Hamiltonian in momentum space is therefore   HCk 3v ¼ ħvt0 sk s0 + msz + DKM t0 sz sz + lBR t0 sx sy − sy sx + lsv tz s0 sz , (12) where the above couplings have a simple correspondence to the spin-hoppings of the TB model (Eq. 8) DKM ¼

lA + lB , 2

lsv ¼

lA − lB , 2

lBR ¼ l :

(13)

The competition between the different spin–orbit energy scales in Eq. (12) influences the energy dispersion while also dictating the topological properties and the spin structure of the eigenstates. Kane–Mele SOC (DKMt0szsz) leads to spin-degenerate bands due to its spin-conserving nature, with the ability to drive the system into a topologically nontrivial phase when it dominates (Qian et al., 2014; Wu et al., 2018; Shi et al., 2019). However, in the honeycomb monolayer systems of interest here, including TMDs and graphene-based vdW heterostructures, the Kane–Mele SOC is negligible. In contrast, the spin–valley coupling (lsvtzs0sz), also known as valley–Zeeman interaction, emerging from sublattice-resolved SOC, is generally significant and yields spin-split bands within each valley. Typically, lsv is of order 100 meV for semiconducting TMDs (intrinsic SOC) (Xiao et al., 2012) and  1 meV for  graphene-TMD heterostructures (proximity-induced SOC) (Tiwari et al., 2022). The Bychkov–Rashba term, lBR t0 sx sy − sy sx , arising at heterostructure interfaces, leads to spin admixture in the spin states, characterized by in-plane spin-momentum locking of the Bloch eigenstates. Similar to spin–valley SOC, this term results in spin-split bands distinguished by spin helicity rather than simple spin. The magnitude of this coupling is typically on the order of 1–10 meV, depending on the TMD used (Tiwari et al., 2022; Wang et al., 2019). The presence of significant Rashba coupling explains the recently observed large charge-to-spin conversion via the ISGE at room temperature (Offidani et al., 2017), as discussed later on in this article. Finally, it is worth noting that the sublattice-resolved spin-flip terms (lAðBÞ nn ) (see Eq. 8) are absent in the low-energy Hamiltonian (i.e., they appear at the next order in the small-k expansion), and as such have little impact on the transport physics of 2D honeycomb layers. The dispersion relation associated with Eq. (12) consists of two pairs of spin-split Dirac bands. A typical energy dispersion relation for a graphene-TMD heterostructure with competing Rashba and spin–valley couplings is shown in Fig. 4a. The general electronic dispersion of C3v invariant systems can be readily seen as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez ðkÞ ¼  ħ2 v2 k2 + D2z ðkÞ, (14) where DKM has been neglected due to its inherently small nature compared to other SOC energy scales, k  |k|, z ¼ 1 is the spin-helicity index, and D2z ðkÞ is the SOC-dependent mass term, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   D2z ðkÞ ¼ m2 + l2sv + 2l2BR + 2z l2BR − mlsv + ħ2 v2 k2 l2BR + l2sv : (15) The spin texture of the eigenstates (see Fig. 4b) can be cast in the following compact form hsiztk ¼ −z RðkÞ ðk^  z^Þ+ mzzt ðkÞ z^,

(16)

where t ¼ 1 is the K valley quantum number. The first term describes the spin winding of the electronic states generated by the BR effect (Rashba, 2009). The second term is due to breaking of sublattice symmetry (lsv 6¼ 0 or m 6¼ 0) and tilts the spins in the z^direction. Because of the Dirac nature of the charge carriers, the spin texture has a strong dependence upon the Fermi energy. In fact, the spins point fully out of the plane at the Dirac point, but acquire an in-plane component as k is increased. For ħvk Dz, one finds rðkÞ ’ cos y and mzzt ðkÞ ’ sin y, with y ¼ −t arctanðlsv =lBR Þ. The tilting angle, y, has opposite signs in different valleys by virtue of time-reversal symmetry. Furthermore, one observes two distinct electronic regimes. For energies within the spin gap (regimes Ia and Ib in Fig. 4), the Fermi surface has a well-defined spin helicity, a feature reminiscent of spin–momentum locking in

Spintronics in 2D graphene-based van der Waals heterostructures

(a)

213

(b)

Hk

II

k

I

Fig. 4 (a) Energy dispersion near the Fermi level. Symmetry breaking SOC opens a spin gap. A small Mexican hat develops in the regime I due to the interplay of spin–valley coupling and Rashba SOC. (b) Tangential winding of the spin texture in regimes I (3D view) and II (top view).

topologically protected surface states (Schwab et al., 2011). Consequently, near-optimal charge-to-spin conversion is observed inside the spin gap via a large ISGE. In contrast, for energies outside the spin gap (regime II in Fig. 4), the spin helicity is no longer a well-defined concept, but the larger Fermi radius of the spin-majority band (z ¼ −1) nevertheless allows for a detectable ISGE (i.e., while the current-induced spin polarization arising from the two subbands have opposite signs, they do not cancel each other). These special features of the electronic and spin structure of proximitized 2D materials are ultimately responsible for the efficient current-driven spin polarization supported by graphene-TMD heterostructures (Offidani et al., 2017).

Relativistic spin–orbit coupled transport phenomena Electronic signatures As alluded to above, the presence of SOC in a material can drastically alter the transport properties of materials due to spin-charge inter–conversion effects. One of the earliest signatures of SOC-affected transport was measured in magnesium films (i.e., 2DEGs) (Bergman, 1982; Sharvin and Sharvin, 1981), where the electrical conductivity was seen to decrease upon the application of a magnetic field (i.e., a negative magneto-conductivity). This behavior can be traced back to the quantum interference between the many paths an electron can take when traveling through a disordered system. In the case of a material where SOC is absent, quantum interference leads to the coherent backscattering of electrons, thus yielding more localized states, and hence reduces the electrical conductivity in a phenomenon known as weak localization (WL). The application of a magnetic field here destroys the coherence of the backscattered electrons and hence inhibits WL. Therefore, a positive magneto-conductivity is a clear signature of WL. In contrast, materials that are SOC-active may exhibit the complete opposite of WL, weak antilocalization (WAL). In this case, the paths of backscattered electrons interfere destructively to yield more delocalized states. This reversal in behavior can be understood to result from the spin DOF becoming an active participant in the Hamiltonian and scattering events. Here, the application of a magnetic field changes the interference of backscattered electrons from destructive to constructive, therefore decreasing the conductivity of a material. The natural signature of SOC-active materials is therefore a negative magneto-conductivity. The role of localization in graphene, however, is more involved due to the presence of other spin-like DOFs, namely, the sublattice and valley DOFs. In the absence of both SOC and inter-valley scattering (i.e., disorder is smooth on the lattice scale), graphene exhibits a WAL phase, the complete opposite of what is seen in 2DEGs. The manifestation of WAL in this system is a result of graphene’s p Berry phase, which precludes electrons from backscattering and thus prevents WL. Another interpretation can be found in considering the pseudospin (sublattice) DOF as an active participant in the Hamiltonian and scattering events, much in a similar manner to how spin becomes an active participant in SOC-active 2DEGs. Upon increasing the concentration of point defects and other short-range scatterers, such that inter-valley scattering no longer remains negligible, the disordered graphene system’s localization phase reverses to exhibit a WL phase instead. Likewise, keeping intervalley scattering weak (i.e., intravalley scattering dominates) and instead introducing a strong SOC also pushes graphene from a WAL to a WL phase. Finally, when both disorder and SOC are strong, graphene moves back to WAL behavior (Sousa et al., 2022). Several experiments on graphene-TMD and bilayer graphene-TMD have revealed WAL phases in these systems (Wang et al., 2016; Völkl et al., 2017; Yang et al., 2017; Wakamura et al., 2018; Amann et al., 2022), indicating the presence of strong symmetry-breaking SOC. However, such observations do not provide a direct spectroscopic probe for the size of the various SOCs present. Only recently has the SOC of these systems been accessed directly (Tiwari et al., 2022; Wang et al., 2019).

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Spintronics in 2D graphene-based van der Waals heterostructures

Spin Hall effect While SOC might affect the electrical conductivity of graphene-based vdW heterostructures through quantum interference, more profound and exotic behavior can be observed when considering other forms of transport. The SHE is one such example of this, whereby the generation of a pure spin current driven solely by electric fields can be achieved without the need for magnetic fields. In pristine graphene—owing to its negligible intrinsic SOC (Sichau et al., 2019)—the SHE is absent. However, this scenario changes drastically in the presence of disorder-induced SOC (Ferreira et al., 2014; Balakrishnan et al., 2014; Milletarì and Ferreira, 2016). Here, the scattering of charge carriers from spin–orbit hot spots aligns spin and orbital angular momentum in opposite directions, resulting in the formation of transverse spin currents; the magnitude of the effect is characterized by the spin Hall angle, g. A direct consequence of the SHE in low-dimensional systems is the accumulation of spin at the system’s boundaries. Specifically, thin films (2D materials) will accrue a collection of oppositely aligned spins at opposite edges, see Fig. 5, due to the resulting spin current pumping up spins toward one boundary and down spins toward the opposite boundary. In thin wires (1D materials), the spin accumulation can be seen to wind around the wire’s surface, much in a similar manner to how a magnetic field appears around a wire carrying an electrical current. Reversing the direction of the applied electrical current reverses the spin accumulation in both cases mentioned here; the up and down spins now accumulate at the opposite boundary to which they had originally found themselves in thin films, and the direction of winding is reversed in thin wires. Broadly speaking, the SHE arises in two forms: intrinsically and extrinsically. The former of these two is a direct result of the material’s band structure, as opposed to being reliant upon external perturbations or extrinsic factors, such as disorder. This manifestation of the SHE can be understood in terms of an internal force experienced by the electrons traveling through the material due to the application of a charge current. This internal force is generated by the SOC experienced by the electrons, and hence is an effect embedded within the band structure of the system. An analogy can be drawn between this spin–orbit force driving the SHE and the Lorentz force responsible for the regular Hall effect. In contrast to the intrinsic SHE is the extrinsic contribution. Here, the generation of a spin current is driven by external mechanisms such as scattering from impurities. In this case, the SHE is the result of asymmetric scattering of electrons based upon their spin; up spins are scattered with a certain directional preference while down spins are scattered with the opposite preference. This type of asymmetric scattering, known as skew scattering, plays a central role in graphene-based vdW heterostructures (Milletarì et al., 2017). To illustrate the importance of disorder-induced corrections to transport for these materials, consider the Hamiltonian for graphene with just Rashba coupling (i.e., lsv ¼ 0 and neglecting DKM, given its small nature in most realistic scenarios). The carrier concentration is typically large enough to place the Fermi energy well above the Rashba pseudogap (i.e., in regime II of the band structure). Assuming only nonmagnetic disorder is present, the system clearly lacks a well-defined quantization axis due to the lack of spin-valley coupling. Therefore, the SHE should not be observable simply because some form of SOC is present. With the inclusion of a spin–valley coupling, it would then be natural to expect the possible occurrence an SHE as a quantization axis will now be well-defined. However, if a theoretical analysis of the SHE for the minimal model without including disorder self-consistently is performed, it would lead one to believe that a finite SHE would exist in the absence of spin–valley coupling. This clearly emphasizes the importance of including disorder in a fully self-consistent manner (Milletarì et al., 2017; Milletarì and Ferreira, 2016). Within the Kubo-Streda formalism of linear response (Streda, 1982; Crépieux and Bruno, 2001), the spin Hall conductivity may sH;I sH;II be written as ssH yx ¼ syx + syx , with (Milletarì and Ferreira, 2016; Milletarì et al., 2017) Z io df n h z + ~ − i 1 h z + ~ + 1 +1 (17a) ssH;I Tr J y Go jx Go − Tr J y Go jx Go + J zy G−o~jx G−o , do yx ¼ − 2p 2 do −1

Fig. 5 Left: Spin textures of spin-majority bands in each valley. The net out-of-plane spin polarization at each valley defines a quantization axis. Right: Spin accumulation at the edges of a thin film due to the SHE. Black arrow: electrical current, orange arrow: spin current, red/blue arrows: up/down spins. The electrons driven by the electrical current are scattered asymmetrically based upon their spin (skew scattering). Here, up spin electrons are scattered preferentially to the left while down spins experience the opposite.

Spintronics in 2D graphene-based van der Waals heterostructures ssH;II ¼ yx

1 2p

Z

+1

−1



∂G+ ∂G+ dof ðoÞRe Tr J zy G+o~jx o − J zy o ~jx G+o , ∂o ∂o

215

(17b)

where J zy is the z-polarized spin current operator in the y-direction, ~jx is the x component of the disorder-renormalized electric current operator, G o are the retarded or advanced disorder-averaged Green’s functions for electrons with energy o, f(o) is the FermiDirac distribution, and the trace is over momentum and all internal DOFs (spin, sublattice, and valley). Given the applied electric field, E, generating an electrical current (assumed to be along the x-axis), the resulting spin Hall current can then be determined sH using J sH y ¼ syx Ex . sH;II The first term, ssH;I yx , is sometimes referred to as the Fermi surface contribution while the second term, syx , is similarly called the Fermi sea contribution. In weakly disordered systems, the type II contribution is higher order in the impurity density and hence may be neglected. Similar reasoning applies to the G+G+ and G−G− terms in the type I contribution. Hence, it turns out that the leading order behavior is dictated solely by the cross term of ssH;I yx . Clearly, the SHE in the metallic regime of disordered materials is essentially a Fermi surface property. A common representation that helps to visualize how disorder is included into linear response is that of Feynman diagrams. Fig. 6 shows the dominant cross term in diagrammatic form, alongside the Dyson series describing the disorder-averaged Green’s functions and the Bethe–Salpeter equation satisfied by the disorder-renormalized vertex. Note, the discussions and analysis here assume that only nonmagnetic (spin-independent) scalar disorder is present. The dashed lines represent an electron (solid line) is evaluated without vertex corrections for the lsv ¼ 0 case, one finds a scattering from an impurity located at the cross. If ssH;I yx nonvanishing result in complete contradiction to what is expected. However, including vertex corrections to any order in the number of scattering events, ssH;I yx can be seen to vanish, thus recovering the expected result for zero spin–valley coupling. Interestingly, if one were to consider a Fermi energy located within the Rashba pseudogap (regime I of the band structure, |e| < 2|lBR|), they would also find a vanishing SHE in the absence of lsv. However, in this case, the type II contribution would no longer be subleading order and hence also needs accounting for (the cross term of the type I contribution remains the dominant part of ssH;I yx ). With this in mind, the computation of Eq. (17) yields (Milletarì et al., 2017) ssH;I yx ¼

e jej ¼ −ssH;II yx : 16p lBR

(18)

Hence, ssH yx ¼ 0 and so the SHE remains absent even in regime I due to the Fermi surface contribution being counteracted by off-surface processes. Shifting focus to the experimentally relevant situation in which lBR 6¼ 0 and lsv 6¼ 0 while the SHE is expected to be observable (due to emergence of an effective spin quantization axis around each valley, see Fig. 5), the role played by disorder changes drastically. If disorder is only accounted for within the Born approximation, where all processes involving three or more scatterings from a single impurity are neglected, then one finds that Eq. (17a) yields a vanishing result. Only upon the inclusion of at least third-order scattering events (at least three scatterings from a single impurity) into the vertex correction does one find a nonzero value for ssH;I yx . By accounting for these higher order processes, a scattering event can now distinguish between left and right based upon the electron’s spin along the well-defined quantization axis courtesy of the spin–valley coupling, that is, skew scattering has been included. Therefore, not only are vertex corrections key to understanding the SHE completely, but so too are the higher order scattering mechanisms allowing for the manifestation of the SHE. In other words, the extrinsic SHE due to scalar disorder effects in graphene-based vdW heterostructures is controlled entirely by skew scattering, which is always active provided the existence of a tilted BR-type spin texture at the Fermi level. As a final note on intrinsic effects, couplings such as electron–phonon and electron–electron interactions, as well as structural defects such as ripples and shears in the individual layers of the heterostructure, are also considered as being intrinsic to the system.

Fig. 6 Diagrammatic representation of the cross term in ssH;I yx (a), the disorder-average Green’s function (b), and vertex correction due to disorder (c). The thick red/blue lines denote retarded/advanced disorder-averaged Green’s functions describing electron propagation, thick orange lines can be either retarded/advanced disorder-averaged electron Green’s functions but must all be of the same type, thin orange lines indicate the free electron Green’s functions, the black circles represent the current and spin current operators (vertices), black crosses represent an impurity, and the orange dashed lines denote an electron scattering from the impurity.

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Spintronics in 2D graphene-based van der Waals heterostructures

Consequently, the intrinsic SHE can potentially play an important role in clean systems, where electron–phonon coupling dominates at room temperature, or in materials with sharp boundaries between structural domains. Just how major these effects are is still a current area of study; hence they shall not be covered in this article.

Inverse spin galvanic effect A natural partner to the SHE is the ISGE: the accumulation of spin upon application of an electric field without an associated spin current. Unlike the SHE, this spin accumulation manifests throughout the whole system. Rather, a nontrivial spin texture, facilitated by the interfacial breaking of inversion symmetry, is enough to allow for a nonzero spin polarization when an electrical current is passed through the system. To demonstrate this, consider, once again, the minimal Dirac–Rashba Hamiltonian (only lBR 6¼ 0). It was shown above that the electron spins are locked in-plane and perpendicular to the electron momentum. In the absence of an electric field (i.e., in equilibrium), the Fermi rings forming the Fermi surface (two in regime II and Ia and one in regime Ib) are perfect circles around each Dirac point. Consequently, the electrons forming these rings yield a net spin polarization of zero (as expected of the nonmagnetic materials discussed here). However, when an electric field is introduced, these Fermi rings are shifted such that they are no longer rotationally symmetric about their respective Dirac points, see Fig. 7. Therefore, the sum of the electron spins from each Fermi ring yields a nonzero spin polarization. The direction of the resulting spin polarization is entirely dependent upon the system’s spin texture. In the case of the Dirac–Rashba model above, the ISGE yields a spin polarization that is perpendicular to the applied electric field. The inclusion of a spin–valley coupling does not change the resulting spin polarization compared to the Dirac–Rashba model (the Onsagerreciprocal phenomena, namely, the SHE and ISGE, were measured in Camosi et al., 2022). This is due to the out-of-plane component gained by the electron spins being opposite in sign between valleys (i.e.,  in the K valleys). Hence, summing over the contribution from both valleys, one finds a vanishing z component of spin accumulation. The role of the Kane–Mele-type SOC in these systems is negligible and hence does not contribute any meaningful changes to the spin texture either. An alternative method to manipulating the spin texture of these materials has been found through the introduction of a twist between a graphene monolayer and a TMD monolayer (Li and Koshino, 2019; David et al., 2019; Péterfalvi et al., 2022; Veneri et al., 2022). By introducing a rotational off-set between the two layers, both the Rashba and spin–valley couplings are affected and acquire a twist-angle, y, dependence. Not only are the magnitudes of these SOCs changed by twisting, but so too is the form of the Rashba term in the Hamiltonian. Upon the introduction of twisting, the spin–valley term maintains the same form as in Eq. (12) but with lsv replaced by l~sv ðyÞ (l~sv ð0Þ ¼ lsv ), and the Rashba term becomes HR ðyÞ ¼ l~BR ðyÞeisz aR ðyÞ=2 ðsx sy − sy sx Þe −isz aR ðyÞ=2 ,

(19)

where l~BR ð0Þ ¼ lBR, and aR(y) is known as the Rashba phase. The exact way in which the Rashba coupling, spin–valley coupling, and Rashba phase vary with twist-angle depends heavily upon the material that graphene is paired with. A guaranteed property, however, is that aR(y) ¼ c1p for y ¼ c2p/6 for c1 , c2 2 . In any case, the spin texture will clearly be affected by the change in the Rashba coupling term’s form. In fact, as the layers are twisted relative to one another, the electron spins can be seen to also rotate away from being locked perpendicular to their momenta. This rotation of the spin texture can be seen in Fig. 8, where the out-ofplane component due to spin–valley coupling has been neglected for ease of illustration. It turns out that, for some materials, there exists a critical angle, yc, where the spin texture is entirely radial (Fig. 8c), sometimes referred to as a Weyl-type or hedgehog spin texture. Clearly, by being the perpendicular analog of the untwisted case, the resulting spin polarization will be perfectly collinear to the applied electric field. For any twist-angles away from y ¼ 0,  yc, p/6, the net spin polarization will be in-plane but neither perpendicular nor collinear to an applied electrical current. The value of yc is sensitive to the partner TMD used, atomic registry, strain distribution, and external perturbations (e.g., perpendicular electric field, Naimer et al., 2021), and hence will vary significantly between materials. Consequently, meaningful predictions for yc are challenging to make. For example, graphene-WSe2

Fig. 7 The manifestation of a nonequilibrium spin polarization with the application of an electric field is pictured above for a system with only Rashba SOC. Focus is placed on a single Fermi ring for ease of illustration. Left: Fermi ring for a system with zero current. The sum of electron spins (orange arrows) yields zero by symmetry. Right: an applied electric field shifts the Fermi ring from its original position (gray), with the electron spins on the Fermi surface rotating to maintain the spin–momentum locking typical of Rashba SOC, resulting in a nonzero spin polarization (red arrow). By inspection, all spins, except those on the x-axis, can be seen to rotate toward alignment with the y-axis. As a result, the net spin polarization is parallel to the y-axis in this case.

Spintronics in 2D graphene-based van der Waals heterostructures

217

Fig. 8 Introduction of a twist between the layers of a graphene-TMD vdW heterostructure can be seen to rotate the spin texture of the Fermi rings. (a) y ¼ 0, all spin are locked in-plane and perpendicular to the momentum. (b) At arbitrary twist angles, y 6¼ c2p/6, yc, the electron spin are still fixed in-plane but not perpendicular of (anti-)parallel to the electron momentum. (c) At critical twist angles, y ¼ yc, the electron spins (anti-)align with their momentum.

has been predicted to host a yc  14 (this prediction is based on an 11-band tight-binding model of the twisted heterostructures informed by both density functional theory calculations (Fang et al., 2015; Gmitra et al., 2016) and available angle-resolved photoemission data (Pierucci et al., 2016; Nakamura et al., 2020). In contrast, even the existence of a critical twist angle for graphene-MoS2 is difficult to ascertain given the variation in the reported material parameters of theory and experiment (Péterfalvi et al., 2022). Defects and structural disorder are also expected to play a significant role. Of particular relevance is twist-angle disorder, a type of spatial inhomogeneity that is ubiquitous in realistic systems (Uri et al., 2020). Its impact on transport properties is expected to depend crucially on the comparative sizes of the spin diffusion length, ls, and the twist-puddle size, x (i.e., typical size of regions with a similar twist-angle). When ls  x, the twist-angle disorder can be considered smooth and hence can be incorporated into linearresponse calculations through an appropriate averaging procedure (e.g., a Gaussian or box weighting could be applied, Veneri et al., 2022). Though it is challenging to make first-principles predictions about the twist-angle behavior of the Rashba phase for realistic systems, the dependence of coupled spin-charge transport phenomena on the Rashba phase can be reliably studied by replacing the standard Rashba SOC in the low-energy Hamiltonian of Eq. (12) with Eq. (19). In this case, the point group symmetry of the system, and therefore Hamiltonian, is once more reduced, going from C3v to C3 for nontrivial twist angles. To understand this from a quantitative perspective, one notes that the ISGE can be written mathematically as Sa ¼ KabEb (assuming Einstein summation), where Kab are the elements of the spin susceptibility tensor. As in the SHE, the spin susceptibility tensor can be determined using linear response theory (Milletarì and Ferreira, 2016; Milletarì et al., 2017), K ab ¼ K Iab + K IIab , Z o df n + ~ −  1 + ~ + 1 +1 (20a) K Iab ¼ − Tr sa Go jx Go − Tr sa Go jx Go + sa G−o~jx G−o , do 4p −1 2 do 

Z ∂G+ ∂G+ 1 +1 K IIab ¼ (20b) dof ðoÞRe Tr sa G+o~jb o − sa o ~jx G+o , ∂o ∂o 4p −1 where the Green’s functions now contain the twisted form of the Rashba Hamiltonian. For disordered materials the G+G+ and G−G− of the type I contribution, as well as the type II contribution, can once again be neglected. Consequently, the results for the in-plane components of the twisted spin susceptibility tensor can be related back to the untwisted ISGE, albeit with modified Rashba and spin–valley couplings, K xx ðl~BR ðyÞ, l~sv ðyÞ; yÞ ¼ K yx ðl~BR ðyÞ, lsv ðyÞ; 0Þ sin aR ðyÞ, K yx ðl~BR ðyÞ, l~sv ðyÞ; yÞ ¼ K yx ðl~BR ðyÞ, lsv ðyÞ; 0Þ cos aR ðyÞ,

(21)

l~3BR ðe2 + l~2sv Þ 4eve K yx ðl~BR , l~sv ; 0Þ ¼ , 2 2 pnu0 e4 ðl~BR + l~2sv Þ − e2 l~4sv + 3l~2BR l~4sv

(22)

where n is the impurity concentration and u0 is the strength of the scalar impurities. These results are strictly valid for perfectly aligned heterostructures. For systems with smooth twist-disorder landscapes (ls  x), an intuitive approximate result can be obtained by taking the convolution of Eq. (21) with a suitable twist-disorder distribution function (Veneri et al., 2022). For twist

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angles y 6¼ yc, c2p/6, both in-plane components of the spin susceptibility tensor will be nonvanishing, thus yielding a nontrivial polarization (Sx, Sy 6¼ 0). At the critical twist angle, the Rashba phase must vanish (aR(yc) ¼ 0), and hence a collinear Edelstein effect (CEE) can be achieved, whereby the resulting spin polarization will be purely (anti-)parallel to the applied electrical current (c.f. Fig. 8c). What makes these twisted graphene-TMD vdW heterostructures appealing is the ease with which the SOC and spin texture can be manipulated; only a simple twist is needed to adjust them. This sets these systems apart from 2DEGs, where both Rashba-type and Dresselhaus-type SOC are required (Trushin and Schliemann, 2007). Control of these SOCs in 2DEGs is achieved via asymmetric doping and the tuning of quantum well widths (Ganichev and Golub, 2014), a set of processes far more complicated and involved than simple twisting. For further details, the reader is referred to Veneri et al. (2022) where the CEE phenomenon was predicted.

Observing spin–charge interconversion To make use of graphene’s large ls while also studying the effects of proximity-induced SOC, spin-valve setups are typically used to measure the ISHE and SGE (the inverse SHE and spin galvanic effect, respectively). In this case, rather than trying to measure a spin current using an FM contact, an electrical current is measured instead by converting an injected spin current into an electrical current (Valenzuela, 2009). A schematic of a spin-valve experiment is shown in Fig. 9. Spin valves operate by using a single FM contact to inject a spin current into graphene. This spin current is then able to flow along the isolated graphene channel until it reaches a T-junction, where it enters a region of high SOC. This region is no longer characterized by just monolayer graphene; instead, it now contains graphene layered on top of a TMD. As a result, the spin current entering this region undergoes the ISHE (due to any out-of-plane spin components) and generates an electrical current perpendicular to the injected spin current. Likewise, the electrons entering the high SOC region naturally also have a component of in-plane spin polarization and so are subject to the SGE, which also generates a perpendicular electrical current. Specifically, the component of the electron’s spin parallel to the spin current generates an electrical current via the SGE while the component of spin in-plane and perpendicular to the spin current yields an electrical current via the collinear SGE (if the material permits this process). The ensuing nonlocal resistance measured across the T-junction is therefore characterized by the spin–charge interconversion effects at the heart of modern spintronics. In order to distinguish between the different electrical signals arising from the ISHE, SGE, and collinear SGE, a combination of magnetic fields must be used in conjunction with various FM orientations. The effect of applying a magnetic field to this setup is to cause spin precession about the applied field. The strength of the measured resistance in the T-junction will then depend on the magnetic field strength (how quickly the spins precess), and the length of the graphene channel (how long the spins have to precess). By combining the spin-valve measurements for various magnetic fields and FM orientations, the electrical signals associated to each of the aforementioned proximity-induced ISHE, and SGE can be isolated by means of a simple symmetry analysis (Cavill et al., 2020). A set of recent spin-valve experiments have revealed graphene-TMD heterostructures to yield room-temperature nonlocal resistances of order 1–10 mO due to the ISHE and SGE (Safeer et al., 2019a; Ghiasi et al., 2019; Benítez et al., 2020). Alternatively, the Onsager-reciprocal phenomena, namely, the SHE and ISGE, can be discerned by measuring the spin accumulation in the direction of the spin current at opposite sides of the high SOC region, as was done in the experiment of Camosi et al. (2022). The latter approach requires a complex multiterminal architecture but has the advantage that it permits isolation of the SHE and ISGE in situations for which the TMD is conducting and thus directly influences the spin–charge conversion processes (due to its high intrinsic SOC). Additionally, spin-valve measurements allow us to distinguish between conventional and anisotropic spin-charge conversion processes. An anisotropic SGE has been recently observed in graphene proximity-coupled to semimetallic (low-symmetry) TMDs (Safeer et al., 2019b). Likewise, an anisotropic ISGE characterized by the presence of spin polarization components parallel and orthogonal to the driving current has been reported experimentally in Camosi et al. (2022). However, an experimental demonstration exploiting twist-angle control in a graphene-semiconducting TMD heterostructures as described by Eq. (21) has yet to be achieved.

Fig. 9 Graphene-based spin valve setup. An electrical current, I, passed through a ferromagnetic contact (blue box) injects a spin current into a graphene sheet. This spin current then travels along a graphene channel and into a region of enhanced SOC (dashed red region), due to proximity with a TMD (red box). This generates a charge accumulation in the arms of the T-junction due to the ISHE and SGE, which yields a nonlocal voltage in the steady state, Vnl, across the bar under the open circuit conditions.

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219

While spin-valve setups are ideally suited for studies of spin dynamics and spin–charge interconversion processes, they do not allow one to discern the size of the SOC present in a graphene-on-TMD heterostructure. To do this, measurement of the Shubnikov–de Haas (SdH) oscillations must be made. The reconstruction of the low-energy electronic structure from SdH data qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi allows for the determination of the average SOC, l ¼ l2BR + l2sv (Tiwari et al., 2022). Experiments have revealed that the typical SOC present in these materials is l ’ 2:51 meV. This average value is in accord to the predictions of microscopic theories for vdW heterostructures (Milletarì et al., 2017; Offidani et al., 2017) regarding the observation of significant spin–charge interconversion efficiencies at room temperature, and is indeed compatible with the spin-valve measurements (Safeer et al., 2019a; Ghiasi et al., 2019; Benítez et al., 2020; Camosi et al., 2022; Li et al., 2020; Hoque et al., 2021), thus demonstrating that proximity-induced SOC is responsible for the observed nonlocal resistances.

Conclusion This article has presented the role of graphene in modern spintronic devices, with a specific emphasis on how its partnership with other 2D materials can allow for bespoke systems with desirable electronic and spin transport properties. By constructing a lowenergy theory of graphene, it becomes clear that transport phenomena in graphene-based systems will be dominated by the behavior of electrons around the Dirac points. From a physical point of view, one of the most striking features of isolated graphene is the description of its electrons as massless chiral Dirac fermions. Furthermore, the decoherence of spins in graphene occurs over large distances, allowing for long-range spin transport, a direct result of the extremely weak intrinsic SOC (Kane–Mele) appearing naturally in graphene. However, despite its ability to host long-range spin diffusion, graphene does not have a natural mechanism allowing for distinct spin control. This can ideally be achieved by pairing it with other materials that enhance its SOC. In particular, the use of TMDs proximity-coupled to graphene gives rise to significant SOCs, of the order of meV, which allows meaningful spin currents and nonequilibrium spin polarizations, via the SHE and ISGE, respectively, to manifest in the graphene layer. Consequently, electronic control of spin transport can be easily achieved and hence amalgamates well with current technological architectures. One major focus of spintronics is the implementation of the SHE and ISGE to generate large spin accumulations and polarizations that can be used to change the magnetization of a ferromagnet. In this case, the magnetization experiences a torque due to the SOC-related phenomena and so is referred to as a spin-orbit torque (SOT). The use of SOTs in technology could allow for the realization of SOT-based magnetic random access memory (SOT-MRAM), removing our reliance upon volatile memory and thus reducing energy consumption. This route away from traditional RAM appears to bear much promise and remains as a current area of interest in research with many facets and subtleties still requiring further study. As a final note, while not covered in this article, interface-induced magnetic exchange interaction is also becoming increasingly important in the field (e.g., as means to allow the manifestation of quantum anomalous Hall phases and spin-dependent Seebeck effect in graphene (Offidani and Ferreira, 2018; Ghiasi et al., 2021)), and the reader is referred to the literature for further details. The same goes for high-temperature topologically nontrivial phases of matter—exhibiting technologically relevant phenomena like the quantum spin Hall effect (Wu et al., 2018)—and the discovery of 2D vdW magnets (Lee et al., 2016; Gong et al., 2017; Huang et al., 2017), which are likely to open up interesting avenues across many emergent fields, including topological quantum computation, spin-orbitronics, magnonics, and antiferromagnetic spintronics (Freedman et al., 2003; Dolui et al., 2020; Xing et al., 2019; Xiao and Tong, 2022).

Acknowledgments The authors acknowledge support from the Royal Society through Grants No. URF/R/191021 (A.F.) and No. RF/ERE/210281 (A.F. and D.T.S.P.). We are indebted to Yue Wang and Robert A. Smith for helpful comments on the manuscript.

References Amann J, Völkl T, Rockinger T, Kochan D, Watanabe K, Taniguchi T, Fabian J, Weiss D, and Eroms J (2022) Counterintuitive gate dependence of weak antilocalization in bilayer graphene/WSe2 heterostructures. Physical Review B 105(11): 115425. https://doi.org/10.1103/PhysRevB.105.115425. Aronov AG and Lyanda-Geller YB (1989) Nuclear electric resonance and orientation of carrier spins by an electric field. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 50(9): 398–400 [JETP Letters 50: 431–434 (1989)]. Aronov AG, Lyanda-Geller YB, and Pikus GE (1991) Spin polarization of electrons by an electric current. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 100(3): 973–981 [Soviet Physics–JETP 73: 537–541 (1991)]. Baibich MN, Broto JM, Fert A, Van Dau FN, Petroff F, Etienne P, Creuzet G, Friederich A, and Chazelas J (1988) Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Physical Review Letters 61(21): 2472–2475. https://doi.org/10.1103/PhysRevLett.61.2472. Balakrishnan J, Koon GKW, Avsar A, Ho Y, Lee JH, Jaiswal M, Baeck S-J, Ahn J-H, Ferreira A, Cazalilla MA, Castro Neto AH, and Özyilmaz B (2014) Giant spin Hall effect in graphene grown by chemical vapour deposition. Nature Communications 5(1): 4748. https://doi.org/10.1038/ncomms5748. Benítez LA, Torres WS, Sierra JF, Timmermans M, Garcia JH, Roche S, Costache MV, and Valenzuela SO (2020) Tunable room-temperature spin galvanic and spin Hall effects in van der Waals heterostructures. Nature Materials 19(2): 170–175. https://doi.org/10.1038/s41563-019-0575-1. Bergman G (1982) Influence of spin-orbit coupling on weak localization. Physical Review Letters 48(15): 1046–1049. https://doi.org/10.1103/PhysRevLett.48.1046.

220

Spintronics in 2D graphene-based van der Waals heterostructures

Binasch G, Grünberg P, Saurenbach F, and Zinn W (1989) Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange. Physical Review B 39(7): 4828–4830. https://doi.org/10.1103/PhysRevB.39.4828. Bychkov YA and Rashba EI (1984) Properties of a 2D electron gas with lifted spectral degeneracy. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 39(2): 66–69 [JETP Letters 39: 78–81 (1984)]. Camosi L, Josef Svetlík J, Costache MV, Torres WS, Aguirre IF, Marinova V, Dimitrov D, Gospodinov M, Sierra JF, and Valenzuela SO (2022) Resolving spin currents and spin densities generated by charge-spin interconversion in systems with reduced crystal symmetry. 2D Materials 9(3): 035014. https://doi.org/10.1088/2053-1583/ac6fec. Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, and Jarillo-Herrero P (2018) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556 (7699): 43–50. https://doi.org/10.1038/nature26160. Castro EV, Novoselov KS, Morozov SV, Peres NMR, Lopes dos Santos JMB, Nilsson J, Guinea F, Geim AK, and Castro Neto AH (2007) Biased bilayer graphene: Semiconductor with a gap tunable by the electric field effect. Physical Review Letters 99(21): 216802. https://doi.org/10.1103/PhysRevLett.99.216802. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, and Geim AK (2009) The electronic properties of graphene. Reviews of Modern Physics 81(1): 109–162. https://doi.org/ 10.1103/RevModPhys.81.109. Cavill SA, Huang C, Offidani M, Lin Y-H, Cazalilla MA, and Ferreira A (2020) Proposal for unambiguous electrical detection of spin-charge conversion in lateral spin valves. Physical Review Letters 124(23): 236803. https://doi.org/10.1103/PhysRevLett.124.236803. Crépieux A and Bruno P (2001) Theory of the anomalous Hall effect from the kubo formula and the Dirac equation. Physical Review B 64(1): 014416. https://doi.org/10.1103/ PhysRevB.64.014416. David A, Rakyta P, Kormányos A, and Burkard G (2019) Induced spin-orbit coupling in twisted graphene-transition metal dichalcogenide heterobilayers: Twistronics meets spintronics. Physical Review B 100(8): 085412. https://doi.org/10.1103/PhysRevB.100.085412. Dean CR, Young AF, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard KL, and Hone J (2010) Boron nitride substrates for high-quality graphene electronics. Nature Nanotechnology 5(10): 722–726. https://doi.org/10.1038/nnano.2010.172. Dolui K, Petrovic MD, Zollner K, Plechác P, Fabian J, and Nikolic BK (2020) Proximity spin-orbit torque on a two-dimensional magnet within van der Waals heterostructure: Currentdriven antiferromagnet-to-ferromagnet reversible nonequilibrium phase transition in bilayer CrI3. Nano Letters 20(4): 2288–2295. https://doi.org/10.1021/acs.nanolett.9b04556. Drögeler M, Volmer F, Wolter M, Terrés B, Watanabe K, Taniguchi T, Güntherodt G, Stampfer C, and Beschoten B (2014) Nanosecond spin lifetimes in single- and few-layer graphenehBN heterostructures at room temperature. Nano Letters 14(11): 6050–6055. https://doi.org/10.1021/nl501278c. Dubois S, Piraux L, George JM, Ounadjela K, Duvail JL, and Fert A (1999) Evidence for a short spin diffusion length in permalloy from the giant magnetoresistance of multilayered nanowires. Physical Review B 60(1): 477–484. https://doi.org/10.1103/PhysRevB.60.477. Dyakonov MI and Perel VI (1971a) Possibility of orienting electrons spins with current. ZhETF Pis. Red. 13(11): 657–660 [Soviet Physics Journal of Experimental and Theoretical Physics Letters 13: 467–469 (1971)]. Dyakonov MI and Perel VI (1971b) Current-induced spin orientation of electrons in semiconductors. Physics Letters A 35(6): 459–460. https://doi.org/10.1016/0375-9601(71) 90196-4. Eda G, Fujita T, Yamaguchi H, Voiry D, Chen M, and Chhowalla M (2012) Coherent atomic and electronic heterostructures of single-layer MoS2. ACS Nano 6(8): 7311–7317. https:// doi.org/10.1021/nn302422x. Edelstein VM (1990) Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Communications 73(3): 233–235. https://doi.org/10.1016/0038-1098(90)90963-C. Fang S, Kuate Defo R, Shirodkar SN, Lieu S, Tritsaris GA, and Kaxiras E (2015) Ab initio tight-binding hamiltonian for transition metal dichalcogenides. Physical Review B 92(20): 205108. https://doi.org/10.1103/PhysRevB.92.205108. Ferreira A, Rappoport TG, Cazalilla MA, and Castro Neto AH (2014) Extrinsic spin Hall effect induced by resonant skew scattering in graphene. Physical Review Letters 112(6): 066601. https://doi.org/10.1103/PhysRevLett.112.066601. Freedman MH, Kitaev A, Larsen MJ, and Wang Z (2003) Topological quantum computation. Bulletin of the American Mathematical Society 40: 31–38. https://doi.org/10.1090/ S0273-0979-02-00964-3. Fulop B, Marffy A, Zihlmann S, Gmitra M, Tovari E, Szentpeteri B, Kedves M, Watanabe K, Taniguchi T, Fabian J, Schonenberger C, Makk P, and Csonka S (2021) Boosting proximity spin-orbit coupling in graphene-WSe2 heterostructures via hydrostatic pressure. npj 2D Materials and Applications 5: 82. https://doi.org/10.1038/s41699-021-00262-9. Ganichev SD and Golub LE (2014) Interplay of Rashba/Dresselhaus spin splittings probed by photogalvanic spectroscopy—A review. Physica Status Solidi B 251(9): 1801–1823. https://doi.org/10.1002/pssb.201350261. Ganichev SD, Ivchenko EL, Bel’kov VV, Tarasenko SA, Sollinger M, Weiss D, Wegscheider W, and Prettl W (2002) Spin-galvanic effect. Nature 417(6885): 153–156. https://doi.org/ 10.1038/417153a. Geim AK and Grigorieva IV (2013) Van der Waals heterostructures. Nature 499(7459): 419–425. https://doi.org/10.1038/nature12385. Ghiasi TS, Kaverzin AA, Blah PJ, and van Wees BJ (2019) Charge-to-spin conversion by the Rashba-Edelstein effect in two-dimensional van der Waals heterostructures up to room temperature. Nano Letters 19(9): 5959–5966. https://doi.org/10.1021/acs.nanolett.9b01611. Ghiasi TS, Kaverzin AA, Dismukes AH, de Wal DK, Roy X, and van Wees BJ (2021) Electrical and thermal generation of spin currents by magnetic bilayer graphene. Nature Nanotechnology 16(7): 788–794. https://doi.org/10.1038/s41565-021-00887-3. Gmitra M, Kochan D, Högl P, and Fabian J (2016) Trivial and inverted Dirac bands and the emergence of quantum spin Hall states in graphene on transition-metal dichalcogenides. Physical Review B 93(15): 155104. https://doi.org/10.1103/PhysRevB.93.155104. Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu ZQ, Cava RJ, Louie SG, Xia J, and Zhang X (2017) Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546(7657): 265–269. https://doi.org/10.1038/nature22060. Guinea F, Castro Neto AH, and Peres NMR (2006) Electronic states and Landau levels in graphene stacks. Physical Review B 73(24): 245426. https://doi.org/10.1103/ PhysRevB.73.245426. Hoque AM, Khokhriakov D, Zollner K, Zhao B, Karpiak B, Fabian J, and Dash SP (2021) All-electrical creation and control of spin-galvanic signal in graphene and molybdenum ditelluride heterostructures at room temperature. Communications on Physics 4(1): 124. https://doi.org/10.1038/s42005-021-00611-6. Huang B, Clark G, Navarro-Moratalla E, Klein DR, Cheng R, Seyler KL, Zhong D, Schmidgall E, McGuire MA, Cobden DH, Yao W, Xiao D, Jarillo-Herrero P, and Xu X (2017) Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546(7657): 270–273. https://doi.org/10.1038/nature22391. Jedema FJ, Filip AT, and van Wees BJ (2001) Electrical spin injection and accumulation at room temperature in an all-metal mesoscopic spin valve. Nature 410(6826): 345–348. https://doi.org/10.1038/35066533. Johnson M and Silsbee RH (1985) Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals. Physical Review Letters 55(17): 1790–1793. https://doi. org/10.1103/PhysRevLett.55.1790. Jung J, DaSilva AM, MacDonald AH, and Adam S (2015) Origin of band gaps in graphene on hexagonal boron nitride. Nature Communications 6(1): 6308. https://doi.org/10.1038/ ncomms7308. Kato YK, Myers RC, Gossard AC, and Awschalom DD (2004) Observation of the spin Hall effect in semiconductors. Science 306(5703): 1910–1913. https://doi.org/10.1126/ science.1105514. Kikkawa JM and Awschalom DD (1999) Lateral drag of spin coherence in gallium arsenide. Nature 397(6715): 139–141. https://doi.org/10.1038/16420. Kochan D, Irmer S, and Fabian J (2017) Model spin-orbit coupling Hamiltonians for graphene systems. Physical Review B 95(16): 165415. https://doi.org/10.1103/PhysRevB.95.165415. Lee J-U, Lee S, Ryoo JH, Kang S, Kim TY, Kim P, Park C-H, Park J-G, and Cheong H (2016) Ising-type magnetic ordering in atomically thin FePS3. Nano Letters 16(12): 7433–7438. https://doi.org/10.1021/acs.nanolett.6b03052.

Spintronics in 2D graphene-based van der Waals heterostructures

221

Li Y and Koshino M (2019) Twist-angle dependence of the proximity spin-orbit coupling in graphene on transition-metal dichalcogenides. Physical Review B 99(7): 075438. https://doi. org/10.1103/PhysRevB.99.075438. Li L, Zhang J, Myeong G, Shin W, Lim H, Kim B, Kim S, Jin T, Cavill S, Kim BS, Kim C, Lischner J, Ferreira A, and Cho S (2020) Gate-tunable reversible Rashba-Edelstein effect in a few-layer graphene/2H-TaS2 heterostructure at room temperature. ACS Nano 14(5): 5251–5259. https://doi.org/10.1021/acsnano.0c01037. Liu Z, Suenaga K, Harris PJF, and Iijima S (2009) Open and closed edges of graphene layers. Physical Review Letters 102(1): 015501. https://doi.org/10.1103/ PhysRevLett.102.015501. Lopes dos Santos JMB, Peres NMR, and Castro Neto AH (2007) Graphene bilayer with a twist: Electronic structure. Physical Review Letters 99(25): 256802. https://doi.org/10.1103/ PhysRevLett.99.256802. Mak KF and Shan J (2016) Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nature Photonics 10(4): 216–226. https://doi.org/10.1038/ nphoton.2015.282. Mak KF, McGill KL, Park J, and McEuen PL (2014) The valley Hall effect in MoS2 transistors. Science 344(6191): 1489–1492. https://doi.org/10.1126/science.1250140. Malajovich I, Kikkawa JM, Awschalom DD, Berry JJ, and Samarth N (2000) Coherent transfer of spin through a semiconductor heterointerface. Physical Review Letters 84(5): 1015–1018. https://doi.org/10.1103/PhysRevLett.84.1015. Mattheiss LF (1973) Band structures of transition-metal-dichalcogenide layer compounds. Physical Review B 8(8): 3719–3740. https://doi.org/10.1103/PhysRevB.8.3719. McCann E (2006) Asymmetry gap in the electronic band structure of bilayer graphene. Physical Review B 74(16): 161403. https://doi.org/10.1103/PhysRevB.74.161403. McCann E and Fal’ko VI (2006) Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Physical Review Letters 96(8): 086805. https://doi.org/10.1103/ PhysRevLett.96.086805. McCann E and Koshino M (2013) The electronic properties of bilayer graphene. Reports on Progress in Physics 76(5): 056503. https://doi.org/10.1088/0034-4885/76/5/056503. Milletarì M and Ferreira A (2016) Quantum diagrammatic theory of the extrinsic spin Hall effect in graphene. Physical Review B 94(13): 134202. https://doi.org/10.1103/ PhysRevB.94.134202. Milletarì M, Offidani M, Ferreira A, and Raimondi R (2017) Covariant conservation laws and the spin Hall effect in Dirac-Rashba systems. Physical Review Letters 119(24): 246801. https://doi.org/10.1103/PhysRevLett.119.246801. Naimer T, Zollner K, Gmitra M, and Fabian J (2021) Twist-angle dependent proximity induced spin-orbit coupling in graphene/transition metal dichalcogenide heterostructures. Physical Review B 104(19): 195156. https://doi.org/10.1103/PhysRevB.104.195156. Nakamura H, Mohammed A, Rosenzweig P, Matsuda K, Nowakowski K, Küster K, Wochner P, Ibrahimkutty S, Wedig U, Hussain H, Rawle J, Nicklin C, Stuhlhofer B, Cristiani G, Logvenov G, Takagi H, and Starke U (2020) Spin splitting and strain in epitaxial monolayer WSe2 on graphene. Physical Review B 101(16): 165103. https://doi.org/10.1103/ PhysRevB.101.165103. Nandkishore R and Levitov L (2010) Dynamical screening and excitonic instability in bilayer graphene. Physical Review Letters 104(15): 156803. https://doi.org/10.1103/ PhysRevLett.104.156803. Offidani M and Ferreira A (2018) Anomalous Hall effect in 2D Dirac materials. Physical Review Letters 121(12): 126802. https://doi.org/10.1103/PhysRevLett.121.126802. Offidani M, Milletarì M, Raimondi R, and Ferreira A (2017) Optimal charge-to-spin conversion in graphene on transition-metal dichalcogenides. Physical Review Letters 119(19): 196801. https://doi.org/10.1103/PhysRevLett.119.196801. Ohno Y, Young DK, Beschoten B, Matsukura F, Ohno H, and Awschalom DD (1999) Electrical spin injection in a ferromagnetic semiconductor heterostructure. Nature 402(6763): 790–792. https://doi.org/10.1038/45509. Peres NMR (2010) Colloquium: The transport properties of graphene: An introduction. Reviews of Modern Physics 82(3): 2673–2700. https://doi.org/10.1103/RevModPhys.82.2673. Péterfalvi CG, David A, Rakyta P, Burkard G, and Kormányos A (2022) Quantum interference tuning of spin-orbit coupling in twisted van der Waals trilayers. Physical Review Research 4 (2): L022049. https://doi.org/10.1103/PhysRevResearch.4.L022049. Pierucci D, Henck H, Avila J, Balan A, Naylor CH, Patriarche G, Dappe YJ, Silly MG, Sirotti F, Johnson ATC, Asensio MC, and Ouerghi A (2016) Band alignment and minigaps in monolayer MoS2-graphene van der Waals heterostructures. Nano Letters 16(7): 4054–4061. https://doi.org/10.1021/acs.nanolett.6b00609. Qian X, Liu J, Fu L, and Li J (2014) Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346(6215): 1344–1347. https://doi.org/10.1126/ science.1256815. Rashba EI (2009) Graphene with structure-induced spin-orbit coupling: Spin-polarized states, spin zero modes, and quantum Hall effect. Physical Review B 79(16): 161409(R). https://doi.org/10.1103/PhysRevB.79.161409. Román RJP, Costa Costa FJR, Zobelli A, Elias C, Valvin P, Cassabois G, Gil B, Summerfield A, Cheng TS, Mellor CJ, Beton PH, Novikov SV, and Zagonel LF (2021) Band gap measurements of monolayer h-BN and insights into carbon-related point defects. 2D Materials 8(4): 044001. https://doi.org/10.1088/2053-1583/ac0d9c. Safeer CK, Ingla-Aynés J, Herling F, Garcia JH, Vila M, Ontoso N, Calvo MR, Roche S, Hueso LE, and Casanova F (2019a) Room-temperature spin Hall effect in graphene/MoS2 van der Waals heterostructures. Nano Letters 19(2): 1074–1082. https://doi.org/10.1021/acs.nanolett.8b04368. Safeer CK, Ontoso N, Ingla-Aynés J, Herling F, Pham VT, Kurzmann A, Ensslin K, Chuvilin A, Robredo I, Vergniory MG, de Juan F, Hueso LE, Calvo MR, and Casanova F (2019b) Large multidirectional spin-to-charge conversion in low-symmetry semimetal MoTe2 at room temperature. Nano Letters 19(12): 8758–8766. https://doi.org/10.1021/acs. nanolett.9b03485. Saito R, Tatsumi Y, Huang S, Ling X, and Dresselhaus MS (2016) Raman spectroscopy of transition metal dichalcogenides. Journal of Physics: Condensed Matter 28(35): 353002. https://doi.org/10.1088/0953-8984/28/35/353002. Schwab P, Raimondi R, and Gorini C (2011) Spin-charge locking and tunneling into a helical metal. Europhysics Letters 93(6): 67004. https://doi.org/10.1209/0295-5075/93/67004. Sharvin DY and Sharvin YV (1981) Magnetic-flux quantization in a cylindrical film of a normal metal. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 34(5): 285–288 [JETP Letters 34: 272–275 (1981)]. Shi Y, Kahn J, Niu B, Fei Z, Sun B, Cai X, Francisco BA, Wu D, Shen Z-X, Xu X, Cobden DH, and Cui Y-T (2019) Imaging quantum spin Hall edges in monolayer WTe2. Science Advances 5(2): eaat8799. https://doi.org/10.1126/sciadv.aat8799. Sichau J, Prada M, Anlauf T, Lyon TJ, Bosnjak B, Tiemann L, and Blick RH (2019) Resonance microwave measurements of an intrinsic spin-orbit coupling gap in graphene: A possible indication of a topological state. Physical Review Letters 122(4): 046403. https://doi.org/10.1103/PhysRevLett.122.046403. Sinova J, Valenzuela SO, Wunderlich J, Back CH, and Jungwirth T (2015) Spin Hall effects. Reviews of Modern Physics 87(4): 1213–1260. https://doi.org/10.1103/ RevModPhys.87.1213. Sousa F, Perkins DTS, and Ferreira A (2022) Weak localisation driven by pseudospin-spin entanglement. Communications on Physics 5(1): 291. https://doi.org/10.1038/s42005022-01066-z. Steenwyk SD, Hsu SY, Loloee R, Bass J, and Pratt WP (1997) Perpendicular-current exchange-biased spin-valve evidence for a short spin-diffusion length in permalloy. Journal of Magnetism and Magnetic Materials 170(1): L1–L6. https://doi.org/10.1016/S0304-8853(97)00061-9. Streda P (1982) Theory of quantised Hall conductivity in two dimensions. Journal of Physics C: Solid State Physics 15(22): L717–L721. https://doi.org/10.1088/0022-3719/15/ 22/005. Tang S, Zhang C, Wong D, Pedramrazi Z, Tsai H-Z, Jia C, Moritz B, Claassen M, Ryu H, Kahn S, Jiang J, Yan H, Hashimoto M, Lu D, Moore RG, Hwang C-C, Hwang C, Hussain Z, Chen Y, Ugeda MM, Liu Z, Xie X, Devereaux TP, Crommie MF, Mo S-K, and Shen Z-X (2017) Quantum spin Hall state in monolayer 1T’-WTe2. Nature Physics 13: 683–687. https://doi. org/10.1038/nphys4174. Tedrow PM and Meservey R (1973) Spin polarization of electrons tunneling from films of Fe, Co, Ni, and Gd. Physical Review B 7: 318–326. https://doi.org/10.1103/PhysRevB.7.318. Tiwari P, Jat MK, Udupa A, Narang DS, Watanabe K, Taniguchi T, Sen D, and Bid A (2022) Experimental observation of spin-split energy dispersion in high-mobility single-layer graphene/WSe2 heterostructures. npj 2D Materials and Applications 6: 68. https://doi.org/10.1038/s41699-022-00348-y.

222

Spintronics in 2D graphene-based van der Waals heterostructures

Trushin M and Schliemann J (2007) Anisotropic current-induced spin accumulation in the two-dimensional electron gas with spin-orbit coupling. Physical Review B 75: 155323. https://doi.org/10.1103/PhysRevB.75.155323. Uri A, Grover S, Cao Y, Crosse JA, Bagani K, Rodan-Legrain D, Myasoedov Y, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, and Zeldov E (2020) Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581(7806): 47–52. https://doi.org/10.1038/s41586-020-2255-3. Vafek O and Yang K (2010) Many-body instability of coulomb interacting bilayer graphene: Renormalization group approach. Physical Review B 81: 041401. https://doi.org/10.1103/ PhysRevB.81.041401. Valenzuela SO (2009) Nonlocal electronic spin detection, spin accumulation and the spin Hall effect. International Journal of Modern Physics B 23(11): 2413–2438. https://doi.org/ 10.1142/S021797920905290X. Veneri A, Perkins DTS, Péterfalvi CG, and Ferreira A (2022) Twist angle controlled collinear Edelstein effect in van der Waals heterostructures. Physical Review B 106: L081406. https://doi.org/10.1103/PhysRevB.106.L081406. Voiry D, Mohite A, and Chhowalla M (2015) Phase engineering of transition metal dichalcogenides. Chemical Society Reviews 44: 2702–2712. https://doi.org/10.1039/C5CS00151J. Völkl T, Rockinger T, Drienovsky M, Watanabe K, Taniguchi T, Weiss D, and Eroms J (2017) Magnetotransport in heterostructures of transition metal dichalcogenides and graphene. Physical Review B 96: 125405. https://doi.org/10.1103/PhysRevB.96.125405. Wakamura T, Reale F, Palczynski P, Guéron S, Mattevi C, and Bouchiat H (2018) Strong anisotropic spin-orbit interaction induced in graphene by monolayer WS2. Physical Review Letters 120: 106802. https://doi.org/10.1103/PhysRevLett.120.106802. Wallbank JR, Mucha-Kruczynski M, Chen X, and Fal’ko VI (2015) Moiré superlattice effects in graphene/boron-nitride van der Waals heterostructures. Annalen der Physik 527(5-6): 359–376. https://doi.org/10.1002/andp.201400204. Wang QH, Kalantar-Zadeh K, Kis A, Coleman JN, and Strano MS (2012) Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nature Nanotechnology 7: 699–712. https://doi.org/10.1038/nnano.2012.193. Wang Z, Ki D-K, Khoo JY, Mauro D, Berger H, Levitov LS, and Morpurgo AF (2016) Origin and magnitude of ‘designer’ spin-orbit interaction in graphene on semiconducting transition metal dichalcogenides. Physical Review X 6: 041020. https://doi.org/10.1103/PhysRevX.6.041020. Wang D, Che S, Cao G, Lyu R, Wantanabe K, Taniguchi T, Lau CN, and Bockrath M (2019) Quantum Hall effect measurement of spin–orbit coupling strengths in ultraclean bilayer graphene/WSe2 heterostructures. Nano Letters 19(10): 7028–7034. https://doi.org/10.1021/acs.nanolett.9b02445. Wang G, Chernikov A, Glazov MM, Heinz TF, Marie X, Amand T, and Urbaszek B (2018) Colloquium: Excitons in atomically thin transition metal dichalcogenides. Reviews of Modern Physics 90: 021001. https://doi.org/10.1103/RevModPhys.90.021001. Weitz RT, Allen MT, Feldman BE, Martin J, and Yacoby A (2010) Broken-symmetry states in doubly gated suspended bilayer graphene. Science 330(6005): 812–816. https://doi.org/ 10.1126/science.1194988. Wu S, Fatemi V, Gibson QD, Watanabe K, Taniguchi T, Cava RJ, and Jarillo-Herrero P (2018) Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359(6371): 76–79. https://doi.org/10.1126/science.aan6003. Xiao F and Tong Q (2022) Tunable strong magnetic anisotropy in two-dimensional van der Waals antiferromagnets. Nano Letters 22(10): 3946–3952. https://doi.org/10.1021/acs. nanolett.2c00401. Xiao D, Liu G-B, Feng W, Xu X, and Yao W (2012) Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Physical Review Letters 108: 196802. https://doi.org/10.1103/PhysRevLett.108.196802. Xing W, Qiu L, Wang X, Yao Y, Ma Y, Cai R, Jia S, Xie XC, and Han W (2019) Magnon transport in quasi-two-dimensional van der Waals antiferromagnets. Physical Review X 9: 011026. https://doi.org/10.1103/PhysRevX.9.011026. Yang Q, Holody P, Lee S-F, Henry LL, Loloee R, Schroeder PA, Pratt WP, and Bass J (1994) Spin flip diffusion length and giant magnetoresistance at low temperatures. Physical Review Letters 72: 3274–3277. https://doi.org/10.1103/PhysRevLett.72.3274. Yang B, Lohmann M, Barroso D, Liao I, Lin Z, Liu Y, Bartels L, Watanabe K, Taniguchi T, and Shi J (2017) Strong electron-hole symmetric Rashba spin-orbit coupling in graphene/ monolayer transition metal dichalcogenide heterostructures. Physical Review B 96: 041409. https://doi.org/10.1103/PhysRevB.96.041409. Yun WS, Han SW, Hong SC, Kim IG, and Lee JD (2012) Thickness and strain effects on electronic structures of transition metal dichalcogenides: 2H-mX2 semiconductors (M ¼ Mo, W; X ¼ S, Se, Te). Physical Review B 85: 033305. https://doi.org/10.1103/PhysRevB.85.033305. Zhang F, Min H, Polini M, and MacDonald AH (2010) Spontaneous inversion symmetry breaking in graphene bilayers. Physical Review B 81: 041402. https://doi.org/10.1103/ PhysRevB.81.041402. Zollner K, Gmitra M, and Fabian J (2020) Swapping exchange and spin-orbit coupling in 2D van der Waals heterostructures. Physical Review Letters 125: 196402. https://doi.org/ 10.1103/PhysRevLett.125.196402.

Optical orientation of spins in semiconductors Mikhail I Dyakonov, Laboratoire Charles Coulomb, Université Montpellier, Montpellier, France; Ioffe Physical-Technical Institute, Saint Petersburg, Russia © 2024 Elsevier Ltd. All rights reserved.

Introduction: Historical background Overview of the subject Band structure of gallium arsenide Spin interactions The Pauli principle Exchange interaction Spin-orbit interaction Hyperfine interaction with nuclear spins Magnetic interaction Photo-generation of carriers and luminescence Angular momentum conservation in optical transitions Optical spin orientation and detection Spin relaxation Generalities otc
1 (most frequent case) otc  1 Spin relaxation mechanisms Elliott-Yafet mechanism Dyakonov-Perel mechanism Bir-Aronov-Pikus mechanism Relaxation via hyperfine interaction with nuclear spins Spin relaxation of holes in the valence band Influence of magnetic field on spin relaxation Spin relaxation of two-dimensional electrons and holes Hanle effect Interconnections between spin and charge Electric current inducing spin current (Spin Hall Effect) Electric current inducing homogeneous spin polarization Conclusion References

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Abstract This brief review presents the historical background together with the theoretical and experimental results on optical spin orientation in semiconductors obtained during half a century of intense studies. Absorption of a circularly polarized photon results in the creation of an electron-hole pair with the total spin equal to the angular momentum of the photon. The degree of circular polarization of the recombination radiation serves as a useful and sensitive indicator of the carriers’ spin state and its changes under the influence of external factors and relaxation processes. Physical phenomena observed with spin-polarized electrons are briefly reviewed.

Key points

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Historical background: spin physics Description of the phenomenon of optical orientation Relevant spin-dependent interactions in semiconductors Role of the spin in optical phenomena Description of spin relaxation Role of spin in transport phenomena

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Introduction: Historical background Maybe the first step toward today’s activity was made nearly a century ago by Robert Wood when even the notion of electron spin was not yet introduced. In a charming paper, Wood and Ellett (1924) describe how the initially observed high degree of polarization of mercury vapor fluorescence (resonantly excited by polarized light) was unexpectedly found to diminish significantly in later experiments. “It was then observed that the apparatus was oriented in a different direction from that which obtained in earlier work, and on turning the table on which everything was mounted through ninety degrees, bringing the observation direction East and West, we at once obtained a much higher value of the polarization.” In this way Wood and Ellet discovered what we now know as the Hanle effect, i.e. depolarization of luminescence by transverse magnetic field (the Earth’s field in their case). It was Hanle (1924) who carried out detailed studies of this phenomenon and provided the physical interpretation. The subject did not receive much attention until 1949 when Brossel and Kastler (1949) initiated profound studies of optical pumping in atoms. (See Kastler’s Nobel Prize award lecture, 1967.) The basic physical ideas and the experimental technique of today’s research originate from these seminal papers: creation of a non-equilibrium distribution of atomic angular momenta by optical excitation, manipulating this distribution by applying dc or ac fields, and detecting the result by studying the luminescence polarization. The relaxation times for the decay of atomic angular momenta can be quite long, especially when hyperfine splitting due to the nuclear spin is involved. A number of important applications have emerged from these studies, such as gyroscopes and hypersensitive magnetometers. The detailed understanding of various atomic processes and of many aspects of interaction between light and matter was pertinent to the future developments, e.g. for laser physics. The first experiment on optical spin orientation of electrons in a semiconductor (Si) was done by Lampel (1968), as a direct application of the ideas of optical pumping in atomic physics. The great difference, which has important consequences, is that now those are the free conduction band electrons (or holes) that get spin-polarized, rather than electrons bound in an atom. This pioneering work was followed by experimental and theoretical studies mostly performed by small research groups at Ioffe Institute in St. Petersburg (Leningrad) and at Ecole Polytéchnique in Paris in the 1970s and early 1980s. At the time, this research met with almost total indifference by the rest of the physics community, and the entire world population of researchers in the field never exceeded 20 persons. Now it can be counted by the hundreds and the number of publications by the thousands. The entire field is reviewed in the “Spin Physics in Semiconductors” book (Dyakonov, 2017).

Overview of the subject In the process of interband absorption of a photon in a semiconductor, an electron in the conduction band and a hole in the valence band are generated, the total spin of the electron and the hole being equal to the angular momentum of the photon absorbed. Photons of right or left circularly polarized light have a projection of the angular momentum on the direction of the wave vector equal to +1 or −1, respectively (in units of the Planck constant ħ). This angular momentum is distributed between the photoexcited electron and hole according to the selection rules determined by the band structure of the semiconductor. The photoexcited electrons and hole live some time t before recombination. During this time the spin orientation of carriers decreases due to various relaxation processes. If the orientation has not entirely disappeared by the time of recombination, the recombination radiation will be partially circularly polarized. Thus the process of optical orientation includes two stages: creation of spin-oriented carriers in absorption of circularly polarized light, and spin relaxation during the carriers’ lifetime. The degree of circular polarization of the recombination radiation serves as a useful and sensitive indicator of the carriers’ spin state and its changes under the influence of external factors and relaxation processes determining the kinetics of the nonequilibrium carriers in a semiconductor. Along with the optical one, other methods of detection of the carriers’ spin orientation are possible. Thus in the experiments of Lampel (1968) nuclear magnetic resonance was used as a detection method. The 29Si nuclei in a silicon crystal were polarized due to their hyperfine interaction with optically oriented electrons. Optical detection was first used by Parsons (1969) in his studies of optical spin orientation in GaSb, see also Parsons (1971). A typical experiment on optical orientation is schematically presented in Fig. 1. The degree of spin orientation of photoexcited carriers is determined by the details of the band structure, the type of optical transition, the relaxation processes, as well as by the influence of various external factors. This is why optical orientation is an effective method of studying a number of physical processes in semiconductors. In the most detailed way optical orientation was studied in GaAs and (GaAl) As solid solutions. The first experiments on optical orientation in these materials were performed by Ekimov and Safarov (1970) and by Zakharchenya et al. (1971). Optical orientation reveals itself in all types of edge luminescence, in particular in recombination radiation of optically oriented excitons. Exciton orientation was discovered in hexagonal crystals of CdSe by Gross et al. (1971). Measurements of the polarization of hot luminescence allow to study the energy and momentum relaxation of hot electrons. A quite special kind of phenomena involves optical effects due to dynamic polarization of the lattice nuclei of a semiconductor. The nuclei, polarized by optically oriented electrons, via hyperfine interaction create an effective magnetic field which, in turn, acts on

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Fig. 1 Schematical drawing of a typical experiment on optical spin orientation. A circularly polarized beam of light (1) induces interband transitions in a semiconductor (2). The degree of circular polarization of the recombination radiation (3) is measured.

the electron spins and thereby on the circular polarization of the luminescence. Studies of associated phenomena began since the work of Ekimov and Safarov (1972) who detected nuclear magnetic resonance (NMR) for the lattice nuclei of a (GaAl)As solid solution by resonant changes in the degree of circular polarization of luminescence. Below the basic theoretical ideas related to optical orientation of electrons and nuclei in semiconductors, as well as the main experimental results, are reviewed. Although most of the considerations are quite general, we shall, for definitiveness, consider semiconductors with the band structure of GaAs, for which processes accompanying optical orientation are best studied experimentally.

Band structure of gallium arsenide Most of the experimental work on optical spin orientation was conducted with GaAs crystals and GaAlAs solid solutions. The band structure of GaAs, a typical zinc-blend semiconductor, is schematically presented in Fig. 2. Near the center of the Brillouin zone there is a simple isotropic conduction band, which is doubly degenerate in spin. The valence band, consists of the sub-bands of light (lh) and heavy (hh) holes which are anisotropic, and the isotropic split-off band (so), all doubly degenerate.

Spin interactions Here we enumerate the possible types of spin interactions that can be encountered in a semiconductor. The existence of an electron spin, s ¼ 1/2, and the associated magnetic moment of the electron, m ¼ eħ/2mc, has many consequences, some of which are very important and define the very structure of our physical world, while others are subtler, but still quite interesting. Below is a list of these consequences in the order of decreasing importance.

The Pauli principle Because of s ¼ 1/2, the electrons are fermions, and so no more than one electron per quantum state is allowed. Together with Coulomb law and Schrödinger equation, it is this principle that is responsible for the structure of atoms, chemical properties, and the physics of condensed matter, biology included. It is interesting to speculate what would our world look like without the Pauli

Fig. 2 Band structure of GaAs: c – conduction band, hh -heavy hole band, lh – light hole band, so – split-off band.

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principle and whether any kind of life would be possible in such a world. Probably, only the properties of the high-temperature, fully ionized plasma would remain unchanged! Note that the Pauli exclusion principle is not related to any interaction: if we could switch off the Coulomb repulsion between electrons (but leave intact their attraction to the nuclei), no serious changes in atomic physics would occur, although some revision of the Periodic Table would be needed. Other manifestations of the electronic spin are due to interactions, either electric (the Coulomb law) or magnetic (related to the electron magnetic moment m).

Exchange interaction It is, in fact, the result of the electrostatic Coulomb interaction between electrons, which becomes spin-dependent because of the requirement that the wavefunction of a pair of electrons be anti-symmetric with respect to the interchange of electron coordinates and spins. If the electron spins are parallel, the coordinate part of the wavefunction should be antisymmetric: ""(r2, r1) ¼ − ""(r1, r2), which means that the probability that two electrons are very close to each other (r1 ¼ r2) is small compared to the opposite case, when the spins are antiparallel and accordingly their coordinate wavefunction is symmetric. Electrons with parallel spins are then better separated in space, so that their repulsion is less and consequently the energy of electrostatic interaction for parallel spins is lower. This exchange interaction is responsible for ferromagnetism. In semiconductors, it is normally not of major importance, except for magnetic semiconductors (like CdMnTe) and for the semiconductor-ferromagnet interfaces.

Spin-orbit interaction If an observer moves with a velocity v in an external electric field E, he will see a magnetic field B ¼ (1/c)E  v, where c is the velocity of light. This magnetic field acts on the electron magnetic moment. This is the physical origin of the spin-orbit interaction, the role of which strongly increases for heavy atoms (with large Z). The reason is that there is a certain probability for the outer s-electron to approach the nucleus and thus to see the very strong electric field produced by the unscreened nuclear charge +Ze at the center. Due to spin-orbit interaction, any electric field acts on the spin of a moving electron. Being perpendicular both to E and v, in an atom the vector B is normal to the plane of the orbit, thus it is parallel to the orbital angular momentum L. The energy of the electron magnetic moment in this magnetic field is  mB depending on the orientation of the electron spin (and its magnetic moment) with respect to B (or to L). Thus the spin-orbit interaction can be written as AL  S, the constant A depending on the electron state in an atom. This interaction results in a splitting of atomic levels (the fine structure), which strongly increases for heavy atoms. In semiconductors, the spin-orbit interaction depends not only on the velocity of the electron (or its quasi-momentum), but also on the structure of the Bloch functions defining the motion on the atomic scale. Like in isolated atoms, it defines the values of the electron g-factors. Spin-orbit interaction is key to our subject as it enables optical spin orientation and detection (the electrical field of the light wave does not interact directly with the electron spin). It is (in most cases) responsible for spin relaxation. And finally, it makes the transport and spin phenomena interdependent.

Hyperfine interaction with nuclear spins This is the magnetic interaction between the electron and nuclear spins, which may be quite important if the lattice nuclei in a semiconductor have non-zero spin (like in GaAs). If the nuclei get polarized, this interaction is equivalent to the existence of an effective nuclear magnetic field acting on electron spins. The effective field of 100% polarized nuclei in GaAs would be several Tesla! Because the nuclear magnetic moment is so small ( 2000 times less than that of the electron) the equilibrium nuclear polarization at the very high magnetic field of 100 T and a temperature of 1 K would be only about 1%. However, much higher degrees of polarization may be easily achieved through dynamic nuclear polarization due to hyperfine interaction with non-equilibrium electrons. Experimentally, non-equilibrium nuclear polarization of several percent is easily achieved. Similar to the spin-orbit interaction, the hyperfine interaction may be expressed in the form AIS (the Fermi contact interaction), where I is the nuclear spin, S is the electron spin, and the hyperfine constant A is proportional to | (0)|2, the square of the electron wave function at the location of the nucleus. Like spin-orbit interaction, the hyperfine interaction strongly increases in atoms with large Z, and for the same reason. An s-electron in an outer shell has a certain probability to be at the center of the atom, where the nucleus is located, and the nearer it is to the center, the less is the nucleus shielded by the inner electrons. Thus the electron wavefunction of an s-electron will have a sharp spike in the vicinity of the nucleus. For example, for the In atom the value of | (0)|2 is 6000 times larger than in the Hydrogen atom! For p-states (angular momentum l ¼ 1), and generally for states with l 6¼ 0, the Fermi interaction does not work, since (0) ¼ 0, and the electron and nuclear spins are coupled by the much weaker dipole-dipole interaction. The dynamic nuclear polarization under conditions of optical orientation of electrons is in fact a result of deep cooling of the nuclear spin system. Convincing experimental proof of such an interpretation was obtained by Fleisher et al. (1976) Optically oriented electrons and polarized lattice nuclei form a strongly coupled system. High sensitivity to small external magnetic fields, long relaxation times and typical nonlinear effects (hysteresis, self-sustained oscillations) are characteristic of this system.

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Magnetic interaction This is the direct dipole-dipole interaction between the magnetic moments of a pair of electrons. For two electrons located at neighboring sites in a crystal lattice this gives energy on the order of 1 K. This interaction is normally too weak to be of any importance in semiconductors.

Photo-generation of carriers and luminescence In the process of interband absorption of a photon with energy ħo > Eg in a semiconductor, an electron in the conduction band and a hole in the valence band are generated. During the process the (quasi)momentum is conserved, however the photon momentum ħk ¼ 2pħ/l, where l is the photon wavelength, is very small (compared, to the electron thermal momentum) and normally may be neglected. In this approximation the optical transitions are vertical: to see what happens, we must simply apply a vertical arrow of length ħo to Fig. 1, so that the arrow touches one of the valence bands and the conduction band. The ends of the arrow will give us the initial energies of the generated electrons and holes. An electron may be created in company with heavy hole, or a light hole; for ħo > Eg + D the electron-hole pair can also involve a hole in the split-off band. Note, that for a given photon energy the initial electron energy will be different for those three processes. The photoexcited carriers live some time t before recombination, which may be radiative (i.e. accompanied by photon emission, which results in luminescence), or non-radiative. In direct-band semiconductors, like GaAs, the recombination is predominantly radiative with a lifetime on the order of 1 ns. It is important to realize that this time is normally very long compared to the carriers’ thermalization time. Thermalization means energy relaxation of carriers in their respective bands due to phonon emission and absorption, which results in an equilibrium Boltzmann (or Fermi, depending on temperature and concentration) distribution function of electrons and holes. Thermal equilibrium between electrons and holes is established by recombination, on the time scale t. Because the recombination time t is so long compared to the energy relaxation time, the luminescence is produced mostly by thermalized carriers and the emitted photons have energies close to the value of Eg, irrespective of the energy of exciting photons.1 It should be noted that semiconductors are normally either intentionally, or non-intentionally doped by impurities. In a p-type semiconductor at moderate excitation power the number of photo-generated holes is small compared to the number of equilibrium holes, so that the photo-created electron will recombine with these equilibrium holes, rather than with photo-generated ones.

Angular momentum conservation in optical transitions Along with energy and momentum conservation, the conservation of the angular momentum is a fundamental law of Physics. Just like particles, electromagnetic waves have angular momentum. Photons of right or left polarized light have a projection of the angular momentum on the direction of their propagation (helicity) equal to +1 or −1, respectively (in units of ħ). Linearly polarized photons are in a superposition of these two states. When a circularly polarized photon is absorbed, this angular momentum is distributed between the photoexcited electron and hole according to the selection rules determined by the band structure of the semiconductor. Because of the complex nature of the valence band, this distribution depends on the value of the momentum of the created electron-hole pair (p and − p). However, it can be shown that if we take the average over the directions of p, the result is the same as in optical transitions between atomic states with j ¼ 3/2, mj ¼ −3/2, −1/2, +1/2, +3/2 (corresponding to bands of light and heavy holes) and j ¼ 1/2, mj ¼ −1/2, +1/2 (corresponding to the conduction band). Possible transitions between these states, as well as between states in the split-off band and the conduction band, for absorption of a right circularly polarized photon with corresponding relative probabilities are presented in Fig. 3. Note, that if we add up all transitions, which is the correct thing to do if the photon energy sufficiently exceeds Eg + D the two spin states in the conduction band will be populated equally. This demonstrates the role of spin-orbit interaction for optical spin pumping.

Optical spin orientation and detection To date, the most efficient way of creating non-equilibrium spin orientation in a semiconductor is provided by interband absorption of circularly polarized light. It can be seen from Fig. 2 that for Eg < ħo < Eg + D this absorption produces an average electron spin along the direction of excitation equal to (−1/2)(3/4) + (+1/2)(1/4) ¼ −1/4 and an average hole spin equal to +5/4, with a sum +1, equal to the angular momentum of the absorbed right circularly polarized photon. Thus in a p-type semiconductor the degree of spin polarization of the photo-excited electrons will be −50%, the minus sign indicating that the spin orientation is opposite to the direction of the angular momentum of incident photons. 1 A small part of the excited electrons can emit photons before losing their energy by thermalization. The studies of the spectrum and polarization properties of this so-called hot luminescence reveal interesting and unusual physics, (Dymnikov, 1976; Mirlin, 1984).

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Fig. 3 Optical transitions between levels with j ¼ 3/2 and j ¼ 1/2 (the bands of light and heavy holes, and the split-off band) and the levels with j ¼ 1/2 (the conduction band) during an absorption of a right-polarized photon. The probability ratio for the three transitions can be calculated to be 3:2:1.

If our electron immediately recombines with its partner hole, a 100% circularly polarized photon will be emitted. However in a p-type semiconductor electrons will predominantly recombine with the majority holes, which are not polarized. Then the same selection rules show that the circular polarization of luminescence should be P0 ¼ 25%, if the holes are not polarized, and if no electron spin relaxation occurs during the electron lifetime t, i.e. if ts  t, where ts is the electron spin relaxation time. Generally, the degree P of circular polarization of the luminescence excited by circularly polarized light is less than P0:   t (1) P ¼ P0 1 + ts In an optical spin orientation experiment a semiconductor (usually p-type) is excited by circularly polarized light with ħo > Eg. The circular polarization of the luminescence is analyzed, which gives a direct measure of the electron spin polarization. Actually, the degree of circular polarization is simply equal to the average electron spin. Thus various spin interactions can be studied by simple experimental means. The electron spin polarization will be measured provided the spin relaxation time ts is not very short compared to the recombination time t, a condition, which often can be achieved even at room temperature.

Spin relaxation Spin relaxation, i.e. disappearance of initial non-equilibrium spin polarization, is the central issue for all spin phenomena. Spin relaxation can be generally understood as a result of the action of fluctuating in time magnetic fields. In most cases, these are not real magnetic fields, but rather “effective” magnetic fields originating from the spin-orbit, or, sometimes, exchange interactions.

Generalities A randomly fluctuating magnetic field is characterized by two important parameters: its amplitude and its correlation time, tc, i.e. the time during which the field may be roughly considered as constant. Instead of the amplitude, it is convenient to use the rms value of the spin precession frequency in this random field, o. Thus we have the following physical picture of spin relaxation: the spin makes a precession around the (random) direction of the effective magnetic field with a typical frequency o and during a typical time tc. After a time tc the direction and the absolute value of the field change randomly, and the spin starts its precession around the new direction of the field. After a certain number of such steps the initial spin direction will be completely forgotten. How this happens, depends on the value of the dimensionless parameter otc, which is the typical angle of spin precession during the correlation time. Two limiting cases may be considered:

vtc
1 (most frequent case) The precession angle is small, so that the spin vector experiences a slow angular diffusion. During a time t, the number of random steps is t/tc, for each step the squared precession angle is (otc)2. These steps are not correlated, so that the total squared angle after a time t is (otc)2 (t/tc). The spin relaxation time may be defined as the time at which this angle becomes of the order of 1. Hence: ts  o2tc. This is essentially a classical formula (the Planck constant does not enter), although certainly it can be also derived quantum-mechanically. Note, that in this case ts  tc.

vtc  1 This means that during the correlation time the spin will make many rotations around the direction of the magnetic field. During the time on the order of 1/o the spin projection transverse to the random magnetic field is (on the average) completely destroyed, while its projection along the direction of the field is conserved. At this stage the spin projection on its initial direction will diminish 3 times. [Let the random magnetic field have an angle y with the initial spin direction. After many rotations the projection of the spin on the initial direction will diminish as (cos y)2. In three

Optical orientation of spins in semiconductors

229

dimensions, the average of this value over the possible orientations of the random field yields 1/3.] After time tc the magnetic field changes its direction, and the initial spin polarization will finally disappear. Thus in the case otc  1 the decay of spin polarization is not exponential in time, and the process has two distinct stages, the first one has a duration 1/o, and the second one has a duration tc. The overall result is: ts  tc. This consideration is quite general and applies to any mechanism of spin relaxation. We have only to understand the values of the relevant parameters o and tc for a given mechanism.

Spin relaxation mechanisms There are several possible mechanisms providing the fluctuating magnetic fields responsible for spin relaxation.

Elliott-Yafet mechanism The electrical field, accompanying lattice vibrations, or the electric field of charged impurities is transformed to an effective magnetic field through spin-orbit interaction. Thus momentum relaxation should be accompanied by spin relaxation (Elliott, 1954; Yafet, 1963). For phonons, the correlation time is on the order of the inverse frequency of a typical thermal phonon. Spin relaxation by phonons is normally rather weak, especially at low temperatures. For scattering by impurities, the direction and the value of the random magnetic field depends on the geometry of the individual collision (the impact parameter). This random field cannot be characterized by a single correlation time, since it exists only during the brief act of collision and is zero between collisions. In each act of scattering the electron spin rotates by some small angle ’. These rotations are uncorrelated for consequent collisions, so the average square of spin rotation angle during time t is on the order of < ’2 > (t/tp), where tp is the momentum relaxation time and < ’ > is the average of ’2 over the scattering geometry. Thus 1/ts  < ’2 >/tp. The relaxation rate is obviously proportional to the impurity concentration.

Dyakonov-Perel mechanism The mechanism is related to the spin-orbit band splitting caused by the absence of inversion symmetry (Dyakonov and Perel, 1971a, 1972). For a given p, let O(p) be the spin precession frequency in this field. It was shown theoretically (Dresselhaus, 1955) that for non-centrosymmetric semiconductors, like GaAs, this frequency is cubic in momentum and given by:


Ox  px py 2 − pz 2 , Oy  py pz 2 − px 2 , Oz  pz px 2 − py 2 , (2) The effective magnetic field changes in time because the direction of p varies due to electron collisions. Thus the correlation time is on the order of the momentum relaxation time, tp, and if Otp
1, which is normally the case, we get: 1/ts  O2tp. In contrast to the Elliott-Yafet mechanism, now the spin rotates not during, but between the collisions. Accordingly, the relaxation rate increases when the impurity concentration decreases (i.e. when tp becomes longer). It happens that this mechanism is often the dominant one, both in bulk AIIIBV and AIIBVI semiconductors, like GaAs, as well as in 2D structures.

Bir-Aronov-Pikus mechanism This is a mechanism of spin relaxation of non-equilibrium electrons in p-type semiconductors due to the exchange interaction between the electron and hole spins or, expressing it otherwise, the exchange interaction between an electron in the conduction band and all the electrons in the valence band (Bir et al., 1976). This spin relaxation rate, being proportional to the number of holes, may become the dominant one in heavily p-doped semiconductors.

Relaxation via hyperfine interaction with nuclear spins The electron spin interacts with the spins of the lattice nuclei, which are normally in a disordered state. Thus the nuclei provide a random effective magnetic field, acting on the electron spin. The corresponding relaxation rate is rather weak, but may become important for localized electrons, when other mechanisms, associated with electron motion, do not work.

Spin relaxation of holes in the valence band The origin of this relaxation is in the splitting of the valence band into sub-bands of light and heavy holes. In this case, ħO(p) is equal to the energy difference between light and heavy holes for a given p and the correlation time is again tp. However, in contrast to the situation for electrons in the conduction band, we have now the opposite limiting case: O( p)tp  1. So, the hole spin relaxation time is on the order of tp, which is very short. One can say that the hole “spin” J is rigidly fixed with respect to its momentum p, and because of this, momentum relaxation leads automatically to spin relaxation. For this reason, normally it is virtually impossible to maintain an appreciable non-equilibrium polarization of bulk holes. However, Hilton and Tang (2002) have managed to observe the spin relaxation (on the femtosecond time scale) of both light and heavy holes in undoped bulk GaAs. The general theory of the

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relaxation of spin, as well as helicity and other correlations between J and p, for holes in the valence band was given by Dyakonov and Khaetskii (1984).

Influence of magnetic field on spin relaxation In the presence of an external magnetic field B, the spins perform a regular precession with a frequency O ¼ gmB/ħ, and one should distinguish between relaxation of the spin component along B and the relaxation, or dephasing, of the perpendicular components. In the magnetic resonance literature it is customary to denote the corresponding longitudinal and transverse times as T1 and T2 respectively. To understand what happens, it is useful to go to a frame rotating around B with the spin precession frequency O. In the absence of random fields, the spin vector would remain constant in the rotating frame. Relaxation is due to random fields in the rotating frame, and obviously these fields now rotate around B with the same frequency O. Thus random fields directed along B are the same as in the rest frame, and cause the same relaxation of the perpendicular spin components with a characteristic time T2  ts. However the perpendicular components of the random field, which are responsible for the relaxation of the spin component along B, do rotate. The importance of this rotation is determined by the parameter Otc, the angle of rotation of the random field during the correlation time. If Otc
1, then rotation is of no importance, since the random field will anyway change its direction after a time tc. However, for Otc  1 the rotating random field will effectively average out during the correlation time, resulting in a decrease of the longitudinal spin relaxation rate. A simple calculation gives: 1 1 1 o2 tc ¼ ¼ : T 1 ts 1 + ðOtc Þ2 1 + ðOtc Þ2

(3)

Interestingly, with increasing magnetic field the longitudinal spin relaxation rate changes from being proportional to tc to becoming proportional to 1/tc. Again, the above classical formula can be also derived quantum-mechanically. From the quantum point of view the longitudinal relaxation is due to flips of the spin projection on B, which requires an energy gmB. Since the energy spectrum of the random field has a width  1/tc, the process becomes ineffective when gmB  ħ/tc, or equivalently, when Otc  1. Ivchenko (1973) has calculated the influence of magnetic field on the Dyakonov-Perel spin relaxation. The result coincides with Eq. (2) with tc ¼tp, except that the spin precession frequency O is replaced by the (greater) electron cyclotron frequency, oc. The reason is that for this case the variation of the vector O(p) is primarily due to the rotation of the electron momentum p in magnetic field.

Spin relaxation of two-dimensional electrons and holes Usually the Dyakonov-Perel mechanism is the dominant one. The relaxation rate is governed by the linear momentum dependence of the effective magnetic field, or the vector O(p), see above. The spin relaxation is generally anisotropic and depends on the growth direction (Dyakonov and Kachorovskii, 1986). As seen from Eq. (7), for quantum wells grown in the h001i direction the effective magnetic field lies in the 2D plane. As a consequence, the spin component perpendicular to the plane decays 2 times faster than the spin in-plane components. More details on spin-orbit interaction in two dimensional systems can be found in Winkler’s book (Winkler, 2003).

Hanle effect Depolarization of luminescence by a transverse magnetic field, first discovered by Wood and Ellett (1924) (in their case it was the magnetic field of the Earth!) and studied in more detail by Hanle (1924), is effectively employed in experiments on spin orientation in semiconductors. The reason for this effect is the precession of electron spins around the direction of the magnetic field. Under continuous illumination, this precession leads to the decrease of the average projection of the electron spin on the direction of observation, which defines the degree of circular polarization of the luminescence. Thus the degree of polarization decreases as a function of the transverse magnetic field. Measuring this dependence under steady state conditions makes it possible to determine both the spin relaxation time and the recombination time. This effect is due to the precession of electron spins in a magnetic field B with the Larmor frequency O. This precession, along with spin pumping, spin relaxation, and recombination is described by the following simple equation of motion of the average spin vector S: dS S S − S0 ¼VS− − dt ts t

(4)

where the first term on the rhs describes spin precession in a magnetic field (O ¼ gmB/ħ), the second term describes spin relaxation, and the third one describes generation of spin by optical excitation (S0/t) and recombination (−S/t). The vector S0 is directed along the exciting light beam, its absolute value is equal to the initial average spin of photo-created electrons. In the stationary state (dS/dt ¼ 0) and in the absence of a magnetic field, one finds:

Optical orientation of spins in semiconductors

Sz ðBÞ ¼

Sz ð0Þ 2, 1 + ðOt ∗ Þ

1 1 1 ¼ + : t ∗ t ts

231

(5)

The effective time t defines the width of the depolarization curve. Thus the spin projection Sz (and hence the degree of circular polarization of the luminescence) decreases as a function of the transverse magnetic field. Combining the measurements of the zero-field value P ¼ Sz(0) and of the magnetic field dependence in the Hanle effect, we can find the two essential parameters: the electron lifetime, t, and the spin relaxation time, ts, under steady-state conditions.

Interconnections between spin and charge Because of spin-orbit interaction, charge and spin are interconnected: there is a number of phenomena where an electrical current produces spin current and/or homogeneous spin polarization and vice versa. Generally, an electric current induces a transverse spin current leading to spin accumulation at the boundaries, or a uniform spin polarization, or both effects simultaneously. Inverse effects exist too. Phenomenologically, all these effects can be derived from pure symmetry considerations, according to the general principle: everything that is not forbidden by symmetry or conservation laws will happen. The microscopic theory should provide the physical mechanism of the phenomenon under consideration, as well as the values of observable quantities.

Electric current inducing spin current (Spin Hall Effect) Spin accumulation at the lateral boundaries of a current-carrying sample (the Spin Hall Effect) was predicted in 1971 by Dyakonov and Perel (1971b, 1971c), together with the Inverse Spin Hall Effect (Dyakonov and Perel, 1971b), which was observed for the first time in 1984 by Bakun et al. (1985). The name “Spin Hall Effect” was given by Hirsch (1999) who re-predicted this phenomenon 28 years later. The first experimental observations of the direct Spin Hall Effect were reported by Kato et al. (2004a) and by Wunderlich et al. (2005). Related phenomena that have attracted much attention are the Spin Hall Magnetoresistance (Dyakonov, 2007) and the swapping effect - a mutual transformation of spin currents, introduced by Lifshits and Dyakonov (2009).

Electric current inducing homogeneous spin polarization In gyrotropic systems (like Te crystals, or III-V quantum wells) the electric current can also generate homogeneous spin polarization. This effect was predicted by Ivchenko and Pikus (1978) for bulk Te crystals and observed by Vorob’ev et al. (1979). In two dimensions this effect was treated theoretically by Vas’ko (1979), Vas’ko and Prima (1979), by Levitov et al. (1985), by Edelstein (1990), and by Aronov et al. (1991). Experimentally it was observed in quantum wells by Silov et al. (2004), by Ganichev et al. (2006), and also, for strained bulk materials, by Kato et al. (2004b). Inversely, spin polarization in such systems can induce electric current as it was shown theoretically by Ivchenko et al. (1989). The first experimental demonstration of this effect, named the spin-galvanic effect, was reported by Ganichev et al. (2002). Furthermore, the interconnection between the spin polarization, spin current, and charge current is particularly pronounced under polarized radiation giving rise to photo-galvanic phenomena. The circular photo-galvanic effect was independently predicted by Ivchenko and Pikus (1978) and by Belinicher (1978). It was first observed in Te crystals by Asnin et al. (1978) and in quantum wells by Ganichev et al. (2001). This kind of effects has been re-discovered theoretically several times. Hence, a large variety of different names exist labelling one and the same thing: appearance of non-equilibrium spin polarization induced by dc electric current in gyrotropic media with linear in p spin splitting of energy bands, and vice versa.2

Conclusion Optical orientation of electron (and nuclear) spins in semiconductors is a well-developed field of research with many hundreds of publications. During more than 50 years since the first experiments, an enormous progress has been made in discovering and understanding the physical mechanisms involving various interactions between electron and nuclear spins in solids with optical radiation, electric and magnetic fields, and a large number of new and sometimes unexpected effects has been discovered. 2 Some of the labels used are: current-induced spin polarization (CISP), inverse spin-galvanic effect (ISGE), current-induced spin accumulation (CISA), magneto-electric effect (MEE), kinetic magneto-electric effect (KMEE), Edelstein effect (EE), inverse Edelstein effect (IEE), and Rashba-Edelstein effect (REE).

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References Aronov AG, Lyanda-Geller YB, and Pikus GE (1991) Spin polarization of electrons by an electric current. Soviet Physics - JETP 73: 537–541. Asnin VM, Bakun AA, Danishevskii AM, Ivchenko EL, Pikus GE, and Rogachev AA (1978) Observation of a photo-emf that depends on the sign of the circular polarization of light. Soviet Journal of Experimental and Theoretical Physics Letters 28: 74–77. Bakun AA, Zakharchenya BP, Rogachev AA, Tkachuk MN, and Fleisher VG (1985) Pis’ma Zh. Eksp. Teor. Fiz. 40, 464 (1984). JETP Letters 40: 1293. Belinicher VI (1978) Space-oscillating photocurrent in crystals without symmetry center. Physics Letters A 66: 213–214. Bir GI, Aronov AG, and Pikus GE (1976) Spin relaxation of electrons due to scattering by holes. Soviet Physics - JETP 42: 705–712. Brossel J and Kastler A (1949) La détection de la résonance magnétique des niveaux excités : l’effet de dépolarisation des radiations de résonance optique et de fluorescence. Comptes rendus de l’Académie des Sciences 229: 1213–1215. Dresselhaus G (1955) Spin-Orbit Coupling Effects in Zinc Blende Structures. Physics Review 100: 580–586. Dyakonov MI (2007) Magnetoresistance due to edge spin accumulation. Physical Review Letters 99: 126601. Dyakonov MI (ed.) (2017) Spin Physics in Semiconductors, 2nd edn Springer International Publications AG. Dyakonov MI and Kachorovskii VY (1986) Spin relaxation of two dimensional electrons in non-centrosymetric semiconductors. Soviet Physics – Semiconductors 20: 110–116. Dyakonov MI and Khaetskii AV (1984) Relaxation of nonequilibrium carrier-density matrix in semiconductors with degenerate bands. Soviet Physics - JETP 59: 1072–1078. Dyakonov MI and Perel VI (1971a) Spin orientation of electrons associated with the interband absorption of light in semiconductors. Soviet Physics - JETP 33: 1053–1059. Dyakonov MI and Perel VI (1971b) Possibility of orienting electron spins with current. Soviet Journal of Experimental and Theoretical Physics Letters 13: 467–469. Dyakonov MI and Perel VI (1971c) Current-induced spin orientation of electrons in semi-conductors. Physics Letters A 35: 459–460. Dyakonov MI and Perel VI (1972) Spin relaxation of conduction electrons in non-centrosymmetric semiconductors. Soviet Physics - Solid State 13: 3023–3026. Edelstein VM (1990) Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Communications 73: 233–235. Ekimov AI and Safarov VI (1970) Optical orientation of carriers in interband transitions in semiconductors. JETP Letters 12: 198–201. Ekimov AI and Safarov VI (1972) Optical detection of dynamic polarization of nuclei in semiconductors. Soviet Journal of Experimental and Theoretical Physics Letters 15: 257–261. Elliott RJ (1954) Theory of the effect of spin-orbit coupling on magnetic resonance in some semiconductors. Physics Review 96: 266–279. Fleisher VG, Dzhioev RI, and Zakharchenya BP (1976) Optical cooling of a nuclear spin system of a conductor in a weak oscillating magnetic field. JETP Letters 23: 18–22. Ganichev SD, Ivchenko EL, Danilov SN, Eroms J, Wegscheider W, Weiss D, and Prettl W (2001) Conversion of spin into directed electric current in quantum wells. Physical Review Letters 86: 4358–4361. Ganichev SD, Ivchenko EL, Bel’kov VV, Tarasenko SA, Sollinger M, Weiss D, Wegscheider W, and Prettl W (2002) Spin-galvanic effect. Nature 417: 153–156. Ganichev SD, Danilov SN, Schneider P, Bel’kov VV, Golub LE, Wegscheider W, Weiss D, and Prettl W (2006) Electric current-induced spin orientation in quantum well structures. Journal of Magnetism and Magnetic Materials 300: 127–131. Gross EF, Ekimov AI, Razbirin BS, and Safarov VI (1971) Optical orientation of free and bound excitons in hexagonal crystals. Soviet Journal of Experimental and Theoretical Physics Letters 14: 70–73. Hanle W (1924) Über magnetische beeinflussung der polarisation der resonanz fluoreszenz. Zeitschrift für Physik 30: 93–105. Hilton DJ and Tang CL (2002) Optical Orientation and Femtosecond Relaxation of Spin-Polarized Holes in GaAs. Physical Review Letters 89: 146601-(1-4). Hirsch JE (1999) Spin hall effect. Physical Review Letters 83: 1834–1837. Ivchenko EL and Pikus GE (1978) New photogalvanic effect in gyrotropic crystals. Soviet Journal of Experimental and Theoretical Physics Letters 27: 604–608. Ivchenko EL (1973) Fiz. Tverd. Tela (Leningrad) 15, 1566 (1973). Soviet Physics. Solid State 15: 1048. Ivchenko EL, Lyanda-Geller YB, and Pikus GE (1989) Photocurrent in structures with quantum wells with an optical orientation of free carriers. Soviet Journal of Experimental and Theoretical Physics Letters 50: 175–177. Kastler A (1967) Optical methods for studying hertzian resonances. Science 158: 214–221. Kato YK, Myers RC, Gossard AC, and Awschalom DD (2004a) Observation of the spin Hall effect in semiconductors. Science 306: 1910–1913. Kato YK, Myers RC, Gossard AC, and Awschalom DD (2004b) Current-induced spin polarization in strained semiconductors. Physical Review Letters 93: 176601-(1-4). Lampel G (1968) Nuclear dynamic polarization by optical electronic saturation and optical pumping in semiconductors. Physical Review Letters 20: 491–493. Levitov LS, Nazarov YV, and Eliashberg GM (1985) Magnetoelectric effects in conductors with mirror isomer symmetry. Soviet Physics - JETP 61: 133–137. Lifshits MB and Dyakonov MI (2009) Swapping spin currents: Interchanging spin and flow directions. Physical Review Letters 103: 186601–186604. Mirlin DN (1984) Optical alignment of electron momenta in GaAs-type semiconductors. In: Meier F and Zakharchenya BP (eds.) Optical Orientation, pp. 133–171. North Holland: Amsterdam. Parsons RR (1969) Band-to-band optical pumping in solids and polarized photoluminescence. Physical Review Letters 23: 1152–1154. Parsons RR (1971) Optical pumping and optical detection of spin-polarized electrons in a conduction band. Canadian Journal of Physics 49: 1850–1860. Silov AY, Blajnov PA, Wolter JH, Hey R, Ploog KH, and Averkiev NS (2004) Current-induced spin polarization at a single heterojunction. Applied Physics Letters 85: 5929–5931. Vas’ko FT (1979) Spin splitting in the spectrum of two-dimensional electrons due to the surface potential. Soviet Journal of Experimental and Theoretical Physics Letters 30: 541–544. Vas’ko FT and Prima NA (1979) Spin splitting of spectrum of 2-dimensional electrons. Soviet Physics - Solid State 21: 994–998. Vorob’ev LE, Ivchenko EL, Pikus GE, Farbstein II, Shalygin VA, and Sturbin AV (1979) Optical activity in tellurium induced by a current. Soviet Journal of Experimental and Theoretical Physics Letters 29: 441–445. Winkler R (2003) Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Berlin: Springer. Wood RW and Ellett A (1924) Polarized resonance radiation in weak magnetic fields. Physics Review 24: 243–254. Wunderlich J, Kaestner B, Inova J, and Jungwirth T (2005) Experimental observation of the spin hall effect in a two-dimensional spin-orbit coupled semiconductor system. Physical Review Letters 94: 047204-(1-4). Yafet Y (1963) g-factors and spin-lattice relaxation of conduction electrons. Solild State Physics 14: 1–98. Zakharchenya BI, Fleisher VG, Dzhioev RI, Veshchunov YP, and Rusanov IB (1971) Effect of optical orientation of electron spins in a GaAs crystal. JETP Letters 13: 137–139.

Raman spectroscopy of graphene and related materials Anna K Otta and Andrea C Ferrarib, aNS3E Laboratory, UMR 3208 ISL/CNRS/UNISTRA, French-German Research Institute of Saint-Louis, Saint Louis, France; bCambridge Graphene Centre, University of Cambridge, Cambridge, United Kingdom © 2024 Elsevier Ltd. All rights reserved.

Introduction Brief overview of light scattering processes Raman scattering process Raman spectroscopy of graphene and graphene layers Counting graphene layers: 2D peak and ultra-low frequency modes The Raman spectrum of strained graphene The Raman spectrum of doped graphene The Raman spectrum of defected graphene Magnetic fields Key issues Conclusion Acknowledgments References

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Abstract Raman spectroscopy is one of the main characterization techniques for graphene and related materials. It is a non-destructive technique that can give insight in the material’s quality, the number of layers, and is sensitive to any changes in electric or magnetic fields, band structure and temperature, making it ideal to probe layered materials.

Key points

• • • •

Brief introduction to Raman scattering Explanation of the Raman spectrum of graphene/graphene layers and how it changes as function of doping, defects, and strain Summary of the capabilities of Raman spectroscopy of graphene Overview of the current challenges and general outlook

Introduction A single layer of graphene (SLG) (Novoselov et al., 2004) can be seen by using a microscope if placed over a Si+SiO2 thickness
100 nm or
300 nm (Casiraghi et al., 2007a). The SiO2 layer acts as a cavity for the light and results in either constructive or destructive interference depending on its thickness (Casiraghi et al., 2007a). Fig. 1 shows the calculated optical contrast as function of laser wavelength and SiO2 thickness with contrast maxima at
100 and
300 nm thickness for commonly used laser wavelengths between
450 and
600 nm. While imaging by optical contrast can give an idea of its thickness, it is not enough to get more quantitative information, such as doping, disorder, strain, etc. Raman spectroscopy is a powerful characterization technique for carbons in general, ranging from fullerenes, nanotubes, graphitic carbons to amorphous and diamond-like carbons (Ferrari and Robertson, 2000; Tuinstra and Koenig, 1970; Dresselhaus et al., 2005; Dresselhaus et al., 1996). In graphene, Raman spectroscopy can now be routinely used to extract the number of layers, N, to estimate the type and amount of doping and strain, as well as to check the quality of graphene as it this spectroscopic technique is also sensitive to defects (Ferrari and Basko, 2013).

Brief overview of light scattering processes Light scattering can be either elastic, i.e. the incident and scattered light have the same frequency, or inelastic, where the scattered light has a different frequency with respect to the incident light. In classical electrodynamics, when light impinges on a medium, electrons within that medium start to oscillate at the same frequency as the incident electromagnetic wave. This periodic perturbation of the electron cloud results in the creation of an oscillating dipole moment leading to the emission of light with the same frequency as the incident light, i.e. elastic scattering of light. Examples are Mie scattering (Mie, 1908) and Rayleigh Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00252-3

233

234

Raman spectroscopy of graphene and related materials

Fig. 1 Calculated optical contrast of graphene on Si/SIO2 as a function of wavelength and SiO2 thickness. Reprinted with permission from Casiraghi C, Hartschuh A, Lidorikis E, Qian H, Harutyunyan H, Gokus T, Novoselov KS, and Ferrari AC (2007a) Rayleigh imaging of graphene and graphene layers. Nano Letters 7: 2711–2717. Copyright 2007 American Chemical Society.

scattering (Strutt, 1871), depending on the particle size in the medium. Light scattering off particles larger than the wavelength and having a refractive index different from that of the continuous medium in which they are embedded is referred to as Mie scattering, which can be observed when sun light scatters from the water droplets within clouds. If the particle size is smaller than the wavelength of light, such as is the case for atoms or molecules, the process is called Rayleigh scattering. An example is the color of the sky, either blue during the day or red at sun rise or sunset, where light is scattered off the molecules in the atmosphere. In the case of inelastic light scattering, the scattered light has a different frequency or energy with respect to the incident light. This process can be more conveniently described by quantum theory, whereas an electron absorbs a photon, leaving behind an empty space (a hole), and as a result is promoted to an energetically higher level. The electron can then either fall directly back to a lower level, while emitting a photon with energy corresponding to the energy level difference, or scatter with the lattice, i.e. create or annihilate/absorb a phonon, before recombining radiatively with a hole. Examples for inelastic light scattering are Brillouin and Raman scattering. While Brillouin scattering involves low frequency lattice vibrations, i.e. acoustic phonons, Raman scattering involves optical phonons in first order scattering processes (Brüesch, 1986). The Raman scattering process was first theoretically predicted in 1923 by Austrian physicist Smekal (Smekal, 1923). The experimental proof was delivered in 1928 by Landsberg and Mandelstam (Landsberg and Mandelstam, 1928) in crystals and by Raman and Krishnan (Raman and Krishnan, 1928) in liquids. For his findings Raman was awarded the Nobel Prize in physics in 1930 (Nobel-Prize). Since then - and even more so after the invention of the laser in 1960 (Schawlow and Townes, 1958; Maiman, 1960) - Raman spectroscopy has become a standard materials characterization tool as it is non-destructive, fast and easy to use (Ferrari and Basko, 2013).

Raman scattering process Raman scattering is the inelastic scattering between light, i.e. a photon, and matter. It involves the exchange of rotational or vibrational quanta of energy in molecules or the creation and annihilation of phonons with frequency O in solids (Cardona and Güntherodt, 1982). If light of frequency oL is incident on a material, most of the light is elastically scattered. This Rayleigh scattering process is shown in Fig. 2. From a classical point of view, light can be described as a propagating electromagnetic wave with E(t) ¼ E0 cos (oL t) with amplitude E0 and frequency oL. The electron cloud responds to the fast changing electric field and this will induce a dipole moment m(t) proportional to the electric field of the incoming light (Yu and Cardona, 1996): mðt Þ ¼ a Eðt Þ ¼ a0 E0 cosðoL t Þ

(1)

where the proportionality constant a is the polarizability. This describes the ability of, e.g., molecules to be polarized, and depends on the relative position of the individual atoms. Atomic vibrations are confined to certain energetic levels and are quantized into phonons. The physical displacement of the atoms dQ about their equilibrium position associated with a phonon can be described as: dQ ¼ Q0 cos (Ot), where O is the phonon frequency and Q0 is the displacement about the equilibrium position. A material’s response to electromagnetic radiation depends on the atomic positions. The atomic displacements change the susceptibility, thus its polarization, depending on the phonon involved. At room temperature (RT) the atomic displacement is small

Raman spectroscopy of graphene and related materials

235

Fig. 2 Energy level diagram showing Rayleigh and Raman scattering (Stokes and anti-Stokes) processes. Blue lines depict absorption of a photon, while red lines depict emission of a photon. Non-resonant processes involve virtual states, while resonant scattering processes involve real states.

compared to the lattice constant. Therefore, the polarizability a can be approximated by a Taylor series expansion around the lattice displacement. Considering only first order terms we get:

∂a

∂a

dQ + . . . ¼ a + Q cosðOt Þ (2) a ¼ a0 + 0 ∂Q
0 ∂Q
0 0 Inserting Eq. (2) in (1) gives: mðt Þ ¼ a0 E0 cosðoL t Þ +


∂a

E Q cosðOt Þ cosðoL t Þ ∂Q
0 0 0

Using a trigonometric identity for the product of the cosine terms leads to
1 ∂a

E Q f cos½ðO −oL Þt  + cos½ðO + oL Þtg mðt Þ ¼ a0 E0 cosðoL t Þ + 2 ∂Q
0 0 0

(3)

€j, where m € is the second derivative of m with respect to time, t: The intensity of the scattered light ISc is proportional to jm ISc ∝ cos 2 ðoL Þ + cos 2 ½ðoL + OÞt  + cos 2 ½ðoL − OÞt  + cross terms

(4)

The first term corresponds to the elastically scattered light, i.e. Rayleigh scattering, since it only depends on oL,while the other two terms describe the inelastically scattered light, i.e. Raman scattering. Here we distinguish between Stokes (S) and anti-Stokes (AS) scattering, both named after Sir G. G. Stokes (Larmor, 1907). In S-scattering a phonon is created by energy transfer from the electron to the lattice resulting in the (oL − O) term, meaning the scattered light has a lower frequency with respect to the incident light, while in AS-scattering a phonon is annihilated (or absorbed), giving the (oL + O) term. The S and AS Raman scattering processes are depicted in Fig. 2. AS Raman scattering involves the absorption of a phonon. Consequently, the scattering process starts directly from a vibrational state as shown in Fig. 2. At RT the AS-Raman peaks have lower intensity compared to S, but their intensity increases with rising temperature (T). This can be explained by considering the Bose-Einstein distribution, whereby the number of phonons in a certain energy state strongly depends on T, and increases with T (Kittel, 2005). If nq is the phonon occupation number, then the S-Raman scattering intensity will be proportional to nq + 1, since a phonon is created during the scattering process. For AS, we start at an excited (vibrational) state and a phonon is annihilated, giving rise to intensity proportional to nq. Thus, the S/AS intensity ratio can be expressed as (Placzek, 1934; Krishnan and Narayanan, 1950):    4 IStokes oL − O ℏO ¼ exp , (5) Ianti −Stokes oL + O kB T where kB is the Boltzmann constant, oL is the frequency of the laser light, O is the phonon frequency, ℏ is the reduced Planck constant. By re-arranging Eq. (5) for T, the sample T can be determined by fitting the peak intensity of a peak on S and AS side of the Raman spectrum. The intensity of the scattered light also strongly depends on whether we have a resonant or non-resonant process (Loudon, 1965). In a resonant process, real electronic states are involved, while in non-resonant Raman scattering virtual states are used to

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describe the process. The intensity of the Raman scattered light depends on the polarization of the incident and scattered light as well as on the phonon involved in the scattering process. The symmetry of the phonon mode is incorporated in the so-called Raman b this term can be expressed as (Yu and Cardona, 1996) tensor R. Introducing a unit vector in direction of the lattice displacement Q
∂a

b Q (6) R¼ ∂Q
0 Assuming b ei and b eSc to be unit vectors of the polarization of the incoming and scattered light, the Raman intensity is proportional to
2 ∂a
eSc j (Yu and Cardona, 1996). Thus, for Raman scattering to occur, the term ∂Q I∝jb ei R b
has to be non-zero, meaning the 0

polarizability has to change. By doing polarization dependent measurements, the symmetry of the Raman tensor, thus that of the corresponding phonon, can be obtained (Cardona and Güntherodt, 1982). This is very useful for correct assignment of the Raman peaks. During the scattering process, energy and momentum conservation have to hold: ℏoL ¼ ℏoSc  ℏO and kL ¼ kSc  q, where ℏ is the reduced Planck constant, oL, oSc, O are the frequency of the incident light, the scattered light and the phonon, respectively, and kL, kSc, q are the corresponding wave vectors and the phonon momentum. The wavevectors can be expressed in terms of wavelength l via k ¼ 2pl−1. The lattice constant of a material is of the order of a few A˚ , thus much smaller than l (in the order of a few hundreds of nm), giving kL, kSc  pa−1, which is the size of the first Brillouin zone (BZ), where a is the lattice constant. Combining this with the equations for energy and momentum conservation we can deduct that q  pa−1 meaning q  0. This is called the fundamental Raman selection rule, and tells us that the phonons contributing to first order Raman scattering stem from the BZ center. For higher P order scattering processes, i.e. processes involving multiple phonons, the selection rule becomes q ¼ 0. So, in a second order Raman process, two phonons with equal but opposite momentum are contributing since q + (−q) ¼ 0.

Raman spectroscopy of graphene and graphene layers In real space, single layer graphene (SLG) has a hexagonal structure resulting in a hexagonal structure in reciprocal space as well, where G marks the BZ center and the high symmetry points at the corner are labelled as K and K0 , Fig. 3A. At these so-called Dirac points, SLG has a unique linear dispersion relation (Dirac cones). In SLG there are two atoms per unit cell giving rise to six normal modes (two being doubly degenerate) at G with irreducible representation: A2u + B2g + E1u + E2g (Nemanich et al., 1977). The E2g and B2g modes are both doubly degenerate, whereas the E2g mode is Raman active and the B2g is silent, i.e. neither Raman nor IR active (Nemanich et al., 1977). The A2u and E1u modes are both IR active modes (Ferrari and Basko, 2013). The Raman spectrum of graphite and multilayer graphene consists of two different sets of peaks: those at higher wavenumbers, such as the D, G, 2D and 2D’ modes due to in-plane vibrations, and those at low frequency, such as the shear (C) and layer breathing modes (LBMs), due to the relative movement of the whole atomic planes, either parallel or perpendicular (Ferrari and Basko, 2013; Tan et al., 2012), Fig. 4. In SLG only the first set of peaks is visible (Ferrari et al., 2006).

Fig. 3 (A) SLG structure in reciprocal space (BZ). The high symmetry points and a schematic of the electronic dispersion at K and K0 (Dirac cones) are indicated. (B) Doubly degenerate E2g mode and A1g breathing mode giving rise to the G and D peak in SLG. (C) SLG Phonon dispersion. The red lines indicate the positions of Kohn anomalies. (C) Reprinted with permission from Piscanec S, Lazzeri M, Mauri F, Ferrari AC, and Robertson J (2004) Kohn anomalies and electron-phonon interactions in graphite. Physical Review Letters 93: 185503. Copyright (2004) by the American Physical Society.

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Fig. 4 Comparison of Raman spectra of pristine graphene, defective graphene and graphite.

The G peak
1580 cm−1 arises from an E2g phonon stemming from the G-point, Fig. 3B and C. The D peak, corresponding to an A1g phonon (Fig. 3B), is due to the breathing mode of six-atom rings and requires a defect for its activation (Ferrari and Robertson, 2000; Thomsen and Reich, 2000; Tuinstra and Koenig, 1970) to fulfill the fundamental Raman selection rule. This peak is activated by an intervalley double resonance process (Thomsen and Reich, 2000), connecting K and K0 , and comes from a transverse optical (TO) phonon at the K point, Fig. 3C. Due to the presence of a Kohn anomaly (Piscanec et al., 2004), i.e. a damping of the phonon dispersion (red lines in Fig. 3C), it is also highly dispersive with excitation energy shifting by
50 cm−1 eV−1 (Pocsik et al., 1998). Similarly, an intravalley double resonance scattering process at K or K0 gives rise to another defect-activated peak, the D0 peak. In defective graphene also the combination mode D + D0 starts to appear. The 2D and 2D’ peaks are the second order modes of the D and D0 peak, where the momentum conservation is fulfilled by scattering of two phonons with opposite wave vectors. Thus, these peaks do not require defects for their activation and are always present in the Raman spectrum of graphene (Ferrari et al., 2006; Basko et al., 2009). This means that only the G peak in the Raman spectrum of graphene fulfills the fundamental Raman selection rule directly as its origin is at the center of the Brillouin zone where q ¼ 0. All scattering processes are depicted in Fig. 5.

Counting graphene layers: 2D peak and ultra-low frequency modes The Raman peaks of graphene can be fitted with a Lorentzian line shape, which arises from a finite, homogeneous lifetime broadening. Here we use the following notation (Ferrari and Basko, 2013) to refer to the peak fitting parameters: ‘I’ for peak height (intensity), ‘A’ for peak area, ‘Pos’ for peak position and ‘FWHM’ for the full-width at half-maximum of the peak. The Raman scattering process in SLG is always resonant due to its linear band structure. The band structure of SLG and multilayer graphene (MLG) is reflected in the shape of the 2D peak (Ferrari et al., 2006). In SLG, the 2D peak is one sharp peak that can be fitted with a single Lorentzian line, while in Bernal stacked bilayer graphene (BLG) it splits into four components following the evolution of the band structure, Fig. 6A and B. Thus, the 2D peak shape can be used to estimate N. However, beyond
5–10 layers, the 2D peak will start looking like that of graphite, Fig. 6A, and the 2D peak shape cannot be used, alone, to estimate N correctly. Low frequency modes are far more sensitive to N, even beyond N ¼ 5, Fig. 6C. For accuracy the resulting peak positions of C and/or LBMs can be calculated as the mean value of the fitted S/AS peak positions. The C peak changes its position with N, Fig. 6D. Following a linear chain model where each layer is estimated as one mass coupled by a spring to the next mass (layer), the relation between the position of the C peak (in cm−1) and N can be written as (Tan et al., 2012): rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 a p 1 + cos (7) PosðCÞN ¼ pffiffiffi N 2pc m pffiffiffiffiffiffiffiffi where the pre-factor ð1=pcÞ a=m corresponds to Pos(C)1 in graphite with N ! 1, a
12.8  1018 Nm−3 is the interlayer coupling constant, m ¼ 7.6  10−27kgA˚ −2 is the SLG mass per unit area and c is the speed of light in cm s−1. Thus, by knowing Pos(C) in bulk graphite one can extract the interlayer coupling constant, thus get fundamental information about the interaction between the layers. This model can be applied to all layered materials (LMs), see e.g. refs. (Pizzi et al., 2021; Zhang et al., 2013).

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Fig. 5 Raman scattering processes in SLG. The red shaded area within the linear dispersion relation shows the occupied states. The G peak arises from one phonon scattering. Depending on doping, some processes will no longer be allowed. The D0 (intravalley scattering), D (intervalley scattering) and D + D0 peak require defects for their activation. The two different processes for the D and D0 peak depicted here give the largest contribution. The two-phonon (second order) modes, 2D and 2D’ peak, do not require a defect. Schematics from Ref. Ferrari AC and Basko DM (2013) Raman spectroscopy as a versatile tool for studying the properties of graphene. Nature Nanotechnology 8: 235–246. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Nanotechnology, COPYRIGHT (2013).

The Raman spectrum of strained graphene SLG can be stretched up to at least
20% without breaking (Bunch et al., 2007; Lee et al., 2008). Strain in graphene has not only a major effect on its structure but also on its Raman spectrum. Strain can occur in different forms: Uniaxial (Mohiuddin et al., 2009; Yoon et al., 2011; Mohr et al., 2010; Huang et al., 2010), i.e. strain along one direction, and biaxial strain, i.e. isotropic strain (Ding et al., 2010; Zabel et al., 2012; Metzger et al., 2010). Both types can either be applied as tensile or compressive.

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Fig. 6 (A) 2D peak as function N. (B) For BLG, the 2D peak splits into 4 components. (C) Ultra-low frequency and G peak region of graphene layers. (D) Fitted C and G peak positions. (A and B) Reprinted figure with permission from Ferrari AC, Meyer JC, Scardaci V, Casiraghi C, Lazzeri M, Mauri F, Piscanec S, Jiang D, Novoselov KS, Roth S, and Geim AK (2006) Raman spectrum of graphene and graphene layers. Physical Review Letters 97: 187401. Copyright (2006) by the American Physical Society. (C and D) From Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Materials, Tan PH, Han WP, Zhao WJ, Wu ZH, Chang K, Wang H, Wang YF, Bonini N, Marzari N, Pugno N, Savini G, Lombardo A, and Ferrari AC (2012) The shear mode of multilayer graphene. Nature Materials 11: 294–300, COPYRIGHT (2012).

In general, compressive strain results in an upshift of all Raman peaks, arising from the shortening of the interatomic bond lengths, while tensile causes a downshift, due to the elongation of the bond lengths resulting in a weakening of the vibrational modes. Biaxial strain preserves the hexagonal symmetry in SLG, i.e. it only leads to an isotropic expansion or compression of the hexagonal lattice, resulting in a linear peak shift. The linear change in peak position or frequency ∂o is ∂oG/∂e  −57 cm−1/% for the G peak and ∂o2D/∂e  −140 cm−1/% for the 2D peak (Zabel et al., 2012; Mohiuddin et al., 2009). Unless induced intentionally it is very unlikely to find biaxial strain in graphene samples. Uniaxial strain also results in linear peak shifts (Fig. 7), however it will break the hexagonal symmetry, which has an effect on the Raman peak shape. The G peak is a doubly degenerate mode and uniaxial strain makes both components visible: The peak splits into two components G+ and G−, one parallel and one perpendicular to the direction of applied strain (Mohiuddin et al., 2009), Fig. 7A–C. The rate of change is (Mohiuddin et al., 2009): ∂oG+/∂e  − 10.8 cm−1/% and ∂oG−/∂e  − 31.7 cm−1/% for the G peak and ∂o2D/∂e  − 64 cm−1/% for the 2D peak. A similar splitting of the 2D peak has been reported when the uniaxial strain exceeds
0.5% and is applied along a high-symmetry axis

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Fig. 7 Effect of uniaxial strain on the SLG Raman spectrum. (A) Splitting of the G peak and a linear peak shift for increasing uniaxial stain. (B) Fitted peak positions as function of strain. (C) Schematic showing the origin of the G peak splitting. Eigenvectors of G+ and G− modes, perpendicular and parallel to the strain direction. (D) Evolution of the 2D peak as function of strain. A splitting of the 2D peak can be observed. (E) Fitted peak positions as function of strain. (F) Schematics showing strain induced distortions of SLG and resulting change in distance between K and R (strained K0 ) points. (A, B, C) Reprinted figures with permission from Mohiuddin TMG, Lombardo A, Nair RR, Bonetti A, Savini G, Jalil R, Bonini N, Basko DM, Galiotis C, Marzari N, Novoselov KS, Geim AK, and Ferrari AC (2009) Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Gruneisen parameters, and sample orientation. Physical Review B 79: 205433. Copyright (2009) by the American Physical Society. (D, E, F) Reprinted figures with permission from Yoon D, Son Y-W, and Cheong H (2011) Strain-dependent splitting of the double-resonance Raman scattering band in graphene. Physical Review Letters 106: 155502. Copyright (2011) by the American Physical Society.

(Huang et al., 2010; Mohr et al., 2010; Yoon et al., 2011), Fig. 7D and E. The 2D peak arises from an intervalley scattering process connecting K and K0 points. Considering the structure of graphene, each K point has three neighboring K0 points, leading to three contributions to the scattering process. Fig.7F shows a schematic depicting the possible scattering processes for the 2D peak in strained graphene. The relative position of the Dirac points has been disturbed such that the three scattering mechanisms connecting K and K0 are no longer identical (Huang et al., 2010; Yoon et al., 2011). In the example shown in Fig. 7F two of the three possible scattering mechanism lead to one 2D peak component, while the remaining one gives rise to the second 2D peak component (red and blue arrows in Fig. 7F), depending on the direction of strain either parallel or perpendicular to the armchair edge in graphene.

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The Raman spectrum of doped graphene Most graphene samples produced by either micromechanical cleavage (MC), chemical vapor deposition (CVD), liquid phase exfoliation (LPE) or carbon segregation from SiC or metal substrates are doped due to the sensitivity of graphene to adsorbates, e.g. moisture, and/or due to the interaction with the underlying substrate (Bonaccorso et al., 2012; Backes et al., 2020). Doping can also be applied on purpose either electrically or chemically and has a marked effect on the Raman spectrum of (Basko et al., 2009; Das et al., 2008; Bruna et al., 2014; Kalbac et al., 2010; Casiraghi et al., 2007b): In doped SLG Pos(G) increases, while FWHM(G) decreases for both electron (e) and hole (h) doping, see Fig. 8A and C. The increase in Pos(G) is due to a non-adiabatic removal of the Kohn anomaly at G, while the decrease of FWHM(G) is due to Pauli blocking of the phonon decay channel into e-h pairs when the e-h gap is higher than the phonon energy, and saturates when the Fermi energy, EF, is bigger than half the phonon energy (Basko et al., 2009; Das et al., 2008). The 2D peak, allows distinguishing between e and h doping, as Pos(2D) increases for h doping and decreases for e doping (Das et al., 2008), Fig. 8B. This can be explained by taking the change of the equilibrium lattice parameter into account with a consequent stiffening/softening of the phonon modes, as a result of h and e doping, respectively. Both, intensity and area ratios, I(2D)/I(G) and A(2D)/A(G) depend on EF. The ratios are maximum for EF ¼ 0, and decrease with increasing EF (Basko et al., 2009; Das et al., 2008), Fig. 8D. Using the fitting parameters Pos(G), FWHM(G), Pos(2D), I(2D)/I(G) and A(2D)/A(G) with the corresponding graphs as function of EF presented in refs. (Basko et al., 2009, Das et al., 2008), EF can be extracted from the Raman spectrum. Additional effects occur for high doping levels
0.9 eV. In this case an enhancement of the G peak intensity I(G) can be observed (Zhao et al., 2011). Doping changes the occupation of electronic states and since transitions from an empty state or to a filled state are impossible (Pauli exclusion principle), this can exclude some of the BZ regions from contributing to the Raman matrix element

Fig. 8 (A) Pos(G), (B) Pos(2D), (C) FWHM(G) and (D) I(2D)/I(G) as function of doping. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Nature/Springer, Nature Nanotechnology, Das A, Pisana S, Chakraborty B, Piscanec S, Saha SK, Waghmare UV, Novoselov KS, Krishnamurthy HR, Geim AK, Ferrari AC, and Sood AK (2008) Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor. Nature Nanotechnology 3: 210–215. COPYRIGHT (2008).

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Fig. 9 Effect of high doping on the Raman spectrum of SLG. (A) The 2D peak is suppressed for certain excitation wavelengths. (B) Schematic of SLG band structure for e doped SLG, and 2D peak Raman scattering process. (A) Reprinted with permission from Zhao W, Tan PH, Liu J, and Ferrari AC (2011) Intercalation of few-layer graphite flakes with FeCl3: Raman determination of fermi level, layer by layer decoupling, and stability. Journal of the American Chemical Society 133: 5941–5946. Copyright 2011 American Chemical Society; (B) Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Nature Nanotechnology, Ferrari AC and Basko DM (2013) Raman spectroscopy as a versatile tool for studying the properties of graphene. Nature Nanotechnology 8: 235–246. COPYRIGHT (2013).

(Zhao et al., 2011). Because of suppression of destructive interference this leads to an enhancement of the G peak when |EF| matches half of the laser excitation energy ℏoL/2 (Zhao et al., 2011). At high doping the 2D peak is suppressed when the conduction band becomes filled at the energy probed by the laser (Zhao et al., 2011), Fig. 9A. There are 3 cases that can occur in highly doped samples (Ferrari and Basko, 2013): (i) the energy of the scattered (subscript ‘Sc’) and incoming photon (subscript ‘L’) are larger than twice EF, i.e. oL , oSc > 2Eℏ F : in this case the 2D peak scattering mechanism is allowed, Fig. 9B; (ii) oSc < 2|EF|/ℏ < oL, the photon absorption is allowed but the phonon emission is excluded by Pauli blocking; (iii) oL, oSc < 2|EF|/ℏ both photon absorption and phonon emission are blocked.

The Raman spectrum of defected graphene There are many production methods for graphene, such as MC, CVD, LPE, etc. (Bonaccorso et al., 2012; Backes et al., 2020) and the quality may vary depending on the production technique. E.g. MC SLG is still thought to give the most intrinsic, defect-free and clean samples (Purdie et al., 2018). High quality samples with RT mobility
70,000 cm2V−1 s−1 in single crystal CVD and 30,000 cm2V−1 s−1 in polycrystalline CVD can be achieved by using optimized transfer processes (De Fazio et al., 2019). A measure for quality is the number of defects. Anything that breaks the hexagonal symmetry in graphene can be called a defect. Defects in graphene can occur as chemical adsorbates, edges or structural defects, such as vacancies. Perfect zig-zag edges in graphene cannot produce a D peak (Casiraghi et al., 2009; Cançado et al., 2004). Raman spectroscopy in SLG is sensitive to defects and additional peaks appear. Ref. (Ferrari and Robertson, 2000) introduced a three-stage model of amorphisation going from stage 1: perfect SLG/graphite to nanocrystalline graphene, stage 2: nanocrystalline graphene to low-sp3 amorphous carbon and stage 3: low-sp3 amorphous carbon to highly disordered carbon, i.e. tetrahedral amorphous carbon. Here we will focus on stage 1, where the Raman spectrum evolves as follows: 1. 2. 3. 4. 5.

The D peak appears and the ratio of D to G peak intensities, I(D)/I(G), increases The D0 peak appears All peaks broaden The D + D0 peak appears At the end of stage 1 the G and D0 peak are so wide that it is sometimes more convenient to consider them as a single, upshifted, wide G peak at 1600 cm−1

Fig. 10 plots the evolution of the Raman spectrum of defected SLG within stage 1. A correlation between the intensity ratio I(D)/ I(G) and the average crystal size La was first suggested by Tuinstra and Koenig (1970): IðDÞ CðlÞ ¼ La IðGÞ

(8)

where C is a proportionality constant depending on the laser wavelength l, and C(514 nm)
4.4 nm (Tuinstra and Koenig, 1970; Knight and White, 1989). Initially this was interpreted in terms of phonon confinement: the intensity of the forbidden process would be ruled by the ‘amount of lifting’ of the selection rule (Tuinstra and Koenig, 1970). Now we known that the D peak is produced only in a small region of the crystal near a defect or an edge (Casiraghi et al., 2009). For a nanocrystallite, I(G) is proportional to the sample area,

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Fig. 10 Evolution of SLG Raman spectrum for increasing amount of defects (stage 1) introduced by Ar+ bombardment. Reprinted with permission from Cançado LG, Jorio A, Ferreira EHM, Stavale F, Achete CA, Capaz RB, Moutinho MVO, Lombardo A, Kulmala TS, and Ferrari AC (2011) Quantifying defects in graphene via Raman spectroscopy at different excitation energies. Nano Letters 11: 3190–3196. Copyright 2011 American Chemical Society.

∝L2a , while I(D) is proportional to the overall length of an edge, which scales as ∝La (Ferrari and Basko, 2013) and thus I(D)/I(G) ∝ 1/La. In stage 1, for a sample with defects, I(D) is proportional to the total number of defects probed by the laser spot. Therefore for an average interdefect distance LD and laser spot size LL there are on average (LL/LD)2 defects in the area probed by the laser spot and I(D) ∝ (LL/LD)2 (Ferrari and Basko, 2013). On the other hand, I(G) is proportional to the total area probed by the laser spot ∝L2L , thus I(D)/I(G) ¼ C00 (l)/L2D. This gives C00 (l)/L2D ¼ I(D)/I(G) ¼ C(l)/La (Ferrari and Basko, 2013). However, for an increasing amount of defects the Tuinstra-Koenig relation will no longer hold (Ferrari and Robertson, 2000), as in this case I(D) starts to decrease. The D peak is due to a breathing mode in 6-atoms rings (Ferrari and Robertson, 2000). With increasing amount of defects these rings will start to break and, as a result, I(D) will decrease, which is the onset of stage 2. Taking the excitation energy dependence (E4L ) of the Raman peak intensities into account, the following relation for the interdefect distance has been extracted (Cançado et al., 2011):    ð4:3  1:3Þ  103 IðDÞ −1  L2D nm2 ¼ (9) IðGÞ E4L eV 4 with the relation nD[cm−2] ¼ 1014/(pL2D), the defect concentration nD can be related to I(D)/I(G) (Cançado et al., 2011):     IðDÞ nD cm −2 ¼ ð7:3  2:2Þ  109 E4L eV4 IðGÞ

(10)

Eqs. (9) and (10) can then be used directly to estimate the interdefect distance and corresponding defect concentration in SLG by calculating I(D)/I(G) from the fitted D and G peaks. These relations are limited to Raman active defects. Perfect zig-zag edges (Beams et al., 2011; Casiraghi et al., 2009), charged impurities (Casiraghi et al., 2007b; Das et al., 2008), intercalants (Zhao et al., 2011) and uniaxial or biaxial strain (Mohiuddin et al., 2009; Lee et al., 2008; Proctor et al., 2009; Zabel et al., 2012) do not generate a D peak. Other Raman signatures can be used for these ‘silent’ defects. Eqs. (8) and (9) are valid for undoped SLG. However, most samples are not intrinsic and show EF
200–500 meV (Bruna et al., 2014). In the combined case of defects and doping a modified formula has to be used to correctly estimate the defect concentration and interdefect distance, due to the EF dependence of I(D). (Bruna et al., 2014). Ref. Bruna et al. (2014) showed that I(D) decreases as EF increases, Fig. 11, which would result in an underestimation of defects in doped SLG. By including the correction for the doping dependence of I(D), Eqs. (9) and (10) become (Bruna et al., 2014):    ð1:2  0:3Þ  103 IðDÞ −1  L2D nm2 ¼ (11) fEF ½eV g −ð0:540:04Þ IðGÞ E4L eV 4 and     IðDÞ nD cm −2 ¼ ð2:7  0:8Þ  1010 E4L eV 4 fEF ½eV gð0:540:04Þ Ið GÞ

(12)

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Fig. 11 (A) I(D)/I(G) as function of Fermi level EF and charge carrier concentration. (B) Contour plot of the intensity of the Raman signal as function of EF. Reprinted with permission from Bruna M, Ott AK, Ijäs M, Yoon D, Sassi U, and Ferrari AC (2014) Doping dependence of the Raman spectrum of defected graphene. ACS Nano 8: 7432–7441. Copyright 2014 American Chemical Society.

The doping dependence of the D peak intensity can be explained by taking the scattering rate or line broadening energy g ¼ ℏt−1 into account, where t is the lifetime of an electronic state (Basko et al., 2009). In the presence of defects and doping the total scattering rate gtot is governed by electron-phonon interactions as well as scattering due to defects and e-e interactions, which in turn strongly depends on the EF. It is given by (Bruna et al., 2014): gtot ¼ ge−ph + gD + gee(EF), where gee(EF) ¼ 0.06|EF| (Basko et al., 2009). Thus, for increasing EF, gee will increase, and so will gtot. A higher scattering rate indicates a shorter lifetime, which, in turn, results in an increase in peak width and decrease in peak intensity. Eqs. (11) and (12) are valid for defects within stage 1 and for EF < EL/2 in order to avoid Pauli blocking (Bruna et al., 2014).

Magnetic fields Magnetic fields have also an effect on the Raman spectrum of graphene. In the presence of a magnetic field, electronic trajectories will become circular, thus the backscattering condition of the Raman scattering process will be modified, such that the emitted phonons have smaller momenta (Ando, 2007; Goerbig et al., 2007). Ref. (Faugeras et al., 2010) observed a shift toward lower wavenumbers and a broadening of the 2D peak in SLG when applying a perpendicular magnetic field up to 30 T. In the presence of such a strong magnetic field the electronic states are quantized into discrete levels, called Landau levels (Kittel, 2005). Raman spectroscopy can also be used to probe those Landau levels and extract the Fermi velocity (Ferrari and Basko, 2013). Without magnetic field, electronic excitations in SLG have a continuous spectrum (Wallace, 1947). In the presence of a magnetic field, as result of this electronic quantization, inelastic scattering of photons occurs due to inter-Landau level transitions (Faugeras et al., 2011) resulting in sharp B-field dependent peaks where, instead of a phonon, an e-h pair is emitted (Kashuba and Fal’ko, 2012; Kashuba and Fal’ko, 2009; Mucha-Kruczy nski et al., 2010). Applying magnetic fields can also be used to estimate the e-phonon coupling and probe the electronic Landau levels (Faugeras et al., 2011; Kim et al., 2012; Kossacki et al., 2011).

Key issues While acquiring Raman spectra is not particularly difficult, the data analysis needs to be done considering the contribution of different influences, such as strain, doping, defects, etc. Instead of taking single point spectra, Raman measurements are often performed as mapping, where spectra are recorded in a grid pattern across areas of several
mm2. Having a high spatial resolution is important for Raman mapping. The laser spot size, thus the spatial resolution, in conventional Raman spectroscopy is dictated by the diffraction limit of light, in turn proportional to the laser wavelength, thus of the order of
1 mm (Cardona and Güntherodt, 1982). To analyze small flakes ( 373 K. Equilibrium configuration Epitaxial film on MgO T < 670 K 22
1, 21
1 Structures also observed 300 < T < 1300 K 300 < T < 1000 K

Au

Cu Pt W

Table 2

(1 1 0) (1 1 1) (1 0 0) (1 1 0) (1 1 1) (1 0 0) (1 1 0) (1 1 1) (1 0 0) (1 1 0)

Metastable, T < 400 K T < 200 K

Surface structure of (some) semiconductors.

Semiconductor

Face

Reconstruction

Model

Preparation and remarks

Diamond

(1 0 0) (1 1 0) (1 1 1) (1 0 0)

2
1 1
1 2
1 2
1 p(2
2)

Dimers

c(4
2)

Ordered arrangement of buckled dimers

16
2 4
5 5
1 2
1 7
7 5
5 2
1 p(2
2) c(4
2) 2
1 c(2
8)

Dimers + adatoms (?) Uncertain Uncertain p-Bonded chains DAS (dimers-adatoms-stacking faults) DAS Dimers Ordered structure of dimers Ordered structure of dimers p-Bonded chains Adatoms

Polishing + ann. T > 1300 K Polishing + ann. Cleavage; Polishing + ann. T > 1200 K IBA/MBE MBE; IBA. Present locally depending on the temperature MBE; IBA. Present locally depending on the temperature Ann. T > 1300 K 32
2 structures also observed IBA+ ann. Impurity stabilized (Ni?) IBA+ ann. Impurity stabilized (Ni?) Cleavage. Transforms into 7
7 at T > 550 K MBE; IBA+ ann.; cleavage + ann. T > 550 K MBE. Present locally near steps IBA; MBE MBE. Present locally MBE. Present locally Cleavage. Transforms into c(2
8) at T > 400 K MBE; cleavage + ann. at T > 400 K

Si

(1 1 0) (1 1 1) Ge

(1 0 0) (1 1 1)

p-Bonded chains Buckled dimers Ordered arrangement of buckled dimers

(Continued )

260 Table 2

Surfaces and Interfaces, Electronic Structure of (Continued)

Semiconductor

Face

Reconstruction

Model

Preparation and remarks

GaAs

(1 0 0)Ga

InSb

Complex structure with dimers in the second layer Dimers Rotation-relaxation Ga vacancies As trimers Rotation-relaxation

MBE; IBA

(1 0 0)As (1 1 0) (1 1 1)Ga (1 1 1)As (1 1 0) (1 1 1)ln (1 1 1)Sb (0001)Zn (0 0 0 1)O

(4
2), c(8
2) 2
4 1
1 2
2 2
2 1
1 2
2 3
3 1
1 1
1

ZnO

MBE Cleavage; IBA MBE MBE Cleavage IBA; MBE IBA; MBE Cleavage; IBA. Faceting present Cleavage. Faceting present

ann. ¼ annealing; IBA ¼ ion bombardment + annealing; MBE ¼ molecular tom epitaxy.

Electronic surface states The structural variations associated with reconstruction have important consequences on the electronic properties of the surface. Regardless of this, however, the reduction of the translational invariance caused by the surface is enough to introduce additional levels in the electronic structure of the solid. This was shown, for a Kronig-Penney potential, by I. Tamm in 1932 and extended to a more realistic case by W. Shockley in 1939. A 1D potential with a surface at z ¼ 0 is shown schematically in Fig. 2a. Inside the solid (z < 0) the wave function is given, according to the Bloch theorem, by Cðz, t Þ ¼ uk ðzÞ exp½iðkz −ot Þ

(1)

where uk(z) is a function with the same periodicity of the lattice, k ¼ 2p/l is the wave number, and E ¼ ħo is the energy of the electron. For an unbound lattice (−1 < z < +1), k must be a real number so that Eq. (1) represents a wave propagating along z. Imaginary values of k are discarded because the wave function cannot be normalized. Since the energy is a function of k, real (imaginary) values of k correspond to allowed (forbidden) bands. In the presence of a surface, however, imaginary values of k cannot be discarded since now, the wave function can be normalized and states localized at the surface, with energies in the forbidden gaps, become possible. If it is assumed that k ¼ im (with m real), the spatial part of the wave function can be written (for z < 0) as cm ðzÞ ¼ um ðzÞe −mz

(2)

When m < 0, the wave function can be matched, at the surface, to that appropriate for z < 0 (a decaying exponential if the potential is given by a step function). This is shown in Fig. 2b that represents schematically a state localized at the surface with energy in the forbidden gap. Such a state is called a “surface state.” The matching, however, can also be done when k is real, giving rise to the state shown in Fig. 2c that represents a bulk electron scattered at the surface. In three dimensions, the translational symmetry is preserved on the surface plane so that the surface state is now represented by a Bloch wave propagating along the surface with wave vector k‖; damped on both sides of the surface as in the 1D example. In this case, however, the energy of the surface state does not necessarily fall into the forbidden gaps because of the energy (kinetic and potential) associated to the motion along the surface. When the degeneracy of the surface state with the bulk bands is not limited to energy but extends to k-values, the state is called a “resonant state.” A 1D example is shown schematically in Fig. 2d. In a naïve picture, such a resonant state corresponds to a bulk electron that is scattered at the surface, though dwelling there for some time. In order to present the correlation between surface and bulk states, it is customary to project the bulk bands on the surface Brillouin zone and plot the surface bands there. This is done for unreconstructed Si(111) in Fig. 3. Surface states are represented by dotted lines, surface resonances by full lines, while projected bulk bands are shown as hatched regions. When the surface is reconstructed, the calculation of the bands is more difficult and can be done only on the basis of a model. A common procedure is to pile up a number of crystallographic planes where atoms are displaced from their ideal positions according to a given model of reconstruction. The slab should be large enough (10–20 layers) so that in the interior, the structure of the bulk is resumed. Each slab exposes two surfaces. The slabs are then arranged periodically with spacing (between each two) large enough to avoid a sizable perturbation. This fictitious periodic structure allows the use of the theorems of the 3D band theory. The results are usually presented in terms of the so called “local density of states.” As it is known, the “total density of states” rtot(E) gives the number of states present in a unit energy range around E: X dðE −Ei Þ (3) rtot ðEÞ ¼ i

Surfaces and Interfaces, Electronic Structure of

V(z)

261

Evac

E

0

(a)

z

<z

z

Surface state

(b)

<z

Bulk state z (c)

<z

Resonant state z (d)

Fig. 2 Schematic representation of a surface in a 1D model: (a) potential energy of an electron, (b) wave function for a surface state with energy in the gap, (c) bulk electron scattered at the surface, and (d) surface resonant state.

where d is the Dirac function and the sum runs over the energy states Ei. In turn, the local density of states is defined as: X rlocal ðx, y, z; EÞ ¼ jCi ðx, y, zÞj2 dðE −Ei Þ

(4)

i

that is, each state is weighed with the probability of finding an electron at x, y, z. Fig. 4 shows the local density of states for the ideal surface Si(1 1 1)1
1. The figure clearly shows the presence of unsaturated bonds (dangling bonds, DB) on atoms of the first layer. An important development for calculating the structural and electronic properties of a surface came from the application of the so-called Car-Parrinello method that consists of simulating the dynamical evolution of a reduced set of atoms (100), starting from an initial configuration that can as well be the ideal surface (Car and Parrinello, 1985). For each configuration, the electronic energy is calculated and, through the Hellman-Feynman theorem of quantum mechanics, the force on the single atoms is determined and so is the dynamical evolution of the system. Eventually, a stable configuration is reached that corresponds to the reconstructed surface. This “ab initio” calculation requires, however, a high computational power and some “educated guesses,” especially if the reconstruction is not that of thermodynamical equilibrium. In metals, one-electron theories are generally inadequate, and calculations must be done on the basis of the many-body theory. A model that has given reliable results for the work function of several metals is the so-called “jellium” (Lang and Kohn, 1970), which assumes that the electrons form a gas of negatively charged particles neutralized by a continuous distribution of fixed positive charge that goes abruptly to zero at the surface. Fig. 5 shows the electron density distribution near the surface for two cases representative of

262

Surfaces and Interfaces, Electronic Structure of

16 ev 14

Si (1 1 1)

12 10

Si(1 1 1)1u1

Back antibond 8

[2 1 1] tvac

6

M

4

e

J

Si s-like

2

Si dangling bond *

0

[0 1 1] K

–2 –4 –6 –8 –10 –12 –14 M

*

K

Fig. 3 Bulk bands projected into the surface Brillouin zone for the ideal (un-reconstructed) surface of Si(1 1 1). Bulk bands are hatched: surface states are shown with dotted lines: surface resonances with full lines. The surface Brillouin zone is shown on the right. After Schlüter M and Cohen ML (1978) Physical Review B 17: 716.

Si(111)

Dangling bond surface state

Fig. 4 Charge density contours in the (110) plane (perpendicular to the (1 1 1) surface) for the un-reconstructed (ideal) surface of Si(1 1 1). The dangling bonds protruding into vacuum are clearly visible on the atoms of the surface layer. After Schlüter M, Chelikowsky JR, Louie GS and Cohen ML (1975) Physical Review B 12: 4200.

Surfaces and Interfaces, Electronic Structure of

263

rs = 5

Electronic density

1.0

Positive background

0.5 rs = 2

–1.0

–0.5

0

0.5

1.0

Distance from the surface (Fermi wavelength) Fig. 5 Surface charge density for the jellium model of a metal, plotted for two values of the electron densities. The radius rs (in atomic units) is that of the sphere that contains one electron. rs ¼ 2, 5 correspond approximately to the cases of Al and Cs, respectively. After Lang ND and Kohn W (1970) Theory of metal surfaces: Charge density and surface energy. Physical Review B 1: 4555.

Al (dashed line) and of K (continuous line). Electrons protrude from the surface giving rise to an electric dipole directed inwards, which reduces the work function. Characteristic oscillations of the electron density (Friedel oscillations) should be noticed. A further peculiarity of metals is that the major contribution to the surface barrier comes from the image force of electrostatics. The potential energy is then, at a large distance from the surface, Coulomb-like. If the electron energy corresponds to a gap, the electron cannot penetrate the solid and remains confined in a sort of a well that extends from z ¼ 0 to the Coulomb barrier. In such a well, quantized levels called “image states” exist (Echenique et al., 1985). Each state is characterized by an integer quantum number, n  1. In this scheme, the Shockley (or Tamm) states are those with n ¼ 0. The states discussed so far are called “intrinsic surface states” and can be observed only on clean surfaces. On real surfaces, electronic states associated with mesoscopic defects such as steps, kinks, islands, terraces, and pits, or with surface impurities are also present. Such states are called “extrinsic surface states” and may modify the properties of technical surfaces. In semiconductors, the presence of surface states (both intrinsic and extrinsic), especially when localized in the gap, modifies many of the properties of the sample significantly. The accumulation of charge into surface states creates a macroscopic potential that alters the position of the Fermi level at the surface and introduces a “band bending” that extends for a length of the order of the Debye length defined as LD sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eb e0 kT (5) LD ¼ e2 ðnb + pb Þ where eb (e0) is the dielectric constant and nb, pb the densities of negative and positive carriers in the bulk. LD spans from a fraction of an A˚ at metal densities to a fraction of a mm in large gap semiconductors. Such a band bending bears a great importance on many functions of the metal oxide semiconductor (MOS), Schottky barrier, and heterostructure devices.

Examples of reconstruction of relevant semiconductor surfaces A few examples of reconstruction of relevant surfaces of semiconductors and some of their associated electronic properties are given below. An outline of the basic experimental evidence is also presented for each surface. Covalent or partially ionic semiconductors such as C (diamond), Si, Ge, GaAs, GaP, GaSb, InAs, InP, and InSb, crystallize in the diamond, zinc blende (or wurtzite) structures in which each atom is tetrahedrally coordinated with four neighbors. The chemical bond is essentially of the directional sp3 type. The presence of the surface introduces a number of unsaturated bonds called dangling bonds (DBs) that play a great role in reconstruction. Since the energy associated with sp3 bonds is rather high, the atoms on the surface tend to rearrange in order to reduce the number of DBs. This can be done in a number of ways: dimerization, formation of adatoms (i.e., atoms that have left their ordinary position to sit in the interstice on top of three atoms), p-bonding, etc. It should be observed in this respect that the sp3 bond is easily deformable, in the sense that little energy is required to change the angles among the bonds. Reconstruction takes advantage of this flexibility, allowing several different structures to be present even on the same surface.

264

Surfaces and Interfaces, Electronic Structure of

Si(100)2
1 The ideal (1 0 0) face exposes two DBs per surface atom and is then highly unstable. Dimerization is a process that produces the 2
1 reconstruction usually observed. The 2
1 reconstructed surface is shown schematically in Fig. 6: (a) in top view and (b), (c) in side view. The dimers are symmetric in (b) and asymmetric (buckled) in (c). A small buckling is believed to stabilize the surface. The saturation of the DB cannot be complete so that other types of reconstruction are observed, namely p(2
2) and c(4
2) that occur on 5% of the surface. Two STM pictures that show the three types of reconstruction on the same surface are presented in Fig. 7: in the upper picture dimers are seen most distinctly in rows 7, 1, 13 while the p(2
2) reconstruction is seen in rows 14–15. On the other hand, the c(4
2) reconstruction is seen in rows 2–3 of the lower picture (Tromp et al., 1985).

Si(1 1 1)2
1 The (1 1 1) face is the cleavage plane of Si and exposes one DB per surface atom. The 2
1 reconstruction is observed after cleavage at T < 550 K at which temperature it transforms irreversibly into the more stable 7
7 structure. The electronic structure consists of two bands (one full and one empty) separated by a gap of 0.5 eV, from which optical transitions are observed (Chiarotti et al., 1971; Chiaradia et al., 1984). The opening of the gap is probably the process that stabilizes the surface. Several models have been proposed for this surface. The only one that satisfies the various experimental findings is that proposed in 1981 by K C Pandey, which consists of p-bonded chains along [110] directions (Pandey, 1981). The model is shown in Fig. 8 in top and side views. It is seen that in order to create the chains some bonds should be broken. This, however, can be done almost adiabatically (with negligible activation energy) as can be seen from Fig. 9, where the local density of states is plotted for various configurations starting from a slightly buckled (almost ideal) surface. The chain model explains, among others, the following experimental results: (1) the great anisotropy of optical transitions that occur only if the electric vector lies on one of the [1 1 0] directions (Chiaradia et al., 1984) and (2) the strong dispersion of the occupied states observed in photoemission (Himpsel et al., 1981). The model has been subsequently verified directly by STM (Feenstra et al., 1986). A small buckling along the chain (Fig. 8c) is necessary to explain the relatively large optical gap.

1st Layer

3rd Layer

2nd Layer

4th Layer

b B

(a)

A

a

(b)

(c) Fig. 6 Stick and ball model of the surface of Si(1 0 0) reconstructed 2
1 through dimerization: (a) top view. The 1
1 unit cell (hatched, basis vectors a, b) and the 2
1 ceil (basis vectors A, B) are shown: (b) side view; (c) side view with buckled dimers.

p(2 u 2) [1 1 0] [1 1 0]

19 18 1a 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

19 18 17 1b 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 25 Å

c(4 u2)

Fig. 7 STM pictures of the Si(1 0 0)2
1 surface. The photographs show locally p(2
2) reconstruction (rows 14, 15 in the upper picture) and c(4
2) reconstruction (rows 2, 3 in the lower picture). From Tromp RM, Hamers RJ and Demuth JE (1985) Si(001) dimer structure observed with scanning tunneling microscopy. Physical Review Letters 55: 1303.

1

3 5

2 6 4

7

(a) 1

2

3

4

5

7

6

(b) 2 1 3

4

(c)

Fig. 8 Stick and ball model of the Si(1 1 1)2
1 surface, according to the p-bonded chain reconstruction: (a) top view; (b) side view; and (c) side view for a buckled chain.

266

Surfaces and Interfaces, Electronic Structure of

B B

B A

A

A

C D

D (a)

(b)

C

C

D

(c)

Fig. 9 Theoretical contour plots of charge density in the (1 1 0) plane of Si(1 1 1)2
1: (a) for a slightly buckled (nearly ideal) surface; (b) for an intermediate geometry; and (c) for the chain model of Fig. 8. After Northrup JE and Cohen ML (1982) Physical Review Letters 49: 1349.

Si(1 1 1)7
7 As already mentioned, this is the equilibrium reconstruction of the (1 1 1) face. The elementary cell contains 49 atoms in the uppermost plane. It has then a complex structure that resisted, for at least two decades, all attempts at theoretical explanation. By combining accurate electron diffraction at high energies (to avoid multiple scattering), STM pictures (Binnig et al., 1983), and theoretical considerations, K. Takayanagi and collaborators proposed in 1985 the reconstruction model shown in Fig. 10 (Takayanaji et al., 1985). The elementary cell is a rhomb with two nonequivalent halves and four vacancies at the corners. Dimers can be seen along the sides and the shorter diagonal. 12 adatoms and 6 so-called “rest atoms” (i.e., atoms that have maintained their original positions) are also visible. A stacking fault is present in the left half of the elementary cell. The model is then called DAS (dimers-adatom-stacking fault) and is universally accepted. The number of DB (which were 49 in the unreconstructed cell) has now been reduced to 19 (12 on the adatoms, 6 on the rest atoms, 1 in the vacancy). A similar 5
5 structure (with 6 adatoms, 2 rest atoms, and 4 corner vacancies) is seen in surfaces prepared by MBE and is often associated with the more common 7
7. From the electronic point of view, the 7
7 structure has a metallic character as shown by photoemission and by the absence of optical transitions in the gap. The distribution of surface states is shown in Fig. 11 where data of photoemission spectroscopy (left curve) and inverse photoemission spectroscopy (right curve) are plotted as a function of energy. The figure shows clearly that there is a continuous distribution of states across the gap, giving rise to the metallic conductivity of the surface.

Dimer Adatom (a)

Rest atom

(b)

Fig. 10 Structure of the Si(1 1 1)7
7 surface according to the DAS model: (a) top view; (b) side view. Notice the different positions of the lower neighbors of the rest atoms in the left and right half of the elementary cell, caused by the presence of the stacking fault.

Surface density of states (arb. un.)

Surfaces and Interfaces, Electronic Structure of

267

Si(111)7x7

Energy (eV) Energy Fig. 11 Schematic distribution of occupied and empty surface states as obtained by photoemission spectroscopy that detects occupied states (left curve) and inverse photoemission that detects empty states (right curve). The energy scale has been referred to the Fermi level (EF) position.

Fig. 12 Idealized band structure for a topological insulator. The Fermi level falls within the bulk band gap which is traversed by topologically-protected surface states, brought down from the conduction band and raised from the valence band. By A13ean - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index. php?curid¼11101461.

A completely different example are the so called Topological Insulators (as for example alpha-Sn(001), or bismuth and antimony chalcogenide based materials (Bi2Se3, Bi2Te3, Sb2Te3, Bi1−xSbx, Bi1.1 Sb0.9Te2S)) in which there are no broken bonds at the surface and the metallic conductivity arises from a shearing deformation that maintains the local topology at the surface (Hasan and Kane, 2010). From the theoretical point of view, the conductivity is due to a crossed or inverted gap states. In the simplified model shown in Fig. 12, states of the conduction band and valence band coexist at the Fermi level thus giving rise to conductivity.

Interfaces The interface is the region that connects two different physical systems, for example, a metal and a semiconductor (Schottky barrier), two different semiconductors (heterojunction) and, in an extended sense, the surface itself, which defines the boundary between the solid and the vacuum (or a different environment, as a liquid). When the interface forms and thermal equilibrium is reached, the Fermi levels of the two sides line up. This alignment produces a charge redistribution to preserve the overall neutrality of the system, changing the electrical properties of the contact zone. If two metals come into contact, the difference between the respective work functions results in a flow of electrons from the metal with lower work function F1 into the metal with higher work function F2. A contact potential difference D ¼ (F2−F1)/e develops between the metals (Volta effect), and a dipole layer appears at the interface. Due to the high charge density in metals, the thickness of the interface region is only a fraction of an angstrom.

268

Surfaces and Interfaces, Electronic Structure of

If the junction connects a metal and a semiconductor, the Fermi level alignment in both materials results in the band scheme of Fig. 13 where the band structures are shown in equilibrium: (a) before and (b) after the contact. An electron moving from the metal to the semiconductor (or vice versa) experiences a barrier (Schottky barrier, FSB) given by FSB ¼ F −w

(6)

where w is the electron affinity of the semiconductor. The space charge layer in the semiconductor has a much larger width than in the metal because of the difference of carrier densities and screening lengths (for the definition of Debye screening length see the section “Examples of reconstruction of relevant semiconductor surfaces”, Eq. (5)). In Fig. 14, the experimental values of the Schottky barrier for different metals deposited onto the Si(1 1 1)2
1 surface are shown. Though the general trend of Eq. (6) is maintained, it is apparent that the simple theory outlined above (dotted line) is inadequate to fit the experiments. The qualitative explanation is that in deriving Eq. (6), the presence of electronic surface states at the interface (interface states) has been neglected. If their density is sufficiently large, they pin the Fermi level at the surface of the semiconductor, making the Schottky barrier independent of the work function of the metal (dashdotted line in Fig. 14). The experimental situation is intermediate between the two hypotheses (i.e., absence of interface states and complete pinning of the Fermi level). The nature of the interface states has been the subject of extended research. An important advancement came from the introduction of the so-called metal-induced gap states (MIGS) (Heine, 1965). Consider again Fig. 2c of the section “Electronic surface states,” where the scattering of a bulk electron at the surface is outlined. If the electron at the metal side of the junction has an energy corresponding to the semiconductor gap, it cannot propagate for z > 0. A situation similar to that of Fig. 2c is then a reasonable approximation for the interface wave function: the electronic charge spills into the semiconductor giving rise to a density of states localized at the interface in the semiconductor side (MIGS). On the other hand, extrinsic states due to defects, mismatching, impurities, and structural changes induced by interface chemical reactions have the same general effect as MIGS. A comprehensive theory of the electronic structure of interfaces is at present not available.

)sc

)M

Evac

Evac Fsc Ec

EFM

eVs

n

)SB

SC

EF

EC EF

Ev

EV

Metal n-Type (a)

(b)

Semiconductor

Fig. 13 Band scheme of a metal-semiconductor (n-type) system: (a) before the contact is established; (b) in equilibrium after the contact. FM (FSC) are the work functions of the metal (semiconductor), wSC the electron affinity of the semiconductor. FSB the Schottky barrier height, and eVS the band bending at the surface.

1.0 Au Pt Pd

Si(111)(2 u1) Barrier height )SB (eV)

0.8 Al

(b)

AgCu Ni Cu

0.6

Cr

Al Mg 0.4

Ca Cs (q = 1)

0.2 (a) 0.0

0

2

4

6

Metal work function, eFM (eV) Fig. 14 Experimental values of Schottky barriers for several metals on top of Si(1 1 1)2
1. The lines represent (a) the Schottky behavior given by Eq. (6) (dotted) and (b) the case of complete pinning of Fermi level at the surface (Bardeen behavior, dash-dotted). After Mönch W (1986) Festkörperprobleme. In: Advances in Solid State Physics, vol. 26. Braunschweig: Vieweg.

Surfaces and Interfaces, Electronic Structure of

269

Evac Semiconductor I

Semiconductor II EC

EF

'EC

EF EF

'EV

EV

(a)

(b)

Interface states

eV1

EF

EC

eV2

EF

EC

EV

(c)

(d)

Ev

Fig. 15 Band scheme of a heterojunction: (a) when the materials are not in contact, (b) after the contact is established, but before equilibrium (Fermi levels not aligned), (c) in equilibrium, (d) in equilibrium, when interface states are present.

If the interface is created between two different semiconductors, it is called a heterojunction. The alignment of the Fermi levels on both sides determines the band scheme shown in Fig. 15. The difference between the forbidden gaps of the two semiconductors is distributed between the valence-band discontinuity (DEv) and the conduction-band discontinuity (DEc). Moreover, band bending (eV1 and eV2) occurs at the two sides of the interface. Given their great influence on carrier transport, they are the important parameters determining the performance of the interface and play a central role in the behavior of real heterojunction devices (diodes, solar cells, photodetectors, lasers, quantum devices, etc.). The presence of electronic states at the interface modifies the distribution of charge producing the situation schematically presented in Fig. 15d. Such states originate, in analogy to MIGS, from the tailing of the semiconductor wave function in the forbidden gaps. In the ideal heterojunction, the two materials have the same (or slightly different) lattice parameters so that there is little strain at the interface, and the materials maintain their original properties up to the abrupt junction. If the lattice parameters are different, strain results at the junction, with the formation of dislocations, defects, and disorder. Sizable interdiffusion and chemical reactions may also occur. Such defects give rise to extrinsic electron states. Three significant examples are given below: 1. The interface resulting after growth of GaAs (lattice parameter a ¼ 5.6531 A˚ ) onto AlAs (a ¼ 5.6622 A˚ ) is a representative example of a nearly ideal heterojunction between materials with a very small lattice mismatch (Da/a ¼ 0.5%). The resulting interface is atomically abrupt, with a very low number of defect states. 2. The interface between Ge (a ¼ 5.6578 A˚ ) and Si (a ¼ 5.4307 A˚ ) presents a lattice mismatch of 4%, too large to be simply accommodated by a strained layer. The interface is then characterized by intermixing, with the formation of a GedSi alloy. 3. In the junction between Ge (a ¼ 5.6578 A˚ ) and GaAs (a ¼ 5.6531 A˚ ), the mismatch is extremely favorable (about 8
10−4). Nevertheless, the chemistry of materials, that is, the high chemical affinity of Ge, Ga, and As produces a great complexity of the interface composition, with possible effects on doping. Ge acts as a donor or an acceptor in bulk GaAs, while Ga and As are, respectively, an acceptor or a donor when the Ge atom is substituted. Heterojunctions and heterostructures (i.e., the sequence of a high number of heterojunctions, as in a superlattice) are usually grown by MBE or metalorganic chemical vapor deposition (MOCVD). These epitaxial techniques allow the production of graded junctions that are important in many semiconductor devices.

Surfaces in real-world environment The investigation of surfaces immersed in liquids is remarkably less common with respect to the case of surfaces in UHV. However, important technologies are based on processes that occur at the solid/liquid interface, like electro-catalysis, grafting, biofunctionalization of surfaces, electrodeposition and electro etching of metals down to the nanometer scale. Also biophysical and biochemical systems are commonly prepared and studied in aqueous solutions, and processes of high scientific relevance take place at the

270

Surfaces and Interfaces, Electronic Structure of

contact with the liquid ambient. These facts urge to bridge the gap between the research on surfaces under ideal conditions (as in UHV) and on surfaces under realistic conditions. Although the cleanliness conditions (under preparation protocols for both the surface and the liquid) allow the maintenance of the clean phase for a surface in solution during the time necessary for experiments, only a few spectroscopies and methods can penetrate a thin layer of liquid and reach the solid/liquid interface for in situ study. Photon-based probes [as optical techniques -e.g. Reflectance Anisotropy spectroscopy (RAS)- or X-ray diffraction methods] do not suffer this limitation, provided that the liquid is not absorbing in the selected photon energy range, as already demonstrated in pristine electroreflectance measurements. Also scanning probes (AFM and STM, the latter after the metal tip has been insulated by an appropriate coating) allow the investigation of immersed surfaces, and real space images of their structure can be acquired with atomic resolution. It has thus been shown that single crystalline surfaces of metals in liquid exhibit clean phases as in UHV (Fig. 16a) (Wilms et al., 1999). Furthermore, in electrolytes the adsorption/desorption of ions -effectively assisted by appropriate voltage values applied to the surface with respect to solution- can induce the reconstruction of the surface, determining differently ordered layers (Fig. 16b). Similarly, if different species (ions and/or molecules) coexist within the solution, competing for adsorption sites on the surface, the choice of the potential applied to the sample can tune processes otherwise not possible in UHV (as the electro-compression/-decompression of iodide layers on Cu(100) and Cu(111)). Electrochemical reactions at the solid-liquid interface can be monitored in real time, as shown in Fig. 17, where the interaction of chloride ions with a (110) copper surface in hydrochloric acid solution is followed in situ by RAS and STM (Goletti et al., 2015).

Conclusion The termination of a crystal with a cutting plane, producing what we call the surface, is not merely the introduction of a geometrical boundary, but creates a new fascinating world having peculiar structural and electronic properties with respect to the underlying bulk. Any surface is then an interface, that is a region separating (with a quite abrupt discontinuity) the solid and a different phase in contact (solid, liquid, gaseous), associated to important physical and chemical processes in sensors, piezoelectric and optoelectronic devices, fuel cells, catalysts, solar cells, nanotechnology, etc.. Surfaces are the skin of the materials, the interface with the external world, and consequently the place where fundamental phenomena happen. Clean surfaces in UHV have represented the playground where nearly all the computational methods and the experimental techniques developed for about 50 years in solid state physics have been applied. In the last years new (sometimes revolutionary) topics have appeared, confirming that the physico-chemical properties of surfaces and interfaces govern the interfacial transport of light, excitons, electrons, ions, energy as well as the transduction of electrons into the molecular language of cells. To mention some important examples: (i) topological materials, with surfaces that are robust against any perturbations as long as respecting symmetry of the system (Moore, 2010); (ii) graphene and 2D van der Waals (vdW) layered semiconductors, truly surface materials providing applications in numerous quantum mechanical devices (Duong et al., 2017); (iii) surfaces of wide bandgap and metal oxide materials (SiC, GaN, Ga2O3, ZnO, diamond, TiO2), with an overwhelming range of physical and chemical applications (Diebold et al., 2010); (iv) biological and organic interfaces, leading to the new challenges of the organic bioelectronics emerging field (Fahlman et al., 2019); (v) superhydrophobic surfaces, whose investigation has become a hot topic for its outstanding properties and wide application values in recent years (Jiaqiang et al., 2018).

Fig. 16 Panel (a) Atomic structure of clean Cu(111) in 5 mM H2S04 (2.7 nm
2.7 nm, lt ¼ 51 nA, Ubias ¼ 286 mV). The potential applied to the surface is −800 mV vs. Hg/Hg2S04. Panel (b) Adsorbate structure on Cu(111) in the same solution, when the applied voltage is −440 mV vs. Hg/Hg2S04 (9.6 nm
9.6 nm, lt ¼ 51 nA, Ubias ¼ 195 mV). The small bright dots in panel b represent individual sulfate or bisulfate anions; the long range intensity modulation indicates a Moire superstructure. From Wilms M, Kruft M, Bermes G and Wandelt K (1999) A new and sophisticated electrochemical scanning tunneling microscope design for the investigation of potentiodynamic processes. Review of Scientific Instruments 70: 3641.

Surfaces and Interfaces, Electronic Structure of

271

Fig. 17 Evolution of the optical anisotropy signal of a Cu(110) surface immersed in hydrochloric acid solution (0.1
10−3 M) as a function of the potential applied to the surface with respect to a reference electrode. The signal is acquired at fixed photon energy (2.5 eV), corresponding to the maximum of the RAS spectrum due to the adsorption of chloride ions on the copper surface. DRAS measures the variation of the RAS signal with respect to the clean copper surface (occurring at −600 mV). The arrows inside the curve indicate the scan direction of the potential (positive and negative direction). Reported STM images (81
81 nm2, VBIAS 30 mV, lt 1.4 nm) have been acquired in liquid in the same conditions: (2) clean copper surface; (6) stripes of adsorbed chlorine running along the [001] direction of the surface; (11) copper terraces reappearing after partial desorption of chlorine. From Goletti C, Bussetti G, Violante A, Bonanni B, Di Giovannantonio M, Serrano G, Breuer S, Gentz K and Wandelt K (2015) Cu(110) surface in hydrochloric acid solution: Potential dependent chloride adsorption and surface restructuring. The Journal of Physical Chemistry C 119: 1782.

References Binnig G, Rohrer H, Gerber C, and Weikel E (1983) 7x7 reconstruction on Si(111) resolved in real space. Physical Review Letters 50: 120. Car R and Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Physical Review Letters 55: 2471. Chiaradia P, Cricenti A, Selci S, and Chiarotti G (1984) Differential reflectivity of Si(111)2x1 surface with polarized light: A test for surface structure. Physical Review Letters 52: 1145. Chiarotti G, Nannarone S, Pastore R, and Chiaradia P (1971) Optical absorption of surface states in ultrahigh vacuum cleaved (111) surfaces of Ge and Si. Physical Review B 4: 3398. Diebold U, Li SC, and Schmid M (2010) Oxide Surface Science. Annual Review of Physical Chemistry 61: 129. Duong DL, Yun SY, and Lee YH (2017) van der Waals Layered Materials: Opportunities and Challenges. ACS Nano 11: 11803. Echenique PM, Flores F, and Sols F (1985) Lifetime of image surface states. Physical Review Letters 55: 2348. Fahlman M, Fabiano S, Gueskine V, Simon D, Berggren M, and Crispin X (2019) Interfaces in organic electronics. Nature Reviews Materials 4: 627. Feenstra RM, Thompson WA, and Fein AP (1986) Real-space observation of p-bonded chains and surface disorder on Si(111)2 1. Physical Review Letters 56: 608. Goletti C, Bussetti G, Violante A, Bonanni B, Di Giovannantonio M, Serrano G, Breuer S, Gentz K, and Wandelt K (2015) Cu(110) surface in hydrochloric acid solution: Potential dependent chloride adsorption and surface restructuring. The Journal of Physical Chemistry C 119: 1782. Hasan MZ and Kane CL (2010) Colloquium: Topological insulators. Reviews of Modern Physics 82: 3045. Heine V (1965) Theory of surface states. Physical Review 138: A1689. Himpsel FJ, Heimann P, and Eastman DE (1981) Surface states on Si(111)-(2x1). Physical Review B 24: 2003. Jiaqiang E, Jin Y, Deng Y, Zuo W, Zhao X, Han D, Peng Q, and Zhang Z (2018) Wetting models and working mechanisms of typical surfaces existing in nature and their application on superhydrophobic surfaces: A review. Advanced Materials Interfaces 5: 1701052. Lang ND and Kohn W (1970) Theory of metal surfaces: Charge density and surface energy. Physical Review B 1: 4555. Moore JE (2010) The birth of topological insulators. Nature 464: 194. Pandey KC (1981) New p-bonded chain model for Si(111)-(2x1) surface. Physical Review Letters 47: 1913. Takayanaji K, Tanishiro Y, Takahashi M, and Takashi S (1985) Structural analysis of Si(111)-7
7 by UHV-transmission electron diffraction and microscopy. Journal of Vacuum Science and Technology A 3: 1502. Tromp RM, Hamers RJ, and Demuth JE (1985) Si(001) dimer structure observed with scanning tunneling microscopy. Physical Review Letters 55: 1303. Wilms M, Kruft M, Bermes G, and Wandelt K (1999) A new and sophisticated electrochemical scanning tunneling microscope design for the investigation of potentiodynamic processes. Review of Scientific Instruments 70: 3641.

Further reading Bechstedt F (2003) Principles of Surface Physics. Berlin: Springer. Bechstedt F and Enderlein R (1988) Semiconductor Surfaces and Interfaces. Berlin: Akademie-Verlag. Chiarotti G (ed.) (1993-1996) Physics of Solid Surfaces: Landolt-Börnstein New Series, Group III/24. In: Physics of Solid Surfaces, vol. 4. Berlin: Springer.

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Heine V (1965) Theory of surface states. Physical Review 138: A1689. Lannoo M and Friedel P (1991) Atomic and Electronic Structure of Surfaces. Berlin: Springer. Lüth H (2001) Solid Surfaces, Interfaces and Thin Films, 4th edn Berlin: Springer. Lüth H (1993) Surfaces and Interfaces of Solid Materials. Berlin: Springer Verlag. Mönch W (1995) Semiconductor Surfaces and Interfaces, 2nd edn Berlin: Springer. Mönch W (1986) Festkörperprobleme. Advances in Solid State Physics. 26: Braunschweig: Vieweg. Wandelt K and Thurgate S (eds.) (2003) Solid−Liquid Interfaces: Macroscopic Phenomena-Microscopic Understanding. In: Solid-Liquid InterfacesBerlin: Springer Verlag. Wandelt K (ed.) (2016) Surface and Interface Science: Solid/liquid and Biological Interfaces. In: Solid-Liquid Interfaces, vol. 7. Wiley VCH. Zangwill A (1988) Physics at Surfaces. Cambridge: Cambridge University Press.

Graphene, electronic properties and topological properties Katsunori Wakabayashi, Department of Nanotechnology for Sustainable Energy, School of Science and Technology, Kwansei Gakuin University, Sanda, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Electronic state of graphene Topological properties of graphene Graphene nanoribbons and edge states Charge polarization and Zak phase Energy spectrum and wave functions: Graphene nanoribbons Armchair nanoribbons Zigzag nanoribbons Electronic states of carbon nanotubes Conclusion References

273 274 277 278 280 281 281 282 284 286 286

Abstract Graphene is a one-atomic thickness carbon sheet with a hexagonal lattice structure. The low-energy electronic states of graphene near the Fermi energy are well-described by massless Dirac equation, which gives conical energy dispersion and nontrivial Berry phase. The analysis of Zak phase (known also as Z 2 Berry phase or one-dimensional version of Berry phase) clarifies the existence of topological edge states at graphene zigzag edges. In this chapter, we provide an overview of electronic and topological properties of graphene. The analytic properties of energy spectrum for graphene nanoribbons and carbon nanotubes are also presented.

Key points

• • • • •

Electronic and topological properties of graphene are presented. Low-energy electronic states of graphene are well-described by massless Dirac equation, resulting in the linear energy dispersion with Berry phase p. Analysis of Zak phase on electronic wave functions of graphene predicts the existence of graphene edge states. Analytic properties of energy spectrum for graphene nanoribbons are presented. Analytic properties of energy spectrum for carbon nanotubes are presented.

Introduction Graphene is a one-atomic thickness monoatomic carbon sheet, which has the hexagonal lattice structure (Geim and Novoselov, 2007; Novoselov et al., 2004, 2005; Zhang et al., 2005). Since carbon atoms in graphene form sp2 hybridization, each carbon atom provides one p-electron which itinerates over the graphene. The p-electron of sp2 carbon atoms governs the electronic properties of graphene near the Fermi energy (Aoki and Dresselhaus, 2014; Foa Torres et al., 2014; Katsnelson, 2012; Neto et al., 2009; Saito et al., 1998; Wallace, 1947). Owing to the honeycomb crystal structure of graphene, there are two inequivalent sublattices (two geometrically nonequivalent carbon atoms) in the unit cell, which work as the internal degree of freedom for electron wave functions. In particular, the motion of electrons in graphene near the Fermi energy is well described by the massless Dirac equation (Weyl equation), which can be obtained from tight-binding model for p-electrons by expanding the wave function at the Fermi points (Ajiki and Ando, 1993; Ando, 2005; Ando et al., 1998). At these points, graphene has the conical energy dispersion, i.e., Dirac cone, where the valence and conduction bands conically touch with each other. Since the wave function at Dirac cone has the same form with the eigenfunction of a spin in magnetic monopole field, it accompanies with the nontrivial topological phase, i.e., Berry phase (Berry, 1984). Further topological aspect can also be observed by inspecting the edge effect on electronic properties of graphene. The presence of edges in graphene has strong implications for the low-energy spectrum of the p-electrons (Fujita et al., 1996; Nakada et al., 1996; Wakabayashi et al., 1999). There are two basic edge shapes, armchair and zigzag, which determine the properties of graphene nanoribbons (GNRs). It has been shown that nanoribbons with zigzag edges (zigzag graphene nanoribbons, ZGNRs) possess localized edge states with energies close to the Fermi level (Fujita et al., 1996; Nakada et al., 1996; Wakabayashi et al., 1999). In contrast, edge states are absent for nanoribbons with armchair edges (armchair graphene nanoribbons, AGNRs). The origin of

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edge states can be understood by the consequences of finite Zak phase of bulk wave function of graphene. Zak phase is one-dimensional version of Berry phase (Zak, 1989). In this chapter, we give an overview of topological aspects on electronic properties of graphene. We start with the basics of p-electronic states of graphene using tight-binding model, and derive the massless Dirac equation to show the singular topological properties at Fermi points. Further, we give the overview of edge effect on p-electronic states of graphene on the basis of GNRs, where it is shown that zigzag edges provide robust topological edge states. Finally, we give analytic properties of energy spectrum for GNRs and carbon nanotubes.

Electronic state of graphene We employ a single-orbital nearest-neighbor tight-binding model for the p-electron network to describe the electronic states of graphene (Neto et al., 2009; Wallace, 1947). Fig. 1A and B show the lattice structure and the first Brillouin zone (BZ) of graphene, respectively. The corners of the first BZ are called K+ or K− points, which are also referred to as Dirac points because the energy spectrum at these corners can be described by the massless Dirac equation (Weyl equation). Let us define the wave functions at RA of a lattice site A and RB of a lattice site B as cA(RA) and cB(RB), respectively. These wave functions can be related by the following Schrödinger equations of the tight-binding model; EcA ðRA Þ ¼ −g0

Unit cell

(A)

3 X l¼1

cB ðRA − t l Þ,

(1)

b2

(B)

A b1

B

a2

M

a1

Γ ky

y

a kx

x

(C)

(D)

3 Energy E/γ0

Energy

E/γ0

2 1 0 -1

3

0

Conduction band

K

Fermi Energy

-2 -3

Γ

M

K

Γ

Density of States D(E)

-3

Valence band

Fig. 1 (A) Graphene sheet in real space, where the black (white) circles denote A(B)-sublattice sites; a is the lattice constant and a1 ¼ (a, 0) and    pffiffiffi    1 p1ffiffi 2p 2 a 2 ¼ −a=2, 3a=2 are the primitive vectors. (B) First BZ of graphene. K + ¼ 2p a 3 , 3 , K − ¼ a 3 , 0 , G ¼ ð0, 0Þ: Note that there are three K+ and K−

points, which can be connected by the reciprocal lattice vectors. The valence and conduction bands are in contact at the degeneracy points, K and K0 . (C) Energy band structure of graphene within the irreducible BZ, together with DOS of a graphene sheet. (D) (left) 3D plot of the energy band structure for p-electrons. (right) Zoom of energy band structure around K-points, where the energy dispersion has conical shape, i.e., Dirac cone.

Graphene, electronic properties and topological properties

EcB ðRB Þ ¼ −g0

3 X l¼1

cA ðRB + tl Þ:

275

(2)

Here E is the eigenenergy. RA ¼ p n1ffiffiffia1 + n2a2 + t 3 and RB ¼ n1a1 + n2a2, where integers n1 and n2 represent the locations of sites A and  B, respectively. t3 ¼ a=2, −a=2 3 is the relative vector between A and B atoms. a ¼ 2.46 A˚ is the lattice constant of graphene. g0 is the transfer integral between nearest-neighbor carbon sites, which has been estimated to be about 2.75 eV in a graphene system. Here we assume a plane wave form of the wave function, i.e., cA(RA) ∝ fA(k) exp (ik  RA) and cB(RB) ∝ fB(k) exp (ik  RB), where k ¼ (kx, ky) is the wavenumber vector. Thus, we obtain the eigenvalue equation     f ðkÞ f ðkÞ HðkÞ A ¼E A , (3) f B ð kÞ fB ðkÞ with the Hamiltonian HðkÞ ¼

0 hðkÞ

! h∗ ðkÞ : 0

(4)

Here   hðkÞ ¼ −g0 1 + e−ika1 + e−ika2 ¼ −g0 |hðkÞ|eifk , and fk is the phase measured from kx axis. Thus, the energy bands are given by ES ¼ s |f(k)|, namely, ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3ky a kx a kx a 2 Es ðkÞ ¼ sg0 1 + 4 cos + 4 cos cos 2 2 2

(5)

(6)

with s ¼  1. Because one carbon site has one p-electron on average, only the E−(k) band is completely occupied. Thus, s ¼ +1 and s ¼ −1 correspond to the conduction and valence bands, respectively. The wave function can be written as     fA ðkÞ 1 1 : (7) ¼ pffiffiffi fB ðkÞ 2 seifk Thus, the Bloch wave function of graphene has a relative phase between A and B sublattice sites. The density of states (DOS) is calculated as ð 1 X 1 DðEÞ ¼ − Im dk , (8) p s¼ 1stBZ E − Es ðkÞ + i where the k-integration is over the first BZ and  is an infinitesimally small real number. Fig. 1C shows the energy band structure of graphene for p-electrons and corresponding DOS. Fig. 1D shows 3D plot of the energy band structure. Near the G point, both valence and conduction bands can be expressed as quadratic functions of kx and ky, i.e., E(k) ¼  g0(3 − 3|k|2/4). At the M points, which are the midpoints of the sides of the hexagonal first BZ, a saddle point appears in the energy dispersion in the vicinity of E ¼  g0, resulting in logarithmic divergence of the DOS as can be seen in Fig. 1C. Near the K point at pffiffiffithe corner of the hexagonal first BZ, the energy dispersion is a linear function of the magnitude of the wave vector, EðkÞ ¼  3g0 a|k|=2, where the DOS linearly depends on energy. Here a(¼|ai|(i ¼ 1, 2)) is the lattice constant. The Fermi energy is located at the K points, and there is no energy gap at these points because E(k) vanishes there owing to the hexagonal symmetry. Owing to the linear dispersion at K points, graphene has the conical dispersion as shown in the zoom of Fig. 1D. The slope of the conical dispersion gives the Fermi velocity vF of graphene, i.e., pffiffiffi 1 3 g0 a vF ¼ rk EðkÞ ¼ , (9) 2 ħ ħ which is approximately vF  106 m/s. Hence, the Fermi velocity of graphene is about 1/300 of the speed of light. The electronic state in the vicinity of a K point can be described by the massless Dirac equation, i.e., Weyl equation (Ajiki and Ando, 1993; Ando et al., 1998). In the continuum limit of lattice coordination, the wave function around K+ point can be written as ( cA ðr Þ ¼ eiK + r fKA + ðr Þ (10) cB ðr Þ ¼ eiK + r fKB + ðr Þ Here fKA+(r) and fKB +(r) are the envelope functions for A and B sublattice at the electron coordinate r ¼ (x, y). Applying the Taylor expansion after we substitute these form into the tight-binding equations, we obtain the following equation, ! 0 k^x − ik^y g (11) FK + ðr Þ ¼ EFK + ðr Þ, 0 k^x + ik^y where FK+(r) is two-component envelope functions

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Graphene, electronic properties and topological properties

FK + ðr Þ ¼

! fKA + ðr Þ : fKB + ðr Þ

(12)

k^x and k^y are wave number operators and are replaced with the differential operators of the coordinates k^x ! −i ∂x∂ and k^y ! −i ∂y∂ : The band parameter g is given by pffiffiffi 3a g¼ (13) g : 2 0 Similarly, for K− point, we have ! 0 k^x + ik^y g (14) FK − ðr Þ ¼ EFK − ðr Þ: 0 k^x − ik^y Eq. (11) is mathematically equivalent to the two-dimensional version of Weyl equation, which describes the motion of massless relativistic particle, and also refereed as massless Dirac equation. The discussion can be also extended to multi-layer graphene and twisted bilayer graphene (Bistritzer and MacDonald, 2011; Guinea et al., 2006; Koshino and Ando, 2007; Koshino et al., 2018; McCann and Fal’ko, 2006; Nilsson et al., 2006). In terms of the Pauli spin matrices       0 1 0 −i 1 0 sx ¼ , sy ¼ , sz ¼ , (15) 1 0 i 0 0 −1 Eq. (11) can be written as 



gs  ^ kFK + ðr Þ ¼ EFK + ðr Þ,

(16)

where s ¼ (sx, sy) and ^ k ¼ k^x , k^y : Similarly, Eq. (14) can be rewritten as gs ∗  ^ kFK − ðr Þ ¼ EFK − ðr Þ,

(17)

where s∗ ¼ (sx, −sy). The energy spectrum and eigenfunction of massless Dirac equation can be obtained by assuming the plane wave form as a solution of the envelope function, namely,   f FK  ðr Þ∝ A exp ðik  r Þ: (18) fB Here it should be noted that the wave number k is measured from either K+ or K− point. From this, we can immediately obtain the linear spectrum of energy as ES ðkÞ ¼ sg|k|, with s ¼ 1:

(19)

1 FKsk ðr Þ ¼ pffiffiffiffiffiffiffiffiffi expðik r ÞFKsk , Lx Ly

(20)

The wave function can be written as

with the spin sector 1 FKsk ¼ pffiffiffi 2



1 seiyk

 :

(21)

Here uk is defined as the angle between wave vector k and the kx-axis, i.e., kx + iky ¼ | k | eiuk. Note that the term “spin” appearing here does not mean the real spin of the electron, since the eigenfunction for the spin sector describes the amplitude of the wave function on A and B sublattices. Thus, it is normally called pseudo-spin to distinguish it from the real electron spin. It should be noted that K+ and K− points have opposite phase uk. Using Eq. (19), Eq. (16) can be rewritten as s^ k K+ F ðr Þ ¼ sFK + ðr Þ: |k|

(22)

^

It should be noted that s|k| k represents the helicity operator. Thus, the helicity of electrons in graphene depends on the band index s. Similarly, the expectation value of pseudo-spin around K+point is given as   (23) hsi ¼ hsx i, sy ¼ sðcos uk , sin uk Þ: These results indicate that the electron (hole) near K+ point has a definite pseudo-spin direction, which is parallel (antiparallel) to the direction of motion. These properties are also refereed as chirality. For the state near K− point, the chirality is obtained by reversing the sign of s, i.e., s ! − s. Thus, K+ and K− have opposite chirality from each other, which is the consequences of

Graphene, electronic properties and topological properties

277

two-dimensional analog of the Fermion doubling (Aoki and Dresselhaus, 2014). In addition, the chiral property of wave function leads to the Klein tunneling effect (Katsnelson, 2012; Katsnelson et al., 2006).

Topological properties of graphene The wave function of pseudo-spin sector contains the nontrivial Berry phase (Ando et al., 1998). To see this fact further, here we briefly summarize the key results for topological aspects (Bernevig and Hughes, 2013; Berry, 1984; Vanderbilt, 2018; Xiao et al., 2010). Now let us consider a physical system in which Hamiltonian H smoothly depends on the parameter R, i.e., H ¼ H(R). This parameter slowly changes with time and moves along an arbitrary enclosed path C in R space. To be more specific, let us denote t ¼ 0 as the initial time, and t ¼ T as the final time, so the R(t) satisfies R(0) ¼ R(T ) for a closed path. Now we consider the time-evolution of wave function. The time-dependent Schrödinger equation is HðRðt ÞÞjc ðt Þi ¼ iħ

∂ jc ðt Þi: ∂t

(24)

For each time, we assume that the wave function satisfies the orthonormal basis set {fn(R(t))} with the eigenvalue equation HðRðt ÞÞjfn ðt Þi ¼ En ðRðt ÞÞjfn ðt Þi:

(25)

By assuming that the wave function adiabatically changes, the wave function at time t can be written as jfn ðt Þi ¼ eign ðt Þ jfn ðRðt ÞÞi:

(26)

In general, the phase gn(t) does not need to be zero. Substituting |fn(t)i into Eq. (24), we obtain   d d En ðRn Þjfn ðRÞi ¼ ħ gn jfn ðRÞi + iħ jfn ðRÞi: dt dt Taking the inner product with hfn(R)|, we obtain i

∂ i ∂ g ¼ − En ðRÞ− hfn ðRÞ| |fn ðRÞi ∂t n ħ ∂t i ∂R ¼ − En ðRÞ− hfn ðRÞjrR jfn ðRÞi  : ħ ∂t

Since we are considering a closed path for the parameter R, the system Hamiltonian returns to the initial state after the time T. However, the phase difference between the initial and last time is as follows. gn ðT Þ−gn ð0Þ ðT ∂gn ðt Þ dt ¼ ∂t 0 ðT I 1 En ðRðt ÞÞdt + i hfðRÞjrR jfðRÞi dR ¼− ħ 0 C

(27)

The first term is the conventional dynamical phase. The second part is the geometric phase which depends on the path C, and called Berry phase. When we define the Berry connection An(R) as An ðRÞ ¼ −ihfðRÞjrR jfðRÞi,

(28)

we can rewrite the Berry phase as I gn ½C ¼ −

C

An ðRÞ dR:

(29)

Physically, the Berry connection can be understood as the vector potential in R space, and its rotation F(R) ¼ r  A(R) can be understood as the magnetic field in R space, which is called Berry curvature. Using Stokes’ theorem, the line integral can be written as surface integral I I gn ½C ¼ − r  An ðRÞ dS ¼ − Fn ðRÞdS (30) S

S

Thus, the Berry phase can be understood as arising from magnetic flux in R space. Let us now consider the electronic states of graphene near K points, i.e., Dirac cone. We consider the Bloch wave vector k as the parameter of Hamiltonian H(k). From Eq. (21), we can immediately calculate the Berry connection as follows,

1 AKs  ðkÞ ¼ hFKsk jrk jFKsk ¼  rk yk : 2

(31)

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Graphene, electronic properties and topological properties

It should be noted that K+and K− points have opposite signs for Berry connection. The Berry phase is obtained by the line integration along a closed path with anti-clockwise direction, i.e., I gKs  ½C ¼ AKs  ðkÞ dk C ( (32) p if C encloses k ¼ 0, ¼ 0 if C dose not enclose k ¼ 0: If we recall that the Berry connection A(k) is the vector potential in k space, the Berry phase indicates the magnetic flux through the area enclosed by the path C. Thus, there is a magnetic flux at k ¼ 0 with the magnitude of p, i.e., F Ks  ðkÞ ¼ pdðkÞ:

(33)

We should note that the Berry curvature only exists at Dirac point. In general, if the system has time-reversal symmetry, Berry curvature satisfies F(k) ¼ − F(−k). If the system has the crystal inversion symmetry, Berry curvature satisfies F(k) ¼ F(−k). Hence, if the system has both time-reversal and crystal inversion symmetries, Berry curvature becomes F(k) ¼ 0. Since K+and K− points are connected with the time-reversal symmetry, they have opposite Berry curvature. However, as can be seen below, at Dirac K+ and K− points, graphene acquires non-zero Berry curvature but opposite sign. Owing to this topological singularity, the Boltzmann conductivity of graphene at E ¼ 0 gives the universal value e2/pħ even in the vanishing of density of states (Ando, 2005). In later section, we will inspect Zak phase (also known as Z 2 Berry phase or one-dimensional version of Berry phase) of graphene. Zak phase illustrates the topological origin of graphene edge states.

Graphene nanoribbons and edge states The presence of a graphene edge has a strong impact on the electronic states of Dirac electrons. In this section, we briefly summarize the electronic structures of GNRs using the tight-binding model to see the edge effect on the electronic states of graphene. There are two types of graphene edges: armchair and zigzag edges. These two edges have a 30 difference in their orientation within the graphene sheet. A large difference in the p-electronic structures is induced by the two types of graphene edges (Fujita et al., 1996). In particular, a zigzag edge exhibits localized states, whereas an armchair edge does not exhibit such localized states. The origin of edge state is the topological properties of bulk wave function, which will be explained in the next section. The lattice structures p offfiffiffiGNRs are shown in Fig. 2A and B. Fig. 2A shows a schematic of AGNR. The primitive vector for AGNR is a ¼ (0, aT), where aT ¼ 3a with a being the lattice constant of graphene. Fig. 2B shows a schematic of ZGNR. Similarly, the primitive vector for this system is given by a ¼ (a, 0). We define the width of GNRs as N, where N is the number of dimer (two carbon sites) lines for AGNRs and the number of zigzag lines for ZGNRs. It is assumed that all dangling bonds at graphene edges are terminated by hydrogen atoms and thus do not contribute to the electronic states near the Fermi level. We employ a single-orbital tight-binding model for the p-electron network. Details of the calculation of the eigenenergy spectrum and eigenfunctions are described in the last section. Here we briefly overview the electronic (A) x

(B)

aT

y

1A 2B

0 1 2 3 4

1B 2A

a

N+1 (y=W)

(x=0)

NB NA

3B

B A

NA

NB

A

unit cell

N N+1 (x=W)

3

3A 2B

B

N

2

2A 1B

1

1A

y

0 x

(y=0)

unit cell

Fig. 2 Structure of graphene nanoribbons with (A) armchair edges (armchair ribbon) and (B) zigzag edges (zigzag ribbon). The dashed rectangles define the unit cells; aT and a are the respective unit cell widths for armchair and zigzag nanoribbons; N defines the ribbon width. The  marks pffiffi indicate the missing carbon atoms for the edge boundary condition. The nanoribbon width is defined as W ¼ 12 ðN+ 1Þa for armchair nanoribbons and W ¼ 23 Na + paffiffi3 for zigzag nanoribbons.

Graphene, electronic properties and topological properties

279

states of GNRs. Experimentally, GNRs can be fabricated by using either bottom-up (Cai et al., 2010) or top-down technology (Baringhaus et al., 2014; Han et al., 2007; Kosynkin et al., 2009). Fig. 3A–C show the DOS and energy band structures of AGNRs for three different ribbon widths. The top of the valence band and the bottom of the conduction band are located at ka ¼ 0. Note that the ribbon width determines whether the system is metallic or semiconducting. As shown in Fig. 3B, the system is metallic when N ¼ 3 m − 1, where m ¼ 1,2,3,. For semiconducting ribbons, the direct band gap decreases with increasing ribbon width and approaches zero in the limit of very large N. For narrow undoped metallic AGNR, an energy gap can be formed by Peierls instabilities at low temperatures (Fujita et al., 1996), which is consistent with density functional theory (DFT) calculations (Son et al., 2006a). For ZGNRs, however, a remarkable feature arises in the band structure, as shown in Fig. 4A–C. The top of the valence band and the bottom of the conduction band are always degenerate at k ¼ p/a, and the degeneracy of the center bands at k ¼ p/a does not originate from the intrinsic band structure of graphene. These two special center bands flatten with increasing ribbon width. A pair of partial flat bands appears within the region 2p/3a | k | p/a, where the bands are located in the vicinity of the Fermi level. The electronic states in the partial flat bands of ZGNRs can be understood as localized states near the zigzag edge by examining the charge density distribution (Fujita et al., 1996; Nakada et al., 1996; Wakabayashi et al., 1999). The emergence of the edge states can be explained by considering a semi-infinite graphene sheet with a zigzag edge. First, we show the distribution of charge density in flat band states for different wave numbers in Fig. 5A–D, where the amplitude is proportional to the circle radius. The wave function has a non-bonding character, i.e., it only has a finite amplitude on one of the two sublattices that include the edge sites. It is completely localized at the edge site when ka ¼ p and starts to gradually penetrate into the inner sites when ka deviates from p, reaching an extended state at k ¼ 2p/3a. Considering the translational symmetry, we can start constructing the analytical solution for the edge state by letting the Bloch components of the linear combination of atomic orbitals (LCAO) wave function be . . ., eika(n−1), eikan, eika(n+1),. . . on successive edge sites, where n denotes a site located on the edge. Then, the mathematical condition necessary for the wave function to be exact for E ¼ 0 is that the sum of the components of the complex wave function over the nearestneighbor sites becomes zero. In Fig. 5E, the above condition implies that eika(n+1) + eikan + x ¼ 0, eikan + eika(n−1) + y ¼ 0 and x + y + z ¼ 0. Therefore, the wave function components x, y and z are found to be Dkeika(n+1/2), Dkeika(n−1/2) and D2k eikan, respectively. Here, Dk ¼ − 2 cos (ka/2). Thus, the charge at each non-nodal site of the m-th zigzag chain from the edge. Then the convergence condition density is proportional to D2(m−1) k |Dk| 1 is required, for otherwise the wave function would diverge in a semi-infinite graphene sheet. This convergence condition defines the region 2p/3a | k | p/a, where the flat bands appear.

(A)

(B)

0

ka

π

3

E/γ0

E/γ0

E/γ0

0

−3

(C) 3

3

0

−3

0

0

D(ε)

ka

π

−3

−π

D(ε)

0

ka

π

D(ε)

Fig. 3 Energy band structure E(k) and DOS D(E) of armchair ribbons of various widths: (A) N ¼ 4, (B) N ¼ 5 and (C) N ¼ 30.

(A)

(B)

(C) 3

E/γ0

3

E/γ0

E/γ0

3

0

−3

0

0

0

ka

π

D(E)

−3

0

ka

π

−3

D(E)

−π

0

ka

Fig. 4 Energy band structure E(k) and DOS D(E) of zigzag ribbons of various widths: (A) N ¼ 4, (B) N ¼ 5 and (C) N ¼ 30.

π

D(E)

280

Graphene, electronic properties and topological properties

Fig. 5 Charge density plot for analytical solution of the edge states in a semi-infinite graphene sheet for several wave numbers: (A) k ¼ p/a, (B) k ¼ 8p/9a, (C) k ¼ 7p/9a and (D) k ¼ 2p/3a. (E) Analytical form of the edge state for a semi-infinite graphene sheet with a zigzag edge, shown by the bold lines. Each carbon site is specified by a location index n on the zigzag chain and by a chain order index m, which increases from the edge. The magnitude of the charge density at each site, such as x, y and z, is obtained analytically (see text). The radius of each circle is proportional to the charge density on each site, and the drawing corresponds to k ¼ 7p/9a.

The graphene edge states give rise to large contribution to DOS at the Fermi energy when the system size has a nanometer scale. Even a weak electron-electron interaction gives the spin polarized states with ferrimagnetic correlation along the zigzag edge (Fujita et al., 1996; Son et al., 2006a). Such magnetic states involve particular interests for the application of spintronics devices (Son et al., 2006b).

Charge polarization and Zak phase Charge polarization can be expressed in terms of geometric center of electronic wave functions (King-Smith and Vanderbilt, 1993; Marzari et al., 2012; Resta, 1994; Vanderbilt and King-Smith, 1993; Zak, 1989). For one-dimensional (1D) crystalline system, the charge polarization of n-th energy band is given as a Wannier center, i.e., Pn ¼ hwn(x)| x| wn(x)i, where wn(x) is a Wannier function of n-th energy band (Marzari et al., 2012). Owing to gauge freedom of Wannier functions, Pn is well-defined up to a lattice constant. The edge states of zigzag edge can be understood as a charge polarization owing to the nontrivial topological phase of bulk wave function (Delplace et al., 2011; Hatsugai, 2009; Ryu and Hatsugai, 2002). In this section, we relate the appearance of edge states at graphene zigzag edges to the finite Zak phase of bulk wave function of graphene. To relate Pn to Berry connection, one can apply Fourier transformation to wn and x, which results in (Marzari et al., 2012) ð 1 2p Pn ¼ (34) hcn ji∂kjcn idk, 2p 0 where cn is the periodic part of Bloch function of n-th energy band, and An ¼ hcn| i∂k| cni is Berry connection. We refer to the integration part of Eq. (34) as Zak phase (Zak, 1989). P In a finite chain, fractional charges P ¼  nnoccPn accumulate at the ends of the chain, where the summation is taken over all the occupied energy bands. When inversion symmetry is present, Pn is quantized to 0 or 1/2. In following discussions, we interchangeably use Zak phase and charge polarization to mean the same quantity. In 2D crystalline system, Eq. (34) has the dependence of wavenumber. Charge polarization is given as I 1 A ðk, k? Þdk? , (35) P n ðkÞ ¼ |P | P n where An(k, k?) ¼ hc n(k) | i∂k? | c n(k)i is Berry connection of n-th energy band in 2D momentum space. Here, k is the wavenumber along longitudinal translational invariant direction of GNR. k? is perpendicular to k, i.e., transverse direction of GNRs. P is a straight path connecting two equivalent k points in momentum space along k? direction. Similar to 1D systems, fractional charge Pnocc n Pn  ni accumulates on the edge if a material possesses finite charge polarization. To calculate Zak phase, we use the periodic part of bulk wave function given as Eq. (7), i.e.,   1 1 : (36) jc k i ¼ pffiffiffi ifk 2 se Thus, the Berry connection along k? for the valence band is 1 Aðk, k? Þ ¼ − ∂k? fk : 2

(37)

Graphene, electronic properties and topological properties

281

The Zak phase is I Z ðkÞ ¼

P

Aðk, k? Þdk? ¼ −

1 2

I P

∂k? f k :

(38)

Thus, the Zak phase is given as the integration of phase fk along the path P : Zak phase becomes zero if the integration path is taken in a simply connected region, otherwise Zak phase becomes finite. Fig. 6A shows reciprocal space of graphene and the integration path for Zak phase calculation. P A ðkÞ and P Z ðkÞ are the integration path for AGNR and ZGNR, respectively. Since the zigzag edge is parallel to a1, k is parallel to kx. Thus, the integration path P Z ðkÞ should be taken along ky for ZGNR. Similarly, armchair edges are parallel to y, i.e., (−a1 + a2)/2, k is parallel to b1 − b2. Hence, the integration path P A ðkÞ should be taken along b1 +b2 for AGNR. Fig. 6B shows the phase mapping of fk in the momentum space. The phase jumps of 2p periodically appear along the direction of b1 +b2. Hence, the integration path P Z ðkÞ crosses the phase jump if 2p/3a < |k| < p/a, resulting in finite Zak phase. However, no such phase jump appear for the path P A ðkÞ for arbitrary k. For AGNR, the Zak phase is identically zero, i.e. Z ðkÞ ¼ 0:

(39)

However, for ZGNR, the Zak phase is  Z ðkÞ ¼

p, 0,

2p=3a < jkj < p=a, jkj < 2p=3a:

(40)

The k-region of finite Zak phase coincides with the region of zigzag edge states. Hence, the edge states are attributed to the Zak phase of bulk wave function. The finite Zak phase tells us the existence of edge states. This discussion can be applied to bearded edges (Delplace et al., 2011) and corner states of graphene (Liu and Wakabayashi, 2021). The reason why the edge states appear at E ¼ 0 is supported by the particle-hole (chiral) symmetry of graphene (Ryu and Hatsugai, 2002).

Energy spectrum and wave functions: Graphene nanoribbons In this section, we briefly summarize the analytic solutions of energy spectrum and wave functions for GNRs on the basis of tight-binding model (Wakabayashi et al., 2010).

Armchair nanoribbons The set of equations of motion for AGNRs based on the nearest-neighbor tight-binding model is described by (A)

(B)

(C)

  pffiffiffi pffiffiffi 2p Fig. 6 (A) Reciprocal space of graphene sheet. b 1 ¼ 2p a 1, 1= 3 , b 2 ¼ a 0, 2= 3 are primitive reciprocal vectors. The magenta (cyan) rectangle zone represents the translational invariance of armchair (zigzag) edges. P A ðk Þ and P Z ðk Þ are the integration path for calculation of Zak phase of armchair and zigzag edges, respectively. (B) Phase mapping of fk which is the phase of bulk wave function. The phase jumps of 2p periodically appears along the direction of b1 +b2. When the integration path for Zak phase crosses the phase jump line, finite Zak phase is obtained. (C) Energy band structure and corresponding Zak phase Z ðk Þ for (top panel) ZGNR and (bottom) AGNR. ZGNR has finite Zak phase p for 2p/3 < |k| p (indicated by yellow rectangles).

282

Graphene, electronic properties and topological properties ðE=g0 Þcm,A ¼ −e−ik=2 c m,B − c m−1,B − c m+1,B ,

(41)

ðE=g0 Þc m,B ¼ −e+ik=2 c m,A − c m−1,A − c m+1,A ,

(42)

where m ¼ 1,2,3, ⋯N. c m, A and c m, B are the wave functions at the mA and mB-sites, respectively. The site indices are given in Fig. 2A. The boundary condition for AGNRs is c 0,A ¼ c 0,B ¼ c N+1,A ¼ c N+1,B ¼ 0:

(43)

The energy spectrum for AGNRs is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   k Es ¼ sg0 1 + 2ep cos + e2p , 2

(44)

where p is the wave number in the transverse direction. Here, ep ¼ 2 cos ( p) and s ¼  1; s ¼ + 1 (s ¼ − 1) corresponds to the conduction (valence) energy band. Note that Es becomes zero at k ¼ 0 and p ¼ 2p/3, which corresponds to a Dirac point. The transverse wave number p is quantized owing to the ribbon boundary condition as p¼

r p, N+1

r ¼ 1,2,3, ⋯, N:

(45)

Note that ES ¼ 0 at k ¼ 0 whenever N ¼ 3 m − 1 (m ¼ 1,2,3,), which is simply the condition for metallic AGNRs. Fig. 7A shows the relation between the BZ of graphene and that of AGNRs. The hexagonal BZ of graphene is mapped onto the bold black line on the ky - axis(−p ky p) as the BZ of AGNRs. The shaded rectangle corresponds to the phase space of variables k and p of Eq. (44). Note the range of p is 0 p p. The five red lines parallel to the ky - axis correspond to the cutting lines of the case for AGNR of N ¼ 5 (metallic). The cutting line of r ¼ 4 clearly passes through the Dirac point, i.e., the nanoribbon is metallic. Fig. 7B shows the energy band structure of AGNR (N ¼ 5) with subband indices. In this case, since the cutting line of r ¼ 4 gives p ¼ 2p/3, the corresponding subband passes through the Dirac point, i.e., the nanoribbon has a metallic band structure with linear dispersion. Thus, the energy band structures of AGNRs can be obtained by slicing the band structure of graphene, as in the case of carbon nanotubes (Saito et al., 1991). The wave function is written as 1 0 −s   ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c m,A C B (46) ¼ Nc @ ep + e+ik=2 A sinðmpÞ: c m,B ep + e−ik=2 Here Nc is the normalization constant.

Zigzag nanoribbons The set of equations of motion for ZGNRs is given by ðE=g0 Þc m,A ¼ −c m−1,B − gk c m,B ,

(A)

(47)

(B)

Fig. 7 (A) Relation between the BZ of graphene and that of AGNRs. The hexagonal BZ of graphene is mapped onto the bold black line on the ky - axis (−p ky p) as the BZ of AGNRs. The shaded rectangle corresponds to the phase space of variables k and p of Eq. (44). Note the range of p is 0 p p. The K and K0 points of graphene BZ are mapped on to the red and blue circles on kx-axis in the reduced BZ, respectively. The five red lines parallel to the ky - axis correspond to the cutting lines of the case for the AGNR of N ¼ 5 (metallic). The cutting line of r ¼ 4 clearly passes through the Dirac point, i.e., the nanoribbon is metallic. (B) Energy band structure of AGNR (N ¼ 5) with subband indices. The shaded area indicates the energy spectrum of bulk graphene.

Graphene, electronic properties and topological properties ðE=g0 Þc m,B ¼ −c m+1,A − gk c m,A :

283 (48)

Here gk ¼ 2 cos (k/2) is originated from the Bloch phase, and the site index is m ¼ 0,1,2, ⋯, N + 1. c m, A and c m, B describe the wave functions at the mA- and mB-sites, respectively. The site indices are given in Fig. 2B. The boundary condition for ZGNRs is given by c 0,B ¼ c N+1,A ¼ 0. The energy spectrum of ZGNRs is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es ¼ sg0 1 + g2k + 2gk cosðpÞ, (49) where s ¼  1, and s ¼ + 1 (s ¼ − 1) corresponds to the conduction (valence) energy band. The transverse wave number p is related with longitudinal wave number k as Fðp, N Þ sin ½pN  + gk sin ½pðN + 1Þ ¼ 0:

(50)

Fig. 8A shows the relation between the BZ of graphene and that of ZGNRs. The shaded rectangle corresponds to the phase space of variables k and p, i.e., |k | < p and 0 p p. The numerically obtained solutions of Eq. (50) are shown in Fig. 8B for the case of ZGNR with N ¼ 5. Let us call the solutions F(p,N) ¼ 0 for fixed N as pr(r ¼ 1, 2, ⋯, pn) within 0 < p < p, i.e., excluding p ¼ 0 and p. All these pi solutions give the extended states. The number of pr, i.e., pn, depends on the region of k,  N, |k| kc , pn ¼ (51) N − 1, kc < |k| p, where kc is given as  kc ¼ 2 arccos

1 1  2 1 + 1=N



2 2 1  p + pffiffiffi  : 3 3 N

(52)

In the limit of large N, kc converges to 2p/3 which corresponds to the Dirac K point. In other words, the cutting lines never go through the Dirac points in ZGNRs. One missing solution of Eq. (50) for the range of kc < | k | p can be obtained by analytic continuation as p ! p + i:

(53)

sinhðN Þ − gk sinhððN + 1ÞÞ ¼ 0,

(54)

Then, Eq. (50) is rewritten as

which gives two solutions of  as shown in the Fig. 8B. The energy spectrum can be obtained by using qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es ¼ sg0 1 + g2k − 2gk cosh ,

(55)

which give the central partial flat bands in the region of kc < | k | p. The wave function for extended states is written as

(A)

(B)

(C)

(A)

Fig. 8 (A) Relation between the BZ of graphene and that of ZGNRs. The shaded rectangle corresponds to the phase space of variables k and p of Eq. (50). The K and K0 points of graphene BZ are mapped on to the red and blue circles on k-axis in the reduced BZ, respectively. (B) The cutting lines of the case for ZGNR of N ¼ 5 in the k–p space. The cutting line of r ¼ 5 has the imaginary component  for |k |  |kc|, which corresponds to the mode of the partial flat bands due to the edge states. (C) Energy band structure of ZGNR (N ¼ 5) with subband indices. The shaded area indicates the energy spectrum of bulk graphene.

284

Graphene, electronic properties and topological properties 

c m,A c m,B



 ¼ Nc

ð−1Þr s sinðpðN + 1 − mÞÞ sinðpmÞ

 (56)

with the normalization constant Nc. Thus, the wave function for the subband with odd (even) r has the even (odd) parity. Similarly the wave function for the edge states can be written as     c m,A ð−1Þr s sinhððN + 1−mÞÞ , (57) ¼ N0c eipm c m,B sinhðmÞ with a normalization constant Nc0 . Note that the wave function for AGNRs given by Eq. (46) has a phase difference between sublattice sites A and B due to the chiral nature of graphene. On the other hand, the wave function for ZGNRs is always real and thus has no phase, irrespective of whether the state is extended or localized. This difference in wave functions between AGRNs and ZGRNs is related to the nature of pseudospin in graphene. Although ZGNRs conserve the total time-reversal symmetry, pseudo-time-reversal symmetry (time-reversal symmetry in each valley) is broken owing to the edge states. This nature gives rise to the perfectly conducting channel of ZGNRs (Wakabayashi et al., 2007). It is also possible to derive the analytic wave function of GNRs from the massless Dirac equation (Brey and Fertig, 2006; Wakabayashi, 2000).

Electronic states of carbon nanotubes The electronic states of GNRs are obtained by imposing the open boundary condition in one direction for the electronic states of graphene, and are characterized by the edge structures. In this section, we shall consider imposing the periodic boundary condition in one direction of graphene, which derives the electronic states of carbon nanotubes (CNTs). CNT is a one-dimensional carbon nanomaterial which can be constructed by rolling a graphene into a cylinder form. It was discovered by S. Iijima in 1991 (Iijima, 1991). The structure of CNT is uniquely determined by the chiral vector Ch, which is defined as Ch ¼ na1 + ma2 ðn, mÞ:

(58)

Here m, n are integers and 0 m n. In order to conform to commonly used notation in the research field of CNT, we have pffiffi  pffiffi  3 3 a a 2 a, 2 , a2 ¼ 2 a, − 2 , as shown in Fig. 9A. It should be noted that lattice

redefined the primitive vectors of graphene as a1 ¼

structure is also rotated by 90 degree compared with Fig. 1A. The shortened notation (n,m) is widely used to characterized the geometry of CNT. In general, CNTs are classified as chiral (0 < m < n) and achiral (m ¼ 0 or m ¼ n 6¼ 0). Especially, m ¼ 0 CNTs are known as zigzag nanotube, and m ¼ n CNTs are known as armchair nanotubes. Fig. 9B shows the atomic structures of CNTs with several different chiral indices. Figs. 9C–F show the energy band structures and corresponding DOS for several different CNTs based on the nearest-neighbor tight-binding model. In general, CNTs are known to be metallic with linear dispersion when m – n is multiple of 3, otherwise semiconducting with energy band gap (Hamada et al., 1992; Mintmire et al., 1992; Saito et al., 1991). Please also see the reviews for further details (Charlier et al., 2007; Laird et al., 2015; Saito et al., 1998). In the rest of this section, we shall explain the relation between the energy spectrum of CNTs and the energy band structure of graphene. The structure of CNT can be characterized by its diameter dt and chiral angle yt from a zigzag axis. Since the length of CNT circumference is |Ch|, the diameter dt of CNT is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt ¼ a n2 + nm + m2 =p: (59) The unit cell of a CNT is a rectangle bounded by the vector Ch and translational vector T as shown in the rectangle of Fig. 9A. The translational vector is given as T ¼ t 1 a1 + t 2 a2 :

(60)

t1 and t2 are integers, which are determined by Ch  T ¼ 0 and gcm(t1, t2) ¼ 1. Here, pffiffiffigcm(m, n) means the great common measure of the two integers m and n. The area of CNT unit cell is calculated as jCh  T j ¼ 3a2 ðn2 + nm + m2 Þ=dR , where dR ¼ gcm(2n + m, of the unit cell of 2m +n). Using dR, t1 and pffiffiffi t2 can be written as t1 ¼ (2m + n)/dR, t2 ¼ − (2n + m)/dR. Dividing |Ch  T| by the area 2 graphene ja1  a2 j ¼ 3a2 =2, the number N of hexagons in unit cell of CNT canbe p obtained, i.e., N ¼ 2(n + nm + m2)/dR. ffiffiffi   pffiffiffi  2p 2p The unit vectors in the reciprocal lattice of graphene can be given as b1 ¼ a 1= 3, 1 , b2 ¼ a 1= 3, −1 : These vectors are derived by the relation ai  bj ¼ 2pdij, where dij is the Kronecker delta. In similar manner, the reciprocal vectors of CNT can be derived by using the following relations Ch K 1 ¼ T  K 2 ¼ 2p,

(61)

Ch K 2 ¼ T  K 1 ¼ 0,

(62)

Graphene, electronic properties and topological properties

(A)

285

(B)

(9,0) (C) (9,0)

(D)

D(E)

(E)

(10,0)

(5,5)

(8,2)

(F) (8,2)

(5,5)

D(E)

D(E)

D(E)

Fig. 9 (A) An unrolled CNT projected onto the graphene lattice. When the CNT is rolled up in a cylinder form, the chiral vector Ch gives the circumference of cylinder and the translation vector T aligns along the cylinder axis. yt is chiral angle measured from the zigzag axis. In this section, we have redefine the primitive vector of pffiffi  pffiffi  graphene as a 1 ¼ 23 a, 2a and a 2 ¼ 23 a, − 2a : (B) Atomic structures of (9,0) zigzag CNT (left), (5,5) armchair CNT (middle) and (8,2) chiral CNT (right). Energy band structure and corresponding DOS for (C) (9,0) zigzag CNT, (D) (10,0) zigzag CNT, (E) (5,5) armchair CNT and (F) (8,2) chiral CNT.

where the vector K1 is the wavevector along the circumferential direction and K2 is along the CNT axis. Explicitly, one can write as K1 ¼

1 ð−t b + t 1 b2 Þ, N 2 1

(63)

1 ðmb1 − nb2 Þ: N

(64)

K2 ¼

1st BZ of CNT is given as − Tp < k Tp along the K2 direction. Since NK1 ¼ − t2b1 + t1b2 corresponds to a reciprocal lattice vectors of graphene, two wave vectors which differ by NK1 are equivalent. In addition, since t1 and t2 lack a common divisor other than unity, none of the N − 1 vectors mK1(m ¼ 1, 2, ⋯, N − 1) are reciprocal lattice vectors of graphene. Thus, the N wave vectors mK1 give rise to N discrete k vectors. These arise from the quantized wave vectors associated with the periodic boundary condition due to Ch. Therefore, the energy spectrum of CNT can be obtained by   K (65) Em ðkÞ ¼ E2D k 2 + mK 1 , jK 2 j where m ¼ 0, 1, ⋯, N−1, −

p p 20 ) and the moiré period becomes the atomic scale, however, the long-range effective theory is not applicable anymore, and the system becomes essentially quasiperiodic in that incommensurate periodicities coexist. In particular, the 30-degree rotated TBG has a feature of 12-fold rotational symmetric quasicrystal, which is intimately related to a quasiperiodic tiling in mathematics. The 30-degree TBG was experimentally fabricated by an epitaxial growth on silicon-carbide surface (Ahn et al., 2018; Moon et al., 2019).

Band structure Fig. 3 shows an example of the band calculation for TBGs with some twist angles (Nam and Koshino, 2017), where k, g, m and k0 in the horizontal axis represent the symmetric points in the moiré Brillouin zone in Fig. 2. The black solid curve and the red dashed curve represent the energy bands with and without the lattice relaxation, which is important in the small angle regime y < 2 . In y ¼ 2.65 (Fig. 3a), the linear band at k, k0 are remnants of the Dirac cones of monolayer graphene. In decreasing the twist angle, the band velocity of the Dirac cone is significantly reduced, and nearly flat bands emerge at
1 , which is called the magic

Twisted bilayer graphene

(A)

(B)

(C)

291

(D)

Fig. 3 Band calculation for TBGs with different twist angles. The label k, g, m and k0 represent the symmetric points in the moiré Brillouin zone (Fig. 2). The black solid curves and the red dashed curves, respectively, are the energy bands with and without the lattice relaxation. From Ref. Nam NNT and Koshino M (2017) Lattice relaxation and energy band modulation in twisted bilayer graphene. Physical Review B 96(7): 075311. Errata: Phys. Rev. B 101: 099901 (2020).

angle (Bistritzer and MacDonald, 2011a). There are two flat bands in each spin/valley sector, and hence eight flat bands in total. The of electrons accommodated in a single flat band (i.e., per spin and valley) is given by n0 ¼ 1/SM, where pffiffiffi number  3=2 L2M is the area of the moiré unit cell. The lattice relaxation effect opens energy gaps between the flat bands and excited SM ¼ bands. When y is further decreased beyond the magic angle, the band width is broadened again, and the band structure changes in a complex manner. The reduction of the band velocity and the formation of the moiré subband can be understood by the following argument. Fig. 4a shows the hybridization of the Dirac cones of monolayers. The layer 1’s cone at the center is hybridized with layer 2’s cones shifted by DKj ‘s in Eq. (7). A band anti-crossing occurs at intersections of the cones, as schematically illustrated in Fig. 4b. The system has two characteristic energy scales: the interlayer hopping energy t (or t0 )
0.1 meV, which determines the gap width in the anti-crossing, and ℏvDK ∝ y, which is the energy distance between the Dirac point and the band crossing point. The structure of the moiré subband is determined by the relative magnitude of t to ℏvDK. When y > 5 , one has t  ℏvDK so that the interlayer hopping can be treated as a perturbation for the Dirac cone, giving a small gap at the intersection. In decreasing y, the ratio t/(ℏvDK) increases in inversely proportional to y, and it exceeds 1 at y < 1 . This is a strong coupling regime, where the interlayer coupling completely reconstructs the Dirac cone into a series of the moiré subbands. An extreme model with t ! 0 is called the chiral limit, as the Hamiltonian becomes chiral symmetric (Tarnopolsky et al., 2019). There the lowest energy bands are shown to be exactly flat at a series of angles and the magic angle argued above (y
1 ) corresponds to the greatest one in the series.

(A)

(B)

Fig. 4 (a) Hybridization of graphene’s Dirac cones in TBG. The central Dirac cone of layer 1 (blue) is coupled with the surrounding Dirac cones of layer 2 displaced by DK. (b) Schematic cross-sectional picture of the Dirac cone hybridization.

292

Twisted bilayer graphene

Superconductivity and correlated phenomena The flat band in the magic angle TBG is expected to be a strongly correlated system governed by the electron-electron interaction, since the kinetic energy (band energy) is quenched. Indeed, it was found that TBG samples near 1 exhibit the superconductivity in the low temperature around 1 K (Cao et al., 2018a, 2018b). The superconducting state is highly sensitive to the twist angle, suggesting that the phenomena is related to the band flatness. Fig. 5 is the schematic phase diagram of the magic angle TBG, where the horizontal axis represents the electron density in units of n0 (the number of electrons per a moiré band) and the vertical axis is temperature. The flat band region corresponds to −4 n/n0 4, where the n ¼ 0 is the charge neutral point. In Fig. 5, thick strips at n/n0 ¼ 4 represent the band insulator phases, where the Fermi energy is located in the band gap above / below the flat bands. In decreasing the temperature, one sees extra insulating phases at n/n0 ¼ 2 due to the electron-electron interaction, which are called the correlated insulating (CI) phases. The superconducting phase emerges by slightly doping carriers to the CI phases. Experimentally, the phase diagram is obtained just from a single TBG sample by controlling the charge density by the gate electric field effect. This is in contrast to 3D materials, where changing carrier density requires chemical doping. It was reported that some TBG samples exhibit insulating phases also at the odd fillings n/n0 ¼ 1, 3 (Yankowitz et al., 2019), and superconducting phases nearby (Lu et al., 2019). The ferromagnetism and anomalous Hall effect were also reported at n/n0 ¼ 3 (Serlin et al., 2020; Sharpe et al., 2019). The detailed structure of the phase diagram, particularly at the odd filling, is highly dependent on samples, presumably due to the non-uniformity of the moiré stacking structure and also the effect of the substrate under the TBG. The mechanism of the superconductivity and correlated insulating states in TBG is still under debate. The insulating behavior at the integer filling suggests an analogy to the quantum Hall ferromagnetism where the spin/valley degenerate Landau levels spontaneously split under electron-electron interaction. Various many-body states, such as the inter valley coherent states, were also proposed as possible candidates of the ground states in the magic angle TBG (Bultinck et al., 2020). The superconductivity and correlated phases were also found in other moiré graphene systems, such as twisted double bilayer graphene (i.e., twisted pair of Bernal bilayer graphenes) (Cao et al., 2019; Liu et al., 2019; Shen et al., 2020) and twisted trilayer graphene (Hao et al., 2021; Lin et al., 2020; Park et al., 2021; Zhu et al., 2020), where the band flatness is supposed to be essential as in TBG.

Magneto spectra and quantum hall effect In strong magnetic fields, the energy spectrum of TBG exhibits a fractal gap structure called Hofstadter’s butterfly (Bistritzer and MacDonald, 2011b; Moon and Koshino, 2012). The Hofstadter spectrum generally occurs in two dimensional systems subject to both magnetic field and periodic electrostatic potential (Hofstadter, 1976), where the fractal gap structure is observed when the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi magnetic length ℏ=ðeBÞ and the potential period are comparable. While conventional atomic lattices with periodicities of less than 1 nm require unfeasibly large magnetic fields (B > 10, 000 Tesla) to reach this condition, the TBG with a periodic modulation of the order of 10 nm enables experimental access to the fractal spectrum at a few 10 Tesla. Fig. 6 shows an example of Hofstadter’s spectrum calculated for TBG of y ¼ 2.65 (Moon and Koshino, 2012). The left panel, Fig. 6a presents the energy spectrum plotted against the magnetic field amplitude, where the black part indicates the energy regions where electron states exist, and the white part is energy gap. The middle panel (b) shows the same spectrum, but with color and number inside each gap indicating the quantized Hall conductivity when the Fermi energy lies in the gap. The right-most panel (c) is the zero-field band structure in the same energy range. In Fig. 6a, the spectrum is characterized by the number of magnetic field

Fig. 5 Schematic phase diagram of the magic angle TBG, against the electron density and temperature. CI stands for correlated insulating phase, and SC stands for superconducting phase. Vertical dashed lines indicate correlated insulating phases at the odd fillings observed in some experiments.

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Fig. 6 (a) Energy spectrum plotted against the magnetic field calculated for TBG of y ¼ 2.65 . The black and white regions indicate energy bands and energy gaps, respectively. (b) The same spectrum, but with color and number inside each gap indicating the quantized Hall conductivity. (c) Corresponding band structure 0 at zero magnetic field. K, G, M, K correspond to k, g, m, k0 in Fig. 2, respectively. From Ref. Moon P and Koshino M (2012) Energy spectrum and quantum hall effect in twisted bilayer graphene. Physical Review B 85: 195458.

quanta per a moiré unit cell, F/F0, where F ¼ BSM is the magnetic flux penetrating the moiré unit area SM, and F0 ¼ h/e is the magnetic flux quantum. The quantized Hall conductivity in the mini-gaps behaves non-monotonically as a function of Fermi energy, as usually expected in the Hofstadter butterfly. The recursive gap structure was first observed experimentally in the moiré superlattice of graphene on hexagonal-boron-nitride (hBN) (Dean et al., 2013; Hunt et al., 2013; Ponomarenko et al., 2013).

Conclusion The electronic properties of TBG were briefly reviewed. It was shown that the moiré period, of typically a few to tens of nanometer, significantly modifies the original band structure of graphene, giving rise to a number of unusual phenomena not observed without the moiré stacking. The energy width of the moiré subbands decreases as the twist angle is reduced, and dispersionless flat bands appear at the magic angle near 1 , where superconducting phases and correlated insulating phases emerge due to the electron-electron interaction. In magnetic fields, the spectrum exhibits a fractal Hofstadter’s structure due to the interference of the magnetic field and the long-range moiré periodicity. The discovery of TBG is followed by extensive research of various moiré superlattices other than graphenes. For instance, a twisted stack of transition metal dichalcogenide (e.g., MoS2, WS2, WSe2, etc.) is a moiré superlattice with unusual optical properties, where an optical excitation gives rise to localized excitons trapped to the moiré pattern (Jin et al., 2019; Seyler et al., 2019; Tran et al., 2019). The search for novel physics in twisted systems is extended to a wide variety of two-dimensional systems, including magnetic and superconducting 2D materials and topological insulators.

References Ahn SJ, Moon P, Kim T-H, Kim H-W, Shin H-C, Kim EH, Cha HW, Kahng S-J, Kim P, Koshino M, et al. (2018) Dirac electrons in a dodecagonal graphene quasicrystal. Science 361(6404): 782–786. Bistritzer R and MacDonald A (2011a) Moiré bands in twisted double-layer graphene. Proceedings of the National Academy of Sciences 108(30): 12233. Bistritzer R and MacDonald A (2011b) Moiré butterflies in twisted bilayer graphene. Physical Review B 84(3): 035440. Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A, and Zaletel MP (2020) Ground state and hidden symmetry of magic-angle graphene at even integer filling. Physical Review X 10(3): 031034. Cao Y, Fatemi V, Demir A, Fang S, Tomarken SL, Luo JY, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Kaxiras E, et al. (2018a) Correlated insulator behaviour at half-filling in magic-angle graphene super-lattices. Nature 556(7699): 80. Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, and Jarillo-Herrero P (2018b) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556(7699): 43. Cao Y, Rodan-Legrain D, Rubies-Bigorda O, Park JM, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2019) Electric field tunable correlated states and magnetic phase transitions in twisted bilayer-bilayer graphene. arXiv preprint arXiv.1903.08596.

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Dean C, Wang L, Maher P, Forsythe C, Ghahari F, Gao Y, Katoch J, Ishigami M, Moon P, Koshino M, Taniguchi T, Watanabe K, Shepard K, Hone J, and Kim P (2013) Hofstadter/’s butterfly and the fractal quantum hall effect in moire superlattices. Nature 497(7451): 598–602. Hao Z, Zimmerman A, Ledwith P, Khalaf E, Najafabadi DH, Watanabe K, Taniguchi T, Vishwanath A, and Kim P (2021) Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371(6534): 1133–1138. Hofstadter D (1976) Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Physical Review B 14(6): 2239. Hunt B, Sanchez-Yamagishi J, Young A, Yankowitz M, LeRoy B, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, and Ashoori R (2013) Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure. Science 340(6139): 1427–1430. Jin C, Regan EC, Yan A, Utama MIB, Wang D, Zhao S, Qin Y, Yang S, Zheng Z, Shi S, et al. (2019) Observation of moiré excitons in wse 2/ws 2 heterostructure superlattices. Nature 567(7746): 76–80. Koshino M, Yuan NF, Koretsune T, Ochi M, Kuroki K, and Fu L (2018) Maximally localized wannier orbitals and the extended hubbard model for twisted bilayer graphene. Physical Review X 8(3): 031087. Lin F, Qiao J, Huang J, Liu J, Fu D, Mayorov AS, Chen H, Mukherjee P, Qu T, Sow C-H, et al. (2020) Hetero-moiré engineering on magnetic bloch transport in twisted graphene superlattices. Nano Letters 20(10): 7572–7579. Liu X, Hao Z, Khalaf E, Lee JY, Watanabe K, Taniguchi T, Vishwanath A, and Kim P (2019) Spin-polarized correlated insulator and superconductor in twisted double bilayer graphene. arXiv preprint arXiv.1903.08130. Lopes dos Santos J, Peres N, and Castro Neto A (2007) Graphene bilayer with a twist: Electronic structure. Physical Review Letters 99(25): 256802. Lu X, Stepanov P, Yang W, Xie M, Aamir MA, Das I, Urgell C, Watanabe K, Taniguchi T, Zhang G, Bachtold A, MacDonald AH, and Efetov DK (2019) Su-perconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574(7780): 653–657. Moon P and Koshino M (2012) Energy spectrum and quantum hall effect in twisted bilayer graphene. Physical Review B 85: 195458. Moon P, Koshino M, and Son Y-W (2019) Quasicrystalline electronic states in 30 rotated twisted bilayer graphene. Physical Review B 99(16): 165430. Nam NNT and Koshino M (2017) Lattice relaxation and energy band modulation in twisted bilayer graphene. Physical Review B 96(7): 075311. Errata: Phys. Rev. B 101: 099901 (2020). Park JM, Cao Y, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2021) Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590(7845): 249–255. Ponomarenko LA, Gorbachev RV, Yu GL, Elias DC, Jalil R, Patel AA, Mishchenko A, Mayorov AS, Woods CR, Wallbank JR, Mucha-Kruczynski M, Piot BA, Potemski M, Grigorieva IV, Novoselov KS, Guinea F, Fal’ko VI, and Geim AK (2013) Cloning of dirac fermions in graphene superlattices. Nature 497(7451): 594–597. Serlin M, Tschirhart C, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, and Young A (2020) Intrinsic quantized anomalous hall effect in a moiré heterostructure. Science 367(6480): 900–903. Seyler KL, Rivera P, Yu H, Wilson NP, Ray EL, Mandrus DG, Yan J, Yao W, and Xu X (2019) Signatures of moiré-trapped valley excitons in mose 2/wse 2 heter-obilayers. Nature 567(7746): 66–70. Sharpe AL, Fox EJ, Barnard AW, Finney J, Watanabe K, Taniguchi T, Kastner M, and Goldhaber-Gordon D (2019) Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365(6453): 605–608. Shen C, Chu Y, Wu Q, Li N, Wang S, Zhao Y, Tang J, Liu J, Tian J, Watanabe K, et al. (2020) Correlated states in twisted double bilayer graphene. Nature Physics 16(5): 520–525. Tarnopolsky G, Kruchkov AJ, and Vishwanath A (2019) Origin of magic angles in twisted bilayer graphene. Physical Review Letters 122(10): 106405. Tran K, Moody G, Wu F, Lu X, Choi J, Kim K, Rai A, Sanchez DA, Quan J, Singh A, et al. (2019) Evidence for moiré excitons in van der waals heterostructures. Nature 567(7746): 71–75. Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young AF, and Dean CR (2019) Tuning superconductivity in twisted bilayer graphene. Science 363(6431): 1059–1064. Zhu Z, Carr S, Massatt D, Luskin M, and Kaxiras E (2020) Twisted trilayer graphene: A precisely tunable platform for correlated electrons. Physical Review Letters 125(11): 116404.

Graphene, transport Michihisa Yamamoto, RIKEN Center for Emergent Matter Science, Saitama, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Electronic structures of graphene Topological properties Carrier scattering in graphene devices Ballistic transport and electron optics Valley Hall transport in the presence of charged impurities Nonlocal transport and edge transport Hydrodynamic transport Conclusion References Further reading

295 296 296 297 298 301 302 304 307 308 308 309

Abstract We describe transport properties of graphene and related experiments. The main source of carrier scattering in graphene is charged impurities at low temperatures. This leads to the linear relationship between the conductivity and the carrier density. High-mobility graphene devices also exhibit unique ballistic transport such as the Klein tunneling. At high temperatures, electron-electron scattering becomes significant in clean graphene devices, and carriers behave as fluids. Topological valley transport also occurs in graphene when the inversion symmetry is broken. The topological valley current has been detected in nonlocal transport experiments.

Key points

•

Charged impurity scattering, ballistic transport, topological transport, nonlocal transport, hydrodynamic transport.

Introduction Graphene is a monolayer of carbon atoms in a honeycomb lattice. Its properties had been theoretically investigated already before its discovery, but monolayer had been thought to be too unstable to exist. In 2004, it was shown by Novoselov, Geim, and coworkers that it is possible to exfoliate a monolayer graphene flake from graphite by a scotch tape and transfer it onto a SiO2 substrate (Novoselov et al., 2004). It also became possible to measure the electrical conductivity of graphene by depositing contact metal electrodes on exfoliated graphene flakes. Owing to the unique combination of physical properties of graphene such as high mobility, high transparency, relativistic natures, great mechanical strength, large thermal conductivity, etc., intense studies have been conducted for revealing the electronic properties of graphene by electrical transport experiments. Graphene on SiO2 has high carrier mobility exceeding 10,000 cm2/Vs. The main source of scattering for carriers in graphene is the charged impurities attached during the fabrication process. By annealing the graphene devices, influence of the charged impurities can be suppressed to some extent. Temperature dependence of the conductivity also revealed influence of scattering by phonons. Graphene devices display ballistic transport properties when the device dimension becomes submicron scale at relatively low temperatures. The quantum Hall effect is also observed for graphene on SiO2 owing to its high mobility (Novoselov et al., 2005; Zhang et al., 2005). The high quality of graphene layers also enabled identification of the band structure of few-layer graphene (Craciun et al., 2011). Several years after the discovery of graphene, even higher-quality graphene devices suffering less from charge disorder than those on SiO2 became available by using a suspended graphene and current annealing (Du et al., 2008; Bolotin et al., 2008). The development of layer-by-layer transfer technique has also allowed for stacking different layered materials while precisely tuning their relative angles. When graphene is encapsulated by an atomically flat insulator, hexagonal boron nitride (h-BN) flakes, prepared high-quality graphene devices become stable (Dean et al., 2010). Once graphene is encapsulated, further contamination is avoided. Electrical contacts to graphene are taken by the “edge contacts.” The transport properties of graphene became able to be measured more reproducibly and intrinsic properties of graphene that can be found only in high-quality devices have been investigated. The mobility of suspended or encapsulated graphene exceeds 100,000 cm2/Vs.

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Even in high-quality graphene, charge disorder is still one of the main sources of carrier scattering at low temperatures. In a clean graphene device at high temperatures, the electron-electron scattering length can become smaller than the mean free path and electron-phonon scattering length. The carrier transport then obeys hydrodynamics. The viscosity of graphene induced by strong electron-electron interaction is indeed larger than those of standard semiconductors such as the high electron mobility transistor (HEMT) of a GaAs/AlGaAs heterostructure. When the symmetry of graphene is broken, topological transport associated with the valley contrasting intrinsic Hall conductivity occurs. The symmetry breaking is induced by aligning relative angle between graphene and h-BN such that graphene is affected by the potential from boron and nitrogen atoms, or by applying a perpendicular electric field to a few-layer graphene. For an encapsulated few-layer graphene, one can independently control the carrier density and perpendicular electric field by the top gate and back gate through h-BN layers. Owing to above-mentioned unique properties, graphene attracts considerable attention. In addition, because graphene appears on the surface and can constitute a “van der Waals heterostructure” (Geim and Grigorieva, 2013), it can be combined with various materials in novel functional devices.

Overview In this article, we mainly focus on transport properties of pristine monolayer and bilayer graphene observed in real graphene devices. While graphene can be combined with other various materials in van der Waals heterostructures and be used for spintronics and superconducting device applications, externally induced properties are out of scope in this article. The quantum Hall effect in graphene is also discussed in the other articles of this encyclopedia (Zhang, 2023; McCann, 2006, 2023). We start with description of the intrinsic electronic structure of graphene and discuss corresponding transport properties. We then discuss carrier scatterings in graphene mostly dominated by charged impurities. We also discuss ballistic transport, where device dimension is smaller than or comparable with the mean free path. We then move to the nonlocal transport that is used to detect the topological transport in real graphene devices. Finally, we describe experimental observations of hydrodynamic transport dominated by the electron-electron scattering in graphene devices.

Electronic structures of graphene Monolayer graphene is a zero-gap semiconductor. Energy band of graphene can be calculated using a tight banding model for the honeycomb lattice (Wallace, 1947; Castro Neto et al., 2009). The unit cell consists of two carbon atoms A and B, forming two sublattices. We only consider the nearest neighbor hopping in the first approximation. It is straightforward to derive effective Hamiltonians for the two energetically degenerate but nonequivalent valleys K and K0
 0 kx − iky HK ¼ ħvF kx + iky 0
 0 kx + iky HK 0 ¼ − ħvF kx − iky 0 operating on (| A⟩, | B⟩)T, where | A⟩ and | B⟩ are, respectively, Bloch wave function components on the A and B sublattices, k ¼ (kx, ky) is the wave vector measured from the valley center, ħ is Planck constant divided by 2p, and vF  1  106 m2/s is the velocity of carriers. Using Pauli matrices, we can rewrite it as   HMLG ¼ ħvF tz kx sx + ky sy , (1) where tz is the valley index, i.e., tz ¼ 1 for valley K and tz ¼ − 1 for valley K0 . This Hamiltonian leads to the linear dispersion eðkÞ ¼ ħvF k

(2)

implying that carriers in graphene are massless with the energy independent velocity vF. Here the + and – denotes those of electrons and holes. The e(k) ¼ 0 point is termed the Dirac point or the charge neutrality point because the carrier density n ¼ ne − nh defined by the density ne and nh of electrons and holes becomes zero. Eq. (4) also indicates that the pseudospin defined by the presence of an electron in either of the sublattices and the wave vector k are coupled. In the absence of atomic-scale short-range scattering, pseudospin should be conserved. This prohibits backscattering of electrons because flip of k direction should accompany flip of the pseudospin. This suggests that the backscattering is suppressed near zero energy in the absence of short-range scattering induced by lattice defects. When k rotates once in the xy-plane, the pseudospin also rotates once. This leads to the Berry phase of p at K valley and − p at K0 valley.

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In AB-staked bilayer graphene, there are four atoms in the unit cell labeled as A1, B1, A2, and B2. A1 and B1 are on the bottom layer and A2 and B2 are on the top layer. There is a vertical hopping g between A2 and B1. The effective Hamiltonian is expressed as a 4  4 matrix 1 0 0 0 MLG C B H B −g 0 C C H¼B C B 0 −g @ MLG A H 0 0 where HMLG is the effective Hamiltonian for monolayer graphene (Eq. 1). For the small energy | e(k) |  g, this Hamiltonian can be reduced into the two-band Hamiltonian for the (| A1⟩, | B2⟩)T basis as  2 ! 0 kx − iky ħ2 HK ¼ 2mBL k + ik 2 0 x

HK 0

ħ2 ¼ − 2mBL



y

0 kx − iky

 2

kx + iky

2 !

0

g/(2v2F )

is the effective mass of the bilayer graphene (McCann, 2006, 2023). The low energy bands are parabolic that where mBL ¼ touch each other at zero energy. Using the azimuthal angle f of k and k2 ¼ k2x + k2y , the effective Hamiltonian is rewritten as HBLG ¼

 ħ2 k2  cosð2fÞsx + tv sinð2fÞsy : 2mBL

(3)

This Hamiltonian indicates the Berry phase 2p at K valley and −2p at K0 valley. The band structure of thicker N-layer graphene (N > 2) depends on the stacking. ABC-stacked N-layer graphene has kN dispersion and Berry phase Np. ABA-stacked multilayer graphene consists of monolayer-like (linear) band and bilayer-like (parabolic) bands. The monolayer-like band exists only for odd N. These bands as well as the Berry phase can be experimentally identified by the measurement of the quantum Hall effect (Zhang, 2023; McCann, 2006, 2023). While both monolayer and bilayer graphene are zero-gap semiconductors, the band gap can be induced by breaking the inversion symmetry. When there is a potential difference 2D between A and B sites of monolayer graphene, the mass term Dsz is added to the effective Hamiltonian. The effective Hamiltonian becomes   HMLG ¼ ħvF tz kx sx + ky sy + Dsz : (4) This results in nonlinear bands qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðkÞ ¼  ðħvF kÞ2 + D2 and the effective mass m∗ ¼ D/v2F at the band bottom. For bilayer graphene, when there is a potential difference 2D between A1 and B2 sites, the effective Hamiltonian becomes HBLG ¼

 ħ2 k 2  cosð2fÞsx + tv sinð2fÞsy + Dsz : 2mBL r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

This results in the non-parabolic bands eðkÞ ¼ 

ħ2 k2 2mBL

2

(5)

+ D2 with the gap opening by 2D. The advantage of bilayer graphene

is that this gap can be induced by the application of a perpendicular electric field D that introduces the potential difference between the two layers. We readily expect insulating behavior at the charge neutrality n ¼ 0 in the presence of D for bilayer graphene. This also applies to ABC-stacked multilayer graphene.

Topological properties The effective Hamiltonian of massive graphene in Eqs. (4) and (5) leads to non-zero Berry curvature for D 6¼ 0 (Xiao et al., 2007). The Berry curvature is defined as VðkÞ ¼ irk  huðkÞjrk uðkÞi,

(6)

using the Fourier transformation u(k) of the periodic part of the Bloch function. Since the Berry phase is given by the integral of i⟨u(k)| rku(k)⟩ along a closed loop in the momentum space, V(k) corresponds to the magnetic flux density in the momentum space. This is in analogy with the Aharonov-Bohm phase given by the integral of the vector potential A along a closed loop in the real space, and the magnetic flux density defined by B(r) ¼ rr  A(r). Indeed, the role of V(k) in the semi-classical dynamics in the ðkÞ in the presence of an in-plane electric field E: Bloch bands is understood as a correction to the group velocity v ¼ r_ ¼ ∂eħ∂k

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r_ ¼

∂eðkÞ _ + k  VðkÞ: ħ∂k

(7)

We see that V(k) works as a magnetic flux density in the momentum space by comparing Eq. (7) with ħk_ ¼ − eE − e_r  B. The second term in Eq. (7) driven by E ¼ − ħe k_ is called anomalous velocity: e vA ¼ − E  VðkÞ: ħ

(8)

This term leads to the intrinsic Hall conductivity: sint H ¼ gS

e2 ħ

Z

d2 k f ðkÞOðkÞ, ð2pÞ2

(9)

where gS is the spin degeneracy (usually gS ¼ 2), O(k) is the z-component of the Berry curvature, and f(k) is the Fermi-Dirac distribution function. It is straightforward to derive the Berry curvature for the two-level system represented in Eqs. (4) and (5). For massive monolayer graphene, the Berry curvature is OðkÞ ¼ tz

g2 D , 2eðkÞ3

(10)

which becomes maximum or minimum at K and K0 point. This leads to the valley-dependent intrinsic Hall effect for low energy 0 carriers. Energy dependence of the intrinsic valley Hall conductivity defined as svxy  sint,K − sint,K becomes H H 8 2 2e D > > ð|ej> DÞ < h jej int,K int,K 0 v sxy ðeÞ  sH − sH ¼ (11) 2 > > : 2e ð|ej< DÞ h at zero temperature. In analogy with the spin Hall effect, where spin current is induced in the transverse direction to the injected charge current, valley current is induced by an applied in-plane electric field E to the transverse direction. This phenomenon is called valley Hall effect. svxy corresponds to the conversion ratio between the induced valley current density and E. For bilayer graphene, the Berry curvature and the zero-temperature valley Hall conductivity become qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ħ2 D eðkÞ − D (12) OðkÞ ¼ tz ∗ 3 m eðkÞ 8 2 4e D > > ð|ej> DÞ < h jej v sxy ðeÞ ¼ (13) 2 > > : 4e ð|ej< DÞ h with D tuned by the perpendicular electric field.

Carrier scattering in graphene devices A typical graphene device is supported by a substrate. Graphene is capacitively coupled with a gate electrode (usually a back gate) that can be used to tune the carrier density n. In some devices, graphene is sandwiched by the back gate and top gate through insulating layers. In such a dual-gated device, the carrier density n and the perpendicular electric field D can be independently controlled. This dual-gated structure is useful to control the electronic state of bilayer graphene. By depositing metal electrodes on graphene, one can measure the electrical transport through graphene. The electrical contact is realized with various metals such as Cu, Ag, Au, Ni, and Pd (Giubileo and Di Bartolomeo, 2017). The deposited metal dopes carriers in the region beneath the metal, allowing the electron transport between the metal and graphene. Transfer length or the effective length of the contact region is of the order of micron. The contact resistance usually depends on the length of the deposited metal contact rather than its area. Electrical contacts can also be taken to the edges of graphene. Even though the improvement of the electrical contact is still an important issue, the contact resistance (typically order of 100 Omm) is usually smaller than the resistance of graphene itself in a wide range of temperature. Therefore, the electrical conductance of graphene can be measured precisely as a function of n using a graphene field effect transistor like device and sweeping the gate voltage. g s g v e2 Because monolayer graphene has the linear dispersion, the carrier density is proportional to the energy squared n ¼ 4pħ 2 2, where vF gS ¼ 2 and gv ¼ 2 are the spin degeneracy and valley degeneracy, respectively. The density of states D(e) ¼ dn/de is proportional to e. If the scattering probability ħ/tm(e) is proportional to D(e) ∝ e of the final state, the momentum relaxation time tm(e) is 2  v t ðe Þ proportional to e−1. Therefore, the electrical conductivity s ¼ e2 DðeF Þ F m2 F ∝DðeF Þtm ðeF Þ should be energy and density

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Fig. 1 Typical conductivity of graphene as a function of the back gate voltage. Inset shows a typical device on a SiO2 substrate. Adopted from Castro Neto AH, et al. (2009) The electronic properties of graphene. Reviews of Modern Physics 81: 109.

independent (Shon and Ando, 1998). Despite this initial expectation, s is proportional to n at relatively low temperatures in experiments (see Fig. 1) (Novoselov et al., 2005; Das Sarma et al., 2011; Castro Neto et al., 2009). This n-linear dependence of s is assigned to the scattering by charged impurities. For the d-function-like scattering with the potential extension smaller than the wavelength l ¼ 2p/k, the scattering probability is proportional to e. However, it is proportional to e−1 for long-range scattering, leading to s ∝ e2 ∝ n, as we will discuss in the following. We consider carrier transport in the Boltzmann formalism by linear response of the distribution function fk ¼ f0 + dfk to the in-plane electric field E, where f0 is the Fermi distribution function and dfk is the E linear term. The scattering probability E    2p D   V 0 2 d eðkÞ − e k0 W k0 , k ¼ (14) k ,k ħ between the initial and final wave vectors k and k0 is determined by the matrix element Vk0 ,k of the scattering potential and its squared average h|Vk0 ,k |2i over the scatterers’ configurations. The solution becomes df k ¼ etm ðek Þ

∂f 0 v∙E ∂ek

(15)

with the relaxation time given by 1 ¼ tm ðek Þ

Z

  0  dk0  0 2 1− cos yk,k W k k , ð2pÞ

(16)

where yk, k0 is the scattering angle. For randomly distributed scatters with identical potential shape V(r − ri) and density ni, the matrix element term becomes D  E    V 0 2 ¼ ni V k0 − k 2 F k0 − k , (17) k ,k where V(q) is the Fourier transform of V(r) and F(k0 − k) ¼ (1 + cos yk,k0 )/2 is the form factor associated with the inner product of the wave functions of the initial and final states. From Eqs. (14) and (16), we obtain the relaxation time as Z p 

 jej 1 n dy  ¼ i (18) 1− cos 2 y jV ðqÞj2 tm ðek Þ ħ ðħvF Þ2 0 2p with q ¼ k0 − k. Here, h| V(q)|2i ¼ |V(q)|2 with q ¼ 2k sin (y/2). In Eq. (18), (1 − cos2y) term originating from F(k0 − k) indicates suppression of the backscattering or large-angle scattering near y ¼ p. This is unique in graphene and related with the coupling of k and pseudospin (sublattice degree of freedom) as we have discussed for Eq. (1).

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For the short-range (d-function-like) scattering, V(q) becomes q-independent and is replaced with a constant. Eq. (18) leads to and s being independent of n. For real devices, however, we need to consider the screening effect to discuss the Coulomb scattering. Following the representation in Ando et al. (1982), the screened potential of the charged impurity is given by

1 tm ðeÞ ∝jej

V ðqÞ ¼

2pe2 , kqeðqÞ

(19)

where k is the effective dielectric constant determined by the system environment and e(q) is determined by the screening by electrons in graphene. The screening effect is represented by the polarization function P(q) defined as eðqÞ ¼ 1 +

2pe2 PðqÞ: kq

(20)

P(q) at zero temperature becomes constant for q < 2kF and is simply replaced with the density of states D(eF) (Ando, 2006; Hwang and Das Sarma, 2007). Therefore, at low enough temperature, the Thomas-Fermi approximation holds with the constant screening constant qs as V ðqÞ ¼

2pe2 , kðq + qs Þ

(21)

where qs is given by qs ¼

2pe2 k

Z
−

 ∂f 0 DðeÞde: ∂e

(22)

This q dependence of V(q) in Eq. (18) leads to the ne−1 ∝ e dependence of the momentum relaxation time. The relaxation time tm(ek) and the resulting conductivity calculated by integral of Eq. (15) becomes tm ðeÞ ¼ s¼

n ħ H pgS gv ni jej

e 2 j nj H, 4p2 ħ ni

(23) (24)

where H is the dimensionless constant determined by k (Ando, 2006; Adam et al., 2007). This result accounts for the experimental observation s ∝ n. The mobility is independent of n and depends on the concentration of charged impurities: m¼

e 1 H: 4p2 ħ ni

(25)

The mobility is experimentally evaluated from the carrier density or gate voltage dependence of the conductivity. At the charge neutrality n ¼ 0, the conductivity becomes minimum. The minimum conductivity is predicted to be smin ¼

gs gv e2 , 2p2 ħ

(26)

for the short-range scattering irrespective of the value of the scattering probability W (Shon and Ando, 1998). However, in the presence of the long-range scattering, smin increases with W (Noro et al., 2010). Experimentally, smin differs sample by sample (Tan et al., 2007), which is consistent with the charged impurity scattering. Around n ¼ 0, electron-hole puddles are locally present owing to inhomogeneity of graphene devices. Using the induced local carrier density np at the conductivity minimum, s ∝ n is modified to qffiffiffiffiffiffiffiffiffiffiffiffi2ffi s  smin 1 + nn2 around n ¼ 0. p

For bilayer graphene, the parabolic dispersion leads to n ∝ e and D(e) ∝ e0. The form factor in Eq. (17) becomes F(k0 − k) ¼ (1 + cos 2yk,k0 )/2, allowing large-angle scattering (no suppression at yk,k0 ¼ p). In a similar argument, we find s ∝ n for the short-range scattering and s ∝ n2 for the long-range scattering. Experimentally, s ∝ n is observed in a wide range of n (Morozov et al., 2008). Therefore, it has been pointed out there is a scattering by a short-range disorder in addition to the screened charged impurities in bilayer graphene. s ∝ n for both monolayer and bilayer graphene indicates that screening is much stronger in bilayer graphene than in monolayer graphene. Indeed, we find qs ∝ n−1/2 in bilayer graphene (as in standard two-dimensional electron gas) while qs ∝ n0 in monolayer graphene. The ratio of the screening constant between the bilayer and monolayer graphene pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qBLG =qMLG  16= n=1010 cm − 2 becomes significantly small (screening becomes significantly stronger in bilayer) as n increases s s (Das Sarma et al., 2011). In the presence of a perpendicular electric field for bilayer graphene, the gap 2D opens (McCann, 2006, 2023; Oostinga et al., 2008). When the Fermi energy is tuned into the gap, the conductivity is suppressed. Such a gapped state can be prepared in a dual-gated device by tuning the carrier density n and the perpendicular electric field D independently. However, the conductivity in the gap is not so suppressed as in conventional semiconductors, where the conductivity is scaled to exp(−D/kBT ) owing to the thermal activation across the gap. The low temperature transport in this regime is described by the variable range hopping through impurity-induced states. The conductivity is proportional to exp(−(T0/T )1/3), where T0 depends on the disorder in the device.

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In general, the conductivity becomes approximately the sum of contributions from the variable range hopping and thermal activation across the gap:G  G1 exp (−D/kBT ) + G2 exp (−(T0/T )1/3) (Miyazaki et al., 2010). The first term becomes more dominant for higher temperatures. At high temperatures, carriers in graphene also suffer from scattering by phonons. Electrons are scattered by longitudinal acoustic (LA) phonons, optical phonons on the substrate, and ripples of graphene lattice. The resistivity rLA induced by LA phonons (major contribution) is proportional to the temperature T, except at low temperature below Bloch-Grüneisen temperature pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T BG  54 nð1012 cm2 Þ K, where the phonon is degenerate and rLA is suppressed as rLA ∝ T4 (Stauber et al., 2007; Hwang and Das Sarma, 2008). Contributions of other phonons are nonlinear with T. The contribution of phonon scattering in carrier transport is usually (in present devices) minor compared to the charge impurity scattering even close to the room temperature. For T < TBG, intrinsic phonon-limited mobility is high and exceeds 106 cm2 V−1 s−1 (Hwang and Das Sarma, 2008).

Ballistic transport and electron optics In typical clean graphene devices, the mean free path lm ¼ tmvF is of the order of several microns when n is sufficiently large. This long mean free path allows for realization of ballistic transport by making the device dimension comparable with or smaller than lm. Ballistic transport has already been observed and investigated in conventional semiconductors that have similar properties. However, unlike conventional two-dimensional electron gas systems, it is possible in graphene to tune the Fermi energy locally between the conduction and valence bands by means of local gates. This allows us to prepare in-plane p-n junctions in the ballistic regime. In this section, we discuss unique ballistic transport in bipolar graphene junctions. We focus on the Klein tunneling and electron optics in monolayer graphene. Unique properties in bipolar graphene devices originate from the pseudospin-momentum coupling as we have discussed. It was predicted that a relativistic particle incident on an infinite potential barrier transforms into an anti-particle inside the barrier and moves freely in this otherwise forbidden region, resulting in perfect transmission through the barrier. This phenomenon is known as the Klein tunneling. In graphene, which has relativistic charge carriers, it is possible to demonstrate this Klein paradox on a chip (Katsnelson et al., 2006). We consider a normal incident charge carrier to a graphene n-p interface with a smooth potential which does not introduce inter-valley scattering by an atomic-scale short-range scattering. According to the pseudospin-momentum coupling in Eq. (1), the pseudospin of an incoming electron in the K point is parallel to the momentum. Since we assume the smooth n-p interface potential, the pseudospin is conserved in the tunneling process. Therefore, the electron impinging perpendicular onto the n-p junction cannot be backscattered into a counter propagating electron state which would be in the K0 point. Instead, it is scattered into a hole state which has opposite momentum direction but with the same pseudospin as the impinging electron. This leads to the perfect transmission through the n-p junction. In general, while normally incident particles perfectly transmit through the n-p interface, obliquely incident particles have a transmission probability which depends on the angle of incidence. There is also an analogy between the transmission of the carriers at the graphene n-p interface and optical refraction at the surface of metamaterials with negative refraction index. Using such refraction, incident particles can be focused by graphene n-p junctions. This is the principle of the electric current lenses in n-p-n or p-n-p structures, known as Veselago lenses in optics, in the ballistic transport regime (Cheianov et al., 2007). The Klein tunneling has been demonstrated in dual-gated graphene p-n-p (n-p-n) junction devices. The junction is ballistic when the mean free path lm is larger than the distance between the two n-p interfaces. The electrical resistance is larger in the ballistic transport regime than in the diffusive regime. This is because the tunneling probability for obliquely impinging electrons to each p-n junction is lower in the ballistic regime than in the diffusive regime due to the Klein tunneling process. In a ballistic n-p-n or p-n-p graphene junction, quantum interference also occurs due to a resonant cavity between the two n-p interfaces (Shytov et al., 2008). The conductivity oscillates as a function of the carrier density of the cavity region. But this interference is suppressed for the normal incidence owing to the perfect transmission at both interfaces. The interference is dominated by modes at incident angles, where neither the transmission nor reflection probabilities are too large. The magnitude and phase of the transmission and reflection coefficients can be evaluated by this quantum interference in a perpendicular magnetic field B. The magnetic field B bends the trajectories of the carriers and modifies the incident angles of the carriers at the n-p interfaces. It also modifies the transmission and reflection coefficients at the interfaces. In a certain B range, contributions of zero transverse (y-direction) momentum modes become dominant, where the incident angles at the two interfaces have the same sign (see Fig. 2 (b)). The trajectory of a particle in a round trip between the two interfaces encloses the origin kx ¼ ky ¼ 0 in the momentum space and the particle thus acquires the Berry phase p. This p phase shift is the hallmark of the Klein tunneling that should lead to the perfect transmission (zero reflection) in the normal incidence. The p phase shift of the conductivity oscillation was first observed by Young and Kim as shown in Fig. 2(d) (Young and Kim, 2009). There are also other experimental works on electron optics in ballistic bipolar graphene junction devices. Chen et al. demonstrated Snell’s law at the n-p interface by a magnetic focusing experiment (Chen et al., 2016). It was experimentally confirmed that refraction is negative at a n-p interface while it is positive at a n-n0 or p-p0 (the same polarity with different carrier densities) interface.

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(A)

(B)

(C)

(D)

Fig. 2 Demonstration of the Klein tunneling in graphene. (a) schematic of interference in a n-p-n or p-n-p junction at B ¼ 0. (b) schematic of interference for B > 0. (c) a color-coded plot of the derivative of the conductance with respect to the charge density induced by the top gate (n2) as a function of top (n2) and back (n1) gate-induced charge densities. The top gated region is between the two n-p interfaces. The oscillations in conductance are apparent when the densities n1 and n2 have opposite signs. (d) Conductance G vs n2 for fixed n1 at different external magnetic field values applied perpendicular to the sample (from the bottom to the top conductance curve B ¼ 0, 200, 400, 600 and 800 mT); the dots represent data, the smooth lines are the result of the simulations. The sudden phase shift of p that signals the presence of perfect transmission is indicated by dotted arrows. We find by tracing the arrows that the phase of oscillation changes by p between the green and purple curves. It corresponds to the change of the incident angle from the one depicted in (a) to that in (b). Curves are offset for clarity. Panels (a) and (b): From Shytov AV, Rudner MS, and Levitov LS (2008) Klein backscattering and Fabry–Perot interference in graphene heterojunctions. Physical Review Letters 101: 156804. Panels (c) and (d): From Young A and Kim P (2009) Quantum interference and Klein tunnelling in graphene heterojunctions. Nature Physics 5: 222–226; Craciun MF, et al. (2011) Tuneable electronic properties in graphene. Nano Today 6: 42–60.

Valley Hall transport in the presence of charged impurities Berry curvature that we discussed in a previous section characterizes the intrinsic valley Hall effect. However, there are always extrinsic effects, i.e., influence of scatterers. It is more convenient to consider velocity operator to investigate influence of the scattering by charged impurities. Furthermore, discussion of the valley Hall effect in terms of the Berry curvature in Eq. (10) is semiclassical. Using the velocity operator and standard Kubo formula in the linear regime, we can investigate valley related quantum transport. The following argument was developed by Ando (2015a). The velocity operator is defined as     v ¼ vx , vy ¼ vF s ¼ vF sx , sy :
 0 1 We consider massive monolayer graphene in Eq. (4). We perform a unitary transformation U ¼ on the Hamiltonian for −1 0 K0 valley (tz ¼ − 1), which interchanges the sublattices. The Hamiltonian becomes   H ¼ ħvF kx sx + ky sy + tz Dsz : (27) The equation of motion of the velocity operator becomes v_ ¼

1 D ½v, H ¼ 2v2F ðk  sÞ − 2tz v  ez ħ ih

(28)

using the commutation relation of the Pauli matrices. The first term is the Zitterbewehgung term. The second term corresponds to the anomalous Hall conductivity. It can be viewed as the Lorentz force due to the valley dependent effective magnetic field in the z-direction
 D (29) Beff ¼ tz ∗ ez : mB Here m∗B  eħ/2m∗ is the effective Bohr magneton for the orbital magnetic moment of the valley. The role of the mass term is the valley Zeeman energy induced by the effective magnetic field in the real space. The valley Hall conductivity for zero inter-valley scattering is calculated using the Kubo formula Z D h  iE  4ħ ∂ def ð e Þ Tr j sKxy ¼ ImG ð e + i0 Þ − ð x $ y Þ , (30) Re G ð e + i0 Þ j x y ∂e piL2 where L2 is the system size, j ¼ − ev is the current operator, G(e) ¼ (e − H)−1 is the Green’s function, and h. . .i denotes average over configurations of scatterers. This also leads to Eq. (11). Note that the Berry curvature characterizes the response of the system to the shift in momentum space, while the effective magnetic field discussed here characterizes that to the charge diffusion. Similarly to that the drift charge current driven by the shift in momentum space and the diffusion current driven by the charge density gradient

Graphene, transport

303

are carried by the same conductivity, the Berry curvature and the effective magnetic field in Eq. (29) lead to the same Hall conductivity. The work of Ando analyzes the influence of charged impurity scattering on the valley Hall effect. It is shown that even the slightest degree of scattering in the “clean limit” significantly modifies the valley Hall conductivity. The analytical result for zero temperature by the self-consistent Born approximation is given as 8     2 2 2 > 2e2 8jdjðG0 − G1 Þ 1 + d G0 − 2d G1 − 1 − d G2 > > ð|ej> DÞ     2 < h 1 + 3d2 G0 − 4d2 G1 − 1 − d2 G2 svxy ðeÞ ¼ (31) > > 2e2 > : ð|ej< DÞ h with d ¼ D/e, and Gn are the broadening parameters that go to zero in the clean limit and are defined by the impurity density and potential profile. We find that the scattering does not modify svxy(e) in the band gap. Close to the gap but within the band continuum, we have 2e2 jej : h D

(32)

2e2 D 8ðG0 − G1 Þ : h jej G0 − G2

(33)

svxy ðeÞ  Far away from the gap (| d|  1), we have svxy ðeÞ 

These show enhancement of svxy(e) in the band continuum. In particular, for the limit of short-range scattering, only G0 remains nonzero. We therefore have   2 2e2 D 8 1 + d svxy ðeÞ  , (34) h jej 1 + 3d2 2 showing 8 times enhancement of svxy(e) for |d|  1 from that in Eq. (11). The results for different potential profiles are shown in Fig. 3. We see double-peak structure in svxy(e), plotted as a function of the energy, which persists unless the band gap is too small. In general, enhancement of svxy(e) in the band continuum is larger for the shorter-range scattering. This enhancement is also predicted for bilayer graphene (Ando, 2015b). Enhancement of svxy(e) in the clean limit may be somewhat surprising. It already indicates that the intrinsic picture based on the Berry curvature cannot fully capture the valley Hall physics. It may have an analogy with the anomalous Hall effect driven by spin-orbit interaction. It is known there that the ultra-clean samples fall into the extrinsic regime, where the influence of external scatterers is dominant. The intrinsic regime is achieved in the moderately dirty regime. Similarly, the valley Hall effect in the clean limit is significantly affected by the profile of scatterers.

Fig. 3 Valley Hall conductivity and the density of states in the clean limit for scatters with Gaussian potential with varying potential range d. eC  ħvF is the cutoff energy. From Ando T (2015) Theory of valley hall conductivity in graphene with gap. Journal of the Physical Society of Japan 84: 114705.

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Nonlocal transport and edge transport The measurement of the nonlocal transport is widely used to detect the Hall conductivity. Once we have both the electrical conductivity sxx and valley Hall conductivity svxy, we can formulate the charge current and valley current in the presence of in-plane electric field semi-classically. The following conductance matrix form allows us to examine the conversion between the electric field E (in the x direction) and the electric current density jc, and that between E and the valley current density jv (in the y direction) on a homogenous sheet: !

 sxx − svxy jc E ¼ , (35) svxy sxx jv Ev where Ev 

1 ∂ dm 2e ∂x v

(36)

is defined as the “valley field” induced by the chemical potential difference dmv ¼ mK − mK0 between the two valleys referred to as the “valley voltage.” Taking Ev ¼ 0, the valley Hall effect is described by jv ¼ svxy E:

(37)

as the conversion of the electric field into the valley current. Taking jc ¼ 0 and the inverse of the conductance matrix in Eq. (35), we find E¼

svxy   2 jv : ðsxx Þ2 + svxy

(38)

This describes conversion of the valley current into the electric field. The main source of valley relaxation is the scattering at armchair edges and atomic defects. To take into account inter-valley scattering, we employ a valley diffusion equation: ∂2 1 dm ¼ dm ∂x2 v ðlv Þ2 v

(39)

This equation assumes homogeneous inter-valley scattering length lv. Combining Eqs. (35) and (38) under an appropriate boundary condition, one can characterize the valley-related transport in a device. Three groups have reported valley-mediated transport in a nonlocal measurement setup as shown in Fig. 4 (Yamamoto et al., 2015). Similar Hall bar devices have also been used for measurements of the spin Hall effect. In this setup, charge current IC is injected into the left side of the device (between the contacts A and D), and voltage V induced in the right side of the device (between the contacts B and C) is measured. Nonlocal resistance is defined as RNL ¼ V/Ic. For svxy  sxx, the valley-mediated transport can be

Fig. 4 Hall bar geometry and nonlocal resistance measurement setup for generation and detection of the pure valley current. The charge current Ic between contacts A and D generates a pure valley current in its transverse direction via the valley Hall effect. This valley current is converted into a voltage between contacts B and C via the inverse valley Hall effect.

Graphene, transport

305

understood by a sequential conversion picture; the charge current Ic on the left is first converted into the transverse valley current with the conversion ratio given by the valley Hall angle a ¼ svxy/sxx (valley Hall effect). This valley current is then converted back to the transverse charge current on the right with the conversion ratio again given by a ¼ svxy/sxx (inverse valley Hall effect). This charge current is finally converted to the voltage V. Taking the conversion ratio of each process, we find that  2 1 ¼ svxy r3 , (40) RNL ∝a2 sxx where r ¼ 1/sxx is the resistivity of the device. Using the horizontal width W of the Hall bar and the lateral distance L between contacts A and C, RNL is given by  2
 v W sxy L : (41) exp − RNL ¼ 3 2lv ðsxx Þ lv v For sxy sxx, on the other hand, we obtain
   0 W 1 L ∝ svxy r1 : exp − (42) RNL ¼ 2lv sxx lv The behaviors expressed by Eqs. (41) and (42) should be compared with the van der Pauw formula describing the contribution to RNL of classical diffusive transport. Since graphene has high crystal quality, inter-valley scattering mainly occurs at the edges and therefore lv  W. The first experimental demonstration of the generation of pure valley current was reported for monolayer graphene aligned on hexagonal Boron Nitride (h-BN) crystal (Gorbachev et al., 2014). Since h-BN also has a hexagonal lattice with its lattice constant close to that of graphene, graphene aligned on h-BN is subjected to the periodic potential fluctuation of the boron and nitrogen atoms with its period observed as a moiré pattern. For appropriate angular alignment, this periodic potential produces mini-bands and associated band gap in the energy bands of graphene. Since the spatial inversion symmetry is broken owing to this periodic potential modulation, the valley Hall effect should occur. In the experiment performed by Gorbachev et al. (2014), such a high-quality monolayer graphene shaped into a Hall bar geometry aligned on single-crystal h-BN was employed to measure nonlocal resistance as discussed in the previous section. Gorbachev et al. found that the nonlocal resistance RNL is orders of magnitude larger than that estimated for the current diffusion. It was also confirmed that nonaligned graphene on h-BN, which is not subjected to the periodic potential modulation from the h-BN, does not show such a large RNL. Observation of large RNL was therefore attributed to valley-mediated transport. In addition, RNL has a much sharper response to the gate voltage (which modulates the carrier density) than the resistivity r. It was interpreted as the consequence of nonlinearity of RNL vs r in Eq. (40). This experiment called forth debate concerning the microscopic picture of the nonlocal transport. In the experiment, the valley Hall effect was analyzed for metallic samples while the gap opening is supposed to be a necessary condition to confirm inversion symmetry breaking. It is now known that commensurate graphene/h-BN regions and incommensurate strained regions can form a domain-like structure (Woods et al., 2014) and thus the gap can be locally present even if the sample is metallic. Then the next question is whether the valley transport is mediated by the edge or bulk. Lensky et al. showed that bulk currents just beneath the gap may have dominant contribution to the valley current (Lensky et al., 2015). However, criticism was made that Eq. (40) is inapplicable for metallic samples because it is based on semi-classical theory. Indeed, theory by Ando indicates that the scattering by charged impurities should modify the theoretical curve of svxy used in the analysis. Based on the quantum transport model, completely different scenario for the experimental observation, where transport at the trivial edges make the dominant contribution to the observed RNL, was suggested (Kirczenow, 2015; Marmolejo-Tejada et al., 2018). Recent observation of the non-topological edge current in charge-neutral graphene by a scanning SQUID (Aharon-Steinberg et al., 2021) also seems to support this scenario. Note, however, that existence of the trivial edge state was only observed at low (liquid helium) temperature and that the edge state is disordered. Sui et al. (2015) and Shimazaki et al. (2015) employed a dual-gated structure, where a bilayer graphene flake is sandwiched by top and back gates through insulating h-BN layer. Independent control of n and D enabled them to investigate the valley Hall effect in the “insulating” regime. Fig. 5 shows the gate voltage dependence of RL and RNL (data presented by Shimazaki et al., 2015). At the charge neutrality point (CNP), RL is enhanced when D is increased due to the increase of band gap. RNL also increases with D around the CNP. This is because sxx is reduced while svxy is enhanced with increasing D at the CNP. The measured RNL is more than three orders of magnitude larger than the calculated ohmic contribution, excluding the ohmic contribution as the origin of the observed large RNL. Fig. 5(c) shows the scaling at the CNP for T ¼ 70 K obtained by Shimazaki et al. (2015). At 70 K, the crossover between the band conduction (thermally activated transport across the band gap) and hopping conduction regimes occurs upon changing D. A cubic scaling (Eq. 40) is obtained in the band conduction regime of small D (green line) whereas saturation of RNL is observed in the hopping conduction regime of large D by sweeping D at a fixed T. Sui et al. also observed similar nonlinear scaling between r and RNL by sweeping T at a fixed D (Sui et al. 2015). Since the device is highly resistive in this case, RNL should be proportional to r if the edge transport is dominant. Sui et al. further excluded the edge contribution by employing a sample with a long edge (Sui et al., 2015). The nonlocal signal was comparable as

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2

(A)

4

6

1

I

3

5

7

2

(B) 8

4

6

5

7

1

I 8

3

V

V

(C)

Fig. 5 Valley Hall effect demonstrated by the nonlocal transport measurement. A dual-gated bilayer graphene Hall bar device was employed, where bilayer graphene is sandwiched by the top gate and the back gate though h-BN flakes. (a) Gate voltage dependence of the local resistance RL. It was measured as a function of the top gate voltage VTG and the back gate voltage VBG. The perpendicular electric field D is modulated along the red axis while the carrier density n is modulated along the green axis. The inset is a schematic description of the measurement configuration. The blue arrow stands for the charge flow. (b) Gate voltage dependence of the nonlocal resistance RNL. (c) Scaling relation between the resistivity r and RNL at the charge neutrality point (CNP). Each data point is extracted from the data in (a) and (b) for D ranging from 0.22–0.85 V/nm at the CNP. The inset shows the scaling relation in the band conduction regime observed with a different sample. Details are presented by Shimazaki et al. (2015).

long as the active sample length and width were the same, irrespective of the length of the sample edge. This result supports occurrence of the valley transport in the bulk. The cubic scaling is the expected scaling for the intrinsic valley Hall effect (constant svxy) for svxy  sxx near the CNP. For v sxy sxx, RNL for the intrinsic regime is described by RNL ∝ r. The observed saturation of RNL for large D and r may imply deviation from the intrinsic regime in the hopping conduction regime influenced by charged impurities. Microscopic picture of the observed valley-mediated transport is, however, not yet well understood. Recently, similar nonlocal transport was reported for the charge neutral encapsulated bilayer graphene at D ¼ 0 in the presence of a perpendicular magnetic field B, i.e., for the hall-filled 0th Landau level (Tanaka et al., 2021). Here, the electronic correlation leads to gap opening and the ground state becomes the interlayer antiferromagnetic state, where the spins align ferromagnetically in the in-plane direction within each layer but become opposite between the layers (Kharitonov, 2012). Such a state appears even at B ¼ 0 for suspended bilayer graphene (Weitz et al., 2010), where the screening effect is weaker than encapsulated graphene. Each spin state is polarized in one of the layers, which is equivalent to the presence of a spin-dependent spontaneous perpendicular electric field that induces the spin-dependent valley Hall effect. This charge-neutral spin-dependent valley current mediates the nonlocal transport. It was shown that the large nonlocal transport in this state is observed even for a long-edge device and that it only appears when there is a quasi-long-range spin order in the bulk, i.e., below the Kosterlitz-Thouless transition temperature (Tanaka et al., 2022). This observation implies that this nonlocal transport is also mediated by the bulk region despite the criticism made for a different regime (Aharon-Steinberg et al., 2021).

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Hydrodynamic transport We have discussed carrier transport mostly in a single particle picture. However, at high temperatures where the electron-electron scattering time becomes shorter than that of scattering by charged impurities or phonons, carriers behave as fluids rather than independent particles or quasi-particles (Levitov and Falkovich, 2016). Electron-electron scattering time in graphene is given by tee  ħeF/(kBT )2 in the degenerate case (far from the charge neutrality point). In a clean graphene device, the mean free path, or more precisely, the momentum relaxation length lm can exceed the electron-electron scattering length lee ¼ vFtee at high temperatures. The system then enters the “viscous” regime. Note that both the pffiffiffi momentum relaxation time tm by the charged impurities and tee are proportional to eF ∝ n. Therefore, lm/lee has a weak dependence on n, and the viscous regime appears in a wide range of n. On the other hand, the temperature dependence is different between lm and lee. While lm decays slowly (slower than T−1) with increasing temperature, lee decays quickly as T−2. The viscous regime lee < lm appears at high temperatures typically above 100 K in a clean graphene device. Electron-phonon scattering length le−p induced by LA phonons is proportional to T−1, decaying slower than lee for increasing temperature. Therefore, scattering by phonons does not become dominant over the electron-electron scattering and the viscous regime persists even close to the room temperature. This is in contrast with conventional semiconductors, where the viscous transport can be observed only at low temperatures ( 0 of TBG with a twist angle of 1.08 . The wavefunctions are concentrated on the AA regions, having a much smaller amplitude in the AB and BA regions. (B) Band energy in the first mini-Brillouin zone of the superlattice of TBG with a twist angle of 1.05 , and the corresponding density of states with the Fermi energy located at the half-filled state of the flat band with E < 0 in which the superconductivity is observed. For comparison, the density of states of two layers of single-layer graphene without interlayer interaction is also included. (C) Four-probe resistance curves of TBG devices with twist angles of 1.05 and 1.16 (an optical image of the latter is shown in the inset), with critical temperatures of 1.7 K and 0.5 K, respectively. Each measurement is performed for the Fermi energy near the half-filled state of the E < 0 energy flat band. (D) Four-probe resistance plotted as a function of the carrier density and temperatures for a TBG device with a twist angle of 1.05 and a critical temperature of up to 1.7 K. The two superconductive domes (in blue) near the half-filled state are located at a carrier density of around n  −1.3  1012 cm−2. (E) Temperature dependence of the four-probe resistance curves of a TBG device with a twist angle of 1.27 at different applied pressures. The high-pressure measurements in (E) were done at the optimal doping of the superconductor. (A) Reproduced with permission Cao Y, Fatemi, V, Demir A et al. (2018a) Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556: 80–84. Copyright © 2018, Nature Publishing Group. (B–D) Reproduced with permission Cao Y, Fatemi V, Fang S et al. (2018b) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50. Copyright © 2018, Nature Publishing Group. (E) Reproduced from Yankowitz M, Chen S et al (2019a) Tuning superconductivity in twisted bilayer graphene. Science 363: 1059–1064.

strength. These fluctuations were used to predict the vanishing of the Fermi velocity in TBG and the occurrence of flat bands near the charge neutrality for certain low twist angles, known as magic angles, with the first one occurring at
1.1 (Cao et al., 2018b; Andrei and MacDonald, 2020). The flat bands are mainly localized in the AA-stacked regions, as evidenced in Fig. 8A, and correspond to levels with energies that are mostly independent of the electron momentum (Cao et al., 2018a). The flat band energy and the corresponding density of states are depicted in Fig. 8B. Interaction between the electrons increases for twists near the magic angles, and a strongly correlated system is formed with Mott-like insulating states at half-filling of the flat bands, i.e., for two charge carriers per moiré unit cell. However, the complete implications of these strong correlations were not clear until experimental results could be obtained a few years later, in 2018. The low carrier densities at which the flat bands occur, slightly over 1012 cm−2, are easily attained by electrostatic doping in conventional gated devices. The main difficulty arises from the complexity to fabricate the devices with the exact required twist angles, while avoiding the tendency of the TBG to relax to AB-stacking. The experimental results revealed the emergence of unconventional superconductive states for TBG with a twist angle of
1.1 (Fig. 8C) (Cao et al., 2018b). The term unconventional refers to the fact that these states are not originated by the weak electron-phonon interactions of conventional superconductors, but are instead supposed to arise from the aforementioned strong correlations between electrons in the flat bands. The magic angle TBG have a low critical temperature (TC) of up to
1.7 K that depends on the twist angle. Although this TC is low for any practical application, the simplicity of TBG is expected to help understanding the mechanism underlying unconventional superconductivity and lead to the development of superconductors with higher TC. The superconductive states were originally observed near the correlated-insulating states at half-filling of the flat bands of the valence band (EF < 0), displayed as the two low-resistance domes at both sides of the insulating states in Fig. 8D. Conversely, no superconductivity was observed near the flat bands of the conduction band. Since then, superconductive states have been found at lower temperatures in a wider range of fillings in both the conduction and valence bands (Lu et al., 2019). It is also worth mentioning the emergence of ferromagnetic states near three-quarters filling of the conduction band, for which the magnetization can be controlled by applying a small current (Sharpe et al., 2019).

van der Waals heterostructures

319

Superconductivity can also be induced in TBG with angles larger than the magic angle and at higher critical temperatures by tuning the interlayer distance, for example by the application pressure (Fig. 8E) (Yankowitz et al., 2019a). This highlights the importance of the interlayer coupling in the properties of the heterostructures and provides an additional parameter to tune the properties of vdW heterostructures. Other than for TBG, the emergence of correlated insulated states and of superconductivity has also been predicted in other twisted bilayer vdW heterostructures, including bilayers of hBN and of TMDs, and for some trilayer or thicker heterostructures. The most recent experimental results show signatures of superconductivity in certain heterostructures, such as in twisted trilayer graphene (TLG) superlattices, in superlattices between ABC-stacked TLG and hBN, and in twisted WSe2. However, the complexity of experimentally realizing such heterostructures is slowing down the confirmation of predictions.

Graphene/hBN heterostructures The case of graphene/hBN heterostructures will be described here in some detail, as it is one of the most studied vdW heterostructures given its technological importance. For a more exhaustive review on the topic, the reader is referred to Yankowitz et al. (2019b). The 2D materials are very susceptible to their immediate environment, given that most of their atoms are in their surface. Hence, some 2D materials can be unstable and damaged when exposed to oxygen, moisture or to light. The environment can also significantly alter the properties of stable 2D materials, as can be seen in the p-doping of graphene exposed to air. It is thus important to properly isolate them by selecting adequate substrates and encapsulating them. One of the most employed substrates for devices based on 2D materials is SiO2, as it allows to easily find isolated layers of certain 2D materials by their optical contrast. However, the surface roughness of the SiO2 and the existence of charged surface states limit the performance of the devices due to the scattering of charge carriers. Historically, this has had a negative impact on the mobility of graphene devices, with measured mobilities being much lower than the predicted values. hBN is a vdW material with a similar structure as that of graphene, but with the two C atoms of the unit cell being replaced by B and N atoms. The distinct B and N sublattices cause a lack of inversion symmetry that results in hBN having properties very different from those of graphene, being a dielectric vdW material with a large bandgap of
6 eV. Moreover, hBN has a flat and inert surface free of charge traps, and with a small lattice mismatch with graphene (
1.8%). All this makes hBN an ideal candidate as a substrate for other 2D materials (Fig. 9A and B) (Decker et al., 2011). In particular, the mobility of graphene devices on hBN increases by almost an order of magnitude compared with devices on SiO2 substrates (Fig. 9C). The performance of the devices can still be improved when the graphene is completely encapsulated by hBN, providing ballistic transport for distances of tens of microns and with record mobilities of
1.000,000 cm2/Vs attained at low temperatures ( 0, ml > 0. Here and below the plots are not to scale. The inset shows the spectrum of transverse waves for mt < 0, with oM being the maximum frequency of the lower polariton branch.

(quasiparticles). They can be of electronic nature, such as Frenkel or Wannier–Mott excitons, or lattice vibrations, such as optical phonons in ionic crystals. The electromagnetic field polarizes the medium, and the polarization, which is high near the resonance frequency, in turn influences the electromagnetic field. One can regard it as an interaction between the ‘bare’ photon and ‘bare’ medium excitation (hence the term ‘light-matter interaction’). When the interaction is strong enough, the excitation spectrum of the medium essentially changes. Near the resonance between the light mode and the medium excitation, their ‘bare’ dispersion curves transform into two – the ‘lower’ and the ‘upper’ – split modes know as polaritonic branches. These hybrid modes show anticrossing behavior, so that a gap of the size 2D, where D is the coupling strength defined below, opens in the excitation spectrum (see Fig. 1). This situation is referred to as the ‘strong-coupling regime.’ Polaritons inherit the properties of both ‘bare’ photons and ‘bare’ medium excitations, and this mixed – half-light, half-matter – character of polaritons leads to many interesting physical properties. In solids, the type of crystal excitation, which participates in the formation of polaritons, is encoded in a prefix: ‘exciton–polaritons,’ ‘plasmon–polaritons,’ and ‘optical phonon–polaritons.’ The quasiparticles, which appear as a result of interaction between the electromagnetic field and the resonances of magnetic permeability, are known as ‘magnetic polaritons.’ In gases, the term ‘polariton’ is widely used to describe the coupled state of electric field and atomic or molecular transitions in the gas.

Theoretical approaches Many processes with the participation of polaritons, as well as their dispersion law, can be described within the macroscopic semiclassical approach using the system of Maxwell’s equations for the electromagnetic fields in the medium. For spatially restricted systems, the problem must be supplemented by the fields outside the crystal and by proper boundary conditions. These equations are combined with the material equation, which connects the fields D and E through the tensor of dielectric permittivity eij(o, k). The tensor eij(o, k) that describes the response of the medium is obtained from the microscopic theory or from experiment. In general, in an infinite homogeneous isotropic medium with the account of spatial dispersion,
 ki kj ki kj (1) eij ðo, kÞ ¼ et ðo, kÞ dij − 2 + el ðo, kÞ 2 k k where o is the frequency, k is the wave vector, et(o, k) and el(o, k) are transverse and longitudinal dielectric permittivities, and dij is the Kronecker symbol. The condition el(o, k) ¼ 0 determines the dispersion o ¼ ol(k) of a longitudinal wave, which does not interact with transverse photons. In turn, the poles o ¼ ot(k) of the function et(o, k) determine transverse ‘bare’ excitations of the medium (excitons, plasmons or optical phonons). As shown below, polaritons are the transverse normal modes (i.e., the solutions of Maxwell’s equations) of the medium near these frequencies. The macroscopic description holds as long as the wavelength of light strongly exceeds the lattice constant and the excitonic Bohr radius. In the framework of the alternative microscopic (quantum) approach, polaritons appear due to account of retardation. Excitons are collective electronic material excitations in the approximation when only the instantaneous Coulomb interaction between the molecules (or atoms) is considered (i.e., in the limit c ! 1). Retardation, as known from quantum electrodynamics, is equivalent to taking into account the interaction with transverse electromagnetic field. Hence, polaritons appear as a result of interaction

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499

between excitons and transverse photons. This approach utilizes the second-quantization formalism, where polaritons are the eigenstates of the full Hamiltonian of the crystal, which is a sum of the Hamiltonians of ‘bare’ transverse photons, ‘bare’ excitons, and the Hamiltonian of their interaction. The polariton operators are linear combinations of the operators of ‘bare’ photons and ‘bare’ excitons. Near the resonance, these operators are ðrÞ

rÞ ðkÞB+k x+k,r ¼ Cphot ðkÞa+k + Cðexc ðrÞ⁎

rÞ⁎ xk,r ¼ Cphot ðkÞak + Cðexc ðkÞBk

(2)

where a+k , ak and B+k , Bk are the creation and annihilation operators of photons and excitons, and r ¼ U, L denote the upper or the (r) (r) (k) and Cexc (k) are the amplitudes of the photon and the exciton in each polaritonic lower polaritonic branch. The coefficients Cphot state, and their squared modules are the weights of the ‘bare’ excitations in the polaritonic states. They vary between 0 and 1 with the change of the wave vector k, and at the resonance 2   2   ðU, LÞ U, LÞ ðkres Þ ¼ 1=2 Cphot ðkres Þ ¼ Cðexc Within linear optics, these two approaches yield identical results, and the choice between them is a matter of convenience. The semiclassical (macroscopic) approach is often used to calculate the optical response, while quantum (microscopic) formalism is convenient for studies of non-linear processes. Quantum description is also used to discuss the collective properties of polaritons and their statistics, and for the description of the eigenfunctions of the crystal (in particular, the polaritonic vacuum state is not equivalent to an independent-particle vacuum). Furthermore, quantum description is valid for low-dimensional structures (such as a monolayer, or a chain of molecules), when a dielectric tensor cannot be introduced, and the macroscopic description is not valid.

Strong coupling, weak coupling, and ultrastrong coupling The strong-coupling regime can be destroyed by various dephasing processes. Many of them (such as exciton–phonon interactions and scattering of polaritons by structural disorder) act on polaritons through their material (excitonic) component. In addition, the lifetime of polaritons is restricted by finite lifetime of the photon within the finite-size crystal. The strong-coupling regime holds as long as the exciton–photon coupling strength D exceeds all the involved dephasing rates. If this condition is violated, the system is in the so-called weak-coupling regime, when the gap in the spectrum of elementary excitations does not form. In the quantum language, the strong-coupling regime is described by vacuum Rabi oscillations (reversible exchange of energy quanta between light and matter ‘bare’ states), whereas the weak-coupling regime corresponds to irreversible spontaneous decay of the collective ‘bare’ material excitations of the medium. In the strong-coupling regime the polariton splitting D is a small fraction of the frequency of the material excitation, ot. Since recently, a new regime – ultrastrong coupling, with the coupling strength D and excitation frequency ot being of the same order – became experimentally available. The impact of phonons can be diminished by decreasing the temperature. For instance, in good GaAs crystals, the strong coupling regime holds below 20 K in bulk, and below 70 K in microcavities. Disorder appears as impurities in bulk crystals, and interface roughness and/or alloy fluctuations for cavity- and surface polaritons. The main effect of disorder is that it introduces local excitonic levels, which trap polaritonic states and eventually can destroy them. In the state-of-the-art optical microcavities, the role of disorder can be brought almost to nothing.

Polaritons in bulk Dispersion equation In the framework of the macroscopic approach, the dispersion equation for polaritons in a bulk (3D) medium can be obtained by solving the wave equation for the fields D and E: DE +

o2 D − grad div E ¼ 0 c2

(3)

combined with a material equation. In a uniform isotropic medium, where the wave vector k conserves, and neglecting the dissipation, the dielectric permittivity Et(o, k) near an electronic transition is et ðo, kÞ ¼ eb +

A o2 ðkÞ − o2

(4)

where eb is the background dielectric constant, and A is proportional to the oscillator strength fosc of the transition. Dropping the resonance term would describe ‘bare’ photon in that medium. Furthermore, ot(k) is the dispersion of the crystal excitation, which, for small k, we can write in the effective-mass approximation: ot ðkÞ ¼ ot +

hk2 , 2mt

ot  ot ðk ¼ 0Þ

(5)

500

Polaritons

Here mt is the effective mass of the material excitation, it determines the curvature of its dispersion. If the dependence of the dielectric permittivity on the wave vector (i.e., spatial dispersion) is neglected, then ot(k) ¼ ot and et(o,k)  e(o). This approximation works well for optical phonon–polaritons and some of Frenkel exciton–polaritons, while for Wannier–Mott exciton–polaritons the dependence of et(o,k) on k is usually important. In an infinite isotropic medium, the wave equation, Eq. (3), has three solutions. One is a longitudinal wave, and its dispersion law o¼ ol(k) is determined by el(o,k) ¼ 0. In addition, there are two twice-degenerate (corresponding to two possible polarizations) transverse waves, which are polaritons. Their dispersion law can be found from the equation c2 k2 A ¼ et ðo, kÞ ¼ eb + 2 o2 ot ðkÞ − o2

(6)

which for o  ot simplifies as   o − ophot ðkÞ ðo − ot ðkÞÞ ¼ D2

(7) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffi where D ¼ A=ð4eb Þ∝ f osc is the light-matter coupling strength, and ophot ðkÞ ¼ ck= eb is the dispersion of the photon in the crystal. Near the resonance (i.e., for o  ot(k)  ophot(k)), the polaritonic branches anticross. The smallest separation between them is at the resonance, and is equal to 2D. Far from the resonance, the polaritonic effect is negligible, and polaritonic dispersion curves approach the dispersion curves of the bare photon and material excitation. The solutions of (7) o ¼ oU,L(k) are  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 ophot ðkÞ − ot ðkÞ + 4D2 (8) oU,L ðkÞ ¼ ophot ðkÞ + ot ðkÞ  2 The dispersion curves of bulk exciton-polaritons are plotted in Fig. 1 together with ‘bare’ excitations and the longitudinal wave, which does not couple to light ((A) without and (B) with spatial dispersion). These modes have been seen in many semiconductors (such as CuCl, GaAs, CdS) in a variety of optical experiments (for instance, two-photon absorption measured at different angles or hyper-Raman scattering measurements).

Anisotropic media Some of Frenkel excitons have large oscillator strength and hence interact strongly with light. Organic molecular crystals, which support Frenkel excitons, are optically anisotropic. The properties of polaritons in anisotropic crystals are determined by the structure of the dielectric tensor eik(o,k) of the crystal (see Eq. 1), which relates the fields D and E as Di ðo, kÞ ¼ eik ðo, kÞEk ðo, kÞ Then the equation analogous to (6), which yields the dispersion of polaritons o ¼ o(k), can be obtained from the zeros of the following determinant:   2  o  eik ðo, kÞ −k2 dik + ki kk  ¼ 0 (9)   c2 For an arbitrary direction of the wave vector k, polaritons in anisotropic crystals have a mixed longitudinal–transverse character.

Spatial dispersion and additional waves Spatial dispersion, which manifests as dependence of dielectric permittivity e on the wave vector, reflects the ability of the material excitations to propagate in the crystal due to internal interaction forces. If spatial dispersion is neglected, then there are no bulk waves in the frequency interval ot < o < ol ¼ ol(0), and for all other frequencies the dispersion equation allows for a single propagating bulk wave. In other words, all the frequencies o and wave vectors k are in one-to-one correspondence (see Fig. 1A). The account of spatial dispersion corresponds to finite (as opposed to infinite) effective mass of the material excitation, mt, and hence results in bending of the dispersion curves of ‘bare’ material excitations, so that near the resonances, at the concave side of the dispersion curve, one-to-one correspondence between the wave vectors and frequencies becomes broken (Fig. 1B). As a result, in narrow frequency intervals near the resonances, more than one wave can propagate at a given frequency. The existence of largewave-vector waves, which appear in addition to “regular” small-wave-vector polaritons, was first pointed out by Pekar (1957). They are called ‘additional waves.’ The spatial dispersion shows up as the wave vector dependence of the excitonic resonance, as in Eq. (5) (isotropic medium is assumed). The curve ot(k) bends up or down, depending on the sign of mt. The curve ol(k) also bends up or down depending on the 6 mt. According to the signs of ml and mt, a number of sign of the effective mass ml of the longitudinal exciton, since in general ml ¼ scenarios emerge. Some examples are given below. For instance, in an isotropic nongyrotropic medium with mt > 0, for o> ol, there always exist two propagating transverse waves, a small-wave-vector upper polariton and a large-wave-vector lower polariton (see Fig. 1B). A lower polariton state exists also in the ‘forbidden’ region (ot < o < ol). In Fig. 1B, ml is taken to be positive, that is typical for inorganic semiconductors. Then for o > ol the longitudinal wave coexists with two transverse polaritons. As another example, consider mt < 0, which holds in some molecular crystals. Now there are two propagating transverse waves (both are lower-branch polaritons) for o < oM < ot (see the inset in Fig. 1B). Out of these two waves, the polariton with the larger wave vector has a negative

Polaritons

501

Photon

ω

(+) ω t (k)

ωt ωG

ωt()(k)

k Fig. 2 Dispersion of bulk polaritons (thick red lines) in a gyrotropic cubic crystal. Dashed blue lines show the dispersion of ‘bare’ excitons and ‘bare’ photon, oG is the maximum frequency of the lower polariton branch.

group velocity, and therefore at interfaces it exhibits unusual refraction known as ‘negative refraction.’ As for the longitudinal wave (not shown in the inset), for ml > 0 it exists at o > ol together with the upper polaritonic branch, while for ml < 0 the longitudinal wave is the third wave at o < oM, and it is the only one wave at oM < o < ol. Yet one more interesting case to consider is a cubic gyrotropic 2 crystal, where the dispersion curve of the exciton splits into two branches with linear dispersions: o() t (k) ¼ ot  ak + O(k ). The corresponding polaritonic spectrum with three propagating waves at o < oG < ot and one propagating wave at o > oG is shown in Fig. 2. Again, the polaritonic branch, which asymptotically follows o(−) t (k), exhibits negative refraction. For spatially restricted media (e.g., when calculating the reflectance of a bulk crystal) at the frequencies where the one-to-one correspondence between o and k is broken and an additional wave or waves exist, one has to introduce the so-called additional boundary conditions (ABCs), since usual Maxwell’s boundary conditions are not sufficient to find the relation between the amplitudes of the waves. From a microscopic point of view, ABCs describe the behavior of the fields close to the surface. The character of ABCs essentially depends on the properties of the medium, its surface, and on the properties of the material excitations (excitons). In the most general case, in the framework of linear optics, ABCs are constructed as some linear combinations of the fields E, P and their derivatives. The first ABC proposed by Pekar (1957) is P ¼ 0 at the boundary of the medium. It is good for molecular crystals, but is often suitable also for semiconductors. A widely used approach proposed by Hopfield and Thomas (1963) introduces a ‘dead layer’ of some thickness l near the surface, where excitons cannot penetrate. The thickness l and other phenomenological constants entering ABCs are to be determined from a microscopic theory or from comparison with experiment.

Surface polaritons Surface polaritons are normal modes that propagate along the surface of the medium, or along an interface between two media, or along interfaces of layered structures. The electromagnetic field in these modes decays exponentially with the distance from the surface. As shown below, surface polaritons require the dielectric permittivity of the medium to be negative, which is possible near a resonance frequency of the medium. Surface polaritons can give us information about the surface of the crystal. Furthermore, they may be important as possible final states in various processes of interaction or scattering of electromagnetic waves in crystals.

Surface polaritons at the interface between two semi-infinite media Let a semi-infinite isotropic medium with the dielectric permittivity e(o) (spatial dispersion is neglected) be in contact with an isotropic half-space with the dielectric constant eout. Let k|| be a two-dimensional wave vector parallel to the surface. The surface mode is TM-polarized (see the inset in Fig. 3) and has the following dispersion relation: ε(ω)

TM-polariza on Ez εout k____x ε(ω)

Hy

ωt

Ex

ωS ωl

ω

εout

Fig. 3 Red solid line: graph of the function e(o). At the interface between the materials with the dielectric permittivities e(o) and eout, surface polaritons exist at the frequency interval ot < o < oS highlighted in blue, where −eout < e(o) < 0. The inset shows the electric and magnetic fields in a TM-polarized mode.

502

Polaritons eðoÞ eout qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 2 kk −o eðoÞ=c kk −o2 eout =c2

(10)

Assuming that, as usual, eout > 0, the dispersion equation of surface polaritons can be satisfied only if e(o) < 0. The dispersion equation (10) can be rewritten in a more convenient form: k2k ¼

o2 eðoÞeout c2 eðoÞ + eout

(11)

From here it is seen that the interval of the frequencies where surface polaritons can exist is narrower than ot < o < ol, since for real k|| the negative dielectric permittivity e(o) must necessarily be less than −eout. Let oS denote the upper boundary of the surface polaritons frequency range: e(oS) ¼ − eout. When Eout approaches Eb, oS ! ol. For a dielectric material, the function e(o) is given by (4); its graph is shown in Fig. 3, and the frequency interval where a surface polariton exists is highlighted. For a metal, when pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e(o) ¼ 1 − o2p /o2 (op is the plasma frequency), surface plasmon–polariton exists for 0 < o < op = 1 + eout . Fig. 4A shows surface pffiffiffiffiffiffiffiffi polaritons neglecting the effects of spatial dispersion (thick magenta line marked as SP). The blue dashed lines o ¼ ckk = eout and pffiffiffiffiffi o ¼ ckk = eb show, respectively, the photonic modes in the adjacent half-space and in the crystal (propagating parallel to the crystal surface), and the thick red lines marked as LP and UP are bulk polariton dispersion curves, shown here for reference. The photonic and bulk polaritonic dispersions do not intersect with the dispersion curve of the surface polariton, hence surface polaritons cannot be excited by an incident photon (or, once excited, cannot transform into bulk polaritons or leave the crystal as photons) – the laws of conservation of energy and wave vector cannot be simultaneously satisfied for such processes. To excite surface polaritons, one has to make use of a prism (attenuated total-reflection technique) or a grating on the crystal surface. Surface polaritons were seen as dips in the total reflection of TM-polarized light in many such experiments. If spatial dispersion is taken into account, the spectrum becomes more complicated, since then (assuming a positive effective mass mt) the large-wave-vector lower ‘bulk’ polariton becomes resonant with the small-wave-vector surface polariton (see Fig. 4B). As discussed above, it requires introduction of ABCs, which mix surface polaritons with bulk lower polaritons. The mixing leads to leakage of the energy of the surface wave into the interior of the crystal, providing a new (nonradiative and nondissipative) damping mechanism for surface polaritons.

Surface polaritons in more complex geometry The presence of a surface transition layer can essentially alter the dispersion of surface polaritons. The transition layer may be either natural or artificial (a thin film on a substrate). As shown by Agranovich and Mal’shukov (1974), if an oscillation in the layer is resonant with the surface polariton, a gap with a magnitude DS opens in the surface polariton spectrum. This is similar to what happens to bulk electromagnetic waves near the resonance frequency with the difference that both the excitation in the layer and the surface electromagnetic wave have a two-dimensional character. While usually the effects of a thin layer are of the order of d/l  1 2 (d is the layer thickness, l is the wavelength of the light), here k2k ¼ oc2 eðeoðÞo+Þeeoutout , and therefore the gap was observed in a number of experiments, both in the infrared spectral region and in the spectral region of electronic transitions. Surface texture, either controllable (a grating) or natural (surface roughness), as well as imperfections of the crystal that supports surface polariton (such as polycrystalline grains) alter the dispersion of surface polaritons in significant ways. As is typical for disordered samples, the surface polariton frequency acquires a shift, which results in the change of the polariton’s group velocity. Furthermore, the mode is attenuated while propagating along the surface, due to its interaction with surface imperfections (for

ωl

b

out

UP

ω c k__ /√ε

b

out

UP

ωS

ωt

ω

ω c k__ /√ε

ωl

(B)

ω ck /√ε __

ω

ω ck /√ε __

(A)

ωS

SP

ωt

LP

k__

SP

LP

k__

Fig. 4 Dispersion of surface polariton (magenta solid line) at an interface between a resonant isotropic crystal with dielectric function e(o) and an isotropic material with a dielectric constant eout (A) without and (B) with spatial dispersion. Thick red lines show the dispersions of bulk polaritons, and thin dashed lines show the dispersion of the photons outside the crystal and of the ‘bare’ photon in the crystal. UP, LP, and SP denote, respectively, the upper, lower, and the surface polariton branches.

Polaritons

503

rough surfaces) or scattering at the grains’ boundaries (for polycrystals). Finally, while still being predominantly localized near the interface, the mode can acquire a radiative component, and hence a finite radiative lifetime. In other words, the imperfections scatter a surface polariton wave into other surface polariton modes and into vacuum radiation.

Polaritons in nanostructures with reduced dimensionality In bulk crystals the interaction of excitons with an electromagnetic field does not give rise to radiative decay of the exciton. Instead, the energy oscillates back and forth between exciton and photon states (vacuum Rabi oscillations). This can be attributed to the conservation of total wave vector of bare excitations: An exciton with a given wave vector interacts with only one photon with the same wave vector, and there is no density of states for irreversible radiative decay. Another situation occurs in a confined geometry, i.e. in two- (2D) and one-dimensional (1D) crystals, where an exciton has less degrees of freedom than a photon. As noted by Agranovich and Dubovsky (1966), in such crystals the radiative decay is restored, since in these cases only two (in 2D) or one (in 1D) of the components of the total wave vector are conserved. Consider polaritonic effects in quasi-2D quantum wells and quasi-1D quantum wires. The translational symmetry is preserved (and, consequently, the wave vector k|| of the exciton is a good quantum number) only in the plane of the quantum well in the first case, and only along the quantum wire in the second case. The component of the total wave vector perpendicular to k|| is denoted by k?. Photons are 3D, as before. It turns out that the system supports two distinct kinds of states. Consider first the states with small values of the wave vector pffiffiffiffi kjj < k0  o eb =c. They are degenerate with photons and hence can decay into them. One can regard it as a coupling of an exciton with a wave vector k|| with a reservoir of photons with the same component k|| parallel to the quantum well or the quantum wire, but with all possible values of k?. These photons act as a dissipative bath, and instead of forming stable hybrid states similar to polaritons in the bulk, the exciton decays irreversibly. The corresponding radiative lifetimes can be calculated treating exciton–light coupling as a perturbation. For Frenkel (F) 2 −9 −8 s is the exciton–polaritons, the decay times in 1D and 2D are t(F) 1D ¼ (2pa/l)t0 and t2D ¼ (2pa/l) t0, where : t0  10 –10 radiative lifetime of the excitation of an isolated molecule, and a is the lattice constant (i.e., the radiative decay is enhanced proportionally to the number of molecules within the wavelength of light, l). The radiative lifetime of exciton–polaritons in 2D is then 10−12 s, and such a ‘superradiant decay’ was observed in anthracene. For Wannier–Mott exciton–polaritons, the lifetime of pffiffiffiffiffi ðWMÞ 2D exciton–polariton for k|| ¼ 0 was found to be t2D ¼ m1 c eb =ð2pe2 f osc Þ and is about 10 ps for the parameter of GaAs. The account of the effects of thermalization and disorder-induced partial localization of polaritons considerably complicates the picture both in 1D and 2D cases. For k|| > k0, the modes are nonradiative. In quantum wells, to some extent, they are similar to surface polaritons, since the electromagnetic field in them decays exponentially outside of the quantum well plane. Similarly, the propagating modes can be excited using a prism or a surface grating. Radiative polaritons with k|| < k0 can be excited directly by incident light and were observed in many semiconductor nanostructures in absorption, photoluminescence, and reflectivity spectra.

Cavity polaritons In nanostructures called optical planar microcavities both excitons and photons have a 2D character. The interaction between them leads to formation of states similar to bulk polaritons known as ‘cavity polaritons.’ These states are stable in the sense that an exciton with a given 2D wave vector k|| interacts with only one cavity photon with the same k||, and, similarly to bulk polaritons, the energy oscillates back and forth between these states. In other words, in contrast to the low-dimensional nanostructures described in the previous section, where photons were 3D and the material excitons had a reduced dimension, in a planar microcavity there is no density of states for irreversible radiative decay of the exciton. After several oscillations, however, photons escape the cavity due to a finite transmission of the cavity mirrors. Cavity polaritons were observed for the first time by Wesbuch et al. (1992) in a GaAs planar microcavity. An optical microcavity modifies the photonic density of states and reduces its quantization volume. As a result of confinement both the binding energy of the Wannier–Mott exciton and the exciton–light coupling are enhanced, so that Wannier–Mott exciton–polaritons are considerably more stable in the 2D geometry than in the bulk, and the polariton splitting 2D is much more pronounced. Furthermore, lower polaritons have a finite (rather than zero) energy at k|| ¼ 0, and this lowest-energy state may, in principle, work as a ‘trap’ when polaritons undergo phonon-assisted relaxation after non-resonant excitation. Since the mirrors of planar microcavities are often fabricated as distributed Bragg reflectors, a detailed description of cavity polaritons requires simulating a multi-layered structure. The dispersion equation of cavity polaritons can be obtained from Maxwell’s equations with the help of the transfer-matrix method, which relates the transmission and reflectance at each interface of the multiple layers. Near the resonance, one obtains the dispersion equation of the type (7) with solutions similar to (8), with the energies of ‘bare’ exciton and photon dependent on the in-plane wave vector k||. A significant difference arises for the cavity photon dispersion, since the perpendicular component of its total wave vector is fixed by the boundary conditions at the microcavity mirrors, leading to parabolic (for small k||) dispersion of the cavity photon, which, in a simplifying assumption of ideal mirrors, reads:

504

Polaritons

ω

1

ωt

ωp ( hot k ) __

ωU(k__)

(L) (k )|2 |Cexc __

0.5 (L) (k )|2 |Cphot __

0

) ω t(k __ A

ωL(k__)

B k__ Fig. 5 Dispersion of cavity polaritons (red solid lines) for zero detuning, i.e., when oc ¼ ot. Blue dashed lines correspond to the dispersions of ‘bare’ cavity (L) (L) photon and ‘bare’ exciton. The inset shows the dependence on the wave vector of the exciton and photon weight coefficients | Cexc (k)|2 and | Cphot (k)|2 defined in Eq. (2) for the lower branch.

ophot ðkÞ ¼ oc +

hk2 2m

(12)

 pffiffiffiffi pffiffiffiffi where oc  ophot ð0Þ ¼ pc= Lc eb , Lc is the microcavity thickness, and m ¼ pℏ eb =ðcLc Þ is its effective mass, which is very small −5 (10 me, me is the electron mass). The dispersion curves of cavity polaritons are plotted in Fig. 5. The detuning between the cavity photon and quantum-well exciton u ¼ oc− ot determines the ratio between exciton and photon weights in the polaritonic states for each k||. Since this ratio determines the properties of the polaritonic states (including their ability to interact with each other and disorder), the detuning serves as a control parameter in designing polaritonic properties. The value of detuning is determined by the cavity thickness and, due to a wedge-shaped microcavity profile, can be varied in experiments. At moderate excitation powers, the relaxation of polaritons into the ground state oL(0) is impeded due to the so-called bottleneck effect observed in photoluminescence experiments. Polaritons accumulate near the ‘knee’ at the lower polariton dispersion curve just below the exciton energy (near the point A in Fig. 5), since there the acoustic phonon-assisted relaxation is not effective due to an abrupt decrease of the density of final states. Qualitatively, at this point lower-branch polaritons change their character from exciton-like to photon-like, which essentially impedes the phonon-assisted relaxation (Tassone et al., 1997). Macroscopic population of excitations in oL(0) state can be reached exploiting the bosonic nature of cavity polaritons. Bosonic effects had first manifested themselves in the experiments on stimulated scattering of cavity polaritons in a GaAs/AlGaAs microcavity (Savvidis et al., 2000). In these experiments, lower-branch polaritons were excited resonantly near the inflection point of their dispersion curve (point B in Fig. 5). A convex–concave character of the dispersion curve at this point allows for the polariton–polariton scattering process, when one polariton is scattered into the state with k|| ¼ 0 (‘signal’), and one is scattered to a higher-energy state (‘idler’). A weak stimulation of k|| ¼ 0 state by a probe beam leads to a giant (100-fold) amplification of the ‘signal’ due to the fact that the scattering rate of bosons is proportional to ni(1 + nf), where ni and nf denote, respectively, the populations of the initial and final states. Several years later, Kasprzak et al. (2006) experimentally demonstrated Bose–Einstein condensation (BEC) of polaritons in a CdTe/CdMgTe microcavity. Above a threshold excitation density, the observed broad emission from a non-resonantly pumped microcavity transformed to a narrow peak centered at k|| ¼ 0. This was accompanied by spontaneous build-up of coherent polarization of the condensed polaritons out of depolarized population below the threshold. The polariton occupancy followed Maxwell-Boltzmann distribution with the effective temperature as high as 19 K. Since then, condensation of cavity polaritons and the polariton lasing were demonstrated by other groups in a variety of settings. The materials with larger oscillator strength, such as GaN (Christopoulos et al., 2007), ZnO (Li et al., 2013), organic materials (Kena-Cohen and Forrest, 2010) and mono-atomic layers of transition metal dichalcogenides (TMDCs; Waldherr et al., 2018) allowed for achieving room-temperature polariton lasing. For practical purposes, polaritonic laser requires not only room temperature operation, but also electrical injection of carriers. Research in this direction is underway. The unprecedentedly high-temperature polaritonic BEC owes to small (compared to atoms) effective mass of cavity polaritons. However, polaritonic BEC differs in many ways from text-book BEC of bosonic particles. Many efforts are done to understand polaritonic BEC in presence of such factors as a finite lifetime of polaritons within the cavity, external pumping, strong polariton-polariton interactions, disorder, polarization degree of freedom, and vibronic excitations (in organic materials). Various scenarios are possible. Generally speaking, two-dimension character of polaritons implies that polaritons undergo BerezinskiiKosterlitz-Thouless (BKT) phase transition, rather than “true” Bose–Einstein condensation. This had been confirmed experimentally by Roumpos et al. (2012), by measuring the first order spatial correlation function of polaritons, which showed typical for BKT superfluid phase power-low decay, however modified by the non-equilibrium character of the polariton condensate: Polaritons undergo decay and reappear via pumping, instead of reaching thermalization in a closed system. At the same time, structural disorder combined with driven-dissipative nature of the polariton condensate may lead to appearance of pinned vortices (Lagoudakis et al., 2008) – which are different from vortices formed spontaneously as a result of thermal fluctuations in BKT transition to a superfluid phase. This direction was further explored by Sanvitto et al. (2011), who demonstrated experimentally

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and modelled theoretically all-optical control of motion of vortices and formation of controllable vortex-antivortex pairs in an optical potential, which played the role of artificial disorder. Quite recently, polaritonic condensation served as a basis for realization of an exciton-polariton topological insulator (Klembt et al., 2018). Polaritonic condensate was also suggested as a candidate system for high-temperature superconductivity (Laussy et al., 2010). Yet another direction of research under the umbrella of cavity polaritons is currently gaining momentum. It is often called ‘molecular polaritons,’ and it studies ensembles of molecules strongly coupled to optical cavity vacuum or to surface plasmons. These systems exploit rich level structure of the molecules, with large oscillator strength of some transitions giving rise to strong and even ultrastrong coupling (both for electronic and for dipole-active vibronic transitions), and with a multitude of dark or weakly coupled to light states that take part in the relaxation dynamics of excitations. In particular, organic molecules are responsible for the observation of room-temperature single-molecule strong coupling in plasmonic nanocavities, which are made of gold nanoparticles separated by a sub-nm spacer from a gold substrate (Chikkaraddy et al., 2016). Furthermore, a pioneering observation of a cavity-induced slowing of photoisomerization rate of a photochrome by Hutchison et al. (2012) gave rise to a rapidly evolving direction conventionally referred to as ‘polariton chemistry.’ The key idea is that the strong coupling modifies the level structure of molecules, and hence the energy landscapes governing reactions pathways. As a result, a cavity-embedded molecule under properly chosen resonance conditions may exhibit a different (with respect to a cavity-free scenario) chemical properties (such as isomerization and reactivity), as well as different energy transfer ability.

Dark-state polaritons in atomic gases Nowadays many efforts are devoted to creating techniques for controllable and reversible trapping of photonic states into atomic degrees of freedom. Photons are perfect carriers of information, while the atomic subsystem may act as a storage facility and simultaneously provide interactions between stored photons. Therefore, possible applications in quantum information and optical communication. Recent experiments demonstrated photon trapping in coherently driven atomic gases, first in an ultracold gas, and soon after in warm alkali vapors. The generic scheme involves two electric fields (‘probe’ and ‘control’) and three atomic levels (which we denote by |g1 ⟩, |g2 ⟩ and | e⟩; the first two are long-lived low-energy states, the third is an excited state of the atom). Strong control field, which resonantly couples the states |g2 ⟩ and |e ⟩, dramatically modifies the propagation of a weak probe field in a narrow spectral window near the frequency of the |g1 ⟩!|e ⟩ transition. The refractive index of the gas experienced by the probe has a zero imaginary part at the resonance, and hence the probe propagates without absorption, owing to which this phenomenon is known as ‘electromagnetically induced transparency.’ In addition, near the resonance the real part of the refractive index exhibits strong variation as a function of frequency, and hence the group velocity of the probe is strongly modified. The group velocity of the probe appears to be vg ¼ 

c  1 + D2 =O2c

(13)

where Oc ¼ Ecd| g2⟩!|e⟩/ℏ is the Rabi frequency of the control field with the amplitude Ec driving the transition |g2 ⟩ !|e, and D is, as before, the vacuum Rabi frequency of the |g1 ⟩ !|e > transition. The expression (13) shows that the group velocity of the probe can be modified by varying the intensity of the control field. Decreasing Ec results in slowing the probe down to velocities as small as several meters per second (‘slow light’), or even brings a light pulse to a full stop. When Ec ¼ 0 (control beam is turned off ), vg vanishes, and the photon is trapped in the atomic gas. The trapped photon is retrieved by turning the control laser on again. The storage is completely coherent as long as the entire process is adiabatic. These dynamics happen because inside the gas the probe light is strongly coupled to the collective excitation of the atomic ensemble driven by the control field. The result of this coupling is called ‘dark-state polariton.’ When Ec vanishes, the polariton is mapped onto purely atomic excitations, and the propagation stops.

Conclusion We have discussed polaritons – excitations that form if the strength of interaction between light and a dipole-active material resonance in a solid medium exceeds existing losses and relaxation rates. Though many ‘types’ of polaritons can be distinguished (as determined by the physical nature of the resonance the light couples with), with some reservations it is correct to say that the properties of polaritons depend more on the geometry of the system than on the specifics of the resonance. This is not surprising, given that polaritons can be viewed simply as a doublet of states originating from coupling between two oscillators. The hybrid polaritonic states possess simultaneously some properties of light (higher tolerance of structural disorder, light-like propagation) and some features of material excitations (non-linear interactions, coupling with phonons). Exploring this duality is what makes polaritons unique, and polaritonic field of research attractive and flourishing.

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References Agranovich VM (1959) Dispersion of electromagnetic waves in crystals. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 37: 430. (1960, Soviet Physics ZhETP 10: 307). Agranovich VM and Dubovsky OA (1966) Influence of retardation on spectrum of excitons in one- and two-dimensional crystals. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 3: 345.. (1966, Journal of Experimental and Theoretical Physics (JETP) Letters, 3: 223). Agranovich VM and Mal’shukov AG (1974) Surface polariton spectra if the resonance with the transition layer vibrations exist. Optics Communications 11: 169. Chikkaraddy R, de Nijs B, Benz F, Barrow SJ, Scherman OA, Rosta E, Demetriadou A, Fox P, Hess O, and Baumberg JJ (2016) Single-molecule strong coupling at room temperature in plasmonic nanocavities. Nature Letters 535: 127. Christopoulos S, Höger GB, von Högersthal AJDG, Lagoudakis PG, Kavokin AV, Baumberg JJ, Christmann G, Butté R, Feltin E, Carlin J-F, and Grandjean N (2007) Room-Temperature Polariton Lasing in Semiconductor Microcavities. Physical Review Letters 98: 126405. Fano U (1956) Atomic theory of electromagnetic interactions in dense materials. Physical Review 103: 1202. Hopfield JJ (1958) Theory of the contribution of excitons to complex dielectric constants of crystals. Physical Review 122: 1555. Hopfield JJ and Thomas DG (1963) Theoretical and experimental effects of spatial dispersion on the optical properties of crystals. Physical Review 132: 563. Huang K (1951) On the interaction between the radiation field and ionic crystals. Proceedings of the Royal Society of London Series A 208: 352. Hutchison JA, Schwartz T, Genet C, Devaux E, and Ebbesen TW (2012) Modifying Chemical Landscapes by Coupling to Vacuum Fields. Angewandte Chemie, International Edition 51: 1592. Kasprzak J, Richard M, Kundermann S, et al. (2006) Bose–Einstein condensation of cavity poalritons. Nature 443: 409. Kena-Cohen S and Forrest SR (2010) Room-temperature polariton lasing in an organic single-crystal microcavity. Nature Photonics 4: 371. Klembt S, Harder TH, Egorov OA, Winkler K, Ge R, Bandres MA, Emmerling M, Worschech L, Liew TCH, Segev M, Schneider C, and Höfling S (2018) Exciton-polariton topological insulator. Nature 562: 552. Lagoudakis KG, Wouters M, Richard M, Baas A, Carusotto I, Aandre R, Dang LS, and Deveaud-Pledran B (2008) Quantized vortices in an exciton—polariton condensate. Nature Physics 4: 706. Laussy FP, Kavokin AV, and Shelykh IA (2010) Exciton-polariton mediated superconductivity. Physical Review Letters 104: 106402. Li F, Orosz L, Kamoun O, Bouchoule S, Brimont C, Disseix P, Guillet T, Lafosse X, Leroux M, Mexis M, Mihailovic M, Patriarche G, Reveret F, Solnyshkov D, Zuniga-Perez J, and Malpeuch G (2013) From excitonic to photonic polariton condensate in z ZnO-based microcavity. Physical Review Letters 110: 196406. Pekar SI (1957) The theory of electromagnetic waves in a crystal in which excitons are produced. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 33: 1022.. (1958, Sov. Phys. ZhETP 6: 785). Roumpos G, Lohse M, Nitsche WH, Keeling J, Szymanska MH, Littlewood PB, Löffler A, Höfling S, Worschech L, Forchel A, and Yamamoto Y (2012) Power-law decay of the spatial correlation function in exciton-polariton condensates. PNAS 109: 6467. Sanvitto D, Pigeon S, Amo A, et al. (2011) All-optical control of the quantum flow of a polariton condensate. Nature Photonics 5: 610. Savvidis PG, Baumberg JJ, Stevenson RM, Skolnick MS, Whittaker DM, and Roberts JS (2000) Angle-resonant stimulated polariton amplifier. Physical Review Letters 84: 1547. Tassone F, Piermarocchi C, Savona V, Quattropani A, and Schwendimann P (1997) Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Physical Review B 56: 7554. Tolpygo KB (1950) Physical properties of a lattice of rock salt type composed from deformable ions. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 20: 497. (in Russian). Waldherr M, Lundt N, Klaas M, Betzold S, Wurdack M, Baumann V, Estrecho E, Nalitov A, Cherotchenko E, Cai H, Ostrovskaya EA, Kavokin AV, Tongay S, Klembt S, Höfling S, and Schneider C (2018) Observation of bosonic condensation in a hybrid monolayer MoSe2-GaAs microcavity. Nature Communications 9: 3286. Wesbuch C, Nishioka M, Ishikawa A, and Arakawa Y (1992) Observation of the coupled exciton–photon mode splitting in a semiconductor quantum microcavity. Physical Review Letters 69: 3314.

Further reading Agranovich VM and Ginzburg VL (1984) Crystal Optics with Spatial Dispersion, and Excitions. Berlin: Springer. Agranovich VM and Mills DL (1982) Surface Polaritons. Amsterdam, North-Holland: Modern Problems in Condensed Matter Sciences. Agranovich VM and Bassani GF (2003) Electronic excitations in nanostructures. Thin Films and Nanostructures. vol. 31, pp. 129–184. Amsterdam: Elsevier. Herrera F and Owrutsky J (2020) Molecular polaritons for controlling chemistry with quantum optics. The Journal of Chemical Physics 152: 100902. Kaganov MI, Pustylnik NB, and Shalaeva TI (1997) Magnons, magnetic polaritons and magnetostatic waves. Uspekhi Fiz. Nauk (Physics-Uspekhi) 167: 191. Kavokin A, Malpuech G, Agranovich VM, and Taylor D (2003) Cavity polaritons. Thin Films and Nanostructures. vol. 32. Amsterdam: Elsevier. Khitrova G, Gibbs HM, Jahnke F, Kira M, and Koch SW (1999) Nonlinear optics of normal-mode-coupling semicondcutor microcavities. Reviews of Modern Physics 71: 1591–1639. Lukin MD (2003) Colloquium: Trapping and manipulating photon states in atomic ensembles. Reviews of Modern Physics 75: 457. Ribeiro RF, Martínez-Martínez LA, Du M, Campos-Gonzalez-Angulo J, and Yuen-Zhou J (2018) Polariton chemistry: controlling molecular dynamics with optical cavities. Chemical Science 9: 6325. Yamamoto Y, Tassone F, and Cao H (2000) Semiconductor Cavity Quantum Electrodynamics. Berlin: Springer.

Semiconductor Heterojunctions, Electronic Properties of M Peressi, University of Trieste, Trieste, Italy © 2005 Elsevier Ltd. All rights reserved. This is an update of M. Peressi, Semiconductor Heterojunctions, Electronic Properties of, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 273–281, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/01136-0.

Introduction Electronic Properties: Band Alignments Measuring Band Offsets Predicting Band Offsets Band Offset Trends Lattice-Matched Semiconductor Interfaces Isovalent semiconductor heterojunctions Heterovalent semiconductor heterojunctions Lattice-Mismatched Semiconductor Interfaces Band Offset Engineering Localized Interface States Semiconductor Heterostructures and Spintronics Further Reading

507 509 509 511 512 512 512 512 513 513 514 514 514

Nomenclature a0 e Ec Eg Ev n DEc DEg DEv DV «

lattice parameter electron charge conduction band fundamental bandgap valence band electron density conduction band difference (with respect to reference levels in the bulks) fundamental bandgap difference valence band difference (with respect to reference levels in the bulks) electrostatic potential lineup dielectric constant

Introduction Semiconductor heterojunctions are built by joining, at a planar face, two semiconductors formed by different chemical elements but usually with similar lattice structure and chemical binding, as schematically indicated in Figure 1. From the simple heterojunction constituted by two thick slabs of different materials, which presents only one interface, different heterostructures can be formed – from the double heterojunction showing confinement properties for the charge carriers to the repeated heterojunctions (multiple quantum wells and superlattices) and new artificial structures based on sequences of alternating different semiconductors of varying thickness (10–1000 Å). Semiconductor heterojunctions show peculiar electronic properties giving rise to physical phenomena of a quantum mechanical nature, some of them not possible even in any “natural” bulk material. These peculiar electronic features are determined by the spatial profile of the electronic bands, which is in turn controlled both by intrinsic characteristic of constituent semiconductors, such as lattice constants, energy gaps, doping, and extrinsic features such as chemical and structural details in the interface region, where the bands exhibit discontinuities (see Figure 2). The spatial profile of the electronic bands can be used to properly tailor the space-charge distribution and the behavior of the charge carriers both in the growth direction (transport properties) and in the potential wells parallel to the interface planes (confinement properties), giving the semiconductor heterojunctions a crucial role in modern electronic and optoelectronic devices. The introduction of heterojunctions dates back to almost 50 years ago. Herbert Kroemer and Zhores I Alferov obtained the Nobel Prize in Physics in 2000 “for developing semiconductor heterostructures used in high-speed- and opto-electronics.” In 1957, Herbert Kroemer published the first proposal for a heterostructure transistor. His theoretical work showed that heterostructure devices could offer superior performance compared to conventional transistors. In 1963, H Kroemer and Zhores I Alferov

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Figure 1 Stick-and-ball model of an abrupt semiconductor heterojunction between two zinc blende semiconductors with no-common ions. The interface is (0 0 1) oriented. (Adapted from Sorba L, et al. (1992) Structure and local dipole of Si interface layers in AlAs–GaAs heterostructures. Physical Review B 46: 6834.)

E c (B) CBO E c (A)

E g (B) EF E v (B)

E g (A)

VBO

E v (A) Figure 2 Schematic spatial profile of the valence and conduction bands along the growth direction for a semiconductor heterojunction, with band bending due to space-charge effects and definition of band offsets, VBO and CBO.

independently proposed ideas to build semiconductor lasers from heterostructure devices, using a double heterostructure to confine carriers. Alferov built the first semiconductor laser from gallium arsenide and aluminum arsenide in 1969. The traditional semiconductor heterojunctions involve elements of the central portion of the periodic table, starting from Si and GaAs. Si- and GaAs-based technology is the most mature, but many other III–V and II–VI binary compounds as well as alloys are used nowadays, the choice depending on the particular application. Among alloys, perhaps the most widely studied and used is AlxGa1−xAs, in particular, in its interface with GaAs. High-quality semiconductor interfaces are easily built from two semiconductors which are isovalent and lattice-matched, but more generally they can be also heterovalent (involving semiconductors of different groups) and lattice-mismatched, with slightly different lattice parameters. With the addition of nitrides of II(IV)–VI compounds and of magnetic semiconductors to the most conventional ones, the range of the lattice constants and of the energy gaps which is covered by the different constituents used to form heterostructures is very large, as indicated by Figure 3, suggesting possibilities for new fundamental physics and device applications. Thanks to the improvement of experimental techniques such as molecular beam epitaxy (MBE) and metal–organic chemical–vapor deposition (MOCVD), it is possible to grow high-quality pseudomorphic heterostructures without misfit dislocations or other defects, also with semiconductors with a rather small lattice mismatch (i.e., less than a few percent), and with different but almost commensurate structures. A heterojunction indicated with A/B typically corresponds to a particular growth order: B is the substrate and A the epilayer. Another specific feature that is usually indicated is the crystallographic growth axis, which determines the interface plane.

Semiconductor Heterojunctions, Electronic Properties of

6

509

AlN

Energy gaps (eV)

5 ZnS

4

MnSe

GaN AlAs 3

ZnSe

CdS

AlP GaP

2

GaAs InN

1

Si

CdSe

GaSb InAs PbTe

PbS GeTe HgS

4.5

5

5.5

CdTe

AlSb

InP

Ge

0 4

MnTe ZnTe

PbSe HgSe 6

InSb

Sn HgTe

6.5

7

Lattice constant (Å) Figure 3 Energy gaps vs. lattice constants for various IV (stars), III–V (closed circles), II(IV)–VI (open squares) zinc blende semiconductors. The visible electromagnetic spectrum is also indicated.

Electronic Properties: Band Alignments The spatial profile of the electronic bands is mostly determined by the band discontinuities at the interface. The bandgap difference DEg, normally existing between the constituent materials, is shared among valence and conduction bands, thus giving rise to the valence and conduction band offsets (VBO and CBO): DEg ¼ VBO +CBO

[1]

There are different types of band lineups which are possible, schematically indicated in Figure 4. The straddling or type-I lineup characterizes, for instance, the GaAs/GaxAl1−xAs interface. Modern optoelectronic devices – including quantum well lasers – are based on such a lineup, typically with a double or repeated heterojunction, where the charge carriers are both confined in the slab of material with the lowest energy gap (see Figure 5). Other types of lineups are the staggered or type-II and the broken-gap or typeII-misaligned lineups, which could be preferred for some specific applications. Extensive theoretical and experimental work has targeted the problem of band alignments since a long time. In particular, an important and general issue widely debated is whether the band discontinuities are essentially determined by the bulk properties of the constituents, or if some interface-specific phenomena, such as crystallographic orientation and abruptness, may affect them in a significant way. A large joint experimental and theoretical/computational effort has been performed to finally clarify, quite recently, the physical mechanisms which give rise to the band alignment. Several books and review articles are available on the subject, as indicated in the “Further reading” section. Some of them are focused on the problem of band offset engineering, that is, on the possibility of controlling the interfaces thus providing a way to manipulate the band lineups and tune the transport properties across the junctions.

Measuring Band Offsets Several experimental techniques are available for measuring energy band discontinuities, including optical and transport measurements. Since the late 1970s, spectroscopic techniques have emerged as a fundamental tool for investigating heterojunctions on a microscopic scale. In photoemission spectroscopy, for instance, a direct measurement of the band discontinuities can be done by comparing specific core-level binding energies Ecl with respect to valence band top edges Ev in bulk samples with those from the heterostructure: VBO ¼ ½Ecl ðGa 3dÞ −Ev ðGa AsÞ −½Ecl ðAl 2pÞ −Ev ðAl AsÞ +DEcl An example is schematically shown in Figure 6 for the AlAs/GaAs(0 0 1) interface.

[2]

510

Semiconductor Heterojunctions, Electronic Properties of

Ec

Ec

Ec

Broken gap Straddling Staggered

Ev Ev

Ev

Figure 4 Schematic possible types of band alignments: (a) straddling, (b) staggered, and (c) broken gap. Flat bands are represented, as the focus is on a region which is of the order of 10 atomic units, and the band bending is negligible at this scale.

Ec

E g (A)

E g (B)

Ev

Photoemission intensity (arb. units)

Figure 5 Confinement of electrons and holes in the slab of a material with the lower gap in a double semiconductor heterostructure with straddling-type band alignment.

Al 2p

Ga 3d Bulks (a)

VBO AlAs/GaAs (0 0 1) heterostructure (b) –56

–55

–54

–53

–2

–1

0

1

2

Core-level binding energy E cl (eV) Figure 6 Example of a photoemission measurement of the band offset at the semiconductor heterojunction AlAs/GaAs(0 0 1). Core-level binding energy distribution curves for Al 2p and Ga 3d states referred to the corresponding AlAs and GaAs valence band top edges are schematically drawn. The zero of the energy scale is arbitrarily taken at the center of the Ga 3d emission peak. The variation of the separation DEcl of the core level peaks from the case of the two bulks, considering the topmost valence bands of the two materials aligned (a) and the heterojunction (b) is a direct measurement of the VBO.

Semiconductor Heterojunctions, Electronic Properties of

511

Predicting Band Offsets Theoretical and numerical investigations based on different approaches have contributed toward predicting band offset values and identifying the basic mechanism responsible for band offsets. These investigations can be divided mainly into two classes: (1) models and approaches making simplifying and sometimes drastic approximations in describing the interface, but retaining important physical concepts, and (2) accurate computational predictions based on fully self-consistent ab initio approaches, which allow one to fully take into account the interface details, such as orientation, abruptness, and defects. In general, no universal energy scale exists on which the band structures of the different semiconductors can be easily and uniquely referred, but simple and old models based on intrinsic reference levels have been widely used and have given successful predictions for a number of heterostructures: from the oldest models by Anderson, Frensley, and Kroemer, based on the difference between the electron affinities of the two semiconductors and the concept of charge neutrality levels, to Tersoff’s model emphasizing the role of interface dipoles. More refined quantum mechanical approaches, based on the envelope function concept or using the tight-binding method have also been widely and successfully applied. They are reviewed in the books cited above. The second class of theoretical/computational investigations, more recent, includes ab initio approaches, which are based on the solution of quantum mechanical equations for the system under consideration without any use of empirical parameters. This seemingly unaffordable task has been made feasible, thanks to the density-functional theory (DFT) proposed by Walter Kohn, who received the Nobel Prize in Chemistry in 1998 for the same, and for its approximations for practical applications, such as the local-density approximation. DFT has proven to yield reliable results, at an acceptable computational cost, on the electronic ground-state properties of complex crystalline systems, allowing a meaningful comparison with experiment or even accurate predictions of quantities not yet accessible experimentally, with an extremely useful predictive power. In ab initio approaches, semiconductor heterostructures are normally modeled using periodically repeated supercells with a limited number of atoms. The relevant effects due to the presence of the interface are confined in a small region, and the bulk features of the charge density distribution and electrostatic potential are completely recovered within a few atomic units from the interface. This justifies the concept of abrupt discontinuities of the electronic bands across the interfaces which is used in the schematic diagrams of electronic band profiles. The supercell self-consistent calculations provide the electronic charge density distribution and the corresponding electrostatic potential, as shown in Figure 7. Filtering out the microscopic oscillations with the bulk-like periodicity, one can unambiguously define an “interface dipole” and extract a potential difference across the interface, DV, which, in principle, depends on the interface structural

(110)

2

10 9

1

8

0

7

–1

6

V(z) (eV)

n(z) (electrons/cell)

(a)

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(c)

0.2

8.04

0.1

8.02 8

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7.98

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7.96

V(z) (eV)

n(z) (electrons/cell)

(b)

–0.2 As

Al

As

Al

As

Al

As

Ga

As

Ga

As

Ga

As

Figure 7 Contour plot of the valence electron density distribution (a) for GaAs/AlAs(0 0 1) heterojunction over a (1 1 0) plane containing the growth axis and centered on the interface anion. Profiles of the electron density and of the electrostatic potential along the growth direction averaged over planes parallel to the   interface ( nðzÞ and VðzÞ) (b), and further averaged along the growth direction with a filter equal to the periodicity of bulks (nðzÞ and VðzÞ) (c).

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Semiconductor Heterojunctions, Electronic Properties of

and chemical details. The final band offsets VBO and CBO are determined by the difference DEv and DEc between the relevant band edges in the two materials, measured with respect to the average electrostatic potential in the corresponding bulk crystals: VBO ¼ DV +DEv

[3]

CBO ¼ DV +DEc

[4]

When the two semiconductors constituting the heterostructure are rather similar (in structure and chemical composition), their differences can be quite minute with respect to the typical bulk variations, and a low-order perturbation approach is appropriate. The state-of-the-art theoretical approaches allow one not only to compute band discontinuities accurately but also to obtain an insight into the atomic-scale mechanisms which determine the band lineups, and interpret and predict their trends. However, the predictive capability of theoretical schemes is actually limited by the difficulty in predicting the actual atomic-scale arrangement at the interfaces, and their kinetic versus thermodynamic character.

Band Offset Trends Some peculiar trends in band offsets at semiconductor heterojunctions have been clarified, thanks to the large joint experimental and computational effort: at lattice-matched isovalent heterojunctions, the band offsets depend only on the bulk properties of the two materials, whereas at heterovalent heterojunctions they crucially depend on the interface orientation and other microscopic details. These results are briefly reviewed and justified in the following section.

Lattice-Matched Semiconductor Interfaces Isovalent semiconductor heterojunctions The GaAs/AlAs heterojunction is the simplest prototype of lattice-matched common-anion heterojunctions. The different cations are isovalent; therefore, the induced potential drop across the interface is only due to the electronic charge. Measurements and accurate numerical investigations show that the VBO and CBO are independent of the interface orientation and abruptness. These results can also be generalized to isovalent no-common-ion heterojunctions such as InAs/GaSb and InP/Ga0.47In0.53As, where the lattice-matching conditions result from a balance between the differences of the cationic and anionic core radii, and consequently an important microscopic and configuration-dependent interfacial strain may establish. Consequences of the bulk-like character of the VBO and CBO are the commutativity and transitivity relationships, which are valid, at least approximately, in the whole class of isovalent semiconductor heterojunctions: VBOðA=BÞ ¼ −VBOðB=AÞ

[5]

VBOðA=BÞ ¼ VBOðA=CÞ +VBOðC=BÞ

[6]

Heterovalent semiconductor heterojunctions For the sake of clarity, the reader is referred to the case of Ge/GaAs as the simplest prototype of heterovalent semiconductor heterojunctions, but similar considerations apply to other heterojunctions characterized by a valence mismatch between the constituent atoms, such as those among ZnSe, Ge, and GaAs. In the (1 1 0) direction, each atomic plane is characterized by the same average ionic charge, so that an abrupt junction does not carry ionic charge contribution and the lineup is due only to the electrons. At variance, ideally abrupt interfaces along a polar direction such as (0 0 1) would be charged and hence thermodynamically unstable, as already emphasized first by Harrison in 1978. The simplest neutral and stable interfaces one can envision are terminated by one mixed plane of anions or cations, As0.5Ge0.5 or Ga0.5Ge0.5 (see Figure 8). These two interfaces are stoichiometrically inequivalent, and, because of ionic point charges contribution due to the different valence of the atoms involved (Ge vs. Ga, Ge vs. As), they also correspond to different band offsets. Elementary electrostatics together with a linear response theory predict that the screened ionic point-charge contribution to the offsets is equal in magnitude and opposite in sign for the two interfaces: DV ionic ¼ pe2 =2a0 hei, where a0 is the lattice parameter involved and hei a proper average of the dielectric constants of the constituents. The maximum predicted possible variation with interface composition of the band offset at these interfaces is thus pe2 =a0 hei  0:8eV for Ge/GaAs. For ZnSe/Ge, where the valence difference between the constituting elements is twice, the maximum possible variation would be 2pe2 =a0 hei  1:3eV. In general, the atomic interdiffusion which may also occur across the interface over several atomic planes, depending on the growth conditions, reduces the point-charge contribution to the offset, and the observed variations of the offsets are smaller. However, for heterovalent heterostructures deviations from the commutativity and transitivity rule definitely beyond the experimental resolution, up to 0.5 eV, are observed, and these are a clear fingerprint of the formation of inequivalent interfaces.

Semiconductor Heterojunctions, Electronic Properties of

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(110)

(001) As Ga Ge

(a)

(b)

(c)

Figure 8 Atomic configurations for the two simplest neutral Ge/GaAs interfaces (a), (b), and for a Ge bilayer embedded in GaAs (c). (Adapted from Sorba L et al. (1992) Physical Review B 46: 6834.)

Lattice-Mismatched Semiconductor Interfaces In lattice-mismatched heterojunctions, the band offsets are affected by the stress state of the two materials, depending on the substrate. In pseudomorphically grown heterostructures, the epilayer accommodates its mismatch with the substrate with an almost homogeneous strain and a lattice constant a? along the growth direction, which essentially depends on its elastic properties and on the mismatch with the substrate. The strain state of the epilayer affects the band edges with a combined effect of shift and split. The former affects the averages of the band edge manifolds, calculated with respect to the reference electrostatic potential, and depends on the hydrostatic component of the strain, that is, on the relative volume change with respect to the cubic unstrained material. Splitting effects depend on the orientation and sign of the strain (tensile or compressive). The total variation of the VBO with strain also includes the variation of the electrostatic potential lineup DV. The simplest case of lattice-mismatched heterojunctions is Si/Ge, which is an example of isovalent homopolar interfaces. The predicted variation of the VBO at this heterojunction is 0.5 eV, changing the substrate from Si to Ge. Larger effects could be, in principle, possible at heterovalent lattice-mismatched heterojunctions, such as Si/GaAs, or in presence of larger lattice mismatches. However, the difficulty of growing high-quality lattice-mismatched heterojunctions drastically limits their formation and their use in practice.

Band Offset Engineering In bulk semiconductor technology, one of the most important and widely used features has been the possibility of intentionally varying the electronic properties by alloying and doping. For semiconductor heterojunctions, a similarly important challenge is to control and artificially modify the band offsets. As already mentioned, the VBO and CBO at lattice-matched isovalent semiconductor heterojunctions are mostly determined by the bulk properties of the constituents, and interface details such as orientation and stoichiometry play a very minor role, whereas at heterovalent heterojunctions they are very sensitive to these extrinsic features. As a consequence, heterovalent heterojunctions seem to be ideal candidates as tunable heterojunctions, while using only isovalent materials, it seems that the only way to tune the offset is to act on their bulk properties with strain or with alloying. Actually, the peculiarity of heterovalent interfaces leads naturally to a practical way for modifying the offset also at isovalent heterojunctions, or even for creating an offset at a homojunction. For instance, joining together the two inequivalent interfaces Ge/GaAs(0 0 1) (Ge–As and Ge–Ga terminated), one has the following sequence of atomic planes: ⋯ As–Ga–As–Ge0.5Ga0.5–Ge0.5As0.5–Ga–As–Ga⋯. Therefore, using the difference between the VBOs of these two inequivalent interfaces, a net potential drop DV ¼ pe2 =a0 hei is predicted across the mixed bilayer. This potential drop is the same that would result from a microscopic capacitor whose plates are placed at a distance a0/4, and carry a surface charge s ¼ e=a20 . The above sequence of atomic planes can also be thought of as due to the transfer of a proton per atomic pair from the As to the Ga planes. This viewpoint is easily generalized to arbitrary concentrations of Ge in a pair of consecutive compensated GaAs planes, GexGa1−x and GexAs1−x, to ensure local charge neutrality. The maximum potential drop would be, in principle, DV max ¼ 2pe2 =a0 hei for the case x ¼ 1, that is, for an entire Ge bilayer embedded (see Figure 8). The above results can be further generalized to the heterojunctions, for example, doping a GaAs/AlAs interface with ultrathin layers of Si or Ge. In this case, the

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Semiconductor Heterojunctions, Electronic Properties of

interlayer contribution can reduce or increase the intrinsic VBO according to the growth sequence. These predictions have been indeed experimentally confirmed for Si interlayers at GaAs/AlAs and AlAs/GaAs (0 0 1) junctions and analogous effects have also been observed in other systems.

Localized Interface States Another important issue in the heterojunction electronic structure concerns the presence of localized interface states in heterojunctions, their origin, and their role in determining the electronic characteristics of the junctions. The existence of localized interface states is generally claimed whenever anomalous features are observed in photoemission, optical, or transport measurements. Their unambiguous identification, however, is a difficult task in experiments. Reflectance anisotropy spectroscopy has been recently applied to a few interfaces and seems to be a promising tool for more direct investigations. The electronic states of a heterojunction can be extended everywhere, localized in one material or the other (acting as a quantum well), or localized at the interface. The true interface states are not degenerate with bulk Bloch states, decay exponentially away from the interface, and exist only in the mutual energy gaps. The comparison of the band structures of the two bulk materials with the spectrum of the heterojunction allows one to identify those states which do not belong to the bulk band structures, and are, therefore, candidates to be interface states. Localized electronic states can be successfully identified in numerical investigations also by the local density of states: far from the interface on the two sides of the junction, it yields the bulk densities of states of the two materials, whereas its deviations with respect to the constituting bulk state densities in the interface region would indicate the presence of states which are localized there. Interface states can have different physical origins. In some cases, they are associated with defects which typically arise at junctions between two morphologically different phases (e.g., cubic and wurtzite). But localized electronic states can also occur, in principle, at semiconductor heterojunctions with constituents having the same or similar structure but different chemical properties. Simple electron-counting schemes indicate that the valence mismatch between neighboring atoms from opposite sides of the junction could result in localized interface states corresponding to donor or acceptor bonds. Therefore, heterovalent heterojunctions, which exhibit such kind of bonds in the interface region, are natural candidates for the existence of localized interface states. Localized states have been experimentally detected at diamond/zinc blende interfaces, for example, Ge/ZnSe, at zinc blende/zinc blende heterovalent interfaces, for example GaAs/ZnSe, and even at isovalent no-common-ion interfaces, for example, BeTe/ZnSe. Accurate calculations predict that interface electronic states critically depend not only on the type of heterojunction (i.e., heterovalent rather than isovalent), but on the particular atomic-scale morphology of the interface.

Semiconductor Heterostructures and Spintronics The word “spintronics” denotes electronic-like heterostructures where the relevant physical quantity is the spin of the carriers and its interactions with external magnetic fields, rather than the charge of holes and electrons and the associated electronic properties. Semiconductor heterostructures using carrier spin as a new degree of freedom offer new functionality with respect to conventional junctions. A large effort is currently devoted to integrate traditional III–V and II–VI semiconductors with ferromagnetic semiconductors or, more generally, with other ferromagnetic materials including metals and half-metals. There are, however, several advantages using all-semiconducting devices: interface properties, such as lattice matching and band offsets, are more understood and controllable, and the integration with the existing conventional semiconductor technology is easier. The diluted ferromagnetic semiconductors are semiconductor compounds where a fraction of the constituent ions is replaced by magnetic ions. This is the case, for instance, of (Ga,Mn)As, which has received particular attention: Mn is a heterovalent substitutional impurity for Ga in GaAs (valence II with respect to the valence III of Ga), and therefore, it can be used as a source of spin-polarized holes. An Mn content up to 5% is sufficient to make (Ga,Mn)As ferromagnetic. Furthermore, there is the possibility of a good integration of half-metals with conventional semiconductors: these are materials that have only one occupied band at the Fermi level, and behave as a metal in one spin channel and as a semiconductor in the other. Current effort, both experimental and theoretical, is nowadays devoted to predict those particular heterostructures where half-metallicity is also maintained in the presence of the interface.

Further Reading Bastard G (1988) Wave Mechanics Applied to Semiconductor Heterostructures. Les Ulis, Cedex: Les Editions de Physique. Brillson LJ (1992) Surfaces and interfaces: atomic-scale structure, band bending and band offsets. In: Landsberg PT (ed.) Handbook on Semiconductors, vol. 1, pp. 281–417. Amsterdam: North-Holland. ch. 7. Capasso F (1987) Band-gap engineering: from physics and materials to new semiconductor devices. Science 235: 172–176. Capasso F and Margaritondo G (eds.) (1987) Heterojunction Band Discontinuities: Physics and Device Application. Amsterdam: North-Holland. Franciosi A and van de Walle CG (1996) Heterojunction band offset engineering. Surface Science Reports 25(1–4): 1–140. Grimmeiss HG (ed.) (1996) Heterostructures in Semiconductors. Proceedings of the Nobel symposia in physics, Physica Scripta, vol T68, Stockholm.

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Harrison WA, Kraut EA, Waldrop JR, and Grant RW (1978) Polar heterojunction interfaces. Physical Review B 18: 4402–4410. http://www.nobel.se/physics/laureates/1973/esaki-lecture.html http://www.nobel.se/physics/laureates/2000/alferov-lecture.html http://www.nobel.se/physics/laureates/2000/kroemer-lecture.html Margaritondo G (ed.) (1988) Electronic Structure of Semiconductor Heterojunctions. Dordrecht: Kluwer. Ohno Y, Young DK, Beschoten B, Matsukura F, Ohno H, and Awschalon DD (1999) Electrical spin injection in a ferromagnetic semiconductor heterostructure. Nature 402(6763): 790–792. Peressi M, Binggeli N, and Baldereschi A (1998) Band engineering at interfaces: theory and numerical experiments. Journal of Physics D: Applied Physics 31: 1273–1299. Yu ET, McCaldin JO, and McGill TC (1992) Band offsets in semiconductor heterojunctions. In: Ehrenreich H and Turnbull D (eds.) Solid State Physics, vol. 46, pp. 1–146. Boston: Academic Press. Vurgaftman I, Meyer JR, and Ram-Mohan LR (2001) Band parameters for III–V compound semiconductors and their alloys. Journal of Applied Physics 89(11): 5815–5875. Wolf SA, Awschalom DD, Buhrman RA, Daughton JM, von Molnar S, et al. (2001) Spintronics: a spin-based electronics vision for the future. Science 294(5546): 1488–1495.

Semiconductors, Impurity and Defect States in H Ikoma, Tokyo University of Science, Tokyo, Japan © 2024 Elsevier Ltd. All rights reserved. This is an update of H. Ikoma, Semiconductors, Impurity and Defect States in, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 330–334, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00470-8.

Introduction Structural and Electrical Properties of Impurities and Crystal Defects Theory of the Electronic States of the Shallow Impurities Experimental Determination of the Shallow Energy Levels Theory of the Electronic States of the Deep Impurities (or Defects) Experimental Determination of the Deep Energy Levels Further Reading

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Introduction The role of mobile charge carriers is a key factor in semiconductor device operations. These free carriers are supplied by doping semiconductors with certain kinds of impurities. The electrical properties of semiconductors can thus be controlled by adding a small amount of impurities. However, some other impurities and crystal defects often have detrimental effects on the performance of the device. These impurities and defects create discrete energy levels in the bandgap of the host semiconductor, and exert various influences (usefully or detrimentally) on the semiconductor characteristics. Therefore, a knowledge of the electronic structure of the impurities and defect states in semiconductors is very important.

Structural and Electrical Properties of Impurities and Crystal Defects Impurity atoms normally substitute the atoms of the host semiconductor (substitutional impurity atom). However, some of them, especially those with small atomic radii, often enter the interstitial site in the host crystal lattice (interstitial impurity atom). On the other hand, the well-known point defects – the vacancies (lattice points where atoms are lacking) and the interstitials (atoms of the host semiconductor occupying the interstitial sites) – are examples of the intrinsic crystal defects. These point defects sometimes generate a complex center such as the Frenkel defect (the pair of vacancy and interstitial atoms in the neighboring lattice sites). There are two types of impurities and defects from the electrical point of view. One of them supplies free electrons and the other provides holes in the host crystal. The former is called donor and the latter, acceptor. For example, when V-column elements such as P and As are doped in Si (IV-element), one excess valence electron is easily liberated (the impurity atom is ionized) in the Si lattice even at room temperature because of the low binding energy between the electron and the impurity atom. Also, when a III-element such as B is introduced into Si, one valence electron is deficient so that one hole is created and easily released in the Si lattice at room temperature due to their very low binding energy. These impurities, having very low binding (ionization) energies are termed shallow impurities (shallow donor and shallow acceptor). If Si is doped with VI-elements (S, Se, and so on) or II-elements (Zn, Cd, and so on), two free electrons or two holes per impurity atom, respectively, are created (double donor and double acceptor). On the other hand, some impurity elements (and crystal defects) have very high binding energies so that electrons or holes are not easily liberated in the host crystal at room temperature. These impurities with high binding energies are called deep impurities. Heavy metals such as Fe, Ni, and Cu in Si, for example, are deep impurities. In compound semiconductors, the same impurity atom behaves both as a donor and an acceptor depending on its occupied site (amphoteric impurities). The Si atom in GaAs behaves as a donor if it occupies the Ga site and as an acceptor when it substitutes the As atom. When IV-elements (Ge, C, etc.) are doped in Si, they also behave as donors or acceptors although there are cases when they are not electrically active. These impurities are called isoelectronic impurities. With doping of the impurities in semiconductors, the discrete energy levels are normally generated in the bandgap of the host semiconductor. For shallow donors and acceptors, these energy levels are located very near the bottom of the conduction band and the top of the valence band, respectively. On the other hand, the energy levels for the deep impurities occur near the middle of the bandgap.

Theory of the Electronic States of the Shallow Impurities Shallow impurities can be theoretically treated using the effective mass approximation and the hydrogen atom model. Consider the case of the donor (the acceptor can also be considered similarly). The excess electron moves in an orbit round the impurity ion due to the Coulomb interaction between them in the host crystal, just like the hydrogen atom. The Coulomb potential is much smaller than that in the hydrogen atom due to the screening effect of the valence electrons (the change of the valence electron distributions around the impurity ion) in the crystal lattice. The screened Coulomb potential is written as U¼Zq2/esr, where Z is the valence number of the impurity ion, q the electronic charge, es the dielectric constant of the host semiconductor and, r is the distance

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between the electron and impurity ion. The screening effect is effectively represented by the dielectric constant es. Thus, the electron is very loosely bound by the impurity ion (very low binding energy) and its wave function extends in the wide range of the crystal lattice. Hence, the electron moves feeling both the periodic potential of the host crystal and the screened Coulomb potential. The effect of the periodic potential on the motion of electrons is well described in terms of the “effective mass” m , that is, electrons in the periodic potential of crystal behave as if they are free electrons with their mass m being different from that (m0) in a vacuum. Therefore, the electrons in the orbital round the shallow impurity ions can be considered as electrons with an effective mass m moving in the screened Coulomb potential U. The fundamental Schrödinger equation for these electrons is written as follows:  
 ℏ r2 +ðV −U Þ CðrÞ ¼ ECðrÞ 2m0 Here, r2 ¼ ð@ 2 =@x2 +@ 2 =@y2 +@ 2 =@z2 Þ is an operator, CðrÞ the wave function of the electron, r ¼ ðx, y, zÞ the position vector of the electron, V and U the periodic potential in the perfect crystal and the screened Coulomb potential, respectively, and E is the energy of the electron. In the case of U¼0 (perfect crystal), the solution of the above equation is given by the Bloch wave function CnkðrÞ ¼ expðikrÞunk ðrÞ, where exp(ikr) is the wave function of the free electron (plane wave) extending to the whole crystal, and unk(r) is the function having the periodicity of the crystal lattice with n and k being the quantum number specifying the energy bands and the wave number vector, respectively. For applying the effective mass approximation to the shallow impurity problem, the wave function CðrÞ is expanded by the Bloch functions or their Fourier transforms (the Wannier functions) and substituted in the equation. Thus, the following equation is obtained:   ℏ 2 r −U FðrÞ ¼ EFðrÞ 2m This equation is the same as that for the hydrogen atom except for the difference in the electron mass. The wave functions of the electrons are given as the product of FðrÞ and the function which has the periodicity of the crystal lattice. The above equation is for the case of the isotropic conduction band (in which the effective mass is the same for all crystal orientations). For example, some compound semiconductors such as GaAs and InP have this type of conduction bands. On the other hand, the conduction bands are anisotropic in semiconductors such as Si, Ge, and GaP, that is, the effective mass is different in different crystal orientations. The valence bands of most semiconductors have degeneracy in which the energies of the “heavy” holes and the “light” holes coincide at the top of the band. For these cases, the above equation should be modified to take the anisotropy or degeneracy of the band structures into consideration. The details of the same are not discussed here. The solution of the above equation is well known for the “hydrogen atom.” The electronic states of the hydrogen atom are specified by the principal (Bohr) quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin quantum number s. The principal quantum number n is the positive integer and the angular momentum quantum number l  n −1 for each n value. The magnetic quantum number m is given as m ¼ l, l −1, l −2, 0, . . ., −l −1, −lðl  m  −lÞ and the spin quantum number s ¼ 1=2. The electronic states corresponding to l¼0, 1, and 2 are called s, p, and d states, respectively. Then the state ofn¼1 and l¼0 is called as the 1s state (the ground state: the state of the lowest energy). The higher energy states (the excited states) are thus called as 2s (n¼2,l¼0), 2p(n¼2, l¼1), 3s(n¼3, l¼0), 3p(n¼3, l¼1), 3d(n¼3, l¼2) states and so on. The energies En of these states obtained from the above equation are given as En ¼ −E0 =n2 ðn¼1, 2, 3, ...Þ E0 ¼ðe0 =es Þ2 ðm =m0 ÞEH

Here E0 is the energy of the ground (1s)  state and,  e0 and es are the dielectric constants of the vacuum and semiconductor respectively. EH is given as EH ¼ ð1=e0 Þ2 q4 m0 =2ℏ 2 ¼ 13:6eV and is the ground-state energy of the real hydrogen atom (the Rydberg constant). From the above equations, it is obvious that the energy En is only dependent on the principal quantum number n (degenerate with respect to the other quantum numbers l, m, and s) in the case of the isotropic conduction band. For the donor and acceptor, respectively, the ionization energies (the binding energies) ED and EA from the ground state can be calculated as ED ¼ EC −E0 and EA ¼ EV −E0, where EC and EV are, respectively, the energies of the bottom of the conduction band and the top of the valence band. In Figure 1, the shallow impurity levels are schematically shown. The wave function is F1s ðr Þ of the ground state is written in the following: F1s ðr Þ ¼ ð1=pÞ1=2 ð1=a Þ3=2 expð −r=a Þ    Here, it is assumed that z¼1a is the Bohr radius given as a ¼ e0 m Þ h2 =m0 q2 and represents the extent of the electronic wave function in the real crystal space. The wave function exhibits a sharp decrease at r > a. The wave functions of some excited states are as follows: F2s ðr Þ¼ð1=pÞ1=2 ð1=2a Þ3=2 ½2 −ðr=a Þexpð −r=2a Þ F2p ðrÞ¼ð1=pÞ1=2 ð1=2a Þ3=2 ðr=√3Þexpð −r=2a Þ

In these excited states, the electron wave functions show a strong decay as r > 2a

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Conduction band

Donor levels

3s, 3p, 3d Excited states 2s, 2p 1s Ground state

Acceptor levels

1s Ground state 2s, 2p 3s, 3p, 3d Excited states

Valence band

Figure 1 Shallow impurity level scheme in the wave number (k) space.

Experimental Determination of the Shallow Energy Levels The energy levels of shallow impurities can be determined accurately using the infrared absorption measurements at very low temperatures. It is well known that electrons bound to a hydrogen atom can be excited to higher energy levels by the absorption of light. Similarly, electrons captured by shallow impurity levels are transferred to higher energy levels by absorbing infrared light. These optical transitions are allowed only when the difference (Dl) in the angular momentum quantum numbers (l) between two related energy levels is 1 ðDl ¼ 1Þ (the selection rule). Thus the electrons can be excited from the 1s (ground) state to the 2p, 3p, 4p, . . . (excited) states. However, the transitions 1s ! 2s, 1s ! 3s, . . . (and so on) are not allowed. The absorption peaks of the infrared light are very sharp because these energy levels are completely discrete. By measuring the infrared absorption spectrum at very low temperature (4.2 K), one can accurately determine the energies between the ground state and many excited states. The excited levels with very large principal quantum numbers n are located very close to the conduction or valence band edge so that one can determine the ground-state (1s) level and any excited (2p, 3p, . . .) levels measured from the band edge (ionization energies). In Table 1, some examples of the measured shallow donor and acceptor levels (ionization energies) in Si are shown together with the calculated ones. The agreement between the measured and calculated values is excellent.

Theory of the Electronic States of the Deep Impurities (or Defects) For the deep impurities, electrons (or holes) are very tightly bound to the impurity atom nucleus due to the highly localized core potential. The motion of the bound electrons (or holes) is strongly affected by this core potential and the wave functions of these carriers are also highly localized around the impurity atoms (or defect centers). Therefore, both the effective mass approximation and the hydrogen atom model cannot be applied to the deep level cases. The tight binding approximations, such as the linear combination of atomic orbitals (LCAO) method, are suitable to treat the deep-level problem. Since the core potentials are not small, the Table 1

Some examples of the ionization energies of the ground states of shallow donors and acceptors

Host semiconductors

Impurities

Experimental value(meV)

Theoretical value(meV)

Si

P (D) As (D) Sb (D) B (A)

45.5 53.7 42.7 45.0

44.3 53.1 31.7 31.6

GaAs

Si (D) Ge (D)

5.84 5.88

5.72 5.72

Te (D)

7.14 0.6

7.14 0.6

InP InSb D: donor; A: acceptor.

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conventional perturbation theory is not applicable and the Green-function approach is useful to solve the fundamental Schrödinger equation. The Schrödinger equation describing the motion of the electron bound to the deep impurity is given as follows: ðH0 +U ÞCðr Þ ¼ ECðr Þ Here, H0 is the Hamiltonian operator for the perfect crystal and U is the localized potential due to the deep impurity atom. In order to solve the above equation, the impurity potential U needs to be determined. However, this is very difficult for the following reason. In many cases of deep impurities, the impurity atoms are slightly displaced from the lattice point of the host semiconductor, and a microstrain is induced around the impurity atoms. It is assumed that if the impurity atom occupies the lattice point (as is the case for shallow impurities), a shallow energy level of ED is generated. When this impurity atom is somewhat displaced from the lattice point (the lattice relaxation occurs), a deep energy level E0 is assumed to be created. Also, a lattice relaxation energy (strain energy) EL occurs. If E0+EL is higher than ED, it is energetically stable to form the deep level with a small displacement of the impurity atom from the lattice point rather than the shallow levels. If the mixed semiconductor crystal AlGaAs is doped with Si, a deep-level center (called the DX center) is created while Si atoms act as shallow donors in GaAs. This DX center is considered to be as the above stated case. In order to determine the impurity potential U, the effect of the lattice relaxation should be taken into account exactly, which, however, is a very difficult problem. Many calculations have been performed by assuming forms of the   various P impurity potential U. The wave function Cðr Þ is expanded in terms of the sp3 atomic orbital functions Fj as C ¼ j aj Fj (LCAO). Then the Schrödinger equation is solved using the Green-function approach under the various assumptions of the impurity potentials. However, quantitative comparisons between the calculated and observed energy levels are difficult because of the difficulty in including the lattice relaxation effect in the analysis. There are some cases of deep levels, in which the energy level merges into the conduction or the valence band of the host semiconductor. This level is called the resonant level. The existence of this level affects the band structure of the host crystal because the density of states of the impurities are superposed to those of the conduction or the valence band. With doping of the isoelectronic impurities, this resonant level is often observed. These resonant energy levels can also be calculated using the above Green-function approach.

Experimental Determination of the Deep Energy Levels The deep level transient spectroscopy (DLTS) is the most powerful tool to determine deep energy levels. This method is based on the measurements of the transient electrical capacitances of the metal–semiconductor (Schottky) contact as the pulsed reverse bias voltage is applied. The transient Schottky capacitance C(t) is written as follows. h   i N −t exp Cðt Þ ¼ C1 1 − 2 T Here N ¼ NT =ðN D +NT Þ , NT and ND are the density of the deep impurity and the shallow donor, respectively. C1 ¼ fðes e0 =2ÞðV +V D ÞðND −NT Þg1=2 , es and e0 are the dielectric constant of the semiconductor and a vacuum, respectively, and VD is the diffusion potential of the Schottky contact. The relaxation time t can be written as en ¼ 1=t, where en is the electron (or hole) emission coefficient of the deep impurity and given as en ∝Cn exp½ −ðEC −ET =kT Þ because electrons (and holes) are emitted from the deep levels by thermal activation. Here, Cn is the capture cross section of the deep impurity, ET and EC are the deep impurity energy level and the bottom of the conduction band, respectively. T and k are the absolute temperature and the Boltzmann constant, respectively. If the transient capacitance C(t) is measured as a function of temperature T, the temperature dependence of en is obtained. From the plot of ln en versus 1/T (the Arrhenius plot), the deep energy level ET can be easily determined. The capture cross section Cn and the deep impurity density NT are at the same time determined. The famous example of the deep level is the EL2 center observed in the semi-insulating GaAs single crystal which is the cause of the high resistivity. The ionization energy of this EL2 center was measured to be 0.70.8 eV (located near the middle of the bandgap of GaAs).

Further Reading Pantelides S (1978) The electronic structure of impurity and defect states in semiconductors. Reviews of Modern Physics 50: 797–858. Pantelides S (1986) Deep Centers in Semiconductors, A State of the Art Approach. New York: Gordon and Breach. Yu PY and Cardona M (1996) Fundamentals of Semiconductors. Berlin: Springer.

Semiconductor Nanostructures B Voigtländer, Forschungszentrun Jülich GmbH, Jülich, Germany © 2005 Elsevier Ltd. All rights reserved. This is an update of B. Voigtländer, Semiconductor Nanostructures, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 290–297, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00589-1.

Introduction Physical Principles of Self-Organized Growth Semiconductor Nanoislands Lateral Positioning of Nanoislands by Growth on Templates Vertical Stacking of Nanoislands Semiconductor Nanowires Nanowires on Facetted Surfaces Monolayer Thick Wires at Step Edges Free-Standing Semiconductor Nanowires Grown by Vapor Liquid Solid Growth Hybrid Systems – Combination of Lithography and Self-Organized Growth Further Reading

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Introduction Two conceptually different methods are used to fabricate semiconductor nanostructures. In the top–down approach, lithography is used to fabricate nanostructures. In the bottom-up approach, small nanostructures form by self-organization during epitaxial growth. The greatest advantage of the top–down approach is the large variety of different structures, which can be generated by lithographic methods. In the case of the bottom-up approach, the variety of structures is much more limited. Islands, wires, rods, and rings have been fabricated bottom-up. However, the bottom-up approach offers the unique opportunity to fabricate very small nanostructures down to the one-digit nanometer range beyond the limits of current lithography techniques. Another advantage of self-organized fabrication of semiconductor nanostructures is the inherent parallel approach, in which billions of nanostructures are generated in parallel. The focus of this article is the self-organized growth of semiconductor structures with lateral dimensions below 100 nm. Semiconductor quantum-well structures are nanostructures in the growth directions and not lateral nanostructures, and will not be considered in this article. The actual growth techniques used to fabricate the nanostructures like molecular beam epitaxy (MBE) and chemical vapor deposition (CVD) will be considered in separate articles. The methods used to analyze the nanostructures are scanning tunneling microscopy (STM), scanning force microscopy (AFM or SFM), transmission electron microscopy (TEM), and low energy electron microscopy (LEEM). Different properties of semiconductor nanostructures, and the applications of self-organized nanostructures in devices are discussed elsewhere in this encyclopedia. As examples for the self-organized growth of nanostructures, material systems such as Si/Ge and GaAs/InAs are used predominantly. After introducing the self-organized semiconductor nanoislands grown on planar substrates, the aligned growth on prestructured samples is discussed. Subsequently, self-organized nanowires grown on facetted surfaces and the growth of monolayer thick wires by step-flow growth are presented. Finally, the growth of nanowires by vapor liquid solid (VLS) epitaxy and hybrid systems, where nanowires aligned to lithographically defined structures are fabricated, are presented.

Physical Principles of Self-Organized Growth Semiconductor nanostructures can be fabricated by self-organization using heteroepitaxial growth, which is the growth of a material B on a substrate of a different material A. In heteroepitaxial growth, the lattice constants of the two materials are often different. The lattice mismatch for the two most commonly used material systems Si/Ge and GaAs/InAs is 4.2% and 7%, respectively (schematically shown in Figure 1a). This lattice mismatch leads to a buildup of elastic stress in the initial two-dimensional (2D) growth in heteroepitaxy. In the case of Ge heteroepitaxy on Si, the Ge is confined to the smaller lattice constant of the Si substrate, that is, the Ge is strained to the Si lattice constant. This results in a buildup of elastic stress (Figure 1b). One way to relax this stress is the formation of 3D Ge islands. In the 3D islands, only the bottom of the islands is confined to the substrate lattice constant. In the upper part of the 3D island, the lattice constant can relax to the Ge bulk lattice constant and reduce the stress energy this way (Figure 1c). This growth mode characterized by the formation of a 2D wetting layer and the subsequent growth of (partially relaxed) 3D islands is called Stranski–Krastanov growth mode. The driving force for the formation of self-organized semiconductor nanoislands in heteroepitaxial growth is the buildup of elastic strain energy in the stressed 2D layer. As a reaction to this, a partial stress relaxation by the formation of 3D islands can lower the free energy of the system. The process of island formation close to equilibrium is a trade-off between elastic relaxation by formation of 3D islands which lowers the energy of the system and an increase of the surface area which increases the energy. Apart

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Figure 1 (a) Schematic representation of Si and Ge crystals with different lattice constants, (b) buildup of elastic strain energy during 2D growth with Ge confined to the Si lattice constant, and (c) elastic relaxation by formation of 3D islands (Stranski–Krastanov growth). In the upper part of the 3D island, the lattice constant relaxes towards the Ge bulk constant. The usual form of the 3D islands is a pyramid and not like the one shown in this schematic sketch.

from the formation of 3D islands, there is another process which can partially relax the stress of a strained 2D layer: the introduction of misfit dislocations. This corresponds to the removal of one lattice plane of a compressively strained 2D layer. If a lattice plane is removed in regular distances in the 2D layer, a misfit dislocation network forms. Depending on the growth parameters, temperature, and growth rate, the self-organized growth can be close to equilibrium or in the kinetically limited regime. Close to equilibrium, that is, at high-growth temperatures or low-deposition rates, the occurring morphology (strained layer or 3D islands or a film with dislocations) is only determined by the energies of the particular configurations. The morphology with the lowest energy will be formed. If the growth is kinetically limited, the activation barriers are important. For instance, an initially flat strained layer can transform to a morphology with 3D islands or to a film with dislocations. What actually happens depends on the kinetics of the growth process, that is, on the activation energy for formation of 3D islands compared to the activation energy for the introduction of misfit dislocations.

Semiconductor Nanoislands Stranski–Krastanov growth occurs in InAs/GaAs growth. An example of InAs nanoislands grown on a GaAs substrate is shown in the atomic force microscopy image in Figure 2. The InAs islands were grown by MBE at a growth temperature of 800 K. The width of the islands is 20 nm and the height 6 nm with a density of 1010 cm−2. The challenges in the growth of these semiconductor islands are to grow islands of the desired size and density and with a high-size uniformity. In general, a higher growth temperature generally leads to the formation of larger islands; a higher growth rate leads to the formation of smaller islands. The size of the islands increases with the coverage. Often the density of the islands saturates at an early stage of the growth. These are general trends which may depend on the material system and the particular deposition technique. In some cases (self-limiting growth), the size of the islands saturates and the density increases with coverage. This kind of growth mode leads to a high-size uniformity of the islands. The size uniformity achieved in self-organized growth of semiconductor islands can be as small as 10%. For optoelectronic applications, the nanoislands have to be capped (i.e., embedded in a semiconductor to prevent oxidation of the islands under ambient conditions where the optic measurements are performed) and care has to be taken that they change their shape and composition

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Figure 2 InAs nanoislands grown on a GaAs surface imaged by AFM. (Courtesy of A Lorke; reproduced with permission from Warburton RJ (2002) Contemporary Physics 44: 351; © Taylor & Francis Ltd. (http:www.tandf.co.uk).)

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during the capping process. The confinement of charge carriers in all three directions gives rise to atomic-like energy levels. Quantum dot lasers operating at room temperature have now been realized. The islands grown on a flat substrate are usually not ordered laterally due to the random nature of the nucleation process. In the following sections, it is shown how nucleation at specific sites can be achieved.

Lateral Positioning of Nanoislands by Growth on Templates On homogenous substrates, the location of the nanostructured islands formed were not predictable due to the stochastic nature of the nucleation process. It is desirable to position islands at specific (preselected) locations. This is desirable if the islands are used as functional units of nanoelectronic devices. Electric contacts can be structured easier when the location of an island is predefined. The approaches are now discussed where island nucleation occurs not randomly, but at specific locations. These predefined locations are usually specific sites on regularly prestructured substrates. Examples of such prestructured substrates are: regularly stepped substrates, facetted substrates, a regular arrangement of dislocations, or long-range surface reconstructions. An example of ordered nucleation at a prestructured substrate is shown in Figure 3. Here Ge islands nucleate above dislocation lines. An SiGe film is grown on an Si(0 0 1) substrate. At the interface between the SiGe film and the substrate, dislocations form. The driving force for the formation of the dislocations is the relief of elastic strain, which arises due to the different lattice constants between the Si substrate and a Ge/Si film on this substrate. Due to the crystal structure of the substrate, the dislocations are aligned in straight lines along the two perpendicular high-symmetry directions of the substrate. During annealing, the dislocations form a relatively regular network, due to a repulsive elastic interaction between the dislocations. Figure 3 shows growth of Ge islands on this prestructured substrate. Rows of islands grow preferentially above the dislocation lines. The nucleation of Ge islands is more favorable at the relaxed areas above the dislocations, which have a lattice constant closer to that of Ge than the strained areas of the SiGe film, where the lattice constant is confined to that of the Si substrate. Islands have a lower energy if they grow with their natural lattice constant compared to growth at a different lattice constant (strained islands). This leads to a preferred nucleation of Ge islands above the dislocations. The nucleation does not occur randomly at the surface, but simultaneously at sites which have the same structure. This can lead to a narrower size distribution than for the growth on unstructured Si(0 0 1) substrates. In the case of the growth on a prestructured substrate, another process of self-organization occurs (before the growth of islands) during the formation of the regular arrangement of the defect structures. Here, it is often a repulsive elastic interaction, which leads to a regular distance between steps or dislocations under equilibrium conditions. Since island nucleation occurs at the defect sites, the distances are determined by the distances of the defects. Nucleation at predefined defect sites has two advantages over the random nucleation. First, the islands are located at specific sites and second, the size distribution is narrower due to simultaneous nucleation at sites with identical environment.

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Pm Figure 3 Ordered nucleation of Ge islands which is prestructured by an underlying network of dislocations. Image size 7 mm. (Reprinted figure with permission from Shiryaev SY, Jensen F, Lundsgaard Hansen J, Wulff Petersen J, and Nylandsted Larsen A (1997) Physical Review Letters 78: 503; © American Physical Society.)

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Vertical Stacking of Nanoislands A nucleation of nanoislands preferentially above existing ones can be achieved when a new layer of Ge (InAs) islands is grown on top of an Si (GaAs) spacer layer. If the spacer layer is sufficiently thin, the nucleation of islands occurs just above pre-existing islands. This vertical stacking of islands is shown in Figure 4 and can be explained as follows: the lattice constant of the top part of the Ge islands is increased close to the bulk constant. Subsequent Si deposition leads to an increased lattice constant of the Si spacer layer just above the Ge islands. Upon the nucleation of the next Ge island layer, Ge islands nucleate preferentially at locations on the Si spacer layer where the lattice constant is closest to the Ge lattice constant. This is just the case above the Ge islands. The residual strain field of the Ge islands in the lower layer seeds the nucleation in the upper layer. It was observed that the size uniformity of the 3D islands in subsequent layers is improved.

Semiconductor Nanowires Nanowires on Facetted Surfaces Some surfaces such as the Si(1 1 3) surface are known to facet, that is, to form alternating regions with a very high step density (step bunches) alternating with flat surface regions. Subsequent growth of Ge can result in preferential growth of nanowires in the step-bunch region. This effect is even more pronounced in Si/Ge multilayers. In Figure 5, a cross-sectional TEM image of a stack of five alternating depositions of Ge and Si is shown. Ge nanowires with a thickness of 4.5 nm and a width of 100 nm are formed. The formation of these nanowires is explained as a strain relaxation effect of Ge growing at the step-bunch regions where it can relax the elastic strain more effectively than on a compact 2D layer. After growth of an Si buffer layer, the next SiGe wire structure nucleates above the previous one. This is a strain-induced effect similar to the one shown for the vertical stacking of the islands.

Monolayer Thick Wires at Step Edges Regular surface steps can be used to fabricate monolayer thick Ge wires using step-flow growth. Pre-existing step edges on the Si(1 1 1) surface are used as templates for the growth of 2D Ge wires at the step edges. When the diffusion of the deposited atoms is sufficient to reach the step edges, these deposited atoms are incorporated exclusively at the step edges and the growth proceeds by a homogenous advancement of the steps (step-flow growth mode). If small amounts of Ge are deposited, the steps advance only some nanometers, and narrow Ge wires can be grown. A key issue for the controlled fabrication of nanostructures consisting of different materials is a method of characterization, which can distinguish between the different materials on the nanoscale. In case of the important system Si/Ge, it has been difficult to differentiate between Si and Ge due to their similar electronic structure. However, if the surface is terminated with a monolayer of Bi, it is possible to distinguish between Si and Ge. Figure 6a shows an STM image after repeated alternating deposition of 0.15 atomic layers of Ge and Si, respectively. Due to the step-flow growth, Ge and Si wires are formed at the advancing step edge. Both elements can be easily distinguished by the apparent heights in the STM images. It turns out that the height measured by the STM is higher on areas consisting of Ge (red stripes) than on areas consisting of Si (orange stripes). The assignment of Ge and Si wires is evident from the order of the deposited materials (Ge, Si Ge, Si and Ge, respectively in this case). The initial step position is located right from the grown Ge wires. The step edge has advanced towards the left after the growth of the nanowires. The reason for the height difference in STM between Si and Ge is the different electronic structure of a Ge–Bi bond compared to an Si–Bi bond. The apparent height of Ge areas is 0.1 nm higher than the apparent height of Si wires (Figure 6b). The width of the Si and Ge wires is 3.5 nm as measured from the cross section (Figure 6b). The nanowires are 2D with a height of only one atomic layer (0.3 nm). Therefore, the cross section of a 3.3 nm wide Ge nanowire contains only 21 atoms (Figure 6c). The height difference arises due to an atomic layer of Bi which was deposited initially. The Bi floats always on top of the growing layer because it is less strongly bound to the substrate than Si or Ge. The Si/Ge wires are homogenous in width over larger distances and have a length of several thousand nanometers. Different widths of the wires can be easily achieved by different amounts of Ge and Si deposited.

100 nm Figure 4 Cross-sectional TEM image of a stack of Ge islands in 40 nm thick Si spacer layers. The islands are aligned on top of underlying islands. (Reproduced from Vescan L (1998) Thin Solid Films 336: 244, with permission from Elsevier.)

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SiGe Si SiGe

100 nm Figure 5 Cross-sectional TEM image of stacked Ge nanowires grown on a facetted Si(1 1 3) surface. (Reproduced with permission from Brunner K (2002) Si/Ge Nanostructures. Report of Progress in Physics 65: 27–72; © IOP Publishing.)

Germanium Silicon Height (nm)

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Si substrate 3.3 nm (c) Figure 6 (a) STM image of 2D Ge/Si nanowires grown by step flow at a pre-existing step edge on a Si(1 1 1) substrate. Si wires (orange) and Ge wires (red) can be distinguished by different apparent heights. (b) The cross section shows the dimensions of the Si and Ge nanowires. The width of the wires is 3.5 nm and the height is only one atomic layer (0.3 nm). (c) Atomic structure of a 3.3 nm wide Ge wire on the Si substrate capped by Bi. The cross section of the Ge wire contains only 21 Ge atoms. (Reproduced with permission from Voigtländer B (2001) Fundamental processes in Si/Si and Ge/Si epitaxy studied by scanning tunneling microscopy during growth, Surface Science Reports 43: 127.)

Free-Standing Semiconductor Nanowires Grown by Vapor Liquid Solid Growth In the VLS method, small catalytic nanoparticles (often gold) induce the growth of a homogenous rod of a semiconductor, with the diameter of the rod determined by the nanoparticle. The gold nanoparticles can be fabricated by heating a thin evaporated gold film, which breaks up and reshapes into nanoscale droplets. Alternatively, gold aerosol nanoparticles with sizes of a few tens of nanometers are used. The gold nanoparticles form a eutectic alloy with the substrate material (for instance, GaAs). Using MBE or metallorganic vapor phase epitaxy, nanowires can be grown. These nanowires are different from the ones discussed so far, in that they are free-standing on the substrate and grow vertically. The previously considered nanowires were epitaxially grown along their length on the substrate or even embedded into the substrate during growth. The VLS method was used for the Si/Ge and the GaAs material system. This growth method can even be used to grow semiconductor heterostructures within the nanowires with atomically sharp interfaces. Figure 7 shows an InAs nanowire (green) with several InP barriers (red). The rapid alternation of the composition is controlled by the supply of precursor atoms to the eutectic melt supplied as molecular beams. Of particular interest is the fact, that the very small cross section allows efficient lateral relaxation of the nanowire, thereby providing freedom to combine materials with very different lattice constants to create heterostructures within the nanowire. The problem of incorporation of misfit dislocations, when a critical thickness is exceeded, does not occur due to the small lateral size of the nanowires. The VLS method is also used to grow semiconductor heterostructures, not along the wire, but radial heterostructures, the so-called core-shell structures.

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Figure 7 Composition profile of an InAs nanowire, containing several InP heterostructures, using reciprocal space analysis of lattice spacings with a TEM. InAs lattice spacings have been color-coded with green and InP spacings with red. (Reproduced with permission from Björk MT, Ohlsson BJ, Sass T, Persson AI, Thelander C, et al. (2002) Nano Letters 2: 88.)

Core Shell (a)

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Figure 8 (a) Schematic of the core-shell growth of a radial Ge/Si structure (b) TEM image of a Ge/Si core-shell nanowire. Scale bar is 50 nm. (c) Elemental mapping along the cross section showing the Ge (red circles) and Si (blue circles) concentrations. (Reproduced with permission from Lauhon LJ, Gudiksen MS, Wang D, and Lieber CM (2002) Epitaxial core-shell and core-multishell nanowire heterostructures. Nature 420: 57.)

The synthesis of Si/Ge core-shell nanowires by chemical vapor deposition is achieved by the following process schematically shown in Figure 8a. Initially a Ge nanowire is grown by vapor phase epitaxy growth from a gold nanoparticle. In this case, the substrate serves only as a support and the nanowires are not epitaxially connected to the substrate. The deposition temperature is chosen so low that no germane (GeH4) is decomposed along the wire and radial growth is suppressed. Only axial growth occurs by the VLS mechanism at the gold nanoparticle. Subsequently, a boron-doped Si shell is grown by CVD. The addition of diborane serves to lower the decomposition temperature of silane (SiH4) and “turns on” the radial growth. Figure 8b shows a TEM image of the Ge core (imaged darker) and the Si shell (lighter gray) with a diameter of 50 nm. The elemental TEM cross-sectional mapping shown in Figure 8c shows the radial chemical composition in a wire (core diameter 26 nm, shell thickness 15 nm). By oxidation of a shell of the radial heterostructure, it is possible to build a coaxially gated nanowire transistor.

Hybrid Systems – Combination of Lithography and Self-Organized Growth In hybrid methods, self-organization is combined with lithographic patterning. In this approach, self-organization is used to form nanostructures on a smaller scale than the one accessible by lithography. Most importantly, the hybrid methods provide a direct contact of nanostructures formed by self-organization to lithographically patterned structures. As an example, the self-organized growth of Ge islands in oxide holes is shown in Figures 9a–9d. The staring surface is a silicon substrate with a thin oxide layer at the surface. Electron lithography is used to remove the oxide and form holes of a diameter of 0.5 mm where the bare Si surface is

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(a)

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Figure 9 (a–d) Growth of Ge islands in holes on an oxidized Si substrate. (Reproduced with permission from Kim ES, Usami N, and Shiraki Y (1999) Semiconductor Science and Technology 14: 257; © IOP Publishing Limited.)

exposed. Self-organized growth of Ge leads to the formation of Ge islands, which can be smaller than the size scale of the electron beam lithography. The gas phase growth of Ge is selective, that is, Ge only grows inside the holes in the oxide and not on the oxide itself. Figure 9 shows the nucleation of Ge islands in the holes in the oxide for different growth temperatures. At lower temperatures, the island density is so large that several islands nucleate in one oxide hole. If the temperature is increased finally, only one Ge island nucleates in each oxide hole. The size of the Ge island is smaller than the lithographically defined oxide hole. This approach is called lithographic downscaling. However, the positioning of the islands is not perfect. As seen in Figure 9d, the position of the Ge island inside the oxide hole is not defined but is rather randomly in the center or at the corner of the oxide hole. Due to the fact that the Ge does not grow on the oxide, the edges of the oxide hole are not sinks for deposited Ge atoms. Therefore, the Ge adatom concentration is homogenous across the hole, and the nucleation of the Ge island is random within the oxide hole. If the edges of the hole were sinks for Ge atoms (for instance, if the edges of the hole consisted of Si), the adatom density would have a maximum at the center of the hole and the nucleation of the Ge islands would occur preferentially at the center of the oxide holes. A simple structure formed by lithographic methods is a mesa (from the Spanish table), which is a flat terrace separated from the rest of the wafer by trenches. A silicon mesa structure is imaged by electron microscopy and shown in Figure 10. It has been observed that the growth of Ge on this mesa leads to preferential nucleation of Ge islands at the mesa edges. Figure 10 shows a regular alignment of Ge islands along the mesa edges. This preferred nucleation of islands at the convex part of the surface profile is quite

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Figure 10 Regular alignment of Ge islands at the edge of a mesa which was fabricated by optical lithography. (Reproduced from Vescan L (1989) Journal of Crystal Growth 194: 173, with permission from Elsevier.)

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Figure 11 Cross-sectional TEM image of vertically stacked GaAs nanowires and AlGaAs barriers aligned along a lithographically defined V-groove. (Reproduced with permission from Gustafsson A, Reinhardt F, Baisiol G, and Kapon E (1995) Applied Physics Letters 67: 3673; © American Institute of Physics.)

unexpected. Considering the surface curvature alone, the chemical potential is lowest at locations with a concave curvature. Atoms diffuse toward the lower chemical potential, that is, to the concave parts of the surface. This reasoning is true for homoepitaxy. In heteroepitaxy, another contribution is important: the strain-dependent contribution to the chemical potential. The convex regions are most favorable for strain relaxation of the strained Ge wetting layer covering the Si substrate. Therefore, the strain contribution to the chemical potential has a minimum at convex edges. In the current case, this strain-induced contribution to the chemical potential overwhelms the curvature contribution and leads to a minimum in the total chemical potential at the convex edges. Therefore, these convex edges with a minimum in the chemical potential provide a favorable nucleation site for Ge islands. Also, nanowires can be fabricated, aligned to lithographically defined structures. V-grooves can be defined by lithography and anisotropic wet chemical etching with a low-etch speed for the crystal plane forming the side facets of the V-groove. The formation of nanowires at the bottom of the V-groove is induced by the self-limiting nature of the growth front of AlGaAs on the grooved substrate. Vertically stacked arrays of GaAs/AlGaAs nanowires exhibiting 5–10% size uniformity have been demonstrated (Figure 11). The layer structure consists of GaAs nanowires separated by Al0.42Ga0.58As barriers. The formation of the crescent-shaped GaAs wires is based on the formation of slow-growing side facets along the V-grooves. The migration of adatoms away from these side-walls toward the V-groove bottom produces the concave surface profiles in the corners of the V-grooves. PACS: 68.65.−k

Further Reading Berbezier I, Ronda A, and Portavoce A (2002) Si/Ge nanostructuresnew insights into growth processes. Journal of Physics: Condensed Matter 14: 8283. Bimberg D, Grundmann M, and Ledentsov NN (1989) Quantum Dot Heterostructures. New York: Wiley. Gustafsson A, Reichhardt F, Baisiol G, and Kapon E (1995) Low-pressure organometallic chemical vapor deposition of quantum wires on V-grooved substrates. Applied Physics Letters 67: 3673. Moriarty P (2001) Nanostructured materials. Reports of Progress in Physics 64: 297. Motta N (2002) Self-assembling of Ge/Si(1 1 1) quantum dotsscanning microscopy probe studies. Journal of Physics: Condensed Matter 14: 8353. Stangl J, Holy V, and Bauer G (2004) Structural properties of self-organized semiconductor nanostructures. Reviews of Modern Physics 76: 725. Teichert C (2002) Self-organization of nanostructures in semiconductor heteroepitaxy. Physics Report 365: 335. Yang B, Feng Liu MG, and Lagally (2004) Local strain-mediated chemical potential control of quantum dot self-organization in heteroepitaxy. Physics Review Letters 92: 025502.

Epitaxy D Maryenko, RIKEN Center for Emergent Matter Science(CEMS), Wako, Japan © 2024 Elsevier Ltd. All rights reserved. This is an update of L. Miglio, A. Sassella, Epitaxy, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 157–166, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00692-6.

Introduction Substrate and lattice matching Modes of film growth Material characterization Epitaxy techniques Chemical vapor deposition Liquid-phase epitaxy Solid-phase epitaxy Molecular beam epitaxy Pumping system Material evaporation source Flux measurement and stoichiometry Oxide compound Pulsed laser deposition Sputtering Conclusion References Further reading

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Abstract Epitaxy refers to the growth of well-oriented crystalline layer on top of the other. The ability to produce such layers with high crystalline quality allows to form novel material combinations that do not exist in nature. The emerging property of artificial is important not only for fundamental studies but also for the application. Therefore various growth techniques have been developed, which are widely applied in different branches of solid state electronics. Here, we describe some aspects of epitaxial growth that appear to be important from the practical standpoint and should make the reader acquainted with epitaxy.

Introduction In 1928, Louis Royer coined the term epitaxy to describe the crystal-oriented growth of one crystalline material on top of the other (Royer, 1928). The term is composed of two Greek words epi (epi), meaning “above”, and taxies (taxiz), meaning “order”. By that time the phenomenon of oriented crystal growth has been known to occur in nature, and many examples have been already documented (Barker, 1906). Pashley et al. refer to the work of Moritz L. Frankenheim in 1836 as the first recorded example of epitaxial growth of sodium nitrate NaNO3 on a cleaved calcite crystal CaCO3 (Frankenheim, 1836; Pashley, 1956). An earlier work by Wakkernagel in 1825 is mentioned to be disputed (Wakkernagel, 1825; Pashley, 1956). Since then the epitaxial growth has evolved tremendously and became a key enabler in the development of modern technologies and synthesis of advanced materials. Epitaxial methods allow to achieve the structural quality that can even be superior to that obtained in the bulk. The ability of epitaxial methods to produce thin layers and multilayers of different materials, sharp interfaces, and spatially resolved doping allowed to obtain entirely novel physical properties. Light-emitting diodes, semiconductor lasers, high frequency transistors, and amplifiers are just few examples of how the development of epitaxial methods affected our daily life. Fundamental discoveries (such as quantum Hall effect, and Giant Magnetoresistance) were also brought by the development of high-quality structures. In the epitaxy, we address two main questions–how the material grows and how to grow the materials. An epilayer (a layer of material formed by epitaxial growth) on a substrate forms from a metastable phase brought by deposition of amorphous compound (SPE), liquid phase (LPE), vapor and gas (vapor-phase epitaxy). The driving force for the formation of an epilayer is the chemical potential difference between metastable and stable phases of material. The crystalline phase of the epilayer material will try to minimize the total energy of the system. Therefore, the surface tensions are important for the epitaxial growth. When two materials grown on top each other are isochemical in their composition, we refer to the homoepitaxial growth process. In an ideal homoepitaxial growth the epitaxial layer is the continuation of a bulk crystal so that any evidence of substrate–film interface is absent. In such a growth, the homoepitaxial layer can have a higher crystal structure quality (e.g., less defects) or the material can be chemically doped, producing a thin conducting layer. When two materials combined together in an epitaxial way are heterochemical, we refer to heteroepitaxial growth and the synthesized structures are called heterostructures.

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Epitaxy Physical Vapour Chemical Vapour Deposition Deposition Hybrid (PVD) (CVD) molecular beam epitaxy pulsed laser deposition sputtering

Fig. 1 Vapor-phase epitaxy is widespread growth method. Other epitaxial techniques include liquid-phase epitaxy (LPE) and solid-phase epitaxy (SPE).

Vapor-phase epitaxy is perhaps the mostly employed and widespread epitaxial method. Depending on the transport mechanism of gaseous species, we distinguish chemical vapor deposition (CVD) and physical vapor deposition (PVD) (Fig. 1). CVD is subclassified according to the chemistry of source gases (MOCVD ¼ metal-organic CVD), growth conditions, etc. PVD refers to the vaporization of source materials in vacuum. The classes of PVD methods are again distinguished by the vaporization methods (thermal evaporation, sputtering). Famous PVD methods are pulsed laser deposition (PLD), where the vapor is produced by laser ablation, and molecular beam epitaxy (MBE), where the source material is vaporized by thermal heating. Metalorganic MBE (MOMBE) is a hybrid method of CVD, and PVD allows to overcome the limitation of MBE and produces high-quality semiconductor and recently even oxide structures. This article touches some aspects of epitaxy that are important from the practical point of view and should help the reader to get acquainted with this topic. An interested reader is referred to the “Further reading” section.

Substrate and lattice matching The growth process of epitaxial layers (also epilayers or epitaxial film) is influenced by the substrate at the very initial stage. The formation of the first epitaxial layer is pivotal for the growth of the next following layers, as its quality will dictate the subsequent growth process. Therefore, the selection of the appropriate substrate is crucial for achieving high crystalline perfection of the epitaxial layers. The substrates may require some pretreatment to obtain a clean crystalline surface with atomically flat terraces before initiating the growth. The preparation steps can include mechanical polishing, chemical treatment, degassing/oxidation, and annealing. The preparation procedure depends on the substrate material and its crystal orientation, whereas specific recipes may already be established. The choices for substrates and especially of those with good crystallinity are limited. Therefore, epitaxial growth happens between materials with not only different chemical compositions but also different structural parameters. The condition on similar crystal structure for epilayer and substrate is not strict. It should be noted that the epitaxial layers and the substrate can have different crystallographic orientation and even different crystal structures. From a pure geometrical consideration there can be such crystal orientation relationships between substrate and epilayer for which the three-dimensional film and substrate lattices coincide most closely at their two-dimensional interface (Balluffi et al., 1982; Zur and McGill, 1984). However, if the size of the common superlattice, formed by lattice coincidence, is large, the chemistry at the interface may play a major role and the epitaxial relation between the substrate and the deposit may not be obvious. There are several ways how the difference in structural parameters (lattice mismatch) can be accommodated at the interface. If the interaction across the interface is weak, such as in van der Waals systems (graphene and other 2D layered materials), the epilayer will not be strained and will lie on the substrate. The weak forces at the interface can maintain the rotational alignment with the substrate. A more common situation occurs when the epilayer is chemically bonded to the substrate. The growth process depends strongly whether the epilayer is coherent or incoherent with the substrate. The growth is incoherent, when the epilayer assumes any in-plane lattice constant to minimize the free energy. In case of coherent growth, the epilayer adopts the in-plane lattice constant of the substrate as shown in Fig. 2A. The resulting elastic strain energy increases the free energy significantly. To quantify the degree of lattice mismatch, a parameter is defined: f ¼

af ilm − asub asub

(1)

where asub is the lattice constant of the substrate and afilm is the lattice constant of unstrained epilayer. Epitaxial layers with larger lattice constants than the substrate are under compressive strain, while the converse results in tensile strain. Since the strain imposed by the substrate is of a 2D character, the epilayer’s lattice parameter perpendicular to the interface will change in such a way that its internal energy is minimized. The change of the lattice constant is by the three-dimensional form of Hooke’s law. The resulting strain components are parallel and perpendicular to the surface. They are opposite in sign since the natural tendency of the strained structure is to preserve the volume of the unit cell. It is important to emphasize that such a crystallographic approach to match the substrate and the epilayer may not always be sufficient. One of the strongest points of epitaxy is that it can also stabilize crystal phases, which are not stable in the bulk form (Bruinsma and Zangwill, 1986). Thermodynamical consideration of the growth

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Epitaxy (a)

(b)

Fig. 2 Crystal lattice accommodation: (A) Substrate stains the epilayer by imposing the in-plane lattice constant of the epilayer. Out-of-plane lattice constant alters so that the volume of the unit cell is preserved. (B) Elastic strain of the epilayer relaxes by forming defects, when a critical layer thickness is exceeded.

dictates that the epitaxial stabilization is governed by the interplay between surface diffusion and the nucleation energy during the growth. Therefore, elemental tin assumes metastable diamond structure on the (001) surface of InSb and CdTe (Farrow et al., 1981), and anatase phase of TiO2 is stabilized on SrTiO3 (001) (Hsieh et al., 2002). Similarly such approach is demonstrated for the growth of several other oxides such as SrMnO3, BaMnO3, and CaCuO2 (Farrow, 1983; Gorbenko et al., 2002). In the coherent growth, it is considered that only the epilayer is strained since it is much thinner than the substrate. However, when the substrate thickness and the epilayer thickness become comparable, the stress is distributed between both of them. Such situation is realized when a free standing membrane (so-called compliant substrate) is employed in the epitaxial growth (Lo, 1991; Teng and Lo, 1993; Jesser et al., 1999; Feng and Liu, 1983). When the elastic energy reaches a certain value, it becomes more favorable for the structure to relieve the stress by breaking and reforming the bonds, such as shown in Fig. 2. These features are called misfit dislocations. The overall strain will be reduced but the dislocation energy increases. The thickness at which the misfit dislocation forms is called critical thickness (der Merve, 1962; Matthews and Blakeslee, 1974; People and Bean, 1986). Such dislocations are local inhomogeneities, which change the heterostructure properties (by acting as a trap, scattering center, etc.) and deteriorate the device performance based on this material. Besides misfit dislocations, other types of defects can form, such as point defects, dislocations, stacking faults, twins, and antiphase boundaries in alloys and compound multilayers. All of them act degrading on the physical properties of the materials. It is the aim of epitaxial growth to avoid the defect formation as much as possible by optimizing the growth parameters, the growth conditions, and improving epitaxial technique.

Modes of film growth Epitaxial growth takes place on a surface of a material, called substrate. Chemical elements of the desired material are supplied toward the surface in a way that depends on a particular epitaxial method. Fig. 3A exemplifies evaporation of atoms of two different species reaching the substrate surface with a step-terrace structure. Upon arrival a series of surface processes are involved in the epitaxial growth, which are schematically illustrated in Fig. 3A. Among them, the following are the most important:

• • • •

immediate re-evaporation or adsorption of species impinging on the substrate surface, surface diffusion and in case of molecules also dissociation, incorporation of the constituent atoms into the crystal lattice of the substrate or the formed epilayer desorption of the species not incorporated into the crystal lattice.

In all thermal processes a characteristic activation energy (for adsorption, re-evaporation, etc.) has to be overcome, whose value depends on the details of the process. Respectively, the number of particle participating in the particular process is given by the Arrhenius exponential law. On a phenomenological level, there are three distinct modes of film growth depicted in Fig. 3B–D. One of them is Frank–van der Merwe growth mode or also known as layer-by-layer growth mode (Frank and Van der Merve, 1949). It is realized when the interaction between the atoms of deposit and the substrate is stronger than the bonding between deposited atoms. Thus the first atoms condense on the substrate surface and form a complete monolayer. It is followed by the formation of a somewhat less tightly bound second layer. Given that the interaction with the substrate decreases with the number of epitaxial layers, the binding tends toward the value for a bulk crystal of the deposit. As a result, each new layer can start to form only when the last layer has been complete. Another growth mode occurs when the atoms of the deposit are stronger bound to each other than to the substrate. In such case the epitaxial material favors the formation of islands. Therefore, this growth mode is called island growth mode or Volmer–Weber growth mode (Volmer and Weber, 1926).

(a)

(b)

(c)

re-evaporation (d)

adsorption intralayer diffusion

diffusion nucleation

(e)

F S int

Fig. 3 Thin film growth. (A) Schematic representation of film growth mechanism on the surface. Two species are supplied to the growth surface. Several processes are possible: adsorption (attaching to the surface), desorption (re-evaporation from the surface), diffusion, nucleation/incorporation. (B) Frank-Van der Merwe growth mode. (C) Volmer–Weber growth mode. (D) Stranski–Krastanov growth mode. (E) Balance of surface tension forces acting on the epitaxial layer defines the growth mode.

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An intermediate growth mode is layer-plus-island growth mode also known as Stranski–Krastanov growth mode (Stranski and Krastanov, 1938). After the formation of one of few monolayers, the subsequent layer growth becomes unfavorable and islands start forming on top of the full layers. Many factors might account for this mixed growth mode. Among them are a large lattice mismatch between substrate and epitaxial film, and the symmetry or orientation of the overlayers with respect to substrate. The occurrence of different growth modes can also be comprised in terms of the relations between surface and interface free energies g, which describes a characteristic free energy needed to create an additional surface area (Grabow and Gilmer, 1988). Since g can also be treated as a force per unit length of boundary, the force equilibrium between the contact point of the 3D film island and the substrate can be written as gS ¼ gS −F + gF cos f

(2)

where gS is the surface free energy of the substrate surface, gF is the surface free energy of the film surface, and gS−F is the free energy of the substrate–film interface. Fig. 3E presents schematically the forces acting on the surface of the film formation. Respectively, the layer-by-layer growth is realized for f ¼ 0, and the condition above becomes gS  gS −F + gF cos f. The other limiting case is the island growth with f > 0 and thus gS < gS −F + gF cos f. This model captures also the mixed Stranski–Krastanov growth mode. At the initial stage of the growth layer-by-layer growth mode is realized. But the formation of intermediate layers alters gamma values, changing subsequently the force balance at the interface. The changes in gamma values can be understood by considering a lattice mismatch between deposited film and substrate. The lattice of the epitaxial layer tries to adjust to the substrate crystal lattice at the cost of elastic deformation energy. The transition from layer to island growth occurs when the spatial extent of the elastic strain field exceeds the range of the adhesion forces within the deposited material. The simple relation does not consider the gas phase of material above the surface of deposited film. When the particle transfers from the vapor phase into condensed phase at some pressure p, the free enthalpy changes by: DG ¼ nDm ¼ nkT lnðp=p0 Þ

(3)

where p0 is the equilibrium vapor pressure, n is the particle number, and Dm is the change of chemical potential between solid and vapor phases. The ratio p/p0 is associated with the degree of supersaturation. Taking the vapor phase into account, the conditions for layer-by-layer or island growth have to be supplemented. They become:

• •

gS  gS −F + gF cos f + CnkT lnðp=p0 Þ, layer growth gS < gS −F + gF cos f + CnkT lnðp=p0 Þ, island growth

The equations indicate that the supersaturation can be the driving force for the formation of epitaxial layers. More importantly, they indicate that the growth mode is not constant material parameter, but can be changed by varying the supersaturation condition. Layer-by-layer growth mode is favored with increasing supersaturation condition. This property is employed in the epitaxial growth of III–V compounds with MBE, where the group V component has relatively high equilibrium vapor pressure. In case when homoepitaxial growth is realized, the balance of free energies of interface, film, and substrate surfaces predicts an epitaxial growth in layer-by-layer mode (Frank–van der Merve). However, the kinetic energy of atoms on the surface of the substrate can add variation in the thin film growth process. For a simplified consideration, interlayer and intralayer diffusion processes are of prime interest. The first process describes the atoms diffusion on a terrace, while the second diffusion process considers the atom motion across the step edge onto a lower terrace (Fig. 3A). Note that an atom traveling on a lower terrace will be incorporated at the step edge due to its higher binding energy. Depending on the relative interlayer and interlayer diffusion rates, three different nucleation process are realized:

• • •

Step-flow mode. It takes place when the intralayer mobility is so high that all atoms reach the steps before the nucleation on the terrace occurs. Layer-by-layer growth. It is realized when the interlayer diffusion dominates. The diffusion length is smaller than the terrace width. The nucleation proceeds on the terraces and subsequent growth of adatom islands. This growth mode leads to a periodic change of surface morphology: the surface flatness recovers after completing the formation of an atomic layer. Multilayer growth. In ideal case it is realized when no atom transport takes place. Every atom stays on the level where it was deposited.

Based on this consideration, it appears reasonable that the growth mode can be tuned by adjusting the substrate temperature at a fixed deposition rate. At lowering the temperature, the step-flow growth mode goes over to layer-by-layer growth mode and reaches finally the multilayer 3D growth mode. Experimentally an ideal layer-by-layer growth is difficult to realize since the nucleation on a layer always takes place before the layer formation is fully completed. Consequently, the epitaxial surface always becomes rough after the deposition of many layers. In practice, one strives to obtain epitaxial layers with a low surface roughness. The surface atoms in the uppermost epitaxial layer differ from those in the bulk due to the absence of chemical bonding on one side. As a result, the surface atoms shift from their positions in the bulk so that a new equilibrium state is achieved. One distinguishes two main rearrangement effects. Relaxation is attributed to the shift of atoms normal to the surface. In this case the lateral periodicity is preserved with respect to the bulk. Usually few top most layers are involved in this rearrangement type. Reconstruction is another

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type of atom rearrangement on the surface. Here the surface energy is minimized when the atoms restore some broken bonds. This is accompanied by the shift of atoms parallel to the surface. Thus the unit cell of the reconstructed surface differs from that in the bulk. The observation of diffraction pattern, such as performed with RHEED (see section below), can detect the surface reconstruction.

Material characterization The quality of the synthesized film can be accessed by various characterization techniques. The main goal of characterization is to help a grower to optimize the growth conditions for improving the crystal structure quality, reducing defects, achieving desirable stoichiometry, etc. Quality of crystal affects the physical properties of the synthesized material. The list of the characterization techniques presented here is by far not complete: Atomic force microscopy (AFM). The technique allows to measure the roughness of the surface down to the atomic level. Scanning probe technique monitors the force between the sharp scanning tip and the sample surface. The change of the force is used to produce the images of surface. The method has also high lateral spatial resolution and can operate at ambient conditions. X-ray diffraction and reflection (XRD). Diffraction of X-ray on the sample surface provides information on the crystal structure of epitaxial layers: lattice parameters (in-plane and out-of-plane), orientation relation between epitaxial layer and substrate, thickness the surface roughness (involving some modeling) can be measured. Scanning electron microscopy (SEM). It is an electron microscope that provides images by scanning focused electron beam across the surface and detecting the signal from secondary or backscattered electrons. It images the surface topology, grain size and shape, film voids, microcracks, etc. Energy dispersive X-ray (EDX). The signal of backscattered electrons in SEM is used to evaluate the chemical composition of the material. The signal comes from the thickness of about 1mm with lateral resolution of a comparable size. EDX’s detection limit is 0.1% and it can detect elements with atomic number higher than 11. Usually EDX unit is installed together with SEM. Secondary ion mass spectroscopy (SIMS). This technique allows to evaluate the composition, of the epilayer. It uses a focused ion beam to sputter the surface of the sample and analyze the ejected secondary ions. The technique allows to obtain the depth profile information on material composition. Own to its high sensitivity, SIMS is used to quantify the incorporated impurities. Transmission Electron Microscope (TEM). It provides high-resolution plan-view lattice images and transverse film section. It can thus visualize lattice orientations, interface quality and defects, atom mixing at interface, superlattices, etc. X-Ray photoelectron spectroscopy (XPS). It is a photoemission spectroscopy analyzing the energy of the electrons emitted from the surface as a result of irradiation by single-energy X-ray photons. XPS is a surface-sensitive method and provides information on surface composition as well as its chemical state. It detects all elements with the exception of hydrogen and helium. Ellipsometry. The technique is based on the analysis of polarized light reflected from the sample. It probes the complex refractive index and allows to extract the information about layer thickness and roughness. Almost all of these techniques characterize the material after the growth. An outstanding technique that can probe the surface of the substrate and monitor the growth processes is Reflection High-Energy Electron Diffraction. Since this technique is compatible with the vacuum environment, it is widely employed to monitor the growth process in PLD and MBE (Hasegawa, 2012; Ichimiya and Cohen, 2004; Haeni et al., 2000). Reflection high-energy electron diffraction (RHEED). As the name implies, the technique exploits diffraction of electrons by surface atoms and thus probes the periodic arrangement of atoms at the surface. In this technique an electron gun (e-gun) generates a monochromatic electron beam with a typical energy of 10–50 keV, which strikes the surface of the sample at a grazing angle of about 1 degree. At this energy the electron wavelength is in sub-angstrom range, which is thus smaller than the thickness of an atomic layer. Furthermore, due to a small incident angle, the penetration depth is only few atomic layers below the surface, which explains the sensitivity of RHEED technique to the surface morphology. The diffracted electrons interfere constructively at specific angles given by the electron wavelength, crystal structure, crystal orientation, and atom arrangement on the surface. The resulting diffraction pattern can instantly be visualized on a photoluminescent screen. It is the subject of analysis since it carries the information on surface processes. When the sample surface is smooth, the diffracted intensity is a maximum. However, when the epitaxial surface is partially covered, the diffracted intensity decreases due to destructive interference in the diffracted electron beam from different areas of the surface. Hence, when the layer-by-layer epitaxial growth mode is realized, the diffracted intensity reveals an oscillatory behavior. This is visualized by plotting the specular spot intensity as a function of time during the epitaxial growth. The oscillations are routinely employed to measure the thickness of the epitaxial films and superlattice layers since each period of oscillations corresponds to the completion of one monolayer. Furthermore, by counting the number of RHEED oscillations, one can calibrate the deposition flux, control alloy composition, and adjust it respectively. If the surface does not become smooth after completion of each monolayer, the amplitude of RHEED oscillations decays gradually due to destructive interference. The RHEED oscillations are characterized by their period, amplitude, phase, damping of the oscillations, their initial behavior at the beginning of growth, and their recovery after growth. These features can provide valuable information about the growth process. Besides the diffraction spots from the surface, which are anticipated from the kinematic scattering analysis, additional intensity lines and curves appear in the diffraction pattern. They are the result of the dynamical scattering and are known as Kikuchi lines.

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They emerge as a result of nonlinear effects and originate from electrons scattered incoherently from the crystal by phonons, plasmons, defects, or electron–electron interactions. The observation of sharp and clear Kikuchi lines is an indication of a flat and crystalline surface (Hasegawa, 2012).

Epitaxy techniques Various deposition techniques can be distinct by the kinetic energy of particles impinging on the substrate surface and can be grouped in three kinetic energy regimes (Cuomo et al., 1991). Low energy process (on the order of 0.1 eV) encompasses chemical vapors deposition and thermal evaporation. Here the particle kinetic energy is on the order of the thermal energy associated with the temperature of the substrate. Atoms impinging on the substrate surface thermalize almost immediately with the substrate, get adsorbed, and diffuse on the surface until they get permanently bonded at a nucleation site. At a given temperature, the condensation rate can be increased by increasing the vapor flux. At elevated substrate temperature the enhanced kinetic energy of atoms facilitates to overcome the surface traps and to find energetically more favorable states. This results in a growth of films with fewer defects. In this kinetic energy regime, the material forms thermodynamically stable state. Another deposition regime with energies above 1eV is realized when the material flux is created by sputtering, laser vaporization, etc. The energy of such species corresponds to the equivalent temperatures exceeding the temperature of the accommodating surface. It is rather associated with the atomic energy and bond strength. In this case the interaction with the surface becomes significant. As a result the impinging species can create defects on the surface, which can become important for the formation of interfaces between materials. In this energy regime metastable material phases can form. In the other deposition regime, the kinetic energies are above 100 eV and the particles can even penetrate the surface and be implanted in the material.

Chemical vapor deposition CVD is a widely used material deposition technique in which thin films are synthesized on a heated substrate via a chemical reaction of supplied precursors. In contrast to PVD, such as MBE and PLD (discussed later), CVD heavily relies on chemical reactions. Therefore, thermodynamics governs the CVD reaction and indicates the direction in which the reaction proceeds. Perhaps the first report on using the CVD technique dates back to 1946, when Teal et al. deposited silicon film on ceramic surfaces by flowing silicon tetrachloride (SiCl4) and hydrogen gas (Teal et al., 1946). Due to the chemical reaction SiCl4 + 2H2 ! Si + 4HCl

(4)

a smooth silicon film forms at 1100C. CVD process consists of several steps which are pictorially depicted in Fig. 4:

• • • • •

Transport of reactants to the deposition region Diffusion of reactants from the main gas stream to the wafer surface. Prior to that, the reactant gases can undergo reaction and form an intermediate reactant. Adsorption on the wafer surface and diffusion on the surface Surface processes (chemical decomposition and reaction, site incorporation, nucleation) lead to the thin film synthesis and growth. Additional stimulators (light, plasma, temperature) can be introduced to promote the chemical reaction. Desorption of byproducts from the surface and their transport away from the deposition region heating zone mass transport diffusion of by-product

diffusion inlet

exhaust

adsorption/ decomposition/ chem. reaction substrate heating zone Fig. 4 Chemical vapor deposition. Schematic representation of processes during CVD. Shown are the sequence of gas transport and reaction contributing to the thin film growth. The molecules are dissociated from the growth surface and undergo a chemical reaction on it. The by products are removed from the active zone.

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The main components of a CVD system are the reaction chamber and gas handling system (gas delivery and exhaust gas treatment system). In a most common configuration the reactor is a horizontal quartz tube, where the substrate is placed on a boat/susceptor made of quartz or graphite. A gas inlet injector is connected to one side of the tube, where the gas supply is regulated by the mass flow controller. In case the precursor is in a solid state form, it can be supplied to the deposition region either by sublimation or by dissolution in an appropriate solvent. The other side of the reactor is connected to the exhaust to remove the by products of thin film growth. Potentially hazardous by products should be removed by introducing suitable traps. The reactor can be connected to the vacuum system to purge the deposition chamber before the deposition and to achieve the necessary pressure for the transport of reactants. The temperature of the substrate can be raised locally by resistance heating or by raising the temperature of the entire reaction camber using an external furnace. For obtaining high quality film it is essential not only to select the proper substrate (for instance due to its catalytic ability or other chemical properties), but requires also the optimization of the growth condition such as temperature, gas flow rate, pressure etc. There are various CVD types which differ from each other by reactor (growth chamber) configuration, operation conditions, substrate heating methods, stimulators to promote the chemical reaction, precursor type, etc. Among them are:

• • •

Metal-organic CVD (MOCVD), also known as MOVPE (metal-organic vapor-phase epitaxy), employs metal-organic precursors to form thin films. The precursors are usually volatile and toxic so that a particular precaution has to be taken. This CVD method is widely used for the growth of III-V semiconductor structures for optoelectronic applications. Plasma-enhanced CVD uses plasma (partially ionized gas) to promote the chemical reactions. Atomic layer deposition (ALD). The precursors are introduced in a pulsed manner. They are absorbed in a self-limiting chemical reaction on the substrate surface. Between pulses the carrier gas removes the remaining precursors. Layer-by-layer growth can be achieved, whereas the film thickness can be controlled by the number of precursor cycles.

Started with the growth of silicon-based coating, CVD became an important technique for semiconductor industry to produce thin films. It also plays an important role in developing and exploring novel materials. Versatile CVD techniques are employed for the large-scale synthesis of graphene and polymeric thin films, and 2D layered materials such as transition metal dichalcogenides (Sun et al., 2021).

Liquid-phase epitaxy LPE is the crystal growth deposition technique of a single crystalline layer from a liquid phase (a solution or melt) on a substrate acting as a seed (Dawson, 1972; Stringfellow, 1982). The idea behind the growth technique is to create a supersaturated solution of a material that has to be deposited. The substrate is then brought in contact with the melt, from which a solid-state layer precipitates as an epitaxial film on the substrate. The growth process occurs close to thermodynamic equilibrium so that the system reaches a thermodynamic state of lower free energy. This distinguishes LPE from deposition techniques such as PLD, MBE, and vapor-phase epitaxy (VPE). Although multiple layer systems can be synthesized from solutions of different compositions, the feasibility to combine them is driven by the thermodynamic processes. Therefore, materials must have, for instance, similar or controllable liquidus temperatures. The growth can also be impeded by the lattice mismatch between seeding and epitaxial layers. Furthermore, the mass transport and diffusion between the layers limit the interface sharpness and can result in unintentional compositional and doping gradings. Because of thermodynamics, LPE is sensitive to temperature changes, solution composition, surface tension, quality of seeding substrate, etc. The great advantages of LPE growth are the high growth rate, exceptional high purity crystals with low defect density (impurities tend to segregate rather than to be incorporated into the epilayer), and high selectivity on patterns, substrates, and facetted growth, which can produce shaped crystals and selectively nucleated crystals with high aspect ratios. This made LPE to the pioneering growth method to produce the optoelectronic devices (LED, laser, solar cells) based on II-VI and III-V semiconductors (Nelson, 1963; Rupprecht et al., 1967; Woodall, 1972; Alferov et al., 1969). Other compounds, including oxides and superconductors, can also be grown with LPE (Ehrentraut et al., 2006; Klemenz and Scheel, 1993; Nagarajan et al., 2011; Yamada et al., 1993). See “Further reading” for more information on this epitaxy method.

Solid-phase epitaxy SPE refers to the type of growth when a material undergoes a transition from amorphous phase to a crystalline phase. Usually the material is deposited in an amorphous form on a crystalline substrate, which serves as a template for crystal growth(Mayer et al., 1968; Cho, 1969a; Marks et al., 2021). When heated, atoms in the amorphous phase reorder by local bond rearrangements at the crystalline–amorphous interface. The material phase transformation can also be induced by laser, electron, or ion irradiation. An annealing step to recrystallize layers during the ion implantation or to heal defects can also be considered as a SPE (Streit et al., 1984; Norga et al., 2006). SPE is also employed for growing a buffer layer to improve heteroepitaxy of lattice-mismatched heterostructures. See “Further reading” for more information on this epitaxy method.

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Molecular beam epitaxy MBE is a physical vapor deposition technique, which is frequently used in research laboratories and industry. The technique was introduced in 1960s by A. Y. Cho and J. R. Arthur (Arthur, 1968; Cho and Arthur, 1975; Cho, 1969a, 1970). MBE is low-energy deposition technique in ultra-high vacuum. The constituent elements of the crystalline solid are supplied from the source to the substrate as a directed ray of neutral molecules and atoms, so called molecular beam. Such transport implies that the mean free path of species is larger than the distance between the source and the substrate. As a result, the atoms experience no collision before they reach the substrate. A thin epitaxial layer crystalizes via reactions between molecular beams, and a substrate surface holds at high temperature. Therefore, the composition of the epilayer depends on the relative arrival rates of the constituent elements. By controlling the evaporation rate of the source, the arrival rate of molecular beam and thus the epilayer composition can be adjusted. Such a control of epilayer stoichiometry is one of the key strengths of MBE growth. A low growth rate (1mm h−1) assures that the species absorbed on the substrate surface can diffuse (Fig. 3A). Therefore, MBE allows to achieve smooth epilayer surface. Each molecular beam source is equipped with a mechanical shutter, which is placed in front of it and is used to start and stop the growth fluxes by blocking the emitting flux from the substrate in a time much shorter than the time needed to grow a monolayer. This allows to establish a sharp interface between two materials. A high purity of original element source and high vacuum help to reduce the contamination of epilayer by impurities. The vacuum environment offers an excellent opportunity to monitor the growth process in situ. RHEED is frequently installed in the growth chambers. Thus MBE can produce high-quality thin film structures with layer thickness precision on the atomic level. Fig. 5 is a schematic view of an MBE chamber that contains essential elements of MBE equipment independent of its purpose. Not shown is a load-lock chamber system that allows to introduce the wafer (or substrates) into the growth chamber without breaking the high vacuum.

Pumping system It is one of the essential components to assure the molecular beam propagation between source and substrate. The mean free path L changes with the pressure P as L ¼ 5  10−3/P, where the distance is given in cm and the pressure in Torr. Thus under high vacuum of 10−9 Torr, the mean free path is about 5  106 cm. The vacuum is generated by pumps and cryoshrouds. The layout of the pumping system is crucial for an MBE setup; it must also assure an effective evacuation of residual source gases and reactant byproduct gases. Depending on the purpose, an MBE system can be equipped with various pump systems, such as turbo-molecular pump, ion pump, and cryogenic pump. Material evaporation source is another important components of an MBE layout. Depending on the material to be evaporated, different sources for producing molecular beam are available.

flange with liquid nitrogen feedthroughs

manipulator with substrate heater

cryopanel

pumping port

wafer transfer port shutter ports

metrology ports: - RHEED - ellipsometry - etc.

effusion cell ports base flange with additional ports: - pyrometer - laser reflectometry - e-beam evaporator - etc.

Fig. 5 MBE Schematic view of the growth chamber of an MBE machine. The drawing is the courtesy of VEECO.

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Material evaporation source

Cells. Knudsen’ cell is by far the most important type of evaporation source in a molecular beam system. It is filled with ultrahigh purity materials, which reduces incorporation of impurities in the epilayers. Basically a Knudsen cell is a tubular crucible (made of pyrolytic boron nitride, quartz, tungsten, or graphite), filled with the desired chemical element and is heated resistively up to the level of desired material flux. The Knudsen cells installed in MBE have large opening and thus they are different from the true Knudsen source (Knudsen, 1915), which should have an orifice size comparable with the mean free path of the gas molecule at the temperature of evaporation. The material flux from the cell and its angular distribution in a real MBE system depend not only on the crucible shape but also on the evaporant fill level. Owing to the large thermal mass the cell temperature and thus the evaporation rate cannot be changed quickly. The temperature is usually controlled with a thermocouple. A special type of effusion cell is a cracker cell. Some elements (such as As and P) evaporate as tetramers, although their dimers show a better sticking coefficient to the substrate surface and thus a better growth performance. Breaking of tetramers (As4) in dimers (As2) is accomplished in a cracker cell, which is a Knudsen cell having two zones of temperature. The low temperature zone produces the tetramer vapor and the high temperature zone cracks the tetramers to dimers. The cracking efficiency depends on the catalyst, whereas Ta and Rh are very effective. Gaseous sources. Alternative material flux source was initially introduced by Panish due to difficulty of growth of semiconductor compounds with elemental source of phosphorus (Panish, 1980; Panish et al., 1985). AsH3 and PH3 gas sources were introduced to grow InGaAsP. The hydrates are injected into the chamber and are dissociated in cracker cell so that dimers of elements are formed. Also metalorganic (MO) compounds are employed as alternative material sources, which thus borrows an idea from chemical vapor deposition technique (MOVPE). Metalorganic compounds are less thermally stable and can dissociate directly on a growing substrate. The by products are removed from the environment, whereas the cryopumps can be employed to maintain a good vacuum condition. MBE system equipped with MO source is often called MOMBE. The advantage of it is that MO needs a low temperature to produce sufficient flux; it is stored outside of MBE chamber and thus can be readily replaced; a high growth rate can be achieved. MO compounds are however toxic and less pure than the elemental source material. E-gun. The operation of Knudsen cells is limited to a temperature of about 1400C. Yet, some elements require a higher temperature to be evaporated and to obtain a sufficient material flux for the epitaxial growth. In this case, the materials such as Ti, Si, Mo, and W. can be evaporated using intense beams of high-energy electrons, so called e-beam gun (Kasper, 1982; Koenig et al., 1979; Ota, 1983). The heating energy is transferred by electrons on just an evaporating surface of target. Although such evaporation technique is more prone to flux instability, the introduction of active feedback loops between evaporation parameters and flux can minimize the fluctuations. Since this technique can evaporate a large number of elements, the e-beam evaporators are frequently employed in the epitaxy (Yamamoto et al., 2013). Laser was also employed for heating the sources in MBE growth of II-VI compounds. With this technique, Hg1−xCdxTe epitaxial layers and superlattices were grown (Cheung and Madden, 1987; Cheung et al., 1986).

Flux measurement and stoichiometry Accurate flux measurements and flux stability are crucial for the growth of high-quality films with well-defined stoichiometry. Fluxes from the effusion cells can be characterized by various techniques. Quartz crystal microbalance (QCM) measures the flux by relating the shift in the resonance frequency to the accumulation of mass on the probe located at the substrate position. Beam equivalent pressure (BEP) can be obtained by inserting an ion gauge in front of the substrate, so called beam flux monitor (BFM). Atomic absorption spectroscopy and cross-beam mass spectroscopy are other examples for monitoring the material flux of each effusion cell. Despite the variety of methods, an excellent control of flux and thus stoichiometry is difficult. Moreover, the flux stability will also depend on the stability of the cell temperature. All these lead to the deviation of the desired compound composition. The desired structure accumulates defects or can even form other crystalline phases. As a result, the physical properties of the desired compound are altered. Yet, the thermodynamics can enable a stoichiometric compound composition within some range of growth parameters (substrate temperature and material flux). Such thermodynamic regime is also known as an adsorption-controlled growth. One talks also about an opening of a growth window. A prerequisite for this is that one of the constituent elements has to be volatile. This requirement is fulfilled for the growth of III-V semiconductors, which has thus significantly contributed to the success of semiconductors’ growth. The respective thermodynamic conditions can be estimated from the equilibrium lines in a pressure–temperature diagram (Arthur, 1968; Cho, 1971). The lower bound is given by As condensation on a Ga-rich GaAs surface, while the upper bound is given by the vapor pressure curve of As. Between two boundaries, sufficient As is supplied to saturate each Ga monolayer, and any excess As desorbs from the surface and enters the vapor phase. Thus, the growth rate is determined by the Ga flux and the self-regulated growth can be obtained. Since the equilibrium lines are separated from each other by orders of magnitude in the As gas pressure, the request on the precise control of the flux and temperature is relaxed. In principal, if the growth conditions are within the growth window, an identical crystalline compound forms. If the parameters are outside of the growth window, the compound decomposes, for instance by forming liquid Ga and solid As phases. The regime of adsorption-controlled growth is also realized for some oxide compounds with volatile elements Pb and Bi, such as PbTiO3, BiFeO3, BiMnO3, and (Rb, Ba)BiO3 (Theis et al., 1998; Ihlefeld et al., 2008; Lee et al., 2010; Hellman and Hartford, 1990).

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Oxide compound The growth of oxides constitutes one of the important fields in the material science, as the oxides show a diverse range of physical phenomena in bulk and thin films and therefore are considered as a promising material platform for realizing novel functionalities (Schlom et al., 2008; Hwang et al., 2012). High-quality structures are realized in an MBE chamber equipped with a suitable oxygen source, which can be molecular oxygen O2 (suitable for easy oxidizing compounds), oxygen plasma (activated by RF plasma source), or ozone, which is the most reactive and the cleanest oxygen source (Schlom et al., 1988; Eckstein et al., 1989; Berkley et al., 1988, 1989). By using a highly reactive oxidizer a long mean free path should be guaranteed to avoid gas-phase reactions. Usually the nozzle of oxygen line is positioned about 10 cm away from the substrate surface. A complete oxidation of epilayers is always a matter of concern in oxide MBE. Yet, it is possible to achieve high-quality oxide structures. ZnO is one of the examples. The heterostructures based on ZnO feature high electron mobility, giving rise to a variety of quantum phenomena (Tsukazaki et al., 2010; Falson et al., 2011, 2016, 2015; Maryenko et al., 2018, 2021). High-quality ternary oxides, such as SrTiO3 perovskite, are grown by employing metalorganic precursor (MO) as the source for Ti element, that is titanium(IV)isopropoxide (TTIP, Ti[OCH(CH3)2]4). A high vapor pressure of TTIP enables an adsorption controlled growth and therefore offers an excellent control over the epilayer stoichiometry (Endo et al., 1991; Jalan et al., 2009a, b, c; Brahlek et al., 2018). Once TTIP molecule reaches the hot substrate surface, it decomposes in nonvolatile TiO2 and by products consisting of volatile C3H6 and H2O. TiO2 reacts with SrO formed on the surface, leading to the formation of a stoichiometric SrTiO3 layer. The structures synthesized in this way allowed to achieve high electron mobility (Son et al., 2010) and to demonstrate the quantum Hall effect for the first time in oxide perovskite structures (Matsubara et al., 2016). Oxide MBE with MO sources is referred as hybrid MBE. The approach to synthesize oxide perovskite with MOs has been extended on the growth on vanadium perovskite, using vanadium-triisopropoxide (VTIP) as a source of vanadium (Moyer et al., 2013; Brahlek et al., 2018) and high mobility BaSnO3, employing hexamethylditin (HMDT) as a precursors to supply tin (Chambers et al., 2016). The reactive environment of oxide growth requires the use of oxidation-resistive materials for components that are particularly exposed to high temperatures. Heating of substrate is also an important part of consideration for oxide MBE. The usage of laser heating system instead of resistive heaters allows not only to achieve even higher growth temperatures but also to reduce the heat load on the growth environment. As a result, the incorporation of undesired impurities collected on the chamber walls of the heating system is reduced. Due to the local heating by laser, the heat capacitance is reduced and the quick change of the growth temperature can be achieved, which can be favorable for growth protocols requiring the temperature changes. High temperature can even prepare in situ the substrate surfaces suitable for the growth (Jaeger et al., 2018; Braun et al., 2020). Fig. 6 is the photograph of an oxide MBE machine installed at RIKEN. It is equipped with several retractable Knudsen cells, supply of metal-organic precursors, and ozone source. Laser heater system allows to achieve temperature well above 1000C. The pumping system, consisting of turbomolecular pump and cryopump, allows to remove the by products of metalorganic

nitrogen supply line

laser for substrate heating

distribution chamber turbo molecular pump

load lock chamber

RHEED gun

beam flux gauge

cryopump

Knudsen cell

cell’s shutter

Fig. 6 Photograph of oxide MBE VEECO GEN10 installed at RIKEN. Additionally it is equipped with the ozone source and gas cell for supply of metalorganic precursor. The laser beam for substrate heating is guided by the optical fiber to the growth chamber.

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precursors effectively. Cryogenic panels cooled with liquid nitrogen minimize the thermal load of the growth chamber and allow to trap impurities from the gas phase during thin film deposition.

Pulsed laser deposition PLD is one of the PVD process techniques, where the material to be deposited is ablated (evaporated) by highly energetic laser beam focused on the starting material, so called target. This growth technique emerged due to the development of laser technology. In fact, the first demonstration of PLD growth technique was in 1965 by Smith and Turner, who used ruby laser to synthesis optical coatings (Smith and Turner, 1965). However, a broad proliferation of PLD growth method has been triggered by the work by Dijkkamp and coworkers demonstrating the thin film growth of high temperature superconductors YBa2Cu3O7 (Dijkkamp et al., 1987). Since then PLD has been employed for synthesis of all kinds of materials, encompassing oxides, nitrides, carbides, organic materials, polymers, and metallic systems. Because of the similarity with MBE growth technique, one finds for PLD growth also the notation as laser-assisted MBE or laser MBE. Fig. 7 shows the main concept of PLD process. A pulsed laser beam passes through an optical window of a vacuum chamber and is focused onto a target, where it is partially absorbed. A dense layer of vapor forms in front of the target, whose temperature and pressure increase during the light absorption process, resulting in a partial ionization of the ablated materials. The formed layer expands from the target surface and appears as a forward-directed luminous plume. During the expansion, internal thermal and ionization energies are converted into the kinetic energy of ablated particles. The ablation plume provides thus the material flux for the film growth. The threshold power of high energy laser needed to produce such a plume depends on the target material, its morphology, and the laser pulse wavelength and duration. A typical laser fluence amounts to 1–10 Jcm−2. Excimer laser or high harmonics of Nd:YAG solid-state laser can provide such high energy laser pulses and are frequently employed in PLD setups. For an efficient ablation, the optical absorption coefficient of the target material should be high at the laser operation wavelength. The target materials do not have to have the same phase as the desired thin film; only the cation stoichiometry needs to be preserved, assuming stoichiometric transfer between target and film. Each ablation pulse provides material amount sufficient for the deposition of a submonolayer of the desired thin film phase. Therefore, PLD deposition is very suitable for multilayer and interface formation where submonolayer control is required (Fig. 8). The kinetic energy of species arriving at the substrate can modify the thin-film growth. Upon ablation the species in the ablated plume interact with the gas atmosphere introduced during the growth. So the particles, kinetic energy at the substrate surface depends on the gas parameter, mass, and pressure. Thus the kinetic energy can range from high initial energy (several hundred electron volts) with no gas atmosphere to low energies (few tenths of electron volts) from thermalization at sufficiently large ambient pressure. Upon arrival at the substrate surface, the diffusion time of the atoms usually exceeds the optical pulse duration, leading effectively to instantaneous deposition. During the time between subsequent optical pulses, the adatoms can rearrange on the surface by diffusion and are subsequently incorporated through nucleation and growth. Such a separation of the growth process in two steps, instantaneous deposition and nucleation, is unique for thin film synthesis with PLD. The two steps can be controlled independently from each other. While the instantaneous deposition rate can be controlled by adjusting the laser energy density

Substrate heating system: IR-Laser / Lamp heater

substrate RHEED gun

RHEED pattern

1000°C florescent screen

oxygen

Laser beam

plume target pump

Vacuum 10-9 atm

Fig. 7 Schematic view of PLD. The laser beam ablates the target and creates a plume that propagates toward the substrate kept at high temperature. RHEED is used to monitor the growth process. Inset shows the RHEED pattern as observed on the fluorescent screen. Spots of RHEED pattern and Kikuchi lines indicate a good film quality.

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Epitaxy halogen lamp with substrate manipulator

mirrors for guiding the laser beam

growth chamber

RHEED gun er

turbomolecular pump

er

las

im

video camera for RHEED pattern

c ex

load-lock chamber

pirometer target carousel with motor Fig. 8 Photograph of a PLD installed at RIKEN.

(pulse energy and laser focus on the target), ambient gas condition, distance between target and substrate, and the growth rate are determined by the laser pulse repetition rate. Moreover, the growth process will depend on the substrate surface preparation quality and the substrate temperature. Besides the UHV optical view port for introducing the laser light into the growth chamber, a usual PLD chamber contains a vacuum pumping system (turbomolecular pumps) to maintain a background pressure between 10−6 and 10−11 Torr. Substrate heater, target manipulator, and gas inlet are also essential components of PLD setup. The substrate heating system can be realized in various ways; it can be a conventional SiC heater as well as halogen lamp or an infrared laser. For the deposition of oxides, halogen lamp and infrared laser are widely used, since they do not oxidize in oxygen reactive atmosphere. The substrate is usually placed about 2–10 cm away from the target. The target manipulator can be a carousel containing several targets. The targets can be in situ exchanged, facilitating the synthesis of multilayer structures consisting of different materials. Also various gases (molecular O2, N2, Ar, etc.) can be introduced into the chamber through the gas inlet during thin film deposition to promote the reaction of the evaporated species. To increase the reaction rate, oxygen plasma, nitrogen plasma, or even ozone can be introduced through the gas inlet. The growth process is monitored with the help of RHEED. The PLD technique is probably one of the most versatile thin film synthesis methods as it offers a number of advantages (Willmott and Huber, 2000):

• • • • •

Almost any material can be grown with PLD. The source of material evaporation (pulsed laser) is outside of the growth chamber, offering the flexibility in geometrical arrangement and material choice. Nearly there is one-to-one stoichiometry transfer from target to the film, even for chemically complex systems. There is synthesis of materials with a chemical composition far from equilibrium. This opens the way for attaining materials with novel electronic states, which cannot be obtained under thermal conditions. There is tunability of the growth rate due to laser operation in a pulse regime.

Besides numerous advantages that PLD can offer, there are several fundamental and technical drawbacks as well:

• • • • •

Ablation target process can also produce ejecta, which will lead to the defect formation in the synthesized films. Target contains impurities either introduced during sintering of the target or from the source material. As a result, impurities can be incorporated into the epitaxial layers. high kinetic energy species in the plume bombard the surface of substrate as well as subsequently formed epitaxial layers. This results in the formation of crystallographic defects, which can be detrimental for the formation of sharp interfaces. The process of plume formation leads to an inhomogeneous distribution of flux and angular distribution within the plume. As a result, the synthesized structures can be inhomogeneous. Stoichiometric transfer from target to the desired materials is not always granted. The stoichiometry transfer can depend on the deposition condition, such as laser fluence or temperature (Groenendijk et al., 2016; Ohnishi et al., 2008; Tomar et al., 2019). For targets with volatile elements, the target has to be enriched with the respective element. The change of a stoichiometric

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composition requires a fabrication of a new target. Moreover, the ablation of the target can modify its surface, which can lead to the variation of the target stoichiometry at the position of the laser focus. In extreme case the change of the target stoichiometry can lead to the decrease of the growth rate.

Sputtering Sputtering is the ejection of particles from the target by its bombardment with energetic particles. Sputtering process relies on production of plasma through electrical discharge and electrostatic acceleration of ions, inert gas ions like argon, towards the source material. The impinging ions transmit energy and momentum to the source material and knock out the particles form the surface of the target. Ejected particles are the source material for films grown by sputter deposition. The particles ejected in sputtering are predominantly neutral atoms in the ground state, although ions and atomic clusters can also be present in the emitted flux. Such process of creating particle flux is similar to that of PLD, except that in PLD the target is ablated by laser light. There are different kinds of sputtering processes, depending on condition (reactive sputtering, bias sputtering) or configuration (ion beam sputtering, DC sputtering, RF sputtering, magnetron sputtering). For instance, magnetron sputtering implies that a magnetic field is created in the vicinity of target holder, which bends the ionized particles on a specific area of a target. This increases the deposition rate and the deposition area. Sputtering deposition technique is cost-effective, shows a high throughput and can produce large-area films. Such characteristics make the deposition technique attractive for industrial growth applications. See “Further reading” for more information on this epitaxy method.

Conclusion Epitaxy is a broad research field concerning with the mechanism and methods to grow-well oriented crystalline layers. Over the last century a various practical methods have been developed that allowed the synthesis of an enormously large pallet of materials. GaAs-based structures are a striking example of the opportunities that open when structures are optimized and the growth technique is refined (Pfeiffer and West, 2003; Pfeiffer et al., 1989; Umansky et al., 2009; Gardner et al., 2016; Chung et al., 2018, 2021). In this context, the continuous improvement of material quality, the development of new materials and their multilayers including hybrid structures is an important direction, that will enable to open new horizon not only in fundamental science but also in applications. Epitaxial methods are therefore essential for traversing along this path.

References Alferov ZI, Andreev VM, Korolkov VI, Portnoi EL, and Tretyakov DN (1969) Coherent radiation of epitaxial heterojunction structures in AlAs-GaAs system. Soviet Phys. Semiconductors 2: 1289. Arthur JR (1968) Interaction of Ga and As2 molecular beams with GaAs surfaces. Journal of Applied Physics 39: 4032. Balluffi RW, Brokman A, and King AH (1982) CSL/DSC lattice model for general crystal-crystal boundaries and their line defects. Acta Metallurgica 30: 1453. Barker TV (1906) Journal of the Chemical Society, Transactions 89: 1120. Berkley DD, Johnson BR, Anand N, Beauchamp KM, Conroy LE, Goldman AM, Maps J, Mauersberger K, Mecartney ML, Morton J, Tuominen M, and Zhang YJ (1988) In situ formation of superconducting yba2cu3o7 thin films using pure ozone oxidation. Applied Physics Letters 53: 1973–1975. Berkley DD, Goldman AM, Johnson BR, Morton J, and Wang T (1989) Techniques for the growth of superconducting oxide thin films using pure ozone vapor. Review of Scientific Instruments 60: 3769–3774. Brahlek M, Gupta AS, Lapano J, Roth J, Zhang H-T, Zhang L, Haislmaier R, and Engel-Herbert R (2018) Frontiers in the growth of complex oxide thin films: Past, present, and future of hybrid MBE. Advanced Functional Materials 28: 1702772. Braun W, Jaeger M, Laskin G, Ngabonziza P, Voesch W, Wittlich P, and Mannhart J (2020) In situ thermal preparation of oxide surfaces. APL Materials 8: 071112. Bruinsma R and Zangwill A (1986) Structural transitions in epitaxial overlayers. Journal de Physique 47: 2055. Chambers SA, Kaspar TC, Prakash A, Haugstad G, and Jalan B (2016) Band alignment at epitaxial BaSnO3/SrTiO3 (001) and BaSnO3/LaAlO3(001) heterojunctions. Applied Physics Letters 108(15): 152104. Cheung JT and Madden J (1987) Growth of HgCdTe epilayers with any predesigned compositional profile by laser molecular beam epitaxy. Journal of Vacuum Science and Technology B 5: 705. Cheung JT, Niizawa G, Moyle J, Ong NP, Paine BM, and Vreeland T (1986) HgTe and CdTe epitaxial layers and HgTe-CdTe superlattices grown by laser molecular beam epitaxy. Journal of Vacuum Science and Technology A 4: 2086. Cho AY (1969a) Epitaxy by periodic annealing. Surface Science 17: 494. Cho AY (1970) Morphology of epitaxial growth of GaAs by a molecular beam method: The observation of surface structures. Journal of Applied Physics 41: 2780. Cho AY (1971) Film deposition by molecular-beam techniques. Journal of Vacuum Science and Technology 8: S31. Cho AY and Arthur JR (1975) Molecular beam epitaxy. Progress in Solid State Chemistry 10: 157. Chung YJ, Baldwin KW, West KW, Shayegan M, and Pfeiffer LN (2018) Surface segregation and the Al problem in GaAs quantum wells. Physical Review Materials 2: 034006. Chung YJ, Villegas Rosales KA, Baldwin KW, Madathil PT, West KW, Shayegan M, and Pfeiffer LN (2021) Ultra-high-quality two-dimensional electron systems. Nature Materials 20: 632–637. Cuomo JJ, Pappas DL, Bruley J, Doyle JP, and Saenger KL (1991) Vapor deposition processes for amorphous carbon films with sp3 fractions approaching diamond. Journal of Applied Physics 70: 1706–1711. Dawson LR (1972) Liquid phase epitaxy. Progress in Solid State Chemistry 7: 117. 039. der Merve JHV (1962) Crystal Interfaces. Part II. Finite Overgrowths. Journal of Applied Physics 34: 123. Dijkkamp D, Venkatesan T, Wu XD, Shaheen SA, Jisrawi N, Min-Lee YH, McLean WL, and Croft M (1987) Preparation of Y-Ba-Cu oxide superconductor thin films using pulsed laser evaporation from high Tc bulk material. Applied Physics Letters 51: 619–621.

542

Epitaxy

Eckstein JN, Schlom DG, Hellman ES, von Dessonneck KE, Chen ZJ, Webb C, Turner F, Harris JS, Beasley MR, and Geballe TH (1989) Epitaxial growth of high-temperature superconducting thin films. Journal of Vacuum Science & Technology B: Microelectronics Processing and Phenomena 7: 319–323. Ehrentraut D, Sato H, Miyamoto M, Fukuda T, Nikl M, Maeda K, and Niikura I (2006) Fabrication of homoepitaxial ZnO films by low-temperature liquid-phase epitaxy. Journal of Crystal Growth 287: 367–371. Endo K, Saya S, Misawa S, and Yoshida S (1991) Preparation of yttrium barium copper-oxide superconducting films by metalorganic molecular-beam epitaxy. Thin Solid Films 206: 143. Falson J, Maryenko D, Kozuka Y, Tsukazaki A, and Kawasaki M (2011) Magnesium doping controlled density and mobility of two-dimensional electron gas in MgxZn1−xO/ZnO heterostructures. Applied Physics Express 4: 091101. Falson J, Maryenko D, Friess B, Zhang D, Kozuka Y, Tsukazaki A, Smet JH, and Kawasaki M (2015) Even-denominator fractional quantum Hall physics in ZnO. Nature Physics 11: 347. Falson J, Kozuka M, Uchida Y, Smet JH, Arima T, Tsukazaki A, and Kawasaki M (2016) MgZnO/ZnO heterostructures with electron mobility exceeding 1x 106cm2V −1cm−1. Scientific Reports 6: 26598. Farrow RFC (1983) The stabilization of metastable phases by epitaxy. Journal of Vacuum Science & Technology B: Microelectronics Processing and Phenomena 1: 222–228. Farrow RFC, Robertson DS, Williams GM, Cullis AG, Jones GR, Young IM, and Dennis PNJ (1981) The growth of metastable, heteroepitaxial films of a-Sn by metal beam epitaxy. Journal of Crystal Growth 54: 507. Feng ZC and Liu HD (1983) Generalized formula for curvature radius and layer stresses caused by thermal strain in semiconductor multilayer structures. Journal of Applied Physics 54: 83. Frank FC and Van der Merve JH (1949) One-dimensional dislocations. I. Static theory. Proceedings of the Royal Society of London A198: 205–216. Frankenheim ML (1836) Annalen der Physik 37: 516. Gardner GC, Fallahi S, Watson JD, and Manfra MJ (2016) Modified MBE hardware and techniques and role of gallium purity for attainment of two dimensional electron gas mobility > 35  106cm2/Vs in AlGaAs/GaAs quantum wells grown by MBE. Journal of Crystal Growth 441: 71–77. Gorbenko OY, Samoilenkov SV, Graboy IE, and Kaul AR (2002) Epitaxial stabilization of oxides in thin films. Chemistry of Materials 14: 4026–4043. Grabow MH and Gilmer GH (1988) Thin film growth modes, wetting and cluster nucleation. Surface Science 194: 333. Groenendijk DJ, Manca N, Mattoni G, Kootstra L, Gariglio S, Huang Y, van Heumen E, and Caviglia AD (2016) Epitaxial growth and thermodynamic stability of sriro3/srtio3 heterostructures. Applied Physics Letters 109: 041906. Haeni J, Theis C, and Schlom D (2000) RHEED intensity oscillations for the stoichiometric growth of SrTiO3 thin films by reactive molecular beam epitaxy. Journal of Electroceramics 4: 385–391. Hasegawa S (2012) Reflection high-energy electron diffraction. In: Characterization of Materials, pp. 1–14. John Wiley & Sons, Ltd. Hellman ES and Hartford EH (1990) Adsorption controlled molecular-beam epitaxy of rubidium barium bismuth oxide. Journal of Vacuum Science and Technology B 5: 332. Hsieh CC, Wu KH, Juang JY, Uen TM, Lin J-Y, and Gou YS (2002) Monophasic TiO2 films deposited on SrTiO3(100) by pulsed laser ablation. Journal of Applied Physics 92: 2518–2523. Hwang H, Iwasa Y, Kawasaki M, Keimer B, Nagaosa N, and Tokura Y (2012) Emergent phenomena at oxide interfaces. Nature Materials 11: 103. Ichimiya A and Cohen PI (2004) Reflection High-Energy Electron Diffraction. Cambridge. Ihlefeld JF, Podraza NJ, Liu ZK, Rai RC, Xu X, and Heeg T (2008) Optical band gap of BiFeO3 grown by molecular-beam epitaxy. Applied Physics Letters 92: 142908. Jaeger M, Teker A, Mannhart J, and Braun W (2018) Independence of surface morphology and reconstruction during the thermal preparation of perovskite oxide surfaces. Applied Physics Letters 112: 111601. Jalan B, Engel-Herbert R, Cagnon J, and Stemmer S (2009a) Growth modes in metal-organic molecular beam epitaxy of TiO2 on r-plane sapphire. Journal of Vacuum Science and Technology A 27: 230. Jalan B, Engel-Herbert R, Wright NJ, and Stemmer S (2009b) Growth of high-quality SrTiO3 films using a hybrid molecular beam epitaxy approach. Journal of Vacuum Science and Technology A 27: 461. Jalan B, Moetakef P, and Stemmer S (2009c) Molecular beam epitaxy of SrTiO3 with a growth window. Applied Physics Letters 95: 032906. Jesser WA, van der Merve JH, and Stoop PM (1999) Misfit accommodation by compliant substrates. Journal of Applied Physics 85: 2129. Kasper E (1982) Growth kinetics of Si-molecular beam epitaxy. Applied Physics A: Materials Science & Processing 28: 129. Klemenz C and Scheel HJ (1993) Liquid phase epitaxy of high-Tc superconductors. Journal of Crystal Growth 129: 421–428. Knudsen M (1915) Die maximale Verdampfungsgeschwindigkeit des Quecksilbers. Annalen der Physik (Leipzig) 47: 697. Koenig U, Kibbel H, and Kasper E (1979) Si-MBE: Growth and Sb doping. Journal of Vacuum Science and Technology B 16: 985. Lee JH, Ke X, Misra R, Ihlefeld JF, Xu XS, and Mei ZG (2010) Adsorption-controlled growth of BiMnO3 films by molecularbeam epitaxy. Applied Physics Letters 96: 262905. Lo YH (1991) New approach to grow pseudomorphic structures over the critical thickness. Applied Physics Letters 59: 2311. Marks SD, Lin L, Zuo P, Strohbeen PJ, Jacobs R, Dongxue D, Waldvogel JR, Liu R, Savage DE, Booske JH, Kawasaki JK, Babcock SE, Morgan D, and Evans PG (2021) Solid-phase epitaxial growth of the correlated-electron transparent conducting oxide SrVO3. Physical Review Materials 5: 083402. Maryenko D, McCollam A, Falson J, Kozuka Y, Bruin J, Zeitler U, and Kawasaki M (2018) Composite fermion liquid to Wigner solid transition in the lowest Landau level of zinc oxide. Nature Communications 9: 4356. Maryenko D, Kawamura M, Ernst A, Dugaev VK, Sherman EY, Kriener M, Bahramy MS, Kozuka Y, and Kawasaki M (2021) Interplay of spin-orbit coupling and Coulomb interaction in ZnO-based electron system. Nature Communications 12: 3180. Matsubara Y, Takahashi K, Bahramy M, Kozuka Y, Maryenko D, Falson J, Tsukazaki A, Tokura Y, and Kawasaki M (2016) Observation of the quantum Hall effect in d-doped SrTiO3. Nature Communications 7: 11631. Matthews JW and Blakeslee AE (1974) Defects in epitaxial multilayers: 1. Misfit dislocations. Journal of Crystal Growth 27: 118. Mayer JW, Eriksson L, Picraux ST, and Davies JA (1968) Ion implantation of silicon and germanium at room temperature. Analysis by means of 1.0-MeV helium ion scattering. Canadian Journal of Physics 45: 663. Moyer JA, Eaton C, and Engel-Herbert R (2013) Highly conductive SrVO3 as a bottom electrode for functional perovskite oxides. Advanced Materials 25: 3578. Nagarajan M, Sudhakar S, Lourdudoss S, and Baskar K (2011) Growth of Zn3As2 on GaAs by liquid phase epitaxy and their characterization. Journal of Crystal Growth 314(1): 119–122. Nelson H (1963) RCA Review 24: 503. Norga GJ, Marchiori C, Rossel C, Guiller A, Locquet JP, Siegwart H, Caimi D, Fompeyrine J, Seo JW, and Dieker C (2006) Solid phase epitaxy of SrTiO3 on (Ba, Sr)O/Si(100): The relationship between oxygen stoichiometry and interface stability. Journal of Applied Physics 99: 084102. Ohnishi T, Shibuya K, Yamamoto T, and Lippmaa M (2008) Defects and transport in complex oxide thin films. Journal of Applied Physics 103: 103703. Ota Y (1983) Silicon molecular-beam epitaxy. Thin Solid Films 106: 3. Panish MB (1980) Molecular beam epitaxy of GaAs and InP with gas sources for arsine and phosphine. Journal of the Electrochemical Society 127: 2729. Panish MB, Temkin H, and Sumski S (1985) Gas source MBE of InP and GaxIn1−xPyAs1−y - material properties and heterostructure lasers. Journal of Vacuum Science and Technology B 3: 657. Pashley DW (1956) The study of epitaxy in thin film surface films. Advances in Physics 5: 173–240. People R and Bean JC (1986) Erratum: Calculation of critical layer thickness versus lattice mismatch for GexSi1−x/Si strained layer heterostructures. Applied Physics Letters 49: 229. Pfeiffer L and West KW (2003) The role of MBE in recent quantum Hall effect physics discoveries. Physica E: Low-dimensional Systems and Nanostructures 20(1): 57–64. Pfeiffer L, West KW, Stormer HL, and Baldwin KW (1989) Electron mobility exceeding 107cm2/Vs in a modulation-doped GaAs. Applied Physics Letters 55: 1888–1890.

Epitaxy

543

Royer L (1928) Recherches expérimentales sur l’épitaxie ou orientation mutuelle de cristaux d’espèces différentes. Bulletin de la Société française de Minéralogie 51: 7–159. Rupprecht H, Woodall JM, and Pettit GD (1967) Efficient visible electroluminescence at 300K from Ga1−xAlxAs p-n junctions grown by liquid-phase epitaxy. Applied Physics Letters 11: 81. Schlom DG, Eckstein JN, Hellman ES, Streiffer SK, Harris JS, Beasley MR, Bravman JC, Geballe TH, Webb C, von Dessonneck KE, and Turner F (1988) Molecular beam epitaxy of layered Dy-Ba-Cu-O compounds. Applied Physics Letters 53: 1660–1662. Schlom DG, Chen L-Q, Pan X, Schmehl A, and Zurbuchen MA (2008) A thin film approach to engineering functionality into oxides. Journal of the American Ceramic Society 91: 2429. Smith HM and Turner AF (1965) Vacuum deposited thin films using a ruby laser. Applied Optics 4: 147–148. Son J, Moetakef P, Jalan B, Bierwagen O, Wright NJ, Engel-Herbert R, and Stemmer S (2010) Epitaxial SrTiO3 films with electron mobility exceeding 30000 cm2V−1s−1. Nature Materials 9: 482. Stranski IN and Krastanov L (1938) Zur theorie der orientierten ausscheidung von ionenkristallen aufeinander. Ber. Akademie der Wissenschaften und der Literatur 146: 797. Streit D, Metzger RA, and Allen FG (1984) Doping of silicon in molecular beam epitaxy systems by solid phase epitaxy. Applied Physics Letters 44: 234–236. Stringfellow GB (1982) Epitaxy. Reports on Progress in Physics 45: 469. Sun L, Yuan G, Gao L, Yang J, Chhowalla M, Gharahcheshmeh MH, Gleason KK, Choi YS, Hong BH, and Liu Z (2021) Chemical vapour deposition. Nature Reviews Methods Primers 1: 5. Teal GK, Fisher JR, and Treptow AW (1946) A new bridge photo-cell employing a photo-conductive effect in silicon. Some properties of high purity silicon. Journal of Applied Physics 17: 879–886. Teng D and Lo YH (1993) Dynamic model for pseudomorphic structures grown on compliant substrates: An approach to extend the critical thickness. Applied Physics Letters 62: 43. Theis CD, Yeh J, Schlom DG, Hawley ME, and Brown GW (1998) Adsorption-controlled growth of PbTiO3 by reactive molecular beam epitaxy. Thin Solid Films 325: 107. Tomar R, Varma RM, Kumar N, Sarma DD, Maryenko D, and Chakraverty S (2019) Conducting lavo3/srtio3 interface: Is cationic stoichiometry mandatory? Advanced Materials Interfaces 7: 1900941. Tsukazaki A, Akasaka S, Nakahara K, Ohno Y, Ohno H, Maryenko D, Ohtomo A, and Kawasaki M (2010) Observation of the fractional quantum Hall effect in an oxide. Nature Materials 9: 889. Umansky V, Heiblum M, Levinson Y, Smet J, Nübler J, and Dolev M (2009) MBE growth of ultra-low disorder 2DEG with mobility exceeding 35106 cm2/Vs. Journal of Crystal Growth 311(7): 1658–1661. Volmer M and Weber A (1926) Keimbildung in fibersattigten Gebilden. Zeitschrift fr Physikalische Chemie 119: 277. Wakkernagel F (1825) Kastner’s Archly. f. gesammte Naturlehre 5: 293. Willmott PR and Huber JR (2000) Pulsed laser vaporization and deposition. Reviews of Modern Physics 72: 315–328. Woodall JM (1972) Solution grown Ga1−xAlxAs superlattice structures. Journal of Crystal Growth 12: 32. Yamada A, Tamada H, and Saitoh M (1993) Liquid phase epitaxial growth of LiNbO3 thin film using Li2O-B2O3 flux system. Journal of Crystal Growth 132(1): 48–60. Yamamoto H, Krockenberger Y, and Naito M (2013) Multi-source mbe with high-precision rate control system as a synthesis method sui generis for multi-cation metal oxides. Journal of Crystal Growth 378: 184–188. Zur A and McGill TC (1984) Lattice match: An application to heteroepitaxy. Journal of Applied Physics 55: 378.

Further reading Asahi H and Horikoshi Y (2019) Molecular Beam Epitaxy: Materials and Applications from Electronics and Optoelectronics. Wiley. Eason R (2007) Pulsed Laser Deposition of Thin Films. Wiley. Harsha KSS (2005) Principles of Physical Vapor Deposition of Thin Films. Elsevier. Henini M (2013) Molecular Beam Epitaxy: From Research to Mass Production. Elsevier. Herman MA and Sitter H (1988) Molecular Beam Epitaxy: Fundamental and Current Status. Springer-Verlag. Herman MA, Richter W, and Sitter H (2004) Epitaxy: Physical principles and technical implementation. Ichimiya A and Cohen PI (2004) Reflection High Energy Electron Diffraction. Cambridge. Koster G, Huijben M, and Rijnders G (2015) Epitaxial Growth of Complex Metal Oxides. Elsevier. Lüth H (2015) Solid Surfaces, Interfaces, and Thin Films. Springer-Verlag. Nishinga T (2015) Handbook of Crystal Growth. Thin Film Epitaxy: Materials, Processes, and Technology. Elsevier. Panish MB and Temkin H (1993) Gas Source Molecular Beam Epitaxy. Springer-Verlag. Pohl UW (2013) Epitaxy of Semiconductors. Springer. Tsao JY (1993) Materials Fundamentals of Molecular Beam Epitaxy. Academic Press.

Epitaxy (classical MBE) Christian Heyn, Center for Hybrid Nanostructures (CHyN), University of Hamburg, Hamburg, Germany; © 2024 Elsevier Ltd. All rights reserved.

Introduction Epitaxy in technology Molecular beam epitaxy (MBE) Nucleation and epitaxial growth modes Strain in epitaxy Self-assembly of nanostructures Strain-induced self-assembly Droplet-based self-assembly Vapor–liquid–solid (VLS) growth of nanowires In situ growth monitoring Conclusion References

544 545 545 546 548 549 549 550 550 551 552 552

Abstract This chapter describes general aspects of epitaxy which denotes the growth of crystalline layers with order given by a crystalline substrate. A focus is on molecular beam epitaxy (MBE) representing a powerful method for epitaxy in particular of ultra-clean two-dimensional semiconductor heterostructures, where the precise control on the composition, size, and roughness of the deposited layers enables an engineering of the band gaps and doping profiles. Using self-assembled growth schemes, also one-dimensional nanowires and zero-dimensional quantum dots can be fabricated without the need for lithography. Discussed topics are technological aspects, basics of crystal growth, epitaxy of strained layers, strain-induced and droplet-based self-assembly, and in situ growth monitoring with focus on electron diffraction.

Introduction Epitaxy denotes a kind of crystal growth where new crystalline layers are formed with order given by a crystalline substrate. The term epitaxy reflects this oriented overgrowth by referring the Greek language with epi meaning above and taxis meaning in an ordered manner. Epitaxy can form small crystalline clusters in contact with a substrate or epitaxial layers covering the substrate at a large lateral extent. Historically, first experimental studies on the oriented overgrowth of a crystalline material on a crystalline substrate are reported 1928 by L. Rover (Poppa, 1964). Depending on the involved materials one distinguishes between homoepitaxy where substrate and new layer are composed of the same material and heteroepitaxy where substrate and new layer are made from different materials. Homoepitaxy is technologically important for example, for the fabrication of ultra-pure and low-defect single crystals, whereas heteroepitaxy is much more flexible and allows for example, the creation of so-called heterostructures, which are stacked crystalline layers of different materials. In particular heterostructures made from semiconductor materials with different band gaps are technologically very important and enable the fabrication of layered quantum structures like quantum wells (QWS) and superlattices by the so-called band-gap engineering (Arthur, 2002). A popular material combination is here GaAs/AlGaAs which allows a significant variation of the band gap whereas the lattice constants are almost equal. In addition to semiconductors, also metallic systems (Bauer, 1982) and even organic materials (Lee et al., 2014) can be grown epitaxially. In an idealized picture, the substrate is a perfect single-crystal characterized by the symmetry of the unit cell and the lattice constant asub. For epitaxial growth, the new layer should have the same crystal symmetry and lattice constant alayer. However, epitaxy is possible also for material combinations with slightly different lattice constants, which is described by the lattice mismatch f¼

jalayer − asub j asub

(1)

A lattice mismatch larger than zero induces strain energy in the crystal which can be relaxed by formation of dislocations or by switching to an island growth mode (see Section “Nucleation and epitaxial growth modes”).

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Epitaxy in technology Epitaxy is widely used for instance in semiconductor technology for the fabrication of devices where layers of elementary or compound semiconductors with controlled thickness, band gap, and doping profile are requested. Several approaches to realize epitaxial growth are applied. One major difference between the methods is the state of matter in which the source material for the new layers is supplied. In solid-phase epitaxy (SPE), an amorphous film in contact with a crystalline substrate is recrystallized during a thermal annealing step. Liquid-phase epitaxy (LPE) is a method to grow crystal layers from a molten material (Fig. 1). Here the substrate is often a small seed crystal which defines the crystallographic order of the new layers. For crystallization, the temperature of the melt is reduced to a value below the melting point or the melting point is reduced by adding a further material. For example, liquid gallium droplets in contact with a GaAs substrate can be crystallized to GaAs by adding arsenic in the so-called droplet epitaxy (see Section “Droplet-based self-assembly”). Both techniques, SPE and LPE, are used for the fabrication of semiconductor crystals with a low defect density. However, a fast change of the source materials for the new layers, which is essential for the fabrication of heterostructures, is difficult to realize using SPE or LPE. Here, methods are advantageous where the source material is supplied as a vapor in the gas phase. In physical vapor deposition (PVD), the source materials in solid or liquid phase are not in contact with the substrate. A vapor is generated from the source materials by thermal heating (evaporation) or by irradiation with light (pulsed-laser deposition), electrons (electron-beam PVD), or ions (sputter deposition). The vapor is transported through a vacuum to the substrate and condenses there (Fig. 1). It should be noted that PVD is often used for the coating of devices where an epitaxial growth is not required. A type of PVD that is optimized for the growth of epitaxial layers is the molecular beam epitaxy (MBE, see Section “Molecular beam epitaxy (MBE)”). Another vapor-based technique for epitaxy is the chemical vapor deposition (CVD) where the source materials are supplied as chemical precursors usually from a gas source. The precursor molecules are dissociated on the substrate and the desired material is incorporated into the growing layer (Fig. 1). This is in contrast to MBE where usually elementary source materials are supplied and directly incorporated. In a rough comparison, MBE requires expensive ultra-high vacuum (UHV) equipment, typically works at a rather slow growth speed of about 1 mm/h, and the growth is directed perpendicular to the evaporation sources. CVD uses reaction chambers with less expensive vacuum requirements, the growth speed is typically higher at about 10 mm/h, and the growth can be undirected. The usage of precursor molecules in a CVD process allows a chemically selective growth that can be used for instance for an area selective deposition (ASD) or for deposition of highly conformal thin films by sequential reaction of mono-layered precursor molecules in atomic layer deposition or epitaxy (ALD, ALE). To summarize, MBE is a rather slow and expensive technique, but allows the fabrication of ultra-pure heterostructures with excellent crystal quality and is often used for projects in fundamental research on quantum systems or as a pioneering technology to evaluate new materials or device concepts. CVD systems are usually cheaper with a higher throughput which suggests this method also for industrial device fabrication.

Molecular beam epitaxy (MBE) The foundations of molecular beam epitaxy were established in the late 1960s by J.R. Arthur and A. Y. Cho starting with focus on III/V compound semiconductors (Cho and Arthur, 1975). MBE takes place in an UHV environment in the 10−10 to 10−12 Torr range to minimize the level of unintentional background doping. This requires a thorough preparation of the vacuum chamber, powerful vacuum pumps, and large usually liquid–nitrogen cooled cryo-shrouds inside the chamber. In solid-source MBE, pure elements are evaporated in thermally heated effusion cells, the vapor travels through the vacuum as a directed beam, and finally condenses on the substrate surface. Kinetic gas theory estimates at a pressure of 1  10−6 Torr inside a vapor beam a rate of gas impingement on the substrate of about 1 monolayer (ML) per s, which corresponds for GaAs to a growth speed of 1 mm/h, typical for MBE. On the other side, assuming a very low background pressure of 10−12 Torr inside the vacuum chamber, the rate of unintentional gas impingement is about 10−6 ML/s. In a worst case scenario where all impinging species are incorporated, this would result in a contaminated

Liquid-Phase Epitaxy, LPE

Physical Vapor Deposition, PVD

Chemical Vapor Deposition, CVD Precursor gas

Vapor Liquid Dissociation

Fig. 1 Illustrations of various methods used for epitaxial growth on a crystalline substrate. In liquid-phase epitaxy (LPE) a molten material in contact with a substrate is crystallized. Vapor-phase-based methods take place in a vacuum and the source material is supplied in an elementary form in physical vapor deposition (PVD) or as a precursor gas in chemical vapor deposition (CVD).

546

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crystal. Fortunately, at usual MBE growth temperatures, only a minor part of the species contributing to the background pressure is incorporated into the growing film. Critical species during MBE of III/V-compound semiconductors are often oxygen or carbon which can significantly degrade the optoelectronic properties of the deposited layers. This indicates the total pressure as an inadequate quantity for the characterization of an MBE system. More meaningful are the partial pressures of the relevant species. A further important point is the stoichiometry of the materials during epitaxy of compound semiconductors. As an example, for MBE of GaAs, the group III element Ga is typically evaporated at a temperature of about TGa ’ 900 C and the group V element As at about TAs ’ 200 C. The different evaporation temperatures are given by the respective vapor pressures of the elements. This yields three regimes for the substrate temperature Tsub. For a substrate temperature Tsub < TAs below the minimum temperature required for evaporation, all deposited material will stick on the substrate and the stoichiometry of the deposit must be maintained by adjusting perfectly equal fluxes of Ga and As. Due to technical limitations, equal fluxes are hard to realize and this regime usually results in the formation of nonstoichiometric films with a poor crystalline quality. The regime with Tsub > TGa is also not applicable since here all deposited material will be re-evaporated. In the intermediate regime, TAs < Tsub < TGa, Ga will stick and As should be re-evaporated. However, due to a strong chemical bond between Ga and As, only excessive As without bond to a Ga atom re-evaporates. That means, for an As flux higher than that of Ga, this mechanism can provide perfectly stoichiometric GaAs layers. This example illustrates that epitaxy of stoichiometric compounds is possible, but requires appropriate temperatures and material combinations. Assuming a sufficient group V element flux to maintain stoichiometry, the thickness and composition of a deposited layer are precisely controlled by the sum of the fluxes of the group III elements. The individual material fluxes toward the substrate are adjusted by the temperatures of the respective effusion cells. However, for abrupt interfaces between layers composed of different materials, a temperature change is not fast enough. This issue is solved by so-called shutters in front of each effusion cell, which allow an abrupt switching of the respective flux within typically 0.1 s. Considering a usual growth speed of 1 ML/s, the shutters enable the deposition of layers with thickness below one ML. For instance, closing and re-opening of an Al shutter can generate a GaAs QW embedded in AlGaAs with atomically sharp interfaces. Other important MBE grown structures are for instance quantum cascade lasers utilizing multiple QWs for intersubband transitions, or high-electron-mobility transistors (HEMTs), which is a special type of a field-effect transistor where the conductive channel is spatially separated from the doped region to reduce the charge-carrier scattering. In the last years, the range of materials grown by MBE was expanded from the initial arsenides to other III/V compounds such as phosphides and nitrides, II/VI compounds, elementary semiconductors, oxides, diluted magnetic semiconductors, topological insulators, and organic semiconductors. Due to the high crystal quality of MBE grown material, the possibility to deposit layers of controlled band gap and doping with atomic precision, and its flexibility regarding the materials, MBE has contributed to several Nobel Prizes in Physics, including the fractional quantum Hall effect (1998), semiconductor heterostructures (2000), and solid state lighting (2014). Recent overviews on MBE are given for instance in Orton and Foxon (2015) and Asahi and Horikoshi (2019).

Nucleation and epitaxial growth modes This section describes the basic mechanisms of epitaxial growth using an MBE configuration as example. Usually, MBE growth takes place far away from equilibrium since new material arriving on the growing surfaces covers already deposited material before this moves to surface positions with a minimal energy. That means energetically unfavorable configurations are possible and kinetic models are required for a description of the growth process. Fig. 2 gives an overview about the basic processes during MBE. Growth is driven by adatoms from the vapor beam which arrive with a rate F on a crystalline substrate. The following description assumes a single-compound material, and an atomically flat substrate surface with discrete adsorption sites given by the crystal lattice. The surface coverage y of the new material is characterized in the units of monolayers (MLs), where a coverage of one ML indicates that in average every surface site is occupied by one adatom. Vapor

Arrival Desorption

Exchange Diffusion

Nucleation Attachment

Escape

Fig. 2 Schematic illustration of the basic processes during MBE growth from a vapor beam which can be described by the rate for arrival from the beam flux F, the desorption rate RD, and the surface diffusion rate D. The inset shows the potential landscape with the energy barrier for surface diffusion ES, and the nearest-neighbor binding energy EN.

Epitaxy (classical MBE)

547

After arrival, the adatoms can be incorporated into the growing film or they can be re-evaporated by desorption with a rate RD. The probability for incorporation is described by the sticking coefficient a ¼ (F − RD)/F. At usual process temperatures T, the sticking coefficient a ’ 1 and the adatoms migrate on the substrate surface which is called surface diffusion. Further possible processes like the exchange of new adatoms with atoms from the substrate crystal are neglected in the following: Kinetic approaches describe surface diffusion by jumps of adatoms to neighboring surface sites with a thermally activated rate D ¼ n exp½−ES =ðkB TÞ, where n is a vibrational frequency, ES is the activation energy for surface diffusion, and kB is Boltzmann’s constant. In this picture, a single adatom on the surface will not be incorporated but instead performs a permanent diffusion. A second adatom arrives after a time interval 1/F and now a nucleus can be formed by a collision between the migrating adatoms. The additional lateral binding energy EN between the neighboring adatoms inside the nucleus strongly reduces the surface diffusion rate where now for the activation energy a value ES + nEN must be considered, with the number n of neighboring atoms. For large values of EN, the lateral bonds inside a nucleus cannot be broken which is called irreversible aggregation. Often reversible aggregation is observed, where small values of EN allow an escape of adatoms from a nucleus. This can result in the dissolution of a nucleus and reduce the density N of nuclei on the surface. Usually the size of a nuclei increases with deposition time and is balanced by the rates of adatom attachment and escape. With larger size, the initial nuclei are often called growth islands or clusters. Models of nucleation and island growth often use coupled rate equations to describe the time-dependent average densities of adatoms and islands (Venables et al., 1984; Zinke-Allmang et al., 1992). With some approximations, the maximum island density can be estimated from N∝ðF=DÞp

(2)

where the scaling parameter is p ’ 1/3.5 for 3D islands and p ’ 1/3 for 2D islands. That means, the island density increases with increasing flux F and decreases with increasing temperature and, thus, increasing diffusion rate D. In a simple picture assuming one adatom and one island on the surface, a high D means a long diffusion length and a high probability that the adatom attaches to the island before a second adatom arrives for a nucleation. On the other side, at a low D or high F, the second adatom will arrive before the first atom can attach to the existing island and the probability for nucleation of a new island is high. As an example for the influence of the substrate temperature on the nucleation, Fig. 3 shows the density of Al and Ga droplets on an AlGaAs substrate. Such droplets are relevant for the so-called droplet epitaxy of semiconductor nanostructures (see Section “Droplet-based self-assembly”). Clearly visible is the tunability of the droplet density over several orders of magnitude. Furthermore, the slope of the droplet density versus T depends on the materials, which indicates material-dependent values of ES and EN. The general trend of the experimental data agrees with Eq. (2), but for a detailed analysis, more elaborated models are required. In Fig. 3, the experimental droplet densities data are quantitatively reproduced by a model based on coupled rate equations (Heyn and Feddersen, 2021). In the literature, three possible epitaxial growth modes are assumed (Venables et al., 1984), which are illustrated schematically in Fig. 4. In the layer or Frank-van der Merwe mode, the deposited material starts with the formation of a complete monolayer on the substrate surface. After its completion, a second monolayer is formed and so on. This growth mode is usually observed for material systems where the deposited adatoms are more strongly bound to the substrate than to each other (ES > EN). Since the layer mode provides very smooth two-dimensional (2D) surfaces or heterostructures with abrupt inner interfaces, it is technologically very important for instance for the semiconductor device fabrication. In the layer plus island or Stranski–Krastanov mode, growth starts similar to the layer mode with the formation of monolayers which form the so-called wetting layer (WL). However, after WL formation, subsequent layer growth is energetically unfavorable and the initial 2D growth front switches to the formation of

T [°C]

Density, N [cm-2]

600

400

200

1x1010 1x109 1x108

Droplets on AlGaAs Al droplets

1x107

Ga droplets 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1/T [1000/K] Fig. 3 Density N of Al and Ga droplets deposited by MBE on an AlGaAs substrate for droplet epitaxy as function of the deposition temperature T. The experimental data (symbols) are taken from atomic force microscopy (AFM) images. The lines are calculated using a model based on coupled rate equations with two material-dependent model parameters, the energy barrier for surface diffusion ES, and the nearest-neighbor binding energy EN (Heyn and Feddersen, 2021).

548

Epitaxy (classical MBE) , strain

> Hc1this magnetic flux exists in the form of a hexagonal lattice of quantized vortices. Each vortex can be regarded as a tube of radius of the London magnetic penetration depth l(T ) in which screening currents circulate around a small non-superconducting core of radius x(T ), where the coherence length x(0) ’ ℏvF/2pkBTc at T ¼ 0 quantifies the size of the Cooper pairs, and vF is the Fermi velocity (Tinkham, 1996). Vortices become energetically favorable at H > Hc1 if the Ginzburg–Landau parameter k ¼ l/x exceeds 2−1/2, as it happens in many superconducting compounds, especially such extreme type-II superconductors as Nb3Sn or layered high-Tc cuprates or Fe-based superconductors (FBS) with k ¼ 20 − 100. The magnetic flux produced by screening currents in a vortex equals the flux quantum f0 ¼ 2  10−15 Wb, so the vortex density B/f0 is proportional to the magnetic induction B. Bulk superconductivity is destroyed as the normal cores of vortices overlap at the upper critical field, Hc2(T ) ¼ f0/2pm0x2(T ). In isotropic superconductors such as NbdTi and Nb3Sn, vortex lines are continuous, but in highly anisotropic layered high-Tc compounds, such as Bi2Sr2Ca2Cu3Ox and Bi2Sr2CaCu2Ox, vortex lines break into stacks of weakly coupled “pancake” vortices whose circulating currents are mostly confined within the superconducting CuO2 planes (Blatter et al., 1994). The current density J exerts the Lorentz force f L ¼ f0 ½J  bz per unit length of the vortex line directed along the unit vector bz , so that fL is perpendicular to both J and bz . Vortices can also exist in thin films of type-I superconductors in a perpendicular magnetic field if the film thickness d is much smaller than l (Pearl, 1964; Blatter et al., 1994; Brandt, 1995). Here we mostly focus on bulk Jc(H, T ) in type-II superconductors with large k and transverse sizes much larger than l at magnetic fields well above Hc1. In this case Jc(T, H) is not affected by demagnetizing effects which make Hc1

Fig. 1 A typical Jc(T, H) surface which defines the region J < Jc on the H–T diagram, where a type-II superconductor can carry weakly dissipative transport currents at electric fields E < Ec with Ec depending on a particular application. Intersection of the Jc(T, H) surface with the T-H plane defines the irreversibility field H∗(T ) at which a resistive transition to a flux flow state occurs.

556

Superconductivity: Critical currents

dependent on the sample geometry, and the magnetic induction B ¼ m0H in a superconductor is directly proportional to the applied field H (Campbell and Evetts, 1972). In addition, the surface superconductivity can occur at Hc2 < H < Hc3 for the field parallel to a perfect surface of a single crystal, where Hc3  1.7Hc2 at T  Tc (Tinkham, 1996). Yet for realistic imperfect surfaces in high-Jc superconductors in inclined magnetic fields, the third critical field Hc3 is very close to Hc2 so the surface superconductivity does not play a significant role in current transport controlled by pinning of vortices in the bulk. Superconductors carry bulk weakly-dissipative (due to thermally-activated flux creep) currents caused by a vortex density gradient (or bending of vortices in thin films) described by the Maxwell equation r  B ¼ m0J. This magnetic flux gradient can only be sustained by pinning of vortices by microstructural defects. Flux pinning is determined by local gains in the free energy of curvilinear vortices due to interactions of their normal cores and screening currents with these microstructural imperfections. Furthermore, a macroscopic current density J exerts the Lorentz force FL ¼ J  B per unit volume of the magnetic flux. The critical current density Jc(T, B) is then determined by the balance of the pinning and Lorentz forces Jc(T, B)B ¼ Fp(T, B), where Fp is the volume pinning force produced by the materials defects (Campbell and Evetts, 1972). Ideally, a type-II superconductor can carry any non-dissipative current density J smaller than Jc but if J exceeds Jc, a superconductor switches into a dissipative flux flow state in which vortices are driven by the Lorentz force. In any case, Jc cannot exceed the depairing current density Jd at which the transport current breaks the Cooper pairs: Jc ¼

f0 33=2 pm0 l2 x

(1)

Here Jd is of the order of the current density circulating near the normal vortex, where Jd(T ) ¼ Jd(0)(1 − T/Tc)3/2 at T  Tc. Pinning defects can be classified by their spatial extent and by mechanisms of the elementary pinning forces fp between vortices and defects (Campbell and Evetts, 1972; Blatter et al., 1994; Brandt, 1995). These defects can be either small pinning centers like non-superconducting precipitates and oxygen vacancies in high-Tc superconductors, or defects extended along vortex lines (grain boundaries, dislocations, columnar radiation tracks of nanorods). In turn, elementary pinning interactions can be due to either a short-range interaction of the vortex cores with point defects or a long-range magnetic interaction with extended planar defects. The core pinning results from a gain in the superconducting condensation energy at the core trapped by a defect, the strongest pinning force corresponding to defect sizes  x (see Fig. 2). Magnetic pinning results from the deformation of vortex screening currents near a planar defect on the scale  l, as shown in Fig. 3. The magnetic pinning force f(x) ¼ f20/4pm0l2x can be rather strong if the distance x between the vortex and the defect is smaller than the interaction length l ¼ f0/2pm0l2Jb. Here, l < l is the distance from the vortex core at which the circulating current density J(x) equals the maximum current density Jb which can pass through the defect. If the distance between the normal cores of bulk vortices and the defect is smaller than l ¼ f0/2pm0l2Jb, vortices are attracted strongly to the defect. Vortices trapped by the defect may have no normal cores which turn into extended regions of length l along the defect in which the superconducting gap is not suppressed by circulating vortex currents. These elongated vortices on the defect have a highly anisotropic pinning force, which is maximum for J parallel to the defect and minimum for J perpendicular to the defect. Identification of the most effective pinning defects in a particular material is a difficult task. Practical limits to flux pinning in high-field superconductors have been inferred from extensive studies of Nb–47%Ti, in which strong pinning by a dense  20–25 vol% lamellar structure of metallic a-Ti ribbons  1 nm (0.2x) thick can produce Jc that approaches 5–10% of Jd at zero field and 4.2 K (Larbalestier et al., 2001). The mechanism of flux pinning is also known for Nb3Sn, in which Jc is determined by the magnetic interaction of vortices with Sn-deficient grain boundaries (Campbell and Evetts, 1972; Durrell et al., 2011). These materials have zero-field Jc values, which can exceed 1010–1011 A m−2 at 4.2 K. In high-Tc cuprates or Fe-based superconductors (FBS) the coherence length x ’ 1.5 − 2 nm is so small that practically any common atomic-size defects such as vacancies or dislocations can pin vortices. The critical temperature of high-Tc superconductors is also sensitive to the carrier density, which, in turn, is determined by local non-stoichiometry, so that even a weak hole-depletion at crystalline defects may locally drive a high-Tc cuprate into an antiferromagnetic insulator. The proximity of the superconducting state in high-Tc cuprates or FBS to antiferromagnetic state, the d-wave pairing symmetry, the short in-plane coherence length x(0) ’ 1.5 − 2 nm and the large Thomas-Fermi screening length lD  x(0), all combine to weaken superconductivity near crystal defects, such as impurities, dislocations, and grain boundaries, thus making them effective pinning centers (Gurevich, 2014). Typical parameters of some high-Jc superconductors with strong pinning are given in Table 1. These parameters can vary significantly, depending on concentrations of impurities and the materials treatments (Larbalestier et al., 2001; Hosono et al., 2018; Yao and Ma, 2021).

Fig. 2 Pinning of vortex cores (vertical lines) by uncorrelated defects (red dots) in a film of thickness d in a magnetic field H. The cylindrical region around the vortex cores depicts the region of circulating screening currents.

Superconductivity: Critical currents

557

Fig. 3 Deformation of vortex current streamlines near a planar defect shown by a horizontal dashed line.

Table 1

Characteristic parameters of some high-Jc type-II superconductors.

Material

Tc, K

l(0), nm

x(0), nm

Hc2, T

H
, T

Jd, A m−2

Jc, A m−2

Nb–Ti Nb3Sn YBa2Cu3O7-x Bi2Sr2Ca2Cu3Ox Ba0.6K0.4Fe2As2

9 18 91 108 38

240 65 150 (H||c) 150 (H||c) 200 (H||c)

4 3 1.7 (ab plane) 1.5 (ab plane) 2 (ab plane)

12 (4.2 K) 27 (4.2 K) >100H||c (4.2 K) >100H||c (4.2 K) >100H||c (4.2 K)

10.5 (4.2 K) 24 (4.2 K) 7–11 (77 K) 0.2 (77 K) >100H||c (4.2 K)

1.6  1011 (4.2 K) 5.2  1012 (4.2 K) 1.5  1011 (77 K) 3  1012 (4.2 K) 1012 (4.2 K)

4  109 (5 T, 4.2 K) 1010 (0 T, 4.2 K) 4  1010 (0 T, 77 K) 1010 (0 T, 4.2 K) 1010 (0 T, 4.2 K)

Single-vortex and collective pinning by point defects Calculation of Jc requires summation of elementary pinning forces acting on flexible, strongly interacting vortex lines (Campbell and Evetts, 1972; Blatter et al., 1994; Brandt, 1995). The basic physics of the pinning force summation can be understood by evaluating Jc for a rigid straight vortex interacting with randomly distributed small non-superconducting precipitates in a film of thickness d (Fig. 2). If d is much greater than the mean spacing li between the precipitates, a vortex interacts with N  r2p dnp pinning centers, adjusting its position so that the net force vanishes. Here rp is the pinning interaction radius, and np ¼ l−3 i is the density of pins. The transport current displaces the vortex from a local minimum in the pinning potential, so that elementary pinning forces do not compensate each other. The balance of the total pinning force from uncorrelated defects in the volume dr2p against the Lorentz force, fpN1/2 ’ df0Jc yields rffiffiffi 1 g Jc ’ (2) f0 d where g ¼ f2p r2p np is the pinning parameter. As follows from Eq. (2), the critical current density Jc(d) decreases as the film thickness increases. If d ! 1, Eq. (2) gives Jc ¼ 0 because the net pinning force of a rigid infinite straight vortex in a random pinning potential averages to zero. Evaluation of bulk Jc must take into account bending of vortices near pinning centers. The effect of vortex elasticity can be understood using the following approach of a theory of weak collective pinning (Blatter et al., 1994; Brandt, 1995). Define a correlation length Lc along the vortex line by the relation ⟨u(Lc)u(0)⟩ ¼ r2p , where ⟨u(z)u(0)⟩ is a correlation function of pinning-induced transverse vortex displacements. The relative displacement of the ends of a vortex segment of length Lc exceeds the pinning interaction radius rp, so a vortex may be regarded as an array of straight segments of length Lc pinned independently. Therefore, Jc can be evaluated from Eq. (2) with d replaced by Lc. In turn, Lc is evaluated by equating a characteristic pinning energy f0Jcrp and an elastic bending energy e(rp/Lc)2, where e ¼ f20/4pm0l2 is the vortex line energy scale. The condition f0Jcrp ’ e(rp/Lc)2 along with Eq. (2) at d ’ Lc yield:

558

Superconductivity: Critical currents 2=3

2

Jc ’

g3

1 1 3 3

f0 e r p

, Lc ’

e2=3 r p g1=3

(3)

Due to the collective interaction of a vortex with many pins, Jc increases as the line energy e decreases because a softer vortex bends more easily to accommodate the pinning centers. Meanwhile, the correlation length Lc increases as the pinning parameter g decreases, resulting in a “dimensional crossover” in films, from the two-dimensional pinning of rigid straight vortices at d < Lc to the three-dimensional pinning of flexible vortices for d > Lc (Blatter et al., 1994; Brandt, 1995). A theory of weak collective pinning in strong fields H  Hc1 involves two characteristic spatial scales: the correlation length along the vortex lines Lc(T, B, g) and the correlation length Rc(T, B, g) perpendicular to the vortex lines. These lengths define the correlation volume Vc ¼ R2c Lc - a mesoscopic region in which pinned vortices approximately maintain the positional hexagonal short-range order of the ideal flux line lattice. The positional long-range order of vortices is destroyed by pinning on scales larger than the correlation lengths Lc and Rc. The critical current density is then defined by the balance of the Lorentz force applied to the correlation volume, BJcVc against the pinning force from spatially uncorrelated defects, fpN1/2 c , where Nc ¼ npVc. Hence, rffiffiffiffiffi fp np Jc  (4) BRc Lc where both Lc(T, B, g) and Rc(T, B, g) depend not only on T and B, but also on the pinning parameter g. The lengths Lc(T, B, g) and Rc(T, B, g) are determined from the conditions ⟨u(Rc, 0)u(0, 0)⟩ ¼ r2p and ⟨u(0, Lc)u(0, 0)⟩ ¼ r2p , where the correlation function of vortex displacements ⟨u(r)u(0)⟩ is calculated using an elasticity theory for the vortex lattice. Depending on the relations between the pinning correlation lengths, the London penetration depth and the vortex spacing a ¼ (f0/B)1/2, the collective pinning theory predicts a variety of behaviors of Lc(T, B, g) and Rc(T, B, g) which manifest themselves in different temperature and field dependencies of Jc(T, B). For layered high-Tc superconductors, the pinning theory takes into account anisotropic elastic response of the vortex lattice. A crossover from a weak collective pinning to a strong single-vortex pinning occurs at lower B for which Rc(T, B, g) becomes smaller than the vortex spacing. Unlike the collective pinning of vortices by many uncorrelated weak pins, interaction of a vortex with a strong pinning center can occur in a hysteretic manner because the elastic vortex line bends toward the pin and jumps into it once the spacing between the vortex and the pin gets smaller than a trapping distance which depends on the pin strength and the flux density B/f0. The transition from the weak collective pinning to the hysteretic strong core pinning occurs if kL ¼ f p =xC > 1. Here kLis the Labusch parameter which quantifies the strength of a single pin in terms of the ratio of the pinning force and bending elasticity of the vortex C (Blatter et al., 2004). At low fields B  Bc2, the vortex elasticity theory gives C ’ 4ð2pB=f0 Þ1=2 e and the criterion of strong pinning kL > 1 takes the form: pffiffiffiffiffiffi f p f0 pffiffiffiffiffiffiffiffiffi > 1 kL  (5) 4xe 2pB Eq. (5) shows that the strong hysteretic pinning tends to occur in the case of large pinning force, soft vortex line and low magnetic field. Increasing the density of pinning centers can push Jc toward the depairing limit, as it can indeed occur in optimized NbTi alloys or some of high-Tc cuprates. For instance, oxide nanoparticles of Y2O3 or BaZrO3 incorporated into YBa2Cu3O7-x films can be tuned by varying the shape, size and distribution of oblate or prolate nanoprecipitates, self-assembled chains of nanoparticles or nanorods, typically spaced by 4–10 nm and being 2–4 nm in diameter to provide the strongest core pining of vortices (Maiorov et al., 2009; Haugan et al., 2020). The artificial pinning centers do enhance Jc at low field B < 1 tesla where Jc(77K, 0)  (5 − 8)  1010 A m-2 in YBa2Cu3O7-x films at self-field can approach 10-20% of Jd. Such “designer” pinning nanostructures also increase Jc at intermediate fields most relevant to magnet applications. For instance, the high values of Jc(0, 77K) ’ 2.7  1010 A m−2 and of Jc(5T, 77K) ’ 109 A m−2 were observed on YBa2Cu3O7-x films with Y2O3 nanoparticles. Technological advances have resulted in high Jc(0, 77K) ≳ 4  1010 A m−2 in FBS as well (Hosono et al., 2018; Yao and Ma, 2021). Shown in Fig. 4 are two examples of strong pinning due to a dense array of spherical dielectric pins (a) and columnar pins with suppressed Tc (b). The very high Jc values in the cuprates and FBS mentioned above imply that pinning centers subdivide vortices into segments nearly as short as the vortex core diameter 23/2x. To illustrate how far could Jc be increased, let us estimate Jc for strong core pinning by insulating nanoprecipitates of radius r0  x which chop vortices into segments of length l  l, as shown in Fig. 4A. If the ends of each vortex segments are fixed by the strong pins, Jc is determined by the pin breaking mechanism due to cutting and reconnection of vortex semi-loops in the limit of kL  1 (Brandt, 1995). In this case vortex segments bow out under the action of flowing current, then reconnect and escape at the critical current density (Gurevich, 2014):    4pr 30 f0 Gl Jc ’ (6) ln 1 − 2x 2pm0 l2 Gl 3xc l3 Eq. (6) describes the limit of Jc in a uniaxial superconductor, where l and x are the field penetration depth and the coherence length in the isotropic ab plane, and the anisotropy parameter G ¼ (mc/m)1/2 > 1 depends on the ratio of electron mass mc along the c-aixs to the mass m in the ab plane (typical values of G in YBa2Cu3O7-x and FBS range from 2 to 7). The factors in front of the brackets in Eq. (6) describe Jc due to pin breaking of a flexible vortex segment of length l. The factor in the brackets of Eq. (6) accounts for the reduction of the current-carrying cross section by randomly distributed dielectric pins, where the percolation threshold xc varies

Superconductivity: Critical currents

559

Fig. 4 Configurations of pinned curvilinear vortices (the cores of vortices of diameter ’ 2x are shown by red) for: (A) strong pinning by dielectric nano precipitates, where the blue region of width ’l in shows the spatial extend of superconducting currents (depicted by black arrows) circulating around the vortex cores. (B) pinning of fluctuating vortices by columnar defects with suppressed Tc.

from 1/2 in the two-dimensional limit of layered materials G  1 to xc  2/3 in the isotropic 3D limit (G ¼ 1). Interplay of pinning and current blocking by pins results in a maximum Jc(l) at lm  (16p/3xc)1/3r0 (Gurevich, 2014). This yields the optimal volume fraction of nanoprecipitates, xm  9 − 12%and the maximum Jc  (0.2 − 0.3)Jd. These numbers are close to the maximum values which have been achieved in NbTi (Larbalestier et al., 2001) or YBa2Cu3O7-x with optimized pinning nanostructure (Haugan et al., 2020; Stangl et al., 2021). The maximum in Jc(x) at an optimum volume fraction of pins has also been obtained in numerical simulations of the time-dependent Ginzburg-Landau equations for vortices pinned by metallic inclusions or regions with reduced Tc (Kwok et al., 2016).

Pinning and current blocking by extended defects Pinning by extended defects, such as grain boundaries or columnar radiation defects shown in Figs. 3 and 4B, respectively differs from pinning by uncorrelated disorder, because a single extended defect parallel to B can pin a long vortex segment greater than the correlation length Lc ¼ x(Jd/Jc)1/2 of the collective pinning theory. Pinning by extended defects can be rather effective and result in high Jc values if the vortex is aligned with the defect. Indeed, a single vortex trapped by a cylindrical cavity of radius r0  x, has the pinning energy Up of the order of the condensation energy of the vortex core f20/4pm0l2, yielding a very high Jc ’ Up/xf0 of the order of the depairing current density. This model has been used to describe experiments on irradiation of high-Tc superconductors with 1–10 GeV heavy ions (Pb, Au, Xe, etc.), which can produce 10–100 mm columnar tracks of about 5–10 nm in diameter comparable to x (see Fig. 4B) The areal density of columnar defects n□ is customary expressed in terms of the matching field Bm ¼ f0n□ which typically ranges from 0.1 to 10 T for different irradiation conditions. For B < Bm, each vortex is trapped by a columnar defect, resulting in a significant enhancement of Jc in high-Tc superconducting films, as has been shown in many experiments. Columnar defects are most effective for vortices aligned with the defects, but Jc decreases strongly if the magnetic field is misaligned, so that Jc becomes very anisotropic with respect to the direction of B. Pinning by extended planar defects such as grain boundaries (GBs) in Nb3Sn, or by thin a-Ti metallic ribbons in NbdTi can also result in high Jc  1010 − 1011 A m−2 at 4.2 K in self field (see Table 1). Chemical nonstoichiometry, lattice disorder, strain, and charging effects associated with GBs cause local suppression of superconductivity in the cuprates and FBS. This can give rise to weakly linked GBs and magnetic granularity especially pronounced in high-Tc cuprates in which the critical current density through the grain boundaries Jb decreases exponentially as a function of the misorientation angle y between the neighboring grains if y > y0  4 − 5∘ (Hilgenkamp and Mannhart, 2002). As a result, Jb through GBs or thin a-Ti ribbons in NbdTi, is smaller than Jd in the bulk. It is the reduction of Jb as compared to Jd, which causes a strong magnetic and core interaction between a vortex and a planar defect, particularly, the radical change in the structure of the vortex core as it moves closer to the defect shown in Fig. 3. This results in two different types of vortices in polycrystals: vortices in the grains pinned by a long-range magnetic interaction with GBs and elongated vortices on GBs. Here Jc can be evaluated by balancing the total Lorentz force Fp  BJcD3 in the grain of size  D

560

Superconductivity: Critical currents

against the total pinning force proportional to the surface area of the grain  D2. This yields a descending dependence of Jc ∝ D−1 on the grain size, as has been observed in Nb3Sn with 1mm ≲ D ≲ 10 mm (Durrell et al., 2011). Grain boundaries can play a dual role in which they both pin vortices and block macroscopic current flow. In this case current loops can form in the grains if the critical current density due to pinning of vortices in the grains, is higher than the current density Jb, which can pass through GBs. This effect of magnetic granularity can limit global critical current density of superconducting polycrystals. In turn, it indicates that there is an optimum value of Jb, which maximizes Jc, because Jc vanishes in the two limiting cases of Jb ¼ Jd (no pinning, since GBs do not disturb currents circulating around vortices) and Jb ¼ 0 (current cannot pass through grain boundaries). Calculation of the optimum Jc, in addition to the summation of vortex pinning interactions, requires taking into account the geometry of the grains and microscopic mechanisms of current transport through the GB. Current transport through GBs is particularly important for high-Tc polycrystals, for which the weak link behavior of high-angle GBs is one of the most serious current-limiting mechanisms. This behavior is due to rapid decrease of Jb(y) ¼ J0 exp (−y/y0) with the misorientation angle y and the resulting increase of the vortex core size, from x to l(y) ¼ xJd/Jb(y) at l(y) ≲ l to the Josephson core length l(y) ¼ [xlJd/Jb(y)]1/2 for l(y) ≳ l. The extended core of the GB vortices leads to their weaker pinning force parallel to the boundary against the Lorentz force of the transport current flowing perpendicular to the boundary. As a result, the global Jc of polycrystals may be limited by depinning of a small fraction of vortices, which can move more easily along a network of GBs. This percolative motion of vortices along GBs is impeded by structural defects along GBs (such as facets, nanoprecipitates) and by magnetic pinning shear stress as the GB vortices move past more strongly pinned vortices in the grain (see Fig. 3). The significant reduction of Jb of GBs in high-Tc superconductors is due to the sensitivity of their critical temperature to a hole depletion, which can drive the high-Tc superconducting state into an antiferromagnetic insulator. The local hole depletion near GBs can be due to local oxygen nonstoichiometry, charging and strain effects produced by the crystalline dislocations which form the GB, or more macroscopic strain fields due to faceting (Hilgenkamp and Mannhart, 2002). The values of Jb can be improved by oxygen overdoping or local doping with appropriate impurities, such as Ca in YBa2Cu3O7 or Sn in Nb3Sn. The current-carrying capability of YBa2Cu3O7 or FSB polycrystals can also be improved by depositing the superconducting films onto “biaxially textured” metallic substrates to eliminate the weakest high-angle GBs.

Global critical currents of superconductors Multiple pinning mechanisms manifest themselves in different dependencies of the critical current density on T and B, which is often described by a scaling relation (Campbell and Evetts, 1972): Jc ðhÞ ¼ J c0 ðT Þh −p ð1 −hÞq

(7)

∗

where h(T ) ¼ H/H (T ) is a reduced field, and Jc0(T ), p and q are the material-dependent parameters. For instance, Eq. (7) with p ¼ 0 and q ¼ 1 describes Jc(B) in optimized Nb–Ti alloys, while Jc(B) in Nb3Sn is usually described by Eq. (7) with p  1/2 and q  2, consistent with pinning of vortices by grain boundaries. In high-Tc superconductors, Jc(T, H) at elevated temperatures T > 20 − 30 K often exhibits an exponential field dependence Jc(H) ¼ Jc0 exp (−H/H0), where the field H0(T ) can be much smaller than Hc2(T ), especially in layered Bi2Sr2CaCu2Ox. The irreversibility field H∗(T ) in layered high-Tc superconductors can be much smaller than Hc2(T ), whereas in low-Tc superconductors, such as NbdTi and Nb3Sn, or “intermediate-Tc” compound MgB2 with Tc ¼ 40 K and some of the FBS, H∗(T )  (0.8 − 0.9)Hc2(T ) is not that different from Hc2(T ), as shown in Fig. 5 (see also Table 1). As a result, layered high-Tc cuprates Bi2Sr2CaCu2Ox and Bi2Sr2Ca2Cu3Ox, despite their very high Hc2, can carry weakly dissipative currents at the liquid nitrogen temperature of 77 K only in a small part of the H–T diagram. This restriction is due to strong magnetic “flux creep” caused by thermally-activated hopping of flexible vortices between neighboring pinning positions at the current density J below Jc. Vortex creep velocity v is parallel to the driving Lorentz force [J  B], causing the electric field E ¼ [v  B] directed along J (in isotropic materials). The E–J characteristic of a superconductor at J < Jc has the form (Blatter et al., 1994):   U ðJ, T, BÞ J E ¼ Ec exp − , (8) J kB T where the Boltzmann factor exp(−U/kBT ) accounts for a fraction of depinned thermally-activated vortices resulting in the global flux motion. The flux creep activation energy U(J, T, B) depends on the current density, which makes the E-J curve highly nonlinear at J < Jc. It is convenient to define the activation energy U(J) in such a way that it vanishes at J ¼ Jc, using one of the conventional approximations: (1) U ¼ (1 − J/Jc)U0, (2) U ¼ U0 ln (J/Jc), (3) U ¼ U0[(Jc/J)m − 1], where U0(T, B) is an energy scale for thermally activated hopping of vortices. These models give an exponential increase of E with J at J  Jc, but different behaviors of E(J) at J  Jc. Low-Tc superconductors typically have U0 ’ 500 − 1000 K, whereas high-Tc superconductors can have considerably smaller U0 ’ 100 − 400 K, since mobile pancake vortices on neighboring ab planes are weakly coupled due to the layered structures of these materials. Because of weak dissipation at J < Jc, the critical current depends on the electric field Ec at which Jc is measured. Generally, Jc is defined as a crossover current density at which a resistive transition from an exponentially small E(J) to a linear flux flow E–J curve occurs as shown in Fig. 6 (a conventional practical criterion of Ec ¼ 1 mV cm−1 is often used). The relation between Jc and Jc0 measured at different electric fields Ec and Ec0 can be obtained from Eq. (8) by linearizing the activation energy U(J) ¼ U0(1 − J/Jc) near Jc:

Superconductivity: Critical currents (A)

(B)

1

HTS

LTS

H

*

Hc2

H/H c2

H/H c2

1

H

H

*

c2

J =0

J >0 c

0

561

c

Jc > 0 T/T

1

c

0

T/Tc

1

Fig. 5 (A) Magnetic phase diagram of low-Tc superconductors, for which H∗(T )  Hc2(T ). (B) Magnetic phase diagram of high-Tc superconductors which cannot carry critical currents in a wide field range H∗(T ) < H < Hc2(T ). Both cases (A) and (B) correspond to extreme type-II superconductors (k  1) for which the lower critical field Hc1  Hc2 is not shown.

E

B

FF E = Ec

Jc

J

Fig. 6 Broadening of the E–J characteristics as B increases. Here the green curve corresponds to B close to but below the irreversibility field B
, while the FF line shows the Ohmic flux flow E–J characteristic at B > B
. The dashed line shows the electric field criterion Ec at which Jc is defined.

J 0c ¼

 1+

 kB T E0 ln c J c Ec U0

(9)

In low-Tc superconductors with kBTc/U0  10−2, the critical current density Jc(Ec) varies only by a few percent in a practically accessible electric field window of 10 nV m−1 < E < 100 mV m−1. By contrast, high-Tc superconductors with kBTc/U0  10−1 have a broader resistive transition at J  Jc and a larger difference between Jc and Jc0 . The ratio kBTc/U0 also defines the rate of slow time decay of currents in superconductors due to thermally-activated flux creep, J(t) ¼ Jc[1 − (kBT/U0) ln (t/t0)]. The condition U0(H∗, T )  kBT determines the irreversibility field H∗(T ) at which a nonlinear E-J curve turns into a nearly linear E(J), as shown in Fig. 6. In layered high-Tc superconductors, such as Bi2Sr2Ca2Cu3Ox, the activation energy of pancake vortices is only a few times greater than kBTc. In this case of H∗(T )  Hc2(T ) at high temperatures pinning becomes ineffective in a significant part of the H–T diagram (Fig. 5B). In low-Tc superconductors, the dissipative region H∗ < H < Hc2 shown in Fig. 5A is rather narrow, as discussed above (see also Table 1). Thermal fluctuations of vortices give rise to the broadening of the resistive transition of a superconductor in a magnetic field and the region of Jc ¼ 0 in Fig. 5. Such thermal depinning of vortices follows from the field and temperature dependencies of the

562

Superconductivity: Critical currents

activation energy U0(T, B) ¼ U00(H0/H)z(1 − T/Tc)s at T  Tc and J ! 0, where the magnitudes of U00 and H0 and the exponents z and s depend on particular pinning mechanisms (Blatter et al., 1994). For weak broadening, the width of the resistive transition DT can be evaluated from the condition U0(H, DT )  kBTc, which yields DT ’ Tc



kB T c U 00

1s 

H H0

zs

(10)

Hence, the width of the resistive transition increases with the magnetic field. For instance, if z ¼ 1 and s ¼ 3/2, Eq. (10) gives DT ∝ H2/3. Thermal wandering of vortices also results in melting of the vortex lattice at H < Hm(T ) < Hc2(T ). The melting field Hm ’ H∗ is controlled by the dispersive line tension of the vortex e(k) for short wavelength bending distortions with the wave number k  l−1 (Blatter et al., 1994; Brandt, 1995): ek ’

f20 e 1 pℏ2 ns ln , e ¼ ¼ 4m kxc G2 4pm0 l2

(11)

where m is the effective electron mass in the ab plane, xc is the coherence length along the c-axis, ns(T ) is the superfluid density, and G ¼ lc/l is the anisotropy parameter, and lc is the magnetic penetration depth along the ab planes. As follows from Eq. (11), high anisotropy and low superfluid density ns can strongly reduce the bending rigidity of a vortex. For instance, ek ’ 104 K nm-1 in Nb with l(0) ’ 40 nm, G ¼ 1, while in YBa2Cu3O7-x with l(77 K) ’ 400 nm, G  5, we have ek ’ 5 K nm-1. Most FBS have ek ’ (200 − 500)(1 − T2/T2c ) K nm-1, while the layered Bi-based cuprates have much lower ek < 1 K nm-1 at 77 K. As a result, vortices in Nb form the rigid Abrikosov lattice, whereas in cuprates at 77 K thermal fluctuations can displace vortex segments by distances larger than the pin spacing thus smearing out the pinning potential and reducing Jc(T, H). For instance, the vortex segments of length  kBT/e  10 − 100 nm shown in Fig. 4A can hop and reconnect at neighboring pins, causing thermally-activated flux creep at J < Jc, no matter how strong the elementary pinning forces may be. The flux creep in arrays of columnar pins occurs via proliferation of vortex kinks between neighboring pins, as shown in Fig. 4B. In this case the flux creep rate can be reduced by splaying the columnar defects, which increases the vortex activation energy as compared to the parallel columnar pins (Maiorov et al., 2009). In addition to the multiscale pinning mechanisms, a global critical current density Jc ¼ Ic/Aeff of a conductor with a critical current Ic can be limited by macroscopic defects such as large second phase precipitates or microcracks which reduce an effective current-carrying cross section Aeff. Here, Aeff is determined by current percolation through arrays of large second-phase precipitates, colonies of high-angle grain boundaries, microcracks, and other inhomogeneities much greater than the spacing between vortices. This results in nonuniform current distributions in superconductors, which have been revealed by magneto-optical imaging or micro-Hall probes (Larbalestier et al., 2001). Macroscopic current flow in superconductors can be described by a nonlinear Maxwell equation r (J(r ’)) ¼ 0 for the electric potential ’, where E(J) is determined by Eq. (8). The strong nonlinearity of E(J) at J < Jc results in extended electric field hot spots near defects on scales much greater than the defect size b. Shown in Fig. 7 is an example of the spatial distribution of electric field around a microcrack of length b calculated for E ¼ Ec(J/Jc)n. In this case the electric field hot pffiffiffi spot has the length L?  nb perpendicular to current flow and Lk  b n parallel to current flow. For typical of low-Tc superconductors values of n ’ 30 − 50, planar defects blocking only a few percent of the sample cross section can cause noticeable voltage drops on the conductor and reduce its current-carrying capability (Friesen and Gurevich, 2001).

Fig. 7 Spatial distribution of the electric field magnitude in a hot spot near a planar cut of length b ¼ 0.05w in a film of width w, calculated for the power-law E–J characteristic E ¼ Ec(J/Jc)n with n ¼ 30. Here, E0 is the uniform field far away from the defect. Shown below the surface of E(x, y) are the corresponding current streamlines in the film, where the red lines depict the boundaries of a flux jet emitted from the defect perpendicular to current flow.

Superconductivity: Critical currents

563

Fig. 8 A coated conductor in which the YBa2Cu3O7-x film is grown on a textured Ni alloyed substrate with a complex buffer layer structure, which enables replication of the low-angle grain structure of the substrate in YBa2Cu3O7-x and protects it from chemical contamination. The stabilizing layers of Ag and Cu on top of the 1–2 mm thick YBa2Cu3O7-x film provide protection of the conductor against thermal runaway. The superconducting current-carrying film takes only a few % of the conductor cross-section.

Strong pinning of vortices and high irreversibility field are not yet sufficient for applications which require long polycrystalline wires. One of the main issues for the cuprates (and to a lesser extent for FBS) is that grain boundaries between misoriented crystallites impede current flow. This is because the GB critical current density Jb ¼ J0 exp (−y/y0) drops exponentially with the misorientation angle y, as it was mentioned above. For the cuprates, y0  3 − 5o so the spread of misorientation angles Dy  40o in polycrystals can reduce Jb by 2–3 orders of magnitude. FBS also have weak-linked GBs but the larger values of y0  7 − 9o in FBS make them less prone to the electromagnetic granularity, (Durrell et al., 2011; Hosono et al., 2018). Discovered in 1988, the current-limiting GBs in YBa2Cu3O7-x bicrystals were immediately recognized as a serious obstacle for applications because, instead of flowing along the wire, current breaks into disconnected loops circulating in the grains. This problem has been addressed by the coated conductor technology in which a fraction of high-angle GBs with y > 5 − 7o is reduced by growing YBa2Cu3O7-x films on textured metallic substrates. Fig. 8 shows an example of the coated conductor, which has made the idea of YBa2Cu3O7-x “single crystal by the mile” a reality available for magnet dc applications. FBS coated conductors have also been developed (Hosono et al., 2018; Yao and Ma, 2021). While the coated conductors can carry high critical currents at 77 K, they utilize only a few % of the current-carrying cross-section, so Jc of the YBa2Cu3O7-x film must be pushed to the limit by incorporating dense arrays of dielectric nanoprecipitates (Maiorov et al., 2009; Haugan et al., 2020). However, the composite tapes shown in Fig. 8 have very high hysteretic remagnetization losses in alternating magnetic fields (Campbell and Evetts, 1972). For this reason, ac applications of superconductors in motors and generators require round composite wires in which twisted superconducting filaments are embedded in a highly conductive Cu or Ag normal matrix. This technology has been developed for multifilamentary NbdTi and Nb3Sn round wires (Larbalestier et al., 2001), as well as for multifilamentary wires of the biaxially-textured Bi2Sr2CaCu2Ox in Ag matrix, and FBS wires (Yao and Ma, 2021).

Conclusion Critical currents in superconductors are determined by multiscale pinning of strongly interacting curvilinear vortices by materials defects. Here non-superconducting defects can play a dual role by both pinning the vortices and blocking current flow, as it happens, for example, in Nb3Sn in which pinning is caused by grain boundaries. As the volume fraction x of pinning centers increases, Jc(x) of low-Tc superconductors can be pushed to a fundamental limit at an optimum value xc determined by interplay of pinning and current blocking effects. The field range in which a superconductor can carry weakly-dissipative critical currents can be widened significantly by increasing Hc2 by nonmagnetic impurities. The physics of critical currents in anisotropic high-Tc cuprates and FBS with small Fermi energies, low superfluid densities and competing anti-ferromagnetic orders is further complicated by strong thermal fluctuations of vortices at high temperatures which can drastically reduce Jc and the irreversibility field. This manifests itself in different mechanisms of Jc(T, H) at low and high temperatures. At T  Tc the effect of vortex fluctuations is weak so Jc is determined by flux pinning in the same way as in

564

Superconductivity: Critical currents

low-Tc superconductors and Jc in the cuprates can be increased greatly by incorporating strong pinning structure of oxide nanoprecipitates. However, as T increases, the effect of vortex fluctuations in layered high-Tc superconductors becomes dominant and the field region Hc1 < H < H∗(T ), where they can carry weakly dissipative currents shrinks much faster than the decrease of Hc2(T ) with T. It turned out that the designer pinning nanostructures that increase Jc by several orders of magnitude at low temperatures in YBa2Cu3O7-x are less effective in increasing H∗(T ) at 77 K, although a combination of random nanoprecipitates and splayed pinning nanorods can reduce the effect of thermally-activated flux creep (Maiorov et al., 2009; Haugan et al., 2020). Understanding the ways by which pinning can be optimized to increase the irreversibility field in anisotropic high-Tc superconductors at 77 K remains an important outstanding problem.

References Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, and Vinokur VM (1994) Vortices in high-temperature superconductors. Reviews of Modern Physics 66: 1125–1388. Blatter G, Geshkenbein VB, and Koopmann JAG (2004) Weak to strong pinning crossover. Physical Review Letters 92: 067009. Brandt EH (1995) The flux-line lattice in superconductors. Reports on Progress in Physics 58: 1465–1594. Campbell AM and Evetts JE (1972) Critical currents in superconductors. London: Taylor and Frances. Durrell JH, Eom CB, Gurevich A, Hellstrom EE, Tarantini C, Yamamoto A, and Larbalestier DC (2011) The behavior of grain boundaries in Fe-based superconductors. Reports on Progress in Physics 74: 124511. Friesen M and Gurevich A (2001) Nonlinear current flow in superconductors with restricted geometries. Physical Review B 63: 064521. Gurevich A (2014) Challenges and opportunities in applications of unconventional superconductors. Annual Reviews of Condensed Matter Physics 5: 35–56. Haugan T, Puig T, Matsumoto K, and Wu J (2020) Artificial pinning centers in (Y,Re)-Ba-Cu-O superconductors: Recent progress and future perspectives. Superconductor Science and Technology 30: 040301. Hilgenkamp H and Mannhart J (2002) Grain boundaries in high-Tc superconductors. Reviews of Modern Physics 74: 485–549. Hosono H, Yamamoto A, Hiramatsu H, and Ma Y (2018) Recent advances in iron-based superconductors toward applications. Materials Today 21: 278–301. Kwok WK, Welp U, Glatz A, Koshelev AE, Kihlstrom KJ, and Crabtree GW (2016) Vortices in high-performance high-temperature superconductors. Reports on Progress in Physics 79: 116501. Larbalestier D, Gurevich A, Feldmann DM, and Polyanskii A (2001) High-Tc superconducting materials for electric power applications. Nature 414: 368–377. Maiorov B, Baily SA, Zhou H, Ugurlu O, Kennison JA, Dowden PC, Holesinger TG, Foltyn SR, and Civale L (2009) Synergetic combination of different types of defects to optimize pinning landscape using BaZrO3-doped YBa2Cu3O7. Nature Materials 8: 398–404. Pearl J (1964) Current distribution in superconducting films carrying fluxons. Applied Physics Letters 5: 65–66. Stangl A, Palau A, Deutscher G, Obradors X, and Puig T (2021) Ultra-high critical current densities of superconducting YBa2Cu3O7-d thin films in the overdoped state. Scientific Reports 11: 8176. Tinkham M (1996) Introduction to Superconductivity, 2nd edn. New York: McGraw-Hill. Yao C and Ma Y (2021) Superconducting materials. Challenges and opportunities for large-scale applications. iScience 24: 102541.

High-temperature superconductors MJ Qina, Xun Xub, and Shi Xue Douc, aInstitute of Materials Engineering, Australian Nuclear Science and Technology Organisation, Menai, NSW, Australia; bInstitute for Superconducting & Electronic Materials, University of Wollongong, Wollongong, NSW, Australia; cInstitute of Energy Materials Science (IEMS), University of Shanghai For Science and Technology, Shanghai, China © 2024 Elsevier Ltd. All rights reserved. This is an update of M.J. Qin, S.X. Dou, Superconductors, High Tc, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 112–120, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00706-3.

Introduction Overview Crystal structures Phase diagram Superconducting properties Critical temperature Tc Coherence length j Penetration depth l Critical fields Pairing symmetry Energy gap Type-II superconductivity Vortex structure Irreversibility field Critical currents and flux pinning Flux creep Melting of the vortex lattice Mixed-state phase diagram Normal-state properties Experimental methods Magnesium diboride (MgB2) Iron-based superconductor Thin films Bulk materials Tapes and wires Coated conductors Conclusion References

566 566 566 567 567 568 568 568 569 570 571 572 572 573 573 574 575 575 576 576 577 577 577 577 578 578 578 579

Abstract This chapter reviews some of the fundamentalproperties of high-temperature superconductors (HTSC), which include crystal structures, phase diagrams, superconducting properties, mechanisms, material processing and some newly developing superconductor. These would show a strong trait behavior of the high-temperature superconductivity that has been the subject of intensive study. It has received much attention due to its industrial applications.

Key points

• • • • • •

Crystal structures of most HTSCs can be obtained by stacking perovskite, rock salt and/or fluorite structure. The optimum Tc values above the liquid nitrogen temperature (77 K). Superconductivity happens when charge carriers overcome their mutual electrostatic repulsion, bind together into Cooper pairs, and condense into a single quantum state below a certain temperature. HTSCs are typical type-II superconductors by varying the external field H, at the lower critical field Hc1. Due to short coherence lengths and high anisotropy in cuprate superconductors, the vortex structures, and critical currents are different from those in conventional superconductors. Due to high-operating temperatures and reduced pinning compared to conventional superconductors, relaxation in HTSCs is very large. Thin films, bulk materials, tapes and wires, and coated conductors have received much attention

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Introduction In 1911, Kamerlingh Onnes discovered superconductivity in mercury with a transition temperature Tc of 4.2 K, at which the DC resistance of mercury dropped to zero. The second characteristic of a superconductor was its perfect diamagnetism, discovered by Meissner and Ochsenfeld in 1933. A theoretical understanding of superconductivity is based on the BCS microscopic theory, proposed by Bardeen, Cooper, and Schrieffer in 1957. Before 1986, the highest Tc in the now so-called low-temperature superconductors (LTSC) was found in Nb3Ge (23K). In 1986, Bednorz and Muller reported superconductivity in (LaBa)2CuO4 with a Tc of 35 K, in contrast to the previous record of 23 K in LTSC. Their discovery opened up the era of high-temperature superconductivity. The discovery of high-temperature superconductors (HTS) materials marked a significant milestone in the field of superconductivity, as it revealed the potential for superconductivity at higher temperatures than had previously been thought possible. Shortly after that, YBa2Cu3O7 (Tc ¼ 90 K), Bi2Sr2Ca2Cu3O10 (Tc ¼ 110 K), and Tl2Ba2Ca2Cu3O10 (Tc ¼ 125 K) were discovered. Since then, hightemperature superconductivity has been the subject of intensive study. In 2001, it was discovered that magnesium diboride (MgB2) exhibits superconductivity with a Tc value of around 40 K, which is relatively high compared to most previously known superconductors. Since 2008, the Hideo Hosono group has been actively studying iron-chalcogenide and iron-pnictide superconductors. This article reviews some of the fundamental properties of high-temperature superconductors (HTSC) (Blatter et al., 1994; Campbell and Evetts, 1972; Cohen and Jensen, 1997; de Gennes, 1966; Ginsberg, 1989; Canfield and Crabtree, 2003; Biswal and Mohanta, 2021; Nagamatsu, 2001; Bednorz and Müller, 1986).

Overview The year 1986 was a remarkable year for superconductivity physics. It was followed by a worldwide superconductivity research boom called the gold rush by superconducting physicists, which led to the discovery of seven series of HTSC materials with Tc above 90 K. The newly discovered HTSC have many exotic properties, such as small superconducting coherence length and anisotropic nature of the coherence length; anisotropic large penetration depth; anisotropic large energy gap; superconducting carriers are all holes; the existence of pseudo-energy gap; anisotropic and nonlinear characteristics of the normal state resistivity with temperature; nonlinear characteristics of the Hall coefficient with temperature; the parent is an antiferromagnetic structure, and the structure of the material is typical of the second class of superconductors with superconducting layers interspersed between charge bank phases. The material is a typical type II superconductor with a charge reservoir layer interspersed with superconducting layers. Since the discovery of HTSC, the BCS theory has been used to explain them, but the Tc of HTSC have far exceeded those predicted by the theory, and the BCS theory has never given results on the phase anisotropy of superconductor parameters. Superconducting physicists have proposed various models of strongly coupled superconducting matter (Fermi liquid model, non-Fermi liquid model) in an attempt to explain the mechanism of superconductivity in HTSC, but until now, no theory has been able to explain all the properties of HTSC. Although proposed by renowned physicists, the various theories that have been proposed to explain the mechanism of high-temperature superconductivity (HTSC) in materials like cuprates have not yet been universally accepted as definitive explanations for the phenomenon. Although the BCS theory has obvious flaws, superconducting physicists still use it as a basis for studying and measuring the results of superconductivity studies. The reason is that the carriers of both LTSC and HTSC are Cooper electron pairs, and in this respect, they are the same. This is the basic premise of most superconducting physicists to acquiesce to the correctness of BCS theory. Therefore, the study of superconducting physics is of great theoretical significance and great practical significance (Huebener, 1970; Jin, 1993; Senoussi, 1992; Silver et al., 2002; Tinkham, 1996). Numerous reviews have been written on the topic of high-temperature superconductors (HTS) materials, covering various aspects such as their properties, theory, and applications. Some of the most well-known and highly cited reviews in the field, which helped establish the field of HTS by presenting the first examples of materials that exhibit superconductivity at temperatures higher than previously thought possible (Anderson, 1987; Bednorz and Müller, 1988; Taillefer, 2010; Armitage and Greene, 2010).

Crystal structures Crystal structures of most HTSCs can be obtained by stacking perovskite, rock salt and/or fluorite structure along the c-axis. As an example, Fig. 1 shows the orthorhombic YBa2Cu3O7−d unit cell. Other families of HTSCs can be constructed in a similar way. The CuO chains are unique to YBa2Cu3O7−d; however, the most important features of the crystal structures of HTSCs are the two-dimensional copper oxide CuO2 conduction planes, which are critical for high-temperature superconductivity. Carriers are doped into the CuO2 conduction planes by transfer of charges resulting from the introduction of foreign ions, interstitial defects, and vacancies in structural blocks between two CuO2 planes. These blocks are called charge reservoirs. The crystal structure of YBa2Cu3O7−d consists of stacked layers of CuO2 planes and CuO chains, with Y and Ba ions located between these layers. The CuO2 planes are the main sites of superconductivity, as they contain the Cu ions in a square planar arrangement that can form strong electron pairing. The CuO chains, on the other hand, act as charge reservoirs and can affect the doping level of the CuO2 planes. In the orthorhombic phase of YBa2Cu3O7−d, the CuO2 planes are perpendicular to the crystallographic c-axis, and

High-temperature superconductors

567

O(1) O(2) Cu(1)

Ba Cu(2) O(3)

O(4)

Y

Ba

Fig. 1 Crystal structure of orthorhombic YBa2Cu3O7−d. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

the unit cell is elongated along the b-axis. The CuO chains run along the a-axis, and each CuO2 plane is separated from the adjacent planes by one CuO chain. This arrangement of CuO2 planes and CuO chains is often referred to as a “bilayer” structure, and it is one of the distinguishing features of the high-temperature superconductors.

Phase diagram Depending on the temperature and the doping level, HTSCs demonstrate a wide variety of behaviors (see the phase diagram for hole-doped HTSCs in Fig. 2). Undoped HTSCs are antiferromagnetic Mott insulators with nearest-neighbor Cu2+–Cu2+ antiferromagnetic exchange interaction in the CuO2 planes. The Néel temperature TN for the antiferromagnetic-to-paramagnetic transition decreases with increasing doping level. At a certain doping level, the antiferromagnetism vanishes and one enters the pseudogap or underdoped region. The normal-state properties in this region are significantly different from those of a Fermi liquid. Some of the most interesting behaviors of HTSCs are observed in this region. Superconductivity arises upon further doping. The transition temperature Tc first increases with increasing doping level, reaching a maximum Tc at an optimal doping level, then decreases and finally vanishes with further increases in doping (overdoped regime). The non-Fermi liquid region above the superconducting region is well described by the so-called marginal Fermi liquid (MFL) hypothesis.

Superconducting properties The most unusual fundamental properties of HTSCs, compared to LTSCs, are high-transition temperature, short coherence length, and high anisotropy. The high anisotropy results from the layered structure shown in Fig. 1. A phenomenological description can be achieved by means of the anisotropic Ginzburg–Landau equation. A phenomenological description of high-temperature superconductors (HTSCs) can be achieved by means of the anisotropic Ginzburg-Landau (AGL) equation, which takes into account the anisotropic nature of these materials due to their layered crystal structure (Blatter et al., 1994). The AGL equation is a generalization of the GL equation, which is a widely used phenomenological model for describing the behavior of superconductors near the critical temperature. The AGL equation has been used to explain various superconducting properties of HTSCs, such as their anisotropic magnetic response and their nonlinear transport behavior (Brandt, 1995; Zhu and Ting, 2009).

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Temperature

T *(d )

TM(d )

Marginal Fermi liquid (MFL)

AFM Fermi liquid SC

d

Hole concentration Fig. 2 Generic temperature (T ) vs. doping level (d) phase diagrams of cuprates in zero magnetic field. (AFM: antiferromagnetic phase; SC: superconducting phase; TN, and T are the Neel and pseudogap transition temperatures, respectively). Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

Critical temperature Tc The critical temperature (i.e., transition temperature from normal state to superconducting state) is one of the most important parameters characterizing superconductors. The optimum Tc values for some of the HTSCs are listed in Table 1. Tc values above the liquid nitrogen temperature (77 K) open the way for many applications. However, higher operating temperatures also introduce large thermal fluctuations in HTSCs. Based on the two principal properties of superconductors (zero-resistance and perfect diamagnetism), the usual methods to obtain Tc are to measure the resistance versus temperature curve R(T) or the AC susceptibility versus temperature w(T ) curve. Other methods such as measurement of DC magnetization versus temperature M(T ) and heat capacity versus temperature C(T ) are also applied to measure Tc.

Coherence length j It determines the distance over which the Cooper pairs are correlated. The short coherence lengths observed in HTSCs give rise to several unusual properties. Because the coherence length is much shorter than the penetration length, high-temperature cuprates are all type-II superconductors (see below). The short coherence length makes it very difficult to fabricate homogeneous hightemperature superconducting samples. Also associated with the short coherence length is the weak pinning of flux lines in HTSCs, compared with the pinning in LTSCs. To measure the coherence length, one usually needs to measure the upper critical fields Hab c2(0) and Hcc2(0), then the coherence lengths can be derived according to the equations based on the anisotropic Ginzburg–Landau equation m0 Hcc2 ð0Þ ¼ F0 =2px2ab ð0Þ m0 Hab c2 ð0Þ ¼ F0 =2pxab ð0Þxc ð0Þ c Here, Hab c2(0) and Hc2(0) indicate the upper critical fields at T ¼ 0 K for fields applied parallel to the ab-plane and the c-axis, respectively. Typical values for the coherence lengths are xab(0) ¼ 1.2–1.6 nm, xc(0) ¼ 0.15–0.3 nm, for YBa2Cu3O7−d, xab(0) ¼ 2.7–3.9 nm, xc(0) ¼ 0.045–1.6 nm for Bi2Sr2CaCu2O8+d, and xab(0) ¼ 2.1 nm, xc(0) ¼ 0.03 nm for Tl2Ba2CaCu2O8+d.

Penetration depth l It determines the distance that the applied DC magnetic field can penetrate exponentially into the superconductor. There are several methods to measure the penetration depth, such as muon spin resonance (mSR), electron paramagnetic resonance (EPR), and the polarized neutron reflection techniques. The penetration depth can also be derived from measurements of susceptibility or magnetization on samples with dimensions similar to the penetration depth. Typical values for the penetration depths are for

High-temperature superconductors Table 1

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Space group, crystal structure, and optimum Tc of selected cuprate superconductors.

Compound

Space group

Structure

Optimum Tc

La2CuO4

I4/mmm P42/ncm Bmab Fmmm I4/mmm P4/mmm P4/mmm Pmmm Ammm Ammm I4/mmm P4/mmm Cmmm Amaa A2/a C2 Fmmm Amaa I4/mmm I4/mmm Fmmm I4/mmm I4/mmm I4/mmm P4/mmm P4/mmm P4/mmm P4/mmm

214-T

40

214-O

40

214-T 214-T 123-T 123-O 124 247 223 2123

25

Bi-2201

10

Bi-2212

95

Bi-2223 Tl-2201

110

Tl-2212 Tl-2223 Tl-2234 Tl-1201 Tl-1212 Tl-1223 Tl-1234

115 125

Nd2CuO4 (Nd,Ce,Sr)2 CuO4 YBa2Cu3O6 YBa2Cu3O7 YBa2Cu4O8 Y2Ba4Cu7O15 (Ba,Nd)2(Nd,Ce)2Cu3O8 Pr2Ysr2Cu3O8 Bi2Sr2CuO6 Bi2Sr2CaCu2O8 Bi2Sr2Ca2Cu3O10 Tl2Ba2CuO6 Tl2Ba2CaCu2O8 Tl2Ba2Ca2Cu3O10 Tl2Ba2Ca3Cu4O12 TlBa2CuO5 TlBa2CaCu2O7 TlBa2Ca2Cu3O8 TlBa2Ca3Cu4O11

0 90 80 40 40 70

Z

xb

xc Y

lc lb

Fig. 3 Schematic diagram of a single vortex when H is parallel to the ab-plane, showing the anisotropic penetration depth and coherence length. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

YBCO, the in-plane penetration depth lab(0) is in the range of 130–180 nm, while the out-of-plane penetration depth lc(0) is in the range of 500–800 nm. In contrast, for BSCCO, lab(0) is greater than or equal to 3700 nm, while lc(0) is in the range of 270–300 nm. Fig. 3 shows schematically the length scales (l and x) of a vortex when the applied field is parallel to the ab-plane.

Critical fields High-temperature cuprates are all type-II superconductors with three characteristic critical fields: the lower critical field Hc1, the upper critical field Hc2, and the thermodynamic critical field Hc (see Fig. 4). m0H2c /2 is the free-energy difference between the superconducting and normal states. When the applied field is lower than Hc1, the superconductor shows perfect diamagnetism (called the Meissner state); when the applied field is larger than Hc1 but lower than Hc2, the applied field penetrates into the superconductor in the form of vortices (the so-called mixed state); when the applied field is larger than Hc2, the superconductor is forced into the normal state (see Fig. 4). As the upper critical fields at low temperatures are experimentally inaccessible, one is normally interested in Hc2 close to Tc; then using the dirty limit equation

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High-temperature superconductors

Normal state

H

Hc2 Mixed state

Hc1 Meissner state Tc

Temperature

Fig. 4 Phase diagram for a type-II superconductor. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

 Hc2 ð0Þ ¼ −0:693 

dHc2 dT

 Tc

Tc

c to obtain Hc2(0). Typical values for YBa2Cu3O7 – d are m0Hab c2(0) ¼ 200 T and m0Hc2(0) ¼ 40 T. The lower critical field Hc1 can be obtained from the virgin magnetization curves; typical values for YBa2Cu3O7 – d at 4.2 K are m0Hcc1 ¼ 90 mT, and m0Hab c1 ¼ 20 mT.

Pairing symmetry The pairing symmetry is very important for the discovery of the pairing mechanism. After the discovery of HTSCs, experiments such as those involving the observation of magnetic flux states of a polycrystalline YBa2Cu3O7−d ring, Andreev reflection, the Shapiro step of a Josephson junction, the high-resolution Bitter pattern technique, and Little–Parks oscillations have been performed to test whether Cooper pairing operates as in conventional superconductors. The results have confirmed electronic pairing in cuprate superconductors. In conventional superconductors, the phonon mediated electron–electron interaction gives rise to spin-singlet pairing with s-wave symmetry. However, the pairing symmetry in cuprate superconductors has been a controversial topic. Phase sensitive techniques (superconducting quantum interference device (SQUID)) interferometry, single Josephson junction modulation, thin film magnetometry, etc., combined with the refinement of several other symmetry-sensitive techniques (penetration depth, specific heat, thermal conductivity, angle-resolved photoemission, Raman scattering, nuclear magnetic resonance, nonlinear Meissner effect), have provided evidence in favor of d-wave symmetry pairing in cuprate superconductors. The energy gap, quasiparticle state, vortex state, and the surface and interfaces of d-wave superconductors are distinctly different from the s-wave case. The pairing symmetry in high-temperature superconductors (HTSC) and low-temperature superconductors (LTSC) can be different. In conventional low-temperature superconductors, such as niobium or lead, the electron pairing is mainly due to electron-phonon interactions and the pairing symmetry is well described by the BCS theory, which predicts a s-wave pairing symmetry. This means that the electron pairs have no angular momentum and the energy gap in the electronic density of states is isotropic. In contrast, the electron pairing mechanism in high-temperature superconductors is still a matter of debate, and several unconventional pairing symmetries have been proposed. Some high-temperature superconductors, such as cuprate superconductors, have a d-wave pairing symmetry, where the energy gap in the electronic density of states has nodes along the directions of the crystal lattice. Other high-temperature superconductors, such as iron-based superconductors, have different types of pairing symmetry, such as s, s++, or dxy-wave symmetry. The difference in the pairing symmetry between HTSC and LTSC arises due to the different nature of the electron pairing mechanism. In HTSC, the electron pairing is believed to arise from the interplay of different competing orders, such as antiferromagnetism and superconductivity, and the symmetry of the superconducting order parameter is strongly influenced by these competing orders. In LTSC, on the other hand, the electron-phonon interaction is the dominant pairing mechanism, and the symmetry of the superconducting order parameter is determined by the properties of the phonon modes in the crystal lattice. Detailed discussions can be found in the references (Tsuei and Kirtley, 2000; Scalapino, 2012; Norman et al., 2011; Lee et al., 2006; Mazin et al., 2008).

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Energy gap The energy gap indicates the minimum energy required to excite a quasiparticle from a phase-coherent Cooper-pair condensate. The BCS theory predicts a universal ratio between the energy gap and the thermal energy 2D(0)/kBTc ¼ 3.53, here D(0) is the energy gap at T ¼ 0 K. For HTSCs, the energy gap can be determined from experiments such as single-electron tunneling, Andreev reflection, Raman scattering, photoemission spectra. Typical values for YBa2Cu3O7−d are 2Dab(0)/kBTc ¼ 6–8, 2Dc(0)/ kBTc ¼ 3–3.5; and for Bi2Sr2CaCu2O8+d are 2Dab(0)/kBTc ¼ 8–11, 2Dc(0)/kBTc ¼ 5–7. For conventional superconductors, the Cooper pairs have an isotropic s-wave symmetry, which means that the energy gap is 2D everywhere in k-space (see Fig. 5). However, for a d-wave symmetry superconductor, in momentum space when kx ¼  ky, the energy gap is zero, which implies that it would change sign at kx ¼  ky (see Fig. 5). Both properties will have important implications from both fundamental and applied points of view.

(a)

ky

2'

kx

(b)

ky –

kx +

+

– Fig. 5 The energy gap for s-wave superconductors (upper panel) and d-wave superconductors (lower panel). The s-wave symmetry has an isotropic value of 2D, while for a d-wave symmetry the gap vanishes and changes sign at kx ¼ ky. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

572

High-temperature superconductors

Type-II superconductivity HTSCs have very short coherence lengths x and large penetration depths, so the Ginzburg–Landau parameter k ¼ l/x  1, and HTSCs are typical type-II superconductors. In this section, some of the properties in the mixed state are discussed (Ginsberg, 1989). In particular, The Lawrence-Doniach model is an important theoretical tool for studying the behavior of type-II superconductors, including 2D superconductors, in the presence of a magnetic field (Lawrence and Doniach, 1971). The LD model was first proposed by Lawrence and Doniach in 1971 and has since been used extensively to study the properties of type-II superconductors, including high-temperature superconductors (HTS) and 2D superconductors. The model takes into account the anisotropy of the superconductor, as well as the effects of the magnetic field on the electronic states and the order parameter of the superconductor. In the LD model, the superconductor is described as a stack of many 2D layers, with the magnetic field applied perpendicular to the layers. Within each layer, the superconducting order parameter is assumed to vary smoothly along the direction parallel to the layers, while it is assumed to be uniform along the perpendicular direction. The LD model also takes into account the anisotropy of the superconductor, which arises from the fact that the coherence length (the characteristic length scale over which the superconducting order parameter varies) is different along different crystallographic directions. The LD model can be used to calculate the irreversibility line, which is the boundary between the reversible and irreversible regimes of the superconductor in the presence of a magnetic field. Below the irreversibility line, the superconductor can expel the magnetic field and maintain its superconducting state. Above the irreversibility line, the superconductor is no longer able to expel the magnetic field and enters a mixed state, where vortices start to penetrate the superconducting layer. Using the LD model, it is possible to calculate the distribution of vortices in the superconductor as a function of the applied magnetic field and the temperature. The model can also be used to calculate the critical current, which is the maximum current that can flow without destroying the superconducting state. In addition, the LD model can be used to study the effects of defects and impurities on the behavior of the superconductor, including the pinning of vortices and the enhancement of the critical current (Brandt, 1995; Campbell and Evetts, 1972; Kim and Beasley, 1975; Dew-Hughes, 1981; Blatter et al., 1994).

Vortex structure Due to the high anisotropy of cuprate superconductors, the vortex structures are different from those in conventional superconductors. When the applied field is along the ab-plane, the screening current flows both along the ab-plane and along the c-axis. There exist insulating layers alternating with conduction layers along the c-axis, the current is the Josephson current in the insulating layers (see Fig. 6). When the applied field is parallel to the c-axis or at an angle to the c-axis, the vortices can be regarded as pancake vortices stacked along the c-axis (see Fig. 7). Each pancake vortex is located in one of the conduction planes. The pancake vortices in the neighboring layers are coupled by means of a Josephson current, which forms a Josephson vortex. The forces between pancake vortices in the conduction plane are repulsive, while the forces between pancake vortices in the neighboring layers are attractive. Depending on the temperature and the magnetic field, the vortex system can go through a 3D to 2D transition. The magnetic properties of 2D superconductors can be described by the Lawrence–Doniach (LD) model.

Z

Y

Fig. 6 Schematic diagram of a single vortex when the applied magnetic field is parallel to the a-axis (x-direction). Gray areas indicate the superconducting layers. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

High-temperature superconductors Z

573

q

Y

Fig. 7 Schematic diagram for pancake vortices when the applied magnetic field is at an angle to the c-axis (z-direction). Gray areas indicate the superconducting layers. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

Irreversibility field After the discovery of cuprate superconductors, it was soon reported that zero-field-cooled (ZFC) and field-cooled (FC) magnetization of La–Ba–Cu–O versus temperature at a fixed applied magnetic field can be separated into two regions. Below Tirr, ZFC and FC magnetization are different and irreversible, above Tirr, ZFC and FC magnetization are the same and reversible. Tirr is dependent on the magnetic field. Tirr(H) or Hirr(T ) is called the irreversibility line. Hirr(T ) is lower than Hc2 but larger than Hc1 and separates the mixed state into two regions, a vortex liquid above Hirr(T ) and a vortex solid below Hirr(T ). The irreversibility line can also be obtained by hysteresis loop measurements and I–V characteristics measurements. Generally, the irreversibility line follows 1 −Tirr/Tc0 ¼ CHqirr, with Tc0 the zero-field transition temperature, and ½ < q < 3/4, depending on different materials. The vortex system depends on three energy scales: the pinning potential Epin, the thermal energy Eth and the vortex–vortex interaction energy Evv. When the Eth  Epin, the system reaches the reversible region (Ullmaier, 1975).

Critical currents and flux pinning Due to short coherence lengths and high anisotropy in cuprate superconductors, critical currents are much more complicated than in conventional superconductors. For an anisotropic superconductor, three critical currents are defined depending on the direction of the applied magnetic field (see Fig. 8). Critical currents result from the interaction between vortices and crystal defects (pinning effect). Twin boundaries and screw dislocations in single crystal YBa2Cu3O7−d and Y2BaCuO5 phase, as well as stacking faults and dislocations in melt-textured-growth (MTG) YBa2Cu3O7−d, have all been reported to act as pinning centers. Heavy ion and particle irradiation have also been used to enhance the critical current density in cuprate superconductors. Due to the short coherence length, point defects such as vacancies and substitution atoms may play an important role in cuprate superconductors (Huebener, 1970). Critical current can be directly obtained from transport I–V measurements. From hysteresis loop measurements, the critical state model is used to derive the current density from the measured hysteresis loop,   20DM

H c Jab,c ¼ að1 − a=3bÞ Because a larger electric field criterion is used in transport measurement, it usually provides a larger critical current density than magnetic measurements. What it means is that when critical current density is measured using transport measurements (such as resistivity measurements), a larger electric field criterion is typically used compared to when it is measured using magnetic measurements. This is because the electric field criterion is related to the resistive behavior of the material, whereas the magnetic measurements are related to the pinning of vortices. The use of a larger electric field criterion typically results in a higher critical current density because it allows for higher current densities to flow without causing the material to become resistive. Magnetic measurements, on the other hand, typically use lower field strengths, resulting in lower critical current densities because the magnetic field has a stronger effect on vortex pinning. In polycrystalline samples, the current density across grains, called the intergrain critical current density, is very small, and shows the properties of a Josephson current. This is due to large angle grain boundaries in polycrystalline samples, and therefore these grain boundaries are called weak links. In order to have high critical current density, weak links should be avoided, by using, for example, melt-textured-growth (MTG) for YBa2Cu3O7−d bulk materials. Intragrain critical current densities up to 106 A cm−2 at 77 K and 0 T has been reported for YBa2Cu3O7−d thin films.

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c

H

Jc-ab Jab-ab a H Jab-c

c

b a Fig. 8 Schematic diagram showing the anisotropic critical current density in cuprate superconductors, when H||ab (upper panel) and H||c (lower panel). Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

Flux creep In conventional superconductors, the current density was observed to decrease logarithmically with time at fixed temperature and field, an effect called magnetic relaxation. Magnetic relaxation specifically refers to the behavior of the current density in a superconductor over time under the influence of a magnetic field. In conventional superconductors, the presence of magnetic vortices can disrupt the flow of current through the material, leading to a decrease in current density over time as vortices become thermally activated and move through the material. This logarithmic decrease in current density with time is known as magnetic relaxation (Brandt, 1995). This effect was explained by vortices thermally activated out of the pinning potential (flux creep) with the hopping frequency.   −U ð jÞ v ¼ v0 exp kB T where v0 is the attempt frequency and U( j) the effective activation energy depending linearly on the current density,   1−j U ð jÞ ¼ U 0 jc0 The current density decreases logarithmically with time as      k T t jðt Þ ¼ jc0 1 − B ln U0 t0 In cuprate superconductors, strong and nonlogarithmic relaxation has been observed (see Fig. 9). Due to high-operating temperatures and reduced pinning compared to conventional superconductors, relaxation in HTSCs is very large. Nonlogarithmic relaxation in HTSCs results from a nonlinear U( j) relationship. A logarithmic U( j) relationship has been proposed to account for the experimental results:   j U ð jÞ ¼ U 0 log c j On the other hand, according to the vortex glass theory, the U( j) relationship is   m  jc U0 −1 U ð jÞ ¼ m j

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0.022 YBCO HIIc 2T T = 70 K

Magnetic moment (emu)

0.020

0.018

0.016

0.014

0.012

0.010 102

103 Time (s)

Fig. 9 Magnetic relaxation of a melt-texture-growth (MTG) YBCO sample at H||c ¼ 2 T and 70 K, showing nonlogarithmic relaxation. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

Here, the exponent m depends on the dimensionality of the vortex system. Magnetic relaxation measurements have been extensively performed on HTSCs to study the flux dynamics. The purpose is to determine the effective pinning potential U(j) and compare it with theoretical predictions. Several methods such as the Maley method, the generalized inversion scheme, and dynamic relaxation and the AC susceptibility technique have been developed for this purpose. However, because too many parameters are used in theoretical predictions, it is impossible to unambiguously determine the effective activation energy from experiments. Most results in this area are not conclusive (Clem and Coffey, 1990; Matsuda and Tsuneto, 1989; Yeshurun et al., 1996; Kes et al., 1991).

Melting of the vortex lattice Thermal fluctuations in HTSCs are much more important than in conventional superconductors. According to the Lindemann melting criterion, as the displacement away from the ideal configuration by thermal fluctuation reaches 10–30% of the average flux-line separation, melting transition from vortex solid-to-vortex liquid (shear modulus equal to zero) might take place. The curve Tm(H) in the H–T phase diagram at which the flux lattice melts is called the melting line. It has been suggested that in clean samples with negligible pinning the vortex solid-to-vortex liquid transition is expected to be of the first order, while static disorders drive the transition to the second order as in the vortex glass theory. Evidence for a first-order melting transition comes from a jump in the reversible magnetization, changes in latent heat, a frequency-independent peak in the AC susceptibility and a sharp, hysteretic, resistive transition. However, the effect of static disorders and their dimensionality on the order of transition is unclear as yet (Wördenweber, 1999).

Mixed-state phase diagram For a type-II superconductor, by varying the external field H, at the lower critical field Hc1, a phase transition from the Meissner phase to the Abrikosov phase (or mixed phase) takes place. The basic unit of the mixed state is the vortex, which contains a flux quantum given by F0 ¼ ch/2e ¼ 2.07  10−7G cm2, penetrating the system and forming a perfect triangular crystal (the Abrikosov lattice). At the upper critical field Hc2, the system is forced to undergo a second phase transition to the normal state. The simple phase diagram shown in Fig. 4 neglects disorders and thermal fluctuations, which is very serious in cuprate superconductors due to higher operating temperatures. When thermal fluctuations are taken into account, the first modification applied to the phase diagram ought to be the inclusion of the “water-like” melting phenomenon discussed above. In clean systems, the melting curve stretches from Hc1 up to Hc2 (see Fig. 10). In case of disordered systems, the melting line terminates at much lower fields (characterized by the first-order transition). In practice, one is often far from the critical fields, leading to the unified phase diagram in Fig. 11. Then, one has to discriminate between a sharp first-order phase transition at lower fields separating Bragg glass and vortex liquid and a more continuous second-order-like transition at higher fields separating vortex glass and vortex liquid.

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Hc2 Melting line Vortex liquid

Vortex solid

Hc1 T Fig. 10 Schematic vortex phase diagram for high-temperature superconductors without disorder. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

B

Vortex liquid Vortex glass Melting line Bragg glass

T Fig. 11 Schematic vortex phase diagram for high-temperature superconductors with disorder. Reproduced from Qin MJ and Dou SX (2005) Superconductors, high Tc. Encyclopedia of Condensed Matter Physics, 1st edn, vol. 6, pp. 112–120. Amsterdam: Elsevier.

Normal-state properties Properties in the normal state of cuprate superconductors have been found to be very useful for the understanding of the mechanism of high-temperature superconductivity. These properties include resistivity, Hall coefficient, thermopower, magnetism, thermal conductivity, and optical properties. Although superconducting properties of cuprate superconductors are quite similar to those of conventional superconductors, the normal-state properties are unique. As an example, some general features for normal-state resistivity are listed: (1) the resistivity is anisotropic with the c-axis resistivity rc two orders of magnitude larger than the in-plane resistivity rab, (2) rab shows metallic behavior, but rc mostly shows semiconductor behavior. (3) rab is close to linear in T, which is the most striking normal-state property of cuprate superconductors. Theoretical explanations for these behaviors and other unique behaviors observed in optical and thermal properties remain controversial.

Experimental methods Here are some examples of experimental methods and studies that are commonly used to determine scaling lengths and energies in HTSCs, along with corresponding references: Angle-resolved photoemission spectroscopy (ARPES): This technique is used to measure the electronic structure of a material, including the superconducting gap, which can provide information about the

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coherence length and energy scale of the superconducting state (Damascelli et al., 2003; Norman, 1998). Scanning tunneling microscopy and spectroscopy (STM/STS): These techniques can be used to directly probe the local density of states and the superconducting gap in a material, which can provide information about the coherence length and energy scale of the superconducting state (Fisher et al., 2011; McElroy et al., 2005). Microwave surface impedance measurements: These measurements can be used to determine the superfluid density of a material, which can provide information about the penetration depth and energy scale of the superconducting state (Prozorov et al., 1998). Magnetic field dependence of critical current density (Jc): This measurement can be used to determine the magnetic field penetration depth and coherence length of a material, which are related to the energy scale and correlation length of the superconducting state (Brandt, 1995).

Magnesium diboride (MgB2) MgB2 has superior properties such as the thermodynamic and transport properties, the isotope effect, band structures, promising critical current density, the ability of doping effect and pressure effect. However, after the discovery of its superconducting properties in 2001 by Akimitsu’s group, it has been a promising compound in superconductivity applications. MgB2 is competitive among high-temperature superconductors (HTS) because it has fascinating advantages such as simple crystal structure, low material cost and small anisotropy. As it is, MgB2 has opened our eyes to previously unrealized possibilities for electron–phonon superconductivity. Higher transition temperatures and other compounds now occupy our dreams and novel basic phenomena such as two-gap superconductivity drive theory and experiments in new directions. Unlike the still controversial cuprates, the physical picture and quantitative theoretical description for MgB2 are now available. The behavior near Tc is determined by the large 2D gap, which produces a step as expected for a conventional one-gap superconductor (Canfield and Crabtree, 2003).

Iron-based superconductor In recent years, Fe-based superconductors have found a lot of unanticipated surprises. These types of superconductors consist of different classes of iron materials, like pnictides, chalcogenides, intermetallics and oxides. Although each group has its own features, some most important properties of the materials are quite similar. In the study of superconductors of iron-chalcogenides and iron-pnictides, Hideo Hosono group is active since 2008 (Hosono et al., 2015). Distinctly HTSC copper oxide superconductors, from magnetic properties, structures and superconducting behavior, the Tc of Fe-based superconductors is as much as 55 K, the electron-phonon interaction is quite too weak. Fe-based superconductors can be spread into a number of types of substances consisting of 3 households of iron-based superconductors such as ‘1111’ system RFeAsO (R: the rare earth element) consisting of LaFeAsO, SmFeAsO, PrFeAsO etc. ‘111’ type LiFeAs, NaFeAs & LiFeP etc., ‘122’ type BaFe2As2SrFe2As2 or LaFe2As2 etc. The iron-based superconductors give a wonderful likelihood of understanding unconventional superconductivity which could be useful to explain the mechanism of High-Tc superconductivity, and also the implementation of iron-based superconductors, especially in tapes and wires that can be fabricated for industry applications (Biswal and Mohanta, 2021).

Thin films YBa2Cu3O7−x(YBCO) thin films are the most studied high-temperature superconducting thin films. Critical current densities up to 106 A cm−2 have been achieved in high-quality YBCO thin films, making them very attractive for applications. There are, in general, two main methods for fabricating high-temperature superconducting thin films: in situ or ex situ film growth. In situ films are oxygen-treated before they are removed from the growth chamber, and therefore are superconducting after growth. Ex situ thin films must be annealed in an oxygen-rich environment after growth to be superconducting. Nearly every growth technique has been tried on high-temperature superconducting thin films, including evaporation, pulsed laser deposition (PLD), metal organic chemical vapor deposition (MOCVD), and on-axis and off-axis sputtering. Every technique has its merits and drawbacks, details of which can be found in the references. The search for substrate materials is an active research area. High-quality YBCO thin films have been deposited on LaAlO3, SrTiO3, MgO, YSZ, etc. BiSrCaCuO and TlBaCaCuO thin films have also been extensively studied. However, it is very difficult to obtain single-phase BiSrCaCuO thin films, while Tl is very toxic, creating problems in fabricating TlBaCaCuO thin films. The main application of superconducting thin films is in the area of superconducting electronics.

Bulk materials Many large-scale applications such as flywheel energy-storage systems need HTSCs in bulk form. Sintered YBCO polycrystalline samples used to have very low critical current densities (100–1000 A cm−2), which was attributed to weak links at grain boundaries, such as structural disorder at grain boundaries, chemical or structural variations at grain boundaries and non-superconducting materials along grain boundaries, as well as microcracking. Texture processing has been developed as a means to avoid or minimize the effects of weak links in polycrystalline YBCO. The initial success with bulks was due to the melt-textured-growth (MTG) process,

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which enhanced the critical current density two to three orders of magnitude over that of sintered materials. Several modified MTG methods were developed later in order to further enhance the critical current density. These include the quench-and-melt-growth (QMG) process, the melt-powder-melt-growth (MPMG) process, the power-melting process (PMP), the floating-zone-melt (FZM) process, the platinum-doped melt-growth (PDMG) process, and the horizontal Bridgman method.

Tapes and wires The majority of the work on superconducting tapes and wires focuses on the Bi2Sr2Ca2Cu3O10−x(BSCCO) system. The reasons are that the BSCCO system has a higher critical current at high magnetic fields, and that the BSCCO system does not suffer from the weak link problems of YBCO. The fabrication of BSCCO tapes and wires is usually carried out by means of the so-called oxide power-in-tube (OPIT) method. The powder is packed in a tube (typically Ag), then the Ag tube is drawn to a small diameter and rolled into a flat tape. The tape is then annealed, re-rolled and re-annealed again to obtain the best superconducting properties. Critical current densities up to 7000 A cm−2 for a 650 m long and 19-filament-rolled tape have been achieved. Tapes and wires are widely used in applications such as motors and generators, magnet, transformers, and transmission lines.

Coated conductors Coated conductors are YBCO thin films deposited onto polycrystalline substrates. The films are biaxially textured, thus overcoming the phenomenon of weak links. This has been achieved by preparing a biaxially textured buffer layer, upon which the YBCO film is subsequently grown epitaxially. The two most advanced techniques are the ion beam-assisted deposition (IBAD) process, and the rolling-assisted biaxially textured substrate (RABiTS) process. In addition, inclined substrate deposition (ISD) has also emerged as a promising method for the fabrication of YBCO conductors. The coated conductors potentially have a performance superior to the first generation bismuth-based HTS compounds (BSCCO), such as high current carrying capacities, strong mechanical properties, and extraordinary magnetic field tolerances. Recently, it has been demonstrated that current densities up to 106 A cm−2 have been achieved in conductors prepared by coating thick films of YBCO onto flexible metallic substrates. One of the biggest hurdles to widespread application of YBCO-coated conductor tape is developing a manufacturing process that will produce it in long lengths and at prices competitive to copper for applications such as motors, generators, transmission cables, and other power systems.

Conclusion In Summary, Superconductivity occurs when charge carriers form Cooper pairs and enter a single quantum state at a temperature below a threshold. In conventional superconductors, phonons are responsible for pairing, while in HTSCs, the mechanism is controversial. Several theories have been proposed, but no comprehensive theory accounts for all experimental observations. HTS materials have unique properties, such as the ability to carry large amounts of electrical current with zero resistance, operate at higher temperatures, and possess unique mechanical and thermal properties that offer potential for various applications. Continued research and development in this field will likely lead to further discoveries and new applications of these materials. The production techniques, performance, and low price are crucial in realizing practical and cheap cryogenic systems. Nanomaterials and superconducting physics are both branches of condensed matter physics, and a breakthrough in either discipline will have a chain reaction leading to rapid development. Certainly. Here are some examples of novel and unconventional functionalities that HTS materials have brought compared to LTS materials: 1. HTS materials have very high critical current densities, allowing them to operate at much higher magnetic fields than LTS materials. This makes them attractive for applications such as magnets for MRI machines, fusion reactors, and particle accelerators. 2. HTS materials can be fabricated as thin films and coatings using techniques such as pulsed laser deposition, sputtering, and metal-organic deposition. This allows for the development of high-performance, flexible, and lightweight superconducting materials for various technological applications. 3. The unique properties of HTS materials have opened up new possibilities for developing novel superconducting devices and systems, such as superconducting microwave filters and resonators, superconducting power transmission lines, and superconducting fault current limiters. 4. HTS materials exhibit improved stability compared to LTS materials, particularly at high magnetic fields and temperatures. This makes them more suitable for practical applications where stability is critical. 5. There have been recent reports of HTS materials exhibiting superconductivity at or near room temperature, which could revolutionize the field of superconductivity and lead to a wide range of new applications.

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Overall, HTS materials have brought a range of novel and unconventional functionalities that have expanded the field of superconductivity and opened up new possibilities for technological applications. However, they also present unique challenges and require continued research and development to fully realize their potential.

References Anderson PW (1987) High-temperature superconductivity: An overview. Science 235(4793): 1196–1198. Armitage NP and Greene RL (2010) High-temperature superconductivity in cuprates: The non-BCS paradigm and its implications. Reviews of Modern Physics 82(1): 242–268. Bednorz JG and Müller KA (1986) Possible high Tc superconductivity in the Ba-La-Cu-O system. Zeitschrift fur Physik B: Condensed Matter 64(2): 189–193. Bednorz JG and Müller KA (1988) Physics of high-temperature superconductors. Nature 332(6162): 283–287. Biswal G and Mohanta KL (2021) A recent review on iron-based superconductor. Materials Today: Proceedings 35: 207–215. Blatter G, Feigel’Man MV, Geshkenbein VB, Larkin AI, and Vinokur VM (1994) Vortices in high-temperature superconductors. Reviews of Modern Physics 66: 1125. Brandt EH (1995) The flux-line lattice in superconductors. Reports on Progress in Physics 58(11): 1465–1594. Campbell AM and Evetts JE (1972) Critical currents in superconductors. Advances in Physics 21(2): 199–428. Canfield PC and Crabtree GW (2003) Magnesium diboride: Better late than never. Physics Today 56(3): 34. Clem JR and Coffey WA (1990) Thermally activated flux flow in superconductors. Physical Review B 42: 6209–6223. Cohen LF and Jensen HJ (1997) Open questions in the magnetic behavior of high-temperature superconductors. Reports of Progress in Physics 60: 1581. Damascelli A, Hussain Z, and Shen ZX (2003) Angle-resolved photoemission studies of the cuprate superconductors. Reviews of Modern Physics 75(2): 473. de Gennes PG (1966) Superconductivity of Metals and Alloys. New York: Benjamin. Dew-Hughes D (1981) Flux pinning and its measurement. Advances in Physics 30(4): 111–159. Fisher O, et al. (2011) Local pairing and pseudogap in the heavily overdoped superconductor Bi2Sr2CaCu2O8+ d. Nature Physics 7(8): 611–616. Ginsberg DM (ed.) (1989) Physical Properties of High Temperature Superconductors I, II, III, IV. Singapore: World Scientific. Huebener RP (1970) Magnetic Flux Structures in Superconductors. Berlin: Springer. Hosono H, Hayashi K, Shimakawa M, and Takagi H (2015) Iron-based superconductors: An overview. Materials Research Society Bulletin 40(8): 646–656. Jin SH (ed.) (1993) Processing and Properties of High Tc Superconductors. Singapore: World Scientific. Kes PH, Tsuei CC, Scharnberg K, Gupta A, Kapitulnik A, Li TW, Schenstrom A, Klemm RA, and Clark WG (1991) Flux creep and magnetic relaxation in high-temperature superconductors. Physical Review Letters 67: 1911–1914. Kim YB and Beasley MR (1975) Critical current measurements in thin-film superconductors. Physical Review B 12(9): 4158–4163. Lawrence WE and Doniach S (1971) Nonlinear conductivity and fluctuation effects in superconductors. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 324(1559): 301–312. Lee PA, Nagaosa N, and Wen XG (2006) Doping a Mott insulator: Physics of high-temperature superconductivity. Reviews of Modern Physics 78(1): 17–85. Mazin II, Singh DJ, Johannes MD, and Du MH (2008) Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1-xFx. Physical Review Letters 101(5): 057003. Matsuda T and Tsuneto T (1989) Flux creep in type-II superconductors. Journal of the Physical Society of Japan 58: 3656–3666. McElroy K, et al. (2005) Spin-density wave suppression of superconductivity in stripe-ordered La1.875Ba0.125CuO4: Scanning tunneling microscopy and spectroscopy studies. Physical Review Letters 94(19): 197005. Nagamatsu J (2001) Superconductivity at 39 K in magnesium diboride. Nature 410(6824): 63–64. Norman MR, Pines D, and Kallin C (2011) The unconventional superconductivity in the heavy fermion compounds. Advances in Physics 60(5): 451–563. Norman MR (1998) Materials science: The gap seen up close. Nature 392(6672): 125–126. Prozorov R, et al. (1998) Muon spin rotation and microwave surface impedance measurements of the superconducting energy gap in YBa2Cu3O7−d. Physical Review B 58(5): R292. Scalapino DJ (2012) A common thread: The pairing interaction for unconventional superconductors. Reviews of Modern Physics 84(4): 1383–1417. Senoussi S (1992) Review of the critical current densities and magnetic irreversibilities in high Tc superconductors. Journal of Physics (III), France 2: 1102. Silver T, Pan AV, Ionescu M, Qin MJ, and Dou SX (2002) Developments in high temperature superconductivity. Annual Report Progress. Chemistry 98(C): 323. Taillefer L (2010) The pseudogap in high-temperature superconductors: An experimental survey. Annual Review of Condensed Matter Physics 1: 51–70. Tinkham M (1996) Introduction to Superconductivity, 2nd edn. New York: McGraw Hill. Tsuei CC and Kirtley JR (2000) Pairing symmetry in cuprate superconductors. Reviews of Modern Physics 72: 969. Ullmaier H (1975) Irreversible Properties of Type II Superconductors. Berlin: Springer. Wördenweber R (1999) Mechanism of vortex motion in high temperature superconductors. Reports of Progress in Physics 62: 187. Yeshurun Y, Malozemoff AP, and Shaulov A (1996) Magnetic flux structures in superconductors with strong pinning. Reviews of Modern Physics 68: 911–941. Zhu JX and Ting CS (2009) Nonlinear electronic transport in high-temperature superconductors. Reports on Progress in Physics 72(2): 026501.

Phase diagrams of high-temperature superconductors Shin-ichi Uchida, Department of Physics, University of Tokyo, Tokyo, Japan; Institute of Advanced Industrial Science & Technology, Tsukuba, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Uniqueness of the cuprates Temperature-doping phase diagram Key issues Spin order Spin fluctuations Unconventional superconductivity Precursor pairing Charge order CDW fluctuations Pseudogap regime Stripe order Overdoped regime Strange metal regime Conclusion Summary Outlook Magnetic phase diagram New spectroscopy and hidden orders References Further reading

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Abstract Since the discovery in 1986, enormous research efforts have remarkably advanced our understanding of the high-Tc cuprates, but some important issues remain unresolved. Primarily strong electronic correlations lead to superconductivity with unconventional d-wave pairing, which emerges upon doping into the Mott insulating state. The temperature-doping phase diagram of the cuprates is extremely complex with several orders or incipient orders showing up. In particular, the pseudogap and strange-metal regimes encompass a wide region and appear to be home not only to the superconducting order but also to various spin and charge orders.

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The cuprate phase diagrams are much more complex than previously anticipated as a standard one with antiferromagnetic, superconducting, Fermi liquid phases, and pseudogap regime as main players. The generic phase diagram further incorporates short-range and incipient spin, charge, and superconducting orders as well as strange metal regime. The short-range spin order –spin-density-wave (SDW)- in the heavily underdoped region is replaced by the short-range charge order –charge-density-wave (CDW)- as doping proceeds, and simultaneously the d-wave superconducting order sets in. The doping dependence of the superconducting transition temperature Tc shows a dome and Tc is controlled mainly by the superfluid density rather than the superconducting gap magnitude. Above Tc, there is a regime of precursory pairing -superconductivity without global phase coherence- in which the superconducting gap is almost unaffected. The precursory pairing state is a two-dimensional superconducting state without interlayer phase coherence and also a vortex liquid state under magnetic fields. The pseudogap regime is a playground for these orders and possibly other hidden orders, being intertwined and sometimes competing with one another. However, its origin remains a puzzle, and whether or not the doping, at which the pseudogap collapses, is a quantum critical point continues to be a matter of intense debate. Both SDW and CDW fluctuations are pervasively present over the entire superconducting doping range, extending to temperatures far above Tc with large energy scales of a few tenths of eV. The stripe order observed specific to the La-based cuprates is a showcase displaying how the spin-charge-pairing orders are intertwined and may thus be viewed as a pseudogap regime in a different appearance.

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The hole overdoped regime is not a simple Fermi liquid as formerly thought. On passing through the optimal doping, the superfluid density starts to decrease despite that the normal-state carrier density continues to increase. Its reason is currently under hot debate. Above the pseudogap temperature in the underdoped region and above Tc in the overdoped region, the metallic state is recognized as a strange or bad metal. In this regime, the charge transport properties are difficult to reconcile with the notion of quasiparticles in the Landau-Boltzmann theory. The mechanism of the charge transport in this regime is at present difficult to be elucidated. However, apparently this regime is a home to both pseudogap and Fermi liquid, and hence is a key to resolve the high-Tc problems.

Introduction The copper oxides (cuprates) stand out in their high Tc values, and are only family that achieved Tc well above the liquid nitrogen temperature at ambient pressure. High-Tc superconductivity (HTS) in the cuprates was discovered by Bednorz and Müller in 1986. Even after intensive research efforts over 35 years unraveling the mechanism of HTS and searching for higher Tc continue to be two grand challenges in condensed matter sciences. The understanding of the high-Tc cuprate materials and their electronic properties has substantially advanced. A consensus that has been achieved is that primarily repulsive electron-electron interactions lead to superconductivity with unprecedentedly high Tc as well as unconventional Cooper pairs. The common structural unit, CuO2 plane, is an insulator driven by strong electron-electron repulsive interaction (a Mott insulator). Unconventional superconductivity emerges upon doping into the insulating CuO2 plane. Another consensus is that characterization of the “normal” state, in particular, the pseudogap regime above Tc and the ‘strange metal’ regime at elevated temperatures in the phase diagram, is indispensable for unraveling the mechanism. Recent progress is the discovery of a “plethora of orders” with broken-symmetry and their fluctuations that manifest in the pseudogap regime. In this chapter, starting from an overview of how the cuprates are unique in their electronic structure and what are necessary ingredients for high Tc, are described the current understanding and the key issues of the doping ( p)-temperature (T ) phase diagram and relationship of various emergent orders with the superconducting order.

Overview Uniqueness of the cuprates High-temperature superconductivity with Tc well exceeding the liquid-nitrogen temperature (77 K) is found only in a class of materials whose basic structural unit is the surprisingly simple CuO2 plane, layers of Cu and O arranged in a square lattice (see Figs. 1A and B). The neighboring CuO2 planes are separated by so-called “charge reservoir” layers/blocks. Roles of the charge reservoir layers are to supply carriers to the CuO2 planes by chemical substitution and/or by adding or reducing oxygen atoms, and to stabilize the cuprate crystal structure the negatively charged CuO2 planes, [Cu2+O2−2]2−, are neutralized by forming an alternating stack with positively charged layers. The undoped CuO2 plane is a “Mott” or charge transfer (CT) insulator due to the strong Coulomb repulsion (U
6–8 eV) between electrons on the 3d orbitals of Cu atoms. As schematically depicted, the energy gap is actually created between occupied O2p band and empty Cu3d band (the upper Hubbard band), so it is called as a charge-transfer (CT) gap ECT (Fig. 1C). A Cu2+ ion has 9 electrons in the 3d orbitals and it is four-fold coordinated in a CuO2 square lattice. As a consequence, the uppermost 3d orbital is 3dx2-y2 and occupied by a single electron (hole) as depicted in Fig. 1D. This lone electron (hole) is localized on Cu due to strong Coulomb repulsion, leaving a magnetic moment associated with the S ¼ 1/2 spin quantum number. The Cu spins align antiferromagnetically on the CuO2 plane in the undoped cuprate due to superexchange interaction J between neighboring Cu spins via an intervening O atom. The singly occupied Cu 3 dx2-y2 level form a single band by the hybridization with the O2px and O2py states which would be half-full, if the Coulomb repulsion were not strong. Since Cu is at the end of the 3d transition metal series in the periodic table, the energies of Cu3d states are incidentally close to those of the O2p states as schematically shown. This makes the CT energy gap small, ECT ¼ 1.5–2.0 eV, compared with that in other transition-metal oxide. In particular, the wavefunction of 3dx2-y2 has large overlap with that of O2px and O2py. The overlap integral or hopping matrix element tpd is large,
0.3–0.4 eV, leading to strong hybridization between the two. The unique features of the cuprates and widely anticipated necessary conditions for high Tc are thus summarized as. (1) (2) (3) (4)

layer structure, strong electronic correlations, S ¼ 1/2 spin state, accidental degeneracy between Cu3d and O2p energies.

No other materials so far known have all these features simultaneously.

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(A)

(B)

La/Sr Cu

O

O

Cu E Cu 3d-band: Upper Hubbard (C)

(D) O 2py

ECT

O 2p band

Cu 3dx2-y2

O 2px U

Cu 3d-band: Lower Hubbard Fig. 1 (A) Crystal structures (schematic) of a representative high-Tc cuprate, La2-xSrxCuO4, and (B) the CuO2 plane. (C) Schematic energy scheme of the insulating undoped cuprate showing a strong electronic correlation (U) on a Cu atom and a charge-transfer (CT) energy gap between O2p and Cu3d upper Hubbard band. (D) Orbital configuration of a CuO2 unit showing relevant Cu3d x2-y2 and O2ps (s ¼ x or y) orbitals, both being strongly hybridized.

The large overlap integral tpd and the closeness of 3d and 2p levels, i.e., small CT gap, give rise to strong covalent nature of the Cu-O bond as well as to the unusually large spin superexchange interaction J
tpd4/ECT3
0.15 eV. The strong covalent bond makes the phonon frequency corresponding to the modulations of the Cu-O bond length (or bond angle) fairly high (ħO0
0.07 eV). The large energy scales of the charge (tpd), spin (J), and phonon (O0) related excitations are probably sources of strong pairing interactions and hence high Tc. The layer structure leads to their highly two-dimensional (2D) character of the electronic state, and S ¼ 1/2, the minimal spin quantum number, makes the spin fluctuation maximal. Because of these two, the otherwise very stable AF order (due to large J) becomes fragile against doping and raising temperature. Finally, the strong correlations favor unconventional pairing with nodes in the superconducting (SC) gap.

Temperature-doping phase diagram In the ‘standard’ phase diagram (Orenstein and Millis, 2000) sketched in Fig. 2, antiferromagnetic (AF), superconducting (SC), and overdoped ‘Fermi liquid’ (FL) phases evolve with increase of doping (either electrons or holes). In the case of the electron-doped Temperature

Cuprates

Antiferromagnetic

FL Electron

Pseudogap

SC

SC Doping

FL Hole

Fig. 2 ‘Standard’ phase diagram of electron- and hole-doped cuprates. With increase of doping, the antiferromagnetic phase transforms to the superconducting (SC) and then the Fermi liquid (FL) phase. A broad antiferromagnetic phase and a wide pseudogap regime are characteristic of electron- and hole doped cuprates, respectively.

Phase diagrams of high-temperature superconductors

583

cuprates, Nd2-xCexCuO4 (Tcmax ¼ 25 K) and so-called infinite-layer Sr1-xLaxCuO2 (Tcmax ¼ 40 K), the AF phase is robust, while the SC phase is realized in a narrow doping range. The superconducting Tc values are distinctly low, although the superconductivity is conceived to be unconventional d-wave. On the other hand, in the phase diagram of the hole doped cuprates pseudogap (PG) regime prevails, and at high temperatures there is a “strange” metal regime in which the in-plane (current flowing along the CuO2 planes) resistivity continues to increase linearly with T without showing a tendency for saturation.

Key issues This chapter focuses on the phase diagram of the hole-doped cuprates. They are genuine high-Tc superconductors, and the understanding of their phase diagram is believed to be a key to unravel the mechanism of high-Tc superconductivity. Recent progress in various advanced spectroscopies has revealed that the phase diagram, shown in Fig. 3, is much more complicated than previously anticipated, and incorporates various orders (broken symmetries). It turns out that various ordering phenomena including various types of fluctuating orders take place in the pseudogap regime.

Spin order Local S ¼ 1/2 spin magnetic moments on Cu atoms align antiferromagnetically in the CuO2 plane (the Néel order with ordering wavevector QAF ¼ (0.5, 0.5) in the unit of 2p/a) by the huge superexchange interaction J between neighboring Cu spins. The long-range antiferromagnetic (AF) order sets on at T ¼ TN (
300 K for the parent compound) in a second order phase transition. The ordering temperature TN is much lower than J
1500 K because the AF order is subject to strong thermal and quantum mechanical fluctuations. The former arises from the two-dimensionality, while the latter from the S ¼ 1/2 minimal spin quantum number. This phase is essentially Mott insulating and the insulating gap is a Cu3d-O2p charge-transfer (CT) gap, ECT ¼ 1.5–2.0 eV, arising from the strong on-Cu site electronic repulsion. The long-range AF order is quite fragile against hole doping. The long-range AF order collapses upon doping holes of about 0.02 per Cu atom, but spin order persists as a short-range spin-density-wave (SDW) order which is incommensurate with the underlying lattice. Different from the long-range AF order, the short-range SDW freezes gradually with lowering temperature. The onset temperature of SDW is low TSDW < 50 K. The spin correlation length is short but finite at all temperatures showing slow dynamics with a wide distribution of the relaxation rates, characterized as a spin-glass. For further doping, typically p ¼ 0.05–0.08, the short-range SDW disappears, and the superconducting order sets on.

Spin fluctuations Even after SDW disappears, dynamical spin fluctuations remain up to very high dopings. The spin excitation spectrum or the dispersion of spin excitations is similar in all cuprate families. The undoped cuprates with commensurate AF order show spin wave (magnon) dispersion from zero at a wavevector (0.5, 0.5) in the unit of 2p/a up to a high energy of
0.3 eV reflecting extremely large superexchange coupling J (100–150 meV). With hole doping, the magnons evolve into heavily damped excitations (called paramagnons) dispersing upward from a characteristic energy or (Fong et al., 2000). Below or, the excitations are less damped with downward dispersion toward incommensurate spin ordering wavevectors (0.5  ds), ds ¼ QSDW. This hourglass-shaped dispersion of spin excitations is in common among the cuprate families. 400 T* Temperature T (K)

300

Strange Metal

Pseudogap TN

200

Tc2D

precursory SC

TCDW short-range charge order

100 AF 0

0

TSDW p1

Tc3D d-SC 0.1

popt p*0.2 p2

FL 0.3

Hole doping p (per Cu) Fig. 3 Generic temperature (T ) - doping ( p) phase diagram of the hole-doped cuprates. Various orders emerge below the T  line. The onset of short-range charge order is indicated by red square, and below TSDW (blue dashed curve) a short-range spin order exists. Signatures of precursory superconductivity are observed within the subdome shadowed by pale green.

584

Phase diagrams of high-temperature superconductors

The low-energy spin excitations are gapless, but when p exceeds 0.08 (around the onset of SC), a gap in the spin excitations (spin gap) starts to develop and simultaneously a short-range charge order (charge-density-wave) appears. The spin fluctuations or dynamic antiferromagnetic spin correlations are still intact up to high doping region (Dean et al., 2013). These dynamic spin fluctuations in cuprates are much stronger than in conventional metals,

Unconventional superconductivity As regards the SC phase, there are many differences from conventional superconductors: 1) Tc is unprecedentedly high. 2) The SC gap function, D(k), has an unconventional d-wave symmetry (Tsuei and Kirtley, 2000), strongly momentum (k) dependent, 2D (k) ¼ D0 [cos(kxa) – cos(kya)], a being the in-plane lattice constant. It changes sign on the Fermi surface (FS) and hence vanishing at some k’s (gap nodes). The gap nodes are along the Brillouin-zone diagonal, and the gap magnitude is maximum (called as “antinodes”) at the zone edges, (0.5, 0) and (0, 0.5) in the unit of 2p/a, as schematically depicted in Fig. 4. 3) The SC transition temperature Tc as a function of hole doping p, Tc ( p), shows non-monotonic variation with p, consisting of a dome, starting to rise from p1, showing a weak cusp (or plateau) at p
1/8 for La2-xSrxCuO4 (YBa2Cu3O7-y), reaching a maximum at so-called “optimal” doping popt, and then dropping to zero at p2. In real cuprate samples both ends, p1 and p2, of the Tc dome are not sharp boundaries. Due to unavoidable inhomogeneity in the local dopant density, the superconducting state near the ends is granular in nature and hence the SC transition is percolative. On the other hand, the amplitude of D(k), typically 30–40 meV, is only weakly dependent on p, so there appears no direct link between Tc and D0 (unlike in conventional s-wave SC D
2kBTc). 4) Tc appears to have a more intimate link to the superfluid density. The superfluid density or SC phase stiffness rs (
ns/m , ns and m being Cooper pair density and carrier effective mass, respectively) in cuprates is by orders of magnitude smaller than that in conventional superconductors. Tc increases linearly with rs in the underdoped region (Uemura et al., 1989), stops increasing or saturates around popt, and then starts to decrease beyond popt. rs is, in principle, a very basic quantity that characterizes the SC order, but in conventional superconductors rs is so high that it does not explicitly appear in the superconducting properties. By contrast, rs in the high-Tc cuprates is anomalously low, and consequently shows up as a relevant parameter in various properties including Tc.

Precursor pairing As the superfluid density or the phase stiffness rs is low and the materials are effectively two-dimensional (2D), the cuprates are expected to subject to strong SC phase fluctuations (Emery and Kivelson, 1995). The best circumstantial evidence for a precursory SC (or phase-disordered SC) state comes from diamagnetism at high magnetic fields (Li et al., 2010) that is observed from Tc up to Tpair (about 1.5 Tc). Though weak compared to full Meissner screening, it is still large compared to that of simple metals. Compelling evidence is in the temperature evolution of the gap itself; The SC gap observed by angle-resolved photoemission spectroscopy (ARPES) does not collapse at Tc but persists up to temperature about Tpair (Chen et al., 2019). The large Nernst effect observed in the temperature range between Tc and Tpair (Xu et al., 2000) is also a signature of precursory SC. It cannot be ascribed to the motion of normal-state charge carriers, but most likely arises from moving vortices in the SC condensate which destroy the global phase coherence without affecting the SC gap. Such Nernst effect is reminiscent of genuine 2D superconductors where the pairs establish a long-range coherence through a Berezinskii-Kosterlitz-Thouless (BKT) transition. The BKT transition is driven by

Fig. 4 Variation of the magnitude and sign of the d-wave superconducting gap in the first Brillouin zone. The Fermi surface for the optimally doped cuprate is superposed on the gap map. The gap nodes are along the zone-diagonal and the gap magnitude is maximal at the zone edges (antinodes).

Phase diagrams of high-temperature superconductors

585

proliferation of vortices. These signatures differ from superconducting fluctuations observed in classic superconductors which are fluctuations of the amplitude of the pair wavefunction and observed only in the close vicinity of Tc. In the case of cuprates, the amplitude remains finite above Tc, and the fluctuations are primarily of the phase. Magnetic field penetrates a type-II superconductor as vortices, and the vortices form a regular array (vortex solid). For a conventional superconductor the vortex lattice is collectively pinned by impurities or defects, so that the full diamagnetic response (Meissner effect) and the zero-resistivity disappears only when the magnetic field (B) reaches the upper critical field Hc2. By contrast, the vortex lattice in cuprates easily melt by increasing B to a field value Hm(T ) much smaller than Hc2. The observed large Nernst signal and the diamagnetic response above Tc indicate that a vortex liquid state exists even above Tc up to the temperature Tpair (Li et al., 2013). The persistence of the vortex liquid is thought to be due to the small interlayer phase stiffness in conjunction with the extremely large pairing energy scale. The former makes the interlayer SC phase coherence to be destroyed at a weak field comparable to the lower critical field Hc1, and the latter does the pairing amplitude |C| robust against raising T above Tc and against increasing B. Therefore, the vortex liquid phase is regarded as a manifestation of the 2D precursory pairing phase.

Charge order Charge order (CO) or periodic spatial modulations of charge density (charge density wave, CDW). CO was first discovered in the La-based cuprates with the K2NiF4 structure (La214), La2-x-yNdySrxCuO4 and La2-xBaxCuO4 (LBCO) at x
1/8 as stripe order (Tranquada et al., 1995). The stripe charge order is a long-range order and coexists with the spin order. Recently, CDW has been discovered for almost all the cuprate families, typically YBa2Cu3O7-y (YBCO). In contrast to the stripe order, the short-range SDW order is not seen in YBCO in the doping range larger than 0.08 where the CDW order is observed. In this regime, spin excitations are gapped, while charge excitations are gapless. The x-ray scattering experiments find a gradual onset of static and short-range CDW order at TCDW ¼ 100–160 K (Chang et al., 2012; Ghiringhelli et al., 2012). The incommensurate CDW has ordering vectors (QCDW) always along one of the Cu-O bond directions and hence unidirectional. QCDW is typically 0.25–0.3 (2p/a), depending on material, doping as well as temperature. The CDW correlation length (x) estimated from the x-ray scattering peak width is about a few CDW periods,
15–25 A˚ in many cuprate families. The microscopic observation of CDW in Bi2Sr2CaCu2O8+d (Bi2212) by scanning-tunneling-spectroscopy (STS) has confirmed a unidirectional charge order (Kohsaka et al., 2007). In the direction perpendicular to the planes, the CDW has almost no correlation between neighboring layers, so it is confined within a CuO2 plane, i.e., two-dimensional (2D). Experimental probes with longer time scales, such as nuclear-magnetic-resonance (NMR) (Wu et al., 2011), have confirmed that the short-range charge order is almost static, presumably due to pinning of correlated charge fluctuations by lattice disorder. As a function of doping, the onset temperature TCDW forms another dome between 0.08 and 0.18 with a maximum at p
1/8 (Blanco-Canosa et al., 2014). Intriguingly, the onset temperature of the precursory pairing, Tpair, is comparable with TCDW, and both show similar doping dependence. Both TCDW- and Tpair-subdomes share nearly the same doping region within the superconducting Tc dome. TCDW is highest near p ¼ 1/8 where Tc shows a shallow dip or a plateau but Tpair is also highest. This indicates that CDW in cuprates is not a mere competitor of d-SC, but it is obviously “intertwined” with d-SC in marked contrast to the known CDW systems in which CDW and SC are competing and mutually exclusive orders.

CDW fluctuations The recent extensive study of the resonant-inelastic-x-ray-scattering (RIXS) of NdBa2Cu3O7-y (and YBCO) has observed a broad peak, being centered near QCDW, attributable to charge density fluctuations signal arising from dynamic CDW correlations. The dynamic CDW correlations are found to exist over a wide region of the T – p phase diagram (Arpaia et al., 2019). They persist up to temperatures well above TCDW and even above the PG temperature T  , and are thought to be precursor to the low-temperature quasi-static CDW. The precursor CDW is also two-dimensional, and temperature independent. The in-plane CDW correlation length xCDW is extremely short, only one CDW period or less and T-independent. Along the doping-axis, they are observed from the lowest end (p
0.08) of the quasi-static CDW to even the overdoped region (p
0.2). Notably, the integrated scattering intensity (spectral weight) of the fluctuating CDW is by about an order of magnitude larger than that of the quasi-static CDW, indicating that only a small fraction of the total CDW correlations condenses into the quasi-static low-temperature CDW order due possibly to pinning by lattice disorder. There are two possibilities on the mechanism of the formation of charge order in cuprates: One is the real-space mechanism based on the strong electronic correlations and the other is the momentum-space mechanism such as Fermi surface nesting assisted by electron-phonon coupling for the classical CDW systems. The universal dynamic CDW correlations with high-temperature and high-energy scales point toward the strong electronic correlations as a primary driving force of the cuprate charge order. Now that both AF (or SDW) and CDW fluctuations are pervasively present in the entire SC doping region, an important question to be answered in future is which fluctuations of either spin or charge or both orders drive the high-Tc superconductivity. The spectrum of CDW fluctuations covers a wide momentum region and extends to the energy scale of 0.2–0.3 eV, comparable to that of the hourglass-shaped spin fluctuations. It is theoretically argued that if the spin fluctuations contributed to the pair formation as a glue, the charge fluctuations would play either as an independent glue to enhance SC order or as a pair breaker for d-wave pairing.

586

Phase diagrams of high-temperature superconductors

Pseudogap regime This, sometimes called as an underdoped regime, is the most mysterious regime in the phase diagram, and the understanding its origin and nature is considered to be a key to resolve the high-Tc mechanism: The pseudogap (PG) regime covers a wide T - p area of the phase diagram (T < T  and p <
popt) in which various spectroscopies indicate a suppression of low energy electronic excitations below the pseudogap temperature T  . There remain puzzles in the origin of PG, but it is a playground for various types of ordering phenomena, SDW, CDW as well as d-SC. As described above, the quasistatic incommensurate spin and charge short-range orders as well as precursory pairing develop within the PG regime. In the PG regime below the T  line various spectroscopies show a suppression of low energy electronic excitations (Homes et al., 1993). An energy scale of the PG is typically
0.1 eV, significantly larger than the SC gap scale (typically 30–40 meV). PG opens in the portion of the Fermi surface near antinodes (the Brillouin-zone edges, k ¼ (0.5, 0) and (0, 0.5) in the unit of 2p/a), leaving behind the Fermi arc centered at nodes on which gapless and coherent quasiparticles reside (Marshall et al., 1996). This anisotropic gap gives rise to a dramatic change in various properties at T  . For example, the c-axis (directed perpendicular to the CuO2 planes) resistivity shows an upturn below T  as T is decreased like an insulator, whereas the in-plane resistivity drops at T  associated with a drop in the carrier scattering rate. The structure of the pseudogap in momentum space was directly mapped by ARPES experiments. In the temperature range between T  and Tc, the charge excitations are gapless on the Fermi arc and a gap is present near antinodes (Ding et al., 1996; Loeser et al., 1996), which crudely mimics the d-wave SC gap (the pseudogap is apparent only in the “antinodal” regions at which the d-wave gap is largest). In the SC phase, PG coexists with d-wave SC gap in non-trivial manner. Below Tc d-wave SC gap starts to open on the Fermi arc, whereas PG in the antinodal region remain almost unchanged with magnitude larger than inferred d-wave gap extrapolated from the Fermi arc region (Tanaka et al., 2006). Given that there is no evidence that the SC gap energy scale is diminished by the presence of PG, it is likely that PG is also home to the SC pairing correlation. Hence, PG is not necessarily a competing order of d-SC, but appears to be home to various orders including SC. It continues to be a matter of debate whether the T  line indicates a phase transition or a crossover. T  rapidly decreases with increasing p, and the T  ( p) line appears to penetrate the Tc dome, extrapolating to a hole density p
0.18. If PG is a phase associated with some broken symmetry, then p would be a quantum critical point (QCP) at which a continuous phase transition takes place at zero temperature. CDW with broken translational symmetry is a candidate for the PG phase, but in the phase diagram the CDW order exists in the doping region narrower than that of PG and TCDW is significantly lower than T . There are experimental observations that are suggestive of broken symmetries below T  other than translational symmetry associated with SDW and CDW. Spin-flip neutron scattering experiments (Fauquc et al., 2006) indicate intra-unit cell time-reversal symmetry breaking (q ¼ 0 staggered magnetic order), supposedly arising from counter-circulating orbital current inside the unit cell (Varma, 2006). Scanning-electron-tunneling spectroscopy (STS) also suggests a rotational symmetry breaking inside the unit cell (q ¼ 0 nematic order) (Lawler et al., 2010). A signature of rotational symmetry breaking was also detected by thermodynamic probes. However, the intra-unit-cell (q ¼ 0) orders cannot open a gap by themselves, and even if a gap opens with a help of some secondary order, it is difficult to account for such a large PG energy scale. The presence of a QCP at p is also challenged by the most recent experiments (Cooper et al., 2009; Chen et al., 2019). Thus, the origin of the PG has yet been identified. Apart from the presence or otherwise of QCP, the PG regime is a playground for various types of q 6¼ 0 short-range orders including SDW, CDW and precursory pairing (or pair-density-wave) which can, in principle, create a gap. Since they are intertwined to each other, it is possible to speculate that the observed pseudogap is a spin-charge-pair entangled gap (Loret et al., 2019).

Stripe order The stripe order in cuprates is a unique spin and charge order and a visible case of an intertwined charge, spin, and electron-pair orders. A plausible structure of the static spin-charge stripe order, illustrated in Fig. 5A, is a periodic array of charge stripes with antiferromagnetically ordered spin stripes between neighboring charge stripes (Tranquada et al., 1995). The periods of the spin and charge stripe array are typically 8a and 4a, respectively, for the doping x ¼ 1/8 in LBCO. The periods change with changing doping, but the charge order wavevector QCO of the charge stripes is locked to that (QSO) of the spin stripes with QCO ¼ 2QSO. In the materials showing stripe order a structural phase transition from high-temperature-orthorhombic (HTO) to low-temperaturetetragonal (LTT) occurs before the stripe order starts to develop. The LTT lattice deformation acts as a pinning potential for the charge stripes. Matching to the LTT structure, the orientation of the stripes changes by 90 as one moves from one layer to the next, forming three-dimensional (3D) spin-charge stripe order. The magnitude of the ordering wavevector QCO of the charge stripes increases with doping as expected in a real space picture, that is, QCO is determined by a distance between charge stripes with the same hole concentration. Though special to La214 cuprates, the stripe order helps in understanding the interplay between SDW, CDW and SC orders (including precursory SC) all of which are important ingredients for shaping the ‘generic’ phase diagram of the cuprates, in particular, the PG regime. The spin stripe order onsets at lower temperature, TSO, than the onset of the charge stripe order, TCO. Triggered by the LTT lattice deformation, segregation of doped holes starts to form charge stripes at TCO. After the holes are fully segregated on the charge stripes, the AF ordered domains, or spin stripe order, are created between neighboring charge stripes at TSO. Distinct from the insulating

Phase diagrams of high-temperature superconductors

(B)

Temperature T (K)

(A)

587

60

La2-xBaxCuO4

TLTT

TCO CO Tc2D

40

TSO

TCO

SO 20

SC Tc

0

0.10

TBKT

SC

p = 1/8

0.15

3D

Hole doping p Fig. 5 (A) A schematic structure of spin-charge stripe order. (B) Phase diagram of the stripe-ordered La2-xBaxCuO4. At x ¼ 1/8 the stripe order is most stable. The onset of charge (spin) stripe is indicated by TCO (TSO). The formation of charge stripe is triggered by the structural phase transition at TLTT, and the onset of spin stripe coincides with the onset (Tc2D) of two-dimensional superconductivity. The three-dimensional superconductivity is suppressed at x ¼ 1/8, but soon recovers with increase or decrease of x from 1/8.

stripes in other doped Mott-insulators such as nickelates and cobaltites. The cuprate stripe ordered state stays metallic and even superconducting. While the stripe order suppresses bulk (three-dimensional, 3D) superconductivity, there is growing evidence that pairing, in a form distinct from the spatially uniform d-wave SC order, is taking place (Li et al., 2007). This superconducting order can be viewed as a spatially modulated SC order, pair-density-wave (PDW), Given that the stripe orientation varies 90 on moving from one layer to the next, and if SC order between one charge stripe and the next is antiphase within a layer, the Josephson coupling between layers is completely frustrated (Himeda et al., 2002; Berg et al., 2007). This interlayer decoupling isolates each superconducting CuO2 plane and thus a two-dimensional superconducting state (2D-SC), likely PDW, emerges in the stripe-ordered state. The phase diagram of LBCO is shown in Fig. 5B. For x ¼ 1/8 (TCO ¼ 55 K) the onset of 2D pairing correlation within the CuO2 plane coincides with the onset of the spin-stripe order (TSO
40 K). At TSO a steep drop of the in-plane resistivity (rab) is observed whereas the c-axis resistivity (rc) continues to increase. With further cooling below
16 K, rab becomes vanishingly small but rc is still finite, evidencing that the pairs establish a long-range coherence within a plane - 2D superconductivity order - through a Berezinskii-Kosterlitz-Thouless (BKT) transition observed at TBKT
16 K (Li et al., 2007). The pronounced nonlinear current-voltage characteristics observed below 16 K yields further support the 2D-SC. Different from other cuprate families, the coexisting spin stripe order in the La214 family is a prerequisite for realizing the supposed PDW state with antiphase SC order between neighboring charge stripes within a layer. In an array of antiphase charge stripes the pair wavefunction can get zero where the amplitude of spin stripe is maximum which matches the empirical rule that static SC order avoids spatial overlap with static AF order. This might be the reason why the onset of 2D-SC coincides with TSO. To achieve three-dimensional superconductivity (3D-SC), it is necessary to lock together the phases of the superconducting CuO2 planes via interlayer Josephson coupling. When the 2D superconducting correlations within the plane sufficiently develop, even a small interlayer Josephson coupling is enough to realize 3D-SC. The frustration of the interlayer Josephson coupling is strongest at x ¼ 1/8. As a consequence, 3D bulk superconductivity does not easily occur. When the Ba content x in LBCO is changed to slightly smaller or larger values than 1/8, the stripe order and hence PDW order gets weaker. As a consequence, the 3D-SC or uniform d-SC order rapidly develops due to weakening of the frustration of interlayer Josephson coupling. The onset of the 3D-SC, Tc3D, shows a steep and deep dip at x ¼ 1/8, forming two Tc-domes. Thus, it is likely that 3D SC in x ¼ 1/8 LBCO below Tc3D
5 K develops due to inhomogeneity in the local hole density, resulting from Josephson coupling between dilute patches in the neighboring layers where the hole density deviates from p ¼ 1/8. The presence of fluctuating charge stripe has been confirmed for the La214 cuprate family (Miao et al., 2017). In LBCO with x ¼ 1/8, static stripe order is absent above TCO ¼ 55 K (¼ TLTT), instead the charge-stripe fluctuations with dynamic stripe correlations are observed by RIXS at temperatures above 55 K. They show up in RIXS as a broad peak centered near QCO via the x-ray resonant scattering process. This high-T scattering is found to comprise spectral weight by about 7 times larger than that of the low-T scattering from static charge stripes, similar to the relation between the precursory CDW and the quasistatic CDW in the YBCO family. Regardless of the stripe order in La214 and the CDW in other cuprates, they show up as dynamic CDW correlations almost ubiquitously observed at high temperatures. At low temperatures, QCO (or QCDW) in La214 would be determined by coupling between CDW and SDW that organizes well-correlated stripe order. On the other hand, in other cuprates this mechanism does not work due to the presence of a spin gap and the absence of the LTT lattice structure. In this case, other material-specific details may be relevant for determining the characteristics of the quasi-static CDW. The PG temperature T  is not clearly defined in the stripe ordered cuprates, but a gap distinct from the d-SC gap opens near antinodes below TSO (¼ Tc2D) (He et al., 2009). Its momentum dependence is almost the same as that for underdoped Bi2212 below Tpair where PG and d-SC coexist. Also, in common with the canonical PG state is that SDW, CDW and pairing orders and their precursors are intertwined. From these, the stripe ordered state may be regarded as a PG state with a different appearance, and the onset of precursory SC at Tpair in the generic phase diagram would correspond to the onset of 2D superconductivity within a

588

Phase diagrams of high-temperature superconductors

CuO2 plane without interlayer phase coherence. In fact, with lowering T below Tpair a BKT-like transition is observed in strongly anisotropic Bi2212 at T slightly above Tc, preempting the onset of 3D superconductivity (Corson et al., 1999; Li et al., 2005). The generic phase diagram suggests that short-range and possibly fluctuating charge, spin, and pairing orders might be intertwined in the PG regime. The 2D superconductivity in the stripe ordered cuprates is best explained by the presence of PDW, a typical spin-charge-pair intertwined order. Also for non-La214 cuprates, a series of the STS experiments on Bi2212 have yielded evidences for PDW with spatially modulated superfluid density and superconducting gap magnitude that coexist with the uniform d-wave SC order below Tc (Hamidian et al., 2016; Edkins et al., 2019; Du et al., 2020) and persists to the PG regime above Tc. PDW appears in common with all the cuprate families and hence another candidate for the origin of PG.

Overdoped regime This regime is roughly divided into two regions at around the higher end p2 of the Tc dome, although p2 is not a sharp boundary and dependent on homogeneity of doped hole density in real materials. One region is lightly overdoped region covering from the doping optimal for SC (popt) or a putative end of PG (p ) to p2 (see Fig. 3). Tc starts to gradually decrease and then is rapidly diminished toward p2. It appears that a Fermi liquid begins to be established with a well-developed large Fermi surface consistent in detail with the prediction of one electron band theory. This is supported by ARPES measurements where now relatively sharp peaks are observed even in the vicinity of the antinodes. In this “lightly overdoped region” CDW and other orders are not seen, but SC apparently survives maintaining fairly high Tc. The T2 component in the in-plane resistivity in the normal state, characteristic of a Fermi liquid, gradually dominates. However, the T-linear component remains, which is characteristic of the resistivity above T  in the PG regime and is interpreted as a sign of ‘strange’ metallicity. Inelastic neutron scattering (INS) data indicate a dramatic suppression of magnetic spectral weight near the antiferromagnetic wavevectors QAF ¼ (0.5, 0.5) (Wakimoto et al., 2007), which may be interpreted as a disappearance of the spin-fluctuation pairing glue, explaining why Tc goes down. Curiously the superfluid density rs(0) at T
0 also starts to decrease for p beyond popt or p , although the normal-state carrier density continues to increase (Uemura et al., 1993; Bernhard et al., 2001; Bozovic et al., 2016). This is quite anomalous, since, in a clean conventional superconductor, all carriers are expected to form the SC condensate. A typical interpretation of the reduction in rs(0) is a dominance of pair breaking due to disorder (high density of dopant atoms) coupled with a diminished pairing amplitude D0 in the overdoped regime (Lee-Hone et al., 2017). Another is based on existence of two fluids in this region; one fluid involves incoherent quasiparticles (QPs), possibly related to SC condensate, and the other does coherent QPs originating from a non-SC Fermi liquid. In a real-space picture, the origin of the decreased superfluid density is a phase separation. With increasing p toward p2, non-SC phase, probably the genuine FL phase well beyond p2 is gradually mixed up with the SC phase due to a significant distribution of local hole density p. If this lightly overdoped region were not a single phase but spatially phase separated, then a single-phase uniform d-SC state might be unstable in the cuprates. It is also possible to speculate a momentum-space “phase separation”. In this doping region, QPs are coherent on a large portion of the Fermi surface, contributing to the charge transport, but there remain incoherent QPs on some small portion which is responsible for the T-linear component in the resistivity and form superfluid below Tc. For further overdoping (p > p2
0.25) in the “heavily overdoped region”, SC is completely suppressed, and quasiparticles are coherent throughout the large FS, which is confirmed by both ARPES and quantum oscillations measurements. Most of the properties in this region are normal except for that the spin wave, characteristic of localized spins, is robustly present without changing its dispersion in the AF ordered phase. While spin fluctuations at QAF fade out, pronounced spin fluctuations remain at smaller wavevectors (Dean et al., 2013), implying that strong electron correlations persist even in heavily overdoped cuprates. Then, it is speculated that a second high-Tc dome may appear in the doping region (e.g., p > 0.4) next to the heavily overdoped FL (Maier et al., 2019). This region is not accessible for the representative cuprate families such as LSCO, YBCO, Bi2212, and even overdoped Tl2201. At present, at least two cuprate materials, Cu0.75Mo0.25Sr2YCu2O7.54 and Ba2CuO4-y synthesized under high pressures, show SC with Tc
80 K and are supposed to have doped hole density in the range between 0.4 and 0.5.

Strange metal regime The metallic state in the high T region above the PG T  and that in the region from the putative critical doping p to near the end of the Tc dome are recognized as “bad” metal and “strange” metal, respectively. In the high-temperature “bad” metal regime, the in-plane resistivity continues to increase linearly with T without a tendency for saturation, while in the low-temperature “strange” metal regime the T-linear resistivity persists even toward T ¼ 0 (Martin et al., 1990; Cooper et al., 2009) when superconductivity is suppressed by strong magnetic fields or by disorder (see Fig. 6). These two are in principle distinct - a system can be bad without being strange and vice versa. However, the cuprates are apparently both bad and strange. In a range of doping, near popt, the resistivity is linear in T all the way from T
0 (when SC is suppressed) up to the highest temperatures as measured (typically
1000 K) (Gurvitch and Fiory, 1987). The slope of the T-linear resistivity is constant, showing no discernible crossover from “strange” to “bad” metal regime. Given that both are not distinguished in the cuprates, the high-temperature “bad metal” regime above T  is labeled by “strange metal” in the phase diagram (Fig. 3), The basic difference of the high-T strange metal from a conventional metal is the absence of quasiparticles. In a normal metal, the resistivity saturates at high temperatures when the mean free path (l) of charge carrier becomes of the order of its de Broglie wavelength lF (lF ¼ 2p/kF where kF is the Fermi wavenumber) – the Mott-Ioffe-Regel (MIR) limit. Since the magnitude of the

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589

1.0 La2-xSrxCuO4 x = 0.2 In-plane resistivity rab (mΩ cm)

isochoric resistivity

Bad Metal

0.5

Mott-Ioffe-Regel limit

Strange Metal 0

0

500

1000

Temperature T (K) Fig. 6 Typical temperature dependent in-plane resistivity measured for La2-xSrxCuO4 with x ¼ 0.20. The resistivity values measured under constant pressure are transformed to the values at constant sample volume (isochoric) using the data of the pressure dependence of resistivity and bulk modulus. The resistivity increases linearly to the highest temperatures without showing saturation at temperatures well exceeding the Mott-Ioffe-Regel limit (shaded region; its numerical value is somewhat arbitrary). The T-linear resistivity persists even toward T ¼ 0 (red dashed line) when the superconducting transition is suppressed by magnetic fields.

cuprate resistivity in the high-T region is larger than that of typical metals by more than an order, the resistivity well exceeds the MIR limit in the high T region (Hussey et al., 2004). Nevertheless, the resistivity shows a “metallic” linear-in-T dependence. Therefore, well-defined quasiparticles do not exist and hence the resistivity is difficult to be interpreted in terms of propagating quasiparticles in the Boltzmann transport theory. The failure of the ordinary physics of good metals has provided strong motivation for proposing quite an exotic theory such as holographic duality (Zaanen, 2019). The origin of this behavior is, however, still a subject of much debate, but is of obvious importance since PG and/or FL appear to get born upon cooling from this regime, perhaps one of the most mysterious aspects of cuprates. PG and FL emerge from totally incoherent ‘strange metal’ by partial restoration of quasiparticle coherence over a range of the Fermi surface (Fermi arc) in the former and a complete restoration in the latter. In the low-T strange metal regime, the T-linear resistivity persistent down to T
0 indicates a scale invariance or a vanishing intrinsic energy scale which is characteristic of quantum critical metals. When subjected to infrared photons with frequency o, the low-energy optical conductivity of scale invariant metals is set by the larger energy scale between kBT and ħo, that is, the inelastic scattering rate of electrons, ħ/t
max [kBT, ħo] (van der Marel et al., 2003). In the same way, under external magnetic fields B, the magnetoresistance, Dr/r0 ¼ [r(B) – r(0)]/r(0), is controlled by ħ/t
max [kBT, mBB], mB being the Bohr magneton. A crossover in magnetoresistance is actually observed at low temperatures from Dr/r0
B2 at low fields to Dr/r0
B at high fields (Ayres et al., 2021). The recent experimental findings challenge the view of the scale invariance in association with QCP. First, experimental effort to unambiguously identify the presence of the QCP is limited by the onset of SC at fairly high temperature. Second, the T-linear resistivity extending to T
0 persists over a broad doping range, from popt to around the end of the Tc dome in contrast to the above QCP scenario. Third, concerning the quantum critical wedge in the higher doping side, p > p , a big question that remains unanswered is how really different the supposed Fermi liquid at lower temperatures is from the strange metal at higher temperatures. ARPES shows there is only a weak crossover line that separates these two regimes, but other experimental probes fail to identify it.

Conclusion Summary Recent progress in various advanced spectroscopies has revealed that the T-p phase diagram of hole-doped cuprates is much more complicated than the widely spread ‘standard’ phase diagram (Fig. 2). The generic phase diagram incorporates various spin, charge and electron-pair orders in addition to the unconventional d-wave SC order. It turns out that various ordering phenomena including their precursory orders predominantly take place in the PG regime. Thus, the PG regime seems a home to these orders. The PG regime appears to also accommodate some intra-unit-cell (q ¼ 0) orders such as circulating orbital current order and nematic order, which are observed exactly below the pseudogap temperature T  , but they cannot create a gap by themselves. These recent findings pose additional questions, which are primary orders and how they are inter-related (or intertwined) to shape the generic phase

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diagram and finally to establish superconductivity at Tc of
100 K with global phase coherence. To this, the spin-charge stripe order realized in the La-based cuprates is an instructive example. The stripe order displays how the spin and charge orders are intertwined with the pairing correlation to form a spatially modulated SC order, PDW, and thereby 2D superconductivity. The high-temperature metallic state above T  is recognized as a strange or bad metal regime which cannot be described by the quasiparticle picture. The quasiparticles are totally absent or completely incoherent, but nonetheless the T-linear resistivity persists up to the highest temperature measured. The origin of this behavior is still a subject of much debate, but the understanding of this regime is of obvious importance since PG in the underdoped region and probably FL in the heavily overdoped region appear to get born upon cooling from this regime. The failure of the ordinary physics of good metals has provided strong motivation for proposing quite a exotic theory such as holographic duality. Another important observation by the advanced spectroscopy is the persistence of spin and charge excitations with SDW (or AF) and CDW correlations up to temperatures even higher than T  and in the entire SC doping region. Both cover a wide momentum region centered at AF and CDW wavevectors and extend to very high energies of order of a few tens of eV. They are not only possible sources of strong electron scattering that makes the electrons completely incoherent but also possible glues for the formation of Cooper pairs. The AF spin fluctuations have been widely believed as a strong candidate of the pairing glue. A question to be answered is whether the CDW fluctuations play a role of an independent or additional glue to enhance the SC Tc or conversely as a pair breaker for d-SC.

Outlook Magnetic phase diagram Since the various orders emerge with doping into the Mott insulating phase, the cuprate phase diagram has been investigated mainly in the temperature-doping plane. In order to better understand the known phases and to search for hidden phases, it would be helpful to explore a phase diagram using a control parameter other than doping. Pressure and magnetic field are typical control parameters for other systems. While pressure, either uniaxial or hydrostatic, is not an effective perturbation to change the completion and/or intertwining among different phases in cuprates, intense magnetic field (B) has recently been found to substantially affect the SC state as well as the competition between SC and CDW orders (Yu et al., 2016). The B-T phase diagram of cuprates with B in the direction perpendicular to the CuO2 planes, has not been fully investigated because of the limitation of available magnetic field strength, and continues to be a matter of debate. One of the unsettled problems is that experiments so far made have never observed the upper critical field Hc2, i.e. the pair-breaking field, convincingly even near Tc. The vortex lattice in cuprates easily melt at a field value Hm(T ) much weaker than the inferred Hc2 (T ), and the vortex liquid state is found to persist above Tc up to Tpair (Li et al., 2013). As a consequence, the vortex liquid phase occupies a surprisingly large region of the B-T phase diagram. Extending to a temperature Tpair and to an experimentally inaccessible high field. Further, the spatially modulated SC state (PDW), a spin-charge-pairing intertwined order typically observed in the stripe order, appears to be stabilized in the region of a vortex halo as evidenced by the STS measurement on Bi2212 (Edkins et al., 2019). However, effects of magnetic field on CDW and SDW orders that probably coexist with the vortex liquid state remain open. Perhaps related open question is whether or not an electron Fermi surface pocket, which is supposed to emerge as a consequence of Fermi-surface-reconstruction in the presence of a static CDW, exists in the vortex liquid region suggested by the observation of quantum oscillations (Doiron-Leyraud et al., 2007).

New spectroscopy and hidden orders Unraveling the origin of the pseudogap (PG) state of the high-Tc cuprates has been an issue of continuous challenge for almost three decades. PG turns out to be a playground of several orders. Among them, SDW order, superconducting order and CDW order are now well known and experimentally confirmed. These three orders and their precursors are found to exist in the PG regime. Nevertheless, it remains unclear how they form the PG state above Tc. This suggests that there might be a hidden order which has not been detected by the currently available experimental probes, and would be a remaining piece for finally resolving the PG puzzle. To uncover hidden orders, a challenge for developing a new spectroscopy may be needed, such as one, for example, that allows for a detection of dipole-forbidden optical excitations or is sensitive to various forms of vortices or topological defects as well as to orders and excitations involving orbital degree of freedom. In the history of the research of high-Tc superconductivity, breakthroughs have always been made by new spectroscopy.

References Arpaia R, et al. (2019) Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-Tc superconductor. Science 365: 906–910. Ayres J, et al. (2021) Incoherent transportacross the strange metal regime of highly overdoped cprates. Nature 595: 661–667. Berg E, et al. (2007) Dynamical layer decoupling in a stripe-ordered high-Tc superconductor. Physical Review Letters 99: 127003. Bernhard C, et al. (2001) Anomalous peak in the superconducting condensate density of cuprate high-Tc superconductors at a unique critical doping state. Physical Review Letters 86: 1614–1617. Blanco-Canosa S, et al. (2014) Resonant x-ray scattering study of charge-density wave correlations in YBa2Cu3O6+x. Physical Review B 90: 054513. Bozovic I, He X, Wu J, and Bollinger AT (2016) Dependence of the critical temperature in overdoped copper oxides on superfluid density. Nature 536: 309–311. Chang J, et al. (2012) Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67. Nature Physics 8: 871–876.

Phase diagrams of high-temperature superconductors

591

Chen SD, et al. (2019) Incoherent strange metal sharply bounded by a critical doping in Bi2212. Science 366: 1099–1102. Cooper RA, et al. (2009) Anomalous criticality in the electrical resistivity of La2-xSrxCuO4. Science 323: 603–607. Corson J, Mallozzi R, Orenstein J, Eckstein JN, and Bozovic I (1999) Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+d. Nature 398: 221–223. Dean MPM, et al. (2013) Persistence of magnetic excitations in La2-xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal. Nature Materials 12: 1019–1023. Ding H, et al. (1996) Spectroscopic evidence for a pseudogap in the normal state of underdoped high-Tc superconductors. Nature 382: 51–54. Doiron-Leyraud N, et al. (2007) Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor. Nature 447: 565–568. Du Z, et al. (2020) Imaging the energy gap modulations of the cuprate pair-density-wave state. Nature 580: 65–70. Edkins SD, et al. (2019) Magnetic field-induced pair density wave in the cuprate vortex halo. Science 364: 976–980. Emery VJ and Kivelson SA (1995) Importance of phase fluctuations in superconductors with small superfluid density. Nature 374: 434–437. Fauqúe B, et al. (2006) Magnetic order in the pseudogap phase of high-Tc superconductors. Physical Review Letters 96: 197001. Fong HF, et al. (2000) Spin susceptibility in underdoped YBa2Cu3O6+x. Physical Review B 61: 14773–14786. Ghiringhelli G, et al. (2012) Long-range incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x. Science 337: 821–825. Gurvitch M and Fiory AT (1987) Resistivity of La1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: Absence of saturation and its implication. Physical Review Letters 59: 1337–1340. Hamidian MH, et al. (2016) Detection of a Cooper-pair-density-wave in Bi2CaCu2O8+x. Nature 532: 343–347. He R-H, et al. (2009) Energy gaps in the failed high-Tc superconductor La1.875Ba0.125CuO4. Nature Physics 5: 119–123. Himeda A, Kato T, and Ogata M (2002) Stripe statewith spatially oscillating d-wave superconductivity in the two-dimensional t-t’-J model. Physical Review Letters 88: 117001. Homes CC, et al. (1993) Optical conductivity of c axis oriented YBa2Cu3O6.70: Evidence for a pseudogap. Physical Review Letters 71: 1645–1648. Hussey NE, Takenaka K, and Takagi H (2004) Universality of the Mott-Ioffe-Regel limit of metals. Philosophical Magazine 84: 2847–2864. Kohsaka Y, et al. (2007) An intrinsic bond-centered electronic glass with unidirectional domains in underdoped cuprates. Science 315: 1380–1385. Lawler MJ, et al. (2010) Intra-unit-cell electronic nematicity of the high-Tc copper-oxide pseudogap states. Nature 466: 347–351. Lee-Hone NR, Dodge JS, and Broun DM (2017) Disorder and superfluid density in overdoped cuprate superconductors. Physical Review B 96: 024501. Li L, Wang Y, Naughton MJ, Ono S, Ando Y, and Ong NP (2005) Strongly nonlinear magnetization above Tc in Bi2Sr2CaCu2O8+d. Europhysics Letters 72: 451–457. Li Q, Hucker M, Gu GD, Tsvelik AM, and Tranquada JM (2007) Two-dimensional superconducting fluctuations in stripe-ordered La1.875Ba0.125CuO4. Physical Review Letters 99: 067001. Li L, et al. (2010) Diamagnetism and Cooper pairing above Tc in cuprates. Physical Review B 81: 054510. Li L, Yayu W, and Ong NP (2013) Reply to “Comment on ‘Diamagnetism and Cooper pairing above Tc in cuprates’”. Physical Review B 87: 056502. Loeser AG, et al. (1996) Excitation gap in the normal state of underdoped Bi2Sr2CaCu2O8+d. Science 273: 325–329. Loret B, et al. (2019) Intimate link between charge density wave, pseudogap and superconducting energy scales in cuprates. Nature Physics 15: 771–775. Maier T, Berlijn T, and Scalapino DJ (2019) Two pairing domes as Cu2+ varies to Cu3+. Physical Review B 99: 224515. Marshall DS, et al. (1996) Unconventional electronicstructure evolution with hole doping in Bi2Sr2CaCu2O8+d: Angle-resolved photoemission results. Physical Review Letters 76: 4841–4844. Martin S, Fiory AT, Fleming RM, Schneemeyer LF, and Waszczak JV (1990) Normal-state transport properties of Bi2+xSr2-yCuO6+d crystals. Physical Review B 41: 846–849(R). Miao H, et al. (2017) High-temperature charge density wave correlations in La1.875Ba0.125CuO4 without spin-charge locking. Proceedings of the National Academy of Sciences of the United States of America 114: 12430–12435. Orenstein J and Millis AJ (2000) Advances in the physics of high-temperature superconductivity. Science 288: 468–474. Tanaka K, et al. (2006) Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212. Science 314: 1910–1913. Tranquada JM, Sternlieb BJ, Axe JD, Nakamura Y, and Uchida S (1995) Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375: 561–563. Tsuei CC and Kirtley JR (2000) Pairing Symmetry in cuprate superconductors. Reviews of Modern Physics 72: 969–1016. Uemura YJ, et al. (1989) Universal correlations between Tc and ns/m in high-Tc cuprate superconductors. Physical Review Letters 62: 2317–2320. Uemura YJ, et al. (1993) Magnetic-field penetration depth in Tl2Ba2CuO6+d in the overdoped regime. Nature 364: 605–607. van der Marel D, et al. (2003) Quantum critical behavior in a high-Tc superconductor. Nature 425: 271–274. Varma C (2006) Theory of the pseudogap state of the cuprates. Physical Review B 73. 155113. Wakimoto S, et al. (2007) Disappearance of antiferromagnetic spin excitations in overdoped La2-xSrxCuO4. Physical Review Letters 98. 247003. Wu T, et al. (2011) Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy. Nature 477: 191–194. Xu ZA, et al. (2000) Vortex-like excitations and the onset of superconducting phase fluctuation in underdoped La2-xSrxCuO4. Nature 406: 486–488. Yu F, et al. (2016) Magnetic phase diagram of underdoped YBa2Cu3Oy inferred from torque magnetization and thermal conductivity. Proceedings of the National Academy of Sciences of the United States of America 113: 12667–12672. Zaanen J (2019) Planckian dissipation, minimal viscosity and the transport in cuprate strange metals. SciPost Physics 6: 061.

Further reading Agterberg DF, et al. (2020) The physics of of pair density waves. Annual Review of Condensed Matter Physics 11: 231–270. Keimer B, Kivelson SA, Norman MR, Uchida S, and Zaanen J (2015) From quantum matter to high-temperature superconductivity in copper oxides. Nature 518: 179–186. Lee PA, Nagaosa N, and Wen X-G (2006) Doping a Mott insulator: Physics of high-temperature superconductivity. Reviews of Modern Physics 78: 17–85. Sachdev S (1999) Quantum Phase Transitions. Cambridge: Cambridge University Press. Scalapino DJ (2012) A common thread: the pairing interaction for unconventional superconductors. Reviews of Modern Physics 84: 1383–1417. Tranquada JM (2020) Cuprate superconductors as viewed through a stripe lens. Advances in Physics 69: 437–509. Zaanen J, Sun Y, Liu Y, and Schalm K (2015) Holographic duality for condensed matter physics. Cambridge: Cambridge University Press.

Superconductors, hydrogen-based Katsuya Shimizu, KYOKUGEN, Center for Science and Technology under Extreme Conditions, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Superconductivity and high pressure The Holy Grail Hydrogen-rich compounds High pressure Theoretical predictions Synthesis of superconducting hydrides Pressure generation and measurement technique Synthesis through molecular dissociation Direct synthesis from element and hydrogen Near room-temperature superconductivity in hydrides Lanthanum hydride USOs (unidentified superconducting objects) Future prospects Toward higher temperatures Toward lower pressure Control of hydrogenation Conclusion References

592 592 592 593 593 594 594 594 595 595 595 595 596 596 596 596 596 597 597

Abstract Superconductivity, which was previously believed to occur only at low temperatures close to zero Kelvin, has recently been believed to occur even at room temperature. Experimental results of superconductivity at high temperatures close to room temperature have been reported in hydrogen-based materials, hydrides, although ultrahigh pressures in excess of 100 GPa are required.

Key points

• • •

Explore the occurrence of superconductivity at higher temperatures, including room temperature. Experimental results of hydrogen-based superconductivity. Requirement of very high pressures for hydrogen-based high-temperature superconductivity.

Introduction Superconductivity, which was previously believed to occur only at low temperatures close to zero Kelvin, has recently been believed to occur even at room temperature. Although it requires extreme pressure conditions of over 1 million atmospheres (about 100 GPa), the RTS (room-temperature superconductivity) is not a dream but has become a concrete possibility, raising hopes for the emergence of superconducting materials without cooling in the near future. For over a century since the discovery of the superconductivity in the early 20th century, researchers have focused on increasing the critical temperature (Tc) for reducing the cooling costs. The historical progression is illustrated in Fig. 1. Following the rapid rise in the Tc observed in the 1980s with the discovery of copper oxide superconductors, the recent surge in research on hydride superconductors (hydrogen-based superconductor) has sparked considerable interest. How did this rapid development occurred?

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Fig. 1 History of superconducting transition temperatures (Tc). Experimentally observed superconducting transition temperatures are shown for the year of discovery. Symbols are classified according to the type of superconductivity. ●: conventional (BCS), ▲: unconventional (not explained by BCS), : iron-based.

▪

Superconductivity and high pressure The Holy Grail There is something often referred to as the Holy Grail by researchers in the field of high-pressure science. It is solid metallic hydrogen. Hydrogen, atomic number 1, is the most fundamental element consisting of one proton and one electron. What would be the nature of hydrogen at extremely high dense condition? Hydrogen is normally a gas, forming molecules (H2), but when compressed, it transitions through a liquid phase and become a solid. With further compression, the molecules are expected to dissociate into monatomic structure, with protons forming a crystalline lattice, resulting in solid metallic hydrogen. According to the BCS theory, which explains the mechanism of superconductivity, metallic hydrogen, with its strong electron-phonon interaction and high lattice vibration frequency, is expected to exhibit superconductivity with very high Tc, room-temperature superconductivity (Ashcroft, 1968). On the other hand, liquid metallic hydrogen exists in the interiors of gas giant planets such as Jupiter, and it is believed to be the source of their geomagnetic field. This is the same principle as the generation of geomagnetism by liquid iron in the Earth’s interior. Liquid metallic hydrogen has already been produced in the laboratory (Weir et al., 1996; Ohta et al., 2015), solid metallic hydrogen has remained elusive due to the lack of sufficient pressure achieved by current high-pressure techniques. This is why it is referred to as the Holy Grail.

Hydrogen-rich compounds As an alternative to metallic hydrogen, hydrogen-rich compounds had been proposed as candidates for room-temperature superconductors. Instead of compressing hydrogen itself, the proposal is to compress hydrogen-rich compounds, where hydrogen within the crystal can approach a state similar to metallic hydrogen at lower pressures than compressing hydrogen itself, thus creating a metallic hydrogen-like state in the hydrogen-rich compound (Ashcroft, 2004). In 2015, a German research group has conducted extensive experiments upon this concept, they discovered superconductivity with a Tc exceeding 200 K by pressurizing hydrogen sulfide (H2S) (Drozdov et al., 2015). This breakthrough led to hydrogen-rich compounds, hydrides, becoming strong candidates for high-temperature superconductors.

High pressure It is fair to say that the properties of matter are determined by the interaction of electrons. This is not limited to superconductivity; changing the distance between atoms with pressure can change this interaction between electrons and significantly alter the properties of matter. When higher pressure is applied, crystals not only shrink but also change their arrangement (crystal structure), leading to the emergence of new properties resulting from the new structure.

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Theoretical predictions It is generally not easy and challenging to theoretically predict the crystal structure and resulting properties of a material under high pressure, which greatly influences its characteristics. However, recent advancements in computational methods have significantly improved the accuracy of prediction of stable structures under pressure. The discovery of superconductivity in hydrogen sulfide (Drozdov et al., 2015) was initiated by theoretical predictions (Li et al., 2014) and later confirmed through experiments, which revealed that the observed crystal structure and Tc (Einaga et al., 2016) were consistent with the theoretical predictions (Duan et al., 2014). This not only proves the accuracy of the theoretical prediction, but also that this superconductivity is based on the BCS theory. The success of this theoretical prediction led to the search for the next hydride superconductor, and calculations were actively carried out. One element was selected from the periodic table to be combined with hydrogen, and the ratio of hydrogen, pressure, and structure at which the hydride would be stable were calculated. They then use these structural parameters to calculate the superconducting properties, proposing various hydrogen-rich compounds as potential candidates for high-temperature superconductivity. In other words, hydride synthesis experiments are being conducted in the computer before actual experiments in laboratory. Table 1 lists some of the candidates that have the potential to realize room temperature superconductivity (Wang et al., 2012; Feng et al., 2015; Liu et al., 2018).

Synthesis of superconducting hydrides Pressure generation and measurement technique Before discussing the specific details of hydride superconductors, the generation of high-pressure environments exceeding 100 GPa and the techniques for measuring superconductivity within these environments are briefly introduced in Fig. 2. See references (Shimizu et al., 2005) for details. High pressure is generated by a Diamond-Anvil Cell (DAC). A photograph of the DAC for low-temperature use is shown in Fig. 2C, and its pressure generation section is shown in Fig. 2B. The DAC is composed of two diamond anvils. The sample and pressure transmitting medium are enclosed in a space between two diamond anvils. Four electrodes are usually placed on the sample to perform a four-terminal electrical resistance measurement. These electrodes can also be deposited on the surface of the upper diamond anvil. External heating of the sample can apply to the sample by irradiating with infrared lasers.

Table 1

Candidate hydrides with potential for room temperature superconductivity.

Hydrides

Tc(K)

P (GPa)

MgH6 CaH6 YH6 YH10 LaH10

260 220– 235 251– 264 305– 326 274

300 150 120 120 200

(A)

(B)

sample

(C)

pressure medium electrode insulation gasket heating laser

0.1 mm

diamond anvil 0.1 mm

1.0 mm

Fig. 2 Preparation of high-pressure experiments. (A) cross section of the heart of DAC (Diamond- Anvil Cell). The sample is pressurized by upper and lower diamond anvil through the pressure medium. The electrical resistance of the sample is measured by four electrodes. (B) schematic drawing of the assembly of the sample and electrodes between diamond anvils. (C) photograph of DAC.

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Synthesis through molecular dissociation Hydrogen sulfide (H2S) is a gas molecule at ambient temperature and pressure. Its boiling point is approximately 200 K, and it can be liquefied by introducing H2S gas into a DAC that has been cooled below its boiling point using a refrigerator. Thus, in the case of H2S, the sample and pressure medium in Fig. 2A are combined and equal to sample. Upon pressurization, H2S liquid immediately solidifies, and further pressurization leads to the metallization, then the superconductor with a Tc of 200 K is achieved (Drozdov et al., 2015). Reproduction experiments and structural analysis using synchrotron X-rays (SPring-8) have revealed that hydride (H3S: sulfur hydride SH3), whose hydrogen content is higher than that of hydrogen sulfide (H2S), is the true entity of the 200 K superconductor. Furthermore, X-ray structural analysis revealed that sulfur is precipitated during the synthesis process, and as shown in Eq. (1), H3S with a higher hydrogen content is formed from parent substance, H2S displacing sulfur under high pressure as Fig. 3. 3H2 S ! 2H3 S + S

(1)

This is an example of hydride synthesis through the molecular dissociation. For the candidate substances listed in Table 1, there is currently no parent substance (such as H2S for H3S) at this moment.

Direct synthesis from element and hydrogen After the discovery of the superconductivity of hydrogen sulfide (Drozdov et al., 2015), a direct synthesis of hydride from sulfur and hydrogen was attempted. By substituting sulfur for the sample and hydrogen for the pressure medium in Fig. 2, this synthesis was performed. After pressurization up to 150 GPa, sulfur transformed to metal, but any reaction with hydrogen did not occur. When the metallic sulfur was heated by an infrared laser, the hydrogen in contact with the sulfur was heated together and the synthesis reaction (2S + 3H2 ! 2H3S) occurred. This was monitored by the powder X-ray diffraction analysis, which was performed simultaneously with this laser heating and electrical resistivity measurements. When the DAC was cooled afterwards, the superconducting transition was observed at about 200 K, as in Drozdov et al., 2015 reported. In comparison to the synthesis through the molecular dissociation mentioned, this direct synthesis method was expected to have an advantage in purity and crystallinity of H3S without the excess sulfur, but the observed Tc remained the same (Nakao et al., 2019). This direct synthesis method served as a model for the syntheses of superconducting hydrides, and most of the later synthesis experiments employed this method.

Near room-temperature superconductivity in hydrides Lanthanum hydride Among the candidates listed in Table 1, the synthesis of lanthanum hydride (LaH10) and its superconductivity with a transition temperature exceeding 250 K were reported nearly simultaneously by research groups from the United States and Germany in 2019. In both cases, the superconductivity was discovered at pressures and temperatures that were approximately equal to the values shown in Table 1 (Somayazulu et al., 2019; Drozdov et al., 2019). LaH10 has a structure in which hydrogen atoms surround lanthanum atoms in a cage-like arrangement as shown in Fig. 3; although the position of the hydrogen element cannot be determined by X-ray structure analysis, the structure formed by lanthanum is consistent with what theoretical calculations predict. The synthesis methods of both groups are based on Fig. 2A, with the German group (Drozdov et al., 2019) using hydrogen and the

Fig. 3 Typical crystal structures of superconducting hydrides. (left) H3S. Hydrogen atoms are arranged in a straight line between the lattices of sulfur atoms. (right) LaH10. Hydrogen atoms surround lanthanum atoms.

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US group (Somayazulu et al., 2019) using ammonia borane (NH3BH3) as a hydrogen source. After pressurizing the sample at room temperature to the theoretically predicted pressure, the sample was heated by an infrared laser and synthesized by reacting the sample with hydrogen in contact with the sample or with hydrogen generated from ammonia borane. The use of ammonia borane as a hydrogen source has become more common in recent years due to the convenience of experimental setup compared to using hydrogen directly.

USOs (unidentified superconducting objects) Many hydride superconductors have been predicted from theoretical calculations and many demonstration experiments have been reported. It is foreseen that the day when the superconducting transition temperature exceeds room temperature is approaching. In this context, in 2020, a report emerged of observing the superconductivity with the Tc exceeding room temperature at 250 GPa in a carbon-sulfur hydride compound (CSH) (Snider et al., 2020). Furthermore, in 2023, the observation of room-temperature superconductivity in nitrogen-doped Lu hydride (NLuH) at very low pressure, 1 GPa was announced (Dasenbrock-Gammon et al., 2023). There on the Web and in newspapers are claiming “room-temperature superconductivity has finally been achieved.” However, at the time of writing this article, no research group has been able to reproduce these experiments, and the paper (Snider et al., 2020) has been retracted. These things remind us the term of “Unidentified Superconducting Object” (USO) that defined by Prof. Kitazawa to refer to materials that failed to satisfy all of the following conditions during the fever of copper oxide superconductors in the 1980s: (1) observation of zero resistance, (2) observation of the Meissner effect, (3) determination of crystal structure, and (4) reproduction by other research groups. At the moment, (3) and (4) are missing from both CSH and NLuH, so a follow-up report will be forthcoming.

Future prospects Toward higher temperatures How can we achieve higher transition temperatures (Tc)? One approach is to substitute or add lighter elements as constituents. In the case of hydrogen sulfide, this would involve replacing sulfur with a lighter element in the same group, such as oxygen, or adding oxygen to the system. Such proposals based on theoretical calculations have been put forward, including the suggestion of adding phosphorus (Ge et al., 2016; Nakanishi et al., 2018). These substitutions and additions have been common strategies employed whenever a new superconductor is discovered. However, as of now, no significant increase in Tc has been observed through these methods in superconducting hydrides.

Toward lower pressure Theoretical calculations and experiments are underway for ternary hydrides with one more parent element. The purpose of this research is to explore superconducting hydrides that can be synthesized at lower pressures, in addition to achieving higher transition temperatures (Tc). The ultimate goal is to synthesize room-temperature, ambient-pressure superconductors that do not require pressure. If the synthesis pressure can be reduced to less than 50 GPa, which can be generated by a high-pressure apparatus such as a KAWAI-type press, it will make possible to synthesize the superconductors with millimeter-sized samples. Further reduction of the pressure to a few GPa could enable the synthesis of centimeter-sized samples. This would enable the neutron diffraction experiments with sufficient intensity and reveal the nature of hydrogen, the key element in hydride superconductors. Then this would contribute to a better understanding of hydride superconductors and the development of pressure-free room-temperature superconductors will become more realistic.

Control of hydrogenation There have been several theoretical proposals for hydrides exceeding room temperature, but experimental verification has been challenging. This is partly due to the high cost of experiments consuming diamonds, but also because successful examples that match the theoretical predictions are limited. For example, different hydrides with different amounts of hydrogen are synthesized. During the synthesis process, it is possible that more stable hydrides with a different hydrogen content exist along the temperature-pressure path, which prevents further hydrogenation. There may be an optimal temperature and pressure pathway that avoids this problem, but it is considered difficult to clarify the synthesis pathway by theoretical calculation due to the large amount of calculation. There is no doubt that hydrogen-rich compounds are strong candidates to realize room-temperature superconductivity. Many candidate materials will be theoretically proposed, experimentally tested, and valuable results will be fed back to theory. This will lead to a more precise selection of candidate materials.

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Conclusion Although an extreme environment of ultrahigh pressure is necessary, hydrogen has been growing expectations not only as an energy source but also as a source of superconductor to conserve energy. However, it is expected to take some more time before we can fully utilize this near room-temperature superconductivity. Current efforts focus on reducing the required pressure and on considering utilizing them even at high pressures. Hydrides are a strong candidate for room-temperature superconductivity, and at the same time, a significant exercise in materials science.

References Ashcroft NW (1968) Metallic hydrogen: A high-temperature superconductor? Physical Review Letters 21: 1748–1749. Ashcroft NW (2004) Hydrogen dominant metallic alloys: High temperature superconductors? Physical Review Letters 92: 187002. Dasenbrock-Gammon N, Snider E, McBride R, Pasan H, Durkee D, Khalvashi-Sutter N, Munasinghe S, Dissanayake SE, Lawler KV, Salamat A, and Dias RP (2023) Evidence of near-ambient superconductivity in a N-doped lutetium hydride. Nature 615: 244–250. Drozdov AP, Eremets MI, Troyan IA, Ksenofontov V, and Shylin SI (2015) Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525: 73–76. Drozdov AP, Kong PP, Minkov VS, Besedin SP, Kuzovnikov MA, Mozaffari S, Balicas L, Balakirev FF, Graf DE, Prakapenka VB, Greenberg ED, Knyazev A, Tkacz M, and Eremets MI (2019) Superconductivity at 250 K in lanthanum hydride under high pressures. Nature 569: 528–531. Duan D, Liu Y, Tian F, Li D, Huang X, Zhao Z, Yu H, Liu B, Tian W, and Cui T (2014) Pressure-induced metallization of dense (H2S)2H2 with high-Tc superconductivity. Scientific Reports 4: 6968. Einaga M, Sakata M, Ishikawa T, Shimizu K, Eremets MI, Drozdov AP, Troyan IA, Hirao N, and Ohishi Y (2016) Crystal structure of the superconducting phase of sulfur hydride. Nature Physics 12: 835–838. Feng X, Zhang J, Gao G, Liu H, and Wang H (2015) Compressed sodalite-like MgH6 as a potential high-temperature superconductor. RSC Advances 5: 59292–59296. Ge Y, Zhang F, and Yao Y (2016) First-principles demonstration of superconductivity at 280 K in hydrogen sulfide with low phosphorus substitution. Physical Review B 93: 224513. Li Y, Hao J, Liu H, Li Y, and Ma Y (2014) The metallization and superconductivity of dense hydrogen sulfide. The Journal of Chemical Physics 140: 174712. Liu H, Naumov II, Hoffmann R, Ashcroft NW, and Hemley RJ (2018) Potential high-Tc superconducting lanthanum and yttrium hydrides at high pressure. PNAS 114: 6990–6995. Nakanishi A, Ishikawa T, and Shimizu K (2018) First-principles study on superconductivity of P- and Cl-doped H3S. Journal of the Physical Society of Japan 87: 124711. Nakao H, Einaga M, Sakata M, Kitagaki M, Shimizu K, Kawaguchi S, Hirao N, and Ohishi Y (2019) Superconductivity of pure H3S synthesized from elemental sulfur and hydrogen. Journal of the Physical Society of Japan 88: 123701. Ohta K, Ichimaru K, Einaga M, Kawaguchi S, Shimizu K, Matsuoka T, Hirao N, and Ohishi Y (2015) Phase boundary of hot dense fluid hydrogen. Scientific Reports 5: 6560. Shimizu K, Amaya K, and Suzuki N (2005) Pressure-induced superconductivity in elemental materials. Journal of the Physical Society of Japan 74: 1345–1357. Snider E, Dasenbrock-Gammon N, McBride R, Debessai M, Vindana H, Vencatasamy K, Lawler KV, Salamat A, and Dias RP (2020) Room-temperature superconductivity in a carbonaceous sulfur hydride. Nature 586: 373–377. Somayazulu M, Ahart M, Mishra AK, Geballe ZM, Baldini M, Meng Y, Struzhkin VV, and Hemley RJ (2019) Evidence for Superconductivity above 260 K in Lanthanum Superhydride at Megabar Pressures. Physical Review Letters 122: 027001. Wang H, Tse JS, Tanaka K, Iitaka T, and Ma Y (2012) Superconductive sodalite-like clathrate calcium hydride at high pressures. PNAS 109: 6463–6466. Weir ST, Mitchell AC, and Nellis WJ (1996) Metallization of fluid molecular hydrogen at 140 GPa (1.4 Mbar). Physical Review Letters 76: 1860–1863.

Unconventional superconductivity Igor I Mazin, Department of Physics & Astronomy, George Mason University, Fairfax, VA, United States © 2024 Elsevier Ltd. All rights reserved.

Abstract The division of the broad field of superconductivity into conventional and unconventional is conditional, just as any binary division based on a subjective notion of “conventionality”. It is often in the eye of the beholder. Still, it is often a useful, even if fuzzy, line between superconductivity that follows, with modifications, the traditional framework established by Bardeen, Cooper, Schrieffer, and Eliashberg, and that which deviates from that path.

Key points

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Defining unconventional superconductivity Higher angular momenta Examples of unconventional superconductors Odd-frequency order parameter Conditions conducive for unconventional superconductivity Some useful theorems Multigap superconductors Further reading

Unconventional superconductivity is an umbrella term that has been used by different researchers with different meaning. Common definitions refer to conventionality in three separate aspects: pairing mechanism, pairing symmetry, and transition temperature. In addition, occasionally conventional in their mechanism superconductors are considered unconventional because of their specific unusual properties. Examples of that include, for instance, Ising superconductors and topological superconductivity. The most restrictive definition of conventional superconductors (CSC) stipulates that CSC are (a) mostly mediated by adiabatic (e.g., respecting the Migdal theorem) harmonic phonons, (b) have isotropic or nearly-isotropic s-wave order parameter and (c) have critical temperature below 20–30 K. Such a limiting definition is rarely called for. Recently discovered superconducting superhydrates are believed to be phonon-mediated and isotropic s-wave, despite having critical temperatures approaching the room temperature and possibly nonadiabatic and anharmonic phonons, yet they are usually classified as CSC. In fact, these criteria are not independent. Indeed, a purely attractive interaction, such as phonon-mediated, cannot generate a superconducting state with a sign-changing order parameter, and, conversely, a pure repulsive (in the momentum space) interaction cannot generate a state that has an order parameter of the same sign everywhere. Most scientists agree that a pairing state that involves an order parameter of lower point group symmetry than the hosting crystal (p-wave, d-wave etc.) should be classified as unconventional, regardless of the transition temperature. Obviously, such states cannot originate from pure phonon attraction, so pairing interaction is by definition unconventional in that sense. On the other hand, repulsive interactions, such as those of magnetic origin, may result in a sign-changing order parameter of the s-wave angular symmetry. Such states are often called s
, and also usually classified as unconventional. At the dawn of superconductivity it was believed that unconventional superconductors are but a theoretical Kunststück that is not supposed to occur in real materials, for the reason that nonmagnetic impurities are pair-breaking for such a state, and—as it was thought—would kill such superconductivity dead except in exceptionally clean samples. It turned out that reality is not that bleak, and, in particular, that unconventional superconductors tend to have short coherence lengths, and therefore less prone to pair breaking (which is controlled by the ratio of the coherence length and the mean free path). In the moment, a number of well-studied superconductors have been convincingly demonstrated to have unconventional pairing symmetries. This includes high-Tc cuprates (d-wave), Fe-based superconductors (s
); a number of heavy fermion systems, where the exact symmetry of the order parameter has not been established above reasonable doubt, are generally believed to host unconventional, possibly triplet, superconductivity. Same can be said about many organic (quasi-1D) superconductors. An interesting case is Sr2RuO4: For at least 20 years it was believed it was a triplet p-wave superconductors, but recent experiments convincingly point toward a singlet state. Yet, there is little doubt that superconductivity there, while still unresolved, is unconventional. A separate, even more exotic case is the so-called odd-frequency superconductivity, whereby the frequency-dependent order parameter is assumed to change sign not (or not only) in the momentum space, but when the frequency argument changes its sign. No realistic candidates among real materials have been identified, albeit there were some proposals of this character, but there is some experimental evidence that such states can be generated by proximity effects near superconductor-ferromagnet interfaces.

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The number of unconventional superconductivity candidates identified in the last decades is considerable, and suggests several empirical rules that are conducive for UCSC. These tend to be low-dimensional (quasi-1D or quasi- 2D), and tend to have a phase diagram where superconductivity emerges near magnetic (usually antiferromagnetic) states. This is not unexpected, given that UCSC is likely to be mediated by some kind of spin fluctuations, and both proximity to magnetism and low dimensionality favor spin fluctuations. Let us now discuss in more details the principal classes of UCSC. High-Tc cuprates emerge in the vicinity of the Mott-Hubbard metal insulator transition. The Cu2+ ions have one hole in the rather localized d-shell, which is therefore subject to strong local Coulomb repulsion, U  t, where t is the one-electron hopping parameters for d-electrons. This generates a nearest neighbor antiferromagnetic superexchange, proportional to t2/U. In parent compounds this results in a checkerboard antiferromagnetic ordering with the wave vector Q ¼ (1,1,0) p/a. Upon doping the magnetic order melts and the insulating state is being transformed into a strongly correlated metal. Instead of the static order, Cu spins now fluctuate with the same wave vector Q. The one-electron band structure of these cuprates is formed by a single x2–y2 orbital, so the corresponding Fermi surface is close to a circular cylinder. It is easy to see that if one postulates the d-wave order parameter with the same symmetry, x2–y2, spin fluctuations with the wave vector Q will be spanning two parts of the Fermi surface with the opposite signs of the order parameter and therefore be pairing. This simple argument has led to the view, shared by a majority of researchers, that superconductivity in cuprates is driven by spin-fluctuations. Similarly, in Fe-based superconductors, the magnetic order in the parent antiferromagnets corresponds to Q ¼ (1,0,0)p/a in the single-Fe unit cell, and the Fermi surface, in most cases, consists of two groups, one centered around q ¼ (0,0,0)p/a and the other around q ¼ (1,0,0)p/a; so assigning different signs to the order parameters in the two sets assures that spin-fluctuations are pairing. Thus, a plurality, and may be even majority of scientists believe these materials, as well, to be fluctuations-driven. These two example are, arguably, the best studied UCSC. Yet, there is a solid body of evidence in favor of unconventional pairing in numerous other materials, including, but not limited to organic metals, heavy fermions, or twisted bilayer graphene. Some useful theorems regarding unconventional pairing symmetries are: (1) A fully local (i.e., contact) Coulomb repulsion, such as the Hubbard interaction, cancels out if the order parameter integrates to zero over the entire Fermi surface (regardless of the nature of the pairing interaction). This can be also reformulated by applying the Fourier transform and considering the structure of a Cooper pair in coordinate space. In this case, the component of the pair are localized on different lattice sites. In case of cuprates (d-wave) these are nearest neighbors in the square lattice, and in case of Fe-based superconductors (s
) on the second neighbors. (2) While the order parameters itself may follow a lower-symmetry representation than the full crystal symmetry (A1g), its amplitude, i.e., the excitation gap, usually does follow the full symmetry (some exceptions to this rule, often called “nematic” superconductivity, have been proposed theoretically). (3) Even if the order parameter does not obey the point group, it is, including the phase, subject to the Bloch theorem, so that D(k) ¼ D (k + G), where G is a reciprocal lattice vector. It is a powerful theorem because it forces order parameter nodes at some special high-symmetry points. Finally, it is worth bringing up yet another class of materials, multigap superconductors, where the magnitude of the order parameter changes exceptionally strongly over the Fermi surface or between different Fermi surface pockets. These are not usually called UCSC now, but they were not appreciated (albeit theoretically discussed already in the late 50s) until the discovery of a two-gap superconductivity in MgB2 in 2001—at that time MgB2 was routinely referred to as an unconventional, but phonon-driven superconductor. Further reading: (Sigrist and Ueda, 1991; Zhou et al., 2021; Norman, 2011; Moriya and Ueda, 2003) (general), (Mazin and Antropov, 2003) (MgB2), (Lee et al., 2006) (cuprates), (Hirschfeld et al., 2011; Hosono and Kuroki, 2015) (Fe-based).

References Hirschfeld PJ, Korshunov MM, and Mazin II (2011) Gap symmetry and structure of Fe-based superconductors. Reports on Progress in Physics 74: 124508. Hosono H and Kuroki K (2015) Iron-based superconductors: Current status of materials and pairing mechanism. Physica C: Superconductivity and Its Applications 514: 399. Lee PA, Nagaosa N, and Wen X-G (2006) Doping a Mott insulator: Physics of high-temperature superconductivity. Reviews of Modern Physics 78: 17. Mazin II and Antropov VP (2003) Electronic structure, electron-phonon coupling, and multiband effects in MgB2. Physica C: Superconductivity 385: 49. Moriya T and Ueda K (2003) Antiferromagnetic spin fluctuation and superconductivity. Reports on Progress in Physics 66: 1299. Norman MR (2011) The challenge of unconventional superconductivity. Science 332: 196. Sigrist M and Ueda K (1991) Phenomenological theory of unconventional superconductivity. Reviews of Modern Physics 63: 239. Zhou X, Lee W-S, Imada M, Trivedi N, Phillips P, Kee H-Y, Törmä P, and Eremets M (2021) High-temperature superconductivity. Nature Reviews Physics 3: 462.

Superconducting density of states from scanning tunneling microscopy Hermann Suderow, Laboratory of Low Temperatures and High Magnetic Fields, Condensed Matter Physics Department, Nicolás Cabrera Institute and Condensed Matter Physics Center (IFIMAC), Associated Unit UAM-CSIC, Autonomous University of Madrid, Madrid, Spain © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Tunneling conductance Distribution of values of the superconducting gap Behavior at defects and impurities Key issues Superconductors with a small gap anisotropy Two-gap superconductivity in MgB2 Layered transition metal dichalcogenides Nickel borocarbides Summary and future directions Summary Future directions Cuprate superconductors Pnictide- and Fe-based superconductors Topological superconductors Highly disordered superconductors Conclusion Acknowledgments References

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Abstract In Bardeen, Cooper and Schrieffer’s (BCS) theory of superconductivity, the usual metallic parabolic nearly-free-electron band is modified by opening a gap at the Fermi level. The electronic density of states is zero within the gap and presents peaks at the gap edges. Many materials have superconducting properties that deviate from BCS theory and, as a consequence, the density of states is very different, too. The differently shaped density of states signals new paradigms in superconductivity, such as two-band, anisotropic, or multigap superconductivity. Here, measurements of the density of states obtained using Scanning tunneling microscopy (STM) in representative compounds are reviewed and future prospects of advances in superconductivity using STM are discussed.

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Present basic aspects of the superconducting density of states. Discuss the tunneling conductance in representative examples. Discuss methods to address new superconductors.

Introduction The superconducting state is characterized by the condensation of pairs of electrons into Cooper pairs described by a macroscopic quantum wave function C(r) which has a smooth dependence on the position r (Leggett, 2005; Larkin, 2005). Consequences are the establishment of dissipationless transport and of flux quantization (Kirtley and Tsuei, 2005; Brandt, 2005), both of great importance for applications (Bishop, 2005a; Gurevich, 2005). The occurrence of superconductivity is described by Bardeen, Cooper and Schrieffer’s (BCS) theory (Bennemann, 2005, 2023). BCS theory describes the condensate as well as its excitations. The latter are located at an energy E above the superconducting gap D. Within BCS theory, the pairing interaction energy is of the order of the Debye temperature and much larger than the superconducting gap D, both also much smaller than the Fermi energy EF. After more than 66 years of BCS theory, numerous compounds and systems have been shown to be superconducting, with critical temperatures ranging from below a mK up to near room temperature (Bishop, 2005b; Qin and Dou, 2005). The pairing

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interaction energy and Fermi temperatures vary across many orders of magnitude, too (Nozières and Schmitt-Rink, 1985; Fazio and van der Zant, 2001; Shibauchi et al., 2020; Blatter et al., 1994). The energy dependence of the excitations of the superconducting state depends on the pairing interaction and on the coexistence with other magnetic or charge ordering phenomena. As a consequence, the electronic excitations can be very different in different superconducting materials. In Fig. 1A, the energy dispersion relation within BCS theory is schematically shown. The superconducting density of states is obtained by integrating over the wavevector (Fig. 1B). In BCS theory, superconductivity modifies the free electron parabolic electronic energy dispersion relation and opens a gap with size
D above and below the Fermi level. This provides sharp peaks in the density of states at the superconducting gap energy
D and no states for |E| < D (Fig. 1B). The superconducting density of states can be probed by many measurements, such as specific heat, thermal conductivity, nuclear magnetic resonance, and penetration depth. All these probes address however the energy integral of the density of states. To obtain the energy dependence of the density of states, one has to measure directly the electronic band structure. This requires injecting or retrieving electronic excitations from the superconductor, which can be done by tunneling or by photoemission (Cleland, 2005; Margaritondo, 2005). Tunneling conductance experiments can be usually made at lower temperatures than photoemission. Furthermore, a magnetic field can be applied. Thus, tunneling conductance measurements have been more widely applied to superconductors. Tunneling conductance can be measured between a superconductor and another electrode through an evaporated insulating barrier (Dragoman and Dragoman, 2005) or by approaching a tip to the surface of a superconductor and using vacuum as the tunneling barrier. Here the focus is on results obtained using the latter approach with a Scanning tunneling microscope (STM). Other aspects of superconductivity and tunneling, such as DC and AC Josephson effects, Andreev reflection or superconducting vortex physics are discussed by Cleland (2005), Fischer et al. (2007), Suderow et al. (2014), Tafuri (2023), and Mikitik (2023).

Overview Tunneling conductance Bringing two metallic electrodes at a sufficiently short distance, of the order of the nm, produces a tunneling current that flows between both electrodes. This is schematically shown in Fig. 2. The tunneling current is given by the overlap between the

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1 (A) The dispersion relation for free electrons is shown by the red dashed line. In a superconductor below Tc, the dispersion relation is modified by the opening of the superconducting gap, following BCS theory. The corresponding band structure is shown by the colored red and blue lines. Red is for electron-like excitations and blue is for hole-like excitations. (B) The BCS density of states as a function of the energy (blue line). (C)–(F) Schematic view of different gap structures is shown as blue lines within a two-dimensional Brillouin zone (black square). Dashed blue lines provide the Fermi wave vector. (C) S-wave BCS single gap. The gap magnitude (D) is constant over all the Fermi surface. (D) Two disk-shaped Fermi surfaces (grey circles) having two different magnitudes of the superconducting gap. (E) A square-shaped Fermi surface, with two different values of the superconducting gap along two directions. (F) Fermi surface of a band structure that crosses the Brillouin zone. The gap magnitude is different on the central pocket as compared to the pockets at the corners of the Brillouin zone.

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Fig. 2 Schematic view of tunneling between tip and sample in an STM. The tunneling gap of width d is shown by the vertical arrow. The electronic wave functions (CT and CS for tip and sample) are schematically represented by the blue and red lines. The bias voltage is shown by a horizontal grey arrow. Work functions of each electrode are shown schematically as FT and FS for tip and sample, respectively.

wavefunctions extending into vacuum. Bardeen’s formalism, modified by Tersoff and Hamann for STM (Bardeen, 1961; Tersoff and Hamann, 1983, 1985; Agrait et al., 2003), describes the tunneling current as Z 4pe 1 IðVÞ ¼ dEjMj2 N s ðE − eVÞN t ðEÞð f ðE − eV − f ðEÞÞ, (1) ħ −1 where e is the electron charge, ħ is the Planck’s constant, E is the energy, V is the applied voltage, M is the tunneling matrix element, Ns(E − eV) and Nt(E) are the densities of states of tip and sample, and f(E) is the Fermi function. Here the focus is on the situation where a normal metal such as Au or Pt is used for the tip.1 At energies approximately below
100 meV of the Fermi level, Nt is approximately flat and constant and can be taken out of Eq. (1). Then one can write for the bias dI : voltage dependence of the tunneling conductance sðVÞ ¼ dV Z 1 ∂f ðE − eVÞ sðVÞ∝ dENs ðEÞ : (2) ∂V −1 At sufficiently low temperatures, ∂f ðE∂V− eVÞ is close to a d function, for which one obtains that the tunneling conductance versus bias voltage V follows the superconducting density of states s(V ) ’ Ns(eV ). In a simple metal, with only a single band crossing the Fermi level and a fully spherical Fermi surface, BCS theory leads to an equally isotropic superconducting gap, schematically shown in Fig. 1C (in two dimensions for clarity). Then, Eq. (2) provides E . an excellent account of the tunneling conductance, which closely follows the BCS relation sðVÞ ’ Ns ðE ¼ eVÞ ∝ pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 E − D

Distribution of values of the superconducting gap Most materials are however not simple metals and can have quite an intricate set of bands crossing the Fermi level. Let us consider a hypothetical metal with two isotropic bands crossing the Fermi level (Fig. 1D). To find the superconducting density of states, one has to integrate the reciprocal wave vector dependence. The result is a sum with two contributions N(E) ¼ Ns,1(E) + Ns,2(E), each Ns,i¼1,2(E) eventually following the BCS relation for a fully opened superconducting gap with generally different values D1 and D2. Similarly, a superconductor with an anisotropic crystal lattice might have a single band crossing the Fermi level, but different values of the superconducting gap along different crystalline directions (Fig. 1E). Alternatively, there can be bands located at different places on the Brillouin zone, showing different values of the superconducting gap (Fig. 1F). The latter cases might again lead to N(E) ¼ Ns,1(E) + Ns,2(E) with different gap values D1,2. Generally there can be many contributions to N(E) from portions of the band structure close to the Fermi level located in places in the Brillouin zone with very different values of the superconducting gap Di.

1 When using a superconducting tip as Al, Pb, or Nb, Nt(E − eV) is strongly featured in a similar energy range as the superconducting counter electrode. In that case, Eq. (1) has to be evaluated in full to understand the tunneling conductance (Rodrigo and Vieira, 2004; Rodrigo et al., 2004a; Proslier et al., 2006; Guillamon et al., 2008; Kohen et al., 2006; Liu et al., 2021a; Pan et al., 1998; Heinrich et al., 2018), with resulting curves that can be very rich, showing, for example, ranges with a negative differential resistance (Hatter et al., 2015; Huang et al., 2020), enhanced resolution in energy (Heinrich et al., 2018; Ruby et al., 2015; Kim et al., 2020), divergent behavior at the gap edge of the tip or situations in which I(V) follows quite closely the superconducting Ns (Crespo et al., 2012).

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In an STM experiment, one measures the tunneling conductance at an atomically well-defined position on the surface of the superconducting sample. To discuss the actual shape of the tunneling conductance measured with an STM at atomic scale, let us go back to Eq. (1). The term |M|2 depends on the wave function overlap between tip and sample, which depends of course on the actual atomic level position of the tip. But in presence of an anisotropic band structure, the different wave function overlap also produces different contributions to Ns(E − eV) from different portions of the band structure close to the Fermi level. The relative weight of these contributions varies at the atomic scale. Thus, both |M|2 and Ns(E − eV) are different at each atomically resolved position over the surface of the sample. This provides a certain spatial dependence of the combined term |M|2Ns(E − eV). To find the values of the superconducting gap in different portions of the band structure from the tunneling conductance, one can write 1 0 X E C B (3) Ns ðEÞ ¼ gi Re@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA i E2 − D2i where gi is the relative weight of the contribution of a certain portion of the band structure and Di is the corresponding value of the superconducting gap. For the schematic gap structures shown in Fig. 1C–F, one finds Ns(E) as a sum of two BCS densities of states, each peaked at D1 and D2. But the actual gap distribution gi in a superconducting material can be smooth, with many Di varying for different values and directions of the reciprocal space wave vector k. Generally, this eliminates the divergent quasiparticle peaks of the BCS expression E NBCS ðEÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi and produces instead Ns(E) with rounded peaks at the values where the gap distribution Di concentrates. 2 2 E − D

Note that rounded peaks are also obtained when taking into account lifetime broadening effects in the calculation of the density of states (Dynes et al., 1978). Lifetime broadening is a consequence of the pairing interaction and can considerably influence the BCS expression for the density of states. When lifetime broadening is large, it generally leads to an increased density of states at zero energy. Eq. (3) instead refers to a gap broadening due to the superconducting anisotropy and leads to a vanishing zero bias conductance as long as Di  kBT for all i. Studying the gi and Di as a function of the position on the surface, one can extract considerable information over the shape of the superconducting gap (Rodrigo and Vieira, 2004; Guillamón et al., 2008). For example, in a two-gap superconductor with Di distributed around two main values centered at D1,avg and D2,avg, tunneling into a certain surface can produce a Ns(E) with a contribution from portions of the band structure highlighting D1,avg or D2,avg, depending on the actual atomic termination at each surface (Cho et al., 2017; Fente et al., 2018). Even when scanning with the tip over a certain surface, one can find different contributions to the tunneling conductance at different atomic positions (Guillamon et al., 2008). This does not mean that the superconducting density of states changes at atomic length scales (usually far below the superconducting coherence length). Instead, it shows that the contribution to the tunneling current at each atomic position depends on the contributions to the tunneling density of states from different places of the Brillouin zone. The overall superconducting density of states is by contrast the sum over the whole band structure and does not change as a function of the position in absence of disturbances of the superconducting condensate.

Behavior at defects and impurities When measuring the surface of a single crystal, there are often impurities or defects. Charge screening implies an oscillatory density of states around impurities or defects, with a wave vector of the order of the Fermi wave vector kF. The usual Friedel oscillations in a metal are a consequence of the spatial dependence of the charge density summed from the bottom of the band to the Fermi energy (Ziman, 1972). STM allows measuring the density of states on the surface at different energies E. Maps of the tunneling conductance s(eV) at a bias voltage eV ¼ E0 provide the spatial dependence of Ns(x, y) at E0. These maps present often oscillations, at each energy E0, created by impurities or defects. The wave vector of the oscillations is related to the value of the electronic wave vector at the energy E0. Thus, the wave vector follows the electron dispersion relation as a function of the energy E0. Note that these are not really Friedel oscillations in the usual sense. One can understand the oscillatory behavior observed by STM by using scattering of electrons confined to surfaces, as opposed to Friedel oscillations which are obtained by integrating the density of states over all energies. At the end, the result is similar, except that surface states scatter electrons and lead to an oscillatory signal, whereas the bulk band structure leads to Friedel oscillations whose projection on the surface is observed, too (Crommie et al., 1993; Heller et al., 1994; Crommie et al., 1995; Fiete and Heller, 2003; Petersen et al., 1998; Hoffman et al., 2002; McElroy et al., 2003; Hoffman, 2011; Pascual et al., 2004; Ortuzar et al., 2022). The wave vector observed in the maps of the tunneling conductance s(eV) is two times the wave vector of the dispersion relation. Elastic scattering joins portions of the band structure at opposite wave vectors, k and −k. One can write for the Fourier transform of the spatial dependence of the superconducting density of states with a modulation: gðE, qÞ ∝ jVðE, qÞjJðE, qÞ,

(4)

where g is the Fourier transform of the tunneling conductance, V (E, q) is the scattering potential, J is the joint density of states, J(E, q) ∝ Ns(E, k1)Ns(E, k2) and q ¼ k2 − k1 the scattering wave vector joining pairs of states at the same energy E (Hoffman et al., 2002; Fente et al., 2018). One can see that the scattering intensity and pattern depend on the scattering properties of the impurity or defect (the scattering potential V (E, q)) and on the density of states. For example, an impurity with a twofold anisotropy can

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produce scattering along the impurity’s anisotropy axis. Quasiparticle peaks located at the same gap values E ¼ Di, j also produce enhanced scattering. Thus, if one considers isotropic impurities or defects, one can obtain the variation of the superconducting gap in reciprocal space.

Key issues In this section, a selection of superconductors with a density of states deviating from most simple s-wave BCS theory is presented. These systems have been widely studied during past years. One can probably safely state that there is an ample consensus about the interpretation of the observed tunneling density of states and its relation to the bulk superconducting properties.

Superconductors with a small gap anisotropy In Fig. 3 the tunneling conductance in superconducting Al and Pb is shown (Martín-Vega et al., 2021). The density of states is zero within the superconducting gap and the quasiparticle peaks are well developed, following closely the BCS density of states NBCS(E). Similar results have been obtained for Nb, V, La or Re (Guillamon et al., 2008; Pan et al., 1998; Assig et al., 2013; Ruby et al., 2015; Fernández-Lomana et al., 2021; Senkpiel et al., 2020; Tonnoir et al., 2013; Senkpiel et al., 2022; Ast et al., 2016; Eltschka et al., 2015; le Sueur et al., 2008; Lo Conte et al., 2022; Kim et al., 2018; Nadj-Perge et al., 2014; Palacio-Morales et al., 2019). In spite of being elemental superconductors, some materials can have several bands crossing the Fermi level. When single crystalline samples are measured, the associated difference in the gap magnitude over the Fermi surface is well resolved. For example, Pb has two well defined values of the superconducting gap, as shown by Ruby et al. (2015). When nonsingle-crystalline samples are measured, there is often a small distribution of values of the superconducting gap Di (see Eq. 3), following a Gaussian shape around the value expected from the single-gap BCS theory (Rodrigo and Vieira, 2004; Rodrigo et al., 2004a, b; Suderow et al., 2002). A similar result is observed in many binary intermetallic superconducting compounds (Herrera et al., 2023b). Some are shown in Fig. 4. In these cases, often the distribution of Di is more extended than in elemental superconductors.

Two-gap superconductivity in MgB2 The best known example for a two-gap superconductor is MgB2 (Nagamatsu et al., 2001; Rubio-Bollinger et al., 2001; Bud’ko et al., 2001). There are different bands crossing the Fermi level with small interband interactions and very different electron–phonon coupling. Bands derived from B s-orbitals have a small overlap to bands from B p-orbitals. At the same time, there is a very strong electron–phonon coupling due to the in-plane B atom vibrations (Kortus et al., 2001; Choi et al., 2002). This leads to large values of the superconducting gap in the s-derived bands, whereas the gap remains small in the p-derived bands. The tunneling conductance (Fig. 5) nicely shows the two superconducting gaps, and there is quite a wide distribution of gap values around the two main gap values (D1avg ¼ Dp  2.2 meV and D2avg ¼ Ds  7.1 meV) (Martinez-Samper et al., 2003; Rubio-Bollinger et al., 2001; Giubileo et al., 2001). (b)

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Fig. 4 Tunneling conductance obtained in b-Bi2Pd (A)and in AuSn4 (B). The tunneling conductance is normalized to its value at voltages well above the superconducting gap. The bias voltage is normalized to the BCS value of the superconducting gap DBCS. (A) Adapted with permission from Herrera E, Guillamón I, Galvis JA, Correa A, Fente A, Luccas RF, Mompean FJ, García-Hernández M, Vieira S, Brison JP, and Suderow H (2015) Magnetic field dependence of the density of states in the multiband superconductor b-Bi2Pd. Physical Review B 92(5): 054507. https://doi.org/10.1103/PhysRevB.92.054507, copyrighted by the American Physical Society. (B) Adapted with permission from Herrera E, Wu B, O’Leary E, Ruiz AM, Águeda M, Talavera PG, Barrena V, Azpeitia J, Munuera C, García-Hernández M, Wang L-L, Kaminski A, Canfield PC, Baldoví JJ, Guillamón I, and Suderow H (2023b) Band structure, superconductivity, and polytypism in AuSn4. Physical Review Materials 7(2): 024804. https://doi.org/10.1103/PhysRevMaterials.7.024804 copyrighted by the merican Physical Society. See also Choi D-J, Fernández CG, Herrera E, Rubio-Verdú C, Ugeda MM, Guillamón I, Suderow H, Pascual JI, and Lorente N (2018) Influence of magnetic ordering between Cr adatoms on the Yu-Shiba-Rusinov states of the b-Bi2Pd superconductor. Physical Review Letters 120(16): 167001. https://doi.org/10.1103/PhysRevLett.120. 167001 and Lv Y-F, Wang W-L, Zhang Y-M, Ding H, Li W, Wang L, He K, Song C-L, Ma X-C, and Xue Q-K (2017) Experimental signature of topological superconductivity and Majorana zero modes on b-Bi2Pd thin films. Science Bulletin 62(12): 852–856. ISSN 2095-9273. https://doi.org/10.1016/j.scib.2017.05.008.

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Layered transition metal dichalcogenides Transition metal dichalcogenides also have Fermi surfaces with different bands crossing the Fermi level (Inosov et al., 2008; Zhao et al., 2021). These compounds consist of blocks of a transition metal and two chalcogen atoms. They have a layered crystalline structure with weak bonding among layers. The arrangement of layers and of the blocks inside each layer is different for each polytype of the crystalline structure. The double hexagonal polytypes, marked as 2H, show superconductivity in 2H-NbSe2, 2H-NbS2, 2H-TaS2, and 2H-TaSe2. Their band structure consists mainly of two transition-metal-derived bands crossing the Fermi level, although there can be bands derived from the chalcogen (Johannes et al., 2006). In all cases except 2H-NbS2, superconductivity coexists with a charge density wave (CDW). The latter leads to a large in-plane anisotropy, with opening of a gap in some portions of the normal-state band structure. The superconducting density of states, shown in Fig. 6, is compatible with a superconducting gap that is different in the two bands crossing the Fermi level in 2H-NbS2 (Guillamón et al., 2008). In 2H-NbSe2, there is in addition a strong gap anisotropy (middle curve of Fig. 6), with a feature in the tunneling conductance pointing out a significant

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Vbias/'BCS Fig. 6 Tunneling conductance obtained in 2H-NbSe2, 2H-NbS2, and 2H-TaS2. The tunneling conductance is normalized to its value at voltages well above the superconducting gap. Curves are displaced along the y-axis by 1.5 units of normalized conductance. The bias voltage is normalized to the BCS value of the superconducting gap DBCS. Adapted with permission from Guillamon I, Suderow H, Guinea F, and Vieira S (2008) Intrinsic atomic-scale modulations of the superconducting gap of 2H-NbSe2. Physical Review B 77(13): 134505. https://doi.org/10.1103/PhysRevB.77.134505; Guillamón I, Suderow H, Rodrigo JG, Vieira S, Rodière P, Cario L, Navarro-Moratalla E, Martí-Gastaldo C, and Coronado E (2011) Chiral charge order in the superconductor 2H-TaS2. New Journal of Physics 13(10): 103020. https://doi.org/10.1088/1367-2630/13/10/103020; Guillamón I, Suderow H, Vieira S, Cario L, Diener P, and Rodière P (2008) Superconducting density of states and vortex cores of 2H-NbSe2. Physical Review Letters 101(16): 166407. https://doi.org/10.1103/PhysRevLett.101.166407, copyrighted by the American Physical Society. See also Noat Y, Silva-Guillén JA, Cren T, Cherkez V, Brun C, Pons S, Debontridder F, Roditchev D, Sacks W, Cario L, Ordejón P, García A, and Canadell E (2015) Quasiparticle spectra of 2H-NbSe2: Two-band superconductivity and the role of tunneling selectivity. Physical Review B 92(13): 134510. https://doi.org/10.1103/PhysRevB.92.134510 and Galvis JA, Chirolli L, Guillamón I, Vieira S, Navarro-Moratalla E, Coronado E, Suderow H, and Guinea F (2014) Zero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by scanning tunneling spectroscopy. Physical Review B 89(22): 224512. https://doi.org/10.1103/PhysRevB.89.224512

reduction of the gap at some portions of the Fermi surface (Guillamon et al., 2008; Noat et al., 2015). Furthermore, there is an intrinsic atomic-scale modulation of the superconducting properties through a pair density wave (Liu et al., 2021b). 2H-TaS2 presents a significant correlation between the CDW and superconductivity, and an enhancement of the superconducting gap magnitude at some locations (Guillamón et al., 2011). 2H-TaSe2 has the smallest Tc (below 0.15 K in the bulk) and presents the most peculiar superconducting properties, with a density of states that strongly varies as a function of the position (Galvis et al., 2013). Possibly, superconductivity is enhanced at the surface but influenced by normal underlying layers. There is a large zero-bias anomaly in single layers, which also appears in 2H-TaS2 and has been associated with unconventional two-dimensional superconductivity (Galvis et al., 2014, 2013; Chen et al., 2019). Further experiments in single layers of dichalcogenides deposited or grown on different substrates show a reduction of the gap anisotropy and strong variations in the superconducting gap magnitude, which might decrease (as in 2H-NbSe2) or increase (as in 2H-TaS2) with the thickness (Navarro-Moratalla et al., 2016; Rubio-Verdú et al., 2020; Valencia-Ibáñez et al., 2022).

Nickel borocarbides The rare-earth-nickel borocarbides RNi2B2C (with R ¼ Gd-Lu, Y) present superconductivity up to about 16 K in nonmagnetic RNi2B2C (R ¼ Lu, Y) and coexisting magnetic order for rare earths R with unfilled 4f shells RNi2B2C (R ¼ Gd-Tm) (Cava et al., 1994; Bud’ko and Canfield, 2006; Canfield et al., 1998; Mazumdar et al., 1993; Müller and Narozhnyi, 2001; Cho et al., 1996, 1995). Superconductivity in nonmagnetic RNi2B2C (R ¼ Lu, Y) arises from a highly anisotropic electron–phonon interaction related to a nesting feature at the Fermi surface (Dugdale et al., 2008; Bullock et al., 1998; Dugdale et al., 1999). The superconducting gap is fully developed, although the quasiparticle peaks are very broad, pointing out a large distribution of values Di (Fig. 7). There is a small feature for energies below the quasiparticle peaks, which could indicate a distribution of gaps around two values (MartínezSamper et al., 2003; De Wilde et al., 1997; Sakata et al., 2000; Kaneko et al., 2012). Measurements in magnetic rare-earth-nickel borocarbides RNi2B2C (R ¼ Tm, Er) show a fully opened gap in the antiferromagnetic phase of TmNi2B2C and a closed gap in the ferromagnetic phase of ErNi2B2C (Fig. 7) (Crespo et al., 2006; Suderow et al., 2001). The latter points out that there is significant pair breaking related to the ferromagnetic phase.

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Summary and future directions Summary From the examples discussed above, one can see that the tunneling conductance, as measured with STM, is a powerful tool to obtain the superconducting density of states and can lead to insightful descriptions of the superconductor. The tunneling conductance provides the energy dependence of the electronic properties and vividly shows the degree of anisotropy and of the difference of properties in multiple bands. This text mostly discusses until now simple and well-understood examples where the distribution of values of the superconducting gap can be readily associated with the band structure or to the appearance of charge or magnetic order. This can be further explored by using the capability of the STM to scan spatially the superconducting density of states, which leads to direct measurements of the superconducting properties in real space, providing vivid images of the vortex lattice or of real-space anisotropic properties (Fischer et al., 2007; Suderow et al., 2014). In addition, the structure of the superconducting gap in reciprocal space can be also obtained. Presently, STM is widely applied to practically any superconducting material as long as the surface can be prepared with sufficient cleanliness to allow for vacuum tunneling. There are many topical compounds and systems in which STM provides new information, which is mostly under debate. The following paragraphs aim to discuss the latter, presenting a brief view of recent advances in a few topical superconductors.

Future directions The systems discussed below can be understood as “unconventional” superconductors (Mazin, 2023). The term “unconventional superconductivity” can be defined as “superconductivity with nontrivial Cooper pairing” (Mineev and Samokhin, 1999). A possible meaning of “nontrivial” is that the symmetry of the superconducting wave function is reduced with respect to the lattice symmetry. The broken gauge symmetry of the superconducting state, analogous to the broken time-reversal symmetry in a ferromagnet, would be the “trivial” situation. The breaking of additional symmetries combined with gauge symmetry might lead to “nontrivial” behavior, in analogy to the many kinds of wavy magnetic order (antiferromagnetism, ferrimagnetism, spiral magnetism, etc.). But the variety of behaviors that do not follow the basic principle of BCS theory of a spin-singlet Cooper pair wave function which is

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isotropic in real and reciprocal space is very large. Thus, there are many authors who use the term “unconventional” for all those systems with an anisotropic or spatially varying Cooper pair wave function, often in materials where one can find enhanced electronic (Coulomb) interactions (Chubukov, 2023).

Cuprate superconductors Cuprate superconductors have been intensively studied by STM (Fischer et al., 2007). The layered structure often leads to high quality surfaces where atomic features are visualized with beautiful detail and/or the superconducting density of states is measured accurately (Renner et al., 1998; Hoogenboom et al., 2001; Maggio-Aprile et al., 1997; Berthod et al., 2017). The superconducting gap anisotropy was measured by tracing the superconducting density of states as a function of the position in presence of scatterers (Hoffman et al., 2002; McElroy et al., 2003). Furthermore, studies of scattering on nonmagnetic pair breaking impurities vividly showed the sign changes of the superconducting wave function over different reciprocal space directions, demonstrating d-wave superconductivity (Pan et al., 2000). Further work includes the observation of a pair density wave (Berg et al., 2009b, a; Agterberg et al., 2020). A uniform superconductor in which superconductivity coexists with a CDW presents oscillations of the density of states at the surface Ns(r) at qCDW, Ns(r) ∝ cos(qCDW r) eventually on top of a background Ns (Guillamon et al., 2008). A pair density wave at qPDW eventually shares the same wavevector as a CDW in the same system (the latter can be actually considered as a secondary order arising from the pair density wave, Agterberg and Garaud, 2015; Dai et al., 2018; Agterberg et al., 2020). The oscillations of the combined pair density and Cooper pair wave function lead to a combined order parameter which changes sign as a function of the position (Berg et al., 2009a, b). The expected resulting oscillatory behavior of the combined order parameter is proportional to cos(qPDW r). It can be located at the same wavevector as the pair density wave although it changes sign (Liu et al., 2021b). Other sign changing modulated superconducting states, due to the interaction with magnetism, are known at ferromagnetic–superconductor interfaces and at high magnetic fields (Casalbuoni and Nardulli, 2004; Buzdin, 2005, 2012; Bergeret et al., 2001; Bergeret and Ilic, 2023). The Ns(r) due to a pair density wave might eventually drop down to zero, if nothing else contributes to Ns except the combined pair density and superconducting order parameters. On the other hand, the expected oscillations of the pair density wave state in the density of states are due to the product of the pair density wave function with its complex conjugate, leading to Ns(r) ∝ cos(2qPDW r) and have different spatial dependencies when oscillations are enhanced at an impurity or at a vortex (Wang et al., 2018). Nevertheless, the CDW oscillations of the density of states are never purely harmonic, for which one can also expect higher order density of states oscillations, purely from coexisting CDW and superconducting order. Thus, both features of pair densities waves, the measurement of the sign change in the oscillatory behavior and the behavior of higher order oscillations, are quite difficult to address and require careful comparisons of tunneling conductance maps made under different conditions. The pair density wave has been studied using density of states as well as Josephson effect measurements (Liu et al., 2021b; Choubey et al., 2020; Edkins et al., 2019). It could well be related to observations of oscillatory behavior in other superconductors that have remained unexplained or unstudied (Agterberg et al., 2020).

Pnictide- and Fe-based superconductors Pnictide- and Fe-based superconductors have been also intensively studied with STM (Hoffman, 2011; Paglione and Greene, 2010; Fernandes et al., 2014; Song et al., 2011). In one specific case, which is FeSe and related doped compounds, the atomic lattice has been observed by many groups, allowing for in-depth investigations of the tunneling conductance and of tunneling conductance maps. One of the defining features of these superconductors, and in particular of FeSe, is the structural transition between a high temperature tetragonal phase and a low temperature orthorhombic phase. While the lattice parameters in both phases are often very similar, the electronic structure turns out to be highly anisotropic in the orthorhombic phase. This leads to very anisotropic electronic features that have been termed “electronic nematic” in analogy to the nematic state of liquid crystals (Wang et al., 2022; Fernandes and Millis, 2013; Li et al., 2017; Chuang et al., 2010; Watson et al., 2015). The nematicity is often associated with an antiferromagnetic order, except in FeSe where it has been linked to magnetic fluctuations and frustrated magnetism (Wang et al., 2015; Glasbrenner et al., 2015; Yu and Si, 2015). The nematicity leads to highly anisotropic scattering features in tunneling conductance maps, both in normal and superconducting phases (Chuang et al., 2010; Allan et al., 2013; Kasahara et al., 2014). Nematicity is intimately related to the orbital structure (Sprau et al., 2017; Rhodes et al., 2022; Kang et al., 2018; Yu et al., 2018; Hu et al., 2018; Liu et al., 2018; Baek et al., 2015; Li et al., 2020). The orbital contribution to the electronic band structure is considerably modified in the low temperature nematic phase. But the microscopic mechanism to enter the nematic phase is still under debate and requires more in-depth investigations of the local structure of superconductivity in FeSe. Recent developments have allowed to address superconductivity in CaKFe4As4. This is a stoichiometric compound with a large critical temperature of Tc ¼ 35 K (Iyo et al., 2016; Meier et al., 2016, 2017; Liu et al., 2020). Other pnictide- and Fe-based superconductors are nonstoichiometric at their highest Tc’s. CaKFe4As4 presents two-gap superconductivity from tunneling conductance measurements (Cho et al., 2017). Studying tunneling conductance maps, one can extract the opening of the superconducting gap in different bands (Fente et al., 2018). Substitution of Ca or K leads to superconductivity coexisting with new

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phases, such as hedgehog magnetism or ferromagnetism (Meier et al., 2018; Bao et al., 2018; Kim et al., 2021; Collomb et al., 2021). The superconducting gap and band structure in each phase are strongly modified (Llorens et al., 2021; Stolyarov et al., 2020).

Topological superconductors Topological superconductors can be defined in analogy to topological insulators. In the latter, the usual orbital sequence of bands is inverted in the bulk. Often in light semiconductors, the s electrons form the conduction band while the valence band is formed by p and d electrons. When one takes a semiconducting compound having a heavy element, the shape of s orbitals is modified due to a stronger attraction potential and it might fall in energy sufficiently that the s-derived bands fall below the bands derived from p and d orbitals. Such an orbital sequence cannot be transferred into vacuum, where the influence of the heavy element is absent. Thus, bands must cross the Fermi level at the surface and lose band inversion in the vacuum. This leads to a surface state which is topologically protected and occurs forcibly in any semiconductor with band inversion. In a superconductor with an orbitally antisymmetric superconducting order parameter (as opposed to the spin antisymmetric usual BCS state), there are also topologically protected surface states. The gap must close in some form at the surface, leading to a considerable modification of the superconducting density of states. This has been analyzed for the Landau framework of superconductivity by Sigrist and Ueda (1991) and Mineev and Samokhin (1999) and, more recently, within the topological classification by Schnyder et al. (2008) and Ryu (2023). Many intermetallic compounds have band structure properties in the normal phase that lead to topologically nontrivial surface states. Some of them are superconducting. Efforts to observe unconventional superconducting properties have been made, for example, in the low temperature superconductor Au2Pb (Martín-Vega et al., 2022). Indeed, a gapless state was observed. However, it is very difficult to separate a modification of the properties at the surface by changes in the band structure from modifications due to the presence of a topologically enforced surface state. The complexity arising from the surface is not yet resolved, being worsened in Au2Pb because of the absence of an atomically flat surface. Further efforts have been carried out in b −Bi2Pd, where the surface state is better defined, although the superconducting gap is fully open (Herrera et al., 2015; Choi et al., 2018; Li et al., 2019; Sakano et al., 2015; Lv et al., 2017; Kacmarcík et al., 2016). Important advances have been made in Te doped FeSe, where a Te p-electron-derived band is located near the Fermi level and can lead to band inversion and thus topological superconductivity (Zhang et al., 2018; Chen et al., 2018). Possibly more promising are systems in which the appearance of unconventional superconducting properties is intimately related to strong electronic correlations (Kohn and Luttinger, 1965), such as heavy-fermion and ferromagnetic superconductor (Coleman, 2007; Aoki et al., 2011; Flouquet, 2005). Electronic correlations at the surface in these systems remain largely unstudied. Very recently, quantized states have been observed on the surface of a heavy fermion superconductor (Herrera et al., 2023a). This is a very promising result, which should help understanding surface features in unconventional superconductors. One system, where topological superconductivity from electronic correlations is suspected since long, is Sr2RuO4 (Mackenzie and Maeno, 2003; Sigrist, 2005; Kallin, 2012). Here, atomically flat surfaces have been observed, leading to the determination of the normal-state band structure by defect scattering (Kreisel et al., 2021; Wang et al., 2017; Firmo et al., 2013). An opened superconducting gap has only been observed in absence of atomically flat surfaces (Suderow et al., 2009). This provides a clue about possible advancement routes. It is possibly important to take into account the influence of electronic correlations and surface states on the tunneling conductance at the surface. Further work on the surface properties of potential topological superconductors might help to disentangle the expectations from theory in the surface state of topological systems, which include for instance Majorana modes presenting non-Abelian statistics.

Highly disordered superconductors In the presence of disorder, electron screening is expected to decrease (Altshuler and Aronov, 1985). The influence of Coulomb correlations thus increases, leading to interesting behaviors in numerous thin film or layered superconducting systems where disorder is be controlled with a tuning parameter. The superconductor-to-insulator transition is a quantum phase transition whose properties have been studied in detail using tunneling conductance (Goldman, 2010; Goldman and Markovic, 1998; Lin et al., 2015; Vinokur, 2023). At both sides of the transition, there is a gap, one is the insulating gap and another one is the superconducting gap (Sacépé et al., 2010; Roy et al., 2019; Valles et al., 1989; Butko et al., 2000; Sherman et al., 2012; Sacépé et al., 2011; Feigel’man et al., 2007). There is an intimate relation among both gap features (Vinokur et al., 2008; Bouadim et al., 2011; Bielejec et al., 2001). For example, the superconducting gap is considerably enhanced with respect to BCS theory when approaching the transition (Sacépé et al., 2010; Sacépé et al., 2008). Furthermore, the electronic properties both in the normal and superconducting phase are inhomogeneous (Mondal et al., 2011; Kamlapure et al., 2013; Ganguly et al., 2017). The inhomogeneity is not ordered, and does not provide modulations of the density of states, which lead to the peaks in the Fourier transform observed in ordered metals. Instead, it produces long wave-length fluctuations without a particular direction (Bouadim et al., 2011; Postolova et al., 2020; Feigel’man et al., 2010; Carbillet et al., 2020). The actual electronic symmetry imposed by the crystal vanishes, which leads to slowly varying normal and superconducting state properties. Then, the fluctuations induced by Coulomb interactions are seen in the part of the Fourier transform of small wave vectors as a blob. Its dependence on the bias voltage is eventually connected to the energy dependence of the density of states, presenting enhanced fluctuations when the density of states increases, for instance, at the quasiparticle peaks (Postolova et al., 2020).

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Conclusion The full extent of STM, tunneling conductance, and its spatial dependence, provide important information about the nature of the superconducting state, helping to identify systems with fundamentally new properties. Here the very basic aspects, connected to BCS theory taught in the classroom, have been reviewed. The first part mostly deals with rather conclusive insights. The second part includes a few lines of work addressing topical problems in superconductivity, which, quite often, provide new questions, sometimes about the technique itself, that point toward the next step. This hopefully serves as an introduction to the study of tunneling spectroscopy in superconductors, with the caveat that the reader should seek for further information, as some important developments have been certainly left out of the list, for the sake of clarity and space.

Acknowledgments The author acknowledges help with figures and intense discussions with Edwin Herrera, Beilun Wu, José Gabriel Rodrigo, and Isabel Guillamón. Discussions, help, and guidance from Prof. Sebastián Vieira are also acknowledged. Support by the Spanish Research State Agency (PID2020-114071RB-I00, TED2021-130546B-I00 and CEX2018-000805-M), by the Comunidad de Madrid through program NanomagCOST-CM (no. S2018/NMT-4321) and EU program Cost CA21144 (Superqumap) is also acknowledged. The SEGAINVEX at UAM is acknowledged for the design and construction of STM electronics and of cryogenic equipment.

References Agrait N, Yeyati AL, and van Ruitenbeek JM (2003) Quantum properties of atomic-sized conductors. Physics Reports 377(2): 81–279. ISSN: 0370-1573. https://doi.org/10.1016/ S0370-1573(02)00633-6. Agterberg DF and Garaud J (2015) Checkerboard order in vortex cores from pair-density-wave superconductivity. Physical Review B 91(10): 104512. https://doi.org/10.1103/ PhysRevB.91.104512. Agterberg DF, Davis JCS, Edkins SD, Fradkin E, Van Harlingen DJ, Kivelson SA, Lee PA, Radzihovsky L, Tranquada JM, and Wang Y (2020) The physics of pair-density waves: Cuprate superconductors and beyond. Annual Review of Condensed Matter Physics 11(1): 231–270. https://doi.org/10.1146/annurev-conmatphys-031119-050711. Allan MP, Chuang T-M, Massee F, Xie Y, Ni N, Bud’ko SL, Boebinger GS, Wang Q, Dessau DS, Canfield PC, Golden MS, and Davis JC (2013) Anisotropic impurity states, quasiparticle scattering and nematic transport in underdoped Ca(Fe1−xCox)2As2. Nature Physics 9(4): 220–224. ISSN: 1745-2481. https://doi.org/10.1038/nphys2544. Altshuler BL and Aronov AG (1985) Chapter 1—Electron–electron interaction in disordered conductors. In: Efros AL and Pollak M (eds.) Electron-Electron Interactions in Disordered Systems, Modern Problems in Condensed Matter Sciences, vol. 10, pp. 1–153. Elsevier. ISSN: 0167-7837. https://doi.org/10.1016/B978-0-444-86916-6.50007-7. Aoki D, Hardy F, Miyake A, Taufour V, Matsuda TD, and Flouquet J (2011) Properties of ferromagnetic superconductors. Comptes Rendus Physique 12(5): 573–583. ISSN: 1631-0705. https://doi.org/10.1016/j.crhy.2011.04.007 (Superconductivity of strongly correlated systems). Assig M, Etzkorn M, Enders A, Stiepany W, Ast CR, and Kern K (2013) A 10 mK scanning tunneling microscope operating in ultra high vacuum and high magnetic fields. Review of Scientific Instruments 84(3): 033903. https://doi.org/10.1063/1.4793793. Ast CR, Jäck B, Senkpiel J, Eltschka M, Etzkorn M, Ankerhold J, and Kern K (2016) Sensing the quantum limit in scanning tunnelling spectroscopy. Nature Communications 7(1): 13009. ISSN: 2041-1723. https://doi.org/10.1038/ncomms13009. Baek S-H, Efremov DV, Ok JM, Kim JS, van den Brink J, and Büchner B (2015) Orbital-driven nematicity in FeSe. Nature Materials 14(2): 210–214. ISSN: 1476-4660. https://doi.org/ 10.1038/nmat4138. Bao J-K, Willa K, Smylie MP, Chen H, Welp U, Chung DY, and Kanatzidis MG (2018) Single crystal growth and study of the ferromagnetic superconductor RbEuFe4As4. Crystal Growth & Design 18(6): 3517–3523. https://doi.org/10.1021/acs.cgd.8b00315. Bardeen J (1961) Tunnelling from a many-particle point of view. Physical Review Letters 6(2): 57–59. https://doi.org/10.1103/PhysRevLett.6.57. Bennemann KH (2005) Superconductivity: BCS theory. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 72–81. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00701-4. Bennemann KH (2023) Superconductivity: BCS theory. In: Tafuri F (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Berg E, Fradkin E, and Kivelson SA (2009a) Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors. Nature Physics 5(11): 830–833. ISSN: 1745-2481. https://doi.org/10.1038/nphys1389. Berg E, Fradkin E, and Kivelson SA (2009b) Theory of the striped superconductor. Physical Review B 79(6): 064515. https://doi.org/10.1103/PhysRevB.79.064515. Bergeret S and Ilic S (2023) Superconductor-ferromagnet hybrid structures. In: Bergeret S and Ilic S (eds.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Bergeret FS, Volkov AF, and Efetov KB (2001) Long-range proximity effects in superconductor-ferromagnet structures. Physical Review Letters 86(18): 4096–4099. https://doi.org/ 10.1103/PhysRevLett.86.4096. Berthod C, Maggio-Aprile I, Bruér J, Erb A, and Renner C (2017) Observation of Caroli-de Gennes-Matricon vortex states in YBa2Cu3O7−d. Physical Review Letters 119(23): 237001. https://doi.org/10.1103/PhysRevLett.119.237001. Bielejec E, Ruan J, and Wu W (2001) Hard correlation gap observed in quench-condensed ultrathin beryllium. Physical Review Letters 87(3): 036801. https://doi.org/10.1103/ PhysRevLett.87.036801. Bishop DJ (2005a) Superconductivity: Applications. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 66–72. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00708-7. Bishop DJ (2005b) Superconductors, metallic. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 121–128. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00705-1. Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, and Vinokur VM (1994) Vortices in high-temperature superconductors. Reviews of Modern Physics 66(4): 1125–1388. https:// doi.org/10.1103/RevModPhys.66.1125. Bouadim K, Loh YL, Randeria M, and Trivedi N (2011) Single- and two-particle energy gaps across the disorder-driven superconductor-insulator transition. Nature Physics 7(11): 884–889. ISSN: 1745-2481. https://doi.org/10.1038/nphys2037. Brandt EH (2005) Superconductivity: Ginzburg-Landau theory and vortex lattice. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 98–105. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00702-6.

Superconducting density of states from scanning tunneling microscopy

611

Bud’ko SL and Canfield PC (2006) Magnetism and superconductivity in rare earth-nickel-borocarbides. Comptes Rendus Physique 7(1): 56–67. ISSN: 1631-0705. https://doi.org/ 10.1016/j.crhy.2005.11.011. Superconductivity and magnetism. Bud’ko SL, Lapertot G, Petrovic C, Cunningham CE, Anderson N, and Canfield PC (2001) Boron isotope effect in superconducting MgB2. Physical Review Letters 86(9): 1877–1880. https://doi.org/10.1103/PhysRevLett.86.1877. Bullock M, Zarestky J, Stassis C, Goldman A, Canfield P, Honda Z, Shirane G, and Shapiro SM (1998) Low-energy phonon excitations in superconducting RNi2B2C. Physical Review B 57(13): 7916–7921. https://doi.org/10.1103/PhysRevB.57.7916. Butko VY, DiTusa JF, and Adams PW (2000) Coulomb gap: How a metal film becomes an insulator. Physical Review Letters 84(7): 1543–1546. https://doi.org/10.1103/ PhysRevLett.84.1543. Buzdin AI (2005) Proximity effects in superconductor-ferromagnet heterostructures. Reviews of Modern Physics 77(3): 935–976. https://doi.org/10.1103/RevModPhys.77.935. Buzdin A (2012) Non-uniform Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Physica B: Condensed Matter 407(11): 1912–1914. ISSN: 0921-4526. https://doi.org/10.1016/j. physb.2012.01.062. Proceedings of the International Workshop on Electronic Crystals (ECRYS-2011). Canfield PC, Gammel PL, and Bishop DJ (1998) New magnetic superconductors: A toy box for solid-state physicists. Physics Today 51(10): 40–46. https://doi.org/ 10.1063/1.882396. Carbillet C, Cherkez V, Skvortsov MA, Feigel’man MV, Debontridder F, Ioffe LB, Stolyarov VS, Ilin K, Siegel M, Roditchev D, Cren T, and Brun C (2020) Spectroscopic evidence for strong correlations between local superconducting gap and local Altshuler-Aronov density of states suppression in ultrathin NbN films. Physical Review B 102(2): 024504. https://doi.org/ 10.1103/PhysRevB.102.024504. Casalbuoni R and Nardulli G (2004) Inhomogeneous superconductivity in condensed matter and QCD. Reviews of Modern Physics 76(1): 263–320. https://doi.org/10.1103/ RevModPhys.76.263. Cava RJ, Takagi H, Zandbergen HW, Krajewski JJ, Peck WF, Siegrist T, Batlogg B, van Dover RB, Felder RJ, Mizuhashi K, Lee JO, Eisaki H, and Uchida S (1994) Superconductivity in the quaternary intermetallic compounds LnNi2B2C. Nature 367(6460): 252–253. ISSN: 1476-4687. https://doi.org/10.1038/367252a0. Chen M, Chen X, Yang H, Du Z, Zhu X, Wang E, and Wen H-H (2018) Discrete energy levels of caroli-de gennes-matricon states in quantum limit in FeTe0.55Se0.45. Nature Communications 9(1): 970. ISSN: 2041-1723. https://doi.org/10.1038/s41467-018-03404-8. Chen W, Zhu Q, Zhou Y, and An J (2019) Topological ising pairing states in monolayer and trilayer TaS2. Physical Review B 100(5): 054503. https://doi.org/10.1103/ PhysRevB.100.054503. Cho BK, Canfield PC, and Johnston DC (1995) Onset of superconductivity in the antiferromagnetically ordered state of single-crystal DyNi2B2C. Physical Review B 52(6): R3844–R3847. https://doi.org/10.1103/PhysRevB.52.R3844. Cho BK, Canfield PC, and Johnston DC (1996) Breakdown of de gennes scaling in r1−xRxNi2B2C compounds. Physical Review Letters 77(1): 163–166. https://doi.org/10.1103/ PhysRevLett.77.163. Cho K, Fente A, Teknowijoyo S, Tanatar MA, Joshi KR, Nusran NM, Kong T, Meier WR, Kaluarachchi U, Guillamón I, Suderow H, Bud’ko SL, Canfield PC, and Prozorov R (2017) Nodeless multiband superconductivity in stoichiometric single-crystalline CaKFe4As4. Physical Review B 95(10): 100502. https://doi.org/10.1103/PhysRevB.95.100502. Choi HJ, Roundy D, Sun H, Cohen ML, and Louie SG (2002) The origin of the anomalous superconducting properties of MgB2. Nature 418(6899): 758–760. ISSN: 1476-4687. https://doi.org/10.1038/nature00898. Choi D-J, Fernández CG, Herrera E, Rubio-Verdú C, Ugeda MM, Guillamón I, Suderow H, Pascual JI, and Lorente N (2018) Influence of magnetic ordering between Cr adatoms on the Yu-Shiba-Rusinov states of the b-Bi2Pd superconductor. Physical Review Letters 120(16): 167001. https://doi.org/10.1103/PhysRevLett.120.167001. Choubey P, Joo SH, Fujita K, Du Z, Edkins SD, Hamidian MH, Eisaki H, Uchida S, Mackenzie AP, Lee J, Davis JCS, and Hirschfeld PJ (2020) Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity. Proceedings of the National Academy of Sciences 117(26): 14805–14811. https://doi.org/10.1073/ pnas.2002429117. Chuang T-M, Allan MP, Lee J, Xie Y, Ni N, Bud’ko SL, Boebinger GS, Canfield PC, and Davis JC (2010) Nematic electronic structure in the parent state of the iron-based superconductor Ca(Fe1−xCox)2As2. Science 327(5962): 181–184. https://doi.org/10.1126/science.1181083. Chubukov A (2023) Superconductivity: Electronic mechanisms. In: Mazin I (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Cleland AN (2005) Superconductivity: Tunneling. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 105–112. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00704-X. Coleman P (2007) Heavy fermions: Electrons at the edge of magnetism. In: Handbook of Magnetism and Advanced Magnetic Materials. John Wiley and Sons, Ltd, ISBN: 9780470022184. https://doi.org/10.1002/9780470022184.hmm105. Collomb D, Bending SJ, Koshelev AE, Smylie MP, Farrar L, Bao J-K, Chung DY, Kanatzidis MG, Kwok W-K, and Welp U (2021) Observing the suppression of superconductivity in RbEuFe4As4 by correlated magnetic fluctuations. Physical Review Letters 126(15): 157001. https://doi.org/10.1103/PhysRevLett.126.157001. Crespo M, Suderow H, Vieira S, Bud’ko S, and Canfield PC (2006) Local superconducting density of states of ErNi2B2C. Physical Review Letters 96(2): 027003. https://doi.org/ 10.1103/PhysRevLett.96.027003. Crespo V, Maldonado A, Galvis JA, Kulkarni P, Guillamon I, Rodrigo JG, Suderow H, Vieira S, Banerjee S, and Rodiere P (2012) Scanning microscopies of superconductors at very low temperatures. Physica C: Superconductivity 479: 19–23. ISSN: 0921-4534. https://doi.org/10.1016/j.physc.2012.02.014 (Proceedings of VORTEX VII Conference). Crommie MF, Lutz CP, and Eigler DM (1993) Imaging standing waves in a two-dimensional electron gas. Nature 363(6429): 524–527. ISSN: 1476-4687. https://doi.org/ 10.1038/363524a0. Crommie MF, Lutz CP, Eigler DM, and Heller EJ (1995) Waves on a metal surface and quantum corrals. Surface Review and Letters 02(01): 127–137. https://doi.org/10.1142/ S0218625X95000121. Dai Z, Zhang Y-H, Senthil T, and Lee PA (2018) Pair-density waves, charge-density waves, and vortices in high-Tc cuprates. Physical Review B 97(17): 174511. https://doi.org/ 10.1103/PhysRevB.97.174511. De Wilde Y, Iavarone M, Welp U, Metlushko V, Koshelev AE, Aranson I, Crabtree GW, and Canfield PC (1997) Scanning tunneling microscopy observation of a square Abrikosov lattice in LuNi2B2C. Physical Review Letters 78(22): 4273–4276. https://doi.org/10.1103/PhysRevLett.78.4273. Dragoman D and Dragoman M (2005) Tunneling devices. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 269–277. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/01143-8. Dugdale SB, Alam MA, Wilkinson I, Hughes RJ, Fisher IR, Canfield PC, Jarlborg T, and Santi G (1999) Nesting properties and anisotropy of the Fermi surface of LuNi2B2C. Physical Review Letters 83(23): 4824–4827. https://doi.org/10.1103/PhysRevLett.83.4824. Dugdale SB, Utfeld C, Wilkinson I, Laverock J, Major Z, Alam MA, and Canfield PC (2008) Fermi surfaces of rare-earth nickel borocarbides. Superconductor Science and Technology 22(1): 014002. https://doi.org/10.1088/0953-2048/22/1/014002. Dynes RC, Narayanamurti V, and Garno JP (1978) Direct measurement of quasiparticle-lifetime broadening in a strong-coupled superconductor. Physical Review Letters 41(21): 1509–1512. https://doi.org/10.1103/PhysRevLett.41.1509. Edkins SD, Kostin A, Fujita K, Mackenzie AP, Eisaki H, Uchida S, Sachdev S, Lawler MJ, Kim E-A, Davis JCS, and Hamidian MH (2019) Magnetic field induced pair density wave state in the cuprate vortex halo. Science 364(6444): 976–980. https://doi.org/10.1126/science.aat1773. Eltschka M, Jäck B, Assig M, Kondrashov OV, Skvortsov MA, Etzkorn M, Ast CR, and Kern K (2015) Superconducting scanning tunneling microscopy tips in a magnetic field: Geometrycontrolled order of the phase transition. Applied Physics Letters 107(12): 122601. https://doi.org/10.1063/1.4931359. Fazio R and van der Zant H (2001) Quantum phase transitions and vortex dynamics in superconducting networks. Physics Reports 355(4): 235–334. ISSN: 0370-1573. https://doi. org/10.1016/S0370-1573(01)00022-9.

612

Superconducting density of states from scanning tunneling microscopy

Feigel’man MV, Ioffe LB, Kravtsov VE, and Yuzbashyan EA (2007) Eigenfunction fractality and pseudogap state near the superconductor-insulator transition. Physical Review Letters 98(2): 027001. https://doi.org/10.1103/PhysRevLett.98.027001. Feigel’man MV, Ioffe LB, Kravtsov VE, and Cuevas E (2010) Fractal superconductivity near localization threshold. Annals of Physics 325(7): 1390–1478. ISSN: 0003-4916. https://doi. org/10.1016/j.aop.2010.04.001. July 2010 Special Issue. Fente A, Meier WR, Kong T, Kogan VG, Bud’ko SL, Canfield PC, Guillamón I, and Suderow H (2018) Influence of multiband sign-changing superconductivity on vortex cores and vortex pinning in stoichiometric high-Tc CaKFe4As4. Physical Review B 97(13): 134501. https://doi.org/10.1103/PhysRevB.97.134501. Fernandes RM and Millis AJ (2013) Nematicity as a probe of superconducting pairing in iron-based superconductors. Physical Review Letters 111(12): 127001. https://doi.org/ 10.1103/PhysRevLett.111.127001. Fernandes RM, Chubukov AV, and Schmalian J (2014) What drives nematic order in iron-based superconductors? Nature Physics 10(2): 97–104. ISSN: 1745-2481. https://doi.org/ 10.1038/nphys2877. Fernández-Lomana M, Wu B, Martín-Vega F, Sánchez-Barquilla R, Álvarez Montoya R, Castilla JM, Navarrete J, Marijuan JR, Herrera E, Suderow H, and Guillamón I (2021) Millikelvin scanning tunneling microscope at 20/22 T with a graphite enabled stick-slip approach and an energy resolution below 8 mev: Application to conductance quantization at 20 T in single atom point contacts of Al and Au and to the charge density wave of 2H-NbSe2. Review of Scientific Instruments 92(9): 093701. https://doi.org/10.1063/5.0059394. Fiete GA and Heller EJ (2003) Colloquium: Theory of quantum corrals and quantum mirages. Reviews of Modern Physics 75(3): 933–948. https://doi.org/10.1103/ RevModPhys.75.933. Firmo IA, Lederer S, Lupien C, Mackenzie AP, Davis JC, and Kivelson SA (2013) Evidence from tunneling spectroscopy for a quasi-one-dimensional origin of superconductivity in Sr2RuO4. Physical Review B 88(13): 134521. https://doi.org/10.1103/PhysRevB.88.134521. Fischer O, Kugler M, Maggio-Aprile I, Berthod C, and Renner C (2007) Scanning tunneling spectroscopy of high-temperature superconductors. Reviews of Modern Physics 79(1): 353–419. https://doi.org/10.1103/RevModPhys.79.353. Flouquet J (2005) On the heavy fermion road. In: Halperin WP (ed.) Progress in Low Temperature Physics, vol. 15, pp. 139–281. Elsevier. https://doi.org/10.1016/S0079-6417(05) 15002-1. Galvis JA, Rodière P, Guillamon I, Osorio MR, Rodrigo JG, Cario L, Navarro-Moratalla E, Coronado E, Vieira S, and Suderow H (2013) Scanning tunneling measurements of layers of superconducting 2H-TaSe2: Evidence for a zero-bias anomaly in single layers. Physical Review B 87(9): 094502. https://doi.org/10.1103/PhysRevB.87.094502. Galvis JA, Chirolli L, Guillamón I, Vieira S, Navarro-Moratalla E, Coronado E, Suderow H, and Guinea F (2014) Zero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by scanning tunneling spectroscopy. Physical Review B 89(22): 224512. https://doi.org/10.1103/PhysRevB.89.224512. Ganguly R, Roy I, Banerjee A, Singh H, Ghosal A, and Raychaudhuri P (2017) Magnetic field induced emergent inhomogeneity in a superconducting film with weak and homogeneous disorder. Physical Review B 96(5): 054509. https://doi.org/10.1103/PhysRevB.96.054509. Giubileo F, Roditchev D, Sacks W, Lamy R, Thanh DX, Klein J, Miraglia S, Fruchart D, Marcus J, and Monod P (2001) Two-gap state density in MgB2: A true bulk property or a proximity effect? Physical Review Letters 87(17): 177008. https://doi.org/10.1103/PhysRevLett.87.177008. Glasbrenner JK, Mazin II, Jeschke HO, Hirschfeld PJ, Fernandes RM, and Valentí R (2015) Effect of magnetic frustration on nematicity and superconductivity in iron chalcogenides. Nature Physics 11(11): 953–958. ISSN: 1745-2481. https://doi.org/10.1038/nphys3434. Goldman AM (2010) Superconductor-insulator transitions of quench-condensed films. Low Temperature Physics 36(10): 884–892. https://doi.org/10.1063/1.3517172. Goldman AM and Markovic N (1998) Superconductor-insulator transitions in the two-dimensional limit. Physics Today 51(11): 39–44. https://doi.org/10.1063/1.882069. Guillamon I, Suderow H, Guinea F, and Vieira S (2008) Intrinsic atomic-scale modulations of the superconducting gap of 2H-NbSe2. Physical Review B 77(13): 134505. https://doi.org/ 10.1103/PhysRevB.77.134505. Guillamón I, Suderow H, Vieira S, Cario L, Diener P, and Rodière P (2008) Superconducting density of states and vortex cores of 2H-NbSe2. Physical Review Letters 101(16): 166407. https://doi.org/10.1103/PhysRevLett.101.166407. Guillamon I, Suderow H, Vieira S, and Rodiere P (2008) Scanning tunneling spectroscopy with superconducting tips of Al. Physica C: Superconductivity and its Applications 468(7): 537–542. ISSN: 0921-4534. https://doi.org/10.1016/j.physc.2007.11.066. Proceedings of the Fifth International Conference on Vortex Matter in Nanostructured Superconductors. Guillamón I, Suderow H, Rodrigo JG, Vieira S, Rodière P, Cario L, Navarro-Moratalla E, Martí-Gastaldo C, and Coronado E (2011) Chiral charge order in the superconductor 2H-TaS2. New Journal of Physics 13(10): 103020. https://doi.org/10.1088/1367-2630/13/10/103020. Gurevich A (2005) Superconductivity: Critical currents. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 82–88. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00737-3. Hatter N, Heinrich BW, Ruby M, Pascual JI, and Franke KJ (2015) Magnetic anisotropy in Shiba bound states across a quantum phase transition. Nature Communications 6(1): 8988. ISSN: 2041-1723. https://doi.org/10.1038/ncomms9988. Heinrich BW, Pascual JI, and Franke KJ (2018) Single magnetic adsorbates on s-wave superconductors. Progress in Surface Science 93(1): 1–19. ISSN: 0079-6816. https://doi.org/ 10.1016/j.progsurf.2018.01.001. Heller EJ, Crommie MF, Lutz CP, and Eigler DM (1994) Scattering and absorption of surface electron waves in quantum corrals. Nature 369(6480): 464–466. ISSN: 1476-4687. https://doi.org/10.1038/369464a0. Herrera E, Guillamón I, Galvis JA, Correa A, Fente A, Luccas RF, Mompean FJ, García-Hernández M, Vieira S, Brison JP, and Suderow H (2015) Magnetic field dependence of the density of states in the multiband superconductor b-Bi2Pd. Physical Review B 92(5): 054507. https://doi.org/10.1103/PhysRevB.92.054507. Herrera E, Guillamón I, Barrena V, Herrera WJ, Galvis JA, Yeyati AL, Rusz J, Oppeneer PM, Knebel G, Brison JP, Flouquet J, Aoki D, and Suderow H (2023a) Quantum-well states at the surface of a heavy-fermion superconductor. Nature 616: 465–469. https://doi.org/10.1038/s41586-023-05830-1. Herrera E, Wu B, O’Leary E, Ruiz AM, Águeda M, Talavera PG, Barrena V, Azpeitia J, Munuera C, García-Hernández M, Wang L-L, Kaminski A, Canfield PC, Baldoví JJ, Guillamón I, and Suderow H (2023b) Band structure, superconductivity, and polytypism in AuSn4. Physical Review Materials 7(2): 024804. https://doi.org/10.1103/PhysRevMaterials.7.024804. Hoffman JE (2011) Spectroscopic scanning tunneling microscopy insights into Fe-based superconductors. Reports on Progress in Physics 74(12): 124513. https://doi.org/ 10.1088/0034-4885/74/12/124513. Hoffman JE, McElroy K, Lee D-H, Lang KM, Eisaki H, Uchida S, and Davis JC (2002) Imaging quasiparticle interference in Bi2Sr2CaCu2O8+d. Science 297(5584): 1148–1151. https:// doi.org/10.1126/science.1072640. Hoogenboom BW, Kadowaki K, Revaz B, Li M, Renner C, and Fischer Ø (2001) Linear and field-independent relation between vortex core state energy and gap in Bi2Sr2CaCu2O8+d. Physical Review Letters 87(26): 267001. https://doi.org/10.1103/PhysRevLett.87.267001. Hu H, Yu R, Nica EM, Zhu J-X, and Si Q (2018) Orbital-selective superconductivity in the nematic phase of FeSe. Physical Review B 98(22): 220503. https://doi.org/10.1103/ PhysRevB.98.220503. Huang H, Padurariu C, Senkpiel J, Drost R, Yeyati AL, Cuevas JC, Kubala B, Ankerhold J, Kern K, and Ast CR (2020) Tunnelling dynamics between superconducting bound states at the atomic limit. Nature Physics 16(12): 1227–1231. ISSN: 1745-2481. https://doi.org/10.1038/s41567-020-0971-0. Inosov DS, Zabolotnyy VB, Evtushinsky DV, Kordyuk AA, Büchner B, Follath R, Berger H, and Borisenko SV (2008) Fermi surface nesting in several transition metal dichalcogenides. New Journal of Physics 10(12): 125027. https://doi.org/10.1088/1367-2630/10/12/125027. Iyo A, Kawashima K, Kinjo T, Nishio T, Ishida S, Fujihisa H, Gotoh Y, Kihou K, Eisaki H, and Yoshida Y (2016) New-structure-type Fe-based superconductors: CaAFe4As4 (A ¼ K, Rb, Cs) and SrAFe4As4 (A ¼ Rb, Cs). Journal of the American Chemical Society 138(10): 3410–3415. https://doi.org/10.1021/jacs.5b12571. PMID: 26943024. Johannes MD, Mazin II, and Howells CA (2006) Fermi-surface nesting and the origin of the charge-density wave in NbSe2. Physical Review B 73(20): 205102. https://doi.org/ 10.1103/PhysRevB.73.205102. Kallin C (2012) Chiral p-wave order in Sr2RuO4. Reports on Progress in Physics 75(4): 042501. https://doi.org/10.1088/0034-4885/75/4/042501.

Superconducting density of states from scanning tunneling microscopy

613

Kamlapure A, Das T, Ganguli SC, Parmar JB, Bhattacharyya S, and Raychaudhuri P (2013) Emergence of nanoscale inhomogeneity in the superconducting state of a homogeneously disordered conventional superconductor. Scientific Reports 3(1): 2979. ISSN: 2045-2322. https://doi.org/10.1038/srep02979. Kaneko S-i, Matsuba K, Hafiz M, Yamasaki K, Kakizaki E, Nishida N, Takeya H, Hirata K, Kawakami T, Mizushima T, and Machida K (2012) Quantum limiting behaviors of a vortex core in an anisotropic gap superconductor. Journal of the Physical Society of Japan 81(6): 063701. https://doi.org/10.1143/JPSJ.81.063701. Kang J, Fernandes RM, and Chubukov A (2018) Superconductivity in FeSe: The role of nematic order. Physical Review Letters 120(26): 267001. https://doi.org/10.1103/ PhysRevLett.120.267001. Kasahara S, Watashige T, Hanaguri T, Kohsaka Y, Yamashita T, Shimoyama Y, Mizukami Y, Endo R, Ikeda H, Aoyama K, Terashima T, Uji S, Wolf T, von Löhneysen H, Shibauchi T, and Matsuda Y (2014) Field-induced superconducting phase of FeSe in the BCS-BEC cross-over. Proceedings of the National Academy of Sciences 111(46): 16309–16313. https:// doi.org/10.1073/pnas.1413477111. Kacmarcík J, Pribulová Z, Samuely T, Szabó P, Cambel V, Šoltýs J, Herrera E, Suderow H, Correa-Orellana A, Prabhakaran D, and Samuely P (2016) Single-gap superconductivity in b-Bi2Pd. Physical Review B 93(14): 144502. https://doi.org/10.1103/PhysRevB.93.144502. Kim H, Palacio-Morales A, Posske T, Rózsa L, Palotás K, Szunyogh L, Thorwart M, and Wiesendanger R (2018) Toward tailoring Majorana bound states in artificially constructed magnetic atom chains on elemental superconductors. Science Advances 4(5): eaar5251. https://doi.org/10.1126/sciadv.aar5251. Kim H, Rózsa L, Schreyer D, Simon E, and Wiesendanger R (2020) Long-range focusing of magnetic bound states in superconducting lanthanum. Nature Communications 11(1): 4573. ISSN: 2041-1723. https://doi.org/10.1038/s41467-020-18406-8. Kim TK, Pervakov KS, Evtushinsky DV, Jung SW, Poelchen G, Kummer K, Vlasenko VA, Sadakov AV, Usoltsev AS, Pudalov VM, Roditchev D, Stolyarov VS, Vyalikh DV, Borisov V, Valentí R, Ernst A, Eremeev SV, and Chulkov EV (2021) Electronic structure and coexistence of superconductivity with magnetism in RbEuFe4As4. Physical Review B 103(17): 174517. https://doi.org/10.1103/PhysRevB.103.174517. Kirtley JR and Tsuei CC (2005) Superconductivity: Flux quantization. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 89–95. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00703-8. Kohen A, Proslier T, Cren T, Noat Y, Sacks W, Berger H, and Roditchev D (2006) Probing the superfluid velocity with a superconducting tip: The Doppler shift effect. Physical Review Letters 97(2): 027001. https://doi.org/10.1103/PhysRevLett.97.027001. Kohn W and Luttinger JM (1965) New mechanism for superconductivity. Physical Review Letters 15(12): 524–526. https://doi.org/10.1103/PhysRevLett.15.524. Kortus J, Mazin II, Belashchenko KD, Antropov VP, and Boyer LL (2001) Superconductivity of metallic boron in MgB2. Physical Review Letters 86(20): 4656–4659. https://doi.org/ 10.1103/PhysRevLett.86.4656. Kreisel A, Marques CA, Rhodes LC, Kong X, Berlijn T, Fittipaldi R, Granata V, Vecchione A, Wahl P, and Hirschfeld PJ (2021) Quasi-particle interference of the van hove singularity in Sr2RuO4. npj Quantum Materials 6(1): 100. ISSN: 2397-4648. https://doi.org/10.1038/s41535-021-00401-x. Larkin AI (2005) Superconductivity: General aspects. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 95–98. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00700-2. le Sueur H, Joyez P, Pothier H, Urbina C, and Esteve D (2008) Phase controlled superconducting proximity effect probed by tunneling spectroscopy. Physical Review Letters 100(19): 197002. https://doi.org/10.1103/PhysRevLett.100.197002. Leggett AJ (2005) Quantum mechanics: Foundations. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 51–56. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00616-1. Li J, Pereira PJ, Yuan J, Lv Y-Y, Jiang M-P, Lu D, Lin Z-Q, Liu Y-J, Wang J-F, Li L, Ke X, Van Tendeloo G, Li M-Y, Feng H-L, Hatano T, Wang H-B, Wu P-H, Yamaura K, TakayamaMuromachi E, Vanacken J, Chibotaru LF, and Moshchalkov VV (2017) Nematic superconducting state in iron pnictide superconductors. Nature Communications 8(1): 1880. ISSN: 2041-1723. https://doi.org/10.1038/s41467-017-02016-y. Li Y, Xu X, Lee M-H, Chu M-W, and Chien CL (2019) Observation of half-quantum flux in the unconventional superconductor b-Bi2Pd. Science 366(6462): 238–241. https://doi.org/ 10.1126/science.aau6539. Li J, Lei B, Zhao D, Nie LP, Song DW, Zheng LX, Li SJ, Kang BL, Luo XG, Wu T, and Chen XH (2020) Spin-orbital-intertwined nematic state in FeSe. Physical Review X 10(1): 011034. https://doi.org/10.1103/PhysRevX.10.011034. Lin Y-H, Nelson J, and Goldman AM (2015) Superconductivity of very thin films: The superconductor-insulator transition. Physica C: Superconductivity and its Applications 514: 130–141. ISSN: 0921-4534. https://doi.org/10.1016/j.physc.2015.01.005. Superconducting Materials: Conventional, Unconventional and Undetermined. Liu D, Li C, Huang J, Lei B, Wang L, Wu X, Shen B, Gao Q, Zhang Y, Liu X, Hu Y, Xu Y, Liang A, Liu J, Ai P, Zhao L, He S, Yu L, Liu G, Mao Y, Dong X, Jia X, Zhang F, Zhang S, Yang F, Wang Z, Peng Q, Shi Y, Hu J, Xiang T, Chen X, Xu Z, Chen C, and Zhou XJ (2018) Orbital origin of extremely anisotropic superconducting gap in nematic phase of FeSe superconductor. Physical Review X 8(3): 031033. https://doi.org/10.1103/PhysRevX.8.031033. Liu W, Cao L, Zhu S, Kong L, Wang G, Papaj M, Zhang P, Liu Y-B, Chen H, Li G, Yang F, Kondo T, Du S, Cao G-H, Shin S, Fu L, Yin Z, Gao H-J, and Ding H (2020) A new Majorana platform in an Fe-As bilayer superconductor. Nature Communications 11(1): 5688. ISSN: 2041-1723. https://doi.org/10.1038/s41467-020-19487-1. Liu X, Chong YX, Sharma R, and Davis JCS (2021a) Atomic-scale visualization of electronic fluid flow. Nature Materials 20(11): 1480–1484. ISSN: 1476-4660. https://doi.org/ 10.1038/s41563-021-01077-1. Liu X, Chong YX, Sharma R, and Davis JCS (2021b) Discovery of a Cooper-pair density wave state in a transition-metal dichalcogenide. Science 372(6549): 1447–1452. https://doi. org/10.1126/science.abd4607. Llorens JB, Herrera E, Barrena V, Wu B, Heinsdorf N, Borisov V, Valentí R, Meier WR, Bud’ko S, Canfield PC, Guillamón I, and Suderow H (2021) Anisotropic superconductivity in the spin-vortex antiferromagnetic superconductor CaK(Fe0.95Ni0.05)4As4. Physical Review B 103(6): L060506. https://doi.org/10.1103/PhysRevB.103.L060506. Lo Conte R, Bazarnik M, Palotás K, Rózsa L, Szunyogh L, Kubetzka A, von Bergmann K, and Wiesendanger R (2022) Coexistence of antiferromagnetism and superconductivity in Mn/ Nb(110). Physical Review B 105(10): L100406. https://doi.org/10.1103/PhysRevB.105.L100406. Lv Y-F, Wang W-L, Zhang Y-M, Ding H, Li W, Wang L, He K, Song C-L, Ma X-C, and Xue Q-K (2017) Experimental signature of topological superconductivity and Majorana zero modes on b-Bi2Pd thin films. Science Bulletin 62(12): 852–856. ISSN: 2095-9273. https://doi.org/10.1016/j.scib.2017.05.008. Mackenzie AP and Maeno Y (2003) The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing. Reviews of Modern Physics 75(2): 657–712. https://doi.org/10.1103/ RevModPhys.75.657. Maggio-Aprile I, Renner C, Erb A, Walker E, and Fischer Ø (1997) Critical currents approaching the depairing limit at a twin boundary in YBa2Cu3O7−d. Nature 390(6659): 487–490. ISSN: 1476-4687. https://doi.org/10.1038/37312. Margaritondo G (2005) Photoelectron spectromicroscopy. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 272–279. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00587-8. Martín-Vega F, Barrena V, Sánchez-Barquilla R, Fernández-Lomana M, Benito Llorens J, Wu B, Fente A, Perconte Duplain D, Horcas I, López R, Blanco J, Higuera JA, Mañas-Valero S, Jo NH, Schmidt J, Canfield PC, Rubio-Bollinger G, Rodrigo JG, Herrera E, Guillamón I, and Suderow H (2021) Simplified feedback control system for scanning tunneling microscopy. Review of Scientific Instruments 92(10): 103705. https://doi.org/10.1063/5.0064511. Martín-Vega F, Herrera E, Wu B, Barrena V, Mompeán F, García-Hernández M, Canfield PC, Black-Schaffer AM, Baldoví JJ, Guillamón I, and Suderow H (2022) Superconducting density of states and band structure at the surface of the candidate topological superconductor Au2Pb. Physical Review Research 4(2): 023241. https://doi.org/10.1103/ PhysRevResearch.4.023241. Martinez-Samper P, Rodrigo JG, Rubio-Bollinger G, Suderow H, Vieira S, Lee S, and Tajima S (2003) Scanning tunneling spectroscopy in MgB2. Physica C: Superconductivity 385(1): 233–243. ISSN: 0921-4534. https://doi.org/10.1016/S0921-4534(02)02296-7. Martínez-Samper P, Suderow H, Vieira S, Brison JP, Luchier N, Lejay P, and Canfield PC (2003) Phonon-mediated anisotropic superconductivity in the Y and Lu nickel borocarbides. Physical Review B 67(1): 014526. https://doi.org/10.1103/PhysRevB.67.014526.

614

Superconducting density of states from scanning tunneling microscopy

Mazin I (2023) Unconventional superconductivity. In: Mazin I (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Mazumdar C, Nagarajan R, Godart C, Gupta LC, Latroche M, Dhar SK, Levy-Clement C, Padalia BD, and Vijayaraghavan R (1993) Superconductivity at 12 K in Y-Ni-B system. Solid State Communications 87(5): 413–416. ISSN: 0038-1098. https://doi.org/10.1016/0038-1098(93)90788-O. McElroy K, Simmonds RW, Hoffman JE, Lee D-H, Orenstein J, Eisaki H, Uchida S, and Davis JC (2003) Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+d. Nature 422(6932): 592–596. ISSN: 1476-4687. https://doi.org/10.1038/nature01496. Meier WR, Kong T, Kaluarachchi US, Taufour V, Jo NH, Drachuck G, Böhmer AE, Saunders SM, Sapkota A, Kreyssig A, Tanatar MA, Prozorov R, Goldman AI, Balakirev FF, Gurevich A, Bud’ko SL, and Canfield PC (2016) Anisotropic thermodynamic and transport properties of single-crystalline CaKFe4As4. Physical Review B 94(6): 064501. https://doi.org/ 10.1103/PhysRevB.94.064501. Meier WR, Kong T, Bud’ko SL, and Canfield PC (2017) Optimization of the crystal growth of the superconductor CaKFe4As4 from solution in the FeAs-CaFe2As2-KFe2As2 system. Physical Review Materials 1(1): 013401. https://doi.org/10.1103/PhysRevMaterials.1.013401. Meier WR, Ding Q-P, Kreyssig A, Bud’ko SL, Sapkota A, Kothapalli K, Borisov V, Valentí R, Batista CD, Orth PP, Fernandes RM, Goldman AI, Furukawa Y, Böhmer AE, and Canfield PC (2018) Hedgehog spin-vortex crystal stabilized in a hole-doped iron-based superconductor. npj Quantum Materials 3(1): 5. ISSN: 2397-4648. https://doi.org/10.1038/s41535017-0076-x. Mikitik G (2023) Superconductivity: Ginzburg-Landau theory and vortex lattice. In: Mikitik G (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Mineev VP and Samokhin KV (1999) Introduction to Unconventional Superconductivity. Gordon and Breach Science Publishers. Mondal M, Kamlapure A, Chand M, Saraswat G, Kumar S, Jesudasan J, Benfatto L, Tripathi V, and Raychaudhuri P (2011) Phase fluctuations in a strongly disordered s-wave NbN superconductor close to the metal-insulator transition. Physical Review Letters 106(4): 047001. https://doi.org/10.1103/PhysRevLett.106.047001. Müller K-H and Narozhnyi VN (2001) Interaction of superconductivity and magnetism in borocarbide superconductors. Reports on Progress in Physics 64(8): 943. https://doi.org/ 10.1088/0034-4885/64/8/202. Nadj-Perge S, Drozdov IK, Li J, Chen H, Jeon S, Seo J, MacDonald AH, Bernevig BA, and Yazdani A (2014) Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346(6209): 602–607. https://doi.org/10.1126/science.1259327. Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y, and Akimitsu J (2001) Superconductivity at 39 K in magnesium diboride. Nature 410(6824): 63–64. ISSN: 1476-4687. https:// doi.org/10.1038/35065039. Navarro-Moratalla E, Island JO, Mañas-Valero S, Pinilla-Cienfuegos E, Castellanos-Gomez A, Quereda J, Rubio-Bollinger G, Chirolli L, Silva-Guillén JA, Agraït N, Steele GA, Guinea F, van der Zant HSJ, and Coronado E (2016) Enhanced superconductivity in atomically thin TaS2. Nature Communications 7(1): 11043. ISSN: 2041-1723. https://doi.org/10.1038/ ncomms11043. Noat Y, Silva-Guillén JA, Cren T, Cherkez V, Brun C, Pons S, Debontridder F, Roditchev D, Sacks W, Cario L, Ordejón P, García A, and Canadell E (2015) Quasiparticle spectra of 2H-NbSe2: Two-band superconductivity and the role of tunneling selectivity. Physical Review B 92(13): 134510. https://doi.org/10.1103/PhysRevB.92.134510. Nozières P and Schmitt-Rink S (1985) Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. Journal of Low Temperature Physics 59(3): 195–211. ISSN: 1573-7357. https://doi.org/10.1007/BF00683774. Ortuzar J, Trivini S, Alvarado M, Rouco M, Zaldivar J, Yeyati AL, Pascual JI, and Bergeret FS (2022) Yu-Shiba-Rusinov states in two-dimensional superconductors with arbitrary Fermi contours. Physical Review B 105(24): 245403. https://doi.org/10.1103/PhysRevB.105.245403. Paglione J and Greene RL (2010) High-temperature superconductivity in iron-based materials. Nature Physics 6(9): 645–658. ISSN: 1745-2481. https://doi.org/10.1038/nphys1759. Palacio-Morales A, Mascot E, Cocklin S, Kim H, Rachel S, Morr DK, and Wiesendanger R (2019) Atomic-scale interface engineering of Majorana edge modes in a 2D magnetsuperconductor hybrid system. Science Advances 5(7): eaav6600. https://doi.org/10.1126/sciadv.aav6600. Pan SH, Hudson EW, and Davis JC (1998) Vacuum tunneling of superconducting quasiparticles from atomically sharp scanning tunneling microscope tips. Applied Physics Letters 73(20): 2992–2994. https://doi.org/10.1063/1.122654. Pan SH, Hudson EW, Lang KM, Eisaki H, Uchida S, and Davis JC (2000) Imaging the effects of individual zinc impurity atoms on superconductivity in Bi2Sr2CaCu2O8+d. Nature 403 (6771): 746–750. ISSN: 1476-4687. https://doi.org/10.1038/35001534. Pascual JI, Bihlmayer G, Koroteev YM, Rust H-P, Ceballos G, Hansmann M, Horn K, Chulkov EV, Blügel S, Echenique PM, and Hofmann P (2004) Role of spin in quasiparticle interference. Physical Review Letters 93(19): 196802. https://doi.org/10.1103/PhysRevLett.93.196802. Petersen L, Sprunger PT, Hofmann P, Lægsgaard E, Briner BG, Doering M, Rust H-P, Bradshaw AM, Besenbacher F, and Plummer EW (1998) Direct imaging of the two-dimensional Fermi contour: Fourier-transform STM. Physical Review B 57(12): R6858–R6861. https://doi.org/10.1103/PhysRevB.57.R6858. Postolova SV, Mironov AY, Barrena V, Benito-Llorens J, Rodrigo JG, Suderow H, Baklanov MR, Baturina TI, and Vinokur VM (2020) Superconductivity in a disordered metal with Coulomb interactions. Physical Review Research 2(3): 033307. https://doi.org/10.1103/PhysRevResearch.2.033307. Proslier T, Kohen A, Noat Y, Cren T, Roditchev D, and Sacks W (2006) Probing the superconducting condensate on a nanometer scale. Europhysics Letters 73(6): 962. https://doi.org/ 10.1209/epl/i2005-10488-0. Qin MJ and Dou SX (2005) Superconductors, high Tc. In: Bassani F, Liedl GL, and Wyder P (eds.) Encyclopedia of Condensed Matter Physics, pp. 112–120. Oxford: Elsevier, ISBN: 978-0-12-369401-0. https://doi.org/10.1016/B0-12-369401-9/00706-3. Renner C, Revaz B, Genoud J-Y, Kadowaki K, and Fischer O (1998) Pseudogap precursor of the superconducting gap in under- and overdoped Bi2Sr2CaCu2O8+d. Physical Review Letters 80(1): 149–152. https://doi.org/10.1103/PhysRevLett.80.149. Rhodes LC, Eschrig M, Kim TK, and Watson MD (2022) FeSe and the missing electron pocket problem. Frontiers in Physics 10. ISSN: 2296-424X. https://doi.org/10.3389/ fphy.2022.859017. Rodrigo JG and Vieira S (2004) STM study of multiband superconductivity in NbSe2 using a superconducting tip. Physica C: Superconductivity 404(1): 306–310. ISSN: 0921-4534. https://doi.org/10.1016/j.physc.2003.10.030. Proceedings of the Third European Conference on Vortex Matter in Superconductors at Extreme Scales and Conditions. Rodrigo JG, Suderow H, and Vieira S (2004a) On the use of STM superconducting tips at very low temperatures. The European Physical Journal B–Condensed Matter and Complex Systems 40(4): 483–488. ISSN: 1434-6036. https://doi.org/10.1140/epjb/e2004-00273-y. Rodrigo JG, Suderow H, Vieira S, Bascones E, and Guinea F (2004b) Superconducting nanostructures fabricated with the scanning tunnelling microscope. Journal of Physics: Condensed Matter 16(34): R1151. https://doi.org/10.1088/0953-8984/16/34/R01. Roy I, Ganguly R, Singh H, and Raychaudhuri P (2019) Robust pseudogap across the magnetic field driven superconductor to insulator-like transition in strongly disordered NbN films. The European Physical Journal B 92(3): 49. ISSN: 1434-6036. https://doi.org/10.1140/epjb/e2019-90488-0. Rubio-Bollinger G, Suderow H, and Vieira S (2001) Tunneling spectroscopy in small grains of superconducting MgB2. Physical Review Letters 86(24): 5582–5584. https://doi.org/ 10.1103/PhysRevLett.86.5582. Rubio-Verdú C, García-García AM, Ryu H, Choi D-J, Zaldívar J, Tang S, Fan B, Shen Z-X, Mo S-K, Pascual JI, and Ugeda MM (2020) Visualization of multifractal superconductivity in a two-dimensional transition metal dichalcogenide in the weak-disorder regime. Nano Letters 20(7): 5111–5118. ISSN: 1530-6984. https://doi.org/10.1021/acs. nanolett.0c01288. Ruby M, Heinrich BW, Pascual JI, and Franke KJ (2015) Experimental demonstration of a two-band superconducting state for lead using scanning tunneling spectroscopy. Physical Review Letters 114(15): 157001. https://doi.org/10.1103/PhysRevLett.114.157001. Ryu S (2023) Superconductors, topological. In: Ryu S (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Sacépé B, Chapelier C, Baturina TI, Vinokur VM, Baklanov MR, and Sanquer M (2008) Disorder-induced inhomogeneities of the superconducting state close to the superconductorinsulator transition. Physical Review Letters 101(15): 157006. https://doi.org/10.1103/PhysRevLett.101.157006. Sacépé B, Chapelier C, Baturina TI, Vinokur VM, Baklanov MR, and Sanquer M (2010) Pseudogap in a thin film of a conventional superconductor. Nature Communications 1(1): 140. ISSN: 2041-1723. https://doi.org/10.1038/ncomms1140.

Superconducting density of states from scanning tunneling microscopy

615

Sacépé B, Dubouchet T, Chapelier C, Sanquer M, Ovadia M, Shahar D, Feigel’man M, and Ioffe L (2011) Localization of preformed Cooper pairs in disordered superconductors. Nature Physics 7(3): 239–244. ISSN: 1745-2481. https://doi.org/10.1038/nphys1892. Sakano M, Okawa K, Kanou M, Sanjo H, Okuda T, Sasagawa T, and Ishizaka K (2015) Topologically protected surface states in a centrosymmetric superconductor b-PdBi2. Nature Communications 6(1): 8595. ISSN: 2041-1723. https://doi.org/10.1038/ncomms9595. Sakata H, Oosawa M, Matsuba K, Nishida N, Takeya H, and Hirata K (2000) Imaging of a vortex lattice transition in YNi2B2C by scanning tunneling spectroscopy. Physical Review Letters 84(7): 1583–1586. https://doi.org/10.1103/PhysRevLett.84.1583. Schnyder AP, Ryu S, Furusaki A, and Ludwig AWW (2008) Classification of topological insulators and superconductors in three spatial dimensions. Physical Review B 78(19): 195125. https://doi.org/10.1103/PhysRevB.78.195125. Senkpiel J, Dambach S, Etzkorn M, Drost R, Padurariu C, Kubala B, Belzig W, Yeyati AL, Cuevas JC, Ankerhold J, Ast CR, and Kern K (2020) Single channel Josephson effect in a high transmission atomic contact. Communications Physics 3(1): 131. ISSN: 2399-3650. https://doi.org/10.1038/s42005-020-00397-z. Senkpiel J, Drost R, Klöckner JC, Etzkorn M, Ankerhold J, Cuevas JC, Pauly F, Kern K, and Ast CR (2022) Extracting transport channel transmissions in scanning tunneling microscopy using superconducting excess current. Physical Review B 105(16): 165401. https://doi.org/10.1103/PhysRevB.105.165401. Sherman D, Kopnov G, Shahar D, and Frydman A (2012) Measurement of a superconducting energy gap in a homogeneously amorphous insulator. Physical Review Letters 108(17): 177006. https://doi.org/10.1103/PhysRevLett.108.177006. Shibauchi T, Hanaguri T, and Matsuda Y (2020) Exotic superconducting states in FeSe-based materials. Journal of the Physical Society of Japan 89(10): 102002. https://doi.org/ 10.7566/JPSJ.89.102002. Sigrist M (2005) Review on the chiral p-wave phase of Sr2RuO4. Progress of Theoretical Physics Supplement 160: 1–14. ISSN: 0375-9687. https://doi.org/10.1143/PTPS.160.1. Sigrist M and Ueda K (1991) Phenomenological theory of unconventional superconductivity. Reviews of Modern Physics 63(2): 239–311. https://doi.org/10.1103/ RevModPhys.63.239. Song C-L, Wang Y-L, Cheng P, Jiang Y-P, Li W, Zhang T, Li Z, He K, Wang L, Jia J-F, Hung H-H, Wu C, Ma X, Chen X, and Xue Q-K (2011) Direct observation of nodes and twofold symmetry in FeSe superconductor. Science 332(6036): 1410–1413. https://doi.org/10.1126/science.1202226. Sprau PO, Kostin A, Kreisel A, Böhmer AE, Taufour V, Canfield PC, Mukherjee S, Hirschfeld PJ, Andersen BM, and Davis JCS (2017) Discovery of orbital-selective cooper pairing in FeSe. Science 357(6346): 75–80. https://doi.org/10.1126/science.aal1575. Stolyarov VS, Pervakov KS, Astrakhantseva AS, Golovchanskiy IA, Vyalikh DV, Kim TK, Eremeev SV, Vlasenko VA, Pudalov VM, Golubov AA, Chulkov EV, and Roditchev D (2020) Electronic structures and surface reconstructions in magnetic superconductor RbEuFe4As4. The Journal of Physical Chemistry Letters 11(21): 9393–9399. https://doi.org/ 10.1021/acs.jpclett.0c02711. PMID: 33095988. Suderow H, Martinez-Samper P, Luchier N, Brison JP, Vieira S, and Canfield PC (2001) Tunneling spectroscopy in the magnetic superconductor TmNi2B2C. Physical Review B 64(2): 020503. https://doi.org/10.1103/PhysRevB.64.020503. Suderow H, Bascones E, Izquierdo A, Guinea F, and Vieira S (2002) Proximity effect and strong-coupling superconductivity in nanostructures built with an STM. Physical Review B 65(10): 100519. https://doi.org/10.1103/PhysRevB.65.100519. Suderow H, Crespo V, Guillamon I, Vieira S, Servant F, Lejay P, Brison JP, and Flouquet J (2009) A nodeless superconducting gap in Sr2RuO4 from tunneling spectroscopy. New Journal of Physics 11(9): 093004. https://doi.org/10.1088/1367-2630/11/9/093004. Suderow H, Guillamón I, Rodrigo JG, and Vieira S (2014) Imaging superconducting vortex cores and lattices with a scanning tunneling microscope. Superconductor Science and Technology 27(6): 063001. https://doi.org/10.1088/0953-2048/27/6/063001. Tafuri F (2023) Josephson junctions. In: Tafuri F (ed.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Tersoff J and Hamann DR (1983) Theory and application for the scanning tunneling microscope. Physical Review Letters 50(25): 1998–2001. https://doi.org/10.1103/ PhysRevLett.50.1998. Tersoff J and Hamann DR (1985) Theory of the scanning tunneling microscope. Physical Review B 31(2): 805–813. https://doi.org/10.1103/PhysRevB.31.805. Tonnoir C, Kimouche A, Coraux J, Magaud L, Delsol B, Gilles B, and Chapelier C (2013) Induced superconductivity in graphene grown on Rhenium. Physical Review Letters 111(24): 246805. https://doi.org/10.1103/PhysRevLett.111.246805. Valencia-Ibáñez S, Jiménez-Guerrero D, Salcedo-Pimienta JD, Vega-Bustos K, Cárdenas-Chirivi G, López-Manrique L, Herrera-Vasco E, Galvis JA, Hernández Y, and Giraldo-Gallo P (2022) Raman spectroscopy of few-layers TaS2 and Mo-doped TaS2 with enhanced superconductivity. Advanced Electronic Materials 8(10): 2200457. https://doi.org/10.1002/ aelm.202200457. Valles JM, Dynes RC, and Garno JP (1989) Temperature dependence of the two-dimensional electronic density of states in disordered metal films. Physical Review B 40(11): 7590–7593. https://doi.org/10.1103/PhysRevB.40.7590. Vinokur V (2023) Superinsulators. In: Vinokur V, Diamantini C, and Trugenberger CA (eds.) Encyclopedia of Condensed Matter Physics. Oxford: Elsevier. Vinokur VM, Baturina TI, Fistul MV, Mironov AY, Baklanov MR, and Strunk C (2008) Superinsulator and quantum synchronization. Nature 452(7187): 613–615. ISSN: 1476-4687. https://doi.org/10.1038/nature06837. Wang F, Kivelson SA, and Lee D-H (2015) Nematicity and quantum paramagnetism in FeSe. Nature Physics 11(11): 959–963. ISSN: 1745-2481. https://doi.org/10.1038/ nphys3456. Wang Z, Walkup D, Derry P, Scaffidi T, Rak M, Vig S, Kogar A, Zeljkovic I, Husain A, Santos LH, Wang Y, Damascelli A, Maeno Y, Abbamonte P, Fradkin E, and Madhavan V (2017) Quasiparticle interference and strong electron-mode coupling in the quasi-one-dimensional bands of Sr2RuO4. Nature Physics 13(8): 799–805. ISSN: 1745-2481. https://doi.org/ 10.1038/nphys4107. Wang Y, Edkins SD, Hamidian MH, Davis JCS, Fradkin E, and Kivelson SA (2018) Pair density waves in superconducting vortex halos. Physical Review B 97(17): 174510. https://doi. org/10.1103/PhysRevB.97.174510. Wang Q, Fanfarillo L, and Böhmer AE (2022) Editorial: Nematicity in iron-based superconductors. Frontiers in Physics 10. ISSN: 2296-424X. https://doi.org/10.3389/ fphy.2022.1038127. Watson MD, Kim TK, Haghighirad AA, Davies NR, McCollam A, Narayanan A, Blake SF, Chen YL, Ghannadzadeh S, Schofield AJ, Hoesch M, Meingast C, Wolf T, and Coldea AI (2015) Emergence of the nematic electronic state in FeSe. Physical Review B 91(15): 155106. https://doi.org/10.1103/PhysRevB.91.155106. Yu R and Si Q (2015) Antiferroquadrupolar and Ising-nematic orders of a frustrated bilinear-biquadratic heisenberg model and implications for the magnetism of FeSe. Physical Review Letters 115(11): 116401. https://doi.org/10.1103/PhysRevLett.115.116401. Yu R, Zhu J-X, and Si Q (2018) Orbital selectivity enhanced by nematic order in FeSe. Physical Review Letters 121(22): 227003. https://doi.org/10.1103/PhysRevLett.121.227003. Zhang P, Yaji K, Hashimoto T, Ota Y, Kondo T, Okazaki K, Wang Z, Wen J, Gu GD, Ding H, and Shin S (2018) Observation of topological superconductivity on the surface of an ironbased superconductor. Science 360(6385): 182–186. https://doi.org/10.1126/science.aan4596. Zhao B, Shen D, Zhang Z, Lu P, Hossain M, Li J, Li B, and Duan X (2021) 2D Metallic transition-metal dichalcogenides: Structures, synthesis, properties, and applications. Advanced Functional Materials 31(48): 2105132. https://doi.org/10.1002/adfm.202105132. Ziman JM (1972) Principles of the Theory of Solids, second ed. Cambridge University Press. https://doi.org/10.1017/CBO9781139644075.

Josephson junctions Francesco Tafuri, Dipartimento di Fisica “E.Pancini", Università di Napoli Federico II, Napoli, Italy © 2024 Elsevier Ltd. All rights reserved.

Introduction The coupling between macroscopic quantum systems and the equations of the Josephson effect From tunnel barriers to transparent interfaces More transparent barriers and the Josephson effect derived from Quasi-particle Andreev Bound States Essential parameters of a Josephson junction Properties of a Josephson junction and imprints of macroscopic quantum phenomena Current-Voltage (I–V) characteristics: From the washboard potential to macroscopic quantum phenomena Hamiltonian of a Josephson junction Temperature and magnetic dependence of the critical current and of I–V curves Temperature dependence of Ic Magnetic dependence of the critical current Emerging trends: From qubits to hybrid JJs Low temperature Josephson junctions for qubits, quantum circuits and superconducting electronics Hybrid Josephson junctions uncovering emergent phenomena: from semiconducting and ferromagnetic barriers to topological insulators and Van der Waals crystals The semiconducting barrier Topological insulators The ferromagnetic barrier vdW crystals integration into JJs and Twisted HTS JJs Conclusion References

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Abstract A Josephson junction has the unique capability of transferring quantum mechanics rules to macroscopic systems and is at the same time a versatile component of all superconducting electronics with its key feature of carrying a dissipationless phase-driven current. Progress in materials science and nanotechnology consolidates expectations for a series of challenging applications, including the realization of multi-qubit quantum processors, while unconventional hybrid and high critical temperature superconductor Josephson junctions keep disclosing novel problems at the frontier of our knowledge. The diversity in Josephson junctions opens ‘horizons’. Much is happening and partly needs to be consolidated. Much remains to do.

Introduction In these 50 years since its discovery (Josephson, 1962), the Josephson effect has played a key role for advances in physics both on fundamental aspects and applications. The key feature is a dissipationless current driven by a phase difference of the macroscopic wavefunction between the two superconducting electrodes. A Josephson junction (JJ) has thus the ability to manipulate and measure this phase difference transferring on a macroscopic circuit the quantum mechanics approach commonly applied on microscopic entities. Quantum tunneling takes place at a macroscopic scale in a JJ. A macroscopic superconducting circuit incorporating JJs will be described by a Hamiltonian built on the charge and phase operators. The states of the circuit will correspond to the eigenfunctions of the Hamiltonian associated to the specific circuit. The Josephson junction is an extremely versatile component of all superconducting circuits providing a series of unique properties and the required nonlinear behavior necessary for a wide range of applications (Barone and Paternò, 1982; Likharev, 1986). The Josephson junction is the main block of the superconducting qubit and thus of the emerging field of superconducting quantum computation engineering. Josephson coupling is not confined to junctions using artificial barriers, but it extends to intrinsic junctions that may naturally build up across superconducting planes or grains or between coherent islands of size characteristic of the material. The Josephson coupling keeps being a formidable tool to disclose new fundamental physics (Barone and Paternò, 1982; Likharev, 1986; Tafuri, 2019). Since the discovery of high-temperature superconductivity in cuprates, for instance, Josephson junction-based phase-sensitive experiments are believed and have been used to provide the most convincing evidence for determining the pairing symmetry (Tsuei et al., 1994; Wollman et al., 1995; Van Harlingen, 1995; Tsuei and Kirtley, 2000). Hybrid systems offer special opportunities by combining the potentials of the superconducting electrodes with those of the barriers, as for instance topological materials, semiconducting nanowires, ferromagnets, graphene. These barriers and the related interfaces are considered as triggers of Majorana fermions, triplet superconductivity and fancy physics. We will give an account of the meaning of the Josephson effect, also

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through its formalism, and of the methods used to demonstrate its nature and to establish its codes. These are indispensable references to be used in recent explorations on the Josephson effect. Much has to be expected in the future.

The coupling between macroscopic quantum systems and the equations of the Josephson effect From tunnel barriers to transparent interfaces Brian Josephson discovered that tunneling effect is not limited to quasi-particles and that also Cooper pairs can tunnel through a thin insulating layer (I) when the electrodes are composed of two superconductors SR and SL (Josephson, 1962). Cooper pair tunneling carries information on the phase difference of the macroscopic wavefunctions of the two superconductors. The Josephson effect is expressed by the two following equations, originally derived for an SR-I-SL junction: Is ¼ Ic sin’

(1)

d’=dt ¼ 2eV=ℏ

(2)

where ’ ¼ ’R − ’L is the phase difference between the two superconducting electrodes ’R and ’L, e and ℏ are the electron charge and the reduced Planck constant respectively. IC is the maximum critical current and depends on the temperature and the magnetic field. Josephson supercurrent survives as long as the macroscopic wave functions of the superconducting electrodes overlap in the barrier region (Fig. 1a). The first tunnel Josephson junctions used soft superconductors as Sn, In, Pb, and thermal oxidation for the barrier. Later on, the “lead-alloy technology” was the first to use self-limiting sputter-oxidation process. A key advance was the development of devices (j)

c

c b

a

a (b)

(a)

(c)

001-TILT

b

(d)

c

100-TILT

b

(e)

b

c a

a (k)

(h) (f)

100-TWIST c (i)

(g) L

W

c b

b a

a

(l)

Fig. 1 (a) Spatial dependence of the macroscopic wave functions of the two electrodes in a Josephson junction in a 3-d and 2-d view. Different junction configurations: window-type geometry for a sandwich junction with (b) insulating or (c) normal metal or semiconducting or ferromagnet barrier. (d) Coplanar variable-thickness bridge; the barrier is grown before the deposition of the superconductor; the barrier can be a flake of graphene or of topological insulator (e) or a nanowire (f ). The coplanar can also have an inverted structure where the barrier is deposited on the top of the barrier afterwards, as shown in (g). In (h) the scheme of a point contact junction is shown. In all these graphs we use grey/black for a superconductor, yellow for a metal or a semiconductor or a ferromagnet. Thin insulating barrier is in violet, while thick insulating layers are in green, and they serve to completely isolate the electrodes. The red arrow approximately indicates the current direction. In (i) scheme of microbridge. L refers to the length of the barrier, while W is the width of the weak link. Tri-dimensional view of a 001 tilt (j), 100 tilt (k) and 100 twist (l) grain boundary junction. In (j) it is also reported the bicrystal substrate, which in the most common technique determines the growth of the thin superconducting layer.

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based on “rigid” superconductors as Nb, with the use of artificial barriers replacing the native Nb oxide barriers. Al turned to be the perfect material solution forming a natural, self-limiting, high quality, insulating oxide (Gurvitch et al., 1983). Other rigid superconductors such as NbN, Nb3Sn, V3Si and Nb3Ge were later used (Wolf et al., 2017). The impact of high critical temperature superconductors (HTS) was also impressive for the development of activities on Josephson devices and to use these properties at unprecedented values of temperature and magnetic fields (Koelle et al., 1999; Tsuei and Kirtley, 2000; Hilgenkamp and Mannhart, 2002; Tafuri and Kirtley, 2005). For a finite voltage V 6¼ 0 the phase ’ varies in time ’ ¼ ’o + 2e/ℏ V t, as derived from Eq. (2), and an alternating current I ¼ Ic sin (’o + 2e/ℏ V t) appears with an angular frequency o ¼ 2eV/ℏ. This is called the a.c. Josephson effect, and the ratio between frequency and voltage is given by: n/V ¼ 483.6 MHz/mV, of the order of 109–1013 Hz at typical voltages 10−3—10 mV. A direct manifestation of the a.c. Josephson effect is the presence of current steps at constant voltages: Vn ¼ nh/(2e) where n ¼  1,  2, . . . in presence of microwave irradiation at a frequency no. The steps are the direct consequence of an interaction between the a.c. Josephson current and the applied microwave signal. These steps first observed by Shapiro (Shapiro, 1963) are commonly reported as Shapiro steps. The a.c. Josephson effect can be also derived from the very general arguments of spontaneous breakdown of electromagnetic gage invariance (Weinberg, 1986; Anderson, 1997). The high precision predictions about superconductors, and for the a.c. Josephson effect the frequency of the alternating current in terms of the fundamental constant e/h, derive from spontaneous breakdown of electromagnetic gage invariance in a S (Weinberg, 1986; Thouless, 1998). The Josephson effect is not restricted to tunnel junctions (Fig. 1b), but it occurs in junctions with more transmissive barriers where the I layer is replaced by a metallic N layer (see for instance Fig. 1c). The Josephson supercurrent will flow ffiin the resulting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SL-N-SR structure as far as the thickness of the barrier is comparable with the coherence length xN ¼ ℏDdiff =ðkB T Þ induced in the barrier (de Gennes, 1964) (T is the temperature and Ddiff ¼ vFℓ/3 is the normal metal diffusion constant, being vF the Fermi velocity, ℓ the electron mean free path and kB the Boltzmann’s constant respectively). xN can be in some cases even of the order of a few microns. A barrier composed by a normal metal dramatically reduces the normal state junction resistance (RN) and significantly modifies the junction capacitance (C). New physical “processes” replace tunneling, and they all fall in the category of effects induced by the proximity effect (PE), which will be specific of the junction configuration depending on the material, the type of barrier, the geometry, the boundary conditions. PE expresses the mutual influence between the superconducting electrode and the adjacent barrier layer, other than insulating (Clarke, 1971,de Gennes, 1964, de Gennes, 1966). The barrier is not limited to a normal metal but can be for instance a semiconductor, a ferromagnet, a topological insulator and so on, with novel effects related to the intrinsic nature of the barrier and of the interfaces. The various materials and their properties influence the geometry and the layout of the junction inspiring alternative ways of placing a barrier as reported in Fig. 1d–g, respectively. An accurate description of systems based on S/N interfaces can be also given in terms of Andreev reflection (AR) (Andreev, 1964), the microscopic process in which a dissipative electrical current is converted at an S/N interface into dissipationless supercurrent. AR is the scattering mechanism describing how an electron excitation slightly above the Fermi level in the normal metal is reflected at the interface as a hole excitation slightly below the Fermi level (Andreev, 1964). In Fig. 2a the energy representation of AR is sketched. The missing charge of 2e is removed as a Cooper pair. This is a branch-crossing process which converts electrons into holes and vice versa, and therefore changes the net charge in the excitation distribution. The reflected hole (or electron) has a shift in phase compared to the incoming electron (or hole) wave-function: ’hole ¼ ’elect + ’superc  arccosðE=DÞ, ’elect ¼ ’hole − ’superc  arccosðE=DÞwhere D and jsuperc are the gap value and the superconducting phase of the S respectively. jsuperc is equivalent to jL or jR used in all previous expressions. The macroscopic phase of the S and the microscopic phase of the quasi-particles are therefore mixed through AR. The Andreev-reflected holes operate as a parallel conduction channel to the initial electron current. The normal state conductance of the S/N interface is doubled for applied voltages less than the superconducting gap eV < D (Blonder et al., 1982). Blonder et al. (1982) (BTK) introduced the dimensionless parameter Z, proportional to the potential barrier at the interface, to describe the barrier transparency. This allows the continuous passage from the tunnel limit to a transmissive barrier since the barrier transparency is defined as D ¼ 1/(1 + Z2). An exact expression for the tunneling current has been also obtained, using standard, many-body, nonequilibrium Green’s function techniques (Arnold, 1985). The Landauer conductance expression has been extended to the case of an S-N interface P 2 2 through scattering matrix theory (Beenakker, 1997) GNS ¼ 2e2 =ðpℏ Þ N n¼1 t n =ð2 − t n Þ , where the tn values are the transmission eigenvalues for the n channel, and N is the total number of channels. In a SL-N-SR structure multiple Andreev reflections occur (see Fig. 2b) (see below). The modern era of Josephson devices is thus strongly influenced by the combined continuous progress in material science and nanotechnologies applied to superconductivity. These aspects are tightly connected. Progress in material science means new materials and new superconductors, and novel abilities in building interfaces and in the precise control of heterostructure in the growth process. Barriers of classical tunnel junctions (employing well-established low critical temperature superconductors (LTS)) benefit as well of the general technological progress and are designed and fabricated with unprecedented precision, opening the route to more performing devices. Advances in nanotechnologies applied to superconductivity is a necessary tool towards the practical implementation of Josephson devices using advanced materials, since they allow to suitably scale the size of the barrier and to handle pre-built barriers as for instance nanowires (NWs) and flakes. Hybrid junctions are an obvious consequence of the combined progress of material science and nanotechnology. Another practical way to form a Josephson junction is through a point contact (Fig. 1h) or by creating a micro-restriction in a superconducting thin film (Fig. 1i). Scanning Tunnel Microscopy and atomic point contacts represent advanced versions of the original point contact technique (Wolf, 1985). For widths of the order of a few times the coherence length of the superconductor xS, the microbridge will behave as a Josephson weak link (Likharev, 1979). This type of junction depends very critically on the dimensions of the microbridge and its characteristic lengths, ranging from a Josephson junction to a bridge hosting phase slip events

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(a)

(c)

(b) (d)

(g) (e)

(f)

Fig. 2 (a) Energy representation of AR process: an electron excitation slightly above the Fermi level in the normal metal is reflected at the interface as a hole excitation slightly below the Fermi level. The electron is reflected as a hole with the same momentum and opposite velocity. The missing charge of 2e is absorbed as a Cooper pair by the superconducting condensate. (b) Energy representation of Andreev reflections in SNS junctions. (c) The d-wave OP symmetry configuration is reported for a 001 tilt GB for two generic misorientation angles y1 and y2. Maximum coupling (maximum Ic) happens for lobes facing each other (y1¼ y2 ¼ 0 ) (d), while minimal coupling (minimal Ic) occurs for a lobe facing a node (y1 ¼ 0 , y2 ¼ 45 ) (e). (f ) “Corner” junction between an HTS d-wave electrode and an ordinary s-wave counter-electrode. (g) Andreev bands for different values of the D parameter: curve a D ¼ 1, curve b D ¼ 0.8; curve c D ¼ 0.5; an opening of a gap in the Andreev band is evident in the presence of a potential barrier.

or Abrikosov vortex motion respectively. A kind of ‘phase diagram’ can be derived as a function of their dimensions (width W, length L) (Likharev, 1979). The Josephson effect only takes place for L < 3.5 xS independently of the width W (Likharev, 1979). Phenomena associated to phase slips (W < xS) or to the motion of Abrikosov vortices (W > xS) will take place for L > 3.5 xS (Likharev, 1979). In the limit of long microbridges, Josephson behavior substantially disappears in favor of a regime of strong superconductivity, where the critical current is controlled by standard depairing processes (Tinkham, 2004). Josephson coupling can also take place at grain boundaries (GBs) and this property has been widely used to produce HTS JJs (Hilgenkamp and Mannhart, 2002; Tafuri and Kirtley, 2005). Any GB can be considered as the result of three fundamental operations as shown in Fig. 1j–l (Chaudari et al., 1988; Dimos et al., 1990): one electrode can be tilted with respect to the other electrode around the c-axis (001 tilt), or tilted around the a or b direction (tilt of the c-axis, 100 tilt) or twisted along the junction interface (100 twist). GBs can realize quite special atomically flat interface configurations. In HTS the d-wave order parameter (OP) symmetry generates quite unconventional features. In Fig. 2c the d-wave OP symmetry configuration is reported for a 001 tilt GB for two generic misorientation angles y1 and y2 (Hilgenkamp and Mannhart, 2002; Tsuei and Kirtley, 2000, Tafuri and Kirtley, 2005). Maximum coupling (maximum Ic) happens for lobes facing each other (y1 ¼ y2 ¼ 0 ) (Fig. 2d), while minimal coupling (minimal Ic) occurs for a lobe facing a node (y1 ¼ 0 , y2 ¼ 45 ) (Fig. 2e). D-wave effects are visible in the whole class of HTS Josephson junctions in opportune conditions. In Fig. 2f for instance a “corner” junction between an HTS d-wave electrode and an ordinary s-wave counter-electrode is depicted. Here it is possible to measure the effects generated by the contemporary presence of two channels probing both lobes of the OP in the kx and ky directions respectively (Wollman et al., 1995; Van Harlingen, 1995). The angular dependence of the Josephson critical currents of c-axis tilt biepitaxial GB YBCO junctions has been measured in remarkable agreement with predictions based on d-wave OP symmetry (Lombardi et al., 2002; Tafuri and Kirtley, 2005; Kirtley, 2019b). On bicrystal GB JJs it is also based the powerful demonstration of the spontaneous generation of half-flux quantum vortices in the appropriate geometry for the cuprates, observed by a scanning SQUID (superconducting quantum interference device) microscope at the tricrystal point for an optimally doped YBCO (Tsuei et al., 1994; Tsuei and Kirtley, 2000). Finally in the layered crystal

620

Josephson junctions

structure of cuprate superconductors (especially in the most anisotropic compounds such as Bi2Sr2CaCu2O8), the current-flow of Cooper pairs from the CuO2 sheets across the blocking layer is of Josephson type and, thus, a single crystal forms a natural stack of Josephson junctions (Kleiner et al., 1992; Kleiner and Wang, 2019). In conclusions, we have never had so many different families of superconducting materials and so many different types of Josephson junctions as nowadays (Tafuri, 2019), with so many fundamental open questions on their nature and on the prevalent mechanisms they exchange phase information.

More transparent barriers and the Josephson effect derived from Quasi-particle Andreev Bound States A more transparent barrier, as occurring for barriers composed of normal metals, modifies the critical current-phase IS(’) relation (CPR): X ðI sin ðn’Þ + Jn cos ðn’ÞÞ (3) IS ð’Þ ¼ n1 n The d.c. Josephson Eq. (1) represents a particular case of this general expression. The In contribution depends on the barrier transparency D as a Dn power-law. The Jn vanish if time-reversal symmetry is not broken. The expression above may include effects related to possible anisotropic OP symmetry as previously mentioned (Tsuei and Kirtley, 2000). In this case In and Jn would depend on the angles y1 and y2 of the crystallographic axes with respect to the junction interface of the left and right electrodes respectively (Tsuei and Kirtley, 2000). The CPR is the major Josephson ‘code’ and is the input to calculate most of dynamical junction properties. Deviations due to higher harmonics occur in JJs with highly transparent barriers also at the meso- and nano-scale. In atomic point contacts single Andreev processes have been counted (Chauvin et al., 2007). Quasi-particle Andreev bound states (ABSs) provide a natural framework to describe the Josephson effect in a large variety of systems traceable to SL-N-SR structures. The electron obtains an extra phase of ’L − ’R + p in each period (see Fig. 2b). The spectrum of the elementary excitations of a N layer in contact with S on both sides is quantized for E < D. The expression of the bound state pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy in the SL-N-SR one-dimensional system in the short junction limit L  xN is (Kulik, 1969): E ¼ D 1 − D sin 2 ð’=2Þ , as shown in Fig. 2g. There is a general relation between the current through the Andreev state and the phase dispersion of the energy of the Andreev dE state, IS ¼ 2e ℏ d’ (Beenakker, 1992; Beenakker, 1997; Golubov et al., 2004; Lofwander et al., 2001). The d.c. Josephson current is a resonant effect, where the Josephson current flows through resonant Andreev levels. Surfaces may hybridize and form bound states in JJs, and Andreev reflection may lead to the formation of zero energy quasi-particle bound states, as for instance in d-wave superconductors (Hu, 1994; Sigrist, 1998; Kashiwaya and Tanaka, 2000; Lofwander et al., 2001). The existence of midgap states enhances the Josephson current at low temperatures. ABSs may have special features in hybrid junctions with unconventional barriers as for instance graphene (G) and topological insulator (TI). Differently from the usual case, where the electron and hole both lie in the conduction band, at a G/S interface for instance specular AR happens if an electron in the conduction band is converted into a hole in the valence band (Beenakker, 2008).

Essential parameters of a Josephson junction The energy associated with the phase difference j across the Josephson junction depends on the critical current Ic and is: E ¼ EJ ð1 − cos’Þ

(4) −15

where EJ ¼ FoIc/(2p) indicates the Josephson energy, being Fo ¼ h/(2e)  2.07 10 Weber the flux quantum. This represents the energy stored in the junction in the superconducting state and comes from the kinetic energy of the charge carriers. The other relevant scaling energy of a JJ is associated to the charge Q which can accumulate on the capacitor C formed by the junction, giving rise to an electrostatic Coulomb energy Ec ¼ Q2/(2C). From the fundamental Josephson relations, it results that a change in the supercurrent modifies the Josephson phase, thus determining a nonzero voltage state. The junction thus acts like an effective nonlinear inductance LJ ¼

ℏ 2eIc cos’

as it can be inferred by differentiating Eq. (1) and then using Eq. (2) (Barone and Paternò, 1982; Likharev, 1986). LJ has the unusual property of taking negative values in opportune intervals of the phase j. A variable inductance is particularly relevant for quantum information manipulation (Clarke and Wilhelm, 2008; McDermott, 2009; Martinis, 2009; Oliver and Welander, 2013; Devoret and Schoelkopf, 2013; Kjaergaard et al., 2020). The maximum critical current Ic, the normal state resistance Rn and the capacitance C of the junctions define the characteristic energy scales of a JJ, the Josephson inductance LJ, the plasma frequency oJ and the whole phase dynamics as it will be described in the next section. They also determine the spatial scale of the magnetic response of the junction by setting the Josephson penetration depth lJ.

Josephson junctions

621

Properties of a Josephson junction and imprints of macroscopic quantum phenomena Current-Voltage (I–V) characteristics: From the washboard potential to macroscopic quantum phenomena In Fig. 3 we report I–V curves when changing the transparency of the junction barrier. The shape of the I–V curves are the very first fingerprints of the nature of a JJ. If we keep constant the cross section of the barrier, a change in the transparency implies a variation of the critical current density Jc. As a reference we also include an I–V curve with a characteristic downward bending (Fig. 3a) measured in microbridges, nanowires and HTS GBs with small misorientation. In a junction with higher Jc, as occurring for S-N-S JJs the bending is in the upward direction (Fig. 3b). This behavior is characteristic of low capacitance junctions and is commonly reported as overdamped regime. At lower values of Jc, I–V curves present hysteresis directly associated to the dielectric nature of the barrier and its capacitance (underdamped regime) (Fig. 3c). Switching events from the S to the R branch (see inset of Fig. 3c) as a function of the bias current are a manifestation of a stochastic process that is fully codified in switching current distributions (SCDs) (see below). Hysteresis can be incomplete with finite retrapping currents depending on dissipation (see for example Fig. 3d). The Resistively Shunted Junction (RSJ) model, first introduced by Mc-Cumber and Stewart (Stewart, 1986; McCumber, 1968) gives an account of the I–V phenomenology in a variety of weak links (Barone and Paternò, 1982; Likharev, 1986). Representing the displacement current by a capacitor C and the sum of the quasi-particle and insulator leakage current by a resistance R, the current conservation for the equivalent circuit of the junction gives the following equation: I + IF ¼ Ic sin ’ +

V + CdV=dt R

The noise source IF is associated with its shunt resistance. In the following we will confine to the case R]Rn (Devoret et al., 1985; Martinis et al., 1987; Massarotti and Tafuri, 2019a). By using Josephson relations, it can be transformed in:  2 2  2 Fo ∂ ’ Fo 1 ∂’ ∂U C 2 + + ¼0 (5) 2p 2p R ∂t ∂’ ∂t where U ¼ − Fo =ð2pÞðIc cos ’ + I’Þ

(6)

P(I)

I (c) (a)

Ic Ic

V

I

Δsw

(b)

(d)

Fig. 3 At very high Jc, as occurring in microbridges and nanowires or in GBs with small misorientation angles (a), the I–V curves present a characteristic down-ward bending. (b) I–V characteristics of overdamped junctions. (c) I–V characteristics of underdamped junctions. This behavior is characteristic of tunnel junctions, and hysteresis is directly associated to the dielectric nature of the barrier. The switch from the S to the R branch is a stochastic process with a peculiar distribution. Hysteresis can be incomplete with finite retrapping currents and the presence of leakage currents (d).

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Josephson junctions

Fig. 4 (a) Washboard potential for different values of the bias current for the standard sinusoidal Is(j) relation. In (b) and (c) two dimensional projections for two different values of the current (0.6 Ic and 1.1 Ic) are given as examples This phase representations correspond to the yellow points in the I–V curve reported in (d). The particle/phase overcomes the barrier in the washboard potential (e) by Thermal Activation (TA) or by Macroscopic Quantum Tunneling (MQT) (f ), then it rolls in the running state. The SCDs are reported as a function of temperature in (g). The standard deviation s of the distributions are plotted as a function of the temperature in (h). Tcr signals the crossover between the TA the MQT regimes and is tuned by the magnetic field. Blue curve and red curve sketch measurements in absence and in presence of an externally applied magnetic field respectively. The magnetic field reduces Ic and thus oJ, changing finally Tcr.

This equation describes the motion of a ball moving on the tilted washboard potential U (Stewart, 1986, McCumber, 1968) and allows a direct understanding of the junction non-linear dynamics. The term involving C represents the mass of the particle, the 1/R term represents the damping of the motion, ’ the position of the particle and the average “tilt” of the washboard is proportional to I. A tridimensional view of U as a function of j and I is shown in Fig. 4a. pffiffiffi When ramping the bias current I, the tilt of the energy potential increases and the height DU ¼ 2=3 EJ ð1 − I=Ic Þ3=2 of the energy barrier between consecutive wells decreases. For values of I < Ic, the particle is confined to one of the potential wells, where it oscillates back and forth at the plasma frequency oJ(I) ¼ (2pIc/(CFo))1/2(1 − (I/Ic)2)1/4 (see Fig. 4b). For I < Ic the average voltage across the junction is zero. When the current I exceeds Ic, the particle rolls down the washboard (see Fig. 4c); in this case a voltage appears across the junction (see Fig. 4d). Due to effects induced by thermal fluctuations and quantum tunneling, the junction may switch to the finite voltage state for values of I < Ic. The relative weight of these two escape processes depends on the temperature of the system (Kramers, 1940; Caldeira and Leggett, 1981; Devoret et al., 1985; Martinis et al., 1987). A wide variety of I–V characteristics can be described through an opportune choice of parameters within the RSJ model and its variants. The McCumber-Stewart parameter bc ¼ 2pIcR2C/Fo is a practical way to estimate the amount of damping of the junction (Barone and Paternò, 1982, Likharev, 1986). The strength of the friction can be also expressed through the junction damping parameter Q ¼ oJRC. Underdamped junctions have hysteretic I–V curves for bc > 1 and are hence latching. For bc < 1 overdamped junctions have non-hysteretic I–V curves and are non-latching. Apart from an obvious capacitive effect, the shape of the subgap currents is very indicative of the dissipative effects (Massarotti and Tafuri, 2019b), which are also strongly influenced by the environment, i.e., the circuitry connected to the junction. Both the nonlinear-Resistive (RSJN) model and the Tunnel-JunctionMicroscopic (TJM) model attempt to include dissipative effects for a better account of subgap leakage currents (Scott, 1970; Likharev, 1986; Massarotti and Tafuri, 2019b). The washboard potential offers a very intuitive picture even on macroscopic quantum phenomena (Leggett, 1980; Caldeira and Leggett, 1983). In the resistive state the particle is rolling down the washboard potential, it is escaping from a generic well by thermal activation following the very general Arrhenius law (Kramers, 1940) (Fig. 4e). In underdamped junctions and at very low temperatures, the dominant contribution to phase escape from the well is due to quantum tunneling (see Fig. 4f ) (Caldeira and Leggett, 1981). A study of the distribution of switching events as a function of bias current and obviously of the temperature is the appropriate tool to quantify contributions to escape due to thermal activation (TA) and macroscopic quantum tunneling (MQT),

Josephson junctions

623

respectively. The SCDs (Fig. 4g) (Fulton and Dunkleberger, 1974) and their first and second momenta (the mean I and the standard deviation s) (Fig. 4h) are neat codes of the junction phase dynamics (Devoret et al., 1985, Martinis et al., 1987). SCD is extracted by repeating the measurement several times (typically 10,000 events). All SCDs are measured at different temperatures as shown in the sketch reported in Fig. 4g. The width s of each SCD is finally reported as a function of T in Fig. 4h. Tcr ¼ ℏoJ/(2pkB) indicates the crossover temperature from the thermal to the quantum regime (Devoret et al., 1985, Martinis et al., 1987). In the quantum regime s does not depend on T (blue curve). To rule out the possibility that the saturation is due to noise or other spurious effects, the same measurements are realized in presence of an externally applied magnetic field (red curve). The magnetic field reduces Ic and thus oJ, changing finally Tcr. The tuning of Tcr demonstrates that the saturation of s is not due to extrinsic effects (Devoret et al., 1985, Martinis et al., 1987). A collection of experiments on the transition from TA to MQT is reviewed in (Massarotti and Tafuri, 2019b), including those on unconventional JJs. MQT and energy level quantization have been observed in YBCO biepitaxial GB JJs (Bauch et al., 2005; Bauch et al., 2006). The observation of MQT even for junctions with a lobe of the OP facing a node demonstrates that the quality factor Q of high critical temperature superconductor grain boundary JJs is however high enough to observe macroscopic quantum behaviors and that low energy quasi-particles in d-wave JJs are less harmful and dissipative than expected (Bauch et al., 2005, Bauch et al., 2006). In later studies (Longobardi et al., 2012), junction parameters have been finely tuned to explore phase dynamics in YBCO JJs in the moderately damped regime, where phase diffusion regime can also occur in an opportune range of temperatures. After the first escape process, the particle can be retrapped in one of the next wells and then released again, and this will be visible through distinctive features in SCDs measurements. This effect has been first clearly observed in low critical temperature superconductor JJs (Kivioja et al., 2005; Männik et al., 2005; Massarotti and Tafuri, 2019b).

Hamiltonian of a Josephson junction The passage from the classical to the quantum description of a Josephson junction or of a superconducting electrical circuit is obtained by replacing classical variables with corresponding quantum operators. The Hamiltonian function is replaced by a function of operators. Operatively, the circuit Hamiltonian will be built by adding the kinetic energy associated with the charging energy of the capacitive elements K ¼ CV2/2 and the potential energy associated with the Josephson inductance UJ ¼ − EJ cos ’ and with the inductance of the superconducting leads UL ¼ F2/(2L), where F is the magnetic flux (Devoret and Martinis, 2004; Wendin and Shumeiko, 2007; Wendin, 2017; Kockum and Nori, 2019; Kjaergaard et al., 2020). All these quantities have to be expressed in terms of j for a given circuit element, which are connected to V and F by the Josephson Eq. (2) and by the relation F ¼ Foj respectively. The commutation relation commonly reported in literature is [F, q] ¼ 2iℏ where F and q are conjugate operators (Likharev, 1986), or more simply [’, n] ¼ i, where n the induced charge on the capacitor measured in units of 2e (Cooper pairs). The charge n and phase ’ operators do not commute, which means that their expectation values cannot be measured simultaneously. P The circuit Hamiltonian is finally constructed by summing up the energies of all circuit elements: H ¼ K(nj) + U(’j) (Wendin and Shumeiko, 2007; Kjaergaard et al., 2020). If several circuit elements are connected in a closed loop, the flux quantization P equation imposes a constraint on the phases of these elements: ’i + ’e ¼ 2pn, where ’e ¼ 2e/ℏ Fe and Fe is the phase associated with the applied magnetic flux. The Hamiltonian of a current-biased Josephson junction will be therefore expressed as: H ¼ Eco n2 − EJ cos’ − ℏI’=ð2eÞ

(7)

where Eco ¼ (2e) /(2C) is the charging energy and I is the applied current. Quantum Josephson junctions with either a well-defined charge or phase variable will depend on the relative magnitude of Ec ¼ Econ2 and EJ (phase for EJ  Ec, charge for EJ  Ec respectively) (Clarke and Wilhelm, 2008). The Hamiltonian of the Josephson junction can be part of a more general Hamiltonian describing a generic circuit (Devoret and Martinis, 2004; Wendin and Shumeiko, 2007; Wendin, 2017; Kockum and Nori, 2019; Krantz et al., 2019; Kjaergaard et al., 2020). The Cooper pair box, which consists of a superconducting island connected to a superconducting reservoir through a small junction, has been the first superconducting qubit whose quantum state was manipulated coherently (Nakamura et al., 1999). Its Hamiltonian is:  2 (8) H ¼ Eco n − ng − EJ cos ’ 2

where ng ¼ − CgVg/(2e) plays the role of external controlling parameter, and Cg is the gate capacitance. Similar expressions describe other quantum circuits, from phase to flux qubits, from the transmon and the quantronium to the fluxonium (Devoret and Martinis, 2004; Koch et al., 2007; Martinis, 2009; Wendin, 2017). Solving the Hamiltonian for each specific circuit will allow to find the energy-level structure of the system.

Temperature and magnetic dependence of the critical current and of I–V curves Temperature dependence of Ic The temperature dependence of I–V curves is a standard key reference to understand the nature of the junction. Accurate predictions exist to evaluate deviations of the IcRn vs T dependence from the tunnel limit represented by the Ambegaokar-Baratoff (AB) regime valid for the SIS configuration (Ambegaokar and Baratoff, 1963). In point contacts in the dirty (KO1), where the mean free path l is

624

Josephson junctions

(b)

(a)

e IcRn/(2πTc)

0.3 L/ξN(Tc) 0 1

2e IcRn/(πΔ)

2 3

Increasing H

0

T/Tc 0 a

10 0

0

T/Tc

'

e IcRn/Δ

6

1

0-state

4

(c)

1 b c 0

0

T/Tc

1

Fig. 5 (a) IcRn (T) is reported in units normalized to the gap value D as a function of the temperature T, for different values of the ratio between the barrier length L and xn. (b) In a S-F-S junction IcRn is reported in units normalized to the critical temperature Tc as a function of T/ Tc, for different values of the exchange field H. Above a critical field an additional minimum in IcRn signals the 0-p transition. (c) IcRn (T) is calculated for different misorientation angles in HTS d-wave GB JJs: curve a d0/d0; curve b d0/dp/8 and curve c d0/dp/4. (a) Adapted from Ref. Likharev KK (1979) Superconducting weak links. Reviews of Modern Physics 51: 101; (b) Adapted from Ref. Golubov AA, Kupriyanov MY and Il’ichev E (2004) The current-phase relation in Josephson junctions. Reviews of Modern Physics 76: 411; (c) Adapted from Ref. Tanaka Y and Kashiwaya S (1997) Theory of Josephson effects in anisotropic superconductors. Physical Review B 56: 892

shorter than the length of the weak link L, (l < L) (Kulik and Omelyanchuk, 1975) and clean (KO2), where (l > L) (Kulik and Omelyanchuk, 1977) limits, the values of Ic at T ¼ 0 K are higher than AB value. If a metallic barrier N replaces the insulator I, the IcRn (T) is quite different and even changes concavity (Likharev, 1986). In Fig. 5a a variety of qualitatively different behaviors, including the transition from the short (L < > xn) regime, is reported (Likharev, 1979; Delin and Kleinsasser, 1996; Golubov et al., 2004). The tail in the exponential growth and the width of the intermediate region substantially depends on L/xn (Barone and Paternò, 1982; Likharev, 1979; Golubov et al., 2004). Finer quantitative deviations depend on details of the proximity effect and on boundary conditions. Radically different IcRn(T) curves are measured, when a 0-p transition is induced by changing the temperature with a characteristic non-monotonous dependence of the IcRn(T). This is visible in Fig. 5b when the exchange field H is increased above a certain threshold in a S-F-S junction (Ryazanov et al., 2001; Golubov et al., 2004; Buzdin, 2005). A peculiar 1/T scaling of the IcRn has been predicted for ideal interfaces in HTS d-wave GB JJs (Barash et al., 1996; Tanaka and Kashiwaya, 1997; Riedel and Bagwell, 1998; Kashiwaya and Tanaka, 2000; Lofwander et al., 2001). This is due to the change in the dispersion relation for the ABSs, and the corresponding angle integrated current-phase relation, which is particularly relevant for high misorientation angles in GB JJs. In the resonant case, the splitting of the levels, and consequently the width of the Andreev band, will be particularly large, proportional to D1/2. The (maximum) IcRn is proportional to D1/2, and is much larger than the AB value in conventional s-wave tunnel junctions, which is proportional to D (Likharev, 1979; Tanaka and Kashiwaya, 1997). This is shown in Fig. 5c, where IcRn(T) is calculated for different misorientation angles in HTS d-wave GB JJs. Using the notation dy1/dy2, by increasing the misorientation angle from d0/ d0 (curve a) to d0/dp/8 (curve b) and finally d0/dp/4 (curve c), dramatic changes occur in the shape of the IcRn(T). In (c) IcRn (T) does not saturate (Golubov et al., 2004).

Magnetic dependence of the critical current The phase variation induced by an externally applied magnetic field has been since early times an invaluable and unambiguous tool to demonstrate the Josephson effect and a key fingerprint for plenty of sensor applications. This is widely documented in all textbooks and especially in (Barone and Paternò, 1982). Geometry of the junction, nature of the electrodes or of the barrier and their possible inhomogeneities determine specific space-dependent phase variations across the barrier, inducing Meissner screening magnetic interference effects, vortex formation and trapping, shielding and spontaneous supercurrents. All these effects cooperate to induce a distinctive spatial distribution of the critical current density across the junction barrier, and thus special features in the magnetic dependence of the Ic and of the I–V curves. The final expression of the dependence of the critical current on the external magnetic flux for an ideal rectangular geometry is:   sin p F Fo (9) Is ðFÞ ¼ Js WL F p Fo In Eq. (10) the product W∙L represents the junction magnetic area. The curve is reported in Fig. 6a.

Josephson junctions

625

1

1

0 -6

Φ/Φο

-4

-2

0

0

-6

6

4

2

(b)

Ic/JcoA

Ic/JcoA

(a)

-4

-2

1 (c)

W= 0.5 ,1,2,3,5,200

W/λJ=4

1

4

2

6 (d)

Ic/JcoA

Ic/JcoA

W/λJ=10

0 Φ/Φο

W/λJ=2 W/λJ=1

0

1

0

Φ/Φο

2

0

3

0

1

1

3

Φ/Φο

(e) 3

Ic/JcoA 0

2

1

5

Φ/Φο

(

4

-2

6

-1

0

1

2

Fig. 6 (a) Magnetic field dependence of the maximum Josephson current for ‘ideal’ JJ; (b) Dependence of the IS on the magnetic field for a d-wave corner junction composed of Pb and YBCO electrodes, two symmetric maxima appear at finite magnetic fields. In (c) self-field effects induced by non-uniform distribution of the critical current occurring in long junctions, are visible in the magnetic pattern for ordinary junctions. The reduction and then the fading of the secondary lobes accompanied by a shift in the position of the minima of IS also occurs in diffusive SNS junctions (d). In spin filter SFS JJs the IS keeps memory of the history of how the magnetic field has been applied (e). The sequence of how the magnetic field is applied in the experiments is: red curve, black curve, blue curve. (b) Adapted from Wollman DA, Van Harlingen DJ, Giapintzakis J, Ginsberg DM (1995) Evidence for d-wave pairing from the magnetic field modulation of in YBa2Cu3O7-d -Pb Josephson junctions. Physical Review Letters 74, 797–800; (c) adapted from Kirtley JR, Moler KA and Scalapino DJ (1997) Spontaneous flux and magnetic-interference patterns in 0-p Josephson junctions. Physical Review B 56: 886; (d) Adapted from Bergeret FS and Cuevas JC (2008) The vortex state and Josephson critical current of diffusive SNS junction. Journal of Low Temperature Physics 153: 304–324; (e) Adapted from Massarotti D, Pal A, Rotoli G, Longobardi L, Blamire MG and Tafuri F (2015) Macroscopic quantum tunnelling in spin filter ferromagnetic Josephson junctions. Nature Communications 6: 7376.

It can be derived the one-dimensional time-independent sine-Gordon equation combining Maxwell’s equation and the d.c. Josephson relation: ∂2 ’ sin ’ ¼ 2 ∂x2 lJ

(10)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where lJ ¼ Fo =ð2pdmo Jc Þ is the Josephson penetration depth and gives a measure of the distance from the edge where d.c. Josephson currents are confined, being d ¼ t + lR + lL, with t the barrier thickness, lR(L) the London penetration depth in the R(L) electrode and mo the magnetic permeability of vacuum respectively. We refer to textbooks and extensive reviews for a detailed account of theory and all deviations (Barone and Paternò, 1982; Kirtley, 2019a; Kirtley, 2019b). The magnetic patterns shown in Fig. 6 give finally a brief overview on some of the most significant deviations from the standard patterns. In all figures on y-axis Ic is always reported. On the x-axis depending on the origin of the curves (simulations or experiments) magnetic flux or magnetic field is reported, respectively. In a d-wave corner junction composed of Pb and YBCO electrodes, two symmetric maxima appear at finite magnetic fields (Fig. 6b) (Wollman et al., 1995). If the d-wave junction has more than the two facets of the corner junction, relative maxima in IS appear between the two absolute maxima (for an applied magnetics flux Fmax ¼ NFo/2) (Smilde et al., 2002). The vanishing IS at F ¼ 0 occurs for an even number of facets N. In Figs. 6c self-field effects induced by non-uniform distribution of IS occurring in long junctions, are briefly sketched for ordinary junctions (Kirtley et al., 1997; Barone and Paternò, 1982). When increasing the width W of the junction with respect to the Josephson penetration depth lJ, lobes at higher fields tend to be smeared out and the characteristic interferometric behavior is even lost when W is larger than 10 lJ. The reduction and then the fading of the secondary lobes accompanied by a shift in the position of the minima of IS also occurs pffiffiffiffiffiffiffiffi in diffusive SNS junctions (see 6d) (Bergeret and Cuevas, 2008). For W comparable or smaller than xH ¼ Fo=H the formation of a linear array of vortices is not favored and the field acts as a strong pair-breaking mechanism monotonically suppressing IS. The monotonic suppression of Ic is also obtained while increasing L, even for fixed W to about xH (Bergeret and Cuevas, 2008). The

626

Josephson junctions

presence of higher harmonics in the current-phase relation also affects the magnetic response (Goldobin et al., 2007; Kirtley, 2019a; Kirtley, 2019b). A distinctive feature is the appearance of oscillations of IS with twice shorter period in H when increasing the intensity of the second harmonic. SFS JJs are expected to generate several anomalous magnetic behaviors due to the magnetic nature of the barrier itself (Ryazanov et al., 2001; Buzdin, 2005). IS keeps memory of the history of how the magnetic field has been applied and the magnetic pattern is hysteretic (see Fig. 6e) (Senapati et al., 2011; Massarotti et al., 2015). Hybrid junctions using almost bi-dimensional flakes are posing new questions also on the magnetic dependence of IS. In addition to extrinsic features as those for instance induced by the way contacts are placed in a typical coplanar junction configuration, the intrinsic nature of the barrier may add special fingerprints. A recent example in a Ti/Al-InAs/GaSb-Ti/Al junction is given by the influence of edge states (Pribiag et al., 2015). Here gate-tuning between edge-dominated and bulk-dominated regimes of superconducting transport is demonstrated through superconducting quantum interference. The edge-dominated regime arises only under conditions of high-bulk resistivity, which has been associated with the two-dimensional topological phase. For wide barriers, supercurrents will flow along distant edges resembling the typical response of a SQUID (Pribiag et al., 2015).

Emerging trends: From qubits to hybrid JJs The new frontiers of the Josephson effect are currently developing along a few exciting and challenging directions.

Low temperature Josephson junctions for qubits, quantum circuits and superconducting electronics JJs are the fundamental cells for quantum bits and quantum circuits with a variety of functionalities. Superconducting qubits encode information in the energy eigenstates of Josephson junction-based electronic circuits (Clarke and Wilhelm, 2008; McDermott, 2009; Martinis, 2009; Oliver and Welander, 2013; Devoret and Schoelkopf, 2013; Krantz et al., 2019; de Leon et al., 2021). The first cornerstone was the experiment in 1999 where time-resolved coherent oscillations in a superconducting qubit were observed (Nakamura et al., 1999). Further key steps were the observation of coherent oscillations in coupled superconducting qubits (Pashkin et al., 2003; Yamamoto et al., 2003) and the significant improvements of the coherence times of these devices (Clarke and Wilhelm, 2008; Vion et al., 2002; Devoret and Schoelkopf, 2013, Krantz et al., 2019; Kjaergaard et al., 2020). There are currently several types of superconducting qubits with different circuit construction. The highly anharmonic Josephson potential allows to selectively excite the states used as basis of the qubit, and to avoid contributions from other energy levels. Thermal energy must be sufficiently low to avoid incoherent mixing of eigenstates (kBT < G4 then reduces to Pph(Q) > Pph(0). For fermions with one circular FS, Pph(q) either stays constant or decreases with q, then this condition is not satisfied. However, in our case, there are two FS’s, separated by Q, one of hole type and the other of electron type. One can easily verify that, in this situation, Pph(Q) is logarithmically enhanced comparable to Pph(0), hence irreducible G3 is necessary larger than irreducible G4. This is lattice analog of KL scenario. full What kind of the pairing state we get? Both Gfull hh and Ghe do not depend on the direction along each of the two pockets, hence the pairing state is necessary s-wave. full At the same time, near the pole Gfull hh ¼ − Ghe , which implies that the two-fermion pair wave function changes sign between hole and electron pockets. Such an s-wave state is often call s+−. The majority of researchers believe that s+− is the correct gap symmetry for FeSCs. It is consistent with STM and neutron scattering data. Nevertheless, a conventional s-wave gap due to phonons and a d-wave gap symmetry for strongly hole doped 111 materials have also been proposed.

Cuprates Cuprates are layered materials with one or more crystal planes consisting of Cu and O atoms (two O per Cu), and charge reservoirs between them. Superconductivity is widely believed to originate from electron-electron interactions in these CuO2 planes. The undoped parent compounds are Mott insulators/Heisenberg antiferromagnets due to very strong Coulomb repulsion which prevents electron hopping from Cu to Cu and therefore localizes electrons near lattice sites (Lee et al., 2006; Keimer et al., 2015). Doping these insulating CuO2 layers with carriers (by adding/removing electrons from/to charge reservoir) leads to a (bad) metallic behavior and to the appearance of high-temperature superconductivity. A schematic phase diagram of doped cuprates is shown in Fig. 3.

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Superconductivity: Electronic mechanisms

Fig. 3 The phase diagram for the cuprates. In similarity with FeSCs, parent compounds are antiferromagnets. Among differences, the most important one is that antiferromagnetism is of Heisenberg type, and is a part of Mott behavior In the region below T∗, the system displays pseudogap behavior. d-wave symmetry of superconducting state holds for both hole and electron dopings. From Armitage N.P., Fournier P., Greene R.L. (2010) Reviews of Modern Physics 82: 2421.

There are several features in the phase diagram, like the pseudogap in hole-doped cuprates, which are still not fully understood, although a substantial progress has been made over the last few years (Basov and Chubukov, 2011; Chubukov and Hirschfeld, 2015; Tremblay and Chubukov, 2017; Armitage et al., 2010; Lee et al., 2006; Keimer et al., 2015). By all accounts, the symmetry of the superconducting state does not change between small doping, where pseudogap physics is relevant, and doping above the optimal one. For these larger dopings, ARPES and quantum oscillation experiments show a large FS, consistent with Luttinger count for fermionic states. In this doping range, it is natural to expect that the pairing symmetry can be at least qualitatively understood by performing the same analysis as for Fe-based systems. The FS for hole-doped cuprates is an is open electron FS shown in Fig. 2 (center). The fermionic density of states is the largest near the points (0, p) and ( p,0), where two FS lines come close to each other (the density of states is logarithmically enhanced and actually diverges when the two FS lines merge at (0, p) and ( p,0)—this is termed as Van Hove singularity). The FS regions with the largest DOS mostly contribute to superconductivity, and, to first approximation, one can consider the FS in Fig. 2 (center) as consisting of four patches. Experiments clearly indicate that bound fermionic pairs are spin singlets. A spin-singlet pair wave function is an even function of momentum. This leaves two non-equivalent patches, which for definiteness we choose to be near (0,p) and (p,0). The resulting two-patch model is in many respects similar to the two-pocket model for Fe-pnictides, only instead of hole-hole, electron-electron, and hole-electron interaction we now have intra-patch and inter-patch interactions for two patches, which we label as 1 and 2. Introducing as before inter-patch repulsion G4 and inter-patch pair-hopping interaction G3, we obtain, in full analogy with (3),   1 G4 + G3 G4 − G3 full + G11 ¼ − 2 1 + ðG4 + G3 ÞPpp 1 + ðG4 − G3 ÞPpp   (5) 1 G4 + G3 G4 − G3 full G12 ¼ − , − 2 1 + ðG4 + G3 ÞPpp 1 + ðG4 − G3 ÞPpp and   G3 − G4 ¼ 2U 2 Pph ðQÞ − Pph ð0Þ

(6)

The two particle-hole polarization bubbles can be straightforwardly calculated for the lattice model of fermionic dispersion with hopping t (t0) between nearest (next-nearest) neighbors The result is that Pph(Q) > Pph(0), hence irreducible G3 again exceeds full irreducible G4, and superconductivity develops. Near the pole, Gfull 11  − G12 . Now, in the two-pocket model for FeSCs, the sign changing pair wave-function changes sign between different FS pockets, but preserves the same sign along a given pocket. In two-patch model for the cuprates, the sign-changing wave function changes sign between the two ends of the same “arc” of the FS, i.e., it changes sign under p/2 rotation from x to y axis and necessary passes through zero in between. The prototype wave function for such a state is coskx − cosky. According to a standard classification scheme, such a wave function belongs to B1g representation and is often called d-wave because it changes sign 4 times along the full FS. The d-wave symmetry of superconducting order parameter (often called gap function) in the cuprates has been firmly established by a number of experiments. In 2011, Juan Carlos Campuzano from Argonne National Laboratory, P. Johnson from Brookhaven National Laboratory, and Z.X. Shen from Stanford University were awarded the prestigious Oliver E. Buckley Condensed Matter Prize by the American Physical Society for innovative ARPES experiments, which established d-wave gap symmetry.

Superconductivity: Electronic mechanisms

637

Doped graphene The same logics can be applied to a system of fermions of a hexagonal lattice, at a so-called van Hove doping, when the system is about to undergo a topological transition between a singly connected large FS and a set of disconnected small FS’s (Gonzalez, 2008; Nandkishore et al., 2012). Such a situation holds in a single-layer graphene at a doping when six separate Fermi pockets that emerge out of Dirac points, merge (Fig. 4). A van Hove doping of a single layer graphene has been achieved by placing Ca and K dopants above and below a graphene layer. Near van Hove instability, low-energy physics is again described by a patch model, but now there are three non-equivalent patches, labeled 1,2 and 3 in Fig. 2 (right). We introduce intra-patch and inter-patch vertices Gij, i, j 2 (1,2,3) with Gij ¼ Gji. Because the three patches are fully symmetric, the number of independent vertices is just two: G11 ¼ G22 ¼ G33 , G12 ¼ G13 ¼ G23

(7)

and there are two irreducible pairing interactions, which like before we call G3 and G4. Summing up ladder diagrams in the particle-particle channel, one obtains the result similar, but not identical to (5):   1 G4 + 2G3 G4 − G3 full +2 G11 ¼ − 3 1 + ðG4 + 2G3 ÞPpp 1 + ðG4 − G3 ÞPpp   (8) 1 G + 2G G4 − G3 full 4 3 G12 ¼ − : − 3 1 + ðG4 + 2G3 ÞPpp 1 + ðG4 − G3 ÞPpp For Hubbard interaction the bare G03 ¼ G04 ¼ U are again equal, i.e., at this level there is no superconductivity, but irreducible interactions are     G4 ¼ U 1 + 3UPph ð0Þ ,G3 ¼ U 1 + 2UPph ðQÞ (9) where now Q is the momenta between van Hove points. Superconductivity develops when Pph(Q) > (3/2)Pph(0). Computations of the polarization bubble shows that Pph(Q) is again logarithmically singular and well exceeds Pph(0). Then superconductivity does develop. and intrapatch Gfull is What is the symmetry of a pair wave function? Near the instability the ratio of inter-patch Gfull ij ii full ∗ D −1=2: ¼ − 2 for i 6¼ j. If we view Gii as the modulus square of the superconducting orderparameter |△i |2 and Gfull ij as Re[Di Dj ], 2p 1 we find that the phase of complex △ must change by  2p 3 between each pair of patches cos 3 ¼ − 2 . In other words, if the order parameter in patch 1 is △1, then D2 ¼ D1e2pi/3 and D3 ¼ D1e4pi/3. The two resulting structures of △ are shown in Fig. 5. Each of the two candidate order parameters has d-wave symmetry, because if we extend it to the whole FS, we find that it changes sign 4 times.

Chiral superconductivity The two order parameters in Fig. 5 represent counter clockwise and clockwise phase winding by 4p along the full FS. These order parameters are completely equivalent, i.e., there is an additional Z2 symmetry of a superconducting wave function, which is different from a conventional U(1) phase symmetry. The analysis of the Landau functional in the superconducting state shows that Z2 symmetry gets broken, i.e., the system spontaneously chooses counter clockwise or clockwise phase winding. Such a state is called d + id or d − id. It breaks time reversal and inversion symmetries and is called a chiral superconductor. The nontrivial topology of the d  id state manifests itself in exceptionally rich phenomenology, including a quantized spin and thermal Hall conductance, and a quantized boundary current in magnetic field (Gonzalez, 2008; Nandkishore et al., 2012). Theoretical analysis of such

Fig. 4 Schematic phase diagram of doped graphene. Tc is the instability temperature toward spin singlet d + id or spin triplet f-wave SC states, or SDW state. Tc is plotted against doping (n). Doped Graphene is expected to be mostly superconducting with competition with the SDW phase near the Van-Hove region. Modified from Kiesel M.L., et al. (2012) Physical Review B 86: 020507(R).

638

Superconductivity: Electronic mechanisms (A)

(B)

1

Δe 2’

Δe

Δ

3

3’

2’

2

3

1’

Δ

Δe 3’

2

Δ

Δ Δe

1

Δ

Δ

Δe

1’

Δe

Fig. 5 The phases of the pair-wave functions at the patches (regions with enhanced density of states). The left and right represent the two Z2 breaking d-wave solutions d + id or d − id. From Maiti S., Chubukov A.V. (2014) In: Bennemann, Ketterson (eds.) “Novel Superfluids”, Chapter 15. Oxford Press.

superconductors and a search for their practical realizations are integral parts of modern studies of superconductivity. The most frequently discussed example is spin-triplet p + ip superconductivity, which has multiple applications for quantum computation (Sato and Ando, 2017). In practical applications, p + ip gap symmetry has been discussed in the context of superfluidity in 3He. It was proposed for Sr2RuO4, but recent experiments cast doubt on this explanation. Understanding of the pairing symmetry in Sr2RuO4, which by all accounts, falls into a weak coupling category, is another actively explored area of modern research on superconductivity.

Twisted bilayer graphene

Superconductivity with highest Tc  3 K and strongly correlated behavior around it have been recently discovered in a twisted bilayer graphene (TBG), consisting of two graphene sheets rotated by a “magic” angle (Cao et al., 2018; Stepanov et al., 2020) (Fig. 6). STM data for TBG show sharp van Hove peaks, and several researchers argued that superconductivity in TBG emerges by the same reason as in a single graphene sheet, and has a chiral d-wave symmetry. Others, however, argued that superconductivity may come from electron-phonon interaction and is likely s-wave. Another proposal is that superconductivity is ultimately a fundamentally novel phenomenon, related to insulating states nearby (Khalaf et al., 2021). Another potential novel feature of superconductivity in TBG is suggested breaking of lattice-rotational symmetry below Tc, in which case the ordered state becomes a nematic superconductor. Theoretical understanding of nematic superconductivity here and in other systems is another rapidly developing field of research, along with the studies of partially melted superconducting states, in which fluctuating Cooper pairs do not establish phase coherence, but already break Z2 time-reversal symmetry or symmetry associated with a rotational invariance.

s-wave superconductivity from repulsion Another issue, first discussed in early studies of superconductivity, but revisited recently in the context of superconductivity in strontium titanate and related systems, is whether one can potentially get an ordinary s-wave superconductivity with momentum-independent order parameter from repulsion. This question has been raised for electron-phonon superconductivity

Fig. 6 The phase diagram of TBG. Reproduced from Cao Y., et al. (2018) Nature 556: 43; Stepanov P., et al. (2020) Nature 583: 375, with permission from Springer Nature.

Superconductivity: Electronic mechanisms

639

as Coulomb repulsion is present in systems, which are believed to be s-wave electron-phonon superconductors, and is larger than electron-phonon attraction. A frequently cited explanation is that in cases where the Fermi energy EF is much larger than the Debye frequency oD, there is a wide range of frequencies where the repulsive Coulomb repulsion is logarithmically renormalized down dy ladder series in the same particle-particle channel that would give rise to pairing in case of attraction (the Tyablikov-McMillan logarithm). If this range is wide enough, the renormalized Coulomb repulsion becomes smaller than the electron-phonon attraction at energies below oD. Upon closer examination, this explanation appears incomplete as Tyablikov-McMillan renormalization holds for the full interaction, i.e., for the sum of electronelectron and electron-phonon interactions. Under the renormalization this full interaction decreases, but does not change sign, i.e., remains repulsive. How one can get superconductivity in this situation? It has been originally discussed quite some time ago for large EF/oD and has been re-analyzed recently in the context of superconductivity in systems with small Fermi energy, like SrTiO3, Pb1−xTlxTe, half-Heusler compounds, where EF and oD are comparable and Tyablikov-McMillan reasoning does not work. It has been realized that the actual reason why electron-phonon superconductivity holds despite larger Coulomb interaction, is that the full interaction on the Matsubara axis (where it is real) is a dynamical one, V (Om), and although a phonon-mediated attraction does not invert the sign of V (Om), it nevertheless reduces it at frequencies below the Debye energy. A convenient way to model V (Om) is to treat it as a sum of two parts: a constant Hubbard repulsion U, and a frequency-dependent attractive part due to electron-phonon interaction: V ðOm Þ∝U −

U el −ph

1 + ðOm =O1 Þ2

,

(10)

where O1 is of order of the Debye energy and Om is a bosonic Matsubara frequency. A similar model has been recently used for dynamically screened electron-electron interaction in single-crystal Bi, where O1 is of the order of plasma frequency. For U > Uel−ph, V (Om) > 0 for all frequencies. Yet, because the interaction is dynamical, one can, to first approximation, replace it by a step-like function, with different values at small at large frequencies (regions 1 and 2) and solve a set of coupled equations for G11, G22, and G12. The set is qualitatively the same as in (3), (5), (8), and the analogs of G3 and G4 are approximately U and U − Uel −ph Without electron-phonon attraction, G3 ¼ G4 ¼ U. Once electron-phonon attraction is added, G3 becomes larger than G4, and superconductivity develops. Like in previous cases, this order parameter necessary changes sign, but here it does this as a function of frequency, i.e., the signs of the gap function at small and large om are different. The McMillan formula Tc  O1e−1/(NFU−l∗), where l∗1/ log EF/O1 is reproduced for EF  O1 once one integrate out fermions with high-frequencies and obtain an effective interaction between low-energy fermions. This interaction is attractive when NFU > l∗, but it incorporates the fact that the sign of the order parameter at large frequencies is opposite to that at small frequencies.

Superconductivity at strong coupling The weak coupling approach, based on second-order renormalization of the Hubbard interaction, is highly useful for the understanding of the symmetry of the superconducting order parameter for a particular material, but at the same time is incomplete as it is based on the assumptions that the bare interaction in the relevant pairing channel is zero, and that fermions are weakly interacting quasiparticles. These assumptions work for the case of a weak Hubbard interaction, but not beyond it. Indeed, a generic screened Coulomb interaction is larger at small momentum transfer, hence, in the notations used above, a bare G4 is larger than a bare G3. To get an attraction, the “2kF” screening must overcome this difference, and to verify this one necessary has to go beyond weak coupling. Modern studies of superconductivity beyond weak coupling can be broadly divided into three classes: (i) The analysis of “2kF” screening beyond second order in perturbation, but still at weak coupling. This is most often done using some version of renormalization group (RG) analysis; (ii) The analysis of the pairing in a metal, mediated by a soft collective boson in either spin or charge channel. The most studied examples here are pairing by spin fluctuations and by Ising-nematic fluctuations. This is valid for an intermediate coupling strength, when on one hand interaction is strong enough to bring the system close to an instability in spin or charge channel, and on the other the carriers retain their itinerant character; (iii) The analysis of the pairing in the truly strong coupling regime, when interaction is strong enough to localize most of the carriers. The most studied example here is pairing of a doped Mott insulator, extensively discussed for the cuprates, and pairing of a doped Chern insulator, recently suggested in the studies of TBG.

Renormalization group analysis The RG approach is an attempt to go beyond the lowest order in perturbation already for weak coupling, when dimensionless g3 ¼ NFG3 and g4 ¼ NFG4 are small. A good starting point here is the BCS approach, in which one assumes that the bare dimensionless pairing interaction g0 is small and neglects all regular corrections in powers of g0, but sums up series of corrections in g0L, where L ¼ logW/T is a Cooper logarithm. Within BCS approximation, the series in g0L are geometrical, and the dressed g ¼ g0/(1 − g0L) (The dressed vertex G ¼ G0(g/g0) ∝/(1 − gL)). The flow of the dressed g with running L (i.e., with changing

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Superconductivity: Electronic mechanisms

temperature) can be cast into the differential equation dotg ¼ g2, where g% ¼ dg/dL and BCS result about the pairing instability at weak coupling can be reformulated as the divergence of the running g as the temperature approaches Tc (or, equivalently, as the divergence of the running vertex G). This reasoning has been extended in the last two decade to extend the analysis of the irreducible vertex beyond second order in perturbation. In all three cases discussed in Sec. III the particle-hole polarization Pph(Q) is logarithmically enhanced by one reason or the other. The idea is then to treat logarithmic terms in the particle-hole channel on equal footing with logarithmic renormalizations in the particle-particle channel and detect the flow of irreducible g3 and g4 simultaneously with their renor-malizations in the particle-particle channel. As long as gi are small and giL  1, this can be done rigorously. The corresponding RG procedure is called it parquet RG as logarithmic renormalizations are extended simultaneously into two particle-hole and particle-particle channels, which can be viewed as the two “perpendicular” directions. A similar but somewhat more involved functional RG has also be applied for this purpose. There are two goals of parquet/functional RG analysis. First to verify whether in the flow the running g3 becomes larger than g4, even if bare g4 was larger. If so, then the system self-generates attraction below some upper cutoff, determined by the system. Second, to understand the interplay between superconductivity and particle-hole orders, which can also develop under RG flow. As an illustration, consider the 2-pocket model for FeSCs. For the RG analysis, we need to add two other interactions between low-energy fermions: interaction between fermionic densities in the two pockets, g1, and intra-pocket density-density interaction g2 (cacafbfb and cafafbcb terms, respectively). At second order, these two interaction renormalize the irreducible g3 and g4. Beyond second order, they also flow with L and can give rise to SDW, charge-density-wave (CDW), nematic and other orders. The full set of parquet RG equations is g_ 1 g_ 2 g_ 3 g_ 4

¼ ¼

g21 + g23 2g2 ðg1 − g2 Þ

¼ ¼

2g3 ð2g1 − g2 − g4 Þ −g23 −g24 

The solution is shown in Fig. 7. We see that g3 does become larger than g4 above some L, even if its bare value was smaller. The competition between superconductivity and other orders is determined by the behavior of different gi near the fixed trajectory, where all gi ∝ 1/(Lcr − L) and Lcr ¼ logW/Tcr, where Tcr is the highest instability temperature.

Pairing by spin and charge fluctuations When gi are not small, neglecting regular corrections in gi is not an option. Because no controlled calculations are possible in this case (except for specific extensions to, e.g., large number of fermionic flavors) one frequently discussed option is to adopt a semi-phenomenological approach based on the experimental phase diagram. Many materials, in which superconductivity is believed to be of electronic origin, the phase diagram shows some other ordered phases close to where superconductivity exists: undoped cuprates display antiferromagnetic order, parent compounds of FeSCs show either SDW or nematic order, in doped graphene there are strong SDW and CDW correlations, etc. In this situation, it is tempting to consider the propagator of a soft boson in either spin or charge channel as a pairing glue for superconductivity, i.e., consider pairing mediated by spin or charge fluctuations.

gi

5

0 g1 g2 g

3

g4 −5

0

0.2

0.4

0.6 0.8 RG scale L

1

1.2

Fig. 7 The flow of dimensionless couplings g1,2,3,4. g3 grows and eventually crosses g4, which becomes negative at a large enough RG scale. From Maiti S., Chubukov A.V. (2014) In: Bennemann, Ketterson (eds.) “Novel Superfluids”, Chapter 15. Oxford Press.

Superconductivity: Electronic mechanisms

641

From theory viewpoint, in this approach one extends the computation of the irreducible vertex functions in the particle-particle channel by restricting to a particular set of diagrams beyond second order. The vertex function can be generally split into spin and charge components: Gab;gd ¼ Gc dagdbd + Gs sagsbd, here si are Pauli matrices. A re-examination of second order results in Sec. III shows that for a repulsive interaction, the “2kF” screening enhances the spin component of the interaction at momentum Q. Collecting ladder series of the same renormalizations, one further enhances Gs. If the enhancement is strong enough, one can keep only this term and neglect the interaction at small momentum transfer and the charge component of the interaction near Q. Then one arrives at the effective 4-fermion spin-spin interaction with momentum transfer near Q. For the 1-band cuprate case X Gs ðqÞc{k,a sag ck+ q,g c{p ,b sbd cp −q ,d (11) Hef f ¼ k,q,p

By construction, this interaction is attractive in the corresponding spatial channel (d-wave for the cuprates, s+− and, in special cases, d-wave for FeSCs, d  id for doped graphene. We illustrate this in Fig. 8. The attractive component has the same spatial structure as in 2nd order calculations, so in this respect the difference between the second order analysis and spin fluctuation approach is only in the magnitude of the attractive interaction. It is customary to use the Ornstein-Zernike form Gs ¼ G/(x−2 + (q − Q)2), where x is a magnetic correlation length. For large x, Gs is large for q near Q. The same strategy has been used to obtain the interaction mediated by SDW or nematic fluctuations.

Superconductivity near a quantum-critical point

At a first glance, the attraction mediated by a soft boson can be made as large as possible by focusing on q  Q and making x larger, i.e., bringing the system closer to a magnetic instability. This is not completely true, however, as in most cases of interest a collective boson can decay into a particle-hole pair, that is the bosonic propagator contains the Landau damping term | Om |/vFq. This leads to new physics, which attracted a substantial interest in recent years (Abanov and Chubukov, 2020; Chubukov et al., 2020; Esterlis and Schmalian, 2019; Wang, 2020). Namely, when projected into the particle-hole channel, boson-mediated interaction gives rise to dynamical self-energy, S(k,om). At x ¼ 1 and in dimension D  3, this self-energy at k ¼ F acquires a non-Fermi liquid form |om |asignom with a < 1 at small frequencies. In other words, the same interaction that gives rise to the pairing, also gives rise to incoherent non-Fermi liquid (NFL) behavior. Whether or not superconductivity occurs in this situation depends on the outcome of the interplay between strong dynamical pairing interaction and strong fermionic incoherence, which acts again pairing. From

Fig. 8 Two routes to superconductivity (a)—Two electrons attract each other when the 1st polarizes the lattice, and the second is attracted to this region. The pair wave function C(r) of the relative electronic coordinate r, has the full symmetry of the crystal and gives rise to a gap function △k of the same sign for all momenta k on the FeSC Fermi surface (green ¼ +). (b) Electrons interact with each other via Coulomb interaction. Shown is an example where the dominant interaction is the magnetic exchange arising between opposite spin electrons due to Coulomb forces. The first electron polarizes the conduction electron gas antiferromagnetically, and an opposite spin electron can lower its energy in this locally polarized region. In this case C(r) has a node at the origin, helping to avoid the Coulomb interaction, and can have either s+/− or dx2− y2 form, as shown. These two possibilities lead to gap functions of opposite sign on the Fermi surface (orange ¼ −). From Chubukov A.V., Hirschfeld P.J. (2015) Physics Today.

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Superconductivity: Electronic mechanisms

theory perspective, fermionic incoherence destroys Cooper logarithm and reduces the tendency to pairing, while the opening of a superconducting gap eliminates scattering at low energies and renders a Fermi liquid behavior. Because incoherence and pairing come from the same interaction, they are intertwined, and in general their characteristic scales comparable. This issue recently attracted strong interest among both condensed-matter and high-energy theorists. Itinerant quantum-critical systems, analyzed analytically in recent years, include fermions in spatial dimensions D  3, two-dimensional (2D) fermions near a SDW and CDW instabilities, 2kF density-wave instability, pair-density-wave instabilities, q ¼ 0 instabilities toward a Pomeranchuk order and toward a state with circulating currents, 2D fermions at a half-filled Landau level, generic quantum-critical models with varying critical exponents, Sachdev-Ye-Kitaev (SYK) and SYK-Yukawa models, extensively studied in recent years, strong coupling limit of electron-phonon superconductivity, and even color superconductivity of quarks, mediated by gluon exchange. These problems have also been studied using various numerical techniques (see literature in Abanov and Chubukov, 2020; Chubukov et al., 2020; Esterlis and Schmalian, 2019; Wang, 2020). To find the outcome of the interplay between SC and NFL, one needs to analyze the set of coupled integral equations for the fermionic self-energy and the pairing vertex., In most metallic quantum-critical models, the dressed low-energy fermions are fast excitations compared to soft bosons for one reason of another. In this situation, which bears parallels with the Eliashberg theory for electron-phonon interaction, the momentum integration can be carried out exactly, after projecting the pairing interaction into the proper irreducible channel, and to resolve the competition between NFL and superconductivity, one has to solve one-dimensional non-linear integral equation for the gap function △(om): Dðom Þ ¼ pT

X Dðom0 Þ − Dðom Þ oom0 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ðom − om0 Þ 2 2 m0 ðom0 Þ + D ðom0 Þ

(12)

where V (Om) is an effective “local” interaction with singular frequency dependence 1/|Om |g. The exponent g > 0 depends on the microscopic realization. A small g ¼ O() holds for models in D ¼ 3—and for pairing of quarks, mediated by gluon exchange, g ¼ 1/3 for 2D models at the onset of a nematic or Ising-ferromagnetic order, and for fermions at a half-filled Landau level, g  1/2 for 2D models at the onset of SDW or CDW order, g  0.68 for pairing in SYK-type modes with equal number of fermion and boson flavors, g ¼ 1 for the pairing mediated by 2D propagating bosons, g  1.2 has been suggested for Fe-pnictides, and g ¼ 2 describes pairing mediated by an Einstein phonon, in the (properly defined) limit of vanishing Debye frequency. The analysis of (12) shows that NFL in the normal state does not prevent pair formation below a certain Tp, whose value depends on g, but in a non-critical way (Fig. 9). It is less certain whether Tp is a true superconducting transition temperature or onset temperature for pairing of still incoherent fermions. In the latter case, the actual Tc is smaller than Tp, and in between Tc and Tp the system displays preformed pair behavior.

Strong coupling limit Superconductivity in the strong coupling limit have been mostly studied for fermions doped into a Mott insulator. These doped fermions form a small Fermi surface, whose area scales as doping x rather than 1 − x expected for a metal with the same number of fermions. A highly non-trivial superconductivity mediate by topological skirmion excitations has been proposed in the context of superconductivity near Chern insulating regime in TBG (Khalaf et al., 2021). Another research direction is superconductivity near the boundary to the Mott phase in systems like C60.

Fig. 9 The onset temperature of the pairing, Tp, as a function of g. From Abanov A.R., Chubukov A. V. (2020) Physical Review B 102: 024524 (and references therein).

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Conclusion Superconductivity is a fascinating subject, which remains at the forefront of research in condensed matter physics in the 21st century. In this Chapter I described several directions of the research over the last two decades, with emphasis on unconventional superconductivity of strongly correlated electrons. Other exciting developments in the field of superconductivity are recent works on Higgs excitations in superconductors, on feedback from superconductivity on electronic spectrum, particularly for strongly correlated electrons (neutron resonance and related stuff ), on highly non-trivial superconducting order parameters, which preserve the Fermi surface (called Bogolyubov Fermi surface for this purpose), on odd-frequency superconductivity, and on the interplay between pair formation and a true superconductivity (often called BCS-BEC crossover). Another rapidly growing area of research is the analysis of topological aspects of superconductivity. This last area includes the analysis of topological aspects of the pairing near a quantum-critical point.

Acknowledgments This work was supported by US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0014402.

References Abanov AR and Chubukov AV (2020) Physical Review B 102: 024524. (and references therein). Armitage NP, Fournier P, and Greene RL (2010) Reviews of Modern Physics 82: 2421. Baranov M, Chubukov AV, and Kagan MY (1992) International Journal of Modern Physics 6: 2471. Basov DN and Chubukov AV (2011) Nature Physics 7: 273. Cao Y, et al. (2018) Nature 556: 43. Chubukov AV and Hirschfeld PJ (2015) Physics Today 68(6): 46. Chubukov AV, Abanov AR, Esterlis I, and Kivelson SA (2020) Annals of Physics 417: 168190. Esterlis I and Schmalian J (2019) Physical Review B 100: 115132. Fay D and Layzer A (1968) Physical Review Letters 20: 187. Gonzalez J (2008) Physical Review B 78: 205431. Keimer B, Kivelson SA, Norman MR, Uchida S, and Zaanen J (2015) Nature 518: 179. Khalaf E, Chatterjee S, Bultinck N, Zaletel MP, and Vishwanath A (2021) Science Advances 7(19). Kiesel ML, et al. (2012) Physical Review B 86: 020507(R). Kohn W and Luttinger JM (1965) Physical Review Letters 15: 524. Lee PA, Nagaosa N, and Wen X-G (2006) Reviews of Modern Physics 78: 17. Maiti S and Chubukov AV (2014) In: Bennemann and Ketterson (eds.) “Novel Superfluids”, Chapter 15. Oxford Press. Nandkishore R, Levitov L, and Chubukov A (2012) Nature Physics 8: 158–163. Sato M and Ando Y (2017) Reports on Progress in Physics 80: 076501. Stepanov P, et al. (2020) Nature 583: 375. Tremblay A-M and Chubukov AV (2017) CERN Courier. Aug. 11. Wang Y (2020) Physical Review Letters 124: 017002.

Electrodynamics of superconductors☆ Vladimir Kozhevnikov, Tulsa Community College, Tulsa, Oklahoma, United States; KU Leuven, Leuven, Belgium © 2024 Elsevier Ltd. All rights reserved.

Introduction Meissner state Definitions Cooper pairs Microscopic whirls Penetration depth Temperature Flux quantization Total current Conclusion Acknowledgments References Further reading

644 645 645 646 651 652 652 653 655 655 656 656 656

Abstract Electrodynamics of superconductors is primarily the electrodynamics of the Meissner state, a state characterized by zero magnetic induction of a superconducting fraction of conduction electrons. Simultaneously, the Meissner state is characterized by zero resistivity and zero entropy of these electrons. The latter means that the temperature of an ensemble of superconducting electrons is zero as well. Understanding properties of the Meissner state provides a key to understand and predict these and other equilibrium and non-equilibrium properties of superconductors, e.g., properties of the intermediate and mixed states, flux quantization, resistanceless transport current, etc. The reader will see that all these properties have a single and very simple origin: quantization of the angular momentum of the conduction electrons combined into Cooper pairs, and that the electrodynamics of superconductors and superconductivity as a whole is a magnificent manifestation of the boundless ingenuity of Nature and of the power of Physics laws.

Key points

• • • • •

The electromagnetic and thermal properties of superconductors are dictated by Cooper pairs. On this reason, the electrodynamics of superconductors is primarily the electrodynamics of an ensemble of Cooper pairs. By definition of Cooper pairs, the paired electrons orbit their center of mass. The stability of Cooper pairs means that the pairs obey the Bohr-Sommerfeld quantization condition. In the Meissner phase (the phase of a superconducting sample with zero magnetic induction B) the magnetic field intensity H is equal to or greater than the intensity of the applied field H0. The properties of superconductors (zero resistivity, zero magnetic induction, zero entropy, flux quantization, etc.) are governed by the precession of Cooper pairs with quantized angular momentum. Due to zero entropy, the temperature of an ensemble of Cooper pairs equal to zero regardless of the temperature of the sample.

Introduction Superconductivity was discovered in the laboratory of Kameringh Onnes as a phenomenon of a sharp drop of electrical resistance in a high purity mercury wire at temperature T below 4.2 K (Kamerlingh Onnes, 1911). An original estimate of the resistance drop was 104 times compare to the resistance just above the transition temperature, called the critical temperature Tc, a constant of the superconducting (S) material. Fairly soon it became clear that resistance of a superconductor is not merely a small, but zero. Namely, the current in a closed S circuit represents a persistent current like that caused by electrons bound in atoms.1 It was also found that the S state exists up to a certain temperature-dependent critical magnetic field Hc(T ), which is another characteristic of

☆ 1

Change History: July 2022. V Kozhevnikov updated the chapter.

Persistent current is a current running in a closed circuit without both thermal and radiation losses.

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the material. While number of superconductors was (and is) steadily growing, at an early stage it seemed that zero resistivity is the only property distinguishing superconductors from the so-called normal (N) metals. The second hallmark of superconductivity, the disappearance of thermoelectric effects, was discovered by Walter Meissner (Meissner, 1927). Lack of thermo-e.m.f. suggests that entropy of the S fraction of conduction electrons (called “superconducting electrons”) is zero. If so, then superconductivity can be a previously unknown thermodynamic state of matter characterized by complete ordering of superconducting electrons. Gorter and Casimir used zero entropy as the base postulate of their two-fluid model (Gorter and Casimir, 1934), a seminal thermodynamic theory addressing properties of superconductors in zero field. On the other hand, a non-zero thermo-e.m.f. looks incompatible with zero resistivity of the S state, since otherwise it would lead to an infinite current and therefore to an instantaneous restoration of the N state. The idea of the new thermodynamic state was brilliantly confirmed in a so-called Meissner effect independently discovered by Meissner and Ochsenfeld2 (Meissner and Ochsenfeld, 1933) and Ryabinin and Shubnikov (Rjabinin and Shubnikow, 1934). It was found that, apart from zero resistivity and zero entropy, superconductivity is also characterized by zero magnetic induction B regardless on the history of the field application. Hence, a sample in the Meissner state (MS) represents a perfect diamagnetic, i.e., its magnetic susceptibility w and permeability mm(¼1 + 4pw) are equal to −1/4p and zero, respectively. Important to note that (i) like in regular diamagnetics, in the MS w and therefore mm do not depend on the sample temperature T and the applied field H0; and (ii) if the first two “big zeroes” of superconductivity take place in samples of any shape, connectivity and purity, the third zero (B ¼ 0 or the Meissner effect) is observed only with sufficiently pure singly connected samples of an ellipsoidal shape. It was found that there are two kinds of superconductors distinguishing by the sign of the S/N interphase energy also called surface tension. Materials with positive and negative surface tension are referred to as type-I and type-II superconductors, respectively. All type-I superconductors are elementary metals, while type-II materials, with very few exceptions, are alloys and multicomponent compounds. A sample of type-I material can be made behaving like type-II superconductor by introducing impurities and/or reducing the sample size, e.g., by decreasing the thickness of a film sample. However, the opposite is not possible. Other really astonishing properties, such as the flux quantization, Josephson effect and others, were discovered in 1960s and later. A search for the source of the superconductivity phenomenon culminated in a ground braking theoretical discovery of Leon Cooper (1956), who shown that even a weak attraction can lead to formation of stable electron pairs called Cooper pairs. The origin of attraction lays in the medium polarization occurring due to electron-lattice interaction, as it was first suggested by Fröhlich (1950) and discussed in the BCS theory (Bardeen, Cooper, Schrieffer, 1957). Cooper’s prediction was irrefutably confirmed in experiments (Deaver and Fairbank, 1961 and many others). Cooper pairs profoundly alter the medium electrodynamics. The physics behind these alterations is the subject of this chapter. The chapter consists of three sections: Meissner state, Flux quantization and Total current, and a brief summary. References to literature for further reading are given at the end. The cgs units are used because these units are the most appropriate for discussing electrodynamics of continuous media.3 In view of the space limit only key references are provided.

Meissner state Definitions The MS is an equilibrium S state defined as the state with B ¼ 0 throughout the sample volume V. Alike, the MS can be defined as the state with mm ¼ 0 or w ¼ − 1/4p all over V. The MS takes place in sufficiently pure massive and simply connected samples of an ellipsoidal shape in the applied field H0 < Hc1(1 − ). Here Hc1 is the lower critical field of type-II superconductors equal to the thermodynamic critical field Hc in the type-I counterparts, and  is the demagnetizing factor4 with respect to an ellipsoidal axis parallel to H0. A sample is considered as massive if its minimal size significantly exceeds a so-called penetration depth l, a width of the surface layer where B decays from the induction of external field near the sample surface Bext to zero in the sample interior. A unique feature of the ellipsoidal bodies in a static magnetic field is uniformity of the field intensity H (also called the magnetizing force, field strength, etc.) throughout their volume, as it was first shown by Poisson (Poisson, 1824) and discussed by Maxwell (Maxwell, 1873). The intensity H is responsible for magnetization I( ¼ wH). Depending on a demagnetizing field Hd(¼4pI), in diamagnetics H(¼ H0 − Hd) is greater than or equal to H0 in magnitude, and its direction can differ from that of the applied field. H is the field acting on “native” (belonging to the body) charges, whereas the induction B is the field acting on a “foreign” (extraneous) charges.5 Contrarily to B(¼ H + 4pI), the average of microscopic fields caused by all native charges, H does not include the field due to the charge in question.6 Respectively, in any medium H differs from B. On the contrary, in a free space (vacuum) I ¼ 0 and, consequently, H ¼ B.7 2

An active promoter of this experiment was von Laue.

Different dimensions of the induction B and the intensity H of the magnetic field in the SI unit system, make this system in its current form inapplicable for this purpose. 3

4

The demagnetizing factors are determined by relationship of the ellipsoidal axes. Respectively, -s are defined only for the ellipsoidal bodies.

5

On that reason the field measured by a prob charge, e.g., in mSR, is B, whereas in magnetic resonances the measured or controlling field is H.

6

Here, the charge should be understood as the current caused by all types of microscopic movement, including the spinning one, of a given charge.

Important that this is not just a mathematical coincidence, but a physical identity. This implies that in any system of physical units the dimensions of B and H must be the same. 7

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Electrodynamics of superconductors (A)

B0=H0

(B)

H0=B0

H

B=0

Fig. 1 Induction B (A) and intensity H (B) of the magnetic field inside and outside of a long cylindrical sample in the Meissner state when the field H0 is applied transversely to its longitudinal axis. For this sample/field configuration  ¼ 1/2. Inside the sample B ¼ 0 and H ¼ H0/(1 − ) ¼ 2H0; outside it H ¼ B and far away H0 ¼ B0. Shadowed circle is a cross-section of the sample oriented perpendicular to the page.

At the sample boundary the normal component of the intensity Hn undergoes a discontinuity DHn ¼ 4pIn, where In is the magnitude of the normal component of magnetization I. Correspondingly, the intensity can be viewed as a potential field created by magnetic charges sitting on the boundary and having a surface density In. On the other hand, the intensity inside the sample, i.e. in any volume element dV not crossed by the sample boundary, represents a solenoidal or divergenceless field, which, therefore, can be described by a vector potential. As an example, Fig. 1 schematically shows the induction B and the intensity H of the magnetic field in and out of a sample with  ¼ 1/2 when it is in the MS.8 If H0 is parallel to an axis of the ellipsoidal sample in the MS, H ¼ H0/(1 − ), where  is the demagnetizing factor about this axis. If H0 is not parallel to either axis, H is the vector sum of the intensities relative to each axis. At any orientation of H0 with respect to the sample in the MS, inside it the magnitude of intensity H is less than Hc1. After all, a very important property of the MS follows from the Law of the symmetry of time reversal, mandatory for any (either classical or quantum) equilibrium state. According to this Law, no total current can occur in the equilibrium state9 and therefore all currents in the MS must be compensated. In samples with  ¼ 0, referred to as the samples of cylindrical geometry, the outer field due to the magnetized sample is absent and respectively H ¼ H0 both inside and outside the sample. Such samples can be, e.g., long cylinders, infinite slabs or wide ribbon-like foils in H0 parallel to their generating line.10 Here our discussion of the MS will be limited by the cylindrical samples. In this case the magnetic moment M of the sample is V H 4p 0

(1)

M  H0 VH20 ¼ : 2 8p

(2)

M  wVH ¼ − and the sample magnetic energy Em is Z Em  −

M  dH0 ¼ −

By virtue of the energy conservation, in the cylindrical samples Em ¼ DT, where the latter is the field-induced change of the kinetic energy of electrons. As an example, graphs for M vs H0 for a type-I superconductor with  ¼ 0 are shown in Fig. 2.

Cooper pairs The Cooper pairs are defined as a correlated state of two conduction electrons with zero net kinetic linear momentum with respect to their center of mass and zero net spin. The latter condition stems from the requirement of thermodynamics since the free energy of a system of units with zero spin is lesser than it would be if the spin is not zero. The same condition justifies the thermodynamic advantage of the electron pairing since it allows to null the spin. Zero net kinetic linear momentum and stability of the pairs mean that the paired electrons orbit their center of mass like, e.g., stars in a binary star. Therefore, each Cooper pair possesses the angular momentum i and the magnetic moment m ¼ gi, where g is the gyromagnetic ratio. As measured by Kikoin and Goobar, 1938, g in superconductors equals its classical value e/2mc. Here m and e are mass and charge of one electron, respectively, and c is the electromagnetic constant of the cgs unit system equal to speed of light. This confirms that the “superconducting” electrons are effectively spinless. 8 The London theory, adopted for description of the MS in the Ginzburg-Landau and BCS theories, is based on an assumption that in superconductors the magnetic susceptibility mm and the dielectric permittivity ee are unity and thereforeB ¼ H. However, this is only true in a vacuum. 9 The time transformation t ! − t changes the direction of magnetic moment of a system with total current and therefore such a system can not be in the equilibrium state by definition. 10

Any sample with  ¼ 0 can be broken for a large number of identical thin circular cylinders parallel to H0.

647

0.0

-2.0x10

-4

-4.0x10

-4

-6.0x10

-4

-8.0x10

-4

2.00 K 2.25 2.50 2.75 3.00 3.20

H0

4 mm

M (emu)

Electrodynamics of superconductors

3 µm

0

50

H0 (Oe) 100

150

200

250

300

Fig. 2 Magnetic moment of a 2.9 − mm thick indium film in the parallel field H0 measured at constant temperatures, as indicated. For this sample  ¼ 0 and H ¼ H0. The data were taken with the sample cooled in zero field (arrow up for 2.0 K) and in the field exceeding Hc (arrow down for 2.0 K). Hysteresis at the S/N transition is the supercooling effect caused by the positive S/N surface tension. The film geometry is shown in the insert. After Kozhevnikov V, Suter A, Prokscha T, and Van Haesendonck C (2020) Journal of Superconductivity and Novel Magnetism 33: 3361; reprinted with permission from Springer Nature.

Stability of the pairs also means that the orbital motion of the paired electrons occurs without energy dissipation. Therefore they must obey the Bohr-Sommerfeld quantization condition. At the same time, one should bear in mind that to meet the law of momentum conservation, in a magnetic field the linear momentum should be taken in a generalized form. Hence, the quantization condition for the single Cooper pair is I ~ cp  dl ¼ 2pRp~cp ¼ 2p~icp ¼ nh, (3) p where p~cp and ~icp are the generalized linear and angular momentum of the pair, respectively, R is the radius of orbital motion of paired electrons, h is the Planck constant, and n is a non-negative integer (n ¼ 0,1,2, . . .). ~ , the generalized linear momentum of the paired electrons p ~ cp is By definition of p     eA eA ~1 + p ~ 2 ¼ mv 1 + 1 + mv2 + 2 ~ cp ¼ p (4) p c c where v is the electron velocity and A is the vector potential of the field acting on the electron, i.e. of the intensity H; subscripts 1 and 2 denote the quantities related to the first and second electron in the pair. ~ cp is the sum of kinetic linear In the ground state, i.e. at zero field and temperature, n in Eq. (3) takes the lowest value n ¼ 0 and p momentums p(=mv) of electrons. Hence,     ~ cp ¼ pcp ¼ p10 + p20 ¼ 0 p (5) 0

0

where subscript 0 denotes the zero field. h  i ~ cp ¼ 0 is a The last part of Eq. (5) is identical to the definition of Cooper pair. The first part of this equation p 0

modified London’s rigidity, originally formulated for a single superconducting electron (London, 1960). Also, Eq. (5) implies that the center of mass of the pair is at rest with respect to the sample. The motion of electrons in one Cooper pair at zero field and temperature is schematically shown in Fig. 3. The orbit diameter 2R0 corresponds to the coherence length x0 of the BCS theory and x of the Ginzburg-Landau (GL) theory (Ginzburg and Landau, 1950), which are close to each other at these conditions. One more important property at zero field is related to the ensemble of Cooper pairs: due to symmetry, the total magnetic moment of all pairs (the magnetic moment of the sample) is zero, i.e.

R0

J =

Fig. 3 Schematics of a single Cooper pair at zero field and temperature. i0 and m0 are the angular momentum and the magnetic moment of the pair, respectively; g is the gyromagnetic ratio; p10 and p20 are the kinetic linear momentums of electrons; J0 is the current caused by the orbital motion of electrons round their center of mass; R0 is the orbit radius; red dot designates the center of mass of the pair.

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Electrodynamics of superconductors M0 ¼

X

m0 ¼ 0

(6)

where summation is taken over all pairs.11 In view of uniformity of the bulk properties of the MS, Eq. (6) holds for the unit volume as well as for a physically infinitesimal volume element dV. The latter in superconductors is defined as a volume, which size is much smaller than the size of the volume taken by the S phase and much larger than the spatial inhomogeneity of microscopic currents, i.e. R0. We remind that Cooper pairs are heavily overlap, but it affects neither stability, no mobility of the pairs. Taking into account the wave side of the electron nature, this fact is similar as the overlapping of myriads of electromagnetic waves does not prevent us from seeing objects or enjoying music broadcast from the other side of the globe. Now it is rather obvious what happens when the magnetic field is turned on: Cooper pairs precess. Let us see how does it go. In the field, each of the paired electrons (as well as all other charges constituting the sample) experiences the action of the Lorentz force.12 In absence of the applied electric field,13 this force is     dp e ∂A e e ∂A e + vH¼ + v  ðr  AÞ ¼ F (7) c ∂t c c ∂t c dt Here t is time, A is the vector potential of the field H(¼r  A), and v is the electron velocity, which magnitude is about the same as the Fermi velocity vF (the maximum difference between v and vF is on the order of a percent). Important to note that since inside real bodies B and H are different, one has to distinguish the vector potential for H and for B. In our notations A is the vector potential of the intensity H inside the sample. Below we will see how A is related to the vector potential AB defined by the relationship B ¼ r  AB. As usual, the supplementary condition r  A ¼ 0 is presumed. The first term in the right-hand side of Eq. (7) is an electric force FE caused by the changing magnetic field. Corresponding vortex electric field E is defined as   F 1 ∂A : (8) E E¼− e c ∂t As with bound electrons in conventional materials, the electric field E (existing while the magnetic field is changing) does the work resulting in a change of kinetic energy of electrons and in the appearance of an induced magnetic moment in each pair. Specifically, when the vector potential changes from zero to A, which corresponds to the intensity change from zero to H, the velocity of electrons in the pair is changed from v0 to v0 . The difference vi ¼ v0 − v0 is the velocity induced due to the applied magnetic field. Integrating Eq. (8) over time, one obtains vi ¼ −e

A : cm

(9)

This is a very important formula. It shows that vi is parallel to A. Therefore, since by definition A lies in the plane perpendicular to H, vi and therefore the induced current also lies in the plane transverse to H. On the other hand, the induced current makes a closed loop (it cannot leave the sample) and the loop must be circular due to rotational symmetry stemming from uniformity of H (all directions in the transverse plane are equivalent); therefore lines of the vector potential make the circular loops either. Hence, an appropriate gauge for the vector potential of H is the circular one, namely A ¼ H  r/2. The lines of this vector potential are shown in Fig. 4.

y

=

−

+

H×r

=

x

~H Fig. 4 Lines of the vector potentials A of the circular gauge for a uniform magnetic field H directed toward the reader (along the z-axis); b x and b y are unit vectors along x and y axes, respectively. r is a radius vector lying in xy plane. As for all vector fields, the line of A is the directional line tangential to the vector A in each of its point.

11 Note that Eq. (6) resembles the definition of a diamagnetic atom, where each electron has a non-zero orbital momentum, while the momentum of the atom is zero. 12 The action of a magnetic field on atomic electrons and nuclei results in the usual diamagnetic response; its action on unpaired conduction electrons causes a weak Landau diamagnetism. 13

A static electric field does not penetrate the superconducting sample due to the screening effect of always present “normal” electrons.

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Thus, since e < 0, Eq. (9) indicates that the current induced in each Cooper pair creates a magnetic moment directed opposite to H, i.e. the induced moment is always diamagnetic, in accord with requirements of thermodynamics and the time reversal symmetry. This also means that diamagnetism (both in normal and superconducting materials) results from the changing magnetic field in full consistency with the laws of the classical physics. Next, like in regular diamagnetics, in Eq. (9) the time has dropped out implying that the induced velocity vi is indifferent to the rate of the field change. After all, Eq. (9) implies that the vector potential in superconductors is not gauge-invariant since a change in A changes the induced moment and because A of any but the circular gauge does not possess the necessary rotational symmetry. However, this is not surprising,14 since the gauge invariance is inapplicable to quantities defined by the vector potential in explicit form. Non-fulfillment of the gauge-invariance in superconductors is an additional confirmation of the fact that the vector potential is a real and primary characteristics of the magnetic field, as demonstrated by the Aharonov-Bohm effect (Ehrenberg and Siday, 1949; Aharonov and Bohm, 1959). The second term in the right hand side of Eq. (7) is termed the magnetic force FM. It is perpendicular to the electron velocity and therefore does no work. Hence, FM does not affect the electron kinetic energy and the magnitude of its magnetic moment. On the other hand, since the uniform magnetic field cannot change position of the pair’s center of mass, FM1 ¼ − FM2 (those are the magnetic forces acting on the 1st and 2nd electron in the pair), and also FM1 and FM2 are non-central forces. Hence, they make a couple resulting, due to non-zero i0, in precession of m0 relative to the vector H, as schematically shown in Fig. 5. The angular velocity of precession, referred to as Larmor frequency, is o ¼ −gH ¼ −

eg L H, 2mc

(10)

where gL is the Lande factor. As mentioned, in superconductors g takes the classical value, i.e. gL ¼ 1. As follows from the Larmor theorem, if vi  v0, precession of the electron orbit is equivalent to the undisturbed orbital motion at the field absence (i.e. with fixed m0 and R0) plus an additional (field induced) circular motion with the angular velocity o and radius r proportional to R0, which leads to appearance of the diamagnetic moment mcp. On the other hand, the invariability of m0 means that Eq. (6) is valid both in the absence and presence of the magnetic field. This can be also understood basing on the fact that precession is an inertia-free motion and therefore all pairs precess synchronously,15 i.e. without changing their mutual orientation. Thus, we arrive to conclusion (following both from consideration of the action of Lorentz force Eq. (7) and from the Larmor theorem) that the net effect of the magnetic field is the induced circular motion of the paired electrons in the planes transverse to H. On the other hand, as it is known for any kind of cyclic motion in magnetic field, the changing | H| changes the magnitude of vi only, i. e. the radius of the induced motion r does not depend of the field provided that vi  v0.

Fig. 5 Schematics of the Cooper pair precession in the magnetic field H. Vectors i0 and m0, current J0 and radius R0 are the same quantities as those at zero field shown in Fig. 3. A dot-dashed circle designates the path of the tip of the precessing i0. A dashed blue circle depicts the field-induced current Jcp; it is also a line of the vector potential directed opposite to the current Jcp. A and vi designate the vector potential and the induced velocity of one electron, respectively; r is the radius of the induced current; mcp is the induced magnetic moment of the pair. Designated by a single arrow A and vi are proportional but not equal. After Kozhevnikov V (2021) Journal of Superconductivity and Novel Magnetism 34: 1979; reprinted with permission from Springer Nature.

14

Recall that the gauge-invariance does not hold in the London and BCS theories.

15

Also, as it is shown below, the induced circular motion of the paired electrons occurs in phase.

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Fig. 6 A cross sectional view of a conventional diamagnetic sample showing induced bound currents caused by precession of the atomic electron orbits; the field is directed into the page. This picture is similar to the induced currents caused by precession of Cooper pairs in a superconducting sample in the Meissner state. After the textbook of Tamm IE, Fundamentals of the Theory of Electricity, 9th edn. Moscow: Mir, 1979, reprinted with permission from Mirtitles.

~ cp does not depend on the magnetic field either, namely One can also show that p ~ cp ¼ p ~1 + p ~2 ¼ p ~ 10 + p ~ 20 ¼ 0: p

(11)

Hence, the modified London’s rigidity holds both in zero and non-zero field. Below we will see that it also holds at non-zero temperature. The induced currents in superconductors in the MS (and in other equilibrium S states) are qualitatively identical to the induced bound currents in regular diamagnetics schematically shown in Fig. 6. In our case, like in the normal materials, the induced currents mutually compensate each other in the sample bulk, leaving an uncompensated surface current caused by electrons bound in Cooper pairs with resting centers of inertia. Then the magnetic moment of the sample is exactly the same as the moment produced by a continuous (circumferential) surface current. Now we are ready to calculate magnetic proprieties of our cylindrical sample in the MS. After turning on the applied field H0, the field intensity and the vector potential inside the sample after a short relaxation time become H and A, respectively. Then, using Eq. (9), we write. e e mvi ¼ − A ¼ − H  r: c 2c

(12)

Hence, mv ¼ −

e H, 2c

(13)

where v ¼ (r  vi)/r2 is the angular velocity of the induced circular motion of the paired electrons. Taking into account that e < 0, Eq. (13) reads that v is parallel to H and its magnitude is o¼

vi e ¼ H: r 2mc

(14)

Comparing with Eq. (10) we see that o equals the classical Larmor frequency (gL ¼ 1). This confirms that we really deal with the precession of paired electrons with zero total spin, since non-paired electrons (as any other charge) in a magnetic field circulate with a so-called cyclotron frequency oc ¼ (e/mc)H.. The induced current per one electron in the pair Ji is Ji ¼

e e2 H: o¼ 4pmc 2p

(15)

As follows from Eqs. (14) and (15), neither induced angular velocity o, nor the induced current Ji depend on r. However, this is not the case for the induced magnetic moment and the corresponding change of kinetic energy of the paired electrons: they both depend on r2. On the other hand, r for Cooper pairs with different orientation of m0 is different. So what we want to know is the mean square hr2i, which we will denote as r2i . Using Eq. (15), the magnitude of an average induced magnetic moment per one electron in Cooper pairs is   e2 r 2i ðr =2Þ2 H 1 e2 ns 4p r i 2 H H¼ ¼ i 2 : mi ¼ J i pr 2i ¼ c mc2 2 4pns 4mc2 lL 4pns

(16)

where ns is a number density of superconducting (i.e. paired) electrons and lL is so-called London penetration depth defined as  lL ¼

mc2 4pns e2

1=2

¼

mcp c2 4pncp q2cp

!1=2 ,

(17)

where mcp ¼ 2m, qcp ¼ 2e and ncp ¼ ns/2 are the mass, charge and number density of Cooper pairs, respectively. Since induced magnetic moments in all Cooper pairs are parallel, the magnitude of the magnetic moment of our sample is M ¼ ncp Vmcp ¼

ðr i =2Þ2 V ðr =2Þ2 V H¼ i 2 H0 : 2 4p lL lL 4p

(18)

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651

Comparing this result with Eq. (1), we see that the rms radius of the induced motion of the paired electrons is r i ¼ 2lL :

(19)

Since r does not depend on the field, ri and therefore lL do not depend on the field either. Next, basing on Eq. (6) one can show that, like in regular diamagnetics, the change of kinetic energy of the superconducting electrons DT is the sum of the field-induced kinetic energies of each of these electrons, i.e. DT ¼ Ek ¼ ei ns V ¼ ecp ncp V, where ecp ¼ 2ei is the average kinetic energy of the induced motion of electrons in one pair. Using Eq. (14), we write  2 2 mv2 e2 H2 r 2i ðr i =2Þ H ei ¼ i ¼ : ¼ 2 lL 2 8pns 8c m Thus, taking into account Eq. (19), the change of kinetic energy of the paired electrons is  2 ðr i =2Þ H2 H2 DT ¼ ei ns V ¼ V¼ V: lL 8p 8p

(20)

(21)

(22)

In our sample H ¼ H0 and, according to the energy conservation, DT ¼ Em ¼ H02V/8p (see Eq. 2). Hence, our description meets the Law of energy conservation. Next, we calculate the gyromagnetic ratio coming from its definition. Using Eqs. (14) and (16) we write g

e2 r 2i H 2c m Mi mi ns V e ¼ ¼ ¼ i ¼ , ii ns V Li mvi r i 4mc2 er 2i H 2mc

(23)

where Mi and Li are magnitudes of the induced magnetic moment and of the induced angular momentum of the sample, respectively; and ii is an average field-induced angular momentum per one electron. We see that g is fully consistent with the experimental result of I. Kikoin and Goobar. After all, calculate the induction in the sample interior. Using Eqs. (16) and (19), and the fact that mi is negative, we obtain B ¼ H + 4pI ¼ H + 4pðmi ns Þ ¼ H−4p

H n ¼ 0, 4pns s

(24)

in agreement with the result of Meissner and Ochsenfeld and of Rjabinin and Shubnikov. Correspondingly, the magnetic permittivity mm  B/H and susceptibility per unit volume w of the S phase are zero and −1/4p, respectively. Naturally, due to microscopic character of the induced currents, there is no problem in establishing the Meissner condition (B¼ 0) in samples/domains of any shape, as soon as the field H is uniform. The latter is indeed so in ellipsoidal samples, regardless whether they are in the MS or in the inhomogeneous equilibrium states, i.e., in the mixed state of type-II and in the intermediate state of type-I superconductors. This explains why the Meissner state is observed only in the ellipsoidal bodies, as well as the plenty domain shapes observed in the intermediate state. The above consideration applies to the samples cooled in zero field (ZFC), whereas the Meissner effect is about both the ZFC and FC (field-cooled) samples. So, what happens in the latter case? Upon lowering temperature below Tc(H0) in the fixed field H0 < Hc1(1 − ), a temperature dependent fraction of conduction electrons condenses forming stable Cooper pairs. This means that speed of these electrons drops from vF to v0, each pair starts orbiting its center of mass and, being in the field, the pairs precess. Like in regular diamagnetics, the latter leads to establishing magnetization I, the field intensity H ¼ H0 − 4pI and the induction B ¼ hhi ¼ H + 4pI, where hhi is the average microscopic field. Hence, after the short relaxation time needed to establish the intensity H, the environment inside the FC sample becomes the same as that in the ZFC sample. Thus, the given description meets the Meissner effect indeed.16 One more remark. As mentioned above, radius of the electron orbit R0 in precessing Cooper pairs is fixed (i.e. it does not depend on the field) and the Larmor theorem is exact if vi  v0. Taking typical value of Hc1  100 Oe and lL  10−6 cm, from Eq. (12) one finds vi  102cm/s, which is six orders of magnitude less than v0  vF  108 cm/s. So, there is no doubt that R0 does not depend on the field. On the other hand, since ri and R0 are quantities of the same or close order of magnitude, this estimate implies that o  o0  v0/R0, i.e. the precession is very slow.

Microscopic whirls Now, let us reconstruct the current structure of the MS. We understand that all induced currents form identical circular loops with the rms radius ri laying in parallel planes transverse to H. How these currents are arranged with respect to each other? 16

The standard description of the Meissner effect is based on the London theory in which this effect is achieved by postulating B ¼ 0.

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Coming from symmetry, one can expect either complete chaos or complete order. Since the system of the ordered currents has lesser free energy, the second option prevails. This is consistent with experimental results of Keesom and Kok (1934) that entropy of the sample in the S state is less than that in the N state. An ordered structure of identical circular currents laying in the planes transverse to H possesses the maximum symmetry if the currents form a 2D hexagonal lattice of cylindrical micro-whirls resembling densely packed and tightly “wound” micro-solenoids parallel to each other and to H. Length of each whirl/solenoid equals the sample size along direction of H and its rms diameter is 2ri ¼ 4lL. Since the solenoids are parallel, they do not interact. In addition, since the currents are given by the field-induced circular motion of the paired electrons, the complete ordering implies that in all Cooper pairs this motion is in phase.17 Just a few facts supporting this statement. (i) The internal energy of the cylindrical sample in the MS is just the sum of kinetic energies ei (Eq. 22) and it does not contain the term(s) responsible for interaction. (ii) In pure type-II superconductors Abrikosov vortices18 make the hexagonal lattice, as was first observed by Essmann and Träuble (Essmann and Träuble, 1967). (iii) An experimental evidence showing that the Abrikosov vortices do not interact with each other in pure samples (Kozhevnikov et al., 2018). Naturally, the complete ordering of the field-induced currents together with Eq. (6) mean that entropy of the ensemble of Cooper pairs Scp is zero, exactly as it takes place for the magnetic part of entropy in regular diamagnetics. Therefore the given theoretical interpretation, referred to as a micro-whirls (MW) model, is consistent with the absence of the thermoelectric effects in superconductors and justifies the postulate of zero entropy of superconducting electrons of the two-fluid model of Gorter and Casimir. One more detail about the micro-whirls. A spacing between current loops D along the whirl axis is a universal quantity19 equal to D¼

2e2 ¼ 5:6  10−13 cm  6 fm: mc2

(25)

So D is about a tripled size of a proton (1.7 fm) and therefore the whirls are similar to very tightly wound solenoids indeed. An important feature of a single whirl is that it is characterized by two microscopic quantities with dimension of length. Those are the radius of the orbital motion R0 and the radius of the field-induced circular motion ri. As mentioned, ri is proportional to R0, both radii do not depend on the field and the ratio ri/R0 is a constant of superconducting material. However, if vector ri always lays in the transverse plane, R0 is symmetrically distributed over 3D space as follows from Eq. (6). So, like in the Langevin theory of diamagnetism, a mean squared projection of R0 on the transverse plane R?2 equals 2R02/3. In other words, R? is an rms radius of a cylinder coaxial to the cylinder with radius ri and filled by orbiting Cooper pairs. Hence, an individual whirl can be viewed as a “double wall” cylinder with radii R? and ri. A parameter of material in the MW model is ℵ (aleph) defined as ℵ¼

ri : R?

(26)

In materials with ℵ < 1 the S/N surface tension is positive and correspondingly such materials represent type-I superconductors. In type-II superconductors the S/N interphase energy is negative and ℵ > 1. The cross section of the whirl structure in type-I and type-II superconductors is shown in Fig. 7.

Penetration depth Another important property of the MS is the penetration depth l, a width of the surface layer where B decades from Bext(¼ B0 ¼ H0 for the cylindrical samples) to zero in the sample interior. According to the London theory, in a sample with a plane boundary (i.e., in the massive cylindrical samples), the induction B(z), where z is the distance from the surface, decays exponentially with the decay constant lL. Correspondingly, l is infinite and an effective penetration depth leff ¼ lL.20 What is l in the MW model? To answer this question one needs to know the angular distribution of m0, which is beyond the scope of the semiclassical consideration employed in the model. One can estimate l assuming that the boundary of the blue cylinders in Fig. 7 is sharp or the induced current Ji is linear. Then for the plane sample boundary l ¼ ri ¼ 2lL and leff ¼ lL/2. This is, of course, an oversimplified estimate, however in any case in the MW model l is finite and close to lL.

Temperature All properties discussed so far are related to those for the samples at T ¼ 0. However, it is quite obvious that nothing will change if T 6¼ 0. 17 Note the direct analogy with the phase of the quantum-mechanical wave function. The wave functions of all Cooper pairs must be in phase since otherwise there would be a total current forbidden by the time reversal symmetry. 18

The Abrikosov vortices represent holes in the network of the ordered currents of the MS.

Since ri2 ¼ (2lL)2 1/ncp, the universality of D is a necessary condition that the whirls, i.e., ordered Cooper pairs, fill the entire volume of the sample in the MS at any change of ncp (occurring when T changes) regardless of the specific material of the sample. 19

20

leff is such a depth over which the flux of the penetrating field is the same as the flux of the real field and the flux density B is constant and equal to Bext.

Electrodynamics of superconductors (A)

(B)

TYPE-I

ri

H

RA

653

TYPE-II

ri

RA

Fig. 7 Schematics of the micro-whirl structure in the transverse cross section of type-I (A) and type-II (B) superconductors. In (A) ri < R? or ℵ < 1; in (B) ri > R? or ℵ > 1. The induced currents (all are in phase with each other) are designated by arrows. Areas filled with Cooper pairs are colored in pink; the cross-sectional areas of the induced currents are colored in blue. R? is the rms projection of the pairs radius R0 onto the plane perpendicular to H; ri is the rms radius of the induced motion of the paired electrons. Boundaries of all cylinders are diffuse (unsharp), the cylinders overlap without leaving voids. The field H is directed toward the reader. After Kozhevnikov V (2021) Journal of Superconductivity and Novel Magnetism 34: 1979; reprinted with permission from Springer Nature.

Indeed, we have seen that entropy of the ensemble of Cooper pairs Scp is zero. Therefore, according to the Third law (Nernst’s theorem), the temperature of the ensemble of Cooper pairs Tcp is zero as well, regardless on the sample temperature T. Hence, all results obtained in this section hold in the whole temperature range of the existence of Cooper pairs. In other words, the paired electrons are in the ground state in the entire temperature and field range of the S state. However, Tcp ¼ 0 does not mean that the sample temperature has no effect on the properties of paired electrons, since otherwise Tc would be infinite. The sample temperature affects the lattice polarization responsible for the electron pairing. Correspondingly, it changes ns(¼2ncp), as shown in the two-fluid model. Therefore, the change of T leads to the change of lL and, correspondingly, to the change of ri, the rms radius of the field-induced motion of electrons in the pairs. On the other hand, ri is proportional to R0 with pffiffiffiffiffiffiffiffi the proportionality coefficient ℵ (more strictly 2=3ℵ) determined by the superconducting material. In its turn, ℵ does not depend on T due to the constancy of Tcp(¼0).

Flux quantization Let us consider a closed macroscopic loop l located inside a sample in the MS away from its surface. For simplicity, take a cylindrical ~ cp sample and the loop lying in the transverse plane as shown in Fig. 8A. Now, calculate the circulation of the linear momentum p over this loop. D E Moving along the loop, we will pass through lots of pairs, so we should consider the average momentum p~cp and the average quantum number hni. Since all Cooper pairs are in identical conditions hni ¼ n. Therefore, the Bohr-Sommerfeld quantization condition for this case is I D E ~ cp  dl ¼ nh, (27) p l

where l is the loop D length. E ~ cp and take into account that in the MS n ¼ 0. Then, Eq. (27) becomes Now, open p (B)

(A)

ℓ

H0

ℓ

Fig. 8 Cross section of solid (A) and hollow (B) cylindrical samples in a field H0 parallel to the long axis of the samples. l is a loop lying away from the wall(s) of the cylinders.

654

Electrodynamics of superconductors I D E I I I 2e ~ cp  dl ¼ hmvi1 i  dl + hmvi2 i  dl + p hAi  dl ¼ 0: c l l l l

(28)

The first two integrals are zero due to mutual compensation of the induced kinetic linear momentums of electrons in neighboring pairs, similar as it takes place in the regular diamagnetics. Now, what is hAi, the average of the vector potential of the intensity H? To answer, we apply Stokes’ theorem; then Eq. (28) is rewritten as I Z 2e 2e ðr  hAiÞ  df ¼ 0 (29) hAi  dl ¼ c l c Fl where Fl is the area of a surface bounded by the loop l and df is a vector element of this surface. From Eq. (29) we see that the integral over the area Fl is the flux of a vector [r  hAi] and this flux equals zero. Therefore, since r  A  H 6¼ 0, hAi 6¼ A. On the other hand, inside our sample the induction B and therefore its flux is zero (Eq. (39)). Therefore, Eq. (29) suggests that hAi is the vector potential of the magnetic flux density (induction) AB defined as B ¼ r  AB. In other words, the vector potential of the flux density AB is a macroscopic average of the vector potential A determining the field induced microscopic currents Ji. This is an exact match with the classical definition of AB as a macroscopic mean of the vector potentials caused by the microscopic currents. So, putting hAi ¼ AB, we rewrite Eq. (29) as Z Z 2e 2e 2e ðr  hAiÞ  df ¼ B  df ¼ F ¼ 0, (30) c Fl c Fl c where f is the magnetic flux through the area Fl. Now, take a hollow (tube-like) thick-wall long cylinder shown in Fig. 8B, apply the field H0(< Hc1) parallel to its longitudinal axis and cool the cylinder below Tc. The loop l encircles the opening of the cylinder and lies inside the wall far (compare to l) from the inner and outer cylinder surfaces. The induction inside the wall is zero, implying that, as in Eq. (28), hmvii ¼ 0. On the other hand, the flux f inside the hollow cylinder is frozen and therefore it is not zero. Then Eq. (27) yields I D E I Z Z 2e 2e 2e 2e ~ cp  dl ¼ r  hAi  df ¼ B  df ¼ F ¼ nh: p hAi  dl ¼ (31) c c c c l l Fl Fl Hence, the magnetic flux passing through the opening (plus surrounding it penetration area) in a multiply connected superconductor is F¼

c nh: 2e

(32)

This is the famous London’s flux quantization but for the paired electrons. Hence, the origin of the superconducting flux quantization is the Bohr-Sommerfeld quantization condition, which also justifies existence of the field induced persistent currents in the MS. Eq. (32) implies that the flux quantum and therefore the flux passing through each so-called Abrikosov vortex in type-II superconductors in the mixed state is F0 ¼

c pcℏ h¼ : 2e e

(33)

As well known, the flux quantization Eq. (32) and the single flux quantum in the Abrikosov vortices Eq. (33) are in full agreement with experiment. Thus, inside the S phase the vector potential AB ¼ hAi, where A is the vector potential of the field H. Now, what is the value of AB inside the sample in the MS? In the plane perpendicular to H we have uniformly distributed identical induced circular currents, i.e. in-plane currents with the same magnitude Jcp and the same rms radius ri circulating clockwise relative to H. Therefore, exactly as it takes place in regular diamagnetics, equal amount of electricity flows in opposite directions throughout an out-of-plane cross section of an arbitrary chosen volume element dV. Hence, an average current density hji ¼ enshvii is zero in any volume element of the sample interior or inside the S phase. So, taking into account Eq. (9), we write AB  hAi ¼ hv i i ¼ hji ¼ 0:

(34)

Thus, inside the sample in the MS, or in general inside the S phase, in spite of non-zero H, an average of its vector potential hAi, as well as the average induced velocity hvii, and the average induced current hji are all equal zero. Out of these three equalities the last one [hji ¼ 0] is the most important: it shows that all currents in the S phase are mutually compensated, as required by the Law of the time reversal symmetry.

Electrodynamics of superconductors

655

Total current As we have seen, the equilibrium magnetic properties of superconductors are qualitatively similar to the properties of conventional diamagnetics. In both cases these properties are due to magnetization arising from precession of the microscopic magnetic moments caused by the orbital motion of electrons bound either in atoms (conventional diamagnetics) or in Cooper pairs (superconductors). A colossal quantitative difference in the magnetic susceptibilities in these materials (up to 5 orders of magnitude!) is due to the differences in the size of the orbits and correspondingly in the radii of the induced microscopic currents of the bound electrons. However, there is also an important qualitative difference in these induced currents. Namely, in conventional diamagnetics the orbiting electrons are bound in motionless atoms (fixed in the crystal lattice), while in superconductors - in movable Cooper pairs. The pairs mobility allows them to compose the whirl structure with zero entropy. On the other hand, since the pairs have electric charge, they can form an electric current, referred to as the total current. It includes the transport current and the field-induced current encircling the openings in multiply connected bodies. In the latter case, as was for the first time observed by Deaver and Fairbank (1961), the total current is quantized due to the flux quantization. The total current in superconductors plays a role similar to that of the transport current in conventional metals, where it is executed by conduction electrons. The latter obey Fermi-Dirac statistics and their energy equals the Fermi energy EF. This implies that the carriers of the transport current in normal metals, or in the N phase of superconductors, are “very hot”: their temperature, the temperature of the ensemble of conduction electrons, equals the Fermi temperature TF ¼ EF/kB  104 K (kB is the Boltzmann constant), i.e., it is about the same as the sun surface temperature. In striking contrast, the charge carriers (Cooper pairs) of the total current in the S phase are “deadly cold.” Since their spins are zero, the pairs obey Bose-Einstein statistics and, since the temperature of the ensemble of the paired electrons Tcp is zero, they form the Bose-Einstein condensate (BEC). This (zero temperature) leads to the disappearance of the thermoelectric effects, as observed in experiments. The amazing transformation of the “very hot” single electrons to the “zero-temperature” electron pairs can be compared with the fact known from the relativity theory: two flying apart massless photons form a massive pair located in their center of mass. Therefore, the total current in superconductors represents the transport current in BEC, known on the properties of superfluid helium. Correspondingly, the total current set in a closed superconducting circuit continues running without energy dissipation provided speed of the charge carriers (density of the total current) is lesser of a definite critical value (Landau, 1941; Feynman, 1955). Thus, in the MW model the electromagnetic properties of superconductors caused by the total current21 resemble the properties of hypothetical perfect conductors. The hell-mark of the perfect conductors is irreversibly of the sample magnetic moment induced by the changing applied magnetic field; its direction is determined by the Lenz law whereas the magnetic moment caused by magnetization is always diamagnetic. On this reason the multiply connected bodies can never be in thermodynamic equilibrium and therefore neither the principal of the free energy minimum nor the time reversal symmetry can be applied to them. However, it should be stressed that even in the presence of the total current, magnetic properties of superconductors in the MW model are different from those of the perfect conductors. Since the total current is accompanied by its own magnetic field, it represents a combination of the whirl and translational motion of the paired electrons regardless of the presence or absence of the applied field. This explains the observation reported by Meissner and Ochsenfeld in their fundamental paper of 1933 (Meissner and Ochsenfeld, 1933): “When the parallel superconductors are connected end-to-end in series and an external current is connected to flow through them above the critical temperature the magnetic field between the superconductors is increased below the transition temperature the external current being unchanged.”

Conclusion In this chapter we presented the basics of the micro-whirls model of superconductivity and a description of main electrodynamic properties as it follows from the model. It is important to underline, that the “big zeroes” of superconductivity listed in the abstract, as well as the zero generalized linear momentum (modified London’s rigidity), the zero average vector potential and the zero average current in the sample bulk, all rise from a single root: quantization of the angular momentum of electrons combined into Cooper pairs. It will not be redundant to note that the fact of zero temperature of Cooper pairs regardless of the sample temperature means that there is no fundamental limit in the critical temperature of superconductors, as it is really observed after the discovery of high-temperature superconductors by Bednorz and Müller in 1986 (Bednorz and Müller, 1986).

21 We are talking about the dc total current. Consideration of the ac current must include contribution of the unpaired electrons, which is significant at frequencies ≳ 103MHz.

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Acknowledgments The author is deeply grateful to Prof. Vladimir Kresin and Prof. Orest Simko for valuable comments after reading the manuscript. After the submission of this chapter, tragic news came: on July 28, Professor Kresin passed away. Vladimir Kresin was a brilliant physicist and real gentleman. This chapter is dedicated to his blessed memory.

References Aharonov Y and Bohm D (1959) Physical Review 115: 485. Bardeen J, Cooper LN, and Schrieffer JR (1957) Physics Review 108: 1175. Bednorz JG and Müller KA (1986) Zeitschrift für Physik B 64: 189. Cooper LN (1956) Physics Review 104: 1189. Deaver BS Jr. and Fairbank WM (1961) Physical Review Letters 7: 43. Ehrenberg W and Siday RE (1949) Proceedings of the Physical Society B 62: 8. Essmann U and Träuble H (1967) Physics Letters 24A: 526. Feynman RP (1955) Progress in Low Temperature Physics. vol. I, Amsterdam: Elsevier, 17. Fröhlich H (1950) Physics Review 79: 845. Ginzburg VL and Landau LD (1950) Journal of Experimental and Theoretical Physics 20(1064). Gorter CJ and Casimir HBG (1934) Physikalishce Zeitschrift 35: 963. Kamerlingh Onnes H (1911) KNAW Proceedings 13: 1274. 14: 113 (1911). Keesom WH and Kok JA (1934) Physica I: 503. Kikoin IK and Goobar SV (1938) C. R. Academy of Sciences USSR 19: 249. Journal of Physics USSR 3: 333 (1940); reprinted in “I. K. Kikoin - Physics and Fate”, Ed. S. S. Yakimov, p. 145, Nauka, Moscow, 2008. Kozhevnikov V, Valente-Feliciano A-M, Curran PJ, Richter G, Volodin A, Suter A, Bending SJ, and Van Haesendonck C (2018) Journal of Superconductivity and Novel Magnetism 31: 3433. Landau LD (1941) Journal of Physics/Academy of Sciences of the USSR 5: 71. London F (1960) Superfluids. vol. I. N.Y: Dover. Maxwell JC (1873) A Treatise on Electricity and Magnetism, 2nd edn. vol. II. Oxford: Clarendon Press. Meissner WZ (1927) Ges. Kältenindustr. 34: 197. Meissner W and Ochsenfeld R (1933) Naturwissenschaften 21: 787. Poisson M (1824) Memoire sur La Theorie Du Magnetisme. Lu a l’Academie Royale des Sciences. 2 Fevrier. Rjabinin GN and Shubnikow LW (1934) Nature 134: 286.

Further reading Details of the micro-whirls model with necessary references to experimental and theoretical works, and a broader description of the superconducting properties are is available in Kozhevnikov V (2021) Meissner effect: History of development and novel aspects. Journal of Superconductivity and Novel Magnetism 34: 1979–2009. A perfect review of experimental and theoretical works up to 1952 is available in Shoenberg D (1962) Superconductivity, 2nd. edn. Cambridge: University Press. For fundamentals of electrodynamics we recommend (these textbooks complement but not replace one another) Landau LD, Lifshitz EM, and Pitaevskii LP (1984) Electrodynamics of Continuous Media, 2nd edn. Oxford: Elsevier. Purcell EM (1985) Electricity and Magnetism, 2nd edn. Boston: McGraw-Hill. Griffiths DJ (2017) Introduction to Electrodynamics, 4th edn. Cambridge: University Press. Tamm IE (1979) Fundamentals of the Theory of Electricity, 9th edn. Moscow: Mir (available online). Thermodynamics of superconductors is discussed in Kozhevnikov V (2019) Thermodynamics of Magnetizing Materials and Superconductors. Boca Raton: CRC Press.

Superconductivity: BCS theory KH Bennemann, Physics Department, FU-Berlin, Berlin, Germany © 2024 Elsevier Ltd. All rights reserved. This is an update of K.H. Bennemann, Superconductivity: BCS Theory, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 72–81, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00701-4.

Introduction BCS theory Novel superconductors Summary and outlook Conclusion Acknowledgments References Further reading

657 660 665 665 668 669 669 669

Abstract Since the discovery by Kamerlingh Onnes in 1911 superconductivity has remained an important phenomenon in solid state physics with continuing discoveries. The explanation of the superconductivity phenomenon was given by the most elegant BCS theory of Bardeen, Cooper, and Schrieffer in 1957. The formation of Cooper pairs results from an effective attractive phononmediated interaction between two electrons with opposite spin near the Fermi surface. The BCS theory provided justification to the successful phenomenological Ginzburg–Landau theory from 1950. We discuss the BCS theory, Josephson tunneling of Cooper pairs between tunnel junctions, and the McMillan equation for the superconducting transition temperature and use this for its pressure dependence and for superconductivity in amorphous metals. We also discuss the Eliashberg theory treating electrons and phonons in superconductors on the same footing. Furthermore, we also discuss the high transition temperature (cuprate) superconductors for which the Cooper pair glue is still debated and for which the repulsive coupling yields an order parameter of d-symmetry. Interesting behavior of various superconducting nanostructures (e.g., system of quantum dots) is discussed. This is a corrected and extended version of the chapter (Bennemann, 2005).

Key points

• • • • • • • •

The BCS theory provided the most elegant microscopic explanation of the superconductivity phenomenon. Formation of Cooper pairs results from an effective attractive phonon-mediated interaction between two electrons with opposite spin near the Fermi surface. The BCS theory provided a justification to the successful phenomenological Ginzburg–Landau theory. Josephson tunneling of Cooper pairs between tunnel junctions is discussed. The McMillan equation for the superconducting transition temperature is used for description of its pressure dependence and for explaining superconductivity in amorphous metals. The Eliashberg theory treats electrons and phonons in superconductors on the same footing. In the high transition temperature (cuprate) superconductors, the Cooper pair glue is still debated and the repulsive coupling yields a d-symmetry of the order parameter. Behavior of various superconducting nanostructures (e.g., system of quantum dots) is discussed.

Introduction For many years, superconductivity has remained one of the most interesting phenomena in physics. After its discovery by Kamerlingh Onnes in 1911, many superconductors have been found, see Fig. 1. The discovery of MgB2 as a superconductor with a transition temperature Tc ¼ 39 K suggests that further surprises can be expected in the future. The basic interdependence of superconductivity and lattice structure still needs to be understood. Fig. 2 shows the transition temperature of some interesting metals. Some of the cornerstones in the history of superconductivity were the observation of (1) a vanishing electrical DC resistivity r(T ) at Tc, (2) the magnetic behavior exhibiting type-I superconductors expelling magnetic flux (Meissner–Ochsenfeld), and type-II superconductors where quantized magnetic flux can penetrate in filaments, (3) the isotope effect, Tc ∝ M−a, for example, a  0.5 for Hg, suggesting that the electron–phonon coupling might be responsible for superconductivity, and (4) Josephson tunneling with an electronic current j ¼ j0 + j1 sin½Df − ð2eV 21 =ℏÞt, where Df is the phase difference and V21 the voltage between two superconductors 1 and 2. There are many additional interesting observations including the occurrence of a gap in the electronic density of states (DOS),

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00255-9

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2s

lA

llA

3

4

Li

Be

11

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lVB

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VlB

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Cl

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Ti

V

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Mn

Fe

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37

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lllA

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llB

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Fig. 1 The occurrence of superconductivity in the periodic table is illustrated. As the history of superconductivity shows, alloys and compounds of the elements play an important role. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

30 Nb3Ge NbAlGe 20 Tc(K)

NbN

10 Pb

V3Si Nb3Sn

NbO Nb

Hg 1910 1920 1930 1940 1950 1960 1970 1980 Year Fig. 2 History of the transition temperature Tc for the first 70 years following the discovery of superconductivity in 1911. The A-15 compounds were of particular interest in the search for higher Tc superconductors. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

gapless superconductors, and the interdependence of atomic structure and Tc, seen in particular in layered structures, and granular superconductors. Most important theories developed for a physical understanding of the superconductivity phenomena were: (1) the London theory in 1935 explaining theMeissner–Ochsenfeld effect by using for the superconducting current driven by the vector potential
the expression js ¼ − c=4pL2L A, with the penetration depth, lL ¼ (mc2/4pe2ns)1/2, where ns is the superfluid density (Abrikosov et al., 1963). (2) the Ginzburg–Landau theory in 1950, which extended the London theory by writing the free energy in terms of an order parameter c(r, t) ¼ |c|ei’(r) as, ( )   2 Z 2  b ℏ2  e⁎  + h , F ¼ F0 + d3 r ajcj2 + jcj4 + r − i A c (1)  2m ⁎  ℏc 8p 2 yielding the current via minimization with respect to c (or c∗): j¼ −i

ℏe ⁎ e ⁎2 ðc ⁎ rc − crc ⁎ Þ − ⁎ jcj2 A: ⁎ 2m m c

(2)

In 1957, the Bardeen–Cooper–Schrieffer (BCS) theory (Bardeen et al., 1957) [see also Parks (1969) and Schrieffer (1964)] derived superconductivity from an electronic Hamiltonian H ¼ H0 + Heph, where Heph describes the electron–phonon interaction. It is to be noted that Eq. (2) immediately yields the Meissner effect, by assuming that the superconducting wave function is rigid and is not

Superconductivity: BCS theory

659

affected by an electromagnetic field (thus, the first term in Eq. (2) is zero), and for c ¼ |c|eif(r) (cc⁎  ns and f are conjugate variables). The Josephson tunnel current between two superconductors 1 and 2 (with phase and voltage difference Df, DV) is given by   e⁎ jJ ¼ j0 sin Df − DVt (3) ℏ Here, Df ¼ ’1 − ’2 + j0 ¼

e ⁎ℏ jcj2 , m ⁎d

2p f0

Z

2 1

dl  A

ðm ⁎ ¼ 2m, e ⁎ ¼ 2eÞ

and d is a parameter referring to the tunnel junction, f0 ¼ hc/2e. Assuming the single-valuedness of c, one obtains the quantization of flux trapped (in a filament) in a superconductor (see gauge transformation c): f ¼ nf0 ,

n ¼ 0,  1, . . .

(4)

These are remarkable general results that hold for different possible mechanisms causing condensation of a Fermi liquid into a superconducting state. The BCS theory is one of the most elegant theories in physics; it effectively explains superconductivity resulting from Cooper pairs (k", −k#) for electrons with opposite spin and momentum k at the Fermi surface. As illustrated in Fig. 3, the exchange of virtual phonons yields an attractive interaction and thus a singlet Cooper pairing. This pairing causes a gap in the single-electron DOS, reflecting the binding energy of the Cooper pairs, D ¼ DðT, h, . . .Þ . The theory correctly predicts the behavior and some other thermodynamical and electrodynamical quantities. Following the BCS theory, the quantum field theory using the Green’s function method, in particular by Gor’kov et al., gave the theory of superconductivity a very elegant form. In general, for phase transitions (ferromagnetism, etc.), the superconducting state cannot be obtained from the normal state by using the perturbation theory. That the electron–phonon interaction described in second-quantization notation by X
 gkk0 l c+k0 ck dl ðqÞ + d+l ð −qÞ + ⋯ (5) Heph ¼ k,k0,l is responsible for superconductivity was suggested earlier by Fröhlich and others and explains, for example, the isotope effect. Here ck and c+k represent the electron annihilation and creation operators, while dl ðqÞ and d+l ðqÞ are the corresponding annihilation and creation operators for phonons of momenta q ¼ k0 − k. The electron–phonon coupling constant is given by X ℏN   0 a k0 jri U ia jk E l ðaÞeiqra (6) gkk0 l ¼ 2o M ql a i,a where Uia is the potential of the ion i, a and E ql are the lattice polarization vectors. Since, as illustrated in Fig. 3, emission and absorption of phonons by electrons at the Fermi surface can lead to an effective attractive interaction, it seemed possible that Cooper pairing (k", −k#) occurs for electrons within an energy shell of 2oD, where oD is the Debye frequency. On general grounds, one expects, for an attractive potential with a range of the order of the interatomic distance (a  pF−1 ), due to the Pauli principle, singlet pairing and pairing of k and −k electrons to avoid kinetic energy of the Cooper pairs and thus to obtain maximal condensation energy. To avoid the destructive repulsive electron–electron interaction, the pairing results from a retarded potential.

g

k2

•

k2+q

Dl(q,w)=

k1

~ g

2wqO 2 w 2 – w qO

k1–q

Fig. 3 Illustration of the interaction between electrons with k1 and k2 via exchange of a phonon with polarization l, wave vector q and energy  oql; oql ’ ek1−q − ek1; then an attractive effective electron–electron interaction g2D < 0 results, if o < oql, g ¼ gG, where G denotes many-body corrections to the electron–phonon coupling. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

660

Superconductivity: BCS theory

The electron k" polarizes the lattice for a time of the order of t  oD−1 , which is slower than the electron motion, and then the other electron −k#, arriving at the polarized lattice within that time might feel an effective attractive electron–electron interaction due to over-screening. The resultant binding energy 2D causes a correlation and coherence length of the order of x  vF/2D. It is to be noted that Coulomb fields are screened at a distance of the order of the interatomic distance. This analysis and the study by P Cooper, showing that for electrons at the Fermi surface the pair wave function c ¼ kkF ak eikr , r ¼ r1 − r2 lowers the energy (ðH0 + V Þc ¼ Ec and V < 0), gave a more definite clue that the Cooper pairing of the electrons within an energy shell of the order of 2ℏoD causes superconductivity. Cooper pairing is the essence of the electronic BCS theory, also for the novel superconductivity of the cuprates like YBCO and HgBCCO, which gives the microscopic justification of the phenomenological Ginzburg–Landau theory. Its formulation was achieved with the help of quantum field-theoretical methods (Tinkham, 1975).

BCS theory The BCS theory begins by using the Hamiltonian H¼

X X xk nks + V kk0 a+k" a+−k# a −k0 # ak0 " k ,s k,k0

(7)

with excitation energies xk ¼ ek − m for electrons near the Fermi energy (m is the chemical potential) and the effective interaction Vkk0 between electrons with wave vectors k and −k, and spin " and #. Note that the remaining interactions between the electrons are assumed to be included in the energies xk. a+ks and aks are the usual creation and annihilation operators for fermions with momentum k and spin s, and nks is the particle number occupation operator. The BCS theory uses the wave function, Q jci ¼ ðuk + vk a+k" a+−k# Þjc0 i, k (8) juk j2 + jvk j2 ¼ 1, to calculate the free energy. The ground state is |c0i. jvk j2 gives the probability of a pair (k", −k#). The diagonalization of H is most elegantly achieved by the Bogoliubov and Valatin canonical transformations b −k+ ¼ uk a−k# + vk a+k" ,

bk − ¼ uk ak" − vk a+−k# ,

u2k + v2k ¼ 1,

(9)

which introduce new fermion operators bks and b+ks. Due to isotropy, the coefficients uk and vk depend only on k and can be chosen to be real and such that, for a given entropy, the electronic energy E is minimal. Then one gets straightforwardly (with Vkk0 ¼ −g/V), " # " !#2 X X
2 X g X 2 2 E¼ xk 2vk + uk − vk N ks − u k vk 1 − Nks , (10) V s¼ s¼ k

k

where Nks ¼ b+ksbks and, from ∂E/∂uk, the result,


 2xk uk vk ¼ Dk u2k − v2k :   Here, using u2k + v2k ¼ 1 and Eq. (9), the order parameter (Dk ∝ a − k# ak" ), X gX Dk ¼ u k vk 1 − Nks V s¼

(11) ! (12)

k

has been introduced. The order parameter can be expressed as   E k0 g X Dk0 tanh V 0 2E k0 2kB T k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E k ¼ x2k + D2k :

Dk ¼

(13)

It is of interest to note that Eq. (13) has solutions not only for attractive interaction (with g > 0) but possibly also for repulsive ones with g < 0. Then the k-dependence of Dk is important. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 The new quasiparticle energies are E k ¼ ðdE=dN k Þuk ,vk ¼ x2k + D2k . Their distribution function is Nk ¼ ðeE k =kB T + 1Þ . These P quasiparticles contribute to the energy the term k,s E k Nks. It is to be noted that 2Dk is the binding energy of the correlated electrons. Thus, the coherence length x ¼ ℏvF =2D yields the correlation distance, for most conventional superconductors, a much larger value than the interatomic distance ℏ=pF . Note that for an isotopic gap Dk ¼ D one gets

Superconductivity: BCS theory 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 x2k + D2 gX 1 A: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh@ 1¼ V 2kB T 2 2 k 2 xk + D

661

(14)

and from this, for D(T ), the expression  1=2 T DðTÞ ¼ 3:2kB T c 1 − : Tc

P R oD  The superconducting transition temperature Tc follows from D(Tc). Thus, k ! −oD de , T c ’ 1:14ℏoD e −1=ðNð0ÞgÞ ∝M −1 2Dð0Þ ¼ 3:52: kB T c

(15)

(16)

As shown by McMillan (1968), Bennemann and Garland (1972), and others, Eq. (16) needs to be improved to calculate realistic Tc values for simple metals, transition metals, A-15 compounds, alloys, etc., and to understand the interdependence of superconductivity and atomic structure better. Taking into account more details of the electron–phonon coupling, the phonon spectrum, the retardation of the pairing potential, and the remaining repulsive electron–electron interaction (∝ m⁎, pseudo-Coulomb potential), one gets the McMillan equation (McMillan, 1968):

1:04ð1 + lÞ o T c ’ D exp − 1:45 l − m ⁎ ð1 + 0:62lÞ (17) Z 1 2 a ðoÞFðoÞ l¼2 : do o 0 Here, the effective coupling constant l depends on the DOS, N(0). First, l ∝ N ð0Þ, then l saturates and decreases again as N(0) increases such as in transition metals. Note that l ¼ (m /m) − 1. Typically m  0.1, but m may also change strongly near magnetic instabilities. The DOS for the quasiparticles is given by e N s ðeÞ ¼ Nn ðeÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 : 2 e −D

(18)

This follows from the one-to-one correspondence of old and new quasiparticles, N s ðeÞde ¼ N n ðxÞdx . For the thermodynamic potential O, which is the appropriate function for a system with given m, all thermodynamic properties can be calculated using the formulas,



D2 ∂O ∂g T,V,m ¼ −V g2 Z D 2 dð1=gÞ Os − On ¼ dD0 D0 dD0 0

(19)

This gives, in particular at Tc, the specific heat (Fig. 4), cs − cn ’

4mpF Tc, 7zð3Þℏ3

cs ðTÞ ∝ e −D0 =kB T ,

(20)

where z(x) is the Riemann–Zeta function, V ¼ 1. To obtain the behavior in electromagnetic fields, one uses the recipe p !p − (e/c)A, P p ! − iℏr, where A denotes the vector potential (A ¼ q aqeiqr). This yields the coupling, eℏ X ðvk A + A vk Þ 2mc k eℏ X k aq c+k+q,s cks : ¼− mc k, q

Hint ¼ i

(21)

Here, the spin is unchanged since A couples dominantly to the orbital motion of pairing electrons. (Note that q  a ¼ 0 for tranversal fields.) Thus, one obtains, as in the Ginzburg–Landau theory for the electric current due to A (j ¼ env) for Cooper pairs (m ! 2m, e ! 2e), j¼− P

ne2 A + j2 , mc

(22)

where the paramagnetic current is j2 ¼ eℏ=m k,q ka+k −q ak , which would vanish if the superconducting wave function c is rigid, that is, not affected by A. It is to be noted that, since ns is the density of Cooper pairs, the first term gives the famous London equation

662

Superconductivity: BCS theory

Hk

C(T ) 'C

'

Ns(w )

k

0 (b)

–'

0

Tc

T

Nn(0)

kF

(a)

1

'

w

(c) Fig. 4 Illustration of (A) the gap D in the elementary excitation spectrum, (B) the jump (CS/Cn)Tc ¼ 2.43 in the specific heat, and (C) the electronic DOS. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

ns e2 c A A¼− mc 4pl2 mc2 : l2 ¼ 4pe2 ns j1 ¼ −

(23)

Hence, by determining the penetration depth l of an external magnetic field h into a superconductor, one may calculate the superfluid density ns, an important quantity reflecting whether all electrons at the Fermi surface pair. Evidently, if x > l, Eq. (22) gives the Meissner effect. Using the response theory, jq ¼ −

c K a , 4p q q

(24)

where Kq ¼ l−2(1 + l2Kq0 ) and Kq0 is calculated within the BCS theory. The two characteristic length scales x and l suggest thatpthere exist two types ffiffiffi pffiffiffiof superconductors characterized by k ¼ l/x. As shown by Abrikosov, they correspond to types I (k < 1= 2 ) and II (k > 1= 2 ). Superconductors of type I repel an external magnetic field, which however may penetrate via an array of quantized flux filaments into type-II ones. For these, one expects vortex lines with D ’ 0 around them. The superconductor compromises both effects in order to profit from the Cooper-pair condensation energy and the magnetic field energy. It is to be noted that the formation of a vortex costs an energy of the order (xh2/8p) due to a variation of D(r) or the wave function c, and reduces the (diamagnetic) field energy by about (lh2/8p). Fig. 5 characterizes the behavior of superconductors in an external magnetic field. From the general formula for the free energy, Fs ¼ Fn −

h2 , 8p

(25)

assuming no field penetration into the superconductor, one gets a critical magnetic field, Hc1 below which a pure Meissner superconductor is present. Above Hc1, the magnetic flux f may penetrate until Hc1 ¼ f/px2 (f ¼ flux of area px2). As already noted, the flux f is quantized due to the macroscopic phase of the superconductor as a whole. Note that for fields h such that pffiffiffi Hc1 < h < Hc2 one gets a mixed state, with Hc2 ¼ 2kHc1. Surfaces may favor Cooper-pair condensation. Then there exists another critical field Hc3 above which superconductivity at the surface is destroyed (Hc3 ’ 1.7Hc2). Obviously, strong magnetic fields may also affect Cooper pairing by destroying the singlet state. This is of relevance for the interdependence of superconductivity and magnetism in antiferromagnets or ferromagnets. As a result of internal molecular fields acting on the electron spins, one may get a spatially inhomogeneous superconducting state, D ¼ D(k, r), and then again a compromise to profit from both Cooper pairing energy and magnetic energy. The spin–orbit coupling and scattering by local spins (paramagnetic impurities) also affect the singlet pairing. Scattering by P paramagnetic impurities with spins via exchange coupling (J), Hex ¼ −J iSi s, will, of course, weaken the singlet Cooper pairing. The interesting analysis by Abrikosov and Gor’kov yields

Superconductivity: BCS theory

Superconductor

663

Normal metal –M

Bint = 0

B

B

(a)

Hc1 Hcth

0

Hc2

(b)

H

j

H

n-phase

jc

HC3 HC2

Normal

Superconducting

Mixed phase

H

HC1

Hc s-phase Tc

T

T (d)

(c)

Fig. 5 (A) Meissner effect for type-I superconductors where, below the superconducting transition temperature Tc, the magnetic flux B is abruptly expelled; (B) Typical behavior of type-II superconductors, where, below Bc2, the flux penetrates, and perfect Meissner effect behavior occurs below Bc1; (C) dependence of type-I superconductor on B and j and (D) temperature dependence of critical magnetic fields. At Hc3, surface superconductivity disappears, B ¼ H + 4pM. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

 ln

Tc T c0

 ¼c

 

 1 1 ℏ 1 −c , + 2 2 4pt T c

(26)

where t is spin scattering time and c(x) ¼ G0 (x)/G(x) is the digamma function. The discovery of gapless elementary excitations was unexpected. The effects on the singlet superconducting state due to spin-mixing via spin–orbit coupling and spin-flip scattering via exchange interaction are observed through the temperature dependence of the Knight shift K(T ), for example, and cause K > 0 for T ! 0. Clearly, the spin splitting of the Fermi surface for spin " and # electrons also weakens pairing of k and −k electrons. For a better understanding of the phonon and electron dynamics due to Heph, one can use the Hamiltonian H ¼ H0 + Heph + Hel, where H0 refers to phonons and electrons in the absence of Heph, and Hel to the electron–electron interactions. The dynamic field operators c, c+, and ’ (in this context ’ represents the phonon field, which is ∝ d + d+ introduced earlier) are treated on the same level. This yields renormalizations of both phonon and electron excitations. The physical situation is illustrated in Fig. 6. Then one obtains a frequency-dependent complex superconducting order parameter D(o) and, for the DOS (see Fig. 7), o NT ðoÞ  Re qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 o − D2 ðoÞ

(27)

 Using matrix Green’s functions, the Nambu notation c+k ¼ c+k" c−k# , and Matsubara frequencies, one gets for Gk ðtÞ ¼ − hT t ck ðtÞc+k ð0Þi, and similarly for Dðq, tÞ, the Dyson equations shown in Fig. 6 where  G¼

G F+

 F , ~ G

 G0 ¼

One gets (for D ¼ (ionZ)2 − (ek + x)2 − f2, on ¼ (2n + 1)pT )

G0 0

 0 , ~0 G

 G0 ¼

D 0

0 ~ D



664

Superconductivity: BCS theory χ =

+

g

0

+

=

g

+

g

gel

gel

g

0

Fig. 6 Illustration of the dynamics of the coupled electron and phonon systems using Feynman diagrams and thermodynamic matrix Green’s functions G, D (Eliashberg-type theory (McMillan, 1968)). The graph(¼¼¼)refers to an additional electron–electron interaction and also to spin excitations (and impurity scattering) (g ¼ eph., gel ¼ el–el.). From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

1.14 0.8

1.12

0.6

Expt.

'(w )/w t1

1.10

NT (w )/N(0)

1.08 1.06

'2

'1

0.4 0.2 0

6 2

1

3

4

–0.2 1.04

7

5

(w–'0)/w t1

–0.4

1.02

–0.6

BCS

1.00 0.98 0.96 0.94

SSW 1

2

(w –'0)/w t1

3

4

5

Fig. 7 Tunneling DOS with structure due to electron–phonon interaction. Also D(o) ¼ D1 + iD2 is given (ot: phonon frequency). From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

Gðk, ion Þ ¼

ion + ek + x , D

Fðk, ion Þ ¼

fðk, ion Þ , D

(28)

for the electron (G) and Cooper pair Green’s function (F), respectively. Note that D ¼ 0 yields the elementary excitations and D ¼ f/Z the order parameter with Z ’ 1. After continuation to the real axis, one obtains Z Nð0Þ oc 0 Dðo0 Þ do Re pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 , DðoÞ ¼ (29) ZðoÞ D0 o0 2 − D with U c ¼ V c =ð1=Nð0ÞV c lnðom =o0 Þ due to electron–electron interactions, (om  eF, o0—cutoff parameter) and (from phonon spectrum Ba(q, z)), Z Z P 2kF dgq 1 K ph ¼ dzBa ðq, zÞjgqa j2 2 a 0 2kF 0 (30)

 1 1 :  0 + o + o + z − id o0 − o + z − id These equations are the essence of the Eliashberg-type theory (McMillan, 1968) and produce, for a simplified phonon spectrum, the results obtained by Scalapino, Schrieffer, and Wilkins (SSW) (Scalapino et al., 1966)) shown in Fig. 7. This analysis is of utmost significance since it shows definitely that for BCS-type superconductors the electron–phonon coupling causes Cooper pairing. Note that phase-coherent Cooper pairs occur at Tc. The phonon frequencies are

Superconductivity: BCS theory

2 jgq j2 1 −1 , o2q ’ o0q + 2o0q V c ðqÞ eðq, oÞ

665

(31)

where e(q, o) is the dielectric function. Presumably, the Dyson equation illustrated in Fig. 6 also applies if D is replaced by the Green’s function of spin excitations, the dynamical spin susceptibility w(q, o), or of other modes that possibly cause Cooper pairing. Fig. 6 describes not only singlet but also triplet Cooper pairing such as in He3, and also suppression of Cooper pairing due to magnetic interactions; see “Further reading” section. Superconductivity in different materials, for example, isotropic lattices, layered lattice structures, nanostructures, and amorphous metals can be described adequately by taking into account details of the coupling constant gkk0 l and its renormalization due to vertex corrections (Ward identities) (Table 1).

Novel superconductors The essence of novel superconductivity in cuprates like LBCO, YBCO, and HgBCCO is the presence of singlet Cooper pairs with d-wave symmetry. Such a symmetry is expected for repulsive coupling g, see the discussion by Manske et al. (2000), as implied by the experimental results for the high superconducting transition temperatures ranging from 100 to 130 K and higher. Manske et al. (2000) have discussed in detail the phase diagram for the high transition temperature cuprates assuming spin fluctuation induced Cooper pairing and including their phase fluctuations expected for underdoped cuprates. Also for positive interaction g, it is argued how the gap equation provides a solution of the d-wave symmetry (Bennemann and Ketterson, 2003, 2014).

Summary and outlook Both BCS theory and its extension, including dynamical effects yielding the Abrikosov–Gor’kov and Eliashberg theory, explain superconductivity resulting from electron–phonon coupling. It still remains a challenge to calculate Tc from first principles and its dependence on atomic structure, lattice instabilities, and magnetic activity. The results shown in Table 1 were obtained by Garland et al. (1968) using the McMillan equation and assuming phonon softening. A very good agreement with experiment by Buckel and Hilsch (1954) was achieved. Changes in the atomic structure affect the phonon spectrum and the coupling constant gql and Tc, D, etc., as observed for disordered and amorphous metals. In transition metals and alloys thereof with large N(0), one may attempt to relate Tc to the cohesive energy, since g hrtiji t, where tij is the hopping integral for electrons (see also studies by Labbé et al. (1967)). For a study of pressure-induced superconductivity in narrow band metals (Bennemann and Garland, 1972) both changes in the phonon spectrum and coupling constant g may be important. Similarly, for transition metal alloys AxB1−x, the different bonds (AA, AB, and BB) may play a role in determining D(x) and Tc(x). For this, the BCS theory has been extended by Bennemann and coworkers (see Kerker and Bennemann, 1974), using the coherent-potential approximation. In case of strong anisotropy of the lattice structure as in layered structures, one may expect approximately two coupling constants gi (gi rti  e), different critical fields Hac2−b and Hcc2 , and two gaps (i ’ p, s, a − b in plane, c ? a − b) Di Zi ’ pT

X

lil − mil⁎ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dl ðmÞ: l,m o2m + D2l ðmÞ

(32)

Fig. 8 shows the results for MgB2. Note that hybridization of p and s electrons gives a common Tc. The interplay of magnetism and superconductivity suggested already by the occurrence of superconductivity in the periodic table is, for several reasons, of utmost interest. This has, for example, been studied for (film) structures N1 |N2 | N3. Results are given in Fig. 9. Organic superconductors display, in particular, interesting dimensional structural effects and those due to charge-wave and magnetic instabilities. Also, astonishingly, superconductivity may be induced by strong magnetic fields. In Fig. 10, the interesting phase diagram of k-(BEDT-TTF)2-Cu[N(CN)2]Br, a typical composition of a charge-transfer salt, is shown. Such molecules are stacked into layers. Planes of Cu[N(CN)2]Br anions separate these layers. Within these layers, the molecules form dimers and Table 1

Estimates of the superconducting transition temperature Tc in disordered and amorphous metals.

Material

(Tc/ Tc0)expt.

(Tc/ Tc0)calc.

Al Pb Ga Sn In

5 0 8  1.3  1.3

4.9 0 8 2 1.2

Courtesy of W. Buckel et al.

666

Superconductivity: BCS theory

c a

b Mg B

(a)

Magnetic field (T)

20 Hè

15 10 5

H^

0

10

20

30

40

Temperature (K)

(b)

Fig. 8 (A) Structure of MgB2 (AlB2, etc.). The boron planes seem to play an important role regarding superconductivity and Cooper pairing. The Mg–B bonds are k ? softer than the B–B ones. (B) Critical upper magnetic fields Hc2 and Hc2 referring to the B-planes and direction perpendicular to it, respectively. The anisotropic behavior of Hc2(T ) suggests two-gap behavior. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, pp. 72–81. first ed., Amsterdam: Elsevier.

1 2d N1

N2

P N3

AP

0.5 Tc /Tco LO

jT (a)

(b)

0

h (dF/dS)

Fig. 9 (A) Illustration of a tunnel junction N1/N2/N3, where Ni may refer to singlet and triplet superconductors and to normal-state metals, including ferromagnets. The tunnel current jT may refer to single electrons or Cooper pairs and transport of charge and spin. Depending on the thickness, 2d, of the tunnel medium proximity effect and Andreev reflection at S/N interfaces plays a role. (B) Phase diagram of a (F1/S/F3) sandwich. S refers to a singlet superconductor and F1, F3 to ferromagnets with parallel (P) or antiparallel (AP) orientation of their magnetization. LO is the Larkin–Ovchinnikov phase, h the exchange field and Tc and Tco, the superconducting transition temperature in the presence and absence, respectively, of the exchange field. dF and dS refer to the thickness of the ferromagnetic and superconducting film, respectively. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

electrons or holes can then hop easily from one molecule to the next one, but not perpendicular to the layers. The salts consist of conducting layers separated by a nonconducting environment (anions). Clearly, bond-length changes, also due to pressure, will sensitively change the electrical properties. Also, a magnetic field perpendicular to the layers (along the c-axis) affects the system due to the formation of Landau levels, which will pass the Fermi energy as the external magnetic field varies (cf. de Haas–van Alphen oscillations). In stronger magnetic fields quantum mechanical interband tunneling also occurs. Note that superconductivity occurs at relatively low temperatures, usually upon applying pressure. Experiments indicate unconventional Cooper pairing (2D0  7kBTc, non-s-symmetry of Dk). Whether phonons are involved in Cooper pairing must be clarified by further analysis. In an external magnetic field, interesting behavior, such as Landau-level formation and spin-split bands, may result due to the layered structure. Possibly, Cooper pairs ((k", −k+q#)) may form. A Larkin–Ovchinnikov (LO) state with Dk  cosðk  rÞ may occur. In general, the response to an external magnetic field is highly anisotropic. Furthermore, it is interesting that superconductivity may

Superconductivity: BCS theory

Paramagnetic insulator

Temperature (K)

40

667

Metallic

30 Antiferromagnetic/ superconductor 20 Antiferromagnetic 10

Unconventional superconductivity

0 0

200

400 Pressure (bar)

600

Fig. 10 Phase diagram of k-(BEDT-TTF)2 Cu[N(CN)2]Br, an organic superconductor, from AC susceptibility and nuclear magnetic resonance. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

be induced in l-(BETS)2FeCl4 by a magnetic field h, which destroys long-range magnetic order of the Fe3+. Field-induced superconductivity may also occur in a-(BEDT-TTF)2KHg(SCN)4 by affecting the charge density wave (CDW) with the external magnetic field h. Further experiments are needed; however, rich behavior may be expected in general since CDW and SDW excitations, Landau levels, and 2D properties are present. Superconductivity in small particles (radius R) due to electronic level discretization ðd2  eF =n  ðℏvF =RÞ ðkF RÞ −2 , x  ðRd=kT c ÞðRkF Þ2 Þ, and charge fluctuations, Q ¼ en , Q  e, as a result of single electron and Cooper pair hopping, exhibit interesting behavior. The charging of small particles is controlled by electrostatic energy. The change of the charge Q ¼ en ! Q  e causes an energy change e2/2C, where C is the capacitance of the small particle, leading to a Coulomb blockade (in tunneling, for example). At temperatures T ’ 0, due to the electrostatic energy, it is (DQ ¼ eDn) En ¼

Q2 Q − C0 V, 2C C

(33)

Q (2e)

(C and C0 are capacitances of a system including voltage contact and V is the external potential), and the blockade is periodically lifted as a function of n and V (En+1 ¼ En). In the superconducting state, one has for T ’ 0 that D > e2/2C. The situation is illustrated in Fig. 11. From En+1 + D ¼ En and En+2 ¼ En, n even and large sea of Cooper pairs, a period doubling results, with respect to the normal state. Note that the equation En+2 ¼ En might not hold for smaller density of Cooper pairs. In view of the strong quantummechanical behavior of small particles, structures consisting of a larger number of small particles seem very interesting (mesoscopic systems).

2e n

n+2

2 Δ> e 2C

V (eV) Fig. 11 Charging-up behavior of a small superconductor with capacitance C and charge Q ¼ en, n even, in a potential V (D > e2/2C). This behavior follows from En ¼ En+2 and reflects Cooper pair formation. One also has En ¼ En+1 + D, where En+1 refers to a state with one unpaired electron. Then Q changes by e at DV ∝ (2n + 1). From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

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1

Tc /zEJ 0.5 d

O

zEJ /Ec 0

1

2

3

4

Fig. 12 Transitions in an ensemble of superconducting quantum dots controlled by the Josephson-energy EJ and the Coulomb-energy Ec ¼ e2/2C. Tc refers to the global superconducting transition temperature of the ensemble. Phase O refers to a globally ordered superconducting state of the ensemble and phase d to superconducting grains with no global Cooper pair phase coherence, z is the coordination number. From Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier.

An ensemble of grains for illustration may exhibit transitions to superconductivity at Tc1, where the single grains get superconducting, and at Tc2, where the whole ensemble becomes superconducting (Tc2 Tc1) globally and phase coherently; see Fig. 12. Generally, an ensemble of grains may resemble an alloy-like array of S/S and S/N junctions (S ¼ superconducting, N ¼ normal state grains). The behavior of an ensemble of grains depends, of course, on the coupling and electron hopping between the grains. Also the structural order of the grains plays a role. A regular lattice-like arrangement of the grains (one-dimensional or two-dimensional topology) is of particular interest. For small distances between the cluster particles, electron tunneling and inelastic Cooper pair tunneling (Josephson coupling) occur. Hence, charge fluctuations are present in such granular superconductors. The resultant interesting Josephson effect can be described by P H ¼ HT − Eg ði, jÞ cos Fij i,j (34) 2e2 X + ðN i − Nj Þ2 + ⋯ C i,j The first term refers to normal electron hopping between the grains, the second term to Cooper pairs tunneling between neighboring superconducting grains (Josephson effect), and the last term is the electrostatic energy due to different charges of neighboring grains i, j. For simplicity, the same capacitances C and EJ (Josephson energy) are used. The phase difference Fij ¼Fi −Fj refers to the phases Fi of the superconducting order parameter Di ¼ jDi jeiFi of grain i. It is important to note that [F, N] ¼ i, where N is the Cooper-pair number operator.

Conclusion In conclusion, BCS superconductivity has been overviewed. While the basis of the BCS theory is well developed, details of the physics may have interesting consequences that have not been understood yet. New important discoveries can be expected. The relationship of phase-coherent and, in particular, phase-incoherent Cooper pairing with the Bose–Einstein condensation (BEC) is of fundamental interest. One expects, in particular, for more local pairing, perfect Boson behavior of the two paired electrons and thus an intimate relationship between Cooper pairing and BEC. Meissner-type superconductivity and BEC should result only for phasecoherent Cooper pairing. The size of the Cooper pairs is of the order of the coherence length. For small superconductivity density, one may reach a state (also possibly achieved geometrically, e.g., in narrow pipes, granular superconductors, etc.) where the singulet Cooper pairs do not overlap and behave like Bosons and a BEC transition is expected (Bennemann and Ketterson, 2014). Near the crossover transition, anomalous behavior may occur, for example, in cuprate underdoped superconductors due to phase fluctuations. Applying the Heisenberg uncertainty relation, one gets that the product of phase and occupation is  h (Bennemann and Ketterson, 2014). While for the cuprates the glue for Cooper pairing Phonons or Spin fluctuations is still debated, it seems that the essential structure of BCS theory is valid (namely Cooper pairs, Josephson current, order parameter equation with g positive or negative) (Bennemann, 2005). Various properties of conventional and unconventional superconductors have been discussed. The general calculation of the superconducting transition temperature remains a problem. Also a discussion of other Cooper pairing mechanisms (spin fluctuation besides phonon-induced superconductivity) as well as triplet pairing like in He-3 is an open issue.

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Acknowledgments The author is grateful to F. Nogueira, J. W. Garland, J. Ketterson, W. Buckel, M. Avignon, L. Tewordt, M. Peter, R. Parks, D. Manske, I. Eremin, J. Schmalian, R. Schrieffer, J. Bardeen, and many more for helping him in understanding superconductivity. A helpful visit at the Aspen Institute of Physics is also gratefully acknowledged. The author is grateful to V. M. Fomin and F. Nogueira, who prepared this Chapter for publication.

References Abrikosov AA, Gor’kov LP, and Dzyaloshinski IE (1963) Quantum Field Theory in Statistical Physics Englewood Cliffs. NJ: Prentice-Hall Inc. Bardeen J, Cooper LN, and Schrieffer JR (1957) Physical Review 106: 162–164. Physical Review 108, 1175–1204. Bennemann KH (2005) Superconductivity: BCS theory. In: Encyclopedia of Condensed Matter Physics, 1st edn., pp. 72–81. Amsterdam: Elsevier. Bennemann KH and Garland JW (1972) In: Douglass DH (ed.) Superconductivity in d- and f-Band Metals, AIP Conference Proceedings, vol. 4, pp. 103–137. Bennemann KH and Ketterson JB (2003) The Physics of Superconductors. Berlin: Springer. Bennemann KH and Ketterson JB (2014) Novel Superfluids. Oxford: Oxford Science Publications. Buckel W and Hilsch R (1954) Zeitschrift für Physik 138: 109–120. Garland JW, Bennemann KH, and Mueller FM (1968) Physical Review Letters 21: 1315–1319. Kerker G and Bennemann KH (1974) Solid State Communications 14: 399–401. Labbé J, Barišic S, and Friedel J (1967) Physical Review Letters 19: 1039–1041. Manske D, Eremin I, and Bennemann KH (2000) Physical Review B 62: 13922. McMillan WL (1968) Physical Review 167: 331–344. Parks RD (1969) Superconductivity. New York: Marcel Dekker. Scalapino DJ, Schrieffer JR, and Wilkins JW (1966) Physical Review 148: 263–279. Schrieffer JR (1964) Theory of Superconductivity. New York: Benjamin. Tinkham M (1975) Introduction to Superconductivity. New York: McGraw-Hill.

Further reading General literature on: Magnetism, history of; Quantum Mechanics: Methods; statistical mechanics: Classical; superconductivity: General aspects; superconductivity: Ginzburg-Landau theory and vortex lattice; superconductivity: Tunneling; superconductors, high Tc. Bassani, F., Liedl, G. L., Wyder, P. (Eds.) Encyclopedia of Condensed Matter Physics. 1st edn., Elsevier, Amsterdam. Brandt EH (2023) Superconductivity: Ginzburg-Landau theory and vortex lattice. In: Encyclopedia of Condensed Matter Physics, 2nd edn. Oxford: Elsevier. Chubukov A (2023) Superconductivity: electronic mechanisms. In: Encyclopedia of Condensed Matter Physics, 2nd edn. Oxford: Elsevier. Gurevich A (2023) Superconductivity: Critical currents. In: Encyclopedia of Condensed Matter Physics, 2nd edn. Oxford: Elsevier. Kozhevnikov V (2023) Electrodynamics of superconductors. In: Encyclopedia of Condensed Matter Physics, 2nd edn. Oxford: Elsevier. Mazin I (2023) Superconductors, conventional vs unconventional. In: Encyclopedia of Condensed Matter Physics, 2nd edn. Oxford: Elsevier.

Quantum fluctuations in superconducting nanowires Andrei D Zaikina and IE Tammb,c, aInstitute for Quantum Materials and Technologies, Karlsruhe Institute of Nanotechnology (KIT), Karlsruhe, Germany; bDepartment of Theoretical Physics, P.N. Lebedev Physical Institute, Moscow, Russia; cNational Research University Higher School of Economics, Moscow, Russia © 2024 Elsevier Ltd. All rights reserved.

Introduction Superconducting fluctuations Quantum phase slips Quantum phase transition Phase-charge duality “Superconducting” regime: Dissipative transport and noise “Insulating” regime: Localization of Cooper pairs Conclusion References

670 671 672 674 675 676 678 680 681

Abstract Low temperature properties of superconducting nanowires are determined by quantum fluctuations. Both Gaussian fluctuations of the superconducting phase—associated with plasma modes propagating along the wire—and non-Gaussian fluctuations of the order parameter—quantum phase slips—are equally important being responsible, e.g., for a “superconductor-insulator” quantum phase transition governed by the wire diameter. Thicker (“superconducting”) nanowires demonstrate non-vanishing resistance and shot noise of the voltage caused by quantum phase slips. Thinner (“insulating”) nanowires become truly insulating only at scales exceeding a non-perturbative localization length for Cooper pairs, whereas at shorter length scales they still exhibit superconducting properties.

Key points

•

Microscopic theory of superconducting fluctuations, proliferation of quantum phase slips in superconducting nanowires, description of quantum properties of “superconducting” and “insulating” phases.

Introduction A vast majority of properties of bulk conventional superconductors is perfectly described by the standard Bardeen-Cooper-Schrieffer (BCS) theory operating with complex order parameter D ¼ |D| exp (i’) within the mean field approach (Tinkham, 1996). Superconducting fluctuations play little role here and, hence, macroscopic phase coherence usually remains well preserved over the whole sample. Unlike in bulk superconductors, fluctuations become far more important in lower dimension where—according to the well known Mermin-Wagner-Hohenberg theorem—fluctuations of the superconducting phase ’ should destroy the long-range order, i.e., long-range phase coherence. Fortunately, this formal observation does not yet imply that low dimensional systems cannot exhibit superconducting properties. This is because any generic superconducting system has a finite size in which case phase coherence can survive at least to a certain extent. For instance, two-dimensional (2D) superconducting films are known to undergo BerezinskiiKosterlitz-Thouless (BKT) topological phase transition. As a result, the decay of superconducting correlations in such films changes from exponential at higher temperatures T to power law at lower ones, implying that at low enough T long range phase coherence remains essentially preserved in samples of a finite size. Hence, generic 2D metallic films can and do turn superconducting. Does superconductivity also survive in quasi-one-dimensional (1D) wires or do phase fluctuations disrupt supercurrent in such systems? The answer to this question is of both fundamental interest and practical importance due to rapidly progressing miniaturization of superconducting circuits employed in a broad range of applications, such as, e.g., quantum nanoelectronics, quantum information and metrology. As we will see below, superconducting fluctuations play a crucial role in 1D wires and essentially determine both their transport and ground state properties. The behavior of superconducting nanowires turns out to be rich, complex and highly interesting.

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https://doi.org/10.1016/B978-0-323-90800-9.00151-7

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Fig. 1 Schematics of small fluctuations of the order parameter (upper panel) and the phase slip process (lower panel) in superconducting nanowires. During the latter process the absolute value of the order parameter D gets locally and temporarily suppressed due to thermal and/or quantum fluctuations while its phase ’ suffers a jump by 2p. Adapted from Semenov AG and Zaikin AD (2022) Superconducting quantum fluctuations in one dimension. Physics-Uspekhi 65: 945–983.

Superconducting fluctuations Thermal and quantum fluctuations may cause deviations of both the modulus and the phase of the superconducting order parameter D(x) (with x being the coordinate along the wire) from their mean field values. One can distinguish (i) small (Gaussian) fluctuations of the order parameter and (ii) non-Gaussian fluctuations, i.e., the so-called phase slips (Arutyunov et al., 2008; Bezryadin, 2013; Zaikin, 2010; Zaikin and Golubev, 2019). These two kinds of fluctuations are schematically illustrated in Fig. 1. The strength of fluctuation effects in superconducting nanowires is essentially controlled by two different parameters, dimensionless conductance gx and dimensionless admittance gZ. These parameters are defined as gx ¼ Rq =Rx ,

gZ ¼ Rq =Zw ,

(1)

where Rq ¼ 2p/e ’ 25.8 KO is the quantum resistance pffiffiffiffiffiffiffiffiffi unit, Rx is the normal state resistance of the wire segment of length equal to the superconducting coherence length x,Zw ¼ L=C is the wire impedance, L ¼ 1=ðpsN D0 sÞ and C are respectively the kinetic wire inductance (times length) and the geometric wire capacitance (per length), sN is the normal state Drude conductance of pffiffi the wire and s is the wire cross section. Both parameters (1) scale with the wire cross section as gx ∝ s and gZ ∝ s: Note that here and below for simplicity and compactness we set the Planck constant ħ, the Boltzmann constant kB and the speed of light c equal to unity. If necessary, these parameters can easily be restored at any stage of our analysis. In generic metallic nanowires the dimensionless conductance gx typically strongly exceeds unity. In this case one can demonstrate that small (Gaussian) fluctuations of both the absolute value and the phase of the order parameter yield only a small (∝1/gx) fluctuation correction to the BCS (mean field) value of this parameter D0. In particular, at T ! 0 one has (Zaikin, 2010; Zaikin and Golubev, 2019) 2

D ¼ D0 − dD0 ,

dD0 1 3=2   Gi1D : D0 gx

(2)

Here Gi1D is the so-called Ginzburg number (Larkin and Varlamov, 2005) which controls the fluctuation region in the immediate vicinity of the BCS critical temperature TC. Eq. (2) demonstrates that at low T the order parameter suppression due to Gaussian fluctuations may become significant only in extremely thin nanowires with Gi1D  1 and gx  1. It turns out that in the limit gx  1 Gaussian phase fluctuations in superconducting wires are practically decoupled from those plasma modes (the of |D|. Such phase fluctuations are controlled by the parameter gZ (1) and are intimately related topsound-like ffiffiffiffiffiffi so-called Mooij-Schön modes) propagating along superconducting wires with the velocity v ¼ 1= LC: As we will see below, the presence of such plasma modes is crucially important and to a large extent determines the physics of superconducting nanowires at low temperatures.

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Turning to non-Gaussian fluctuations of the order parameter illustrated in lower panel of Fig. 1, we observe that the phase slip process corresponds to temporal suppression of |D(x)| down to zero in some point x ¼ x0 inside the wire. As soon as the modulus of the order parameter |D(x0)| vanishes, the phase ’(x0) becomes unrestricted and can jump by 2p. After that |D(x0)| gets restored, the phase becomes single valued again and the system returns to its initial state accumulating the net phase shift 2p. According to the Josephson relation each such phase jump—depending on its sign—implies positive or negative voltage pulse _ dV ¼ ’=2e: In the absence of any bias current the average number of “positive” voltage pulses is exactly equal to that of the “negative” ones. Hence, the net voltage drop across the wire remains zero. Applying the current I ∝ |D|2 r ’ one creates nonzero phase gradient along the wire, thereby breaking the symmetry between “positive” and “negative” phase slips and making the former to prevail over the latter (or vice versa). As a result, the net voltage drop V across the wire now differs from zero. Hence, phase slips should cause non-zero resistance R ¼ V/I of 1D superconducting wires at temperatures below TC. At T just slightly below TC phase slips occur predominantly due to thermal fluctuations of the order parameter. Creation rate of such thermally activated phase slips (TAPS) and, hence, the wire resistance R below TC are determined by the activation exponent, i.e., RðT Þ ∝ exp ð −dF=T Þ, dF 

N0 D20 ðT Þ sxðT Þ, 2

(3)

where the free energy difference dF plays the role of an effective potential barrier for TAPS determined by the superconducting condensation energy N0D02/2 (N0 being the normal electron density of states) within the volume  sx(T ) of the wire segment where superconductivity is destroyed by thermal fluctuations. At temperatures close to TC Eq. (3) yields appreciable resistivity which was indeed detected in experiments performed on small superconducting whiskers with typical diameters  0.5 mm. However, as the temperature is lowered considerably below TC the TAPS rate decreases exponentially and no measurable wire resistance is predicted by Eq. (3) in the limit of low T. In this limit quantum fluctuations of the order parameter should take over. As we already indicated, significant non-Gaussian quantum fluctuations in such wires are quantum phase slips (QPS). Each QPS event involves suppression of the order parameter in the phase slip core and a winding of its phase around this core. This configuration describes quantum tunneling of the order parameter field D(x) through an effective potential barrier  dF. The probability of such tunneling process turns out to be negligible for thicker wires. However, recent progress in nanolithographic technique allowed to fabricate samples with much smaller diameters down to—and even below—10 nm. In this case the potential barrier dF ∝ s becomes much lower and the QPS tunneling rate increases dramatically enabling quantum phase slips to dominate low temperature behavior of superconducting nanowires.

Quantum phase slips While the TAPS rate at T close to TC can be estimated employing just a simple Ginzburg-Landau-type of approach, theoretical analysis of QPS effects requires a much more complicated theory describing full quantum dynamics of the superconducting order parameter field at any temperature down to T ¼ 0. This theory is constructed in the spirit of the celebrated Feynman-VernonCaldeira-Leggett influence functional technique (Zaikin, 2010; Zaikin and Golubev, 2019). b for electrons in a superconductor that includes a short range attractive BCS The starting point is the microscopic Hamiltonian H and a long range repulsive Coulomb interactions as well as electromagnetic fields. With the aid of the Hubbard-Stratonovich transformations one decouples these interaction terms and introduces the fluctuating order parameter field D and the fluctuating scalar potential V. Integrating out all electronic degrees of freedom, one arrives at the grand partition function Z expressed in terms of the path integral Z   ^ Z ¼ Tr exp −H=T ¼ D2 D DV D A e−Seff : (4) The gauge invariant effective action reads Seff ¼ −Tr ln G −1 + S0 ½V, A, D:

(5)

Here A is the electromagnetic vector potential and 0 G −1

1 mv 2s i C B ∂t + xðrÞ − ieF + 2 − 2 fr, vs g |D| C, ¼B A @ 2 mvs i |D| ∂t − xðrÞ + ieF − − fr, vs g 2 2

where x(r)  − r2/2m − m + U(x) describes a single conduction band with quadratic dispersion and also includes an arbitrary impurity potential U(x) and the curly brackets denote the anticommutator. We also introduce the gauge invariant linear combinations   1 2e ’_ F ¼ V − , vs ¼ r’ − A , (6) 2m c 2e which define the so-called chemical potential of Cooper pairs F and the superconducting velocity vs. Quantum phase slip configuration represents a nontrivial saddle point of the action (5). It is a formidable task to explicitly find this saddle point for the full nonlinear effective action Seff. Obviously, this task can hardly be accomplished in practice. On the other

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hand, the problem becomes simpler provided one is interested in estimating the QPS effective action Sqps up to a numerical prefactor of order one. For that purpose we can resort to several approximations. Specifically, we perform a perturbative expansion of the action up to the second order in F, vs and the fluctuating part of the order parameter dD(x, t) ¼ |D(x, t)| − D0 around its equilibrium value D0. In addition, we average over the random potential of impurities. After this step the effective action becomes translationally invariant both in space and in time. Integrating out V-and A-fields we identically rewrite our partition function and the effective action only in terms of the fluctuating variables dD and ’. Then performing the Fourier transformation with respect to both x and t, expanding the action in powers of o and q2 up to terms o2q2 and rewriting the action again in the space-time domain, we obtain  Z    2 C ∂’ 2 1 ∂’ + 2 Seff ’ dxdt 8e2 ∂t 8e L ∂x  2 2 psN s ∂ ’ + sN 0 dD2 + (7) 64e2 D0 ∂x∂t  2  2 ) sN 0 ∂dD psN 0 D ∂dD + + , 8D0 ∂x 12D20 ∂t where D is the normal state diffusion constant. The first two terms under the integrals of Eq. (7) describe phase fluctuations corresponding to Mooij-Schön plasma modes, the third term  sN ðr’_ Þ2 has to do with “gapped” dissipation inside the wire and is reminiscent of that describing capacitance renormalization due to retardation effects in Josephson junctions, while the last three terms account for the loss of the condensation energy due to fluctuations of |D|. It is convenient to separate the action for a single QPS Sqps into a core part Score and that originating from outside the core Sout, which depends on the hydrodynamics of the electromagnetic fields, i.e., Sqps ¼ Score + Sout :

(8)

Let us denote the QPS core size as x0 and the (imaginary) time duration of the QPS event as t0. At this stage both these parameters are not yet known and remain to be determined from our subsequent analysis. We first evaluate the hydrodynamic part Sout. Outside the QPS core only the phase variable ’ fluctuates both in space and in time ~ ðx, tÞ corresponding to a single QPS event should satisfy the identity while |D| ¼ D0 does not. Then the saddle point solution ’ ~ − ∂t ∂x ’ ~ ¼ 2pdðt, xÞ ∂x ∂t ’

(9)

implying that after a wind around the QPS center (assumed to be located at t ¼ 0 and x ¼ 0) the phase should change by 2p. We observe that QPS is just equivalent to a vortex in space-time with the phase distribution ’(x, t) described by the saddle point solution ~ ðx, tÞ ¼ −arctan ðx=vtÞ: ’

(10)

Substituting the solution (10) into the action (7), we obtain Sout ¼ lln½minðL,v=T Þ=max ðx0 ,vt0 Þ,

(11)

where the parameter l ¼ gZ/8 sets the scale for the hydrodynamic part of the QPS action, L is the wire length. Let us now turn to the core term Score. As we are anyway aiming at evaluating this term up to a numerical prefactor of order one, without loss of generality we may approximate the order parameter field inside the QPS core by two simple functions   dDðx, tÞ ¼ D0 exp −x2 =2x20 −t2 =2t20 , (12) ’ðx, tÞ ¼ −ðp=2Þ tanh ðxt0 =x0 tÞ:

(13)

These functions obey all necessary requirements, the most important of which are: (i) the magnitude of the order parameter D(x, t) vanishes at x ¼ 0 and t ¼ 0 and coincides with the mean field value D0 outside the QPS core at |x| ≳ x0/2, |t| ≳ t0/2 and (ii) the phase ’(x, t) flips at x ¼ 0 and t ¼ 0 in a way to provide the change of the net phase difference across the wire by 2p. Note that the choice of the trial functions (12), (13) is by no means restrictive for our purposes, almost any other sufficiently smooth function obeying the above requirements can be used to estimate Score within the required accuracy. Substituting the functions (12), (13) into the action (7), setting sN ¼ 2e2N0D and minimizing the resulting expression for Score with respect to x0 and t0, we arrive at the final results both for the core parameters pffiffiffiffiffiffiffiffiffiffiffiffi x0  x ¼ D=D0 , t0  1=D0 (14) and for the core action pffiffiffiffiffiffiffiffiffi Score ¼ ag x  N0 s DD0 :

(15)

Note that during this minimization procedure we neglected the contribution Sout (11) to the total QPS action Sqps (8). This is appropriate because the two contributions to the QPS action depend differently on the wire thickness, i.e., Score ∝ N and pffiffiffiffiffi Sout ∝ N , where N ¼ p2F s=4p defines the total number of conducting channels in the wire and pF is the Fermi momentum. Hence, for generic metallic wires with N  1 one usually has Score  Sout. In addition, one can verify that the results (14) and (15)

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hold for not too long wires with lengths L ≲ xN : The latter condition is satisfied in a vast majority of experiments with superconducting nanowires. We also point out that minimization of the “condensation energy” contribution dD2 (the last three terms in Eq. (7)) alone pffiffiffiffiffiffiffiffiffi already yields the correct estimate for Score  N0 s DD0 : This observation, however, by no means implies that phase fluctuations terms in Eq. (7) can be neglected. Just on the contrary, phase fluctuations yield exactly the same order-of-magnitude estimate for Score provided one keeps “gapped” dissipation term  sN ðr’_ Þ2 with the Drude expression sN ¼ 2e2N0D. At the same time, e.g., ignoring this term by formally setting sN ! 0 in Eq. (7) during our minimization procedure would immediately yield meaningless results for x0 and t0 as well as the QPS action parametrically different from that in Eq. (15). The result (15) allows to recover the QPS tunneling amplitude gqps within the exponential accuracy. Evaluating the contribution of fluctuations around the QPS saddle point one gets (Zaikin and Golubev, 2019)     gqps  gx D0 =x exp −ag x , (16) where gx is supposed to be much larger than one and the prefactor a  1 can be regarded as a fit parameter, which can be extracted, e.g., from the available experimental data. The QPS amplitude (16) will play a prominent role in our subsequent analysis.

Quantum phase transition Although the hydrodynamic part of the action (11) and the parameter l do not enter directly into the expression for the QPS amplitude gqps (16), they also plays an important role, as they determine interactions between different quantum phase slips mediated by plasmon modes propagating along the wire. Consider the configuration of n phase slips essentially equivalent to n vortices in space-time with the corresponding core coordinates (xn, tn). Provided the phase slip cores do not overlap with each other the core contributions are independent and simply add up. In order to evaluate the hydrodynamic part we substitute the superposition of the saddle point solutions for n quantum P ~ i ðx − xi , t − ti Þ into the action. Then we get SQPS(n) ¼ nScore + Sint(n), where phase slips ’ ¼ ni ’  X rij F X ðnÞ + 0 I ni ti : (17) Sint ¼ −l ni nj ln x0 c i i6¼j

Here rij ¼ (v (ti − tj) + (xi − xj) ) defines the distance between the i-th and j-th QPS in the (x, t) plane and ni ¼  1 are QPS topological charges which fix the sign of the phase change after a wind around the QPS center. In Eq. (17) for simplicity we assumed x0  vt0 and included an additional term which keeps track of the total current I possibly flowing across the wire. We observe that different quantum phase slips interact logarithmically in space-time attracting (repelling) each other in the case of opposite (equal) topological charges. The grand partition function of the wire Z is represented as a sum over all possible QPS configurations: Z L 1   Z X 1 y 2n L dx1 dx2n ... Z¼ x x0 2n! 2 0 x0 x0 n¼0 (18) Z 1=T Z 1=T X ð2nÞ dt1 dt2n  ... e−Sint , t0 t0 n ¼1 t0 t0 2

2

2 1/2

i

P where y ¼ x0t0gqps is an effective QPS fugacity. The sum in Eq. (18) runs over neutral QPS configurations with 2n i ni ¼ 0, which is a direct consequence of the boundary condition ’(x, t) ¼ ’(x, t + 1/T ) in the corresponding path integral for Z: In the limit I ! 0 Eqs. (17) and (18) define the standard model for a 2D gas of logarithmically interacting charges ni. Making use of the standard BKT renormalization group (RG) analysis of logarithmically interacting 2D Coulomb gas we arrive at the scaling equations for the interaction parameter l and the charge fugacity y: ∂L l ¼ −4p2 l2 y2 ,

∂L y ¼ ð2 − lÞy:

(19)

Here L ¼ ln (r/x0) is the scaling parameter. Following the standard line of reasoning we immediately conclude that a quantum phase transition (QPT) for phase slips occurs in a long superconducting wire at T ! 0 and l ¼ 2 + 4py ’ 2:

(20)

This is nothing but BKT-like phase transition for topological charges ni in space-time (Zaikin and Golubev, 2019). The difference from the standard BKT transition in 2D superconducting films is that in our case the transition is driven by the interaction parameter pffiffi l ¼ gZ =8∝ s and not by temperature. For thicker wires with l > 2 (corresponding to the “ordered” phase) QPS with opposite topological charges are bound in pairs (dipols) thereby limiting the effect of quantum fluctuations, whereas in thinner ones with l < 2 (“disordered” phase) the density of free (unbound) QPS always remains nonzero implying that long range phase coherence gets destroyed in this case. Below we will demonstrate that this QPT can be interpreted as a superconductor-insulator phase transition with rather non-trivial features of both “superconducting” and “insulating” phases.

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Phase-charge duality Before we turn to this issue let us establish an important property of superconducting nanowires which is the so-called phase-charge duality. To begin with, we note that duality between the phase and the charge variables is a well known property of Josephson junctions. In particular, under a certain duality transformation the effective actions for Josephson tunnel junctions in the phase and in the charge representations are exactly transformed onto each other (Schön and Zaikin, 1990). Furthermore, in the absence of an external circuit the Josephson junction can be described in terms of an effective Hamiltonian for a “quantum particle” in the periodic potential in the (quasi)-charge space. Within this picture, the charge Q and the flux F are canonically conjugate variables fully analogous to the momentum and coordinate variables in quantum mechanics. Similar duality ideas can be worked out and applied to superconducting nanowires (Semenov and Zaikin, 2022). To this end, let us for simplicity assume that our superconducting wire is isolated from any external circuit and ignore for a moment all kinds of fluctuations of the absolute value of the order parameter including, of course, quantum phase slips. In this case only phase fluctuations remain and the wire effective action with a high accuracy can be described by the terms in the first line of Eq. (7). Then the effective Hamiltonian of the wire reduces to that of a transmission line  2 # Z L " ^2 ^ ðxÞ Q ð xÞ 1 ∂x ’ ^ TL ¼ + dx H , (21) 2C 2e 2L 0 ^ ðxÞ and ’ ^ ðxÞ are canonically conjugate local charge and phase operators obeying the commutation relations where Q

 ^ ðxÞ, ’ ^ ðx0 Þ ¼ −2iedðx − x0 Þ: Q

(22)

These operators can also be expressed as ^ ðxÞ, ^ ð x Þ ¼ 1 ∂x ^ ^ ðxÞ ¼ 2eF Q wðxÞ, ∂x ’ F0 ^ ðxÞ represent some new (dual) operators. Combining Eqs. (21) and (23), we obtain where ^wðxÞ and F  2 # Z L "^2 F ðxÞ wðxÞ 1 ∂x ^ ^ HTL ¼ dx + : 2L F0 2C 0

(23)

(24)

^ ðxÞ is the local flux operator, while ^ The physical meaning of both dual operators is transparent from Eqs. (23) and (24), i.e., F wðxÞ is proportional to the operator for electric charge passed through the point x. Let us now include quantum phase slips into our consideration. With the aid of a rigorous calculation one can demonstrate that the total effective wire Hamiltonian in the presence of QPS takes the form Z L ^ TL − gqps ^ wire ¼ H dx cos ð^ wðxÞÞ, (25) H 0

  ^ TL is defined in Eq. (24). Evaluating the partition function Z ¼ Tr exp −H ^ wire =T with the Hamiltonian (25) and where H expanding in powers of gqps one immediately recovers Eq. (18). Let us now take a look at Fig. 2 displaying two complementary superconducting devices. One of them is an arbitrarily long superconducting nanowire surrounded by the vacuum or an insulator (upper part of the figure). In the device depicted in the lower part of the figure the superconductor is interchanged with the vacuum/insulator, thus forming a spatially extended Josephson junction between two superconductors. A superconducting nanowire in Fig. 2 is described by the Hamiltonian (25), whereas the Hamiltonian corresponding to a Josephson junction (of length L) in Fig. 2 has the well known form !2 # Z L Z L " ^2 ^ ðxÞ j Q ðxÞ 1 ∂x f ^ ðxÞ, ^ dx + dx cos f (26) HJJ ¼ − C 2CJ 2e 2e 0 2LJ 0 ^ ðxÞ are the local charge and phase difference operators, CJ and LJ represent respectively the Josephson junction where Q^ðxÞ and f capacitance and inductance per unit junction length and, finally, jC is the Josephson critical current density. Under the transformation of the operators ^ ðxÞ $ Q^ðxÞ, F

^ ð xÞ ^ wðxÞ $ f

(27)

^ JJ are exactly dual to each other provided we interchange ^ wire and H the Hamiltonians H F0 $ 2e,

gQPS $

jC , 2e

L $ CJ ,

C $ LJ :

(28)

The above duality transformations—on one hand—interchange magnetic and charging energies in these two Hamiltonians and—on the other hand—establish the correspondence between the last term in Eq. (25) describing the effect of QPS and the Josephson coupling energy in the second line in Eq. (26) that accounts for Cooper pair tunneling across the junction.

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Quantum fluctuations in superconducting nanowires

Fig. 2 Dual tunneling processes for a quantum fluxon (that tunnels through a superconducting nanowire) and for a Cooper pair (which tunnels across a Josephson junction). Reproduced from Arutyunov KYu, Lehtinen JS, Radkevich A, Semenov AG, and Zaikin AD (2021) Superconducting insulators and localization of Cooper pairs. Communications Physics 4: 146. This work is licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons. org/licenses/by/4.0/.

Superconductor

Fig. 3 The system under consideration. Adapted from Semenov AG and Zaikin AD (2022) Superconducting quantum fluctuations in one dimension. PhysicsUspekhi 65: 945–983.

Hence, tunneling of a Cooper pair with charge 2e between two superconductors is a dual process to a QPS event that can also be viewed as tunneling of a flux quantum F0 across a superconducting wire. Indeed, the last term in the Hamiltonian (25) contains a     linear combination of creation ei^w and annihilation e −i^w operators for the flux quantum F0. Each QPS event corresponds to the R _ net phase jump by 2p associated with voltage pulse dV ¼ ’=2e and magnetic flux |dV(t)| dt ¼ F0 passing through the wire in the direction normal to its axis, as it is illustrated in Fig. 2.

“Superconducting” regime: Dissipative transport and noise Let us examine the properties of superconducting nanowires in the “ordered” phase l > 2 where QPS are bound in pairs and, hence, phase fluctuations are largely suppressed. These properties turn out to differ drastically from those of bulk superconductors (Semenov and Zaikin, 2022; Zaikin and Golubev, 2019). Let us consider the circuit depicted in Fig. 3. It consists of a superconducting nanowire, an external capacitance C switched in parallel to this wire and the voltage source Vx applied via resistor Rx. The voltage V(t) at the left end of the wire x ¼ 0 fluctuates and can be measured by a detector while its right end (x ¼ L) is grounded. An effective Hamiltonian for this system can be written in the form  2 ∂ ^¼H ^ wire −I’=2e + 1 −i H +Q , (29) 2C ∂ð’=2eÞ ^ wire is defined in Eqs. (24) and (25) and describes the superconducting wire, whereas the second and third terms in where the term H the right-hand side of Eq. (29) account respectively for the potential energy tilt produced by an external current I ¼ Vx/Rx and for the charging energy. The variable ’(t)  ’(0, t) represents the phase of the order parameter field D(x, t) at x ¼ 0 and the charge Q equals to Q(t) ¼ w(0, t)/F0. Here we also set ’(L, t)  0. In order to proceed it is convenient to make use of the Keldysh path integral technique, thus defining our variables of interest on the forward (F) and backward (B) time branches of the Keldysh contour, i.e., we now have ’F, B(t) and wF, B(x, t). We also introduce

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“classical” and “quantum” variables, respectively ’+(t) ¼ (’F(t) + ’B(t))/2 and ’−(t) ¼ ’F(t) − ’B(t) (and similarly for the w-fields). Making use of the Josephson relation between the voltage and the phase one can formally express the expectation value for the (symmetrized with respect to time moments) product of n voltage operators in the form

’_ + ðt 1 Þ’_ + ðt 2 Þ . . . ’_ + ðt n ÞeiSqps 0 , hV ðt 1 ÞV ðt 2 Þ . . . V ðt n Þi ¼ (30) ð2eÞn where Z Sqps ¼ −2gqps

ZL dx sin ðw+ Þ sin ðw − =2Þ

dt

(31)

0

and Z h. . .i0 ¼

D2 ’ðt ÞD2 wðx,t Þð. . .ÞeiS0 ½’, w

(32)

^ ð29Þ with gqps ¼ 0. The above formally implies averaging with the Keldysh effective action S0 corresponding to the Hamiltonian H exact expressions describe complete statistics of voltage fluctuations across superconducting nanowires. Provided l > 2 the expectation values (30) can be evaluated perturbatively in gqps. In the zero order in gqps all averages can be handled exactly. Expanding Eq. (30) up to the second order in gqps and performing all necessary averages we arrive at the results for voltage correlators of an arbitrary order. In particular, for the expectation value of the voltage operator we obtain   F0 Lvg2qps 2 F0 I F0 I B sinh , (33) hV i ¼ 4 2 2T where BðoÞ ¼ tl0 ð2pT Þl−1

  l io G 2l − 2pT G 2 + GðlÞ

io 2pT



and G(x) is the Euler Gamma-function. Evaluating the wire resistivity r from the above equations we get ( 2 2l−3 gqps T , T  F0 I, r ¼ hV i=ðILÞ∝ 2l−3 2 T  F0 I: gqps I ,

(34)

(35)

These results imply that due to QPS effects charge transport across our nanowires remains dissipative even in the “superconducting” phase l > 2: The linear wire resistance tends to zero only at T ! 0, whereas its nonlinear resistance remains non-zero down to T ¼ 0. This effect remains weak only for sufficiently thick nanowires with very large values of gx (in which case gqps is vanishingly small) but becomes progressively more important for thinner nanowires. It is also worth pointing out that the key dissipation mechanism in this case is associated with Mooij-Schön plasma modes playing the role of an effective quantum bath for electrons inside the wire. Quantum phase slips cause not only non-vanishing wire resistance but also voltage fluctuations. In order quantify our description of such fluctuations we will analyze the voltage-voltage correlation function which can also be recovered from Eqs. (30)–(32) within the framework of the same perturbative in gqps approach. Let us define the noise power spectrum SO which perturbative expression consists of three different contributions: Z ð0Þ SO ¼ dteiOt hV ðt ÞV ð0Þi ¼ SO + SrO + SaO : (36) The first of these contributions SO(0) has nothing to do with QPS and just defines equilibrium voltage noise for a transmission line. The other two terms are due to QPS effects. In the zero bias limit I ! 0 the term SaO vanishes, and the equilibrium noise spectrum SO ¼ SO(0) + SrO is determined from the fluctuation-dissipation theorem. At non-zero bias values the QPS noise turns non-equilibrium. In the zero frequency limit O ! 0 the terms SO(0) and SrO tend to zero, and the voltage noise SO!0  S0 is determined solely by SaO which yields  F I S0 ¼ F0 coth 0 hV i, (37) 2T where hVi is specified in Eqs. (33) and (34). Hence, we obtain ( T 2l−2 , S0 ∝ 2l−2 I ,

T  F0 I, T  F0 I:

(38)

At higher temperatures T  F0I Eq. (38) reduces to equilibrium voltage noise S0 ¼ 2TR of a linear Ohmic resistor R ¼ rL. In the opposite low temperature limit T  F0I it accounts for QPS-induced shot noise S0 ¼ F0hVi obeying Poisson statistics with an effective “charge” equal to the flux quantum F0.

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Quantum fluctuations in superconducting nanowires

This result sheds light on the physical origin of shot noise in superconducting nanowires: It is produced by coherent tunneling of magnetic flux quanta F0 across the wire. As we have already learned, such flux quanta can be viewed as charged quantum “particles” (fluxons) passing through (and being scattered at) an effective spatially extended “tunnel” barrier which role is played by a nanowire. It is also interesting to analyze voltage noise produced by QPS at non-zero frequencies. In the limit of sufficiently high frequencies and/or long wires v/L  O  D0 we obtain O O coth 2T l ð0Þ : (39) SO ¼ 8pe2 ðO=2EC Þ2 + ðl=pÞ2 This contribution does not depend on the wire length L. At low T and O/l ≳ EC ¼ e2/2C we have SO(0) ∝ 1/O, i.e., in this regime the wire may generate 1/f voltage noise. Evaluating the QPS terms SrO and SaO we observe that the latter scales linearly with the wire length L whereas the former does not. Hence, the term SrO can be safely neglected in the long wire limit. For the remaining QPS term SaO we get SaO ¼

    Ll2 vg2qps F0 I F0 I B − O − B + O 2 2 4e2   F0 I F0 I sinh B 2T 2  :    O ðO=2EC Þ2 + ðl=pÞ2 sinh 2T

(40)

At T ! 0 from Eq. (40) we find ( SaO ∝

Il −1 ðI −2O=F0 Þl −1 , 0,

O < F0 I=2, O > F0 I=2:

(41)

In order to interpret this result we note that at T ¼ 0 each QPS event excites (at least) two plasmons with total energy E ¼ F0I propagating in the opposite directions along the wire. One of those plasmons (with energy E/2) gets dissipated at the grounded end of the wire while another one (also with energy E/2) reaches its opposite end causing voltage fluctuations (emits a photon) with frequency O measured by a detector. Obviously, at T ¼ 0 this process is only possible at O < E/2 in the agreement with Eq. (41). In order to complete our analysis of voltage fluctuations we remark that Eqs. (30)–(32) allow to also recover all higher order correlators (cumulants) of the voltage operator, thus enabling one to construct full counting statistics of quantum phase slips. In the zero frequency limit this statistics is Poissonian, and it becomes very complicated beyond this limit. To conclude, already in the “superconducting” regime l > 2 the nanowires exhibit physical properties dramatically different from those of bulk superconductors. Such unusual behavior includes not only non-vanishing resistance of superconducting nanowires down to T ¼ 0 but also coherent voltage fluctuations generated by quantum phase slips. In the presence of a current bias I quantum tunneling of magnetic fluxons across the wire causes Poissonian shot noise with a non-trivial power law dependence of its spectrum on both I and frequency O.

“Insulating” regime: Localization of Cooper pairs Let us now turn to the analysis of quantum fluctuations in superconducting nanowires in the “insulating” regime l < 2. For simplicity we start by considering two somewhat different configurations displayed in Fig. 4. One of them (left plot) corresponds to a nanowire in the form of a ring with a perimeter L ¼ 2pR. For brevity in what follows we will denote it as a QPS ring meaning that

Fig. 4 Left: Quantum phase slip ring, i.e., an ultrathin superconducting ring threaded by an external magnetic flux. Right: Quantum phase slip junction embedded in a thick superconducting ring. Reproduced from Semenov AG and Zaikin AD (2013) Persistent currents in quantum phase slip rings. Physical Review B 88: 054505.

Quantum fluctuations in superconducting nanowires

679

the nanowire remains thin enough for QPS effects to be pronounced there. Another configuration (right plot) also corresponds to a superconducting ring which now has thicker and thinner parts. Quantum fluctuations of the order parameter are negligible in the bulk part of the ring but proliferate inside its thinner part (a nanowire of length L) which we will call a QPS junction. In both cases the superconducting ring is pierced by an external magnetic flux Fx. For the sake of definiteness below we will only consider the case of a QPS ring, exactly the same results will also apply to a QPS junction. We first assume that the QPS ring perimeter L is small enough enabling us to disregard the coordinate dependence of our fluctuating variables of interest. Specifically, it implies that we can ignore the last term in the Hamiltonian (24). In this limit the effective Hamiltonian of our QPS ring that follows from Eqs. (24) and (25) takes the form.  2 ^−f ^ ¼ EL f − gqps L cos ^ w, (42) H x 2 ^ ¼ F=F ^ 0 and fx ¼ Fx/F0. where EL  gx Dx=L, f The Hamiltonian (42) describes a fictitious quantum particle on a ring in the presence of a cosine external potential. The solution of this problem is well known from any standard course in quantum mechanics. One can distinguish two different regimes by comparing the “kinetic” and “potential” energies EL and gqpsL or, equivalently, by comparing the ring perimeter L and the parameter (Zaikin, 2010)   (43) Lc0  x exp agx =2 : For smaller rings with L  Lc the flux-dependent ground state energy of the problem reads E0(fx) ’ (EL/2)minn(n + fx)2 except in the vicinity of the crossing points fx ¼ 1/2 + n where the gap to the first excited energy band dE01 ¼ gqpsL opens up due to level repulsion. Evaluating the supercurrent I flowing around the ring at T ! 0 by means of the formula I¼

e ∂E0 ðfx Þ , p ∂fx

(44)

one recovers the standard sawtooth-like dependence which—not too close to the points fx ¼ 1/2 + n—is not affected by QPS. For larger ring perimeters L ≳ Lc0 the bandwidth shrinks while the gaps become bigger. Then, combining the corresponding solution for the ground state energy E0(fx) with Eq. (44), at T ! 0 we obtain rffiffiffi L −3a4 gx −L=Lc0 e e : (45) I ¼ I0 sin ð2pfx Þ, I0  eg x D x This equation demonstrates that quantum phase slips yield exponential suppression of the supercurrent even at T ¼ 0 provided the ring perimeter L exceeds the critical value Lc0 (43). As we already pointed out, the above analysis holds only for sufficiently small values of the ring perimeter L. In a general case it is necessary to employ the full sine/Gordon Hamiltonian (24), (25) instead of the reduced one (42). This amounts to taking into account logarithmic interactions between quantum phase slips, cf., e.g., Eq. (17). This task can be accomplished by means of the RG approach employing (19). As before, assuming for simplicity that v/Dx0  x and starting renormalization at the shortest scale L  x we proceed to bigger scales. Ignoring weak renormalization of the parameter l we obtain the solution of Eq. (19) in the form gqps(L) ¼ gqps(x/L)l. Our RG procedure should be stopped at the scale corresponding to the ring perimeter L  L. As a result, we arrive at the renormalized QPS amplitude ~gqps ¼ gqps ðxc =LÞl :

(46)

This result allows to conclude that inter-QPS interaction effects remain weak and can be disregarded only for very small values of l  1/ ln (L/x) or, equivalently, for ring perimeters obeying the inequality   1 : (47) L  x exp l This inequality naturally restricts both the wire length and cross section values at which it can be treated as a zero-dimensional object described by the Hamiltonian (42). Substituting the renormalized QPS amplitude (46) instead of the bare one and extracting the correlation length Lc from the condition EL  ~gQPS L, we get (Semenov and Zaikin, 2022)  ag  x , (48) Lc  x exp 2−l where l is supposed not to exceed 2. We observe that the correlation length Lc (48) increases with increasing l and eventually diverges at the quantum phase transition point l ¼ 2. In the limit of small l  1/ ln (L/x) the parameter Lc reduces to Lc0 (43). As before, for smaller rings with L  Lc and at T ! 0 the supercurrent I is almost insensitive to QPS, whereas in the opposite limit L  Lc we again reproduce Eq. (45) for I, where now

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Quantum fluctuations in superconducting nanowires

Fig. 5 Localization of Cooper pairs in the ground state of a uniform superconducting nanowire due to strong zero-point fluctuations of magnetic flux. Reproduced from Arutyunov KYu, Lehtinen JS, Radkevich A, Semenov AG, and Zaikin AD (2021) Superconducting insulators and localization of Cooper pairs. Communications Physics 4: 146. This work is licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

I0  egx D

 1=2−3l=4 1 3a 1−l=2 R e− 4 gx e−ðL=Lc Þ : x

(49)

We observe that the supercurrent gets essentially suppressed by quantum fluctuations as soon as L exceeds Lc. The same conclusions hold also for superconducting nanowires displayed, e.g., in Fig. 3. For such wires we again apply the sine-Gordon Hamiltonian (24), (25) but, unlike in the case of a ring geometry of Fig. 4, we do not anymore impose any periodic boundary conditions at the wire ends. It is well established that for l < 2 the relevant excitations of our sine-Gordon theory are 1 ~ ∝ g2−l kinks and anti-kinks (as well as their bound states) with an effective gap in the spectrum D qps : This gap, in turn, gives rise to the ~ defined in Eq. (48). same correlation length Lc ∝ 1=D The appearance of this fundamental length scale in our problem is directly related to the flux-charge duality. It can be interpreted as a result of spontaneous tunneling of magnetic fluxons F0 back and forth across the wire, as it is illustrated in Fig. 5. These strong quantum fluctuations of magnetic flux wipe out phase coherence at distances ≳Lc and yield effective localization of Cooper pairs at such ~ as an effective Coulomb gap for the wire segment of length scales. Accordingly, one may interpret the energy  xDgx =Lc ∝ D length  Lc. To conclude, the properties of the “insulating” phase realized at T ! 0 in thinnest wires and rings with l < 2 turn out to be highly non-trivial. In this phase, quantum phase slips remain unbound and determine a nonperturbative localization length of Cooper pairs Lc beyond which the supercurrent gets suppressed by quantum fluctuations. Accordingly, for T ! 0 such nanowires turn insulating only at length scales exceeding Lc, whereas at shorter length scales they may still exhibit superconducting properties.

Conclusion In a sharp contrast to bulk superconductors, low temperature properties of quasi-one-dimensional nanowires are essentially determined by quantum fluctuations which are, in turn, controlled by two different parameters, the dimensionless normal state conductance of the wire segment gx and the dimensionless wire admittance gZ. Both these parameters (1) decrease with decreasing wire diameter, thus making quantum fluctuations progressively more pronounced. Provided gx remains very large, the low temperature behavior of the system is determined by small (Gaussian) quantum fluctuations of the phase of the order parameter controlled by the parameter gZ. Such fluctuations are directly related to sound-like plasma modes propagating along superconducting nanowires. As soon as the dimensionless conductance gx becomes not too large quantum phase slips come into play forming a logarithmically interacting “gas” in 1 + 1 (space + time) dimensions. Such interactions between different QPS are mediated by plasma modes and cause a “superconductor-insulator” quantum phase transition at T ! 0 and l  gZ/8 ’ 2. The “superconducting” phase l > 2 is characterized by bound QPS pairs and weaker decay of superconducting correlations in space-time. Nevertheless, even in this regime current-biased superconducting nanowires acquire a non-zero resistance and exhibit shot noise of the voltage. These phenomena can be interpreted in terms of tunneling of quantum fluxons across the nanowire. In the context of flux-charge duality, such fluxons (“charged” by the flux quantum F0) can be treated as effective quantum “particles” exactly dual to Cooper pairs with charge 2e. Such “particles” obey complicated full counting statistics which reduces to Poissonian one in the zero frequency limit. In thinner nanowires with l < 2 quantum phase slips remain unbound and superconducting correlations decay exponentially at length scales exceeding the size of a “superconducting domain” Lc (48). At such length scales and at T ! 0 superconducting

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nanowires exhibit an insulating behavior, whereas at shorter length scales they may demonstrate superconducting properties albeit possibly affected by quantum fluctuations of the phase. Hence, the correlation length Lc may be interpreted as an effective localization length for Cooper pairs. These and other non-trivial quantum properties of superconducting nanowires remain under intensive investigation and also form a basis for a variety of applications of such nanowires in quantum information technology, metrology and nanoelectronics.

References Arutyunov KY, Golubev DS, and Zaikin AD (2008) Superconductivity in one dimension. Physics Reports 464: 1–70. Bezryadin A (2013) Superconductivity in Nanowires. Weinheim: Wiley-VCH. Larkin AI and Varlamov AA (2005) Theory of Fluctuations in Superconductors. Oxford: Clarendon. Schön G and Zaikin AD (1990) Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions. Physics Reports 198: 237–412. Semenov AG and Zaikin AD (2022) Superconducting quantum fluctuations in one dimension. Physics-Uspekhi 65: 945–983. Tinkham M (1996) Introduction to Superconductivity. New York: McGraw-Hill. Zaikin AD (2010) uperconducting nanowires and nanorings. In: Sattler KD (ed.) Handbook of Nanophysics: Nanotubes and Nanowires, pp. 40-1–40-24. Boca Raton: CRC Press. Zaikin AD and Golubev DS (2019) Dissipative Quantum Mechanics of Nanostructures: Electron Transport, Fluctuations and Interactions. Singapore: Jenny Stanford.

Superconductor-ferromagnet hybrid structures F Sebastiana,b and Stefan Ilica, aCentro de Física de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, Donostia-San Sebastián, Spain; b Donostia International Physics Center (DIPC), Donostia-San Sebastián, Spain © 2024 Elsevier Ltd. All rights reserved.

Introduction Metallic S/F structures Manifestations of oscillatory condensate function in observables Odd-frequency triplet superconductivity in S/F structures The role of spin-orbit coupling Magnetic proximity effect Superconductor-ferromagnetic insulator structures F/S/F spin valves Non-equilibrium properties of S/F structures Conclusion References

682 683 684 686 687 687 688 689 689 691 691

Abstract This chapter presents an overview of the physics of superconductor-ferromagnet (S/F) hybrid structures. In addition to the important role in the development of superconducting spintronics, topological superconductors, and radiation detectors, these structures present a variety of exciting effects resulting from the interaction between superconductivity and magnetism. The superconducting proximity effect produces an oscillatory electron pair wave function inside the ferromagnet. This has several striking consequences, such as non-monotonic dependence of the critical temperature on the thickness of the ferromagnet and the “p” Josephson effect. Moreover, an odd-frequency triplet component of the superconducting condensate is generated in such systems. Since it is also even in momentum, the triplet is insensitive to non-magnetic disorder. When the magnetization of the ferromagnet is non-homogeneous, the triplets can penetrate over large distances in a ferromagnet even with a strong exchange field. We also discuss the inverse proximity effect in S/F structures, which induces a finite magnetic moment in the superconductor and may lead to spin-splitting of the density of states. We finish the chapter with an overview of the non-equilibrium properties of S/F structures.

Key points

• • • • •

The superconductor/ferromagnet (S/F) structures provide a platform to explore the interplay and coexistence of two antagonistic phenomena - ferromagnetism and superconductivity. The proximity effect at the S/F interface produces an oscillatory electron-pair wave-function inside the ferromagnet. This leads to several striking phenomena, such as non-monotonic dependence of the critical temperature on the thickness of the ferromagnet, and the “p” Josephson effect. An odd-frequency triplet component of the superconducting condensate is generated in S/F structures. Since it is also even in momentum, the triplet is insensitive to non-magnetic disorder. In the case when magnetization of the ferromagnet is non-homogeneous, the triplets can penetrate across large distances in the ferromagnet and are not suppressed even by strong exchange fields. Magnetism may be induced in a superconductor in a S/F structure - the magnetic proximity effect. This manifests as suppression of the critical temperature and the spin-splitting of the density of states of the superconductor. S/F structures also provide a good platform to explore novel non-equilibrium phenomena related to the coupling of different electronic degrees of freedom.

Introduction The interplay between superconductivity and exchange fields has been a topic of interest practically since the publication of the BCS theory in 1957. In a very early work, Abrikosov and Gor’kov (1960) demonstrated how a small amount of random magnetic impurities suppresses superconductivity, by breaking the local time-inversion symmetry due to the exchange field between impurities and conduction electrons. If the impurities are ferromagnetically aligned the situation is even more detrimental for conventional superconductivity, since the effective exchange field generated by the impurities tries to align the spins of the Cooper pair, which are in a singlet state. This leads to a pair-breaking mechanisms and hence suppression of the superconducting state.

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00130-X

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An interesting consequence of the interplay between superconductivity and an exchange field was predicted separately by Fulde and Ferrell (1964) and Larkin and Ovchinnikov (1965), who showed that the exchange field in a superconductor can lead to an inhomogeneous superconducting state, the so called FFLO state. This state is very sensitive to disorder and therefore very difficult to observe in conventional superconductors. There is, however, another platform that allows the study of the interplay between superconductivity and exchange fields at the mesoscopic scale: the hybrid superconductor/ferromagnet S/F structures, which show interesting emerging states. In the case of S/F metallic structures, the superconducting proximity effect allows superconducting pair correlations to penetrate into the ferromagnet, in analogy to the proximity effect in superconductor/normal metal systems. The difference with the latter is that the electrons of the pair, which in conventional superconductors are in a singlet state, in the ferromagnet experience a different potential depending on their spin. This means that at the Fermi level, each electron in the pair acquires a different momentum, and the total momentum of the pair is no longer zero. Quantitatively, in an ideal 1D case, the momentum at the Fermi level for spin-up (down) electrons is p
¼ pF
h/vF, where h is the exchange field in the ferromagnet, and vF and pF are the Fermi velocity and momentum, respectively. Thus, the total momentum acquired by the electron pair is Q  p+ − p− ¼ 2h/vF, which determines the oscillation period of the superconducting correlations in the ferromagnet, in a sense analogous to the FFLO state. In real metallic S/F structures, the oscillations of the superconducting correlations in the F layer is accompanied by an exponential decay away from the S/F interface. Specifically, the electron pair wave function has the form CðxÞ  e −x=xF ,

(1)

where x is the spatial coordinate normal to the S/F interface and xF is a complex length defined in a metallic system by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DF xF  : 2ðpT + ihÞ

(2)

weffiffiffiffiffiffiffiffiffiffiffiffi use ffithe units where ℏ ¼ kB ¼ 1. In a non-magnetic Here DF is the diffusion coefficient of the F metal, T the temperature, and p metal (N), h ¼ 0 and the decay of C into N is over the thermal length  DN =T , which at low temperature is only limited by inelastic scattering [see Fig. 1A]. In a ferromagnet h is finite and the pair correlation function oscillates and decays in space according to Eqs. (1), (2), see Fig. 1B. This results in a several remarkable phenomena discussed in detail in Section “Metallic S/F structures”: non-monotonic dependence of the critical temperature on the thickness of the F layers in S/F bilayers, oscillation of the local electronic density of states in S/F bilayers, and a phase shift of p in the current-phase relation of a S/F/S Josephson junctions, for certain thicknesses of the F-layer.

Metallic S/F structures Most of the experimental research on S/F structures is done on metallic systems. Typical superconductors are Al, Nb, V, whereas F layers consist of Fe, Ni, Co, Py, Ho, etc. All these materials can be well described using the diffusion equation, which in the case of superconductivity was derived by Usadel (1970). For a qualitative description of the proximity effect and calculation of observables, such as the critical current or the density of states, it is most often sufficient to consider the linearized Usadel equation. It is an equation determining the anomalous quasiclassical Green’s function, fb, describing the superconducting pair condensate in the F layer: n o −DF r2 fb + jon j + ih  ssgn on , fb ¼ 0: (3) Here on ¼ pT(2n + 1) are the Matsubara frequencies, h the exchange field in the ferromagnet, and s ¼ (sx, sy, sz) is the vector of spin Pauli matrices. The condensate function depends on the spatial coordinate and the frequency, fbðr, on Þ, and it is a matrix in spin space. Eq. (3) is valid when superconducting correlations in F are weak, which is true at large temperatures T  D, as well as for the case of a weak proximity effect due to non-ideal S/F interfaces. If these conditions are not met, the non-linear Usadel equation must be used. In order to determine the function fb, Eq. (3) needs to be supplemented with a boundary condition at the S/F interface ∂x fb ¼ −gf BCS :

(4)

Here, g is a parameter describing the transparency of the interface (g ¼ 0 corresponds to infinitely resistive interface, whereas g ! 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi describes a fully transparent interface), x is the direction perpendicular to the interface, and f BCS ¼ D= o2n + D2 :. One can solve Eq. (3) for the case of an homogeneous magnetization and a very long F layer (dF  xF), where we set h  s ¼ hsz without loss of generality. fb is then a diagonal matrix with elements 2

2

f
¼ gfBCS x
e −xxr =jx+ j eixxi =jx+ j ,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x
¼ DF =ð2jon j  2ihsgnon Þ ¼ xr
isgnon xi , and

(5)

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Superconductor-ferromagnet hybrid structures

Fig. 1 Schematic representation of the spatial dependence of the pair correlation function C in a (A) S/N structure, (B) S/F structure with homogeneous magnetization, and (C) S/F structure with inhomogeneous magnetization. In panel (C) SRT and LRT stand for short-range and long-range triplets, respectively.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u jon j2 + h2
jon j2 xr,i ¼ tDF   : 4 jon j2 + h2

(6)

Eq. (5) confirms the decay and oscillations of the condensate function in the F layer. For a strong ferromagnet with h  D, the pffiffiffiffiffiffiffiffiffiffiffi decaying length and oscillation period is xF  DF =h, which for usual ferromagnets (Fe, Co) corresponds to a sub-nanometer scale. This length scale is too short to be probed in an experiment, and therefore, one needs to use weaker ferromagnets with h ≲ D, or as we discuss below, to create pair correlations with equal spin projection.

Manifestations of oscillatory condensate function in observables The oscillations of the condensate function in the ferromagnet lead to several theoretical predictions successfully tested in experiments. For example, the critical temperature Tc of the S/F bilayer shows non-monotonic behavior as a function of the thickness dF of the ferromagnetic layer (Buzdin and Kupriyanov, 1990; Radovic et al., 1991). Furthermore, Tc can be calculated by combining the solution of the linearized Usadel equation with the self-consistency condition for the order parameter D. Fig. 2A shows the measurement of Tc by Lazar et al. (2000), which is in good agreement with theoretical predictions.

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8

(A)

Tc(K)

6

4

2 1 10

0

20 dFe(Å)

30

40

(B)

Normalized conductance

1.010

7 75 Å

6

1.005

5 1.000

4 50 Å

0.995

3

PdNi/Nb A

0.990

SiO

2

0.985 –4

2 Nb

B

1

–2

1

Al

0 2 Energy (meV)

4

0

(C)

80 IcRn(mV)

60

40

20

0 40

60

80

100

120

2df (Å)

Fig. 2 Main results of several experiments on S/F structures. (A) Measurement of non-monotonic critical temperature in Pb/Fe bilayers as a function of ferromagnet thickness dF. The solid and dashed lines are fits to the theory assuming a perfect and imperfect interface, respectively. (B) Measurements of the differential conductance in PdNi/Nb bilayer for different thicknesses of the ferromagnet PdNi, illustrating oscillating density of states. (C) Critical current of a S/F/S Josephson junction as a function of the thickness of the ferromagnet PdNi. The point where critical current vanishes (dF  65A˚ ) corresponds to the transition from “0” to “p” phase. (A) Adapted with permission from Lazar L, Westerholt K, Zabel H, Tagirov L, Goryunov YV, Garif’yanov N, and Garifullin I (2000) Superconductor/ ferromagnet proximity effect in Fe/Pb/Fe trilayers. Physical Review B 61(5): 3711. Copyright (2000) by the American Physical Society. (B) Reproduced with permission from Kontos T, Aprili M, Lesueur J, and Grison X (2001) Inhomogeneous superconductivity induced in a ferromagnet by proximity effect. Physical Review Letters 86(2): 304. Copyright (2001) by the Americal Physical Society. (C) Adapted with permission from Kontos T, Aprili M, Lesueur J, Genêt F, Stephanidis B, and Boursier R (2002) Josephson junction through a thin ferromagnetic layer: Negative coupling. Physical Review Letters 89(13): 137007. Copyright (2002) by the American Physical Society.

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Another example are the oscillations of the local quasi-particle density of states (DoS) N(E) (Buzdin, 2000), which can be calculated using Eq. (5) as N(E) ¼ N0 + dN(E), where dN ðEÞ ¼

 N0  2 Re f + + f 2− io !E : n 4

j

(7)

Here, N0 is the DoS in the normal state of the F layer, dN is the correction due to the superconducting proximity effect, and E is the energy. In S/N structures, induced superconductivity suppresses the density of states around the Fermi level, E ¼ 0. In other words, dN < 0. If one goes beyond the linearized limit, such suppression can be significant and induce a gap in the density of states of the N layer. In contrast, the spatial oscillations of the condensate in S/F structures can lead to different signs of dN, and hence to a situation where the DoS of S/F is higher than F alone at E ¼ 0 for some dF. This has been seen in the tunneling spectroscopy experiment by Kontos et al. (2001) [see Fig. 2B]. The oscillations of the superconducting condensate also manifest in the Josephson effect of S/F/S junctions. The Josephson effect establishes that the supercurrent can flow in the junction of two superconductors in the presence of a phase difference ’ between them IS ¼ Ic sin ’:

(8)

Eq. (8) is known as the current-phase relation, where Ic is the critical current of the junction. The ground state of the junction corresponds to a zero-current state, which is either at ’ ¼ 0 or ’ ¼ p. In conventional tunneling or S/N/S Josephson junctions, ’ ¼ 0 corresponds to a minimum of energy; therefore, it is the ground state of the junction. Moreover, in an S/N/S junction, Ic monotonically decreases as the N thickness increases. In a S/F/S junction, because of the oscillation of the condensate function, Ic has an oscillatory behavior. In the case of large exchange field h  T and large temperatures the critical current is given by Ic  e −dF =xF cosðdF =xF Þ,

(9) pffiffiffiffiffiffiffiffiffiffiffi where xF  DF =h: Therefore, depending on the thickness dF, Ic may change its sign. If Ic < 0, then ’ ¼ 0 corresponds to a maximum of the Josephson energy, which is now minimized by ’ ¼ p. Junctions with positive and negative Ic are called “0” and “p” junctions, respectively. The “0-p” transition in S/F/S junctions was first predicted by Buzdin, Bulaevskii, and Panyukov in 1982 (Buzdin, 2005; Buzdin et al., 1982) and observed in several experiments (see e.g. Ryazanov et al. (2001), Kontos et al., (2002)). In the former experiment, a ferromagnet CuxNi1−x has been used, whose exchange field is weak, h  Tc. In this case, the penetration of the condensate function in the F region is long-range. The complex length x+ depends on temperature and leads to oscillations of Ic as a function of the temperature, as observed in the experiment. Oscillations of Ic as a function of dF have also been reported in experiments [see Fig. 2C].

Odd-frequency triplet superconductivity in S/F structures In 2001, the same year that the 0-p transition was experimentally verified, Bergeret, Volkov, and Efetov (Bergeret et al., 2001, 2005; Kadigrobov et al., 2001) predicted the existence of a long-range triplet component of the superconducting condensate in S/F structures with inhomogeneous magnetization. Such component corresponds to equal-spin Cooper pairs, which are insensitive to the exchange field. To understand how this component appears it is sufficient to consider the linearized theory of the previous section. If the magnetization is homogeneous there is a single spin quantization axis, and the condensate matrix fb is diagonal with components f
, see Eq. (5). If the magnetization of the F layer is inhomogeneous, for instance due to magnetic domains and domain walls, the condensate matrix can be decomposed into singlet and triplet components: fb ¼ f s s0 + f t s,

(10)

where ft ¼ (fx, fy, fz) is the triplet condensate vector. In the previous example of a homogeneously magnetized ferromagnet, only the singlet and fz component are finite, with fs ¼ ( f+ + f−)/2 and fz ¼ ( f+ − f−)/2. The latter corresponds to the triplet component with zero total spin projection, i.e. to the Cooper pair state (" # + # "). It follows from Eq. (3) that: (i) The exchange field term always leads to singlet-triplet conversion. (ii) The singlet component is an even function of the Matsubara frequency, whereas the triplet component is odd in frequency. This is a consequence of the Pauli exclusion principle. Namely, the components of ft are even in momentum (s-wave symmetry) and even in spin (triplet state), and therefore they need to be odd in frequency. This type of superconductivity is denoted as odd-frequency triplet superconductivity. (iii) Since these odd-frequency triplets are even in momentum, they are insensitive to non-magnetic disorder (Anderson theorem). This is in contrast to triplets associated with unconventional superconductivity with p-wave symmetry, which are suppressed by disorder. field h are not affected by it. Namely, those components (iv) Components of the triplet vector ft orthogonal to the local exchange pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decay in F without oscillating, over the thermal length xT ¼ DF =ð2jon jÞ: In a ferromagnet with a strong exchange field, xT is much larger than the magnetic length defined in Eq. (5). Thus, this component is long-range [see the sketch in Fig. 1C].

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Long-range triplets correspond to Cooper pairs in a triplet state with finite spin projection, i.e. combination of ("") and (##). Because the spins of the electron pairs are parallel, they induce spin-polarized supercurrents in a S/F/S junction. Observation of long-range Josephson coupling attributed to triplet supercurrents has been reported in a wide variety of magnetic materials, including half-metals (Keizer et al., 2006; Singh et al., 2015), multilayered F-structure (Khaire et al., 2010), and magnets with conical magnetic texture such as Ho (Robinson et al., 2010).

The role of spin-orbit coupling The physical picture described above may be modified in an S/F system with spin-orbit coupling (SOC). Namely, SOC in diffusive systems leads to spin relaxation, which in the superconducting state manifests as an additional relaxation of the triplet component of the condensate. Specifically, the characteristic length over which the long-range triplet component of the condensate decays in a ferromagnet is given by (Demler et al., 1997) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 4 −1 , (11) xt ¼ jon j + DF tso where tso is the scattering time between collisions at spin-orbit impurities. In S/F systems with a lack of inversion symmetry, due to either non-centrosymmetric materials, or hybrid interfaces, SOC can be of the Rashba type. In such a case the effect of SOC can be introduced in the hdiffusion equation, Eq. (3), as a SU(2) vector potential, i b by substituting the derivatives by SU(2)-covariant ones: r ! f b  : This substitution leads to new terms describing A, r ¼ r −i A, precession and relaxation of the triplet component. In particular, the precession term may convert the short-range triplet component into a long-range one. Moreover, Rashba-like SOC in superconducting structures may lead to an additional source of singlet-triplet conversion. The SOC-mediated singlet-triplet conversion in combination with a finite magnetization leads to a variety of magnetoelectric phenomena in S/F structures, meaning that a spin response can be achieved by passing a current and vice-versa. Some prominent examples of such phenomena are the anomalous Josephson effect, where an arbitrary phase shift ’0 can appear in the current-phase relation, and the supercurrent diode effect, where the critical current depends on the direction of current flow.

Magnetic proximity effect While the superconducting proximity effect is responsible for the effects described above, the inverse proximity effect, or magnetic effect, is also important. Namely, magnetism may be induced in a superconductor in an S/F structure. The first manifestation of this phenomenon is the strong suppression of the superconducting critical temperature of thin S films in contact with F metal with a strong exchange field. However, superconductivity survives if the superconductor is thick enough or the exchange field is not particularly strong. The triplet correlations induced in the F layer may penetrate the S layer and change its spectrum. Indeed, as predicted by Bergeret et al. (2004) and verified experimentally by Xia et al. (2009), the triplet correlations induce a magnetic moment in the superconductor opposite to the average magnetization of the itinerant electrons in the ferromagnet. For example, the magnetic moment of an F particle embedded in a superconductor is screened via the inverse proximity effect, as illustrated in Fig. 3. Such screening can be understood by recalling the paramagnetic response of a superconductor. In the pure paramagnetic case a magnetic field B induces in a superconductor a magnetic moment:

Fig. 3 Ferromagnetic grain embedded in a superconductor. The magnetic moment of the grain is screened by the superconductor due to the magnetic proximity effect.

688

Superconductor-ferromagnet hybrid structures S ¼ w0 B + dSðT Þ:

(12)

The first term is the usual Pauli paramagnetic term, with w0 ¼ −2mBN0 and mB being the Bohr magneton. The second term is the deviation from the Pauli paramagnetic response due to the superconducting correlations. In a homogeneous case and at zero temperature dS(T ) ¼ −w0B, which reflects the fact that superconductors exhibit zero paramagnetic response at T ¼ 0. In the presence of SOC dS(0) 6¼ −w0B, and the superconductor may exhibit a finite paramagnetic susceptibility at T ¼ 0. In a hybrid S/F structure, the ferromagnet is usually described by free electrons in an effective exchange field h acting as a paramagnetic field. It is only finite in the F region and induces a local magnetic moment equivalent to the Pauli term in Eq. (12), proportional to N0h(r). The second term on the right-hand side of Eq. (12) describes the response of the superconductor to the exchange field. It is a non-local magnetic moment with an opposite sign extending over a distance of the order of the superconducting coherence length away from the F region (see Fig. 3). While the local magnetic moment is finite, after spatial integration, the total one is zero at T ¼ 0, as in the homogeneous case. The local magnetic moment can be probed using the polar Kerr effect or low-energy muons.

Superconductor-ferromagnetic insulator structures Beside all-metallic S/F structures, the magnetic proximity effect can also be observed in ferromagnetic insulator/superconductor structures (FI/S) (Hijano et al., 2021; Meservey and Tedrow, 1994). Ferromagnetic insulators, such as EuS, EuO, or GdN, when put in contact with a thin superconducting film, such as Al or Nb, may induce in the latter a spin-split BCS density of states. This is equivalent to the Zeeman splitting induced in a thin superconducting film in a large in-plane magnetic field [see Fig. 4A]. At the same time, since the electrons from S cannot penetrate the insulating FI, there is no superconducting proximity effect. In other

Fig. 4 (A) Spin-split density of states in a FI/S structure. (B) Measurements of the differential conductance of a EuS/Al structure, as a function of externally applied magnetic field B and voltage, from Hijano et al., (2021). Well defined spin-splitting can be observed even at B ¼ 0. Adapted from Hijano A, Ilic S, Rouco M, González-Orellana C, Ilyn M, Rogero C, Virtanen P, Heikkilä T, Khorshidian S, Spies M. et al. (2021) Coexistence of superconductivity and spin-splitting fields in superconductor/ferromagnetic insulator bilayers of arbitrary thickness. Physical Review Research 3(2): 023131. This work is licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

Superconductor-ferromagnet hybrid structures

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words, the Cooper pairs do not “leak” from S to FI, resulting in robust superconductivity in S and sharp coherence peaks in the density of states. The origin of spin-splitting is the interfacial effective exchange coupling between the localized magnetic moments of the FI and the conduction electrons of the S layer. If the thickness of the latter is smaller than the  coherence length, then the density of states can be assumed to be homogeneous in space and given by NðEÞ ¼ 12 N" ðEÞ + N # ðEÞ , with E
he f f , N",# ðEÞ ¼ Re qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 E
he f f −D2

(13)

where N", # is the density of states for spin-up and -down. The effective spin-splitting field heff is given by the interfacial exchange field multiplied by a/dS, where a is an atomic length scale over which the exchange interaction takes place, and dS the thickness of the superconducting layer. Since the end of the 1980s, numerous spectroscopy experiments on EuS/Al have demonstrated large spin-split density of states even at zero magnetic field [see Fig. 4]. Recently, interest in these systems has been renewed because of their potential for application in detectors, spintronics, and topological superconducting qubits. For some of these applications, an important feature of FI is that it can also act as a spin-filtering tunneling barrier, as spin-filtering combined with spin-splitting leads to electron-hole (e-h) symmetry breaking. Namely, the total DoS N(E) is e-h symmetric, N(E) ¼ N(−E), however if one spin species is filtered-out the symmetry is broken, N", #(E) 6¼ N", #(−E), as seen from Eq. (13). Such spin-dependent transport leads to several interesting phenomena in FI/S structures, such as thermoelectric effects and rectification of AC currents which can be used for detectors or thermometers.

F/S/F spin valves Another manifestation of the magnetic proximity effect is the change of the critical temperature of F/S/F structures, called spin-valves, when the relative magnetization of the two F layers changes between parallel (P) and antiparallel (AP) configurations (see Fig. 5). The effective exchange field in the superconducting layer is an average of the exchange fields induced at each S/F interface. In the P configuration the averaged exchange field induced in the superconductor is larger as in the AP case, and therefore leads to lower critical temperature. This behavior was theoretically predicted by De Gennes (1966) on the example of FI/S/FI structures and verified in experiments by Hauser (1969). In later works, these claims were extended to metallic samples: theoretically by Buzdin et al. (1999) and Tagirov (1999), and experimentally by Gu et al. (2002) and Moraru et al., (2006). Another type of spin valve, namely S/F/F/S junctions, has been used to accurately control the 0-p transition by Gingrich et al. (2016).

Non-equilibrium properties of S/F structures So far we considered the spectral and equilibrium properties of S/F hybrid structures. There is also a large experimental and theoretical activity on the transport properties of such structures under non-equilibrium conditions (Beckmann, 2016; Bergeret et al., 2018; Heikkilä et al., 2019). A typical experimental setup is sketched in Fig. 6: a superconducting wire is contacted to various

Fig. 5 Resistance measurement by Moraru IC, Pratt W Jr, and Birge NO (2006) in a Ni/Nb/Ni spin-valve, showing that the superconducting transition temperatures in the AP state is higher than in the P state. Adapted with permission from Moraru IC, Pratt W Jr, and Birge NO (2006) Magnetization-dependent Tc shift in ferromagnet/superconductor/ferromagnet trilayers with a strong ferromagnet. Physical Review Letters 96(3): 037004. Copyright (2006) by the American Physical Society.

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Fig. 6 Sketch of a setup for detection of the non-equilibrium modes in a superconductor (blue wire) via transport measurements. Injector and detector are normal or ferromagnetic metals.

electrodes, which can be non-magnetic or ferromagnets. One can, for example, inject a current from the injector by applying a voltage Vinj, and measure the current Idet in the detector circuit at zero bias (Hübler et al., 2012). The injected current drives the superconductor to a nonequilibrium state. Then, the current in the detector is given by the expression Idet ¼ Gdet ðm + P det ms Þ,

(14)

where Gdet is the tunneling conductance and Pdet is the effective spin polarization of the detector. The charge imbalance m and spin accumulation ms are related to the different components of the distribution function matrix via [see Silaev et al., 2015]: Z 1 1 m¼ − dEðN+ FT + N− F Ls Þ, (15) 2 0 Z   1 1  ms ¼ − dE N + F Ts + N− F L−F eq , (16) 2 0 where N
¼ N"
N#, and Feq(E) ¼ tanh (E/2T ) is the equilibrium distribution function. The different functions F describe the different non-equilibrium modes. We use here the conventional notation denoting by the label T/L the transverse/longitudinal modes, which are symmetric/antisymmetric functions of the energy E. FT describes the charge imbalance mode, which quantifies the imbalance between quasiparticles above and below the Fermi surface. This mode can be excited by charge injection from a normal electrode to the superconductor. The mode FL describes local changes in the effective temperature. Any heating mechanism can generally generate it. In the absence of ferromagnets, only these two modes can be excited and have been widely studied in the 70s and 80s. In the presence of ferromagnetic injectors/detectors, or spin-splitting fields in the superconductor, another two modes appear. The spin imbalance mode, FTs, describes the spin accumulation induced, for example, by a spin-polarized current injected from a ferromagnetic electrode. And finally the spin-energy mode, FLj, describes a non-equilibrium state with antisymmetric differences in the electron-hole and spin-up/down distributions. Nonequilibrium modes are determined by the Keldysh-Usadel kinetic equations presented in Silaev et al. (2015). Fig. 7A shows the calculated non-local conductance defined as gnl ¼ dIdet/dVinj for different values of the spin-splitting field h. At zero field, only the terms proportional to N+ in Eqs. (15) and (16) contribute to the detector current, Eq. (14). The contribution from the spin accumulation, Eq. (16), decays over the spin relaxation length. Therefore, if the distance between injector and detector is much larger than this length, the signal is dominated by the charge imbalance. This explains the almost symmetric signal, gnl(Vinj)  gnl(−Vinj), observed in experiments on charge imbalance (Hübler et al., 2012; Quay et al., 2013). In the case of a finite spin-splitting field N− 6¼ 0, so the second terms in Eqs. (15) and (16) also contribute to gnl. If the spin-splitting is caused by an external magnetic field, the latter suppresses the charge imbalance via the orbital depairing effect. The non-local signal is then dominated by the energy mode FL, the second term in Eq. (16). This results in almost antisymmetric gnl(Vinj) curves, as shown in Fig. 7A. This contribution is independent of the elastic spin-dependent scattering, since the FL mode relaxes only by inelastic scattering. At low temperatures, inelastic scattering becomes weak and therefore the spin accumulation can be long-range, as observed in experiments by Hübler et al. (2012), Wolf et al. (2013) and Quay et al. (2013). The comparison between theory and the experiment by Hübler et al. (2012), Fig. 7B, shows a very good agreement. Interestingly, according to the theory one can also generate a spin accumulation even if the injector is a nonmagnetic metal. The simple injection of a current at voltages larger than the superconducting gap generates energy imbalance FL. If the spin-splitting field is finite, a long-range contribution from the second term of Eq. (16) appears. This is also in accordance with the observations of Wolf et al. (2013). Long-range spin-dependent signals can have direct applications in the field of spintronics (Eschrig, 2011; Linder and Robinson, 2015).

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Fig. 7 (A) Calculated non-local conductance for different values of the spin-splitting field. (B) Comparison between theory (red curve) and experiment of Hübler et al. (2012) (blue curve). Adapted with permission from Silaev M, Virtanen P, Bergeret F, and Heikkilä T (2015) Long-range spin accumulation from heat injection in mesoscopic superconductors with Zeeman splitting. Physical Review Letters 114(16): 167002. Copyright (2015) by the American Physical Society.

Finally, the spin-energy mode, FLs, has been recently detected by Kuzmanovic et al. (2020) in a setup as the one sketched in Fig. 6. The experiment did not use ferromagnetic materials and hence only the charge imbalance, Eq. (15), contributes to the non-local signal, Eq. (14). Using the kinetic theory developed by Silaev et al. (2015) and Heikkilä et al. (2019), Kuzmanovic et al. (2020) could unambiguously identify the spin energy mode FLs.

Conclusion The S/F structures provide an opportunity to study the interplay and coexistence of two antagonistic phenomena - ferromagnetism and superconductivity. As such, they have been a subject of vigorous investigation in the last few decades, both on the experimental and theoretical front. In this chapter we summarized the main results of these efforts, using a simple diffusion-like Usadel equation to qualitatively illustrate the main points. The interest in these structures persists to this day, as they have been identified as a building block for many applications ranging from superconducting spintronics, topological superconductivity, to radiation detectors. Recent advances in sample preparation even allow to build atomically thin S/F structures in Wan der Vaals heterostructures, using monolayer ferromagnets such as NiBr2.

References Abrikosov AA and Gor’kov LP (1960) Contribution to the theory of superconducting alloys with paramagnetic impurities. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 39. Beckmann D (2016) Spin manipulation in nanoscale superconductors. Journal of Physics: Condensed Matter 28(16): 163001. Bergeret F, Volkov A, and Efetov K (2001) Long-range proximity effects in superconductor-ferromagnet structures. Physical Review Letters 86(18): 4096. Bergeret F, Volkov AF, and Efetov K (2004) Induced ferromagnetism due to superconductivity in superconductorferromagnet structures. Physical Review B 69(17): 174504. Bergeret F, Volkov AF, and Efetov KB (2005) Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures. Reviews of Modern Physics 77(4): 1321. Bergeret FS, Silaev M, Virtanen P, and Heikkilä TT (2018) Colloquium: Nonequilibrium effects in superconductors with a spin-splitting field. Reviews of Modern Physics 90(4), 041001. Buzdin A (2000) Density of states oscillations in a ferromagnetic metal in contact with a superconductor. Physical Review B 62(17): 11377. Buzdin AI (2005) Proximity effects in superconductorferromagnet heterostructures. Reviews of Modern Physics 77(3): 935. Buzdin AI and Kupriyanov MY (1990) Transition temperature of a superconductor-ferromagnet superlattice. JETP Letters 52: 1089. Buzdin AI, Bulaevskii L, and Panyukov S (1982) Critical-current oscillations as a function of the exchange field and thickness of the ferromagnetic metal (F) in an SFS Josephson junction. JETP Letters 35(4): 178–180. Buzdin AI, Vedyayev A, and Ryzhanova N (1999) Spin-orientation–dependent superconductivity in F/S/F structures. Europhysics Letters 48(6): 686. De Gennes P (1966) Coupling between ferromagnets through a superconducting layer. Physics Letters 23(1): 10–11. Demler EA, Arnold G, and Beasley M (1997) Superconducting proximity effects in magnetic metals. Physical Review B 55(2): 15174. Eschrig M (2011) Spin-polarized supercurrents for spintronics. Physics Today 64(1): 43. Fulde P and Ferrell RA (1964) Superconductivity in a strong spin-exchange field. Physical Review 135(A): A550. Gingrich E, Niedzielski BM, Glick JA, Wang Y, Miller D, Loloee R, Pratt W Jr., and Birge NO (2016) Controllable 0–p josephson junctions containing a ferromagnetic spin valve. Nature Physics 12(6): 564–567. Gu J, You C-Y, Jiang J, Pearson J, Bazaliy YB, and Bader S (2002) Magnetization-orientation dependence of the superconducting transition temperature in the ferromagnetsuperconductor-ferromagnet system: CuNi/Nb/CuNi. Physical Review Letters 89(26): 267001. Hauser J (1969) Coupling between ferrimagnetic insulators through a superconducting layer. Physical Review Letters 23(7): 374. Heikkilä TT, Silaev M, Virtanen P, and Bergeret FS (2019) Thermal, electric and spin transport in superconductor/ferromagnetic-insulator structures. Progress in Surface Science 94(3): 100540. Hijano A, Ilic S, Rouco M, González-Orellana C, Ilyn M, Rogero C, Virtanen P, Heikkilä T, Khorshidian S, Spies M, et al. (2021) Coexistence of superconductivity and spin-splitting fields in superconductor/ferromagnetic insulator bilayers of arbitrary thickness. Physical Review Research 3(2): 023131.

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Superconductor-ferromagnet hybrid structures

Hübler F, Wolf M, Beckmann D, Löhneysen H, and v. (2012) Long-range spin-polarized quasiparticle transport in mesoscopic Al superconductors with a Zeeman splitting. Physical Review Letters 109(20): 207001. Kadigrobov A, Shekhter R, and Jonson M (2001) Quantum spin fluctuations as a source of long-range proximity effects in diffusive ferromagnet-super conductor structures. Europhysics Letters 54(3): 394. Keizer RS, Gönnenwein ST, Klapwijk TM, Miao G, Xiao G, and Gupta A (2006) A spin triplet supercurrent through the half-metallic ferromagnet CrO2. Nature 439(7078): 825–827. Khaire TS, Khasawneh MA, Pratt W Jr., and Birge NO (2010) Observation of spin-triplet superconductivity in Co-based Josephson junctions. Physical Review Letters 104(13), 137002. Kontos T, Aprili M, Lesueur J, and Grison X (2001) Inhomogeneous superconductivity induced in a ferromagnet by proximity effect. Physical Review Letters 86(2): 304. Kontos T, Aprili M, Lesueur J, Genêt F, Stephanidis B, and Boursier R (2002) Josephson junction through a thin ferromagnetic layer: negative coupling. Physical Review Letters 89(13): 137007. Kuzmanovic M, Wu B, Weideneder M, Quay C, and Aprili M (2020) Evidence for spin-dependent energy transport in a superconductor. Nature Communications 11(1): 1–7. Larkin A and Ovchinnikov YN (1965) Nonuniform state of superconductors. Soviet Physics - JETP 20(3): 762. Lazar L, Westerholt K, Zabel H, Tagirov L, Goryunov YV, Garif’yanov N, and Garifullin I (2000) Superconductor/ferromagnet proximity effect in Fe/Pb/Fe trilayers. Physical Review B 61(5): 3711. Linder J and Robinson JW (2015) Superconducting spintronics. Nature Physics 11(4): 307–315. Meservey R and Tedrow P (1994) Spin-polarized electron tunneling. Physics Reports 238(4): 173–243. Moraru IC, Pratt W Jr., and Birge NO (2006) Magnetization-dependent Tc shift in ferromagnet/superconductor/ferromagnet trilayers with a strong ferromagnet. Physical Review Letters 96(3): 037004. Quay C, Chevallier D, Bena C, and Aprili M (2013) Spin imbalance and spin-charge separation in a mesoscopic superconductor. Nature Physics 9(2): 84–88. Radovic Z, Ledvij M, Dobrosavljevic-Grujic L, Buzdin AI, and Clem JR (1991) Transition temperatures of superconductor-ferromagnet superlattices. Physical Review B 44(2): 759. Robinson J, Witt J, and Blamire M (2010) Controlled injection of spin-triplet supercurrents into a strong ferromagnet. Science 329(5987): 59–61. Ryazanov V, Oboznov V, Rusanov AY, Veretennikov A, Golubov AA, and Aarts J (2001) Coupling of two superconductors through a ferromagnet: Evidence for a p junction. Physical Review Letters 86(11): 2427. Silaev M, Virtanen P, Bergeret F, and Heikkilä T (2015) Long-range spin accumulation from heat injection in mesoscopic superconductors with Zeeman splitting. Physical Review Letters 114(16): 167002. Singh A, Voltan S, Lahabi K, and Aarts J (2015) Colossal proximity effect in a superconducting triplet spin valve based on the half-metallic ferromagnet CrO2. Physical Review X 5(2): 021019. Tagirov L (1999) Low-field superconducting spin switch based on a superconductor/ferromagnet multilayer. Physical Review Letters 83(10): 2058. Usadel KD (1970) Generalized diffusion equation for superconducting alloys. Physical Review Letters 25(8): 507–509. Wolf MJ, Hübler F, Kolenda Sv, Löhneysen Hv, and Beckmann D (2013) Spin injection from a normal metal into a mesoscopic superconductor. Physical Review B 87(2): 024517. Xia J, Shelukhin V, Karpovski M, Kapitulnik A, and Palevski A (2009) Inverse proximity effect in superconductor-ferromagnet bilayer structures. Physical Review Letters 102(8): 087004.

Superconductivity: Ginzburg-Landau theory and vortex lattice☆ EH Brandt⁎, Max-Planck-Institut fur Metallforschung, Stuttgart, Germany © 2024 Elsevier Ltd. All rights reserved. This is an update of E.H. Brandt, Superconductivity: Ginzburg–Landau Theory and Vortex Lattice, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 98–105, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00702-6.

Introduction Ginzburg-Landau theory Critical fields Vortex lattice London theory Vortices near surfaces and in films Elasticity of the vortex lattice Conclusion References

693 694 694 695 697 698 699 701 701

Abstract Some properties of the flux-line lattice in conventional and high-Tc superconductors are reviewed, with particular emphasis on phenomenological theories, nonlocal elasticity, irreversible magnetization curves, and influence of the specimen shape on the electromagnetic response.

Key points

• • •

Main concepts and results of Ginzburg-Landau and London theories Description of vortex lattices in type-II superconductors within the Ginzburg Landau and London theories Elasticity of the vortex lattice in type-II superconductors

Introduction After the discovery of superconductivity in 1912 by Heike Kamerlingh-Onnes in Leiden, it took almost 50 years until this fascinating phenomenon was understood microscopically, when in 1957 Bardeen, Cooper, and Schrieffer established their BCS theory. But long before this electron-pairing theory, there were powerful phenomenological theories which were able to explain most electromagnetic and thermodynamic observations on superconductors, and which are very useful today also (Brandt, 1995; De Gennes, 1966; Tinkham, 1975). The London theory, conceived by Fritz and Heinz London in 1935, is particularly useful for the description of the high-Tc superconductors, which were discovered in 1987 by Bednorz and Muller in Zurich. Perhaps the most useful and elegant phenomenological theory of superconductors was established by Vitalii Ginzburg and Lew Landau in 1951. The Ginzburg-Landau (GL) theory generalizes Landau’s theory of second-order phase transitions to spatially varying charged systems in a magnetic field and should thus be applicable near the superconducting transition temperature Tc, but in many cases it yields a qualitatively correct behavior also at lower temperatures. The GL theory can be derived from the microscopic BCS theory, and it reduces to the London theory in situations when the GL function (the superconducting order parameter) has nearly constant magnitude. The theory of Ginzburg and Landau was also generalized to the cases of multiband and unconventional superconductors (Orlova et al., 2013; Sigrist and Ueda, 1991). The GL theory has high predictive power. It predicts that superconductors can be of type-I (with positive energy of the wall between normal conducting and superconducting domains) or of type-II (with negative wall energy, pointing to an instability) and that superconductivity can be suppressed by a high magnetic field and by a high current density, namely, by the depairing current which breaks the Cooper pairs. Its most spectacular success was the prediction of spontaneous nucleation and penetration of magnetic vortices in type-II superconductors by Alexei Abrikosov in 1957. The properties of these Abrikosov vortices (or fluxons, flux lines, carrying one quantum of magnetic flux F0 ¼ h/2e ¼ 2.07  10−15Tm2) and their motion and pinning by material inhomogeneities, are important topics both in research and in the practical application of superconductors. Most applications ☆ Change History: September 2022. B Grigorii Mikitik added two new references to Introduction. Updated abstract, conclusions, composed the list of references, and corrected misprints.

Deceased. The present version has been redrafted by Grigorii Mikitik, B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00079-2

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require type-II superconductors with high critical magnetic fields and currents, but high loss-free supercurrents require that the vortices are pinned so that they cannot move under the action of this current and dissipate energy.

Ginzburg-Landau theory The GL theory introduces a complex, spatially varying order parameter c(r) in addition to the magnetic field B(r) ¼ r  A(r) or vector potential A. The GL function c(r) later turned out to be proportional to the BCS energy-gap function D(r), and its square |c(r)|2 to the density of Cooper pairs. It defines two characteristic lengths: the superconducting coherence length x sets the scale over which c(r) can vary, while l governs the variation of the magnetic field as in the London theory. Both l and x diverge at the superconducting transition temperature Tc according to l ∝ x ∝ (Tc − T )−1/2, but their ratio, the GL parameter k ¼ l/x, is nearly independent of the temperature T. The GL theory reduces to the London theory (which is valid down to T ¼ 0) in the limit x  l, which means constant magnitude |c(r) | ¼ const (except in the vortex cores, where c vanishes, see below). The GL equations are obtained by minimizing a free-energy functional F{c, A} pffiffiffi  pffiffiffiwith respect to the GL function c(r) and the vector potential A(r). With the length unit l and magnetic field unit 2Bc Bc ¼ F0 = 8pxl is thermodynamic critical field), the GL functional reads #   2 Z "   ir B2c 1 2 2 4   −jCj + jCj +  − −A C + ðr  AÞ d3 r: (1) FfC, Ag ¼ m0 V 2 k Here the integral is over the volume V of the superconductor. The first two terms are the superconducting condensation energy, which is minimum when |c|2 ¼ 1. The third term is the energy cost of the spatial variation of c; this gauge-invariant gradient also introduces A and thus the magnetic field. The last term is the magnetic energy density B2/2m0. The GL functional and the resulting GL equations may be expressed in terms of the real-valued function |c| and the gauge-invariant supervelocity r’/k − A, where ’(r) is the phase of c ¼ | c | exp (i’). From the variation dF/dc ¼ 0 follows the first GL equation determining the amplitude and phase of c. From the variation dF/dA ¼ 0 follows the supercurrent density −1 2 2 J ¼ m−1 0 l [(F0/2p) r ’ − A]|c| , which has to be supplemented by the Maxwell equation J ¼ m0 curl curl A to obtain an equation for A or J alone. The minimization of F yields not only the functions c(r) and A(r), but also the boundary condition that the current density and supervelocity do not have a component perpendicular to the free surface of the superconductor. In an external field Ha, one has to minimize not the free energy F but the Gibbs free energy G ¼ F−BHa , where B is the spatial average of the magnetic induction B in the superconductor. The condition ∂G=∂B ¼ 0 yields the equilibrium field Ha ¼ ∂F=∂B: In sufficiently low Ha, the superconductor expels the applied magnetic field completely, that is, one has B ¼ 0 inside the superconductor (Meissner state). More precisely, the parallel applied field penetrates into a thin surface layer of thickness l. In a superconductor filling the half space x  0, one has B(x) ¼ m0Ha exp (−x/l). This screening of Ha is caused by a surface current of density j(x) ¼ (Ha/l) exp (−x/l) flowing perpendicular to Ha. Such an exponential screening, strictly spoken, occurs only for London superconductors with large GL-parameter k  1, while for smaller k values both |c| and B vary near the surface and have to be obtained by solving the GL equations. Moreover, for superconductors of realistic shape, such as short cylinders or rectangular plates, the penetration of a magnetic field into the surface layer has to be calculated numerically; only for spheres and long cylinders with k  1, analytical solutions were obtained by London.

Critical fields pffiffiffi In type-I superconductors (defined by k < 1= 2), complete screening occurs when the applied magnetic field Ha is less than the −1 Bc; in larger fields Ha  Hc, these superconductors are normal conducting. For type-II thermodynamic critical field Hc ¼ mp 0 ffiffiffi superconductors (defined by k  1= 2 ), full screening occurs up to the lower critical field Hc1 ¼ m−1 0 Bc1 where penetration of Abrikosov vortices begins. With a further increasing of Ha, more vortices penetrate in the form of a more or less regular flux-line

Fig. 1 The triangular vortex lattice. The dots mark the positions of vortex cores. From Brandt E.H. (2005) Superconductivity: Ginzburg-Landau theory and vortex lattice. Encyclopedia of Condense Matter Physics, 1st edn., pp. 98–105. Elsevier.

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lattice (Fig. 1), which has an average induction B ¼ nF0 , where n is the area-density of vortices and F0 ¼ 2.07  10−15Tm2 is the quantum of magnetic flux. When Ha reaches the upper critical field Hc2 ¼ m−1 0 Bc2, the vortex cores overlap so strongly that the superconductor turns normal and one has B ¼ m0 Ha > Bc2 : The GL theory yields F0 ð ln k + aÞ , 4pl2

F F Bc ¼ pffiffiffi 0 , Bc2 ¼ 0 2 (2) 2px 8plx pffiffiffi pffiffiffi   with aðkÞ  0:5 + ð1 + ln 2Þ= 2k− 2 + 2 (Brandt, 2003). At k ¼ 1= 2, one has exactly Bc1 ¼ Bc ¼ Bc2, and for type-II superconductors Bc1  Bc  Bc2. In both types of superconductors, Bc determines the area under the magnetization curve m0M(Ha); this area equals the superconducting condensation energy B2c /m0. For ideal pin-free superconductors, the magnetization is M ¼ m−1 0 B−H a  0, see Fig. 2. For type-II superconductors with an ideal planar surface, Hc is also the field up to which the so-called Bean-Livingston surface barrier may prevent a vortex penetration (“overheating”). Interestingly, as discovered theoretically by De Gennes in 1963, even above Bc2, a thin surface sheath of thickness x remains superconducting up to a third critical field, Bc3 ¼ 1.695Bc2 (De Gennes, 1966). The above scenario applies to long superconductors in the parallel magnetic field Ha. In realistic samples like spheres, platelets, or films, demagnetization effects lead to flux penetration at lower fields. In ideal ellipsoids, the penetration fields, Hc or Hc1, are reduced by a factor 1−N, where N is the demagnetization factor. For spheres, one has N ¼ 1/3, and for long cylinders, N ¼ 0 in a parallel field and N ¼ 1/2 in a perpendicular field, while for thin plates in a perpendicular field, one has 1 − N  1, and thus very small fields will force flux penetration from the edges. In type-I superconductors with N > 0, demagnetization effects lead to partial penetration of flux in the form of planar domains or more complicated structures, while in type-II superconductors they just shear the magnetization curve, but the vortex lattice remains uniform if the specimen is an ellipsoid. For other shapes such as platelets, strips, or disks with constant thickness, the demagnetization effects are more complicated and lead to a geometrical barrier for flux penetration, with a penetration field which is approximately Hc1 times the square root of the aspect ratio thickness/width (Brandt et al., 2013; Zeldov et al., 1994). For specimens much larger than l, this result can be derived by the continuum theory without considering individual vortex lines. But for small mesoscopic superconductors with size comparable to or smaller than l, the full GL theory has to be solved numerically to see the detailed pattern of penetrated flux lines and the shape of the magnetization curve M(Ha) that may exhibit jumps at certain values of Ha. Bc1 ¼

Vortex lattice Abrikosov (1957) found a solution of the GL theory that exhibits a two-dimensional (2D) regular lattice of zeros in the order parameter c(x, y) and a periodic magnetic field B(x, y). This solution describes a lattice of parallel vortex lines along z. The ideal lattice is triangular, that is, each vortex has six nearest neighbors, see Fig. 1. The vortices start penetrating at Ha ¼ Hc1. The profiles |c(r)|2 and B(r) of one isolated vortex line are shown in Fig. 3 (r is the radial coordinate). As Ha is increased, more vortices penetrate and form a vortex lattice as shown in Fig. 4. First, the magnetic fields of the flux lines overlap, such that the amplitude of the periodic B(x, y) decreases. Then, the cores also overlap, such that the amplitude of the order parameter |c|2 decreases until it vanishes at Ha ¼ Hc2. For b ¼ B=Bc2 > 0:5, good approximations are (Brandt, 1995) jCðrÞj2 ¼

X 1−B=Bc2 a cos Kr, ½1−1=ð2k2 ÞbA + 1 K K

(3)

Fig. 2 Magnetization curves M(H) of long type-II superconductors in parallel magnetic fieldp H ffiffi¼ ffi Ha (demagnetization factor N ¼ 0) calculated from Ginzburg-Landau theory for various Ginzburg-Landau parameters k ¼ 0.75. . .3. For k ¼ 1= 2,−M jumps vertically from −M ¼ H to M ¼ 0 at H ¼ Hc1 ¼ Hc2.

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Fig. 3 Magnetic field B(r) and order parameter |c(r)|2 of an isolated flux line calculated from the Ginzburg-Landau theory for Ginzburg-Landau parameters k ¼ 2, 5, and 20.

Fig. 4 Two profiles of the magnetic field B(x, y) and order parameter |C(x, y)|2 along the x-axis (a nearest-neighbor direction) for flux-line lattices with lattice spacing a ¼ 4l (bold lines) and a ¼ 2l (thin lines). The dashed line shows the magnetic field of an isolated flux line from Fig. 3. Calculations from the Ginzburg-Landau theory for k ¼ 5.

BðrÞ ¼ m0 Ha −

F0 jCðrÞj2 : 4pl2

(4)

Here r ¼ (x, y) and the sum is over all reciprocal lattice vectors Kmn ¼ (2p/x1y2)(my2; −mx2 + nx1) of the periodic flux-line lattice (FLL) with fluxline positions Rmn ¼ (mx1 + nx2; ny2) (m, n ¼ integer). The unit-cell area x1y2 yields the mean induction B ¼ F0 =x1 y2 : The first Brillouin zone has the area pk2BZ with k2BZ ¼ 4pB=F0 : For general lattice symmetry, the Fourier coefficients aK and the Abrikosov parameter bA are:   aK ¼ ð−1Þm+mn+n exp −K 2mn x1 y2 =8p , (5) < jCj4 > X 2 ¼ aK : (6) < jCj2 >2 K P P −ffiffiffi1. In vortex lattice with spacing From c(r ¼ 0) ¼ 0 follows KaK ¼ 0, or with aK¼0 pffiffiffi pffiffiffi¼ 1, K6¼0aK ¼p pffiffiffi  particular, for the triangular 1=2 , one has x1 ¼ a,x2 ¼ a=2,y2 ¼ 3a=2,B ¼ 2F0 = 3a2 ,aK ¼ ð−1Þp exp −pp= 3 with p ¼ m2 + mn + n2 ¼ R2mn/ a ¼ 2F0 = 3B R210. This yields bA ¼ 1.15960 (the reciprocal-lattice sum converges very rapidly) and the useful relationships bA ¼

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pffiffiffi K 10 ¼ 2p=y2 ,K 210 ¼ 16p2 =3a2 ¼ 8p2 B= 3F0 : For square vortex lattice, one has x1 ¼ y2 ¼ a,x2 ¼ 0,B ¼ F0 =a2 , and bA ¼ 1.18034. This means the energy of the square vortex lattice is only slightly larger than that of the triangular lattice, by 2% at most. A vortex lattice with square symmetry has been observed experimentally, for example, when the underlying square symmetry of the atomic lattice couples to the vortex lattice such that the square vortex lattice has lower energy, but also other effects may stabilize the square lattice, for example, an Fermi surface or a deviation from the GL theory.  anisotropic   The free energy F B per unit volume and the negative magnetization M ¼ Ha −m−1 0 B  0 with Ha ¼ ∂F=∂B are (still for B=Bc2 > 0:5 and from the GL theory):  2 2   Bc2 −B =2m0 B F B ¼ , (7) − 2m0 ð2k2 −1ÞbA + 1     Bc2 −B F −m0 M B ¼ (8) ¼ 0 < jCj2 > : ð2k2 −1ÞbA + 1 4pl2 P  and ¼ (4pl2/F0)m0 | M|. For the periodic field B(r) ¼ KBK cos Kr, the Fourier coefficients are BK6¼0 ¼ m0 MaK ,BK¼0 ¼ B, At lower induction B=Bc2 < 0:5, the vortex lattice and its magnetization curve have to be computed from the GL theory numerically, see Figs. 2–4. However, when k  1, one may also use the London theory, which is a good approximation for small inductions B=Bc2 < 0:2.

London theory The London theory follows from the GL theory by putting |c | ¼ 1, but it may also be obtained by minimizing the sum F of the energy of the magnetic field B(r) and the kinetic energy of the supercurrent density J(r) ¼ m−1 0 r  B(r). This yields the London energy functional Z  2 1 F ðBÞ ¼ B + l2 ðr  BÞ2 d3 r: (9) 2m0 V Minimizing this with respect to B ¼ r  A, one obtains the homogeneous London equation B − l2r2B ¼ 0 or J ¼ −2 − m−1 0 l A where the Maxwell equations r B ¼ 0 and r  B ¼ m0J were used, and the vector potential A was chosen in the “London gauge,” which requires that r A ¼ 0 and A is parallel to the surface everywhere. In presence of vortices, one has to add singularities that describe the vortex core, which may be straight or curved. For straight parallel vortex lines along the unit vector z, one gets the modified London equation X (10) B−l2 r2 B ¼ zF0 d2 ðr−rn Þ: n

and d2(r) ¼ d(x)d(y) is the Here rn ¼ (xn, yn) are the 2D vortex positions, which now do not have to form a periodic lattice, R R 2D delta function. This linear equation may be solved by Fourier transform using the relations exp (ikr)d2k ¼ 4p2d2(r) and exp (ikr) (k2 + l−2)−1d2k ¼ 2pK0(| r| /l). Here, K0(x) is a modified Bessel function with the limits K0(x)  ln (1.123/x) for x  1 and K0(x)  (p/2x)1/2 exp (−x) for x  1. The resulting magnetic field of any arrangement of parallel vortices is the sum of individual London vortex fields centered at the positions rn,   jr−rn j F X K0 : (11) BðrÞ ¼ z 0 2 l 2pl n The energy F2D of this 2D arrangement of vortex lines with length L (the specimen height) is obtained by inserting Eq. (10) into Eq. (9). Integrating over the delta function, one finds that the London energy is determined by the magnetic field values at the vortex positions,   rm −rn  F20 XX F X   : (12) B rm ¼ L K F2D ¼ L 0 0 l 2m0 m 4pm0 l2 m n This expression shows that the energy is composed of the pairwise interaction energy (terms m 6¼ n) and the self-energy of the vortices (terms m ¼ n). To avoid the divergence of the self-energy, one has to cut off the logarithmic infinity of B at the vortex centers rn by introducing a finite radius of the vortex core of order x, the coherence length of the GL theory. This cutoff may be achieved by

1=2 , and multiplying B by a normalization factor 1 replacing, in Eqs. (11) and (12), the distance rmn ¼ |rm − rn| by ~r mn ¼ r 2mn + 2x2 to conserve the flux F0 of the vortex. This analytical expression suggested by John Clem for a single vortex and later generalized to the vortex lattice (Hao et al., 1991), is an excellent approximation, as was shown numerically by solving the GL equation for the pffiffiffi periodic FLL in the entire ranges of B and k for 0  B  Bc2 and k  1= 2 (Brandt, 1997). For curved vortices at arbitrary positions rn(z) ¼ [xn(z), yn(z), z], the 3D modified London equation reads

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B−l2 r2 B ¼ F0

XZ

drn d3 ðr−rn Þ:

(13)

n

Here, the integral is along the vortex lines and d3(r) ¼ d(x)d(y)d(z). The resulting magnetic field and energy are, with h i1=2 2 ~r mn ¼ rm −rn  + 2x2 , BðrÞ ¼ F 3D ¼

F0 X 2m0 m

Z

F0 X 4pl2 n

  drm B rm ¼

Z drn

   exp −~r mn rm ¼ r =l   , ~r mn rm ¼ r

F20 XX 8pm0 l2 m n

Z

Z drm

drn

(14)

  exp −~r mn =l : ~r mn

This means all the vortex segments interact with each other similar to magnetic dipoles or tiny current loops, but the magnetic  long-range interaction ∝1/r is screened by a factor exp −~r mn =l : The 3D interaction between curved vortices is visualized in Fig. 5.

Vortices near surfaces and in films The above London solutions apply to vortices in the bulk. Near the surface of the superconductor, these expressions have to be modified. In simple geometries, for example, for superconductors with one or two planar surfaces surrounded by vacuum, the magnetic field and energy of a given vortex arrangement is obtained by adding the field of appropriate images (in order to satisfy the boundary condition that no current crosses the surface) and a magnetic stray field which is caused by a fictitious surface layer of magnetic monopoles. This stray field makes the total magnetic field B(r) continuous across the surface. Fig. 6 shows an example for this. The magnetic field and interaction of straight vortices oriented perpendicular to a superconducting film of arbitrary thickness was calculated by Carneiro and Brandt (2000) from the London theory and by Brandt (2005) from the GL theory. In films of thickness d  l in a perpendicular magnetic field, the short vortices interact mainly via their magnetic stray field outside the 2 ofthe 2D vortex superconductor over  penetration depth L ¼ 2l /d. With decreasing −1thickness d, the Fourier transform  R an2 effective −1 2 ~ for d  l to V~ ðkÞ ¼ E0 k2 + kL−1 for d  l, interaction V ðr Þ ¼ d k=4p V ðkÞ exp ðikrÞ changes from V~ ðkÞ ¼ E0 k2 + l−2 2 2 where E0 ¼ dF0/(m0l ).

Fig. 5 Pairwise interaction between all line elements (arrows) of curved flux lines within the London theory (schematic).

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Fig. 6 Magnetic field lines of a straight vortex in and near a superconducting half space. From the London theory, for k ¼ 20. The dashed lines give the radial field lines of a magnetic charge of size 2F0 positioned on the vortex axis at a depth −z0 ¼ l (left half, the far field) and −z0 ¼ 1.27l (right half ), which gives a better fit to the near field. The lower plot enlarges the center of the upper plot.

As shown by Clem (1991), a similar (but 3D) magnetic interaction  −2 exists  between the 2D “pancake vortices” in the super2 −1 l + k , where now d is the distance between the layers, conducting Cu-O layers of high-Tc superconductors, V~ðkÞ ¼ E0 dk23 k−2 2 3 k22 ¼ k2x + k2y , k23 ¼ k22 + k2z , and l ¼ lab is the penetration depth for the supercurrents flowing in these layers.

Elasticity of the vortex lattice Small flux-line displacements caused by pinning forces or by thermal fluctuations may be calculated using the linear elasticity theory of the vortex lattice. Fig. 7 visualizes the three basic distortions of the triangular vortex lattice: shear g, uniaxial compression e, and tilt a, defining the three elastic moduli c66, c11, and c44 and the naive (local) elastic energy F elast ¼

V c e2 + c66 g2 + c44 a2 : 2 11

(15)

The linear elastic energy Felast of the vortex lattice is obtained by expanding its free energy F with respect to small displacements un(z) ¼ rn(z) − Rn ¼ (unx, uny) of the vortices from their ideal parallel lattice positions Rn and keeping only the quadratic terms. This yields (Brandt, 1995)

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Superconductivity: Ginzburg-Landau theory and vortex lattice

Fig. 7 The three basic homogeneous elastic distortions of the triangular vortex lattice. The full dots and solid lines mark the ideal lattice, and the hollow dots and dashed lines the distorted lattice.

Felast ¼

1 2

Z

d3 k ∗ 3 ua ðkÞFab ðkÞub ðkÞ, 8p BZ

(16)

where u(k) is the Fourier transform of the displacement field un(z), (a, b) ¼ (x, y), and k ¼ (kx, ky, kz). The k-integral in Eq. (16) is over the Brillouin zone (BZ) of the FLL since the “elastic matrix” Fab(k) is periodic in the kx, ky plane; the finite vortex core radius restricts the kz integration to |kz|  x−1. For an elastic medium with uniaxial symmetry, the elastic matrix reads h

 i (17) Fab ðkÞ ¼ ðc11 −c66 Þka kb + dab k2x + k2y c66 + k2z c44 : Here the coefficients c11, c66, and c44 are the elastic moduli of uniaxial compression, shear, and tilt, respectively. For the vortex lattice, Fab(k) was calculated from the GL and London theories. The result, a sum over  reciprocal 1=2 lattice vectors, should coincide is the radius of the circularized with expression (17) in the continuum limit, that is, for small |k |  kBZ, where kBZ ¼ 4pB=F0 (actually hexagonal) Brillouin zone of the triangle vortex lattice with area pk2BZ. In the London limit, one finds for isotropic superconductors, the elastic moduli (Blatter et al., 1994; Brandt, 1995) 2

c11 ðkÞ 

B =m0 BF0 =m0 , c66  , 16pl2 1 + k2 l2 2

c44 ðkÞ 

B =m0 BF0 =m0 k2 + ln : 8pl2 1 + k 2 l2 1 + k2z l2

(18)

 2 2   The GLtheory yields 1=2 an additional factor 1−B=Bc2 in c66, that is, c66 ∝BðB−Bc2 Þ , and replaces l in c11 and c44 (first term) by ’ (Brandt, 1995). l ¼ l= 1−B=Bc2 The k dependence (dispersion) of the compression and tilt moduli c11(k) and c44(k) means that the elasticity of the vortex lattice is nonlocal, that is, strains with short wavelengths, 2p/k  2pl, have a much lower elastic energy than a homogeneous compression or tilt (corresponding to k ! 0) with the same amplitude. This elastic nonlocality comes from the fact that the magnetic interaction between the flux lines typically has a range l much longer than the flux-line spacing a0; therefore, each flux line interacts with many other flux lines. Note that a large l causes a small shear stiffness since c66 ∝ l−2, and a smaller c11(k > l−1) at short wavelengths, but the uniform compressibility c11(k ¼ 0) is independent of l. As a consequence of nonlocal elasticity, vortex displacement un(z) caused by local pinning forces, and also the space- and time-averaged thermal fluctuations of the vortex positions, , are much larger than they would be if c44(k) had no

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2

dispersion, that is, if it were replaced by c44 ð0Þ ¼ BHa  B =m0 : The maximum vortex displacement u(0) ∝ f caused at r ¼ 0 by a point force of density fd3(r), and the thermal fluctuations ∝ kBT, are given by similar expressions (Brandt, 1995), Z k2BZ l 2uð0Þ < u2 > d3 k 1

   : (19)  3 1=2 2 2 2 kB T f 8p BZ kx + ky c66 + kz c44 ðkÞ 8p½c66 c44 ð0Þ In this result, a large factor [c44(0)/c44(kBZ)]1/2  kBZl  pl/a  1 originates from the elastic nonlocality. In anisotropic superconductors with B k c (the crystalline c axis), the thermal fluctuations, Eq. (19), are enhanced by an additional factor G ¼ lc/lab  1, where lab and lc are the two penetration depths of uniaxially anisotropic superconductors.

Conclusion The phenomenological London and Ginzburg-Landau theories are effective tools for investigating both low-Tc and high-Tc superconductors. These theories permit one to understand the thermodynamic properties of superconducting materials and to describe the vortex lattice in type-II superconductors even without any information on the microscopic mechanism of the superconductivity. In particular, the lattices of straight and curved vortices, the vortices in films, and the vortex-lattice elasticity, which is a necessary constituent of an analysis of the vortex pinning, have been considered in the framework of this approach.

References Abrikosov AA (1957) On the magnetic properties of superconductors of the second group. Soviet Physics—JETP 5: 1174–1182. Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, and Vinokur VM (1994) Vortices in high-temperature superconductors. Reviews of Modern Physics 66: 1125–1388. Brandt EH (1995) The flux-line lattice in superconductors. Reports on Progress in Physics 58: 1465–1594. Brandt EH (1997) Precision Ginzburg-Landau solution of ideal vortex lattices for any induction and symmetry. Physical Review Letters 78: 2208–2211. Brandt EH (2003) Properties of the ideal Ginzburg-Landau vortex lattice. Physical Review B 68(054506): 1–11. Brandt EH (2005) Ginzburg-Landau vortex lattice in superconducting films of finite thickness. Physical Review B 71(014521): 1–21. Brandt EH, Mikitik GP, and Zeldov E (2013) Two regimes of vortex penetration into platelet-shaped type-II superconductors. Journal of Experimental and Theoretical Physics 117: 439–448. Carneiro GM and Brandt EH (2000) Vortex lines in films: Fields and interactions. Physical Review B 61: 6370–6376. Clem JR (1991) Two-dimensional vortices in a stack of thin superconducting films: A model for high-temperature superconducting multilayers. Physical Review B 43: 7837–7846. De Gennes PG (1966) Superconductivity of Metals and Alloys. New York: Benjamin. Hao Z, Clem JR, Mc Elfresh MW, Civale L, Malozemov AP, and Holtzberg F (1991) Physical Review B 43: 2844–2852. Orlova NV, Shanenko AA, Milocevic MV, Peeters FM, Vagov AV, and Axt VM (2013) Ginzburg-Landau theory for multiband superconductors: Microscopic derivation. Physical Review B 87(134510): 1–8. Sigrist M and Ueda K (1991) Phenomenological theory of unconventional superconductivity. Reviews of Modern Physics 63: 239–311. Tinkham M (1975) Introduction to Superconductivity. New York: McGraw-Hill. Zeldov E, Larkin AI, Geshkenbein VB, Konczykowski M, Majer D, Khaykovich B, Vinokur VM, and Shtrikman H (1994) Geometrical barriers in high-Tc superconductors. Physical Review Letters 73: 1428–1431.

Nanodimensional superconducting quantum interference devices MI Faley, ER-C-1(Physics of Nanoscale Systems) Forschungszentrum Jülich, Jülich, Germany © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview Key issues Conclusion References

702 703 708 709 710

Abstract The fabrication methods, principle of operation, microstructural, and electron transport properties, as well as the application of nanoscale superconducting quantum interference devices (nanoSQUIDs) are considered. NanoSQUIDs by definition have a submicrometer loop size, which limits dimensions of Josephson junctions to about 100 nm or less. This makes the use of tunnel junctions problematic, but encourages the use of nanobridges and other types of Josephson junctions with high critical current densities. The use of nitride superconductors helps to optimize the superconducting parameters and enhances the corrosion resistance of the nanobridge Josephson junctions and nanoSQUIDs. Additional passivation by a Si layer improves thermal sink and protects against the mechanical and electrical fragility of the devices. Nanosculpturing of cantilevers makes it possible to position the nanoSQUID at a distance of 10–100 nm from the objects under study in the nanoSQUID scanning measurement system. Several promising applications of nanoSQUIDs are briefly reviewed.

Key points

• • • • • •

NanoSQUIDs with a sub-micrometer loop size and dimensions of Josephson junctions to about 100 nm or less. The use of nanobridges and other types of Josephson junctions with high critical current densities. The use of nitride superconductors to optimize the superconducting parameters and to enhance the corrosion resistance of the Josephson junctions and nanoSQUIDs. Passivation by a Si layer to improve thermal sink and protect against the mechanical and electrical fragility of the devices. Nanosculpturing of cantilevers makes it possible to position the nanoSQUID at a distance of 10–100 nm from the objects under study in the nanoSQUID scanning system. Several promising applications of nanoSQUIDs are considered.

Introduction A nanoscale Superconducting QUantum Interference Device (nanoSQUID) is defined as a submicron loop of superconducting material that is interrupted by at least one Josephson junction. Josephson junctions are weak links connecting two superconducting leads. These weak links can be made by many different methods: using tunnel barriers, normal conducting or semiconducting interlayers, by superconducting interlayer with lower Tc as in the electrodes, or using short narrow constrictions (nanobridges). NanoSQUIDs are among the most sensitive tools for measuring very weak magnetic fields with high spatial resolution down to a few nanometers (Anahory et al., 2020; Granata and Vettoliere, 2016; Foley and Hilgenkamp, 2009). The submicron size of the nanoSQUID loop imposes serious limitations on the size of Josephson junctions: Typical Josephson junction sizes for implementation in nanoSQUIDs are around 300 nm, or more often below 100 nm when better spatial resolution is required. This makes the use of tunnel junctions problematic, but encourages the use of single layer short narrow constrictions (nanobridges) and other types of Josephson junctions with high critical current densities. Tunnel types of Josephson junction are traditionally used in superconducting electronics and currently also in superconducting quantum bits (qubits). The resistively shunted tunnel Josephson junctions, which have been designed for most applications in superconducting electronics, are typically a few micrometers in size (together with the shunt resistor) and do not fit inside nanoSQUIDs. In the case of submicron junctions, the realization of the three-layer structure with internally shunted junctions in the form of thinner tunnel junctions that are used for qubits or with a normal or a semiconductor conductor barrier is technologically relatively complex and associated with an increasing scattering of the junction parameters. An exceptional example is a nanosculpturing of tunnel junctions using a focused ion beam (FIB) that makes it possible to achieve high critical current modulation depth, non-hysteretic I(V)-characteristics at 4.2 K and a record energy resolution of about 1.3 h, where h is Planck’s constant (Schmelz et al., 2017). The renaissance of nanobridge Josephson junctions (nJJs) is observed in the case of nanoSQUIDs after more than 40 years of dominance of tunneling Josephson junctions for other applications in superconducting electronics. The great advantage of nJJs especially for preparation of planar nanoSQUIDs is

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that their size is limited only by the spatial resolution of the single layer patterning methods, and the possibility to achieve the highest critical current density that approaches the depairing current densities in the superconducting films. The operating temperature of nanoSQUIDs is lower than the superconducting transition temperature Tc of the superconducting material, and it is typically liquid helium temperature 4.2 K or lower for low-temperature superconductors such as Nb, NbN, TiN, etc., or liquid nitrogen temperature 77.4 K or lower for high-temperature superconductor YBa2Cu3O7−x. The use of nitride superconductors helps to optimize the superconducting parameters and enhances the corrosion resistance of the Josephson junctions and nanoSQUIDs. Additional passivation by a Si layer can improve thermal sink and protect against the mechanical and electrical fragility of the devices. Nanosculpturing of cantilevers makes it possible to position the nanoSQUID at a distance of 10–100 nm from the objects under study in the nanoSQUID scanning system. Attocube’s low temperature positioning and scanning system allows extremely large scanning of areas up to 5
5 mm with sub-nm spatial resolution at 4.2 K. NanoSQUIDs are designed to study small magnetic systems like magnetic nanoparticles or current distribution at low temperatures with nanoscale spatial resolution down to 40 nm and high sensitivity to tiny magnetic moments that is expressed in the units of spin sensitivity mB/√Hz, where mB ffi 9.27
10−24 J/T is Bohr magneton. Currently, the best nanoSQUIDs are able to measure distribution of magnetic fields of single atomic spins with spin sensitivity 2IC similar voltage pulses U(F,t) proceed on DC SQUID but they depend also on magnetic flux F passing through the SQUID loop due to the phase difference between the Josephson oscillations on individual junctions. Averaging of U(F,t) over the period t of the Josephson oscillations results in the DC voltage V across the DC SQUID: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Z 1 t RN IB 2IC pF V¼ , (2) U ðF, t Þdt  cos 1− 2 IB t 0 F0 where RN is the normal state resistance of the individual Josephson junction in the DC SQUID. The DC voltage V across the DC SQUID is thus a periodic function of the magnetic flux F passing through the SQUID loop. Eq. (2) was obtained under the assumption of a negligible inductance of nanoSQUIDs. The non-zero inductance Ls reduces modulation of critical current of DC SQUID to DIs(max)  F0/Ls. By increasing Ls, the optimal modulation parameter bL ¼ 2LsIc/F0 ffi 1 for DC SQUIDs corresponds to still large modulation of critical current and already good magnetic coupling to the measured signal. Responsible for magnetic coupling, geometrical inductance of nanoSQUID is typically below 0.5 pH and typical critical current is limited to Ic  20 mA even for nJJs in order to avoid hysteretic I(V)-characteristics. Taking into account only the geometrical inductance corresponds to the modulation parameter bL ffi 0.01. On the other hand, kinetic inductance of the nanoSQUID loop LkS ffi 2pm0lL2R/A ffi 60 pH, where (for example) R ffi 300 nm is the nanoSQUID’s loop radius, lL ffi 500 nm is London penetration depth in superconducting film of the loop and A ffi 0.01 mm2 is the estimated cross sectional area of the loop. This corresponds to the boundary value of the modulation parameter bL ffi 1.1. The task of optimizing nanoSQUIDs includes taking into account its kinetic inductance, limiting it, and finding ways to reduce it through the correct design, technology, and choice of , where ℓ and w are length and width of material. The theoretical expectation for kinetic inductance of a nanobridge Lkn ffi g wl kℏR B Tc nanobridge, g ffi 0.2 and R▪ is the normal-state sheet resistance of the superconducting film. The Josephson current-phase relation Is ¼ Ic sin(D’) is realized in Josephson junctions made by different methods in spite of variety of used electron-transport mechanisms: tunneling effect in insulator barrier, proximity effect in normal conducting and semiconducting barriers or spatially constricted superconducting current in nJJs. Operation principle of tunnel and proximity effect Josephson junctions is well described in many current textbooks and handbooks (see, for example, (Mangin and Kahn, 2017) and (Ruggiero, 2015)) as a standard explanation of Josephson effect because such 3-layer junctions are the most widely used type of Josephson junction in conventional superconducting electronics. Tunnel junctions are currently also effectively used in qubits (e.g., flux qubits and transmons) thanks to their low dissipation levels and large subgap resistances. NanoSQUID are in most cases based on nJJs because nJJs provide the highest critical current density and a better reproducibility in the sub-100 nm size range as compared with three-layer junctions. Description of the operation principle of nJJ can be made according to the theory of (Aslamazov and Larkin, 1968) that is using the Ginzburg–Landau (GL) equation. The GL equation describes Josephson effect in nJJs and it is valid only at temperatures T sufficiently close to the electrode critical temperature Tc: 0.9 T < T < Tc. In a one-dimensional approximation the GL equation can be written in the form: x2

d2 C + C − CjCj2 ¼ 0, dx2

(3)

where x is the coherence length and C is the order parameter. When nanobridge length l is shorter than coherence length, ideally 2 ℓ < < x, the first term dominates and the Eq. (3) is reduced to the Laplace equation ddxC2 ¼ 0, for which the solution is the linear function C ¼ a + bx, with x ranging from 0 to l:     x x iD’ e C1 , C ¼ 1 − C1 + (4) l l where D’ is the phase difference between the wave functions in the electrodes and C1 is the order parameter in the electrodes. Insertion of the latter equation for C into the GL equation for the superconducting current Is  Im(C rC) results in the Josephson current-phase dependence Is ¼ Ic sin(D’).  −1=2 that simplifies the condition The coherence length diverges at Tc according to the temperature dependence xðT Þ∝ 1 − TTc ℓ 4.2 K represent a challenge for the structuring of the nJJs using these materials. The current-phase relationship becomes multivalued when the length ℓ of nJJ exceeds the value 3.49x(T), for example, at temperatures T T  . The symmetrization of the switching distribution due to the interplay between escape and retrapping events can be clearly measured by considering the skewness of the distributions g as a function of the temperature. g is the ratio m3/s3 where m3 is the third central moment of the distribution. In the MQT and TA regimes, where retrapping processes are negligible, g ¼ −1, which is consistent with a typical escape rate which follows the Arrhenius law (Massarotti and Tafuri, 2019; Longobardi et al., 2011a; Fenton and Warburton, 2008; Murphy et al., 2013). As the temperature increases and the junction falls in the PD regime, the distributions become more and more symmetric as g tends to zero. Therefore, the switching events on the left side of the distribution (ascending side), which occur for lower values of the bias current, are prevented by phase retrapping, with a consequent symmetrization of the SCDs (Massarotti and Tafuri, 2019; Longobardi et al., 2011a; Fenton and Warburton, 2008; Murphy et al., 2013). The possibility to have extremely low Ic can be functional to investigate phase dynamics at extreme conditions. In recent experiments on submicron Nb/AlOx/Nb junctions (Yu et al., 2011) an anomalous s(T ) dependence with a negative ds/dT over the

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Critical current fluctuations in Josephson junctions

entire temperature range has been observed. This regime can be achieved by engineering junctions with lower critical current and junction capacitance such that the ratio Ic0/C, which regulates Tcross, is constant and the transition temperature T  , which scales with the Josephson energy (Krasnov et al., 2005), is lower or comparable to the quantum crossover temperature Tcross. Another example is given by the work by Yoon et al. (2011). They have engineered Al/AlOx/Al JJs in order to obtain low critical currents, of about 400 nA, and low capacitance, of about 40 fF, at the same time. In this way they have observed that TA is completely suppressed since T  is lower than Tcross. On the other hand, TA regime is recovered by adding a shunting capacitance in the device circuit, and this demonstrates that the capacitance can be used in order to tune the phase dynamics. Similar phenomenology of the SCDs, as reported in Fig. 3A and B, can be achieved by using unconventional HTS Josephson devices, such as YBCO grain boundary (GB) biepitaxial JJs (Bauch et al., 2005, 2006; Stornaiuolo et al., 2013). In particular, it is possible to engineer devices with T  < Tcross (Longobardi et al., 2012), to which correspond a direct transition from MQT to the PD regime, and TA is completely suppressed. For temperatures well below Tcross, MQT contributions to escape rates are larger than those coming from both thermal escape and retrapping processes (see Fig. 3C), while above Tcross the retrapping rate dominates over thermal and quantum escape rates (see Fig. 3D). The typical temperature behavior of s in this regime is shown in Fig. 3A. Below Tcross, histograms overlap and s saturates, which is a typical signature of a quantum activation regime. In analogy to what commonly done to prove MQT in underdamped junctions (Devoret et al., 1985; Clarke et al., 1988; Massarotti and Tafuri, 2019), the magnetic field can be used to tune in situ Tcross to unambiguously prove MQT as source of the saturation of s. In the presence of magnetic field, a lower value of Tcross and lower values of s are observed, as schematically sketched in Fig. 3C and expected from the theory (see Eq. 7). In Fig. 3A, increasing the temperature above Tcross, the switching current histograms shrink rather than broaden, which corresponds to the negative temperature derivative of s in Fig. 3B. This behavior is consistent with a diffusive motion due to multiple escape and retrapping processes in the potential wells. The experimental results and the numerical outcomes can be condensed in a phase diagram, which includes the various activation regimes (Kivioja et al., 2005; Longobardi et al., 2012). The phase diagram is a function of the characteristic scaling energies EJ and kBT and of the dissipation parameter Q; therefore it is valid in a large range of dissipation conditions and constitutes a functional guide to classify the switching behavior of a JJ. Moreover, it is a reference for phase dynamics of novel types of junctions and systems for which the nature of the current-induced transition from the superconducting to the normal state has not been completely clarified.

The very high critical current density regime The temperature behaviors of critical current fluctuations reported in the previous sections are typical of JJs characterized by critical current density Jc in the range 10–103 A/cm2. Recently, measurements of SCD have been performed on a series of different nanostructures (Murphy et al., 2013; Li et al., 2011; Sahu et al., 2009; Murphy et al., 2015; Massarotti et al., 2015b; Baumans et al., 2017; Ejrnaes et al., 2019; Salvoni et al., 2022; Zgirski et al., 2020, 2021; Zhang et al., 2022). Some of them are junctions and they can be classified in the schemes described in the previous sections. Some of them are simple nanowires (Zaikin, 2023). A subtle path exists between these different systems with analogies and distinctive features. When considering that a micro-bridge of width of the order of the coherence length behaves as a JJ (Barone and Paternò, 1982; Likharev, 1979), measurements of SCDs will turn to be a direct (a)

Increasing the temperature

(b)

Tcross

(c)

H=0

Tcross

H 0

(d)

Fig. 3 (A) SCDs as a function of T when the PD transition temperature T  is lower than the quantum crossover Tcross and TA is suppressed. (B) Temperature behavior of s for H ¼ 0 (black points) and in the presence of magnetic field (red triangles): the magnetic field reduces both Tcross and the values of s, thus confirming that spurious noise can be disregarded as the origin of s saturation. Schematic escape rates of MQT, TA, and retrapping processes at T < Tcross (panel (C), MQT escape rate dominates) and at T > Tcross (panel D, retrapping processes dominate).

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way of discriminating the phase dynamics and the transport in nontrivial cases, which are going to be more and more common with advances in nano-patterning superconductors at extreme scales. Recent experiments on superconducting nanowires have shown an anticorrelation between the bath temperature and the width of the switching distributions (Li et al., 2011; Sahu et al., 2009; Murphy et al., 2015; Baumans et al., 2017; Ejrnaes et al., 2019; Salvoni et al., 2022). A numerical model in order to study the stochastic aspects of the switching dynamics in these devices has been developed by Shah et al. (2008). The model is based on a freestanding wire of effective length L and cross-sectional area A, the ends of which are at the bath temperature Tb. All heat generated locally in the wire can be taken away only through the ends. The stochasticity inherent in the switching process is based on the fact that the resistive fluctuations of the superconducting nanowire consist of discrete phase-slip events that occur at random instants and are centered at random spatial locations along the wire (Little, 1967; Zaikin, 2023). Given that edge effects favor phase-slip locations away from the wire ends, the source term is restricted to the region at the center of the wire of length W (Shah et al., 2008). Stochastic phase slips that heat the wire and heat dissipation that cools the wire are the dominating mechanisms. Since the phase-slip rate depends on the local temperature of the wire, heating by phase slips can create a runaway cascade that eventually overheats the wire. More details can be found in Massarotti and Tafuri (2019). The stochastic phase-slip dynamics is reduced to the following ordinary differential equation for the time evolution of the temperature of the central segment (Shah et al., 2008): X dT ¼ −aðT, T b ÞðT − T b Þ + ðT, IÞ dðt − t i Þ dt i

(10)

The relaxation coefficient a(T, Tb) depends on the thermal conductivity K(T ), on the thermal capacity Cv(T ) of the phase slip volume, and on Tb (Shah et al., 2008). The second term on the right-hand side corresponds to stochastic heating by phase slips, whose rate G(I, T ) is temperature and current dependent and follows the Arrhenius law: GðI, TÞ ¼ OðI, TÞ exp −

DFðI, TÞ kB T

(11)

where O(I, T ) is the attempt frequency (Shah et al., 2008; Massarotti et al., 2015b) and DF(I, T ) is the free energy barrier given by (Little, 1967): pffiffiffi  6f0 Ic ðTÞ DFðI, TÞ ¼ (12) ð1 − I=Ic0 Þ5=4 , kB T If Ti and Tf are the temperatures before and after a phase slip, respectively, the temperature jump (I, T ) due to a phase slip can be expressed as Z T f ¼T i + AW Cv ðyÞdy ¼ hI=2e: (13) Ti

The first term in Eq. (10) represents the cooling mechanism as a result of conduction of heat from the central segment to the external bath via the two end segments, each of length (L − W). The temperature-dependent cooling rate is given by Z T 4 1 dyKðyÞ (14) aðT, T b Þ ¼ WðL − WÞCv ðTÞ T − T b T b In addition to restrictions on length scales, in the equations above the time for a phase slip and the quasiparticle thermalization time are both assumed to be smaller than the heat diffusion time (Shah et al., 2008). Solutions of Eq. (10) have been performed numerically by Shah et al. (2008) in a wide range of temperature and current. At low temperatures the occurrence of just one phase-slip is sufficient to cause the nanowire to switch from the superconducting to the resistive state. In this temperature range, the width of the switching histograms increases when increasing the temperature, following the usual TA expressed by the Arrhenius law. Such a broadening in the distribution width is indeed obtained up to a crossover temperature. At higher temperatures, the distribution width shows an anomalous decrease since histograms begin to shrink rather than broaden. Therefore, one phase slip is not sufficient to induce switching and multiple phase slips are needed. Such striking behavior of the distribution width may be understood by the following argument: the larger the number of phase slips in the sequence inducing the transition from the superconducting to the resistive state, the smaller the stochasticity in the switching process and, hence, the sharper the distribution of switching currents (Shah et al., 2008; Sahu et al., 2009; Murphy et al., 2015; Baumans et al., 2017; Ejrnaes et al., 2019). As in the case of moderately damped JJs, the change in the sign of the derivative of s(T ) is related to a change in the phase-slips dynamics from single to multiple phase-slips fluctuations in the wire. Typical measurements of SCDs on high Jc JJ are reported in Fig. 4A (Massarotti et al., 2015b). In this case, the critical current of the junction is about 50 mA and the junction width is about 200 nm; thus the resulting Jc is of the order 105 A/cm2, from one to two orders of magnitude larger than the junctions analyzed in the previous sections, thus Jc high values close to those observed in nanowires (Li et al., 2011; Sahu et al., 2009; Murphy et al., 2015). For high values of Jc, the phase dynamics is radically different

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Critical current fluctuations in Josephson junctions (a)

(b) Increasing the temperature

Fig. 4 (A) SCDs measured on high Jc JJ in a wide range of temperatures. (B) Temperature dependence of s of high Jc JJ (black circles) is compared to those calculated through numerical simulations within the RCSJ model based on different values of Q (the lines are a guide for the eye). The transition temperature T  and the shape of s(T ) in the PD regime are both affected by the junction quality factor Q, while the temperature behavior of s of high Jc JJ cannot be reproduced within the RCSJ model.

above the transition temperature, defined as the maximum of s(T ). In particular, the rate of decrease of s(T ) above its maximum value turns to be a distinctive marker of the phase dynamics. In the high Jc JJ, the slope of s(T ) is much smaller when compared to those of moderately damped JJs, as shown in Fig. 4B where the temperature behavior of s is reported as a result of numerical simulations for different values of the Q factor. The smooth decrease of s(T ) cannot be described in terms of the intermediate dissipation regime: according to the RCSJ model, keeping all the other junction parameters fixed, an increase of Jc leads to an enhancement of Ic and Q. The increase of Q moves T  to higher values, and the negative slope of s(T ) above T  becomes steeper and steeper, as shown by the numerical simulations for Q ranging between 1 and 5 reported in Fig. 4B. The switching dynamics of JJs in the Jc interval 104–105 A/cm2 can be consistently reproduced by considering a transition driven by local heating events, as shown in Massarotti et al. (2015b) and Baumans et al. (2017) possibly induced by intrinsic inhomogeneous composition unavoidable for high Jc junctions. Large values of Joule power density deposited in the weak link can induce a self-heating process during the switch to the resistive branch (Courtois et al., 2008). The absence of a set of self-consistent electrodynamics parameters to describe the temperature behavior of the SCDs is a strong indication of the failure of the standard Josephson dynamics. For larger values of Jc, heating driven mechanisms become dominant with a transition to the normal state locally in the junction area. These events can be modeled as heating events, as confirmed by the details of the simulations which are reported by Massarotti et al. (2015b), Baumans et al. (2017), and Shah et al. (2008), in the sense that they are local processes, break the coherence of the phase information, and are described by a heat diffusion-like equation. As it occurs in superconducting nanowires, the temperature jump (T ) depends on temperature because the specific heat is strongly temperature-dependent. At low temperatures, the specific heat is quite low; thus with each heating event there is a considerable increase in the temperature. In addition, the thermal conductivity and the thermal capacity are both quite low as the system is deeply into the superconducting phase. The junction is rather isolated from the environment and the heating event is destructive for the superconducting state. For this reason a single heating event can induce a direct jump to the resistive state at low temperatures: it induces a large local heating that is difficult to dissipate (Massarotti and Tafuri, 2019; Massarotti et al., 2015b). At high temperatures, the temperature jump (T ) per heating event is rather small. In addition both the thermal conductivity and the thermal capacity increase with increasing temperature as well. Thermal diffusion and contact with the environment is more effective and multiple heating events are required for switching. This occurs at higher temperatures, where the derivative ds/dT is negative. Therefore, it turns out that SCDs allow to distinguish different dissipation processes: a JJ cannot sustain an unlimited increase in the critical current Ic, and thus in the quality factor Q, through larger critical current density Jc while still preserving all the properties of the Josephson effect and all the features of the underdamped regime (Massarotti and Tafuri, 2019). The classical Josephson phase dynamics, which takes place in junctions characterized by lower critical current densities Jc, is replaced at high Jc values by a regime driven by local heating events where phase information is lost. Nonequilibrium phenomena produce hysteretic I–V characteristics and modify the influence of dissipation, thus becoming measurable through modeling of the SCDs in terms of heating modes (Shah et al., 2008; Massarotti et al., 2015a, b)

Conclusion Every experiment or application using a superconducting weak link is based on how the phase difference ’ between the electrodes evolves in time and space. The large variety of barriers now available between the superconducting electrodes offer novel functionalities and efficient tuning of physical processes occurring at the nanoscale and at different interfaces. Measurements of SCDs focus on the moment at which resistance originates in superconducting weak links; thus they constitute a direct way of discriminating the phase dynamics and the transport also in nontrivial cases of Josephson nanodevices and hybrid systems (Feofanov et al., 2010; Inomata et al., 2005; Massarotti et al., 2015a; Goldobin et al., 2007; Sickinger et al., 2012; Goldobin et al., 2013; Ahmad et al., 2020). These junctions are going to be more and more common with advances in nanopatterning superconductors and in materials science; thus the range of the energy dynamical parameters is significantly enlarged. In this chapter

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we have represented well-defined criteria to identify different regimes of the phase dynamics. We believe that further progress in the field will lead to novel insights and a wide vision on macroscopic quantum phenomena in a variety of complementary systems, thus promoting novel arguments on the interplay of coherence and dissipation in solid state systems. Issues on a more detailed understanding of coherence, dissipation, and noise in the various devices play a significant role in the progress of quantum circuits (Larsen et al., 2015; de Lange et al., 2015; Wiedenmann et al., 2016; Ahmad et al., 2022).

References Ahmad H, Caruso R, Pal A, Rotoli G, Pepe GP, Blamire M, Tafuri F, and Massarotti D (2020) Physical Review Applied 13: 014017. Ahmad HG, Brosco V, Miano A, Di Palma L, Arzeo M, Montemurro D, Lucignano P, Pepe GP, Tafuri F, Fazio R, and Massarotti D (2022) Physical Review B 105: 214522. Bae MH, Sahu M, Lee HJ, and Bezryadin A (2009) Physical Review B 79: 104509. Barone A and Paternò G (1982) Physics and Applications of the Josephson Effect. John Wiley and Sons. Bauch T, Lombardi F, Tafuri F, Barone A, Rotoli G, Delsing P, and Claeson T (2005) Physical Review Letters 94: 087003. Bauch T, Lindstrom T, Tafuri F, Rotoli G, Delsing P, Claeson T, and Lombardi F (2006) Science 311: 56. Baumans XDA, Zharinov VS, Raymenants E, Alvarez SB, Scheerder JE, Brisbois J, Massarotti D, Caruso R, Tafuri F, Janssens E, Moshchalkov VV, Van de Vondel J, and Silhanek AV (2017) Scientific Reports 7: 44569. Ben-Jacob E, Bergman DJ, Matkowsky BJ, and Schuss Z (1982) Physical Review A 26: 2805. Bergeret FS and Ilic S (2023) ECMP, 2nd ed. Oxford: Elsevier. Borzenets IV, Coskun UC, Jones SJ, and Finkelstein G (2011) Physical Review Letters 107: 137005. Caldeira AO and Leggett AJ (1981) Physical Review Letters 46: 211. Clarke J and Wilhelm FK (2008) Nature (London) 453: 1031. Clarke J, Cleland AN, Devoret MH, Esteve D, and Martinis JM (1988) Science 239: 992. Courtois H, Meschke M, Peltonen JT, and Pekola JP (2008) Physical Review Letters 101: 067002. de Lange G, van Heck B, Bruno A, van Woerkom DJ, Geresdi A, Plissard SR, Bakkers EPAM, Akhmerov AR, and Di Carlo L (2015) Physical Review Letters 115: 127002. der Boer W and de Bruyn Ouboter R (1980) Physica B: Condensed Matter 98: 185. Devoret MH and Schoelkopf RJ (2013) Science 339: 1169. Devoret MH, Martinis JM, Esteve D, and Clarke J (1984) Physical Review Letters 53: 1260. Devoret MH, Martinis JM, and Clarke J (1985) Physical Review Letters 55: 1908. Dmitrenko IM, Khlus VA, Tsoi GM, and Shnyrkov VI (1985) Fizika Nizkikh Temperatur 11: 146 (Sov. J. Low Temp. Phys. 11, 77 (1985)). Ejrnaes M, Salvoni D, Parlato L, Massarotti D, Caruso R, Tafuri F, Yang XY, You LX, Wang Z, Pepe GP, and Cristiano R (2019) Scientific Reports 9: 8053. Fenton JC and Warburton PA (2008) Physical Review B 78: 054526. Feofanov AK, Oboznov VA, Bol’ginov VV, Lisenfeld J, Poletto S, Ryazanov VV, Rossolenko AN, Khabipov M, Balashov D, Zorin AB, Dmitriev PN, Koshelets VP, and Ustinov AV (2010) Nature Physics 6: 593–597. Fulton TA and Dunkleberger LN (1974) Physical Review B 9: 4760. Goldobin E, Koelle D, Kleiner R, and Buzdin AI (2007) Physical Review B 76: 224523. Goldobin E, Kleiner R, Koelle D, and Mints RG (2013) Physical Review Letters 111: 057004. Grabert H, Olschowski P, and Weiss U (1987) Physical Review B 36: 1931. Iansiti M, Johnson AT, Smith WF, Rogalla H, Lobb CJ, and Tinkham M (1987) Physical Review Letters 59: 489. Inomata K, Sato S, Nakajima K, Tanaka A, Takano Y, Wang HB, Nagao M, Hatano H, and Kawabata S (2005) Physical Review Letters 95: 107005. Jackel LD, Gordon JP, Hu EL, Howard RE, Fetter LA, Tennant DM, Epworth RW, and Kurkijarvi J (1981) Physical Review Letters 47: 697. Josephson BD (1962) Physics Letters 1: 251. Kautz RL and Martinis JM (1990) Physical Review B 42: 9903. Kivioja JM, Nieminen TE, Claudon J, Buisson O, Hekking FWJ, and Pekola JP (2005) Physical Review Letters 94: 247002. Kockum AF and Nori F (2019) Quantum Bits With Josephson Junctions, pp. 703–741. Cham: Springer International Publishing. Koshino M (2023) ECMP, 2nd ed. Oxford: Elsevier. Kramers HA (1940) Physica (Utrecht) 7: 284. Krasnov VM, Bauch T, Intiso S, Hürfeld E, Akazaki T, Takayanagi H, and Delsing P (2005) Physical Review Letters 95: 157002. Larsen TW, Petersson KD, Kuemmeth F, Jespersen TS, Krogstrup P, Nygard J, and Marcus CM (2015) Physical Review Letters 115: 127001. Li P, Wu PM, Bomze Y, Borzenets IV, Finkelstein G, and Chang AM (2011) Physical Review Letters 107: 137004. Likharev KK (1979) Reviews of Modern Physics 51: 101. Little WA (1967) Physics Review 156: 396. Longobardi L, Massarotti D, Rotoli G, Stornaiuolo D, Papari G, Kawakami A, Pepe GP, Barone A, and Tafuri F (2011a) Physical Review B 84: 184504. Longobardi L, Massarotti D, Rotoli G, Stornaiuolo D, Papari G, Kawakami A, Pepe GP, Barone A, and Tafuri F (2011b) Applied Physics Letters 99: 062510. Longobardi L, Massarotti D, Stornaiuolo D, Galletti L, Rotoli G, Lombardi F, and Tafuri F (2012) Physical Review Letters 109: 184504. Männik J, Li S, Qiu W, Chen W, Patel V, Han S, and Lukens JE (2005) Physical Review B 71: 220509. Massarotti D and Tafuri F (2019) Phase Dynamics and Macroscopic Quantum Tunneling, vol. 286. Cham: Springer. Massarotti D, Longobardi L, Galletti L, Stornaiuolo D, Montemurro D, Pepe GP, Rotoli G, Barone A, and Tafuri F (2012) Journal of Low Temperature Physics 38: 263. Massarotti D, Pal A, Rotoli G, Longobardi L, Blamire MG, and Tafuri F (2015a) Nature Communications 6: 7376. Massarotti D, Stornaiuolo D, Lucignano P, Galletti L, Born D, Rotoli G, Lombardi F, Longobardi L, Tagliacozzo A, and Tafuri F (2015b) Physical Review B 92: 054501. Murphy A, Weinberg P, Aref T, Coskun U, Vakaryuk V, Levchenko A, and Bezryadin A (2013) Physical Review Letters 110: 247001. Murphy A, Semenov A, Korneev A, Korneeva Y, Gol’tsman G, and Bezryadin A (2015) Scientific Reports 5: 10174. Ono RH, Cromar MW, Kautz RL, Soulen RJ, Colwell JH, and Fogle WE (1987) IEEE Transactions on Magnetics MAG-23: 1670. Prance RJ, Long AP, Clarke TD, Widom A, Mutton JE, Sacco J, Potts MW, Negaloudis G, and Goodall F (1981) Nature 289: 543. Sahu M, Bae MH, Rogachev A, Pekker D, Wei TC, Shah N, Goldbart PM, and Bezryadin A (2009) Nature Physics 5: 503. Salvoni D, Ejrnaes M, Gaggero A, Mattioli F, Martini F, Ahmad HG, Di Palma L, Satariano R, Yang XY, You L, Tafuri F, Pepe GP, Massarotti D, Montemurro D, and Parlato L (2022) Physical Review Applied 18: 014006. Shah N, Pekker D, and Goldbart PM (2008) Physical Review Letters 101: 207001. Sickinger H, Lipman A, Weides M, Mints RG, Kohlstedt H, Koelle D, Kleiner R, and Goldobin E (2012) Physical Review Letters 109: 107002. Stewart WC (1968) Applied Physics Letters 12: 277. Stornaiuolo D, Rotoli G, Massarotti D, Carillo F, Longobardi L, Beltram F, and Tafuri F (2013) Physical Review B 87: 134517.

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Tafuri F (2023) ECMP, 2nd ed. Oxford: Elsevier. Tinkham M (1996) Introduction to Superconductivity. New York: McGraw-Hill. Vion D, Gotz M, Joyez P, Esteve D, and Devoret MH (1996) Physical Review Letters 77: 3435–3438. Voss RF and Webb RA (1981) Physical Review Letters 47: 265. Washburn S, Webb RA, Voss RF, and Farris SM (1985) Physical Review Letters 54: 2712. Wiedenmann J, Bocquillon E, Deacon RS, Hartinger S, Herrmann O, Klapwijk TM, Maier L, Ames C, Brüne C, Gould C, Oiwa A, Ishibashi K, Tarucha S, Buhmann H, and Molenkamp LW (2016) Nature Communications 7(1): 10303. Yoon Y, Gasparinetti S, Mottonen M, and Pekola JP (2011) Journal of Low Temperature Physics 163: 164. Yu HF, Zhu XB, Peng ZH, Tian Y, Cui DJ, Chen GH, Zheng DN, Jing XN, Lu L, Zhao SP, and Han S (2011) Physical Review Letters 107: 067004. Zaikin A (2023) ECMP, 2nd ed. Oxford: Elsevier. Zgirski M, Foltyn M, Savin A, Naumov A, and Norowski K (2020) Physical Review Applied 14: 044024. Zgirski M, Foltyn M, Savin A, Naumov A, and Norowski K (2021) Physical Review B 104: 014506. Zhang Y, Liu G, and Lau CN (2008) Nano Research 1: 145–151. Zhang B, Zhang L, Chen Q, Guan Y, He G, Fei Y, Wang X, Lyu J, Tan J, Li H, Dai Y, Li F, Wang H, Yu S, Tu X, Zhao Q, Jia X, Kang L, Chen J, and Wu P (2022) Physical Review Applied 17: 014032. Zonda M, Belzig W, and Novotny T (2015) Physical Review B 91: 134305.

Fast dynamics of vortices in superconductors Oleksandr V Dobrovolskiy, Faculty of Physics, Nanomagnetism and Magnonics, Superconductivity and Spintronics Laboratory, University of Vienna, Vienna, Austria © 2024 Elsevier Ltd. All rights reserved.

Introduction Flux-flow instability models Flux-flow instability mechanisms Quasiparticle scattering mechanisms Larkin-Ovchinnikov model of FFI Refinements of the Larkin-Ovchinnikov model Quasiparticle relaxation in dirty and clean regimes High-Tc superconductors and ultraclean regime Pinning effects on the flux-flow instability Effects of uncorrelated disorder on FFI Low-field crossover in the instability velocity Pinning effects on self-heating and FFI Vortex lattice commensurability effects on FFI Fast dynamics of guided magnetic flux quanta From global to local instability models Local FFI model for thin films near Tc Rearrangement of vortices in the TDGL model Edge-controlled FFI in the weak-pinning regime Edge barrier effects on the fluxon speed limits Edge defects as gates for fast-moving fluxons Fast vortex dynamics under a microwave ac stimulus Enhancement of superconductivity at microwaves Condensate and quasiparticles under dc/ac biasing Microwave stimulation in the vortex state Braking of instabilities by a microwave stimulus Future directions Superconductivity in curved 3D nanoarchitectures Cryogenic magnonics and magnon fluxonics Conclusion Acknowledgments References

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Abstract The last decade has been marked by great interest in the dynamics of vortices moving at high (>10 km/s) velocities in superconductors. However, the flux-flow instability (FFI) prevents its exploration and sets practical limits for the use of vortices in various applications. Even so, FFI has turned into a valuable tool for studying the quasiparticle energy relaxation in superconductors, with the view of enhancement of the single-photon detection capability of micrometer-wide strips operated at large bias currents. In this context, the failure of “global” FFI models for materials with strong intrinsic and edge vortex pinning has urged the elaboration of “local” FFI models. This chapter outlines the recent advances in the research on FFI and highlights superconductors with perfect edges and weak volume pinning as prospective materials for studying fast vortex dynamics and nonequilibrium superconductivity.

Key points

• • • •

Flux-flow instability at high vortex velocities. Pinning effects on the flux-flow instability. Paradigm shift from global to local instability models. Fast vortex motion at a microwave ac stimulus.

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Introduction Fast vortex dynamics and related nonequilibrium phenomena are essential subjects of research in modern superconductivity (Bezuglyj et al., 2019; Budinska et al., 2022; Buh et al., 2015; Cherpak et al., 2014; Dobrovolskiy et al., 2020a; Dobrovolskiy et al., 2019a; Dobrovolskiy et al., 2020c; Embon et al., 2017; Grimaldi et al., 2015; Gurevich, 2014; Kogan and Nakagawa, 2021; Kogan and Nakagawa, 2022; Kogan and Prozorov, 2020; Lara et al., 2017; Lyatti et al., 2020; Madan et al., 2018; Pathirana and Gurevich, 2021; Puica et al., 2012; Sheikhzada and Gurevich, 2017; Sheikhzada and Gurevich, 2020; Wördenweber et al., 2012). High velocities of magnetic flux quanta (Abrikosov vortices or fluxons) attract great interest because of the fundamental questions regarding their stability as topological excitations of the superconducting order parameter c and the ultimate speed limits for magnetic flux transport via vortices at intense transport currents. Furthermore, fast dynamics of fluxons determines the vortex-assisted mechanism of the initial dissipation of superconducting microstrip single-photon detectors (Korneeva et al., 2020; Vodolazov, 2017) and makes accessible novel phenomena in hybrid superconductor-based systems, such as the generation of sound (Bulaevskii and Chudnovsky, 2005; Ivlev et al., 1999) and spin (Bespalov et al., 2014; Dobrovolskiy et al., 2021) waves, as well as emerging functionalities, e.g., in microwave radiation/detection (Bulaevskii and Chudnovsky, 2006; Dobrovolskiy et al., 2018; Dobrovolskiy et al., 2020b; Lösch et al., 2019). Probing the upper speed limits for vortices represents a valuable approach for studying the energy relaxation processes in superconductors (Attanasio and Cirillo, 2012; Knight and Kunchur, 2006; Leo et al., 2011; Leo et al., 2020a). This method allows one to quantify the lifetimes of quasiparticles (unpaired electrons) when their energy distribution is out of equilibrium. The physical phenomenon underlaying this approach is termed flux-flow instability (FFI) (Bezuglyj and Shklovskij, 1992; Kunchur, 2002; Larkin and Ovchinnikov, 1975; Larkin and Ovchinnikov, 1986). In the current-voltage (I–V) curve of superconductors, FFI becomes apparent as a discontinuous jump to a highly resistive state at some instability current I∗ which corresponds to the instability voltage V ∗, see Fig. 1A. Note that I∗ is usually smaller than the maximal current a superconductor can carry without dissipation—the depairing current Id, thereby setting the actual limit for the use of superconductors in applications. The instability (A)

(C)

(B)

(D)

Fig. 1 (A) Typical I–V curve of a superconductor in the mixed state, with the indicated instability current I∗ and instability voltage V ∗, and the regimes of pinned vortices (I), depinning transition (II), flux flow (III), nonlinear conductivity (IV), flux-flow instability (V), and highly-resistive state (VI). (B) Predictions of different FFI models for v∗(B). LO: (Larkin and Ovchinnikov, 1975), BS: (Bezuglyj and Shklovskij, 1992), D: (Doettinger et al., 1995), K: (Kunchur et al., 2001), G: (Grimaldi et al., 2010), S: (Shklovskij, 2017), V: (Vodolazov, 2019). (C) Different FFI regimes on the B-T plane. Bt: overheating field in the BS model (Bezuglyj and Shklovskij, 1992). (D) Temperature dependences of the inelastic electron-electron and electron-photon scattering rates. Tx: crossover temperature.

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voltage allows one to quantify the maximal vortex velocity (instability velocity) v∗ by using the standard relation v∗ ¼ V ∗/(BL), where B is the magnitude of the applied magnetic field and L the distance between the voltage leads. The understanding of the microscopic mechanisms determining the maximal vortex velocities and the behavior of v∗(B) is an essential task in the FFI research, see Fig. 1B. Reviews and discussions of FFI and nonlinear effects occurring during fast vortex motion are given in Attanasio and Cirillo (2012), Dean and Kunchur (2018), Huebener (2019), Kunchur and Knight (2003), Misko et al. (2007) and Vodolazov (2020). This chapter presents (i) a summary of the different FFI models with emphasis on the links between them, (ii) an outline of the vortex-pinning effects on FFI and the associated paradigm shift from global (Bezuglyj and Shklovskij, 1992; Larkin and Ovchinnikov, 1975; Larkin and Ovchinnikov, 1986) to local (Bezuglyj et al., 2019; Vodolazov, 2019) FFI models, and (iii) some of the prospective directions in the fast dynamics research.

Flux-flow instability models Flux-flow instability mechanisms Theoretical foundations for understanding of the electronic nonequilibrium effects were laid by Eliashberg (1970) and Eliashberg and Ivlev (1986). Due to external energy input, the quasiparticle energy distribution is modified with respect to equilibrium, which affects the superconducting parameters such as critical temperature Tc, critical current Ic, and superfluid density, and can even result in their enhancement (Wyatt et al., 1966). For instance, the quasiparticle energy distribution can be modified via irradiation with light, microwaves, phonons, injection of quasiparticles, etc. (Gray, 1981). Experimental studies of the photon absorption response and the effects of high-frequency irradiation of superconductors are usually performed at zero external magnetic field (Natarajan et al., 2012; Zolochevskii, 2013). By contrast, nonequilibrium effects in the mixed state and the vortex motion under intense transport currents and FFI are studied in the presence of external magnetic fields (Gray, 1981). The FFI mechanisms are distinct at high (T  Tc) and low (T  Tc) temperatures, see Fig. 1C. Close to Tc, FFI can be described by the Larkin-Ovchinnikov (LO) theory (Larkin and Ovchinnikov, 1975; Larkin and Ovchinnikov, 1986). In the LO theory, FFI is associated with a shrinking of the moving vortex and a reduction of the viscosity of the superconducting medium due to the diffusion of quasiparticles from the vortex core to the surroundings. Well below Tc, the LO mechanism is ineffective since in this case the superconducting energy gap D does not depend strongly on small changes of the electron distribution function. At T  Tc, FFI is described within the framework of the Kunchur (K) hot-electron model in which the electronic temperature Te is elevated due to dissipation. The higher Te stipulates the creation of additional quasiparticles, which diminishes D. This leads to an expansion of the vortex core (Kunchur et al., 2001; Kunchur et al., 2002), which has the effect of reducing the viscous drag because of the softening of gradients of the vortex profile (Kunchur, 2002). The LO model (Larkin and Ovchinnikov, 1975) is justified when the inelastic electron-electron scattering time tee is much larger than the electron-photon scattering time tep, tee  tep, which ensures a non-thermal electron energy distribution. By contrast, tee  tep in the K model implies a thermal-like electron distribution so that the electronic ensemble exhibits a shift of the electronic temperature Te with respect to the phonon temperature Tp (Kunchur et al., 2001).

Quasiparticle scattering mechanisms 2 3 3 −1 2 Standard estimates (Abrikosov, 1988; Kaplan et al., 1976) for the scattering rates t−1 ee ¼ kBT /rħeF and tep ¼ kBT /r ħTD [eF: Fermi 2 2 energy, TD: Debye temperature] give a crossover temperature Tx ¼ r kBTD/eF. Here, r < 1 is the phonon reflection coefficient at the film-substrate interface, arising from their acoustic mismatch. Thus, one has tee > tep above Tx which can be viewed as a crossover temperature between the K and LO models, see Fig. 1D. For the classical superconductors, the electron-phonon interaction is usually dominant compared to the electron-electron interaction (Aronov and Spivak, 1978). By contrast, the electron-electron interaction is more important in materials with a high TD and in high-Tc superconductors (Doettinger et al., 1997). For instance, Tx ’ 1 K for Nb makes accessible both K and LO regimes (Leo et al., 2011). By contrast, Tx ’ 100 K for YBCO points to the dominating electron-electron scattering in the entire temperature range of the superconducting state (Knight and Kunchur, 2006). Finally, Tx ’ 0.5 K for amorphous superconductors like MoGe (Liang and Kunchur, 2010) suggests that the LO mechanism dominates in a broad range of temperatures of the superconducting state.

Larkin-Ovchinnikov model of FFI The key point of the LO theory (Larkin and Ovchinnikov, 1975; Larkin and Ovchinnikov, 1986) is illustrated in Fig. 2A. At high vortex velocities, the quasiparticles within the vortex core are accelerated toward the core boundary, where they undergo Andreev reflection (Andreev, 1964; Blonder et al., 1982; Octavio et al., 1983) and alternate between electrons and holes. As a result, their energy increases and they eventually escape from the vortex core. This leads to a shrinking of the vortex core and the vortex viscosity  becomes a nonmonotonic function of the velocity v, namely (v) ¼ (0)/[1 + (v/v∗)2] (Larkin and Ovchinnikov, 1975; Larkin and Ovchinnikov, 1986), so that the viscous force Fv ¼ (v)v has a maximum at v∗. At velocities exceeding v∗, Fv decreases, leading to an even further increase of v, causing an instability in the vortex motion. The nontrivial nature of the LO effect should be emphasized especially. If one uses the Bardeen-Stephen expression (Bardeen and Stephen, 1965) for  ¼ F20/(2px2rn) [F0: magnetic flux quantum, x: coherence length, rn: resistivity in the normal state], one finds that it actually increases if the size of the vortex core (2x) decreases. By contrast, in the LO model, the vortex core shrinks and this

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(A)

(B)

Fig. 2 Schematics of the vortex core shapes in the LO (Klein et al., 1985; Larkin and Ovchinnikov, 1975) and VP (Vodolazov and Peeters, 2007) models assuming spatially uniform and nonuniform nonequilibrium quasiparticle distributions, respectively. (A) The electric field generated by fast vortex motion raises the energy of the quasiparticles trapped in the core. During this process the particle character alternates between electron-like and hole-like due to Andreev reflections. The vortex core shrinks in consequence of the quasiparticle escape. (B) Deformation of the vortex core due to vortex motion. The color depth indicates the density of the quasiparticles. In case the quasiparticle diffusion length is smaller than x, diffusion of the quasiparticles is not strong and locally there is an effective cooling and heating of the quasiparticles.

results in a decrease of . This contradiction can be explained by the failure of the Bardeen-Stephen model for the vortex as a normal cylinder with radius of the order of x when the vortex is moving with a high enough velocity v. The essential assumption of the LO theory is a spatially uniform nonequilibrium quasiparticle distribution function f(E) in the superconductor. Accordingly, while the vortex core shrinks as the vortex is accelerated to high velocities, its shape remains axially symmetric in the LO model, see Fig. 2A. Moreover, FFI nucleates at all points of the superconductor at the same time and this leads to a field-independent v∗ at which FFI occurs, see Fig. 1B. Thus, all refinements of the LO model dealing with the spatially uniform nucleation of FFI can be termed global FFI models.

Refinements of the Larkin-Ovchinnikov model Early experiments on low-Tc superconductors confirmed the LO prediction of FFI (Klein et al., 1985; Musienko et al., 1980; Volotskaya et al., 1992). It was revealed that FFI occurs at magnetic fields B ≲ 0.7Bc2 (Babic et al., 2004), where Bc2 is the upper critical field of the superconductor. From the good quantitative agreement between theory and experiment in large magnetic fields, it was concluded (Klein et al., 1985; Musienko et al., 1980) that the condition of spatially uniform f(E) is well satisfied when the pffiffiffiffiffiffiffiffiffiffiffi ∗ distance between vortices aðBÞ ’ pF 0 =B  Le  v te [Le: quasiparticle diffusion length, te: quasiparticle energy relaxation time]. ffiffiffiffiffiffiffiffi However, at B  Bc2, v ∗ ðBÞ  1=B was observed in many experiments (Doettinger et al., 1994; Doettinger et al., 1995; Lefloch et al., 1999), see Fig. 1B. The observed dependence was explained as a consequence of the crossover from a uniform to a nonuniform pffiffiffiffiffiffiffiffi distribution of nonequilibrium quasiparticles when Le ¼ Dte [D: electron diffusion coefficient] becomes comparable with the intervortex distance, v∗ te ’ a(B). Doettinger et al suggested (Doettinger et al., 1995) that at smaller fields the system can be recovered to a spatially homogeneous state by allowing v∗ to grow accordingly to the increase of a with decrease of the applied pffiffiffi 1=2  pffiffiffiffiffiffiffiffi . The LO expression was complemented (Doettinger et al., 1995) with the term a= Dte, yielding magnetic field, a ¼ 2F0 = 3B " #1=2   ð1 −t Þ1=2 D½14zð3Þ1=2 a v ¼ 1 + pffiffiffiffiffiffiffiffi , (1) pte Dte pffiffiffi where t ¼ T/Tc and z(x) is the Riemann function. Note that a v ∗ ðBÞ∝1= B dependence appears in other FFI models as well. In the original LO theory, the nonlinear flux-flow behavior is due only to the electric field-induced change in f(E) and not Joule heating. Bezuglyj and Shklovskij (BS) extended the LO theory in the thin-film configuration by taking into account Joule heating effects (Bezuglyj and Shklovskij, 1992). BS found that even if the Joule heating of the lattice is negligible, quasiparticles can still experience an overheating due to the finite heat removal rate of the power dissipated in the sample which can trigger FFI. Thus, in the BS model, f(E) depends on the vortex density and the rate of heat removal from the film to the substrate. BS used the electronic temperature approximation, i.e. the assumption that the quasiparticles temperature Te is uniform throughout the film thickness due to a high electron thermal conductivity. Thus, if the film thickness d is much smaller than the electron-phonon scattering length (that usually occurs in thin films), the nonequilibrium phonons can escape from the film without re-absorption leading to the electron overheating (Bezuglyj and Shklovskij, 1992). BS introduced a transition magnetic field Bt, see Fig. 1C, which depends on the heat transfer coefficient between the film and the substrate (Bezuglyj and Shklovskij, 1992). At fields B < Bt the hypothesis of a uniform quasiparticles distribution in the LO model remains justified. By contrast, at B  Bt dissipation during the flux flow raises the electronic temperature Te and the onset of FFI is governed by thermal effects. The BS model yields v∗(B) ∝ B−1/2, see Fig. 1B. Note just the same dependence v∗(B) ∝ B−1/2 is found in the K model (Kunchur et al., 2001) at T  Tc in which thermal effects diminish the superconducting order parameter and lead to an expansion of the vortex cores. However, in the low-temperature regime, the quasiparticles heating reduces Bc2 to Bc2(Te) < Bc2. ∗

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Quasiparticle relaxation in dirty and clean regimes The schematics of the vortex core in Fig. 2A allows one to consider distinct quasiparticle energy relaxation processes for dirty and clean superconductors (Peroz and Villard, 2005). When the mean electron free path l is shorter than the coherence length (dirty limit, l < x), the energy relation occurs via the scattering process inside the vortex core. In the opposite case (clean limit, l > x) the quasiparticles can relax their energy by recombination outside the vortex core only after covering a distance within the core equivalent to several x. In the microscopic picture where the quasiparticle energy rise comes from Andreev reflections (Andreev, 1964; Blonder et al., 1982; Octavio et al., 1983) at the vortex core boundary, see Fig. 2A, an increase of the quasiparticles energy occurs well below v∗ in clean samples so that v∗ values in the clean limit are noticeably smaller than in the dirty limit (Peroz and Villard, 2005). Indeed, in the dirty limit, more energy is needed to induce the depletion of quasiparticles of the core. The shrinking of the vortex core occurs at higher energy and te is substantially temperature independent. In the clean limit, the quasiparticle distribution function is influenced by the recombination of quasiparticles to Cooper pairs. This means that in the clean limit te is strongly temperature dependent since the quasiparticles recombination mechanism obeys an exponential law (Doettinger et al., 1997; Liang and Kunchur, 2010; Peroz and Villard, 2005).

High-Tc superconductors and ultraclean regime FFI was also extensively investigated for high-temperature superconductors (HTSs). These studies were performed for epitaxial (Cherpak et al., 2014; Doettinger et al., 1994; Doettinger et al., 1995; Xiao and Ziemann, 1996), vicinal (Puica et al., 2012) and He ion-irradiated (Xiao et al., 1996) YBa2Cu3O7− d films, MgB2 (Kunchur et al., 2003), Bi2Sr2CaCu2O8+ d (Xiao et al., 1998), La1.85Sr0.15CuO4− x (Doettinger et al., 1997), Nd2− xCexCuOy (Huebener et al., 1999; Stoll et al., 1999; Stoll et al., 1998) films and YBa2Cu3O7− d nanowires (Lyatti et al., 2018; Rouco et al., 2018). The distinct effects of intrinsic (nonequilibrium) and extrinsic (thermal runaway) mechanisms driving FFI were discussed for YBa2Cu3O7− d films (Maza et al., 2008) and for iron based superconductors (Leo et al., 2016). Signatures of FFI in the anisotropic FeSeTe have allowed for its treatment halfway between low-Tc and high-Tc superconductors. In the dirty limit, the energy distribution of the quasiparticles in the vortex core is strongly smeared out due to their high scattering rate. The separation between the quasiparticles energy levels and between the Fermi energy xF and the lowest bound state in the vortex core is proportional to x−2 (Huebener, 2019). With an estimate x ≲ 10 nm for disordered low-temperature superconductors (Córdoba et al., 2019; Korneeva et al., 2020; Porrati et al., 2019), the core can be treated as an energy continuum of quasiparticles. By contrast, the extremely small x ≲ 1 nm in the HTS cuprates makes the electron quantum structure of the vortex core essential. For its experimental observation, the energy smearing de ¼ ħ/t due to the mean electronic scattering time t must be sufficiently small, de  D2/eF, corresponding to the ultraclean superconducting limit. Such a regime has been experimentally realized in Huebener et al. (1999), Stoll et al. (1999) and Stoll et al. (1998). Proceeding from an isolated vortex to the vortex lattice, the discrete energy levels within the vortex core interact between neighboring vortices and broaden into mini-bands (Huebener, 2019). In this case, vortex motion at high velocities is accompanied by emerging phenomena, such as Bloch oscillations of the quasiparticles, negative differential resistance, and multi-step instabilities (Huebener, 2019; Huebener et al., 1999; Stoll et al., 1999; Stoll et al., 1998).

Pinning effects on the flux-flow instability Effects of uncorrelated disorder on FFI The LO model considered a moving vortex lattice in an infinite superconductor in the Wigner-Seitz approximation and ignored collective effects related to vortex pinning and lattice transformations. The assumption of no pinning corresponds to a d-functional velocity distribution in the vortex ensemble and a free flux flow with zero critical current (Liang et al., 2010). In this case, an increase of current produces a shift of the velocity distribution to higher mean values hvi until the vortex instability occurs at v∗ for all vortices simultaneously. Note, it is the mean value hvi which is deduced from the dc voltage measured experimentally. In disordered systems, a rather narrow distribution of vortex velocities, see Fig. 3A, can be realized in a weak-pinning regime or at high enough fields and currents such that the vortex-vortex interaction is stronger than the vortex-pinning interaction (Blatter et al., 1994). In particular, an increase of the current I above Ic produces a shift of the velocity distribution toward higher mean values hvi and progressively narrows the distribution peak in consequence of the enhanced long-range order in the moving vortex lattice (Koshelev and Vinokur, 1994). The FFI jump occurs at the end of the quasilinear flux-flow regime, see Fig. 3B. The commonly occurring vortex dynamics regimes are indicated in Fig. 1A. The introduction of strong pinning by uncorrelated disorder substantially modifies the previous picture (Leo et al., 2010; Ruck et al., 2000; Shklovskij et al., 2017; Silhanek et al., 2012). As pointed out by Silhanek et al. (2012), on the one hand, the presence of disorder broadens the velocity distribution and extends the nonlinear regime for currents above the critical value Ic (Grimaldi et al., 2009b). This is schematically shown in Fig. 3C and D. On the other hand, strong pinning increases Ic and causes FFI to occur within the depinning-mediated nonlinear section of the I–V curve. Importantly, the broadening of the velocity distribution implies a sizable separation between hvi and the maximal attainable vortex velocity v∗. This means that it is the vortices which move faster than other vortices in the ensemble which actually trigger FFI. Once FFI is triggered in the regions of faster-moving vortices it is then spread over the entire volume of the superconductor.

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Fast dynamics of vortices in superconductors

(A)

(B)

(C)

(D)

Fig. 3 Schematic representation of different dynamic regimes after Silhanek et al. (2012). Velocity distribution for the case of flux flow (A) and plastic flow (C). Expected I–V curves for weak (B) and strong (D) pinning strengths. The critical current density Ic indicates the onset of dissipation. The instability current I∗ and voltage V ∗ indicate where FFI takes place.

Thus, in this plastic vortex flow regime, the system reaches the instability point at a lower vortex velocity than expected, practically when vmax ’ v∗. In this case, the LO assumption of homogeneous quasiparticle distribution in the whole sample has to be dropped and instead homogeneous quasiparticle distribution only along the paths where the vortices move has to be considered (Silhanek et al., 2012). As the magnetic field increases, the distance between the vortices decreases and the vortex-vortex interaction increases, which in turn effectively diminishes the role of pinning. This narrows the vortex velocity distribution so that v∗ increases. With a further increase of the magnetic field, the assumption of homogeneous quasiparticle distribution becomes justified again, and the LO scenario is retained. This gives rise to qualitatively different dependences of v∗(B), see Fig. 1B, and enables controlling the flux-flow dissipation via vortex pinning engineering in superconductors (Grimaldi et al., 2012; Ruck et al., 2000).

Low-field crossover in the instability velocity A crossover to a different behavior of v∗(B) at magnetic fields B < Bcr1 was reported by Grimaldi et al. (2009a) and Grimaldi et al. (2010) for Nb films with moderately strong intrinsic pinning. This field Bcr1 corresponds to a crossover from the regime where v∗ increases with increasing magnetic field to the regime v∗ ∝ B−1/2, see Fig. 1B. Magneto-optical imaging of the flux penetration into the films at B  Bcr1 revealed that the flux does not penetrate with a smooth advancing front, but instead as a series of irregularly shaped protrusions. This causes the formation of an inhomogeneous magnetic field distribution and induces preferential stationary channels for the moving vortices with a total effective width l (Grimaldi et al., 2010). An inspection of the films by field emission scanning electron microscopy revealed that these channels are grouped together in clusters along the strip (Grimaldi et al., 2010). This means that the voltages measured across the distance L between the voltage leads are actually produced across an effective distance l < L, so that in the low-field range B < Bcr1, the flux-flow channels fill only parts of the whole strip length. On the contrary, when the flux penetration becomes smooth, above some threshold temperature T∗ (’ 0.75Tc for the films studied in (Grimaldi et al., 2010)), the pinning strength decreases so the vortex flow becomes uniform as well. In this way, below T∗, the decreasing low-field dependence v∗(B) is preserved, whereas above T∗, the crossover field disappears, leading to a monotonous decrease of v∗(B) ∝ B−1/2, see Fig. 1B and C. Herewith, Bcr1 corresponds to a crossover from the channel-like vortex motion to a uniform flux-flow state with increase of the magnetic field. Accordingly, at T < T∗, the flux-flow resistance rff assumes lower values than the Bardeen-Stephen prediction while the expected linear rff(B) ∝ B is retained at T > T∗ (Grimaldi et al., 2010).

Pinning effects on self-heating and FFI The effects of pinning and self-heating on FFI in thin films were considered theoretically by Shklovskij et al for the cases T  Tc (Shklovskij, 2017) and T  Tc (Shklovskij et al., 2017). The problem was considered in the single-vortex approximation, upon the nonlinear I–V curves derived for a saw-tooth (Shklovskij and Dobrovolskiy, 2006) and cosine (Shklovskij and Dobrovolskiy, 2008) pinning potentials.

Fast dynamics of vortices in superconductors

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In the low-temperature regime, within the framework of the K model (Kunchur, 2002), the presence of pinning has been revealed (Shklovskij, 2017) to not modify the magnetic field dependences of the electric field E∗(B), current density j∗(B) and resistivity r∗ at the instability point, which remain monotonic. By contrast, the derivative of the instability velocity dv∗/dB may change its sign twice, as observed in experiments (Dobrovolskiy et al., 2017b; Grimaldi et al., 2009a; Grimaldi et al., 2011; Grimaldi et al., 2012; Grimaldi et al., 2010; Leo et al., 2010; Leo et al., 2020b; Silhanek et al., 2012). In particular, experiments on weak-pinning 5 mm-wide Mo3Ge thin films (Leo et al., 2020b) revealed that the observed nonmonotonic v∗(B) dependence can be described well with the theoretical dependences of Shklovskij (2017). For wider films, that is in the absence of the surface pinning caused by the mesoscopic geometry, the experimentally observed (Leo et al., 2020b) dependence v∗(B) ∝ B−1/2 fits the K model (Kunchur et al., 2001). In the high-temperature regime, within the framework of the LO theory (Larkin and Ovchinnikov, 1975) and its generalization by BS (Bezuglyj and Shklovskij, 1992), the increase of the pinning strength does not change the field dependences of the electric field strength E∗(B) and the current density j∗(B) qualitatively. At a fixed magnetic field value, E∗ and v∗ decrease while j∗ increases with increasing magnetic field. In this way, the theoretical analysis of Shklovskij (2017) and Shklovskij et al. (2017) corroborates that the presence of pinning leads to a reduction of v∗, with exception of highly-correlated, periodic pinning-induced regimes in the vortex dynamics as outlined next.

Vortex lattice commensurability effects on FFI In the presence of a periodic pinning, the spatial commensurability between the vortex lattice with the pinning landscape leads to magnetoresistance minima and critical current maxima (Cuadra-Solis et al., 2014; Dobrovolskiy et al., 2020a; Dobrovolskiy et al., 2012; Dobrovolskiy et al., 2018; Dobrovolskiy et al., 2017b; Jaque et al., 2002; Lu et al., 2007; Villegas et al., 2003). In this regime, v∗ attains a maximum when the spacing between the rows of vortices driven across an array of nanogrooves is commensurate with the distance between them (Dobrovolskiy et al., 2017b). The location of this matching peak can be tuned through the nanostructure period variation. The considered case can be realized for nanopatterned superconductors at the fundamental matching field when vortex dynamics is in a coherent regime, i.e., the entire vortex ensemble behaves as a vortex crystal (Cuadra-Solis et al., 2014; Dobrovolskiy et al., 2020a; Lu et al., 2007; Villegas et al., 2003). At the same time, the background pinning due to undesired random disorder must be weak to ensure the long-range order in the vortex lattice in the vicinity of the depinning transition. This condition can be realized, e.g., in weak-pinning amorphous Mo3Ge (Grimaldi et al., 2015), Nb7Ge3 (Babic et al., 2004), MoSi (Budinska et al., 2022), nanocrystalline Nb–C (Dobrovolskiy et al., 2020c), as well as Al (Silhanek et al., 2012) and epitaxial Nb films in the clean superconducting limit (Dobrovolskiy and Huth, 2012).

Fast dynamics of guided magnetic flux quanta An interesting possibility to enhance the long-range order in the moving vortex lattice is offered by the vortex guiding effect (Dobrovolskiy et al., 2010; Niessen and Weijsenfeld, 1969; Shklovskij and Dobrovolskiy, 2006). This effect consists in the noncollinearity of the velocity of vortices with the driving force exerted on them by the transport current. In the regime of guiding under the action of the driving Lorentz-type force FL directed at a small tilt angle a with respect to linearly extended “pinning sites”, see Fig. 4A, all vortices move along the pinning channels and the distribution of their velocities is close to the d-functional shape. The small angle a ensures that the normal component of the driving force FL sin a is not large enough to let vortices overcome the potential troughs, while the tangential component FL cos a is large enough to drive the vortices at a few km/s velocities (Dobrovolskiy et al., 2019a). The presence of the vortex velocity component along the sample length in the guiding regime leads to the appearance of a transverse voltage. This voltage vanishes upon a transition from the vortex guiding regime to a regime in which vortices start overcoming the pinning potential barriers (Reichhardt and Reichhardt, 2008; Shklovskij and Dobrovolskiy, 2006). By guiding magnetic flux quanta at a tilt angle of 15 degrees with respect to Co nanostripe arrays deposited on top of Nb films, a fivefold enhancement of v∗ (up to 5 km/s at low magnetic fields) was observed experimentally (Dobrovolskiy et al., 2019a). In a different experiment, I–V measurements on tracks perpendicular to the vicinal (8 degrees) step direction of YBa2Cu3O7− d films (current crossed between ab planes) provided evidence for the sliding motion along the ab planes (vortex channeling) at velocities exceeding 8 km/s (Puica et al., 2012).

From global to local instability models Local FFI model for thin films near Tc Distinct from the nonlinear conductivity regime considered by LO, FFI jumps were also observed in linear I–V sections (Grimaldi et al., 2011; Volotskaya et al., 1992). A broad distribution of vortex velocities caused by the presence of regions with different pinning strengths is a likely cause for such a regime. Namely, on average, an essentially larger number of slowly moving vortices can make a larger contribution to the measured voltage as compared to the contribution of a much smaller number of faster moving vortices, allowing the I–V curve to maintain a linear shape up to the instability point. The respective local FFI theory was developed

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Fast dynamics of vortices in superconductors

(A)

(B)

Fig. 4 (A) Schematic of a vortex lattice which is commensurate with a periodic washboard pinning landscape. When the vortices are guided at some tilt angle a with respect to the pinning channels, the long-range order in the vortex lattice is enhanced (Dobrovolskiy et al., 2020a; Dobrovolskiy et al., 2019a). (B) The combination of a weak intrinsic pinning, close-to-depairing critical current and fast relaxation of nonequilibrium allows for ultra-high vortex velocities (Dobrovolskiy et al., 2020c). The nucleation of FFI occurs at the film edge of the film and is decisively controlled by its quality (Budinska et al., 2022; Vodolazov, 2019).

by Bezuglyj et al. (2019), based on the assumption that FFI occurs upon reaching I∗ not in the entire superconducting film but in a narrow strip with weak pinning across the film width in which vortices move much faster than outside it. As a consequence of the overheating of the local areas with faster moving vortices, normal domains can be formed across the superconducting film. Once the current density exceeds some threshold current density (determining the equilibrium of the non-isothermal normal/superconducting boundary) (Bezuglyj and Shklovskij, 1984; Gurevich and Mints, 1984) these domains begin to grow and the whole sample transits to the normal state (Bezuglyj et al., 2019). In this case, the standard relation v∗ ¼ V ∗ /(BL) can no longer yield the instability velocity quantitatively. Nevertheless, the functional relations between the LO instability parameters in the case of a local FFI remain the same as in the case of a global FFI, but require renormalization of the sample length (Bezuglyj et al., 2019).

Rearrangement of vortices in the TDGL model The LO model assumes that the vortex lattice does not exhibit any structural changes upon the transition of the superconductor to a state with a resistance close to the normal value. However, experiments on low- and high-temperature superconductors suggested that some kind of phase transition may occur in the fast-moving vortex lattice and regions with fast and slow vortex motions may appear in the sample (Adami et al., 2015; Grimaldi et al., 2015). An alternative view of the moving vortex at high velocities, in which it loses the shape of a rigid cylinder, was used by Vodolazov and Peeters (2007). Reducing the problem to a two-dimensional one, VP solved numerically the generalized time-dependent Ginzburg-Landau (TDGL) equations (Kramer and Watts-Tobin, 1978; Watts-Tobin et al., 1981) which take into account the spatially nonuniform distribution of quasiparticles as the longitudinal (odd in energy E) part fL(E) ¼ f(−E)− f(E) of the nonequilibrium f(E) distribution is localized only in the region where the time derivative ∂| c|/∂ t is finite, that is only near the moving vortex core, see Fig. 2B. Indeed, the motion of a vortex implies a suppression of the order parameter c in front of the vortex and its recovery behind it. If the vortex velocity is large enough, v ’ x/te, the number of quasiparticles in front of the vortex becomes smaller than the equilibrium value and it becomes larger behind the vortex due to the finite relaxation time of f(E). Effectively, this can be viewed as cooling of the quasiparticles in front of the vortex and heating behind the vortex. Since the relaxation time of c depends on temperature as t|c |1/(Tc −T ), the healing time of the order parameter behind the vortex is long while the time of the order parameter suppression in front of the vortex is short. This leads to an elongated shape of the vortex core with a point where | c | ¼ 0 displaced in the direction of the vortex motion.

Fast dynamics of vortices in superconductors

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The elongated shape of the core of a fast-moving vortex is in line with the field distribution around it as derived analytically based on the time-dependent London (TDL) equation (which ignores the vortex core) (Kogan and Prozorov, 2020). The flux quantum of a moving vortex is redistributed: The back-side part of the flux is enhanced, whereas the in-front part is depleted (Kogan and Nakagawa, 2021). Distortions of the field distribution of single moving vortices lead to distorted intervortex interactions and therefore to a change in the vortex lattice structure (Kogan, 2018). At large velocities, the moving vortex lattice adopts the structure with one of the lattice vectors along the velocity, the same result as obtained within TDGL (Li et al., 2004; Vodolazov and Peeters, 2007).

Edge-controlled FFI in the weak-pinning regime In the Knight and Kunchur (2006), Kunchur (2002), Shklovskij (2017), Bezuglyj and Shklovskij (1992) and Shklovskij et al. (2017) models the temperature of the electrons Te was calculated from the balance between the spatially- and time-averaged Joule dissipation and the heat removal to the substrate, which ignores the discrete nature of the moving vortices. Vodolazov (2019) solved the TDGL equation in conjunction with the heat conductance equation for Te and the energy balance equation to find the phonon temperature Tp, which were derived from the kinetic equations for the electron and phonon distribution functions (Vodolazov, 2017). In the V model, in the low-resistive state, there is a temperature gradient across the width of the strip with maximal local temperature near the edge where vortices enter the sample (Vodolazov, 2019). The higher temperature at the edge is caused by the larger current density in the near-edge area due to the presence of the edge barrier for vortex entry and, hence, the locally larger Joule dissipation. With increase of the current, there is a series of transformations of the moving vortex lattice in the V model. At I ≲ I∗, localized areas with strongly suppressed superconductivity and closely spaced vortices appear near the hottest edge (left edge in Fig. 4B). Upon reaching I∗, these areas begin to grow in the direction of the opposite edge and form highly resistive Josephson SNS-like links—vortex rivers—along which the vortices move. Such vortex rivers were observed experimentally by scanning local probes (Embon et al., 2017; Silhanek et al., 2010). Due to the increasing dissipation, vortex rivers evolve into normal domains which then expand along the strip (Dobrovolskiy et al., 2020c). Distinct from the modified LO model (Doettinger et al., 1995), in the V model v∗  B−1/2 only at comparatively large magnetic in the moving fields (see Fig. 1B), when the intervortex distance at I ’ Ic and I ’ I∗ is almost the same despite the transformations pffiffiffi vortex lattice. By contrast, at relatively small magnetic fields, a in the vortex rows is smaller than (2F0 =B 3)1/2 at I ’ I∗ and the number of vortices n is smaller than follows from nF0 ¼ BS [S: sample area]. This leads to a weaker dependence v∗(B) than v∗  B−1/2 following from “global” instability models. This behavior is confirmed experimentally (Dobrovolskiy et al., 2020c).

Edge barrier effects on the fluxon speed limits The prediction (Vodolazov, 2017) and experimental confirmation (Charaev et al., 2020; Chiles et al., 2020; Korneeva et al., 2020; Korneeva et al., 2018) of the single-photon detection capability of micrometer-wide strips has turned FFI into a widely used method for judging whether a superconducting material could be potentially suitable for single-photon detection (Caputo et al., 2017; Cirillo et al., 2021; Dobrovolskiy et al., 2020c; Lin et al., 2013). This assessment is based on the deduction of the quasiparticle relaxation times from FFI. However, while many dirty superconductors (NbN (Korzh et al., 2020), MoSi (Korneeva et al., 2020), NbRe (Cirillo et al., 2020), NbReN (Cirillo et al., 2021), etc.) possess good single-photon detection capability, v∗ and te deduced from FFI are not always consistent with the fast relaxation processes implied by photon-counting experiments (Budinska et al., 2022). The edge quality turns out to be decisive for the nucleation of FFI and, hence, quantification of the relaxation times from the I–V measurements (Budinska et al., 2022). By investigating FFI in wide MoSi strips differing only by the edge quality, a factor of 3 larger critical currents Ic, a factor of 20 higher maximal vortex velocities of 20 km/s, and a factor of 20 shorter te have been revealed for MoSi films with perfect edges. Thus, for the deduction of the intrinsic te of the material from the I–V curves, utmost care should be taken regarding the edge and sample quality, and such a deduction is justified only if the field dependence of Ic points to the dominating edge pinning of vortices (Budinska et al., 2022).

Edge defects as gates for fast-moving fluxons The requirement of close-to-depairing critical currents Ic in superconductor strips is related to blocking of the penetration of vortices via the strip edges and knowledge of the effects of various edge defects on the penetration and patterns of Abrikosov vortices (Aladyshkin et al., 2001; Glazman, 1986; Sivakov et al., 2018; Vodolazov et al., 2015; Vodolazov et al., 2003). Once the Lorentz force exerted on a vortex by the transport current exceeds the force of attraction of the vortex to the sample edge, the edge barrier is suppressed. This suppression can be local in the case of a local increase of the current density—current-crowding effect (Adami et al., 2013), and it can be realized, e.g., in strips with an edge defect (Budinska et al., 2022; Dobrovolskiy et al., 2020c). In this case, the

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defect acts as a gate (Aladyshkin et al., 2001) for vortices entering into the superconductor strip and crossing it under the competing action of the Lorentz and vortex-vortex interaction forces. If the size of the defect is much larger than x, the defect can serve as a nucleation point for several vortex chains (Budinska et al., 2022; Embon et al., 2017). Such chains form a vortex jet with the apex at the defect and expanding due to the repulsion of vortices as they move to the opposite film edge. If the defect size is ’ x, the vortices enter into the strip consequentially. However, in the presence of fluctuations and inhomogeneities in the strip, the vortex chain evolves into a diverging jet because of the intervortex repulsion. Vortices penetrating into a superconducting Pb film at rates of tens of GHz and moving with velocities of up to tens of km/s were observed by scanning SQUID-on-tip (SOT) microscopy (Embon et al., 2017). Though these vortices move faster than the perpendicular current superflow which drives them, no excessive changes in the structure of the Abrikosov vortex core has been revealed even at velocities of the order of 10 km/s (Embon et al., 2017). Because of the large current density gradient across the constriction with narrowing, a vortex chain splits up into a fan-like pattern of slipstreaming vortices. The TDGL modeling suggested that the slipstreaming vortices could further evolve to Josephson-like vortices moving with velocities as high as 100 km/s (Embon et al., 2017). Spatial modulation of the order parameter between these vortices is rather weak and the channel behaves effectively as a self-induced Josephson junction, which appears without materials weak links. Such flux channels in thin films can be interpreted as phase-slip lines (Sivakov et al., 2003). Vortex jet shapes have recently been analyzed based on the dynamic equation (Bezuglyj et al., 2022). For wide strips (width w much larger than the penetration depth l), the analytically calculated vortex jet shapes reproduce qualitatively the experiment (Embon et al., 2017). For narrow strips (x  w < l), the discrete TDGL approach reveals dynamic transitions from a vortex chain to a vortex jet and, at I ≲ I∗ ’ Id, from the vortex jet to a vortex river (Bezuglyj et al., 2022).

Fast vortex dynamics under a microwave ac stimulus Enhancement of superconductivity at microwaves Microwave irradiation is being used to control the quantum properties of different systems, ranging from supercurrents in superconductors to mechanical oscillators (Barzanjeh et al., 2022; Bergeret et al., 2010; Clerk et al., 2020; McIver et al., 2012; Palomaki et al., 2013). Nowadays, using nonequilibrium pumping for cooling is a hot topic (Bhadrachalam et al., 2014; Fornieri and Giazotto, 2017; José Martínez-Pérez and Giazotto, 2014; Maillet et al., 2020; Schneider et al., 2020; Uroš et al., 2020). In 1966, microwave stimulated superconductivity (MSSC) was discovered in superconducting bridges (Wyatt et al., 1966) and later confirmed for different type I superconducting systems (Beck et al., 2013; Dayem and Wiegand, 1967; de Visser et al., 2014; Heslinga and Klapwijk, 1993; Pals and Dobben, 1979; Pals and Dobben, 1980; Tolpygo and Tulin, 1983). It was later found that also acoustic waves (Tredwell and Jacobsen, 1975) and tunneling injection currents may, under certain conditions, cause an increase of the gap in the excitation spectrum, the order parameter, the critical current and the critical temperature Tc (Gray, 1981). The phenomenon of MSSC was explained by Eliashberg (1970) as a consequence of an irradiation-induced redistribution of quasiparticles away from the gap edge. Namely, in the BCS theory, there is a fundamental connection between the gap value D and the occupation of the quasiparticle states (Bardeen et al., 1957; Tinkham, 2004). When the temperature is increased from zero, more and more quasiparticles are excited blocking states which were previously available for Cooper pair formation. Ultimately this process leads to the disappearance of superconductivity at Tc. Extraction of quasiparticles leads to enhancement of superconductivity but is elusive to realize. However, there is another possibility. Electron states near the Fermi surface are more important to Cooper pair formation than states further away from it. If a quasiparticle in a state near eF is removed to a state with higher energy, the energy gap increases, although the total number of quasiparticles remains constant. This is the essential basis of the E mechanism. In a microwave field, the quasiparticles are “pumped up”, away from the gap. Even if the average energy of the quasiparticles increases, the gap is enhanced. Recent reviews on MSSC are given in Klapwijk and de Visser (2020) and Tikhonov et al. (2020).

Condensate and quasiparticles under dc/ac biasing The vortex state is characterized by a spatially modulated superconducting order parameter which vanishes in the vortex cores and attains a maximal value between them. Accordingly, a type II superconductor can be regarded as a continuum medium consisting of bunches of quasiparticles in the vortex cores surrounded by a bath of the superconducting condensate formed by the superfluid of Cooper pairs. So far there is no available theory addressing the complex interaction of the superimposed dc and microwave currents with the quasiparticles and the condensate at the same time. However, the essential ingredients of this interaction have been studied separately. The effect of a dc current on the superconducting condensate is well known (Anthore et al., 2003). With an increase of a dc current, the absolute value of the order parameter is decreasing and the peak in the density of states at the edge of the superconducting gap is smeared, see Fig. 5A. This is because of gaining a finite momentum by the Cooper pairs that form a coherent excited state that plays a central role in the explanation of the gauge invariance of the Meissner effect (Anderson, 1958). A general theory of depairing by a microwave field has been formulated recently (Semenov et al., 2016), under the condition of rather small ac

Fast dynamics of vortices in superconductors

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(A)

(B)

Fig. 5 (A) Schematics of the modification of the electronic distribution function f(E) in the presence of a microwave excitation after (Tikhonov et al., 2018). The redistribution of the quasiparticles between the D− ħo and D+ ħo states leads to a higher inclination of f(E) at the Fermi energy level eF, which can be viewed as an effective “cooling”. (B) Time-averaged density of states in a dirty superconductor at low temperature (T  Tc) under different biasing after (Semenov et al., 2016; Tikhonov et al., 2020). The value of the current or its amplitude is 0.25Ic.

field amplitudes. It has been shown that the ground state of a superconductor is altered qualitatively in analogy to the depairing due to a dc current. However, in contrast to dc depairing, the density of states acquires steps at multiples of the microwave photon energy and shows an exponential-like tail in the subgap regime (Semenov et al., 2016). Additionally, depending on temperature, one can consider two regimes in which the response of a superconductor is dominated either by the response of the superfluid (T  Tc) or by the quasiparticles (T  Tc). It is also known that at T  Tc, microwave radiation can be absorbed by quasiparticles, leading to a nonequilibrium distribution over the energies (Eliashberg, 1970), see Fig. 5B. In general, the response of the condensate to an external microwave field becomes apparent via a change of the kinetic impedance (imaginary part of the complex resistivity) while the quasiparticles give rise to the microwave loss (real part of the complex resistivity). The known effects of stimulation of superconductivity by a microwave stimulus in the absence of an external magnetic field include an enhancement of the Ginzburg-Landau depairing current (in narrow channels) implying a transition to a resistive state due to the formation of phase-slip centers (Zolochevskii, 2013) or the Aslamazov-Lempitskii maximum current (in wide films) at which the vortex structure induced by the self-field evolves into the first phase-slip line (Aslamazov and Lempitskii, 1982). In the presence of an external magnetic field inducing vortices in the superconductor, the vortex-induced microwave losses dominate the response of the superconductor (Pompeo and Silva, 2008).

Microwave stimulation in the vortex state From Fig. 5A it follows that a dc current leads to a smearing of the BCS peak at the gap edge. Further, the presence of vortices leads to a strongly spatially non-uniform order parameter (with normal-state regions inside the vortex cores). Moreover, the motion of current-driven vortices leads to dissipation (Brandt, 1995; Pompeo and Silva, 2008). All together, these circumstances diminish the stimulation effect of the microwave excitation. Nonetheless, in the regime of low vortex densities and for superconductors with

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rather slow relaxation of nonequilibrium quasiparticles, the stimulation of superconductivity under a microwave ac stimulus can be observed in the presence of vortices (Dobrovolskiy et al., 2019b; Lara et al., 2015). Note, the dynamics of vortices under a microwave ac stimulus differs qualitatively from the dc-driven vortex motion. Namely, while a moderately strong dc transport current (i.e., when the current-induced Lorentz-type force exceeds the pinning force (Brandt, 1995)) causes a translational motion of vortices, a microwave ac current shakes the vortices. Accordingly, in the presence of pinning sites, one can distinguish between the low-frequency and high-frequency regimes in the vortex dynamics. Namely, when the ac half-period is long enough, a vortex can move over many pinning sites, while in the high-frequency regime a vortex is shaken in vicinity of one pinning site. The crossover between the two regimes occurs at the so-called depinning frequency fd (Coffey and Clem, 1991; Dobrovolskiy et al., 2015; Gittleman and Rosenblum, 1966; Pompeo and Silva, 2008; Shklovskij and Dobrovolskiy, 2008; Shklovskij et al., 2014). In the low-frequency regime, the pinning forces dominate and the vortex response is weakly dissipative. By contrast, in the high-frequency regime, the oscillating vortices cease to feel the detail of the pinning potential, the frictional forces prevail, and the response is strongly dissipative. Note, this consideration is justified for short vortices in thin films and it is inapplicable to thick films where long vortices can get elastically distorted extending over many pinning sites (Pathirana and Gurevich, 2021). The depinning frequency depends on temperature fd(T, H, I) ’ fd(0, H, I)[1−(T/Tc)4] (Alimenti et al., 2020; Zaitsev et al., 2003), magnetic field fd(T, H, I)−fd(T, 0, I)[1−(H/Hc2)2] (Alimenti et al., 2020; Janjuševic et al., 2006) and the dc current as fd(T, H, I) ’ fd(T, H, 0)[1 −(I/Id)m]n with the exponents m between 3/2 and 4 and n between 1/4 and 2/3, depending on the details of the pinning potential (Dobrovolskiy et al., 2017a). At relatively low excitation power levels, for a commensurate vortex lattice in a periodic pinning landscape, ac losses due to ac-driven vortex oscillations are minimal because of the enhanced efficiency of the pinning (Dobrovolskiy et al., 2020a; Dobrovolskiy and Huth, 2015). By contrast, a qualitatively different behavior is observed when the microwave power increases beyond a certain, frequency-dependent power level: the microwave losses are maximal for pinned vortices, and they are minimal for depinned vortices (Dobrovolskiy et al., 2019b; Lara et al., 2015). In this nonlinear regime, the vortices act as “relaxators” rather than rigid “oscillators”. Since quasiparticles escape from the vortex cores, their escape can be seen as an effective decrease of the amplitude of the vortex oscillations with increase of the ac frequency.

Braking of instabilities by a microwave stimulus The FFI onset can be advanced or delayed in the presence of superimposed dc and ac current drives (Cherpak et al., 2014; Dobrovolskiy et al., 2020b). Namely, in the presence of a dc current, an abrupt transition to a high-loss state occurs at lower microwave power levels (Cherpak et al., 2014). The addition of an ac stimulus on top of a dc current allows for tuning the FFI onset (Dobrovolskiy et al., 2020b), see Fig. 6A, depending on the ac power P and frequency f. Thus, when the ac power exceeds some threshold value Pst, at high enough (GHz) ac frequencies the quasilinear flux-flow regime can be extended up to a higher current ∗ I∗ mw, whereas the FFI occurs at a smaller Imw in the presence of a low-frequency ac current (Dobrovolskiy et al., 2020b). With a (A)

(B)

Fig. 6 (A) Evolution of the instability current f ≶ fd for a Nb film at T ¼ 0.998Tc and H ¼ 10 mT in a broad range of frequencies and power levels of the ac stimulus. I∗: instability current value without microwave stimulus. (B) The sketch illustrates the overlap of the regions where vortices move under a microwave ac field of different powers and frequencies. As the vortices move, their radii change. This could lead the heated zones to overlap and produce avalanches. (A) Adapted under the terms of the CC-BY Creative Commons Attribution 4.0 International license Dobrovolskiy OV, González-Ruano C, Lara A, Sachser R, Bevz VM, Shklovskij VA, Bezuglyj AI, Vovk RV, Huth M and Aliev FG (2020b) Moving flux quanta cool superconductors by a microwave breath. Communications on Physics 3(1): 64. doi: 10.1038/s42005-020-0329-z. Copyright 2020, The Authors, published by Springer Nature; (B) Lara A, Aliev FG, Moshchalkov VV and Galperin YM (2017) Thermally driven inhibition of superconducting vortex avalanches. Physical Review Applied 8: 034027. doi: 10.1103/PhysRevApplied.8.034027. Copyright 2017, American Physical Society.

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further increase of the ac power, the heating will dominate the cooling effect and the FFI will occur at a smaller I∗ mw. This behavior can be explained, qualitatively, based on a model of “breathing mobile hot spots”, implying a competition of heating and cooling of quasiparticles along the trajectories of moving fluxons whose core sizes vary in time. This breathing mode appears owing to the variation of the vortex core sizes in time due to the periodically modulated quasiparticle escape from the cores. Accordingly, the strong oscillations of vortices lead to the formation of “clouds” of quasiparticles around the vortex cores, whose relaxation takes place in a larger volume as compared to the non-excited case. Thus, dissipation is decreasing due to the combined effects of the redistribution of quasiparticles away from the gap edge via the Eliashberg mechanism and the relaxation of the quasiparticles in larger volumes around the vortex cores. A theory of MSSC at high vortex velocities is yet to be elaborated. Instabilities can also occur in superconductors when the flux is getting redistributed upon temperature or magnetic field variation (Altshuler and Johansen, 2004; Aranson et al., 2005; Colauto et al., 2021; Mints and Rakhmanov, 1981). Herewith, vortex penetration may occur either smoothly or in sudden bursts, giving rise to thermomagnetic instabilities. In the case of smooth penetration, the flux density inside the superconductor changes gradually, being described by a critical state model (Bean, 1962; Kim et al., 1963; Zeldov et al., 1994). However, when the penetration is abrupt, narrow branches of flux, in the form of dendrites, invade the sample. The critical state model is no longer appropriate to describe this case. In a typical critical state type of flux distribution, millions of vortices self-organize, under the competing vortex-vortex repulsion, Lorentz and pinning forces. This delicate balance results in metastable states between which sudden flux redistributions occur. These flux avalanches occur typically during a slow ramping of the applied magnetic field, and at temperatures below a certain fraction of Tc. The flux patterns are never reproduced when experiments are repeated, thus ruling out possible explanations based on material defects guiding the flux motion (Colauto et al., 2021). The thermomagnetic instability or flux jumping arises because of two fundamental reasons (Denisov et al., 2006): (i) motion of magnetic flux releases energy, and hence increases the local temperature; (ii) the temperature rise decreases flux pinning, and hence facilitates the flux motion. This positive feedback can result in thermal runaways and global flux redistributions jeopardizing superconducting devices. According to the model based on thermomagnetic instabilities (Rakhmanov et al., 2004), an avalanche is triggered by a thermal fluctuation (hot spot), which facilitates more flux motion toward the hot place, with a subsequent heat release. Under quasi-equilibrium conditions, the avalanches can also be triggered by microwave pulses (Cuadra-Solis et al., 2013) or ac signals at near resonant frequencies of superconducting cavities (Ghigo et al., 2007). Under a broadband microwave field sweep, flux avalanches are triggered at different depinning frequencies due the distribution in strength of individual vortex pinning sites (Awad et al., 2011). At I∗ mw, flux avalanches exhibit a thermally driven behavior since higher temperatures facilitate the entrance of magnetic flux and enhance the vortex mobility. Accordingly, the critical microwave power Pc for triggering an avalanche monotonously decreases with approaching Tc. However, at f ’ fd, the behavior of Pc(T ) inverts, demonstrating a thermally driven avalanche inhibition (Lara et al., 2017). This effect can be understood as a consequence of the dependence of the vortex core size on P and f. Namely, at f ≶ fd, the LO effect manifests itself only in the regions of maximal velocity (i.e., close to displacement minima), see panel 1 in Fig. 6B. On the other hand, at f ’ fd, the vortices become more mobile. As a result, the LO reduction of the vortex core manifests itself during the whole displacement cycle, see panel 2 in Fig. 6B. Below fd and at low P, local “normal” regions created by the driven vortices do not overlap and are not sufficient to trigger the avalanche process. Increasing the microwave power below fd enhances the absolute changes in the vortex core size, inducing the overlap between different extended core areas and triggering the avalanche at P ’ Pc, see panel 3 in Fig. 6B. This process is thermally activated since the vortex diameter strongly increases as T ! Tc. However, at f ’ fd and in the temperature region where vortices become depinned, the LO effect reduces the vortex core ending up in only a small variation in the vortex core size during periodically driven motion. This reduces the probability of the overlap between normal regions and, therefore, the avalanche is triggered at higher microwave powers for a given temperature (Lara et al., 2017).

Future directions Superconductivity in curved 3D nanoarchitectures 3D nanoarchitectures have become of increasing importance across various domains of science and technology (Fomin, 2018; Fomin, 2021; Makarov et al., 2022), including magnetism (Fernández-Pacheco et al., 2017; Skjærvø et al., 2020), photonics (von Freymann et al., 2010), magnonics (Gubbiotti, 2019) and plasmonics (Winkler et al., 2017). From the applications viewpoint, the extension of nanoscale superconductors into the third dimension allows for the full-vector-field sensing in quantum interferometry (Martínez-Pérez et al., 2018), noise-equivalent power reduction in bolometry (Lösch et al., 2019) and reduction of footprints of fluxonic devices (Fourie et al., 2011). Extending quasi-1D or -2D superconductor manifolds into the third dimension allows for their thermal decoupling from the substrate and the development of multi-terminal devices and circuits with complex interconnectivity. Herewith, curved geometries bring about the smoothness of conjunctions, which allows for avoiding undesired weak links at sharp turns (Porrati et al., 2019) and minimizing current-crowding effects (Clem and Berggren, 2011). The complex interplay between Meissner screening currents, Abrikosov vortices and slips of the phase of the superconducting order parameter in curved 3D nanoarchitectures gives rise to topological modes which do not occur in planar 2D films (Fomin and Dobrovolskiy, 2022). Nowadays, topological transitions between the different regimes in the vortex and phase-slip dynamics in curved 3D nanoarchitectures are a subject of extensive investigations (Córdoba et al., 2019; Fomin et al., 2022; Fomin et al., 2012; Fomin et al., 2020). The stability of these regimes under intense dc transport currents and high ac frequencies requires experimental and theoretical exploration.

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Cryogenic magnonics and magnon fluxonics The interplay between the fundamental excitations in superconducting and magnetic systems has recently given birth to the research domains of cryogenic magnonics and magnon fluxonics (Dobrovolskiy and Chumak, 2022; Dobrovolskiy et al., 2019c; Golovchanskiy et al., 2019; Golovchanskiy et al., 2018). Magnonics is one of the most rapidly growing research fields in modern magnetism (Barman et al., 2021). It is concerned with the dynamics of spin waves (and their quanta—magnons), which are precessional excitations of ordered spins in magnetic materials, and their use for wave-based information processing (Chumak et al., 2022). In this regard, Meissner screening currents offer interesting possibilities to control the magnetization dynamics in superconductor-based hybrids at low temperatures (Golovchanskiy et al., 2019; Golovchanskiy et al., 2018). Furthermore, the periodic modulation of the local magnetic field emanating from the vortex cores induces Bloch-like bandgaps in the magnon frequency spectrum while a current-driven vortex lattice acts as a moving Bragg grating (Barnes et al., 1996), leading to Doppler shifts of the magnon bandgap frequencies (Dobrovolskiy and Chumak, 2022; Dobrovolskiy et al., 2019c). The fast motion of the vortex lattice opens up new prospects for the magnon-fluxon interaction: the phenomenon of Cherenkov radiation of magnons (Bespalov et al., 2014; Bulaevskii et al., 2005; Shekhter et al., 2011), when the vortex lattice velocity reaches the phase velocity of the spin wave. The experimentally observed magnon emission is unidirectional (spin wave propagates in the direction of motion of the vortex lattice), monochromatic (magnon wavelength is equal to the vortex lattice parameter), and coherent (spin-wave phase is self-locked due to quantum interference with eddy currents in the superconductor) (Dobrovolskiy et al., 2021). The magnon radiation is accompanied by a magnon Shapiro step in the I–V curve of the superconductor, reduces the dissipation (Bespalov et al., 2014; Bulaevskii et al., 2005; Shekhter et al., 2011), and inhibits the FFI onset (Dobrovolskiy et al., 2021). Thus, the generation of collective modes by a fast-moving vortex lattice can be viewed as an interesting approach for the inhibition of FFI and studies of fast vortex dynamics. Further research should address magnetic flux transport mechanisms in trancient Abrikosov-Josephson vortex regimes (Embon et al., 2017) and ultra-fast vortex motion in systems exhibiting a superconductor-insulator transition (Porrati et al., 2022).

Conclusion Summing up, the exploration of ultrafast vortex dynamics in superconductors has recently turned into a major research avenue. These studies are motivated by the enhancement of current-carrying capacity of superconductors, reduction of microwave loss therein, and improvement of superconducting single-photon detectors. From the viewpoint of basic research, the fast-moving ensemble of quantized magnetic flux lines entails emerging nonequilibrium phenomena and complex interactions, and it allows for the generation of collective modes in superconductor-based systems. It is anticipated that in the years to come, research along these lines will fuel the domain of vortex matter under nonequilibrium conditions and will lead to the development of novel applications.

Acknowledgments This work is based on the research funded by the German Research Foundation (DFG) through Grant Nos. DO1511/2-1, DO1511/2-4, DO1511/3-1 and the Austrian Science Fund (FWF) through Grant Nos. I 4889 (CurviMag) and I 6079 (FluMag). Further, support by the European Cooperation in Science and Technology via COST Actions CA16218 (NANOCOHYBRI), CA19108 (HiSCALE), and CA21144 (SUPERQUMAP) is acknowledged.

References Abrikosov AA (1988) Fundamentals of the Theory of Metals, North-Holland, Amsterdam. Adami O-A, Cerbu D, Cabosart D, Motta M, Cuppens J, Ortiz WA, Moshchalkov VV, Hackens B, Delamare R, Van de Vondel J, and Silhanek AV (2013) Current crowding effects in superconducting corner-shaped Al microstrips. Applied Physics Letters 102(5): 052603 1–5. https://doi.org/10.1063/1.4790625. Adami O-A, Jelic ZL, Xue C, Abdel-Hafiez M, Hackens B, Moshchalkov VV, Milosevic MV, Van de Vondel J, and Silhanek AV (2015) Onset, evolution, and magnetic braking of vortex lattice instabilities in nanostructured superconducting films. Physical Review B 92: 134506. https://doi.org/10.1103/PhysRevB.92.134506. Aladyshkin A, Mel’nikov AS, Shereshevsky IA, and Tokman ID (2001) What is the best gate for vortex entry into type-II superconductor? Physica C 361(1): 67–72. Alimenti A, Pompeo N, Torokhtii K, Spina T, Flükiger R, Muzzi L, and Silva E (2020) Microwave measurements of the high magnetic field vortex motion pinning parameters in Nb3Sn. Superconductor Science and Technology 34(1): 014003. https://doi.org/10.1088/1361-6668/abc05d. Altshuler E and Johansen TH (2004) Colloquium: Experiments in vortex avalanches. Reviews of Modern Physics 76: 471–487. https://doi.org/10.1103/RevModPhys.76.471. Anderson PW (1958) Coherent excited states in the theory of superconductivity: Gauge invariance and the Meissner effect. Physics Review 110: 827–835. https://doi.org/10.1103/ PhysRev.110.827. Andreev AF (1964) Thermal conductivity of the intermediate state of superconductors. Soviet Physics - JETP 19: 1228. Anthore A, Pothier H, and Esteve D (2003) Density of states in a superconductor carrying a supercurrent. Physical Review Letters 90: 127001. https://doi.org/10.1103/ PhysRevLett.90.127001. Aranson IS, Gurevich A, Welling MS, Wijngaarden RJ, Vlasko-Vlasov VK, Vinokur VM, and Welp U (2005) Dendritic flux avalanches and nonlocal electrodynamics in thin superconducting films. Physical Review Letters 94: 037002. https://doi.org/10.1103/PhysRevLett.94.037002. Aronov AG and Spivak BZ (1978) Soviet Physics - JETP 50: 328.

Fast dynamics of vortices in superconductors

749

Aslamazov LG and Lempitskii SV (1982) Superconductivity stimulation by a microwave field in superconductor-normal metal-superconductor junctions. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 82: 1671–1678. Attanasio C and Cirillo C (2012) Quasiparticle relaxation mechanisms in superconductor/ferromagnet bilayers. Journal of Physics. Condensed Matter 24(8), 083201. http://stacks.iop. org/0953-8984/24/i¼8/a¼083201. Awad AA, Aliev FG, Ataklti GW, Silhanek A, Moshchalkov VV, Galperin YM, and Vinokur V (2011) Flux avalanches triggered by microwave depinning of magnetic vortices in Pb superconducting films. Physical Review B 84: 224511. https://doi.org/10.1103/PhysRevB.84.224511. Babic D, Bentner J, Sürgers C, and Strunk C (2004) Flux-flow instabilities in amorphous Nb0.7Ge0.3 microbridges. Physical Review B 69: 092510 1–4. https://doi.org/10.1103/ PhysRevB.69.092510. Bardeen J and Stephen MJ (1965) Theory of the motion of vortices in superconductors. Physics Review 140: A1197–A1207. https://doi.org/10.1103/PhysRev.140.A1197. Bardeen J, Cooper LN, and Schrieffer JR (1957) Microscopic theory of superconductivity. Physics Review 106: 162–164. https://doi.org/10.1103/PhysRev.106.162. Barman A, et al. (2021) The 2021 magnonics roadmap. Journal of Physics. Condensed Matter 33(41): 413001. https://doi.org/10.1088/1361-648x/abec1a. Barnes SE, Cohn JL, and Zuo F (1996) The possibility of flux flow spectroscopy. Physical Review Letters 77: 3252–3255. https://doi.org/10.1103/PhysRevLett.77.3252. Barzanjeh S, Xuereb A, Gröblacher S, Paternostro M, Regal CA, and Weig EM (2022) Optomechanics for quantum technologies. Nature Physics 18(1): 15–24. https://doi.org/ 10.1038/s41567-021-01402-0. Bean CP (1962) Magnetization of hard superconductors. Physical Review Letters 8: 250–253. https://doi.org/10.1103/PhysRevLett.8.250. Beck M, Rousseau I, Klammer M, Leiderer P, Mittendorff M, Winnerl S, Helm M, Gol’tsman GN, and Demsar J (2013) Transient increase of the energy gap of superconducting NbN thin films excited by resonant narrow-band terahertz pulses. Physical Review Letters 110: 267003. https://doi.org/10.1103/PhysRevLett.110.267003. Bergeret FS, Virtanen P, Heikkilä TT, and Cuevas JC (2010) Theory of microwave-assisted supercurrent in quantumpoint contacts. Physical Review Letters 105: 117001. https://doi. org/10.1103/PhysRevLett.105.117001. Bespalov AA, Mel’nikov AS, and Buzdin AI (2014) Magnon radiation by moving Abrikosov vortices in ferromagnetic superconductors and superconductor-ferromagnet multilayers. Physical Review B 89: 054516. https://doi.org/10.1103/PhysRevB.89.054516. Bezuglyj AI and Shklovskij VA (1984) Thermal domains in inhomogeneous current-carrying superconductors. current-voltage characteristics and dynamics of domain formation after current jumps. Journal of Low Temperature Physics 57(3): 227–247. https://doi.org/10.1007/BF00681190. Bezuglyj A and Shklovskij V (1992) Effect of self-heating on flux flow instability in a superconductor near Tc. Physica C 202(314): 234. http://www.sciencedirect.com/science/article/ pii/0921453492901659. Bezuglyj AI, Shklovskij VA, Vovk RV, Bevz VM, Huth M, and Dobrovolskiy OV (2019) Local flux-flow instability in superconducting films near Tc. Physical Review B 99: 174518. https:// doi.org/10.1103/PhysRevB.99.174518. Bhadrachalam P, Subramanian R, Ray V, Ma L-C, Wang W, Kim J, Cho K, and Koh SJ (2014) Energy-filtered cold electron transport at room temperature. Nature Communications 5(1): 4745. https://doi.org/10.1038/ncomms5745. Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, and Vinokur VM (1994) Vortices in high-temperature superconductors. Reviews of Modern Physics 66: 1125–1388. https://doi. org/10.1103/RevModPhys.66.1125. Blonder GE, Tinkham M, and Klapwijk TM (1982) Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. Physical Review B 25: 4515–4532. https://doi.org/10.1103/PhysRevB.25.4515. Brandt EH (1995) The flux-line lattice in superconductors. Reports on Progress in Physics 58(11): 1465–1594. http://stacks.iop.org/0034-4885/58/i¼11/a¼003. Budinska B, Aichner B, Vodolazov DY, Mikhailov MY, Porrati F, Huth M, Chumak AV, Lang W, and Dobrovolskiy OV (2022) Rising speed limits for fluxons via edge-quality improvement in wide MoSi thin films. Physical Review Applied 17: 034072. https://doi.org/10.1103/PhysRevApplied.17.034072. Buh J, Kabanov V, Baranov V, Mrzel A, Kovic A, and Mihailovic D (2015) Control of switching between metastable superconducting states in d-MoN nanowires. Nature Communications 6: 10250. https://doi.org/10.1038/ncomms10250. Bulaevskii LN and Chudnovsky EM (2005) Sound generation by the vortex flow in type-II superconductors. Physical Review B 72: 094518. https://doi.org/10.1103/ PhysRevB.72.094518. Bulaevskii LN and Chudnovsky EM (2006) Electromagnetic radiation from vortex flow in type-II superconductors. Physical Review Letters 97: 197002. https://doi.org/10.1103/ PhysRevLett.97.197002. Bulaevskii LN, Hruška M, and Maley MP (2005) Spectroscopy of magnetic excitations in magnetic superconductors using vortex motion. Physical Review Letters 95: 207002. https:// doi.org/10.1103/PhysRevLett.95.207002. Caputo M, Cirillo C, and Attanasio C (2017) NbRe as candidate material for fast single photon detection. Applied Physics Letters 111(19), 192601. https://doi.org/ 10.1063/1.4997675. Charaev I, Morimoto Y, Dane A, Agarwal A, Colangelo M, and Berggren KK (2020) Large-area microwire MoSi single-photon detectors at 1550 nm wavelength. Applied Physics Letters 116(24), 242603. https://doi.org/10.1063/5.0005439. Cherpak NT, Lavrinovich AA, Gubin AI, and Vitusevich SA (2014) Direct-current-assisted microwave quenching of YBa2Cu3O7− x coplanar waveguide to a highly dissipative state. Applied Physics Letters 105(2), 022601. https://doi.org/10.1063/1.4890123. Chiles J, Buckley SM, Lita A, Verma VB, All-maras J, Korzh B, Shaw MD, Shainline JM, Mirin RP, and Nam SW (2020) Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared. Applied Physics Letters 116(24), 242602. https://doi.org/10.1063/5.0006221. Chumak AV, et al. (2022) Roadmap on spin-wave computing. In: IEEE Transactions on Magnetics xx, xx . Cirillo C, Chang J, Caputo M, Los JWN, Dorenbos S, Esmaeil Zadeh I, and Attanasio C (2020) Superconducting nanowire single photon detectors based on disordered NbRe films. Applied Physics Letters 117(17), 172602. https://doi.org/10.1063/5.0021487. Cirillo C, Granata V, Spuri A, Di Bernardo A, and Attanasio C (2021) NbReN: A disordered superconductor in thin film form for potential application as superconducting nanowire single photon detector. Physical Review Materials 5: 085004. https://doi.org/10.1103/PhysRevMaterials.5.085004. Clem JR and Berggren KK (2011) Geometry-dependent critical currents in superconducting nanocircuits. Physical Review B 84: 174510. https://doi.org/10.1103/ PhysRevB.84.174510. Clerk AA, Lehnert KW, Bertet P, Petta JR, and Nakamura Y (2020) Hybrid quantum systems with circuit quantum electrodynamics. Nature Physics 16(3): 257–267. https://doi.org/ 10.1038/s41567-020-0797-9. Coffey MW and Clem JR (1991) Unified theory of effects of vortex pinning and flux creep upon the rf surface impedance of type-II superconductors. Physical Review Letters 67: 386–389. https://doi.org/10.1103/PhysRevLett.67.386. Colauto F, Motta M, and Ortiz WA (2021) Controlling magnetic flux penetration in low-Tc superconducting films and hybrids. Superconductor Science and Technology 34(1), 013002. https://doi.org/10.1088/1361-6668/abac1e. Córdoba R, Mailly D, Rezaev RO, Smirnova EI, Schmidt OG, Fomin VM, Zeitler U, Guillamón I, Suderow H, and De Teresa JM (2019) Three-dimensional superconducting nanohelices grown by He+-focused-ion-beam direct writing. Nano Letters 19: 8597. Cuadra-Solis P-D-J, Hernandez JM, García-Santiago A, Tejada J, Vanacken J, and Moshchalkov VV (2013) Avalanche-like vortex penetration driven by pulsed microwave fields in an epitaxial LaSrCuO thin film. Journal of Applied Physics 114(23), 233902. https://doi.org/10.1063/1.4848997. Cuadra-Solis P-D-J, Garcia-Santiago A, Hernandez JM, Tejada J, Vanacken J, and Moshchalkov VV (2014) Observation of commensurability effects in a patterned thin superconducting Pb film using microwave reflection spectrometry. Physical Review B 89: 054517. https://doi.org/10.1103/PhysRevB.89.054517. Dayem AH and Wiegand JJ (1967) Behavior of thin-film superconducting bridges in a microwave field. Physics Review 155: 419–428. https://doi.org/10.1103/PhysRev.155.419.

750

Fast dynamics of vortices in superconductors

de Visser PJ, Goldie DJ, Diener P, Withington S, Baselmans JJA, and Klapwijk TM (2014) Evidence of a nonequilibrium distribution of quasiparticles in the microwave response of a superconducting aluminum resonator. Physical Review Letters 112: 047004. https://doi.org/10.1103/PhysRevLett.112.047004. Dean CL and Kunchur MN (2018) Dissipative-regime measurements as a tool for confirming and characterizing near-room-temperature superconductivity. Quantum Studies: Mathematics and Foundations 5(1): 111–121. https://doi.org/10.1007/s40509-017-0123-0. Denisov DV, Shantsev DV, Galperin YM, Choi E-M, Lee H-S, Lee S-I, Bobyl AV, Goa PE, Olsen AAF, and Johansen TH (2006) Onset of dendritic flux avalanches in superconducting films. Physical Review Letters 97: 077002 1–4. https://doi.org/10.1103/PhysRevLett.97.077002. Dobrovolskiy OV and Chumak AV (2022) Nonreciprocal magnon fluxonics upon ferromagnet/superconductor hybrids. Journal of Magnetism and Magnetic Materials 543: 168633. https://www.sciencedirect.com/science/article/pii/S0304885321008726. Dobrovolskiy OV and Huth M (2012) Crossover from dirty to clean superconducting limit in dc magnetron-sputtered thin Nb films. Thin Solid Films 520: 5985–5990. http://www. sciencedirect.com/science/article/pii/S0040609012005718. Dobrovolskiy OV and Huth M (2015) Dual cut-off direct current-tunable microwave low-pass filter on superconducting Nb microstrips with asymmetric nanogrooves. Applied Physics Letters 106: 142601 1–5. https://doi.org/10.1063/1.4917229. Dobrovolskiy OV, Huth M, and Shklovskij VA (2010) Anisotropic magnetoresistive response in thin Nb films decorated by an array of Co stripes. Superconductor Science and Technology 23(12): 125014 1–5. https://doi.org/10.1088/0953-2048/23/12/125014. Dobrovolskiy OV, Begun E, Huth M, and Shklovskij VA (2012) Electrical transport and pinning properties of Nb thin films patterned with focused ion beam-milled washboard nanostructures. New Journal of Physics 14: 113027 1–27. http://stacks.iop.org/1367-2630/14/i¼11/a¼113027. Dobrovolskiy OV, Huth M, and Shklovskij VA (2015) Alternating current-driven microwave loss modulation in a fluxonic metamaterial. Applied Physics Letters 107: 162603 1–5. http:// scitation.aip.org/content/aip/journal/apl/107/16/10.1063/1.4934487. Dobrovolskiy OV, Huth M, Shklovskij V, and Vovk RV (2017a) Mobile fluxons as coherent probes of periodic pinning in superconductors. Scientific Reports 7: 13740. https://doi.org/ 10.1038/s41598-017-14232-z. Dobrovolskiy OV, Shklovskij VA, Hanefeld M, Zörb M, Köhs L, and Huth M (2017b) Pinning effects on flux flow instability in epitaxial Nb thin films. Superconductor Science and Technology 30(8): 085002. http://stacks.iop.org/0953-2048/30/i¼8/a¼085002. Dobrovolskiy OV, Bevz VM, Mikhailov MY, Yuzephovich OI, Shklovskij VA, Vovk RV, Tsindlekht MI, Sachser R, and Huth M (2018) Microwave emission from superconducting vortices in Mo/Si superlattices. Nature Communications 9(1): 4927. https://doi.org/10.1038/s41467-018-07256-0. Dobrovolskiy OV, Bevz VM, Begun E, Sachser R, Vovk RV, and Huth M (2019a) Fast dynamics of guided magnetic flux quanta. Physical Review Applied 11: 054064. https://doi.org/ 10.1103/PhysRevApplied.11.054064. Dobrovolskiy OV, Sachser R, Bevz VM, Lara A, Aliev FG, Shklovskij VA, Bezuglyj A, Vovk RV, and Huth M (2019b) Reduction of microwave loss by mobile fluxons in grooved Nb films. Physica Status Solidi (RRL) 13: 1800223. https://doi.org/10.1002/pssr.201800223. Dobrovolskiy OV, Sachser R, Brächer T, Böttcher T, Kruglyak VV, Vovk RV, Shklovskij VA, Huth M, Hillebrands B, and Chumak AV (2019c) Magnon-fluxon interaction in a ferromagnet/ superconductor heterostructure. Nature Physics 15: 477. https://www.nature.com/articles/s41567-019-0428-5. Dobrovolskiy O, Begun E, Bevz V, Sachser R, and Huth M (2020a) Upper frequency limits for vortex guiding and ratchet effects. Physical Review Applied 13: 024012. https://doi.org/ 10.1103/PhysRevApplied.13.024012. Dobrovolskiy OV, González-Ruano C, Lara A, Sachser R, Bevz VM, Shklovskij VA, Bezuglyj AI, Vovk RV, Huth M, and Aliev FG (2020b) Moving flux quanta cool superconductors by a microwave breath. Communications on Physics 3(1): 64. https://doi.org/10.1038/s42005-020-0329-z. Dobrovolskiy OV, Vodolazov DY, Porrati F, Sachser R, Bevz VM, Mikhailov MY, Chumak AV, and Huth M (2020c) Ultra-fast vortex motion in a direct-write Nb-C superconductor. Nature Communications 11(1): 3291. https://doi.org/10.1038/s41467-020-16987-y. Dobrovolskiy OV, Wang Q, Vodolazov DY, Budinska B, Sachser R, Chumak A, Huth M, and Buzdin AI (2021) Cherenkov radiation of spin waves by ultra-fast moving magnetic flux quanta. arXiv:2103.10156 https://arxiv.org/abs/2103.10156. Doettinger SG, Huebener RP, Gerdemann R, Kühle A, Anders S, Träuble TG, and Villégier JC (1994) Electronic instability at high flux-flow velocities in high-Tc superconducting films. Physical Review Letters 73: 1691–1694. https://doi.org/10.1103/PhysRevLett.73.1691. Doettinger S, Huebener R, and Kühle A (1995) Electronic instability during vortex motion in cuprate superconductors regime of low and high magnetic fields. Physica C 251(3): 285–289. http://www.sciencedirect.com/science/article/pii/0921453495004114. Doettinger SG, Kittelberger S, Huebener RP, and Tsuei CC (1997) Quasiparticle energy relaxation in the cuprate superconductors. Physical Review B 56: 14157–14162. https://doi. org/10.1103/PhysRevB.56.14157. Eliashberg GM (1970) Film superconductivity stimulated by a high-frequency field. JETP Letters 11: 186. http://www.jetpletters.ac.ru/ps/1716/article_26086.pdf. Bezuglyj AI, Shklovskij VA, Budinska B, Aichner B, Bevz VM, Mikhailov MY, Vodolazov DY, Lang W, and Dobrovolskiy OV (2022) Vortex jets generated by edge defects in current-carrying superconductor thin strips. Physical Review B 105: 214507. https://doi.org/10.1103/PhysRevB.105.214507. Eliashberg GM and Ivlev BI (1986) Nonequilibrium superconductivity, North-Holland, Amsterdam. p. 211. Embon L, Anahory Y, Jelic ZL, Lachman EO, Myasoedov Y, Huber ME, Mikitik GP, Silhanek AV, Milosevic MV, Gurevich A, and Zeldov E (2017) Imaging of super-fast dynamics and flow instabilities of superconducting vortices. Nature Communications 8(1): 85. https://doi.org/10.1038/s41467-017-00089-3. Fernández-Pacheco A, Streubel R, Fruchart O, Hertel R, Fischer P, and Cowburn RP (2017) Three-dimensional nanomagnetism. Nature Communications 8: 15756. https://doi.org/ 10.1038/ncomms15756. Fomin VM (2018) Topology-Driven Effects in Advanced Nanoarchitectures. Springer International Publishing195–220. https://doi.org/10.1007/978-3-319-90481-8-10. Fomin VM (2021) Self-Rolled Micro- and Nanoarchitectures: Effects of Topology and Geometry. Berlin/Boston: De Gruyter. Fomin VM and Dobrovolskiy OV (2022) A perspective on superconductivity in curved 3D nanoarchitectures. Applied Physics Letters 120(9): 090501. https://doi.org/ 10.1063/5.0085095. Fomin VM, Rezaev R, and Dobrovolskiy OV (2022) Topological transitions in ac/dc-driven superconductor nanotubes. Scientific Reports 12: 10069. https://doi.org/10.1038/s41598022-13543-0. Fomin VM, Rezaev RO, and Schmidt OG (2012) Tunable generation of correlated vortices in open superconductor tubes. Nano Letters 12: 1282. https://doi.org/10.1021/nl203765f. Fomin VM, Rezaev RO, Smirnova EI, and Schmidt OG (2020) Topological transitions in superconductor nanomembranes in a magnetic field with submicron inhomogeneity under a strong transport current. Communications on Physics 3: 144. https://www.nature.com/articles/s42005-020-00411-4. Fornieri A and Giazotto F (2017) Towards phase-coherent caloritronics in superconducting circuits. Nature Nanotechnology 12(10): 944–952. https://doi.org/10.1038/ nnano.2017.204. Fourie CJ, Wetzstein O, Ortlepp T, and Kunert J (2011) Three-dimensional multi-terminal superconductive integrated circuit inductance extraction. Superconductor Science and Technology 24(12): 125015. https://doi.org/10.1088/0953-2048/24/12/125015. Ghigo G, Laviano F, Gozzelino L, Gerbaldo R, Mezzetti E, Monticone E, and Portesi C (2007) Evidence of RF-driven dendritic vortex avalanches in MgB2 microwave resonators. Journal of Applied Physics 102(11): 113901. https://doi.org/10.1063/1.2816257. Gittleman JI and Rosenblum B (1966) Radio-frequency resistance in the mixed state for subcritical currents. Physical Review Letters 16: 734–736. https://doi.org/10.1103/ PhysRevLett.16.734. Glazman LI (1986) Vortex induced transverse voltage within film. Journal of Low Temperature Physics 12: 389. Golovchanskiy IA, Abramov NN, Stolyarov VS, Bolginov VV, Ryazanov VV, Golubov AA, and Ustinov AV (2018) Ferromagnet/superconductor hybridization for magnonic applications. Advanced Functional Materials 28(33): 1802375. https://doi.org/10.1002/adfm.201802375.

Fast dynamics of vortices in superconductors

751

Golovchanskiy IA, Abramov NN, Pfirrmann M, Piskor T, Voss JN, Baranov DS, Hovhannisyan RA, Stolyarov VS, Dubs C, Golubov AA, Ryazanov VV, Ustinov AV, and Weides M (2019) Interplay of magnetization dynamics with a microwave waveguide at cryogenic temperatures. Physical Review Applied 11: 044076. https://doi.org/10.1103/ PhysRevApplied.11.044076. Gray KE (ed.) (1981) Nonequilibrium Superconductivity, Phonons, and Kapitza Boundaries. New York/London: Plenum Press. Grimaldi G, Leo A, Cirillo C, Attanasio C, Nigro A, and Pace S (2009a) Magnetic field and temperature dependence of the critical vortex velocity in type-II superconducting films. Journal of Physics: Condensed Matter 21(25): 254207. http://stacks.iop.org/0953-8984/21/i¼25/a¼254207. Grimaldi G, Leo A, Nigro A, Pace S, and Huebener RP (2009b) Dynamic ordering and instability of the vortex lattice in Nb films exhibiting moderately strong pinning. Physical Review B 80: 144521 1–10 https://doi.org/10.1103/PhysRevB.80.144521. Grimaldi G, Leo A, Zola D, Nigro A, Pace S, Laviano F, and Mezzetti E (2010) Evidence for low-field crossover in the vortex critical velocity of type-II superconducting thin films. Physical Review B 82: 024512. https://doi.org/10.1103/PhysRevB.82.024512. Grimaldi G, Leo A, Cirillo C, Casaburi A, Cristiano R, Attanasio C, Nigro A, Pace S, and Huebener R (2011) Non-linear flux flow resistance of type-II superconducting films. Journal of Superconductivity and Novel Magnetism 24(1–2): 81–87. https://doi.org/10.1007/s10948-010-0902-x. Grimaldi G, Leo A, Nigro A, Silhanek AV, Verellen N, Moshchalkov VV, Milosevic MV, Casaburi A, Cristiano R, and Pace S (2012) Controlling flux flow dissipation by changing flux pinning in superconducting films. Applied Physics Letters 100(20). http://scitation.aip.org/content/aip/journal/apl/100/20/10.1063/1.4718309. Grimaldi G, Leo A, Sabatino P, Carapella G, Nigro A, Pace S, Moshchalkov VV, and Silhanek AV (2015) Speed limit to the Abrikosov lattice in mesoscopic superconductors. Physical Review B 92: 024513. https://doi.org/10.1103/PhysRevB.92.024513. Gubbiotti G (ed.) (2019) Three-Dimensional Magnonics: Layered, Micro- and Nanostructures. Jenny Stanford Publishing. Gurevich A (2014) Reduction of dissipative nonlinear conductivity of superconductors by static and microwave magnetic fields. Physical Review Letters 113: 087001. https://doi.org/ 10.1103/PhysRevLett.113.087001. Gurevich AV and Mints RG (1984) Soviet Physics Uspekhi 27: 19. Heslinga DR and Klapwijk TM (1993) Enhancement of superconductivity far above the critical temperature in double-barrier tunnel junctions. Physical Review B 47: 5157–5164. https://doi.org/10.1103/PhysRevB.47.5157. Huebener RP (2019) The abrikosov vortex lattice: Its discovery and impact. Journal of Superconductivity and Novel Magnetism 32(3): 475–481. https://doi.org/10.1007/s10948-0184916-0. Huebener RP, Stoll OM, and Kaiser S (1999) Electronic vortex structure and quasiparticle scattering in the cuprate superconductor Nd2− xCexCuOy. Physical Review B 59: R3945–R3947. https://doi.org/10.1103/PhysRevB.59.R3945. Ivlev BI, Mejía-Rosales S, and Kunchur MN (1999) Cherenkov resonances in vortex dissipation in superconductors. Physical Review B 60: 12419–12423. https://doi.org/10.1103/ PhysRevB.60.12419. Janjuševic D, Grbií MS, Požek M, Dulcic A, Paar D, Nebendahl B, and Wagner T (2006) Microwave response of thin niobium films under perpendicular static magnetic fields. Physical Review B 74: 104501 1–7 https://doi.org/10.1103/PhysRevB.74.104501. Jaque D, Gonzalez EM, Martin JI, Anguita JV, and Vicent JL (2002) Anisotropic pinning enhancement in Nb films with arrays of submicrometric Ni lines. Applied Physics Letters 81: 2851–2854. http://link.aip.org/link/?APL/81/2851/1. José Martínez-Pérez M and Giazotto F (2014) A quantum diffractor for thermal flux. Nature Communications 5(1): 3579. https://doi.org/10.1038/ncomms4579. Kaplan SB, Chi CC, Langenberg DN, Chang JJ, Jafarey S, and Scalapino DJ (1976) Quasiparticle and phonon lifetimes in superconductors. Physical Review B 14: 4854–4873. https:// doi.org/10.1103/PhysRevB.14.4854. Kim YB, Hempstead CF, and Strnad AR (1963) Magnetization and critical supercurrents. Physics Review 129: 528–535. https://doi.org/10.1103/PhysRev.129.528. Klapwijk TM and de Visser PJ (2020) The discovery, disappearance and re-emergence of radiation-stimulated superconductivity. Annals of Physics 417: 168104. https://www. sciencedirect.com/science/article/pii/S0003491620300373. Klein W, Huebener RP, Gauss S, and Parisi J (1985) Nonlinearity in the flux-flow behavior of thin-film superconductors. Journal of Low Temperature Physics 61(5): 413–432. https:// doi.org/10.1007/BF00683694. Knight JM and Kunchur MN (2006) Energy relaxation at a hot-electron vortex instability. Physical Review B 74: 064512. https://doi.org/10.1103/PhysRevB.74.064512. Kogan VG (2018) Time-dependent London approach: Dissipation due to out-of-core normal excitations by moving vortices. Physical Review B 97: 094510. https://doi.org/10.1103/ PhysRevB.97.094510. Kogan VG and Nakagawa N (2021) Current distributions by moving vortices in superconductors. Physical Review B 103: 134511. https://doi.org/10.1103/PhysRevB.103.134511. Kogan VG and Nakagawa N (2022) Dissipation of moving vortices in thin films. Physical Review B 105: L020507. https://doi.org/10.1103/PhysRevB.105.L020507. Kogan VG and Prozorov R (2020) Interaction between moving Abrikosov vortices in type-II superconductors. Physical Review B 102: 024506. https://doi.org/10.1103/ PhysRevB.102.024506. Korneeva YP, Vodolazov DY, Semenov AV, Florya IN, Simonov N, Baeva E, Korneev AA, Goltsman GN, and Klapwijk TM (2018) Optical single-photon detection in micrometer-scale NbN bridges. Physical Review Applied 9: 064037. https://doi.org/10.1103/PhysRevApplied.9.064037. Korneeva YP, Manova N, Florya I, Mikhailov MY, Dobrovolskiy O, Korneev A, and Vodolazov DY (2020) Different single-photon response of wide and narrow superconducting MoxSi1− x strips. Physical Review Applied 13: 024011. https://doi.org/10.1103/PhysRevApplied.13.024011. Korzh B, Zhao Q-Y, Allmaras JP, Frasca S, Autry TM, Bersin EA, Beyer AD, Briggs RM, Bumble B, Colangelo M, Crouch GM, Dane AE, Gerrits T, Lita AE, Marsili F, Moody G, Peña C, Ramirez E, Rezac JD, Sinclair N, Stevens MJ, Velasco AE, Verma VB, Wollman EE, Xie S, Zhu D, Hale PD, Spiropulu M, Silverman KL, Mirin RP, Nam SW, Kozorezov AG, Shaw MD, and Berggren KK (2020) Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector. Nature Photonics 14(4): 250–255. https://doi. org/10.1038/s41566-020-0589-x. Koshelev AE and Vinokur VM (1994) Dynamic melting of the vortex lattice. Physical Review Letters 73: 3580–3583. https://doi.org/10.1103/PhysRevLett.73.3580. Kramer L and Watts-Tobin RJ (1978) Theory of dissipative current-carrying states in superconducting filaments. Physical Review Letters 40: 1041–1044. https://doi.org/10.1103/ PhysRevLett.40.1041. Kunchur MN (2002) Unstable flux flow due to heated electrons in superconducting films. Physical Review Letters 89: 137005. https://doi.org/10.1103/PhysRevLett.89.137005. Kunchur MN and Knight JM (2003) Hot-electron instability in superconductors. Modern Physics Letters B 17(10 − 12): 549–558. https://doi.org/10.1142/S0217984903005573. Kunchur MN, Ivlev BI, and Knight JM (2001) Steps in the negative-differential-conductivity regime of a superconductor. Physical Review Letters 87: 177001. https://doi.org/10.1103/ PhysRevLett.87.177001. Kunchur MN, Ivlev BI, and Knight JM (2002) Shear fragmentation of unstable flux flow. Physical Review B 66: 060505. https://doi.org/10.1103/PhysRevB.66.060505. Kunchur MN, Wu C, Arcos DH, Ivlev BI, Choi E-M, Kim KHP, Kang WN, and Lee S-I (2003) Critical flux pinning and enhanced upper critical field in magnesium diboride films. Physical Review B 68: 100503. https://doi.org/10.1103/PhysRevB.68.100503. Lara A, Aliev FG, Silhanek AV, and Moshchalkov VV (2015) Microwave-stimulated superconductivity due to presence of vortices. Scientific Reports 5: 9187. https://doi.org/10.1038/ srep09187. Lara A, Aliev FG, Moshchalkov VV, and Galperin YM (2017) Thermally driven inhibition of superconducting vortex avalanches. Physical Review Applied 8: 034027. https://doi.org/ 10.1103/PhysRevApplied.8.034027. Larkin AI and Ovchinnikov YN (1975) Nonlinear conductivity of superconductors in the mixed state. Journal of Experimental and Theoretical Physics 41: 960. http://jetp.ras.ru/cgi-bin/ e/index/e/41/5/p960?a¼list. Larkin AI and Ovchinnikov YN (1986) Nonequilibrium Superconductivity. Amsterdam: Elsevier493.

752

Fast dynamics of vortices in superconductors

Lefloch F, Hoffmann C, and Demolliens O (1999) Nonlinear flux flow in TiN superconducting thin film. Physica C 319(3): 258–266. http://www.sciencedirect.com/science/article/pii/ S092145349900297X. Leo A, Grimaldi G, Nigro A, Pace S, Verellen N, Silhanek A, Gillijns W, Moshchalkov V, Metlushko V, and Ilic B (2010) Pinning effects on the vortex critical velocity in type-ii superconducting thin films. Physica C 470(19): 904–906. http://www.sciencedirect.com/science/article/pii/S0921453410001607. Leo A, Grimaldi G, Citro R, Nigro A, Pace S, and Huebener RP (2011) Quasiparticle scattering time in niobium superconducting films. Physical Review B 84: 014536 1–7 https://doi. org/10.1103/PhysRevB.84.014536. Leo A, Marra P, Grimaldi G, Citro R, Kawale S, Bellingeri E, Ferdeghini C, Pace S, and Nigro A (2016) Competition between intrinsic and extrinsic effects in the quenching of the superconducting state in fe(se,te) thin films. Physical Review B 93: 054503. https://doi.org/10.1103/PhysRevB.93.054503. Leo A, Nigro A, Braccini V, Sylva G, Provino A, Galluzzi A, Polichetti M, Ferdeghini C, Putti M, and Grimaldi G (2020a) Flux flow instability as a probe for quasiparticle energy relaxation time in Fe-chalcogenides. Superconductor Science and Technology 33(10): 104005. https://doi.org/10.1088/1361-6668/abaec1. Leo A, Nigro A, and Grimaldi G (2020b) Critical phenomenon of vortex motion in superconductors: Vortex instability and flux pinning. Low Temperature Physics 46(4): 375–378. https://doi.org/10.1063/10.0000870. Li D, Malkin AM, and Rosenstein B (2004) Structure and orientation of the moving vortex lattice in clean type-II superconductors. Physical Review B 70: 214529. https://doi.org/ 10.1103/PhysRevB.70.214529. Liang M and Kunchur MN (2010) Vortex instability in molybdenum-germanium superconducting films. Physical Review B 82: 144517. https://doi.org/10.1103/PhysRevB.82.144517. Liang M, Kunchur MN, Hua J, and Xiao Z (2010) Evaluating free flux flow in low-pinning molybdenum-germanium superconducting films. Physical Review B 82: 064502 1–5. https:// doi.org/10.1103/PhysRevB.82.064502. Lin S-Z, Ayala-Valenzuela O, McDonald RD, Bulaevskii LN, Holesinger TG, Ronning F, Weisse-Bernstein NR, Williamson TL, Mueller AH, Hoff-bauer MA, Rabin MW, and Graf MJ (2013) Characterization of the thin-film NbN superconductor for single-photon detection by transport measurements. Physical Review B 87: 184507. https://doi.org/10.1103/ PhysRevB.87.184507. Lösch S, Alfonsov A, Dobrovolskiy OV, Keil R, Engemaier V, Baunack S, Li G, Schmidt OG, and Bürger D (2019) Microwave radiation detection with an ultra-thin free-standing superconducting niobium nanohelix. ACS Nano 13: 2948–2955. https://doi.org/10.1021/acsnano.8b07280. Lu Q, Reichhardt CJO, and Reichhardt C (2007) Reversible vortex ratchet effects and ordering in superconductors with simple asymmetric potential arrays. Physical Review B 75: 054502. https://doi.org/10.1103/PhysRevB.75.054502. Lyatti M, Wolff MA, Savenko A, Kruth M, Ferrari S, Poppe U, Pernice W, Dunin-Borkowski RE, and Schuck C (2018) Experimental evidence for hotspot and phase-slip mechanisms of voltage switching in ultrathin YBa2Cu3O7− x nanowires. Physical Review B 98: 054505. https://doi.org/10.1103/PhysRevB.98.054505. Lyatti M, Wolff MA, Gundareva I, Kruth M, Ferrari S, Dunin-Borkowski RE, and Schuck C (2020) Energy-level quantization and single-photon control of phase slips in YBa2Cu3O7–x nanowires. Nature Communications 11(1): 763. https://doi.org/10.1038/s41467-020-14548-x. Madan I, Buh J, Baranov VV, Kabanov VV, Mrzel A, and Mihailovic D (2018) Nonequilibrium optical control of dynamical states in superconducting nanowire circuits. Science Advances 4(3). http://advances.sciencemag.org/content/4/3/eaao0043. Maillet O, Subero D, Peltonen JT, Golubev DS, and Pekola JP (2020) Electric field control of radiative heat transfer in a superconducting circuit. Nature Communications 11(1): 4326. https://doi.org/10.1038/s41467-020-18163-8. Makarov D, Volkov OM, Kákay A, Pylypovskyi OV, Budinská B, and Dobrovolskiy OV (2022) New dimension in magnetism and superconductivity: 3D and curvilinear nanoarchitectures. Advanced Materials 34: 2101758. https://doi.org/10.1002/adma.202101758. Martínez-Pérez MJ, Pablo-Navarro J, Müller B, Kleiner R, Magén C, Koelle D, de Teresa JM, and Sesé J (2018) NanoSQUID magnetometry on individual as-grown and annealed Co nanowires at variable temperature. Nano Letters 18(12): 7674–7682. https://doi.org/10.1021/acs.nanolett.8b03329. Maza J, Ferro G, Veira JA, and Vidal F (2008) Transition to the normal state induced by high current densities in YBCO thin films: A thermal runaway account. Physical Review B 78: 094512. https://doi.org/10.1103/PhysRevB.78.094512. McIver JW, Hsieh D, Steinberg H, Jarillo-Herrero P, and Gedik N (2012) Control over topological insulator photocurrents with light polarization. Nature Nanotechnology 7(2): 96–100. https://doi.org/10.1038/nnano.2011.214. Mints RG and Rakhmanov AL (1981) Critical state stability in type-II superconductors and superconducting-normal-metal composites. Reviews of Modern Physics 53: 551–592. https://doi.org/10.1103/RevModPhys.53.551. Misko VR, Savel’ev S, Rakhmanov AL, and Nori F (2007) Negative differential resistivity in superconductors with periodic arrays of pinning sites. Physical Review B 75: 024509. https:// doi.org/10.1103/PhysRevB.75.024509. Musienko LE, Dmitrenko IM, and Volotskaya VG (1980) Nonlinear conductivity of thin films in a mixed state. JETP Letters 31: 567. Natarajan CM, Tanner MG, and Hadfield RH (2012) Superconducting nanowire single-photon detectors: Physics and applications. Superconductor Science and Technology 25(6), 063001. http://stacks.iop.org/0953-2048/25/i¼6/a¼063001. Niessen AK and Weijsenfeld CH (1969) Anisotropic pinning and guided motion of vortices in type-II superconductors. Journal of Applied Physics 40(1): 384–393. http://link.aip.org/ link/?JAP/40/384/1. Octavio M, Tinkham M, Blonder GE, and Klapwijk TM (1983) Subharmonic energy-gap structure in superconducting constrictions. Physical Review B 27: 6739–6746. https://doi.org/ 10.1103/PhysRevB.27.6739. Palomaki TA, Harlow JW, Teufel JD, Simmonds RW, and Lehnert KW (2013) Coherent state transfer between itinerant microwave fields and a mechanical oscillator. Nature 495(7440): 210–214. https://doi.org/10.1038/nature11915. Pals JA and Dobben J (1979) Measurements of microwave-enhanced superconductivity in aluminum strips. Physical Review B 20: 935–944. https://doi.org/10.1103/ PhysRevB.20.935. Pals JA and Dobben J (1980) Observation or order-parameter enhancement by microwave irradiation in a superconducting aluminum cylinder. Physical Review Letters 44: 1143–1146. https://doi.org/10.1103/PhysRevLett.44.1143. Pathirana WPMR and Gurevich A (2021) Effect of random pinning on nonlinear dynamics and dissipation of a vortex driven by a strong microwave current. Physical Review B 103: 184518. https://doi.org/10.1103/PhysRevB.103.184518. Peroz C and Villard C (2005) Flux flow properties of niobium thin films in clean and dirty superconducting limits. Physical Review B 72: 014515 1–6. https://doi.org/10.1103/ PhysRevB.72.014515. Pompeo N and Silva E (2008) Reliable determination of vortex parameters from measurements of the microwave complex resistivity. Physical Review B 78: 094503 1–10. https://doi. org/10.1103/PhysRevB.78.094503. Porrati F, Barth S, Sachser R, Dobrovolskiy OV, Seybert A, Frangakis AS, and Huth M (2019) Crystalline niobiumcarbide superconducting nanowires prepared by focused ion beam direct writing. ACS Nano 13(6): 6287–6296. https://doi.org/10.1021/acsnano.9b00059. Porrati F, Jungwirth F, Barth S, Gazzadi G C, Frabboni S, Dobrovolskiy O V, and Huth M (2022) Highly-packed proximity-coupled DC-Josephson junction arrays by a direct-write approach. Advanced Functional Materials. 2203889. https://doi.org/10.1002/adfm.202203889. Puica I, Lang W, and Durrell JH (2012) High velocity vortex channeling in vicinal YBCO thin films. Physica C 479: 88–91. https://www.sciencedirect.com/science/article/pii/ S0921453411005594. Rakhmanov AL, Shantsev DV, Galperin YM, and Johansen TH (2004) Finger patterns produced by thermomagnetic instability in superconductors. Physical Review B 70: 224502. https://doi.org/10.1103/PhysRevB.70.224502. Reichhardt C and Reichhardt CJO (2008) Moving vortex phases, dynamical symmetry breaking, and jamming for vortices in honeycomb pinning arrays. Physical Review B 78: 224511. https://doi.org/10.1103/PhysRevB.78.224511.

Fast dynamics of vortices in superconductors

753

Rouco V, Massarotti M, Stornaiuolo D, Papari GP, Obradors X, Puig T, Tafuri F, and Palau A (2018) Vortex lattice instabilities in YBa2Cu3O7− x nanowires. Materials 11: 211. Ruck BJ, Trodahl HJ, Abele JC, and Gesel-bracht MJ (2000) Dynamic vortex instabilities in ta/ge-based multilayers and thin films with various pinning strengths. Physical Review B 62: 12468–12476. https://doi.org/10.1103/PhysRevB.62.12468. Schneider M, Brächer T, Breitbach D, Lauer V, Pirro P, Bozhko DA, Musiienko-Shmarova HY, Heinz B, Wang Q, Meyer T, Heussner F, Keller S, Papaioannou ET, Lägel B, Löber T, Dubs C, Slavin AN, Tiberkevich VS, Serga AA, Hille-brands B, and Chumak AV (2020) Bose-Einstein condensation of quasiparticles by rapid cooling. Nature Nanotechnology 15(6): 457–461. https://doi.org/10.1038/s41565-020-0671-z. Semenov AV, Devyatov IA, de Visser PJ, and Klapwijk TM (2016) Coherent excited states in superconductors due to a microwave field. Physical Review Letters 117: 047002. https:// doi.org/10.1103/PhysRevLett.117.047002. Sheikhzada A and Gurevich A (2017) Dynamic transition of vortices into phase slips and generation of vortex-antivortex pairs in thin film josephson junctions under dc and ac currents. Physical Review B 95: 214507. https://doi.org/10.1103/PhysRevB.95.214507. Sheikhzada A and Gurevich A (2020) Dynamic pair-breaking current, critical superfluid velocity, and nonlinear electromagnetic response of nonequilibrium superconductors. Physical Review B 102: 104507. https://doi.org/10.1103/PhysRevB.102.104507. Shekhter A, Bulaevskii LN, and Batista CD (2011) Vortex viscosity in magnetic superconductors due to radiation of spin waves. Physical Review Letters 106: 037001. https://doi.org/ 10.1103/PhysRevLett.106.037001. Shklovskij VA (2017) Pinning effects on hot-electron vortex flow instability in superconducting films. Physica C 538: 20–26. http://www.sciencedirect.com/science/article/pii/ S0921453417300953. Shklovskij VA and Dobrovolskiy OV (2006) Influence of pointlike disorder on the guiding of vortices and the Hall effect in a washboard planar pinning potential. Physical Review B 74: 104511 1–14. https://doi.org/10.1103/PhysRevB.74.104511. Shklovskij VA and Dobrovolskiy OV (2008) AC-driven vortices and the Hall effect in a superconductor with a tilted washboard pinning potential. Physical Review B 78: 104526. https:// doi.org/10.1103/PhysRevB.78.104526. Shklovskij VA, Sosedkin VV, and Dobrovolskiy OV (2014) Vortex ratchet reversal in an asymmetric washboard pinning potential subject to combined dc and ac stimuli. Journal of Physics. Condensed Matter 26(2), 025703. http://stacks.iop.org/0953-8984/26/i¼2/a¼025703. Shklovskij VA, Nazipova AP, and Dobrovolskiy OV (2017) Pinning effects on self-heating and flux-flow instability in superconducting films near Tc. Physical Review B 95: 184517. https://doi.org/10.1103/PhysRevB.95.184517. Silhanek AV, Miloševic MV, Kramer RBG, Berdiyorov GR, Van de Vondel J, Luccas RF, Puig T, Peeters FM, and Moshchalkov VV (2010) Formation of stripelike flux patterns obtained by freezing kinematic vortices in a superconducting Pb film. Physical Review Letters 104: 017001. https://doi.org/10.1103/PhysRevLett.104.017001. Silhanek AV, Leo A, Grimaldi G, Berdiyorov GR, Milosevic MV, Nigro A, Pace S, Verellen N, Gillijns W, Metlushko V, Ilic B, Zhu X, and Moshchalkov VV (2012) Influence of artificial pinning on vortex lattice instability in superconducting films. New Journal of Physics 14(5), 053006. http://stacks.iop.org/1367-2630/14/i¼5/a¼053006. Sivakov AG, Glukhov AM, Omelyanchouk AN, Koval Y, Müller P, and Ustinov AV (2003) Josephson behavior of phase-slip lines in wide superconducting strips. Physical Review Letters 91: 267001 1–4. https://doi.org/10.1103/PhysRevLett.91.267001. Sivakov AG, Turutanov OG, Kolinko AE, and Pokhila AS (2018) Spatial characterization of the edge barrier in wide superconducting films. Low Temperature Physics 44(3): 226–232. https://doi.org/10.1063/1.5024540. Skjærvø SH, Marrows CH, Stamps RL, and Heyderman LJ (2020) Advances in artificial spin ice. Nature Reviews Physics 2(1): 13–28. https://doi.org/10.1038/s42254-019-0118-3. Stoll OM, Kaiser S, Huebener RP, and Naito M (1998) Intrinsic flux-flow resistance steps in the cuprate superconductor Nd2− xCexCuOy. Physical Review Letters 81: 2994–2997. https://doi.org/10.1103/PhysRevLett.81.2994. Stoll OM, Huebener RP, Kaiser S, and Naito M (1999) Magnetic field and temperature dependence of the intrinsic resistance steps in the mixed state of the cuprate superconductor Nd2− xCexCuOy. Physical Review B 60: 12424–12428. https://doi.org/10.1103/PhysRevB.60.12424. Tikhonov KS, Skvortsov MA, and Klapwijk TM (2018) Superconductivity in the presence of microwaves: Full phase diagram. Physical Review B 97: 184516. https://doi.org/10.1103/ PhysRevB.97.184516. Tikhonov KS, Semenov AV, Devyatov IA, and Skvortsov MA (2020) Microwave response of a superconductor beyond the eliashberg theory. Annals of Physics 417: 168101. https:// www.sciencedirect.com/science/article/pii/S0003491620300348. Tinkham M (2004) Introduction to Superconductivity. Mineola, New York. Tolpygo SK and Tulin VA (1983) Influence of microwave irradiation on high-frequency absorption by thin superconducting films (superconductivity stimulation). Soviet Physics - JETP 57: 123. Tredwell TJ and Jacobsen EH (1975) Phonon-induced enhancement of the superconducting energy gap. Physical Review Letters 35: 244–247. https://doi.org/10.1103/ PhysRevLett.35.244. Uroš D, Manuel R, Kahan D, David G, Vladan V, Nikolai K, and Markus A (2020) Cooling of a levitated nanoparticle to the motional quantum ground state. Science 367(6480): 892–895. https://doi.org/10.1126/science.aba3993. Villegas JE, Savel’ev S, Nori F, Gonzalez EM, Anguita JV, Garcia R, and Vicent JL (2003) A superconducting reversible rectifier that controls the motion of magnetic flux quanta. Science 302(5648): 1188–1191. http://www.sciencemag.org/content/302/5648/1188.abstract. Vodolazov DY (2017) Single-photon detection by a dirty current-carrying superconducting strip based on the kinetic-equation approach. Physical Review Applied 7: 034014. https:// doi.org/10.1103/PhysRevApplied.7.034014. Vodolazov DY (2019) Flux-flow instability in a strongly disordered superconducting strip with an edge barrier for vortex entry. Superconductor Science and Technology 32(11): 115013. https://doi.org/10.1088/1361-6668/ab4168. Vodolazov DY (2020) Flux flow instability in type II superconducting strips: Spatially uniform versus nonuniform transition. Low Temperature Physics 46(4): 372–374. https://doi.org/ 10.1063/10.0000869. Vodolazov DY and Peeters FM (2007) Rearrangement of the vortex lattice due to instabilities of vortex flow. Physical Review B 76: 014521. https://doi.org/10.1103/ PhysRevB.76.014521. Vodolazov DY, Maksimov IL, and Brandt EH (2003) Vortex entry conditions in type-II superconductors: Effect of surface defects. Physica C 384(1): 211–226. https://www. sciencedirect.com/science/article/pii/S0921453402018774. Vodolazov DY, Ilin K, Merker M, and Siegel M (2015) Defect-controlled vortex generation in current-carrying narrow superconducting strips. Superconductor Science and Technology 29(2): 025002. https://doi.org/10.1088/0953-2048/29/2/025002. Volotskaya VG, Dmitrenko IM, Koretskaya OA, and Musienko LE (1992) A study of the effect of superheating on the Larkin-Ovchinmkov instability evolution in thin Sn films. Fizika Nizkikh Temperatur 18: 973. von Freymann G, Ledermann A, Thiel M, Staude I, Essig S, Busch K, and Wegener M (2010) Three-dimensional nanostructures for photonics. Advanced Functional Materials 20(7): 1038–1052. https://doi.org/10.1002/adfm.200901838. Watts-Tobin RJ, Krähenbühl Y, and Kramer L (1981) Nonequilibrium theory of dirty, current-carrying superconductors: Phase-slip oscillators in narrow filaments near Tc. Journal of Low Temperature Physics 42(5): 459–501. https://doi.org/10.1007/BF00117427. Winkler R, Schmidt F-P, Haselmann U, Fowlkes JD, Lewis BB, Kothleitner G, Rack PD, and Plank H (2017) Direct-write 3d nanoprinting of plasmonic structures. ACS Applied Materials & Interfaces 9(9): 8233–8240. https://doi.org/10.1021/acsami.6b13062. Wördenweber R, Hollmann E, Schubert J, Kutzner R, and Panaitov G (2012) Regimes of flux transport at microwave frequencies in nanostructured high-Tc films. Physical Review B 85: 064503 1–6 https://doi.org/10.1103/PhysRevB.85.064503.

754

Fast dynamics of vortices in superconductors

Wyatt AFG, Dmitriev VM, Moore WS, and Sheard FW (1966) Microwave-enhanced critical supercurrents in constricted tin films. Physical Review Letters 16: 1166–1169. https://doi. org/10.1103/PhysRevLett.16.1166. Xiao ZL and Ziemann P (1996) Vortex dynamics in YBCO superconducting films: Experimental evidence for an instability in the vortex system at high current densities. Physical Review B 53: 15265–15271. https://doi.org/10.1103/PhysRevB.53.15265. Xiao ZL, Häring J, Heinz B, and Ziemann P (1996) Influence of He ion bombardment at medium energies on the vortex instability at high current densities observed in YBaCuO films. Journal of Low Temperature Physics 105(5): 1159–1164. https://doi.org/10.1007/BF00753856. Xiao ZL, Andrei EY, and Ziemann P (1998) Coexistence of the hot-spot effect and flux-flow instability in high-Tc superconducting films. Physical Review B 58: 11185–11188. https:// doi.org/10.1103/PhysRevB.58.11185. Zaitsev AG, Schneider R, Linker G, Ratzel F, Smithey R, and Geerk J (2003) Effect of flux flow on microwave losses in YBa2Cu3O7− x superconducting films. Physical Review B 68: 104502. https://doi.org/10.1103/PhysRevB.68.104502. Zeldov E, Larkin AI, Geshkenbein VB, Konczykowski M, Majer D, Khaykovich B, Vinokur VM, and Shtrikman H (1994) Geometrical barriers in high-temperature superconductors. Physical Review Letters 73: 1428–1431. https://doi.org/10.1103/PhysRevLett.73.1428. Zolochevskii IV (2013) Stimulation of superconductivity by microwave radiation in wide tin films. Low Temperature Physics 39: 571.

Non-Abelian anyons and non-Abelian vortices in topological superconductors Yusuke Masakia,b , Takeshi Mizushimac , and Muneto Nittab,d,e , aDepartment of Physics, Tohoku University, Sendai, Miyagi, Japan; bResearch and Education Center for Natural Sciences, Keio University, Yokohama, Kanagawa, Japan; cDepartment of Materials Engineering Science, Osaka University, Toyonaka, Osaka, Japan; dDepartment of Physics, Keio University, Hiyoshi, Japan; eInternational Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Higashi-Hiroshima, Hiroshima, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Non-Abelian anyons Braid group and quantum statistics Non-Abelian anyons Ising anyons Fibonacci anyons Yang-Lee anyons Vortex anyons Non-Abelian anyons in topological SCs Majorana zero modes as Ising anyons Majorana zero modes Braiding Majorana zero modes Platforms for Majorana zero modes HQVs with Majorana zero modes in SF 3He HQVs in SCs Superconducting nanowires Vortices in Fe(Se,Te) Topological quantum computation Symmetry-protected non-Abelian anyons Mirror and unitary symmetries Multiple Majorana zero modes and Coxeter group Non-Abelian vortex anyons Non-Abelian vortices Examples of non-Abelian vortices Nematic liquid crystals UN liquid crystals D2-BN liquid crystals (Poenaru and Toulouse, 1977; Volovik and Mineev, 1977; Mermin, 1979; Balachandran et al., 1984) D4-BN liquid crystals Spinor BECs BN phases in spin-2 BEC (Song et al., 2007; Uchino et al., 2010; Kobayashi et al., 2012; Borgh and Ruostekoski, 2016) Cyclic phase in spin-2 BEC (Semenoff and Zhou, 2007; Kobayashi et al., 2009, 2012) Other examples Fusion rules for non-Abelian vortex anyons Non-Abelian vortex anyons in D2-BN Non-Abelian vortex anyons in spin-2 BECs 3 P2 Topological SFs Overview of 3P2 topological SFs Nematic state Cyclic state Ferromagnetic state Magnetized nematic state Vortices in 3P2 SFs Half quantum vortex in D4-BN state Vortices in D4-BN state Axisymmetric singly quantized vortex Nonaxisymmetric singly quantized vortex HQVs Isolated HQV (k,n) ¼ (1/2, +1/4) Isolated HQV (k,n) ¼ (1/2, −1/4) Molecule of HQVs (stability) Non-Abelian anyon in non-Abelian vortex Existence of zero energy states Discrete symmetry of vortices

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Topology of the vortex-bound states Conclusion Acknowledgments References

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Abstract Anyons are particles obeying statistics of neither bosons nor fermions. Non-Abelian anyons, whose exchanges are described by a non-Abelian group acting on a set of wave functions, are attracting a great attention because of possible applications to topological quantum computations. Braiding of non-Abelian anyons corresponds to quantum computations. The simplest non-Abelian anyons are Ising anyons which can be realized by Majorana fermions hosted by vortices or edges of topological superconductors, n ¼ 5/2 quantum Hall states, spin liquids, and dense quark matter. While Ising anyons are insufficient for universal quantum computations, Fibonacci anyons present in n ¼ 12/5 quantum Hall states can be used for universal quantum computations. Yang-Lee anyons are nonunitary counterparts of Fibonacci anyons. Another possibility of non-Abelian anyons (of bosonic origin) is given by vortex anyons, which are constructed from nonAbelian vortices supported by a non-Abelian first homotopy group, relevant for certain nematic liquid crystals, superfluid 3 He, spinor Bose-Einstein condensates, and high density quark matter. Finally, there is a unique system admitting two types of non-Abelian anyons, Majorana fermions (Ising anyons) and non-Abelian vortex anyons. That is 3P2 superfluids (spin-triplet, p-wave paring of neutrons), expected to exist in neutron star interiors as the largest topological quantum matter in our universe.

Key points

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Unlike fermions and bosons, the exchange of non-Abelian anyons is described by unitary transformations that operate on topologically protected degenerate ground states. The adiabatic exchanges of two or more non-Abelian anyons manipulate quantum information stored in a set of degenerate ground states, leading to topological quantum computations. The simplest non-Abelian anyons are Ising anyons which are predicted to exist as Majorana zero modes in topological superconductors, fractional quantum Hall states, Kitaev materials, and dense quark matter. Another non-Abelian anyons are vortex anyons, which are constructed from non-Abelian vortices supported by a nonAbelian first homotopy group. Such vortex anyons exist in liquid crystals, superfluid 3He, spinor Bose-Einstein condensates, and high-density quark matter. 3 P2 superfluids, which are expected to be realized in neutron stars, provide unique systems simultaneously admitting two kinds of non-Abelian anyons, that is, Ising anyons and vortex anyons.

Introduction In three spatial dimensions, all particles are either bosons or fermions in quantum physics, that is, a wave function of multiparticle states is symmetric (antisymmetric) under the exchanges of two bosons (fermions). On contrary, in two spatial dimensions, there exist exotic particles classified to neither bosons nor fermions, anyons. A wave function of two anyons receives a nontrivial phase factor under their exchanges (Leinaas and Myrheim, 1977; Wilczek, 1982). Such exotic particles play essential roles in fractional quantum Hall states (Halperin, 1984; Arovas et al., 1984), and have been experimentally observed for n ¼ 1/3 fractional quantum Hall states (Bartolomei et al., 2020; Nakamura et al., 2020). Recently, yet exotic particles attracted great attention, that is, non-Abelian anyons. Non-Abelian anyons are described by a set of multiple wave functions, and the exchanges of two non-Abelian anyons lead to unitary matrix operations on a set of wave functions. They have been theoretically predicted to exist in n ¼ 5/2 fractional quantum Hall states (Moore and Read, 1991; Nayak and Wilczek, 1996), topological superconductors (SCs) and superfluids (SFs) (Read and Green, 2000; Ivanov, 2001; Kitaev, 2001), and spin liquids (Kitaev, 2006; Motome and Nasu, 2020), and experimental observation is pursued. Non-Abelian anyons are attracting significant interests owing to the possibility to offer a platform of topologically protected quantum computations realized by braiding of non-Abelian anyons (Kitaev, 2003; Kitaev and Laumann, 2008; Nayak et al., 2008; Pachos, 2012; Sarma et al., 2015; Field and Simula, 2018). Since the Hilbert space and braiding operations are topologically protected, they are robust against noises in contrast to the conventional quantum computation methods. Recently, it has been reported that non-Abelian braiding and fusions have been experimentally realized in a superconducting quantum processor, where the fusion and braiding protocols are implemented using a quantum circuit on a superconducting quantum processor (Andersen et al., 2022), thereby opening a significant step to realize topological quantum computations. One of the main routes to realize non-Abelian anyons is based on Majorana fermions in topological SCs (Ivanov, 2001; Kitaev, 2001; Alicea, 2012; Leijnse and Flensberg, 2012; Beenakker, 2013; Silaev and Volovik, 2014; Elliott and Franz, 2015; Sato and

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Fujimoto, 2016; Mizushima et al., 2016; Sato and Ando, 2017; Beenakker, 2020; Marra, 2022). Majorana fermions were originally proposed in high energy physics to explain neutrinos; they are particles that coincide with their own antiparticles (Majorana, 1937). In condensed matter physics, Majorana fermions are localized at vortices or edge of materials for which several protocols of nonAbelian braiding were proposed. Non-Abelian anyons constructed from Majorana fermions are so-called Ising anyons. They are not enough for universal quantum computations, and thus some nontopological process should be included (Nayak et al., 2008). In contrast, another type of anyons called Fibonacci anyons (Trebst et al., 2008) can offer a promising platform for universal topological quantum computation that all quantum gates are implemented by braiding manipulation in a topologically protected way (Preskill, 2004; Bonesteel et al., 2005; Hormozi et al., 2007). Such Fibonacci anyons are proposed to exist in n ¼ 12/5 quantum Hall states, a junction made of a conventional SC and a n ¼ 2/3 fractional quantum Hall state (Mong et al., 2014), interacting Majorana fermions realized in a septuple-layer structure of topological SCs (Hu and Kane, 2018), and Rydberg atoms in a lattice (Lesanovsky and Katsura, 2012). Yang-Lee anyons are also proposed as nonunitary counterparts of Fibonacci anyons, obeying nonunitary non-Abelian statistics (Ardonne et al., 2011; Freedman et al., 2012; Sanno et al., 2022). The aforementioned anyons are all quasiparticle excitations composed of fermions. On the other hand, a different type of nonAbelian anyons can be composed of bosons in certain ordered states accompanied with symmetry breakings G ! H. They are called non-Abelian vortex anyons, whose exchange statistics are non-Abelian due to non-Abelian vortices, that is quantum vortices supported by a non-Abelian first homotopy (fundamental) group of order parameter manifolds, p1(G/H) (Bais, 1980; Wilczek and Wu, 1990; Bucher, 1991; Brekke et al., 1993; Lo and Preskill, 1993; Lee, 1994; Brekke et al., 1997).1 Such non-Abelian vortices exist in liquid crystals (Poenaru and Toulouse, 1977; Volovik and Mineev, 1977; Mermin, 1979; Lavrentovich and Kleman, 2001), 3 He SFs (Balachandran et al., 1984; Salomaa and Volovik, 1987; Volovik, 2003), spinor Bose-Einstein condensates (BECs) (Semenoff and Zhou, 2007; Kobayashi et al., 2009, 2012; Borgh and Ruostekoski, 2016), and high-density quark (QCD) matter (Fujimoto and Nitta, 2021a, b, c; Eto and Nitta, 2021). Non-Abelian braiding of vortex anyons in spinor BECs and its application to quantum computations were proposed (Mawson et al., 2019). In addition to these systems admitting one type of non-Abelian anyons, there is the unique system simultaneously admitting two kinds of non-Abelian anyons, Ising anyons based on Majorana fermions and non-Abelian vortex anyons. It is a 3P2 SF, spintriplet and p-wave pairing with the total angular momentum two (Hoffberg et al., 1970; Tamagaki, 1970; Takatsuka and Tamagaki, 1971; Takatsuka, 1972; Richardson, 1972). Such 3P2 SFs are expected to be realized by neutrons, relevant for neutron star interiors (Sedrakian and Clark, 2018). 3P2 SFs are the largest topological SFs in our universe (Mizushima et al., 2017) and admit non-Abelian vortices (Masuda and Nitta, 2020). Non-Abelian vortices host Majorana fermions in their cores (Masaki et al., 2022), thus behaving as non-Abelian anyons. The purpose of this article is to summarize these non-Abelian anyons of various types. After introducing basics of non-Abelian anyons in Section “Non-Abelian anyons,” we describe non-Abelian anyons in fermionic and bosonic systems, based on Majorana fermions and non-Abelian first homotopy group in Sections “Non-Abelian anyons in topological SCs” and “Non-Abelian vortex anyons,” respectively. In Section “3P2 Topological SFs,” we introduce 3P2 SFs as the unique system simultaneously admitting two kinds of non-Abelian anyons. We summarize this article in Section “Summary.”

Non-Abelian anyons Braid group and quantum statistics Here we consider pointlike topological defects (e.g., vortices in two-dimensional spinless SCs), which behave as identical particles in a two-dimensional plane. The exchange of n particles in three or higher dimension is described by the symmetric group Sn (Leinaas and Myrheim, 1977). There are two one-dimensional representations of Sn,  1, due to even or odd permutation and + 1 (− 1) corresponds the Bose (Fermi) statistics. Two dimension is special and the exchange of particles is given by the braid group Bn (Wu, 1984). The braid of particles is expressed as a set of operators Tk(1  k  n − 1) that exchange the neighboring kth and (k + 1)th particles in an anticlockwise direction. The operators obey the relations (see Fig. 1) TiTjTi ¼ TjTiTj, TiTj ¼ TjTi,

for ji −jj ¼ 1, for ji −jj  2:

(1) (2)

The exotic statistics of particles represented by the braid group stems from the relation T i−1 6¼ T i . In the one-dimensional representation, the generator of Ti is given by a phase factor that a wave function under the exchange of particles acquires, tj  tðT j Þ ¼ eiyj (0  yj < 2p). The relation in Eq. (1) implies that the exchange operation of any two particles induces the same phase factor t1 ¼ t2 ¼ ⋯ ¼ tn −1 ¼ eiy. The phase factor characterizes the quantum statistics of particles (Wu, 1984), and the absence of the relation T 2i ¼ 1 allows for the fractional (anyon) statistics with neither y ¼ 0 (bosons) nor y ¼ p (fermions). In addition to the one-dimensional representation, the braid group has non-Abelian representations. In Section “Non-Abelian anyons in topological 1 The term “non-Abelian” on vortices depends on the context. In the other contexts (in particular in high energy physics), vortices in a symmetry breaking G ! H with non-Abelian magnetic fluxes are often called non-Abelian even though p1(G/H) the first homotopy group is Abelian (Hanany and Tong, 2003; Auzzi et al., 2003; Eto et al., 2006; Shifman and Yung, 2007; Eto et al., 2014). In condensed matter physics, vortices with Majorana fermions in their cores are also sometimes called non-Abelian vortices (because they are non-Abelian anyons). In this article, the term “non-Abelian” on vortices is used only for vortices with non-Abelian first homotopy group p1.

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Fig. 1 Schematic of the braid relations in Eqs. (1) and (2): TiTj ¼ TjTi for |i − j | 2 (left) and TiTjTi ¼ TjTiTj for |i − j | ¼ 1 (right).

SCs,” we will show that the generators of the braid of Majorana zero modes are noncommutative as [ti, tj] ¼ 6 0 for |i − j| ¼ 1 and the pointlike defects hosting Majorana modes behave as non-Abelian anyons. Although the braid group is trivial in three dimensions, a three-dimensional model with pointlike topological defects, which host Majorana modes and obey the non-Abelian statistics, has been proposed (Teo and Kane, 2010a). The non-Abelian statistics of the defects, which behave as hedgehogs, can be interpreted as the projective ribbon permutation statistics (Freedman et al., 2011b, a). Such non-Abelian statistics enables to construct three-dimensional networks of topological superconducting wires supporting Majorana modes.

Non-Abelian anyons In three spatial dimensions, the statistics of particles is determined by their intrinsic spins. According to the spin statistics theorem, all particles with integer (half-integer) spin are bosons (fermions). A pair formed by two fermions with the spin 1/2 behaves as a boson, and the spin of the composite particle obeys the “fusion rule” 12  12 ¼ 0  1, where 0 and 1 denote the spin singlet and triplet states, respectively. When particles are trapped in a two-dimensional plane, however, there is another possibility that is neither fermions nor bosons, that is, anyons. In general, anyons can be characterized by the topological charge, the fusion rule (Ncab ), the associative law (the F-matrix), and the braiding operation (the R-matrix) (Preskill, 2004; Nayak et al., 2008; Pachos, 2012). Let 1 and fa, b, . . .g be the vacuum and the different species of particles, respectively. Consider the anyon model spanned by M ¼ f1, a, b, . . .g and bring two anyons a and b together. The fused particle also belongs to M. The fusion rule is represented by X Ncab c, (3) ab¼ c2M

Ncab

where fusion coefficients are nonnegative integers. Fig. 2A shows the diagrammatic expression of the fusion of two anyons with topological charges a and b to an anyon with charge c. When Ncab is not zero for only one value of c, the fusion of paired a and b anyons is uniquely determined and the anyon is called the Abelian anyon. The non-Abelian anyons are characterized by two or more coefficients that satisfy Ncab 6¼ 0. In the context of quantum computation, the fusion rule determines the Hilbert space that encodes quantum information, and quantum computation is implemented by the braiding manipulation of non-Abelian anyons in a topologically protected way. Fig. 2B shows the diagrammatic expression of the R-matrix. When one anyon moves around the other, the pairwise anyons acquire a phase. As mentioned below, the R-matrix describes a phase resulting from the exchange of anyons a and b which fuse to a anyon c. Let us also consider the fusion of three anyons a, b, and c into an anyon d. The outcome of (a)

(b)

(c)

Fig. 2 (A) Diagram of the fusion of two anyons a and b to an anyon c. Diagrammatic expressions of the R-matrix (B) and the F-matrix (C).

Non-Abelian anyons and non-Abelian vortices in topological superconductors

759

the fusion process is independent of order in which the anyons are to be fused. This implies that the fusion process obeys the associative law, (a  b)  c ¼ a  (b  c), which is characterized by the F-matrix. The F matrix represents the transformation between different fusion bases or the choice of order of fusion, which is expressed as shown in Fig. 2C. The R-matrix and the F-matrix are the building blocks for constructing the braid group in multiple anyon systems (Preskill, 2004).

Ising anyons An example of the non-Abelian anyons is the Ising anyon (Kitaev, 2006; Nayak et al., 2008; Pachos, 2012). The Ising anyon model consists of the vacuum 1, Ising anyons s, and Dirac (complex) fermions c, which obey the fusion rules s  s ¼ 1  c, s  c ¼ s, c  c ¼ 1, 1  x ¼ x,

(4)

where x 2{1, s, c}. Consider three Ising anyons, where the two left-most anyons fuse into either 1 or c (see Fig. 2C with (a, b, c) ! s and i, j ¼ {1, c}). The R-matrix and the F-matrix are given, respectively, by !   Rss 0 1 0 1 R¼ , (5) ¼ eip=8 ss 0 Rc 0 i 1 Fssss ¼ pffiffiffi 2



1 1

 1 : 1

(6)

The diagonal components of the R-matrix are the phases resulting from the counterclockwise exchange of two left-most Ising anyons (s) fusing to 1 or c while the twice exchange operation of two right-most Ising anyons is represented by the unitary matrix, F−1R2F ¼ e−ip/4sx. Hence, braiding two anyons corresponds to the implementation of the quantum gates acting on the quantum states spanned by 1 and c. In general, the anyons are described by conformal field theory, corresponding to gapless edge states residing in the boundary of two-dimensional gapped topological phases. For Ising anyons, the theory is the conformal field theory with the central charge c ¼ 1/2, which describes critical Ising models such as the two-dimensional Ising model at the point of second-order phase transition (Di Francesco et al., 1997). The Moore-Read state in the fractional quantum Hall state at the filling factor n ¼ 5/2 supports this type of non-Abelian anyons (Nayak et al., 2008). The Ising anyons can also be realized in the Kitaev’s honeycomb model, which is an exactly solvable model of quantum spin liquid states (Kitaev, 2006). In this model, the spins are fractionalized to Majorana fermions coupled to Z2 gauge fields. The Ising anyons appear as Majorana zero modes bound to the Z2 flux. Materials including 4d or 5d atoms with a strong spin–orbit coupling have been proposed as candidates of Kitaev magnets, and the half-integer thermal Hall effect was reported in a-RuCl3 (Kasahara et al., 2018; Yamashita et al., 2020; Yokoi et al., 2021; Bruin et al., 2022), which is a signature of Majorana fermions in the chiral quantum spin liquid phase (Vinkler-Aviv and Rosch, 2018; Ye et al., 2018). Apart from materials, Google team reported the demonstration of fusion and braiding rules of non-Abelian Ising anyons on a superconducting quantum processor, where the fusion and braiding protocols are implemented using a quantum circuit on a superconducting quantum processor (Andersen et al., 2022). Another platform to realize Ising anyons is a topological SC. In the context of topological SCs, the Ising anyons (s) appear as Majorana zero modes bound at their boundaries or topological defects, such as the surface, interface, and vortices (Read and Green, 2000; Kitaev, 2001; Nayak et al., 2008; Alicea, 2012; Sato and Fujimoto, 2016; Mizushima et al., 2016). Pairwise Majorana zero modes form a complex fermion that can define either the unoccupied state (1) or the occupied state (c) of the zero energy eigenstate, implying s  s ¼ 1  c. The vacuum 1 corresponds to a condensate of Cooper pairs, while c represents a Bogoliubov quasiparticle which can pair into a condensate, that is, c  c ¼ 1. The detailed properties and realization of Majorana zero modes in topological SCs are described in Sections “Majorana zero modes as Ising anyons” and “Platforms for Majorana zero modes.”

Fibonacci anyons

Another example is the Fibonacci anyons (Trebst et al., 2008). The Fibonacci anyon model consists of the vacuum 1 and the nontrivial anyon t, which obey the fusion rules t  t ¼ 1  t,

1  x ¼ x,

(7)

where x ¼ 1, t. The first rule implies that the fusion of two anyons may result in either annihilation or creation of a new anyon, and thus the Fibonacci anyon may be its own antiparticle. Repeated fusions of the n + 1 t-anyons result in either the vacuum or the t anyon as t  t  ⋯  t ¼ an 1  bn t, where an ¼ 1 for n ¼ 2 and an ¼ n − 2 for n  3. The coefficient bn grows as the Fibonacci series, and the first few values in the sequence are b2 ¼ 1, b3 ¼ 2, b4 ¼ 3, b5 ¼ 5, . . .. The R-matrix and the F-matrix are given, respectively, by !  tt  R1 0 ei4p=5 0 R¼ ¼ , (8) 0 Rtt 0 ei2p=5 t

760

Non-Abelian anyons and non-Abelian vortices in topological superconductors

F tttt

¼

f1 f1=2

f1=2 f1 ,

! (9)

pffiffiffi where f ¼ ð1 + 5Þ=2 is the golden ratio. The Fibonacci anyons are described by the level-1 G2 Wess–Zumino–Witten theory with the central charge c ¼ 14/5 (Mong et al., 2014), where G2 is the simplest exceptional Lie group. While Ising anyons are not sufficient for universal quantum computation, the Fibonacci anyon systems can offer a promising platform for universal topological quantum computation that all quantum gates are implemented by braiding manipulation in a topologically protected way (Preskill, 2004; Bonesteel et al., 2005; Hormozi et al., 2007). The existence of the Fibonacci anyons is predicted in the n ¼ 12/5 fractional quantum Hall state that is described by the Read-Rezayi state (Read and Rezayi, 1999). It is also proposed that a junction made of a conventional SC and the n ¼ 2/3 fractional quantum Hall state supports the Fibonacci anyons (Mong et al., 2014). Fibonacci anyons can also be made from interacting Majorana fermions realized in a septuple-layer structure of topological SCs (Hu and Kane, 2018) and from Rydberg atoms in a lattice (Lesanovsky and Katsura, 2012).

Yang-Lee anyons There are nonunitary counterparts of Fibonacci anyons, which are referred to as Yang-Lee anyons. The conformal field theory corresponding to Yang-Lee anyons is nonunitary and Galois conjugate to the Fibonacci conformal field theory (Ardonne et al., 2011; Freedman et al., 2012). Because of nonunitarity, the central charge and the scaling dimension for the one nontrivial primary field are negative, c ¼ −22/5 and D ¼ −2/5, respectively. As shown in Eq. (9), the F-matrix for Fibonacci anyons is given by the golden ratio f, that is, one solution of the equation x2 ¼ 1 + x, which is an algebraic analog of the fusion rule. The F-matrix for Yang-Lee anyons is obtained from Eq. (9) by replacing f ! −1/f as pffiffiffiffi ! f i f Ftttt ¼ , (10) pffiffiffiffi i f f as − 1/f is the other solution of the equation x2 ¼ 1 + x. In Eq. (10), the bases of the F-matrix are spanned by the vacuum state 1 and a Yang-Lee anyon t. The R-matrix are given by !  tt  R1 0 ei2p=5 0 R¼ ¼ : (11) 0 Rtt 0 eip=5 t Unlike the Fibonacci anyons, braiding two Yang-Lee anyons is represented by a combination of the R-matrix and the nonunitary matrix, F−1RF. While Yang-Lee anyons obey the same fusion rule as that of Fibonacci anyons given by Eq. (7), the F-matrix in Eq. (10) is the nonunitary and the Yang-Lee anyons obey the nonunitary non-Abelian statistics. The nonunitary conformal field theory with c ¼ −22/5 describes the nonunitary critical phenomenon known as the Yang-Lee edge singularity (Cardy, 1985). Let us consider the Ising model with an imaginary magnetic field ih (h 2 R). For temperatures above the critical temperature, the zeros of the partition function in the thermodynamic limit, which are referred to as the Lee-Yang zeros, accumulate on the line h > hc and the edge of the Lee-Yang zeros corresponds to the critical point hc (Lee and Yang, 1952). As h (> hc) approaches the edge, the density of zeros has a power-law behavior as |h − hc|s, which characterizes the critical phenomenon (Kortman and Griffiths, 1971; Fisher, 1978). For instance, the magnetization exhibits singular behavior with the same critical exponent s. The Yang-Lee edge singularity is also realized by the quantum Ising model with a real transverse field and a pure-imaginary longitudinal field (von Gehlen, 1991). The quantum Ising model with an imaginary longitudinal field, which supports the Yang-Lee anyons, can be constructed from Majorana zero modes in a network of topological superconducting wires coupled with dissipative electron baths (Sanno et al., 2022). The Majorana modes bound at the end points of one-dimensional topological SCs constitute spin 1/2 operators. A coupling of Majorana zero modes with electrons in a metallic substrate plays a role of the pure-imaginarily longitudinal field, while the tunneling of Majorana zero modes between neighboring superconducting wires induces a transverse magnetic field. Schemes for the fusion, measurement, and braiding of Yang-Lee anyons are also proposed in Sanno et al. (2022). As mentioned above, the Yang-Lee anyons obey the nonunitary non-Abelian statistics. The nonunitary evolution of quantum states has been discussed in connection with measurement-based quantum computation (Terashima and Ueda, 2005; Usher et al., 2017; Piroli and Cirac, 2020; Zheng, 2021). Although the Yang-Lee anyons with the nonunitary F-matrix are not suitable for application to unitary quantum computation, they can be the building blocks for the construction of measurement-based quantum computation. In addition, as the nonunitary quantum gates can be implemented by braiding manipulations, the Yang-Lee anyon systems may offer a quantum simulator for nonunitary time evolution of open quantum systems in a controllable way.

Vortex anyons In this article, we also discuss non-Abelian anyons made of bosonic (topological) excitations in ordered states. The nontrivial structure of the order parameter manifold appears in the liquid crystals, spin-2 BECs, the A phase of SF 3He, dense QCD matter, and 3 P2 SFs. The line defects in such ordered systems, such as vortices, are represented by non-Abelian first homotopy group and their topological charges are noncommutative. Such topological defects with noncommutative topological charges behave as

Non-Abelian anyons and non-Abelian vortices in topological superconductors

761

non-Abelian anyons, called the non-Abelian vortex anyons. In Section “Non-Abelian vortex anyons,” we demonstrate that order parameter manifolds in nematic liquid crystals and spin-2 BECs admit the existence of non-Abelian vortices and show the fusion rules of such non-Abelian vortex anyons.

Non-Abelian anyons in topological SCs Majorana zero modes as Ising anyons Majorana zero modes An elementary excitation from superconducting ground states is a Bogoliubov quasiparticle that is a superposition of the electron and hole. The quasiparticle excitations are described by the Bogoliubov-de Gennes (BdG) Hamiltonian !  cj X { c i , c i Hij , (12) H¼ c {j ij Hij ¼

hij

Dij

D{ij

h ij

! :

(13)

Here, c j is the N-component vector of the electron field operator and H is the 2N  2N Hermitian matrix, where N is the sum of the spin degrees of freedom and the number of the lattice sites and so on. The N  N Hermitian matrix h describes the normal state Hamiltonian and the superconducting pair potential D obeys Dt ¼ −D because of the Fermi statistics, where at denotes the transpose of a matrix a. The BdG Hamiltonian naturally holds the particle-hole symmetry CHC −1 ¼ −H,

(14)

where the particle-hole operator C ¼ YK is an antiunitary operator composed of the unitary operator Y and the complex conjugation operator K. The self-conjugate Dirac fermions are called Majorana fermions, where the quantized field C  (c, c {)t obeys C ¼ CC,

C2 ¼ +1:

(15)

We expand the quantized field C in terms of the energy eigenstates. The energy eigenstates are obtained from the BdG equation, X Hij ðwE Þj ¼ EðwE Þi , (16) j

which describes the quasiparticle with the energy E and the wave function wE. Eq. (14) guarantees that the quasiparticle state with E > 0 and wE is accompanied by the negative energy state with − E and w −E ¼ CwE. Thus, the negative energy states are redundant as long as the particle-hole symmetry is maintained. Let E be the quasiparticle operator which satisfies the anticommutation relations, fE , {E0 g ¼ dE,E0 and fE , E0 g ¼ f{E , {E0 g ¼ 0 (E, E0 > 0). The self-charge conjugation relation (15) then implies that the quasiparticle annihilation operator with a positive energy is equivalent to the creation with a negative energy as E ¼ {−E, and C is expanded only P in terms of positive energy states as CðrÞ ¼ E>0 ½wE E + CwE {E . The condition (15) can be fulfilled by odd-parity SCs. In the absence of spin–orbit coupling, the spin-singlet pair potential is always invariant under the spin rotation, and the particle-hole exchange operator is given by C2 ¼ −1 in each spin sector. Hence, spin-singlet SCs cannot satisfy Eq. (15). Spin–orbit coupling, however, enables even spin singlet SCs to host Majorana fermions (Sato and Ando, 2017; Alicea, 2012; Sato and Fujimoto, 2016). Now, let us suppose that a single zero-energy state exists, and w0 is its wave function. Then, we can rewrite the quantized field to i Xh (17) C ¼ w0 g + wE E + CwE {E : E>0

We have introduced g, instead of E¼0, to distinguish the zero mode from other energy eigenstates. Owing to Eq. (14), the zero-energy quasiparticle is composed of equal contributions from the particle-like and hole-like components of quasiparticles, that is, Cw0 ¼ w0 . The self-conjugate constraint in Eq. (15) imposes the following relations: g{ ¼ g, fg, {E>0 g

(18)

and (g) ¼ 1 and fg, E>0 g ¼ ¼ 0. The quasiparticle obeying this relation is called the Majorana zero mode. The zero energy states appear in a topological defect of topological SCs, such as chiral SCs. In Fig. 3A, we show the spectrum of Andreev bound states bound at a vortex in a chiral SC, where the level spacing between the zero mode and the lowest excitation state is DE D20 =eF (Kopnin and Salomaa, 1991; Volovik, 1999; Read and Green, 2000; Matsumoto and Heeb, 2001). In many vortices, the hybridization between neighboring Majorana modes gives rise to the formation of the band structure with the width DEM e−D/x (Cheng et al., 2009; Mizushima and Machida, 2010), where D and x are the mean distance of neighboring vortices and superconducting coherence length, respectively (see Fig. 3B). The Majorana zero modes exhibit the non-Abelian anyonic behaviors (Ivanov, 2001). To clarify this, we start with two Majorana zero modes residing in an SC. Using two Majorana operators, g1 and g2, we define the new fermion operators c and c{ as 2

762

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Fig. 3 Schematic of the quasiparticle states bound at a single vortex (A) and energy spectrum of many vortices (B) in a spinless SC, where the level spacing of vortex bound states is DE D20 =eF and DEM denotes the band width of Majorana states bound at each core. (C) Operations of the braiding matrices U12 and U23 and the F-matrix in four Majorana modes.

1 c ¼ ðg1 + ig2 Þ, 2

1 c{ ¼ ðg1 −ig2 Þ, 2

(19)

which obey the anticommutation relations, {c, c{} ¼ 1 and {c, c} ¼ {c{, c{} ¼ 0. The two degenerate ground states are defined as the vacuum |0i and the occupied state of the zero energy state |1i ¼ c{|0i, respectively, where the former (latter) state is the even (odd) fermion parity. We note that as the BdG Hamiltonian for superconducting states is generally commutable with the parity operator, the fermion parity remains as a good quantum number. For the even (odd) parity sector, the Hilbert space is spanned by using |0i (|1i) and excited states that are constructed as {E {E0 {E00 ⋯j0i ({E {E0 {E00 ⋯j1i). The Majorana operators, g1, g2, and ig1g2, act on the Hilbert space as the Pauli matrices sx, sy, and sz, respectively. The eigenstates of the Majorana operators g1 and g2 are given by the superposition of the degenerate states with different fermion parity, |0i and |1i. Hence, the eigenstate of a single Majorana zero mode cannot be a physical state. Consider 2N Majorana zero modes denoted by gj (j ¼ 1, ⋯ , 2N), where N complex fermions are constructed by the fusion of ith and jth Majorana zero modes as cij ¼ (gi + igj)/2. We define the occupation number operator of the complex fermion, 1 nij  c{ij cij ¼ ð1 + igi gj Þ: 2

(20)

In a basis that diagonalizes paired Majorana modes igigj, two eigenvalues of the complex fermion, nij ¼ 0 and 1, correspond to the fusion channels 1 and c, respectively. Hence, the Majorana zero mode is referred to as the Ising anyon. 2N degenerate ground states are expressed in terms of the occupation numbers as jn12 , n34 , ⋯ i, which are separated to the sectors of the even/odd fermion parity. Here we assume that the temperature of the system is much lower than the level spacing (DE) between the Majorana zero mode and the lowest excitation (non-Majorana) state. Then, the 2N−1 degenerate ground states in each fermion parity sector can be utilized as topological qubits, where quantum information is stored in a topologically protected way.

Braiding Majorana zero modes

Here we discuss the braiding statistics of Majorana zero modes gi and gj and show the non-Abelian statistics of Majorana zero modes (Ivanov, 2001; Alicea et al., 2011; Clarke et al., 2011). While we consider the exchange of Majorana zero modes residing in vortices, the theory is also applicable to Majorana zero modes bound at the end points of one-dimensional topological SCs. Let Tij be the braid operators that satisfy Eqs. (1) and (2) and transform gi and gj to eyi gj and eiyj gi , respectively. The unitary time evolution of Majorana zero modes is governed by the Heisenberg equation, i dtd gj ðtÞ ¼ ½gj ðtÞ, HðtÞ . The positions of two Majorana modes are adiabatically exchanged in the time interval [0, T]. The adiabatic condition defines the lower bound for the time scale of the braiding operation, T, so that T is much longer than the inverse of the level spacing between the Majorana zero mode and the lowest excitation (non-Majorana) state, DE. In addition, the upper bound is associated with the band width of Majorana modes DEM e−D/x (see Fig. 3B), that is, DE −1  T  DEM−1 . Within the adiabatic condition, the braiding dynamics of Majorana zero modes can be regarded as the unitary time evolution, gj ðtÞ ¼ U {ij ðtÞgj ð0ÞU ij ðtÞ. After the braiding operation, gi (gj) changes to gj (gi) with an additional phase shift. Then, the braiding operation is represented by U {ij gi U ij ¼ eiyi gj ,

U {ij gj U ij ¼ eiyj gi ,

(21)

where Uij  Uij(T ) is the unitary operator which describes the exchange operation of two Majorana zero modes g and g , that is, the i

2

2

j

representation of Tij. According to the conditions, ðeiyi gj Þ ¼ ðU {ij gi U ij Þ ¼ 1 and (gj)2 ¼ 1, the phase shifts obey yi ¼ np and yj ¼ mp,

Non-Abelian anyons and non-Abelian vortices in topological superconductors

763

where n, m 2 Z. The braiding operations must not change the parity of the occupation number defined in Eq. (20) and thus satisfy U {ij nij U ij ¼ nij , which imposes the condition, y1 + y2 ¼ (2n + 1)p, on the phase shift. As a result, the exchange operation of two Majorana zero modes obtains the following braiding rules: U {ij gi U ij ¼ gj ,

U {ij gj U ij ¼ −gi :

(22) p ffiffiffi Consider four Majorana zero modes denoted by g1, g2, g3, and g4, which form two complex fermions as c12  ðg1 + ig2 Þ= 2 and pffiffiffi 3 4 c34  ðg + ig Þ= 2 (Fig. 3C). When the Majorana mode “3” adiabatically encircles the Majorana mode “2,” both Majorana modes operators acquire the p phase shift, g2 7! − g2 and g3 7! − g3, corresponding to the twice operation of Eq. (22). Therefore, the braiding operation changes the occupation numbers of the complex fermion n12  c{12 c12 and n34  c{34 c34. For example, the above braiding generates a pair of the complex fermions |11i from their vacuum |00i. Here these two states are orthogonal, h11|00i ¼ 0. The braiding rule can be generalized to 2N Majorana modes. The 2N Majorana modes are fused to N complex fermions, leading to the 2N−1-fold degeneracy of ground states while preserving fermion parity. As discussed above, when the ith and jth Majorana modes are exchanged with each other, their operators behave as gi 7! gj and gj 7! − gi. The representation of the braid operator Tij that satisfies Eq. (22) is given in terms of the zero mode operators as (Ivanov, 2001)     p 1 U ij ¼ eiy exp gj gi ¼ eiy pffiffiffi 1 + gj gi : (23) 4 2 From now on, we omit the overall Abelian phase factor eiy as it is not important for quantum computation. Eq. (23) also holds in the case of the Moore-Read state (Nayak and Wilczek, 1996). For N ¼ 1, there is only a single ground state in each sector with definite fermion parity, and the exchange of two vortices results in the global phase of the ground state by eip/4. One can easily find that for N  2, the exchange operators Uij and Ujk do not commute to each other, [Uij, Ujk] 6¼ 0, implying the non-Abelian anyon statistics of the Majorana zero modes. For four Majorana zero modes (N ¼ 2), twofold degenerate ground states exist in each fermion-parity sector: |00i  |vaci and j11i ¼ c{12 c{34 jvaci in the sector of even fermion parity, and j10i ¼ c{12 jvaci and j01i ¼ c{34 jvaci in the sector of odd fermion parity. For the even-parity sector, the representation matrix for the exchange of 1 $ 2 and 3 $ 4 (Fig. 3C) is given by p

p

U 12 ¼ U 34 ¼ e −i4 j00ih00j+ei4 j11ih11j:

(24)

This merely rotates the phase of the ground state as in the N ¼ 1 case. In contrast, the representation matrix for the intervortex exchange (2 $ 3 in Fig. 3C) has the mixing terms of the two degenerate ground states |00i and |11i, 1 U 23 ¼ pffiffiffi ½j00ih00j −ij00ih11j + j11ih11j −ij11ih00j : 2

(25)

We note that the choice of the pairing to form the complex fermion is arbitrary. The change of the fused Majorana modes corresponds to the change of the basis from one which diagonalizes ig1g2 and ig3g4 to another which diagonalizes ig1g3 and ig2g4. The basis transformation is represented by the F-matrix as   1 1 1 F ¼ pffiffiffi : (26) 2 1 −1 The braiding matrix in Eq. (24) implies that the exchange of two Majorana zero modes fusing to c (|11i) acquires an additional p/2 phase compared to the fusion channel to 1 (|00i). Hence, Eq. (24) satisfies the property of the R-matrix in Eq. (5), that is, ss Rss 1 ¼ −iRc .

Platforms for Majorana zero modes The realization of non-Abelian anyons requires to freeze out the internal degrees of freedom of Majorana modes. The simplest example is spinless p-wave SCs/SFs, which emerge from the low-energy part of spinful chiral p-wave SCs. To clarify this, we start with spin-triplet SCs, whose pair potential is given by a 2  2 spin matrix (Leggett, 1975)   D"" ðkÞ D"# ðkÞ ^ DðkÞ ¼ (27) ¼ ism sy dm ðkÞ ¼ ism sy Ami k^i , D#" ðkÞ D## ðkÞ where k^i  ki =kF is scaled with the Fermi momentum kF, and the repeated Greek/Roman indices imply the sum over x, y, z. Here we omit the spin-singlet component. Owing to the Fermi statistics, the spin-triplet order parameter, d(k), obeys d(k) ¼ −d(−k). For spintriplet p-wave pairing, the most general form of the order parameter is given by a 3  3 complex matrix, Ami 2 C , where the components are labelled by m, i 2 {x, y, z}.

HQVs with Majorana zero modes in SF 3He We consider the Anderson-Brinkman-Morel (ABM) state (Anderson and Morel, 1961; Anderson and Brinkman, 1973), as a prototypical example of chiral p-wave states hosting Majorana zero modes. The ABM state is realized in the A-phase of the SF

764

Non-Abelian anyons and non-Abelian vortices in topological superconductors

3

He, which appears in high pressures and high temperatures (Vollhardt and Wölfle, 1990; Volovik, 2003). At temperatures above the SF transition temperature, T > Tc  1–2 mK, the normal Fermi liquid 3He maintains a high degree of symmetry G ¼ SOð3ÞL  SOð3ÞS  Uð1Þ,

(28)

where SO(3)L, SO(3)S, and U(1) are the rotation symmetry in space, the rotational symmetry of the nuclear spin degrees of freedom, and the global gauge symmetry, respectively. The tensor Ami transforms as a vector with respect to index m under spin rotations, and, separately, as a vector with respect to index i under orbital rotations. The order parameter of the ABM state is then given by the complex form ^ j + in^j Þ: Amj ¼ Dei’ d^m ðm

(29)

The ABM state is the condensation of Cooper pairs with the “ferromagnetic” orbital, and spontaneously breaks the timereversal symmetry. The orbital part of the order parameter is characterized by a set of three unit vectors forming the triad ^ n, ^ ^lÞ, where ^l ¼ m ^  n^ denotes the orientation of the orbital angular momentum of Cooper pairs. The remaining symmetry ðm, in the ABM state is HA ¼ SOð2ÞSz  SOð2ÞLz−’  Z2, where SOð2ÞSz is the two-dimensional rotation symmetry in the spin space. The ABM state is also invariant under SOð2ÞLz−’ which is the combined gauge-orbital symmetry, where the U(1) phase ^ 0j + in^0j Þ. In ^ j + in^j ! e −id’ ðm rotation, ’ ! ’ + d’, is compensated by the continuous rotation of the orbital part about ^l, m ^ m, ^ −m, ^ nÞ ^ ! ð −d, ^ −nÞ. ^ The manifold of the order parameter degeneracy is addition, Z2 is the mod-2 discrete symmetry ðd, then given by RA ’ G=HA ’ S2S  SOð3ÞLz ,’ =Z2 :

(30)

^ The degeneracy space has an extra Z2 symmetry that the change from d^ to −d^ The two-sphere, with the variation of d. can be compensated by the phase rotation ’ ! ’ + p. The topologically stable linear defects in the ABM state are characterized by the group of the integers modulo 4 (Vollhardt and Wölfle, 1990; Volovik, 1992; Salomaa and Volovik, 1987; Volovik, 2003), S2S, is associated

p1 ðRA Þ ’ p1 ðSOð3Þ=Z2 Þ ’ Z4 :

(31)

There exist four different classes of topologically protected linear defects in the dipole-free case. The four linear defects can be categorized by the fractional topological charge, NA ¼ 0,

1 3 , 1, , 2 2

(32)

where NA ¼ 3/2 is topologically identical to NA ¼ −1/2. The representatives of NA ¼ 0 and NA ¼ 1/2 classes include continuous vortex such as the Anderson-Toulouse vortex and half quantum vortex (HQV), respectively. Owing to NA ¼ 2 ¼ 0, a pure phase vortex with winding number 2 is continuously deformed into a nonsingular vortex without a core, that is, the Anderson-Toulouse vortex (Anderson and Toulouse, 1977). The NA ¼ 1/2 vortex is a combination of the half-wound d-disgyration with a half-integer value of the U(1) phase winding (Fig. 4). The extra Z2 symmetry allows us to take the half-integer value of the topological charge, because the p-phase jump arising from the half-winding of the U(1) phase (’ ¼ y/2) can be canceled out by the change in the ^ The NA ¼ 1 class includes a pure phase vortex with odd winding number and the radial/tangential orientation of d^ (d^ ! −d). disgyrations without phase winding. The latter was originally introduced by de Gennes as l-textures with a singularity line (De Gennes, 1973; Ambegaokar et al., 1974). The vortex with the fractional charge NA ¼ 1/2 is a harbor for spinless Majorana zero modes. We introduce the center-of-mass pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ^ RÞ, where r ¼ x2 + y2. The vortex core is located at r ¼ 0. The vortex state is coordinate of Cooper pairs, R ¼ (r, y, z), as DðkÞ ! Dðk,

subject to the boundary condition at r ! 1 where the orbital angular momentum of the Cooper pair is aligned to the z-axis (^l k z^) and the U(1) phase ’ continuously changes from 0 to 2pk along the azimuthal (y) direction, Ami ðr ¼ 1, yÞ ¼ Deiky d^m ðyÞð^ xj + i^ yj Þ:

(33)

Owing to the spontaneous breaking of the gauge-orbital symmetry, there are two classes for the vorticity k: Integer quantum vortices with k 2 Z and HQVs with k 2 Z=2. In the HQVs, both the U(1) phase and d^ rotate by p about the vortex center (see Fig. 4). In ^ general, the d-texture for HQVs is obtained as ^ ¼ cosðksp yÞ^ x + sinðksp yÞ^ y, dðyÞ

(34)

^ The HQV is characterized by (k, ksp) ¼ (1/2, 1/2) while the integer quantum vortex has where ksp denotes the winding of d. (k, ksp) ¼ (1, 0). It is remarkable to notice that since the ABM state is the equal spin pairing state, the order parameter for the HQV is recast into the representation in the spin basis as h i ^ ¼ D eiðk −ksp Þy j""i + eiðk+ksp Þy j##i : D (35)

Non-Abelian anyons and non-Abelian vortices in topological superconductors

765

Fig. 4 Schematic of the HQV realized in the ABM state. The color map shows the U(1) phase winding from ’ ¼ 0 (red ) at y to p (blue) at y ¼ 2p, and the arrows represent the texture of the d^-vectors shown in Eq. (34) with ksp ¼ 1/2.

For the HQV with (k, ksp) ¼ (1/2, 1/2) the |""i Cooper pair possesses the spatially uniform phase while the |##i pair has the phase winding of 2p around the vortex as in a conventional singly quantized vortex. Thus, the vortex-free state in the " spin sector exhibits fully gapped quasiparticle excitations, while the low-lying structures in HQV are effectively describable with a singly quantized vortex in the # spin sector, that is, the spin-polarized chiral p-wave SC. An odd-vorticity vortex in the spin-polarized chiral system hosts a single spinless Majorana zero mode that obeys non-Abelian statistics. The existence of non-Abelian anyonic zero modes in half quantum vortices was first revealed by Ivanov, who developed the non-Abelian braiding statistics of vortices with spinless Majorana zero modes (Ivanov, 2001). In the bulk A-phase of the SF 3He, an obstacle to realizing the HQVs is the formation of continuous vortices with the ^l-texture, which are characterized by the topological charge N ¼ 0. As the orientation of the ^l-vector is associated with the orbital motion of the Cooper pair, the ^l-texture can be uniformly aligned in a parallel plate geometry with thickness D. The motion of Cooper pairs is confined in the two-dimensional plane and ^l is locked perpendicular to the plates. Applying a magnetic field further restricts the orientation of d^ to the two-dimensional plane perpendicular to the applied field, which is a favorable situation to stabilize the ^ HQVs. In the parallel plate geometry with D ¼ 12.5mm, the measurements of NMR frequency shift observed that the d-vectors are confined to the two-dimensional plane perpendicular to ^l (Yamashita et al., 2008). The experiment was performed in a rotating cryostat at ISSP, University of Tokyo. The parallel plates are rotated at the angular velocity ≲12 rad=s but no conclusive evidence of HQVs was observed (Yamashita et al., 2008). Although the SF 3He-A thin film provides an ideal platform for HQVs with Majorana zero modes, the realization of HQVs remains as a challenging task. We also note that point-like solitons in a superfluid 3He film, such as spin disclinations, were shown to behave as anyons with fractional charges (Volovik and Yakovenko, 1989). Another promising route to realize HQVs with Majorana zero modes is to artificially introduce well-controlled disorders with high-porosity aerogel. In particular, the polar phase was observed in anisotropic aerogels consisting of uniaxially ordered alumina strands, called the nematic aerogels (Dmitriev et al., 2015; Halperin, 2019) (see Fig. 5A). The order parameter of the polar phase is zi ) is confined by the uniaxially anisotropic disorders. As in the given by Ami ¼ Dei’ d^m z^i , where the orbital state of the Cooper pair (^ ^ concomitant with the half-integer vorticity can realize HQVs in the polar phase. ABM state in the bulk 3He, the texture of the d-vector In the polar phase, HQVs are energetically preferable to integer quantum vortices at zero magnetic fields and magnetic fields applied along the uniaxial anisotropy (Nagamura and Ikeda, 2018; Mineev, 2014; Regan et al., 2021). Indeed, the HQVs were experimentally observed in nematic aerogels under rotation (Autti et al., 2016). The HQVs were also created by temperature quench via the Kibble-Zurek mechanism (Rysti et al., 2021). As shown in Fig. 5A, the superfluid phase diagram in nematic aerogels is drastically changed from that of the bulk 3He without disorders, where the polar-distorted A and B (PdA and PdB) phases are stabilized in addition to the polar phase. The NMR measurements performed in Mäkinen et al. (2019) observed that the HQVs survive across the ^ phase transition to the PdA phase. As shown in Fig. 4, the HQV is accompanied by the d-soliton in which the d^ orientation rotates. Fig. 5B shows the observed NMR spectra which have satellite peaks in addition to the main peak around the Larmor frequency. The main peak is a signal of the bulk PdA phase. The satellite peak originates from the spin excitation localized at the d^ -solitons

^ + ienÞ ^ i , where m ^ is aligned along the connecting pairs of HQVs. The order parameter of the PdA phase is given by Ami ¼ Dei’ d^m ðm axis of nematic aerogels and e 2 (0, 1) is the temperature- and pressure-dependent parameter on the distortion of Eq. (29) by nematic aerogels. Although the HQVs in the polar phase host no Majorana zero modes, the low energy structure of the HQVs in the PdA phase is essentially same as that of HQVs in the ABM state and each core may support the existence of spinless Majorana zero modes (Mäkinen et al., 2019; Mäkinen et al., 2022). Hence, the survival of the HQVs in the PdA phase is important as a potential platform for topological quantum computation. The elucidation of the nature of the fermionic excitations bound to the HQVs remains as a future problem.

HQVs in SCs

In contrast to the SF 3He, it is difficult to stabilize HQVs in SCs. The difficulty is attributed to the fact that the mass (charge) current is screened exponentially in the length scale of the penetration depth l, while such a screening effect is absent in the spin current

766

Non-Abelian anyons and non-Abelian vortices in topological superconductors (a)

(b)

Fig. 5 (A) The experimental setup and the phase diagram in the liquid 3He with nematic disorders and (B) NMR spectra at pressure P ¼ 7 bar and T ¼ 0.60Tc in the presence of a magnetic field perpendicular to the anisotropy direction of nematic disorders (Mäkinen et al., 2019), where Tc is the critical temperature of the bulk SF 3He without disorders. The disorders consist of nearly parallel Al2O3 strands, where the diameter and mean distance are d2  8nm and d1  50nm, respectively. In (B), the main peaks with green and magenta colors correspond to the signal of the bulk PdA phase while the satellite peaks originate from the spin excitation bound to the d^-solitons connecting pairs of HQVs. The satellite peak remains unchanged after the thermal cycling illustrated by purple arrows in (A). Taken from Mäkinen JT, Dmitriev VV, Nissinen J, Rysti J, Volovik GE, Yudin AN, Zhang K, and Eltsov VB (2019) Half-quantum vortices and walls bounded by strings in the polardistorted phases of topological superfluid 3He. Nature Communications 10 (1): 237. https://doi.org/10.1038/s41467-018-08204-8

Non-Abelian anyons and non-Abelian vortices in topological superconductors

767

(Chung et al., 2007). The HQV is accompanied by the spin current in addition to the mass current, while in the integer quantum ^ vortex, the integer vorticity (k 2 Z) and uniform d-texture in Eq. (33) induce no spin current. Therefore, the absence of the screening effect on the spin current is unfavorable for the stability of the HQVs relative to the integer quantum vortices, where the later may host spin-degenerate Majorana zero modes. It has also been proposed that configurational entropy at finite temperatures drives the fractionalization of the integer quantum vortices into pairwise HQVs (Chung and Kivelson, 2010). However, no firm experimental evidence for HQVs has been reported in SCs.

Superconducting nanowires Here we briefly mention recent progress on the search of Majorana zero modes bound at the end points of superconducting nanowires, although we mainly focus on non-Abelian anyons emergent in vortices in this article. The first idea was proposed by Kitaev that a spinless one-dimensional SC is a platform to host a Majorana zero mode (Kitaev, 2001). Although it was initially thought to be a toy model, it was recognized that a system with the same topological properties can be realized by a semiconductor/SC heterostructure (Alicea, 2012; Sato and Ando, 2017). The key ingredients are a semiconductor nanowire and a large Zeeman magnetic field in addition to the proximitized superconducting gap D (Sato et al., 2009, 2010). Consider the Hamiltonian of the semiconductor nanowire under a magnetic field B, h(p) ¼ p2/2m − m + lpsy − Bsz, which appears in the diagonal part of Eq. (13), where m is the mass of an electron and l is the strength of the spin–orbit coupling. The spin–orbit coupling is not only a source of nontrivial topology, but is also necessary to realize spinless electronic systems together with the large Zeeman field (B > m). In fact, if the proximitized superconducting gap is taken into account, a combination of spin–orbit coupling with the superconducting gap induces topological odd-parity superconductivity. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The topological phase with the spinless Majorana zero mode appears in the higher magnetic field satisfying B > Bc ¼ D2 + m2 . A direct signature of Majorana zero mode is the quantization of a zero-bias peak in differential conductance, where the height of the zero-bias peak stemming from the Majorana zero mode is predicted to be quantized to the universal conductance value of 2e2/h at zero temperature (Akhmerov et al., 2011; Fulga et al., 2011). Consider a charge-transfer process through a normal metal–SC junction. When an electron with the incident energy E enters from the normal metal to the SC, it forms the Cooper pair with an electron in the Fermi sea at the interface and a hole is created in the normal side as a consequence of momentum conservation. Such ¼ ½uE , vE t be the two-component a process in which electrons are retroreflected as holes is called Andreev reflection. Let fin;out E wavefunctions of the incident and reflected particles, respectively, where uE (vE) accounts for the electron (hole) component. We in now consider the scattering problem fout E ðxÞ ¼ SðEÞfE ðxÞ. The S-matrix is defined in the particle-hole space as ! r ee ðEÞ r eh ðEÞ SðEÞ ¼ , (36) r he ðEÞ r hh ðEÞ which is a unitary matrix S 2 U(2). The coefficient ree (reh(E)) is the amplitude of the normal (Andreev) reflection, and the others are the reflection coefficients of incident holes. The conductance G(E) at the normal/SC interface is obtained with the conductance quantum G0 ¼ e2/h per spin as G(E) ¼ 2G0|reh(E)|2. The particle-hole symmetry imposes on the S matrix the constraint, CSðEÞC −1 ¼ Sð− EÞ, leading to det Sð0Þ ¼ jr ee ð0Þj2 − jr eh ð0Þj2 at E ¼ 0. Let V be the unitary matrix defined by V f0 ¼ (Reu, Imu). In this basis, the scattering matrix S(0) is 0 0 0 transformed as S0 (0) ¼ V S(0)V{, where S0 (0) 2 O(2) is an orthogonal matrix satisfying S { ¼ S −1 ¼ S tr and det Sð0Þ ¼ det S0 ð0Þ ¼ 1. Hence, there are two different processes, a perfect normal reflection process (det Sð0Þ ¼ +1) and a perfect Andreev reflection process (det Sð0Þ ¼ −1). This discrete value, det Sð0Þ ¼ 1, is a topological invariant representing the parity of the number of Majorana zero modes residing at the interface. When det Sð0Þ ¼ −1, there exists at least one Majorana zero mode, which involves the perfect Andreev reflection process and the quantized conductance, Gð0Þ ¼

2e2 : h

(37)

The analytical expression of the conductance is obtained by solving the BdG equation and quasiclassical Usadel equation at an interface of the normal metal and unconventional SCs (e.g., p- and d-wave SCs) (Tanaka and Kashiwaya, 1995; Kashiwaya and Tanaka, 2000; Tanaka et al., 2004, 2005), where the conductance per spin reaches 2e2/h. The quantized conductance is also related to the Atiyah–Singer index when the Hamiltonian holds the chiral symmetry (Ikegaya et al., 2016). The conductance has recently been measured in a junction system of InSb-Al hybrid semiconductor-SC nanowire devices. The differential conductance yields the plateau behavior where the peak height reaches values 2e3/h (Zhang et al., 2021). In the device used in the experiment, the tunnel barrier is controlled by applying a gate voltage to the narrow region between the electrode and the region where the proximity effect occurs in the semiconductor wire. In the vicinity of the barrier, the unintentionally formed quantum dot or nonuniform potential formed in the vicinity of the interface gives rise to the formation of nearly zero-energy Andreev bound states. Such Andreev bound states mimic the plateau of G(0) ¼ 2e2/h even in the topologically trivial region of the magnetic field (B < Bc) (Liu et al., 2017; Prada et al., 2020; Cayao and Burset, 2021; Yu et al., 2021; Valentini et al., 2021). From the plateau of G(0), it is not possible to identify whether the observed G(0) 2e2/h is due to Majorana zero mode or the effect of accidentally formed s bound states. Alternative experimental schemes to distinguish the trivial and topological bound states have been proposed (Liu et al., 2018; Ricco et al., 2019; Yavilberg et al., 2019; Awoga et al., 2019; Zhang and Spånslätt, 2020; Schulenborg and Flensberg, 2020; Pan et al., 2021; Liu et al., 2021; Ricco et al., 2021; Thamm and Rosenow, 2021; Chen et al., 2022; Sugeta et al., 2022).

768

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Vortices in Fe(Se,Te) The iron-based SC Fe(Se,Te) is a candidate of topologically nontrivial superconducting state supporting Majorana zero modes. In the parent material FeSe, the 3d electrons of Fe near the Fermi level mainly contribute to superconductivity, and the pz orbitals of Se appear on the higher energy. Substitution of Se with Te causes a band inversion between the pz and 3d orbitals. As a result of the band inversion, the normal state of FeTe1−xSex is topologically nontrivial, and accompanied by the surface Dirac fermions (Zhang et al., 2018). Below the superconducting critical temperature, the superconducting gap is proximitized to the surface Dirac states. It was theoretically pointed out that the topological phase with a spinless Majorana zero mode can be realized when the superconducting proximity effect occurs in the Dirac fermion system (Sato, 2003; Fu and Kane, 2008). Therefore, the surface state of Fe(Se,Te) is a topological superconducting state. When a magnetic field is applied to Fe(Se,Te), the vortex lines penetrate to the surface state and the zero energy state appears. The wave function of the zero mode is tightly bound at the intersection of the vortex line and the surface (Hosur et al., 2011; Kawakami and Hu, 2015). When a magnetic field is applied to an SC, a quantized vortex penetrates. The magnetic flux is accompanied by the 2pm winding of the U(1) phase and the superconducting gap vanishes. In other words, a quantized vortex can be regarded as a quantum well with a radius of x and a height of D. Hence, the Andreev bound states are formed and the level spacing between them is on the order of D2/eF, where the level spacing is about 100–200 meV in Fe(Se,Te). In the topological phase, the lowest level has the exact zero energy, and the quasiparticle behaves as Majorana fermion. Ultra-low temperature STM/STS with high energy resolution observed a pronounced zero-bias conductance peak (Machida et al., 2019). If a Majorana zero mode is bound to the vortex, the conductance should be quantized to the universal value 2e2/h independent of tunnel barriers. In the tunneling spectroscopy performed while changing the distance between the sample surface and the STM tip, the height of the zero-bias conductance peak is approximately 0.6 times as large as the universal value stemming from the Majorana zero mode (Zhu et al., 2020). Although 2e2/h has not been reached, this plateau structure strongly suggests the existence of Majorana zero modes because it cannot be explained by non-Majorana vortex bound states.

Topological quantum computation Quantum computation based on the braiding of Majorana zero modes is basically along the two directions: implementation of quantum gates and measurement based quantum computation. The unitary evolution of the qubits is controlled by a set of discrete unitary operations, that is, quantum gates. The simple example of the quantum gates acting on a single qubit is the set of the Pauli gates, which are given by the three Pauli matrices X  sx, Y  sy, and Z  sz. The multiqubit gate operation can be implemented by a combination of a set of single-qubit gates and the controlled-NOT (CNOT) gate. According to the Solovay–Kitaev theorem (Nielsen and Chuang, 2010),pan pffiffiffiffi single-qubit pffiffiffigate can be approximated by a sequence of the discrete gate operations (H, S, T ), ffiffiffi arbitrary where H ¼ ðX + ZÞ= 2 , S ¼ Z , and T ¼ S are the Hadamard gate, the single-qubit 4p rotation, and the T-gate (p/8-gate), respectively. Universal quantum computation can be implemented by a sequence of the single-qubit gates (H, S, T ) and the CNOT gate. In Majorana qubits, the Pauli gates are expressed in terms of the braiding operations as X ¼ iU 223 , Y ¼ iU 231 , and Z ¼ iU 212 , and the Hadamard and S gates are implemented by a combination of such operations as H ¼ iU12U23U12 and S ¼ eip/4U12. In addition, the CNOT gate can be implemented by a combination of the measurement and braiding operations in two Majorana qubits (i.e., 8 Majorana zero modes) with a single ancilla Majorana qubit (Bravyi, 2006). All the Clifford gates (H, S, CNOT) are realized as a composition of braiding operators in a topologically protected way. According to the Gottesman–Knill theorem, however, quantum circuits only using the elements of Clifford group can be efficiently simulated in polynomial time on a classical computer (Nielsen and Chuang, 2010). Indeed, degenerate quantum states composed of multiple Ising anyons offer tolerant storage of quantum information, and their braiding operations provide all necessary quantum gates in a topologically protected way, except for the nonClifford T-gate. The T-gate can be implemented only in a topologically unprotected way. For instance, Karzig et al. proposed a scheme to implement the T-gate in a Y-shaped junction accompanied by 4 Majorana zero modes (Karzig et al., 2016). As mentioned in Section “Non-Abelian anyons,” Fibonacci anyon systems can offer a platform for universal topological quantum computation, where all the single-qubit gates (H, S, T ) and the CNOT gate are implemented by the braiding manipulations of the anyons in a topologically protected way (Preskill, 2004; Bonesteel et al., 2005; Hormozi et al., 2007). Another way to realize universal quantum computation in Majorana qubits is measurement-based quantum computation. It was realized that all braiding manipulations can be implemented in a topologically protected manner by the measurements of topological charges of pairwise anyons (Bonderson et al., 2008, 2009). The process of the quantum information encoded to Majorana qubits can take place by a series of simple projective measurements without braiding Majorana modes. The building block is a Coulomb-blockaded Majorana box which is made from even numbers of Majorana modes in a floating topological SC (Plugge et al., 2017; Karzig et al., 2017; Oreg and von Oppen, 2020).

Symmetry-protected non-Abelian anyons Symmetry-protected non-Abelian anyons in spinful SCs and SFs were discussed for class D (Ueno et al., 2013; Sato et al., 2014; Fang et al., 2014) and for class DIII (Liu et al., 2014; Gao et al., 2016). In the former (latter) case, mirror reflection symmetry (time reversal symmetry) plays an essential role in the topological protection of non-Abelian nature. In addition, unitary symmetry

Non-Abelian anyons and non-Abelian vortices in topological superconductors

769

protected non-Abelian statistics has been theoretically proposed (Hong et al., 2022). The non-Abelian statistics of vortices supporting multiple Majorana zero modes has also been discussed in Yasui et al. (2011) and Hirono et al. (2012) in the context of high-energy physics.

Mirror and unitary symmetries As the simple example of symmetry protected non-Abelian anyons, we consider integer quantum vortices in chiral p-wave SCs and 3 He-A (Sato et al., 2014), where each vortex core supports spinful Majorana zero modes. Contrary to the standard wisdom, a pair of Majorana zero modes in integer quantum vortices can be protected by the mirror symmetry and obey the non-Abelian statistics (Ueno et al., 2013; Sato et al., 2014). Let us consider two-dimensional superconducting film and assume that the normal state holds the mirror symmetry with respect to the xy-plane, Mxy hðkÞM{xy ¼ hðkÞ, where k ¼ (kx, ky) is the two-dimensional momentum and the mirror operator is defined as Mxy ¼ isz. The nontrivial topological properties are characterized by the first Chern number in each mirror subsector. When D(k) obeys MxyD(k)Mtxy ¼ D(k) ( ¼ ), the BdG Hamiltonian is commutable with the mirror reflection operator in the particle-hole space, M , as ! 0 Mxy   ½M , HðkÞ ¼ 0, M ¼ : (38) 0 M xy Thus, the eigenstates of HðkÞ are the simultaneous eigenstates of M . Let |un(l)(k)i and EðlÞ n be the wavefunction and eigenenergy of the Bogoliubov quasiparticles in each mirror subsector, where the l ¼ i are the eigenvalues of M. Then, the first Chern number is defined in each mirror subsector, Z i ðlÞ Ch1 ¼ F ðlÞ 2 Z, (39) 2p where F ðlÞ ¼ dAðlÞ is the Berry curvature in each mirror subsector and AðlÞ m ðkÞ ¼

P

ðlÞ

En < B H ’ D2 ¼ 13 , Ix ¼ @ 0 > : 0

0 −1 0

1 0 0 −1 C B 0 A, Iy ¼ @ 0 −1 0

0 +1 0

1 0 0 −1 C B 0 A, Iz ¼ @ 0 −1 0

0 −1 0

19 0 > = C 0 A : > ; +1

(55)

Note that D2 ’ Z2  Z2 is Abelian. The order parameter manifold is G=H ¼ SOð3Þ=D2 ’ SUð2Þ=Q,

(56)

where the quaternion group Q are universal (double) covering group of D2:   Q ¼ 12 ,  isx ,  isy ,  isz :

(57)

Note that H ’ Q is a non-Abelian group while H ’ D2 is an Abelian group. This breaking can occur for instance by five real-scalar order parameters belonging to spin-2 representation of SO(3), which is a traceless symmetric 3  3 tensor A with real components, taking a form of A ¼ diagð1, r, −1−rÞ:

(58)

The first homotopy group is isomorphic to the quaternion group Q in Eq. (57) p1 ðSUð2Þ=QÞ ’ Q:

(59)

The elements in Eq. (57) correspond to the ground state (12), spin vortices of 2p rotation (−12), and spin vortices of p rotation about the x, y, z-axes ( isx, y, z). These vortex configurations at the large distance can be asymptotically written as A OðyÞAy¼0 OT ðyÞ,

OðyÞ 2 SOð3Þ

(60)

with the azimuthal angle y. The concrete forms of O(y)’s are given in Table 1. Corresponding SU(2) elements g(y) can be defined as a double covering of SO(3) with g(2p) 2 p1(G/H). A spin vortex of 2p rotation (−12) and a spin vortex of p rotation ( isa) (a ¼ x, y, z) are given by   y y y gðyÞ ¼ exp i s n ¼ cos 12  is n sin , 2 2 2 (61)   y y y gðyÞ ¼ exp i sa ¼ cos 12  isa sin , 4 4 4 respectively, with a unit three-vector n, giving the elements of p1(G/H) at y ¼ 2p: g(2p) ¼ −12 and g(2p) ¼ isa, respectively. While the former is Abelian because [g(2p), H ] ¼ 0 (even though g(2p) 6¼ g(0)), the latter is non-Abelian because [g(2p), H ] 6¼ 0 with the universal covering H ’ SU(2),3 and H is topologically broken to K a ¼ f12 ,  isa g.

3

Note that [O(2p), H] ¼ 0, and thus the criterion of non-Abelian property should be considered in the universal covering group H but not in H.

774

Non-Abelian anyons and non-Abelian vortices in topological superconductors Table 1 Asymptotic configurations of spin vortices in D2-BNs: the fundamental group elements g(2p) 2 p1(G/H), the SU(2) elements g(y) 2 SU(2), O(2p), and the SO(3) group elements O(y) 2 SO(3), acting on the order parameter A as Eq. (60). The group actions O(y) and g(y) for a spin vortex of 2p rotation (−12) can be a rotation around any axis along the unit vector n: Li are 3  3 spin-1 matrices [Li ]jk ¼ −ieijk. g(2p) 2 p1(G/H) g(y) O(2p) O(y)

12

−12

 isx

 isy

 isz

12 13 13



exp i y2 n s 13 expðiyn LÞ



exp i y4 sx Ix

exp  i y2 Lx



exp i y4 sy Iy

exp  i y2 Ly



exp i y4 sz Iz

exp  i y2 Lz

The elements in Eq. (57) are grouped to the conjugacy classes, Eq. (53), consisting of five elements: f12 g, f−12 g, fisx g, fisy g, fisz g:

(62)

This follows for instance from (isy)−1(isx)(isy) ¼ −isx, implying that a vortex corresponding to isx becomes − isx when it travels around a vortex corresponding to isy. Thus, the vortices corresponding to isx and − isx are indistinguishable although they are not the same. From (isa)(−isa) ¼ 12 (a ¼ x, y, z), vortices belonging to the same conjugacy class are antiparticles of each other, and thus they are similar to Majorana fermions discussed in the last section. In three spatial dimensions, this phenomenon of noncommutativity appears as follows: when two vortex lines corresponding to isx and isy collide in three spatial dimensions, the third vortex bridging them is created (Poenaru and Toulouse, 1977; Mermin, 1979).

D4-BN liquid crystals The order parameter of D4 BNs is a set of two unoriented indistinguishable rods of the same lengths orthogonal to each other. In this case, the (unbroken) symmetry is a dihedral group D4 keeping a square invariant: G=H ¼ SOð3Þ=D4 ’ SUð2Þ=D 4 ,

(63)

where M denotes a universal covering group of M. The fundamental group is given by p1 ðSUð2Þ=D 4 Þ ’ D 4

(64)

with the sixteen elements   1 12 ,  isx ,  isy ,  isz ,  C4 ,  C1 4 ,  isx C4 ,  isx C4

with C4  e

ip4sz

(65) pffiffiffi 2 4 ¼ ð1= 2Þð12 + isz Þ. Note (C4) ¼ isz and (C4) ¼ −12. The conjugacy classes consist of the following seven elements 1 1 f12 g, f12 g, fisx ,  isy g, fisz g, fC4 , C1 4 g, fC4 ,  C4 g, fisx C4 ,  isx C4 g:

C34

(−C4−1

C4−3 ),

(66)

C4−1 .

¼ vortices of these elements have more energy than those of C4 and This symmetry breaking Since −C4 ¼ of D4 nematics can be realized by a higher rank tensor order parameter (Mietke and Dunkel, 2022), in which more generally Dn nematic liquid crystals were also discussed.

Spinor BECs Next we provide examples of spinor BECs (Kawaguchi and Ueda, 2012). The order parameter of spin-2 BECs is a traceless symmetric 3  3 tensor A with complex components transforming under the symmetry G ¼ U(1)  SO(3) as A ! eiy gAgT ,

eiy 2 Uð1Þ,

g 2 SOð3Þ:

(67)

Phases of condensations with total angular momentum two are classified by Mermin (Mermin, 1974). They are nematic, cyclic, and ferromagnetic phases. All phases are theoretically possible in the case of spin-2 BECs. The spin-2 BECs are experimentally realized by 87Rb atoms for which the phase is around the boundary between cyclic phase and ferromagnetic phase, see for example, Tojo et al. (2009). Here, we discuss nematic and cyclic phases which host non-Abelian vortex anyons. In the next section, we discuss 3 P2 SF which is in the nematic phase.

BN phases in spin-2 BEC ( Song et al., 2007; Uchino et al., 2010; Kobayashi et al., 2012; Borgh and Ruostekoski, 2016) The nematic phase consists of three degenerate phases: the UN, D2- and D4-BN phases. The order parameters of the UN, D2-, and D4-BN phases are given by

Non-Abelian anyons and non-Abelian vortices in topological superconductors

775

UN : A diagð1, −1=2, −1=2Þ, D2 -BN : A diagð1, r, −1 −rÞ,

(68)

D4 -BN : A diagð1, −1, 0Þ, respectively, where r 2 R and −1 < r < −1/2. In the D2-BN phase, the limits r ! −1/2, −1 correspond to UN and D4-BN phases, respectively. The symmetry breaking patterns and the order parameter manifolds are UN :

SOð3Þ G ’ Uð1Þ  RP 2 , ¼ Uð1Þ  H Oð2Þ

D2 -BN :

SOð3Þ SUð2Þ G , ’ Uð1Þ  ¼ Uð1Þ  D2 Q H

D4 -BN :

G Uð1Þ  SOð3Þ Uð1Þ  SUð2Þ ’ : ¼ D4 D 4 H

(69)

These phases are continuously connected by the parameter r interpreted as a quasi-Nambu-Goldstone mode. The degeneracy is lifted by quantum effects and either of these phases remains as the ground state (Uchino et al., 2010). The D2-BN and D4-BN admit non-Abelian vortices. The order parameters of the UN and D2-BN phases of spin-2 BECs are merely products of the U(1) phonon and the order parameters of the corresponding nematic liquids. Thus, we concentrate on the D4-BN phase hereafter. The fundamental group of the D4-BN phase is given by   Uð1Þ  SUð2Þ ffi Zh D 4 (70) p1 D 4 where h is defined in Kobayashi et al. (2012). This consists of the following sixteen elements  ðN,  12 Þ, ðN,  isx Þ, ðN,  isy Þ, ðN,  isz Þ,        o 1 1 1 1 N + ,  C4 , N + ,  isx C4 , N + ,  C1 , N + ,  isx C1 4 4 2 2 2 2

(71)

where the first and second elements of a pair pffiffiffi( , ) denote the circulation k (U(1) winding number) and an SU(2) element, p respectively with N 2 Z. Here C4  ei4sz ¼ ð1= 2Þð12 + isz Þ satisfying C44 ¼ −12. The conjugacy classes of Eq. (71) are composed of seven elements for each N: ðIÞ ðIIÞ ðIIIÞ ðIVÞ ðVÞ ðVIÞ ðVIIÞ

fðN, 12 Þg, fðN,  12 Þg, fðN,  isx Þ, ðN,  isy Þg, fðN,  isz Þg, n   o 1 1 , N + , C4 , N + , C1 4 2 2 n   o 1 1 N + ,  C4 , N + ,  C1 , 4 2 2 n   o 1 1 N + ,  isx C4 , N + ,  isx C1 : 4 2 2

(72)

They describe (I) integer vortices (N ¼ 0 corresponds to the vacuum), (II) spin vortices of 2p rotation, (III),(IV) spin vortices of p rotation around the x, y, and z axes, and (V)–(VII) non-Abelian HQVs. In Section “3P2 Topological SFs,” we see that 3P2 SFs are in the D4-BN phase in a strong magnetic field. There, a singly quantized vortex (1, 1) splits into two HQVs (1/2, C4) and ð1=2, C4−1 Þ both in the conjugacy class (V).4 When they fuse, they go back to the singly quantized vortex (1, 1). However, one of them, for example, ð1=2, C4−1 Þ can transform to the other (1/2, C4) once some other vortex passes through between them because they belong to the same conjugacy class. If they fuse after that, they become (1, isz) because C24 ¼ isz which is a composite belonging to (IV), of a singly quantized vortex and a spin vortex of p rotation.

Cyclic phase in spin-2 BEC ( Semenoff and Zhou, 2007; Kobayashi et al., 2009, 2012) The order parameter of the cyclic phase is A diagð1, e2pi=3 , e4pi=3 Þ,

4

Topologically a decay into a pair (1/2, −C4) and ð1=2, −C4−1 Þ in the conjugacy class (VI) is also possible, but energetically disfavored.

(73)

776

Non-Abelian anyons and non-Abelian vortices in topological superconductors

yielding the symmetry breaking to the tetrahedral group H ’ T and the order parameter manifold, given by G Uð1Þ  SOð3Þ Uð1Þ  SUð2Þ ’ ¼ T T

H with the universal covering group T of T. The fundamental group in the cyclic phase is given by   Uð1Þ  SUð2Þ ffi Zh T

p1

T

(74)

(75)

with the 24 elements  ðN,  12 Þ, ðN,  isx Þ, ðN,  isy Þ, ðN,  isz Þ,         1 1 1 1 N + ,  C3 , N + ,  isx C3 , N + ,  isy C3 , N + ,  isz C3 , 3 3 3 3        o 1 1 1 1 N  ,  C23 , N  ,  isx C23 , N  ,  isy C23 , N  ,  isz C23 , 3 3 3 3

(76)

with the same notation with Eq. (71). Here C3  (1/2)(12 + isx + isy + isz) satisfying (C3)3 ¼ −12. The conjugacy classes of Eq. (76) are composed of the following seven elements for each N (Semenoff and Zhou, 2007): ðIÞ ðIIÞ ðIIIÞ ðIVÞ ðVÞ ðVIÞ ðVIIÞ

fðN, 12 Þg, fðN,  12 Þg, fðN,  isx Þ, ðN,  isy Þ, ðN,  isz Þg, n       o 1 1 1 1 N + , C3 , N + ,  isx C3 , N + ,  isy C3 , N + ,  isz C3 , 3 3 3 3 n       o 1 1 1 1 N + ,  C3 , N + , isx C3 , N + , isy C3 , N + , isz C3 , 3 3 3 3 n       o 1 1 1 1 N  , C23 , N  , isx C23 , N  , isy C23 , N  , isz C23 , 3 3 3 3 n       o 1 1 1 1 N  ,  C23 , N  ,  isx C23 , N  ,  isy C23 , N  ,  isz C23 : 3 3 3 3

(77)

They describe (I) integer vortices (N ¼ 0 corresponds to the vacuum), (II) spin vortices of 2p rotation, (III) spin vortices of p rotation around the x, y, z axes, and (IV)–(VII) 1/3-quantum vortices.

Other examples

Another example in condensed matter physics can be found in vortex lines in the dipole-free A-phase of SF 3He (Balachandran et al., 1984; Salomaa and Volovik, 1987). An example in high energy physics can be found in color flux tubes (“non-Abelian” vortices) in high-density QCD matter (Fujimoto and Nitta, 2021a, b, c; Eto and Nitta, 2021).

Fusion rules for non-Abelian vortex anyons Two non-Abelian vortices belonging to the same conjugacy class are indistinguishable even if they are not identical. For instance, a vortex a 2 p1(G/H) and a vortex b ¼ cac−1 with c 2 p1(G/H) are not the same and thus a and antivortex of b cannot pair annihilate. Nevertheless they are indistinguishable with a help of c. This leads a class of non-Abelian statistics (Brekke et al., 1993; Lo and Preskill, 1993; Lee, 1994; Brekke et al., 1997). Such non-Abelian anyons are called non-Abelian vortex anyons. In order to discuss non-Abelian vortex anyons, first we define anyons {1, s, t} appropriately by conjugacy classes of non-Abelian vortices. Then, the fusion rule (3) for non-Abelian vortex anyons can be written as (Mawson et al., 2019) t  t ¼ N 1tt 1  N stt s  Nttt t, t  s ¼ t, s  s ¼ 1, x  1 ¼ x,

(78)

with x 2{1, s, t}.

Non-Abelian vortex anyons in D2-BN Let us discuss the simplest case. For D2-BN liquid crystals or D2-BN phases of spin-2 BECs, the conjugacy class is given in Eq. (62). We then define 1 ¼ f+12 g, Ncab

s ¼ f12 g,

t ¼ fisa g

(79)

with a being either x, y, or z. The coefficient counts how many ways anyons a and b fuse to an anyon c. The relations (−isa)(+isa) ¼ (+isa)(−isa) ¼ +12 and (−isa)(−isa) ¼ (+isa)(+isa) ¼ −12 lead N1tt ¼ 2 and Nstt ¼ 2, respectively. Furthermore, t anyons do not fuse to t in this case and then Nttt ¼ 0. We thus reach

Non-Abelian anyons and non-Abelian vortices in topological superconductors t  t ¼ 21  2s, t  s ¼ t, s  s ¼ 1,

777

(80)

x  1 ¼ x,

with x 2{1, s, t}. Since two t-anyons fuse to two different anyons 1 and s, t anyons are non-Abelian anyons. Comparing this fusion rule with that of the Ising anyons in Eq. (4), we find that these non-Abelian vortex anyons are similar to the Ising anyons. In fact, a t anynon coincide with its antiparticle, and so similar to a Majorana fermion. Gathering all ta ¼ {isa} (a ¼ x, y, z) together, we obtain the following fusion formula from the relation (isa)(isb) ¼ −dab + eabc(−isc): ta  tb ¼ 2dab 1  2dab s  2eabc tc ta  s ¼ ta , s  s ¼ 1, x  1 ¼ x,

(81)

with x 2{1, s, tx, ty, tz}. In this case, there are the three non-Abelian anyons ta coupled to each other.

Non-Abelian vortex anyons in spin-2 BECs

Non-Abelian vortex anyons in spin-2 BECs were studied in Mawson et al. (2019). For the cyclic phase in a spin-2 BEC, {1, s, t} are defined by the conjugacy classes (I), (II) and (III) with N ¼ 0 in Eq. (77). Then, we have (see Table 2) N1tt ¼ 6, Nstt ¼ 6, Nttt ¼ 4: t  t ¼ 61  6s  4t, t  s ¼ t, s  s ¼ 1,

(82)

x  1 ¼ x,

with x 2{1, s, t}. The non-Abelian 1/3-quantum vortices in (IV)–(VII) are also non-Abelian anyons but one cannot restrict to the N ¼ 0 sector, and needs infinite numbers of anyons for a closed algebra. The fusion rule for vortex anyons in the D4-BN phase of a spin-2 BEC was also obtained by the conjugacy classes (I)–(IV) in Eq. (72) (Mawson et al., 2019). In this case, t in the cyclic phase is split into t1 ¼ III0 and t2 ¼ IV0. Then, we have N1t1 t1 ¼ 4, Nst1 t1 ¼ 4, Ntt11 t1 ¼ 4, N1t2 t2 ¼ 2, Nst2 t2 ¼ 2 with the other components zero: t1  t1 ¼ 41  4s  4t1 , t2  t2 ¼ 21  2s, ti  s ¼ ti ,

s  s ¼ 1,

(83) x  1 ¼ x,

with i ¼ 1, 2 and x 2{1, s, ti}. See Table 3. The non-Abelian HQVs in (V)–(VII) are also non-Abelian anyons, but again infinite numbers of anyons are necessary for a closed algebra. The same fusion rule in Eq. (83) should hold for a D4-BN liquid crystal, as seen in Eq. (66). Including Bogoliubov modes, the algebra of vortex anyons is extended to a quantum double (Koornwinder et al., 1999; Mawson et al., 2019). An application to quantum computation was also proposed in Génetay Johansen and Simula (2022). Table 2 Fusion rule for vortex anyons in a cyclic spin-2 BEC (Mawson et al., 2019). The subscript on the conjugacy class denotes the U(1) winding number N.

1 ¼ I0 s ¼ II0 t ¼ III0 IV0 V0 VI−1 VII−1

I0

II0

III0

IV0

V0

VI−1

VII−1

I0 II0 III0 IV0 V0 VI−1 VII−1

II0 I0 III0 V0 IV0 VII−1 VI−1

III0 III0 6I0  6II0  4III0 3IV0  3V0 3IV0  3V0 3VI−1  3VII−1 3IV−1  3VII−1

IV0 V0 3IV0  3V0 3VI0 VII0 VI0  3VII0 4I0  2III0 4II0  2III0

V0 IV0 3IV0  3V0 VI0  3VII0 3VI0 VII0 4II0  2III0 4I0  2III0

VI−1 VII−1 3VI−1  3VII−1 4I0  3III0 4II0  2III0 3IV−1 V−1 IV−1  3V−1

VII−1 VI−1 3VI−1  3VII−1 4II0  2III0 4I0  2III0 IV−1  3V−1 3IV−1  2V−1

Table 3 Fusion rule for vortex anyons in a D4-BN spin-2 BEC (Mawson et al., 2019). The subscript on the conjugacy class denotes the U(1) winding number N.

1 ¼ I0 s ¼ II0 t1 ¼ III0 t2 ¼ IV0 V0 VI0 VII0

I0

II0

III0

IV0

V0

VI0

VII0

I0 II0 III0 IV0 V0 VI0 VII0

II0 I0 III0 IV0 VI0 V0 VII0

III0 III0 4I0  4II0  4IV0 2III0 2VII0 2VII0 4IV0  4V0

IV0 IV0 2III0 2I0  2II0 V0 VI0 V0 VI0 2VII0

V0 VI0 2VII0 V0 VI0 2I1 IV1 2II1 IV1 2III1

VI0 V0 2VII0 V0 VI0 2II1 IV1 2I1 IV1 2III1

VII0 VII0 4V0  4VI0 2VII0 2III1 2III1 4I1  4II1  4IV1

778

Non-Abelian anyons and non-Abelian vortices in topological superconductors

P2 Topological SFs

3

Finally we discuss the twofold non-Abelian object in a novel topological SF called a 3P2 SF: The twofold non-Abelian nature is attributed to the fermionic part (Section “Non-Abelian anyons in topological SCs”) and the vortex part (Section “Non-Abelian vortex anyons”). The 3P2 SF is a condensate of spin-triplet (S ¼ 1) p-wave (L ¼ 2) Cooper pairs with total angular momentum J ¼ 2. In the following, we briefly review 3P2 SFs in Section “Overview of 3P2 topological SFs.” In Section “Half quantum vortex in D4-BN state,” the stability of a pair of HQVs compared with a singly quantized vortex is discussed. In Section “Non-Abelian anyon in nonAbelian vortex,” we show that a Majorana fermion, a non-Abelian Ising anyon, exists in the core of each HQV.

Overview of 3P2 topological SFs Study of 3P2 SFs was initiated since 1970s (Hoffberg et al., 1970; Tamagaki, 1970; Takatsuka and Tamagaki, 1971; Takatsuka, 1972; Richardson, 1972; Sedrakian and Clark, 2018). On the basis of the analysis of the phase shifts from nucleon-nucleon scattering in a free space, spin-singlet s-wave 1S0 symmetry is a dominant attractive channel in the inner crust region (r≲0:5r0 , where r0 ¼ 0.17fm−3 is called the nuclear density) (Tamagaki, 1970). The critical temperature was estimated as 108–1010 K, which is two orders of magnitude smaller than the Fermi energy. For further increase in density 0:7r0 ≲r≲3r0 in the inner core region, the 1S0 superfluidity is suppressed, and instead the attractive channel of the 3P2 grows. Because of the attractively strong spin–orbit force, the 3P2 Cooper pair channels with high angular momentum become more attractive than other 3P pairing symmetry in contrast to the atomic physics where the lower total angular momentum state is favored as in the SF 3He-B phase. The anisotropic form of the Cooper pair is thought to affect the Cooling rate of neutron stars. Extreme conditions of neutron stars are not only high density but also rapid rotation, and a strong magnetic field. Especially, neutron stars accompanied by magnetic field ranging from 1015 to 1018 G are called magnetars. Rich structures of exotic condensates and vortices explained below may be the origin of the observed pulser glitches, that is, the sudden increase of the angular momentum of neutron stars. Among them, the existence of non-Abelian HQVs in high magnetic fields is responsible for the scaling law of the glitches without any fitting parameters (Marmorini et al., 2020). In terms of the symmetry point of view, 3P2 SFs spontaneously break the global gauge symmetry U(1)’ and the simultaneous rotational symmetry in the spin-momentum space SO(3)J which originally exist in the normal state inside neutron stars. The order parameter of 3P2 SFs is spanned by 3 by 3 traceless symmetric tensor Ami, where the 2 by 2 gap function in momentum space can be ^ given by DðkÞ ¼ Ami ki s^m is^y . For later convenience, we introduce a representation based on the total angular momentum along the quantization axis, M ¼ −2, ⋯ , 2: Ami ¼

2 X M¼ −2

gM ½GM mi ,

(84)

where gM is a complex wave function of the Cooper pair in the angular momentum sector M. A basis set GM is given for the ^ by quantization axis w ½G2 mi ¼

(85)

^ m ð^ ^i + w ð^ u  i^ vÞ m w u  i^ vÞi , 2

(86)

^m w ^i ^ um u^i  ^ vm ^ vi + 2 w pffiffiffi , 6

(87)

½G1 mi ¼  ½G0 mi ¼

ð^ u  i^ vÞm ð^ u  i^ vÞi , 2

^ constitute the orthonormal triad. where u^, v^, and w From a phenomenological aspect, the Ginzburg-Landau (GL) energy functional invariant under the SO(3)J and U(1)’ symmetry was obtained for 3P2 SFs as (Richardson, 1972; Fujita and Tsuneto, 1972; Sauls and Serene, 1978; Muzikar et al., 1980; Sauls et al., 1982) F ¼ atr½AA +b1 jtrA2 j2 + b2 ðtr½AA Þ2 + b3 tr½A2 A 2 :

(88)

It should be remarked that spin-2 spinor BECs (L ¼ 0, S ¼ 2) and d-wave superconductivity (L ¼ 2, S ¼ 0) have similar order parameter structures. The GL functional for the 3P2 Cooper pairs was minimized by Sauls and Serene by using the correspondence to the L ¼ 2 GL functional which was solved by Mermin (Sauls and Serene, 1978; Mermin, 1974). The ground-state solutions are classified into three types of phases determined by bi¼1, 2, 3, as shown in Fig. 8A: nematic phases with time reversal symmetry and the cyclic and ferromagnetic phases as nonunitary state. The microscopic derivation of the GL parameters clarifies that the ground state is the nematic phases in the weak coupling limit (Sauls and Serene, 1978; Sauls, 1980). The order parameter tensor of ^m w ^ i with a real number r 2 [−1, −1/2] (see also Eq. 68). The nematic phase um u^i + r^ vm v^i −ð1 + rÞw a nematic phase is given by Ami ¼ D½^ has a continuous degeneracy which is lifted by either a magnetic field or sixth-order terms in the GL free energy. For the GL parameters derived from the microscopic model, the UN phase for r ¼ −1/2 is favored at zero magnetic field while D2-BN (− 1 < r < −1/2) and D4-BN r ¼ −1 phases are favored for moderate and strong magnetic fields, respectively (Masuda and Nitta, 2016),

Non-Abelian anyons and non-Abelian vortices in topological superconductors

(b)

(a)

-0.5 2

U

Ferro.: Weyl

/B

1

-0.6

-0.7

-0.8

topo.

topo. UN

-0.9

-1.0

(c)

(d)

0.12 0.10

BN

topo.

BN

0.08 0.06

0

1

weak coupling limit

2

0.04

Cyclic: Weyl

0.02 0

779

0

0.2

0.4

0.6

0.8

1.0

0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0 1.0

0

1.0

Fig. 8 (A) Phase diagram without magnetic field based on the GL theory. (B) Gap structure of the nematic state. (C) Phase diagram with magnetic field in the weak coupling limit. (D) The momentum resolved local density state at the surface perpendicular to the z^-axis. The magnetic field is applied parallel (perpendicular) to the surface [top (bottom) panel]. Adapted from Mizushima T, Masuda K, and Nitta M (2017) 3P2 Superfluids are topological. Physical Review B 95 (14): 140503. https://doi.org/10.1103/PhysRevB.95.140503. ©2022 American Physical Society.

relevant for magnetars. The schematic images of the gap structures for the UN and D4-BN states are shown in Fig. 8B, where the P arrows account for the momentum-dependent direction of the d-vector, defined by dm(k) ¼ iAmiki. We will discuss other possible states later in the context of the fermionic topology. The microscopic model of 3P2 SFs in terms of the fermion degrees of freedom was constructed by Richardson (1972), Tamagaki (1970), Takatsuka and Tamagaki (1971), and Takatsuka (1972): The starting microscopic Hamiltonian is composed of the onebody term H1 and the interaction term H2 given, respectively, in the following forms: Z !{ ! (89) H1 ¼ drc ðrÞ½h^N ðirÞ cðrÞ, Z H2 ¼ 

dr

X g { T ðrÞT ab ðrÞ: 2 ab ab¼x, y, z

(90)

!

In the first line, ½cðrÞ s ¼ cs ðrÞ and ½h^N ð −irÞ s,s0 is the sum of the kinetic energy h0 ð −irÞds,s0 ¼ ð −r2 =2m −mÞds,s0 measured from ^ for a magnetic field B ¼ Bn. The 3P2 force with coupling strength g > 0 is the chemical potential m, and the Zeeman energy −B s P  s0 ðrÞ cs ðrÞ , where r   k −1 r . represented by H2 with the pair-annihilation operator T defined by T ab ðrÞ ¼ ss0 ½t ab,ss0 ð −irÞc F ^ We have introduced a spin–momentum ffiffiffi force via the 2  2 matrix in spin space t ab defined by pffiffifficoupling in theppair   ¼ is^y s^a ð −ir  b Þ+s^b ð −ir  a Þ =2 2 −dab s ð  ^t ab ð −irÞ ^ −irÞ=3 2 . Within the mean field approximation, the BdG equation is derived as ! !  Hðir, RÞu n ðRÞ ¼ en u n ðRÞ,

 Hðir, RÞ ¼

h^N ðirÞ

^ ½Dðir, RÞ

^ Dðir, RÞ

½h^N ðirÞ

(91)

! ,

(92)

^ −ir, RÞ ¼ P is^a s^y f2Aab ðRÞð −irb Þ + ½ð −irb ÞAab ðRÞ g. This model was also used to determine where the gap matrix is given by Dð a,b 2kF the aforementioned GL parameters in the weak coupling limit and was directly investigated within a quasiclassical approximation recently in Mizushima et al. (2017). Even in the presence of the magnetic field, the quasiclassical approximation, where the size of the Fermi surface is treated as an infinite one, predicts that a nematic state is the most stable with parameter r determined by the strength of the magnetic field as in the GL theory. The T-B phase diagram is shown in Fig. 8C. As explained later, the ferromagnetic state appears near Tc when the finite-size correction of the Fermi surface is taken into account. The microscopic analysis reveals the existence of the tricritical point on the phase boundary between the D4-BN and D2-BN states in the T-B phase diagram shown in Fig. 8C: at temperatures below the tricritical point, the phase transition becomes discontinuous (Mizushima et al., 2017). The existence of the tricritical point was later confirmed in Mizushima et al. (2020) also in the GL theory with higher order terms (up to the eighth order (Yasui et al., 2019a)). In addition to such drastic change in critical phenomena, the advantage of the microscopic model lies in studies of the fermionic topology of the superfluidity. In the following, we explain the fermionic topology of the possible phases of the 3P2 SFs.

Nematic state In the weak coupling limit without a magnetic field, the UN state is the ground state. The order parameter tensors of the UN state and the BN state under a magnetic field are already explained above. On the basis of the symmetry of the BdG Hamiltonian, the nematic state is revealed as a topological SF with time reversal symmetry (a class DIII in the classification of topological insulators and SCs). The analysis of the BdG equation in the presence of the boundary clarifies the existence of the gapless surface bound Majorana fermion, the hallmark character of the topological states. The surface Majorana fermion has an Ising spin character

780

Non-Abelian anyons and non-Abelian vortices in topological superconductors

(Chung and Zhang, 2009; Nagato et al., 2009; Volovik, 2010; Mizushima and Machida, 2011), that is, the only external field coupled to the Ising spin gives a mass gap to the Majorana fermion as shown in Fig. 8D. The magnetic field direction giving the gap to the Majorana fermion is the direction perpendicular to the surface, denoted by n?. This is because the surface Majorana fermion is protected not only by the time reversal symmetry but also another key symmetry, which is the magnetic p-rotation symmetry about n?. The magnetic p-rotation symmetry is also called the P3 symmetry.5 The magnetic p-rotation is the combined operation of the time-reversal and the p rotation. The chiral operator, defined by the combination of particle-hole operation C and the magnetic protation P3 as G ¼ CP3, commutes with the Hamiltonian Hðk? Þ with magnetic field B n? ¼ 0 and the chiral symmetric momentum k? ¼ k?n? Rsuch that G : k?!k?. Using this commutation relation, a one-dimensional winding number can be introduced as 1 w1d ¼ − 4pi dk? Tr½GHðk? Þ∂k? Hðk? Þ ¼ 2, which is unchanged unless the symmetry is broken or the bulk gap is closed. The P3 symmetry and thus the chiral symmetry is broken by the magnetic filed B n? 6¼ 0 which gives a mass gap to the Majorana fermion (the bottom panel of Fig. 8D, where n? k z^).

Cyclic state

For − 6b1 < b3 < 0 based on the basis of SO(3)J invariant GL functional, the cyclic state is the ground state. The order parameter ^m w ^ i with o ¼ e2pi/3 [see also Eq. (73)], which is unique except for trivial SO(3)J and tensor has a form of Ami ¼ D½^ um u^i + o^ vm v^i + o2 w U(1)’ rotations. In the absence of magnetic field, the quasiparticle excitation energy consists of two branches E(k) ¼ [h0(k)2 + |d(k)|2  |d(k)d (k)|]1/2, where E+(k) is full gap and E−(k) has eight nodal points ka¼1,⋯ ,4 , each of which is a Weyl point. The subscript a denotes the four vertices of a tetrahedron, and  represents the monopole charge of the Weyl point. Because of the topological nature of Weyl SCs/SFs, there exist surface zero energy states which connect two oppositely charged Weyls points on the projected momentum space normal to the surface direction.

Ferromagnetic state

For |b1|− b1 < b3, the ferromagnetic state described by Ami ¼ Dð^ um + i^ vm Þð^ ui + i^ vi Þ minimizes the GL functional. This nonunitary state is also unique except for trivial SO(3)J and U(1)’ rotations. The order parameter is equivalent to the A1 state of the SF 3He. The bulk quasiparticle spectrum consists of two parts: one is a normal fluid and the other is a Weyl SF with a pair of oppositely charged Weyl fermions. The ferromagnetic state is also realized in the limit of a strong magnetic field or the vicinity of the critical temperature even in the weak coupling limit owing to the finite-size correction of the Fermi surface.

Magnetized nematic state Recently, even in the weak coupling limit, nonunitary states are predicted in a magnetic field (Mizushima et al., 2021). As mentioned above, quasiclassical approximation, which treats the size of the Fermi surface as infinity, allows only the nematic states, which is unitary. By contrast, in the presence of the finite-size correction of the Fermi surface, a magnetic field along the ^ -direction induces the nonunitary component into the nematic state with r 2 (−1, −1/2) as Ami ¼ D½^ um u^i + r^ v m v^i + w ^mw ^ i or equivalently ikð^ um v^i + ^v m u^i Þ − ð1 + rÞw 1 0 1 ik 0 C B (93) Ami ¼ D@ ik r 0 A: 0 0 −ð1 + rÞ For simplicity, Mizushima et al. (2021) studied the case of r ¼ −1, that is, in a strong magnetic field. In the quasiclassical limit, k ¼ 0 and the D4-BN state is realized. The order parameter tensor in the angular momentum representation is reduced to Ami ¼ D[G2 + G−2]mi, which shows the same gap amplitudes for M ¼ 2. The finite size correction of the Fermi surface induces a finite k. In the angular momentum representation, the order parameter tensor becomes Ami ¼ D[(1 + k)G2 + (1 − k)G−2]mi. Thus, k represents the imbalance between the M ¼ 2 sectors. In terms of the gap matrix in the spin basis, the Cooper pairs are decomposed into two spin polarized sectors |""i and |##i with different gap amplitudes. Each spin sector has a polarized orbital angular momentum state, and thus, a pair of the Weyl fermions appears at the north pole and the south pole in each spin sector. The orbital angular momentum between these two spin states is opposite, which means that two Weyl fermions with opposite helicity at each pole. This is in contrast to the A2 phase proposed for the SF 3He. In the limit of a strong magnetic field or the vicinity of the critical temperature, k ! 1, the ferromagnetic state is realized. However, we note that this imbalance originates from the finite size effect of the Fermi surface. The effect is parametrized by D/eF, which is usually small for conventional neutron stars, and thus the imbalance k is also expected to be small.

Vortices in 3P2 SFs

The vortices admitted in 3P2 SFs are basically the same as those in spin-2 BECs, which are discussed in Section “Examples of nonAbelian vortices.” Table 4 summarizes remaining symmetry, order parameter manifold, and fundamental group of the topological defects of the above-discussed possible phases. Non-Abelian vortex anyons are included in the D2- and D4-BN states and the cyclic state as discussed in Section “Fusion rules for non-Abelian vortex anyons.” Non-Abelian vortex anyons in the D2-BN states are spin vortices of p rotation, as shown in Eq. (81). The D4-BN states admit HQVs (V)–(VII) in Eq. (72) in addition to spin vortices (III) and 5

Here we use the bar to distinguish the magnetic p-rotation about the x-axis introduced later.

Non-Abelian anyons and non-Abelian vortices in topological superconductors

781

Table 4 Summary of order parameters (r, k in Eq. (93)), remaining symmetry (H ), (R ’ G/H ), and topological vortices p1(R ) in possible phases. Phase

r & k in Eq. (93)

H

R ’ G/H

p1(R)

UN

r ¼ −1/2 & k ¼ 0

D1 ’ O(2)

Z  Z2

D2-BN D4-BN Cyclic Mag. D2-BN Mag. D4-BN FM

r 2 (−1, −1/2) & k ¼ 0 r ¼ −1 & k ¼ 0 r ¼ ei2p/3 & k ¼ 0 r 2 (−1, −1/2) & k 2 (0, 1) r ¼ −1 & k ¼ −1 r ¼ −1 & k ¼ 1

D2 D4 T 0 C4 Uð1ÞJz +2F

Uð1Þ  RP2 [U(1)  SO(3)]/D4 [U(1)  SO(3)]/D4 [U(1)  SO(3)]/T U(1)  SO(3) ½Uð1Þ  SOð3Þ =Z4 SOð3ÞJz −2F =Z2

ZQ Zh D 4 Zh T

Z  Z2 Zh C 4 Z4

Source: Taken from Mizushima T, Yasui S, Inotani D, and Nitta M (2021) Spin-polarized phases of 3P2 superfluids in neutron stars. Physical Review C 104 (4): 045803. ISSN 2469-9985. https://doi.org/10.1103/PhysRevC.104.045803. See also the references therein.

(IV) in Eq. (72) as non-Abelian vortex anyons (see also Table 3). The cyclic states admit 1/3-quantum vortices (IV)–(VII) in Eq. (77) as well as spin vortices (III) in Eq. (77) as non-Abelian vortex anyons (see also Table 2). Especially, the non-Abelian vortex in the D4-BN state is the main target in the remaining part. We explain the vortices in the D4-BN state in the beginning of the next subsection.

Half quantum vortex in D4-BN state Vortices in D4-BN state The D4-BN state, which can be thermodynamically stabilized by a large magnetic field, has a pair of point nodes at the north and south poles along the direction of the magnetic field (Masuda and Nitta, 2016; Mizushima et al., 2017; Yasui et al., 2019b; ^ ¼ z^. The homogeneous order Mizushima et al., 2020). Hereafter, we focus on the D4-BN state with a pair of point node along w parameter tensor has a diagonal form (Sauls and Serene, 1978): A ¼ Ddiag(1, −1, 0) (Eq. 68), which is invariant under a D4 group. (A similar order parameter form is taken in the planar state of SF 3He (Makhlin et al., 2014; Silaev and Volovik, 2014), but different topological defects are resulted from different order parameter manifolds.) As already discussed in Section “Non-Abelian vortex anyons,” the order parameter manifold in the D4-BN state is then characterized by the broken symmetry, R ’ [U(1)  SO(3)]/D4, see Eq. (69). The topological charges of line defects are characterized by the first homotopy group, Eq. (75) p1 ðRÞ ’ Zh D 4

(94)

where h denotes a product defined in Kobayashi et al. (2012). This ensures two different classes of topological line defects: Vortices with commutative topological charges and vortices with noncommutative topological charges. The former includes integer vortices (I) in Eq. (72) with or without internal structures, while an example of the latter is the non-Abelian HQV, (V)–(VII) in Eq. (72). Integer vortices are characterized by vorticity quantized to an integer number because of the singlevaluedness of the order parameter. Fractionally quantized case such as a HQV usually has a phase jump (p-phase jump for a HQV) along the contour surrounding the vortex. In the D4-BN state, however, the phase discontinuity is compensated by the phase originating from the discrete rotation of the D4 symmetry. Thereby HQVs are topologically allowed. Here, we particularly focus on the case where the vorticity is along the z^-direction accompanying the p-phase jump compensated by the C4 rotation about the z^-axis, as indicated in Figs. 8B, 10A and F, and 11A. Such HQVs belong to the group (V) or (VI) in Eq. (72). Hereafter, we review possible vortices when the vorticity and the point nodes of the D4-BN state are along the z^-direction. As a basis set of the order parameter tensor Ami, the eigenstates of the z component of the total angular momentum, denoted by JzGM ¼ P MGM, are convenient: Ami ðr, yÞ ¼ 2M¼ −2 gM ðr, yÞ½GM mi , where the cylindrical coordinate system R ¼ (r, y) is introduced with assumption of uniformity along the z-direction. In the region far from the vortex core r ! 1, there is no radial dependence, and the order parameter modulation is described by the U(1) phase rotation characterized by vorticity k and the simultaneous rotation in the spin–orbit space of Cooper pairs. For the latter rotation by angle ’ about the z^-axis, the other components of the triad, x^, and y^, are transformed as Rð’Þ : x^ ! cos ’^ x + sin ’^ y,

(95)

Rð’Þ : ^ y !  sin ’^ x + cos ’^ y:

(96)

Using GM, the U(1) phase winding and the spin–orbit rotation for nonzero bulk components (i.e., r ! 1) are expressed as Ami ðyÞ ¼

2 X M¼2

gM,1 eikyM’ ½GM mi 

2 X M¼2

gM,1 eiðknMÞy ½GM mi :

(97)

782

Non-Abelian anyons and non-Abelian vortices in topological superconductors Table 5 Internal structures of axisymmetric vortices. The second and third columns show order parameters of the core structure. The fourth column is coreless (singular) whether the vortex core is occupied (unoccupied) by some SF components. The last column accounts for the preserved symmetry. Vortex

gM¼0(r)

gM¼1(r)

Core

Symmetry

o vortex u vortex v vortex w vortex uvw vortex

Real Complex Real Real Complex

Zero Zero Real Imaginary Complex

Singular Singular Coreless Coreless Coreless

P1, P2, P3 P1 P2 P3 –

Especially, in the D4-BN state with the point nodes along the z^-axis, |g2, 1| ¼ |g−2, 1| and g1, 1 ¼ g0.1 ¼ 0. Here in the second line, we parametrize ’ ¼ ny along the contour surrounding the vortex, where the spin–orbit rotation can be regarded as the n-fold J-disgyration.6 Note that n is not necessarily an integer. The boundary condition of a vortex characterized by (k, n) is given by Eq. (97).

Axisymmetric singly quantized vortex

The integer vortices belong to a class k 2 Z. Even in the singly quantized case k ¼ 1, however, there are many possibilities because of differences in the internal structure and the disgyration in the spin–orbit space. As a simple case, the axisymmetric vortex can be P described as Ami ðr, yÞ ¼ 2M¼ −2 gM ðrÞeiðk −MÞy ½GM mi in a whole region, which includes complex radial functions gM(r) to be determined. The disgyration of the axisymmetric vortex is one-fold (n ¼ 1 in Eq. 97), that is, the 2p rotation of the triad around the vortex. Thus, the axisymmetric vortex belongs to the conjugacy class (II) in Eq. (72).7 Although the loss in the kinetic energy becomes large because of the J-disgyration, there is a nice analogy with vortices in SF 3He-B phase. As in the SF 3He-B phase, the symmetry of the vortices can be characterized by three discrete symmetries called P1,2,3 symmetries (Salomaa and Volovik, 1985b, 1987), which are discussed in the next subsection.8 The internal structures g−1,0,1(r) characterize a classification based on these discrete symmetries, which allows several vortices as summarized in Table 5. Within an axisymmetric condition, only o vortex or v vortex is realized as a self-consistent solution. The o vortex has a singular core and is the most symmetric one so that it meets all the discrete symmetries. By contrast, the core of the axisymmetric v vortex is occupied by the SF component g1 for k ¼ 1, and it has the P2 (magnetic mirror reflection) symmetry, but does not have the P1 (inversion) or P3 (magnetic p-rotation about the x-axis) symmetries. Energetically, the v vortex is more stable than the o vortex because of a gain in the condensation energy in the core region. However, the induced component of the v vortex which occupies the core is suppressed by a magnetic field .

Nonaxisymmetric singly quantized vortex

Without the axisymmetric condition, the lower energy condition for a singly quantized vortex is given by (k, n) ¼ (1, 0) to reduce the loss of the gradient energy. Such a vortex belongs to the conjugacy class (I) in Eq. (72). In this case, so-called double-core vortex (d vortex) can be a self-consistent solution (Masaki et al., 2022) as in the SF 3He-B phase (Thuneberg, 1986; Salomaa and Volovik, 1986). The d vortex is also called the nonaxisymmetric v vortex because it has the P2 symmetry but does not have the P1 and P3 symmetries (Tsutsumi et al., 2015). Note that the axial symmetry is spontaneously broken for the d vortex in the SF 3He-B phase, while the symmetry is broken by the boundary condition for the d vortex in a 3P2 SF. Fig. 9 shows the order parameter structure of the d vortex in the D4-BN state for T ¼ 0.4Tc and B ¼ 0.5Tc with critical temperature Tc. In the panels, the unit of length is the coherence length defined by x0 ¼ vF/2pTc with Fermi velocity vF, where h⋯iF accounts for the Fermi surface average. The bulk components g2 have a conventional vortex structure with a single phase winding [see panels (A) and (B)]. The double core h i1=2 ^ { Di ^ in panel (F). Such a structure is due to the structure can be clearly observed in the total amplitude defined by 32 TrhD F occupations of the core by gM¼1 as shown in panels (C) and (D). As can be seen below, the d vortex is not the lowest energy state among the vortex states with (k, n) ¼ (1, 0) in the D4-BN state and it splits into two HQVs. The d vortex in the SF 3He-B phase is the most stable vortex solution in the low pressure region (Kasamatsu et al., 2019; Regan et al., 2019). A d vortex is realized as the lowest energy state in the UN and the D2-BN states.

HQVs

The HQVs can be characterized by k ¼ 1/2. The possibility of the half-quantized vortex in the D4-BN state was pointed out long back in 1980 (Sauls, 1980) on the basis of the topological consideration of the form of Ami. Since k is a half odd integer, the U(1) phase part gives the minus sign under the spatial rotation y ! y + 2p. By rotating the triad by angle  p/2 while going around the vortex, which is denoted by n ¼ 1/4 in Eq. (97), the above minus sign can be compensated by the other minus sign stemming from the disgyration. This topological consideration is represented by the boundary condition Eq. (97) with (k, n) ¼ (1/2, 1/4), which 6 The term “J-disgyration” accounts for a singularity in the spin–orbit space, that is, the total angular momentum (J) space. Spin disgyrations and orbital disgyrations are introduced after Eq. (32). 7

In Section “Examples of non-Abelian vortices,” we discuss the spin vortex of 2p rotation in spin-2 BECs.

8

The P3 symmetry is already introduced when defining the one-dimensional winding number in the bulk nematic state.

Non-Abelian anyons and non-Abelian vortices in topological superconductors

783

Fig. 9 Gap structure of the d vortex in the D4-BN state (T ¼ 0.4Tc and B ¼ 0.5Tc). (A)–(E) Amplitude of each angular momentum sector. Red arrows indicate the h i1=2 ^ ^ { Di . phase winding structures. (F) Total amplitude of the order parameter defined by 32 TrhD F

Fig. 10 HQV with (k, n) ¼ (1/2, +1/4) [(A)–(E)] and (k, n) ¼ (1/2, −1/4) [(F)–(J)]. The boundary conditions are sketched in (A) and (F). The amplitudes of gM for M ¼ −2 and 0 (+ 2 and 0) are shown in (B) and (D) [(G) and (I)], respectively. The phases of gM for M ¼ −2 and 0 (+ 2 and 0) are shown in (C) and (E) [(H) and (J)], respectively. Adapted from Masaki Y, Mizushima T, and Nitta M (2022) Non-Abelian half-quantum vortices in 3P2 topological superfluids. Physical Review B 105 (22): L220503. ISSN 2469-9950. https://doi.org/10.1103/PhysRevB.105.L220503. ©2022 American Physical Society.

belongs to the conjugacy class (V) or (VI) in Eq. (72). The gradient energy of an isolated HQV far from the core is a half of that of an singly quantized vortex described by (k, n) ¼ (1, 0), namely, the energy of two HQVs is the same as that of the singly quantized vortex except for contributions from the vortex cores. In other words, whether the singly quantized vortex can be split into two HQVs depends on the energy of their internal structures and the interaction energy between two HQVs.

784

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Fig. 11 (A) Schematic image of a molecule of non-Abelian HQVs in a D4-BN state at a cross section perpendicular to the two parallel vortex lines, characterized by (k, n) ¼ (1/2, +1/4) at x ¼ dv/2 and (1/2, −1/4) at x ¼ −dv/2. Its spin-momentum structure is shown by objects with color arrows representing d vectors. The color map on the surface shows the U(1) phase, and the bottom plot shows the induced component also shown in panel (C-1). (B) Interaction energy of two HQVs as a function of their separation dv. The inset shows the total amplitude of the order parameter for the HQV molecule whose intervortex distance is indicated by the arrow. The free energy is scaled as Jsn ¼ Jsn =ðnn T 2c x20 Oz Þ, where Oz is the length of the system in the z direction, and nn is the density of states at the Fermi energy in the normal state. The energy at dv ¼ 0 is the free energy of the d vortex shown in Fig. 9. (C-1) and (C-2) The amplitude and the phase of the induced component g0. The intervortex distance dv is the same as that in the inset of panel (B). Adapted from Masaki Y, Mizushima T, and Nitta M (2022) Non-Abelian half-quantum vortices in 3P2 topological superfluids. Physical Review B 105 (22): L220503. ISSN 2469-9950. https://doi.org/10.1103/PhysRevB.105. L220503. ©2022 American Physical Society.

Although the above topological consideration does not take into account the core structure, recently, the whole structures of isolated HQVs were studied using the phenomenological GL theory (Masuda and Nitta, 2020; Kobayashi and Nitta, 2022a, b) and the microscopic quasiclassical theory (Masaki et al., 2022). It is proposed that an integer vortex with k ¼ 1 can split into two HQVs with (k, n) ¼ (1/2, 1/4) and (k, n) ¼ (1/2, −1/4) (Masuda and Nitta, 2020). Here we assume that k > 0 without loss of generality. The configuration of the two HQVs given as one for n1 ¼ +1/4 with its core at R1 and the other for n2 ¼ −1/4 with its core at R2 has the following boundary condition at R: X eiðk−n1 MÞy1 +iðk −n2 MÞy2 gM GM , (98) AðRÞ M¼2

where y1 (y2) is the angle of R −R1 (R −R2). Since the difference between y1 and y2 is negligible for |R|!1, the phase part behaves as expð2ikyÞ [y ’ (y1 + y2)/2], which means the boundary conditions for the two HQVs and the singly quantized vortex are equivalent in the limit r ! 1. In the following, we explain the internal structure of each HQV, and the interaction energy of the two HQV vortex through the comparison of the singly quantized vortex (Masaki et al., 2022). Particularly, the nonaxisymmetric internal structure of each HQV is important to the interaction energy.

Isolated HQV (k,n) ¼ (1/2, +1/4)

The two disgyrations of the triads, given by n ¼ 1/4, are inequivalent because one is parallel to the vorticity while the other is antiparallel. Here we first explain the antiparallel case (n, k) ¼ (1/2, 1/4). In the above notation, by setting R1 ¼ (0, 0) and R2 ¼ (−1, 0), and thus setting y2 ! 0, the boundary condition is obtained from Eq. (98). In this case, the relative phase of g2 is zero, and the schematic image of the disgyration is drawn in Fig. 10A. Each circular object shows the directions of d-vectors by colored arrows, whose color bar is the same as in panel (C). There are also the cyan and magenta lines, which represent the directions of the eigenvalues +1 and −1 of the order parameter A, respectively. This HQV is (1/2, C4) in the conjugacy class (V) in Eq. (72). The self-consistent solutions of the nontrivial components are shown in panels (B)–(E). Here g2, not shown, is almost uniform without phase winding, as expected from the boundary condition k − n1M ¼ 1/2 − 1/4 2 ¼ 0. The other bulk component g−2 is a conventional singular vortex with the single-phase winding shown in panel (C). The amplitude and the phase of the internal structure of g0 are, respectively, shown in panels (D) and (E). The phase of the internal structure is oppositely winding against g−2, which can be understood by analogy with a vortex in a spinless chiral p-wave SC. We assume, for simplicity, kz ¼ 0, and then write kw ¼ ðkx + iwky Þ=kF ¼ eiwa with the sign of the angular momentum w ¼ . With this notation, the gap function is written as pffiffiffi pffiffiffi ^ F , RÞ ¼ ½ −g ðRÞk+ + g ðRÞ= 6k− j""⟩ + ½g ðRÞk− − g ðRÞ= 6 k+ j##⟩: (99) Dðk 2 0 −2 0

Non-Abelian anyons and non-Abelian vortices in topological superconductors

785

In each spin sector, two opposite orbital-angular-momenta, k , are mixed owing to the induced component g0. In a spinless chiral p-wave SC, the induced component has the opposite angular momentum relative to the component which has a nonzero order parameter far from the vortex core, that is, wi ¼ −wb (Heeb and Agterberg, 1999; Matsumoto and Heeb, 2001). Here we call the latter component the bulk component, and wb (wi) represents the sign of the angular momentum for the bulk (induced) component. The vorticity of the induced component, ki, is given by ki ¼ kb + wb − wi, where kb is the vorticity of the bulk component. Here the wb − wi accounts for the angular momentum difference between the bulk and induced components, and is reduced to 2wb. For the (1/2, +1/4)-HQV, the bulk component g−2 has the vorticity kb ¼ k − n(−2) ¼ 1, and wb ¼ −1. Thereby ki ¼ kb + 2wb ¼ −1 successfully explains the vorticity of the induced component shown in panel (E). More striking feature is the three-fold symmetry in the amplitude of the induced component |g0|. Reflecting this discrete symmetry, the phase evolution of the induced component surrounding the vortex is nonlinear. As seen from Eq. (99), the main structure of this HQV is in the spin-down sector.

Isolated HQV (k,n) ¼ (1/2, −1/4)

The other HQV, characterized by (1/2, −1/4), at R2 ¼ (0, 0), is obtained by R1 ! (+1, 0), that is, y ! p in Eq. (98), in which the relative phase of g2 is opposite because of the phase factor exp½ið1=2 + M=4Þp from the (1/2, +1/4) HQV. This HQV is ð1=2, C4−1 Þ in the conjugacy class (V) in Eq. (72), and its disgyration is schematically drawn in Fig. 10F. In contrast to the previous case, a conventional vortex structure appears in the M ¼ +2 sector shown in panels (G) and (H), while an almost uniform structure without phase winding is in the M ¼ −2 sector. In panel (J) the phase winding of the induced component is +3, which can be understood as in the previous case: ki ¼ kb + 2wb ¼ 3 for kb ¼ k + M/4 ¼ 1 and wb ¼ +1. The phase evolution surrounding the vortex is again nonlinear reflecting the discrete symmetry of the amplitude |g0|, which is fivefold as shown in panel (H). The major vortex structure of this HQV is in the spin-up sector.

Molecule of HQVs (stability)

The two types of internal structures in the M ¼ 0 component induced for the HQVs with n ¼ 1/4 are modulated by the connection of these two HQVs. This modulation causes an interaction between the two HQVs. The interaction energy of the two HQVs + ðR1 Þ −J sn− ðR2 Þ based on the Luttinger–Ward energy functional J sn (Vorontsov and Sauls, is defined by DJ sn ðdv Þ ¼ J sn ðR1 , R2 Þ −J sn 2003). Here the two HQVs are at R1 ¼ (dv/2, 0) and R2 ¼ (−dv/2, 0), schematically shown in Fig. 11A, whose energy is denoted by + J sn ðR1 , R2 Þ. The energies of the isolated HQVs (k, n) ¼ (1/2, +1/4) at R1 and (1/2, −1/4) at R2 are denoted by J sn ðR1 Þ and J sn− ðR2 Þ, respectively. Fig. 11B shows the interaction energies of the two HQVs as a function of intervortex distance dv. When dv ¼ 0, the d vortex (the singly quantized vortex with double core structure) is realized. The gap structure is shown in Fig. 9. Importantly, the positive energy of the d vortex means that two isolated HQVs (dv ! 1) are more stable than the d vortex. The actual interaction energy takes the minimum at finite dv, indicating the instability of the d vortex into a bound state of the two HQVs like a molecule. As discussed in Section “Examples of non-Abelian vortices,” a singly quantized vortex (k, n) ¼ (1, 0) in the conjugacy class (I) in Eq. (72) splits into two HQVs (1/2, +1/4) and (1/2, −1/4) in the class (V). The stabilization mechanism in this molecule state is a deformation in the nonaxisymmetric-induced component g0, as shown in Fig. 11C-1 and C-2, realized in the strongly spin–orbit-coupled Cooper pairs (Masaki et al., 2022). This mechanism is purely intrinsic and possible even in the weak coupling limit, different from those in other systems such as the SF 3He (Salomaa and Volovik, 1985a) and unconventional SCs (Chung et al., 2007): In the SF 3He-A phase, Volovik and Salomaa phenomenologically introduced some corrections in spin mass to stabilize the HQV (Salomaa and Volovik, 1985a); This correction might be regarded as a kind of strong coupling effect through the Fermi liquid correction, but another strong coupling effect is known to destabilize the HQV (Kawakami et al., 2009, 2010, 2011; Mizushima et al., 2016). Vakaryuk and Leggett unveiled that the HQVs in equal spin pairing states, such as 3He-A and D4-BN, are accompanied by a nonzero spin polarization even in the absence of external Zeeman coupling (Vakaryuk and Leggett, 2009). The coupling of such spin polarization to external magnetic fields may affect the stability of HQVs. Another example of the stabilization mechanism is an extrinsic one in the polar and polar-distorted phases of the SF 3He because of strongly anisotropic impurities, as discussed in Section “Platforms for Majorana zero modes.”

Non-Abelian anyon in non-Abelian vortex In this section, first the zero energy-bound states in above-discussed HQVs are demonstrated. Second we summarize the discrete symmetries in the presence of vortices, and finally discuss the topological protection of the zero energy states and their non-Abelian nature.

Existence of zero energy states To investigate fermionic bound states at discrete energy levels, it is necessary to solve the BdG Eq. (92). By assuming the spatial uniformity along the z-direction, the quantum number is labeled as n ¼ (a, kz), and only the kz ¼ 0 sector is focused on to seek for zero energy bound states. The direct numerical calculation gives us a spectral function including the discrete bound states around the HQV-cores, as shown in Fig. 12A. There are two zero energy states localized in both of the cores. Here, the molecule state of the

786

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Fig. 12 Local density of states nkz ¼0 ðR; oÞ at kz ¼ 0 and y ¼ 0 for a pair of HQVs located at (x, y) ¼ (dv/2, 0) with dv ’ 10.7x0. The spatial distributions of the two zero energy states are different. Adapted from Masaki Y, Mizushima T, and Nitta M (2022) Non-Abelian half-quantum vortices in 3P2 topological superfluids. Physical Review B 105 (22): L220503. ISSN 2469-9950. https://doi.org/10.1103/PhysRevB.105.L220503. ©2022 American Physical Society.

two HQVs with dv ’ 10.7x0 is used for a gap function in the BdG equation, and the spectral function, also known as the local density P of state, is defined as nkz ¼0 ðR; oÞ ¼ a,s jua,kz ¼0,s ðRÞj2 dðo −ea,kz ¼0 Þ along R ¼ (x, 0), where ua,kz ,s ðRÞ is the particle component with !

spin s of the eigenfunction u a,kz ðRÞ. In the calculation, the parameter kFx0, characterizing the discreteness of the bound states, is set to 5. The Majorana condition, ua,kz ¼0,s ðRÞ ¼ ½va,kz ¼0,s ðRÞ , for the wave functions of the zero energy state can be confirmed (Masaki et al., 2022).

Discrete symmetry of vortices We utilize the semiclassical approximation to clarify the symmetry and topology of vortices, following Tsutsumi et al. (2015), Mizushima et al. (2016), and Shiozaki and Sato (2014). In the semiclassical approximation, the spatial modulation due to a vortex line is treated as adiabatic changes as a function of the real-space coordinate surrounding the defect with the angle y. Then, the BdG Hamiltonian in the base space, (k, y), is obtained from Eq. (92) as ! ^ yÞ Dðk, h^N ðkÞ  ðk, yÞ ¼ H , (100) ^ { ðk, yÞ −h^ ðkÞ D N where the 3P2 pair potential within a semiclassical approximation is given by ^ yÞ ¼ is^m Ami ðyÞki =kF s^y : Dðk,

(101)

Let us now summarize the discrete symmetries of the BdG Hamiltonian in Eq. (100). In the presence of vortex, the BdG Hamiltonian breaks the time-reversal symmetry T^ ¼ −is^y K, but holds the particle-hole symmetry   yÞC−1 ¼ −Hð−k, yÞ, CHðk,

(102)

where C ¼ tx K and K is the complex conjugation operator. In addition to the particle-hole symmetry, three discrete symmetries, the {P1, P2, P3}, are pointed out to be relevant to vortices of the SF 3He-B phase (Salomaa and Volovik, 1985b, 1987). Their representations are given by ^2,x , ^ P^3 ¼ T^ C P^1 ¼ PU p 1,

(103)

−i^J x p

^2:x ¼ e and P^2 ¼ P^1 P^3 . Here, P, Up ¼ e ,C are the spatial inversion, the discrete phase rotation, and the p-rotation in the spin–orbit space about the x-axis. The physical meanings of P1, P2, and P3 symmetries are the inversion, magnetic reflection, and magnetic p-rotation symmetries, respectively. From the definition, P1P2P3 ¼ 1. Under these symmetry operations, a momentum k and a spin s are transformed as i(k+1)p

P 1 : k ! ðkx ,  ky ,  kz Þ, s ! ðsx , sy , sz Þ,

(104)

P 2 : k ! ðkx ,  ky ,  kz Þ, s ! ðsx , sy , sz Þ,

(105)

P 3 : k ! ðkx , ky , kz Þ, s ! ðsx , sy , sz Þ:

(106)

Under these transformation, the order parameters are transformed as P 1 : gM ðr, yÞ ! gM ðr, y + pÞ,

(107)

787

Non-Abelian anyons and non-Abelian vortices in topological superconductors P 2 : gM ðr, yÞ ! ð1Þk+M ½gM ðr, p  yÞ , M



P 3 : gM ðr, yÞ ! ð1Þ ½gM ðr,  yÞ :

(108) (109)

For axisymmetric vortices, we confirm the nonzero components of gM(r) summarized in Table 5 by using the relation gM(r, y) ¼ gM(r)ei(k−M)y. For example, P 2 : gM ðrÞeiðk −MÞy ! g M ðrÞeiðk −MÞy , and when the P2 symmetry is preserved, gM ðrÞ ¼ g M ðrÞ can be obtained. In the case of the d vortex, the P3 symmetry is broken because of the nonzero g1 components, and the P2 symmetry is the only preserved symmetry among these three. In the case of isolated HQVs, even though g1 ¼ 0, because of the threefold, and fivefold symmetry in the gap amplitude of g0, the P1 and P2 symmetries are broken, but the P3 symmetry is preserved. Another discrete symmetry is the mirror reflection symmetry about the plane perpendicular to the vortex line, that is, the xyplane, which is already discussed in Section “Symmetry-protected non-Abelian anyons.” The mirror refection symmetry Mxy transforms the momentum and the spin as Mxy : k ! ðkx , ky ,  kz Þ, s ! ðsx ,  sy , sz Þ,

(110)

and thus the order parameter is transformed as Mxy : gM ðr, yÞ ! ð−1ÞM+1 gM ðr, yÞ:

(111)

Therefore, in the presence of Mxy, even M and odd M cannot be mixed. In the case of the D4-BN state with the point nodes along the z-direction, gM¼2 6¼ 0, and the only possible mirror operation that commutes with the BdG Hamiltonian at the mirror invariant momentum kM ¼ (kx, ky, 0) is  ¼ − in Eq. (38) when g1 ¼ 0. The above-discussed vortices of a 3P2, SF which preserve the Mxy symmetry, are the axisymmetric o and u vortices, and the HQVs.

Topology of the vortex-bound states On the basis of the above-discussed discrete symmetries, the zero energy state of each HQV is shown to be protected by two topological numbers: one based on the chiral symmetry, a combined symmetry between the particle-hole symmetry and the P3 symmetry, and the other based on the mirror reflection symmetry. When the P3 symmetry is preserved, the semiclassical BdG Hamiltonian is transformed under the combined symmetry operation G ¼ CP 3 with G2 ¼ +1 as  x , −ky , −kz , −yÞ:  yÞG−1 ¼ −Hðk GHðk,

(112)

Particularly, for ky ¼ kz ¼ 0 and y ¼ 0 or p, the BdG Hamiltonian anticommutes with G, which can be regarded as chiral symmetry (Sato et al., 2011; Mizushima et al., 2012; Tsutsumi et al., 2013; Shiozaki and Sato, 2014; Tsutsumi et al., 2015; Masaki et al., 2020; Mizushima et al., 2016). As in the bulk nematic state, as long as the chiral symmetry is preserved, one can define the onedimensional winding number for kx ¼ (kx, 0, 0) and y ¼ 0 or p as Z h i 1  x , yÞ : H  −1 ðkx , yÞ∂k Hðk w1d ðyÞ ¼ − dkx tr G (113) x 4pi The vortices that preserve the P3 symmetry are o and w vortices, and the non-Abelian HQVs. Among them, the winding numbers (w1d(0), w1d(p)) of o and w vortices are (2, −2), and thus the topological invariant w1d ¼ (w1d(0) − w1d(p))/2 ¼ 2 ensures the existence of two zero energy states. In the case of non-Abelian HQVs, (w1d(0), w1d(p)) is (2, 0) for (k, n) ¼ (1/2, +1/4) and (0, −2) for (1/2, −1/4).9 In both cases, the topological invariant w1d becomes 1, which ensures the existence of one zero energy state in each HQV core. Next we explain the topological invariant based on the mirror reflection symmetry Mxy. As discussed above, the BdG    for  ¼ − in Eq. (38). Hamiltonian HðkM , yÞ for the mirror invariant momentum kM commutes with the mirror operator M  Thus, HðkM , yÞ can be block diagonalized with respect to the mirror eigenstates with eigenvalues l ¼ i as  M , yÞ ¼ Hðk

 H~ l ðkM , yÞ: l

(114)

~ l, the reduced particle-hole symmetry C~ is preserved, though Significantly, even for the BdG Hamiltonian in each mirror subsector, H  ¼+. Therefore, H ~ l belongs to the class D as well as spinless chiral SCs (Schnyder it is not for the other mirror symmetry operator M et al., 2008), and the Z2 invariant, nl can be constructed in each mirror subsector (Teo and Kane, 2010b; Qi et al., 2008). Nontrivial (odd) value of nl ensures that the vortex has a single Majorana zero mode which behaves as a non-Abelian (Ising) anyon (Ueno et al., 2013; Sato et al., 2014; Tsutsumi et al., 2013). The Z2 invariant nl is defined on the base space (S2  S) composed of the two dimensional mirror invariant momentum space kM and the angle in the real space surrounding the vortex y, and constructed by the dimensional reduction of the second Chern number defined in the four-dimensional base space (S3  S) composed of k and y. The Z2 invariant in each mirror subsector is given by the integral of the Chern–Simons form: 9

The choice of the relative phase between g2 as p by y1 ! p leads to (w1d(0), w1d(p)) ¼ (0, −2) rather than (2, 0) for the HQV of (k, n) ¼ (1/2, −1/4).

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nl ¼

 2 Z i ~ 3,l mod 2, Q p S2 S1

~ 3,l ¼ tr½A~l ^d A~l + 2 A~l ^A~l ^A~l : Q 3

(115) (116)

A non-Abelian Berry connection A~l is given by i½A~l nm ¼

X~ ~ ⟨ul,n ðkM , yÞj∂a ul,m ðkM , yÞ⟩da, a

(117)

~ l ðk, yÞ. Within the semiclassical approximation in the D4-BN where a denotes (kx, ky, y), and j~ ul,m ðkM , yÞi is the mth eigenstate of H state, the Z2 invariant is calculated as nl ¼ ℓl kl mod 2,

(118)

where ℓl and kl are the first Chern number and the vorticity in the mirror subsector l. The former characterizes the bulk topology and ℓl¼i ¼ 1. The latter is given by kl ¼ k − nM for the bulk component M in the l subsector. In the case of the isolated HQVs, (n+i, n−i) ¼ (0, −1) [(+1, 0)] for (k, n) ¼ (1/2, +1/4) [(1/2, −1/4)]. The two HQVs host Majorana fermions in different mirror sectors, and thus the Z2 invariant of the molecule state is given by (n+i, n−i) ¼ (+1, −1). The Z2 invariant of the o vortex has the same one, and the pairwise Majorana fermions are localized at the same core (Masaki et al., 2020). The zero energy state bound in each HQV core is protected by the two topological invariants based on the mirror reflection symmetry and the chiral symmetry, and it is identified as non-Abelian (Ising) anyon of the Majorana fermion. Therefore, the HQV in the D4-BN state has twofold non-Abelian nature: one from the non-Abelian Ising anyon and the other from non-Abelian first homotopy group.

Conclusion We have presented various types of non-Abelian anyons in topological SCs/SFs and other systems. Non-Abelian anyons are attracting a great attention because of possible applications to topological quantum computations, where quantum computations are realized by braiding of non-Abelian anyons. The simplest non-Abelian anyons are Ising anyons realized by Majorana fermions hosted by vortices (or edges) of topological superconductors, n ¼ 5/2 quantum Hall states, spin liquids, and QCD matter. These are, however, insufficient for universal quantum computations. The other anyons that can be used for universal quantum computations are given by Fibonacci anyons, which exist in n ¼ 12/5 quantum Hall states. There are also Yang-Lee anyons, which are nonunitary counterparts of Fibonacci anyons. Another class of non-Abelian anyons of the bosonic origin can be given by non-Abelian vortex anyons realized by non-Abelian vortices supported by a non-Abelian first homotopy group. These vortex anyons exist in BN liquid crystals, cholesteric liquid crystals, spin-2 spinor BECs, and QCD matter. There is a unique system simultaneously admitting two kinds of non-Abelian anyons, which is the Majorana fermions (Ising anyons) and non-Abelian vortex anyons. That is 3P2 SFs, spintriplet, p-wave paring of neutrons, expected to exist in neutron star interiors as the largest topological quantum matter in universe.

Acknowledgments This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Quantum Liquid Crystals” (Grant No. JP22H04480) from JSPS of Japan and JSPS KAKENHI (Grants No. JP18H01217, No. JP19K14662, No. JP20K03860, No. JP20H01857, No. JP21H01039, and No. JP22H01221), and the WPI program “Sustainability with Knotted Chiral Meta Matter (SKCM2)” at Hiroshima University.

References Akhmerov AR, Dahlhaus JP, Hassler F, Wimmer M, and Beenakker CWJ (2011) Quantized conductance at the Majorana phase transition in a disordered superconducting wire. Physical Review Letters 106(5): 057001. https://doi.org/10.1103/PhysRevLett.106.057001. Alford MG, Benson K, Coleman SR, March-Russell J, and Wilczek F (1990) The interactions and excitations of nonabelian vortices. Physical Review Letters 64: 1632. https://doi.org/ 10.1103/PhysRevLett.64.1632 [Erratum: Phys.Rev.Lett. 65, 668 (1990)]. Alford MG, Benson K, Coleman SR, March-Russell J, and Wilczek F (1991) Zero modes of nonabelian vortices. Nuclear Physics B 349: 414–438. https://doi.org/10.1016/0550-3213 (91)90331-Q. Alford MG, Lee K-M, March-Russell J, and Preskill J (1992) Quantum field theory of non-Abelian strings and vortices. Nuclear Physics B 384: 251–317. https://doi.org/ 10.1016/0550-3213(92)90468-Q. hep-th/9112038. Alford MG, Schmitt A, Rajagopal K, and Schäfer T (2008) Color superconductivity in dense quark matter. Reviews of Modern Physics 80: 1455–1515. https://doi.org/10.1103/ RevModPhys.80.1455. Alicea J (2012) New directions in the pursuit of Majorana fermions in solid state systems. Reports on Progress in Physics 75(7): 76501. Alicea J, Oreg Y, Refael G, von Oppen F, and Fisher MPA (2011) Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nature Physics 7: 412. https://doi.org/10.1038/nphys1915.

Non-Abelian anyons and non-Abelian vortices in topological superconductors

789

Ambegaokar V, deGennes PG, and Rainer D (1974) Landau-Ginsburg equations for an anisotropic superfluid. Physical Review A 9(6): 2676–2685. https://doi.org/10.1103/ PhysRevA.9.2676. Andersen TI, et al. (2022) Observation of non-Abelian exchange statistics on a superconducting processor. arXiv:2210.10255. Anderson PW and Brinkman WF (1973) Anisotropic superfluidity in 3He: A possible interpretation of its stability as a spin-fluctuation effect. Physical Review Letters 30(22): 1108–1111. https://doi.org/10.1103/PhysRevLett.30.1108. Anderson PW and Morel P (1961) Generalized Bardeen-Cooper-Schrieffer states and the proposed low-temperature phase of liquid He3. Physical Review 123(6): 1911–1934. https:// doi.org/10.1103/PhysRev.123.1911. Anderson PW and Toulouse G (1977) Phase slippage without vortex cores: Vortex textures in superfluid 3He. Physical Review Letters 38(9): 508–511. https://doi.org/10.1103/ PhysRevLett.38.508. Ardonne E, Gukelberger J, Ludwig AWW, Trebst S, and Troyer M (2011) Microscopic models of interacting Yang-Lee anyons. New Journal of Physics 13(4): 045006. https://doi.org/ 10.1088/1367-2630/13/4/045006. Arovas D, Schrieffer JR, and Wilczek F (1984) Fractional statistics and the quantum Hall effect. Physical Review Letters 53: 722–723. https://doi.org/10.1103/PhysRevLett.53.722. Autti S, Dmitriev VV, Mäkinen JT, Soldatov AA, Volovik GE, Yudin AN, Zavjalov VV, and Eltsov VB (2016) Observation of half-quantum vortices in topological superfluid 3He. Physical Review Letters 117(25): 255301. https://doi.org/10.1103/PhysRevLett.117.255301. Auzzi R, Bolognesi S, Evslin J, Konishi K, and Yung A (2003) NonAbelian superconductors: Vortices and confinement in N¼2 SQCD. Nuclear Physics B 673: 187–216. https://doi.org/ 10.1016/j.nuclphysb.2003.09.029. hep-th/0307287. Awoga OA, Cayao J, and Black-Schaffer AM (2019) Supercurrent detection of topologically trivial zero-energy states in nanowire junctions. Physical Review Letters 123(11): 117001. https://doi.org/10.1103/PhysRevLett.123.117001. Bais FA (1980) Flux Metamorphosis. Nuclear Physics B 170: 32–43. https://doi.org/10.1016/0550-3213(80)90474-5. Balachandran AP, Lizzi F, and Rodgers VGJ (1984) Topological symmetry breakdown in cholesterics, nematics, and 3He. Physical Review Letters 52: 1818–1821. https://doi.org/ 10.1103/PhysRevLett.52.1818. Balachandran AP, Digal S, and Matsuura T (2006) Semi-superfluid strings in high density QCD. Physical Review D 73: 074009. https://doi.org/10.1103/PhysRevD.73.074009. Bartolomei H, Kumar M, Bisognin R, Marguerite A, Berroir J-M, Bocquillon E, Plaçais B, Cavanna A, Dong Q, Gennser U, Jin Y, and Fève G (2020) Fractional statistics in anyon collisions. Science 368(6487): 173–177. https://doi.org/10.1126/science.aaz5601. Beenakker CWJ (2013) Search for Majorana fermions in superconductors. Annual Review of Condensed Matter Physics ISSN: 1947-5454. 4(1): 113–136. https://doi.org/10.1146/ annurev-conmatphys-030212-184337. Beenakker CWJ (2020) Search for non-Abelian Majorana braiding statistics in superconductors. SciPost Physics Lecture Notes ISSN: 2590-1990. 15: 15. https://doi.org/10.21468/ SciPostPhysLectNotes.15. Bonderson P, Freedman M, and Nayak C (2008) Measurement-only topological quantum computation. Physical Review Letters 101(1): 010501. https://doi.org/10.1103/ PhysRevLett.101.010501. Bonderson P, Freedman M, and Nayak C (2009) Measurement-only topological quantum computation via anyonic interferometry. Annals of Physics ISSN: 0003-4916. 324(4): 787–826. https://doi.org/10.1016/j.aop.2008.09.009. Bonesteel NE, Hormozi L, Zikos G, and Simon SH (2005) Braid topologies for quantum computation. Physical Review Letters 95(14): 140503. https://doi.org/10.1103/ PhysRevLett.95.140503. Borgh MO and Ruostekoski J (2016) Core structure and non-Abelian reconnection of defects in a biaxial nematic Spin-2 Bose-Einstein condensate. Physical Review Letters 117(27): 275302. https://doi.org/10.1103/PhysRevLett.117.275302 [Erratum: Physical Review Lett. 118, 129901 (2017)]. Bravyi S (2006) Universal quantum computation with the n ¼ 5/2 fractional quantum Hall state. Physical Review A 73(4): 042313. https://doi.org/10.1103/PhysRevA.73.042313. Brekke L, Imbo TD, Dykstra H, and Falk AF (1993) Novel spin and statistical properties of nonabelian vortices. Physics Letters B 304: 127–133. https://doi.org/10.1016/0370-2693 (93)91411-F. hep-th/9210131. Brekke L, Collins SJ, and Imbo TD (1997) Non-Abelian vortices on surfaces and their statistics. Nuclear Physics B 500: 465–485. https://doi.org/10.1016/S0550-3213(97)00409-4. hep-th/9701056. Bruin JAN, Claus RR, Matsumoto Y, Kurita N, Tanaka H, and Takagi H (2022) Robustness of the thermal Hall effect close to half-quantization in a-RuCl3. Nature Physics 18: 401. https://doi.org/10.1038/s41567-021-01501-y. Bucher M (1991) The Aharonov-Bohm effect and exotic statistics for nonabelian vortices. Nuclear Physics B 350: 163–178. https://doi.org/10.1016/0550-3213(91)90256-W. Bucher M, Lee K-M, and Preskill J (1992) On detecting discrete Cheshire charge. Nuclear Physics B 386: 27–42. https://doi.org/10.1016/0550-3213(92)90174-A. hep-th/ 9112040. Cardy JL (1985) Conformal invariance and the Yang-Lee edge singularity in two dimensions. Physical Review Letters 54(13): 1354–1356. https://doi.org/10.1103/ PhysRevLett.54.1354. Cayao J and Burset P (2021) Confinement-induced zero-bias peaks in conventional superconductor hybrids. Physical Review B 104(13): 134507. https://doi.org/10.1103/ PhysRevB.104.134507. Chen W, Wang J, Wu Y, Qi J, Liu J, and Xie XC (2022) Non-Abelian statistics of Majorana zero modes in the presence of an Andreev bound state. Physical Review B 105(5): 054507. https://doi.org/10.1103/PhysRevB.105.054507. Cheng M, Lutchyn RM, Galitski V, and Das Sarma S (2009) Splitting of Majorana-fermion modes due to intervortex tunneling in a px + ipy superconductor. Physical Review Letters 103 (10): 107001. https://doi.org/10.1103/PhysRevLett.103.107001. Chung SB and Kivelson SA (2010) Entropy-driven formation of a half-quantum vortex lattice. Physical Review B 82(21): 214512. https://doi.org/10.1103/PhysRevB.82.214512. Chung SB and Zhang S-C (2009) Detecting the Majorana fermion surface state of 3He−B through spin relaxation. Physical Review Letters ISSN: 0031-9007. 103(23): 235301. https://doi.org/10.1103/PhysRevLett.103.235301. Chung SB, Bluhm H, and Kim E-A (2007) Stability of half-quantum vortices in px + ipy superconductors. Physical Review Letters 99: 197002. https://doi.org/10.1103/ PhysRevLett.99.197002. Clarke DJ, Sau JD, and Tewari S (2011) Majorana fermion exchange in quasi-one-dimensional networks. Physical Review B 84(3): 035120. https://doi.org/10.1103/ PhysRevB.84.035120. De Gennes PG (1973) Long range distortions in an anisotropic superfluid. Physics Letters A ISSN: 0375-9601. 44(4): 271–272. https://doi.org/10.1016/0375-9601(73)90916-X. Di Francesco PD, Mathieu P, and Sénéchal. (1997) Conformal Field Theory. New York: Springer-Verlag. Dmitriev VV, Senin AA, Soldatov AA, and Yudin AN (2015) Polar phase of superfluid 3He in anisotropic aerogel. Physical Review Letters 115(16): 165304. https://doi.org/10.1103/ PhysRevLett.115.165304. Elliott SR and Franz M (2015) Colloquium: Majorana fermions in nuclear, particle, and solid-state physics. Reviews of Modern Physics ISSN: 0034-6861. 87(1): 137–163. https://doi. org/10.1103/RevModPhys.87.137. Eto M and Nitta M (2009) Color magnetic flux tubes in dense QCD. Physical Review D 80: 125007. https://doi.org/10.1103/PhysRevD.80.125007. Eto M and Nitta M (2021) Chiral non-Abelian vortices and their confinement in three flavor dense QCD. Physical Review D 104(9): 094052. https://doi.org/10.1103/ PhysRevD.104.094052. 2103.13011. Eto M, Isozumi Y, Nitta M, Ohashi K, and Sakai N (2006) Solitons in the Higgs phase: The moduli matrix approach. Journal of Physics A: Mathematical and General 39: R315–R392. https://doi.org/10.1088/0305-4470/39/26/R01. hep-th/0602170.

790

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Eto M, Hirono Y, Nitta M, and Yasui S (2014) Vortices and other topological solitons in dense quark matter. Progress of Theoretical and Experimental Physics 2014(1): 012D01. https:// doi.org/10.1093/ptep/ptt095. 1308.1535. Fang C, Gilbert MJ, and Bernevig BA (2014) New class of topological superconductors protected by magnetic group symmetries. Physical Review Letters 112(10): 106401. https://doi. org/10.1103/PhysRevLett.112.106401. Field B and Simula T (2018) Introduction to topological quantum computation with non-Abelian anyons. Quantum Science and Technology 3: 045004. Fisher ME (1978) Yang-Lee edge singularity and f3 field theory. Physical Review Letters 40(25): 1610–1613. https://doi.org/10.1103/PhysRevLett.40.1610. Freedman M, Hastings MB, Nayak C, and Qi X-L (2011a) Weakly coupled non-Abelian anyons in three dimensions. Physical Review B 84(24): 245119. https://doi.org/10.1103/ PhysRevB.84.245119. Freedman M, Hastings MB, Nayak C, Qi X-L, Walker K, and Wang Z (2011b) Projective ribbon permutation statistics: A remnant of non-abelian braiding in higher dimensions. Physical Review B 83(11): 115132. https://doi.org/10.1103/PhysRevB.83.115132. Freedman MH, Gukelberger J, Hastings MB, Trebst S, Troyer M, and Wang Z (2012) Galois conjugates of topological phases. Physical Review B 85(4): 045414. https://doi.org/ 10.1103/PhysRevB.85.045414. Fu L and Kane CL (2008) Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Physical Review Lett. 100(9): 096407. https://doi.org/ 10.1103/PhysRevLett.100.096407. Fujimoto Y and Nitta M (2021a) Non-Abelian alice strings in two-flavor dense QCD. Physical Review D 103: 054002. https://doi.org/10.1103/PhysRevD.103.054002. 2011.09947. Fujimoto Y and Nitta M (2021b) Topological confinement of vortices in two-flavor dense QCD. Journal of High Energy Physics 09: 192. https://doi.org/10.1007/JHEP09(2021)192. 2103.15185. Fujimoto Y and Nitta M (2021c) Vortices penetrating two-flavor quark-hadron continuity. Physical Review D 103(11): 114003. https://doi.org/10.1103/PhysRevD.103.114003. 2102.12928. Fujita T and Tsuneto T (1972) The Ginzburg-Landau equation for 3P2 pairingsuperfluidity in neutron stars. Progress in Theoretical Physics 48(3): 766–782. https://doi.org/10.1143/ PTP.48.766. Fujiwara T, Fukui T, Nitta M, and Yasui S (2011) Index theorem and Majorana zero modes along a non-Abelian vortex in a color superconductor. Physical Review D 84: 076002. https:// doi.org/10.1103/PhysRevD.84.076002. 1105.2115. Fulga IC, Hassler F, Akhmerov AR, and Beenakker CWJ (2011) Scattering formula for the topological quantum number of a disordered multimode wire. Physical Review B 83(15): 155429. https://doi.org/10.1103/PhysRevB.83.155429. Gao P, He Y-P, and Liu X-J (2016) Symmetry-protected non-Abelian braiding of Majorana Kramers pairs. Physical Review B 94(22): 224509. https://doi.org/10.1103/ PhysRevB.94.224509. Génetay Johansen E and Simula T (2022) Prime number factorization using a spinor Bose–Einstein condensate-inspired topological quantum computer. Quantum Information Processing 21(1): 31. https://doi.org/10.1007/s11128-021-03366-9. Haim A and Oreg Y (2019) Time-reversal-invariant topological superconductivity in one and two dimensions. Physics Reports ISSN: 0370-1573. 825: 1–48. https://doi.org/10.1016/j. physrep.2019.08.002. Time-reversal-invariant topological superconductivity in one and two dimensions. Halperin BI (1984) Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Physical Review Letters 52: 1583–1586. https://doi.org/10.1103/ PhysRevLett.52.1583 [Erratum: Phys.Rev.Lett. 52, 2390 (1984)]. Halperin WP (2019) Superfluid 3He in aerogel. Annual Review of Condensed Matter Physics 10(1): 155–170. https://doi.org/10.1146/annurev-conmatphys-031218-013134. Hanany A and Tong D (2003) Vortices, instantons and branes. Journal of High Energy Physics 07: 037. https://doi.org/10.1088/1126-6708/2003/07/037. hep-th/0306150. Heeb R and Agterberg DF (1999) Ginzburg-Landau theory for a p-wave Sr2RuO4 superconductor: Vortex core structure and extended London theory. Physical Review B ISSN: 01631829. 59(10): 7076–7082. https://doi.org/10.1103/PhysRevB.59.7076. http://arxiv.org/abs/cond-mat/9811190. Hirono Y, Yasui S, Itakura K, and Nitta M (2012) Non-Abelian statistics of vortices with multiple Majorana fermions. Physical Review B 86: 014508. https://doi.org/10.1103/ PhysRevB.86.014508. Hoffberg M, Glassgold AE, Richardson RW, and Ruderman M (1970) Anisotropic superfluidity in neutron star matter. Physical Review Letters 24(14): 775. https://doi.org/10.1103/ PhysRevLett.24.775. Hong J-S, Poon T-FJ, Zhang L, and Liu X-J (2022) Unitary symmetry-protected non-Abelian statistics of Majorana modes. Physical Review B 105(2): 024503. https://doi.org/ 10.1103/PhysRevB.105.024503. Hormozi L, Zikos G, Bonesteel NE, and Simon SH (2007) Topological quantum compiling. Physical Review B 75(16): 165310. https://doi.org/10.1103/PhysRevB.75.165310. Hosur P, Ghaemi P, Mong RSK, and Vishwanath A (2011) Majorana modes at the ends of superconductor vortices in doped topological insulators. Physical Review Letters 107(9): 097001. https://doi.org/10.1103/PhysRevLett.107.097001. Hu Y and Kane CL (2018) Fibonacci topological superconductor. Physical Review Letters 120(6): 066801. https://doi.org/10.1103/PhysRevLett.120.066801. Ikegaya S, Suzuki S-I, Tanaka Y, and Asano Y (2016) Quantization of conductance minimum and index theorem. Physical Review B 94(5): 054512. https://doi.org/10.1103/ PhysRevB.94.054512. Ivanov DA (2001) Non-abelian statistics of half-quantum vortices in p-wave superconductors. Physical Review Letters 86(2): 268. https://doi.org/10.1103/PhysRevLett.86.268. Karzig T, Oreg Y, Refael G, and Freedman MH (2016) Universal geometric path to a robust Majorana magic gate. Physical Review X 6(3): 031019. https://doi.org/10.1103/ PhysRevX.6.031019. Karzig T, Knapp C, Lutchyn RM, Bonderson P, Hastings MB, Nayak C, Alicea J, Flensberg K, Plugge S, Oreg Y, Marcus CM, and Freedman MH (2017) Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes. Physical Review B 95(23): 235305. https://doi.org/10.1103/ PhysRevB.95.235305. Kasahara Y, Ohnishi T, Mizukami Y, Tanaka O, Ma S, Sugii K, Kurita N, Tanaka H, Nasu J, Motome Y, Shibauchi T, and Matsuda Y (2018) Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559: 227. https://doi.org/10.1038/s41586-018-0274-0. Kasamatsu K, Mizuno R, Ohmi T, and Nakahara M (2019) Effects of a magnetic field on vortex states in superfluid 3He−B. Physical Review B 99(10): 104513. https://doi.org/10.1103/ PhysRevB.99.104513. Kashiwaya S and Tanaka Y (2000) Tunnelling effects on surface bound states in unconventional superconductors. Reports on Progress in Physics 63(10): 1641–1724. https://doi.org/ 10.1088/0034-4885/63/10/202. Kashuba O and Timm C (2015) Topological Kondo effect in transport through a superconducting wire with multiple Majorana end states. Physical Review Letters 114(11): 116801. https://doi.org/10.1103/PhysRevLett.114.116801. Kawaguchi Y and Ueda M (2012) Spinor Bose-Einstein condensates. Physics Reports 520: 253–381. https://doi.org/10.1016/j.physrep.2012.07.005. Kawakami T and Hu X (2015) Evolution of density of states and a spin-resolved checkerboard-type pattern associated with the Majorana bound state. Physical Review Letters 115(17): 177001. https://doi.org/10.1103/PhysRevLett.115.177001. Kawakami T, Tsutsumi Y, and Machida K (2009) Stability of a half-quantum vortex in rotating superfluid 3He−A between parallel plates. Physical Review B ISSN: 1098-0121. 79(9): 092506. https://doi.org/10.1103/PhysRevB.79.092506. Kawakami T, Tsutsumi Y, and Machida K (2010) Singular and half-quantum vortices and associated Majorana particles in superfluid 3 He-A between parallel plates. Journal of the Physical Society of Japan ISSN: 0031-9015. 79(4): 044607. https://doi.org/10.1143/JPSJ.79.044607. Kawakami T, Mizushima T, and Machida K (2011) Zero energy modes and statistics of vortices in spinful chiral p-wave superfluids. Journal of the Physical Society of Japan ISSN: 0031-9015. 80(4): 044603. https://doi.org/10.1143/JPSJ.80.044603.

Non-Abelian anyons and non-Abelian vortices in topological superconductors

791

Kells G, Meidan D, and Brouwer PW (2012) Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement. Physical Review B 86(10): 100503. https://doi.org/10.1103/PhysRevB.86.100503. Kitaev AY (2001) Unpaired Majorana fermions in quantum wires. Physics-Uspekhi 44(10S): 131–136. https://doi.org/10.1070/1063-7869/44/10S/S29. Kitaev AY (2003) Fault tolerant quantum computation by anyons. Annals of Physics 303: 2–30. https://doi.org/10.1016/S0003-4916(02)00018-0. Kitaev A (2006) Anyons in an exactly solved model and beyond. Annals of Physics (New York) ISSN: 00034916. 321(1): 2–111. https://doi.org/10.1016/j.aop.2005.10.005. Kitaev A and Laumann C (2008) Topological Phases and Quantum Computation. Lecture Notes of the Les Houches Summer School. No. 89. , pp. 101–125. Oxford University Press, ISBN: 0-19-957461-8. Knapp C, Chew A, and Alicea J (2020) Fragility of the fractional Josephson effect in time-reversal-invariant topological superconductors. Physical Review Letters 125(20): 207002. https://doi.org/10.1103/PhysRevLett.125.207002. Kobayashi M and Nitta M (2022a) Core structures of vortices in Ginzburg-Landau theory for neutron 3P2 superfluids. Physical Review C 105(3): 035807. https://doi.org/10.1103/ PhysRevC.105.035807. 2203.09300. Kobayashi M and Nitta M (2022b) Proximity effects of vortices in neutron 3P2 superfluids in neutron stars: Vortex core transitions and covalent bonding of vortex molecules. arXiv:2209.07205. Kobayashi M, Kawaguchi Y, Nitta M, and Ueda M (2009) Collision dynamics and rung formation of non-Abelian vortices. Physical Review Letters 103: 115301. https://doi.org/ 10.1103/PhysRevLett.103.115301. Kobayashi S, Kobayashi M, Kawaguchi Y, Nitta M, and Ueda M (2012) Abe homotopy classification of topological excitations under the topological influence of vortices. Nuclear Physics B 856: 577–606. https://doi.org/10.1016/j.nuclphysb.2011.11.003. Koornwinder TH, Schroers BJ, Slingerland JK, and Bais FA (1999) Fourier transform and the Verlinde formula for the quantum double of a finite group. Journal of Physics A: Mathematical and General 32: 8539–8549. https://doi.org/10.1088/0305-4470/32/48/313. math/9904029. Kopnin NB and Salomaa MM (1991) Mutual friction in superfluid 3He: Effects of bound states in the vortex core. Physical Review B 44(17): 9667–9677. https://doi.org/10.1103/ PhysRevB.44.9667. Kortman PJ and Griffiths RB (1971) Density of zeros on the Lee-Yang circle for two Ising ferromagnets. Physical Review Letters 27(21): 1439–1442. https://doi.org/10.1103/ PhysRevLett.27.1439. Lavrentovich OD and Kleman M (2001) Cholesteric liquid crystals: Defects and topology. In: Kitzerow H-S and Bahr C (eds.) Chirality in Liquid Crystals, pp. 115–158. New York, NY: Springer New York. Lee K-M (1994) Vortices on higher genus surfaces. Physical Review D 49: 2030–2040. https://doi.org/10.1103/PhysRevD.49.2030. hep-th/9307139. Lee TD and Yang CN (1952) Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Physical Review 87(3): 410–419. https://doi.org/10.1103/ PhysRev.87.410. Leggett AJ (1975) A theoretical description of the new phases of liquid 3He. Reviews of Modern Physics 47(2): 331–414. https://doi.org/10.1103/RevModPhys.47.331. Leijnse M and Flensberg K (2012) Introduction to topological superconductivity and Majorana fermions. Semiconductor Science and Technology 27(12): 124003. http://stacks.iop.org/ 0268-1242/27/i¼12/a¼124003. Leinaas JM and Myrheim J (1977) On the theory of identical particles. Nuovo Cimento B 37: 1–23. https://doi.org/10.1007/BF02727953. Lesanovsky I and Katsura H (2012) Interacting Fibonacci anyons in a Rydberg gas. Physical Review A 86(4): 041601. https://doi.org/10.1103/PhysRevA.86.041601. Liu X-J, Wong CLM, and Law KT (2014) Non-Abelian Majorana doublets in time-reversal-invariant topological superconductors. Physical Review X 4(2): 021018. https://doi.org/ 10.1103/PhysRevX.4.021018. Liu C-X, Sau JD, Stanescu TD, and Das Sarma S (2017) Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks. Physical Review B 96(7): 075161. https://doi.org/10.1103/PhysRevB.96.075161. Liu C-X, Sau JD, and Das Sarma S (2018) Distinguishing topological Majorana bound states from trivial Andreev bound states: Proposed tests through differential tunneling conductance spectroscopy. Physical Review B 97(21): 214502. https://doi.org/10.1103/PhysRevB.97.214502. Liu D, Cao Z, Liu X, Zhang H, and Liu DE (2021) Topological Kondo device for distinguishing quasi-Majorana and Majorana signatures. Physical Review B 104(20): 205125. https://doi. org/10.1103/PhysRevB.104.205125. Lo H-K and Preskill J (1993) Non-Abelian vortices and non-Abelian statistics. Physical Review D 48: 4821–4834. https://doi.org/10.1103/PhysRevD.48.4821. Machida T, Sun Y, Pyon S, Takeda S, Kohsaka Y, Hanaguri T, Sasagawa T, and Tamegai T (2019) Zero-energy vortex bound state in the superconducting topological surface state of Fe(Se,Te). Nature Materials 18: 811. https://doi.org/10.1038/s41563-019-0397-1. Majorana E (1937) Teoria simmetrica dell’elettrone e del positrone. Nuovo Cimento 14: 171–184. https://doi.org/10.1007/BF02961314. Makhlin Y, Silaev M, and Volovik GE (2014) Topology of the planar phase of superfluid 3He and bulk-boundary correspondence for three-dimensional topological superconductors. Physical Review B 89(17): 174502. https://doi.org/10.1103/PhysRevB.89.174502. Mäkinen JT, Dmitriev VV, Nissinen J, Rysti J, Volovik GE, Yudin AN, Zhang K, and Eltsov VB (2019) Half-quantum vortices and walls bounded by strings in the polar-distorted phases of topological superfluid 3He. Nature Communications 10(1): 237. https://doi.org/10.1038/s41467-018-08204-8. Mäkinen JT, Zhang K, and Eltsov VB (2022) Vortex-bound solitons in topological superfluid 3He. arXiv:2211.17117. Marmorini G, Yasui S, and Nitta M (2020) Pulsar glitches from quantum vortex networks. arXiv:2010.09032. Marra P (2022) Majorana nanowires for topological quantum computation. Journal of Applied Physics 132: 231101. Masaki Y, Mizushima T, and Nitta M (2020) Microscopic description of axisymmetric vortices in 3P2 superfluids. Physical Review Research 2(1): 013193. https://doi.org/10.1103/ PhysRevResearch.2.013193. Masaki Y, Mizushima T, and Nitta M (2022) Non-Abelian half-quantum vortices in 3P2 topological superfluids. Physical Review B ISSN: 2469-9950. 105(22): L220503. https://doi.org/ 10.1103/PhysRevB.105.L220503. Masuda K and Nitta M (2016) Magnetic properties of quantized vortices in neutron 3P2 superfluids in neutron stars. Physical Review C 93(3): 035804. https://doi.org/10.1103/ PhysRevC.93.035804. Masuda K and Nitta M (2020) Half-quantized non-Abelian vortices in neutron 3P2 superfluids inside magnetars. Progress of Theoretical and Experimental Physics 2020(1): 013D01. https://doi.org/10.1093/ptep/ptz138. Matsumoto M and Heeb R (2001) Vortex charging effect in a chiral px  ipy-wave superconductor. Physical Review B ISSN: 0163-1829. 65(1): 014504. https://doi.org/10.1103/ PhysRevB.65.014504. Mawson T, Petersen TC, Slingerland JK, and Simula TP (2019) Braiding and fusion of non-Abelian vortex anyons. Physical Review Letters 123(14): 140404. https://doi.org/10.1103/ PhysRevLett.123.140404. Mermin ND (1974) d-Wave pairing near the transition temperature. Physical Review A 9: 868–872. https://doi.org/10.1103/PhysRevA.9.868. Mermin ND (1979) The topological theory of defects in ordered media. Reviews of Modern Physics 51: 591–648. https://doi.org/10.1103/RevModPhys.51.591. Mietke A and Dunkel J (2022) Anyonic defect braiding and spontaneous chiral symmetry breaking in dihedral liquid crystals. Physical Review X 12(1): 011027. https://doi.org/ 10.1103/PhysRevX.12.011027. 2011.04648. Mineev VP (2014) Half-quantum vortices in polar phase of superfluid 3He. Journal of Low Temperature Physics 177: 48. https://doi.org/10.1007/s10909-014-1189-2. Mizushima T and Machida K (2010) Splitting and oscillation of Majorana zero modes in the p-wave BCS-BEC evolution with plural vortices. Physical Review A 82(2): 023624. https:// doi.org/10.1103/PhysRevA.82.023624. Mizushima T and Machida K (2011) Effects of Majorana edge fermions on dynamical spin susceptibility in topological superfluid 3He-B. Journal of Low Temperature Physics ISSN: 00222291. 162(3-4): 204–211. https://doi.org/10.1007/s10909-010-0297-x.

792

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Mizushima T, Sato M, and Machida K (2012) Symmetry protected topological order and spin susceptibility in superfluid 3He−B. Physical Review Letters 109(16): 165301. https://doi. org/10.1103/PhysRevLett.109.165301. Mizushima T, Tsutsumi Y, Kawakami T, Sato M, Ichioka M, and Machida K (2016) Symmetry-protected topological superfluids and superconductors-from the basics to 3He-. Journal of the Physical Society of Japan 85(2): 022001. https://doi.org/10.7566/JPSJ.85.022001. Mizushima T, Masuda K, and Nitta M (2017) 3P2 superfluids are topological. Physical Review B 95(14): 140503. https://doi.org/10.1103/PhysRevB.95.140503. Mizushima T, Yasui S, and Nitta M (2020) Critical endpoint and universality class of neutron 3P2 superfluids in neutron stars. Physical Review Research 2: 013194. https://doi.org/ 10.1103/PhysRevResearch.2.013194. Mizushima T, Yasui S, Inotani D, and Nitta M (2021) Spin-polarized phases of 3P2 superfluids in neutron stars. Physical Review C ISSN: 2469-9985. 104(4): 045803. https://doi.org/ 10.1103/PhysRevC.104.045803. Mong RSK, Clarke DJ, Alicea J, Lindner NH, Fendley P, Nayak C, Oreg Y, Stern A, Berg E, Shtengel K, and Fisher MPA (2014) Universal topological quantum computation from a superconductor-Abelian quantum Hall heterostructure. Physical Review X 4(1): 011036. https://doi.org/10.1103/PhysRevX.4.011036. Moore GW and Read N (1991) Nonabelions in the fractional quantum Hall effect. Nuclear Physics B 360: 362–396. https://doi.org/10.1016/0550-3213(91)90407-O. Moore C, Stanescu TD, and Tewari S (2018a) Two-terminal charge tunneling: Disentangling Majorana zero modes from partially separated Andreev bound states in semiconductorsuperconductor heterostructures. Physical Review B 97(16): 165302. https://doi.org/10.1103/PhysRevB.97.165302. Moore C, Zeng C, Stanescu TD, and Tewari S (2018b) Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Majorana zero modes. Physical Review B 98(15): 155314. https://doi.org/10.1103/PhysRevB.98.155314. Motome Y and Nasu J (2020) Hunting Majorana fermions in Kitaev magnets. Journal of the Physical Society of Japan ISSN: 0031-9015. 89(1): 012002. https://doi.org/10.7566/ JPSJ.89.012002. Muzikar P, Sauls JA, and Serene JW (1980) 3P2 pairing in neutron star matter: Magnetic field effects and vortices. Physical Review D 21: 1494–1502. https://doi.org/10.1103/ PhysRevD.21.1494. Nagamura N and Ikeda R (2018) Stability of half-quantum vortices in equal-spin pairing states of 3He. Physical Review B ISSN: 2469-9950. 98(9): 094524. https://doi.org/10.1103/ PhysRevB.98.094524. Nagato Y, Higashitani S, and Nagai K (2009) Strong anisotropy in spin susceptibility of superfluid 3 He-B film caused by surface bound states. Journal of the Physical Society of Japan ISSN: 0031-9015. 78(12): 123603. https://doi.org/10.1143/JPSJ.78.123603. Nakamura J, Liang S, Gardner GC, and Manfra MJ (2020) Direct observation of anyonic braiding statistics. Nature Physics 16: 931–936. https://doi.org/10.1038/s41567-020-1019-1. Nakano E, Nitta M, and Matsuura T (2008) Non-Abelian strings in high density QCD: Zero modes and interactions. Physical Review D 78: 045002. https://doi.org/10.1103/ PhysRevD.78.045002. Nayak C and Wilczek F (1996) 2n-Quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states. Nuclear Physics B ISSN: 0550-3213. 479(3): 529–553. https://doi.org/10.1016/0550-3213(96)00430-0. Nayak C, Simon SH, Stern A, Freedman M, and Das Sarma S (2008) Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80: 1083–1159. https:// doi.org/10.1103/RevModPhys.80.1083. Nielsen M and Chuang I (2010) Quantum Computation and Quantum Information. Cambridge University Press. Oreg Y and von Oppen F (2020) Majorana zero modes in networks of Cooper-Pair boxes: Topologically ordered states and topological quantum computation. Annual Review of Condensed Matter Physics 11(1): 397–420. https://doi.org/10.1146/annurev-conmatphys-031218-013618. Pachos JK (2012) Introduction to Topological Quantum Computation. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511792908. Pan H and Das Sarma S (2020) Physical mechanisms for zero-bias conductance peaks in Majorana nanowires. Physical Review Research 2(1): 013377. https://doi.org/10.1103/ PhysRevResearch.2.013377. Pan H, Cole WS, Sau JD, and Das Sarma S (2020) Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures. Physical Review B 101(2): 024506. https://doi.org/10.1103/PhysRevB.101.024506. Pan H, Sau JD, and Das Sarma S (2021) Three-terminal nonlocal conductance in Majorana nanowires: Distinguishing topological and trivial in realistic systems with disorder and inhomogeneous potential. Physical Review B 103(1): 014513. https://doi.org/10.1103/PhysRevB.103.014513. Piroli L and Cirac JI (2020) Quantum cellular automata, tensor networks, and area laws. Physical Review Letters 125(19): 190402. https://doi.org/10.1103/PhysRevLett.125.190402. Plugge S, Rasmussen A, Egger R, and Flensberg K (2017) Majorana box qubits. New Journal of Physics 19: 012001. https://doi.org/10.1088/1367-2630/aa54e1. Poenaru V and Toulouse G (1977) The crossing of defects in ordered media and the topology of 3-manifolds. Journal de Physique (Paris) 38: 887. https://doi.org/10.1051/ jphys:01977003808088700. Prada E, San-Jose P, de Moor MWA, Geresdi A, Lee EJH, Klinovaja J, Loss D, Nygård J, Aguado R, and Kouwenhoven LP (2020) From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires. Nature Reviews Physics 2: 575. https://doi.org/10.1038/s42254-020-0228-y. Preskill J (2004) Lecture Notes for Physics 219: Quantum Computation. http://www.theory.caltech.edu/preskill/ph219/ph219_2004.html. Preskill J and Krauss LM (1990) Local discrete symmetry and quantum mechanical hair. Nuclear Physics B 341: 50–100. https://doi.org/10.1016/0550-3213(90)90262-C. Qi X-L, Hughes TL, and Zhang S-C (2008) Topological field theory of time-reversal invariant insulators. Physical Review B 78(19): 195424. https://doi.org/10.1103/ PhysRevB.78.195424. Read N and Green D (2000) Paired states of fermions in two-dimensions with breaking of parity and time reversal symmetries, and the fractional quantum Hall effect. Physical Review B 61: 10267. https://doi.org/10.1103/PhysRevB.61.10267. Read N and Rezayi E (1999) Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level. Physical Review B 59(12): 8084–8092. https://doi.org/10.1103/PhysRevB.59.8084. Regan RC, Wiman JJ, and Sauls JA (2019) The vortex phase diagram of rotating superfluid 3He-B. Physical Review B ISSN: 2469-9950. 101(2): 024517. https://doi.org/10.1103/ PhysRevB.101.024517. Regan RC, Wiman JJ, and Sauls JA (2021) Half-quantum vortices in nematic and chiral phases of 3He. Physical Review B 104(2): 024513. https://doi.org/10.1103/ PhysRevB.104.024513. Ricco LS, de Souza M, Figueira MS, Shelykh IA, and Seridonio AC (2019) Spin-dependent zero-bias peak in a hybrid nanowire-quantum dot system: Distinguishing isolated Majorana fermions from Andreev bound states. Physical Review B 99(15): 155159. https://doi.org/10.1103/PhysRevB.99.155159. Ricco LS, Sanches JE, Marques Y, de Souza M, Figueira MS, Shelykh IA, and Seridonio AC (2021) Topological isoconductance signatures in Majorana nanowires. Scientific Reports 11: 17310. https://doi.org/10.1038/s41598-021-96415-3. Richardson RW (1972) Ginzburg-Landau theory of anisotropic superfluid neutron-star matter. Physical Review D 5: 1883–1896. https://doi.org/10.1103/PhysRevD.5.1883. Rysti J, Mäkinen JT, Autti S, Kamppinen T, Volovik GE, and Eltsov VB (2021) Suppressing the Kibble-Zurek mechanism by a symmetry-violating bias. Physical Review Letters 127(11): 115702. https://doi.org/10.1103/PhysRevLett.127.115702. Salomaa MM and Volovik GE (1985a) Half-quantum vortices in superfluid 3He-A. Physical Review Letters 55(11): 1184–1187. https://doi.org/10.1103/PhysRevLett.55.1184. Salomaa MM and Volovik GE (1985b) Symmetry and structure of quantized vortices in superfluid 3B. Physical Review B 31(1): 203–227. https://doi.org/10.1103/PhysRevB.31.203. Salomaa MM and Volovik GE (1986) Vortices with spontaneously broken axisymmetry in 3B. Physical Review Letters ISSN: 0031-9007. 56(4): 363–366. https://doi.org/10.1103/ PhysRevLett.56.363. Salomaa MM and Volovik GE (1987) Quantized vortices in superfluid 3He. Reviews of Modern Physics 59(3): 533–613. https://doi.org/10.1103/RevModPhys.59.533. Sanno T, Yamada MG, Mizushima T, and Fujimoto S (2022) Engineering Yang-Lee anyons via Majorana bound states. Physical Review B 106(17): 174517. https://doi.org/10.1103/ PhysRevB.106.174517.

Non-Abelian anyons and non-Abelian vortices in topological superconductors

793

Sarma SD, Freedman M, and Nayak C (2015) Majorana zero modes and topological quantum computation. npj Quantum Information 1(1): 15001. https://doi.org/10.1038/ npjqi.2015.1. http://www.nature.com/articles/npjqi20151. Sato M (2003) Non-Abelian statistics of axion strings. Physics Letters B ISSN: 0370-2693. 575(1): 126–130. https://doi.org/10.1016/j.physletb.2003.09.047. Sato M and Ando Y (2017) Topological superconductors: A review. Reports on Progress in Physics 80(7): 076501. https://doi.org/10.1088/1361-6633/aa6ac7. Sato M and Fujimoto S (2016) Majorana fermions and topology in superconductors. Journal of the Physical Society of Japan ISSN: 0031-9015. 85(7): 072001. https://doi.org/ 10.7566/JPSJ.85.072001. Sato M, Takahashi Y, and Fujimoto S (2009) Non-Abelian topological order in s-wave superfluids of ultracold fermionic atoms. Physical Review Letters 103(2): 020401. https://doi.org/ 10.1103/PhysRevLett.103.020401. Sato M, Takahashi Y, and Fujimoto S (2010) Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors. Physical Review B 82(13): 134521. https://doi. org/10.1103/PhysRevB.82.134521. Sato M, Tanaka Y, Yada K, and Yokoyama T (2011) Topology of Andreev bound states with flat dispersion. Physical Review B 83(22): 224511. https://doi.org/10.1103/ PhysRevB.83.224511. Sato M, Yamakage A, and Mizushima T (2014) Mirror Majorana zero modes in spinful superconductors/superfluids non-Abelian anyons in integer quantum vortices. Physica E 55: 20. Sauls, J. A., 1980. Anisotropic Superfluidity in Neutron Stars and Strong Coupling Effect in Superfluid 3He (Ph.D. thesis). SUNY Stony Brook. Sauls JA and Serene JW (1978) 3P2 Pairing near the transition temperature in neutron-star matter. Physical Review D 17(6): 1524–1528. https://doi.org/10.1103/ PhysRevD.17.1524. Sauls JA, Stein DL, and Serene JW (1982) Magnetic vortices in a rotating 3P2 neutron superfluid. Physical Review D 25(4): 967–975. https://doi.org/10.1103/PhysRevD.25.967. Schnyder AP, Ryu S, Furusaki A, and Ludwig AWW (2008) Classification of topological insulators and superconductors in three spatial dimensions. Physical Review B 78(19): 195125. https://doi.org/10.1103/PhysRevB.78.195125. Schulenborg J and Flensberg K (2020) Absence of supercurrent sign reversal in a topological junction with a quantum dot. Physical Review B 101(1): 014512. https://doi.org/ 10.1103/PhysRevB.101.014512. Schwarz AS (1982) Field theories with no local conservation of the electric charge. Nuclear Physics B 208: 141–158. https://doi.org/10.1016/0550-3213(82)90190-0. Sedrakian A and Clark JW (2018) Superfluidity in nuclear systems and neutron stars. arXiv:1802.00017. Semenoff GW and Zhou F (2007) Discrete symmetries and 1/3-quantum vortices in condensates of F¼2 cold atoms. Physical Review Letters 98: 100401. https://doi.org/10.1103/ PhysRevLett.98.100401. Shifman M and Yung A (2007) Supersymmetric solitons and how they help us understand non-Abelian gauge theories. Reviews of Modern Physics 79: 1139. https://doi.org/10.1103/ RevModPhys.79.1139. hep-th/0703267. Shiozaki K and Sato M (2014) Topology of crystalline insulators and superconductors. Physical Review B 90(16): 165114. https://doi.org/10.1103/PhysRevB.90.165114. Silaev MA and Volovik GE (2014) Andreev-Majorana bound states in superfluids. Journal of Experimental and Theoretical Physics ISSN: 1063-7761. 119(6): 1042–1057. https://doi. org/10.1134/S1063776114120097. Song JL, Semenoff GW, and Zhou F (2007) Quantum fluctuation-induced uniaxial and biaxial spin nematics. Physical Review Letters 98: 160408. https://doi.org/10.1103/ PhysRevLett.98.160408. cond-mat/0702052. Sugeta M, Mizushima T, and Fujimoto S (2022) Enhanced 2p-periodic Aharonov-Bohm effect as a signature of Majorana bound states probed by nonlocal measurements. https://doi. org/10.48550/ARXIV.2205.07506. Takatsuka T (1972) Superfluid state in neutron star matter. III tensor coupling effect in 3P2 energy gap. Progress in Theoretical Physics 47(3): 1062–1064. https://doi.org/10.1143/ PTP.47.1062. Takatsuka T and Tamagaki R (1971) Superfluid state in neutron star matter. II properties of anisotropic energy gap of 3P2 pairing. Progress in Theoretical Physics 46(1): 114–134. https://doi.org/10.1143/PTP.46.114. Tamagaki R (1970) Superfluid state in neutron star matter. I generalized Bogoliubov transformation and existence of 3P2 gap at high density. Progress in Theoretical Physics 44(4): 905–928. https://doi.org/10.1143/PTP.44.905. Tanaka Y and Kashiwaya S (1995) Theory of tunneling spectroscopy of d-wave superconductors. Physical Review Letters 74(17): 3451–3454. https://doi.org/10.1103/ PhysRevLett.74.3451. Tanaka Y, Nazarov YV, Golubov AA, and Kashiwaya S (2004) Theory of charge transport in diffusive normal metal/unconventional singlet superconductor contacts. Physical Review B 69(14): 144519. https://doi.org/10.1103/PhysRevB.69.144519. Tanaka Y, Kashiwaya S, and Yokoyama T (2005) Theory of enhanced proximity effect by midgap Andreev resonant state in diffusive normal-metal/triplet superconductor junctions. Physical Review B 71(9): 094513. https://doi.org/10.1103/PhysRevB.71.094513. Tanaka Y, Sanno T, Mizushima T, and Fujimoto S (2022) Manipulation of Majorana-Kramers qubit and its tolerance in time-reversal invariant topological superconductor. Physical Review B 106(1): 014522. https://doi.org/10.1103/PhysRevB.106.014522. Teo JCY and Kane CL (2010a) Majorana fermions and non-Abelian statistics in three dimensions. Physical Review Letters 104(4): 046401. https://doi.org/10.1103/ PhysRevLett.104.046401. Teo JCY and Kane CL (2010b) Topological defects and gapless modes in insulators and superconductors. Physical Review B 82(11): 115120. https://doi.org/10.1103/ PhysRevB.82.115120. Terashima H and Ueda M (2005) Nonunitary quantum circuit. International Journal of Quantum Information 3(4): 633. https://doi.org/10.1142/S0219749905001456. Thamm M and Rosenow B (2021) Transmission amplitude through a Coulomb blockaded Majorana wire. Physical Review Research 3(2): 023221. https://doi.org/10.1103/ PhysRevResearch.3.023221. Thuneberg EV (1986) Identification of vortices in superfluid 3HeB. Physical Review Letters 56(4): 359–362. https://doi.org/10.1103/PhysRevLett.56.359. Tojo S, Hayashi T, Tanabe T, Hirano T, Kawaguchi Y, Saito H, and Ueda M (2009) Spin-dependent inelastic collisions in spin-2 Bose-Einstein condensates. Physical Review A 80(4): 042704. https://doi.org/10.1103/PhysRevA.80.042704. Trebst S, Troyer M, Wang Z, and Ludwig AWW (2008) A short introduction to Fibonacci anyon models. Progress of Theoretical Physics Supplement ISSN: 0375-9687. 176: 384–407. https://doi.org/10.1143/PTPS.176.384. Tsutsumi Y, Ishikawa M, Kawakami T, Mizushima T, Sato M, Ichioka M, and Machida K (2013) UPt3 as a topological crystalline superconductor. Journal of the Physical Society of Japan 82(11): 113707. https://doi.org/10.7566/JPSJ.82.113707. Tsutsumi Y, Kawakami T, Shiozaki K, Sato M, and Machida K (2015) Symmetry-protected vortex bound state in superfluid 3He−B phase. Physical Review B 91(14): 144504. https:// doi.org/10.1103/PhysRevB.91.144504. Uchino S, Kobayashi M, Nitta M, and Ueda M (2010) Quasi-Nambu-goldstone modes in Bose-Einstein condensates. Physical Review Letters 105: 230406. https://doi.org/10.1103/ PhysRevLett.105.230406. Ueno Y, Yamakage A, Tanaka Y, and Sato M (2013) Symmetry-protected Majorana fermions in topological crystalline superconductors: Theory and application to Sr2RuO4. Physical Review Letters 111(8): 087002. https://doi.org/10.1103/PhysRevLett.111.087002. Usher N, Hoban MJ, and Browne DE (2017) Nonunitary quantum computation in the ground space of local hamiltonians. Physical Review A 96(3): 032321. https://doi.org/10.1103/ PhysRevA.96.032321. Vakaryuk V and Leggett AJ (2009) Spin polarization of half-quantum vortex in systems with equal spin pairing. Physical Review Letters 103(5): 057003. https://doi.org/10.1103/ PhysRevLett.103.057003.

794

Non-Abelian anyons and non-Abelian vortices in topological superconductors

Valentini M, Peñaranda F, Hofmann A, Brauns M, Hauschild R, Krogstrup P, San-Jose P, Prada E, Aguado R, and Katsaros G (2021) Nontopological zero-bias peaks in full-shell nanowires induced by flux-tunable Andreev states. Science 373(6550): 82–88. https://doi.org/10.1126/science.abf1513. Vinkler-Aviv Y and Rosch A (2018) Approximately quantized thermal Hall effect of chiral liquids coupled to phonons. Physical Review X 8(3): 031032. https://doi.org/10.1103/ PhysRevX.8.031032. Vollhardt D and Wölfle P (1990) The Superfluid Phases of Helium 3. London: Taylor & Francis, ISBN: 9780850664126. https://books.google.co.jp/books?id¼t0Rw75gMuwIC. Volovik GE (1992) Exotic Properties of Superfluid 3He. Singapore: World Scientific. Volovik GE (1999) Fermion zero modes on vortices in chiral superconductors. JETP Letters 70: 609. https://doi.org/10.1134/1.568223. Volovik GE (2003) The Universe in a Helium Droplet. Oxford, U.K.: Clarendon. Volovik GE (2010) Topological superfluid 3He-B in magnetic field and Ising variable. JETP Letters ISSN: 0021-3640. 91(4): 201–205. https://doi.org/10.1134/ S0021364010040090. Volovik GE and Mineev VP (1977) Investigation of singularities in superfluid He3 and in liquid crystals by the homotopic topology methods. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 72: 2256 [Sov. Phys. JETP 45, 1186 (1977)]. Volovik GE and Yakovenko VM (1989) Fractional charge, spin and statistics of solitons in superfluid 3He film. Journal of Physics. Condensed Matter 1(31): 5263. https://doi.org/ 10.1088/0953-8984/1/31/025. von Gehlen G (1991) Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field. Journal of Physics A: Mathematical and General 24(22): 5371. https:// doi.org/10.1088/0305-4470/24/22/021. Vorontsov AB and Sauls JA (2003) Thermodynamic properties of thin films of superfluid 3He-A. Physical Review B 68(6): 064508. https://doi.org/10.1103/PhysRevB.68.064508. Wilczek F (1982) Quantum mechanics of fractional spin particles. Physical Review Letters 49: 957–959. https://doi.org/10.1103/PhysRevLett.49.957. Wilczek F and Wu Y-S (1990) Space-time approach to holonomy scattering. Physical Review Letters 65: 13–16. https://doi.org/10.1103/PhysRevLett.65.13. Wölms K, Stern A, and Flensberg K (2014) Local adiabatic mixing of Kramers pairs of Majorana bound states. Physical Review Letters 113(24): 246401. https://doi.org/10.1103/ PhysRevLett.113.246401. Wölms K, Stern A, and Flensberg K (2016) Braiding properties of Majorana Kramers pairs. Physical Review B 93(4): 045417. https://doi.org/10.1103/PhysRevB.93.045417. Wong CLM and Law KT (2012) Majorana Kramers doublets in d x 2 −y 2 -wave superconductors with Rashba spin-orbit coupling. Physical Review B 86(18): 184516. https://doi.org/ 10.1103/PhysRevB.86.184516. Wu Y-S (1984) General theory for quantum statistics in two dimensions. Physical Review Letters 52(24): 2103–2106. https://doi.org/10.1103/PhysRevLett.52.2103. Yamashita M, Izumina K, Matsubara A, Sasaki Y, Ishikawa O, Takagi T, Kubota M, and Mizusaki T (2008) Spin wave and vortex excitations of superfluid 3He-A in parallel-plate geometry. Physical Review Letters 101(2): 025302. https://doi.org/10.1103/PhysRevLett.101.025302. Yamashita M, Gouchi J, Uwatoko Y, Kurita N, and Tanaka H (2020) Sample dependence of half-integer quantized thermal Hall effect in the Kitaev spin-liquid candidate a−RuCl3. Physical Review B 102(22): 220404. https://doi.org/10.1103/PhysRevB.102.220404. Yasui S, Itakura K, and Nitta M (2010) Fermion structure of non-Abelian vortices in high density QCD. Physical Review D 81: 105003. https://doi.org/10.1103/PhysRevD.81.105003. 1001.3730. Yasui S, Itakura K, and Nitta M (2011) Majorana meets coxeter: Non-Abelian Majorana fermions and non-Abelian statistics. Physical Review B 83: 134518. https://doi.org/10.1103/ PhysRevB.83.134518. Yasui S, Itakura K, and Nitta M (2012) Dirac returns: Non-abelian statistics of vortices with dirac fermions. Nuclear Physics B 859: 261. https://doi.org/10.1016/j. nuclphysb.2012.02.007. Yasui S, Hirono Y, Itakura K, and Nitta M (2013) Non-Abelian statistics of vortices with non-Abelian dirac fermions. Physical Review E 87(5): 052142. https://doi.org/10.1103/ PhysRevE.87.052142. 1204.1164. Yasui S, Chatterjee C, Kobayashi M, and Nitta M (2019a) Reexamining Ginzburg-Landau theory for neutron 3P2 superfluidity in neutron stars. Physical Review C 100(2): 025204. https://doi.org/10.1103/PhysRevC.100.025204. Yasui S, Chatterjee C, and Nitta M (2019b) Phase structure of neutron 3P2 superfluids in strong magnetic fields in neutron stars. Physical Review C 99(3): 035213. https://doi.org/ 10.1103/PhysRevC.99.035213. Yavilberg K, Ginossar E, and Grosfeld E (2019) Differentiating Majorana from Andreev bound states in a superconducting circuit. Physical Review B 100(24): 241408. https://doi.org/ 10.1103/PhysRevB.100.241408. Ye M, Halász GB, Savary L, and Balents L (2018) Quantization of the thermal hall conductivity at small Hall angles. Physical Review Letters 121(14): 147201. https://doi.org/10.1103/ PhysRevLett.121.147201. Yokoi T, Ma S, Kasahara Y, Kasahara S, Shibauchi T, Kurita N, Tanaka H, Nasu J, Motome Y, Hickey C, Trebst S, and Matsuda Y (2021) Half-integer quantized anomalous thermal Hall effect in the Kitaev material candidate a-RuCl3. Science 373(6554): 568–572. https://doi.org/10.1126/science.aay5551. Yu P, Chen J, Gomanko M, Badawy G, Bakkers EPAM, Zuo K, Mourik V, and Frolov SM (2021) Non-Majorana states yield nearly quantized conductance in proximatized nanowires. Nature Physics 17: 482. https://doi.org/10.1038/s41567-020-01107-w. Zhang G and Spånslätt C (2020) Distinguishing between topological and quasi Majorana zero modes with a dissipative resonant level. Physical Review B 102(4): 045111. https://doi. org/10.1103/PhysRevB.102.045111. Zhang F, Kane CL, and Mele EJ (2013) Time-reversal-invariant topological superconductivity and Majorana Kramers pairs. Physical Review Letters 111(5): 056402. https://doi.org/ 10.1103/PhysRevLett.111.056402. Zhang P, Yaji K, Hashimoto T, Ota Y, Kondo T, Okazaki K, Wang Z, Wen J, Gu GD, Ding H, and Shin S (2018) Observation of topological superconductivity on the surface of an ironbased superconductor. Science 360(6385): 182. https://doi.org/10.1126/science.aan4596. Zhang H, de Moor MWA, Bommer JDS, Xu D, Wang G, van Loo N, Liu C-X, Gazibegovic S, Logan JA, Car D, het Veld RLMO, van Veldhoven PJ, Koelling S, Verheijen MA, Pendharkar M, Pennachio DJ, Shojaei B, Lee JS, Palmstrøm CJ, Bakkers EPAM, Das Sarma S, and Kouwenhoven LP (2021) Large zero-bias peaks in InSb-Al hybrid semiconductorsuperconductor nanowire devices. arXiv:2101.11456. Zheng C (2021) Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm. Scientific Reports 2021: 3960. https://doi.org/10.1038/s41598021-83521-5. Zhu S, Kong L, Cao L, Chen H, Papaj M, Du S, Xing Y, Liu W, Wang D, Shen C, Yang F, Schneeloch J, Zhong R, Gu G, Fu L, Zhang Y-Y, Ding H, and Gao H-J (2020) Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor. Science 367(6474): 189. https://doi.org/10.1126/science.aax0274.

Topological superconductors Shinsei Ryu, Department of Physics, Princeton University, Princeton, NJ, United States © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview One-dimensional topological superconductors Two-dimensional topological superconductors Three-dimensional topological superconductors Key issues Periodic table of topological insulators and superconductors Bulk-boundary correspondence and the effect of disorder The effect of interactions Conclusion Acknowledgments References

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Abstract This article provides a concise description of topological superconductors. After reviewing representative examples of fully gapped topological superconductors in various dimensions, we discuss the basic concepts including the symmetry classification scheme of generic fermionic systems and topological classifications. The bulk-boundary correspondence, one of the hallmarks of topological superconductors, is also discussed, addressing disorder and interaction effects.

Key points

• • •

Topological superconductors are quantum phases of matter that cannot continuously be deformable to trivial states of matter, such as a collection of atoms without any interatomic interaction. Topological superconductors host exotic Andreev bound states at their boundaries and vortex cores. They are stable against perturbations and disorder. In some cases, the boundary states are zero-energy Majorana modes. The close-knit relation between the bulk topological properties and protected boundary states is called the bulk-boundary correspondence. Examples of topological superconductors include one-dimensional topological superconductors (the Kitaev chain), twodimensional chiral p-wave superconductors, and the B phase of 3He in three dimensions. Topological superconductors in different dimensions and symmetry classes can be summarized in the periodic table of topological insulators and superconductors.

Introduction The concept of topology has prevailed in condensed matter physics to date. The quantum Hall effect (both integer and fractional ones) is one of the first examples that made us realize the pivotal role played by topology in many-body quantum physics (Prange and Girvin, 1990). Since then, the topological concepts in condensed matter physics have been developed in great depth and breadth. The list of topological phenomena and topological systems has also been expanding vastly. In particular, the discovery of topological insulators fueled the development of topological condensed matter physics (Hasan and Kane, 2010; Qi and Zhang, 2011; Bernevig and Hughes, 2013). Topology is a branch of mathematics that deals with properties that do not change under a continuous deformation of objects (which can be large as long as tearing and gluing are avoided). Likewise, quantum phenomena that have a topological origin or that occur within topological phases of matter can be robust against perturbations such as disorder and interactions. Furthermore, topological phenomena in many-body quantum systems are of purely quantum mechanical origin and go beyond what we expect from classical physics. Topological concepts can be applied to superconductors (and superfluids). In fact, at the mean-field level, topological superconductors can be thought of as a close cousin of (integer) quantum Hall states and topological band insulators. Within the mean-field theory, superconductors are described in terms of (almost noninteracting) fermionic Bogoliubov quasiparticles and bosonic Cooper pairs. From the point of view of quasiparticles, Cooper pairs can be viewed as a nondynamical background. We can

Encyclopedia of Condensed Matter Physics, Second Edition

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then discuss the band structure of quasiparticles using the Bogoliubov-de Gennes (BdG) Hamiltonian. In the quantum Hall effect and topological insulators, single-particle (Bloch) wave functions below the energy gap have certain topological properties that are characterized by some topological invariants, such as the Chern number (quantized Hall conductivity). Similarly, in topological superconductors, wave functions of fermionic quasiparticles have topological characters. Furthermore, like their nonsuperconducting cousins, topological superconductors support gapless modes at their boundaries. These are Andreev bound states that appear within the superconducting gap. For certain types of topological superconductors, the Andreev bound states are Majorana fermion modes—one of the reasons why topological superconductors have become the subject of intense research. Majorana fermions are emergent excitations in condensed matter systems and are known for the fact that they are their own antiparticles. While Majorana fermions have appeared in various branches of physics, such as Neutrino physics, the recent surge of interest came from their promise for robust quantum computation. Due to their topological origin, Majorana fermions can be used as stable qubits (Nayak et al., 2008; Alicea, 2012; Beenakker, 2013).

Overview Some of the defining properties of gapped topological (ground) states are: (i) Topological states are not adiabatically connected to trivial states. Here, by trivial states, we mean states that can be written as a product state, which does not have spatial correlations and entanglement, such as a collection of atoms without any interatomic interactions. In other words, in the phase diagram, topological phases are necessarily separated from trivial phases by quantum phase transitions. (ii) Topological states are characterized by quantized topological invariants. These topological invariants are often related to quantized responses of topological systems to an external background. For example, for particle number-conserving systems, an external electromagnetic field can be used to detect, for example, the quantized Hall conductivity. It can also take the form of an unoriented spacetime topology that can detect topological phases protected by symmetry that reverses the orientation of spacetime, such as time-reversal symmetry (TRS). Quantized topological invariants should be contrasted with local order parameters that can detect and characterize more conventional phases of matter with (spontaneous) symmetry-breaking. (iii) In the presence of a spatial boundary, d-dimensional topological systems necessarily support excitations localized at the boundary. The appearance of these (d − 1)-dimensional boundary states is a direct manifestation of the bulk topological properties (topological invariants). These boundary states are said to be anomalous in the sense that they cannot exist on their own as (d − 1)-dimensional isolated systems, that is, they cannot be realized as (d − 1)-dimensional isolated lattice systems. This close-knit relation between bulk topological properties and anomalous boundary states is called the bulk-boundary correspondence. As alluded to in (ii), all these can be discussed in the presence of proper symmetry, such as TRS. For example, the adiabatic (dis) continuity of ground states can be discussed in the presence of symmetry. Because of the symmetry, states that are otherwise connected to trivial states can be made nontrivial. We say that these states are symmetry-protected (from being trivial). We note that while we focused above on fully gapped topological phases, gapless topological states are also possible, and need to be discussed separately. Also, there is the concept of topologically ordered states (phases), examples of which include fractional quantum Hall states (Wen, 2007). While the above descriptions apply to topologically ordered phases, they require further characterizations such as topological ground-state degeneracy and exotic anyonic excitations. The above descriptions (i–iii) apply to a broad class of topological systems and topological superconductors are no exceptions. To illustrate the concept of topological superconductors, we will discuss three representative examples: one-dimensional topological superconductivity realized in the so-called Kitaev chain, two-dimensional chiral p-wave superconductors, and the threedimensional topological superconductivity (superfluidity) realized in the B-phase of 3He. (In the following, we will work with the natural unit ℏ ¼ c ¼ 1 unless otherwise stated.)

One-dimensional topological superconductors First, let us discuss a one-dimensional topological superconductor using the Kitaev chain (Kitaev, 2001). The Kitaev chain is defined by the Hamiltonian   X { X { ^ ¼t ^ci ^ci+1 + ^c{i+1 ^ci  m ^ci ^ci  1=2 H 2 i i (1)   1X ⁎ { { D ^ci ^ci+1  D^ci ^ci+1 , + 2 i

Topological superconductors

797

where ^c{i (^ci ) is a fermion creation (annihilation) operator at site i on the one-dimensional lattice, and t, D, and m are parameters representing hopping amplitude, pairing potential, and chemical potential, respectively. Note that we here consider spinless fermions. This type of BdG Hamiltonian may arise in proximity-induced superconductivity in a quantum wire with strong spin–orbit coupling, and ferromagnetic atomic chains on a superconductor (Lutchyn et al., 2010; Oreg et al., 2010; Nadj-Perge et al., 2014). For simplicity, we set t ¼ D 2  in the following. The Kitaev chain in momentum space is written as " # i ^ck Xh { ^ ¼1 ^ck , ^c−k HðkÞ { , H 2 (2) ^c k −k

where HðkÞ ¼ ðt cos k − mÞt3 − t sin k t2 , where k 2 [−p, p) is the one-dimensional wave number, and t1,2,3 are the Pauli matrices acting on the particle-hole space. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From the quasiparticle band structure, e ðkÞ ¼  ðtcos k − mÞ2 + t 2 sin 2 k, we see that there are two gapped phases realized in this model, for m > t and m < t. At m ¼ t, the gap in the quasiparticle band structure closes (quantum phase transition). These two gapped phases are topologically distinct. Namely, there is no adiabatic path connecting these two phases. This can be checked explicitly for the above Hamiltonian. No matter how we change t, D, and m; there is no way to connect these two phases without going through a quantum phase transition. This turns out to be true even when we consider a more generic class of deformations of the Hamiltonian. This can be argued at least in two ways. First, we can construct a (bulk) quantized topological invariant. It is given by a Wilson loop of the Berry connection for the lower quasiparticle band, I p  (3) W ¼ exp i dk A− ðkÞ : −p

−

Here, A (k) is the Berry connection constructed from the quasiparticle wave function for the lower band, HðkÞju− ðkÞi ¼ e− ðkÞju− ðkÞi, as A− ðkÞ ¼ ihu− ðkÞj ∂k∂ ju − ðkÞi: It can be shown that the topological invariant is quantized and takes only two possible values, W ¼ +1 and W ¼ −1. The quantization follows from the condition t1 Hð−kÞT t1 ¼ − HðkÞ,

(4)

that is satisfied for generic BdG Hamiltonians. For the above model, W ¼ +1 for m > t while W ¼ −1 for m < t. The above formula for the topological invariant can properly be generalized when there are more than one quasiparticle band for negative quasiparticle energy. Also, it can be written in different forms, for example, by using the Pfaffian of the BdG Hamiltonian (Kitaev, 2001; Chiu et al., 2016). The topological invariant can also be formulated in terms of the fermion number parity of the many-body ground state |GSi; ^

when periodic boundary condition is imposed, |GSi is fermion number parity odd/even, hGSjð−1ÞN jGSi ¼ 1 when we are in the P { ^ ^^ topological/trivial superconducting phase, respectively. Here, ð−1ÞN ¼ ð−1Þ i ci ci is the total fermion number parity operator. This is the quantized response of one-dimensional topological superconductors to the twisted boundary condition. Second, in the presence of boundaries (open boundary condition), the topological superconducting phase supports Majorana fermion modes localized at the boundaries. These states appear within the bulk quasiparticle energy gap, and are exactly at zero energy (in the limit where the ratio between the bulk correlation length and the system size goes to zero). To see this, let us introduce Majorana fermion operators as ^ lj :¼ ^c{j + ^cj ,

l^0j :¼ ð^cj  ^c{j Þ=i,

(5)

which correspond to the “real” and “imaginary” parts of the complex fermion operators ^cj and ^c{j . They satisfy fl^j , l^k g ¼ l j Þ2 ¼ ð^ l 0 Þ2 ¼ 1. In the Majorana basis, fl^0 , l^k g ¼ 2djk , fl^j , l^0 g ¼ 0: In particular, ð^ j

k

j

^ ¼ − it H 2

X j

im l^0j l^j+1 − 2

X l^ l^0 : j

(6)

j j

P ^ ¼ −it l^0 l^ . If we consider the Kitaev chain with L sites and with open Deep inside the topological superconductor phase, m ¼ 0, H j j j+1 2 ^ does not include Majorana operators l^1 ¼ ^c + ^c{1 and l^0 ¼ ð− iÞð^c − ^c{L Þ . Hence, trivially, we have boundary conditions, H L 1 L ^ l^1  ¼ ½H, ^ l^0  ¼ 0 . This means that the modes l^1 and l^0 correspond to zero-energy eigenstates of the single-particle BdG ½H, L

L

Hamiltonian localized at j ¼ 1 and j ¼ L, respectively. They exist within the bulk quasiparticle energy gap. Due to these zero-energy modes, there is a twofold degeneracy for the many-body ground states. They are characterized by the fermion number { { of the fermion operators, f^ :¼ ðl^1 + il^0 Þ=2 and f^ :¼ ðl^1 − il^0 Þ=2, f^ f^ −1=2 ¼ −il^1 l^0 =2 ¼ 1=2. L

L

L

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Two-dimensional topological superconductors As a second example, let us consider a two-dimensional spinless chiral p-wave superconductor (Read and Green, 2000). The BdG Hamiltonian is given by   Z h i ^c ^¼1 H d2 k ^c{k , ^c −k HðkÞ k{ , ^c − k 2   x Dk where HðkÞ ¼ k , (7) D k − xk k2 xk ¼ − m, Dk ¼ jDjðkx − iky Þ: 2m Here, k ¼ (kx, ky) is two-dimensional momentum, and m, m, |D|, representing the mass of fermions (electrons), chemical potential, and the amplitude of pairing potential, respectively, are real parameters. As we can see from the quasiparticle band structure, there are two phases in this model for m > 0 and m < 0 (the so-called weak and strong pairing phases). These two phases are topologically distinct. As before, this can be argued in two different ways. First, we can define a bulk topological invariant. To this end, we define a ^ three-dimensional vector R(k) as HðkÞ ¼ RðkÞt: We can further normalize R, RðkÞ ¼ RðkÞ=jRðkÞj. Deep inside the strong pairing ^ phase, m  1, the vector RðkÞ is localized around the R^z axis. On the other hand, in the weak pairing phase, the trajectory or the ^ ^ surface swept by the vector RðkÞ is more nontrivial; as k changes in the two-dimensional momentum space, RðkÞ wraps the unit sphere once. The topological invariant (the Chern number) is then defined by Z 2  d k ^ ^ R ∂x R ∂y R^ , Ch ¼ (8) 4p where ∂i ¼ ∂ki . It can be shown that the Chern number is quantized to be integral values as far as there is a gap in the quasiparticle band structure. From the direct calculation, Ch ¼ 1 for m > 0 and Ch ¼ 0 for m < 0. These values reflect the nontrivial/trivial behavior of the R^ vector mentioned before. For generic quasiparticle band structures with more than two quasiparticle bands, we can use the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula to define and calculate the topological invariant, Z X 1 d2 k Ch ¼ eij ∂i Aaj ðkÞ, (9) 2p a where the sum runs over all quasiparticle bands below zero energy and Aai ðkÞ is the Berry connection for the eigenfunction |ua(k)i of the a-th quasiparticle band, Aai ðkÞ ¼ ihua ðkÞj∂i jua ðkÞi . The nontrivial topological character is also reflected in the many-body ground-state wave function. In the weak pairing phase, the ground-state wave function (projected to a sector with a fixed number of particles) is given by the Pfaffian, Pf (zi−zj)−1, where zi ¼ xi + iyi is the complexified real-space coordinate of the i-th fermion. The wave function decays algebraically for the distance between fermions. On the other hand, in the strong pairing phase, the many-body ground-state wave function is short-ranged. Second, in the topologically nontrivial phase, the system supports a chiral edge state. At low energies, the dispersion of the edge state is linear—it is a chiral Majorana (Majorana-Weyl) fermion. At finite temperatures (T ), the chiral edge state carries nonzero thermal current, I ¼ Ch 2

p2 k2B 6h

T 2 . This leads to the thermal Hall conductivity kxy ¼

2 Ch p2 kB T, 2 3h

(10)

where we restored the Planck constant h and the Boltzmann constant kB . This quantization is an analog of the quantized electrical Hall conductivity in the quantum Hall effect. The universal coefficient Ch/2 is called chiral central charge. The chiral central charge is an integer for noninteracting fermion systems (e.g., integer quantum Hall states). The fractional value of the central charge is indicative of electron fractionalization and nontrivial topological order. Similarly, a midgap zero-energy state appears in the presence of a superconducting vortex. Similar to the end state of the one-dimensional topological superconductor discussed above, this zero-energy state is a Majorana state.

Three-dimensional topological superconductors Besides the above examples, there are other kinds of topological superconductors. First, topological superconductors may exist in the presence of symmetries. In the above examples, the only relevant symmetry is the conservation of the fermion number parity. (Here, the fermion number parity conservation simply means Hamiltonians must be bosonic operators. Quite often in the literature, this is not phrased or regarded as symmetry.) We can have other symmetries such as time reversal, spin rotation, and crystallographic symmetries. In some cases, some symmetry is necessary to have a topological distinction between different quantum ground states—they may belong to a symmetry-protected phase. Second, topological superconductors are possible other than one and two dimensions discussed above. For example, topological superconductors protected by TRS can exist in

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three spatial dimensions. The B phase of 3He is an example (the so-called Balian-Werthamer (BW) state). It is described by the BdG Hamiltonian 3 2 ^ck" 7 6 6 ^ck# 7 Z h i 7 6 1 7 ^¼ d3 k ^c{k" , ^c{k# , ^c−k" , ^c−k# HðkÞ6 H 6 ^c{ 7, 2 6 −k" 7 5 4 ^c{−k# (11) " # xk Dk where HðkÞ ¼ , D{k − xk xk ¼

k2 − m, 2m

Dk ¼ jDjðisy Þk  s,

where the Pauli matrices s1,2,3 act on spin degrees of freedom. In addition to the particle-hole condition (4), the BdG Hamiltonian also satisfies s2 Hð− kÞ s2 ¼ HðkÞ, that is, TRS. Combined together, we see that the BdG Hamiltonian (11) anticommutes with G  s2 t1, GHðkÞG ¼ − HðkÞ —we say HðkÞ has chiral symmetry (CS). This implies, in the basis where G is diagonal, the Hamiltonian takes a block off-diagonal form,

0 DðkÞ HðkÞ ¼ : (12) D{ ðkÞ 0 We note that CS pairs up (nonzero) eigenvalues and eigenvectors of HðkÞ : It is straightforward to see that when wa(k) is an eigenvector with eigenvalue ea(k), HðkÞwa ðkÞ ¼ ea ðkÞwa ðkÞ , Gwa(k) is another eigenvector with eigenvalue −ea(k). CS and the off-diagonal structure are also useful to introduce the topological invariant of the three-dimensional, time-reversal symmetric topological superconductor (superfluid). We first note that HðkÞ2 ¼ diag ðDD{ , D{ DÞ, and hence the eigenvalues of HðkÞ,  ea(k), are obtained by solving DD{ ua ¼ e2a ua , D{ Dva ¼ e2a va : (13) P We then introduce a k-dependent matrix, qðkÞ :¼ a ua ðkÞv{a ðkÞ (where ua(k) and v{a ðkÞ are column and row vectors, respectively). The topological invariant is then given by (Schnyder et al., 2008) Z 3 d k ijk  1 n¼ e Tr ðq ∂i qÞðq1 ∂j qÞðq1 ∂k qÞ 2 , (14) 24p2 i, j, k ¼ x, y, z, ∂i ¼ ∂ki : For m > 0, the topological invariant is n ¼ 1 while for m < 0, it is zero.

Key issues Periodic table of topological insulators and superconductors A systematic way to discuss topological superconductors (and topological insulators) in different dimensions and with different symmetries is to utilize the Altland–Zirnbauer symmetry classification (Altland and Zirnbauer, 1997). In this classification, quadratic fermionic Hamiltonians (including the BdG Hamiltonians) are classified in terms of the presence/absence of timereversal and particle-hole symmetries (spin rotation symmetry also plays a role). The Altland–Zirnbauer classification concerns BdG Hamiltonians given in the following form: " # h D H¼ , − D − hT (15) D ¼ − DT :

h{ ¼ h,

Here, h is a normal part (e.g., lattice tight-binding Hamiltonian), and D represents superconducting order parameters. For singleparticle (BdG) Hamiltonians, we can consider three discrete symmetries, time-reversal, particle-hole, and chiral symmetries. (When the total Hamiltonian H is invariant under unitary symmetry (e.g., spin-rotation symmetry) and block-diagonalizable, we focus on each irreducible block of H.) TRS is defined by U T H U {T ¼ H, 

(16) y

where UT is a unitary matrix. For example, using the y component of the spin operator S , UT is given by U T ¼ expðipS Þ. Note that, when the time-evolution of a state c is given by i∂cðtÞ=∂t ¼ HcðtÞ , c0 ¼ UTc obeys the time-reversed Schrödinger equation, − i∂c0 ðtÞ=∂t ¼ Hc0 ðtÞ, if H satisfies (16). Next, particle-hole symmetry (PHS) is defined by y

800

Topological superconductors U C HT U {C ¼ −H,

(17) 0



where UC is a unitary matrix. We note that if we have an eigenstate c with eigenvalue e, c ¼ UCc is also an eigenstate with energy − e. When H is invariant under both time-reversal and particle-hole, there is a unitary matrix that anticommutes with H, H G ¼ −G H,

G ¼ U T U {C :

(18)

(See the previous example presented around (12).) When such a unitary matrix exists, we say H is chiral symmetric. It is however possible that a Hamiltonian is chiral symmetric without having time-reversal and particle-hole symmetries. Furthermore, time-reversal and particle-hole symmetries can be typified into two classes. As well known, time reversal for integer spin particles squares to +1 while that for half-odd integer particles squares to −1. We can make a similar distinction for particle-hole transformations—those that square to +1 and −1. To summarize, by considering the presence/absence of time-reversal and particlehole symmetries, and when they exist, their types, we have 3 3 ¼ 9 possible cases. Adding the case where we have neither timereversal nor PHS, we have 10 different cases in total. These 10 symmetry classes were identified by Altland and Zirnbauer around 1996 in the context of disordered BdG Hamiltonians and random matrix theory. The 10 symmetry classes are summarized in Table 1. In the leftmost column of the table, A, AI, etc., are nomenclature for the symmetry classes. This is based on the fact that the symmetry classes of Hamiltonians are in one-to-one correspondence with the classification of symmetric spaces by Cartan. The ensemble of generic BdG Hamiltonians satisfying (4) is called symmetry class D. By imposing various symmetries, BdG Hamiltonians can realize five other symmetry classes: DIII, A, AIII, C, and CI. First, imposing TRS leads to symmetry class DIII. If we impose the conservation of Sz, we realize symmetry class A, and, with further requiring TRS, symmetry class AIII. On the other hand, if we impose SU(2) spin rotation symmetry, we realize symmetry class C (without TRS) and CI (with TRS). For the 10 Altland–Zirnbauer symmetry classes, and a given spatial dimension d, Table 1 summarizes the presence/absence of topological insulators and superconductors, and when they exist, the type of topological invariants (Schnyder et al., 2008; Ryu et al., 2010; Kitaev, 2009). Here, ∅ means that there are no topological insulators and superconductors, while , 2 , and 2 mean topological invariants are integer-valued, 2 valued, and even integer-valued, respectively. The Kitaev chain and chiral p-wave examples both belong to class D, in d ¼ 1 and d ¼ 2, respectively. The BW state is a topological superconductor (superfluid) in symmetry class DIII in d ¼ 3 dimensions.

Bulk-boundary correspondence and the effect of disorder In all the three examples we have discussed, the topological superconductors in d ¼ 1, 2, 3 spatial dimensions, gapless states (zero-energy Majorana mode for the case of 1d topological superconductor) appear when the system is terminated by a boundary. These states localize near the boundary and are stable against gap-opening by perturbations that respect the symmetries of the system—they are said to be ingappable or ungappable. The presence of these stable boundary states near a boundary is a direct consequence of nontrivial bulk topology. Furthermore, these boundary states are anomalous in the sense that they cannot be realized on their own in an isolated (d − 1)-dimensional lattice system. In this sense, topological bulk and anomalous boundary systems are inseparable. The close Table 1

Classification of topological insulators and superconductors.

class

TRS

PHS

CS

d¼1

d¼2

d¼3

A AIII AI BDI D DIII AII CII C CI

0 0 +1 +1 0 −1 −1 −1 0 +1

0 0 0 +1 +1 +1 0 −1 −1 −1

0 1 0 1 0 1 0 1 0 1

;  ;  2 2 ; 2 ; ;

 ; ; ;  2 2 ; 2 ;

;  ; ; ;  2 2 ; 2

The leftmost column (A, AIII, . . ., CI) represents 10 symmetry classes of single-particle Hamiltonians classified in terms of the presence or absence of time-reversal symmetry (TRS) and particle-hole symmetry (PHS), as well as chiral symmetry (CS). In the table, the absence of symmetries is denoted by 0. The presence of these symmetries is denoted either by +1 or −1, depending on whether the (antiunitary) operator implementing the symmetry at the level of the single-particle Hamiltonian squares to + 1 or − 1. d is the spatial dimension. The symbols 2 ,  and 2 indicate the relevant topological invariant is 2 -valued, integral-valued, or even integral-valued, respectively [; means there is no topological invariant (no topologically-nontrivial state)]. The table is partitioned into two groups, classes A and AIII (“complex classes”) which do not preserve TRS and PHS, and the rest (“real classes”). If we extend the classification table to arbitrary spatial dimension d, the complex (real) classes exhibit two (eight)-fold periodicity in d.

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relation between them is known as bulk-boundary correspondence. Some of the salient features of the bulk-boundary correspondence are as follows. First, as mentioned already, a boundary of a topological bulk system cannot be trivially gapped (while preserving symmetries). For noninteracting systems, this means boundary modes have to be gapless. (In interacting systems and when (d − 1) ¼ 2, boundary systems can be topologically ordered instead of being gapless (Vishwanath and Senthil, 2013).) Also, boundary states avoid the so-called fermion doubling theorem (the Nielsen-Ninomiya theorem and its variants), which claims that, in lattice systems, Weyl points should always appear in pairs with zero net chirality. Finally, when the topological properties of a bulk system are protected by some symmetry, the symmetry is not strictly preserved on its boundary. Even though the boundary system may look symmetric classically (i.e., at the level of Hamiltonian or Lagrangian), the symmetry is violated through quantum mechanical effects. This type of symmetry breaking is called quantum anomaly in quantum field theory. All in all, these features of boundary states are not compatible with isolated (fully-regularized) quantum many-body systems, and they have to be realized on a boundary of a higher-dimensional (topological) system. We have so far assumed that perturbations and deformations in boundary systems are spatially homogeneous. The bulkboundary correspondence however holds even in the presence of disorder—Gapless boundary states on a boundary of topological systems are stable against disorder. That is, the boundary modes do not Anderson localize even in the presence of disorder, as long as symmetry conditions (if any) are preserved or enforced, and as long as disorder does not alter the bulk topological properties. This is in sharp contrast with regular (lattice) systems, where Anderson localization is ubiquitous and inevitable when disorder is strong enough (as shown originally by Anderson (Anderson, 1958)). The lack of Anderson localization indicates the absence of atomic limit, and, once again, that the boundary system cannot be realized as a (d − 1)-dimensional lattice system. See, for example, Nomura et al. (2007) which demonstrates the lack of Anderson localization on the surface of a d ¼ 3-dimensional topological insulator. In Schnyder et al. (2008) and Ryu et al. (2010), the lack of Anderson localization at boundaries was hypothesized to be the defining property of topological insulators and superconductors and used as a guiding principle to search and classify topological insulators and superconductors.

The effect of interactions All of the aforementioned examples can be understood in terms of noninteracting or mean-field Hamiltonians. With interactions, some (noninteracting) topological superconductors are stable, while others cease to be topological, that is, once interaction effects are included, they are adiabatically connected to trivial states. Interactions can also give rise to new topological phases that do not exist without interactions. For example, the topological superconductor phase in the Kitaev chain and the weak paring phase of the chiral p-wave superconductor turn out to be stable even with interactions. On the other hand, the topological classification in symmetry class BDI in d ¼ 1 dimension is altered in the presence of interactions. The topological classification at the noninteracting level is  (c.f. Table 1). A simple example of topological superconductors in this class is obtained from the Kitaev chain by imposing TRS. This classification, however, in the presence of interactions is known to be modified to 8 (Fidkowski and Kitaev, 2010, 2011). Namely, noninteracting topological superconductors with topological invariant being an integer multiple of 8 can be turned into a topologically trivial state by turning on interactions. We say the integral topological classification collapses or is reduced to 8 . Beyond the above one-dimensional example, a similar reduction of topological classifications has been established in other symmetry classes in various spatial dimensions. For example, the integer classification of three-dimensional time-reversal symmetric topological superconductors is reduced to 16 . This has been argued, following the spirit of the bulk-boundary correspondence, by studying surface states of 3d topological superconductors (Fidkowski et al., 2013; Wang and Senthil, 2014; Metlitski et al., 2014; You and Xu, 2014; Kitaev, 2015; Morimoto et al., 2015). Furthermore, topological quantum field theory approaches have been developed (along with the developments of symmetry-protected topological phases in general) (Kapustin et al., 2015; Witten, 2016; Freed and Hopkins, 2021).

Conclusion Topological superconductors have become an active field of research, with exciting new discoveries being made constantly. In this article, we mainly focused on fully gapped ones within the BdG (mean-field) formalism. The theory of topological superconductors and related topological systems has been vastly expanding, and there are many topics that we cannot discuss in this article. One notable omission is the topic of gapless topological superconductors. Nodal systems, such as semimetals and nodal superconductors, can exhibit nontrivial band topology, even though the bulk gap closes at certain points in the Brillouin zone. The Fermi surfaces (superconducting nodes) of these gapless materials are topologically protected by topological invariants, which are defined in terms of an integral along a surface enclosing the gapless points. Similar to fully gapped topological systems, the topological characteristics of nodal materials manifest themselves at the surface in terms of gapless boundary modes. Depending on the symmetry properties and the dimensionality of the bulk Fermi surface, these gapless boundary modes form Dirac cones, Fermi arcs, or flat bands. Topological nodal systems can be protected by nonspatial symmetries (i.e., time-reversal or PHS) as well as spatial lattice symmetries, or a combination thereof. Examples of gapless topological superconductors include two-dimensional dx2 − y2 -wave superconductors (Hu, 1994) and the A phase of superfluid 3He (Volovik, 2011, 2003).

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Other topics that we did not have space to discuss include, for instance, topological classification in the presence of crystalline symmetries (topological crystalline superconductors), higher-order topology (Benalcazar et al., 2017), and topological superconductors that can emerge in dynamical settings such as topological Floquet superconductors (Jiang et al., 2011). In terms of material realization, a variety of systems have been discussed in search of topological superconductors and Majorana fermions. Despite the abundance of examples of topological insulators, topological superconductors remain rare, and to conclude decisively the realization of topological superconductors has been more subtle and difficult. This is partly because an unconventional pairing symmetry is required for a topologically nontrivial state. However, zero-energy Majorana bound states have also been explored in heterostructures involving s-wave superconductors. Examples of this type of setup are a superconductor-magnet domain wall along the edge of a two-dimensional quantum spin Hall insulator (Fu and Kane, 2009), or a Chern insulator interface (Qi et al., 2010; Cheng, 2012; Lindner et al., 2012; Vaezi, 2013; Clarke et al., 2013; Barkeshli et al., 2013; Mong et al., 2014). Another example is a flux vortex across a superconducting interface between three-dimensional topological and trivial insulators (Fu and Kane, 2008). In these examples, s-wave superconductors are not topological. Nevertheless, these systems still support zero-energy Majorana bound states, and nontrivial topological invariants can be defined for the heterostructures (defects) themselves (Teo and Kane, 2010). In an even broader context, beyond the regular superconductors, the physics of topological superconductors and Majorana fermions are relevant to n ¼ 5/2 fractional quantum Hall states (paired states of composite fermions) (Moore and Read, 1991; Greiter et al., 1992), and the Kitaev spin liquid (Kitaev, 2006).

Acknowledgments SR is supported by a Simons Investigator Grant from the Simons Foundation (Award No. 566116).

References Alicea J (2012) New directions in the pursuit of Majorana fermions in solid state systems. Reports on Progress in Physics 75(7): 076501. https://doi.org/10.1088/0034-4885/75/ 7/076501. 1202.1293. Altland A and Zirnbauer MR (1997) Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Physical Review B 55(2): 1142–1161. https://doi.org/ 10.1103/PhysRevB.55.1142. cond-mat/9602137. Anderson PW (1958) Absence of diffusion in certain random lattices. Physical Review 109: 1492–1505. https://doi.org/10.1103/PhysRev.109.1492. Barkeshli M, Jian C-M, and Qi X-L (2013) Twist defects and projective non-Abelian braiding statistics. Physical Review B 87(4): 045130. https://doi.org/10.1103/ PhysRevB.87.045130. 1208.4834. Beenakker CWJ (2013) Search for majorana fermions in superconductors. Annual Review of Condensed Matter Physics 4: 113. https://doi.org/10.1146/annurev-conmatphys030212-184337. Benalcazar WA, Bernevig BA, and Hughes TL (2017) Quantized electric multipole insulators. Science 357(6346): 61–66. https://doi.org/10.1126/science.aah6442. 1611.07987. Bernevig BA and Hughes T (2013) Topological Insulators and Topological Superconductors. Princeton University Press. Cheng M (2012) Superconducting proximity effect on the edge of fractional topological insulators. Physical Review B 86(19): 195126. https://doi.org/10.1103/PhysRevB.86.195126. 1204.6084. Chiu C-K, Teo JCY, Schnyder AP, and Ryu S (2016) Classification of topological quantum matter with symmetries. Reviews of Modern Physics 88(3): 035005. https://doi.org/10.1103/ RevModPhys.88.035005. 1505.03535. Clarke DJ, Alicea J, and Shtengel K (2013) Exotic non-Abelian anyons from conventional fractional quantum Hall states. Nature Communications 4: 1348. https://doi.org/10.1038/ ncomms2340. 1204.5479. Fidkowski L and Kitaev A (2010) Effects of interactions on the topological classification of free fermion systems. Physical Review B 81(13): 134509. https://doi.org/10.1103/ PhysRevB.81.134509. 0904.2197. Fidkowski L and Kitaev A (2011) Topological phases of fermions in one dimension. Physical Review B 83(7): 075103. https://doi.org/10.1103/PhysRevB.83.075103. 1008.4138. Fidkowski L, Chen X, and Vishwanath A (2013) Non-Abelian topological order on the surface of a 3D topological superconductor from an exactly solved model. Physical Review X 3(4): 041016. https://doi.org/10.1103/PhysRevX.3.041016. 1305.5851. Freed DS and Hopkins MJ (2021) Reflection positivity and invertible topological phases. Geometry and Topology 25: 1165–1330. https://doi.org/10.2140/gt.2021.25.1165. 1604.06527. Fu L and Kane CL (2008) Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Physical Review Letters 100(9): 096407. https://doi.org/ 10.1103/PhysRevLett.100.096407. 0707.1692. Fu L and Kane CL (2009) Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction. Physical Review B 79(16): 161408. https://doi.org/ 10.1103/PhysRevB.79.161408. 0804.4469. Greiter M, Wen XG, and Wilczek F (1992) On paired Hall states. Nuclear Physics B 374: 567–614. https://doi.org/10.1016/0550-3213(92)90401-V. Hasan MZ and Kane CL (2010) Colloquium: Topological insulators. Reviews of Modern Physics 82: 3045–3067. https://doi.org/10.1103/RevModPhys.82.3045. 1002.3895. Hu C-R (1994) Midgap surface states as a novel signature for d2xa-x2b-wave superconductivity. Physical Review Letters 72: 1526–1529. https://doi.org/10.1103/PhysRevLet-72-1526. Jiang L, Kitagawa T, Alicea J, Akhmerov AR, Pekker D, Refael G, Cirac JI, Demler E, Lukin MD, and Zoller P (2011) Majorana fermions in equilibrium and in driven cold-atom quantum wires. Physical Review Letters 106(22): 220402. https://doi.org/10.1103/PhysRevLett.106.220402. 1102.5367. Kapustin A, Thorngren R, Turzillo A, and Wang Z (2015) Fermionic symmetry protected topological phases and cobordisms. Journal of High Energy Physics 2015: 52. https://doi.org/ 10.1007/JHEP12(2015)052. 1406.7329. Kitaev AY (2001) 6. QUANTUM COMPUTING: Unpaired Majorana fermions in quantum wires. Physics Uspekhi 44(10S): 131. https://doi.org/10.1070/1063-7869/44/10S/S29. cond-mat/0010440. Kitaev A (2006) Anyons in an exactly solved model and beyond. Annals of Physics 321(1): 2–111. https://doi.org/10.1016/j.aop.2005.10.005. cond-mat/0506438. Kitaev A (2009) Periodic table for topological insulators and superconductors. In: Lebedev V and Feigel’Man M (eds.). Advances in Theoretical Physics: Landau Memorial Conference, vol. 1134, American Institute of Physics Conference Series, pp. 22–30. 0901.2686. Kitaev A (2015) Homotopy-theoretic approach to SPT phases in action: Z16 classification of three-dimensional superconductors. http://www.ipam.ucla.edu/abstract/?tid¼12389& pcode¼STQ2015.

Topological superconductors

803

Lindner NH, Berg E, Refael G, and Stern A (2012) Fractionalizing Majorana fermions: Non-Abelian statistics on the edges of Abelian quantum Hall states. Physical Review X 2(4): 041002. https://doi.org/10.1103/PhysRevX.2.041002. 1204.5733. Lutchyn RM, Sau JD, and Das Sarma S (2010) Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Physical Review Letters 105 (7): 077001. https://doi.org/10.1103/PhysRevLett.105.077001. 1002.4033. Metlitski MA, Fidkowski L, Chen X, and Vishwanath A (2014) Interaction effects on 3D topological superconductors: Surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets. arXiv e-prints. arXiv:1406.3032. Mong RSK, Clarke DJ, Alicea J, Lindner NH, Fendley P, Nayak C, Oreg Y, Stern A, Berg E, Shtengel K, and Fisher MPA (2014) Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure. Physical Review X 4(1): 011036. https://doi.org/10.1103/PhysRevX.4.011036. 1307.4403. Moore GW and Read N (1991) Nonabelions in the fractional quantum Hall effect. Nuclear Physics B 360: 362–396. https://doi.org/10.1016/0550-3213(91)90407-O. Morimoto T, Furusaki A, and Mudry C (2015) Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions. Physical Review B 92(12): 125104. https://doi.org/10.1103/PhysRevB.92.125104. 1505.06341. Nadj-Perge S, Drozdov IK, Li J, Chen H, Jeon S, Seo J, MacDonald AH, Bernevig BA, and Yazdani A (2014) Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346(6209): 602–607. https://doi.org/10.1126/science.1259327. 1410.0682. Nayak C, Simon SH, Stern A, Freedman M, and Das Sarma S (2008) Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80: 1083–1159. https:// doi.org/10.1103/RevModPhys.80.1083. Nomura K, Koshino M, and Ryu S (2007) Topological delocalization of two-dimensional massless dirac fermions. Physical Review Letters 99(14): 146806. https://doi.org/10.1103/ PhysRevLett.99.146806. 0705.1607. Oreg Y, Refael G, and von Oppen F (2010) Helical liquids and Majorana bound states in quantum wires. Physical Review Letters 105(17): 177002. https://doi.org/10.1103/ PhysRevLett.105.177002. 1003.1145. Prange RE and Girvin SM (1990) The Quantum Hall effect. Springer-Verlag, ISBN: 9780387971773. Qi X-L and Zhang S-C (2011) Topological insulators and superconductors. Reviews of Modern Physics 83(4): 1057–1110. https://doi.org/10.1103/RevModPhys.83.1057. 1008.2026. Qi X-L, Hughes TL, and Zhang S-C (2010) Chiral topological superconductor from the quantum Hall state. Physical Review B 82(18): 184516. https://doi.org/10.1103/ PhysRevB.82.184516. 1003.5448. Read N and Green D (2000) Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Physical Review B 61(15): 10267–10297. https://doi.org/10.1103/PhysRevB.61.10267. cond-mat/9906453. Ryu S, Schnyder AP, Furusaki A, and Ludwig AWW (2010) Topological insulators and superconductors: Tenfold way and dimensional hierarchy. New Journal of Physics 12(6): 065010. https://doi.org/10.1088/1367-2630/12/6/065010. 0912.2157. Schnyder AP, Ryu S, Furusaki A, and Ludwig AWW (2008) Classification of topological insulators and superconductors in three spatial dimensions. Physical Review B 78(19): 195125. https://doi.org/10.1103/PhysRevB.78.195125. 0803.2786. Teo JCY and Kane CL (2010) Topological defects and gapless modes in insulators and superconductors. Physical Review B 82(11): 115120. https://doi.org/10.1103/ PhysRevB.82.115120. 1006.0690. Vaezi A (2013) Fractional topological superconductor with fractionalized Majorana fermions. Physical Review B 87(3): 035132. https://doi.org/10.1103/PhysRevB.87.035132. 1204.6245. Vishwanath A and Senthil T (2013) Physics of three-dimensional bosonic topological insulators: Surface-deconfined criticality and quantized magnetoelectric effect. Physical Review X 3(1): 011016. https://doi.org/10.1103/PhysRevX.3.011016. 1209.3058. Volovik GE (2003) Universe in a Helium Droplet. Oxford, UK: Oxford University Press, ISBN: 0521670535. Volovik GE (2011) JETP Letters 93: 66. Wang C and Senthil T (2014) Interacting fermionic topological insulators/superconductors in three dimensions. Physical Review B 89(19): 195124. https://doi.org/10.1103/ PhysRevB.89.195124. 1401.1142. Wen XG (2007) Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons. Oxford: Oxford University Press. Witten E (2016) Fermion path integrals and topological phases. Reviews of Modern Physics 88(3): 035001. https://doi.org/10.1103/RevModPhys.88.035001. 1508.04715. You Y-Z and Xu C (2014) Symmetry-protected topological states of interacting fermions and bosons. Physical Review B 90(24): 245120. https://doi.org/10.1103/ PhysRevB.90.245120. 1409.0168.

Superinsulation MC Diamantinia, CA Trugenbergerb, and VM Vinokurc, aNiPS Laboratory, INFN and Department of Physics and Geology, University of Perugia, Perugia, Italy; bSwissScientific Technologies SA, Geneva, Switzerland; cTerra Quantum AG, Terra Quantum AG, St. Gallen, Switzerland © 2024 Elsevier Ltd. All rights reserved.

Introduction Gauge theories of planar superconductors Superconducting films Josephson junction arrays The quantum phases in the vicinity of the SIT Quantum phase diagram at T ¼ 0 Finite temperature phase diagram The nature of the superinsulator Asymptotic freedom and size effects Electric Meissner effect and its manifestation in the I(V ) characteristics Dynamical exponents and the linear potential Oblique superinsulators in 3D The role of disorder Conclusion References

804 806 807 808 808 808 809 809 811 811 813 813 814 815 815

Abstract Superinsulation is a state of matter in which the electrical conductivity vanishes in the finite-temperature range between zero and the superinsulating transition temperature. This state is dual to superconductivity, where the resistance vanishes within a similar finite-temperature interval. The phenomenon of superinsulation arises in certain quantum materials possessing either inherent or self-induced granularity. In these materials the effective electromagnetic action contains magnetic monopoles. When these proliferate, the Coulomb interactions dominate magnetic interactions. In two dimensions (2D), magnetic monopole instantons turn the Coulomb interaction between two charges of the opposite signs into a linear attractive potential causing the same charge confinement that holds quarks together in hadrons and, therefore, leading to an infinite electric resistance even at finite temperatures. In three dimensions (3D), a superinsulator harbors a condensate of magnetic monopoles resulting in a similar charge confinement. The phenomenon of superinsulation has been experimentally discovered and studied in nearly 2D thin superconducting films of materials with a large normal-state dielectric permittivity and in generic granular superconductors.

Key points The article gives an introductory description of the superinsulation, specifically focusing on:

• • • • • • •

The role of magnetic monopoles in materials Showing that planar superconductors are governed by a topological gage field theory Deriving the quantum and thermal phase structure of planar superconductors Illustrating the magnetic–electric duality between the superconducting and superinsulating phases, with zero and infinite electric resistance, respectively Discussing the nature of superinsulators and showing that they realize an Abelian version of the confinement that holds quarks together in hadrons Proposing that an oblique version of superinsulation may be realized in the pseudogap state of high temperature superconductors Discussing the role of disorder

Introduction Superinsulation, a state of matter mirror dual to superconductivity, refers to the complete disappearance of the electrical conductivity in certain materials when they are cooled below a certain critical transition temperature. Superconductivity is a phenomenon whereby an electric current flows through a material, called superconductor, even in the absence of applied voltage hence without resistance. Accordingly, a superinsulator is a material that does not conduct electricity despite applied voltage, hence having infinite resistance.

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Superinsulation emerges because of the electric–magnetic symmetry of laws of nature. To place electric and magnetic fields on equal footing, one has to introduce magnetic charges qg as a dual mirror of electric charges qe. Then electric–magnetic symmetry manifests as a symmetry under simultaneous interchange of electric and magnetic fields, E ! B, B ! −E, and electric and magnetic charges, qe , qg. Magnetic charges are perfectly consistent in a quantum theory of electromagnetism, provided the product of electric and magnetic charges is quantized, as shown in a seminal work by Dirac in the 1930s (Dirac, 1931). Magnetic charges, called magnetic monopoles, appear naturally in the most grand-unified theories (GUTs) of particle interactions (Goddard and Olive, 1978) and are a cornerstone of the superinsulation phenomenon. Despite good 40 years of intensive search, however, they have never been detected in any high-energy experiment. While the GUT monopoles are elusive, other Abelian U(1) monopoles can lead to notable effects. As Polyakov pointed out (Polyakov, 1975), there are two possible kinds of electromagnetism, the noncompact one, with gage group  and the compact one, with the gage group U(1). In classical physics, where only equations of motion matter, there is no real distinction between the two. At the quantum level, however, things become different: it is the compact quantum electrodynamics (QED) that admits magnetic monopoles, see Polyakov (1987). There are two ways of obtaining compact QED: either one considers it as a surviving low-energy theory after spontaneous symmetry breaking of a larger compact group, or one formulates it on a lattice. Either way, compact QED is an effective theory, with an ultraviolet (UV) cutoff. Already in the 1970s it was realized that dual superconductivity and related Abelian magnetic monopoles in the U(1)N−1 subgroup of SU(N) offer a model for quark confinement via the dual Meissner effect that squeezes color-electric fields into thin tubes, strings, linearly binding quarks into hadrons (Mandelstam, 1976; Nambu, 1974; ‘t Hooft, 1978). This mechanism can also take place in other settings. Integrating out the matter degrees of freedom, the nonrelativistic Maxwell action becomes the effective action for insulators. If this effective action is compact, it admits magnetic monopoles, which can lead to confinement, see Greensite (2011) of electric charges by squeezing electric flux into thin tubes as shown in Fig. 1. This is a new, superinsulating phase of insulators. The idea that the electric–magnetic duality, resulting in charge confinement, is realized in condensed matter physics was first proposed in 1996 in the context of Josephson junction arrays (JJAs) (Diamantini et al., 1996) and the ensuing new state of matter was called superinsulator since it is characterized by an infinite electrical resistance (Diamantini et al., 1996). The duality in terms of charges and vortices was subsequently surmised in numerical simulations (Krämer and Doniach, 1998). In 2008, superinsulation was independently proposed as a new state of matter emerging in thin superconducting films (Vinokur et al., 2008; Baturina and Vinokur, 2013) as a manifestation of the uncertainty principle relating amplitude and phase of the superconducting order parameter. Experimentally, superinsulation has been detected in TiN (Baturina et al., 2007; Vinokur et al., 2008) and NbTiN (Mironov et al., 2018) films and in InO (Sambandamurthy et al., 2005) and NbSi (Humbert et al., 2021) films. The superinsulating state, detected again later in InO (Ovadia et al., 2015), was referred there as “finite-temperature insulator” and was associated with charge carriers overheating (Basko et al., 2007; Altshuler et al., 2009). Here, however we would like to dispel the view that overheating is a “description of the nature” of this new state of matter. It is most certainly not. Overheating is a possible model of how this state is destroyed in a dynamical phase transition when a sufficiently strong applied voltage is applied. A model of the new state must explain why the charges become immobile below a transition temperature in the first place. The idea behind overheating is that this is due to many-body localization (MBL) (Altshuler et al., 2006), a model that has been studied in 1D but for which there is no established support in 2D, let alone for the essentially 3D InO films, in which the coherence length is quite smaller than the thickness. In the section below dedicated to the role of disorder, we show how recent experimental data essentially rule out this idea. Superinsulation appears as an inherent feature of the superconductor–insulator transition (SIT) at low temperatures. In superconducting films with a large normal-state dielectric permittivity, the Coulomb interaction becomes logarithmic over the entire system and competes with the magnetic interaction between vortices (Baturina and Vinokur, 2013). When two interactions are

Fig. 1 Confinement of Cooper pairs by a magnetic monopole condensate. Left panel illustrates the usual 3D Coulomb interaction between two charges q1 and q2 of opposite signs. Right panel illustrates that, in full analogy with the charge condensate currents squeezing magnetic field into thin filaments, Abrikosov vortices, the magnetic condensate currents squeeze electric field lines into electric strings with the linear tension s. As a result, the attracting interaction between charges grows linearly with the separation distance r. Reproduced from Baturina TI and Vinokur VM (2013) Superinsulator-superconductor duality in two dimensions. Annalen der Physik 331: 236–257.

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Fig. 2 Sheet resistance as a function of temperature for a TiN film (Diamantini et al., 2019). The dashed straight line in this logarithmic scale is the activated Arrhenius behavior typical of insulators. The deviation at very low temperatures shows the hyperactivated behavior indicating superinsulation. The deviation becomes stronger when a magnetic field is applied. Reproduced from Diamantini MC et al. (2019) The superconductor-superinsulator transition: S-duality and the QCD on the desktop. Journal of Superconductivity and Novel Magnetism 32: 47–51.

comparable, the superconducting condensate breaks down into separate superconducting granules connected via Josephson links. This granular array hosts superconducting islands (Sacépé et al., 2008, 2011; Kowal and Ovadyahu, 1994; Fistul et al., 2008) intertwined with the coreless Josephson vortices (Diamantini et al., 2020a, 2021b). Superinsulation arises due to instability of these vortices upon further increase of the 2D Coulomb interaction. Magnetic monopole instantons (Polyakov, 1987), corresponding to vortex tunneling events, cause the Coulomb interaction between charges to become linear (Diamantini et al., 2018), see Fig. 1 so that only neutral “electric pions” survive at large distances and charge transport is suppressed (Diamantini et al., 2020b). The result is the hyperactivated behavior of the sheet resistance as function of temperature, the exemplary R□ ð1=TÞ dependence being shown in Fig. 2. In the following exposition, we use natural units c ¼ 1, ℏ ¼ 1, e0 ¼ 1.

Gauge theories of planar superconductors The vicinity of the SIT is governed by the interplay of two topological phenomena, mutual statistics Aharonov–Bohm–Casher (ABC) interactions (Aharonov and Bohm, 1959; Aharonov and Casher, 1984) between Cooper pairs and Josephson vortices and their Berezinskii–Kosterlitz–Thouless (BKT) (Berezinskii, 1971; Kosterlitz and Thouless, 1972, 1973) topological phase transitions. In the critical vicinity of the SIT the films acquire a self-induced electronic granular structure, and they have to be viewed as arrays of superconducting condensate droplets with the characteristic size ℓ ’ OðxÞ, where x is the superconducting coherence length, coupled via Josephson links, that is, as a JJA. The vortices in the system are Josephson vortices which, unlike Abrikosov vortices, do not have a normal core. Above the vortex–BKT transition, that is, above the vortex unbinding temperature TVBKT, the Josephson vortices have topological mutual statistics interactions with the out-of-condensate Cooper pairs that emerge there. Moreover, as Josephson vortices do not have normal cores, their dynamics acquires ballistic character (van Otterlo et al., 1994; van der Zant et al., 1992). As a consequence, the description of the SIT and planar superconductivity is achieved by the topological gage theory presented below. En passant we remark that the XY model, adapted to describe 2D superfluids and viewed as an equivalent to the Coulomb gas picture (Minnhagen, 1987), is often applied also to charged superconductors. However, although it captures several experimental features fairly well, it does not account for the mutual statistics topological interactions and ballistic dynamics of Josephson vortices, hence misses topological aspects of 2D superconductivity that are essential near the SIT. The ABC topological interactions are encoded in the phase acquired by the wave function of one of the types of the excitation, say a charge, when it encircles the other, say a vortex, Aharonov–Bohm effect (Aharonov and Bohm, 1959), and vice versa, Aharonov–Casher effect (Aharonov and Casher, 1984). These interactions (Wilczek, 1990) are infrared (IR)dominant since they have an infinite interaction range. In the Euclidean partition function describing point charges and vortices they must be accounted for by the topological Gauss linking number of their trajectories. This topological invariant is nonlocal, but, as noted by Wilczek (1992), can be turned local by coupling the point charge and vortex currents, Qm and Mm to

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two fictitious gage fields am and bm with mixed Chern–Simons term (Jackiw and Templeton, 1981; Deser et al., 1982a, b), resulting in the Euclidean partition function Z P Dam Dbm e − S , Z ¼ fQm , Mm g (1) Z i S ¼ d3 x am eman ∂a bn + iam Qm + ibm Mm : 2p This same model can also be derived from another point of view. Following Wen and Zee (1992) in their derivation of the longdistance effective field theories of incompressible quantum Hall states, one can define topologically conserved charge and vortex currents in terms of the curl of a fictitious pseudovector field bm and a fictitious vector field am, am: 1 m 1 man f ¼ e ∂a bn , 2p 2p 1 m 1 man fm ¼ g ¼ e ∂a an : 2p 2p jm ¼

(2)

The fields am and bm are gauge fields, since physical currents are left invariant under the gauge transformations am ! am + ∂mΥ and bm ! bm + ∂mC, where Υ and C are arbitrary scalar functions. The effective action involving charges and vortices must therefore be a gauge theory. The IR-dominant gauge invariant term which preserves the parity (P) and time-reversal (T ) symmetries is the mixed Chern–Simons term. If the gauge group is U(1), the gauge fields are compact. In a lattice formulation, as appropriate for a granular system of the typical size ℓ, the gauge functions are angular variables defined in the domain [−p, +p]. To enforce this periodicity we must introduce integer-valued link variables Qm and Mm, and we are back to Eq. (1). The Chern–Simons term describes only the topological interactions. To include the dynamics of both charges and vortices, one adds all the possible local gauge invariant terms. This results in the most general (Euclidean) effective action which encodes all the physics of a 2D system of interacting charges and vortices (Diamantini et al., 1996, 2020a) S¼

X ℓ0 ℓ2 X ℓ ℓ2 ℓ ℓ2 e ℓ ℓ2 ℓ ℓ2 e i iℓ0 a0 Q0 + iℓai Qi + iℓ0 b0 M0 + iℓbi Mi , am kmn bn + 0 2 f 20 + 0 2 f 2 + 0 2 g20 + 0 2 g2 + 2p 2e 2e 2e m 2e m v v q q x x, i

(3)

where ℓ0 ¼ ℓ/v is the lattice spacing in the Euclidean time direction, v is the speed of light in the material, and kmn is the lattice version of the Chern–Simons term (Diamantini et al., 1996), whose exact form is not of importance for what follows. There are four phenomenological parameters in this action. The couplings e2v and e2q distinguish between the electric and magnetic sectors of the theory and have canonical dimension [1/length]. The dimensionless parameters e and m distinguish between “electric” and “magnetic” components within the sector and represent the relative dielectric permittivity and magnetic permeability. pffiffiffiffiffi Correspondingly, v ¼ 1= me is the light velocity in the material. In materials that manifest superinsulating behavior v  1 so that the “magnetic” components are suppressed with respect to “electric” ones. The two extreme cases, m ¼ 1, e  1 and e ¼ 1, m  1 correspond to superconducting thin films and to fabricated JJAs respectively (Fazio and van der Zant, 2001). In this latter case the effective field theory (3) becomes exact in the limit m ! 1 when the ballistic term for vortices is included with a particular mass that makes the model self-dual.

Superconducting films Superconducting films are characterized by two spatial scales, the thickness, d, and the London penetration depth, lL, of the material. The couplings of the effective gage theory are expressed in terms of these two scales,  2  e e2q ¼ O , 2pd !   (4) p pd ¼O 2 2 , e2v ¼ O 2 e l? e lL where l? ¼ l2L /d is the so-called Pearl length (Tinkham, 1996) and represents the electric and magnetic energies of a fundamental granule. The topological interaction induces a spectrum gap (Jackiw and Templeton, 1981; Deser et al., 1982a, b). Both charge and vortex fluctuations acquire the dispersion relation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ m 2 v4 + v2 p 2 , (5) with the topological mass m¼

  eq ev 1 , ¼O 2pv vlL

(6)

The parameters e2q and e2v can be traded for this topological mass m ¼ eqev/2pv and one dimensionless parameter g ¼ ev =eq ¼ Oðd=ðalL ÞÞ, where a ¼ e2/4p is the fine structure constant. The parameter g plays the role of a dimensionless conductivity.

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Josephson junction arrays The Hamiltonian for an exemplary square JJA comprising superconducting islands with spacing ℓ and characterized by phases ’, voltages V and nearest-neighbor Josephson coupling of strength EJ is given by (Fazio and Schön, 1991)    X C0 X C 2 H¼ (7) V 2x + V y − V x + EJ 1 − cos ’y − ’x : 2 2 x

Here each island has a self-capacitance C0 and C are mutual capacitances to its nearest neighbors. This Hamiltonian can be also expressed in terms of the integer number qx of Cooper pairs on the islands H¼

X x

4EC qx

X 1 EJ ð1 − cos ðDi ’x ÞÞ, 2 qx + C0 =C − r x, i

(8)

where EC ¼ e2/2C is the scale of the charging energy and Di denotes the finite difference operator in direction i. In the self-capacitance limit C0 ! 0, and adding the vortex kinetic term with the topological mass of Eq. (6) (Diamantini et al., 1996), one finds that the zero-temperature partition function corresponding to this Hamiltonian has precisely the action of the effective gage theory (3) in the limit e ¼ 1, m ! 1, with the substitutions ve2v ! 4p2 EJ , ve2q ! 8EC ,

(9)

so that rffiffiffiffiffiffiffiffiffi p2 EJ , 2EC p ffiffiffiffiffiffiffiffiffiffiffiffi vev eq ¼ mv2 ¼ ! 8EC EJ ¼ oP , 2p g!

Dtop

(10)

where oP is the plasma frequency of the array. Again, the dimensionless coupling governing the phase structure of the array is the ratio of the two magnetic and electric energy scales in the problem. Here, it becomes explicit that this parameter indeed governs the conductivity of the array. The role of the topological energy gap, Dtop, is taken by the plasma frequency, describing small phase oscillations at the superconducting islands of the array.

The quantum phases in the vicinity of the SIT Quantum phase diagram at T ¼ 0 Given that e2q and e2v have the canonical dimension [1/length] and, correspondingly, the two kinetic terms in (3) are IR-irrelevant, one might be tempted to leave them out and consider only the IR-dominant mixed Chern–Simons theory. Unfortunately, this leads to wrong results and the origin of the problem lies exactly in the role of the quantized charges and vortices associated with the compactness of the two gage groups. The important point is that the “topological limit” e2q ! 1 and e2v ! 1 is not well defined without specifying the value of g in this limit. The behavior of quantized charges and vortices depends crucially on this value and, as a consequence one obtains very different ground states when g is varied, typically by changing the film thickness, applying a magnetic field or gate voltages, as shown in Fig. 3. In the framework of the gauge theory, the SIT is a material realization of the field-theoretic S-duality which expresses the above mentioned invariance of Maxwell’s equation under the exchange of electric and magnetic fields and charges. The structure of the phase diagram is governed by the dimensionless conductivity g and the additional dimensionless parameter  describing the strength of quantum fluctuations. Near the SIT, where eq  ev,  acquires the form (in physical cgs units)   1 pℓ ℓ , (11) G ¼ a me l? 2al? where a ¼ e2 =ℏc  1=137 is the fine structure constant, me ¼ Oð1Þ is the positional entropy contribution to the free energy of the system, and G is the diagonal element of the 3D Green function describing electromagnetic interactions screened by the CS mass: Z p 1 1 GðmℓvÞ ¼ d3 k (12)  i 2 : P2 2 ð2pÞ3 −p ðmℓvÞ + i¼0 4 sin k2 Identifying the UV cutoff with the superconducting coherence length x yields the geometric factor as d/(klL), where k ¼ lL/x is the Landau–Ginzburg parameter. The parameter  plays the role analogous to the role of the inverse of the Ginzburg–Landau parameter of 3D superconductors. For  < 1, there is a direct transition from a superconductor to a superinsulator, whose exact nature is the subject of the next section. This direct phase transition is the first-order quantum transition in the sense that it goes via a region of coexistence of both states

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Fig. 3 The possible quantum phases (T ¼ 0) of an interacting system of charges and vortices as a function of the dimensionless conductance g which controls the relative strength of the vortex interaction with respect to the Coulomb interaction. The parameter  is a function of the ratio ℓ/lL of the granule size to the penetration depth of the bulk superconductor and controls the presence/absence of an intermediate state in the transition.

around the value g ¼ 1. At this self-dual point g ¼ 1, the system is a universal metal with exactly the quantum of sheet resistance, R□ ¼ RQ ¼ h=4e2 (Fisher and Grinstein, 1988; Fisher, 1990a; Fisher et al., 1990b; Girvin et al., 1992). For  > 1, which is realized when the granule size and the London penetration depths are comparable, quantum fluctuations destroy both Cooper pairs and monopole condensates and the universal metal point opens up to a full fledged metal phase made of bosons (Diamantini et al., 1996), the so-called Bose metal (Das and Doniach, 1999, 2001; Dalidovich and Phillips, 2002; Phillips and Dalidovich, 2003), see for a review (Kapitulnik et al., 2019). This Bose metal is nothing else but the bosonic topological insulator (Lu and Vishwanath, 2012; Yang and Senthil, 2013; Chen et al., 2013), with a bulk frozen by the topological gap caused by the mutual statistics interactions and symmetry-protected edge states giving rise to the metallic behavior (Diamantini et al., 2020a). The two quantum transitions between superconductor and bosonic topological insulator and between bosonic topological insulator and superinsulator are quantum BKT transitions (Diamantini et al., 2020a). The point where all three transitions meet is a quantum tri-critical point. In the vicinity of this tri-critical point, the geometric scales of the problem satisfy the relation ℓ ¼ d/a. This indicates that the transition from superconducting to superinsulating behavior is caused by strong quantum fluctuations.

Finite temperature phase diagram Fig. 4 shows the finite-temperature phase diagram for  > 1. Both the phase transitions are BKT transitions, one corresponding to vortex unbinding (for the superconductor) and one to charge unbinding (for the superinsulator) (Mironov et al., 2018). The two critical conductances satisfy the self-dual relation g1g2 ¼ 1 around the quantum resistance value g ¼ 1. In the phase diagram presented in Fig. 4 we have sketched the behavior of the topological gap D(g) in the bosonic topological insulator phase (red dashed line) and we have marked by horizontal dashed lines what experiments, necessarily done at various finite temperatures, would see. For temperatures sufficiently above the topological gap, charges are essentially free in the intermediate state and we have the usual superconductor-to-metal transition. When the temperature is below the topological gap, an experiment would detect simply the transition from a superconductor to an activated insulator. This is the origin of the by now paradigmatic name “superconductor-to-insulator” transition (SIT) (Fazio and Schön, 1991; Fisher and Grinstein, 1988; Fisher, 1990a; Efetov, 1980; Haviland et al., 1989; Hebard and Paalanen, 1990). At extremely low temperatures the bulk becomes effectively frozen and the Bose metal (Kapitulnik et al., 2019) conduction by edge modes starts to be detected. This behavior is seen also in fabricated devices (Bøttcher et al., 2018). However, only at temperatures and conductivities g low enough can the transition to a superinsulator be detected. At T ¼ 0 the full quantum regime is realized and an experiment would record a transition from a superconductor to a superinsulator with a Bose metal intermediate phase (Diamantini et al., 2020a).

The nature of the superinsulator Describing quark interactions within hadrons, ‘t Hooft (1978) put forth an appealing confinement mechanism and coined the term “superinsulator” for the confined quark matter with infinite chromo-electric resistance (‘t Hooft, 1978). Polyakov showed (Polyakov, 1987) that this confinement mechanism occurs also in Abelian gage theories, provided they are compact and hence support topological excitations and magnetic monopoles, which are instantons in 2D and solitons in 3D.

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Fig. 4 The finite-temperature phase diagram of an interacting charge–vortex system for the case  > 1. The transitions are vortex and charge BKT transitions with critical temperatures TVBKT and TCBKT, respectively. The red dashed line marks the behavior of the topological gap D(g) in the bosonic topological insulator phase while the horizontal black dashed lines sketch the transitions seen by an experiment done at the respective temperatures. Reproduced from Trugenberger CA et al. (2020) Magnetic monopoles and superinsulation in Josephson junction arrays. Quantum Reports 2: 388–399.

Remarkably, this Abelian confinement emerges in a condensed matter realization of superinsulators. These electric superinsulators constitute a state of matter with infinite resistance at finite temperatures due to electric strings binding Cooper pairs P into “electric mesons.” Coupling the matter charge current jm to the real electromagnetic gauge potential Am, S ! S + i xℓ0ℓ2Amjm in the Euclidean action (with ℓ0 being the minimal length along the time axis), and integrating out all matter variables, one obtains the effective action describing the electromagnetic response of a system. For the bosonic topological insulator, this is   1 X 2 Seff Am ¼ 2 E , 2eeff x x

(13)

 2    1 e lL e2eff ¼ O , ¼O g d

(14)

with the effective coupling constant

This is the electromagnetic response of a Mott insulator with high dielectric permittivity so that the electric field E dominates the action. The effective coupling e2eff characterizes the strength of the Coulomb interaction, which grows with decreasing g, in particular, with decreasing film thickness d. In the superinsulating phase the effective action becomes   1 X 2 2p2 X 1 Seff Am ¼ 2 E + 2 mx mx , 2eeff x x eeff x −r2

(15)

where mx are dynamic magnetic monopole instantons, as shown in Fig. 5, over which one has also to sum in the partition function. Contrary to charges, whose conservation is ensured by the Noether theorem, Josephson vortices are topological excitations, characterized by the topological quantum number labeled by the homotopy group P1 ðUð1ÞÞ ¼ , which in this case is the winding number of the longitudinal component of the gauge field. Such topological quantum numbers can vary in local tunneling events, called instantons, which appear as particles in the Euclidean space-time lattice (Coleman, 1985). These Euclidean particles are (nonrelativistic) magnetic monopoles (Polyakov, 1975). In the relativistic limit, magnetic monopoles interact via the 3D Euclidean Coulomb potential and are, thus, always in a plasma phase (Polyakov, 1975). In the present case, since only electric fields enter the game, the interaction is the logarithmic 2D Coulomb potential (Trugenberger et al., 2020), to which the standard treatment of the 2D Coulomb gas (Minnhagen, 1987) with the temperature replaced by the effective coupling e2eff applies. In particular, at small e2eff, that is, large g, the monopoles are confined and the electromagnetic response of the system coincides with the electromagnetic response of the bosonic topological insulator. However, monopoles unbind if e2eff is sufficiently large, that is, if g is sufficiently small, and the Coulomb interaction becomes stronger than its critical value. The resulting monopoles’ “endogenous disorder” squeezes electric fields created by neighboring electric charges of opposite signs into thin electric flux tubes, Polyakov’s electric strings, dual to superconducting Abrikosov vortices, see, for example, Tinkham (1996). This induces a linear electric potential Vstring ¼ sL between the two probe charges at distance L. Hence, the charges can be viewed as bound by a string with the tension (Diamantini et al., 2020b) pffiffiffi 2 8eeff − ep2eff G2 ð0Þ s¼ , (16) e pℓ0 ℓ where G2(0) is the IR-regularized 2D lattice Coulomb potential at the coinciding points. pffiffiffi When the two probe charges are separated by distances larger than the typical string length ds ¼ 1= s on which the tension starts to be felt, a couple Cooper pair-Cooper hole emerges out of the vacuum to form two short strings replacing the long one. This is the

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Fig. 5 A (nonrelativistic) magnetic monopole instanton. The flux 2p on a plaquette formed by 4 granules disappears in a tunneling event between times t and t + ℓ0. . Reproduced from Diamantini MC et al. (2020b) Direct probe of the interior of an electric pion in a Cooper pair superinsulator. Communications on Physics 3: 142

property of confinement, the mechanism holding quarks together in color-neutral hadrons (Greensite, 2011). This, of course, assumes dynamical matter but indeed, confinement by U(1) monopoles survives when dynamical matter is coupled to the gage fields, at least for one species of relativistic fermions (Nogueira and Kleinert, 2008). Confinement means that, on large scales, the spectrum of the theory does not contain charged Cooper pairs anymore but only neutral electric “pions.” As a consequence, the system, if large enough, acquires infinite resistance, which holds up to the temperature of the charge BKT transition, TCBKT.

Asymptotic freedom and size effects Confining strings are characterized by two scales, the typical length after which tension is felt, ds, and the width, w. The latter is set by the photon mass, the gap characterizing the superinsulating phase, w ¼ 1/mph (Polyakov, 1975, 1987). Contrary to strings in quantum chromodynamics, the width and length of these Abelian strings scale differently with the coupling constant (Caselle et al., 2016) so that in the physical regime, the Abelian strings are thin and long. pffiffiffi The confinement scale ds ¼ 1= s sets the dimension of the neutral bound states. In quantum chromodynamics, the strong interaction binding quarks implies that mesons and hadrons are extremely small and it is impossible to “look inside” them. Quarks can be observed only indirectly, by smashing hadrons in colliders and observing the jets formed in these collisions. The electric forces in superinsulators are much weaker than the strong interactions; hence electric pions are much larger than their elementary particle cousins and it is conceivable that one can measure their interior simply by experimenting on small enough samples. What would one expect in this interior? Non-Abelian gage theories are characterized by the phenomenon of asymptotic freedom (Gross, 1999), the coupling constant, and thus the strength of the interaction, flows to zero at short scales. While this is the original point of view from the vicinity of the Gaussian UV fixed point, asymptotic freedom can be understood also from the point of view of the IR string theory, where gauge interactions become nonperturbative. This is the statement that the string tension decreases, and eventually vanishes, when approaching short scales, where perturbative gauge interactions take over. This is particularly important in the present case, where we have two clearly distinct scales. At intermediate observation scales w ¼ 1/mph < robs < ds, Cooper pairs feel neither the string tension nor the standard Coulomb potential, which is screened by the photon mass on scales larger than w. One can thus expect that the hyperactivated resistance behavior caused by the string should give way directly to a saturated metallic resistance when the sample size decreases below the critical threshold ds. This is fully confirmed in experiment (Diamantini et al., 2020b), as shown in Fig. 6.

Electric Meissner effect and its manifestation in the I(V ) characteristics Exactly as superconductivity can be destroyed by the application of a strong enough magnetic field, superinsulation does not survive the application of a sufficiently strong electric field, that is, an external voltage. An electric field E decreases the effective linear potential of Cooper pairs separated by the distance L to U eff ¼ ðs − 2eEÞL. Below the critical value Vc1 ¼ s(T )L/2e, however, the effective string tension seff(T ) remains positive and the I(V ) characteristics is given by (Diamantini et al., 2020b)   s ðTÞL sðTÞL , V < V c1 ¼ , (17) I ∝ V exp − eff T 2e which, in the thermodynamic limit L ! 1, implies an infinite resistance. This is the electric Meissner state of the superinsulator. When the applied voltage V reaches the critical value Vc1 ¼ s(T )L/2e, the effective tension vanishes. This, however, does not entail complete destruction of the superinsulating state, the photon mass gap is still present. This means that, above this threshold, thin, long electric strings can penetrate the sample from one end to the other and current starts to pass. This is the transition from the

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Fig. 6 Size dependence of the sheet resistance of a NbTiN film. In this logarithmic plot the dashed straight line corresponds to the usual activated behavior of a normal insulator. The largest samples show the hyperactivated behavior characteristic of superinsulation. The resistance of the shortest sample size 0.2 mm, however, saturates at the lowest temperatures, denoting a metallic behavior. For comparison, the string scale in this experiment was estimated as ds  0.13 mm. Reproduced from Diamantini MC et al. (2020b) Direct probe of the interior of an electric pion in a Cooper pair superinsulator. Communications on Physics 3: 142

electric Meissner state of a superinsulator, in which electric fields are completely expelled, to the mixed state in which electric flux tubes penetrate the sample. In this regime, the I(V) characteristics acquire a power law form (Diamantini et al., 2020b),

at V c1

I ∝ ðV − sðTÞL=2eÞ1+T CBKT =16T ,

L T CBKT , < V < V c2  sðTÞ + 2:718r 0 2e

(18)

where TCBKT is the charge BKT transition temperature. Above the second critical threshold Vc2, the photon mass vanishes, superinsulation is completely destroyed, and the system becomes a normal insulator. The three possible states, Meissner state, mixed state, and normal insulator, are separated by two kinks in the I(V ) characteristics of a superinsulating material (Diamantini et al., 2020b), as shown in Fig. 7. For an infinite sample, the current I(V ) vanishes identically in the Meissner state, realized for V < Vc1.

Fig. 7 The two kinks in the I(V ) characteristics of a NbTiN film. For an infinite sample the I(V ) curve vanishes identically for V < Vc1, reflecting the electric Meissner state.

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Fig. 8 Characteristic current time dependence upon application of a rectangular voltage pulse with amplitude V ¼ 0.155 V and duration 50 ms (left panel) to a NbTiN film at the temperature of 20 mK. In the superinsulating state as compared to the same response of the normal insulator at 300 mK. The right panel displays the log–log plot of the current response time tsh as function of the amplitude of the rectangular voltage pulse. The experimental data are shown by symbols. The solid lines present the fitting functions revealing the power law scaling of tsh(V − Vp) dependence. Reproduced from Mironov A-Y et al. (2022) Relaxation electrodynamics of superinsulators. Scientific Reports 12: 19918.

Dynamical exponents and the linear potential The charge-hole potential in the superinsulating state can be measured directly by applying a voltage pulse and measuring the delay for current passage (Mironov et al., 2022). The results are presented in Fig. 8, in which the scaling of the current response time as a function of the pulse strength (V − Vc1) (here denoted as Vp) is shown for a rectangular pulse of strength V and duration 50 ms applied to a NbTiN film at 20 mK in the superinsulating state, and compared to the current response for the normal insulator at 300 mK. The two kinks described in the previous section are again clearly visible and correspond to two different dynamical exponents m in the scaling t sh ∝ ðV − V c1 Þ − m :

(19)

Consider two charges  2e of mass m bound by the linear potential V ðx1 , x 2 Þ ¼ sjx1 − x2 j, where s is the string tension of the confining electric string and xi are the coordinates of the two particles. The critical value of the potential when current starts to pass is Vc1 ¼ sL, with L being the sample length, corresponding to a configuration in which the two opposite charges reach the sample boundaries. In the center of mass coordinates, one can reduce the problem to a single particle of reduced mass m/2 at the origin, subject to the potential energy U(r) ¼ 2esr, with r denoting the separation distance. A positive linear potential results in an attractive constant force Fa ¼ 2es ¼ 2eVc1/L. An external potential V amounts to an additional repulsive force Fr ¼ 2eV/L. The total acceleration of separation is thus a ¼ (2/m)Ftot ¼ (4e/mL)(V − Vc1). The corresponding equation of motion describing the separation distance of the charge-hole system is rðtÞ ¼

2e ðV − V c1 Þt 2 : mL

(20)

Current starts to pass when charges of opposite sign reach the sample boundaries, that is when r ðt c1 Þ ¼ L. The delay for current passage is thus given by inverting Eq. (20), rffiffiffiffiffiffiffiffiffi mL2 t cr ¼ ðV − V c1 Þ−1=2 : (21) 2e The observed dynamical exponent m ¼ 1/2 is thus a direct confirmation of the linear potential binding charges of opposite sign.

Oblique superinsulators in 3D Superconductors with emergent granularity also arise in three dimensions (Diamantini et al., 2021a). Similarly to their 2D counterparts, they are natural hosts for quantized Josephson vortices that comprise supercurrent rings spanning over adjacent granules in the same plane. The difference is that now since the current rings pile on top of each other to minimize the interaction energy, these Josephson vortices become 1D extended excitations. Since the resulting liner tension is low, nothing prevents the vortex cores to tear making them open and having magnetic monopole-antimonopole pairs at their ends. In the 3D case, magnetic

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monopoles are no more instantons but become real, particle-like excitations. If the vortex tension vanishes, the quantized vortices turn into the unobservable Dirac strings (Dirac, 1931; Goddard and Olive, 1978): there is no energy price to pay to make one vortex go from one monopole to infinity and then return back to antimonopole. In this case magnetic monopoles form a Bose condensate, that is, 3D superinsulator (Diamantini et al., 2021c). All the described properties of superinsulators, including the confining electric string between Cooper pairs, the Meissner state, and the nonlinear I(V ) curves can be derived, mutatis mutandis, as in the 2D case. The 3D superinsulators are characterized by dual London equations for the magnetic monopole current jm, p 1 ∂t jm ¼ 2 B, e lE p 1 r ^ jm ¼ 2 E: e le where the electric penetration depth is set by the photon mass, lE ¼ 1/mph. The new effect, arising exclusively in 3D, is the possible presence of the topological y-term (Treiman et al., 1985) Z y Sy ¼ 2 d4 x EB: 4p

(22)

(23)

in the electromagnetic action, giving rise to the so-called axion electrodynamics (Wilczek, 1987). If the gauge group is compact, the new parameter y is an angle. The correct periodicity is 2p for fermionic systems and 4p for bosonic ones (Metlitski et al., 2013; Vishwanath and Senthil, 2013). The y-term changes the character of superinsulators since it induces the electric charge y/2p on magnetic monopoles (Witten, 1979) which become dyons, the particles carrying both electric and magnetic charge (Goddard and Olive, 1978). In the presence of the y-term one arrives at the oblique superinsulator, a Bose condensate of both electric and magnetic charges, in which the corresponding oblique London equations hold:   p L2 2e2 y ∂t jm ¼ 2 B + E , p 2p e 4e (24)   2 p L 2e2 y r ^ jm ¼ 2 E − B : p 2p e 4e Here, L denotes the gap energy scale. Combining these equations with the (static) Ampère equations in the presence of the magnetic charge r ^ B ¼ 2e je , p r ^ E ¼ − jm , e

(25)

and using the fact that the electric current is je ¼ (y/2p)jm since it is carried by charged magnetic monopoles one obtains (for sourceless currents) ! 1 r2 − 2 je ¼ 0, (26) ly where ly ¼

p 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : eL  p 2  y 2 + 2p 2e2

(27)

Currents are screened in the interior of the dyon condensate and survive only in the surface strips of the width set by the oblique penetration depth ly. While in normal superinsulators the penetration depth increases in the strong coupling limit, the contrary is true for oblique superinsulators, where for e ! 1 we have ly ! 0. One can show that in this limit, oblique superinsulators form a topological state of matter in which only edge states survive while the bulk charges are frozen by the diverging gap (Diamantini et al., 2021b). The y-term residing on closed manifolds is equivalent to the Chern–Simons term on the boundary (Treiman et al., 1985). This induces statistical transmutations for the surviving edge states (Wilczek, 1990). In particular, for y ¼ 2p, the edge states become symmetry-protected charge 2e fermions and thus are expected to behave as an ideal Fermi liquid, with the resistance scaling as the square of temperature. The experiments reported precisely this R ∝ T2 resistance behavior in the pseudogap state of high-Tc superconductors (Zhou, 2021). This observation in conjunction with the nematicity and magnetoelectric effects observed in the pseudogap state, set the base for putting forth the proposal that the pseudogap state, proclaimed in Zhou (2021) as an unanswered issue, is actually the oblique superinsulator (Diamantini et al., 2021b).

The role of disorder The first quantitative descriptions of the superinsulating state were derived in the context of JJAs (Diamantini et al., 1996; Fistul et al., 2008) and revealed that, in this context, the tuning parameter driving the SIT is g ’ EJ/EC that is, the relative strength of the Josephson coupling and the Coulomb interaction. At the same time, experimentally, superinsulation was discovered

Superinsulation

815

(Baturina et al., 2007) by driving a TiN film across the SIT via tuning the room temperature sheet resistance R□ ðT Room Þ, which is a measure of disorder. Indeed, disorder enhances the Coulomb interaction by reducing the charge screening (Finkel’shtein, 1987). The main parameter driving the SIT is the relative strength of the Coulomb interaction: disorder is indeed one of the possible ways to affect this strength, as is the thickness of the film or an applied magnetic field. A widespread view at the dawn of SIT studies was that disorder plays also another role, governing the structure of the phase diagram near the SIT via Anderson localization of the Cooper pairs (Fisher, 1990a; Fisher et al., 1990b) or its interacting version, referred to as many-body localization (Altshuler et al., 2006; Basko et al., 2007; Altshuler et al., 2009), see Sacépé et al. (2020) for a review. A consistent theory of the SIT (Diamantini et al., 2020a, 2021a) demonstrates, however, that localization does not play a role in the critical vicinity of the SIT because, on one hand, of the dominance of topological interactions in the intermediate Bose metal state and, on the other hand, the presence of long range linear Coulomb interactions in the superinsulating state. Exactly as in the quantum Hall effect (QHE), localization can help to pin the quasi-particles forming around the topological ground state (Pu et al., 2021) corresponding to the Bose metal, although, strictly speaking not even this is necessary, neither for the formation of plateaus in the QHE (Kim and Kivelson, 2021) nor for the emergence of the topological Bose metal phase (Diamantini et al., 2021a). On the other side, many-body localization does not survive linear interactions, not even in 1D, where they are kinematic (Nandkishore and Sondhi, 2017; Akhtar et al., 2018), and may become effective when only short-range interactions are present. Only after one has identified the correct neutral physical states in the spectrum and only short-range interactions are left, can one even speak of localization. The observation of two kinks (rather than only one), see Fig. 6, in the I(V ) curves in the superinsulating state (Diamantini et al., 2020b), is a strong experimental evidence that Cooper pairs are confined by Polyakov’s strings, rather than localized by disorder. Localization by disorder is further strongly disfavored by the measurement of a linear potential between charges of opposite sign in the superinsulating state (Mironov et al., 2022). Notably, the confinement by this linear potential induces breaking of thermalization and ergodicity. These phenomena are typically associated with many-body localization in the presence of disorder (James et al., 2019; Mazza et al., 2019). Note that while the many-body localization phenomenon is far from being established in 2D and 3D, the phenomena associated with it are straightforward and typical consequences of the established existence of the confinement phase. Therefore, while disorder may appear as an important tuning parameter for the SIT, it is irrelevant in the renormalization group sense in the critical vicinity of the SIT, since it does not influence the character of the phases emerging near the superconductor-superinsulator transition. The interpretation of this effect as a finite-temperature insulating behavior due to many-body localization proposed in Ovadia et al. (2015) is thus disfavored by recent experimental evidences.

Conclusion We have shown that condensed matter systems that admit magnetic monopoles lead to a material realization of the S duality under exchange of electric and magnetic fields and charges. Under this duality, a superconductor is mapped into a superinsulator, a phase in which the system exhibits an infinite electric resistance (even at finite temperatures). Contrary to the superconducting phase, where vortices experience logarithmic interactions, in superinsulators charges are bound into neutral pairs by linear potentials due to the “endogenous disorder” created by the monopoles. Superinsulation has been experimentally detected in TiN, InO, NbTiN, and NbSi films. In bulk materials, a particular, “oblique” version of superinsulation may be realized in the pseudogap state of hightemperature superconductors.

References Aharonov Y and Bohm D (1959) Significance of electromagnetic potentials in quantum theory. Physics Review 115: 485–491. Aharonov Y and Casher A (1984) Topological quantum effects for neutral particles. Physical Review Letters 53: 319–321. Akhtar AA, et al. (2018) Symmetry breaking and localization in a random Schwinger model with commensuration. Physical Review B 98: 115109. Altshuler BL, et al. (2006) Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of Physics 321(5): 1126–1205. Altshuler BL, et al. (2009) Jumps in current-voltage characteristics in disordered films. Physical Review Letters 102: 176803. Basko DM, et al. (2007) Possible experimental manifestations of the many-body localization. Physical Review B 76: 052203. Baturina TI and Vinokur VM (2013) Superinsulator-superconductor duality in two dimensions. Annalen der Physik 331: 236–257. Baturina TI, et al. (2007) Localized superconductivity in the quantum-critical region of the disorder-driven superconductor-insulator transition in TiN thin films. Physical Review Letters 99: 257003. Berezinskii VL (1971) Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Soviet Physics–JETP 32: 493–500. Bøttcher CGL, et al. (2018) Superconducting, insulating and anomalous metallic regimes in a gated two-dimensional semiconductor-superconductor array. Nature Physics 14: 1138–1144. Caselle M, et al. (2016) Width of the flux tube in compact U(1) gauge theory in three dimensions. Journal of High Energy Physics 02: 180. Chen X, et al. (2013) Symmetry protected topological orders and the group cohomology of their symmetry group. Physical Review B 87: 155114. Coleman S (1985) Aspects of Symmetry. Cambridge: Cambridge University Press. Dalidovich D and Phillips P (2002) Phase glass is a Bose metal: A new conducting state in two dimensions. Physical Review Letters 89: 27001. Das D and Doniach S (1999) Existence of a Bose metal at T ¼ 0. Physical Review B 60: 1261–1275. Das D and Doniach S (2001) Bose metal: Gauge field fluctuations and scaling for field-tuned quantum phase transitions. Physical Review B 64: 134511. Deser S, et al. (1982a) Three-dimensional massive gauge theories. Physical Review Letters 48: 975–978. Deser S, et al. (1982b) Topologically massive gauge theories. Annals of Physics 140: 372–411. Diamantini MC, et al. (2021c) Quantum magnetic monopole condensate. Communications on Physics 4: 25. Diamantini MC, et al. (1996) Gauge theories of Josephson junction arrays. Nuclear Physics B 474: 641–677.

816

Superinsulation

Diamantini MC, et al. (2018) Confinement and asymptotic freedom with Cooper pairs. Communications on Physics 1: 77. Diamantini MC, et al. (2019) The superconductor-superinsulator transition: S-duality and the QCD on the desktop. Journal of Superconductivity and Novel Magnetism 32: 47–51. Diamantini MC, et al. (2020a) Bosonic topological intermediate state in the superconductor-insulator transition. Physics Letters A 384: 126570. Diamantini MC, et al. (2020b) Direct probe of the interior of an electric pion in a Cooper pair superinsulator. Communications Physics 3: 142. Diamantini MC, et al. (2021a) Superconductor-to-insulator transition in absence of disorder. Physical Review B 103: 174516. Diamantini MC, et al. (2021b) Topological nature of high-temperature superconductivity. Advanced Quantum Technologies 4: 2000135. Dirac PAM (1931) Quantized singularities in the electromagnetic field. Proceedings of the Royal Society of London A 133: 60–72. Efetov KB (1980) Phase transition in granulated superconductors. Soviet Physics–JETP 51: 1015–1022. Fazio R and Schön G (1991) Charge and vortex dynamics in arrays of tunnel junctions. Physical Review B 43: 5307–5320. Fazio R and van der Zant H (2001) Quantum phase transitions and vortex dynamics in superconducting networks. Physics Reports 355: 235–334. Finkel’shtein AM (1987) Superconducting transition temperature in amorphous films. JETP Letters 45: 46–49. Fisher MPA (1990a) Quantum phase transitions in disordered two-dimensional superconductors. Physical Review Letters 65: 923–926. Fisher MPA and Grinstein G (1988) Quantum critical phenomena in charged superconductors. Physical Review Letters 60: 208–211. Fisher MPA, et al. (1990b) Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition. Physical Review Letters 64: 587–590. Fistul MV, et al. (2008) Collective Cooper-pair transport in the insulating state of Josephson-junction arrays. Physical Review Letters 100: 086805. Girvin SM, et al. (1992) Universal conductivity as the superconductor-insulator transition in two dimensions. Progress of Theoretical Physics Supplement 107: 135–144. Goddard P and Olive DI (1978) Magnetic monopoles in gauge field theories. Reports on Progress in Physics 41: 1357–1437. Greensite J (2011) An Introduction to the Confinement Problem. Berlin: Springer-Verlag. Gross D (1999) Twenty five years of asymptotic freedom. Nuclear Physics B, Proceedings Supplements 74: 426–446. Haviland D, et al. (1989) Onset of superconductivity in the two-dimensional limit. Physical Review Letters 62: 2180–2183. Hebard A and Paalanen MA (1990) Magnetic-field-tuned superconductor-insulator transition in two-dimensional films. Physical Review Letters 65: 927–930. Humbert V, et al. (2021) Overactivated transport in the localized phase of the superconductor-insulator transition. Nature Communications 12: 6733. Jackiw R and Templeton S (1981) How super-renormalizable interactions cure infrared divergences. Physical Review D 23: 2291–2304. James AJA, et al. (2019) Nonthermal states arising from confinement in one and two dimensions. Physical Review Letters 122: 130603. Kapitulnik A, et al. (2019) Anomalous metals–failed superconductors. Reviews of Modern Physics 91: 011002. Kim K-S and Kivelson S-A (2021) The quantum Hall effect in the absence of disorder. Quantum Matter 6: 22. Kosterlitz J-M and Thouless D-J (1972) Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory). Journal of Physics C: Solid State Physics 5: L124. Kosterlitz J-M and Thouless D-J (1973) Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C: Solid State Physics 6: 1181–1203. Kowal D and Ovadyahu Z (1994) Disorder induced granularity in an amorphous superconductor. Solid State Communications 90: 783–786. Krämer A and Doniach S (1998) Superinsulator phase of two-dimensional superconductors. Physical Review Letters 81: 3523–3526. Lu Y-M and Vishwanath A (2012) Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach. Physical Review B 86: 125119. Mandelstam S (1976) Vortices and quark confinement in non-Abelian gauge theories. Physics Reports 23: 245–249. Mazza P, et al. (2019) Suppression of transport in non-disordered quantum spin chains due to confined excitations. Physical Review B 99: 180302. Metlitski M-A, et al. (2013) Bosonic topological insulator in three dimensions and the statistical Witten effect. Physical Review B 88: 035131. Minnhagen P (1987) The two-dimensional coulomb gas, vortex unbinding and superfluid-superconducting films. Reviews of Modern Physics 59: 1001–1066. Mironov A-Y, et al. (2018) Charge Berezinskii-Kosterlitz-Thouless transition in superconducting NbTiN films. Scientific Reports 8: 408. Mironov A-Y, et al. (2022) Relaxation electrodynamics of superinsulators. Scientific Reports 12: 19918. Nambu Y (1974) Strings, monopoles and gauge fields. Physical Review D 10: 4262–4268. Nandkishore R-M and Sondhi S-L (2017) Many-body localization with long-range interactions. Physical Review X 7: 041021. Nogueira F-S and Kleinert H (2008) Compact quantum electrodynamics in 2+1 dimensions and spinon confinement: A renormalization group analysis. Physical Review B 77: 045107. Ovadia M, et al. (2015) Evidence for a finite-temperature insulator. Scientific Reports 5: 13503. Phillips P and Dalidovich D (2003) The elusive Bose metal. Science 302: 243–247. Polyakov A-M (1975) Compact gauge fields and the infrared catastrophe. Physics Letters 59: 82–84. Polyakov A-M (1987) Fields and Strings. Chur, Switzerland: Harwood Academic Publisher. Pu S, et al. (2021) Anderson localization in fractional quantum Hall effect. arXiv.2109.00362. Sacépé B, et al. (2008) Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition. Physical Review Letters 101: 157006. Sacépé B, et al. (2011) Localization of preformed Cooper pairs in disordered superconductors. Nature Physics 7: 239–244. Sacépé B, et al. (2020) Quantum breakdown of superconductivity in low-dimensional materials. Nature Physics 16: 734–746. Sambandamurthy G, et al. (2005) Experimental evidence for a collective insulating state in two-dimensional superconductors. Physical Review Letters 94: 017003. ‘t Hooft G (1978) On the phase transition towards permanent quark confinement. Physical Review B 138: 1–25. Tinkham M (1996) Introduction to Superconductivity. New York: Dover Publications. Treiman SB, et al. (1985) Current Algebra and Anomalies. Singapore: World Scientific. Trugenberger CA, et al. (2020) Magnetic monopoles and superinsulation in Josephson junction arrays. Quantum Reports 2: 388–399. van der Zant H, et al. (1992) Ballistic motion of vortices in Josephson junction arrays. Europhysics Letters 18: 343–512. van Otterlo A, et al. (1994) Quantum vortex dynamics in Josephson junction arrays. Physica B 203: 504–512. Vinokur VM, et al. (2008) Superinsulator and quantum synchronization. Nature 452: 613–615. Vishwanath A and Senthil T (2013) Physics of three-dimensional Bosonic topological insulators: Surface-deconfined criticality and quantized magnetoelectric effect. Physical Review X 3: 011016. Wen XG and Zee A (1992) Classification of Abelian quantum Hall states and matrix formulation of topological fluids. Physical Review B 46: 2290–2301. Wilczek F (1987) Two applications of axion electrodynamics. Physical Review Letters 58: 1799–1802. Wilczek F (1990) Fractional Statistics and Anyon Superconductivity. Singapore: World Scientific. Wilczek F (1992) Disassembling anyons. Physical Review Letters 69: 132–135. Witten E (1979) Dyons of charge y/2p. Physics Letters 86: 283–287. Yang C and Senthil T (2013) Boson topological insulators: A window into highly entangled quantum phases. Physical Review B 87: 235122. Zhou X (2021) High-temperature superconductivity. Nature Review Physics 3: 462–465.

ENCYCLOPEDIA OF CONDENSED MATTER PHYSICS SECOND EDITION

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ENCYCLOPEDIA OF CONDENSED MATTER PHYSICS SECOND EDITION

EDITOR-IN-CHIEF Tapash Chakraborty⁎ Professor, Tier-I Canada Research Chair, University of Manitoba, Winnipeg, MB, Canada VOLUME 3

⁎

Retired.

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge MA 02139, United States Copyright © 2024 Elsevier Ltd. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers may always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-32-390800-9 For information on all publications visit our website at http://store.elsevier.com

Publisher: Oliver Walter Acquisitions Editor: Oliver Walter Content Project Manager: Naman Negi Associate Content Project Manager: Nandhini Mahendran Designer: Mark Rogers

EDITOR-IN-CHIEF BIOGRAPHY Tapash Chakraborty is a retired professor of physics from the University of Manitoba, Canada, where he was also the Canada Research Chair in Nanoscale Physics (2003–17). His research focus has been on many-body theories of various quantum materials, such as nuclear matter, helium liquids, and electron-hole liquids, that he studied under the tutelage of late Professor Manfred Ristig, at the University of Köln (1979–81). Just a year after the discovery of the fractional quantum Hall effect, he (working with his coworkers) demonstrated that electron–electron interactions could actually lead to quantum Hall states with nontrivial spin configurations, and that the low-lying excitations could be described as “spin-reversed quasiparticles.” This prediction motivated much experimental work from laboratories around the world, as a result of which the concept of spin-reversed quasi-particles was established. Similarly, his work on the quantum Hall states in double layer systems has received much attention from the community. Furthermore, Dr. Chakraborty (working with Dr. P. Maksym and others) made pioneering contributions to the many-electron theory of quantum dots. Dr. Chakraborty has also contributed to the theory of electronic properties of various nanoscale systems, such as spin-orbit coupled quantum dots, quantum rings, and single- and double-layer graphene (primarily with Dr. V. Apalkov). He has authored many books and reviews: the book with Dr. Pietiläinen on the quantum Hall effect and the single-authored book on quantum dots have become standard references in their respective fields.

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EDITORIAL ADVISORY BOARD Ana M. Sanchez Department of Physics, University of Warwick, Coventry, United Kingdom Hideo Aoki Department of Physics, University of Tokyo, Tokyo, Japan Robert H. Blick Center for Hybrid Nanostructures, University of Hamburg & DESY, Hamburg, Germany Roberto Raimondi Mathematics and Physics Department, Roma Tre University, Rome, Italy Rudolf A. Roemer Department of Physics, University of Warwick, Coventry, United Kingdom Vladimir Mihajlovic Fomin Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany

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INTRODUCTION Electrons in orbits contained by the quantum, And atoms conjoined by the dipolar force, Gravity balanced by pressure And flap of a wing. Here, I remember the angles and curves, Calculus, I learned it all – Each bird bit and moon fracture Fixed by the symmetries, Airborne and airless, aloft. – Alan Lightman: Song of Two Worlds

Condensed matter physics (CMP) is the largest discipline in physics, involving a wide variety of interesting concepts and challenges. To quote Leggett (1992), CMP embraces topics “as diverse as traditional solid-state physics, neutron stars and the physics of biological matter.” He further added that, by a liberal definition “just about all of physics outside atomic, nuclear and particle physics and cosmology” fall in this vast enterprise1. One of the primary goals of CMP is to explore the fundamental properties of matter—solids, liquids or gasses—comprising a large number of interacting constituent particles (Laughlin and Pines, 2000). The quantum-mechanical nature of these many-body systems makes condensed-matter highly nontrivial, sometimes counterintuitive and even astonishing. Superconductivity and the fractional quantum Hall effect (FQHE) naturally fall within that category. In fact, for the past four decades, the quantum Hall effects (QHEs) have been the epitome of elegant CMP (von Klitzing et al., 2020). Our fundamental system of units has been redefined following the discovery of the QHE when the “von Klitzing constant” RK ¼ 25, 812.8074593045 O came into being (von Klitzing, 2019; Nawrocki, 2019). Observation of the QHE in graphene2 was crucial in determining the Dirac nature of charge carriers there (see for example, Jiang et al., 2007). Another effect that outwardly resembles the QHE but is conceptually quite different is the FQHE, which is a quintessential manifestation of electron correlations governed by the Coulomb interaction (Störmer, 1992; Eisenstein and Stormer, 1990). To explain this fascinating effect, Laughlin (1983) introduced his eponymous wave function that not only provided the foundation for understanding the effect but also triggered a new wave of ideas with impact far beyond the realm of CMP3. Who would have imagined that a large collection of interacting electrons confined on a plane in a perpendicular magnetic field would result in elementary excitations (quasiholes and quasiparticles) (Chakraborty and Pietiläinen, 1988, 1995; Halperin, 1983; Clark and Maksym, 1989) carrying fractional electron charge and obeying the exotic exchange statistics of anyons (Halperin, 1984; Arovas et al., 1984)? These quasiparticles and quasiholes of the Laughlin state still remain an enigma (Laughlin, 1984; Chakraborty, 1985; Morf and Halperin, 1986). Fractional statistics is a truly pathbreaking concept in CMP that was introduced by Leinaas and Myrheim (1977), and has been recently confirmed experimentally (Bartolomei et al., 2020; Nakamura et al., 2020). Much of the history of CMP can be viewed as a series of paradigm shifts4. The field evolved rapidly through frequent ground-breaking discoveries that challenged existing paradigms and opened up new applications, and this evolution is expected to continue (Leggett, 2018; Cohen, 2008). For example, our current understanding of 1 Interestingly, some similarities do exist between the mathematical frameworks that describe low-temperature CMP and early-universe cosmology. See Kibble and Srivastava (2013). 2

An isolated single layer of graphite consisting of carbon atoms bonded together in a hexagonal structure. See, e.g., Abergel et al. (2010).

3

For example, an important aspect of Laughlin's theory is its seamless concinnity of classical plasma physics and the planar electron gas that has inspired others to use a similar approach to quark-gluon plasma (Lu, 2022). 4

A detailed exposition of the meaning and examples of “paradigm shifts” (Kuhn-type) can be found in Leggett (2018).

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the FQHE required deep mathematical analysis and a wide range of experimental exploration, which revealed a treasure trove of unique phenomena that are yet to be fully understood or explained. Other studies of electron correlations in low spatial dimensions have ushered in many groundbreaking notions, such as Pfaffian states, Majorana fermions in condensed matter, Chern insulators, and topological quantum materials that promise a fault tolerant means to perform quantum computation. The presence of incompressible FQHE states in monolayer graphene (Apalkov and Chakraborty, 2006), bilayer graphene (Apalkov and Chakraborty, 2010, 2011), and double-layer graphene (Liu et al., 2019) has demonstrated the robustness of the effect, and provides invaluable insight into the nature of this effect. The dynamics of an electron in a periodic potential subjected to a perpendicular magnetic field has been an intriguing problem for more than half a century (Obermair and Wannier, 1976; Osadchy and Avron, 2001)5. Within the nearest neighbor tight-binding description of the periodic potential the electron energy spectrum is described by the Harper equation (Harper, 1955). Numerical solution of this equation (Hofstadter, 1976) has revealed that the applied field splits the Bloch bands into subbands and gaps. As the field is varied, the splitting of band continues indefinitely, leading to bands within bands within bands. When plotted as a function of the magnetic flux per lattice cell, the energy spectrum depicts a fractal pattern (a self-similar pattern that repeats at every scale) and is known in the literature as Hofstadter’s butterfly because of the resemblance of the pattern to butterflies6. Observation of the butterfly spectra in real systems is an arduous task. For example, in ordinary crystalline lattices, the required magnetic field to see the butterflies exceeds 1000 Tesla, which is far beyond the reach of present-day technology. Graphene seems to be the ideal system in the quest of those fractal butterflies (Weiss, 2013). It is interesting to note that a mathematical solution of the Harper equation, in contrast to the numerical solution mentioned above, remained a challenging problem (the so-called Ten Martini Problem) for mathematicians until it was solved (Avila and Jitomirskaya, 2006) in 2006. Magnetism has been known to mankind since ancient times, and it remains a major topic in CMP. The road to understanding magnetism has been long and tortuous. At various times in history, magnets were claimed to have the healing power to cure a myriad of ailments that included epilepsy, arthritis, gout, and baldness. Magnets were the prime ingredients for the “elixir of life” by Paracelsus (1493–1541). Then there was the healing process of “animal magnetism” proposed by Franz Anton Mesmer (1734–1815) (mesmerism)7, which was famously mocked in Mozart’s Così fan tutte (Steptoe, 1986). Our current understanding of magnetism is deeply rooted in quantum mechanics as Van Vleck explained (Van Vleck, 1978): “Quantum mechanics is the key to understand magnetism. When one enters the first room with this key there are unexpected rooms beyond, but it is always the master key that unlocks each door.” For centuries, the magnetism of certain rocks and metallic iron was only used for the practical purposes of direction finding, particularly navigation with a compass. Motors, generators, and electrical distribution networks appeared only in the 19th century after the discovery of the deep connection between electricity and magnetism. Magnetic order is a collective phenomenon akin to superconductivity and superfluidity that involves a very large number of particles. The rich variety of magnetically ordered states in solids, not only ferromagnetism but also antiferromagnetism, ferrimagnetism, and a range of noncollinear structures, was confirmed only in the middle of the 20th century, with the availability of neutron beams in research reactors. A range of new magnetic materials are taking their place as active layers in thin film sensors and memory devices (Baltz et al., 2018). Besides information storage (Blundell, 2001), important technical applications include magnetic resonance imaging (MRI) and the use of magnetic nanoparticles in medicine. Many outstanding discoveries in magnetism and magnetic materials have resulted in rapid advances in information storage technology. Over the last two decades spintronics has emerged as an important field of research both from the point of view of fundamental science and advanced technological applications (Yamaguchi et al., 2021; Pinarbasi and Kent, 2022). The giant magnetoresistance (GMR) effect, the spin Hall effect, and the spin galvanic effects are typical examples of developments in the field. Spintronics aims at understanding how the spin degree of freedom of the electrical carriers can be controlled in order to obtain devices with new functionalities and better performance. This endeavor often combines concepts from different aspects of condensed matter systems, giving the topic a fascinating interdisciplinary flavor. Spintronic phenomena are now being studied in an impressive range of different materials ranging from metals to half-metals to semiconductors, graphene and other two-dimensional systems such as transition metal dichalcogenides. 5

For a recent review on butterflies in graphene, see for example Chakraborty and Apalkov (2015).

6

However, as one of the authors of this Encyclopedia has (somewhat humorously) pointed out, this butterfly has Lebesgue measure zero, that is, it cannot fly!.

7

A wonderful account of the use of magnetism in medicine since ancient times can be found in Mourino (1991).

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The GMR effect and the related tunnel magnetoresistance effect (TMR) found their first commercial use in magnetic field sensors for hard-disk read-heads, before broadening their field of application to include magnetic sensors for the automobile industry, solid-state compasses, biosensors, and nonvolatile magnetic memory (Hartmann, 2000). The ability to manipulate the magnetic state of ferromagnetic structures led to initial developments in spintronics, where the spin direction was controlled by externally applied magnetic fields. However, the push for downscaling and faster timescales has prompted research advances in the past decade where the manipulation of magnetic configurations is achieved by local electric currents (the spin torque effect) (Brataas et al., 2012; Locatelli et al., 2014). In recent years, various novel concepts, such as chiral spintronics, skyrmionic spin textures, and magnetic domain walls in spintronics, have been actively pursued. The rate of progress in spintronics is truly astounding (Editorial, 2022). In the past century, superconductivity has raised many theoretical challenges and triggered numerous technical developments. Discovered in mercury in 1911, this unique phenomenon occurs when the electrical resistance of many metals and alloys drops to zero when cooled below about 4 K, the temperature of liquid helium. More significantly, the material expels magnetic flux and becomes perfectly diamagnetic. Superconductivity has been adopted or evaluated for a wide range of technical applications (Rogalla and Kes, 2012). These include electric power generation, electric power transmission, high-speed maglev transportation, and ultra-strong magnetic field generation in high-resolution MRI and other nuclear magnetic resonance (NMR) systems. Interestingly, it took almost 50 years after the original discovery for the phenomena to be explained by the BCS theory (Tinkham, 2004) of reciprocal-space electron pairing, via the electron–phonon interaction. This electron–phonon mediated superconductivity became known as “conventional superconductivity” after the discovery of “high-temperature superconductivity” (Müller and Bednorz, 1987) at liquid nitrogen temperatures in tetragonal copper oxides in 1986. This threw open a new challenge to explain the properties of strongly correlated anisotropic electronic systems. Further new classes of superconductors that remain to be properly understood include the iron-based pnictides discovered in 2008 (Hosono et al., 2018). Recently, a two-dimensional superlattice made from a twisted bilayer of “magic angle” graphene was found to exhibit unconventional superconductivity (Cao et al., 2018), driven by electron–electron interactions. The phenomenon of spontaneous symmetry breaking (SSB) is ubiquitous in physics, and plays a fundamental role in CMP, particle physics, and even in cosmology. A classic illustration of SSB being Heisenberg’s 1928 paper on ferromagnetism (Heisenberg, 1928). It is responsible for various collective modes, such as the famous Nambu-Goldstone (NG) and Higgs modes. Named by Baker and Glashow (Baker and Glashow, 1962), SSB corresponds to the case where the ground or lowest-energy state does not have the symmetry of the underlying dynamics. Instead, multiple degenerate ground states are available and the symmetry breaking is spontaneous because it is not clear, a priori, which one of those ground states will be chosen. In this well-trodden territory a simple analogy of this abstruse concept given by A. Salam is illuminating (CERN, 1977): “Imagine a banquet where guests sit at round tables. A bird’s eye view of the scene presents total symmetry, with serviettes alternating with people around each table. A person could equally well take a serviette from his right or from his left. The symmetry is spontaneously broken when one guest decides to pick up from his left and everyone else follows suit.” Over the years, development of the concept of SSB has taken a circuitous route (Gaudenzi, 2022): “from the land of particle physics to the physics of many bodies (solids and nuclei), from there to the province of superconductivity, and from the latter back to particle physics ... .” SSB’s significance extends beyond the physics of the Higgs boson to the study of low-energy phenomena such as superconductivity, superfluidity, magnetism, and others. Historically, the particle physics community has been guided by this exploration in their quest to understand and discover particles like the Higgs boson. More to the point, when a continuous symmetry is spontaneously broken, a gapless mode, the NG mode, appears which governs the low-energy behavior of the system. As an example, in a solid, the acoustic phonon is the NG mode associated with translational symmetry breaking and determines the behavior of the specific heat at low temperature (the Debye T3 law). Recently, signatures of Higgs and Goldstone modes have been proposed to exist in a perovskite oxide (Marthinsen et al., 2018), in other crystalline structures (Vallone, 2019), and also in various other quantum systems (Léonard et al., 2017). On a lighter note, it is the symmetry breaking that is claimed to have prevented the fabled donkey (Weintraub, 2012) in the parable by Jean Buridan (1300–50) from dying of starvation (Fubini, 1974). Common to the theoretical descriptions of these phenomena across CMP and high-energy physics is Quantum field theory (QFT). QFT was originally developed to describe the fundamental processes in high-energy physics, but has now become a valuable tool to address problems across physics, in particular, in CMP (Fradkin, 2013; see also, Mustafa, 2023), statistical physics, modern quantum chemistry, and even in

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pure mathematics (Folland, 2008). Advances in our understanding of strongly correlated electron systems in CMP such as high-temperature superconductivity and the FQHE have been aided by the QFT. It has facilitated the treatment of spontaneous symmetry broken states, such as superfluids. Interestingly, many of the conceptual developments in CMP have, in return, had a reciprocal impact in QFT. A spectacular example being Nambu’s work in dynamical symmetry breaking (Weinberg, 1986; Nambu, 2009) in particle physics, based on the BCS theory of superconductivity. A glass is a noncrystalline solid obtained by quenching a liquid. It has a liquid-like structure, but a transition from liquid to solid behavior occurs on cooling. It may have one of a continuum of possible ground states that exhibit frozen-in liquid-like disorder. The term “spin glass” was coined by B.R. Coles (Mydosh, 1993) in 1970 for a diluted magnetic material with frozen-in paramagnetic-like spin disorder where the magnetic moments interact with randomly varying positive or negative exchange interactions leading to a multitude of possible ground states with different, random orientations of the spins. As in a normal glass, the energy barriers between the continuum of metastable states prevent it from ever reaching equilibrium. This phase of condensed matter has been an important and productive area of research. Spin-glass models are conceptually simple, but embody sophisticated physics. These models have become a paradigm for the solution of complex optimization problems (Mézard, 2022): much like the undulating flight patterns of starlings around the skies over Rome (Parisi, 2023). After cooling from the paramagnetic phase, the spin glass remains out of equilibrium, and it slowly evolves. This aging phenomenon corresponds to the growth of a “spin-glass order” parameter, whose correlation length can be measured (Joh et al., 1999). A cooling temperature step during aging causes a partial “rejuvenation,” while the “memory” of previous aging is stored and can be retrieved (Refregier et al., 1987; Bouchaud et al., 2001). Many glassy materials present aging, and rejuvenation and memory effects can be found in other cases, but they are usually less pronounced. Numerical simulations of these phenomena are under active investigation using custom-built computers (Baity-Jesi et al., 2023; Vincent, 2023). A general understanding of glassy systems, to which spin glasses bring some special insight, is being pursued. It was realized decades ago that the physics of spin glasses has deep connections with the behavior of complex biological systems. In fact, spin-glass type states in neural network models offer interesting insights into states of high and low brain activity, as observed in mammals (Recio and Torres, 2016). These analogies are potentially useful for applications in fields such as robotics and artificial intelligence (AI) (Andriuschenko et al., 2022). One of the most spectacular phenomena in CMP, discovered in the last century, was Bose-Einstein condensation (BEC). This macroscopic quantum phenomenon had its genesis in the work of Satyendra Nath Bose (Bose, 1924; Blanpied, 1972; Tomczyk, 2022) (whose name is forever associated with bosons), and has fascinated physicists for decades, ever since the first observation of this exotic state of matter in an ultracold gas of rubidium atoms (Georgescu, 2020). Among the advances made in this field, we highlight only a few: simulation of correlated electron states, such as the FQHE state and a bosonic analog of the Laughlin state, and the famous BEC-BCS crossover from a BEC to a BCS superfluid of weakly bound Cooper pairs, as the interparticle interaction strength is varied. Noteworthy is the recent measurement of the dynamic structure factor in an ultracold Fermi gas of lithium-6 atoms in the “unitary” limit, where a linear phonon mode for low momentum transfer can be clearly discerned (Biss et al., 2022). Perhaps such a method might one day shed light on the Laughlin quasiparticle gap and the collective modes that are still largely unresolved (Chakraborty and Pietiläinen, 1988, 1995; Halperin, 1983; Clark and Maksym, 1989). Liquid helium, 4He, and the less common isotope 3He are unique quantum systems (Volovik, 2009). They remain liquid even at absolute zero temperature, a direct manifestation of Heisenberg’s uncertainty principle and zero-point energy. The light mass of helium atoms and their weak interatomic attraction leads to a substantial zero-point kinetic energy and the system remains in the liquid state at ambient pressure. Solidification occurs when a pressure of about 25 MPa is applied. The different quantum statistics of the two helium isotopes (4He is a boson while 3He is a fermion) are responsible for the different physical characteristics of the two systems. Microscopic approaches to understand the ground state and low-energy excitations of the liquid state have occupied researchers for many decades (see for example, Ristig and Clark, 1976; Lam et al., 1977). These highly correlated quantum fluids have been established as systems where the efficacies of different many-body theories can be tested (Kallio and Smith, 1977; Lantto et al., 1977). These systems have been described as Ariadne’s thread in this Encyclopedia, where the strongly interacting constituent particles are treated by a variety of ingenious theoretical and experimental approaches. Interestingly, liquid helium has been proposed to be a medium to design multiexcitation detectors to probe dark matter (Schutz and Zurek, 2016) in the keV mass limit. Although this looks far afield from CMP, the interface between CMP and high-energy particle physics could inspire new horizons in CMP, as discussed earlier.

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CMP also has an enormous impact on our daily lives with unceasing technological innovation stemming directly from the advances in CMP research. It is also closely associated with the progress of our understanding of materials science and mechanical, chemical, and electrical engineering that deals with properties of novel materials and devices. As some authors have very aptly pointed out (Chaikin and Lubensky, 1995): The use and understanding of matter in its condensed (liquid or solid) state have gone hand in hand with the advances of civilization and technology since the first use of primitive tools. So important has the control of condensed matter been to man that prehistoric ages—the Stone Age, the Bronze Age, the Iron Age—are named after the material dominating the technology of the time. In fact, modern civilization is entirely dependent on our deep understanding and practical use of materials, old and new: Steel, glass, and concrete have shaped our modern living with comfort, health, longevity, and material wealth. All around us, from a tiny paper clip to giant skyscrapers, from jet planes to modern medicines, and the ubiquitous plastics that are integrated into nearly all aspects of human life, are testaments to our control over the myriad behavior of materials (Miodownik, 2014). Today, a large number of materials are designed by using principles of quantum mechanics (Freeman, 2002). The Silicon chip that launched the information age is a great example of the effect of CMP on our daily lives. The scaling of transistors down to the nanometer range has enabled them to be embedded in greater numbers in the all-pervasive electronic devices that have transformed every facet of life all around the world. The transition from room-size to pocket-size computers with ever increasing computational power and memory, that occurred in only a few decades, epitomizes the unrelenting progress made in this direction. Advances in nanoscale research have opened the door to exploration of the natural world at the ultra-small scale (Chakraborty, 2022), with the potential to shape our future world. Quantum dots (QDs), the quasi-zero-dimensional electron systems, are one of the most extensively studied nanostructures in the past three decades. They represent the ultimate reduction in dimensionality of a semiconductor device. Here the electrons are confined (either electrostatically or structurally) in all three directions and occupy spectrally sharp energy levels similar to those found in atoms (Chakraborty, 1999), hence the popular moniker artificial atoms (Maksym and Chakraborty, 1990). QDs and other similar nanostructures (Chakraborty, 2003) offer a rich variety of fundamental phenomena that result from manipulation of single electrons (single electronics) or single spins at the nanoscale. The study of interacting electrons in a QD requires large-scale numerical computations involving matrices of dimensions in the millions (“monster matrices”) (Chakraborty and Pietiläinen, 2005; Pietiläinen and Chakraborty, 2006). QDs are also thought to have vast potential for future technological applications. One prominent example being quantum cryptography where the QDs are used in single photon emitters (Nowak et al., 2014), and in single photon detectors (Shields et al., 2000). Quantum cryptography, in turn, forms the basis of the quantum Internet (Liao et al., 2018). Other applications include lasers, and spintronics in quantum computing, entertainment industry (Bourzac, 2013), and even in biology (Bruchez and Holz, 2007). Research in recent years has increasingly revealed that quantum phenomena are also ubiquitous in the natural world (Ball, 2011). They govern how birds are able to navigate around the globe using Earth’s magnetic field or how plants use photosynthesis to harvest sunlight for conversion to chemical energy. All depend on subtle quantum effects that are now coming to light, thereby ushering the topic of quantum biology into CMP. Deoxyribonucleic acid (DNA), the molecule responsible for storage of genetic information in the cells of all living organism, is entrusted with the task of preserving, copying, and transmitting information that are necessary for living beings to grow and function. It is often referred to as the body’s hereditary material as DNA is replicated and transmitted from parents to their offspring. This “molecule of life” was discovered in 1869 by Friedrich Miescher in Tübingen, Germany, who called it “nuclein,” as it was isolated from the nucleus of white blood cells (Dahm, 2005; Thess et al., 2021). It took more than eighty years from that momentous event, to discover that DNA has a right-handed double-helix structure with appropriate base pairing (Watson and Crick, 1953; Klug, 2004). That discovery was so important that it has been compared with the discovery of the atomic nucleus in physics (Frank-Khamenetskii, 1997). Understanding the structure of the atom heralded quantum physics, while understanding of the structure of DNA brought forth the field of molecular biology. DNA has several remarkable properties that make this molecule a prime candidate for nano-electronic materials (Chakraborty, 2007). These include molecular recognition and the ability to self-assemble via the complementarity of the base sequences on the two strands, which means that DNA can be integrated error-free in current semiconductor technology. DNA can also detect information about its own integrity that can trigger its eventual repair mechanism. All these properties are perfectly suitable for developing nanoscale electronics involving DNA

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(Taniguchi and Kawai, 2006). Despite the promise of harnessing biology to realize integrated molecular electronics remains fascinating, much work remains to be done to make it a reality (Braun and Keren, 2004). Protein molecules play a vital role in all living organisms. They are described as Nature’s robots (Tanford and Reynolds, 2001) because for every imaginable task required in a living organism, and for every step of that task, there is a protein designed to carry it out. These proteins are even programmed to know when to turn on or off. Just as the twenty-six letters in the alphabet can spell out hundreds of thousands of words, there are twenty different naturally occurring amino acids that can be combined into many more different proteins than there are atoms in the known universe! Protein’s biological function is determined by its unique and native three-dimensional structures that are characteristic of individual proteins (Gomes and Faísca, 2019; Echenique, 2007). The question of how proteins fold to their unique, compact, and highly organized functional states has remained an enduring challenge (“The protein-folding problem”) (Chan and Dill, 1993; Dill and MacCallum, 2012; Wolynes and Eaton, 1999). Recently, there has been a tremendous development in this area of research. In November 2020, an AI program called AlphaFold, designed to tackle the problem of protein folding, predicted the three-dimensional structures of almost every protein known to science—about 200 millions in all (Jumper et al., 2021). AlphaFold uses a machine learning methodology to bring together information from preexisting knowledge of protein structures, and other geometric and physical constraints to predict protein structures. While this has been heralded a watershed moment in structural biology (Perrakis and Sixma, 2021), there are many key questions that AlphaFold cannot address yet. These include aspects of protein behavior that cannot be understood from a static structure, such as how proteins respond to their environment or how intrinsically disordered proteins actually are organized. Some authors (Nasser et al., 2021) have likened it to solving a crime story: while AlphaFold presents a snapshot in time, the question about how we get there remains unanswered. Biological systems have developed elaborate mechanisms to ensure that proteins fold correctly, or otherwise, they are detected and degraded before any serious harm can result to the host organism. However, protein misfolding does happen and that misfolding in the cell is linked to an array of diseases, including cancers, cardiovascular disease, type II diabetes, and numerous neurodegenerative diseases, such as Alzheimer’s, Parkinson’s, and Huntington’s disease. NMR spectroscopy has been an important technique to study protein folding where valuable insights are gained into many aspects of the folding process (Jane Dyson and Wright, 1996). In this Encyclopedia, we strive to present a taste of all the developments in CMP described above (and much more), written by experts in the field. We are keenly aware that many important and interesting topics are not covered here. However, that is not for want of trying. As some authors had rightly pointed out (Hoddeson, 1992): The field is huge and varied and lacks the unifying features beloved of historians, neither a single hypothesis or set of basic equations underpin the field, as in quantum mechanics or relativity theory, nor a single spectacular and fundamental discovery such as uranium fission for nuclear technology or the structure of DNA for molecular biology. CMP owes what unity it has simply to a common concern with solid and liquid materials. This encompasses such a large range of theories and methods that the field resembles a confederation of interest groups rather than a single entity. Clearly, the task of collecting everything in one place is a monumental endeavor, but we do hope that our compilation of a large number of chapters in this five-volume work, covering a broad range of in-depth descriptions of varied phenomena in CMP will be a treasured source of facts and ideas for the community. It is undeniably true that there would have been no Encyclopedia without the extraordinary contributions of the large number of authors whose valuable time and efforts brought this project to fruition. My sincere thanks to all of them. I also wish to extend my deep appreciation to the staff at Elsevier for their skillful handling of this huge project. The cover image was created by Hong-Yi Chen from National Taiwan Normal University. The image below was created by Elisabetta Collini, one of the authors. My deepest appreciation to them for taking the time to do this work. Finally, my sincere thanks to the Section editors, Vladimir Fomin, Ana Sanchez, Roberto Raimondi, Hideo Aoki, Rudolf Römer, and Robert Blick for the effort they all expended, that has contributed to the success of this venture. Special thanks go to Vladimir Fomin and Ana Sanchez for securing a large share of the chapters and for their enduring help in providing valuable advice and support to bring the project to fruition. Thanks are also due to many of my friends and colleagues, in particular, Michael Coey, Jürg Fröhlich, Sinéad Griffin, Peter Maksym, Eric Vincent, and Jakob Yngvason for their careful and critical reading of the introduction. Tapash Chakraborty St. Catharines Ontario, Canada

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References Abergel DSL, Apalkov V, Berashevich J, Ziegler K, and Chakraborty T (2010) Properties of graphene: A theoretical perspective. Advances in Physics 59: 261. https://doi.org/10.1080/ 00018732.2010.487978. Andriuschenko P, Kapitan D, and Kapitan V (2022) A new look at the spin glass problem from a deep learning perspective. Entropy 24: 697. https://doi.org/10.3390/e24050697. Apalkov VM and Chakraborty T (2006) Fractional quantum Hall states of Dirac electrons in graphene. Physical Review Letters 97: 126801. https://doi.org/10.1103/ PhysRevLett.97.126801. Apalkov VM and Chakraborty T (2010) Controllable driven phase transitions in fractional quantum Hall states in bilayer graphene. Physical Review Letters 105: 036801. https://doi.org/ 10.1103/PhysRevLett.105.036801. Apalkov VM and Chakraborty T (2011) Stable pfaffian state in bilayer graphene. Physical Review Letters 107: 186803. https://doi.org/10.1103/PhysRevLett.107.186803. Arovas D, Schrieffer JR, and Wilczek F (1984) Fractional statistics and the quantum Hall effect. Physical Review Letters 53: 722. https://doi.org/10.1103/PhysRevLett.53.722. Avila A and Jitomirskaya S (2006) Solving the ten martini problem. Lecture Notes in Physics 690: 5. https://doi.org/10.1007/3-540-34273-7_2. Baity-Jesi M, et al. (2023) Memory and rejuvenation effects in spin glasses are governed by more than one length scale. Nature Physics 19: 978–985. https://doi.org/10.1038/ s41567-023-02014-6. Baker M and Glashow SL (1962) Spontaneous breakdown of elementary particle symmetries. Physics Review 128: 2462. https://doi.org/10.1103/PhysRev.128.2462. Ball P (2011) Physics of life: The dawn of quantum biology. Nature 474: 272. https://doi.org/10.1038/474272a. Baltz V, et al. (2018) Antiferromagnetic spintronics. Reviews of Modern Physics 90: 015005. https://doi.org/10.1103/RevModPhys.90.015005. Bartolomei H, et al. (2020) Fractional statistics in anyon collisions. Science 368: 173. https://doi.org/10.1126/science.aaz5601. Biss H, et al. (2022) Excitation spectrum and superfluid gap of an ultracold Fermi gas. Physical Review Letters 128: 100401. https://doi.org/10.1103/PhysRevLett.128.100401. Blanpied WA (1972) Satyendranath Bose: Co-founder of quantum statistics. American Journal of Physics 40: 1212. https://doi.org/10.1119/1.1986805. Blundell S (2001) Magnetism in Condensed Matter. New York: Oxford U.P. Bose SN (1924) Plancks gesetz und lichtquantenhypothese. Zeitschrift für Physik 26: 178. https://doi.org/10.1007/BF01327326. Bouchaud J-P, Dupuis V, Hammann J, and Vincent E (2001) Separation of time and length scales in spin glasses: Temperature as a microscope. Physical Review B 65: 024439. https://doi.org/10.1103/PhysRevB.65.024439. Bourzac K (2013) Quantum dots go on display. Nature 493: 283. https://doi.org/10.1038/493283a. Brataas A, Kent AD, and Ono H (2012) Current-induced torques in magnetic materials. Nature Materials 11: 372. https://doi.org/10.1038/nmat3311. Braun E and Keren K (2004) From DNA to transistors. Advances in Physics 53: 441. https://doi.org/10.1080/00018730412331294688. Bruchez MP and Holz CZ (2007) Quantum Dots: Applications in Biology. New Jersey: Humana Press. https://doi.org/10.1385/1597453692. Cao Y, et al. (2018) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556: 43. https://doi.org/10.1038/nature26160. CERN. (1977) Gauge theories with the lid, partially, off. CERN Courier 17: 271. https://cds.cern.ch/record/1730245/files/vol17-issue9-p271-e.pdf. Chaikin PM and Lubensky TC (1995) Principles of Condensed Matter Physics. Cambridge University Press. https://doi.org/10.1017/CBO9780511813467. Chakraborty T (1985) Elementary excitations in the fractional quantum Hall effect. Physical Review B 31: 4026. https://doi.org/10.1103/PhysRevB.31.4026. Chakraborty T (1999) Quantum Dots: A Survey of the Properties of Artificial Atoms. Amsterdam: North-Holland. https://doi.org/10.1016/B978-0-444-50258-2.X5000-7. Chakraborty T (2003) Nanoscopic quantum rings: A new perspective. Advances in Solid State Physics 43: 79. https://doi.org/10.1007/978-3-540-44838-9_6. Chakraborty T (ed.) (2007) Charge Migration in DNA. Springer. https://doi.org/10.1007/978-3-540-72494-0. Chakraborty T (2022) Nanoscale Quantum Materials: Musings on the Ultra-Small World. CRC Press. https://doi.org/10.1201/9781003090908. Chakraborty T and Apalkov VM (2015) Fractal butterflies of Dirac fermions in monolayer and bilayer graphene. IET Circuits, Devices and Systems 9: 19. https://doi.org/10.1049/ietcds.2014.0275. Chakraborty T and Pietiläinen P (1988) The Fractional Quantum Hall Effect. Springer Verlag. https://doi.org/10.1007/978-3-642-97101-3. Chakraborty T and Pietiläinen P (1995) The Quantum Hall Effects, 2nd. Springer. https://doi.org/10.1007/978-3-642-79319-6. Chakraborty T and Pietiläinen P (2005) Optical signatures of spin-orbit interaction effects in a parabolic quantum dot. Physical Review Letters 95: 136603. https://doi.org/10.1103/ PhysRevLett.95.136603. Chan HS and Dill KA (1993) The protein folding problem. Physics Today 46: 24. https://doi.org/10.1063/1.881371. Clark R and Maksym P (1989) Fractional quantum Hall effect in a spin. Physics World 2: 39. https://doi.org/10.1088/2058-7058/2/9/23.

xvi

Introduction

Cohen ML (2008) Fifty years of condensed matter physics. Physical Review Letters 101: 250001. https://doi.org/10.1103/PhysRevLett.101.250001. Dahm R (2005) Friedrich miescher and the discovery of DNA. Developmental Biology 278: 274. https://doi.org/10.1016/j.ydbio.2004.11.028. Dill KA and MacCallum JL (2012) The protein-folding problem, 50 years on. Science 338: 1042. https://doi.org/10.1126/science.1219021. Echenique P (2007) Introduction to protein folding for physicists. Contemporary Physics 48: 81. https://doi.org/10.1080/00107510701520843. Editorial (2022) New horizons in spintronics. Nature Materials 21: 1. https://doi.org/10.1038/s41563-021-01184-z. Eisenstein JP and Stormer HL (1990) The fractional quantum Hall effect. Science 248: 1510. https://doi.org/10.1126/science.248.4962.1510. Folland GB (2008) Quantum Field Theory: A tourist guide for Mathematicians. American Mathematical Society. https://doi.org/10.1090/surv/149. Fradkin E (2013) Field Theories of Condensed Matter Physics. Cambridge University Press. https://doi.org/10.1017/CBO9781139015509. Frank-Khamenetskii MD (1997) Unraveling DNA: The Most Important Molecule of life. Basic Books. Freeman AJ (2002) Materials by design and the exciting role of quantum computation/simulation. Journal of Computational and Applied Mathematics 149: 27. https://doi.org/ 10.1016/S0377-0427(02)00519-8. Fubini S (1974) Present trends in particle physics. In: Proc. Intl. Conf. on High Energy Physics. Didcot, England: Rutherford Laboratory. https://cds.cern.ch/record/417271/files/ CM-P00060491.pdf. Gaudenzi R (2022) Historical Roots of Spontaneous Symmetry Breaking: Steps Towards an Analogy. Springer. https://doi.org/10.1007/978-3-030-99895-0. Georgescu I (2020) 25 Years of BEC. Nature Reviews Physics 2: 396. https://doi.org/10.1038/s42254-020-0211-7. Gomes CM and Faísca PFN (2019) Protein Folding: An Introduction. Springer. https://doi.org/10.1007/978-3-319-00882-0_1. Halperin BI (1983) Theory of the quantized Hall conductance. Helvetica Physica Acta 56: 75. https://doi.org/10.5169/seals-115362. Halperin BI (1984) Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Physical Review Letters 52: 1583. https://doi.org/10.1103/PhysRevLett.52.1583. Harper PG (1955) Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A 68: 874. https://doi.org/10.1088/03701298/68/10/304. Hartmann U (2000) Magnetic Multilayers and Giant Magnetoresistance: Fundamentals and Industrial Applications. Berlin: Springer Verlag. https://doi.org/10.1007/978-3-66204121-5. Heisenberg W (1928) Zur theorie des ferromagnetismus. Zeitschrift für Physik 49: 619. https://doi.org/10.1007/BF01328601. Hoddeson L (1992) Out of the Crystal Maze: Chapters From the History of Solid-State Physics. N.Y.: Oxford University Press. Hofstadter D (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14: 2239. https://doi.org/10.1103/ PhysRevB.14.2239. Hosono H, et al. (2018) Recent advances in iron-based superconductors toward applications. Materials Today 21: 278. https://doi.org/10.1016/j.mattod.2017.09.006. Jane Dyson H and Wright PE (1996) Insights into protein folding from NMR. Annual Review of Physical Chemistry 47: 369. https://doi.org/10.1146/annurev.physchem.47.1.369. Jiang Z, et al. (2007) Quantum Hall effect in graphene. Solid State Communications 143: 14. https://doi.org/10.1016/j.ssc.2007.02.046. Joh YG, Orbach R, Wood JJ, Hammann J, and Vincent E (1999) Extraction of the spin glass correlation length. Physical Review Letters 82: 438. https://doi.org/10.1103/ PhysRevLett.82.438. Jumper J, et al. (2021) Highly accurate protein structure prediction with alphafold. Nature 596: 583. https://doi.org/10.1038/s41586-021-03819-2. Kallio A and Smith RA (1977) How simple can HNC be and still be right? Physics Letters B 68: 315. https://doi.org/10.1016/0370-2693(77)90483-X. Kibble T and Srivastava A (2013) Condensed matter analogues of cosmology. Journal of Physics. Condensed Matter 25: 400301. https://doi.org/10.1088/0953-8984/25/ 40/400301. Klug A (2004) The discovery of the DNA double helix. Journal of Molecular Biology 335: 3. https://doi.org/10.1016/j.jmb.2003.11.015. Lam PM, Clark JW, and Ristig ML (1977) Density matrix and momentum distribution of helium liquids and nuclear matter. Physical Review B 16: 222. https://doi.org/10.1103/ PhysRevB.16.222. Lantto LJ, Jackson AD, and Siemens PJ (1977) HNC variational calculations of boson matter. Physics Letters B 68: 311. https://doi.org/10.1016/0370-2693(77)90482-8. Laughlin RB (1983) Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50: 1395. https://doi.org/10.1103/ PhysRevLett.50.1395. Laughlin RB (1984) Primitive and composite ground states in the fractional quantum Hall effect. Surface Science 142: 163. https://doi.org/10.1016/0039-6028(84)90301-7. Laughlin RB and Pines D (2000) The theory of everything. Proceedings of the National Academy of Sciences of the United States of America 97: 28. https://doi.org/10.1073/ pnas.97.1.28. Leggett A (1992) On the nature of research in condensed-state physics. Foundations of Physics 22: 221. https://doi.org/10.1007/BF01893613. Leggett A (2018) Reflections on the past, present and future of condensed matter physics. Science Bulletin 63: 1019. https://doi.org/10.1016/j.scib.2018.07.004. Leinaas JM and Myrheim J (1977) On the theory of identical particles. Il Nuovo Cimento B 37: 1. https://doi.org/10.1007/BF02727953. Léonard J, et al. (2017) Monitoring and manipulating higgs and goldstone modes in a supersolid quantum gas. Science 358: 1415. https://doi.org/10.1126/science.aan2608. Liao S-K, et al. (2018) Satellite-relayed intercontinental quantum network. Physical Review Letters 120: 030501. https://doi.org/10.1103/PhysRevLett.120.030501. Liu X, Hao Z, Watanabe K, Taniguchi T, Halperin BI, and Kim P (2019) Interlayer fractional quantum Hall effect in a coupled graphene double layer. Nature Physics 15: 893. https://doi. org/10.1038/s41567-019-0546-0. Locatelli N, Cros V, and Grollier J (2014) Spin-torque building blocks. Nature Materials 13: 11. https://doi.org/10.1038/nmat3823. Lu W (2022) Quark-gluon plasma and nucleons à la Laughlin. International Journal of Geometric Methods in Modern Physics 19: 2250061. https://doi.org/10.1142/ S021988782250061X. Maksym P and Chakraborty T (1990) Quantum dots in a magnetic field: Role of electron-electron interactions. Physical Review Letters 65: 108. https://doi.org/10.1103/ PhysRevLett.65.108. Marthinsen A, et al. (2018) Goldstone-like phonon modes in a (111)-strained perovskite. Physical Review Materials 2: 014404. https://doi.org/10.1103/PhysRevMaterials.2.014404. Mézard M (2022) Spin glasses and optimization in complex systems. Europhysics News 53: 15. https://doi.org/10.1051/epn/2022105. Miodownik M (2014) Stuff Matters: Exploring the Marvelous Materials That Shape Our Man-Made World. N.Y.: Houghton Mifflin Harcourt. Morf R and Halperin BI (1986) Monte Carlo evaluation of trial wave functions for the fractional quantized Hall effect: Disk geometry. Physical Review B 33: 2221. https://doi.org/ 10.1103/PhysRevB.33.2221. Mourino MR (1991) From thales to lauterbur, or from the loadstone to MR imaging: Magnetism and medicine. Radiology 180: 593. https://doi.org/10.1148/radiology.180.3.1871268. Müller KA and Bednorz JG (1987) The discovery of a class of high-temperature superconductors. Science 237: 1133. https://doi.org/10.1126/science.237.4819.1133 Mustafa MG (2023) An introduction to thermal field theory and some of its application. The European Physical Journal Special Topics 232: 1369–1457. https://doi.org/10.1140/epjs/ s11734-023-00868-8. Mydosh JA (1993) Spin Glasses: An Experimental Introduction. CRC Press. Nakamura J, et al. (2020) Direct observation of anyonic braiding statistics. Nature Physics 16: 931. https://doi.org/10.1038/s41567-020-1019-1. Nambu Y (2009) Nobel lecture: Spontaneous symmetry breaking in particle physics: A case of cross fertilization. Reviews of Modern Physics 81: 1015. https://doi.org/10.1103/ RevModPhys.81.1015. Nasser R, Dignon GL, Razban RM, and Dill KA (2021) The protein folding problem: The role of theory. Journal of Molecular Biology 433: 167126. https://doi.org/10.1016/j. jmb.2021.167126. Nawrocki W (2019) Introduction to Quantum Metrology, 2nd. Springer. https://doi.org/10.1007/978-3-030-19677-6. Nowak AL, et al. (2014) Deterministic and electrically tunable bright single-photon source. Nature Communications 5: 3240. https://doi.org/10.1038/ncomms4240.

Introduction

xvii

Obermair GM and Wannier GH (1976) Bloch electrons in magnetic fields: Rationality, irrationality, degeneracy. Physica Status Solidi (B) 76: 217. https://doi.org/10.1002/ pssb.2220760122. Osadchy D and Avron JE (2001) Hofstadter butterfly as quantum phase diagram. Journal of Mathematical Physics 42: 5665. https://doi.org/10.1063/1.1412464. Parisi G (2023) In a Flight of Starlings: The Wonders of Complex Systems. Penguin Press. Perrakis A and Sixma TK (2021) Ai revolutions in biology. EMBO Reports 22: e54046. https://doi.org/10.15252/embr.202154046. Pietiläinen P and Chakraborty T (2006) Energy levels and magneto-optical transitions in parabolic quantum dots with spin-orbit coupling. Physical Review B 73: 155315. https://doi. org/10.1103/PhysRevB.73.155315. Pinarbasi M and Kent AD (2022) Perspectives on spintronics technology: Giant magnetoresistance to spin transfer torque magnetic random access memory. APL Materials 10: 020901. https://doi.org/10.1063/5.0075945. Recio I and Torres JJ (2016) Emergence of low noise frustrated states in E/I balanced neural networks. Neural Networks 84: 91. https://doi.org/10.1016/j.neunet.2016.08.010. Refregier P, Vincent E, Hammann J, and Ocio M (1987) Ageing phenomena in a spin-glass: Effect of temperature changes below Tg. Journal of Physics (France) 48: 1533. https://doi. org/10.1051/jphys:019870048090153300. Ristig ML and Clark JW (1976) Density matrix of quantum fluids. Physical Review B 14: 2875. https://doi.org/10.1103/PhysRevB.14.2875. Rogalla H and Kes PH (2012) 100 Years of Superconductivity. Boca Raton: Taylor & Francis Group. https://doi.org/10.1201/b11312. Schutz K and Zurek KM (2016) Detectability of light dark matter with superfluid helium. Physical Review Letters 117: 121302. https://doi.org/10.1103/PhysRevLett.117.121302. Shields AJ, et al. (2000) Detection of single photons using a field-effect transistor gated by a layer of quantum dots. Applied Physics Letters 103: 3673. https://doi.org/ 10.1063/1.126745. Steptoe A (1986) Mozart, mesmer and ‘cosi fan tutte’. Musics and Letters 67: 248. https://doi.org/10.1093/ml/67.3.248. Störmer HL (1992) Two-dimensional electron correlation in high magnetic fields. Physica B: Condensed Matter 177: 401. https://doi.org/10.1016/0921-4526(92)90138-I. Tanford C and Reynolds J (2001) Nature’s Robots: A History of Proteins. Oxford University Press. Taniguchi M and Kawai T (2006) Dna electronics. Physica E: Low-dimensional Systems and Nanostructures 33: 1. https://doi.org/10.1016/j.physe.2006.01.005. Thess A, et al. (2021) Historic nucleic acids isolated by Friedrich Miescher contain RNA beside DNA. Biological Chemistry 402: 1179. https://doi.org/10.1515/hsz-2021-0226. Tinkham M (2004) Introduction to Superconductivity. N.Y.: Dover. Tomczyk H (2022) Did Einstein predict Bose-Einstein condensation? Studies in History and Philosophy of Science 93: 30. https://doi.org/10.1016/j.shpsa.2022.02.014. Vallone M (2019) Higgs and goldstone modes in crystalline solids. Physica Status Solidi 257: 1900443. https://doi.org/10.1002/pssb.201900443. Van Vleck JH (1978) Quantum mechanics—The key to understanding magnetism. Reviews of Modern Physics 50: 181. https://doi.org/10.1103/RevModPhys.50.181. Vincent E (2023) Rejuvenated but remembering. Nature Physics 19: 926–927. https://doi.org/10.1038/s41567-023-02097-1. Volovik GE (2009) The Universe in a Helium Droplet. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199564842.001.0001. von Klitzing K (2019) Essay: Quantum Hall effect and the new international system of units. Physical Review Letters 122: 200001. https://doi.org/10.1103/PhysRevLett.122.200001. von Klitzing K, Chakraborty T, Kim P, Madhavan V, Dai X, McIver J, Tokura Y, Savary L, Smirnova D, Rey AM, Felser C, Gooth J, and Qi X (2020) 40 years of the quantum Hall effect. Nature Reviews Physics 2: 397. https://doi.org/10.1038/s42254-020-0209-1. Watson JD and Crick FHC (1953) Molecular structure of nucleic acids. Nature 171: 737. https://doi.org/10.1038/171737a0. Weinberg S (1986) Superconductivity for particular theorists. Progress of Theoretical Physics Supplement 86: 43. https://doi.org/10.1143/PTPS.86.43. Weintraub R (2012) What can we learn from Buridan’s ass? Canadian Journal of Philosophy 42: 281. https://doi.org/10.1353/cjp.2012.0013. Weiss D (2013) The butterfly emerges. Nature Physics 9: 395. https://doi.org/10.1038/nphys2680. Wolynes PG and Eaton WA (1999) The physics of protein folding. Physics World 12: 39. https://doi.org/10.1088/2058-7058/12/9/24. Yamaguchi A, Hirohata A, and Stadler BJH (2021) Nanomagnetic Materials: Fabrication, Characterization and Application. Elsevier. (Chapter 5). https://doi.org/10.1016/C2018-005445-6.

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LIST OF CONTRIBUTORS FOR VOLUME 3 Kyosuke Adachi RIKEN Center for Biosystems Dynamics Research, Kobe, Japan; RIKEN InterdisciplinaryTheoretical and Mathematical Sciences Program, Wako, Japan Jimmie Adriazola Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States M Agio Università degli Studi di Pavia, Pavia, Italy Farzin Akbar Micro- and NanoBiomedical Engineering Group (MNBE), Institute for Integrative Nanosciences, Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany M-C Amann Technische Universität Munchen, Garching, Germany LC Andreani Università degli Studi di Pavia, Pavia, Italy SN Aqida Automotive Engineering Centre, Universiti Malaysia Pahang, Pekan, Pahang, Malaysia Giorgio Baccarani Department of Electrical Electronic and Information Engineering (DEI), and Advanced Research Center on Electronic Systems E. de Castro (ARCES), University of Bologna, Bologna, Italy M Basini Department of Physics, Stockholm University, Stockholm, Sweden J Bass Michigan State University, East Lansing, MI, United States K Behnia Ecole Supérieure de Physique et de Chimie Industrielles, Paris, France

Antonio Bianconi Institute of Crystallography, CNR, Rome, Italy; RICMASS Rome International Center for Materials Science, Rome, Italy Christopher Bolton Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States D Bossini Department of Physics and Center for Applied Photonics, University of Konstanz, Konstanz, Germany K Bowman Purdue University, West Lafayette, IN, USA Stéphane Brûlé Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France Gaetano Campi Institute of Crystallography, CNR, Rome, Italy A Cano Université Grenoble Alpes, CNRS, Institut Néel, Grenoble, France David Castellanos Robles Micro- and NanoBiomedical Engineering Group (MNBE), Institute for Integrative Nanosciences, Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany Tapash Chakraborty Department of Physics, Brock University, St. Catharines, ON, Canada; Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada JR Chelikowsky University of Texas at Austin, Austin, TX, USA Kikuo Cho Emeritus of Osaka University, Kobe, Japan Alexander T Clark Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States

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List of Contributors for Volume 3

John W Clark Department of Physics, Washington University, St. Louis, MI, United States; Centro de Investigação em Matemática e Aplicações, University of Madeira, Funchal, Portugal

Elena Gnani Department of Electrical Electronic and Information Engineering (DEI), and Advanced Research Center on Electronic Systems E. de Castro (ARCES), University of Bologna, Bologna, Italy

Rosa Córdoba Institute of Molecular Science, University of Valencia, Paterna, Spain

M Goerdeler Institut für Metallkunde und Metallphysik, Aachen, Germany

Francisco AG de Lira Department of Physics, Federal University of Maranhão, São Luí s, Maranhão, Brazil

H Julian Goldsmid School of Physics, University of New South Wales, Sydney, NSW, Australia

Alexander L Efros Center for Computational Material Science, Naval Research Laboratory, Washington, DC, United States

G Gottstein Institut für Metallkunde und Metallphysik, Aachen, Germany

Christoph Eigen Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom

M Grundmann Universität Leipzig, Leipzig, Germany

NG Einspruch University of Miami, Coral Gables, FL, USA SR Elliott University of Cambridge, Cambridge, United Kingdom Stefan Enoch Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

Sébastien Guenneau The Blackett Laboratory, Department of Physics, Imperial College, London, United Kingdom Ralf Hielscher Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany Y Hirayama NTT Basic Research Laboratories, Kanagawa, Japan

Leonardo Fallani Department of Physics and Astronomy, University of Florence, Florence, Italy; LENS European Laboratory for Nonlinear Spectroscopy, Firenze, Italy

Massimo Inguscio LENS European Laboratory for Nonlinear Spectroscopy, Firenze, Italy; Campus Bio-Medico University of Rome, Rome, Italy

John F Federici Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States

Hajime Ishihara Department of Materials Engineering Science, Osaka University, Osaka, Japan; Center for Quantum Information and Quantum Biology, Osaka University, Osaka, Japan

AM Finkel’stein Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot, Israel; Department of Physics and Astronomy, Texas A&M University, College Station, TX, United States Paola Gallo Dipartimento di Matematica e Fisica, Università “Roma Tre”, Roma, Italy Jorge M García Instituto de Micro y Nanotecnologí a, IMN-CNM, CSIC, (CEI UAM + CSIC), Isaac Newton, Madrid, Spain Ian Gatley Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States Samuel Gatley Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States

EL Ivchenko Ioffe Institute, Russian Academy of Sciences, St. Petersburg, Russia Yoshihiro Iwasa Department of Applied Physics and Quantum-Phase Electronics Center, The University of Tokyo, Tokyo, Japan; RIKEN Center for Emergent Matter Science, Wako, Japan Wataru Izumida Tohoku University, Sendai, Japan Alexey Kavokin International Center for Polaritonics, School of Science, Westlake University, Hangzhou, Zhejiang Province, China; Russian Quantum Center, Skolkovo, Moscow Region, Russia; Moscow Institute of Physics and

List of Contributors for Volume 3

Technology, Institutskiy Pereulok, Dolgoprudny, Moscow Oblast, Russia AV Kimel Institute for Molecules and Materials, Radboud University, Nijmegen, The Netherlands Albert Kirakosyan Department of Solid State Physics, Yerevan State University, Yerevan, Armenia D Klenam Academic Development Unit & School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Johannesburg, South Africa

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LB Ma Institute for Integrative Nanosciences, Leibniz IFW Dresden, Dresden, Germany J Maier Physical Chemistry of Solids, Max Planck Institute for Solid State Research, Stuttgart, Germany Aram Manaselyan Department of Solid State Physics, Yerevan State University, Yerevan, Armenia Euclydes Marega Jr Instituto de Fí sica de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil

Alexander M Korsunsky Trinity College, Oxford, United Kingdom

G Mattei CNR, Istituto dei Sistemi Complessi, Rome, Italy

EE Krasovskii Donostia International Physics Center DIPC, P. Manuel de Lardizabal 4, San Sebastián, Spain; Departamento de Polí meros y Materiales Avanzados: Fí sica, Quí mica y Tecnologí a, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Donostia-San Sebastián, Spain; IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

JD Maynard The Pennsylvania State University, University Park, PA, United States

Wolfgang Lang Faculty of Physics, University of Vienna, Vienna, Austria Fabrice P Laussy Faculty of Science and Engineering, University of Wolverhampton, Wolverhampton, United Kingdom; Russian Quantum Center, Skolkovo, Moscow Region, Russia Jean-Pierre Leburton University of Illinois at Urbana-Champaign, Urbana, IL, United States AP Levanyuk Department of Physics, University of Washington, Seattle, WA, United States A-M Leventi-Peetz Federal Office for Information Security (BSI), Bonn, Germany R Livi Dipartimento di Fisica e Astronomia, Università di Firenze and Sezione INFN di Firenze, Sesto Fiorentino, Italy; ISC-CNR, Sesto Fiorentino, Italy AC Lund Massachusetts Institute of Technology, Cambridge, MA, USA Wenchen Luo School of Physics and Electronics & Hunan Key Laboratory of Nanophotonics and Devices, Central South University, Changsha, China

F McBagonluri Department of Mechanical Engineering, Academic City University College, Accra, Ghana Mariana Medina-Sánchez Micro- and NanoBiomedical Engineering Group (MNBE), Institute for Integrative Nanosciences, Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany; Chair of Micro- and NanoSystems, Center for Molecular Bioengineering (B CUBE), Dresden University of Technology, Dresden, Germany Vram Mughnetsyan Department of Solid State Physics, Yerevan State University, Yerevan, Armenia Yuli V Nazarov Department of QuantumNanoscience, Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands David Neilson CMT Group, Department of Physics, University of Antwerp, Antwerp, Belgium; ARC Centre of Excellence for Future Low Energy Electronics Technologies, School of Physics, The University of New South Wales, Sydney, NSW, Australia Jessy Nemati Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States Yoji Ohashi Department of Physics, Keio University, Yokohama, Japan Michael O’Donovan Photonics Theory Group, Tyndall National Institute, Cork, Ireland; School of Physics, University College Cork, Cork, Ireland

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List of Contributors for Volume 3

Eoin P O’Reilly Photonics Theory Group, Tyndall National Institute, Cork, Ireland; School of Physics, University College Cork, Cork, Ireland

Stefan Schulz Photonics Theory Group, Tyndall National Institute, Cork, Ireland; School of Physics, University College Cork, Cork, Ireland

J M Parker Department of Materials Science and Engineering, Sheffield University, Sheffield, United Kingdom

G Schwiete Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL, United States

G Pastore Dipartimento di Fisica, Università di Trieste, Trieste, Italy

Armen Sedrakian Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany; Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland

Shenglin Peng School of Information Science and Technology, Northwest University, Xi’an, China Luís Fernando C Pereira Department of Physics, Federal University of Maranhão, São Luí s, Maranhão, Brazil Jan Petzelt Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic GVSS Prasad Institut für Metallkunde und Metallphysik, Aachen, Germany Nick P Proukakis Joint Quantum Centre (JQC) Durham–Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, United Kingdom Alessio Recati Pitaevskii BEC Center, CNR-INO and Department of Physics, University of Trento, Trento, Italy Gregory S Rohrer Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA, United States Rudolf A Römer Department of Physics, University of Warwick, Coventry, West Midlands, United Kingdom Mauro Rovere Dipartimento di Matematica e Fisica, Università “Roma Tre”, Roma, Italy Ivan Rychetský Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic W Schattke Donostia International Physics Center DIPC, P. Manuel de Lardizabal 4, San Sebastián, Spain; Institut für Theoretische Physik und Astrophysik der ChristianAlbrechts-Universität, Leibnizstraße 15, Kiel, Germany CA Schuh Massachusetts Institute of Technology, Cambridge, MA, USA

F Seitz Rockefeller University, New York, NY, USA V Silkin Donostia International Physics Center DIPC, P. Manuel de Lardizabal 4, San Sebastián, Spain; Departamento de Polí meros y Materiales Avanzados: Fí sica, Quí mica y Tecnologí a, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Donostia-San Sebastián, Spain; IKERBASQUE, Basque Foundation for Science, Bilbao, Spain Edilberto O Silva Department of Physics, Federal University of Maranhão, São Luí s, Maranhão, Brazil Robert P Smith Clarendon Laboratory, University of Oxford, Oxford, United Kingdom W Soboyejo Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA, United States EB Sonin Racah Institute of Physic, Hebrew University of Jerusalem, Jerusalem, Israel Sandro Stringari Pitaevskii BEC Center, CNR-INO and Department of Physics, University of Trento, Trento, Italy BA Strukov Department of Physics, Lomonosov Moscow State University, Moscow, Russia T Takahashi University of Tokyo, Tokyo, Japan Yoshiro Takahashi Department of Physics, Graduate School of Science, Kyoto University, Kyoto City, Japan EV Thuneberg QTF Centre of Excellence, Department of Applied Physics, Aalto University, Espoo, Finland

List of Contributors for Volume 3

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V Yu Timoshenko Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia

Nickolas Warholak Department of Physics, New Jersey Institute of Technology, Newark, NJ, United States

MP Tosi Scuola Normale Superiore, Pisa, Italy

Stuart I Wright EDAX, St George, UT, United States

V Unikandanunni Department of Physics, Stockholm University, Stockholm, Sweden

Michihisa Yamamoto RIKEN Center for Emergent Matter Science, Wako, Japan

M Van Rossum IMEC, Leuven, Belgium

Kosuke Yoshioka Photon Science Center, School of Engineering, The University of Tokyo, Tokyo, Japan

JW Wang School of Electronic and Information Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen, China

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CONTENTS OF VOLUME 3 Superfluidity

1

EV Thuneberg

BEC-BCS crossover in ultracold atomic gases and neutron stars

10

Yoji Ohashi

Superconductivity and superfluidity in neutron stars

22

Armen Sedrakian and John W Clark

BEC-BCS crossover, condensed matter experiments

31

Kyosuke Adachi and Yoshihiro Iwasa

Excitonic superfluidity in electron-hole bilayer systems

38

David Neilson

Spin superfluidity

51

EB Sonin

Bose–Einstein condensation

68

Leonardo Fallani, Massimo Inguscio, Alessio Recati, and Sandro Stringari

Universality of Bose–Einstein condensation and quenched formation dynamics

84

Nick P Proukakis

Interacting Bose-condensed gases

124

Christoph Eigen and Robert P Smith

Cold-atom systems as condensed matter physics emulation

135

Yoshiro Takahashi

Ionic and mixed conductivity in condensed phases

145

J Maier

Electronic structure of liquids

161

G Pastore and MP Tosi

Supercooled liquids

171

Paola Gallo and Mauro Rovere

Glasses

181

J M Parker

Disordered solids and glasses, electronic structure of

191

SR Elliott

Acoustics: Physical principles and applications to condensed matter physics

200

JD Maynard

Numerical methods for localization

212

Rudolf A Römer

Disordered electron liquid with interactions: Theoretical aspects

220

AM Finkel’stein and G Schwiete

xxv

xxvi

Contents of Volume 3

Basics of simulations and carrier localization effects in semiconductor materials

236

Eoin P O’Reilly, Michael O’Donovan, and Stefan Schulz

Conductivity, electrical

251

J Bass

Coulomb blockade

260

Yuli V Nazarov

Conductivity, thermal

273

K Behnia

Transport properties: Mass transport

278

R Livi

Ferroelectricity

284

AP Levanyuk, BA Strukov, and A Cano

Planar quantum dots: Theoretical approaches

297

Aram Manaselyan, Vram Mughnetsyan, and Albert Kirakosyan

Quantum dots: Optical properties

308

V Yu Timoshenko

The physics of quantum dots

325

Jean-Pierre Leburton

Self-organized semiconductor nanostructures (quantum dots, quantum rings and their arrays)

337

Euclydes Marega Jr

Optical properties in rolled-up structures

354

JW Wang and LB Ma

Nanostructured superconductors

368

Wolfgang Lang

Kondo effects in quantum dots: Theory

381

Wataru Izumida

Kondo effects in quantum dots: Experiment

388

Michihisa Yamamoto

Spin textures in quantum dots and quantum rings

400

Wenchen Luo, Shenglin Peng, and Tapash Chakraborty

Quantum rings: Electronic properties

415

Luí s Fernando C Pereira, Francisco AG de Lira, and Edilberto O Silva

Aharonov-Bohm effect in self-assembled InAs/GaAs Quantum Rings: Fabrication, formation mechanism and optical characterization

426

Jorge M Garcí a

Superstripes landscape in perovskites high Tc superconductors

437

Gaetano Campi and Antonio Bianconi

Additive nanofabrication using focused ion and electron beams

448

Rosa Córdoba

Microstucture and macrostructure

465

Gregory S Rohrer

Structures: Orientation texture

481

Stuart I Wright and Ralf Hielscher

Nanostructures, Electronic Structure of

500

JR Chelikowsky

THz light and manipulations of matter M Basini and V Unikandanunni

509

Contents of Volume 3

The importance of full-scale experiments for the study of seismic metamaterials

xxvii 519

Stéphane Brûlé, Stefan Enoch, and Sébastien Guenneau

Quantum devices of reduced dimensionality

529

M Grundmann

Thermoelectric energy conversion devices

534

H Julian Goldsmid

Geometry matters: Gamete transport using magnetic microrobots

540

David Castellanos Robles, Farzin Akbar, and Mariana Medina-Sánchez

Memory devices—Non-volatile memories

552

Elena Gnani and Giorgio Baccarani

Memory devices – Volatile memories

576

Elena Gnani and Giorgio Baccarani

Residual stress and strain evaluation across the scales

592

Alexander M Korsunsky

Terahertz nondestructive evaluation of additively manufactured and multilayered structures

601

Alexander T Clark, Jessy Nemati, Christopher Bolton, Nickolas Warholak, Jimmie Adriazola, Ian Gatley, Samuel Gatley, and John F Federici

Excitons in nanoscale semiconductor structures

629

Alexander L Efros

Bose-Einstein Condensation of Excitons in a bulk semiconductor

644

Kosuke Yoshioka

Nonlocal optical response of nanostructures

653

Hajime Ishihara and Kikuo Cho

Optical properties of dielectrics and semiconductors

665

EL Ivchenko

Polarizability and its generalization

683

Kikuo Cho

Ultrafast spectroscopy of correlated materials

694

D Bossini and AV Kimel

Excitons in crystals

706

Fabrice P Laussy and Alexey Kavokin

Optical properties of surface enhanced Raman scattering

728

G Mattei and SN Aqida

Photonic Bandgap Materials, Electronic States of

739

M Agio and LC Andreani

Integrated Circuits

748

M Van Rossum

Semiconductor Devices

757

T Takahashi

Semiconductor Lasers

770

M-C Amann

Transistors

781

Y Hirayama

Silicon, History of

791

F Seitz and NG Einspruch

Local field effects A-M Leventi-Peetz, EE Krasovskii, V Silkin, and W Schattke

802

xxviii

Contents of Volume 3

Dielectric function

812

Jan Petzelt and Ivan Rychetský

Mechanical properties: Fatigue

818

D Klenam, F McBagonluri, and W Soboyejo

Mechanical Properties: Elastic Behavior

838

K Bowman

Mechanical Properties: Plastic Behavior

844

G Gottstein, M Goerdeler, and GVSS Prasad

Mechanical Properties: Strengthening Mechanisms in Metals AC Lund and CA Schuh

853

Superfluidity EV Thuneberg, QTF Centre of Excellence, Department of Applied Physics, Aalto University, Espoo, Finland © 2024 Elsevier Ltd. All rights reserved. This is an update of E.V. Thuneberg, Superfluidity, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 128–133, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00710-5.

Introduction Occurrence Microscopic origin Hydrodynamics Quantization of circulation Rotating superfluid and vortex lines Phase slip, Josephson effect and critical velocity Conclusion References

1 1 3 5 6 7 8 9 9

Abstract This chapter gives an introduction to basic properties of superfluids. After defining the subject, we list systems where it occurs: liquid 4He, liquid 3He, dilute atomic gases, neutron stars and superconducting metals. The microscopic basis for the formation of the superfluid phase is considered in boson and fermion systems. The two-fluid model is introduced and some consequences are discussed. The quantization of circulation is presented. Quantized vortex lines are introduced and they are used to explain the structure of a rotating superfluid. Phase slip, Josephson effect and critical velocity are discussed.

Key points

• • • • • •

Definition of superfluidity Occurrence of superfluidity Microscopic basis of superfluidity Two-fluid model Quantized circulation, vortices and rotating superfluid Phase slip, Josephson effect and critical velocity

Introduction Fluids (gases and liquids) are distinguished from solids by the property that they can flow. In almost all cases there is viscosity associated with the flow. Due to viscosity, the flow energy is gradually dissipated into heat. Contrary to this common situation, there is a special class of fluids, which can flow without viscosity. These are called superfluids and the phenomenon is called superfluidity (London, 1954; Tilley and Tilley, 1990; Guénault, 2003; Annett, 2004; Leggett, 2006; Bennemann and Ketterson, 2013; Barenghi and Parker, 2016). As a concrete example, consider a ring-shaped container filled with superfluid, see Fig. 1. Once the fluid is put into circular motion, it will continue to circulate and no energy is dissipated. The flow can continue as long as the external conditions remain unchanged, in particular the temperature is kept low enough. Superfluids show many spectacular phenomena, which are discussed in Sections “Hydrodynamics,” “Quantization of circulation,” “Rotating superfluid and vortex lines,” “Phase slip, Josephson effect and critical velocity.” Before going into these we discuss the systems where superfluidity occurs (Section Occurrence) and the microscopic basis of superfluidity (Section Microscopic origin). While Section Microscopic origin gives deeper insight, it is not absolutely necessary for discussing the phenomena in Sections “Hydrodynamics,” “Quantization of circulation,” “Rotating superfluid and vortex lines,” “Phase slip, Josephson effect and critical velocity.”

Occurrence Superfluidity occurs only in certain substances under special conditions. Superfluidity occurs at temperatures T below a transition temperature Tc, which strongly depends on the substance.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00045-7

1

2

Superfluidity

vs

Fig. 1 Once generated, the circulation of a superfluid (with velocity vs) persists as long as the experiment can be continued.

pressure (MPa)

4 solid (bcc)

3

normal liquid

2 superfluid

1

gas

0 0

1

2 3 4 temperature (K)

5

6

Fig. 2 Phase diagram of 4He at low temperatures. 4He remains liquid at zero temperature if the pressure is below 2.5 MPa (approximately 25 atm). The liquid has a phase transition to a superfluid phase, also known as He-II, at the temperature of 2.17 K (at vapor pressure).

4

pressure (MPa)

solid 3

superfluid A phase

2 superfluid B phase 1 normal fluid 0

0

1 2 temperature (mK)

3

Fig. 3 The phase diagram of 3He at low temperatures. Note that the temperature is in units of millikelvin, in contrast to Fig. 2. Two superfluid phases, A and B, are shown. The figure is based on data by Greywall (1986).

As a first case we discuss liquid helium. Under standard pressure and temperature helium is a gas. It liquifies at temperatures around 4 K. Cooling further down, it enters the superfluid phase at temperatures around 2 K, depending on pressure (Wilks, 1967). The phase diagram of natural helium at low temperatures is shown in Fig. 2. Natural helium consists essentially of isotope 4He. A 4 He atom is a boson since both the electronic spin and the nuclear spin vanish. Helium was first liquefied in 1908 and the superfluid phase was discovered in 1938. Helium has another stable isotope, 3He. A 3He atom is a fermion because the nuclear spin is 1/2. At temperatures below a few kelvin, its behavior is radically different from the isotope 4He. It also becomes superfluid, but at temperatures that are a factor of one thousand smaller than for 4He (Wheatley, 1975; Leggett, 1975; Vollhardt and Wölfle, 1990; Dobbs, 2001). The phase diagram of 3 He at low temperatures is shown in Fig. 3. 3He has three different superfluid phases, A, A1, and B. The A1 phase only appears in magnetic field, and therefore is not visible in Fig. 3. Recently, also the polar phase has been demonstrated by adding dilute impurity, aerogel, into the liquid (Dmitriev et al., 2015). The superfluid phases A and B of 3He were found in 1972. With laser cooling it is possible cool certain dilute atomic gases like 1H, 6Li, 7Li, 23Na, 40Ka, 87Rb to very low temperatures. These gases condense to superfluid state at temperatures on the order of 1 mK, depending on the density and on atomic interactions (Pethick and Smith, 2002; Pitaevskii and Stringari, 2003; Leggett, 2006; Bloch et al., 2008). This state shows many properties that are similar to the helium superfluids, although it is not a thermodynamically stable state, and therefore the flow cannot last for ever. Most of the discussion in this article applies also to condensed gases. An important difference is that instead of container walls for helium liquids, one has to consider the confining potential of the gas, which can be generated either magnetically (by field

Superfluidity

3

gradients) or optically (by laser beams). The gas atoms 1H, 7Li, 23Na, 87Rb are bosons while 6Li and 40Ka are fermions. Superfluid condensation in a gas was first found in 87Rb in 1995. Superfluidity is expected to occur also in astrophysical objects. The neutron liquid in a neutron star is believed to be in a superfluid state (Page et al., 2014; Gezerlis et al., 2014; Chamel, 2017). Also protons in a neutron star are expected to condense to a superfluid state. Both neutrons and protons are fermions. The superfluidity has been suggested as an explanation for the observed sudden changes in the rotation velocity of pulsars. Superfluidity is closely related to superconductivity (Tinkham, 1996; Bennemann and Ketterson, 2008). Superconductivity means that electric current can flow without resistance. This phenomenon appears in several elemental metals like Al, Sn, and Nb at temperatures on the order of 10 K or below. It also appears in several alloys and compounds. Superconductivity arises from resistanceless motion of the conduction electrons in a metal. Therefore, superconductivity can be understood as Superfluidity of the conduction electrons, which are fermions. Part of the discussion in this article applies also to superconductivity, but there are differences caused mainly by two reasons. (a) Electrons have electric charge and therefore their motion is essentially coupled with magnetic field. (b) The crystal lattice of the ions constitutes a preferred frame of reference, which does not exist for helium liquids. Superconductivity was first found in Mercury in 1911.

Microscopic origin In short, superfluidity can be explained as a quantum mechanical effect that shows up on a macroscopic scale. Quantum mechanics is crucial in understanding the microscopic world. It explains that electrons in atoms have only discrete energies. There is no friction on the atomic scale, and the electrons can circulate the nucleus without losing energy. We know that quantum mechanics rarely shows up on macroscopic scale. Instead of quantum mechanics, macroscopic objects obey the rules of classical physics. The reason is that a macroscopic sample consists of large number of particles and, instead of individual particles, one can only observe their average behavior. Usually the particles are in different quantum states, and an average over them obeys classical laws of physics. Examples of these laws are the Navier-Stokes equations for fluids and Ohm’s law for electrical conduction. Superfluidity is an exception to this general rule. In superfluids a macroscopic number of particles is in the same quantum state. It follows that summing over particles does not lead to averaging, but produces a macroscopic wave function. Consider a free particle with mass m and momentum p. Its kinetic energy is E ¼ p2/2 m. Its state is represented by the single-particle wave function   1 i cðr Þ ¼ pffiffiffiffi exp pr , (1) ℏ V where V is the volume of the system and h ¼ 2pℏ the Planck constant. Below we also use the wave vector k so that p ¼ ℏk. The wave function of a many body system C (r1, r2, . . .) is more general and depends on the coordinates ri of all particles, i ¼ 1, 2, . . ., N, where N is the number of particles. Further analysis depends essentially whether the particles are bosons or fermions. We now assume the particles are bosons. This means that the total wave function must be symmetric when exchanging any pair of particles. In the case of two particles this means C (r1, r2) ¼ C (r2, r1). Further we assume that there is no interaction between the bosons. It can be shown that the occupation of the lowest energy state becomes macroscopic, if the temperature T is less than T BE ¼

 2=3 h2 N 2pmkB 2:612V

(2)

Here kB the Boltzmann constant. This is known as Bose-Einstein condensation (BEC). The wave function (1) of the lowest energy state (p ¼ 0) becomes macroscopic. At zero temperature, all particles are in this state. While the ideal gas model explains Bose-Einstein condensation, it is quite insufficient in other respects. The interactions between particles are essential for the system to show superfluidity (see Section “Phase slip, Josephson effect and critical velocity”). In interacting system the macroscopically occupied state c(r) need not be the lowest energy state, and thus the macroscopic wave function can be nontrivial. In Bose gases (87Rb, etc.) the interactions are weak, and a quantitative description can be achieved by the relatively simple Gross-Pitaevskii equation. In liquid 4 He the interactions are much stronger, and a quantitative theory is not easily achieved. But even in that case, formula (2) gives 3.1 K, which is not too far from the measured value. A simple phenomenological description of 4He superfluid is obtained by considering its elementary excitations, see Fig. 4A. The excitation spectrum is liner, E( p) ¼ cp at small momenta, where c is the velocity of sound. These excitations are known as phonons. At larger momenta E( p) starts to decrease and it attains a minimum. The excitations near this minimum are known as rotons. In Bose gases only phonon excitations are visible because of weaker interactions. Let us now turn to fermions. The fermions have spin, which has to be described by an additional index s. Here we consider only spin-half particles, where the spin projection s takes two values, s ¼  12. The wave function of a fermion system is C (r1, s1, r2, s2, . . .), and it has to be antisymmetric in the exchange of any pair of particles. For a two-particle state this means

4

Superfluidity

(A)

(B)

E

E/kB(K)

10

Δ k(1/nm) 10

k

20

0

kF

Fig. 4 The energy of elementary excitations as a function of the wave number, E(k). (A) Elementary excitations in liquid 4He at vapor pressure (black solid line). The free atom kinetic energy is given by dashed line. The figure is based on neutron-scattering data by Henshaw and Woods (1961). (B) Sketch of elementary excitations of a degenerate fermion system. The excitation energy in the normal state (dashed lines) vanishes at the Fermi surface, k ¼ kF. The excitations are particle type at k > kF and hole type at k < kF. In the superfluid state (black solid line) an energy gap D opens up. For visibility of the figure, the scales are far from those ones in superconducting metals and liquid 3He. In both panels the sloped straight lines (red) correspond the critical velocity according to Landau criterion (17).

Cðr 1 , s1 , r 2 , s2 Þ ¼ −C ðr 2 , s2 , r 1 , s1 Þ:

(3)

This implies that the occupation of any single-particle state only can be zero or one. This is known as the Pauli exclusion principle. Thus macroscopic occupation of a single-particle state (1) is not possible. Superfluidity in a fermion system can appear as a result of an attractive interaction between particles. Such an interaction can cause formation of pairs. Each pair has to satisfy the antisymmetry condition (3). However, a pair is a unit that behaves like a boson. In particular, it is not excluded that several pairs are in the same pair state. Superfluidity in fermion systems can be understood as a macroscopic occupation of a single pair state. Thus superfluidity can be understood as Bose condensation of pairs. The spin part of the pair wave function has four different possibilities. These can be classified as a singlet state "# − #"

(4)

(which is a compact notation for ds1,1/2 ds2,−1/2 − ds1,−1/2 ds2,1/2) and three triplet states, which can be chosen as − "" + ## , ið"" + ##Þ, "# + #" : Let us first study the case of spin singlet (4). The pair wave function in this case is assumed to be of the form   r + r2 w ðr 1 −r 2 Þð"# − #"Þ C ðr 1 , s1 , r 2 , s2 Þ ¼ c 1 2

(5)

(6)

where we have separated the orbital wave function to a center of mass part c and a relative part w. The singlet spin state (4) is antisymmetric in the exchange of the two spins. In order to satisfy pair antisymmetry (3), the corresponding orbital part w(r1 − r2) has to be symmetric, w(r) ¼ w(−r). In most superconductors the pair wave function is of the form (6). In majority of them (Al, Sn, Nb, . . .) w is approximately independent of the direction of r1 − r2. This is called s-wave pairing in analogy with s, p, d, etc., atomic orbitals. In high-Tc superconductors there is strong evidence of d-wave symmetry of w. In Fermi gases s-wave pairing has been observed with the spin being pseudo spin formed by two atomic hyperfine states (Zwierlein, 2014). Another alternative is that the spin state of a pair is triplet (5). This case is realized in 3He and possibly in some superconductors. In 3He the orbital wave function is of p type. There are three degenerate p-wave states px, py and pz. The pair wave function can be written as Cðr 1 , s1 , r 2 , s2 Þ ¼

3 X 3 X j¼1 m¼1

cmj

  r1 + r2 ^2 : pj ðr 1 −r 2 Þi^ sm s 2

(7)

Here ib sm b si . s2 denotes the same three spin states as in Eq. (5), but expressed using Pauli spin matrices b The macroscopic wave function of bosons is called order parameter, since it describes ordering of the particles and it vanishes in the normal fluid phase. For fermions the same role is played by the center of mass part of the pair function. This is the soft degree of freedom, which can change as a function of time and location, whereas the other parts in the pair wave function (6)–(7) are fixed. We see that the order parameter in 4He and in most superconductors is a complex-valued scalar c, but in 3He it is a 3  3 matrix cm j. The order parameter in condensed boson gases is a scalar in the simplest case, but including atoms in different hyperfine states allows more complicated forms. In neutron stars both spin singlet and triplet pairing has been predicted to occur. An important energy scale in a Fermi system is Fermi energy. It is the energy up to which all noninteracting single particle states would be filled at zero temperature. It is EF ¼ pF2/2m, where the Fermi momentum pF is related to the particle density by

Superfluidity

5

N/V ¼ p3F /3p2ℏ3. Compared with thermal energy kBT, it corresponds to Fermi temperature TF ¼EF / kB. For 3He TF  1 K and for elemental superconductors TF  104 K. We see that for both 3He and superconductors, the transition temperature is small compared to the Fermi temperature, Tc / TF  10−3 or less. This implies that only excitations that have low energy (EF) close to the Fermi surface p ¼ pF play role in Superfluidity. Such excitations are described by Landau’s Fermi liquid theory. The normal state excitations are quasiparticles which are free-particle like and weakly interacting, see Fig. 4B. In the superfluid state pairs are formed. They are weakly bound and called Cooper pairs. The pair condensation opens an energy gap in the excitation spectrum. Quantitative theory of fermion superfluids is based on the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (Bardeen et al., 1957). The extension to cover superfluid 3He is discussed by Serene and Rainer (1983). Above we have discussed the boson and fermion cases separately. But in fact, one can think to go continuously from one case to the other. By increasing the attractive interaction of the fermions in a Cooper pair, one could transform it to a strongly bound pair, which could be considered as a single Bose particle. This BCS-BEC crossover has been experimentally studied in fermion gases.

Hydrodynamics Many properties of superfluids can be understood in terms of the two-fluid model. The basic assumption is that the liquid consists of two parts. These are called the superfluid and normal components. The current density j can be represented as a sum j ¼ rs v s + rn v n :

(8)

Here rs and vs are the density and velocity of the superfluid component and rn and vn are the corresponding quantities for the normal component. The liquid density is the sum of the two densities, r ¼ rs +rn. The superfluid component can flow without viscosity and it carries no heat or entropy. Moreover it is curl free, r  vs ¼ 0:

(9)

(This is valid only in uncharged superfluids.) The normal component behaves more like a usual viscous fluid. The two-fluid model can be justified from the microscopic theory discussed in Section Microscopic origin. The superfluid component corresponds to particles in the macroscopic wave function, and the normal component to particles in the excited states. The densities of the two components depend on temperature. At zero temperature all particles are in the condensate, rn ¼ 0 and rs ¼ r. With increasing temperature more particles are excited out of the condensate to the excited states. Thus rn(T ) grows and rs(T ) drops, see Fig. 5. The superfluid transition Tc is the temperature at which the condensate ceases to exist, and corresponds to rs ¼ 0 and rn ¼ r. In application of the two-fluid model we can distinguish two cases. Consider first the case that only the superfluid component moves, vn ¼ 0. This can be realized in the ring-shaped container (Fig. 1). Once the superfluid component is set into motion, it can

ρn /ρ 1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

T/Tc

Fig. 5 The normal fluid fraction rn/r as a function of temperature relative to Tc. The lowest curve is for superfluid 4He at vapor pressure and is based on data in Peshkov (1946). The other curves are for superfluid 3He-B at melting pressure (upper curve) and vapor pressure (middle curve) according to weak coupling theory (BCS theory generalized to triplet p-pairing and including Fermi-liquid interactions). The superfluid fraction rs/r ¼ 1−rn/r.

6

Superfluidity z

vs T+Δ T

p+Δ p

vn

J = ∫ (ρ svs+ρ nvn)dz = 0

T

p

Fig. 6 A difference in temperature DT generates flow of normal and superfluid components in opposite directions, and a pressure difference Dp appears. The viscosity of the normal component causes vn to vanish at walls. The superfluid velocity vs(z) has to be constant in order to be curl free (9). In a channel with closed ends the total mass flow J has to vanish.

persists because flow is ideal with no viscosity. This explains that superfluid can flow through narrow pores or as a thin film in response to small or vanishing pressure difference. The other type of motion is that also the normal component participates in it. This takes place, for example, when an object moves in the liquid. Because of motion of the normal component, viscous forces and dissipation appear. This explains that nonvanishing viscosity is measured with a viscometer, where the superfluid is placed between two plates or coaxial cylinders that move at different velocities. The measured viscosity can still be small at low temperatures, where the normal component density is vanishingly small. Superfluids show peculiar mixing of thermal and mechanical properties. Consider superfluid in a channel which is heated at one end, see Fig. 6. The superfluid component is attracted to the hot region because the chemical potential is lower there. As a consequence a pressure difference appears. This drives the normal component in the direction of decreasing temperature and convects the heat away from the source. Assuming the geometry does not allow net mass transfer, the mass transported by the normal and superfluid components in opposite directions are equal in magnitude. Such counter flow can lead to exceptionally large heat conductance. In addition to usual sound wave, superfluids have another propagating mode. This second sound is an oscillation where normal and superfluid components move in opposite directions. This leads to oscillation of temperature whereas the density remains nearly constant. Second sound can be generated by heating the superfluid periodically, and standing waves of temperature have been demonstrated experimentally. In addition to the mass current j, there can be persistent spin currents. This is possible in superfluids whose order parameter is more complicated than scalar (3He). Spin current is described by a tensor jmjspin The index m ¼ x, y, z indicates the direction of spin angular momentum that is flowing, and j ¼ x; y; z indicates the direction of the flow. Even in equilibrium the order parameter of 3He has nontrivial spatial variation called texture. This is associated with persistent spin currents and, in case of 3He-A, also with persistent mass currents.

Quantization of circulation Consider a superfluid with order parameter c. (Assume an uncharged superfluid, c can be either scalar or matrix.) The superfluid velocity vs can be expressed as a function of the order parameter as vs ¼

ℏ rf: M

(10)

Here f(r) is the phase of the order parameter, c (r) ¼Aeif(r), and the amplitude A is assumed constant. M is the boson mass, i.e., the mass of a particle in a boson superfluid and the mass of a pair in a fermion superfluid. Eq. (10) can be justified starting from the expression of current in quantum mechanics. An alternative form of Eq. (10) is obtained by taking line integral along a closed path, I h vs dl ¼ N : (11) M Here we have used the property that f is defined modulo 2p, and N is an integer. Eq. (11) is known as quantization of circulation. The curl-free condition (9) is a direct consequence of Eq. (10) or (11). Consider again superfluid in a ring-shaped container (Fig. 1). We can apply Eq. (11) to a path in the ring. We see that, in addition to being persistent, the superfluid velocity can only have discrete values. A similar phenomenon in superconductors is flux quantization. It says that magnetic flux through a ring shaped superconductor is an integral multiple of flux quantum F0 ¼h/2e, where e is the elementary charge.

Superfluidity

(A)

(B)

7

vs

φ → φ +2π

r

Fig. 7 (A) Schema of a vortex line, where the phase of the order parameter changes by 2p when encircling the line. (B) Dependence of the azimuthal velocity field (12) on the distance r from the vortex line.

vn= Ωr vs

velocity

r R Fig. 8 In a rotating container the vortex lines form a regular array so that the superfluid and normal fluid velocities, vs and vn, are equal on the average. In equilibrium the vortex array rotates rigidly with the container.

Rotating superfluid and vortex lines Let us consider superfluid in a container that is rotated with angular velocity O. The normal component will follow this motion because of its viscosity. In equilibrium it rotates uniformly with the container, vn ¼ V  r. This is not possible for the superfluid component because it has to be curl free (9). [Eq. (9) should be compared to r  vn ¼ 2 V.] The rotating state of a superfluid is most commonly realized by vortex lines (Andronikashvili and Mamaladze, 1966; Donnelly, 1991). On a path around the vortex line, the phase f changes by 2p (or an integral multiple of it). This is illustrated in Fig. 7. Equivalently, the circulation of superfluid velocity (11) around the vortex line is h/M. Assuming cylindrical symmetry, the phase f is the same as the azimuthal angle in the cylindrical coordinate system (r, f, z). The velocity field can be calculated from Eq. (10): vs ¼

ℏ ^ f, Mr

(12)

b is a unit vector in the azimuthal direction. where f The structure of the rotating state is determined by minimum of free energy. The rotation of the container is taken into account by minimizing F ¼ F0 - L  V, where F0 is the free energy functional in the stationary case and L the angular momentum. In the two-fluid model this reduces to Z 1 F ¼ d3 r rs ðvs −vn Þ2 + constant: (13) 2 Thus the optimal solution corresponds to vs as equal as possible to vn ¼ V  r, but subject to condition (10). This is achieved by a regular array of vortex lines, see Fig. 8. The number of vortex lines n per unit area is determined by the condition that the circulations of normal and superfluid velocities are the same over an area containing many vortex lines. This yields n¼

2MO : h

(14)

There are approximately 1000 vortex lines in a circular container of radius 1 cm that is rotating 1 round per minute. Vortex lines in an uncharged superfluid are analogous to flux lines, which occur in type II superconductors. Flux lines of superconductors appear in magnetic field, which is analogous to rotation of an uncharged superfluid. The velocity field (12) of a vortex diverges at the vortex line. Thus there must be a vortex core, where the two-fluid description is insufficient. A finite energy in the vortex core is achieved if the amplitude of the order parameter vanishes at the vortex line. This is

8

Superfluidity

the case for a scalar order parameter. For a matrix order parameter it is not necessary that all components of the matrix vanish at the line. Such vortex lines are realized in superfluid 3He-B. The quantization of the superfluid velocity (11) is not always true for uncharged superfluids. This happens when there is an additional contribution to the superfluid velocity (10) coming from the matrix form of the order parameter. Such a case is realized in superfluid 3He-A, and careful reanalysis of the rotating state is needed. It turns out that, in addition to one-dimensional vortex lines, the vorticity may be arranged as two-dimensional vortex sheets and three-dimensional textures. All these have been confirmed experimentally (Lounasmaa and Thuneberg, 1999). In any case, a homogeneous rotation of the superfluid is excluded. Many more topological objects in quantum liquids and their parallels in particle physics and cosmology are discussed by Volovik (2003). Besides rotating superfluid, vortex lines appear also in other circumstances. Vortices are important in limiting the maximal flow velocity, as will be discussed in the following section. By strong mechanical driving or by large thermal counterow (Fig. 6) one can generate a large number of vortices. This is the quantum version of turbulence (Skrbek et al., 2021). In three dimensions the energy of vortex lines is high relative to the quasiparticle excitations, and therefore they are not important for thermodynamics. This changes in a thin liquid film, which forms essentially a two-dimensional system. The transition from superfluid to normal state can be seen as formation of free vortices that destroy the superfluid coherence, which is known as Kosterlitz-Thouless transition.

Phase slip, Josephson effect and critical velocity Let us study superow in a channel under thermal equilibrium (vn ¼ 0). The maximum supercurrent is determined by a process called phase slip. Consider that a short piece of vortex line is nucleated at a surface on one side of the channel. This vortex expands, goes through the whole cross section of the channel, and finally disappears on the other side. As a result of this process, the phase difference Df between the ends of the channel has changed by 2p. Part of the superfluid kinetic energy is dissipated in the motion of the vortex. This means that the flow ceases to be dissipationless above a critical velocity for phase slips. Phase slips preferentially take place in constrictions of the flow channel, where the superfluid velocity has its maximum value. A special type of phase slip takes place in very short constrictions, where Eq. (10) ceases to be valid. An ideally short constriction shows the Josephson effect, where the supercurrent Js depends on the phase difference Df as Js ¼ Jc sin ðDfÞ,

(15)

and Jc is a constant. Moreover, the time derivative of Df is proportional to the difference of the chemical potential Dm on the two sides of the constriction, dDf 2Dm ¼ − : ℏ dt

(16)

Combining the two equations, one sees that a constant Dm generates an oscillating current at the frequency 2Dm/h. The Josephson effect takes place in all superfluids. It has extensively been studied in superconductors, where it is straightforward to fabricate barriers through which electrons can tunnel. For helium one has to use sufficiently small constrictions. Josephson effect in 3He has been discussed by Davis and Packard (2002). Let us consider a macroscopic moving object in a stationary superfluid. As discussed above, there generally is viscous drag from the normal component, but restricting to low temperatures, say below 0.2Tc, the normal fraction is vanishingly small. In this case the motion is nearly dissipationless as long as the object does not generate new excitations. The creation of excitations is limited by energy and momentum conservation. A simple calculations shows that no excitations are generated below the velocity vc ¼ min

EðpÞ , p

(17)

which means the minimum of E( p)/p over all excitations. In connection of superfluidity, the velocity (17) is known as Landau velocity. It is more general, however, because the condition v < vc ¼ min o(k)/k means relatively low dissipation in any system, where the medium has waves with angular frequency o and wave number k. The critical velocity (17) can also be derived from the condition that the waves generated by the object are stationary in the frame where the object is at rest. This is familiar to us, for example, from waves generated by a ship on the surface of water. When the object velocity exceeds the Landau velocity vc, a steep increase of the drag force is expected. Let us examine this in special cases. For 4He the critical velocity is determined by the roton minimum [sloped line in Fig. 4A]. This gives vc  60 m/s. Landau critical velocity has been observed for ions moving in superfluid 4He under pressure. In most experiments, the measured critical velocity is much lower. This is commonly interpreted to be caused by vortices, which are hard to avoid in a macroscopic set up. For 3He-B the Landau velocity vc ¼ D/pF, which corresponds to 27 mm/s at vapor pressure. Again, critical velocity of this magnitude has been seen with moving ions. For macroscopic objects the measured critical velocity is smaller than this. However, a recent experiment observes very little dissipation up to velocity  2vc (Bradley et al., 2016). At the time of writing, this is not understood theoretically (Kuorelahti et al., 2018). Landau velocity has also been applied to cases where only the superfluid component is in motion. For example, in ideal boson gas and in normal state fermions (dashed lines in Fig. 4) it gives vanishing critical velocity, vc ¼ 0. This is consistent with other

Superfluidity

9

arguments that these systems are not superfluids. However, for non-s-wave pairing of fermions, it is possible that the energy gap D depends on the point on the Fermi surface, and can have nodes, leading to vc ¼ 0. Such a case appears 3He-A. In this case the vanishing of the Landau velocity does not imply absence of superfluidity.

Conclusion We have given introduction to the basic properties of superfluids. Several more advanced topics are mentioned, but all details of them are left to be read from the given references and other literature.

References Andronikashvili EL and Mamaladze YG (1966) Quantization of macroscopic motions and hydrodynamics of rotating helium II. Reviews of Modern Physics 38: 567–625. https://doi.org/ 10.1103/RevModPhys.38.567. Annett JF (2004) Superconductivity, Superfluids, and Condensates. Oxford: Oxford. Bardeen J, Cooper LN, and Schrieffer JR (1957) Theory of superconductivity. Physical Review 108: 1175–1204. https://doi.org/10.1103/PhysRev.108.1175. Barenghi CF and Parker NG (2016) A Primer on Quantum Fluids. Springer. Bennemann KH and Ketterson JB (eds.) (2008) Superconductivity. In: vol. I–II. Heidelberg: Springer. Bennemann KH and Ketterson JB (eds.) (2013) Novel Superfluids. In: vol. I. Oxford: Oxford. Bloch I, Dalibard J, and Zwerger W (2008) Many-body physics with ultracold gases. Reviews of Modern Physics 80: 885–964. https://doi.org/10.1103/RevModPhys.80.885. Bradley DI, Fisher SN, Guénault AM, Haley RP, Lawson CR, Pickett GR, Schanen R, Skyba M, Tsepelin V, and Zmeev DE (2016) Breaking the superfluid speed limit in a fermionic condensate. Nature Physics 12: 1017–1021. https://doi.org/10.1038/nphys3813. Chamel N (2017) Superfluidity and superconductivity in neutron stars. Journal of Astrophysics and Astronomy 38: 43. https://doi.org/10.1007/s12036-017-9470-9. Davis JC and Packard RE (2002) Superfluid 3He Josephson weak links. Reviews of Modern Physics 74: 741–773. https://doi.org/10.1103/RevModPhys.74.741. Dmitriev VV, Senin AA, Soldatov AA, and Yudin AN (2015) Polar phase of superfluid 3He in anisotropic aerogel. Physical Review Letters 115: 165304. https://doi.org/10.1103/ PhysRevLett.115. 165304. Dobbs ER (2001) Helium Three. Oxford: Oxford. Donnelly RJ (1991) Quantized Vortices in Helium II. Cambridge: Cambridge. Gezerlis A, Pethick CJ, and Schwenk A (2014) Pairing and superfluidity of nucleons in neutron stars. In: Bennemann KH and Ketterson JB (eds.) Novel Superfluids. vol. 2, pp. 580–615. Oxford: Oxford. (chapter 22). Greywall DS (1986) 3He specific heat and thermometry at millikelvin temperatures. Physical Review B 33: 7520–7538. https://doi.org/10.1103/PhysRevB.33.7520. Guénault T (2003) Basic Superfluids. London: Taylor & Francis. Henshaw DG and Woods ADB (1961) Modes of atomic motions in liquid helium by inelastic scattering of neutrons. Physical Review 121: 1266–1274. https://doi.org/10.1103/ PhysRev. 121.1266. Kuorelahti JA, Laine SM, and Thuneberg EV (2018) Models for supercritical motion in a superfluid Fermi liquid. Physical Review B 98: 144512. https://doi.org/10.1103/ PhysRevB.98.144512. Leggett AJ (1975) A theoretical description of the new phases of liquid 3He. Reviews of Modern Physics 47: 331–414. https://doi.org/10.1103/RevModPhys.47.331. Leggett AJ (2006) Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems. Oxford: Oxford. London F (1954) Superfluids. vol. II. New York: Wiley. Lounasmaa OV and Thuneberg E (1999) Vortices in rotating superfluid 3He. Proceedings of the National Academy of Sciences of the United States of America 96: 7760–7767. https:// doi.org/10.1073/pnas.96.14.7760. arXiv:https://www.pnas.org/content/96/14/7760.full.pdf. Page D, Lattimer JM, Prakash M, and Steiner AW (2014) Stellar superfluids. In: Bennemann KH and Ketterson JB (eds.) Novel Superfluids. volume 2, pp. 505–579. Oxford: Oxford. chapter 21. Peshkov VP (1946) Determination of the velocity of propagation of the second sound in helium II. Journal of Physics (Moscow) 10: 389. Pethick CJ and Smith H (2002) Bose-Einstein Condensation in Dilute Gases. Cambridge: Cambridge. Pitaevskii L and Stringari S (2003) Bose-Einstein condensation. Oxford: Oxford. Serene J and Rainer D (1983) The quasiclassical approach to superfluid 3He. Physics Reports 101: 221–311. https://doi.org/10.1016/0370-1573(83)90051–0. Skrbek L, Schmoranzer D, Midlik S, and Sreenivasan KR (2021) Phenomenology of quantum turbulence in superfluid helium. Proceedings of the National Academy of Sciences of the United States of America 118. https://doi.org/10.1073/pnas.2018406118. arXiv:https://www.pnas.org/content/118/16/e2018406118. full.pdf. Tilley DR and Tilley J (1990) Superfluids. vol. II. Bristol: IOP Publishing. Tinkham M (1996) Introduction to Superconductivity, 2nd ed. New York: McGraw-Hill. Vollhardt D and Wölfle P (1990) The Superfluid Phases of Helium 3. London: Taylor & Francis. Volovik GE (2003) The Universe in a Helium Droplet. Oxford: Clarendon. Wheatley JC (1975) Experimental properties of superfluid 3He. Reviews of Modern Physics 47: 415–470. https://doi.org/10.1103/RevModPhys. 47.415. Wilks J (1967) The Properties of Liquid and Solid Helium. Oxford: Clarendon. Zwierlein MW (2014) Superfluidity in ultracold atomic fermi gases. In: Bennemann KH and Ketterson JB (eds.) Novel Superfluids. volume 2, pp. 269–422. Oxford: Oxford. chapter 18.

BEC-BCS crossover in ultracold atomic gases and neutron stars Yoji Ohashi, Department of Physics, Keio University, Yokohama, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Overview of BEC-BCS crossover physics CFB model and BCS model BEC-BCS crossover at T ¼ 0 BEC-BCS crossover at Tc Nambu-Goldstone mode Tan’s contact Unitary Fermi gas and universal thermodynamics Neutron superfluid in the crust regime of neutron star Conclusion Acknowledgments References

10 11 11 13 14 16 17 18 19 20 20 20

Abstract This chapter reviews how the BCS-type Fermi superfluid discussed in superconductivity continuously changes to the Bose-Einstein condensation of tightly bound molecules, as the strength of a pairing interaction between fermions increases. Theoretical treatment and basic properties of this quantum many-body phenomenon are explained. Realization in ultracold 40 K and 6Li Fermi gases with a Feshbach resonance, as well as a similar phenomenon predicted in the low-density crust regime of neutron stars, are also addressed.

Key points

• • • • •

The BEC-BCS crossover provides a unified description of the Fermi and the Bose superfluids. This phenomenon has been realized in 40K and 6Li Fermi atomic gases, by using a tunable pairing interaction associated with a Feshbach resonance. A similar phenomenon is predicted in the low-density crust regime of a neutron star. In this system, the increase of the neutron number density as one goes inside the star effectively tunes the strength of a pairing interaction. The mean-field BCS theory can describe the BEC-BCS crossover at T ¼ 0, when strong-coupling corrections to the Fermi chemical potential are taken into account. To evaluate Tc in the BEC-BCS crossover region, one needs to include pairing fluctuations, going beyond the mean-field BCS theory.

Introduction The BEC (Bose-Einstein condensation)-BCS (Bardeen-Cooper-Schrieffer) crossover is a quantum many-body phenomenon, where the character of a Fermi superfluid continuously changes from the weak-coupling BCS-type (which is discussed in superconductivity) to the BEC of tightly bound molecules, as one increases the strength of a pairing interaction between fermions. Here, ‘crossover’ means the absence of any phase transition during the change from the BCS to BEC region. The BEC-BCS crossover was originally proposed in the field of superconductivity in 1960s (Eagles, 1969); however, since the experimental tuning of interaction strength was not easy, no one could realize this crossover phenomenon in the 20th century. This situation drastically changed in 2004, when the superfluid phase transition was experimentally achieved in an ultracold 40 K Fermi atomic gas (Regal et al., 2004). Soon later, a superfluid 6Li Fermi gas was also realized (Zwierlein et al., 2004; Kinast et al., 2004; Bartenstein et al., 2004). These Fermi atomic gases are experimentally prepared by trapping vaporized atoms in a harmonic potential produced by magnetic/optical manner, and then cooling the system down to below micro-Kelvin by using laser cooling, as well as evaporative cooling. While the pairing interaction in conventional s-wave superconductivity (such as Al) is medicated by phonon (Fig. 1(a)), superfluid 40K and 6Li Fermi gases use a pairing interaction associated with a Feshbach resonance (Chin et al., 2010) (Fig. 1(b)). This pairing mechanism has the unique property that one can tune the interaction strength by adjusting an external magnetic field. Indeed, by using this advantage, the superfluid phase transition and the BEC-BCS crossover were simultaneously realized in 40K and 6Li Fermi gases with a Feshbach resonance. In the low-density crust regime of a neutron star, neutron liquid is considered to be in the s-wave superfluid state (Gandolfi et al., 2015) by an attractive interaction with the s-wave scattering length as ’ – 18.5 fm (Stoks et al., 1993). Using the typical value of the

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00100-1

BEC-BCS crossover in ultracold atomic gases and neutron stars

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(B)

(A)

Feshbach resonance

phonon

Fig. 1 Illustration of pairing mechanism. (a) Phonon-mediated pairing interaction in superconductivity, working between spin-" and spin-# electrons. (b) Feshbach-induced tunable pairing interaction in ultracold Fermi atomic gases. Pseudospin-", # represent two atomic hyperfine states contributing to the Cooper-pair formation.

Fermi momentum kF  1 fm−1 in the neutron-stare crust, the scaled interaction strength is estimated as (kFas)–1  1, indicating that the system is close to the unitary limit ((kFas)−1 ¼ 0). Before 2004, any experiment could not access such a strong-coupling superfluid state, because all the conventional superconductors were in the weak-coupling BCS regime ((kFas)–1  – 1). However, since the superfluid 40K and 6Li Fermi gases can widely cover various interaction strengths, ranging from (kFas) ≲ − 1 to (kFas) ≳ + 1, we can now experimentally examine physical properties of a nearly unitary Fermi superfluid. In addition, although the value of the neutron s-wave scattering is fixed, the scaled interaction strength (kFas)−1 varies as one goes inside the neutron star, due to the increase of the neutron number density rN ∝ k3F . Thus, the tunable pairing interaction associated with a Feshbach resonance is also convenient to simulate this situation in terrestrial experiments. In the followings, we overview the BEC-BCS crossover physics discussed in 40K and 6Li Fermi gases, as well as its application to the s-wave neutron superfluid predicted in the crust regime of a neutron star. For simplicity, we set ħ ¼ kB ¼ 1 and the system volume V is taken to be unity.

Overview of BEC-BCS crossover physics CFB model and BCS model A two-component ultracold Fermi gas with a Feshbach resonance is described by the following coupled fermion-boson (CFB) model Hamiltonian (Holland et al., 2001; Ohashi and Griffin, 2002): P P HCFB ¼ xp c{p,s cp,s + xBq b{q bq q p,s i Ph { +g bq c − p + q=2,# cp + q=2," + h:c: (1) p,q P { { cp0 q=2," c −p + q=2,# c − p0 + q=2,# cp0 + q=2," : −U p,p0 ,q

c{p, s

is the creation operator of a Fermi atom with pseudospin s ¼ ", #, describing two atomic hyperfine states forming Cooper Here, pairs. xp ¼ ep − m ¼ p2/(2 m) − m is the kinetic energy of a Fermi atom with an atomic mass m, measured from the Fermi chemical potential m. bq describes a molecular boson appearing in the intermediate state of a Feshbach resonance shown in Fig. 1(b). The molecular kinetic energy is given by xBq ¼ eBq − mB ¼

q2 + 2n − mB , 2M

(2)

where 2n is the threshold energy of a Feshbach resonance, and mB is the Bose chemical potential. Since one Feshbach molecule consists of two Fermi atoms, we take M ¼ 2 m and mB ¼ 2 m. The second line in Eq. (1) describes a Feshbach resonance with the coupling constant g, where two Fermi atoms form a quasi-molecular boson and it dissociates into two Fermi atoms again. –U is a residual weak inter-atomic interaction which has nothing to do with the Feshbach resonance. The CFB Hamiltonian in Eq. (1) makes us expect two kinds of superfluid states: One is characterized by the BEC order parameter, fm  hbq¼0i, and the other is characterized by the BCS superfluid order parameter, X  c −p,# cp," : DU (3) p

However, imposing the condition that the Bose order parameter fm has no time dependence in the thermal equilibrium state (∂ fm/∂ t ¼ 0), one exactly obtains the following relation (Ohashi and Griffin, 2003):

12

BEC-BCS crossover in ultracold atomic gases and neutron stars

fm ¼ −

g D : 2n −2m U

(4)

Thus, only the superfluid phase with fm 6¼ 0 and D 6¼ 0 is actually possible. ~ has the ~  D −gf , one finds that the mean-field gap equation to determine D Introducing the composite order parameter D m form,  X Ep g2 1 tanh 1 ¼ U res + , (5) 2n − 2m p 2Ep 2T where Ep ¼

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 ~ ep − m + D

(6)

describes Bogoliubov single-particle excitations in the superfluid phase. From the comparison of Eq. (5) with the ordinary BCS gap equation (see Eq. (9)), we find that the Feshbach-induced pairing interaction ( UFR) is the second term in [. . .] in Eq. (5), that is, U FR ¼ −

g2 : 2n − 2m

(7)

Since the threshold energy 2n of a Feshbach resonance depends on an external magnetic field B, Eq. (7) is tunable by adjusting B. We briefly note that this tuning mechanism uses the fact that hyperfine states of Fermi atoms confined in a Feshbach molecule (closed channel) are different from those of the incident and the outgoing atoms (open channel) shown in Fig. 1(b) (Chin et al., 2010). Their Zeeman energies then become different from each other. Because the threshold energy 2n is directly related to the energy difference between the open channel and the closed channel, B-dependent 2n is realized. In the current experiments on superfluid 40K and 6Li Fermi gases, the so-called broad Feshbach resonance is always used. This resonance type is characterized by a large coupling constant g. In this case, the pairing interaction UFR in Eq. (7) can be strong even when 2n  2m. Then, while a molecular boson virtually appears in the intermediate state of the Feshbach resonance (see Fig. 1(b)), the occupation number of the Feshbach-molecular band eBq ¼ q2/(2M) + 2n ( 2m), which is located much higher than the bottom energy of the atomic band ep, is negligibly small, as far as we consider the interesting BEC-BCS crossover region (Partridge et al., 2005). As a result, for the study of 40K and 6Li Fermi gases with a broad Feshbach resonance, one may use the simpler BCS model Hamiltonian, X xp c{p,s cp,s HBCS ¼ p,s

−U

X

c{p+q=2," c{−p+q=2," c −p0 +q=2," cp0 +q=2," ,

(8)

p,p0 ,q

instead of the CFB model in Eq. (1). In Eq. (8), –U is not the fixed residual weak interaction involved in Eq. (1), but a tunable pairing interaction associated with a broad Feshbach resonance. Then, the superfluid state is characterized by the ordinary BCS order parameter D in Eq. (3). In the mean-field approximation, it obeys the BCS gap equation (Schrieffer, 1964), 1¼U

X 1 Ep tanh , 2T 2E p p

(9)

~ is replaced by the ordinary BCS one D. where Ep is given in Eq. (6) where the composite order parameter D The above simplification is not valid for a narrow Feshbach resonance, being characterized by a small coupling g. In this case, in order for the Feshbach-induced interaction UFR to be strong, one needs to lower the threshold energy 2n down to be close to 2 m so that the denominator in Eq. (7) is much smaller than g2. Such a situation naturally brings about the sizable occupation of the molecular band, so that one must deal with the CFB model. The momentum summation in the gap equation (9) exhibits ultraviolet divergence without a cutoff, because the contact-type interaction –U is used in Eq. (8). Regarding this, in conventional s-wave superconductivity, the phonon-mediated pairing interaction works only near the Fermi surface within the energy range [eF − oD, eF + oD], where eF is the Fermi energy and oD ( eF) is related to the Debye frequency. Thus, one actually does not need to worry about this divergence, because the momentum summation in Eq. (9) is physically restricted to this energy range (which is also referred to as the BCS window in the literature). On the other hand, such a physical cutoff is absent, or at least unclear, in the case of the Feshbach-induced pairing interaction. In this case, to remove the ultraviolet divergence from the gap equation (9), it is convenient to measure the interaction strength in terms of the observable s-wave scattering length as, which is related to the bare interaction –U as (Randeria, 1995) 4pas U P : ¼ − m 1 −U p 2e1p

(10)

Although the momentum summation in Eq. (10) diverges without a cutoff, we actually do not explicitly evaluate this equation, but ‘borrow’ the observed value of as. In this scale, the weak-coupling BCS regime and the strong-coupling BEC regime are specified as (kFas)–1 ≲ – 1 and 1 ≲ (kFas)–1, respectively. The region, –1 ≲ (kFas)–1 ≲ 1, is called the BCS-BEC crossover region. Using as, we obtain the renormalized gap equation,

BEC-BCS crossover in ultracold atomic gases and neutron stars

1¼ −

  Ep 4pas X 1 1 : tanh − m p 2Ep 2T 2ep

13

(11)

The momentum summation in Eq. (11) now converges without a cutoff.

BEC-BCS crossover at T ¼ 0 The BCS gap equation (11) was originally derived for the weak-coupling superconducting state. However, it can also describe the BEC-BCS crossover phenomenon at T ¼ 0, at least qualitatively, if one solves Eq. (11) together with the equation for the number N of fermions in the BCS approximation,  X xp Ep 1 − tanh N¼ , (12) Ep 2T p to self-consistently determine D and m for a given interaction strength. This approach is sometimes referred to as the BCS-Leggett theory (Leggett, 2006). In the weak-coupling BCS limit ((kFas)–1  – 1), Eqs. (11) and (12) give (Pethick and Smith, 2008) mðT ¼ 0Þ ¼ eF ð> 0Þ, DðT ¼ 0Þ ¼

(13)

p 1 8 e e2kF as ð eF Þ: e2 F

(14)

In the strong-coupling BEC limit ((kFas)–1  + 1), on the other hand, we obtain 1 mðT ¼ 0Þ ¼ E2b ð< 0Þ, 2 bound rffiffiffiffiffiffi 16 1=4 3=4 DðT ¼ 0Þ ¼ jmj eF ð jmjÞ: 3p

(15) (16)

Here, E2b bound ¼ −

1 ma2s

(17)

is the energy of a two-body bound state, obtained from the following two-particle Schrödinger equation when as > 0:  −

 r2r1 r2r2 − − Udðr 1 − r 2 Þ Cðr 1 − r 2 , s1 , s2 Þ ¼ E2b bound Cðr 1 − r 2 , s1 , s2 Þ: 2m 2m

(18)

In Eq. (18), the bound-state wavefunction C(r1 − r2, s1, s2) has the form, 1 1 e −|r 1 − Cðr 1 − r 2 ; s1 ; s2 Þ ¼ pffiffiffiffiffiffiffiffiffiffi 2pa2 |r 1 − r 2 |

r 2 |as

|0, 0 > ,

(19)

where | 0, 0i is the spin-singlet state. Since the contact interaction –U d (r1 − r2) works only when two fermions come to the same spatial position, the spin-triplet bound state is not obtained from Eq. (18). Since the chemical potential physically means the energy to add one fermion to the system, Eq. (13) reasonably indicates the presence of a Fermi sphere in the weak-coupling limit. We also find from Eq. (15) that, when two fermions are added to the system deep inside the BEC regime, they immediately form a two-body bound state given in Eq. (19), so that the system energy is lowered by 2  m ¼ E2b bound ( 2.0 n 0

Inner core r < 6 km S-wave hyperonic condensates, pion and kaon Bose condensates Color superconducting phases of dense quark matter Fig. 1 Schematic interior of a neutron star of mass M ¼ 1.4M⊙ with its four basic regions: outer and inner crusts and outer and inner cores. The four shells are identified along with the values of their respective outer radii r. Also indicated are the corresponding transition densities in units of n0, as well as the particle content and condensates that may exist within each shell. Note that the figure is not to scale, low-n/large-r domains being strongly expanded.

proton quasiparticle spectra, its role is less crucial compared to the situation in metals, because of the attractive interaction between nucleons in the free space as opposed to e–e repulsion. Neutron-proton pairing is strongly suppressed by the large difference in the Fermi energies of neutrons and protons. The nucleon-nucleon potentials are analyzed in terms of their partial-wave components: in the low density (low energy) regime the dominant attractive interaction among neutrons, as well as protons, is in the 1 S0 partial-wave channel. This remains the case roughly up to the saturation density of nuclear matter, n0 ’ 0.16 fm−3. At higher densities the most attractive channel is the coupled 3P2−3F2 channel. It is worth noting that in isospin-symmetric nuclear matter with equal number of protons and neutrons, the dominant interaction channels in free space are the 3S1−3D1 and 3D2 channels at low and high energies. The characteristic scale of pairing gaps is of the order of 1 MeV, which is much larger than the characteristic temperature of mature neutron stars (T ’ 108 K, 1 MeV ¼ 1.16 1010 K). Accordingly, neutron and proton superfluids can be treated effectively in the low-temperature limit. Fig. 1 shows schematically the structure of a neutron star and the location of various superfluid and superconducting phases. There exists significant observational evidence for superfluidity of nucleonic components in neutron stars. The pulsed emission of pulsars (with periods of from milliseconds to several seconds) is caused by the periodic rotation of the star. This makes them nearly perfect clocks, but energy losses increase the rotational periods of pulsars over secular timescales. However, some pulsars undergo abrupt increases (glitches) in their rotation and spin-down rates that are followed by slow relaxation toward their pre-glitch values (in some cases with some permanent shift of either sign in the asymptotic values). These relaxations take place on a time scale of order weeks to years. A regularly glitching pulsar is the Vela pulsar, with typical changes in the spin Dn/n ’ 10−6 and spin-derivative D_n=_n ’ 10 −2 : Another prominent pulsar in the Crab nebula shows smaller glitches with Dn/n ’ 10−8 (Espinoza et al., 2011). The long relaxations after glitches are commonly attributed to components within the star that are only weakly coupled to the rigidly rotating non-superfluid component, to which the emission of pulsed radiation is locked. Candidates for such a phase are various neutron superfluid components the core or in the crust (Haskell and Melatos, 2015; Haskell and Sedrakian, 2018; Andersson, 2021). Another (indirect) source of evidence for superfluidity of neutron star interiors is their cooling evolution over long time-scales of order of millions of years. Modeling of this process shows that the suppression of the specific heat of the nucleonic component by the gaps in the neutron and proton quasiparticle spectra and additional neutrino emission due to Cooper pair-breaking processes are necessary for a proper account of the available data on the surface temperatures of neutron stars (Yakovlev et al., 2001; Page et al., 2013; Sedrakian, 2007; Schmitt and Shternin, 2018). Natural units ħ ¼ c ¼ kB ¼ 1 will be used throughout.

Overview Microscopic theories The main challenge of computation of parameters of nucleonic superfluids and superconductors lies in the fact that nucleons form a strongly interacting system. Variety of microscopic methods have been employed to compute, most notably, the gap in the quasiparticle spectrum. At the mean-field BCS level and in the energy range where the interactions are well constrained by the elastic nucleon-nucleon scattering data, i.e., for laboratory energies Elab. 1= 2, indicating that the proton superconductor is type II. This implies that electromagnetic vortices are formed with a density (Sedrakian et al., 1995) ðV Þ

np

¼

B , f0

p f0 ¼ , e

(4)

where fO is the flux quantum and B is the mean magnetic-field induction. As in metallic type II superconductors, the B-field will penetrate the proton superconductor if the field strength is between a lower critical field Hc1 and an upper critical field Hc2. However, since neutron stars are expected to feature a magnetic field before proton superconductivity sets in and the timescales for the expulsion of the magnetic field from the star are very long, the flux remains frozen in the matter (Baym et al., 1969a). This implies that local regions of with high-intensity magnetic field can be formed with H > Hc1, which will allow quantum flux tubes to pffiffiffi form. Computations show that in the high-density regime kGL < 1= 2, which implies a type I superconductivity (Bruk, 1973; Sedrakian et al., 1997; Wood and Graber, 2022). As known from condensed matter systems, flux tubes are unstable in this case. The field is then arranged in superconducting and normal domains. The size of the domains depends on the mesoscopic physics, but is expected to be significantly larger than the size of a flux tube, which is of the order of l  100 fm. In magnetars the fields are so strong that their characteristic energy can be of the order of the gap. Indeed, we note that interior fields of magnetars could be of the order of B  1016–1017 G without affecting the hydrostatic stability of the star. The interaction between the nucleon spin and the magnetic field is given by the term mNB, where mN ¼ eħ/2mp is the nuclear magneton. Because this interaction will induce an imbalance in the numbers of spin-up and spin-down particles, one can anticipate that imbalance pairing patterns will arise in strongly magnetized superfluids in neutron stars. The fields at which the Chandrasekhar-Clogston limit is achieved are of the order Hc2  1016–1017 G for neutrons (Stein et al., 2016) and Hc2  1015 G for protons (Sinha and Sedrakian, 2015). Microscopic calculations show that the proton pairing gap and Hc2 depend substantially on the density and it is plausible to expect that in some

Superconductivity and superfluidity in neutron stars

27

regions of the star the condition B
Hc2 will be fulfilled, i.e., the superconductivity will be quenched by the field. It should be noted that the mechanism described above does not work in the case of spin-parallel P-wave pairing (Mizushima et al., 2021). The protonic superconductor will form quantum vortex arrays which are much more dense than those in the neutron superfluid for typical fields of the order of 1012 G and typical rotation frequencies of neutron stars. The number of flux tubes per unit area of a single neutron vortex is of the order of 1013. There exist several scenarios on how the networks of neutron and proton vortices interact, and evolve in time. The “frozen-in” scenario assumes that the proton vortex array is tightly bound to the global field of the star and neutron vortices move through pinning barriers formed by proton flux-tubes. This resistive flow would then determine the rate at which the superfluid will react to the changes in the rotational rate of the star. The problem is complicated by the fact that (a) the topology of the magnetic field could be quite complicated (e.g., poloidal, toroidal, etc. fields) and therefore the structure of the flux-tube networks may be intricate; (b) the force between neutron and proton vortices strongly depends on their relative orientation, therefore their coupled dynamics will depend on the assumed configurations. As a consequence of the entrainment, a neutron vortex may carry a whole cluster of proton flux tubes colinear with it (Sedrakian et al., 1995). The reason is that in the vicinity of the core of the neutron vortex the field intensity is larger than Hc1 (ignoring for the moment any “primordial” field), so the proton superconductor is inevitably forced to create flux tubes. A cluster “dressing” a neutron vortex provides a much more effective scattering target than a single neutron vortex magnetized by entrainment. This has interesting implications for the response time scales of the superfluid core to sudden glitches (Sedrakian and Cordes, 1999). The interaction between neutron and proton components in the case of type-I superconductivity of protons is complicated due to a number of factors. In some cases, when simplifying assumptions about the structure of the domains are made, one can compute the dynamics of a neutron vortex embedded in a type-I superconductor. In this case the coupling of the superfluid to the normal matter turns out to be less efficient (Sedrakian, 2005).

Thermal evolution: Effects of superfluidity The thermal evolution of neutron stars can be tentatively divided into three stages: (a) An initial fast transient of 100 years during which the star’s temperature drops to about 108 K at which the star’s core becomes isothermal. (b) The following t ≲ 105 yr are characterized by neutrino emission from the star’s interior. This stage determines the subsequent evolution of the star and is important for interpretation of the X-ray data obtained from the observations by the satellites. (c) The last stage is characterized by the dominance of the photon emission from the surface of the star (Page et al., 2006). The pairing of nucleonic components has significant effect on the cooling evolution of neutron stars and, therefore, the values of the gaps are crucial input in the simulations. Firstly, the pairing reduces the specific heat of nucleonic component in full analogy with the reduction of the specific heat of a superconductor as observed under terrestrial conditions. This reduces the “thermal inertia” of the star, which makes it easier to change its thermal content for a given energy loss mechanism. Secondly, because pairing blocks fermion excitations with energies smaller than twice the gap value, the neutrino radiation processes are exponentially suppressed for temperatures T  D. This is quite analogous to the disappearance of resistivity in an ordinary superconductor due to the suppression of scattering by the gap. Thirdly, close to the critical temperature of pairing, close to which a star spends substantial time, pair-breaking and formation bremsstrahlung processes occur, in which Cooper pairs are thermally broken and recombined, with emission of neutrino-anti-neutrino pairs (Leinson and Pérez, 2006; Kolomeitsev and Voskresensky, 2008, 2010; Sedrakian, 2012).These processes are analogs of the condensed-matter processes of radio-frequency absorption by superconductors, which allow one to measure the magnitude of the gap in a variety of systems, including ultracold gases. Thus, one of the aims of research on thermal evolution of neutron stars is to place constraints on the pairing patterns and magnitudes of the gaps through a comparison of theoretical cooling curves with the observed surface temperature of neutron stars.

Selected key issues Neutron gap, induced interactions, non-adiabatic effects As should be clear from the discussion above, the magnitude of the pairing gap in neutron superfluid and proton superconductor is of crucial importance for modeling an array of problems in compact stars, including rotational dynamics (glitches, post-glitch relaxations), thermal evolution (specific heats, pair-breaking radiation rates), etc. While there is a convergence in the literature with regard to the mean-field and low-density behavior of the neutron gap, and some consensus about the suppression of the gap by the polarization effects, the ultimate role of various medium effects such as the polarization, off-shell contributions from the self-energy, and short-range correlations are still unsettled. Neutron pairing at subnuclear densities thus remains a topical issue where sophisticated many-body methods are called for. Non-adiabatic effects, well-known from the strong-coupling Eliashberg theories in condensed matter have rarely been considered in the case of nucleonic matter, but may offer an avenue to assess the impact of the induced interactions on the pairing gap beyond the Fermi-surface approximation.

The Proton gap and the role of the environment The unbound proton component is commonly embedded in a much denser neutron component, whose effects on the proton pairing may be more significant than in the case of neutrons. For example, the polarization of the medium in this case and the contribution from the induced interaction may be larger than in the case of neutron matter. Short-range correlations and non-Fermi

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liquid effects are expected to be more pronounced for protons, which fill a much smaller Fermi-sphere than neutrons. It is plausible that protons cannot be approximated, in a reasonable manner, as well-defined quasiparticles.

Higher-spin pairing The P-wave superfluidity of neutrons at high densities is uncertain for several reasons. Firstly, the interactions are not fixed by phase-shifts above a certain density, so the results can vary widely. Secondly, the gap has many components characterized by various quantum numbers and, in addition, there is an important tensor coupling of the P to the F-wave. Thus, an accurate solution of the pairing problem even at the mean-field level poses a significant problem.

Superfluid fraction in neutron star crusts The fraction of neutron superfluid in the crust can be reduced if some of it is entrained by the lattice component. The resulting reduction of the superfluid moment of inertia can be prohibitive for explanation of glitches by crust-based models only. There is an ongoing debate about the actual amount of entrained neutrons, the key issue being the inclusion of pairing effects in the band structure calculations (Watanabe and Pethick, 2017; Chamel, 2017; Kashiwaba and Nakatsukasa, 2019).

Conclusion We have briefly reviewed some topics related to nucleonic superfluidity in neutron stars. It is evident that there are profound parallels to condensed matter systems, in particular to cold atomic gases where many features discussed can be studied experimentally under controlled conditions. At the same time, a closer look demonstrates that the physics of neutron stars requires answers to questions that have not been or in the future may not be in the focus of condensed matter theory. At the microscopic scale, it can be stated that the pairing problem at the level of mean-field BCS theory, with the pairing interaction extracted directly from free-space nuclear interactions, is essentially solved within the density range corresponding to energies where the scattering phase shifts are known. Various microscopic many-body calculation predict pairing properties that vary to certain degree, specifically with regard to the issue of suppression of S-wave pairing in neutron matter by long-range collective fluctuations in the nuclear medium. Recent progress has been in the area of treating fluctuation corrections and the effects of short-range correlations due to the repulsion of the two-nucleon potential at short distances. The ultimate goal of achieving a convergent result at least in the low-density regime is within reach. More challenging problems include the studies of pairing at higher densities, in particular pairing gaps in the 3P2−3F2 channel, where the increasingly role of the three-nucleon forces cannot be neglected. Furthermore, the non-adiabatic effects (through the frequency dependence of the gap) and their impact on the phenomenology of neutron stars remains unexplored. At meso-scales, it has been well established that the neutron superfluid rotates by forming a lattice of neutron vortices, protons are mostly type-II superconductors with characteristic critical fields within the range encountered in neutron stars, and pinning effects may play a prominent role both in the crust (vortex-nucleus pinning) and in the core (vortex-flux-tube) pinning. It has been shown that neutron vortices which are embedded in a proton superconductor acquire a non-quantized magnetic flux due to the entrainment effect; interestingly this observation does not have analogs in condensed matter theory to date. Open questions at the meso-scale involve (a) the structure of the pairing matrix for the 3P2−3F2 paired condensate and its vortex solutions, (b) type-I superconductivity in the deep core of a neutron stars, in its domain structure and dynamics, (c) the spatial form and structure of proton vortex flux tubes, their possible clustering, and their interaction with neutron vortices; (d) the mutual friction coefficients in each specific scenario of neutron pairing and type of superconductivity, (e) the amount of entrainment in the crust, which remains a significant problem for crust-only glitch models in spite of convergent results for pinning forces that have emerged in recent years. At the macroscopic scales, modeling of neutron-star thermal evolution and dynamics and comparison with available and emerging data is needed to gain control over the physics at smaller scales. Open questions include the location of glitches (crust vs. core vs. mixed location models), the components of the star responsible for post-glitch relaxation, and understanding the diversity of post-glitch behavior in theoretical models. Accounting for the wide diversity of measured temperatures of neutron stars remains a challenge to theory, and the role played by pairing is far from being explored exhaustively, although our understanding of the basic process and their effects on cooling are solid. In closing, we can be confident that a significant number of observational programs in the electromagnetic spectrum and in gravitational waves—those planned and those already operating—will provide the community with new information on neutron stars, which is so crucial for the feedback and development of the theory. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) grant No. SE1836/5-2, and the Polish National Science Centre (NCN) grant 2020/37/B/ST9/01937. The authors acknowledge the support of the European COST Action PHAR0S(CA16214).

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References Abrikosov A (1988) Fundamentals of the Theory of Metals. North-Holland, Amsterdam: Courier Dover Publications. Allard V and Chamel N (2022) 1S0 pairing gaps, chemical potential and entrainment matrix in superfluid neutron-star cores for the Brussels-Montreal functionals. arXiv e-prints, art. arXiv:2203.08778. Alpar MA, Langer SA, and Sauls JA (1984) Rapid postglitch spin-up of the superfluid core in pulsars. ApJ 282: 533–541. Anderson PW and Itoh N (1975) Pulsar glitches and restlessness as a hard superfluidity phenomenon. Nature 256(5512): 25–27. Andersson N (2021) A superfluid perspective on neutron star dynamics. Universe 7(1): 17. Andreev AF and Bashkin EP (1976) Three-velocity hydrodynamics of superfluid solutions. Soviet Physics - JETP 42: 164. Baldo M and Schulze HJ (2007) Proton pairing in neutron stars. Physical Review C 75(2): 025802. Baym G, Pethick C, and Pines D (1969a) Superfluidity in neutron stars. Nature 224: 673–674. Baym G, Pethick C, Pines D, and Ruderman M (1969b) Spin up in neutron stars: The future of the Vela pulsar. Nature 224: 872–874. Bedaque PF, Caldas H, and Rupak G (2003) Phase separation in asymmetrical fermion superfluids. Physical Review Letters 91(24): 247002. Bruk YM (1973) The intermediate state in a superconducting neutron star. Astrofizika 9: 237–256. Bulgac A (2019) Time-dependent density functional theory for fermionic Superfluids: From cold atomic gases—To nuclei and neutron stars crust. Physica Status Solidi B: Basic Research 256(7): 1800592. Cao LG, Lombardo U, and Schuck P (2006) Screening effects in superfluid nuclear and neutron matter within Brueckner theory. Physical Review C 74(6): 064301. Carlson J, Gandolfi S, and Gezerlis A (2012) Quantum Monte Carlo approaches to nuclear and atomic physics. Progress of Theoretical and Experimental Physics 2012(1): 01A209. Caroli C, De Gennes PG, and Matricon J (1964) Bound Fermion states on a vortex line in a type II superconductor. Physics Letters 9: 307–309. Carter B, Chachoua E, and Chamel N (2006) Covariant Newtonian and relativistic dynamics of (magneto)-elastic solid model for neutron star crust. General Relativity and Gravitation 38: 83–119. Chamel N (2012) Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids. Physical Review C 85(3): 035801. Chamel N (2013) Crustal entrainment and pulsar glitches. Physical Review Letters 110(1): 011101. Chamel N (2017) Entrainment in Superfluid Neutron-Star Crusts: Hydrodynamic Description and Microscopic Origin. Journal of Low Temperature Physics 189: 328–360. Clark JW (2013) CBF in BCS. In: Broglia RA and Zelevinsky V (eds.) Fifty Years of Nuclear BCS: Pairing in Finite Systems, pp. 360–375. Singapore: World Scientific Publishing. Clark JW, Källman C-G, Yang C-H, and Chakkalakal DA (1976) Effect of polarization on superfluidity in low density neutron matter. Physics Letters A 61: 331–334. de Gennes P-G (1999) Superconductivity of Metals and Alloys. New York, NY: Advanced Book Program, Perseus Books. Dean DJ and Hjorth-Jensen M (2003) Pairing in nuclear systems: From neutron stars to finite nuclei. Reviews of Modern Physics 75: 607–656. Ding D, Rios A, Dussan H, Dickhoff WH, Witte SJ, Carbone A, and Polls A (2016) Pairing in high-density neutron matter including short- and long-range correlations. Physical Review C 94(2): 025802. Espinoza CM, Lyne AG, Stappers BW, and Kramer M (2011) A study of 315 glitches in the rotation of 102 pulsars. MNRAS 414(2): 1679–1704. Fan HH and Krotscheck E (2019) Variational and parquet-diagram theory for strongly correlated normal and superfluid systems. Physics Reports 823: 1–59. Fulde P and Ferrell RA (1964) Superconductivity in a strong spin-exchange field. Physics Review 135: 550–563. Gandolfi S, Palkanoglou G, Carlson J, Gezerlis A and Schmidt KE (2022) The 1S0 pairing gap in neutron matter. arXiv e-prints, art. arXiv:2201.01308. Ginzburg VL (1969) Superfluidity and superconductivity in the universe. Journal of Statistical Physics 1572-96131(1): 3–24. Hall HE and Vinen WF (1956) The rotation of liquid Helium II. II. The theory of mutual friction in uniformly rotating Helium II. Proceedings of the Royal Society of London Series A 238: 215–234. Haskell B and Melatos A (2015) Models of pulsar glitches. International Journal of Modern Physics D: Gravitation; Astrophysics and Cosmology 24: 1530008. Haskell B and Sedrakian A (2018) Superfluidity and superconductivity in neutron stars. Astrophysics and Space Science Library 457: 401–454. Jackson AD, Lande A, and Smith RA (1982) Variational and perturbation theories made planar. Physics Reports 86: 55–111. Jackson AD, Lande A, and Smith RA (1985) Planar theory made variational. Physical Review Letters 54: 1469–1471. Kashiwaba Y and Nakatsukasa T (2019) Self-consistent band calculation of the slab phase in the neutron-star crust. Physical Review C 100(3): 035804. Khalatnikov I (1989) An Introduction to the Theory of Superfluidity. Advanced Books Classics SeriesAddison-Wesley Publishing Company. ISBN: 9780201095050. Kobayashi M and Nitta M (2022) Core structures of vortices in Ginzburg-Landau theory for neutron 3P2 superfluids. Physical Review C 105(3): 035807. Kobyakov D and Pethick CJ (2013) Dynamics of the inner crust of neutron stars: Hydrodynamics, elasticity, and collective modes. Physical Review C 87(5): 055803. Kolomeitsev EE and Voskresensky DN (2008) Neutrino emission due to Cooper-pair recombination in neutron stars reexamined. Physical Review C 77(6): 065808. Kolomeitsev EE and Voskresensky DN (2010) Neutral weak currents in nucleon superfluid Fermi liquids: Larkin-Migdal and Leggett approaches. Physical Review C 81(6): 065801. Krotscheck E and Wang J (2021) Variational and parquet-diagram calculations for neutron matter. III. S-wave pairing. Physical Review C 103(3): 035808. Krotscheck E and Wang J (2022) Variational and parquet-diagram calculations for neutron matter. IV. Spin-orbit interactions and linear response. Physical Review C 105: 034345. Larkin AI and Ovchinnikov Y (1965) Inhomogenous superconductivity. Soviet Physics - JETP 20: 762. Leinson LB and Pérez A (2006) Vector current conservation and neutrino emission from singlet-paired baryons in neutron stars. Physics Letters B 638: 114–118. Lombardo U and Schulze H-J (2001) Superfluidity in neutron star matter. In: Blaschke D, Glendenning NK, and Sedrakian A (eds.) Physics of Neutron Star Interiors. vol. 578, p. 30. Berlin: Springer Verlag. Lombardo U, Nozières P, Schuck P, Schulze H-J, and Sedrakian A (2001) Transition from BCS pairing to Bose-Einstein condensation in low-density asymmetric nuclear matter. Physical Review C 64(6): 064314. Martin N and Urban M (2016) Superfluid hydrodynamics in the inner crust of neutron stars. Physical Review C 94(6): 065801. Masaki Y, Mizushima T, and Nitta M (2022) Non-Abelian half-quantum vortices in 3P2 topological superfluids. Physical Review B 105(22): L220503. Mizushima T, Yasui S, Inotani D, and Nitta M (2021) Spin-polarized phases of 3P2 superfluids in neutron stars. Physical Review C 104(4): 045803. Müther H and Sedrakian A (2002) Spontaneous breaking of rotational symmetry in superconductors. Physical Review Letters 88(25): 252503. Müther H and Sedrakian A (2003) Phases of asymmetric nuclear matter with broken space symmetries. Physical Review C 67(1): 015802. Page D, Geppert U, and Weber F (2006) The cooling of compact stars. Nuclear Physics A 777: 497–530. Page D, Lattimer JM, Prakash M, and Steiner AW (2013) Pairing and superfluidity of nucleons in neutron stars. In: Bennemann KH and Ketterson JB (eds.) Novel Superfluids, International Series of Monographs on Physics, p. 505. Oxford: Oxford University Press. Pethick CJ, Chamel N, and Reddy S (2010) Superfluid dynamics in neutron star crusts. Progress of Theoretical Physics Supplement 186: 9–16. Rios A, Polls A, and Dickhoff WH (2017) Pairing and short-range correlations in nuclear systems. Journal of Low Temperature Physics 189(5–6): 234–249. Schmitt A and Shternin P (2018) Reaction Rates and Transport in Neutron Stars. In: Rezzolla L, Pizzochero P, Jones DI, Rea N, and Vidaña I (eds.) Astrophysics and Space Science Library, Volume 457 of Astrophysics and Space Science Library, p. 455. Schulze H-J, Polls A, and Ramos A (2001) Pairing with polarization effects in low-density neutron matter. Physical Review C 63(4): 044310. Schunck CH, Shin Y, Schirotzek A, Zwierlein MW, and Ketterle W (2007) Pairing without superfluidity: The ground state of an imbalanced Fermi mixture. Science 316(5826): 867. Sedrakian A (2005) Type-I superconductivity and neutron star precession. Physical Review D 71(8): 083003. Sedrakian A (2007) The physics of dense hadronic matter and compact stars. Progress in Particle and Nuclear Physics 58: 168–246. Sedrakian A (2012) Vertex renormalization of weak interactions in compact stars: Beyond leading order. Physical Review C 86(2): 025803.

30

Superconductivity and superfluidity in neutron stars

Sedrakian A and Clark JW (2019) Superfluidity in nuclear systems and neutron stars. European Physical Journal A 55(9): 167. Sedrakian A and Cordes JM (1999) Vortex-interface interactions and generation of glitches in pulsars. MNRAS 307(2): 365–375. Sedrakian DM and Shakhabasian KM (1980) On a mechanism of magnetic field generation in pulsars. Astrofizika 16: 727–736. Sedrakian AD, Sedrakian DM, Cordes JM, and Terzian Y (1995) Superfluid core rotation in pulsars. II. Postjump relaxations. ApJ 447: 324. Sedrakian DM, Sedrakian AD, and Zharkov GF (1997) Type I superconductivity of protons in neutron stars. MNRAS 290(1): 203–207. Sedrakian A, Kuo TTS, Müther H, and Schuck P (2003) Pairing in nuclear systems with effective Gogny and Vlow–k interactions. Physics Letters B 576: 68–74. Sedrakyan AD and Sedrakyan DM (1995) Dynamics of rotating superfluid systems with pinning. Soviet Journal of Experimental and Theoretical Physics 81(2): 341–346. Sinha M and Sedrakian A (2015) Magnetar superconductivity versus magnetism: Neutrino cooling processes. Physical Review C 91(3): 035805. Stein M, Sedrakian A, Huang X-G, and Clark JW (2014) BCS-BEC crossovers and unconventional phases in dilute nuclear matter. Physical Review C 90(6): 065804. Stein M, Sedrakian A, Huang X-G, and Clark JW (2016) Spin-polarized neutron matter: Critical unpairing and BCS-BEC precursor. Physical Review C 93(1): 015802. Su RK, Yang SD, and Kuo TTS (1987) Liquid-gas and superconducting phase transitions of nuclear matter calculated with real time Green’s function methods and Skyrme interactions. Physical Review C 35: 1539–1550. Takatsuka T and Tamagaki R (1971) Superfluid state in neutron star matter. II—Properties of anisotropic energy gap of 3P2 pairing. Progress of Theoretical Physics 46(1): 114–134. Takatsuka T and Tamagaki R (1993) Chapter II. Superfluidity in neutron star matter and symmetric nuclear matter. Progress of Theoretical Physics Supplement 112: 27–65. Urban M and Ramanan S (2020) Neutron pairing with medium polarization beyond the Landau approximation. Physical Review C 101(3): 035803. Volovik G (2009) The Universe in a Helium Droplet. International Series of Monographs on PhysicsOxford: Oxford University Press. ISBN: 9780199564842. Wambach J, Ainsworth TL, and Pines D (1993) Quasiparticle interactions in neutron matter for applications in neutron stars. Nuclear Physics A 555: 128–150. Watanabe G and Pethick CJ (2017) Superfluid density of neutrons in the inner crust of neutron stars: New life for pulsar glitch models. Physical Review Letters 119(6): 062701. Wlazlowski G, Sekizawa K, Magierski P, Bulgac A, and Forbes MM (2016) Vortex pinning and dynamics in the neutron star crust. Physical Review Letters 117(23): 232701. Wood TS and Graber V (2022) Superconducting phases in neutron star cores. Universe 8(4): 228. Yakovlev DG, Kaminker AD, Gnedin OY, and Haensel P (2001) Neutrino emission from neutron stars. Physics Reports 354: 1–155. Yu Y and Bulgac A (2003) Spatial structure of a vortex in low density neutron matter. Physical Review Letters 90(16): 161101.

BEC-BCS crossover, condensed matter experiments Kyosuke Adachia,b and Yoshihiro Iwasac,d, aRIKEN Center for Biosystems Dynamics Research, Kobe, Japan; bRIKEN InterdisciplinaryTheoretical and Mathematical Sciences Program, Wako, Japan; cDepartment of Applied Physics and Quantum-Phase Electronics Center, The University of Tokyo, Tokyo, Japan; dRIKEN Center for Emergent Matter Science, Wako, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Density-controlled BCS-BEC crossover New materials Graphene-based systems Iron-based systems Organic systems Conclusion References

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Abstract The BCS-BEC crossover indicates the qualitative change in physical properties of Fermions with attractive interactions from the weakly to strongly interacting regime, offering an important guideline for maximizing Tc with given interactions. Though Uemura et al. (1991) pointed out that high-Tc cuprates and other exotic superconductors can be close to the crossover regime, the superconducting gap, D, is typically more than two orders of magnitude smaller than the Fermi energy, EF. This suggests that these materials are still far from the crossover regime. Recently, several classes of new superconductors with comparable D and EF have been discovered. Furthermore, the density-controlled BCS-BEC crossover has been achieved in gate-controlled two-dimensional superconductors. We review the recent progress of superconductor research from the viewpoint of the BCS-BEC crossover.

Key points

• •

With the development of experimental techniques, the BCS-BEC crossover has been investigated in several classes of new superconductors. Material research substantiated by theoretical approaches should be essential in exploring new physical properties in the BCS-BEC crossover.

Introduction Fermions with attractive interactions, like electrons in a superconductor, show qualitatively different collective properties depending on the interaction strength. The BCS-BEC crossover indicates the change in physical properties from weak to strong interaction regions, and several classes of superconductors have recently been investigated in the context of the BCS-BEC crossover. In the Introduction, we review the concept of the BCS-BEC crossover, a short history on the experimental side, and a simple theoretical expectation. Then we move to the subsequent sections that focus on the recent material research. We first explain the basic concept of the BCS-BEC crossover using Fig. 1 (Randeria and Taylor, 2014). For Fermions with weak attractive interactions (left side in Fig. 1), the formation and condensation of pairs with opposite spins occur at the same temperature (T ⁎ ¼ Tc), and superconductivity (or superfluidity for electrically neutral Fermions) appears below Tc. In such a weak-coupling region, which is known as the BCS regime, superconductivity can be described by the BCS theory. In contrast, in the strong-coupling region called the BEC regime (right side in Fig. 1), pair formation temperature T ⁎ is much higher than condensation temperature Tc, and superconductivity is regarded as BEC of tightly bound pairs. The superconducting states in the BCS and BEC regimes are connected without any phase transition through a crossover, which is called the BCS-BEC crossover. Typically, the crossover appears around the unitarity, which is the threshold for the creation of a two-body bound state in the vacuum. Importantly, Tc could be maximized around the unitarity, and thus approaching from the BCS to the BEC regime possibly provides us with some hint toward room temperature superconductivity. The history of research on the BCS-BEC crossover is rather old. Early on, researchers had superconductors (Eagles, 1969; Nozières and Schmitt-Rink, 1985) or liquid helium (Leggett, 1980) in mind, but since the BEC and BCS-BEC crossover were experimentally discovered in ultracold atoms of 40K (Regal et al., 2004) and 6Li (Zwierlein et al., 2004), the major target was changed to ultracold Fermion atoms. The biggest experimental advantage of ultracold atoms is that the interaction between two Fermions can be tuned

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Fig. 1 Typical phase diagram for the BCS-BEC crossover (Randeria and Taylor, 2014). The horizontal axis represents the interaction strength measured by 1/kFa, where kF and a are the Fermi wavenumber and s-wave scattering length, respectively. From Randeria M and Taylor E (2014) Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein condensation and the unitary Fermi gas. Annual Review of Condensed Matter Physics 5: 209.

by the application of an external magnetic field through the Feshbach resonance. This allows the ultracold atom systems to access the crossover regime from the BEC side. For example, recent experimental techniques have enabled direct measurements of the excitation spectrum of Fermions through the BCS-BEC crossover (Biss et al., 2022). The BCS-BEC crossover was first discussed from the material viewpoint by Uemura et al. (1991). They found that many new superconductors discovered in the 1980s, including high-Tc cuprates, organics, heavy Fermions, and even fullerenes, have low carrier density and are positioned rather close to the BCS-BEC crossover regime with kBTc/EF  0.05, where EF is the Fermi energy, as shown in the well-known Uemura plot. However, the superconducting temperatures of these superconductors do not reach the BEC temperature estimated from statistical mechanics, and reaching this BEC regime remained a challenge for the material science of superconductors in the last century. In contrast to the ultracold atom experiments, systematic control of the Fermion interaction strength is usually difficult in experiments of superconductors. Instead, carrier density can be tuned by doping or in more sophisticated ways explained in the next section, and thus the carrier density-induced BCS-BEC crossover is the realistic target in superconductors in contrast to the case of ultracold atoms. Here we briefly discuss the importance of dimensionality in inducing the BCS-BEC crossover by the change in carrier density. In Fig. 2, we show the ground-state superconducting gap divided by the Fermi energy, D/EF, as a function of the Fermion density, which is calculated by the mean-field theory for (Fig. 2A) two-dimensional (2D) and (Fig. 2B) three-dimensional (3D) Fermion gas models with contact attractive interactions. Here, we roughly regard the regions with D/EF < 0.1 and D/EF > 1 as the BCS and BEC regimes, respectively. For the 3D case (Fig. 2B), we show the curves for positive, negative, and zero inverse scattering lengths, corresponding to different interaction strengths. We can see the qualitative difference

Fig. 2 Ratio of the superconducting gap to the Fermi energy, D/EF, as a function of density for the (A) 2D and (B) 3D Fermion gas models with contact attractive interactions, calculated by the mean-field theory. For (A), density is normalized by mEB/2p, where m is the particle mass and EB is the two-particle bound-state energy. For (B), a is the s-wave scattering length, where the unitarity corresponds to a−1 ¼ 0; density is normalized by |a|−3 and k3F for a−1 6¼ 0 and a−1 ¼ 0, respectively, where kF is the Fermi wavenumber.

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between the 2D and 3D systems: reducing the density in the 2D system can induce the crossover from the BCS regime to the BEC regime, whereas the change of density in the 3D system can only achieve either the BCS or BEC side, depending on the interaction parameter a. This notable difference indicates that the unitarity limit (i.e., a−1 ¼ 0) is moved to the BCS limit in the 2D case, suggesting that 2D systems have a great advantage for the experimental realization of the density-controlled BCS-BEC crossover in superconductors.

Density-controlled BCS-BEC crossover Following the considerations above, it is natural for experimentalists to choose 2D or layered materials to realize the carrier density-induced BCS-BEC crossover in superconductors. The field effect transistor, or more broadly, the “gating” technique has been utilized in this direction since it is a powerful tool for controlling the carrier density in superconductors, especially in the low carrier density regime. Recently, the gate-controlled BCS-BEC crossover has been reported on two different materials. One is the twisted bior tri-layer graphene (Cao et al., 2018; Park et al., 2021), and the other is the Li-intercalated ZrNCl (Nakagawa et al., 2021; Heyl et al., 2022). The former graphene systems are well-known 2D superconductors, where superconductivity appears in the extremely low carrier density regime owing to its moire superlattice. The carrier density is continuously controlled by the gate voltage. The latter ZrNCl is a band insulator and a van der Waals layered material, which consists of double honeycomb ZrN layers sandwiched by Cl layers. Intercalated Li ions go into the van der Waals gap between the Cl layers and supply electrons to the ZrN layer, which leads to conducting and superconducting properties. The carrier density is changed through the amount of the Li ions, which were controlled by an ionic gating device with LiClO4/polyethylene glycol as an electrolyte. Fig. 3 displays an electronic phase diagram for LixZrNCl. Here, Li amount x is supposed to be identical to the carrier density. In the high-density region, Tc is basically insensitive to the carrier density, whereas it forms a peak at x  0.01. An important feature is that the pseudogap phase, detected by the tunneling spectroscopy, appears below x ¼ 0.1, and its onset temperature T ⁎ keeps increasing as x is decreased. The open triangles in Fig. 3 are the data obtained from the experiments on bulk polycrystalline samples, where x ¼ 0.05 has been the limit (Kasahara et al., 2009). The ionic gating device has allowed us to go below this limit and reach the BCS-BEC crossover, as shown with the blue diamonds. Here we note that LixZrNCl is a layered material, and thus the system is highly two dimensional. Indeed, the resistive transitions exhibit the Berezinskii-Kosterliz-Thouless (BKT) behavior with the BKT transition temperature TBKT slightly lower than Tc. However, the difference between TBKT and Tc is small and negligible in the scale of Fig. 3. Fig. 4 is the so-called Uemura plot, which shows the relation between Tc and Fermi temperature TF, as mentioned in the Introduction. The important feature of LixZrNCl is that it starts from the deep inside of the BCS regime and traverses all the way to the theoretical BEC line for 2D systems by reducing the carrier density by almost two orders of magnitude. When compared with the phase diagram in Fig. 3, one realizes that the region on the left side of the peak of Tc reaches the theoretical limit for 2D systems. This behavior is a clear demonstration of the density-induced BCS-BEC crossover.

Fig. 3 Electronic phase diagram of LixZrNCl (Nakagawa et al., 2021). Blue triangles show the superconducting transition temperature Tc defined by the half resistance. Red circles show the gap-opening temperature T⁎ from the tunneling spectroscopy.

34

BEC-BCS crossover, condensed matter experiments

Fig. 4 Uemura plot. The magenta and green dashed lines are the theoretical predictions for the BEC limits in 2D and 3D systems, respectively. Two plots designated by e and h for FeSe show the points for the electron and hole pockets, respectively. From Nakagawa Y, Kasahara Y, Nomoto T, Arita R, Nojima T, and Iwasa Y (2021) Gate-controlled BCS-BEC crossover in a two-dimensional superconductor. Science 372: 190.

Furthermore, comparing the experimental (Fig. 3) and theoretical (Fig. 1) phase diagrams, we find that these two show qualitatively the same behavior by appropriately exchanging the horizontal axis. The pseudogap phase disappears in the BCS regime of Fig. 3, whereas it rapidly grows as the carrier density is reduced. Presumably, the layered structure of LixZrNCl shifts the unitarity toward the BCS side and enables the density-controlled BCS-BEC crossover, just like the 2D case in Fig. 2A. Uemura et al. (1991) pointed out that many exotic superconductors discovered in the 1980s and 1990s fell on the green belt in Fig. 4, but the value of Tc/TF was still one order of magnitude smaller than the theoretical predictions for the BEC limit. However, new materials discovered in the past decades are indeed very close to the limit for 2D systems. In Fig. 4, we can find FeSe systems and twisted bilayer graphene. Interestingly, these are 2D or quasi-2D systems, but we should point out that the BCS-BEC crossover in 3D is not prohibited, but only the density-controlled crossover in purely 3D systems with fixed interaction strength is prohibited by the theory. In the following, we briefly address such newly emerging superconductors.

New materials Graphene-based systems The BCS-BEC crossover was first pointed out in the discovery paper of superconductivity in magic-angle twisted bilayer graphene (Cao et al., 2018). The twisted bilayer graphene is also a 2D system and the crossover was approached also by the gating technique, but in this case, by the standard solid gate dielectric of exfoliated hexagonal BN. Amazingly, superconductivity appeared at the extreme carrier density of 1.5  1011 cm−2 as optimal, which was determined by the quantum oscillation. This carrier density is more than two orders of magnitude smaller than the optimal value in LixZrNCl. They realized the value of Tc/TBEC ¼ 0.37, where the 3D value for the BEC limit (TBEC ¼ 0.218TF) was used for comparison. This means that they have achieved Tc/TF ¼ 0.08, which is close enough to the BEC limit for 2D systems (TBEC ¼ 0.125TF), as seen in Fig. 4. In Park et al. (2021), the same group reported that the twisted trilayer graphene also exhibits the BCS-BEC crossover and reaches Tc/TF ¼ 0.125, the limiting value for 2D systems.

Iron-based systems Fig. 4 indicates that two groups of FeSe are placed near the 2D BEC line. One is the FeSe single crystal with Tc ¼ 9 K, and the other is the monolayer FeSe on the SrTiO3 substrate. The former is a semimetal with a small carrier density, whereas, for the latter, many

BEC-BCS crossover, condensed matter experiments

35

electrons are doped possibly from the SrTiO3. Despite the big difference in Tc, which is around 9 K (Hsu et al., 2008) for the single crystal and 100 K for the monolayer (Ge et al., 2015), Tc/TF is close to the 2D BEC limit of 0.125 for both systems. Since Tc in FeSe/ SrTiO3 has not been reproduced since the first paper and remains controversial, we focus on the superconductivity of the FeSe single crystal in the following. One of the distinctive features common in iron-based superconductors is the multiband electronic structure. In FeSe, the cylindrical hole and electron bands contribute to the Fermi surface and the low-energy excitations (Terashima et al., 2014). According to the ARPES and STS experiments (Kasahara et al., 2014; Hashimoto et al., 2020), the values of Tc/TF are around 0.05 and 0.1 for the hole and electron bands, respectively, and D/EF > 0.1 for both the hole and electron bands, suggesting that FeSe undergoes superconductivity between the BCS and BEC regimes. Despite the large values of Tc/TF and D/EF, the pseudogap has not been observed in the STS measurements, which may be explained by the theoretically expected suppression of the pseudogap region due to the nature of compensated semimetal (Chubukov et al., 2016). In the BCS-BEC crossover, since the pairing interaction energy becomes comparable to the kinetic energy, the effect of superconducting fluctuation is expected to be significant. In FeSe, enhancement of the diamagnetic susceptibility has been observed well above Tc (Kasahara et al., 2016). Fig. 5 shows the temperature dependence of the diamagnetic susceptibility in FeSe (light-blue and red points), compared with the prediction of the Gaussian fluctuation theory (blue line), which is applicable to superconductors in the BCS regime with small fluctuation effects. The large gap between the observed susceptibility and the theoretical prediction has been attributed to the enhanced superconducting fluctuation in the crossover regime, though a smaller fluctuation effect has also been reported on FeSe using a different measurement technique (Takahashi et al., 2019). The multiband features in the BCS-BEC crossover have also been studied using the S-doped FeSe systems (Hashimoto et al., 2020). As the S doping is increased, the orthorhombic-tetragonal structural transition is known to occur, and the pseudogap has been observed in the tetragonal phase in ARPES experiments. In addition, the Bogoliubov quasiparticles show a relatively flat dispersion relation in the S-doped systems, which may be attributed to the interband coupling between multiple hole bands near the G point. A comprehensive understanding of the multiband effects in the BCS-BEC crossover will require further experimental and theoretical investigations.

Organic systems Suzuki et al. (2022) announced the discovery of pressure-driven BCS-BEC crossover in an organic superconductor, k-(BEDT-TTF)4Hg2.89Br8 (k-HgBr hereafter). In this compound, dimers of BEDT-TTF molecules constitute a nearly isotropic triangular lattice, separated by insulating Hg2.89Br8 layers. The nonstoichiometry of Hg arises from its sublattice incommensurate with the host lattice formed by BEDT-TTF and Br. The triangular lattice structure is almost identical to that of the Mott-insulating spin liquid candidate k-(BEDT-TTF)2Cu2(CN)3, in which the spin-1/2 localized on each BEDT-TTF dimer interacts with its neighbors through an antiferromagnetic exchange energy of 250 K but does not order down to 30 mK. Due to the nonstoichiometry in k-HgBr, this compound is regarded as a hole-doped Mott insulator and exhibits superconductivity at Tc  5 K under pressure above 0.24 GPa.

Fig. 5 Temperature dependence of the diamagnetic susceptibility above Tc (around 9K) for two values of the external magnetic field in FeSe (Kasahara et al., 2016). The vertical axis represents the difference between the ab-plane and c-axis susceptibilities with the subtraction of the corresponding value at a high field (7T). The blue line is the prediction of the Gaussian (Aslamasov-Larkin) fluctuation theory. T⁎ is the temperature at which the temperature derivative of resistivity takes the minimum, and the field dependence of the susceptibility is significant below T⁎. From Kasahara S et al. (2016) Giant superconducting fluctuations in the compensated semimetal FeSe at the BCS-BEC crossover. Nature Communications 7: 12843.

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BEC-BCS crossover, condensed matter experiments

Fig. 6 Pressure dependence of Tc and kFx|| for k-(BEDT-TTF)4Hg2.89Br8 (Suzuki et al., 2022). Regarding the points indicated by 80% and 90% in the kFx|| plot, x|| is the coherence length derived from the out-of-plane upper critical field, which is defined by the magnetic field showing 80% and 90% of the normal-state resistivity, respectively. From Suzuki Y, Wakamatsu K, Ibuka J, Oike H, Fujii T, Miyagawa K, Taniguchi H, and Kanoda K (2022) Mott-driven BEC-BCS crossover in a doped spin liquid candidate kappa-(BEDT-TTF)4Hg2.89Br8. Physical Review X 12: 011016.

Fig. 6 shows the pressure evolution of Tc and kFx||, where x|| is the in-plane coherence length. kFx|| was estimated as 3 at low pressure, indicating that the material is already in the BCS-BEC crossover regime. As pressure is increased, kFx|| increases by a factor of 20 associated with the peaking of Tc, which indicates the crossover from the BEC to the BCS regime. According to the analysis, the pressure-induced change in kFx|| is dominated by the change in x||, which indicates that the crossover is predominantly driven by the interaction strength. This poses an interesting contrast with the density-controlled BCS-BEC crossover in LixZrNCl and graphene systems.

Conclusion In this chapter, we have reviewed recent studies on the BCS-BEC crossover in materials. According to a simple theory on Fermion gas systems, 2D materials are expected to have the advantage of realizing the BCS-BEC crossover by tuning the carrier density. Indeed, the density-controlled crossover has been found in 2D-like ZrNCl and graphene-based systems using the gating technique. More broadly, several kinds of materials have been reported as candidates in the crossover or even BEC regime. From the viewpoint of the BCS-BEC crossover, we have explained the essential features of graphene-based, iron-based, and organic systems. Material research supported by theoretical approaches will be crucial in elucidating the guiding principles for realizing the BCS-BEC crossover and exploring new physical properties.

References Biss H, Sobirey L, Luick N, Bohlen M, Kinnunen JJ, Bruun GM, Lompe T, and Moritz H (2022) Excitation spectrum and superfluid gap of an ultracold fermi gas. Physical Review Letters 128: 100401. Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, and Jarillo-Herrero P (2018) Unconventional superconductivity in magic-angle graphene superlattices. Nature 556: 43. Chubukov AV, Eremin I, and Efremov DV (2016) Superconductivity versus bound-state formation in a two-band superconductor with small Fermi energy: Applications to Fe pnictides/ chalcogenides and doped SrTiO3. Physical Review B 93: 174516. Eagles DM (1969) Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors. Physics Review 186: 456. Ge J-F, Liu Z-L, Liu C, Gao C-L, Qian D, Xue Q-K, Liu Y, and Jia J-F (2015) Superconductivity above 100 K in single-layer FeSe films on doped SrTiO3. Nature Materials 14: 285. Hashimoto T, et al. (2020) Bose-Einstein condensation superconductivity induced by disappearance of the nematic state. Science Advances 6: eabb9052.

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Heyl M, Adachi K, Itahashi YM, Nakagawa Y, Kasahara Y, List-Kratochvil EJW, Kato Y, and Iwasa Y (2022) Vortex dynamics in the two-dimensional BCS-BEC crossover. Nature Communications 13: 6986. Hsu F-C, et al. (2008) Superconductivity in the PbO-type structure a-FeSe. Proceedings of the National Academy of Sciences of the United States of America 105: 14262. Kasahara Y, Kishiume T, Takano T, Kobayashi K, Matsuoka E, Onodera H, Kuroki K, Taguchi Y, and Iwasa Y (2009) Enhancement of pairing interaction and magnetic fluctuations toward a band insulator in an electron-doped LixZrNCl Superconductor. Physical Review Letters 103: 077004. Kasahara S, et al. (2014) Field-induced superconducting phase of FeSe in the BCS-BEC cross-over. Proceedings of the National Academy of Sciences of the United States of America 111: 16309. Kasahara S, et al. (2016) Giant superconducting fluctuations in the compensated semimetal FeSe at the BCS-BEC crossover. Nature Communications 7: 12843. Leggett AJ (1980) In: Pekalski A and Przystawa JA (eds.) Modern Trends in the Theory of Condensed Matter, pp. 13–27. Berlin, Heidelberg: Springer Berlin Heidelberg. Nakagawa Y, Kasahara Y, Nomoto T, Arita R, Nojima T, and Iwasa Y (2021) Gate-controlled BCS-BEC crossover in a two-dimensional superconductor. Science 372: 190. Nozières P and Schmitt-Rink S (1985) Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. Journal of Low Temperature Physics 59: 195. Park JM, Cao Y, Watanabe K, Taniguchi T, and Jarillo-Herrero P (2021) Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590: 249. Randeria M and Taylor E (2014) Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein condensation and the unitary Fermi gas. Annual Review of Condensed Matter Physics 5: 209. Regal CA, Greiner M, and Jin DS (2004) Observation of resonance condensation of fermionic atom pairs. Physical Review Letters 92: 040403. Suzuki Y, Wakamatsu K, Ibuka J, Oike H, Fujii T, Miyagawa K, Taniguchi H, and Kanoda K (2022) Mott-driven BEC-BCS crossover in a doped spin liquid candidate kappa-(BEDT-TTF) 4Hg2.89Br8. Physical Review X 12: 011016. Takahashi H, Nabeshima F, Ogawa R, Ohmichi E, Ohta H, and Maeda A (2019) Superconducting fluctuations in FeSe investigated by precise torque magnetometry. Physical Review B 99: 060503. Terashima T, et al. (2014) Anomalous Fermi surface in FeSe seen by Shubnikov-de Haas oscillation measurements. Physical Review B 90: 144517. Uemura YJ, et al. (1991) Basic similarities among cuprate, bismuthate, organic, chevrel-phase, and heavy-fermion superconductors shown by penetration-depth measurements. Physical Review Letters 66: 2665. Zwierlein MW, Stan CA, Schunck CH, Raupach SMF, Kerman AJ, and Ketterle W (2004) Condensation of pairs of fermionic atoms near a Feshbach resonance. Physical Review Letters 92: 120403.

Excitonic superfluidity in electron-hole bilayer systems David Neilsona,b, aCMT Group, Department of Physics, University of Antwerp, Antwerp, Belgium; bARC Centre of Excellence for Future Low Energy Electronics Technologies, School of Physics, The University of New South Wales, Sydney, NSW, Australia © 2024 Elsevier Ltd. All rights reserved.

Introduction Theory of electron-hole superfluidity Superfluid equations Screening in a superfluid Superfluid crossover physics Superfluid gap and superfluid onset density Superfluid transition temperature Identifying superfluidity Systems Gallium arsenide double quantum wells Double monolayer graphene Double bilayer graphene Double transition metal dichalcogenide monolayers Indium arsenide–gallium antimonide double quantum wells Future systems Other quantum wells Silicon-germanium double quantum wells Other atomically thin layers Double monolayer graphene in a periodic magnetic field Nanoribbons from double monolayer graphene Hybridization of double bilayer graphene Double trilayer and quadlayer graphene Double van der Waals layers with Type-III interfaces Superlattice of transition metal dichalcogenide monolayers Conclusion Acknowledgments References

39 39 40 40 41 42 42 43 44 44 44 45 46 46 46 46 46 47 47 47 47 48 48 48 48 48 48

Abstract There is accumulating evidence for the existence of quantum condensation of excitons in double layer semiconductor heterostructure devices, with the electrons and holes spatially separated in adjacent parallel conducting layers. This has sparked a lot of interest in excitonic superfluidity at equilibrium. The electron-hole pairing attraction is strong, promising high transition temperatures. Because the attraction is long-ranged, screening controls many superfluid properties. The systems offer the exciting prospect of a tunable electronic device that can sweep across the superfluid BCS-BEC crossover regimes. Pros and cons of different candidate heterostructures are discussed, accompanied by a brief overview of the current experimental status.

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Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00138-4

Excitonic superfluidity in electron-hole bilayer systems

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Key points

• • • • • • • •

How to generate equilibrium excitonic superfluidity and Bose Einstein condensation Systems are double conducting-layer semiconductor electronic devices Electrons and holes are separated in adjacent layers to avoid recombination Electron-hole pairing attraction is strong Pairing attraction is long-ranged so screening controls many superfluid properties Systems can be tuned across all the superfluid BCS-BEC crossover regimes Advantages and disadvantages of different material configurations Current experimental status

Introduction The electrons and holes in semiconductors can form bound pairs called excitons. The attraction between the charged electrons and holes is Coulombic, with strong exciton binding energies that can exceed thermal energies even at room temperature. In the phenomenon of conventional superconductivity, weakly bound pairs of electrons quantum condense into a single quantum state. This allows the condensate to flow through the metallic conductor without resistance. The electron pairs are bound by a residual attraction that is weak, so the dissipation-less flow of electrical current is generally limited to very low temperatures. Based on the strong electron-hole Coulomb attraction, Keldysh and Kopaev (1965) predicted many years ago that, not only should excitons quantum condense, but the energy gap of the resulting superfluid should be large. However, realization of superfluidity was blocked by the very short exciton lifetimes, as the electrons and holes recombine and destroy the exciton. The short lifetimes make it difficult for the excitons to quantum condense in bulk semiconductors. One workaround is to couple the excitons to light confined in a cavity. Hybrid exciton-polaritons can be formed with long enough lifetimes to quantum condense (Kasprzak et al., 2006). However, to maintain a constant exciton-polariton population it must be optically pumped, leading to a non-equilibrium steady-state. Lozovik and Yudson (1975, 1976) and Shevchenko (1976) proposed instead to generate equilibrium electron-hole superfluidity by separating the electrons and holes in a semiconductor heterostructure, confining them in two adjacent quantum wells. This suppresses electron-hole recombination and stabilizes the exciton lifetimes indefinitely. This article focuses on realizations of this idea. Implementation of the idea remained technically unfeasible for many years. The quantum well subbands form quasi two-dimensional conducting layers with typical average interparticle spacings r0
10 nm. To generate a population of excitons with strong electron-hole attraction and weak screening, the quantum wells must be fabricated no more than a perpendicular distance r0 apart (Neilson et al., 1993; Liu et al., 1996, 1998). This is no mean task. When the layer separation is less than the effective Bohr radius typical of semiconductors, the exciton binding energies can approach
1000 K, leading to large superfluid energy gaps,
300 K (Perali et al., 2013). However in two-dimensional systems, large superfluid gaps do not in general translate into high transition temperatures. The reason is that strong fluctuations, together with the topological nature of the two-dimensional Berezinskii-Kosterlitz-Thouless transition (Berezinsky, 1972; Kosterlitz and Thouless, 1973), make the transition temperature only dependent on the density. The transition temperature does not depend on the strength of the pairing interaction. A further major challenge in electron-hole superfluidity is screening of the long range pairing attraction. We recall that most superconductors and superfluids involve contact or short-range pairing interactions, so screening is not important. With the pair interaction for electron-hole superfluidity that is Coulombic and long-range, screening effects are unavoidable and strong. In fact, screening can weaken the pairing attraction to such a degree that superfluidity would occur only at impractically low temperatures (Kharitonov and Efetov, 2008). However at low enough densities, the existence of a large superfluid gap in the excitation energy spectrum can block low-lying single-particle excitations near the Fermi surface sufficiently to allow strong-coupled superfluidity (Lozovik et al., 2012; Perali et al., 2013). But even when strong-coupled superfluidity does stabilize, the maximum transition temperature is in general capped at low values. This is because of the joint effect of the transition temperature being determined by the density and the superfluidity being restricted to low densities.

Theory of electron-hole superfluidity The electron-hole bilayer system consists of an n-doped and a p-doped conducting layer separated by a thin insulating layer of thickness d and dielectric constant k (Fig. 1). V eh is the electron-hole attractive screened interaction between the layers, and V ee(V hh) are the repulsive Coulomb interactions for the electrons (holes) interacting within their layers. Voltages applied to the top and bottom metal gates, VTG and VBG, and a voltage across the layers, VBB, are used to adjust the electron and hole densities n (Lee et al., 2014; Zeng and MacDonald, 2020). Changing the densities, changes the average strength of the electron-electron (hole-hole) 
 pffiffiffiffiffiffi Coulomb interactions, V ee V hh ¼ e2 =kr 0 ¼ e2 pn=k: The most favorable condition for superfluidity is an equilibrium exciton gas with equal electron and hole densities (Pieri et al., 2007).

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Excitonic superfluidity in electron-hole bilayer systems

Fig. 1 Schematic representation of double electron-hole layer system. An upper conducting layer of electrons and a lower conducting layer of holes are separated a distance d by a thin insulating layer. Average interparticle spacing within each layer is r0. The interactions are electron-hole between the layers, V eh, and electron-electron (hole-hole) within the layers, V ee(V hh ). The potentials VTG and VBG are from top and bottom metal gates which, together with the bias between the layers VBB, tune the electron and hole densities.

Superfluid equations At zero temperature, a BCS mean-field description is a good approximation for strong-coupling superfluidity as well as for the familiar weak-coupling case (Nozières and Pistolesi, 1999). For the p-doped bilayer, the standard particle-hole transformation is used to map the negative-energy valence band to a positive-energy conduction band, with positively charged holes filling their conduction band states up to a Fermi level EF > 0. The BCS normal Green function Gðk, ien Þ and anomalous Green function ℱ(k, ien) for symmetric electron and hole bands with dispersion ek are given by (Lozovik and Sokolik, 2010), u2k v2k + ien − Ek ien + Ek (1) uk vk uv ℱ ðk, ien Þ ¼ − k k , ien − Ek ien + Ek qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with energies Ek ¼ x2k + D2k , xk ¼ ek − m, and Bogoliubov factors, vk ¼ 12 ð1 − xk =Ek Þ and uk ¼ 12 ð1 + xk =Ek Þ: For a given value Gðk, ien Þ ¼

of the chemical potential in the superfluid, m  EF, the equations for the momentum-dependent electron-hole superfluid energy gap Dk and density n are, P 1 eh Dk0 eh ; Dk ¼  k0 ℱ kk0 V kk 0 2Ek0 2

n ¼ 2gv

X

v2 : k k

(2)

eh 0 is a screened interaction between the electrons and holes that depends on the momentum exchange in the scattering process. Vk−k eh 0 accounts for the overlap between the electron and This gives the superfluid gap its momentum dependence. The form factor ℱk−k hole single-particle wave functions. gv is the valley degeneracy. There have been extensions of the mean-field approach to non-symmetric electron and hole bands and to unequal densities (Pieri et al., 2007; Subasi et al., 2010; Conti et al., 2021a).

Screening in a superfluid eh 0 in Eq. (2) must be determined self-consistently. In the static When there is a superfluid energy gap, the screened interaction Vk−k limit, it is given by the linear-response expression (Lozovik and Sokolik, 2010),

V qeh ¼

1

− 2V ee 0 ðqÞP0 ðqÞ

+

h


V 0eh ðqÞ 2
eh 2 i V ee − V 0 ðqÞ ðP0 ðqÞÞ2 0 ðqÞ

(3)

2 −qd where Veh /(kq) is the bare electron-hole Coulomb attraction, and V0ee(q) (V0hh(q)) ¼ 2pe2/(kq) the bare 0 (q) ¼ −2pe e electron-electron (hole-hole) Coulomb repulsion within a layer. For Fermi liquids the linear-response polarizability P0(q) is the static Lindhard function for a two-dimensional layer (Stern, 1967), but in a superfluid, P0(q) ¼ P0(n)(q) + P0(a)(q), a sum of the normal (n) and anomalous (a) linear response polarizabilities (Lozovik et al., 2012; Perali et al., 2013). The polarizabilities P0(n)(q) and P0(a)(q) for the superfluid are constructed from the normal and anomalous Green functions (Fig. 2). Summing the fermionic Matsubara frequencies en of the particle-hole loop, and taking the static limit (Lozovik et al., 2012),

ðnÞ

P0 ðqÞ ¼ −2gv ðaÞ

P0 ðqÞ ¼ 2gv ℱqee is the electron-electron form factor.

X ee u2k v2k −q + v2k v2k −q ℱq Ek + Ek −q

(4)

k

X k

eh

ℱq

2uk vk uk −q vk −q : Ek + Ek −q

(5)

Excitonic superfluidity in electron-hole bilayer systems

41

Fig. 2 Normal and anomalous polarizabilities P0(n)(q) and P(a) 0 (q) made up of two normal and two anomalous Green functions, respectively.

P0(n)(q) and P0(a)(q) in the superfluid state are suppressed for small momentum transfers q for two reasons. The presence of a superfluid gap in the excitation energy spectrum increases the magnitude of the energy denominators in the polarizabilities, which weakens the screening. In addition, for any non-zero energy gap, there is an exact cancelation of the sum P0(n)(q) + P0(a)(q) at q ¼ 0, rigorously eliminating screening in the long-wave limit (Lozovik et al., 2012). Increasing the strength of the superfluid gap expands the range of q over which the screening is suppressed. When D  EF, screening in the superfluid state is completely suppressed by D. The physical picture is that, with such strong coupling, the electron-hole pairs are compact relative to their spacing, quasi-neutral excitons which mutually interact through weak dipole repulsion. Since the pairs closely resemble weakly interacting bosons, screening is negligible (Perali et al., 2013). Nilsson and Aryasetiawan (2021) have extended the self-consistent approach to include dynamic screening (see also Sodemann et al., 2012). Intralayer interaction contributions to the quasiparticle self-energies, and hence to Ek, have been considered by Debnath et al. (2017). These generalizations introduce corrections that renormalize the coupling strength and the superfluid gap.

Superfluid crossover physics Eagles (1969), Leggett (1980), and Nozières and Schmitt-Rink (1985) developed a formalism for the BCS–BEC crossover, in which the spontaneous symmetry broken Bardeen–Cooper–Schrieffer (BCS) (Bardeen et al., 1957) superfluidity smoothly connects with Bose–Einstein condensation (BEC) (Schafroth et al., 1957) through a continuous reduction in the spatial size of the fermion pairs, here electron-hole pairs. As the pair coupling interaction is strengthened, the electron-hole pairs evolve from weakly-coupled, diffuse, interpenetrating Cooper pairs, the BCS regime, to pairs comparable in diameter to their spacing in the layers, the BCS-BEC crossover regime, and finally to pairs that are compact, non-overlapping composite bosons, the BEC regime. For a review, see Strinati et al. (2018). An important feature of a semiconductor bilayer heterostructure is that the average coupling strength hV eei/EF can be tuned by changing the density. In the example of parabolic bands, the Fermi energy EF
1/r20, so hV eei/EF
r0. Thus the electron-hole layers can be swept continuously across the BCS-BEC crossover regimes of the superfluid, starting from the weak coupled BCS regime at high densities (small r0) to the strong-coupled BEC regime at low densities (large r0) (Pieri et al., 2007; Perali et al., 2013; Liu et al., 2022). The nature of the superfluidity in each of the regimes can be characterized by the dependence on k of the superfluid gap Dk. These properties are illustrated in Fig. 3, showing Dk in the three BCS to BEC crossover regimes. The regimes have been identified independently using the superfluid condensate fraction CF (Guidini and Perali, 2014). For the weak-coupled BCS limit, corresponding to CF  0.2, the Dk  EF is narrowly peaked at k ¼ kF, reflecting the interpenetrating nature of the Cooper pairs.

Δ k /EF

10

BEC (CF=0.9)

1 CROSSOVER (CF=0.5)

0.1

BCS (CF=0.1)

1

2

3

4

5

6

7

8

9

k/kF Fig. 3 Contrasting momentum dependence of the superfluid gap Dk for the different BCS-BEC crossover regimes as labeled, with the corresponding value of the condensate fraction CF. Dk is scaled with the Fermi energy EF, k with the Fermi momentum kF.

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Excitonic superfluidity in electron-hole bilayer systems

MF-US

2

Δ max (Ry*)

1.6 1.2

MF-SCSS

0.8 MF-SCDS 0.4 0

1

2

3

4

5

6

*

r0 /aB Fig. 4 Superfluid gap Dmax, the maximum of Dk, in effective Rydbergs Ry∗ as a function of the average interparticle distance r0. a∗B is the effective Bohr radius. The curves compare the mean-field results for self-consistent static screening (MF-SCSS), for self-consistent dynamic screening (MF-SCDS), and in the absence of screening (MF-US). With self-consistent screening, there is a superfluid onset density (small arrows) at which Dmax drops by orders of magnitude.

In the BCS-BEC crossover regime, 0.2 < CF < 0.8, the maximum in Dk is of the order of EF and has shifted to k < kF, with a broad peak. In the BEC regime, CF  0.8, the maximum is at k ¼ 0, and Dk  EF is almost constant out to k  kF, corresponding to pairs that are compact in real space compared with the interparticle spacing.

Superfluid gap and superfluid onset density Fig. 4 demonstrates a central rôle played by screening for the long-range pair interaction. The system is double bilayer graphene with a 1 nm thick hexagonal boron nitride insulating barrier. Dmax is the maximum of Dk. In the weak-coupled regime at high densities, small r0, the value of Dmax calculated without screening (MF-US) is much larger than Dmax calculated with self-consistent screening, both static (MF-SCSS) and dynamic (MF-SCDS). While this difference is to be expected, the figure reveals a striking additional result: in the presence of screening there exists a superfluid onset density n0, above which there is a sharp orders of magnitude decrease in Dmax. For densities n > n0, the superfluidity is negligible, with the superfluid gap lying in the mKelvin energy range (Kharitonov and Efetov, 2008). In real systems, residual disorder would more than likely destroy such weakly bound pairs. While screening is responsible for completely suppressing superfluidity at high densities, by contrast in the strong-coupled region at low densities, r0/a⁎B > 4, it is the superfluidity that suppresses the effect of screening. In this case, Dmax calculated with or without screening is the same. This occurs because Dmax  EF, blocking the low-lying states that are responsible for screening. Fig. 4 also shows the effect of dynamics on the screening (Nilsson and Aryasetiawan, 2021). At low densities, screening is unimportant. However, at higher densities the magnitude of the superfluid gap is increased by the dynamic corrections to static screening and the onset density is higher. The results shown in Figs. 3 and 4 were determined within the mean-field approximation with self-consistent screening. Effects such as self-energy insertions, vertex corrections, etc., are not considered. For this reason, it is useful to compare these results with results determined from a full diffusion quantum Monte-Carlo calculation which includes all these additional effects. Fig. 5 compares results calculated using diffusion quantum Monte-Carlo (López Ríos et al., 2018), with a mean-field calculation for the same system using the static self-consistent screened pairing interaction (Neilson et al., 2014). The system is the same as for Fig. 4. Fig. 5(a) shows that the superfluid gap Dmax from mean-field with self-consistent screening is substantially in agreement with the full quantum Monte Carlo result. This demonstrates that the mean-field calculation with self-consistent screening captures the essence of the phenomenon, including the remarkable existence and the location of an onset density. Fig. 5(b) characterizes the BCS-BEC crossover regimes, comparing the diffusion quantum Monte-Carlo superfluid condensate fraction CF with the condensate fraction from mean-field. In most of what should be the BCS regime, the density is already above the onset density, so in most of the BCS regime the weak superfluidity cannot overcome the strong screening (Perali et al., 2013).

Superfluid transition temperature The binding energies of the pairs in these double layer systems can be very large, resulting in large superfluid gaps Dmax > 300 K, but disappointingly, these huge values do not usually translate into room temperature superfluidity. This is because in two-dimensional systems, transition temperatures are tightly capped by the Mermin-Wagner theorem (Mermin and Wagner, 1966; Hohenberg, 1967). The maximum transition temperature is limited by the topological properties of the Berezinskii-Kosterlitz-Thouless (BKT) is, up to a constant, equal to the superfluid stiffness transition (Kosterlitz and Thouless, 1973). The transition temperature TBKT c rS(T ). It depends only on the density and it is not affected by the coupling strength. For parabolic bands, it is linearly proportional h 2i ℏ n, where M is the mass of the electron-hole pair. Consequently, the maximum transition temperature ¼ p2 2M to the density, T BKT c Tcmax occurs at the superfluid onset density, n ¼ n0.

Excitonic superfluidity in electron-hole bilayer systems

1.2

43

(a) MF-SCSS

Δ max (Ry*)

0.9 QMC 0.6

0

(b)

BEC

0.3

MF-SCSS

0.8

CROSSOVER

QMC CF

0.6 0.4

0

BCS

0.2

1

2

3 4 * r0 /aB

5

6

Fig. 5 Comparison of (a) superfluid gap Dmax in effective Rydbergs Ry∗ and (b) superfluid condensate fraction CF, calculated within mean-field using self-consistent static screening (MF-SCSS) and by a full diffusion quantum Monte Carlo (QMC) calculation. r0/a∗B is the average interparticle distance scaled with the effective Bohr radius. In (b) the BCS-BEC crossover regimes are indicated on the right axis.

The complete phase diagram is shown in Fig. 6. Above the onset density n > n0, the superfluid transition temperature is in the mKelvin range, above which the system is a Fermi liquid. For n < n0 and temperatures above TBKT , signatures of the superfluid state c should persist arising from the pseudogap. The pseudogap arises from localized dynamic fluctuations of the pairing field (Franz, 2007), and is a precursor of the superfluid gap in the normal degenerate electron gas. It extends up to the degeneracy temperature TD (Butov, 2004). As it can be significant up to temperatures T
Dmax (Perali et al., 2002; Gaebler et al., 2010), in the present case the pseudogap could survive to room temperature.

Identifying superfluidity To identify clear signatures of electron-hole superfluidity remains a major challenge. Although the superfluid current is neutral, nevertheless because the charged electrons and holes are spatially separated, an electrical current can be generated by passing counterflow currents in the two layers in opposite directions (Sivan et al., 1992; Su and MacDonald, 2008). Since in a superfluid

TD

Pseudogap in degenerate exciton gas

Fermi Liquid

max

Tc

Classical exciton gas

T

T BK c T

Superfluid 0

n0

∨~mK ∧ n

Fig. 6 Temperature-density T-n phase diagram. TBKT is the superfluid transition temperature. n is the equal density of electrons and holes. The maximum c occurs at the onset density n0. For densities n > n0, the superfluid transition temperature is in the mKelvin range. TD is the degeneracy transition temperature Tmax c temperature for the exciton gas.

44

Excitonic superfluidity in electron-hole bilayer systems

the counter-currents will couple to the dissipationless neutral exciton current, the counter-currents are expected also to be dissipationless (Eisenstein and MacDonald, 2004; Fil and Shevchenko, 2018). Properties of the electron-hole drag resistance, the bilayer transresistivity, is another signature of condensation (Conti et al., 1998; Hu, 2000). The transresistivity is directly related to the interlayer scattering rate (Narozhny and Levchenko, 2016), and using a minimum dissipation principle for the superfluid current, Vignale and MacDonald (1996) showed that superfluidity can be characterized by a discontinuous jump in the drag resistance at the transition temperature. As the temperature is further decreased, the drag resistance diverges for temperatures approaching zero. Enhanced electron-hole tunneling between the layers is a further indicator of spontaneous interlayer phase coherence (Eisenstein and MacDonald, 2004; Efimkin et al., 2020). However, Zeng and MacDonald (2020) showed that a strong interlayer tunneling current between the layers influences the transport and can lead to a steady state that is not in equilibrium. Josephson junction-like behavior across a weak link could also be used in identifying superfluidity (Zenker et al., 2015; Pascucci et al., 2022).

Systems Although the electron-hole double layer system is a multiband system, with a hole band and an electron band, it is possible to have just a single condensate with the one gap parameter. This is diametrically opposite to multiband superconductors like magnesium diboride in which the pairing occurs within the bands and for which multiple coupled condensates must appear, each condensate with a gap parameter associated with a different band (Komendová et al., 2012).

Gallium arsenide double quantum wells Electron-hole superfluidity and BCS-BEC crossover physics was first proposed for an excitonic system in a conventional semiconductor heterostructure of double quantum wells in gallium arsenide (Comte and Nozières, 1982). Technical advances in the early 1990s in the fabrication of double quantum wells in III–V semiconductors offered an apparently unobstructed path (Balatsky et al., 2004) to experimentally realize superfluidity in a double layer semiconductor system (Sivan et al., 1992; Kane et al., 1994; Pohlt et al., 2002). Charge-separated excitons were produced by confining electrons and holes in adjacent quantum wells consisting of two disconnected thin slabs of gallium arsenide, with a very thin slab of aluminum gallium arsenide between them as an insulating barrier (Butov et al., 2002). Superfluidity in gallium arsenide double quantum wells differs in important ways from superfluidity in coupled atomically flat layers like graphene (Section “Double monolayer graphene”). Advantages include a large bandgap, and low-energy bands that are insensitive to gate potentials. Pieri et al. (2007) showed that the large difference in the electron and hole effective masses in gallium arsenide, m⁎h/m⁎e
5, as well as any density imbalance, have further intriguing consequences. The large mass difference should lead to the existence of the exotic superfluid phases, Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) (Fulde and Ferrell, 1964; Larkin and Ovchinnikov, 1965) and Sarma (Sarma, 1963). However, a significant drawback of gallium arsenide is that in double quantum wells, the electron–hole pairing interactions are weak since it is difficult to fabricate the wells sufficiently close to each other. A limitation with transport in quantum wells is that the wells must not be fabricated too narrow ( 2.35 is the pair coupling strong enough for a superfluid gap to suppress the screening, allowing the superfluidity to exist. For the band parameters in graphene, hVi/EF
0.7, so the pair coupling strength would always be too weak. Thus at any density in monolayer graphene, superfluidity is suppressed by screening. Because the electrons and holes are massless, localized excitons cannot form (Lozovik and Sokolik, 2008). If any remnant superfluidity survived in the presence of such strong screening, its transition temperature Tc would be impractically low (Kharitonov and Efetov, 2008). Lozovik and Sokolik (2010) and Sodemann et al. (2012) generalized the static screening theory to dynamic screening. Dynamic screening did not alter the key conclusion that superfluidity is not to be expected in double monolayer graphene. Electron-hole Coulomb drag measurements of Gorbachev et al. (2012) bear this conclusion out. No jump or divergence in the drag resistance was observed, and the T2 temperature behavior of the drag resistance across the full density and temperature range was consistent with Fermi liquid properties.

Double bilayer graphene With the difficulties faced by double monolayer graphene, Perali et al. (2013) proposed an alternate strategy: to substitute two graphene bilayers for the two graphene monolayers while retaining the hexagonal boron nitride insulating barrier. In graphene bilayers, the energy bands e(k) resemble the bands of conventional semiconductors and are approximately parabolic. Then, as in the semiconductors, reducing the density increases hVi/EF and strengthens the pair coupling. With strong electron-hole coupling, the superfluidity can overcome the screening. The suppression of screening is more dramatic for the quadratic bands of bilayer graphene than for linear bands of monolayer graphene. This is because with the larger density of states for quadratic bands, the corresponding gaps from Eq. (2) are larger, while at the same time EF is smaller. A larger D/EF ratio is more effective at suppressing the screening. The bandgap for bilayer graphene is very small and tunable, typically Eg
5 − 100 meV, up to a maximum Eg
250 meV. The bandgap is determined by the gate voltages. For such small bandgaps, the valence band provides a second superfluid gap, and the superfluid is multicomponent. The two superfluid components are coupled through Josephson-like transfers of electron-hole pairs between the valence and conduction band condensates (Conti et al., 2017). This should reinforce the superfluid gaps, but with small bandgaps there are additional interband contributions to the screening coming from valence to conduction band excitations (Conti et al., 2019). These two effects act in opposing directions. In the end, the effect of the additional screening that weakens the superfluid gaps turns out to be the stronger effect, making Josephson-like transfers negligible (Conti et al., 2019). The reason is the following. These transfers would leave in their trail a population of free vacancies in the valence-band, adding to the screening. This makes the resulting superfluid gaps always smaller than the bandgap, so Josephson-like pair transfers are suppressed. Without the pair transfer, the net result is that the conduction and valence band condensates are decoupled, with the conduction band condensate dominant. For small bandgaps, the conduction band condensate is weakened by the additional interband screening arising from single-particle excitations out of the valence band. As the bandgaps get bigger, interband screening weakens and the conduction band superfluid gap becomes stronger. A further advantage of large bandgaps comes from the fact that the bilayer graphene conduction band is flatter than a parabolic band for low energies (Castro et al., 2010). This results in a larger density of states, so the Fermi energy EF at a given density is reduced, thus permitting a superfluid gap of a given magnitude to block a wider range of low-lying excitations. This more efficiently suppresses the screening, and the superfluid gap is increased. In summary, in double bilayer graphene the superfluid onset density is sensitive to the variable bandgap. When the bandgap Eg < EF, the interband contributions reinforce the screening, and the onset density is low. Conversely, when Eg  EF the onset density is larger. For a separation between the bilayers d ¼ 1 nm and bandgap Eg ≳ 35 meV, the onset density n0 ≳ 2 1011 cm−2. This puts strong-coupled superfluidity in the BCS-BEC crossover and BEC regimes within experimental reach in double bilayer graphene. Burg et al. (2018) observed strongly enhanced interlayer tunneling at equal electron and hole densities in double bilayer graphene separated by a d ¼ 1.4 nm tungsten diselenide insulating barrier. Below a temperature Tc
1.5 K, the interlayer tunneling current had a vertical onset as a function of interlayer voltage, suggesting a transition to a coherent state at that temperature. Such tunneling enhancement is a strong indicator of exciton condensation. The interlayer current Iint was measured as a function of the interlayer voltage VBB, at fixed top and back gate voltages, VTG and VBG (see Fig. 1).

46

Excitonic superfluidity in electron-hole bilayer systems

The vertical onset of the interlayer tunneling current and the tunneling enhancement were observed at densities only below an onset density n0 ¼ 8 1011 cm−2, a value consistent with theory (Perali et al., 2013; López Ríos et al., 2018; Nilsson and Aryasetiawan, 2021).

Double transition metal dichalcogenide monolayers To circumvent the detrimental effect of the valence band screening encountered in double bilayer graphene, Conti et al. (2020b) suggested double monolayers from the transition metal dichalcogenide (TMD) family, specifically, molybdenum disulfide, molybdenum diselenide, tungsten disulfide, and tungsten diselenide. This group have large bandgaps, Eg ≳ 1 eV (Mak et al., 2010; Jiang, 2012). The conduction bands in this class of transition metal dichalcogenides are split by spin-orbit coupling, leading to two condensates in the conduction bands and hence multicomponent superfluidity. The two superfluid components reinforce the superfluidity in this case (Conti et al., 2017). In addition, the electron and hole effective masses are large, increasing the binding energies compared to bilayer graphene. This further strengthens the electron-hole pairing interactions (Fogler et al., 2014). Lagoin and Dubin (2021) pointed out that the presence of moiré potentials in transition metal dichalcogenides heterobilayers will further enhance the effective masses. As a result of the large bandgaps, the reinforcing multicomponent effects, and the large effective masses, the superfluid onset densities in double layer transition metal dichalcogenides are high, with a 1 nm barrier, n0
1 1013 cm−2 (Conti et al., 2020b). These are much larger than in double bilayer graphene, n0
8 1011 cm−2 (Burg et al., 2018; Conti et al., 2019), phosphorene, n0
4 1012 cm−2 (Saberi-Pouya et al., 2018), and gallium arsenide, n0
5 1010 cm−2 (Saberi-Pouya et al., 2020). With such high onset densities, the transition metal dichalcogenides hold great promise for exciton superfluids at high-temperature, with
100 K (Conti et al., 2020b). transition temperatures predicted as high as TBKT c This family of materials hosts additional curious possibilities. Conti et al. (2020a) showed that by exploiting a change in the energy band alignments, certain combinations of transition metal dichalcogenides can provide a doping-dependent switch, to switch from one-component to two-component superfluidity (Conti et al., 2020b). The switch in doping also affects the spin alignment of the electron-hole pairs, leading to an intriguing possibility of tuning from a system purely of dark excitons to a system purely of bright excitons (Combescot et al., 2017). Enhanced interlayer tunneling and enhanced electroluminescence intensity have been observed in molybdenum diselenide–tungsten diselenide separated by 1–2 nm of hexagonal boron nitride (Wang et al., 2019; Chaves and Neilson, 2019; Ma et al., 2021). Again there was an onset density, below which a large tunneling current was observed between the monolayers. At exactly equal electron–hole densities, the electroluminescence intensity from electron–hole recombination triggered by inter-layer tunneling was strongly enhanced, a property again consistent with exciton condensation. The tunneling enhancement and the electroluminescence intensity shared the same onset density, and the properties persisted up to remarkably high temperatures, Tc
100 K (Fogler et al., 2014).

Indium arsenide–gallium antimonide double quantum wells Indium arsenide–gallium antimonide double quantum wells have an inverted band structure with finite overlap of the conduction and valence bands. With electrons in the indium arsenide well and holes in the gallium antimonide well, the electrons and holes are in equilibrium without photoexcitation, and they are spatially separated without the need for a barrier. Du et al. (2017) reported optical spectroscopic and electronic transport evidence for the formation of an excitonic insulator gap in this system at relatively high densities. Terahertz transmission spectra exhibited two absorption lines which are quantitatively consistent with predictions from the pair-breaking excitation dispersion, calculated based on the BCS superfluid gap equation (Eq. 2). Transport measurements show an insulating gap of D
25 K, with a transition temperature of Tc
10 K which, with quantized edge conductance, suggests the existence of a topological excitonic insulator phase.

Future systems Other quantum wells Silicon-germanium double quantum wells Conti et al. (2021b) have proposed electron-hole quantum wells in a strained silicon-germanium double layer heterostructure as an interesting alternative system for superfluidity. Because of the Type-II band alignment of the silicon-germanium interface (Virgilio and Grosso, 2006), there is no requirement for an insulating barrier to keep the electrons and holes apart, as is needed in gallium arsenide double quantum wells. The resultant close proximity of the electrons and holes greatly enhances the electron-hole pair interaction, leading to much higher onset densities than in gallium arsenide. A compressively strained 3 nm epitaxial germanium layer and tensile strained 3 nm epitaxial silicon layer ensures that the layers are lattice matched. The low-lying bands result in a double quantum well structure, with the electrons confined in the silicon well and the holes in the germanium well. In the presence of the strain, the effective mass imbalance is large, m⁎h/m⁎e ¼ 0.26 (Lodari et al., 2021; Schäffler, 1997; Zwanenburg et al., 2013).

Excitonic superfluidity in electron-hole bilayer systems

47

The system offers some unique advantages. One is the ability to independently contact the layers since there exists no quantum well for holes in the silicon layer and there is no quantum well for electrons in the germanium layer that is energetically accessible. This makes it straightforward with a simple and robust process, to independently electrically contact just one layer and not the other. In contrast, to fabricate independent contacts for the systems already discussed requires painstaking multi-stage processing. A second distinct advantage is scalability. The silicon-germanium heterostructures would be grown on 300 mm silicon wafers using mainstream chemical vapor deposition (Pillarisetty, 2011). In this way, these heterostructures can take advantage of stateof-the-art semiconductor manufacturing, promising high device yield and integration. A silicon-germanium bilayer is readily integrated with advanced quantum technologies (Scappucci et al., 2020), including long-lived electron spin qubits in silicon (Watson et al., 2018), hole spin qubit arrays in germanium (Hendrickx et al., 2020), and superconducting contacts to holes (Vigneau et al., 2019). By contrast, device fabrication for double graphene bilayers and double transition metal dichalcogenide bilayers involves layer by layer assembly using pick and transfer techniques. These are prone to layer wrinkling, contamination, and misorientation (Frisenda et al., 2018). Device yields tend to be low and prospects for scalability limited. III–V semiconductor based devices such as gallium arsenide, offer more promise for scalability, but the molecular beam epitaxy growth technique is not compatible with standard complementary metal-oxide semiconductor technology, thus limiting the prospects for advanced manufacturing and large scale device integration.

Other atomically thin layers Double monolayer graphene in a periodic magnetic field

To boost the value of the ratio hV i/EF, Dell’Anna et al. (2015) proposed applying a periodically modulated external magnetic field perpendicular to the two graphene monolayers. The magnetic field preserves the linear dispersion of the graphene monolayer energy bands, but it reduces the constant Fermi velocity vF, flattening the gradient of the bands. This reduces EF, increases hV i/EF, and strengthens the superfluid pair coupling. A magnetic field B? ≳ 1 T with a periodicity ≲10 nm, would boost hV i/EF above 2.35 (see Section “Double monolayer graphene”). The resulting superfluid gaps are large, of the order of hundreds of Kelvin. on the magnetic field strength. At Bmax There is an upper limit Bmax ? ? , the Fermi energy passes through the top of the band of the magnetic field. Once the Fermi energy is in the bandgap, the assumption that the spectrum is linear is no longer valid.

Nanoribbons from double monolayer graphene Zarenia et al. (2016) predicted strong electron-hole superfluidity in coupled monolayer graphene armchair-edge terminated nanoribbons, separated by a hexagonal boron nitride insulating barrier (Fig. 7). The nanoribbons would be independently contacted, with top and bottom metal gates to tune the densities. In contrast to graphene monolayers, the bands of the nanoribbons are parabolic at low energies and there is a bandgap. As a result, the screening of the electron-hole pairing interaction is much weaker compared with coupled graphene monolayers. The one-dimensional quantum confinement of the electrons and holes further enhances the pairing attraction. The one-dimensional van Hove enhancement of the density of states magnifies superfluid shape resonances from the confinement (see (Perali et al., 1996; Bianconi et al., 1997, 1998)), acting non-linearly through the superfluid gap equation to enhance the magnitude of the gaps. The resulting superfluidity is strong-coupled with very large superfluid gaps that can be in excess of D
1000 K. The superfluidity is multicomponent, with different condensates for the multiple subbands of the nanoribbon.

Hybridization of double bilayer graphene

In double bilayer graphene with separations between the bilayers less than d ¼ 1 nm, the electron-hole pairing interaction would become very strong, potentially increasing superfluid onset densities and hence transition temperatures. However, for separations d < 1 nm, electron-hole recombination rates will be extremely rapid, thus making equilibration uncertain. When the separation

Fig. 7 Electron-doped and hole-doped armchair-edge terminated graphene nanoribbons of width W separated by a hexagonal boron nitride insulator (h-BN) of thickness d. Top and back metal gates tune the densities.

48

Excitonic superfluidity in electron-hole bilayer systems

between the graphene bilayers is comparable to the thickness of each bilayer (0.37 nm (Nilsson et al., 2008)), strong hybridization will occur across all four monolayers making up the double graphene bilayer system. Su and MacDonald (2017) demonstrated that there could exist a rich diagram of novel phases, including spin-density-wave states and mixed excitonic and spin order states. In practice, with tunneling rates that grow exponentially as the bilayer separations decrease, whether these new phases could be realized is unclear.

Double trilayer and quadlayer graphene Zarenia et al. (2014) extended the idea of increasing the density of states, and thus reducing the Fermi energy for a given density, to double trilayer and double quadlayer graphene. The additional layers depress the Fermi energy at a given density, which further strengthens the coupling in the superfluid. The strong coupling increases the onset density and hence the maximum transition
40 K are predicted. temperature. For double quadlayer graphene, transition temperatures as high as TBKT c

Double van der Waals layers with Type-III interfaces Gupta et al. (2020) identified lattice-matched van der Waals double monolayer heterostructures with broken-gap Type-III interface band alignment, in which the top of the valence band in the upper layer lies above the bottom of the conduction band in the lower layer. This band alignment permits electrons to spontaneously transfer from the top of the upper-layer valence band to the bottom of the lower-layer conduction band, leaving a hole behind in the upper-layer valence band. In this way, the Type-III interface spontaneously generates without gate voltages, spatially separated densities of electrons and holes that are in equilibrium. As with Type-II interfaces, Type-III interfaces need no insulating barrier to separate the electrons and holes, leading to strong electron-hole pairing interactions. The density range, 1011 − 1013 cm−2, is experimentally accessible. To sweep across the different BCS-BEC crossover regimes, the densities can be tuned with gate voltages or with strain.

Superlattice of transition metal dichalcogenide monolayers Van der Donck et al. (2020) proposed overcoming the topological limitations on the Berezinskii-Kosterlitz-Thouless transition temperature, by replacing the double electron-hole layers with a superlattice of alternating electron-doped and hole-doped layers. Since a superlattice is a three-dimensional structure, the transition is no longer topological, opening a new way for the strong electron-hole pairing interactions and the correspondingly large superfluid gaps to be exploited, so as to drive the transition to much higher temperatures. A superlattice of alternating tungsten disulfide and tungsten diselenide monolayers, was demonstrated to be optimal, with transition temperatures close to room temperature, Tc
270 K.

Conclusion This chapter has presented an overview to a phenomenon that recently has received impetus from accumulating experimental evidence that equilibrium electron-hole superfluidity, encompassing Bose-Einstein condensation of excitons and BCS-BEC crossover physics, is realizable in semiconductor heterostructure devices in which electrons and holes are kept spatially separated in adjacent parallel conducting layers. The electron-hole pairing attraction is strong, promising high transition temperatures. Because the attraction is long-ranged, screening determines many superfluid properties. Heterostructures currently receiving much attention include gallium arsenide and indium arsenide–gallium antimonide double quantum wells, and graphene and transition metal dichalcogenide double layers. Advantages, disadvantages, and the current experimental status of these and other promising systems have been discussed. To identify clear signatures of electron-hole superfluidity remains a major challenge. Current efforts focus on searching for dissipationless counterflow currents in the two layers, electron-hole drag resistance, and enhanced electron-hole tunneling between the layers combined with electroluminescence from the ensuing recombination. An ability to investigate crossover physics in electronic devices at equilibrium offers exciting prospects for the future.

Acknowledgments I am grateful to Dr. Sara Conti for a great deal of assistance in the preparation of this article.

References Balatsky AV, Joglekar YN, and Littlewood PB (2004) Dipolar superfluidity in electron-hole bilayer systems. Physical Review Letters 93: 266801. Bardeen J, Cooper LN, and Schrieffer JR (1957) Theory of superconductivity. Physics Review 108(5): 1175. Berezinsky VL (1972) Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Soviet Physics—JETP 34: 610. (Zh. Eksp. Teor. Fiz. 61, 1144 (1972)). Bianconi A, Valletta A, Perali A, and Saini N (1997) High Tc superconductivity in a superlattice of quantum stripes. Solid State Communications 102(5): 369–374. Bianconi A, Valletta A, Perali A, and Saini NL (1998) Superconductivity of a striped phase at the atomic limit. Physica C 296: 269–280. Britnell L, Gorbachev RV, Jalil R, Belle BD, Schedin F, Mishchenko A, Georgiou T, Katsnelson MI, Eaves L, Morozov SV, Peres NMR, Leist J, Geim AK, Novoselov KS, and Ponomarenko LA (2012) Field-effect tunneling transistor based on vertical graphene heterostructures. Science 335(6071): 947–950.

Excitonic superfluidity in electron-hole bilayer systems

49

Burg GW, Prasad N, Kim K, Taniguchi T, Watanabe K, MacDonald AH, Register LF, and Tutuc E (2018) Strongly enhanced tunneling at total charge neutrality in double-bilayer graphene-WSe2 heterostructures. Physical Review Letters 120: 177702. Butov LV (2004) Condensation and pattern formation in cold exciton gases in coupled quantum wells. Journal of Physics. Condensed Matter 16(50): R1577–R1613. Butov LV, Lai CW, Ivanov AL, Gossard AC, and Chemla DS (2002) Towards Bose–Einstein condensation of excitons in potential traps. Nature 417(6884): 47–52. Castro EV, Novoselov KS, Morozov SV, Peres NMR, Dos Santos JMBL, Nilsson J, Guinea F, Geim AK, and Castro Neto AH (2010) Electronic properties of a biased graphene bilayer. Journal of Physics. Condensed Matter 22(17): 175503. Chaves A and Neilson D (2019) Two-dimensional semiconductors host high-temperature exotic state. Nature (London) 574(7776): 39. Combescot M, Combescot R, and Dubin F (2017) Bose-Einstein condensation and indirect excitons: A review. Reports on Progress in Physics 80(6): 066501. Comte C and Nozières P (1982) Exciton Bose condensation: The ground state of an electron-hole gas. I. Mean field description of a simplified model. Journal de Physique 43: 1069. Conti S, Vignale G, and MacDonald AH (1998) Engineering superfluidity in electron–hole double layers. Physical Review B 57: R6846–R6849. Conti S, Perali A, Peeters FM, and Neilson D (2017) Multicomponent electron-hole superfluidity and the BCS-BEC crossover in double bilayer graphene. Physical Review Letters 119: 257002. Conti S, Perali A, Peeters FM, and Neilson D (2019) Multicomponent screening and superfluidity in gapped electron-hole double bilayer graphene with realistic bands. Physical Review B 99: 144517. Conti S, Neilson D, Peeters FM, and Perali A (2020a) Transition metal dichalcogenides as strategy for high temperature electron-hole superfluidity. Condensed Matter 5: 22. Conti S, Van der Donck M, Perali A, Peeters FM, and Neilson D (2020b) Doping-dependent switch from one- to two-component superfluidity in coupled electron-hole van der Waals heterostructures. Physical Review B 101: 220504(R). Conti S, Perali A, Peeters FM, and Neilson D (2021a) Effect of mismatched electron-hole effective masses on superfluidity in double layer solid-state systems. Condensed Matter 6(2). Conti S, Saberi-Pouya S, Perali A, Virgilio M, Peeters FM, Hamilton AR, Scappucci G, and Neilson D (2021b) Electron-hole superfluidity in strained Si/Ge type II heterojunctions. npj Quantum Materials 6(1): 1–7. Croxall AF, Das Gupta K, Nicoll CA, Thangaraj M, Beere HE, Farrer I, Ritchie DA, and Pepper M (2008) Anomalous Coulomb drag in electron-hole bilayers. Physical Review Letters 101: 246801. Das Gupta K, Croxall AF, Waldie J, Nicoll CA, Beere HE, Farrer I, Ritchie DA, and Pepper M (2011) Experimental progress towards probing the ground state of an electron-hole bilayer by low-temperature transport. Advances in Condensed Matter Physics 2011: 727958. Debnath B, Barlas Y, Wickramaratne D, Neupane MR, and Lake RK (2017) Exciton condensate in bilayer transition metal dichalcogenides: Strong coupling regime. Physical Review B 96: 174504. Dell’Anna L, Perali A, Covaci L, and Neilson D (2015) Using magnetic stripes to stabilize superfluidity in electron-hole double monolayer graphene. Physical Review B 92: 220502. Du L, Li X, Lou W, Sullivan G, Chang K, Kono J, and Du R-R (2017) Evidence for a topological excitonic insulator in InAs/GaSb bilayers. Nature Communications 8(1): 1971. Eagles DM (1969) Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors. Physics Review 186: 456–463. Efimkin DK, Burg GW, Tutuc E, and MacDonald AH (2020) Tunneling and fluctuating electron-hole Cooper pairs in double bilayer graphene. Physical Review B 101: 035413. Eisenstein JP and MacDonald AH (2004) Bose–Einstein condensation of excitons in bilayer electron systems. Nature 432(7018): 691. Fil DV and Shevchenko SI (2018) Electron-hole superconductivity. Low Temperature Physics 44(9): 867–909. Fogler MM, Butov LV, and Novoselov KS (2014) High-temperature superfluidity with indirect excitons in van der Waals heterostructures. Nature Communications 5: 4555. Franz M (2007) Importance of fluctuations. Nature Physics 3: 686. Frisenda R, Navarro-Moratalla E, Gant P, Pérez De Lara D, Jarillo-Herrero P, Gorbachev RV, and Castellanos-Gomez A (2018) Recent progress in the assembly of nanodevices and van der Waals heterostructures by deterministic placement of 2D materials. Chemical Society Reviews 47: 53–68. Fulde P and Ferrell RA (1964) Superconductivity in a strong spin-exchange field. Physics Review 135: A550. Gaebler JP, Stewart JT, Drake TE, Jin DS, Perali A, Pieri P, and Strinati GC (2010) Observation of pseudogap behaviour in a strongly interacting Fermi gas. Nature Physics 6: 569. Gorbachev RV, Geim AK, Katsnelson MI, Novoselov KS, Tudorovskiy T, Grigorieva IV, MacDonald AH, Morozov SV, Watanabe K, Taniguchi T, and Ponomarenko LA (2012) Strong Coulomb drag and broken symmetry in double-layer graphene. Nature Physics 8(12): 896. Guidini A and Perali A (2014) Band-edge BCS–BEC crossover in a twoband superconductor: Physical properties and detection parameters. Superconductor Science and Technology 27(12): 124002. Gupta S, Kutana A, and Yakobson BI (2020) Heterobilayers of 2D materials as a platform for excitonic superfluidity. Nature Communications 11(1): 1–7. Hendrickx NW, Franke DP, Sammak A, Scappucci G, and Veldhorst M (2020) Fast two-qubit logic with holes in germanium. Nature 577(7791): 487–491. High AA, Leonard JR, Remeika M, Butov LV, Hanson M, and Gossard AC (2012) Condensation of excitons in a trap. Nano Letters 12(5): 2605. Hohenberg PC (1967) Existence of long-range order in one and two dimensions. Physics Review 158: 383. Hu BY-K (2000) Prospecting for the superfluid transition in electron-hole coupled quantum wells using Coulomb drag. Physical Review Letters 85: 820. Jiang H (2012) Electronic band structures of molybdenum and tungsten dichalcogenides by the GW approach. Journal of Physical Chemistry C 116(14): 7664. Kane BE, Eisenstein JP, Wegscheider W, Pfeiffer LN, and West KW (1994) Separately contacted electron–hole double layer in a GaAs/AlxGa1−x As heterostructure. Applied Physics Letters 65(25): 3266–3268. Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P, Keeling JMJ, Marchetti FM, Szymanska MH, André R, Staehli JL, Savona V, Littlewood PB, Deveaud B, and Dang LS (2006) Bose–Einstein condensation of exciton polaritons. Nature 443(7110): 409–414. Keldysh LV and Kopaev YV (1965) Possible instability of semimetallic state toward Coulomb interaction. Soviet Physics—Solid State USSR 6(9): 2219. (Fizika Tverdogo Tela, 6, 2791-2798 (1964)). Kharitonov MY and Efetov KB (2008) Electron screening and excitonic condensation in double-layer graphene systems. Physical Review B 78: 241401. Komendová L, Chen Y, Shanenko AA, Miloševic MV, and Peeters FM (2012) Two-band superconductors: Hidden criticality deep in the superconducting state. Physical Review Letters 108(20): 207002. Kosterlitz JM and Thouless DJ (1973) Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C: Solid State Physics 6: 1181–1203. Lagoin C and Dubin F (2021) Key role of the moiré potential for the quasi-condensation of interlayer excitons in van der Waals heterostructures. Physical Review B 103: L041406. Larkin AI and Ovchinnikov YN (1965) Quantum theory of defects in crystals. Soviet Physics—JETP 20: 762–770. (Zh. Eksp. Teor. Fiz. 47, 1136 (1964)). Lee K, Fallahazad B, Xue J, Dillen DC, Kim K, Taniguchi T, Watanabe K, and Tutuc E (2014) Chemical potential and quantum Hall ferromagnetism in bilayer graphene. Science 345(6192): 58. Leggett AJ (1980) Diatomic molecules and Cooper pairs. In: Modern Trends in the Theory of Condensed Matter, pp. 13–27. Springer. Liu L, Swierkowski L, Neilson D, and Szymanski J (1996) Static and dynamic properties of coupled electron-electron and electron–hole layers. Physical Review B 53: 7923–7931. Liu L, Swierkowski L, and Neilson D (1998) Exciton and charge density wave formation in spatially separated electron–hole liquids. Physica B: Condensed Matter 249-251: 594–597. Liu X, Li JIA, Watanabe K, Taniguchi T, Hone J, Halperin BI, Kim P, and Dean CR (2022) Crossover between strongly coupled and weakly coupled exciton superfluids. Science 375(6577): 205–209. Lodari M, Hendrickx NW, Lawrie WIL, Hsiao T-K, Vandersypen LMK, Sammak A, Veldhorst M, and Scappucci G (2021) Low percolation density and charge noise with holes in germanium. Materials for Quantum Technology 1(1): 011002. López Ríos P, Perali A, Needs RJ, and Neilson D (2018) Evidence from quantum Monte Carlo simulations of large-gap superfluidity and BCS-BEC crossover in double electron-hole layers. Physical Review Letters 120: 177701. Lozovik YE and Sokolik AA (2008) Coherent phases and collective electron phenomena in graphene. Journal of Physics Conference Series 129(1): 012003. Lozovik YE and Sokolik A (2010) Ultrarelativistic electron-hole pairing in graphene bilayer. European Physical Journal B 73(2): 195.

50

Excitonic superfluidity in electron-hole bilayer systems

Lozovik YE and Yudson VI (1975) Feasibility of superfluidity of paired spatially separated electrons and holes. JETP Letters (USSR) 22(11): 274–276. (Pis’ma Zh. Eksp. Teor. Fiz. 22, 556-559 (1975)). Lozovik YE and Yudson VI (1976) A new mechanism for superconductivity: Pairing between spatially separated electrons and holes. Soviet Physics—JETP 44: 389–397. (Zh. Eksp. Teor. Fiz. 71, 738-753 (1976)). Lozovik YE, Ogarkov SL, and Sokolik AA (2012) Condensation of electron-hole pairs in a two-layer graphene system: Correlation effects. Physical Review B 86: 045429. Ma L, Nguyen PX, Wang Z, Zeng Y, Watanabe K, Taniguchi T, MacDonald AH, Mak KF, and Shan J (2021) Strongly correlated excitonic insulator in atomic double layers. Nature 598(7882): 585–589. Mak KF, Lee C, Hone J, Shan J, and Heinz TF (2010) Atomically thin MoS2: A new direct-gap semiconductor. Physical Review Letters 105(13): 136805. Mermin ND and Wagner H (1966) Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Physical Review Letters 17: 1133. Min H, Bistritzer R, Su J-J, and MacDonald AH (2008) Room-temperature superfluidity in graphene bilayers. Physical Review B 78(12): 121401. Narozhny BN and Levchenko A (2016) Coulomb drag. Reviews of Modern Physics 88: 025003. Neilson D, Swierkowski L, Szymanski J, and Liu L (1993) Excitations of the strongly correlated electron liquid in coupled layers. Physical Review Letters 71: 4035–4038. Neilson D, Perali A, and Hamilton AR (2014) Excitonic superfluidity and screening in electron-hole bilayer systems. Physical Review B 89: 060502(R). Nilsson F and Aryasetiawan F (2021) Effects of dynamical screening on the BCS-BEC crossover in double bilayer graphene: Density functional theory for exciton bilayers. Physical Review Materials 5: L050801. Nilsson J, Castro Neto AH, Guinea F, and Peres NMR (2008) Electronic properties of bilayer and multilayer graphene. Physical Review B 78: 045405. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, and Firsov AA (2004) Electric field effect in atomically thin carbon films. Science 306(5696): 666. Nozières P and Pistolesi F (1999) From semiconductors to superconductors: A simple model for pseudogaps. European Physical Journal B 10(4): 649. Nozières P and Schmitt-Rink S (1985) Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. Journal of Low Temperature Physics 59(3): 195. Pascucci F, Conti S, Neilson D, Tempere J, and Perali A (2022) Josephson effect as signature of electron-hole superfluidity in bilayers of van der Waals heterostructures. Physical Review B. In press. Perali A, Bianconi A, Lanzara A, and Saini N (1996) The gap amplification at a shape resonance in a superlattice of quantum stripes: A mechanism for high Tc. Solid State Communications 100(3): 181–186. Perali A, Pieri P, Strinati GC, and Castellani C (2002) Pseudogap and spectral function from superconducting fluctuations to the bosonic limit. Physical Review B 66: 024510. Perali A, Neilson D, and Hamilton AR (2013) High-temperature superfluidity in double-bilayer graphene. Physical Review Letters 110: 146803. Pieri P, Neilson D, and Strinati GC (2007) Effects of density imbalance on the BCS-BEC crossover in semiconductor electron-hole bilayers. Physical Review B 75(11): 113301. Pillarisetty R (2011) Academic and industry research progress in germanium nanodevices. Nature 479(7373): 324–328. Pohlt M, Lynass M, Lok JGS, Dietsche W, Klitzing KV, Eberl K, and Mühle R (2002) Closely spaced and separately contacted two–dimensional electron and hole gases by in situ focused–ion implantation. Applied Physics Letters 80(12): 2105–2107. Saberi-Pouya S, Zarenia M, Perali A, Vazifehshenas T, and Peeters FM (2018) High-temperature electron-hole superfluidity with strong anisotropic gaps in double phosphorene monolayers. Physical Review B 97(17): 174503. Saberi-Pouya S, Conti S, Perali A, Croxall AF, Hamilton AR, Peeters FM, and Neilson D (2020) Experimental conditions for the observation of electron-hole superfluidity in GaAs heterostructures. Physical Review B 101: 140501(R). Sarma G (1963) On the influence of a uniform exchange field acting on the spins of the conduction electrons in a superconductor. Journal of Physics and Chemistry of Solids 24(8): 1029. Scappucci G, Kloeffel C, Zwanenburg FA, Loss D, Myronov M, Zhang J-J, De Franceschi S, Katsaros G, and Veldhorst M (2020) The germanium quantum information route. Nature Reviews Materials 1–18. Schäffler F (1997) High-mobility Si and Ge structures. Semiconductor Science and Technology 12(12): 1515–1549. Schafroth MR, Butler ST, and Blatt JM (1957) Quasichemical equilibrium approach to superconductivity. Helvetica Physica Acta 30: 93–134. Seamons JA, Morath CP, Reno JL, and Lilly MP (2009) Coulomb drag in the exciton regime in electron-hole bilayers. Physical Review Letters 102: 026804. Shanenko AA, Croitoru MD, Vagov AV, Axt VM, Perali A, and Peeters FM (2012) Atypical BCS-BEC crossover induced by quantum-size effects. Physical Review A 86(3): 033612. Shevchenko SI (1976) Theory of superconductivity of systems with pairing of spatially separated electrons and holes. Soviet Journal of Low Temperature Physics 2(4): 251. (Fiz. Nizk. Temp. 2, 505 (1976)). Sivan U, Solomon PM, and Shtrikman H (1992) Coupled electron-hole transport. Physical Review Letters 68: 1196. Sodemann I, Pesin DA, and MacDonald AH (2012) Interaction-enhanced coherence between two-dimensional Dirac layers. Physical Review B 85: 195136. Stern F (1967) Polarizability of a two-dimensional electron gas. Physical Review Letters 18: 546–548. Strinati GC, Pieri P, Röpke G, Schuck P, and Urban M (2018) The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems. Physics Reports 738: 1–76. Su J-J and MacDonald AH (2008) How to make a bilayer exciton condensate flow. Nature Physics 4(10): 799–802. Su J-J and MacDonald AH (2017) Spatially indirect exciton condensate phases in double bilayer graphene. Physical Review B 95: 045416. Subasi AL, Pieri P, Senatore G, and Tanatar B (2010) Stability of Sarma phases in density imbalanced electron-hole bilayer systems. Physical Review B 81: 075436. Van der Donck M, Conti S, Perali A, Hamilton AR, Partoens B, Peeters FM, and Neilson D (2020) Three-dimensional electron-hole superfluidity in a superlattice close to room temperature. Physical Review B 102: 060503. Vignale G and MacDonald AH (1996) Drag in paired electron-hole layers. Physical Review Letters 76: 2786. Vigneau F, Mizokuchi R, Zanuz DC, Huang X, Tan S, Maurand R, Frolov S, Sammak A, Scappucci G, Lefloch F, and Franceschi SD (2019) Germanium quantum-well Josephson field-effect transistors and interferometers. Nano Letters 19(2): 1023–1027. Virgilio M and Grosso G (2006) Type-I alignment and direct fundamental gap in SiGe based heterostructures. Journal of Physics. Condensed Matter 18(3): 1021–1031. Wang Z, Rhodes DA, Watanabe K, Taniguchi T, Hone JC, Shan J, and Mak KF (2019) Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature (London) 574(7776): 76–80. Watson TF, Philips SGJ, Kawakami E, Ward DR, Scarlino P, Veldhorst M, Savage DE, Lagally MG, Friesen M, Coppersmith SN, Eriksson MA, and Vandersypen LMK (2018) A programmable two-qubit quantum processor in silicon. Nature 555(7698): 633–637. Zarenia M, Perali A, Neilson D, and Peeters F (2014) Enhancement of electron-hole superfluidity in double few-layer graphene. Scientific Reports 4: 7319. Zarenia M, Perali A, Peeters FM, and Neilson D (2016) Large gap electron-hole superfluidity and shape resonances in coupled graphene nanoribbons. Scientific Reports 6(1): 24860. Zeng Y and MacDonald AH (2020) Electrically controlled two-dimensional electron-hole fluids. Physical Review B 102: 085154. Zenker B, Fehske H, and Beck H (2015) Excitonic Josephson effect in double-layer graphene junctions. Physical Review B 92: 081111. Zwanenburg FA, Dzurak AS, Morello A, Simmons MY, Hollenberg LCL, Klimeck G, Rogge S, Coppersmith SN, and Eriksson MA (2013) Silicon quantum electronics. Reviews of Modern Physics 85: 961–1019.

Spin superfluidity EB Sonin, Racah Institute of Physic, Hebrew University of Jerusalem, Jerusalem, Israel © 2024 Elsevier Ltd. All rights reserved.

Introduction Concept of superfluidity Spin superfluidity in ferromagnets Spin superfluidity in antiferromagnets Superfluid spin transport without spin conservation law Long-distance superfluid spin transport Experiments on detection of spin superfluidity Conclusion References

51 52 56 59 62 63 64 65 66

Abstract The phenomenon of superfluidity (superconductivity) is a possibility of transport of mass (charge) on macroscopical distances without essential dissipation. In magnetically ordered media with easy-plane topology of the order parameter space the superfluid transport of spin is also possible despite the absence of the strict conservation law for spin. The article addresses three key issues in the theory of spin superfluidity: topology of the order parameter space, Landau criterion for superfluidity, and decay of superfluid currents via phase slip events, in which magnetic vortices cross current streamlines. Experiments on detection of spin superfluidity are also surveyed.

Key points

• • • • •

Spin superfluidity is the possibility to transport spin with essentially suppressed dissipation on long distances. Spin superfluidity is possible if the magnetic order parameter space has the topology of a circumference. The necessary topology is provided by easy-plane anisotropy in ferromagnets or by magnetic field in antiferromagnets. Metastability of spin superfluid current states is restricted by the Landau criterion. Decay of spin superfluid currents is realized via phase slips, in which magnetic vortices cross current streamlines.

Introduction The phenomenon of superfluidity (superconductivity in the case of charged fluids) is known more than a hundred years. Its analog for spin (spin superfluidity) occupies minds of condensed matter physicists from 70s of the last century. The term superfluidity is used in the literature to cover a broad range of phenomena in superfluid 4He and 3He, Bose-Einstein condensates of cold atoms, and solids. In this article superfluidity means only the possibility to transport a physical quantity (mass, charge, spin, . . .) without dissipation (more accurately, with essentially suppressed dissipation). This corresponds to the original hundred-years old meaning of the term from the times of Kamerlingh Onnes and Kapitza. The superfluidity is conditioned by the special topology of the order parameter space (vacuum manifold). Namely, this topology is that of a circumference in a plane. The angle of rotation around this circumference is the order parameter phase describing all degenerate ground states. In superfluids the phase is the phase of the macroscopic wave function. In magnetically ordered systems (ferro- and antiferromagnets) the necessary topology is provided by easy-plane magnetic anisotropy, and the phase is the angle of rotation around the axis (further the axis z) normal to the easy plane. Currents of mass (charge) or spin are proportional to phase gradients and are called supercurrents. In early discussions of the spin supercurrent it was considered as a counterflow of superfluid currents of particles with different spins in superfluid 3He (Vuorio, 1974), i.e., spin was transported by itinerant spin carriers. Later it was demonstrated that spin superfluidity is a universal phenomenon, which does not require mobile spin carriers and is possible in magnetic insulators (Sonin, 1978a, 1982). It can be described within the framework of the standard Landau–Lifshitz–Gilbert (LLG) theory. However, the publications conditioning spin superfluid transport by the presence of mobile carriers of spin continued to appear in the literature (Bunkov, 1995; Shi et al., 2006). According to Shi et al. (2006), it is a critical flaw of spin-current definition if it predicts spin currents in insulators. Strictly speaking the analogy of spin superfluidity with superfluids is complete only if there is invariance with respect to any rotation in the spin space around the axis z. Then according to Noether’s theorem the spin component along the axis z is conserved.

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Spin superfluidity

But there is always some magnetic anisotropy, which breaks the rotational invariance. Correspondingly, there is no strict conservation law for spin, while in superfluids the gauge invariance is exact, and the conservation law of mass (charge) is also exact. In the past there were arguments about whether superfluidity is possible in the absence of the conservation law. This dispute started at discussion of superfluidity of electron-hole pairs or excitons. The number of electron-hole pairs can vary due to interband transitions, and the degeneracy with respect to the phase of the pair condensate is lifted. Guseinov and Keldysh (1972) called this effect “fixation of phase.” They demonstrated that spatially homogeneous stationary current states are impossible, and concluded that there is no analogy with superfluidity. However, later it was demonstrated that phase fixation does not rule out the existence of weakly inhomogeneous stationary current states analogous to superfluid current states (Kulik and Shevchenko, 1976; Lozovik and Yudson, 1977; Sonin, 1977). This analysis was extended on spin superfluidity (Sonin, 1978a,b, 1982). In magnetism violation of the spin conservation law usually is rather weak because it is related with relativistically small (inversely proportional to the speed of light) processes of spin-orbit interaction. In fact, the LLG theory itself is based on the assumption of weak spin-orbit interaction (Landau and Lifshitz, 1980). Above we discussed supercurrents generated in the equilibrium (ground state) of the magnetically ordered medium with easy plane topology. But in magnetically ordered system this topology is possible also in non-equilibrium coherent precession states, when spin pumping supports spin precession with fixed spin component along the magnetic field. Such non-equilibrium coherent precession states, which are called nowadays magnon BEC, were experimentally investigated in the B phase of superfluid 3He and in yttrium-iron-garnet (YIG) films (Bunkov, 1995; Demokritov et al., 2006). Spin superfluid transport is possible as long as the spin phase gradient does not exceeds the critical value determined by the Landau criterion. The Landau criterion checks stability of supercurrent states with respect to elementary excitations of all collective modes. The Landau criterion determines a threshold for the current state instability, but it tells nothing about how the instability develops. The decay of the supercurrent is possible only via phase slips. In a phase slip event a magnetic vortex crosses current streamlines decreasing the phase difference along streamlines. Below the critical value of supercurrent phase slips are suppressed by energetic barriers. The critical value of the supercurrent at which barriers vanish is of the same order as that estimated from the Landau criterion. This leads to a conclusion that the instability predicted by the Landau criterion is a precursor of the avalanche of phase slips not suppressed by any activation barrier. The superfluid spin transport on macroscopical distances is possible only if the spin phase performs a large number of full 2p rotations along current streamlines (large winding number), and phase slips are suppressed by energetic barriers. On the other hand, small phase variations less than 2p are ubiquitous in magnetism. They emerge in any spin wave, any domain wall, or due to disorder. Their existence is confirmed by numerous experiments in the past. Spin currents generated by these small phase differences transport spin only on small distances and oscillate in space or time, or both. Their existence is not a manifestation of spin superfluidity. In last decades the interest to spin superfluidity was revived (Bunkov and Volovik, 2013; Chen and MacDonald, 2017; Evers and Nowak, 2020; Iacocca et al., 2017; Qaiumzadeh et al., 2017; Sonin, 2010, 2017, 2019b, 2020; Sun et al., 2016; Takei et al., 2014; Takei and Tserkovnyak, 2014; Tserkovnyak and Kläui, 2017) in connection with the emergence of spintronics. The present article reviews the three essentials of the spin superfluidity concept: topology, Landau criterion, and phase slips. The article focuses on the qualitative analysis avoiding details of calculations, which can be found in original papers. After the theoretical analysis the experiments searching for spin superfluidity are shortly discussed.

Concept of superfluidity Let us remind the concept of superfluidity for the transport of mass (charge). In superfluid hydrodynamics there are the Hamilton equations for the pair of the canonically conjugate variables “phase-particle density”: ħ

d’ dH ¼ − , dn dt

dn dH ¼ : dt ħd’

(1)

Here dH ∂H ∂H ¼ −r  , dn ∂n ∂rn

dH ∂H ∂H ¼ −r  d’ ∂’ ∂r’

(2)

are functional derivatives of the Hamiltonian H¼

ħ2 n r’2 + E0 ðnÞ, 2m

(3)

’ is the phase of the wave function describing the Bose-Einstein condensate (BEC) in Bose liquids or the Cooper pair condensate in Fermi liquids, and E0(n) is the energy of the superfluid at rest, which depends only on the particle density n. Taking into account the gauge invariance [U(1) symmetry] ∂H/∂’ ¼ 0, the Hamilton equations are reduced to the equations of hydrodynamics for an ideal liquid:

Spin superfluidity

53

dv ¼ −rm, dt

(4)

dn ¼ −r  j: dt

(5)

m

In these expressions m¼

∂E0 ħ2 + r’2 ∂n 2m

(6)

is the chemical potential, and j ¼ nv ¼

∂ℋ ħ∂r’

(7)

is the particle current. We consider the zero-temperature limit, when the superfluid velocity coincides with the center-of-mass velocity of the whole liquid: v¼

ħ r’: m

(8)

The continuity equation Eq. (5) satisfies the conservation law of mass (charge), which follows from the gauge invariance. A collective mode of the ideal liquid is a plane sound wave ∝eikr−iot with the wave vector k, the frequency o, and the linear spectrum o ¼ usk. The sound velocity is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ∂2 E 0 : (9) us ¼ m ∂n2 In the sound wave the phase varies in space, i.e., the wave is accompanied by mass currents [Fig. 1(a)]. An amplitude of the phase variation is small, and currents transport mass on distances of the order of the wavelength. A real superfluid transport on macroscopic distances is possible in current states, which are stationary solutions of the hydrodynamic equations with finite constant currents, i.e., with constant nonzero phase gradients. In the current state the phase rotates through a large number of full 2p-rotations along streamlines of the current [Fig. 1(b)]. These are supercurrents or persistent currents. The crucial point of the superfluidity theory is why the supercurrent in Fig. 1(b) is a metastable state and does not decay a long time. The first explanation of the supercurrent metastability was the well known Landau criterion (Landau, 1941). According to this criterion, the current state is stable as far as any quasiparticle in a moving liquid has a positive energy in the laboratory frame and therefore its creation requires an energy input. Let us suppose that elementary quasiparticles of the liquid at rest have an energy e(p) ¼ ħo(k), where p ¼ ħk is the quasiparticle momentum. If the liquid moves with the velocity v the quasiparticle energy in the laboratory frame becomes ~eðpÞ ¼ eðpÞ + pv: This is the Doppler effect in the Galilean invariant fluid. The current state is stable if the energy ~e is never negative. This condition is the Landau criterion:

(a)

(b)

Fig. 1 Phase (in-plane rotation angle) variation in space at the presence of mass (spin) currents. (a) Oscillating currents in a sound (spin) wave. (b) Stationary mass (spin) supercurrent. From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

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Spin superfluidity

v < vL ¼ min

eðpÞ oðkÞ ¼ min : p k

(10)

If quasiparticles are phonons (quanta of sound waves) the Landau critical velocity vL is the sound velocity us. In superfluid 4He the Landau critical velocity vL is determined by the roton part of the spectrum. It is a few times less than the sound velocity. The Landau criterion checks the stability with respect to elementary microscopic perturbations of the current state, but does not provide an information how the instability would develop. The whole process of the supercurrent decay is connected with generation of macroscopic perturbations. These perturbations are quantum vortices. If the vortex axis (vortex line) coincides with the z axis, the phase gradient around the vortex line is given by vv ¼

k ½b z  r ħ , r’v ¼ m 2pr 2

(11)

where r is the position vector in the xy plane and k ¼ h/m is the velocity circulation quantum. Creation of the vortex requires some energy. The vortex energy per unit length (line tension) is determined mostly by the kinetic (gradient) energy in the area not very close to the vortex axis where the particle density does not differ essentially from its equilibrium value n0 (the London region): Z ħ2 ðr’v Þ2 pħ2 n0 r (12) ev ¼ n0 d2 r ¼ ln m : 2m m rc The upper cut-off rm of the logarithm is determined by geometry. For the vortex shown in Fig. 2(a) it is the distance of the vortex line from a sample border. The lower cut-of rc is the vortex-core radius. It determines the distance r at which the phase gradient is so high that the density n starts to decrease compared to the equilibrium density n0 at large r. The energy E0(n) of the resting superfluid at small n − n0 is determined by the fluid compressibility: E 0 ð nÞ ¼ E 0 ð n0 Þ +

m ðn0 −nÞ2 u2s : 2n0

(13)

At the distance rc from the vortex axis the energy mn r ’2v /2 becomes of the order of the compressibility energy at n0 − n  n0. This happens when the velocity vv(r) induced by the vortex becomes of the order of the sound velocity us. This yields rc  k/us. Inside the core the density vanishes at the vortex axis eliminating the singularity in the kinetic energy. For the weakly non-ideal Bose-gas rc is on the order of the coherence length. Phase slips are impeded by energy barriers determined by topology of the order parameter space (vacuum manifold). The order parameter of a superfluid is a complex wave function ¼ 0ei’, where the modulus 0 of the wave function is a positive constant determined by minimization of the energy and the phase ’ is a degeneracy parameter. Any degenerate ground state in a closed annular channel (torus) with some constant phase ’ maps on some point at the circumference | | ¼ 0 in the complex plane , while a current state with the phase change 2pN around the torus maps onto a path [Fig. 3(a)] winding around the circumference N times. The integer winding number N is a topological charge. The current can relax if it is possible to change the topological charge.

(a)

π/4

W

0 2π

7π/4

π/2

rm 3π/2

3π/4

5π/4 π

(b) 0

W −π/2

2π

rm

3π/2

π/2 π

5π/2

Fig. 2 Mass and magnetic vortices. (a) Vortex in a superfluid or magnetic vortex in an easy-plane ferromagnet without in-plane anisotropy. (b) magnetic vortex at small average spin currents (hr’i  1/l ) for fourfold inplane symmetry. The vortex line is a confluence of four 90o domain walls (solid lines). From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

Spin superfluidity

55

Fig. 3 Topology of the uniform mass current and the vortex states. (a) The current state in a torus maps onto the circumference | | ¼ 0 in the complex —plane, where 0 is the modulus of the equilibrium order parameter wave function in the uniform state. (b) The current state with a vortex maps onto the circle | |  0. From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

A change of the topological charge from N to N − 1 is possible, if a vortex generated at one border of a channel with a moving superfluid moves across current streamlines “cutting” the channel cross-section, and annihilates at another border as shown in Fig. 2 (a). This is a phase slip. In the phase slip event the distance rm of the vortex from a border varies from zero to the width W of the channel. The energy of the vortex in a moving superfluid is determined by a sum of the constant gradient r’0, which determines the supercurrent, and the phase gradient r’v induced by the vortex. The vortex energy consists of the energy of the vortex in a resting fluid given by Eq. (12) and of the energy from the cross terms of the two gradient fields r’0 and r’v: ~ev ¼

pħ2 n0 r 2pħ2 n0 L ln m − Sr’0 , m rc m

(14)

where L is the length of the vortex line and S is the area of the cut, at which the phase jumps by 2p. For the 2D case shown in Fig. 2 (a) (a straight vortex in a slab of thickness L normal to the picture plane) S ¼ Lrm. The vortex motion across the channel (growth of rm) is impeded by the barrier determined by the maximum of the energy ~ev as a function rm. The height of the barrier is the vortex energy at rm ¼ 1/2 r ’0: eb 

pħ2 n 2 : L ln m r c r’0

(15)

The barrier disappears at gradients r’0  1/rc, which are of the same order as the critical gradient determined from the Landau criterion. On the other hand, in the limit of small velocity v ∝ r ’0 the barrier height grows and at very small velocity v  ħ/mW reaches the value em 

pħ2 n W : L ln m rc

(16)

Thus, in the thermodynamic (macroscopic) limit W ! 1 the barrier height becomes infinite. Since the phase slip probability exponentially decreases on the barrier height (whether the barrier is overcome due to thermal fluctuations or via quantum tunneling) the life time of the current state in conventional superfluidity diverges when the velocity (phase gradient) decreases. This justifies calling superfluidity macroscopic quantum phenomenon. In the 3D geometry the phase slip is realized with expansion of vortex rings. For the ring of radius R the vortex-length and the area of the cut are L ¼ 2pR and S ¼ pR2 respectively, and the barrier disappears at the same critical gradient 1/rc as in the 2D case. The state with the vortex in a moving superfluid maps on the full circle | |  0 [Fig. 3(b)]. The area outside the vortex core maps on the circumference | | ¼ 0, while the core maps on the interior of the circle. In a weakly interacting Bose-gas the structure of the core is determined by solution of the Gross—Pitaevskii equation (Pitaevskii and Stringari, 2003; Sonin, 2016). The details of the core structure are not important for the content of the present article. For better understanding of the superfluidity phenomenon it is useful to consider a mechanical analog of superfluid current (Sonin, 1982). Let us twist a long elastic rod so that a twisting angle at one end of the rod with respect to an opposite end reaches values many times 2p. Bending the rod into a ring and connecting the ends rigidly, one obtains a ring with a circulating persistent angular-momentum flux (Fig. 4). The flux is proportional to the gradient of twisting angle, which plays the role of the phase gradient in the supercurrent. The deformed state of the ring is not the ground state of the ring, but it cannot relax to the ground state

56

Spin superfluidity

Torque

Superflow of angular momentum Fig. 4 Mechanical analog of a persistent current: A twisted elastic rod bent into a closed ring. There is a persistent angular-momentum flux around the ring. From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

via any elastic process, because it is topologically stable. The only way to relieve the strain inside the rod is plastic displacements. This means that dislocations start to move across rod cross-sections. The role of dislocations in the twisted rod is similar to the role of vortices in the superfluid.

Spin superfluidity in ferromagnets For a ferromagnet with magnetization density M the LLG equation is (Landau and Lifshitz, 1980)   ∂M ¼ g H eff  M , ∂t

(17)

where g is the gyromagnetic ratio between the magnetic and mechanical moment. The effective magnetic field is determined by the functional derivative of the Hamiltonian H: H eff ¼ −

dH ∂H ∂H ¼ − + ri dM ∂M ∂ri M

(18)

According to the LLG equation, the absolute value M of the magnetization does not vary. The evolution of M is a precession around the effective magnetic field Heff. If spin-rotational invariance is broken and there is uniaxial crystal magnetic anisotropy the phenomenological Hamiltonian is H¼

GM2z A − HM: ri Mri M + 2 2M2

(19)

The parameter A is stiffness of the spin system determined by exchange interaction. If the anisotropy parameter G is positive, it is the “easy plane” anisotropy, which keeps the magnetization in the xy plane. If the external magnetic field H is directed along the z axis, the z component of spin is conserved because of invariance with respect to rotations around the z axis. For the sake of simplicity we ignore the magnetostatic energy, which depends on sample shape. Since the absolute value M of magnetization is fixed, the magnetization vector M is fully determined by the z magnetization component Mz and the angle ’ showing the direction of M in the easy plane xy: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mx ¼ M? cos ’, My ¼ M? sin ’, M? ¼ M2 − M2z : (20) In the new variables the Hamiltonian is H¼

M2 AM2? ðr’Þ2 + z − HMz : 2 2w

(21)

Here we neglected gradients of Mz. The magnetic susceptibility w ¼ M2/G along the z axis is determined by the easy-plane anisotropy parameter G. The LLG equation reduces to the Hamilton equations for a pair of canonically conjugate continuous variables “angle–moment”:

Spin superfluidity

57

1 d’ dH ∂H ¼ − , ¼ − g dt dMz ∂Mz

(22)

1 dMz dH ∂H ¼ ¼ −r , g dt d’ ∂r’

(23)

where functional derivatives are taken from the Hamiltonian Eq. (21). Using the expressions for functional derivatives the Hamilton equations are 1 d’ M − wH ¼ AMz ðr’Þ2 − z , g dt w

(24)

1 dMz + rJ s ¼ 0, g dt

(25)

where Js ¼ −

∂H ¼ −AM2? r’ ∂r’

(26)

is the spin current. Although our equations are written for magnetization, but not for the spin density, Js is defined as a current of spin with the spin density Mz/g. There is an evident analogy of Eqs. (24) and (25) with the hydrodynamic equations (4) and (5) for the superfluid. This analogy of magnetodynamics with hydrodynamics was pointed out by Halperin and Hohenberg (1969) for spin waves in antiferromagnets. The analogy is also important for spin superfluidity. There are linear solutions of Eqs. (24) and (25) describing the plane spin-wave mode ∝eikr−iot with the sound-like spectrum: o ¼ csw k,

(27)

where csw ¼ gM?

rffiffiffi A w

(28)

is the spin-wave velocity in the ground state. The variation of the phase in the space is the same as in the sound mode propagating in the superfluid and shown in Fig. 1(a). However, as well as the mass current in a sound wave, the small oscillating spin current in the spin wave does not lead to long-distance superfluid spin transport, which this article addresses. Spin superfluid transport on long distances is realized in current states with spin performing a large number of full 2p-rotations in the easy plane as shown in Fig. 1(b). In the current state with a constant gradient of the spin phase K ¼ r’, there is a constant magnetization component along the magnetic field (the axis z): Mz ¼

wH : 1 − wAK 2

(29)

Like in superfluids, the stability of current states is connected with topology of the order parameter space. In isotropic ferromagnets (G ¼ 0) the order parameter space is a spherical surface of radius equal to the absolute value of the magnetization vector M [Fig. 5(a)]. All points on this surface correspond to the same energy of the ground state. Suppose we created the spin current state with monotonously varying phase ’ in a torus. This state maps on the equatorial circumference in the order parameter space. The topology allows to continuously shift the circumference and to reduce it to a point (the northern or the southern pole). During this process shown in Fig. 5(a) the path remains in the order parameter space all the time, and therefore, no energetic barrier resists to the transformation. Thus, in isotropic ferromagnets the metastable current state are not expected. In a ferromagnet with easy-plane anisotropy (G > 0) the order parameter space reduces from the spherical surface to a circumference parallel to the xy plane. It is shown in Fig. 5(b) for zero magnetic field when the circumference is the equator. This order parameter space is topologically equivalent to that in superfluids. Now the transformation of the circumference to the point costs the anisotropy energy. This allows to expect metastable spin currents (supercurrents). The magnetic field along the anisotropy axis z shifts the easy plane either up [Fig. 5(d)] or down away from the equator. In order to check the Landau criterion, one should know the spectrum of spin waves in the current state with the constant value of the spin phase gradient K ¼ r’ and with the longitudinal (along the magnetic field) magnetization given by Eq. (29). The spectrum is determined by solving the Hamilton equations Eqs. (24) and (25) linearized with respect to weak perturbations of the current state. We skip the standard algebra given elsewhere (Sonin, 2019b). Finally one obtains (Iacocca et al., 2017; Sonin, 2019b) the spectrum of plane spin waves: o + wk ¼ ~csw k:

(30)

Here ~csw ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 −wAK 2 csw

(31)

is the spin-wave velocity in the current state, and the spin wave velocity csw in the ground state is given by Eq. (28). The velocity w ¼ 2gMz AK

(32)

58

Spin superfluidity

Fig. 5 Mapping of spin current states on the order parameter space of the ferromagnet. (a) Spin current in an isotropic ferromagnet. The current state in torus maps on an equatorial circumference on the sphere of radius M (top). Continuous shift of mapping on the surface of the sphere (middle) reduces it to a point at the northern pole (bottom), which corresponds to the ground state without currents. (b) Spin current in an easy-plane ferromagnet at Mz ¼ 0. The easy-plane anisotropy reduces the order parameter space to an equatorial circumference in the xy plane topologically equivalent to the order parameter space in superfluids. (c) Spin current state with a magnetic vortex in an easy-plane ferromagnet at Mz ¼ 0. The states map on the surface of the upper or the lower hemisphere. (d) Spin current in an easy-plane ferromagnet at Mz 6¼ 0. Spin is confined in the plane parallel to the xy plane but shifted closer to the northern pole. A nonzero Mz appears either in the equilibrium due to a magnetic field parallel to the axis z, or due to spin pumping. (e) Spin current state with a magnetic vortex in an easy-plane ferromagnet at Mz 6¼ 0. The state maps on the surface of the upper or the lower parts of the sphere. Two options of mapping are not degenerate, and the phase slip with the magnetic vortex of a smaller energy (a smaller area of mapping) is more probable. From Sonin EB (2020) Superfluid spin transport in magnetically ordered solids (review article). Fizika Nizkikh Temperatur 46: 523–535, [Low Temp. Phys. 46, 436 (2020)].

can be called Doppler velocity because its effect on the frequency is similar to the Doppler effect in a Galilean invariant fluid [see the text before Eq. (10)]. However, our system is not Galilean invariant (Iacocca et al., 2017), and this is the pseudo Doppler effect. Because of it, the gradient K proportional to w is present also on the right-hand side of the dispersion relation Eq. (30). We obtained the gapless Goldstone mode with the sound-like linear in k spectrum. The current state becomes unstable when at k antiparallel to w the frequency o becomes negative. This happens at the gradient K equal to the Landau critical gradient M? 1 pffiffiffiffiffiffi : K L ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4M − 3M? wA

(33)

In the limit of weak magnetic fields when Mz  M the Landau critical gradient is 1 gM K L ¼ pffiffiffiffiffiffi ¼ : wA wcsw

(34)

In this limit the pseudo-Doppler effect is not important, and at the Landau critical gradient KL the spin-wave velocity ~csw in the current state vanishes. In the opposite limit Mz ! M (M? ! 0) the Landau critical gradient, KL ¼

M? 1 pffiffiffiffiffiffi , 2M wA

(35)

decreases, and the spin superfluidity becomes impossible at the phase transition to the easy-axis anisotropy (M? ¼ 0). Deriving the sound-like spectrum of the spin wave we neglected in the Hamiltonian terms dependent on gradients rMz. One should take into account these terms at the wavelength on the order of the coherence length rffiffiffiffi M pffiffiffiffiffiffi M2 A x0 ¼ wA ¼ : (36) M? G M? The Landau critical gradient KL is on the order of the inverse coherence length 1/x0. The current states relax to the ground state via phase slips events, in which magnetic vortices cross spin current streamlines. At Mz ¼ 0 the current state with a magnetic vortex maps on a surface of a hemisphere of radius M either above or below the equator (Nikiforov and Sonin, 1983) as shown in Fig. 5(c). The vortex core has a structure of a skyrmion. The skyrmion mapping on a hemisphere is called meron. At Mz ¼ 0 two magnetic vortices with opposite spin polarizations have the same energy, and both can participate in phase slips. But at Mz 6¼ 0 the magnetic vortex with a smaller mapping area [Fig. 5(e)] has a smaller energy, and phase slips with its participation are more frequent. Since inside the core the gradients rMz cannot be ignored, the core radius is on the order of the coherence length x0. Variation of the magnetization direction in space inside the skyrmion core is schematically shown

Spin superfluidity

59

(a)

θ0 (b)

θ0 (c)

Fig. 6 Skyrmion cores of magnetic vortices. Variation of magnetization vectors (M in a ferromagnet, M1 and M2 in an antiferromagnet) in the vortex core as a function of the distance r from the vortex axis is shown schematically. Horizontal directions of magnetizations correspond to the direction of the z axis in the spin space. (a) The magnetic vortex in the ferromagnet corresponding to the gapless Goldstone spin wave mode with the coherence length x0 given by Eq. (36). (b) The magnetic vortex in the antiferromagnet corresponding to the Goldstone spin wave mode with the coherence length x0 given by Eq. (59). (c) The magnetic vortex in the antiferromagnet corresponding to the gapped spin wave mode with the coherence length x given by Eq. (61). From Sonin EB (2019a) Interplay of spin and mass superfluidity in antiferromagnetic spin-1 Bose–Einstein condensates and bicirculation vortices. Physical Review Research 1: 033103.

in Fig. 6(a). One can find more details and numerical calculations of the structure of skyrmion cores of magnetic vortices in ferromagnets elsewhere (Sonin, 2018). The estimation of barriers for phase slips in spin-superfluid ferromagnets is similar to that in the case of mass superfluidity. The spin phase gradient in the current state with a straight magnetic vortex parallel to the axis z is r’ ¼

½b z  r + K: r2

(37)

We consider a 2D problem of the straight magnetic vortex at the distance rm from the plane border. The gradient K is parallel to the border. Substituting the phase gradient field Eq. (37) into the kinetic energy and integrating the energy over the London region, where M? is close to its value in the ground state, one obtains the energy of the magnetic vortex per unit length:   r ~ev ¼ pAM2? ln m − 2Kr m : (38) rc The magnetic vortex energy has a maximum at rm ¼ 1/2K. The energy at the maximum is a barrier preventing phase slips: eb ¼ pAM2? ln

1 : 2Kr c

(39)

The barrier vanishes if the gradient K becomes of the order of the inverse vortex core radius rc. This gradient is on the order of the Landau critical gradient KL. Considering mapping of current states with nonzero magnetization Mz [Fig. 5(d) and (e)], we had in mind the equilibrium magnetization Mz ¼ wH produced by an external magnetic field H. In the equilibrium there is no precession of the magnetization M around the axis z. However, non-equilibrium states with non-equilibrium magnetization Mz, which makes the magnetization (spin) to coherently precess, are also possible. One can create them by pumping of magnons, which bring spin and energy into the system. Spin pumping compensates inevitable losses of spin due to spin relaxation. However, usually the process of spin relaxation is weak, and one may treat the coherent precession state as a quasi-equilibrium state at fixed Mz. The coherent precession state does not requires the crystal easy-plane anisotropy for its existence. The easy-plane topology of Fig. 5(d) is provided dynamically, and, as a result, metastable spin current states are also possible. Spin superfluidity in the quasi-equilibrium coherent precession state was investigated theoretically and experimentally in the B phase of superfluid 3He (Bunkov, 1995). Later the coherent precession state in 3He was rebranded as magnon BEC (Bunkov and Volovik, 2013). Coherent spin precession state (also under the name of magnon BEC) was revealed also in YIG magnetic films (Demokritov et al., 2006). Spin superfluidity in YIG was discussed by Sun et al. (2016, 2017) and Sonin (2017). In the quasi-equilibrium coherent precession state demonstration of the long-distance superfluid spin transport is problematic (see Section “Experiments on detection of spin superfluidity”). Semantic dilemma “coherent precession state, or magnon BEC” was discussed by Sonin (2010).

Spin superfluidity in antiferromagnets The dynamics of a bipartite antiferromagnet can be described by the LLG equations for two spin sublattices coupled via exchange interaction (Keffer and Kittel, 1951):

60

Spin superfluidity dMi ¼ g½H i  Mi : dt

(40)

Here the subscript i ¼ 1, 2 indicates to which sublattice the magnetization Mi belongs, and Hi ¼ −

dH ∂H ∂H ¼ − + rj dMi ∂Mi ∂rj Mi

(41)

is the effective field for the ith sublattice determined by the functional derivative of the Hamiltonian H. For an isotropic antiferromagnet the Hamiltonian is H¼

Aðri M1 ri M1 + ri M2 ri M2 Þ M1 M2 + 2 w + A12 rj M1 rj M2 − Hm:

(42)

The total magnetization is m ¼ M1 + M2 ,

(43)

L ¼ M1 − M2

(44)

and the staggered magnetization

is normal to m. In the LLG theory absolute values of sublattice magnetizations M1 and M2 are equal to constant M, and in the uniform state without gradients the Hamiltonian is H¼ −

M2 m2 + − Hmz : w 2w

(45)

Here and later on we assume that the magnetic field is applied along the z axis. The constant term M2/w can be ignored. In the ground state the total magnetization m is directed along the magnetic field (the z axis), and the staggered magnetization L is confined to the xy plane. The order parameter for an antiferromagnet is the unit Néel vector l ¼ L/L. The order parameter space for the isotropic antiferromagnet in the absence of the external magnetic field is a surface of a sphere, as for isotropic ferromagnets. But in ferromagnets the magnetic field produces an easy axis for the magnetization, while in the antiferromagnet the magnetic field produces the easy plane for the order parameter vector l. Thus, the easy-plane topology necessary for the spin superfluidity in antiferromagnets does not require the crystal easy-plane anisotropy. In the analogy to the ferromagnetic case, one can describe the vectors of sublattice magnetizations Mi with the constant absolute value M by the two pairs of the conjugate variables (Miz, ’i), which are determined by the two pairs of the Hamilton equations: 1 d’i dH ∂H ¼ − , ¼ − g dt dMiz ∂Miz

(46)

1 dMiz dH ∂H ∂H ¼ − r : ¼ g dt d’i ∂’i ∂r’i

(47)

Let us consider the magnetization distribution with axial symmetry around the axis z: mz ¼ M sin y0 , mx ¼ my ¼ 0, 2 L M1x ¼ −M2x ¼ cos ’, 2

M1z ¼ M2z ¼

M1y ¼ −M2y ¼

L sin ’, 2

(48) Lz ¼ 0: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here ’ ¼ ’1 ¼ p − ’2 is the angle of rotation of L around the z axis, L ¼ 4M2 − m2z ¼ 2M cos y0 , and y0 is the canting angle. The Hamiltonian Eq. (42) for the axisymmetric case becomes the Hamiltonian H¼

m2z A L2 ðr’Þ2 − Hmz + − 2w 4

(49)

for the pair of the canonically conjugate variables (mz, ’). The Hamilton equations for this pair are 2

Here A− ¼ A − A12, and

1 d’ A − mz ðr’Þ m − wH − z ¼ , 2 g dt w

(50)

1 dmz + rJ s ¼ 0: g dt

(51)

Spin superfluidity

Js ¼ −

A − L2 r’ 2

61

(52)

is the superfluid spin current. These equations are identical to Eqs. (24)–(26) for the ferromagnet after replacing the spontaneous magnetization component Mz by the total magnetization component mz, A by A− /2, and M? by L. In the stationary spin current state there is a constant gradient K ¼ r’ of the spin phase and a constant total magnetization mz ¼

wH : 1 − wA− K 2 =2

(53)

For checking the Landau criterion we must know the spectrum of all collective modes. Solution of Eqs. (50) and (51) linearized with respect to plane wave perturbations m0 ∝ eikr−iot and ’0 ∝ eikr−iot of the stationary spin current state yield the spectrum of the Goldstone gapless mode: o + wk ¼ ~csw k:

(54)

Here the spin-wave velocity ~csw in the current state and the Doppler velocity w are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wA − K 2 ~csw ¼ csw 1 − , w ¼ gmz A − K, 2

(55)

while csw

rffiffiffiffiffiffi rffiffiffiffiffiffi A− A− ¼ 2gM cos y0 ¼ gL 2w 2w

(56)

is the spin-wave velocity in the ground state without spin currents. The difference between gapless modes in a ferromagnet and an antiferromagnet is that in the former the total magnetization precesses around the z axis, while in the latter there is the precession of the staggered magnetization. Oscillations of sublattice magnetizations M1 and M2 in the gapless mode are illustrated in Fig. 7(a). In antiferromagnets there is another mode in which the total magnetization m and the staggered magnetization L perform rotational oscillations around the axis normal to the axis z. Without spin supercurrents this axis does not vary in space, and one can choose it to be the axis y. The oscillations of the sublattice magnetizations are illustrated in Fig. 7(b). In the spin current state L rotates around the axis, which itself rotates in the easy-plane xy along the current streamlines. The magnetic field breaks invariance with respect to rotations around axes normal to the field, and the mode spectrum is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (57) o + wk ¼ o20 + csw k2 , where the gap is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 m2z − c2sw K 2 : o0 ¼ w2

(58)

More details of the derivation are given by Sonin (2019b). Applying the Landau criterion to the gapless mode at small canting angles y0 (weak magnetic fields), one obtains the critical gradient KL and the correlation length x0, rffiffiffiffiffiffiffiffi 1 wA − K L ¼ , x0 ¼ , (59) 2 x0 similar to those obtained for a ferromagnet [Eqs. (35) and (36)]. However, in contrast to a ferromagnet where the susceptibility w is connected with weak anisotropy energy, in an antiferromagnet the susceptibility w is determined by a much larger exchange energy

(a)

(b)

Fig. 7 The schematic picture of the two spin wave modes in the bipartite antiferromagnet in the plane xz. (a) The gapless Goldstone mode. There are oscillations of the canting angle determining the magnetization component mz and rotational oscillations around the axis z. (b) The gapped mode. There are rotational oscillations of the two magnetizations around the axis y. From Sonin EB (2020) Superfluid spin transport in magnetically ordered solids (review article). Fizika Nizkikh Temperatur 46: 523–535, [Low Temp. Phys. 46, 436 (2020)].

62

Spin superfluidity

and is rather small. As a result, in the antiferromagnet the gapped mode loses its stability at much lower values of K than the gapless mode. This happens at the Landau critical gradient KL ¼

1 x

(60)

when the gap given by Eq. (58) vanishes and the mode frequency becomes complex. Here we introduced a new correlation length x¼

cs wc ¼ s : gH gmz

(61)

As usual, the instability with respect to phase slips with magnetic vortices starts at the gradients of the same order as the Landau critical gradient. Two modes in antiferromagnets have different correlation lengths, and, correspondingly, there are two types of magnetic vortices with different structure and size of the skyrmion core. Fig. 6(b) shows schematically spatial variation of the sublattice magnetizations in the skyrmion core of a magnetic vortex connected with the Goldstone gapless mode. The core radius is of the order of the correlation length x0 given by Eq. (59). Inside the core the canting angle y0 grows and reaches p/2 at the vortex axis. This transforms an antiferromagnet to a ferromagnet with the magnetization 2M. The transformation increases the exchange energy, and at weak magnetic fields creation of magnetic vortices connected with the gapped mode starts earlier. Its core size is determined by the larger correlation length x determined by the Zeeman energy and given by Eq. (61). The skyrmion core connected with the gapped mode is illustrated in Fig. 6(c).

Superfluid spin transport without spin conservation law Though processes violating the spin conservation law are relativistically weak, their effect is of principal importance and cannot be ignored in general. Here we consider the effect of broken rotational symmetry in the easy plane in ferromagnets. Its extension on antiferromagnets requires insignificant modifications. One should add the s-fold in-plane anisotropy energy ∝Gin to the Hamiltonian (21), which becomes H¼

M2z AM2? ðr’Þ2 − gMz H + + Gin ½1 − cosðs’Þ: 2w 2

(62)

Then the spin continuity equation (25) becomes   sinðs’Þ dMz , ¼ −rJ s + sGin sinðs’Þ ¼ AM2? r2 ’ − dt l2

(63)

where sffiffiffiffiffiffiffiffiffiffiffi AM2? l¼ sGin

(64)

is the thickness of the wall separating domains with s equivalent easiest directions in the easy plane. In the stationary spin current states dMz/dt ¼ 0, and the phase ’ is a periodical solution of the sine-Gordon equation parametrized by the average phase gradients hr’i. At small hr’i  1/l the spin structure constitutes the chain of domains with the period 2p/s hr’i. Spin currents (gradients) inside domains are negligible but there are spin currents inside domain walls. The spin current has a maximum in the center of the domain wall equal to rffiffiffi 2A Jl ¼ : (65) s l The spin transport in the case hr’i  1/l hardly reminds the superfluid transport on macroscopic scales: spin is transported over distances on the order of the domain-wall thickness l. With increasing hr’i the density of domain walls grows, and at hr’i 1/l the domains coalesce. Deviations of the gradient r’ from the constant average gradient hr’i become negligible. This restores the analogy with the superfluid transport in superfluids (Sonin, 1978a,b, 1982), and spin non-conservation can be ignored. The transformation of the domain wall chain into a weakly inhomogeneous current state at growing hr’i is illustrated in Fig. 8. According to the Landau criterion, spin current states become unstable at r’  1/x0, where the correlation length x0 is given by Eq. (36). Thus, one can expect metastable nearly uniform current states at 1/l  hr’i  1/x0. This is possible if the easy plane anisotropy energy G essentially exceeds the in-plane anisotropy energy Gin. In the limit r’  1/l of strongly nonuniform current states the decay of the current is also possible only via phase slips, but the structure of magnetic vortices is essentially different from that in the opposite limit r’ 1/l. The magnetic vortex is a line defect at which s domain walls end. The phase slip with the magnetic vortex crossing the streamlines in the channel leads to annihilation of s domain walls. This process is illustrated in Fig. 2(b) for the fourfold in-plane symmetry (s ¼ 4). An important difference with conventional mass superfluidity is that while the existence of conventional superfluidity is restricted only from above by the Landau critical gradients, the spin superfluidity is restricted also from below: gradients should

63

Spin superfluidity ϕ

ϕ

1 l

x ϕ ϕ

1 l

x Fig. 8 The nonuniform spin-current states with hr’i  1/l and hr’i 1/l. From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

not be less than the value 1/l. Thus, the gradient K ¼ hr’i in the expression Eq. (39) barrier height cannot be less than 1/l, and the height of the barrier cannot exceed em ¼ pAM2? ln

l : rc

(66)

In contrast to the maximal phase slip barrier Eq. (16) in mass superfluidity, in spin superfluidity the maximal phase slip barrier does not become infinite in the macro-scopic limit W ! 1 (König et al., 2001; Sonin, 1978a,b, 1982).

Long-distance superfluid spin transport The absence of the strict spin conservation law also leads to the dissipative process, which is impossible in the mass superfluidity and very important for long-distance superfluid spin transport (Sonin, 1978a, 2010; Takei et al., 2014; Takei and Tserkovnyak, 2014): longitudinal spin relaxation characterized in the magnetism theory by the Bloch time T1. Taking the Bloch relaxation into account, the equation for the non-equilibrium magnetization M0z ¼ Mz − wH becomes. M0 1 dM0z ¼ −rJ − z : gT 1 g dt

(67)

Here the total current J ¼ Js + Jd includes not only the spin supercurrent Js given by Eq. (26), but also the spin diffusion current Jd ¼ −

D rMz : g

(68)

In the absence of spin superfluidity (Js ¼ 0) Eq. (67) describes pure spin diffusion [Fig. 9(a)]. Its solution, with the boundary condition that the spin current J0 is injected at the interface x ¼ 0, is rffiffiffiffiffiffi T 1 −x=Ld J d ¼ J0 e−x=Ld , M0z ¼ gJ0 , (69) e D where Ld ¼

pffiffiffiffiffiffiffiffiffi DT 1

(70)

is the spin-diffusion length. Thus, the effect of spin injection exponentially decays at the scale of the spin-diffusion length, and the density of spin accumulated at the other border of the medium decreases exponentially with growing distance d. However, if spin superfluidity is possible, the spin precession equation Eq. (24) becomes relevant. According to this equation, in a stationary state the magnetization M0z cannot vary in space [Fig. 9(b)] since the gradient rM0z leads to the linear in time growth of the gradient r’. The right-hand side of Eq. (24) is an analog of the chemical potential, and the requirement of constant in space magnetization Mz is similar to the requirement of constant in space chemical potential in superfluids, or the electrochemical potential in superconductors. As a consequence of this requirement, spin diffusion current is impossible in the bulk since it is simply “short-circuited” by the superfluid spin current. The bulk spin diffusion current can appear only in AC processes. If the spin superfluidity is possible, the spin current can reach the spin detector at the plane x ¼ d opposite to the border where spin is injected. As a boundary condition at x ¼ d, one can use a phenomenological relation Js(d) ¼ M0z (d)vd connecting the spin

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(a) Spin injection J0

Jd Mz

(b)

Js

Spin injection

Spin injection

J0 Mz

(c) Spin injection

Js

J0

x 0

d

Fig. 9 Long distance spin transport. (a) Injection of the spin current J0 into a spin-non-superfluid medium. (b) Injection of the strong spin current J0 Jl into a spin-superfluid medium. (c) Injection of the weak spin current J0 < Jl into a spin-superfluid medium. From Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255.

current with the non-equilibrium magnetization at the border with a non-magnetic medium. Here vd is a phenomenological constant. This boundary condition (Sonin, 1978a) was confirmed by the microscopic theory of Takei and Tserkovnyak (2014). Together with the boundary condition Js(0) ¼ J0 at x ¼ 0 this yields the solution of Eqs. (24) and (67):   T1 x : (71) gJ0 , J s ðxÞ ¼ J 0 1 − M0z ¼ d + vd T 1 d + vd T 1 Thus, the spin accumulated at large distance d from the spin injector slowly decreases with d as 1/(d + C) [Fig. 9(b)], in contrast to the exponential decay ∝e−d/Ld in the spin diffusion transport [Fig. 9(a)]. The constant C is determined by the boundary condition at x ¼ d. The long-distance superfluid spin transport is possible only if the injected spin current is not too small. If the injection spin current J0 is less than the current Jl determined by Eq. (65) the superfluid spin current penetrates into the medium only at distances not longer than the width l of a domain wall in a non-uniform spin current state at a very small average gradient hr’i  1/l [Fig. 9(c)]. This threshold for the long-distance spin superfluid transport is connected with the absence of the strict conservation law for spin (Section “Superfluid spin transport without spin conservation law”).

Experiments on detection of spin superfluidity Experimental detection of spin superfluidity does not reduce to experimental evidence of the existence of spin supercurrents proportional to gradients of spin phase. As pointed out in Introduction (Section “Introduction”), such supercurrents produced by spin phase difference smaller than 2p emerge in any non-uniform spin structure. Numerous observations of spin waves and domain structures during the more than half-a-century history of modern magnetism cannot be explained without these microscopic spin currents. Only detection of macroscopical spin supercurrents produced by the phase difference many times larger than 2p is evidence of spin superfluidity. The experimental evidence of macroscopical spin supercurrents was obtained in the past in the B phase of superfluid 3He (Borovik-Romanov et al., 1987). A spin current was generated in a long channel connecting two cells filled by 3He - B. The quasi-equilibrium state of the coherent spin precession was supported by spin pumping. The magnetic fields applied to the two cells were slightly different, and therefore, the spins in the two cells precessed with different frequencies. A small difference in the frequencies leads to a linear growth of difference of the precession phases in the cells and a phase gradient in the channel. When the gradient reached the critical value, 2p phase slips were detected. This was evidence of non-trivial spin supercurrents. This experiment was done in the dynamical state of coherent spin precession (non-equilibrium magnon BEC). The states require pumping of spin in the whole bulk for their existence. In the geometry of the experiment on long-distance spin transport (Section “Long-distance superfluid spin transport”) this would mean that spin is permanently pumped not only by a distant

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Fig. 10 Long distance spin transport in the experiment by Yuan et al. (2018). Spin is injected from the left Pt wire and flows along the Cr2O3 film to the right Pt wire, which serves as a detector. The arrowed dashed line shows a spin-current streamline. From Sonin EB (2020) Superfluid spin transport in magnetically ordered solids (review article). Fizika Nizkikh Temperatur 46: 523–535, [Low Temp. Phys. 46, 436 (2020)].

injector but also all the way up to the place where its accumulation is probed. Thus, the spin detector measures not only spin coming from the distant injector but also spin pumped close to the detector. Therefore, the experiment cannot demonstrate the existence of long-distance superfluid spin transport, but can provide, nevertheless, indirect evidence that long-distance superfluid spin transport is possible in principle. The experiment on detection of long-distance superfluid spin transport (Section “Long-distance superfluid spin transport”) was recently done by Yuan et al. (2018) in antiferromagnetic Cr2O3. The spin was injected from a Pt injector by heating [the Seebeck effect (Seki et al., 2015)] on one side of the Cr2O3 film and spin accumulation was probed on another side of the film by the Pt detector via the inverse spin Hall effect (Fig. 10). In agreement with the theoretical prediction, they observed spin accumulation inversely proportional to the distance from the interface where spin was injected. In the experiment of Yuan et al. (2018) spin injection required heating of the Pt injector, and the spin current to the detector is inevitably accompanied by a heat flow. Lebrun et al. (2018) argued that Yuan et al. (2018) detected a signal not from spin coming from the distant injector but from spin generated by the Seebeck effect at the interface between the heated antiferromagnet and the Pt detector. If true, Yuan et al. (2018) observed not long-distance spin transport but long-distance heat transport. A resolution of this controversy requires further experimental and theoretical investigations. In particular, one could check long-distance heat transport scenario by replacing the Pt spin injector in the experiment of Yuan et al. (2018) by a heater, which produces heat but no spin. Observation of the long-distance superfluid spin transport was also reported by Stepanov et al. (2018) in a graphene quantum Hall antiferromagnet. However, the discussion of this report requires an extensive theoretical analysis of the n ¼ 0 quantum Hall state of graphene, which goes beyond the scope of the present article. A reader can find this analysis in Takei et al. (2016).

Conclusion The article addressed the basics of the spin superfluidity concept: topology, Landau criterion, and phase slips. Metastable (persistent) superfluid current states are possible if the order parameter space (vacuum manifold) has the topology of a circumference like in conventional superfluids. In ferromagnets it is the circumference on the spherical surface of the spontaneous magnetizations M, and in antiferromagnets it is the spherical surface of the unit Néel vector L/L, where L is the staggered magnetization. The topology necessary for spin superfluidity requires the magnetic easy-plane anisotropy in ferromagnets, while in antiferromagnets this anisotropy is provided by the Zeeman energy, which confines the Néel vector in the plane normal to the magnetic field. The Landau criterion was checked for the spectrum of elementary excitations, which are spin waves in our case. In ferromagnets there is only one gapless Goldstone spin wave mode. In bipartite antiferromagnets there are two modes: the Goldstone mode in which spins perform rotational oscillations around the symmetry axis and the gapped mode with rotational oscillations around the axis normal to the symmetry axis. At weak magnetic fields the Landau instability starts not in the Goldstone mode, but in the gapped mode. In contrast to superfluid mass currents in conventional superfluids, metastable spin superfluid currents are restricted not only by the Landau criterion from above but also from below. The restriction from below is related to the absence of the strict conservation law for spin. The Landau instability with respect to elementary excitations is a precursor for the instability with respect to phase slips. The latter instability starts when the spin phase gradient reaches the value of the inverse vortex core radius. This value is on the same order of magnitude as the Landau critical gradient. Magnetic vortices participating in phase slips have skyrmion cores, which map on the upper or lower part of the spherical surface in the space of spontaneous magnetizations in ferromagnets, or in the space of the unit Néel vectors in antiferromagnets. It is worthwhile to note that in reality it is not easy to reach the critical gradients discussed in the present article experimentally. The decay of superfluid spin currents is possible also at subcritical spin phase gradients since the barriers for phase slips can be overcome by thermal activation or macroscopic quantum tunneling. This makes the very definition of the real critical gradient rather ambiguous and dependent on duration of observation of persistent currents. Calculation of real critical gradients requires a detailed

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dynamical analysis of processes of thermal activation or macroscopic quantum tunneling through phase slip barriers, which is beyond the scope of the present article. One can find examples of such analysis for conventional superfluids with mass supercurrents in Sonin (2016). Experimental evidence of the existence of metastable superfluid spin currents in the B phase of superfluid 3He was reported long ago (Borovik-Romanov et al., 1987). The experiment was done in the non-equilibrium state of coherent spin precession, which requires permanent spin pumping in whole bulk for its existence. This does not allow to check true long-distance superfluid spin transport without any additional spin injection on the way from an injector to a detector of spin. The experiment demonstrating the long-distance transport of spin in the solid antiferromagnet was reported recently (Yuan et al., 2018). But the interpretation of this experiment in the terms of spin superfluidity was challenged (Lebrun et al., 2018), and experimental verification of the long-distance superfluid spin transport in magnetically ordered solids convincing many (if not all) in the community is still wanted. Mechanical analogy of the mass superfluidity discussed in the end of Section “Concept of superfluidity” is valid also for spin superfluidity. The “superfluid” flux of the angular momentum in a twisted elastic rod is similar to the superfluid spin current in magnetically ordered solids. Of course, it is not obligatory to discuss the twisted rod in terms of angular-momentum flux. More usual is to discuss it in the terms of the elasticity theory: deformations, stresses, and elastic stiffness. On the same grounds, one can avoid to use the terms “spin current” and “spin superfluidity” and consider the spin current states as metastable helicoidal spin structures determined by “phase stiffness.” This stance was quite popular in early disputes about spin superfluidity. Nowadays in the era spintronics the terms “spin supercurrents” and “spin superfluidity” are widely accepted. In this article the essentials of spin superfluidity were discussed for simpler cases of a ferromagnet and of a bipartite antiferromagnet at zero temperature. Spin superfluidity was also investigated in antiferromagnets with a more complicated magnetic structure (Li and Kovalev, 2021). At finite temperates the presence of the gas of incoherent magnons was taken into account in the two-fluid theory (Flebus et al., 2016) similar to the two-fluid theory in the theory of mass superfluidity. The present article focused on spin superfluidity in magnetically ordered solids. In superfluid 3He spin superfluidity coexists with mass superfluidity. Recently investigations of spin superfluidity were extended to spin-1 BEC, where spin and mass superfluidity also coexist and interplay (Armaitis and Duine, 2017; Lamacraft, 2017; Sonin, 2018, 2019a). This interplay leads to a number of new nontrivial features of the phenomenon of superfluidity. The both types of superfluidity are restricted by the Landau criterion for the softer collective modes, which usually are the spin wave modes. As a result, the presence of spin superfluidity diminishes the possibility of the conventional mass superfluidity. Another consequence of the coexistence of spin and mass superfluidity is phase slips with bicirculation vortices characterized by two topological charges (winding numbers) (Sonin, 2019a).

References Armaitis J and Duine RA (2017) Superfluidity and spin superfluidity in spinor Bose gases. Physical Review A 95: 053607. Borovik-Romanov AS, Bunkov YM, Dmitriev VV, and Mukharskii YM (1987) Observation of phase slippage during the flow of a superfluid spin current in 3He-b. Pis0 ma v Zhurnal Èksperimental0 noy i Teoreticheskoy Fiziki 45: 98–101. [JETP Lett. 45, 124–128 (1987)]. Bunkov YM (1995) Spin supercurrent and novel properties of NMR in 3He. Progress of Low Temperature Physics 14: 68. edited by W. P. Halperin (Elsevier). Bunkov YM and Volovik GE (2013) Spin superfluidity and magnon Bose-Einstein condensation. In: Bennemann KH and Ketterson JB (eds.) Novel Superfluids. International Series of Monographs on Physics, vol. 1, pp. 253–311. Oxford University Press. Chap. IV. Chen H and MacDonald AH (2017) Spin–superfluidity and spin–current mediated nonlocal transport. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose–Einstein Condensation, pp. 525–548. Cambridge University Press. Chap. 27. arXiv:1604.02429. Demokritov SO, Demidov VE, Dzyapko O, Melkov GA, Serga AA, Hillebrands B, and Slavin AN (2006) Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443: 430–433. Evers M and Nowak U (2020) Transport properties of spin superfluids: Comparing easy-plane ferromagnets and antiferromagnets. Physical Review B 101: 184415. Flebus B, Bender SA, Tserkovnyak Y, and Duine RA (2016) Two-fluid theory for spin superfluidity in magnetic insulators. Physical Review Letters 116: 117201. Guseinov RR and Keldysh LV (1972) Nature of the phase transition under the conditions of an “exitonic” instability in the electronic spectrum of a crystal. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 63: 2255. [Sov. Phys.–JETP 36, 1193 (1972)]. Halperin BI and Hohenberg PC (1969) Hydrodynamic theory of spin waves. Physics Review 188: 898–918. Iacocca E, Silva TJ, and Hoefer MA (2017) Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films. Physical Review Letters 118: 017203. Keffer F and Kittel C (1951) Theory of antiferrornagnetic resonance. Physics Review 85: 329–337. König J, Bønsager MC, and MacDonald AH (2001) Dissipationless spin transport in thin film ferromagnets. Physical Review Letters 87: 187202. Kulik IO and Shevchenko SI (1976) Exciton pairing and superconductivity in layered systems. Fizika Nizkikh Temperatur 2: 1405–1426. [Sov. J. Low Temp. Phys. 2, 687 (1976)]. Lamacraft A (2017) Persistent currents in ferromagnetic condensates. Physical Review B 95: 224512. Landau LD (1941) Theory of superfluidity of helium II. Journal of Physical (USSR) 5: 71. Landau LD and Lifshitz EM (1980) Statistical Physics. Part II. Pergamon Press. Lebrun R, Ross A, Bender SA, Qaiumzadeh A, Baldrati L, Cramer J, Brataas A, Duine RA, and Kläui M (2018) Tunable long-distance spin transport in a crystalline antiferromagnetic iron oxide. Nature 561: 222–225. Li B and Kovalev AA (2021) Spin superfluidity in noncollinear antiferromagnets. Physical Review B 103: L060406. Lozovik YE and Yudson VI (1977) Interband transitions and the possibility of current states in systems with electron-hole pairing. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 25: 18–21. [JETP Lett. 25, 14–17 (1977)]. Nikiforov AV and Sonin EB (1983) Dynamics of magnetic vortices in a planar ferromagnet. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 85: 642–651. [Sov. Phys.–JETP 58, 373 (1983)]. Pitaevskii L and Stringari S (2003) Bose–Einstein Condensation. Oxford University Press. Qaiumzadeh A, Skarsvåg H, Holmqvist C, and Brataas A (2017) Spin superfluidity in biaxial antiferromagnetic insulators. Physical Review Letters 118: 137201. Seki S, Ideue T, Kubota M, Kozuka Y, Takagi R, Nakamura M, Kaneko Y, Kawasaki M, and Tokura Y (2015) Thermal generation of spin current in an antiferromagnet. Physical Review Letters 115: 266601. Shi J, Zhang P, Xiao D, and Niu Q (2006) Proper definition of spin current in spin-orbit coupled systems. Physical Review Letters 96: 076604.

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Sonin EB (1977) On superfuidity of Bose condensate of electron-hole pairs. Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 25: 95–98. [JETP Lett. 25, 84–87 (1977)]. Sonin EB (1978a) Analogs of superfluid currents for spins and electron-hole pairs. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 74: 2097–2111. [Sov. Phys.–JETP, 47, 1091–1099 (1978)]. Sonin EB (1978b) Phase fixation, exitonic and spin superfluidity of electron-hole pairs and antiferromagnetic chromium. Solid State Communications 25: 253–255. Sonin EB (1982) Superflows and superfluidity. Uspekhi Fiziologicheskikh Nauk 137: 267. [Sov. Phys.–Usp., 25, 409 (1982)]. Sonin EB (2010) Spin currents and spin superfluidity. Advances in Physics 59: 181–255. Sonin EB (2016) Dynamics of Quantised Vortices in Superfluids. Cambridge University Press. Sonin EB (2017) Spin superfluidity and spin waves in YIG films. Physical Review B 95: 144432. Sonin EB (2018) Spin and mass superfluidity in ferromagnetic spin-1 BEC. Physical Review B 97: 224517. Although the subject of the paper is the BEC of cold atoms with spin 1, Sec. VII of the paper investigates vortices in a solid ferromagnet in order to compare them with vortices in the spin-1 BEC. Sonin EB (2019a) Interplay of spin and mass superfluidity in antiferromagnetic spin-1 Bose–Einstein condensates and bicirculation vortices. Physical Review Research 1: 033103. Sonin EB (2019b) Superfluid spin transport in ferro- and antiferromagnets. Physical Review B 99: 104423. Sonin EB (2020) Superfluid spin transport in magnetically ordered solids (review article). Fizika Nizkikh Temperatur 46: 523–535. [Low Temp. Phys. 46, 436 (2020)]. Stepanov P, Che S, Shcherbakov D, Yang J, Chen R, Thilahar K, Voigt G, Bockrath MW, Smirnov D, Watanabe K, Taniguchi T, Lake RK, Barlas Y, MacDonald AH, and Lau CN (2018) Long-distance spin transport through a graphene quantum Hall antiferromagnet. Nature Physics 14(9): 907–911. Sun C, Nattermann T, and Pokrovsky VL (2016) Unconventional superfluidity in yttrium iron garnet films. Physical Review Letters 116: 257205. Sun C, Nattermann T, and Pokrovsky VL (2017) Bose–Einstein condensation and superfluidity of magnons in yttrium iron garnet films. Journal of Physics D: Applied Physics 50(14), 143002. Takei S and Tserkovnyak Y (2014) Superfluid spin transport through easy-plane ferromagnetic insulators. Physical Review Letters 112: 227201. Takei S, Halperin BI, Yacoby A, and Tserkovnyak Y (2014) Superfluid spin transport through antiferromagnetic insulators. Physical Review B 90: 094408. Takei S, Yacoby A, Halperin BI, and Tserkovnyak Y (2016) Spin superfluidity in the n ¼ 0 quantum Hall state of graphene. Physical Review Letters 116: 216801. Tserkovnyak Y and Kläui M (2017) Exploiting coherence in nonlinear spin-superfluid transport. Physical Review Letters 119: 187705. Vuorio M (1974) Condensate spin currents in helium-3. Journal of Physics C: Solid State Physics 7(1): L5–L8. Yuan W, Zhu Q, Tang S, Yao Y, Xing W, Chen Y, Ma Y, Lin X, Shi J, Shindou R, Xie XC, and Han W (2018) Experimental signatures of spin superfluid ground state in canted antiferromagnet Cr2O3 via nonlocal spin transport. Science Advances 4(4): eaat1098.

Bose–Einstein condensation Leonardo Fallania,b, Massimo Ingusciob,c, Alessio Recatid, and Sandro Stringarid, aDepartment of Physics and Astronomy, University of Florence, Florence, Italy; bLENS European Laboratory for Nonlinear Spectroscopy, Firenze, Italy; cCampus Bio-Medico University of Rome, Rome, Italy; dPitaevskii BEC Center, CNR-INO and Department of Physics, University of Trento, Trento, Italy © 2024 Elsevier Ltd. All rights reserved. This is an update of M. Inguscio, S. Stringari, Bose–Einstein Condensation, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 131–141, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00759-2.

Introduction How to reach BEC in dilute atomic gases Imaging the new macroscopic quantum state Role of interactions Fermi gases Coherence and superfluidity Quantum mixtures of ultracold atomic gases Further directions Optical lattices Disorder Synthetic fields and synthetic matter Low-dimensional physics Long-range interactions Supersolidity Conclusion References

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Abstract Bose–Einstein condensation embodies some of the most fascinating aspects of quantum mechanics. Predicted 100 years ago, its experimental realization in 1995 opened up unexpected directions in both fundamental and applied quantum physics. This chapter describes the main features of Bose–Einstein condensation, focusing on its experimental realization in ultracold atomic gases. We will present the theoretical description of the effect and the main laboratory techniques used to produce and probe ultracold quantum gases, discussing both milestone experiments and emerging research directions that the theoretical and experimental investigation of Bose–Einstein condensation has opened in the recent past.

Key points

• • • • • • •

Describe the phenomenon of Bose–Einstein condensation (BEC). Illustrate the experimental techniques to achieve BEC in ultracold atomic gases. Explain the role of atom–atom interactions in BEC. Discuss the connection between BEC and fundamental phenomena such as superfluidity and quantum coherence. Show how BEC and superfluidity can be achieved for interacting spin mixtures of ultracold Fermi gases. Show seminal experiments that highlighted the main properties of atomic BECs. Discuss emerging research directions stemming from the experimental realization of BEC in ultracold gases.

Introduction At room temperature the behavior of a gas is governed by the laws of classical mechanics. In fact the thermal de Broglie wavelength sffiffiffiffiffiffiffiffiffiffiffiffi 2pℏ 2 , (1) lT ¼ mkB T which describes the smearing of the position of the atoms due to the Heisenberg uncertainty principle (ℏ is the Planck’s constant divided by 2p, kB is the Boltzmann constant, m is the atomic mass, and T is the temperature of the gas), is much smaller than the average spacing between atoms, which then behave as classical objects. As the gas is cooled, however, the smearing increases and the

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wavefunctions of adjacent atoms overlap, causing the atoms to lose their identity. If the overlap is large, the effects of quantum mechanics cannot be ignored any longer. When quantum effects become important, it is crucial to distinguish between Bose and Fermi statistics. Bosons are particles with even total spin, and their many-body wave function is symmetric with respect to the exchange of two particles. As a consequence, bosons like to occupy the same state. Conversely fermions (particles with odd spin) are described by an antisymmetric wave function and cannot occupy the same state due to the Pauli exclusion principle. At very low temperatures, bosons and fermions behave quite differently (see Fig. 1), giving rise to distinct dynamic and thermodynamic behaviors. Bosons are known to undergo a phase transition below a critical temperature Tc. This transition is characterized by the macroscopic occupation of a single particle state and is called Bose–Einstein condensation. A unique peculiarity of the transition is that it can occur even in the absence of interactions, being driven by genuine quantum statistical effects. Albert Einstein predicted the occurrence of this transition in 1924 (Einstein, 1924), on the basis of a paper of the Indian physicist S. N. Bose (Bose, 1924), devoted to the statistical description of the quanta of light. For a long time, Einstein’s predictions had no practical impact and only after the discovery of superfluidity in liquid helium in 1938, the phenomenon of Bose–Einstein condensation became again the object of theoretical investigation, with the pioneering works by London, Bogoliubov, Landau, Lifshitz, Penrose, Onsager, and Feynman. When the temperature is low enough, all interacting systems, with the exception of helium, undergo a phase transition to the solid phase. This behavior is illustrated in Fig. 2, where we show a typical pressure–temperature phase diagram. In the figure we also draw the pressure–temperature line characterizing the BEC phase transition of an ideal gas. Above this line, a dilute gas would be Bose–Einstein condensed. However, this configuration is unstable since thermodynamic equilibrium would correspond to the solid

Fig. 1 Schematics of the level occupancy for a harmonically trapped gas at ultralow temperature. The ground state of the many-body system is completely determined by the quantum statistics. At zero temperature bosons (even-spin particles) all occupy the same single-particle ground state, forming a Bose–Einstein condensate. Fermions (odd-spin particles), obeying the Pauli exclusion principle, pile up until they reach the Fermi energy EF.

P

BEC line

solid

liquid

gas T Fig. 2 A typical pressure–temperature phase diagram. The dashed line corresponds to the BEC phase transition for an ideal gas.

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phase for such values of pressure and temperature. This shows that BEC can be actually achieved only in conditions of metastability and that the density of the gas should be small enough in order to suppress the collisional processes (three-body collisions) responsible for bringing the system into the thermodynamically stable solid phase. In practice, typical densities reached in the BEC phase are 1013 − 1015 cm−3, so the predicted values of Tc are extremely low (a few microKelvin or even less). This explains why Bose–Einstein condensation was experimentally realized in atomic gases only seventy years after Einstein’s historical paper. Bose–Einstein condensation was achieved for the first time in ultracold gases of alkali atoms in 1995 and was recognized with the Nobel Prize for Physics in 2001 to E. A. Cornell, W. Ketterle, and C. E. Wieman (Cornell and Wieman, 2002; Ketterle, 2002). This achievement is the result of extraordinary efforts made in atomic physics during the two last decades of 1900s (Inguscio and Fallani, 2013; Inguscio et al., 1999), through the development of advanced techniques of laser cooling and atom trapping, which were recognized with the award of the Nobel Prize for Physics in 1997. At present BEC has been achieved in atomic gases made by isotopes of a variety of different chemical elements: alkali atoms (Li, Na, K, Rb, and Cs), alkaline-earth atoms (Ca and Sr) and other two-electron atoms (Yb) including metastable Helium (He ), inner-shell transition metals and lanthanide atoms (Cr, Dy, and Er), and in atomic hydrogen (H) as well.

How to reach BEC in dilute atomic gases The first experimental studies on BEC were focused on spin-polarized hydrogen, which was considered the most natural candidate because of its light mass, which naturally leads to a more pronounced quantum behavior (see Eq. 1). To this purpose cryogenic and evaporative cooling techniques were developed since the early 1970s. Eventually, BEC was first realized in alkali atoms, despite their relatively large masses, because of their suitability to be cooled with laser techniques. Indeed, all the experimental realization of BECs in atomic gases (except hydrogen) do involve some form of laser cooling, where the radiation pressure force, arising from the exchange of momentum between the photons of a laser field and the atoms, is used to reduce the atomic velocity (Inguscio and Fallani, 2013; Inguscio et al., 1999). The most common configuration is that of a magnetooptical trap (MOT), where three pairs of orthogonal, counterpropagating laser beams, red-detuned with respect to an atomic transition, result in a viscous force (also called “optical molasses”) slowing the motion of the atoms. In addition, the presence of a magnetic field gradient allows the atoms to be spatially confined around the point where the magnetic field vanishes. In the case of rubidium, one can typically trap  109 atoms with a temperature of 10 mK in a region of 1 cm3. The density and temperature achievable in a MOT are far from the values required for Bose–Einstein condensation, mostly because of laser-induced collisions between the atoms, which limit the maximum density achievable in a MOT. The next stage consists in transferring the laser-cooled sample from the MOT to a conservative trap, either a magnetic trap or an optical dipole trap, as shown in Fig. 3. In both the cases the trapping potential arises from the interaction of an atomic dipole moment with a spatially varying field: magnetic traps are based on the Zeeman interaction −m  B(r) between the magnetic dipole moment m (proportional to the angular momentum of the atom) and an inhomogeneous magnetic field B(r); optical dipole traps are based on the optical dipole potential −d  E(r) which arises when a far-detuned laser field with electric field amplitude E(r) induces an electric dipole

Fig. 3 Atom trapping. In a magnetic trap the atoms are trapped in a local minimum of the magnetic field B(r) created by sets of electromagnets. In an optical dipole trap the atoms are trapped in local minima or maxima of the laser intensity I(r) ∝ |E(r)|2 (maxima, for red-detuned optical traps).

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moment d (proportional to the electric field) in the atom. In both the cases the atoms are trapped in a local minimum of the confining potential, corresponding to a magnetic field minimum for magnetic traps or to a minimum/maximum of light intensity (according to the sign of the laser detuning) for optical dipole traps. Despite the intrinsic robustness of magnetic traps (typically created with arrangements of electromagnets), optical dipole traps have become increasingly popular over the last 20 years, as they allow for a much larger variety of trapping potentials. Exploiting the interference between counterpropagating laser beams, it is possible to produce for example, optical lattices, that is, periodic arrays of microscopic optical dipole traps corresponding to the antinodes of the laser standing wave (see Section “Further directions”). Furthermore, by shaping the trapping laser with spatial light modulators, it is possible to create almost any trapping configuration, including “box-like potentials” where BEC in a uniform gas can be studied. The final cooling stage is the so called evaporative cooling, that consists in the selective removal of the most energetic atoms from the trap. This process, allowing a reduction of the average energy per atom, can be accomplished either by using frequency-selective radiofrequency/microwave transitions toward spin states that are untrapped by magnetic traps, or by lowering the trap depth in optical dipole traps. If this forced evaporation is slow enough the remaining atoms have time to collide and restore a thermodynamic equilibrium at lower temperatures. At the end of this process, if the density of the sample is high enough, the phase transition to a Bose–Einstein condensate can take place (typical values for rubidium are n  1014 cm−3 and T  100 nK). Bose–Einstein condensation in a trap occurs both in momentum and coordinate space, with the atomic density distribution reflecting the shape of the trapping potential. All these cooling stages are performed in a ultra-high-vacuum environment (typical background pressure  10−11 torr), in order to ensure an optimal thermal isolation and a long lifetime (1 min) of the trapped atomic samples against collisions with the room-temperature background gas.

Imaging the new macroscopic quantum state Images of the ultracold atomic sample can be obtained by shining a resonant laser beam. The absorption of light by the atoms creates a shadow that is recorded by a CCD camera. The typical size of a condensate is of the order of a few micrometers and a very high resolving power is hence necessary for in situ imaging. Most frequently, images are taken by switching the trap off and allowing the gas to expand to larger sizes. Typical images obtained after expansion are shown in Fig. 4. At temperatures higher than Tc (left), the sample expands ballistically and quickly reaches an isotropic distribution, well described by a Maxwellian distribution, from which the temperature of the gas can be measured. At the onset of the BEC transition (center), the shape of the cloud is characterized by a pronounced increase of the density in the center and by a characteristic bimodal distribution: the non condensed atoms determine the wings of the distribution, still given by a Maxwellian form, while the condensed atoms give rise to a narrow central peak, which becomes more and more pronounced as the temperature is lowered (right). The shape of the central peak is no longer a

Fig. 4 (Top) Absorption images of an expanded cloud of 87Rb bosons recorded at LENS by varying the temperature T across the BEC transition at TC. From left to right, the images show a thermal cloud, a partially condensed cloud and a pure BEC. (Bottom) Cross sections of the density distributions. The lines are fits of the experimental points with a gaussian, a bimodal distribution and an inverted parabola, respectively.

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(B) 1.0

1000

0.8

800

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Fig. 5 (A) Condensate fraction N0/N as a function of T/TC: circles are early experimental results from JILA, while the dashed line is the theoretical prediction of Eqs. (2) and (3). (B) BEC density profile: circles are experimental results from Harvard University while the dashed line is the theoretical prediction for a noninteracting BEC. Reproduced from Ensher JR, Jin DS, Matthews MR, Wieman CE, and Cornell EA (1996) Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation. Physical Review Letters 77: 4984 and Hau LV, Busch BD, Liu C, Dutton Z, Burns MM, and Golovchenko JA (1998) Near-resonant spatial images of confined Bose-Einstein condensates in a 4-Dee magnetic bottle. Physical Review A 58: R54 with permission from the authors.

Maxwellian and depends on the properties of the confining potential: in the most common experimental realization, the trap can be approximated as harmonic and the shape of the condensate is an inverted parabola, as predicted by the Gross–Pitaevskii theory (see Section “Role of interactions”). The analysis of the images also provides the number N of atoms in the sample and the condensed fraction. In the presence of harmonic confinement, theory predicts precise values for both the critical temperature Tc and the fraction of condensed atoms for T  Tc. For an ideal gas trapped by a harmonic potential one finds (Dalfovo et al., 1999) kB T c ’ 0:94ℏo0 N1=3

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where o0 is the geometric average of the trapping frequencies and N0 is the number of atoms in the condensate. One of the first theory–experiment comparisons (Ensher et al., 1996) is shown in Fig. 5A, where the value of Tc is a few hundreds of nanoKelvin. Despite the relatively small number of atoms used in this experiment ( 104) the evidence for the phase transition is very clear. The good agreement with the predictions of the ideal gas model indicates that the effects of atom–atom interactions on the condensate fraction and on the value of Tc are small. This is the consequence of the extreme diluteness of the trapped gas.

Role of interactions Interactions between atoms play an important role in the experimental realization of ultracold atomic gases. Most importantly, they are responsible for the thermalization of the gas following the selective removal of the hottest atoms in evaporative cooling. As discussed in the previous section, this is a crucial step for the achievement of the low temperatures that are required for the realization of Bose–Einstein condensation. Besides their importance in the cooling process, interactions play a crucial role in determining some of the equilibrium properties of the Bose–Einstein condensed gas (Dalfovo et al., 1999; Pethick and Smith, 2002; Pitaevskii and Stringari, 2016). For example, the size of the condensate, which, in the absence of interactions, is fixed by the harmonic-trap oscillator length, can become dozens of times larger as a consequence of the repulsion between atoms, as shown in Fig. 5B (Hau et al., 1998). Interactions also affect other dynamic properties, like the propagation of collective excitations and the expansion of the BEC after the removal of the confining potential. Both the equilibrium and the dynamic properties of interacting dilute Bose–Einstein condensed gases are well described by the Gross–Pitaevskii equation, which then provides the basic theoretical framework to describe several many-body features of these systems,  2  ∂c ℏ 4pℏ 2 a 2 (4) r2 + VðrÞ + ¼ − jcj c iℏ 2m ∂t m

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where c is the BEC wavefunction and V (r) is the external confining potential. The last term in parenthesis, leading to a nonlinear character of the equation, describes atom–atom interactions in a mean-field approximation. Remarkably, interactions in ultracold dilute gases are described by a single parameter a, which fixes the most relevant many-body properties of the system and whose sign characterizes the nature of the system in a unique way: when a > 0 interactions have a repulsive character, when a < 0 they are attractive. In the case of Bose gases a  0 is required to ensure the stability of large Bose–Einstein condensed gases against collapse. When a becomes negative the nonlinearity in the Gross–Pitaevskii equation is also responsible for solitonic solutions (bright solitons) which propagate without dispersion in 1D configurations. An exciting feature of ultracold-atom experiments is the possibility to control the scattering length by taking advantage of the so-called Feshbach resonances (see Fig. 6A). This phenomenon occurs when the energy of two colliding atoms in a given spin configuration (open channel) is quasiresonant with the energy of a molecular state that can be formed by those atoms in a different spin configuration (closed channel). The resonance condition can be controlled by tuning the energy separation D(B) between the two channels thanks to the differential Zeeman shift induced by an external magnetic field B. When the resonance condition is met, the scattering length exhibits a divergence and changes sign from one resonance side to the other, eventually crossing a point where it can be zeroed, as shown in Fig. 6B (Inouye et al., 1998). This magnetic tuning of the scattering length allows for unique possibilities in the physics of interacting ultracold gases, as described further in the next sections.

Fermi gases Differently from the case of bosons, an ideal Fermi gas does not exhibit a phase transition at low temperature. The system, however, is still characterized by important quantum phenomena originating from the Pauli exclusion principle. This is well illustrated in Fig. 1, where one sees that at very low temperatures the gas exhibits single occupancy of the single-particle states up to a maximum energy, called Fermi energy. The experimental procedure to reach quantum degeneracy in Fermi gases requires an additional effort compared to the case of bosons. As a matter of fact, in a Fermi gas of identical particles, the antisymmetrization requirement forbids s-wave collisions, which is the only collisional channel available at low temperature, and this causes the absence of those thermalization processes which are crucial for the mechanism of evaporative cooling. This difficulty can be overcome by cooling mixtures of fermions in two spin states, which are distinguishable and thus can interact also at low temperature. Alternatively, one can use mixtures of fermions and bosons, either of the same chemical species or of different species: bosons are cooled down with the standard evaporative cooling technique and then fermions thermalize sympathetically. Fig. 7 shows this second type of approach, achieved with mixtures of potassium and rubidium (Roati et al., 2002). Despite the complication of the cooling procedure, the absence of interactions in a gas of spin-polarized fermions can provide important advantages. An example is given by the application of polarized Fermi gases to accurate measurements, that is, in atom interferometry, or in high-precision spectroscopy, where the absence of atom–atom interactions prevents unwanted decoherence effects or interactions shifts, respectively. On the other side, the possibility of trapping fermions in different spin states has opened new perspectives related to the physics of interacting systems. In the first decade of 2000s, experimentalists were able to realize superfluid configurations in interacting Fermi gases, profiting of the possibility of tuning the interaction between pairs of atoms in different spin states in proximity of

Fig. 6 (A) A Feshbach resonance occurs when the energy of two colliding atoms (open channel) matches the energy of a molecular bound state for the same atoms in a different spin configuration (closed channel), and can be tuned by applying a magnetic field that shifts the two molecular potentials V (R), where R is the interatomic distance. (B) First observation of a Feshbach resonance in a 23Na Bose–Einstein condensate reported in 1998 at MIT. Reproduced from Inouye S, Andrews MR, Stenger J, Miesner H-J, Stamper-Kurn DM, and Ketterle W (1998) Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392: 151 with permission from the authors.

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Fig. 7 Sympathetic cooling of fermionic 40K with bosonic 87Rb, as first reported at LENS in 2002. The bosons are directly cooled with standard evaporative cooling techniques and the fermions thermalize sympathetically at lower temperatures via collisions with the bosons. The figure shows the simultaneous onset of Fermi degeneracy for 40K (left) and of Bose–Einstein condensation for 87Rb (right). The absorption images are taken for decreasing temperatures (from top to bottom) and the sections show the correspondent momentum distributions. Figure reproduced from Roati G, Riboli F, Modugno G, and Inguscio M (2002) Fermi-Bose quantum degenerate 40K-87Rb mixture with attractive interaction. Physical Review Letters 89: 150403.

Feshbach resonances (see Inguscio et al. (2007) for a review of the first years of investigations). In the case of Fermi gases, thanks to the repulsive effect of the Fermi pressure, the system can support a < 0 values. At zero temperature, dilute Fermi gases interacting with small and negative scattering length exhibit a superfluid phase described by the traditional BCS theory, which was first developed to describe the phenomenon of superconductivity. If the scattering length becomes large or positive, new scenarios for fermionic superfluidity take place. In particular, by adjusting the scattering length from a < 0 to a > 0 across a Feshbach resonance, it is possible to produce ultracold molecules made by two fermions in different spin states. Since these molecules have a bosonic nature, being composed of an even number of fermions, they can undergo a phase transition to BEC, similar to that exhibited by a gas of bosonic atoms. Molecular BECs were first demonstrated in 2003 starting from 6Li and 40K fermionic atoms (see Fig. 8 for the realization of a molecular 40K2 BEC reported in Greiner et al. (2003)).

Fig. 8 Bose Einstein condensation of 40K2 molecules, as reported in 2003 at JILA. The (bosonic) dimers are produced starting with quantum degenerate 40K fermions in two spin states and sweeping the value of an applied magnetic field across a Feshbach resonance. (A) Absorption images of the molecular density distribution after expansion. When the initial temperature of the fermionic cloud is lowered across a critical value (from left to right) a narrow peak appears in the momentum distribution. (B) Cross section and fit with a bimodal distribution. The change in the density profile is a signature that a BEC of molecules has formed. Reproduced from Greiner M, Regal CA, and Jin DS (2003) Emergence of a molecular Bose-Einstein condensate from a Fermi gas. Nature 426: 537 with permission from the authors.

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In the following years, important efforts led to the first experimental investigations of the BEC-BCS crossover (see Fig. 9A), which occurs by varying continuously the scattering length a across a Feshbach resonance from positive to negative values (Zwerger, 2014). In particular, at resonance, where the scattering length becomes much larger than the average distance between atoms, the system exhibits new challenging features and acquires a universal behavior which is independent from the actual value of the scattering length. This is the so called unitary Fermi gas, of special interest also in other domains of physics like in neutron matter and characterized by peculiar superfluid features. In particular, the value of the critical temperature at unitarity, characterizing the transition from the normal to the superfluid phase, is a significant fraction of the Fermi temperature (see Fig. 9B), much higher than in the case of the presently available high-Tc superconductors. The theoretical investigation of the unitary regime cannot be simply described using the theory of weakly interacting gases and requires the use of more advanced many-body approaches.

Coherence and superfluidity Bose–Einstein condensates are characterized by a complex order parameter (the so called wavefunction of the condensate) pffiffiffiffiffi F ¼ n0 eiS :

(5)

The modulus is fixed by the condensate density n0, which, in dilute gases and at temperatures much smaller than Tc, practically coincides with the density of the sample. The phase S of the order parameter is at the origin of important properties of the system, both concerning coherence and superfluid effects. Coherence phenomena make the physics of these systems similar to that of a laser, if one replaces photons with atoms. Indeed, the laser field can be described in terms of photons all occupying the same mode of the electromagnetic field, similarly to the atoms of a BEC all sharing the same wavefunction. Bright sources of atoms, the so called atom lasers, can be obtained outcoupling the atoms from the condensate, that is the analog of the laser cavity in which coherent photons are stored, with the eventual demonstration of atom lasers operating in continuous-wave mode (Chen et al., 2022). Another example of this analogy is revealed by the interference phenomena observed with Bose–Einstein condensates. In Fig. 10 we show the interference fringes produced by two separate condensates overlapping after expansion (Andrews et al., 1997). This experiment, which first demonstrated the coherence of the condensed state, is the analog of the most famous double slit experiment with light. An important consequence of the complex order parameter and of the presence of interactions concerns the phenomenon of superfluidity. In fact the gradient of the phase S is proportional to the superfluid velocity: vs ¼

ℏ rS m

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Before 1995, superfluid phenomena had been observed only in dense liquids, like 4He. The possibility of investigating superfluidity in dilute gases has allowed a better understanding of its microscopic origin and its relationship to the phenomenon of BEC (Pitaevskii and Stringari, 2016). Among the various manifestations of superfluidity, it is worth mentioning the rich variety of collective oscillations exhibited by these confined systems, which have been the object of intense theoretical and experimental work in the last decades. The images in Fig. 11 show the shape oscillation of a Bose–Einstein condensate, from which one can extract the frequency of the collective modes with high precision. The dynamic behavior of the condensate is well described by the hydrodynamic equations of superfluids, which predict values for collective frequencies in excellent agreement with the experiment.

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Fig. 10 Interference of two BECs overlapping after expansion. In the central region, one clearly sees interference fringes, similar to the ones observed in optics in the famous Young experiment. This image was recorded in 1997 at MIT in the first experiment where interference of two BECs was observed. Reproduced from Andrews MR, Townsend CG, Miesner HJ, Durfee DS, Kurn DM, and Ketterle W (1997) Observation of interference between two Bose condensates. Science 275: 637 with permission from the authors.

Fig. 11 Collective excitations of a Bose–Einstein condensate, as detected in early MIT experiments. The excitations were produced by modulating the magnetic field used to trap the condensate and then letting the cloud evolve freely. The field of view in the vertical direction is about 600 mm and the time step is 5 ms per frame. The measured frequency of these collective excitations is a clear signature of the BEC superfluidity. From D. Stamper-Kurn and W. Ketterle (private communication), reproduced from Dalfovo F, Giorgini S, Pitaevskii LP, and Stringari S (1999) Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Physics 71: 463.

An even more spectacular prediction of superfluidity concerns the rotational properties of ultracold atomic gases. Superfluids cannot rotate like classical objects because of the irrotationality constraint imposed by Eq. (6). A striking consequence is that angular momentum can be carried only by quantized vortices. If the confining trap rotates slowly, vortices cannot be formed because they are energetically unfavorable and the sample does not carry any angular momentum. However, when the angular velocity increases, quantized vortices are formed (see Fig. 12A). Eventually, at high angular velocities, one can generate a vortex lattice of regular geometrical shape. Quantized vortices have been observed also across the BCS–BEC crossover of an interacting Fermi gas, reflecting the universality of the superfluid behavior exhibited by interacting quantum gases (see Fig. 12B). Another important manifestation of coherence is given by the Josephson oscillations characterized by the coherent tunnelling of atoms through the barriers generated by external double-well (Albiez et al., 2005) or many-well potentials (Cataliotti et al., 2001), as shown in Fig. 13. This peculiar effect is associated with an oscillating behavior of the relative phase between the two condensates occupying the wells and has been observed also through the interference patterns measured after expansion. The Josephson effect plays a crucial role also in the coherent oscillation between condensates in two different internal state (so-called internal Josephson effect), where all the atoms of the cloud oscillate in sync between the two states. The Josephson effect was originally predicted for superconductors, where Cooper pairs can move by coherent tunnelling through a small insulating region. Recently, at LENS, such fascinating phenomenon has been studied in superfluid Fermi gases across the BEC–BCS crossover, allowing for the observation of effects beyond the perturbative BCS theory. Collective modes of Josephson-like nature have been investigated also by exciting the center of mass oscillations of a BEC gas confined in the combined potential created with a harmonic trap and an optical lattice produced by laser fields. Fig. 13 (right) shows that only the superfluid condensed component is able to tunnel coherently while the thermal component is localized. Theoretically one can predict the frequency of the oscillation, which turns out to be in good agreement with the experiment.

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Fig. 12 (A) Formation of vortices in a Bose–Einstein condensate, as observed in 2000 at LKB-ENS: as the rotation frequency of the trap increases (from left to right), a higher number of vortices is formed (Madison et al., 2000). (B) Vortex arrays in two-component strongly interacting Fermi gases, demonstrating a superfluid behavior all the way from the BCS to the BEC limit, as first observed in 2005 at MIT. Reproduced from Madison KW, Chevy F, Wohlleben W, and Dalibard J (2000) Vortex formation in a stirred Bose-Einstein condensate. Physical Review Letters 84: 806 and Inguscio M, Ketterle W, and Salomon C (eds.) (2007) Ultracold fermi gases. In: Proceedings of the International School of Physics “Enrico Fermi”, Varenna (Italy)-Course CLXIV. IOS Press (chapter by W. Ketterle and M. Zwierlein) with permission from the authors.

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Fig. 13 Josephson effect in atomic superfluids. (Left) Josephson oscillations induced by tunnelling in a double-well potential, as reported in 2005 at Heidelberg University. (Right) Josephson dynamics in an optical lattice made by many potential wells, as reported in early experiments at LENS: while the BEC performs coherent oscillations, in analogy with the double-well Josephson effect, the center of mass of the thermal cloud is stuck in the initial position. Left: Reproduced from Albiez M, Gati R, Fölling J, Hunsmann S, Cristiani M, and Oberthaler MK (2005) Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction. Physical Review Letters 95: 010402 with permission from the authors.

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Fig. 14 Propagation of sound modes in a fermionic 6Li superfluid, as reported in 2013 at Innsbruck University. (A) A focused laser beam creates an excitation in a localized region of the superfluid. (B) Propagation of density waves (ordinary sound). (C) Propagation of entropy/temperature waves (second sound). The two sounds clearly propagate at a different speed. Reproduced from Sidorenkov LA, Tey MK, Grimm R, Hou Y-H, Pitaevskii L, and Stringari S (2013) Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498: 78.

Important superfluid phenomena take place also at finite temperature, where Landau’s theory predicts the occurrence of peculiar dynamic features described by two-fluid hydrodynamics. The most spectacular manifestation is the propagation of a new type of sound, the so-called second sound, that was extensively investigated in superfluid helium. Differently from usual sound, which corresponds to a pressure wave, second sound can be regarded as an entropy wave and the corresponding velocity is intimately related to the superfluid density of the system. Second sound was measured in a strongly interacting Fermi gas close to unitarity (Fig. 14) allowing for the first determination of the temperature dependence of the superfluid fraction of a Fermi superfluid (Sidorenkov et al., 2013). More recently, second sound was also measured in two-dimensional Bose gases.

Quantum mixtures of ultracold atomic gases The possibility of reaching ultracold temperatures in both Bose and Fermi gases and in a variety of different atomic species has opened an exciting development of experimental and theoretical activities in the field of the so called quantum mixtures, where gases of different atomic species can co-exist and interact in the quantum degenerate limit, giving rise to novel phenomena (Grimm et al., 2023). This possibility was long sought in the past in the case of mixtures of liquid 4He and 3He, where, however, the possibility of obtaining two interacting superfluids is inhibited by the lack of miscibility of the two helium fluids at very low temperature. New possibilities have now become available in the case of ultracold atomic gases, both with Bose–Bose and Bose–Fermi superfluid mixtures, including the interdisciplinary topic of polarons, which corresponds to extreme population unbalanced mixtures, where essentially only one or few impurities of a certain atomic species are immersed in a bath formed by the other component.

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The studies of polarons in a clean ultracold-gases setup has already allowed for a better understanding and direct measurement of the concept of dressed particles by the bath excitations, as well as the back action of the impurities on the bath itself. The phenomenology specific of a certain mixture is very rich and very diverse depending on the “ingredients” of the selected mixture. In particular for Bose mixtures, one can distinguish between mixtures where the number of atoms of each species is conserved and those in which only the total number of atoms is conserved. To the first class belong mixtures of different atomic species while the second class concerns mixtures of hyperfine levels where, due to either spin-changing collisions or external fields, the number conservation of the single components is broken. Among the most interesting effects in mixtures we can count the existence of a collisionless drag between the two superfluid components and, even more remarkably, of self-bound droplets and of a liquid-like (incompressible) phase. The description of the above-mentioned phenomena is not included in a simple two-component Gross–Pitaevskii equation and requires the inclusion of the so-called quantum fluctuations. For non-number-conserving mixtures, it is remarkable the presence of magnetic-like quantum phase transitions (i.e., phase transitions that occur at zero temperature), where the magnetic language is used to describe the spinor or vector order parameter of the system. It is therefore possible to study magnetic superfluids and the interplay between superfluidity and magnetism. In the same framework, it was possible to realize condensates with spin–orbit coupling, allowing for new states of matter, complementary to the electronic states in solid-state physics.

Further directions The possibility of tuning the interactions among atoms and the large variety of trapping conditions reachable by combing magnetic and optical methods has allowed for the realization of a number of configurations involving Bose and/or Fermi gases (see Bloch et al. (2008) and Inguscio et al. (2016) for a review of major developments). The resulting scenarios are still evolving and cover topics of high interdisciplinary interest, as briefly listed below.

Optical lattices Optical lattices can be used to provide perfect periodic potentials where the atoms can be trapped, in very close analogy to what happens in the solid state, where the electrons are bound to the periodic arrangement of atoms of the crystalline structure. Since the first years of this century, optical lattices have been used as a powerful instrument to probe the coherence and superfluid properties of Bose–Einstein condensates: notable examples are the observation of many-BEC interference with longrange coherence or the realization of Josephson junctions arrays (Cataliotti et al., 2001). Optical lattices are also an ideal tool to study key aspects of quantum transport in defect-free solid-state analogs, such as the modification of the transport properties caused by the emerging band structure (Fallani et al., 2003), according to Bloch theorem (Fig. 15A). Fundamental effects of solid-state physics were cleanly observed in this setting, the most prominent one being the occurrence of undamped Bloch oscillations activated by a constant force (Fig. 15B), which are also a valuable resource for precision measurements in the context of atom interferometry (Roati et al., 2004). This analogy with the solid state has been exploited intensively to provide the first realizations of the idea of quantum simulation pioneered by R. Feynman in the 1980s. Optical lattices were used to provide a clean realization of bosonic and fermionic Hubbard models, originally introduced in condensed-matter physics to describe interacting quantum particles hopping between the sites of a deep lattice. By an accurate experimental control of the coupling constants of the Bose–Hubbard model, a quantum phase transition between a superfluid state and a Mott insulator was observed (Greiner et al., 2002), as shown in Fig. 16: when the depth of the optical lattice is small, interactions effects are weak and the system exhibits long-range coherence, which is witnessed by the sharp peaks in the matter-wave interference; when the lattice depth is increased above a critical value, repulsive interactions between particles dominate over the hopping and the system localizes in a Mott state with no particle number fluctuations and no coherence, and the peaks in the matter-wave interference disappear. Experimental studies of the Fermi-Hubbard model have also evidenced the antiferromagnetic ordering of the Mott localized phase for a mixture of spin-up/spin-down fermions, which is believed to provide a key mechanism for the explanation of high-Tc superconductivity in cuprates, also opening new paths for the exploration of spin lattice models.

Disorder Not only light can be used to create optical lattices where solid-state models can be simulated in a defect-free way. It can also be used to realize controllable disordered or quasi-disordered potentials, via the use of speckle patterns or multichromatic optical lattices, respectively. In this settings, Anderson localization of matter waves was first observed by studying the expansion of a Bose–Einstein condensate in a disordered potential (Billy et al., 2008; Roati et al., 2008): when the amount of disorder was strong enough, the BEC stopped its free expansion in the disordered potential, in correspondence with the transition of the eigenstates from extended to localized (see Fig. 17). Since the first observations of localization in disordered BECs, the interplay between disorder and interactions has been studied extensively, also in the context of the topic of many-body localization, where localization is accompanied by nontrivial quantum effects concerning thermodynamics and equilibrium.

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Fig. 15 Quantum transport in an optical lattice, as studied in early experiments at LENS: (A) Experimental measurements of the velocity and effective mass of a BEC in an optical lattice as a function of its quasimomentum, compared with the predictions of the Bloch theory describing electrons in a crystal lattice. (B) Ultracold fermions trapped in a vertical optical lattice, performing long-lived Bloch oscillations under the effect of gravity: measuring the frequency of the oscillation allows an accurate determination of the local gravity acceleration g and can be used for high-precision measurements of forces on a short distance scale. Reproduced from Fallani L, Cataliotti FS, Catani J, Fort C, Modugno M, Zawada M, and Inguscio M (2003) Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice. Physical Review Letters 91: 240405 and Roati G, De Mirandes E, Ferlaino F, Ott H, Modugno G, and Inguscio M (2004) Atom interferometry with trapped Fermi gases. Physical Review Letters 92: 230402.

Fig. 16 Quantum phase transition from a superfluid to a Mott insulator for ultracold bosons in a 3D optical lattice, as first observed in 2002 at MPQ. In the Mott phase, the atoms are localized in the lattice sites, phase fluctuations increase and the long-range coherence of the superfluid phase is lost. This is evidenced by the disappearance of the matter-wave interference pattern when the lattice depth is increased (from A to H). Reproduced from Greiner M, Mandel O, Esslinger T, Hänsch TW, and Bloch I (2002) Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415: 39 with permission from the authors.

Synthetic fields and synthetic matter Starting from the 2010s new experimental approaches involving the manipulation of BECs with laser light have allowed researchers to engineer new kind of interactions for quantum simulation experiments. Pioneering experiments performed at JQI/NIST demonstrated the possibility of realizing light-induced spin–orbit interactions, by which the motion of the atoms is coupled to their spin, or synthetic magnetic fields, in which the phase imprinted by the interaction with the lasers emulates a magnetic vector potential, thus mimicking the effect of a synthetic magnetic field on atoms with a synthetic charge (see Goldman et al. (2014) for a review). These first demonstrations triggered a new field of research, in which BECs and Fermi gases are used to study novel quantum phases and topological states of matter.

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Fig. 17 Evidences of Anderson localization with ultracold atoms, as first reported in 2008 at Institut d’Optique and LENS. (A) The wavefunction of a weakly  interacting 87Rb BEC expanding in a speckle potential shows exponentially decreasing tails. (B) The momentum distribution jCðpÞj2 of a noninteracting 39K BEC in an incommensurate bichromatic lattice becomes broad as the wavefunction C(x) changes from extended to Anderson-localized when the amplitude of the quasidisorder is above a localization threshold. Reproduced from Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clément D, Sanchez-Palencia L, Bouyer P, and Aspect A (2008) Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453: 891 and Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, and Inguscio M (2008) Anderson localization of a noninteracting Bose-Einstein condensate. Nature 453: 895. with permission from the authors.

Low-dimensional physics Since the first realizations of optical lattices, it was clear that the strong confinement of the atoms in the lattice sites also represented a practical tool to study BEC in low-dimensional settings. As a matter of fact, a strong 1D optical lattice can be used to restrict the atomic motion on a 2D plane while a strong 2D lattice can create quantum wires, where the motion of the atoms is restricted along 1D only. The dimensionality of space has important consequences on the collective properties of many-body quantum systems, with dramatic effects that emerge from the interplay of interactions and reduced dimensionality. In an interacting 2D Bose gas superfluidity can arise even in the absence of Bose–Einstein condensation, as a result of a Berezinskii–Kosterlitz–Thouless (BKT) transition. The collective excitations of an interacting 1D Bose gas can be described in terms of the theory of Luttinger liquids and the system can even realize a “fermionized” Tonks–Girardeau gas, where the strong repulsion between the bosons force them to acquire a fermionic behavior, playing the role of an effective Pauli repulsion.

Long-range interactions The vast majority of experimental and theoretical works on ultracold gases deals with atoms whose interaction—at the energy and temperatures of interest—can be described as contact-like and described by s-wave scattering lengths. More recently it has been possible to trap and manipulate gases where the dipolar interaction is strong enough to give rise to new phenomena, due to its longrange and anisotropic nature. A first approach to make the dipolar interaction relevant for the phenomenology of the system is to consider atoms with a relatively large dipolar interaction, as rare-earth atoms, and reduce the strength of the contact interaction by exploiting Feshbach resonances. In this way physicists were able to obtain the quantum equivalent of ferrofluids, to measure rotonlike spectra, and, for Fermi gases, to characterize Fermi sphere deformations, just to mention a few. In particular it was observed that, when the dipolar interaction becomes large enough, the gas can break into a matrix of self-bound droplets (Kadau et al., 2016), as shown in Fig. 18. The nature of these droplets is very similar to that of the droplets found in Bose–Bose mixtures, since they are both stabilized by quantum fluctuations. Even more importantly, such droplets can be made coherent among each other, allowing for the very first observation of the so-called supersolid state (see below). Magnetic dipolar atomic gases are not the only platforms where long-range interaction can play a major role in the context of ultracold gases. Another possibility is to exploit the electric dipole moment of heteronuclear binary molecules. In this case the dipole–dipole interaction can be very strong and much attention has been devoted to long-range interaction effects in molecular gases loaded in deep optical lattices, with the realization of important solid-state Hamiltonians. Other approaches to synthesize effective long-range interactions include atoms in optical cavities, Rydberg atoms, and trapped ions. The latter platforms are considered to be good candidates also for the realization of atom-based quantum computers.

Supersolidity In the last few years supersolidity has emerged as an exciting and active frontier of research in the field of ultracold atoms. Supersolidity is an intriguing phenomenon, where both superfluid and crystalline properties coexist as a consequence of the simultaneous breaking of phase and translational symmetry. After unsuccessful attempts in solid helium, supersolidity was

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Fig. 18 Observation of self-bound quantum droplets in a dipolar gas of 164Dy, as observed in 2016 at University of Stuttgart. Reproduced from Kadau H, Schmitt M, Wenzel M, Wink C, Maier T, Ferrier-Barbut I, and Pfau T (2016) Observing the Rosensweig instability of a quantum ferrofluid. Nature 530: 194 with permission from the authors.

experimentally reported in 2017 in Bose–Einstein condensates with spin–orbit coupling (MIT) and inside optical resonators (ETH). Remarkably, in 2019 the supersolid state was realized in dipolar Bose gases almost at the same time in three different laboratories (at LENS-Pisa, Innsbruck and Stuttgart). The experimental and the theoretical research activity has then become particularly rich (see Recati and Stringari (2023)) for a review). In particular in the case of dipolar gases, phase coherence, the spatial modulations of the density profile, and the superfluid rotational properties, as well as the emergence of the Goldstone modes associated with the superfluid and crystal behavior, were observed.

Conclusion The experimental realization of Bose–Einstein condensation in ultracold atomic gases in 1995 opened completely new research directions that were inconceivable before then. BEC experiments have become “quantum laboratories,” where quantum theories can be tested for the first time “on demand,” providing a natural embodiment of the “quantum simulation” approach envisioned by R. Feynman. Almost three decades of exciting research have led us to a much better understanding of fundamental physical effects, in a continuous synergy between theoretical studies and experimental investigations, always going hand-in-hand, one prompting each other in a virtuous circle. What could the developments of the next three decades be? In the last chapter we have identified some of the major trending topics, but the list is far from conclusive and it is difficult to foresee what the next breakthroughs will be. What we can certainly say is that a growing, enthusiastic community of scientists will keep working, trying to achieve a better understanding of the quantum world, also with the aim of translating this knowledge into new quantum-based technologies for the benefit of society at large.

References Albiez M, Gati R, Fölling J, Hunsmann S, Cristiani M, and Oberthaler MK (2005) Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction. Physical Review Letters 95: 010402. Andrews MR, Townsend CG, Miesner HJ, Durfee DS, Kurn DM, and Ketterle W (1997) Observation of interference between two Bose condensates. Science 275: 637. Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clément D, Sanchez-Palencia L, Bouyer P, and Aspect A (2008) Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453: 891. Bloch I, Dalibard J, and Zwerger W (2008) Many-body physics with ultracold gases. Reviews of Modern Physics 80: 885. Bose SN (1924) Plancks Gesetz und Lichtquantenhypothese. Zeitschrift für Physik 26: 178. Cataliotti FS, Burger S, Fort C, Maddaloni P, Minardi F, Trombettoni A, Smerzi A, and Inguscio M (2001) Josephson junction arrays with Bose-Einstein condensates. Science 293: 843. Chen C-C, Escudero RG, Minárˇ J, Pasquiou B, Bennetts S, and Schreck F (2022) Continuous Bose-Einstein condensation. Nature 606: 683.

Bose–Einstein condensation

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Cornell EA and Wieman CE (2002) Nobel lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Reviews of Modern Physics 74: 875. Dalfovo F, Giorgini S, Pitaevskii LP, and Stringari S (1999) Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Physics 71: 463. Einstein A (1924) Quantentheorie des einatomigen idealen gases. Sitzungsberichte der Preuischen Akademie der Wissenschaften: 261. (1925) Quantentheorie des einatomigen idealen Gases. Sitzber. Kgl. Preuss. Akad. Wiss. 3. Ensher JR, Jin DS, Matthews MR, Wieman CE, and Cornell EA (1996) Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation. Physical Review Letters 77: 4984. Fallani L, Cataliotti FS, Catani J, Fort C, Modugno M, Zawada M, and Inguscio M (2003) Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice. Physical Review Letters 91: 240405. Goldman N, Juzeliunas G, Öhberg P, and Spielman IB (2014) Light-induced gauge fields for ultracold atoms. Reports on Progress in Physics 77: 126401. Greiner M, Mandel O, Esslinger T, Hänsch TW, and Bloch I (2002) Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415: 39. Greiner M, Regal CA, and Jin DS (2003) Emergence of a molecular Bose-Einstein condensate from a Fermi gas. Nature 426: 537. Grimm R, Inguscio M, Stringari S, and Lamporesi G (eds.) (2023) Proceedings of the International School of Physics “Enrico Fermi”, Varenna (Italy)–Course 211. Quantum Mixtures With Ultracold Atoms. IOS Press. In press. Hau LV, Busch BD, Liu C, Dutton Z, Burns MM, and Golovchenko JA (1998) Near-resonant spatial images of confined Bose-Einstein condensates in a 4-Dee magnetic bottle. Physical Review A 58: R54. Inguscio M and Fallani L (2013) Atomic Physics: Precise Measurements and Ultracold Matter. Oxford University Press. Inguscio M, Stringari S, and Wieman CE (eds.) (1999) Proceedings of the International School of Physics “Enrico Fermi”, Varenna (Italy)–Course CXL. Bose-Einstein Condensation in Atomic Gases. IOS Press. Inguscio M, Ketterle W, and Salomon C (eds.) (2007) Proceedings of the International School of Physics “Enrico Fermi”, Varenna (Italy)–Course CLXIV. Ultracold Fermi gases. IOS Press. Inguscio M, Ketterle W, Stringari S, and Roati G (eds.) (2016) Proceedings of the International School of Physics “Enrico Fermi”, Varenna (Italy)–Course CXCI. Quantum Matter at Ultralow Temperatures. IOS Press. Inouye S, Andrews MR, Stenger J, Miesner H-J, Stamper-Kurn DM, and Ketterle W (1998) Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392: 151. Kadau H, Schmitt M, Wenzel M, Wink C, Maier T, Ferrier-Barbut I, and Pfau T (2016) Observing the Rosensweig instability of a quantum ferrofluid. Nature 530: 194. Ketterle W (2002) Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser. Reviews of Modern Physics 74: 1131. Madison KW, Chevy F, Wohlleben W, and Dalibard J (2000) Vortex formation in a stirred Bose-Einstein condensate. Physical Review Letters 84: 806. Pethick CJ and Smith H (2002) Bose-Einstein Condensation in Dilute Gases. Cambridge University Press. Pitaevskii LP and Stringari S (2016) Bose-Einstein Condensation and Superfluidity. Oxford University Press. Recati A and Stringari S (2023) Supersolidity in ultra-cold dipolar gases. Nature. In press. Roati G, Riboli F, Modugno G, and Inguscio M (2002) Fermi-Bose quantum degenerate 40K-87Rb mixture with attractive interaction. Physical Review Letters 89: 150403. Roati G, De Mirandes E, Ferlaino F, Ott H, Modugno G, and Inguscio M (2004) Atom interferometry with trapped Fermi gases. Physical Review Letters 92: 230402. Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, and Inguscio M (2008) Anderson localization of a non-interacting Bose-Einstein condensate. Nature 453: 895. Sidorenkov LA, Tey MK, Grimm R, Hou Y-H, Pitaevskii L, and Stringari S (2013) Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498: 78. Zwerger W (2014) The BCS-BEC Crossover and the Unitary Fermi Gas. Berlin, Heidelberg: Springer.

Universality of Bose–Einstein condensation and quenched formation dynamics Nick P Proukakis, Joint Quantum Centre (JQC) Durham–Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, United Kingdom © 2024 Elsevier Ltd. All rights reserved.

Introduction “Universality” of Bose–Einstein condensation Dynamical phase transition stages Relevant coherence measures Correlation functions Penrose–Onsager condensation vs. quasi-condensation Workhorse model of Bose–Einstein condensation Universality during phase transition crossing: The Kibble–Zurek scaling law Early ideas Numerical phase transition model Criticality at equilibrium Criticality in a dynamical setting The Kibble–Zurek scaling law Analyzing Kibble–Zurek in an elongated ultracold atomic system Beyond homogeneous Kibble–Zurek considerations Universality during phase-ordering dynamics Scaling hypothesis and late-time coarsening Evolution around nonthermal fixed points Relation of quenched turbulence dynamics to “controlled” turbulence experiments Selected further considerations Beyond continuous single-component atomic gases Driven-dissipative condensation in exciton-polariton systems Bose–Einstein condensation on astrophysical/cosmological scales? Conclusion Acknowledgments References

85 85 88 89 89 90 92 93 93 94 95 95 97 99 102 104 104 106 108 109 109 110 114 117 117 117

Abstract The emergence of macroscopic coherence in a many-body quantum system is a ubiquitous phenomenon across different physical systems and scales. This chapter reviews key concepts characterizing such systems (correlation functions, condensation, and quasi-condensation) and applies them to the study of emerging nonequilibrium features in the dynamical path toward such a highly coherent state: particular emphasis is placed on emerging universal features in the dynamics of conservative and open quantum systems, their equilibrium or nonequilibrium nature, and the extent that these can be observed in current experiments with quantum gases. Characteristic examples include symmetry-breaking in the KibbleZurek mechanism, coarsening and phase-ordering kinetics, and universal spatiotemporal scalings around nonthermal fixed points and in the context of the Kardar–Parisi–Zhang equation; the chapter concludes with a brief review of the potential relevance of some of these concepts in modeling the large-scale distribution of dark matter in the universe.

Key points

• • • • •

84

Demonstration of ubiquitous nature of Bose–Einstein condensation and key observables characterizing it. Unified overview of the dynamical phase transition crossing from an incoherent initial condition to a (highly/fully) coherent final state, introducing key universal scaling laws applicable during such process. Detailed explanation of the Kibble–Zurek mechanism for a driven dynamical phase transition crossing through an external time-dependent parameter. Discussion of phase-ordering process and universal scaling at late evolution times following an instantaneous quench from an incoherent state. Applications of above frameworks to the dynamics of ultracold atoms and exciton-polaritons, with concrete examples and experimental observations.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00253-5

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Discussion of other highly nonequilibrium features of quantum gases originally stemming from cosmology (nonthermal fixed points) and classical systems (Kardar–Parisi–Zhang). Review of cosmological condensation model for dark matter, and its analogies to laboratory condensates.

Introduction Bose–Einstein condensation (BEC), first predicted nearly 100 years ago, is a phenomenon associated with the emergence of a highly coherent state which manifests itself across vastly different physical systems: Interestingly, not only the existence of the effect but also the manner in which it emerges dynamically across such systems can exhibit the same characteristic features (Proukakis et al., 2017). This chapter reviews such universality of the existence and the dynamical emergence of a Bose–Einstein condensate starting from an incoherent (or thermal) state: in particular, attention is paid to universal scaling laws and (dynamical) self-similarity associated with the initial growth stages when the system breaks the Hamiltonian symmetry to randomly choose phase domains in a potentially “turbulent” manner, and with the subsequent relaxation of such a nonequilibrium state to the corresponding final equilibrium state (or steady state). The main aim of this chapter is to give an overview of key concepts of (selected) universal scaling laws which have developed over decades across distinct branches of physics, and focus on those with recent applications to experimentally controlled quantum gases; as such, this chapter is only intended to give a “flavor” and not a thorough overview of such topics across other systems, for which the reader is referred to suitable review articles. Nonetheless, before concluding, I also briefly discuss a less widely known emerging new example characterizing the interplay between condensed matter physics, nonequilibrium statistical physics and cosmology/high-energy physics, in the context of how quasi-condensation, BEC and turbulence, characterized through relevant correlation functions, could potentially relate to our understanding of large-scale cosmological structures.

“Universality” of Bose–Einstein condensation BEC is a fundamental manifestation of many-particle coherence in a quantum system, occurring when a system of many particles acquires wave-like properties. Such behavior emerges when the de Broglie wavelength l
h/p (where h is Planck’s constant and p is momentum) of the underlying particles becomes comparable to, or exceeds, the typical interparticle separation in the medium (Fig. 1). The textbook discussion of this effect (Huang, 1987; Dalfovo et al., 1999; Pitaevskii and Stringari, 2016; Pethick and Smith, 2002; Ueda, 2010) typically focusses on the case of a three-dimensional (3D) system of bosonic particles (i.e., particles of integer spin obeying Bose–Einstein statistics), for which such condensation emerges when the dimensionless phase-space density nl3 ≳ Oð1Þ (where n ¼ N/V is the number density of particles in the system) exceeds a value of order 1 (in a 3D noninteracting system, the exact value is z(3/2)  2.612). This occurs at a characteristic, or critical, value (Ec) of the relevant external parameter, E, which varies across, and thus induces, the transition. For example, in the well-studied case of particles being cooled (and noting that for such systems l
(mT )−1/2), BEC emerges at a critical temperature, Tc—a well-known characteristic in superfluidity/condensation phase diagrams of liquid helium and ultracold quantum gases.

3 Incoherent Regime

D

a1

Transition Regime

D

Coherent Regime

D

=0

Control Parameter

Fig. 1 Schematic of the transition from a system of incoherent, distinguishable (potentially composite) bosonic particles having a mean interparticle distance D
n−1/3 (left), through the critical region in which the de Broglie wavelength, l, becomes comparable to D (when nl3
1) (middle), to the quantum-degenerate (highly coherent) regime in which the typical wavelength much exceeds the mean interparticle distance (right) in terms of an external control parameter E (e.g., corresponding to decreasing system temperature). Depicted arrows indicate particle velocities, which typically start aligning in the transition region (when the external control parameter, E acquires a critical value Ec, which is typically set to zero for convenience) eventually forming a coherent matter wave, known as a Bose–Einstein condensate, with a unique (randomly chosen) direction; the reduced vector magnitudes shown here indicate cooling.

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However, in general, the role of the critical external control parameter could be played by many different physical variables (Snoke et al., 2017): for example, for the systems considered in this chapter, beyond a critical temperature or a critical density, such a phenomenon could also arise in terms of a critical interaction—whose magnitude can also be controlled by the system confinement/dimensionality—or a critical pumping rate, although more broadly a system could be induced to criticality through a critical magnetic field or pressure. The bosonic particle undergoing such change of state can be fundamental (e.g., photons (Klaers et al., 2010)), or composite— noting that an even number of fermions also acquires integer spin, thus behaving bosonically. The simplest example of a composite bosonic particle is that of a Hydrogen 1H atom, which comprises two fermions: a proton in the core, surrounded by a single electron. As it turns out, significant success in laser cooling and trapping (Chu, 1998; Cohen-Tannoudji, 1998; Phillips, 1998) has facilitated BEC to be achieved in a range of dilute (and thus typically weakly interacting) alkali gases (Cornell and Wieman, 2002; Ketterle, 2002): in fact the first observations of a weakly interacting BEC across any physical system were made in ultracold atomic gases of 87Rb and 23Na (prior to its observation in spin-polarized 1H, which followed a few years later (Fried et al., 1998; Greytak and Kleppner, 2017)). Helium is the first known superfluid (and the only one known to exist in thermodynamic equilibrium). Although the well-studied bosonic 4He, which is also a composite boson, exhibits a clear transition at some characteristic temperature and reaches 100% superfluidity at T ¼ 0, the fraction of condensed particles in such a system nonetheless remains under 10% due to the relatively strong atomic interactions. Composite bosonic quasiparticles can also be formed from an even number of fermions: the most striking example is that of electron-hole Cooper pairs in superconductors or pairs of fermionic 3He atoms exhibiting superfluidity. Although the detailed microscopic mechanism here is different (the well-known Bardeen–Cooper–Schrieffer, or BCS, mechanism (Cooper and Feldman, 2010)), ultimately the composite bosonic quasiparticle still effectively undergoes a BEC-type transition to a state of macroscopic coherence (Leggett, 2006). In fact, the exquisite controlled setting of ultracold atomic gases enables one to probe, even experimentally, the transition from a BEC to a BCS coherent system (Zwerger, 2011). Another example receiving significant current attention over the past 15 years is the condensation of a composite half-matter-half-light quasiparticle known as an exciton-polariton condensate, which manifests itself in pumped semiconductor microcavities (Sanvitto and Timofeev, 2012; Deng et al., 2010; Carusotto and Ciuti, 2013). Signatures of BEC (and macroscopic coherence in general) in fact arise across an immensely broad range of length and energy/ temperature scales (see Fig. 2), spanning from nuclear and atomic physics, through optical and condensed matter systems to astrophysical (neutron star cores) and even cosmological (fuzzy dark matter) settings (Proukakis and Burnett, 2013)—a more extended discussion on such diverse manifestations can be found in Proukakis et al. (2017). Although BEC was initially discussed in the context of a homogeneous, noninteracting 3D system, realistic systems are in fact of finite size and do exhibit some interactions—which can mask the effects of BEC, as in the case of superfluid 4He. This led to a manydecade-long search for experimentally controllable weakly interacting systems which could exhibit BEC: ultracold atomic experiments have provided a natural setting for achieving and observing nearly pure condensates, despite their (typically) inhomogeneous nature and finite system size. Current experiments in such systems in fact facilitate a detailed understanding of the modifications to both equilibrium coherence properties and dynamical formation induced by inhomogeneous/trapped and finite-size configurations, the influence of (realistic) weak interactions, the transition to more strongly interacting settings, and the confinement to lower effective dimensionalities, alongside studies of mixtures, superfluid turbulence, quantum phase transitions and even long-range interactions and supersolidity. Moreover, the last 2 decades have facilitated another well-controlled system, namely that of excitonpolaritons, which additionally offer glimpses to nonequilibrium phase transition features. Beyond a classification of equilibrium properties of such systems, a fascinating generic underlying question with a history of significant contributions spanning many decades and across diverse physical settings concerns the fundamental question of how such systems form, and what common—or universal—features may exist during such a process. In this chapter, I discuss in a broad sense the main aspects of the dynamical path toward such a highly coherent state, focusing primarily on the context of trapped ultracold (bosonic) atomic gases, with some emphasis on realistic inhomogeneous settings that facilitate direct links to controlled experiments: applicability and extension of such concepts to other physical settings and systems—most notably exciton-polariton condensates—are also discussed at relevant points within this chapter. The presentation is kept at a broad introductory level, geared to a nonexpert audience, with the intention to give a flavor of such rich questions—for a more complete picture, and historical overview, the reader is referred to a number of excellent reviews on such topics (Proukakis et al., 2017; del Campo and Zurek, 2014; del Campo et al., 2013; Bray, 1994; Rutenberg and Bray, 1995a; Schmied et al., 2019; Chantesana et al., 2019; Jos, 2013; Bloch et al., 2022). This chapter is structured as follows: The rest of Section “Introduction” gives a unified qualitative overview of the dynamical stages in the transition process from an initial incoherent state to the emerging final equilibrium or steady-state of the system (Section “Dynamical phase transition stages”), discussing the key physical quantities used to characterize such a system (Section “Relevant coherence measures”), with a brief introduction to the basic phenomenological description of a BEC (Section “Workhorse model of Bose–Einstein condensation”). Section “Universality during phase transition crossing: The Kibble–Zurek scaling law,” which forms the core of this chapter, addresses how coherence grows dynamically during an external parameter quench through spontaneous symmetry breaking and defect formation in the context of the so-called Kibble–Zurek mechanism giving rise to universal scaling laws: after giving a brief historical introduction (Section “Early ideas”) and presenting a minimal numerical model capturing such effects (Section “Numerical phase transition model”), the key underlying principles and scaling law of Kibble–Zurek are described mathematically by extending the ideas of equilibrium criticality (which are thus also briefly reviewed here in Section “Criticality at equilibrium”) to dynamical transition crossing (Section “Criticality in a dynamical setting” and

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Fig. 2 Examples of very different systems exhibiting macroscopic quantum coherence and Bose–Einstein condensation, facilitated through a high dimensionless phase-space density nld
1 (where d the system dimensionality). For a more extended discussion of typical control parameters, probes, observables, and related characteristic properties and dimensionalities across such diverse systems, see Snoke et al. (2017). Adapted with permission from Proukakis NP and Burnett K (2013) Quantum gases: Setting the scene. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics: 1 (Cold Atoms), London: Imperial College Press

“The Kibble–Zurek scaling law”) and then applying them to recent experiments (Section “Analyzing Kibble–Zurek in an elongated ultracold atomic system”) which highlight the need for a more extended framework to account for inhomogeneous systems and their interplay with causality, which is also presented here (Section “Beyond homogeneous Kibble–Zurek considerations”). Section “Universality during phase-ordering dynamics” discusses universal phase-ordering dynamics following an instantaneous quench, demonstrating the emergence of universal scaling of the spatial correlation function in characteristic late-time evolution stages of a relaxing quantum gas (Section “Scaling hypothesis and late-time coarsening”), with particular emphasis on highly nonequilibrium quenches giving rise to “turbulent” states evolving in the vicinity of nonthermal fixed points (Section “Evolution around nonthermal fixed points”). Section “Selected further considerations” presents a highly-relevant (but heuristic) selection of applications of such concepts to other systems, briefly mentioning for completeness other effects and manifestations in ultracold quantum gases not discussed here (Section “Beyond continuous single-component atomic gases”). Section “Driven-dissipative condensation in exciton-polariton systems” focuses on the emergence of dynamical features discussed in Sections “Universality during phase transition crossing: The Kibble–Zurek scaling law” and “Universality during phase-ordering dynamics” and the related spatiotemporal scaling according to a (quantum) mapping of phase evolution in an open, driven-dissipative, quantum gas of exciton-

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polaritons to the well-known (classical) Kardar–Parisi–Zhang equation characterizing interfacial growth. Section “Bose–Einstein condensation on astrophysical/cosmological scales?” discusses the potential relevance of some of the discussed characterizations to the context of a cosmological-scale BEC, which is shown to exhibit many characteristics similar to harmonically confined ultracold quantum gases. General concluding remarks and outlook are given in Section “Conclusion.”

Dynamical phase transition stages Condensate formation is a fundamental nonequilibrium problem with a very long history dating over many decades (see, e.g., selected influential early works (Kagan, 1995; Stoof, 1995, 1999), discussions (Gardiner et al., 1998; Bijlsma et al., 2000) of the first quantitative ultracold atomic growth experiments (Miesner et al., 1998; Köhl et al., 2002), and the review (Davis et al., 2017), which contains many more highly relevant references). The main question to be addressed here is how a system, originally in a disordered/incoherent state, transitions—through the breaking of a continuous symmetry—to an equilibrium state (or steady state) exhibiting a high degree of macroscopic coherence, the extent of which is set by the particular system under consideration (e.g., system geometry and dimensionality, nature and strength of interactions between constituent particles, mass of constituent boson). A schematic of the main steps during such a process is shown in Fig. 3 in terms of the external control parameter E. From an equilibrium perspective, such a transition can be visualized as a gradual passage through different equilibrium states, with a well-identified critical region (around E
Ec
0) separating purely incoherent from emerging coherent states. But how is such a picture affected when the external control parameter is dynamically varied, that is, E ! E(t)? Let us consider here by means of an example the specific case of a temperature quench, for which E ¼ (Tc − T )/Tc. During the first stage of cooling, and while the system is still (sufficiently) above the critical temperature, particle dynamics can be described in the usual (classical) phase-space picture, based on a Boltzmann equation with collisions (Lifshitz and Pitaevskii, 1995). Under continuous cooling toward the critical point, particles are gradually redistributed into lower energy states (with elastic collisions typically occurring on a much faster timescale than that governing the variation of the external control parameter, thus providing the required thermalization). The second, and arguably most interesting, stage in the evolution addresses the question of how the system dynamically crosses the phase transition, and what the nature of the nonequilibrium state generated in this vicinity of the critical point is. This is discussed in detail in Section “Universality during phase transition crossing: The Kibble–Zurek scaling law”: for now, it suffices to say that this leads to a delay in the crossing of the phase transition, and the emergence of interesting (potentially “chaotic,” or “turbulent-like”) nonequilibrium states displaying a (potential) abundance of phase defects and relevant universal properties: such emerging transient states depend, in general, on both the dimensionality/geometry of the system and the rate of crossing of the phase transition (e.g., adiabatically vs. instantaneously). The final stage in the equilibration/thermalization of the system is intricately related to the subsequent relaxation of such phase defects, and the related emergence of (a certain degree of ) macroscopic coherence across the system: universal behavior is again

Dynamical Phase Transition Schematic

0

Phase-Ordering

Equilibrium Time Critical Delay Point ̂

Control Parameter

ϵ=(

− )/

Fig. 3 Dynamical phase transition schematic: As the external control parameter, E, is dynamically varied across the transition (i.e., E ! E(t )), the system exhibits a delayed response compared to the corresponding equilibrium critical point at E ¼ Ec (corresponding, for example, to T ¼ Tc), with Ec typically set to 0, thus separating the incoherent (E < 0) from the coherent (E > 0) side of the phase transition. As soon as E(t ) dynamically crosses to a positive value, and still within the critical regime (i.e., E(t ) ! 0+), it acquires (in general) a chaotic, turbulent nonequilibrium configuration containing phase defects: such physics is dominated by a universal Kibble–Zurek scaling law governing the dynamical phase transition. The subsequent gradual relaxation of such defects leads to the emergence of macroscopic coherence over much longer timescales; this includes a self-similar “phase ordering” process obeying universal scaling laws. Other universal features discussed in this chapter (such as spatio-temporal evolution around nonthermal fixed points and Kardar–Parisi–Zhang scaling) may also appear at appropriate scales and evolutionary stages during the extended process, depending on the particular system, initial state preparation, and quenching protocol. Specific equilibrium density snapshots shown here sufficiently before/after the transition are for the specific cases of an elongated inhomogeneous 3D, and a homogeneous 2D, ultracold atomic gas—both analyzed within this chapter.

Universality of Bose–Einstein condensation and quenched formation dynamics

89

expected to emerge here in some dynamical scenarios/regimes (e.g., for homogeneous systems, or very rapidly quenched, highly nonequilibrium states), but the specific system details can potentially also leave their imprint on this process, thus masking (or even potentially modifying the details of ) such universality. Such discussion can be formulated in terms of two commonly considered types of quenched regimes: instantaneous quenches, where the system is prepared, or induced, into a highly nonequilibrium state, and finite-duration (i.e., noninstantaneous) quenches starting from an incoherent equilibrium configuration, in which the system is typically modeled on the basis of a smoothly varying external parameter inducing such transition. Both such cases lead to interesting universal physical laws, associated with the dynamical phase transition crossing and the subsequent phase-ordering process, as discussed in more detail below (Sections “Universality during phase transition crossing: The Kibble–Zurek scaling law” and “Universality during phase-ordering dynamics”), starting from the finite-duration phase transitions.

Relevant coherence measures Classification of the system equilibrium and dynamical properties is intricately related to appropriate characterization of the degree of system coherence.

Correlation functions

{

^ tÞ and C ^ ðr, tÞ, a classification of the coherence in both the spatial and For a quantum system described by Bose field operators Cðr, temporal domains can be obtained through the nth-order correlation function
 GðnÞ r1 , r01 , r2 , r02 , ⋯ , t1 , t10 , t2 , t20 , ⋯

^ { ðr1 , t1 ÞC ^ { ðr2 , t2 Þ⋯C ^ { ðrn , tn ÞCðr ^ 0 , t 0 Þ⋯Cðr ^ 0 , t 0 ÞCðr ^ 0 , t 0 Þi ¼ hC n n 2 2 1 1

(1)

measured across n different spatial locations and at n different times (with . . . intended to reflect remaining terms such that there are ^  ðrn , t n Þ Cðr ^ 0 , t 0 Þ contained within the above expectation value expression). n pairs of C n n In practice, the most elementary—and thus most commonly discussed—correlation functions are those associated with spatial and temporal one-body correlations, respectively defined as Fixed t :

^ { ðr, tÞCðr ^ 0 , tÞi Cðr, r0 Þ ¼ Cðr, r0 , t, tÞ ¼ hC

(2)

Fixed r :

^ { ðr, tÞCðr, ^ t 0 Þi, Cðt, t 0 Þ ¼ Cðr, r, t, t 0 Þ ¼ hC

(3)

as these contain the primary information about spatial and temporal coherence. The typical definition of equilibrium BEC is in the form of so-called off-diagonal-long-range order (ODLRO), associated with the ^ { ðr, tÞ Cðr ^ 0 , tÞi (Pitaevskii and Stringari, nonvanishing asymptotic (spatial) behavior of the one-body density matrix rðr, r0, tÞ ¼ hC 2016; Ueda, 2010; Leggett, 2006). In particular, ODLRO is defined by {

^ ðr, tÞCðr ^ 0, tÞi 6¼ 0, Limjr − r0 j!1 hC

(4)

with its value tending to some nonzero constant limit  (r, t) (r0 , t), consistent with the existence of off-diagonal correlations in the one-body density matrix. This introduces the quantity (r, t) as the so-called order parameter of the system. To highlight the existence of a nonzero value, this is usually expressed as ¼ | |eiy, a form that facilitates an analogy to classical fluids through the mass density r ¼ mj j2 and superfluid velocity v ¼ ðħ=mÞry (Pitaevskii and Stringari, 2016). Moreover, one often connects the BEC order parameter with the expectation value of the Bose field operator through ^ Such an argument is motivated by the analogous classical description of the electromagnetic field, in which one ðr, tÞ ¼ hCi. introduces a complex “classical” electric field as the expectation value of the corresponding (quantum-mechanical) electric field operator. While somewhat useful, there are some subtleties in following such procedure for massive particles, like atoms, which (unlike photons) cannot be created/destroyed from the vacuum by single-particle creation/annihilation operators: an insightful discussion of this issue, and the definition of BEC can be found in the book by Leggett (2006). While models explicitly maintaining the quantum-mechanical nature of the operator also exist, it is not essential to discuss such an extension here, with the reader instead referred to recent reviews (Proukakis and Jackson, 2008; Proukakis et al., 2013). For the purposes of this discussion, we approximate the actual field operator by a fluctuating classical field (obeying a classical probability distribution in phase space), thus accommodating both density and phase fluctuations into the subsequent description in a manner which significantly simplifies the direct numerical modeling (see later). All discussion, from this point onwards are thus expressed in terms of the complex/multimode classical field, which in this work will be denoted by F(r, t) and also referred to occasionally as the order parameter. When studying the correlation in a given system, it is often useful to deal with appropriately normalized correlation functions, denoted by g(n): for example, the above general expression (Eq. 1) (written now in terms of the classical field F(r, t)) can be

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Universality of Bose–Einstein condensation and quenched formation dynamics

normalized by the product Pi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hjFðri , t i Þj2 ihjFðr0i , t 0i Þj2 i . This is useful as it implies that the same-location (all ri, ri0 the same),

equal-time (all ti, ti0 the same) normalized correlation function g(n)(0) has the elegant limiting cases (Cohen-Tannoudji and Robilliard, 2001)   1 ðPurely CoherentÞ gðnÞ ðri ⋯ri ; t i ⋯t i Þ ¼ : (5) n! ðIncoherentÞ In what follows, I will use interchangeably the (unnormalized) correlation functions C(r, r0 ) and C(t, t0 ), or their normalized versions respectively defined by hF ðr, tÞFðr0 , tÞi gð1Þ ðr, r0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , hjFðr, tÞj2 i hjFðr0 , tÞj2 i

(6)

hF ðr, tÞFðr, t 0 Þi gð1Þ ðt, t 0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : hjFðr, tÞj2 i hjFðr, t 0 Þj2 i

(7)

and

Note that although Eqs. (6) and (7) are often considered independently, a combined analysis of these two quantities can provide further crucial information about the extent of the “equilibrium” (or “nonequilibrium”) nature of the system, or the emerging spatiotemporal scaling relations (see subsequent discussion on exciton-polaritons in Section “Driven-dissipative condensation in exciton-polariton systems”). As previously mentioned, the hallmark of BEC in a 3D system is the striking emergence of ODLRO, signifying that (some degree of ) coherence is maintained across all space, that is, g(1)(r, r0 ; t) tends to a nonzero-valued plateau (having a value between 0 and 1) as |r − r0 | ! 1: for a finite-size system, it is of course only meaningful to study the dependence of g(1) on |r − r0 | over the spatial extent of the (coherent part of the) system. In fact, the extent of deviation of the plateau value from the starting value of 1 (taken at r ¼ r0 ) provides information about the fraction of incoherent particles coexisting with the condensed part (CohenTannoudji and Robilliard, 2001). Key emerging concepts utilized throughout this chapter, and described in more detail later, are those of self-similarity and scaling hypothesis. In a nutshell, the underlying idea is that under appropriate conditions (e.g., in a restricted spatial and/or temporal window) the microscopic details of the system are not relevant, with the physical description of key system observables acquiring a universal description in terms of appropriately rescaled variables and a specific functional form. Elimination of nonessential details at short scales (times) in favor of emphasizing large-scale (long-time) features—a key step in the universal description of a physical system associated with the analysis of critical phenomena and the concept of renormalization group theory (see, e.g., Nishimori and Ortiz, 2011; Binney et al., 1992; Bruce and Wallace, 1992)—is possible when there exists a characteristic lengthscale (timescale), which is much larger (longer) than all other relevant system scales (but smaller than the system size/evolution time). This in turn allows us to express relevant correlation functions over a limited spatial and/or temporal range in terms of a unique (but not a priori known) functional form whose arguments are appropriately rescaled spatial (temporal) variables, combined with a small number of critical exponents characterizing such behavior (Nishimori and Ortiz, 2011; Binney et al., 1992). Details of such scalings will become more apparent at relevant points throughout this chapter. Given the existence of both density and phase fluctuations in a realistic system, it is natural to examine their interplay and relative importance. In general, as a system approaches quantum degeneracy from the incoherent regime, density fluctuations start becoming suppressed at an earlier point (larger value of |E|) than phase fluctuations. In the specific context of cooling (thermal phase transition), this implies the emergence of two distinct characteristic temperatures for suppression of density and phase fluctuations. This is particularly pronounced in lower effective dimensionalities (i.e., in quasi-1D and quasi-2D geometries) (Petrov et al., 2004), but such effect is still present in 3D geometries, even if largely obscured (and so, subdominant). In order to also investigate the emergence of density fluctuations, we thus introduce here the second-order, or density–density correlation function (taken here for simplicity at equal times) gð2Þ ðr, r0 , tÞ ¼

hF ðr, tÞF ðr0 , tÞFðr0 , tÞFðr, tÞi hnðr, tÞ nðr0 , tÞi ¼ , 2 2 0 hnðr, tÞi hnðr0 , tÞi hjFðr, tÞj ihjFðr , tÞj i

(8)

with local density–density correlations accessible by evaluating such quantity at r0 ¼ r. As a direct consequence of Eq. (5), a pure BEC can be identified through g(1)(r, r, t) ¼ g(2)(r, r, t) ¼ 1, whereas an incoherent (thermal) field has g(2)(r, r, t) ¼ 2: this simple identification provides a further probe for testing the degree of coherence of a system at a given time t.

Penrose–Onsager condensation vs. quasi-condensation The above discussion has introduced the concept of ODLRO for characterizing a condensate at equilibrium: However, one is typically dealing with a finite-size system, and so such discussion has to be necessarily limited to within the spatial extent of the

Universality of Bose–Einstein condensation and quenched formation dynamics

91

system (which should much exceed all other characteristic system lengthscales to make this nontrivial), and as limiting behavior in the thermodynamic limit. In practice, for a finite-sized interacting system, one can instead proceed with a different characterization, following a seminal contribution by Penrose and Onsager (1956). In particular, one can diagonalize the off-diagonal one-body density matrix r(r, r0 , t) P to calculate its eigenvalues Nn through rðr, r0 , tÞ ¼ n Nn ðtÞfn ðr, tÞfn ðr0 , tÞ, where fn(r, t) denotes the corresponding n-th eigenfunction. If in doing so one finds a macroscopically dominating eigenvalue, which is on the same order as the total particle number, then one can identify the mode corresponding to such eigenvalue with the Bose–Einstein condensate mode (Leggett, 2006). Mathematically, a dominant eigenvalue (if one exists) and corresponding eigenfunction of the density matrix can be identified from the solution of Z dr0 Rðr, r0 Þfn ðr0 Þ ¼ Nn fn ðrÞ: (9) In practice one often uses a slightly relaxed criterion associated with the existence of a single eigenvalue which is much larger than all other eigenvalues of the system, but does not necessarily need to be on the same order as the total number of particles in the system. Such a mode is nowadays routinely referred to as the Penrose–Onsager mode of the system (and the above condition known as the Penrose–Onsager criterion) (Blakie et al., 2008). In the context of numerical simulations of a fluctuating classical field, it is often numerically prohibitive to perform such a calculation by means of an ensemble/trajectory averaging. Thus, for purely practical reasons, one often resorts to time-averaging of a single trajectory as a numerical trick to achieve the same effect: the idea here is to compute the off-diagonal density matrix by averaging over a large number of samples spread over a timescale which is both longer than that governing the evolution of intrinsic fluctuations and shorter than any typical evolution of interest in the system (e.g., collective mode, large-scale motion of defects): in such a scenario, fluctuations in the field quickly randomize a single trajectory, thus making an average over different time evolutions of a single trajectory statistically analogous to averaging over many different independent trajectories: in other words, with careful choice of parameters, averaging over different time samples can also mimic the effect of averaging over different trajectories not only in a dynamical steady state in which the order parameter fluctuates about some mean value but also in a dynamical setting (Blakie et al., 2008). Note that global observables depending critically on the dynamical path additionally require averaging over (at least) a small number of trajectories to ensure the result is indeed statistically independent (and does not contain information about the system history): this is for example the case when probing correlation functions and defect numbers later in the context of quenched (Kibble–Zurek) dynamics in an inhomogeneous geometry (Section “Analyzing Kibble–Zurek in an elongated ultracold atomic system”). At this stage it is important to also comment on the role of system dimensionality: As well known, ODLRO—and so BEC—is strictly speaking restricted to 3D (or higher) configurations, with corresponding homogeneous ideal 2D systems (for all T 6¼ 0) only exhibiting a reduced degree of coherence, known as quasi-long-range (or topological) order; such distinction arises despite the fact that both systems exhibit superfluidity (Pitaevskii and Stringari, 2016; Ueda, 2010). Reduced coherence in 2D arises as a result of the dominant role of long-wavelength excitations destroying the overall phase coherence of the system, according to the wellknown Hohenberg–Mermin–Wagner theorem. In such a case, the system exhibits a distinct phase transition, known as the Berezinskii–Kosterlitz–Thouless (BKT) transition (Kosterlitz, 2017; Jos, 2013): this is associated for E ≳ 0 with the binding of the free vortices which typically proliferate above the critical temperature (E > 0)—and whose presence randomizes the system phase—into vortex–antivortex pairs; such binding effectively screens their existence, which can be visualized as annihilation. Thermal fluctuations in 2D at any nonzero temperature reduce the coherence of the system compared to the corresponding 3D case, with the corresponding spatial profile of the correlation exhibiting instead in 2D a power-law decay and quasi-long-range order (as opposed to ODLRO). Given that the dominant factor in the behavior of the spatial correlation function g(1)(r, r0 ) is the distance between the two points, that is, |r − r0 |, and to simplify the expressions, here I arbitrarily choose to label one of these points as 0, and the distance between them as r, such that the relevant normalized spatial correlation function takes the form g(1)(r) ¼ g(1)(0, r). Below the critical temperature (E ! 0+)—and in the specific context of an infinite homogeneous system—this takes in the r ! 1 limit the dimensionality-dependent form   constant ð3DÞ gð1Þ ð0, rÞj0 0 the coherent side. Our understanding of the dynamics of the phase transition can be cast in terms of the relative magnitude of these two times (correlation time and systemadjustment timescale). As the system moves toward the critical point (starting from an incoherent equilibrium state), the time taken for the system to adjust to its new instantaneous equilibrium solution grows, and starts doing so very rapidly as it approaches closer to the (equilibrium) critical point (where it diverges). Thus, necessarily, as the system is dynamically induced through the phase transition, the situation will arise, in which the rate at which the system is driven through the phase transition parameter space is so fast that its corresponding inverse timescale is shorter than the time required for the system to equilibrate over an increasing lengthscale; this happens when the correlation time t becomes (in the vicinity of the critical point) larger than the magnitude of the instantaneous _ associated with the rate of quenching. Beyond such time, the system no longer has enough time to adjust as a whole timescale jE=Ej,

Universality of Bose–Einstein condensation and quenched formation dynamics  

Impulsive

! 

Symmetry Breaking

Symmetric State



 AdiabaƟc

)



97

AdiabaƟc



Quench Direction

) z  )

_) _

Fig. 6 Schematic of the “cartoon” Kibble–Zurek scenario, describing the phase transition from an incoherent to a coherent state through a dynamically induced linear crossing of the distance to criticality E(t ) in time. Shown in terms of E(t ) are the correlation time, which diverges from both sides as E(t ) ! 0 [dashed red line] and an instantaneous timescale E=E_ set by the rate at which the quench is induced. Regarding the latter, I note here that when plotting _ thus allowing the system to reach, in principle, arbitrarily close to the critical point before it can things in terms of E(t ), slower quenches have a steeper slope of jE=Ej, no longer adiabatically follow equilibrium states in its crossing through the phase transition. During its crossing, the system can only adiabatically follow _ otherwise, the characteristic correlation length setting the size of the domains becomes frozen to its corresponding equilibrium states for as long as jtj < jE=Ej; value, x^ ¼ xð^ EÞ evaluated at E^ ¼ Eð−^t Þ and the system behaves impulsively between −^t and ^t. The symmetry of the prequench (symmetric, hFi ¼ 0) and postquench (symmetry-broken, hFi 6¼ 0) equilibria of the system are depicted in terms of the thermodynamic potential with the radial direction depicting the absence (left) or existence (right) of a nonzero order parameter: in the latter case, and while the azimuthal angle in this “Mexican hat” potential can in principle take any value with equal probability, the merging dynamics of domains of spontaneously generated random phases ultimately leave the system at sufficiently large values of E > 0 (and in this case at sufficiently late, or asymptotically long, times such that all critical transient dynamics have died out) with a particular spontaneously chosen direction, or phase y (whose gradient sets the superfluid velocity).

to the external quench, and so different parts of the system necessarily evolve independently of each other, no longer controlled by the external drive (e.g., cooling ramp); in other words, the system is allowed to evolve independently on a local scale, and so the global symmetry is broken. This implies that the thermodynamic potential of the system transitions suddenly from a single minimum at the origin corresponding to the absence of any identified order parameter, that is, hFi ¼ 0, to a typical “Mexicanhat” potential with hFi 6¼ 0 (see Fig. 6); in such a potential, the radial location of the minima indicates a nonzero expectation value of the system order parameter F, which thus acquires a meaning on this side of the phase transition while the azimuthal direction contains information about the phase of F. Note that, at this stage, all minima in such a potential, corresponding to different azimuthal orientation (at fixed radial distance), become equally probable, that is, all local phase choices are equally likely; this in turn implies that the phase varies randomly (following a random walk) between different domains, the size of which is (typically much) smaller than the system size. In fact, the typical size of such domains is set by the equilibrium correlation length of the system evaluated at the distance to criticality E corresponding to the time when the symmetry is broken.

The Kibble–Zurek scaling law Having introduced the key concepts, one can now formulate the fundamental equations governing the Kibble–Zurek mechanism of quenched phase transitions. The underlying idea is to use the equilibrium behavior of the system near criticality to make specific predictions regarding the nonequilibrium consequences of the dynamics of symmetry breaking. Mathematically, this relies on _ [diagonal lines in Fig. 6] becomes equal to the identifying the characteristic time at which the instantaneous quench timescale jE=Ej diverging relaxation timescale t: such time, typically denoted by ^ t , signifies the so-called freeze-out time, that is, the time during the quench when the system behavior seizes being identified by parameters adiabatically following the equilibrium properties of the system as a whole (at a rate set by the quench rate). The typical size of subsequent independent domains—which have (in this simplistic scenario) no causal connection between them—is then set by the instantaneous value of the system correlation length at the freeze-out time. The discussion below follows closely the review articles (del Campo et al., 2013; del Campo and Zurek, 2014) based on the considerations by Kibble and Zurek. Mathematically, one equates the diverging relaxation time t, evaluated at the freeze-out time −^ t , to the time j −^ t j set by _ − ^t Þj. The fundamental Kibble–Zurek relation can thus be written as jEð −^t Þ=Eð tð−^ t Þ ¼ j −^ t j,

(19)

As the above equation simply labels the two points of intersection of the solid and dashed lines in Fig. 6, which are the same in this (presumed symmetric) schematic across both sides of the phase transition, one often quotes this as the more “memorable” and algebraically equivalent form tð^ tÞ ¼ ^ t . Since the quench rate E(t) depends on the quench timescale, tQ, then so do the abovementioned intersection points, thus characterizing the dependence of correlation time—and, by extension, of correlation length (during the quenched evolution)—on tQ. In its most basic (and most commonly presented) form (discussed here), the Kibble–Zurek scenario considers the system dynamics as being “frozen” between −^ t and + ^ t (i.e., when t > j^ t j ), with such temporal window—during which the system remains fragmented into a number of smaller domains, exhibiting no correlation between them—labeled as the “impulsive” regime. To understand the implications of such relation further, let us now consider a specific case—namely the example of a 3D homogeneous system, known to obey the equilibrium critical scalings of Eqs. (16) and (17). In that case, one can re-express Eq. (19) [or its equivalent relation tð^ tÞ ¼ ^ t ] as

98

Universality of Bose–Einstein condensation and quenched formation dynamics  −nz t^ ¼^ tð^ t Þ ¼ t0 jEð^ t Þj−nz ¼ t0 t: tQ

(20)

Such a relation allows us to obtain a characteristic scaling of the freeze-out time, ^t, in terms of tQ, with a corresponding relation for x^ obtained through ^ t ¼ x^z . Ignoring microscopic prefactors, which are not relevant for the universal system dynamics, one thus deduces that (for a homogeneous 3D system) nz=ð1+nzÞ ^ , t
tQ

(21)

−n n=ð1+nzÞ x^ ¼ xðEð^t ÞÞ ¼ x0 jEð^ t Þj
tQ :

(22)

and correspondingly

Although the Kibble–Zurek relation (Eq. 19) will always specify a characteristic scaling for ^t and x^ in terms of tQ around criticality, the exact form (and arising exponents) of such emerging relations will depend on the dimensionality and universality class of the system—with these results requiring revisiting in an inhomogeneous context (see later). ^ as the dominant relevant scales in the system in the critical region. As We have thus identified the timescale t^ and length scale x, such, within such limited regime, one can formulate the temporal and spatial dependence of physical variables in the system in a universal manner by a simple rescaling of the form Time

t ! t=^ t r ! r=x^

Distance Wavevector k ! kx^

Having identified the relevant scales in the system allows us to formulate the corresponding scaling hypothesis for the system correlations (applicable in the long-wavelength, low-frequency limit); in the discussion below I also ignore for simplicity the anomalous critical exponent (denoted by ), as its value is typically zero or very close to it (Nishimori and Ortiz, 2011; Binney et al., 1992). The Kibble–Zurek scaling hypothesis thus states that in the critical region ½ −^ t, ^ t , the correlation function can be cast in the form

 1 t r Cðr, tÞ
d−2 F , , (23) ^ t x^ x^ where F is some system-specific function, and d is the system dimensionality. In general, this implies that the correlation function is dynamically evolving in a manner that changes both its shape and its range. However, the discussion so far has been focused on the (adiabatic-impulsive-adiabatic) “cartoon” Kibble–Zurek scenario of Fig. 6; it is thus pertinent to note here that while such a “cartoon” picture is indeed consistent with the scaling function of Eq. (23), it nonetheless further constraints it by implying a particular (time-independent) form for F. Setting this aside for now, I note that the Kibble–Zurek relation of Eq. (19) enables us to predict the scaling of the number of defects spontaneously generated within such system during the linearly quenched phase transition crossing with the quench timescale tQ. Based on the typical size of independent domains x^ in the broken-symmetry phase (set by the equilibrium correlation length x evaluated at the freeze-out time −^ t ), and information about the system dimensionality, d, and defect dimensionality, D, this takes the form (del Campo et al., 2013)

D x^
ðtQ Þ−a , (24) ndefect
x^d where the exponent a > 0 in general depends on the critical exponents (n, z) and the system/defect dimensionalities (d, DÞ. For example, ðd − DÞ ¼ 1 for a 2D planar soliton in a 3D system or a point-like solitonic defect in a 1D system, while ðd − DÞ ¼ 2 in both cases of a linear vortex filament in a 3D system, or a point-like vortex defect in a 2D system, because two dimensions are needed to accommodate the phase profile around such a vortex. As expected, one finds that—compared to ^ lead to _ has a smaller slope, and thus yields a smaller value of x, slow quenches—faster quenches (i.e., small tQ) for which jE=Ej the generation of more defects; this is because more independent regions of smaller size can be accommodated in a given system size. In the specific 3D homogeneous example, a ¼ ðd − DÞn=ð1 + nzÞ, whereas this becomes modified in the presence of harmonic confinement (see later). The power of the presented “cartoon” Kibble–Zurek mechanism is that it predicts the correct scalings with tQ for a linearly quenched homogeneous system. Nonetheless, an immediate shortcoming of such adiabatic-impulsive-adiabatic scenario, which can be identified, is that it ignores the temporal growth of correlations within the critical region; moreover, realistic experiments (besides being performed in finite-size systems) are often conducted in spatially varying potentials, thus requiring the extension of such mechanism to inhomogeneous settings. Both these issues are addressed in Sections “Analyzing Kibble–Zurek in an elongated ultracold atomic system” and “Beyond homogeneous Kibble–Zurek considerations.” In the context of ultracold atom experiments, beyond the previously-mentioned ring-trap geometry (Corman et al., 2014; Aidelsburger et al., 2017), the Kibble–Zurek phenomenon across a thermal phase transition has also been studied in different

Universality of Bose–Einstein condensation and quenched formation dynamics

99

settings, including box-like geometries in 3D (Navon et al., 2015) and 2D (Chomaz et al., 2015), elongated 3D (Lamporesi et al., 2013; Donadello et al., 2016; Liu et al., 2018; Wolswijk et al., 2022) and oblate (pancake-like) (Goo et al., 2021, 2022; Kim et al., 2022) inhomogeneous traps, and even in the context of strongly interacting fermionic superfluids (Ko et al., 2019) (Tuning the interactions across the BEC-BCS crossover through the unitarity limit in a strongly-interacting superfluid revealed a near-constant, that is, interaction-strength-independent, critical exponent a.). The insightful experiment at Cambridge in a 3D box-like geometry (Navon et al., 2015) probed the system spatial correlation function at different times, by interfering the system with a displaced copy of itself: they used such measurements to explicitly demonstrate the existence of critical slowing down; by determining the scaling of the correlation length with tQ—and using the expected (and previously experimentally extracted (Donner et al., 2007) value of n  2/3 discussed earlier)—they were able to show that the dynamical critical exponent z ¼ 3/2, consistent with the expectation for the 3D XY model universality class. Many of these early ultracold experiments also probed the number of generated vortices during a controlled quench, and its dependence on the quench timescale tQ (Corman et al., 2014; Chomaz et al., 2015; Donadello et al., 2016) (see subsequent discussions for more recent works). On the whole, good overall agreement with the expected behavior was obtained for sufficiently slow quenches as long as other appropriate factors, such as dimensionality and type of confinement (e.g., harmonic, ring-trap), were taken into account; however, a common potential limitation in such cases arises from the fact that experimental defect counting is typically only possible at times significantly longer than + ^ t (which can generally be rather restrictive for most defect types due to their relatively short lifetimes), or after allowing the system to relax and/or expand for some time to make defect detection easier; such aspects are further discussed below.

Analyzing Kibble–Zurek in an elongated ultracold atomic system To gain further direct microscopic insight into the complex system dynamics over all timescales and physical observables, and probe the physics beyond the “cartoon” Kibble–Zurek picture presented so far, I briefly review here key results obtained in the context of a particular sequence of experiments conducted at Trento in an elongated 3D harmonic geometry (Lamporesi et al., 2013; Donadello et al., 2016; Liu et al., 2018; Wolswijk et al., 2022). In such experiments, the system was cooled through the phase transition at _ with approximately fixed initial and final states. different rates T, A schematic of such process—based on extensive numerical simulations (Liu et al., 2018) via Eq. (15)—is shown in Fig. 7. An initially incoherent gas is dynamically quenched through the critical point (which is itself shifted by the combination of fluctuations and mean-field effects). As predicted, the system reacts (“unfreezes” its dynamics) with a certain delay time + ^ t after crossing the equilibrium critical point, at which point the highly symmetry-broken state looks rather “chaotic”: here the randomly distributed and tangled purple lines of variable lengths and orientations depict spontaneously generated defects in the form of highly excited vortices in a strongly turbulent state. Notice that the defects only emerge within an elliptical volume—consistent with the

Delay Ɵme ~ t Equilibrium CriƟcal Point High Velocity field 3% peak density 0.1% peak density

24 Time Fig. 7 Schematic of entire condensate growth process during a driven cooling quench from an initial equilibrium thermal configuration: Shown are density isosurfaces of the most populated (Penrose–Onsager) mode based on numerical simulations of Eq. (15) for the parameters of a recent elongated 3D harmonic trap experiment of Lamporesi et al. (2013), Donadello et al. (2016), as discussed in Liu et al. (2018). After the characteristic Kibble–Zurek delay time of
^t, one sees the spontaneous emergence of a “turbulent” quasi-condensate state (green) containing a large number of spontaneously generated defects (shown in purple) whose density decreases as one moves away from the trap center: these tangled defects gradually decay to a small number of well-formed vortices which are rather long-lived and become trapped in the growing macroscopically occupied condensate mode. During such evolution, the relative weight of the dominant eigenvalue (compared to the next largest one) increases significantly, revealing clear evidence of the transition from a quasi-condensate exhibiting multiple equally likely populated modes (around ^t) to a pure BEC (with one dominant eigenvalue) in the latter density snapshots, in which the relative volume occupied by the defects has significantly decreased.

100

Universality of Bose–Einstein condensation and quenched formation dynamics

Fig. 8 Numerical demonstration of the Kibble–Zurek scaling hypothesis revealing self-similar evolution for the parameters of Fig. 7 in terms of appropriately rescaled (i) zero-momentum mode occupation f(k ¼ 0), and (ii) coherence length lcoh for a range of different tQ. Main plots show rescaled evolution of insets, with time measured in terms of (t − tc) (where tc ¼ t(E ¼ 0)) and scaled to ^t. Rescaled vertical axes respectively plot (i) ðt0 =^tÞfðk ¼ 0Þ (where t0 is a nonuniversal ^ In the unscaled insets (plotted in terms of (t − tc), the value of the quench timescale tQ decreases from the lower (slower quench, constant time) and (ii) l coh =x. green) to the upper (faster quench, red) curves. Adapted with permission from Liu I-K, Dziarmaga J, Gou S-C, Dalfovo F, and Proukakis NP (2020) Kibble-Zurek dynamics in a trapped ultracold Bose gas. Physical Review Research 2: 033183. https://doi.org/10.1103/PhysRevResearch.2.033183. Copyright (2020) by the American Physical Society.

imposed anisotropic harmonic confinement—and that the defect density is highest at the trap center, where the particle density is also highest: I will comment later on the significance of such observations. Based on the Kibble–Zurek scaling hypothesis, all system observables become universal during the time interval t 2 ½−^ t, ^ t , thus uniquely specifying the temporal window during which one should be investigating universal Kibble–Zurek scaling effects for _ constant-rate cooling quenches tQ ¼ T c =jTj. The emergence of such universal features can be seen in Fig. 8 which showcases the unscaled (insets) and appropriately rescaled dynamical evolution of (i) the zero-momentum mode occupation f(k ¼ 0) [left], and (ii) the evolving system correlation length lcoh [right]. Unscaled images are plotted in terms of the time (t − tc) lapsed since crossing the equilibrium phase transition, whereas such shifted time has been additionally scaled by ^ t in the rescaled plots [main plots]. Moreover, in the rescaled plots, the vertical-axis ^ These simulations (Liu et al., 2020) reveal universal features up t f ðk ¼ 0Þ and lcoh =x. variables have been respectively rescaled as t0 =^ to a timescale of
^ t after the system crosses the equilibrium critical point. Although a delayed response was also found in more recent experiments with the same system, at such early times there was neither direct access to the spatially resolved density distribution nor the vortex number, with such delay instead attributed to other factors masking the ability to extract ^ t directly (Wolswijk et al., 2022). As the system is further cooled (i.e., E > 0 increases), one obtains a clear glimpse of its quasi-condensate characteristics: in particular, one sees the emergence of scattered localized regions of randomly selected near-constant phase (and so high coherence locally), separated by chaotically evolving and highly excited defects (vortices): at this stage there are many highly occupied modes competing against each other, and such information is revealed by the study of the largest, approximately equally dominant, eigenvalues (Liu et al., 2018)—providing direct evidence of quasi-condensation. The state of the system can also be thought of as “turbulent” (in the broad interpretation of the word). As time evolves, and the system is further externally cooled (at the same constant rate), the random quasi-coherent regions grow in size, with the defects separating such regions interacting with other defects, thus dissipating their energy and shrinking in size. Naturally, the process of defect generation through the symmetry-breaking process and defect decay through coarsening coexist temporally, making it hard to isolate the relative importance of such competing mechanisms. However, at sufficiently late times after the dynamical phase transition crossing (i.e., for E(t) 0), and within a specific temporal window, such dynamics are generally expected to exhibit a new set of self-similar (universal) scaling behavior associated with the phase-ordering process (Bray, 1994; Damle et al., 1996). The complicated inhomogeneous and anharmonic elongated 3D geometry of this system largely conceals such features, which are therefore discussed in more detail later on in the “cleaner” context of a homogeneous 2D quantum gas in Section “Scaling hypothesis and late-time coarsening.” The net effect of this lengthy process is the gradual establishment of domains of increased volume and coherence, and thus the gradual transition from a state of “quasi-condensation” to that of true condensation, with properties set by the external system parameters (T and m). During such dynamics, the tight transverse direction rapidly constrains the system evolution, and dictates the specific type of defect preferentially forming in such a system; this corresponds to the macroscopic excitation with the lowest energy in the given geometry (In this case, such defect is a so-called “solitonic vortex” which exhibits a 2p phase winding characteristic of quantized vorticity, but with a nonuniform gradient, such that its phase profile resembles the characteristic planar jump of a dark soliton from larger distances (Donadello et al., 2014).). As the defect density gradually decreases, one sees the emergence of a phase-coherent condensate within which are embedded few long-lived defects; this is in fact a very clear dynamical demonstration of the trapping of defects emerging as relics of the phase transition in the growing highly coherent system. The particular geometrical set-up constrains their interactions such that they only interact rather infrequently when few defects remain. However, when they do, the defects can reconnect and dissipate away further energy (Serafini et al., 2017). After significant such evolution, the system eventually expels all of its defects: at this point, the competition between the various quasi-condensate modes is complete, and one arrives at a fully

Universality of Bose–Einstein condensation and quenched formation dynamics

−

/2

101

/2

=0 Quenched EvoluƟon

Post-Quench EvoluƟon

Fig. 9 Dependence of dynamical growth on quench rate for the parameters of Fig. 7 based on characteristic single-trajectory temporal evolution from a fixed initial (thermal) to a fixed final state (with a
75% BEC fraction) for three different values of tQ: schematic at the top shows symmetric quench protocol from a m < 0, T > Tc initial thermal state to one with m > 0, T < Tc indicating corresponding temperatures and atom numbers characteristic of such controlled quench experiment. The quench rate increases from quasi-adiabatic (top) to fast (bottom), with middle trajectory showcasing a typical case where Kibble–Zurek scaling is expected to be fully applicable. Remarkably, faster quenches (bottom) contain (on average) more defects at a very late evolution time that slower quenches. Adapted with permission from Liu I-K, Donadello S, Lamporesi G, Ferrari G, Gou S-C, Dalfovo F, and Proukakis NP (2018) Dynamical equilibration across a quenched phase transition in a trapped quantum gas. Communications Physics 1(1): 24. ISSN 2399-3650. https://doi.org/10.1038/s42005-018-0023-6. Copyright (2018) by the Nature Publishing Group

phase-coherent Bose–Einstein condensate. Such a transition is further confirmed by a study of the ratio of the largest to secondlargest eigenvalue of the system as a function of time (Liu et al., 2018, 2020). Density profiles (green) in Figs. 7 and 9 showcase the evolution of the spatial distribution of the most populated eigenstate, even though one only becomes genuinely dominant (the Penrose–Onsager mode) at late times. Obviously, details of such process are heavily dependent on the quench protocol (e.g., temperature quench, interaction quench, geometry/phase-space quench), system geometry and dimensionality and interaction strength (which respectively set the typical number, type, and size of the defects). A characteristic illustration of how the quenching rate (which for given initial and final states is simply a function of the inverse quench timescale tQ) affects the dynamics can be found in Fig. 9, which displays the system evolution over an extended fixed total evolution time for three different quench rates (Liu et al., 2018). As evident, more defects are generated on average early on for more rapid quenches (bottom); moreover, because of this—and despite rapid defect annihilation and expulsion—typically more defects remain visible at a later absolute time when performing rapid cooling (i.e., for systems that cross the equilibrium critical point E ¼ 0 faster), than in more gradual cooling schemes. For completeness, I note that while the example trajectories chosen to be displayed here are characteristic of the average behavior, each single cooling realization (for a fixed value of tQ) can lead to some variation in the number (and distribution) of generated defects; therefore, one needs to perform a number of distinct numerical/experimental realizations to appropriately extract relevant averaged parameters (e.g., defect number evolution). Performing such averaging over different realizations, both experiments (Donadello et al., 2016) and simulations (Liu et al., 2018) found a power law decay in the defect number with tQ, with an exponent broadly consistent with the analytically predicted exponent a for this setting (del Campo and Zurek, 2014); while such features/measurements were only experimentally accessible at rather late evolution times, much exceeding ^ t , numerical simulations demonstrated that the actual overall scaling behavior was rather insensitive to evolution time; this is presumably because after defects have formed, they primarily propagate “freely” in the confining potentials for significant fractions of time, thus being rather long-lived; this is particularly true when the number of defects is not too large and they are spread over a large volume, as is the case for not-very-fast quenches. Importantly—and rather t Þ): this arises when the maximum intuitively—such Kibble–Zurek scaling was not however seen for very fast quenches (tQ
Oð^ number of defects that can be accommodated in the system volume (or the maximum number that can be experimentally, or numerically, detected) has been reached in a given geometry, thus giving rise to a plateau in the defect number for small tQ; this was

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clearly observed experimentally (at late times) (Donadello et al., 2016), and found to be consistent both with a simple model and with numerical simulations (Liu et al., 2018). We also note that the decay of fluctuations was found to obey a universal power law, with a distinct exponent to that corresponding to the growth of condensation (Liu et al., 2018; Wolswijk et al., 2022). The quenching of defect number for very rapid quenches (i.e., the deviation from the Kibble–Zurek scaling law ndefect
(tQ)−a) was also observed in strongly interacting fermionic superfluids (Ko et al., 2019) and in more recent bosonic experiments in an oblate geometry (Goo et al., 2021, 2022; Kim et al., 2022). The latter series of experiments also reported observational evidence for universal coarsening dynamics during the early stages of defect formation. Moreover, they were able to observe spatial variations in the generated defect number: both these features are in qualitative agreement with the presented numerical simulations (Liu et al., 2018). All above features indicate that—rather unsurprisingly—there is more to driven phase transition crossing than the simple (but still very elegant and powerful) adiabatic-impulsive-adiabatic Kibble–Zurek model of Fig. 6. Below I address more systematic extensions of such a model, as highly relevant in light of such recent experimental developments.

Beyond homogeneous Kibble–Zurek considerations The preceding discussion has highlighted two limitations of the adiabatic-impulsive-adiabatic Kibble–Zurek scenario of Fig. 6 which I address here. The first one, concerns the “freezing” of the correlation length x during the entire evolution t 2 ½ −^ t, ^ t , to a constant (timeindependent) value corresponding to its instantaneous value at the time −^ t when symmetry-breaking occurs; in other words x ! x^ ¼ xðEð−^t ÞÞ, which would imply a constant size of domains during ½ −^ t, ^ t , which is somewhat counter-intuitive. In fact, during such a temporal period, causality implies the existence of a characteristic velocity for the growth of correlations which—in this context—can be defined as (Dziarmaga et al., 1999; Zurek, 2009; Sadhukhan et al., 2020) x^ v^ ¼ : ^ t

(25)

Given that the typical locally coherent regions grow with such a velocity even during t 2 ½ −^ t, ^ t , the emerging effective size of domains at ^t is in fact larger than previously suggested, with xð+ ^ t Þ typically a few times its corresponding value at −^ t (Sadhukhan et al., 2020); a schematic of such enhanced picture is shown in Fig. 10. Although such considerations evidently decrease the total number of spontaneously generated defects, rather remarkably they do not actually invalidate the previous arguments in terms of the existence of scalings of the defect number with quench timescale (Eq. 24), provided of course that the system inhomogeneity determining the shape of the region of increasing density is appropriately accounted for in the exponent a. In this context, it should be noted that the “cartoon” Kibble–Zurek scenario discussed in Section “The Kibble–Zurek scaling law” (Fig. 6) was formulated for a homogeneous system and thus assumed that the system reacts, in a statistical sense, in the same way irrespective of which part of the system one looks at. However, as clearly visible from Fig. 7—and commented upon earlier—the density is highest at the center of a harmonically confined gas, implying that coherence grows from the trap center radially outwards. As a result, both the critical temperature of the system and hence the distance to criticality E become position-dependent (Zurek, 2009; del Campo et al., 2011, 2013; del Campo and Zurek, 2014), that is, Tc ! Tc(r) and E(t) ! E(r, t). A natural question then arises, as to the extent to which the previous “cartoon” picture holds under inhomogeneous trapping, and how to appropriately generalize this.

^ ^t. Unlike the “cartoon” Fig. 10 Extended view of the Kibble–Zurek mechanism explicitly accounting for a sonic horizon propagation with a critical speed of x= ^ ^ regime of impulsive dynamics for t 2 ½−t, t shown in Fig. 6, during which the correlation length is assumed to remain frozen to its value at −^t, this improved picture allows the “sonic cone” to grow during such time window by a constant rate set by the instantaneous slope of xð−^tÞ=^t. As a result, the typical radius of the ^ Reprinted with permission from Sadhukhan D, Sinha A, Francuz A, Stefaniak J, Rams MM, region over which broken symmetry emerges grows from x^ to few x. Dziarmaga J, and Zurek WH (2020) Sonic horizons and causality in phase transition dynamics. Physical Review B 101: 144429. https://doi.org/10.1103/PhysRevB. 101.144429. Copyright (2020) by the American Physical Society.

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In the context of the homogeneous Gross–Pitaevskii equation (with repulsive interactions g > 0), symmetry-breaking is induced simultaneously across the entire system when the chemical potential transitions from negative values (corresponding to a purely thermal cloud) to positive values. Due to the presence of the trap, one can instead define an effective spatially dependent chemical potential m(r, t) ¼ (m(t) − V (r)); thus outlining at any given time the region with m(r, t) > 0 where symmetry-breaking takes place. Such region grows radially outwards from the trap center (here in an elliptical manner), thus defining a phase transition front which propagates in the system with a characteristic speed

 T ð0Þ dT c ðrÞ : (26) vF ðrÞ ¼ c tQ dr As such, the simultaneous emergence of symmetry breaking in a homogeneous system can be described in terms of an infinite phase transition front velocity vF, and this is also (locally) the case at the center of a harmonic trap where local evolution mimics that of a homogeneous system. The extent to which the emerging symmetry-breaking features in the phase transition of a system have a global or a local character thus depends on the competition between the characteristic velocity v^ for the growth of correlations set by causality and the velocity vF for the growth of the critical front as determined by the system geometry, thus also accounting for the shape of the growing volume of the coherent density front. The generalization of the Kibble–Zurek effect, which accounts for such spatial variations and local features, is known as the inhomogeneous Kibble–Zurek mechanism, and is extensively discussed in Zurek (2009), del Campo et al. (2011), del Campo et al. (2013), del Campo and Zurek (2014), Sadhukhan et al. (2020), and Dziarmaga et al. (1999). As a result, the previously discussed “cartoon” picture of Section “The Kibble–Zurek scaling law,” known as the homogeneous Kibble–Zurek picture remains qualitatively valid as long as (and within the regions where) vF > v^,

(27)

that is, the growth of the critical front happens on a faster rate than such causality-induced coherence growth associated with the sonic horizon. In such a regime, the phase of each newly formed domain is chosen locally. As explained above, this is always the case in the center of the harmonic trap, and in a region near the propagating front (due to critical slowing down) (del Campo et al., 2013). As a result, defects can in general only form spontaneously over a restricted volume in an inhomogeneous system, thus leading to a reduced number of emerging defects in comparison to the homogeneous predictions. In the opposite limit of vF < v^, the growing condensate can rapidly communicate its choice of phase to the neighboring domains, and so no further defect formation can occur. For an evolving system to enter a regime where it can probe its true inhomogeneous nature, the sonic horizon should be growing faster than the evolution of the critical front. This, in turn, places constraints on the values of tQ required in a given system for a quench to allow the inhomogeneous nature of the phase transition to be fully probed; applying such simple arguments to the above elongated 3D experiment suggested the need for significantly slower quenches, thus also requiring longer evolution times and enhanced system stability. Although such a regime was not reached in the experiments of Donadello et al. (2016), it should nonetheless be noted that (i) the corresponding numerical simulations discussed in Section “Analyzing Kibble–Zurek in an elongated ultracold atomic system” clearly demonstrated the spatial dependence of spontaneous defect generation over a restricted volume, and that (ii) both observed and numerically computed scalings of the defect density with tQ were found to be consistent with predictions based on critical exponents for a harmonic trap (Donadello et al., 2016; Liu et al., 2018) (leading to an amended power a ¼ ðd − DÞð1 + 2nÞ=ð1 + nzÞ (del Campo et al., 2011; del Campo and Zurek, 2014) when such geometrical features of the growth were taken into account). Furthermore, it has been noted that a quasi-2D geometry may be a more favorable setting for observing such spatial signatures of the inhomogeneous nature of the dynamical phase transition. This was indeed shown in very recent experiments in a quasi-2D geometry; specifically, following experiments highlighting the anticipated importance of coarsening dynamics during the early stages of spontaneous defect formation (Goo et al., 2021, 2022) observations of the spatial distribution of defects in such a geometry found (after allowing the system to relax further and expand) a local suppression of spontaneous defect generation with tQ, which was more pronounced in the outer region (Kim et al., 2022). In other words, the local value of the power law exponent −aðrÞ a(r) dictating the defect number through the power law ndefect
tQ was inferred after expansion to increase locally as one moves away from the trap center. While such an observation is consistent with the interplay of causality and inhomogeneity, they consistently found higher exponents than might be anticipated. Although various factors could be at play here, associated with the fact that observations were made after relaxation and expansion (and so not at ^ t ), the ongoing defect dynamics and coarsening, and intricate details of the trapping potential; this does also pose interesting questions meriting further investigation regarding the applicability of the inhomogeneous Kibble–Zurek effect. Irrespective of the precise details just discussed of corrections to the basic Kibble–Zurek scheme, which allow for evolution of the characteristic lengthscale and the interplay between inhomogeneity and causality, as the system exits the critical region in the E > 0 side of the phase transition at t ¼ +^ t, it will be left in a rather nonequilibrium state (the faster the quench, the more nonequilibrium the state is likely to be—see Fig. 9); such a nonequilibrium state features a random mosaic of domains of of different, near-constant, phase, separated by defects of variable length, orientation, and curvature, in a way which accounts for the differences in phase between the establishing micro-domains. The relaxation of such a state through the gradual decay of the number/length of defects will ultimately lead to the final coherent state for the given system parameters. Here, one can distinguish two cases: Either one continues driving the system to the final desired equilibrium and studies its relaxation in an “open” (driven) environment or one

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leaves the system in such a highly chaotic/turbulent state and observes its dynamical evolution toward relaxation in a “closed” manner: the fundamental dynamics governing these two cases are different. Nonetheless, in both cases, universal laws can (and do, in general) emerge in certain temporal (and momentum) windows. Below I discuss the general concept of self-ordering by means of a scaling hypothesis, paying attention to nonequilibrium features, particularly in terms of universal late-time dynamics and other more general nonequilibrium features facilitating a bidirectional dynamic scaling, both of which can leave a significant imprint in the decay of superfluid turbulence. Although very distinct dynamical regimes with different scaling laws, the dynamically evolving x^ discussed above does bear some notional analogy to the dynamically growing lengthscale Lc(t) emerging in the scaling regime discussed below.

Universality during phase-ordering dynamics So far, I have discussed the role of finite-duration quenches in a system, focusing on an incoherent to coherent thermal phase transition. We have discussed a broad dynamical range from quasi-adiabatic to near-instantaneous transitions, and explained how universal Kibble–Zurek dynamics emerges in the critical region provided the quench is not too fast. As discussed, the relaxation process in the system is governed by the decay of defects giving rise to growing regions of constant phase, with the quasi-condensate giving way to a macroscopic condensate (Witkowska et al., 2011; Liu et al., 2018), a process known as phase-ordering. In order to present the key features of such process in its most pure form, it is convenient to consider the idealized limit of instantaneous quenches. Moreover, based on the above findings on the role of inhomogeneous trapping, and in order to suppress transversal (e.g., Kelvin-wave-like) excitations on the defects, I initially restrict the discussion to two-dimensional homogeneous systems, before returning to more general nonequilibrium settings.

Scaling hypothesis and late-time coarsening The dynamics through which the system expels all defects and orders its phase is known as phase-ordering kinetics, and has a long history. Here I do not attempt to give a full account—referring the reader to excellent reviews (Bray, 1994; Rutenberg and Bray, 1995b, a)—but rather to discuss its features in the specific context of quenched quantum gases. Naturally, even when quenched instantaneously, the system requires some time to order itself: as argued before, defects—which emerge at the interfaces across regions of different phases—gradually disappear, thus allowing adjacent regions of particular (randomly chosen) phases to merge, leading to the temporal growth of the typical domain size. A key point to note here is that this is also a scaling phenomenon. From a statistical point of view, the distribution of such domains—each with a constant phase, yet no phase coherence between them—looks at late times remarkably similar to the one at earlier times, with the growth of the typical domain size fully accommodated through a (simple) global change of scale. We recall at this stage that a dynamical scaling hypothesis was previously considered in the context of the Kibble–Zurek scenario [Eq. (23), Section “The Kibble–Zurek scaling law”]. The scaling function was generally argued to change in time both in terms of its shape (through its explicit dependence on t=^ t ) and its spatial extent. In stark contrast to this, I note here that during the late-time self-similar dynamics and in the scaling limit r x, the shape of the correlation function remains unchanged (i.e., there is no explicit dependence on time), with its spatial range changing in time only through a dynamically growing domain size Lc(t), that is, through a dependence of the form r/Lc(t); such an increase in the characteristic length scale is consistent with the number of independent domains and defect number decreasing. The existence of a single (dynamical) dominant lengthscale in the system allows us to formulate the correlation function in the form (Bray, 1994)

 r (28) Cðr, tÞ
F Lc ðtÞ with the corresponding momentum distributions taking the form nðk, tÞ
½Lc ðtÞd GðkLc ðtÞÞ:

(29)

Here, F is an (unspecified) universal function and G is its Fourier transform, with temporal variation fully accounted for by the change in scale. To calculate how such a characteristic lengthscale varies for a nonconserved field (i.e., in the presence of external dissipation), one can utilize energetic considerations, taking into account the system symmetries and the fact that the defects are not isolated (thus introducing a lengthscale beyond which a defect becomes screened by other defects). By estimating a critical velocity for the growth of such domains u ¼ dLc(t)/dt and integrating, one can obtain the functional form of Lc(t), as explicitly discussed in Bray (1994). This is generally found to exhibit a power law scaling of the form Lc(t)
tb, with the critical exponent b related to the dynamical critical exponent z through b ¼ 1/z. We note that in the special case of point vortices in a purely 2D geometry the presence of external dissipation (e.g., through coupling the system to an external bath) leads to well-known logarithmic corrections such that Lc(t)
(t/log(t))1/z.

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105

The topic of phase-ordering has been extensively discussed in the literature (see, e.g., Yurke et al., 1993; Rutenberg and Bray, 1995a; Damle et al., 1996; Jelic and Cugliandolo, 2011; Nam et al., 2012; Karl and Gasenzer, 2017; Groszek et al., 2021). To make this point clear, I briefly discuss here recent extended numerical simulations (Groszek et al., 2021) in a 2D box with periodic boundary conditions in the context of both the conservative and the dissipative Gross–Pitaevskii equations, respectively via Eqs. (14) and (15). In both cases, phase-ordering during the postquench evolution takes place through the annihilation of vortex-antivortex pairs (white/black squares in temporal evolution snapshots shown in Fig. 11A–C). Spatial correlations at a general time t are probed in ð1Þ terms of the equilibrium correlation function geq ðr, tÞ—the latter obtained after very lengthy numerical propagation—through the generic relation

 gð1Þ ðr, tÞ r ¼F : (30) ð1Þ Lc ðtÞ geq ðr, tÞ Correlation functions in terms of the scaled time-dependent variable r/Lc(t) are found to collapse perfectly onto a single function (Fig. 11D) within a certain time window, which thus identifies both the evolutionary stage when the scaling hypothesis (Eq. 28) holds and the manner in which Lc(t) grows, directly confirming an evolution of the form Lc(t)
tb. Correspondingly, directly counting the decay in the vortex number, one confirms a scaling for the vortex density of the form nV
1=L2c
t −2b . Identifying b ¼ 1/z allows the extraction of the dynamical critical exponent z through these fits during the scaling interval. The behavior shown in Fig. 11 is generic, broadly applicable to both the conservative and the dissipative Gross–Pitaevskii equations. Nonetheless, a detailed numerical comparison of the predictions between the conserved and dissipative regimes leads to slightly different behavior and modifications in the precise value of the critical exponent. This can be traced back to the fundamental question of “isolated” (conservative) vs. ‘open’ (dissipative) quantum systems (Hohenberg and Halperin, 1977); in the purely dissipative limit (i.e., Eq. (15) without the 1 in (1 − ig)), the existence of vortices leads to logarithmic corrections to the system dynamics, such that Lc(t)
(t/log(t))1/z with z ¼ 2 exactly, and such behavior is explicitly reproduced in the simulations [see also subsequent discussion in the context of instantaneously quenched driven-dissipative exciton-polariton condensate systems (Section “Driven-dissipative condensation in exciton-polariton systems”)]. However, in the purely conservative limit (and without

Fig. 11 Phase-ordering kinetics of a 2D homogeneous Bose gas following an instantaneous quench across the BKT phase transition: (A)–(C) Evolutionary snapshots of the phase, showing the reduction in the number of vortices/antivortices (white/black squares respectively). (D) Self-similarity through the collapse (main plot) of the spatial correlation functions evaluated at different times t within the “collapse window,” scaled to ħ=m (inset) onto a universal scaling function F in terms of the scaled variable r/Lc(t ) according to Eq. (28). (E) Postquench growth of the characteristic lengthscale Lc(t ), clearly highlighting the time window when all self-similar correlation functions perfectly collapse onto one another. (F) Corresponding decreasing evolution of the vortex number density nV. Although both (E) and (F) plot the variation across the entire temporal domain, note that the very late-time evolution is somewhat sensitive to numerical boundary effects (as can be seen from the slight deviation of both behaviors at very late times). Adapted with permission from Groszek AJ, Comaron P, Proukakis NP, and Billam TP (2021) Crossover in the dynamical critical exponent of a quenched two-dimensional Bose gas. Physical Review Research 3: 013212. https://doi.org/10.1103/ PhysRevResearch. (3.013212). Copyright (2021) by the American Physical Society.

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including such logarithmic corrections), the value of z was found to lie approximately 15–20% below the z ¼ 2 value (Groszek et al., 2021). While a possible explanation for such a change in z is likely to arise from a power-law vortex mobility due to vortex-sound interactions, a detailed understanding of such subtleties (as well as its extension to inhomogeneous systems) is still lacking.

Evolution around nonthermal fixed points At this point, the reader is briefly reminded (in rather basic terms) of the notion of “fixed points” in renormalization group theory—for more details the reader is referred to Nishimori and Ortiz (2011), Binney et al. (1992), and Bruce and Wallace (1992). In such a language, one typically scales out microscopic details in favor of global features of macroscopic observables; as such, one deals with a parameter space of available many-body configurations of the system expressed in terms of a limiting set of relevant “coupling” parameters: long-time evolution between possible many-body system configurations is then modeled through flowing trajectories within such space, in a manner governed by the system Hamiltonian. Such a picture can be characterized by a small number of fixed points, which define the nature of the system evolution for different configurations. Specifically, flow around (toward/away from) such points qualitatively indicates the system evolution in parameter space. Naively, one can think of a fixed point as arising when a change of the scaling parameter, for example, Lc(t), leaves the correlation function unchanged (Binney et al., 1992). For example, for a system undergoing a single-phase transition, one can directly identify certain fixed points: in the common case of a thermal phase transition, one identifies a low-temperature fixed point (denoting the system equilibrium at T ¼ 0, e.g., in the fully coherent state, or more generally in the E ! +1 limit) and a high-temperature fixed point (denoting the system in the T !1, or E !−1 limit); ultimately the system flows to either of those, with all equilibria below/above the critical point formally mapping onto the former (condensed), or latter (thermal) by an appropriate rescaling of relevant parameters. Such trajectories are separated by a critical line/surface, which contains a “mixed” critical fixed point, such that it is attractive along all the directions within the critical surface, and repulsive along those pointing away from the critical surface (Binney et al., 1992). Note that exactly at the critical point, the system exhibits fluctuations at all lengthscales, making it impossible to eliminate (as argued above) short-scale behavior through a sequence of transformations, thus implying that a critical point is a fixed point. More generally, a fixed point is characterized by a pure scaling form and a coupling which no longer changes under a rescaling. The previously discussed phaseordering dynamics is a simple example of a scaling phenomenon controlled by a (strong coupling) fixed point (Bray, 1994). In general, various different classes of fixed points may exist. In particular, some attention is currently being devoted to the context of so-called ‘nonthermal’ fixed points, which arise as a fixed point in a typical far-from-equilibrium scenario: as such, they can be thought of as nonequilibrium configurations exhibiting scaling in both space and time, in terms of a limited number of properties. These were originally introduced by Berges and coworkers in the context of reheating after early-universe inflation (Berges et al., 2008; Berges and Hoffmeister, 2009; Piñeiro Orioli et al., 2015; Berges, 2016), and have been increasingly investigated across different physical systems, including heavy-ion collisions (Berges et al., 2015, 2021) and quantum gases (Nowak et al., 2011, 2012, 2014; Karl and Gasenzer, 2017; Chantesana et al., 2019; Schmied et al., 2019), also finding applications in a range of experiments (Prüfer et al., 2018; Erne et al., 2018; Glidden et al., 2021; García-Orozco et al., 2022; Madeira and Bagnato, 2022). The emergence of such points builds upon the concepts of self-similarity and critical behavior, generalizing them to far from equilibrium dynamics. Under appropriate conditions, a system may spend a significant amount of time around such a nonthermal fixed point. The discussion below is largely based on recent reviews (Chantesana et al., 2019; Schmied et al., 2019; Madeira and Bagnato, 2022). In a nutshell, the dynamics of a system in the vicinity of a nonthermal fixed point acquires a generalized evolution according to !

a

b t t n k, t 0 , (31) nðk, tÞ ¼ t0 t0 where a and b are two scaling parameters which are, in general, distinct and which, combined, identify the universality class of the system. The relation between a and b is in fact set by the relevant conservation laws of the system. Essentially, these define—within a broad momentum range—how particles are redistributed in the self-similar regimes, and relate the scaling exponents a and b. In the simplest case of particle number conservation (and for a particle with quadratic dispersion relation), this leads to a ¼ bd (in the long-wavelength, or infrared, regime), where d is the system dimensionality. Further noting that the characteristic lengthscale Lc(t)  tb, and b ¼ 1/z practically reduces such expression to that of Eq. (29) governing the self-similar phase-ordering kinetics previously discussed. As such the preceding discussion of Section “Scaling hypothesis and late-time coarsening” can be seen as a special case of nonthermal fixed points under total particle number conservation. However, the concept of nonthermal fixed points is more general. In particular, I note that the scaling function (Eq. 31) can behave differently in different momentum limits, namely, the previously discussed “inverse” particle transfer toward lower momenta arises from the conservation of particle number in the infrared (low-momentum) limit. Scalings due to energy conservation instead affect the ultraviolet (high-momentum) limit, leading to energy transport toward higher momenta and a ¼ (d + z)b. As a result, evolution around a nonthermal fixed point leads to a bidirectional flow (see Fig. 12). As a highly nonequilibrium feature, it is not surprising that the potential emergence of a nonthermal fixed point following a thermal quench would require a rather specific engineered nonequilibrium set of initial conditions. In this context, I note the seminal numerical work of Berloff and Svistunov (2002). They considered a quench in a 3D box, starting with a strongly P nonequilibrium initial condition (Svistunov, 1991; Kagan et al., 1992; Kagan and Svistunov, 1994, 1997) F(r, t ¼ 0) ¼ k akeik r, for which the magnitudes of the complex amplitude ak were obtained from the self-similar solution of the (wave/semiclassical)

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107

Fig. 12 Schematic of the bidirectional flow exhibited by the spectrum n(k, t) vs. k [shown here in terms of p ¼ ħk] around a nonthermal fixed point: A highly nonequilibrium initial distribution with equal population across all modes up to a cutoff, and random phases—shown here by the dashed red lines—evolves in a manner such that particles flow in an inverse manner toward lower k (through conservation of particle number: a, b > 0), while energy flows toward higher k (a, b < 0), carried by few particles in a manner which conserves total system energy. Reprinted with permission from Chantesana I, Orioli AP, and Gasenzer T (2019) Kinetic theory of nonthermal fixed points in a Bose gas. Physical Review A 99(4): 043620. https://doi.org/10.1103/PhysRevA.99.043620. Copyright (2019) by the American Physical Society

Boltzmann equation describing the system dynamics sufficiently before the critical phase transition region, and their phases were uniformly distributed between 0 and 2p. Evolving by means of the classical field Eq. (14), they observed, as would be expected, particle build-up toward lower momenta, with most particles restricted to modes below a certain cutoff. This is a direct indication of the gradual build-up of a quasicondensate, arising from the macroscopic occupation of, and competition between, numerous lowmomentum modes. Significantly, however, filtering out higher momentum modes revealed a tangle of vortices, a characteristic of a “strongly” turbulent state. As the governing equation (Eq. 14) is conservative, this tangle was found to gradually decay, following the relaxation process of superfluid turbulence in a nondriven system through the generation of sound waves and shrinking/annihilation of defects via a typical energy cascade. Similar features, revealing also information about the vortex structure and its decay products, were found when doing a limited time averaging, thus connecting to the previous discussion on the application of short-time averaging during evolutionary dynamics (Section “Penrose–Onsager condensation vs. quasi-condensation”). We also note here that probing the state of an initially nonequilibrium system after very long classical field evolution in a separate numerical work revealed the existence of vortex tangles, with the total vortex linelength increasing significantly as the system energy, and thus temperature, was increased (Davis et al., 2002). Considering a similar scenario, Nowak et al. (2014) studied the system evolution following a sudden thermal quench by removing high-momentum particles from the distribution. When a relatively small amount of energy was removed through the quench, they also found the system to evolve toward a coherent condensate, with clear evidence of a tangle of vortices after filtering short-wavelength fluctuations. However, when a stronger quench was performed by instantaneously removing all the particles above some momentum cutoff value, they found a long-standing strong wave turbulence inverse cascade toward lower energies, with a rather distinct power law power spectrum n(k)
k−5; such behavior, which is consistent with so-called “Porod” high-momentum tails n(k)
1/kd+2 (Schmied et al., 2019; Bray, 1994) corresponds to the case of randomly distributed vortices at momenta exceeding the inverse lengthscale corresponding to the distance between defects, a feature that is a direct consequence of ^ around a vortex line. Nonetheless, excitations in the high-momentum part of the spectrum still the velocity field (v
ð1=rÞ f) revealed n(k)
k−2, leading to a bimodal spectrum. Importantly, the above-discussed features were now found to become observable without the need to filter out short-wavelength fluctuations. Ultimately, the overall momentum distribution takes the form n(k)
k−2, leading to the relaxation of the system—but this now happens on a much longer timescale. This was considered as evidence of the system spending significant time around a nonthermal fixed point. This example clearly shows the distinct features that a thermal phase transition can exhibit depending on the nature and details of the strength of the cooling quench, that is, in this case on the initial nonequilibrium conditions for the classical field evolution. Further evidence of the emergence and decay of the arising vortex tangle and properties of the order parameter were also discussed in Kobayashi and Cugliandolo (2016b), Kobayashi and Cugliandolo (2016a), and Stagg et al. (2016). Dynamical properties of evolution consistent with Eq. (31) were recently observed in experiments in an isolated quench-cooled atomic gas (Glidden et al., 2021). By removing a significant fraction of the atoms and energy of a system initially just above the critical temperature and doing this process with interactions switched off, the experiment was able to generate a highly nonequilibrium state, and observe its relaxational dynamics, thus verifying the expected distinct scaling relations across the ultraviolet and infrared limits, corresponding to different conservation laws (Glidden et al., 2021) (see Fig. 13). Moreover, this experiment was also able to confirm the emergence of a quasicondensate in the early evolution stages, with the coherence across the system gradually growing to span the whole system, in a manner consistent with the system equilibrium for the given atom number and total energy. For completeness, I note here that manifestations of evolution consistent with Eq. (31) have also been reported (prior to the above experiment) in a quasi-one-dimensional spin-1 BEC (Prüfer et al., 2018), and a single-component BEC (Erne et al., 2018) while also bearing close analogies to the inverse cascade evolutions observed in 2D experiments (Johnstone et al., 2019; Gauthier

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Fig. 13 Experimental demonstration of bidirectional flow according to Eq. (31) in an isolated d ¼ 3 ultracold atomic gas in a box quenched from just above the critical temperature to a highly nonequilibrium situation by removing significant fractions of atoms and energy such that the system ultimately relaxes to a state with an  40% BEC fraction: The unscaled momentum distribution shown in (B) at different evolution times following the nonequilibrium quench is scaled differently in the limiting (A) infrared case (for which one finds a, b > 0, with a/b  3, consistent with the condition a/b ¼ d for particle-conserving transport), and (C) ultraviolet case (for which one finds a/b  5 with a, b < 0 consistent with the prediction a/b ¼ d + 2 for energy-conserving transport). Adapted with permission from Glidden JAP, Eigen C, Dogra LH, Hilker TA, Smith RP, and Hadzibabic Z (2021) Bidirectional dynamic scaling in an isolated Bose gas far from equilibrium. Nature Physics 17: 457–461. Copyright (2021) by the Nature Publishing Group

et al., 2019). We refer the reader to these works and excellent recent reviews on nonthermal fixed points—which also address in some detail the relation of nonthermal fixed points to weak-wave turbulence and strong turbulence (Schmied et al., 2019; Chantesana et al., 2019)—and their experimental study in ultracold quantum matter (Madeira and Bagnato, 2022).

Relation of quenched turbulence dynamics to “controlled” turbulence experiments

In this chapter, I have focused on the emergence of “turbulent” features through the instantaneous (Section “Universality during phase-ordering dynamics”) or driven (Section “Universality during phase transition crossing: The Kibble–Zurek scaling law”) cooling across a thermal phase transition, starting from an incoherent state, that is, quenching from E < 0 to E > 0, and studying the relaxation of the emerging nonequilibrium states generated, which typically include some type of vortex tangle. As well known (Tsatsos et al., 2016; Barenghi et al., 2014; Tsubota et al., 2017), an open, dynamically driven, system can reach a steady state under continuous driving, subject to a dissipation mechanism at a different scale from that where energy is being injected, thus acquiring a momentum spectrum exhibiting universal features in a certain “inertial” range. On the other hand, a closed turbulent nonequilibrium system will gradually see its vorticity decay in a manner which can also display universal scaling laws; for completeness, I briefly consider both these cases below.

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First, if the system is left isolated shortly after its quench into such a turbulent state, one could also compare its subsequent relaxation evolution to studies characterizing the decay of superfluid turbulence. The topic of superfluid turbulence has a rich history (Barenghi et al., 2001; Vinen and Niemela, 2002), with an extensive body of literature on liquid helium supplemented by recent work on ultracold quantum gases which gives access to different lengthscales (Barenghi et al., 2014; Tsatsos et al., 2016; Tsubota et al., 2017). The relaxation dynamics can be different, depending on whether the system is primarily a superfluid, or has a significant (potentially dominant) thermal fraction. Current studies, based on experimental observations (Walmsley and Golov, 2008; Bradley et al., 2006) and detailed numerical simulations (Baggaley et al., 2012; Barenghi et al., 2016), make a distinction between two types of superfluid turbulence at relatively low temperatures when a well-formed condensate exists (see also recent reviews (Barenghi et al., 2014; Tsatsos et al., 2016)). The most common form of turbulence discussed is that of Kolmogorov, or “quasi-classical” turbulence, which can exhibit (under continuous injection of energy) a cascade of kinetic energy from large to small eddies, thus giving rise to the well-known classical Kolmogorov spectrum Ek
k−5/3 for k  1/l (where l is the typical intervortex spacing); this form of turbulence—which is thus a property of the motions at lengthscales larger than the intervortex spacing—is now understood to be associated with the existence of turbulent bundles of vortices. Once the injection of energy is discontinued, this leads to a total vortex linelength decay (characterizing the intensity of turbulence), which scales as LV
t−3/2. On the other hand, some systems can exhibit a (potentially transient) turbulent state, somewhat similar to random flow, which is dominated by the presence of individual randomly oriented tangled vortices (with most of the energy contained in the intermediate scales): in such a regime, known as “ultraquantum,” or “Vinen” turbulence, the vortex linelength decays instead according to LV
t−1. Early controlled low-temperature turbulence experiments with ultracold atoms focused on the generation of tangled vorticity through either shaking across two distinct axes (Henn et al., 2009) or intense periodic driving (Navon et al., 2016) in both cases starting from a T  0 pure condensate. As such dynamics are generated by initiating the dynamical quenching from the coherent side, it is clearly a very distinct excitation process to the thermal quenches starting from an incoherent system discussed in this chapter. Among other findings, this has led to the observation of the expected power-law spectrum n(k)
k−g in a box geometry, consistent with (Kolmogorov-Zakharov) weak-wave turbulence of compressible superfluids (Navon et al., 2016); further characterization of the flux of such a cascade through dissipation, has enabled the probing of the propagation of the cascade front in momentum space at both short- and long-time evolution (Navon et al., 2019); very recent experiments from the same group have also facilitated a direct measurement of the equation of state in experiments. The concept or reversibility was also probed in such quenches. Experiments in a box revealed that when such a turbulent system was left undriven for a significant period, it would reform into a near-pure condensate consistent with the initial state, that is, the process was found to be fully reversible (Navon et al., 2016). Nonetheless, I also note here a recent related experiment (García-Orozco et al., 2022) studying the evolution of n(k, t) from a nonequilibrium initial state generated by the application of a misaligned sinusoidally varying magnetic field on an initial highly coherent harmonically confined BEC. The excitation was performed in such a manner so as to perturb the system through dynamical rotations and distortions. Rather interestingly, such an experiment observed a behavior broadly consistent with the scaling relation of Eq. (31), but in this case both a and b exponents were found to be negative (beyond the expected a, b < 0 values in the ultraviolet regime) also in the infrared regime: this should be contrasted to the previously discussed dynamics observed following a strong nonequilibrium quench in a box-like geometry (Glidden et al., 2021) for which in the infrared regime both a, b > 0. For completeness, I mention also here related studies of such an inverse excitation process starting from a coherent equilibrium BEC and leading to a strongly nonequilibrium symmetric state, proceeding through a sequence of states with spontaneously arising topological defects: while this effect has been loosely named an “inverse Kibble–Zurek” scenario (Yukalov et al., 2015)—in the sense of dynamics in the opposite direction to the established (and previously discussed) Kibble–Zurek mechanism—this is simply a qualitative analogy, as no corresponding detailed scaling laws have been obtained. Although a detailed understanding of all above dynamical issues is still an open matter, the ongoing experiments evidence the important role that an inhomogeneous geometry can play, and the richness of the dynamical features it can generate, which may have a nonnegligible impact on the types of turbulence generated and their subsequent relaxation. Moreover, differences in the dynamical properties of turbulent states generated starting from a well-formed superfluid, or an incoherent thermal cloud also highlight the important role that the thermal cloud can play on such dynamics. For a more in-depth discussion of the current understanding of the relation between nonthermal fixed points, weak-wave turbulence and strong turbulence, and the relevance of such fixed points to other dynamical aspects, such as prethermalization (Berges et al., 2004; Gring et al., 2012), the reader is referred to the review articles (Schmied et al., 2019; Chantesana et al., 2019).

Selected further considerations Beyond continuous single-component atomic gases The properties of ultracold atomic gases can be controlled experimentally by a range of different tunable parameters. In this chapter, I have focused on the role of temperature in inducing a transition across criticality between two single-component thermalized states with very different characteristic properties. Relevant comments have also been made regarding the role of geometry and dimensionality in single atomic species confined in continuous potentials, such as harmonic traps and box-like potentials.

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The exquisite control offered in experiments with ultracold atoms has opened up the way for a plethora of other types of quenches— with, no doubt, new physics yet to be explored in them, although significant progress toward various probes of universality have already been performed. While this section cannot do justice to such exciting developments, for completeness I note in passing a few such areas of relevance below. A transition across criticality can also be engineered at fixed temperature by changing the interaction energy, which can also control the type of emerging equilibrium phase transition. In this vein, Smith et al. (2011) demonstrated the smooth crossover from an interaction-driven BKT transition to a BEC transition in the limit of vanishing interactions. Such issues are further discussed in Smith (2017). For example, the interaction energy can be tuned by changing either the atom number/density or the effective strength g of contact interactions appearing in the Gross–Pitaevskii equation—which provides another exciting knob for probing universal physics. Moreover, one can also control the dominant type and range of interactions in dipolar atomic condensates, which offer tuneability in the relative strength of local and long-range interactions (Chomaz et al., 2022). Furthermore, degenerate Fermi gas settings provide an added “dimension” to such topics through control of the interaction strength, as they encompass different physical regimes depending on the value of (kF a)−1, where kF is the Fermi wavevector and a the s-wave scattering length. In particular, such systems facilitate a study of the BEC limit (in the sense of atoms pairing up to form closely bound molecules which then Bose-condense), the BCS limit (where the formation of Cooper pairs of atoms in momentum space leads to familiar BCS-like effects), and—even more interestingly—the unitarity regime between these two, exhibiting infinite-range interactions. The physics of the unitarity limit is however also accessible in Bose gases. Moreover, the coexistence of mixtures of quantum-degenerate gases and components with a spin degree of freedom (or, so-called, spinor gases) enable such studies to be generalized to quantum mixtures. In the vein of the main topics covered in this chapter, I note a significant (but not exhaustive) body of literature on Kibble–Zurek-type quenches, including, for example, through interaction-induced quenches in mixtures (Sabbatini et al., 2011) and spinor condensates (Sadler et al., 2006; Saito et al., 2007;  Swisłocki et al., 2013; Witkowska et al., 2013; Williamson and Blakie, 2016; Anquez et al., 2016; Jiménez-García et al., 2019), prethermalization quenches in 1D (Langen and Schmiedmayer, 2017; Gring et al., 2012), superfluid quenches across unitarity (Eigen et al., 2018, 2017), and quantum quenches (Dziarmaga, 2010; Polkovnikov et al., 2011) in periodic potentials (optical lattices) (Greiner et al., 2002; Chen et al., 2011; Braun et al., 2015; Clark et al., 2016), which could facilitate applications in quantum simulations (Bloch et al., 2012; Gardas et al., 2018; Bando et al., 2020), or in the presence of disorder (Meldgin et al., 2016), with open realms for studying Kibble–Zurek physics in systems such as dipolar gases and photon condensates. An important feature of quantum quenches is that they are performed under conditions of negligible thermal dissipation, thus potentially facilitating better control and cleaner scalings while allowing for the possibility of the emergence of new interesting physical regimes. Clearly these, and other settings not mentioned here, demonstrate the power of controlled quantum gas experiments with ultracold atomic gases. Nonetheless, new physics corresponding to the dynamics in an open quantum system can be probed in another experimentally well-controlled quantum gas platform, namely that of exciton-polariton systems, which I discuss next.

Driven-dissipative condensation in exciton-polariton systems A somewhat distinct physical system, that of exciton-polariton condensates, can be studied by embedding a 2D quantum well in a semiconductor microcavity driven by external pumping (laser) (Deng et al., 2010). The presence of the cavity allows the emergence of a new type of quasiparticle, which is a mixture of a photon and a matter excitation; under sufficiently strong pumping, this leads to “hybridization” and a new dressed-state bosonic quasiparticle whose spectrum facilitates the emergence of a quasi-ordered state with a high degree of coherence (Kasprzak et al., 2006; Carusotto and Ciuti, 2013; Bloch et al., 2022). As this is a strictly 2D system, technically one should not speak of “condensation,” but rather of a “BKT”-type phase transition to a “quasi-condensate” state exhibiting quasi-long-range order, as discussed in more detail below. Experimentally, there are various distinct pumping schemes that can be used to generate such an exciton-polariton system; one of these used rather commonly is that of incoherent pumping, which allows the quasi-ordered state to form spontaneously, through relaxation in the hybridized energy spectrum (relaxation in the so-called “lower polariton” branch). Under the assumption that the exciton bath can be adiabatically eliminated—which is valid in many experimental set-ups—the fundamental equation describing such a scenario can take the form of a generalized Gross–Pitaevskii equation already alluded to in Section “Workhorse model of Bose–Einstein condensation” (Wouters and Carusotto, 2007; Keeling and Berloff, 2008). When further adding stochastic noise to this (and in the usual convention of setting ħ ¼ 1), this takes the form of a stochastic complex Ginzburg–Landau equation (Chiocchetta and Carusotto, 2013; Carusotto and Ciuti, 2013; Gladilin et al., 2014; Comaron et al., 2018): ! ∂F r2 i P 2 − g F + dW (32) i + gjFj + ¼− 2m ∂t 2 1 + jFj2 =ns where m is the polariton mass (which is many orders of magnitude smaller than the electron mass), P is the strength of the homogeneous external drive, g is the polariton–polariton interaction strength, ns is a (phenomenological) saturation density, and dW a complex-valued noise term satisfying hdW (r, t)dW(r0 , t)i ¼ [(P + g)/2]dr, r0 , where g here is the inverse of the polariton lifetime.

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I note here that, despite the decaying nature of the quasiparticle, the system does nonetheless enter a nonequilibrium steady-state (provided external pumping is maintained). Different variants of this equation exist in the literature, including extensions that account for a frequency-selective pumping mechanism favoring relaxation to low-energy modes (Chiocchetta and Carusotto, 2013; Wouters et al., 2010; Comaron et al., 2018). For such systems, the control parameter is not temperature, but pumping; in fact the quasi-coherent regime sets in when pumping exceeds a threshold “critical pumping” value. In other words, E ¼ (P − Pc)/Pc such that E > 0 occurs in the high pumping limit P > Pc (recall that in the case of cold quantum matter E > 0 labeling the coherent side is a low-temperature case T < Tc). In the mean-field limit, when E > 0, Eq. (32) supports a nontrivial steady-state solution in the form of |F|2 ¼ ns(P/g − 1). Based on the previous discussion, one would expect a BKT transition associated with vortex–antivortex binding with increasing pumping strength; however the driven-dissipative nature of the system can modify such picture of “equilibrium condensation” previously discussed in the context of ultracold atomic gases (For the sake of clarity, I note here that while ultracold atomic condensates can be thought of as equilibrium, that is, fully-relaxed, during their probed long timescale, the true thermodynamic equilibrium of such systems is in fact a solid, facilitated by three-body loss processes which are not usually dominant during experimentally-probed timescales in most cold atom settings, thus enabling “long” lifetimes of up to a few minutes), and this has led to significant discussion in the community about the nature of the phase transition (Altman et al., 2015; Dagvadorj et al., 2015; Keeling et al., 2017). In effect, the extent that the phase transition from disordered to (quasi-)ordered state is an “equilibrium” phase transition or not—and the nature of the arising correlation functions on the quasi-ordered state—depend on the system parameters, which can be controlled by the quality of the sample on which experiments are performed (i.e., the polariton lifetime), but also by the spatial pump profile, details of the pumping scheme, and any engineered anisotropy in the effective polariton mass. Depending on such parameters, these systems can also exhibit a broader range of universal behavior, as discussed below. We start with revisiting the universal dynamics during a gradual driven transition (Kibble–Zurek) and the late-time phaseordering for parameters corresponding to recent exciton-polariton experiments (Nitsche et al., 2014). The equilibrium criticality phase diagram can be characterized by looking at the divergence of the characteristic coherence length, x on the incoherent (here below threshold) pumping side E < 0 and the corresponding behavior of the exponent as of the spatial coherence power law decay on the E > 0 (quasi)ordered side (Eq. (10) with a(T ) ! as). Such behavior is shown in Fig. 14A as a function of the ratio of the pumping power to the corresponding critical value predicted by mean-field theory (respectively by green and red lines) (Comaron et al., 2021). First, it should be noted here that due to the inclusion of fluctuations (via the noise term), the transition is in fact shifted from the predicted mean-field value by a few %. It is instructive to also consider the decay of the temporal correlations as one approaches the critical region from the E > 0 (quasi-)ordered side; these are shown by the red line. The diverging behavior on the (quasi)ordered side is the same qualitatively for spatial and temporal correlations. However, it clearly becomes evident (see also inset) that as ¼ 2at (Comaron et al., 2021), with such a relation between their respective exponents giving the transition a genuine nonequilibrium flavor (Szyma nska et al., 2006, 2007). Moreover, various experiments observed the universal scaling exponent characteristic of the spatial correlation function decay on the (quasi)ordered side to exceed the expected equilibrium value of (1/4) at the critical point (Roumpos et al., 2012; Nitsche et al., 2014): although this has also been seen in numerical simulations (Dagvadorj et al., 2015), the results presented here—while broadly consistent with such a picture—cannot provide a conclusive answer because the precise value is also expected to be somewhat sensitive to finite-size effects (Comaron et al., 2021). Having identified the precise location of the critical point, one can now consider the dependence of the system properties as a function of the distance to criticality E, the first step toward performing and characterizing linear quenches across the phase transition. As expected the number of vortices in this 2D system at equilibrium decreases very dramatically in the vicinity of E ¼ 0 (dashed red line in Fig. 14B). Quenching across the transition at different rates leads to such vortices surviving for significant amounts of time after the system parameters have been quenched to the (quasi-)ordered side (Zamora et al., 2020); this is here visible by the slower decay of vortices with E, where E ¼ E(t) ¼ t/tQ following the quenched system evolution through the externally imposed parameter. The slower decay of vortices in time as the system adapts to the external quench across the phase transition leads to the expected divergence in the relaxation time (red line, t(E), in bottom left inset to Fig. 14B). The faster one quenches across the phase transition (i.e., the smaller that tQ is), the deeper into E(t) > 0 values that the vortices can be observed on the (quasi)ordered side. Of course, ultimately such vortices will decay to the corresponding equilibrium value at such value of E, marked by the dashed red line. _ (dashed blue line in bottom left inset to Fig. 14B), and so the exact location I recall here that the quench rate affects the slope of EðtÞ=EðtÞ of the intersection point of these two curves which, according to the fundamental Kibble–Zurek equation, defines the timescale −^ t. Numerical simulations (Zamora et al., 2020) have confirmed the validity of the Kibble–Zurek scenario, by revealing the expected proportionality between the observed relaxation time of vortex dynamics (essentially the time when the quenched solid lines in Fig. 14B _ ^ deviate from the equilibrium dashed red line) and the time predicted by matching tð−^ t Þ to Eð−^ t Þ=Eð− t Þ (up to some irrelevant microscopic nonuniversal prefactor); such proportionality can be seen in the top right inset. In the long-time limit, the vortices decay. As discussed in Section “Scaling hypothesis and late-time coarsening,” this decay (following instantaneous quenching into the (quasi-)ordered state) is expected to enter another scaling window, associated with phase-ordering (Jelic and Cugliandolo, 2011). Already from Fig. 14B one can see that the vortex decay slows down as their number decreases for a given value of tQ, that is, along a particular curve shown in Fig. 14B. This is because vortices move (on average) further and further apart; thus the frequency of their annihilations slows down. Then, at some late time, when there are only relatively few vortices left in the system, the behavior changes drastically. Such behavior (following now an instantaneous quench) has been mapped out in Fig. 14C; after a very rapid decay in the number of vortices (blue) and corresponding increase in the

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Fig. 14 Universal features of a nonequilibrium driven-dissipative exciton-polariton condensate for typical experimental parameters. (A) Dependence of the coherence length, x, defined by g(1)(r)
e−r/x on the incoherent side of the phase transition, and of the spatial (temporal) exponents as (at) defined on the quasi-ordered side by gð1Þ ðrÞ ¼ r −as (gð1Þ ðtÞ ¼ t −at ). Horizontal axis depicts the external pump power in units of the critical value predicted by mean-field theory, with the transition occurring above the critical pump threshold (here Pcr
1.033); fluctuations clearly shift this by (up to) a few %, toward a higher pump value. Inset: zoomed-in version clearly revealing as ¼ 2at, as direct evidence of the nonequilibrium nature of the phase transition in such parameter regime. (B) Demonstration of validity of the Kibble–Zurek relation (Eq. 19). Main panel shows the dependence of vortex number on E ¼ (P − Pc)/Pc for the equilibrium case (dashed red line) and the corresponding dependence for hNv (t )i in terms of E(t ) for a linear external quench in the pump via E(t ) ¼ t/tQ, with tQ decreasing between curves as one moves from left to right. For each tQ, there is a corresponding value of E ¼ Eð^t num Þ at which the (dynamical) curve determining the postquench number of vortices (solid) as a function of E deviates from the corresponding equilibrium one (dashed red)—these values are shown by the dashed vertical lines, and _ in the vicinity of the yield the numerical estimate ^t num . Insets: (bottom left) Relation between diverging relaxation time t and characteristic timescale EðtÞ=EðtÞ critical point, with their intersection defining the freeze-out time −^t KZ . (top right) Demonstration of j^t num j
j^t KZ j validating the expected Kibble–Zurek proportionality relation (up to some microscopic, nonuniversal prefactor). (C) Evolution of decreasing number of vortices/antivortices (blue curve) and growing characteristic lengthscale (green curve) highlighting the dramatic early postquench coarsening evolution, and their corresponding dependence on time in the scaling regime, allowing us to extract the dynamical critical exponent z ¼ 2. Note that all axes are plotted on logarithmic scale, with the temporal axis also including the anticipated logarithmic correction to the system evolution. (A) Adapted with permission from Comaron P, Carusotto I, Szymanska, MH, and Proukakis NP (2021) Nonequilibrium Berezinskii-Kosterlitz-Thouless transition in driven-dissipative condensates(a). Europhysics Letters 133(1): 17002. https://doi.org/10.1209/02955075/133/17002. Copyright (2021) by the Institute of Physics. (B) Adapted with permission from Zamora A, Dagvadorj G, Comaron P, Carusotto I, Proukakis NP, and Szyma nska MH (2020) Kibble-Zurek mechanism in driven dissipative systems crossing a nonequilibrium phase transition. Physical Review Letters 125: 095301. https://doi.org/10.1103/PhysRevLett.125.095301. Copyright (2020) by the American Physical Society. (C) Adapted with permission from Comaron P, Dagvadorj G, Zamora A, Carusotto I, Proukakis NP, and Szymanska MH (2018) Dynamical critical exponents in driven-dissipative quantum systems. Physical Review Letters 121 (9): 095302. Copyright (2018) by the American Physical Society

characteristic lengthscale describing the typical size of the emerging quasi-ordered regions (green), the system enters the scaling regime (Comaron et al., 2018). In this regime, properties of the system can be characterized by a single growing lengthscale and thus all correlation functions collapse onto a single function, in terms of such an appropriately rescaled variable Lc(t) (see Eq. 30). Interestingly, one now sees very clearly both the existence of the expected logarithmic corrections in the presence of external coupling (dissipation) (previously discussed in Section “Scaling hypothesis and late-time coarsening”), and the anticipated value of the dynamical critical exponent (z ¼ 2) (Jelic and Cugliandolo, 2011). While this behavior is expected to continue at later times, finite size effects make it harder to obtain concrete predictions of Lc(t) from the spatial correlation function. The above results have painted the picture of a “nonequilibrium” BKT phase transition, and an anticipated late-time phaseordering law, with significant discussion in the literature addressing the nature of this phase transition (Dagvadorj et al., 2015; Altman et al., 2015; Keeling et al., 2017). Parallel to this, there has been quite some interest in searching for a different kind of universal behavior on the quasi-ordered (E > 0) side of the transition, associated with a so-called Kardar–Parisi–Zhang (KPZ) universality class (Kardar et al., 1986).

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The KPZ equation describes the stochastic growth of the height of an interface and is established in diverse (classical) physical settings, including, for example, polymer growth, and spreading of wildfires. In the present context, I note that the KPZ equation arises pffiffiffiffiffiffiffiffiffiffiffiffi as an equation modeling the phase of the superfluid y, obtained from Fðr, tÞ ¼ nðr, tÞeiyðr,tÞ in the presence of a momentumdependent damping rate under the assumption that the density fluctuations (dn(r, t) ¼ n(r, t) − hn(r, t)i) vary slowly both in space (rdn  0) and in time (∂n/∂t  0). Such an assumption makes the KPZ physics more relevant in regimes characterized by weak-phase fluctuations, namely, deep in the quasi-ordered phase E > 0. In that case, one obtains the KPZ equation in the form (Altman et al., 2015; Gladilin et al., 2014; Ji et al., 2015; Wachtel et al., 2016; He et al., 2015; Squizzato et al., 2018; Deligiannis et al., 2021) pffiffiffiffi ∂y l ¼ nr2 y + ðryÞ2 + D : ∂t 2

(33)

The terms on the right-hand side describe the diffusion leading to smoothening, a nonlinear contribution leading to critical roughening, and a stochastic noise term, with the constants n, l, and D depending on microscopic system parameters. This equation maps the evolution of the phase of the polariton mean field to a growing interface with the same internal dimension. Let us consider the general normalized first-order correlation functions in space and time in the form g(1)(r, r + Dr, t, t + Dt) and denote this by the short-hand notation g(1)(Dr, Dt). In general (upon neglecting density–density and density–phase correlations), one can show that (Fontaine et al., 2022) gð1Þ ðDr, DtÞ ¼ hexpðiDyðDr, DtÞÞi

(34)

where Dy denotes the change in phase from some reference value. To lowest order in the fluctuations, this would thus give the dominant behavior that

2w ! 1 Dr ð1Þ Fixed Dt 0 : jg ðDr, Dt 0 Þj
exp − 2 l (35)

2b ! 1 Dt ð1Þ Fixed Dr 0 : jg ðDr0 , DtÞj
exp − 2 t thus introducing the two KPZ critical exponents w and b for space and time, respectively, denoting the roughness and growth critical exponents (with l and t, the corresponding nonuniversal parameters). As a result, KPZ scaling would show up as “stretched exponential” decay in the correlation functions in the limit of a relatively well-formed mean field, that is, for “sufficiently large” E > 0 values (such behavior is termed “stretched exponential” here because 2w, 2b 6¼1 as would be the case in a “normal” exponential). In order to unequivocally observe such behavior, one would generally need to observe very large systems (and for very long times) because of the existence of a characteristic minimum lengthscale for the onset of KPZ physics (Altman et al., 2015). Notwithstanding the earlier comments about a possible nonequilibrium BKT phase transition, an experiment conducted with high-quality samples (and so longer polariton lifetimes, thus facilitating a less nonequilibrium regime) found two interesting results (Caputo et al., 2018) (see Fig. 15A); first, although spatial and temporal correlation functions could be fitted by both stretched exponential and power-law decays in the vicinity of the critical region, the power-law decay always fits better on the quasi-ordered side of the phase transition, thus explicitly verifying BKT-type correlations—consistent with BKT being the transition mechanism. Second, unlike the previous discussion, the power-law exponents for the spatial and temporal decay on the quasi-ordered side were found to exactly match each other, both having a value clearly below 0.25. All this corroborated to an equilibrium BKT phase transition in this particular experimental setting, demonstrating that exciton-polariton condensates can control the extent of the “equilibrium” or “nonequilibrium” nature of the phase transition through the sample quality characterizing the polariton lifetime. From a theoretical point of view, the universal scaling function for KPZ takes the form (Deligiannis et al., 2021) CðDr, DtÞ ¼ hðDyðDr, DtÞÞ2 i − hDyðDr, DtÞi2 
¼ −2ln gð1Þ ðDr, DtÞ

 jDrj ¼ C0 ðDtÞ2b F y0 1=z , Dt

(36)

where F(y) is a universal scaling function (whose asymptotics are known), z ¼ w/b and C0 and y0 are nonuniversal constants. Such emergence of KPZ scaling was conclusively demonstrated in an appropriately engineered (discrete) 1D setting (Fontaine et al., 2022), constituting the first realization of KPZ scaling in a quantum system. In addition to identifying a spatial/temporal region where Eqs. (35) hold (Fig. 15B(i)–(ii)), they were able to convincingly demonstrate the emergence of a universal spatiotemporal KPZ scaling function by plotting C(Dr, Dt)/(Dt)2b as a function of Dr/Dt1/z, for the particular exponent values of b ¼ 1/3 and z ¼ 3/2, thus corresponding to w ¼ 1/2. Extending such analysis to the 2D limit, the same team also managed to numerically demonstrate—sufficiently above threshold so that vortices do not overwhelm KPZ effects (Deligiannis et al., 2022)—the emergence of scaling according to Eq. (35), with critical exponents consistent with those anticipated in the 2D KPZ universality class. This opens up the possibility of observing—for the first time—2D KPZ physics in an experiment, a long-standing goal in nonequilibrium statistical physics. The above brief discussion highlights driven-dissipative exciton-polaritons as ideal systems for tuning both the extent of equilibrium (or nonequilibrium) nature of the BKT phase transition, and the prevailing universality class through carefully engineered experiments (Zamora et al., 2017; Diessel et al., 2022).

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Fig. 15 (A) Regime of 2D equilibrium BKT. Dependence of the spatial and temporal correlation function exponents as a function of the pumping (here d) scaled to the critical value (dBKT) for the experimental parameters of Caputo et al. (2018). Depicted values correspond to exponents labeling correlation decay in the form of either stretched exponentials (left part of plot) or algebraic decay (right part of plot) according to which one of these fits works best in each regime. Here one sees clearly that algebraic decay (characteristic of the 2D XY model) dominates (overstretched exponential decay) for all points above threshold, and that, importantly, the emerging spatial (a) and temporal (b) exponents are equal and do not exceed the value of 0.25 at the critical point. (B) and (C) Experimental demonstration of nonequilibrium KPZ scaling in 1D. (B)(i)–(ii) Identification of spatiotemporal regime where both spatial and temporal correlations exhibit their expected stretched exponential scalings of Eq. (35) on the quasi-ordered side. (C) Appropriately rescaled plot confirming the KPZ picture of Eq. (36) for all data points in (B)(i)–(ii) within the corresponding scaling regimes for which |g(1)| takes the values shown within the annulus in the bottom right inset. Note that such scaling function shows a plateau at small y ¼ y0Dr/(Dt)2/3 and linear growth at large y. Insets: (top left) Corresponding analysis by means of numerical simulations; (bottom right) Coherence map showing value of |g(1)| (see colorbar) in terms of Dr (denoted here by Dx) and Dt. (A) Adapted with permission from Caputo D, Ballarini, D, Dagvadorj G, Munoz CS, de Giorgi M, Dominici L, West, K, Pfeiffer LN, Gigli G, Laussy FP, Szymanska MH, and Sanvitto D (2018) Topological order and thermal equilibrium in polariton condensates. Nature Materials 17: 145–151. Copyright (2018) by the Nature Publishing Group. (B) Adapted with permission from Fontaine Q, Squizzato D, Baboux F, Amelio I, Lemaître A, Morassi M, Sagnes I, Gratiet L, Harouri A, Wouters M, Carusotto I, Amo A, Richard M, Minguzzi A, Canet L, Ravets S, and Bloch J (2022) Kardar-Parisi-Zhang universality in a one-dimensional polariton condensate. Nature 608: 687–691. Copyright (2022) by the Nature Publishing Group

Bose–Einstein condensation on astrophysical/cosmological scales? Due to its universal nature, BEC has even been hypothesized to be at play across very different astrophysical/cosmological settings. Perhaps the most broadly known example is that of neutron stars (Pethick et al., 2017), whose core is expected to exhibit both superfluidity and superconductivity. However, various other forms of condensation have been discussed in the literature (see, e.g., early discussions in Griffin et al., 1995), including rather interestingly condensation of gravitons (Dvali and Gomez, 2017). Below I briefly outline an alternative idea gaining increasing attention currently in the literature in the context of dark matter modeling. The possibility of the existence of a cosmological-scale gravitating BEC of axionic particles has been proposed as a plausible alternative picture of the nature of dark matter (Hu et al., 2000) (see also the related scenario of thermalizing QCD axions (Banik

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and Sikivie, 2017)). More recently, the focus of such theoretical ideas appears to be in the context of ultralight axion-like particles of pffiffiffiffi mass 10−22 eV/c2: due to their low mass, these can give rise to galactic-size de Broglie wavelengths (recall that l
1= m ), thus facilitating condensation on such unprecedented scales. While such systems cannot be controlled in any way (and so the question of phase transition crossing becomes less relevant), nonetheless, predictions of such models in the current (late-time, virialized) state could ultimately be related to cosmological observations. As well known, there is currently an established model of cold dark matter (CDM), which explains the large-scale structures of the universe very well through N-body numerical simulations (Frenk and White, 2012). Nonetheless, some discrepancies have been recently noted in relation to the small-scale structure (Weinberg et al., 2015; Bullock and Boylan-Kolchin, 2017; Del Popolo and Le Delliou, 2017) (most notably the inferred distribution of the dark matter density at the very center of galaxies). One very elegant way (but admittedly not the only one) to resolve such discrepancies is to describe such features through BEC in the context of the so-called FDM model (Schive et al., 2014; Mocz et al., 2017; Marsh, 2016; Hui, 2021; Ferreira, 2021). The underlying idea here is that dark matter consists of ultralight axionic particles, of mass 10−22eV/c2. Due to their high density and low mass, such particles would have a very high phase-space density allowing the emergence of BEC on galactic scales. The power of such an approach is that it describes “short-range” (on cosmological scales) features by the emergence of a so-called gravitationally self-bound soliton of dark matter while still reproducing the very successful large-scale predictions of the usual CDM model—most notably the Navarro-FrenkWhite (NFW) radial density profiles at large distances from the galactic core (Navarro et al., 1996). The equation governing such cosmological condensates is a Schrödinger equation, embedded within the gravitational field (Widrow and Kaiser, 1993; Seidel and Suen, 1990; Schive et al., 2014; Mocz et al., 2017). In general, one could also envisage the existence of interactions between such axionic particles (although some constraints on their maximum strength do exist) (Chavanis, 2011; Desjacques et al., 2018; Mocz et al., 2023; Delgado and Muñoz Mateo, 2022), giving rise to the following system of coupled equations, loosely termed the Gross–Pitaevskii–Poisson equations:  2 2  ∂Fðr, tÞ ħ r ¼ − + grðr, tÞ+mVðr, tÞ Fðr, tÞ (37) iħ ∂t 2m where the Newtonian potential V (r, t) follows the Poisson equation, r2 Vðr, tÞ ¼ 4pGðrðr, tÞ − rÞ

(38)

with the mass density r(r, t) ¼ |F(r, t)| , and r its spatially averaged constant value (and G denotes the gravitational constant). (Interestingly this same set of equations has also been recently used to model glitches in pulsars (Verma et al., 2022).) While the validity of the FDM model has yet to be tested, even in the absence of interactions (and we may still be a long way away before some of the emerging predictions can be directly tested against observational data), there is a rapidly growing body of work (see, e.g., recent reviews (Marsh, 2016; Hui, 2021; Ferreira, 2021)) addressing the model’s predictions both on the cosmological large scale (Fig. 16A) (Schive et al., 2014; Mocz et al., 2017; May and Springel, 2021) and on the level of a single (isolated) cored-halo regime corresponding to the distribution of dark matter on the level of a typical galaxy (Fig. 16B and C) (Mocz et al., 2017; Schwabe et al., 2016; Chan et al., 2022; Liu et al., 2023). The important new contribution from the FDM picture discussed in the above works (and many references therein) is that the density at the center of the galaxy behaves very differently to CDM predictions, with the emergence of a well-formed soliton whose density remains finite as r ! 0. Essentially this leads to a bimodal profile of a central soliton surrounded by fluctuating dark matter density satisfying the NFW profile; this is somewhat reminiscent of the density profiles of ultracold atomic gases in harmonic traps, with gravitational attraction in the galactic case playing a role analogous to the harmonic confinement of ultracold atomic gases—such an important and interesting analogy has been explicitly discussed in Liu et al. (2023). Addressing the coherent properties of such a (virialized) cored-halo system, Liu et al. (2023) demonstrated that the discussed solitonic core is in fact fully coherent in the sense of a dominant Penrose–Onsager mode, with a nearly constant spatial correlation function in the central solitonic region exhibiting g(1)(r)  g(2)(r)  1 for r ≲ r c, where rc labels a typical soliton radius. Beyond this, there is an intermediate region of gradually decreasing coherence spatially, with global-phase coherence across the soliton core lost between the soliton core and an outer “transition” region rt, at which point g(1)(r)  0 and g(2)(r)  2, consistent with a chaotic field. Nonetheless, closer inspection demonstrates that the outer halo region (r > rt) contains randomly distributed regions of near constant density, which locally feature enhanced phase coherence. Such domains are separated by randomly oriented vortices that scramble the phase between them; in the cosmological context, such regions are termed “granules.” Such a picture closely resembles a cosmological-size quasi-condensate, as shown schematically—based on actual numerical data—in Fig. 16C (Liu et al., 2023); as such vortices are randomly oriented/distributed and move around (albeit on rather cosmologically-slow timescales), there is no overall phase coherence after radial, or temporal averaging. Moreover—at least in the absence of self-interactions—this system exhibits familiar scaling laws of (ultraquantum) turbulence, with such states in fact rather similar qualitatively to the early dynamical phases of condensate formation discussed in Section “Analyzing Kibble–Zurek in an elongated ultracold atomic system” (Liu et al., 2023); interestingly recent works have shown such a structure to be very long-lived, unlike the rapidly decaying turbulent states generated in nondriven ultracold atomic systems. The predictions of the FDM model may end up not having a direct bearing on real-world observables. Nonetheless, the mere fact that a cosmological model of a BEC can actually generate features seemingly consistent with our current understanding of large-scale structure in the universe is truly remarkable in its own right, and demonstrates the power and universality of such a physical phenomenon. 2

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Fig. 16 Cosmological condensation in the fuzzy dark matter (FDM) model (also known as cDM in Schive et al. (2014)): (A) Large-scale comparison of FDM predictions (left) to the established cold dark matter (CDM) model based on N-body simulations (right) from a given set of initial cosmological conditions; large-scale features are excellently reproduced. (B) FDM simulations focusing on an isolated (single) galactic halo, which in the FDM model contains a characteristic solitonic core at its center (left image), reveal a number of interesting features shown in the right image; first, the radial density profile (black line) both exhibits a core at the center (unlike the CDM model predictions that lead to an “unphysical” 1/r divergence as r ! 0, which is not shown here) and reproduces the expected NFW profiles (dashed brown line) at larger distances (r > rt); importantly, the solitonic core exhibits near-perfect coherence g(1)(r)  g(2)(r)  1 for r  rc , with coherence only gradually lost with increasing radial distance up to rt, at which point the NFW density profile sets in; in fact, the solitonic core can be fully described as a Penrose–Onsager condensate mode (green line) all the way up to rt, with the remaining (incoherent) density concentrated primarily outside the solitonic region, but with a nonzero (albeit small) component still present at smaller radii; this is directly analogous to the condensate-thermal cloud bimodal distribution of Hartree-Fock theories for ultracold atomic gases (see, e.g., Proukakis and Jackson, 2008) and the corresponding experimentally obtained density distributions. (C) Example schematic of the density distribution of the solitonic core (yellow core—with both rc and rt radii shown to give a sense of scale) surrounded by regions of decreasing density filled with random vortices (purple lines) which scramble the phase between different domains of locally similar phase—constituting a galactic-size example of a quasi-condensate state. (A) Reprinted with permission from Schive H-Y, Chiueh T, and Broadhurst T (2014) Cosmic structure as the quantum interference of a coherent dark wave. Nature Physics 10(7): 496–499. https://doi.org/10.1038/nphys2996.1406.6586. Copyright (2014) by the Nature Publishing Group. (B) and (C) Adapted with permission from Liu I-K, Proukakis NP, and Rigopoulos G (2023) Coherent and incoherent structures in fuzzy dark matter haloes. Monthly Notices of the Royal Astronomical Society 521(3): 3625–3647. ISSN 0035-8711. https://doi.org/10.1093/mnras/stad591. Copyright (2023) by Oxford University Press.

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Conclusion In this chapter, I have given a broad overview of manifestations of macroscopic coherence across a diverse range of physical systems, spanning from the very tiny (exciton-polaritons, atoms) to the very large (cosmological). While the specific physical manifestation across such systems might be different, with BEC associated with a single macroscopic observable and off-diagonal long-range order giving way—in some limits—to related behavior in terms of a Berezinskii–Kosterlitz–Thouless transition to a quasi-ordered state, or even to a state exhibiting stretched exponential correlation functions consistent with those of the Kardar–Parisi–Zhang model over a restricted spatiotemporal regime, such systems nonetheless exhibit evidence of universal behavior in appropriate temporal domains and/or spatial regions, with some common observable features. This facilitates an exciting cross-fertilization of ideas, which have borne out of seemingly distinct fields such as low-temperature physics; statistical physics; condensed matter; and atomic, molecular, and optical physics. Interestingly, many of the ideas discussed in this chapter have their origins in cosmological or theoretical physics studies, even if perhaps some of those features are now becoming more the playground of controlled quantum matter experiments. For example, the concept of defects arising during a dynamical phase transition becoming embedded into, and thus observable within, the long-time evolution of the system—first proposed in the context of a hot Big Bang model—has not only found significant validation in a diverse range of coherent quantum systems but also led to exciting studies pushing new frontiers in quantum gas experiments conducted with an unprecedented flexibility and accuracy—such as experiments with ultracold atoms. In this context, one is now investigating the modifications to this Kibble–Zurek scenario of driven quenching across the phase transitions imposed by inhomogeneities and the interplay with causality. Moreover, the concept of highly nonequilibrium scenarios, associated with the dynamics around nonthermal fixed points emerged initially from studies of reheating after early universe inflation, but recent years have seen a number of distinct experiments with ultracold atoms revealing evidence of dynamics consistent with predicted scalings around such nonthermal-fixed points. Such nonequilibrium scenarios have a close connection with the emergence of (strong) turbulence, in a generalized manner, that is, with the state of a system filled with random tangled defects, which can also exhibit various types of universal scaling laws in appropriate regimes. Beyond the familiar and already observed direct energy cascade Kolmogorov turbulence of a driven system and associated decay of bundles of vortices once driving is removed, experiments with liquid helium and ultracold atomic systems have revealed a new “type” of turbulence, known as “ultraquantum” (or “Vinen”) turbulence, which is dominated by the tangling and relaxation of individual vortices. The rapid quenching of a phase transition creates a strongly “turbulent” state with broadly similar features qualitatively, whose subsequent evolution ultimately follows universal scaling laws—whether in the context of spatially self-similar solutions during phase-ordering, the stronger spatiotemporal evolution around nonthermal-fixed points, or even evolution related to the Kardar–Parisi–Zhang spatiotemporal self-similarity. Interestingly, such features also connect the driven or instantaneous breaking of a continuous symmetry with phenomena investigated in fluids, including classical fluids and growth of interfaces. As such, systems exhibiting macroscopic quantum coherence in controlled quantum matter experiments give access to a range of different universal types of behaviors, and the emergence of a range of scaling laws having wide applicability to very different physical systems across both the classical and the quantum realm. Combined with the enhanced quantum control in these systems, which facilitates the on-demand construction of system Hamiltonians—thus making them ideal systems for quantum simulations—such systems collectively open up a very exciting, highly interdisciplinary realm for a deeper understanding of the nonequilibrium physical world.

Acknowledgments The unified presentation given here has been made possible through collaborations (or extended discussions) on these topics with many excellent colleagues and friends over a period of years, including specifically for the presented context and concepts Tom Billam, Tom Bland, Jerome Beugnon, Iacopo Carusotto, Paolo Comaron, Leticia Cugliandolo, Franco Dalfovo, Jacek Dziarmaga, Jean Dalibard, Piotr Deuar, Gabriele Ferrari, Andrew Groszek, Giacomo Lamporesi, Fabrizio Larcher, Rob Smith, Henk Stoof, Marzena Szymanska, and Alex Zamora, to whom I am grateful. In such a context, I would particularly like to highlight Dr I-Kang (Gary) Liu and Dr Paolo Comaron who have been pivotal to my respective understanding of Kibble–Zurek physics and polariton condensation, as well as Dr Gerasimos Rigopoulos and Dr I-Kang Liu for our recent collaboration into coherent cosmological endeavors. During the preparation of this chapter, I have benefited enormously from discussions with, and direct feedback on this manuscript from, Vanderlei Bagnato, Carlo Barenghi, Paolo Comaron, Jacek Dziarmaga, Thomas Gasenzer, Gary Liu, Gerasimos Rigopoulos, and Marzena Szymanska. I also thank Jacek Dziarmaga, Thomas Gasenzer, Anna Minguzzi, Hsi-Yu Schive, Rob Smith, and Wojciech Zurek for explicit permissions to reprint key figures from their earlier works, and the UK EPSRC, Leverhulme Trust, and the Horizon-2020 framework for funding—with particular note to the Quantera grant “Nonequilibrium dynamics in Atomic systems for QUAntum Simulation” (NAQUAS) which refocused my interest on such topics.

References Adams A, Carr LD, Schäfer T, Steinberg P, and Thomas JE (2012) Strongly correlated quantum fluids: Ultracold quantum gases, quantum chromodynamic plasmas and holographic duality. New Journal of Physics 14(11): 115009. https://doi.org/10.1088/1367-2630/14/11/115009. Aidelsburger M, Ville JL, Saint-Jalm R, Nascimbène S, Dalibard J, and Beugnon J (2017) Relaxation dynamics in the merging of n independent condensates. Physical Review Letters 119: 190403. https://doi.org/10.1103/PhysRevLett.119.190403.

118

Universality of Bose–Einstein condensation and quenched formation dynamics

Al Khawaja U, Andersen JO, Proukakis NP, and Stoof HTC (2002) Low dimensional Bose gases. Physical Review A 66(1): 013615. https://doi.org/10.1103/PhysRevA.66.013615. Altman E, Sieberer LM, Chen L, Diehl S, and Toner J (2015) Two-dimensional superfluidity of exciton polaritons requires strong anisotropy. Physical Review X 5: 011017. https://doi. org/10.1103/PhysRevX.5.011017. Anquez M, Robbins BA, Bharath HM, Boguslawski M, Hoang TM, and Chapman MS (2016) Quantum Kibble-Zurek mechanism in a spin-1 Bose-Einstein condensate. Physical Review Letters 116: 155301. https://doi.org/10.1103/PhysRevLett.116.155301. Baggaley AW, Barenghi CF, and Sergeev YA (2012) Quasiclassical and ultraquantum decay of superfluid turbulence. Physical Review B 85: 060501. https://doi.org/10.1103/ PhysRevB.85.060501. Bando Y, Susa Y, Oshiyama H, Shibata N, Ohzeki M, Gómez-Ruiz FJ, Lidar DA, Suzuki S, del Campo A, and Nishimori H (2020) Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond. Physical Review Research 2: 033369. https://doi.org/10.1103/PhysRevResearch.2.033369. Banik N and Sikivie P (2017) Cosmic axion Bose-Einstein condensation. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Barenghi CF, Donnelly RJ, and Vinen WF (2001) Quantized Vortex Dynamics and Superfluid Turbulence. Lecture Notes in Physics. , 571. Berlin: Springer. Barenghi CF, Skrbek L, and Sreenivasan KR (2014) Introduction to quantum turbulence. Proceedings of the National Academy of Sciences of the United States of America 111: 4647–4652. supplement 1. Barenghi CF, Sergeev YA, and Baggaley AW (2016) Regimes of turbulence without an energy cascade. Scientific Reports 6: 35701. Bäuerle C, Bunkov YM, Fisher SN, Godfrin H, and Pickett GR (1996) Laboratory simulation of cosmic string formation in the early Universe using superfluid 3He. Nature 382(6589): 332–334. https://doi.org/10.1038/382332a0. Berges J (2016) Nonequilibrium quantum fields: From cold atoms to cosmology. In: Strongly Interacting Quantum Systems out of Equilibrium. Lecture Notes of the Les Houches Summer School: Volume 99, August 2012. Oxford University Press. https://eur03.safelinks.protection.outlook.com/?url¼https%3A%2F%2Fdoi.org%2F10.1093%2Facprof%3Aoso%2F9780198768166. 003.0002&data¼05%7C01%7Cnikolaos.proukakis%40newcastle.ac.uk%7C90a05c0408694589ee1f08db0b714766%7C9c5012c9b61644c2a91766814fbe3e87%7C1%7C0% 7C638116353390136069%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata¼0gz0T 2ycHAJ1uYfQ1hcBciuW2C%2BvomBpm5gk7OyAN%2F8%3D&reserved¼0. Berges J and Hoffmeister G (2009) Nonthermal fixed points and the functional renormalization group. Nuclear Physics B813: 383. https://doi.org/10.1016/j.nuclphysb.2008.12.017. 0809.5208. Berges J, Borsanyi S, and Wetterich C (2004) Prethermalization. Physical Review Letters 93: 142002. https://doi.org/10.1103/PhysRevLett.93.142002. hep-ph/0403234. Berges J, Rothkopf A, and Schmidt J (2008) Non-thermal fixed points: Effective weak-coupling for strongly correlated systems far from equilibrium. Physical Review Letters 101: 041603. https://doi.org/10.1103/PhysRevLett.101.041603. 0803.0131. Berges J, Boguslavski K, Schlichting S, and Venugopalan R (2015) Universality far from equilibrium: From superfluid Bose gases to heavy-ion collisions. Physical Review Letters 114: 061601. https://doi.org/10.1103/PhysRevLett.114.061601. Berges J, Heller MP, Mazeliauskas A, and Venugopalan R (2021) Qcd thermalization: Ab initio approaches and interdisciplinary connections. Reviews of Modern Physics 93: 035003. https://doi.org/10.1103/RevModPhys.93.035003. Berloff NG (1999) Nonlocal nonlinear Schrödinger equations as models of superfluidity. Journal of Low Temperature Physics 116: 359–380. Berloff NG and Svistunov BV (2002) Scenario of strongly nonequilibrated Bose-Einstein condensation. Physical Review A 66(1): 013603. https://doi.org/10.1103/ PhysRevA.66.013603. Berloff NG, Brachet M, and Proukakis NP (2014) Modeling quantum fluid dynamics at nonzero temperatures. Proceedings of the National Academy of Sciences of the United States of America 111(supplement 1): 4675–4682. https://doi.org/10.1073/pnas.1312549111. Beugnon J and Navon N (2017) Exploring the Kibble-Zurek mechanism with homogeneous Bose gases. Journal of Physics B: Atomic, Molecular and Optical Physics 50(2): 022002. https://doi.org/10.1088/1361-6455/50/2/022002. Bijlsma MJ, Zaremba E, and Stoof HTC (2000) Condensate growth in trapped Bose gases. Physical Review A 62: 063609. https://doi.org/10.1103/PhysRevA.62.063609. Billam TP, Brown K, and Moss IG (2022) False-vacuum decay in an ultracold spin-1 Bose gas. Physical Review A 105: L041301. https://doi.org/10.1103/PhysRevA.105.L041301. Binney J, Dowrick NJ, Fisher AJ, and Newman MEJ (1992) The Theory of Critical Phenomena: An Introduction to the Renormalization Group. Oxford Science Publ. Clarendon Press, ISBN: 9780198513933. https://books.google.co.uk/books?id¼lvZcGJI3X9cC. Blakie PB, Bradley AS, Davis MJ, Ballagh RJ, and Gardiner CW (2008) Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Advances in Physics 57(5): 363–455. https://doi.org/10.1080/00018730802564254. 0809.1487. Bland T, Marolleau Q, Comaron P, Malomed B, and Proukakis NP (2020) Persistent current formation in double-ring geometries. Journal of Physics B: Atomic, Molecular and Optical Physics 53: 115301. http://iopscience.iop.org/10.1088/1361-6455/ab81e9. Bloch I, Dalibard J, and Nascimbène S (2012) Quantum simulations with ultracold quantum gases. Nature Physics ISSN: 1745-2473. 8(4): 267–276. https://doi.org/10.1038/ nphys2259. http://www.nature.com.ezproxy.lib.monash.edu.au/nphys/journal/v8/n4/abs/nphys2259.html?foxtrotcallback¼true. Bloch J, Carusotto I, and Wouters M (2022) Non-equilibrium Bose-Einstein condensation in photonic systems. Nature Reviews Physics 4: 470–488. Bradley DI, Clubb DO, Fisher SN, Guénault AM, Haley RP, Matthews CJ, Pickett GR, Tsepelin V, and Zaki K (2006) Decay of pure quantum turbulence in superfluid 3He-B. Physical Review Letters 96: 035301. https://doi.org/10.1103/PhysRevLett.96.035301. Bradley AS, Gardiner CW, and Davis MJ (2008) Bose-Einstein condensation from a rotating thermal cloud: Vortex nucleation and lattice formation. Physical Review A 77: 033616. https://doi.org/10.1103/PhysRevA.77.033616. Braun S, Friesdorf M, Hodgman SS, Schreiber M, Ronzheimer JP, Riera A, del Rey M, Bloch I, Eisert J, and Schneider U (2015) Emergence of coherence and the dynamics of quantum phase transitions. Proceedings of the National Academy of Sciences of the United States of America ISSN: 0027-8424. 112(12): 3641–3646. https://doi.org/10.1073/ pnas.1408861112. Bray AJ (1994) Theory of phase-ordering kinetics. Advances in Physics ISSN: 0001-8732. 43: 357–459. https://doi.org/10.1080/00018730110117433. Bruce A and Wallace D (1992) Critical point phenomena: Universal physics at large length scales. In: Davies P (ed.) The New Physics. Cambridge: Cambridge University Press. Bullock JS and Boylan-Kolchin M (2017) Small-Scale Challenges to the LCDM Paradigm. Annual Review of Astronomy and Astrophysics 55(1): 343–387. https://doi.org/10.1146/ annurev-astro-091916-055313. 1707.04256. Caputo D, Ballarini D, Dagvadorj G, Munoz CS, de Giorgi M, Dominici L, West K, Pfeiffer LN, Gigli G, Laussy FP, Szymanska MH, and Sanvitto D (2018) Topological order and thermal equilibrium in polariton condensates. Nature Materials 17: 145–151. Carmi R, Polturak E, and Koren G (2000) Observation of spontaneous flux generation in a multi-Josephson-junction loop. Physical Review Letters 84: 4966–4969. https://doi.org/ 10.1103/PhysRevLett.84.4966. Carusotto I and Ciuti C (2013) Quantum fluids of light. Reviews of Modern Physics 85(1): 299. Chae SC, Lee N, Horibe Y, Tanimura M, Mori S, Gao B, Carr S, and Cheong S-W (2012) Direct observation of the proliferation of ferroelectric loop domains and vortex-antivortex pairs. Physical Review Letters 108: 167603. https://doi.org/10.1103/PhysRevLett.108.167603. Chan HYJ, Ferreira EGM, May S, Hayashi K, and Chiba M (2022) The diversity of core-halo structure in the fuzzy dark matter model. Monthly Notices of the Royal Astronomical Society ISSN: 0035-8711. 511(1): 943–952. https://doi.org/10.1093/mnras/stac063. Chantesana I, Orioli AP, and Gasenzer T (2019) Kinetic theory of nonthermal fixed points in a Bose gas. Physical Review A 99(4): 043620. https://doi.org/10.1103/ PhysRevA.99.043620. Chavanis P-H (2011) Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions. I. Analytical results. Physical Review D 84: 043531. https://doi.org/10.1103/PhysRevD.84.043531.

Universality of Bose–Einstein condensation and quenched formation dynamics

119

Chen D, White M, Borries C, and DeMarco B (2011) Quantum quench of an atomic Mott insulator. Physical Review Letters 106: 235304. https://doi.org/10.1103/ PhysRevLett.106.235304. Chiocchetta A and Carusotto I (2013) Non-equilibrium quasi-condensates in reduced dimensions. EPL (Europhysics Letters) 102(6): 67007. Chomaz L, Corman L, Bienaimé T, Desbuquois R, Weitenberg C, Nascimbène S, Beugnon J, and Dalibard J (2015) Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas. Nature Communications 6(1): 6162. https://doi.org/10.1038/ncomms7162. Chomaz L, Ferrier-Barbut I, Ferlaino F, Laburthe-Tolra B, Lev BL, and Pfau T (2022) Dipolar physics: A review of experiments with magnetic quantum gases. Reports on Progress in Physics 86(2): 026401. https://doi.org/10.1088/1361-6633/aca814. Chu S (1998) Nobel lecture: The manipulation of neutral particles. Reviews of Modern Physics 70: 685–706. https://doi.org/10.1103/RevModPhys.70.685. Chuang I, Durrer R, Turok N, and Yurke B (1991) Cosmology in the laboratory: Defect dynamics in liquid crystals. Science 251(4999): 1336–1342. https://doi.org/10.1126/ science.251.4999.1336. Clark LW, Feng L, and Chin C (2016) Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition. Science 354(6312): 606–610. https://doi. org/10.1126/science.aaf9657. Cockburn SP, Negretti A, Proukakis NP, and Henkel C (2011) Comparison between microscopic methods for finite-temperature Bose gases. Physical Review A 83: 043619. https:// doi.org/10.1103/PhysRevA.83.043619. Cohen-Tannoudji CN (1998) Nobel lecture: Manipulating atoms with photons. Reviews of Modern Physics 70: 707–719. https://doi.org/10.1103/RevModPhys.70.707. Cohen-Tannoudji C and Robilliard C (2001) Wave functions, relative phase and interference for atomic Bose-Einstein condensates. Comptes Rendus Physique 2: 445. Comaron P, Dagvadorj G, Zamora A, Carusotto I, Proukakis NP, and Szymanska MH (2018) Dynamical critical exponents in driven-dissipative quantum systems. Physical Review Letters 121(9): 095302. Comaron P, Carusotto I, Szymanska MH, and Proukakis NP (2021) Non-equilibrium Berezinskii-Kosterlitz-Thouless transition in driven-dissipative condensates(a). Europhysics Letters 133(1): 17002. https://doi.org/10.1209/0295-5075/133/17002. Cooper LN and Feldman D (2010) Bcs: 50 Years. World Scientific Publishing Company, ISBN: 9789814360869. https://books.google.co.uk/books?id¼spnFCgAAQBAJ. Corman L, Chomaz L, Bienaime T, Desbuquois R, Weitenberg C, Nascimbene S, Dalibard J, and Beugnon J (2014) Quench-induced supercurrents in an annular Bose gas. Physical Review Letters 113: 135302. Cornell EA and Wieman CE (2002) Nobel lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Reviews of Modern Physics 74: 875–893. https://doi.org/10.1103/RevModPhys.74.875. Dagvadorj G, Fellows JM, Matyjaskiewicz S, Marchetti FM, Carusotto I, and Szymanska MH (2015) Nonequilibrium phase transition in a two-dimensional driven open quantum system. Physical Review X 5: 041028. https://doi.org/10.1103/PhysRevX.5.041028. Dalfovo F, Giorgini S, Pitaevskii LP, and Stringari S (1999) Theory of Bose-Einstein condensation in trapped gases. Reviews of Modern Physics 71: 463–512. Damle K, Majumdar SN, and Sachdev S (1996) Phase ordering kinetics of the Bose gas. Physical Review A 54: 5037–5041. https://doi.org/10.1103/PhysRevA.54.5037. Das A, Sabbatini J, and Zurek WH (2012) Winding up superfluid in a torus via Bose Einstein condensation. Scientific Reports 2: 352. Davis MJ, Morgan SA, and Burnett K (2002) Simulations of thermal Bose fields in the classical limit. Physical Review A 66: 053618. https://doi.org/10.1103/PhysRevA.66.053618. Davis MJ, Wright TM, Gasenzer T, Gardiner SA, and Proukakis NP (2017) Formation of Bose-Einstein condensates. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. del Campo A and Zurek WH (2014) Universality of phase transition dynamics: Topological defects from symmetry breaking. International Journal of Modern Physics A ISSN: 0217-751X. 29(08): 1430018. https://doi.org/10.1142/S0217751X1430018X. del Campo A, Retzker A, and Plenio MB (2011) The inhomogeneous Kibble-Zurek mechanism: Vortex nucleation during Bose-Einstein condensation. New Journal of Physics 13(8): 083022. https://doi.org/10.1088/1367-2630/13/8/083022. del Campo A, Kibble TWB, and Zurek WH (2013) Causality and non-equilibrium second-order phase transitions in inhomogeneous systems. Journal of Physics. Condensed Matter 25(40): 404210. https://doi.org/10.1088/0953-8984/25/40/404210. Del Popolo A and Le Delliou M (2017) Small scale problems of the LCDM model: A short review. Galaxies 5(1): 17. https://doi.org/10.3390/galaxies5010017. 1606.07790. Delgado V and Muñoz Mateo A (2022) Self-interacting superfluid dark matter droplets. Monthly Notices of the Royal Astronomical Society ISSN: 0035-8711. 518(3): 4064–4072. https://doi.org/10.1093/mnras/stac3386. Deligiannis K, Squizzato D, Minguzzi A, and Canet L (2021) Accessing Kardar-Parisi-Zhang universality sub-classes with exciton polaritons(a). Europhysics Letters 132(6): 67004. https://doi.org/10.1209/0295-5075/132/67004. Deligiannis K, Fontaine Q, Squizzato D, Richard M, Ravets S, Bloch J, Minguzzi A, and Canet L (2022) Kardar-Parisi-Zhang universality in discrete two-dimensional driven-dissipative exciton polariton condensates. Physical Review Research 4: 043207. https://doi.org/10.1103/PhysRevResearch.4.043207. Deng H, Haug H, and Yamamoto Y (2010) Exciton-polariton Bose-Einstein condensation. Reviews of Modern Physics 82(2): 1489. https://doi.org/10.1103/RevModPhys.82.1489. Desjacques V, Kehagias A, and Riotto A (2018) Impact of ultralight axion self-interactions on the large scale structure of the universe. Physical Review D 97: 023529. https://doi.org/ 10.1103/PhysRevD.97.023529. Diessel OK, Diehl S, and Chiocchetta A (2022) Emergent Kardar-Parisi-Zhang phase in quadratically driven condensates. Physical Review Letters 128: 070401. https://doi.org/ 10.1103/PhysRevLett.128.070401. Donadello S, Serafini S, Tylutki M, Pitaevskii LP, Dalfovo F, Lamporesi G, and Ferrari G (2014) Observation of Solitonic Vortices in Bose-Einstein Condensates. Physical Review Letters 113(6): 065302. https://doi.org/10.1103/PhysRevLett.113.065302. Donadello S, Serafini S, Bienaimé T, Dalfovo F, Lamporesi G, and Ferrari G (2016) Creation and counting of defects in a temperature-quenched Bose-Einstein condensate. Physical Review A 94(2): 023628. Donner T, Ritter S, Bourdel T, Ottl A, Köhl M, and Esslinger T (2007) Critical behavior of a trapped interacting Bose gas. Science 315(5818): 1556–1558. https://doi.org/10.1126/ science.1138807. Drummond LV and Melatos A (2017) Stability of interlinked neutron vortex and proton flux tube arrays in a neutron star: Equilibrium configurations. Monthly Notices of the Royal Astronomical Society ISSN: 0035-8711. 472(4): 4851–4869. https://doi.org/10.1093/mnras/stx2301. Dvali G and Gomez C (2017) Graviton BECs: A new approach to quantum gravity. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Dziarmaga J (2010) Dynamics of a quantum phase transition and relaxation to a steady state. Advances in Physics 59(6): 1063–1189. Dziarmaga J, Laguna P, and Zurek WH (1999) Symmetry breaking with a slant: Topological defects after an inhomogeneous quench. Physical Review Letters 82: 4749–4752. https:// doi.org/10.1103/PhysRevLett.82.4749. Eigen C, Glidden JAP, Lopes R, Navon N, Hadzibabic Z, and Smith RP (2017) Universal scaling laws in the dynamics of a homogeneous unitary Bose gas. Physical Review Letters 119: 250404. https://doi.org/10.1103/PhysRevLett.119.250404. Eigen C, Glidden JAP, Lopes R, Cornell EA, Smith RP, and Hadzibabic Z (2018) Universal prethermal dynamics of Bose gases quenched to unitarity. Nature 563: 221–224. Erne S, Bücker R, Gasenzer T, Berges J, and Schmiedmayer J (2018) Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium. Nature 563: 225–229. Ferreira EGM (2021) Ultra-light dark matter. Astronomy and Astrophysics Review 29: 7. Fialko O, Opanchuk B, Sidorov AI, Drummond PD, and Brand J (2015) Fate of the false vacuum: Towards realization with ultra-cold atoms. Europhysics Letters 110(5): 56001. https:// doi.org/10.1209/0295-5075/110/56001. Fontaine Q, Squizzato D, Baboux F, Amelio I, Lemaître A, Morassi M, Sagnes I, Gratiet L, Harouri A, Wouters M, Carusotto I, Amo A, Richard M, Minguzzi A, Canet L, Ravets S, and Bloch J (2022) Kardar-Parisi-Zhang universality in a one-dimensional polariton condensate. Nature 608: 687–691.

120

Universality of Bose–Einstein condensation and quenched formation dynamics

Frenk CS and White SDM (2012) Dark matter and cosmic structure. Annalen der Physik 524(9-10): 507–534. https://doi.org/10.1002/andp.201200212. 1210.0544. Fried DG, Killian TC, Willmann L, Landhuis D, Moss SC, Kleppner D, and Greytak TJ (1998) Bose-Einstein condensation of atomic hydrogen. Physical Review Letters 81: 3811. García-Orozco AD, Madeira L, Moreno-Armijos MA, Fritsch AR, Tavares PES, Castilho PCM, Cidrim A, Roati G, and Bagnato VS (2022) Universal dynamics of a turbulent superfluid Bose gas. Physical Review A 106: 023314. https://doi.org/10.1103/PhysRevA.106.023314. Gardas B, Dziarmaga J, Zurek WH, and Zwolak M (2018) Defects in quantum computers. Scientific Reports 8: 4539. https://doi.org/10.1038/s41598-018-22763-2. Gardiner CW and Davis MJ (2003) The stochastic Gross-Pitaevskii equation: II. Journal of Physics B: Atomic, Molecular and Optical Physics ISSN: 0953-4075. 36(23): 4731. https:// doi.org/10.1088/0953-4075/36/23/010. Gardiner CW, Lee MD, Ballagh RJ, Davis MJ, and Zoller P (1998) Quantum kinetic theory of condensate growth: Comparison of experiment and theory. Physical Review Letters 81: 5266. Gauthier G, Reeves MT, Yu X, Bradley AS, Baker MA, Bell TA, Rubinsztein-Dunlop H, Davis MJ, and Neely TW (2019) Giant vortex clusters in a two-dimensional quantum fluid. Science 364(6447): 1264–1267. https://doi.org/10.1126/science.aat5718. Gladilin VN, Ji K, and Wouters M (2014) Spatial coherence of weakly interacting one-dimensional nonequilibrium bosonic quantum fluids. Physical Review A 90: 023615. https://doi. org/10.1103/PhysRevA.90.023615. Glidden JAP, Eigen C, Dogra LH, Hilker TA, Smith RP, and Hadzibabic Z (2021) Bidirectional dynamic scaling in an isolated Bose gas far from equilibrium. Nature Physics 17: 457–461. Goo J, Lim Y, and Shin Y (2021) Defect saturation in a rapidly quenched Bose gas. Physical Review Letters 127: 115701. https://doi.org/10.1103/PhysRevLett.127.115701. Goo J, Lee Y, Lim Y, Bae D, Rabga T, and Shin Y (2022) Universal early coarsening of quenched Bose gases. Physical Review Letters 128: 135701. https://doi.org/10.1103/ PhysRevLett.128.135701. Greiner M, Mandel MO, Esslinger T, Hänsch T, and Bloch I (2002) Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature 415: 39. Greytak T and Kleppner D (2017) A history of Bose-Einstein condensation of atomic hydrogen. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Griffin A, Snoke DW, and Stringari S (1995) Bose-Einstein Condensation. Cambridge: Cambridge University Press. Gring M, Kuhnert M, Langen T, Kitagawa T, Rauer B, Schreitl M, Mazets I, Smith DA, Demler E, and Schmiedmayer J (2012) Relaxation and prethermalization in an isolated quantum system. Science 337(6100): 1318. https://doi.org/10.1126/science.1224953. Groszek AJ, Comaron P, Proukakis NP, and Billam TP (2021) Crossover in the dynamical critical exponent of a quenched two-dimensional Bose gas. Physical Review Research 3: 013212. https://doi.org/10.1103/PhysRevResearch.3.013212. He L, Sieberer LM, Altman E, and Diehl S (2015) Scaling properties of one-dimensional driven-dissipative condensates. Physical Review B 92: 155307. https://doi.org/10.1103/ PhysRevB.92.155307. Hendry PC, Lawson NS, Lee RAM, McClintock PVE, and Williams CDH (1994) Generation of defects in superfluid 4He as an analogue of the formation of cosmic strings. Nature 368: 315–317. Henn EAL, Seman JA, Roati G, Magalhães KMF, and Bagnato VS (2009) Emergence of turbulence in an oscillating Bose-Einstein condensate. Physical Review Letters 103: 045301. https://doi.org/10.1103/PhysRevLett.103.045301. Hohenberg PC and Halperin BI (1977) Theory of dynamic critical phenomena. Reviews of Modern Physics 49: 435–479. https://doi.org/10.1103/RevModPhys.49.435. Holzmann M, Fuchs J-N, Baym GA, Blaizot J-P, and Laloë F (2004) Bose-Einstein transition temperature in a dilute repulsive gas. Comptes Rendus Physique ISSN: 1631-0705. 5(1): 21–37. https://doi.org/10.1016/j.crhy.2004.01.003. Bose-Einstein condensates: Recent advances in collective effects. Hu W, Barkana R, and Gruzinov A (2000) Fuzzy cold dark matter: The wave properties of ultralight particles. Physical Review Letters ISSN: 00319007. 85(6): 1158–1161. https://doi. org/10.1103/PhysRevLett.85.1158. Huang K (1987) Statistical Mechanics. New York: Wiley. Hui L (2021) Wave dark matter. Annual Review of Astronomy and Astrophysics 59: 247–289. https://doi.org/10.1146/annurev-astro-120920-010024. 2101.11735. Jacquet MJ, Weinfurtner S, and König F (2020) The next generation of analogue gravity experiments. Philosophical Transactions of the Royal Society A 378: 20190239. Jelic A and Cugliandolo LF (2011) Quench dynamics of the 2d XY model. Journal of Statistical Mechanics: Theory and Experiment 2011(2): P02032. Ji K, Gladilin VN, and Wouters M (2015) Temporal coherence of one-dimensional nonequilibrium quantum fluids. Physical Review B 91: 045301. https://doi.org/10.1103/ PhysRevB.91.045301. Jiménez-García K, Invernizzi A, Evrard B, Frapolli C, Dalibard J, and Gerbier F (2019) Spontaneous formation and relaxation of spin domains in antiferromagnetic spin-1 condensates. Nature Communications 10: 1422. Johnstone SP, Groszek AJ, Starkey PT, Billington CJ, Simula TP, and Helmerson K (2019) Evolution of large-scale flow from turbulence in a two-dimensional superfluid. Science 364 (6447): 1267–1271. https://doi.org/10.1126/science.aat5793. Jos JV (2013) 40 Years of Berezinskii-Kosterlitz-Thouless Theory. EBL-Schweitzer. World Scientific, ISBN: 9789814417648. https://books.google.co.uk/books?id¼roO6CgAAQBAJ. Kagan Y (1995) Kinetics of Bose-Einstein condensate formation in an interacting Bose gas. In: Griffin A, Snoke DW, and Stringari S (eds.) Bose-Einstein Condensation, p. 202. Cambridge: Cambridge University Press. Kagan Y and Svistunov BV (1994) Kinetics of the onset of long-range order during Bose condensation in an interacting gas. Soviet Physics–JETP 78: 187. Kagan Y and Svistunov BV (1997) Evolution of correlation properties and appearance of broken symmetry in the process of Bose-Einstein condensation. Physical Review Letters 79: 3331. Kagan Y, Svistunov BV, and Shlyapnikov GV (1992) Kinetics of Bose condensation in an interacting Bose gas. Soviet Physics–JETP 75: 387. Kardar M, Parisi G, and Zhang Y-C (1986) Dynamic scaling of growing interfaces. Physical Review Letters 56: 889–892. https://doi.org/10.1103/PhysRevLett.56.889. Karl M and Gasenzer T (2017) Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas. New Journal of Physics 19(9): 093014. https://doi.org/ 10.1088/1367-2630/aa7eeb. Kasprzak J, et al. (2006) Bose-Einstein condensation of exciton polaritons. Nature (London) 443(7110): 409–414. https://doi.org/10.1038/nature05131. Keeling J and Berloff NG (2008) Spontaneous rotating vortex lattices in a pumped decaying condensate. Physical Review Letters 100: 250401. https://doi.org/10.1103/ PhysRevLett.100.250401. Keeling J, Sieberer LM, Altman E, Chen L, Diehl S, and Toner J (2017) Superfluidity and phase correlations of driven dissipative condensates. In: Proukakis N, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Ketterle W (2002) Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser. Reviews of Modern Physics 74: 1131–1151. https://doi.org/ 10.1103/RevModPhys.74.1131. Kibble TWB (1976) Topology of cosmic domains and strings. Journal of Physics A: Mathematical and General 9(8): 1387. Kim M, Rabga T, Lee Y, Goo J, Bae D, and Shin Y (2022) Suppression of spontaneous defect formation in inhomogeneous Bose gases. Physical Review A 106: L061301. https://doi. org/10.1103/PhysRevA.106.L061301. Klaers J, Schmitt J, Vewinger F, and Weitz M (2010) Bose-Einstein condensation of photons in an optical microcavity. Nature 468: 545–548. Ko B, Park JW, and Shin Y (2019) Kibble-Zurek universality in a strongly interacting Fermi superfluid. Nature Physics 15: 1227–1231. https://doi.org/10.1038/s41567-019-0650-1. Kobayashi M and Cugliandolo LF (2016a) Quench dynamics of the three-dimensional U(1) complex field theory: Geometric and scaling characterizations of the vortex tangle. Physical Review E 94: 062146. https://doi.org/10.1103/PhysRevE.94.062146. Kobayashi M and Cugliandolo LF (2016b) Thermal quenches in the stochastic Gross-Pitaevskii equation: Morphology of the vortex network. Europhysics Letters 115(2): 20007. https:// doi.org/10.1209/0295-5075/115/20007. Köhl M, Davis MJ, Gardiner CW, Hänsch TW, and Esslinger T (2002) Growth of Bose-Einstein condensates from thermal vapor. Physical Review Letters 88: 080402. https://doi.org/ 10.1103/PhysRevLett.88.080402.

Universality of Bose–Einstein condensation and quenched formation dynamics

121

Kosterlitz JM (2017) Nobel lecture: Topological defects and phase transitions. Reviews of Modern Physics 89: 040501. https://doi.org/10.1103/RevModPhys.89.040501. Lamporesi G, Donadello S, Serafini S, Dalfovo F, and Ferrari G (2013) Spontaneous creation of Kibble-Zurek solitons in a Bose-Einstein condensate. Nature Physics 9(10): 656. Langen T and Schmiedmayer J (2017) Quenches, relaxation and prethermalization in an isolated quantum system. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Leggett AJ (2006) Quantum Liquids. Oxford University Press. Lifshitz EM and Pitaevskii LP (1995) Physical Kinetics. Course of Theoretical Physics. , vol. 10. Elsevier Science, ISBN: 9780750626354. https://books.google.co.uk/books? id¼h7LgAAAAMAAJ. Liu I-K, Donadello S, Lamporesi G, Ferrari G, Gou S-C, Dalfovo F, and Proukakis NP (2018) Dynamical equilibration across a quenched phase transition in a trapped quantum gas. Communications Physics ISSN: 2399-3650. 1(1): 24. https://doi.org/10.1038/s42005-018-0023-6. Liu I-K, Dziarmaga J, Gou S-C, Dalfovo F, and Proukakis NP (2020) Kibble-Zurek dynamics in a trapped ultracold Bose gas. Physical Review Research 2: 033183. https://doi.org/ 10.1103/PhysRevResearch.2.033183. Liu I-K, Proukakis NP, and Rigopoulos G (2023) Coherent and incoherent structures in fuzzy dark matter haloes. Monthly Notices of the Royal Astronomical Society ISSN: 0035-8711. 521(3): 3625–3647. https://doi.org/10.1093/mnras/stad591. Madeira L and Bagnato VS (2022) Non-thermal fixed points in Bose gas experiments. Symmetry 678: 14. Marsh DJE (2016) Axion cosmology. Physics Reports 643: 1–79. https://doi.org/10.1016/j.physrep.2016.06.005. 1510.07633. May S and Springel V (2021) Structure formation in large-volume cosmological simulations of fuzzy dark matter: Impact of the non-linear dynamics. Monthly Notices of the Royal Astronomical Society 506(2): 2603–2618. https://doi.org/10.1093/mnras/stab1764. 2101.01828. Meldgin C, Ray U, Russ P, Chen D, Ceperley DM, and DeMarco B (2016) Probing the Bose glass-superfluid transition using quantum quenches of disorder. Nature Physics 12: 646–649. https://doi.org/10.1038/nphys3695. Miesner H-J, Stamper-Kurn DM, Andrews MR, Durfee DS, Inouye S, and Ketterle W (1998) Bosonic stimulation in the formation of a Bose-Einstein condensate. Science 279(5353): 1005–1007. https://doi.org/10.1126/science.279.5353.1005. Mocz P, Vogelsberger M, Robles VH, Zavala J, Boylan-Kolchin M, Fialkov A, and Hernquist L (2017) Galaxy formation with BECDM - I. Turbulence and relaxation of idealized haloes. Monthly Notices of the Royal Astronomical Society 471(4): 4559–4570. https://doi.org/10.1093/mnras/stx1887. 1705.05845. Mocz P, Fialkov A, Vogelsberger M, Boylan-Kolchin M, Chavanis P-H, Amin MA, Bose S, Dome T, Hernquist L, Lancaster L, Notis M, Painter C, Robles VH, and Zavala J (2023) Cosmological structure formation and soliton phase transition in fuzzy dark matter with axion self-interactions. Monthly Notices of the Royal Astronomical Society ISSN: 0035-8711. 521(2): 2608–2615. https://doi.org/10.1093/mnras/stad694. Monaco R, Mygind J, Rivers RJ, and Koshelets VP (2009) Spontaneous fluxoid formation in superconducting loops. Physical Review B 80: 180501. https://doi.org/10.1103/ PhysRevB.80.180501. Nam K, Baek W-B, Kim B, and Lee SJ (2012) Coarsening of two-dimensional XY model with Hamiltonian dynamics: Logarithmically divergent vortex mobility. Journal of Statistical Mechanics: Theory and Experiment 2012(11): P11023. https://doi.org/10.1088/1742-5468/2012/11/P11023. Navarro JF, Frenk CS, and White SDM (1996) The structure of cold dark matter Halos. The Astrophysical Journal 462: 563. Navon N, Gaunt AL, Smith RP, and Hadzibabic Z (2015) Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas. Science 347(6218): 167–170. https://doi. org/10.1126/science.1258676. Navon N, Gaunt A, Smith R, and Hadzibabic Z (2016) Emergence of a turbulent cascade in a quantum gas. Nature 539: 72–75. Navon N, Eigen C, Zhang J, Lopes R, Gaunt AL, Fujimoto K, Tsubota M, Smith RP, and Hadzibabic Z (2019) Synthetic dissipation and cascade fluxes in a turbulent quantum gas. Science 366(6463): 382–385. https://doi.org/10.1126/science.aau6103. Nishimori H and Ortiz G (2011) Elements of Phase Transitions and Critical Phenomena. Oxford Graduate Texts. OUP Oxford, ISBN: 9780199577224. https://books.google.co.uk/ books?id¼qiEUDAAAQBAJ. Nitsche WH, Kim NY, Roumpos G, Schneider C, Kamp M, Höfling S, Forchel A, and Yamamoto Y (2014) Algebraic order and the Berezinskii-Kosterlitz-Thouless transition in an excitonpolariton gas. Physical Review B 90(20): 205430. https://doi.org/10.1103/PhysRevB.90.205430. Nowak B, Sexty D, and Gasenzer T (2011) Superfluid turbulence: Nonthermal fixed point in an ultracold Bose gas. Physical Review B 84: 020506. https://doi.org/10.1103/ PhysRevB.84.020506. Nowak B, Schole J, Sexty D, and Gasenzer T (2012) Nonthermal fixed points, vortex statistics, and superfluid turbulence in an ultracold Bose gas. Physical Review A 85: 043627. https://doi.org/10.1103/PhysRevA.85.043627. Nowak B, Schole J, and Gasenzer T (2014) Universal dynamics on the way to thermalization. New Journal of Physics 16(9): 093052. https://doi.org/10.1088/1367-2630/16/ 9/093052. Penrose O and Onsager L (1956) Bose-Einstein condensation and liquid helium. Physical Review 104: 576. Pethick CJ and Smith H (2002) Bose-Einstein Condensation in Dilute Gases. Cambridge University Press. Pethick CJ, Schafer T, and Schwenk A (2017) Bose-Einstein condensates in neutron stars. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Petrov DS, Gangardt DM, and Shlyapnikov GV (2004) Low-dimensional trapped gases. Journal de Physique IV 116: 5–44. Phillips WD (1998) Nobel lecture: Laser cooling and trapping of neutral atoms. Reviews of Modern Physics 70: 721–741. https://doi.org/10.1103/RevModPhys.70.721. Pickett GR (2017) A simulated cosmological metric: The superfluid 3he condensate. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Piñeiro Orioli A, Boguslavski K, and Berges J (2015) Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points. Physical Review D 92(2): 025041. https://doi.org/10.1103/PhysRevD.92.025041. 1503.02498. Pitaevskii L and Stringari S (2016) Bose-Einstein Condensation and Superfluidity. Oxford: Oxford University Press. Polkovnikov A, Sengupta K, Silva A, and Vengalattore M (2011) Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Reviews of Modern Physics 83: 863–883. https://doi.org/10.1103/RevModPhys.83.863. Prokof’ev N and Svistunov B (2002) Two-dimensional weakly interacting Bose gas in the fluctuation region. Physical Review A 66: 043608. https://doi.org/10.1103/ PhysRevA.66.043608. Proukakis NP (2006) Spatial correlation functions of one-dimensional Bose gases at equilibrium. Physical Review A 74(5): 053617. https://doi.org/10.1103/PhysRevA.74.053617. cond-mat/0606617. Proukakis NP and Burnett K (2013) Quantum gases: Setting the scene. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics: 1 (Cold Atoms). London: Imperial College Press. Proukakis NP and Jackson B (2008) Finite-temperature models of Bose-Einstein condensation. Journal of Physics B: Atomic, Molecular and Optical Physics ISSN: 0953-4075. 41(20): 203002. https://doi.org/10.1088/0953-4075/41/20/203002. Proukakis N, Gardiner S, Davis M, and Szymanska M (2013) Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics: 1 (Cold Atoms). ICP. Proukakis NP, Snoke DW, and Littlewood PB (2017) Universal Themes of Bose-Einstein Condensation. Cambridge University Press. Prüfer M, Kunkel P, Strobel H, Lannig S, Linnemann D, Schmied C-M, Berges J, Gasenzer T, and Oberthaler MK (2018) Observation of universal dynamics in a spinor Bose gas far from equilibrium. Nature 563: 217–220. Roumpos G, et al. (2012) Power-law decay of the spatial correlation function in exciton-polariton condensates. Proceedings of the National Academy of Sciences 109(17): 6467–6472. https://doi.org/10.1073/pnas.1107970109.

122

Universality of Bose–Einstein condensation and quenched formation dynamics

Rutenberg AD and Bray AJ (1995a) Energy-scaling approach to phase-ordering growth laws. Physical Review E 51(6): 5499–5514. https://doi.org/10.1103/PhysRevE.51.5499. Rutenberg AD and Bray AJ (1995b) Phase ordering of two-dimensional XY systems below the Kosterlitz-Thouless transition temperature. Physical Review E 51(3): R1641. https://doi. org/10.1103/PhysRevE.51.R1641. Sabbatini J, Zurek WH, and Davis MJ (2011) Phase separation and pattern formation in a binary Bose-Einstein condensate. Physical Review Letters 107: 230402. https://doi.org/ 10.1103/PhysRevLett.107.230402. Sadhukhan D, Sinha A, Francuz A, Stefaniak J, Rams MM, Dziarmaga J, and Zurek WH (2020) Sonic horizons and causality in phase transition dynamics. Physical Review B 101: 144429. https://doi.org/10.1103/PhysRevB.101.144429. Sadler LE, Higbie JM, Leslie SR, Vengalattore M, and Stamper-Kurn DM (2006) Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose-Einstein condensate. Nature 443: 312–315. Saito H, Kawaguchi Y, and Ueda M (2007) Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate. Physical Review A 76(4): 043613. Sanvitto D and Timofeev V (2012) Exciton Polaritons in Microcavities: New Frontiers. Springer Series in Solid-State Sciences. Springer Berlin Heidelberg, ISBN: 9783642241864. https://books.google.co.uk/books?id¼3ybfbv8d3NUC. Schive H-Y, Chiueh T, and Broadhurst T (2014) Cosmic structure as the quantum interference of a coherent dark wave. Nature Physics 10(7): 496–499. https://doi.org/10.1038/ nphys2996. 1406.6586. Schmied C-M, Mikheev AN, and Gasenzer T (2019) Non-thermal fixed points: Universal dynamics far from equilibrium. International Journal of Modern Physics A ISSN: 0217-751X. 34(29): 1941006. https://doi.org/10.1142/S0217751X19410069. Schwabe B, Niemeyer JC, and Engels JF (2016) Simulations of solitonic core mergers in ultralight axion dark matter cosmologies. Physical Review D 94(4): 043513. https://doi.org/ 10.1103/PhysRevD.94.043513. 1606.05151. Seidel E and Suen W-M (1990) Dynamical evolution of Boson stars: Perturbing the ground state. Physical Review D 42(2): 384–403. https://doi.org/10.1103/PhysRevD.42.384. Serafini S, Galantucci L, Iseni E, Bienaimé T, Bisset RN, Barenghi CF, Dalfovo F, Lamporesi G, and Ferrari G (2017) Vortex reconnections and rebounds in trapped atomic Bose-Einstein condensates. Physical Review X 7: 021031. https://doi.org/10.1103/PhysRevX.7.021031. Smith RP (2017) Effects of interactions on Bose-Einstein condensation. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Smith RP, Campbell RLD, Tammuz N, and Hadzibabic Z (2011) Effects of interactions on the critical temperature of a trapped Bose gas. Physical Review Letters 106: 250403. https:// doi.org/10.1103/PhysRevLett.106.250403. Snoke DW, Proukakis NP, Giamarchi T, and Littlewood PB (2017) Universality and Bose-Einstein condensation: Perspectives on recent work. In: Proukakis NP, Snoke DW, and Littlewood PB (eds.) Universal Themes of Bose-Einstein Condensation. Cambridge: Cambridge University Press. Squizzato D, Canet L, and Minguzzi A (2018) Kardar-Parisi-Zhang universality in the phase distributions of one-dimensional exciton-polaritons. Physical Review B 97: 195453. https:// doi.org/10.1103/PhysRevB.97.195453. Stagg GW, Parker NG, and Barenghi CF (2016) Ultraquantum turbulence in a quenched homogeneous Bose gas. Physical Review A 94: 053632. https://doi.org/10.1103/ PhysRevA.94.053632. Stoof HTC (1995) Condensate formation in a Bose gas. In: Griffin A, Snoke DW, and Stringari S (eds.) Bose-Einstein Condensation, pp. 226–245. Cambridge: Cambridge University Press. Stoof HTC (1999) Coherent versus incoherent dynamics during Bose-Einstein condensation in atomic gases. Journal of Low Temperature Physics 114(1/2): 11. Stoof HTC and Bijlsma MJ (2001) Dynamics of fluctuating Bose-Einstein condensates. Journal of Low Temperature Physics 124: 431–442. https://doi.org/10.1023/ A:1017519118408. Svistunov BV (1991) Highly nonequilibrium Bose condensation in a weakly interacting gas. Journal of the Moscow Physical Society 1: 373–390. Swisłocki T, Witkowska E, Dziarmaga J, and Matuszewski M (2013) Double universality of a quantum phase transition in spinor condensates: Modification of the Kibble-z˙urek mechanism by a conservation law. Physical Review Letters 110: 045303. https://doi.org/10.1103/PhysRevLett.110.045303. Szymanska MH, Keeling J, and Littlewood PB (2006) Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Physical Review Letters 96: 230602. https://doi.org/10.1103/PhysRevLett.96.230602. Szymanska MH, Keeling J, and Littlewood PB (2007) Mean-field theory and fluctuation spectrum of a pumped decaying Bose-Fermi system across the quantum condensation transition. Physical Review B 75: 195331. https://doi.org/10.1103/PhysRevB.75.195331. Tsatsos MC, Tavares PES, Cidrim A, Fritsch AR, Caracanhas MA, dos Santos FEA, Barenghi CF, and Bagnato VS (2016) Quantum turbulence in trapped atomic Bose-Einstein condensates. Physics Reports ISSN: 0370-1573. 622: 1–52. https://doi.org/10.1016/j.physrep.2016.02.003. Quantum turbulence in trapped atomic Bose-Einstein condensates. Tsubota M, Fujimoto K, and Yui S (2017) Numerical studies of quantum turbulence. Journal of Low Temperature Physics 188: 119–189. Ueda M (2010) Fundamentals and New Frontiers of Bose-Einstein Condensation. World Scientific, ISBN: 9789812839596. https://books.google.co.uk/books?id¼iix3_pqy6ysC. Ulm S, Roßnagel J, Jacob G, Degünther C, Dawkins ST, Poschinger UG, Nigmatullin R, Retzker A, Plenio MB, Schmidt-Kaler F, et al. (2013) Observation of the Kibble-Zurek scaling law for defect formation in ion crystals. Nature Communications 4: 2290. Verma AK, Pandit R, and Brachet ME (2022) Rotating self-gravitating Bose-Einstein condensates with a crust: A model for pulsar glitches. Physical Review Research 4: 013026. https://doi.org/10.1103/PhysRevResearch.4.013026. Vinen WF and Niemela JJ (2002) Quantum turbulence. Journal of Low Temperature Physics 128: 167–231. Volovik GE (2009) The Universe in a Helium Droplet. International Series of Monographs on Physics. Oxford University Press. https://books.google.co.uk/books?id¼VuLJrQEACAAJ. Wachtel G, Sieberer LM, Diehl S, and Altman E (2016) Electrodynamic duality and vortex unbinding in driven-dissipative condensates. Physical Review B 94: 104520. https://doi.org/ 10.1103/PhysRevB.94.104520. Walmsley PM and Golov AI (2008) Quantum and quasiclassical types of superfluid turbulence. Physical Review Letters 100: 245301. https://doi.org/10.1103/ PhysRevLett.100.245301. Warszawski L, Melatos A, and Berloff NG (2012) Unpinning triggers for superfluid vortex avalanches. Physical Review B 85: 104503. https://doi.org/10.1103/PhysRevB.85.104503. Weiler CN, Neely TW, Scherer DR, Bradley AS, Davis MJ, and Anderson BP (2008) Spontaneous vortices in the formation of Bose-Einstein condensates. Nature 455: 948–951. https:// doi.org/10.1038/nature07334. 0807.3323. Weinberg DH, Bullock JS, Governato F, Kuzio de Naray R, and Peter AHG (2015) Cold dark matter: Controversies on small scales. Proceedings of the National Academy of Science 112 (40): 12249–12255. https://doi.org/10.1073/pnas.1308716112. 1306.0913. Widrow LM and Kaiser N (1993) Using the Schrodinger equation to simulate collisionless matter. Astrophysical Journal Letters 416: L71. https://doi.org/10.1086/187073. Williamson LA and Blakie PB (2016) Universal coarsening dynamics of a quenched ferromagnetic spin-1 condensate. Physical Review Letters 116(2): 025301. https://doi.org/ 10.1103/PhysRevLett.116.025301. Witkowska E, Deuar P, Gajda M, and Rza˛ Z˙ ewski K (2011) Solitons as the early stage of quasicondensate formation during evaporative cooling. Physical Review Letters 106: 135301. https://doi.org/10.1103/PhysRevLett.106.135301. Witkowska E, Dziarmaga J, Swisłocki T, and Matuszewski M (2013) Dynamics of the modified Kibble-z˙urek mechanism in antiferromagnetic spin-1 condensates. Physical Review B 88: 054508. https://doi.org/10.1103/PhysRevB.88.054508. Wolswijk L, Mordini C, Farolfi A, Trypogeorgos D, Dalfovo F, Zenesini A, Ferrari G, and Lamporesi G (2022) Measurement of the order parameter and its spatial fluctuations across Bose-Einstein condensation. Physical Review A 105: 033316. https://doi.org/10.1103/PhysRevA.105.033316. Wouters M and Carusotto I (2007) Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons. Physical Review Letters 99: 140402. https://doi.org/10.1103/ PhysRevLett.99.140402.

Universality of Bose–Einstein condensation and quenched formation dynamics

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Wouters M, Liew TCH, and Savona V (2010) Energy relaxation in one-dimensional polariton condensates. Physical Review B 82: 245315. https://doi.org/10.1103/ PhysRevB.82.245315. Yukalov VI, Novikov AN, and Bagnato VS (2015) Realization of inverse Kibble-Zurek scenario with trapped Bose gases. Physics Letters A ISSN: 0375-9601. 379(20): 1366–1371. https://doi.org/10.1016/j.physleta.2015.02.033. Yurke B, Pargellis AN, Kovacs T, and Huse DA (1993) Coarsening dynamics of the XY model. Physical Review E 47: 1525–1530. https://doi.org/10.1103/PhysRevE.47.1525. Zamora A, Sieberer LM, Dunnett K, Diehl S, and Szymanska MH (2017) Tuning across universalities with a driven open condensate. Physical Review X 7: 041006. https://doi.org/ 10.1103/PhysRevX.7.041006. Zamora A, Dagvadorj G, Comaron P, Carusotto I, Proukakis NP, and Szymanska MH (2020) Kibble-Zurek mechanism in driven dissipative systems crossing a nonequilibrium phase transition. Physical Review Letters 125: 095301. https://doi.org/10.1103/PhysRevLett.125.095301. Zurek WH (1985) Cosmological experiments in superfluid helium? Nature 317(6037): 505–508. Zurek WH (1996) Cosmological experiments in condensed matter systems. Physics Reports 276(4): 177–221. Zurek WH (2009) Causality in condensates: Gray solitons as relics of BEC formation. Physical Review Letters 102: 105702. https://doi.org/10.1103/PhysRevLett.102.105702. Zwerger W (2011) The BCS-BEC Crossover and the Unitary Fermi Gas. Lecture Notes in Physics. Springer Berlin Heidelberg, ISBN: 9783642219771. https://books.google.co.uk/ books?id¼VjdUd5JbRooC.

Interacting Bose-condensed gases Christoph Eigena and Robert P Smithb, aCavendish Laboratory, University of Cambridge, Cambridge, United Kingdom; bClarendon Laboratory, University of Oxford, Oxford, United Kingdom © 2024 Elsevier Ltd. All rights reserved.

Introduction Noninteracting Bose gases Weak repulsive contact interactions Excitations and sound Quantum depletion Interaction effects near Tc Unitary contact interactions Attractive contact interactions and dipolar interactions Attractive contact interactions Interacting Bose mixtures Dipolar interactions Conclusion Acknowledgments References

124 125 126 126 127 128 129 130 130 131 131 132 132 133

Abstract We provide an overview of the effects of interactions in Bose-condensed gases. We focus on phenomena that have been explored in ultracold atom experiments, covering both tuneable contact interactions and dipolar interactions. Our discussion includes: modifications to the ground state and excitation spectrum, critical behavior near the Bose–Einstein condensation temperature, the unitary regime where the interactions are as strong as allowed by quantum mechanics, quantum droplets in mixtures, and supersolids in dipolar gases.

Key objectives

• • • •

Summarize ideal-gas Bose–Einstein condensation. Review the consequences of weak repulsive contact interactions on the ground state, excitations, and thermodynamics of Bose condensed gases. Showcase recent work on the unitary Bose gas. Highlight novel effects in condensates that experience attractive or dipolar interactions, including the formation of quantum droplets and supersolids.

Introduction Bose–Einstein condensation (BEC) plays an important role across a variety of quantum systems and phenomena, from superconductivity (Onnes, 1911), and exciton (Deng et al., 2002, 2006; Kasprzak et al., 2006; Balili et al., 2007) and magnon (Demokritov et al., 2006) condensation in the solid state, to superfluidity in liquid helium (Kapitza, 1938; Allen and Misener, 1938), to photon condensates in optically pumped dye-filled mircocavities (Klaers et al., 2010), and to its most direct demonstration in ultracold dilute gases (Anderson et al., 1995; Davis et al., 1995). A unique aspect of the Bose condensed state is that the (phase) order it displays is not primarily the result of interparticle interactions, but can occur in an ideal (noninteracting) gas solely due to quantum statistics. In this article we discuss how the interplay of interparticle interactions with the underlying phase order affects the Bosecondensed state, and the host of fascinating phenomena that ensue. While such questions first arose in the context of liquid helium (a strongly interacting fluid), here we focus our discussion on dilute ultracold atomic gases. In these systems the interaction strength can readily be tuned, allowing controlled access to both subtle and dramatic interaction effects. This enables tests of fundamental many-body theories and the progressive exploration toward strong interactions, where a first principles description is intractable. We further restrict our discussion to continuous Bose gases in three dimensions (3D), leaving aside fermionic systems where BEC can occur as a consequence of pairing (Inguscio et al., 2007; Zwerger, 2011; Zwierlein, 2014), Bose gases confined in optical lattices (Bloch, 2005; Morsch and Oberthaler, 2006), and lower-dimensional systems where the enhanced role of fluctuations often precludes BEC (Cazalilla et al., 2011; Hadzibabic and Dalibard, 2011). Note that we provide only a brief overview here; many of the topics are covered in much greater detail elsewhere (see, e.g., Pethick and Smith, 2002; Pitaevskii and Stringari, 2016). We also refer the reader to

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other related articles within this volume; Stringari (2023) gives a more general introduction to BEC in ultracold atom systems, and while here we focus mainly on equilibrium systems Proukakis (2023) explores non-equilibrium effects on BEC formation. The article is organized as follows. In “Noninteracting Bose gases” we briefly review Bose–Einstein condensation in a gas of noninteracting particles, before exploring the effect of different interparticle interactions on the Bose condensed state: “Weak repulsive contact interactions” and “Unitary contact interactions” consider repulsive contact interactions in, respectively, the weak and strong limit. “Attractive contact interactions and dipolar interactions” explores phenomena which occur due to either attractive or dipole-dipole interactions. Finally, we conclude and provide a brief outlook in “Conclusion.”

Noninteracting Bose gases BEC is a phase transition associated with the onset of long-range phase order, which results from the macroscopic occupation of a single-particle state. This is a consequence of Bose–Einstein statistics and occurs at a critical temperature Tc far above the singleparticle energy-level spacing. Qualitatively, the transition can be understood by considering lengthscales in a quantum gas (see Fig. 1(a)). The two relevant lengthscales are the interparticle spacing, n−1/3, set by the gas density n, and the (quantum-mechanical) thermal wavelength l ¼ [2pℏ2/(mkBT )]1/2 set by the temperature T and the mass m of the gas particles. Condensation occurs when the thermal wavelength is large enough to be comparable to the interparticle spacing (l ≳ n−1/3). More quantitatively, we consider the occupation of a momentum state p in a homogeneous ideal (noninteracting) gas, which is given by the Bose distribution fB ¼

1 , eðe − mÞ=ðkB T Þ −1

(1)

where m is the chemical potential and e ¼ p2/(2m) the energy of the state. The total number of particles in the gas is found by summing over all possible states: X ge N¼ , (2) ðe − mÞ=ðkB T Þ − 1 e e where the degeneracy factor ge accounts for the number of states with a given e. Demanding all terms in the sum to be real positive numbers requires m  e0, where e0 is the ground-state energy. As m approaches e0 from below (practically achieved by, e.g., increasing N) the ground-state occupation can become arbitrarily large (see Eq. (1)), whereas the occupation of all the other states (the excited states) sums to a finite (critical) number in 3D. This statistical saturation of excited states (graphically demonstrated in Fig. 1(b)) is, in essence, the mechanism for BEC. At fixed temperature, as particles are added to a Bose gas they populate the excited states until their population saturates at the critical atom number. From this point on, any additional particles must enter the ground state, which thus becomes macroscopically occupied. In the thermodynamic limit, in which both N and the volume of the system V are large, while the particle density n is finite, we may (semiclassically) approximate the sum over excited states by an integral. The density of excited states n0 (also known as the thermal density), is given by:   Z g3=2 em=ðkB T Þ dp 1 0 ¼ , (3) n ¼ 2 l3 ð2pℏ Þ3 eðp =ð2mÞmÞ=ðkB T Þ  1

Fig. 1 Ideal Bose gases. (a) Noninteracting ultracold Bose gases feature two relevant lengthscales: the interparticle spacing n−1/3 and the thermal wavelength l. (b) Excited- and ground-state phase space densities vs. total phase space density. Above a critical Dc (dotted vertical line), n0 l3 saturates and n0l3 grows. (c) Condensed fraction n0/n vs. T/Tc (Eq. (5)).

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P l 3/2 where the polylogarithm g3/2(x) ¼ 1 . We can re-express this result in terms of the phase space density n0 l3 ¼ g3/2(em/(kBT )), l¼1 x /l which reaches a maximum (critical) value when m ¼ 0 (which is allowed as e0 ! 0), such that, n0 l3 ¼ Dc ¼ g3=2 ð1Þ  zð3=2Þ  2:612, where z is the Riemann function. For a fixed n, the BEC transition temperature is given by kB T c ¼

 2=3 2pℏ 2 n , m zð3=2Þ

(4)

and the condensed fraction n0 =n ¼ 1 − ðT=T c Þ3=2 ,

(5)

where n0 is the condensate density (see Fig. 1(c)). In the following section we discuss how this picture changes in the presence of weak repulsive contact interactions, a regime readily accessible in ultracold-gas experiments.

Weak repulsive contact interactions For a dilute atomic gas the effective low-energy interaction between two atoms at r and r0 can be approximated as a contact interaction gd(r − r0 ) with g ¼ 4pℏ2a/m, where a is the s-wave scattering length. The interactions are thus quantified by a single lengthscale a, which can be tuned experimentally using so-called Feshbach resonances (Chin et al., 2010). The consequences of interactions are often grouped into mean-field (MF) and beyond-mean-field (BMF) effects. Simplistically speaking, MF effects result from the effective potential felt by a particle due to the average local particle density. Within the Hartree–Fock approximation (Dalfovo et al., 1999) the excited-state atoms feel an effective potential Vint0 ¼ g(2n0 + 2n0 ) and, owing to the lack of the exchange interaction for particles in the same state, the ground-state atoms experience a different interaction potential Vint,0 ¼ g(n0 + 2n0 ). In a homogeneous system, these interaction potentials lead to a uniform energy offset, i.e., e ¼ p2/2m + Vint in Eq. (1). This is accompanied by a simple shift of the chemical potential such that it is only the factor of two difference between the condensate-condensate and condensate-thermal interaction potentials that leads to non-trivial interaction effects at the MF level. By contrast, in inhomogeneous systems the entire interaction potential is important as it effectively modifies the form of the trap. This often results in MF effects being more prominent in inhomogeneous systems; see Smith (2017) for a detailed discussion. Note that, at the MF level, the condensate behavior can be described by a macroscopic wavefunction that obeys the Gross–Pitaevski equation (see Pitaevskii and Stringari, 2016), a nonlinear Schrödinger equation. Beyond-mean-field effects arise due to changes of the many-body wavefunction, sometimes referred to as quantum fluctuations. These effects become significant if the interaction energy gn is no longer negligible compared to the relevant (kinetic) energy. In particular, significant effects occur whenever low-energy (long-wavelength) excitations play a central role. We now consider the effect of interactions on the elementary excitations of a Bose-condensed system, on its ground state, and on its behavior near Tc.

Excitations and sound In a homogeneous system, at the MF-level the excitation spectrum is simply given by e(k) ¼ p2/(2m) + Vint0 − Vint,0 ¼ ℏ2k2/(2m) + gn0, where k ¼ p/ℏ is the excitation wavenumber. For low-energy (low-k) excitations this MF approach is no longer adequate; to next order the effect of interactions can be calculated using the Bogoliubov transformation (Bogoliubov, 1947), which enables a description in terms of noninteracting quasiparticles; this approach predicts an excitation spectrum sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ℏ2 k2 ℏ2 k2 eðkÞ ¼ + 2gn : (6) 2m 2m pffiffiffiffiffiffiffiffiffiffiffi Now at low-k there are collective excitations with a sound-like form e/ℏ ¼ c0k, where c0 ¼ gn=m: This is crucial for superfluidity in Bose condensed systems, as it ensures (unlike in an ideal gas) a nonzero superfluid critical velocity (according to the Landau criterion). The sound-like excitation spectrum was first seen in liquid 4He (Nozières and Pines, 1990). In ultracold atomic gases, following initial studies of collective excitations (Jin et al., 1996; Mewes et al., 1996; Andrews et al., 1997), a direct comparison to Eq. (6) was made possible by using Bragg spectroscopy on an ultracold Bose–Einstein condensate (see Steinhauer et al., 2002, Kozuma et al., 1999, Stenger et al., 1999 and Fig. 2(a)). At finite temperature the situation is further complicated by the presence of thermal excitations, which lead to two distinct sound modes, the (faster) first and (slower) second sound. The two modes can be understood, in the hydrodynamic limit, within the two-fluid model (Landau, 1941), which separately considers the normal and superfluid components. The nature of the sound modes changes depending on how compressible the Bose fluid is. In the almost incompressible liquid helium, first and second sound are, respectively, density and temperature (or entropy) waves. Instead, in a dilute Bose fluid (which is highly compressible) the two modes are excitations of mainly the normal and superfluid components (see Hilker et al., 2022 and Fig. 2(b)). Quantized vortices are another important type of excitation that feature in interacting Bose-condensed gases. Such vortices carryffi pffiffiffiffiffiffiffiffiffiffi angular momentum quantized in units of ℏ/m and have vanishing density along the vortex core over a healing length 1= 8pna (see Madison et al., 2000 and Fig. 2(c)). In a system containing many vortices the interactions between them can result in the

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Fig. 2 Excitations in interacting degenerate Bose gases. (a) Excitation spectrum of a weakly interacting Bose–Einstein condensate, measured in a harmonic trap (Steinhauer et al., 2002). The solid line shows the fitted Bogoliubov form, with a linear spectrum at low momenta (dotted line), the slope of which gives the speed of sound. (b) At finite temperature a Bose fluid supports two sound modes, here measured in a box-trapped ultracold gas (Hilker et al., 2022); the speeds are normalized by the Bogoliubov speed and the solid lines show the two-fluid model predictions. (c) Vortices are another important type of excitation; the image shows a single vortex generated via stirring a Bose–Einstein condensate with a laser beam (Madison et al., 2000). (d) When large numbers of vortices are generated the interactions between them lead to the formation of an Abrikosov lattice (Abo-Shaeer et al., 2001). Panels adapted from: (a) Steinhauer, R Ozeri, N Katz, and N Davidson (2002) Excitation spectrum of a Bose–Einstein condensate, Physical Review Letters 88:120407, (b) Hilker, LH Dogra, C Eigen, JAP Glidden, RP Smith, and Z Hadzibabic (2022) First and second sound in a compressible 3D Bose fluid, Physical Review Letters 128:223601, (c) Madison, F Chevy, W Wohlleben, and J Dalibard (2000) Vortex formation in a stirred Bose–Einstein condensate, Physical Review Letters 84:806, (d) Abo-Shaeer, C Raman, JM Vogels, and W Ketterle (2001) Observation of vortex lattices in Bose–Einstein condensates, Science 292:476.

formation of vortex lattices (see Abo-Shaeer et al., 2001 and Fig. 2(d)). Note that these lattices are closely related to those that occur in type-II superconductors in the presence of a magnetic field; rotation can be thought of as an artificial magnetic field, which is an example of a synthetic gauge field (see, e.g., Goldman et al., 2014; Zhai, 2015; Aidelsburger et al., 2018).

Quantum depletion In addition to modifying the excitation spectrum, interparticle interactions also modify the ground state, an effect known as quantum depletion. This entails the coherent expulsion of particles from the single particle (p ¼ 0) ground state, even at T ¼ 0; in essence, it is energetically favorable for some higher momentum states to be occupied. The most striking example of quantum depletion occurs in liquid 4He, where even though at T ¼ 0 the system is 100% superfluid, the condensed fraction n0/n is only around 10%. However, liquid 4He is a strongly interacting fluid, and so theoretical predictions are challenging. Instead, for a weakly interacting gas [(na3)1/2  1], a theoretical description is more tractable, with T ¼ 0 Bogoliubov theory predicting (Lee et al., 1957) n0 8  1=2 : ¼ 1 − pffiffiffi na3 n 3 p

(7)

This famous prediction was experimentally verified in a box-trapped (homogeneous) ultracold gas (see Lopes et al., 2017 and Fig. 3). Within the same theoretical framework, the ground-state energy is also modified, as first measured by Navon et al. (2011).

Fig. 3 Measurement of quantum depletion. The maximal diffracted fraction  vs. the interaction parameter (na3)1/2; here a Bragg filtering technique was used to spatially separate the condensate from the high-k components of the gas (including the quantum depletion; see top inset). Assuming both perfect k-space separation and filtering, corresponds to the condensed fraction. A linear fit (solid line) gives a slope g ¼ 1.5(2), quantitatively confirming the Bogoliubov prediction  g ¼ 8= 3p1=2  1:5: The fact that the intercept is only close to unity, arises from an imperfect k-space separation and a non-zero initial temperature. The cartoon (bottom inset) depicts the coherent excitations out of the (blue) condensate, which occur as pairs of atoms with opposite momenta. Figure adapted from Lopes, C Eigen, N Navon, D Clément, RP Smith, and Z Hadzibabic (2017) Quantum depletion of a homogeneous Bose–Einstein condensate, Physical Review Letters 119:190404.

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Interaction effects near Tc We now turn to the effect of interactions on the thermodynamics of a Bose condensed gas, focusing on the behavior close to Tc; these effects are summarized in Figs. 4(a, b). First, in the (partially) condensed phase, interactions cause n0 l3 to no longer be saturated at Dc (cf. Figs. 1(b) and 4(a)), but instead it decreases for increasing nl3. This effect comes about (at the MF level) because an atom can reduce its interaction energy by entering the condensate. Note that for a harmonically trapped gas the opposite effect occurs, with the excited-state population increasing as the system goes more deeply into the condensed state (Tammuz et al., 2011). Secondly, condensation occurs at Dc < D0c (or equivalently for Tc > T0c at fixed n); here the superscript 0 denotes the ideal-gas results. This effect can only be understood at the BMF level, as long-wavelength fluctuations become increasingly important close to Tc. Calculating the shift was a significant theoretical challenge and it took several decades to reach a consensus (see Andersen, 2004; Arnold and Moore, 2001; Baym et al., 2001; Holzmann et al., 2004 for a review); the shift is predicted to be (Arnold and Moore, 2001; Kashurnikov et al., 2001) DT c a  1:8 , l T 0c

(8)

but this result has yet to be confirmed experimentally. In a harmonic trap, a relatively simple geometric (MF) effect results in a negative Tc shift, with the more interesting BMF effects expected to be pushed to higher order in a/l. As shown in Fig. 4(c), measurements in a harmonic trap (Smith et al., 2011) did show an upward shift from the MF prediction (dashed line), but making a quantitative comparison with Eq. (8) (dotted line) was not possible. Thirdly, interactions modify the critical exponents of the BEC transition, changing the universality class to that of the 3D XY model. In particular, the correlation length critical exponent is predicted to change from v ¼ 1 to v  0.67 (Burovski et al., 2006a); see Fig. 4(b). This exponent has been measured by locally probing a harmonically trapped ultracold gas (see Donner et al., 2007 and Fig. 4(d)), and also with much higher precision in 4He (see Burovski et al., 2006b).

Fig. 4 Thermodynamics near Tc. In (a) and (b) we illustrate the effects of interactions near Tc in a homogeneous system, while (c) and (d) highlight related experimental results. (a) Excited- and ground-state phase-space densities as a function of the total phase-space density for an interacting Bose gas (cf. Fig. 1(b)). For nl3 > Dc , n0 l3 is no longer saturated at Dc , but decreases as nl3 increases. The critical phase-space density also undergoes a small shift. (b) The correlation length x vs. reduced temperature t ¼ (T − Tc)/Tc near the BEC transition. The interacting Bose gas is in the 3D XY Model universality class, which has x  t −v with v  0.67. (c) Measured Tc shift vs. interaction strength for a harmonically trapped Bose gas Smith et al. (2011); an upward shift from the predominant MF geometric effect (dashed line) is seen, but quantitatively comparing this to the homogeneous system result (dotted line) was not possible. (d) Measurement of the correlation length critical exponent in a ultracold Bose gas Donner et al. (2007); the line is a fit to the data, which gives v ¼ 0.67(13) consistent with the 3D XY Model. Panels adapted from: (c) Smith, RLD Campbell, N Tammuz, and Z Hadzibabic (2011) Effects of interactions on the critical temperature of a trapped Bose gas, Physical Review Letters 106:250403, (d) Donner, S Ritter, T Bourdel, A Ottl, M Köhl, and T Esslinger (2007) Critical behavior of a trapped interacting Bose gas, Science 315:1556.

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Unitary contact interactions Here we consider Bose-condensed systems in the strongly interacting limit (na3  1), experimentally accessible at a Feshbach resonance, where a diverges. In this so-called unitary regime, the interactions between particles are as strong as allowed by quantum mechanics. The fact that the value of the diverging a can no longer matter leads to the universality hypothesis, in which the interparticle spacing n−1/3 is the only relevant lengthscale for a degenerate gas (see Cowell et al., 2002, Ho, 2004 and Fig. 5(a)). This also sets the natural momentum, energy, and time scales:  1=3 ℏkn ¼ ℏ 6p2 n , En ¼ ℏ2 k2n =ð2mÞ, and t n ¼ ℏ=En :

(9)

In contrast to their widely studied Fermi counterparts (Inguscio et al., 2007; Zwerger, 2011; Zwierlein, 2014), unitary Bose gases feature dramatically enhanced particle loss1 and associated heating, which makes their study an inherently dynamical problem and raised the question whether such a gas can ever exist in equilibrium. The experimental approach has been to forsake equilibrium and quench into the unitary regime (by rapidly increasing a), to then observe the ensuing dynamics. These experiments have shown a remarkable degree of universality (see Makotyn et al., 2014, Eigen et al., 2017, 2018, Klauss et al., 2017 and Fig. 5(b, c)). The post-quench dynamics also revealed that the gas, even though it is not in true thermal equilibrium, does attain a quasi-equilibrium (prethermal) state (Berges et al., 2004; Makotyn et al., 2014; Yin and Radzihovsky, 2016; Eigen et al., 2018). Eventually, the gas decays and heats, ultimately _ ðEn =EÞ2 , where E is the kinetic energy per particle following the dimensionless loss rate for a thermal unitary gas −t n n=n∝ (dashed line in Fig. 5(b); see also thermal unitary-gas measurements Rem et al., 2013, Fletcher et al., 2013, Eismann et al., 2016).

Fig. 5 Universal behavior in unitary Bose gases. (a) In a weakly interacting Bose gas (left) the interactions, characterized by a, set the relevant lengthscale (size of the circles in the cartoon), so that when “zooming in” on a high-density gas it looks different to a low-density gas. In the unitary regime (right), upon zooming in, the high-density gas instead looks the same as the low-density one, exhibiting scale invariance. (b, c) Examples of experimental verification of the universality hypothesis. For initially degenerate Bose gases of initial density n0 quenched to unitarity, both (b) the dimensionless loss rate vs. E/En (Eigen et al., 2017) and (c) the momentum-dependent prethermal relaxation timescale t/tn0 (Eigen et al., 2018) are universal functions that depend solely on the density. Panels adapted from: (b) Eigen, JAP Glidden, R Lopes, N Navon, Z Hadzibabic, and RP Smith (2017) Universal scaling laws in the dynamics of a homogeneous unitary Bose gas, Physical Review Letters 119:250404, (c) Eigen, JAP Glidden, R Lopes, EA Cornell, RP Smith, and Z Hadzibabic (2018) Universal prethermal dynamics of Bose gases quenched to unitarity, Nature 563:221.

1 3i −5 i −1 _ _ For i-body loss, dimensional analysis gives n=n∝ℏ=ma n (Thøgersen et al., 2008; Mehta et al., 2009), which for na3  1 gives n=n∝1=t n for all i. Also note that in Bose gases Efimov physics further modulates the atom loss (Efimov, 1970; Braaten and Hammer, 2007; Naidon and Endo, 2017).

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Intriguingly, the observed post-quench prethermal state featured: (1) a momentum-dependent relaxation timescale approximately given by ℏ/e(k) (see Eq. (6)) with gn replaced by En (see Fig. 5(c)), and (2) a universal momentum distribution, which implies a non-zero condensed fraction. This hints at the possibility of a novel superfluid state consisting of Efimov trimers,2 which has also been theoretically predicted (Piatecki and Krauth, 2014; Musolino et al., 2022).

Attractive contact interactions and dipolar interactions So far, we have only discussed the effect of repulsive contact interactions on a single component Bose gas, which for na3  1 leads to relatively modest changes (see “Weak repulsive contact interactions”). We now turn to scenarios where the interactions are (at least partially) attractive. This can lead to far more dramatic consequences and also to situations where BMF effects play a critical role. In the following, we in turn discuss: single component condensates with weak attractive contact interactions, quantum mixtures, and situations in which the atoms interact via both contact and dipolar interactions.

Attractive contact interactions For attractive contact interactions (a < 0) in the thermodynamic limit, Bose–Einstein condensates are not stable, and are susceptible to collapse. This can be seen by examining the Bogoliubov excitation spectrum (Eq. (6)) and noting that negative values of g lead to imaginary excitation energies at low k, signifying unstable (exponentially growing) modes; this is sometimes referred to as a phonon instability. Trapped (finite-sized) 3D condensates can be stabilized by the ground-state kinetic energy (quantum pressure), such that for a given atom number N, collapse only occurs above a critical attractive |ac|. Note that in lower-dimensional systems the situation is somewhat different, e.g., in 1D the balance of kinetic and interaction energy permits the formation of solitons (Strecker et al., 2002; Khaykovich et al., 2002). Condensate collapse was originally explored for atoms confined in harmonic traps (Gerton et al., 2000; Roberts et al., 2001; Donley et al., 2001) and later also for those in 3D box potentials (Eigen et al., 2016). In Fig. 6(a, b) we show, respectively, experimental images of the collapse in harmonically and box-trapped gases. In both cases, the critical point is given by Nac ∝ L, where L is the linear size of the single-particle ground state (see Gerton et al., 2000, Roberts et al., 2001, Donley et al., 2001, Eigen et al., 2016 and Fig. 6(c)). The basic picture can again be understood from Eq. (6): Equating 2gn to ℏ2k2/(2m) with (the minimum) k  p/L (using n  N/L3) predicts a critical value of Nac/L  − 0.2 (cf. Fig. 6(c)). More quantitatively, the collapse of such matter waves can be well described using the Gross–Pitaevskii equation. In single-component condensates with attractive contact interactions, the BMF effects are typically negligible as a is close to zero. However, in situations where two (potentially) large MF effects cancel, BMF effects can become important and even lead to qualitatively new effects; examples of such scenarios are explored in the two following sections.

Fig. 6 Condensate collapse. (a, b) Absorption images of ultracold quantum gases following wave collapse for (a) a harmonically trapped (Donley et al., 2001) and (b) a box-trapped (Eigen et al., 2016) cloud. (c) Measured critical ac (in units of the Bohr radius a0) vs. 1/N for collapse of a condensate confined in a cylindrical box trap of size L (length equal to diameter) (Eigen et al. (2016)). The solid line shows a fit to the data, which gives −ac ¼ 0.16(4)L/N consistent with the Gross–Pitaevskii equation prediction −ac ¼ 0.17L/N. Panels adapted from: (a) Donley, NR Claussen, SL Cornish, JL Roberts, EA Cornell, and CE Wieman (2001) Dynamics of collapsing and exploding Bose–Einstein condensates, Nature 412:295, (b, c) Eigen, AL Gaunt, A Suleymanzade, N Navon, Z Hadzibabic, and RP Smith (2016) Observation of weak collapse in a Bose–Einstein condensate, Physical Review X 6:041058.

2

The Efimov effect is a quantum three-body effect first discussed in nuclear physics (Efimov, 1970).

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Fig. 7 Realizing quantum droplets in Bose mixtures. (a) Illustrative phase diagram for a11 ¼ a22 > 0 showing the miscible, immiscible, collapse, and droplet pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi regimes. At the MF level for a homogeneous gas, the miscible region is given by − a11 a22 < a12 < a11 a22 (between the dotted lines). Quantum droplets can be pffiffiffiffiffiffiffiffiffiffiffiffi formed in a sliver of the phase diagram near da ¼ a12 + a11 a22 ¼ 0, where BMF effects can become important (see text). (b) Images demonstrating the existence of a droplet phase (Cabrera et al., 2018); for da ¼ − 3.2a0 (left) the cloud remains self-bound after release from the trap, compared to da ¼ 1.2a0 (right), which shows the typical expansion of a miscible gas. Panel (b) adapted from Cabrera, L Tanzi, J Sanz, B Naylor, P Thomas, P Cheiney, and L Tarruell (2018) Quantum liquid droplets in a mixture of Bose–Einstein condensates, Science 359:301.

Interacting Bose mixtures In a two-component quantum mixture with contact interactions between the constituents, there are three relevant scattering lengths: two intrastate ones (a11 and a22), and an interstate one (a12). A sketch of the zero-temperature phase diagram of such a mixture is shown in Fig. 7(a) for the case a11 ¼ a22 > 0. pffiffiffiffiffiffiffiffiffiffiffiffiffi At the MF level, for a12 > a11 a22 the interstate repulsion causes the two components to separate (the system is immiscible), pffiffiffiffiffiffiffiffiffiffiffiffiffi whereas for a12 < − a11 a22 the interstate attraction overwhelms the intrastate repulsion, leading to collapse. In between, the two components are miscible—forming overlapping condensates. The miscible-immiscible transition has been studied since the early days of atomic-gas condensates (see, e.g., Hall et al., 1998; Stenger et al., 1998; Papp et al., 2008). pffiffiffiffiffiffiffiffiffiffiffiffiffi At the collapse boundary, where the MF terms (almost) cancel, the residual mean-field interaction, da ¼ a12 + a11 a22 , is small. However, in contrast to a single-component gas near the collapse boundary, the BMF effects are not necessarily negligible. The BMF effects are typically repulsive in nature and scale more strongly with density than the MF effects that drive the collapse, opening the possibility of quantum liquid droplets where (at an appropriate density) the attractive MF attraction can be balanced by BMF repulsion (Petrov, 2015). Such droplets were realized in 2018 (see Cabrera et al., 2018, Semeghini et al., 2018 and Fig. 7(b)). The “liquid” label refers to the fact that, in principle, the droplets are stabilized at a fixed density and are self bound; in practice, the high densities involved lead to (three-body) losses that limit the droplet lifetime. In the special case where da  0, a so-called “LHY fluid” is formed, which only features BMF interactions (Skov et al., 2021).

Dipolar interactions In this section we briefly review novel effects of dipole-dipole interactions on Bose-condensed gases (see, e.g., Lahaye et al., 2009, Norcia and Ferlaino, 2021, Chomaz et al., 2023 for a more detailed treatment). Such interactions, which arise for particles with a permanent dipole moment, are long-range and anisotropic; the interaction potential for two polarized dipoles a distance r ¼ |r| apart is given by V dd ¼

Cdd 1 − 3 cos 2 y , 4p r3

(10)

where y is the angle between r and the polarization axis, and Cdd the dipole coupling constant, which is given by m0m2(d2/e0) for magnetic (electric) dipoles. The strength of dipolar interactions are also commonly captured by the dipolar length add ¼ Cddm/ (12pℏ2). For a homogeneous gas with both contact and dipole-dipole interactions, the Bogoliubov excitation spectrum (Eq. (6)) is modified such that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi     ℏ2 k2 ℏ2 k2 1 n , (11) eðkÞ ¼ + 2gn + 2Cdd cos 2 a − 2m 2m 3 where a is the angle between k and the polarization direction. Several consequences of dipolar interactions can directly be inferred from Eq. (11). Even for weak dipolar interactions the excitation spectrum is anisotropic, which leads to anisotropic sound waves (Bismut et al., 2012). Moreover, for Cdd/3 > g (or equivalently add > a) unstable phonon modes exist and (at least from a MF perspective) the gas

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Fig. 8 Dipolar Bose gases. (a) Absorption image revealing the effects of dipolar interactions following the collapse and subsequent explosion of a condensate (Lahaye et al., 2008). (b) Observation of dipolar quantum droplets (Kadau et al., 2016). (c) Sketch of the roton excitation spectrum seen in dipolar gases that are confined along their polarization direction. As the relative strength of dipole-dipole interactions increases the roton minimum deepens and then triggers (red line) an instability near krot. (d) Cartoon of the phase diagram of a dipolar gas in a cigar-shaped trap. Starting with a condensate, reducing a/add triggers the roton instability resulting in an array of droplets. For a sliver of the phase diagram a supersolid state exists, in which the droplets are phase coherent (Tanzi et al., 2019; Böttcher et al., 2019; Chomaz et al., 2019). (e) Observation of a 2D supersolid (Norcia et al., 2021; Bland et al., 2022); (left) an in-situ image revealing a hexagonal density modulationand (right) a time-of-flight (ToF) image, averaged over multiple realizations, which demonstrates the global phase coherence of the system. Panels adapted from: (a) Lahaye, J Metz, B Fröhlich, T Koch, M Meister, A Griesmaier, T Pfau, H Saito, Y Kawaguchi, and M Ueda (2008) d-Wave collapse and explosion of a dipolar Bose–Einstein condensate, Physical Review Letters 101:080401, (b) Kadau, M Schmitt, M Wenzel, C Wink, T Maier, I Ferrier-Barbut, and T Pfau (2016) Observing the Rosensweig instability of a quantum ferrofluid, Nature 530:194,, (e) Bland, E Poli, C Politi, L Klaus, MA Norcia, F Ferlaino, L Santos, and RN Bisset (2022) Two-dimensional supersolid formation in dipolar condensates, Physical Review Letters 128:195302.

can undergo collapse, with the angular dependence of the interactions leading to spectacular collapse dynamics (see Lahaye et al., 2007 and Fig. 8(a)). As in the case of a Bose mixture, collapse occurs at a nonzero g, opening up the possibility of the collapse being arrested by BMF effects, which again can result in the formation of quantum droplets (see Kadau et al., 2016, Schmitt et al., 2016, Ferrier-Barbut et al., 2016, Chomaz et al., 2016 and Fig. 8(b)). The presence of a trap tends to stabilize the system. A particularly interesting situation arises when a dipolar gas is tightly confined along the polarization direction, which leads to the development of an in-plane roton-like excitation spectrum (see Petter et al., 2019 and Fig. 8(c)). Note that although similar to the roton excitation spectrum originally seen in liquid 4He (see, e.g., Donnelly et al., 1981), its microscopic origin is different, occurring as the consequence of both confinement and long-range anisotropic interactions. When the roton gap closes, an instability occurs at finite k ¼ krot, such that without any stabilization mechanism a runaway density modulation would ensue. However, the collapse can yet again be halted by BMF effects, leading to the formation of a droplet array. For a narrow range of parameters close the stability boundary, these droplets remain phase coherent, signaling the existence of a supersolid state—a state with both long-range phase order (and hence superfluidity) and long-range spatial order (the density modulation) (see Tanzi et al., 2019; Böttcher et al., 2019; Chomaz et al., 2019; Norcia et al., 2021; Bland et al., 2022 and Fig. 8(d, e)).

Conclusion In summary, we have reviewed effects of both contact and dipolar interactions on Bose-condensed gases. Many of the observed phenomena have been long predicted, if only more recently confirmed in ultracold gases, while others, such as the observation of supersolidity, were largely unexpected. Looking forward, further breakthroughs in both categories are within reach. A major longstanding task is to experimentally resolve the simple question of how weak interactions shift the BEC transition temperature. On the more exploratory side, the realms of strong contact and dipolar interactions offer the possibility of novel many-body physics.

Acknowledgments We thank Zoran Hadzibabic for comments on the manuscript. C. E. acknowledges support from Jesus College (Cambridge).

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References Abo-Shaeer JR, Raman C, Vogels JM, and Ketterle W (2001) Observation of vortex lattices in Bose–Einstein condensates. Science 292: 476. Aidelsburger M, Nascimbene S, and Goldman N (2018) Artificial gauge fields in materials and engineered systems. Comptes Rendus Physique 19: 394. Allen JF and Misener AD (1938) Flow of liquid helium II. Nature 141: 75. Andersen JO (2004) Theory of the weakly interacting Bose gas. Reviews of Modern Physics 76: 599. Anderson MH, Ensher JR, Matthews MR, Wieman CE, and Cornell EA (1995) Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269: 198. Andrews MR, Kurn DM, Miesner H-J, Durfee DS, Townsend CG, Inouye S, and Ketterle W (1997) Propagation of sound in a Bose–Einstein condensate. Physical Review Letters 79: 553. Arnold P and Moore G (2001) BEC transition temperature of a dilute homogeneous imperfect Bose gas. Physical Review Letters 87: 120401. Balili R, Hartwell V, Snoke D, Pfeiffer L, and West K (2007) Bose-Einstein condensation of microcavity polaritons in a trap. Science 316: 1007. Baym G, Blaizot J-P, Holzmann M, Laloë F, and Vautherin D (2001) Bose–Einstein transition in a dilute interacting gas. European Physical Journal B: Condensed Matter and Complex Systems 24: 107. Berges J, Borsányi S, and Wetterich C (2004) Prethermalization. Physical Review Letters 93: 142002. Bismut G, Laburthe-Tolra B, Maréchal E, Pedri P, Gorceix O, and Vernac L (2012) Anisotropic excitation spectrum of a dipolar quantum Bose gas. Physical Review Letters 109: 155302. Bland T, Poli E, Politi C, Klaus L, Norcia MA, Ferlaino F, Santos L, and Bisset RN (2022) Two-dimensional supersolid formation in dipolar condensates. Physical Review Letters 128: 195302. Bloch I (2005) Ultracold quantum gases in optical lattices. Nature Physics 1: 23. Bogoliubov NN (1947) On the theory of superfluidity. Journal of Physics (USSR) 11(23). Böttcher F, Schmidt J-N, Wenzel M, Hertkorn J, Guo M, Langen T, and Pfau T (2019) Transient supersolid properties in an array of dipolar quantum droplets. Physical Review X 9: 011051. Braaten E and Hammer H-W (2007) Efimov physics in cold atoms. Ann. Phys. 322: 120. Burovski E, Machta J, Prokof’ev N, and Svistunov B (2006a) High-precision measurement of the thermal exponent for the three-dimensional x y universality class. Physical Review B 74: 132502. Burovski E, Prokof’ev N, Svistunov B, and Troyer M (2006b) Critical temperature and thermodynamics of attractive fermions at unitarity. Physical Review Letters 96: 160402. Cabrera CR, Tanzi L, Sanz J, Naylor B, Thomas P, Cheiney P, and Tarruell L (2018) Quantum liquid droplets in a mixture of Bose–Einstein condensates. Science 359: 301. Cazalilla MA, Citro R, Giamarchi T, Orignac E, and Rigol M (2011) One dimensional bosons: From condensed matter systems to ultracold gases. Reviews of Modern Physics 83: 1405. Chin C, Grimm R, Julienne P, and Tiesinga E (2010) Feshbach resonances in ultracold gases. Reviews of Modern Physics 82: 1225. Chomaz L, Baier S, Petter D, Mark MJ, Wächtler F, Santos L, and Ferlaino F (2016) Quantum-fluctuation-driven crossover from a dilute Bose–Einstein condensate to a macrodroplet in a dipolar quantum fluid. Physical Review X 6: 041039. Chomaz L, Ferrier-Barbut I, Ferlaino F, Laburthe-Tolra B, Lev BL, and Pfau T (2023) Dipolar physics: A review of experiments with magnetic quantum gases. Rep. Prog. Phys. 86: 026401. Chomaz L, Petter D, Ilzhöfer P, Natale G, Trautmann A, Politi C, Durastante G, van Bijnen RMW, Patscheider A, Sohmen M, Mark MJ, and Ferlaino F (2019) Long-lived and transient supersolid behaviors in dipolar quantum gases. Physical Review X 9: 021012. Cowell S, Heiselberg H, Mazets IE, Morales J, Pandharipande VR, and Pethick CJ (2002) Cold Bose gases with large scattering lengths. Physical Review Letters 88: 210403. Dalfovo F, Giorgini S, Pitaevkii LP, and Stringari S (1999) Theory of Bose–Einstein condensation in trapped gases. Reviews of Modern Physics 71: 463. Davis KB, Mewes MO, Andrews MR, van Druten NJ, Durfee DS, Kurn DM, and Ketterle W (1995) Bose–Einstein condensation in a gas of sodium atoms. Physical Review Letters 75: 3969. Demokritov SO, Demidov VE, Dzyapko O, Melkov GA, Serga AA, Hillebrands B, and Slavin AN (2006) Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443: 430. Deng H, Weihs G, Santori C, Bloch J, and Yamamoto Y (2002) Condensation of semiconductor microcavity exciton polaritons. Science 298: 199. Deng H, Press D, Götzinger S, Solomon GS, Hey R, Ploog KH, and Yamamoto Y (2006) Quantum degenerate exciton-polaritons in thermal equilibrium. Physical Review Letters 97: 146402. Donley EA, Claussen NR, Cornish SL, Roberts JL, Cornell EA, and Wieman CE (2001) Dynamics of collapsing and exploding Bose–Einstein condensates. Nature 412: 295. Donnelly RJ, Donnelly JA, and Hills RN (1981) Specific heat and dispersion curve for helium II. Journal of Low Temperature Physics 44: 471. Donner T, Ritter S, Bourdel T, Ottl A, Köhl M, and Esslinger T (2007) Critical behavior of a trapped interacting Bose gas. Science 315: 1556. Efimov V (1970) Energy levels arising from resonant two-body forces in a three-body system. Physics Letters B 33: 563. Eigen C, Gaunt AL, Suleymanzade A, Navon N, Hadzibabic Z, and Smith RP (2016) Observation of weak collapse in a Bose–Einstein condensate. Physical Review X 6: 041058. Eigen C, Glidden JAP, Lopes R, Navon N, Hadzibabic Z, and Smith RP (2017) Universal scaling laws in the dynamics of a homogeneous unitary Bose gas. Physical Review Letters 119: 250404. Eigen C, Glidden JAP, Lopes R, Cornell EA, Smith RP, and Hadzibabic Z (2018) Universal prethermal dynamics of Bose gases quenched to unitarity. Nature 563: 221. Eismann U, Khaykovich L, Laurent S, Ferrier-Barbut I, Rem BS, Grier AT, Delehaye M, Chevy F, Salomon C, Ha L-C, and Chin C (2016) Universal loss dynamics in a unitary BOSE gas. Physical Review X 6: 021025. Ferrier-Barbut I, Kadau H, Schmitt M, Wenzel M, and Pfau T (2016) Observation of quantum droplets in a strongly dipolar BOSE gas. Physical Review Letters 116: 215301. Fletcher RJ, Gaunt AL, Navon N, Smith RP, and Hadzibabic Z (2013) Stability of a unitary BOSE gas. Physical Review Letters 111: 125303. Gerton JM, Strekalov D, Prodan I, and Hulet RG (2000) Direct observation of growth and collapse of a Bose–Einstein condensate with attractive interactions. Nature 408: 692. Goldman N, Juzeliunas G, Öhberg P, and Spielman IB (2014) Light-induced gauge fields for ultracold atoms. Reports on Progress in Physics 77: 126401. Hadzibabic Z and Dalibard J (2011) Two-dimensional Bose fluids: An atomic physics perspective. Rivista del Nuovo Cimento della Societa Italiana di Fisica 34: 389. Hall DS, Matthews MR, Ensher JR, Wieman CE, and Cornell EA (1998) The dynamics of component separation in a binary mixture of Bose–Einstein condensates. Physical Review Letters 81: 4531. Hilker TA, Dogra LH, Eigen C, Glidden JAP, Smith RP, and Hadzibabic Z (2022) First and second sound in a compressible 3D Bose fluid. Physical Review Letters 128: 223601. Ho T-L (2004) Universal thermodynamics of degenerate quantum gases in the unitarity limit. Physical Review Letters 92: 090402. Holzmann M, Fuchs JN, Baym G, Blaizot JP, and Laloë F (2004) Bose–Einstein transition temperature in a dilute repulsive gas. Comptes Rendus Physique 5: 21. Inguscio M, Ketterle W, and Salomon C (2007) Ultracold Fermi Gases, Proceedings of the International School of Physics “Enrico Fermi", Course CLXIV. Jin DS, Ensher JR, Matthews MR, Wieman CE, and Cornell EA (1996) Collective excitations of a Bose–Einstein condensate in a dilute gas. Physical Review Letters 77: 420. Kadau H, Schmitt M, Wenzel M, Wink C, Maier T, Ferrier-Barbut I, and Pfau T (2016) Observing the Rosensweig instability of a quantum ferrofluid. Nature 530(194): 1508.05007. Kapitza P (1938) Viscosity of liquid helium below the l - Point,. Nature 141: 74. Kashurnikov VA, Prokof’ev NV, and Svistunov BV (2001) Critical temperature shift in weakly interacting Bose gas. Physical Review Letters 87: 120402. Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P, Keeling JMJ, Marchetti FM, Szymanska MH, Andre R, Staehli JL, Savona V, Littlewood PB, Deveaud B, and Dang LS (2006) Bose–Einstein condensation of exciton polaritons. Nature 443: 409. Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr LD, Castin Y, and Salomon C (2002) Formation of a matter-wave bright soliton. Science 296: 1290.

134

Interacting Bose-condensed gases

Klaers J, Schmitt J, Vewinger F, and Weitz M (2010) Bose–Einstein condensation of photons in an optical microcavity. Nature 468: 545. Klauss CE, Xie X, Lopez-Abadia C, D’Incao JP, Hadzibabic Z, Jin DS, and Cornell EA (2017) Observation of Efimov molecules created from a resonantly interacting Bose gas. Physical Review Letters 119: 143401. Kozuma M, Deng L, Hagley EW, Wen J, Lutwak R, Helmerson K, Rolston SL, and Phillips WD (1999) Coherent splitting of Bose–EINSTEIN condensed atoms with optically induced bragg diffraction. Physical Review Letters 82: 871. Lahaye T, Koch T, Frohlich B, Fattori M, Metz J, Griesmaier A, Giovanazzi S, and Pfau T (2007) Strong dipolar effects in a quantum ferrofluid. Nature 448: 672. Lahaye T, Metz J, Fröhlich B, Koch T, Meister M, Griesmaier A, Pfau T, Saito H, Kawaguchi Y, and Ueda M (2008) d-Wave collapse and explosion of a dipolar Bose–Einstein condensate. Physical Review Letters 101: 080401. Lahaye T, Menotti C, Santos L, Lewenstein M, and Pfau T (2009) The physics of dipolar bosonic quantum gases. Reports on Progress in Physics 72: 126401. Landau LD (1941) The theory of superfluidity of helium II. Journal of Physics (USSR) 5(71). Lee TD, Huang K, and Yang CN (1957) Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Physics Review 106: 1135. Lopes R, Eigen C, Navon N, Clément D, Smith RP, and Hadzibabic Z (2017) Quantum depletion of a homogeneous BOSE–EINSTEIN condensate. Physical Review Letters 119: 190404. Madison KW, Chevy F, Wohlleben W, and Dalibard J (2000) Vortex formation in a stirred Bose–Einstein condensate. Physical Review Letters 84: 806. Makotyn P, Klauss CE, Goldberger DL, Cornell EA, and Jin DS (2014) Universal dynamics of a degenerate unitary Bose gas. Nature Physics 10: 116. Mehta NP, Rittenhouse ST, D’Incao JP, von Stecher J, and Greene CH (2009) General theoretical description of n-body recombination. Physical Review Letters 103: 153201. Mewes M-O, Andrews MR, van Druten NJ, Kurn DM, Durfee DS, Townsend CG, and Ketterle W (1996) Collective excitations of a Bose–Einstein condensate in a magnetic trap. Physical Review Letters 77: 988. Morsch O and Oberthaler M (2006) Dynamics of Bose–Einstein condensates in optical lattices. Reviews of Modern Physics 78: 179. Musolino S, Kurkjian H, Van Regemortel M, Wouters M, Kokkelmans SJJMF, and Colussi VE (2022) Bose–Einstein condensation of EFIMOVIAN triples in the unitary BOSE gas. Physical Review Letters 128(020401). Naidon P and Endo S (2017) Efimov physics: A review. Reports on Progress in Physics 80: 056001. Navon N, Piatecki S, Günter K, Rem B, Nguyen TC, Chevy F, Krauth W, and Salomon C (2011) Dynamics and Thermodynamics of the low-temperature strongly interacting Bose gas. Physical Review Letters 107: 135301. Norcia MA and Ferlaino F (2021) Developments in atomic control using ultracold magnetic lanthanides. Nature Physics 17: 1349. Norcia MA, Politi C, Klaus L, Poli E, Sohmen M, Mark MJ, Bisset RN, Santos L, and Ferlaino F (2021) Two-dimensional supersolidity in a dipolar quantum gas. Nature 596: 357. Nozières P and Pines D (1990) The Theory of Quantum Liquids. Westview Press. Onnes HK (1911) Further Experiments With Liquid Helium. D. On the Change of Electrical Resistance of Pure Metals at Very Low Temperatures, etc. V. The Disappearance of the Resistance of Mercury. Proc. K. Ned. Akad. Wet. , Vol. 14, p. 113. Papp SB, Pino JM, and Wieman CE (2008) Tunable miscibility in a dual-species Bose–Einstein condensate. Physical Review Letters 101: 040402. Pethick CJ and Smith H (2002) Bose–Einstein Condensation in Dilute Gases. Cambridge University Press. Petrov DS (2015) Quantum mechanical stabilization of a collapsing Bose-Bose mixture. Physical Review Letters 115: 155302. Petter D, Natale G, van Bijnen RMW, Patscheider A, Mark MJ, Chomaz L, and Ferlaino F (2019) Probing the roton excitation spectrum of a stable dipolar BOSE gas. Physical Review Letters 122: 183401. Piatecki S and Krauth W (2014) Efimov-driven phase transitions of the unitary Bose gas. Nature Communications 5: 3503. Pitaevskii L and Stringari S (2016) Bose–Einstein Condensation and Superfluidity. Oxford University Press. Proukakis NP (2023) Universality of Bose-Einstein condensation and quenched formation dynamics. Encyclopedia of Condensed Matter Physics. 2nd ed. Elsevier. Rem BS, Grier AT, Ferrier-Barbut I, Eismann U, Langen T, Navon N, Khaykovich L, Werner F, Petrov DS, Chevy F, and Salomon C (2013) Lifetime of the Bose gas with resonant interactions. Physical Review Letters 110: 163202. Roberts JL, Claussen NR, Cornish SL, Donley EA, Cornell EA, and Wieman CE (2001) Controlled collapse of a BOSE–EINSTEIN condensate. Physical Review Letters 86: 4211. Schmitt M, Wenzel M, Böttcher F, Ferrier-Barbut I, and Pfau T (2016) Self-bound droplets of a dilute magnetic quantum liquid. Nature 539: 259. Semeghini G, Ferioli G, Masi L, Mazzinghi C, Wolswijk L, Minardi F, Modugno M, Modugno G, Inguscio M, and Fattori M (2018) Self-bound quantum droplets of atomic mixtures in free space. Physical Review Letters 120: 235301. Skov TG, Skou MG, Jørgensen NB, and Arlt JJ (2021) Observation of a Lee-Huang-Yang fluid. Physical Review Letters 126: 230404. Smith RP (2017) Effects of interactions on Bose–Einstein condensation. In: Proukakis N, Snoke D, and Littlewood P (eds.) Universal Themes of Bose–Einstein Condensation. Cambridge University Press. Smith RP, Campbell RLD, Tammuz N, and Hadzibabic Z (2011) Effects of interactions on the critical temperature of a trapped Bose gas. Physical Review Letters 106: 250403. Steinhauer J, Ozeri R, Katz N, and Davidson N (2002) Excitation spectrum of a Bose–Einstein condensate. Physical Review Letters 88: 120407. Stenger J, Inouye S, Stamper-Kurn DM, Miesner H-J, Chikkatur AP, and Ketterle W (1998) Spin domains in ground-state Bose–Einstein condensates. Nature 396: 345. Stenger J, Inouye S, Chikkatur AP, Stamper-Kurn DM, Pritchard DE, and Ketterle W (1999) Bragg spectroscopy of a Bose–Einstein condensate. Physical Review Letters 82: 4569. Strecker KE, Partridge GB, Truscott AG, and Hulet RG (2002) Formation and propagation of matter-wave soliton trains. Nature 417: 150. Stringari S (2023) Bose–Einstein condensation. Encyclopedia of Condensed Matter Physics. 2nd ed. Elsevier. Tammuz N, Smith RP, Campbell RLD, Beattie S, Moulder S, Dalibard J, and Hadzibabic Z (2011) Can a Bose gas be saturated? Physical Review Letters 106: 230401. Tanzi L, Lucioni E, Famà F, Catani J, Fioretti A, Gabbanini C, Bisset RN, Santos L, and Modugno G (2019) Observation of a dipolar quantum gas with metastable supersolid properties. Physical Review Letters 122: 130405. Thøgersen M, Fedorov DV, and Jensen AS (2008) N-body EFIMOV states of trapped bosons. EPL (Europhysics Letters) 83(30012). Yin X and Radzihovsky L (2016) Postquench dynamics and prethermalization in a resonant Bose gas. Physical Review A 93: 033653. Zhai H (2015) Degenerate quantum gases with spin–orbit coupling: A review. Reports on Progress in Physics 78: 026001. Zwerger W (2011) BCS-BEC Crossover and the Unitary Fermi Gas. Springer. Zwierlein MW (2014) Superfluidity in ultracold atomic Fermi gases. In: Bennemann K-H and Ketterson JB (eds.) Novel Superfluids: Volume 2. Oxford University Press.

Cold-atom systems as condensed matter physics emulation Yoshiro Takahashi, Department of Physics, Graduate School of Science, Kyoto University, Kyoto City, Japan © 2024 Elsevier Ltd. All rights reserved.

Introduction Key issues Overview Optical lattice minimum Controllability of parameters Measurement methods Quantum gas microscopy Ultracold Fermions as Fermi-Hubbard-Model emulation Cold atom SU(N) fermion system Cold atom realization of dissipative Hubbard model as open quantum system emulation Open quantum system Dissipation in cold atom experiment Cold atom realization of Thouless pump as topological physics emulation Thouless pump in cold atom experiments Two-orbital cold atom system as quantum transport emulation Two-orbital two-electron atom system as Kondo-effect emulation Cold atom system as mesoscopic quantum transport emulation Non-standard optical lattice system as novel energy-band emulation Optical Lieb lattice system as novel flat-band emulator Conclusion Acknowledgments References

135 136 136 137 137 137 138 138 138 139 139 139 139 140 140 140 141 141 141 142 142 142

Abstract Ultracold-atom systems are an ideal quantum simulator or emulator of condensed matter physics, providing a unique approach to study strongly correlated quantum many body physics. In particular, a system of ultracold atoms loaded into an optical lattice is well described by a Hubbard model, an important minimal model in the study of condensed matter physics. High controllability of system parameters is a key to this new approach. In this chapter, we describe in detail the basic techniques as well as the recent developments of this rapidly growing field.

Key points

• • • • • •

Ultracold atom systems, especially those loaded into an optical lattice are an ideal quantum emulation platform of condensed matter physics, enabling the exploration of quantum many-body problems. Quantum emulation of a Fermi-Hubbard model as well as its SU(N) extension made possible by the use of two-electron atom system is important since classical computer cannot efficiently solve these models. Open quantum system can be studied with cold-atom quantum emulators by introducing the dissipation in a well-controlled manner. Quantum emulator of ultracold atom systems can be utilized to study topological phenomena, Thouless pump being a representative example. Quantum transport phenomena are studied by exploiting the state-of-the-art technologies of cold-atom systems. The flexibility of optical lattice setups for cold atoms enables the study of novel phenomena originating from unique band dispersions.

Introduction Recently, atomic physicists can prepare ultracold atoms in a trap formed by a laser beam or a magnetic field in a high vacuum condition. Cooling to quantum degenerate regimes like a Bose-Einstein condensates or Fermi degenerate gases at a temperature of nano kelvin is achieved by laser cooling and evaporative cooling techniques (Ketterle and Druten, 1996). They are a closed quantum system, well isolated from the environment. Ultracold atom systems are regarded as an ideal quantum emulator or simulator of

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quantum many body systems owing to the high controllability of system parameters (Bloch et al., 2008). In particular, among many, an inter-atomic interaction can be finely and widely controlled by a Feshbach resonance technique, which is not available by condensed matter systems. This leads to the realization of the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover, for example. Other illustrative examples of quantum emulation studies using ultracold atoms include the physics of universal few body bound states like Efimov trimers (Naidon and Endo, 2017). Of particular importance and interest in the study of quantum emulation of condensed matter systems is a system of ultracold atoms loaded into a periodic potential with lattice constant of the order of the laser wavelength, called an optical lattice (Bloch et al., 2008, 2012; Gross and Bloch, 2017; Schäfer et al., 2020). It is well described by a Hubbard model, which is crucially important in condensed matter physics, and is regarded as an important minimal model in the study of strongly-correlated quantum many body systems like high-temperature cuprate superconductors (Lee et al., 2006). Although only the nearest neighbor hopping t and on-site interaction U are involved, the Hubbard model captures the essence and provides the explanation of diverse phenomena observed in condensed-matter physics. Quantitative understandings that can be obtained from the quantum emulation of strongly correlated many-body systems will offer unique guidelines for synthesis of novel materials.

Key issues Here, we note that it is obvious that experiments using ultracold atoms are not always regarded as quantum emulation. Then, in what conditions should the experiments performed using cold atoms be termed as quantum emulation? Here, we provide a brief comment on the requisite for quantum emulation using ultracold atoms (Qiu et al., 2020). We think that, at least, the Hamiltonian or model of the target system should be clearly defined with reasonable grounds, and only the difficulty to describe an observed behavior by a mean-field theory, or the complexity to calculate static or dynamic properties of the system, for example, should be insufficient. One illustrative example of quantum emulation research from the engineering point of view is the computationally hard problem of Fermi-Hubbard model (FHM) where the sign-problem prevents the efficient numerical approach with quantum Monte-Carlo methods. From the viewpoints of scientific importance, on the other hand, one can target the conceptually important phenomena like topological quantum phenomena, no matter how computationally hard or not. One of the important advantages of quantum emulation approach of condensed matter physics using ultracold atoms is rich varieties of interesting research topics. Fig. 1 illustrates some examples of interesting optical lattice quantum emulation research topics. These include non-equilibrium dynamics, potential disorder, SU(N) spin symmetry, non-standard lattice, topological physics, open quantum dynamics, quantum mixtures, quantum transport, synthetic gauge field, and so on.

Overview In this article, we describe in detail the basic techniques as well as several interesting research topics of optical lattice quantum emulation research. In Section “Optical lattice minimum,” the basic knowledges and techniques required to understand the optical lattice system are summarized. In particular, we pay special attention to a powerful technique of quantum gas microscopy. The following sections are devoted to the focused discussion on the selected topics. In Section “Ultracold Fermions as Fermi-HubbardModel emulation,” we describe in detail a system of ultracold fermions as Fermi-Hubbard-model emulator, with the special emphasis on the unique possibility of studying SU(N) Fermi-Hubbard model. Section “Cold atom realization of dissipative Hubbard model as open quantum system emulation” describes the cold-atom realization of dissipative Hubbard model as an open quantum system emulator, which is of recent interest. In Section “Cold atom realization of Thouless pump as topological physics emulation,” we describe the cold-atom realization of Thouless pump as an illustrative example from the viewpoint of the

SU(N) spin symmetry

…

disorder/impurity dissipation nonequilibrium dynamics

Quantum Emulator Using Ultracold Atoms In an Optical lattice

non-standard lattice

long-range interaction transport

mixture topology

artificial gauge field

Fig. 1 Few examples of interesting optical lattice quantum emulation research topics.

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emulation of topological physics which is intensively and extensively studied in condensed matter physics. Section “Two-orbital cold atom system as quantum transport emulation” is devoted to the discussion on the quantum transport emulation of the Kondo effect as well as mesoscopic quantum transport. The final focused topic described in Section “Non-standard optical lattice system as novel energy-band emulation” is a non-standard optical-lattice system as novel-energy-band emulation, with special emphasis on a Lieb lattice. We conclude this article by describing the summary and future directions in the final Section “Conclusion.”

Optical lattice minimum Fig. 2 shows the schematic view of an optical lattice. A variety of atomic species have been successfully cooled to quantumdegenerate regimes, such as rather standard alkali-metal, alkali-earth-like, and magnetic lanthanide atoms (Schäfer et al., 2020). These ultracold atoms are first prepared in a weak harmonic trap. Then, adiabatic ramping-up of an optical lattice potential results in a successful loading of the ultracold atoms into an optical lattice with no significant heating effect. Sufficiently deep optical lattice potential enables us to treat the atoms in an optical lattice by the tight-binding model where an atom is localized on each lattice site and undergoes hopping between adjacent lattice sites. A single-band Hubbard model, then, describes the behavior of the atoms by a tunneling term characterized by a tunneling energy t and an on-site interaction term characterized by an energy U when more than two particles occupy the same lattice site. In addition, a weak external confinement is superimposed to a periodic optical lattice potential and is described as a site-dependent energy offset ei where i represents the site index. Multiple occupancy sites are created, depending on the total number of atoms and the external confinement.

Controllability of parameters As is mentioned, the Hubbard model is characterized by important parameters of a tunneling energy t and an on-site interaction energy U. Different from the ordinary condensed matter system, the ratio of these two parameters is precisely controlled experimentally by tuning the depth of the optical lattice potential. Individual control of these two is also possible. Feshbach resonances (Chin et al., 2010) provide the novel possibility of controlling the strength as well as sign of the on-site interaction, while the strength, sign, and even the complex phases (Peierls phases) of the tunneling energy t can also be controlled by lattice shaking and Raman-assisted tunneling methods. As in the condensed matter system, filling factor, temperature, and the total atom number can be finely controlled.

Measurement methods Here, we review some basic measurement methods. The time-of-flight (TOF) method, in which the optical lattice potential is suddenly turned off and the atom imaging is taken after a certain time, TOF, provides the information of atomic coherence over the lattice sites, which is the key for the observation of the superfluid-to-Mott insulator transition of the Bose-Hubbard model, where the sharp interference peaks of TOF images is an important signature of superfluid phase and the vanishing of the interference Mott-insulator (Greiner et al., 2002). In the slight modification of TOF method, known as band-mapping method, TOF images are taken after adiabatic ramp-down of the optical lattice, enabling the measurement of quasi-momentum distributions of the atoms (Bloch et al., 2008). In addition, the application of a spin-dependent potential gradient just before TOF measurement enables the spin-resolving imaging through a Stern-Gerlach effect. High-resolution spectroscopy using radio-frequency and optical clock transition resolves the spectra from different occupancies (Schäfer et al., 2020). Spin correlation is the observable of great interest in the study of quantum magnetism. Similar to the solidstate quantum dot systems, a singlet-triplet-oscillation protocol is developed for cold atom systems to reveal the spin correlation between nearest-neighbor sites by combining the methods of superlattice, photo-association, Feshbach resonance, or a band mapping (Trotzky et al., 2010; Greif et al., 2013; Ozawa et al., 2018). Long-range coherences of a superfluid phase of bosons are successfully detected by exploiting the Talbot effect incorporating the in-trap atom expansion (Santra et al., 2017). Berry curvature

on-site interaction energy U

laser

laser cold atom

Fig. 2 Schematic view of an optical lattice.

tunneling energy t

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and topological invariants are successfully measured through the excitation rate measurement to higher bands after amplitude modulation, revealing a quantum geometric tensor or through a shift of an atom cloud in the case of Thouless pump, for example.

Quantum gas microscopy Ultimately, the atom distribution in a 2D optical lattice is measured by single-site resolving imaging of quantum gas microscopy (QGM), enabling the study of a quantum many-body system in a direct manner (Gross and Bloch, 2017). The QGM method is first developed for a bosonic atom of 87Rb by utilizing the favorable features of cooling and imaging. The quantum phase transition of superfluid to Mott insulator states has been successfully demonstrated from the spatial distribution of atoms. In particular, the density plateau is directly confirmed as well as the particle-hole correlations in the Mott insulating state. The ability to observe the single-site resolving imaging is also utilized to a single-atom addressing technique, enabling the studies of non-equilibrium dynamics like a quantum walks, spin wave propagations, quantum thermalization, and quantum many-body localization. From the quantum information research perspective, entanglement entropy is directly measured. The QGM is also applied to fermionic atoms by developing efficient cooling methods, which is described in the next section.

Ultracold Fermions as Fermi-Hubbard-Model emulation In the context of condensed matter physics emulation, ultracold fermions in an optical lattice play a fundamental role, as strongly correlated many-body systems of fermionic particles of electrons do in condensed-matter physics (Auerbach, 2012). A new approach made possible by using ultracold fermionic atoms in optical lattices will give important insights for strongly correlated electron systems. Here, we describe in more detail the recent experiments using ultracold fermions as Fermi-Hubbard-Model emulation. As is mentioned, a system of ultracold fermions in an optical lattice is well described by the Fermi-Hubbard model (FHM), HFHM ¼ −t

X hi,ji,s

c{i,s cj,s + U

X

ni," ni,# + ei

i

X i,s

ni,s

(1)

where ci,s is fermionic annihilation operator for site i and spin s ¼ + 1/2(") or − 1/2(#), ni,s ¼ c{i,sci,s is the number operator, and ei represents the weak potential offset superimposed to the optical lattice potential. In this case of two-component or SU(2) FHM, similar to the electrons, as a higher temperature phase, a paramagnetic Mott insulator state emerges when the interaction is strongly repulsive. However, at a lower temperature below the Néel temperature, quantum magnetism manifests itself as an antiferromagnetic ordered state, as a result of the super-exchange interaction (Moriya and Ueda, 2003). Quantum emulation of the Fermi Hubbard model is especially important since the understanding of the behaviors of doped 2D SU(2) FHM at low temperatures, key for high-Tc superconductivity, still escapes complete understanding in spite of intensive studies in both condensed matter theory and experiments (Lee et al., 2006). Mott insulating phases of cold atom FHM have been realized and studied by various techniques. In particular, the QGM is quite useful for studying the cold atom FHM, and especially QGM with a spin resolving ability is a powerful method for the short- as well as long-range spin correlation measurements of FHM. In fact, the QGM method enables the observation of an antiferromagnetic correlation and ordered phase (Mazurenko et al., 2017). More recent studies explore the doped region of Fermi-Hubbard model, revealing the non-local correlations of string orders. Toward the goal of realization of the d-wave superfluidity of FHM, further cooling of fermions in optical lattices is crucially important. Local manipulation of the optical lattice potential using elaborate spatial light modulating devices enables the successful demonstration of powerful strategy of entropy redistribution, leading to the temperature of 0.25(2)/kBt (Mazurenko et al., 2017). Here kB is the Boltzmann constant.

Cold atom SU(N) fermion system Beyond the scope of the emulation of traditional condensed matter systems, a new state of matter can be realized by utilizing novel possibilities available in ultracold atomic systems. As one illustrative example, we can here describe in detail the realization of SU(N) Fermi Hubbard model. Originally, the study of SU(N) Hubbard model has started from a purely theoretical interest as an extension of the conventional SU(2) Fermi Hubbard model (Affleck, 1985), and now reveals rich quantum phases (Wu et al., 2003; Cazalilla and Rey, 2014). The enhancement of quantum fluctuation causes qualitatively different behaviors from those of SU(2) model. The SU(N) FHM is described by the following Hamiltonian Eq. (2) HSUðNÞFHM ¼ −t

X hi,ji,s

c{i,s cj,s + U

X i,s6¼s0

ni,s ni,s0 + ei

X i,s

ni,s

(2)

where s ¼ 1, 2, . . . N. Note that the hopping matrix element t and on-site interaction U do not depend on the spin s, which assures the SU(N) symmetry.

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Implementation of SU(N) Fermi Hubbard model is realized by using fermionic isotopes of alkali-earth-like atoms, such as ytterbium (173Yb) and strontium (87Sr). In these atoms, the electron and nuclear spin angular momenta are decoupled in the ground 1S0 and metastable 3P0 states. This results in the independence of the inter-atomic interaction with respect to the spin degrees of freedom s, here corresponding to the nuclear spin I, which is key for realizing the SU(N ¼ 2I + 1) FHM. A high temperature phase of a paramagnetic SU(N) Mott insulator is realized by loading the SU(N) fermions into an optical lattice. This is confirmed by several signatures like the charge excitation gap, suppressed compressibility ∂n/∂m, low temperature estimated by doubloon-production-rate measurement (Taie et al., 2012) and in situ observation of a Mott plateau (Hofrichter et al., 2016). Quantum magnetism or spin correlation expected at a lower temperature phase is studied for various lattice geometries including 1D, 2D, 3D, and a dimerized lattice. An STO technique, first developed for quantum-dot system and later applied to cold atom system of bosons (Trotzky et al., 2010) and fermions (Greif et al., 2013), is successfully applied to observe the nearestneighbor antiferromagnetic spin correlations of SU(N) fermions of 173Yb (Ozawa et al., 2018). In particular, the experimental data for a system of SU(6) fermions in a 1D lattice is compared with the theoretical calculations based on exact diagonalization and determinantal quantum Monte-Carlo methods, indicating the achieved temperature of about 0.1 of the tunneling energy t (Taie et al., 2022). This corresponds to the lowest temperature ever achieved for ultracold fermions. The observed significant cooling effect for the SU(6) system is attributed to the enhanced Pomeranchuk cooling effect, first observed in solid 3He. Note that even the stateof-the-art theoretical calculations are not powerful to quantitatively explain the data for higher dimensions, illustrating the importance of the SU(N) fermion system as quantum emulator.

Cold atom realization of dissipative Hubbard model as open quantum system emulation Open quantum system One of the recent considerable interests in condensed matter physics, especially in theory, is the study of open quantum systems. This is reasonable because the dissipation is ubiquitous in real condensed matter systems. Theoretically, an introduction of coupling between a closed quantum system with the environment calls the necessity to describe the system dynamics by Liouvillian dynamics, which is qualitatively different from a unitary dynamics of an isolated, closed quantum mechanical system. The novel role of the dissipation as a tool for preparation of certain quantum states is recently discussed (Daley, 2014; Müller et al., 2012). As an example, here we consider the case of a dissipative Bose-Hubbard model with two-body dissipation. The master equation for density operator r describing this system is given as,   d r ¼ −i HBHM , r + L2 ðrÞ, dt X X HBHM ¼ −t b{i bj + U ni ðni − 1Þ=2, i hi,ji   X ℏG L2 ðrÞ ¼ −b{i b{i bj bj r − rb{i b{i bj bj + 2bj bj rb{i b{i , 4 i ℏ

(3) (4) (5)

where bj represents boson annihilation operator and G denotes the strength of the two-body dissipation.

Dissipation in cold atom experiment Ultracold atoms in an optical lattice is in itself a well-defined closed quantum many-body system, isolated in a vacuum chamber. Therefore, when one artificially introduces dissipation processes in a well-controlled manner, one has a novel possibility of realizing open quantum system with engineered dissipation. In fact, various types of dissipation, such as a one-body dissipation via an electron beam (Labouvie et al., 2015) and well-controlled photon scattering process (Patil et al., 2015; Takasu et al., 2020), two-body loss via a molecular inelastic collision (Syassen et al., 2008; Yan et al., 2013) and photo-association (Tomita et al., 2017), and three-body loss via a Feshbach resonance (Mark et al., 2012) are successfully introduced to a cold atom system. The successful implementation of dissipation process reveals the novel behaviors due to the dissipation, such as the localization of particle, quantum Zeno effect, stabilization of the Mott insulator, realization of PT-symmetric state, formation of ferromagnetic spin correlation indicating the negative spin temperature (Honda et al., 2023), and so on.

Cold atom realization of Thouless pump as topological physics emulation Another example of the research topics of considerable interests in condensed matter physics is topological physics. Especially in recently years, there has been impressive progress in topological solid-state material synthesis. The theoretical backgrounds of these recent studies can be found in the seminal paper (Thouless et al., 1982) by Thouless, Kohmoto, Nightigale, and den Nijs (TKNN) in which the quantization of the Hall conductance for the integer quantum Hall effect is revealed as a topological invariant known as a

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Chern number. High controllability of ultracold atoms in optical lattices enables the researchers to realize various topologicallynon-trivial systems (Goldman et al., 2016). In another seminal 1983’s Thouless paper (Thouless, 1983), the quantization of the charge transferred in each pumping cycle for a periodically driven electron gas in an infinite one-dimensional periodic potential is revealed as a topological invariant. Importantly, these seemingly different phenomena share the same topological origin. Topological robustness inherent in these phenomena are also crucially important from the viewpoints of the application to the standard: the quantized Hall conductance and Thouless pump can serve as very accurate standards for electric resistance and electric current, respectively.

Thouless pump in cold atom experiments Successful demonstration of the Thouless charge pump (Nakajima et al., 2016; Lohse et al., 2016) is achieved by constructing a controllable one-dimensional periodic potential consisting of an optical superlattice and loading ultracold bosonic or fermionic atoms. These experiments are based on the specific lattice model of a Rice-Mele model given in Eq. (6),   X (6) HRM ¼ −ðJ + dÞa{i bi − ðJ − dÞa{i bi+1 + h:c: + D a{i ai − b{i bi , i

where ai and bi are annihilation operators of an atom in the two sublattices of the i-th unit cell, J  d is the tunneling amplitude within and between unit cells, and D denotes a staggered on-site energy offset. Temporal periodic change of the dimerized hopping amplitude d and staggered energy D, forming a particular trajectory in the d-D parameter space, drives Thouless pumping, the topology of which is determined by the trajectory that encircles the degeneracy point of d ¼ D ¼ 0. The quantization of Thouless pumping manifests itself as the quantized shift of the atomic cloud. In particular, the experiments with various pumping sequences reveal the topological natures of this pumping: relevance on the winding number of the trajectory around the degeneracy point as well as the irrelevance on the details of the trajectory.

Two-orbital cold atom system as quantum transport emulation From the viewpoints of quantum emulation of condensed matter physics, two-electron atoms, or alkali-earth-like atoms are unique in that their long-lived metastable states like the 3P0 and 3P2 electronic states, combined with the ground state 1S0 enables the study of the multi-orbital physics. These metastable states are accessible from the 1S0 state through a coherent excitation utilizing very stable lasers with a Hz-level linewidth, which also attracts much attention for the application to an optical frequency standard. In addition to the inter-orbital spin-exchange interaction for nuclear spin degrees of freedom, the existence of a magnetic (orbital) Feshbach resonance between the ground 1S0 and 3P2 (3P0) states offers the powerful control of the interatomic interaction between the atoms in these states. Among the spin-orbital-related physics so far studied in condensed matter physics, the Kondo effect (Kondo, 1964), arising from an antiferromagnetic spin-exchange interaction between conduction electrons and magnetic impurities, is quite important, since it is one of the central problems of the strongly-correlated quantum many-body physics. In the case of periodically aligned localized spins, the corresponding system is described by the Kondo lattice model, exhibiting rich quantum phases like a paramagnetic phase and the Ruderman-Kittel-Kasuya-Yoshida ordered phase in a Doniach phase diagram (Doniach, 1977). The system is described by the following Hamiltonian HKLM, X { X { { HKLM ¼ −t c ci,c,s cj,c,s + V ex ci,c,s ci,l,s0 ci,c,s0 ci,l,s, (7) i,s6¼s0 hi,ji,s where tc is the tunneling amplitude in the conduction band, V ex the spin-exchange energy between the conduction electron and localized impurity, and the symbols c and l the conduction and localized orbitals, respectively. We note that, for a system of alkali atoms, there are also several proposals of cold atom quantum emulator of the Kondo effect.

Two-orbital two-electron atom system as Kondo-effect emulation Owing to the existence of two-orbital as well as spin degrees of freedom, as is mentioned, it is quite natural to consider a system of two-electron atoms as an experimental platform of Kondo-effect or Kondo lattice model emulator (Gorshkov et al., 2010). We can expect that the application of a quantum gas microscope technique for two-electron atoms will reveal the Kondo screening behavior, for example. However, this does not mean that any two-electron atoms can be used for this emulation. In order to study the Kondo effect, the interorbital spin-exchange interaction should be antiferromagnetic, which makes only a fermionic isotope of 171Yb a natural candidate (Ono et al., 2019), while an interesting possibility of controllability of spin-exchange interaction is theoretically proposed and experimentally demonstrated for 173Yb atoms with a ferromagnetic spin-exchange interaction. Toward this goal, progresses are obtained like a construction of localization-itinerant mixed-dimensional two-orbital optical lattice system and a successful observation of the spin-exchange dynamics between 171Yb atoms in the 1S0 and 3P0 states (Ono et al., 2021a).

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Cold atom system as mesoscopic quantum transport emulation In solid state mesoscopic materials, there have been extensive studies on quantum transport of electrons between terminals through narrow channels (Ihn, 2010; Imry, 2002). The quantized conductance, predicted by the Landauer formula, is one illustrative example. High controllability and state-of-the-art optical trapping technique enable the realization of narrow conduction channels for quantum transport experiments using ultracold atoms (Krinner et al., 2017). This quantum emulator using ultracold atoms, extended to mesoscopic quantum transport research, is called atomtronics. A series of elaborate experiments have established a cold-atom analog of mesoscopic quantum point contact, verifying conductance quantization, for example (Krinner et al., 2015). More recently, a novel scheme of a quantum transport emulation using the spin degrees of freedom is proposed (You et al., 2019, and Nakada et al., 2020), and experimentally realized (Ono et al., 2021b). In particular, the use of a multicomponent spin system enables the realization of a multiterminal quantum point contact system.

Non-standard optical lattice system as novel energy-band emulation Recent advance of highly sophisticated laser technologies makes not only standard but also various types of optical lattice geometries available for quantum emulation experiments. In fact, various geometries of optical lattices are realized (Windpassinger and Sengstock, 2013). In addition to standard 1D, 2D, or 3D lattices, non-standard lattices of superlattices (Fölling et al., 2007), triangular lattices (Becker et al., 2010), Kagomé lattices (Jo et al., 2012), Lieb lattices (Taie et al., 2015), honeycomb lattices (Tarruell et al., 2012), quasi-crystal potentials (Viebahn et al., 2019) are successfully formed, for example, enabling the study of geometric frustration, topology, and so on. Novel geometries of lattice structures lead to interesting energy bands which will play crucial roles in the manifestation of the novel quantum phases. For example, a honeycomb lattice shows a linear dispersion of Dirac cone, and the Kagomé lattice structure exhibits a totally vanishing dispersion, called a flat band with the macroscopic degeneracy.

Optical Lieb lattice system as novel flat-band emulator Here, we describe in detail a Lieb lattice as an illustrative example of optical-lattice realization of interesting non-standard lattice structure. Cold-atom systems provide the possibility to study exotic phases like supersolids using bosons loaded into a Lieb lattice as a result of interplay between frustrated kinetic energy and inter-atomic interactions, as well as the ferromagnetism using fermions based on the Lieb’s theorem. The Lieb lattice supports the flat band as well as the Dirac cone, and is also called a d-p model since it is identical to the lattice structure of the CuO2 plane of high-Tc superconductors. The lattice structure of the Lieb lattice involves three sublattices A, B, and C, in which A-sites forms a square lattice and B- and C-sites are located on every line of the square lattice. Taking only the nearest-neighbor tunneling t into consideration and considering the quasi-momentum eigenstates |q, A⟩, |q, B⟩, and |q, C⟩ as basis sets, the tight-binding Hamiltonian can be described by transfer matrix M, given by  1 0 0 −2tcosðqx d=2Þ −2tcos qy d=2 C B C B 0 0 (8) M ¼ B −2tcosðqx d=2Þ C, A @   0 0 −2tcos qy d=2 where d denote the lattice constant. The diagonalization of this transfer matrix gives the energy-bands and the eigenvalues. Importantly, the second energy-band of the Lieb lattice exhibits the flat band with no energy dispersion, with the eigenstates given by a superposition of only the B- and C- sites, jq, FB⟩ ¼ cos yq jq, B⟩ − sin yq jq, C⟩

(9)

where tanyq ¼ cos(qxd/2)/cos(qyd/2). Note that the flat band structure is not originated from a zero hopping energy but from the destructive interference between the hopping from B- to A- sites and that from C- to A- sites. Optical-lattice realization of the Lieb lattice is done by superimposing three different kinds of optical lattices by exploiting the high flexibility of laser phases and wavelengths. The controllability of the optical lattice parameters enables the direct observation of the localization of a matter wave within the B- and C- sites in the flat band, different from the delocalization in a usual dispersive band. In addition, the structure of the transfer matrix M suggests the analogy between the phenomenon of Stimulated Raman Adiabatic Passage (STIRAP) process (Bergmann et al., 1998), studied both theoretically and experimentally in quantum optics, and the Lieb lattice, studied in condensed matter physics. In fact, this is the nice example in which the phenomena discussed in different fields of physics can be understood on the same grounds. This close analogy provides the interesting manner of understanding of the flat band. Namely, the eigenstate of the flat band is nothing but a dark state which does not couple with electromagnetic fields applied in the STIRAP process.

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This analogy between the Lieb lattice and laser-coupled three-level system considered in a STIRAP process not only provides the deeper understanding but also the novel possibility of realization of a matter-wave analog of the STIRAP, called spatial adiabatic passage, or transport without transit (Rab et al., 2008). According to this analogy, we can consider novel quantum transport of a quantum-mechanical particle between well-separated positions of B- and C- sites in real space, with no possibility of finding the atoms at the intermediate position of A-site. In fact, this novel transport is realized experimentally (Taie et al., 2020).

Conclusion In summary, after the basic techniques of an optical lattice are introduced, we described several interesting research topics of quantum emulation of condensed matter physics. The discussed topics include quantum emulation of Fermi-Hubbard-model, dissipative Hubbard model, Thouless pump, quantum transport, and a Lieb lattice model. Here, we briefly mention other important topics which are not involved in the former sections. The first is the study of the nonequilibrium dynamics of a quantum many-body system. It is noted that an optical lattice quantum emulator is ideal for this purpose, owing to the high controllability, especially for the lattice potential and interatomic interaction in a time-dependent manner, allowing for the study of Kibble-Zurek mechanism across phase transitions (Braun et al., 2015) and the dynamical spreading of quantum information and the Lieb-Robinson bound (Cheneau et al., 2012). This is particularly important since the long-term dynamics of quantum many-body system after quantum quench is in general quite hard to simulate with classical computers, and thus the experimental results can serve as a versatile benchmark for developing a new numerical technique. In addition to the regular periodic lattice potentials, introducing disorder potentials enables the quantum emulation of disorderinduced phenomenon such as Anderson localization (Billy et al., 2008; Roati et al., 2008), many-body localization (Abanin et al., 2019), and so on. Disordered potentials are introduced by a number of ways like imprinting speckle patterns, superpositions of optical lattices at incommensurable lattice spacings, and introducing atomic impurities. In spite of the fact that ultracold atoms are neutral in charge, recent developments of cold-atom quantum emulation technologies successfully introduce artificial gauge fields, spin-orbit coupling and topologically non-trivial bands in optical lattices (Cooper et al., 2019). The notion of synthetic dimensions (Ozawa and Price, 2019) is also actively applied to the study of quantum emulation of condensed matter physics. A rich variety of parameter spaces spanned by time, internal degrees of freedom of atoms, and well-defined atomic momentum are available in a cold-atom system. Note that already explained example of Thouless pump utilizes the time as one of the two dimensions required for defining the Chern number. Furthermore, Raman-coupling of internal states can implement spin-orbit coupling in cold atom systems, enabling the various research related with condensed matter physics. Here, we also briefly note on the recent development of the novel optical trapping system, called an atom tweezer array system (Saffman et al., 2010; Kaufman and Ni, 2021), which offers rather complementary possibilities with the optical lattice platform. By tightly focusing laser beams created by spatial light modulating devices with a high-numerical aperture objective lens, single atoms are trapped in defect-free atomic arrays with spacings of about a few micrometer with rather arbitrary geometries in one, two, and three dimensions. Moreover, Rydberg state excitation can induce strong interaction between the atoms in neighboring sites of a tweezer array. The Rydberg atom tweezer array system is now extensively used for quantum emulation of Ising spin model and reveals quantum thermalization problem like a quantum many-body scar, for example. The final remark is on an interesting future possibility of performing remote quantum emulation experiment (Heck et al., 2018) by anyone, including condensed matter theorists, who is not an expert of cold-atom experiments that require the knowledge and experience to operate very complicated machines. Such a remote quantum emulation system will enhance the development of new numerical calculations.

Acknowledgments We greatly thank many experimental and theoretical collaborators to understand the works described in this article. Especially, we thank Y. Takasu, S. Taie, S. Nakajima, H. Ozawa, R. Yamamoto, Y. Takata, N. Kitamura, K. Honda, Y. Nakamura, T. Kusano, K. Ono, T. Ishiyama, and T. Takano, for experiment, and I. Danshita, S. Goto, M. Yamashita, K. Inaba, Y. Kuno, S. Uchino, Y. Nishida, L. Wang, K. Hazzard, R. T. Scalettar, and E. Ibarra-García-Padilla for theory. The work was partially supported by the Grant-in-Aid for Scientific Research of JSPS (Nos. JP17H06138, JP18H05405, and JP18H05228), the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) program, JST CREST (No. JP-MJCR1673), MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant No. JPMXS0118069021, and JST Moonshot R&D – MILLENNIA Program (Grant No. JPMJMS2269).

References Abanin DA, Altman E, Bloch I, and Serbyn M (2019) Colloquium: Many-body localization, thermalization, and entanglement. Reviews of Modern Physics 91: 021001. Affleck I (1985) Large-N limit of SU(N) quantum “spin” chains. Physical Review Letters 54: 966. Auerbach A (2012) Interacting Electrons and Quantum Magnetism. Graduate Texts in Contemporary Physics. Springer New York.

Cold-atom systems as condensed matter physics emulation

143

Becker C, et al. (2010) Ultracold quantum gases in triangular optical lattices. New Journal of Physics 12: 065025. Bergmann K, Theuer H, and Shore BW (1998) Coherent population transfer among quantum states of atoms and molecules. Reviews of Modern Physics 70: 1003–1025. Billy J, et al. (2008) Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453: 891–894. Bloch I, Dalibard J, and Zwerger W (2008) Many-body physics with ultracold gases. Reviews of Modern Physics 80: 885–964. Bloch I, Dalibard J, and Nascimbène S (2012) Quantum simulations with ultracold quantum gases. Nature Physics 8: 267–276. Braun S, et al. (2015) Emergence of coherence and the dynamics of quantum phase transitions. Proceedings of the National Academy of Sciences of the United States of America 112: 3641–3646. Cazalilla MA and Rey AM (2014) Ultracold Fermi gases with emergent SU(N) symmetry. Reports on Progress in Physics 77: 124401. Cheneau M, et al. (2012) Light-cone-like spreading of correlations in a quantum many-body system. Nature 481: 484–487. Chin C, Grimm R, Julienne P, and Tiesinga E (2010) Feshbach resonances in ultracold gases. Reviews of Modern Physics 82: 1225. Cooper NR, Dalibard J, and Spielman IB (2019) Topological bands for ultracold atoms. Reviews of Modern Physics 91: 015005. Daley AJ (2014) Quantum trajectories and open many-body quantum systems. Advances in Physics 63: 77–149. Doniach S (1977) The Kondo lattice and weak antiferromagnetism. Physica B+ C 91: 231. Fölling S, et al. (2007) Direct observation of second-order atom tunnelling. Nature 448: 1029–1032. Goldman N, Budich JC, and Zoller P (2016) Topological quantum matter with ultracold gases in optical lattices. Nature Physics 12: 639–645. Gorshkov AV, Hermele M, Gurarie V, Xu C, Julienne PS, Ye J, Zoller P, Demler E, Lukin MD, and Rey AM (2010) Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nature Physics 6: 289–295. Greif D, Uehlinger T, Jotzu G, Tarruell L, and Esslinger T (2013) Short-range quantum magnetism of ultracold fermions in an optical lattice. Science 340: 1307–1310. Greiner M, Mandel O, Esslinger T, Hansch TW, and Bloch I (2002) Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415: 39–44. Gross C and Bloch I (2017) Quantum simulations with ultracold atoms in optical lattices. Science 357: 995–1001. Heck R, Vuculescua O, Sørensena JJ, Zoller J, Andreasena MG, Basone MG, Ejlertsena P, Elíassona O, Haikkaa P, Laustsena JS, Nielsena LL, Maoa A, Mullera R, Napolitanoa M, Pedersena MK, Thorsena AR, Bergenholtza C, Calarcoc T, Montangeroc S, and Shersona JF (2018) Remote optimization of an ultracold atoms experiment by experts and citizen scientists. Proceedings of the National Academy of Sciences of the United States of America 115: E11231–E11237. Hofrichter C, Riegger L, Scazza F, Höfer M, Rio Fernandes D, Bloch I, and Fölling S (2016) Direct probing of the mott crossover in the SU(N) Fermi-Hubbard model. Physical Review X 6: 021030. Honda K, Taie S, Takasu Y, Nishizawa N, Nakagawa M, and Takahashi M (2023) Observation of the sign reversal of the magnetic correlation in a driven-dissipative fermi gas in double wells. Physical Review Letters 130: 063001. Ihn T (2010) Semiconductor Nanostructures: Quantum States and Electronic Transport. Oxford: Oxford University Press. Imry Y (2002) Introduction to Mesoscopic Physics. Oxford: Oxford University Press. Jo G-B, Guzman J, Thomas CK, Hosur P, Vishwanath A, and Stamper-Kurn DM (2012) Ultracold atoms in a tunable optical Kagome lattice. Physical Review Letters 108: 045305. Kaufman AM and Ni K-K (2021) Quantum science with optical tweezer arrays of ultracold atoms and molecules. Nature Physics 17: 1324–1333. Ketterle W and Druten NJV (1996) Evaporative cooling of trapped atoms. In: Bederson B and Walther H (eds.) Advances In Atomic, Molecular, and Optical Physics, vol. 37, pp. 181–236. Academic Press. Kondo J (1964) Resistance minimum in dilute magnetic alloys. Progress of Theoretical Physics 32: 37. Krinner S, Stadler D, Husmann D, Brantut J-P, and Esslinger T (2015) Observation of quantized conductance in neutral matter. Nature 517: 64–67. Krinner S, Esslinger T, and Brantut J-P (2017) Two-terminal transport measurements with cold atoms. Journal of Physics: Condensed Matter 29: 343003. Labouvie R, Santra B, Heun S, Wimberger S, and Ott H (2015) Negative differential conductivity in an interacting quantum gas. Physical Review Letters 115: 050601. Lee PA, Nagaosa N, and Wen X-G (2006) Doping a Mott insulator: Physics of high-temperature superconductivity. Reviews of Modern Physics 78: 17–85. Lohse M, Schweizer C, Zilberberg O, Aidelsburger M, and Bloch I (2016) A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nature Physics 12: 350–354. Mark MJ, Haller E, Lauber K, Danzl JG, Janisch A, Büchler HP, Daley AJ, and Nägerl H-C (2012) Preparation and spectroscopy of a metastable mott-insulator state with attractive interactions. Physical Review Letters 108: 215302. Mazurenko A, Chiu CS, Ji G, Parsons MF, Kanász-Nagy M, Schmidt R, Grusdt F, Demler E, Greif D, and Greiner M (2017) A cold-atom Fermi–Hubbard antiferromagnet. Nature 545: 462–466. Moriya T and Ueda K (2003) Antiferromagnetic spin fluctuation and superconductivity. Reports on Progress in Physics 66: 1299–1341. Müller M, Diehl S, Pupillo G, and Zoller P (2012) Engineered open systems and quantum simulations with atoms and ions. Advances in Atomic, Molecular, and Optical Physics 61: 1–80. Naidon P and Endo S (2017) Efimov physics: A review. Reports on Progress in Physics 80(056001): 1–78. Nakada S, Uchino S, and Nishida Y (2020) Simulating quantum transport with ultracold atoms and interaction effects. Physical Review A 102: 031302. Nakajima S, Tomita T, Taie S, Ichinose T, Ozawa T, Wang L, Troyer M, and Takahashi Y (2016) Topological Thouless pumping of ultracold fermions. Nature Physics 12: 296–300. Ono K, Kobayashi J, Amano Y, Sato K, and Takahashi Y (2019) Antiferromagnetic interorbital spin-exchange interaction of 171Yb. Physical Review A 99: 032707. Ono K, Amano Y, Higomoto T, Saito Y, and Takahashi Y (2021a) Observation of spin-exchange dynamics between itinerant and localized 171Yb atoms. Physical Review A 103: L041303. Ono K, Higomoto T, Saito Y, Uchino S, Nishida Y, and Takahashi Y (2021b) Observation of spin-space quantum transport induced by an atomic quantum point contact. Nature Communications 12: 6724. Ozawa T and Price HM (2019) Topological quantum matter in synthetic dimensions. Nature Reviews Physics 1: 349–357. Ozawa H, Taie S, Takasu Y, and Takahashi Y (2018) Antiferromagnetic spin correlation of SU(N) Fermi gas in an optical superlattice. Physical Review Letters 121: 225303. Patil YS, Chakram S, and Vengalattore M (2015) Measurement-induced localization of an ultracold lattice gas. Physical Review Letters 115: 140402. Qiu X, Zou J, Qi X, and Li X (2020) Precise programmable quantum simulations with optical lattices. npj Quantum Information 87: 1–8. Rab M, Cole JH, Parker NG, Greentree AD, Hollenberg LCL, and Martin AM (2008) Spatial coherent transport of interacting dilute Bose gases. Physical Review A 77: 061602. Roati G, et al. (2008) Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453: 895–898. Saffman M, Walker TG, and Mølmer K (2010) Quantum information with Rydberg atoms. Reviews of Modern Physics 82: 2313–2363. Santra B, et al. (2017) Measuring finite-range phase coherence in an optical lattice using Talbot interferometry. Nature Communications 8: 15601. Schäfer F, Fukuhara T, Sugawa S, Takasu Y, and Takahashi Y (2020) Tools for quantum simulation with ultracold atoms in optical lattices. Nature Reviews Physics 2: 411–425. Syassen N, Bauer DM, Lettner M, Volz T, Dietze D, García-Ripoll JJ, Cirac JI, Rempe G, and Dürr S (2008) Strong dissipation inhibits losses and induces correlations in cold molecular gases. Science 320: 1329–1331. Taie S, Yamazaki R, Sugawa S, and Takahashi Y (2012) An SU(6) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling. Nature Physics 8: 825–830. Taie S, Ozawa H, Ichinose T, Nishio T, Nakajima S, and Takahashi Y (2015) Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice. Science Advances 1: e1500854. Taie S, Ichinose T, Ozawa H, and Takahashi Y (2020) Spatial adiabatic passage of massive quantum particles in an optical Lieb lattice. Nature Communications 11: 257. Taie S, Ibarra-García-Padilla E, Nishizawa N, Takasu Y, Kuno Y, Wei H-T, Scalettar RT, Hazzard KRA, and Takahashi Y (2022) Observation of antiferromagnetic correlations in an ultracold SU(N) Hubbard model. Nature Physics 18: 1356–1361.

144

Cold-atom systems as condensed matter physics emulation

Takasu Y, Yagami T, Ashida Y, Hamazaki R, Kuno Y, and Takahashi Y (2020) PT-symmetric non-Hermitian quantum many-body system using ultracold atoms in an optical lattice with controlled dissipation. Progress of Theoretical and Experimental Physics 2020: 12A110. Tarruell L, Greif D, Uehlinger T, Jotzu G, and Esslinger T (2012) Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483: 302–305. Thouless DJ (1983) Quantization of particle transport. Physical Review B 27: 6083–6087. Thouless DJ, Kohmoto M, Nightingale MP, and den Nijs M (1982) Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters 49: 405–408. Tomita T, Nakajima S, Danshita I, Takasu Y, and Takahashi Y (2017) Observation of the Mott insulator to superfluid crossover of a driven-dissipative Bose-Hubbard system. Science Advances 3: e1701513. Trotzky S, Chen Y-A, Schnorrberger U, Cheinet P, and Bloch I (2010) Controlling and detecting spin correlations of ultracold atoms in optical lattices. Physical Review Letters 105: 265303. Viebahn K, Sbroscia M, Carter E, Yu J-C, and Schneider U (2019) Matter-wave diffraction from a quasicrystalline optical lattice. Physical Review Letters 122: 110404. Windpassinger P and Sengstock K (2013) Engineering novel optical lattices. Reports on Progress in Physics 76: 086401. Wu C, Hu J-P, and Zhang S-C (2003) Exact SO(5) symmetry in the spin 3/2 fermionic system. Physical Review Letters 91: 186402. Yan B, Moses SA, Gadway B, Covey JP, Hazzard KRA, Rey AM, Jin DS, and Ye J (2013) Observation of dipolar spin-exchange interactions with lattice-confined polar molecules. Nature 501: 521–525. You J-S, Schmidt R, Ivanov DA, Knap M, and Demler E (2019) Atomtronics with a spin: Statistics of spin transport and nonequilibrium orthogonality catastrophe in cold quantum gases. Physical Review B 99: 214505.

Ionic and mixed conductivity in condensed phases☆ J Maier, Physical Chemistry of Solids, Max Planck Institute for Solid State Research, Stuttgart, Germany © 2024 Elsevier Ltd. All rights reserved. This is an update of J. Maier, Ionic and Mixed Conductivity in Condensed Phases, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 9–21, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00432-0.

Introduction: Hopping process Charge carrier concentrations in pure compounds and dilute bulk Doping effects High charge carrier concentrations—Interactions Boundary layers (Heterogeneous doping) Nano-Ionics Partial equilibrium—Bridge between high-temperature and low-temperature situation Mobility: From low to high concentrations Chemical diffusion in mixed conductors and battery storage Ion transport in biology Conclusion References

145 146 149 152 153 154 156 156 157 159 159 160

Abstract Ion transport plays a key role in solid state science. It is not only conceptually important, it is also key to solid state devices in energy research. Moreover, the kinetics of solid state chemical processes typically rely on it. If electron transport is involved as well, redox processes can occur, e.g., storage processes in lithium or sodium batteries. Of particular significance are ionic and mixed conductivity processes in interfacially dominated and/or confined systems (Nanoionics). The latter establishes natural ties to biology. The article traces back all these phenomena to the respective mechanistic centers (e.g., point defects in ionic solids) and to the basic thermodynamic and kinetic relations.

Key points

• • • • •

Basics and mechanisms of ion transport. Ionic charge carriers in pure, doped and partially frozen states. Transport in bulk, at interfaces, and in confined systems. Ion transport enabling electrolyte function and nerve signals. Mixed conductivity and ambipolar transport enabling electrode storage and compositional changes.

Introduction: Hopping process While a quantum mechanical particle such as quasi-free-electron can tunnel through a barrier by interference of the states on both sides of the barrier (denoted by x and x0 ), that is, by seizing the hindrance in a wave-like way, the transport of a heavy particle such as an ion has to rely on the ensemble fraction with energy high enough to reach the final state (hopping) (Allnatt and Lidiard, 1993; Maier, 2004). On the one hand, under favorable circumstances protons may also tunnel, while on the other hand electrons that strongly polarize their environment (large polarons) behave almost classically as ions. In most cases of interest (consider A as an ion with charge zA), the hopping of A from x to x0 (x0 ¼ x + △x) may be written as. A ðxÞ + V 0 ðx0 Þ Ð V ðxÞ + A 0 ðx0 Þ

(1) 0

(The dash takes account of the possibility for A to have a different structural environment at x ). In cases where A denotes an ion on a regular site in a crystal, V is the vacancy that serves as a jump partner, while in cases where A refers to an excess ion sitting on interstitial sites, V denotes the vacant interstitial site. In crystals, V (in the first case) or A (in the second case) are usually diluted with the energy levels of these sites being well defined. The concentration of the respective reaction partner is then approximately constant and the corresponding rate equation for the hopping process is pseudo-monomolecular. In amorphous solids, the sites are ☆ Change History: September 2022. J Maier updated sections Nano-Ionics, Chemical diffusion in mixed conductors and battery storage, Ion transport in biology, Summary and Figures 16 and 17.

Encyclopedia of Condensed Matter Physics, Second Edition

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less defined, which also holds for the spatial energy distribution; in polymers, melts or liquids, these sites and energy distributions fluctuate. In these cases, one would better speak of smeared-out density of states, as far as thermal excitations are concerned. The rate constant for the hopping rate equation is decisively influenced by the activation free enthalpy. The parameters k and DG6¼, for example, for the forward reaction (index*) related through (prefactor k0 is proportional to the attempt frequency G0) 0 6¼ 1 * * DG A (2) k ¼ k0 exp @ RT refer to the built-in part of the  forward rate constant and its activation free energy, respectively. This expression has to be * complemented by the factor exp −a DfF=RT in order to take account of the applied electric field. The parameter Df is that * part of the external voltage that drops from x to x0 and the term a Df measures the portion of Df that is relevant for the forward reaction (distance between saddle point and initial state).   * ( The relations for the backward reaction are analogous a + a ¼ 1 . This leads to the Butler-Volmer equation which describes the charge transfer from x to x0 for an asymmetric free-energy profile. In the homogeneous bulk region (and also in the space charge (

*

*

(

region sufficiently far from the interface, i.e., A ¼ A0 ), the built-in profile is symmetrical, that is, ( k ¼ k ¼ k, a ¼ a ¼ 1=2) and *

(

DG ¼ DG. The result for the rate represented by the current density ij of carrier j in x-direction ij ¼ −2kj cj zj F sinh ’ −sj

zj FDf 2RT

(3)

Df Dx

can be simplified to Ohm’s law (bottom part) for Df  RT/zjF; the conductivity sj of the carrier is proportional to its mobility uj(uj ∝ kj), (molar) concentration (cj) and (molar) charge (zjF) and given as sj ¼ zj Fuj cj

(4)

Allowing also for a concentration variation during the process and generalizing to three dimensions, but restricting to small effects the Nernst-Planck equation is arrived at ij ¼ −Dj zj Frcj −sj rf

(5)

where the diffusion coefficient (D) and conductivity (s) are related through the Nernst-Einstein equation. Close to equilibrium, this equation corresponds to the result obtained by linear irreversible thermodynamics, ij ¼

sj r~ mj zj F

(6)

~ ¼ m + zFf (chemical potential + which assumes the current to be proportional to the gradient in the electrochemical potential m (molar charge x electrical potential)). The latter relation is more general concerning the concentration range, while the former (as regards the diffusion term) is a little more general as far as the magnitude of this driving force is concerned. In cases where the current of a given species is also generated by secondary driving forces (e.g., by r~ m of a different particle), cross terms occur for which the Onsager-Casimir symmetry relations are valid. Also Eq. (6) can be generalized towards higher driving forces. Maier (2004) and Riess and Maier, 2008 give a symmetrized relation in terms of the electrochemical potential drop over the jump distance appearing as exponent in a sinh-function (similar as in Eq. (3)). As regards the determination of the conductivity, the first task is to determine the carrier concentration as a function of the control parameters. This is feasible by solving the defect model for the system under consideration.

Charge carrier concentrations in pure compounds and dilute bulk Fig. 1 displays the fundamental charge carrier formation reaction in an “energy level” diagram for a fluid (H2O) and an ionic crystal (AgCl) (Maier, 1993a). These levels refer to effective standard (electro) chemical potentials or—in the case of the “Fermi levels” that are positioned within the gap—full (electro) chemical potentials. (As long as the bulk is considered, zFf can be neglected and one ~ .) As long as—in pure materials—the gap remains large compared to RT, the Boltzmann form of the can refer to m instead of m chemical potential of the respective charge carrier (defect) is valid which has the form   cj (7) mj ¼ m∘j + RT ln ∘ cj As outlined below, this relation holds largely independent of the charge carrier situation and the form of the energy level distribution. If the levels are smeared out or if there is a band of levels, mj refers to an effective level (i.e., band edges in nondegenerate semiconductors or an appropriately averaged energy when dealing with fluids or disordered solids). The coupling of the ionic level picture and the electronic level picture for AgCl (Fig. 2) is established via the thermodynamic relation

Ionic and mixed conductivity in condensed phases

147

Fig. 1 Electronic and ionic disorder in ionic solids and water in “physical” (top) and “chemical” language (bottom).

Fig. 2 Coupling of ionic and electronic levels in AgCl.

~e− ¼ mAg, and hence via the component potential which reflects the precise position in the phase diagram. The meaning of ~Ag+ + m m m becomes obvious, if for simplicity one considers an elemental crystal (with defect j). For the formation of Nj identical defects (among N regular positions), a local free enthalpy of NjDgj is required in the interaction-free case; including also the configurational contribution, the Gibbs energy of the defective crystal (GP refers to the perfect crystal) reads   (8) G ¼ Gp + Nj Dg∘j −kB T ln N Nj The configuration term describes the number of combinations of Nj elements out of N choices without repetition. The calculation of  Dg , the free energy to form a single defect, requires an atomistic consideration (Allnatt and Lidiard, 1993; Kröger, 1964). If one refers to ionic defects in crystals, it is essentially composed of lattice energy, polarization energy, and vibrational entropy terms. In more general terms, Nj is the number of defects and N the number of particles that can be made defective. Owing to the

Fig. 3 Contributions to the free enthalpy of the solid by defect formation with a constant total number of sites. From Maier J (2004) Physical chemistry of ionic materials. In: Ions and Electrons in Solids. John Wiley & Sons, Ltd: Chichester.

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infinitely steep decrease of the configurational term with Nj, a minimum in G(Nj) (see Fig. 3) is observed in the elemental crystal; here the chemical potential of j, that is, mj  ∂ G/∂(Nj/Nm), vanishes. Generally it holds that Nj N −Nj Nj ’ m∘j + RT ln N nj ∘ mj + RT ln n

mj ¼ m∘j + RT ln

(9)

(n  N/Nm,nj  Nj/Nm,Nm ¼ Avogadro0 s number; mj ¼ NmDgj ). It is noteworthy that the strict result (top) which is formally valid for higher concentrations also is of the Fermi-Dirac type. This is due to the fact that one refers to combinations without repetition, that is, double occupancy is excluded and hence the sites are exhaustible similarly as it is the case for quantum states in electronic problems (Maier, 2005; Kirchheim, 1988). (Combinations with repetitions lead to Bose-Einstein distribution.) In a situation in which the levels are broadened to a more or less continuous zone (e.g., electrons in semiconductors and ions in fluids), the parameters Nj and mj of the Fermi-Dirac form have to be attributed to an infinitely small level interval. Denoting the partial free energy level by e, this integral ranges from ej to ej + dej and the total (molar) concentration follows from integration:   Z D 2j d2j   (10) nj ¼  zone 1 + exp 2j −m =RT j whereby the molar density of states D(ej) appears in the integrand. At equilibrium, mj is independent of ej. Considering without restriction of generality, the almost empty zone and the lower edge of this zone designated as ej, one can neglect the unity in the denominator of the above for all levels more distant from mj than ej0 which leads to the Boltzmann expression  equation    0 nj ’ nj ðT Þ exp −2j −mj =RT . The effective value nj is given by 13 2 0  Z 2j −20j    A5 (11) nj ð T Þ ¼ d2j 4D 2j exp @ RT zone nj being usually weakly temperature dependent. It is worth noting that, owing to the high dilution, changes of the parameters with occupation have been ignored (for electrons in semiconductors, this is called the “rigid band model”). As anticipated above, the  Boltzmann form of the chemical potential results with m representing an effective value. If n is identified with nj, the effective value  0 mj is given by ej . As long as one refers to dilute conditions, considerable freedom of normalization is left with respect to the concentration measure. In ionic compounds, charged defects occur pairwise, the formation of which is subject to disorder reactions P (and electroneutrality conditions). Then, only the reaction combination of chemical potential vanishes, that is j vrjmj ¼ 0 P P ~j ¼ 0 since j vrjzj ¼ 0), where vrj is the stoichiometric coefficient of j in the respective disorder reaction r (identical to j vrj m (cf. Fig. 1, bottom). Boltzmann expressions can be written for the individual carriers, if the formation energies and the configuration entropies are considered to be independent. The application of chemical thermodynamics elegantly permits treatment of a variety of ionic and electronic disorder processes that simultaneously occur in a given solid. This is not only possible for the internal disorder equations (such as Schottky, Frenkel, anti-Frenkel, anti-Schottky, electron-hole formation) but also for external reactions such as the interaction of an oxide with the oxygen partial pressure and hence the stoichiometric variation. Under Boltzmann (i.e., dilute) and Brouwer (i.e., two majority carriers) conditions (Kröger, 1964), an adequate expression is arrived at for any charge carrier concentration as a function of the control parameters (temperature T, doping content, component partial pressure P), namely Y b cj ðT, P Þ ¼ aj j P Nj K r ðT Þgrj (12) r

((Nj, grj, aj, bj) being characteristically simple rational numbers; the total pressure is assumed to be constant.) For multinary compounds (i.e., m components) at complete equilibrium (m −1) component partial pressures are to be considered. This solution is only a sectional solution with a window in which the majority carriers situation (Brouwer condition) does not change. The powerlaw equation defines van’t Hoff diagrams characterized by −

X ∂ ln cj grj Dr H∘ ¼ ∂1=RT r

(13)



according to which, for not too extended T-ranges (i.e., DrH , the reaction enthalpy of the defect reaction g, is constant), straight lines are observed in the ln cj versus 1/T representation, as well as Kröger-Vink diagrams characterized by straight lines in the plots ln cj versus lnP with slopes ∂ ln cj ¼ Nj ∂ ln P

(14)

Fig. 4 displays the defect chemistry within the phase width of the oxide MO. At low oxygen partial pressure (PO2), vacant oxygen ions (V´O ) and reduced states (e0 ) are in the majority (2[V´O ] ’ [e0 ]  2[Oi00 ], [h]) (N − regime) whilst oxygen interstitials (Oi00 ) and oxidized states (h∙) dominate for very high PO2(2[Oi00 ] ’ [h˙] [e0 ],[V∙∙O]) (P-regime). At intermediate PO2, usually ionic disorder

Ionic and mixed conductivity in condensed phases

149

Fig. 4 Internal redox and acid-base chemistry in a solid MO1+d within the phase width at a given temperature T (disorder in the M-sublattice assumed to be negligible).

prevails ([V∙∙O] ’ [Oi00 ]  [e0 ],[h∙]) (I-regime); the alternative situation that electronic disorder predominates ([e0 ] ’ [h∙]  [V∙∙O], [Oi00 ]) is usually scarce. Exactly at the Dalton composition (i.e., at PO(i2 ) where 2[O00 i] ¼ 2[V∙∙O] ¼ [e0 ] ¼ [h∙]), the “nonstoichiometry” d in MO1+ d (i.e.,[O00 i] − [V∙∙O] ¼ 12 ½h:  − 12 ½e0 ) is precisely zero. For d  0, p-conductors (mobility of h∙ usually much greater than for O00 i) are referred to, for d  0, one refers (mobility of e0 is usually much greater than for V∙∙O) to n-conductors, while at d ’ 0, mixed conduction is expected (only if the ionic concentrations are much larger than the electric ones, pure ion conduction results). Usually, the entire diagram is not observed, as there are limits of experimentally realizing extreme PO2 values as well as limits with respect to the phase stability (formation of higher oxides or lower oxides, if not of the elemental metal). Fig. 5 gives three examples: SnO2 as an oxygen-deficient material, La2CuO4 as an example of a p-type conductor, and PbO as an example of I-regime exhibiting mixed conduction. (In PbO, most probably, the counter defect to Oi00 is Pb´i rather than V∙∙O, the results, however, are not different then.) The slopes directly follow from simple mass action considerations The model example of AgCl shall be considered in more detail. It also exhibits predominant ionic disorder, but now the decisive disorder reaction is the Frenkel reaction of the Ag sublattice (i.e., the formation of silver ion vacancy, VAg0 , and interstitial silver ion, Ag∙i). The role of PO2 is played by the chlorine partial pressure. The mass action law for the interaction with Cl2 reads [Agi] being [Agi][VAg0 ] ¼ K F and that for the band-band transfer PCI2+1/2[e0 ]¼ constant, the mass action constant for the Frenkel reaction

pffiffiffiffiffiffiffi −1=2 +1=2 ∙ ˙ ’ 0 0 0 [e ][ℎ ] ¼ KB. Since [Agi] ’ [VAg ]  [e ], [h ], it follows as solution V Ag ¼ ½Ag i   ¼ K F , e’ ∝P CI2 , ½h: ∝P CI2 .

Doping effects Dopants, i.e. irreversibly introduced structure elements as Cd2+ substituting Ag+ to form the defect Cd∙Ag, do not significantly influence the relevant mass action laws, but appear in the electroneutrality relation, here : h : i h 0 i (15) Agi + CdAg ’ V Ag Coupling the above relation with the Frenkel equation [Agi][VAg0 ] ¼ KF leads to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i C C2 0 V Ag ¼ + + KF 4 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :

C C2 Agi ¼ − + + KF 4 2

(16) (17)

pffiffiffiffiffiffi where C stands for [CdAg]. Consider the Brouwer conditions again. For C  K F , the intrinsic result [VAg0 ] ¼ [Ag∙i] ¼ √ KF is pffiffiffiffiffiffi obtained. For C  K F , one arrives at

V Ag ¼ C (18)

150

Ionic and mixed conductivity in condensed phases

(a)

(b)

Fig. 5 Three experimental examples of Kroeger-Vink diagrams in a pure oxide MO with ideal defect chemistry. (a) SnO2 as an n-type conductor, (b) PbO as a mixed conductor, and (c) La2CuO4 as a p-type conductor. Reproduced with permission from Maier J (2003a) Ionic and mixed conductors for electrochemical devices. Radiation Effects and Defects in Solids 158: 1–10; © Taylor and Francis.

½Ag∙ i  ¼ K F C −1

(19)

which are immediately obtained by neglecting [Ag∙i] from the electroneutrality relation. Power-law equations are then also valid for the electronic minority carriers in AgCl. Fig. 6 refers to AgCl and displays the solutions for the ionic defect arrived at by the more accurate equations for both concentrations and conductivities. The different behaviors of defect concentrations and conductivities stem from the fact that the interstitials are more mobile than the vacancies. The introduction of the dopant increases the concentration of the oppositely charged defect and depresses the concentration of the counterdefect according to electroneutrality and mass action. This leads to a decreased ionic conductivity, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi before the counter defect starts defining the overall conductivity. The conductivity minimum lies approximately at K F ui =uv . Figs. 7 and 8 display the dependence of conductivity on temperature in pure samples and on the doping concentration at given T with regard to positive and negative doping. While the response to Cd2+ doping (Cd∙Ag) follows exactly the predictions (see Figs. 7 and 8), the effect of S2− doping (SCI0 ) suffers from interaction effects (see below, the absence of a minimum in Fig. 8, LHS, is in qualitative agreement with the fact that [Ag∙i] is increased). It is clear that for doped samples under Brouwer and Boltzmann conditions, the concentration of any carrier is not only a function of P, T but also of C, and reads (Maier, 2004) ! ! Y bj Y Npi grj Mj cj ðT, P, CÞ ¼ aj Pp K r ðT Þ C (20) p

Nj

r

pi Pp P N p

is replaced by in order to allow for multinary equilibria labeled by p). Since. unlike (the in situ parameters) P and T, the variation of C requires a new high temperature preparation, C is designated as an ex situ parameter. Now this also predicts sectionally constant slopes in diagrams of the type log cj versus log C according to (p

∂ ln cj ¼ Mj ∂ ln C For a simple defect chemistry, the sign of Mj is determined by (“rule of heterogeneous doping”)

(21)

Ionic and mixed conductivity in condensed phases

151

Fig. 6 Defect concentrations and conductivities as a function of temperature for a fixed Cd-content (a) and (b) and as a function of Cd-content for a fixed temperature (c) and (d).

Fig. 7 Experimental conductivity data for nominally pure and doped AgCl as a function of 1/T. Here log(sT ) is plotted instead of log s to take account of the slight T-dependence of the prefactor. However, this does not alter the slope noticeably (DFH 0: formation enthalpy of Frenkel defects; DHj 6¼: migration enthalpy). Reproduced from Corish J, Jacobs PWM (1972) Ionic conductivity of silver-chloride single-crystals. Journal of Physics and Chemistry of Solids 33: 1799–1818, with permission from Elsevier.

152

Ionic and mixed conductivity in condensed phases

Fig. 8 The dependence of the conductivity increase brought about by the impurities on the S and Cd content of AgBr. The RHS solid curves are calculated according to the ideal defect chemistry. Reproduced with permission from Teltow J (1949) Zur Ionenleitung und Fehlordnung von Silberbromid mit Zusätzen zweiwertiger Kationen. I. Leitfähigkeitsmessungen und Zustandsdiagramme. Annalen der Physik 6: 63–70; © WileyVCH.; Teltow J (1950) Zur Ionen-, Elektronenleitung und Fehlordnung von Silberbromid mit Zusätzen von Silber-, Cadmium- und Bleisulfid. Zeitschrift für Physikalische Chemie 195: 213–224; & Oldenbourg Wissenschaftsverlag; © Wiley-VCH.

zj Mj tobs, where tobs is the typical observation time of experiments. The transition from liquid to glass is not a conventional phase transition between equilibrium states, however as long as the condition tlife >tobs is satisfied it is possible to use thermodynamic concepts, as we will see below. Many systems show a transition from the supercooled liquid to a glassy state. In a glass the atoms are not located in precise positions but the local arrangement of the atoms, called short range structure, sometimes resembles that of the corresponding crystal. The terms amorphous solid is often used as synonymous of glass. There are also systems, like water, where it is difficult to drive directly the supercooled liquid to the glassy state, the amorphous ice. Given the importance of water we will discuss its case later. In Fig. 2 it is represented the behavior of the enthalpy H ¼ U + pV as function of temperature. This thermodynamic potential is the most used in experiment on melting that are performed at constant pressure since it gives a measure of the work required for changing volume. As usual in the liquid phase the enthalpy decreases at decreasing temperature and at the melting point Tm there is a change DH related to the latent heat of fusion (or solidification). If the system is maintained in the supercooled liquid state the

Fig. 1 Schematic representation of a possible Gibbs free energy profile below melting as determined by the system configuration. From the upper stable liquid state it is possible upon cooling to reach the crystal stable state but with a very rapid quench the system can be driven to a metastable state. The higher the barrier DG between the higher and the lower minima the more the lifetime of the metastable state is increased.

Supercooled liquids

173

Fig. 2 Enthalpy change upon cooling. On the left Tm is the melting temperature. Tg is the glass transition temperature. On the right side the transitions to Glass1 takes place at a higher temperature with respect to Glass2 due to the different quenching rate.

enthalpy keeps to decrease as a continuous function. At the transition to the glassy state, T ¼ Tg, it is usually observed a change in the slope and the enthalpy decreases in similar way as in the crystalline phase. It has to be noted that while the melting temperature has a fixed value, since the liquid-glass transition does not occur between two equilibrium phases the glass transition temperature Tg depends on the quenching rate. By slowing down the cooling rate it is found a lower glass transition temperature. A schematic example is shown in the right side of Fig. 2. So by slowing down the quenching rate it is possible to decrease Tg. Kauzmann explored the possibility of predicting an asymptotic glass transition temperature. He observed from the phenomenology that the entropy of the supercooled liquid decreases more rapidly than the entropy of the crystal and he deduced that it could exists a temperature where the entropy of the supercooled liquid could reach and cross the curve of the crystal entropy. The temperature TK, that we call now Kauzmann temperature, is defined as the temperature where Sl ðT K Þ − Scr ðT K Þ ¼ 0:

(2)

where Sl and Scr are respectively the liquid and crystal entropy. Kauzmann realized that for T < TK the more disordered liquid state would have an entropy lower with respect to the ordered crystal. In order to avoid this paradox Kauzmann suggested that a crystallization would occur in approaching TK. Estimations of TK require extrapolations of experimental data well below Tg. The Kauzmann hypothesis has been widely discussed in literature. It has been suggested that the Kauzmann temperature would correspond to a limit of very slow supercooling process with the liquid attaining an ideal glass transition.

Dynamics of supercooled liquids Diffusion The motion of the atoms in a liquid is characterized by different regimes of diffusion. Each atom is surrounded by nearest neighbor but in the initial time it moves freely. This is called ballistic regime. After a period of time the atom starts to collide with the others and the diffusion enters in a more complex regime. This type of motion with frequent collisions is related to the old studies performed by the botanist Robert Brown in 1827 on the motion of pollen suspended in water. In 1905 Einstein reconsidered the work done by Brown and he developed a more general theory of the diffusion based on statistical mechanics. The diffusion of atoms can be studied by considering the displacement of an atom r(t) from the initial position at t ¼ 0. Since the motion is completely random the average on the time indicated with h. . .i will give hr(t)i ¼ 0. For this reason to study the phenomenon we must consider the mean square displacement(MSD). The average on the time obtained in experiments is replaced in the theoretical approach by the statistical mechanics average, realized by averaging on a great number of trajectories. The MSD is one of the important quantities that connects directly our macroscopic measure to the microscopic behavior. Einstein introduced the diffusion coefficient D as a phenomenological parameter and he found that along each direction in the long time limit hx2(t)i ! 2Dt. So the MSD increases linearly with the time following in three dimensions the Einstein relation < r 2 ðt Þ > ¼ 6Dt:

(3)

The diffusion of atoms in the liquid can be studied by experiments and simulations. It is observed that the diffusion is strongly determined by the thermodynamic conditions as represented in Fig. 3. In a liquid above melting after the ballistic regime for the frequent collisions the diffusion process enters in the Brownian regime described by the Einstein relation. From the long time slope the diffusion coefficient can be determined. Upon supercooling it is observed a strong modification. After the ballistic regime it appears a plateau. Below melting, even if it is avoided the transition to a crystalline structure, there is a formation of a cage of nearest neighbors around each atom. At variance

174

Supercooled liquids

Diffusion

102

n nia

ime

reg

w

Bro

above melting

100

-2

10

tic

lis

l Ba

e

m

gi

re

supercooled liquid

10-4 10-2

100

t

102

103

Fig. 3 Mean square displacement as function of time for decreasing temperatures.

with crystal the cage persists for a short time, but the region of the plateau corresponds to the period in which the atom is trapped in the cage. This is called cage effect. After a certain amount of time the cage relaxes and the diffusion starts again in the Brownian regime. The length of the plateau increases upon decreasing temperature.

Fragile and strong liquids Related to the diffusion in liquid it is also the viscosity, indicated as . The viscosity is intended as the friction due to the collisions that slows down the velocity of the moving particles. An higher viscosity reduces the diffusion. For spherical particles of radius r0 they are related through the Stokes-Einstein formula D¼

kB T : 6pr 0 

(4)

It is well known that a large increase of the viscosity is observed in approaching a transition to a solid state. By collecting the experimental measures of a large number of glass formers Angell found evidence that the viscosity increases of many order of magnitude in the supercooled liquid and it reaches a common value at the glass transition. He proposed as definition of Tg the temperature at which the viscosity  is found to be  ¼ 1013 poise

(5)

By considering the behavior of the viscosity in approaching the common value at Tg Angell introduced a precise classification of the glass former materials. Two types of phenomenology are found, as shown schematically in Fig. 4. There are systems, classified as strong glass formers, where log() increases linearly. This behavior is called à la Arrhenius and it is the most simple to explain  ðT Þ ¼ 0 eEA =kB T

(6)

where EA is the activation energy and 0 is the limit at high temperature. In an Arrhenius process it is considered that a particle can move by overcoming a potential barrier with EA the energy required. In the category of strong liquids there are systems like SiO2 and GeO2. This glass formers are characterized by a tetrahedral local order that does not change going to the glassy state. As a consequence it is found a small change in the specific heat across Tg. Other systems, called fragile, show a so called super Arrhenius behavior.  approximately follows the Vogel-Fulcher-Tamman (VFT) phenomenological formula
 BT 0 (7)  ¼ 0 exp T − T0 where T0 is an hypothetical temperature below Tg where  would diverge. The parameter B is related to the degree of fragility, since smaller values of B correspond to higher fragility. Fragile liquids, as for instance glycerol or ethanol, show a complete change in their local structure with a large change in the specific heat across Tg. The motion of the particles in fragile liquids upon supercooling is determined by a sequence of overcoming barriers in a configurational space more complex with respect to the case of strong liquids.

Supercooled liquids

175

Fig. 4 Behavior of log  as function of Tg /T: strong behavior as Eq. (6), fragile behavior as Eq. (7).

Slow dynamics The thermodynamic conditions determine the macroscopic density of the fluid r0. At the microscopic level the atoms move continuously. The local density of the particles r(r, t) fluctuates around the equilibrium value r0. During fluctuations the particles usually move from a zone at higher density to a zone with lower density to partially restore the equilibrium. In order to study more in details the density fluctuations we have to consider that the microscopic fluctuation in a position r at time t d r ðr, t Þ ¼ rðr, t Þ −r0 is correlated to the fluctuation in another position r0 at a different time t0 of another or the same particle. In the study of the diffusion we follow the motion of a single particle. In similar way we can follow the fluctuations of the density of a single particle rs(r, t) and look how the fluctuation at a certain time and position is related to a previous fluctuation in a different position and time. For this purpose we can introduce the statistical average of the product of the single particle fluctuations Cs ðr −r 0 , t −t 0 Þ ¼ hdrs ðr, t Þ drs ðr 0 , t 0 Þi

(8) 0

0

This is the self correlation function that in an homogeneous system depends from the differences r—r and t—t . The Fourier transforms of density correlation functions are accessible to neutron scattering experiments. By considering the Fourier transform of the single particle density fluctuations drs(k, t) it is possible to define the self intermediate scattering function as Fs ðk, t Þ ¼

1 hdrs ðk, t Þ drs ð −k, 0Þi N

(9)

From its definition Fs(k, t ¼ 0) ! 1. In liquids above melting the behavior of the Fs(k, t) is represented in Fig. 5 (black curve) for a wave vector k ¼ k0 where k0  2p/r0 with r0 the nearest neighbor distance between the atoms, the most interesting distance to explore for this phenomenon. After a short period corresponding to the ballistic regime, the function decays exponentially with the Debye equation F s ðk, t Þ ¼ e −t=t

(10)

where it is introduced the relaxation time t, that measures the time needed to the system to reach the equilibrium. So t is connected to diffusion and viscosity. In particular it results that t ¼ cost

T ∝ D

(11)

according to Eq. (4) t is proportional to . In the supercooled liquid it is observed a change in the behavior of Fs(k, t) due to the cage effects. As shown in Fig. 5, a plateau appears in correspondence to the period in which the atom is trapped in the cage. The long time decay is modified in the form of a stretched exponential

176

Supercooled liquids

1

ballistic regime Supercooled liquid

FS(k0,t)

0.8

plateau

0.6 KWW decay

0.4

Normal liquid

0.2 0

10-2

100

102

t

Fig. 5 Behavior of Fs(k0, t) for a liquid above melting and for a supercooled liquid. In the normal liquid (black curve) after the ballistic regime the function decays exponentially. In the supercooled liquid (red curve) after the ballistic regime the function shows a plateau and it decays with the KWW function. b

Fs ðk, t Þ  e −ðt=tÞ

(12)

where it is found that the exponent b < 1. This type of decay is called Kohlrausch-Williams-Watts (KWW) function. The relaxation time t increases of orders of magnitude with decreasing temperature with a different behavior expected for fragile and strong liquids since from Eq. (11) t ∝ . The dynamical behavior of supercooled liquids has been studied and interpreted by several theories and models. One of the most famous theoretical approach is the Mode Coupling Theory for the evolution of glassy dynamics (MCT). The MCT is a microscopic theory that provides an unified description of the dynamical properties of supercooled liquids. The cage effect and the long time decay of the correlation functions are derived from first principles starting from the only input of the static structure of the liquid. The glass transition is interpreted in terms of a transition from an ergodic regime where the system can reach an equilibrium state after the cages are dissolved to a non ergodic regime where the dynamic of the liquid is arrested. In particular the plateau in both the case of the diffusion and the dynamic correlation functions extends to infinity. The crossover would take place at a specific temperature TC. The relaxation time t introduced above will become infinite with a power law t  ðT − T C Þ −g :

(13)

where g is an exponent related to the microscopic properties of the liquid. According to MCT, by considering Eq. (11) the diffusion constant goes asymptotically to zero for T ! TC. In summary the MCT predicts the glass transition in terms of an ideal ergodic to non ergodic cross over taking place at a temperature TC that results higher with respect to the conventional glass transition temperature Tg. The MCT predictions can be tested by experiments in terms of asymptotic behavior of the diffusion and the long time decay of the density correlation functions. Often MCT works only in the regime of mild supercooling where extended structural relaxation drives the system toward ergodicity. In the deep supercooled region for many glass formers when the structural relaxations are frozen hopping (activated) processes restore ergodicity. In this regime the system explores the valley of the free energy landscape. Adam and Gibbs connected the relaxation to equilibrium of supercooled liquid in the deep supercooled regime with the entropy of the system. In particular they consider the role of the configurational entropy Sc that is approximately identified as the difference between the liquid entropy and the crystal entropy. Sc is related to the number of available glassy states at the given temperature and pressure conditions. Adam and Gibbs found for the relaxation time the equation
 a : (14) t ∝ exp TSc ðT Þ In the supercooled liquid approaching the glassy state Sc decreases, as consequence of the decrease of the possible configurations for the system. Sc can be derived from phenomenological observations of quantities like the specific heat in approaching the glass transition, in particular the configurational part of the heat capacity can be approximated as proportional to the inverse temperature Cc 

c , T

(15)

this can be integrated to obtain Sc. By assuming that it exists a temperature T0 for which Sc (T ! T0) ¼ 0 the configurational entropy can be calculated as Z T c dT 0 , (16) Sc ð T Þ ¼ 0 2 T 0 ðT Þ and substituting in Eq. (14) we obtain

Supercooled liquids
t ¼ t0 exp

 BT 0 : ðT −T 0 Þ

177

(17)

the VFT formula introduced before as obtained from experimental data.

Supercooled water Among all glass formers water singles out because its behavior is characterized by anomalous properties in comparison with other substances. The most well known anomaly is the fact that ice is less dense than liquid water. Upon decreasing temperature the density of liquid water increases as usual but at a temperature of 4  C at ambient pressure the density starts to decrease. The change of slope of the density vs. temperature defines a temperature of maximum density (TMD). The line of the TMD as function of pressure extends just above the melting line. At melting it is found that the density of the crystalline phase rcrys is less than the density of rliq. The reason is connected to the particular structure of liquid water and ice. The structure is based on the formation of hydrogen bond (HB) between water molecules. As shown in Fig. 6 the water molecule is formed by the bond of an oxygen and two hydrogens. In the H2O molecule the oxygen remains with two lone pairs of electrons and it has an effective negative charge while each hydrogen in the bond with the oxygen results to have a positive charge. The negative lone pair can attract the positive hydrogen of another molecule to form the so called hydrogen bond (HB). If enough molecules are present each molecule results to be connected to other four as shown in the figure. The pentamer shown in the figure is the nucleus of the crystalline structure of ice.

No man’s land and glassy states of water Because of its strongly stable pentameric structure water has a very strong tendency to crystallize. Even the ice cubes contained in our freezer are polycrystalline and made of small single crystal grains of hexagonal ice, which is the stable state of ice at ambient pressure. As a consequence the quenching of liquid water below melting to enter in the supercooled region is more difficult than in other liquids. Beside, the presence of impurities favors the formation of ice nuclei and the crystallization. By appropriate experimental techniques the concentration of impurities can be restricted, in this way the temperature of the onset of crystallization can be reduced. By extrapolating, it is possible to define for any pressure a limiting value for the temperature of supercooling. This temperature is called the homogeneous nucleation temperature, indicated with TH. The curve of TH as function of pressure has been for many years the lowest limit for performing experiments on supercooled water. In particular they were left without explanation the experimental observations of the anomalous increase of quantities like the isothermal compressibility KT and the isobaric specific heat Cp upon supercooling in approaching the TH line, as shown in Fig. 7. This type of strong increase of KT and Cp is observed when a liquid approaches the liquid-gas transition as a precursor effect of the divergence of those quantities at the critical point. New more recent methods have been successful in supercooling below the homogeneous nucleation temperature. Amorphous ice systems can be obtained by overtaking the region of supercooled liquid for instance by rapid quenching directly from the vapor. Two different amorphous ice states have been found, see Fig. 8. At lower pressure the low density amorphous (LDA) that coexists with the high density amorphous (HDA) at higher pressure. The coexistence line is close to 200 MPa and a first order transition is observed between the two states with changes in volume and enthalpy. LDA and HDA are characterized by different local structures. LDA shows a more ordered short range structure similar to a tetrahedral network, while in HDA the molecules are arranged in a more disordered structure where the tetrahedral short range order is lost.

Fig. 6 (a) water molecule, (b) H-bond between two molecules, (c) HB between four water molecules.

178

Supercooled liquids 110

KT (GPa-1)

1.25

CP (J mol-1 K-1)

100

Isothermal compressibility

1 0.75 0.5

90

240

260

280 T (K)

300

Isobaric specific heat

80

240

260

280 T (K)

300

Fig. 7 Isobaric specific heat and isothermal compressibility of water upon supercooling.

300 T (K) Supercooled Liquid 250

TH

LLCP 200

No man's land LDL

TX

150

Amorphous ices 0

0.1

LDA

p (GPa)

HDL

HDA 0.2

0.3

Fig. 8 Phase diagram of supercooled liquid water and amorphous ice states. The LDA/HDA coexistence evolves in the LDL/HDL coexistence in the no man’s land, where it would be present a liquid-liquid critical point (LLCP).

LDA and HDA undergo a glass transition at around 136 K, above 136 K it is found an high viscous liquid that at ambient pressure crystallizes in a cubic ice at 150 K. In analogy with the definition of the homogeneous nucleation line TH it is possible to define a temperature of spontaneous crystallization TX. The lines of TH and TX as function of pressure delimit from above and below respectively the so called no man’s land, where supercooled water is very difficult to study by experiments.

The second critical point and two forms of liquid water In the no man’s land region recent extensive computer simulations and experiments gave access to a rich phenomenology, see Fig. 8. The LDA and HDA upon increasing temperature above the line TH undergo a transition to two forms of liquid, called for analogy with the glassy states, low density liquid (LDL) and high density liquid (HDL). LDL and HDL keep local structures very similar to the corresponding glassy states. The great difference is that the coexistence line LDL/HDL would terminate in a critical point, called liquid-liquid critical point (LLCP), hypothesized by Poole, Sciortino, Essmann, and Stanley in 1992. On the basis of results of

Supercooled liquids

179

computer simulations and extrapolation of experiments the LLCP was estimated to be in the no man’s land at around 220 K at a pressure of 100 MPa approximately. The presence of this second critical point could explain the experimental observations of the anomalous increase of the isothermal compressibility KT and the isobaric specific heat Cp in approaching the border of the no man’s land. It is found that the LDL configuration is characterized by a lower entropy and energy with respect to the HDL structure. In LDL in fact the short range order is similar to the tetrahedral arrangement observed in ice. In the HDL structure instead there is a collapse of the tetrahedral structure making possible to reach an higher density. We have already seen that in a fluid the microscopic fluctuation of the density in a position r is correlated to the fluctuation in another position r0 . In normal thermodynamic conditions the density fluctuations in different points are correlated in a range determined by the correlation length x. Upon approaching the critical point the correlation length increases and it diverges at the transition. This induces the large increase and the divergence of KT and Cp since they become proportional to powers of x. As in the case of the liquid-gas transition also for the LLCP we can consider more in details the behavior of the system in approaching the critical point from the single phase region. It is found that both the isothermal compressibility and the specific heat start to show maxima along lines that finally collapse on the so called Widom line, the locus of the maxima of the correlation length that extends from the critical point toward the single phase region. In the case of the liquid-gas transition the Widom line close enough to the critical point separates liquid-like states from gas-like states in the single phase region. In computer simulations the Widom line can be located by searching the maxima of the specific heat or the isothermal compressibility in the single phase region. At the crossing of the critical point the Widom line will be connected to the coexistence line in the two phase region. We will show below that the Widom line has an important role in the classification of the dynamical behavior of supercooled water. In order to validate the results obtained with different water models in computer simulations, the liquid-liquid transition in supercooled water has been studied also with very refined methods of calculation of the free energy already typically used for the analysis of the second order phase transitions. The free energy was considered as function of two order parameters: the density and also a bond orientational order parameter that indicates the degree of crystalline order to exclude the formation of ice upon cooling. Recent interpretations of experiments and computer simulations start to consider liquid water as a mixture of the two competing states with different short range order HDL and LDL. With refined numerical analysis it is possible to evidence that water show a bimodal distribution of the two components. At ambient conditions the HDL component prevails but upon supercooling the LDL component appears and at low enough temperature starts to prevails. The liquid behaves similarly to a non ideal binary mixtures dominated by an interplay between energy and entropy. In this respect the critical point will be determined by an entropy driven unmixing process. A number of recent experiments evidenced the presence of LDL/HDL states in agreement with the theoretical predictions. A clear support to the two state model and the liquid-liquid transition derives also from the study of the dynamical behavior of supercooled water.

Dynamics of supercooled liquid water If we want to study the dynamics of translational motion of the water molecules we have to consider the diffusion and the density fluctuations of the center of mass of the molecules that corresponds approximately to the position of the oxygen. Supercooled water show a dynamical behavior very similar to the other supercooled liquids but with a very important peculiarity related to the presence of the LDL/HDL components. The self intermediate scattering function of the oxygen’s obtained by computer simulation upon supercooling, shows a long time KWW decay characterized by the stretched exponential b

−ðt=tÞ FOO s ðk, t Þ  e

(18)

In a range of temperature below melting the relaxation time t follows the MCT prediction of a power law divergence, Eq. (13). However at given pressure in a range of density by going down in temperature it is found a crossover to an Arrhenius behavior, by recalling Eq. (6) and that t   t ðT Þ ¼ t0 eEA =kB T

(19)

In Fig. 9 it is shown the data obtained in the simulation of supercooled water with the use of the TIP4P/2005 model. The data can be fitted with the MCT until a certain temperature then it appears a clear deviation and the remaining points follow an Arrhenius behavior. According to the classification introduced above this crossover from a fragile (MCT type) to a strong (Arrhenius type) behavior can be interpreted by considering that HDL and LDL have a more fragile and a more strong character respectively. A detailed analysis in all the range of the supercooled region shows that the fragile to strong crossover (FSC) takes place at the crossing of the Widom line as shown in the scheme in Fig. 10. These phenomenologies were observed with quasi elastic neutron scattering and optical Kerr effect measurements.

180

Supercooled liquids

Supercooled liquid water Relaxation time τ Density: 0.98 g/cm3

103

τ (ps)

Simulation data MCT fit Arrhenius fit

101

100

200

250 T (K)

300

Fig. 9 Relaxation time of water (in logarithmic scale) at the indicated density. The black points are the data from computer simulation with the TIP4P/2005 model, the MCT fit is obtained with Eq. (13) by assuming TC as a fitting parameter, the Arrhenius fit is obtained with Eq. (19).

Coexistence line

HDL p

LDL

LLCP Fr

ag

St

ro

ng

ile

Widom Line

T Fig. 10 Schematic representation of the phase diagram of supercooled water close to the LLCP. The coexistence line separates the LDL and HDL forms of water. The Widom line in the single phase region separates the HDL like from the LDL like regions.

Conclusion Supercooled liquids represents an important metastable state characterized by very peculiar connections between thermodynamics and dynamical properties. The microscopic processes that produce the slowing down of dynamics and the corresponding increase of viscosity in approaching the glassy state are widely studied with various techniques in experiment and in computer simulation. Supercooled liquids are in fact important for applications, developing technologies and development of new materials.

Further reading Debenedetti PG (1996) Metastable Liquids. Concepts and Principles. Princeton: Princeton University Press. Debenedetti PG and Stillinger FH (2001) Supercooled liquids and the glass transition. Nature 410: 259–267. Gallo P and Rovere M (2021) Physics of Liquid Matter. Cham, Switzerland: Springer International Publishing. Gallo P, et al. (2016) Water: A Tale of Two Liquids. Chemical Reviews 116: 7463–7500. Götze WG (2009) Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory. Oxford: Oxford University Press. Poole PH, Sciortino F, Essmann U, and Stanley HE (1992) Phase behaviour of metastable water. Nature 360: 324–328.

Glasses J M Parker, Department of Materials Science and Engineering, Sheffield University, Sheffield, United Kingdom © 2024 Elsevier Ltd. All rights reserved. This is an update of J.M. Parker, Glasses, Editor(s): Franco Bassani, Gerald L. Liedl, Peter Wyder, Encyclopedia of Condensed Matter Physics, Elsevier, 2005, Pages 273–280, ISBN 9780123694010, https://doi.org/10.1016/B0-12-369401-9/00538-6.

Introduction What is a glass? How are glasses made? Glass structure Short-range order Medium-range order Defects Nonsilicate glass-forming systems Crystal growth and phase separation Phase separation Glass formation Viscosity Processing Transformation range effects Thermal properties Mechanical behavior Optical properties Diffusion and electrical behavior Conclusion Further reading

181 182 182 182 182 183 185 185 185 186 186 186 187 187 187 188 188 189 190 190

Abstract This overview of glassy materials begins with production methods, highlighting how the homogeneity achieved, the creation of flat surfaces and the elimination of bubbles leads to their inherent transparency. The links between composition, structure and behavior are explained, leading on to an extended discussion of the main properties, ranging from mechanical and chemical durability through thermal, optical, diffusion and electrical characteristics. These properties are linked to common applications.

Key points

• • • • • •

What differentiates glasses from other materials? How are they made? What are their compositions and structures Their thermal, chemical and mechanical properties Their optical properties Diffusion and electrical behavior

Introduction Glasses feature widely in everyday life and also in science. Their transparency and chemical durability, combined with ease of fabrication, ensure a role in windows, containers, table- and laboratory-ware; they are key components in optical instrumentation, for example, microscopes, cameras, telescopes and optical fibers. Some devices utilize specific interactions between light and glass, for example, lasers. Other applications use their electrical and magnetic behavior, strength, and thermal characteristics, including crystallization and phase separation. The range and flexibility of known glass compositions allow different properties to be independently tailored. Combining glasses with other materials or modifying their behavior by coating has extended their use further. Understanding glass structure and its consequences is also challenging the boundaries of condensed matter physics.

Encyclopedia of Condensed Matter Physics, Second Edition

https://doi.org/10.1016/B978-0-323-90800-9.00238-9

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What is a glass? Glasses have been defined as inorganic products of fusion cooled to a solid without crystallizing. This encapsulates key properties: glasses have a liquid-like structure, but cooling has caused a transition from liquid to solid behavior. It also implies features that are now disputed: organic glasses exist, and melting is not the only route for glass making. A characteristic implicit in the definition is that crystallization rates are low; this is often associated with high melt viscosities below the liquidus (TL). Glasses differ from amorphous solids, in that they can be reheated to the liquid state; the latter crystallize on heating before melting (Doremus, 1994; Scholze, 1988; Uhlman and Kreidl, 1990; Varshneya, 1994; Zarzycki, 1991; Ossi, 2003).

How are glasses made? Often glasses are made from solids by heating until fully melted, bubbles have dissipated, and the melt is homogeneous. Commercially, temperatures may reach 1600
C and melting may take days. Laboratory melts need a few hours. Density separation of batch components, differential melting rates, contamination by the container, and volatilization losses can limit the glass quality. High melt viscosities limit diffusion and militate against homogeneity; mechanical or convection-driven stirring may be necessary. The oxidation state of any multivalent ions depends on the atmosphere around the batch during initial melting; subsequently, equilibrium with the furnace atmosphere is achieved only slowly. Rapid quenching may be needed, for example, roller quenching, splat cooling onto a rotating copper wheel, or fiber pulling. Alternative production routes include: condensation of vapor phase reaction products on cold substrates; hydrolysis of organic derivatives of silicon (e.g., Si(OC2H5)4) or other components to give a gel that is dried and sintered to a fully dense product; and extended grinding or applying pressure to crystals.

Glass structure Short-range order Glass structures have been studied by X-ray or neutron diffraction, various spectroscopic techniques, and computer modeling. Their short-range structure often mirrors that of corresponding crystalline compounds, particularly higher temperature phases. Zachariasen postulated that single-component glass-forming systems have cations with low coordination numbers (3 or 4) surrounded by bridging anions, each linking two coordination polyhedra at corners rather than edges. SiO2 consists of a matrix of SiO4 tetrahedra (Fig. 1) that defines its low thermal expansion coefficient (TEC), high viscosity at TL, and high melting point. Other single-component glass-forming systems include GeO2, BeF2, B2O3, and As2O3.

Fig. 1 A schematic representation of the structure of vitreous silica. The tetrahedral SiO4 units in silica are represented by triangular units.

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A second component can extend glass formation, for example, to 58 mol% Na2O in the Na2O–SiO2 system. Each oxygen added breaks one corner linkage in the network, creating two nonbridging oxygens (Fig. 2). Silicons are called “network formers” and sodiums, which enter interstices surrounded by bridging and nonbridging oxygens are termed “network modifiers.” A silicon linked to three bridging and one nonbridging oxygen is denoted by Q3; the spread of Qn species depends on composition and is generally narrow. Adding soda (Na2O) increases the TEC and reduces viscosity by depolymerizing the network, while increasing density by filling interstices. Various cations adopt a network-forming role (e.g., Ge, P, Pb). The melt structure, being flexible, can accommodate many modifier ions despite different radii and preferred coordination, for example, Li+, K+, Rb+, Mg2+, Ca2+, Fe2+. Intermediate ions (Al3+, Fe3+) can take either role in the structure. Some oxide combinations form glasses even though the components do not—the network-forming ion is called a “conditional” glass former, for example, Al in CaO–Al2O3 glasses. The oxygens can also be (partially) replaced by other anions, for example, F, S, N (Scholze, 1988). Properties can often be modeled using expressions that are linear functions of composition and the viscosity of industrial glasses is usually calculated rather than measured. Anomalies occur if a cation changes role with composition (e.g., B coordination depends on alkali content); the TECs of M2O–B2O3 glasses display a minimum with composition (the “boric oxide anomaly”), requiring a [B2O3]2 term in composition models. Liquidus temperatures, which are discontinuous across phase boundaries, are best modeled using the underlying free energy functions. Components can also interact: Na2O decreases network connectivity, while Al2O3 is a network former at low concentrations and, being oxygen deficient, removes nonbridging oxygens. If Na2O:Al2O3 ¼ 1, a highly connected, viscous system results; excess Na2O or Al2O3 decreases viscosity, in the latter case because Al coordination changes from [4] (former) to [6] (modifier).

Medium-range order Diffraction experiments give structural data in the form of radial distribution functions, representing the summed distribution of atoms with distance around all species present (Fig. 3); network-forming ions sit in remarkably regular sites but longer distance correlations are harder to resolve because of peak overlap. Extended X-ray absorption fine structure (EXAFS) experiments allow study of the environment of individual species and suggest that network modifiers also sit within polyhedra of notable regularity, that is, all cations optimize their local structure. Such distribution functions show that the network-forming polyhedra can also form extended structures, for example, vitreous B2O3 consists of BO3 triangles, that form planar groupings, termed boroxol rings (B3O6),

Fig. 2 A schematic representation of the structure of an Na2O–SiO2 glass illustrating the break-up of the network by the introduction of nonbridging oxygens (linked to only one Si), and the sites taken by the sodium ions.

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Glasses 250

pr2 g (r)

200 150 Si–O(2)

100 Si–O

Si–Si O–O

50 0

0

1

2

3 r (Å)

4

5

6

Fig. 3 An idealized radial distribution function for SiO2. The first four peaks give the distribution of nearest neighbor Os around Si, the Si–Si distribution, the O–O distribution, and the distribution of second nearest neighbor oxygens around Si. The peak areas give coordination numbers and the breadths indicate thermal and structural disorder. A random distribution would give a parabolic curve. Experimental curves show some ripple at low r caused by data termination errors. Many authors now plot correlation functions (pr g(r)) rather than r.d.f.s.

Fig. 4 A schematic representation of the modified random network model for an Na2O–SiO2 glass showing the tendency for modifier ions to segregate.

not found in B2O3 crystals. The modified random network model for alkali silicate glasses is based on segregation of network modifying atoms into layers surrounding SiO2-rich regions (Fig. 4). The diffraction patterns for many glasses have a relatively sharp low-angle first diffraction peak, consistent with extended ordering, as confirmed by modeling and electron microscopy. IR spectroscopy supports such structural models by accounting for the network vibrational modes. In Raman and Brillouin scattering experiments, glasses have, as a common feature, a boson peak at 100 cm−1 that provides additional insight into nanometer-scale inhomogeneities. Models incorporating medium-range order are distinct from older microcrystallite theories of glass, discounted by the lack of small angle X-ray or neutron scattering beyond that predicted from incipient random fluctuations in density. These conclusions underpin the thermodynamic modeling of glass properties based on contributions from components with appropriate compositions in defined proportions (Zarzycki, 1991).

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185

Defects Glasses can contain bonding and electronic defects induced by radiation or high temperatures. Their concentrations are low but they influence optical properties by introducing visible or ultraviolet (UV) absorptions and causing refractive index (r.i.) changes. Defects may involve unpaired electrons ( ), e.g., the nonbridging oxygen hole center, ^SidO ; E´ center, ^Si ; peroxy radical, ^Si2dOdO ; or species such as Zr3+, F−2. Some defects can be eliminated by optical or thermal bleaching, while others survive for years under normal conditions.

Nonsilicate glass-forming systems Silicate glasses have three-dimensional (3D) network structures with intermediate ionic/covalent bonds. Alkali phosphate glasses have similar bonding but are often better described as chains (Ropp, 1992) Chalcogenide glasses are largely covalent and based on networks, sheets, chains, and rings. Hydrogen-bonded molecular glasses are common in organic systems (e.g., sugars), while some highly ionic or metallic systems adopt almost random close-packed structures. Various salts (nitrates, sulfates, acetates) also form glasses. Even crystals can show glassy behavior. Spin glasses may form in magnetic systems, based on disordered arrangements of unpaired electron spins, or molecules may have random orientations on a regular lattice as in KBr–KCN. Polymers usually consist of covalently bonded chains, which may have irregular monomer unit sequences or cross-link randomly along their lengths, giving glassy materials at low temperatures that cannot untangle into crystalline forms. In extreme cases, oxide glass melts have been described with two or more configurations that transform only slowly from one to the other or may coexist in equilibrium (Ropp, 1992).

Crystal growth and phase separation Once a melt is cooled below TL, crystal growth is possible. The driving force (difference in free energy between crystalline and liquid state) increases with undercooling and initially growth rates rise as temperature falls (Fig. 5). Subsequently kinetic factors, for example diffusion through the melt to the growing crystal and atomic rearrangement at the interface, limit growth. In stable glass-forming systems, growth rates may be only mm h−1 but may reach m s−1 in glassy metals. For large undercoolings the fastest growing phase is not necessarily the primary phase defined by the phase diagram; metastable phases also often form. Crystallization requires a nucleation step. Homogeneous nucleation is an intrinsic property. The free energy gain on producing a crystalline nucleus

Liquidus temperature, upper limit for crystal growth

TL Crystal growth rate

Temperature

Metastable zone of supercooling

Homogeneous nucleation rate

Tg

Crystal growth and crystal nucleation rates Fig. 5 A schematic representation of the crystal growth and nucleation curves for a single phase precipitating from a glass-forming melt. In reality other phases may also appear at different temperatures with their own growth rate curves.

186

Glasses

must balance the energy penalty of generating its surface. Surface effects dominate for small particles and so nuclei must exceed a critical size before they are viable. Small undercoolings require a large critical nucleus whose spontaneous formation is improbable, so melts can exist metastably for long periods just below TL. At lower temperatures, the critical nucleus size decreases and nucleation rates rise rapidly to a maximum just above Tg, subsequently falling as kinetic factors dominate. Homogeneous nucleation is, however, rare. Nucleation starts more easily from pre-existing discontinuities, for example, “dirty” surfaces or container walls. Nucleation can be engineered by dissolving at high temperatures a component that becomes insoluble on cooling and precipitates (by homogeneous nucleation or phase separation). Examples include halides (NaF, CaF2, CuCl), oxides (TiO2, ZrO2, P2O5), or metals (Ag, Au). If homogeneous nucleation rates are high, two-step heat treatment in the nucleation and growth regions can give a finely crystalline structure. The resulting “glass ceramics” are made by standard melting and forming technology but, once crystallized, can have enhanced properties, while lacking the porosity intrinsic to conventional ceramics. Glass ceramics can remain transparent if the crystals are smaller than the wavelength of light and r.i. differences are minimized (Bach, 1995; Lewis, 1989).

Phase separation Phase separation is the chemical segregation of a melt into two liquids, driven by an associated reduction in free energy, for example, Na2O–SiO2 melts, with Na2O < 20 mol%, separate into SiO2 and sodium silicate; the SiO2 structure is relatively intolerant to impurities. The minor phase may occur as droplets or a continuously connected phase, whose scale depends on heat treatment time and whose microstructure can be preserved by quenching. The initial step may involve homogeneous nucleation or spinodal decomposition (no energy barrier). Borosilicate glasses also phase separate; the connected phase can be removed by solution, allowing manufacture of finely porous SiO2.

Glass formation First, the flexibility of linkages in network systems results in only small energy differences between glassy and crystalline states (cf. the many SiO2 polymorphs). This is a key thermodynamic factor in glass formation. Second, the network connectivity and bond strength cause the high melt viscosity and slow crystallization rates. Finally, the high melting temperatures destroy, by reaction or solution, foreign particles that might otherwise catalyze crystallization. Mathematical models, based on crystal growth and nucleation behavior, and their degree of overlap with temperature, can predict critical cooling rates for glass formation. They suggest that many systems, including some metal alloys, can form glasses. Cooling rates of 106
C s−1 may be needed, so heat transfer limits their production to thin artifacts, for example, ribbons or wires. New glass-forming systems have been identified, based on TeO2, Al2O3, Ga2O3, ZrF4, InF3, by using faster quenching, smaller samples, and purer materials melted under clean conditions to eliminate nucleating agents. Glasses are often associated with eutectics where liquidus temperatures are low and the melts differ most in composition (and structure) from crystalline compounds. Adding more components can also improve glass-forming ability—the so-called “confusion” or “garbage-can” principle (Doremus, 1994; Uhlman and Kreidl, 1990; Varshneya, 1994; Babcock, 1977; Rawson, 1991).

Viscosity

Length change or Cp

Melt viscosities (Z) typically follow the Vogel–Tammann–Fulcher equation (VTF) with temperature: logZ ¼ A + B/(T − T0) where A, B, and T0 are fitted constants. At the temperature where Z ¼ 1012 Pa s−1, deformation rates are slow, the material is effectively solid over short times (102s), and this provides one definition of the “glass transition temperature, Tg.” By the strain point (Z ¼ 1013.6 Pa s−1), stress release takes hours. First-order thermodynamic properties such as volume are continuous through the glass transition, in contrast to phase changes such as crystallization, but second-order thermodynamic properties (specific heat (Cp) or TEC) are stepped (Fig. 6). Tg can therefore also be measured from Cp or length changes using a differential scanning calorimeter or Further peaks, troughs associated with crystallization and melting Cp

Turnover associated with onset of flow

'l l

Tg Temperature Fig. 6 An idealized graph of the specific heat of a glass as a function of temperature during a constant heating rate experiment and the length of a glass specimen as a function of temperature under similar heating conditions. There is a step in Cp at Tg and a change of slope (corresponding to a step in the TEC) in the specimen length.

Glasses

187

a dilatometer, but the values depend on the heating rate used, since the transition is kinetically defined, and are not necessarily equal for different properties. Some have concluded that glass flows at room temperature, it being a liquid, and have perpetuated a myth that old windows are thicker at their base. Neither measurements nor calculations substantiate such conclusions, flow having become immeasurably slow. Where old panes are thicker at one edge, it is a manufacturing defect. A glass artifact, cooled quickly, will solidify with large temperature gradients present; the differential contraction on reaching room temperature can induce significant stresses. These stresses depend on cooling rate, TEC, elastic properties, and thickness; bulk articles are therefore annealed, by equilibration above Tg, followed by slow cooling to the strain point. Angell has compared glass-forming systems using normalized plots of log Z against Tg/T; those displaying Arrhenius behavior most closely he termed “strong” glasses while those with large deviations he termed “fragile.” Fragility implies cooperative breakdown of the melt structure into increasingly small units above Tg. The VFT formalism predicts Z ¼ 1 at T − T0. Some studies have attached physical significance to T0, relating it to the temperature where extrapolation of experimental data predicts that the entropy of a glass becomes equal to that of the corresponding crystal, that is, the configurational entropy falls to zero, a concept underpinning the “Kauzmann paradox” (Varshneya, 1994).

Processing High melt viscosities, along with gravity and surface tension, aid shaping as the melt cools. Sheet glass achieves optical flatness by gravity-driven flow, while floating on a molten tin bath with a 7 mm equilibrium thickness defined by surface tension and density. Optical fibers maintain their r.i. profiles during drawing. Containers and laboratory ware are blown and pressed by complex machinery to give the desired glass distribution. Strain rates may be high. A few glasses display non-Newtonian flow by structural alignment during forming (Cable and Parker, 1992).

Transformation range effects Glass structure does not depend only on composition but also on thermal history; densities may vary by 1%. Fast quenching freezes in a structural state corresponding to equilibrium at a higher temperature than slow cooling. The effective quenched-in equilibrium temperature is called the “fictive temperature.” Control of cooling through the transformation range allows fine tuning of an optical glass r.i., and low cooling rates ensure the whole sample has the same thermal history (and minimum stress). Optical glass manufacturers recommend that their products should not be heated above Tg ¼ 150
C because of potential property changes. Similar considerations apply to the processing of glass sheets for LCD displays. The electronic circuitry used to address individual pixels is printed at high temperatures in several layers; precise registry demands dimensional stability. A glass heated in the transformation range may, after an initial instantaneous change, relax exponentially to a new equilibrium state (Maxwell spring and dashpot model). Stress release rates can be measured and accurate data as f(t, T) are needed for high-precision fabrication. Detailed analysis suggests a stretched exponential formulation: St ¼ S0 exp. (−[t/t]n) where St is the stress at time t, S0 the initial stress, and t is proportional to viscosity, n ranges typically from 0.5 near Tg to 1. Many other properties behave similarly although not necessarily with the same constants for a given glass. Study of relaxation behavior and structural dynamics has given fruitful insights into the nature of the glass transition. At lower temperatures, bulk flow ceases but atoms can slide into new relative positions corresponding to different local energy minima, associated with the disordered structure. Such effects are manifest in internal friction experiments, for example, measurement of the damping of a pendulum suspended from a glass fiber under torsion. Damping involves transfer of mechanical energy to thermal energy and is observed down to room temperature in silicate glasses. Different mechanisms, termed a and b relaxations, dominate as temperature rises. These are attributed to mechanisms from modifier ion motion and relative movement of nonbridging oxygens, to motion of whole structural units. A related phenomenon is semipermanent densification of v-SiO2 at high pressures (GPa). Deformations of several percent do not fully relax on return to normal atmospheric conditions. Kinetic studies of relaxation suggest a low activation energy process, which is therefore not bond breakage; Raman spectroscopy confirms this. Early mercury-in-glass thermometers also displayed delayed elasticity. After calibration at 100
C, the mercury failed to return immediately to its 0
C calibration point on cooling. This relaxation behavior was minimized in mixed alkali glasses. Similar effects have been observed in rare-earth doped glasses; optical pumping into particular excited states, and the consequent energy transfer to the lattice, has excited structural changes that only recover at raised temperatures. The associated r.i. changes allow holographic image storage (Brawer, 1985).

Thermal properties Specific heats are consistent with Dulong and Petit’s law near Tg but show a positive step at Tg, the size of which indicates the level of structural breakdown occurring. Thermal conductivities are lower than for equivalent crystalline phases and decrease with falling temperature, consistent with the short phonon mean free paths associated with structural disorder. At high temperatures, heat transport by photons becomes significant—glass is semitransparent in the near infrared allowing absorption and re-emission as a heat transport mechanism, termed radiation conductivity. Radiation conductivity varies as T3 and is significant in commercial glass melting (Babcock, 1977).

188

Glasses

Mechanical behavior Glasses are brittle, with strengths dominated by their surface state. In microhardness tests, they exhibit limited flow under compression, attributed to densification and faulting. No flow occurs under tensile stress however; surface flaws remain atomically sharp at high loads and are effective stress raisers (stress at crack tip is enhanced by (r/r)1/2 where r is crack length and r tip radius). Flaws are induced by impact, friction, adhering particl