Quantum Hybrid Electronics and Materials (Quantum Science and Technology) 9811912009, 9789811912009

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Quantum Science and Technology

Yoshiro Hirayama Kazuhiko Hirakawa Hiroshi Yamaguchi   Editors

Quantum Hybrid Electronics and Materials

Quantum Science and Technology Series Editors Raymond Laflamme, University of Waterloo, Waterloo, ON, Canada Daniel Lidar, University of Southern California, Los Angeles, CA, USA Arno Rauschenbeutel, Vienna University of Technology, Vienna, Austria Renato Renner, Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland Jingbo Wang, Department of Physics, University of Western Australia, Crawley, WA, Australia Yaakov S. Weinstein, Quantum Information Science Group, The MITRE Corporation, Princeton, NJ, USA H. M. Wiseman, Griffith University, Brisbane, QLD, Australia Section Editor Maximilian Schlosshauer, Department of Physics, University of Portland, Portland, OR, USA

The book series Quantum Science and Technology is dedicated to one of today’s most active and rapidly expanding fields of research and development. In particular, the series will be a showcase for the growing number of experimental implementations and practical applications of quantum systems. These will include, but are not restricted to: quantum information processing, quantum computing, and quantum simulation; quantum communication and quantum cryptography; entanglement and other quantum resources; quantum interfaces and hybrid quantum systems; quantum memories and quantum repeaters; measurement-based quantum control and quantum feedback; quantum nanomechanics, quantum optomechanics and quantum transducers; quantum sensing and quantum metrology; as well as quantum effects in biology. Last but not least, the series will include books on the theoretical and mathematical questions relevant to designing and understanding these systems and devices, as well as foundational issues concerning the quantum phenomena themselves. Written and edited by leading experts, the treatments will be designed for graduate students and other researchers already working in, or intending to enter the field of quantum science and technology.

More information about this series at https://link.springer.com/bookseries/10039

Yoshiro Hirayama · Kazuhiko Hirakawa · Hiroshi Yamaguchi Editors

Quantum Hybrid Electronics and Materials

Editors Yoshiro Hirayama Tohoku University Sendai, Miyagi, Japan National Institute for Quantum and Radiological Science and Technology (QST) Takasaki, Gunma, Japan

Kazuhiko Hirakawa Institute of Industrial Science The University of Tokyo Tokyo, Japan

Hiroshi Yamaguchi NTT Basic Research Laboratories Atsugi, Kanagawa, Japan

ISSN 2364-9054 ISSN 2364-9062 (electronic) Quantum Science and Technology ISBN 978-981-19-1200-9 ISBN 978-981-19-1201-6 (eBook) https://doi.org/10.1007/978-981-19-1201-6 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Together with about 50 scientists in Japan, we pursued the basic science of hybrid quantum systems as part of the “Science of Hybrid Quantum Systems” project (HQS project). This collaborative effort started in July 2015 as one of the projects accepted for a Grant-in-Aid for Scientific Research on Innovative Areas, organized by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. At the time we started the project, the main studies in the field of quantum science and technology were aiming to demonstrate quantum information processing, especially quantum cryptography and quantum computations, on the basis of multiple qubit integration. Various physical systems such as superconductive devices, electron spins, atom, and ion traps were investigated as candidates for scalable integration. In contrast to such studies, which were conducted by integrating the same type of qubits, our project took a different approach: finding novel applications by integrating different types of quantum systems. We studied the coupling between different quantum excitations, such as charges, Cooper pairs, electrons and nuclear spins, photons, and phonons, with the aim of impacting a wide range of science and technology fields spanning fundamental physics to biological applications. As a number of joint studies progressed, some new research directions came about. Over the course of our research, we noticed that attempts to hybridize different physical systems often yielded highly beneficial results even in the classical regime. The 100% quantum regime is not a necessary condition to produce innovative technologies, but many interesting technological demonstrations are available in the intermediate domains between the purely classical and the 100% quantum. It is, therefore, one of the most significant conclusions of this project that the hybridization of “quantum” and “classical” is also an important approach in quantum technologies. Various high-impact and intriguing results were obtained during the five years this project spanned, and many collaborations are being pursued even after its conclusion. We are planning to publish two books (Hybrid Quantum Systems and Quantum Hybrid Electronics and Materials, both published by Springer Nature) to summarize our activities and to broadly discuss the importance of research based on hybridizing the systems constructed from different physical excitations as well as different quantum regimes. v

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This book, Quantum Hybrid Electronics and Materials, focuses on the hybridization of charges, Cooper pairs, electron and nuclear spins, photons, and phonons. As described in Chapter “Diversity of Hybrid Quantum Systems,” our focus is on technology improvement, material development, and advances in new metrological techniques with an eye toward future hybrid quantum systems. The many diverse chapters herein discuss various levels of hybridization ranging from 0 to 100% quantum, all of which will play a key role in future hybrid quantum systems. We hope that this book will contribute to an appreciation of the interests, importance, and diversity of “Hybrid Quantum Systems.” Finally, we are grateful to the members of the Editorial Office of Springer-Nature Publishing for their excellent help. We would also like to thank all the individual authors for their significant efforts. Sendai/Takasaki, Japan Tokyo, Japan Atsugi, Japan

Prof. Yoshiro Hirayama Prof. Kazuhiko Hirakawa Hiroshi Yamaguchi Senior Distinguished Researcher

Contents

Diversity of Hybrid Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yoshiro Hirayama

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Phonon Engineering for Quantum Hybrid Systems . . . . . . . . . . . . . . . . . . . Roman Anufriev and Masahiro Nomura

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Phonon Engineering of Graphene by Structural Modifications . . . . . . . . . Takayuki Arie and Seiji Akita

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On-Chip Wave Manipulations Enabled by Electromechanical Phononic-Crystal Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daiki Hatanaka, Megumi Kurosu, and Hiroshi Yamaguchi

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Electron and Phonon Transport Simulation for Quantum Hybrid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nobuya Mori and Gennady Mil’nikov

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Suspended Carbon Nanotubes for Quantum Hybrid Electronics . . . . . . . Yoshikazu Homma, Takumi Inaba, and Shohei Chiashi

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Quantum Effects in Carbon Nanotubes: Effects of Curvature, Finite-Length and Topological Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Wataru Izumida Synthesis and Transport Analysis of Turbostratic Multilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Ryota Negishi and Yoshihiro Kobayashi Quantum Anomalous Hall Effect in Magnetic Topological Insulator . . . . 181 Minoru Kawamura Transport Properties and Terahertz Dynamics of Single Molecules . . . . . 209 Shaoqing Du and Kazuhiko Hirakawa

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Novel Phonon Generator and Photon Detector by Single Electron Transport in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Mikio Eto and Rin Okuyama Hyperfine-Mediated Transport in a One-Dimensional Channel . . . . . . . . 257 Mohammad Hamzah Fauzi and Yoshiro Hirayama Microscopic Properties of Quantum Hall Effects . . . . . . . . . . . . . . . . . . . . . 277 Katsushi Hashimoto, Toru Tomimatsu, and Yoshiro Hirayama Semiconductor Chiral Photonic Crystal for Controlling Circularly Polarized Vacuum Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Satoshi Iwamoto, Shun Takahashi, and Yasuhiko Arakawa Hybrid Structure of Semiconductor Quantum Well Superlattice and Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Kouichi Akahane

Diversity of Hybrid Quantum Systems Yoshiro Hirayama

Abstract Quantum systems have received much interest from the view point of not only large-scale quantum computation/simulation but also quantum-enabled technologies (QETs), such as a highly-sensitive metrology based on quantum coupling. A variety of QETs require the hybridization of charges, spins, nuclear spins, photons, and/or phonons. Such hybridization also provides a way to make a quantum coupling across various scales of energy and distance. We need sophisticated control techniques and new materials to achieve such hybridization and reach to the goal of QET. This chapter discusses the outline and future direction of these activities as an introduction of hybrid quantum systems, especially quantum hybrid electronics and materials. Keywords Hybrid quantum systems · Quantum technology · Quantum sensing · Quantum materials

1 Introduction “Quantum systems” is an ambiguous term, but it has always been used in the cutting edge studies. The two-dimensional systems in SiO2 /Si interfaces or AlGaAs/GaAs quantum wells are characterized by a quantum confinement described by the Schrödinger equation. All low-dimensional (two-, one-, and zero-dimensional) devices require quantum physics to understand their behaviors. Mesoscopic research, which started in the 1990s or shortly before, developed fabrication technologies for sub-micron scale systems and low-dimensional devices, such as the quantum Hall Y. Hirayama (B) Center for Science and Innovation in Spintronics (CSIS), Tohoku University, 2-1-1 Katahira, Aoba, Sendai 980-8577, Miyagi, Japan e-mail: [email protected] Center for Spintronics Research Network (CSRN), Tohoku University, 2-1-1 Katahira, Aoba, Sendai 980-8577, Miyagi, Japan Takasaki Advanced Radiation Research Institute, National Institute for Quantum and Radiological Science and Technology (QST), 1233 Watanuki, Takasaki 370-1292, Gunma, Japan © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_1

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system, quantum point contact, and quantum dots. The operations of these devices can be understood with the help of quantum mechanics. This was the first generation of quantum systems. In the second generation, quantum mechanics was used not only to understand the background physics of device operations, but also to govern the operations themselves. The devices are controlled on the stage of quantum mechanics, now called “quantum technology”. A typical research target of quantum technology is quantum information processing [1]. Quantum cryptography can realize completely safe communication without risk of eavesdropping. A quantum computer can solve problems that are difficult for present-day super-computers to solve. Development toward quantum computing progressed with the introduction of Shor’s factorization algorithm (1994), which verifies the powerful features of quantum computing from a mathematical viewpoint. In contrast with quantum technology, existing technologies are called “classical” even if they include quantum aspects. In quantum computing, there is a new direction to develop quantum simulators even if they are incomplete as a quantum computer. Quantum simulators with thousands of qubits have been demonstrated and are now commercially available. To accumulate qubit structures and realize a quantum simulator or quantum computer is one important direction in the development of quantum technology. However, as shown in Fig. 1, other directions exist, including that to use small numbers of qubits to realize highly-sensitive quantum metrology. For example, for magnetic sensors, √ classical N times measurements improve sensitivity by N , but entangled N states can improve it N times in the optimal case. The improvement in this case is for example three-times improvement. This number is not huge; however, if we can finish a medical check in a third of the time or obtain deceased signals on a onethird scale, we can receive tremendous advantages. Therefore, control of quantum coupling without large-scale qubit integration is an attractive target. Furthermore, developing a quantum coupling between different physical systems is also considered an attractive target in many studies. Achieving full quantum control is still in its early stages, so it is natural to consider such a direction, sometimes called quantum enabled technologies (QETs).

Quantum Computer

Qubit integration

Fig. 1 Research directions of quantum science and technology. One direction is integration of the same qubit toward quantum computer. The completely different direction is the small-number quantum coupling of different qubits toward evolutional metrology and other purposes. We can expect plenty of different combinations there

Quantum Network

Quantum Simulator

Quantum Enabled Technology Quantum Sensor

Qubit application

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Fig. 2 Quantum hybridization between different physical systems. We can combine charges, spins, nuclear spins, photons, phonons, and so on. Development of cutting-edge technologies is needed for each system, including the sophisticated manipulation of photons and phonons to realize good hybridization

Hybrid quantum systems are based on QETs. Although it looks different from qubit integration, it has enabled the development of quantum techniques connecting charges, spins, nuclear spins, photons, and phonons (see Fig. 2). Quantum hybridization is one of the key technologies to realize quantum networks and/or a quantum Internet in the future. As shown in Fig. 1, reaching to the quantum network, there is the means to integrate different qubits, and then moving onto qubit numbers. In particular, quantum coupling between a qubit and a communication photon would have a huge impact. Other types of couplings will also contribute to short distance quantum connections, quantum memory, and various kinds of quantum sensing. The realization of a quantum network requires both quantum integration and hybridization.

2 Various Hybridizations As mentioned in the introduction and shown in the schematic view of Fig. 2, one direction of quantum hybridization is to achieve the quantum coupling between charges, spins, nuclear spins, photons, and phonons. Figure 3 provides different viewpoints of such hybridization. Spin–spin and Coulomb interactions are limited to a distance scale less than the μm range and an energy scale in the μeV to meV range. On the other hand, photon interactions cover a wide energy range from the THz regime to visible light. Photons can connect over very long distances (km). Phonons have small energy; but propagate to a distance scale in the μm to mm range. Therefore, quantum hybridization including charges, spins, nuclear spins, photons,

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Fig. 3 Different viewpoints of quantum hybridization. The hybridizations including charges, spins, nuclear spins, photons, and phonons correspond to establishing hybridization over a wide range of energy and distance scales

and phonons corresponds to a hybridization over a wide range of energy and distance scales. We can select different types of coupling depending on necessity and demand. It is important for the future quantum network to cover a wide range of energy and distance scales to meet various demands. In the coupled quantum dot, we can coherently control the charge (a single electron charge) and spin of a single electron as a charge qubit [2] and spin qubit [3], respectively. We can hybridize these coherent operations of charge and spin in semiconductor quantum systems. Electron and nuclear spins create a hyperfine coupling, and it is possible to realize hybridization between them. Recently, coherent coupling between the electron and nuclear spins at its individual level has become possible in the nitrogen-vacancy (NV) center of diamond and conventional Si quantum dots [4, 5]. It is possible to quantum couple an electron spin and the quantum state of a single photon toward spin and photon quantum hybridization, especially in diamond. It is also possible by using a semiconductor quantum dot. For superconductor qubits, they can be coupled with microwave photons. The conversion of microwave and optical communication photons has also been extensively studied toward developing a quantum interface between photons and superconductor qubits. In cavity quantum systems, quantum coupling between photons and trapped atoms (or ions) are essential. Quantum coupling between photons and atoms via evanescent coupling has been recently demonstrated by trapping atoms near carefully shape-controlled optical fibers [6]. There are many different quantum schemes where quantum features are finally converted to photons. In such situations, a coherent conversion between different photon wavelengths is important. This is because the manipulation of atoms, ions, and confined spins in solid-state systems has respectively different optimum wavelengths, and such wavelengths are different from the best ones in optical communication. Polarization insensitive quantum frequency conversion (QFC) was recently demonstrated using periodically poled lithium niobate (PPLN) crystal, and will therefore be a key device in future quantum hybrid systems [7, 8]. Although the quantum control of phonons is very difficult and requires extremely low temperatures, recent progress in micromechanical/nanomechanical systems has

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provided novel hybridization schemes based on phonons. For example, in cuttingedge semiconductor micromechanical/nanomechanical structures, we can place quantum devices on a mechanical bridge, resulting in a hybridization of charges, spins, and phonons [9, 10]. Coupling with nuclear spins and phonons has also been demonstrated [10]. It is also interesting to make a quantum coupling between photons and phonons by using an optical fiber and its mechanical vibration. Under appropriate conditions, evanescent coupling and its vibration can make a hybrid system between photons and phonons [8]. Although quantum computers and simulators will be achieved on the basis of special types of qubit, there will be many attractive quantum transducers connecting many different types of quantum systems. The various quantum couplings will be important in quantum sensing and quantum networks in the future. In the last of this section, it is noteworthy to point out a different concept of quantum hybridization. That is, hybridization between quantum and classical systems. The term of “hybridization” for many people conjures up images of “hybrid car”. The concept of a hybrid car is the hybridization of a newly developed electric car and traditional old-style gasoline car. Although the hybrid car is not 100% electric, its fuel efficiency has largely improved and has had a tremendous impact on society. The ratio of “quantum” varies from 0 to 100%. We have experienced that attempts to hybridize different physics quantities can sometimes yield highly-influential results even at 0% quantum. One good example is the fast, sensitive thermomechanical THz bolometer proposed by Zhang et al. [11]. Attempts to hybridize phonons and photons reached a combination of a mechanical bridge system and THz absorber, resulting in a novel THz detector with unprecedented performance. This detector is still a 100% classical system, and the introduction of quantum operations will further improve its performance in the future. In a number of sensing systems, the hybridization of “quantum” and “classical” results in higher sensitivity than existing systems. Even though a number of simulators do not include a fully quantum approach, it is possible to expect improved performance compared with the present-day super computers for a number of mathematical problems. Examples are coherent Ising machine with measurement feedback coupling [12]. In situations where fully fault-free quantum computing still requires a lot of development time, there is an approach called noisy intermediate-scale quantum (NISQ). The major trials of NISQ programs keep the weakness of quantum computer, noise accumulation, in mind and use quantum-classical hybridized systems. The data are exchanged between quantum and classical systems to compensate for the weaknesses of present-day quantum systems. How to include “classical” is a really important issue for “quantum” systems. Many different types of hybridization will open up new directions where many different types of sensors can be coupled with optical communication networks. Such directions include changing how medical systems operate, such as daily health checks and changing many social activities, such as autonomous driving, which is based on sensing. In these fields, a number of applications need a 100% quantum system, while others can sufficiently operate with a 100% classical system. The 100% quantum is not a necessary condition to produce evolutional systems and there are

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100% classical

100% quantum

Fig. 4 Analogy with a hybrid car (or train) that provides a different concept of hybrid quantum systems, where the hybridization changes from 100% quantum (0% classical) to 0% quantum (100% classical). It is noteworthy that 100% quantum is not a necessary condition to produce evolutional impacts

many intermediate situations. The hybridization of “quantum” and “classical” is a very important concept in quantum hybridization (Fig. 4).

3 Sharpening of Each Technology The diversity and hybridization of quantum systems are supported by the sophistication of technologies to manipulate charges, spins, nuclear spins, photons, and phonons. Quantum-dot systems and low-dimensional devices have been developed on the basis of GaAs, InGaAs, Si, and SiGe by using electric-field manipulation from surface nanoscale gates. The coherent manipulation of charge and spin has been demonstrated not only for electrons but also for holes. In a spin qubit, a radiofrequency (RF) magnetic field is usually applied by using a microwave antenna. However, to improve spatial selectivity, a micro-magnet is used to provide a local magnetic field gradient [13]. Electrons acquire an effective oscillating magnetic field when they spatially oscillate in the microscopic scale in a tilted magnetic field. The high speed and high fidelity Larmor oscillation of electron spin has been demonstrated by using such technology in combination with isotope engineering discussed in the next section [14]. In the field of the spin qubits, nuclear spins are regarded as obstacles, making the background nuclear spin fluctuation as the noise source of a spin qubit. The widelyaccepted method to remove obstacles is isotope engineering to remove nuclear spins discussed in the next section. However, nuclear spins can also be positively used as a quantum memory of the electron spin. The mutual coherent interaction between electron spin and nuclear spins is well studied for the NV center and surrounding nuclear spins in diamond [15]. Studies have shown that coherent coupling between a single electron spin and many nuclear spins can be manipulated in an InGaAs quantum dot [16]. The development of sophisticated crystal growth also provides us with a new paradigm. One example is an attempt to grow ultra-high-quality AlGaAs/GaAs quantum well structures to form a high-mobility two-dimensional electron system beyond 107 cm2 /Vs [17]. Enhanced mobility means less scattering and electrons have a stronger interaction with each other without disturbance of disorder. Historically, increased mobility changes the transport characteristics from a classical Hall

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system, integer quantum Hall system, and fractional quantum Hall system. Ultrahigh mobility has enabled novel fractional quantum Hall effects have been demonstrated. The most important achievement here is the finding of the 5/2 state [18]. This even denominator state is expected to have non-Abelian statistics and may be used for error-free quantum manipulation. Recently, the 3/2 state was also found in the confined regime in the high mobility two-dimensional system in GaAs [19]. Although it is a completely different story, the sophisticated crystal growth contributes to high-quality, density-controllable, and uniform quantum-dot structures [20]. Needless to say, they have made a great contribution to the research of hybrid quantum systems using quantum dots. Photons always play an essential role in hybrid quantum system because they can be controlled coherently and, more importantly, can connect over a long distance as already repeatedly described in this chapter. Naturally, sophisticated techniques in the photonic field will be a key technology of hybrid quantum systems in the future. Photonic crystals have been developed to manipulate photon propagation in solid-state systems. By using photonic crystals, like sophisticated three-dimensional photonic crystals, we can confine photons in the microscopic area and provide efficient coupling between photons and quantum dots where charge and spin can be well manipulated [21]. Recently, a topological idea was introduced for photonic crystals resulting in optical propagation along a sharp corner without loss [22]. Finally, it is noteworthy to introduce new advances in phonon manipulation. The cutting-edge technology will result in the detection of a single phonon, although demonstrations require extremely low noise measurements in ultra-low temperatures. Similar to photonic crystals, phononic crystals have been proposed and demonstrated. Phonon propagation is well manipulated in these systems [23], and phonons will play an important role as quantum connectors of different quantum systems on a relatively short distance scale.

4 Importance of Material Development The development of material science and engineering is also essential to the success of hybrid quantum systems as schematically shown in Fig. 5. One direction is material engineering based on already developed materials. This direction will push the hybrid quantum systems to cooperate with existing devices. Naturally, silicon is the most important material for present-day electronics. Differently from GaAs-based materials having nuclear spins for all isotopes, the only 29 Si with a natural abundance of about 5% has a nuclear spin of I = 1/2. By using isotope engineering and removing 29 Si, it is possible to create artificial materials without fluctuation arising from hyperfine interaction. Eliminating the background fluctuation of nuclear spins in the Si-based quantum systems results in long time coherence in the Si-based quantumdot spin qubit [14]. Recently, the combination of 29 Si density control and strongly confined quantum dots enable the demonstration of coherent coupling between a

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Op cal fiber, micro-reasonater, Photon, phonon, spin-wave, ---

Semiconductor quantum systems

Superconductor Nanowire quantum systems

Graphene Carbon-nanotube Diamond NV

Nanocarbons

Atoms & ions

Novel Materials

Fig. 5 Diversity of materials and systems for hybrid quantum systems

single electron spin and nuclear spin in a silicon metal-oxide-semiconductor (MOS) system [5]. Isotope engineering is also important in carbon-based materials, such as carbon nanotubes, graphene, and diamond. 13 C (I = 1/2 nuclear spin) has a 1% natural abundance and its nuclear spins near the NV center in diamond can be coherently coupled with the electron spin in the NV center [4, 15]. In such an approach, controlling the 13 C abundance in diamond is important. The periodic structures of 12 C and 13 C can be realized in nanotubes and graphene by using sophisticated crystal growth [24]. They are interesting from the viewpoint of phononic crystals based on nanocarbons, including thermal propagation engineering. Although strongly associated with the sophisticated crystal growth, nanowire structures, especially InAs and InSb nanowires, provide us with exciting directions. On one hand, they are extensions of existing narrow-gap semiconductors, and on the other hand, they can be considered as completely new materials. The narrow diameters release lattice-matching conditions. High quality heterostructures and even wire crossing can be made as designed by using crystal growth techniques [25]. More importantly, superconducting metal like Al can be grown directly on InAs (or InSb) nanowires, providing a most promising testbed of Majorana fermion physics [26], where one can expect a contribution to error-free qubit manipulation. These nanowires also provide us with opportunities to experiment with microscopic mechanical systems. The suspended carbon nanotube is also important for quantum hybrid electronics [27]. From the viewpoint of Majorana physics and novel spin manipulation, a topological insulator is an exciting emerging material. In the ideal topological insulator, the bulk is an insulator and only the surface channel holds carriers. These transports are

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topologically protected and the carrier-flow direction is associated with spin polarization. A Josephson-junction made of a topological insulator like HgTe showed topological superconductivity characteristics suggesting Majorana zero modes [28]. There are many other unique approaches, for example, magnetic doping in the topological insulators produces novel quantum anomalous Hall effects [29]. The von Klitzing constant, h/e2 , can be measured even in zero magnetic fields [29]. Although it is not directly related to quantum hybridization, it may change the situation of resistance standards in the future. Other important materials for hybrid quantum systems are layered materials represented by graphene. As well as graphene, many layered materials such as transition metal dichalcogenide monolayers have been developed. The layered structure operating as a good insulator like BN enables us to realize sophisticated functional layered structures based on van der Waals epitaxy, where layered structures can be made without lattice-constant limitation [30]. Atomically-thin floating films are also attractive as a novel mechanical bridge, where hybridization between phonons and other physics quantities can be expected. Despite its importance, high-quality layer formation still relies on mechanical peering, requiring the development of growth techniques for high-quality and wide-area atomic-layer films for future applications. Finally, the importance of defects and impurities should be addressed because they can act as an atomic-scale quantum system. The NV center is one of the wellknown defects in diamond and now the key component of highly-sensitive quantum sensing [15, 31]. Tremendous efforts are being made to efficiently couple nitrogen and vacancy, control the density of the NV center, arrange the orientation of the NV center, and control the charge in the NV center. There are similar defects in SiC. For example, VSi defects operate as a good qubit [32], have sensitivity to magnetic fields [33], and their operation wavelengths match better to optical fiber communication than those of the diamond NV center. SiC has now become a key material for highpower electronics; therefore, a combination of quantum sensing and already existing classical devices can be expected. Certain deep defects in Si behave as atomic-scale quantum dots because the donor level can attract electrons and operate as a single electron transistor [34]. Reflecting its small size, operating qubits can be extended to higher temperature regions than conventional qubits in silicon. Matching with already existing Si transistor technologies is also an attractive point in this approach.

5 Development of New Metrology For quantum sensors (for example, diamond NV sensors), one of the most important issues is√ how the sensitivity can be improved by quantum sensing. It has been shown that the T improvement is expected in principle for quantum sensing with a given total measurement time T when comparing quantum and classical sensing, but it is unclear what happens in a realistic noise environment. It has been theoretically √ shown that novel methods including quantum teleportation can demonstrate T improvement even in realistic noise environments [35, 36].

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The similar improvement in sensitivity can be expected for the quantum coupling. One of the best systems for investigating the effect of quantum coupling on quantum sensing would be superconducting quantum systems. Superconducting qubits are the most advanced systems toward quantum computers (simulators) and the coupling of thousands of qubits has already been demonstrated through microwave reasonator [37]. In such a system, the highly-sensitive sensing of a magnetic field is possible and electron- spin- resonance (ESR) has been demonstrated [38]. It may provide a good test system on how the control of quantum coupling improves the quantum metrology. Needless to say, diamond NV sensors are also a key device of quantum sensing. They will change our daily lives from performing health checks to autonomous driving. The capabilities of such a sensor, especially the nanoscale resolution of magnetic properties [31] is important as the metrology to probe condensed matter physics for hybrid quantum systems. It will be especially powerful to clarify the characteristics of layered structures and the surface channel of topological insulators. It is noteworthy that apart from novel nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) based on diamond NV, diamond NVs also contribute to largely improve already existing NMR and MRI. The optically-induced dynamic polarization of electron spins in diamond NVs can enhance polarization of nearby nuclear spins, improving the nuclear polarization level of materials used in conventional NMR and MRI [39]. NMR provides a versatile tool to study characteristics of not only chemical and biological systems but also semiconductor quantum systems. Although conventional NMR suffers from insensitivity and diamond NV sensing has limitations in the characterization of buried quantum structures, the resistive- detection of nuclear resonance can be applied to a single two-, one-, and zero-dimensional system embedded in semiconductors, especially GaAs-based semiconductors. In spite of their importance, the spin characteristics (such as spin polarization and its fluctuation) are difficult to clarify from the conventional transport measurements in semiconductor quantum systems. The resistively-detected NMR (RDNMR) measurements, such as Knight shift and nuclear relaxation measurements, clarify spin characteristics without ambiguity [40]. For example, the Knight shift measurement of the 5/2 state confirmed full-spin-polarized feature, leading to strong support for the 5/2 state being non-abelian and encouraging attempts to apply an even-denominator state to topologically- protected quantum operations [41]. The interesting features of RDNMR can be extended to one-dimensional quantum point contacts, including the successful observation of the strain distribution in a narrow channel [42]. Without a doubt, nanoprobe techniques are very important for not only the microscopic characterization of quantum systems but the future manipulation of quantum states in hybrid quantum systems. A combination of nanoprobe and diamond NVs was reported to be successful [31]. Novel scanning gate technology has been developed to visualize edge channels (incompressible stripe) in the integer quantum Hall effect regime [43]. This technique may also be effective to study the characteristics of edge channels in topological materials. Furthermore, a combination of a nanoprobe scanning gate and RDNMR enables to realize the MRI of semiconductor quantum

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structure versions [43, 44] Nanoprobing is also powerful in the THz regime, providing unprecedented control of electrons and sophisticated measurements of the dynamic properties for ultrafast atomic- scale electronics and metrology [45]. Finally, it is exciting to show a novel metrologies in the THz regime. THz matches well with the vibration mode of a molecule. THz spectroscopy has been developed to study the vibration of a sigle molecule trapped between an extremely narrow metal gap [46, 47]. The single molecule operates as a quantum dot, where the charge and spin (both electron and nuclear spins) play an important role. The vibrational and rotational modes can be affected by electron (or nuclear) spin polarization. Therefore, the developed system provide a new way to perform quantum hybridization including charges, spins, nuclear spins, THz photons, and phonons in the molecule.

6 Concluding Remarks Hybrid quantum systems include charges, spins, nuclear spins, photons, phonons, and more, and are diverse. Naturally, many different sciences and technologies, including different materials, are essential for the development of hybrid quantum systems. The following chapters of this book “Quantum Hybrid Electronics and Materials” provides us with cutting- edge examples of the sharpening of existing technologies, development of materials, and new metrologies, which are all helpful for the future development of hybrid quantum systems. The electronics and materials discussed here are also important as peripheral technologies of quantum information technologies. Acknowledgements I thank all the members of the MEXT project, Grant-in-Aid for Scientific Research on Innovative Areas “Science of Hybrid Quantum Systems” (2015-2020), especially K. Ishibashi, K. Hirakawa, H. Yamaguchi, and K. Nemoto for their valuable inputs and discussions. I acknowledge support from JSPS (KAKENHI 15H0587and 15K217270) and Tohoku University’s GP-Spin (Graduate Program in Spintronics) program.

References 1. Nielsen, M. A., & Chuang, I. L. (2000). Quantum computation and quantum information. Cambridge University Press. 2. Hayashi, T., Fujisawa, T., Cheong, H. D., Jeong, Y. H., & Hirayama, Y. (2003). Coherent manipulation of electronic states in a double quantum dot. Physical Review Letters, 91, 226804. 3. Koppens, F. H. L., Buizert, C., Tielrooij, K. J., Vink, I. T., Nowack, K. C., Meunier, T., Kouwenhoven, L. P., & Vandersypen, L. M. K. (2006). Driven coherent oscillations of a single electron spin in a quantum dot. Nature, 442, 766–771. 4. Yang, S., Wang, Y., Hien Tran, T., Momenzadeh, S. A., Markham, M., Twitchen, D. J., Stohr, R., Neumann, P., Kosaka, H., & Wrachtrup, J. (2016). High fidelity transfer and storage of photon states in a single nuclear spin. Nature Photonics, 10, 507–511.

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5. Hensen, B., Wei Huang, W., Yang, C. H., Wai Chan, K., Yoneda, J., Tanttu, T., Hudson, F. E., Laucht, A., Itoh, K. M., Ladd, T. D., & Morello, A. (2020). A silicon quantum-dot-coupled nuclear spin qubit. Nature Nanotechnology, 15, 13–17. 6. Aoki, T. Cavity quantum electrodynamics with laser-cooled atoms and optical nanofibers, Chapter 12. In Hybrid quantum systems (Springer-Nature, 2021) ISBN : 978-981-16-6679-7. 7. Ikuta, R., Kobayashi, T., Kawakami, T., Miki, S., Yabuno, M., Yamashita, T., Terai, H., Koashi, M., Mukai, T., Yamamoto, T., & Imoto, N. (2018). Polarization insensitive frequency conversion for an atom-photon entanglement distribution via a telecom network. Nature Communications, 9, 1997. 8. Kobayashi, T., Asano, M., Ikuta, R., Ozdemir, S. K., & Yamamoto, T. Photonic quantum interfaces among different physical systems, Chapter 9. In Hybrid Quantum Systems (SpringerNature, 2021) ISBN : 978-981-16-6679-7. 9. Okazaki, Y., Mahboob, I., Onomitsu, K., Sasaki, S., & Yamaguchi, H. (2016). Gate-controlled electromechanical backaction induced by a quantum dot. Nature Communications, 7, 11132. 10. Okazaki, Y., & Yamaguchi, H. Phonon-electron-nuclear spin hybrid systems in an electromechanical resonator, Chapter 11. Hybrid Quantum Systems (Springer-Nature, 2021) ISBN : 978-981-16-6679-7. 11. Zhang, Y., Hosono, S., Nagai, N., Song, S. H., & Hirakawa, K. (2019). Fast and sensitive bolometric terahertz detection at room temperature through thermomechanical transduction. Journal of Applied Physics, 125, 151602. 12. Yamamoto, Y., Aihara, K., Leleu, T., Kawarabayashi, K. I., Kako, S., Fejer, M., Inoue, K., & Takesue, H. (2017). Coherent Ising machines—optical neural networks operating at the quantum limit. npj Quantum Information, 49. 13. Pioro-Ladriere, M., Obata, T., Tokura, Y., Shin, Y.-S., Kubo, T., Yoshida, K., Taniyama, T., & Tarucha, S. (2008). Electrically driven single-electron spin resonance in a slanting Zeeman field. Nature Physics, 4, 776–779. 14. Yoneda, J., Takeda, K., Otsuka, T., Nakajima, T., Delbecq, M. R., Allison, G., Honda, T., Kodera, T., Oda, S., Hoshi, Y., Usami, N., Itoh, K. M., & Tarucha, S. (2018). A quantumdot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%. Nature Nanotechnology 13, 102–106. 15. Mizuochi, N. Control of spin coherence and quantum sensing in diamond, Chapter 1. In Hybrid quantum systems (Springer-Nature, 2021) ISBN : 978-981-16-6679-7. 16. Gangloff, D. A., Éthier-Majcher, G., Lang, C., Denning, E. V., Bodey, J. H., Jackson, D. M., Clarke, E., Hugues, M., Le Gall, C., & Atatüre, M. (2019). Quantum interface of an electron and a nuclear ensemble. Science, 364, 62–66. 17. Chung, Y. J., Villegas Rosales, K. A., Baldwin, K. W., Madathil, P. T., West, K. W., Shayegan, M., & Pfeiffer, L. N. (2021). Ultra-high-quality two-dimensional electron systems. Nature Materials, 20, 632–637. 18. Willett, R., Eisenstein, J. P., Störmer, H. L., Tsui, D. C., Gossard, A. C., & English, J. H. (1987). Observation of an even-denominator quantum number in the fractional quantum Hall effect. Physical Review Letters, 59, 1776. 19. Fu, H., Wu, Y., Zhang, R., Sun, J., Shan, P., Wang, P., Zhu, Z., Pfeiffer, L. N., West, K. W., Liu, H., Xie, X. C., & Lin, X. (2019). 3/2 fractional quantum Hall plateau in confined two-dimensional electron gas. Nature Communications, 10, 4351. 20. Akahane, K. Hybrid structure of semiconductor quantum well superlattice and quantum dot, Chapter 14, this book. 21. Iwamoto, S., Takahashi, S., & Arakawa, Y. Semiconductor chiral photonic crystal for controlling circularly polarized vacuum field, Chapter 13, this book. 22. Iwamoto, S., Ota, Y., & Arakawa, Y. (2021). Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform. Optical Materials Express, 11, 319–337. 23. Hatanaka, D., Kurosu, M., & Yamaguchi, H. On-chip wave manipulations enabled by electromechanical phononic-crystal waveguides, Chapter 3, this book.

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24. Arie, T. & Akita, S. Phonon engineering of graphene by structural modifications, Chapter 2, this book. 25. Barrigón, E., Heurlin, M., Bi, Z., Monemar, B., Samuelson, L. (2019). Synthesis and applications of III–V nanowires. Chemical Reviews, 119, 9170–9220. 26. Krogstrup, P., Ziino, N. L. B., Chang, W., Albrecht, S. M., Madsen, M. H., Johnson, E., Nygård, J., Marcus, C. M., & Jespersen, T. S. (2015). Epitaxy of semiconductor–superconductor nanowires. Nature Materials, 14, 400–406. 27. Homma, Y., Inaba, T., & Chiashi, S. Suspended carbon nanotubes for quantum hybrid electronics, Chapter 5, this book. 28. Bocquillon, E., Deacon, R. S., Wiedenmann, J., Leubner, P., Klapwijk, T. M., Brüne, C., Ishibashi, K., Buhmann, H., Molenkamp, L. W. (2017). Gapless Andreev bound states in the quantum spin Hall insulator HgTe. Nature Nanotechnology, 12, 137 29. Kawamura, M. Quantum anomalous Hall effect in magnetic topological insulator, Chapter 8, this book. 30. Novoselov, K. S., Mishchenko, A., Carvalho, A., & Castro Neto, A. H. (2016). 2D materials and van der Waals heterostructures. Science, 353, aac9439. 31. Casola, F., van der Sar, T., & Yacoby, A. (2018). Probing condensed matter physics with magneto. Nature Reviews/Materials, 3, 1–13. 32. Wolfowicz, G., Heremans, F. J., Anderson, C. P., Kanai, S., Seo, H., Gali, A., Galli, G., & Awschalom, D. D. (2021). Quantum guidelines for solid-state spin defects. Nature Reviews Materials, 6. 33. Ohshima, T., Satoh, T., Kraus, H., Astakhov, G. V., Dyakonov, V., & Baranov, P. G. (2018). Creation of silicon vacancy in silicon carbide by proton beam writing toward quantum sensing applications. Journal of Physics D: Applied Physics, 51, 333002. 34. Ono, K. High-temperature spin qubit in silicon tunnel field-effect transistors, Chapter 7. In Hybrid quantum systems (Springer-Nature, 2021) ISBN : 978-981-16-6679-7. 35. Matsuzaki, Y., Benjamin, S., Nakayama, S., Saito, S., & Munro, W. J. (2018). Quantum Metrology beyond the Classical Limit under the Effect of Dephasing. Physical review letters, 120, 140501. 36. Matsuzaki, Y. Robust quantum sensing, Chapter 13. In Hybrid quantum systems (SpringerNature, 2021) ISBN : 978-981-16-6679-7. 37. Kakuyanagi, K, Matsuzaki, Y, Déprez, C, Toida, H, Semba, K, Yamaguchi, H, Munro, W. J., & Saito, S. (2016). Observation of collective coupling between an engineered ensemble of macroscopic artificial atoms and a superconducting resonator. Physical Review Letters, 117, 210503. 38. Budoyo, R. P., Toida, H, & Saito, S. Electron spin resonance detected by superconducting circuits, Chapter 5. In Hybrid quantum systems (Springer-Nature, 2021) ISBN : 978-981-166679-7. 39. Schwartz, I., Scheuer, J., Tratzmiller, B., Müller, S., Chen, Q., Dhand, I., Wang, Z. Y., Müller, C., Naydenov, B., Jelezko, F., & Plenio, M. B. (2018). Robust optical polarization of nuclear spin baths using Hamiltonian engineering of nitrogen-vacancy center quantum dynamics. Science Advances, 4, eaat8978. 40. Hirayama, Y., Yusa, G., Hashimoto, K., Kumada, N., Ota, T., & Muraki, K. (2009). Electronspin/nuclear–spin interactions and NMR in semiconductors. Semiconductor Science and Technology, 24, 023001 [Topical Review]. 41. Tiemann, L., Gamez, G., Kumada, N., & Muraki, K. (2012). Unraveling the spin polarization of the ν = 5/2 fractional quantum Hall state. Science, 335, 828. 42. Fauzi, M. H., & Hirayama, Y. Hyperfine-mediated transport in a one-dimensional channel, Chapter 12, this book. 43. Hashimoto, K., Tomimatsu, T., & Hirayama, Y. Microscopic properties of quantum hall effects, Chapter 13, this book. 44. Hashimoto, K., Tomimatsu, T., Sato, K., & Hirayama, Y. (2018). Scanning nuclear resonance imaging of a hyperfine-coupled quantum Hall system. Nature Communications, 9, 2215.

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45. Yoshioka, K., Katayama, I., Arashida, Y., Ban, A., Kawada, Y., Konishi, K., Takahashi, H., & Takeda, J. (2018). Tailoring single-cycle near field in a tunnel junction with carrier-envelope phase-controlled terahertz electric fields. Nano Letters, 18, 5198–5204. 46. Du, S., Yoshida, K., Zhang, Y., Hamada, I., & Hirakawa, K. (2018). Terahertz dynamics of electron–vibron coupling in single molecules with tunable electrostatic potential. Nature Photonics, 12, 608–612. 47. Du, S., & Hirakawa, K. Transport properties and terahertz dynamics of single molecules, Chapter 10, this book.

Phonon Engineering for Quantum Hybrid Systems Roman Anufriev and Masahiro Nomura

Abstract Thermal management is essential for efficient semiconductor-based quantum systems. In this chapter, we discuss fundamental principles of heat conduction engineering via wave properties of phonons and review recent advances in this field. In phononic crystals—acoustic analogs of photonic crystals—phonons that remain coherent upon scattering on periodic boundaries can experience wave interference. In turn, the interference changes phonon dispersion relation and thus affects the thermal properties of the structure. In the past few years, researchers theoretically demonstrated that phononic crystal nanostructures could suppress or enhance heat conduction depending on the design of the structure. However, experiments could demonstrate such coherent control of thermal conductance only either at low temperatures or in superlattices with the periodicity and roughness at atomic scale. Thus, we discuss the advantages and limitations of coherent control of heat conduction, as well as possible strategies to overcome these limits in the future. Keywords Phonons · Phonon engineering · Thermal conductivity · Phononic crystals · Superlattices

1 Introduction A qubit, a two-state quantum mechanical system, is the basic unit of information in quantum computing. The coherence of a quantum system is important because the coherent superposition of two states is the source of unique functionality of the quantum computing. In a quantum system, qubits are placed distantly and coupled by quantum channels. Photons are the best flying qubits for long distance quantum state transfer. Semiconductor spin qubits are good candidates for quantum information storage due to long coherence time. However, several processes induce spin relaxation in semiconductors, and one of them is the phonon mediated spin-flip. Therefore, it is important to control the phonon R. Anufriev · M. Nomura (B) Institute of Industrial Science, The University of Tokyo, Tokyo, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_2

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motion in crystals for solid-based quantum hybrid systems to maintain coherence by reducing phonon-induced scattering. There are two main research directions that use semiconductor nanostructures for hybrid quantum systems; using nano/micro mechanical oscillators around the quantum ground state via optomechanics [1, 2] or cryogenic methods [3, 4], and preparing a prohibited area for phonons by phononic crystal structures [2, 5]. In addition to these research directions, we are interested in thermal phonon engineering, which controls broad spectral range of phonons at high temperatures, by nanostructuring. In this Chapter, we focus on the fundamentals of phonon engineering in semiconductor thin films based on phononic crystal nanostructures and recent theoretical and experimental studies on phonon and heat transport control. We explain the basic principles of phonon manipulation technique using the wave nature of phonons. Phonon band engineering by phonic crystals is based on the band folding effect, which changes the phonon dispersion relation by the additional periodicity. These wave phenomena are sometimes counterintuitive and result in unexpected phonon and heat transport properties.

2 Fundamentals In semiconductors, heat is primarily conducted by phonons—the quasi-particles representing quanta of lattice vibrations. As such, heat conduction in a given material depends on the phonon properties in this material. In turn, the phonon properties are mainly determined by the phonon dispersion relation—the dependence of phonon frequency on the wavevector. In the phonon dispersion, the slope of the branches determines the phonon group velocity and density of states, thus dictating how much and how quickly heat can be carried by the phonon gas. For this reason, control over the phonon dispersion ultimately leads to control over the heat conduction properties of materials or structures. Since phonons are essentially waves in the atomic lattice, phonon dispersion engineering is heavily inspired by methods established during the past decades for manipulations of electromagnetic waves in photonics. The core idea of this approach is to introduce the second-order periodicity, thus creating an artificial crystal, which would have properties different from the original bulk material. Similarly to photons in photonic crystals, phonons in such periodic structures would reflect from the periodic boundaries and experience constructive and destructive wave interference. As a rule, the wave interference changes the original dispersion in the given material into the phonon dispersion of artificial periodic structure. In turn, any changes in the phonon dispersion change the heat conduction properties. Thus, by engineering the artificially periodic structures, researchers can obtain the desired thermal properties of the material. Due to the similarity between these periodic structures for phonon manipulations and photonic crystals for light manipulations, the phonon analogs are called phononic crystals. Let us now consider examples and the working principles of such phononic crystal.

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(a) Pillars Membrane

(b)

PnC

(c)

Fig. 1 Schemes of a hole- and b pillar-based phononic crystals. c Phonon dispersion showing flattening of the branches as compared to the dispersion of the membrane

To illustrate, Fig. 1a, b shows typical phononic crystals based on thin membranes. The structure is essentially a suspended membrane with a periodic array of reflectors. The reflectors or the scattering centers are typically either holes in the membrane, pillars on the surface of the membrane, or inclusions inside the membrane. Correspondingly, such phononic crystals are called hole-, pillar, or inclusion-based. Phonon dispersion ω(k) of such periodic structures can be obtained by simulating one unit cell with periodic boundary conditions using the finite element method. By numerically solving the elastodynamic wave equation: μ∇ 2 u + (μ + λ)∇(∇ · u) = −ρω2 u

(1)

with u as a displacement vector, ρ as material density, λ and μ as Lamé parameters, we obtain ω(k) in a given geometry. Figure 1c shows that without the periodic reflectors, the membrane has a wellknown dispersion relation of a two-dimensional system and consists of three types of modes: box modes, symmetrical, and anti-symmetrical modes. In contrast, in the membrane with the periodic holes or pillars, the branches of phonon dispersion become flattened. The flattening of dispersion branches is caused by two main reasons. On the one hand, the phonon interference is induced by the reflection of phonons from the periodic holes or pillars. On the other hand, in the case of pillarbased phononic crystals, phonons can resonate in the pillars, which produces flat modes in the phonon dispersion. Such flat resonant modes have nearly zero group

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velocity and thus remain localized in the pillars. For this reason, such modes are called local resonances. The flat dispersion branches can result in the formation of phononic bandgaps— ranges of frequency at which phonon dispersion has no available states. In other words, phonons with the frequency inside the bandgap cannot propagate through the structure. The phononic bandgaps are used to create phononic cavities and phonon waveguides, similar to those used in photonics. Yet, the most profound effect of phonon interference in phononic crystals comes not from the bandgaps, which cover relatively narrow ranges of frequency, but from overall changes of the phonon dispersion. Figure 2a shows that the phonon group velocity in phononic crystals with arrays of holes is reduced as compared to the membrane without holes due to the flattening of the dispersion branches [6]. Likewise, the phonon density of states is affected by the interference. The density of states can be increased as well as reduced depending on the period of phononic crystal (Fig. 2b). As a result, the heat flux that is roughly proportional to the product of the group velocity and density of states is reduced in phononic crystals as compared to the membrane without holes (Fig. 2c) The strength of this reduction depends on the periodicity of phononic crystal, and as we will see further can even inverse into enhancement. Thus, phononic crystals can control heat conduction in the coherent regime using the periodicity as a tuning parameter. However, the interference and corresponding changes in phonon dispersion occur only as long as phonons reflect elastically. Such elastic phonon scattering is called coherent or specular, meaning that phonons preserve their phase and remain coherent after such scattering events. The probability of such specular scattering depends on phonon wavelength (λ), surface roughness (σ ) and phonon incident angle (α) as: p = exp(−16π 2 σ 2 cos2 α/λ2 )

(2)

Conversely, if the phonon wavelength is too short as compared to the surface roughness, the probability of specular scattering is low. In this case, the scattering will likely to be diffusive and coherence of phonons will be lost, and the conduction regime becomes incoherent. In this incoherent regime, phonons cannot develop the interference patterns and the phonon dispersion remains unchanged. Also, in the dense phonon gas, phonons frequently collide with each other, and these phonon-phonon scattering events destroy the coherence of phonons too. For this reason, phononic crystals are expected to be more efficient at lower temperatures, at which the phonon gas is rarefied, and the phonon-phonon scattering events become less frequent. Thus, the main challenge of thermal engineering via wave properties of phonons is to preserve the phonon coherence over long distances.

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Fig. 2 Calculated phonon a group velocity, b density of states, and c heat flux in phononic crystals of different periods at 0.5 K [6]

3 Advanced Researches In this section, we will discuss recent advances in the theoretical and experimental research on the heat conduction control using the wave properties of phonons. Theoretical works predicted several remarkable effects that normally cannot be achieved within a traditional paradigm of heat conduction engineering. To give an example, Fig. 3 shows the thermal conductance calculated from the dispersion relations of phononic crystals at 0.5 K simulated under a purely coherent phonon transport assumption, as was shown above. Under these assumptions, phonons behave as mechanical waves that reflect elastically from the boundaries and interfere.

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Fig. 3 Normalized thermal conductance of two-dimensional phononic crystals based on holes and pillars as a function of the crystal period [10]

Thus, the thermal conductance is determined by the interference patters created in the structure and can be calculated from as the phonon dispersion ω(k) as: F B Z ∂ω ∂ f (ω, T ) 1  dk, G= ωm (2π )2 m ∂k ∂T

(3)

0

where f is the Bose-Einstein distribution, and the integral is evaluated over the first Brillouin zone for each phonon mode m. In this purely coherent model, the calculated thermal conductance turns out to be inversely proportional to the periodicity of the phononic crystal. In other words, less porous phononic crystal yields lower thermal conductance. This result contrasts with a classical reduction of heat conduction in porous materials, in which the reduction is strongest for the highest, not the lowest, density of pores. Remarkably, when the periodicity becomes shorter than 50 nm, the phononic crystal becomes surprisingly more conductive than the membrane without holes. That is, porous material is more conductive than the material without pores at all. This counterintuitive effect is called the thermal conductance boost and is caused by an increase in the density of states due to the band flattening [7, 8]. Thus, in the purely coherent picture of phonon transport, heat conduction can be enhanced or suppressed by choosing specific periods of the phononic crystal. Moreover, the same effect enables controlling the heat capacity of the structure. For example, phononic patterning of thin membranes can increase or decrease the heat capacity depending on the design of phononic crystal [9]. Together these two effects can control the thermal conductance and heat capacity almost independently by simultaneously increasing one and decreasing the other. The effects of band flattening in planar phononic crystals were experimentally observe using the Brillouin light scattering technique. Experiments demonstrated that when the thin membrane is patterned with pillars or holes, they create flat branches in

Ordered

Fig. 4 Scanning electron microscopy images of ordered and disordered phononic crystals from Maire et al. [16] (left) and Lee et al. [17] (right). Plot shows estimated coherence of heat conduction in two-dimensional phononic crystals as a function of temperature. The data from [15–18]

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300 nm

600 nm

Disordered

Phonon Engineering for Quantum Hybrid Systems

300 nm

600 nm

phonon dispersion at frequencies up to 30–40 GHz [11, 12]. However, since heat is conducted by phonons in a wide range of frequencies, the impact of flattening of the low-frequency dispersion branches on the heat conduction depends on temperature. At low temperatures, the flattened low-frequency branched may cover a sufficient part of phonon spectrum and impact heat conduction, whereas at high temperature, their spectral contribution, and thus their impact on heat conduction, can be negligible. Figure 4 shows the estimated degree of coherence observed at different temperatures in two-dimensional phononic crystals [13]. Experiments at sub-kelvin temperatures demonstrated that heat conduction in phononic crystals is strongly reduced as compared to the membranes without holes [14, 15]. Moreover, the measured thermal conductance shows an accurate agreement with the predictions of the modeling based on purely coherent approximation. Thus, at sub-kelvin temperatures, heat conduction is close to the fully coherent regime. At higher temperatures, experiments detected only the much weaker impact of the phonon interference on the heat conduction [16]. By comparing heat conduction in ordered and disordered phononic crystals at 4 K, shown in Fig. 4, the difference presumably caused by wave effects seamed to affect less than 20% of the total thermal conductivity. This implies that at low temperatures, heat conduction is in an intermediate state between coherent and regimes. In this intermediate regime, lowfrequency phonons with long wavelengths and mean free paths can maintain their

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coherence whereas high-frequency phonons are likely to transport heat incoherently [16]. At temperatures above 15 K, the difference between ordered and disordered phononic crystals seems to disappear entirely. Studies at even higher temperatures could not detect any impact of the phonon interference effects in two-dimensional phononic crystal nanostructures [17, 18]. Thus, heat conduction seem to pass into purely incoherent regime at temperatures higher than few tens of kelvins. This comparison shows that the current state of nanofabrication enables creating twodimensional phononic crystals that can efficiently control heat conduction only at low temperatures. Figure 4 also indicates that surface roughness is expected to play a key role in preservation of phonon coherence, as we discussed above. Although currently fabricated nanostructures typically have surface roughness of about 2 nm, which allows to preserve phonon coherence only up to few tens of kelvins, nanostructures with smoother surfaces may support coherent conduction at higher temperatures. Indeed, in nanostructures with short enough periodicity and smooth interfaces, the phonon coherence can be preserved even at room temperature. Such conditions are met, for example, in one-dimensional superlattices—systems of periodic layers of different materials. Figure 5a, b and show examples of such crystalline superlattices. Studies on superlattices revealed that heat conduction exhibits a crossover from a coherent to incoherent regime as superlattice period becomes longer than a few nanometers [19–21]. Figure 5c shows the thermal conductivity measured for different superlattices as a function of their period [19–21]. In the coherent regime that corresponds to short periods, the thermal conductivity decreases as the period is increased, in agreement with the simulations under the elastic approximation. Conversely, in the incoherent

(a)

(c) 10

1 nm

(b) Sn Ti Ni Hf

1 nm

Incoherent

Thermal conductivity (W m-1K-1)

(CaTiO3)2 (SrTiO3)2

Coherent

1

Ravichandran et al. Saha et al. Holuj et al.

1

10

100

Superlattice period (nm)

Fig. 5 a Transmission electron microscopy images of a (STO)m /(CTO)n and b TiNiSn/HfNiSn superlattices [19, 21]. c Thermal conductivity of superlattices reported in the literature [19–21] as a function of their period at 300 K. The dependences show the crossover from incoherent to coherent heat conduction. Reproduced from [22]

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23

regime, when the period of the superlattice is too long, the thermal conductivity decreases as the period is decreased because the denser interface introduces more frequent diffuse scattering events. The characteristic superlattice period that separates the coherent heat conduction regime from the incoherent depends on the temperature and becomes longer at lower temperatures [19]. Yet, despite some success in the demonstration of coherent heat conduction in superlattices, the one-dimensional nature of these nanostructures limits their possible applications in heat conduction engineering. Another system, in which the coherent modifications of phonon dispersion are expected to impact heat conduction, consists of a thin membrane with arrays of nanoscale pillars. Such pillar-based phononic crystals have been extensively studied over the past decade using various simulation techniques [23]. Simulations suggest that pillars might cause a reduction in the thermal conductivity as substantial as orders of magnitude, even at room temperature. However, the experiments so far failed to demonstrate any modifications of the thermal properties that could be clearly attributed to the coherent effects induced by the pillars. Nevertheless, coherent control of the phonon transport found applications even at room temperatures in the field of optomechanics. In optomechanical cavities, the light is coupled with low-frequency mechanical vibrations that have sufficiently long wavelengths to preserve the coherence. This effect can be used for cooling [24] or amplification [25] of phonon modes in the optomechanical cavity. Future studies should aim for expanding the working range of phononic crystals to make them applicable both at higher temperatures and frequencies. This can be achieved by improving the fabrication quality of phononic crystals. Advances in nanofabrication should enable lowering the surface roughness and miniaturizing the phononic crystals themselves, making them a viable technology of future thermal management.

References 1. Kleckner, D., & Bouwmeester, D. (2006). Sub-kelvin optical cooling of a micromechanical resonator. Nature, 444(7115), 75–78. 2. Chan, J., Alegre, T. P., Safavi-Naeini, A. H., Hill, J. T., Krause, A., & Göblacher, S., Aspelmeyer, M., & Painter, O. (2011). Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature, 478(7367), 89–92. 3. O’Connell, A. D., Hofheinz, M., Ansmann, M., Bialczak, R. C., Lenander, M., Lucero, E., Neeley, M., Sank, D., Wang, H., Weides, M., Wenner, J., Martinis, J. M., & Cleland, A. N. (2010). Quantum ground state and single-phonon control of a mechanical resonator. Nature, 464(7289), 697–703. 4. Rocheleau, T., Ndukum, T., Macklin, C., Hertzberg, J. B., Clerk, A. A., & Schwab, K. C. (2009). Preparation and detection of a mechanical resonator near the ground state of motion. Nature, 463(7277), 72–75. 5. Pennec, Y., Vasseur, J. O., Djafari-Rouhani, B., & Dobrzy´nski, L., & Deymier, P. A. (2010). Two-dimensional phononic crystals: Examples and applications. Surface Science Reports, 65(8), 229–291.

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6. Anufriev, R., & Nomura, M. (2016). Reduction of thermal conductance by coherent phonon scattering in two-dimensional phononic crystals of different lattice types. Physical Review B, 93, 045410. 7. Anufriev, R., & Nomura, M. (2015). Thermal conductance boost in phononic crystal nanostructures. Physical Review B, 91(24), 245417. 8. Puurtinen, T. A., & Maasilta, I. J. (2016). Low-temperature coherent thermal conduction in thin phononic crystal membranes. Crystals, 6(6), 72. 9. Puurtinen, T. A., & Maasilta, I. J. (2016). Low temperature heat capacity of phononic crystal membranes. AIP Advances, 6(12), 121902. 10. Anufriev, R., & Nomura, M. (2017). Heat conduction engineering in pillar-based phononic crystals. Physical Review B, 95(15), 155432. 11. Graczykowski, B., Sledzinska, M., Alzina, F., Gomis-Bresco, J., Reparaz, J. S., Wagner, M. R., & Sotomayor Torres, C. M. (2015). Phonon dispersion in hypersonic two-dimensional phononic crystal membranes. Physical Review B, 91(7), 075414. 12. Yudistira, D., Boes, A., Graczykowski, B., Alzina, F., Yeo, L. Y., Sotomayor Torres, C. M., & Mitchell, A. (2016). Nanoscale pillar hypersonic surface phononic crystals. Physical Review B, 94(9), 094304. 13. Nomura, M., Shiomi, J., Shiga, T., & Anufriev, R. (2018). Thermal phonon engineering by tailored nanostructures. Japanese Journal of Applied Physics, 57, 080101. 14. Zen, N., Puurtinen, T. A., Isotalo, T. J., Chaudhuri, S., & Maasilta, I. J. (2014). Engineering thermal conductance using a two-dimensional phononic crystal. Nature Communications, 5, 3435. 15. Maasilta, I. J., Puurtinen, T. A., Tian, Y., & Geng, Z. (2015). Phononic thermal conduction engineering for bolometers: From phononic crystals to radial casimir limit. Journal of Low Temperature Physics, 184, 211–216. 16. Maire, J., Anufriev, R., Yanagisawa, R., Ramiere, A., Volz, S., & Nomura, M. (2017). Heat conduction tuning using the wave nature of phonons. Science Advances, 3, e1700027. 17. Lee, J., Lee, W., Wehmeyer, G., Dhuey, S., Olynick, D. L., Cabrini, S., Dames, C., Urban, J. J., & Yang, P. (2017). Investigation of phonon coherence and backscattering using silicon nanomeshes. Nature Communications, 8, 14054. 18. Wagner, M. R., Graczykowski, B., Reparaz, J. S., El Sachat, A., Sledzinska, M., Alzina, F., & Sotomayor Torres, C. M. (2016). Two-dimensional phononic crystals: disorder matters. Nano Letters, 16, 5661–5668. 19. Ravichandran, J., Yadav, A. K., Cheaito, R., Rossen, P. B., Soukiassian, A., Suresha, S. J., Duda, J. C., Foley, B. M., Lee, C.-H., Zhu, Y., Lichtenberger, A. W., Moore, J. E., Muller, D. A., Schlom, D. G., Hopkins, P. E., Majumdar, A., Ramesh, R., & Zurbuchen, M. A. (2014). Crossover from incoherent to coherent phonon scattering in epitaxial oxide superlattices. Nature Materials, 13(2), 168–72. 20. Saha, B., Koh, Y. R., Feser, J. P., Sadasivam, S., Fisher, T. S., Shakouri, A., & Sands, T. D. (2017). Phonon wave effects in the thermal transport of epitaxial tin/(al,sc)n metal/semiconductor superlattices. Journal of Applied Physics, 121(1), 015109 21. Hołuj, P., Euler, C., Balke, B., Kolb, U., Fiedler, G., Müller, M. M., Jaeger, T., Angel, E. C., Kratzer, P., & Jakob, G. (2015). Reduced thermal conductivity of TiNiSn/HfNiSn superlattices. Physical Review B, 92(12), 125436. 22. Anufriev, R., Maire, J., & Nomura, M. (2021). Review of coherent phonon and heat transport control in one-dimensional phononic crystals at nanoscale. APL Materials, 9(7), 070701. 23. Anufriev, R., & Nomura, M. (2018). Phonon and heat transport control using pillar-based phononic crystals. Science and Technology of Advanced Materials, 19(1), 863–870. 24. Chan, J., Mayer Alegre, T. P., Safavi-Naeini, A. H., Hill, J. T., Krause, A., Gröblacher, S., Aspelmeyer, M., & Painter, O. (2011). Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature, 478(7367), 89–92. 25. Kippenberg, T. J., & Vahala, K. J. (2008). Cavity optomechanics: back-action at the mesoscale. Science, 321(5893), 1172–1176.

Phonon Engineering of Graphene by Structural Modifications Takayuki Arie and Seiji Akita

Abstract A phonon is a quasiparticle responsible for hybrid quantum systems. Since it is the main heat carrier of graphene, phonon transport in graphene can be evaluated by measuring the thermal properties, which are modulated by structural modifications such as doping and defects. This chapter reports the impact of introducing isotope atoms and defects on the electronic and thermal properties of graphene. Controlling the phonon transport properties induced by the structural modifications plays a key role not only in fundamental physics but also in various applications of graphene such as thermal management devices. Keywords Graphene · Phonon · Thermal conductivity · Isotope · Defect

1 Introduction In carbon nanomaterials such as carbon nanotubes and graphene, each carbon atom binds to form sp2 hybrid orbitals. This arrangement provides in-plane honeycomb lattices of a graphite sheet. In addition to its extraordinary electronic [1–3] and optical properties [4, 5], graphene shows extremely high thermal transport properties [6, 7]. The thermal conductivity κ is expressed as κ = κe + κl , where κ e and κ l are the electron and lattice (phonon) contributions to κ, respectively. κ e is normally described by the Wiedemann–Franz law as κe /σ = L T , where σ is the electrical conductivity   and L = π 2 /3 (k B /e)2 is the Lorentz number with the Boltzmann constant k B and the elementary charge e. In graphene, the contribution of κ e to κ is quite small ( 108 [26], and the effective mass has been reduced to m eff ∼10−20 kg by optimizing the device host medium and geometry [31]. This enables us to resolve an extremely tiny force at a zepto-newtons-level (12 × 10−21 N Hz−1/2 ) in a carbon nanotube resonator [31] and detect the spin [8, 11, 12] and charge of a single electron [6, 7] and thousands of nuclear spins [9, 10]. Mechanical resonating systems have also been recently explored for possible application to RF signal processing. The low dissipation and short wavelength of acoustic waves compared with those of electromagnetic waves at a given frequency can be exploited to build compact processing elements such as filters, oscillators and switches with low consumption power [13, 14]. Surface-acoustic-wave (SAW) and bulk-acoustic-wave (BAW) filters and oscillators have been used practically in communications devices. However, these devices have passive structures and are thus static components that lack dynamic functionality.

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Fig. 1 a Illustrations of a cantilever (left), a beam (middle), and a membrane structure (right). b Left: SEM image of a mechanical parametron device based on a GaAs/AlGaAs-based electromechanical resonator [33]. Middle: illustration of a child’s swing sustained by the parametric modulation. A father (blue) pushes his child (red) at frequency f 0 . The swing oscillation is amplified with the modulation in a center mass position of the child at 2 f 0 . Right: temporal response of a harmonic driving at f 0 (blue) and a parametric pump at 2 f 0 (red) and the resultant amplified harmonic oscillation (black)

To overcome this drawback, a parametric modulation scheme has been proposed and demonstrated [32], where a secondary external stimulus is used to modulate the mechanical characteristics dynamically as shown in Fig. 1b [33]. For instance, in a parametric amplification technique, which is familiar to a child’s swing, driving the device at 2 f 0 , with f 0 being the resonance frequency, induces strain modulation twice during a single vibration cycle, enabling the displacement amplitude of the f 0 vibration to be amplified by the modulation. In this way, a variety of dynamic control schemes have been developed based on the parametric approach, such as frequency conversion [34, 35], mode coupling [36, 37], squeezing [32, 38], and lasing [39]. In addition, this has also led to the exploration of complex nonlinear dynamics [40–45] and even the study of a mechanical analogue of the Ising Hamiltonian [46]. However, since most available acoustic oscillations are localized vibrations, the acoustic devices are placed at the terminals of an integrated circuit, so their practical use is limited to only sensing and filtering. Therefore, the dynamics of acoustic phonons has not yet been fully investigated and utilized. This is in stark contrast to photons, where optical waves are employed not only to process information but to transmit it over long distances as well, which has enabled the formation of all optical networks. The key to expanding the availability of phononic technology is the PnC, which is described in the next subsection.

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1.2 Phononic Crystal (PnC) A PnC is an artificial acoustic material in which different elastic composites–air and a solid material–are periodically arrayed on an acoustic wavelength scale [15, 16]. Acoustic waves traveling in this structure experience a periodic elastic potential resulting from the difference in the acoustic velocity between the air and solid. This induces wave reflection at the air-solid interface. When the acoustic wavelength (λ) is equal to the periodicity (a), namely the Bragg condition nλ = 2a, where n = 1, 2, 3..., the reflected waves destructively interfere with incident waves, thus preventing the later from propagating in this acoustic lattice [47]. This phenomenon is a phononic bandgap, which is an acoustic analogue of the bandgap in a semiconductor crystal, where the de-Broglie wavelength of an electron and atomic lattice constant satisfy the Bragg condition [see Fig. 2a] [48]. Similar to semiconductor-based superlattices grown by molecular beam epitaxy, the dispersion relation of acoustic waves can be designed by engineering the periodicity. As a result, functional components such as an acoustic resonator (cavity) and a waveguide can be realized by locally

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Fig. 2 a Left: a schematic of a two-dimensional PnC, which is composed of two different elastic media (red/gray) with a periodic constant a. Right: dispersion relation of acoustic waves traveling in an unstructured membrane (dashed line) and a PnC membrane (red line). A bandgap opens when the Bragg condition is satisfied, as highlighted in yellow. b A schematic of a sonic crystal constructed from a periodic arrangement of metal pillars that prevent sound at a specific frequency from propagating. c Schematic of a hypersonic PnC circuit on a suspended membrane where linedefect waveguides and cavities are integrated

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modifying the PnC lattice so that the acoustic propagation can be spatially trapped and guided in an efficient manner. Thus the PnC platform is suitable for making use of both traveling and localized acoustic vibrations. Research on the PnC began in 1979 [49] with the intent to apply them for sound management, e.g., sound insulators, as shown in Fig. 2b [50]. In recent years, this has been used for chip-scale control of high-frequency and ultrahigh-frequency acoustic phonons such as ultrasound (kilohertz to megahertz) and hypersound (gigahertz range) as shown in Fig. 2c [51–55]. The ability to modulate the band structure of acoustic waves is useful in designing an RF filter function and realizing ultralowloss and noise-free resonators for sensitive sensors and stable oscillators [56, 57]. Furthermore, a cavity and a waveguide can be sustained in the PnC lattice [53, 58– 61], which has led to an attempt to develop all-phononic circuits [62]. However, the conventional PnC architectures have also been passive, the mechanical characteristics of which are restricted to only the lattice design. Thus, realizing dynamic properties in PnC remains challenging.

1.3 Phononic Technology for Hybrid Quantum Systems Alternatively, one of the most intriguing and important applications of phononic systems is a hybrid quantum system. This emerging field seeks to exploit acoustic phonons for quantum technology and information processing. Thanks to the coherent and CMOS-compatible mechanical structures, they have been exploited in the research fields of photonics, magnonics, and electronics as illustrated in Fig. 3a. In particular, interactions between mechanical and optical fields through radiation pressure and photoelastic effects can be enhanced in such a hybrid optomechanical

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Fig. 3 a Phononic technology enabling different fields of photonics, magnonics and electronics to be hybridized. Representative interactions, such as magnetostriction, radiation pressure, and the piezoelectric effect, and conceptual images of hybrid systems are also shown

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system because the simultaneous integration of mechanical and photonic systems on a tiny chip and the energetically stable and high-quality optical/mechanical oscillations drastically increases the coupling strength [63, 64]. To that end, the ability to efficiently manipulate the vibration energy in N/MEMS and PnCs by using light and vice versa has finally led to the observation of the mechanical quantum ground state [65–67]. This demonstration opened new opportunities for using the mechanical system as a quantum mediator capable of bridging information between various quantum subsystems, such as flux, charge and spin qubits, and photonic crystals [68, 69]. The mechanical vibrations offer not only efficient transduction between microwavefrequency signals and optical-wavelength signals (photonic circuits) but also intricate on-chip manipulations of quantum information thanks to the slow propagation speed and short wavelength [70–79]. Thus, increasing the mechanical functionality is important for developing quantum-enabled technology such as quantum sensing and realizing quantum networks/circuits. Moreover, hybrid mechanical systems are recently becoming important even in classical (non-quantum) applications [80–84]. For instance, the advent of a phoxonic crystal, which is the simultaneous localization of a photonic crystal and a PnC, has enabled the realization of an efficient ultrahigh-frequency acousto-optical modulator [81, 82] and wavelength converter [84]. The ultrasonic and hypersonic vibrations can be used to generate or modulate the dynamics of exciton [85, 86] and spin waves [87–92] rather than external electromagnetic waves as shown in Fig. 2c. Thus, the development of the acoustic controls with high versatility is essential and highly desired.

1.4 Electromechanical PnC As described above, two major acoustic platforms have been used in phononics– N/MEMS and PnC–each of which has advantages and disadvantages. For the former, various techniques for dynamic control of acoustic vibrations have been proposed and demonstrated, whereas we have lacked the ability to spatially control them. In contrast, the PnC platform provides spatial control of acoustic wave transmission but not dynamic control. Namely, the functionality of these two different systems is in a complementary relationship. Here, we introduce a novel acoustic platform called an electromechanical PnC, which is realized by combining these two platforms [17]. The platform consists of a one-dimensional array of electromechanical membrane resonators [93, 94]. The unit cell of the membrane resonators are peizoelectrically active and can be driven by N/MEMS technology. In addition, the periodic arrangement gives rise to phononic bandgap that enables us to tailor the dispersion relation of acoustic waves [18]. By investigating the fundamental properties of traveling ultrasound waves, such as their group velocity and dispersion relation, a method of acoustic waveform engineering was developed using the dispersion effect [19]. Moreover, nonlinear propagation

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dynamics was investigated that allow the observation of acoustic nonlinear phenomena and the demonstration of dynamic wave control with N/MEMS technology [20]. The demonstrations show that the electromechanical PnC platforms could greatly enhance the usability of acoustic phonons for signal processing [95–97] and hybrid quantum system applications.

2 Device and Properties In this section, we describe the device structure and the fabrication. Then, we cover the acoustic transmission properties based on the dispersion relation and the group velocity with respect to various lattice designs.

2.1 Fabrication The membrane resonators based on the device are fabricated from GaAs (5 nm)/ Alx Ga1−x As (95 nm) (x = 0.27, 0.30)/n-GaAs (100 nm)/Al y Ga1−y As (3.0 μm) (y = 0.65) heterostructure as shown in Fig. 4 [17]. A periodic array of circular holes is formed with a pitch (a) in the membrane heterostructure by photolithography and wet etching with a mixed solution of hydrogen peroxide (H2 O2 ) and phosphoric acid (H3 PO4 ). We adjust the etching time so that the hole depth reaches the Al y Ga1−y As layer. Subsequently, the sample is immersed in a diluted hydrofluoric acid (HF, 5%)

Membrane resonator array

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Fig. 4 Schematic of an electromechanical PnC WG composed of a one-dimensional membrane resonator array. The membranes are made by a GaAs/Alx Ga1−x As/n-doped GaAs heterostructure (x = 0.27, 0.30) with dimensions of hole pitch a and width w, and they are suspended by selectively etching the Al y Ga1−y As sacrificial layer (y = 0.65). Acoustic waves are piezoelectrically excited by applying an oscillating voltage to an electrode at a WG end. They are detected with an optical Doppler interferometer at room temperature in a vacuum. The figure is partially reproduced from Ref. [17]. © Springer Nature (2014)

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to selectively etch the Al y Ga1−y As layer through the holes. This results in a suspended GaAs thin film, the diameter (w) of which is determined by the etching time. Ultrasonic acoustic waves are piezoelectrically excited by applying an alternating voltage between the n-GaAs layer and one of the electrodes (Cr (5 nm)/Au (80 nm)) at both ends of the array. The resultant vibrations travel down the membrane-arrayed waveguide (WG) and are measured with an optical laser Doppler interferometer. The dispersion relation of the acoustic propagation can be adjusted by varying hole pitch a and WG width w. All measurements on the devices are performed at room temperature in a vacuum.

2.2 Spectral Transmission Properties The fundamental propagation dynamics of the device can be investigated by activating continuous waves from one end and measuring them at the other end. The spectral response of a PnC WG with (w, a) = (30 μm, 10 μm) is shown in Fig. 5 [17]. In the WG composed of ten membranes (N = 10), multiple resonance peaks appear in the spectrum (red solid line in the bottom panel). However, the free spectral range (FSR) of the peaks (Δf FSR ) gets smaller as N increases to 20 (orange) and 50 (green) and finally, a continuous transmission spectrum is formed at N = 100 (blue). The spectrum is generated in the entire range from 2.2 MHz to 8 MHz except at about 4.5 MHz, where the acoustic vibrations are significantly suppressed as highlighted in yellow. This is ascribed to the emergence of a phononic bandgap that results from

N = 100 (1.0 mm)

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Fig. 5 Transmission spectra of acoustic propagation in a PnC WG with membrane number constituting WG N = 10 (red), 20 (orange), 50 (green) and 100 (blue). The vibrations are activated at the right end and measured at the left end. Bandgaps caused by Bragg reflection at the solid/air-hole interfaces are highlighted in yellow. The FSR between neighboring peaks is denoted as Δf FSR . Illustrations of PnC WGs with various lengths are shown on the right. Reproduced from Ref.[17]. © Springer Nature (2014)

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the Bragg reflection of the acoustic waves at the interface between a membrane and air hole. There is a slight discrepancy in the spectral position of the bandgap between N = 50 and 100. This could be caused by fabrication imperfections in WG width and hole diameter. Thus, the transition from the spectrally discrete to a continuous one indicates the formation of a PnC in the membrane WG. This PnC property relies on the device geometry, namely WG width (w) and hole pitch (a). In other words, the band structure of the membrane WG can be modified by changing the periodic geometry. We prepared three PnC WGs, which are labeled Device 1, 2, and 3 and have (w, a) = (22, 10), (29, 12) and (34, 8) in micrometers, respectively [18]. The transmission spectrum of acoustic waves in Device 1 is shown in the right panel of Fig. 6a, and the corresponding dispersion relation calculated by finite-element method (FEM) is shown in the left panel. The vibrations are observed above 3.8 MHz, the lowest frequency (cut-off frequency) below which the vibrations are prohibited from propagating in the WG. Additionally, they are strongly suppressed between 5.6 and 6.2 MHz and then, emerge again beyond 6.2 MHz. The experimental results can be understood from the FEM simulation as shown in the left panel of Fig. 6a. The calculated dispersion relation reveals that a first phononic band is formed from 3.8 MHz (blue), and this is consistent with the experimental cut-off

Fig. 6 a–c Calculated dispersion relation (left panel) and the experimental spectral response of acoustic waves (right panel) in a PnC WG with (w, a) = (22, 10), (29, 12) and (34, 8) in μm, respectively. Bandgaps are highlighted in yellow. The insets in (b) and c show the enlarged spectral responses. d Displacement mode profiles in various phononic branches labeled in a–c. Reproduced from Ref.[18]. © Institute of Physics (2015)

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frequency. This band sustains an acoustic waveguiding mode where one vibrational antinode is located in the center of the WG width, thus resulting in the experimental observation of the transmission of acoustic waves. As the operation frequency quadratically increases, the wavenumber (k) (wavelength (λ)) of the acoustic waves increases (decreases). When k is equal to π/a satisfying the Bragg condition λ = 2a, the reflected waves at the hole/solid interfaces destructively interfere with the incident waves, giving rise to a phononic bandgap. This spectral width is determined by the frequency difference between mode A and B, where both wavelengths are the same but the antinode locations are different, namely between holes or on holes respectively, as shown in Fig. 6d. As a result, no available eigenmodes are around 6 MHz and the acoustic transmission is suppressed. Then, another phononic branch is generated above 6.2 MHz. This guides the acoustic waves, thus allowing the vibrations to be experimentally observed again. The dashed line shows a second order phononic branch carrying a guiding mode containing two antinodes [see mode C in Fig. 6d]. This mode cannot be excited from the piezoelectric transducer because this overlaps both of its anti-phase vibrations. In this way, the experimentally obtained spectral transmission properties can be understood. By changing to Device 2 with (w, a) = (29 μm, 12 μm), the cut-off frequency and the bandgap spectral position are reduced to 2.0 and 4.0 MHz as shown in Fig. 6b, respectively, because the WG width w and pitch a become larger than those of Device 1. An additional bandgap also occurs around 7 MHz, which results from the antimode crossing between the third and fifth bands. Finally, in Device 3 with (w, a) = (34 μm, 8 μm), a further increase in w lowers the cut-off frequency to 1.8 MHz, whereas the frequency at the first Brillouin zone edge is raised to 6.5 MHz as shown in the left panel of Fig. 6c. However, the bandgap effect is not observed in the experimental transmission spectrum [see the right panel of Fig. 6c]. This can be attributed to the existence of an alternative band at the frequency that guides acoustic waves in the WG and thus obscures the bandgap effect. Additionally, two excitable bands exist beyond 6 MHz, making equally distant resonance peaks unclear. This is in contrast to Device 2, where a single mode supports the transmission [see the insets of Fig. 6b, c]. Thus, the acoustic transmission characteristics can be modified by engineering the PnC geometry.

2.3 Temporal Transmission Property A time domain analysis of traveling acoustic wavepackets gives us a further understanding of the transmission properties in the devices. This allows us to measure not only the group velocity of acoustic waves (vg ) and group velocity dispersion (GVD) but also to resolve spectrally degenerated waves sustained by different phononic bands. Figures 7a–c show the temporal responses of ultrasound wavepackets measured at one end of the WG while injecting them with an input pulse width T0 = 8 μs from the other end at various frequencies in Device 1, 2, and 3, respectively [18]. The vibrations are firstly observed after a finite transit time and subsequently

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Fig. 7 a–c Temporal response of acoustic wavepackets with T0 = 8 μs in the frequency range of 2–9 MHz in Device 1, 2, and 3 respectively. The acoustic propagation is suppressed at the bandgaps (black arrows). The propagation sustained by mode D in the third branch in Device 3 are highlighted by a pink dotted box in (c). Reproduced from Ref.[18]. © Institute of Physics (2015)

multiple vibration fringes are found. This is due to multiple round trips caused by the reflection at both ends. This propagation is suppressed at the bandgap frequencies denoted by the black arrows. On the other hand, another vibration fringe is observed between 5 and 9 MHz in Device 3 as highlighted in the pink dotted box. This transmission originates from the degenerated third band carrying the waveguiding mode D in Fig. 6c. The time domain measurements enable us to clarify the fundamental transmission properties of the WG, which is difficult to do in frequency domain measurements. Note that one round-trip time varies with frequency, indicating that the group velocity vg has a frequency dependence. By using a simple calculation vg = 2L/Δt, where L and Δt are the WG length and the duration for one round trip, respectively, vg can be experimentally estimated as shown in Fig. 8a–c [18]. It monotonically increases with increasing frequency except near the bandgap frequencies, where it reveals a drastic reduction. The group velocity can be theoretically determined from its slope, namely vg = 2π ∂ f /∂k, and thus it decreases as the frequency approaches the bandgap where the dispersion curve becomes flattened. Indeed, the experimental results well reproduce the FEM calculated ones. The GVD is a key parameter determining the waveform of an acoustic pulse in such a dispersive system. The GVD coefficient (k2 ) is given by, k2 ≡

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(1)

and this is plotted as a function of frequency in Fig. 8d–f. Also shown, by solid lines, is the calculated k2 . The sign of the GVD coefficient changes with frequency. For instance, it is negative (positive) below (beyond) 5.5 MHz in first band of Device 1. This indicates that acoustic waves with higher frequencies travel faster (slower)

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than those with lower ones in negative (positive) k2 . In contrast, k2 = 0 is obtained at specific frequencies in this PnC WG, for instance 6.5 MHz in Device 2, where acoustic waves propagate at the same speed. Therefore, this frequency is suitable for sending waves long distances while maintaining the pulse shapes, whereas finite GVD coefficients k2 = 0 become useful for tuning the pulse shapes as described later.

3 Temporal and Dynamic Control of Acoustic Waves Based on the acoustic transmission characteristics tailored by the PnC, a variety of acoustic wave controls can be demonstrated using the dispersive and nonlinear effects, such as for temporal pulse compression and frequency conversion. This has led to the development of dynamic functionality combined with a local geometric modulation technique, such as memory operation [17, 95] and routing [97]. Here we introduce several key experimental demonstrations on dynamic wave controls using this electromechanical PnC.

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3.1 Waveform Engineering via Group Velocity Dispersion The GVD permits waves with different frequencies to travel at different speeds. This usually broadens the temporal width of an acoustic pulse, so it should be avoided for the purpose of wavepacket transmission. From a different perspective, however, this phenomenon is useful in that enables the pulse waveform to be modulated. In this experiment, we attempted to demonstrate pulse compression of an acoustic wavepacket using the GVD. Before moving to experimental results, we theoretically explain the underlying physics. Assuming a simple one-dimensional beam with an infinite length, the vibration displacement z(x, t) at the coordinate along the propagation direction x and at time t, can be predicted by the Euler-Bernoulli equation as [98] EI

∂2z ∂4z + ρ S + α1 z + α2 z 2 + α3 z 3 = 0, ∂x4 ∂t 2

(2)

where E, I , S, and ρ are Young’s modulus, the moment of inertia, the area of the cross section, and the density, respectively. αn (n = 1,2,3) is an acoustic coefficient. In particular, α2 and α3 express acoustic nonlinearities of the beam arising from a static displacement (z 0 ) and an elongation of the beam, respectively. The origin of these nonlinearities are described in detail elsewhere [99]. Here, we use the slowly varying amplitude approximation, where the envelope of a pulse A(x, t) centered around wavenumber k and angular frequency ω changes slowly in time and space, and it is given by z(x, t) = A(x, t) exp {i(kx − ωt)} . (3) By substituting Eq. (3) into Eq. (2), the nonlinear Schr¨ondinger equation (NLSE) for a traveling acoustic vibration is obtained as [98] η k2 ∂ 2 A ∂A =− A−i + iξ A|A|2 , ∂x 2 2 ∂T 2

(4)

where ξ is a nonlinear parameter and η is a propagation loss that is necessary to take into account to the propagation dynamics in a real device. T = t − x/vg is a new time coordinate moving with a pulse at vg . The NLSE is fundamental for describing wave propagation dynamics in a dispersive and nonlinear medium.

3.1.1

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The NLSE can be a powerful tool to predict the waveform of traveling waves in such a dispersive WG [19, 100]. By considering pulse amplitude given by A(x, T ) = A0 exp(−ηx/2)U (x, T ),

(5)

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and substituting Eq. (5) into Eq. (4) when the nonlinear coefficient ξ = 0, the NLSE can be simplified as ∂U k2 ∂ 2 U . (6) = −i ∂x 2 ∂T 2 This can be solved by introducing the Fourier transform of U (x, T ) given by 1 U (x, T ) = 2π



∞ −∞

U˜ (x, ω) exp(−iωT )dω,

(7)

where a Gaussian pulse input is defined as U˜ (x, ω) = U˜ (0, ω) exp(i k22 ω2 x). U˜ (0, ω) is the Fourier transform of the incident pulse at x = 0, and it is obtained as U˜ (0, ω) =



∞ −∞

U (0, T ) exp(iωT )dT.

(8)

Then, the evolution of a traveling wavepacket is predicted as U (x, T ) = 

T0 T02 − ik2 x

 exp −

 T2 , 2(T02 − ik2 x)

(9)

where T0 is the half-width at the 1/e-intensity point. This expresses the waveform of an acoustic Gaussian pulse propagating in a dispersive medium, indicating that the GVD affects the pulse width during the propagation. The resultant temporal width T1 can be estimated as   2 x T1 = T0 1 + , (10) xD T2

where x D ≡ |k02 | is the dispersion length. This suggests that a propagating acoustic pulse more widely broadens with larger k2 and smaller input pulse width in spite of the sign of the k2 . To confirm the temporal dynamics experimentally, an electromechanical PnC with (w, a) = (22 μm, 8 μm) is used as shown in Fig. 9a [19]. This WG geometry gives us single-mode transmission in a broad range between 3.5 and 10 MHz except for the bandgap at 8 MHz as shown in Fig. 9b. This allows vg and k2 to be estimated experimentally as shown in Fig. 9c, d, respectively. In order to investigate the spatial and temporal evolution of the acoustic wavepackets, acoustic Gaussian pulses with T0 = 2.5, 1.2, and 0.7 μs are excited in the 3–8 MHz range from the right edge of the WG and measured in the time domain at the left edge as shown in Figs. 10a–c, respectively. Simultaneously, the resultant pulse widths (T1 ) as function of frequency is extracted by fitting the Gaussian function to the experimental result, and they are shown in Fig. 10d–f. The acoustic pulses at T0 = 2.5 μs maintain the waveforms in

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reference

Lock-in amplifier Oscilloscope

b

c

d

Fig. 9 a Schematic of the device with N = 100 and measurement configuration. Optically detected acoustic signals are measured by a lock-in amplifier for phase-sensitive detection followed by an oscilloscope. The scale bar in the inset is 5 μm. b The dispersion relation calculated by the FEM (left) and the spectral transmission (right) in the device (w, a) = (22, 8) in μm, where a bandgap is highlighted in yellow. (c) and d The frequency dependence of the group velocity vg and GVD coefficient k2 , respectively. Reproduced from Ref.[19]. © Springer Nature (2018)

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Fig. 10 a–c Temporal transmission of acoustic Gaussian wavepackets in the frequency range from 3 to 8 MHz with T0 = 2.5, 1.2 and 0.7 μs, respectively. d–f Temporal width of output wavepackets (T1 ) at various frequencies, measured at propagation distances x = 0 (pink), 1 (red), 3 (green), and 5 mm (blue). The theoretically predicted output widths at these distances are shown as the solid lines. Reproduced from Ref.[19]. © Springer Nature (2018)

the 6–7 MHz range even at propagation length x = 5 mm as shown in Fig. 10d. These frequencies are around the center of the phononic band, where the GVD coefficients k2 are negligibly small, namely vg is invariant in frequency, as shown in Fig. 9d. On the other hand, the width T1 drastically increases and expands with propagation length around the band edges, especially at 4 to 5 MHz. The absolute k2 is large at these frequencies and vg differs between different wave frequencies. Therefore, differentfrequency wave components in a wavepacket travel at different speeds, resulting in the envelope being stretched. This pulse broadening effect can also be confirmed

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more significantly at T0 = 1.2 and 0.7 μs as shown in Fig. 10e, f, respectively. This is because a shorter pulse contains wider spectral wave components, and thus the difference in vg of acoustic waves in the pulse becomes larger. This behavior is reasonable and consistent with Eq. (10) derived from the NLSE. The experimental results are well predicted by the theoretical ones as shown by solid lines in Fig. 10d–f. Thus, the acoustic propagation dynamics including the dispersion effect follows the NLSE.

3.1.2

Pulse Compression

The existence of GVD allows acoustic waves with different frequencies to be carried with different group velocities, thereby resulting in broadening of the pulse. In other words, utilizing this GVD also makes it possible to compress the wavepacket [19, 100]. To that end, a chirped Gaussian pulse is used as an input, and it is given by   1 + iC T 2 U (0, T ) = U0 exp − , 2 T02

(11)

where C is a chirp parameter. When C > 0 (C < 0), the frequency increases (decreases) with time. By using Eqs. (6)–(8), the output waveform at position x and T reads   T 2 (1 + iC)(T02 + k2 C x + ik2 x) U0 T0 , exp − U (x, T ) =  (T02 + k2 C x)2 + (k2 x)2 T02 + ik2 x(1 + iC) (12) and then the temporal pulse width T2 results in  T2 = T0

Ck2 x 1+ T02

2

 +

x xD

2 .

(13)

This indicates that acoustic pulses are being compressed during propagation when Ck2 < 0, whereas they are being stretched when Ck2 > 0. This waveform modulation becomes significant in large k2 and C. Also note that Eq. (13) becomes equal to Eq. (10) assuming an unchirped input C = 0 is assumed. Based on the above theoretical approach, the propagation dynamics of such a chirped acoustic pulse is investigated in the PnC WG. Figure 11a–c show the temporal response of traveling pulses measured at the left edge of the WG by injecting a chirped pulse from the right edge with C = -3, 0, and 3 at 5.8 MHz, respectively, where k2 = −2.8 × 10−10 s2 /m [19]. Monotonic pulse broadening is observed with increasing x at C = 0 [Fig. 11b] as described in the previous section. This effect is further enhanced by using a chirp pulse with C = −3 as an input because Ck2 is positive at this frequency [Fig. 11a]. On the other hand, the output pulse is compressed

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and its peak amplitude is amplified during propagation when a chirp pulse with C = 3 is injected, that is Ck2 < 0 as shown in Fig. 11c. The resultant pulse width can be extracted by fitting a Gaussian function to it, and the propagation distance dependence at various Cs between -3 and 3 is shown in Fig. 12 [19]. As confirmed in Fig. 11, the output pulse width T2 monotonically increases with the distance with negative C, whereas it drastically decreases with positive C. Although the pulse compression at C = 3 provides the minimum value of T2 = 1.0 μm at x = 6 mm, it returns to broadening as seen by the red squares. This can be understood as follows. A positive chirp pulse (C > 0) has low-frequency wave components in the leading part of the wavepacket and high-frequency components in the tailing part. Negative GVD (k2 < 0) allows the high-frequency waves to be guided faster than low-frequency ones, whereby the pulse is compressed in the beginning up to x = 6–7 mm. Once all the frequency wave components are at the same position, the high-frequency waves travels in advance of the low-frequency ones, resulting in pulse broadening at x > 7 mm. Moreover, this dispersive modulation is enhanced as C increases, so the pulse width is significantly reduced from T0 = 1.9 μs to T2 = 0.6 μs at C = 9.7 as shown in Figs. 12 and 13 [19]. The most compressed pulse contains nearly twocycle oscillations in the wavepacket. The measurable minimum width T2 is now limited by the finite bandwidth of a lock-in amplifier. Therefore, the optimization of the measurement set up can realize the observation of stronger compression. These experimental results can be well reproduced by the theoretical approach as shown by solid lines of Fig. 12, indicating the usability of the NLSE to predict the propagation dynamics of acoustic waves in dispersive phononic WGs.

3.1.3

Phase Modulation Due to Nonlinear Acoustic Effect

In addition, the dynamics of the acoustic propagation are affected by the acoustic nonlinearity of a medium. This allows the dynamic modulation in a traveling wavepacket itself and a copropagating wavepacket at a different frequency, which are referred to as self-phase modulation (SPM) and cross-phase modulation (XPM), respectively. In the NLSE [Eq. (4)], this nonlinear effect originates from the third term on the right hand side and can be activated when a large displacement amplitude

a

b = -3

c =0

=3

Fig. 11 a–c Temporal evolution of acoustic chirped pulses at one WG end which are excited from the other end at 5.8 MHz with T0 = 2.5 μs and C = -3, 0 and 3 respectively. The GVD coefficient is k2 = -2.8 × 10−10 s2 m−1 at 5.8 MHz. Reproduced from Ref.[19]. © Springer Nature (2018)

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Fig. 12 Propagation distance (x) dependence of output pulse width (T2 ) excited at 5.8 MHz with T0 = 2.5 μs and C = -3 ∼ 3. The solid lines are the theoretical results by Eq. (13). Reproduced from Ref.[19]. © Springer Nature (2018)

Fig. 13 Strong compression of a traveling acoustic wavepacket with C = 9.7 at 5.8 MHz measured at a WG edge. Reproduced from Ref.[19]. © Springer Nature (2018)

= 9.7

is excited in the system. It is essential for generating solitons and four-wave mixing (FWM) and thus for the ability to dynamically control acoustic waves. To understand the underlying physics, the NLSE is used to derive the basic equation of nonlinear acoustic propagation [19, 100]. Assuming the GVD coefficient k2 = 0 and an acoustic input pulse given by ηx

U (x, T ), A(x, T ) = U0 exp − 2

(14)

they are substituted into Eq. (4), which then reads  ηx

exp − ηx2 ∂U 2 2 = iξU0 exp − |U | U = i |U |2 U, xNL ∂x 2

(15)

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where xNL ≡ 1/|ξ |U02 is defined as the nonlinear length. To solve it, U (x, T ) is decomposed into the amplitude R(x, T ) and the phase term φNL (x, T ), which is written as U (x, T ) = R(x, T ) exp (iφNL (x, T )). This is substituted to Eq. (15), and by comparing the real and imaginary parts in the equation, we finally obtain the following equations  exp − ηx2 ∂φ N L ∂R = 0, = |R|2 . ∂x ∂x xNL

(16)

The resultant nonlinear phase φ N L at x = L is given by φ N L (L , T ) = |U (0, T )|2

xeff , xNL

(17)

where xeff ≡ 1−expη(−ηL) is the effective propagation length. This indicates that the SPM effect increases as the wave propagates and the pulse amplitude increases. Additionally, the temporal variation in the phase induces an instantaneous frequency change: xeff ∂ 1 ∂φNL =− |U (0, T )|2 . (18) Δf = − 2 ∂T xNL ∂ T For instance, a negative frequency shift is generated in the leading (tailoring) edge of the pulse when ξ > 0 (ξ < 0). This SPM process can be used to make a chirped acoustic pulse from an unchirped pulse without distortion of the envelope. By optimizing the sign and magnitude of the nonlinear parameter ξ and the GVD coefficient k2 , advanced nonlinear phenomena such as solitons can be realized. Moreover, this acoustic nonlinearity also enables the phase of copropagating waves to be modulated simultaneously. Two continuous waves at frequency of f s and f p , called the signal and pump, are injected into such a nonlinear medium, and they are expressed by Us, p =

  1 u s, p exp i(ks, p x − 2π f s, p T ) + c.c.. 2 s, p

(19)

These waves interact through the third-order nonlinear term [see Eq. (15)] during propagation. As a result, this process generates several waves with different frequencies. Especially in case of our experimental configuration as shown later where f p > f s , the terms containing the following frequencies remain fs , f p , 2 f p ± fs , 2 fs ± f p , 3 f p , 3 fs , .

(20)

The first and second terms in Eq. (20) are XPM and SPM processes, respectively, when the pump is excited strongly enough to induce the nonlinearity. They dynamically modulate the phase of propagating waves. The third and fourth terms are FWM processes that generate a new wave, called an idler. Additionally, the phase-matching

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Fig. 14 Schematic of a long 33-mm PnC WG (purple). The WG consists of approximately 4100 membranes arrayed with a = 8 μm and w = 27.7 μm. It meanders, but the curves are designed to mitigate undesired acoustic reflection. An IDT is formed at the left edge, and it comprises 100 finger-pair electrodes, each of which has an electrode width of 5 μm and a pitch 20 μm as shown in the insets. The scale bars are 5 μm (left) and 20 μm (right). On the other end, a single electrode is formed. Reproduced from Ref.[20]. © American Physics Society (2020)

condition should be satisfied to realize efficient FWM, for instance, ki = 2k p ± ks . The last two terms are third-harmonic generation (THG), which also requires the phase matching. In general, FWM is easier to generate than THG because the frequencies of all the waves involved are close to each other. Thus, energy and phase conservation can be achieved in these processes, and they hold promise for developing functional acoustic devices such as frequency converters and amplifiers. To experimentally demonstrate these nonlinear wave phenomena, excitation of acoustic waves with displacement large enough to induce the nonlinear effect is necessary. However, the piezoelectric transduction efficiency of GaAs is poor compared with the commonly used piezoelectric materials such as LiNbO3 and ZnO. To overcome this limitation, an inter-digit transducer (IDT) is incorporated into a PnC WG as shown in Fig. 14 [20]. The IDT consists of 100 electrode-finger pairs (NIDT = 100) with a width and pitch ( p) of 5 μm and 20 μm, respectively. Acoustic waves excited from each finger electrode in the IDT interfere with each other constructively, thereby amplifying the vibrations at the frequencies determined by the finger pitch and acoustic wave velocity (v0 ), f 0 = v0 / p. The actuated amplitude increases with increasing NIDT , whereas the operation bandwidth is reduced as Δf = f 0 /NIDT . In addition, the WG of the device is elongated to 33 mm to observe the spatial evolution of the wave dynamics clearly.

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Fig. 15 a Spectral response of a 33-mm-long PnC WG. The vibrations excited in an IDT (blue) and a single electrode (red) are measured at propagation distances x = 5 and 1 mm respectively. b Enlarged transmission spectra. c Dispersion relation of the device calculated by the FEM. d Experimental and calculated GVD coefficients k2 as function of frequency, denoted by filled circles and a solid line, respectively. Reproduced from Ref.[20]. © American Physics Society (2020)

Figure 15a shows the spectral response of the device by exciting acoustic waves from the IDT and a single-finger electrode (the same transducer used in the previous experiments) and measuring them at x = 5 and 1 mm as denoted by blue and red solid lines, respectively [20]. Indeed, the vibration enhancement due to the IDT is observed in the frequency range of 5–6 MHz, where the amplitude is two orders of magnitudes larger than that in the single electrode as shown in detail in Fig. 15b, and the single-mode transmission is available that allows the waveform of acoustic waves to be observed clearly as shown in Fig. 15c. The bandwidth of the vibration amplification (Δf ) is spectrally limited due to NIDT = 100. Therefore, the input pulse width T0 = 40 μs is chosen, so that wave components of the pulse are mostly within Δf . The operating frequency is set to f 0 = 5.38 MHz, where k2 = 2 × 10−10 s2 /m as shown in Fig. 15d. From this experimental condition, the dispersion length is estimated to be xD = 8 m. This value is much longer than the WG length, so the dispersion effect on wave propagation can be ignored. Thus, the experimental results reveal the usability of the IDT and device configuration for nonlinear measurements. The spatial evolution of the amplitude and phase of an acoustic Gaussian wavepacket with a temporal width of T0 = 40 μs excited from the IDT, measured at propagation distances from x = 0 to 19.5 mm, is shown in the left and right panels of Fig. 16a–f, respectively [20]. At x = 0 mm, the amplitude of the excited wave proportionally increases with increasing excitation voltage from 0 to 1.2 Vrms , whereas the

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Fig. 16 a–c Temporal response of acoustic pulse amplitude excited at various voltages and measured at x = 0 mm, 8 mm and 19.5 mm respectively. d–f The temporal response of acoustic pulse phase (φNL ) excited at various voltages and measured at x = 0, 8, and 19.5 mm respectively. Reproduced from Ref.[20]. © American Physics Society (2020)

phase (φNL ) is invariant in time [see Fig. 16a, d]. However, when the wave propagates to x = 8 mm, the phase is temporally changed and the maximum phase modulation becomes significant as the excitation voltage increases, in spite of the reduction of wave amplitude due to the propagation loss [see Fig. 16b, e]. Then, it reaches 60 degrees at x = 19.5 mm and 1.2 Vrms as shown in Fig. 16c, f. This experimental phase evolution with respect to the propagation distance is plotted at the voltages of 0.6, 0.8, 1.0, and 1.2 Vrms as denoted by the solid circles in Fig. 17 [20]. The phase evolution due to SPM is calculated using Eq. (17) as denoted by the solid line, where the experimentally-obtained η = 0.29 dB/mm is used. They show good agreement with the experimental results, which confirms that the observed phase modulation is caused by the SPM process originating from the device nonlinearity. The theoretical fitting also allows the nonlinear parameter to be estimated as ξ = -16 nm−2 m−1 for the first time. Moreover, the instantaneous frequency shift (Δ f ) is estimated as shown

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Fig. 17 Spatial evolution of the maximum phase modulation due to the SPM effect at 0.6 (blue), 0.8 (green), 1.0 (orange), and 1.2 Vrms (red). The theoretical fitting of Eq. (17) is shown by solid lines. Reproduced from Ref.[20]. © American Physics Society (2020)

Fig. 18 Temporal response of the instantaneous frequency (Δf ) in an acoustic wavepacket propagating at x = 0 (blue), 10 (green), and 19.5 mm (red). Also shown are the calculated results based on Eq. (18) as the solid lines. Reproduced from Ref.[20]. © American Physics Society (2020)

in Fig. 18. As expected, negative chirped pulses can be generated, and this observation is consistent with the theoretical prediction in Eq. (18). Thus, the experimental results provide detailed information on the nonlinear wave dynamics in the PnC WG, which is useful in designing nonlinear phononics experiments and predicting their results. The nonlinear effect induces spectral modulation not only in a traveling acoustic wave through SPM but also in copropagating waves through the XPM process. Then, a new acoustic wave (idler) also emerges through the FWM process. To investigate the nonlinear interaction, a continuous signal wave at f s and acoustic pump pulse at f p are injected from the IDT electrode simultaneously, and the resultant transmission spectra are measured at various distances from x = 2 to 27 mm as shown in Fig. 19a–f [20]. Here, the signal wave is excited with 0.4 Vrms at f s = 5.264 MHz, and the pump wave is excited with 2.0 Vrms and T0 = 40 μs at f p = 5.383 MHz, whose excitation amplitude is strong enough to induce the nonlinear effect. At x = 2 mm, the signal and pump waves are observed as shown in Fig. 19a. The pump has a Gaussian waveform that is spectrally broadened around f p . On the other hand, the signal is a continuous wave, so a single tone is observed at f s , but its bottom is also broadened. This could be a consequence of XPM through the pump. This nonlinear effect also induces wave mixing between the signal and pump and thus result in the generation of a new wave (idler) at f i = 2 f p − f s = 5.502 MHz via the FWM process. The idler amplitude increases to more than 10 pm at x = 7 mm [see Fig, 19b]. However, it decreases

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Fig. 19 a–f Spectral response of acoustic wave propagation due to FWM at distances x = 0 (a), 7 (b), 12 (c), 17 (d), 22 (e), and 27 mm (f) when the signal and pump from an IDT are simultaneously excited. The signal has a continuous wave at 5.264 MHz and 0.4 Vrms . The pump is a Gaussian waveform with T0 = 40 μs and at 5.383 MHz and 2.0 Vrms . In the experiment, the pump pulse was injected with a repetition rate 500 Hz and measured with a resolution bandwidth 10 Hz (red). The spectrum predicted by the NLSE is shown by the solid blue line, where reflections at WG edges are taken into account. Reproduced from Ref.[20]. © American Physics Society (2020)

with further increasing distance x = 12, 17, 22, and 27 mm [see Fig. 19c–f]. The spectral transmission response at various distances x can be numerically reproduced by using the NLSE, and the calculation results are plotted by blue solid lines. In the calculation, the GVD coefficient k2 = -2×10−10 s2 /m and the nonlinear parameter ξ = -16 nm−2 m−1 are used, and the propagation losses of the signal and pump are extracted by fitting the experimental results as shown in Fig. 20 [20]. It should be noted that the calculation can also predict the interference fringes appearing in the experimental result, which results from the reflection of the waves at the WG edge. The spatial evolution of the peak amplitude of the signal, pump, and idler wave is plotted in Fig. 20, and they are denoted by blue, red, and green, respectively. Also shown is the amplitude of the spectrally broadened spectrum due to XPM, as denoted by yellow. In particular, the idler monotonically increases with increasing propagation distance up to x = 7–10 mm, but it starts to reduce after x = 10 mm. At the beginning of the propagation, the pump pulse is large enough to induce the FWM process, so the idler generation evolves with distance. However, the propagation loss in the WG reduces the pump amplitude, which weakens the idler generation efficiency. As a result, the propagation loss of the idler overwhelms the FWM process, thus suppressing the idler amplitude. Simultaneously, the spatial evolution of the amplitude of the XPM-induced spectrum at f s can also be reproduced by the theoretical prediction of the NLSE. It should also be noted that the pump amplitude at

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Fig. 20 Propagation dynamics of FWM, where the amplitudes of a signal (blue), pump (red), idler (green) and XPM-induced spectrum (yellow) at the given frequencies are plotted as a function of distances x between 0 and 31 mm. The numerical simulation results are shown by solid lines. The gray line indicates the averaged noise level of the measurement setup. Reproduced from Ref.[20]. © American Physics Society (2020)

f p is drastically decreased near x = 30 mm because of the destructive interference between the incident wave and the wave reflected at the WG edge, and this is also shown in Fig. 19f. Thus, the results indicate the possibility of dynamic control of acoustic waves, for instance frequency conversion and wave amplification, which holds promise for the generation of novel nonlinear phononic phenomena such as soliton [101] and rogue waves [102].

4 Conclusion A novel electromechanical system comprising N/MEMS and PnC has been proposed and demonstrated. This system makes it possible to engineer and exploit both dispersion and nonlinear acoustic effects for waveform engineering and nonlinear acoustic wave propagation. By using the GVD, a compressed acoustic pulse is generated from a chirped acoustic wave in the PnC WG. Additionally, spectral modulation in a traveling wavepacket via SPM and XPM is observed, which allows nonlinear parameters to be experimentally estimated in a phononic WG platform for the first time. Then, the frequency conversion and generation of a new idler wave is also realized, and this nonlinear control capability has led to the development of dynamic and intricate wave manipulation techniques on a chip [95–97, 103, 103]. As one future prospect, the use of atomic-layer materials such as graphene in a PnC medium holds promise for enhancing the dynamic modulation acoustic properties and thus improving acoustic wave control capabilities [104, 105]. These experimental demonstrations offer the opportunity to investigate novel nonlinear phononics, such as acoustic solitons and rogue waves, and point to potential applications of acoustic waves in hybrid quantum technology.

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Acknowledgements We thank Dr. Imran Mahboob for fruitful discussions and helpful comments on this study. We are also grateful to Dr. Koji Onomitsu for support with device fabrication. The authors acknowledge the financial support from Grant-in-Aid for Scientific Research on Innovative Areas No. JP15H05869.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

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Electron and Phonon Transport Simulation for Quantum Hybrid System Nobuya Mori and Gennady Mil’nikov

Abstract To facilitate design and analysis of quantum hybrid devices it is highly desirable to develop a quantum device simulator for predicting the quantum transport properties of coupled systems. This chapter focuses on the non-equilibrium Green’s function (NEGF) simulation for electron, phonon, and coupled electronphonon transport. The NEGF method treats rigorously the quantum nature of the carrier transport which play an important role in the hybrid systems. Firstly, the NEGF simulation of phonon transport in the ballistic case without inelastic scatterings is discussed. The R-matrix method is then introduced, which reduces the computational burden in evaluating Green’s functions. Secondly, the NEGF simulation of electron transport in the ballistic case is discussed. After briefly describing the basic NEGF equations for the electron transport simulation, the equivalent model is introduced, which substantially reduces the system Hamiltonian size and speeds up the NEGF simulation. Finally, coupled electron-phonon transport simulation is briefly discussed. In each section, the theory is introduced, followed by simple numerical examples including phonon transport through one-dimensional harmonic chain and isotopically disordered graphene nanoribbons, electron transport in semiconductor nanowire transistors, and coupled electron-phonon transport in one-dimensional resonant tunneling structures. Keywords Quantum transport · NEGF · Simulation · Electron · Phonon

1 Introduction In this chapter we consider electron and phonon quantum transport simulation in hybrid systems where the inelastic electron-phonon interaction may become essential causing cross-correlation between the two transport mechanisms. In order to facilitate

N. Mori (B) · G. Mil’nikov Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_5

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the hybrid device design and analysis it is highly desirable to develop a quantum device simulator for predicting the transport properties of such systems. In practical applications, it is important to choose the most suitable methodology for a specific system and transport properties to be analyzed. Commonly used methods for electron transport simulations in realistic devices include drift-diffusion model [1], Boltzmann transport equation (BTE) [2, 3], Monte Carlo (MC) simulation [4–7], Wigner function [8–10], and non-equilibrium Green’s function (NEGF) method [11–15]. Phonon transport simulations have been also performed by various methodologies such as the Fourier law, BTE [16, 17], MC simulation [10, 18, 19], molecular dynamics (MD) [20–22], and the NEGF [23–29]. Naturally each method has its own pros and cons, and there is no universal best methodology. One has to choose the most suitable computational scheme case by case. In this chapter, we focus on the NEGF simulation for electron, phonon, and coupled electron-phonon transport. The NEGF method treats rigorously the quantum nature of the carrier transport which play an important role in the hybrid systems, and it is suitable to a source-channel-drain type of the device structure. The NEGF formalism also provides a suitable framework for coupled electron-phonon transport [30–37], which is an essential feature of the quantum hybrid systems. The NEGF method, however, requires substantial computational resources and the efficiency of numerical algorithm is of utmost importance for simulating realistic complex device structures. In this chapter, we introduce the basic equations for the NEGF method and summarize two algorithms which substantially reduce the computational burden: R-matrix method [38–40] and equivalent model [41, 42]. The chapter is organized as follows. In Sect. 2, the NEGF simulation of phonon transport in the ballistic case without inelastic scatterings is discussed. Adopting a simple one-dimensional harmonic chain model, basic equations for the NEGF method for phonon transport are introduced. The R-matrix method is then introduced, which reduces the computational burden in evaluating Green’s functions. In Sect. 3, the NEGF simulation of electron transport in the ballistic case is discussed. After briefly describing the basic NEGF equations for the electron transport simulations, the equivalent model is introduced, which substantially reduces the system Hamiltonian size and speeds up the NEGF simulation. In Sect. 4 coupled electron-phonon transport simulation is briefly discussed. In this chapter we merely outline the main equations of the NEGF method. For the details of the formalism and full derivation of the NEGF equations, the readers should refer to Refs. [11–15, 23–35].

2 Phonon Transport Simulation 2.1 NEGF Method In this section we introduce the basic equations for the non-equilibrium Green’s function (NEGF) method and present the R-matrix propagation algorithm by con-

Electron and Phonon Transport Simulation … kL

ML

kL

–2

ML

kL

–1

ML

kL

0

M1

75 M2

k1

MN

k2

2

1

L

N

kR

MR

N+1

C

kR

MR

kR

N+2

R

Fig. 1 One-dimensional harmonic chain. The system is divided into three regions: semi-infinite left reservoir (L), channel region consisting of N atoms (C), and semi-infinite right reservoir (R). The atoms are enumerated by an integer n: n ≤ 0 for the left reservoir, 1 ≤ n ≤ N for the channel, and n ≥ N + 1 for the right reservoir. The mass of the n-th atom is Mn and the spring constant between the n-th atom and the (n + 1)-th atom is kn . The reservoirs are assumed to be uniform; Mn = ML for n ≤ 0, Mn = MR for n ≥ N + 1, kn = kL for n ≤ 0, and kn = kR for n ≥ N

sidering a simple example of ballistic phonon transport thorough a one-dimensional harmonic chain. The system is shown schematically in Fig. 1 and it can be divided into three regions: the finite channel region (C) of N atoms and two semi-infinite reservoirs (L and R) on the left and right side. The atom numeration is chosen as n ≤ 0 for the left reservoir, 1 ≤ n ≤ N for the channel, and n ≥ N + 1 for the right reservoir. The mass of the n-th atom is designated as Mn and the spring constant between the n-th atom and the (n + 1)-th atom is designated as kn . The reservoirs are assumed to be uniform i.e. Mn = ML for n ≤ 0, Mn = MR for n ≥ N + 1, kn = kL for n ≤ 0, and kn = kR for n ≥ N . Without loss of generality we can assume the spring constants within the reservoirs to be the same as the coupling between the reservoirs and the channel. This condition can always be satisfied by the appropriate choice of the interface position. The equation of motion for the whole system is given as follows. The potential energy V (x) of the system is given by V (x) =

1 n

2

kn (xn − xn+1 )2 ,

(1)

where xn is the n-th atomic displacement. The equations of motion then read Mn

 d 2 xn = − Cnn  xn  , dt 2  n

(2)

where the dynamical matrix Cnn  is given by

Cnn 

⎧ ⎪ ⎪ kn−1 + kn ∂ 2 V (x) ⎨ −kn−1 = = ⎪ −kn ∂ xn ∂ xn ⎪ ⎩ 0

(n  = n) (n  = n − 1) . (n  = n + 1) (otherwise)

(3)

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By introducing the mass normalized displacement u i = motion becomes u¨ = −u with

nn  = √

Cnn  Mn Mn 

⎧ kn−1 + kn ⎪ ⎪ ⎪ ⎪ Mn ⎪ ⎪ ⎪ ⎨ − √ kn−1 = Mn Mn  ⎪ kn ⎪ ⎪ ⎪ −√ ⎪ ⎪ Mn Mn  ⎪ ⎩ 0

√

Mi xi , the equation of (4)

(n  = n) (n  = n − 1)

.

(5)



(n = n + 1) (otherwise)

The retarded Green’s function plays a central role in the NEGF method. Specifically, in the ballistic case without inelastic scattering one can compute physical quantities of interest, such as phonon density-of-states and thermal conductance, only from the retarded Green’s function. In numerical simulations, it is convenient to work with a matrix form of the Green’s function. The Green’s function for the total system, however, is represented by an infinite matrix, which cannot be directly used in numerical applications. As explained below, one can obtain the retarded Green’s function of the central channel region in the finite matrix form by making use of the spatial periodicity in the reservoirs which can be used in numerical simulations of phonon transport properties of the system. We now derive the compact form of the retarded Green’s function of the central channel region. In frequency domain, the equation of motion for the harmonic modes u = Ue−iωt becomes ⎤⎡ . ⎤ ⎡ . ⎤ .. .. .. . . . . ⎢ ⎥⎢ ⎢ . ⎥ ⎥ . . .. ⎢ ⎥ ⎢ U−1 ⎥ ⎢ U−1 ⎥ dL sL .. ⎢ ⎥⎢ ⎢ ⎥ ⎥ .. .. ⎢ ⎥ ⎢ U0 ⎥ ⎢ U0 ⎥ † sL dL . s0 . ⎢ ⎥⎢ . . ⎥ ⎢ .. ⎥ ................ ⎢ ⎥⎢ ⎢ ⎥ ⎥ . .. ⎢ ⎥⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ U1 ⎥ ⎢ U1 ⎥ s0† .. d1 s1 . ⎢ ⎥⎢ U ⎥ ⎢ U ⎥ .. † .. ⎢ ⎥⎢ 2 ⎥ ⎢ 2 ⎥ . s1 d2 . ⎢ ⎥ ⎢ . ⎥ = ω2 ⎢ . ⎥ , .. .. .. ⎢ ⎢ ⎥ ⎢ .. ⎥ ⎥ . . . . ⎢ ⎥⎢ . ⎥ ⎢ ⎥ .. .. ⎢ ⎢ ⎥ ⎢ UN ⎥ ⎥ . dN . sN ⎢ ⎥ ⎢ UN ⎥ ⎢ ⎥ ................ ⎢ ⎥⎢ . . ⎥ ⎢ .. ⎥ .. . ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢U N +1 ⎥ ⎢U N +1 ⎥ . s N† .. dR sR .. .. † ⎢ ⎥ ⎢U ⎢U ⎥ ⎥ ⎣ ⎦ ⎣ N +2 ⎦ ⎣ N +2 ⎦ . . sR dR .. .. .. .. .. . . . . .

(6)

2kR kn−1 + kn 2kL , dR = , dn = , (n = 1, 2, . . . , N ), ML MR Mn

(7)

⎡

..

where dL =

Electron and Phonon Transport Simulation …

sL = −

77

kL kR kn , sR = − , sn = − √ , (n = 0, 1, . . . , N ). ML MR Mn Mn+1

(8)

Note that in the simple one-dimension model sn is just a real number but we keep the symbol of conjugate transport †, so that our equations are valid in more general case. The dotted lines in Eq. (6) represent the interface positions of the system, and Eq. (6) can be written in the corresponding block matrix form ⎤⎡ ⎤ ⎡ ⎤ . . UL UL L .. LC .. 0 ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ † .... . . . . .... ⎢  . C . CR ⎥ ⎢ UC ⎥ = ω2 ⎢ UC ⎥ . ⎦⎣ . ⎦ ⎣ . ⎦ ⎣ LC . . . . . . . . . . UR UR 0 .. †CR .. R

    ⎡

(9)

For example, C is a matrix of size N × N defined as ⎡

⎤

d1 s1 ⎢ s1† d2 ⎢ C = ⎢ .. ⎣ .

⎥ ⎥ ⎥. ⎦

(10)

dN  represents the dynamical matrix for the total system In Eq. (9), the infinite matrix  and the corresponding retarded Green’s function Dr (ω) is defined as ⎤−1 ω2 − L −LC 0 = ⎣ −†LC ω2 − C −CR ⎦ , 0 −†CR ω2 − R ⎡

]−1 Dr (ω) = [ω2 − 

(11)

where the frequency should be regarded as ω + i0. Shifting the frequency by an infinitesimal amount +i0 corresponds to the outgoing boundary conditions in the retarded Green’s function which is essential for the correct choice of the periodic solutions in Eqs. (16), (17), and (19) below. By representing Dr (ω) as a block matrix form of ⎡

⎢ r ⎣ DCL Dr

r r DRCL DRC

⎤

⎡ 2 ⎤−1 ω − L −LC 0 † r ⎥ DCR ⎦ = ⎣ −LC ω2 − C −CR ⎦ 0 −†CR ω2 − R Dr

r r DLr DLC DLCR

(12)

R

and by performing some matrix manipulations, we obtain the Green’s function of the channel region as Dr (ω) = [ω2 − C − rL (ω) − rR (ω)]−1 .

(13)

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Here all the quantities are square matrices of size N × N and ω2 should be regarded as the diagonal N × N matrix with elements ω2 . The contact self-energies, rL (ω) and rR (ω), are given by rL (ω) = †LC dLr (ω) LC , rR (ω) = CR dRr (ω) †CR ,

(14)

where the retarded Green’s functions dLr (ω) and dRr (ω) of the left and right reservoirs are defined as dLr (ω) = [ω2 − L ]−1 , dRr (ω) = [ω2 − R ]−1 .

(15)

Although both dLr (ω) and dRr (ω) are infinite matrices, we only need a finite portion of dLr (ω) and dRr (ω) for evaluating rL (ω) and rR (ω) because both LC and CR have only one non-zero element in the lower-left corner. More specifically, it suffices to compute only the lower-right element of dLr (ω) and the upper-left element of dRr (ω), which can be done by making use of the spatial periodicity of the reservoirs. The result reads    1 (16) [dLr (ω)]lower-right = 2 (ω2 − dL ) ± (ω2 − dL )2 − 4sL2 , 2sL    1 [dRr (ω)]upper-left = 2 (ω2 − dR ) ± (ω2 − dR )2 − 4sR2 , (17) 2sR which gives the contact self-energies, rL (ω) and rR (ω) ⎡ ⎢ rL (ω) = ⎢ ⎣

πLr (ω)

⎤

⎡

⎤

⎥ ⎢ ⎥ , r (ω) = ⎢ R ⎦ ⎣

⎥ ⎥ ⎦

(18)

πRr (ω) where (for positive ω) ⎧    ⎪ 2 ⎪ for ηα2 (ω) ≤ 1 ⎨ sα ηα (ω) + i 1 − ηα (ω) r   , πα (ω) =  ⎪ ⎪ ⎩ sα ηα (ω) − sgn(ηα (ω)) ηα2 (ω) − 1 for ηα2 (ω) > 1 and ηα (ω) =

ω2 − dα , (α = L, R). 2sα

(19)

(20)

Here sgn(x) represents a sign function defined as sgn(x) = 1 for x > 0, −1 for x < 0, and 0 for x = 0. From Eqs. (10), (13), and (18), we can calculate the channel Green’s function Dr (ω) by a N × N matrix inversion.

Electron and Phonon Transport Simulation …

79

Once the retarded Green’s function Dr (ω) is known, one can compute various physical quantities of interest. For example, the phonon density of states, ρ(ω), is given by 2ω Tr Im Dr (ω). (21) ρ(ω) = − π Here Tr M represents the trace of a matrix M and Im z the imaginary part of z. The phonon transmission function, T (ω), through the channel region is given in terms of Dr (ω) and rα (ω) (α = L, R) as follows: T (ω) = Tr[ L (ω)Dr (ω) R (ω)Dr † (ω)],

(22)

where α (ω) is defined as

α (ω) = i[rα (ω) − rα† (ω)], (α = L, R).

(23)

When the left and right reservoirs are maintained at a constant temperature TL and TR , respectively, the thermal current, Q, form the left reservoir to the right reservoir is given by ∞ ω T (ω)[ f BE (TL ; ω) − f BE (TR ; ω)]dω. (24) Q= 2π 0

Here f BE (T ; ω) is the Bose-Einstein distribution function given by f BE (T ; ω) = [exp(ω/kB T ) − 1]−1 . By setting TL = T + T and TR = T and taking the limit of T → 0, we obtain the thermal conductance, K , at temperature T as follows: Q = T →0 T

∞

K = lim

=

π kB 6

0

∂ f BE (T ; ω) ω T (ω) dω 2π ∂T

∞ T (ω)w(ω/kB T )dω,

(25)

0

where w(x) is defined as w(x) =

x 2 ex 3 . π 2 (ex − 1)2

(26)

By introducing a normalized variable x = ω/kB T , Eq. (25) can be written as 

∞ K = KQ

T 0

 kB T x w(x)d x, 

(27)

80

N. Mori and G. Mil’nikov

where K Q = π kB2 T /6 = π kB2 T /3h. In the low temperature limit (T → 0), Eq. (27) reduces to ∞ (28) K zero = K Q T (0)w(x)d x = K Q T (0). 0

This implies that the ballistic thermal conductance at the low temperature limit is determined by the low frequency transmission function. On the other hand, by introducing a normalized variable y = ω/ω0 with a characteristics frequency of the system of ω0 (such as the maximum phonon frequency), Eq. (25) can be written as 1 K = K0 2

∞ T (yω0 )

  π2 ω0 w y dy, 3 kB T

(29)

0

where K 0 = kB ω0 /π . In the high temperature limit (kB T  ω0 ), Eq. (29) reduces to ∞ ∞ 1 kB T (ω) dω. (30) K ∞ = K 0 T (yω0 ) dy = 2 2π 0

0

This implies that the phonon transmission for the entire frequency range equally contributes to the ballistic thermal conductance at high temperature limit.

2.2

R-matrix Method

In realistic simulation, one often needs an effective method to reduce the computational burden since the direct application of the NEGF equations may become numerically too expensive. One of the bottlenecks of the simulation is the matrix inversion for computing the retarded Green’s function in Eq. (13). The R-matrix method presents an effective way for reducing the computational cost of the NEGF simulation. Here we summarize the R-matrix method by using the one-dimensional harmonic chain model from the previous subsection. We first define the matrix R(ω) = [ω2 − C ]−1 ,

(31)

which represents the Green’s function for the isolated channel (i.e. decoupled from the reservoirs). Here ω is a real number without an infinitesimal shift +i0 and in the present example R(ω) is a real N × N matrix. From R(ω), we can compute the retarded Green’s function of the connected channel (i.e. coupled to the reservoirs) as follows: (32) Dr (ω) = R(ω)[I − r (ω)R(ω)]−1 ,

Electron and Phonon Transport Simulation …

81

where r (ω) = rL (ω) + rR (ω). Since the coupling with the reservoirs is localized at the channel ends (r11 and rN N are the only nonzero terms in the self-energy r ), the surface part of R(ω) is sufficient to perform phonon transport simulation. We thus define the R-matrix R(ω) ≡ R(ω)  surface by 

   RLL (ω) RLR (ω) R11 (ω) R1N (ω) R(ω) = = , RRL (ω) RRR (ω) R N 1 (ω) R N N (ω)

(33)

where Ri j (ω) with i, j = 1, N represent the surface part of the total Green’s function R(ω) for a system of N atoms. Note that Eq. (33) differs from the original definition of the R-matrix of the atomic theory in the sign convention [43–46]. Importantly, for any short range dynamical matrix, the R-matrix can be calculated recursively by adding new atoms to the system one by one. Thus, given the Rmatrix R(ω) for a segment of n atoms, the R-matrix rn+1 (ω) = [ω2 − dn+1 ]−1 for the isolated (n + 1)-th atom and the spring connection sn , one can obtain the new R-matrix (i.e. the surface part of the Green’s function for the segment of (n + 1) atoms)   LR (ω) LL (ω) R R  (34) R(ω) =  RR (ω) , RRL (ω) R where RR = [1 − rn+1 sn† R22 sn ]−1rn+1 , R RR sn† RRL , RL = R R LL = RLL + RLR sn R RL , R

(35)

LR = RLR sn R RR . R

(38)

(36) (37)

In the present example, the propagation algorithm requires no matrix operations but we retain the matrix notation in order that the above equation are applicable in general case. The numerical algorithm can be further simplified by reducing the number of matrix operations and using the same memory for the R-matrix before and after adding the new atom. The flow of the final R-matrix propagation algorithm proceeds as follows. 1. Compute r1 (ω) = [ω2 − d1 ]−1 and define the R-matrix for the isolated first atom with the elements RLL = RLR = RRL = RRR = r1 (ω). 2. Repeat the iteration RRR = [ω2 − dn+1 − sn† RLL sn ]−1 ,

(39)

RRL = RRR sn RRL , RLL = RLL + RLR sn RRL ,

(40) (41)

T RLR = RRL .

(42)

†

for n = 1, 2, . . . , N − 1 to obtain the R-matrix for the channel of N atoms.

82

N. Mori and G. Mil’nikov

In practical simulations, all the quantities in the above equations become matrices and the R-matrix propagation algorithm requires one matrix inversion at each iteration. For a channel of N sites with m × m matrix diagonal elements of the dynamical matrix the algorithm requires ∼ N times matrix inversions of a matrix of size m × m, whose computational time scales as ∼ N m 3 . If we use a direct matrix inversion of the whole channel matrix, whose size is (N m) × (N m), the computational time scales as ∼ N 3 m 3 , which is ∼ N 2 times longer than the computational time of the R-matrix method. The recursive Green’s function (RGF) algorithm [47, 48] also reduces the computational time by a ∼ N −2 factor. There is, however, an essential difference between the R-matrix and the RGF methods: the propagated unit is a closed system in the R-matrix method while the RGF method essentially considers a growing open system. As a result, in the RGF method we have to deal with complex-valued matrices while in the R-matrix method we deal with real-valued matrices (for a system which can be described by a real-valued dynamical matrix/Hamiltonian). We can also greatly reduce memory usage. In the above algorithm, the required computer memory size is independent of the channel length N and we can simulate a longer system without suffering from the insufficient memory situation. There is, however, a drawback of the R-method originating in the fact that the propagated unit is a closed system. If a frequency mesh point of the ω-integration coincides with one of the eigen-frequencies of the closed system, a divide-by-zero error will occur. However, for a simple system in which we know all the eigen-frequencies, we can exclude the frequency points in advance. On the other hand, for realistic complex systems, the divide-by-zero error seldom happens in practice. If it happens, we just disregard the point or repeat the calculation at a nearby frequency. Note that this divide-by-zero situation occurs only in numerical computation and the singularities can be removed in theory [49].

2.3 Simulation Examples 2.3.1

One-Dimensional Harmonic Chain

Here we calculate the channel length dependence of the thermal conductance in an isotopically disordered one-dimensional harmonic chain [50–53] by using the model considered in Sect. 2.1. The system is presented in Fig. 1: The left and right reservoirs consist of the same kind of atoms, called A-atom, and the channel region mainly consists of A-atom but contains isotope impurities, called B-atom, with a concentration of C. The mass of A-atom (B-atom) is MA (MB ) and the number of A-atoms (B-atoms) in the channel is NA (NB ). The total number of atoms in the channel is, therefore, N = NA + NB and the impurity concentration is given by C = NB /(NA + NB ). We assume that the spring constants are the same throughout the system, and are designated as k. We compute the thermal conductance, K , of the channel in the high temperature limit; i.e. K is calculated by Eq. (30).

Fig. 2 The thermal conductance K of the isotopically disordered one-dimensional harmonic chains in the high temperature limit for the mass ratio R = 1.1 and the impurity concentration C = 0.1. Open circles show K for periodically arranged impurities and closed circles for randomly arranged impurities. The thermal conductance is normalized by K 0 = kB ω0 /π

83

Thermal Conductance / K0

Electron and Phonon Transport Simulation …

100

10−1

10−2

10−3 2 10

C = 0.1 R = 1.1

N 4

10

6

10

Length, N

−½ 8

10

Figure 2 shows the channel length dependence of the thermal conductance K for the mass ratio R = MB /MA = 1.1 and the impurity concentration C = 0.1. K 0 = kB ω0 /π with ω0 = (k/MA )1/2 . We compare two cases in the spatial distribution of impurities: periodically arranged impurities (open circles in Fig. 2) and randomly arranged impurities (closed circles in Fig. 2). For the periodically arranged impurities, the thermal conductance is nearly independent of the channel length N . This implies that the wave nature of phonons is properly incorporated in the present simulation method. Since the atomic structure in the channel region is different from that in the reservoirs, the phonon reflection occurs at the interfaces between the channel and reservoirs, which causes a tiny change in K vs. N (which, however, cannot be seen in Fig. 2). For the randomly arranged implies, the thermal conductance decreases as the channel length increases exhibiting a super-diffusive behavior K ∝ N −1/2 (the closed circles in Fig. 2) which can be explained in terms of the existence of lowfrequency non-localized modes [50, 51]. Figure 3 shows the transmission functions of particular samples where the channel length N ranges from 102 (bottom panel) up to 106 (top panel). The horizontal axis has a linear scale in Fig. 3a and a log scale in Fig. 3b. As the channel length becomes longer, higher frequency modes are cut off. The demarcation frequency, ωd , between high frequency localized modes and low frequency non-localized modes is given by  ωd = 4

k M 1 √ , δ M 2 N

(43)

where M = (1 − C)MA + C MB is the average mass and δ M 2 = C(1 − C)(MA − MB )2 is the variance [51]. The demarcation frequencies are plotted as vertical dashed lines in Fig. 3b. Since ωd is inversely proportional to N 1/2 , the thermal conductance exhibits the super-diffusive nature of K ∝ N −1/2 . Matsuda and Ishii [50] derived an analytical expression of K for N  1 based on the Kubo formula:

N. Mori and G. Mil’nikov 1

N = 10 6

1

N = 10 6

0 1

10

5

0 1

10 5

0 1

10

4

0 1

10 4

0 1

10 3

0 1

10 3

0 1

10 2

0 1

10 2

0 0

0.5

1

1.5

Transmission

Transmission

84

0 −3 10

2

/

−2

−1

10

10

/

(a)

0

10

(b)

Fig. 3 Transmission functions of the isotopically disordered one-dimensional harmonic chains. The channel length N ranges from 102 (bottom panel) to 106 (top panel). The mass ratio is R = 1.1 and the impurity concentration is C = 0.1. The horizontal axis has a linear scale in (a) and a log scale in (b). The vertical dashed lines in (b) indicate the demarcation frequencies

Thermal Conductance / K0

Fig. 4 The thermal conductance for various R (solid circles) compared with the thermal conductance calculated by the analytical expression based on the Kubo formula (dotted lines)

C = 0.1

100

R = 1.1 R = 1.2 R = 0.5 R = 2.0

−1

10

10−2 N

10−3 2 10

4

10

−½ 6

10

Length, N

8

10

 K MI

kB = √ 2 π

k M 1 √ . δ M 2 N

K MI is also shown in Fig. 4 for comparison purposes.

(44)

Electron and Phonon Transport Simulation …

85 0.2

reservoir

L

0.15

Energy (eV)

W 3 2

x

1 1

Ny = 10

reservoir

Ny

y

in-plane mode out-of-plane mode

0.1

0.05 2

Nx

0 1

0.5

0

0.5

1

Wavevector / q0 (a)

(b) √ Fig. 5 a Armchair-edge graphene nanoribbon (AGNR) of length L (= N x 3a) and width W (= N y a) connected to left and right reservoirs. b Phonon dispersion of a pristine AGNR of N y = 10 (W = 2.46 nm) calculated with the force-constant model up to fourth-nearest-neighbor interactions. Left panel shows the in-plane dispersion, √ and right panel the out-of-plane dispersion. The phonon wave vector is normalized by q0 = π/( 3a)

2.3.2

Isotopically Disordered Armchair-Edge Graphene Nanoribbons

The simulation method of Sect. 2.1 is applied to isotopically disordered graphene nanoribbons (GNRs) [54, 55]. As discussed in Sect. 2.3.1, the thermal conductance, K , of a one-dimensional isotopically disordered harmonic chain exhibits superdiffusive nature, K ∝ L −1/2 as L → ∞, where L is the length of the chain. This anomalous diffusion is attributed to the phonon transmission of lower frequency modes below the demarcation frequency. In the lower frequency region, a onedimensional harmonic chain has a linear dispersion, ω(q) ∝ q. For graphene, inplane phonons have a linear dispersion while out-of-plane phonons have a quadratic dispersion. It is, therefore, interesting to investigate whether there is a difference between the thermal conductance of the in-plane mode and that of the out-of-plane mode as L → ∞ in isotopically disordered GNRs. We consider an armchair-edge GNR (AGNR) whose schematic diagram is given in Fig. 5a. The central channel region of L × W is connected to the left and right reservoirs. The x (y) axis is taken to be parallel (perpendicular) to the√phonon transport direction. We choose the unit structure as a rectangle of size 3a × a (a is the lattice constant) as shown in dotted lines in Fig. 5, and the channel dimensions √ L and W are expressed in terms of the number of the unit structures: L = N x 3a and W = N y a. In the channel, isotope impurities, 13 C, are randomly distributed with a concentration of C. We calculate the phonon modes of AGNRs using the force-constant model up to fourth-nearest-neighbor interactions [56]. The phonon

N. Mori and G. Mil’nikov

15

Nx = 1000 Ny = 10 C = 0.1

w/o imp

Transmission Function

Transmission Function

86

10

5

0 0

w/ imp

0.05

0.1

0.15

15

Nx = 1000 Ny = 10 C = 0.1

10 w/o imp 5 w/ imp 0 0

0.2

0.05

0.1

0.15

0.2

Energy (eV)

Energy (eV)

(a)

(b)

Fig. 6 a Transmission function for the in-plane mode in an AGNR of N x = 103 and N y = 10 (L = 430 nm and W = 2.46 nm) without impurities (gray line) and with 10% impurities (black line). b The same as (a) but for the out-of-plane mode

−½

Thermal Conductance / K0

Fig. 7 The thermal conductance K of the isotopically disordered AGNR with N y = 10 (W = 2.46 nm) in the high temperature limit for the impurity concentration C = 0.1. Closed circles show K for the in-plane mode and open circles for the out-of-plane mode

Nx 10 0

in-plane

10−1 out-of-plane 10−2

10−3

Nx−1

Ny = 10 C = 0.1 1

10

2

10

3

10

4

10

Length, Nx

5

10

6

10

transmission function is then calculated by the NEGF method combined with the R-matrix propagation algorithm. Figure 5b shows the phonon dispersion, ω(q), of a pristine AGNR of N y = 10 (W = 2.46 nm). The force constant parameters of Ref. [57] are used. The lowest branch has a linear dispersion for the in-plane mode and a quadratic dispersion for the out-of-plane mode. Figure 6a and b show the transmission functions, T (ω), of an AGNR of N x = 103 and N y = 10 (L = 430 nm and W = 2.46 nm) for the in-plane and out-of-plane phonons, respectively. The gray lines shows T (ω) for the pristine AGNR and the black lines correspond to the isotopically disordered AGNR with C = 0.1. We find

Electron and Phonon Transport Simulation …

87

that the transmission at higher frequencies is almost suppressed by the impurity scattering which may be attributed to the fact that the phonon modes are well localized only for the frequencies above the demarcation value ωd as has been discussed above for the case of one-dimensional harmonic chain. Figures 7 shows the AGNR length dependence of the thermal conductance in the high temperature limit calculated by Eq. (30). The in-plane thermal conductance exhibits the super-diffusive nature of K ∝ L −1/2 similar to the one-dimensional harmonic chain. On the other hand, the out-of-plane thermal conductance does not show a clear power-law behavior in the calculated L range up to 430 μm.

3 Electron Transport Simulation 3.1 NEGF Method In this subsection we briefly describe the NEGF method for electron transport simulation in the ballistic case without inelastic scatterings, where it becomes equivalent to the Landauer formalism [58]. Let us consider an electronic system consisting of the channel coupled to the reservoirs (or source and drain electrodes) similar to the phononic system from the previous section. The system Hamiltonian, H, is written as (45) H = HL + HLC + HC + HCR + HR , where HC describes the central channel region of finite size, HL (HR ) is the Hamiltonian for the infinite left (right) reservoir, and HLC (HCR ) represents the coupling term which is assumed to be localized at the interface between the channel and the left (right) reservoir. Without inelastic scatterings, the ballistic electron current I is given by 2e I = h

 T (E) [ f FD (μL ; E) − f FD (μR ; E)] d E

(46)

with the transmission function T (E) = Tr[ L (E)G r (E) R (E)G r † (E)].

(47)

Here the retarded Green’s function G r (E) is given by G r (E) = [E − HC − Lr (E) − Rr (E)]−1 ,

(48)

αr (E) is the contact self-energy for the electrode α, αr (E) = i[ αr (E) − αr † (E)], f FD (μ; E) = [exp((E − μ)/kB T ) + 1] is the Fermi-Dirac distribution function, and r μα is the Fermi level of the electrode α (α = L, R). The contact self-energies L,R (E)

N. Mori and G. Mil’nikov

Energy, E

Energy, E

Transport Window

88

EM Basis

Wavevector, k

(b) EM Band

mode space

extra

Wavevector, k

AM Basis

(a) AM Band

(c) EM Construction

Fig. 8 Equivalent model (EM) reproduces a transport window of the band structure calculated by an atomistic model (AM), such as density-functional theory and tight-binding model. a The AM describes a wide energy region from deep inside valence bands to higher conduction bands. b The EM basis size is smaller than the AM basis size, but the EM band can correctly reproduce the AM band within the transport energy window. c The EM basis can be constructed from primary low-dimensional atomistic basis extracted from a set of Bloch states (α, β, γ ) and extra basis states through a variational procedure to minimize the EM density-of-states in the transport window

can be calculated from the reservoir Green’s function for HL,R and the corresponding coupling terms HLC , HCR by making use of the spatial periodicity in the electrodes. All the matrices in Eq. (48) have the same finite size and we can use the above equations to simulate the electronic transport properties of the system. However, numerical studies in realistic systems often require significant computational resources, especially in the case of atomistic transport simulation. The R-matrix method can be used to reduce the computational burden. For electron transport simulation, the equivalent model offers further improvements, as explained in the next subsection.

3.2 Equivalent Model The equivalent model (see Fig. 8a and b) significantly reduces the computational burden for the atomistic NEGF simulation of electron transport. As discussed in the previous section, the phonon transport generally involves all the modes in the full phononic band structure except for the cases of very low temperatures. On the contrary, the electronic transport properties are determined by an energy interval which is normally much narrower than the full electronic-band width. In the equivalent model, we make use of this fact and extract from the original large basis set of the

Electron and Phonon Transport Simulation …

W

89 T

W

H0 Channel

Source

Drain

Fig. 9 Schematic diagram of a nanowire structure with tridiagonal block Hamiltonian

channel Hamiltonian a relatively small portion of most relevant modes which mainly contribute to the transport process. We consider a nanowire structure whose Hamiltonian is written as the form of tridiagonal block matrix (see Fig. 9): Hnn = H0 + Vn ,

Hn,n+1 = W,

Hn+1,n = W T ,

(49)

where H0 is the diagonal block of the channel Hamiltonian and W is the transfer Hamiltonian connecting the nearest neighbor blocks. We assume that H0 and W are given by an atomistic model, such as density functional theory (DFT) or tight-binding model, and their matrix size N × N is large. The size of the original Hamiltonian in Eq. (49) can be reduced in the equivalent model representation: (50) AM =  ψEM with a rectangular basis transform matrix  of size N × N in such way that Eq. (50) remains valid for any scattering solutions within a narrow transport window. Here AM is a column vector of size N and represents the electronic states of the original atomistic-model Hamiltonian, while ψEM is a column vector of size N and represents the electronic states of the equivalent-model Hamiltonian. Note that we can drastically reduce the channel Hamiltonian size for N  N . The basis transform matrix  can be constructed as follows (Fig. 8c). The equivalent model is based on a simple idea that a smooth set of Bloch Hamiltonians, H (k), can be well represented by a moderate number of representative points in the Brillouin zone under condition that the resulting model ensures minimum density-of-states. The equivalent model construction, therefore, can be reduced to minimization of a certain functional in the form:  ε1 (k),ε2 (k) (εnk ), (51) N F[] = nk

ε1 ,ε2 (ε) represents a smoothened where n is the band index, k is the wave vector, and N number of Bloch energies εnk in the interval [ε1 , ε2 ] and is, for example, given by

90

N. Mori and G. Mil’nikov

~

Fig. 10 Smoothened number of Bloch energies ε1 ,ε2 (ε). Dashed, dotted, N and solid lines correspond to n z = 2, 5, and 10, respectively

N( ) 1

2 5 10 0 1

2n z  z j − εc ε1 ,ε2 (ε) = 1 , N 2n z j=1 z j − ε

2

(52)

where εc = (ε1 + ε2 )/2, z j = εc + ρ exp[(iπ/n z )( j − 21 )], and ρ = (ε2 − ε1 )/2. As shown in Fig. 10, n z is a parameter which controls a smoothness of the func(ε). tional N The primary low-dimensional atomistic basis is first extracted from a set of Bloch states computed at a set of k-points in the Brillouin zone. This set of a mode-space basis, however, normally allows for additional unphysical branches leading to extra density-of-state components within the transport window. To eliminate these unphysical branches, extra basis states are then further constructed by minimizing F[], which ensures minimum density-of-states within the transport window. More details on this variational calculation can be found in Refs. [41, 42].

3.3 Simulation Example Here we present an example of the atomistic NEGF simulation of an n-type Si nanowire (NW) metal-oxide-semiconductor field-effect transistor (MOSFET). The atomistic device Hamiltonian for the transport simulation is obtained by a firstprinciples method based on the real-space density functional theory (RSDFT) [59, 60]. Compared to the conventional plane wave basis representation in the firstprinciples methods, the RSDFT approach does not require the time-consuming fast Fourier transform and effective parallel algorithms can be implemented for solving the Kohn-Sham sparse matrix equations with smooth Troullier-Martins nonlocal pseudopotentials [61]. The method has been proven to be suitable for large scale parallel DFT calculations in nanoscale systems of tens of thousands atoms [62]. We performed the RSDFT-NEGF simulation of Si NW MOSFETs with nonequilibrium polarization charge effects. The atomistic description is usually used only for calculating mobile charges of current-carrying states in the device and the polarization response of the channel material is incorporated into the Poisson equa-

Electron and Phonon Transport Simulation …

Drain Current (μA)

10 1 10

91

MFA

0

DFPT

10−1 10−2 10−3

D = 1 nm Lg = 6 nm T = 300 K

10−4 10−5 0

0.2

0.4

0.6

Gate Voltage (V) Fig. 11 Transfer characteristics of an n-Si nanowire MOSFET with D = 1 nm, L g = 6 nm, and tox = 0.2 nm at T = 300 K. The first-principles RSDFT method is only used in the silicon core region. Open circles are obtained from standard NEGF simulation within a mean field approximation (MFA) using Si = 11.9 in the Poisson equation. Close circles are obtained by directly computing the polarization charge in the silicon core from the density functional perturbation theory (DFPT)

tion within a mean-field approximation. The dielectric response in a strongly confined nanostructure is, however, expected to deviate from the bulk material. Moreover, in the nanoscale regime, we can no longer assume the local charge neutrality of the induced polarization density. The density functional perturbation theory [63] provides a well established approach for calculating the dielectric response in bulk media. We have generalized the equivalent model as to become applicable to the perturbative calculations with the DFT Hamiltonian in the RSDFT. We assume that the polarization response is local i.e. it is independent of the potential outside the device area. This assumption is consistent with the Neumann boundary conditions in the Poisson equation. We use the equivalent-model representation in two different ways. First, an equivalent model for the conduction band is implemented in order to facilitate the first-principles NEGF calculation of the mobile carrier and charge distribution. Second, to evaluate the polarization charge, we use a set of many small equivalent models for a sequence of energy intervals which cover the whole valence band. Figure 11 shows the calculated transfer characteristics of a Si NW MOSFET with the silicon diameter D = 1 nm, the gate length L g = 6 nm, and the oxide thickness tox = 0.2 nm at T = 300 K. Open circles correspond to the mean field approximation (Si = 11.9) which is compared with the result of the first-principles calculations of polarization charge (solid circles). A noticeable decrease of the drain current is caused by the nonzero polarization charge and it cannot be explained simply by the reduced value of dielectric constant  in the nanowire. The results show that standard mean-field approximation with fixed dielectric parameters may fail since

92

N. Mori and G. Mil’nikov

it cannot reproduce the induced macroscopic polarization charge separation in the device channel.

4 Electron and Phonon Transport Simulation 4.1 NEGF Method In this section, we consider coupled electron-phonon transport simulation. In Sects. 2 and 3, the ballistic electron and phonon transport without mutual interaction is considered and physical quantities of interest are calculated from the retarded Green’s functions. For ballistic case, an injected carrier (phonon or electron) travels through the channel to reservoirs without changing its energy. The carrier energy distribution in the channel is, therefore, determined solely by the carrier distribution in the reservoirs, where the thermal equilibrium is assumed to be maintained, and a Landauertype formula can be obtained with the transmission function being given from the retarded Green’s functions. When we include the electron-phonon interaction, the injected carrier could change its energy in the channel, and we can no longer work only with the retarded Green’s function. The so-called correlation functions, which describe the carrier distribution in the channel, will play an important role under such condition. In the following, the relevant formalism is sketched by considering a simple one-dimensional system. We consider a one-dimensional coupled electron-phonon system. For electrons, a one-dimensional nearest-neighbor tight binding model is adopted with a single atomic orbital at each lattice cite. The uncoupled electronic Hamiltonian is then written as  †  † εi ci ci − t0 (ci† ci+1 + ci+1 ci ), (53) He = i

i

where i is the site index, ci† and ci are the electron creation and annihilation operators, respectively, εi is the electron onsite energy, and t0 is the transfer integral between the nearest neighbor atoms which is assumed to be constant throughout the system. For phonons, a one-dimensional harmonic chain model is adopted in which each atom is connected by uniform springs to the nearest neighbor atoms. The phononic Hamiltonian is then given by Hp =

1 2 1 u˙ + k0 (u i+1 − u i )2 , 2 i i 2 i

(54)

where u i is the mass-normalized displacement of the i-th atom, u˙ i = du i /dt, and k0 is the spring constant. The electron-phonon interaction is assumed to take place only in the channel region with the interaction Hamiltonian of the form

Electron and Phonon Transport Simulation …

Hep =



93

Mikj ci† c j u k .

(55)

i jk

Here Mikj represents the interaction strength. The electron-phonon interaction, Hep , brings additional self-energies. When we neglect the electron-phonon interaction, the retarded Green’s functions can be obtained from the channel Hamiltonian/dynamical matrix and the contact selfenergies, and we have Eq. (48) for electrons and Eq. (13) for phonons. When we include the electron-phonon interaction, the retarded Green’s functions take the form r (E)]−1 G r (E) = [E − HC − Lr (E) − Rr (E) − ep

(56)

for electrons and Dr (ω) = [ω2 − C − rL (ω) − rR (ω) − rep (ω)]−1

(57)

r (E) and rep (ω), are the retarded scattering for phonons. The additional terms, ep self-energies representing the effect of the electron-phonon interaction on the carrier dynamics. Since the carriers change their energies via the electron-phonon interaction when traveling through the channel, we keep counting the number of carriers with different energies, which can be done by calculating the correlation functions, or lesser Green’s functions defined by < (E)]G a (E) G < (E) = G r (E)[ L< (E) + R< (E) + ep

(58)

for electrons and < < a D < (ω) = Dr (ω)[< L (ω) + R (ω) + ep (ω)]D (ω)

(59)

for phonons. Here G a (E) (= [G r (E)]† ) and D a (ω) (= [Dr (ω)]† ) are the advanced Green’s functions and the self-energies α< (E), < α (ω) for the reservoir (α = L, R) are given by (60)

α< (E) = − f FD (μα ; E){ αr (E) − αa (E)} and

r a < α (ω) = f BE (Tα ; ω){α (ω) − α (ω)},

(61)

< (E) and < respectively. The self-energies, ep ep (ω), represent the carrier scattering rates due to the electron-phonon interaction. The electron current and the phonon thermal current can be calculated from the correlation functions. The electron current transported out of the reservoir α is calculated as    2e (62) d E Tr G > (E) α< (E) − G < (E) α> (E) , Iα = h

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Phonon Thermal Current (nW)

Electron Current (μA)

T = 300 K 0.5

0

b

I

w

b

U

−0.5 eV −0.4

−0.2

0

0.2

Voltage (V) (a)

0.4

0.1

T = 300 K

−QR +QL

0

QL

QR

−0.1 −0.4

−0.2

0

0.2

0.4

Voltage (V) (b)

Fig. 12 a Electron current, I , as a function of applied voltage for a one-dimensional resonant tunneling structure with well width w = 2.4 nm, barrier width b = 1.2 nm, and barrier height U = 0.5 eV. Positive electron current corresponds to electrons flowing from the left electrode to the right electrode. b Phonon thermal current, +Q L and −Q R , as a function of applied voltage. Q α is defined as thermal current transported out of the reservoir α (α = L, R)

where the factor 2 accounts for two spin orientation and the phonon thermal current transported out of the reservoir α is given by  Qα = −

  dω < > ω Tr D > (ω)< α (ω) − D (ω)α (ω) . 4π

(63)

Here the greater self-energies α> (ω), > α (ω) and the corresponding correlation functions G > (E), D > (E) are given by the equations akin to Eqs. (60, 61) and Eqs. (58, 59): > (E)]G a (E), G > (E) = G r (E)[ L> (E) + R> (E) + ep

α> (E) >

= D (ω) =

> α (ω) =

(64)

{1 − f FD (μα ; E)}{ αr (E) − αa (E)}, > > a Dr (ω)[> L (ω) + R (ω) + ep (ω)]D (ω), {1 + f BE (Tα ; ω)}{rα (ω) − aα (ω)}.

(65) (66) (67) β

For the detailed explanations of the inelastic part of the self-energies, ep (E) and β ep (ω) (β = r, a, ), the readers should refer to Refs. [30–35]. When we switch β β off the electron-phonon interaction, i.e. Hep = 0, ep (E) and ep (ω) vanish and Eqs. (62) and (63) reduce to the Landauer-type formula of Eqs. (46) and (24), respectively.

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4.2 Simulation Example We show an example of the coupled electron-phonon transport in a one-dimensional resonant tunneling structure, whose schematic diagram is given in the insets of Fig. 12. The system is described by Eqs. (53), (54), and (55). For electronic system, the active region of width L consists of a symmetric double barrier structure whose well and barrier widths are w and b, respectively. Under applied bias V , the electron onsite energy is given by ia + Ei , εi = −eV L

 Ei =

U (i ∈ barrier region) , 0 (otherwise)

(68)

where a is the lattice constant and U is the barrier height. For phononic system, we assume that the system is uniform and characterized by an atomic mass, M, and a spring constant, k0 . For electron-phonon interaction, we consider the case where the electron transfer is affected by the lattice distortion so that the interaction Hamiltonian is given by  † t0 i,i+1 (u i+1 − u i ) (ci† ci+1 + ci+1 ci ) (69) Hep = i

with a constant strength λ of the coupling matrix elements i,i+1 . The electronphonon interaction is assumed to take place only in the active region. Figure 12a shows the current-voltage characteristics at T = 300 K. The parameters used for the calculation are as follows: the lattice constant a = 0.2 nm, the transfer integral t0 = 2 /(2 ma 2 ) with m = 0.2 m 0 , the atomic mass M = 28 u, the √ spring constant k0 = 32 N/m, and the electron-phonon coupling strength λ = ( M)−1 with  = 2.38 nm. The Fermi level in the electrode is set at the bottom of the conduction band. Clear resonant structures appear on the I -V characteristics at the resonant biases V ≈ ±0.2 V, which originate in the resonant tunneling through a confined electron level in the well region. Figure 12b shows the phonon thermal current as a function of the applied voltage. Although the left- and right-reservoirs are maintained at the same temperature T = 300 K, a net phonon flow between the reservoirs is generated under a finite applied bias. This is due to the electron-current-induced forces on the phonons. Away from the resonant biases, a net energy exchange between electrons and phonons absents, i.e. Q L + Q R = 0. At around the resonant biases, a net energy transfer from electrons to phonons occurs enhancing the phonon thermal current.

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Suspended Carbon Nanotubes for Quantum Hybrid Electronics Yoshikazu Homma, Takumi Inaba, and Shohei Chiashi

Abstract A single-walled carbon nanotube (SWCNT), suspended between microstructures, is an excellent system for probing responses to molecular adsorption/encapsulation, optical/thermal excitation, etc., because of the quasi 1D electronic structure of the SWCNT allowing highly resonant optical transitions, and sensitivity to the surrounding environment with the outer surface and the inner space. In this chapter, we will focus on the synthesis and investigation of phononic properties of suspend SWCNTs for the applications of SWCNTs for future quantum hybrid electronics. Keywords Carbon nanotubes · Phonon · Exciton · Raman scattering

1 Introduction A single-walled carbon nanotube (SWCNT) [1] is a graphene cylinder with a nanometer diameter. SWCNTs have a quasi 1D electronic states because of confinement of electron wavefunctions of the 2D electronic states of graphene in the circumstance direction [2]. They are semiconducting or metallic depending on the chirality, the atomic arrangement in the tube wall. SWCNTs are a natural quantum hybrid system of electron, photon and phonon, because excitation by photons creates electron–hole pairs, excitons [3], which are stable even at elevated temperatures due to the strong confinement effect; excitons are relaxed by the interaction with phonons, and finally emit photons if the SWCNT is semiconducting having a band gap (Fig. 1). The electronic states of SWCNTs have been used for quantum dots and single electron transistors [4], and photonic transitions have been used for single photon sources [5, 6]. However, although the phonon properties of SWCNTs have been intensively investigated, phonons of SWCNTs have not been applied to quantum information Y. Homma (B) · T. Inaba Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan e-mail: [email protected] S. Chiashi The University of Tokyo, Tokyo, Japan © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_6

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Fig. 1 SWCNT as a hybrid quantum system

technologies. Phonons in SWCNTs are sensitively accessed with Raman scattering spectroscopy resonant to the quasi 1D electronic states [7]. Together with the robust exitonic states, the resonant Raman transitions in SWCNTs are attractive as a hybrid system of electron, photon and phonon. However, technological applications of phonons in SWCNTs remain unexplored. For such applications, we need to investigate deeply the intrinsic properties of phonons in SWCNTs and their interactions with photons and electrons. Raman scattering spectroscopy (hereafter referred to Raman spectroscopy) has been widely used to characterize SWCNT samples [8]. In Raman scattering spectra (Fig. 2), the G-band, D-band, and radial breathing mode (RBM) peaks appear as the characteristic features of SWCNTs [8]. The G-band is an in-plain tangential shear mode of carbon atoms in the graphene lattice, and commonly observed around

Fig. 2 Raman scattering spectrum from a suspended (9, 7) SWCNT

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the wavenumber1 of 1590 cm−1 in sp2 carbon allotropes, such as graphene, graphite, SWCNTs and multiwalled CNTs. For SWCNTs, due to the large coverture effect, the two tangential modes of graphene split into two peaks, the stronger and weaker peaks, called G+ and G− peaks, respectively. Semiconducting SWCNTs exhibit sharp G+ and G− peaks. The D-band is generated by a double resonance process involving elastic scattering by defects or disorder of graphene/graphite lattice. This mode usually appears at 1330–1360 cm−1 depending on the laser excitation energy. The D-band is present in all carbon allotropes, including sp2 and sp3 disordered carbon. The RBM has frequencies lower than 300 cm−1 and is associated with the total symmetric movement of carbon atoms in the radial direction. The RBM is of special interest because it is a nanotube-specific feature for which the phonon frequency varies with the diameter of the SWCNT. The RBM has therefore been used to determine diameter distributions in nanotube samples. Evaluating nanotube diameters is crucial to the practical application of SWCNTs because many physical properties depend on the diameter. In addition to the RBM and the G-mode, SWCNTs generate additional and less intense Raman peaks that is often referred to as the intermediate frequency mode (IFM) in the frequency ranges from 300 to 1000 cm−1 [9]. The peak at around 855 cm−1 is assigned to the out-of-plane transverse optical (oTO) phonon mode [9, 10].

2 Synthesis and Evaluation of Suspended Carbon Nanotube 2.1 Synthesis An individual SWCNT suspended between a pair of micropillars or over a trench is free from other solid substance except for the contact parts [11]. Because singly suspended SWCNTs are not suffered from substrate effects or bundle effects, they are useful to investigate intrinsic properties of SWCNTs. In order to use suspended SWCNT for optical spectroscopy, obtaining singly suspended SWCNTs is critically important because bundling causes alternation of emission and excitation energies or even quenching photoemission when metallic nanotubes are included. Thus, growth conditions need to be precisely controlled by knowing the formation mechanism of suspended structure. Suspended nanotubes can be formed by chemical vapor deposition (CVD) using metal nanoparticle catalysts and carbon baring gases. When SWCNTs grow from nanoparticle catalysts, the growing SWCNT does not extend straightly but swings around and tends to touch neighboring substances such as other nanotubes, the substrate surface, and, if any, protrusions on the surface [12]. When they grow from the top of a tall pillar, they have a chance to touch the top of another pillar. To enhance 1

The Raman spectra are expressed with the abscissa in wavenumber shift, (excitation laser wavenumber – wavenumber of Raman peak) in the unit of cm−1 . In this chapter, Raman wavenumber shift is referred to as Raman frequency.

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the formation probability, the aspect ratio of the pillar height to the pillar spacing should be high. When the aspect ratio is smaller than 0.5, the yield of suspended nanotubes decreases [13]. Another factor is just to grow SWCNTs from the pillar top area, i.e., selectively form active catalyst particles at the pillar top. We used silica pillars fabricated by photolithography and coated them with a thin silicon film (≈40 nm-thick) except for the pillar-top area [11]. We deposited cobalt to form cobalt nanoparticles (≈3 nm-diameter). Cobalt reacts with silicon forming CoSi2 , which has a cubic lattice with almost the same lattice constant as the silicon crystal and incorporated in the silicon crystal [14]. Thus, cobalt nanoparticles are only formed on the pillar-top area. Figure 3a shows scanning electron microscopy (SEM) images of suspended SWCNTs grown between silica micropillars. To obtain a long suspension, the pillar spacing was set 7–10 μm with the pillar height of 5–12 μm. The tapered shape was taken to deposit the silicon thin layer on the side wall of the pillar. The darker contrast of the pillar top is due to the absence of the silicon layer. Ethanol CVD was performed at 850–900 ˚C. In addition to the suspended SWCNTs, several SWCNTs are seen along the pillar wall. It is just accidental to obtain such a nice suspension. Furthermore, to obtain singly suspended SWCNTs, the growth time should be short, 3–5 min, otherwise bundling occurs with an increase in growth time. Figure 3b is a transmission electron microscopy (TEM) image demonstrating that the suspended SWCNT is composed of just an SWCNT with a diameter of ≈2 nm [15]. Because of vibration of the suspended string, an atomic layer resolution image can be obtained at the very vicinity of the pillar. The problem of direct growth of singly suspended SWCNTs by CVD is a low yield. The bridge formation is just accidental and attempts to increase the yield tend to result in bundle formation. Another possible way to fabricate singly suspended

Fig. 3 a SEM images, and b TEM image of suspended SWCNT (After [15]). L is the spacing between pillars and H is the height of the pillars

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SWCNTs is manipulation of individual long SWCNTs. Mechanical extraction [16] or transfer [17] of SWCNT can be used.

2.2 Evaluation For the characterization of suspended SWCNTs, photoluminescence (PL) spectroscopy and Raman scattering spectroscopy are widely used. PL spectroscopy is applicable to semiconducting SWCNTs [18]. Usually, the excitation energy is chosen to the gap of the 2nd singular states between conduction band (c2 ) and valence band (v2 ), E 22 , as shown in Fig. 4a. The emission energy corresponds to the band gap, E 11 . However, the actual scheme of optical transitions in SWCNT is different from the simple single-particle picture depicted in Fig. 4a [19]. Since the excited electrons and holes are confined in a 1D nanotube, they interact strongly, forming hydrogen-atom like binding states, or excitons. Thus, the effective transition energies are decreased as schematically shown in Fig. 4b. Because of dielectric screening effect of the Coulomb interaction between electron and holes, the exciton binding energy changes with the dielectric environment of SWCNT, i.e. both the E 11 and E 22 are changed by adsorbing or encapsulating materials [20–22]. Although a semiconducting SWCNT has a direct band gap, PL is only observed for an isolated SWCNT: either isolated in solution by wrapping with surfactant [18] or suspended between microstructures [23]. Because the suspended SWCNTs are free from the surfactant, substrate and other SWCNTs, the intrinsic properties of them are easily accessed. Raman scattering, on the other hand, can be detected both for metallic and semiconducting SWCNTs and SWCNTs on the substrate as well, but much intense signals can be obtained with suspended SWCNTs [11]. We used a microscopic

Fig. 4 PL of semiconducting SWCNT. a Density of states of single-electron picture. b Excitonic picture of optical transition in SWCNT

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spectrometer: PL and Raman scattering from suspended SWCNTs were excited with a Ti:sapphire tunable laser. The excitation wavelength for PL spectroscopy ranged from 700 to 850 nm, whereas that for Raman scattering spectroscopy was fixed at 785 nm [24]. The PL and Raman spectra were detected by InGaAs and chargecoupled device photodetectors, respectively. We used PL imaging spectroscopy to measure spatially resolved PL spectra [25]. The PL images from the SWCNTs were acquired with an InGaAs 2D photodetector, with changing the bandpass wavelength of an acousto-optical tunable filter.

3 Lattice Vibration Properties of Suspended Carbon Nanotubes 3.1 Radial Breathing Mode RBM peaks are used for the determination of tube diameter d tube and chirality (n,m) by the resonant Raman scattering effects. The frequency of RBM peaks ωRBM is basically inversely proportional to d tube , but influenced by other factors, such as the effects of substrate and bundle formation (so-called environment effect). A variety of formula has been used to relate ωRBM to d tube . Commonly used ones are ωRBM =

A dtube

+B

(1)

or ωRBM =

A dtube



2 1 + Cdtube .

(2)

Experimentally, sets of the A, B, and C values have been reported for various types of SWCNT samples [26–30]. Those extra terms are due to the interaction with other SWCNTs (“bundling effect”) [27], substrates [31], strain effects [32], etc. Although A and B of Eq. 1 are just fitting parameters, Eq. 2 is derived from a cylinder model subjected to an inward pressure [30, 33]. However, the values of A, B, and C are just empirical ones depending on the environment of each situation. A singly suspended SWCNT is free from the effect of substrate and other nanotubes. Nevertheless, it suffers from the effect of adsorption. We found that water molecules physically adsorb on the SWCNT surface and stably form a cylindrical adsorption layer around SWCNTs in water vapor or even in ambient air [21]. The adsorption layer affects the RBM frequency and the optical transition energy (E ii ). Despite the weak interaction between water molecules and SWCNT, the change observed with the water adsorption is sizable, suggesting that even weak physisorption has a significant influence on the vibration properties of SWCNTs.

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Figure 5 shows the effect of water adsorption on Raman scattering and PL spectra simultaneously observed using a (9, 4) suspended SWCNT with controlling vapor pressure at room temperature [34]. The water vapor pressure dependence of the PL emission wavelength and ωRBM is shown in Fig. 5c. Both PL emission wavelength and ωRBM drastically shifted at a transition pressure Pt . The cause of the drastic changes is the adsorption and desorption of water molecules on the SWCNT surface [21]. Water molecules on the outer surface of SWCNTs increase the dielectric constant surrounding SWCNTs and decrease the E ii [35]. The intensity of the RBM peak decreased below Pt , as shown in Fig. 5a, because of the blue shift of E 22 . Above Pt , both the wavelength of PL emission and ωRBM slightly increased with an increase in the water pressure. The RBM vibration of the SWCNTs and the adsorption layer system has been analyzed as harmonic oscillation of the tube coupled with a cylindrical adsorption layer [34]. The interaction between a pair consisting of a carbon atom of SWCNT and an adsorption molecule is represented by the Lennard–Jones potential. A general expression for ωRBM with water adsorption effect was obtained and plotted in Fig. 6. The solid red line represents ωRBM without adsorption layer (in vacuum) and the dashed blue line indicates ωRBM with water adsorption. The tube diameter dependence of ωRBM experimentally measured in vacuum (open diamond) and water vapor (open circle) are shown with open symbols. Corresponding data were also calculated with molecular dynamics (MD) simulations. The experimental data (open diamond and circle) and the MD simulation results (filled diamond and circle) exhibited good

Fig. 5 Water vapor pressure dependence of a RBM peak and b PL emission spectra simultaneously measured in different water vapor pressure. c Water vapor pressure dependence of ωRBM and PL emission peak wavelength (After [34])

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Fig. 6 Tube diameter dependence of ωRBM of SWCNTs in vacuum, water-molecule adsorbed SWCNTs, and bundled SWCNTs (After [34]). The experimental and MD simulation results are shown. ωRBM of bundled SWCNT (▴) calculated by a MD simulation are also shown

agreement both for in vacuum and water vapor conditions. Importantly, ωRBM in vacuum are perfectly fitted with ωRBM = 228/d t + 0 (solid red line). The zero B clearly indicates that the ωRBM in vacuum is the frequency of the intrinsic radial ex from the adsorption model breathing oscillation of SWCNTs. The calculated ωtube (dashed blue curve) also fitted with ωRBM measured in water vapor. These good quantitative agreements among the results of the experiments, MD simulation, and the adsorption model, as shown in Fig. 6, suggest that the interaction between the water molecules and SWCNTs is the van der Waals interaction, which can be described by the Lennard–Jones potential, and the vibrational coupling between SWCNT and the adsorption layer modulates ωRBM . When SWCNT bundling is regarded as SWCNT adsorption for each other, the adsorption model for ωRBM is also applicable to a bundle structure. We assumed the bundle structure where an SWCNT was surrounded by six neighboring SWCNTs with the same diameter in a 2D hexagonal crystal [34]. The interaction between a pair of parallel SWCNTs is described by the integral of the interaction between carbon atoms of each SWCNT [36]. The MD simulation results (filled triangle) and the calculation with the harmonic oscillation model [34] (dotted curve) also showed good agreement. Interestingly, ωRBM of bundled SWCNT was almost the same as that of SWCNTs in water vapor in this d tube range. As-grown SWCNT samples are generally composed of bundled and unbundled (isolated) SWCNTs. Water molecules readily adsorb on isolated SWCNTs [21] and on the exterior surface of SWCNT bundles in ambient condition at room temperature, which means that all the as-grown SWCNTs are wrapped with water molecules or other SWCNTs. Therefore, bundled and unbundled SWCNTs are hard to distinguish only by Raman shifts of RBM peaks.

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3.2 Intermediate Frequency Mode The intermediate frequency mode of SWCNTs appears in the frequency range from 300 to 1000 cm−1 , as an intermediate range between the RBM and G-mode. These peaks have been investigated using ensemble samples [8] and individual SWCNTs on a substrate [9]. As demonstrated in the previous section, individually suspended SWCNTs are instrumental to elucidate intrinsic vibration properties of SWCNTs and thus are useful to investigate the intermediate frequency range Raman peaks as well. The major Raman modes of SWCNTs, RBM and G-mode, are associated with zero momentum ( momentum) phonons. The IFM is attributed to K-momentum phonons, which is associated with low frequency phonons (i.e., acoustic phonons) having nonzero momentum (Fig. 7b) [37]. In this respect, the IFM is quite interesting, because photons with nearly-zero momentum can couple with simultaneously acoustic and nonzero momentum phonons. Although applications of non-zero momentum phonons remain challenging, investigation into intrinsic properties of the IFM could provide possible future applications of phonons in SWCNTs. The Raman spectra of IFM in the frequency range from 300 to 500 cm−1 were investigated using individually suspended SWCNTs, whose chiralities were assigned based on PL spectroscopy [38]. Owing to the introduction of defects due to the use of an intense excitation laser, Raman peaks originating from K-momentum phonons (IFM and D-mode peaks) were enhanced, while those from -momentum phonons (RBM, oTO and G-mode peaks) were decreased. Since the PL intensity is more sensitively affected by the defect density, the Raman peak intensities are compared with the PL intensity simultaneously obtained. Furthermore, simultaneous collection of the Raman and PL spectra compensates for the effect of optical absorption and demonstrates the influence of exciton − phonon coupling on the peak intensities.

Fig. 7 Schematic illustrations of a electronic energy dispersion in a semiconducting SWCNT and b phonon dispersion of graphene. The separation between K and K’ in a correspond to the magnitude of the K momentum vector

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The intensities of each Raman and PL peak from a (11, 3) SWCNT are plotted as a function of the duration of intense laser illumination in Fig. 8. The intensities of -momentum phonons decrease slowly as the PL intensity decreases (Fig. 8b). By contrast, the IFM and D-mode intensities increase rapidly and then saturated. Synchronous changes in the IFM and D-mode intensities are clearly seen in Fig. 8b. The Raman scattering intensities are plotted against the PL intensity in Fig. 8c. The intensities of -momentum phonons, RBM, oTO mode and G-mode, linearly correlate with the PL intensity. For K-momentum phonons, presence of defects is essential because K-momentum phonons need to be elastically scattered by disorder or defect of the graphene lattice. Thus, the intensity change rate sensitively inverselycorrelates with the PL intensity. However, heavy damage of the SWCNT causes grovel extinction of phonon modes as well as PL. The intensives of K-momentum phonons exhibit a maximum at ≈30% of PL intensity. Therefore, the K-momentum phonon intensity can be modulated by introducing defects. The IFM-mode frequency and intensity depend on the chirality. Raman spectra obtained from six chirality SWCNTs are shown in Fig. 9 [39]. In these spectra, the intensities are normalized relative to the intensity of the G-mode for each sample. The effect of chirality on the IFM intensity appears to be similar to that for the RBM: both intensities decrease on going from top to bottom of Fig. 9. The IFM peak of the (12, 5) nanotube is very weak and appears slightly below 500 cm−1 . The Raman spectra of (9, 7), (9, 8) and (12, 5) nanotubes contain peaks at higher wavenumbers,

Fig. 8 a Raman spectrum of suspended (11,3) SWCNT. b Raman and PL intensities as a function of the laser irradiation time. c Relationship between Raman and PL intensities (After [38])

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Fig. 9 Raman spectra from suspended SWCNs with various chiralities (After [39])

around 550–600 cm−1 , which are ascribed to another IFM originate from different phonon branch [9]. As shown in Fig. 8, the IFM and D-mode peak intensities showed almost the same behavior upon the introduction of defects, while the RBM and G-mode peaks behave similarly, too. Thus, the ratios I IFM /I D and I RBM /I G are compared for assessing the effect of chirality on the Raman scattering intensity. Note that the ratio I RBM /I G is used when evaluating the chiral angle dependence of the RBM intensity [40]. The black plots in Fig. 10a, in which the error bars represent standard deviations, summarize the I IFM /I D values for each chirality. To allow a comparison with the RBM intensities, I RBM /I G data (gray plot) are also shown as a function of the chiral angle. The I IFM /I D ratio clearly depends on chirality and exhibit the same general trend as I RBM /I G . The dashed curves in Fig. 10a show the electron phonon matrix element for the RBM [41]. The type 2 nanotubes, i.e., (n-m)mod3 = 2, (12, 1), (11, 3), (10, 5) and (9, 7), follow the dashed curve nicely, while the type 1 nanotube, (n-m)mod3 = 1, (12, 5) and (9, 8), seem to deviate from the dashed curve. For those two nanotubes, a weak Raman intensity close to the noise level makes the data unreliable. The RBM has its origin at the out-of-plane transverse acoustic mode of graphene. When graphene is rolled up to form a tube, the atomic deformation vector has a radial component. The IFM is also associated with the out-of-plane acoustic-like phonon mode as shown in Fig. 10b.2 Different from the RBM, the atomic deformation vector propagates to the tube axis direction. 2

An animation of IFM can be found at: https://www.researchgate.net/publication/337706741_ IFM_Animation.

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Fig. 10 a Effect of the chiral angle on the Raman intensity ratios IIFM /ID and IRBM /IG . Dashed curves show electron phonon matrix element for RBM (Ref. 41) (After [39]). b Illustration of atomic deformation vectors for IFM of (11,3) SWCNT

3.3 G Mode The G-band of graphene and graphite [42] comes from two degenerate E 2g symmetry phonons, which are clearly separated under strain [43]. Semiconducting SWCNTs exhibit sharp G+ and G− peaks, and the origin of G+ and G− peaks are LO (longitudinal optical) and TO (transverse optical) phonons, respectively [44, 45]. Group theory and numerical calculations state that the G-band consists of six different modes, namely two A, two E 1 and two E 2 symmetries in irreducible representation [46]. Based on the theoretical prediction, six different components of the G-band were experimentally assigned, and their polarization characteristics were discussed [47–49]. However, in the early stage, Raman scattering spectra were obtained from the mixture of SWCNTs with various chiralities, which made the assignment unreliable. Furthermore, it was pointed out that polarization characteristics of Raman spectra of SWCNTs is not explained by the argument of group theory along. Polarization dependence of Raman intensity I is represented by 2  I ∝ e i · R · e s  ,

(3)

where ei and es are incident and scattered polarization, respectively, and R is Raman tensor. At the same time, light absorption spectra drastically change with changing the polarization direction of incident light due to the selection rule for energy transition. For example, the optical transition is arrowed for energy E 22 with parallel polarization light to the tube axis, while that is arrowed for energy E 12 with perpendicular polarization light to the tube axis [50]. Therefore, in order to determine the phonon symmetry of Raman peaks from SWCNTs, it is important to take account of the selection rule for energy transition and the polarization dependence of light

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absorption in investigation of the polarization characteristics of Raman scattering spectra from individual SWCNTs. We experimentally determined that the phonon symmetry of G+ and G− using a singly suspended (9, 8) SWCNT and polarized Raman scattering spectroscopy [51]. Figure 11 shows the PL intensity depending on the direction of SWCNT axis. The (9, 8) SWCNT is resonant to the excitation laser with 785 nm in wavelength. PL intensity was maximum when the polarization direction of the incident laser was parallel to the SWCNT axis [52]. The fitting curve is expressed by cos2 (φ − φ 0 ) [52], where φ is the polarization direction of the incident laser in the x–z plane, and φ 0 is the tube axis direction. When the SWCNT direction was φ 0 = 12.8˚, the PL intensity was maximum. Hereafter, θ denotes the angle between the SWCNT tube axis and the polarization direction of the excitation laser and the angle where the PL intensity maximum is defined to be θ = 0. The G-band of (9, 8) SWCNT was measured with different θ as shown in Fig. 12. The measurements were performed in two configurations, where the polarization direction of the excitation light was parallel (VV configuration) or perpendicular (VH configuration) to that of scattering light. In VV configuration, G+ and G− peaks were fitted with two Lorentzian functions as shown in (a). The Raman frequencies of G+ and G− peaks were 1593 and 1561 cm−1 , respectively, and their peak width (FWHM) were approximately 8 and 12 cm−1 , respectively. From such data, θ-dependence of G+ and G− intensity in VV and VH configurations were obtained and shown in Fig. 13. In the VV configuration, G+ and G− intensities were maximum at θ = 0, while they were minimum at θ = 90˚. On the other hand, in the VH configuration, θ-dependence of G+ intensity clearly shows nearly four-times symmetry. Because the intensity of G− peak was extremely weak, the polarization angle dependence of G− peak was

Fig. 11 a PL spectra measured at different angle of the laser polarization. b Laser polarization angle dependence of PL emission intensity (After [51])

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Fig. 12 G-band measured at different angles of the laser polarization in a VV and b VH configurations (After [51])

not obtained. Although both the G+ and G− peaks exhibit maxima at θ = 0, the intensity ratio between G− and G+ peaks shows a different θ-dependence in the VV configuration, as shown in Fig. 13c. The intensity ratio is minimum and maximum at θ = 0 and 90˚, respectively. The possible symmetries of the G-band are A, E 1 and E 2 [46], whose Raman tensors are ⎛ ⎞ a00 A =⎝ 0 a 0 ⎠, 00b ⎛ ⎞ ⎛ ⎞ 000 0 0 −c E 1 =⎝ 0 0 c ⎠, ⎝ 0 0 0 ⎠, 0c0 −c 0 0 ⎛ ⎞ ⎛ ⎞ 0d0 d 0 0 (4) E 2 =⎝ d 0 0 ⎠, ⎝ 0 −d 0 ⎠, 000 0 0 0 where a, b, c and d are Raman tensor elements. The θ-dependence of G+ and G− intensity was not simply expressed by Eq. 3 with Raman tensors in Eq. 4, as the previous study mentioned [53]. Therefore, the two effects were considered; one is the resonance effect due to the selection rule of energy transition [50]; another is the depolarization effect [54] on the polarized

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Fig. 13 Laser polarization angle dependence of a G+ and G− peak intensities in VV configuration and b G+ intensities in VH configuration. c Laser polarization angle dependence of IG− /IG+ in VV configuration (After [51])

Raman spectroscopy. Because the resonance effect and depolarization effects are the same as those for the PL polarization angle dependence (Fig. 11b), they can be simply expressed by multiplying a factor, cos2 θ [50]. The present data were represented by fitting curves, as shown in Fig. 13, where the Raman tensor of A symmetry was applied. The θ-dependence of the G+ and G− peak intensities in the VV and VH configurations were expressed by IVV ∝ (a sin2 θ + b cos2 θ )2 × cos2 θ,

(5)

IVH ∝ (b − a)2 sin2 θ cos2 θ × cos2 θ,

(6)

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where, (a, b) for G+ and G− peaks were (6.5, 32.8) and (3.5, 7.3), respectively. The ratio b/a = 5.05 for G+ and 2.09 for G− . Equations 5 and 6 well describe not only the laser polarization angle dependence of both I G − and I G + but also their intensity ratio (I G − /I G + ) in VV configuration as shown in Fig. 13 with the solid lines. Therefore, by correcting the resonance and depolarization effects in the θ-dependence, the Raman tenser of the G+ and G− phonons could be successfully determined to have A symmetry.

4 Thermal Conductivity Measurement 4.1 Method SWCNTs are expected to have high thermal conductivity along the tube axis [55]. However, experimental measurements of the thermal conductivity for individual SWCNTs are extremely difficult, and their thermal properties are still unclear. The thermal conductivity measurement of suspended SWCNTs has been performed by Joule-heating method [56], steady-state method [57, 58], and the temperature dependence of the G-band [59]. That of SWCNTs on substrates has been also performed by 3ω method [60]. The obtained values of the thermal conductivity vary greatly because they are affected by the assumed temperatures distribution [56, 59, 60], the contact thermal resistance [57, 58], the temperature dependence of the electric resistance [56, 60] etc. In addition, the characterization of SWCNTs for the thermal property measurement was insufficient and chirality assigned SWCNTs were rarely used. We used a long suspended SWCNT that was fully characterized by optical spectroscopic methods and measured the temperature distribution and thermal conductivity along the tube axis by PL imaging [25]. The advantages of our method are that no electrodes are needed, and the temperature distribution can be measured optically without contacting SWCNTs. The suspended SWCNTs used were characterized by PL and Raman spectroscopies. An example is shown in Fig. 14. The PL map and Raman RBM peak consistently show the SWCNT is a (9, 8) nanotube with the diameter of 1.18 nm. The D-band peak was not detected indicating the high quality of the nanotube. Furthermore, the SWCNT should be perfectly suspended from a top of pillar to another. It should not be bundled. Because these are very strict requirements, one can find few suspended SWCNTs suitable to the thermal conductivity measurement. The whole part of the suspended (9, 8) SWCNT was illuminated with a defocused laser beam of 720 nm wavelength, and PL emission images were measured by imaging spectroscopy. Figure 15a shows the PL images from the SWCNTs through the tunable band-pass filter with different wavelengths (1300–1400 nm). The emission intensity changed along the tube axis depending on the band-pass wavelength.

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Fig. 14 Characterization of suspended SWCNT. a SEM image, b PL map, and c Raman scattering spectra from the (9, 8) suspended SWCNT (After [25])

Fig. 15 a PL images measured through the band-pass filter with different wavelengths. The scale bar of PL images is 5 µm. b Relationship between the intensity and the band-pass wavelength of PL images at the edge and the center of SWCNT. c Pseudo color image of suspended SWCNT constituted from PL images in a (After [25])

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Figure 15b shows the relationship between the intensity and the band-pass wavelength at the edge and center positions of the SWCNT. Each corresponds to the local PL emission spectrum and was fitted with a Lorentzian function. At the edge and center, PL emission wavelength was 1339.7 and 1355 nm, respectively. Figure 15c shows a pseudo-color image representing the emission wavelength distribution along the tube axis. The color represents the emission wavelength and the intensity of the local PL emission spectra at each pixel. The emission wavelength distribution reflects the temperature distribution along the tube axis. The relationship between the emission energy shift and temperature has been evaluated for six different chiralities of SWCNTs from 300 to 760 K. The amount of energy shift E is well described by the empirical formula of Varshni [61], E = −

aT 2 T + T0

(7)

where a is the constant and T 0 corresponds to the Debye temperature, and a = 0.177 meV/K and T 0 = 1800 K, which is the Debye temperature of graphite at 600 K [62]. The PL imaging measurements were performed at about 10 Pa, where the heat transfers between air molecules and the SWCNT could be ignored. The heat loss of the thermal radiation from the SWCNT is also ignored because of the small surface area and low temperature. The diameter of SWCNT is approximately 1 nm, which is much smaller than the lateral resolution of our optical measurement system. Thus, SWCNT is regarded as an ideal line light source with an infinitesimal width. A point spread function (PSF) was calculated from blurring along the perpendicular direction to the SWCNT axis and the sharpening processing was performed with the PSF. The exciton generated by the incident photon gradually relaxes with the scattering of optical phonons in a few femtosecond and acoustic phonons in a few picosecond [63], and then it reaches thermal equilibrium. On the other hand, the group velocity of acoustic phonons is ∼104 m/s, and the thermal energy travels ahead ∼101 nm in a few picosecond. Therefore, it is not necessary to consider the non-thermal equilibrium state nor thermal transports under laser irradiation.

4.2 Analysis of Thermal Conductivity The distribution of emission energy obtained by PL imaging were converted into the temperature distribution along the tube axis [25]. Figure 16 shows the temperature distribution of SWCNT measured with different laser powers. The pillars were located at x = − 6 and 6 μm, where the SWCNT was fixed. The temperature difference between the center and edge of SWCNT was about 80 K at I 0 = 2.03 μW, and it increased to about 500 K at I 0 = 4.15 μW. If the laser power density is uniform along the tube axis and thermal conductivity of SWCNT, κ, is independent of temperature,

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Fig. 16 a Temperature distribution along the tube axis with different irradiation laser powers (I0). b Temperature dependence of the thermal conductivity of (9, 8) SWCNTs with different lengths (After [25])

the temperature distribution is exactly parabolic. However, the temperature distribution is non-parabolic. Therefore, the power density distribution of the excitation laser and the temperature dependence of κ were considered. The power distribution 1D (x). of the excitation laser is expressed by 1D Gaussian, Plaser The κ of SWCNT was calculated based on the heat equation S

 dT (x) d 1D κ(T (x)) + α Plaser (x) = 0, dx dx

(8)

where T (x) is the temperature distribution and S is the cross-sectional area of SWCNT, S = π d tube l with the van der Waals distance, l (3.4 Å). α is the absorption coefficient. In the case of (9, 8) SWCNT, α at 720 nm was calculated to be 0.0560 [64]. In the calculation of α, E 11 = 0.9270 eV and E 22 = 1.590 eV, which are the optical transition energies of (9, 8) in vacuum, were used. Here, off resonant excitation energy (1.72 eV) was used to avoid the α change with temperature. The second derivative of T with respective to x was calculated from the temperature distribution, and κ was calculated based on Eq. 8. Figure 16b shows the temperature dependence of κ obtained for the temperature distribution at various excitation laser powers. The κ clearly depends on the local temperature along the tube axis. Four (9, 8) SWCNTs with different lengths (10 and 12 μm) were measured. They showed the similar temperature dependence of κ, and the thermal conductivity continuously decreases as the temperature increases. In general, Umklapp phonon − phonon scattering (3-phonon scattering) is a dominant process on the thermal conductivity at high temperatures. As the temperature increases, in addition to that of 3-phonon scattering, the contribution of higher order phonon scattering processes to the thermal conductivity appears. The temperature dependence of thermal conductivity at high temperatures follows a simple Matthiessen’s rule [56],

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κ(T ) = −

1 . c1 T + c2 T 2

(9)

The terms of T −1 and T −2 correspond to 3-phonon and higher order phonon scattering processes, respectively. The experimental data in Fig. 15b are well-fitted to this equation with c1 = (1.90 ± 0.10) × 10−6 and c2 = (6.11 ± 2.10) × 10−10 . We can see that the term of T −1 is dominant in the measurement temperature range. The thermal conductivity of (9, 8) SWCNTs is estimated to be 1166 ± 243 W/(m·K) at 400 K from the plot. The thermal conductivity obtained here is lower than some of the previously reported ones [56, 57]. However, our measurement was performed for the fully characterized and multiplicate (9, 8) SWCNTs with high precision (the measurement error was estimated 21%). Therefore, we believe that our measurement and results are reliable. Note that the temperature dependent emission energy shift has been measured above the room temperature, and it was confirmed that water did not adsorb on SWCNTs by PL spectroscopy. It is possible that the adsorption and desorption of water make a difference in the temperature dependence of PL emission wavelength in the low temperature range.

5 Summary In this chapter, the intrinsic phononic properties of SWCNTs measured by Raman spectroscopy using singly suspended SWCNTs are described. The RBM frequency of suspended SWCNTs in vacuum (i.e., without water adsorption) is exactly inversely proportional to the tube diameter without environmental terms. The G+ and G− bands of SWCNTs have the Raman tensors of A symmetry. The IFM shows the same chirality dependence as the RBM, both of which have the out-of-plane deformation vectors. Accurate measurements of thermal conductivity are crucial for the thermoelectric application of SWCNTs. A PL imaging method has been developed using a long suspended SWCNT. For applications of these phonon modes, the modification of the characteristic properties is necessary. The RBM frequency has been shown to be shifted by adsorption of water on the outer surface of SWCNTs. The IFM intensity can be enhanced by introducing defects. Furthermore, it has been shown that PL emission/excitation energies of SWCNTs are changed by the phase of water encapsulated in the inner space of SWCNTs [65]. As demonstrated in 3.1 and 3.2, phnonons and excitons are strongly coupled in SWCNTs, controlling the adsorption and encapsulation enables modification of photonic and excitonic responses of SWCNTs. The application of those phononic properties of SWCNTs to quantum information technologies is still challenging, but the understanding them deeply will hopefully bring a breakthrough in this field.

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Quantum Effects in Carbon Nanotubes: Effects of Curvature, Finite-Length and Topological Property Wataru Izumida

Abstract In this article, quantum effects in low energy electronic phenomena in the single-wall carbon nanotubes are overviewed based on our recent theoretical studies. These effects play dominant roles in the quantum transport phenomena in the nanotube hybrid systems. The nanotube can be metallic or semiconducting depending on its chirality because of the discretization of wavevector in circumference direction. The curvature of nanotube surface induces the energy gap in the metallic nanotubes and spin-orbit interaction. Finiteness of the nanotube length leads to the discretization of energy levels. Bulk-edge correspondence for edge states in the gap reveals another feature of the nanotubes as topological matters. Keywords Quantum transport · 1D model · Edge states

1 Introduction Single-wall carbon nanotubes (SWNTs) have been extensively studied as ideal onedimensional (1D) conductors in nanoscale. Hybrid structures, in which individual ultraclean SWNTs are embedded in semiconductor electronic devices, have been fabricated in the last two decades. Quantum transport measurements have revealed that there exist fine structures. These have been reviewed in detail in the recent review article [1]. The nanotubes can be either metallic or semiconducting depending on their chirality [2]. This property has been well-known and it can be understood from the electronic structure of the graphene by employing the boundary condition in the circumference direction. However, the observed fine structure cannot be captured by this simple picture. It exhibits the unique quantum effect of the nanotubes. In this chapter, our recent theoretical works focusing the quantum effects unique in the nanotubes are overviewed. W. Izumida (B) Tohoku University, Sendai 980-8578, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_7

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The nanotube surface possesses the finite curvature. The curvature-induced effect has modified the simple picture on the electronic property. It had been pointed out that the nanotubes classified into the metallic nanotubes have a small but finite energy gap due to the finite curvature effect [3–7]. Further, the finite curvature combined with the spin degrees of freedom can be the origin of the spin-orbit interaction [6, 8–10]. Microscopic theories that enables quantitative discussions on these curvatureinduced phenomena have been developed [11, 12]. The bound-state spectrum, the discrete energy levels, naturally arises due to the finiteness of the nanotube length. There exist two valleys in the energy bands of the nanotubes. For the bound-state, the picture of the valley as a good quantum number has been widely used [1]. To explore the role of the valleys on the bound state, it has been carried out the construction of microscopic theory; reconstruction of the cutting lines using the rotational symmetry of the system and modelling the nanotubes to simplified 1D models [13, 14]. Based on the theory and the numerical calculation for the finite-length tight-binding model, the bound state has been investigated including the evanescent modes near the boundary. It has been shown that the degeneracy of the bound states is not only lifted by the curvature-induced spin-orbit interaction but also by a valley mixing from the edges [13, 15]. Furthermore, open-ended nanotubes commonly host edge states which energies lie in the bulk band gap [13, 16, 17]. The theory for the finite-length has been expanded to the edge states. The exact relation of one-to-one correspondence between the number of edge states and a winding number topological invariant, the bulk-edge correspondence, has been shown [14]. The topological considerations can give a new perspective on the nature of these localized states [14, 18–20]. Further, for the hybrid system of a nanotube proximity coupled to a superconductor, emergence of the topological states localized at the edges including the Majorana quasiparticles has been investigated, by utilizing the developed 1D model, employing the numerical tight-binding calculations and showing three-dimensional profile of the bound states [21–23]. In the following sections, the basic property of the nanotubes, which also include the recently developed theories on the cutting lines and the 1D lattice model, and the subjects on the curvature-induced effect, bound states in the finite-length nanotubes, and the topological edge states, are discussed.

2 Basics of Electronic Property of Carbon Nanotubes In this section, the basics of electronic property of carbon nanotubes are discussed. The discussion includes not only the well-known conventional ones [1, 2] but also the recently developed theories on the cutting lines and the 1D lattice model [13, 14] which will be utilized in the later discussion.

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

125

2.1 Geometrical Structure of Nanotubes Geometrical structure of a SWNT is defined by rolling up a graphene sheet in the direction of the chiral vector [2], Ch = na1 + ma2 ,

(1)

√ into √ a seamless cylinder as shown in Fig. 1. Here a1 = ( 3/2, 1/2)a and a2 = ( 3/2, −1/2)a are the unit vectors of graphene, a = 0.246 nm is the lattice constant. The set of the two integers (n, m), called chirality, defines the geometrical structure of the nanotube. In this chapter, we consider the rolling of the graphene from the front to the back as shown in Fig. 1, and the case of n ≥ m ≥ 0. This corresponds to the zigzag-right handedness, except for the zigzag (m = 0) and armchair (n = m) nanotubes [24]. Related to the geometrical structure, let us also introduce the following quantities which appears later. √ The diameter of the nanotube is given as dt = |Ch |/π = a n 2 + m 2 + nm/π , which is the order of 1 nm. For example, dt  0.95 nm for the (n, n) = (7, 7) armchair nanotube. The √ chiral angle is defined as the angle between a1 and Ch : θ = arccos(2n + m)/2 n 2 + m 2 + nm, which takes 0◦ < θ < 30◦ for the chiral (n > m > 0), θ = 0◦ for the zigzag and θ = 30◦ for the armchair nanotubes.

(a)

(b)

a1 a2

T

na1

ma2

Ch y

Ch dt

x

(n,m)=(6,3)

Fig. 1 a Geometrical structure of a SWNT. b The nanotube is defined by rolling up a graphene in the direction of Ch . The shadowed areas repeat the structure in the un-shadowed area. The figure shows the case of the chirality (n, m) = (6, 3)

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W. Izumida

Fig. 2 The graphene hexagonal lattices in the real and the reciprocal spaces. a A and B sublattices, unit vectors a1 , a2 and vectors δ j ( j = 1, 2, 3) from the A site to the three nearest-neighbor B sites. b Reciprocal lattice vectors b1 and b2 , and the hexagonal first Brillouin zone

A property which plays an important role in the later part of this article is the Cd rotational symmetry with respect to the nanotube axis, where d is the greatest common divisor of n and m, d = gcd(n, m). We shall introduce n  = n/d and m  = m/d, which will frequently appear later. The 1D unit vector of the nanotube, which directs to the axis of nanotube and is perpendicular to the chiral vector, is given by T = t1 a1 + t2 a2 , where t1 = (2 m + n)/d R , t2 = −(2n + m)/d R and d R = gcd(2n  + m, 2m + n). Therefore, the lattice constant of the 1D nanotube is T = |T | = a 3(n 2 + m 2 + nm)/d R . As shown in Fig. 2, the two-dimensional (2D) Brillouin zone (BZ) of the graphene in the 2D reciprocal √ lattice space is defined√by the two reciprocal lattice vectors, b1 = (2π/a)(1/ 3, 1) and b2 = (2π/a)(1/ 3, −1). Conventionally the first BZ is defined by the hexagon. The center of the hexagon is Γ point, and the corners of the hexagon are K and K  points. The two vectors K 1 = (−t2 b1 + t1 b2 )/N and K 2 = (mb1 − nb2 )/N are introduced to be reciprocal lattice vectors for the two vectors Ch and T . That is, K 1 is parallel to Ch and its length is |K 1 | = 2π/|Ch | = 2/dt , K 2 is parallel to T and |K 2 | = 2π/|T |. Here N = |Ch × T |/|a1 × a2 | = 2(n 2 + m 2 + nm)/d R is the number of graphene unit cell in the rectangle defined by Ch and T .

2.2 Electronic Property of a Graphene Here the electronic structure of a graphene is reviewed to consider that of a nanotube. As shown soon later, the electronic structure near the K and K  points play dominant role for the low-energy properties. We shall consider the following nearest-neighbor tight-binding Hamiltonian, H=

3  r

j=1

† γ j cAr cBr+δ j + H.c.,

(2)

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

127

where cσ r is the annihilation operator of an electron of the π orbital at sublattice  σ (= A, B) at site r. The sum r runs over all Nc A site in the system, and δ1 = (a1 + a2 ) /3, δ2 = (a1 − 2a2 ) /3, δ3 = (−2a1 + a2 ) /3 are the vectors point to the three nearest-neighbor B sites from the A site as shown Fig. 2. γ j = Ar|H |Br + δ j  is the hopping integral between the nearest-neighbor orbitals, where |σ r represents the π orbital at the atom position r. For a moment we consider the constant hopping γ1 = γ 2 = γ 3 ≡ γ

(3)

and the spinless model for simplicity. The Bloch function with the wavevector k for the σ sublattice is given by 1  ik·r e |σ r. |σ k = √ Nc r

(4)

Using this basis, the Hamiltonian matrix is given by  H (k) =

 0 f (k) , f ∗ (k) 0

where f (k) = Ak|H |Bk = γ

3 

eik·δ j .

(5)

(6)

j=1

At the K point we have the relation 3 

eik·δ j = 1 + e−i 3 + ei 3 = 0 2π

2π

(7)

j=1

reflecting the symmetry at the point. Then the function holds f (k) = 0. The same is true at the K  points. Therefore, the function is linearly expanded with the wavevector near the K and K  points. The Hamiltonian matrix is expressed with a unitary transformation [11] as  H (kc , kt ) = vF

0 kc − iτ kt kc + iτ kt 0



  = vF kc σˆ x + τ kt σˆ y ,

(8)

√ where vF = − 3aγ /2 is the Fermi velocity, which has been estimated as vF  8.32 × 105 m/s in the numerical calculation [12]. σˆ x (σˆ y ) is the x (y) component of the Pauli matrix. kc (kt ) is the component of the circumference, Ch (axis, T ), direction of the wave vector, measured from the K (K  ) point. The index τ denotes the K (τ = 1) or K  (τ = −1). Equation (8) gives the following energy bands known as the Dirac cones,

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Fig. 3 The Dirac cones at the K and K  points in the 2D BZ

ε(kc , kt ) = ±vF kc2 + kt2 ,

(9)

where + (−) denotes conduction (valence) band. As shown in Fig. 3 the energy bands show cone shapes (known as the Dirac cones). The electronic states near these points are called K and K  valleys. The coefficient vF  550 (meV · nm) connects the typical energy and the wavelength in each subject.

2.3 Cutting Lines and Electronic Property of a Nanotube Now let us consider the electronic property of a nanotube. Limited states in the 2D BZ are allowed for the nanotubes, reflecting the discretizing the wave number in the circumference direction under the periodic boundary condition k · Ch = 2π μ where μ ∈ Z. The 1D BZs of the nanotube plotted in the graphene 2D BZ are called cutting lines. The cutting lines are expressed by the following equation K2 , (10) k = μK 1 + k |K 2 | where k ∈ R. As shown in Fig. 4, the cutting lines are separated by K 1 and the lines are parallel to K 2 . The energy for the given states in the nanotube is obtained with Eqs. (9) and (10).

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

(a)

(b)

kt (6,3)

kt (6,4)

kc

129

kc

K2 ky kx

K1

Fig. 4 The cutting lines plotted in the hexagonal BZ for a (n, m) = (6, 3) and b (n, m) = (6, 4) nanotubes

Fig. 5 Schematics of the energy bands for the nanotubes. a The metallic nanotubes exhibit gapless linear energy bands. b The semiconducting nanotubes exhibit the hyperbola energy bands with the energy gap

As shown in Fig. 4a, there exist nanotubes in which a cutting line pass the K or K  points. Such a nanotube is metallic since there exist gapless linear energy bands at the Fermi energy. As discussed above, the nanotube is metallic if there exists a cutting line passing through the K or K  points, whereas the nanotube is semiconducting if no cutting line passes through the K or K  points. Figure 5 shows the Schematics of the energy bands for the metallic and the semiconducting nanotubes. In the following the condition for the chirality that the nanotube is metallic or semiconducting is discussed. At the K point, the following identity t1 − t2 n−m −−→ 1 K1 − K2 Γ K = (−b1 + b2 ) = − 3 3 3

(11)

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W. Izumida

holds. Therefore, there exists a cutting line which passes the K point if (n − m)/3 is an integer. That is, for a nanotube in which the relation mod(n − m, 3) = 0 holds the nanotube is metallic, whereas for the nanotube mod(n − m, 3) = 1, 2 the nanotube is semiconducting. The distance to the closest cutting line from the K point is 2/3dt for the semiconducting nanotubes. The same is true for the K  points. Therefore, the semiconductor energy gap is inversely proportional to the diameter,  4vF /3dt , which is the order of 1 eV. It will be shown in later that the nanotubes classified into the metallic nanotubes possess a small energy gap because of the finite-curvature of the nanotube surface. For the wavenumber kc measured from the K and K  points, the cutting lines can be expressed by the equation, kc =

2(μ − ντ/3) dt

(12)

where μ is an integer counted from the K (K  ) point. And an integer ν is introduced to indicate the metallic [ν = 0 for mod(2n + m, 3) = 0], type-1 semiconducting [ν = 1 for mod(2n + m, 3) = 1] or type-2 semiconducting [ν = −1 for mod(2n + m, 3) = 2] nanotubes.

2.4 Angular Momentum and Cutting Line The geometrical structure of the nanotube has Cd rotational symmetry with respect to the nanotube axis. The angular momentum around the nanotube axis L t = μ is a conserved quantity. The quantity is characterized by the integer μ, which is the integer specifying the cutting line. In the sense of the angular momentum, the minimal set of the cutting lines is defined with the d lines as follows [13, 14], − where

π π ≤k< and μ = 0, . . . , d − 1, az az √ 3ad Td = √ az = 2 N 2 n + m 2 + nm

(13)

(14)

is the shortest distance between two A (B) atoms in the axis direction (see Fig. 7a). Note that the angular momenta μ and μ are equivalent if mod(μ − μ , d) = 0, thus, e.g., μ = −1 is equivalent to μ = d − 1. The nanotubes can be alternatively classified into two classes according to the angular momentum of the two valleys [13–15, 25]: (i) zigzag class, which includes metal-1 (metallic nanotubes with d R = d) and semiconducting nanotubes with d ≥ 4, in which the two valleys have different angular momenta, (ii) armchair class, which includes metal-2 (metallic nanotubes with d R = 3d) and semiconducting nanotubes

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

Zigzag-class (9,0)

Armchair-class (6,3)

(6,6)

K K’

K’

K

K’

131

(6,4)

K K’

K

Fig. 6 The cutting lines defined by Eqs. (10) and (13) for the nanotubes in the zigzag- and the armchair-classes. The red lines shows the cutting lines passing the K or K  points for the metallic nanotubes [(n, m) = (9, 0), (6, 3), (6, 6)], and these closest to the K or K  points for the (6, 4) semiconducting nanotube Table 1 Cutting lines μ closest to K and K  points. The value of μ is given in μ = 0, . . . , d − 1. The sign − and + in μ1 and μ1 applied for type-1 [mod(2n + m, 3) = 1] and type-2 [mod(2n + m, 3) = 2] semiconducting nanotubes, respectively Type Cutting line closest to K and K  Semiconducting nanotubes [mod(2m + n, 3) = 1, 2]     , d , μ1 = mod − 2n+m∓1 ,d μ1 = mod 2n+m∓1 3 3 d=1 μ1 = μ1 = 0 μ1 = μ1 = 1 d=2 d≥4 μ1 = μ1 Metallic nanotubes [mod(2m + n, 3) = 0]    2n+m   μ K = mod 2n+m 3 , d , μ K = mod − 3 , d Metal-1 (d R = d) μ K = μ K  Metal-2 (d R = 3d) μK = μK  = 0

with d ≤ 2, in which the two valleys have the same angular momentum. Here the angular momentum μτ of valley τ (= K , K  ) is defined as follows. For the metallic nanotubes, μτ is the angular momentum at the τ point. For the semiconducting nanotubes, the corresponding angular momenta are given by the ones which are closest to the τ point (see Fig. 6) Their explicit expressions are given in Table 1 [14]. As will be discussed in later, for the finite-length nanotubes in the zigzag class the valley can be a good quantum number whereas in the armchair class the two valleys could mix even the nanotubes maintains the Cd rotational symmetry.

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W. Izumida

(a)

(b) µ=0

µ=1

µ=2

az

(c) (n,m)=(6,3) l–m’ l

l+n’

Fig. 7 Construction of the 1D lattice model for (n, m) = (6, 3). a The nanotube and its mapping to b d chains of the 1D lattice where d = gcd(n, m) = 3 c structure of the 1D lattice model. The solid lines show the nearest-neighbor carbon-carbon bond and hopping between the atoms

2.5 1D Lattice Model As explained in the previous subsection, the angular momentum is a well-defined quantity reflecting the rotational symmetry. This symmetry allows us to decompose the Hamiltonian into the μ = 0, . . . , d − 1 subspaces. The decomposition is performed by a partial Fourier transform in the circumference direction to the field operators. To achieve this, it is convenient to use the helical-angular construction [14, 26]. The decomposition constructs the 1D lattice model as depicted in Figs. 7. Then the Hamiltonian is written as [14], H=

d−1 

Hμ ,

(15)

μ=0

Hμ =γ

3 

2π

ei d

Δν j μ †  cA(μ, ) cB(μ, +Δ

 j)

+ H.c.

(16)

j=1

where is the integer denoting the lattice position in the nanotube axis direction. (The lattice position in the axis direction is measured in units of az .) The hopping distance δ j and the phase factor δν j are given as Δ 1 = 0, Δ 2 = n  ,

Δ 3 = −m  ,

(17)

Quantum Effects in Carbon Nanotubes: Effects of Curvature … (n,m)=(6,3)

εμ (eV)

8 (a) 6 4 2 0 μ=2 –2 –4 –6 –8 –3 –2 –1

133 (6,4)

(b)

μ=0 μ=1

0

k

1

μ=1

μ=0

2

3 –3

–2

–1

0

1

k

2

3

Fig. 8 Conduction and valence bands of a (n, m) = (6, 3) metallic, b (n, m) = (6, 4) semiconducting nanotubes calculated from Eq. (22). p = 1, q = 0 for (a) and p = 2, q = 1 for (b)

and

Δν1 = 0, Δν2 = − p,

Δν3 = q.

(18)

They are determined from δ j = δν j Ch /d + δ j H, where H = pa1 + q a2 is the vector defined in the helical–angular construction, in which the atomic position r is expressed by the alternative unit vectors Ch /d and H. The integers p and q are chosen to satisfy the relation (19) m  p − n  q = 1. then we have C × Ch /d = a1 × a1 . As shown in Fig. 7c, the Hamiltonian in each μ subspace represents a ladder-type 1D lattice model. This simplification of the model enables us the microscopic and detailed analysis for the electronic structure in each subspace [13, 14, 18–21]. As a simple example, we shall demonstrate to calculate the energy bands of the nanotubes using the 1D lattice model. Exploiting translational invariance, the Hamiltonian (16) in the Bloch basis is written as  Hμ (k) =

 0 f μ (k) , f μ∗ (k) 0

where f μ (k) = Akμ|Hμ |Bkμ = γ

3 

2π

ei d

(20)

Δν j μ ikΔ j

e

,

(21)

j=1

|σ kμ is the Bloch state of σ sublattice with wavenumber k and the angular momentum μ. The conduction and the valence bands of the nanotubes are given by diagonalizing the matrix of Eq. (20),

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W. Izumida

εμ (k) = ±| f μ (k)|,

(22)

where the signs + and − correspond to the conduction and the valence bands, respectively. Figure 8 shows the numerically calculated energy bands for the (n, m) = (6, 3) and (n, m) = (6, 4) nanotubes with the value γ = −2.57 eV. The metallic and semiconducting features are clearly observed in the calculations. For the later discussion we shall consider the 1D lattice model for the (n, n) armchair nanotubes with μ = 0 angular momentum. For this case, as depicted in Fig. 9a the model can be further simplified, decoupled into the two chain models. The Hamiltonian is given by Hμ=0 = Ha + Hb , Ha = −γ



a † a − γ



Hb = γ



b † b





a † a +1 + H.c. ,



+γ





(23)



(24)



b † b +1

+ H.c. ,

(25)



with the recombination of the orbitals to the anti-bonding and bonding orbitals,   1  1  a = √ cA(μ=0, ) − cB(μ=0, ) , b = √ cA(μ=0, ) + cB(μ=0, ) . 2 2

(26)

The energy bands for each chain are given by εa (k) = γ (1 + 2 cos k) , εb (k) = −γ (1 + 2 cos k) .

(27)

As shown in Fig. 9b they cross at k = ±2π/3 which corresponding to the K and K  points.

(a)

(b) 3 2 1 –3

–2

–1

0

1

2

3

k

–1 –2 –3

Fig. 9 a Transformation of the 1D model for the armchair nanotubes to the decoupled two chain models. b Energy bands Eq. (27) for each chain model

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

135

Fig. 10 a Curvature of the nanotube surface modifies the hopping integrals between π orbitals, and b induces the hopping integrals of π -σ and π -s orbitals. c The spin-orbit (SO) interaction mixes the π - and σ -orbitals

3 Curvature Induced Effects Curvature of the nanotube surface modifies the electronic structure from that of the graphene. As schematically shown in Fig. 10, the curvature modifies the hopping integrals between π orbitals, and induces the hopping integrals of π -σ and π -s orbitals. In the tight-binding method the curvature effect can be taken into account with the Slater-Koster type projection [6, 11]. In this section, the curvature-induced effects on the Dirac cones given in Sect. 2.2 are discussed. The curvature-induced effects discussed in this section can also be taken into account in the 1D lattice model shown in Sect. 2.5 by modifying the three nearestneighbor hopping integrals and including the second nearest neighbor hopping terms. The treatments, which are not addressed in this chapter, has been given in the Ref. [18].

3.1 Curvature-Induced Energy Gap in the Metallic Nanotubes First let us discuss the spinless effect of the curvature-induced shifting of the Dirac point from the K (K  ) point. The modification of the hopping integrals by the curvature is differ among the three hopping integrals; γ j = γ j  where j = j  . Therefore, the condition f (k) = 0 no longer holds at the K and K  points. This results the shift of the Dirac points from these points. The effect is expressed by adding the following correction term [5, 6, 11]  Hcv = vF

0 −τ Δkc + iΔkt −τ Δkc − iΔkt 0



  = vF −τ Δkc σˆ x − Δkt σˆ y , (28)

to the Hamiltonian Eq. (5). The shifts Δkc and Δkt have the diameter dt and chiral angle θ dependences as Δkc = β 

cos 3θ sin 3θ , Δkt = ζ , dt2 dt2

(29)

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W. Izumida

(n,m)=(6,3)

Fig. 11 Curvature-induced Dirac points shifts by Eq. (29) (the spin-independent shifts shown in the 30 times enlarged circles), Eq. (34) (spin-orbit effect shown in the further 100 times enlarged circles), for the case of the (n, m) = (6, 3) nanotube

The coefficients of β  and ζ are obtained as β  = 0.0436 nm, ζ = −0.185 nm with the ab initio based calculation on the extended tight-binding model [11], in which π and σ orbitals of carbon atom are taken into account and the hopping and the overlap integrals between two atoms evaluated by the ab initio calculation for interatomic distance up to 10 bohr (∼ 5 Å) in the three-dimensional structure are included. The term Δkc in Eq. (28) simply shifts the Dirac points i.e. kc → kc − τ Δkc in Eq. (8), which gives a small energy gap, 2vF Δkc , for the metallic nanotubes except for the armchair nanotubes (θ = 30◦ ). The energy gap for the metallic nanotubes can be the order of 10 meV. The term Δkt shifts the Dirac points along the cutting line. The shifts by these terms are shown in Fig. 11.

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3.2 Spin-Orbit Interaction Next we shall discuss the effect on the electron spin, the spin-orbit interaction. Under the atomic potential in each site, the spin-orbit interaction is expressed by Hso,Catom =

 1  r · s, Vso 2 r

(30)

where  r is the angular momentum operator acts on the atomic orbitals at the atomic site r, s is the Pauli matrix for the spin, and Vso = 6 meV is the spin-orbit coupling constant. Therefore, as schematically shown in Fig. 10c, the spin-orbit interaction mixes the π - and σ -orbitals on the atomic site. The effect of the spin-orbit interaction combined with the curvature-induced mixing appears on the Dirac cones. The effect is given by [11],  Hso =

−vF sΔkso τ sεso −vF sΔkso τ sεso

 = τ sεso σˆ I − vF sΔkso σˆ x ,

(31)

where s indicate the spin up (s = 1) and down (s = −1) in the direction to the nanotube axis, and σˆ I is the unit matrix. The term Δkso and εso have the diameter and chiral angle dependences as 1 , dt cos 3θ εso = α2 Vso , dt

Δkso = α1 Vso

(32) (33)

where α  = α1 /vF and the coefficients are obtained by the numerical calculation [11] as α1 = 0.048 nm, and α2 = −0.045 nm. The spin-orbit interaction appears as the two effects on the Dirac cones; Dirac point shifting depending on the spin direction expressed by Eq. (32), and the valley dependent Zeeman-like contribution expressed by Eq. (33). The both terms modify the energy bands in the similar energy scale. As schematically shown in Fig. 12, the model naturally explains the experimentally observed asymmetric splitting between the election and hole [10]. The spin-orbit splitting is the same between the electron and hole if the diagonal term in (31) is missed [6]. As shown in Fig. 12, the splitting at the bottom of the conduction band is larger than that at the top of the valence band when a cutting line is located at a side, whereas the opposite is true when a cutting line is located at another side. The relative position of the closest cutting line to the Dirac points depends on the chirality [11]. Together with the spin-independent curvature-induced term of Eq. (28), the curvature-induced spin-orbit term of (31) and the unperturbed term of Eq. (5), the modified energy bands for each spin s and the valley τ are calculated as

138

W. Izumida

2εso

kc

2Δkso

Fig. 12 Schematics of the shift of the Dirac cones and asymmetric spin-orbit splitting. The center figure shows the shift of the Dirac cones by the terms Eqs. (32) and (33). Left and right figures show the energy band splitting depending on the relative position of the cutting lines

εsτ (kc , kt ) = τ sεso ± vF (kc − τ Δkc − sΔkso )2 + (kt − τ Δkt )2 .

(34)

As Fig. 11 shows, the effect of the spin-orbit interaction is an energy scale of sub-milli-electron volt, 100 times smaller than 324 the spin-independent curvature effect. It appears in the low temperature quantum transport phenomena [1].

3.3 Asymmetric Velocities In addition to the curvature effects discussed above, there exist asymmetric velocities of Dirac particles as another curvature-induced effect [12]. As shown in Fig. 13 the numerical calculation exhibits the tilting of the metallic linear energy band. The behavior is interpreted that the left- and right-going Dirac particles have the different velocities in a valley, vL(τ ) = vR(τ ) . The effect has been explained by the additional terms explaining the tilting as well as warping of the Dirac cones as the curvature effect. The asymmetric velocities appear as the vernier-like spectrum in the discrete energy levels in the finite-length nanotubes as discussed later.

Quantum Effects in Carbon Nanotubes: Effects of Curvature … 0.3

(a)

(b)

K

K’

0.2 0.1

E (eV)

Fig. 13 Energy bands for the metallic nanotubes around the Fermi energy, near a the K point, b the K  point [12]. Inset in a shows tilted Dirac cones near the K point for (4,4) nanotube

139

0.0

-0.1

(7,1) (9,0) (4,4) E

-0.2 -0.3 -0.5

kc kt

(6,6) 0.0

0.5

k-τΔk (nm-1)

0.0

0.5

4 Electrons in Finite-Length Nanotubes The typical length of the nanotubes is of the order of micrometer L NT ∼ 1 µm and less. Due to the confinement in the finite-length, energy levels of electrons are quantized. The energy level spacing ∼ vF π/L NT is of the order of milli-electron-volt. The quantum transport measurements on the carbon nanotube quantum dots have observed the discrete energy levels [1]. Fourfold degeneracy of the discrete energy levels has been shown by the several groups. It has been considered that the fourfold degeneracy is an intrinsic property of the nanotubes reflecting the two degenerate valleys together with two spins degrees of freedom. After the observation of the spin-orbit splitting in the ultraclean nanotube [10] the picture has been modified. However, the picture of the decoupling of two valleys, in which the valley is treated as a good quantum number, has been widely used for the bound states. In this section we shall consider the electronic states in the finite-length nanotubes to discuss the microscopic condition to justify this picture. The theory for the finite-length has been expanded to the edge states lying in the bulk band gap and the topological nature of the nanotubes has been revealed [14, 18, 19].

4.1 Discrete Energy Levels For a free electron confined in a 1D finite-length box, the standard particle-in-a1D-box model explains the discrete energy levels of the bound states. A left-going wave is scattered at the left boundary to a right-going wave, and then the right-going wave is scattered at the right boundary to the left-going wave. As the result of these processes, the bound state consists of a left-going wave and a right-going wave. The pair of the left- and right-going waves is uniquely determined for a given energy.

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W. Izumida

Fig. 14 Schematics under the decoupling of two valleys. a Formation of the discrete energy levels in each valley, and b lifting the fourfold degeneracy by the spin-orbit splitting in each valley

For the nanotubes the particle-in-a-1D-box model cannot be simply applied because there exist two left-going waves and two right-going waves from the two valleys. In general, the ratios of these traveling waves in a bound state are determined by microscopic conditions such as the chirality and the boundary condition. First let us consider what justifies the picture of valley to be a good quantum number in the bound states. In this picture the two valleys are decoupled and wellseparated, then the particle-in-a-1D-box model is applied in each valley: the bound state is constructed from the left- and right-going waves in the same valley. This model explains the observation of the fourfold degeneracy of the energy levels, reflecting the two valley states with two spin states. The degeneracy is lifted by the spin-orbit interaction. The schematics for this picture are given in Fig. 14. What could be the microscopic condition for this picture? We shall suppose an ideal ultraclean finite-length nanotube, which has the same Cd rotational symmetry with the corresponding bulk system. Since the conservation of angular momentum, the states with different angular momenta are well-decoupled, whereas the states with the same angular momentum are coupled by the scattering at the boundaries. As already discussed in Sect. 3.3, the nanotubes are classified into two classes in the sense of the angular momentum of the valley. The decoupling of two valleys is expected for the ideal finite-length zigzag-class nanotubes. On the other hand, the two valleys can couple for the finite-length armchair-class nanotubes even both ends keep the same rotational symmetry with the bulk system. The above feature has been confirmed in the numerical calculation of the extended tight-binding model for the finite-length nanotubes [13, 14]. Figures 15a, b show the numerical calculation of the discrete energy levels εl in the (6, 3) zigzag-class nanotube with 50 nm length, where l is the energy level index numbered in ascending order of the energy and l = 0 corresponds to the highest occupied molecular orbital (HOMO) level. As shown in the level separation εl+1 − εl , they show nearly fourfold degeneracy and its small lift by the spin-orbit interaction as expected. On the other hand, Figs. 15c, d show the numerical calculation for (6, 4) armchair-class nanotube with 50 nm length. Except near the bottom or the top of the energy gap, the energy levels show twofold degeneracy reflecting the coupling of two valleys.

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141

Fig. 15 Numerically calculated energy levels εl and level separation εl+1 − εl for 50 nm length finite-length nanotubes [13, 14]. a, b show these for (n, m) = (6, 3), and c, d show these for (6, 4)

There exists alternatively a completely different picture for the bound states, strong valley coupling. The bound states are formed from a left-going wave in a valley and a right-going wave in another valley in this alternative picture. The picture could hold for the finite-length armchair nanotubes since the low energy lying channel can be decoupled into two channels as discussed in Sect. 2.5 with the 1D model. The schematics for the formation of the bound states and expected energy levels are shown in Fig. 16. Reflecting the asymmetric velocities between left- and rightgoing waves in the same valley vL(τ ) = vR(τ ) as pointed out in Sect. 3.3, the vernier-like structure in the discrete energy levels from two different level spacings, vL(K ) π/L NT and vR(K ) π/L NT , is expected.

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W. Izumida

Fig. 16 a Schematics of strong valley coupling in the eigenfunctions. b Vernier structure of quantized energies with two different level spacings

The numerical calculation has shown that the picture of the strong valley coupling holds for some armchair-class nanotubes with some conditions [13, 14]. Figure 17 shows the numerical calculation for the (7, 4) zigzag-class nanotube with 50 nm length. The calculated energy levels and the level separation for the boundary shape depicted in Fig. 17a are shown in Figs. 17b, c. The level separation shows oscillatory behavior between nearly fourfold and twofold degeneracies, showing the vernier-like spectrum. The Fourier transform of the wavefunctions has confirmed that the strong valley coupling for the formation of the bound states [13]. Another calculation shows that the coupling between two valleys is quite sensitive to factors such as the atomic structure at the boundary. Surprisingly, removing a single atom at the boundary leads to the very different behavior in the energy levels as shown in Figs. 17d, e, in which each of levels shows nearly fourfold degeneracy. The boundary is known as the minimal boundary, in which the edge has minimum numbers of empty sites and dangling bonds, and these numbers are the same [27]. The degeneracy is lifted in the small energy scale. The lift of the degeneracy is observed, and it is regardless the spin-orbit interaction in some energy regions in which left of the degeneracy by the valley mixing overcomes the spin-orbit interaction. The calculations have shown the varieties on the role of the valley on the formation of the bound states. To have more comprehensive microscopic picture on the role of the valley on the bound state, further study including such as the electron correlation which may smear the sensitivity on the atomic shape at the boundary, would be required. The 1D lattice model has been investigated analytically to explore the role of the valley on the bound state [13, 14]. For the chiral nanotubes the simplification of the model as shown for the armchair nanotubes has not been achieved. It has been revealed that the evanescent modes under the boundary conditions play dominant role to construct the bound states, and, the bound states are quite sensitive to the

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143

Fig. 17 Boundary shape and numerically calculated energy levels for 50 nm length (7, 4) nanotube [13]. a Unfolded tube near the left end. The empty sites are represented by the dashed circles, and the carbon atoms at the boundary are marked by the filled circles. b Energy levels εl and c level separation, εl+1 − εl . d and e show the energy levels and level separation calculated for the different boundary shape in which a single atomic site indicated by red filled circle with label K in (a) is removed

ρ(z)

W. Izumida

ρ(z)

144

0 10 20 30 40 50

0 10 20 30 40 50

Fig. 18 Numerically calculated edge states localized at the left and right ends at the zero energy for the (6, 4) nanotube [14]

Fig. 19 a Solutions λ for Eqs. (35) and (36) in the complex plane for (n, m) = (6, 4), μ = 1, p = 2, q = 1 at the zero energy ε = 0. Dashed circle C1 shows the unit circle centered at the origin. Blue filled circles and red filled squares are the solutions for A and B modes, respectively. b The boundary for the model, which is terminated at = 1. It shows that the number of the boundary condition φσ ( ) = 0 is n  for the A sublattice, and m  for the B sublattice

atomic structure at the boundary shape, as shown in the numerical calculations. The analysis on the 1D model for the bound states has been extended to that for edge states whose energies lie in the bulk band gap with the topological discussion. The subject on the edge states is discussed in the next subsection.

4.2 Nanotubes as Topological Matters The majority of the nanotubes exhibit edge states in the energy gap at the boundary [13, 14, 16, 17]. The bulk-edge correspondence for the edge states has revealed another feature of nanotubes as topological matters [14]. Here we discuss this subject. Figure 18 shows the the zero energy edge states localized at the left and right ends with the decay length ∼ 2vF /εgap , where εgap is the semiconductor energy gap, for the (6, 4) nanotube observed in the same numerical calculation with Figs. 15c, d. For this case the total number of the edge states is 4, one edge state per end and per spin.

Quantum Effects in Carbon Nanotubes: Effects of Curvature …

145

The 1D lattice model is utilized to analyze an eigenstate for a given energy. For a state φσ ( ) with energy ε which satisfy the relations φσ ( + t) = λt φσ ( ), φB ( ) = ηφA ( ), we have the following simultaneous equations for λ and η, 2π

1 + ei d

pμ −n 

1 + e−i d

2π

λ

+ e−i d

2π

pμ n 

2π

λ + ei d

qμ m 

λ

qμ −m 

λ

ε η, γ ε1 = . γ η =

(35) (36)

The equations have 2(n  + m  ) sets of solutions for (λ, η). A mode with |λ| < 1 (|λ| > 1) is an evanescent mode having larger amplitude at left (right), and a mode with |η| < 1 (|η| > 1) is polarized at the A (B) sublattices, and a mode with |λ| = |η| = 1 is a traveling mode which extends in whole of the system. The numerically solved solutions λ for (n, m) = (6, 4), μ = 1, p = 2, q = 1 at the zero energy ε = 0 are shown in Fig. 19a, which shows the number of left evanescent modes polarized at the A sublattice NA = 2 and that at the B sublattice NB = 3. (These for μ = 0 channel are NA = 3 and NB = 2, not shown.) The analytical solutions of NA and NB for given (n, m) have been given in the reference [14]. The wavefunction near the left end is determined as the linear combination of the relevant modes with |λ| ≤ 1 under the boundary conditions. The number of edge states is evaluated by the difference between the number of independent evanescent modes and the number of boundary conditions. As shown in Fig. 19b, the number of the boundary condition φσ ( ) = 0 is given as n  for the A sublattice and m  for the B sublattice. Therefore, the numbers of edge states which are localized at A and B sublattices at the left end are given by, (L,A) = 0, Nedge

(L,B) Nedge = NB − m  ,

(37)

respectively, for each μ and each spin state. For the case of n > m ≥ 0 the number of edge states for the A sublattice is zero because the number of boundary conditions is greater than or equal to that of the relevant modes for the A sublattice. For (6, 4) (L,B) (L,B) = 1 for μ = 1, and that for μ = 0 is Nedge = 0 from the values nanotube, Nedge shown above. The numbers estimated by Eq. (37) give the edge states for each spin and each end, therefore total number of edge states is 4 which is consistent with the numerical calculation shown in Figs. 15c and 18. The above discussion can be connected to a topological discussion for the bulk states. For the bulk Hamiltonian (20) one can introduce the winding number 1 wμ = 2π

2π dk 0

∂ arg f μ (k) , ∂k

(38)

where f μ (k) is the function given in Eq. (21). Since the integral of Eq. (21) gives phase accumulation from k = 0 to k = 2π for the function with 2π periodicity, the winding number of Eq. (38) is an integer. The winding number is well-defined value

146 Fig. 20 Phase of f μ as a function of k plotted on the torus surface showing the region of 0 ≤ k ≤ 2π and 0 ≤ arg f m u ≤ 2π for the (6, 4) nanotube [14]

W. Izumida

μ=1

μ=0 arg(fμ)

k

for a system with a finite energy gap, in which f μ (k) is finite value for any k and then arg f μ (k) is well-defined value for any 0 ≤ k < 2π . Note that if energy gap closes at k, at which f μ (k) = 0, arg f μ (k) cannot be defined for the metallic case as shown in Fig. 8a. The topological argument can be carried out for the metallic nanotubes since they possess small but finite energy gap, by including the gapped feature in the 1D lattice model [18, 19]. Figure 20 shows arg f μ (k) for (n, m) = (6, 4). It is plotted on a torus, in which circumferential direction corresponds to argument of f μ (k) as a function of k in the direction around the torus, to show the “winding” property clearly, in 0 ≤ k < 2π and 0 ≤ arg f μ < 2π . The winding numbers, how many times the curves wind around the circumference direction, are wμ=0 = 0 and wμ=1 = 1. The numbers agree with (L,B) estimated in Eq. (37). This agreement is not the number of the edge states Nedge accidental. The bulk-edge correspondence (L,B) wμ = Nedge

(39)

is exactly given for the nanotubes [14, 19]. We shall introduce our recent studies related to the subject in this section. For the hybrid system of a nanotube proximity coupled to a superconductor, the topological zero energy bound states localized at the edges, including the Majorana quasiparticles, has been investigated [21–23]. Furthermore, formation of end spins on the topological edge states and their antiferromagnetic and ferromagnetic interaction have demonstrated using the density matrix renormalization group [20]. Acknowledgements The author acknowledges JSPS KAKENHI Grants JP15K05118, JP15KK0147, JP16H01046, JP18H04282. The parts of the studies overviewed in this article have been done with the collaboration with K. Sato, A. Vikström, R. Okuyama, A. Yamakage, R. Saito, M. Eto, M. Marganska, L. Milz, C. Strunk, M. Grifoni, C. P. Moca, B. Dóra, Ö. Legeza, J. K. Asbóth, G. Zaránd.

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References 1. Laird, E. A., Kuemmeth, F., Steele, G. A., Grove-Rasmussen, K., Nygård, J., Flensberg, K., & Kouwenhoven, L. P. (2015). Quantum transport in carbon nanotubes. Reviews of Modern Physics, 87, 703. 2. Saito, R., Dresselhaus, G., & Dresselhaus, M. S. (1998). Physical properties of carbon nanotubes. Imperial College Press. 3. Hamada, N., Sawada, S., & Oshiyama, A. (1992). New one-dimensional conductors: Graphitic microtubules. Physical Review Letters, 68, 1579–1581. 4. Saito, R., Fujita, M., Dresselhaus, G., & Dresselhaus, M. S. (1992). Electronic structure of graphene tubules based on c60. Physical Review B, 46, 1804–1811. 5. Kane, C. L., & Mele, E. J. (1997). Size, shape, and low energy electronic structure of carbon nanotubes. Physical Review Letters, 78(10), 1932. 6. Ando, T. (2000). Spin-orbit interaction in carbon nanotubes. Journal of the Physical Society of Japan, 69, 1757. 7. Ouyang, M., Huang, J.-L., Cheung, C. L., & Lieber, C. M. (2002). Energy gaps in “metallic” single-walled carbon nanotubes. Science, 292, 702. 8. Chico, L., Lopez-Sancho, M. P., & Munoz, M. C. (2004). Spin splitting induced by spin-orbit interaction in chiral nanotubes. Physical Review Letters, 93. 9. Huertas-Hernando, D., Guinea, F., & Brataas, A. (2006). Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Physical Review B, 74. 10. Kuemmeth, F., Ilani, S., Ralph, D. C., & McEuen, P. L. (2008). Coupling of spin and orbital motion of electrons in carbon nanotubes. Nature, 452, 448. 11. Izumida, W., Sato, K., & Saito, R. (2009). Spin-orbit interaction in single wall carbon nanotubes: Symmetry adapted tight-binding calculation and effective model analysis. Journal of the Physical Society of Japan, 78. 12. Izumida, W., Vikström, A., & Saito, R. (2012). Asymmetric velocities of Dirac particles and Vernier spectrum in metallic single-wall carbon nanotubes. Physical Review B, 85, 165430. 13. Izumida, W., Okuyama, R., & Saito, R. (2015). Valley coupling in finite-length metallic singlewall carbon nanotubes. Physical Review B, 91. 14. Izumida, W., Okuyama, R., Yamakage, A., & Saito, R. (2016). Angular momentum and topology in semiconducting single-wall carbon nanotubes. Physical Review B, 93. 15. Marganska, M., Chudzinski, P., & Grifoni, M. (2015). The two classes of low-energy spectra in finite carbon nanotubes. Physical Review B, 92, 075433. 16. Sasaki, K., Murakami, S., Saito, R., & Kawazoe, Y. (2005). Controlling edge states of zigzag carbon nanotubes by the Aharonov-Bohm flux. Physical Review B, 71, 195401. 17. Marga´nska, M., del Valle, M., Jhang, S. H., Strunk, C., & Grifoni, M. (2011). Localization induced by magnetic fields in carbon nanotubes. Physical Review B, 83, 193407. 18. Okuyama, R., Izumida, W., & Eto, M. (2017). Topological phase transition in metallic singlewall carbon nanotube. Journal of the Physical Society of Japan, 86(1). 19. Okuyama, R., Izumida, W., & Eto, M. (2019). Topological classification of the single-wall carbon nanotube. Physical Review B, 99, 115409. 20. Moca, C. P., Izumida, W., Dóra, B., Legeza, Ö, Asbóth, J. K., & Zaránd, G. (2020). Topologically protected correlated end spin formation in carbon nanotubes. Physical Review Letters, 125, 056401. 21. Izumida, W., Milz, L., Marganska, M., & Grifoni, M. (2017). Topology and zero energy edge states in carbon nanotubes with superconducting pairing. Physical Review B, 96. 22. Marganska, M., Milz, L., Izumida, W., Strunk, C., & Grifoni, M. (2018). Majorana quasiparticles in semiconducting carbon nanotubes. Physical Review B, 97, 075141. 23. Milz, L., Izumida, W., Grifoni, M., & Marganska, M. (2019). Transverse profile and threedimensional spin canting of a Majorana state in carbon nanotubes. Physical Review B, 100, 155417.

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Synthesis and Transport Analysis of Turbostratic Multilayer Graphene Ryota Negishi and Yoshihiro Kobayashi

Abstract In this chapter, after a brief introduction of the lattice and electronic structures of turbostratic graphene with random stacking, we summarize our recent research on the fabrication of the turbostratic graphene samples and the analysis of their transport properties. Turbostratic graphene was mainly prepared by overlayer growth on graphene flakes or graphene nanoribbons used as structural templates. Chemical vapor deposition process without metal catalyst was employed for the graphene growth. Thermal reduction of graphene oxide (GO) at ultrahigh temperature was also utilized for producing the turbostratic multilayer graphene. Carrier transport properties observed from these various turbostratic multilayer graphene materials were anomalously different from those of multilayer graphene with thermodynamically stable AB stacking because a linear dispersion should maintain in each layer of the turbostratic graphene. The screening effect due to the multilayer turbostratic stacking effectively reduces the carrier scattering caused by charge impurities present on the substrate, resulting in significant enhancement of carrier mobility. The reduced GO thin-film prepared by ultrahigh temperature process restores graphitic honeycomb structures and gives like to a band-like transport with high carrier mobility, which was completely different from the variable range hopping mechanism previously observed from reduced GO. These results indicate that multilayer graphene with turbostratic stacking should offer the enormous advantage to improve the performance of graphene devices and will pave the way for practical quantum device applications. Keywords Turbostratic graphene · Twisted stacking · CVD growth · Graphene oxide · Transport properties R. Negishi · Y. Kobayashi (B) Department of Applied Physics, Osaka University, 2-1 Yamadaoka, Suita 565-0871, Osaka, Japan e-mail: [email protected] R. Negishi e-mail: [email protected] Present Address: R. Negishi Toyo University, 2100 Kujirai, Kawagoe 350-8585, Saitama, Japan © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_8

149

150

R. Negishi and Y. Kobayashi

1 Introduction Due to the excellent physical properties, monolayer graphene composed of sp2 hybridized carbon atoms with a single atomic thickness has attracted considerable attention since it was discovered [1]. One of the most interesting aspect of monolayer graphene is the linear dispersion of the energy band structure near the Fermi level. This linear dispersion relation is drastically different from the energy dispersion in the 2D free-electron system, and gives rise to the exotic carrier transport behaviors, such as extremely high mobility caused by massless fermion behavior [2], Klein tunneling effect [3, 4], long ballistic transport [5], and an unusual halfinteger quantum Hall effect [6]. These features are attractive for practical targets such as field-effect transistors (FETs) [7, 8], flexible electrodes [9], and battery materials [10]. The electrical transport properties of graphene are seriously degraded for multilayer graphene with the most stable AB (Bernal) stacking due to the interlayer coupling, which causes the linear dispersion character to disappear [11, 12]. Multilayer graphene with random stacking (hereafter denoted as “turbostratic” or “twisted” graphene) should be a promising candidate to overcome this issue since theoretical research predicts that multilayer turbostratic graphene preserves the linear dispersion relation similar to monolayer graphene due to the weak interlayer coupling (Fig. 1)

Fig. 1 Calculated electronic band structures of bilayer graphene with Bernal AB stacking and turbostratic stacking. (Reproduced with permission from Latil et al. [13]. Copyright 2007 by the American Physical Society)

Synthesis and Transport Analysis of Turbostratic Multilayer Graphene

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[13]. Linear dispersion like monolayer graphene has also been confirmed experimentally using angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy/spectroscopy (STM/STS) [14–16]. The linear dispersion feature in turbostratic graphene should completely differ from that in typical AB stacking graphene. It is expected to exhibit exotic electrical transport behaviors beyond monolayer graphene electronics as well as monolayer characteristics such as massless fermion carriers. Consequently, the effect of the stacking structure in multilayer graphene on the carrier transport properties is an important subject and should be explored. Generally, turbostratic multilayer graphene is constructed by stacking graphene layers with random twist angles, resulting in the breaking of the symmetry in the stacking direction. Figure 2a schematically depicts a lattice model of twisted bilayer graphene with twist angle θ, which is the building block for multilayer graphene. The first and second layers are drawn in black and red, respectively. For simplicity, it is assumed that the two layers have a common origin and are commensurate, leading to the unit cell generated according to the procedure described in the literature [17–19]. The lattice vectors of the unit cell are given by T1 = ma1 + na2 = na1 + ma2 T2 = R(60◦ )T1 = (n + m)a1 − na2

(1)

where n and m are integers (0 < m < n) satisfying gcd(n, m) = 1 (gcd: greatest common divisor), a1 , a2 (a’1 , a’2 ) are the primitive lattice vectors for the first (second) graphene layer. The lattice constant T and twist angle θ are given by  |m − n|a T = |T1 | = |T2 | = a n 2 + nm + m 2 = 2 sin(θ/2) cos(θ ) =

n 2 + 4nm + m 2 2(n 2 + nm + m 2 )

(2)

(3)

where a = |a1 | = |a2 |~0.246 nm is the lattice constant of graphene. Bilayer graphene with a twist angle of θ is identical with that for θ + 120º due to the C3 symmetry, and the graphene with - θ is the mirror image of that with θ. Hence, we should consider the range of 0º < θ < 60º for the twist angle in the structural analysis. Even when the lattice structure is nonperiodic and incommensurate, a periodic and hexagonal moiré pattern is observed due to interference between the graphene layers (Fig. 2b). The period L of the moiré pattern (i.e., the distance between two neighboring AA stacking regions indicated by the arrow) is related to the twist angle θ and is given by L=

a 2 sin(θ/2)

(4)

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Fig. 2 Atomic structure models of twisted bilayer graphene. a The unit cell for (n, m) = (3, 2) twisted bilayer graphene. The second graphene layer (red line) is rotated by θ relative to the AB stacking. b Typical moiré pattern in TBG schematically drawn for θ = 10º. The twist angle θ can be obtained from the periodicity of the moiré pattern L

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for relatively small θ. In the case of the commensurate bilayer graphene discussed above, the moiré period L and the unit cell size T have a simple relation of L=

T |m − n|

(5)

Moiré patterns are often observed experimentally from multilayer, especially bilayer graphene in atomic-scale images by STM, transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM) [20–22]. Several examples will be provided in Sect. 2 of this chapter. Evaluation of the stacking order is an essential analysis of multilayer graphene. Practically, the average (on the submicron scale) fraction of turbostratic stacking to the AB stacking in multilayer graphene is an important factor to investigate the correlation between the stacking structure and the transport properties. Raman spectroscopy is a powerful tool for this purpose, and is a complementary approach to moiré analysis at the atomic scale. According to Cançado et al. [23], the G’(2D) band in the Raman spectrum from multilayer graphene can be fitted by three components of the Lorentzian peaks: G’3DA , G’2D , and G’3DB . A typical example of the fitting analysis for multilayer graphene sample is shown in Fig. 3 [24]. The two Raman components of G’3DA and G’3DB originate from the 3D graphite phase with ordered AB stacking, and the other one, G’2D , is from the 2D layer phase with turbostratic (disordered) stacking. In the peak-fitting analysis, the center positions of the G’3DA , G’2D , and G’3DB signals are fixed around 2680, 2700 and 2720 cm−1 . The intensity (=integrated area) ratio of the peaks G’3DA and G’3DB is confirmed to be constant with a value of I(G’3DB )/I(G’3DA ) ~ 2. Hence, the relative volume ratio R of the AB stacking 3D graphite phase to all graphitic phases, including turbostratic graphene, can be evaluated as R=

I (G 3D B ) I (G 3D B ) + I (G 2D )

(6)

Fig. 3 Peak-fitting analysis of the 2D bands observed from processed GOs: a Processed in N2 at 1700 ºC for 10 min. b Processed in an ethanol environment at 1800 ºC for 10 min

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and the fraction of the 2D (turbostratic) phase should be obtained from 1-R. As shown in Fig. 3, the formation of the turbostratic graphene sensitively depends on the reaction condition, and Raman analysis clearly discriminates the structural difference of the stacking in the graphene samples. The detailed formation process is described in Sect. 2.

2 Multilayer Graphene Fabricated by the Solid-Template Process In the general investigation of the carrier transport properties in graphene, isolated graphene flakes must be incorporated into the device structures via electron beam and/or photolithography processes. To fabricate the FET with a graphene channel, an isolated graphene flake by mechanical exfoliation from highly oriented pyrolytic graphite (HOPG) and/or chemical vapor deposition (CVD) growth using Cu foil is transferred onto a Si wafer capped with SiO2 [1, 25]. This situation is far from ideal graphene because the SiO2 surface in contact with graphene is not atomically flat and many charged impurities naturally exist on the SiO2 surface. The structural modulation of graphene induced by the surface roughness of the SiO2 layer leads to the formation of an electron–hole puddle with various carrier densities (Fig. 4a). As shown in Fig. 4b, the random potential induced by the electron–hole puddle and residual charged impurities in graphene leads to remote Coulomb scattering [26], indicating that the polarity in the graphene can easily be modulated at the Dirac point, as shown in Fig. 4c. This situation completely degrades the intrinsic electrical transport properties in graphene on the device substrate. Carrier scattering induced by the surrounding environment is dramatically improved by the screening effect of multilayer graphene [27, 28]. As described in the previous section, in multilayer graphene with Bernal stacking, which occurs naturally in HOPG, the electronic band structure is dramatically modulated by strong interlayer coupling between the layers in Bernal stacking, and the electronic band structure approaches a parabolic dispersion as the number of layers increases. This means that the exotic electrical properties of monolayer graphene disappear in a multilayer graphene system. On the other hand, for a system with randomly oriented turbostratic stacking in the in-plane rotation direction, the electrical property is isolated in each layer due to weak interlayer coupling. This results in an electronic band structure similar to that in monolayer graphene as shown in Fig. 1. The most common method of synthesizing multilayer graphene is mechanical exfoliation from bulk graphite, which consists of predominant fractions of the Bernal stacking graphene layers. The carrier mobility in relatively thick multilayer graphene (~10 L) is higher than that of few-layer graphene (2–10 L) due to screening of the environmental effects from the substrate. It has been reported that the screening length is ~1 nm [28–30]. Unfortunately, the carrier mobility in multilayer graphene

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(b)

Ideal graphene

Real graphene on the device substrate

SiO2 surface

(c) Sheet resistance

EF EF

Carrier density

Fig. 4 Illustration of a the interface between graphene and the SiO2 surface, b potential mapping of holes (red) and electron (blue) due to the charge fluctuation and c sheet resistance versus carrier density in graphene

is lower than that in intrinsic monolayer graphene observed in suspended graphene [2] since the electronic band structure in orderly stacked multilayer graphene approaches the parabolic dispersion at the Dirac point as the number of layers increases. The CVD method using metal catalysis with a high carbon solubility can also fabricate multilayer graphene [31–33]. In this liquid–solid growth process, graphene layers are easily rotated in the liquid phase of metal catalysis, which tends to form lowest-energy stable AB stacking structure. As for fabricating the metastable structures using CVD, additional graphene layer growth on graphene as a solid growth template is a promising approach [8, 34, 35]. In this vapor–solid growth process, the grown graphene layer should preferentially form a metastable stacking structure such as random stacking because it is difficult to rotate the grown graphene layer in a solid environment. Figures 5a and b show the optical microscope and the enlarged atomic force microscope images observed from a mechanically exfoliated graphene flake before and after CVD growth. Figure 5c shows the height profiles along the L-L’ lines indicated in the atomic force microscope (AFM) images. Before CVD growth, two terraces with different heights are observed, as indicated by the arrows in the height profiles. Their heights in the graphene flake gradually increase as the CVD growth proceeds. Figure 6 shows the AFM images

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Fig. 5 a Optical microscope and b AFM images of graphene before and after CVD growth at 730 ºC, and c height profiles along the L-L’ line measured by AFM. (Reprinted with permission from R. Negishi et al. [35], Copyright 2011 by Elsevier)

observed from the surfaces of the grown graphene layers prepared at a higher process temperature (1300 ºC). A two-dimensional graphene island with a monolayer height and lateral growth at the step edge are clearly observed as indicated by the height profile along the M-M’ line. Figure 7 shows typical Raman spectra measured at the mechanically exfoliated graphene flake after CVD growth at various growth temperatures. Three primary features, which are well-known as the Raman bands from graphene, are observed: the G-band at ~1580 cm−1 due to the double degenerate (iTO and LO) phonon mode at the Brillouin zone center, the D-band at ~1350 cm−1 due to a double-resonance process induced by defects and the 2D-band at ~2700 cm−1 due to a double-resonance process sensitive to the layer stacking. The D-band intensity is extremely reduced as the process temperature increases. This indicates that a higher process temperature around 1300 ºC is a major parameter for improving the quality of the synthesized graphene layer by reducing the defect density. Figure 8 shows the thickness dependence of the Raman spectra observed from the graphene layers grown at 1300 ºC [36]. The growth time controls the number of grown graphene layers. The number of layers efficiently increases to 2, 6, and 11 as the growth time increases. The intensity ratio (I D /I G ) of the D-band and the Gband, and the full width at half maximum (FWHM;  D ) of the D-band are ~0.36 and ~49 cm−1 for 6-layer graphene. The grain size estimated from the I D /I G according

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Fig. 6 AFM images of the surfaces a before and b after growth at 1300 ºC. The height profiles along M-M’ lines are also shown in c

to reference [37] is about 60 nm. The values of the I D /I G and  D are smaller than those values (I D /I G = ~0.75,  D = ~201 cm−1 ) observed from multilayer graphene grown at a relatively low temperature (720 °C) [38]. The stacking structures in the multilayer graphene are mainly divided into two types: ordered stacking (e.g., AB stacking) and turbostratic stacking. In multilayer graphene with ordered stacking, the peak shape in the 2D-band spectrum is decomposed by two Lorentzian peaks at ~2680 and ~2720 cm−1 [39, 40]. On the other hand, the peak shape observed from turbostratic multilayer graphene with completely random stacking shows a single Lorentzian at ~2700 cm−1 , which is similar to the

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Fig. 7 Raman spectra observed from graphene prepared at different growth temperatures

Fig. 8 a Raman spectroscopy observed from monolayer and multilayer graphene and b the 2Dband fitted by the three Lorenz peaks for the evaluation of the turbostratic ratio. (Reprinted with permission from C. P. Wei et al. [36]. Copyright 2019 by The Japan Society of Applied Physics)

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monolayer graphene as described in Sect. 1 [23, 40]. The fitting curve analysis of the actual Raman spectrum for the 2D-band region shows three Lorentzian components at ~2680, ~2700 and ~2720 cm−1 . From the analyzed intensities, the volume ratio of the turbostratic fraction (R’) in multilayer graphene can be evaluated according to (6) [23] as R = 1 − R = 1 −

I (G 3D B ) + I (G 2D )

I (G 3D B )

(7)

The R’ values in grown multilayer graphene with 2, 6, and 11 layers are about 92, 80 and 69%, indicating that the grown multilayer graphene forms a turbostratic stacking at a high ratio. Figure 9a–c shows STEM images taken from pristine monolayer graphene, which was used as a growth template [41]. Periodic bright spots are clearly observed. Figure 9d also shows the two-dimensional fast Fourier transform (2D-FFT) pattern obtained from the STEM image. The six-fold symmetry of the graphene lattice results in the 2D-FFT pattern with six hexagonally distributed spots. Typical multilayer graphene with pure AB Bernal stacking also gives rise to a hexagonal lattice structure [42]. On the other hand, multilayer graphene grown on the monolayer graphene template partly exhibits the moiré patterns, as shown in the STEM images (Fig. 9eg)). Note that the moiré patterns indicate the formation of rotational stacking faults in the grown multilayer graphene [43] and lead to the appearance of super lattice structures with a large unit cell (Fig. 9g). The unit cells observed in the moiré patterns show a non-uniform size and primitive translation vector with the different directions. These features produce the ring-like shape in the 2D-FFT pattern, as shown in Fig. 9h. This means that the grown graphene layers are randomly oriented, and the stacking structure of the grown multilayer graphene forms a turbostratic configuration. Peak analysis of the Raman spectra in the 2D-band region indicates that the grown graphene layers have a turbostratic stacking structure with weak interlayer coupling [34, 35]. The moiré patterns of grown multilayer graphene are also observed in the STM images, as shown in Fig. 9i. The twist angle is estimated to be ~4.6º from the observed period of the moiré pattern and Eq. (4). This result confirms the formation of bilayer graphene with turbostratic stacking [44]. Figure 9j is the STEM image of a cross-section of the grown multilayer graphene. The grown graphene layers are uniformly stacked, and their spacing (0.32~0.34 nm) matches the interlayer space of graphene or graphite materials. The continuous graphene layers suggest that graphene layer growth proceeds by a layer-by-layer mode. We confirmed that the grain size evaluated from the intensity ratio (I G /I D ) of the G- and D-band in the Raman spectra [37] significantly increases for a high growth temperature above 1300 ºC. This means that highly crystalline multilayer graphene with a turbostratic stacking structure forms on the graphene solid template via CVD growth at a high temperature above 1300 ºC. Figure 10a shows the dependence of the sheet resistance for the graphene layers on their thickness [41]. The sheet resistance has been evaluated in the configuration of FET devices. The carrier concentration in the graphene channel is modulated by

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(b)

(a)

(c)

(d)

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5 nm

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(e)

(h)

0.5 nm

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(i) (j) 1 nm 2 nm

100 nm

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(nm)

0

0.4 nm

Fig. 9 STEM images and their FFT patterns observed from a–d the suspended monolayer graphene and e–h the grown multilayer graphene. i STM image obtained in air from bilayer graphene grown on a monolayer graphene template. j A cross-sectional TEM image of the grown multilayer graphene. (Reprinted with the permission from R. Negishi et al. [41], Copyright 2020 by John Wiley and Sons)

applying a gate voltage, showing ambipolar conduction of a typical graphene material. The highest sheet resistance value (minimum conductance) corresponds to the Dirac point. Although the sheet resistance differs in the absolute value, the ambipolar property is preserved as the number of layers increases. The sheet resistance of the monolayer graphene is 2.8 k/ (Fig. 10b). It has been reported that the typical sheet resistance of monolayer graphene on the SiO2 /Si substrate is 1~3 k/ [45, 46]. The slightly higher sheet resistance than that previously reported is due to the top gate structure in which the graphene channel is sandwiched between the fused quartz substrate and the Si3 N4 insulator layer and has contact with both layers. Charge impurities should also exist at the interface between graphene and insulator surfaces [26]. This significantly degrades the conductance of graphene by remote Coulomb

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Fig. 10 a Sheet resistance of the grown multilayer graphene as a function of the gate voltage, and b layer number dependence of the sheet resistance at the Dirac point. (Reprinted with the permission from R. Negishi et al. [41], Copyright 2020 by John Wiley and Sons)

scattering [27]. However, the sheet resistance at the Dirac point observed from the grown multilayer graphene channel is significantly reduced as the number of layers increases (Fig. 10b). The resistance of multilayer graphene can be given by a simple model in which the plane direction resistance of the layers is connected in parallel between sourcedrain electrodes, assuming that the conductance between graphene layers connected by van der Waals force can be ignored. Then, the total resistance (Rtotal ) for n layers of graphene can be described as 1 Rtotal

=

1 Rbottom

+

1 1 1 + ··· + + R1 Rn−2 Rtop

(8)

where Rbottom and Rtop mean the sheet resistances of the monolayer at the bottom and top in contact with the insulator layer and are evaluated to be 2.8 k/ for the monolayer. The R1 , · · · , Rn-2 are the resistance of each layer in inner graphene. Assuming the monolayer graphene and inner graphene have equal sheet resistance values, the calculated curve of the sheet resistance with respect to the number of layers should be the solid line in Fig. 10b. However, the observed resistance value deviates from the calculated curve, as it shows a lower resistance. The sheet resistance of the inner layer estimated from the experimental value of 7 layers is 630 / per layer. This value is much lower than the typical resistance (1~3 k/) observed from monolayer graphene on a SiO2 /Si substrate [45, 46]. The remarkably low resistance in the grown multilayer graphene should be caused by the screening effect where the influence of the charge impurities existed between the graphene and insulator is quickly diminished within a few layers [28]. Figure 11 shows the dependence on the number of layers in the carrier mobility evaluated from the FET transconductance (Fig. 11a) and from the Hall-effect

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Fig. 11 Layer number dependence of the field effect mobility evaluated from a the transfer characteristics in the FETs with grown graphene. b Layer number dependence of the carrier mobility and carrier concentration observed in the Hall-effect measurement using the van der Pauw geometry. (Reprinted with the permission from R. Negishi et al. [41], Copyright 2020 by John Wiley and Sons)

measurement (Fig. 11b). The Hall carrier mobility is slightly smaller than that evaluated from the transconductance of the FET. In the Hall-effect measurement, the Fermi level is not controlled by the gate electrode. Graphene is mostly doped by electrons or holes due to the environmental effects. The carrier mobility in the Halleffect measurement is evaluated at a position shifted from the Dirac point, resulting in a lower carrier mobility. Note that the maximum carrier mobility is observed at 2–4 layers. This interesting trend is observed in both the FET and Hall-effect measurements. In the case of graphene with AB stacking, the carrier mobility of a few-layer graphene degrades as the number of layers increases because their electronic band structure near the Femi level approaches a parabolic dispersion [27]. The degradation of carrier mobility is evident between monolayer and bilayer graphene [47]. However, multilayer graphene grown by the solid template method shows the opposite behavior in the layer number dependence of the carrier mobility. Figures 12 shows the temperature dependence of the sheet resistance (ρ) at the Dirac point normalized to that observed at 300 K. The data indicated by the open symbols are observed from mechanically exfoliated graphene [38]. In a multilayer graphene device with five layers, the ρ(10 K)/ρ(300 k) varies by a factor of two from that measured at room temperature. This strong temperature dependence is related to the band overlap between the conduction and valence bands. The band overlap region increases as the layer number of multilayer graphene with well-ordered AB stacking increases [48]. Accordingly, the carrier density thermally excited in the multilayer graphene becomes higher as the number of layers increases. Note that the temperature dependence of the sheet resistance for the grown multilayer graphene (filled square symbols) is closer to that of monolayer graphene (open square symbols) than that of mechanically exfoliated multilayer graphene with AB stacking (open circle symbols). This indicates that the electronic band structure in the graphene layers obtained by the solid growth template method preserves the linear dispersion as a quasi-monolayer due to a weak interlayer coupling.

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Fig. 12 Temperature dependence of the sheet resistance (ρ) at the Dirac point. The sheet resistance is normalized to that observed at 300 K. White circle and square represent the data from the graphene obtained from HOPG by mechanical exfoliation method. Black square indicates the multilayer graphene grown by the solid template method

Monolayer graphene (open square symbols) is nearly independent of temperature because the carrier density induced by the charge impurities is larger than the density of thermally excited carriers around the Dirac point due to the linear dispersion. The important feature is that the sheet resistance observed from the grown multilayer graphene slightly depends on the temperature compared to monolayer graphene. The weak but significant temperature dependence of the sheet resistance is represented as an inherent property in monolayer graphene [49] since the effect of charge impurities is reduced by the screening effect. This quasi-monolayer properties of the grown multilayer graphene with turbostratic stacking leads to improved electrical performances such as conductivity and carrier mobility compared with monolayer graphene on a SiO2 /Si substrate [50].

3 Multilayer Graphene Nanoribbon Since theoretical calculations using a tight binding model have predicted that a quasione dimensional monolayer graphene nanoribbon (GNR) exhibits a finite energy band gap at the Fermi level [51], its anomalous electrical properties such as the bandgap opening [52, 53] and electron confinements [54, 55] have been experimentally investigated in GNR materials. The edge structures and widths of GNR play important roles in its electrical band structures [56, 57]. Therefore, various syntheses of GNR using electron beam lithography [58, 59], chemical cutting or etching of graphene [60, 61], nickel nanobars [62], and precursor monomers as a bottom-up approach [63] have been studied to precisely control the structures. In addition to the edge structure and the width of the GNR, fabrication of multilayer GNR and

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its stacking are crucial issues for practical GNR applications. In this section, we will show recent results on multilayer turbostratic GNR formation using vapor–solid growth processes as shown in the Sect. 2, and the analysis of its electrical properties. Figures 13a and b show AFM images observed from the same location for samples of pristine GNR and multilayer GNR grown by the solid template method [8]. The height distributions of the GNRs before/after growth were analyzed from these Fig. 13 AFM images of GNRs a before and b after CVD growth. c Height and d width distributions of GNRs before and after CVD growth. (Reprinted with the permission from R. Negishi et al. [8], Copyright 2017 by AIP Publishing)

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images (Fig. 13c). Pristine GNRs exhibit a height distribution ranging from ~0.4 to 1.8 nm, meaning that the GNRs are formed by an unzipping process because the distribution is obviously smaller than the diameters of the double-walled carbon nanotubes (DWCNTs) used as starting material (~3–15 nm) [64]. The height of the monolayer graphene sheet on the SiO2 /Si substrate is ~0.6–0.8 nm [65]. This value is larger than the interlayer distance (0.34 nm) in bulk graphite due to the difference in the interface structure. Since the monolayer height is ~0.6–0.8 nm, pristine GNRs are a mixture of monolayer and bilayer structures. The height distribution analyzed from GNRs after growth significantly differs from that for pristine GNRs. The 95% confidence intervals obtained by statistical analysis are 0.98–1.17 nm for the pristine GNRs and 2.56–2.81 nm for the GNR after growth. The ranges of the confident intervals do not overlap, indicating the formation of multilayer GNRs by overlayer growth. The height profiles can be fitted by a Gaussian function. It is assumed that the center position of the Gaussian distribution is the representative height of GNR. The height difference (H) of ~1.6 nm evaluated from the center position is compatible with the statistical analysis of the confidence intervals, indicating the formation of an additional 4–5 layers of graphene on the pristine GNRs. The full width at half maximum (FWHM) of the distribution has almost the same value between the pristine (0.79 nm) and grown (0.93 nm) GNRs. This suggests that the graphene growth proceeds uniformly in a layer-by-layer mode on each GNR template. Figure 13d shows the change in the width distributions of the GNRs by CVD growth. If the size scale of the observation target material is comparable to the curvature radius of the AFM tip, the tip shape strongly influences the target’s height profile. Therefore, an accurate evaluation of the GNR width by AFM requires corrections in the observed topographic images that take the tip shape into account. Figure 14

Fig. 14 Schematic drawing for the evaluation of the GNR width from the height profile obtained by AFM measurements. (Reprinted with the permission from R. Negishi et al. [8], Copyright 2017 by AIP Publishing)

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shows a schematic drawing of the evaluation of the GNR width from a height profile obtained by AFM measurements. The radius of the curvature of the ATM tip strongly affects the observed value of the width because GNR is very small. The actual width (W) can be evaluated by the following formula  W = W  − 2D = W  − 2 (r + a)2 − (r + a − h)2

(9)

where W’ is the width obtained from the height profile in the AFM measurement, r is the curvature radius of the AFM tip, a is the distance between the tip and sample, and h is the sample height. 2D is the difference in the width between W and W’. To simplify the formula, we compensated for the width of GNR obtained from the height profiles, assuming that the tip shape is a sphere with a 7-nm curvature radius and that the tip is in direct contact with the surfaces of the sample and the substrate (a = 0). The distribution of the GNR widths can be fitted by a Gaussian function, and the 95% confidence intervals for the distribution before and after growth are 8.81–10.06 nm and 12.73–13.78 nm, respectively. The width distribution obviously changes from the pristine to the grown GNR. The difference in the width (W~3.2 nm) indicates that the graphene layers grow along the lateral direction. Figure 15 shows the transfer characteristics (conductance vs gate voltage) in the FETs using a single channel of pristine GNR and grown multilayer GNR with 8 layers measured at 10 K and 300 K [66]. The conductance (σ) is defined as the inverse of the sheet resistance (Rs ) as σ =

1 = RS



Isd Vsd



L ch Wch

 (10)

At a low temperature, the minimum conductance (σmin ) as the OFF state region reaches the detection limit of current (~10 pA) in our source-measure apparatus, and semiconducting characteristics are observed. The strong suppression of the conductance in the OFF state region is due to the formation of transport gaps associated with the disordered structures [67]. For measurements at room temperature, the transfer characteristics for the grown multilayer GNR slightly depend on the temperature, resulting in the degradation of the ION /IOFF ratio. This means that the suppression effect in the grown multilayer GNR is significantly reduced with increasing measurement temperature due to the thermal excitation. Figure 16 shows the change in the sheet resistance as a function of the number of layers [66]. The sheet resistance is evaluated from the maximum conductance (σmax ) value in the ON state region. Note that the sheet resistance decreases as the number of layers increases. We consider a simple resistance model in which the plane direction resistance of the layers is connected in parallel between the source and drain electrodes, assuming that the conductance between graphene layers connected by van der Waals force can be ignored. This situation is similar to that of multilayer graphene described in Sect. 2, except the width is very narrow. The total sheet resistance (Rtotal ) of the multilayer GNR composed of n layers is given by

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Fig. 15 Transfer characteristics in the a,·b monolayer and c,·d grown multilayer GNR-FETs measured at 10 and 300 K, respectively. (Reprinted from R. Negishi et al. [66]. Copyright 2021 by Springer Nature) Fig. 16 Sheet resistance measured in the ON state region vs the number of layers in the GNR-FETs. The solid line indicates the sheet resistance calculated by a simple resistance model. (Reprinted from R. Negishi et al. [66]. Copyright 2021 by Springer Nature)

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Fig. 17 Self-consistent potential profiles at a the bottom layer and b the top layer in the multilayer GNR composed of eight layers on the SiO2 substrate when the density of the charged impurity is 2 × 1011 cm−2 . (Reprinted from R. Negishi et al. [66]. Copyright 2021 by Springer Nature)

1 Rtotal

=

1 1 1 1 + + ··· + + R1 R2 Rn−1 Rn

(11)

where Rn and R1 are the sheet resistance of monolayer GNR in the n-th layer and pristine monolayer GNR used as a growth template. Assuming that the sheet resistance of each layer in the grown GNR is equal to that of the pristine monolayer GNR as a growth template, the solid line in Fig. 16 shows the calculated sheet resistance as a function of the number of layers. However, the observed resistance deviates from the calculated curve and shows a remarkably lower resistance, as indicated by the arrows. A similar behavior with a drastically improved conductance as the number of layers increases has also been observed in turbostratic multilayer graphene, as shown in Fig. 10b. The improvement is explained by reducing the carrier scattering due to the screening effect of the stacked multilayer from the charged impurities on the SiO2 /Si substrate. Figures 17a and b show the results of Monte Carlo simulation for the selfconsistent potential profiles of the bottom layer and the top layer in the multilayer graphene with 8 layers on the SiO2 substrate when the density of charged impurity is 2 × 1011 cm−2 , respectively [66]. The potential profile is calculated within the Thomas–Fermi approximation. In the bottom layer attached on the substrate, the surface potential is strongly modulated by the charged impurities, indicating that the electron–hole puddles, which are localized states near the charge neutrality point, are formed by the potential barrier of the charged impurities [68]. In this case, the carrier transport property is dominated by two-dimensional variable range hopping (2D-VRH) conduction via the localized states. On the other hand, the modulation of the surface potential due to the charged impurities is dramatically reduced in the top layer because the charges on the substrate are screened by the interlayers between the top and bottom layers in the multilayer GNR. Since the electrical band structure in the top layer of the multilayer GNR is not modulated by charged impurities, carrier transport is dominated by thermally activated (TA) conduction via a continuous band

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structure without any localized states as indicated by Fig. 16. The same screening effects are also observed in multilayer graphene, as described in Sect. 2 [27, 29]. Hence, it is reasonable that the improved conductance in the turbostratic multilayer GNR is caused by the efficient screening effect.

4 Multilayer Graphene Synthesized from Graphene Oxide Materials Graphene oxide (GO), which is produced by chemical oxidation of graphite, has attracted great interest in large-area synthesis of graphene due to its cost-effectiveness and mass production. Since GO has rich oxygen-containing groups, many studies have explored reduction processes such as thermal [69–72] and chemical reductions [73–75]. The most straight-forward goal of the reduction process is to produce graphene-like materials that resemble highly crystalline graphene prepared by mechanical exfoliation of the graphite crystal. Although these processes can efficiently remove oxygen-containing groups, it is difficult to repair defects such as vacancies and an amorphous-like π network composed of a mixture of sp2 and sp3 carbons in the GO produced by the oxidation of graphite. An ultrahigh temperature process under a reactive ethanol environment efficiently promotes the reduction and restoration of the graphitic structure in a defective graphene oxide [24]. Figures 18a–c and e show the Raman spectra obtained from the GO samples reduced by the thermal annealing under a reactive ethanol environment. Figure 18d shows the Raman spectrum obtained from the GO sample prepared by thermal annealing under an inert gas (N2 ) environment. The GO samples after the thermal processes produce distinct 2D bands in their Raman spectra as shown in Fig. 18 a–e, while a 2D band is not observed from the pristine GO. This result indicates that thermal annealing at high temperature is very effective for the reduction and restoration of graphitic structures in the defective GO. The crystallinity of the processed GO can be evaluated from the intensity ratio of the D and G-bands (I(D)/I(G)), and the 2D and G-bands (I(2D)/I(G)) in the Raman spectra. The GO processed in ethanol at 1800 ºC exhibits excellent features of the D and 2D-bands in the Raman spectrum, and its crystallinity is comparable to that of Cu-CVD grown graphene. In addition to the ultrahigh temperature, the ethanol environment plays an important role in this marked restoration process. Similar I(D)/I(G) ratios are observed in Fig. 18b and d, indicating that a comparable crystallinity is achieved under an ethanol environment at a much lower temperature. A comparison of Fig. 18a and d indicates that the process in an ethanol environment provides much lower I(D)/I(G) ratios than that in an N2 environment for a given temperature. In the defect healing process, it should be noted that carbon species decomposed from ethanol are provided to defect sites, resulting in a detectable weight increase of GO. Since the amount of the ethanol supplied by the thermal process is extremely small, the deposition of an additional carbon layer described in Sects. 2 and 3 should be negligible

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Fig. 18 Raman spectra observed from the multilayer reduced GO prepared by thermal annealing under various conditions and I(D)/I(G), I(2D)/I(G), and FWHM of the 2D band for the evaluation of the crystallinity and stacking structure. a–c Processed for 10 min in an ethanol environment using a solar furnace. Process temperatures are indicated. d Processed for 10 min in a N2 environment at 1700 °C using the solar furnace. e Processed for 120 min in an ethanol environment at 1100 °C using a conventional CVD apparatus. f Pristine GO without any thermal process for comparison. Raman spectra were acquired using 532 nm laser excitation. (Reprinted with permission from T. Ishida et al. [24]. Copyright 2016 by The Japan Society of Applied Physics)

for the process conditions in the reduction and restoration of the graphitic structure in the defective GO. A GO thin film on a substrate is prepared by spin coating of an aqueous GO dispersion to a substrate. Figure 19 shows optical microscope images of a GO thin film prepared by spin-coating of (a) 1 time and (c) 7 times and subsequent thermal treatment (1300 ºC) in an ethanol environment. The white and black regions mean the coated GO thin film and the exposed substrate. When spin coating is only applied once, the black areas are observed in some places, indicating that the graphene oxide

(a)

(b)

10 µm

10 µm

Fig. 19 Optical images of the reduced GO thin films prepared with different spin-coating times: a 1 time and b 7 times

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film is a discontinuous inhomogeneous film. On the other hand, when the spin coating process is increased, a continuous GO thin film is successfully obtained as shown in Fig. 19b. Figure 20 shows the Raman spectra obtained from the Cu-CVD monolayer graphene and the reduced graphene oxide (rGO) film prepared by different spincoating times. The spectra are normalized by the peak intensity (I sub. (obs.)) from the fused quartz substrate (~460 cm−1 ) as indicated by the arrow in Fig. 20. The number of layers in graphene is determined by the intensity ratio (I G (obs.)/I sub. (obs.)) of the peaks observed at the G-band region (1580 cm−1 ) and the substrate (460 cm−1 ) in the Raman spectra, assuming that the laser power decreases by 2.3% through the monolayer graphene layer due to the self-absorption effect [76]. As described in Sect. 1, the peak-shape analysis of the 2D-band in the Raman spectra of graphene and reduced GO provide information about the stacking structure in the multilayer graphene. As previously mentioned, Fig. 3 shows typical results of the peak-fitting analysis of the 2D bands observed from multilayer reduced GO thin films. Table 1 summarizes the analyzed results [24]. For an inert N2 environment, the volume ratio of 3D graphite rapidly increases from 30 to 60% as the process temperature rises from 1500 to 1700 °C. However, the situation completely differs for the reactive ethanol environment. Even after the thermal process at 1800 °C, R maintains a low value similar to that for 1500 °C. This result means that turbostratic stacking structures are formed in ethanol environments by the ultrahigh-temperature process of single-layer GO aggregates with random orientations, suggesting that ethanol suppresses graphitization of the processed graphene even at ultrahigh temperatures. Fig. 20 Raman spectra observed from the pristine GO and the reduced GO prepared by thermal annealing at 1300 ºC under an ethanol environment

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Table 1 Summary of the 2D-band peak-fitting analysis. (Reprinted with permission from T. Ishida et al. [24]. (Copyright 2016 by The Japan Society of Applied Physics) Environment

Temperature (°C)

I(D)/I(G)

I(2D)/I(G)

R(%)

N2

1500

0.6

0.3

30

N2

1700

0.3

0.3

60

Ethanol

1500

0.3

0.6

10

Ethanol

1800

0.1

0.7

20

Fig. 21 Carrier mobilities in multilayer reduced graphene devices as a function of the number of layers. (Reprinted with permission from R. Negishi et al. [77]. Copyright 2016 by Springer Nature)

The formation of a turbostratic structure at an ultrahigh temperature, which is in striking contrast to graphene processed in N2 environments without ethanol, is an anomalous phenomenon. Figure 21 shows the carrier mobilities of rGO films prepared by thermal treatment in reactive ethanol and inert H2 /Ar gas environments (hereafter, H2 /Ar treatment), which are indicated by red circles and green diamonds, respectively [77]. The carrier mobilities of the rGO films prepared by ethanol treatment as a function of the process temperatures are evaluated from the Hall-effect measurements using the van der Pauw method. The carrier mobilities of the rGO films prepared by H2 /Ar treatment are evaluated from the source-drain current (I sd ) as a function of the gate voltage (V g ) in FETs using a standard formula [78]. The carrier mobilities improve as the process temperature increases in each treatment. Note that the carrier (hole) mobilities in the rGO films prepared by ethanol treatment show significantly higher values than in the rGO films prepared by H2 /Ar treatment. The highest observed mobility reaches ~210 cm2 /Vs at room temperature (~270 cm2 /Vs at 77 K), which is higher than the carrier mobilities (0.06–95 cm2 /Vs) of the rGO films prepared by other reduction methods [73, 79–83]. Hence, the ethanol treatment at a high process temperature effectively improves the carrier mobility of the rGO films.

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We measured the temperature dependence of the conductivity to identify the carrier transport mechanism. In the 2D-VRH model, the temperature dependence of the conductivity (σ (T )) has the following form [84]  σV R H (T ) ∝ exp

B



T 1/3

(12)

Hopping parameter B is expressed as  B=

3 k B N (E F )L 2

1/3 (13)

where k B is the Boltzmann constant, L is the localization length of the electronic wave function for the conjugated π-electron system in the rGO, and N(E F ) is the density of states (DOS) near the Fermi level. Figure 22 plots ln(σ ) as a function of T −1/3 . The ln(σ ) measured in the rGO films prepared by ethanol treatment at 900 ºC is well-fitted by the 2D-VRH conduction (Fig. 22a). On the other hand, Eq. (12) does not provide well-fitted curves for the ln(σ ) measured for the rGO films prepared by ethanol treatment at a process temperature above 1000 ºC. The contribution of thermally activated (TA) conduction reflects the fact that phonon scattering [85] is observed in the rGO films. The characteristics of ln(σ ) measured from the rGO films prepared by ethanol treatment at a process temperature above 1000 °C (Fig. 22b and c) can be fitted by the sum of the 2D-VRH and TA conductions (green solid line), which is expressed by.

Fig. 22 Analysis of the conductance in rGO films. ln(σ ) vs. T −1/3 are observed from rGO films prepared by ethanol treatment at a 900, b 1000, and c 1130 °C. Experimental data are fitted by 2DVRH (blue dashed lines) or the sum of TA and 2D-VRH (green solid line) conductions. (Reprinted with permission from R. Negishi et al. [77]. Copyright 2016 by Springer Nature)

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  Ea σT A (T ) ∝ exp − kB T

(14)

The carrier transport properties that show a combination of the 2D-VRH and TA conductions are commonly observed in disordered systems, such as amorphous semiconductors in bulk materials [86]. This is the first observation of carrier transport properties explained by the sum of the 2D-VRH and TA conductions in graphene thin-films synthesized from GO as single-atomic-layer materials. The contribution of TA conduction in the carrier transport mechanism suggests that band-like transport occurs by forming continuous conduction/valence bands. Therefore, our findings indicate that our reduction process is a key technique for the synthesis of largearea graphene thin films showing the intrinsic electrical properties of graphene from defective GO materials. From the fitting analysis, we can estimate L and thermal activation energy (E a ) as shown in Table 2. E a is defined as the energy required for exciting carriers into non-localized states beyond the mobility edges [86]. A higher process temperature in ethanol treatment increases L. This trend agrees with the results in our previous report where the process temperature was limited to 950 °C [87]. Note that the values of L in the rGO films increase extraordinarily by an ethanol treatment at a process temperature above 1000 °C. E a decreases with an increasing process temperature from 1000 to 1130 °C. Since an increasing L denotes the expansion of the conjugated π-electron system in the rGO, it is expected that the origin of E a is the electrical structures with non-localized states due to the formation of large sp2 domains composed of the conjugated π-electron system. A process temperature below approximately 950 °C in an ethanol environment is insufficient for restoration of the graphitic structure in rGO, although the oxygencontaining functional groups are efficiently removed under this condition [88–90]. Figure 23 shows STEM images obtained from the suspended monolayer rGO sheet prepared by ethanol treatment at (a) 900 ºC and (b) 1100 ºC. The yellow circle and green arrows indicate the vacancy and domain boundaries. The TEM image in Fig. 23a indicates that many defects such as an amorphous-like π network composed of a mixture of sp2 and sp3 carbon structures are observed in the whole region. On the other hand, Fig. 23b shows the periodic spots, indicating an improved graphene Table 2 E a and L evaluated from the fitting analysis of the conductivity. (Reprinted with permission from R. Negishi et al. [77]. Copyright 2016 by Springer Nature) H2 /Ar treatment at 1130°C

Ethanol treatment at 900°C

1000°C

1130°C

E a /meV

–

–

84.6

16.3 ± 4.0

L/ nm

4.3 ± 1.0

6.3 ± 1.3

30

184.0 ± 39.4

Transport mechanism

VRH

VRH

TA + VRH

TA + VRH

Number of data

3

5

1

4

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175

(a)

4 nm

1 nm

4 nm

1 nm

(b)

Fig. 23 Transmission electron microscopy (TEM) images obtained from the suspended monolayer rGO sheet prepared by ethanol treatment at a 900 ºC and b 1100 ºC. (Reprinted with permission from R. Negishi et al. [77]. Copyright 2016 by Springer Nature)

crystallinity. Considering that the contribution of TA conduction with E a is observed in rGO films prepared by ethanol treatment at a process temperature above 1000 °C (Fig. 23), the origin of TA conduction depends on the change in the band structures near the Fermi level with structural restoration of the graphitic structure in GO. Ethanol vapor treatment at a high process temperature above 1100 ºC improves the crystallinity of GO materials due to the efficient structural restoration. Raman spectra in the 2D band region observed from the reduced multilayer graphene reveal that the stacking structure in the system is formed by turbostratic stacking due to weak interlayer coupling between layers. Moreover, we observed improved carrier transport properties due to the screening effect [44, 66].

5 Conclusion and Future Perspectives In this chapter, we introduced anomalous carrier transport properties of various turbostratic multilayer graphene materials in FETs. Graphene with a single atomic layer thickness has a very low carrier density at the Dirac point. Hence, the material

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is extremely sensitive to the environment such as the substrate. This is both an advantage and a disadvantage of graphene. On the other hand, twisted few-layer graphene has rotational degrees of freedom between the layers, and exhibits a variety of carrier transport behaviors such as insulation and superconductivity depending on the rotation angle. Hence, few-layer graphene controlling the rotational degrees should realize a new platform for atomic layer physics and quantum device applications. In particular, multilayer graphene controlled by turbostratic stacking shows completely different carrier transport properties than graphitized AB stacking because a linear dispersion is maintained in each layer. The screening effect due to the multi-stacking enables us to reduce the carrier scattering caused by charge impurities present on the substrate. This should pave the way for practical quantum device applications utilizing the intrinsic properties of graphene. The behavior of carrier conduction at low-temperature in strong magnetic fields will provide more detailed information about the carrier transport mechanism, such as the elastic and inelastic scattering length and should be explored in future works. The preliminary result is exhibited in Fig. 24. The magnetic field dependence of the carrier transport behavior is examined for turbostratic multilayer graphene prepared by the solid growth template method described in Sect. 2 of this chapter. The negative magnetoresistance due to the weak localization state is observed near 0 T. The elastic scattering length (le ) and phase relaxation length (lp ) can be evaluated from the negative magnetoresistance observed in a Hall-bar device using fitting analysis based on random potential scattering theory. le and lp increase as the number of layers increases up to three layers. When the number of layers is further increased, le hardly changes, but the lp decreases significantly. One dominant factor for lp is phonon scattering. Lattice defects tend to increases as the number of layers increases as shown in Fig. 8. le strongly depends on the remote Coulomb scattering due to the Fig. 24 Magnetic field dependence of the resistance of the grown multilayer graphene using the solid template method

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charge impurities fixed on the substrate. As shown in the surface potential simulation in Sect. 3, the surface potential is strongly modulated by the charged impurities and the screening effect because interlayer graphene dramatically reduces remote Coulomb scattering on the substrate. This indicates that Ie is efficiently improved by the screening effect in multilayer graphene. Unfortunately, we have not been able to observe the half-integer quantum Hall effect, which is one of the most important features of a linear dispersion in monolayer graphene. The grain size of the grown multilayer graphene evaluated from the D-band in the Raman spectra is about 60 nm, and further improvement of the crystallinity by optimizing the growth conditions is necessary to observe the half-integer quantum Hall effect.

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Quantum Anomalous Hall Effect in Magnetic Topological Insulator Minoru Kawamura

Abstract The quantum anomalous Hall effect (QAHE) is one of the hallmark phenomena associated with topological properties of the electronic band structure. When an electric current flows in a ferromagnetic topological insulator, the Hall resistance orthogonal to both current and magnetization is quantized to the von Klitzing constant h/e2 , where h is the Planck’s constant and e is the elementary charge. The QAHE shares many phenomenological features with the quantum Hall effect in two-dimensional electron systems under strong magnetic fields despite the different microscopic origins of the phenomena. In this chapter, we review the basic concept and emerging properties of the QAHE. In particular, we discuss the transport properties in the engineered magnetic heterostructure films of topological insulators which have provided a unique platform to study emergent topological phenomena.

1 Introduction The band theory in solids has successfully explained physical properties of materials, such as electric transport properties and optical properties [1]. Recent studies in condensed-matter physics have unveiled importance of topological aspects of the band structures to the physical properties. Exploring quantum phenomena arising from topological properties of the electronic band structures is one of the central research topics in contemporary condensed-matter physics [2–4]. The topological properties of the band structure in a momentum-space are usually characterized by integer topological invariant numbers, such as the Chern number or the Z 2 Chern number [2–4]. Because the integer topological invariant numbers cannot be changed by continuous deformation, electronic properties associated with the topological invariant numbers attain robustness against external perturbations or disorder in the materials.

M. Kawamura (B) RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako 351-0198, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_9

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The integer quantum Hall effect (QHE) [5] is one of the examples of the topologically protected electronic phenomenon where relevance of the band topology was first recognized. The QHE can be observed in high-mobility two-dimensional electron systems (2DESs) formed at the semiconductor interfaces under strong perpendicular magnetic fields. When an electric current (I ) and a magnetic field (B) perpendicular to a 2DES are applied, a voltage transverse to both I and B is produced, which is known as a Hall voltage (Vy ) [6]. The Hall resistance (R yx = Vy /I ) increases linearly with the magnetic field when the magnetic field is weak. However, as the magnetic field is further increased, R yx turns to increase stepwise accompanied by Hall resistance plateaus. The values of the plateaus are quantized to the von Klitzing constant h/e2 divided by integer numbers N . Here, h is the Planck’s constant and e is the elementary charge. The quantization of R yx is accompanied by vanishing of the longitudinal resistance. The QHE is originated from the formation of discrete Landau levels generated by cyclotron motion of electrons under strong perpendicular magnetic fields. As the magnetic field is increased, the density of states in each Landau level is increased. Consequently, the Fermi energy crosses the Landau levels, resulting in stepwise changes in R yx . Theoretical studies by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) [7, 8] showed that the Hall conductivity σx y measured in the unit of e2 / h is equivalent to the summation of the Berry curvature [9] in the momentum space over the occupied states. In the case of QHE, the summation of the Berry curvature becomes an integer number N which is equal to the number of the occupied Landau levels. This integer N is a topological invariant number called the Chern number, which cannot be changed by the continuous deformation of the system. The TKNN theory relates the topological invariant number to the experimentally measurable transport coefficient, Hall conductivity, by σx y = N e2 / h. The topological robustness assures the precise quantization of R yx to the values of h/e2 /N . The quantum anomalous Hall effect (QAHE) is another transport phenomenon associated with the topological property of the band structure. It was first proposed theoretically [10–15] and later confirmed experimentally [16]. The QAHE possesses many common features with the QHE but the QAHE can be observed in magnetic materials. Therefore, the QAHE can be regarded as a quantized version of the anomalous Hall effect (AHE) of magnetic materials. The AHE was first discovered by Hall in 1881 using nickel and cobalt foils [17]. The magnitude of the Hall voltage is usually more than ten times larger than that of non-magnetic metal foils. Despite the clear and distinguish phenomenon, its origin has been an issue of a long debate [18]. One of the mechanisms for the AHE is an intrinsic mechanism which originates from the spin-orbit interaction in the spin-polarized electronic bands [19]. It was later recognized that the topological property of the band structure, namely the Berry curvature, plays a central role in the intrinsic mechanism [20–22]. In metallic ferromagnets with moderate conductivity, the intrinsic mechanism tends to dominate the AHE, while the other mechanisms, such as skew scatterings and side jumps, are relevant to highly conductive ferromagnets [23]. The Berry connection An (k) and the Berry curvature n (k) are respectively defined as

Quantum Anomalous Hall Effect in Magnetic Topological …

     ∂  An (k) = i u nk   u nk ∂k n (k) = ∇k × An (k),

183

(1) (2)

where |u nk  represents the periodic part of the Bloch wave function with energy εn (k), n the band index, and k the wave vector. Using n (k), semiclassical equations of motion for the center of a wave packet can be written as [18, 24] 1 ∂εn (k) ˙ + k × n (k)  ∂k e k˙ = − E,  x˙ =

(3) (4)

where E is the externally applied electric field. The second term in Eq. (3), known as the anomalous velocity, becomes nonzero when the inter-band matrix elements of the current operator are relevant. Because the second term is orthogonal to the electric field E, it can produce the Hall voltage. According to the TKNN theory [7], the Hall conductivity σx y can be described by using nz (k) as  f (εn (k))nz (k), (5) σx y = e2 nk

where f (ε) is the Fermi distribution function. When both time-reversal symmetry and inversion symmetry are preserved, the Berry curvature vanishes at all the points in the Brillouin zone. In a system where either time-reversal symmetry or inversion symmetry is broken, nz can be nonzero. In a ferromagnetic material, the time-reversal symmetry is broken, and/therefore the Berry curvature is enhanced near the avoided crossing points due to the spin-orbit coupling [25, 26]. Because of the enhancement, the AHE becomes large as the Fermi energy E F approaches to the crossing points by tuning the carrier density. Therefore, the QAHE, which is an extreme case of the AHE, had been expected to be observed in ferromagnetic insulators [27]. However, despite intensive research for the QAHE in ferromagnetic insulator materials, the quantized AHE had not been discovered. To realize the quantized AHE, the emergence of topological insulators (TIs) had to wait. This chapter is organized as follows. We first review the topological insulators mainly focusing on the experimental aspects of the studies. There are several excellent review papers on TIs [2–4] and magnetic TIs (MTIs) [28–31]. Therefore, in this chapter, we pick up some topics related to the QAHE. Next, we review the experimental observation of the QAHE in magnetically-doped TIs and discuss its robustness, highlighting the development of engineered magnetic heterostructure films of MTIs. Then, we discuss some topologically distinct phases on the surface of MTIs. These include an insulating phase called an axion insulator which is theoretically predicted to host the topological magnetoelectric effect. We also discuss the topological quantum phase transition between the QAH insulator and the trivial insulator originating

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from the hybridization of the surface states. Finally, future outlooks are discussed, introducing recent observation of the QAHE of the magnetic proximity effect origin and the QAHEs in various materials platforms.

2 Topological Insulator Topological insulator is a new kind of insulator with an inverted band structure due to strong spin-orbit interaction [2–4, 32–34]. In a TI, a gapless metallic state is formed at the edge or surface of a sample while its bulk part is insulating with an energy gap. Because of the energy gap, electric current cannot flow through the bulk part but can flow through the edge or surface state where the energy gap is closed. A TI can be distinguished from an ordinary insulator by a topological invariant number of the band structure called the Z 2 Chern number, which takes values of either 0 or 1. The edge or surface states are protected by the time-reversal symmetry, meaning that they exist robustly so far as the time-reversal symmetry is preserved. The concept of TI was first introduced to 2D systems [32, 33] and later extended to three-dimensional (3D) systems [34]. Hereafter, we consider the 3D TIs. The 2D surface state of a 3D TI in the low-energy regime is described by a relativistic Dirac electron model with a linear dispersion relation.   H = v p y σx − px σ y

(6)

where v is the velocity, p = ( px , p y ) is the momentum vector, and σ = (σx , σ y , σz ) are the Pauli matrices. The electronic band structure of the Hamiltonian forms a conical dispersion relation as shown in Fig. 1a with a band touching point at E = 0 called the Dirac point. Importantly, the direction of a spin eigenstate is always orthogonal to the direction of momentum as depicted by red arrows in Fig. 1a, which is known as the spin-momentum locking. A theoretical study based on the numerical band calculation has identified more than 3000 TIs among about 27,000 stoichiometric materials [35]. Some of them are synthesized and are experimentally identified as 3D TIs using spectroscopic methods, such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy/spectroscopy (STM/STS). Among the currently discovered 3D TIs, Bi2 Se3 , Bi2 Te3 , and Sb2 Te3 with tetradymite structure are the most well-studied group of materials [36–40]. These TIs have a rather simple surface state described by a single Dirac cone at the  point of the Brillouin zone on each surface. These materials have a layered crystal structure where a unit layer contains Se(Te)-Bi(Sb)-Se(Te)Bi(Sb)-Se(Te) stacking sequence along the c axis and the unit layers (1 quintuple layer (1QL)) are coupled to each other by the van der Waals force. Theoretically, these compounds were predicted to be 3D TIs with a non-zero Z 2 Chern number based on the first-principle band calculation [36]. In the case of Bi2 Se3 , the 4 p orbital of Se and the 6 p orbital of Bi at the  point are inverted due to the strong spin-orbit

Quantum Anomalous Hall Effect in Magnetic Topological … (a)

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(b)

E ky kx (c) z

(d) e

y

e

e e

x

e

e

e e

Fig. 1 a, b Schematics of dispersion relations on the surface of a 3D topological insulator without (a) and with (b) the mass term mσz . The band touching point in (a) is the Dirac point. An energy gap opens at the Dirac point when the mass term mσz is introduced as shown in (b). The directions of the spin eigenstates are shown by red arrows. c, d Real space image of the topological surface state without (c) and with (d) the mass term. Magnetic moments to produce the mass term is depicted by green arrows in (d). The surface states are gapless at all the surfaces in (c). The surface states perpendicular to the magnetic moments are gapped in (d) while the side surfaces remain gapless

interaction. The linear dispersion relation of the surface Dirac cone [37–40] and its spin texture in the momentum space [40] were confirmed by the surface-sensitive spin-resolved ARPES. The linear-dispersion relation of the surface Dirac cone was also demonstrated by the Landau level spectroscopy using STM/STS under magnetic fields [41, 42]. When bulk crystals of Bi2 Te3 and Sb2 Te3 are synthesized, the crystals tend to become p-type semiconductors. To observe intriguing effects arising from the topological surface state using transport measurement, E F has to be tuned in the bulk band gap. Considerable experimental efforts have been paid for the Fermi energy tuning [43–45]. Thin-film growth is one of the promising ways to achieve a bulk insulating topological insulator with E F in the bulk band gap. The E F tuning to the bulk band gap was accomplished in the (Bi,Sb)2 Te3 (BST) thin films grown by molecular beam epitaxy (MBE) by carefully tuning the Bi/Sb beam flux ratio [46]. Magnetic topological insulator When magnetic moments are introduced to a TI, a band gap can open at the band touching point in the dispersion relation of the surface band as shown in Fig. 1b. Formation of the energy gap is essential to the emergence of the QAHE [10–15]. The energy gap is induced by the exchange interaction between the itinerant electron

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spins and the local magnetic moments. The exchange interaction is described by J n S S¯z σz , where J is the exchange coupling constant, n S the density of the local spins with averaged spin S¯z normal to the surface, and σz the z component of electron spin. By introducing m ≡ J n S S¯z , the model Hamiltonian can be described as   H = v p y σx − px σ y + mσz .

(7)

As a consequence of the mass term mσz , the time-reversal symmetry is broken and the dispersion relation is changed to  E(k x , k y ) = ± 2 v 2 (k x2 + k 2y ) + m 2 ,

(8)

which has an energy gap in the range −m < E < m. The opening of the exchange gap is accompanied by modification in spin orientations as shown in Fig. 1b. At the top (bottom) of the valence (conduction) band, spin is down (up) for m > 0. As the energy is shifted away from the band edge, the spins tend to align in the k x -k y plane, making a hedgehog-like spin texture in the momentum space. In the presence of the mass term mσz , the z component of the Berry curvature ±,z (k) can be written as [22] ±,z (k x , k y ) = ∓

±2 v 2 m , 2[2 v 2 (k x2 + k 2y ) + m 2 ]3/2

(9)

where ± denotes the upper (+) and the lower (-) bands. |±,z (k)| becomes large as E approaches ±m, resulting in a large anomalous velocity. Using the TKNN formula, the Hall conductivity σx y at T = 0 is calculated as

σx y =

⎧ 2 e m ⎪ ⎪ (E F < −m, m < E F ) ⎪ ⎨ 2h  2 2 2  v k + m2 ⎪ ⎪ e2 m ⎪ ⎩ 2h |m|

F

(10)

(−m < E F < m)

where kF = E F /v is the Fermi wave number. The Hall conductivity is quantized to σx y = ±e2 /2h for a single Dirac cone when E F resides in the exchange gap. This is the quantum anomalous Hall effect which has been theoretically proposed for TIs with broken time-reversal symmetry [10–15]. The sign of σx y depends on the sign of m but is independent of the sign of the carrier charge. The formation of the exchange gap and the E F tuning to the energy gap are the two important requisites for the QAHE. Because the QAHE is induced by the exchange gap, the QAHE can occur even at zero magnetic field in ferromagnetic materials with a sizable remanent magnetization. The zero-magnetic-field realization of the quantized Hall conductivity is one of the most remarkable features of the QAHE. The idea of zero-magnetic-field quantization of the Hall resistance itself was originally discussed by Haldane [47]

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using a different theoretical model. The fact that the quantized Hall conductivity does not vanish in the limit of m → 0 is referred to as the parity anomaly [48]. Because two equivalent surfaces perpendicular to the magnetization are simultaneously measured in a standard experimental setup, the experimentally measured σx y are doubled to ±e2 / h, as we see below. The coexistence of two equivalent surfaces can be viewed as a manifestation of Nielsen-Ninomiya’s fermion doubling theorem in lattice systems [49]. To induce a spontaneous magnetization in a TI, chemical doping with 3d transition metal elements is one of the effective approaches [14, 50–53]. Two mechanisms are discussed as a possible origin of the ferromagnetic order in a magneticallydoped topological insulator. These are Ruderman-Kittel-Kasuya-Yoshida (RKKY) mechanism [54] and the local valence-electron mediated van Vleck mechanism or the Rowland-Bloembergen (RB) mechanism [55]. The RKKY mechanism plays a dominant role in the ferromagnetic order in Mn-doped (Bi, Sb)2 Te3 [56]. As demonstrated experimentally, the ferromagnetic order temperature TC is rather sensitive to the carrier density in this system. The other type of ferromagnetic order is realized in Cr-doped (Bi, Sb)2 Te3 . When carrier density and carrier type are changed by tuning the Bi/Sb ratio x in Cr0.22 (Bix Sb1−x )1.78 Te3 , the coercive field does not change largely with x, and TC is not so sensitive to x, either. These results indicate relevance of the RB mechanism to the ferromagnetic order in Cr-doped (Bi, Sb)2 Te3 [57]. The formation of the exchange gap in the magnetically-doped topological insulator was directly verified by ARPES in Fe-doped Bi2 Se3 [58, 59] and Mn-doped Bi2 Se3 [60]. The size of the exchange gap is reported as about 30 meV in Mn-doped Bi2 Se3 [60]. The exchange gap on the cleaved surface of bulk Cr0.08 (Bi0.1 Sb0.9 )1.92 Te3 (TC = 18 K) was measured by STM/STS [61]. They successfully measured the spatial distribution of the exchange gap. Despite the large magnetization gap 30 meV on average, the exchange gap distributes between 9 and 48 meV. The spatial mapping of the exchange gap showed that the gap size strongly correlates with the randomly distributed Cr density.

3 Experimental Observation of Quantum Anomalous Hall Effect After the theoretical proposal [14], experimental searches for the QAHE were pursued using magnetically-doped topological insulators. The first experimental demonstration of the QAHE was conducted using thin films of Cr0.15 (Bi0.1 Sb0.9 )1.85 Te3 with a nominal thickness of 5 QL (∼ 5 nm) grown on SrTiO3 (111) substrates by MBE [16]. The sample film was cooled down to 30 mK using a dilution refrigerator for the transport measurements. When E F was changed by applying a gate voltage Vg to the SrTiO3 substrate, the Hall resistance R yx became h/e2 in a certain range of Vg near the charge neutrality point as show in Fig. 2c. The value of R yx decreased as the gate voltage was detuned from the charge neutrality point. The Hall resis-

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Fig. 2 a, b Magnetic field dependence of Hall resistivity ρ yx (a) and longitudinal resistivity ρx x (b) under several gate voltages Vg . ρ yx reaches the von Klitzing constant h/e2 when Vg = −1.5 V c Gate voltage dependence of ρx x (red) and ρ yx (blue) at B = 0 T. d Gate voltage dependence of σx x 2 ) (red) and σx y (blue) calculated from ρx x and ρ yx using the tensor relation σx x = ρx x /(ρx2x + ρ yx 2 2 and σx y = ρ yx /(ρx x + ρ yx ). Measurement temperature was 30 mK. From Ref. [16]. Reprinted with permission from AAAS

tance switched between +h/e2 and −h/e2 when the magnetization direction was reversed at the coercive field Hc ∼ ±0.15 T (Fig. 2a). The quantized Hall resistance survived even at the zero magnetic field due to the remanent magnetization. The zero-magnetic-field quantization of the Hall resistance makes a sharp contrast to the QHE of the cyclotron motion origin which requires high magnetic fields. As E F crossed the charge neutrality point by varying Vg , the sign of the anomalous Hall resistance did not change. This again contrasts to the QHE where the sign of the Hall resistance is switched by the change in the carrier type. Despite the successful observation of the quantized Hall resistance, the zero longitudinal resistance was not achieved in this experiment. The lowest value of Rx x was around 2 k (Fig. 2b). Several scenarios have been discussed to explain the residual longitudinal resistance, including the residual bulk conductivity and presence of helical edge states [62]. After this pioneering work, the experimental observations of the QAHE were reported from several groups using Cr-doped (Bi, Sb)2 Te3 (CBST) films grown on GaAs substrates [63] and InP substrates [64]. The QAHE was also demonstrated in thin films of V-doped (Bi, Sb)2 Te3 (VBST) [65, 66]. Compared to CBST, VBST has a

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large Hc of about 1 T. The quantized R yx was robustly observed in macroscopic-scale samples ranging from 200 μm to 2 mm [63], demonstrating scale-invariant quantization of the QAHE. This contrasts to the quantum spin Hall effect in HgTe/MgHgTe heterostructure which requires μm-scale samples [67]. The macroscopic scale of the sample size indicates relevance of the chiral edge channels to the QAHE as similar to the QHE. The similarity between the QAHE and the QHE is also seen in the quantum criticality and delocalization behavior of the QAHE [64, 66]. As temperature is lowered, the conductivity tensor components (σx y , σx x ) converge either (e2 / h, 0) or (0, 0) depending on magnetic field and gate voltage. The temperature-dependent trajectory is similar to the theoretically calculated renormalization group flow [68]. The agreement indicates that the phase transition between the QAHE phase and the adjacent phase belongs to the same category as the plateau-plateau transitions in the QHE [68–70]. Edge channel picture The quantized Hall resistance of the QHE is often explained using the edge channel picture [71–73]. In the case of the QHE in a 2D electron system, each Landau level is bent up near the sample boundary due to the electrostatic confinement potential, and the Fermi energy crosses the Landau levels to form one-dimensional edge channels. In the case of the QAHE, the top and bottom surface states of a magnetic topological insulator are smoothly connected by the side surfaces and therefore there is no confinement potential to bent the energy band. However, the side surfaces plays the role of the edge channel. While the surface state perpendicular to the magnetization is gapped, the surface state parallel to the magnetization remains gapless. Therefore, these gapless surfaces can carry the electric current along the sample edge as schematically depicted in Fig. 1d. Because electron spins are aligned to the magnetization direction on the side surface, electron momentum in one direction ( px > 0) is allowed due to the spin-momentum locking. Thus, electrons in an edge channel can travel only in one direction without backscatterings. To be backscattered, electrons have to be scattered to the opposite side of the sample edge which is separated by a macroscopic distance. Thus, the backscattering of electrons is strongly suppressed in the quantum anomalous Hall (QAH) state. When the electric current is carried by the backscattering-free edge channels, the Hall conductance is readily calculated as σx y = e2 / h using the Landauer-Buttiker formalism as similar to the QHE [71–73]. Formation of the chiral edge channel can be considered as a manifestation of the edge-bulk correspondence, which is one of the most important concepts of topological phases of matter [74, 75]: A gapless edge channel is formed at the boundary between the gapped areas with different topological invariant numbers. To understand the edge channels, an unfolded view of the surface state [76] is convenient. In the unfolded view, the top and bottom surfaces are subjected to the magnetizations with the opposite directions, resulting in opposite signs of m. Therefore the top and the bottom surfaces have different Chern numbers of C = +1 and C = −1. Consequently, a gapless edge state is formed at the domain boundary between the gapped regions with C = ±1. Thus, the edge channels on the side surface of the QAH system can be understood as the boundary states between two domains with different Chern

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numbers. The dispersion relation on the side surface consists of a gapless chiral edge mode and gapped helical modes [77]. When the Fermi energy is located between the energy gap of the helical edge modes, (−v/d < E F < v/d), only the chiral edge mode contributes to the electron transport, resulting in the quantization of the Hall resistance. The formation of the chiral edge channel at the magnetic domain boundary and its control has been demonstrated by an experiment using a magnetic force microscope (MFM) [78]. In this study, magnetic domain structures were artificially created in the QAH system by closely approaching a ferromagnetic tip of the MFM, and by scanning it. The resultant change in the resistance was monitored while writing the magnetic domains as shown in Fig. 3a. Figure 3c shows the change in the resistances associated with the domain wall motion. When the domain wall was located between the electrodes 1 and 2, the edge channel ran along the domain wall from the bottom edge to the top edge. As a consequence, the resistance between the electrodes 1 and 2 became 2h/e2 while the resistance between the electrodes 3 and 4 remained almost zero. The measured resistance values agreed with those calculated using the Landauer Büttiker formalism assuming that the back-scattering-free one-dimensional chiral edge channels ran around each magnetic domain. Thus, the existence of the chiral edge channel at the domain wall was well demonstrated by the experiment. The reconfigurable chiral edge channel may be useful for future electronics applications of the QAHE. Magnetic heterostructure Although the QAHE was successfully observed in magnetically-doped topological insulators, the observable temperatures in the early stage of the research [16, 63, 64] were below 100 mK, which was much lower than the temperature expected from the exchange gap measured by ARPES [58–60] or STS/STM [61, 79]. Spatially modulated exchange gap due to the inhomogeneous distribution of Cr or V atoms has been considered as one of the origins of the low observable temperature [61]. The presence of such a spatial distribution of magnetic elements was revealed by the STM/STS measurements [61, 79] and a scanning SQUID magnetometory [80– 82]. The size of the exchange gap was reported to vary between 9 meV to 48 meV correlating with the Cr atom density in Cr-doped (Bi,Sb)2 Te3 [61]. The influence of disorder was also pointed out from the transport studies[83, 84]. As shown in Fig. 4a, b, the temperature dependence of σx x showed a crossover from the thermal activation transport to the variable range hopping transport as temperature was decreased. In the thermal activation regime, electrons in the topological surface states were thermally excited beyond the exchange gap to contribute to the dissipative transport with a finite value of σx x . As the temperature was lowered, the excitation beyond the exchange gap was suppressed. Instead, the variable-range hopping conduction among the mid-gap localized states turned to dominate the transport. The temperature dependence of σx x followed σx x ∝ exp[−(T0 /T )1/3 ] in the low temperature regime. The exponent 1/3 was consistent with the variable range hopping of a non-interacting 2D system.

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Fig. 3 a Schematic of the measurement procedure. The up-spin domain (pink) is expanded by scanning the ferromagnetic MFM tip. Consequently, the domain wall (DW) moves in the direction indicated by an arrow. b An optical micrograph of the sample. The device size is 24 μm × 48 μm. c Resistance change as a function of the DW position x.The shaded areas denote the three typical cases of the DW position relative to the voltage contacts corresponding to the schematics above the graph. The Hall resistance on the left (R13 ) and right (R24 ) pairs of electrodes are shown on the upper panel and the longitudinal resistance on the lower (R12 ) and the upper (R34 ) edges are shown on the lower panel. From Ref. [78]. Reprinted with permission from AAAS

The localized states may also account for the small critical current for the breakdown of the QAHE [83, 84]. The critical current is proportional to the width of the Hall-bar shaped devices as shown in the inset of Fig. 4c. This observation indicates that the Hall electric field across the Hall bars is relevant to the breakdown of the QAHE. Assuming that the electric field induced by the current is evenly applied between the localized states, the spatial extent of a localized state can be estimated by comparing the temperature dependence and the current dependence of σx x . In this particular work [83], the shortest localization length near the charge neutrality point was estimated as 5 μm which is more than 100 times longer than the typical localization length in a 2DES at a GaAs/AlGaAs interface under the QHE. The long localization length indicates that the electrons can penetrate through the surface gapped region, leading to the small critical current for the breakdown of the QAHE. These transport properties are consistent, at least qualitatively, with the observations of the spatially inhomogeneous exchange gap.

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Fig. 4 a, b Temperature dependence of the longitudinal conductivity σx x at B = 0 T and various gate voltages VG in a homogeneously Cr-doped (Bi,Sb)2 Te3 thin film sample. The horizontal axis is 1/T in (a) and 1/T 0.33 in (b). c Gate voltage dependence of the critical current I0 for the breakdown of the QAHE measured in three Hall-bar shaped samples with different widths. The inset shows I0 at VG = 0 V as a function of the Hall-bar width. Reprinted figure with permission from Ref. [83]. Copyright (2017) by the American Physical Society

To improve the stability of the QAHE, magnetic heterostructure films of CBST/ BST/CBST were developed [85], exploiting the layer-by-layer growth technique of MBE. In the heterostructure shown in Fig. 5c, Cr atoms were doped only in the 1-nmthick layers embedded 1 nm below and above the surfaces to suppress disorder at the topological surface states. A relatively high Cr concentration of 20 % was adopted to open a sizable exchange gap at the surface states. As shown in Figs. 5d–g, the QAHE was observed even at T = 0.5 K which was much higher than the earlier studies. The residual longitudinal resistance decreased to almost zero when temperature was lowered to 50 mK. The robustness of the QAHE in the heterostructured film was also seen in the gate voltage dependence of a field-effect-transistor type device where the heterostructure film was covered with the AlOx dielectric layer and Ti/Au gate electrode was formed on top of it. The gate voltage range for the quantized Hall resistance in the heterostructure film was much wider than in the homogeneously doped film. This observation indicates the formation of a larger exchange gap in the heterostructure film. It was also reported that the observable temperature of the QAHE was increased by co-doping of Cr and V into (Bi, Sb)2 Te3 [86]. The

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Fig. 5 a–c Schematic layer structure of the uniformly doped film (a) and modulation-doped magnetic heterostructure films (b, c); x = 0.10, y = 0.78 in (a), x = 0.46, y = 0.78 in (b), and x = 0.46, y = 0.78 in (c). d–g Gate voltage (VG ) dependence of R yx [(d) and (f)] and R x x [(e) and (g)] at B = 0 T. Temperatures are 0.5 K [(d) and (e)] and 50 mK [(f) and (g)], respectively. Reprinted from Ref. [85], with the permission of AIP publishing

simultaneous doping of Cr and V atoms was reported to improve the homogeneity of the ferromagnetism. Possible applications of QAHE Because of the remarkable properties of the chiral edge channel transport at zero magnetic field, possible applications of the QAHE has been discussed since the discovery of the QAHE. These include a meteorological resistance standard [87–89] and a microwave circulator utilizing the chiral propagation of the edge-magentoplasmon [90]. Although the chiral edge channel transport has been realized in the QHE for decades, its manifestation at the zero magnetic field is attractive from the view point of practical applications. Here, we review the precision measurement of the quantized Hall resistance in the QAHE. Since the redefinition of the SI units based on the fundamental physical constants in 2019, the electric resistance has been

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defined by the von Klitzing constant h/e2 [91, 92]. Currently, the standard value of h/2e2 is materialized by the Hall resistance plateau of the QHE at around the Landau level filling factor ν = 2 with a relative uncertainty below one part in 1011 [93]. The QHE-based resistance standard requires strong magnetic fields and low temperatures of typically 10 T and 1 K, respectively. If the quantization accuracy of the QAHE exceeds or is comparable to that of the QHE, a quantum resistance standard can be achieved without using a superconducting solenoid. Even though the accuracy is not so high, the realization of standard resistor with more than 9 digits without a superconducting solenoid would be useful to meet the increasing demand for commercial resistance standards. Figure 6 shows the schematic measurement circuit used for the high precision measurement of the QAHE in a magnetic heterostructure film CBST/BST/CBST [89]. The sample of magnetic heterostructure film was cooled down to T = 15 mK and the Hall resistance was compared with a commercially available calibrated reference resistor placed at room temperature. The gate voltage was tuned to an optimum value so that the Fermi energy lies in the exchange gap. The measured deviation of R yx from the von Klitzing constant h/e2 was reported to be less than 2 ppm when the measurement current was 30 nA. The longitudinal resistance was as small as 1  at the zero magnetic field. This result verifies the very high-accuracy quantization of the Hall resistance in the QAHE. The main source of the uncertainty in this study originated from the statistical error of the measurement. Employment of Wheatstone bridge [93] or cryogenic current comparator [87, 88] would reduce the statistical error. The measurement current is limited by the critical current Ic for the breakdown of the QAHE. For further improvement, an increase in Ic in necessary. Similar uncertainty of 0.1 ∼ 1 ppm has been also reported in homogeneously-doped CBST [87] and VBST [88]. The uncertainty reported so far, however, does not reach the criteria of the primary quantum resistance standard which is an uncertainty less than 1 parts per billion. Optimization of the heterostructure and the growth conditions as well as employing other materials would lead to the more robust QAHE with a larger Ic and a higher operating temperature.

4 Various Phases of Magnetic Topological Insulator Axion insulator Besides the QAH insulator phase, a magnetic topological insulator can host various exotic phases. One interesting phase among them is the axion insulator where an unusual crossed electromagnetic response is expected to occur [11, 15, 76, 94, 95]. The electromagnetic response in a 3D TI is described by the Lagrangian 1 L= 8π

 

α  1 2 2 εE − B + θ E · B, μ 4π 2

(11)

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Fig. 6 a Schematic of the measurement setup for the precise comparison of the Hall resistance RQAHE with a calibrated reference resistor Rref . b The normalized deviation rQAHE = RQAHE /(h/e2 ) − 1 plotted as a function of gate voltage (VG ) at μ0 H = 0 T. The excitation current ISD = 30 nA. Reprinted from Ref. [89], with the permission of AIP publishing

where E and B are electric and magnetic fields, ε and μ are the dielectric constant and magnetic permeability of the media, respectively, and α = e2 /c is the fine structure constant. The first term describes the conventional Maxwell’s equation. The second term, called the axion term, is related to the topological nature of the 3D TI; θ = π for a topological insulator and θ = 0 for a conventional insulator. Both the quantized anomalous Hall effect and the quantized Faraday and Kerr rotation angles [96–99] are indeed consequences of the axion term. When the time-reversal symmetry is broken, the crossed electromagnetic response takes place: Namely, an electric polarization P is induced by magnetic field and a magnetization M is induced by electric field, P= M =

e2 B, 2h 2 e E. 2h

(12) (13)

This crossed response is known as the topological magnetoelectric (TME) effect. The surface circulating electric field induced by B produces the Hall current which produces the electric polarization P in the direction parallel to B. One of the remarkable features of the TME effect is that the coefficients are quantized to half of the quantum conductance e2 /2h which is described only by the fundamental physical constants being independent of materials parameters. The universal coefficients in the TME are distinct from the conventional magnetoelectric effect in multiferroic materials [100]. To observe the TME effect, it is necessary to break the time-reversal symmetry and to make the surface state insulating. As a result, the whole system becomes an insulator, which is termed as the axion insulator, while keeping the non-trivial topology in the bulk part. Theorists have proposed to open an energy gap on all surface states by surrounding a topological insulator with a ferromagnetic materials and by aligning their magnetic moments to point normally to the surface [11, 15]. However, such a magnetic moment alignment is difficult to realize. In the case of thin films, the requirement for the magnetic moment alignment is rather relaxed [76,

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Fig. 7 a Schematic of layer structure of the Cr- and V-doped tricolor heterostructure film. The composition (x, y, z, w) = (0.24, 0.74, 0.15, 0.80). b Magnetic field dependence of Hall conductivity (σx y , blue) and longitudinal conductivity (σx x , red) of the Cr- and V- doped film at T = 60 mK. c Magnetic field dependence of σx y in the Cr-doped (x = 0.24, green) and V-doped (z = 0.15, pink) bicolor heterostructure films at 500 mK. From Ref. [101]. Reprinted with permission from AAAS

95]. In a thin film with a thickness d vF / , the side surfaces can be gapped due to the quantum confinement effect when the magnetizations on the top and bottom surfaces are aligned anti-parallel to each other. Because the top and bottom surfaces are already gapped by the magnetization, the energy gap is formed on all the surfaces. If the Fermi energy is tuned to the energy gap, the whole system becomes an insulator with vanishing σx x and σx y . Thus, the condition to realize the axion insulator is fulfilled in a magnetic topological insulator thin film with the anti-parallel magnetization alignment. The axion insulator was experimentally demonstrated by Mogi et al. [101] using designed magnetic heterostructure thin films. They grew films having a CBST layer near the top surface and a VBST layer near the bottom surface with a few-nm-thick non-magnetic BST separation layer as shown in Fig. 7a. The total film thickness was 9 nm which is thin enough to open an energy gap on the side surfaces under anti-parallel magnetization. Due to the difference between the coercive fields in CBST (0.2 T) and VBST (0.8 T), the anti-parallel magnetization alignment was realized between 0.2 T < B 100 kV/cm). Note that E THz in the free space is of the order of a few V/cm under the experimental condition. This means that the THz field at the molecule is enhanced by a factor of ~105 from its free-space value by the plasmonic effect of the metal nanogap electrodes. This field enhancement factor is close to λ/d (λ: THz wavelength, d: nm-scale gap width); it can be understood that the THz radiation of λ ~ 100 μm is squeezed into a space d ~ 1 nm, as shown in Fig. 13b.

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Fig. 13 a n-photon conductance J n for the photon-assisted tunneling. b Top view of the sample used for the photon-assisted tunneling experiment. Figure reused with permission from: b, ref. [36], APS

3.3 Excitation of Molecular Vibrations by Terahertz Radiation in Single Molecule Transistors We can gain various information on excitations in molecules, either electronic or vibronic, from THz spectroscopy. However, as mentioned in the introduction of this section, it is very difficult to perform THz spectroscopy on single molecules, because (i) there is a huge mismatch between the THz wavelength (~100 μm) and the size of single molecules (1 nm) (the “diffraction limit”), and (ii) the THz absorption by single molecules is extremely small. To overcome the diffraction limit, we can use nanogap electrode to concentrate THz fields on a single molecule, as described in the previous section. Concerning the extremely weak THz absorption by a single molecule, the problem can be resolved by measuring the THz induced change in current that goes through the molecule by utilizing the nanogap electrodes. This is the great advantage of using the SMT geometry for single molecule THz spectroscopy. For spectroscopic measurements, we need a broadband THz radiation. The most conventional broadband THz radiation is the blackbody radiation from heated materials, such as SiC glowbars. Another recently developed broadband THz source is pulsed THz bursts; when a semiconductor surface is illuminated with femtosecond laser pulses, instantaneous current pulses J(t) or polarization pulses P(t) are created and pulsed THz fields whose amplitude is proportional to dJ(t)/dt or d2 P(t)/dt 2 are generated. The power spectrum of the emitted THz radiation extends from 100 GHz up to several ~ several tens THz. We will show an example of THz spectroscopy of a single C60 molecule. After confirming from transport measurements that a target C60 molecule is captured in the nanogap [see Fig. 14a], we illuminate the sample with THz pulses and search for the THz induced photocurrent. By sweeping V G while applying a small sourcedrain voltage (V DS = 0.1 mV), the THz-induced photocurrent can be measured as

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Fig. 14 Vibron-assisted tunneling in a C60 single molecule transistor. a Tunnel-in process: when the lowest unoccupied molecular orbital (LUMO) level is above the Fermi level of the electrodes, an electron in the electrode can absorb a vibron and jump into the molecule when molecular vibration is excited by THz. b Tunnel-out process: when the highest occupied molecular orbital (HOMO) level in the molecules is below the Fermi level of the electrode, the electron can absorb a vibron and jump out of the molecule when molecular vibration is excited by THz. Figure reused with permission from: ref. [26], Springer Nature

a function of V G . The black curve in Fig. 14b is the sample conductance in the dark condition, whereas the red trace is the THz-induced photocurrent. The peak of the black curve corresponds to the charge degeneracy point of the C60 SMT, which corresponds to the crossing points of the ground-state conductance lines shown in Fig. 14a. When we look at the change in current by THz illumination (red curve in Fig. 14b), a very small, but finite THz-induced photocurrent of the order of 100 fA is observed near the charge degeneracy point. Let us discuss how the photocurrent is generated in the C60 SMT. Because of the broadband THz excitation, the vibration of the C60 molecule is excited and creates new tunneling paths for electrons; i.e., the vibron-assisted tunneling processes as schematically shown by dotted lines in Figs. 14c and d. When the lowest unoccupied molecular orbital (LUMO) level is above the Fermi level of the electrodes, an electron in the electrodes cannot enter the molecule in the dark condition. However, when the vibration of the C60 molecules is excited by the THz radiation, an electron in the electrode can absorb a vibron and tunnel into the molecule (vibron-assisted tunneling). In this case, the electron number on the molecule increases from N to

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N + 1; i.e., the molecule changes its state from C60 to C60 − . Although N cannot be determined only from the present experiment, N is most likely to be zero. Figure 14d shows a vibron-assisted tunneling-out process when the highest occupied molecular orbital (HOMO) level is below the Fermi level of the electrode. In this case, the electron on the HOMO level cannot leave the molecule in the dark condition, but, by THz illumination, it can escape the molecule via the vibron-assisted tunneling. Therefore, THz-induced vibron-assisted tunneling signal appears near the Coulomb peak measured in the I-V measurements in the dark condition.

3.4 Ultrafast Vibration of a Single Fullerene Molecule For spectroscopic measurements, we face a problem that the SMTs are very slow devices due to their high tunnel resistances. It is, therefore, impossible to read an ultrafast current change in a single shot measurement. To overcome this problem, we can use the time-domain THz autocorrelation measurements, as shown in Fig. 15a. Using a beam splitter, we split the laser pulses into two parts and create double laser pulses. A surface of an InAs wafer is consecutively pumped by these femtosecond laser pulses and the time-correlated THz double pulses are generated. By recording the photocurrent induced by the THz double pulses as a function of the time interval between the THz pulses, τ, we can obtain an interferogram of the photocurrent (an interference pattern of the two photocurrent traces generated by two THz pulses) induced in the single molecule by THz radiation. Figure 15b shows an interferogram of the THz-induced photocurrent measured for a C60 SMT when the gate voltage is set at the peak position of the photocurrent (V G = 0.024 V). A clear center peak and interference feature can be seen. By calculating the Fourier spectrum of the interferogram, a THz spectrum of a C60 SMT can be obtained. Two peaks are observed at around 2 and 4 meV, as shown in Fig. 15c. It should be noted that a THz vibrational spectrum can be obtained even for a single molecule when the SMT geometry is used. The low-energy excitations observed for a C60 SMT originate from the vibronassisted tunneling promoted by the THz-induced center-of-mass oscillation of the C60 molecule. Park et al. observed vibron-assisted inelastic tunneling in the tunneling spectroscopy measurements. In the transport experiments, the molecular vibration is excited by injected tunnel electrons and the excited state lines are visible only inside the single electron tunneling region of the Coulomb stability diagrams. In the present THz spectroscopy, molecular vibrations are excited not only by tunneling electrons but also by impulsive THz fields generated by femtosecond laser pulses. Therefore, vibron-assisted tunneling signal appears even in the Coulomb blockaded regions. We can roughly estimate the quality (Q)-factor of the C60 molecular vibration to be approximately 3-5. This means that an electron that hops on the C60 molecule resides on the molecule at least for about 3–5 cycles of vibration and leaves the molecule. We can estimate the electron tunneling time, τ T , through the C60 molecule to be 8 ps from the sample conductance in the dark condition (~7 μS), using the

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Fig. 15 a Schematic illustration of the experimental setup. Femtosecond laser pulses (center wavelength: 810 nm, pulse duration: ~10 fs, repetition rate: 76 MHz) are split into two parts and focused on a THz emitter (InAs wafer). The generated THz bursts are collected and focused onto the single molecule transistor (SMT) mounted in a 4 He cryostat. b Quasi-autocorrelation trace (interferogram) of the THz induced photocurrent measured at the peak of the photocurrent (V G = 0.024 V, V DS = 0.1 mV). The quasi-autocorrelation trace was obtained by averaging data for ten scans. c Fourier spectrum of the interferogram shown in b. The spectrum exhibits sharp peaks at ~2 meV and ~4 meV. Figure reused with permission from: ref. [26], Springer Nature

following relationship; G max =

l r e2 , 4k B T l + r

(1)

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and assuming Γ l ≈ Γ r ≈ 1/τ T . Here, e is the elementary charge, k B the Boltzmann constant, T the temperature, and Gmax the conductance of the Coulomb peak. Γ l (Γ r ) is the tunnel coupling between the molecule and the left (right) electrode. The obtained τ T is consistent with the electron dwell time on the molecule determined from the Q-factor.

3.5 Effect of Single Electron Charging on the Electronic Structures of Single Molecules A very interesting observation is that the vibron peaks around 2 meV and 4 meV are finely split into two. The magnitudes of the peak splittings at around 2 meV and 4 meV are 0.8 meV and 0.6 meV, respectively. The excited-state lines due to molecular vibrations in the Coulomb stability diagrams originate from the Franck-Condon effect. As schematically shown in Fig. 16a, the overlap of the vibrational wave functions gives the tunneling probability of an electron for the transition between the Nand N + 1-states (the Franck-Condon principle). When the charge-state changes from the N− to N + 1-electron state, the equilibrium position of the molecule may shift by δ. This shift induces not only the diagonal transitions but also the off-diagonal transitions between the vibrational states of the N− and N + 1-charge states, giving rise to multiple excited-state lines in the Coulomb stability diagrams. In this discussion, the vibrational frequencies of the molecule for the N− and N + 1-charge states were assumed to be the same. However, this may not be the case in actual systems; i.e., the van der Waals potential felt by a C60 molecule on the gold surface may depend on the charge state of the molecule. To gain insight into the vibrational and electronic states of the C60 molecule in the SMT geometry, let us see the result of van der Waals inclusive density functional theory (DFT) calculations of neutral and negatively charged C60 on a Au(111) surface, which correspond to the N- and N + 1-charged states, respectively. In Fig. 16b, the interaction energy curves for neutral and negatively charged C60 on a Au (111) surface are plotted. The equilibrium C60 -surface distance is 0.243 nm and the vibrational energy, èω, is 4.1 meV. The calculated vibrational energy is larger than the value obtained by the THz spectroscopy (~2 meV), presumably because the counter electrode is lacking in the present calculation. When C60 is negatively charged, the system gets destabilized, because the anti-bonding state between the LUMO of C60 and the substrate state is partially occupied. Accordingly, the equilibrium distance becomes longer by 0.011 nm and the vibrational energy is lowered by 0.7 meV. This result is opposite to a simple expectation when only the image-charge force in the metal electrode is considered. The calculations suggest that the vibrational energy depends on the charge state and, thus, the splitting of the vibron-assisted tunneling peak is expected. Indeed, the magnitude of the observed splitting (0.6 ~ 0.8 meV) is in good agreement with the calculated change in the vibrational energy (0.7 meV).

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Fig. 16 a Schematic illustration of the vibronic energy diagram of the C60 molecule in the nanogap. The C60 molecule experiences the van der Waals potential on the surface of the gold electrode and performs the centre-of-mass oscillation. The parabolas and the wavefunctions in the diagram schematically illustrate the vibronic energy states for the N− and N + 1-electron states. When an electron is added to the C60 molecule, the equilibrium position of the molecule shifts by δ. The vibronic energies for the N− and N + 1-electron states are expressed as E(N) and E(N + 1), respectively. Due to the resonant absorption of THz photons, electron tunnelling via the GS(N) ↔ ES(N + 1) and ES(N) ↔ GS(N + 1) transitions become possible and generate THz-induced photocurrent even in the Coulomb gap, where GS and ES denote the ground state and the excited state, respectively. b Calculated interaction energy for a C60 molecule on a gold surface. Open diamond and filled diamond display the interaction energies calculated as a function of the moleculesurface distance for C60 and C60 − on a Au(111) surface, respectively. The molecule-surface distance is defined as the difference between the average z-coordinates of the bottom carbon atoms of C60 and the surface gold atoms, as denoted by an arrow in the inset. For the C60 molecule on the gold surface (the N-electron state), the characteristic vibronic energy, èω, is 4.1 meV. When an electron is added to the C60 molecule (the N + 1-electron state), the equilibrium distance becomes larger by 0.011 nm. At the same time, the characteristic vibronic energy becomes smaller by 0.7 meV. Figure reused with permission from: ref. [26], Springer Nature

Here, we show that THz spectroscopy can detect an ultrafast oscillatory motion of a single C60 molecule. Furthermore, it is also found that the observed THz peaks are finely split into two, reflecting the difference in the van der Waals potential profile experienced by the C60 molecule on the metal surface when the number of electron on the molecule fluctuates between N and N + 1 during the single electron tunneling process. Such an ultrahigh-sensitivity to the electronic/vibronic structures of a single molecule upon adding/removing a single electron has been achieved by using the THz spectroscopy in the SMT geometry.

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3.6 Rattling Motion of a Single Atom Encapsulated in a Fullerene Cage It is known that fullerene molecules can accommodate a single atom in their cage structure and such encapsulated atoms can significantly alter the properties of the fullerene molecules [38–42]. Therefore, it is of prime importance to clarify the dynamics of such complex molecules. Recently, ultrafast motion of a single Ce atom encapsulated in a fullerene cage (Ce@C82 ) was studied [43]. A Ce@C82 fullerene molecule was placed in a sub-nm gap of metal electrodes and excited by THz radiation to investigate ultrafast dynamics of the Ce atom, as shown in Fig. 17a. As seen in Fig. 17b, one broad dip and one broad peak are observed in the THz photocurrent spectra around 5 meV and 15 meV, respectively. From a comparison with a reference experiment on an empty C84 molecule, the two vibrational modes around 5 meV and 15 meV are attributed to the ultrafast motion of the Ce atom in the cage. It is predicted that the Ce atom in the C82 cage has two vibrational modes [44], as schematically illustrated in the inset of Fig. 17b; the vibrational mode at ~5–6 meV is the lateral vibration of the Ce atom with respect of the C82 cage wall

Fig. 17 a Schematic illustration of the present experimental setup for Fourier transform infrared spectroscopy. Broadband THz radiation (blackbody light source at ~ 1100 K) is focused onto a SMT mounted in a 4 He cryostat. b Spectrum of the THz-induced photocurrent. Two broad peaks at around ~5 meV and ~15 meV with opposite polarities are observed. The peak around 5 meV is due to the lateral vibration of the encapsulated Ce atom and the one around 15 meV is due to the longitudinal vibration. Figure reused with permission from: ref. [43], IOP

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and the peak at ~15-16 meV is due to the longitudinal Ce-C82 vibration. Previous THz measurements on an ensemble of Ce@C82 molecules observed only one broad peak [44] extending from 2 to 12 meV and could not resolve the two peaks, most likely due to inhomogeneous broadening. The observation of the two rattling modes has become possible only after inhomogeneous broadening has been removed by measuring a single molecule. In addition, it is noted that these two modes exhibit rather broad peaks, indicating the coherence time of the Ce atom motion is very short. The short coherence time probably reflects chaotic interaction between the Ce atom and the cage vibrational modes [45–47]. Furthermore, the longitudinal vibrational mode exhibits a broad, nonLorentzian spectrum, suggesting that the confinement potential for the Ce atom in the direction perpendicular to the cage wall is very anharmonic [46–48]. The present results have demonstrated that the THz spectroscopy using nanogap electrodes has an extremely high-sensitivity that can sense ultrafast motion of a single atom.

3.7 Single Molecule Terahertz Spectroscopy Using Scanning Tunneling Probes Another approach for investigation of nm-scale systems in the THz range is the combination of THz spectroscopy with scanning tunneling microscopy (STM). A sharp tip of an STM can be used as an antenna to focus THz radiation on to nm-scale systems. This approach was pioneered by Knoll and Keilmann [49]. They have shown that the scanning near-field optical microscope can reveal sub-wavelength detail because it uses near-field probing rather than beam focusing. They demonstrated the use of the aperture-less approach to scanning near-field optical microscopy to obtain contrast in vibrational absorption on a scale of about 100 nm, about one-hundredth of a wavelength. More recently, attempts of combining STM with THz spectroscopy have been intensively pursued [50–53]. Such measurements have made it possible to take images not only with a sub-nm spatial resolution but also with a sub-picosecond time resolution. Cocker, et al. performed THz spectroscopy of a single pentacene molecule by using THz scanning tunneling microscopy (THz-STM), as shown in Fig. 18a. They used an STM tip to focus the THz radiation on a single pentacene molecule on a conductive substrate. Between the substrate and the molecule, they inserted a very thin NaCl layer to realize a double-barrier resonant tunneling geometry. At low bias voltages, tunneling electrons cannot access empty molecular orbitals and must tunnel from the tip to the metal substrate in a single step. However, when the voltage reaches a threshold and either LUMO or HOMO enters the bias window, the electron localizes in the corresponding molecular state after having tunneled through the first and before tunneling through the second barrier. The alignment of the LUMO or HOMO

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Fig. 18 a Terahertz pulses propagating through free space are focused onto a nm-region by a STM tip. b Concept of steady-state and ultrafast tunneling out of the HOMO of a single molecule. Figure reused with permission from: ref. [50], Springer Nature

transport level with the Fermi energy of the tip thus opens up a new tunneling channel and, consequently, appears as a peak in the differential conductance (Fig. 18b). This tunneling process can be induced also in a dynamic manner by replacing the steady state bias voltage with a sub-picosecond THz transient pulse (Fig. 18b); the THz waveform can be viewed as a transient bias voltage and an ultrafast modulation in the level alignment takes place. Thus, it is expected to temporarily open a tunneling channel through the HOMO when the peak of the waveform matches the corresponding transport level. In the experiment, they positioned the STM tip over a pentacene molecule, turned off the bias voltage, and measured the net THz-induced tunnel current through the molecule as a function of the THz peak field. They observed a rectified current approaching one electron per THz pulse. The threshold peak field is about 0.25 kV/cm, corresponding to a transient peak bias voltage of about −1.65 V.

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Fig. 19 a Steady-state, constant-current STM image of the HOMO at V DC = − 1.7 V and I = 0.83 pA. Greyscale range = 2.3 Å. b THz-STM image taken at the same time as the topography image, with the V THz-peak ≈ − 2.05 V. The THz-induced current was in the direction of electron tunneling from the HOMO to the tip, even though the DC bias was positive and far from either molecular resonance. c THz-STM image (close-up of the blue rectangle in b) measured at constant height, with V DC = 0 V, V THz-peak ≈ –2.05 V, and a time resolution of 115 fs. The THz-induced current is calibrated in units of rectified electrons per THz pulse. d DFT-derived HOMO contours of the free pentacene molecule. Figure reused with permission from: ref. [50], Springer Nature

The great advantage of using an STM is that we can search for target molecules by STM. Furthermore, STM can visualize the shape of molecular orbitals in a single pentacene molecule by mapping the THz-induced current, as shown in Fig. 19. However, as discussed in Fig. 19, one drawback is that the control of the electrostatic potential and the number of electrons in the molecule is not possible.

4 Summary We have reviewed recent research activities on physics of sub-nm scale systems in the THz frequency range. Information on electronic and vibronic properties of individual molecules provides key insights to future technological developments. We have shown that nanogap electrodes, both STM tips and electromigration-fabricated ones, are very useful for investigating not only dc transport properties but also THz dynamics of single molecules and single atoms. In particular, we have shown that the single molecule transistor geometry provides us a unique opportunity to study THz dynamics of single molecules with controlled number of conduction electrons as well as an ultrahigh-sensitivity to the electronic/vibronic structures of a single molecule. This novel measurement scheme provides a new opportunity for investigating ultrafast THz dynamics in sub-nm scale systems.

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Novel Phonon Generator and Photon Detector by Single Electron Transport in Quantum Dots Mikio Eto and Rin Okuyama

Abstract We theoretically study the electron-phonon and electron-photon interactions in quantum dots to propose a single phonon generator and sensitive photon detector. First, we study the electron-LO-phonon interaction in double quantum dot in series. When the energy level spacing between the quantum dots matches the energy of LO phonons, single electron transport is accompanied by the emission of single LO phonon, analogously to cavity-QED for atoms in a photon cavity. This yields a single phonon generator on demand. We also comment on the possibility of phonon lasing and antibunching. Second, we examine a photocurrent through an array of quantum dots in parallel. In the geometry of single electron transistor (SET), the irradiation of THz light excites an electron from an energy level below the Fermi level (E F ) to a level above E F in the quantum dot, which results in the electric current without applying a bias voltage. In an array of quantum dots in parallel in SET, the photocurrent can be enhanced by the Dicke effect due to the formation of entangled states if the dot-dot distance is smaller than the photon wavelength. This could be applied to a sensitive detector of THz light. Keywords Quantum dot · Carbon nanotube · Electron-phonon interaction · Photocurrent · Dicke effect

1 Introduction Quantum dot is a small box fabricated on semiconductors, to confine a number of electrons [1]. It possesses discrete energy levels and large Coulomb interaction. When a quantum dot is connected to source and drain electrodes through tunnel barriers, M. Eto (B) Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan e-mail: [email protected] R. Okuyama Department of Physics, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki 214–8571, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_11

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L L

R

R

Vgate

eVgate Fig. 1 Schematic view of quantum dot in the single electron transistor (SET). A quantum dot is connected to source (L) and drain (R) electrodes through tunnel barriers, in which discrete energy levels are electrostatically tunable using a gate electrode. The single electron transport takes place when an energy level appears between the Fermi levels in electrodes L and R. In the right panel, discrete levels in the quantum dot involve the charging energy, or mean field by the Coulomb interaction

eVbias

Fig. 2 Schematic view of double quantum dot in series, under a large bias voltage between source (L) and drain (R) electrodes. The energy level spacing between the quantum dots,  = ε L − ε R , is tunable by applying a gate voltage. When  is changed, the current I shows a large peak at  = 0. In the presence of electron-LO-phonon coupling, the current also shows subpeaks at  = ωph , 2ωph , . . ., where ωph is the phonon energy, reflecting a single electron transport accompanied by the emission of single phonon, two phonons, . . .

single electron transport takes place when an energy level is put between the Fermi levels E F in these electrodes. The energy levels are electrostatically tunable using the gate electrode attached to the quantum dot, as depicted in Fig. 1. This system is called single electron transistor (SET). The number of electrons is fixed by the Coulomb blockade when the energy levels are absent between the E F ’s. In double quantum dot in series, the single electron transport is observed when the energy levels are matched in the two quantum dots under a large bias voltage [2], as shown in Fig. 2. Such systems of quantum dot(s) are suitable for the study of electron-phonon and -photon interactions owing to the single electron nature and tunability. This chapter is devoted to our theoretical studies on these interactions. We propose a single phonon generator and sensitive detector of THz photon. In the first part of this chapter, we pay attention to the longitudinal optical (LO) phonons in double quantum dot connected in series. When the energy level spacing  between the quantum dots is tuned to match the phonon energy ωph , the single

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electron transport is accompanied by the emission of single phonon. This can be applied to a single phonon generator, analogously to cavity- or circuit-QED for atoms in a cavity. We also mention the possibility of phonon lasing and antibunching. The energy of LO phonons is so large as ωph = 36 meV in GaAs that the phonons are usually irrelevant to the transport through the quantum dot within the energy scale of a few meV. However, the transport with the LO phonon emission was observed experimentally in a double quantum dot of vertical structure [3]. Further, we examine the interaction between the electron and phonon (vibration mode) in a suspended carbon nanotube, where the electron-phonon interaction is much stronger than in the GaAs quantum dot. Then we observe the Franck-Condon effect on the transport through the double quantum dot. In the second part, we examine a photocurrent through a quantum dot in the SET by irradiating the terahertz (THz) light. The THz light induces the transition of an electron between discrete energy levels in the quantum dot (see Fig. 4 in Sect. 3). In the Coulomb blockade regime, the excitation from a level below E F in the source and drain leads to that above E F , results in an electric current without applying a bias voltage. This could be applied to a sensitive detector of THz light [4–6]. We formulate the photocurrent through a quantum dot, considering the electronphoton interaction in the case of classical THz light. Then we examine an array of quantum dots in parallel in the SET. When the dot-dot distance is much smaller than the wavelength of light, the absorption of photon makes entangled states among the quantum dots. This could enhance the photocurrent by the Dicke effect, similarly to the superradiance [7, 8], and in consequence, increase the sensitivity of the THz-light detector.

2 Single Phonon Generator by Double Quantum Dot The subject of this section is the electron-LO-phonon in double quantum dot (DQD). In Sect. 2.1, we begin with the interaction in single quantum dot. We show that an electron in the quantum dot is coupled to a single mode of LO phonons. Since the mode of LO phonons does not diffuse owing to their dispersionless nature, the LO phonons by themselves play a role of natural cavity. The DQD interacting with LO phonons can be regarded as a two-level system in a cavity, like the cavity quantum electrodynamics (QED) for atoms in a photon cavity [9]. Here we mention the cavity-QED utilizing artificial atoms such as quantum dots, superconducting qubits, etc. They are being intensively studied for wide application to, e.g., quantum information processing [10], single photon source [11], microlaser by so-called circuit QED [12]. In these systems, the vacuum Rabi splitting is observed in the strong coupling regime, g  γ , κ, (1) where g is the coupling constant of the electron-photon interaction, γ is decay rate for electron from upper to lower level, and κ is the escape rate for photon from the

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cavity.1 Recently, the stronger coupling regimes were realized [13, 14]: In addition to Eq. (1), the condition of λ  1 is satisfied, where λ=

g ωph

(2)

with the photon energy in the cavity ωph . In the case of electron-LO-phonon coupling in quantum dots, λ = 0.01 ∼ 0.1 for the DQD fabricated on GaAs and λ  1 for the DQD on carbon nanotube (CNT), where ωph in Eq. (2) is the energy of LO phonons or vibration mode in CNT. Thus the systems of electron-phonon coupling may be suitable for the study of various coupling regimes of cavity-QED. We study the DQD with LO phonons in two situations: In Sect. 2.2, the tunnel coupling Vc within the DQD is smaller than the linewidths  L ,R in quantum dots (L, = R) owing to the tunnel coupling to a lead. This is the case of a relevant experiment [3]. We formulate a single phonon generation by single electron transport through the DQD made on GaAs. We also comment on the Franck-Condon effect in the DQD on CNT. In Sect. 2.3, the case of Vc >  L ,R is considered. We review our previous work that proposed the phonon laser and phonon antibunching using the DQD on GaAs [15, 16]. These coherent phenomena are prohibited by the Franck-Condon effect in the DQD on CNT.

2.1 Electron-Phonon Interaction in Single Quantum Dot Let us begin with the electron-LO-phonon interaction in a quantum dot. No tunnel couplings to external leads are considered in this subsection. For an electron confined in the quantum dot with an energy level ε0 , the Hamiltonian is given by H = He + Hph + Hep , He = ε0 d † d,  ωq aq† aq , Hph = q

Hep =



  † d † d, gq aq + a−q

(3) (4) (5)

q

1

When Eq. (1) is satisfied, the two-level system coupled to photons is well described by the JaynesCummings model in Eq. (26). Then the upper level with no photon, |e|0ph , is coherently coupled with the lower level with one photon, |g|1ph , when the level spacing  = εe − εg matches the photon energy ωph . This results in a level splitting between them, which is called vacuum Rabi splitting.

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using creation and annihilation operators (d † , d) for an electron in the quantum dot and those (aq† , aq ) for a phonon with wavenumber q and energy ωq . The electron spin is irrelevant and omitted. In Hep , an electron couples to the LO phonons via the Fröhlich interaction,  gq =

e2 ωq 2V



  1 1 1 − |ψ(r)|2 eiq·r dr, (∞) (0) q

(6)

where ψ(r) is the electron wavefunction, (∞) [ (0)] is the dielectric constant at high [low] frequency, and V is the system volume. When ψ(r) is confined in a quantum dot of radius R, which is usually much larger than the lattice constant (∼ 1 Å), only the phonons of q  1/R are coupled to the electron because of an oscillating factor in the integral in Eq. (6). Thus we can safely restrict ourselves to the phonons in the long wavelength regime, where ωq = ωph is independent of q in a good approximation. Hep indicates that an electron in the quantum dot is coupled to a single mode of LO phonon, i.e., a linear combination of LO phonons |q, |LO =

1 g−q |q g q

(7)

with the coupling constant g=



|gq |2 .

(8)

q

The phonon mode |LO corresponds to the dipole excitation in the crystal when an electron is added to the quantum dot. By the unitary transformation for phonons from |q to |LO and other modes |q  orthogonal to |LO (LO|q  = 0), we obtain ⎛ Hph = ωph ⎝a † a +



⎞ aq† aq ⎠ ,

(9)

q



Hep = gd † d a + a † ,

(10)

where a † and a (aq† and aq ) are the operators for phonon |LO (|q ). Disregarding the decoupled modes, we have 

H = ε0 d † d + ωph a † a + gd † d a + a † .

(11)

Therefore, an electron in the quantum dot is coupled to a single mode of LO phonons with energy ωph . Note that the LO phonons work as a natural cavity since they do not diffuse owing to the flat dispersion (see Sect. 2.3).

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Polaron Formation The Hamiltonian in Eq. (11) is diagonalized by the unitary transformation T = e−λ(a

†

−a)

(12)

with the dimensionless coupling constant λ in Eq. (2). The eigenstates and energies are as follows. • |0 ⊗ |nph with E 0,n = nωph , for no electron in the quantum dot, |0, and n phonons, |nph . • |1 ⊗ T |nph with E 1,n = ε0 + (n − λ2 )ωph , for one electron in the quantum dot, |1, and renormalized state of n phonons, |n ˜ ph = T |nph . For example, ˜ ph = T |0ph = e−λ2 /2 |0

∞  λ2n √ |nph . n! n=0

The phonon states, |n ˜ ph = T |nph , indicate the polaron formation when an electron is added to the quantum dot.

2.2 LO Phonon Generator by Double Quantum Dot We examine the DQD in series, as depicted in Fig. 2, in the presence of electron-LOphonon interaction. We assume single energy level ε L (ε R ) in quantum dot L (R). The Hamiltonian for electrons is    (13) εα dα† dα + Vc d R† d L + d L† d R He = α=L ,R

where dα† and dα are the creation and annihilation operators for an electron in quantum dot α (= L, R). Vc is the tunnel coupling between the quantum dots. There are two relevant modes of LO phonons, |LO, L coupled to quantum dot L and |LO, R coupled to quantum dot R. Using operators aα† and aα (α = L, R) for them,  

 ωph aα† aα + gα dα† dα aα + aα† , (14) Hph + Hep = α=L ,R

with coupling constants g L and g R after the decoupled modes are neglected. We examine the transport from lead L to R under a sufficiently large bias. The tunnel coupling Vα to lead α results in the linewidth in quantum dot α (= L, R) α = 2π να (Vα )2

(15)

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with density of states in the lead να .2 We evaluate the current by the perturbation with respect to the tunnel coupling Vc between the quantum dots. We simply assume that LO, L|LO, R = 0 in this subsection. Current Through DQD on GaAs The current through the DQD shows a peak structure as a function of level spacing  = ε L − ε R , with main peak at  = 0 and subpeaks at  = nωph (n = 1, 2, · · · ), at low temperatures of kB T ωph . When λ 1, the phonons do not take part in the electron transport at the main peak ( ≈ 0). The current is estimated to be I =

/2 2e 2 V  c 2 + (/2)2

(16)

with  =  L +  R . At the first subpeak of  ≈ ωph , the current is accompanied by the single phonon emission. It is I =

˜ 2e 2 g L2 + g 2R /2 Vc 2 2 ˜  ωph ( − ωph )2 + (/2)

(17)

with ˜ =  + ph . Here, ph = /τph is the level broadening of LO phonon by the finite lifetime τph . The main peak and two subpeaks were observed in the transport experiment [3]. At the first subpeak, single electron transport is accompanied by the emission of single LO phonon although the peak height is much smaller than the main peak when λ = g L ,R /(ωph ) 1. Current Through DQD on Carbon Nanotube In a suspended CNT, the electron is coupled with a longitudinal stretching mode of the vibration. Then the coupling is so strong (λ ∼ 1) that the polaron formation influences the transport through the DQD. The main peak of the current is given by   /2 2e 2 g L2 + g 2R V exp − I = , 2 ˜ 2  c  + (/2)2 ωph

(18)

˜ = ε L − g L2 /(ωph ) − [ε R − g 2R /(ωph )] is the spacing between the renorwhere  malized levels by the electron-LO-phonon interaction. The difference of the polarons The linewidth α is not changed by the polaron formation under a large enough bias voltage since ˜ ph to lead R with n phonons all the phonon states can take part in the transport, e.g., from |R ⊗ |0 ∞  ˜ 2 = 0|T † T |0 = 1. Note that, for a finite bias voltage, the linewidth (n = 0, 1, 2, · · · ): |n|0|

2

n=0

is renormalized to be less than α by the polaron formation. This is called Franck-Condon blockade [17], which was observed in CNT [18].

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in quantum dot L and R reduces the current by a factor of exp[−(g L2 + g 2R )/(ωph )2 ]. This is the Franck-Condon effect. Similarly, the first subpeak of the current is written as   ˜ g L2 + g 2R 2e 2 g L2 + g 2R /2 Vc . (19) I = 2 exp − 2 2 ˜ ˜  ( − ωph )2 + (/2) ωph ωph In this case, more than one LO phonons are emitted by the single electron transport at the first subpeak. This is because some phonons are created by the polaron formation in the tunneling processes between the leads and quantum dots, besides the tunneling between the quantum dots.

2.3 Phonon Lasing and Antibunching In the previous subsection, we have examined the case of Vc <  L ,R by the perturbative calculation with respect to Vc . In this subsection, we investigate the opposite case of Vc >  L ,R . The Hamiltonian for the DQD is the same as before, H = He + Hph + Hep , given in Eqs. (13) and (14). First, we introduce two modes of LO phonons |LO, S and |LO, A from |LO, L and |LO, R: aL + a R aL − a R , aA = √ , (20) aS = √ 2(1 + S) 2(1 − S) with S = LO, L|LO, R. The Hamiltonian in Eq. (14) is rewritten as  Hph + Hep = ωph N S + λ S (a S + a S† )(n L + n R )

 +N A + λ A (a A + a †A )(n L − n R ) ,

(21)

using N S = a S† a S , N A = a †A a A , and dimensionless coupling constants λ S , λ A . As discussed in our previous papers [15, 16], A-phonons play a crucial role for phononassisted tunneling between the quantum dots and thus phonon lasing, whereas Sphonons do not since it couples to the total number of electrons in the DQD. Both phonons are relevant to the Franck-Condon effect. The tunnel couplings to the leads result in the linewidth α in Eq. (15). Under a large bias, an electron tunnels from lead L to quantum dot L with tunneling rate  L / and tunnels out from quantum dot R to lead R with  R /. Hereafter, α is redefined by α . We also introduce the phonon decay rate ph to take into account a natural decay of LO phonons due to the lattice anharmonicity [19]. We describe the dynamics of density matrix ρ for the DQD with phonons. The basis set consists of |DQD ⊗ |n S  S ⊗ |n A  A with |0, |L, |R for the DQD state |DQD and phonon numbers n S , n A = 0, 1, 2, 3, · · · . We assume a large Coulomb interaction in the DQD to avoid the doubly occupied state. The Liouville equation is given by

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1 ∂ ρ = [H, ρ] +  L D[d L† ]ρ +  R D[d R ]ρ + ph (D[a S ] + D[a A ])ρ, ∂t i

(22)

using the Lindblad dissipator 1 D[A]ρ = Aρ A† − {ρ, A† A}. 2

(23)

We adopt the Born-Markov-Secular approximation to Eq. (22) by diagonalizing the Hamiltonian H . We evaluate the electric current in the stationary state by numerical calculations [15, 16]. Phonon Lasing We show the calculated results in Fig. 3 in the case of  L ,R  ph . Panel (a) shows the electric current as a function of level spacing  = ε L − ε R for the DQD fabricated on GaAs (λ A = 0.1; solid line) and on CNT (λ A = 1; dotted line). λ S = 0 for simplicity. The current shows a peak structure with the main peak at  ≈ 0 and subpeaks at

1

(a)

0.5

0 3

(b)

2

1 -1

0

1

2

Fig. 3 Calculated results for the electron transport through double quantum dot in series, as depicted in Fig. 2, in the presence of electron-LO-phonon coupling. The phonon decay rate ph is much smaller than the tunneling rates  L ,R . a Electric current and b autocorrelation function of Aphonons g (2) A (0) in Eq. (24), as a function of energy level spacing  between the quantum dots in units of the phonon energy ωph . The dimensionless coupling constant is λ A = 0.1 (solid line) and 1 (dotted line) while λ S = 0 (see text for A- and S-phonons). I0 = e R /(2 +  R /  L ) is the current at  = 0 in the absence of electron-phonon coupling.  L =  R = 100ph and Vc = 0.1ωph

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 ≈ nωph (n = 1, 2, · · · ), as discussed in Sect. 2.2 for Vc <  L ,R . However, I () shows different properties in the present case of Vc >  L ,R : (i) All the peaks are the same in height, whereas the first subpeak is smaller than the main peak by a factor of (g L2 + g 2R )/(ωph )2 in Sect. 2.2. The electron transport takes place through energy levels in the whole DQD in the former. It is determined by the tunneling between the quantum dots in the latter, which is influenced by the phonon states with electron in either quantum dot. (ii) In the case of strong coupling (λ  1), the Franck-Condon effect suppresses the transport between the quantum dots and hence reduces the current peaks in Sect. 2.2. This suppression is not seen in the present case because the Franck-Condon effect does not take place for the tunneling between the leads and DQD at the resonance under a large bias voltage (see footnote 2 in this chapter). Note that peak widths of I () are decreased by the larger electron-phonon coupling in Fig. 3a. Now we discuss the possibility of phonon lasing. As mentioned previously, our system is analogous to the cavity-QED since the LO phonons act as a natural cavity. The pumping to the upper level is realized by an electric current through the DQD under a finite bias, analogously to the circuit-QED [12]. To examine the amplification of A-phonons, we calculate the phonon autocorrelation function g (2) A (τ ) =

: N A (0)N A (τ ) : , N A 2

(24)

which is the probability of phonon emission at time τ on the condition that a phonon is emitted at time 0. A value of g (2) A (0) = 1 indicates Poissonian distribution of phonons, which is a criterion of phonon lasing. We plot g (2) A (0) in Fig. 3b as a function of . We observe that the lasing condition is satisfied at the current subpeaks,  ≈ ωph and 2ωph , in the case of DQD on GaAs (solid line). In this case, a phonon is emitted by the single electron transport through the DQD. When  L ,R  ph , the emitted phonons remain around the DQD, or in a natural cavity, which results in the stimulated emission of phonons. Note that this lasing is not affected by the presence of coupling λ S to S-phonons [16]. In the DQD on CTN, the strength of the electron-phonon interaction is comparable to the phonon energy (λ A ∼ 1). In this case, the lattice distortion by the FranckCondon effect seriously disturbs the coherent coupling between an electron and phonons in the DQD and, as a result, suppresses the phonon lasing. Indeed, g (2) A (0) > 1 at the current subpeaks, indicating the phonon bunching or thermalization [dotted line in Fig. 3b].3 Phonon Antibunching We briefly mention our calculated results for  L ,R  ph . In this case, the phonon lasing cannot be expected because the phonons decay before the stimulated emission. Then we observe another phenomenon, i.e., antibunching of LO phonons for the (2)

We obtained an analytical expression, g A (0) = (ν + 4λ2A )/(ν + 2λ2A ) + O(ph /  L ,R ), at the 2 νth subpeak of current [15]. This equation indicates g (2) A (0) 1 (phonon lasing) for λ A ν and (2) g A (0) 2 (thermalized phonons by the lattice distortion) for λ2A  ν in the case of small ph .

3

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DQD on GaAs. At the current subpeaks, we find g (2) A (0) 1, representing a strong antibunching of phonons [15, 16]. This is because the phonon emission is regularized by the electron transport through the DQD. Note that the phonon antibunching does not takes place for Vc <  L ,R in Sect. 2.2: The electron transport is governed by the tunneling between the quantum dots, which results in the Poisson process for electron transport. For Vc >  L ,R , on the other hand, the electron motion is regularized by the sequential tunnelings, from lead L to quantum dot L, quantum dot L to R, and quantum dot R to lead R. This results in the antibunching of electron transport and thus the antibunching of phonon emission. In the DQD on CNT (λ A ∼ 1), neither phonon antibunching nor phonon lasing can be observed because of an effective phonon thermalization due to the Franck-Condon effect. Realization of Phonon Lasing We discuss possible experimental realizations to observe the LO phonon lasing in semiconductor-based DQDs. In GaAs, an LO phonon around the  point decays into an LO phonon and a TA phonon around the L point, which are not coupled to the DQD. These daughter phonons can be detected by the transport through another DQD fabricated nearby, which may enable the observation of phonon lasing. With a decay rate ph ∼ 0.1 THz in GaAs [19], however, the lasing condition  L ,R  ph might be hard to realize. Other materials with longer lifetime of optical phonons, such as ZnO [20], may be preferable to realize the phonon lasing. Our research of LO phonons is also applicable to a free-standing semiconductor membrane as a phonon cavity [21, 22], in which a resonating mode plays a role of LO phonons. Our theory implies that a DQD could generate various quantum states of mechanical oscillators.

3 THz Photon Detector by Quantum Dot Array In this section, we examine the photocurrent through an array of quantum dots in parallel. The observation of the photocurrent through a single quantum dot was reported under the irradiation of terahertz (THz) light, using the SET in the Coulomb blockade regime [4–6]. Let us consider two energy levels in a quantum dot, |g below the Fermi level E F in the leads and |e above E F , as depicted in Fig. 4. The irradiation of the THz light excites an electron from |g to |e. Then the excited electron goes out to lead L or R with the tunneling rate eL ,R while another electron comes into |g from either lead with gL ,R . This results in an electric current without applying a bias voltage. The current direction is determined by an asymmetry of tunnel couplings to the leads, or the ratios among eL ,R and gL ,R . In Sect. 3.1, we give fundamental explanations for the electron-photon interaction in a quantum dot and the Dicke effect in an array of quantum dots. In the next subsection, we formulate the photocurrent through single quantum dot. In Sect. 3.3, we study the photocurrent through an array of quantum dots and discuss the condition for the enhanced photocurrent by the Dicke effect.

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v

v

Fig. 4 Schematic view of photocurrent through a quantum dot in the SET in the Coulomb blockade regime. No bias voltage is applied between leads L and R for source and drain electrodes, respectively. The irradiation of a THz light excites an electron from an energy level below the Fermi level (E F ) to a level above E F in the quantum dot, which results in the electric current in either direction. The current direction is determined by an asymmetric factor in Eq. (36), using the tunneling rates gα from lead α to the lower level in the quantum dot and eα from the upper level to lead α (= L, R)

3.1 Electron-Photon Interaction in Quantum Dot Let us begin with the electron-photon interaction in a single quantum dot with two energy levels. We do not consider the tunnel coupling to the leads in this subsection. Using a pseudo-spin operator of S = 1/2 for the two levels, the Hamiltonian is written as (25) H1 = S z + g(a + a † )(S + + S − ) + ωa † a, where  = εe − εg is the spacing between the upper level (|e = |S z = 1/2) and lower level (|g = |S z = −1/2). S ± = S x ± i S y is the raising and lowering operators to describe the transition from |g to |e and vice versa, respectively. The electron spin is irrelevant and omitted. a † and a are creation and annihilation operators of a photon with energy ω. g is the coupling constant of the electron-photon interaction. We consider a single photon mode for simplicity. The rotating wave approximation yields H1 = S z + g(aS + + a † S − ) + ωa † a,

(26)

which includes an electron transition from |g to |e with photon absorption and that from |e to |g with photon emission. This is called Jaynes-Cummings model, which is applicable when  ∼ ω. Next, consider N equivalent quantum dots coupled to a single photon mode. The Hamiltonian reads HN = 

N  i=1

Siz +

N  i=1

gi (a + a † )(Si+ + Si− ) + ωa † a,

(27)

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using pseud-spin operator Si for the ith quantum dot with two energy levels, |ei  and |gi . The coupling constant is gi ∝ ei |eiq·r e · p|gi  where q and e are the wavenumber and polarization of the photon, respectively, while p is the momentum operator for electron. gi ≈ geiq·Ri with Ri being the position of the ith quantum dot when the size of the quantum dot is much smaller than the wavelength 2π/q of the photon (dipole approximation). Besides, if the whole system of N quantum dots is within the wavelength, eiq·Ri can be safely disregarded in gi , which results in HN = S z + g(a + a † )(S + + S − ) + ωa † a. Here, Sz =

N  i=1

Siz , S ± =

N 

(28)

Si±

i=1

are operators of the total pseudo-spin. In the rotating wave approximation, it becomes HN = S z + g(aS + + a † S − ) + ωa † a.

(29)

The Hamiltonian in Eq. (29) explains the superradiance by the Dicke effect. It enhances the spontaneous emission of photons from N identical two-level systems when all the systems are initially excited, |e1 |e2 |e3  · · · . The initial state is equal to the state of total pseudo-spin S = N /2 and its z component S z = N /2, |S = N /2, S z = N /2. Accompanied by the photon emission [a † in Eq. (29)], the lowering operator S − makes the states of S z = N /2 and S z = (N /2) − 1, (N /2) − 2, · · · . The emission rate from |S = N /2, S z = M is remarkably enhanced by a factor of (J + M)(J − M + 1) with J = N /2, particularly for large N and M ∼ 0. This is the superradiance proposed by Dicke [7, 8]. Let us consider the simplest case with N = 2 in Fig. 5. The initial state is expressed by |e1 |e2  = |S = 1, S z = 1. The emission of one photon results in the state of 1 |S = 1, S z = 0 = √ (|e1 |g2  + |g1 |e2 ) . 2

(30)

This is the entangled state between the two quantum dots. The emission rate for the next photon is twice as large as that from non-entangled states, |e1 |g2  or |g1 |e2 . This state is optically bright [|B in Fig. 5b], whereas the other entangled state

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(a)

(b)

Fig. 5 Optical transition in two identical quantum dots. Each quantum dot has two energy levels, |gi  and |ei  (i = 1, 2). In panel a, the optical transition takes place in either quantum dot, with a common transition rate γ . Panel b shows the Dicke effect when the dot-dot distance is much smaller than the wavelength of photon. From the ground √ state |g1 |g2 , the absorption of a photon results in the entangled state, |B = (|e1 |g2  + |g1 |e2 )/ 2, which is optically √ bright. The transition rate is enhanced twice, whereas the dark state |D = (|e1 |g2  − |g1 |e2 )/ 2 is excluded by the optical transition

1 |S = 0, S z = 0 = √ (|e1 |g2  − |g1 |e2 ) 2 is optically dark (|D). The Dicke effect stems from the formation of entangled states as in Eq. (30). We are interested in this Dicke effect on the photocurrent. This issue will be discussed in Sect. 3.3.

3.2 Formulation of Photocurrent Through Single Quantum Dot Before discussing an array of quantum dots, we formulate the photocurrent through single quantum dot. An analytical expression is given in the case of classical THz light. In the Jaynes-Cummings Hamiltonian H1 in Eq. (26), operators a and a † are ¯ iωt with constant a, ¯ respectively, for the classical light of replaced by ae ¯ −iωt and ae frequency ω. Besides, we replace the pseudo-spin operators by creation (dg† , de† ) and annihilation operators (dg , de ) for an electron in the quantum dot. Introducing the electron-electron interaction U in the quantum dot, the Hamiltonian becomes H1 =

 † (d de − dg† dg ) + g(e−iωt de† dg + eiωt dg† de ) + U de† de dg† dg , 2 e

(31)

where g has been redefined by g/a. ¯ In the rotation frame, it is reduced to H1 =

 − ω † (de de − dg† dg ) + g(de† dg + dg† de ) + U de† de dg† dg . 2

(32)

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The tunnel coupling between the quantum dot and leads is described by the tunnel Hamiltonian HT =

(ε k E F ) 

 (e) † Vα,k de cα,k + h.c. , (33)

k

where cα,k annihilates an electron at state k in lead α. Here, we assume that level |g couples to |k below E F , whereas level |e couples to |k above E F . This assumption is justified in the Coulomb blockade regime where /2 − E F is much larger than α [given below in Eq. (35)] and temperature kB T . The tunnel Hamillinewidths g,e tonian in Eq. (33) is not changed in the rotation frame if the states below (above) E F in the leads are rotated in the same direction as |g (|e) in the quantum dot. Now we calculate the photocurrent using the Hamiltonian H1 + HT . We examine the transport in the Coulomb blockade regime with N = 1 electron in the quantum dot, using two methods. (Method 1) Density Matrix Method The density matrix ρ is used for states |g, |e, and |0 in the quantum dot as a basis set, where |0 is the empty state. We assume that the doubly occupied state |g|e is excluded by a large Coulomb interaction U in the quantum dot. HT is taken into account by the second-order perturbation. The Liouville equation is given by 1 ∂ ρ = [H1 , ρ] + g D[dg† ]ρ + e D[de ]ρ, ∂t i g

(34)

g

where g =  L +  R and e =  Le +  eR with4 α g/e

(εk ≶E F ) 2π  (g)/(e) = |Vα,k |2 δ(εk − εg/e ).  k

(35)

D[dg† ] and D[de ] are the Lindblad dissipators given in Eq. (23). Equation (34) yields ∂ ρ00 ∂t ∂ ρgg ∂t ∂ ρee ∂t ∂ ρeg ∂t

= −g ρ00 + e ρee , g (ρeg − ρge ) + g ρ00 , i g = − (ρeg − ρge ) − e ρee , i  e 1  ( − ωph )ρeg + g(ρgg − ρee ) − ρeg , = i 2 =

4 Note that symbols ’s are used for the linewidth or broadening of energy levels in Eq. (15) in Sect. 2.2. They express the transition rates by the tunnel coupling to leads in Sects. 2.3, 3.2, and 3.3. The latter corresponds to the former divided by .

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and ρge = [ρeg ]∗ . Solving the equations in the stationary state (∂ρ/∂t = 0), we evaluate the electric current as I1 = (eR − eL )ρee − (gR − gL )ρ00   gR − gL eR − eL g 2 e = − .(36) eR + eL gR + gL (ω − )2 + (e /2)2 + (2 + e / g )g 2 The current direction is determined by a prefactor in Eq. (36), which is a combination α of the ratios of g/e in Eq. (35). As a function of photon energy ω, the photocurrent shows the Lorentzian peak: It is maximal when ω matches  = εe − εg . Note that the diagonal elements of density matrix, ρ j j ( j = 0, g, and e), represent the probability of the states, p j ( p0 + pg + pe = 1). The current in Eq. (36) can be obtained by master equations for p j : d p0 = −g p0 + e pe , dt d p g = γ ( pe − p g ) +  g p0 , dt d pe = −γ ( pe − pg ) − e pe , dt if the optical transition rate is given by5 γ =

g 2 e . (ω − )2 + (e /2)2

(Method 2) Master Equation for Transport with Virtual States In method 1, state |0 is treated as a real state as well as states |g and |e. In the Coulomb blockade regime, however, states |0 (and |g|e) take part in the transport as virtual states. In method 2, they are taken into account as virtual states in the second-order perturbation with respect to HT . It is expressed by the effective Hamiltonian (εk >E F ) (εk E F ) (εk E F ) (εk E F ) (εk E F ) (εk E F ) (εk E F ) (εk E F ) (εk 1, the response

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would be the opposite namely the down spin electron would see the potential barrier gets lowered. An early study by Kane et al.in 1990s used a spin-diode method instead to prove hyperfine-mediated transport [20]. They observe hysteretic feature in the I–V curve and attributed the effect as a signature of dynamic nuclear polarization. The idea was then further extended to a double and single quantum point contact by Wald et al. from which similar feature was obtained as depicted in Fig. 5 [19]. Although all experimental demonstrations in the point contact have undoubtly pointed out to the hyperfine-mediated transport, however its theoretical studies are still lagging behind. Only recently Anirban et al. [21] has successfully reproduced the main feature observed in Fig. 6 by casting the hyperfine interaction directly into Fig. 5 Conductance traces through a single quantum point contact as a function of dc Wald et al. source-drain voltage with two opposing sweep directions for a vQPC > 1 and b vQPC < 1. Reprinted with permission from[19]

36.6

Rd (kΩ)

Fig. 6 75 As RDNMR of a point contact as a function of rf frequency. The spectrum is swept with increasing the frequency at a rate of 100 Hz/s and amplitude of − 30 dBm. The spectrum is measured at B = 4.5 T and T = 300 mK. Reprinted with permission from Fauzi et al. [18]

36.5

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Landauer-Buttiker equation. Similar footstep is used as well by Stano et al. but in a parallel magnetic setting [22].

5 RDNMR Lineshapes in a Quantum Point Contact Although we have discussed some of the signature of hyperfine-mediated transport in the previous section, however a definite smoking-gun evidence must be coming from NMR spectrum. In a typical experiment, dynamic nuclear polarization is immediately followed by sweeping radio frequency magnetic field across Larmor frequency of a nuclei, in this particular case to Ga or As nuclei. The rf is applied to a home-made 0.3 mm thick copper coil wounded around the device multiple times. The rf sweep itself operates in a continuous wave mode where the rf irradiation time far exceeds nulear spin coherence time on the order of milisecond. Note that coherent oscillation of nuclear spin can be measured using various rf pulse protocols as demonstrated in a number of reports [23–25]. A typical RDNMR signal of a point contact is displayed in Fig. 6. The generated in-plane rf magnetic field couples to the nuclear spin in such a way that when the rf hits the Larmor frequency, the resistance drops of about 200  since the nuclear polarization is destroyed. The nuclear spin gets repolarized when the rf passes the resonance point. One thing worth mentioning that stands out from and in contrast to RDNMR in 2D systems is that it responses well even to a very low rf power of −30 dBm or less. Similar responsiveness is also observed in Nuclear Electric Resonance of a quantum point contact [26]. Since the hyperfine field directly influences the transmission probability of a given spin-edge channel through the point contact as we explain in the previous section, one can easily understand how various RDNMR lineshapes emerged as well as their associated spin-flip scattering events in the point contact. This is surprisingly in contrast to the 2D systems, where the lineshapes are still poorly understood to date. Motivated by the lack of understanding of the lineshapes, Fauzi et al. study various RDNMR lineshapes in the point contact and the findings are summarized in Fig. 7. The measurements are carried out in a strong tunneling regime and observe four different lineshapes. In the vicinity of vqpc = 1 plateau, a dispersive (inverted dispersive) RDNMR lineshape is emerged, resembling those found in the 2D system [28]. Outside this regime, the resonance turns into an usual dip or peak structure. The peculiar dispersive lineshapes emerge due to simultanous occurence of two spinflip scattering events, giving rise to two localized regions with opposite nuclear spin polarization. Although both of them are in contact with electrons in the point contact, but they polarize in a region with different degree of electron spin polarization.

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Fig. 7 Various emerging RDNMR lineshapes observed in the point contact in the so-called strong tunneling regime. The spectrum is measured at B = 4.5 T and T = 300 mK. Adapted with permission from Fauzi et al. [18]

6 Structural Lattice Deformation Another important aspect of a high-spin nucleus (I > 1/2) is that it has a quadrupole moment that can sensitively sense electric field gradient (EFG) from the surrounding environment. One major source of EFG is coming from structural deformation that forcibly disturbs charge arragements in the host crystal. Anisotropic strain variation of less than 10–4 can be detected by a quadrupole nuclei [29]. The electric field gradient V (EFG) and strain ε are linked by elastic tensor matrix S as follow. ⎞ ⎛ ⎞⎛ ⎞ ⎛ S11 S12 S12 εx x Vx x ⎝ Vyy ⎠ = ⎝ S12 S11 S12 ⎠⎝ ε yy ⎠. (1) Vzz S12 S12 S11 εzz

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Fig. 8 a–b Differential thermal contraction between metal and semiconductor when cooled from room (RT) to low (LT) temperature. c Energy diagram of I = 3/2 nuclear spin system in the presence of external magnetic field (left), non-zero quadrupole splitting (middle), and Knight fields (right). Reprinted with permission from Noorhidayati et al. [27]

Due to GaAs crystal symmetry, 2S 12 = − S 11 [30]. First-order quadrupole splitting ( Q  E Z ), with the magnetic field oriented parallel to the principal z-axis, is given by  

eQ S11 eQ S11 1 εzz − εxx + εyy , Q = Vzz = 2h 2h 2

(2)

here the new notation S 11 is the elastic tensor component. The first order quadrupole splitting modifies the 3/2-spin nuclear energy level into three non-equidistance levels as depicted in Fig. 8b, namely central and two sattelite transitions. It is important to note that for the first oder perturbation, the central transition is not affected by the quadrupole splitting but the Knight field. We will come back again to this important point and put to use in Sect. 9. A common example of structural deformation involves nano-meter metal gate patterning on GaAs semiconductor surface used to restrict electron/hole movement into reduced dimension. Since the metal and semiconductor have different coefficient of thermal expansion, naturally strain would develop at the interface when the device is cooled to cryogenic temperature as schematically displayed in Fig. 8a. The strain then propagates down to the active 2DEG layer embedded in the semiconductor and may alter its electrical and optical properties. However, the imprinted strain distribution in the nano-gate/semiconductor platform is largely unknown, partly due to the fact that the initial strain from differing thermal expansion coefficients has to be experimentally determined. For a practical purpose, let consider the case for a triple-gated quantum point contact device shown in Fig. 1a. An elastic model calculation allows us to evaluate the strain pattern imprinted 175-nm below the surface where the channel is located as depicted in Fig. 9a. The initial strain developed at the interface is taken to be 5 × 10–3 to match the experimental data shown in Fig. 10. Since the strain profile is mirror symmetric, one can target only half the section (x < 0) and deplete the other half. We identify three representative regions to focus on, (i)–(iii), as indicated in

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Fig. 9 a Simulated lateral strain profile using COMSOL mutliphysics and the corresponding quadrupole splitting at the center of quantum point contact located 175 nm below the surface. Reprinted with permission from Fauzi et al. [31]. b Race track-like schematic of spin edge current. The yellow dashed line indicates the gate layout hovering above the 2DEG. The black region indicates the depleted region

Fig. 10 (top panels) magneto-conductance traces and the corresponding quadrupole splitting along the conductance marked by the red line. The spectra are measured at B = 4.5 T and T = 300 mK. Inset in each bottom panel shows a represented RDNMR spectrum. Reprinted with permission from Fauzi et al. [31]

Fig. 9a. To guide the edge channel to pass through the selected region, we can use gate bias tuning technique as schematically displayed in Fig. 9b. Since this is kind of a blind experiment, many gate bias tuning trials are needed. Region (ii) can be accessed by tuning V CG = −0.45 V and V SG2 = −1.65 V fixed as displayed in Fig. 10a–b. The quadrupole splitting progressively increases from 10 kHz at V SG1 = −0.7 V to 25 kHz at V SG1 = −1.1 V. Now to access region (iii), where we expect a constant strain field far underneath split metal gate 1, tuning V CG = −0.65 V and V SG2 = −1.1 V are needed. There, the quadrupole splitting of

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about 10 kHz is detected and unchanged throughout the V SG1 bias range of interest. Finally, when we set V CG = −0.2 V and V SG2 = −2.52 V, we can access region (i) half-way in between center metal gate and split metal gate 1 where a maximum strain field is expected. The detected quadrupole splitting is about 45 kHz with all resonance peaks are clearly separated. Another lesson than can be drawn from this experiment by Fauzi et al. [31] is that a slight change in the bias voltage condition can consideralby change the strain in the channel. We expect similar thing to happen for a device with more complex gate architecture.

7 Overtone RDNMR Overtone NMR, first proposed by Tycko and Opella, is a non-linear effect arising due to interaction between a quadrupole nuclei and rf magnetic field [32]. The transition is observed at twice the Larmor frequency and is excited when the rf power is high enough so that a second-order perturbation starts kicking in. The selection rule m = ±2 normally forbidden in a normal setting is now relaxed in the overtone NMR due to the non-linearity effect. Overtone is categorically different in its origin from nuclear electric resonance, although both of them measure double quantum transition (m = ±2). Overtone is magnetic interaction while nuclear electric resonance is electrical interaction. Figure 11 compares the fundamental and overtone RDNMR spectra with the rf power delivered at the top of cryostat of −30 dBm (1 μW). The central transition normally seen in the fundamental spectrum now disappears in the overtone RDNMR. Although the power is seemingly very low, the effective power at the sample could be much lower than that due to the impedance mismatch, surpisingly it still excites the overtone transition. Further investigation on power dependence is clearly needed to know where the overtone spectrum vanishes and whether the fundamental RDNMR spectrum would follow similar fate as well.

8 Spin Dynamics in 1D Semiconductor Devices Nuclear spin relaxation rate (1/T1) measures the rate at which nuclear spins dumps away their energy to the surrounding (e.g. crsytal lattice and electrons) and equilbriate with. At low temperature where the lattice is nearly frozen, the relaxation is predominanly assisted by electrons. One then can learn a great deal of electron spin dynamics by looking at how fast the nuclear spin relaxes towards equilbrium. Skyrmions Versus Lateral Confinement Shondi et al. predicted in his seminal paper that spin excitation at filling factor v = 1 quantum Hall ferromagnet involves not only a single spin reversal but collective spin reversal known as Skyrmions [34].

270 31.0 30.9

Rd (kΩ)

Fig. 11 75 As RDNMR spectra measured at a fundametal f 0 = 33.049 Hz (upper panel) and twice the fundamental frequency (lower panel). Inset in each panel shows its possible quantum transition. The spectra are measured at B = 4.5 T and T = 300 mK

M. H. Fauzi and Y. Hirayama

SG -1.23 V

30.8 30.7 30.6 -80

-40

0

40

80

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f - f0 (kHz)

Rd (kΩ)

31.1

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0

2(f - f0) (kHz)

The existence of Skyrmion spin excitation was later confirmed by Barret et al. using optically pumped NMR technique [35]. They observe a sudden drop in the electron spin polarization as the filling factor is tuned slightly away from the exact filling factor v = 1. At sufficiently low temperature, Skyrmion is also predicted to get crystalized whose energy is gapless at long wavelength limit. A strong coupling to the nuclei is then expected in this case. Indeed a rapid nuclear spin relaxation rate observed in a 2D system by a number of independent studies confirmed the prediction [36–38]. Kobayashi et al. takes on the idea further to a quasi-one dimensional channel [33]. The lateral confinement has a similar effect to raising the temperature, namely breaking the crystal long-range ordering. In a 2D system, there are many pathways to transmit information from one end to another end, but in a 1D system the pathways is restricted and the ordering is therefore prone to fluctuations. The Skyrmion crystal is then expected to melt by the confinement. If the assertion is true, the change from crystalline to liquid state should be reflected in the nuclear spin relaxation rate measurement. To prove the hypothesis, they use a 3 μm-long split metal gates in a 2D electron gas embedded beneath the surface (inset of Fig. 12a). The device is also conveniently

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Fig. 12 a Longitudinal resistance Rxx as a function of back gate bias voltage with (V SG = −1.2 V) and without (V SG = 0 V) constriction formed. Fractional filling 2/3 is used to initialize and readout the nuclear spin polarization. b Nuclear spin relaxation rate, T 1 −1 , as a function of calculated wire filling factor under various V SG bias voltages. The measurements are carried out at B = 8.8 T and a base temperature T = 50 mK. Reprinted with permission from Kobayashi et al. [33]

equipped with back gate to control global electron density. Similar to the previous report by Hashimoto et al. [37], they use fractional filling factor v = 2/3 to initialize and readout resistively the remaining nuclear spin polarization due to interaction with Skyrmions confined in the channel as displayed in Fig. 12a. The nuclear spin relaxation rate as a function of calculated local filling factor measured at several bias voltage condition to the metal split gate is depicted in Fig. 12b. The more negative bias applied to the split metal gate V SG , the stronger the confinement becomes. For a less confined wire with V SG = -0.6 V, the relaxation rate is very fast at around vwire = 0.9 of about 0.3 s−1 , in agreement with previous reports in a 2D system. But then with increasing the confinement strength, the relaxation rate gets suppressed about 0.1 s−1 at V SG = −1.5 V. The study provides the first observation of the confinement effect on the nuclear spin relaxation rate due to interaction with the Skyrmion. However, given that there are no rigorous theoretical calculation to date, it is still not clear whether the suppression by a factor of 3 is expected at all when the Skyrmion is changed its phase from crystalline to liquid state. The 0.7 Conductance Anomaly The 0.7 conductance anomaly collectively refers to a zero-field conductance structure developed below the last integer plateau [39, 40]. The observed features are believed due to a manifestation of electron–electron interactions in a quasi 1D channel. However, what kind of electronic state is emerged is still debated to date. The community is roughly divided into two distinct camps. First is a spin camp with its variance namely Kondo-related effect [40, 41] and spontaneous spin polarization [39]. A second camp is a non-spin camp whose stories are revolved around a modified van-Hove singularity by Coulomb interaction [42]. Cooper and Triphati [43] gives a theoretical prediction that measuring the nuclear spin relaxation rate would give a promising test bed for discriminating a number of possible scenario leading to the 0.7 anomaly. However the prediction is still yet to be

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verified experimentally since generating non-equlibrium nuclear spin polarization close to zero-field in a quantum point contact is notoriously difficult to do. Recently, Fauzi et al. proposes a method to generate non-equlibrium nuclear spin polarization that might be suited for the 0.7 anomaly study using a higher Landau level state [14].

9 Electron Spin Polarization in a 1D System We have discussed the reciprocality of the hyperfine interaction in Sect. 4. The reciprocality causes a finite electron spin polarization P to be picked up by the nuclear spin as an extra magnetic field known as Knight shift. For a 3/2 nuclear spin system, its energy level will be modified as schematically displayed in Fig. 8b. Lets take a closer look at the central transition (1/2 → −1/2) so that we can exclude the influence from the first-order quadrupole interaction. The central resonance will be Knight shifted to a lower frequency by the amount given by Ks =

Au 0 n P 2h w y wz

here the new notation u0 is the semiconductor unit cell, w is the effective confined region. The normalized electron spin polarization P is proportional to the difference between the spin-up and spin-down electron density (transmittion) probability P=

T↑ − T↓ n↑ − n↓ = n↑ + n↓ T↑ + T↓

Due to a direct proportionality, one can extract the electron spin polarization from the collected Knight shift RDNMR spectra. In an attempt to solving the enigmatic 0.7 anomaly, Kawamura et al. successfully applies RDNMR pump–probe technique to measure electron spin polarization in a quantum point contact under parallel magnetic field B = 4.7 T and T = 20 mK. They collects RDNMR spectra at several bias voltages where electrons only occupy the last 1D subband as displayed in Fig. 13b and then plot the extracted Knight shift as a function of bias voltage in Fig. 13c. The data reveal that the Knight shift reaches a maximum value of around 10 kHz at the 0.7 regime and then drops down to zero when the conductance is pinched off (G = 0) since there is no electrons to participate. The Knight shift drops as well when the conductance hits the integer plateau (G = 2e2 /h). Converting the Knight shift to the polarization, they are able to estimate the polarization reaching 70% at the 0.7 regime. Moreover, the collected Knight shift profile is continously evolved in agreement with numerical simulation without bound state formation depicted in Fig. 13d. Note that the Knight shifts itself are collected in a condition where the spin degeneracy is almost fully lifted by the action of magnetic field of B = 4.7 T. It

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Fig. 13 a Schematic sequence for the pump-probe RDNMR measurement. b 75 As RDNMR spectra collected at several bias voltages. c Knight shift K plotted as a function of split gate bias below the first integer plateau. d Calculated magnetization density at the QPC center. Reprinted with permission from Kawamura et al. [44]

remains to be seen whether the conclusion of no bound state formation holds up still at zero magnetic field or close to. Similar RDNMR pump–probe technique has been used to investigate breakdown mechanism of electrons at a quantum point contact filling factor v = 1 quantum Hall effect [45, 46]. The electrons are subjected to various bias conditions from weak to strong bias regime. The collected Knight shift monotonically decreases with increasing bias voltage. This is reasonable since the inter-edge spin flip scattering (forward or backward) probability increases as well, making the effective electron spin polarization reduces. Interestingly, they found that the electron spin polarization exctracted from the Knigh shift and shot noises differs at high bias regime, namely the shot noise is saturated. They suggest that the discrepancy is due to additional breakdown mechanism namely the closure of exchange-enhanced spin gap. In addition to measuring the Knight shift, one can also measure the central transition linewidth. This would give a new piece of information on how electrons distribute themselves in the point contact when subjected to a perpendicular magnetic field. Figure 14 displays the central transition linewidth measured at 1.25 up to 7 T. Naively speaking, one would expect the central transition linewidth to scale linearly with the field indicated by the black dashed line in Fig. 14. This is because to maintain the bulk filling factor v = 2 at different magnetic field, one has to tune the electron density n. While this is true for relatively low to moderate field below 3 T, it deviates substantially from a linear fit when the field is ramped above.

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Fig. 14 75 As central transition resonance linewidth as a function of magnetic field. The black dashed line is the expected linear trend and the blue horizontal dashed line is the nuclear dipole limit of 1.5 kHz. Reprinted with permission from Noorhidayati et al. [27]

When the field is ramped up, the Coulomb interaction between particles increases in proportion to the square root of magnetic field strength. To minimize the Coulomb interaction, the particles have to distance themselves from each other. Acknowledgements We would like to thank Katsushi Hashimoto, Bhaskaran Muralidharan, and Tomosuke Aono for fruitful discussions on the subject and collaborations. We would like to thank (late) Katsume Nagase and Ken Sato for their assistance in fabrication and technical processes. We would like to thank a number of Master and Ph.D. students directly involved in the project throughout the years M. F. Sahdan, S. Maeda, M. Takahashi, A. Noorhidayati, and T. Sobue.

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Microscopic Properties of Quantum Hall Effects Katsushi Hashimoto, Toru Tomimatsu, and Yoshiro Hirayama

Abstract Macroscopic quantum phenomena are often governed by the microscopic nature of quantum systems. An example of this is the quantum Hall (QH) effect, which emerges in semiconductor quantum structures under strong magnetic fields and exhibits non-dissipative electronic transport under the QH chiral edge state. This transport is protected by the QH insulating phase, which contains the so-called localised state, and is strongly affected by the spin subsystems. In this chapter, we review the microscopic aspects of electronic and spin-related QH systems by describing various scanning-probe experiments that have been performed by our group and other researchers. We focus on non-uniform QH states involving bulk-localised and edge states and non-uniform QH spin states that couple with nuclear spins through hyperfine interaction. In particular, the latter were addressed in our recent study in which we used a scanning probe applying a nuclear resonance (NR) technique to visualise the spatial distributions of both nuclear and electron spin polarisation. We demonstrate how the microscopic structures of these states are correlated with the QH effect. Keywords Quantum Hall effect · Nuclear magnetic resonance · Scanning probe

K. Hashimoto (B) · T. Tomimatsu · Y. Hirayama Graduate School of Sciences, Tohoku University, Sendai 980-8578, Japan e-mail: [email protected] Y. Hirayama e-mail: [email protected] K. Hashimoto · Y. Hirayama Centre for Spintronics Research Network, Tohoku University, Sendai 980-8578, Japan Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai 980-8577, Japan Y. Hirayama Takasaki Advanced Radiation Research Institute QST, Watanuki, Takasaki, Gunma 370-1292, Japan © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_13

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1 Introduction 1.1 Outline The quantum Hall (QH) effect, a macroscopic quantum mechanical phenomenon occurring in solid-state materials, exhibits peculiar transport properties originating from the microscopic configuration of the QH system, which comprises electronic and nuclear spin subsystems. Although understanding these fundamental microscopic structures is important for uncovering the origin of the macroscopic QH phenomena, their properties cannot be resolved through simple transport measurement. To a certain extent, this limitation can be overcome through the use of versatile and widely used scanning probe-based electronic imaging tools to visualise quantum phenomena at high spatial resolutions. Recent scanning probe measurements have revealed QH behaviour at the microscopic level, thereby significantly improving the understanding of the QH effect. In this chapter, we will review the results of recent scanning probe experiments as follows. The remainder of this section will comprise a description of the properties of electronic transport, including effects arising from electron and nuclear spin polarisation. In Sect. 2, we introduce various electrical imaging tools based on scanning probes that have been used to visualise quantum phenomena at high spatial resolutions. Then, in Sect. 3 we present one of the key microscopic features of QH transport, that is, the incompressible phase. In particular, we discuss how nonequilibrium-transport-assisted scanning gate imaging can be used to demonstrate the robustness of the microscopic structures of incompressible QH phases that contribute to topologically quantised and non-dissipative transport. In Sect. 4, we discuss the spatial variations in the electron and nuclear spin polarisations in QH regimes. In particular, we focus on the unique scanning-probe measurements that can be obtained via scanning nuclear resonance imaging to demonstrate the microscopic origin of QH-breakdown-related phenomena. Finally, we summarise and discuss the outlook for future scanning probe measurement applications in Sect. 5.

1.2 Quantum Hall Effect When an electron confined to a plane is subjected to a strong √ magnetic field (e.g., B = 10 T), it undergoes cyclotron motion with a radius, rc = /eB (e: the electron charge, h: Planck’s constant), of approximately 8 nm. This additional magnetic confinement quantises the electron’s kinetic energy into discrete levels, i.e., Landau levels (LLs) (left panel of Fig. 1), with quantised wave functions (right panel of Fig. 1) [1]. Because the energy gaps between the LLs are proportional to the magnetic field, increasing the magnetic field strength causes the Fermi level to cross LLs. When the Fermi energy moves into a gap between LLs, the electrons essentially cannot move, resulting in the formation of a QH insulating phase also known as the incompressible

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Fig. 1 Landau quantisation under a strong magnetic field. Left: Splitting of continuous-density 2DES states into discrete Landau levels (LLn; n: integer numbers), which are further split into spin up and down states with Zeeman gaps. Right: Corresponding two-dimensional LL wave functions √ √ (along X- and Y-axes) with unit magnetic length l B = /eB ≈ 25.7/ B(tesla) nm [1]

phase. By contrast, electrons for which the Fermi energy coincides with the centre of an LL can extend over a sample, allowing the associated two-dimensional system (2DES) to enter a QH metallic phase, i.e., the compressible phase. The transition from the insulating to the metallic phase characterises the QH effect, which can be macroscopically observed via transport measurement using QH devices. Typical devices used to observe the QH effect are shown in Fig. 2a–c. The so-called Corbino disk (Fig. 2a) and Hall bar (Fig. 2b) define the lateral region of 2DES states within quantum wells (Fig. 2c); this produces electric channels that are confined either to the space between the source and drain electric contacts within the bulk region of the 2DES (Corbino disk) or the edge and bulk regions (Hall bar). The bulk transport measurements obtained using the Corbino disk device (Fig. 2d) reveal the oscillating magnetoresistance that directly accounts for the alternation between the QH insulating and metallic phases induced by the magnetic field. When each LL is fully occupied near the integer filling factors, ν = i (i = 1, 2, 3, . . .), the insulating phase prevents current flow, thereby increasing the resistance. However, when the Fermi level is pinned within an LL at a half filling factor (e.g., ν = 1.5), the metallic phase carries the current, thereby reducing the resistance. As shown in Fig. 2e the magnetoresistance measured by the Hall bar device (Fig. 2b) exhibits the opposite behaviour, with the resistance vanishing near ν = i but increasing near the half filling factors. This discrepancy can be attributed to edge transport through the so-called edge channels that connect the source and drain electrodes and contribute to the macroscopic transport in the Hall bar. Near the edge of the sample, the confinement potential caused by the physical edge induces strong LL bending, which in turn causes the Fermi level to cross the LL near the edge despite the

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Fermi level pinning at the LL gap in the bulk region. Along the metallic edge channels, the electrons drift in one direction based on the combined contributions of the cyclotron motion and electric field (E-field) induced by the imposed magnetic field and edge confinement potential, respectively. The resulting counter-propagating edge channel along both sides of the 2DES edges, namely, the chiral edge channel (indicated with arrows in Fig. 2b), demonstrates non-dissipative transport except when backscattering occurs between the edge channels [2]. This is the key microscopic phenomenon causing the non-dissipative chiral transport of the integer QH effect and is characterised by a vanishing of the longitudinal resistance and the generation of a universal quantised Hall conductance that is protected by a topological invariance [3, 4]. The robustness of the non-dissipative transport around the integer filling factors relies on the presence of the insulating area between the chiral edge channels. As previously mentioned, this bulk insulating phase emerges when the Fermi energy (E F ) is located at the mid-gap between the LLs at ν = i. When ν is varied from ν = i, E F deviates from the mid-gap and begins to overlap with the tail of the LL, which practically possesses a finite width in the energy due to broadening induced by the electrostatic-potential disorder. The QH bulk insulating phase survives as long as the resulting electronic states bound to the disorder potential form so-called localised states that are spatially isolated from each other. The bulk insulating phase prevents backscattering between the two side chiral edge channels (upper panel of Fig. 2h). Further variation of the filling factor from ν = i causes the localised states to grow and eventually connect with each other, resulting in so-called extended states that can be directly confirmed via scanning tunnelling spectroscopy, as shown in Fig. 2f, g [6]. The extended states connect the two side edge channels, thereby inducing backscattering (lower panel of Fig. 2h), and as a result, finite resistance for Hall-bar transport. In addition to the bulk insulating phase, backscattering is further suppressed by the insulating strip that forms along the edge at which the filling factor is slightly higher than the integer filling factor [7]. The bulk insulating phase (shown schematically as the light grey area in the upper panel of Fig. 3b) that emerges at ν = i splits into two insulating strips and moves to the two side edges at ν > i (shown schematically in the upper panels of Fig. 3a). These so-called incompressible strips also prevent backscattering through the bulk extended phase between the two side chiral edge channels. Consequently, backscattering is suppressed by the edge insulating phase at ν > i as well as by the bulk insulating phase, including the localised states, at ν ≤ i. The incompressible barrier can be broken by a high imposed source-drain voltage. Such an electrically induced breakdown of QH topological protection can originate from backscattering between the counter-propagating edge channels through the incompressible region [8, 9]. The bottom panels of Fig. 3a, b depict Hall potential profiles that are strongly tilted within the incompressible region by the imposed high magnitude E-fields at ν > i and ν ≤ i, respectively. In particular, the potential slopes near both edges become asymmetric across the Hall bar at ν > i, thereby inducing an asymmetric LL slope (Fig. 3e) and localised nonequilibrium phenomena such as

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Fig. 2 Transport measurements of quantum Hall (QH) effect. a Measurement setup using a Corbino disk device. b Measurement setup using a Hall bar device. Arrows indicate the metallic edge channel. c Quantum well structure in which a two-dimensional electron gas is confined to a width of 20 nm within an InSb quantum well. d Measured two-terminal magnetoresistance of Colbino disk device. e Longitudinal resistance Rxx and Hall resistance Rxy obtained via fourterminal measurement using a Hall bar. f, g QH localised and extended states probed via scanning tunnelling microscope. The measurements were performed at B = 12 T, T = 300 mK on a surface 2DES induced by deposition of Cs onto InSb. h Schematic of current flow in a Hall bar. Top: Edge channels and bulk localised states at ν = i. Bottom: Backscattering through the bulk extended states at ν = i. a–e are copied from Ref. [5] and used in accordance with the Creative Commons Attribution (CC BY) license. f, g Ref. [6].

quasi-inelastic tunnelling, heating, and phonon and/or photon emission [10]. These scenarios can effectively explain the results of transport measurements [11], used to determine the threshold voltage of the E-field QH breakdown (Fig. 3c) that is characterised by backscattering through the QH insulating phase. The threshold voltage, which is plotted in the figure as a function of the filling factor, exhibits different Hall-bar width dependencies above and below ν = 2.0. At ν ≤ 2, the threshold voltage strongly depends on the Hall-bar width; at ν > 2, this dependence weakens. The threshold-voltage curves normalised by Hall-bar width in Fig. 3d overlap at ν ≤ 2, indicating that the threshold increases linearly with the Hall-bar width. This can be explained by the widening of the bulk incompressible (insulating) phase (Fig. 3b)

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Fig. 3 QH breakdown induced by inter LL scattering. a, b Upper panel: Schematics of compressible and incompressible areas across the Hall bar at ν > i (a) and ν ≤ i (b). Lower panels: Evolution of Hall potential profiles from equilibrium (low source-drain voltage) to nonequilibrium (high source-drain voltage) conditions across the Hall bar along the horizontal line marked in (a), (b). c, d Dependence of threshold voltage for QH effect breakdown on filling factor. The vertical axis indicates the threshold voltage (c) and the value normalised by Hall bar width (d). The filling factor is varied by sweeping the magnetic field over the range B = 8–10 T at a constant electron density. The measurement temperature is 1.3 K. e Schematic LL profile across Hall bar at nonequilibrium condition. a–d are copied from Ref. [11] and used in accordance with the Creative Commons Attribution (CC BY) license. e Ref. [10]

with increasing Hall-bar width, which eventually increases the threshold voltage for scattering through the bulk phase. By contrast, the width of the edge incompressible strip (Fig. 3a) is insensitive to the Hall-bar width, resulting in a weaker relationship between the threshold voltage and the Hall-bar width at ν > 2.

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Fig. 4 Restively detected nuclear magnetic resonance (NMR) in 2DES confined within a GaAs quantum well. The NMR signals are captured at T = 340 mK and ν = 2/3 (B = 5.8 T). The inset schematically depicts the setup of the detection system

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1.3 Resistively Detected Nuclear Magnetic Resonance in QH Effect A QH system under a strong magnetic field comprise a wide variety of electronic spin states that often couple with the nuclear spins of the host material through hyperfine interactions. Examples of hyperfine-coupled systems are found in integer [12] and fractional [13, 14] QH systems in the nonequilibrium condition induced by high-magnitude imposed electric currents. The accompanying E-field imposed on the 2DES drives electron tunnelling between energetically degenerate spin LLs (LL↑ and LL↓). The resultant electron-spin flip transfers its angular momentum to the nuclei, thereby driving the dynamic nuclear polarisation (DNP). Current-driven DNPs of 85 and 15–20 % have been achieved using spin-resolved edge channels [15] and fractional QH spin domains [16, 17], respectively. Nuclear spin polarisation in hyperfine-coupled QH systems can be resistively detected with high resistive sensitivity using nuclear resonance (NR) measurements. Nuclear magnetic resonance (NMR) measurement has been carried out via resistive detection and current-driven DNP in a number of systems, including spin-resolved QH edge channels [18] and integer and fractional QH bulk states [12, 14, 19]. To perform NMR measurement in these cases, radio frequency (RF) magnetic fields were applied to the samples by wrapping several turn of coil around them, as shown in the inset of Fig. 4. The resistively detected NMR results are shown in Fig. 4. The resistance, pre-enhanced by DNP, decreases owing to NMR-induced depolarisation at the Larmor frequency of the host nuclear elements of the GaAs quantum well.

1.4 Nuclear Electric Resonance of Quadrupolar Nuclei RF magnetic fields coupled with nuclear magnetic dipole moments are generally used for NMR. The RF E-field at the Larmor frequency can also couple with the nuclear

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| −3⁄2⟩

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| −1⁄2⟩ | +1⁄2⟩ | +3⁄2⟩

Fig. 6 Schematic energy-level diagram of the nuclear spin states of 75 As for I = 3/2. The electric quadrupole coupling (red arrows) can drive both m = ±1 and m = ±2 transitions between levels that have different directions of charge distribution (blue ellipsoid shown to the right) [24, 25]. By contrast, the magnetic dipole coupling induces m = ±1 transitions (black arrows)

quadruple moments that are intrinsic to quadrupolar nuclear spins such as the total spin I > 1/2 [20]. The RF E-field dynamically distorts the electronic environment of the nucleus—and, therefore, the E-field gradient—eventually producing nuclear electric resonance (NER) transition. This NER method works at most nuclear spins, as indicated in blue in Fig. 5. For instance, all host nuclei elements of the GaAs quantum well, i.e., 75 As, 69 Ga, and 71 Ga have I = 3/2 quadrupolar nuclei containing I = 3/2 levels labelled by the magnetic quantum number m = ±1/2, ±3/2 (Fig. 6). The electric quadrupole transition, which is driven by the E-field gradient, involves m = ±1 and ±2 transitions (marked in the figure by red arrows). This contrasts with the effect of the RF magnetic field, which drives a single m = ±1 transition (marked by black arrows). The selection rule of electric quadrupole transition is based on the non-spherical charge distributions of high-spin nuclei (I > 1/2), which are schematically shown as ellipsoids on the right side of Fig. 6. Only transitions between different charge-distribution levels are allowed, resulting in m = ±1 and ±2 transitions. This NER has been demonstrated in the bulk [21, 22] and in quantum wells [23, 24] of GaAs. The NER method can be conveniently applied for producing local NR for scanning probe imaging, which will be discussed later in Sect. 4.

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1.5 Resistively Detected Knight-Shift Measurement in QH Effect Resistively detected NR provides an essential probe of electronic spin polarisation. In particular, in materials that have the s-type conduction band electrons, the nuclei experience an effective magnetic field under a spin-polarised electron system via hyperfine Fermi-contact interaction, which leads to shifts in the NR frequency. The absolute value of this NR frequency shift is called the Knight shift and is proportional to the electron spin polarisation. Direct measurement of electron spin polarisation has been used for the GaAs 2DES, which has the s-type conduction band electrons, to study various exotic QH states, including QH ferromagnets at ν = 1 [26], (and also at 2/3 [27]), quantum fluids at ν = 5/2 [28], and canted antiferromagnets in the bilayer at ν = 2 [29]. Figure 7a shows a typical Knight shift between two 75 As nuclear resonance spectra captured in fully spin-polarised (black spectrum), ν = 1, and unpolarised QH states (red spectrum), ν = 2, respectively. The nuclear resonance peak measured at the fully spin-polarised QH state is clearly shifted down by K s = 43.7 kHz relative to the resonance frequency obtained at the unpolarised QH state. Figure 7b shows the spectra obtained at ν = 1.0 as the source-drain current, Isd , is increased to the QH breakdown regime. The spectrum captured at Isd = 1.7 µA shows an apparent peak near the NR frequency for a maximum Knight shift K s = 43.7 kHz. As Isd increases, the spectral peaks broaden and develop accompanying tails on the lowerfrequency side. The observed broadening indicates that the fully spin polarised QH state undergoes breakdown, which can occur in either a homogeneously or inhomogeneously spatial manner, which must be determined using microscopic observation of the electron spin polarisation.

2 Scanning Probe Techniques As discussed in Sect. 1.2, the QH effect is caused by the microscopic configuration of QH states. In particular, the QH effect for a high mobility 2DES confined within, e.g., a GaAs quantum well can be affected by a spatially inhomogeneous distribution of nuclear and electron spin polarisations (Sect. 1.5). For 2DESs that are exposed on a surface, e.g., graphene or an adsorbate-induced 2DES [6], scanning tunnelling spectroscopy is a powerful tool for accessing nanoscale QH behaviour (see Fig. 2f, g). However, this approach cannot be applied to a high-mobility 2DES in the quantum well that is embedded beyond a few hundred nanometres below a surface and covered by an insulating capped layer. To probe the local properties of transport (involving electron/nuclear spin polarisation) of such quantum-well 2DESs, it is necessary to apply a scanning probe along the insulating surface using a method based on atomic force microscopy (AFM). In this section, several AFM-based scanning probe methods—in particular, a unique scanning gate imaging method—are introduced.

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2.1 Scanning Gate Imaging Scanning gate imaging allows for the successful visualisation of the transport properties of a high-mobility 2DES [31]. In this technique, a metallic probe is used as a local gate to carry out scanning using AFM. The local gating influence is detected using an electrical transport measurement, and the resulting images show quantised transport channels flowing through the quantum point contact. This technique has also been applied to the detection of high-magnetic-field phenomena such as the QH effect [32–36] and the quantum spin Hall effect [37]. Other imaging techniques, including single-electron transistor [38], Hall-potential [7, 39], microwave impedance [40], and capacitance imaging [41], have been also used for the study of the QH effect.

2.2 Nonequilibrium-Transport-Assisted Scanning Gate Imaging The scanning gate technique has been further combined with nonequilibrium transport techniques to map the QH transport property. A schematic of a detection setup applying this approach is shown in Fig. 8a. In this design, a low tip voltage of

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Vtip ∼ 0.2 V is used to compensate for the contact potential mismatch between the tip and sample; this contrasts with conventional scanning gate imaging approaches [32, 34–36], in which highly negative tip voltages are applied to deplete electrons. To address the QH phase at ν  i and obtain a local signal, nonequilibrium transport is used to induce backscattering between the chiral edge channels through an incompressible region. Figure 8b depicts the alternating compressible and incompressible regions that form along an edge of the Hall bar at ν ≥ i. The local ν (νL ) of an incompressible strip is maintained at νL = i, whereas the bulk ν is modified by sweeping B or n s . To achieve the nonequilibrium condition, the source-drain current (Isd ) is increased until the Hall voltage deviates from the QH condition. The imposed Hall voltage predominantly enhances the potential slope within the innermost incompressible region located at the boundary between the edge and bulk phases [7], as previously shown in Fig. 3a and e, thereby inducing inter-LL tunnelling from the edge to the bulk through the innermost incompressible strip. Electrons are then scattered to the opposite counter-propagating edge channel through bulk compressible or directional-hopping channels along the Hall E-field. This backscattering process produces a dissipative current [10, 42] and, in turn, a nonzero longitudinal resistance. The tip provides a small local electric perturbation to the 2DES as a result of the rearrangement of the effective potential mismatch by the imposed excess Hall voltage. This eventually bends the LL locally, thereby increasing the inter-LL tunnelling rate, as shown in Fig. 8b. The tip-induced inter-LL tunnelling further enhances backscattering and, as a result, the longitudinal voltage (Vx ). By mapping the resulting V , the innermost incompressible region can be visualised.

2.3 Scanning Nuclear Resonance Imaging The scanning-gate-imaging setup explained in the previous subsection Sect. 2.2 can also be used in imaging based on NR scanning. For local NR excitation, a scanning metallic tip is used to apply an RF E-field to the nuclear spins, which is dynamically polarised by the current in the GaAs quantum well. The E-field dynamically distorts the electronic environment of the nucleus. This induces a time-varying E-field gradient that can couple [24, 25] with the quadrupolar nuclear spins, which eventually produces the NR, as mentioned in Sect. 1.4. The induced local NR is detected using an electrical transport measurement technique, namely, resistive detection (Sect. 1.3). This detection scheme relies on the Overhauser field of nuclear spin polarisation, which modifies electronic Zeeman splitting to vary the resistance in the spin-split QH regimes.

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Fig. 8 Local NR measurements. a Schematic of experimental setup for scanning gate and NR imaging. Vx is recorded as the tip is scanned at Vtip with Isd and B. In the scanning NR microscope, the RF E-field-mediated NR is excited by applying an RF voltage to the metallic tip via a continuously pumping current (Isd )-induced DNP in the QH breakdown regime near ν = 1. The filling factor is tuned by the B-field and the back-gated electron density. NR is detected using the resistive detection technique, i.e., measuring the longitudinal voltage Vx . b Schematic of inter-LL tunnelling (marked by red arrows) between two LL sub-bands near the edge of the higher μchem side under the nonequilibrium condition. This condition is derived from the deviation between the Fermi energies in the edge (E f,edge ) and bulk (E f ) compressible regions, respectively. Tip-induced LL bending, indicated by the blue dashed line, enhances the inter-LL tunnelling which accompanies DNP. The incompressible and compressible strips are indicated by ‘IS’ and ‘CS’, respectively. ‘Edge’ and ‘Bulk’ indicate the edge strips and 2DES bulk region, respectively. The magnitudes of νL with respect to the exact integer i are also shown for each region. b Copied from Ref. [43] and used in accordance with the Creative Commons Attribution (CC BY) license

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3 Microscopic Properties of Quantum Hall Electronic System As mentioned in Sect.1.2, the QH transport occurring in a Hall bar is determined by the edge and bulk transport phenomena. Near the edges of the sample, the confinement potential induced by the physical edge bends the LLs to form spatially alternating incompressible and compressible strips. In particular, the incompressible strips play an important role in the suppression of backscattering between the counterpropagating edge channels. The filling factor dependence of the incompressible strips can be identified via Hall-potential imaging [7, 39]. Figure 9 shows Hall-potential

Fig. 9 Filling-factor dependence of incompressible region obtained via Hall potential imaging. The Hall potential curves along the same line across the Hall bar are plotted at different filling factors around ν = 2. The slope region in the Hall potential corresponds to the incompressible region that appears near the edge at ν = 2.5 and distributes into the bulk region as ν reduces to the integer value i. After Ref. [44] (©Clearance Center’s RightsLink)

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profiles captured across a Hall bar at around ν = 2. As shown in Fig. 3a and b, the Hall potential exhibits a slope in the incompressible region but becomes flat in the compressible region owing to screening effect. The potential slope is predominantly enhanced at the innermost edge of the incompressible strip (Fig. 3a) at ν > i, which merges into the bulk incompressible region (Fig. 3b) at ν  i. The expected filling factor dependence of the Hall potential is seen in the experimentally observed Hallpotential profile in Fig. 9, in which the innermost edge incompressible strip observed at ν = 2.50 moves and spreads to the bulk as the filling factor reduces to ν ≤ 2.00. The filling-factor-dependent incompressible strip/region can be further imaged along the x-y plane via nonequilibrium-transport-assisted scanning gate imaging, as described in Sect. 2.2. In the nonequilibrium regime, scanning-gate imaging can pinpoint the areas susceptible to the breakdown of topological protection, thereby providing access to the local breakdown occurring in the incompressible region. The resulting images (Fig. 10a) exhibit distinct line-like patterns extending in the x-direction along a Hall bar edge (left dashed lines) that correspond to the side with the higher chemical (lower Hall) potential across the y-direction of the Hall bar, as shown in Fig. 3e. Decreasing ν from 2.17 shifts the position of the line patterns and widens to the bulk of the 2DES (Fig. 10a). In Fig. 10b, the positions (dots) and widths (bars) of the line patterns are compared with the extent of the innermost QH

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Fig. 10 Scanning-gate microscopy (SGM) measurements near ν = 2; copied from Ref. [43] and used in accordance with the Creative Commons Attribution (CC BY) license. a Representative V images at different ν tuned by electron density at B = 4 T near ν = 2. The scale bar is 4 µm. Dashed lines denote Hall bar edges. b Positions (dots) and widths (bars) of V peaks as a function of ν. The positions and widths are respectively determined by the distance from the Hall bar edge (yk ) and the full width at half maximum (WFWHM ). The grey area denotes the incompressible regions determined using local spin-density approximation (LSDA) calculation for ν = 2 at the equilibrium condition

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incompressible region (grey area) calculated via local spin-density approximation (LSDA) [45, 46]. It is seen that the ν evolution of the incompressible phases that is observed in the nonequilibrium condition is in good agreement with the LSDA calculation. This indicates the robust QH nature corresponding to the evolution of the innermost incompressible QH phase from the edge strip into the bulk localisation. The same tendency is also observed in the spin-gap incompressible region that emerges at odd ν, as shown in Fig. 11a–c. Figure 11c shows remarkable closed-loop patterns (around the white crosses) over the entire Hall bar in the x-direction. The observed loop structure can be attributed to an incompressible barrier encircling compressible puddles [47] in which electrons accumulate to screen the potential valleys, as depicted in Fig. 11f. Increasing the Isd used to obtain the low-current images in Fig. 11a–c by a factor of 2–5 (as indicated by the Vx − Isd curves in Fig. 11g) causes the local incompressible patterns near the edge (Fig. 11a) and in the bulk region (Fig. 11b) to gradually expand into the compressible region with increasing Isd until they cover the entire region and exhibit a dense filament pattern independent of ν (Fig. 11d, e). The observed vanishing of the ν dependency in the patterns clearly indicates a global breakdown of the ν-dependent QH effect.

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4 Microscopic Observation of Hyperfine-Coupled Quantum Hall Systems Various QH electron spin states have been microscopically studied, primarily through the use of optical methods. Spin-polarised edge states can be resolved by applying circularly polarised near-field optical microscopy [46], whereas the electron spin distributions of the respective domains can be visualised in the QH ν = 2/3 domain system using scanning photoluminescence microscopy [48]. A unique scanningprobe-based method has also been developed to probe QH spin polarisation without the influence of photo-excited carriers. In this section, the spatial variation of the nuclear and electron spin polarisation in one of the hyperfine-coupled systems—the QH breakdown regime—is demonstrated via the scanning NR imaging technique. [30].

4.1 Local NR and Resistive Detection As described in Sect. 2.3, local NR can be driven by an RF E-field irradiated from a metallic tip and the resulting NR spectrum can be resistively detected. Figure 12a shows raw spectral data for a 75 As NR recorded using the sweeping continuous-wave frequency of an RF E-field at B = 8 T, along with the ν = 1.05 QH breakdown condition. At the point marked by notation ‘i’ near the left-side edge of the Hall bar (inset of Fig. 12a), an NR signal (red curve) is clearly observed at a frequency of ∼115.725 MHz. However, the intensity of the NR spectrum is strongly suppressed at the centre (point ‘ii’, black curve) and right-side edge (point ‘iii’, blue curve) of the Hall bar. The NR intensity (Vx ) is mapped as an area (dashed square in Fig. 12b) and is displayed in Fig. 12c. The clear line pattern extends over a distance of 25 µm in the x-direction along a Hall bar mesa edge marked by dash lines at ν = 1.10. Reducing the filling factor to ν = 1.00 causes the pattern to eventually cover the interior of the Hall bar (Fig. 12d). The observed ν-dependent patterns can be interpreted as regions in which DNP is driven by inter-LL scattering [12] between the oblique LL0↑ and LL0↓ in the incompressible QH strip (Fig. 3e), which exhibits the same ν dependence shown in Fig. 11a–c. In Fig. 13a, the DNP region delineated by the line pattern switches to another edge side when the source-drain current direction is reversed. This indicates that DNP is driven at the side of the lower-Hall (higher-chemical)-potential edge. Photoemission is observed along the opposite edge in Fig. 13b [49]. These experimental results microscopically confirm the proposed local nonequilibrium phenomena that are depicted in Fig. 3e.

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Fig. 12 NR-intensity mapping; copied from Ref. [30] and used in accordance with the Creative Commons Attribution (CC BY) license. a Local 75 As NR spectra in the ν = 1.05 QH breakdown region captured using a sweeping-frequency continuous-wave E field at B = 8 T, T < 300 mK and Isd = 0.6 µA. Spectra i, ii, and iii were captured at the points marked on the topographic profile of the Hall bar (inset). The obtained spectra are displayed in their corresponding colours after subtracting the position-dependent Vx offset from the raw spectra. b Optical micrograph of Hall bar, with directions of the magnetic field and electron drift indicated. c, d NR-intensity images taken at B = 7 T near the critical current of QH breakdown at different values of ν in the area marked by the broken square on the Hall bar (b). The dashed lines indicate the Hall bar mesa edges

4.2 NR Spectroscopy Mapping Using the DNP induced over the bulk region, as shown in Fig. 14b, NR-spectroscopic mapping was performed at Isd = 2.6 µA. Figure 14c shows the representative spectra (circles) with curves fitted (red curves) at the spatial points marked by crosses along the expected path of electron drift (Fig. 14b). The spectrum peak shift from 115.74 to 115.76 MHz between points ‘i’ on the electron injection side (the upper-left voltage probe) and ‘v’ on the lower side of the Hall bar demonstrates that K s decreases spatially from 35.5 to 27.5 kHz. This trend is strikingly clear in the K s image in Fig. 14f, which indicates a spatial reduction in the electron spin polarisation Pe . The maximum reduction in K s is K s  9.2 kHz, which corresponds to the variation Pe  21%. The observed spatial variation of electron depolarisation can be explained by the electron heating of the fully spin-polarised QH state. Although the spatial variation in Pe observed at Isd = 2.6 µA is already substantial at Isd = 2.0 µA (Fig. 14e), it weakens near the critical current. At Isd = 1.6 µA (Fig. 14d), Pe appears to be nearly constant and close to 100% over most of the region. This indicates that the fully spin-polarised QH state is robust near the critical current of QH breakdown. The observed Isd -induced evolution of spatial gradient in Pe evolves with Isd (e.g., by a factor of 1.6 when Isd is increased from 2.0 to 2.6 µA), thereby broadening the K s distribution. The spatial variation in K s accounts for the broadening of the global NR spectra that is shown in Fig. 7b.

294 Fig. 13 Comparison between DNP and photo emission regions. NR-intensity images (a) and photo-emission mapping (b) obtained when the source-drain current direction is reversed. b After Ref. [49] (©American Physical Society)

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5 Summary and Outlook Scanning probe imaging can be used to produce microscopic depictions of QH transport that reflect electron/nuclear spin polarisation. The results obtained from this imaging provide essential microscopic information that complements our understanding of the QH phenomena obtained via conventional global measurements. Scanning probe methods are constantly evolving, allowing them to tackle topical issues of solid-state physics, e.g., the suppression of the quantised conductance of the quantum spin Hall effect [37, 50] and the nonzero longitudinal resistance of the anomalous quantum Hall effect [51, 52]. In particular, the unique technique of scanning nuclear resonance imaging can be utilised in the direct microscopic examination

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of various hyperfine-coupled quantum systems such as QH skyrmions [14, 26] chiral QH edge states [5], helical nuclear magnetism [53], and spin-helical topological surface states [54]. Furthermore, local nuclear electric resonance can be utilised in the imaging and manipulation of single-atomic nuclear spin [55]. These tools for the microscopic observation of novel solid-state properties might enable the uncovering of new nanoscale science and technology applications.

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Semiconductor Chiral Photonic Crystal for Controlling Circularly Polarized Vacuum Field Satoshi Iwamoto, Shun Takahashi, and Yasuhiko Arakawa

Abstract Three-dimensional photonic crystal (3D PhC) structure is one of the fascinating platforms providing versatile possibilities to manipulate photons and their interaction with materials. In particular, chiral 3D PhCs can modify the circularly polarized vacuum field inside the structures enabling the control of the interaction between materials and circularly polarized photons. Here, we will discuss a type of chiral 3D PhCs consisting of semiconductor-air gratings stacked with different angles. The chiral 3D PhC can exhibit giant optical activity without using any chiral materials. Moreover, when quantum dots are introduced inside the structure, highly circularly polarized emission can be realized reflecting the modification of circularly polarized vacuum field inside the chiral PhC. We will also discuss the future prospects of the potential applications of the chiral PhC for hybrid quantum systems. Keywords Photonic crystal · Chirality · Circularly polarized light · Photonic density of states · Vacuum field · Quantum dot

1 Introduction Light-matter interaction is a fundamental physical process that underlies all optical phenomena including refraction, reflection, light emission process, and others. The interaction depends not only on the properties of materials but also on the properties S. Iwamoto (B) Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan e-mail: [email protected] Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan S. Iwamoto · Y. Arakawa Institute for Nano Quantum Information Electronics, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan S. Takahashi Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_14

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of the light field (electromagnetic field) in which the materials are placed. Purcell effect [1] is a well-known example as a consequence of light-matter interaction in modified electromagnetic environments. The spontaneous emission rate of an atom placed in a cavity can become higher than that in free space. The cavity increases the density of states of photons and enhances the vacuum field strength at the position of the atom, that is, changes the electromagnetic vacuum from that in free space. This results in the enhancement of the interaction of the atom with cavity photons, leading to the acceleration of emission rate. Recent advances in nanophotonic structures make it possible to engineer the electromagnetic vacuum to observe the Purcell effect even in the optical frequency range. Single-photon sources utilizing the Purcell effect on solid-state quantum emitters like semiconductor quantum dots (QDs) and color centers are now intensively studied for the application to quantum information technology [2, 3]. A representative structure of photonic nanostructures is photonic crystals (PhCs) [4–6]. PhCs are artificial optical materials having a periodic structure of dielectric constant whose periodicity is of the order of optical wavelength. Due to the periodicity, photons in PhCs follow the band structure (photonic band structure), which defines the properties of the vacuum field within the structures, as electrons in solid do. Photonic band engineering produces unique light propagation phenomena such as negative refraction [7, 8] and slow light [9]. Besides, controlling defects artificially introduced in regular PhCs allows one to realize high-Q nanocavities with small mode volume V [10], which have been applied to such as filters, nanolasers, and so on. PhC nanocavities coupled with single QDs are a good testbed for the fundamental study on solid-state quantum electrodynamics [11, 12]. Most of these studies are based on controlling the vacuum fields for linearly polarized light. On the other hand, circularly-polarized (CP) light has been used in various applications, for instance, biomolecular sensing and three-dimensional (3D) display. CP light is also playing important roles in spintronics and in quantum information devices such as spin-photon interfaces, which are one of the key quantum hybrid systems enabling quantum memories and long-distance quantum networks. A typical conventional method to get CP light is to convert linearly polarized light by using birefringence or optical activity in natural materials. The generation of CP photons directly from spin-polarized electrons in materials has been investigated, which usually requires an external magnetic field or special techniques to inject spinpolarized carriers into active media. Engineering the vacuum field for CP photons and tailoring the interaction between materials and CP photons will be a new approach for controlling and generating CP photons. Chiral PhC is one of the photonic nanostructures enabling the engineering of CP vacuum field. Chiral PhC, which is usually made of non-chiral material, consists of a periodic array of a unit structure without mirror and spatial inversion symmetry such as chiral and helical structures. As for chiral materials in nature, chiral PhCs show different optical responses to CP light depending on their handedness, which realizes artificial optical rotation and circular dichroism. As reviewed in the next section, there are several structures regarded as chiral PhCs having their periodicity in one-, two-, and three-dimension. Among them, since the electromagnetic

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field of a CP state inherently has 3D nature, 3D chiral PhCs have wide possibilities for controlling CP light and its interaction with matter in an artificial manner. In particular, semiconductor-based chiral 3D PhCs are a fascinating platform. Large refractive indices of semiconductor materials enable the stronger modulation of the vacuum field for CP light. The optical responses can be potentially controlled in an electrical manner. More importantly, it is easy to introduce high-quality light emitters including QDs into the structures, which can make them applicable to efficient CP-light emitting devices, spin-photon interfaces, and so on. In this Chapter, we discuss semiconductor-based chiral 3D PhCs. In Sect. 2 we will briefly review some of the chiral PhCs. Then, in Sect. 3, we will explain 3D chiral woodpile structures in more detail, which is followed by a detailed description of the fabrication scheme. Artificial chirality realized in such structures will be discussed. Section 4 is devoted to the discussion about the control of CP light emission from QDs in the chiral 3D PhC without an external magnetic field. As well as numerical analysis, we will show the experimental results indicating the influences of CP vacuum altered by the chiral 3D PhC on CP light emission from QDs. Finally, we will provide future prospects in Sect. 5.

2 Chiral Photonic Crystals Chiral optical responses and chiral light-matter interactions have been investigated in various types of nanophotonic structures such as plasmonic structures [13], metamaterials [14], and nanophotonic waveguides including 2D PhC waveguides [15]. Here we briefly review some of chiral PhCs having chiral or helical structures. Cholesteric liquid crystal is in one of the self-assembled liquid–crystal phases under a particular condition. In the cholesteric phase, the direction of the molecular axis changes gradually from layer to layer and a helical structure with a certain pitch is formed. Since the in-plane layers are homogeneous, the cholesteric liquid crystals behave as a one-dimensional chiral PhC. As a consequence of Bragg reflection due to the periodicity along the helical axis, a stop gap for light propagating along the axis is opened. A unique feature of the stop gap is the strong polarization dependence. Within the stop gap, light with a particular CP state will be selectively reflected depending on the handedness of the helical structure. Thus, such a stop gap is referred to as a polarization gap. In addition to the observations of the polarization gap, lasing oscillations using liquid–crystal chiral PhCs doped with dyes have been demonstrated. By utilizing a large difference in density of states for two CPs at the polarization band edges, band-edge lasers with a strong CP contrast were realized [16]. Defect-mode lasing using a cavity structure consisting of a defect layer sandwiched by two liquid–crystal chiral PhCs was also demonstrated [17]. Details and other demonstrations of liquid–crystal lasers can be found in a review paper [18]. Quasi-2D PhCs with chiral patterns can also show strong chiral responses. Huge optical rotation was observed in a quasi-2D PhC where gammadion structures are patterned on top of a dielectric planar waveguide [19]. Ellipticity measured in the

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transmitted light shows the opposite sign for left- and right-twisted gammadion patterns, indicating that the optical activity in the structures is induced by the chiral patterns. Similar structures embedding semiconductor QDs in the waveguide layer have been reported. The anisotropy of the vacuum fields between two opposite CP states at the plane of QDs results in the CP light emission from QDs with a high degree of polarization [20, 21]. Furthermore, using such a quasi-2D chiral PhC made on top of a Bragg microcavity with GaAs quantum wells, CP lasing has been demonstrated [22]. In those structures, the presence of the waveguide layer or the Bragg mirror below the chiral patterns induces structural anisotropy which is essential to exhibit the optical activity. 2D chiral patterns without such anisotropy along the direction perpendicular to the plane don’t show any polarization effects [23]. This is the reason why these structures are called “quasi”-2D structures. 3D PhCs, which can fully utilize the third dimension, are expected to manipulate chiral responses and to engineer the CP vacuum over a broad spectral and spatial range. 3D PhCs whose unit cell contains several helices overlapped each other with a spiral unit structure were investigated theoretically as possible structures with a wide complete photonic bandgap, at first [24, 25 ]. Following these theoretical proposals, helical 3D PhCs were fabricated and investigated in terms of their photonic bandgap properties [26, 27]. However, chiral optical responses were not discussed at that time. After a while, it was found that a PhC with helices can have a polarization gap along the helical axis [28]. Separating the helices rather than overlapping them as in [28] opens a significant polarization gap while the complete photonic gap closes. The theoretical prediction was soon experimentally confirmed by using a polymetric chiral 3D PhC fabricated by direct laser writing (DLW) technique [29]. Another kind of 3D chiral PhCs is a rotationally stacked woodpile PhC or a chiral woodpile structure which is discussed in the following sections. These chiral PhCs were firstly discussed theoretically as a possible structure to obtain circularly polarized thermal radiation. Significant contrast in the density of states for two opposite CP states allows a highly-circularly polarized thermal emission [30]. The theoretical study stimulated the experimental realization of the structure [31]. The polymetric chiral PhC showed strong circular dichroism at the telecommunication wavelength range. Apart from these 3D chiral PhCs showing uniaxial chirality, bi-chiral structures [32] and gyroid networks [33, 34] have been investigated as chiral 3D PhCs with cubic symmetries. Chiral PhCs having such high spatial symmetry are expected to realize novel applications. In fact, a chiral beam splitter, which splits two opposite CP light as a conventional polarization beam splitter, has been demonstrated using a gyroid PhC fabricated by DLW technique [35]. These chiral 3D PhCs have different band structures for two CP states of light propagating along the helical/chiral axis of the structures. Consequently, they exhibit giant optical birefringence and broadband circular dichroism compared with those in optically active materials in nature. Furthermore, the difference also creates a large unbalance in the density of states for two CP states, which enables the control of light emission properties of the materials embedded in the structure.

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So far, most of the demonstrations of the chiral 3D PhCs have been limited for the polymer-based structure, because of the complexity of the chiral structures besides the requirement of sophisticated techniques for fabricating 3D PhCs. As discussed in the previous section, the semiconductor platform has some advantages particularly for the application to CP light-emitting devices, spintronics, and quantum information. Thus, the development of semiconductor-based 3D chiral PhCs is expected to pave the way to wider applications of chiral PhCs. In the following Sections, we will discuss the recent progress on semiconductor-based chiral 3D PhCs.

3 Three-Dimensional Chiral Woodpile Photonic Crystals In this section, we show the structures of the studied chiral PhCs in detail. Then those fabrication processes are presented from semiconductor crystal growth to a micromanipulation technique. As a proof of chirality, we also show the experimental results of optical activity for the chiral PhCs.

3.1 Chiral Woodpile Structures The studied chiral PhCs were layer-by-layer structures in which each layer has a periodical rod array, as shown in Fig. 1. The in-plane angle of the rods between the neighboring layers was changed by 60° or 45°. In the case of 60° (45°) rotation, three (four) layers construct a single helical unit. Similar to the conventional Bragg reflection, circularly polarized light, propagating in the helical axis direction and having the same effective wavelength and chirality as the helical period, can be reflected by the chiral PhC, called as circular Bragg reflection. The in-plane period a and the helical period p are both in a sub-micron scale, and the dielectric material constructing the PhCs is GaAs which refractive index nGaAs ~ 3.4, so that the PhCs are expected to work for near-infrared light. In fact, in the case of a = 500 nm, p = 675 nm, and the width of rods, w = 130 nm, the structural averaged refractive index is nave = 1.6, and the central wavelength for circular Bragg reflection can be roughly estimated as λ = 1.6 × 675 = 1080 nm [36]. About the handedness of the chirality, we need to pay lots of attention. In the fabrication process, we stack the layers from bottom to top as explained in the next sub-section. Thus, in Fig. 1a, b, we stack the layers in left-handed 60° rotation. On the other hand, incident light propagating from top to bottom feels 60° right-handed chirality. So as for the light propagating from bottom to top, due to the structural C 2 rotation symmetry around the x or y axis. Throughout this Chapter, we define the handedness of the structural chirality by the propagating light, not by the fabrication process. Furthermore, we should be aware that the right-handed 60° rotation can also be accounted for the left-handed 120° rotation, as shown in Fig. 2. This dual chirality

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Fig. 1 Schematic diagrams of the studied chiral woodpile PhCs. a, b Chiral PhC formed by an in-plane rotation angle of 60°. c, d Chiral PhC formed by an in-plane rotation angle of 45°. For clarity, a part of the structure is removed in (a) and (c), and each layer is spatially separated in (b) and (d). Adapted from Ref. [65]

Fig. 2 Handedness of the structural chirality for the PhC formed by 60° in-plane rotation. Opposite chirality can be counted. Reprinted from Ref. [53] with permission of AIP Publishing

can be distinguished by their frequency response, because the helical unit differs for each sense of rotation. Indeed, 60° rotation requires 6 layers for 360° full rotation, while 120° rotation needs 3 layers. The circular Bragg reflection actually appears in different frequency regions for different handedness of CP, as shown in the following sub-section. In addition to the structural chirality, there are two definitions for the handedness of circular polarization. The observer judges the handedness of circularly polarized

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light propagating “away from” the observer or propagating “towards” the observer. Throughout this Chapter, we adopt the latter definition of the handedness of CP.

3.2 Fabrication of Semiconductor-Based Chiral Photonic Crystals Fabrication of 3D structure in nano or sub-micron scale is still challenging, although the first concept of PhCs was proposed for 3D PhCs [4]. Recent 3D nano-fabrication technique has provided several methods, such as a wafer fusion technique [37], multi-directional etching [38], DLW [39], a micro-manipulation technique [40, 41], and a chemical self-assembled method for microspheres [42], inverse opal [43]. Particularly for chiral structures, metal or polymer helices fabricated by DLW [29, 31, 44] have been studied intensely. Among these fabrication methods, we adopt the micro-manipulation technique due to the following two reasons. One is that the highest quality factor of a 3D PhC nano-cavity has been achieved (Q ~ 93,000) [45], indicating high accuracy in this fabrication method. The other reason is that this method can be applicable to semiconductor materials which have relatively large refractive indices ~3 and are compatible to the current electric circuits. In addition, semiconductor materials provide a good platform for electron-photon conversion including spin/orbital angular momentum. In fact, the first coherent transfer of single spin between a single electron (hole) and a single photon was performed in a semiconductor quantum dot [46, 47]. The semiconductor-based 3D chiral PhC controlling circularly polarized light, spins in photons, can be applied to quantum information technology and spintronics. In our fabrication method for the GaAs-based chiral PhCs, first, we grew a sacrificial AlGaAs layer having a thickness of 1.5 µm on a commercial GaAs substrate, followed by a growth of a GaAs slab layer having a thickness of t ~ 150 nm by metal–organic chemical vapor deposition. The in-plane patterns of rods having a width w ~ 130 nm and a period a = 500 nm were drawn on the GaAs slab by electron beam (EB) lithography. Note that these structural parameters are typical values which slightly vary for target wavelength in each experiment. Figure 3a, b show scanning electron micrograph (SEM) images of the in-plane pattern and the cross-section of the periodic rods, respectively, after the lithography process. The patterns were then transferred to the GaAs slab by inductively coupled plasma reactive ion etching (ICP-RIE). In this dry etching process using Cl2 and Ar gases, the GaAs slab uncovered by the residual resist in the former lithography process is anisotropically etched, as shown in Fig. 3c, d. In order that the sidewalls of the rods in the GaAs slab are expected to be vertical. We carefully tuned the etching parameters such as the flow rate of the gases, the ICP power, and the bias power accelerating the ions and the radicals to attack the sample. The vertical etching was performed to reach the AlGaAs sacrificial layer which was removed by the following wet etching

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Fig. 3 Typical SEM images of the various patterned plates from several viewpoints, after the EB lithography process (a), (b), after the dry etching process (c), (d), after the wet etching process (e), (f). (a) and (e) show top views of the stripe patterns rotated by 60° and 45° in the in-plane direction, respectively. (c), (f) and (b), (d) show tilt-angled views and cross-section of the pattern. In (d), the color difference indicates each layer of EB resist, GaAs slab layer, and AlGaAs sacrificial layer. The depth of the dry etching is ~200 nm, reaching to the AlGaAs sacrificial layer for the next wet etching process. In (f), the patterned GaAs slab is suspended in air due to the removal of the AlGaAs sacrificial layer

process using a hydrofluoric acid solution to form suspended structures. Figure 3e, f show some of the patterned plates after the wet etching process. For the PhCs in Sect. 4, apart from these simple GaAs plates, called as passive plates, we also grew another GaAs slab layer containing three-layer-stacked InAs self-assembled QDs on another GaAs substrate. The dots having a diameter of 10 nm,

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a height of 3 nm, and a density of 1.0 × 1010 /cm2 were used as near-infrared light emitters in the following experiment. This wafer was processed in the same way as for the passive plates to fabricate suspended active plates. These passive and active plates were stacked one by one using a micromanipulation technique in the appropriate stacking order for each experiment. Under SEM observation, the suspended plates were mechanically separated from the substrate by a metal needle, and brought to another substrate having vertically standing posts. As shown in Fig. 4a, these GaAs-based posts having a height of 5–7 µm are also fabricated by the similar dry etching process for the plates. The plates were placed along the corner of the guide posts, as shown in Fig. 4b. The taper of the posts is typically 20 nm/1 µm-height, which mainly determines the stacking accuracy. Some of the chiral PhCs measured in the following sections are shown in Fig. 4c–f. The top views in (d) and (f) show the well-aligned triangular and square lattices, respectively, indicating the high accuracy of the stacking technique. In order to compare the chiral and non-chiral structure, we also prepared a non-chiral (achiral) structure which was composed of the plates having the same in-plane pattern. This micro-manipulation technique is advantageous for the fabrication of 3D structures in terms of various alternatives of in-plane patterns, stacking orders, and materials with/without emitters. Thus, this method has realized 3D woodpile PhC nanocavities made of GaAs [40, 41] as well as Si and Ge [48, 49]. Besides the conventional woodpile structure, a -layered diamond structure [50, 51] and chiral PhCs [52, 53] have also been realized [54]. By introducing QD infrared emitters in the woodpile structure with a high density ~1.0 × 1010 /cm2 , lasing oscillation has been achieved [45, 55]. Also, by introducing QDs with a low density ~1.0 × 108 /cm2 , the nanocavity possesses only a single QD, demonstrating the Purcell effect between a single exciton in the single QD and a single photon in the nanocavity [12, 56]. Recently, integration of a nanocavity and waveguides in a woodpile PhC has been achieved [57].

3.3 Optical Activity In order to verify the chirality of the chiral PhCs, we firstly measured optical activity, circular dichroism (CD) and optical rotation (OR) [52, 53]. Note that the CD in this section does not indicate the conventional phenomena of absorption for CP light, but circular Bragg reflection. On the other hand, OR is measured for linear polarization whose angle of the polarization plane is tilted after the transmission for an optically active media. We also note that the difference in the phase velocity between LCP and RCP actually causes OR. The 3D chiral PhC studied for CD is composed of 16 plates with 60° in-plane rotation. The structural parameters are a = 500 nm, w = 160 nm, and t = 225 nm, thus p = 225 × 3 = 675 nm. The PhC shows left-handed chirality for 120° in-plane rotation. Firstly, we calculated the photonic band structure for the unit cell of the chiral PhC by a plane wave expansion (PWE) method. The left panel in Fig. 5 shows

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Fig. 4 a SEM image of the vertical posts as guides for the stacking. b SEM image in the staking process. The post sample and the metal needle are tilted in the SEM system by ± 45°, respectively, so that the needle is parallel to the posts. c, e SEM images of the fabricated chiral PhC formed by 60° and 45° in-plane rotation. d, f Top view of (c) and (e). The rods show well-aligned triangular and square lattice, respectively. (c), (d) Reprinted from Ref. [53] with permission of AIP Publishing. (e), (f) Adapted with permission from Ref. [52] © The Optical Society

the band structure only in the direction along the helical axis, indicated from the Γ point to the Z point. As reported in Ref. [53], parts of the photonic bands are strongly polarized in each CP, and a polarization bandgap for RCP appears at around p/λ = 0.28 and that for LCP appears at around p/λ = 0.45. These two polarization bandgaps for orthogonal CP were caused by the dual-chirality in the chiral PhC as shown in Fig. 2.

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Fig. 5 Photonic band structure in the direction along the helical axis (left panel) and transmittance spectra for each CP incidence (right panel). In the right panel, dots show the experimental results and solid lines show the FDTD calculation results. (Left panel) Reprinted from Ref. [53] with permission of AIP Publishing

Secondly, we performed the transmission calculation for the chiral PhC composed of 16 plates by a finite difference time domain (FDTD) method. Note that a periodic boundary condition was imposed in the in-plane direction. The obtained numerical transmittance spectra are shown in the right panel in Fig. 5. The frequency regions of the polarization bandgaps for each CP are consistent with the suppressed transmission for corresponding CP components. In the measurement of the CD for this sample, infrared laser light whose wavelength can be tuned from 1300 to 1620 nm was focused on the sample by an objective lens having a numerical aperture of NA = 0.27. The transmitted laser light was collimated again by another objective lens having the same NA. Note that the objective lenses were selected so that the focal depth calculated from the NA is larger than the total thickness of the chiral PhC. Indeed, the focal depth ~4 µm was larger than the sample height, 0.225 × 16 = 3.6 µm. Therefore, a quasi-plane wave can be applied to the sample. The colored dots in the right panel in Fig. 5 shows the measured transmittance spectra for each CP incidence. The transmittance was obtained from the intensity of the transmitted light with and without the chiral PhC. The transmittance of LCP light is larger than that of RCP throughout the measured frequency range, and the transmission ratio between LCP and RCP reached as large as 6 at p/λ = 0.52 corresponding to 1300 nm wavelength. The measured transmittance spectra show good agreement with the numerical curves. The slight deviation is probably caused by the quasi-plane wave in the measurement, while a plane wave was applied in the calculation. This wideband CD is due to the large refractive index of the adopted semiconductor material. When the index contrast between the material and air is small, the CD region gets small [36, 44]. We also measured the polarization of the transmitted light for various wavelengths and various incident polarization. Throughout the measured wavelength region, elliptical polarization was obtained, and there is no dependence on the incident polarization, indicating asymmetric transmission [58, 59]. The measured polarization in the transmitted light was one of the eigen-polarization in the chiral PhC at

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each wavelength. The components of the other orthogonal eigen-polarization were reflected. For the OR measurement, we prepared a 3D chiral PhC composed of 9 passive plates with 45° in-plane rotation. The structural parameters are a = 500 nm, w = 100 nm, and t = 150 nm, thus p = 150 × 4 = 600 nm. The PhC shows left-handed chirality for 135° in-plane rotation. In this measurement, the same setup as in the CD measurement was used, but the linearly polarized laser light was applied onto the top of the sample (forward incidence). In order to eliminate the birefringence effect, we changed the angles of the incident polarization plane and measured the angles of the transmission polarization plane [19, 23, 52, 60]. Figure 6a, b show the OR angle and the ellipticity as a function of the wavelength after eliminating the birefringence effect. The OR angle reached as high as 52.5° for 1.35 µm thickness of the sample at 1.3 µm wavelength. On the other hand, the ellipticity was maintained to be zero throughout the measured wavelength region, indicating linearly polarized light output. We also performed this measurement in the opposite direction, applying the laser light from the bottom substrate to the sample top (backward incidence). The red dots in Fig. 6a, b show the coincidence between the results for the forward and the backward incidence, which is totally different from the Faraday effect using magnetized materials. The FDTD calculation results in the similar calculation condition for the CD also show good agreement with the experimental results. The slight deviation is probably caused by the quasi-plane wave in the measurement.

Fig. 6 a, b OR angle and ellipticity of the transmitted light as a function of the incident wavelength after eliminating the birefringence effect. The experimental results for both forward and backward transmission as well as the calculation results show good agreement. c Photonic band structure in the direction along the helical axis. (a), (b) Adapted with permission from Ref. [52] © OPTICA

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Figure 6c shows the photonic band structure for the chiral PhC formed by 45° rotation. As for Fig. 5, the bands are strongly polarized in each CP, and a large difference in the wave number between the LCP and the RCP band appears in the measured frequency region (shaded part in Fig. 6c). This large difference in phase velocity for LCP and RCP caused the large OR. Indeed, the difference becomes large for high frequency, which is consistent with the large OR angle in Fig. 6a. Different from Fig. 5 for the chiral PhC formed by 60° rotation, Fig. 6c shows only a small CD bandwidth due to the low symmetry in the chiral PhC formed by 45° rotation. In order to align crossing points in the top view, as shown √ in Fig. 4f, the in-plane period is a for plates stacked in the odd number, while a/ 2 for the even number plates. Due to this small CD bandwidth, the ellipticity was negligibly small in the measured wavelength region for linearly polarized incidence.

4 QD Light Emission in Engineered Circularly Polarized Vacuum Field In general, one of the main concepts of PhCs is to modify the linear dispersion of light and form photonic band structures [6]. This artificially tuned dispersion of light leads to plenty of novel photonic properties due to the modification of group velocity of light (wave packet) and photonic density of states (DOS) in the vacuum field. The suppressed DOS causes the prohibition of transmission/emission in the photonic bandgaps, and the enhancement of DOS at the band edge leads to band edge lasing. This is also the case in the CP photonic band structure in Fig. 5. Therefore, when emitters are placed inside the chiral PhC, emission itself can be controlled even at the beginning of the radiative recombination process due to the modified DOS, according to Fermi’s golden rule. In this section, we show the measurement of the CP vacuum field in the chiral PhCs by using QDs as a probe [61].

4.1 Numerical Analysis on the Emission Properties In this section, we mainly discuss a chiral PhC sample with 60° in-plane rotation having structural parameters of a = 500 nm, w = 130 nm, and t = 225 nm, thus p = 675 nm. We stacked 16 plates in the order of 6 passive plates, 3 active plates, and 7 passive plates from bottom to top, keeping the helical symmetry. The PhC shows left-handed chirality for 120° in-plane rotation. Before the optical measurements, we performed numerical calculations for this sample. First, the photonic band structure in the Γ -Z direction is shown in Fig. 7. These structural parameters were actually determined so that a part of the polarization bandgap at around p/λ = 0.48–0.60 as well as band edge at p/λ

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Fig. 7 a Photonic band structure in the direction along the helical axis. The shaded frequency region indicates the horizontal axis in (b) and (c), corresponding to the emission wavelength of the ensemble QDs. b DOCP obtained from the calculated transmitted light and degree of polarized DOS obtained from (a) as a function of wavelength. c Numerically obtained lifetime ratio between LCP and RCP point sources in the chiral PhC. d Schematic diagram of the order of the stacked plates in the numerically studied chiral PhC. The colored 3 plates correspond to (e)–(g). e–g Schematic top view of the 7th, 8th, and 9th plate in the chiral PhC, respectively. We set a circularly polarized radiative source at one of the 12 points in each plate as shown by the white circles. An in-plane unit cell is shown as the black solid lines. Adapted with permission from Ref. [61], Copyright (2017) by American Physical Society

= 0.60 are both included in the typical QD emission wavelength range, 1050– 1200 nm, corresponding to p/λ = 0.563–0.643. Here, we introduce “mode DOS” ρ LCP/RCP (ω) of the vacuum field for the modes polarized in LCP/RCP in Fig. 7a at an angular frequency ω. Since Fig. 7a only shows the particular direction, Γ -Z, this ρ LCP/RCP (ω) is only a part of the total DOS which includes DOS of the other modes propagating in other 3D directions. In the polarization bandgap at around p/λ = 0.48–0.60, ρ LCP (ω) of the vacuum field is suppressed. On the other hand, at the polarization band edge at p/λ = 0.60, since DOS is inversely proportional to dω/dk, ρ LCP (ω) is strongly enhanced. Therefore, the degree of polarized DOS, (ρ LC P/RC P (ω) − ρ LC P/RC P (ω))/(ρ LC P/RC P (ω) + ρ LC P/RC P (ω)), can be plotted as the blue curve in Fig. 7b. Compared with this spectrum of the degree of polarized DOS, we performed FDTD calculations for the transmission of each CP. We applied a pulsed plane wave for each CP in the helical axis direction. The pulse duration was a single cycle for the central wavelength of 1270 nm. The boundary conditions were periodic in the inplane direction, and in the helical axis direction, perfectly matched layers were placed 10 µm away from the center of the PhC. The time-dependent electromagnetic field of the transmitted light was monitored for each CP incidence, which was analyzed in

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frequency space by a Fourier transform. From the transmitted intensity for each CP, I LCP/RCP , the degree of circular polarization (DOCP), (I LC P − I RC P )/(I LC P + I RC P ), can be calculated at each wavelength, as shown by the green curve in Fig. 7b. Even if ρ LCP (ω) is enough large at the polarization band edge, the LCP transmittance reaches unity at most, the same as the RCP transmittance, resulting in the zero DOCP. This difference between the DOCP in the transmission and the degree of polarized DOS is significant to distinguish whether the CP vacuum field was actually modified. In the following measurements, if the QD emission shows DOCP spectrum as the blue curve, then the CP vacuum field is actually modified. On the contrary, if the experimental DOCP spectrum is similar to the green curve, then the chiral PhC works as a simple CP filter without any modification of the vacuum. As another numerical investigation of the modified DOS in the vacuum, we calculated a classical dipole radiation power by the FDTD method [62]. First, we defined the whole calculation domain. The in-plane area of the rod pattern was 9 µm2 corresponding to 6 in-plane periods, owing to computational time restrictions. (In the following experiment, the area was 49 µm2 corresponding to 14 in-plane periods.) As the measured sample, this limited area was numerically surrounded in the in-plane directions by GaAs. We set perfectly matched layers at 2 periods (pitches) away from the edge of the pattern in the in-plane (z) directions. Also, we set spatial domains as 1/16 period (1/48 pitch) in the in-plane (z) directions. Then we put a current source having each CP at a particular position. Figure 7e– g shows a schematic top view of a helical unit composed of the 7th, 8th, and 9th plates out of the 16 plates in total, as shown in Fig. 7d. The 7th, 8th, and 9th plates are colored by red in (e), blue in (f), and green in (g), respectively, and the region surrounded by the black lines represents an in-plane unit cell. Although QDs in the experiment were densely contained in the rods, we put a light source at one of the 12 points shown by the white circles in each plate, owing to computational time restrictions. Since the size of the light source was equal to the spatial domain, the source was assumed to be a point source. The electromagnetic field was generated by a circular dipole in the in-plane direction, because the studied QD having a diameter of 10 nm and a height of 3 nm was almost flat. Under these conditions, we calculated the Poynting vectors at a surface which was set at one period/pitch away from the edge of the pattern and enclosed the light source, then integrated the obtained Poynting vectors over the whole surface and the whole evolution time. A single simulation was performed for a single light source which had a single wavelength and one of the CP, and was located at a single position. Thus, the total number of the simulation, for 36 different source positions, 8 different wavelengths, 2 orthogonal components of the CP, and the chiral PhC existence/absence, was 1152. Then the integrated power with the chiral PhC was normalized by the power without the structure, and we obtained the modification coefficients of the spontaneous emission rate. The modification coefficients were finally averaged over the 36 source positions. Figure 7c shows the ratio of the averaged coefficients for each CP as a function of wavelength. As expected from the frequency distribution of DOS, τ LCP is larger (smaller) than τ RCP in the polarization bandgap (band edge). The difference

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between τ LCP and τ RCP is significant but small because the DOCP is influenced only by ρ LCP/RCP (ω) for the modes propagating in the Γ -Z direction. The other modes propagating in the in-plane or tilted angle directions do not show any difference between each CP, resulting in the small τ LCP /τ RCP after the averaging process. The limited structural size in the calculations is another reason for the small τ LCP /τ RCP .

4.2 Measurement Setups After the fabrication of the chiral PhC whose dimension is the same as the simulated structure in the last sub-section, we performed the following two optical experiments in the absence of an external magnetic field. First, we performed a micro-photoluminescence (PL) measurement for the QDs in the chiral PhC. Since the QD emission intensity gets larger in low temperature due to suppression of non-radiative recombination processes, the sample was cooled down to 20 K in a liquid helium cryostat. The linearly polarized excitation laser light with 980-nm wavelength was focused on the sample by a 60 × aspheric lens (NA = 0.54) placed in the cryostat. The power of the excitation light is 40 µW and the spot size on the sample was about 3 µm which is smaller than the sample area of 10 µm square. The QD emitted light was collimated by a 50 × objective lens (NA = 0.55) placed in the cryostat. The CP components were selected by a following quarter-wave plate and a polarizer, then detected by a spectrometer and a 2D array of InGaAs photodiodes. In order to select the light emitted in the helical axis direction, the back focal plane of the objective lens was relayed by a 4f system using two additional lenses with 400-mm focal length. By selecting a single chip of the photodiode array, the center of the back focal plane was selectively detected. As the second experiment for the emission lifetime of the QDs, we performed a time-resolved PL (TRPL) measurement at 20 K. We used a linearly polarized Ti:sapphire mode-locked laser as the excitation laser. The central wavelength, pulse duration time, repetition rate, and time-averaged power are 890 nm, 1 ps, 80 MHz, and 1.5 µW, respectively. This laser light was focused on the sample by a 50× objective lens (NA = 0.65) with ~3-µm spot size. The QD emitted light was collimated by the same objective lens, then the CP components and the wavelength were selected by a following quarter-wave plate, a polarizer, and a bandpass filter. Finally, the PL intensity decay curve was measured by a superconducting single-photon detector (SSPD) with ~25-ps timing resolution.

4.3 Degree of Circular Polarization in Emission Figure 8a shows the PL spectra from the QDs in the chiral PhC for each CP. The LCP intensity was stronger than the RCP intensity in a narrow wavelength region at around 1120 nm, while the LCP intensity was weaker than the RCP intensity in

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Fig. 8 a, b PL spectra for each CP component in the chiral and non-chiral PhC, respectively. c DCOP spectra for the chiral (black) and non-chiral (red) PhCs. d Schematic diagram of the stacked 16 plates with and without QDs. Adapted with permission from Ref. [61], Copyright (2017) by American Physical Society

a broad wavelength region at around 1200 nm. These LCP sharp peak and RCP broad peak were consistent with a LCP band edge and a LCP bandgap, respectively, in a numerically obtained photonic band structure. Therefore, in the polarization bandgap, the DOS for the LCP mode in the vacuum was suppressed, and the QDs preferred to emit RCP. At the polarization band edge, the DOS for the LCP mode in the vacuum was largely enhanced, and LCP light was strongly emitted from the QDs. On the other hand, as shown in Fig. 8b, the non-chiral structure did not show an intensity difference between LCP and RCP, indicating no DOS modification for CP modes. These results are clearly understood when DOCP spectrum is plotted, as shown in Fig. 8c. The obtained DOCP spectrum for the chiral PhC shows the similar tendency as the degree of polarized DOS in Fig. 7b; the positive DOCP at the polarization band edge and the negative DOCP in the polarization bandgap. Therefore, we confirm that Fig. 8c does not mean that the chiral PhC works as a simple CP filter, but means that the CP vacuum field is modified. This tendency of the DOCP spectrum for the chiral PhC was reproduced in other chiral PhCs having different structural parameters. Figure 9a, b show the DOCP spectra for other chiral PhCs having (a, p, w) = (500 nm, 675 nm, 150 nm) and (500 nm, 720 nm, 120 nm), respectively. Both figures show the positive DOCP at

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Fig. 9 DOCP spectra for other chiral PhCs having different structural parameters, (a, p, w) = (500 nm, 675 nm, 150 nm) for (a), and (a, p, w) = (500 nm, 720 nm, 120 nm) for (b). Adapted with permission from Ref. [61], Copyright (2017) by American Physical Society

around 1150 nm wavelength, and the negative DOCP in the wavelength region larger than 1150 nm. These results indicate that the polarization band edge is at around 1150 nm, which is consistent with the numerical calculations for each structure.

4.4 Radiative Lifetime of Circularly Polarized Components By performing the TRPL measurements for the chiral PhC at 1200 nm wavelength in the polarization bandgap, we obtained the time decay of the PL intensity for each CP, as shown in Fig. 10a. After fitting the decay with a single exponential function, the radiative lifetime for LCP (RCP) component τ LCP (τ RCP ) was obtained to be 1.06 ns (0.94 ns). Since radiative lifetime is proportional to the total DOS ρ total , this lifetime difference indicates that ρ LCP is smaller than ρ RCP in the polarization bandgap. On the other hand, for a single active plate without any in-plane pattern (inset of Fig. 10b), there is no lifetime difference between each CP component, as shown in Fig. 10b. Therefore, we confirm that the significant lifetime difference in Fig. 10a was caused by the chiral PhC. We performed the same TRPL measurement for various wavelengths and plotted the lifetime ratio τ LCP /τ RCP , as shown in Fig. 10c. A significant lifetime difference appears in the long wavelength region >1120 nm which is consistent with the polarization bandgap region in Figs. 7a and 8c. However, τ LCP /τ RCP ~ 1 at the polarization band edge, although ρ LCP must be larger than ρ RCP and τ LCP /τ RCP < 1 is expected. This inconsistency is discussed in the following sub-section.

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Fig. 10 a, b Time decay of the PL intensity for the chiral PhC and a single active plate without in-plane pattern, respectively. The black (red) curve indicates the LCP (RCP) component. Inset in (b) shows an SEM image of the single active plate without in-plane pattern. c τ LCP /τ RCP ratio as a function of the emission wavelength. Reproduced with permission from Ref. [61], Copyright (2017) by American Physical Society

4.5 Discussion From the last sub-sections, the measured CP light was revealed to be directly influenced by the modified DOS of the CP vacuum field. The DOCP spectrum shows the suppression of ρ LCP in the polarization bandgap as well as the enhancement of ρ LCP at the polarization band edge, which is consistent with the numerical results of the degree of mode DOS in Fig. 7b. Also, the suppression of ρ LCP in the polarization bandgap is confirmed by τ LCP /τ RCP > 1 in the TRPL measurement. Furthermore, this lifetime difference is qualitatively consistent with the numerical results in Fig. 7c. However, there are several inconsistencies between the two experimental results as well as the numerical result; (i) the difference between the 50% CP contrast in the DOCP measurement and the only 10% CP contrast in the TRPL measurement, (ii) the difference of τ LCP /τ RCP at the polarization band edge between the numerical and experimental results. In this sub-section, we discuss these inconsistencies as well as (iii) the spin relaxation in the QDs.

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First, both numerically and experimentally obtained lifetime difference between orthogonal CP components is only ~10% which is much smaller than the DOCP ~50% in Fig. 8c. This is because the DOCP is influenced only by ρ LCP or ρ RCP for the selected modes propagating in the z direction. On the other hand, radiative lifetime is influenced by total DOS ρ total which includes all modes propagating in all 3D directions and having all polarization. Therefore, the 50% difference in the DOCP measurement was averaged by the DOS for other modes, and only ~10% difference in the lifetime was obtained. Second, at the polarization band edge, τ LCP /τ RCP should be smaller than unity due to the enhancement of the DOS for LCP, while the experimental result in Fig. 10c does not show such result. Even in the numerical result shown in Fig. 7c, the difference between τ LCP and τ RCP at the band edge is smaller than that in the bandgap. This little difference between τ LCP and τ RCP is caused by the spatial dispersion of the electromagnetic field, as shown in the following numerical calculation. We calculated the electric field intensity for LCP, |E LCP |2 , in the studied chiral PhC at the wavelength of the polarization band edge and the bandgap by the FDTD method. A continuous plane wave having LCP at each wavelength was applied along the helical axis to the chiral PhC composed of 16 plates. Periodic boundary conditions were imposed in the in-plane directions and perfectly matched layers were attached at 10 µm away from the center of the structure in the helical direction. We investigated spatial distributions of |E LCP |2 in an in-plane slice at the center of the 7th plate in a steady state. Figure 11a, b show the obtained |E LCP |2 distribution for the band edge wavelength of 1120 nm and the bandgap wavelength of 1250 nm, respectively. In what follows, we discuss the |E LCP |2 distribution only on the rods, since the QDs were distributed in the dielectric rods. In Fig. 11a, |E LCP |2 at the band edge was strongly localized at the center of the figure, corresponding to the corner of the in-plane unit √ cell indicated by the dotted lines. At points away from the center by 500/ 3 nm in the y-direction, |E LCP |2 was almost zero. In Fig. 11b, on the other hand, |E LCP |2 in the bandgap was weakly localized at the center, and |E LCP |2 showed non-zero values at

Fig. 11 Spatial distribution of |E LCP |2 in an in-plane slice at the center of the 7th plate for the band edge wavelength (a) and for the bandgap wavelength (b). Reprinted with permission from Ref. [61], Copyright (2017) by American Physical Society

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all points on the rods. Note that due to the polarization bandgap for LCP, the whole intensity in Fig. 11b was three times smaller than that in Fig. 11a. The non-zero intensity even in the bandgap was caused by the finite number of the helical period. From these different |E LCP |2 distributions, the local DOS is also strongly localized in the rods at the band edge, whereas it spreads over the rods in the bandgap. Therefore, τ LCP /τ RCP at the band edge is sensitive to averaging for all QD positions in the plates and reaches unity, whereas τ LCP /τ RCP > 1 is maintained in the bandgap, as shown in Fig. 10c. However, if a single QD could be positioned accurately in the chiral PhC, the band edge effect would potentially show a large reduction of τ LCP . In fact, at the corner of the in-plane unit cell in the 7th plate, where |E LCP |2 showed the maximum value, τ LCP /τ RCP was shown to be as low as 0.83 at the polarization band edge due to the large local DOS for LCP. By averaging this result with other 35 results at the different source positions, this significant reduction of LCP became obscure, resulting in Fig. 10c. Third, the spin relaxation in the QDs is taken into account. Since CP corresponds to spin angular momentum of electrons and heavy holes in the solid state as represented by spin-photon interfaces [46, 47], the obtained results of CP emission could be influenced by spin relaxation in the QDs. Rate equations for the population of electron–hole pairs emitting LCP (RCP), PLCP (PRCP ), including spin relaxation in the absence of magnetic field are written by [63], P LC P P LC P − P RC P d P LC P = − LC P − dt τ τ spin P RC P d P RC P P LC P − P RC P = − RC P + dt τ τ spin Here, τ spin is electron/hole spin relaxation time, corresponding to spin dephasing time in ensemble self-assembled QDs, T 2 * ~ 1–10 ns both for electrons and holes in such QDs [64]. This time scale is comparable to the obtained radiative lifetime for both CP light. In this condition with the same population between CPs at t = 0 due to the linearly-polarized excitation pulse, the rate equations still give a significant difference of the radiation lifetime between LCP and RCP. Note that in the assumption of τ spin < < τ LCP (τ RCP ), the time-decay of PLCP and PRCP coincides with each other, which is not the case in this study.

5 Summary In this Chapter, we discussed the recent progress on semiconductor-based chiral 3D PhCs enabling engineering the circularly polarized vacuum field. The chirality formed by the 3D periodic structures as well as the large refractive index of semiconductors can largely modify the photonic band structures for two CP light propagating along the chiral axis. This modification of photonic bands leads to the observations

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of circular dichroism with broader bandwidth than the polymetric counterparts and of giant optical rotation. Besides, taking advantage of the semiconductor platform, the modification of CP vacuum was probed by QDs embedded in the chiral PhC through the measurement of the degree of polarization and the emission dynamics. In particular, the successful observation of the change in lifetime depending on the handedness of emitted CP photons is the first direct proof showing the influence of the modified CP vacuum on the emission dynamics. The results shown in this Chapter indicate that the semiconductor-based chiral PhC could provide a platform for the quantum hybridization of CP photons and spins in solid states. The chiral PhC itself extracts particular CP light from unpolarized light by circular dichroism, or generates spin polarization from unpolarized current injection by asymmetric spontaneous emission of CP light based on the modified chiral density of states. Because many emitters such as QDs are charged with electrons or holes having spin degree of freedom, such control of spin-related light-matter interactions would allow spin pumping and initialization without external magnetic field or CP light excitation, which is useful for spin-based quantum memories and quantum computation. Polarized spin sources from unpolarized light/current excitation/injection are another possible application for semiconductor-based spintronic devices or magnetic materials. As an application for a photonic device, compact spin lasers without spin injection in semiconductor systems could be realized by the chiral PhC. In addition, the chiral 3D PhC can also host CP topological edge states at the surface of the structure [65], which can be used as robust waveguides for CP photons or robust spin sources, or as highly sensitive CP-photon/spin sensors through the quantum hybridization. For some of these applications, a chiral optical cavity embedded in the 3D chiral PhC should be developed in the future. Novel fabrication technologies for the semiconductor-based 3D chiral PhCs are also expected to be developed. They would enable the realization of more complex chiral structures as demonstrated in polymeric structures and would open the door to various new applications. Acknowledgements The authors acknowledge Y. Ota, T. Tajiri, A. Tandaechanurat, J. Tatebayashi for technical support and fruitful discussions. The work introduced in this Chapter was supported by Grant-in-Aid for Scientific Research (16H06085, 26889018), Grant-in-Aid for Specially Promoted Research (15H05700), Grant-in-Aid for Scientific Research on Innovative Areas (15H05868).

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Hybrid Structure of Semiconductor Quantum Well Superlattice and Quantum Dot Kouichi Akahane

Abstract Hybrid structures of InGaAs/InAlAs two-dimensional quantum well superlattices and self-assembled InAs quantum dots (QD) were fabricated via digital embedding method with a molecular beam epitaxy. In this chapter, the structural and optical properties of these structures will be discussed in which unique features appeared these hybrid sutructures. Keywords Quantum well superlattice · Quantum dot · Photoluminescence · Self-assembling · Molecular beam epitaxy Hybrid structures of InGaAs/InAlAs two-dimensional quantum well superlattices and self-assembled InAs quantum dots (QD) were fabricated on an InP(311)B substrate via digital embedding method with a molecular beam epitaxy. In this chapter, the structural and optical properties of these structures will be discussed. In the 1980’s to 1990’s, there was marked progress in crystal growth technology. Techniques such as molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOVPE) enable us to fabricate well-defined heterostructures. Currently, depositions that are only one atom thick can be created [1, 2]. The artificial structure where different material layers are grown alternately is called a superlattice or quantum well (QW) superlattice (Fig. 1). There are two periodicities in the superlattice: a crystal periodicity, which is related to the material itself; and an artificial periodicity, which is longer than the crystal periodicity, and controlled by changing the growth parameters of materials, such as growth duration. Ideally, the QW superlattice should be called one-dimensional superlattice. Two-dimensional electron gases and two-dimensional excitons can also be contained in these one-dimensional superlattices, and this is a subject of investigation in novel physics and device applications [3]. With regards to the investigation into one-dimensional superlattices, the next step involves attempts to increase confinement dimension. Superlattices can form confinement structures using methods such as modification of potential energy in the K. Akahane (B) NICT, Tokyo, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 Y. Hirayama et al. (eds.), Quantum Hybrid Electronics and Materials, Quantum Science and Technology, https://doi.org/10.1007/978-981-19-1201-6_15

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material 2 ~ nm

material 1 Ev z material 2

z

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Fig. 1 The material structures and the band diagrams for QW and super-lattice material 2 Ec z ~ nm Ev y material 1

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Fig. 2 The material structure and the band diagram for QD

epitaxial plane. The most direct scheme is the fabrication of three-dimensional nano structures that produce the quantum effect. These structures are called quantum wires (QWR: two-dimensional confinement) and quantum dots (QD: three-dimensional confinement, as shown in Fig. 2) [4]. In the QD, the energy of carriers is completely quantized (forming completely discrete energy levels) because the carriers are confined three-dimensionally. When QDs were used as the gain medium of semiconductor laser diodes, various improvements of performance were observed such as lower threshold current, higher speed modulation and temperature-independent operation. Additionally, carrier transport is more easily controlled in the one-dimensional superlattice. In this chapter, the characteristics of superlattice and QD will be introduced and the combination of

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these structures, the fabrication technique and the resulting optical properties will also be discussed.

1 Quantum Well and Quantum Well Superlattice When several QWs draw closer to each other in the one-dimensional superlattice, a mini band is formed through the interaction of each quantized energy level in QWs. This situation is similar to the formation of the band structure in semiconductors where atoms draw closer to each other. To form minibands in superlattices, the material systems of III–V compound semiconductors were used, as the band gap energy and lattice constant can be controlled by changing the composition. Therefore, a high-quality hetero interface can be achieved in this material systems. MBE can be for layer-by-layer growth so that highly defined QWs and superlattices can be fabricated. GaAs and AlAs material systems are well researched. In this case, the GaAs is used for the QW region because its band gap energy is less than that of AlGaAs. If a repeated structure is fabricated using these QWs, a superlattice is formed. This material system is advantageous because the lattice constant does not change if the composition of AlGaAs shifts from GaAs formation to AlAs formation. Although the lattice constant of GaAs is smaller than that of AlAs, the mismatch of lattice constant is less than 1%. Therefore, various band alignments can be fabricated while maintaining high crystal quality. The GaAs have a bandgap energy of ~1.42 eV, making it suitable for the study on electronic properties and optical properties. The example for electronic property is the negative resistance in resonant tunneling diodes or superlattice structures [5]. The negative resistance is the phenomenon in which the current across the circuit decreases as the voltage increases. When the Fermi level outside of QW resonates with the energy level of QW, the current– voltage dependence (I–V curve) exhibits a peak structure. After the current peak, the negative resistance appears with increasing bias voltage. In this stage, the Fermi level is produced from the resonant state of the energy level in QW. The change of energy band is schematically shown in Fig. 3. Meanwhile, an analogous optical property is observed in asymmetric quadruple QW superlattices in [6]. In this study, the asymmetric QW superlattice consisted of quadruple QWs and barriers with different thickness that were repeated 20 times and an AlGaAs p–n junction. The quadruple QWs were fabricated with GaAs while AlAs was used for barriers. Therefore, the energy levels in the QWs can be controlled by applying a bias voltage to p–n junction. In this case, photoluminescence spectra shifted or split through the carrier transport from one QW to the other. This transport was associated not only with the resonance of energy levels between two QWs, but also with the resonance of energy levels between QWs and the X-point of barrier material. Interesting PL branches were observed which were caused by interferences among higher-energy sub-bands, long-range -X transfers, and various types of -X mixings.

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Another practical application involved a quantum cascade laser (QCL) which used carrier transports in superlattice and sub-band transition of electron in the QW energy levels [7]. In conventional laser diodes, the recombination of electron and hole occurs among conduction band of electrons and valence band of holes generating photons. In a QCL, only the electrons play a role in generating photons thorough the intersub-band transition. In this case, the energy of photon is determined by the difference in sub-band energies. After generating a photon, the electron accelerates due to the electric field in the superlattice and moves to the next QW (Fig. 4). Repeating this process, one electron can generate many photons which is utilized for optical gain. Although it requires the precise designing of the energy levels of superlattice and Fig. 4 Conduction band energy diagram of a part of a quantum cascade laser photon

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QW to fabricate many layers, high performance QCLs in the mid infrared or far infrared (or terahertz frequency) wavelength region are widely used.

2 Quantum Dot To create a QD structure, the three-dimensional confinement of carriers is needed. It is more difficult to create than QWs which can be fabricated using layer structuring. The early research stage of QD involved etching QW structures to form cylindrical shape, however, damage at the interface presented a problem. A method of potential energy modification for two-dimensional electron gas using external electric field was later developed and it did not possess the same problem of interface damage. In this scheme, high quality QDs were achieved with energy levels that can be changed using external electric fields. Its application in electron transport and other various phenomena were shown experimentally. However, although these top-down methods can control the position of QDs precisely, they cannot create high density QDs. To obtain high density QDs, a self-assembling technique was developed. In this method, high density QDs were formed using only crystal growth in a lattice mismatched material system. However, the lattice mismatch generates a strain which sometimes degrades the quality of the semiconductor crystal. Therefore, a lattice matched material system was chosen to achieve layer-by-layer growth which produces high quality crystals. This growth mode is called the Frank-van der Merwe (FM) mode (Fig. 5a). A large lattice mismatch, however, ensures that the threedimensional growth occurs from the initial stage, but this leads to the introduction of defects and the dislocations in the crystal which degrade crystal quality. This growth mode is called the Volmer–Weber (VW) mode (Fig. 5b). Meanwhile, when the lattice mismatch is of an intermediate degree between these two growth modes, the two-dimensional growth occurs initially followed by three-dimensional growth.

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Fig. 5 Schematic diagram of the growth mode: Frank-van der Merwe (FM), Volmer-Weber (VW), and Stranski–Krastanow (SK)

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In this case, the coherency of the crystal is maintained in the initial stage of threedimensional growth. Therefore, if the growth process stops at the appropriate stage, high quality three-dimensional nanostructures without defects and dislocations may be obtained. This growth mode is called the Stranski–Krastanow (SK) mode (Fig. 5c). The self-assembled QDs obtained by SK growth mode is widely used because of the high density and highly quality QDs obtained from this process. The research on self-assembled QDs was applied to GaAs related compound semiconductor material systems. In this case, InAs was used for the QD because InAs has smaller band gap energy and form confined band alignment compared to (Al)GaAs in both the conduction band and the valence band. In addition, the lattice mismatch between InAs and GaAs was compatible with the SK growth mode [8]. Based on these considerations, QD laser diodes were created using InAs QD grown on GaAs. The diodes possessed good properties such as low threshold current, high modulation speed, and temperature independent threshold current which was predicted in the theoretical investigation. Furthermore, other attractive devices were developed, such as single photon emitters, semiconductor optical amplifiers, and solar cells. With the spread of fiber optic communications, other material systems used to fabricate QD, such as InGaAsP and InGaAlAs on InP, have garnered increasing attention. The lattice constant of these material matches the InP substrate and the bandgap is suitable for the emission along the near-infrared wavelength region, especially the 1550 nm telecom band. In terms of strain, it is surmised that the self-assembled QD can be formed on InP substrate with ease. However, there are other problems present in the growth in these material systems. If the InAs is grown on InP(001) substrate, the formed nanostructure becomes a quantum dash (QDH) which is an elongated structure along the [1–10] direction. For example, the atomic force microscope (AFM) image of a 6 ML-thick InAs on InP(001) surface is shown in Fig. 6a. The wire-like structures are QDHs which formed on the surface and possess widths of approximately 30 nm and lengths of several hundred nanometers. The origin of these

Fig. 6 InAs nanostructure grown a on InP(001) substrate: QSH, b on InP(311)B substrate: QD

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structures is attributed to the large difference of surface diffusion of In adatoms in the MBE growth on InP(001) surface [9–11]. Specifically, the In adatoms diffused largely along the [1–10] direction and not the [110] direction, forming elongated structures along the [1–10] direction. Therefore, an additional scheme should be introduced to form QD structures such using As2 flux and/or using off angle substrates [12]. For example, when InAs is grown on an InP(311)B substrate, round-shaped and uniform QDs were obtained as shown in Fig. 6b [13]. The (311)B surface structure exhibits large anisotropy [14, 15] which suppresses the large diffusion of In adatoms along the [1–10] direction, forming circular InAs QD structures on the surface.

3 Hybrid Structure of Semiconductor Quantum Well Superlattice and Quantum Dot The QW and QD have both advantages and disadvantages over each other. For example, although the QD has strong carrier confinement, it has smaller effective volume. Therefore, QD density should be high to create high performance QD laser diodes. A combination of QW and QD is occasionally adopted to produce improved optical and the electrical properties. For example, a QW was used as a reservoir to inject carriers into a QD through the tunnel barrier in the laser diode. In this case, the modulation frequency was better compared to direct injection [16]. Similar examples are reported involving double QWs where spin polarized electrons were transferred from one QW to the other in a II–VI compound semiconductor [17]. In this case the two QWs has different well widths to control their energy levels. Another example is the dot in well (DWELL) structure where the QDs are embedded in a QW [18, 19]. Its application in InAs QDs grown on GaAs substrate is widely investigated. As mentioned before, the InAs QD structure is grown on GaAs substrate using the SK growth mode. In this case, the emission wavelength of QDs was restricted to around 1000 nm because the lattice mismatch between InAs and GaAs was larger than that of InAs and InP. Therefore, InAs QDs receive stronger compressive strain from the GaAs substrate compared to the InP substrate. This strong compressive strain increases the bandgap of InAs and emission of InAs QDs appears around 1000 nm. The DWELL structure was developed to extend the emission wavelength of InAs QDs to the 1300 nm band which is one of the most important optical bands used in fiber optic communications systems. The schematic structure of DWELL is shown in Fig. 7a. The InAs QDs are embedded in the InGaAs QW which has a smaller bandgap than that of bulk GaAs. There are many kinds of DWELL structures. One involves InAs QDs embedded in the InGaAs QW only above the (Fig. 7b). Another involves a QD embedded in a GaAsSb QW [20, 21]. In this scheme, the InGaAs layer plays two roles. It reduces the confinement strength of InAs QDs by using InGaAs which has a smaller bandgap. In this case, the emission wavelength shifts longer direction. It also reduces the compressive strain which InAs receives from GaAs. As the InGaAs has a larger lattice constant, the compressive

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Fig. 7 Schematic structure of DWELL: a InAs QDs covered by InGaAs QW, b InAs QDs embedded in InGaAs QW

strain becomes smaller from InGaAs embedding as compared to GaAs. As mentioned above, the compressive strain makes the InAs bandgap larger. Therefore, reducing compressive strain also reduces this effect. It also contributes to extend the emission wavelength. These innovations have ensured that high performance 1300 nm band QD laser are being developed thus far [22, 23]. In similar fashion, the embedded structure of InAs QDs in InGaAs/InAlAs superlattice was fabricated [24] on InP(311)B substrate. The aim of this structure was to improve the controllability of barrier height. In addition, a change in carrier dynamics around QD was expected. Usually, the InAs QD is embedded in InGaAlAs. The InGaAlAs quaternary has a wide range of energy gaps and can be lattice matched to InP by changing its composition as shown in Fig. 8. Therefore, the confinement strength of the InAs QD for carriers can be controlled. However, the control of the composition is not easy because the exchange of Ga atoms and Al atoms is not sensitive to the lattice constant. This makes it difficult to identify the accurate composition using non-destructive measurement such as X-ray diffraction (XRD). Therefore, additional measurements which are destructive such as secondary ion mass spectroscopy are needed. This identification is important because the exchange 2.5 AlAs

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of Ga atoms and Al atoms leads to a large change in the InGaAlAs bandgap. On the other hand, as the composition of InGaAs and InAlAs can be determined easily using XRD, the optimization of the growth condition of InGaAs and InAlAs is also easy. As the short period InGaAs/InAlAs superlattices form energy minibands, they can be used to fabricate a confinement structure from InAs QDs and can control energy level precisely. The schematic structure is shown in Fig. 9. First, the InGaAs/InAlAs superlattice is grown on InP(311)B substrate. Subsequently, the self-assembled QDs are formed on the InAlAs layer of superlattice using the SK growth mode. Finally, the InAs QDs are embedded on the InGaAs/InAlAs superlattice. The embedding superlattice is starts from the InAlAs layer. This results in the well-defined energy level of the superlattice which is a repetition of two-dimensional structures. However, the structure around InAs QD is not well known. The actual structure of InAs embedded in the InGaAs/InAlAs superlattice is shown in Fig. 10 as an image from a cross-sectional scanning transmission electron microscope (STEM). In this sample, the superlattice was grown from 4 ML-thick InGaAs and 4ML-thick InAlAs. Before growing the QD layer, the superlattice layer was stoppered using InAlAs. Subsequently, the 5 ML-thick InAs was grown, forming the QD structure self-assembly. After that the InGaAs/InAlAs superlattice was grown to embed InAs QD with the same thickness of the underlying superlattice in which started from the InAlAs layer to realize strong confinement. In the cross-sectional STEM image, well defined InAs QDs and superlattice are observed. This method is called “digital embedding method” [24]. Figure 11 shows InAs QDs structure without an embedding layer grown on (a) bulk InGaAlAs, and (b) InGaAs/InAlAs superlattice. The QDs grown on InGaAs/InAlAs superlattice have similar sizes and densities compared to those grown on bulk InGaAlAs. The upper superlattice is not significantly disordered near the InAs QDs and flatness was displayed on the superlattice above the QDs instead of a corrugated structure. The growth of the superlattice seems promoted around the QDs at the initial stage of embedding because the lattice constant on the top of QDs was expanded where the InAlAs growth suppressed from the view point

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Fig. 10 The cross-sectional STEM image of InAs QDs embedded in In-GaAs/InAlAs superlattice

Fig. 11 The AFM image of InAs QDs a grown on InGaAlAs surface and b grown on InGaAs/InAlAs superlattice

of strain energy unstability. The structure of the superlattice was almost perfect. The miniband of the superlattice acted as a barrier for the QDs. The energy level of QDs was evaluated using photoluminescence (PL) spectroscopy. The PL energy (or wavelength) corresponds to the sum of the quantized energies of the electrons, holes, and bandgaps. In addition, the PL linewidth is related to the size distribution of the QDs. Therefore, the measurement of PL spectrum is important for the evaluation of QDs. In the digital embedding structure, the determination of superlattice miniband is important. The ideal superlattice miniband energy (E) is calculated using the Kronig–Penny model [25] as

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2 cos k(L w + L b ) = 2 cos αL w cosh β L b α2 − β 2 sin αL w sinh β L b , αβ  2m∗ E , α= 2  2m∗ (V0 − E) β= , 2 −

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where Lw , Lb , V0 , and m* denote well width, barrier width, barrier height, and effective carrier mass, respectively [26–28]. The energy levels or minibands determined from this method are shown in Fig. 12. In this case, the condition of Lw = Lb was used and changed from 2 to 16 ML ((a) to (d)). The In0.52 Ga0.24 Al0.24 As band diagram is also shown as a reference. The violet region shows the energy band gap of InGaAs, InAlAs and InGaAlAs, and the yellow regions show energy levels or minibands of the superlattice. The minimum energy of the miniband can be considered as an effective barrier height of the QW. When the thickness of superlattice increases, the lowest energy of miniband decreases. Moreover, the lowest energy of the superlattice with 2ML thickness is of almost the same value as that of the band edge of InGaAlAs.

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Fig. 13 PL spectra of InAs QDs embedded in InGaAs/InAlAs superlattice with a thickness of a 2 ML, b 4 ML, c 8 ML, d 16 ML, and e embedded in InGaAlAs

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The results of PL measurement are shown in Fig. 13. Spectra (a) to (e) show the PL spectra of QDs embedded by InGaAs/InAlAs superlattice with thicknesses ranging from 2 to 16 ML and a reference sample embedded by bulk InGaAlAs. The actual energy levels of QD evaluated using PL measurement (PL peak) shifted to a higher energy region, from 0.763 to 0.777 eV. This is the opposite of what was expected from the theoretical calculation. In this case, the effect of the InAlAs thickness just above/under the QDs layer might be higher than that of the energy level of superlattice. Indeed, when the thickness of InGaAs layer of superlattice increase in which the energy of miniband decrease, the PL peak shifts to a lower energy region as shown in Fig. 13. Therefore, the thickness of InAlAs around the QD is an important parameter used to control the energy levels of QDs and of the superlattice as well. The energy levels of the superlattice also affect the QD energy levels. Figure 14 show the PL spectra of QDs embedded in InGaAs/InAlAs superlattice where the InGaAs thickness is changed from 2 to 8 ML while the InAlAs thickness is fixed at 2 ML. In this case, the PL peak energy shifts to a lower energy region with increasing InGaAs layer thickness. This result agrees with the result of the band energy calculation

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Fig. 15 The temperature dependence of PL spectra for QDs embedded by InGaAs/InAlAs superlattice with a thickness of a 2 ML and b 16 ML

of the superlattice. In addition, another emission different from QD is observed at 0.90 eV in Fig. 13d which corresponds to the lowest energy level of the superlattice. This energy level is located close to QD so that various interactions between the superlattice and QD are to be expected. The primary interaction between the superlattice and the QDs is carrier transfer. The evaluation of PL dependence on temperature will give various information such as the activation energy of the carrier from the confining nanostructure and the crystal quality. The change of the PL spectra of QDs are shown in Fig. 15. Usually, the PL intensity decreases two to three orders of magnitude with increasing temperature from 10 to 300 K. The relationship between PL intensity (I) and temperature (T) are described as I =

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Fig. 16 Temperature dependence of PL intensity embedded by a 2 ML-thick InGaAs/InAlAs superlattice, b 16 ML-thick superlattice, and c InGaAlAs. Circles and square indicate QD and superlattice emissions, respectively. Red lines are fitting curves calculated using Eq. (4)

shows the temperature dependence of the PL intensities of various structures. The circles and squares indicate the results of the QDs and the superlattice, respectively, and the red lines are fitting curves calculated using Eq. (4). The activation energy of carriers from the QD embedded by InGaAlAs (conventional structure) is 201 meV which is sufficiently larger than thermal energy of room temperature. This results in strong emissions observed at room temperature as shown in Fig. 13e. As compared to a conventional structure, a similar activation energy of 194 meV is observed in the sample where InAs QDs are embedded in the 2 ML-thick InGaAs/InAlAs superlattice. Although there are many hetero-interfaces in this sample, the confinement quality was better maintained compared to a conventional structure. The activation energy for the QD emission in the sample with 16 ML-thick superlattice is 169 meV which is slightly smaller than those of the two activation energies mentioned above. This is because the energy level of the superlattice is closer to the QD energy level, allowing the carriers confined in the QDs to escape to the superlattice relatively easily. However, since this value is also larger than the thermal energy of room temperature, a strong emission is observed at room temperature. On the other hand, the emission of the superlattice gradually decreases with increasing temperature. This yields an activation energy of 38 meV which is significantly smaller than that of QD. This is because the carrier in the superlattice can move two dimensionally which leads to its capture in non-radiative recombination centers. In addition, the PL peak intensity of QD and superlattice compete with each other as the temperature changes. This implies that the direction of carrier transport can be altered by changing the temperature. Actual carrier dynamics can be evaluated using time resolved PL measurement. If the carriers are excited by a very short optical pulse, the generated carriers move

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to radiative or non-radiative recombination centers depending on time. Therefore, the time evolution of the PL spectrum gives us the carrier dynamics of the target energy levels. The combination of a monochromator and a streak camera was used to detect high speed changes in luminescence. The PL spectra and time evolution of each peak wavelength measured at 10 K are shown in Figs. 17 and 18. In Fig. 17a–c show the PL spectra from the sample where QDs embedded in InGaAlAs (black), QDs embedded by 2 ML-thick superlattice (red), and QDs embedded by 16 MLthick superlattice (blue), respectively. The QDs emissions at around 0.83 eV and the Fig. 17 (a) PL spectra of the samples where the InAs QDs embedded in bulk InGaAlAs (black), 2 ML-thick superlattice (red), and 16 ML-thick superlattice (blue)

Fig. 18 The time evolution of PL peak intensities in Fig. 14: a measured at 0.805 eV (QDs) in Fig. 14a; b measured at 0.830 eV (QDs) in Fig. 14b; c measured at 0.845 eV (QDs) in Fig. 14c; and d measured at 0.967 eV (superlattice) in Fig. 14c

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SL emission at 0.97 eV were observed. The time evolution of peak energy of the black line spectrum at 0.805 eV is shown in Fig. 18a The PL decay (the relationship between PL intensity (I) and time (t)) is fitted using following equation: 

t I (t) = I1 exp − τ1



  t + I2 exp − , τ2

(5)

where τ1 and τ2 are PL decay times and I1 and I2 are constants. If the origin of PL decay is one, the second term can be ignored. The decay time τ1 = 2.0 ns was obtained in this analysis for the QD embedded in InGaAlAs which is the normal value for a QD in this material system. In this case, the energy of excitation was 1.59 eV (with a pulse width of approximately 200 fs), so that this decay time included the carrier relaxation time from barrier to QDs, the carrier relaxation time from excited states to ground states of QDs and the radiative recombination lifetimes. In the case of QD emission embedded in a 2ML-thick superlattice (red line) the decay curve changed slightly as shown in Fig. 18b. The decay time was estimated to be 1.4 ns. In this case, the generated carriers moved in the superlattice and were captured by the QDs. This occurred more efficiently than in the conventional structure of QDs. Meanwhile, the emission of InAs QDs embedded by 16 ML-thick superlattice had a decay time of 2.8 ns, longer than that of the one embedded by bulk InGaAlAs. In this sample, since the emission of superlattice was also observed, some amount of excited carrier must have been captured for a duration in the superlattice before it moved to the QDs. This is one of the factors of the longer decay time of the QDs. The time evolution of superlattice emission is shown in Fig. 18d which describes a decay much faster than that of the QDs. In the superlattice, the confinement of carriers occurs in one direction. Therefore, the carriers can move along the plane of the layer, especially InGaAs, which leads to enhanced recombination with and without emission, and to escape to other energy levels, such as QD excited states. As the carrier dynamics can be changed by changing superlattice period, superlattices can be useful for devices. For example, if the device need fast responses, such as a high-speed modulation laser diode, the carrier decay time at certain energy level should be small. On the other hand, longer decay times are needed for absorption type devices such as solar cells and photodiodes. The digital embedding method can be used to respond to these requirements.

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