Advances in Optical Physics: Volume 7 Advances in Condensed Matter Optics 9783110307023, 9783110306934

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Liangyao Chen, Ning Dai, Xunya Jiang, Kuijuan Jin, Hui Liu, Haibin Zhao Advances in Condensed Matter Optics

Advances in Optical Physics

| Editor-in-Chief Jie Zhang

Volume 7

Liangyao Chen, Ning Dai, Xunya Jiang, Kuijuan Jin, Hui Liu, Haibin Zhao

Advances in Condensed Matter Optics | Edited by Liangyao Chen

Physics and Astronomy Classification Scheme 2010 78.20.-e, 78.20.Bh, 78.20.Ci, 78.20.Ls, 78.67.Pt, 78.90.+t Editor Prof. Liangyao Chen Kexue Building Fudan University 220 Handan RD 200433, Yangpu District Shanghai China

ISBN 978-3-11-030693-4 e-ISBN (PDF) 978-3-11-030702-3 e-ISBN (EPUB) 978-3-11-038818-3 Set-ISBN 978-3-11-030703-0 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Shanghai Jiao Tong University Press and Walter de Gruyter GmbH, Berlin/Munich/Boston Cover image: Ellende/iStock/Thinkstock Typesetting: PTP-Berlin, Protago TEX-Production GmbH Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

The series: Advances in Optical Physics Professor Jie Zhang, Editor-in-chief, works on laserplasma physics and has made significant contributions to development of soft X-ray lasers, generation and propagation of hot electrons in laser-plasmas in connection with inertial confinement fusion (ICF), and reproduction of some extreme astrophysical processes with laser-plasmas. By clever design to enhance pumping efficiency, he and his collaborators first demonstrated saturation of soft-X-ray laser output at wavelengths close to the water window. He discovered through theory and experiments that highly directional, controllable, fast electron beams can be generated from intense laser plasmas. Understanding of how fast electrons are generated and propagated in laser plasmas and how the resulting electron beams emit from a target surface and carry away laser excitation energy is critical for understanding of the fast-ignition process in ICF. Zhang is one of the pioneers on simulating astrophysical processes by laser-plasmas in labs. He and his collaborators used high-energy laser pulses to successfully create conditions resembling the vicinity of the black hole and model the loop-top X-ray source and reconnection overflow in solar flares. Because of his academic achievements and professional services, Professor Zhang received Honorary Doctors of Science from City University of Hong Kong (2009), Queen’s University of Belfast (2010), University of Montreal (2011) and University of Rochester (2013). He was elected member of CAS in 2003, member of German Academy of Sciences Leopoldina in 2007, fellow of the Third World Academy of Sciences (TWAS) in 2008, foreign member of Royal Academy of Engineering (FREng) of the UK in 2011 and foreign Associate of US National Academy of Sciences (NAS) in 2012. He is the President of Shanghai Jiao Tong University, and also a strong advocate and practitioner of higher education in China.

Preface After a three years’ effort by many top-tier scientists, the book series Advances in Optical Physics (English version) is completed. Optical physics is one of the most active fields in modern physics. Ever since lasers were invented, optics has permeated into many research fields. Profound changes have taken place in optical physics, which have expanded tremendously from the traditional optics and spectroscopy to many new branches and interdisciplinary fields overlapping with various classical disciplines. They have further given rise to many new cutting-edge technologies: – For example, nonlinear optics itself is an interdisciplinary field, which has been developing since the advent of lasers and it is significantly influenced by various technological advances, including laser technology, spectroscopic technology, material fabrication and structural analysis. – With the rapid development of ultra-short intense lasers in the past 20 years, high field laser physics has rapidly developed into a new frontier in optical physics. It contains not only rich nonlinear physics under extreme conditions, but also has the potential of many advanced applications. – Nanophotonics, which combines photonics and contemporary nanotechnology, studies the mechanisms of light interactions with matter at the nanoscale. It enjoys important applications such as in information transmission and processing, solar energy, and biomedical sciences. – Condensed matter optics is another new interdisciplinary field, which is formed due to the intersection of condensed matter physics and optics. Here, on the one hand, lasers are used as probes to study the structures and dynamics of condensed matter. On the other hand, discoveries from condensed matter optics research can be applied to produce new light sources, detectors, and a variety of other useful devices. In the last 20 years, with the increasing investment in research and development in China, the scientific achievements by Chinese scientists also become increasingly important. These are reflected by the greatly increased number of research papers published by Chinese scientists in prestigious scientific journals. However, there are relatively few books for a broad audience – such as graduate students and scholars – devoted to this progress at the frontiers of optical physics. In order to change this situation, three years ago, Shanghai Jiao Tong University Press discussed with me and initiated the idea to invite top-tier scientists to write the series of “Advances in Optical Physics”. Our initial plan was to write a series of introductory books on recent progresses in optical physics for graduate students and scholars. It was later expanded into its current form. The first batch of the series includes eight volumes:

viii | Preface – – – – – – – –

Advances in High Field Laser Physics Advances in Precision Laser Spectroscopy Advances in Nonlinear Optics Advances in Nanophotonics Advances in Quantum Optics Advances in Ultrafast Optics Advances in Condensed Matter Optics Advances in Molecular Biophotonics

Each volume covers a number of topics in the respective field. As the editor-in-chief of the series, I sincerely hope that this series is a forum for Chinese scientists to introduce their research advances and achievements. Meanwhile, I wish these books are useful for students and scholars who are interested in optical physics in general, one of these particular fields, or a research area related to them. To ensure these books could reflect the rapid advances of optical physics research in China, we have invited many leading researchers from different fields of optical physics to join the editorial board. It is my great pleasure that many top tier researchers at forefronts of optical physics accepted my invitation and made their contributions in the last three years. Almost at the same time, De Gruyter learned about our initiative and expressed their interest in introducing these books written by Chinese scientists to the rest of world. After discussion, De Gruyter and Shanghai Jiao Tong University Press reached the agreement in co-publishing the English version of the series. At this moment, on behalf of all authors of these books, I would like to express our appreciation to these two publishing houses for their professional services and supports to sciences and scientists. Especially, I would like to thank Mr. Jianmin Han and his team for their great contribution to the publication of this book series. At the end of this preface, I must admit that optical physics itself is a rapidly expanding forefront of science. Due to the nature of the subject area, this series can never cover all aspects of optical physics. However, what we can do – together with all authors of these books – is to try to pick up the most beautiful “waves” from the vast science ocean to form this series. By publishing this series, it is my cherished hope to attract minds of younger generation into the great hall of optical physics research. Professor Jie Zhang Editor-in-chief

Contents Preface | vii Ning Dai 1 Optoelectronic properties of narrow band gap semiconductors | 1 1.1 Introduction | 1 1.2 Fundamental properties of NGSs | 3 1.2.1 Electronic states and band structures | 4 1.2.2 Structural characteristics | 8 1.2.3 Crystal growth | 9 1.2.4 Electronic properties | 12 1.2.5 Optical properties | 15 1.3 Narrow band gap semiconductors and their basic characteristics | 21 1.3.1 Mercury cadmium telluride (Hg1−x Cdx Te) | 21 1.3.2 Indium antimonide (InSb), Indium arsenide (InAs), Indium arsenide antimonide (InAs1−x Sbx ) | 25 1.3.3 Lead telluride (PbTe), lead selenide (PbSe), lead sulfide (PbS) and tellurium tin-lead (Pb1−x Snx Te) | 27 1.3.4 Heterojunctions, quantum wells, and superlattices | 30 1.4 Basic principles and applications of infrared optoelectronic devices | 31 1.4.1 Basic principles of infrared detectors | 31 1.4.2 Parameters for characterizing the performance of infrared detectors | 35 1.4.3 Photoconductive infrared detectors | 37 1.4.4 Photovoltaic infrared detectors | 39 1.4.5 Quantum well infrared photodetectors | 43 1.4.6 Infrared light sources: infrared light emitting devices and infrared lasers | 44 Xunya Jiang, Wei Li, Zheng Liu, Xulin Lin, Xianggao Zhang, Zixian Liang, Peijun Yao 2 The group velocity picture: the dynamic study of metamaterial systems | 51 2.1 Introduction | 51 2.2 Hyperinterface, the bridge between radiative and evanescent waves | 54 2.2.1 Introduction | 54 2.2.2 Model | 55 2.2.3 Hyperbola dispersion and compressing light pulses effect at HI | 56 2.2.4 Analysis of abnormal optical properties of HI with group velocity | 57

x | Contents 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 2.8 2.8.1 2.8.2 2.8.3 2.8.4

Numerical experiments and results | 59 Section summary | 60 Methods for detecting vacuum polarization by evanescent modes | 61 Study model | 62 The phase change and delay time of evanescent waves in a tiny dissipative medium | 64 Vacuum polarization and refraction index deviations of a vacuum | 65 Detecting vacuum polarization: phase change and delay time | 66 Section summary | 68 The temporal coherence gain of the negative-index superlens image | 68 Introduction | 68 Model | 69 Unusual phenomena | 70 Physical images | 71 Our theory | 72 Section summary | 74 Dynamical process for dispersive cloaking structures | 75 Introduction | 75 Study model | 76 The physical dynamical picture of invisible cloaking | 77 The key factor for the dynamics of invisible cloaking | 78 Section summary | 81 Limitation of the electromagnetic cloak with dispersive material | 82 Introduction | 82 The group velocity and physical limitation of invisible cloaking | 83 Numerical results and discussion | 85 Section summary | 87 Confining the one-way mode at a magnetic domain wall | 88 Introduction | 88 Model | 89 Confining the one-way mode | 90 Robustness against roughness | 92 Photonic splitters and benders | 92 Section summary | 94 Bullet-like light pulse in linear photonic crystals | 95 Introduction | 95 The condition for the existence of bullet-like light pulses | 95 The bullet-like light pulse in PCs | 96 Numerical validation | 96

Contents

2.8.5 2.8.6 2.9

| xi

The effect of high-order dispersion | 98 Section summary | 99 Summary | 100

Liangyao Chen, Yuxiang Zheng, Songyou Wang 3 Study of the characteristics of light propagating at the metal-based interface | 107 3.1 Introduction | 107 3.2 The free-electron gas model and optical constants of metal | 108 3.3 Light refraction properties of a metal-based interface | 112 3.3.1 Normal refraction | 112 3.3.2 Calculations of effective refractive index and refraction angle | 113 3.3.3 Negative refraction of metal-based artificial materials | 116 3.3.4 Measurement of the effective refractive index and refractive angle of light in metal | 120 3.3.5 Influence of variable refractive indices on light velocity | 127 3.4 Affect of surface plasma waves on light propagation in metals | 130 3.5 Conclusion | 134 Haibin Zhao 4 Photo-induced spin dynamics in spintronic materials | 139 4.1 Introduction | 139 4.2 Theory of magnetization dynamics | 140 4.2.1 The Landau–Lifshitz–Gilbert (LLG) equation | 140 4.2.2 The Landau–Lifshitz–Bloch (LLB) equation | 143 4.3 Optical techniques in studies of spin dynamics | 144 4.3.1 Time-resolved magneto-optical spectroscopy | 144 4.3.2 Time-resolved magnetic second-harmonic-generation (TR-MSHG) | 151 4.4 Photo-induced demagnetization and magnetic phase transition | 154 4.4.1 Demagnetization in transition ferromagnetic (FM) metals | 154 4.4.2 Demagnetization in other FM materials | 160 4.4.3 Ultrafast magnetization generation and FM phase transition | 161 4.5 Photo-induced spin precession | 162 4.5.1 Uniform spin precession and spin wave in FM materials | 162 4.5.2 Spin waves in ferromagnetic materials | 165 4.5.3 Mechanisms of spin precession excitation | 167 4.6 Photo-induced spin reversal | 174 4.6.1 Spin switching and reversal in FM materials | 174 4.6.2 Spin reversal in ferromagnetic materials | 176 4.7 Spin dynamics at interfaces and in antiferromagnets | 180 4.7.1 MSHG and magnetism at interfaces | 180

xii | Contents 4.7.2 4.8

Spin dynamics in antiferromagnets | 182 Conclusions and outlook | 183

Kuijuan Jin, Chen Ge, Huibin Lu, Guozhen Yang 5 Research on the photoelectric effect in perovskite oxide heterostructures | 191 5.1 Introduction | 191 5.2 Perovskite oxide | 192 5.2.1 Crystal structure | 192 5.2.2 Electron structure | 193 5.2.3 Mechanism for photoelectric effects in bulk perovskite oxides | 195 5.3 Growth of perovskite oxide films | 200 5.3.1 A brief introduction to the film-growth techniques | 200 5.3.2 Laser molecular beam epitaxy | 201 5.4 Logitudinal photoelectric effects in perovskite oxide heterostructures | 203 5.4.1 Light-generated carrier injection effects | 203 5.4.2 Photovoltaic effect | 205 5.4.3 Theoretical study on longitudinal photoelectric effects | 210 5.5 Lateral photoelectric effect in perovskite oxide heterostructures | 216 5.5.1 Background | 216 5.5.2 Unusual lateral photoelectric effect in perovskite oxide heterostructures | 217 5.5.3 Theoretical study | 219 5.6 Summary | 222 Hui Liu, Shining Zhu 6 Magnetic resonance and coupling effects in metallic metamaterials | 231 6.1 Background | 231 6.2 Magnetic metamolecules | 234 6.2.1 Plasmon hybridization effect | 234 6.2.2 Hybridization effect in magnetic metamolecules | 235 6.2.3 Stereometamaterial | 237 6.2.4 Optical activity in magnetic metamolecules | 237 6.2.5 Radiation of magnetic metamolecules | 239 6.2.6 Other designs of magnetic metamolecules | 239 6.3 One-dimensional magnetic resonator chains | 240 6.3.1 Periodic magnetic resonator chain | 240 6.3.2 Nonperiodic chain of magnetic resonators | 245 6.3.3 Nonlinear and quantum optics of magnetic resonators | 247 6.4 Magnetic plasmon crystal | 248

Contents

6.4.1 6.4.2 6.4.3 6.5

| xiii

Two-dimensional fishnet structure | 249 Two-dimensional nanosandwich structures | 253 Quantum interference in a three-dimensional magnetic plasmon crystal | 261 Summary and outlook | 265

Index | 271

Ning Dai

1 Optoelectronic properties of narrow band gap semiconductors 1.1 Introduction Narrow band gap semiconductors, meaning those with their energy gaps in the infrared wavelength range, are an important branch of semiconductor materials. The narrow band gap gives rise to strong thermal excitation of intrinsic carriers. Compared to their wide band gap counterparts, narrow gap semiconductors (NGSs) have the characteristics of high carrier concentration, difficulty in controlling p-type and n-type doping, and being soft mechanically. All these characteristics determine that NGSs behave quite different from semiconductors with a relatively wide band gap, especially in infrared opto-electronic properties. The peculiar opto-electronic properties of semiconductors lie in their tremendous change in conductivity by even very slight doping and their strong wavelengthdependent optical absorption coefficients. A semiconductor is transparent at photon energies smaller than its band gap and opaque at photon energies larger than the band gap. With excellent opto-electronic properties characterized by strong optical response to light, semiconductors are the best choice for opto-electronic detection. Unfortunately, NGSs gradually loose the features of semiconductors by behaving more and more like metals with decreasing band gap. As a result, infrared devices (detectors and light emitting diodes), especially long wavelength infrared devices, have low opto-electronic performance. The major material problems for NGSs are the following. (1) The bonding between atoms in NGSs is rather weak, leading to high defect density and poor crystalline completeness. It is very difficult to grow the “soft” materials, since they often contain high density of defects and dislocations. Large size single crystals are difficult to grow. For instance, single crystal Si ingots of more than 20 inches in diameter can now be prepared, but the growth of 2 inches Hg1−x Cdx Te already becomes very difficult. In addition, defect density in Si is about 1–2 orders of magnitude lower than that of Hg1−x Cdx Te. (2) Controllable doping, as the most distinct property of semiconductors, becomes rather difficult in NGSs. It is hard to achieve dual doping with excellent performance for a given NGS. Wide band gap semiconductors show properties of insulators. A given wide band gap semiconductor, if it can be easily p-type doped, is hard to be n-type doped, or vice versa. For example, the wide band gap ZnO (band gap is 3.34 eV) and ZnSe (band gap is 2.71 eV) can be easily doped with n-type, while p-type doping is extremely difficult.

2 | Ning Dai Wide band gap ZnTe with a band gap of 2.24eV, on the other hand, can be easily ptype doped, but n-type doping is difficult. Materials with the band gap in the range of 1∼2 eV exhibit typical semiconductor characteristics, and they can be doped with both n-type or p-type, such as Si (band gap = 1.12 eV) and GaAs (band gap = 1.4 eV). For NGSs, the narrower the band gap, the closer their properties to metals. For example, it is difficult to grow intrinsic Hg1−x Cdx Te with a narrow band gap. The carrier concentration in Hg1−x Cdx Te is generally high, due to all kinds of possible contaminations. Although the n-type doping is relatively easy to obtain, good p-type doping is hard to realize. In Hg1−x Cdx Te, p-type doping is often achieved by introduction of defects (such as Hg vacancies), and the carriers have low hole mobility and a low lifetime of the minority carriers. (3) The narrower the semiconductor band gap, the closer the band gap energy to the characteristic energies of impurities, defects, phonon and other low-energy excitation. As a result, the low-energy excitations have a more and more significant impact on the optoelectronic devices. The phonon effect can be suppressed by reducing the operating temperature of devices. For example, the operating temperature of a Hg1−x Cdx Te detector with 15 μm cutoff wavelength is often around 30 K to suppress the intrinsic thermal excitation. Such a low operating temperature is not convenient for the practical use of the device. Low-energy excitation such as defects and impurities cannot be eliminated by lowering temperature. (4) In long-wavelength infrared devices, the fundamental physical processes depend sensitively on the band gap. For a Hg1−x Cdx Te infrared detector, the pn junction leakage current could increase by 2 orders of magnitude under the same operating temperature, for a 15 % band gap variation from 12.5 to 14.5 μm! The above basic physical problems are the essential reasons that cause infrared devices, especially long-wavelength infrared region devices, to have low detectivity, large leakage current, large dark current, big device noise, serious crosstalk, and low operating temperature. The fundamental drawbacks of NGS affect device performance seriously and, sometimes even make devices unusable. For space applications, in particular for signal detection from deep space, the temperature of targeted objects could be extremely low, so that objects emit a very long wavelength and weak signal. Detectors with very high sensitivity and very long cutoff wavelength are thus required. Given the fact that material performance degrades for long wavelength infrared semiconductors, it is thus challenging to obtain detectors with a very long cutoff wavelength. More information for infrared materials and devices can be found in [1–4]. The characteristic wavelength of a cold object could be 15 μm or longer, which is the same order of magnitude as the pixel size of an infrared focal plane array that

1 Optoelectronic properties of narrow band gap semiconductors

|

3

is typically around 20–50 μm. Thus, the optical coupling process is possibly affected by some unknown factors. The interaction between light and matter is an important scientific issue and the interaction between low energy photons and low-dimensional semiconductor structure is a hot topic receiving a great deal of attention. The interaction process can only be well-described by quantum mechanics. A long wavelength focal plane array is a typical system in which a photosensitive pixel is a low-dimensional structure to infrared photons. The details of the interaction processes involve light absorption, reflection, transmission, and energy exchange when light travels in structures of the subwavelength dimension or much smaller than the wavelength dimension. Interaction of light with matter is related to some core issues including photoelectric coupling and photoelectric conversion. Studying interaction mechanism aims to find ways to tune the interaction. The study of NGSs began with PbS in the 1940s. InSb then became a typical narrow band gap semiconductor material after its discovery in 1952. From 1959, studies were focused on HgTe- and CdTe-based ternary alloy – Hg1−x Cdx Te. The energy gap of Hg1−x Cdx Te can be tuned continuously from 1.42 to − 0.3 eV with Cd composition x varying from 1 to 0. The band gap of Hg1−x Cdx Te becomes zero at the Hg content of 0.84. In the past, the studied ternary mixed crystals of NGSs include IV– VI Pb1−x Snx Te and Pb1−x Snx Se, II–VI Hg1−x Cdx Te, Hg1−x Cdx Se and Hg1−x Mnx Te, as well as III–V In1−x Asx Sb and InAs1−x−y Nx Sby . This chapter focuses on the infrared optoelectronic properties and applications of narrow band gap semiconductors in terms of the fundamental material properties, experimental methods, as well as various infrared devices.

1.2 Fundamental properties of NGSs One of the most important parameters that determine the optical and electrical properties of semiconductors is the band gap of materials, and the band gap of crystals may vary from zero (or even negative band gap) to nearly 10 eV depending on materials. In terms of band gap value Eg , crystals with Eg greater than 2.5 eV are generally referred as insulators, and those with Eg between 0–2.5 eV are semiconductors. Materials of negative Eg are semimetals. Metals have energy bands with their conduction band being half filled with electrons. In narrow band gap materials, the conduction band and the valence band are very close in energy, so that the interaction between neighboring bands is very strong. Consequently, the dispersion relation (electron energy E and the wave vector k dependence) exhibits strong nonparabolic characteristics, and the effective mass around the extreme points of dispersion relation is no longer a constant. Effective mass for electrons and holes is approximately proportional to the band gap. A material with a narrow band gap has small electron and hole effective masses, and its band parameters are sensitive to the quantum confinement effect. The effective g factor of a narrow band gap material is rather large, and its magnetic properties are

4 | Ning Dai relatively abundant. Hg1−x Cdx Te, InSb, etc. are popular nonmagnetic materials that show interesting magnetic properties. In fact, those materials are extensively studied for their magnetic quantum transport and spin-dependent phenomena. For example, the Rashba spin-orbit splitting of Hg1−x Cdx Te is as large as 30 meV under zero magnetic fields [5]. In addition, the defect density and background impurity concentration of NGSs are high, and their intrinsic thermal excitation is significant. Thus, the carrier concentration in a NGS is relatively high, and it is relatively easy for the material to become degenerate, as its Fermi level moves into the conduction band.

1.2.1 Electronic states and band structures There are many ways to describe and calculate the band structure of semiconductors, including the tight-binding method based on electron orbital wave functions in free atoms, primitive cell, augmented plane wave, the nearly free electron approximation based on free electron plane wave functions, orthogonal plane wave and pseudopotential method. By the use of high performance computers, energy bands described in the reciprocal space (wave vector space) can be calculated at high accuracy. The envelope function approximation has proven to be very effective. It is a full description of electron and hole states that often gives analytical results for relatively simple systems. The envelope function approximation is particularly suitable for crystals with man-made structures, including periodic and quasi-periodic quantum wells, superlattices. The envelope function approximation is able to describe electronic states in the presence of an electromagnetic field in strain conditions. The simplest envelope function approximation is the effective mass approximation in which the complicated crystal potential is included in the effective masses of electrons and holes, so that the corresponding Schrödinger equation looks like that of a free electron. Assuming isotropic materials and single parabolic band edges, the dispersion relation in the effective mass framework takes the form of Ec,v (k) = ± (

Eg 2

+

ℎ2 k2 ). 2m∗c,v

(1.1)

In equation (1.1), Ec and Ev represents the energy levels at the bottom of the conduction band and the top of the valence band (zero energy is chosen in the middle of the band gap), respectively. Eg is the band gap energy, k the wave vector, m∗c and m∗v are the effective mass of electrons and holes, respectively. Here “+” is chosen for the conduction band and “−” for the valence band. Since both m∗c and m∗v are proportional to Eg which is relatively small for infrared materials, effective masses in NGSs are small compared to materials with large band gaps. Although the effective mass approximation can describe the fundamental properties of electronic states in heterojunction, quantum well, and superlattice, it cannot explain the nonparabolic behavior at the band edges, as well as the coupling between bands. A so-called k ⋅ p method was de-

1 Optoelectronic properties of narrow band gap semiconductors

|

5

veloped and proven to be very effective in dealing with interactions at perturbation approximation. As a semiempirical approach, the k ⋅ p method allows extrapolation of the band structure of materials over the entire Brillouin zone from a restricted set of parameters evaluated for the energy gaps and matrix elements. Although the k ⋅ p method cannot give a global description of the band structure in reciprocal space, it gives information and useful parameters (such as effective masses) concerning the energy band extremes of high symmetry points in the Brillouin zone shown in Figure 1.1. Particularly, electron and hole effective masses and the corresponding wave functions at the high symmetry points can be derived by the k ⋅ p method. These high symmetry points determine the optical and electrical properties of a semiconductor.

z Ga As Λ Г x

L U Q ∑X ∑ Δ Z K W

y

Fig. 1.1. The unit cell of GaAs (left) and the first Brillouin zone in k-space (right).

The motion of electrons in a periodic crystalline potential follows Bloch’s law, i.e. the wave function of electrons has the form eik⋅r uvk (r) (v is the index of the bands). Thus, the corresponding Schrödinger equation is [

p2 + V0 (r)] eik⋅r uvk (r) = Ev (k) eik⋅r uvk (r) , 2m0

(1.2)

where m0 is the effective mass of the free electrons and V0 (r) is the periodic crystalline potential. Note that the spin degree of freedom for electrons has been ignored. Acting the operators on the wave function, and cancelling the plane wave part eik⋅r on both sides of the equation, we then have [

ℎ2 k2 p2 ℎ + V0 (r) + + k ⋅ p] uvk (r) = Ev (k)uvk (r). 2m0 2m0 m0

(1.3a)

Or using simple Dirac operators for the wave function eik⋅r uvk (r) ≡ eik⋅r ⟨r|vk⟩, [

ℎ2 k2 ℎ p2 + V0 (r) + + k ⋅ p] |vk⟩ = Ev (k)|vk⟩. 2m0 2m0 m0

(1.3b)

6 | Ning Dai Including the Pauli spin-orbit interaction Hamiltonian ∇Hso = − 4mℎ2 c4 σ ⋅ p × ∇V(r) 0

equation (1.3) becomes [

p2 ℎ2 k2 ℎ ℎ + V0 (r) + + k⋅π + p ⋅ σ × ∇V(r)] |vk⟩ = Ev (k)|vk⟩, 2m0 2m0 m0 4m20 c2

with π =p+

ℎ σ ×V(r). 4m0 c2

(1.4)

(1.5)

Since {|vk⟩} forms a complete and orthogonal set for any k, the Bloch functions ei0⋅r uv0 (r) = {v0⟩} (at k = 0 are also orthogonal and complete. We can then expend |vk⟩ in terms of {|v0⟩} to get |vk⟩ = ∑ Cv,n (k)|n0⟩. (1.6) n

Multiplying equation (1.6) by ⟨m0| on the left side, and using the orthogonal property of {|v0⟩}, the following algebra equation ∑ {[En (0) + n

ℎ2 k2 ℎ ] δm,n + k ⋅ pm,n } Cv,n (k) = Ev (k)Cv,m (k) 2m0 m0

(1.7)

is obtained, where the matrix element pm,n = ⟨m0 | p | n0⟩ .

(1.8)

To be specific, {|m0⟩} is a complete set with infinite degree of freedom. Thus equation (1.7) represents an n × n matrix with n → ∞. In the matrix formula described by equation (1.7), mℎ k ⋅ pm,n is the nondiagonal term that causes overlapping of band 0 edge wavefunctions |m0⟩. With increasing k and closing energy bands, the overlap integral become large. Theoretically the dispersion relation Ev (k) and the expansion coefficient Cv,m (k) of the system can be extracted by diagonalizing equation (1.7) so as to obtain the wave function |vk⟩. In NGSs, the conduction band Γ6 , the heavy hole and light hole band Γ8 as well as the spin-orbit splitting band Γ7 are very close in energy. In the Kane model, the Hamiltonian corresponding to Γ6 , Γ8 , and Γ7 bands are diagonalized accurately, while the effect of remote bands are treated as perturbations. This gives rise to an 8×8 matrix, instead of dealing with an infinite matrix, making it possible for practical calculation [6]. For instance, the band gap of InSb is 0.23 eV, so that Γ6 , Γ8 , Γ7 are very close in energy. From the 8-band edge wave functions with spin inclusion, i.e. s-state Bloch functions |S, +12 ⟩ and |S, − 12 ⟩, p-state Bloch functions |X, + 12 ⟩, |X, − 12 ⟩, |Y, + 12 ⟩, |Y, − 12 ⟩, |Z, + 12 ⟩, and |Z, − 12 ⟩, where ± 21 represents spin-up and spin-down electron states, respectively, equation (1.7) for the 8 bands can be diagonalized through proper linear combination in terms of those basis functions. After diagonalizing equation (1.7), we obtain a set of new bases – a new set of band edge wave functions that have definite total angular momentum J = L + σ and Jz (the projection of J = L + σ along the z axis). Hence, the sum of the angular momenta L = 0 and σ = 12 yields the total angular momentum

1 Optoelectronic properties of narrow band gap semiconductors

| 7

quantum number J = 12 for the s edge and the sum of L = 1 and σ = 12 results in the total angular momentum quantum number J = 32 and J = 12 for the p band edge. J = 32 for the p band edge are four-fold degenerate, corresponding to Jz = ± 32 heavy hole bands and Jz = ± 12 light hole bands (Γ8 band), while the doublet J = 12 corresponds to the spin-orbit splitting Γ7 band. In most III–V and II–VI compound semiconductors, Γ7 is below Γ8 in energy. The 8 new bases and k ⋅ p matrix elements given by the Kane model are listed in [7]. In terms of k ⋅ p calculation, energy band overlapping, band nonparabolicity and inhomogeneity, and effective masses can be calculated with good accuracy. For instance, equation (1.7) can be diagonalized through second-order perturbation to give En (k) = En (0) +

ℎ2 k2 ℎ2 k2 | ⟨m0 | p | n0⟩ |2 ∑ + 2m0 m20 m En (0) − Em (0)

(1.9)

when the spin degree of freedom is dropped. In equation (1.9) we get the reciprocal of effective mass 1 1 2 | ⟨m0 | p | n0⟩ |2 ∑ = + . (1.10) m∗n m0 m20 m En (0) − Em (0) Including the energy band overlapping, k ⋅ p is able to give the En ∼ k dispersion relation as in equation (1.9), as well as the inhomogeneous band edges and nonparabolic effective masses described by equation (1.10). k ⋅ p is powerful for specific points in the Brillouin zone. A global description of the energy bands relies on other methods, such as tight binding and empirical pseudopotential. Figure 1.2 presents the calculated energy bands of the narrow band gap InSb using empirical pseudopotential [8]. 0 -2

Energy(eV)

-4 -6 -8 -10

mL=2.45 m0

-12

mX =3.90 m0

mΓ =0.016 m0

-14

L

Γ

X

Fig. 1.2. Energy band of InSb calculated by empirical pseudopotential.

8 | Ning Dai 1.2.2 Structural characteristics Most III–V and II–VI NGSs have a zincblende crystalline structure, as shown in Figure 1.1, with their crystal unit cell formed by displacing the face-centered cubic lattice of the positive ions for 1/4 length along the body diagonal of the unit cell with respect to the face-centered lattice of the negative ions. Each positive (negative) ion has four nearest negative (positive) ion neighbors. The lattice constant of a III–V and II– VI group semiconductor is larger than those of Si and Ge of the diamond structure. In particular, the lattice constant of II–VI materials is generally 20 % larger than that of Si. The physical properties of different crystal plane are quite different. For wafers of zincblende semiconductors, if the top surface of the (111) crystalline plane is positiveion-terminated, the bottom surface must be negative-ion-terminated, from the atomic bonding geometry. Although the NGSs and wide band gap semiconductors (such as GaAs and ZnSe) are the same in their crystalline structures, their lattice constants differ quite a lot. Both II–VI HgCdTe and III–V InAsSb have much larger lattice constants than those of Si and Ge, which is partially related to their large atom sizes and weak ionic bonding. Zincblende semiconductors are both ionic and covalent in their atomic bonding, while the bonding characteristics, particularly the bond strength varies greatly with the materials. Hg1−x Cdx Te, for instance, is a ternary compound formed by Hg–Te bonds and Cd–Te bonds. Studies show that the bonding energy of the Hg–Te bonds is roughly one order of magnitude smaller than that of the Cd–Te bonds. In fact, the Hg–Te bonds become unstable at about 100∘ C, while the Cd–Te bonds are still very stable at 300 ∘ C. As a result, it is relatively easy to generate Hg vacancies in Hg1−x Cdx Te, and the growth of high quality Hg1−x Cdx Te is challenging. In Hg1−x Cdx Te, Hg and Cd atoms occupy the cation lattice points with a certain ratio. Since the occupation is random, the atomic potential in those ternary semiconductors is not periodic, although they show crystal properties of materials having long-range order. Characterization of semiconductor crystal structure includes x-ray diffraction, Raman spectroscopy, atomic force microscopy (AFM), as well as scanning tunneling microscopy (STM) for surface studies, etc. In general, the bonds of III–V materials are weaker than those of group IV Si and Ge, but they are more stable than those of II–VI semiconductors. Thus, II–VI NGSs are often grown with relatively high defect densities. Since the energy of infrared photons is close to the characteristic energies of phonon and defects in narrow band gap materials, defects and temperature affect optical properties of NGS materials and, as a result, device performances significantly. Typical defects in Hg1−x Cdx Te include Hg vacancies, dislocations, and other point defects.

1 Optoelectronic properties of narrow band gap semiconductors

| 9

1.2.3 Crystal growth Narrow band gap semiconductors include the binary compounds InSb and PbSe and ternary compounds such as Hg1−x Cdx Te, InAs1−x Sbx and Pb1−x Snx Te. One of the great advantages for the ternary compounds is their tunable band gaps corresponding to their tunable compositions. Among various crystal growth techniques including the Bridgman method [9, 10], solid state recrystallization (SSR), and other bulk growth approaches [11], crystals are grown at thermodynamic equilibrium conditions. Most thin film crystalline growth technologies, namely, molecular beam epitaxy (MBE) [12–14], hot wall epitaxy (HWE) [15, 16], metal organic chemical vapor deposition (MOCVD), nucleation occurs in nonthermodynamic equilibrium conditions [17, 18]. Both the socalled epitaxial growth and vapor deposition are thin film growth techniques in which epitaxial films are grown on the surface of substrates, and the substrate surface is required to be atomically smooth and lattice-matched to the epilayers. As an example of the Bridgman method, the apparatus and growth process is shown in Figure 1.3, where a typical vertical furnace and a temperature distribution are shown. Through proper control of heating, the desired temperature distribution consists of the high temperature, the low temperature, and the transient zones. The transient temperature region between the high and the low temperature zones is critical for high quality material growth. Thus, very often the furnace is designed to have three independent heaters for easy adjustment of temperature distributions. During the crystal growth, the source materials in the quartz tube melt in the high temperatures zone, and the quartz tube is then lowered down to pass the temperature gradient zone slowly to the low temperature zone. At the temperature gradient zone, a liquid/solid interface forms where the crystal growth occurs. As the quartz tube is lowered down, the crystal growth process continues, with the isothermal zone of the liquid/solid interface being always in the temperature gradient zone. The crystal growth is completed when all of the melt in the quartz tube pass the temperature gradient zone. Note that the lower end of the quartz tube is tapered, since small crystals are relatively easy to grow in the initial stage of crystal growth. Through the control on thermodynamic process on liquid/solid interface between small crystal and liquid, the growth of large-size crystals is made possible. The Bridgman method is a relatively simple

T

furnace quartz tube high T zone

melt crystal

Gradient zone low T zone

Fig. 1.3. Schematic diagram of the Bridgman growth technique for bulk crystals.

10 | Ning Dai and practical crystal growth technique suitable for the growth of a variety of crystal materials, including binary, ternary, and even quaternary compound semiconductors. There are numerous factors, including the cleaning of quartz tubes and purity of raw materials, which affect both the quality and the size of crystals, the growth conditions playing a decisive role in the growth of large single crystals of high quality. Growth conditions include not only the temperature and the temperature distribution (i.e. the temperature gradient of the middle zone), but also the solid-liquid interface shape and stability in the growth process, the lowering speed of quartz tubes, the quartz crucible purity, shape, thermal expansion coefficient, the inner surface smoothness, etc. Taking the example of a solid-liquid interface, a flat interface is always beneficial for high quality crystal growth, because in this case the composition of the melt is relatively uniform, and interface stress is small, and dislocations and defects are not easy to generate. If the liquid/solid interface bulges (in the quartz tube, the liquid/solid is high in the middle and low on the edge), though easy for the formation of dislocations, the dislocations are often located in the crystal edges. If the liquid/solid interface is hollow, growth occurs first at the edge, and stress is in the middle of zone. As a result, defects are mostly located in the middle of the crystal ingot bar. As a nonthermodynamic equilibrium growth technique, epitaxial growth offers the flexibility to fabricate quantum wells, superlattices, and other artificial heterostructures. Due to advances in modern material characterization techniques, especially the development of in situ measurement instruments, epitaxial growth systems are able to detect the whole process during the growth. For example, most molecular beam epitaxy chambers are equipped with reflection high energy electron diffraction (RHEED), so crystal growth conditions can be monitored in real time. Figure 1.4 shows the basic scheme of the growth chamber of a molecular beam epitaxial system. In the process of epitaxial growth, source materials in the effusion cells are heated, and solid source materials are directly sublimated into a gas phase at low pressures and deposited on the substrate, where epitaxial crystal nucleation occurs in nonequilibrium thermodynamic conditions [19]. In the case of single element crystal growth such as Si, one effusion cell is used together with possible doping elements installed in other effusion cells for selective use. For the growth of binary semiconductors such as GaAs or ternary compounds as Hg1−x Cdx Te, quaternary compounds as ZnCdMgSe, etc, it is necessary to use more effusion cells and a combination of proper beam equilibrium pressures for these elements. The growth rate of molecular beam epitaxy is rather slow, typically in the order of 0.1–1 nm per second. In addition to the beam and beam pressure ratio, each effusion cell has a shutter for quickly turning on and off the beam. The substrate conditions have a big impact on the quality of the epitaxial crystal layers. High quality epitaxial growth requires atomic smooth and clean substrate surfaces. Currently, cost-effective, large size, and commercially available substrates include Si, Ge, GaAs, InP, etc. However, lattice constants of the substrates are much smaller than those of most II–VI materials. Large area and high quality II–VI semi-

1 Optoelectronic properties of narrow band gap semiconductors

RGA screen isolation valve to the pretreatment chamber

RHEED screen

| 11

beam sources Al Ga

substrate holder

In Si

transmission lever

beam flux monitoring

DRS

Be N2

liquid nitrogen shroud

P4,P2 As4,As2

ion gauge Sb4,Sb2 RHEED electron gun operating lever Fig. 1.4. Schematic description of the growth chamber of a MBE thin film growth system.

conductors is thus extremely challenging to grow epitaxially. An MBE growth system usually consists of three vacuum chambers – a substrate loading chamber, an analysis chamber, and a growth chamber. Various kinds of damage are still left on a substrate after mechanical/chemical polishing, so that a certain chemical etching is needed to remove the damage layer, followed by a rigorous cleaning. The cleaned substrate has to be exposed to atmosphere in the process of being loaded into the loading chamber. Although the loading time is controlled to be very short, it is still long enough for the formation of a surface oxide layer on the substrate. When the loading chamber is pumped down to a certain vacuum degree, a gate valve separating the loading and the analysis chambers is opened to deliver samples to analysis in the analysis chamber. The substrate surface is measured and analyzed before it is sent to the growth chamber. Prior to epitaxial growth, the substrate is heated to a high temperature for deoxidation. Sometimes an ion beam is also used to clean the substrate surface. The surface should then be atomically smooth and free from foreign contamination. Governed by nucleation thermodynamics, the completeness of the epitaxial crystal is closely related to the selection of the growth temperature (substrate temperature during epitaxial growth). In general, epitaxial growth is a process where atoms nucleate continuously on the atomically-stepped substrate surface. Thus, the growth can be viewed microscopically as atomic step motion. The atomic steps is due to the direction deviation of the substrate surface from a high symmetry plane, such as the (100), (110), and

12 | Ning Dai (111) planes. A substrate surface exactly along with the high symmetry plane has no atomic steps, so that it is not suitable for epitaxial growth. The atomic step movement corresponds to so-called two-dimensional (2-D) nucleation, which is favored for high quality epitaxial growth. The 2-D growth occurs within a narrow growth temperature range. If the substrate temperature is too low, the atoms deposited on the substrate are unable to gain enough kinetic energy to move to the atomic step edges, and subsequent atoms would pile up on atomic stairs leading to a 3-dimensional (3-D) growth mechanism. The 3-D mechanism is not conducive to growth. An excessively high substrate temperature, on the other hand, makes deposited atoms too active to nucleate at the step edges. According to the fundamental principles of epitaxial growth, the direction and condition of the substrate surface, growth temperature, and beam flux ratio are essential parameters for high quality growth. Note that for a compound substrate, not only the direction of the substrate but also the kind of the surface atoms is involved. For a (111) GaAs substrate, for instance, one has to define whether the surface is Ga- or As-terminated. There are in general always one or two surface directions on a given substrate suitable for crystal growth. Modern molecular beam epitaxy systems are equipped with high-precision in situ crystal growth monitoring tools. Besides reflection high energy electron diffraction (RHEED), other commonly used real-time monitoring tools include reflectance difference spectroscopy (RDS), photoreflectance (PR), as well as spectroellipsometry (SE).

1.2.4 Electronic properties Having small electron and hole effective masses, the band edges of NGSs are highly nonparabolic, which is often problematic for the accurate calculation and analysis of experimental results. A small carrier effective masses generally correspond to a large carrier mobility. However the supposed high mobilities do not show up due to the relatively poor crystal completeness in narrow band gap Hg1−x Cdx Te, PbTe, etc., in comparison to Si, Ge, GaAs, and InP. Similar to wide band gap semiconductors, strong crystal potentials exist in NGSs. The translational invariant of the periodic crystal potentials gives rise to free-carrier behavior for carriers with certain energies and momenta at which the multiple scattering to the carriers by the periodic potentials cancels each other. A typical example is the electrons in the bottom of the conduction band. The electron motion follows the classical Drude model [20], while at low temperatures and for low carrier concentrations the transport behavior of the carriers can be described quite accurately by the Boltzmann equation [21, 22]. NGSs have a high carrier concentration, so that their Fermi levels are located close to the conduction bands for n-type semiconductors. Fermi–Dirac statistics is required to describe the transport behavior of electrons and holes. As fermions obeying the Pauli principle, the probability for a carrier to occupy an eigen state at energy E is given, according to

1 Optoelectronic properties of narrow band gap semiconductors

|

13

the theory of quantum statistics, by f0 (E)c =

1

(1.11a)

F 1 + exp ( E−E ) k T B

for electrons and f0 (E)v = 1 − f0 (E)c =

1 1 + exp (

EF −E ) kB T

(1.11b)

for holes, where EF is the Fermi energy, kB the Boltzmann constant, and T the absolute temperature. Equations (1.11a) and (1.11b) are the distribution functions of electrons and holes in equilibrium, respectively. Apparently, the occupancy probability of holes is equal to the unoccupancy probability of electrons. From momentum space, the density of states can be easily derived to be 3

1 (2m∗c ) 2 (E − Ec ) 2 G (E)c = 4π V h3

(1.12a)

3 2

G(E)v = 4π V

1 (2m∗v ) (Ev − E) 2 , h3

(1.12b)

assuming the validity of parabolic band edges, where G(E)c (G(E)v ), Ec (Ev ), and m∗c (m∗v ) are the density of states, the energy position at the bottom of the conduction band (at the top of the valence band), and effective mass of electrons in the conduction band (holes in the valence band), respectively. In equation (1.12) h is the Planck constant, and V is the crystal volume. External fields, such as an electric field, magnetic field or temperature gradient, could change the distribution function. An electromagnetic field applies a force on electrons in the Lorentz form (F = −q(ε + v × B)), making the electron wave vector time-dependent, i.e. ℎdk/dt = −q(ε + v × B). Assuming that at a given time t an electron with a wave vector k and a position coordinate r has a distribution function of f (r, k, t), at t + dt the external force changes the function into f (r + dr, k + dk, t + dt) ̇ ̇ or f (r + rdt, k + kdt, t + dt). Thus, the Boltzmann equation describing the change of the distribution function can be written as df 𝜕f 𝜕f 󵄨󵄨󵄨󵄨 (1.13) = + 󵄨 − k̇ ⋅ ∇k f − r ̇ ⋅ ∇, dt 𝜕t 𝜕t 󵄨󵄨󵄨s In (1.13), the first term on the right side is the partial derivative with respect to time, which is nonzero only when the distribution function depends on t explicitly (in the case when the system is in a transient state). The second term is due to the internal scattering and/or collision. The third and the fourth terms are contributed by an externally applied electromagnetic field and temperature gradient. At equilibrium, the external force is time-independent, and then 𝜕f 󵄨󵄨󵄨󵄨 󵄨 = k̇ ⋅ ∇k f + r ̇ ⋅ ∇f . 𝜕t 󵄨󵄨󵄨s

(1.14)

14 | Ning Dai Furthermore, equation (1.14) can be written as f − f0 𝜕f 󵄨󵄨󵄨󵄨 , 󵄨󵄨 = k̇ ⋅ ∇k f + r ̇ ⋅ ∇f = − 󵄨 𝜕t 󵄨s τ

(1.15)

using relaxation time approximation. Equation (1.15), the Boltzmann equation, is useful for the calculation of many physical parameters associated with transport processes where the relaxation time approximation is valid. When an electromagnetic field is the only external field, ℎdk/dt= −q(ε + v × B), and equation (1.15) takes the form of f − f0 q (ε + v × B) ⋅ ∇k f − r ̇ ⋅ ∇f = ℎ τ

(1.16a)

f − f0 q , (ε + v × B) ⋅ ∇v f − r ̇ ⋅ ∇f = m∗ τ

(1.16b)

where m∗ is the effective mass of electrons or holes. From equation (1.16a) and (1.16b) we have used ℎk̇ = m∗ v, where v is the velocity of electrons or holes. According to equation (1.11), f0 depends on energy and temperature. In an isothermal condition, f is only the function of energy (or speed). If initially the carriers in the system are at zero velocity, the distribution function can be expanded so that f = f0 +

𝜕f0 (∇ E) ⋅ v. 𝜕E v

(1.17)

For electrons, v̄ = −eε̄ τ /m∗c , v̄ and ε ̄ are the average drift velocity and the average electric field intensity, respectively. The distribution function is then f = f0 −

𝜕f0 eε ̄ τ (∇v E) ⋅ ∗ . 𝜕E mc

(1.18)

For an electric field applied in the x direction, the current density is ∞

J = −e ∫ vx f (E)G(E)c dE.

(1.19)

0

Substituting equation (1.18) into (1.19) yields ∞

J = e2 ∫ vx 0

𝜕f0 𝜕E ε x τ G(E)c dE, 𝜕E 𝜕vx m∗c

(1.20)

since the first term, the integration with respect to f0 , apparently vanishes. vx and G(E)c (see equation (1.12a)) are the electron velocity in the x direction and the density of states in the bottom of the conduction band. According to Ohm’s law σ = J/ε x , we have ∞ 𝜕f 𝜕E τ 2 σ = e ∫ vx 0 G(E)c dE. (1.21) 𝜕E 𝜕vx m∗c 0

1 Optoelectronic properties of narrow band gap semiconductors

|

15

Simply using E = (1/2)m∗c (vx2 + vy2 + vy2 ) and (1/2)m∗c vx2 = E/3 in the x direction for homogeneous energy distribution, as well as equation (1.12a) for the density of states in the bottom of the conduction band at which the zero energy is chosen, conductivity can then be calculated by 1

∞

16π Vτ e2 (2m∗c ) 2 E − EF ) dE σ= ∫ E3/2 exp (− kB T 3h3 kB T

(1.22)

0

by means of the distribution function with the Boltzmann approximation. Other parameters, such as carrier mobility, can be obtained using the same algebra process. In NGSs, however, the mechanisms for carriers are extremely complicated, because the overall conductivity is often contributed by several kinds of carriers. To separate the contributions, the so-called mobility spectrum was proposed and has been proven to be effective in certain conditions [23]. In addition, the scattering processes are also complicated in NGSs where scattering to carriers comes collectively from neutral impurities, ionized impurities, LO and TA phonons, deformation potentials, alloy fluctuation, piezo-scattering, piezoelectric scattering, and various kind of defects. Although those scattering mechanisms could behave quite differently, there is a lack of effective tools to probe their individual contribution in the scattering processes [22].

1.2.5 Optical properties The band gap of an NGS is in the spectral range of infrared or longer wavelength. Most NGSs are direct band gap materials, so that the materials interact strongly with light and light propagation is forbidden in the materials when the photon energy of light is greater than the band gap energies. The most remarkable feature of semiconductors is the distinct absorption edge in the absorption spectrum corresponding to the band gap. Figure 1.5 presents the optical absorption edge and its temperature dependence for Hg1−x Cdx Te at x = 0.362 [24]. The band gap of elemental (unary) semiconductors and binary semiconductors is a constant at a given temperature and stress. For example, the band gap of InSb is 0.23 eV, corresponding to midwavelength infrared. The absorption edge of ternary and quaternary semiconductors is composition-dependent. The band gap of Hg1−x Cdx Te increases from −0.3 to 1.42 eV, as the composition x varies from 0 to 1 correspondingly. Si, Ge are indirect band gap semiconductors with their band gaps being 1.12 and 0.82 eV, respectively, corresponding to short-wave infrared. Besides band gaps, the main parameters characterizing the optical properties of semiconductor materials include the dielectric function and extinction coefficient. The former describes the propagation characteristic of light, and the latter shows the attenuation behavior. the dielectric function and extinction coefficient are correlated and described by a complex dielectric function. The dielectric function of materials can be determined by the absorption spectra, spectroscopic ellipsometry,

16 | Ning Dai

4

v(cm-1) 3 000

3 500

Eg

LOG (α)

3

2

T(K) 300 250 200 150 100 77

Eg(eV) 0.394 0.386 0.378 0.371 0.365 0.360

X=0.362 d=7 μ 1

0.350 0.400 hν (eV)

Fig. 1.5. The absorption edges of Hg0.638 Cd0.362 Te at several temperatures.

etc. Absorption spectra and spectroscopic ellipsometry characterize the linear interaction between light and semiconductors, while the nonlinear optical interactions include Raman spectroscopy, etc. Raman spectroscopy has a good complementarity with the infrared absorption spectrum, since some transition modes that are infrared forbidden might be Raman activated, or vice versa. As high frequency electromagnetic waves, light propagating in substance is described by the Maxwell equation. The wave equations of electric and magnetic components of light propagating in space in the absence of free charges can be derived from Maxwell equations as follows: 𝜕𝜖 𝜕2 𝜖 − εr ε0 μ0 2 = 0, 𝜕t 𝜕t

(1.23a)

𝜕H 𝜕2 H − εr ε0 μ0 2 = 0. 𝜕t 𝜕t

(1.23b)

∇2 E − σμ0 ∇2 H − σμ0

In the above formulas, σ is the electrical conductivity, εr the relative permitivity, and ε0 and μ0 are the vacuum permitivity and permeability, respectively. Obviously, the electric component E and the magnetic component H of light in the above formula exhibit wave characteristics. The most important parameters to describe optical properties of materials are the dielectric function and refractive index, which are complex number generally, i.e. ε ̃ = ε1 + iε2

(1.24a)

ñ = n + iκ

(1.24b)

1 Optoelectronic properties of narrow band gap semiconductors

| 17

where ε1 and ε2 are the real and the imaginary parts of the complex dielectric function, respectively, n is the refractive index, and κ is known as the extinction coefficient. The relationship of those parameters is given by ε1 = n2 −κ 2

(1.25a)

ε2 = 2nκ

(1.25b)

and n=

2 2 √ √ε1 + ε2 + ε1 2

(1.26a)

κ=

2 2 √ √ε1 + ε2 − ε1 . 2

(1.26b)

The dielectric function generally depends on the intensity and the frequency of incident light. However, the dielectric function is nearly independent of light intensity when the light intensity is not very strong. For very strong light intensity, the dielectric function varies with light intensity, and the variation is generally not reversible, since it is companied with phase transition or burning-out. For a material, the dielectric function is essentially the response function of a material with respect to incident light, and it varies with the frequency of light. Physically, the real part ε1 of the dielectric function characterizes the polarization of material caused by the propagating light, polarization corresponding to the nondissipative “screening” effect on light propagation. The imaginary part ε2 is, on the other hand, the dissipation response of moving electrons to the incident electromagnetic wave so that electromagnetism in materials propagates with a loss due to the dissipation. The complex dielectric function is often written as the product of the real number vacuum permitivity and the complex relative dielectric constant εr̃ , i.e. ε ̃ = ε0 εr̃ = ε0 (εr1 + iεr2 ).

(1.27)

Here εr1 and εr2 are the real and imaginary parts of the complex relative dielectric function, respectively. The real and the imaginary parts are not independent of each other, but linked by the KK (Kramers–Keldish) transformation, i.e. ∞

εr1

ε (ω 󸀠 ) 1 = 1 + P ∫ r2󸀠 dω 󸀠 π ω −ω

(1.28a)

−∞

∞

εr2 =

εr1 (ω 󸀠 ) − 1 󸀠 1 dω , P ∫ π ω − ω󸀠

(1.28b)

−∞

where P represents the Cauchy principal value integrals, and ε ̃ is a function of the incident light frequency.

18 | Ning Dai ε ̃ is thus also a function of the wavelength of the incident light. From the microscopic point of view, the fact that ε ̃ is the function of frequency reflects the response speed of microscopic particles in materials to incident electromagnetic waves in time domain, while ε ̃ as a function of wavelength characterizes the space uniformity of the response to electromagnetic waves. With a wavelength of at least a few microns, the variation of the infrared electromagnetic wave is ignorable within the atomic scale (in the order of a few nanometers). ε ̃ can then be regarded as the function of frequency only. Ions and electrons are charged particles in materials, and their polarization response can keep up with the electromagnetic waves at low frequencies. At high frequencies, however, only the polarization response of electrons could follow the electromagnetic waves, since ions are too heavy. If ε∞ represents the contribution to the polarization response by electrons, the total contributions to dielectric function derived from polarizations classified by different physical mechanisms can be written as ε (ω ) = ε∞ + εinter (ω ) + εintra (ω ) + εphonon (ω ), (1.29) where ε∞ is the high-frequency dielectric constant, εinter (ω ), εintra (ω ), εphonon (ω ) are contributed by electronic interband transitions, intraband transitions, and phonon to the dielectric function, respectively. The photoelectric effect is the most interesting phenomenon among interaction processes between light and matter. An important process in photoelectric effect is the transition of an electron from a low energy level (mostly the ground state) to a high energy level (the excited state) upon absorbing a photon, especially the transition of an electron from the valence band to the conduction band in a semiconductor. The transitions might involve electrons (holes), photons, phonons, as well as other microscopic particles. An optical transition process can be well described within the framework of quantum mechanics. In the single electron case, the Hamiltonian of the photon-electron interaction can be, in the first-order approximation, described by HI = −

e eℎi A⋅p= A ⋅ ∇, m m

(1.30)

where the vector potential A of the electromagnetic field has the form A(q, r, t) = A0 x̂ {exp [i (ω t − q ⋅ r)] + exp [−i (ω t − q ⋅ r)]} .

(1.31)

Here x̂ is the unit vector of the direction of vector potential A, and A0 is the amplitude. Thus, H I can be written as e e A exp [i (ω t − q ⋅ r)] x̂ ⋅ p − A0 exp [−i (ω t − q ⋅ r)] x̂ ⋅ p m 0 m = HI+ exp (iω t) + HI− exp (−iω t, )

HI = −

(1.32)

1 Optoelectronic properties of narrow band gap semiconductors

| 19

where HI+ and HI− are the time-independent parts in the interaction Hamiltonian. If the electronic initial and final state wave functions are Ei t − k ⋅ r)] ui (k, r) ℎ Ef ϕ f (k, r, t) = exp [−i ( t − k ⋅ r)] uf (k, r) . ℎ ϕ i (k, r, t) = exp [−i (

(1.33a) (1.33b)

The electron transition probability from the initial state ϕ i to the final state ϕ f per unit of time is, according to the golden rule, 󵄨󵄨2 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 Θ (k, k󸀠 , ω , t) = 󵄨󵄨󵄨 ∫ dt󸀠 ∫ φf (k󸀠 , r, t)HI φi∗ (k, r, t) dr 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨0 r

(1.34)

From equations (1.32) and (1.33), we have 󵄨󵄨2 󵄨󵄨 t Ef − Ei ± ℎω 󸀠 󵄨󵄨 󵄨󵄨 Θ (k, k󸀠 , ω , t) = 󵄨󵄨󵄨 ∫ exp ( t ) dt󸀠 ∫ ϕf (k󸀠 , r) HI φi∗ (k, r) dr󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 ℎ 󵄨 󵄨0

(1.35)

where ϕf (k 󸀠 , r) and ϕi (k, r) are the space dependent parts of the initial and the final states of electrons, respectively. Moreover, because the photon momentum is small, the momenta of the initial and the final states satisfy the condition of k󸀠 ≅ k for direct transition. The time integral of the function (1.35) gives a δ function so that Θ (k, k󸀠 , ω ) =

2π ℎ

󵄨2 󵄨󵄨 󵄨󵄨∫ ϕ (k󸀠 , r)H ± ϕ ∗ (k, r)dr󵄨󵄨󵄨 δ (E − E ± ℎω ), 󵄨󵄨 󵄨󵄨 f f i I i 󵄨 󵄨

(1.36a)

or, using simple Kronic–Delta notation, Θ (k, k󸀠 , ω ) =

2π 󵄨󵄨 󵄨2 ± 󵄨⟨f | HI | i⟩󵄨󵄨󵄨 δ (Ef − Ei ± ℎω ). ℎ 󵄨

(1.36b)

Here “−“ corresponds to absorbing a photon, “+” to emitting a photon. Furthermore, the transition probability corresponding to the second-order perturbation is Θ 2 (k, k󸀠 , ω ) =

󵄨󵄨 ⟨f | H ± | θ ⟩ ⟨θ | H ± | i⟩ 󵄨󵄨2 2π 󵄨󵄨 󵄨 I I 󵄨󵄨 δ (Ef − Ei ± ℎω ). ∑ 󵄨󵄨󵄨󵄨 󵄨󵄨 ℎ θ 󵄨󵄨 Eθ − Ei ∓ ℎω 󵄨

(1.37)

The total transition probability can be obtained by taking the summation over all initial and final states, which leads to ΘTotal (k, k󸀠 , ω ) = ∑ Θi,f (k, k󸀠 , ω , t) i,f

2π 󵄨 󵄨2 ∑ 󵄨󵄨⟨f | HI± | i⟩󵄨󵄨󵄨 δ (Ef − Ei ± ℎω ), = ℎ i,f 󵄨

(1.38)

20 | Ning Dai where |α⟩ and |β⟩ denote the low- and the high-energy states, respectively. The probability of photon absorption and emission can then be written separately as Θab (k, k󸀠 , ω ) =

2π 󵄨 󵄨2 ∑ 󵄨󵄨⟨β | HI− | α ⟩󵄨󵄨󵄨 δ (Eβ − Eα − ℎω ) ℎ α ,β 󵄨

(1.39a)

Θem (k, k󸀠 , ω ) =

2π 󵄨 󵄨2 ∑ 󵄨󵄨⟨α | HI+ | β ⟩󵄨󵄨󵄨 δ (Eα − Eβ + ℎω ). ℎ α ,β 󵄨

(1.39b)

Since ℎω is a positive number, from the variables in the above δ -functions, one can see that equation (1.39a) describes a process in which an electron transits from the low energy state |α ⟩ to the excited energy state |β ⟩ through absorbing a photon. Equation (1.39b) corresponds to a process where an electron jumps from the high-energy state |β ⟩ to the low-energy state |α ⟩ accompanied by emitting a photon. δ (Eβ −Eα −ℎω ) and δ (Eα − Eβ + ℎω ) represent energy conservation in such a photon absorption process and a photon emission process, respectively. Note that the probability of photon absorption and emission given by equations (1.39a) and (1.39b) is caused by the interaction Hamiltonian HI . If HI is absent in the case of zero light intensity, Θab and Θem must be zero. In fact, the absorption and emission due to the interaction Hamiltonian between electrons in matter and photons in a light field are called stimulated absorption and stimulated emission. In nature there exists a process called spontaneous emission, during which an electron jumps down from a high-energy to a low-energy state and, at the same time, emits a photon. A spontaneous emission is not stimulated by a light field. There are many causes of electrons occupying high-energy states, such as thermal excitation and electron injection, etc. According to electromagnetic field theory, the energy density of a radiation field is related to the square modulus of the vector potential through 󵄨󵄨 󵄨󵄨2 2Nω ℎ , (1.40) 󵄨󵄨A0 󵄨󵄨 = εω where N ω is the photon density of the radiation field with frequency ω , corresponding to the stimulated absorption and emission process without including the spontaneous emission. According to quantum electrodynamics, it has been proven that the total photon density of the radiation field, including spontaneous emission, can be obtained simply by replacing Nω with Nω + 1 in equation (1.40), i.e. 󵄨󵄨 em 󵄨󵄨2 2ℎ (1.41) (N + 1). 󵄨󵄨A0 󵄨󵄨 = εω ω Using the above formula and equation (1.32), equations (1.39a) and (1.39b) can be rewritten as Θab (k, k󸀠 , ω ) =

4π e2 Nω 󵄨 󵄨2 ∑ 󵄨󵄨⟨β | exp (iq ⋅ r) x̂ ⋅ p | α ⟩󵄨󵄨󵄨 δ (Eβ − Eα − ℎω ) ε m2 ω α ,β 󵄨

(1.42a)

Θem (k, k󸀠 , ω ) =

4π e2 Nω 󵄨 󵄨2 ∑ 󵄨󵄨⟨α | exp (iq ⋅ r) x̂ ⋅ p | β ⟩󵄨󵄨󵄨 δ (Eα − Eβ + ℎω ). ε m2 ω α ,β 󵄨

(1.42b)

1 Optoelectronic properties of narrow band gap semiconductors

| 21

Note that equation (1.42) does not explicitly include the light-field mode. The complete probability of absorption and emission can be obtained by summation over all modes of the radiation field. Moreover, according to the principle of detailed balance in statistical mechanics, an electron transition from the initial state |i⟩ to the final state |f ⟩ must meet the condition of |i⟩ being the occupied state and, at the same time, |f ⟩ being the unoccupied state. Detailed discussion on optical properties of semiconductor materials can be found in [25, 26], etc.

1.3 Narrow band gap semiconductors and their basic characteristics Narrow band gap semiconductors are essential for infrared photoelectric conversion and fabrication of infrared photoelectric devices. There are many narrow band gap semiconductors with rather divergent physical properties. Some important NGSs are introduced in the following, including HgCdTe (MCT), InSb, InAs, InAsSb, PbTe, PbSe, PbS, and PbSeTe.

1.3.1 Mercury cadmium telluride (Hg1−x Cdx Te) Hg1−x Cdx Te is the alloy of binary compounds CdTe and HgTe. The band gaps of CdTe and HgTe are 1.42 eV and −0.3 eV, respectively. The band gap of alloy compound Hg1−x Cdx Te covers from −0.3 eV to 1.42 eV with composition x increasing from 0 to 1. As a result, the optical and electrical properties of Hg1−x Cdx Te are strongly dependent on the alloy composition. Furthermore, the band edges of Hg1−x Cdx Te is strongly nonparabolic, due to its narrow band gap feature [27]. Hg1−x Cdx Te is the only narrow band gap alloy with the band gap covering all the three major infrared atmospheric windows, making the material extremely unique. In fact, Hg1−x Cdx Te receives most of the research attention in infrared photoelectric materials [28, 29]. One of the experimentally certified relationships between the band gap and the composition x for Hg1−x Cdx Te is Eg(x, T) = −0.295 + 1.87x − 0.28x2 + (6 − 14x + 3x2 )10−4 T + 0.35x4 .

(1.43)

This formula is applicable for 0.19 ≤ x ≤ 0.433 and is known as the CXT formula [30]. Similar to GaAs, the crystalline structure of Hg1−x Cdx Te is zincblende. The lattice constants of HgTe and CdTe are 0.64614 nm and 0.64809 nm, respectively, which are relatively large compared to most wide band gap semiconductors. The lattice constant of Hg1−x Cdx Te shows a weak nonlinear dependence on composition [28], following a = 6.4614 + 0.0084x + 0.0168x2 − 0.0057x3 .

(1.44)

22 | Ning Dai The lattice mismatch between Hg1−x Cdx Te and alternative substrates Si and GaAs is around 19 % and 14 %, respectively. The large lattice mismatch makes it challenging to grow high quality Hg1−x Cdx Te on Si and GaAs substrates epitaxially. In Hg1−x Cdx Te, the Hg–Te bond energy is rather weak, only about one-tenth of the Cd–Te bond energy. Therefore, the growth of high quality Hg1−x Cdx Te with high Hg content is extremely difficult. Bulk growth techniques for Hg1−x Cdx Te include Te solvent, solid state recrystallization, Bridgman, and others. The prevailing epitaxial growth techniques are molecular beam epitaxy (MBE), metal organic chemical vapor deposition (MOCVD), liquid phase epitaxy (LPE), and hot wall epitaxy (HWE) [31, 32]. Epitaxial growth requires large area substrates. Si, GaAs, and Cd1−x Znx Te (with 4 % Zn) are commonly used substrates for the epitaxial growth of Hg1−x Cdx Te. MBE, MOCVD, and HWE are nonthermodynamic equilibrium growth techniques that tolerate a relative large lattice mismatch. A near-thermodynamic equilibrium growth such as LPE, however, requires closely lattice-matched substrates. The Cd1−x Znx Te substrate is thus the choice for LPE-growing Hg1−x Cdx Te. Unfortunately, large area and high quality Cd1−x Znx Te substrates are difficult to prepare, and they are extremely expensive. Doping is one of the most important factors in determining the performance of photoelectric devices, since doping has a direct impact on the optical and electrical properties of materials. The doping effect is judged by the carrier mobility and the minority carrier lifetime and its influence on lattice completeness. In semiconductors, not all doping elements are activated. For example, Au is the p-type doping element in Hg1−x Cdx Te, but its activation rate is only 3 %. In other words, 97 % of Au impurities do not contribute holes for the NGS, and the neutral impurities often have a strong scattering effect on carriers, resulting in low carrier mobility. The doping property of a semiconductor relies on its band gap, and the relative positions between the conductive band (or the valence band) and the vacuum energy level of the electron. Si has a band gap around 1 eV and shows outstanding doping properties for both p-type and ntype doping. As a typical NGS, Hg1−x Cdx Te has a high intrinsic carrier concentration, and it is hard to get the carrier concentration under control. In addition to thermal excitation, unintentional doping is often inevitably introduced during crystal growth due to the impurities in source materials and other contamination. A typical example is the impurity elements contained in a quartz tube, such as Al, Cu, Mg, Ca, and Ti, although not all of the elements are activated in Hg1−x Cdx Te. In Hg1−x Cdx Te achieving good p-type doping properties remains a challenge. The most commonly used n-type doping elements for Hg1−x Cdx Te are In, I, and O. As a trivalent element, In substitutes for bivalent Hg or Cd and then produces an extra electron. In Hg1−x Cdx Te, indium has moderately high diffusivity and indium doping is controllable. Indium doping in Hg1−x Cdx Te can be realized by thermal diffusion or during epitaxial growth. However, the pn junction formed by In doping is often not steep, due to the large diffusion coefficient of In. As a group VII element, an I atom substitutes a group VI Te and contributes a residual electron. While some doping properties of I in Hg1−x Cdx Te are better than those of In, I may easily form I2 and become inactive.

1 Optoelectronic properties of narrow band gap semiconductors

| 23

From the valence electron number, p-type doping elements in Hg1−x Cdx Te include P, As, Sb, and Ag. The group-V element P or Sb could substitute for Te so that a complete valence shell of the tetrahedral coordinated structure lacks an electron, leading to ptype doping. Those p-type dopants usually have low diffusivity and agglomerate to form clusters. Currently, one has not yet mastered the p-type doping process for those elements. It is also worth pointing out that As, a mysterious element, was theoretically expected to be a very good p-type dopant in group II–VI materials. In fact, As has several bond configurations, while As shows a stable p-type doping feature only under a specific bond configuration [33]. The bonding configuration of the dopants depends on the growth method and technique. Some defects may show certain conductive features. In fact, it is easy to generate Hg vacancies in Hg1−x Cdx Te due to weak Hg–Te bonding energy, and Hg vacancies behave like p-type dopants. However, the p-type electrical property of the Hg vacancies is not stable, as shown by a low hole mobility and short carrier lifetime, due to the defect nature of the Hg vacancies. Although the Hg vacancies are used as p-type doping agents in many Hg1−x Cdx Te detectors, the fundamental way out for upgrading the device performance lies in the substantial improvement of p-type doping techniques. The As doping technique has been improved quite a lot in recent years, which is good news for Hg1−x Cdx Te infrared detectors, especially for long wavelength Hg1−x Cdx Te ones [34]. 3.50 3.45 3.40

x=0.276 x=0.309 x=0.378

n,k

3.35 3.30 3.25 3.20 0.4 0.3 0.2 0.1 0.0 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Energy(eV)

Fig. 1.6. Refractive indices and extinction coefficients of Hg1−x Cdx Te for x = 0.276, 0.309, and 0.378.

Hg1−x Cdx Te with a large Hg content has a small band gap and small electron effective mass. Its electron mobility could be quite large at room temperature, up to several hundred thousand cm2 ⋅ V−1 ⋅ s−1 at 77 K, corresponding to an electron mean free path of a few microns. Transmissivity of Hg1−x Cdx Te is about 50–60 % due to surface reflection, when incident photon energy is less than the band gap. When the incident photon energy is larger than the band gap, the absorption coefficient of Hg1−x Cdx Te is about the order of 104 cm−1 . Thereby, the penetration depth of infrared photons is

24 | Ning Dai shorter than 1 μm. The refractive index of Hg1−x Cdx Te is about 3.0∼3.5 and shows a peak on the curve of the refractive index versus photon energy. The peak is located at the band gap energy of Hg1−x Cdx Te. The refractive indices versus photon energies nearby Hg1−x Cdx Te band edge are present in Figure 1.6 for x = 0.276, 0.309 and 0.378 [35]. Although the optoelectronic properties of Hg1−x Cdx Te are excellent, its thermodynamic stability is rather poor, due mainly to the fragile Hg–Te bonding that tends to break above 100 ∘ C. Besides, the heat conductivity is rather low, so that it is not suitable for a material to be used for high-power devices if the heat dissipation problem is not solved. The very small band gap of Hg1−x Cdx Te gives rise to strong intrinsic thermal excitation in the material, with intentional doping processes being hard to control and unintentional doping being difficult to eliminate. Consequently, often several carrier mechanisms jointly take part in conducting, making control on carrier concentration in Hg1−x Cdx Te extremely difficult. However, good control on carrier concentration is vital for device fabrication. Intrinsic carrier concentration in Hg1−x Cdx Te is closely related to composition, which has been demonstrated by a number of works reported in the literature. Figure 1.7 presents intrinsic carrier concentration versus composition (Figure 1.7a) and temperature (Figure 1.7b) in Hg1−x Cdx Te at various x [36, 37]. The relationship between the composition and the mobility is presented in Figure 1.8 [38].

2

ni(cm-3)

1017 8 6 4

300 K

2 16

10 8 6 4

200 K

2 1015 8 6 4

100 K

x(MOLE FRACTION)

0.3

0.28

0.26

0.24

0.22

0.20

0.18

0.16

0.14

0.10

0.12

2 1014

x 0.16 0.20 0.24 0.28 0.30 0.35 0.40

1017 INTRINSIC CARRIER CONCENTRATION VS MOLE FRACTION OF CADMIUM

INTRINSIC CARRIER CONCENTRATION,ni(cm-3)

1018 8 6 4

1016

1015

0.45 0.50

1014

0.55 10

0.60

13

0.65 1012

0.70 0.75

1011 1010 50

0.80 100 150 200 250 TEMPERATURE(*K)

300

350

Fig. 1.7. Intrinsic carrier concentration of Hg1−x Cdx Te as a function of composition and temperature.

1 Optoelectronic properties of narrow band gap semiconductors

| 25

105

MOBILITY(CM2/ VOLT SEC)

T=300 K

104

103 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 HgTe MOLE FRACTION CdTe CdTe

Fig. 1.8. Mobility vs, mole fraction of HgTe (CdTe) in Hg1−x Cdx Te.

1.3.2 Indium antimonide (InSb), Indium arsenide (InAs), Indium arsenide antimonide (InAs1−x Sbx ) InSb is a III–V compound semiconductor with its internal quantum efficiency close to 100 percent, making it an important midwavelength (3–5 μm) infrared material for high performance detectors [39, 40]. In 1954, the growth of InSb crystal was first reported. The growth methods for InSb include Czochralski, LPE, HWE, MBE, MOCVD, etc. The band gap of InSb is 0.17 eV at 300 K and 0.24 eV at 77 K, with an empirical temperature dependent band gap of Eg = 0.24 −

6 × 10−4 T 2 . T + 500

(1.45)

The lattice constant of InSb is 6.4794 Å at 300 K and it is nearly lattice-matched to CdTe and Hg1−x Cdx Te. At room temperature the electron mobility of intrinsic InSb is up to 80,000 cm2 /(V ⋅ s), which is almost the largest in all semiconductors, with a mean free path of electrons approaching 1 μ m [41]. In addition, the In-Sb bond is stable up to 500 ∘ C, so that the thermal stability of InSb is much better than that of Hg1−x Cdx Te. With a refractive index of 4.0 in optical frequencies, the low-frequency dielectric constant of InSb is 16.8, and the high-frequency dielectric constant is 15.7. Se, S, and Te are now in common use as n-type dopants in InSb. Ge, Cd, Zn, and Cr are often used for p-type doping. Similar to GaAs and InSb, InAs has a zincblende crystal structure. At 300 K the lattice constant of InAs is 0.60583 nm, much smaller than those of InSb, CdTe, and Hg1−x Cdx Te. Useful for fabricating photovoltaic short wave (1–3.8 μm) infrared detectors, InAs is also a direct band gap semiconductor with the band gap of 0.354 eV, and

26 | Ning Dai its carrier mobility is very high. In the short infrared wave band, InAs detectors could work at room temperature, although the device performance is much better upon cooling attributed to significant noise reduction. InAs can also be used to fabricate high power devices such as a semiconductor laser. InAs is also widely used as Terahertz radiation source material. The low- and high-frequency dielectric constants of InAs are 15.15 and 12.3, respectively. According to the periodic table, one could find that selective combinations of group III and group V elements make quite a few III–V binary compound semiconductors, and, furthermore, the binaries may alloy to form ternary and even quaternary alloys. Many of the ternaries and quaternaries are useful or potentially useful for the fabrication of high-performance infrared detectors [42, 43]. For instance, InAs is often mixed with InP to form InP1−x Asx and alloys with GaAs to form an important optoelectronic material in optical communications – In1−x Gax As. It is noteworthy to note that InP1−x Asx is made up of two kinds of anions, in which the material properties (such as lattice constant, band gap, etc.) are tuned by varying the ratio of anions P and As. For In1−x Gax As, the material composition is varied by changing the ratio of cations In and Ga. Since in most case the anion is bigger in geometric size than the corresponding cation, the crystal structure depends mainly on the anion, so that the structural completeness of a ternary alloy (ZnSe1−x Tex and InP1−x Asx , for instance) comprising of different anions is usually not as good as that of the ternary alloy (such as In1−x Gax As, etc.) made up of different cations. Furthermore, some important material parameters of the ternaries with different anions depend on their composition in a strong nonlinear way. For instance, when two binaries InAs and InSb combine to form InAs1−x Sbx ternary alloy, the band gap dependence of the ternary on x shows a distinct bowing behavior at low As concentration, rather than a linear relationship. As a long wavelength infrared material [44], when As content is about 5 % the band gap of InAs1−x Sbx reaches to 12 μm. Particularly, the band gap can be further tuned up to 13 μm or more after incorporating N elements in InAs1−x Sbx [45]. Figure 1.9 shows absorption spectra of InNAsSb with three N contents at 13 %, 14 % and 19 %.

Transmittance(a.u.)

80 a

60

b

c

40 20 0

5

10 15 Wavelength(μm)

20

Fig. 1.9. In NAsSb absorption spectra with 13 % (line a), 14 % (line b), and 19 % (line c) N content.

1 Optoelectronic properties of narrow band gap semiconductors

|

27

Apparently, the absorption edge shifts to longer wavelengths and the crystal qualities degrade with increasing N content.

1.3.3 Lead telluride (PbTe), lead selenide (PbSe), lead sulfide (PbS) and tellurium tin-lead (Pb1−x Snx Te) This section discusses the IV–VI compound semiconductors, in which cations are group IV elements and anions are group VI elements. IV–VI semiconductor materials are the first developed and widely used narrow band gap semiconductor materials [46]. Some commonly used IV–VI infrared semiconductors are lead telluride (PbTe), lead selenide (PbSe), leads sulfide (PbS), etc., in which the metal cations are all Pb. Those IV–VI compounds are thus known as lead-salt semiconductors. Strikingly different from other semiconductors, the electron-effective mass of the lead-salt semiconductors is almost the same as the hole effective mass, i.e. the valence band and the conduction band are mirror symmetric in those compounds, resulting in large overlap integral of optical transition and making them suitable for the fabrication of light detectors or emitters. IV–VI compounds are strongly ionic crystal and have a NaCl crystal structure. Another property of the lead-salt semiconductors is their large refractive index, resulting in slow photoelectric response speed. This limits the device applications in some cases. Figure 1.10 gives the refractive indices of PbTe, PbSe, and PbS at 77 K and room temperature from 0.1 eV to 0.6 eV [46]. The narrow band gaps of the lead-salt semiconductors lead to small effective masses of electrons and holes, 6.6 77 K

6.4 6.2

PbTe

300 K

6.0 5.8 Index of refraction

5.6

373 K

5.4

77 K

PbSe

5.2 5.0

300 K

4.8 4.6 4.4

77 K PbS

373 K 300 K

4.2 4.0 3.8 0.1

373 K 0.2 0.3 0.4 0.5 Photon energy(eV)

0.6

Fig. 1.10. Refractive indices of PbTe, PbSe, and PbSin the infrared range at 77K and room temperature.

28 | Ning Dai as well as strong non-parabolic characteristics of the energy band edges. All being direct band gap semiconductors, the band gaps of PbTe, PbSe, and PbS are 0.31, 0.27 and 0.42 eV at room temperature, and 0.19 eV, 0.15eV, and 0.29 eV at 4 K, respectively. The band gaps of the lead-salt semiconductors decrease with decreasing temperature sensitively. As a result, temperature could be used to alter the band gaps to adjust the response range of infrared spectra of the lead-salt detectors. At room temperature the lattice constants of PbTe, PbSe, and PbS are 0.6462, 0.6117, and 0.5936 nm, respectively. The electron mobilities of PbTe, PbSe, and PbS are typically 300, 1, and 500 cm2 V−1 s−1 , respectively, at room temperature. Interestingly, the electron mobility of PbSe is very low, compared to other lead-salt counterparts. PbTe, PbSe and PbS are mid- or short-wavelength infrared materials. PbSe, for instance, is often used to fabricate infrared detectors operating at the wavelength range of 1.5–5.2 μm, where the devices show the best performance in 3.7–4.7 μm. The devices may work at room temperature, although they perform better at low temperatures. The study of the PbSe detector dates back to the early 1930s. In fact, the first infrared detector used for military purposes was made of PbSe. After the World War II, PbSe became quite popular for use as the material for infrared detectors. The main features of the PbSe detector are its high response sensitivity and fast response speed at room temperature compared to thermal detectors. The primary preparation methods for the material are chemical bath deposition (CBD) and vapor phase deposition (VPD). Just as PbSe, PbS was also an early semiconductor material studied for infrared detectors. Its wavelength range of infrared response is 1–2.5 μm in the short-wave infrared range and the response wavelength extends to longer wavelength (2–4 μm) at low temperature. PbS infrared detectors work either in the photocurrent mode or in the resistance variation induced by light absorption. However, the large dielectric constant of PbS leads to slow device response speed. As another important material from the lead-salt family, PbTe is also a good thermoelectric material, in addition to its usefulness in photoelectric infrared detectors. The good thermoelectric property stems in part from PbTe’s low thermal conductivity. The above-mentioned binary lead-salt semiconductors may combine to form ternaries and quaternaries with tunable material parameters in a certain range by varying the alloy composition. For example, PbSe1−x Tex is a ternary lead-salt semiconductor material formed by PbSe and PbTe alloying In fact, Pb1−x Snx Te is probably the most intensively studied and widely used ternary alloy for infrared detectors among all the IV–VI semiconductors. Before the 1980s, Pb1−x Snx Te received a great deal of attention for its long-wavelength infrared properties. Due to the rise of Hg1−x Cdx Te, which is also a long-wavelength infrared material with a large Hg content, Pb1−x Snx Te gradually gave way to Hg1−x Cdx Te. These ternary semiconductors, which are made up of two kinds of cations and one kind of anions, have better crystal structure completeness, and their doping qualities are relatively easy to control. Ternary alloys Pb1−x Snx Te and Pb1−x Snx Se are also widely used for infrared laser, luminescent tube, thermoelectric devices, etc. Figure 1.11 presents the lattice constant and the band gap dependence on compositions of some well-

1 Optoelectronic properties of narrow band gap semiconductors

eV

Euse

Cds

μm

YbTe E .046Te T .964Se S .036 Pb.954Eu

Pb.94Cd.06S

3 0.4

Bandgap at 77 K

Pb.98Eu.02Se 4

7

0.2

PbS.6Se.4 PbS.3Se.7 PbSe

10 15 30

Pb.96 eV E .04Te Eu 0.5

0.4

Pb.99Eu.01Te.964Se.036

5

Pb.95Sn.05Se

0.1

PbTe.92Se.08 PbTe.4Se.6 Pb.88Sn.12Te Pb.85Sn.15Te.98Se.02

P bTe PbTe

0.3

Pb.76Sn S .24Te

Pb.88Sn.22Se SnSe 5.9

2.5

6.0

3

4

7 0.1

0.1

μm

0.2

Pb.9Sn.1Se

30

2

5

0 15

0.6

∗

PbS

0.3

Pb.95Yb.05Te

EuTe

Bandgap at 300 K

2.5 0.5

| 29

15

Sn.5.Te Pb.5S SnTe

6.1 6.2 6.3 6.4 Lattice constant at 300 K(Å)

6.5

10 30

0

Fig. 1.11. The relationship between the lattice constant and band gap of some ternary and quaternary IV–VI group semiconductor alloy.

known IV–VI group materials at 77 and 300 K [47]. From the figure, the band gap dependence of PbSe1−x Tex on the composition, for instance, shows a strong nonlinear relationship, similar to some III–V ternaries made up of two kinds of cations and one kind of anions. Compared with III–V or II–VI group narrow gap semiconductors (such as InSb or Hg1−x Cdx Te), another special characteristic of the lead-salt compounds is their very large joint density of states at point L in the Brillouin zone, which results in a comparable effective mass along transverse direction and a much larger longitudinal effective mass than those of III–V and II–VI compounds. Therefore, lead-salt semiconductors have a very strong absorption to the incident photons with energies above their band gaps. From Figure 1.11, the alloys formed by some chalcogenides or lead salts offer the tunable infrared absorption band gaps and the fluorescence emission wavelength in a wide range, making it possible to obtain infrared materials with wavelengths longer than 6 μm. By varying the contents of Sn, for instance, the band gaps of Pb1−x Snx Te and Pb1−x Snx Se may extend to long-wavelength infrared above 14 μm, which is very unusual. Particularly, as Sn content in Pb1−x Snx Te and Pb1−x Snx Se approaches 40 % and 20 %, respectively, and the band gaps of the ternary semiconductors become zero at 77 K. Currently, Pb1−x Snx Te and Pb1−x Snx Se are still used for long wavelength in-

30 | Ning Dai frared detectors. The general band structure features of Pb1−x Snx Te and Pb1−x Snx Se are similar to those of the binary lead-salt semiconductors.

1.3.4 Heterojunctions, quantum wells, and superlattices Above we have been describing homogeneous bulk or film materials. A heterojunction refers to an atomically smooth interface formed by two kinds of semiconductors. A quantum well (quantum barrier) is composed of a semiconductor layer with relatively low (high) potential barrier sandwiched between two semiconductor layers of relatively high (low) potential. In addition, the thickness of the quantum well (barrier) layer should be comparable to or smaller than the exciton Bohr radius, in order for the quantum confinement effect to become prominent. Obviously, a quantum well (barrier) consists of two heterojunctions which do not have to be symmetric. The quantum confinement and the optical transition energies of electrons and holes can be tuned by varying the quantum well (barrier) width and height (by using different well and/or barrier materials). A superlattice is formed by the stacking of many quantum wells in which the barrier layers are narrow enough to allow for the overlap of electron/hole wave functions in adjacent wells. It is worth to point out that electrons moving in a superlattice do not really feel the details of distinct potential discontinuities at the interfaces between barriers and wells. Rather, the electrons (holes) “see” only a potential distribution averaged over the size of the electron wave packet that is typically about 5 to 50 nm (the size of the exciton Bohr radius). In addition, in order to achieve long wavelength infrared detection, type II superlattices are utilized, where optical transitions occur between adjacent layers, as shown in Figure 1.12. In this case, the two semiconductors forming type II band alignment could have relatively wide band gaps, so as to avoid the use of narrow band gap materials of poor performance [48, 49]. In recent years, the investigation on type II superlattices and corresponding detectors attracts increasing attention due to relatively mature processing technologies of the wide band gap III–V semiconductors such as GaAs and GaSb. Figure 1.12 depicts the type II band alignment of InAs/GaSb superlat-

GaSb

GaSb Eg

Eg InAs

InAs

InAs

Fig. 1.12. The potential alignment of the InAs/GaSb type II superlattices, as well as the electron optical transition (the inclined arrow) that occurs at the interface of the two kinds of semiconductors after photon absorption.

1 Optoelectronic properties of narrow band gap semiconductors

|

31

tice, indicating that the electrons and holes are confined in neighboring layers. The tilted arrow in the figure shows the interband transition at the interface, corresponding to smaller transition energy than the band gaps of GaSb and InAs.

1.4 Basic principles and applications of infrared optoelectronic devices Infrared detectors, light emitting diodes (LED), and laser diodes (LD) are the most typical and the most widely used infrared optoelectronic devices. The basic physical process inside these devices is photon-electron interaction. The principle process of infrared photons detection is converting incident photons into an electrical signal, while infrared LED and LD operate the opposite way.

1.4.1 Basic principles of infrared detectors Electromagnetic wave covers the wavelength range from 103 m (long wavelength radio) to 10−10 m (gamma rays) and the wavelength of visible light falls in 450–700 nm. As part of electromagnetic wave, the infrared light has a wavelength longer than visible light but shorter than that of terahertz radiation. An infrared detector is a device used to detect infrared signals human eyes are unable to “detect”. The investigation on electromagnetic waves can be classified into three major categories: electromagnetic radiation, radiation propagation, and radiation detection. Since the wavelength of spontaneous emission waves from most interesting objects on the earth is in the infrared region, infrared detection takes a key position in studies, especially in the electromagnetic spectral transmission “windows”, also known as infrared atmospheric windows, located in 1–3 (short wavelength), 3–5 (mid wavelength), and 8–14 (long wavelength) μ m. For practical purposes, the infrared signals to be measured are generally very weak and the infrared signals are often accompanied by strong background radiation. Thus, the essential requirements for infrared detectors are of high sensitivity and low noise. The temperature of objects on the earth is generally around 30∘ C, and the temperature of space targets is even much lower, with their characteristic wavelengths of the objects located at long wavelength infrared and very long wavelength infrared. Therefore, the detection of space targets requires infrared detectors with very a long cutoff wavelength and extremely high sensitivity. On the other hand, the response speed of the device needs to be very fast, in order to detect fast moving objects. Infrared target detection, imaging, and recognition have an important role in the field of aerospace and national security. Figure 1.13 presents the response wavelengths and performances of the detectors fabricated based on a variety of materials. From the figure, a general tendency can be found, i.e. infrared detector performance

32 | Ning Dai

10

12

* 1/2 -1 D (cm. Hz . W )

InAs

10

11

10

10

10

10

9

(PC

)77

K

Ideal photovoltaic

3K

InAs(PV)193 K

2π FOV 300 K background

9 C)1 HgCdTe(PC)77 K Ideal photoconductor S(P 7 K b P )7 C Ideal thermal detector P S( K Pb 5 9 HgCdTe(PV)77 K C)2 )77 K 93 K V S(P 1 K ) P b ( C 7 Te P e(P V)7 PbSn PbS )28 K b(P S K K g(PC n H K I : 7 e K 7 7 G )4.2 ) PC )4.2 C)7 ( (PC C n P ( P b :Z Se Cu( Ge InS Pb Ge: Si:As(PC)4.2 K )2 (PV

95

K

s 9 InA C)2 e(P S b P

Golay cell Radiation thermocouple

5K

GaAs QWIP 77 K InSb(PEM)295 K-Ge:Au(PC)77 K

8

1

1.5

2

3

4

5

6

7 8 9 10

TGS Pyro Thermistor bolometer Thermopile

15

20

30

40

Wavelength(μm)

Fig. 1.13. Response wavelength and performance of a variety of infrared detectors.

decreased significantly, and the working conditions become more demanding, as the increase of cutoff wavelengths of the devices. In 1800 Herschel discovered the infrared part of solar radiation which was unable to be seen by human eyes. The blackened mercury thermometer he used in his experiment can be considered as the first infrared detector. The earliest practical infrared detectors appeared in 1930, which was based on the thermoelectric effect. In 1880, Langley invented what was called the radiometer, bolometer infrared detectors. The bolometer measures the change in resistance of a small metal wire due to infrared radiation. The invention brought the quantitative measurement of infrared radiation to a high level of accuracy. In the 1940s there was a substantial increase in sensitivity since the metal wires were replaced by semiconductors whose resistance-temperature coefficients were much greater than those of metals. This kind of radiometers is often called a “thermistor”. In the late 1960s, the so-called “pyroelectric infrared detectors” experienced a rapid development period. The pyroelectric infrared detectors are based on the spontaneous polarization of the ferroelectric that changes with temperature. The most sensitive infrared detector is, however, of the the photoelectric type. At a nonzero absolute temperature, objects in nature spontaneously radiate electromagnetic waves, and all objects in the earth’s temperature environment radiate infrared light. Due to temperature differences between the objects, surface emissivity differences even at the same temperature (emissivity is equal to the ratio of the radiation flux of an object to that of an ideal blackbody at the same temperature) or shape

1 Optoelectronic properties of narrow band gap semiconductors

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33

differences, the intensity and wavelength of infrared radiation of the objects are different. Despite the fact that visible light and the microwave radar are widely used for signal detection, in many cases infrared detection is the only option. Therefore, for target detection (particularly in a global environment), infrared has a particularly important place. Compared with other means, the infrared detection has the following advantages: 1. Infrared detectors, working in a passive detection way and detecting the spontaneous emission of objects, are easily concealed. 2. Compared to visible detection, infrared detection has good adaptability to the environment. Infrared detection is nearly unaffected by bad weather and is especially suitable for night use. 3. Infrared detectors often detect the emissivity of targets and backgrounds. They are more able to identify the targets than visible light. 4. Compared to radar systems, with infrared systems it is very easy to achieve small, low-power, and high-resolution. An infrared detector is a sensing device that converts an infrared light signal into a measurable physical quantity. Strictly speaking, the physical parameters of any material will change after the absorption of infrared photons. The material can then be used to fabricate infrared detectors if the change of the physical parameter is measurable with high enough sensitivity. Thus, the infrared radiation-induced change of the material parameter can be utilized to measure infrared radiation intensity, as long as the two physical quantities have a one-to-one correspondence. For example, a rise in the temperature of a materials due to the absorption of infrared radiation might induce changes of resistance and spontaneous polarization, or generate a thermoelectromotive force inside the material, or change the material volume, etc. The infrared sensors are called pyroelectric infrared detectors if the absorption of infrared photons results in variation of the temperature of a material and, consequently, the temperature variation gives rise to measurable changes of the physical parameters. The electrical properties of materials may change directly due to the interaction between incident infrared light and materials, in which the materials remain at a constant temperature and infrared radiation is directly converted into electrical signals. This kind of detector is known as a photoelectric infrared device working either in photovoltaic or photoconductive mode, in which the infrared radiation is converted into an electric voltage or electric current signals, respectively. Therefore, classified according to the detection processes, infrared detectors can either of the infrared thermal type or the photoelectric type. Both the thermal type and the photoelectric type have in common that changes of the material parameters have a one-to-one relationship with the infrared radiation intensity of measured objects. The major pyroelectric infrared detectors include:

34 | Ning Dai 1.

resonance detector – utilizing the dependence of the resonant frequency of materials on temperature; 2. thermocouple (or thermoelectric stack) – utilizing the thermoelectric effect, i.e. electrodynamic potential caused by temperature difference at both sides of materials; 3. mercury thermometer and the pneumatic Golay cell – based on the thermal expansion and contraction phenomenon; 4. bolometer – using the dependence of the resistance or dielectric constant of materials on temperature; 5. pyroelectric detector – taking advantage of the dependence of spontaneous polarization on temperature. Lead zirconate titanate (PZT), barium strontium titanate (BST), and barium strontium niobate (BSN) have spontaneous polarization. With the increase in temperature caused by infrared radiation, the spontaneous polarization of a crystal along a certain direction is subjected to change. Thus, the voltage between the pre- and post-surfaces of the crystals perpendicular to the direction of spontaneous polarization varies accordingly. The infrared radiation can be detected by measuring the voltage. The electric properties, such as voltage or conductance, of optoelectronic materials can change directly after absorbing photons. The optoelectronic type of infrared detectors is based on the optoelectronic change of materials and the major optoelectronic detectors include the following: 1. photoconductive detectors – using the photoelectric effect, i.e. the conductivity of the material changes after electrons jump to the conduction band from the valence band after absorbing photons. Both bulk semiconductor materials and thin films could be used to fabricate photoconductive detectors; 2. photovoltaic detectors – utilizing p- and n-type doping to form a pn junction with a strong built-in electric field. When electron-hole pairs are generated after valence electrons in the pn junction vicinity absorb photons with energies higher than the band gap, the built-in electric field in the pn junction region drives the holes into the p-type region and the electrons into the n-type region, so that a measurable voltage at both ends of the detector is established. The infrared signal can be detected through measuring the voltage; 3. Schottky-barrier detectors – using the metal/semiconductor contact. By depositing a thin metal layer on a semiconductor substrate, the contact region between the metal and the semiconductor can form a Schottky barrier. When incident infrared photons are absorbed at the potential barrier region, the photon-excited electrons are swept by the barrier potential into the semiconductor (assuming ntype) and free holes penetrate through the potential barrier. A photon-induced voltage cross the metal and the semiconductor is then established. The Schottky infrared detectors are essentially similar to the pn junction photovoltaic detectors;

1 Optoelectronic properties of narrow band gap semiconductors

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35

4. quantum well IR photodetectors (QWIPs) – utilizing the artificial crystal structures as superlattices and quantum wells grown by MBE or MOCVD. When electrons in QWs jump from a low- to a high-energy level after absorbing infrared photons, the electric properties of the QWs change. QWIPs are designed to measure the change. Governed by the selection rule of the electron transition, optical transitions are allowed only if the electric polarization of light has a component perpendicular to the quantum well layer. Hence the utilization of photons in QWIP is not efficient. Secondly, limited by doping level, the ground state electron concentration in QWs is low and, consequently, the quantum efficiency is also low. In addition, the operating temperature of QWIPs is very low so that it is inconvenient for practical use. The major advantage of QWIPs is the cutoff wavelength which can be extended to very long. Compared to thermal detectors, photon detectors have very high sensitivity and a fast response rate, but the operation requires harsh working conditions. Generally, the detectors work at low temperature, particularly for long wavelength devices. Thermal detectors have no wavelength selectivity, while the photon detectors have a cutoff wavelength. PbS is the earliest modern infrared detector material, followed by the development of PbSe and PbTe. In the past 30 years, Hg1−x Cdx Te infrared detectors have undergone a rapid period of development. Although there have been many technological challenges, the detector sensitivity have been continuously improved, and the longest cutoff wavelength now exceeds 16 μm. The scale of focal plane arrays (FPAs) reaches 4096 × 4096 or greater, and the dual- and multicolor FPAs are also fabricated. Through changing the composition, the band gap of Hg1−x Cdx Te continuously varies from −0.3 to 1.42 eV, and the corresponding detectors cover all three infrared atmospheric windows of 1–3 μm, 3–5 μm and 8–14 μm [51]. Hg1−x Cdx Te is the most important photoelectric infrared detector.

1.4.2 Parameters for characterizing the performance of infrared detectors As devices for high detectivity, the performance of infrared detectors is not only related to the material properties and device structures, but also to the working mode and environment. Therefore it is difficult for the parameters that characterize the device performance to be simple and accurate [52, 53]. Since the detectors work to transfer the radiation into measureable physical signals, the first step in the detection process can be regarded as the response of a material to radiation. If the response to radiation in the photoelectric detector is in the form of voltage and current, one can define a voltage response rate Vs Rv (λ , f ) = (1.46a) Φs (T, λ )δλ

36 | Ning Dai and a current response rate Ri (λ , f ) =

Is , Φs (T, λ )δλ

(1.46b)

where δλ is the bandwidth of the signal and f the signal modulation frequency. Vs and Is are the voltage and the current in the device, respectively, which are induced by the radiation source with a radiation flux equal to 1 at a given wavelength. Note that the units of Rv (λ , f ) and Ri (λ , f ) are different. In contrast to (1.46a) and (1.46b), which are the device responses to a given wavelength radiation, one can also define a response rate with respect to the blackbody radiation, i.e. Rv (T, f ) =

Vs Vs . = ΦB (T) ∫∞ ΦB (T, λ )dλ 0

(1.47)

Here the explicit temperature dependence of Rv is clearly indicated. The signal-tonoise ratio (SNR) is often used to judge our spectral response, which is defined as 󵄨󵄨 󵄨󵄨2 󵄨I 󵄨 SNR = 󵄨 s 󵄨 , 󵄨󵄨 󵄨󵄨2 󵄨󵄨In 󵄨󵄨

(1.48)

where In is the noise current. Apparently, SNR depends on signal strength, so that it is not suited to characterize detector performance. A widely used parameter to measure the sensitivity of the detector is the noise equivalent power (NEP), which is numerically equal to the voltage or current response that gives a signal-to-noise ratio of one in a one Hertz output bandwidth, i.e. NEP = [Φs (T, λ )δλ ]V =V = s n

Vn Vn = , Rv (λ , f ) Vn / [Φs (T, λ )δλ ]V =V s

(1.49)

n

where Vn is the noise voltage. Equation (1.49) indicates that a detector reaches its lowest detection limit when the signal voltage is equal to the noise voltage, although modern scientific instruments are often able to amplify a weak signal below the noise level. Therefore, NEP characterizes the performance of detectors in some extent. Similarly, we have In NEP = (1.50) Ri (λ , f ) for current response. Obviously, the smaller the NEP, the more sensitive the detector. Thus, the detectivity of the detector can be defined as the reciprocal of NEP: D=

1 . NEP

(1.51)

For many detectors, NEP is proportional to the square root of the detector area, and noise current is usually proportional to the square root of the detector bandwidth. Thus, a normalized detectivity is written as D∗ (λ , f ) = D√Aδ f =

√Aδ f √Aδ f √Aδ f Rv (λ , f ) = Ri (λ , f ). = NEP Vn In

(1.52)

1 Optoelectronic properties of narrow band gap semiconductors

| 37

Here λ indicates a specific wavelength. Similarly, D∗ can be defined as ∞

∗

D (T, f ) =

∫0 D∗ (λ , f ) ΦB (T, λ )dλ ∞

∫0 ΦB (T, λ )dλ

.

(1.53)

D∗ (λ , f ) and D∗ (T, f ) are the most widely used parameters to characterize the sensitivity of infrared detectors.

1.4.3 Photoconductive infrared detectors There are two types of detectors based on optoelectronic effect: photovoltaic and photoconductive. In photoconductive detectors, an electron in the valence band absorbs a photon and makes a transition to the conduction band, while a hole is created in the valence band. The creation of the electron-hole pair leads to a change of the material’s conductivity, i.e. the radiation response of the device is manifested in the form of current. Figure 1.14 depicts four kinds of photoelectric conversion mechanisms due to photon absorption: intrinsic absorption, nonintrinsic (impurity) absorption, free electron absorption, and intraband internal absorption. Intrinsic absorption has the mechanism of the excitation of an electron in the valence band into the conduction band through absorbing a photon, resulting in photon-excited electron-hole pairs. Free electron absorption is the process where free electrons absorb photons and become more energetic. In a nonintrinsic absorption process, electrons originally bounded in impurity levels absorb photons and are excited into the conduction band. In the intraband absorption process, electrons in a low energy sublevel in a quantum well absorb photons and jump to a higher energy sublevel in the same band. The sublevels (minibands) are formed due to the quantum confinement applying on electrons by the barrier layers that sandwich the well layer. Intrinsic absorption, free-electron absorption, nonintrinsic (impurity) absorption, and intraband absorption all lead to a change in the electrical properties of materials. The detection of the infrared signal is done by measuring the change. The conductivity in semiconductors can be expressed by σ = q[μn (nn + δ nn ) + μp (np + δ np )] = q(μn nn + μp np ) + q(δ np )(μn + μp ).

(1.54a) (1.54b)

Here nn (np ) is the electron (hole) concentration under thermal equilibrium conditions in the absence of external signal, δ nn (δ np ) the excess electron (hole) concentration excited by external signals, and μ n (μp ) the mobility of electrons (holes). In equation (1.54) we used δ nn = δ np , since the excess electrons and holes are always excited in pairs. Obviously, the second part on the right of equation (1.54b) is the contribution from an external signal on the conductivity. The basic configuration of a photoconductive detector is shown in Figure 1.15. Taking an n-type semiconductor as an example,

38 | Ning Dai

free electron absorption

impurity absorption

intrinsic absorption intraband absorption Fig. 1.14. Three basic absorption processes in a semiconductor. The intraband absorption in aquantum well is shown on the right.

hν electrode

electrode

h

w

Fig. 1.15. The basic structure of a photoconductive detector.

l

we assume that the minority carriers do not recombine during their transport between the electrodes, i.e. the diffusion length of the minority carriers is longer than the device length. If the radiation flux density is Φs , quantum efficiency η , and minority carrier lifetime τp , radiation will stimulate δ nn = δ n p =

ηΦs τp

(1.55) hν excess carriers. Here Φs is the radiant flux density into the device, rather than that of the object to be detected, because the radiation signal may be attenuated by atmosphere and reflected by the surface of the device. Thus, there are ways for an antireflection measurement on the photosensitive surfaces of infrared detectors. The maximum quantum efficiency η was 1, corresponding to the case where each photon entering the device excites a one electron-hole pair. Of course, if the photon energy is twice the band gap, or even three times the band gap, one photon could excite two or more electron-hole pairs, and η is, in these cases, greater than 1. However, the probability for multiple exciton excitation is very low for low radiation signals and in ordinary materials. A bias voltage V applied across the device corresponds to a field of E = V/l, where l is the device length. The current excited by the signal is then Is = q(δ np )(μn + μp )Ehw μp ) Gp Ehw. = qτp μn (1 + μn

(1.56)

In equation (1.56) Gp = δ np /τp is the generation rate of excess holes excited by the radiation signal, h and w are the thickness and the width of the device, respectively,

1 Optoelectronic properties of narrow band gap semiconductors

|

39

and μn E is the electron velocity. The required time for electrons to pass through the device is tn = l/μ n E, so that (1.56) can be rewritten as Is = q

τp tn

(1 +

μp μn

) Gp lhw.

(1.57)

The optical gain is then ξ = =

Is qGp lhw τp tn

(1 +

μp μn

)

(1.58)

As can be seen for the definition ξ , photoconductive gain is equal to the ratio of the total charges collected by the electrode and the charges excited by the radiation signal. ξ is usually a few hundreds to a few thousands in magnitude. It should be pointed out that the above description is a great simplification of the real situation. In practical application, the device noise must be considered. Precise experimental studies and theoretical modeling on noise generation are a difficult task, since the mechanisms of noise generation are usually unclear. One of the most important parameters is the detectivity D∗ (λ , f ) (or D∗ (T, f )) for infrared detectors, as described using (1.52) (or (1.53)), but the detectivity is mainly limited by noise. For photoconductive detectors, the major noise sources are the generation-recombination noise, thermal noise, and 1/f noise. The effects of the noise depend on the device configuration of the detector. In addition, since a target being detected is subjected to certain background conditions and, in many cases, background infrared radiation could be stronger than that of the target, the radiation signal received by a detector is Φs + Φb (Φb is the background radiation flux density). The detectivity of photoconductive detectors is closely related to the noise level that limits detector sensitivity. For example, when only the thermal and the generation-recombination noise need to be taken into consideration, the spectral detectivity of a photoconductive detector is D∗λ = Ri √

Is Aδ f Aδ f = . √ 2 2 + V2 2 Φ (T, λ )δλ Vth V + Vgr s gr th

(1.59)

Here Vth and Vgr are the voltages dropped on detectors caused by thermal and generation-recombination noises.

1.4.4 Photovoltaic infrared detectors One of the unique properties for semiconductors is the ability to change their conductivity by p-type and n-type doping. Although the materials remain charge neutral, p-type semiconductors have high hole concentrations and those of the n-type have

40 | Ning Dai high electron concentrations. The contact of a p-type and an n-type semiconductor gives rise to the formation of a pn junction. In accordance with the laws of thermodynamics, holes in the p region will diffuse into the n region, and electrons in the n region will spread into the p region when the p-type and the n-type semiconductors are brought into contact with each other. The diffusion movement of the electrons and holes generates electron accumulation in the p region in the vicinity of the interface and hole accumulation in the n region near the interface, forming the space charge region and, as a result, a space electric field, referred as the built-in electric field. The built-in electric field directs from the n region to the p region, guiding electrons and holes for their drift movement in the direction opposite to that of the diffusion movement of the corresponding electrons and holes. The formation of the space charge region is due to the dynamic equilibrium between the diffusion and drift of the holes and the electrons, as shown in Figure 1.16, which schematically presents the pn junction and the corresponding electron and hole concentration, electric field, electric potential, and charge distribution. space charge region charge neutral region charge neutral region

(a)

p region

−− −− −− −− −−

++ ++ ++ ++ ++

n region

nn np electron concentration (b) hole concentration x E (c)

x V

(d)

build-in electric field x Q

(e) x

Fig. 1.16. A schematic pn junction (a), electron and hole concentrations (b), electric field intensity (c), voltage (d), and charge distribution (e). ⊕ and ⊝ represent fixed charges in the space charge region.

1 Optoelectronic properties of narrow band gap semiconductors

| 41

The core part of the photovoltaic detectors is the diode shown in Figure 1.16. The space charge region has a built-in electric field, and the strongest electric field is in the vicinity of the interface: the pn junction region. When the electrons in the valence band absorb photons and jump into the conduction band, the electrons and the holes drift to the n and p regions, respectively, driven by the built-in electric field. When the device is in the open circuit case, an open circuit voltage can be measured. If the device is connected to an external circuit, a current can be measured directly and the measured voltage and current correspond to the infrared radiation signal of the target detected. If an electron and a hole recombine in the drift process, they do not contribute to the signal detection. In fact, the electric field is very weak away from the junction area. If the probability of carrier recombination is high during the drift process due to poor material quality, the performance of the device could not be good. Thus, materials for high performance detectors need to have a low defect density. There are usually a large number of defects in the surface and the interface areas in devices where recombination of photon-excited carriers might occur. High-performance detectors require high-quality surface passivation and atomically smooth interfaces. In addition, a high built-in electric field in the space charge region corresponds to a large junction voltage that is beneficial for the effective separation of electron hole pairs. Figure 1.16b–e gives the distributions of the electron and hole concentrations, electric field intensity, build-in electric field intensity, as well as the charge, corresponding to the pn junction in Figure 1.16a. If the radiation flux of a radiation source incident onto the detector at a given wavelength is Φs (λ ), the photocurrent of the detector response should be Is (λ ) = qη

AΦs (λ )δλ , hc/λ

(1.60)

where η is the quantum efficiency, and A is the light sensitive area of the detector. If the radiation signal energy Φs (λ )δλ A is 1, equation (1.60) is then the device responsivity. Thus, the current responsivity has the form of Ri (λ ) = qη (λ /hc).

(1.61)

dI −1 ) is introduced, the voltage response If the dynamic pn junction resistance rj = ( dV of the detector can be written as

Rv (λ ) = qη rj (λ /hc).

(1.62)

A bias voltage is generally used for photovoltaic detectors, so that the energy band distribution (Figure 1.16a) is varied and the photoelectric response behavior of the electrons and the holes to radiation signal is tuned. Figure 1.17 shows the energy band distribution of a pn junction at zero bias (top panel), reverse bias (middle panel), and forward bias (bottom panel). The voltage and current characters of an ideal diode is I = I0 [exp (qV/kB T) − 1] ,

(1.63)

42 | Ning Dai where I0 is the reverse saturation current and V the applied voltage. Assuming the low frequency case, the corresponding mean square noise current is In2 = 2q(I + 2Is )δ f = 2qI0 δ f [exp(qV/kB T) + 1] .

(1.64)

When the background radiation flux is Φb , the current due to the background radiation is qηΦb A, and the current-voltage relationships of the detector is I = I0 [exp(qV/kB T) − 1] + qηΦb A.

(1.65)

This corresponds to the noise current of In2 = 2q(I + 2I0 )δ f = 2qδ f {I0 [exp(qV/kB T) + 1] + qηΦb A} .

(1.66)

If the devices work under zero bias, the junction impedance under zero bias is r0 = (

k T dI −1 ) = B dV V=0 qI0

(1.67)

Thus, the noise current under zero bias is In2 = 2qδ f (

2kB T + qηΦb A) . qr0

(1.68)

Based on the above equation, as well as equations (1.61), (1.62) and (1.52), i.e. D∗(λ , f ) = Ri (λ , f )√Aδ f /In , −1/2 qηλ kB T 1 2 . (1.69) D∗ (λ , f ) = ( + q ηΦb ) 2hc r0 A 2 When the background noise is negligible (thermal noise is dominant), the 12 q2 ηΦ b term in the parentheses can be omitted, so that D∗ (λ , f ) =

qηλ r0 A 1/2 ( ) . 2hc kB T

(1.70)

Thus, the detectivity is proportional to √r0 A. The larger the photosensitive surface area and the junction impedance, the higher the sensitivity of the detector. √r0 A is known as the figure of merit for photovoltaic detectors. Compared to photoconductive detectors, the detectivity of photovoltaic detectors under the same background limitation is about 40 % higher. The operation of photovoltaic detectors does not require an external bias and a load resistor, so that there is no power consumption. The impedance of photovoltaic detectors can be made very high. In addition, device integration is much easier for photovoltaic detectors than for photoconductive detectors. In fact, two-dimensional large-scale focal plane arrays (FPAs) are all based on photovoltaic configurations.

1 Optoelectronic properties of narrow band gap semiconductors

43

eVB

P zero bias

N wD

EF

|

EC EF EV

P e(VB+V)

reverse bias eV

EC EFn N

wD

EV

e(VB+V) forward bias EFp

P

eV

EC EFn

N wD

EV

Fig. 1.17. A pn junction at zero bias (top panel), the reverse bias (middle panel), and forward bias (bottom panel).

1.4.5 Quantum well infrared photodetectors Modern thin film growth technologies make it possible to grow crystals of single atomic layers and the atomic flat heterojunction interface. Thus, the superlattice can be prepared by alternative growth of two semiconductors and a quantum well (QW) structure be grown by a thin layer of a narrow band gap semiconductor sandwiched between two layers of wide band gap materials. In a QW, electrons and holes are confined within the layer with low potential due to the potential energy discontinuities in the interfaces between the two semiconductors. As shown in Figure 1.18, in the case of the type I band gap alignment, the hole potential well in the valence band and the electron potential well in the conduction band are in the same layer. The electrons and the holes are then confined in the same region (see Figure 1.18a). In the case of the type II band alignment, the electrons and holes are confined in separate material layers (see Figure 1.18b). Note that the energy levels for the electrons and the holes confined in the quantum well are discrete. After absorbing photons, electrons in a quantum well might gain energy through interband transition between the conduction band and the valence band as in the bulk material and through intraband transitions between the sublevels within the conduction band or the valence band. As shown in Figure 1.18, the gap between the sublevels could be quite narrow, although the QW might be made from wide band gap semiconductors. The gap between the sublevels can be adjusted by the width of the well,

44 | Ning Dai

Fig. 1.18. A schematic picture of the type I (a) and the type II (b) band gap alignments. The thin solid lines represent the split sublevels in the quantum well. The vertical arrows indicate the transition of electrons to higher levels after absorbing photons.

although it is also influenced by the bulk parameters. Therefore, a QW with narrow sublevel energy splitting can be grown by using wideband gap materials such as GaAs and AlGaAs, i.e. wide band gap materials can be useful in making infrared detectors through the device configuration of QWs [54]. In recent years there have been many reports concerning quantum dot infrared detectors [55]. In principle, the electron density of states in quantum dots that exhibits the form of a δ function should have better device performance than QW detectors.

1.4.6 Infrared light sources: infrared light emitting devices and infrared lasers Laser diodes (LD) and light emitting diodes (LED) are electrical-optical conversion devices which transform input electrical power into output optical power. The main difference between LD and LED is that the emitted light of the former is due to stimulated emission with selected modes, while that of the later is caused by spontaneous emission. Therefore, the monochromaticity, linearity, and intensity of LDs are much better than those of LEDs. Limited by material properties and device techniques, infrared LDs are difficult to produce and up to now there is no mature long wavelength infrared LDs available commercially. Commercial mid-wavelength infrared LDs are produced using IV–VI materials. As for shortwave infrared LDs, the production technologies are mature, and the LDs are based on III–V GaAs, InGaAs, etc. Generally, there are three fundamental requirements in order to generate laser action: 1. The electron density in a high energy state (excited state) is higher than that in a low energy state (ground state), which is called population inversion. 2. There is strong coupling (interaction) between light and electrons in the excited state, ensuring that a sufficient number of electrons return to the ground state via stimulated emission rather than spontaneous emission. In other words, light in the laser devices should have excellent positive feedback route so that all LDs are designed to have a waveguide cavity for guiding light propagation.

1 Optoelectronic properties of narrow band gap semiconductors

3.

|

45

There is always light loss in the devices. Thus light gain must be larger than light loss to ensure light amplification. This explains why there is a lasing threshold current. The injection current must be larger than the threshold current to generate lasing.

It should be noted that a semiconductor laser diode, especially a quantum cascade laser diode, which will be discussed later, does not need to meet all the abovementioned requirements to produce laser emission. The earliest infrared laser diodes contained only a pn junction. Due to the lack of good confinement to the electric and optic fields, the devices had a large threshold current and small output power and required low temperature operation. The rapid development of modern high-precision and highly controllable film growth techniques such as MBE and MOCVD makes it possible to grow atomically smooth heterojunctions and high quality thin film down to a single atomic layer. Now, a variety of high-performance laser diodes with complex structures have been designed and, with the growth techniques of MBE, MOCVD, etc., a number of high-performance semiconductor LDs have been fabricated. Figure 1.19 shows the basic structure of a semiconductor laser. Driven by an externally applied electric field, electrons and holes are injected into the quantum well from the left (n region) and the right (p region), respectively. The quantum well is quite narrow, typically several nanometers to several tens of nanometers, which is easy for MBE and MOCVD with the abilities for controllable growth down to a single atomic layer. If the injected current is large enough, a high concentration of electrons could be accumulated in the conduction band of the quantum well and high density of holes in the valence band of the quantum well at the same time. The injected electrons and holes are confined to the narrow quantum well region by the confinement barriers of the barrier material. The meeting of a large number of electrons and holes in the quantum well gives rise to strong radiation recombination, i.e. a strong optical field. Since the light is confined to the waveguide (as labeled in Figure 1.19) and the quantum well, where there are high concentration of electrons and holes in excited states, as shown in Figure 1.19, the optical field interacts strongly with the electrons and holes. Stimulated by the confined optical field in the waveguide, the electrons in the excited

ground state

excited state

quantum well optical waveguide

Fig. 1.19. A schematic description of the energy band alignment in a semiconductor laser diode.

46 | Ning Dai state recombine with holes in the quantum well via stimulated emission. A laser cavity is usually constituted by two mirrors. In a semiconductor LD, the mirrors are the crystal facets formed by cleaving the material. The reflective facets of the laser cavity may be perpendicular to or parallel to the plane of the quantum well, corresponding to parallel-emitting (edge-emitting) or vertical-emitting, respectively. Thus, the laser beams emitted by the parallel cavity surface emitting laser are parallel to the quantum well (perpendicular to the growing direction of the quantum well), while the laser beams emitted by the vertical cavity surface emitting laser (VCSEL) is perpendicular to the quantum well (parallel to the growing direction of the quantum well). Limited by the device structure, the output power of the parallel cavity surface laser is small. The VCSEL, however, could have very strong output laser power, since it utilizes the entire cavity surface. In addition, the VCSEL is easy to integrate. In a laser diode, the electrons and holes in the excited state are generated by current injection through an external field, or by light excitation, i.e. by using light to pump electrons from the ground state to the excited state. The electrons can also be excited by an electron beam. All in all, the excited states must be populated by a large number of electrons in order to generate laser emitting. For a quantum well LD, the laser emission wavelength can be tuned within a certain range by varying the well width. The wavelength can also be dynamically adjusted within a small range by an external electric field that acts to tilt the quantum well. The quantum confinement effect leads to strong electron and hole accumulation in the quantum well, and achieving such a localized high density of electrons and holes in the excited states thus requires a relatively small current injection, which is beneficial for the initiation of laser emitting. The threshold current of a quantum well laser is thereby much smaller than that of an ordinary pn junction LD. Note that there are two sets of quantum wells in Figure 1.19. The wider well in Figure 1.19 acts as an optical waveguide which confines the distribution of the optical field, i.e. a “light quantum well”, since a semiconductor with a large band gap has a small dielectric constant. The designed device configuration in which the “optical quantum well” is wider than the electric quantum well for the best confinement for both electrons and photons. The high lasing efficiency, low threshold current, and low driving voltage are the fundamental measurements of high performance LDs. The excellent performance is realized through the optimized device structures and precise fabrication techniques. For example, enhancing the confinement to electrons and photons promotes the optical-electric coupling and improves the quality of the cavity. To take a variety of factors into account, the actual structure of a semiconductor laser is very complicated, and the production processes need to be precisely controlled. Figure 1.20 presents a schematic diagram of a GaInSb/AlGaInSb quantum well laser with an output lasing wavelength of 3.5 μm of mid-wavelength infrared [56]. The fabrication of long-wavelength infrared lasers requires semiconductors with very narrow band gaps. Narrow band gap materials are relatively soft and have a large number of intrinsic de-

1 Optoelectronic properties of narrow band gap semiconductors

+

18

-3

17

-3

|

47

3.6 μm Al0.26In0.74Sb p 3×10 cm

400 nm Al0.26In0.74Sb p 2×10 cm 1.75 μm Al0.12Ga0.12In0.76Sb 10 nm Ga0.16In0.84Sb(QW) 20 nm Al0.12Ga0.12In0.76Sb 10 nm Ga0.16In0.84Sb(QW) 1.62 μm Al0.12Ga0.12In0.76Sb +

18

-3

8 μm Al0.26In0.74Sb n 2×10 cm AlzIn1-zSb Interfacial Layer

Semi-insulating GaAs substrate

Fig. 1.20. The structure of a typical GaInSb/AlGaInSb quantum well laser.

fects. Also, doping in narrow band gap semiconductors is difficult to control precisely, and the corresponding device techniques are immature as well. Consequently, long wavelength infrared lasers are hard to fabricate. About 20 years ago, a new concept laser diode, called quantum cascade laser (QCL), was proposed with its working mode very different from traditional semiconductor lasers. QCLs are characterized by ordered steps corresponding to electron energy levels. An electron continuously jumps from one step to the next with lower energy, during which the electron gives away its energy by emitting a photon. QCL process involves only electrons, no holes, which is totally different from the traditional semiconductor LDs. Figure 1.21 shows the energy band distribution of a typical QCL. The figure presents only part of the whole QCL structure, which already looks very complicated. The position of the electron energy levels, the widths of the layers, etc., require careful band gap engineering design and well-controlled device fabrication techniques. A real QCL might contain more than a hundred, even a few hundred layers, which is a big challenge for the epitaxial material growth. The precise thickness of the layers and the material composition relies on a well-controlled growth rate and molecular beam ratios for all the beam source furnaces, and the interfaces (there are so many!) between layers need to be atomically smooth. The quality of the material growth has a great impact on the device performance of the QCL [58]. In a QCL, the steps for electron jumping are in the conduction band, within which optical intraband transitions occur. The energy spacing of the sublevels can be made very small, even though wide band gap semiconductors are adopted for making the

48 | Ning Dai

1.2

Injector region Active region

Energy(eV)

1.0

Injector region

0.8 0.6

λ=5.7 .7 μm

0.4 0.2 0.0

T=300 K Electric Field=67 kV/cm 0

20

40 60 Distance(nm)

80

100

Fig. 1.21. Band potential profile of a typical QCL.

QCL. It is relatively easy for a QCL to work in infrared and even terahertz wavelength [59]. Recently, QCLs have achieved room temperature operation [60].

Bibliography [1]

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50 | Ning Dai [37] J. L Schmit. Intrinsic carrier concentration of Hg1−x Cdx Te as a function of x and T using k ⋅ p calculations, J. Appl. Phys. 41(7) (1970), 2876. [38] W. Scott, Electron mobility in Hg1−x Cdx Te, J. Appl. Phys. 43(3) (1972), 1055. [39] L. Shkedy, T. Markovitz, Z. Calahorra, et al., Megapixel digital InSb detector for midwave infrared imaging, Opt. Engin. 50(6) (2011), 061008. [40] S. H. Park, H. S. Kim, H. S. Shin, et al. Development of InSb semiconductor detector for high resolution radiation measurement, Korean Phys. Soc. 58(6) (2011), 1577–1580. [41] T. Zhang, S. K. Clowes, M. Debnath, et al., High-mobility thin InSb films grown by molecular beam epitaxy, Appl. Phys. Lett. 84(22) (2004), 4463. [42] M. A. Kinch, Fundamentals of Infrared Detector Materials, SPIE Tutorial Texts, vol. TT76, Washington, SPIE Press, 2007. [43] M. Razeghia, Overview of antimonide based III-V semiconductor epitaxial layers and their applications at the center for quantum devices, Eur. Phys. J. AP 23(3) (2003), 149–205. [44] Y. Z. Gao, H. Kan, F. S. Gao, et al. Improved purity of long-wavelength InAsSb epilayers grown by melt epitaxy in fused silica boats, J. Crys. Growth 234(1) (2002), 85–90. [45] Y. Z. Gao, T. Yamaguchi, X. Y. Gong, et al. InNAsSb single crystals with cutoff wavelength of 11-13.5 .μ m grown by melt epitaxy, Jpn. J. Appl. Phys. 42(1) (2003), 4203–4206. [46] J. N. Zemel, J. D. Jensen, and R. B. Schoolar, Electrical and optical properties of epitaxial films of PbS, PbSe, PbTe, and SnTe, Phys. Rev. 140(1A) (1965), A330–A342. [47] J. Webster (ed.), Wiley encyclopedia of electrical and electronics engineering, New York, John Wiley & Sons, Inc. 1999. [48] C. H. Grein, P. M. Young, M. E. Flatt, et al., Long wavelength InAs/InGaSb infrared detectors: Optimization of carrier lifetimes, J. Appl. Phys. 78(12) (1995), 7143 [49] E. H. Steenbergen, B. C. Connelly, G. D. Metcalfe, et al. Significantly improved minority carrier lifetime observed in a long-wavelength infrared III–V type-II superlattice comprised of InAs/InAsSb, Appl. Phys. Lett. 99(25) (2011), 251110. [50] A. Rogalski, Infrared detectors, Amsterdam, Gordon and Breach Science Publishers, 2000. [51] R. Willardson and A. Beer, Mercury cadmium telluride, Semiconductor and Semimetals 18, New York, Academic Press, 1981. [52] M. Henini and M. Razeghi, Handbook of infra-red detection technologies, Kidlington Oxford, UK, Elsevier, 2002. [53] A. Rogalski, K. Adamiec, and J. Rutkowski, Narrow-gap semiconductor photodiodes, Washington, SPIE Press, 2000. [54] E. Costard, P. Bois, X. Marcadet, and A. Nedelcu, QWIP and third-generation IR imagers, Proc. of SPIE 5978 (2005), 59781C. [55] P. Martyniuk and A. Rogalski, Quantum-dot infrared photodetectors: Status and outlook, Prog. Quantum Electron, 32(3–4) (2008), 89–120. [56] G. R. Nash, S. J. Smith, S. D. Coomber, et al., Midinfrared GaInSb/AlGaInSb quantum well laser diodes grown on GaAs, Appl. Phys. Lett. 91(13) (2007), 131118. [57] J. Faist, F. Capasso, D. L. Sivco et al., Quantum cascade laser, Science 264(5158) (1994), 553– 556. [58] S. Slivken, A. Evans, J. David, et al., High-average-power, high-duty-cycle (.λ ∼6 .μ m) quantum cascade lasers, Appl. Phys. Lett. 81(23) (2002), 4321–4323. [59] J. C. Cao, H. Li, Y. J. Han, et al., Terahertz quantum cascade laser at 3.39 THz, Chin. Phys. Lett. 25(3) (2008), 953–956. [60] M. A. Belkin, F. Capasso, F. Xie, et al., Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation, Appl. Phys. Lett. 92(20) (2008), 201101.

Xunya Jiang, Wei Li, Zheng Liu, Xulin Lin, Xianggao Zhang, Zixian Liang, Peijun Yao

2 The group velocity picture: the dynamic study of metamaterial systems 2.1 Introduction Metamaterial is a very fresh concept in modern photonics, which are a new class of electromagnetic media whose permittivity or permeability is beyond traditional values. Fifty years ago a kind of new materials whose permittivity ε and permeability μ are simultaneously negative was theoretically predicted to possess a negative refractive index n with many unusual properties [1]. In past decade, negative-n metallic resonating composites and two dimensional (2D) isotropic negative-n material have been constructed [2, 3], and negative light refraction was observed [4]. The unconventional properties of such materials, for example that an evanescent wave could be amplified by negative-n so that the subwavelength resolution could be achieved [5], have drawn an increasing amount of attention in both science and engineering [6]. After negative-n material, more such unconventional materials have been found, so that a new concept “metamaterial” was created, which refers to an effective medium with very special permittivity εeff , or permeability μeff , or both, over a certain finite frequency band. Such physical media are composed of distinct elements (photonic atoms) which are generally made of subwavelength metallic structure and their size scale is much smaller than the wavelengths in the frequency range of interest. Thus, the effective composite media could be considered homogeneous at the wavelengths under consideration. Since their abnormal properties and related totally new phenomena can even go beyond the traditional physical limit, metamaterial has become one of hottest topics in modern photonics. However, from the beginning of metamaterial research, there are many arguments for a lot of topics, such as Pendry’s famous pioneer work on the superlens [5] was commended several times. One main reason for so many arguments is that the light beams in different metamaterials seem to be too strange (even weird) to be acceptable. So it is natural to argue whether these beams can be real. Another main reason is a general weakness of current metamaterial studies, which mainly focus on the single frequency properties and neglect dispersion. Actually these two reasons are related. We know that dispersion, in the framework of classical electrodynamics, means the electromagnetic response of a material to an external field, and plays the key role in the metamaterial abnormal properties. For these strange beams, such as negative refraction beams, with dispersion we can obtain the group velocity (energy velocity) which determines the beams propagating direction. So the group velocity should be the basic picture for us to understand these strange beams and help us to design related

52 | Xunya Jiang et al. devices. More seriously, the dispersion is related with some very basic limitations of this world, e.g. the causality limitation that the group velocity in metamaterial should be less than speed of light in a vaccum. If our design of devices is based on the metamaterials which violate these basic limitations, the design surely cannot work, since such metamaterial could not exist in this world. From this viewpoint, the group velocity picture is not only needed in understanding and explanation, but also required in research of some topics and design of devices. For instance, in the research of the limitation of the cloak [7, 8] and abnormal phenomena on the interface of the hyperbolic metamaterial [9], if we neglect the dispersion of material, from the group velocity picture we will immediately find that we have fallen into a superluminal trap, since the energy velocity in such artificial systems is divergent. So, the group velocity picture can help us avoid such traps. As a basic value for revealing the complex propagating process, abnormal group velocity has been studied for decades. In the early 1960’s group velocity in material was studied by Brillouin [10], and group velocity in strongly scattering media was investigated by J. H. Page, Ping Sheng et al. in 1996 [11], which indicate that the physical origin of the remarkably low velocities of propagation lies in the renormalization of the effective medium by strong resonant scattering. So far, metamaterials generally are composite of “photonic atoms” which can scatter light coherently. And all abnormal properties of metamaterials, e.g. these strange beams, are from this complex coherent scattering. Another byproduct of these scattering is the (abnormal) group velocity. In other words, the strange beam and abnormal group velocity are two sides of the same coin. Furthermore, with some abnormal group velocity, such as extremely slow light, we can design new signal-processing devices or new detecting devices. Hence, exploring the group velocity in metamaterial is very vital for revealing the emechanism and the design of real optical devices. The numerical simulation takes an important role in research for modern photonics. For metamaterial, since the difficulties of experimental realization, the numerical tools become very essential for researchers. But, in some frequency domain simulation software, the dispersion is totally neglected. As we discussed above, we think such software can mislead researchers to some imaginary metamaterial which cannot exist in this world. For example, for the study of cloaking, this software could present perfect invisibility very easily, but from our study [7, 8], this is misleading. We strongly recommend time-domain software, such as the finite-difference time-domain (FDTD) method or finite-element time-domain (FETD) method. Their simulation results are much more convinciing, since they are generally with physical dispersion in the simulation and suitable for metamaterial studies. This paper is organized as following. Section 2.1 is the introduction, where we generally introduce the group velocity picture of metamaterial study. As we have discussed, the group velocity is the key for understanding these abnormal properties of metamaterials and also can help us to avoid some traps of basic physical limit. We have also commended software suitable for metamaterial studies.

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53

In Sections 2.2–2.8 we will show our study on the unusual properties of metamaterials by the group velocity picture. In Section 2.2, the optical properties of the interface between hyperbolic metamaterial (with anisotropic hyperbolic dispersion) and common dielectric is investigated. With material dispersion, a comprehensive theory is constructed, and the hyperlens effect that the evanescent wave can be converted into the radiative wave is confirmed. At the inverse process of hyperlens, we find a novel mechanism to compress and stop (slow) light at wide frequency range, which can be used as a removable memory or a light trap. All theoretical results are demonstrated by finite-difference time-domain simulation. In Section 2.3 we propose general evanescent-mode-sensing methods to probe the quantum electrodynamics (QED) vacuum polarization. The methods are based on the phase change and the energy time delay of evanescent wave caused by small dissipation. From our methods, high sensitivity can be achieved, even though the external field, realizable in contemporary experiments, is much smaller than the Schwinger critical field. In Section 2.4 the image field of the negative-index superlens with the quasimonochromatic random source is discussed, and a dramatic temporal-coherence gain of the image in the numerical simulation is observed, even if there is almost no reflection and no frequency filtering effects. From the new physical picture, a theory is constructed to obtain the image field and demonstrate that the temporal coherence gain is from different “group” retarded the time of different optical paths. Our theory agrees excellently with the numerical simulation and strict Green’s function method. These studies should have important consequences in the coherence studies in the related systems and the design of novel devices. In the fifth section, the dynamical processes of dispersive cloak by finite-difference time-domain numerical simulation are carried out. It is found that there is a strong scattering process before achieving the stable state, and its time length can be tuned by the dispersive strength. The pointing-vector directions show that a stable cloaking state is constructed locally while an intensity front sweeps through the cloak. Deeper studies demonstrate that the group velocity tangent component is the dominant factor in the process. This study is helpful not only for clear physical pictures but also for designing better cloaks to defend passive radars. In Section 2.6, the limitation of the electromagnetic cloak with dispersive material is investigated based on causality. The results show that perfect invisibility cannot be achieved because of the dilemma that either the group velocity diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method is performed to validate the analysis. In Section 2.7, an approach to control the direction of the group velocity of light is proposed in which light can only propagate unidirectionally. This kind of oneway mode of light can be well-confined at the magnetic domain wall by the pho-

54 | Xunya Jiang et al. tonic band gap of gyromagnetic bulk material. Utilizing the well-confined one-way mode at the domain wall, we demonstrate the photonic one-way waveguide, splitters and benders can be realized with simple structures, which are predicted to be high-efficiency, broadband, frequency-independent, reflection-free, crosstalk-proof, and robust against disorder. Additionally, we find that the splitter and bender in our proposal can be transformed into each other with magnetic control, which may have great potential applications in all photonic integrated circuit. In Section 2.8, we show the bullet-like propagation of light pulses in a particularly designed two-dimensional (2D) photonic crystal. Unlike a traditional light bullet supported by nonlinear materials, this bullet-like propagation is achieved only by a 2D photonic crystal, where the diffraction and the group velocity dispersion of a light pulse are eliminated naturally by combining two distinct properties of photonic crystal, i.e. self-collimation and zero group velocity dispersion. Moreover, we studied the influence of third order dispersion on the propagation of light bullets and found that it can be greatly suppressed by an improved structure of photonic crystal. In Section 2.9, we give a summary of our work.

2.2 Hyperinterface, the bridge between radiative and evanescent waves 2.2.1 Introduction Many new phenomena are observed at the interfaces between metamaterial and common dielectric material, such as the negative refraction which is found in the lefthanded material (LHM) surface. More interestingly, the evanescent wave (EW) cocan be amplified at LHM interface so that superresolution can be achieved [14]. Besides the LHM, there is another class of anisotropic meta-material, the so-called “hyperbolic medium” (HM), in which one of the diagonal permittivity tensor components is negative and results in a hyperbolic dispersion. For convenience, we call the interface between a HM and a common dielectric material “hyper-interface” (HI). Some surprising electromagnetic properties of HI have been intensively studied recently [15, 16]. For instance, HI can convert an EW into a radiative wave (RW) so that the subwavelength information can be observed at far-field, which is called “hyperlens” effect [15]. Very recently it was found that when HI is perpendicular to one asymptote of HM dispersion, abnormal omnidirectional transmission occurs [17]. Although some theoretical and experimental work [16] has demonstrated that EWs really can be converted into RWs by HI of the layered cylindrical HM, a full theory involving the “material dispersion” (which will be explained later) has not been given so far. For metamaterial systems, if there is no physical dispersion, some abnormal optical properties cannot be clearly explained. and the dynamical study of wave propagation cannot be carried out [20]. Even more seriously, the causality violation because of the superluminal group

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55

velocity (vg > c) in HM has been pointed out [18], which makes the observed hyperlens effect doubtful. On the other hand, to compress and to stop (slow) light pulses are very essential for modern optical/photonics research and signal processing. Hence, a new mechanism which can compress and stop (slow) light pulses and is frequency and direction insensitive would induce wide interest in related directions. In this section, we theoretically and numerically investigate the novel optical properties of flat HI [22], in which, unlike the cylindrical HI, the translational symmetry guarantees the simple physical picture for intuitive understanding, the quantitative study of the conversion between EW and RW, etc. On a HI, not only the conversion from EW to RW (CER) of a hyperlens is confirmed, when RW is incident from HM to dielectric (the inverse process of hyperlens), but also the almost total conversion from RW to EW (CRE) can occur, i.e. there is “no transmission and no reflection” (NTNR). More important we find that this is a new mechanism to compress and stop (slow) light pulses in the wide frequency and direction range with many potential applications. Theoretically and numerically we demonstrate that the superluminal group velocity in hyperlens is artificial, since the HM material dispersion is neglected in previous study.

2.2.2 Model Our model is as follows. Assuming two plane waves are incident to HI from HM and isotropic dielectric, and scattered from HI, as shown in the upper-right insert of Figure 2.1. The HI is in the x–z plane, while the incident surface and both HM optical axes lie in the x–y plane. The incident waves are chosen as TM wave with field components (Ex , Ey , Hz ) and the same “parallel wave-vector” kx . The HM is with the permittivity tensor as ε (ω ) 0 εp̂ = ( 1 ) (2.1) 0 ε2 in its principle axes coordinate, where ε1 < 0 and ε2 > 0 are assumed. And the permittivity of isotropic dielectric material is ε . The essential point of our model is that the negative diagonal component is dispersive ε1 = ε1 (ω ), which is called material dispersion in our study. It is well known that dispersion is physically required for real metamaterials with abnormal effective constitutive coefficients, such as negative permittivity. We will see that the material dispersion will help us to obtain a selfconsistent explanation of the abnormal optical properties of HI and to avoid causality violation [18].

56 | Xunya Jiang et al.

kxa/2π

ky kr

0.2 0.1 0.0 -0.1 -0.2

h

h

Hi

HM

Hs

d

d

Hs

Hi -0.4-0.2 0.0 0.2 0.4 -0 0 0.4-0.2 kya/2π

θ kx kt n2 ω n1 ω c c n′2

kx kk′t

ki

kt

Fig. 2.1. The frequency contour of HM and isotropic dielectric material in k space. Inset: upperright, the schematic figure of our model; lower-right, the structures of real HM: the periodic metaldielectric layers and the periodic metal nanowires embedded in a dielectric matrix; upper-left, The frequency contour of periodic metal-dielectric layers; lower-left, the contrast of the conversion between radiative wave and evanescent wave by normal medium and HM.

2.2.3 Hyperbola dispersion and compressing light pulses effect at HI For simplicity, the HM is nonmagnetic and Gaussian unit is employed throughout the section. We define the angle between the HI (or the x-axis) and the positive ε principle axis of HM is θ ,then the most general frequency contour in k space of HM is (kx cos θ − ky sin θ )2

(ky cos θ − kx sin θ )2 ω 2 ( ) . (2.2) = 󵄨 󵄨 󵄨󵄨ε1 󵄨󵄨 ε2 c 󵄨 󵄨 From the standard boundary conditions, the transmission and reflection coefficients are worked out to be 2k̃iyd tdh = − ch ε − k̃ d −

sx

rdh =

−chix ε + k̃iyd ch ε − k̃ d sx

where ⃗ ⃗ k̃ = kc/ω ,

chi(s)x =

iy

iy

h α 2 + k̃x α 𝛾 k̃i(s)y

|ε1 |ε2

,

d k̃i(s)y = ±√ε − k̃x2 ,

2 The group velocity picture: the dynamic study of metamaterial systems |

57

and the factors α , 𝛾 are defined as α = (ε1 sin2 θ − ε2 cos2 θ )1/2 ,

𝛾 = (|ε1 | + ε2 ) sin 2θ /2α .

When k̃x2 > ε , the incident and the reflected waves in the isotropic dielectric are EWs with y-component wave vectors as k̃ d = i√k̃ 2 − ε = −k̃ d = iκ . We note that, two EWs, iy

x

sy

i.e. the incident and reflected EWs, can carry net energy current S⃗ iy in an isotropic dielectric medium, |S⃗ iy | = κε Im(rdh )|S⃗ ty | = − 12 |t|2 ctx . In fact, the conversion between radiative and evanescent waves will also occur at the interface of a normal medium, as shown in lower left inset of Figure 2.1. If the evanescent wave in medium n1 can be converted into a radiative wave in medium n2 , the parallel wave-vector should satisfy n1 ω /c < kx < n2 ω /c, but for HM n󸀠2 , the conversion will happen in n1 ω /c < kx < ∞, so more information can be extracted. Next we will study the inverse process of a hyperlens, i.e. the RW is incident from the HM, and the transmitted field is in the dielectric. For such inverse processes, there 󵄨 󵄨 is a critical condition θ = θc = arctan √ε2 / 󵄨󵄨󵄨ε1 󵄨󵄨󵄨, which means HI (x-axis) is perpendicular to one of hyperbola-dispersion asymptotes, as shown in Figure 2.1. At this critical condition, especially when the transmitted wave is EW, we will find CRE with NTNR, compressing and stopping light pulses, etc. If the incident angle is large enough, k̃x > ε , so that the transmitted field is EW, and since a single EW cannot carry energy current, NTNR is the only possible choice, and we expect that CRE will occur on HI. But if angle θ approaches θc continuously, we will obtain a different result. We first suppose θ ≠ θc , as shown by the dashed lines in Figure 2.1, so the finite k̃ry of a reflected field for a fixed k̃x can be found. Next we let the angle θ approach θc continuously (which can be realized physically by choosing a different direction of HI), then we find that k̃ry → ∞, when θ → θc ,

lim rhd =

θ →θc

󵄨 󵄨 ε k̃x − √󵄨󵄨󵄨ε1 󵄨󵄨󵄨 ε2 (ε − k̃x2 )

󵄨 󵄨 ε k̃x + √󵄨󵄨󵄨ε1 󵄨󵄨󵄨 ε2 (ε − k̃x2 )

is not zero, and the reflected energy current is also not zero. So the theoretical results seem different from our prediction.

2.2.4 Analysis of abnormal optical properties of HI with group velocity To explain the conflicting results, we need to calculate the group velocity inside HM with material dispersion, which will also show that the superluminal group velocity is artificial: vgx =

2cε1 ε2 (α 𝛾k̃yh + k̃x β 2 ) ε22 ε1 (ω )󸀠 ω k̃yp2 − ε12 ε2 ω k̃xp2 − 2ε12 ε22

58 | Xunya Jiang et al.

0.4

vgx

5

vgy

0.2

vgy vg /c

vgx

0

0.0 -0.2

-5

-0.4 (b)

(a) -0.4

-0.2

0.0 δθ

0.2

0.4

-0.4

-0.2

0.0 δθ

0.2

0.4

Fig. 2.2. Two components of group velocity of a reflected field in HM: (a) without material dispersion of ε1 ; (b) with material dispersion.

vgy =

2cε1 ε2 (α 𝛾k̃x − k̃yh α 2 ) ε22 ε1 (ω )󸀠 ω k̃yp2 − ε12 ε2 ω k̃xp2 − 2ε12 ε22

where k̃yp = k̃yh cos θ + k̃x sin θ and k̃xp = k̃x cos θ − k̃yh sin θ are the “k-components” in the principal-axes coordinate of HM and β = (ε2 sin2 θ − ε1 cos2 θ )1/2 . We find that, 𝜕ε 𝜕ε if the material dispersion of HM is neglected, 𝜕ω1 = 𝜕ω2 = 0, then we will obtain the superluminal group velocity, as shown in Figure 2.2a. But with material dispersion, the x- and y-components of vg is recalculated, and we find that there is no vg > c in all cases, as shown in Figure 2.2b, in which two components of vg versus θ − θc , with the ω2

parameters ε2󸀠 = 0, ε2 = 1, ε1 (ω ) = (1 − ωp2 )ω = ωp /√2, and θc = π /4. When approaching the critical angle θ → θc , two components can be approxikx h −1 mated as limθ →θc vgx ∼ ε 󸀠 1(ω ) (kry ) , limθ →θc vgy ∼ ε 󸀠 (ω (kh )−2 . ) ry 1 1 Since k̃ h → ∞ at the critical angle θ , the group velocity of the reflected wave ry

c

should be zero at the critical angle, as also shown in Figure 2.2b. What does the zerogroup velocity of the reflected wave mean? The analysis will give us a clear answer. As we have pointed out, since the reflected energy current S⃗ r is not zero and S⃗ r = vg W, where W is the energy density of reflected wave, the energy density W must be infinitely large at the critical angle. From the dispersion relation we can obtain that the electric field of a reflected wave is really divergent at the critical angle. The divergent field strength means that it needs an infinitely long time to accumulate energy at HI for the reflected field. In other words, there is no physically reflected wave, just as we suspected. When the incident angle is large enough, k̃x2 > ε , since the energy of incident RW cannot be transmitted and can also not be reflected, the only answer is that the energy is stored at the occurance of HI or CRE.

2 The group velocity picture: the dynamic study of metamaterial systems |

59

2.2.5 Numerical experiments and results To confirm our theoretical discussion at θc , the FDTD simulation [23] with strict physical HM dispersion (Drude mode) which satisfies the Kramar–Kronig relation, is done. The parameters of HM and dielectric are ε1 = −3, ε2 = 3, εl = 1.1. For the case (k̃x > ε ), as shown Figure 2.3a, a light beam is incident from HM to HI at a 45∘ angle, as we predicted, and there is no reflection and no transmission, and the field energy is accumulated at HI and stopped there. More detailed observation shows that at the boundary the field energy is mainly at the dielectric side, as shown in Figure 2.3b. We have also checked the group velocities of hyperlens cases and find no violation of the causality. Actually, in FDTD simulation, if there is superluminal group velocity, the program will be numerically unstable. position/λ

6

Normalization of I 4

4 3

2 (a) 0 8

4

8

(b) 0.0 0.2 0.4 0.6 0.8 1.0

2 1

6

0

4 2 0

-1 (c)

-2

4

-3

2 (d) 0

-4 4

8

12 16 position/λ

20

24

Fig. 2.3. (a) The magnetic field Hz distributions for a Gaussian beam incident on the interface with |kx̃ | > √ε . (b) The averaged field intensity versus the vertical distance to the HI. (c) Two pulses are arriving at HI at different times. (d) The fields of two pulses, which stay at the incident positions, at 14 periods after the pulse arriving.

A dynamical study, such as with the pulse incidence, can reveal more interesting phenomena of HI. Since the group velocity along HI is zero in the NTNR case, as discussed, we expect that the pulse energy will accumulate on HI and stay at the incident position until it is dissipated because of absorption of HM. Numerical experiments with incident pulses by FDTD have also been done. As shown in Figure 2.3c,d, two pulses arrive at the HI at different times, and then they stop at the incident positions on HI. The pulse vertical length is compressed to almost zero, but their width stays the same, so that they are still well separated in Figure 2.3d. We emphasize here that this is a novel mechanism to compress and stop (slow) light pulses with special advantages. The first advantage is that this mechanism works in a very wide frequency and wide incident-angle range, which is confirmed by FDTD simulation in Figure 2.3, with the incidence of rather short pulses. The frequency and incident-angle insensitivity is a

60 | Xunya Jiang et al. result of the fact that the mechanism is from a simple geometric property, i.e. the HI (x-axis) perpendicular to one of hyperbola-dispersion asymptotes. The second is that the decay (because of dissipation) of a trapped field on HI is much slower than for common metallic materials, since the trapped field energy is mainly in the dielectric side, as shown in Figure 2.3b. The third is that the trapped signals are easy to take out (read) since they are on the interface. Because of these advantages, HI could be used as a removable recorder (dynamical memory) in optical/photonic signal processing, or as a wide-frequency wide-angle light trapper in photovoltaic devices. However, we should point out that the above theoretical and numerical studies have assumed an ideal hyperdispersion. In reality, such HM does not exist, so that we need to study the limits of hyperdispersion for realizable HM. HM can be realized by many structures, i.e. one-dimensional (1D) periodic metal-dielectric binary layers [14, 24, 25] or two-dimensional (2D) periodic metallic lines [26], as shown in Figure 2.1. For these structures, the dispersion relation can be calculated exactly. In Figure 2.1, the calculated frequency contour of a 1D metal-dielectric binary layers is shown, from which we can see that the effective HM medium is no longer available when |k| approaches π /a. Based on this limit, we can roughly estimate the slow limit of group kx h −1 velocity by vgx ∼ ε 󸀠 1(ω ) (kry ) ∝ 1/𝛾s , vgy ∼ ε 󸀠 (ω (kh )−2 ∝ 1/𝛾s2 , where 𝛾s = kry /k0 ) ry 1 1 is the slowing coefficient. For the 2D metallic-line structure, from the modern technical limit we assume the smallest lattice constant to be a = 10 nm. If the incident is the microwave ω = 5.8 GHz, 𝛾s ∼ 107 , ε1󸀠 (ω ) = 6.9 × 10−10 s, as in [27–29], we obtain vgx ∼ 4.6 m/s, which means considerably slow light, although it is not totally stopped, and vgx ∼ 7.07 × 10−8 m/s, which means that the strongly-compressed light pulses can be easily achieved. It is worth mentioning that for real HM, nontransmission and nonreflection can also be attained. Although the light pulses will finally be absorbed, owing to the low group velocity the light pulses can stay at the interface for a long time. This new effect can be used in optical memory devices.

2.2.6 Section summary In summary, we have investigated theoretically and numerically the optical properties of HI. A theory for the dispersion of metamaterial was constructed, and the hyperlens effect of CER was confirmed. At the inverse process of hyperlens, the abnormal phenomena of CRE with NTNR and a novel mechanism to compress and stop light in wide frequency range were revealed. Based on the calculated group velocity, we demonstrated that the superluminal group velocity in HM previously pointed out is artificial, since material dispersion is neglected. FDTD simulations confirmed that HI has a potential to be a removable optical/photonic recorder, or a wide-frequency wide-angle light trapper. Finally, the realizability of these phenomena on the real metallic structures was discussed. Obviously, the new mechanism works not only for electromagnetic waves, but also for acoustic or matter waves if hyperbolic dispersion is available,

2 The group velocity picture: the dynamic study of metamaterial systems

| 61

so that more interesting phenomena and applications are waiting to be discovered in further theoretical and experimental research.

2.3 Methods for detecting vacuum polarization by evanescent modes A vacuum is one of the most fundamental concepts in all quantum fields of highenergy physics [30], condensed-matter physics [31], statistical optics [32], etc., since all excitations are from a vacuum and are determined by the vacuum properties in some way. The modern concept of a vacuum originated from quantum electrodynamics (QED) [33–38], which describes the interaction between light and matter (including in vacuum), and has been widely and continually studied both experimentally and theoretically. According to QED, a vacuum becomes a weakly anisotropic, dispersive, dissipative, and even nonlinear optical medium when there is an external electric field and its strength is approaching the Schwinger critical value. In other words, the real and imaginary parts of a vacuum refractive index could deviate from the unit and zero [34, 35], respectively. Physically, the deviation of the imaginary part is mainly from the generation of the electron-positron pair. However, the geneeration of electronpositron pairs, also generally referred to as vacuum polarization (VP) processes [34], which is schematically shown in Figure 2.4, has not been directly observed for more than half a century.

Fig. 2.4. The schematic picture of vacuum polarization processes with electron-positron pair generation, with which the vacuum becomes dissipative and anisotropic. The insert is the Feynman diagram of the vacuum polarization processes. The fermionic bold line represents the coupling to all orders to the external electromagnetic field.

62 | Xunya Jiang et al. VP processes are very important in order to understand the basic quantum processes in many fields, e.g. in condensed matter physics, where the “electron-hole” pair generation is widely used in calculating electron self-energy and electron-phonon interaction. For a QED vacuum, one obstacle in the observation of VP is the very high critical electric field, which is beyond the our present technical limits. Therefore, it is natural to wonder if we can find an approach to probe VP with an external field much smaller than the critical electric field.

2.3.1 Study model The idea of this work stems from the notion of the “dual roles” of real and imaginary parts of a refractive index n for radiative or evanescent waves. Supposing a medium with complex dielectric constant n = n󸀠 + n󸀠󸀠 , where n󸀠 and n󸀠󸀠 are extremely small, our goal is to detect the very tiny change of n󸀠 and n󸀠󸀠 . For radiative waves, n󸀠 determines the real part of the wave vector k ≈ n󸀠 ω /c, and we can easily measure the phase change or group delay to detect the change of n󸀠 . So, it is natural to choose the radiative wave as the probing light for measuring the index real part change, as has been done in experiments to detect the vacuum birefringence effect. On the other hand, for radiative waves, very tiny n󸀠󸀠 only causes an extremely small intensity decay, which is very hard to measure in a limited laboratory space. However, for the evanescent waves, the roles of n󸀠 and n󸀠󸀠 are totally reversed, i.e., n󸀠 dominates the decay rate, while the tiny n󸀠󸀠 introduces a real part of a wave vector in the decay direction and causes a phase change which is much easier to detect. Furthermore, we will demonstrate that the tiny imaginary part n󸀠󸀠 can also introduce an energy propagation whose energy velocity ve ∝ n󸀠󸀠 is extremely slow. Such a slow wave can be detected by measuring the delay time τ at a very short distance. The time delay τ can be used for sensitive detecting, especially for the QED VP. We would like to emphasize the mechanism difference between our work and previous work [39–41], which is based on systems with at least “two interfaces” (such as a slab). Such “two-interface” structured will generate both evanescent modes exp (±κ x) at the same time, and these two evanescent modes can carry electromagnetic energy current [9], which is the essence of the “tunneling effect”. Thus, even the dissipation of material can be neglected (dissipation is truly neglected in that work), but the propagation of electromagnetic energy is still available based on that tunneling mechanism. However, in our model the mechanism is totally different, since in it there is only one single interface (Figure 2.5; details can be seen in the following). The obvious evidence is that, without dissipation in our “single-interface” model, there will be no energy current at all in the nondissipation medium [42]; then, the phase change is zero and the energy delay time makes no sense. But we know there is tiny dissipation in the QED vacuum because of VP, so that the phase change and the energy delay time

2 The group velocity picture: the dynamic study of metamaterial systems |

63

reflective light

L θi evanescent wave

probing light

n2=1+δ+in′′

n1=2

region I

x O

region II

z

Fig. 2.5. The schematic diagram of our model.

in our model are not because of the tunneling effect, but purely from QED vacuum dissipation. Here we note that, because the probing light is much weaker than the external field in our study, the nonlinear effect is negligible. Since this is a linear problem, all dynamical processes such as the propagation of envelope fluctuation of the transmitted evanescent wave can be solved by the sum of multifrequency components. Based on the linear property, we can use Green’s function [42] of multifrequency components to obtain the strict numerical results, which can be compared with our analytical ones on the dynamical process of evanescent waves. Our model is schematically shown in Figure 2.5, based on the total internal reflection (TIR) at the interface between a dielectric media n1 (region I) and vacuum (region II). When the incident angle θi > θc = arcsin (1/n1 ), the TIR will occur and the transmitted wave in the vacuum is the evanescent wave. We choose θi a bit larger than θc to make sure that almost all frequency components are totally reflected when the incidence is the slowly-varying quasi-monochromatic wave. An interferometer or a photon detector is set at a distance L from the interface, so that the phase information and intensity change can be detected.

64 | Xunya Jiang et al. 2.3.2 The phase change and delay time of evanescent waves in a tiny dissipative medium The time-dependent Maxwell equations are given by ∇ × E = −μ (z)μ0 𝜕H/𝜕t and ∇ × H = ε (z)ε0 𝜕E/𝜕t, where ε (z) and μ (z) are the relative permittivity and the relative permeability, respectively, and c = 1/√ε0 μ0 . To obtain the concrete results, the system parameters are chosen as follows. The incident angle θi = 0.1667π , the refractive index of region I n1 = √ε1 = 2, and the vacuum refractive index of region II n2 = √ε2 μ2 = 1 + δ + in󸀠󸀠 , where δ ≪ 1, and n󸀠󸀠 ≪ 1, are the real and imaginary index deviations of a vacuum, because of the VP processes caused by a strong external field. If the incident probing light is a plane wave, the transmitted wave in the vacuum region can be generally and written in the form E(x, z, t) = E exp(ikz z + ik// r// − iω t), where k// = n1 sin θi ω /c and kz = √(n2 ω /c)2 − k‖2 are the wave vectors parallel and perpendicular to the interface. For the evanescent wave, kz is described as kz = i√(n1 sin θi )2 − (1 + δ )2

ω ω n󸀠󸀠 + c 2 2 √(n1 sin θi ) − (1 + δ ) c

(2.3)

The physical meaning of kz is very clear, that the imaginary part Im(kz ) = κz corresponds to the exponential decay of the field, and the real part Re(kz ) causes a phase change because of VP. The phase change at distance z = L is Δϕ = Re(kz )L ∝ n󸀠󸀠 L,

(2.4)

which can be measured by interferometers [43]. Besides the phase change Δϕ , with the same model as shown in Figure 2.5, there is another way to detect the tiny n󸀠󸀠 , by measuring the time delay of irradiance [44] fluctuation of the evanescent wave. The physical process can be explained in the following way. First, we suppose that the incident probing light is not a plane wave anymore, but has a slow intensity fluctuation proportional to the irradiance fluctuation, as shown in Figure 2.6a; then the question is: what will happen for the evanescent wave in region II? From the strict Green’s function method with physical dissipation and dispersion, it is found that the fluctuation will propagate on the evanescent wave from the interface to far away, as shown in Figure 2.5b. Thus we can measure the time delay τ of the fluctuation propagation on the evanescent wave to detect the VP effect. The propagation speed of irradiance fluctuation can be obtained by the energy velocity ve which is defined as: ve = |Sz |/W, where Sz = 1/2Re(E × H ∗ )|z is the averaged Poynting vector along the z direction, and W ≈ 14 (ε0 |E|2 + μ0 |B|2 ) is the local energy density of the electromagnetic wave. In our model, one can obtain the energy velocity as ve = χ ⋅ n󸀠󸀠 , (2.5) with χ ≃ c/[(n1 sin θi )2 √(n1 sin θi )2 − (1 + δ )2 ,

2 The group velocity picture: the dynamic study of metamaterial systems |

1

10-4 (b)

50

0.8

40

0.6 I(a.u.)

I(a.u.)

60 (a)

30

65

d1 d2 d3

0.4

20 0.2

10 0 4

4.5

5

5.5 t(fs)

6

6.5

7 ×104

0 4

4.5

5

5.5 t(fs)

6

6.5

7 ×104

Fig. 2.6. The irradiance of light I versus time t: (a) the incident probing light at the interface; (b) the transmitted evanescent wave in region II.

when the dissipation and dispersion are very weak. The physical meaning of ve can be understood as the “propagation” speed of electromagnetic wave irradiance fluctuation of the evanescent wave, which can be measured using irradiance measurement technology [44]. Here we note that with tiny n󸀠󸀠 the expected energy velocity is much smaller than , and hence causality is not violated. Hence, experimentally the time delay τ of the irradiance fluctuation at distance L can be measured: τ = L/ve ∝ 1/n󸀠󸀠 . (2.6) Since it is near the field phenomena, detection should be very near the interface. For the VP effect, since n󸀠󸀠 is extremely small, the “propagation” speed of the irradiance fluctuation is so slow that τ is long enough for detecting even in a very short distance L, e.g. τ reaches the picosecond level when the distance is one tenth of the wavelength L = 60 nm. Therefore, either the phase change Δϕ or the time delay τ are very sensitive for n󸀠󸀠 , and the evanescent wave is a good candidate for probing the VP effect. Here, we note that the famous Kramers–Kronig relations, which show the confinements of the causality limit, still fit for QED vacuums [34]. Hence, the direct observation of the imaginary part of a vacuum index also confirms the dispersive property of QED vacuum.

2.3.3 Vacuum polarization and refraction index deviations of a vacuum Next, we will quantitatively study the detection of QED VP using our methods. Supposing that an external homogeneous constant electric field Eext , which is perpendicular to the xz-plane and smaller than the Schwinger critical electric field Ec , is applied to

66 | Xunya Jiang et al. the vacuum (region II) only, as shown in Figure 2.5, then, the optical properties of the vacuum can be described by the Euler–Heisenberg Lagrangian Leff [34, 37]. Physically, the imaginary part of the Euler–Heisenberg Lagrangian Leff is related to the imaginary part of the VP operator, and therefore corresponds to the generation of electronpositron pairs. This result of QED is well justified not only at zero temperature but also in cases with finite temperature. Consequently, the vacuum refractive index can be deduced from the Lagrangian. In our model, the contribution of the transmitted evanescent wave to Leff is negligible, since its electric field is much weaker than Eext , and furthermore, the external magnetic field is assumed to be zero; thus the vacuum refractive index is only determined by the external homogeneous constant electric field Eext . We use n// and n⊥ to refer to the effective refractive index of vacuums when the electric field of probing lights are parallel and perpendicular to the field Eext , respectively. n// and n⊥ can be obtained from the reference [38]: n// = 1 +

∞ 2α 2 α π 11 ∑ ( + y +i⋅ ) exp (−nπ /y) 45π 4π n=1 y2 n y

n⊥ = 1 +

∞ 7α 2 α 2 11 1 2 ∑ ( π+ y +i⋅ + ) exp (−nπ /y), 90π 4π n=1 3 n y n2 π

and

where y = |Eext |/Ec , and α ≈ 1/137 is the fine-structure constant. Therefore we have δ = Re(n//(⊥) ) − 1, n󸀠󸀠 = Im(n//(⊥) ).

2.3.4 Detecting vacuum polarization: phase change and delay time The parameters of our model in Figure 2.5 are chosen as follows. The wavelength of the probing light is λ0 = 600 nm, the dielectric constant in the region I is ε = 4, and the incident angle is θinc = 0.1667π > θc , which is a little larger than the critical angle of TIR, so that the field in vacuum is evanescent. The distance L for the phase detecting is Lp = 6 μm = 10×λ0 , while for the time delay detecting is Lτ = 60 nm = 0.1×λ0 , respectively. The QED theoretical results of real and imaginary part ofn// and n⊥ are shown in Figures 2.7a and 2.7b, respectively. Inserting these results into equations (2.4) and (2.6), the phase change and the delayed time can be obtained, as shown in Figures 2.7c and 2.7d, respectively. Numerically, the phase change with plane wave incidence and the time delay of local amplitude maximum are calculated using the Green’s function method, which is also shown in Figures 2.7c and 2.7d. Comparing the analytical results from equations (2.4) and (2.6) with the numerical results, we can find that they agree with each other very well. Next we will analyze the possibility for observing the VP effect under experimental conditions. Recent experimental advances [45] have raised hopes that lasers may achieve fields just one or two orders of magnitude below the Schwinger critical field

2 The group velocity picture: the dynamic study of metamaterial systems |

1.000 5

4

×10-4

n

n

n

n

3

1.000 3

Im(n)

Re(n)

1.000 4

1.000 2

2 1

1.000 1 (a) 1 0

0.5

1

(b) 0 0

1.5

0.5

Eext/Ec

1

1.5

Eext/Ec

15

15 (c)

Analytical n

10

(d)

Analytical n

Analytical n

5

Analytical n

Numerical n

0

Numerical n

10

Numerical n

Numerical n

τ/ps

Δφ/mrad

67

15

5

10 5 0

0 0

0.5

1 Eext/Ec

1.5

0.4

0.5 Eext/Ec

0.6 6

Fig. 2.7. The refractive index, the phase change and time delay in the QED vacuum with an external electric field Eext : (a) The real part of n// and n⊥ versus the external electric field strength; (b) the imaginary part of n// and n⊥ versus the external electric field strength; (c) the phase change versus the external electric field strength when the distance from the interface Lp = 6 μm; (d) the time delay versus the external electric field strength when the distance from the interface . Both the results from theory and from Green’s function are shown in (c) and (d).

strength. In this case , and from our numerical and analytical results in Figure 3.4, we can see the Δϕ can reach the order of 10−1 mrad, which is within measuring limit of contemporary interferometers [43]. Very recently it was supposed that the electric field could be effectively amplified 4 times larger by the coherent constructive interference of laser beams [36]. If Eext can reach 0.5Ec by this method, not only Δϕ can be one order larger, but also the delay time τ can get to the subpicosecond level and may be measured using present-day photon detectors.

68 | Xunya Jiang et al. 2.3.5 Section summary In summary, we have investigated the evanescent-mode-sensing methods to directly detect the QED VP based on evanescent waves. Theoretically, we clearly demonstrated that the imaginary part of QED vacuum index, caused by QED VP processes, can generate a phase and time delay of irradiance fluctuation propagation on an evanescent wave. Using the Green’s function method, we obtain the numerical results, which agree with our analytical ones very well. The possibility of directly observing the effects of VP based on evanescent waves is discussed, and it was found that the required external electric field could be much smaller than the Schwinger critical field and may be realizable in present-day experiments. Our methods can also be used for other extremely sensitive detections.

2.4 The temporal coherence gain of the negative-index superlens image 2.4.1 Introduction Veselago predicted that the negative-index material (NIM) has some unusual properties, such as that a flat slab of the NIM can function as a lens for electromagnetic (EM) waves [1]. This research direction was further advanced in the work of Pendry and others [4, 5, 46–58], who showed that a lens with such a NIM (i.e. ε = μ = −1 + δ ) could be a superlens whose image resolution can go beyond the usual diffraction limit. After that, several beyond-the-limit properties of NIM systems were found, such as, the subwavelength cavity [59] and the waveguide [60]. Some of the theoretical results have been confirmed by experiments [4, 46, 49]. These beyond-the-limit properties present us with us new physical images and opportunities to design devices. Recently, the new numerical [50, 51] and theoretical Green’s function [52] methods were used to understand the phenomena in such systems. But so far almost all studies have been done with strictly single-frequency sources, so that the coherent properties of EM waves (or photons) in NIM systems have to the best of our knowledge not been studied. Even more seriously, there is no theory for the propagation of the coherent functions in NIM systems. The importance of coherence research cannot be over-estimated, since coherence is essential in wave interference, imaging, signal processing, and telecommunication [61, 62]. Can we discover these new frontier to go beyond at the coherent properties in NIM systems? If so, can we develop a simple theoretical method to deal with the image coherence of superlenses? In this section, the finite difference time domain (FDTD) method is used in twodimensional (2D) numerical experiments to study the temporal coherence of superlens images with random quasi-monochromatic sources. We observe the dramatic temporal-coherence gain of the superlens image even if the reflection and frequency-

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filtering effects are very weak. Based on the new physical picture of the signal (the fluctuation of random source) propagation in NIM, we construct a theory to obtain the image field and derive the equation of the temporal-coherence relation between the source and its image. The new mechanism of the temporal-coherent gain can be explained by the key idea that the signals on different paths have different “group” delay times. Our theory agrees very well with the numerical and strict Green’s function results.

2.4.2 Model The setup of the 2D system is shown in Figure 2.8. The thickness of the infinite-long NIM slab is d. To realize the negative ε and negative μ , the electric polarization → 󳨀 → 󳨀 density P and the magnetic moment density M are phenomenologically introduced in FDTD simulation [63]. The effective permittivity and permeability of the NIM are εr (ω ) = μr (ω ) = 1 + ωp2 /(ωa2 − ω 2 − i𝛾). In our model, ωa = 1.884 × 1013 /s, 𝛾 = ωa /100, ωp = 10 × ωa . The quasi-monochromatic field is expressed as E(x, t) = U(x, t) exp (−iω0 t), where U(x, t) is a slowly-varying random function, ω0 = π /20δt is the central frequency of our random sources, and δt = 1.18 × 10−15 s is the smallest time-step in FDTD simulation. Atω0 , we have εr = μr = −1.00 − i0.0029. There, we emphasize that in our FDTD simulation the smallest space-step δx = λ0 /20(λ0 = 2π c/ω0 ) and the distance (d/2 = λ0 ) of the source from the lens are too large to excite strong evanescent modes of NIM [50, 51, 53]. Actually the evanescent field in our simulation can be neglected when comparing with the radiating field, and what we are actually studying is the property dominated by the radiating field.

A

A O B d/2

B d

d/2

Fig. 2.8. The schematic diagram of our model with ray paths(left); and the typical snapshot of electric field in our FDTD simulation (right).

70 | Xunya Jiang et al. The source constitutes random plane wave pulses, where the average length of the plane wave pulses is tp , and the initial phases and initial time are random. The fields of the source and image are recorded in an interval of time 4 × 105 δt for analyses. We τ define E(ω ) = limτ →∞ ∫−τ E(t) exp (−iω t) as the field spectrum (FS).

2.4.3 Unusual phenomena At first, the FS width of the random source is a little too large (Δωs ≈ ω0 /20). When we observe the image temporal-coherence gain, we also find that the FS width of the image is sharper than the source (Δωi < Δωs ). It is obvious that there are frequencyfiltering effects because of the NIM dispersion, such as the frequency-dependent interface reflection and focal length. After increasing the pulse-length tp of the source, we reduce the source FS width to Δωs ≈ ω0 /100; then the reflection and focal-length difference are very small [64]. With such a source, the FS widths of source and image

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are almost the same, Δωi ≈ Δωs , as shown in Figure 2.9a. The difference between two widths is < 5 %, which is our criterion for a quasi-monochromatic source. Even so, the dramatic gain in temporal coherence is still observed. In Figure 2.9b the source field (upper) and the image field (lower) versus the time of FDTD simulation are compared. Their profiles are generically similar, but the image profile is much smoother. The normalized temporal-coherence function g(1) (τ ) = ⟨E∗ (t)E(t + τ )⟩/⟨E∗ (t)E(t)⟩ (⟨ ⟩ means the ensemble average) of the source (dark solid line) and the image (gray solid line) from the FDTD simulation are shown in Figure 2.9d. The temporal coherence of the image field is obvious better than the source. From g(1), the image coherent time is obtained: Tico = ∫ gi(1) (τ )dτ = 1268δt , which is about 50 % longer than the source coherent time Tsco = 860δt . Although the increase in spatial coherence only by propagation is well known [62], the dramatic increase in temporal coherence is generally from the high-Q cavities, contrary to our case, which have strong filtering effects. To uncover the new mechanism of the temporal coherence gain in NIM systems, we also have done more numerical experiments in which only the ray near a certain incident angle (shown in Figure 2.8), such as only paraxial rays (θ ≈ 0), can pass through the superlens. Then the image field profile versus time looks very much like the source field and has no gain of coherence anymore. Therefore, the increase in temporal coherence of the superlens image is not from one ray with a certain incident angle, but probably from the interference between the rays with different incident angles. Then, what is the difference between the rays with different incident angles? After carefully checking the field profiles of different-incident-angle cases, we find that the profiles have different delayed time. The larger incident angle the longer the delayed time.

2.4.4 Physical images For a deeper understanding of the new mechanism for the increase in coherence gain and the formulation of our theory, we need make two physical pictures clear. The first one is about the optical path length (OPL) ∫ nds,which determines the wave phase and the refracted “paths” of the rays in Figure 2.8 according to Fermat’s principle (or Snell’s law). Based on ray optics, the superlens and traditional lenses have the same focusing mechanism, that all focusing rays have the same OPL paths ∫paths nds = const (∫paths nds = 0 for superlenses) from the image source [1]. But this picture is so well known that it suppresses another important picture. Because the temporal-coherence information is in the fluctuation signals of a random field, the signal propagating picture should be essential for our study. The optical signals propagate in the group velocity vg which is always positive. Obviously, if the path (in Figure 2.8) is longer (larger incident angle), the signal needs a longer propagating time, which is called the group retarded time (GRT) in this section. Inside the NIM, the GRT of a path should be cos(θd )υ g

(this is confirmed by our numerical experiments), where θ is the incident angle and

72 | Xunya Jiang et al. υg = c/3.04 is the group velocity of NIM around ω0 [65]. The total GRT from source to image is τr = τ0 / cos(θ ), where the τ0 = d/c + d/υg is the GRT of the paraxial ray. Now, the new propagating picture for a signal through superlens is that a signal, generated at ts from the source, will propagate on all focusing paths and arrive at image position at very different time ts + τ0 / cos(θ ) from different paths (this is schematically shown in Figure 2.8). This picture is totally different from traditional lenses, whose images do not have an obvious temporal-coherence increase, because their focusing rays have the same OPL and a similar GRT.

2.4.5 Our theory Based on this analysis, we suppose that the superlens image field of a random quasimonochromatic source is the sum of all the signals from different paths with different GRT. This is the key point of our theory, and then the image field can be obtained: π

Ei (t) =

τ0 1 −iω0 t 1 −iω0 t 2 ∑ Us (t − τr ) = e e ∫ π Us (t − )dθ , U0 U0 cos(θ ) −2 paths

(2.7)

where Us (t) is the slowly-varying profile function of the source, and U0 is the normalization factor. In Figure 2.9c (top), we show the result of the image field based on equation (2.7), we can see it is in excellent agreement with the FDTD result in Figure 2.9b (bottom). To show the interference effect of different paths, we assume there are only two paths (such as A and B in Figure 2.8). Based on equation (2.7) the image field is Ei = e−iω0 t (Us (t − τrA ) + Us (t − τrB )), then the temporal coherence of image is G(τ ) = ⟨Ei∗ (t)Ei (t + τ )⟩ = ⟨Us∗ (t − τrA )Us (t − τrA + τ ) + Us∗ (t − τrB )Us (t − τrB + τ ) + Us∗ (t − τrA )Us (t − τrB + τ ) + Us∗ (t − τrB )Us (t − τrA + τ )⟩. The first two terms are same as the source field (just a time-shift) so they do not contribute to the coherence gain. The last two terms are from interference between two paths. The third (or the fourth) term could be very large at the condition τ ≈ τrB − τrA (or τrA − τrB ). This condition can always be satisfied between any two paths since τ is a continuous variable. So the interfering terms between the paths are responsible for the image temporal-coherence gain. From equation (2.7), after the variable transformation ts = t − τ0 / cos θ and some algebra, the relation of the temporal coherence between the image and the source can be obtained: Gi (τ ) = ⟨Ei∗ (t)Ei (t + τ )⟩ =

−τ0

−τ0 +τ

−∞

−∞

1 ∫ dt1 ∫ dt2 h∗i (t1 )h(t2 + τ )Gs (t2 − t1 ), U02

(2.8)

where hi (t) = (τ0 /t)2 /√1 − (τ0 /t)2 is the response function of different incident angles, and Gs (t2 −t1 ) = ⟨Es∗ (t1 )Es (t2 )⟩ is the temporal-coherence function of the source. Equation (2.8) can also explain the temporal-coherence gain of the image. Even if the source

2 The group velocity picture: the dynamic study of metamaterial systems | 73

field is totally temporal incoherent Gs (t2 − t1 ) ∝ δ (t2 − t1 ), based on equation (2.8) we can find that Gi (τ ) is no longer a δ -function, so the image is partial temporal coherent. The product of h∗i (t1 )h(t2 + τ ) includes the interference between paths. According to our theory, we calculate the image coherence function g(1) versus time (Figure 2.9d) which agrees with our FDTD result (Figure 2.9d) pretty well (we will discuss the deviation later). To further confirm our theory and the FDTD results, the strict Green’s function method [52] is engaged to check our results. We only include the radiating field (no evanescent wave) in Green’s function. The strict image field versus time is shown in Figure 2.9c (bottom), and the image temporal-coherence function g(1) versus time is shown in Figure 2.9d. In Figure 2.9d, we can see that the FDTD result is almost exactly the same as when using the strict Green’s function method. But our theory deviates from the strict result by a lot, τ > 3000δt which corresponds to a very long path (or very large incident angle). This is understandable, since in our theory we neglect the dispersion of NIM totally and only use υg (ω0 ). For the very-large-angle rays, a small index difference (from the dispersion of NIM) can cause a large focal length difference. Hence the deviation is from the focus filtering effect. When we reduce the FS width of source to an even smaller value (i.e. Δωs = ω0 /500), the deviation of our theory is smaller. Although our theory is only a good general approximationy, owing to the picture simplicity and clarity, the theory can help us to study more complex systems qualitatively and quantitatively. The finitely-long 2D superlens is a good example which is hard to handle using the Green’s function method. In Figure 2.10 we plot the coherent time Tico versus superlens length L of the FDTD simulation and of our theory, respectively. They coincide with each other pretty well (the reason for deviation has been discussed). The increase of the Tico with the increase of L can be explained simply ac1 600 1 400 1 200

Ti

co

1 000 800 600 400 200 0

200

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1000

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Fig. 2.10. The coherent time versus the superlens length L, from the FDTD simulation and from our theory

74 | Xunya Jiang et al. cording to our theory. Since the image field is θmax

τ 1 −iω0 t Ei (t) = e ∫ Us (t − 0 )dθ , U0 cos θ θmin

the large-angle paths (θ > θmax and θ < θmin ) and their contribution to the temporalcoherence gain are missed in the short superlens. Obviously, equation (2.7) is suitable not only for a random quasi-monochromatic source, but also for all quasi-monochromatic fields, such as slowly-varying Gaussian pulses and the slowly switching-on process mentioned in [52]. Our theory can also be easily extended to 3D systems. A typical example is shown in Figure 2.11. And owing to the fact that what we find is from the radiating field, the temporal-coherence gain is not the near-field property. Actually, the new mechanism of temporal-coherence gain is not limited to the n ≈ −1 superlens, but is also applicable to other superlenses, such as the photonic crystal superlens in [46, 49, 58]. But the specialties of n ≈ −1 superlens, such as almost no frequency-filtering (no frequency loss) and no reflection (no energy loss), can be used to design novel optical/photonic coherence-gain devices. 1 Source FDTD Theory FDTD paraxial rays

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2.4.6 Section summary In summary, for the first time we have numerically and theoretically studied the temporal coherence of the superlens image with a quasi-monochromatic source. Numerically we observe that the temporal coherence of the image can be improved

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75

considerably even almost without reflection and filtering effects. Based on new physical picture, we construct a theory to calculate the image field and temporal-coherence function, which excellently agree with the FDTD results and strict Green’s function results. The mechanism of the temporal coherence gain is theoretically explained by the different GRT of different paths. Although the evanescent wave is very weak in this study, the coherence of evanescent wave in NIM systems is a very interesting topic, which will be discussed elsewhere [66]. Other related topics, such as the spatial coherence which is very essential for the image quality of the superlens, can also be studied using similar methods. Although our study is within the confinement of classic optics, a similar investigation can be extended to quantum optics [62], and interesting results can be expected. Obviously, the temporal-coherence gain of a superlens is a another piece of evidence that the NIM phenomena are consistent with the causality [49]. We suppose that the temporal-coherence gain phenomena could be observed in microwave experiments [4, 46]. Therefore, this study should have important consequences in future studies of coherence in NIM systems. The no-reflection and no-frequency filtering coherence gain of the superlens has some potential applications in the imaging, the coherent optical communication, and the signal processing.

2.5 Dynamical process for dispersive cloaking structures 2.5.1 Introduction Recently, a theory [67, 68] has been developed based on the geometry transformation to realize a cloaking structure (CS) in which objects become invisible from outside. Then a two dimensional (2D) cylindrical CS [69] and a nonmagnetic optical CS [70, 71] are designed. More surprisingly, the experiment [72] demonstrates that such a 2D CS really works with a “reduced” design made of split-ring resonators. This pioneering work is really interesting and opens new possibilities for realizing the invisibility of which humans have dreamt. However, so far almost all theoretical [67–71, 73–75] studies of CS have been done in the frequency domain, and the geometry transformation idea is supposed to work only for a single frequency, so that the effects of dispersion have not been intensively studied. As pointed out in [68] and in the quantitatively study of our recent work [7], dispersion is required for the cloaking material in order to avoid divergent group velocity. For the dispersive CS, new topics such as the dynamical process can be introduced. Dynamical study is essential for the cloaking study since without it we cannot answer questions such as how the field can get to its stable state, or whether there is any strong scattering or oscillation in the process, or how long the process is, etc. More important, because real radar is generally pulsive, the dynamical process is critical for the cloaking effect around the goal frequency. So a dynamical

76 | Xunya Jiang et al. study not only gives us the whole physical picture of cloaking, but also helps us to design more effective cloaks. In this section, the dynamical process of the electromagnetic (EM) CS is investigated with finite-difference time domain (FDTD) numerical experiments. In our simulation, the Lorentzian dispersion relations are introduced into the permittivity and permeability models, then the real dynamical process can be simulated [76–78]. Based on numerical simulation, we can follow the details of the dynamical process, such as the time-dependent scattered field, the building-up process of the cloaking effect, and the final stable cloaking state. By tuning the dispersion parameters and observing their effects on the dynamical process and the scattered field, we can find the essential elements which dominate the process. Theoretical analysis of these essential elements can help us to have a deeper physical picture beyond the phenomena and to design more effective cloaks.

2.5.2 Study model The setup of the system is shown in Figure 2.12a, similar to the one in [69]. R1 and R2 = 2R1 are the inner and the outer cylindrical radii of the CS, respectively. A perfect electric conductor (PEC) shell is pressed against the inner surface of the CS. The CS is surrounded by the free space with 𝜖 = 1.0 and μ0 = 1.0. From the left side, an

θ

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electric field(Ez) Fig. 2.12. (a) The setup of the system, and the distribution of the electric field at different moments during the process. Parameters are chosen as Ar = 0.4, Aθ = 1.6, Az = 0.4 (in the position where 𝜖z > 1) or Az = 1.6 (where 𝜖z < 1), and 𝛾 = 0.012ω0 ; (a) t = 2.28T ; (b) t = 3.60T ; (c) t = 4.92T ; (d) t = 7.20T ; (e) t = 9.00T ; (f) stable state. T is the period of the incident EM wave.

2 The group velocity picture: the dynamic study of metamaterial systems | 77

incident plane wave with working frequency ω0 is scattered by the CS, and the total field and the scattered field can be recorded inside and outside B1 , respectively, by the numerical technique [79]. So the scattering cross section can be easily calculated. Our study is focused on the E-polarized modes, for which only the permittivity and permeability components 𝜖z , μr , and μθ need be considered (for H-polarized modes, considering the corresponding components μz , 𝜖r , and 𝜖θ , we can obtain the same numerical results in the dynamical process). All of these are supposed to have the form 1 + Fj (r) × fj (ω ), where subscript jcould be z, r, and θ for𝜖z , μr and μθ respectively. The filling factors Fj (r) are only r-dependent, and fj (ω ) = (ωp2 /ωaj2 − ω 2 − iω 𝛾) are the Lorentzian dispersive functions, where the plasma frequency which set to be a constant ωp = 10ω0 , 𝛾 is the “resonance width” or is referred to as the “dissipation factor”, ωaj are the resonant frequency of “atom” resonant units in metamaterials. For the study of the dispersive CS, we suppose that the real parts of the 𝜖z , μr , and μθ always satisfy the geometry transformation of [69] at ω0 : Re[μ (r, ω0 )] = (r − R1 )/r, 2 Re[μθ (r, ω0 )] = r/(r − R1), and Re[𝜖z (r, ω0 )] = R22 (r − R1 )/[(R2 − R1 ) r]. Then the filling factors Fj (r) at different r can be obtained: Fr (r) = {Re[μ (r, ω0 )] − 1}/Re[fr (ω0 )], Fθ (r) = {Re[μθ (r, ω0 )] − 1}/Re[fθ (ω0 )], and Fz (r) = {Re[𝜖z (r, ω0 )] − 1}/Re[fz (ω0 )]. To investigate the dispersive effect on the dynamical process, we tune the dispersion parameters ωaj in our numerical experiments. We use the working frequency ω0 as the frequency unit, since it is the same for all cases in this section, so the ratio Aj = ωaj /ω0 represents ωaj . Obviously, for the Lorentzian dispersive relation, the dispersion is stronger when ω0 and ωaj are closer to each other (the working frequency is near the resonant frequency), or in other words, when Aj approaches 1. Since there are singular values of real part of 𝜖 and μ , in our numerical simulation, we have done some approximations [80], such as we set the maximum and the minimum for 𝜖 and μ . Although such approximations will affect the cloaking effect of stable state [74], we find that the influence of these approximations on the dynamical process is very small and can be disregarded.

2.5.3 The physical dynamical picture of invisible cloaking First, we show an example of evolving electronic field during the dynamical process in Figure 2.12 with concrete parameters of Ar , Aθ , Az , and 𝛾. In Figure 2.12a, the plane wave arrives at the left side of the CS and is ready to enter the CS. From Figures 2.12b–e, the cloaking effect is built up step by step; at last, the field gets to the stable state shown in Figure 2.12f. Because of the dispersion, there is an obvious time delay in the cloaking effect and the strong scattered field is observed. We introduce a time-dependent scattering cross-section σ (t) to quantitatively study the dynamical process, which is defined as ̄ (t)/S̄ σ (t) = Jscat inct

,

(2.9)

78 | Xunya Jiang et al. where t = nT, n = 0, 1, 2, . . . , T is the period of the incident wave, Jscat (t) is the oneperiod-average energy flow of scattered field, and Sinct is the averaged energy flow density of incident field. To observe the dispersive effect on σ (t) during the dynamical process, at the first step, we keep Az and 𝛾 constant and change Ar and Aθ ; the results are shown in Figure 2.13a. From the σ versus t curves we can find the general properties of the dynamical process. First, there is strong scattering in the dynamical process. At the beginning, σ increases rapidly when the wave gets to the CS, then reaches its maximum (at about the 9th period). After that, σ starts to decay until it gets to the stable value (of the stable cloaking state). Secondly, unlike other systems, there is no oscillation in the process. This property will be discussed later. Thirdly, the time length of dynamical process, called the “relaxation time”, generally, can be tuned by the dispersion. From Figure 2.13a we can see that the main dispersive effect is on the decaying process. From case 1 to case 5, Ar and Aθ become closer to 1, so that the dispersion is stronger. We find that the stronger the dispersion, the longer is the relaxation time. For comparing with the cloaking cases, we also show the σ (t) of the naked PEC shell in the case 6. From the definition of σ , we know that the area covered by these curves in Figure 2.13a is proportional to the total scattered energy in the dynamical process. So the CS with the weaker dispersion will scatter less fields (better cloaking effect) in the dynamical process. But such a general conclusion is still not enough for us to get a clear physical picture in order to understand the cloaking dynamical process.

2.5.4 The key factor for the dynamics of invisible cloaking Next we check whether the absorption of the CS is important in the process. The absorption is determined by the imaginary part of 𝜖 and μ . To study this effect, we hold Ar , Aθ , and Az constant, but modify the dispassion factor 𝛾. We modify the filling factors Fj simultaneously, so that the real parts of 𝜖 and μ are kept unchanged at ω0 . In such way, we can keep the dispersion strength almost unchanged, but with the imaginary parts of 𝜖 and μ changed. The results in Figure 2.13b show that stronger absorption only leads to a larger stable value of σ , leaving the relaxation time nearly unchanged. Thus, we can exclude the absorption from the relevant parameter list, since it only influences the σ (t) of the stable state considerably. To obtain a deeper insight into the dynamical process we need to examine it more closely. From Figure 2.12b–e we can see that the “field intensity” (shown by different symbols in the figures) propagates more slowly inside the CS than in the outside vacuum. And when the inside field intensity “catches up” with the outside one in Figure 2.12f, the field in the CS reaches a stable state, and the cloaking effect is built up. In fact, this catching-up process of the field intensity can be shown more clearly by the direction of the Poynting vectors during the dynamical process. In Figure 2.14a–d we show the direction of Poynting vectors in moments of Figure 2.12c–f, respectively. In Figure 2.14 we see that there is the “intensity front” (shown by dashed curve) which

2 The group velocity picture: the dynamic study of metamaterial systems |

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Fig. 2.13. The σ versus t curves. From (a) to (d), Az = 0.4, where 𝜖z < 1, or Az = 1.6, where 𝜖z > 1. (a) Keep 𝛾 = 0.012ω0 unchanged, choose Ar and Aθ as Case 1: Ar = 0.4, Aθ = 1.6, Case 2: Ar = 0.5, Aθ = 1.5, Case 3: Ar = 0.6, Aθ = 1.4, Case 4: Ar = 0.7, Aθ = 1.3, case 5: Ar = 0.8, Aθ = 1.2, case 6: only PEC shell without CS. (b) Keep Ar = 0.4, Aθ = 1.6 unchanged, choose 𝛾1 = 0.012ω0 , 𝛾2 = 0.024ω0 , 𝛾3 = 0.048ω0 , 𝛾4 = 0.096ω0 , 𝛾5 = 0.192ω0 . (c) Keep Aθ = 1.6 and 𝛾 = 0.012ω0 unvaried, and change Ar . (d) Keep Ar = 0.4 and 𝛾 = 0.012ω0 unvaried, and change Aθ .

separates two regions of the CS. At the right side of the front, the field intensity in the CS is much weaker than the outside, and the Poynting vector directions are not regular (especially near the front). But at the left-side region which is swept by the intensity front, the Poynting vectors are very regular and nearly along the “cloaking rays” which was predicted at the coordinate transformation [68]. Since the cloaking effect can be interpreted by the mimic picture that the light runs around the cloaking area through these curved cloaking rays, it is not surprising to find that the stable cloaking state is achieved when the intensity front sweeps through the whole CS and these optical rays are well constructed. The surprising thing is that the stable cloaking state seems to be constructed locally. We believe this property is related to the original cloaking recipe [68], which makes that the cloaking material is almost impedance, matched layer by layer. This also explains why there is generally no oscillation in the cloaking dynamical process. This picture also can interpret the strong scattered field in the dynamical process, since these “irregular rays” at the right-hand region of the intensity front must be strongly scattered. Further, we can use this picture to analysis

80 | Xunya Jiang et al.

(a)

(b)

(c)

(d)

Fig. 2.14. Direction of the Poynting vectors and the intensity front shown by dashed curves at moments during the dynamical process. Parameters are chosen as in Figure 2.1: (a) t = 4.92T ; (b) t = 7.20T ; (c) t = 9.00T ; (d) stable state.

the dynamical process of other incident waves, such as the Gaussian beams, which are composed of different plane wave components. Having understood this, we are now ready to find the correlation between the relaxation time and the CS dispersion. It is well known that the field intensity (or energy) propagates at the group velocity Vg , which is controlled by the material dispersion. So the intensity front, which determines the dynamical process, should move in Vg . Thus, we can explain the results in Figure 2.13, since our modification of the dispersive parameters can cause the Vg change. But, because the cloaking material is a strong anisotropic material, the Vg at different directions could be very different. Can we predict more precisely which component dominates the relaxation time? The answer is “yes.” In Figure 2.14d we can see that the stable energy flow in the CS is nearly along the Vg direction in most regions of the CS. Then it is reasonable to argue that it is the component along the θ direction Vgθ and not the component along the r direction Vgr which dominates the relaxation time and the total scattered energy in the dynamical process.

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For the anisotropic cloaking material, the Vgθ and Vgr can be expressed as 2c ) / (2 + √εz μr 2c ) / (2 + νgr = [∇k ω (k)]r = ( √εz μθ

νgθ = [∇k ω (k)]θ = (

ω εz ω εz

dεz ω dμr + ) dω μr dω dεz ω dμθ + ), dω μθ dω

where c is the velocity of light in vacuum. In order to illustrate our prediction, the σ (t) under different Vgθ and Vgr are investigated, respectively. First, we keep the Vgr unvaried by holding Aθ , Az , and 𝛾 constant (keep d𝜖z /dω and dμθ /dω unchanged), only modify Ar to change the Vgθ . The results are shown in Figure 2.13c, when Ar is closer to 1, the Vgθ becomes smaller (with larger dμr /dω ), the relaxation time is longer, and more energy is scattered in the dynamical process. So the larger Vgθ means a better cloaking effect in the dynamical process. On the other hand, when we keep the Vgθ unvaried and change Vgr by holding Ar , Az , and 𝛾 constant and modifying Aθ , the results are shown in Figure 2.13d. We find that the relaxation time is almost unchanged with the change of Vgr . Obviously, Vgθ is the dominant element in the dynamical process. This conclusion can help us to design a better CS to defend the pulsive radars. In the expression of Vgθ , it is also shown how to tune Vgθ by modifying the dispersion parameters. It seems that the larger Vgθ , the better the cloaking effect in the dynamical process. However, since the Vg (and its components) can generally not exceed c, there is a minimum limit for the relaxation time of the cloaking dynamical process. We can estimate it by dividing the mean length of the propagation rays by Vgθ . In our model, the mean length is (R1 + R2 )/2, about three wavelengths. So the relaxation time can not be shorter than three periods. Figure 2.13 shows that our estimate coincides with our simulation results. Actually, here we are facing a very basic conflict for making a “better” CS, which is discussed more in our other works [7, 81]. The conflict is from the fact that fairly strong dispersion is required to realize a good stable cloaking effect at a certain frequency [7, 68], but in this research we show that the weaker dispersion can realize a better cloaking effect in the dynamical process. For a real design of the CS, there should be an optimized trade-off.

2.5.5 Section summary Based on causality, the limitation of the electromagnetic cloak with dispersive material was investigated in this section. The results show that perfect invisibility cannot be achieved because of the dilemma that either the group velocity Vg diverges or a strong absorption is imposed on the cloaking material. This is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of the finite difference time domain method was performed to validate our analysis.

82 | Xunya Jiang et al.

2.6 Limitation of the electromagnetic cloak with dispersive material 2.6.1 Introduction Through the ages, people have dreamed of having a magic cloak whose user cannot be seen by others. For this fantastic dream, much work has been done by scientists all over the world. For example, some researchers diminished the scattering or the reflection from objects by absorbing screens [82] and small, nonabsorbing, compound ellipsoids [83]. More recently, based on the coordinate transformation, Pendry et al. theoretically proposed a general recipe for designing an electromagnetic cloak to hide an object from the electromagnetic (EM) wave [68]. An arbitrary object may be hidden, because it remains untouched by external radiation. Meanwhile, Leonhardt described a similar method, where the Helmholtz equation is transformed to produce similar effects in the geometric limit [67, 75]. Soon, Cummer et al. simulated numerically (COMSOL) the cylindrical version of this cloak structure using ideal and nonideal (but physically realizable) electromagnetic parameters [69]. Especially, Schurig et al. experimentally demonstrated such a cloak by split-ring resonators [72]. In addition, Cai et al. proposed an electromagnetic cloak using high-order transformation to create smooth moduli at the outer interface, and presented a design of a nonmagnetic cloak operating at optical frequencies [70, 71]. According to the general recipe, the electromagnetic cloak is supposed to be perfect, or “fully functional”, at a certain frequency, as long as we can get very close to the ideal design, although there is a singularity in the distribution, which has been elucidated further in several works [84, 85]. However, in all these pioneering works, interest is mainly focused on single-frequency EM waves, so that the effects of the dispersion, which is related with very basic physical laws, are not well studied. If dispersion is introduced into the study, can we attain a deeper insight into the physics of cloaking? In this section we will show that ideal cloaking cannot be achieved because of another more basic physical limitation – the causality limitation (based on the same limitation. Chen et al. obtained a constraint of the band width that limit the design of an invisibility cloak [86]. Starting from dispersion relation and combining with the demand of causality, we will demonstrate that the ideal cloaking will lead to the dilemma that either the group velocity Vg diverges or a strong absorption is imposed on the cloaking material. Our derivation and numerical experiments based on the finite-differencetime domain (FDTD) methods will show that the absorption cross section will be fairly large and dominates the total cross section for a dispersive cloak, even with very small imaginary parts of permittivity and permeability.

2 The group velocity picture: the dynamic study of metamaterial systems | 83

2.6.2 The group velocity and physical limitation of invisible cloaking Let us consider a more general coordinate transformation on an initial homogeneous medium with 𝜖i = μi in r space, r󸀠 = f (r)θ 󸀠 = θφ 󸀠 = φ . Following the approach in [87] and [89], we get the following radius-dependent, anisotropic relative permittivity and permeability: 𝜖r󸀠 = μr󸀠 = 𝜖i ([r/f (r)])2 [df (r)/dr], 𝜖θ 󸀠 = μθ 󸀠 = 𝜖i /[df (r)/dr] and 𝜖φ 󸀠 = μφ 󸀠 = 𝜖i /[df (r)/dr]. We emphasize that since the transformationis directly acted on the Maxwell equations, the above equations are also suited for the imaginary parts of constitutive parameters, and all physical properties of wave propagation in r space should be inherited in r󸀠 space, such as absorption. It is very important for us to have consistent physical pictures in both spaces. At working frequency ω0 , for a propagating mode with k vector as kr󸀠 kθ 󸀠 kz󸀠 inside the cloak, we have the dispersion relation 2 of the anisotropic material [87] as (kr2󸀠 /n2r󸀠 ) + (kt2󸀠 /n2t󸀠 ) = ωc2 , where kt2󸀠 = kθ2 󸀠 + kψ2 󸀠 , ni r nr󸀠 = √𝜖ψ 󸀠 μ θ 󸀠 = (ni /df (r)), nt󸀠 = nθ 󸀠 = nψ 󸀠 = √𝜖r󸀠 μ θ 󸀠 = f (r) and ni = √𝜖i μ i . Then we can define kr󸀠 = (ω /c)nr󸀠 cos α and kt󸀠 = (ω /c)nt󸀠 sin α , and the group velocity can be obtained as 1 c (cos α )2 (sin α )2 2 ] [ + n2r󸀠 n2t󸀠 vg = , (2.10) (sin α )2 mt󸀠 (cos α )2 mr󸀠 [ ] + nr 󸀠 nt 󸀠 where mr󸀠 = nr󸀠 + ω dnr󸀠 /dω and mt󸀠 = nt󸀠 + ω dnt󸀠 /dω . If the transformation has the following characteristics: f (r = 0) = R1 , f (r = R2 ) = R2 , then when r󸀠 → R1 (or r0 → 0), nt󸀠 will tend to zero, and the group velocity is approximated as c vg ≈ . (2.11) dn 󸀠 |sin α | ω t dω We will discuss equation (2.11) in two cases. The first case is with finite dnt /dω . Obviously, Vg will diverge when sin a → 0 for any finite dnt󸀠 /dω . Such divergence is shown in Figure 2.15 for a concrete example in which the transformation is r = f (r) = ((R2 − R1 )r/R2 ) + R1 , as in [68], R2 = 2R1 , thus nr󸀠 = 2 and nt󸀠 = 2 ∓ 4/( Rr + 2). The 1 dispersion parameters are set as mr󸀠 = 2.5, ω (dnt󸀠 /dω ) = 4 at working frequency. In Figure 2.15, the curves of Vg versus α are plotted for different R1 /r values. We can see that for large R1 /r (r → 0) the group velocity (more precisely, the tangential component of Vg ) will diverge at both peaks around α = 0. Because of the causality limitation, it is well known that the group velocity cannot exceed c except in the “strong dispersion” frequency range (also called the “resonant range”). However, if the working frequency is in the strong dispersion range of the cloaking material, the absorption must be very strong, and it will obviously destroy the ideal cloaking. So perfect invisibility cannot be achieved for finite dnt󸀠 /dω , because this will lead to superluminal velocity or strong absorption.

84 | Xunya Jiang et al.

R1/r =0.5 1.4

R1/r =1.5 R1/r =3.5

Vg /c

R1/r =7.5 R1/r =15.5

1

R1/r =18.5 R1/r =31.5 R1/r =63.5

0.6

0.2 -1.5

-0.5

-1

0 α(rad)

0.5

1

1.5

Fig. 2.15. The group velocity Vg versus α for different R1 /r values.

In addition, the curves with the criterion condition Vg = c on the plane [R1 /r, α ] are plotted for different ω dnt󸀠 /dω in Figure 2.16. The region to the left of the curves corresponds to Vg < c, and the region to the right corresponds to Vg > c. There exists a maximum of (R1 /r) − max R1 /r for each curve in order that Vg ≤ 0 can hold for all the α . Especially, for the no-dispersion case (ω dnt󸀠 /dω ) = 0, we can see that Vg > 0 at all R1 /r for large α values, which means that the whole cloak is not physical if there is no dispersion. This “dispersion-is-required” conclusion can be generally derived from 1.5

dnt′ dω =0 ω dnt′=1 dω dn ω t′=2 dω dn ω t′=4 dω ω

1

α(rad)

0.5 0

-0.5 -1 -1.5 0

10

20

30

40 R1/r

50

60

70

80

Fig. 2.16. The relation between R1 /r and α when Vg = c for different ω dnt 󸀠 dω .

2 The group velocity picture: the dynamic study of metamaterial systems | 85

equation (2.10), and it is consistent with the analysis in [68]. From Figure 2.16, we know that the larger (ω dnt󸀠 /dω ), the larger is max R1 /r. But in any case, for arbitrary finite ω dnt󸀠 /dω , max R1 /r cannot be infinite, so that the superluminal range always exists. The second case of equation (2.11) is with divergent ω dnt󸀠 /dω . From the previous discussion, we know that if the ideal cloak exists, the cloak must be dispersive and ω dnt󸀠 /dω must be divergent when r → 0. Actually, when r → 0, since √𝜖r󸀠 ∝ r, dnt󸀠 /dω ∝ (d𝜖t󸀠 /dω )√𝜖r󸀠 is really divergent for nonzero d𝜖r󸀠 /dω . From equation (2.11), we can see vg → 0 for a finite d𝜖t󸀠 /dω (generally true) at all α values except α = 0 or α = π , so that the group velocity difficulty seems to be overcome. However, because of the causality limitation, the nonzero d𝜖t󸀠 /dω means the nonzero imaginary part of permittivity (nonzero dissipation). The nonzero dissipation and the almost zero group velocity will result in very strong absorption. This means that the energy of rays near the inner cloaking radius R1 is almost totally absorbed by the cloaking material. As we pointed out at the beginning, the absorption in r󸀠 space should also appear in r space because of the consistence between two spaces. The strong absorption in r space can be interpreted in the following way. From the transformation (which is also suited for imaginary part), we can find that when r → 0, the finite imaginary part in r󸀠 space corresponds to the infinite imaginary part in r space, which also means very strong absorption in the initial homogeneous medium. So the perfect cloaking is still impossible because of the strong absorption which is enforced by the causality limitation. For a two-dimensional coordinate transformation, r󸀠 = f (r), θ 󸀠 = θ , and z󸀠 = z, the same conclusions of the causality limitation can be obtained through the similar analysis, although the coordinate transformation and the singularities are different from the three-dimensional case.

2.6.3 Numerical results and discussion Next, we will discuss the physical meaning of the dilemma that either the group velocity Vg diverges or a strong absorption is imposed on the cloaking material. First, this is an intrinsic conflict which cannot be solved by such methods as, for example, “the system is imbedded in a medium” [68]. We believe that the ideal cloaking is impossible because of the causality limitation, and this conclusion is consistent with the statement of previous studies [89] that perfect invisibility is unachievable because of the wave nature of light. Secondly, we have to face the question: “Why is the causality violated for ideal cloaking based on simple coordinate transformation?” Our answer is that the causality is only guaranteed by the Lorentz covariant transformation, but the coordinate transformation for ideal cloaking is not Lorentz covariant. Such a violation is obvious if we suppose that the initial medium in r space is not dispersive, such as a vacuum, but as we have pointed out (also mentioned in [68]), the cloaking material (in r󸀠 space) must be dispersive to avoid the group velocity over c. Such a Lorentz covariant is generally true for “transformation optics”, since material param-

86 | Xunya Jiang et al. eters are nonrelativistic, so the causality limitation should be widely checked. Third, from equation (2.10), we can find that not only the inner layers of the cloak (r󸀠 → R1 ) but also the other layers r󸀠 > R1 must be dispersive. For every layer, a certain dispersive strength is needed to avoid Vg > c. In the following, we will validate that the total cross section can be reduced drastically, and that perfect cloaking cannot be achieved because of strong absorption by FDTD numerical experiments. Compared with other frequency-domain simulation methods, such as the finite element methods or the transfer-matrix methods, the FDTD simulation can better reflect the real physical process of cloaking. For example, we note that the FDTD calculation will be numerically unstable when the dispersion is not included in the cloak’s material. For simplicity, the simulation is limited to a twodimensional cloak [69]. Without lost of generality, only TE modes are investigated in this study (TE modes have the electric field perpendicular to the two-dimensional plane of our mode). Thus the constitutive parameters involved here are 𝜖z󸀠 , μr󸀠 , and μθ 󸀠 . The dispersionis introduced into our FDTD by standard Lorentz model, 𝜖z󸀠 (r󸀠 , ω ) = 1 + μr󸀠 (r󸀠 , ω ) = 1 + μθ 󸀠 (r󸀠 , ω ) = 1 +

Fz󸀠 (r󸀠 )ωpz󸀠 ωaz󸀠 (r󸀠 )2 − ω 2 − iω 𝛾z󸀠 Fr󸀠 (r󸀠 )ωpr󸀠 ωar󸀠 (r󸀠 )2 − ω 2 − iω 𝛾r󸀠

, ,

Fθ 󸀠 (r󸀠 )ωpθ 󸀠 ωaθ 󸀠 (r󸀠 )2 − ω 2 − iω 𝛾θ 󸀠

(2.12)

where ωpz󸀠 , ωpr󸀠 , and ωpθ 󸀠 are plasma frequencies; ωaz󸀠 , ωar󸀠 , and ωaθ 󸀠 are atom resonated frequencies; 𝛾z󸀠 , 𝛾r󸀠 , and 𝛾θ 󸀠 are damping factors; and Fz󸀠 , Fr󸀠 and Fθ 󸀠 are filling factors. In our simulation, an E-polarized time-harmonic uniform plane wave whose wavelength ω0 in vacuum is 3.75 cm is incident from left to right. The real parts of the constitutive parameters at ω0 = 2π c/λ0 satisfy the cloaking coordinate transformation [69, 91], and they are μr󸀠 = r󸀠 − R1 /r󸀠 , μθ 󸀠 = μ1󸀠 , 𝜖z󸀠 = [R2 /R2 − R1 ]2 (r󸀠 − R1 )/r󸀠 , where R1 r is 0.665λ0 and R2 is 1.33λ0 . The dispersive parameters are set as follows: if 𝜖z󸀠 > 1, then ωaz󸀠 = 1.4ω0 , else ωaz󸀠 = 0.6ω0 , ωar󸀠 = 0.6ω0 , ωaθ 󸀠 = 1.4ω0 , 𝛾z󸀠 = 𝛾r󸀠 = 𝛾θ 󸀠 = ω0 /100, 2 2 2 2 2 2 ωpz󸀠 = ωpr󸀠 = ωpθ 󸀠 = 4ω0 , Fz󸀠 = (𝜖z󸀠 − 1)[(ωaz + ω02 𝛾z2󸀠 ]/(ωaz 󸀠 − ω0 ) 󸀠 − ω0 )ωpz 󸀠 , 2 2 2 2 2 2 2 2 2 Fr󸀠 = (μr󸀠 − 1)[(ωar + ω02 𝛾r2󸀠 ]/(ωar + 󸀠 − ω0 ) 󸀠 − ω0 )ωpr󸀠 , Fθ 󸀠 = (μθ 󸀠 − 1)[(ωaθ 󸀠 − ω0 ) 2 2 2 2 2 ω0 𝛾θ 󸀠 ]/(ωaθ 󸀠 − ω0 )ωpθ 󸀠 . In fact, these parameters have many possible choices. The different groups of parameters correspond to different dynamic processes, which we will discuss in another section [92]. Figure 2.17 shows the snapshots of the electric-field distribution in two cases: the cloak with the perfect electric conductor (PEC) at radius R1 (left), and the naked PEC with radius R1 (right). Obviously, the cloak is very effective. Quantitatively, we calculate the absorption cross section and the scattering cross section of the cloak at the stable state, and they are 0.67λ0 and 0.24λ0 , respectively, while the scattering cross section of the naked PEC is 3.14λ0 . So, with the dispersive cloak, the total cross section

2 The group velocity picture: the dynamic study of metamaterial systems | 87

(a)

(b)

Fig. 2.17. The snapshots of the electric-field distribution in the vicinity of PEC: (a) the cloaking structure with a PEC at radius R1 , and (b) the naked PEC with radius R1 .

is three times smaller, and the absorption cross section dominates, as predicted. To emphasize the huge absorption of the cloak, we use a common homogeneous isotropic media, with 𝜖 = μ = 1.1, but all other parameters are the same as the cloak, to replace the cloaking material. Then we find that the absorption cross section is only 0.089λ0 , which is about one order smaller. The reason for the strong absorption has been discussed before. Now we can have a full view of a cloaking recipe based on coordinate transformation. First, the cloaking material must be dispersive, and the strong absorption cannot be avoided because of the causality limitation. Thus it is not perfectly invisible. Second, the scattering cross section of the dispersive cloak could be small, so that the scattered field is weak. Although the ideal invisibility is impossible, the cloaking recipe still has a main advantage. The “strong absorption and weak scattering” property means that the cloak can almost not be observed except from the forward direction, so that such a cloak can well defend the “passive radars” which detect the perturbation of the original field. It is well known that for the Rayleigh scattering case, where the radius of the scatterer is much smaller than the wavelength, the absorption cross section could be larger than the scattering cross section because of the diffraction. The cloaking can be thought as a giant Rayleigh scattering case, where the light rays are forced to “diffract” around the cloaked area.

2.6.4 Section summary In this section, the properties of the dispersive cloak were investigated, and the limitation of causality is revealed. Our study showed that the superluminal velocity or strong absorption cannot be overcome because of the intrinsic conflict between the coordinate transformation to obtain the cloaking and the causality limitation. In addition, we validated the results using a numerical simulation performed in an FDTD algorithm

88 | Xunya Jiang et al. with physical parameters. The numerical experiments show that the absorption cross section is dominant and the scattering cross section can be reduced significantly. The study gives us a full view of the cloaking recipe based on the coordinate transformation, and this will have further profound influence on related topics.

2.7 Confining the one-way mode at a magnetic domain wall 2.7.1 Introduction One-way waveguides have attracted much attention in recent years [98–108] due to ther their great potential application in all photonic integrated circuits. A popular solution for one-way waveguides is to utilize the gyromagnetic photonic crystal [104, 105], which can support a chiral edge state that exhibits an anomalous unique directionality. Due to the unidirectionality of the chiral edge state, a one-way waveguide can be realized via confinement of the surface states [106, 107]. Such one-way photonic crystal waveguides with a chiral edge state has been proven to exhibit strong robustness against disorder. However, the one-way photonic crystal waveguides need complex structures, which may bring more challenges in integrated circuits. In addition, the existing one-way photonic crystal waveguides are all frequency-sensitive. To avoid these difficulties, very recently, another solution for one-way propagation has been proposed [109], in which a broadband one-way mode (OWM) is predicted to propagate along a magnetic domain wall. However, the cost of the solution is the broadband OWM may be not suitable for one-way waveguide, since the mode extends into the bulk rather than confinement at the domain wall. In reviewing these existing efforts, we feel it is desirable to find a design for a one-way waveguide that includes at least three characteristics: (1) simple structure, (2) broadband working frequency; and (3) well-confined mode in the waveguide. In this section, we will present our design for this purpose. Our design is based on the domain wall. The key of this work is to find a confinement mechanism to localize the broadband OWM at the domain wall. We find the OWM can be confined well at the domain wall via the photonic band gap of bulk material [110, 111], unlike the one-way photonic crystal waveguides confining the OWM in a waveguide by the photonic band gap of a gyromagnetic photonic crystal. In this section we will show a one-way waveguide with a simple structure and broadband working frequency, by utilizing the OWM well-confined at the domain wall. Such OWM makes the one-way waveguide highly efficient and very robust against disorder. Besides one-way waveguides, the OWM localized at the domain wall can also be used to design high-efficiency, crosstalk-proof splitters and benders. Additionally, with magnetic controlling, the splitters and benders can be transformed into each other in our proposal, which may have great potential application for all photonic integrated circuits.

2 The group velocity picture: the dynamic study of metamaterial systems |

89

2.7.2 Model Our model for OWM is schematically shown in Figure 2.18a, in which a domain wall exists at the interface between two gyromagnetic media such as yttrium-iron-garnet (YIG) at the xy-plane with antiparallel dc magnetic fields along +z and −z. We use a dark gray area and a light gray area to denote −z and +z dc magnetic fields respectively in this figure and subsequent figures in this section. According to real YIG material, the relative permittivity 𝜖 = 15 and relative permeability would have a gyromagnetic form: μ1 ↔ μ = (∓jμ2 0 where μ1 = 1 +

ωm (ω0 − iαω ) (ω0 − iαω )2 − ω 2

±jμ2 μ1 0

0 0) , 1

and μ2 =

ωm ω . (ω0 − iαω )2 − ω 2

ω0 = 𝛾H0 is the resonance frequency, with as the gyromagnetic ratio, and H0 the sum of the external dc magnetic field. is the damping coefficient, and ωm is the characteristic circular frequency corresponding to a wavevector km = ωm /c and describes the direction of H0 . In this structure, only TE waves can be formed as an OWM propagating along the domain wall.

3.0

1 ωc

0.8

2

Domain wall

ωa

1.5 1.0 0.5

Y z

(a) X

0.6

4

0.4

2

F

ω / ωm

2.0

B

2.5

1

0.2 (b)

ωb

0.0 -20 -10

(c) 0 10 kx / k m

20

0

0.5

×10-3

0 0.5 1 1.5 2 2.5 3 ω / ωm 1

1.5 2 ω / ωm

2.5

3

Fig. 2.18. (a) The model of the domain wall. The external dc magnetic field for y > 0 (dark gray area) and y < 0 (light gray area) are applied along −z and + z, respectively. (b) The dispersion relation of TE mode and the projected band structure of bulk modes. The light gray and dark gray regions represent the photonic band gap and the extended modes of the bulk respectively. Different OWMs including positive propagation OWM-I, positive propagation OWM-II and negative propagation OWM-II are represented by the lines labeled ωa, ωb, and ωc , respectively. (c) The forward transmission F (solid lines) and backward transmission B (dot-dashed lines in the inset box) of OWM-I and OWM-II versus frequency. The distance between the detector and the source is 100 mm.

90 | Xunya Jiang et al. 2.7.3 Confining the one-way mode

-20 (b) -40 40 20 0

y/a y/a

40 20 0 -20 (a) -40 40 20 0

y/a

In order to illustrate how to confine an OWM at the domain wall with a photonic band gap, we study the dispersion relation of bulk modes and OWMs in our structure, by exactly solving the Maxwell equations, and the results are shown in Figure 2.18b. In this figure, the solid lines correspond to the OWMs of domain wall. The brown and gray regions correspond to the projected band structure of bulk modes. For the bulk modes, a gap (gray region) obviously exists. Therefore, we can group the OWMs into two types: OWM-I and OWM-II, whose frequency range are respectively in the gap and in the band of bulk modes. Comparing OWM-I with OWM-II, we see that their forward transmission [109] F and backward transmission [109] B are clearly different, as shown in Figure 2.18c. In this figure, we can see the F and B of OWM-I (in the gray region) are almost a unit and zero, respectively, and they are nearly frequency-independent; while for OWM-II, the forward transmission becomes smaller and the backward transmission becomes larger, and both are frequency-dependent. These results indicate that OWM-I is a much more efficient unidirectional mode. In addition, we can also find that OWM-I is broadband, with working frequency range 1.28ω m to 1.88ω m (here ωm = 5.26 GHz). Physically, the reason that OWM-I exhibits a more efficient unidirectionality than OWM-II can be explained as follows. Only OWM-I can be localized at the domain wall, since it becomes evanescent in the bulk with the intensity decay exponen-

-20 (c) -40 -100 -50

×105 4

90° 120°

3 2

60°

150°

30°

1 0

180°

0°

-1 -2

210°

×42

×1 ×30

330°

-3 0 x/a

50

100

-4

(d)

240°

300° 270°

Fig. 2.19. (a–c) Steady-state scattered fields of OWMa , OWMb and OWMc scattered by a PEC particle located at the center of domain wall. (d) Radiation pattern. Angular dependence of the energy flux is plotted as a function of polar angle. The 0∘ and 180∘ mark the + x and −x direction, respectively. The The solid, dash-dotted and dotted lines correspond to the scattered energy flux of OWMa , OWMb and OWMc , respectively.

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tially away from the domain wall. On the contrary, OWM-II is much easier to extend to the bulk, because their photon energies degenerate with the bulk modes. To show this, three typical OWMs (i.e., OWMa , OWMb and OWMc with frequency ωa = 0.3ωm , ωb = 1.5ωm , and ωc = 2.5ωm , respectively) are studied, which are excited by a current source at the domain wall. Obviously, OWMa belongs to OWM-I, while OWMb and OWMc are the type of OWM-II. With a single perfect electric conductor (PEC) particle scatter located at the domain wall at a distance d = 2 mm from the source (the scatter is on the right of the source for OWMa and OWMc , and on the left for OWMb ), we calculate the scattered fields of the three OWMs and the radiation patterns of the scatter by the finite differential time domain (FDTD) method, which are shown in Figure 2.19. From Figure 2.19a, we can see that the scattered field of OWMa is still well-confined at the domain wall, but others extend into the bulk, as shown in Figure 2.19b,c. The fact that the confinement quality of OWM-I is much better than that of OWM-II can also be illustrated by the radiation pattern of the scatter, as shown in Figure 2.19d. In this figure, we can see that only OWMa can concentrate almost all energy flux propagating along the domain wall, without reflection and diffraction; while OWMb and OWMc have quite a bit of energy flux scattered to other directions. With respect to the directional coefficient [114] of the three modes along the domain wall (0∘ for OWMa and OWMc , 18∘ for OWMb ), we find OWMa is about 42 and 30 times larger than that of OWMb and OWMc , respectively.

Fig. 2.20. (a) The structure of the domain wall with roughness. (b) Steady-state Ez field distribution of OWMa

92 | Xunya Jiang et al. 2.7.4 Robustness against roughness Furthermore, such OWM-I is very robust against disorder, since the disorder-induced backscattering is suppressed, due to its unidirectionality. This phenomenon is very useful for keeping the hight one-way waveguide efficiency when the domain wall is rough. To show this in simple form, as shown in Figure 2.20a, a raised portion is chosen to represent the roughness, with h1 = 10a, h2 = 10a, where a = 1 mm is the length unit in this work. The OWM of this domain wall with roughness is roused and propagates only along one direction, as shown in Figure 2.20b.

2.7.5 Photonic splitters and benders As discussed above, OWM-I is very suitable to attain a high-efficiency one-way waveguide. Due to the unidirectionality and good confinement of OWM-I at the domain wall, it can also be used to design more devices with simple structures in the realm of all photonic integrated circuits, such as splitters and benders, which are predicted to exhibit high-efficiency, broadband, frequency-independent, reflection-free, crosstalkproof and robustness against disorder. As typical examples, Figures 2.21 and 2.22 show our proposal of splitters and benders, respectively. Our simulation results show that all these devices have the same broadband as the one-way waveguide in Figure 2.18. In Figure 2.21a, a one-way cross-domain wall splitter based on the domain wall is constituted by four YIG bulks. Utilizing OWM-I, a beam (OWMa ) from the “input” port is split into two equal intensity beams and exit from two “output” ports, as shown in Figure 2.21b. It is proved that the splitter is a high-efficiency 50 % splitter, since the beams propagating along the domain wall are unidirectional and reflection-free. Furthermore, the beams at different domain walls are immune from crosstalk, because the OWM-I is well-confined. Similarly, based on the simple structure of a domain wall with OWM-I, other oneway cross-domain wall splitters can be designed. For instance, an any-angle splitter and a multiple-beam splitter with simple structures are presented in Figures 2.21c and e, respectively. And their field distributions are shown in Figure 2.21d and f, respectively. From the field distributions, we can find that all the splitters exhibit high efficiency, without reflection, diffraction, and crosstalk. Based upon OWM-I, we find another important application, such as a sharp bender. In Figures 2.22a and c, a 90∘ bender and an any-angle splitter are illustrated, respectively. Due to OWM-I, both benders are high efficiency, which can be seen from the field distributions as shown in Figures 2.22b and d. Actually, the 90∘ bender can be transformed from the splitter in Figure 2.21a just by reversing the dc magnetic field direction of area-3 in Figure 2.21a, and the reverse process is also feasible. Following this method, the any-angle bender and the any-angle splitter in Figure 2.21c can also be transformed into each other. The realization of this transformation requires

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93

Fig. 2.21. The structures of splitters and their Ez field distribution of OWMa . (a) 50 % beam splitter, and (b) its field distribution. (c) Any-angle beam splitter, and (d) its field distribution. (e) Multiplebeam splitter, and (f) its field distribution. The dashed arrow lines indicate the direction of energy flow.

some magnetic control techniques [104, 115] such as homogenization and exact location technique. Therefore, with magnetic control, there is a reversible transformation between splitters and benders in our proposal, which may have great potential applications in all photonic integrated circuits.

94 | Xunya Jiang et al.

Fig. 2.22. Benders based on domain wall with OWMa . (a) 90∘ bender, and (b) its Ez field distribution. (c) Any-angle bender, and (d) its Ez field distribution of the any-angle bender. The dashed arrow lines indicate the direction of energy flow.

2.7.6 Section summary In summary, in this section we have proposed a design for a one-way mode which can be well confined at the magnetic domain wall by the photonic band gap of gyromagnetic bulk material. Utilizing the well-confined one-way mode at the domain wall, we demonstrated that a photonic one-way waveguide, splitter and bender can be realized with simple structures, which are predicted to be highly efficiency, broadband, frequency-independent, reflection-free, crosstalk-proof and robust against disorder. Additionally, we have shown that the splitter and bender in our proposal can be transformed into each other with magnetic control, which may have great potential applications in all photonic integrated circuits.

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2.8 Bullet-like light pulse in linear photonic crystals 2.8.1 Introduction With the development of all-optical communication and computation [116, 117], there is a growing interest in achieving a stable light pulse in on-chip information transmission and signal processing. The stable pulse of light not only provides a natural carrier of binary information, but also can be processed in all-optical systems where self-guiding is required. Traditionally, stable light pulses have been obtained through the use of nonlinear materials [118, 119], where the diffraction and/or the group velocity dispersion (GVD) are compensated by nonlinear self-focusing and self-phase modulation. Thus pulses or beams of light are localized temporally, spatially, or spatiotemporally, which are referred to as temporal solitons, spatial solitons, or light bullets, respectively. However, the nonlinearity strength is energy dependent. Solitons based on nonlinear materials may spread out when their energy is below the critical energy threshold. Moreover, special nonlinear materials are requested to allow stable soliton propagation. These requirements limit the application of solitons. Is there any other approach that can lead to a stable propagation of light pulse? In this section we present that light can stably propagate without temporal, spatial, or spatiotemporal broadening in suitable photonic crystals (PCs), where the diffraction and/or the GVD can be eliminated naturally. Unlike solitons, the stable propagation of light in desirable PCs is independent of pulse energy and does not depend on nonlinear materials. Herrnx we focus on the bullet-like propagation of a light pulse in two-dimensional (2D) PCs, while the same concept can be extended to three-dimensional cases.

2.8.2 The condition for the existence of bullet-like light pulses The behavior of light pulses in PCs is determined by the dispersion relations [118, 119]. Suppose the wave vector components parallel or perpendicular to the propagation direction are defined as k‖ or k⊥ , respectively. The dispersion relation in the crystal is ω (k‖ ,k⊥ ). Given an operating frequency ω0 = ω (k0 ,0), in the vicinity of k0 = (k0 , 0), ω (k‖ , k⊥ ) can be expressed by ω (k‖ , k⊥ ) = ω0 +

1 𝜕2 ω 𝜕ω ) k2 + ( ) (k − k0 ) + ⋅ ⋅ ⋅ . ( 2 𝜕k⊥2 k ⊥ 𝜕k‖ k ‖ 0

(2.13)

0

In equation (2.13), (𝜕ω /𝜕k⊥ )k0 vanishes due to mirror symmetry, and (𝜕2 ω /𝜕k⊥2)k0 is a coefficient related to diffraction. Typically, if (𝜕2 ω /𝜕k⊥2)k0 = 0 are satisfied in a range of k⊥ , it means a beam self-collimation (SC) propagation, which has been theoretically and experimentally demonstrated [122–127]; (𝜕ω /𝜕k‖ )k0 = vg is the group velocity. It is the different vg that results in the temporal broadening of a pulse. For the investigation of the temporal dispersion, we expand k‖ (ω ) in a Taylor series about ω0 , which is

96 | Xunya Jiang et al. represented as k‖ (ω ) = k0 + (ω − ω0 )β1 + where βm = (

1 1 (ω − ω0 )2 β2 + (ω − ω0 )β3 + ⋅ ⋅ ⋅ , 2 6

dm k‖ ) (m = 1, 2, . . .). dω m ω =ω 0

The cubic and higher-order terms in this expansion are generally negligible if (ω − ω 0) ≪ ω 0. The parameter β1 = 1/vg indicates that the envelope of a pulse moves at vg . Parameter β2 is the GVD and responsible to the temporal broadening of a pulse. Particularly, β2 = 0 implies a stable pulse propagation without temporal broadening. Therefore, if SC and zero GVD can be satisfied simultaneously, the aim of realizing bullet-like propagation of light pulse is attained.

2.8.3 The bullet-like light pulse in PCs We will demonstrate that both the SC and the zero GVD can be satisfied simultaneously in a PC structure, where we focus on the structural dispersion and neglect the material dispersion. By numerical calculation, we find such PCs do exist. For example, we consider a 2D crystal consists of a square lattice of air holes in high index material (n0 = 3.46). The hole radius is r = 0.35a, where a is the lattice constant. The isofrequency contour of the band diagram for the TE mode is shown in the inset in Figure 2.23. We can see that the crystal supports a wide-angle SC propagation along direction in the frequency range ω = 0.286–0.302(2π c/a) [125]. The corresponding GVD curve along the direction is presented in Figure 2.23. There is a zero GVD at ω0 = 0.2915(2π c/a), which implies a stable pulse propagation without temporal broadening. Since the zero GVD occurs in the frequency range of SC, bullet-like propagation of light pulse can be maintained in the crystal.

2.8.4 Numerical validation A numerical FDTD simulation is performed to verify the bullet-like propagation mentioned above. We assume that a spatiotemporally Gaussian pulse with central wavelength λ0 = 1.55 μm is incident to the crystal along direction. The transverse width [full width at half maximum (FWHM)] of the pulse is h = 16a, and its time duration (FWHM) w = 0.3 ps, corresponding to a frequency range of ν = (1.922–1.950) × 1014 Hz. Such a light pulse is chosen to meet the requirement of SC and zero GVD. In Figure 2.24, the transverse FWHM width of the pulse during propagation is presented. No transverse broadening is observed for a long propagation distance, indicated by the constant width of the propagating pulse. The pulse keeps the same shape,

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97

500 400 300

β2(a/2πc2)

200

0.2915

100 0 -100 -200 -300 -400 -500 0.26

0.28

0.3 ω(2πc/a)

0.32

0.34

Fig. 2.23. The GVD curve of the second photonic band along ky = 0 in a 2D PC. The corresponding frequency contour presentation of the band diagram is presented in the inset. The light gray solid curve represents the SC frequency contour.

Fig. 2.24. The transverse FWHM width of the pulse during propagation. A snapshot of the magnetic field of the pulse is shown in the inset.

as shown in the inset in Figure 2.24. The time evolution of the pulse is investigated by recording the magnetic field at different propagation distances, which is shown in Figure 2.25a. Evidently, in the concerned monitoring range, the pulse almost keeps the same envelope shape. We then explore the transmitted spectra at several different monitor points, and the results are plotted in Figure 2.26. It is shown that the transmission efficiencies are above 95 % in the frequency range Δν = (1.89–1.99) × 1014 Hz within a propagation distance of 334a. This implies that a pulse shorter than 0.3 ps can remain stable for such a distance.

98 | Xunya Jiang et al.

Fig. 2.25. (a) Time evolution of a pulse in the PC. The fields are recorded at the propagation distances of L = 8.5a, 86.5a, 150.5a, 214.5a, 278.5a, and 342.5a, respectively. (b) Oscillation evolution near the trailing edge of the pulse due to the TOD.

Fig. 2.26. Transmitted spectra at different monitor points. The solid, the dashed, and dot-dashed lines represent the spectra at propagation distances of L = 150.5a, 278.5a, and 342.5a, respectively. These values are normalize by the value at L = 8.5a.

2.8.5 The effect of high-order dispersion Although the bullet-like propagation can be ascribed to the SC and the zero GVD, the higher-order diffraction and dispersion may influence its stability. As an example, we consider the influence of the third-order dispersion (TOD), which is the most important dispersion order after GVD. We estimate the dispersion length of our PC by Ld = w3 /|β3 |, where w is the pulse’s duration. The propagation can be regarded as stable if the propagation distance is shorter than Ld. For our PC, β3 = −1528a2 /4π2 c3 at zero GVD, so Ld = 6.25×104a, which is a tremendously long length for on-chip systems. Although the absolute value of β3 is small enough for the long length of bulletlike propagation, it is very interesting to strictly check the influence of TOD on the pulse’s evolution. It is well known that the TOD may cause an asymmetric oscillation near the leading or trailing edge of the pulse as the pulse evolves [116]. In our case, an unexpected oscillation develops indeed near the trailing edge, which can be seen in Figure 2.25a. For clearness, the trailing edge of the magnetic field Hz at propagation distance L = 85.5a is plotted in detail in Figure 2.25b. It is evident that a distinct

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oscillation develops. Since the TOD determines the stable propagation distance of a bullet-like pulse, a much smaller TOD is required for a long distance transmission. The TOD can be suppressed by choosing a GVD curve whose variation is much smoother and more flat in the vicinity of the zero GVD. It is known that the variation of the GVD curve stems from Bragg scattering. Therefore, by reducing the index contrast or changing the hole radius, the strength of Bragg scattering can be reduced, so the TOD is suppressed. As an example, we change the index and the hole radius of the crystal into n0 = 2.83 and r = 0.32a. The corresponding frequency contour and the GVD curve are shown in Figure 2.27. A much smoother and more flat GVD curve is obtained, and the zero GVD occurs in the frequency range of wide-angle SC, i.e. ω = 0.323–0.34(2π c/a). Clearly, bullet-like propagation of light pulse is supported by the crystal, while the TOD is greatly suppressed.

500 400 300

β2(α/2πc2)

200

0.33

100 0 -100 -200 -300 -400 -500

0.3

0.32

0.34 0.36 ω(2πc/a)

0.38

0.4

Fig. 2.27. The GVD curve of the second photonic band along ky = 0 in a 2D PC, in which a much more flat curve is obtain. The corresponding frequency contours are presented in the inset. The light gray solid curve represents the SC frequency contour.

2.8.6 Section summary In summary, we have demonstrated that bullet-like propagation can be sustained in particularly designed PCs, which have both the properties of SC and zero GVD. The diffraction and the GVD can be eliminated naturally in a suitable PC structure, and so a light pulse can propagate in it without temporal and spatial broadening. Compared to the solitons based on nonlinear materials, this new type of bullet-like propagation is independent of the pulse energy and nonlinear materials are not required. Further-

100 | Xunya Jiang et al. more, the TOD was examined, and we found it can be greatly suppressed by the design of the PC structures.

2.9 Summary In summary, we have investigated the metamaterial systems from the group velocity (energy velocity) viewpoint. From these topics, we demonstrated the importance of group velocity in metamaterial studies. From group velocity, we can find the physical origin of abnormal optical phenomena of metamaterials, such as the “no-transmission no-reflection” on the hypermedium surface which is from a zerogroup-velocity re?ecting mode, and the coherence gain of superlens image which is from the different group delay on different paths. From group velocity, we can avoid some traps of violating basic physical limitation, such as the violation of causality limitation in cloaking study. These traps are very serious, since the metamaterial of our imagination could exist in this world if basic limitations are violated. From group velocity, we can find the key parameter of the cloaking dynamical process and help to optimize the design of a cloak design. From the group velocity of evanescent waves, new detecting methods could be found, for example the detecting of QED vacuum polarization by phase change or delay time of evanescent wave. We believe that only with a well-constructed group velocity view is a deeper understanding of the abnormal optical/photonic properties of metamaterials possible. All this research also shows that the group velocity study of metamaterials can lead us to many new interesting topics, which are still waiting for further research to be done.

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2 The group velocity picture: the dynamic study of metamaterial systems

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Liangyao Chen, Yuxiang Zheng, Songyou Wang

3 Study of the characteristics of light propagating at the metal-based interface 3.1 Introduction Metal-based microstructures and materials play a significant role in influencing the properties of optical and optoelectronic devices. They can, for example, be used to make new devices with high-efficient photovoltaic and solar-thermal conversion abilities achieved in the green energy field [1], high-performance optical reflectors and mirrors [2], conductive transparent windows that would have important applications in the new flat panel display devices [3], stealth devices with characteristics of ultralow electromagnetic radiations [4], perfect optical lenses for super-resolution imaging, biosensors based on surface plasma resonance technologies, etc. [5]. The application covers a wide spectral region from microwave, infrared, visible to ultraviolet light, showing widespread application prospects of metal-based microstructure and materials in various new-technology areas, such as information, energy, national defense, and biotechnology. Since metals have strong optical absorption properties, their optical constants are presented as a complex form whose imaginary part with higher values corresponds to strong optical absorption. This has a significant impact on the light propagation characteristics in the metal, resulting in experiment difficulties for studies of the optical properties of metals with poorer knowledge about the propagation characteristics of light in metal-based materials. Since the optical refractive index represents the characteristics of electromagnetic wave traveling in the metal is a complex number, the electromagnetic field of light in metals is very unevenly distributed. The refractive electric field of light at the metal interface is not uniformly distributed, implying that the surfaces of the light wave with the same phase and amplitude will not be along the same direction in propagation. Under certain conditions, the incident light on the metal surface will excite the so-called surface plasmon polaritons (SPPs), a hybrid excitation formed by quasi-free electrons interacting with photons on the metal surface. The properties of the surface plasma will strongly depend on the micro structure of the metal surface with dispersion features related to the excitation mode. Therefore, in order to attain a profound understanding of the interaction mechanism between surface plasma and optical fields, the propagation characteristics of light associated with the surface plasma excitation can be well controlled at the metal interface. However, there are still many unsolved questions concerning the propagation characteristics of light in the metal-based structures and materials which need to be understood more deeply. Some of those optical phenomena based on the theoretical prediction which happened at the metal interface have not yet been fully confirmed by rigorous

108 | Liangyao Chen et al. experiment measurements. Thus, here we will try to give an introduction by describing the reflection and refraction properties of light with its traveling path measured at the metal interface, as well as the application and mechanism of the surface plasma interacting with light in the metal-based optical structures and materials.

3.2 The free-electron gas model and optical constants of metal There are mainly two mechanisms for the interaction between light and charged particles during the propagation of light in metal. One occurs by the condition in which the photon energy E is greater than the energy gap (Eg ) between two bands, i.e. under the condition of E > Eg , the electrons are excited from the lower energy band to the up band having higher energy. This process is called the interband transitions and has a significant impact on the optical properties of the metal. For noble metals copper, gold and silver with the energy band structure in which the lower d-band is fully occupied by the d elections and the conduction band is filled by quasi-free s electrons, the value of Eg between the s and d bands are about 2.1 eV, 2.5 eV, and 3.9 eV [6–9], respectively. Another mechanism occurs by the condition in which the photon energy E < Eg , implying that only the quasi-free electrons in the conduction band of metal can absorb the energy through the process interacting with photons. This is called the Drude intraband transition, dominating the optical properties of metal within the Drude spectral region [8, 9]. In the optical frequency region of electromagnetic waves, since the magnetic dipole moment is difficult to keep up with the rapid oscillation of the electromagnetic field, the permeability will in assumption be equal to one (μ = 1) [9, 10]. In the Drude spectral region, the dielectric constant of metal materials origins mainly from the interaction between electromagnetic waves and charged microparticles (such as quasi-free-electrons and ions), the movement of the charged particles can thus be described by the classic damped elastic oscillator model [9, 10]: d2 r m dr + (3.1) + mω02 r = −eEloc , τ dt dt2 where m, e, τ and ω0 are the effective mass, charge, lifetime and oscillation frequency of the damped elastic particles, and r is the displacement of charged particles under the local electric field Eloc as a function of time t. Under the electric field of light with frequency ω , the average displacement vector r and dielectric constant ε for N electrons are resolved as m

r=

−eEloc /m , (ω02 − ω 2 ) − iω /τ

ε =1+

ωp2 (ω02 − ω 2 ) − iω /τ

,

(3.2)

respectively, where ωp is the plasma frequency: ωp2 =

4π Ne2 . m

(3.3)

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In the free electron gas model approximation, ω0 ≈ 0. In the higher energy spectral region, for example, in the near-infrared and visible regions, the condition ωτ ≫ 1 can be satisfied to get the complex dielectric constant ε in the Drude spectral region as ε = ε1 + iε2 = 1 − ε1 = 1 −

ωp2 ω

, 2

ωp2 ω2

ε2 =

+i

ωp2 ω 3τ

ωp2

(3.4)

ω 3τ

.

(3.5)

Thus, according to equations (3.4) and (3.5), at the photon energy where E = ℎω = Ep = ℎωp , the plasma resonance will occur at the specific position of the wavelength where ε1 = 0 and ε2 = (1/ωτ ) ≈ 0. Generally, since the plasma resonance frequency for noble metals has the quite high value of Ep ≈ 9 eV, it cannot be measured and observed in the infrared and visible regions directly [6–9]. For actual metal materials, since the electrons and ions at the core energy levels will present extra contributions to ε1 , equation (3.5) is modified as follows: ε1 = 1 + εb −

ωp2 ω2

= 1 + εb −

Ep2 E2

,

(3.6)

where εb (εb ≥ 0) is the contribution to ε1 from the bound charges of the core energy levels. For metal materials with different electronic state structures, such as noble metals copper, gold, and silver, the values of εb are different and will significantly affect the dispersion feature of the plasma resonance taking place at the condition of ε1 = 0. In addition, the real and imaginary parts of dielectric constant also obeys the famous Kramers–Kronig relations (the causal dispersion relation) [9, 11]: ∞

ω 󸀠 ε (ω 󸀠 ) 2 ε1 (ω ) = 1 + ∫ 󸀠 22 dω 󸀠 , π (ω ) − ω 2 0

∞

ε (ω 󸀠 ) − 1 2ω ∫ 1󸀠 2 ε2 (ω ) = − dω 󸀠 . π (ω ) − ω 2

(3.7)

0

Therefore, the strong dispersion relationship of the real part of the dielectric constant as a function of optical frequency will also influence the imaginary part, resulting in the dispersive response to the electric field in the imaginary part. The relationship between the lost energy 𝜛 per second, which is generated from the interaction of charged particles with electromagnetic field in the metal, and the imaginary part of the dielectric constant ε2 as well as the plasma frequency is given by [9, 12] ∞ πωp2 dD 2 ) = ωε2 |Eloc | , ∫ ωε2 dω = 𝜛 = Re (Eloc ⋅ , (3.8) dt 2 0

where D = ε Eloc is the electric displacement vector. Thus, for metal materials, the imaginary part of the dielectric constant ε2 is so important that it cannot be ignored,

110 | Liangyao Chen et al. which represents the photon energy absorption during the light-substance interaction. The complex refractive index ñ and dielectric constant ε satisfy the following relationship: ε = ñ 2 , ñ = n + ik, ε1 = n2 -k2 , ε2 = 2nk, (3.9) where n and k are the real (refractive index) and imaginary (extinction coefficient) part of the complex refractive index. The relationships of reflectivity R or absorption coefficient α with n, k and wavelength λ are such that [9, 10] R=

(1 − n)2 + k2 , (1 + n)2 + k2

α=

4π k , λ

(3.10)

where α = 1/δ , δ is the penetration depth of light in the metal. Apparently, the absorbed photon energy must be greater than zero, satisfying α > 0, k > 0, ε2 > 0, which is restricted by the physical law of energy conservation and was rigorously verified in experiments up to date, not allowing arbitrary assumptions for its value selections. When the plasma resonance occurs at the condition where ε1 = 0 and ε2 ≈ 0, D ≈ 0, this means the internal electric field is strongly shielded by the effect of the plasma resonance. Further, according to the Maxwell equations [9, 10] ∇ ⋅ D = 0,

∇ ⋅ B = 0,

∇ × Eloc = −

1 dB , c dt

∇×H =

1 dD −iωε Eloc = , c dt c

(3.11)

where B and H (H = μ B) are the magnetic flux density and the magnetic field, respectively. For ε ≈ 0, we get ∇ × H = 0,

∇ ⋅ H = 0,

H = 0,

∇ × Eloc = 0.

(3.12)

This shows the characteristics of a longitudinal field to obey the wave vector conservation, indicating that only the longitudinal component of the electromagnetic field that parallels the wave vector is allowed to couple with the plasma wave to induce the plasma resonance [13, 14]. Therefore, the condition trying to use transverse electromagnetic field in the optical frequency range to excite the plasma oscillation is extremely limited. For example, using p-polarized light at a large incident angle, only the limited longitudinal component of the electric field on the surface of silver film can be applied to excite the high transmission plasma resonance peak near 3.8 eV [14]. According to equations (3.5) and (3.6), due to the presence of factor εb , the position of ε1 = 0 will not occur at E = Ep , but on the lower energy side. Johnson and Christy once made systematic experimental measurements and studies on the transmission and reflection spectra of the noble metals copper, gold, and silver. They found that the occurrence of plasma resonance is only observed in silver, but not in gold and copper, due to the significant difference of their band gap Eg and εb [6]. In the experiment, the ellipsometer was used to measure the dielectric constant spectra of copper, gold, and silver [15], as shown in Figures 3.1–3.3. As for the silver sample, since the region of the Drude tail is influenced by εb as well as the interband transition

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111

Fig. 3.1. The experimentally measured complex dielectric function ε (ε = ε1 + i ε2 ) of a copper film sample using the ellipsometer. The condition of ε1 = 0 and ε2 ≈ 0 is not observed.

Fig. 3.2. The experimentally measured complex dielectric function ε (ε = ε1 + i ε2 ) of a gold film sample using the ellipsometer. The condition of ε1 = 0 and ε2 ≈ 0 is not observed.

Fig. 3.3. The experimentally measured complex dielectric function ε (ε = ε1 + i ε2 ) of a silver film sample using the ellipsometer. The condition of ε1 = 0 and ε2 ≈ 0 occurred around E = 3.8 eV.

that happens when the photon energy E is greater than the band gap Eg , the energy position for ε1 = 0 and ε2 ≈ 0 is moved down to about 3.8 eV in the near-ultraviolet region, while, as seen from the dielectric constant spectrum of copper and gold, the condition for the occurrence of a plasma resonance (ε1 = 0 and ε2 ≈ 0) is not satisfied. According to equation (3.5), at the lower energy side of the Drude region, ε2 decreases monotonously with the photon energy E3 , while at the higher energy side of the interband transition absorption edge, ε2 shows a sharp increase with photon energy. Therefore, as seen from the dielectric spectra in Figures 3.1–3.3, due to the influence of Eg and εb there exists some place within the photon energy region of 1–4 eV where the Drude tail in the intraband transition region overlaps with the absorption

112 | Liangyao Chen et al. edge in interband transition region. For copper and gold, since their band gap (Eg are about 2.1 eV and 2.5 eV, respectively) is much smaller than that of silver (Eg is about 3.9 eV), the overlap region is placed at a much lower energy side than that of the silver sample, and thus it is difficult to meet the condition of ε1 = 0 and ε2 ≈ 0. As for the silver sample, assuming ωτ = 10 ≫ 1, Ep ≈ 9 eV, at E = Eg ≈ 3.9 eV, and according to equation (3.5), ε2 ≈ 0.5, which is in good agreement with the results shown in Figure 3.3.

3.3 Light refraction properties of a metal-based interface 3.3.1 Normal refraction In the first half of the 17th century, Snell and Descartes discovered the refraction law of light passing through an interface formed by two different media, later known as Snell’s law of refraction [10]. Snell’s law has been widely used to understand the propagation behavior of light in nonabsorbing transparent materials. For the simplest case, assume that an ideal interface consists of two transparent media with refractive indexes n1 and n2 , respectively, if an incident beam passes through the interface from medium n1 to medium n2 at the incident angle θ1 , the emergent light will be refracted along θ2 according to Snell’s law such that [16] n1 sin θ1 = n2 sin θ2 .

(3.13)

So far, the refractive index for natural transparent materials are all positive real numbers, as are the incident and refractive angles θ1 and θ2 . In the experiment, the values of refractive indexes of materials at specific wavelengths could be strictly determined by measuring the incident and refractive angles. Using these optical constants in research and applications, we are able to accurately predict the resulting refractive angle within the full angle range from 0∘ to 90∘ . This is the fundamental optical principle presented by Snell’s law for its wide applications in research and industry for the design and manufacturing of various optical materials and devices. As discussed above, the metal-based materials have strong optical absorption properties with the refractive index which is a complex number, ñ = n + ik. According to Snell’s law and Maxwell’s equations, the refraction of light at the interface between the transparent medium with positive n1 and the light-absorbing materials with complex ñ 2 will no longer have the simple form as presented by normal Snell’s law, but rather a more general one [10]: n1 sin θ1 = ñ 2 sin θ2̃ .

(3.14)

In equation (3.14), both the refractive index ñ 2 and the refractive angle θ2̃ (θ2̃ = θr + iθi ) will be the complex number. However, the complex form of refractive angle is only a

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113

mathematical presentation; the actual path and direction of the light propagating in light-absorbing materials all have to be the real numbers. Therefore, in the framework of Snell’s law, equation (3.14) can be rewritten to compare with the experimental result in the form that n1 sin θ1 = ñ 2 sin(θ2̃ ) = (n2 + ik2 ) sin(θ2r + iθ2i ) = (n2 a − k2 b) + i(n2 b + k2 a) = n2e sin θ2e ,

(3.15)

where a = sin(θ2r ) cosh(θ2i ), b = cos(θ2r ) sinh(θ2i ), n2 b + k2 a = 0

(3.16)

Equation (3.15) can be used in experiments to measure the effective refractive index and angle of light-absorption materials such as metal, and so on. Given n1 and θ1 , the experimental value and sign of effective refractive index n2e thus can be determined by measuring the effective refractive angle θ2e. Obviously, there are very complicatedly tangled relationships between the effective refractive index n2e and angle θ2e with the optical constant and incident angle of the metal material. Generally n2e ≠ n2 , whereby the simple relationship of n2e = n2 exists only for a nonabsorbing medium for which k2 = 0 and ε2 = 0.

3.3.2 Calculations of effective refractive index and refraction angle The propagation of light on the metal-based interface has always been very complicated, and there is still no conclusive understanding of these issues, despite the studies carried out in recent years. For example, the experimentally measured complex refractive index of silver at the wavelength of 532 nm is ñ = 0.201 + i3.132. Using the same wavelength of laser beam incident from air into the silver interface with incident angle of 70∘ , however, there is still no reliable theoretical and experimental results which can be used to calculate and predict the actual propagation paths and refractive angles of light at the silver side. On the basis of the Maxwell equations and boundary conditions, Born and Wolf have given a set of formulas to calculate the real refractive angle θm at the metal side

114 | Liangyao Chen et al. for light incident from dielectric into metal, such that [10] ñ sin θt = na sin θa q2 cos 2𝛾 = 1 − n2a sin2 θa (n2 − k2 )/(n2 + k2 )2 q2 sin 2𝛾 = 2nkn2a sin2 θa /(n2 + k2 )2 nm = [n2a sin2 θa + q2 (n cos 𝛾 − k sin 𝛾)2 ]1/2 nm sin θm = na sin θa ,

(3.17)

where θa and θt are the incident and complex refractive angles at the dielectric and metal side, respectively, na is the refractive index of the medium, nm and θm are effective real refractive index and angle of the metal, respectively, and q and 𝛾 are intermediate parameters introduced in the process of the calculation. The complex refractive ̃ = 632.8 nm) = 0.216 + indexes of silver at three experimental wavelengths are n(λ ̃ ̃ i3.881, n(λ = 532.0 nm) = 0.201 + i3.132, n(λ = 473.0 nm) = 0.179 + i2.578, respectively. By assuming that the incident medium is air (na =1), according to equation (3.17) the real refractive angle θm at the silver side as a function of incident angle θa is calculated, as shown in Figure 3.4. For a small incident angle θa , θm changes almost linearly with θa , and the effective refractive index nm ≈ n, independent of the imaginary part of the complex refractive index. For large incident angles, the real refractive angle θm hardly changes with a changing incident angle θa . Also, the effective refractive index and real refractive angle at the metal interface as functions of wavelength, optical constant, and incident angle are both positive numbers, and will not change its sign in the entire spectral region.

Fig. 3.4. The calculated real refractive angle θm as a function of the incident angle θa at the silver side for light incident from air into silver according to equation (3.17).

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On the other hand, based on the same principle and boundary conditions, GarciaPomar and Nieto-Vesperinas tried with another set of formulas to calculate the refractive angle θa at the dielectric side for light incident from metal into dielectric, given the real incident angle θm at the metal side, as [17] na sin θt = ñ sin θm q2 cos 2𝛾 = 1 − sin2 θm (n2 − k2 )/n2a q2 sin 2𝛾 = −nk sin2 θm /n2a nm = n/(n2 sin2 θm /n2a + q2 cos2 𝛾)1/2 na sin θa = nm sin θm .

(3.18)

Assuming a light beam is incident from silver into air (na = 1) with a real incident angle θm , using the same optical constants at three experimental wavelengths as those in the previous discussions, according to equation (3.18), the refractive angle θa at the air side as a function of incident angle θm is calculated, as shown in Figure 3.5.

Fig. 3.5. The calculated refractive angle θa at the air side as a function of the real incident angle θm at the silver side for light incident from silver into air.

The refraction path of light in the plane silver film is shown in Figure 3.6. The results of calculation with respect to equations (3.17) and (3.18) show that the final emergent refraction angle at the lower part of the air side is not equal to the initial incident angle at the upper part of the air side because of the different expressions of effective refractive index nm . For example, at the wavelength of 532 nm, an initial incident angle of 0.6 rad in the upper air side would result in a real refractive angle of 1.2 rad at the up part of the air-to-silver interface, as shown in Figure 3.4. However, at the same wavelength, the incident angle of 1.2 rad at the low part of the silver-to-air interface would

116 | Liangyao Chen et al.

Fig. 3.6. The schematic diagram of a refraction path for light which is incident from medium I into the plane metal film II with incident angle θa , then emergent from the same kind of medium I, where the incident angle should be equal to the emergent angle.

result in a final emergent refraction angle of about 0.06 rad at the lower part of the air side, as shown in Figure 3.5, which is about 10 times smaller than the initial incident angle at the upper part of the air side. This is inconsistent with experimental observations in which the incident angle should be equal to the emergent angle at both sides of the plane silver film, as shown in Figure 3.6. The difficulties and limitations in our understanding of these phenomena still remain to be further studied in the future.

3.3.3 Negative refraction of metal-based artificial materials The macroscopic electromagnetic properties of natural substances can be presented by those functions, namely the dielectric constant ε and permeability μ , respectively, which are responsible to the electromagnetic field interacting with charged particles in the materials. For normal dielectric materials, both the dielectric constant and permeability are positive real numbers. On the basis of the Maxwell equations, for electromagnetic waves that propagates in normal dielectric materials, the group of vectors, consisting of electric field strength E, magnetic field strength H, and wave vector K, complies with the right-hand rule, as shown in Figure 3.7a. The direction of propagation of the electromagnetic energy is characterized by the Poynting vector S = E × H. For normal dielectric materials, therefore, the direction of wave vector K is consistent with that of the photonic energy flowing [18]. In 1968, Veselago studied an imaginary substance that had both a negative dielectric constant ε and negative permeability μ [19]. The propagation of electromagnetic waves in this substance was expected to comply with the Maxwell equations as normal materials, while the vector group of the electric field strength E the magnetic field strength H, and the wave vector K follows the left-hand rule, as shown in Figure 3.7b. As a result, the wave vector K and the Poynting vector S are antiparallel. Veselago

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Fig. 3.7. Schematic diagram for the right-hand and left-hand rules.

called this substance a left-handed material (LHM) and the conventional dielectric materials a right-handed material (RHM). Veselago further proved that if a light beam is incident from the vacuum or other common media into the left-handed material, the negative refraction phenomenon will occur at the left-handed material side, that the incident and the refracted light beams are located on the same side referring to the surface normal direction. In this case, in order to ensure the effectiveness of Snell’s law, a negative refractive index has to be introduced, and so this material is also known as the negative index material (NIM). Meanwhile, there are also other novel properties for the negative index material, such as the reversed Doppler effect, the reversed Cerenkov radiation, the negative optical pressure, etc. On the basis of the relationship of s refractive index with the dielectric constant ε and permeability μ , both positive and negative solutions of the refractive index n are mathematically allowed: n = ±√εμ . (3.19) This gives rise to the keen interest on the research concerning the concept and phenomenon of negative refraction which may have potential applications in the field of photonics. Since there will be both positive and negative zones crossing over the zero point of the refractive index of the material in different spectral regions, the dielectric constant and permeability are no longer considered to be simply larger than zero, but have complicated relationships with the microstructure of material and the wavelength of light, thereby affecting the propagation characteristics of light, such as producing phenomena like the negative Goos–Hänchen shift and reverse propagating surface wave, etc. In fact, the current research on negative index materials has been developed in some situations in which arbitrary assumptions, definitions, and value selection about the dielectric constant and permeability are allowed. At the same time, many new left-handed materials with various microstructures have been prepared in intensive studies requiring further exploration of their properties and applications. The study of negative index materials has attracted much attention and was listed by

118 | Liangyao Chen et al. Science as one of the top ten areas of scientific progress in 2003, with expectations and promising prospects of these super-natural propagation characteristics of light for application in wide areas such as the advanced optical filter, power splitters, optical antennas, stealth materials [20–30], etc. The relationship of the complex refractive index ñ (ñ = n + ik) with the complex dielectric constant ε (ε = ε1 + i ε2 ) and the permeability μ (μ = μ1 + i μ2 ) could be further discussed and analyzed in the complex plane. For metal materials, equation (3.19) can be described as ñ = |n|̃ eiθn = √εμ = √εe ,

󵄨 󵄨 εe = 󵄨󵄨󵄨εe 󵄨󵄨󵄨 eiθεe = εe1 + iεe2 ,

(3.20)

where εe is the effective complex dielectric constant, as εe1 = ε1 μ1 − ε2 μ2 ,

εe2 = ε1 μ2 + ε2 μ1 ,

󵄨 󵄨 |n|̃ = √󵄨󵄨󵄨εe 󵄨󵄨󵄨,

θn =

θεe 2

.

(3.21)

Thus, on the complex plane in Figure 3.8, the variable range for the phase angle θεe of the effective dielectric constant εe is 0 ∼ 2π , while that of the complex refractive index ñ is merely 0 ∼ π , which is limited within the upper half of the complex plane. In quadrant I, the refraction of materials is normal, i.e. n > 0 and k ≥ 0. In quadrant II, the refraction of materials is abnormal, namely n < 0 and k ≥ 0, in which case the phase angle θεe of the effective dielectric constant εe is limited in quadrant III and IV with π − 2π .

Fig. 3.8. The relationship between the effective complex dielectric constant εe (εe = |εe |eiθεe ), and the complex refractive index n(̃ ñ = √εe = |n|eiθn ) in the complex plane.

In quadrant IV, εe1 = ε1 μ1 −ε2 μ2 ≥ 0, εe2 = ε1 μ2 + ε2 μ1 ≤ 0 with the physical condition of ε2 > 0 and μ2 > 0 as mentioned above [12]. For the case of εe1 ≥ 0, thus, both ε1 and μ1 are required to have the same sign, i.e. either ε1 < 0 and μ1 < 0, or ε1 > 0 and μ1 > 0. However, the above condition is not be satisfied for the case of εe2 ≤ 0, due to positive ε2 and μ2 in metal materials to make nonvalidation of the condition for which ε1 > 0 and μ1 > 0. As εe changes continuously to the positive real axis of the complex

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plane, namely: θεe = 2π , θn = π , εe2 = ε1 μ2 + ε2 μ1 = 0, we get k = 0 and ñ = −|n|, which means the refractive index is a pure negative real number [29]. However, the condition of εe2 = ε1 μ2 + ε2 μ1 = 0 requires ε1 and μ1 to have the opposite sign, which disagrees with the condition of εe1 = ε1 μ1 − ε2 μ2 > 0, which requires ε1 and μ1 to have the same sign. Therefore, εe is not allowed in quadrant IV of the complex ̃ be allowed, plane, nor will the existence of pure negative refractive index (ñ = −|n|) despite the introduction of the complex permeability hypothesis. Further, according ̃ the reflective index R is expressed as to equation (3.10), for k = 0 and ñ = −|n|, R=

̃ 2 (1 + |n|) > 1. ̃ 2 (1 − |n|)

(3.22)

This would be contrary to the law of energy conservation for the electromagnetic fields. On the negative imaginary axis of quadrant III, θεe = 3π /2, θn = 3π /4, εe1 =0, εe2 = ε1 μ2 + ε2 μ1 < 0, which means that k = n, ε1 /ε2 = μ2 /μ1 , ε1 and μ1 are required to have the same sign, namely ε1 < 0 and μ1 < 0. For noble metals, in the Drude free-electron transition zone, E < Eg , |ε1 /ε2 | ≫ 1, and thus μ2 ≫ |μ1 |, which means that the imaginary part of the complex permeability is a large value, and the material is characteristic of strong electromagnetic absorption. As εe changes continuously to the negative real axis, θεe = π , θn = π /2, εe2 = ε1 μ2 + ε2 μ1 = 0, it is obtained that ε1 and μ1 should have opposite signs, and ñ = ik (n = 0), which means the refractive index is a pure imaginary number. In this case, according to equation (3.10), the reflective index R would be 1 + k2 R= = 1, (3.23) 1 + k2 which means that the electromagnetic waves are forbidden to propagate in the metal. Finally, although the complex refractive index ñ are mathematically allowed to be present anywhere in the upper half of the complex plane (quadrants I–II), the ones at the positive imaginary axis and the negative real axis are forbidden by the constraints of physical laws. Also, in quadrant II, ñ = |n| + ik, according to equation (3.10), the reflective index R would be R=

(1 + |n|)2 + k2 > 1. (1 − |n|)2 + k2

(3.24)

This would also be contrary to the law of energy conservation of the electromagnetic fields. In summary, therefore, ñ is restricted only to quadrant I with n > 0 and k ≥ 0. In the past ten years, the problem of the positive-negative refractive index as seen in equation (3.19) induced by Veselago has been studied in depth both theoretically and experimentally in order to understand the phenomena and laws of the supernormal-natural refraction in metal-based artificial materials, whereby most studies were tried mainly in the framework of Snell’s law. However, based on Maxwell’s equations, the refractive index, including the real part and imaginary parts,

120 | Liangyao Chen et al. not only influence the optical properties such as the refractive index and refractive angle of the metal-based interface, but also profoundly affect optical properties such as transmission T, reflection R and absorption A, and so on. In terms of the internal relationships between other electromagnetic properties and physical quantities, the law of energy conservation will require that T + R + A = 1.

(3.25)

This will make us think more of the real sense that the negative refractive index phenomena with its zero-crossing point could possibly have happened but not been observed yet in the physical spectral region. By means of detailed experimental measurement and analysis of those unique phenomena of negative refraction, including other relevant physical quantities, a deeper understanding of these phenomena will be reached and support the design and application of metal-based artificial materials in the future.

3.3.4 Measurement of the effective refractive index and refractive angle of light in metal Many experimental methods have been developed to measure the refractive index and refractive angle of light in a medium. For metal based materials, due to their strong absorption of light, the effective refractive index and its sign are determined by applying equation (3.17) with the measured real refractive angle and its sign. The experiment framework is still mainly based on Snell’s law. As mentioned above, the effective refractive index obtained from the measured data by using equation (3.17) is not the same as the one defined in optics theory, but depends strongly on the various structures of the material, and has a complicated relation with the optical constants and wavelength. The existence of complicated periodical or nonperiodical microstructures in artificially constructed metamaterials will cause multiple internal refraction and reflection, interference, scattering, diffraction, absorption, etc. to affect the light propagation behavior. For example, the periodical microstructures obtained by deep etching in oxide materials may cause apparent negative refraction with efficiency up to 87.1 % [31, 32], and this phenomenon is ascribed to the mechanism of light interference or diffraction caused by the periodical structure in artificially constructed metamaterials. The results reported by Valentine et al. show that the 21-layered stack of silver-based MgF2 fish-net structure demonstrated a positive refraction in the range of 1200–1450 nm and a negative refraction in the range of 1450–1800 nm [33]. The results by Yao et al. show that the metamaterial structure with silver wires embedded in alumina has a positive refractive index for TM mode and a negative refractive index for TE mode at 780 nm [34]. When the artificially formed metamaterial structure consists of a noble metal Ag and transparent materials with close values of refractive indices, such as magnesium fluoride (n ≈ 1.38) and alumina (n ≈ 1.6) etc., it can

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be demonstrated to have either a negative refractive index or a positive refractive index in the range of 600–1450 nm, indicating the apparent negative or positive sign of the refractive index depends strongly only on the difference of the microstructure of the metamaterials. Obviously, the abnormal refraction and propagation behavior of the light caused by the structure of metal-based artificially constructed metamaterials still lack a reasonable explanation of the physical mechanism. Thus, in order to understand the mechanism of negative refraction behavior of the light interacting with the metal-based and artificially formed metamaterial structure, it is necessary to exclude those side effects and study the light refraction behavior in the frame of Snell’s law at the simplest metal/air interface in the experiment [35]. A set of prism-like noble metal film samples were deposited by using the high purity (99.99 %) noble metal source-target and RF (radio frequency) sputtered in partial pressures of Ar+ onto the double-side-polished plane glass substrate in the Leybold600SP chamber at room temperature. During deposition, the applied voltage, current, power, and base pressure were fixed at 600 V, 10 mA, 50 W, and 7 × 10−6 mbar, respectively. The noble metal film growth rate was calibrated using the Kosaka Surfcorder ET300 and weight measurement. Sputtered noble metal films were grown at a constant thickness rate of about 0.128, 0.121 and 0.110 nm/s for Au, Ag, and Cu [36–38], respectively. To make the sample have a variable and wedge-shaped angle of θm which also acts as the incident angle at the metal side of the metal/air interface, a microstepping motor working in the vacuum chamber was used to in situ control the angle of θm by adjusting the speed of the linear movement of the mask over the sample. Three sets of noble metal film samples, including ten Au samples, seven Ag samples and seven Cu samples, were prepared with various wedge-shaped angles of θm . The angles of θm for the ten Au samples were controlled at about 15.5, 31.0, 46.5, 62.0, 77.6, 94.1, 108.5, 124.1, 139.5, and 155.0 μrad, respectively. The angles of θm for the seven Ag samples were controlled at about 14.7, 29.4, 44.1, 58.8, 74.5, 88.1, and 102.8 μrad, respectively. The angles of θm for the seven Cu samples were controlled at about 24.2, 36.3, 48.4, 60.4, 72.5, 84.6, and 96.7 μrad, respectively. The SEM measurements used to investigate the translucent area at the thinner edge of prism-like samples show that the film samples have a continuous and uniform surface composed by nanosized metal particles, as shown in Figure 3.9 [36–38]. The spectra of the complex dielectric functions of the Au, Ag and Cu film samples were measured at the incident angle of 70∘ by a spectroscopic ellipsometer in the photon energy range of 1.5–4.5 eV [15], and the results are shown in Figures 3.1–3.3. In Figure 3.2, the higher ε2 values in the interband transition region (ε2 = 6.667 at 3.8 eV) indicate that the RF-sputtered samples have a higher film density [39]. On the basis of Snell’s law, the refraction behaviors of the wedge-shaped film of the noble metals (Au, Ag, Cu) were measured. The experiment setup is shown in Figure 3.10. Since the noble metals have strong absorption of light in the visible band, the angles of the wedge-shaped films were limited less than 160 μrad. In order to minimize the laser beam divergence effect, a 20 × beam expander was used to reduce the

0

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122 | Liangyao Chen et al.

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Fig. 3.9. The SEM (a) and AFM (b) images for the Cu sample (θm = 60.4 μrad) at the thinner edge. Both show that the sample has a continuous and uniform surface with surface roughness less than ± 4 nm.

1250 mm Mirror 3 Aperture 1

Beam Expander Laser ×20 Source

nm=1 n =0 m θr

θa

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CCD Camera Screen Precision Computer Auto Mirror 1 Sample Stage

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Glass x Y x=0 x=1 mm

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x=2 mm 200 mm

Fig. 3.10. The experimental setup measuring the refraction of light which was transmitted through the wedged-shaped noble metal film sample. The CCD camera was used to record the transmitted beam spot which was imaged on the screen with the light path extended by ten multiple reflections of the light between two plane mirrors.

beam deviation down to less than 0.3 mrad. Two apertures with diameters of 2 mm were used to let the laser beam pass through. The laser beam was incident onto the glass side of the sample normal to the metal/glass interface and was out at the oblique air/metal interface. According to the experimental sketch shown in Figure 3.10, at the

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metal/air interface, the light beam is incident from metal (ñ = ñ m ) into air (ñ = 1); for the sake of simplicity, equation (3.17) is modified as sin θa = nm sin θm ,

(3.26)

where nm and θm are the effective refraction index and refraction angle at the metal side, respectively, and θa is the refraction angle at the air side. Obviously, the angle of the wedge-shaped film is just the incident angle θm . Once the refraction angle θa at the air side is measured, the effective refraction index nm of the metal will be obtained. For the small angles of θa and θm , Snell’s law will be approximated as θa ≈ nm θm , or in other words, nm ≈ θa /θm . For the incident angle and the refractive angle are very small in this experiment, the output beam transmitted through the film was multiply reflected between two plane mirrors to extend the optical path by tenfold and make the image appear on the screen. The size and shape of the beam image were measured by a CCD (charge-coupled device) camera with a space resolution of 27.4 pixels/mm on the screen. Four lasers with different wavelength, i.e. one He–Ne (λ = 632.8 nm) and three solid lasers (λ = 780 nm, 532 nm, and 473 nm), were used as the light source. In the experiment, the sample, which was mounted on a precision displacement stage controlled by a computer, was moved to the position x = 0, where the laser beam just touches the thinner edge of the wedge-shaped film. The laser beam spot on the screen placed at the far field was recorded by the CCD camera. Then the sample was moved to the position x = 2.2 mm, where the laser beam transmits fully through the film, and the laser beam spot on the screen was recorded by the CCD camera again. Measurements at each sample movement position were repeated ten times to reduce the error and enhance the data reliability. The one-dimensional beam intensity along the x-direction was obtained by integration of the beam intensity in the y-direction. The peak shift of the beam position was analyzed at the center of full-width at half maximum (FWHM) of the light intensity distribution along the x-direction. Figure 3.11 shows the measurement results for an Au sample with wedge angle 108.5 μrad at wavelength 632.8 nm. One may easily find that the beam spot on the screen at far field shifts 163 pixels to a negative direction. The attenuation of the light intensity depends on the thickness of the absorptive film. For the sample with thickness varying linearly along the x-direction, the peak of the intensity distribution of the emergent light beam will shift due to the nonuniform attenuation. However, this effect will not refract the light beam but cause an error in determining the position of the light beam spot. Therefore, in the experiment, a CCD camera was used to record the beam spot close to the sample at the initial position x = 0 mm (the laser beam just touches the thinner edge of the wedge-shaped film) and the spot position x = 2.2 mm (the laser beam transmits fully through the film). The peak position shift of the laser beam caused by the attenuation of the light intensity should be deducted from the total shift of the emerging laser beam spot to obtain the net shift of the beam spot caused by diffraction of the laser beam passing through

124 | Liangyao Chen et al.

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λ=632.8 nm 22

Intensity(a.u.)

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t=2 s L=0.2 m L=0.2 m

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Fig. 3.11. The measured light intensity distributions along the x-direction show negative light refraction as recorded with the pixel numbers by the CCD camera for the wedged Au film sample with wedge angle 108.5 μrad at wavelength 632.8 nm.

the wedged thin film. As shown in Figure 3.11, considering a 22 pixel beam spot shift caused by the attenuation of the light intensity, the net shift of the beam spot at the far field was 141 pixels, corresponding to the negative displacement of 5.15 mm along the x-direction. Beam expansion through the wedged film was observed. This is not due to microscattering effects, unlike the situations in which the light beam goes through the glass substrate and plane Au thin film, respectively, where no beam size expansion effect took place. The expansion effect can be attributed to a nonuniform distribution of the electrical field at the wedged air/metal interface with narrowed beam size. For the Au film sample with a wedge angle of 77.5 μrad, the far field light intensity distribution at λ = 473.0 nm, 532 nm, and 632.8 nm was shown in Figure 3.12. The light beam spots at far field show a negative shift at wavelengths 632.8 nm and 532 nm, but a positive shift at 473 nm. As a comparison, the measurement also was made on a wedge-shaped amorphous Si (a-Si) film sample with a wedge angle of 27.3 μrad. The result shows the pure positive shift of the peak position at the far field, corresponding to the positive angle of refraction. Figure 3.13 shows the measurement results for the Ag sample with a wedge angle of 58.8 μrad at wavelength 632.8 nm, and the position of laser beam spot demonstrates negative shifts of 194 pixels at far field and 13 pixels at the near field. As seen in the light path configuration in Figure 3.10, the light beam refraction angle, which is the angle between the emerging light beam from the metal/air interface and the incident light beam, is θr ≈ Δx/L, where Δx is the net displacement of the light spot along the x-direction with elimination of the effect of nonuniform attenuation, and L is the distance between the sample and the beam spot at the far field. The

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Intensity(a.u.)

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30 3

Au,θm=77.5 μrad Beam position x=0 λ=632.8 nm x=2.2 mm λ=632.8 nm λ=532 nm λ=473 nm Si,θi=27.3 μrad x=2.2 mm λ=632.8 nm

26 t=1 s

t=3 s

t=1 s

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Intensity,arbitrary unit(x105)

Fig. 3.12. The measured light intensity distributions along the x-direction show the positive and negative light refraction as recorded with the pixel numbers by the CCD camera for the wedged a-Si and Au film samples, respectively. The left peak shifts by 26 pixels for a-Si (θm,a-Si = 27.3 μrad, λ = 632.8 nm, t = 0.1 s) and by 30 pixels for Au (θm,Au = 77.5 μrad, λ = 473.0 nm, t = 1 s); the right peak intensity shifts by 30 pixels (λ = 532.0 nm, t = 3 s) and 124 pixels (λ = 632.8 nm, t = 1 s with the intensity amplified by ten times) for the same Au sample.

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Fig. 3.13. The intensity distribution for sample Ag (θm,Ag = 58.8 μrad, λ = 632.8 nm) along the xdirection shows the net refraction light emerging from the Ag/air interface of the film sample. The spots of the transmitted laser beam were measured by the CCD camera at both far (L = 20.55 m) and near (L = 0.2 m) field positions.

126 | Liangyao Chen et al. refraction angles θa of a light beam emerging from the Cu, Au, and Ag wedged films at the air side were determined by the relation of θa = θr − θm . The effective refraction indices were further obtained by using equation (3.26), and the results are shown in Figures 3.14–3.16. 150 473 nm refraction angle(μrad)

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Fig. 3.14. The measured results of the refraction angle θa changing with the incidence angle θm at four different wavelengths for Cu sample to show clearly that the refraction angle is positive at the shorter wavelengths λ = 473 nm and 532 nm (E = 2.62 eV and 2.33 eV), and changes to negative at longer wavelengths λ = 632.8 nm and 780 nm (E = 1.96 eV and 1.59 eV).

(λ=632.8 nm) 50 100 150 Incidence angles θm(μrad)

200

Fig. 3.15. In terms of Snell’s law, the measured net refraction for the pure air/Au interfaces show that the refractive index nm of Au changes from negative to positive, with the dashed lines giving the average values by taking linear regression of the data, i.e. nm−Au = −1.56 ± 0.18, −0.34 ± 0.10, 1.488 ± 0.11 at λ = 632.8, 532.0, and 473.0 nm, respectively.

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Fig. 3.16. The measured results of the refraction angle θa changing with the incidence angle θm at three different wavelengths (λ = 632.8 nm, 532 nm, 473 nm) for an Ag sample. The group refractive index ng is numerically calculated using the data from Palik’s handbook and this work, respectively. The inset shows the numerically calculated spectrum of the group refractive index ng , based on the Drude model.

3.3.5 Influence of variable refractive indices on light velocity The velocity of light propagating in condensed matter will have a variable value due to the light interaction with the medium. There are mainly two kinds of light velocity: the phase velocity vp charactering the phase propagation, and the group velocity vg [10, 40, 41], charactering the energy propagation. The phase velocity vp and the group velocity vg are given by c c (3.27) vp = , v g = , n ng where ng is the group refractive index, which is related to the photon frequency ω or energy E by dn dn ng = n + ω =n+E . (3.28) dω dE Various mechanisms on the refraction of light propagating at the interface of two transparent media have been extensively studied previously [40]. On the basis of Maxwell’s equations, the mechanism of light velocity change is characterized by the refractive index of the media. For most transparent materials with low dispersion or for the materials with low absorption in certain wavelength region, the variation of refractive index with wavelength is quite small, i.e. dn/dω ≈ 0, therefore, ng ≈ n, implying that there is less difference between the phase and group velocities. In equation (3.13), the measured refractive index n is regarded as to have the same value as the group refractive index ng . However, for materials with strong absorption and dispersion, the refractive index varies dramatically with frequency or wavelength, dn/dω ≠ 0 and ng ≠ n, resulting in a large difference between the phase and group velocities. The direction of light refracting from an interface indicates the direction of the energy propagation. As known from the analysis above, with the restriction of

128 | Liangyao Chen et al. physics condition, the value of the complex refractive index with n > 0 is limited in the quadrant I of the complex plane, but the group refractive index ng depends on the dispersion properties and its value may be positive or negative, which is determined by the value of dn/dω . According to equation (3.27), as long as dn/dω < 0 and |dn/dω | > n/ω , then ng < 0. For the noble metal Au, the dispersion property mainly involves two mechanisms: (1) in the Drude intraband transition region with energy less than 2.5 eV, the dominating mechanism is the transition of the free electrons in the s-band [7, 9, 39]; (2) in the spectral band with energy greater than 2.5 eV, the dominating one is due to the interband transitions. The refractive index n is connected with the real and imaginary parts of the dielectric function by n = √ (√ε12 + ε22 + ε1 ) /2.

(3.29)

In the tail of the Drude region with energy less than 2.5 eV, the real and imaginary parts of the dielectric function can be obtained from equations (3.5) and (3.6), and this gives ε1 = ε∞ − Ep2 /E2 ,

ε2 = Ep2 ℎ/E3 τ ,

(3.30)

where ε∞ , Ep , and τ are the dielectric constant at the high energy limit, plasma energy, and relaxation time, respectively. By using the Drude parameters of Cu ε∞ = 5.93, Ep = 8.82 eV, and τ = 2.6×10−14 s [6–9], the dispersion of ng is numerically calculated with the result shown in the inset of Figure 3.17 to indicate that ng can be indeed negative in the Drude region of E < Eg 8

ng

7

ng 0.00

6

-0.05

5

-0.10

4

-0.15

3

-0.20 0.5

2 1

Drudemodel

1.0 1.5 2.0 E(eV) 2.0 eV

0 -1 1.4

1.6

1.8

2.0 E(eV)

2.2

2.4

2.6

Fig. 3.17. On the basis of the Drude model (inset), the dispersion of the refractive index for a Cu sample is numerically calculated and shows that ng changes from negative to positive with the photon energy in the visible region.

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(≈ 2.1 eV), and is close to zero near Eg . The numeric calculations of ng are also made by using the complex refractive index of Cu measured in this work, with the result shown in Figure 3.17. It can be seen that ng changes from negative in the Drude region, by crossing zero at about 2.1 eV, to positive in the interband transition region with increasing photon energy in the visible region. We take the measured data of the Au film sample, ε∞ = 8.13, Ep = 9.33 eV, and τ = 2.7 × 10−14 s [39]. The spectrum of the group refractive index ng was numerically calculated from equations (3.27)–(3.30) in the 1.0–2.5 eV range, with the result shown in Figure 3.18. Numerical calculations also were made based on the data of the refractive index given in [7] as well as those measured in this chapter, with the results shown in Figure 3.18. It can be seen that ng does change from negative to positive with increasing photon energy in the visible region. The zero of ng occurs at around 1.7 eV for the calculation based on the data given in [7] and the data measured in this chapter, and at about 2.32 eV for the calculation based on the Drude model. 1.0

ng

0.05

0.5

0.00

-0.05

ng

-0.10 1.0

2.32 eV 1.5 2.0 E(eV)

2.5

0.0 Palik’s handbook This work Drude model

-0.5 1.0

1.5

E(eV)

2.0

2.5

Fig. 3.18. The group refractive index ng for an Au sample is numerically calculated by using the data from Palik’s handbook and this chapter, respectively. The inset shows the numerically calculated spectrum of the group refractive index ng , based on the Drude model.

The group refractive index ng of the Au sample changes from negative to positive with increasing photon energy, and the variation tendency of ng is similar to that of the effective refractive index nm shown in Figure 3.15, i.e. it goes from more negative nm = −1.56 (E = 1.96 eV) in the Drude region to less negative nm = −0.34 (E = 2.33 eV) at the tail of the Drude region, and passing through zero to be positive nm = 1.488 (E = 2.62 eV) in the interband transitions region. As for the Ag sample, the interband transition is greater than 4.0 eV, and the dispersion property in visible band is dominated by the intraband transitions of free electrons in the Drude region. By taking the measured data of the Ag film sample:

130 | Liangyao Chen et al. ε∞ = 2.71, Ep = 8.97 eV, and τ = 4.5 × 10−14 s [8], the spectrum of the group refractive index ng was numerically calculated from equations (3.27)–(3.30) in the 1.0–4.0 eV region and the result is shown in the inset of Figure 3.16. In the visible band, dn/dω < 0, |dn/dω | > n/ω , ng < 0, and with increasing photon energy, ng goes from more negative to less negative and further to positive with energy greater than 4.0 eV where interband transitions occur. The tendency is in agreement with the variation of effective refractive index, which changes from nm = −5.78 (λ = 632.8 nm), to nm = −2.32 (λ = 532 nm), further to nm = −1.68 (λ = 473 nm), and is expected to pass through the zero point and become positive at short wavelength in the interband transition region. Variation of the light velocity controlled by changing the refractive index in gas or solid media has been observed in some optical measurement experiments [41, 43–45], e.g. the group refractive index ng = −4000, resulting from the strong dispersion dn/dω was observed in Er-doped optical fiber [43] and ng = −23000 in Rb gas [44]. But, as shown in equation (3.7), the real part and imaginary part of optical constant satisfy strictly the Kramers–Kronig dispersion relations [9, 11], the real part n of complex refractive index varying with frequency dn/dω has changed from negative to positive by several orders, and leads to a large variation of the imaginary part k (i.e. the extinction coefficient) with frequency by the same orders, as well as a sharp increase in the variation of the absorption coefficient with frequency by the same orders according to equation (3.10). In optical principles, the dispersion spectra of n (λ ) and k (λ ) are fundamental for studying and analyzing the variation of group velocity and group refractive index. To date, the character of Kramers–Kronig dispersion relations for n(λ ) and k(λ ), which show great variation of wavelength or energy by several orders in related spectral region, there is still a lack of experimental confirmation and analysis. This is the great difficulty and challenge in the experimental study to explore the mechanism of light velocity variation with wavelength or energy controlled by a changing refractive index.

3.4 Affect of surface plasma waves on light propagation in metals According to the Maxwell equations (3.11) and (3.12), as light interacts with the free electrons in the metal and causes strong plasma resonance, only the longitudinal component of electric fields can couple with the plasma resonance. This is the condition for the excitation of plasma resonance [9, 13, 14]. Due to the strong skin effect occurring at a metal surface as light propagates through it, the excited surface plasmon will affect the light propagation properties and result in a surface plasmon polariton (SPP), which is a localized hybridized excitation formed by the interaction of light with the free electrons at the metal surface. When the surface structure of metal changes, the properties of SPP, such as the dispersion relations, excitation mode, etc., will change, and the light propagation properties are modified [46, 47].

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The energy Eps of a surface plasmon excited at a metal surface is related to the energy Ep of a bulk plasmon by [48] Eps = ℎωps =

ℎωp √1 + εd

=

Ep √1 + εd

,

(3.31)

where εd is the dielectric constant of the medium adjacent to a metal medium. At the noble-metal/dielectric interface, most of the dielectric media are transparent in the near infrared and visible regions with the refractive index nd in the 1.3 ∼ 2.5 range and εd (εd = n2d ) in the 1.7 ∼ 6.3 range, and Esp ≈ (0.37 ∼ 0.61)Ep is therefore obtained from equation (3.31). For noble metals, the energy Ep obtained from the measured data is 8.82 eV, 9.33 eV, and 8.97 eV for Cu, Au, and Ag samples, respectively [6–9], and Esp ≈ 3.2 ∼ 5.7 eV. The estimated values of the surface frequency ωsp under those conditions are in agreement with those shown in Figure 2.3 of Maier’s book [37], and thus the surface plasmon will not be observed in the near-infrared and visible regions where E < 3.1 eV [49]. The study of other metals shows similar results, and further study and detailed analysis with the application of other experimental techniques like energy loss spectra on the plasma resonance are necessary [50–52]. The dispersion relation of surface plasmon is given by [49, 53–55] ks = ko √

εd ε ε (ε + iε2 ) , = ko √ d 1 εd + ε εd + ε1 + iε2

(3.32)

where ks is the wave vector of the surface plasmon, k0 (k0 = ω /c) is the vacuum wave vector. For the simplest case to let εd = 1 at the metal/air interface, equation (3.32) can be rewritten as ε + iε2 ω ω ε ks = √ . (3.33) = √ 1 c 1+ε c 1 + ε1 + iε2 If the imaginary part of the dielectric function is omitted in the Drude region near Eg where ε1 = −1, the wave vector ks will be a pure infinitive imaginary quantity to forbid the propagation of the field along the interface. In omitting the core election polarization and the interband transition effect, i.e., assuming εb = 0 and by substituting equation (3.5) into equation (3.33) and applying the Drude parameters of Cu (Ep = 8.82 eV and τ = 2.6 × 10−14 s [8]), the dispersion relation of the photon energy E (ηω ) versus the complex surface wave vector ks (ks = ks1 + iks2 ) was obtained and plotted in Figure 3.19. It can be clearly seen that the sharp resonance peaks of the real and imaginary surface wave vectors happen at the nearly same frequency ωsp or energy Esp (ηωsp ) = Ep /√2 ≈ 6.2 eV where (1 + ε1 ) ≈ 0, but there is no surface plasma resonance occurring in the near infrared and visible Drude region where E < Esp . For the attenuated field of the surface wave which will propagate a distance d in the absorptive material, the intensity of the wave will decay to exp(−2ks2 d). The high value (∼ 4×105 cm−1 ) of the imaginary wave vector ks2 means that the surface wave along the interface can propagate only a very short distance d about 12.5 nm at the energy E = Esp .

132 | Liangyao Chen et al.

10 Esp

Ep

E(eV)

8

Ep 2

6 4 2

Real Imaginary

0 0

10

20 30 ks(104cm-1)

40

50

Fig. 3.19. The dispersion relation of the photon energy E(ω ) versus the real (solid line) and imaginary (dashed line) parts of the complex surface wave vector ks (ks = ks1 + iks2 ) in the Drude region for a Cu sample.

To consider more practically the situation in which both intraband and interband transitions occur in the region E < Ep , experimental data for the noble metals of Cu, Au, and Ag [6] are used to calculate the dispersion curves of E(ηω ) versus ks by using equation (3.22), and the result is shown in Figure 3.20. There are indeed some resonancelike features occurring at about 2.1, 2.5, and 3.9 eV for Cu, Au, and Ag, respectively. In terms of the data analysis, it can be seen clearly that these resonance-like features are actually not attributed to the surface plasma resonance but to the dip and smaller values of the dielectric functions which change dramatically in the cross region of the intraband and interband transitions near Eg as shown in Figures 3.1–3.3. The resonance-like feature of Cu gets much weaker due to the onset of interband transitions closer to the tail of the intraband transition in the cross region where the magnitude of the dielectric function of Cu is large, as seen in Figure 3.20c. When light with frequency near the plasma frequency excites plasma resonance, the reflectivity will drop sharply. Turbadar first observed the resonance-like dip of the internal reflection of the p-polarized light in the visible region for the structure in which the thin metal films were deposited on the bottom of a glass prism, and did not attribute the phenomena to the surface plasma resonance, since the feature could be well explained by the conventional film equations [56, 57]. More recently, Otto [58] and Kretschmann [59] carried out similar measurements by introducing the plasma polarization concept to explain the phenomena with resonance-like features that change with the wavelength and incidence angles. It is understood that the surface plamon resonance with frequency far less than that of nonirradiational plasma resonance can

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6 6

Ag

Au

5

4

4 Eg

E(eV)

E(eV)

5

3

3

Johnson&Christy’s work 2

0

1

2 3 ks(104cm-1) (a)

4

This work

1

Real Imaginary

0

Real Imaginary

2

This work

1

Johnson&Christy’s work

Eg

Real Imaginary

0

5

Real Imaginary

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 ks(104cm-1) (b)

6 Cu

5

E(eV)

4 3 Johnson&Christy’s work Eg

2

Real Imaginary

This work 1 0

Real Imaginary

0

1

2 3 ks(104cm-1) (c)

4

5

Fig. 3.20. In terms of the dielectric functions measured in this chapter and in Johnson and Christy’s work [6] for the noble metals, dispersion curves of E(eV) versus ks are plotted for (a) Ag, (b) Au, and (c) Cu. The resonance-like features occurred at about 2.1 eV, 2.5 eV, and 3.9 eV for Cu, Au, and Ag, respectively, are not attributed to the surface plasma resonance, but to the smaller values of the dielectric functions, which change dramatically in the cross region of the intraband and interband transitions near Eg .

only be excited by the p-polarized electromagnetic wave, which has components parallel to the wave vector of the surface plamon resonance [55]. In the intraband transition region of quasi-free electrons with low energy, the relations given by equations (3.1)–(3.12) based on the Drude model have been experimentally and theoretically well studied for a long time. The theoretical and experimental results are in good agreement with each other. However, the interpretation of the new phenomena and optical properties of metal based materials with microstructures using the principles mentioned above will face difficulty caused by the complex boundary conditions. The theory and concepts about surface plasmon resonance are often vaguely used to explain the enhancement of the optical effect in metal-based

134 | Liangyao Chen et al. materials of microstructures. Although the wave vector ks of surface plasmon can be calculated in principle by equations (3.32) and (3.33), the actually measured wave vector is ko , ks ≠ ko , the indirect analysis on the dispersion relations of surface plasmon vector ks with photon energy wavelength still strongly depends on the dispersion relations of bulk materials, and it is difficult to effectively isolate many side effects caused by the complicated structure for measurement and study. Therefore, we still need to develop new methods for our optical experiments. Through measuring the dispersion relations of resonance energy (ωps ) and the wave vector ks of the surface plasmon and its dependence on the microstructures, we can understand more about the excitation conditions and rules of the surface plasmon in complex metal based materials with microstructures and its effect on optical properties.

3.5 Conclusion For many years, to explore the mechanisms of light interacting with condensed matter, the Drude free electron gas model based on the classical Maxwell equations has provided the most detailed and successful description of the optical properties of metal materials, especially in the visible and infrared region with lower energy, and gives some theoretical expectations in accordance with experimental results. It is also the main content in this chapter to mention the fact that, whether in the past or in the future, this is still the fundamental part of the optical principle for understanding the optical properties of many metal-based materials and structures with their applications. Even though for those metal-based materials with artificially-made microstructure which have been studied for many years, there is still a need for inputting reliable optical constants of metal materials as measured by experiments into a variety of modeling, data simulation, and calculations in order to obtain better and more meaningful results [60]. In the study of complex structure, there are still many difficulties in treating boundary conditions, and the results obtained by ignoring or arbitrarily assuming optical constants of metal are of less importance and less meaningful. Unlike the transparent medium, because of the strong optical absorption of electromagnetic fields and the interaction with the large number of quasi-free electrons in metal, light penetration depth is very short in metal. The optical constants of metals are complex, and will be discussed and analyzed more clearly on a complex plane. The experiments to measure the propagation path of light in metals, including the measurement of refractive angles changing from negative to positive for light propagating across the interface of metal-based materials, are still very difficult. In this chapter, the issues of those subjects current study have been described. In metals, the collective motion of free electron gas driven by electromagnetic fields may induce plasma resonance, including surface plasmon resonance, etc., which is one of the peculiar properties of metals. In experiments, the excited bulk or surface plasmon polaritons need to be appropriately characterized, and the physical causality with the wave vec-

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tor conservation should be strictly obeyed based on Maxwell’s equations. New experimental methods are especially needed in order to effectively measure the dispersion of the surface plasma resonance energy with the wave vector and other physical parameters by exploring more deeply the mechanism induced by the microstructure of metals. Therefore, through careful research on the optical properties of metal-based materials and structures, a profound understanding of the micro-/macromechanisms and laws arising from light interaction with metals can be achieved. This will surely provide a solid foundation for the development and application of novel metal-based optoelectronic materials and devices in the modern information society.

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Haibin Zhao

4 Photo-induced spin dynamics in spintronic materials 4.1 Introduction In the field of microelectronics, keeping Moore’s law has become challenging due to the increased power assumption and heat effect before the fundamental limit of the quantum effect is reached. Meanwhile, spintronics (short for spin electronics) has become an emerging technology utilizing spin as a degree of freedom to achieve many new functionalities complementing the traditional charge-based semiconductor devices. More importantly, spintronic devices have advantages of low power assumption, fast operation speed, high integration, and nonvolatile storage. These advantages may help circumvent the heat sink bottleneck of increasing the speed and integration in the modern semiconductor integrated circuits. The practical implementation of spintronic devices relies on developing new spintronic materials and structures allowing effective control of spin transport and switching, and efficient detection of spin orientation. Thus, understanding the spin dynamics forms the basis of the spintronics. Recently, the ultrafast laser has become an important tool to investigate the spin dynamics in the time domain with femtosecond time resolution. The large nonlinear effect generated by the femtosecond laser pulses also allows probing the complicated magnetic structures and spins at the interfaces, and their dynamics. The switching of the magnetization is normally achieved by applying an external magnetic field with its direction opposite to the magnetization. Under such interaction, the nucleation occurs in a magnetic storage bit, followed by the formation and expansion of the magnetic domain, which ultimately leads to the reversal of the magnetization. This process takes about several nanoseconds or even longer, which is very slow compared to the GHz clock frequency in the state-of-the-art integrated circuits. However, if a pulsed magnetic field is applied perpendicular to the magnetization, a uniform spin precession can be generated, and the magnetization may rotate to the opposite direction. Such magnetization reversal through coherent spin rotation only needs 10–100 picoseconds. Recently, the fast reversal scheme by pulsed magnetic fields has been implemented in the magnetic random access memory (MRAM) devices. Besides the pulsed magnetic fields, many other techniques for spin manipulation have been developed in the recent years. These techniques include spin polarized current, microwave field, magneto-electric coupling, etc. Normally, the magnetization reversal time is considered to be limited by the period of spin precession, which is typically in the range of 10–1000 ps. However, in 1996, it was found that a ferromagnetic medium under the interaction of a femtosecond laser pulse exhibits a fast demagneti-

140 | Haibin Zhao zation that occurs in the subpicosecond time regime. Since then, an ultrafast spin manipulation via femtosecond laser pulse has become an important scientific issue. The time duration of ultrafast laser pulses is close to the spin exchange interaction time, and much smaller than the spin orbit coupling time (0.1–1 ps) and the magnetization precession period. Therefore, the ultrafast laser pulse can be used to investigate the spin dynamics on the time scale of elementary magnetic interaction, allowing exploration of the possibilities of magnetization reversal faster than the precession period, or even beyond the time scale of the elementary magnetic interaction. When a magnetic medium is under the interaction of ultrafast laser pulses with high peak energy, the temperature of the electrons, lattice, and spins will rapidly rise, driving the system into a highly nonequilibrium state. The system then slowly recovers to the equilibrium state by their mutual interactions and subsequent heat diffusion. However, the fundamental theory of magnetism cannot describe how the interactions between spin and orbit, spin and lattice, and electron and lattice affect the rapid change of the magnetic property driven by an ultrafast laser pulse. Therefore, it is extremely important to conduct time-resolved spin dynamics studies to understand the fundamental principles of these interactions and their influences on the spin evolution, which will in turn provide the basis to achieve efficient control of spins on ultrafast time scales. This chapter will present the progress of photo-induced spin dynamics studies in spintronic materials by time-resolved magneto-optical techniques. After the discovery of laser induced ultrafast demagnetization, many other interesting magnetic dynamics have been discovered, including laser-induced magnetization precession, ultrafast magnetization reversal, and fast magnetic phase transition. The underlying physical mechanisms related to these ultrafast magnetic dynamics are still unclear. The important mechanisms include phonon-induced spin scattering, magnon generation, modulation of magnetic anisotropy and exchange interaction, inverse Faraday effect, etc. This chapter aims to introduce and summarize the various magnetic dynamics, organized as follows: Section 4.2 – theory of magnetization dynamics; Section 4.3 – optical techniques in studies of spin dynamics; Section 4.4 – photo-induced demagnetization and magnetic phase transition; Section 4.5 – photo-induced spin precession; Section 4.6 – photo-induced spin reversal; and Section 4.7 – spin dynamics at interfaces and in antiferromagnetic materials.

4.2 Theory of magnetization dynamics 4.2.1 The Landau–Lifshitz–Gilbert (LLG) equation In the presence of a magnetic field H, m experiences a torque T = m × H that induces a precession of m around H. This precession corresponds to the periodical variation of angular momentum L = m/𝛾 under interaction of the torque. Here, 𝛾 represents

4 Photoinduced spin dynamics in spintronic materials | 141

the gyromagnetic ratio, i.e. the ratio of the total magnetic moment and total angular momentum with the spin orbit coupling. The change of angular momentum with time can be expressed as dL d m = = m × H, (4.1) dt dt 𝛾 where the magnetic field H includes the external field Hext , the magneto-crystalline and magneto-elastic anisotropy field, the demagnetization field, due to the shape anisotropy, the spin exchange field, and the dipolar field, due to the inhomogeneous distribution of spins. Thus, the total effective field inducing the torque can be expressed as Heff = Hext + Ha + Hd + Hex + Hdip . By introducing in the toque equation (4.1), a damping term which describes the motion of the magnetization towards the direction of the effective field Heff , we then obtain the Landau–Lifshitz–Gilbert equation: dm α dm [m × ], = 𝛾[m × Heff ] + (4.2) dt m dt where α is the dimensionless Gilbert damping constant. α can also be expressed as λ , where λ is the Landau–Lifshitz damping coefficient. Due to the damping α = |𝛾|m term, the magnetization is eventually aligned parallel to Heff . For precessions with a small angle between the magnetization and the equilibrium position, the LLG equation can give an analytical solution of the time evolution of the magnetization and the corresponding precession frequency. The solution can be obtained by following three steps: (1) find the magnetization direction at the equilibrium; (2) establish the torque equation for magnetization deviating from the equilibrium; (3) linearize the torque equation, and integrate the Maxwell equations, electromagnetic boundary condition, and exchange boundary condition. In the following we give an example of deriving the uniform spin precession in a ferromagnetic thin film. For the film which exhibits the in-plane fourfold and uniaxial magnetic anisotropies, and Hext parallel to the film plane, its free energy can be written as Ea =

K// 4

sin2 (2ϕ ) + Ku sin2 (ϕ − π /4) + K⊥ sin2 θ + 2π M 2 sin2 θ ,

(4.3)

where K// and Ku represent the in-plane fourfold and uniaxial anisotropy constant, respectively. K⊥ denotes out-of plane anisotropy. ϕ and θ are the angles with respect to the in-plane [001] axis for the direction of magnetization M and Hext , respectively. The directions of Hext and M with respect to the coordinate axes are illustrated in Figure 4.1. The direction of the magnetization at equilibrium is determined by −

K// Ms

sin 2ϕ cos 2ϕ +

Ku cos 2ϕ + He sin(δ − ϕ ) = 0, Ms

(4.4)

142 | Haibin Zhao

Fig. 4.1. Coordinate system used for calculating the magnetization precession frequency in a ferromagnetic thin film.

where Ms is the saturated magnetization. The torque equation for M deviating the equilibrium position can be expressed as ⇀ 󵄨󵄨 󵄨󵄨 x 󵄨󵄨 dmx ⇀ dmy ⇀ dmz ⇀ 󵄨 x+ y+ z = −𝛾 󵄨󵄨 Ms + mx 󵄨󵄨 dt dt dt 󵄨󵄨 Hx 󵄨

⇀

y my Hy

⇀

z mz Hz

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨󵄨󵄨 󵄨

(4.5)

where Hx , Hy , Hz are the effective fields along the x-, y-, and z-axes, respectively: dEa + Hext cos(δ − ϕ ) dMx dE Hy = − a + Hext sin(δ − ϕ ) dMy dE Hz = − a . dMz Hx = −

(4.6)

Rewrite Ea as Ea = K//

(Mx cos ϕ − My sin ϕ )2 × (Mx sin ϕ + My cos ϕ )2

+ Ku

Ms4 [Mx (sin ϕ − cos ϕ ) + My (cos ϕ + sin ϕ )]2 2Ms2

M2 + K⊥ z2 + 2π Mz2 . Ms

(4.7)

The torque equation becomes K 2K⊥ K// 1 dmy + sin2 (2ϕ ) + u (sin ϕ − cos ϕ )2 − Hext cos(δ − ϕ )] mz = [−4π Ms − 𝛾 dt Ms Ms Ms 2K// 2Ku 1 dmz cos(4ϕ ) + sin(2ϕ ) + Hext cos(δ − ϕ )] my =[ 𝛾 dt Ms Ms

(4.8)

For uniform spin precession, the precessing magnetization component my and mz have the forms my = my0 eiω t+φy mz = mz0 eiω t+φz

(4.9)

4 Photoinduced spin dynamics in spintronic materials |

143

Insert equation (4.9) into (4.8), and we obtain the frequency ω = 𝛾 [(Hext cos(δ − ϕ ) + H α ) (Hext cos(δ − ϕ ) + H β )] where H α = 4π Ms +

1/2

,

(4.10)

K// 2K⊥ Ku − (sin ϕ − cos ϕ )2 − sin2 (2ϕ ) Ms Ms Ms

and Hβ =

2K// 2Ku cos(4ϕ ) + sin(2ϕ ). Ms Ms

The LLG equation can also be used in the atomistic limit to calculate the evolution of the spin system using Langevin dynamics [2], which has proved to be a powerful approach to modeling ultrafast magnetization processes. However, such simulation needs to take into account the time evolution of each spin, so it is time consuming and is only suitable for systems with a limited number of spins.

4.2.2 The Landau–Lifshitz–Bloch (LLB) equation In the LLG equation, the magnitude of M is fixed such that the longitudinal relaxation of M is neglected. Therefore, the LLG equation is not suitable for describing the spin dynamics in the ultrafast time scale when the magnitude of M changes dramatically due to the large temperature variation under the ultrafast laser interaction. To describe the spin dynamics at elevated temperatures, one may use Langevin simulation or Landau–Lifshitz–Bloch (LLB) equation [3, 4]. The LLB equation combines the Landau–Lifshitz equation at low temperatures and Bloch equation at high temperatures, and it can be expressed as dm (m ⋅ H eff )m [m × [m × H eff ]] + 𝛾λ⊥ , = 𝛾[m × H eff ] − 𝛾λ|| 2 dt m m2

(4.11)

where λ|| and λ⊥ are the dimensionless longitudinal and transverse damping coeffi2T cient. When the temperature T is lower than the Curie temperature TC , λ|| = λ 3T , c

2T . Here, λ|| and λ⊥ depend on λ⊥ = λ (1 − 3TT ); when T is higher than TC , λ|| = λ⊥ = λ 3T c c the Langevin fields ζ (t) [5]. The LLB equation reflects the complex physics revealed by the atomistic model with Langevin dynamics, in particular the change of the magnitude of M during its reversal and the increasing of damping at the elevated temperatures [6]. The longitudinal and transverse relaxation time agree well with those calculated from the Langevin dynamics. Therefore, the LLB equation can be used as a basis for micromagnetic simulations at elevated temperatures. Accordingly, the spin dynamic simulations can be extended from the nanometer length scale in the atomistic model to the micrometer scale with the LLB equation.

144 | Haibin Zhao

4.3 Optical techniques in studies of spin dynamics 4.3.1 Time-resolved magneto-optical spectroscopy (1) Magneto-optical Kerr effect (MOKE) A linear polarized light undergoes a change of polarization state upon reflection from a magnetic medium. This is known as the magneto-optic Kerr effect. Magneto-optics is presently described in the context of either microscopic quantum theory [7] or macroscopic dielectric theory [8]. Microscopically, the coupling between light and electron spin occurs through the spin-orbit coupling, (∇V ×p)⋅s, which results from the interaction of the electron spins with the effective magnetic field the electron “sees” as it moves through the electric field −∇V with momentum p inside a medium. Such an interaction gives rise to offdiagonal components of the dielectric tensor, which is determined by the motion of electrons in a magnetic medium with imbalanced populations of spin-up and spindown electrons. Macroscopic descriptions of the magneto-optic effect are based on the dielectric properties of a magnetic medium, represented by a dielectric tensor: 1 ε = ε0 (−iQz iQy

iQz 1 −iQx

−iQy iQx ) , 1

(4.12)

where Q⃗ = (Qx , Qy , Qz ) is the Voigt vector. From the Maxwell’s equations, the normal modes of light propagating in such medium are left-circular polarized light and right-circular polarized light, with refraction index 1 ⃗ , nL = n (1 − Q⃗ ⋅ k) 2 (4.13) 1 ⃗ ⃗ nR = n (1 + Q ⋅ k) , 2 where n = √ε is the average refraction of index, and k⃗ is the unit vector along the 0

direction of the light propagation. A linear polarized light propagating in the medium will then decompose into left- and right-circular polarized light modes. The difference between the real parts of nL and nR results in phase shifts of the two normal modes due to their different propagating velocities, leading to a rotation of polarization plane of the light. At the same time, the difference between the imaginary parts of nL and nR causes the different absorption rates for the two normal modes, affecting the ellipticity of the light. The magneto-optic Kerr effects are categorized according to the geometry of the magnetization in relation to the plane of incidence and the film plane: (1) longitudinal geometry, in which the applied field lies in the plane of the sample and in the incident

4 Photoinduced spin dynamics in spintronic materials |

145

plane; (2) transverse geometry, in which the applied field lies in the plane of the sample but is perpendicular to the incident plane; and (3) polar geometry, in which the applied field is perpendicular to the plane of the sample and in the incident plane. For longitudinal and polar MOKE, the light polarization is altered by the magnetization, but for the transverse MOKE effect, the magnetization only alters the light intensity and the light polarization is unaffected. Table 4.1 shows the reflectivity of the p- and s-polarized light incident on the surface of a magnetic medium. Table 4.1. The Fresnel coefficients for p- and s-polarized light incident on a magnetic layer having dielectric constant of ε . The incident angle and the refracted angle in the magnetic layer is given by θ1 and θ2 . The substitutions α1 = cos θ1 and α2 = cos θ2 are used. The Voigt coefficients QP , QL , and QT are proportional to the polar, longitudinal, and transverse magnetizations, respectively. rpp

rsp (−rps )

rss

Polar

√ε α1 − α2 √ε α1 + α2

Qp √ε α1 i (√ε α1 + α2 ) (α1 + √ε α2 )

α1 − √ε α2 α1 + √ε α2

Longitudinal

√ε α1 − α2 √ε α1 + α2

QL √ε α1 tan θ2 i (√ε α1 + α2 ) (α1 + √ε α2 )

α1 − √ε α2 α1 + √ε α2

0

α1 − √ε α2 α1 + √ε α2

Transverse

√ε α1 √1 − Q2T /α22 − α2 − i√ε QT α1 tan θ2 √ε α1 √1 −

Q2T /α22

+ α2 − i√ε QT α1 tan θ2

In conventional MOKE measurement, the light is incident on the sample surface after passing through a polarizer, and the reflected light transmits an analyzer before being detected by a photodiode. The different magnetization components can be detected by rotating the polarizer and analyzer. For detection of polar and longitudinal magnetization, the polarization of the analyzer is set nearly 90 ∘ with respect to the polarized plane of the incident light; whereas the polarizations of the analyzer and polarizer are parallel to the incident plane for detection of the transverse magnetization, i.e. θa = θp = 0 ∘ . For p-polarized incident light, the ratio of reflected and incident light can be expressed as I/I0 = (A + BMt /Ms ) cos2 θa + (CMl /Ms + DMp /Ms ) sin θa cos θa ,

(4.14)

where Mt , Ml and Mp represent the transverse, longitudinal, and polar magnetizations, respectively. Ms depicts saturated magnetization. The coefficients A, B, C, and D are determined by the refractive index n = √ε , Voigt constant Q and incident angle.

146 | Haibin Zhao These coefficients can be expressed as [10, 11] 󵄨󵄨 nα − α 󵄨󵄨2 󵄨 2 󵄨󵄨 A = 󵄨󵄨󵄨 1 󵄨 󵄨󵄨 nα1 + α2 󵄨󵄨󵄨 󵄨󵄨 nα − α 󵄨󵄨2 in2 Q sin(2θ1 ) 󵄨 2 󵄨󵄨 B = 󵄨󵄨󵄨 1 + c.c.) 󵄨󵄨 ( 2 2 󵄨󵄨 nα1 + α2 󵄨󵄨 n (n cos2 θ1 − 1) + sin2 θ1 C=

in2∗ Q∗ sin(θ1 )α1 nα1 − α2 ) + c.c. ( 2∗ ∗ ∗ nα1 + α2 n α2 (n α1 + α2∗ )(α1 + n∗ α2∗ )

D=

nα1 − α2 in∗ Q∗ α1 ( ∗ ) + c.c. nα1 + α2 (n α1 + α2∗ )(α1 + n∗ α2∗ )

where α1 and α2 are defined in Table 4.1. The Voigt constant is normally much less than 1, and therefore only the linear term of Q is retained in equation (4.14). From equation (4.14), we note that the transverse MOKE remains the same for the opposite analyzer angle θa , whereas the polar and longitudinal MOKE change sign upon reversing θa . Therefore, we may separate the transverse effect from the polar and longitudinal effects by measuring the light intensity I(θa ) and I(−θa ) at the opposite analyzer angles θa and −θa , respectively. In addition, the longitudinal MOKE formula is an odd function of θa , whereas the polar one is an even function of θa ; thus we may also separate the longitudinal effect from the polar effect by measuring light intensity I(θ1 ) and I(−θ1 ) at opposite incident angles θ1 and −θ1 , respectively. (2) Time resolved magneto-optical Kerr effect (TR-MOKE) Three different configurations for TRMOKE will be discussed in this section [12]: (a) Crossed polarizer configuration In this configuration, a polarizer is placed in front of the sample to polarize the incident light to a desired orientation indicated by an angle of αP . A second polarizer, denoted as analyzer, at an angle of αA , is placed in the optical path after reflection from the sample. Using the Jones matrix, the ratio of the intensity of the reflected light to that of the incident light can be written as 󵄨󵄨 r 󵄨 I/I0 = 󵄨󵄨󵄨󵄨(sin αA cos αA ) ( ss −rsp 󵄨󵄨

󵄨2 rsp sin αP 󵄨󵄨󵄨 )( )󵄨󵄨󵄨 ; rpp cos αP 󵄨󵄨

(4.15)

if αP = 90 ∘ and αA ≪ 1, the intensity of the reflected light is I = R(αA2 − 2αA θ + θ 2 + ε 2 ),

(4.16)

where R = I0 |rss |2 , and θ ̃ = θ + iε = rsp /rss is the complex Kerr angle for s-polarized incident light. Normally, we have |θ ̃ | ≪ αA . This configuration can be used to measure the polarization angle modulation by the longitudinal and polar MOKE. It can also serve to measure the ellipticity by inserting a quarter wave plate in the optical path

4 Photoinduced spin dynamics in spintronic materials |

147

after reflection. In such case, the intensity of the reflected light is I = R(αA2 − 2αA ε + θ 2 + ε 2 ).

(4.17)

Under the excitation of pump laser pulses or other stimuli, the reflectivity R and Kerr angle θ vary with time, denoted as ΔR(t) and Δθ (t). If ΔR ≪ 1, and Δθ ≪ 1, the R θ intensity of the reflected light change by ΔI(t) = 2R0 αA Δθ (t) + αA2 ΔR(t).

(4.18)

Thus, both reflectivity and magnetization variations will affect the detected MOKE signal. The simplest way to separate the two contributions is to take measurements at opposite magnetic field directions (reversed magnetization orientations). If Δθ (t) is the odd function of M, then we obtain Δθ (t) = ((ΔI(t, H) − ΔI(t, −H))/4R0 αA . However, ΔM(t) may not change sign upon reversing M under some circumstances. The alternative way is to measure ΔI(t) at αA and −αA ; then we obtain Δθ (t) = ((ΔI(t, αA ) − ΔI(t, −αA ))/4R0 αA . If we use p-polarized incident light, i.e. αP = 0 ∘ , and αA ≈ 90 ∘ , the variation of the reflected light is similar to that using s-polarized incident light. In this case, both the longitudinal and polar magnetization modulation will contribute to Δθ (t). For αA ≈ 0 ∘ , we then obtain ΔI(t) = I0 ΔA + I0 B ⋅ ΔMT (t)/Ms . This can in principle be used to detect the variation of the transverse magnetization ΔMT (t); however, the second term may be much smaller than the first term, which hinders the measurement of ΔMT (t). (b) Balanced detection with polarization beam splitter This method may enhance the weight of polarization induced change in the detected signal and reduce the contribution from reflectivity change. Thus, it improves the detection of the magnetization dynamics in the time domain. In this configuration, a polarizer is placed in the front of the sample to polarize the incident light, a half wave plate, followed by a polarization beam splitter, is placed in the optical path after reflection. The polarization beam splitter split the light into two beams with different polarizations. The split beams are received by a balanced detector, generating a differential signal. If we choose αA = 90 ∘ , the half wave plate rotates the polarization plane by an angle of α , and the angle between the optical axis and incident plane is set at αA = 90 ∘ , we then obtain the ratio of the intensity of the reflected light to that of the incident light: 󵄨2 󵄨󵄨󵄨 rsp 1 󵄨󵄨 sin α cos α r ) ( )󵄨󵄨󵄨󵄨 ) ( ss I/I0 = 󵄨󵄨󵄨󵄨( 1 0 ) ( 0 󵄨󵄨 −rsp rpp cos α −siα 󵄨󵄨 󵄨2 󵄨󵄨 rsp 1 󵄨󵄨 sin α cos α r 󵄨 ) ( )󵄨󵄨󵄨󵄨 ) ( ss − 󵄨󵄨󵄨󵄨( 0 1 ) ( (4.19) 0 󵄨󵄨 −rsp rpp cos α − sin α 󵄨󵄨

148 | Haibin Zhao Thus, the intensity of the reflected light is I = R[4θ sin α cos α + (cos2 α − sin2 α )(θ 2 + ε 2 − 1)].

(4.20)

∘

If we rotate the half plate to α = 45 , then under the transient excitation, the light intensity varies with time as ΔI(t) = 2R0 Δθ (t) + 2θ0 ΔR(t),

(4.21)

where R0 and θ0 represent the static reflectivity and Kerr rotation, respectively. In practice, often ΔR(t)/R0 ≪ Δθ (t)/θ0 , so that ΔI(t) directly display the magneto-optical dy1−θ 2 −ε 2

namics. If we rotate the half wave plate to α = 12 arctan( 2θ0 0 ), yielding zero signal 0 in the balanced detector for the static case, so we obtain I(t) = 2R(t)[θ (t) − θ0 ]. In the limit of small response, the intensity of the reflected light change by ΔI(t) = 2R0 Δθ (t). Therefore, in this configuration, the change of reflectivity has negligible influence on the light intensity. (c) Polarization modulation Polarization modulation is achieved by using a photo-elastic modulator (PEM) which modulates the polarization of the incident light at a high frequency (typically ω ∼ 50 KHz). The modulated light is reflected from the sample and passes through an analyzer before it is received by the detector. A lock-in amplifier is used to record the signals at the fundamental modulated frequency and doubled frequency, which provide the information of the Kerr ellipicity and rotation. This method efficiently removes the contribution of reflectivity variation to the light intensity change and captures the absolute values of static Kerr angle, thus it may provide a genuine signal of magneto-optical dynamics. To effectively detect MOKE signal, the polarization of the polarizer in the incident optical path is normally set at 45 ∘ with respect to the incident plane, the optical axis of the PEM lies within the incident plane, and the polarization of the analyzer is set at a small angle αA from the incident plane. This configuration results in a reflected light intensity at the detector: 󵄨󵄨 󵄨󵄨 r I(t󸀠 ) = I0 󵄨󵄨󵄨(cos αA sin αA ) ( ss 󵄨󵄨 −r sp 󵄨

1 rsp )( rpp 0

󵄨2 1/√2 󵄨󵄨󵄨󵄨 ) ( ) 󸀠 󵄨 , 1/√2 󵄨󵄨󵄨󵄨 eiδ (t ) 0

(4.22)

where δ (t󸀠 ) = δ0 cos(ω t󸀠 ), and δ0 can be varied between 0 and π . The calculation of the above equation yields I(t󸀠 ) = R[1/2 + (θ + ραA ) cos δ (t󸀠 ) − (ε + ηαA ) sin δ (t󸀠 )] within the lowest order of αA and θ ̃ = θ + iε , where ρ̃ = ρ + iη = rpp /rss . Using expansion of cos δ (t󸀠 ) and sin δ (t󸀠 )in terms of spherical harmonics cos(nω t󸀠 ), the amplitude of the three lowest harmonics (n = 0, 1, and 2) can be written as I0 = R[1/2 + J0 (δ0 )(θ + ραA )] ≈ R/2 I1ω = J1 (δ0 )R(ε + ηαA ) I2ω = J2 (δ0 )R(θ + ραA ),

4 Photoinduced spin dynamics in spintronic materials | 149

where Jn (δ0 ) is the n-th order Bessel function. At αA = 0, the Kerr ellipticity ε and rotation θ can be obtained. Under excitation, the reflectivity and MOKE experience weak perturbations, yielding ΔI1ω (t) ΔR(t) = 2J1 (δ0 ) [Δε (t) + (ε0 + η0 αA ) + Δη (t)αA )] I0 R0

(4.23a)

ΔI2ω (t) ΔR(t) = 2J2 (δ0 ) [Δθ (t) + (θ0 + ρ0 αA ) + Δρ (t)αA )] . I0 R0

(4.23b)

At proper αA , the change of reflectivity contributed to the signal can be removed, and thus we may obtain the genuine magneto-optical dynamics.

(3) Setup of TR-MOKE Figure 4.2 shows a schematic TR-MOKE setup. The typical femtosecond laser source includes a Ti:sapphire oscillator with an 80 MHz repetition rate, a Ti:sapphire amplifier with 250 KHz and 1 KHz repetition rate. The laser beam is split into pump and probe beam by a beam splitter. The pump beam is modulated by a chopper or acoustic optical modulator (AOM) before it is incident on the sample, and the plane-polarized probe beam is incident on the sample after passing through a delay line. The reflected probe light then passes the polarization analyzer before received by the detector. A lock-in amplifier is used to record the small signal modulated by the pump beam.

Ti:Sapphire Oscillator 80 MHz 100fs Pump Beam

Probe Beam 313 Hz

57.605 cm Balance Detector Δt Lock-in Amplifier

Fig. 4.2. Schematic illustration of the TRMOKE setup.

The signal detection in the transient MOKE measurement is depicted schematically in Figure 4.3. (a) The pump beam is blocked: the probe pulses are detected with a photodiode generating current pulses with peak intensity I(0), corresponding to the orientation and modulus of the magnetization at equilibrium. The pulsed photocurrent is

150 | Haibin Zhao

Fig. 4.3. Schematic diagram of transient MOKE signal detection in TRMOKE experiments. The top and bottom panels indicate the pulsed photocurrent from the detector and signal after gated integrator and boxcar averager.

̄ by a gated integrator and boxcar averager, which elimiconverted to a DC signal I(0) nates the void signal and noise. (b) The pump beam is unblocked: the pump beam induces a change of the magnetization, altering the intensity of the time-delayed probe beam that passes through the analyzer, which is represented by I(t) at a time delay t with respect to the pump beam. (c) To measure the intensity difference ΔI(t) = I(0)−I(t) (typically much smaller than I(0)) by the lock-in amplifier, the pump beam is modulated by a chopper or AOM to cause the intensity of the probe beam to alternate at the modulated frequency. A time-resolved MOKE spectrum is then obtained by sweeping the delay time. If the signal produced by the detector has a long duration close to the time interval between two consecutive laser pulses, it can be directly routed to the lock-in amplifier without boxcar integration. The 80 MHz repetition rate of the Ti:sapphire oscillator is often larger than the bandwidth of the detector, generating a DC signal, so that the boxcar is also not needed. For TR-MOKE measurements using polarization modulation, two lock-in amplifiers are used. The detector is connected to a first lock-in amplifier with its reference channel synchronized to either the fundamental modulation frequency (F1 ∼ 50 KHz) or second harmonic frequency (F1 ∼ 100 KHz) of the PEM, corresponding to the Kerr ellipticity and rotation measurements, respectively. The first lock-in is set to a time constant T1 ≫ 1/F1 , and T1 ≪ 1/F2 , where F2 is the chopper frequency. This lock-in generates a magneto-optical signal which varies periodically due to the modulation of the pump laser pulse. The analog output of this lock-in is then connected to the input of a second lock-in, synchronized to the chopper frequency (F2 ∼ 50 Hz). Thus, the change of the magneto-optical signal due to the pump laser excitation is obtained.

4 Photoinduced spin dynamics in spintronic materials | 151

The advantage of this method is the efficient removal of impact by the pump scattering light, and achievements of absolute value of the magneto-optical signal.

4.3.2 Time-resolved magnetic second-harmonic-generation (TR-MSHG) (1) MSHG A laser beam with its electric field E(ω ) = E0 cos(ω t) is incident on a magnetic medium, generating a electric current J(2ω ) = ∇ × M(2ω ) + 𝜕P(2ω )/𝜕t at a doubled frequency 2ω . This current produces electric field E(2ω ) at the frequency of 2ω , i.e. a radiation at second harmonic frequency. Its intensity can be written as I(2ω ) = |E(2ω |2 ∝ |𝜕P2 (2ω )/𝜕t2 + ∇ × 𝜕M(2ω )/𝜕t|2 , where the nonlinear polarization P(2ω ) and magnetization M(2ω ) are (retained with the lowest order) p Q E−SHG Pi (2ω ) = χijk Ej (ω )Ek (ω ) + χijkl Ej (ω )∇k El (ω ) + χijkl Ej (ω )Ek (ω )ElDC

Mi (2ω ) =

M−SHG + χijkl Ej (ω )Ek (ω )MlDC

(4.24a)

m χijk Ej (ω )Ek (ω ).

(4.24b)

The first term on the right side of equation (4.24a) represents the contribution of the p electric dipole, where χijk is the third rank polar tensor; the second, third, and fourth terms represent the contributions of electric quadruple, static electric field, and magM−SHG netization, respectively. χijkl is a fourth-rank axial tensor. Equation (4.24b) depicts m the contributions of the magnetic dipole, where χijk is the third rank axial tensor [13, 14]. This term is often comparable to the second term in equation (4.24a), but much less than the electric dipole contribution in the first term. For a particular magneto-optical configuration, equation (4.24a) may be simplified by neglecting higher order small quantities irrelevant to the magnetization, to one third-rank tensor with different components, which are either even or odd in M, describing the crystallographic or the magnetization-induced contribution, respectively: + − Pi (2ω , ±M) = [χijk (±M) ± χijk (±M)]Ej (ω )Ek (ω ),

(4.25)

+ − represents contributions from the lattice, and χijk is originated from the magwhere χijk − netization M. χijk is an odd function of M, and it scales linearly with M. Under symme+ − try operation, χijk and χijk can be converted as + χijk = ∑ Tii󸀠 Tjj󸀠 Tkk󸀠 χi+󸀠 j󸀠 k󸀠

(4.26a)

− χijk

(4.26b)

=

− ∑ Tii󸀠 Tjj󸀠 Tkk󸀠 χi−󸀠 j󸀠 k󸀠 ,

where T is the coordinate conversion matrix under symmetry operations including reversing M. Equation (4.26a) indicates that M remains the same under symmetry operation, whereas equation (4.26b) depicts the case when M is reversed under the symmetry operation T. Figure 4.4 shows the conversion of M under some commonly used symmetry operation.

152 | Haibin Zhao

M

inversion

M

M

M

M

-M

M

-M

M

M

mirror

Two fold rotation

Fig. 4.4. The transformations of magnetization under the operations of inversion, mirror, and twofold rotation symmetries.

For a medium with inversion symmetry, its T matrix under inversion operation can be written as −1 0 0 T = ( 0 −1 0 ). 0 0 −1 Therefore, all third rank polar tensor components are identically zero, i.e. no second harmonic signal is generated in the bulk of the medium under electric dipole approximation. Because M is an axial vector (pseudo-vector), it remains unchanged under inversion operation. Thus M does not break the inversion symmetry, and it does not contribute to the second harmonic generation. However, the surface and interface break the inversion symmetry, and therefore the polar tensor components become nonzero, generating a second harmonic signal. Furthermore, M alters the symmetry at the surface and interface and forms the time inversion symmetry, thus leading to the magnetization-induced second harmonic signal. Similar to MOKE, the MSHG response depends on the geometry of magnetization in relation to the incident light and sample plane. Some nonzero third-rank polar tensor components for magnetic mediums with body center cubic (bcc) and face center cubic (fcc) symmetry are listed in Table 4.2. Here, the surface of the medium corresponds to the (001) plane, and it lies within coordinate x̂ − ŷ plane with x̂ axis parallel to [100]. The measured intensity in a fixed experimental geometry with opposite magnetization direction can, in general, be written as a sum of effective tensor components: 󵄨 + 󵄨2 − I ± (2ω ) ∝ 󵄨󵄨󵄨χeff (2ω ) ± χeff (2ω )󵄨󵄨󵄨 ,

(4.27)

+ − and χeff are the linear superposition of the multiplications of Fresnel cowhere χeff efficient αijk with even tensor components and odd tensor components, respectively,

4 Photoinduced spin dynamics in spintronic materials |

153

Table 4.2. The nonzero elements of the third-rank electric-dipole susceptibility tensor for the cubic ⃗ ⌣ ⃗ ⌣ ⃗ ⌣z) configurations. The surface structure in the longitudinal (M// x), transverse (M// y), and polar (M// ⌣ ⌣ + − is in the x − y plane. χ and χ denote the even and odd elements with the magnetization. Magnetization direction

χ−

Incident polarization

yyy yxx, yzz zyy xzx = xxz, zxx, zzz

s p s p

s s p p

zyy xzx = xxz, zxx, zzz

s p s p

s s p p

s p s p

s s p p

χ+

⇀

M//x ̂

⇀

M//ŷ

xyy xxx, xzz, zzx yzx

⇀

M//ẑ

zyy xzx = xxz, zxx, zzz

SHG polarization

denoted as + + = ∑ αijk χijk χeff i,j,k − − χeff = ∑ αijk χijk . i,j,k

The magnetic asymmetry can then be defined as A=

I+ − I− . I+ + I−

(4.28)

Because the asymmetry A is normalized to the total SH intensity, it does not depend on the intensity of the fundamental light. Thus, a large asymmetry A can be expected, since the magnitude of the odd components may be in the same order of that of even components, leading to a large nonlinear Kerr rotation which can be 3 orders of magnitude stronger than the linear one [16].

(2) TR-MSHG Similar to the TR-MOKE technique, TR-MSHG exploits an ultrafast pump beam to trigger the magnetization dynamics of a medium, which is then tracked in the time domain by a time-delayed linearly polarized probe beam. However, a special detection system is required to measure the SHG, which is much weaker than the fundamental light reflected from the sample.

154 | Haibin Zhao Several features in the TR-MSHG setup need to be highlighted which are important to accomplish such measurements. (1) A color filter is placed in front of the sample to eliminate the SH light, generated from the laser system and optics, from the fundamental light of the probe beam. Thus, only a beam with pure fundamental light is incident on the sample and interacts with it. (2) The reflected beam from the sample passes through a prism separating the propagation directions of the fundamental light and SHG generated from the sample. (3) The fundamental light needs to be blocked, and the SHG passes through another color filter eliminating the scattered fundamental light and illuminates on a photomultiplier tube. (4) To detect SHG with different polarization, a polarizer, denoted as the analyzer, is placed on a rotatable mount in the optical path after beam separation. The weak SHG signal at wavelength shorter than 600 nm is normally detected by a photomultiplier tube (PMT), whereas an avalanche photodiode is often used for longer wavelengths. The detection can be in photon counting mode or in current mode, depending on the yield of the SHG signal. To detect an extremely weak SHG signal which is even lower than the dark current of the detector, a reference beam at the same wavelength must be introduced to interfere with the SHG signal generated from the sample, thus amplifying the latter and improving the signal to noise ratio by averaging the collected data. For SHG detection using lasers with low repetition rate at 1 KHz, a gated detection system needs to be used to remove the impact of the dark current.

4.4 Photo-induced demagnetization and magnetic phase transition 4.4.1 Demagnetization in transition ferromagnetic (FM) metals (1) Experimental studies on ultrafast demagnetization Beaurepaire and coauthors reported the first work on the ultrafast demagnetization observed by TR-MOKE [17]. They used an ultrafast laser with 60 fs pulse width to excite a ferromagnetic Ni film, and found that the contrast of the magnetic hysteresis loop is significantly reduced within sub-picoseconds after laser excitation, as shown in Figure 4.5. The authors attribute this change to the fast reduction of the magnetization strength, i.e. fast demagnetization. They found that the magnetization reduces by ∼50 % when the pump laser intensity reaches 7 mJ/cm2 . To describe this fast demagnetization, the authors proposed a three temperature model in which electron temperature Te , lattice temperature Tl , and spin temperature Ts are used to describe the thermal dynamics of the each system. Due to strong electron-electron interactions, the electron system reaches its own thermal equilibrium very rapidly. The rising time of Te is approximately 260 fs. The rise of Te leads to strong spin scattering resulting in spin flip. Thus Ts increases rapidly within subpicoseconds and it reaches its maximum at about 2 ps. The rising time of Ts is shorter than the electron-lattice interaction time, so

4 Photoinduced spin dynamics in spintronic materials |

Kerr signal(a.u.)

Normalized remanence

1.0

0.9

0.8

no pump

Δt=2.3 ps

-50

0.7

0 50 H(Oe)

0.6

0.5 0

5

155

Δt(ps)

10

15

Fig. 4.5. Remanent magneto-optical contrast measured for a thin nickel film as a function of time after exciting by a 60 fs laser pulse. The results demonstrate the ultrafast loss of the magnetic order of the ferromagnetic material within a picosecond after laser excitation [17].

that the authors ascribe the direct interaction between electron and spin as the main mechanism for fast demagetization. A few years later, Koopmans and coauthors carefully examined the transient MOKE signal induced by the ultrafast laser pulses. They found that the real and imaginary parts of the Kerr angle have different profiles in the first hundreds of femtoseconds. Therefore, they point out that the magneto-optical signal in this time region is not linearly proportional to the magnitude of the magnetization. The signal is also influenced by the dichroic bleaching effect due to the energy band filling after laser pulse excitation [18]. The authors calculated the strength of the bleaching effect from the excited electron density corresponding to the pump laser pulse energy density in the experiments, and they found it is comparable to the peak value of the transient MOKE signal. The authors assume that the band filling is independent of the temperature, and the magnetization strength follows its temperature dependence in the static case; thus they infer from the transient MOKE spectra at different temperatures that the demagnetization occurs at a time scale of 0.7 ps or shorter. Koopmans et al. point out that the total angular momentum is conserved in the magnetization dynamics, so only electron spin interaction cannot lead to demagnetization. Moreover, the optical dichroic effect is insufficient to provide enough angular momentum for demagnetization. Therefore, the authors believe that the electron lattice interaction plays an important role in fast demagnetization, i.e. the angular momentum is transferred to the lattice. However, the probability of spin flip due to electron-phonon scattering is low at the equilibrium, and the corresponding spin relaxation is at a time scale of tens picoseconds; therefore, the subpicosecond demagnetization time is extraordinary.

156 | Haibin Zhao To examine the transient MOKE signal within 0.5 ps as a true signature of the magnetization dynamics, Guidoni and coauthors conducted further experiments [19]. They utilized an ultrafast laser with a 20 fs pulse width to measure the transient transmission and MOKE signal, and separated the real and imaginary part. The results show that the Kerr rotation (real part) and ellipticity (imaginary part) have an identical time dependence after 150 fs. Moreover, their relative change is one order of magnitude larger than the change of the diagonal component of the dielectric tensor. It was pointed out that the transient MOKE signal due to the bleaching effect scales with the change of the diagonal component of the dielectric function, and therefore these results indicate that the magnetization dynamics dominates the transient MOKE signal. Meanwhile, the authors suggested that the difference between the real and imaginary part within 150 fs cannot be simply ascribed to the bleaching effect, since the electron system is in a highly nonequilibrium state with its temperature rapidly rising due to the strong electron-electron scattering at such a time scale. Combining this effect and the transient MOKE spectra at delay time longer than 150 fs, the authors concluded that it is reasonable to regard the transient MOKE signal within 100 fs as a true reflection of the magnetization dynamics. Several follow-up experiments confirm the ultrafast demagnetization in the ferromagnetic transition metals. In particular, Stamm and coauthors conducted photoinduced magnetization dynamics in Ni films using time resolved x-ray magnetic circular dichroism (XMCD) [20]. They used x-ray pulses with a 100 fs time duration to probe the magnetization dynamics. They found that under femtosecond laser pulse (800 nm) excitation, the difference of XMCD signals at L3 absorption edge for right and left circular polarized x-ray pulses drops about 70 % within 120 fs. From the sum rules, this XMCD signal is proportional to S + 32 L, where S and L denote the spin angular momentum and orbital angular momentum, respectively. At equilibrium, the value of L is about 20 % of that of S; therefore, the 70 % drop of the XMCD signal indicates that the spin angular momentum is quenched within 120 fs, and the angular momentum is transferred to the lattice rather than to the orbit. Fast demagnetization on a subpicosecond time scale will lead to generation of terahertz radiation according to Maxwell’s equations. If one assumes a time-varying magnetization M(t) after coherent excitation of a magnetic medium by ultrafast laser pulses, the electric field in the far field can be expressed as E(t) =

μ0 𝜕2 M(t) (t − r/c), 4π 2 r 𝜕t2

(4.29)

where r is the distance to the magnetic dipole. This equation indicates that the transient terahertz field depends on the second derivative of the magnetization. In addition, the field direction is perpendicular to the magnetization vector. Beaurepaire et al. were the first to measure the profile of a terahertz wave generated by fast demagnetization in Ni films under ultrafast laser excitation [21]. The measured electric field profile is identical to that of the second derivative of the magnetization, and its polarization

4 Photoinduced spin dynamics in spintronic materials |

157

direction is perpendicular to the magnetization orientation, independent on the polarization of the pump laser pulse. Hilton et al. also performed similar experiments in Fe films [22]. They found that the terahertz emission contains two contributions. One part of the terahertz signal, depending on the polarization of the pump laser, is due to the nonlinear effect of optical rectification. Another part which is independent on the polarization of pump laser likely originates from the fast demagnetization. These experiments further confirmed that the demagnetization occurs on the subpicosecond time scale after coherent excitation of femtosecond laser pulses.

(2) Mechanism of ultrafast demagnetization The fact that demagnetization occurs in a very short time scale is extraordinary, since the optical dichroic absorption is not strong enough to generate large demagnetization (> 50 %) observed in the experiments. The real mechanism of fast demagnetization is not yet known. Possible mechanisms include Elliot–Yafet spin flip scattering, magnon generation, and Stoner excitation. Beaurepaire et al. were the first to propose the three-temperature model to explain the ultrafast demagnetization [17], i.e. the energy transfers between electron, spin, and lattice such that each system has its own temperature. Under laser excitation, the electron responds to the light immediately by absorbing a photon and transition to a higher energy level at a time scale of 1 fs. Following that is the collision of the electrons resulting in a thermal equilibrium at electron temperature Te , lasting about 50–500 fs. After that and also accompanied with electron- electron interaction, electrons have coupling with the lattice, leading to the rise of the lattice temperature Tl . For metals, the time scale to finish this process is about 1 ps. In addition, the authors also introduced spin temperature Ts . The spin system couples to the electron and lattice systems and transfers energy into them. Thus, the temperatures of three systems vary with time as Ce d(Te )/dt = −Gel (Te − Tl ) − Ges (Te − Ts ) + P(t), Cs d(Ts )/dt = −Ges (Ts − Te ) − Gsl (Ts − Tl ), Cl d(T l )/dt = −Gel (Tl − Te ) − Gsl (Tl − Ts ),

(4.30)

where: Ce , Cs , and Cl denote the heat capacity of the electron, spin, and lattice, respectively; Gel , Ges , and Gsl represent the coupling between electron and lattice, electron and spin, and spin and lattice; P(t) represents the energy input into the electron system from laser pulses. From simulation of the transient transmission and MOKE spectra, the authors obtain the temperature dependence of the electron, spin, and lattice, as shown in Figure 4.6. The electron can reach a very high temperature because of its small heat capacity. Moreover, because the interaction between electron and spin is very strong, the spin temperature rises more rapidly than the increase of the lattice temperature, giv-

158 | Haibin Zhao

Gel lattice

Ges

Gsl

spins

Temperature

electrons

Te

Ts Tl

0

2

4 Δt(ps)

6

Fig. 4.6. Schematic illustration of interacting reservoirs (carriers, spins, and lattice) in the threetemperature model and their temperature evolution [17].

ing rise to fast demagnetization. Apparently, this model is a phenomenological description in which only energy transfer is considered, whereas angular momentum transfer is neglected. Since the angular momentum is conserved in the dynamics of the entire system, the reduction of the spin angular momentum due to the demagnetization must be accompanied by the enhancement of the orbital or lattice angular momentum. In addition to this, a small part of the angular momentum may transfer to the photons. To understand the essence of fast demagnetization, Koopmans et al. propose that Elliot–Yafet spin flip due to scattering of electrons by phonons and defects is the main mechanism [23, 24]. In this process, the angular momentum is transferred directly from the spin to the lattice. It needs to be pointed out that the spin flip due to the electron scattering occurs only if the spin orbital coupling is involved to hybridize the spin-up and spin-down states in one electron. Although ab initio calculations show that spin-up and spin-down states are highly hybridized due to spin-orbital coupling in transition metals which favors the spin flip scattering, Koopmans et al. did not consider the electron phonon interaction matrix and the phonon dispersion in their calculations. Carva et al. then developed a model to include the interaction matrix and phonon dispersion, and calculate the probability of Elliot–Yafet spin flip in Ni under excitation of femtosecond laser pulses [25]. They found that hot and noneqilibrium electrons may enhance the spin flip probability caused by electron phonon scattering, but it cannot account for the strength of the observed fast demagnetization. Therefore, they infer that other mechanisms may also contribute to the fast demagnetization. Carpene et al. considered the electron-magnon interaction as the major mechanism for demagnetization [26]. They performed TRMOKE measurements and used probe beams with different wavelength to separate the contributions from the elec-

4 Photoinduced spin dynamics in spintronic materials |

159

trons and the lattice to the transient signal. Meanwhile, they obtained the electronmagnon interaction time τem ∼ 50–75 fs, much shorter than the electron lattice interaction time τep ∼ 240 fs. In their model, the hot electrons may excite low energy spin waves (magnon) via spin orbital coupling, thus reducing the magnetization. Here, the spin angular momentum is transferred to become the orbital angular momentum, and the latter is rapidly quenched by the crystal field. The authors pointed out that the quenching of the orbital angular momentum is extremely fast so that Stamm et al. did not observe the angular momentum transfer to the orbits using x-ray pulses with 100 fs duration time. Schmidt et al. found a strong dependence of the lifetime of the photo-excited electrons on their spin and energy from spin-, time-, energy-, and angle-resolved twophoton photoemission [27]. Ab initio calculations as well as many-body treatment indicated that the magnetic dynamics involves magnon emission. Therefore, the authors pointed out that magnon emission by hot electrons occurs on the femtosecond time scale, and thus it provides a significant source for the fast demagnetization. For spin flipping due to electron-magnon interaction, the spin angular momentum is ultimately transferred to the lattice. Although the spin-orbital interaction energy for the atoms of the transition metals is of the order of 50 meV, the spin lattice coupling energy is very small, typically ∼ 100 μeV, approximately of the order of the magneto crystalline energy. This small coupling energy is due to the quenching of the orbital angular momentum by the strong crystal fields, and thus the corresponding spin lattice relaxation time is on the time scale of several hundreds of picoseconds, much longer than the observed demagnetization time. Therefore, from the perspective of the energy scale, the fast demagnetization can be divided into two stages: (1) the spin angular momentum is transferred to atom-like orbits, resulting in magnon generation; (2) the orbital angular momentum is quenched by the crystal fields. The first process needs tens to one hundred femtoseconds, and the second process occurs much faster, and thus the entire demagnetization process is completed within 100 fs. The Stoner excitation needs much larger energy compared to the magnon generation. This is because a single spin flip significantly increases the exchange energy by ∼ 1–2 eV. Thus, the Stoner excitation has negligible contributions to the fast demagnetization. In addition to the above mechanisms, Bigot et al. suggested a coherent ultrafast demagnetization [28]. They measured the light intensity dependence of the Faraday rotation in Ni and CoPt3 using a single laser beam, and found that the magnetization is altered within the time duration of the laser pulse with high energy density. Moreover, the authors performed pump probe measurements and found that a pump laser with different polarization may lead to different transient MOKE signals within the interaction time of a laser pulse. Therefore, they pointed out that the polarized electric field of the strong laser pulse modifies the potential profile in the medium, yielding the alteration of spin-orbital coupling which alters the magnetization. It needs to be

160 | Haibin Zhao pointed out that the coherent interference between the pump and probe light within the pulse width may superpose on the real transient MOKE signal, making the interpretation of the data difficult. In a theoretical work, Zhang et al. utilized the density matrix approach to calculate the transient polarization and magnetization within the time duration of a laser pulse. They considered the coherent interaction, but neglected the heat induced effect, and found that the magnetization may have significant change within the interaction time of a laser pulse [29]. Furthermore, Battiato et al. proposed a superdiffusive spin transport as the major mechanism of ultrafast demagnetization [30]. They considered that electrons in s and p energy bands have very high mobility (1 nm/fs), and the mean free paths of the minority and majority carriers are different. Therefore, electrons excited from the d band to sp-like bands after absorption of photons become spin polarized and diffuse rapidly into the adjacent nonmagnetic layer, giving rise to fast demagnetization. In this model, the spin angular momentum is conserved without transferring to the orbit or lattice.

4.4.2 Demagnetization in other FM materials Besides Fe, Co, and Ni, Gd is also a metallic element with room temperature of FM order. Different from the 3d transition metals where the itinerant electrons dominate the magnetic moment, the FM order in Gd is originated from the localized electrons in 4f inner shell (half filled). In addition, electrons in 5d and 6sp orbits are partially spin polarized due to the exchange interaction with the spins in the inner shell, yielding a atomic magnetic moment μsat = 7μB . Gd undergoes a transition to a FM state at the Curie temperature of Tc = 295 K. Vaterlaus et al. were the first to conduct photo-induced spin dynamics in FM Gd metals. They determined its spin lattice relaxation time of ∼ 10 ps, which corresponds to the time scale required for the heat transfer from the lattice to the spin system. In the later experiments, it was confirmed that the demagnetization time (∼ 50–100 ps) of Gd is much slower than that of Fe, Co, and Ni [31–34]. However, recent experiments revealed that partial demagnetization in Gd occurs at the much faster time scale of ∼ 1 ps. Koopmans et al. developed a microscopic three-temperature model based on the Elliot–Yafet spin flip mechanism to explain why the demagnetization of Gd is slower than that of 3d transition metals [24]. The model indicates that the demagnetization speed is determined by Tc /μsat . This value for metallic Gd is 25 times smaller than that for Ni or Co, so that only a small number of spins are flipped during the electron thermalization and the following heat transfer to the lattice in Gd. As a result, only partial demagnetization occurs within the electron-lattice relaxation time (∼ 1 ps). After that, the electron temperature decreases, but the lattice temperature is still higher than the spin temperature, leading to a slower demagnetization process.

4 Photoinduced spin dynamics in spintronic materials | 161

Muller et al. systematically studied the photo-induced spin dynamics in FM materials with different polarizations at the Fermi level [35]. Their measurements indicate that the demagnetization becomes slower with increasing spin polarized density of states (DOS) at the Fermi level. In particular, half metals exhibit very long demagnetization time. For example, Ni has spin polarized DOS of about 46 %, and a demagnetization time of 100–140 fs. In comparison, CrO2 , a class I half metal, has a spin polarized DOS close to ∼100 % and a demagnetization time of 200–300 ps; Fe3 O4 , a class II half metal, has a spin polarized DOS of 100 % and a demagnetization time of 1000 ps; and La0.66 Sr0.33 MnO3 , a class III half metal, has a spin polarized DOS of more than 90 % and a demagnetization time of 400–600 ps. The slow demagnetization of half metals may be intuitively attributed to the large difference of density of states for spin-up and spin-down directions, yielding a low probability of spin flip. This half-metallic nature causes spins to be thermally insulated from the electrons; thus no fast demagnetization can occur. However, recent observations of fast demagnetization (∼1 ps) in FM GaMnAs and InMnAs with similar band structure to that of the half metals seriously challenge this description in terms of the half metallicity [36–38].

4.4.3 Ultrafast magnetization generation and FM phase transition Although most of the FM metals show a fast demagnetization under interaction by the ultrafast laser pulses, Wang et al. discovered that the FM order of magnetic semiconductor GaMnAs is quickly enhanced under ultrafast laser interaction [39]. In this experiment, GaMnAs was excited by linearly polarized laser pulses with photon energy of 3.1 eV, pulse width of 120 fs, and energy density of ∼ 10 μJ/cm2 . The Kerr rotation was found to decrease within 1 ps after pump laser interaction, indicating a fast demagnetization. Following that, the Kerr rotation gradually increases and becomes positive, reaching a maximum at ∼ 200 ps. This indicates that the magnetization becomes larger due to the enhanced FM order after laser interaction. Wang et al. suggested that the photo-generated hole carriers strengthen the FM interaction of the d electrons of Mn ions, thus leading to the photo-induced enhancement of ferromagnetism. This interpretation is consistent with the mechanism of intrinsic FM originality of GaMnAs. It is widely accepted that the spins of free holes in valence p band are coupled with the spins of d electrons of Mn ions, causing the spins if Mn ions to form an FM order. Therefore, the FM order in GaMnAs is strongly dependent on the hole density. After laser interaction, more holes are generated in the p band, enhancing the hole-induced FM coupling. As a result, both the magnetization and the Curie temperature Tc increases. At the temperature of 10 K higher than Tc , an apparent transient magnetization was also observed. Although the enhancement of ferromagnetism in GaMnAs under interaction by continuous laser was observed before the

162 | Haibin Zhao dynamic measurements [40], Wang et al. were the first to determine the time scale of the enhancement of ferromagnetism. Recently two groups independently showed a phase transition from the antiferromagnetic (AFM) order to FM order on the sub-picosecond time scale in FeRh films [41, 42]. FeRh displays an AFM phase below a phase transition temperature around 370 K. In the AFM phase, Fe has a magnetic moment of 3μB , but the spin moments of neighboring Fe atoms are antiparallel, and Rh atom has a negligible magnetic moment. Above the transition temperature, FeRh displays a FM phase, where Fe and Rh contribute ∼ 3μB and ∼ 1μB spin moments, respectively. Ju et al. studied the laser induced transient MOKE of 100 nm thick Fe45 Rh55 , and they found that a Kerr signal appears within 1 ps after laser interaction at the incident pump density above ∼ 6 mJ/cm2 . This Kerr signal increases with increasing power of the pump laser. This indicates a macroscopic magnetic moment due to the transition from AFM order to FM order in some regions where the electrons are rapidly heated up by the laser pulses. The time scale for the magnetization generation is approximately of the same order as that for fast demagnetization in 3d transition FM metals, thus indicating a significant impact of hot electrons on this process. The time for magnetization enhancement to maximum value becomes longer (∼ 30 ps) with increasing pump power. This indicates a phase transition in larger regions and a possible domain expansion due to the heating of the lattice. The phase transition is accompanied by the lattice expansion, consistent with the equilibrium state where the lattice constant of the FM phase is larger than that of the AFM phase.

4.5 Photo-induced spin precession 4.5.1 Uniform spin precession and spin wave in FM materials The interaction of ultrafast laser pulse with magnetic materials will drive coherent spin precessions as well as demagnetization. The time resolution capability of the ultrafast laser pulses enables measurement of spin precessions with a frequency up to tens of terahertz, thus providing the parameters of the Landau g factor, saturated magnetization, magnetic anisotropy, and exchange coupling energy of a magnetic material. The dynamics of spin precession in the time domain may also provide a precession damping constant, which otherwise can only be achieved indirectly from the broadening of the resonant absorption spectra in the frequency domain by ferromagnetic resonance (FMR) technique. After discovery of ultrafast demagnetization, several groups reported spin precession excitation in FM materials. For example, Kampen et al. reported observations of coherent spin precession triggered by the demagnetization after laser interaction in Ni films [43].

4 Photoinduced spin dynamics in spintronic materials |

163

Fig. 4.7. (a) Schematic pump-probe setup. The magnetization is measured by the polarization state of the reflected probe pulse. (b) Typical measurement on a 7 nm Ni film (open circles: data; thick line: fit) displaying the perpendicular component of the magnetization, Mz , as a function of delay time Δt. The different stages are indicated by numbers. (c) The stages of the excitation process: ⇀ (I) Δt < 0, the magnetization M points in equilibrium direction (dotted line), (IIa) Δt = 0, the mag⇀ nitude of M and the anisotropy change due to heating, thereby altering the equilibrium orientation, ⇀ (IIb) 0 < Δt < 10 ps, M starts to precess around its new equilibrium, (III) Δt > 10 ps, heat has dif⇀ fused away, the magnitude of M and anisotropy are restored, but the precession continues because ⇀ of the initial displacement of M [43].

Figure 4.7 shows a schematic pump-probe setup to measure photo-induced spin dynamics in a 7 nm thick polycrystalline Ni film. In the experiments, the magnetic field is applied nearly perpendicular to the film plane to form a noncollinear geometry of the applied field and magnetization. In such geometry, the fast demagnetization after laser interaction (region IIa) alters the demagnetization field: thus the direction of the overall effective field is modified. This triggers the magnetization to rotate around the new effective field. During this rotation, the magnetization gradually recovers, and the overall effective field returns to its original direction at equilibrium. This process lasts about 10 ps (region IIb) and after that the magnetization deviating from the equilibrium direction continuously precesses around the original effective field, giving rise to the damped oscillating signal in the TR-MOKE spectra (region III). This magnetization oscillation corresponds to the uniform spin precession by comparison with the FMR results. The precession frequencies and damping constants obtained from two different types of experiments are very close. This indicates the all-optical technique as an alternative approach for studying spin wave properties. Unlike the FMR technique where the resonance frequency is limited by the microwave source, the all-optical technique can cover the frequencies of 1–1000 GHz: thus the Landau g factor and demagnetization field can be easily achieved from the field dependence of the uniform precession frequency.

164 | Haibin Zhao In thicker Ni films (> 30 nm), Kampen et al. also observed the first-order standing spin wave. The frequency of the spin wave depends on the spin exchange field which is inversely proportional to the film thickness. From the thickness dependence of precession frequency, the spin wave exchange constant D = 430 meV Å2 was obtained. Later, Zhao et al. studied the in-plane spin wave excitation by applying the magnetic field within the surface plane of a single crystalline Fe film epitaxially grown on a GaAs substrate [44, 45]. Using ultrafast laser pulses to modulate in-plane anisotropy, they observed the uniform spin precession as well as the standing spin waves. Because the single crystalline Fe exhibits in-plane magnetic anisotropy, the spin precessions have different frequencies for different orientations of the applied fields, as shown in Figure 4.8.

Fig. 4.8. (a) Transient Kerr signal, and (b) Fourier transforms of (a) after picosecond excitation of a 10 nm thick epitaxial Fe film on GaAs with magnetic field H = 560 Oe applied along [110], [100], and [1-10] directions. The oscillations with lower frequencies correspond to uniform spin precessions. The oscillations with higher frequencies independent on the field direction are caused by the propagation of coherent longitudinal acoustic phonons in GaAs [44].

From the fit of the field dependent frequency using equation (4.10), the four-fold and two-fold anisotropy were obtained. In addition, fitting the damped Kerr signal oscillations with A exp(−t/τ ) sin(ω t + φ ) gives the damping time τ which is related to the Gilbert damping constant by α β 1 α 𝛾 [(Hext cos(δ − ϕ ) + H ) + (Hext cos(δ − ϕ ) + H )] = . τ 2(1 + α 2 )

(4.31)

In a 50 nm thick Fe film, Zhao et al. observed an oscillating MOKE signal with three different frequencies, corresponding to the three spin wave modes. The spin waves of the lowest, middle, and highest frequencies correspond to the uniform precession, first order, and second order standing spin waves, respectively, as indicated in Figure 4.9. From the LLG equation, the standing spin wave dispersion is written as ω = 𝛾 [(Hext cos(δ − ϕ ) + H α +

2A 2 2A 2 1/2 q ) (Hext cos(δ − ϕ ) + H β + q )] , (4.32) Ms Ms

where q = nπ /l represents the wave vector of the standing spin wave, n is an integer denoting the order of the spin wave, and l is the film thickness. Fitting the field-

4 Photoinduced spin dynamics in spintronic materials |

165

Fig. 4.9. (a) Time-resolved MOKE signal after excitation with a magnetic field H = 370 Oe applied along the [110] direction of a 50 nm thick Fe film grown on GaAs. (b) Fourier transformation of (a). (c) Uniform magnetization precession (n = 0), and first (n = 1) and second (n = 2) order standing spin wave profiles. (d) Field dependence of spin wave frequencies [45].

dependent frequency using this equation gives the spin exchange stiffness constant A = 0.8 × 10−6 erg/cm. Walowski et al. reported observations of Damon–Eshbach (DE)-type surface spin waves in very thick Ni films (d > 100 nm) [46, 47]. The DE spin waves propagate near the surface of the film and their frequencies depend on the magnetic dipolar interaction as well as the exchange field. Since the DE spin waves have wave vectors k parallel to the surface plane and |1/k| is much smaller than the diameter of the laser beam spot on the sample, the appearance of the DE spin waves on the transient MOKE signal is very surprising.

4.5.2 Spin waves in ferromagnetic materials Typical ferromagnetic materials have two sublattices such that its magnetization and angular momentum can be tuned independently. This provides a special opportunity

166 | Haibin Zhao to study laser-induced spin dynamics. For example, In the ferromagnetic compound of rare earth (RE) and 3d transition metal (TM), the magnetization of the RE sublattice (MRE ) is antiparallel to that of the TM sublattice (MTM ). MRE and MTM have different temperature dependences, so that they cancel out at the compensation temperature TM , giving rise to zero total magnetization. However, the gyromagnetic ratios of the two sublattices are different, so the compensation temperature TA at which the angular momentum cancels each other is different from TM . In ferromagnetic materials, the magnetization dynamics of the each sublattice can be described by the LLG equation. However, each equation is coupled via the exchange ex field, which can be written as HRE,TM = −λex MTM,RE , where λex represents the strength of the exchange field. Under the interaction of this exchange field, the spin precessions exhibit two modes. One corresponds to the FMR mode, which can be described by a single LLG equation. It has a precession frequency of ωFMR = 𝛾eff H eff , where 𝛾eff is the effective gyromagnetic ratio with a temperature dependence of 𝛾eff (T) =

MRE (T) − MTM (T) M(T) 󵄨 󵄨 󵄨 󵄨 = A(T) . MRE (T)/ 󵄨󵄨󵄨𝛾RE 󵄨󵄨󵄨 − MTM (T)/ 󵄨󵄨󵄨𝛾TM 󵄨󵄨󵄨

(4.33)

The Gilbert damping constant of this precession mode can be written as αeff (T) =

󵄨 󵄨2 󵄨 󵄨2 λRE / 󵄨󵄨󵄨𝛾RE 󵄨󵄨󵄨 + λTM / 󵄨󵄨󵄨𝛾TM 󵄨󵄨󵄨 A0 , 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = MRE (T)/ 󵄨󵄨𝛾RE 󵄨󵄨 − MTM (T)/ 󵄨󵄨󵄨𝛾TM 󵄨󵄨󵄨 A(T)

(4.34)

where M(T) and A(T) denote net magnetization and angular momentum, respectively. 𝛾RE and 𝛾TM represent the gyromagnetic ratios of the RE and TM sublattices, respectively. λRE and λTM represent the Landau–Lifshitz damping coefficients of the RE and TM sublattices, respectively. If assuming that λRE and λTM are independent of the temperature, A0 is a constant. Besides the FMR mode, another precession corresponds to the exchange resonance mode [48] with a precession frequency of 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 ωex = λex (󵄨󵄨󵄨𝛾TM 󵄨󵄨󵄨 MRE − 󵄨󵄨󵄨𝛾RE 󵄨󵄨󵄨 MTM ) = λex 󵄨󵄨󵄨𝛾RE 󵄨󵄨󵄨 󵄨󵄨󵄨𝛾TM 󵄨󵄨󵄨 A(T).

(4.35)

Equations (4.33) and (4.34) indicate that the frequency and the Gilbert damping constant of the FMR mode diverge at temperatures approaching to TA , and the frequency approaches to zero around TM . In contrast, the exchange resonance mode softens at temperatures approaching to TA . Stanciu et al. were the first to conduct TRMOKE to study the spin dynamics of RE-TM compounds and its temperature dependence [49]. Figure 4.10 shows the spin precession frequency and Gilbert damping constant as a function of the temperature in Gd22 Fe74.6 Co3.4 films. The experimental results show a peak for both the FMR frequency ωFMR and the effective Gilbert damping constant αeff near the angular momentum compensation temperature. This indicates that 𝛾eff is strongly dependent on the temperature, demonstrating the nonequivalent character of the gyromagnetic ratios of the two magnetic

4 Photoinduced spin dynamics in spintronic materials |

167

Fig. 4.10. (a) Temperature dependence of coherent precession of the magnetization in GdFeCo, measured at an external field Hext = 0.29 T. Around 160 K magnetic compensation TM of the ferromagnetic system occurs. The inset shows the alignment of the RE-TM system under an external applied field, below and above TM . (b) Temperature dependence of the magnetization precession frequencies ωFMR and ωex . The insets show schematically the two modes. The solid lines are a qualitative representation of the expected trend of the two resonance branches as indicated by equations (4.33) and (4.35). (c) Temperature dependence of the Gilbert damping parameter. Lines are guides to the eye [49].

sublattices in GdFeCo. The inequivalence of 𝛾RE and 𝛾TM causes the difference between TM and TA . The frequency of the exchange mode decreases when temperature approaches TA , opposite to the FMR mode, in agreement with the theoretical prediction. However, the frequency of FMR mode does not approach zero near TM . This may be due to the excitation of precessions with broad bandwidths in this highly temperaturedependent material. In addition, the frequencies of the two modes are close in this temperature region, so that they mix with each other, and the frequency of the FMR mode increases.

4.5.3 Mechanisms of spin precession excitation (1) Demagnetization and anisotropy field modulation As discussed in Section 4.4.1, demagnetization alters the shape anisotropy, thus triggering magnetization precession. Moreover, demagnetization may alter the effective

168 | Haibin Zhao magnetic field due to the magneto-crystalline anisotropy (Heff ∝ K/Ms ) to induce spin precessions. Actually, laser pulses may directly cause rapid change of the magnetocrystalline anisotropy energy to drive spin precessions. Zhang et al. were the first to reveal such a mechanism of the photo-induced spin precessions [50]. They observed pronounced magnetization precession in a half-metallic CrO2 film, which displays very weak demagnetization. However, CrO2 has a strongly temperature-dependent magneto-crystalline anisotropy. This anisotropy is significantly reduced due to the rapid rise of the electron and spin temperatures under laser interaction: thus the effective anisotropy field is largely modified, driving the rotation of the magnetization. Later, Tomimoto et al. discovered that photo-generated carriers may lead to magnetic anisotropy alteration, driving spin precession, in a FM manganese oxide film (La2−2x Sr1+2x Mn2 O7 ) [51]. For this manganite thin film, tuning of the concentration ratio of La and Sr atoms may alter its carrier density, resulting in a directional change of the magneto-crystalline anisotropy from within the film plane to perpendicular to the plane. The laser interaction may strongly modify the carrier density which causes anisotropy modulation triggering spin precession. This mechanism of carrier-induced anisotropy modulation is apparently a nonthermal effect, different from the heat effect due to the rise of the electron and spin temperatures in CrO2 . The magnetization precession excitation by the magnetic anisotropy alteration due to the nonthermal effect may also occur in magnetic garnet films. Hansteen et al. utilizes linearly polarized ultrafast laser pulses to excite magnetization precessions in Lu3−x−y Yx Biy Fe5−z Gaz O12 films and found the precession amplitude depends on the polarization direction. This anisotropy modulation originates from the transfer of electrons among Fe ions with different valences. Since the different Fe ions are located in octahedral or tetrahedral structures, the electron transfer changes the distribution of the ions, thus altering the symmetry of the material, and as a result, the magnetic anisotropy is modified [52, 53]. Scherbakov et al. revealed another nonthermal effect on magnetic anisotropy modulation in the FM semiconductor GaMnAs [54]. They fabricated a sample with GaMnAs/GaAs/Al tri-layers and directed the laser pulses on the Al surface to generate coherent acoustic phonon pulse, which propagates through the GaAs layer and reaches GaMnAs. The phonon pulse modifies the stress and the resultant magnetoelastic energy in the GaMnAs layer; thus a short effective magnetic pulse is generated to drive the magnetization precession. In this experiment, the GaAs layer is thick enough to prevent the direct laser interaction on the GaMnAs, and thus the heat effect is insulated. Ultrafast laser pulse may also alter the magnetic anisotropy in AFM materials. For example, because the spin orientation of the AFM TmFeO3 is strongly dependent on the temperature, the interaction of the laser pulses generates hot electrons and the resultant phonon excitation alters the magnetic anisotropy. As a result, the FM vector (M1 − M2 ) oscillates in the plane spanned by the new effective field HA and spin vector S1 (S2 ) [55].

4 Photoinduced spin dynamics in spintronic materials | 169

In AFM NiO, the nonthermal effect under laser interaction may affect the magnetic anisotropy. Because the magnetic anisotropy is originated from the dipolar and quadruple interactions between spins of the Ni2+ ions, the change of 3d orbital wave function due to d–d transition by laser excitation alters the magnetic anisotropy, leading to the change of the easy axis from along the [112] to the [111] direction. Duong et al. observed spin related oscillations due to such excitation in NiO by utilizing the TRMSHG technique.

(2) Exchange field modulation Ju et al. revealed the laser induced magnetization precession via the exchange biasing field modulation in FM/AFM bilayer [57]. They studied a NiFe/NiO bilayer structure in which an exchange biasing field of ∼ 100 Oe is formed due to the pinning of the FM spins by the NiO AFM spins near the interface. The laser interaction destroys the AFM order, reducing the pinning strength and the resultant exchange biasing field. Thus, the total effective magnetic field change its direction, triggering the magnetization rotation. The coherent spin precession excitations discussed above require canting of the magnetization, i.e. an angle between the magnetization and the applied field. This is because the total effective field will keep its direction for a geometry where the applied field is along the magnetic easy axis even if the magnitude of the anisotropy undergoes a change by laser interaction. In such a case, the magnetization stays along the equilibrium direction and no rotation occurs. The geometry of canted magnetization requires proper strength of the applied field. Both simulations and experiments indicate that the precession amplitude is small at low fields, and it reaches a maximum value at field approximately equal to the anisotropy field and decreases with further increasing field. Zhao et al. discovered that laser interaction may drive pronounced magnetization precession even for applied fields along the magnetic easy axis in a manganese film La0.67 Ga0.33 MnO3 (LCMO). Moreover, the precession amplitude decreases monotonically with increasing magnetic field, as indicated in Figure 4.11 [58]. This characteristic is markedly different from the behavior of the demagnetization or anisotropy driven spin precessions. The demagnetization was confirmed to be very weak at low temperatures. Although demagnetization appears at temperatures approaching to Tc , it is very slow (> 100 ps). In contrast, the precessions start immediately after the laser interaction, and the starting time has no dependence on the temperature. The FM order in LCMO depends on the double exchange interaction of the 3d electrons in the neighboring Mn ions, and this interaction is strongly correlated with the concentration ratio of the Mn3+ and Mn4+ ions. The laser interaction alters the number of the 3d electrons and their energy as well, promoting AFM interaction. Such transient exchange field becomes the driving force of the spin precession.

170 | Haibin Zhao (3) Photo-induced current and transient magnetic fields Besides the conventional method using an opto-electric switch to direct a large electric current flowing through a small circuit to generate a pulsed magnetic field for spin precession excitation, Acremann et al. found that the photo-induced current, driven by the internal electric field near the interface in the CoFe/GaAs Schottky diode, may directly trigger the magnetization precession in CoFe [59]. Recently, a similar photo-induced spin precession was observed in an Fe/GaAs Schottky diode with narrow depletion layer. The excited spin precession is very weak when the pump and probe beams are spatially overlapped, but it exhibits the largest amplitude when the probe spot is separated from the pump area by 30–50 μm. More strikingly, the precessions show opposite phases when the probe beam is moved from the left side to the right side of the pump location within the incident plane. Moreover, the precessions show opposite phases when reversing the applied field. All these features cannot be explained by the demagnetization or transient anisotropy field, and they point to the precessions excited by the photocurrent. Pump wavelength dependence of the precession amplitude shows that the fast drift of the optically excited free carriers in the narrow depletion layer of GaAs is the key mechanism to generate the significant transient magnetic field triggering spin precessions. In particular, the spin precession can be excited at a very low energy density of the pump pulse (< 1 μJ/cm2 ) in such a Schottky structure, whereas a laser pulse with energy density of ∼1 mJ/cm2 is typically required for demagnetization or anisotropy modulation in 3d transition metals. The magnetization precession excitation at such a low energy density also occurs in the FM semiconductor GaMnAs. For example, Wang et al. reported magnetization precessions driven by pump laser pulse of 0.2 μJ/cm2 [60]. Later, Hashimoto

0.9

H//Easy Axis (a)

(b)

LCMO easy axis

0.6

0.3 T

0.5 T 1.5 T

0.3 Amplitude(a.u.)

Δθ(a.u.)

0.05 T

1.2 0.8 0.4 1.2 0.8

2.5 T 0

50 100 150 Time(ps)

200

0.4 0 1 2 Magnetic Field(T)

Fig. 4.11. (a) Transient Kerr rotation Δθ of a LCMO film with magnetic field H = 0 T applied along the in-plane easy axis. (b) Amplitude of magnetization precession of 60 nm and 100 nm thick LCMO films with field applied along the easy axis at 20 K [58].

4 Photoinduced spin dynamics in spintronic materials | 171

reported precession excitation at an energy density of 3.4 μJ/cm2 [61, 62]. However, the underlying excitation mechanism is yet unknown. The possible interpretations include anisotropy modulation due to photo-induced carriers, or transient fields by laser interactions with Mn ions [63, 64].

(4) Inverse Faraday effect The thermo-dynamical potential Φ of an isotropic, nonabsorbing, magnetically ordered medium with static magnetization M in a monochromatic light field E(ω ) includes a term: Φ = αijk Ei (ω )Ej∗ (ω )Mk , (4.36) where αijk is the magneto-optical susceptibility [65–68]. In the electrical dipole approximation, the linear optical response of a medium to a field E(ω ) is defined by the optical polarization P(ω ) = 𝜕Φ /𝜕E∗ (ω ). Thus, the optical polarization P(ω ) is linearly proportional to the magnetization M: Pi (ω ) = αijk Ej (ω )Mk . Therefore, when a light is transmitted through a medium, its polarization undergoes a rotation proportional to αijk and Mk . From equation (4.36) one can find that the electric field E(ω ) acts as an effective magnetic field Hk = −𝜕Φ /𝜕Mk = αijk Ei (ω )Ej∗ (ω ). αijk is a fully antisymmetric tensor with a single independent element α . Therefore, the effective field can be written as H = α [E(ω ) × E∗ (ω )] .

(4.37)

From this equation we can see that left- and right-handed circularly polarized light produces effective magnetic fieldswith the opposite signs, and the effective field is oriented parallel to the optical wave vector. Thus, the circularly polarized light may affect the magnetization via the inverse Faraday effect, which is determined by the same susceptibility in the traditional magneto-optical Faraday effect. Kimel et al. reported spin precession excitation in magnetic materials by inverse Faraday effect for the first time [69]. They demonstrated this effect in the AFM DyFeO3 cystal, where the AFM-coupled Fe spins are slightly canted due to Dzyaloshinskii– Moriya interaction, yielding a spontaneous magnetization Ms ∼ 8 G. Although this saturated magnetization is very small, DyFeO3 has very strong spin-orbital coupling, and it displays a huge magneto-optical Faraday effect. Its Faraday angle may reach 3000 ∘ cm−1 , enabling observations of precessions with very small amplitudes. Figure 4.12 shows the time-resolved magneto-optical Faraday spectra in DyFeO3 excited by left- and right-handed circularly polarized laser pulses. We can see oscillations at ∼ 200 GHz, which correspond to the spin precessions of Fe ions. The left- and right-handed circularly polarized light excites precessions with the opposite phases. This result clearly indicates a direct coupling of the spin and photon via the inverse Faraday effect, where the angular momentum of the photon determines the precession phase. The precession amplitude displays a linear dependence on the pump laser in-

172 | Haibin Zhao

0.2

σ

σ+ σ-

+

δH+ δH

0.1

0.0 σ0

15

(b) Amplitude(arb.units)

Faraday rotation(deg.)

1.0 (a)

-

DyFeO3 T=95 K

30 45 Time delay(ps)

60

0.5

0.0 0 25 50 75 Pulse fluence(mJ/cm2)

Fig. 4.12. (a) Magnetic excitations in DyFeO3 probed by the magneto-optical Faraday effect. The circularly polarized pumps of opposite helicities excite oscillations of opposite phase. Inset shows the geometry of the experiment. Vectors δ H+ and δ H− represent the effective magnetic fields induced by right-handed σ + and left-handed σ − circularly polarized pumps, respectively. (b) The amplitude of the spin oscillations as a function of pump fluence [69].

tensity, in agreement with the expression of the inverse Faraday effect. The extrapolation of the intensity dependence shows that the photo-induced effect on the magnetization would reach the saturation value of Ms at a pump fluence of about 500 mJ/cm2 , and the effect of such a laser pulse would be equivalent to the application of a magnetic field pulse of about 5 T. Hansteen et al. observed a similar spin precession driven by the inverse Faraday effect in magnetic Garnet films [52, 53]. In their experiments, the magnetic field is applied within the plane of a Lu3−x−y Yx Biy Fe5−z GaO12 film to align the magnetization in the plane. The excitation of left- and right-handed circularly polarized light incident perpendicularly on the film generate oscillating magneto-optical Faraday signals with opposite phases. The initial phases of the precessions in the two cases indicate that the magnetization moves towards opposite directions perpendicular to the plane. This reveals that both the magnetization and the effective field are within the film plane, although they are not parallel to each other. During the presence of the laser pulses, the transient field H F along the k vector of light due to the inverse Faraday effect reaches a magnitude of 0.6 T, much stronger than both the applied field and the anisotropy field. Therefore the magnetization rotates around H F , in the plane of the film, to a new direction. After the laser pulse is gone, the magnetization precesses around the in-plane effective field, leading to an out-of plane magnetization component. The left- and right-handed circularly polarized light generates H F of opposite directions, leading to 180 ∘ phase difference for the two cases. In Lu3−x−y Yx Biy Fe5−z GaO12 , the magnetization precession excitation is also affected by the magnetic anisotropy modulation, as well as by the inverse Faraday effect. Besides the spin precessions by the inverse Faraday effect in the nearly transparent oxides, Stanciu et al. found a similar effect in metallic GdFeCo with ferromagnetic

4 Photoinduced spin dynamics in spintronic materials | 173

order [70]. Because the absorption of light in the metal may result in demagnetization and magnetization precession, a special approach was adopted to remove the heat effect. In this approach, both in-plane and out-of-plane magnetic fields were applied to create a multidomain state in GdFeCo, so that half domains have a magnetization component pointing upwards in the film plane and the other half pointing downwards. All domains have a magnetization component parallel to the applied field. Although the heat effect may drive the magnetization precession in such geometry, the magnetization of the two different domains will precess in opposite phases, thus averaging out the transient magneto-optical signal resulting from such precession. However, the magnetization precessions of different domains excited by the inverse Faraday effect have identical phases, so that they can be detected. Despite the multidomain state, the GdFeCo film may not have exactly same number and size for the two different types of domain, and thus the heat effect cannot be completely removed. The magnetization precession due to the heat effect can be obtained by adding up the precessing signals excited by the left- and right-handed circularly polarized lights, and the subtraction of the two signals represents the precession due to the inverse Faraday effect. The observed oscillations have very fast damping as a result of the averaging from the slightly different domains in this multi-domain state.

(5) Magnon and spin precession Djordjevic and Munzenberg proposed a micromagnetic model for the remagnetization process by assuming an ultrafast demagnetization process equivalent to the generation of magnons with a broad distribution of wave vectors [71]. Their micromagnetic simulations using LLG equations reveal a relaxation channel in which the energy is transferred from short wavelength magnons to long wavelength spin waves via a spinwave relaxation chain. This relaxation process occurs in a time scale of several ps, during which the spin waves also diffuse tens of nanometers from the surface into the bulk of the film, as indicated in Figure 4.13. Because of the finite bandwidth of the long wavelength spin waves, it is not straightforward to distinguish them in the time domain by TRMOKE. The above simulation assumes a spin temperature in the system much lower than the Curie temperature, and thus it is only suitable for spin dynamics at a time scale of longer than 1 ps after laser interaction. At the shorter time scale, Chubykalo-Fesenko [6] and Kazantseva [2, 72] presented the LLG equation on the atomic level. Their models incorporate Heisenberg exchange interaction and Langevin dynamics to include the effect of varied temperature on the evolution of the coupled spins.

2

Mx My Mz

t=0 ps

Mx,y,z(Z)/M0

0

t=0.7 ps

-2 t=6.7 ps -4 t=16.7 ps -6 t=125 ps

xs

-8 0

10

20 30 z(nm)

40

50

Power FFT(Mx,y,z/M0)/Power FFT(250 ps)

174 | Haibin Zhao

t =0 ps 1013 108

t =0.7 ps

103 t =6.7 ps 10

-2

10

-7

t =16.7 ps

10

t =125 ps -12

0.0

0.1

0.2 0.3 kz /2π(nm-1)

0.4

0.5

Fig. 4.13. Cut through the micromagnetic simulation for the Ni film with t = 50 nm and 55 % demagnetization. On the left: the evolution of spin wave emission from the excited area is shown in real space. On the right: the corresponding Fourier transform is shown as a function of the spatial frequency [71].

4.6 Photo-induced spin reversal 4.6.1 Spin switching and reversal in FM materials In spin dynamics, including demagnetization and spin precession exited by the laser pulses discussed in the previous chapters, the magnetization relaxes to its initial state before the interaction of the next pulse, and hence the evolution of the magnetization can be obtained by simply averaging the transient MOKE signals generated by consecutive laser pulses. However, if the magnetization permanently changes its direction without relaxation back to its original state under interaction of a single laser pulse, an external stimulus is required to recover the magnetization before the next pulse interacts with spins for the averaging measurements. Bruce et al. studied the photo-induced magnetization reversal in the CoPt multilayers with perpendicular magnetic anisotropy [73], using an ultrafast laser with 1 KHz repetition rate. To ensure that the magnetization recovers to its original state before the interaction of the each laser pulse, they used a magnetic disk rotating at a high speed (7000 rounds per minute) combined with a permanent magnet restoring the spin state. In the time scale of the studied spin dynamics (< 1 ns), the disk moves much less than 1 μm, an dthus the pump and probe light can be considered to be overlapping in the same region on the disk. The CoPt film used in this experiment has a coercive field of 0.8 T in the static measurement. However, under the interaction of a laser pulse with 0.3 mJ/cm2 , the

4 Photoinduced spin dynamics in spintronic materials |

175

magnetization is reversed at the time scale of ∼1 ns by a reversing field of 0.52 T. This indicates that the magnetization reversal may occur at the applied field significantly smaller than the static coercive field. A micromagnetic model using the LLG equation under incorporation of the longitudinal and transverse relaxation time was developed to describe the underlying physical mechanism. The simulation indicates that the demagnetization accompanied by the increasing of the electron temperature occurs in a time scale of picoseconds, and then the magnetization slowly recovers. In this process, the magnetic domains are randomly oriented, and the lattice has high temperature which lowers the magnetic anisotropy, and therefore the spins are gradually aligned parallel to the external field. The results of the simulation agree well with the experimental observations. Carpene et al. studied the photo-induced spin switching in Fe films on MgO (001) substrate with four-fold in-plane magnetic anisotropy [74]. The magnetic free energy for field H applied within the film plane can be written as E = K 1 sin(2ϕ )2 /4 − M ⋅ H, where K 1 represents the magneto-crystalline anisotropy, and ϕ denotes the angle between the magnetization and the in-plane [100] axis. We can see from this equation two nearly equivalent energy minima at directions close to the [100] and [010] axes when the external field is applied approximately along the [110] axis. In this case, the magnetization may switch from the one minimum to the other via precessing motion under the interaction of a single laser pulse. In this experiment, pulsed magnetic fields were generated to synchronize with the ultrafast laser system of 1 kHz repetition rate. To obtain the magnetization dynamics after interaction of one, two, and three laser pulses, the repetition rate was reduced to 250 Hz. At H = 130 Oe, it was confirmed that a single laser pulse (12 mJ/cm2 ) may trigger the magnetization switching from close to the [100] axis to near the [001] axis. Moreover, the interaction of two consecutive laser pulses does not affect the magnetization direction, whereas three consecutive laser pulses lead to a magnetization switching, identical to that of the interaction of one pulse. The switching time was identified to be about 100 ps from the time evolution of the longitudinal magnetization component. The trajectory of the magnetization during switching in the time domain corresponds to the precessing motion as derived from the LLG equation. The transient anisotropy field was considered to be the dominant force driving a large angle precession which results in the magnetization switching. Astakhov et al. reported that in FM GaMnAs a single laser pulse (150 mJ/cm2 ) may drive the magnetization to rotate 90 degrees [75]. Later, Reid et al. found a rapid reduction of the coercive field of GaMnAs under the interaction of the laser pulse (∼ 0.1 mJ/cm2 ) [76]. The coercive field gradually recovers within about 1.5 ns. The time dependence of the coercive field is obtained from the quasi-static hysteresis loops measured at different time intervals of the two consecutive pump laser pulses with the same energy. Before the coercive field recovers to the static value, a reversal field weaker than the static coercivity may drive the magnetization reversal.

176 | Haibin Zhao 4.6.2 Spin reversal in ferromagnetic materials (1) Spin reversal by linearly polarized light For ferromagnetic GdFeCo, its gyromagnetic ratio approaches infinity at the angular momentum compensation temperature TA , where the frequency of its magnetization precession [49] and the velocity of its domain wall motion [77] increase significantly. Therefore it can be expected that the magnetization may reverse at a very fast speed. In other words, the magnetization has negligible inertia near TA , and it may respond with a large angle rotation under the interaction of a very slight torque. Aeschlimann et al. reported at the earliest time the change of the static spin structure of GdFeCo with the temperature [78, 79]. In their experiments, a constant magnetic field was applied along the magnetization. When the temperature rises above the magnetization compensation temperature TM and the angular momentum compensation temperature TA , the relative magnitude of the magnetizations of FeCo and Gd is reversed, and thus their spin directions are flipped to keep the net magnetization along the external field. Thus a femtosecond laser pulse heating of the sample may act as an instantaneously applied field to reverse the magnetization at a very short time scale. On this basis, Stanciu et al. systematically studied the spin reversal dynamics in GdFeCo under the interaction of the femtosecond laser pulses [80]. In their experiments, an ultrafast laser of 1 KHz repetition rate at the wavelength of 800 nm was used. The detected MOKE signal at this wavelength was originated from the spins of FeCo. It was expected that spin reversal may occur for pump intensity above a threshold, and the experiments indeed showed the spin reversal of FeCo within 700 fs for pump laser intensity higher than 6.3 mJ/cm2 . If the external field is stronger than the static coercive field, the spin will cool down and switch back to its initial direction before the next pump pulse arrives. In contrast, it will stay along the opposite direction after reversal by the pump laser pulse if the field is weaker than the static coercivity.

(2) Spin reversal by circularly polarized light In Section 4.4.3 we discussed some examples of spin precessions driven by the inverse Faraday effect, and pointed out that GdFeCo is not transparent, but has such an effect. So this causes the question to arise: Can the inverse Faraday effect drive the magnetization to reverse its direction? Stanciu et al. were the first to conduct experiments to answer this question [81]. In the experiments, a magnetic alloy Gd22 Fe74.6 Co3.4 was placed under a polarizing microscope, where the dark and bright regions correspond to the magnetization “up” and “down” states, respectively. The circularly polarized laser pulses of a 1 KHz repetition rate were perpendicularly incident on the surface of the sample, generating effective magnetic fields collinear to the magnetization.

4 Photoinduced spin dynamics in spintronic materials | 177

To detect the magnetization reversal by a single femto-second laser pulse, the laser beam was rapidly swept across the sample surface to ensure that each pulse is incident on a different sample position. For regions under the interaction of the righthanded circularly polarized light, the dark positions become bright, and the bright positions remain unchanged. This indicates that the magnetization with a “down” direction is reversed, and the magnetization with an “up” direction keeps the original state, as shown in Figure 4.14. The left-handed circularly polarized light has the opposite effect on the magnetization. These experiments clearly show the transfer of the angular momentum of the photons into the magnetic recording medium and the reversal of the magnetization by a single laser pulse of 40 fs duration time without the aid of an external field.

150 μm σ+

σ-

Fig. 4.14. The effect of single 40-fs circular polarized laser pulses on the magnetic domains in Gd22 Fe74.6 Co3.4 . The domain pattern was obtained by sweeping at high-speed (∼ 50 mm/s) circularly polarized beams across the surface so that every single laser pulse landed at a different spot. The laser fluence was about 2.9 mJ/cm−2 . The small size variation of the written domains is caused by the pulse-to-pulse fluctuation of the laser intensity [81].

Although this experiment demonstrates that a single laser pulse may drive the magnetization reversal, the reversal time scale and dynamics are unclear. To clarify the magnetization dynamics, Vahaplar et al. further conducted TR-MOKE imaging measurements combined with the LLB simulations [82]. In their experiments, a single circularly polarized laser pulse (∼ 0.25 mJ/cm2 ) was used to excite a Gd24 Fe66.5 Co9.5 alloy sample, and a time-delayed linearly polarized light provided that the TR-MOKE imaging captures the magnetic domain structure. After each excitation and detection, a pulsed magnetic field was used to recover the pumped region to its initial state, so that the complete magnetization dynamic can be achieved by taking imaging at different time delays. The experimental results show that the magnetic state of the pumped region (∼ 40 μm) is completely destroyed within the first several hundred femtoseconds after the laser pulse interaction, so that no net magnetization can be detected. In the following tens of picoseconds, the magnetic state of this pumped region is gradually

178 | Haibin Zhao

(a)

Negative 1 ps delay

12 ps

58 ps

σ-

Time 91 ps delay Final state …

σ

′′up′′

+

… σ

-

… σ

′′down′′

+

… (b)

20 μm

1.0 σ-

Mz/Ms

0.5 0.0

~80% ~63%

-0.5

σ+

-1.0 0

τsw

100 200 300 τw−r Delay time(ps)

400

Fig. 4.15. (a) The magnetization evolution in Gd24 Fe66.5 Co9.5 after the excitation with right-handed (σ + ) and left-handed (σ − ) circularly polarized pulses at room temperature. The domain is initially magnetized up (white domain) and down (black domain). The last column shows the final state of the domains after a few seconds. The circles show areas actually affected by pump pulses. (b) The averaged magnetization in the switched areas (∼ 5 μm) after σ + and σ − laser pulses, as extracted from the images in (a) for the initial magnetization up [82].

recovered. Particularly the direction of the magnetization in a small portion of the region (∼ 5μm) depends on the helicity of the pump laser pulse. If excited by the righthanded circularly polarized light, its magnetization nevertheless becomes the “up” direction of its initial state, and vice versa for excitation by the left-handed circularly polarized light, as shown in Figure 4.15. This result indicates that the helicity of the pump laser pulse determines the final direction of the magnetization. In addition, the fast reversal of the magnetization does not follow the conventional route of precessional motion, since the spins are in random directions at the highly non-equilibrium state within the first several tens of picoseconds. The spin dynamics of GdFeCo under the interaction of the femtosecond laser pulse was simulated using the LLB equation. The results show that the magnetization reversal can only occur in situations with proper electron temperature and an effective pulsed magnetic field. The shortest pulsed field required for the magnetization reversal has a time duration of ∼ 250 fs and a magnitude of 20 T at the corresponding electron temperature of 1130 K. Under such condition, the spins are oriented along dif-

4 Photoinduced spin dynamics in spintronic materials |

179

ferent directions at ∼ 250 fs after the interaction of pump laser pulses of different helicities, and the magnetic medium displays a nearly complete demagnetized state. To examine the validity of the simulated results, the energy of the pump laser pulse is varied to alter the temperature of the hot electrons, and the experimental results indeed show that magnetization can only occur with a proper pump intensity, in agreement with the simulation.

(3) Ultrafast magnetization reversal and time-resolved magnetic circular dichroism The TRMOKE technique enables investigation at the subpicosecond time scale wth the dynamics of the net magnetization in ferromagnetic materials, but it cannot resolve the spin dynamics of the sublattice systems composed of different elements. Radu et al. utilized time-resolved x-ray magnetic circular dichroism (XMCD) to study the spin dynamics of Gd and Fe sublattices in GdFeCo alloy under the interaction of linearly polarized laser pulses with a duration time of 60 fs and wavelength of 800 nm [83]. The alloy was excited by pump laser pulses with 4 mJ/cm2 and detected by the x-ray pulses of 100 fs duration time. Figure 4.16 shows the results measured by the time resolved XMCD at the applied field of 0.5 T. We can see that the magnetizations of both Fe and Gd rapidly decrease and reverse their directions immediately after the interaction of the pump laser pulse, and then gradually recover to their initial states. However, the reversal of the two magnetizations occurs at markedly different times. The Fe spins are flipped at about 300 fs, whereas Gd spins take 1.5 ps to reverse. The different reversal time leads to a parallel alignment of the Fe and Gd spins within a (a)

(b)

Normalized XMCD(%)

100 Gd

Gd

Fe

Fe

50 0 -50

-100 -1

0 1 2 Pump−probe delay(ps)

3

-3

0 3 6 9 Pump−probe delay(ps)

12

Fig. 4.16. Element-resolved dynamics of the Fe and Gd magnetic moments measured by timeresolved XMCD with femtosecond time-resolution. (a) Transient dynamics of the Fe (open circles) and Gd (filled circles) magnetic moments measured within the first 3 ps. (b) As (a), but on a 12 ps timescale. The measurements were performed at a sample temperature of 83 K for an incident laser fluence of 4.4 mJ/cm−2 . The solid lines are fits according to a double exponential fit function. The dashed line in both panels depicts the magnetization of the Fe sublattice taken with the opposite sign (that is, opposite with respect to the sign of the measured Fe data) [83].

180 | Haibin Zhao (a)

(c)

(b)

(d)

M ×

M

Fig. 4.17. The magneto-optical images of a Gd24 Fe66.5 Co9.5 continuous film obtained after the action of a sequence of N 100 fs laser pulses. (a) and (b) Initial homogeneously magnetized state of the film with magnetizations “up” and “down” as represented by the circled dot and cross respectively. The light grey region represents magnetization pointing “down” and the darker grey “up”. (c) and (d) The film after an excitation with N (N = 1, 2, . . . , 5) pulses with a fluence of 2.30 mJ/cm−2 . Each laser pulse excites the same circular region of the film and reverses the magnetization within it. The scale bar on the right corresponds to 20 μm [84].

short period of time. This ferromagnetic spin alignment is extraordinary in the magnetic materials with antiferromagnetic exchange interaction. A simulation using the LLG equation based on atomic spins and Langevin dynamics gave the spin dynamics in agreement with the experimental results. Ostler et al. further studied the photo-induced spin dynamics in GdFeCo by the single pulse excitation and detection technique [84]. The TRMOKE imaging revealed net magnetization reversal by a single linearly polarized laser pulse, as shown in Figure 4.17. The XMCD measurements confirmed that both Gd and Fe spins flip their directions. Simulations indicated that the spin reversal occurs at a time scale of 1 ps, and a ferromagnetic state exists within a short period of time due to the different spin reversal times of the two sublattice.

4.7 Spin dynamics at interfaces and in antiferromagnets 4.7.1 MSHG and magnetism at interfaces According to the theory in Section 4.3, MSHG is sensitive to the magnetic properties at the surfaces and interfaces with broken inversion symmetry [15, 85]. This sensitivity was confirmed by two independent groups who detected the MSHG signals from the clean surface of a FM film [86] and buried interfaces in magnetic multilayers [87–89]. From the measured MSHG signals at opposite magnetization directions, the magnetic asymmetry A was calculated to be larger than 50 %. Later, Koopmans et al. measured a nonlinear magneto-optical Kerr rotation of ∼22 degrees in an Fe/Cr multilayer, where the linear Kerr rotation is less than 0.1 degrees.

4 Photoinduced spin dynamics in spintronic materials |

181

Most of the MSHG measurements of the magnetization at surfaces and interfaces were conducted in materials with inversion symmetry. Zhao et al. investigated the interface magnetism of Fe films grown on the GaAs substrate with no inversion symmetry [90]. A strong SHG signal may be generated from the bulk of the GaAs, so that a proper polarization combination is required to detect the slight MSHG signals under suppression of the large bulk SHG. It was found that the combination of s-polarized incident light and s-polarized reflected SHG yields high magnetic asymmetry A for nearly all directions of the in-plane magnetization. The authors found that the switching of the interface magnetization measured from MSHG is markedly different than the switching of the bulk magnetization in the Fe/GaAs (001) heterostructure. This result indicates a noncolinear alignment between the interface and bulk spins due to their different magnetic anisotropies and the weak FM exchange coupling along the perpendicular direction near the interface. To further understand this behavior, the time-resolved MSHG technique was utilized to investigate the interface spin dynamics after the laser interaction [91]. Figure 4.18 shows that the ultrafast laser may excite the coherent spin precessions at the interface as well as in the bulk, but their precession frequencies and phases are quite different. The difference in precession frequency is a direct consequence of the stronger uniaxial anisotropy at the interface than that in the bulk. Since the precession phase is strongly affected by the magnetization orientation, the different switching behaviors also explain the phase difference. This experiment demonstrates that the interfacial spins can be manipulated independently and at higher precession frequencies than that in the bulk. ω

2ω (b)

Bulk

Interface

Fe Mbulk

Δt50 ps M H eff

0

5 10 f(GHz)

Minterface y

z

H=560 Oe

AlGaAs x H

0

500 Time Delay(ps)

1000

Fig. 4.18. (a) Experimental configuration of MSHG measurements of the interface magnetization in Fe/GaAs (001). (b) Time-resolved MSHG signal after ultrafast laser excitation at a magnetic field H = 560 Oe applied along [1–10] direction. The inset shows the Fourier transformation [91].

182 | Haibin Zhao The independent spin precessions at the surface different from that in the bulk was also observed in a Ni81 Fe19 film using the time-resolved MSHG technique [92]. In this experiment, the spin precession was triggered by a pulsed magnetic field perpendicular to the magnetization. The surface spins were found to precess at a lower frequency than the spins in the bulk. This difference was ascribed to the smaller modulus of magnetization and g factor near the surface. The excitation of interface spin precessions by the femtosecond laser pulse may be the result of the fast demagnetization and anisotropy modulation. Shortly after the observation of the fast demagnetization by linear MOKE effect [17], Hohlfeld et al. studied the demagnetization in metallic Ni using time resolved MSHG [93]. They measured the SHG signals for opposite magnetization directions and obtained the evolutions of the electron temperature and the magnetization strength from the sums and differences of the two signals at the time delay longer than 0.3 ps. The curve of the magnetization versus the electron temperature obtained from the dynamic measurements is nearly overlapped with the curve of the saturated magnetization versus temperature in the static case. Therefore it was concluded that the electron and electron interactions drive the electron system into the thermal equilibrium at about 0.3 ps after femtosecond laser pulse excitation. Starting from this time, the electron temperature Te can be used to describe both the electron and the spin systems. Within the shorter time (t < 0.3 ps), the electron system is in a nonequilibrium state, where the curve of the magnetization versus electron temperature deviates from the curve in the static case.

4.7.2 Spin dynamics in antiferromagnets Compared with FM spins, AFM spins may be manipulated more effectively, because the conservation of angular momentum has much weaker influence in the AFM spin dynamics. Since the AFM materials have no net magnetization, the linear magnetooptical Kerr and Faraday effects can not be used to observe their spin dynamics. However, the nonlinear optical techniques can be employed to effectively detect the AFM spins [94–96]. Duong et al. studied the photo-induced spin dynamics in AFM NiO utilizing the time resolved MSHG technique [56]. For NiO, its AFM order may generate a SHG signal due to the magnetic dipole contribution [13, 97, 98]. Since NiO has a relatively high Neel temperature (523 K), the MSHG measurements can be easily performed at room temperature. The measurements reveal oscillations with two frequencies of ∼ 54 GHz and 108 GHz, as shown in Figure 4.19. The lower frequency is close to the magnetic anisotropy energy of 0.1 meV. The third harmonic generation (THG) signal reflecting the lattice state does not show any oscillations. The magnetic anisotropy in NiO originates from the dipolar and quadrupolar interactions between the Ni2+ spins. It is therefore easily modified by the alteration of

20

Delay(ps) 40 60

80

Change of intensity(%)

0

100

0 -5

THG~Iattice SHG ~

Amplitude(a.u.) Change of intensity(%)

4 Photoinduced spin dynamics in spintronic materials | 183

-10 spins -15 (a) (b)

(c)

2

0.0 -0.5

1

-1.0 THG

0 0.0 0.1 0.2 0.3 Frequency(meV/h)

-1.5

0 20 40 60 80100 Delay(ps)

Fig. 4.19. (a) Change of reflected SHG and THG intensities in dependence of delay t between pump beam (ℎωP = 1.55 eV) and probe beam (ℎω = 1.03 eV) at 6 K for a photonic excitation density of 1.02 × 1020 cm−3 . (b) Fourier transform of the SHG data after subtraction of the steplike decrease at t = 0. Dashed and straight lines: fitted spectral contributions and envelope. (c) THG signal from (a) [56].

the 3d orbital wave functions accompanying the excitation of d–d transitions by an optical pump pulse. This may lead to a change in the easy direction of the magnetic anisotropy from [112]̄ to [111]. These two magnetic states have a energy difference D1 = 0.1 meV. For the probe light with the frequency of ω , the SHG in NiO with populated [112]̄ and [111] states can lead to emission of coherent light waves at 2ω + D1 , 2ω , and 2ω − D1 , the interference of which would lead to quantum beats at frequencies D1 and 2D1 . Fiebig et al. further claimed that a modification of the anisotropy is able to trigger an ultrafast 90 ∘ switching of the AFM vector [99]. Although the SHG revealed the coherent oscillations correlated with the magnetic anisotropy, it does not show any oscillations of the AFM spin resonance. Since the SHG may also originate from the lattice, for example the acoustic phonons, the investigation of the ultrafast spin dynamics in antiferromagnets remains an interesting challenge for the future.

4.8 Conclusions and outlook Studies over the past decades have revealed rich spin dynamics in a variety of spintronic materials and structures under the interaction of the ultrafast laser pulses [100]. At the time scale of subpicoseconds, the materials may display demagnetization, magnetic phase transition, and magnetization reversal. At a longer time scale, magnetization precessions and spin waves may be observed. Three major mechanisms have been considered to be responsible for these excitations: (1) spin scattering induced spin flip, and alteration of magnetic anisotropy and exchange coupling due to the rapid rising of the electron temperature; (2) modification of magnetic anisotropy and exchange in-

184 | Haibin Zhao teraction due to the photoexcited carriers; (3) strong magnetic fields generated by the inverse Faraday effect. Under the interaction of ultrafast laser pulses, the magnetic medium undergoes a highly nonequilibrium state, which is beyond the scope of the conventional magnetic theory. Following that, the thermal equilibrium is approached gradually due to the interactions of spin and orbit, spin and lattice, and electron and lattice. The time scale of these interactions varies from subpicoseconds to nanoseconds, whereas the spin exchange interaction, as well as the electron and electron interaction may occur within a few femtoseconds. The photo-excited electrons possess high kinetic energy, and they exchange energy with other cold electrons to form thermal equilibrium state. A series of experiments has shown that this process is accompanied by fast demagnetization. There are even some indications of identical magnetization when the electron temperature in the dynamics process is equal to the temperature in the static case. In other words, the electron temperature determines the modulus of the magnetization. For a dielectric medium, it normally exhibits a large energy bandwidth but has no itinerant electron; thus its electron temperature rises slightly after the laser pulse interaction, corresponding to a slow demagnetization which, on the other hand, depends strongly on the spin lattice interaction time. In metals, the rapid increase of the electron temperature after laser interaction will be followed by a relatively slow rise in the lattice temperature. Both will affect the magnetic anisotropy and spin exchange interaction. The modifications of these fundamental magnetic properties may also occur via the transition of electrons between different energy levels. This process mainly exists in magnetic compounds, in particularly when their magnetism is strongly correlated with the density of the carriers caused by the doping. In magnetic mediums with a strong magneto-optical effect, the femtosecond laser pulse will generate a magnetic field pulse of a few tens of Tesla via the inverse Faraday effect. This pulsed field can drive the magnetization to precess or even reverse its direction. Since the right- and left-handed circularly polarized light generate magnetic pulses with opposite directions, the rotating trajectory of the magnetization can be controlled by the helicity of the circularly polarized light. Because the inverse Faraday effect has no heat effect, it possesses a unique advantage in the ultrafast spin manipulation at high repetition rate. Although recent studies have shown the capability of ultrafast laser pulses with a fast spin manipulation, many fundamental interaction processes are unclear. In particular, understanding the questions about through which channel and by what mechanism the angular momentum is rapidly transferred among the spin, orbit, lattice, and photons during the time of laser pulse interaction with the medium and the followed highly nonequilibrium state remains challenging. At the moment, the modification of spin-orbital coupling and spin exchange interaction was indirectly inferred from magnetic excitations such as spin precessions. Only when these problems are addressed,

4 Photoinduced spin dynamics in spintronic materials | 185

can a laser-controlled magnetism be implemented in fast spin manipulation for practical application. In the future, utilizing femtosecond x-ray pulses to study photo-induced spin dynamics will become an important research direction. This technique allows detection of element-resolved spin dynamics, and it can also resolve the spin and orbital angular momentum. In addition, x-ray beams provide a high spatial resolution of the nanometer scale. Currently, the slicing technique in the synchronized radiation facility can generate femtosecond x-ray pulses, which has already been employed in the time-resolved magnetic circular dichroism for studying the spin dynamics. Moreover, utilizing shorter optical pulses, for example attosecond pulses, will enable studies of spin dynamics at a faster time scale, during which spin-orbital interaction and spin exchange interaction dominate. In the meantime, attosecond pulse shaping will create a new degree of freedom for spin manipulation, since the interaction of the light with a medium drastically depends on the shape and duration of the laser pulse. In summary, ultrafast laser pulses can be utilized not only to investigate the fundamental principles of spin dynamics but also to manipulate the spin state at high speed. The future advancement of the compact ultrafast laser technique will potentially revolutionize the dada storage and information processing technologies.

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Kuijuan Jin, Chen Ge, Huibin Lu, Guozhen Yang

5 Research on the photoelectric effect in perovskite oxide heterostructures 5.1 Introduction The perovskite oxides, as a class of multifunctional materials, have a very long history of being studied. A large range of physical properties have been discovered in these materials, such as ferroelectricity, colossal magnetoresistance, and high temperature superconductivity. The lattice, charge, spin, and orbital degrees of freedom couple with each other in the perovskite oxides, so that novel phenomena are very likely to emerge. The strain and charge transfer at the interface of the heterostructures can lead to new phases or electronic states, which adds more degrees of freedom in the system. The size effect can also play an important role in the properties of the heterostructures. Thus, a richer set of novel phenomena may be found at the interfaces. The heterostructure has been used as a very basic structure of electronic devices. The underlying physics and possible applications have attracted scientists, and much progress has recently been made in this field. The irradiation of lasers which can produce photocarriers is one of the effective methods to modulate the physical properties in perovskite oxide heterostructures. Recently, research on the photo-induced effect in perovskite oxide heterostructures has emerged as one of the focus topics in solid state physics with the development of the oxide film fabrication technique and lasers. Many novel phenomena and intriguing functionality have been explored in these systems. Thus, experimental and theoretical investigations into the photoelectric effect in perovskite oxide heterstructures are of particular interest for both physics and engineering. Photoelectric effects can be classified as lateral and longitudinal photoelectric effects by the direction of the photovoltage relative to the heterostructure, as shown in Figure 5.1. In this chapter, recent progress in the field of the photoelectric effect in perokvkite oxide heterostructures will be introduced. The text is organized as follows. In Section 5.1, we provide background on the basic properties in the perovskite oxides, including lattice structure, electron structure, and photoelectric properties. After presenting the main growth technique of the perovskite oxide thin films, we introduce the recent advances in the longitudinal photoelectric effects and the lateral photoelectric effects in the perovskite oxide heterostructures in Sections 5.3 and 5.4, respectively. A summary and a future outlook will be presented in the last section.

192 | Kuijuan Jin et al.

Fig. 5.1. Shematic diagram of the measurement of lateral and longitudinal photoelectric effects.

5.2 Perovskite oxide 5.2.1 Crystal structure The perovskite is composed of three elements with a chemical formula ABO3 , in which A and B are cations and O is an oxygen anion. As Figure 5.2 shows [29], the A-site ion is in the corners of the cube, while the B-site ion in the body center and the O ions in the face centers form a octahedron with the B-site ion in the center. The perovskite bulk is built up of a stacking of such structures. Most of the perovskite materials have a deviation to the ideal cubic structure [108]. The mismatch can be measured by the tolerance factor, which is a function of the lengths of the A–O and B–O bonds [116]: rA + rO t= , (5.1) √2(rB + rO ) in which, rA , rB , and rO are the radii of the A-site ion, the B-site ion, and the O ion, respectively. Generally, the material is stable if 0.89 < t < 1.02 [31].

Fig. 5.2. The ideal perovskite crystal structure. The largest atoms at the corners, the medium-sized atom in the body center, and the smallest atoms in the face centers are the A cations, B cation, and the O anions, respectively [29].

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5.2.2 Electron structure In the perovskite structure, the B cations at the center of the oxygen octahedron are quite often transitional metal elements with d orbitals. The electronic structure depends crucially on the octahedral environment. The d orbitals fall into two groups, the t2g orbitals and the eg orbitals. The t2g orbitals point between the x-, y-, and zaxes, namely the dxy , dyz and the dxz orbitals, and the eg orbitals point along the axes, namely the dx2 −y2 and the dz2 orbitals, as Figure 5.3 shows. Because the 2p orbitals of the oxygen ions point along the axes, the t2g orbitals have lower overlap with the neighboring 2p orbitals of the oxygen ions, leading to a lower energy of the t2g orbitals because the Coulomb energy is lower. Here we take the 3d orbitals of the Mn ion in the LaMnO3 as an example. The five orbitals split into two groups, the t2g group at lower energy and the eg orbitals at higher energy. Due to the spin-spin interactions between the electrons, the electrons do not always fill the lower t2g levels before they fill the eg levels. The energy cost of putting two electrons in the same orbital is called paring energy. If this energy is higher than that of the difference of the two levels, the electrons will first occupy each energy levels before double filling any orbital if the pairing energy is higher than the difference of the orbital energies. z

x

3d orbital

Oxygen Transition metal y Perovskite Perov structure

x2 - y

2

3z2-r2

eg t2g

xy

yz

zx

Fig. 5.3. The eg -t2g split of the 3d orbitals in an octahedron structure [106].

Another important phenomenon that affects the electron structure of the perovskite oxide is the Jahn–Teller effect in the octahedra with the center of the so-called Jahn– Teller ions such as Mn3+ . For Mn3+ ions with four 3d orbitals, only one eg orbital is occupied. Hence, if the two eg levels degenerate with the average energy of the two levels unchanged, only the lower level will be occupied , which leads to a lower total energy. Since this is energy favorable, the octahedron is distorted spontaneously. The change in the structure results in the split of the t2g levels. The dxy level is raised, and dyz and the dxz levels are lowered, while the total energy keeps exactly the same. The distortion of the octaheron is called the Jahn–Teller distortion. This happens to the Mn3+ ions, since the eg orbitals are partially filled. As for the Mn4+ ions, there is no

194 | Kuijuan Jin et al. Jahn–Teller distortion. This example demonstrates that in the perovskite structure, it is possible to tune the electron structure by the crystal structure, and vice versa.

5.2.2.1 Cation ordering, charge ordering, and magnetic ordering In perovskite structures, the A-site or B-site ion can be partially replaced with other elements without changing the overall crystal structures. Here we take the La1−x Srx MnO3 (LSMO) as an example. Because of the difference in the valences and the sizes of the La3+ ions and the Sr2+ ions, the A-site cations align in an order such that La3+ and Sr2+ are aligned layer by layer when x is 0.5. This phenomenon is called the cation ordering. The Mn ion in LaMnO3 has the valence of +3. Since there is one electron occupying the eg level on each Mn site, the main exchange interaction between the nearest Mn sites is the superexchange, which results in an antiferromagntic order in the structure. However, if some La ions are substituted with Sr ions, the Mn ions can have a different valence of +4. As shown in Figure 5.4, the Mn3+ ion has one electrons on the eg orbitals. There is an oxygen anion which has electrons on the 2p orbitals overlapping with the 3d orbitals of the neighboring Mn4+ and Mn3+ ions. The electrons on the eg orbitals can hop to the unoccupied eg orbitals by indirect exchanging with the electrons on the 2p orbitals of the O-site. If the spins of the 3d electrons in the two Mn ions are parallel, this is energetically favorable, because the hopping can extend the wavefunction of the electrons, which lowers the Coulomb energy. Consequently, at room temperature when LaMnO3 is doped with Sr up to a level of x = 0.175, the dominant exchange mechanism is the double exchange, which results in ferromagnetic alignment of the 3d electrons on the Mn sites. The hopping among the Mn sites can be viewed as the

eg S=1/2 ~1 eV Mn3+(3d4) eg

t2g S=3/2

ferromagnetic interaction S=2

hole doping

t2g LaMnO3 Mott−Insulator(antiferromagnet)

La1-xSrxMnO3 Ferromagnetic Metal

Fig. 5.4. The superexchange interaction and double exchange interaction. [101]

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hopping of the holes in the unoccupied eg level. The possibility of the electrons hopping through the crystal makes the material metallic. If the A-site atoms in the perovskite manganese oxide are partially replaced, it can be written as R1−x Ax MnO3 (0 < x < 1). With various x, the order of the charge, spin, and orbital, can be quite different. If the double exchange is the dominant exchange interaction between electrons in the Mn ions, the structure is in a ferromagnetic metallic state (FM). If the dominant exchange interaction mechanism is not the double exchange, the electrons in the 3d orbitals align antiparallel, while the charge and orbitals on the Mn sites are aligned periodically. Thus the structure is in the chargeordered, orbitally-ordered insulating state (CO/OOI). There might be another state in between, namely the spin glass insulating state (SGI), which means that the alignments of the charge and orbitals are ordered in a short range while the order breaks down in a long range, as shown in Figure 5.5. However, if the temperature is high enough, the entire order breaks down, and the charge, orbital, and spin align randomly in the lattice. The structure gets into a paramagnetic, insulating state. The state of the material depends not only on the constituent, the temperature, but also on many other parameters, such as the radius of the substitution atom, the external field. GdSm Nd Pr

La

TC,TCO,TN and TG(K)

Eu 300 TCO TC

200 CO/OOI 100

0

TN

FM TG

SGI

R0.55Sr0.45MnO3

1.28 1.32 1.36 Averaged Radius,rA(Å)

1.40

Fig. 5.5. The phase diagram of the R0.55 Sr0.45 MnO3 [78].

5.2.3 Mechanism for photoelectric effects in bulk perovskite oxides There is a class of materials which presents a rich set of anomalous properties due to the strong interaction of d, f electrons with each other and with itinerant electrons in the material. Such materials with strong electron-electron interaction are called strongly correlated systems. The perovskite oxides belong to a class of the strongly correlated systems. Due to the strong interaction of electrons and the stability of the oxygen octahedra, the properties of charge, orbital, spin, and lattice in the perovskite oxides are strongly related to each other. A small variation in one of the properties may

196 | Kuijuan Jin et al. have large influence on other properties. A large class of anomalous properties can be presented in the perovskite oxides with an applied external field. Due to the strong Coulomb interaction of the electrons in the narrow d bands of the transitional metal ions, the electrons tend to be localized. However, the bonding effect with the other sites makes the electrons more itinerant. Because of the competition between the two effects, the perovskite oxides are ideal materials for studying the metal-insulator transition under external field. Optical field can be applied to study the transient phase transition process.

5.2.3.1 Photo-induced phase transition It is very difficult to alter the magnetic structure in most materials by irradiation. It is often that the only effect of the light is to heat the system, so that the system is demagnetized. In perovskite oxides, it is possible to change the magnetic structure because it is strongly related to other properties such as charge, orbital, and lattice, especially if the material is close to the phase transition point. For example, In the Pr0.7 Ca0.3 MnO3 (PCMO3) bulk, when the material is in the charge orbitally ordered, insulating (COO/I) phase, it can transfer to the ferromagnetic, metal (FM) phase [58, 79]. Since the magnetic structure and the Jahn–Teller effect are related to each other in manganese oxides, a change in lattice structure can also be observed in this process. In 1997, B. Kiryukhin et al. found a persistent increase of conductivity in PCMO3 induced by x-rays [58]. As shown in Figure 5.6, at a temperature below 40 K, the conductivity of the PCMO3 can increase about six orders of magnitude. The system does not reverse to the high resistance state spontaneously even after a few hours. In the same year, K. Miyano et al. found a photo-induced insulator-metal phase transition in the PCMO3 bulk [79]. As shown in Figure 5.7, after laser pulses with a pulse width of 5 ns and photon energies of 0.6–3.5 eV are irradiated on the PCMO3 bulk, the resistance reduced from 1 GΩ to 5 Ω. Further research shows that the energy of photons is the key factor of the insulator-metal transition in the PCMO3 bulk. 10-1 10-2

0.8

10-3 0.6 0.4

10-4 X−rays off

10-5 10-6

0.2 0 0

X−rays on 2000

4000 6000 Time(s)

Conductance(Ω-1)

Peak intensity(a.u.)

1.0

10-7 8000

10000

Fig. 5.6. X-ray induced increase of conductivity in PCMO3 bulk [58].

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Fig. 5.7. Photo-induced insulator-metal transition process in PCMO3 [79].

The research on magnetic phase transition by applying external field is very valuable in both a theoretical and application aspect. The magnetic properties can be manipulated by using an external strain or magnetic field. Manipulation using an optical field is of great interest because it can be very fast. The time scale COO/I-FM phase transition can be in picoseconds. In 2007, M. Matsubara et al. studied the dynamic process of the charge and spin in Gd0.55 Sr0.45 MnO3 (GSMO) after an irradiation of a pico-second-pulse laser with a time-resolved reflection spectrum and a time-resolved magneto-optical Kerr effect [78]. The distribution of the intensity of the probe laser with the frequency ω is Iin (ω ), and that of the reflected laser is Iout (ω ). The distribution of the reflectivity of the sample can be written as R(ω )=Iout (ω )/Iin (ω ). The reflectivity depends on the electronic structure of the material. A pulse laser beam was split into two, one as a pump laser and the other as a probe laser, after frequency conversion. There is a delay td between the probe laser and the pump laser. The change of the reflection spectrum with time can be obtained by changing the time of delay. This technology is called time-resolved light reflection spectrum. The accuracy of the time of delay can be of approximately pico-seconds. Therefore it is possible to get the pico-second dynamic process of the electronic structure in the samples. If the incident light is polarized, the polarization state of the emergent light depends on the state of the magnetization of the sample. Because the difference between the speeds of the two components of the linear polarized light, there is an additional rotation angle of the plane of the polarization in the emergent light, namely the Kerr rotation angle (θKerr ). The ellipticity of the light is also changed because of the difference in the reflectivity of sample to the two components of the light, namely the Kerr el-

198 | Kuijuan Jin et al. lipticity (𝜖Kerr ). Both the two variables θKerr and 𝜖Kerr depend on the magnetic structure of the sample and the magnetic field intensity. Thus we can use the magneto-optic Kerr effect (MOKE) to detect the magnetic structure in the samples. There are three types of configuration of the MOKE: (1) if the magnetization direction is perpendicular to the surface of the sample and parallel to the plane of the incident light, it is called the polar MOKE; (2) if the magnetization direction is parallel to both the surface of the sample and the plane of the incident light, it is called the longitudinal MOKE; (3) if the magnetization direction is parallel to the surface of the sample and perpendicular to the plane of the incident light, it is called the polar MOKE. It is possible to get the revolutionary process of the magnetic structure with time by using the time-resolved surface magneto-optic Kerr effect (TR-SMOKE). By using the TR-SMOKE technology, M. Matsubara et al. found that the photoinduced phase transition can be represented as follows. At a temperature of 10 K, the GSMO is in the charge-ordered, orbitally-ordered insulating phase (CO/OOI). In the very short time of less than 200 fs when the pump laser is irradiated on the sample, the 3d electrons are excited. Because of the double exchange interaction, the local ferromagnetic phase is formed, while the directions of the magnetization are aligned randomly. No magnetic domain is formed at this stage. The state is close to the spin glass insulating (SGI) state. In the following 0.5 ps, magnetic domains are formed so that the sample is in the ferromagnetic state. Then, as the excited electrons jump back to the ground state, the dominant exchange interaction reverses to the superexchange interaction. The magnetic structure of the sample transfers to the antiferromagntic (AF) or CO/OOI phase. In about 10 ps, the sample is in the AF or CO/OOI phase. As the temperature increases in the process, the sample gets to the paramagnetic insulating phase. Here we can see that the light-induced CO/OOI-FM transition process is very fast. Therefore this may be a way for fast magnetic manipulation.

5.2.3.2 Ultraviolet photodetector Ultraviolet (UV) photodetectors are a class of devices for detecting ultraviolet light by using the photoelectric effect. The UV photodetectors can be classified into two groups by the response band: the visible blind (response wavelength smaller than 380 nm) and solar blind (response wavelength smaller than 290 nm). Although photodetectors based on conventional semiconductors [43, 68, 107] have already been built into applications, the performance is unsatisfactory. It is still necessary to search for new materials, study new underlying physics, design new structures, and develop new devices. Many perovskite oxides have very wide band gaps. For example, the band gaps of STO and LAO are 3.2 eV and 5.6 eV, respectively, making them good candidates for making UV photo-detectors. J. Xing et al. deposited Au films with a thickness of 100 nm on a LAO single crystal substrate of the size of 5 × 10 × 0.5 mm3 with the thermal evaporation technique, and UV lithography and etching were performed to fabricate interdigital electrode pat-

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Photocurrent(μA)

0.10 0.08 Iamp off

Light

0.06

VB

LaAlO3 wafer

0.04 0.02

Oscilloscope

Iamp on 0.00 0

3

6

9 Time(s)

12

15

18

Fig. 5.8. The steady state response of the LAO UV photo-detector with interdigital electrodes of interval of 5 μm. [120]

terns. The LAO crystal can absorb the photons with energy higher than the band gap and photo-induced electron-hole pairs generated, resulting in an increase in the density of the carriers and an increase in conductivity. The photoelectric responses are shown in Figure 5.8. The largest responsitivity is 71.8 mA/W. The cut-off wavelength is 200 nm. The effective noise power is about 7.1 nW, as shown in Figure 5.9. In addition, a few perovskite oxide single crystals have also been used in UV detectors besides LAO, such as SrTiO3 , LiNbO3 , and LiTaO3 [28, 113, 122].

75

(b) 10-4

Photocurrent(mA)

4.0×10-2

60

under sunlight

3.0×10-2 45

2.0×10-2 1.0×10-2

30

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.02 Light intensity (mW/cm )

15 0 200

220 240 260 Wavelength(nm)

280

300

Current(μA)

Responsivity(mA/W)

(a)

10-5

In dark 10-6 0

5

10 VB(V)

15

20

Fig. 5.9. (a) Spectrum response and (b) electric current response of the LAO UV photo-detector [120].

200 | Kuijuan Jin et al.

5.3 Growth of perovskite oxide films 5.3.1 A brief introduction to the film-growth techniques The growth of the complex oxide thin films is more challenging compared to that of the conventional semiconductor films [75]. The research of the high-temperature superconductivity has spurred the advances in the techniques of growing complicated oxide thin films [95] . Recently, many techniques have yielded oxide thin films with abrupt interfaces [75]. The existing thin film deposition techniques include magnetron sputtering (MS), chemical vapor deposition (CVD), molecular beam epitaxy (MBE), pulsed laser deposition (PLD), etc. Magnetron is for high speed sputtering at lower operating pressure, and this technique makes use of the fact that a magnetic field configured parallel to the target surface can constrain particles with electron motion to the vicinity of the target and increase the sputtering rate [89]. The substrates are immersed in Argon and biased to a high voltage, Argon ions and secondary electrons will be generated when the speeding electrons moving to the substrates collides with the argon atom. The target is bombarded by argon ions, and the bombardment process causes the removal, i.e. sputtering, of target atoms which may condense on a substrate as a thin film. Secondary electrons are also emitted from the target surface as a result of ion bombardment and will be influenced by the electric and magnetic field. If the magnetic field is closed, the electrons move in a circle on the surface of the target. Multiple argon ions in this area are generated and bombard the target, which in turn leads to a high sputtering rate. Owing to the high deposition and target utilization rates, many materials have been successfully deposited using this technique. However, it is difficult to fabricate complicated materials. Chemical vapor deposition (CVD) is a chemical process which is often used in the semiconductor industry to produce nanomaterials. In a typical CVD process, a substrate is exposed to one or more volatile precursors, which react and/or decompose on the substrate surface to produce the desired deposit [13, 89]. CVD is practiced in a variety of formats, including plasma-enhanced CVD (PECVD) and metal organic chemical vapor deposition (MOCVD). The advantage of CVD is its low temperature, easy control of the composition and high-quality thin films, but it is a must to find appropriate gas materials. Molecular beam epitaxy (MBE) is applied to the growth of high-quality thin film on a crystal substrate [19, 30, 89]. MBE, as the name suggests, uses localized beams of atoms or molecules in an ultra-high vacuum (UHV) environment to provide a source of constituents to the growing surface of a substrate crystal. The beams impinge on the crystal kept at a moderately elevated temperature which provides sufficient thermal energy to the arriving atoms for them to migrate over the surface to the lattice sites. During operation, reflection high energy electron diffraction (RHEED) is often used for monitoring the growth of the crystal layers. Moreover, the UHV environment

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minimizes contamination of the growing surface, so it yields high-quality epitaxial thin films with atomic-scale thickness control. However, the process is limited by low growth speed and high expense, which make it difficult to grow complicated thin films. With the development of the high-energy pulse laser, the PLD method has been of more importance for its unique advantages and potential [14]. Pulsed laser deposition is a physical vapor deposition process, carried out in a vacuum system. In PLD, a pulsed laser, going through an optical window, is focused onto a target of the material to be deposited. For a sufficiently high laser energy density, each laser pulse vaporizes or ablates a small amount of the material to create a plasma plume. A thin film will be formed if the plasma plume contacts the surface of the substrate. Pulsed laser deposition has several characteristics which made it remarkably competitive in the complex oxide thin-film research. The attractive features consist of its high deposition speed and broad applicability. But since the thickness of the film is hard to take control of, it is difficult to fabricate ultra-thin epitaxial film of an atomic scale.

5.3.2 Laser molecular beam epitaxy The laser molecular beam epitaxy is a refined method of PLD which combines the advantages of both the conventional PLD and MBE. The method is applicable for obtaining both compounds with complicated compositions and high-quality epitaxial thin films [119]. The concept of laser MBE was first proposed by J. T. Cheung, et al. in 1983 [10–12] , but these authors just substituted the source oven with a laser target, which was only a mixed system of conventional MBE and PLD. In 1991, M. Kanai designed and developed a brand new laser MBE system without the conventional source oven [53]. The principle of the LMBE is similar to that of PLD: a high-energy laser beam going through an optical window hits the surface of a target in a vacuum growth chamber, and the electrons can absorb the photons and be excited. The atoms with a positive charge are thus ionized into plasma. The surface of the target will be instantly heated to 103 –104 ∘ C. Thus, even if a target contains elements with different melting points, it will evaporate simultaneously and result in plasma plume formation. When the substrate heated to a certain degree contacts the plasma plume, the atoms in the plasma plume deposit on the surface of the substrates. We can adjust the growth speed of the thin film by adjusting the pressure in the vacuum chamber and the distance between the substrate and the target. The difference in the vacuum degree is the most striking difference between LMBE and PLD. Generally, for LMBE the vacuum degree before deposition is 10−5 –10−8 Pa, and higher than 10−3 Pa in the growth process. When we grow thin films with PLD, the granule will easily be formed, while LMBE can avoid this problem for the above reason. Reflection high-energy electron diffraction (RHEED) is a widely used method for in situ monitoring and analyzation of the growth of thin films [4, 6]. The principle of

202 | Kuijuan Jin et al. RHEED is as follows [59]. The electron gun emits a high-energy electron beam (10– 50 KV) with a diameter between 0.5–1 mm; then the electron beam hits the surface of the sample at near-grazing incidence angle of 1–3∘ , and is reflected onto a screen after diffracted by a periodic crystal lattice at the surface. The diffraction pattern of RHEED shows the two-dimensional reciprocal lattice, and the surface lattice constant can be determined by measuring the distance between the diffraction spots and the parameters of the instrumentation. According to the geometric theory of RHEED [74], the ideal diffraction pattern is spots, but streaks have frequently been observed in experiments. This occurs because the incident electron beam diverges and the electrons in the beam have a range of energies. The energy variation remakes the sphere into a thin spherical shell. Phonon scattering, surface disorder, and other factors make the spots elongated or streaked [60]. The intensities of individual spots on the RHEED pattern fluctuate in a periodic manner which is called the RHEED intensity oscillation [86]. Generally, the RHEED intensity oscillation refers to the fluctuation of the intensity of the specular spots of the specular beam (the zero-th order diffraction beam) [56], and the period of the intensity oscillation corresponds to the time of the formation of a single atomic layer. Figure 5.10 shows in a two-dimensional epitaxial growth mode [88] the changes in the surface morphology and the corresponding RHEED intensity oscillation. The intensity is at a maximum when the surface of the substrate is complete and smooth; When the thin film begin to grow, the atoms deposited on the surface of the substrate begin to nucleate, leading to a decrease in the reflectivity for the electron beam being being damped by the diffraction of the nucleus. The intensity decreases to a maximumg of 1/2 when a fourth of the surface is covered; the intensity continues to decrease as more MONOLAYER GROWTH ELECTRON BEAM [001] [110] [110]

RHEED SIGNAL ~ θ=0 ~ θ=0.25

~ θ=0.5

~ θ=0.75 ~ θ=1.0

Fig. 5.10. The oscillation of RHEED intensity. θ is the coverage rate [88].

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nuclei form and reaches its lowest point when half of the surface is covered; It then increases as the two-dimensional nuclei connect with each other and the smoothness of the surface recovers, and a maximum intensity indicates the completion of a new layer. Therefore, intensity oscillation not only reflects the layer-by-layer growth process of the thin films, but it also gives accurate information about the number of epitaxial unit cells, i.e. the thickness of the epitaxial thin film.

5.4 Logitudinal photoelectric effects in perovskite oxide heterostructures When light is irradiated on perovskite oxide heterostructures, photons are absorbed and electron-hole pairs are generated if the energy of the photons is higher than the band gap of one or some of the materials in the heterostructure. The logitudinal movement of these carriers in the heterostructures can bring some photoelectric functionalities, such as the photovoltaic and photoconductive effects. Several decades ago, researchers used continuous and pulse lasers to study the logitudinal photoelectric properties in the superconducting YBa2 Cu3 O7−δ thin films. A photoelectric response of picoseconds was observed, with which they tried to probe the underlying physics in the superconductors [15, 17, 34–36, 61, 69, 100]. From then on, more and more research on photoelectric effects in perovskite oxide heterostructures was carried out, and this has become one of the frontiers of the interdisciplinary objects of condensed matter physics and optics.

5.4.1 Light-generated carrier injection effects Light-generated carriers on one side of a heterostructure can transfer to the other side, and change the properties of the whole system. This effect is called the photocarrier injection effect. The phenomenon was observed in many heterostructures, such as La0.8 Sr0.2 MnO3 /SrTiO3 ,

La0.8 Sr0.2 MnO3 /SrTiO3 : Nb,

La0.8 Sr0.2 MnO3 /SrTiO3 : Nb, CaCuO2 /SrTiO3 : Nb,

VO2 /TiO2 : Nb,

YBa2 Cu3 O7−δ /SrTiO3 : Nb

[42, 54, 80, 80–84]. We introduce the carrier injection effect with the work of H. Katsu et al. as an example. A La0.8 Sr0.2 MnO3 (LSMO2) film with a thickness of 30 nm was grown on a (001) surface of the substrate of a STO single crystal. The resistance of the LSMO2 film was measured at various temperatures. As the results in Figure 5.11a show, there is a metalinsulator transition at a temperature of 240 K. With the light of a xenon lamp irradiated on the system, the resistance increases, and the temperature of the metal-insulator transition decreases to 230 K. More interestingly, the resistance below 100 K drops sig-

204 | Kuijuan Jin et al. (a)

Light

Au electrode

Al electrode

Current Density/(Acm-2/10-4)

(La,Sr)MnO3 STO(100) substrate 4

Light on Light off

(b)

~500 mV

2

0

-2 -3

-2

-1 Applied Voltage(V)

(c) LSMO

STO EC

Ef

0

1 electron hole

~0.5 eV

EV

Back direction

Zero bias

Forward direction

Fig. 5.11. (a) Schematic diagram of the LSMO2/STO heterostructure. (b) The I-V with light on and off at 30 K. (c) Schematic diagram of the band structure [54].

nificantly. The reason is possibly that there is a structural phase transition of the STO substrate causing the change in the electronic structure. All these results can only be obtained if the energy of the photons is above 3.2 eV, which is the band gap of STO. This reveals that the change of the properties of the LSMO2 film is relevant to the process of carrier generation in the STO. Carriers generated in the STO substrate were injected into the LSMO film. According to the phase diagram of LSMO, the reduction in concentration of holes leads to the weakening of the ferromagnetic double-exchange interaction. Thus the electrons in the eg orbitals become more localized, which causes an increase in the resistance of the system and a decrease of the Tc [108]. In that way, with light of which the photon energy is higher than the band gap of STO is irradiated on STO, the light-generated carriers transport to the LSMO2 film and reduce the concentration of the holes in LSMO2. The weakening of the double exchange interaction increases the resistance and reduces the Tc; below 100 K, the efficiency of the carrier generation in STO increases. With a large number of electrons injected in, the transport behavior of LSMO2 film changes from the metallic to the insulating phase.

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To verify that the phenomenon is caused by the injection of light-generated carriers, H. Katsu et al. measured the electric current with and without light. As shown in Figure 5.11b, with the light off, STO is a good insulator, while with the light on, the electric current in the system is not zero under bias voltage of zero, which indicates that the electrons flow from the STO substrate to the LSMO2 film. The magnetic measurement on the LSMO2 film clearly shows the magnetization of the LSMO2 film decreases with the light on, as shown in Figure 5.12.

light off Magnetization(emu/g)

30

35 light on

20

30 25 0

10

20

40

60

80 100

500 G 0

0

100 200 Temperature(K)

300

Fig. 5.12. Magnetization of an LSMO2 film with light on and off[54].

5.4.2 Photovoltaic effect If light with photon energy higher than the band gap of one or more perovskite materials in the heterostructure irradiates on the system, the light-generated carriers are separated by the built-in electric field. The electrons drift to the n-type region and the holes to the p-type region. Then a potential difference caused by the light is formed in the heterostructure: this is called the photovoltaic effect [104]. The photovoltaic effect in conventional semiconductors has been studied and applied in solar cell, photoelectric detection, etc. Because of the rich features of the perovskite oxides, the photovoltaic effect in those materials can be tuned in many ways, making it possible to produce multifunctional photovoltaic devices. Recently, photovoltaic effects in perovskite heterostructures have attracted extensive attention. P. X. Zhang et al. fabricated the La0.9 Ca0.1 MnO3 thin film on a SrTiO3 substrate and found a photoelectric response with the full width of half the maximum of about 2 μs on it [131]. J. R. Sun et al. have done a series of research projects on the photovoltaic effect in the heterostructures of the manganites [22, 72, 97, 98, 102, 103, 114]. According to their results, there is a large enhancement of the photovoltage as the temperature decreases, because the magnetic properties vary with the temperature, and then the photovoltage changes. They also studied the influence of the Schottky interface of Au/SiTiO3 : 0.05wt%Nb on the photovoltaic effect. They found that the photoelectric current varies little with the junction in various resistance states. They

206 | Kuijuan Jin et al. attribute this phenomenon to the filamentous conductive path formed in the Schottky interface area. LMO, an antiferromagnetic insulating material, was grown between manganite and SNTO as a buffer layer to form a p-i-n structure. They found that with a decrease in thickness of the buffer layer, the photoelectric current decreases significantly. This is because the barrier height of the interface of the oxide heterostructure changes due to the buffer layer. Photovoltaic effects have also been observed in heterostructures of doped manganites and conventional semiconductors. These results can be represented with the electronic band theory [46, 50]. J. Xing et al. studied the influence of the oxygen content of a thin film in the heterostructure of BaTiO3−δ /Si to the photovoltaic effect. According to their results, with a decrease in oxygen content in the film, the rectification characteristic becomes worse, the reverse leakage current increases, and the photoelectric current increases. This phenomenon is thought to be related to the impurity energy level introduced with the oxygen vacancies [121]. E. J. Guo et al. studied the photoelectric effects in the perovskite oxide heterostructures of the controlled with external stress and magnetic field [25–27]. In the following, the progress of the research on the photovoltaic effects in perovskite oxide heterostructures will be discussed under three aspects: the ultrafast photoelectric response, the influence of the thickness of the substrates to the photoelectric response, and the influence of the thickness of films to the photovoltage.

5.4.2.1 Ultrafast photoelectric response H. B. Lu et al. grew a p-type LSMO3 film with a thickness of 300–600 nm on an n-type Si substrate [71]. Nd : YAG laser pulse with a width of 25 ps and wavelength of 1064 nm was used to irradiate on the heterostructure of LSMO3/Si. As Figure 5.13 shows, the rise time is about 10 ns, and the full width of the half maximum is about 12 μs. With a small resistor of 0.2 Ω parallel with the heterostructure, the rise time decreases to about 210 ps, and the full width of the half maximum decreases to about 650 ps, as shown

Fig. 5.13. The time dependence of the photovoltage in the heterostructure of LSMO3/Si. The wavelength of the laser pulse is 1064 nm. The insert figure is the schematic diagram of the measuring circuit [71].

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70

Voltage(mV)

60

hν

50 40 30 20 10 0 0.5

1.0

1.5

2.0 2.5 Time(ns)

3.0

3.5

4.0

Fig. 5.14. The time dependence of the photovoltage in the heterostructure of LSMO3/Si, with a small resistor of 0.2 Ω in parallel. The wavelength of the laser pulse is 1064 nm. The inserted figure is the scheme of the schematic diagram of the measuring circuit [71].

in Figure 5.14. The heterostructure of LSMO3/Si has a high photoelectric sensitivity to about 435 mV/mJ. There are two main factors influencing the rise time and the half width of the photoelectric response. The first is the quality of the interface. The defects have a great influence on the generation and recombination process of the photo-induced carriers. The second is the resistance of the circuit. A heterostructure of the LSMO3/Si with the sectional area of 30 mm2 is about 30 pF. With the resistance of the oscilloscope being 1 MΩ, the discharge time of the circuit is about 30 μs, which is approximately the full width at half-maximum measured in the experiment. With a small resistor in parallel, the discharge time can be reduced. To make sure that the voltage is not due to the thermal electric process, a laser with a wavelength of 10.6 μm was irradiated on the same heterostructure, and no photoelectric response was detected. As the energy of the photons is smaller than the band gaps of the LSMO3 and Si, the influence of the thermal-electric effect can be eliminated. Ultrafast photoelectric response has also been found in other perovskite oxide heterostructures [47, 70, 115, 132, 133]. By reducing the thickness of the substrate to 0.1 mm, photoelectric response time of 86 ps was found in the heterostructure of LaAlO3−δ /Si. A photoelectric response with a rise time of 9 and 23 ns were found in the heterostructure of SrTiO3 /Si, and LSMO1/SNTO, respectively. From these results, we can see that the photoelectric response speed is comparable to that of the photoelectric devices of the conventional semiconductors. Therefore, there are promising prospects for the ultrafast photoelectric detector for the perovskite oxide heterostructures.

208 | Kuijuan Jin et al.

Fig. 5.15. Photoelectric responses in LAO/Si with various thicknesses of substrates. The light source is a XeCl pulse laser with wavelength 308 nm. The insert figure shows the rise time [115].

5.4.2.2 Influence of the thickness of the substrates on photoelectric response J. Wen et al. has grown LaAlO3−δ (LAO) films with a thickness of 400 nm on a p-type substate of Si with thicknesses of 0.71 mm, 0.44 mm, 0.19 mm, and 0.10 mm. These samples are labeled as F1, F2, F3, and F4, respectively [115]. The result of the Hall effect measurement shows that the thin film in the samples are of the n-type. Thus n-p heterostructures are formed between n-LAO and p-Si. Measurements of the photoelectric response were taken on the four samples F1, F2, F3, and F4. The rise times were 38.3, 21.4, 13.9, and 13.3 ns, respectively. The full widths at half maximum are 285.2, 180.6, 134.7, and 79.2 ns, respectively, as shown in Figure 5.15. The photoelectric response time decreases with the decrease in the substrate thicknesses. The reason is that photo-induced carriers can be generated by electrodes at the surfaces of the substrates in a shorter time with a decrease in the substrate thicknesses. The results show that it is an effective way to improve the performance of the ultraviolet photoelectric device by reducing the thickness of the substrates.

5.4.2.3 Influence of the thickness of the film on the photovoltage As well as the thickness of the substrate, the thickess of the film also has a great influence on the photovoltage [93, 111]. C. Wang et al. deposited a group of La0.9 Sr0.1 MnO3 (LSMO1) thin films with various thicknesses on substrates of n − Si and SrTiO3 : wt0.8%Nb. The group of samples of LSMO1/SNTO grown at an oxygen pressure of 5 × 10−2 Pa is labeled as #1. The group of samples of LSMO/Si grown at an oxygen pressure of 5 × 10−2 Pa is labeled as #2. As LSMO1 is a p-type material and Si and SNTO are n-type materials, all three groups of samples form p-n heterostructures.

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3.5 (a1) 3.0 2.5 2.0 1.5 1.0 0.5

Photovoltage(mV)

Photovoltage(mV)

A built-in field can be formed in the heterostructures and drive the photo-induced carriers apart. Consequently, a photovoltaic effect can be detected in these samples. It was found that the photovoltages in the heterostructures with Si as the substrate are much larger than those in the heterostructures with SNTO as the substrate. This is because the mobility of the carriers is higher, the recombination rate of the minority carriers is lower, and the widths of the depletion regions are larger than those in the LSMO1/SNTO heterostructures in the LSMO1/Si heterostructures. It is found that in these samples, the photovoltages vary largely with the thicknesses of the thin films. As the thicknesses of the films decrease, the photovoltages first increase and then decrease. The maximum values are much larger than those in films with larger thicknesses, as shown in Figure 5.16.

100 10 Thickness(nm) -4

-3.6

(b1)

-5 -6 -7

LSMO

SNTO

Energy(eV)

Energy(eV)

-3

350 (a2) 300 250 200 150 100 50 0 10 100 Thickness(nm)

-8

(b2)

-4.0 -4.8 -5.2

LSMO

Si

-5.6 -30 -20 -10 0 10 20 30 X(nm)

-30 -20 -10 0 10 20 30 X(nm)

Fig. 5.16. the dependence of (a) the maximum of the photovoltage on the thicknesses of the films, and (b) the band diagram of the heterostructures. The labels 1 and 2 refer to the group of samples #1 and #2 [111].

The mechanism of the dependence of the maximum photovoltage on the thickness were studied with one dimentional drift-diffusion model, which will be discussed in Section 5.4.3.

210 | Kuijuan Jin et al. 5.4.3 Theoretical study on longitudinal photoelectric effects The self-consistent calculation of the drift-diffusion model, first proposed by Scharfetter and Gummel [94] , is a powerful and effective method for describing carrier transport behavior in semiconductor devices and is still widely used today [24]. In oxide heterostructures, some early theoretical works confirmed that numerical calculations based on the drift-diffusion model could be employed to depict not only the steady transport behavior, but also the transient transport behavior. Q. L. Zhou et al [135] and J. Qiu et al. [92] studied the transport properties of STO homostructures and LSMO3/Si heterostructures, respectively. The band structures, the distribution of carriers, the distribution of the electric field, and the current-voltage curves based on this model fit well with the experimental data. P. Han et al. [32, 33] analyzed the transport mechanism in LSMO1/SNTO heterostructures. C. L. Hu et al. introduced the spin polarized current and phase separation model into the drift-diffusion model to study the dependence of the positive magnetoresistance on the temperature and the bias voltage [44, 45]. C. Wang et al. [111] studied the the mechanism of the dependence of the photovoltage on the thickness of the film by using the drift-diffusion model. L. Liao et al. introduced the time-dependent one-dimentional drift diffusion model to study the longitudinal photoelectric effects [64, 66, 67].

5.4.3.1 One-dimentional steady drift-diffusion model The one-dimentional steady drift-diffusion consists of the Poisson equation and the current continuity equations of the carries. The Poisson equation can be written as 𝜕2 φ (x) e = − [p(x) − n(x) + N], ε 𝜕x2

(5.2)

in which, x, φ (x), e are the position, the electric potential ,and the unit electron charge, respectively; ε denotes the permittivities of the materials; p(x) and n(x) are the concentrations of holes and electrons, respectively; N is the net doping density. The current continuity equations of the electrons and holes are dJn + q(Gn (x) − Rn (x)) = 0 dx −

dJp dx

+ q(Gp (x) − Rp (x)) = 0,

(5.3) (5.4)

in which, Gn and Rn are the generation rate and recombination rate of the electrons, respectively; Gp and Rp are the generation rate and recombination rate of the holes, respectively; Jn and Jp are the current densities of the electrons and hole, respectively. In the Shockley–Read–Hall model [99], the recombination rate is R(x) =

n(x)p(x) − n2i , τn [p(x) + ni ] + τp [n(x) + ni ]

(5.5)

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|

in which, τn and τp are the lifetime of the electrons and holes, respectively; ni is the intrinsic carrier density. The current densities in the bulk regions can be expressed in terms of drift and diffusion as follows, Jn = −e [μn n(x)

dφ (x) dn(x) − Dn ] dx dx

(5.6)

Jp = −e [μp p(x)

dφ (x) dp(x) ], + Dp dx dx

(5.7)

in which, μn and μp are the mobilities of the electrons and holes, respectively; Dn and Dp are the diffusion coefficients of the electrons and holes, respectively, which have the k T Einstein relation with the mobilities D = Be × μ . Here, kB is the Blotzmann constant, and T is the temperature. Some of the parameters of the perovskite oxide are listed in Table 5.1. Table 5.1. The parameters for calculations in the drift-diffusion model [5, 21, 33, 85, 111].

Relative permittivity (ε0 ) Affinity energy (eV) Intrinsic carrier (cm−3 ) Doping concentration (cm−3 ) Electron mobility (cm2 V−1 s−1 ) Hole mobility (cm2 V−1 s−1 )

La0.9 Sr0.1 MnO3

Nb:SrTiO3

Si

65 3.98 4.82 × 109 2.25 × 1019 10 1.8

100 4.00 1.04 × 10−4 2.12 × 1020 8 0.1

11.9 4.05 1.5 × 1010 9.00 × 1016 1350 500

Thermionic emission model developed by Richardson was introduced to describe the current densities at the interfaces, which are: JRich−n = −

A∗n1 T 2 |ΔEc | exp (− ⋅ θ (ΔEc )) n(xinterf− ) Nc1 kB T

A∗ T 2 |ΔEc | ⋅ θ (−ΔEc )) n(xinterf+ ) + n2 exp (− Nc2 kB T JRich−p =

A∗p1 T 2 Np1 −

exp (−

A∗p2 T 2 Nv2

−|ΔEv | ⋅ θ (−ΔEv )) p(xinterf− ) kB T

|ΔEv | exp ( ⋅ θ (ΔEv )) n(xinterf+ ) kB T

(5.8)

(5.9)

The band diagram of the heterostructure can be obtained by solving the equations above. Valence bands and conduction bands near the interfaces of the heterostructures, which show the depletion region, are shown in Figure 5.16c. We can see that the thicknesses of the films for the maximum photovoltages are approximately equal to the widths of the depletion regions in LSMO1/SNTO and LSMO1/Si heterostructures.

212 | Kuijuan Jin et al. Electron and hole pairs can be generated and then be driven apart by the built-in field when the heterostructures are irradiated under light with photon energy larger than the band gap of the materials in the heterostructures. The holes drift to the p layer, and the electrons drift in the opposite direction. Thus the electric field is formed in the heterostructure, which leads to a photovoltage in the heterostructure. The photovoltage is largely related to two processes: the separation of the photoinduced carriers and their recombination. If the thickness of the film is reduced below the width of the depletion region in the film, the distance of the photo-induced carriers driven apart decreases. The photovoltage decreases consequently. On the contrary, a large amount of photo-induced carriers are recombined when they diffuse to the surface of the heterostructure if the thickness of the film is much larger than the width of the depletion region.

5.4.3.2 One-dimentional time-dependent drift-diffusion model The one-dimentional time-dependent drift-diffusion model is employed to study the dynamic process of the photoelectric response in the heterostructures of perovskite oxides. Based on this model we can obtain the distribution of the carriers, which are functions of time. It consists of the Poisson equation and the current continuity equations of the electrons and holes: 𝜕2 φ (x, t) e = − [p(x, t) − n(x, t) + N] 2 ε 𝜕x 𝜕n(x, t) 1 𝜕Jn (x, t) = + (G(x, t) − U(x, t)) = 0 𝜕t q 𝜕x 1 𝜕Jp (x, t) 𝜕p(x, t) =− + (G(x, t) − U(x, t)) = 0. 𝜕t q 𝜕x

(5.10) (5.11) (5.12)

The expressions of the densities of the currents are almost the same as those in the one-dimentional steady drift-diffusion models, except that the potentials and densities of the carriers are time dependent. L. Liao et al. [67] calculted the dynamic process in the heterostructures in LSMO1/ SNTO. Their results show that the model can describe the experiment results very well, as shown in Figure 5.17. The energy band profile and electric field distribution in LSMO1/SNTO heterostructure were obtained, as given in Figure 5.17a,b, respectively. The two boundaries of space-charge region locate on x = 25nm and 45 nm. When light irradiates on the heterostructure, the electrons and holes are created in the bottom of the conduction band and the top of the valence band, respectively. Subsequently, these photo-induced electron-hole pairs diffuse into the depletion region. In this region, the electron hole pairs are separated by a built-in field. The electrons and holes drift into the p-LSMO and n-SNTO regions, respectively. The movement of carriers induces

Photovoltage(V) Photovoltage(V)

5 Research on the photoelectric effect in perovskite oxide heterostructures

0.04 (a) Experimetal 0.03 data Laser 0.02 0.01 FWHM=120 ns 0.00 0.04 (b) Calculated data 0.03 0.02 0.01 0.00 0 200 400 600 X(ns)

|

213

50 Ohm

800

1000 1200

Fig. 5.17. The photovoltage in the heterostructure of LSMO1/SNTO (a) experimental data (b) calculated data [67].

the redistribution of charge in the depletion region and the change of the electric potential difference between the n- and p-type regions. Figures 5.18 and 5.19 exhibit the evolution of the carrier concentration and potential distribuion in a period of 0–20 ns, respectively. The light irradiates on the heterostructure, and the electron-hole pairs are created. When the carriers diffuse into

Fig. 5.18. Variation of the concentration of (a) holes and (b) electrons with time. The region between the two dashed lines is the depletion region [67].

Fig. 5.19. The distribution of electric potential. The insert exhibits the electric potential distribution in LSMO1/SNTO without irradiation [67].

214 | Kuijuan Jin et al. the depletion region, they are driven apart to the SNTO and LSMO1 regions, respectively. The effect of diffusion reaches equilibrium with that of drift when the photoinduced holes and electrons drift into the boundaries of the depletion region. Therefore, the photo-induced holes and electrons cannot move out of the depletion region, and the carriers accumulate in this region. As shown in Figure 5.18, the concentration of holes and electrons increases in the right and left sides of the depletion region, respectively. The variation of carrier concentration in the homogeneous region is much smaller than that in the depletion region. This result indicates that the photoinduced carriers appear mainly in the depletion region. Moreover, the movement of photo-induced carriers reduces the charge polarization in the depletion region. This phenomenon is explained by the fact that both positive and negative charges in the nand p-type regions is decreased by these carriers. Therefore, the potential difference between the p-type and n-type regions decreases with time while laser is on. As shown in Figure 5.19, the electric potential in p-LSMO1 region increases with time. Figures 5.20 and 5.21 exhibit the evolutions of carrier concentration and potential distribution ina period of 100–500 ns with the laser off, when the photovoltage is decaying. Thus the effect of photo-induced carriers diffusing into the depletion region is weakened. Therefore, the equilibrium of the drift and diffusion near the boundaries of

Fig. 5.20. Variation of the concentration of (a) holes and (b) holes with time when the laser is off. The region between two dashedlines is the depletion region [67].

Fig. 5.21. The distribution of electric potential when the laser is off. The insert exhibits the electric potential distribution in LSMO1/SNTO without irradiation [67].

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the depletion region is broken. The state of the system gradually relaxes to that with the original state. Inserting a parallel resistor can have a great influence on the photoelectric response in the perovskite oxide heterostructure. Theoretical analysis of the effect of the parallel resistor can help in the designing of photoelectrical devices. As shown in Figure 5.22, the rising and decaying process are both faster, with resistors of smaller resistance in parallel. However, the maximum value of the photovoltage is smaller in this case. This phenomenon can be interpreted by the RC electric circuit model. Photoinduced carriers are separated by the build-in electric field when the laser is on. Then these carriers diffuse to the surface of the surface of the heterostructure. These two processes can be seen as the charging and discharging process of the capacitor. Thus a resistor in parallel can reduce the relax time of the circuit.

0.5 Ohm 5 Ohm 50 Ohm 500 Ohm 5000 Ohm

0.06

0.04 a.u.

Photovoltage(a.u.)

0.08 0.02 0.00 0

0.04

10

20

30 0

Time(ns) 0.02 0.00 0

400 Time(ns)

800

1200

Fig. 5.22. Photoelectric response of heterostructures with various resistors in parallel. The insert figure is the rising edge of the photovoltages [66].

The photon energy can also influence the response time of the perovskite oxide heterostructure. The photo-induced carriers have higher kinetic energy if the photon energy is higher, so that the effective mobility of the carriers is higher. Figure 5.23 shows that light with larger photon energy can reduce the response time of the LSMO1/SNTO heterostructure. Photovoltage can be increased by reducing the doping concentration of LSMO in the LSMO/SNTO heterostructures [65]. The width of the depletion region is larger if the doping concentration is smaller. This is to say that the distance separating the photoinduced carriers is larger. Thus the induced electric potential is larger. The mobility of

216 | Kuijuan Jin et al.

Photovoltage(a.u.)

0.25

0.23 ns

0.20 0.15

no change of carrier mobilities carrier mobilities increase 10 times carrier mobilities increase 100 times

Laser

0.49 ns

0.10 0.05

0.2 Ω

0.90 ns

0.00

Laser is off at 0.02 ns 0

2

4 6 Time(ns)

8

10

Fig. 5.23. The influence on photovoltage in the heterostructure of various carrier mobilities [66].

carriers in LSMO with smaller doping concentration is higher. Therefore the response speed is increased.

5.5 Lateral photoelectric effect in perovskite oxide heterostructures Electron-hole pairs are generated in the heterostructures under the irradiation of inhomogenous light with photon energy larger than the band gap of one or more of the materials. Because of the gradient of the carrier pairs, the drift and diffusion of the carriers lead to a lateral electric field. Thus a lateral photovoltage (LPV) is generated. This phenomenon is called lateral photoelectric effect (LPE).

5.5.1 Background Research on LPV was begun in 1930 by W. Schottky [96]. Then in 1957, J. T. Wallmark made a preliminary theoretical analysis on the effect [110]. G. Lucovsky gave a detailed theoretical description of the LPE in the p-n junction with one side heavily doped [73]. In 1976, H. Niu et al. made an improvement so that the LPE in junctions without heavy doping can be described [87]. In 1991, S. Amari et al. generalized the theory to two dimensions [3]. These theories have the following common viewpoint. The light-induced electron-hole pairs are separated by the built-in field. Thus most of the electrons are in the n-type material, and most of the holes are in the p-type material. Light-induced carriers diffuse from the light-irradiated area. Thus the electric currents in the n-type material point to the light-irradiated area while the direction of the p-type material is opposite. Therefor, the electric potential is higher in the light-irradiated area than that in the area with no light-irradiated on in the p-type material. In the n-type material, the electric potential is lower in the irradiated area. This is to say that the LPV has the

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opposite sign in p- and n-type materials. In this chapter we call this phenomenon the conventional LPE . Position-sensitive detectors (PSD), which can detect very small shifts of positions [1, 2, 8, 9, 16, 20, 76, 77], based on LPV are widely applied in various fields [7, 55, 57, 91]. The advantage of the LPE-based PSD devices is that it can get signals in a large continuous area. LPE has been found in many structures, such as the heterostructures of AlGaAs/GaAs [105], superlattice of Ti/Si [62, 63, 117, 118], organic semiconductor polymers [51, 52], two-dimentional electronic system[109], hydrogenated amorphous silicon Schottky junction [37–41], perovskite oxide heterostructure [48, 49, 134], metalsemiconductor-oxide-semiconductor structure [18, 123–130], etc. A typical LPV distribution in two dimension is shown in Figure 5.24. Alloyed indium contact

Co3Mn2O SiO2

A

100

LPV(mV)

B

Laser yA x

50 0 -50 3 2 1 0 -2 -1 -1 y(mm) 0 1 -2 x(mm) 2 3 4 -3 (b) B

Si

(a)

-100 -4 -3

Fig. 5.24. (a) Schematic diagram of the measure circuit of LPV. (b) LPV distribution in two dimensions [112].

5.5.2 Unusual lateral photoelectric effect in perovskite oxide heterostructures In 2007, Jin et al. observed the phenomenon that the signs of the LPV measured on both sides of p-n junctions is the same in perovskite oxide heterostructures with intense pulse laser irradiated on [48, 49], which is different from the conventional LPE. Dember effect [90], induced by the difference of the mobilities of photo-induced electrons and holes, was introduced to the quantitatively explain the unusual LPV in the oxide heterostructures. LSMO1/SNTO and LSMO3/Si heterostructures were fabricated by growing a p-type LSMO1 (LSMO3) layer on a n-type SNTO (Si) substrate, with the computer-controlled laser MBE technique. The schematic setup for LPV measurements is shown in the inserts of Figure 5.25. A small area of 0.5 mm diameter on the p-LSMO surface was illuminated by a 308 nm XeCl excimer laser beam (pulse width of 20 ns, pulse energy of 0.15 mJ, and repetition rate of one pulse every 5 minutes to avoid the heating effect).

218 | Kuijuan Jin et al.

B

D

E

m VBA

20 V 0 -20 -40

m ED

(b)

-6 -4 -2 0 2 4 6 Laser spot position X(mm)

200

hν

A

B

D

E m

VBA

100 V 0 -100 -200

m ED

(c)

photovoltage(mV)

40

hν A

photovoltage(mV)

photovoltage(mV)

(a)

-6 -4 -2 0 2 4 6 Laser spot position X(mm)

20 10

hν

A

B

Si SNTO

0 -10 -20 -6 -4 -2 0 2 4 6 Laser spot position X(mm)

Fig. 5.25. The peak values of LPV as a function of the positions of the laser spot in the x-direction in (a) LSMO1/SNTO and in (b) LSMO3/Si heterostructures, the upper panel displays the schematic setup for the LPV measurement. A (−3 mm), B (3 mm), D (−3 mm), and E (3 mm) denote the electrodes. (c) The peak values of LPV for n-SNTO (circles) and Si (squares) substrates [48].

The samples were moved in the lateral direction, and the LPV values were recorded by a sampling oscilloscope of 500 MHz terminated into 1 MΩ at room temperature. Here, the photovoltage, which denotes the peak value of the LPV between the indium electrodes A (x = −3 mm) and B (x = 3 mm) on the LSMO1 (LSMO3) surface, depends on the laser spot position in the LSMO1/SNTO (LSMO3/Si) heterostructures. In addition, the diameter of indium electrodes is about 1 mm. Particularly, the electrodes were always kept in the dark to prevent any electrical contact effects. Figure 5.25a exhibits the measured LPV values in a LSMO1/SNTO heterostructure. Figure 5.25b depicts the experimental LPV values in a LSMO3/Si heterostructure. From Figures 5.25a,b it can be seen that the signs of LPV are the same on both sides of the heterostructures. This phenomenon can be explained as follows. As both electrons and holes are induced by photons and flow out on two sides of oxide heterostructures, the mobility difference of the electrons and holes causes the same sign of LPV on both sides of oxide heterostructures. It should be noted that only strong light can make the Dember-effect induced LPV large enough to be observed. To compare the LPV in the heterostructures and that in the substrate, the LPV measurements for n-SNTO and n-Si substrates were also carried out, and the results are shown in Figure 5.25c. Evidently, it can be concluded that a one-order-of-magnitude enhancement of the LPV was observed, as compared with those of the substrates. In the experiments described above, the LPV signals are mainly photoelectric signals. However, the thermoelectric effect cannot be avoided, although it is not the dominant effect in the setup. In p-type and n-type regions, the gradient of the temperature is the same, while the charge of the carriers in the two regions are different. Conse-

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quently, the thermoelectric effect leads to opposite signs of the LPV in the two regions, which is not the case in the experiment.

5.5.3 Theoretical study 5.5.3.1 Two dimensional drift-diffusion model The conventional theories of the LPE are suitable for stable, weak light-induced LPE. The unusual LPE is transient, strong light-induced LPE, which cannot be solved with the conventional theories. A basic model is required to describe the phenomenon. The time-dependent drift-diffusion model is an effective way to study the dynamic photoelectric process in heterostructures. The two dimensional time-dependent driftdiffusion model consists of the Poisson equation and the continuity equations of the electrons and the holes: 𝜕2 φ (x, y, t) 𝜕2 φ (x, y, t) e + = − [p(x, y, t) − n(x, y, t) + N] ε 𝜕x2 𝜕y2 𝜕n(x, y, t) 1 = ∇ ⋅ jn⃗ (x, y, t) + (G(x, y, t) − R(x, y, t)) 𝜕t e 1 𝜕p(x, y, t) = − ∇ ⋅ jp⃗ (x, y, t) + (Gp (x, y, t) − Rp (x, y, t)), 𝜕t e

(5.13) (5.14) (5.15)

in which x and y are the longitudinal and lateral positions,respectively; t represents the time; jp⃗ and jn⃗ are the current densities of the holes and electrons, respectively; ix⃗ and iy⃗ are the unit vector in the x- and y-directions, respectively. The recombination process is described using the SRH model. The current densities in the bulk regions can be expressed in terms of drift and diffusion as follows, 𝜕φ (x, y, t) 𝜕n(x, y, t) ] ⋅ ix⃗ − Dn 𝜕x 𝜕x 𝜕φ (x, y, t) 𝜕n(x, y, t) − e [μn n(x, y, t) − Dn ] ⋅ iy⃗ 𝜕y 𝜕y

(5.16)

𝜕φ (x, y, t) 𝜕p(x, y, t) + Dp ] ⋅ ix⃗ 𝜕x 𝜕x 𝜕φ (x, y, t) 𝜕p(x, y, t) ] ⋅ iy⃗ . − e [μp p(x, y, t) + Dp 𝜕y 𝜕y

(5.17)

jn⃗ = − e [μn n(x, y, t)

jp⃗ = − e [μp p(x, y, t)

The electric currents at the interface are described using the Richardson thermalemission model, which are presented in equations (5.8) and (5.9). From Figure 5.26, the theoretical results based on the two-dimensional timedependent drift-diffusion model agree well with the experimental data. This result shows that the model can give a good description of the unusual LPV.

220 | Kuijuan Jin et al.

(a)

40

1.0 Photovoltage(a.u.)

m

Photovoltage(mV)

VBA m

20

VED

0 -20

(b) m

VBA

0.5

m

VED

0.0 hν -0.5 -1.0

A

B

D

E

-40 -4

-2

0

2

4 -4 Laser spot position x(mm)

-2

0

2

4

Fig. 5.26. LPV observed in LSMO1/SNTO heterostructures: (a) experimental data [49]; (b) calculated results [64].

5.5.3.2 Unified description of conventional and unusual LPV In order to find out the reasons why the unusual LPV differs considerably from the conventional LPV, Liao et al. carried out a thorough investigation of the dependence of the LPV on the laser pulse energy [64]. As shown in Figure 5.27, the corresponding calculated LPV on the SNTO side exhibits an inversion of the sign of LPV with the increase in the laser pulse energy and lateral modulated behaviors under the laser pulse energy of 0.015 mJ, while calculated LPV on the LSMO1 side becomes larger and larger with the increase in the laser pulse energy as shown in Figure 5.27. This laterally modulated LPV effect can be explained by the competition between the Dember and conventional LPV processes. If the energy of the laser pulse is small, the concentration of the photo-induced carriers can be separated by the built-in field. Most of the carriers are holes in the ptype region and electrons in the n-type region, which results in conventional LPV. If the energy of the laser pulse is very large, the minority carriers also contribute to the

1.0

(a)

(b)

1.0 Energy of laser pulse 0.004 mj 0.015 mj 0.030 mj

0.0

0.5

hν

-0.5 -1.0 -6

-4

-2

0

A

B

D

E

2

4

m (a.u.) VED

m VBA (a.u.)

0.5

0.0

-0.5 -1.0 6

-6

-4

-2

0

2

4

6

Laser spot position(mm)

Fig. 5.27. LPV under irradiations if various laser pulse energies (a) on the LSMO1 side and (b) on the SNTO side [64].

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photovoltage. Both electrons and holes diffuse in the same direction. The mobility of the electrons are larger than that of the holes, which leads to an electric field pointing from the holes to the faster electrons. This is called the Dember effect. Consequently, the signs of LPVs in both p- and n-type regions are the same. Under critical laser pulse energy, the conventional LPV effect and the Dember effect are comparable to each other. Hence, neither of them can dominate the LPV all over the region. In the region near the irradiation center (−2.0 mm, 2.0 mm) where the carrier density is high due to the strong laser pulse irradiation, the Dember effect is stronger than the conventional LPV effect, as shown in Figure 5.27. Therefore, in this region, the farther the position is away from the irradiation center, the smaller the electric potentials are on both sides. While in the region far away from the irradiation center (2.0 mm, 7.5 mm) and (−7.5 mm, 2.0 mm) where the carrier density is low, the Dember potential is weak. As a result, the conventional LPV effect is the main contributor to the LPV. Thus, the calculated results unified the description of the conventional LPV and the Dember effect into the drift-diffusion model. The evolution and the competition process of the conventional and Dember LPV effect have been revealed theoretically. With the increase in the irradiated laser pulse energy, the Dember effect plays a more and more important role in the lateral photoelectric effect. In conclusion, the laser pulse energy is the key factor in determining whether the conventional or the Dember LPV effect dominate the LPV.

5.5.3.3 The enhancement of LPV in heterostructures The unusual LPV induced by the Dember effect can have a one-order-of-magnitude greater enhancement in the heterostructures than that in the substrates. There are mainly two physical origins based on the numerical model. Firstly, the Dember-effectinduced LPV of p-type materials is larger than that of n-type materials with the same carrier concentration. This can be totally ascribed to the difference between the mobilities of electrons and holes. The number of photo-generated electrons and holes are the same in both the p-type and n-type material under the same condition. Consequently, the lateral diffusion current densities are the same. However, the main drift carrier for p-type material are holes, while the main drift carriers for n-type material are electrons. Assuming that the system is in a steady state where the drift current densities can just balance the diffusion current densities, the lateral drift current densities are of the same value for both the p-type and n-type material. Thus, the drift electric field of the p-type material should be larger than that of the n-type material, as the mobility of holes is smaller than that of electrons. Hence, it can be concluded that the Dember-effect-induced LPV of the p-type material is larger than that of the n-type material. Secondly, the built-in field at the interface between the thin film and the substrate also plays an important role in the LPV effect. For revealing the influence of the built-in electric field, the potential difference between the p-type region and the n-type region

222 | Kuijuan Jin et al.

0.5

Built-in field

-3 Energy(eV)

Photovoltage(a.u.)

1.0

-4

LSMO

SNTO

-5 with built-in field without built-in field

-6 180

190

0.0

200 210 220 X(nm)

Y A

-0.5 X -1.0 -6

B P N

-4

2 4 -2 0 Laser Spot Position(mm)

6

Fig. 5.28. The calculated LPVs between the electrodes A and B in the same heterostructures with built-in field and without built-in field, respectively [23].

was zero and 0.52 Volt for the structure without and with the built-in electric field in the calculations, respectively, as shown in Figure 5.28. From Figure 5.28 it can be estimated that the heterostructure with a small built-in field of 0.52 Volt can produce a five times larger LPV than that of the heterostructure without the built-in field. For the structure with the built-in electric field, the photo-generated electron-hole pairs can be separated by the built-in electric field. Thereby, the photo-generated holes are swept into the p-type layer, and the potential of the irradiation region is raised relative to the situation without the built-in electric field. As a result, the Dember-effectinduced LPV for the structure with the built-in electric field is enhanced compared to the one without the built-in electric field. The combination of the above two mechanisms can well explain the one-order-of-magnitude enhancement of the LPV in the oxide heterostructures compared with that in the substrates. The understanding of the mechanisms for the enhanced LPV in oxide heterostructures should be useful in further designing of the structures of potential applications in novel functional devices, for instance high sensitive PSDs and powerful THz sources based on the Dember effect.

5.6 Summary In this chapter, firstly we briefly introduced the basic properties of the perovskite oxides and the photoelectric effects in the bulk. Then we discussed the progress in experimental and theoretical research of the longitudinal and lateral photoelectric effect in

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the perovskite oxide heterostructures. The recent research on the longitudinal photoelectric effect shows that the photo-induced carriers which are injected onto the perovskite oxide film can largely change the electric and magnetic properties of the film, and even lead to phase transitions. The photoelectric response in the heterostructures of perovskite oxide can be of the order of picoseconds. The thickness of the substrates can largely affect the photoelectric response. With the decrease in the thickness of the substrates, carriers can be collected by the electrodes within a shorter time, thus leading to a shorter response time. The thickness of the film in the heterostructures can also affect the photovoltage. As the thickness of the film decreases, the distance of the photo-induced carriers moving to the depletion region gets shorter, which means less recombination and larger photovoltage. Further decrease of the thickness can reduce the distance between the photo-induced carriers, and hence the photovoltage decreases. The parallel resistance also has an impact on the photoelectric response. The smaller the resistance, the faster the response will be, and the larger the resistance, the larger is the photovoltage. Recent research on LPE shows that the power of the light is the key factor which makes the unusual LPE different from conventional LPE. If the intensity of the light is large enough, the minority carrier can largely affect the LPV; thus the unusual LPE is dominant. The LPV observed in the heterostructures of perovskite oxides is much larger than that in the bulk. There are two reasons for this phenomenon. The drift electric field in p-type material can be stronger than that in n-type material. The interface plays an important role in the enhancement of the LPE. Although such advances have been made in the field of the photoelectric process of the perovskite oxide heterostructures, there still remain many open questions. In terms of experimental investigations, the design and optimization of the high-quality low dimensional structures may be very useful for understanding of the mechanism of photoelectric process and developing new photoelectric devices. Novel phenomena could be observed under the irradiation of a stronger laser. Using shorter later pulses to detect the dynamic process might be a way to study the coupling of many degrees of the freedom in the perovskite oxide heterstructures. In terms of theoretical studies, the current drift-diffusion model can only describe the effect of the photo-induced carriers. The development of new models which can be used to describe the coupling of other degrees of the freedom in the oxide structures remains a challenge. Therefore, we expect further studies on both the experimental and theoretical aspects.

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Hui Liu, Shining Zhu

6 Magnetic resonance and coupling effects in metallic metamaterials Although the invention of the metamaterials has stimulated the interest of many researchers and possesses many important applications, the basic design idea is very simple: composing effective media from many small structured elements and controlling its artificial EM properties. According to the effective-media model, the coupling interactions between the elements in metamaterials are somewhat ignored; therefore, the effective properties of metamaterials can be viewed as the “averaged effect” of the resonance property of the individual elements. However, the coupling interaction between elements should always exist when they are arranged into metamaterials. Sometimes, especially when the elements are very close, this coupling effect is not negligible and will have a substantial effect on the metamaterials’ properties. In recent years, it has been shown that the interaction between resonance elements in metamaterials could lead to some novel phenomena and interesting applications that do not exist in conventional uncoupled metamaterials. In this chapter we will give a review of these recent developments in coupled metamaterials. For the “metamolecule” composed of several identical resonators, the coupling between these units produces multiple discrete resonance modes due to hybridization. In the case of a “metacrystal” comprising an infinite number of resonators, these multiple discrete resonances can be extended to form a continuous frequency band by strong coupling. This kind of broadband and tunable coupled metamaterial may have interesting applications. Many novel metamaterials and nanophotonic devices could be developed from coupled resonator systems in the future.

6.1 Background All classical electromagnetic (EM) phenomena in various media are determined by the well-known Maxwell’s equations. To describe the EM properties of a material, two important parameters are introduced, that is, electric permittivity ε and magnetic permeability μ . In principle, if the ε and μ of materials are known, then the propagation of EM waves inside materials, or the EM phenomena at the surface between two materials can be well predicted. For example, the refraction of an EM wave at the interface is described by Snell’s law, sin θi / sin θr = nr /ni , which states that the relation between the incident angle (θi ) and the refracted angle (θr ) is determined by the refractive index, n = √εμ of the two media involved. Clearly, if we can modify ε and μ artificially, then the propagation behavior of EM waves in the material can be manipulated at will. For instance, in 1967 when Veselago first theoretically studied the EM properties of a material with a negative refractive

232 | Hui Liu, Shining Zhu index (simultaneously negative ε and μ ), he found that light will be refracted negatively at the interface between such a material and a normal positive index material [1]. This so-called “negative refraction” phenomenon does not violate the laws of physics; yet it challenges our physical perception and intuition. In such negative index media (NIM), a number of other surprising phenomena were also predicted, such as the reversed Doppler shift and Cerenkov radiation. However, Veselago’s work was ignored for a long time, because no such double negative materials (i.e. where both ε < 0 and μ < 0) are obtainable in nature, making negative refraction seemingly impossible. Indeed, we are limited by the natural material properties. Most dielectrics only have positive permittivities. For most metals, ε < 0 can be met at the optical range, and the plasma frequency can be moved downwards into microwave range by replacing the bulk metal with a rodded medium [2–4], yet permeability is always positive. Negative μ is accessible in some ferromagnetic materials in the microwave region, but they are difficult to find the above terahertz frequencies in the natural world. In recent years, to achieve designable EM properties, especially negative μ at high frequencies, people have invented novel artificial materials known as metamaterials. The basic idea of a metamaterial is to design artificial elements that possess electric or magnetic responses to EM waves. Many such elements can work as artificial “atoms” to constitute a metamaterial “crystal.” The geometric size of these atoms and the distances between them are much smaller than the wavelength of EM waves. Then, for an EM wave, the underlying metamaterial can be regarded as a continuous “effective medium.” Correspondingly, the property of a metamaterial can be described by two effective parameters: effective permittivity εeff and permeability μeff . In 1999, Pendry first designed a metallic magnetic resonance element: a split-ring resonator (SRR) [5]. When an SRR is illuminated by light, the magnetic component of the EM wave induces the faradic current in this structure, giving rise to a magnetic dipole. Using SRRs as structure elements, Pendry constructed a new kind of magnetic metamaterial. The effective permeability, μeff of this metamaterial has the form μeff = 1 −

Fω 2 2 + i𝛾ω ω 2 − ωmp

(6.1)

where F is the fractional volume of the cell occupied by the SRR. Equation (6.1) suggests that μeff follows a Drude–Lorentz resonance, and μeff can be negative around the frequency ωmp if the damping term 𝛾 is not so large. Such plasmon resonance in SRR is caused by a magnetic field, so the corresponding resonance frequency, ωmp here is called the magnetic plasmon (MP) frequency. Motivated by Pendry’s work, D. R. Smith combined SRR and metallic wires to construct a metamaterial with simultaneously negative εeff and μeff [6]. The negative refraction proposed by Veselago was finally experimentally verified in the microwave region [7]. One of the most important applications of NIMs is a superlens, which allows imaging resolution beyond the diffraction limit [8–13]. Considering its significant applications in the visible region, increasing the MP frequency ωmp to obtain negative refrac-

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tion for visible light is a very valuable and challenging task. Given that ωmp arises from an inductor-capacitor circuit (LC) resonance in SRRs and is determined by the geometric size of this structure, it can be increased by shrinking the size of the SRRs. In 2004, X. Zhang and colleagues fabricated a planar structure composed of SRRs. The size of the SRR was just a few micrometers, and ωmp was around 1 THz [14]. Immediately after X. Zhang’s work, Soukoulis and colleagues fabricated an SRR sample with a unit cell of several hundred nanometers, and ωmp was raised to 100 THz [15]. Another result is obtained around 1.5 μm, which is the telecommunication wavelength in the infrared range [16]. As the structure of an SRR is so complex, it is very difficult to decrease its geometric size any further with the existing nanofabrication technique. To obtain MP resonance at higher frequencies, researchers began to seek other simple MP structures. In fact, inductive coupled rod pairs are very simple structures that Zheludev and colleagues proposed as constituting chiral metamaterials [17]. Shalaev and colleagues found that such nanorod pairs could also be used to produce MP resonance and negative refraction at the optical communication wavelength of 1.5 μm [18, 19]. At almost the same time, S. Zhang et al. proposed a double-fishnet structure to obtain negative refraction at about 2 μm [20]. Although Shalaev and S. Zhang verified that their structures possessed a negative refraction index by measuring the phase difference of the transmitted waves, they could not directly observe the negative refraction in their monolayer metamaterial structures. Until quite recently, direct negative refraction was observed by X. Zhang and colleagues in the three-dimensional bulk metamaterials of nanowires [21] and fishnet structures [22] in the optical region. Besides the aforementioned important works on negative refraction, many other studies from recent years provide a good introduction to the rapid progress that has taken place in this field [23–27]. In addition to negative refractions, MP resonance has also been applied to another metamaterial that has attracted considerable attention, namely, cloaking materials [28–31]. Although the invention of metamaterial has stimulated the interest of many researchers, and its various applications have been widely discussed, the basic design idea is very simple: composing effective media from many small structured elements and controlling its artificial EM properties. According to the effective-media model, the coupling interactions between the elements in metamaterials are somewhat ignored; therefore, the effective properties of metamaterials can be viewed as the “averaged effect” of the resonance property of the individual elements. However, the coupling interaction between elements should always exist when they are arranged into metamaterials. Sometimes, especially when the elements are very close, this coupling effect is not negligible and will have a substantial effect on the metamaterial’s properties. Under such circumstances, the uncoupling model is no longer valid, and the effective properties of the metamaterial cannot be regarded as the outcome of the averaged effect of a single element (see Figure 6.1). Many new questions arise: How do we model the coupling in metamaterials? What new phenomena will be introduced by this coupling effect? Can we find any new interesting applications in these coupled systems?

234 | Hui Liu, Shining Zhu

Fig. 6.1. Classification of coupled metamaterials.

Up to now, coupled magnetic metamaterials consisting of resonance elements with a strong coupling interaction have already developed into an important branch of metamaterial research. The “hybridization effect” caused by these magnetic coupling interactions between resonators in metamaterials is attracting increased interest. Some multiple hybrid modes or continuum collective hybrid modes were found in metamaterials after including this hybridization effect (see Figure 6.1). Quite a number of papers have already reported this new kind of coupled resonance modes. Various novel phenomena and properties have been explored, and these have led to many new interesting applications that do not exist in uncoupled metamaterials.

6.2 Magnetic metamolecules 6.2.1 Plasmon hybridization effect In 2003, Halas and colleagues introduced a hybridization model to describe the plasmon response of complex nanostructures. It was shown that the resonance modes of a complex metallic nanosized system could be understood as the interaction or hybridization result of the elementary geometries. The hybridization principle provides a simple conceptual approach to designing nanostructures with desired plasmon resonances. In their subsequent work, this method was successfully used to describe the plasmon resonance in a nanoshell [32], nanoparticle dimers [33], nanoshell dimers [34, 35], and nanoparticles near metallic surfaces [36].

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6.2.2 Hybridization effect in magnetic metamolecules In fact, the hybridization model could also be applied to deal with the EM wave response of metamaterials that comprise many resonance elements. SRR is the best known magnetic “atom” of metamaterials. Therefore, the investigation of how SRRs interact with each other is both a fundamental and typical study. Apparently, a magnetic dimer (MD) made of two SRRs is the simplest system with which to study the coupling effect [37]. In Figure 6.2, we present the general configuration of an MD, composed of two identical SRRs separated by a finite distance, D. To study the magnetic response of this MD, an MP hybridization model was established. In our approach, we use the Lagrangian formalism, first calculating the magnetic energy of a single SRR and later expanding the theory for a system of two coupled SRRs. For simplicity, in the analysis we consider each SRR an ideal LC circuit composed of a magnetic loop (the metal ring) with inductance L and a capacitor with capacitance C (corresponding to the gap). The resonance frequency of the structure is given by ω0 = 1/√LC, and the magnetic moment of the SRR originates from the oscillatory behavior of the currents induced in the resonator.

Fig. 6.2. (a) Structure of a magnetic dimer; (b) equivalent LC circuit; (c) metamaterial made of identical dimer elements (from [37]).

If we define the total charge, Q, accumulated in the slit as a generalized coordinate, the Lagrangian corresponding to a single SRR is written as L = LQ̇ 2 /2−Q2 /(2C), where Q̇ is the induced current, LQ̇ 2 /2 relates to the kinetic energy of the oscillations, and Q2 /(2C) = Lω02 Q2 /2 is the electrostatic energy stored in the SRR’s gap. Similarly, the Lagrangian that describes the MD is a sum of the individual SRR contributions with

236 | Hui Liu, Shining Zhu an additional interaction term L=

L ̇2 2 2 L (Q −ω Q ) + (Q̇ 22 − ω02 Q22 ) + M Q̇ 1 Q̇ 2 , 2 1 0 1 2

(6.2)

where Qi (i = 1, 2) are the oscillatory charges, and M is the mutual inductance. By substituting I in the Euler Lagrange equations, 𝜕L d 𝜕L )− ( = 0 (i = 1, 2), dt 𝜕Q̇ i 𝜕Qi

(6.3)

it is straightforward to obtain the magnetic plasmon eigenfrequencies, ω+/− = ω0 / √1 ∓ κ , where κ = M/L is a coupling coefficient. The high energy or anti-bonding mode, |ω+ ⟩, is characterized by anti-symmetric charge distribution (Q1 = −Q2 ), while the opposite is true for the bonding or low energy |ω− ⟩ magnetic resonance (Q1 = Q2 ). Naturally, the frequency split Δω = ω+ − ω− ≈ κω 0 is proportional to the coupling strength. The hybridization of the magnetic response in the case of a dimer is mainly due to inductive coupling between the SRRs. If each SRR is regarded as a quasi-atom, then the MD can be viewed as a hydrogen-like quasi-molecule with energy levels ω− and ω+ originating from the hybridization of the original (decoupled) state, ω0 . The specific nature of the MP eigenmodes is studied in Figure 6.3 in which the local magnetic field distributions are depicted for the low energy (ω− ) and high energy (ω+ ) states, respectively. In accordance with the prediction based on the Lagrangian approach, the SRRs oscillate in-phase for the bonding mode |ω− ⟩ and out of the phase

Fig. 6.3. The local magnetic field profiles for the (a) bonding and (b) antibonding MP modes; The dependence of the resonance frequencies (c) and the frequency gap (d) on the distance between two SRRs (from [37]).

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for the antibonding mode |ω+ ⟩. Since the mutual inductance M decreases dramatically with distance, a strong change in the resonance frequencies ω± is expected. This phenomenon is demonstrated in Figure 6.3c,d, where the MP eigenfrequencies ω± and the frequency change Δω = ω+ − ω− are calculated. With a decreasing separation between the SRR, an increase in the frequency gap Δω is observed. The opposite effect takes place at large distances where the magnetic response is decoupled. This result has already been experimentally proven in the microwave range [38].

6.2.3 Stereometamaterial Recently, the coupling mechanism between two stacked SRRs was found not only to be determined by the distance between the two elements, but also to depend on the relative twist angle, ϕ [39]. The Lagrangian of such a twisted structure is a combination of two individual SRRs with the additional electric and magnetic interaction terms L=

L ̇2 2 2 L (Q −ω Q ) + (Q̇ 22 − ω02 Q22 ) + MH Q̇ 1 Q̇ 2 2 1 0 1 2 + ME ω02 Q1 Q2 [cos ϕ − α (cos ϕ )2 + β (cos ϕ )4 ].

(6.4)

In fact, magnetic and electric coupling coexist in the system when ϕ ≠ 90∘ , 270∘ . When ϕ is changed, although magnetic coupling maintains the same value, electric coupling will change significantly. Magnetic and electric interactions contribute oppositely and positively for ϕ = 0∘ and 180∘ twisted structures, respectively. By solving the Euler–Lagrange equations, the eigenfrequencies of these coupled systems can be obtained as 1 ∓ κE [cos ϕ − α (cos ϕ )2 + β (cos ϕ )4 ] ω± = ω 0 √ , (6.5) 1 ∓ κH where κE = ME /L and κH = MH /L are the coefficients of the overall electric and magnetic interactions respectively. These results lead to an exciting new concept of plasmonic structures: stereometamaterials, which will have profound application potentials in biophotonics, pharmacology, as well as diagnostics.

6.2.4 Optical activity in magnetic metamolecules In the simulations, a plain EM wave is incident on the system and the amplitude and phase change of the transmission wave is detected in far field. Although the incident ⇀ ̂ the transmission wave is found to acquire both light is linearly polarized (E = Ey y), x, and y electric field components in the resonance frequency range and some phase difference between the two orthogonal components. This change in polarization and phase delay origins from the specific 3-D chiral arrangement of two SSRRs: one SSRR is shifted a distance from the other and rotates 90 ∘ . The electric field in the slit of the

238 | Hui Liu, Shining Zhu first SSRR is aligned along the y-axis, and thus a y-polarized incident wave is electrically coupled into the system. At resonance, strong magnetic interaction between the SSRRs helps to transfer the energy from the front resonator to the back SSRR. Since the electric field in the slit gap of the second SSRR is along x,̂ the electric dipole radiation carries the same polarization. Thus the transmitted wave, detected in far field, is a superposition of x- and y-polarized light. The change in polarization is easily understood by observing the time evolution of the end point of the electric field vector as it travels through space. For an observer facing the approaching wave, the track of the end point is described by the well-known relationship (

Ey 2 2Ex Ey Ex 2 ) +( ) − cos δ = sin2 δ , |Ex | |Ey | |Ex | |Ey |

(6.6)

where the phase difference δ between the components of the electric field is calculated for the transmitted wave. The polarization state of the EM wave is thus determined by δ . The angle θ between the major polarization axis and ŷ can be calculated from the equation tan 2θ = 2|Ex ||Ey | cos δ /(|Ex |2 − |Ey |2 ). Therefore, we can obtain the polarization state if we know the amplitude and phase of x- and y-component of the transmission wave. Firstly, the optical activity of the stereometamaterial is measured in the microwave range. The sample is fabricated according to the design given in Figure 6.2. It is composed by two square copper rings with their slits perpendicular to each other. These two rings are separated by a distance D which is filled with polystyrene with permittivity of 4. Some samples are fabricated with different thickness of middle polystyrene layer. For different thicknesses, the magnetic resonance coupling strength is different. In the experiment setup, a linearly polarized microwave incidents from the source and perpendicularly passes through the plan of our magnetic dimer array. A detector (Agilent E8363A vector network analyzer) is set on the other side of the plan to receive the transmitted signal. As the detector is also polarized, we can measure the different polarization components of the transmission wave through rotating the detector. The experiment results show that the polarization state is changed when a linearly polarized wave passes through the sample. For an incident wave with different frequency, we can obtain different polarized transmission waves. A right-handed elliptical polarized state is produced at the lower frequency of the symmetry mode. A left-handed elliptical polarized state is produced at the higher frequency of the antisymmetry mode. Between these two resonance frequencies, the polarization state is changed continuously from the right-handed to the left-handed state. Also, for stereometamaterial, the left-handed and right-handed polarized waves possess different effective indices; this will show as a circular dichroism [40].

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6.2.5 Radiation of magnetic metamolecules For the structure given in Figure 6.2a, the hybridization effect can change the far-field radiation properties of the SRR dimmer. Through simulations, we can calculate the radiation power of such a coupled structure. We found that radiation loss is suppressed at a lower frequency symmetry mode and is enhanced at a higher frequency antisymmetry mode [41]. The differing radiation behaviors of the hybrid modes can be explained using a visual representation of the dipole model. At resonance frequencies, the electromagnetic wave strongly coupled with the SRRs and generated an oscillating current in the structure. As a result, a strong electric field was generated within the split gap, and a magnetic field was induced inside the loop. Thus, the equivalent radiation structure of one SRR can be viewed as an electric dipole in the split gap in the y-direction and a magnetic dipole in the loop in the z-direction. Accordingly, for a single SRR, the radiation pattern is a combination of an electric dipole and a magnetic dipole with their directions perpendicular to each other. According to classical electrodynamics theory, the radiation power of an electric dipole is much stronger than that of a magnetic dipole, resulting in the latter being dominated by the former. However, the radiation behavior of the two hybrid modes is quite different for the two coupled SRRs. At the lower frequency of the bonding mode, the system is composed of two magnetic dipoles in the same direction and two electric dipoles in opposite directions. As the electric dipoles cancel each other, the coupled SRRs can be regarded as a magnetic dipole. At the higher frequency of the antibonding mode, the system becomes composed of two magnetic dipoles in opposite directions and two electric dipoles in the same direction. As the magnetic dipoles cancel each other, the coupled SRRs can thus be regarded as an electric dipole. Accordingly, it is obvious that the coupled structure behaves like an equivalent magnetic dipole in the bonding mode and an equivalent electric dipole in the antibonding mode. In the above analysis, it can be seen that the hybridization effect of the coupled SRRs system greatly affected the farfield radiation pattern of the coupled SRR structure. The radiation losses in the bonding and antibonding modes also changed dramatically due to this coupling effect.

6.2.6 Other designs of magnetic metamolecules In another work by Giessen and colleagues, the hybridization effect of magnetic resonance was also observed in four-stacked SRRs [42]. In addition to in identical resonators, hybrid modes were found in coupled structures composed of different resonators, including SRR pairs [43], cut-wire pairs [44], trirods [45, 46], and nanosandwiches in defective photonic crystals [47]. These hybrid modes could lead to some new interesting and useful properties, such as optical activity [37, 40] and omnidirectional broadband negative refraction [45] and plasmonic sensors [48].

240 | Hui Liu, Shining Zhu

6.3 One-dimensional magnetic resonator chains Linear chains of closely spaced metal nanoparticles have been intensely studied in recent years. Due to the strong near-field coupling interaction among these nanoparticles, a coupled electric plasmon propagation mode can be established in this chain and can be used to transport EM energy in a transverse dimension considerably smaller than the corresponding wavelength of illumination [49–54]. As this system can overcome the diffractive limit, it can function as a novel kind of integrated subwavelength waveguide. In the previous section we presented the hybridization effect of hybrid modes among several coupled resonators. In this section, we will generalize the theoretical model to one-dimensional metachains of coupled magnetic resonators. We will show that the collective excitation of infinite magnetic atoms in metamaterials can induce a new kind of coherent collective waves, namely, a magnetic plasmon (MP) wave.

6.3.1 Periodic magnetic resonator chain According to classical electrodynamics theory, the radiation loss of a magnetic dipole is substantially lower than the radiation of an electric dipole of a similar size [55]. Thus, using MP to guide EM energy over long distances has great potential for direct application in novel subdiffraction-limited transmission lines without significant radiation losses.

6.3.1.1 Magnetic inductive waves Indeed, MP resonance has been already applied to a one-dimensional subwavelength waveguide in the microwave range [56–58]. Shamonina et al. proposed a propagation of waves supported by capacitively loaded loops using a circuit model in which each loop is coupled magnetically to a number of other loops [56]. Since the coupling is due to induced voltages, the waves are referred to as magneto-inductive waves (MI). MI waves propagating on such 1-D lines may exhibit both forward and backward waves, depending on whether the loops are arranged in an axial or planar configuration. Moreover, the band broadening could be obtained due to the excitation of MI waves, and the bandwidth changes dramatically as we vary the coupling coefficient between the resonators [59]. A kind of polariton mode could be formed by the interaction between the electromagnetic and MI waves, resulting in a tenability of the range where becomes negative [58]. In a biperiodic chain of magnetic resonators, the dispersion of the MI wave will be split into two branches analogous to acoustic waves in solids, and this can be used to obtain specified dispersion properties [60, 61]. In addition to this kind of MI wave, electroinductive (EI) waves were also reported in the microwave range [62]. Further, the coupling may be either of the magnetic or electric type, de-

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pending on the relative orientation of the resonators. This causes the coupling constant between resonators to become complex and leads to even more complicated dispersion [63]. Up to now, a series of microwave devices based on MI waves have been proposed, such as magneto-inductive waveguides [64], broadband phase shifters [65], parametric amplifiers [66], and pixel-to-pixel subwavelength imagers [67, 68].

6.3.1.2 One-dimensional chain of SRR However, in the optical range, the ohmic loss inside metallic structures is much higher than in the microwave range. The MI coupling between the elements is not strong enough to transfer the energy efficiently. In order to improve the properties of the guided MP wave, the exchange current interaction between two connected SRRs is proposed [69], which is much stronger than the corresponding MI coupling. Figure 6.4b shows one infinite chain of SRRs constructed by connecting the unit elements (see Figure 6.4a) one by one. The magnetic dipole model can be applied to investigate this structure. If a magnetic dipole, μm , is assigned to each resonator and only nearest neighbor interactions are considered, then the Lagrangian and the dissipation function of the system can be written as 1 1 2 Lq̇ − (q − qm+1 )2 + M q̇ m q̇ m+1 2 m 4C m 1 R = ∑ 𝛾q̇ 2m . m 2 L=∑ m

(6.7)

Fig. 6.4. (a) Structure of a single SRR; (b) one-dimensional chain of SRRs; (c) equivalent circuit of the chain; (d) FDTD simulation of MP wave propagation along the chain; (e) dispersion curve of the MP wave (from [69]).

242 | Hui Liu, Shining Zhu Substitution of equation (6.7) in the Euler–Lagrangian equations yields the equations of motion for the magnetic dipoles μ̈m + ω02 μm + Γ μ̇m =

1 + 2μm + μm+1 ) − κ2 (μ̈m−1 + μ̈m+1 ), κ ω 2 (μ 2 1 0 m−1

(6.8)

where κ1 and κ2 are the coefficients of the exchange current and MI interactions, respectively. The general solution of equation (6.8) corresponds to an attenuated MP wave: μm = μ0 exp l(−mα d) exp (iω t − imkd), where ω and k are the angular frequency and wave vector, respectively, α is the attenuation per unit length, and d is the size of the SRR. By substituting μm (t) into equation (6.8) and working in a small damping approximation (α d ≪ 1), simplified relationships for the MP dispersion and attenuation are obtained: 1 − κ1 [1 + cos (kd)] ω 2 (k) = ω02 . (6.9) 1 + 2κ2 cos (kd) In Figure 6.4e, numerically and analytically estimated MP dispersion properties are depicted as dots and solid curves respectively. In contrast to the electric plasmon modes in a linear chain of nanosized metal particles where both transverse and longitudinal modes can exist, the magnetic plasmon is exclusively a transversal wave. It is manifested by a single dispersion curve that covers a broad frequency range, ω ∈ (0, ωc ), with the cutoff frequency ℎωc ≈ 0.4 eV. Finally, it is important to mention that the MP properties can be tuned by changing the material used and the size and shape of the individual SRRs. Besides, though changing the connection configuration between SRRs, we can obtain different collective excitation of two different magnetic plasmon modes (see Figure 6.5a and b) [70]. The dispersion curves of these magnetic plamson modes are provided in Figure 6.5c and d. This study provides another method to construct metachains with wide bands that accommodate the MP wave propagation within the preferred characteristics.

6.3.1.3 One-dimensional chain of slit-hole resonators Generally, the MP resonance frequency increases linearly with the decrease in the overall SRR size. However, the saturation of the magnetic response of the SRR at high frequencies prevents this structure from achieving high-frequency operation [71]. In addition, the complicated shape and narrow gap of the SRRs make experiments very challenging. The slit-hole resonator (SHR) [72] is considered to be a good alternative to make subwavelength waveguides because of their simple structures and high working-frequency regime. Figure 6.6a presents the geometry model of an SHR structure. The corresponding LC transmission line model is shown in Figure 6.6b. We also fabricate a sample of a diatomic SHR structure with an FIB (focused ion beam) method, whose image is presented in Figure 6.6c.

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Fig. 6.5. Subwavelength waveguides constituted by the SRR chains with (a) homoconnection and (b) anti-connection. (c) and (d) are the Fourier transformation map in the ω − k space corresponding to the waveguides (a) and (b), respectively (from [70]).

Fig. 6.6. (a) Structure of a diatomic chain of SHRs; (b) equivalent LC circuit of the chain; (c) FIB image of the fabricated sample. The local current distributions are calculated for the (d) optical and (e) acoustic modes; (f) measured transmission map and the calculated angular dependence curve of the optical MP mode (from [72]).

244 | Hui Liu, Shining Zhu For the infinite diatomic chain of SHRs, the Lagrangian equation of this system is expressed as L = ∑( m

L1 Q̇ 2m L2 q̇ 2m (Qm − qm−1 )2 (Qm − qm )2 ). + − − 2 2 2C 2C

(6.10)

Here, we define the oscillating charges in the m-th unit cell as Qm for the bigger SHR with an inductor L1 , and qm for the smaller SHR with an inductor L2 (m = 0, ±1, ±2, ±3, . . .). The two corresponding magnetic dipoles, Um and μm , are defined as Um = Q̇ m /S, μm = q̇ m /s, where S and s are the areas of bigger and smaller SHRs, respec𝜕L 𝜕L tively. From the Euler–Lagrangian equations dtd ( 𝜕𝜕L ) − 𝜕U = 0 and dtd ( 𝜕𝜕L ) − 𝜕μ =0 μ̇ U̇ m

m

m

m

(m = 0, ±1, ±2, ±3, . . .), we obtain the oscillation equations of the m-th bigger SHR and smaller SHR as Ü m + ω12 (2U m − μm − μm−1 ) = 0 { (6.11) μ̈m + ω22 (2μ m − Um − Um+1 ) = 0, where ω1 = 1/√L1 C, and ω2 = 1/√L2 C. Solving the Eigen equations, the MP dispersions are attained as ω±2 = (ω12 + ω22 ) ± √(ω14 + ω24 ) + 2ω12 ω22 cos (kd).

(6.12)

The dispersion relations are numerically depicted as two solid black curves in Figure 6.6d and e. There are two separate dispersion branches for the diatomic chain: the upper branch ω+ (k) and the lower branch ω− (k). For these two branches, the respective resonant manners of the m-th unit cell are quite different. Our simulations show that for the lower branch, ω− (k), Um , and μm oscillate in the same phase (see Figure 6.6d), whereas for the upper branch, ω+ (k), they oscillate in the antiphase (see Figure 6.6e). Using the analogy of the diatomic model of crystal lattice wave, we can refer to the upper curve ω+ (k) as the optical branch and the lower curve ω− (k) as the acoustic branch. Compared with the monatomic chain, which possesses only the acoustic dispersion branch, the optical branch is a new kind of MP mode found in the diatomic chain. Finally, the measured transmittance at different incident angle and different frequency is plotted in Figure 6.6f, which shows an evident MP mode band consistent with the optical branch in the theoretical results.

6.3.1.4 One-dimensional chain of nanosandwiches In the category of magnetic plasmon structures, the nanosandwich structure is also a good choice to reach high resonant frequency, even to light frequency region. Figure 6.7a presents the geometry of a single nanosandwich, composed of two metallic nanodiscs and a dielectric middle layer [73]. The antiparallel currents in the metallic slabs induce a high intensity and confined magnetic field at a certain frequency, thus it can be seen as a magnetic atom. The frequency spectrum and field distribution of such nanosandwich is shown in Figure 6.7b–d, respectively. Such a magnetic

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Fig. 6.7. (a) Structure of a single nanosandwich; (b) one-dimensional chain of the nanosandwich; (c) electric field and (d) magnetic field of the MP wave propagation along the chain; (e) dispersion curve of the MP wave (from [73]).

atom can be used to construct a linear magnetic chain. Due to the near-field electric and magnetic coupling interactions, an MP propagation mode is established in this 1-D system. When excited by an EM wave, a strong local magnetic field is obtained in the middle layer at a specific frequency (Figure 6.7c). For this magnetic plasmon resonance mode, the corresponding electric fields are given in Figure 6.7d. It should also be mentioned that such an MP waveguide is a subwavelength, the energy flow cross section of which is plotted in Figure 6.7e. The field is confined in a small area smaller than the wavelength scale. Through a Fourier transform method, the wave vectors of this MP wave at different EM wave frequencies are calculated. Then, the MP wave dispersion property is obtained (shown as a white line in Figure 6.7e). The light line in free space is also given as a black dotted line in the figure. The MP curve is divided into two parts by the light line. The part above the light line corresponds to bright MP modes whose energy can be radiated out from the chain, while the part below the light line corresponds to dark MP modes whose energy can be well confined within the chain. It is easy to see that the bright MP modes are much weaker than the dark MP modes for their leaky property. Therefore, only those EM waves in the frequency range of the dark MP modes can be transferred efficiently without radiation loss.

6.3.2 Nonperiodic chain of magnetic resonators We also do some structure engineering on the nanosandwich chain. Once the above results for a monoperiodic chain of nanosandwiches have been generalized to graded structures [74], some new interesting properties, such as slow group velocity and a new type of field distribution, are found in these more complex structures. Here, we

246 | Hui Liu, Shining Zhu consider a chain with 41 nanosandwiches, and the spacing obeys the following rule: dm = 225 + 100[(m − 1)/39].

(6.13)

The dispersion relation of the graded chain is shown in Figure 8a. The MP modes can be divided into three parts: gradon (the special mode belonging to the graded structure), extended mode, and evanescent mode. The field distributions of these three types are quite different and the location the field of gradon is strongly depended on the frequency, which is presented in Figure 6.8b. By employing this property, one can manage a wavelength selective switch. Three-port and four-port switches can be realized in this graded nanosandwich chain. The field distributions of the magnetic field corresponding to different mode of these switches are presented in Figure 6.8c. Some

Fig. 6.8. (a) Dispersion relation and propagating length of the graded nanosandwich chain; (b) filed distributions of three different modes; (c) field distributions corresponding to different modes in both three-port and four-port switches. (from [74])

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new interesting properties, such as slow group velocity and band folding of MP waves, are found in these more complex structures.

6.3.3 Nonlinear and quantum optics of magnetic resonators The nonlinear and quantum optics of one-dimensional magnetic plasmon is a very frontier topic. The nonlinear optical properties of metal is very weak, which can be ignored in the usual case. In order to produce an efficient nonlinear optical effect, we can combine the magnetic resonator chain with other nonlinear materials. In one recent work [75] we designed a metallic nanosandwich cavit, which can possess two magnetic plasmon resonance modes at two prescribed resonance wavelengths. Combined with the ytterbium-erbium codoped gain material Er:Yb:YCOB, this cavity can have resonance not only at the lasing wavelength but also at the pumping wavelength, which leads to efficient pump-laser conversion. On the other hand, a quantum dot is also a kind of nonlinear medium which has attracted much research interest. A magnetic metachain can also be combined with quantum dots to obtain nonlinear magnetic plasmon waves [76]. The Hamiltonian of a coupled metamaterial is expressed as follows: M L 1 2 Mh ̇ ̇ H(Qm Q̇ m ) = ∑ [ Q̇ 2m + Qm + (Qm Qm+1 + Q̇ m Q̇ m−1 ) + e (Qm Qm+1 + Qm Qm−1 )] 2 2C 2 2 m (6.14) In this equation, m is the number of unit cells; L and C are the inductor and the capacity of the single resonator, respectively; and Mh and Me describe the magnetic and electric coupling between the unit cells, respectively. Meanwhile, Q and Q̇ correspond to the charge and current on the unit cell, respectively. The terms formed by Q̇ are the current-induced kinetic energies, and those consisting of Q denote the charge-induced energies belonging to the potential energy. If we use the Fourier expansion, Qm = √1M ∑k Qk eikRm . Since the charge Qk has a canonically conjugate variable P = 𝜕L = ( L + M cos(kd))Q̇ , by using the Hamiltonian canonical equations k

𝜕Q̇ k

2

h

−k

Q̇ k = 𝜕H/𝜕Pk and Pk = −𝜕H/𝜕Qk . Considering the quantum condition, Q̂ m and P̂ m possess the commutation relation [Q̂ m P̂ m ] = iℎ. After some derivation, the commutator between and can be derived as [Q̂ P̂ ] = iℎ. In the equation, we used the unitary con󸀠

k k

dition M1 ∑m ei(k+k )md = δk,−k . We also performed a Bogoliubov transformation to the Hamiltonian in equation (6.3) by introducing a set of creation and annihilation operators, â k = Uk Q̂ k + iVk P̂ −k and a+k = Uk Q̂ −k − iVk P̂ k , with parameters Uk = (ℎ)−1/2 √ξ 1 and Vk = (ℎ)−1/2 /√ξ , and ξ = √[ 2C + Me cos(kd)][ 2L + Mh cos (kd)] [16, 17]. The Hamiltonian of a coupled metamaterial in number representation can thus be obtained as follows: 1 H = ∑ (a+k ak + ) ℎωk . (6.15) 2 k

248 | Hui Liu, Shining Zhu We take the compound system composed of metamaterial and quantum dot material as an example. The interaction Hamiltonian of the system can be expressed as Hint = ∑r E ⋅ d; where E denotes the electric field, d refers to the dipole moment of exciton in the quantum dot, and the summation corresponds to all quantum dots in the system [18]. After some derivation, the quantized interaction Hamiltonian can be obtained as follows: k + − − + (6.16) Hint = ℎ[Gk (ak σk + ak σk )]. In this equation, the coupling constant Gk is equal to √ ∫ (ρ2 (α ) − ρ1 (α ))(ϕk (α ) ⋅ d1,2 )2 dα 3 /ℎ, which is of crucial importance in describing the interaction between the meton and the exciton. ϕk (α ) is the eigenstate of the meton corresponding to the electric field distribution with the energy normalized to ℎωk /2, with α being the position of the quantum dot in the unit cell. ρ2 and ρ1 are the population densities of the two levels. The transition operators σk+ and σk− indicate the creatiion and annihilation, respectively, of the quantum dots belonging to the whole system with momentum ℎk. The rotating wave approximation was used to eliminate the energy nonconversing terms. Here, the quantum model is not limited to the magnetic metachain: it can also be used for other quantum metamaterials.

6.4 Magnetic plasmon crystal In addition to the abovementioned 1-D structures, the MP mode introduced by the coupling effect in 2-D systems is also an interesting topic. For 2-D metamaterials, the most important applications are negative refraction, focusing, and superlensing. How do the coupling interactions between elements affect the above processes? They cannot be handled by the conventional effective medium theory. In the microwave range, MI wave theory has been proposed to deal with the coupling effect in the 2-D system. An MI superlens was proposed based on employing the coupling between resonators, which eliminates the weakness of Wiltshire’s first Swiss-roll superlens and has potential for MRI applications. The focusing of indefinite media, originally treated by Smith [77], was investigated by Kozyrev with the aid of MI wave theory. It was found that partial focusing and multiple transmitted beams can be formed by the excitation of MI waves [78]. A further comparison between effective medium theory and MI wave theory was also given by Shadrivov [58], in which the reflection and refraction of MI waves on the boundary of two different effective media were studied. It was shown that both positive and negative refraction may occur under some configurations of the elements [79]. Another interesting finding is that spatial resonances could be formed by the propagation of MI waves on a 2-D array of

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magnetic resonators [80]. Different boundary conditions will produce different current and magnetic field distributions.

6.4.1 Two-dimensional fishnet structure In the optical frequency range, fishnet structures are well known magnetic metamaterials [20] constructed by a metal/insulator/metal (MIM) sandwich with perforated periodic nanohole arrays, schematically shown in Figure 6.9. This structure can somewhat be regarded as an extension of the metal film perforated with nanoholes that exhibits well-known extraordinary optical transmission (EOT) discovered by T. W. Ebessen in 1998 [81]. However, the greatest significance of this fishnet structure is owing to its novel negative refraction property, as revealed by S. Zhang in 2005 [82]. The fundamental physics to realize the negative index in this structure is based on the artificial “magnetic atoms” consisting of the LC resonance between the two coupled metallic segments and the “electric atoms” from continuous metallic strips parts, which produce simultaneous negative effective permeability and permittivity, correspondingly. Based on this kind of design, the performance of the negative index property was subsequently improved by means of structural optimizations [83–85]. Simultaneous negative phase and group velocity of the light in this fishnet metamaterial was obtained via the method of interference [86]. Even more, by further elaborate minimization, the negative index mode was pushed to the red color wavelength [87].

Fig. 6.9. Schematics of fishnet metamaterial (from [91].)

On the other hand, some attempts have been made with regard to considering the effect by stacking multilayer fishnet structures. It was numerically shown that the optical transmission loss for this negative refraction mode can even decrease by stacking more fishnet elements in the propagation direction [88]. However, the underlying physics remains obscure. Focusing on this point, our group made a detailed investigation of the coupling effect not only in the stacked fishnets but also extending the orig-

250 | Hui Liu, Shining Zhu inal double-metal-layer (DML) to a three-metal-layer (TML) fishnet and even two more multilayers [89], in which the coupling effect may be more remarkable. Two distinct modes were found in this TML structure (denoted as R1 and R2), which is actually due to the coupling of two magnetic resonances, shown in Figure 6.10a,b. This coupling effect was subsequently explained by a coupled circuit model associated with a metallic skin-depth-related layer thickness. The improved transmission property can be realized at a certain range of metal layer thickness, as shown in Figure 6.10c, indicating a reduced loss in such a coupled system. This result to a certain extent explained the former results [88], and was later proved by our experimental results. Moreover, this coupling mechanism provides an encouraging vision for realizing a real 3-D negative index metamaterial, which was ultimately fulfilled by the X. Zhang group in 2008 [22]. They fabricated a prism consisting of multilayered fishnets via the focus ion beam etching, and the negative refraction phenomena was definitely revealed by evaluating the position of the transmitted light beam at a particular wavelength.

Fig. 6.10. (a) Detected Hx intensities of TML structure with the frequency ranging from 200 to 300 THz by two probes, which are located at the neighboring layer gaps. (b) The corresponding simulated Hx distributions at the frequencies of R1 and R2. (c) Transmission intensities of mode R0 and R1 versus the layer thickness for the DML and TML structures, respectively (from [89]).

It has been revealed that the coupling effect plays an important role in improving the NIM performance. Actually, the coupling effects are still presented in the in-plane dimensions. G. Dolling et al. noticed this effect, and they found magnetization waves in such fishnet structure at the end of 2006 [90]. However, what they observed is only the lowest mode of a kind of collective magnetic excitation due to the in-plane coupling, which was later called magnetic plasmon polaritons (MPPs). Actually, our group made a detailed investigation on the plasmonic modes of the artificial “magnetic atoms”. Multiple MPP modes associated with reciprocal vectors of the lattice was convinc-

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Fig. 6.11. (a) Simulated magnetic field distribution map for mode 1 (a) and mode 2 (b), in the xyplane (z = 0) in the center of the middle an SiO2 layer. (c) Calculated transmittance map of samples as middle layer thickness s ranging from 25 to 55 nm, where the MPP modes and SPP mode can be clearly observed. Measured (d) and calculated (e) transmission spectra of the fishnet samples with various periods of the hole lattice in the x-direction (from [91]).

ingly demonstrated [91]. Figure 6.11a,b show the magnetic field distributions of two mentioned MPP modes associated with reciprocal vectors of G(0, 1) and G(1, 1). Figure 6.11c is a calculated transmission map with the SiO2 layer thickness ranging from 25–55 nm, where two MPP modes and SPP modes are clearly exhibited. Using this plasmonic property, we also can easily modulate the eigen frequency of the MPP modes via adjusting the structure parameters, as Figure 6.11d,e shows. Afterwards, the dispersion properties of the MPP modes in fishnet structures with a rectangular hole array was studied in succession [92]. By carefully investigation of the transmission property on the oblique incidence for the s- and p-polarization cases, we found a polarization-dependent dispersion property of the involved MPP modes, which was indicated in transmission maps for these two polarizations, as shown in Figure 6.12. From that, we can see that the MPP(1, 1) mode in an s-polarization actually exhibits much larger dispersions than the lowest mode, resulting in two split modes MPP(+ 1, 1) and MPP(−1, 1) with the degeneration broken up. This is very similar as the property of SPP. As for the anisotropic property, we attributed them to different coupling intensities among the artificial “magnetic atoms” with the 2-D plane. Ultimately, we used a formula to describe the dispersion property of MPPs in the fishnet structure as 2π c 1 λMPP(m,n) = ), ( (6.17) |(δy ⋅ kx ± mGx )x̂ + (δx ⋅ ky ± nGy )y|̂ ωLC where δx and δy are introduced to describe the polarization. When the incident light is x-polarized (y-polarized), δx = 1 and δy = 0 (δy = 1 and δx = 0). Recently, the optical transmission property in this fishnet metamaterial has attracted more attention, due to the discussions about the origin of the magnetic re-

252 | Hui Liu, Shining Zhu

Fig. 6.12. Measured transmission maps on an intuitive gray scale versus wavelength and wavevectors for (a) s-polarization and (b) p-polarization (from [92]).

sponse. the Garcia–Vidal group proposed a relative general theory to explain the negative refractive response in this system [93]. They considered that the electrical response of these structures is dominated by the cutoff frequency of the hole waveguide whereas the resonant magnetic response is due to the excitation of gap surface plasmon polaritons (gap-SPP) propagating along the dielectric slab. Furthermore, they predicted that the negative index mode is dispersive with the parallel momentum of the incident light. This conclusion is somewhat consistent with our experimental results [88], although further particular consideration may still theoretically be supplied concerning the different polarizations. Actually, our description of the term of MPP may not conflict with the explanation of the gap-SPP, because the magnetic resonance is identical with the antisymmetric electric resonance mode in a coupled (MIM) structure. Later the explanation of gap-SPP on the negative refractive response in fishnet metamaterial was further confirmed by more detailed analysis [94–96]. Ortuno et al. provided an elaborate interpretation of the role of surface plasmon in the transmission properties, and discovered contributions of coupled external SPP and internal SPP [94], which is privately considered to be another kind of description of the artificial electromagnetic excitations with respect to our findings about the optical magnetism induced by SPP excitations [97]. In fact, our own recent studies have shown that the MPP mode described in [91, 92] is surely identical to the gap-SPPs, due to the same lattice modulation, but only from a different point of view. Untill now, most researchers have come to the consensus that the multiple magnetic modes presented in the fishnet metamaterial with negative refractive response and the underlying physics is due to the internal coupled SPP excitations. It is exactly this coupling effect which has aroused so much interest and so many discussions. In summary, the coupling effect endows the fishnet structure with fruitful, interesting properties, and the complicated mechanism has been recognized more and

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more clearly. In the following, this novel structure in a quasi-2-D system may not only exhibit interesting properties in the broad field of optics, but also let us expect expected some interesting features in near-field optical modulations and integrate applications.

6.4.2 Two-dimensional nanosandwich structures In the above we introduced the one-dimensional chain of nanosandwiches. Such a structure can also be used to constitute two-dimensional magnetic plamson crystals. As the localized plasmon resonance has a strong field enhancement, a twodimensional nanosandwich lattice can be used for nonlinear optics and optical force.

6.4.2.1 Nanolaser in two-dimensional magnetic plasmon crystals Reducing mode volume in cavities is very favorable for achieving strong light-matter interaction processes. These strong interactions are of great benefit for applications in light-emitting devices. Recently, in order to further miniaturize the mode volume and physical structure size, researchers have begun using metal instead of dielectrics to bound the mode field. For example, a metallic-coated cavity formed by encapsulating a semiconductor heterostructure in a thin gold film is used to obtain mode volume far smaller than conventional dielectric cavities. In addition, due to the strong localized surface plasmon (SP) effect, electromagnetic energy is also able to be highly confined in the nanoscale. We designed a metallic nanosandwich cavity which can possess two magnetic plasmon resonance modes at two prescribed resonance wavelengthes [98]. Combined with the Ytterbium-erbium co-doped gain material, this cavity can have resonance not only at lasing wavelengths but also at pumping wavelengtsh, which leads to efficient pump-laser conversion. We selected the Er:Yb:YCa4 O(BO3 )3 (Er:Yb:YCOB) crystal as the gain medium, which is widely used to generate light in an optical communication wavelength region around 1.5 μm. Figure 6.13a presents the proposed laser arrays constructed by immersing a regular array of rectangular silver slab in gain medium Er:Yb:YCOB, which is supported by a silver film and SiO2 glass substrate. When magnetic resonance occurs, the electromagnetic energy is mainly stored in the space between the silver slab and film, so the structure can be considered as a cavity of confining light. One main characteristic of magnetic resonance mode is that the induced current forms loops and strong magnetic fields inside them. For different magnetic resonance mode, different numbers of loops are formed in the cavity. So, we can label one mode as MPij , where i and j denote the formed loop numbers along the x- and y-directions, respectively. It was found that the electric fields of mode MP10 and MP22 have a quite large spatial overlap, so these two modes were selected as the operation modes in our calculations. For the carefully designed geometry size in Figure 6.13a,

254 | Hui Liu, Shining Zhu

Fig. 6.13. Schematic illustration of the operation principle of the laser: (a) Pumping light incident; (b) resonantly pumping cavity mode 980 nm; (c) energy level diagram for the Er–Yb co-doped system; (d) lasing cavity mode (1550 nm) (from [98]).

the resonance wavelengths of the two modes are 980 nm and 1550 nm, respectively. Figures 6.13b and d show respectively the electric field component Ey distributions for the two modes. As the 980 nm and 1550 nm are close to the absorption peak and emission peak of Er:Yb:YCOB, respectively, we can select the higher order MP22 mode as the pumping mode (980 nm), and the lower MP10 mode as the lasing mode (1550 nm). Due to the fact that both the pumping and lasing modes are resonant in the cavity, the pumping efficiency can be greatly enhanced. The FDTD method and a set of rate equations were introduced to model the operation of the laser and predict the lasing condition. The results proved that the proposed double-resonance laser can work as a compact nanolaser with a quite low threshold value. Besides nanosandwich structures, we can also use split-hole resonator structures to obtain nanolasers based on magnetic resonance modes [99].

6.4.2.2 Nonlinear Fano resonance in two-dimensional magnetic plasmon crystals Two-dimensional magnetic plasmon crystals can also be used to enhance third harmonic generation (THG). Here, we use a two-dimensional array of nanosandwichs in the experiment [100]. Besides the local magnetic plasmon resonance (MPR) mode, surface plasmon waves can also be excited in this system (see Figure 6.14a). By finely tuning the interference between the localized MPR and propagating SPP modes at near infrared wavelength, the surface polarization of the pump laser on metamate-

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Fig. 6.14. (a) Both localized MPR mode and propagating SPP mode are excited by a TM wave. G⃗ = 2π /period is the reciprocal vector. (b) Reflection ratio of ETM /ETE on sample B under different incident angles (θ = 48 ∘ , 52 ∘ , 56 ∘ ). (c) Experimental setup of the third harmonic generation. Near-infrared was incident on the sample at angle (θ = 52 ∘ ). (d) Experimental (line with circle symbol) and calculated (solid line) THG efficiency of samples A and B at incident angle of θ = 52 ∘ (from [100]).

rial has Fano-resonant modulation. Spectral resolved third harmonic generation on this metal/dielectric hybrid nanostructure was demonstrated to show a Fano-resonant modulation. This indicates that third-order and other nonlinear optical processes can be tailored by engineering the coupling of the MPR and SPP mode; the interference between them provides us with a more flexible way to design nonlinear optical nanodevices. The linear optical property of this metamaterial was characterized using a spectroscopic ellipsometer. At the resonant wavelength, the induced currents in a disc and slab are antiparallel, which corresponds to MPR modes as previously reported. In ellipsometry measurement, the reflection coefficient ratio of TM and TE waves η = ETM /ETE was retrieved using rotating polarizer technique. In Figure 6.14b, spectrally resolved η was measured at different incident angles. A Fano-type optical response at MPR wavelength was observed. This interesting phenomenon means that the res-

256 | Hui Liu, Shining Zhu onance is not a pure MPR mode. The angle-resolved ellipsometry measurement (incident angle θ was tuned from 48∘ to 56∘ ) was then used to study the mechanism behind this phenomenon. For sample A, magnetic resonance mode is not sensitive to the incident angles, and the reflection dip remains almost at the same wavelength. Angle-resolved reflection ellipticity of the sample is shown in Figure 6.14b. The Fanoresonance is very sensitive to the incident angle and shifts to a longer wavelength with an increasing incident angle. As MPR is a kind of localized mode with a definite resonance frequency, and it should be not sensitive to the incident angle. From our previous study on plasmonic ellipsometry, the angle-dependent shift of the Fano peak in sample B is related to the excitation of propagating surface plasmon polaritons. The excitation of propagating SPP wave is governed by the following momentum conservation condition: 󸀠 k// + kspp − G󸀠 = 0 (Figure 6.4). Here, k// = (ω /c) sin θ is the x-component of the 󸀠 󸀠󸀠 vacuum wavevector; kspp = kspp + ikspp is the wave vector of SPP waves, with the imag󸀠󸀠 inary part kspp representing the ohmic loss of the SPP wave; G = 2π /period + iG󸀠󸀠 is the first order of the reciprocal vector, and the G󸀠󸀠 is the radiative loss. For the sample (with a period of 910 nm), the wavelength of first order SPP is ∼ 1.63 μm at incident an∘ gle θ = 52 , which is very close to the MPR mode with wavelength of ∼1.62 μm. Thus both SPP and MPR modes can be excited, and the interference between MPR and SPP mode shows a Fano-type shape in the reflection spectra. When propagating surface plasmon polaritons are excited, the reflected TM waves show a resonant dip at the resonant frequency, and the reflected TE wave varies slowly on the spectrum. The plasmonic excitation can be characterized from the spectrum of ellipticity (η ). Around the resonant wavelength of surface plasmon polaritons, the TM-polarized reflection spectra for SPP is defined as 󸀠󸀠

rspp = rp0 (1 −

2iG ); k// + kspp − G

here, rp0 = (ngold cos θ − cos θgold )/(ngold cos θ + cos θgold ) is the reflection coefficient on the air/gold surface (θgold is the refractive angle in gold film); G0 = ( 2π )(1 + iΓ ) = p

G󸀠 + iG󸀠󸀠 is the reciprocal vector and Γ is radiative damping rate of the surface plasmon polariton; kspp (ω ) = k0 (ω )√εgold εair /(εgold + εair )

is the wave vector of the SPP wave at the interface between gold and air. The nanosandwich can be regarded as an equivalent LC circuit and described by 1 2 a Lagrangian as L = 2L Q̇ 2 − 2C Q . Here, L and C are the inductance and capacitance of the LC circuit, Q is the net charge in the capacitor, and Q̇ is the induced current in the inductance. In the presence of ohmic dissipation and an external driving field, the Euler–Lagrange equation can be written as d 𝜕L 𝜕L 𝜕R + Em ( )− =− ̇ dt 𝜕Q 𝜕Q 𝜕Q̇

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where R = Reff Q̇ 2 /2 (Reff is the effective resistance in the system) and the e.m.f. (for dB normal incident B0 field) is given by Em = − dt0 Seff (Seff is the effective cross-section area between the patch and slab). If we define an effective magnetic dipole as m = Seff ⋅ Q̇ = αm ⋅ B0 (αm is the effective magnetic polarizability of magnetic resonator), the Euler–Lagrange equation has the following solution: αm =

2 Seff ω2 . 2 L (ω − ω02 ) + i𝛾ω

At the MPR frequency (estimated from Figure 6.2b), the electromagnetic energy is absorbed by the magnetic resonators and finally converted to heat loss. For a pure MPR mode, a Lorenz-shape absorption dip will be observed in the reflected spectrum. Based on the above Lagrangian model, the reflection coefficient of MRC can be described by i𝛾 ω rMPR = (1 − CMPR ), where the absorption coefficient is CMPR = fMPR (ω 2 −ωm2 )+i𝛾 ω [100], m 0 and fMPR is the light coupling efficiency of magnetic resonators for TM polarization, as the magnetic field is always perpendicular to the incident plane. The electromotive force is not sensitive to the incident angle, so the reflection coefficient can be given by rMPRTM = rp0 (1 − CMPR ). In comparison, rMPRTE = rs0 (1 − CMPR cos θ ) is used to calculate the reflection coefficient for TE polarization, as the electromotive force is dedB pendent on the incident angle θ as Em = − dt0 Seff cos (θ ) in this case. The total reflection coefficient for the TM wave is given by the interference of SPP and MPR modes: rTM = (fspp rspp + (1 − fspp )rmpr ), where fspp and (1 − fspp ) are the weight factors to TM describe the ratio of SPP and TM polarized MPP mode respectively. It can be found that Fano resonances appear in the calculated results and agree with experimental measurement (see Figure 6.14b). The nonlinear optical property of Fano-resonant metamaterial was studied by using a third harmonic generation process. A femtosecond-laser pumped optical parametric amplifier (OPA, TOPAS-C) system was used in this experiment. The THG signal was collected by a silicon-based CCD detector and spectrometer (Ocean Optics 4000) after filtering the fundamental wave. The output power was measured by a photodiode (Newport 818-UV). Two linear polarizers were used to control the power and polarization of the pump laser, and the Newport power detector (818-IR) was used to monitor the power variation of the pump laser (NIR) from the side beam of beam splitter. The relative efficiency of the THG (intensity ratio of the THG and pump laser) on samples A and B are given in Figure 6.14c,d. For sample A, the third harmonic generation efficiency decreases when the pump wavelength is tuned from 1.61 to 1.8 μm (Figure 6.14c). However, the spectrum of third harmonic generation on sample B is quite different, in which a Fano-resonant modulation was obtained (Figure 6.14d). The integration of Fano resonance into nonlinear optical metamaterial will provides many more opportunities for realizing nonlinear optical nanodevices. Although we only studied THG here, this kind of nonlinear Fano resonant effect can also be applied to other nonlinear optical processes, such as high harmonic generation, surface enhanced Raman, four-wave mixing, and etc.

258 | Hui Liu, Shining Zhu 6.4.2.3 Negative optical pressure in two-dimensional magnetic resonance cavities The fact that electromagnetic (EM) radiation exerts pressure on any surface was deduced theoretically by Maxwell in 1871, and demonstrated experimentally by Lebedev in 1900 [101]. Recently there has increasing interest in optical forces in various resonance structures, such as photonic crystals, plasmonic structures, microcavities, and waveguides. We propose that instead of the usual positive pressure, one can induce a strong negative pressure between the walls of an open metallic cavity by utilizing the kinetic energy of the electrons inside the metallic cavity walls [102]. Consider a planar structure consisting of a square metal patch placed above another bigger square metal slab, as illustrated in Figure 6.15a. For the structure depicted in Figure 6.15a, the parameters are chosen as A = 200 nm, D = 30 nm, and t = 10 nm in the simulations. Open boundary conditions are used in all three directions. The incident plane EM propagates upward along the y-direction. The E field of incident wave is set as E0 = 1.0 V/m. When the incident frequency is swept from 150 to 350 THz, the frequencydependence of the E field between two patches is plotted in Figure 6.15c. One LC resonance mode is excited at 255 THz. The profiles of the magnetic and electric field at the LC resonance mode are shown in Figure 6.15e and f. We see that the field strongly localizes in the space between the patch and the slab at this resonance frequency. After obtaining the EM fields, the time-averaged optical force ⟨F⟩t between the patch and slab can then be calculated rigorously using the Maxwell’s stress tensor via a surface integral, ⟨F⟩t = ∯s ⟨T⟩t ⋅ ds, where Tαβ = ε0 (Eα Eβ −E ⋅Eδαβ /2) + μ0 (Hα Hβ −H ⋅Hδαβ /2) and S is the integration surface enclosing one patch. Figure 6.15d shows the calculated optical force per unit area acting on the surface of patch as a function of frequency. At the resonance frequency, a strong negative (attractive) pressure on the wall of cavity could be obtained (about −698 Pa/(mW/μm2 )).

Fig. 6.15. (a–b) Schematic of nanocavity with two gold patches; the frequency dependence of (c) local electric field between two patches and (d) optical pressure between two patches (with A = 200 nm, D = 30 nm); (e) magnetic field (on the y-cut middle layer) and (f) electric field (on the z-cut middle layer) at the resonance frequency (from [102]).

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Metamaterials exhibit artificial resonant responses, and such responses can be realized in the visible or infrared frequency range based on the LC resonance effect. Here, the patch and slab forms an LC resonance cavity. We propose a simple Lagrangian model to qualitatively describe the resonance property of the equivalent LC circuit for this structure. The circuit’s Lagrangian could be expressed as. L=

L ̇2 1 2 Q − Q . 2 2C

(6.18)

Here, L = Lm + Le and C = Cm + Ce . Lm = μ0 D/2 and Cm are the external inductance and capacitance, Le = 1/(ωp2 ε0 t) and Ce are the internal inductance and capacitance, Q is the net charge in the capacitor, Lm Q̇ 2 /2 is the magnetic field energy stored between the two patches, Ukinetic = Le Q̇ 2 /2 is the kinetic energy of the electrons due to the induced current inside the metal. We will ignore the internal capacitance which is much smaller than the external capacitor, i.e. C ≈ Cm . Then Uelectric = Q2 /(2Cm ) is the potential energy stored in the cavity. For a finite-sized patch, the edge effect due to field leakage on both ends of the patch has to be considered, and Cm is modeled 2 as Cm = ε0 π22 AD ⋅ (1 + α DA ), where the empirical coefficient α gives the magnitude of the edge effect. In the presence of Ohmic dissipation and an external driving field, L the Euler–Lagrange equation can be written as dtd ( 𝜕𝜕L ) − 𝜕𝜕Q = − 𝜕𝜕R + Em , where R = Q̇ Q̇ 2 R Q̇ /2 (R is the effective resistance in the system) and the E e.m.f. due to the eff

eff

m

(1) (2) (1) (2) external field E0 is Em = (− ∮ E ⋅ dℓ) = −(ℓeff ⋅ E0 − ℓeff ⋅ E0 ⋅ eikD ) with ℓeff and ℓeff being the effective length of the dipole on the patch and the slab, respectively, and the factor eikD = eiDω /c represents the retardation effect. To simplify the expressions, we (2) (1) introduce the notation τ = ℓeff /ℓeff . Solving the Euler–Lagrange equation gives the resonance frequency of the cavity:

ω0 =

1 √(Le + Lm ) ⋅ Cm

1

= √( ε

1 2 0 ωp t

+

μ0 D ) 2

⋅ ε0 π22

. A2 D

⋅ (1 + α DA )

The optical force exerted on the patches can be calculated as a generalized force corresponding to the coordinate D: F=

𝜕L 1 𝜕Lm 𝜕Le 1 Q2 𝜕Cm ( ) Q̇ 2 + ). = ( + 2 𝜕D 2 𝜕D 𝜕D 2 Cm 𝜕D

As the internal inductance does not depend on the gap distance, we have 𝜕Le /𝜕D = 0. Approximating our system by a parallel plate transmission line, we have Lm ∼ D ⇒ 𝜕C C A 𝜕Lm /𝜕D = Lm /D. We also have 𝜕Dm = − Dm A+α . At the resonance frequency ω0 , if time D averaging is performed for one oscillation period, one could obtain L⟨Q̇ 2 ⟩t = (Lm + Le ) ω02 ⟨Q2 ⟩t = ⟨Q2 ⟩t /Cm . Then the time average of optical force can be obtained as ⟨F⟩t = −

1 Le ⟨Q̇ 2 ⟩t 1 ⟨Q2 ⟩]t 1 αD + ( ). 2 D 2 Cm D A + α D

(6.19)

260 | Hui Liu, Shining Zhu Here, ⟨⋅⟩t denotes the time averaging. We can see that the total optical force given in this equation includes a term 1 L ⟨Q̇ 2 ⟩t − e , 2 D which is the attractive force from kinetic energy of electron, and a term 1 ⟨Q2 ⟩t 1 αD ( ), 2 Cm D A + α D which is the repulsive force from the E field edge effect. If the edge effect is small, the optical force will be dominated by an attractive force −

1 Le ⟨Q̇ 2 ⟩t 2 D

derived from kinetic energy of the electron in the induced current. At the resonance frequency, ω0 =

1 √(Lm + Le )Cm

(k0 =

ω0 ), c

1 ⟨Q2 ⟩t 1 (Lm + Le )⟨Q̇ 2 ⟩t = , 2 Cm 2 D ⟨Q̇ 2 ⟩t = (

(1) ℓeff

Reff

2

) ⋅[

1 + τ2 − τ ⋅ cos(k0 ⋅ D)] ⋅ E02 . 2

(1) The effective length of patch could be approximately taken as ℓeff ≃ A, and the area 2

(1) ) . The optical pressure normalized by the incident intensity of patch is S = A2 ≃ (ℓeff 1 2 I0 = 2 ε0 cE0 could be obtained as [36]

P=

η 1 + τ2 αD [ − τ ⋅ cos (k0 ⋅ D)] ⋅ [−1 + (1 + β D) ⋅ ], D 2 A + αD

where η=

2 t ωp ⋅ 2, c 𝛾m

β =

2 t ωp ⋅ 2. 2 C

The result suggests that the optical pressure between the the patch and the slab is induced by the LC resonance is determined by competition between negative pressure from the kinetic energy of electrons and positive pressure from the edge effect of the E field. In the following numerical simulations, we will show that the negative pressure from kinetic energy is much stronger than the positive pressure from edge effect. Then the total optical pressure P is negative. The above mechanism and theoretical model could have potential applications in many other subwavelength optomechanical plasmonic structures.

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In the above discussion, we focused on the optical pressure in a single plasmonic cavity. For a periodic array of many such plasmonic cavities, the total optical force could be seen as the sum of optical force from many such plasmonic resonators if the coupling interaction between them is small, and coherent coupling may further enhance the effect. This could produce a substantial optical force in an extended system.

6.4.3 Quantum interference in a three-dimensional magnetic plasmon crystal Due to recent progress in nanofabrication technology, 3-D optical metamaterials can be realized, exhibiting a bulk negative index, extremely high refractive index, and even a 3-D cloaking. Also, the variety of spatial distributions and symmetries of the arrangement of “metaatoms” or “metamolecules” in the 3-D metamaterials enables various coupling effects, which has attracted much attention, due to their interesting properties. Many novel applications have been introduced by these couplings, such as stereometamaterials, Fano resonance, toroidal dipolar modes, optical Mobius symmetry, etc. Especially in 3-D magnetic metamaterials, the strong interaction between magnetic resonators leads to magnetic plasmon waves (MPWs), which is analog to the spin waves in natural ferromagnetic materials. However, due to the excitation at the low frequency range, the spin waves cannot interact with near infrared photons. MPWs in metamaterials can be coherently excited by photons in the microwave and even optical region. Therefore, the MPWs can be considered as “optical spin waves” which can possibly be tried in mimic spin-photon coupling quantum processes in metamaterials. Up to now, although some new progress in MPWs in optical metamaterials has been recently achieved in the classical regime, there is a lack of experimental work on their quantum counterpart to reveal the quantum properties of the waves in these materials, and their excellent properties have not been utilized in quantum research. Therefore, the investigation on the quantum characteristics of the excitation of MPWs in metamaterials is quite necessary and essential. In one recent work [103], we carried out a two-photon interference experiment to verify the quantum characteristic of the MPWs in a 3-D optical metamaterial by using the photon pairs produced from the spontaneous parametric down-conversion (SPDC) process. As a typical structure of optical metamaterial, the fishnet structure has been widely used to provide negative a phase and negative index. By introducing the vertical coupling between adjacent magnetic resonators, the MPWs, the collective magnetic resonances, composed by the resonators in the multilayer fishnet structure, can be formed. The diagram of this structure is depicted in Figure 6.16a. The sample is fabricated on a 17-layer metal/dielectric stack on the quartz substrate by using focused ion beam milling (FIB) (FEI Co., USA). Silver and SiO2 are chosen to work as the metallic and dielectric components in the 3-D optical metamaterial. An FIB image of the sample is shown in Figure 6.16b. A cross-section of this multilayer fishnet sample

262 | Hui Liu, Shining Zhu

Fig. 6.16. The diagram and FIB image of the multilayer fishnet sample are presented in (a) and (b), respectively. (c) The transmittance in both experimental and numerical simulation of the multilayer fishnet sample are shown, with the transmittance of experiment being multiplied by two for clarity. (d) and (e) The magnetic field distributions of the first order and second order of MPWs remarked by the arrows are plotted, with solid arrows denoting the numerical results and hollow arrows denoting the experimental results (from [103]).

is presented in the inset, in which an evident 17-layered structure can be observed. First, we measured the transmittance of the 3-D optical metamaterial sample. Figure 6.16c depicts the measured transmittance with normal incidence (black line), as well as the numerical simulation result (dot-dashed line) by using the commercial software package CST Microwave Studio. It should mentioned that the imperfections introduced from the thickness of each layer during the deposition and from the size of the holes milled in FIB etching may be the main causes leading to the broadening of the band width of the magnetic plasmon resonance and increasing of the the transmittance at the frequency where we supposed having low transmittance. In both experimental and simulated results, an evident transmittance peak resulted from the MPW formed by vertical coupling between magnetic resonators in the sample can be observed around 1064 nm marked by the left arrows in Figure 6.16c. This peak corresponds to the first MPW mode with the phase differences through the whole thickness being π . The magnetic field distribution of this MPW mode obtained from numerical simulation is plotted in Figure 6.16d, which gives the phase relation of the field in the multilayer sample. The wavelength difference of the transmittance peaks between experimental and simulated result is smaller than 10 nm, indicating that the

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263

sample is able to provide the required properties belonging to the designed model. Additionally, a small peak can also be found at about 1155 nm, pointed out by the right arrows in both experimental and stimulated results. This mode corresponds to the second MPW with the phase difference equal to 2π , which can also be regarded as the second excited spin wave-like mode. Its magnetic field distribution is shown in Figure 6.16e. The observation of such mode can further confirm the MPW properties of the main peak around 1064 nm. In order to reveal the quantum nature of the 3-D optical metamaterial, we then verify the quantum characteristic of the MPWs in it by measuring the two-photon interference of the reradiating photons from the sample. If this quantum properties survive, quantum characteristic of the mediated state, MPWs in 3-D optical metamaterial, can be confirmed because the states before and after both have the quantum characteristic. This method has been used to demonstrate the quantum property of surface plasmons in 1-D and 2-D plasmonic structures. The experimental setup is shown in Figure 6.17a. 1 cm long KTiOPO4 (KTP) cut for type II spontaneous parametric down-conversion (SPDC) is pumped by a 532 nm continuous laser to produce a pair of orthogonally polarized photons at 1064 nm, horizontally (H) and vertically (V) polarized photons. The SPDC photons are divided into two beams by a polarizing beam splitter (PBS). One beam irradiates the 3-D optical metamaterial sample and excites the MPWs in it. At the other side of the sample, the

Fig. 6.17. (a) Experimental set-up. The coincidence counts without and with 3-D optical metamaterial sample are shown in (b) and (c), respectively. The black dots correspond to the experimental data and the line is the fitting result. Error bars are plus or minus one standard deviation derived from the Poisson distributed counting statistics (from [103]).

264 | Hui Liu, Shining Zhu MPWs reradiates as photons, which are collected by a single mode fiber. The other beam is directly collected by a single mode fiber. Then the two fibers are coupled into a single mode fiber coupler (Thorlabs) which serves as a 50 : 50 beam splitter. The time delay between the two beams can be adjusted by a time delay line (OZ delay line driver) with a single step of 0.7 μm. The difference of the location of coincidence dips in Figure 6.17b and c is due to the thickness of 0.5 mm SiO2 substrate. Together with the careful modulation of polarization controller, the best overlap of the two beams can be achieved. After the coupler, the photons are detected by two silicon APDs (Perkin Elmer). The coincidence measurement is performed by using a photon correlator card (DPC-230, Becker & Hickl GmbH), with the coincidence time window of 3 ns. To get better visibility we use a 2 nm bandwidth interference filter IF1 centered at 1064 nm before the PBS. We also use two 10 nm bandwidth interference filters IF2 centered at 1064 nm before the fiber couplers to block the background light. The result of the coincidence measurement after substracting noise background versus time delay in the absence of the sample is presented in Figure 6.17b, with the least-squares fit to a Gaussian function (solid curve). A Hong-Ou-Mandel dip is obtained with the visibility of 89 ± 3.0 %, indicating the quantum properties of the photons produced by SPDC process. The noise background is measured by shifting the relative delay of the signals given by the two APDs to avoid detection of photon pairs. So the noise background subtracted is the sum of the dark count rate and the double pair emission rate of the SPDC source, which can lead to uncorrelated detections. The uncorrelated fiber collection positions can also induce unwanted coincidence detections, resulting in reduced visibility. Furthermore, the very low single-photon detection efficiency of the silicon APD at 1064 nm (∼ 1 %) leads to a low single-to-noise ratio, contaminating the experimental results. However, the visibility both with and without 3-D metamaterial samples surpasses the upper bound of the 50 % predicted by classical light field theory. And, visibilities higher than 71 % can be used to test quantum nonlocality in the experiment. Therefore, the quantum property of the two-photon interference of the photon pairs produced from SPDC has been confirmed, which is easy to understand. Then, the 3-D optical metamaterial sample is inserted into one of the two arms of the interferometer. The coincidence measurement result after the noise background is subtracted is shown in Figure 6.17c. A solid curve fitted with a Gaussian function gives a visibility of 86 ± 6.0 %, which means that the reradiated photons from the sample also present the quantum property of the two-photon interference. Moreover, the profile of the coincidence curve is almost the same as that of the free space in Figure 6.17b. This indicates the almost complete survival of the quantum property in the conversion from the twophoton state to MPWs to a two-photon state. Therefore, the mediated state, MPWs in 3-D optical metamaterial, is shown to carry the quantum characteristic, and it consequently could be described in a quantum language.

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6.5 Summary and outlook Although metamaterials composed of uncoupled magnetic resonance elements have been successfully applied to produce intriguing effects such as negative refraction, cloaking, and superlensing, all of these were devised within a very narrow frequency range around a specific resonance frequency. This disadvantage restricts the practical applications of metamaterials. In addition to the above-mentioned applications of the linear optical effect, due to the great enhancement of the local field inside magnetic resonators, magnetic metamaterials have also been proposed for use in nonlinear optical processes. However, the nonlinear optical processes that occur between waves of several different frequencies typically require a broad frequency bandwidth. The narrow single resonance property of conventional metamaterials is a considerable disadvantage for their potential nonlinear optical applications. The MP modes introduced by the coupling effect in metamaterials may provide a possible way to overcome the above-mentioned obstacle. As described in this chapter, hybrid MP resonance modes can be attained in a system with several coupled resonators. This hybridization effect results in multiple discrete resonance frequencies of magnetic metamaterials. When all the resonance elements in metamaterials are coupled together through a particular method, the multiple resonance levels will be extended to a continuous frequency band. Therefore, the excitation of MP modes in such metamaterials can be continually tuned within a rather wide range. Compared with conventional metamaterials made from uncoupled elements, this kind of broadband tunable magnetic metamaterial based on the coupling effect will have much more interesting and prospective applications, especially on the nonlinear optical effect. Based on this discussion, we anticipate that many novel metamaterials and nanophotonic devices will be developed from coupled resonator systems.

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Index absorption, 120, 127 – coefficient, 110, 130 – edge, 111 absorptive film, 123 absorptive material, 131 AFM spins, 182 anisotropy field modulation, 167 antiferromagnets, 180 antiferromagntic order, 194 apertures, 122 attenuation, 123, 124 balanced detection, 147 beam – expander, 121 – expansion, 124 benders, 92 biosensors, 107 boundary conditions, 113, 115, 133, 134 bullet-like light pulse, 95, 96 cation ordering, 194 causal dispersion, 109 CCD (charge-coupled device), 123 CCD camera, 123 charge ordering, 194 charge-coupled device (CCD), 123 chemical formula, 192 chemical vapor deposition, 200 circularly polarized light, 176 cloaking structures, 75 coherent ultrafast demagnetization, 159 complex – dielectric constant, 109, 118 – dielectric functions, 121 – form of refractive angle, 112 – permeability, 119 – refractive angles, 114 – refractive index, 110, 113, 114, 118, 119, 128–130 condensed matter, 127 conduction band, 108 conventional and unusual LPV, 220 conventional LPV effect, 221 crystal structure, 192

Damon–Eshbach, 165 damped elastic oscillator, 108 Dember effect, 221 deposition, 121 dielectric – constant, 108–110, 116, 117, 128, 131 – constant spectra, 110 – function, 128, 131, 132 – spectra, 111 diffraction, 120 dispersion, 128, 130, 135 – curves, 132 – features, 107 – property, 129 – relation, 131 dispersion – relations, 130, 131, 134 – spectra, 130 dispersive, 53, 55, 61, 65, 75, 77, 80, 82, 85 – response, 109 double exchange, 194 drift-diffusion model, 210 Drude – model, 129, 133 – region, 128, 131 – spectral region, 109 effective – complex dielectric constant, 118 – constant, 118 – dielectric constant, 118 – media, 233 – -media model, 231 – refraction index, 123 – refractive index, 113–115, 120, 129, 130 electric – displacement vector, 109 – field strength, 116 electromagnetic absorption, 119 electron structure, 193 electron-magnon interaction, 158 Elliot–Yafet spin flip scattering, 157 ellipsometer, 110 emergent light, 112

272 | Index energy – band structure, 108 – gap, 108 – loss spectra, 131 – propagation, 127 erovskite oxide heterostructures, 191 evanescent wave, 54 exchange field modulation, 169 excitation mode, 107, 130 extinction coefficient, 110 Fano resonance, 254 femtosecond laser, 149 ferromagnetic, 194 – materials, 176 film-growth techniques, 200 fishnet structure, 249 flat panel display devices, 107 full-width at half maximum (FWHM), 123 FWHM (full-width at half maximum), 123 Gilbert damping constant, 141 group – index, 130 – refractive index, 127–130 – velocity, 51, 57, 83, 95, 127, 130 gyromagnetic, 88, 89 high-order dispersion, 98 hybridization effect, 234 hyper-interface, 54 hyperbolic medium, 54 ime delay, 64 incident – angle, 110, 112, 113, 115, 116, 121 – beam, 112 – light, 107 – medium, 114 influence – of the thickness of the film, 208 – of the thickness of the substrates, 208 interband transitions, 108, 112, 121, 128–132 interfaces, 180 interference, 120 internal refraction, 120 intraband – transition, 108, 128, 129, 133 – transition region, 111

inverse Faraday effect, 171 invisible cloaking, 77, 78, 83 Jahn–Teller effect, 193 Kramers–Kronig – ispersion relations, 130 – relations, 109 Landau–Lifshitz–Bloch (LLB) equation, 143 Landau–Lifshitz–Gilbert (LLG) equation, 140 laser molecular beam epitaxy, 201 lateral photoelectric effect, 216 left-hand rule, 116 left-handed material, 54, 117 light – bullets, 95 – intensity, 123, 124 – propagation, 107 – velocity, 127, 130 light-absorption materials, 113 light-generated carrier injection, 203 linearly polarized light, 176 longitudinal photoelectric effects, 203, 210 magnetic – anisotropy, 164 – asymmetry, 153 – circular dichroism, 179 – dimer (MD), 235 – dipole, 108 – domain wall, 88 – field strength, 116 – inductive waves, 240 – metamolecules, 237, 239 – ordering, 194 – plasmon, 232 – plasmon crystal, 248 – resonance cavities, 258 – resonator chains, 240 magnetization dynamics, 140 magneto-optical Kerr effect (MOKE), 144 magnetron sputtering, 200 magnon, 173 Maxwell’s equations, 110, 112, 113, 116 metamaterial structure, 120 metamaterials, 120, 121, 231 MOKE, 144

Index |

molecular beam epitaxy, 200 MSHG, 151 nanosandwiches, 244 negative – Goos–Hänchen shift, 117 – index, 68 – index material, 117 – optical pressure, 117 – refraction, 117, 121, 232 – refraction phenomenon, 117 – refractive index, 117, 119–121 noble metals, 121, 120 nonabsorbing medium, 113 one-way mode, 88 optical – absorption, 107, 112, 134 – antennas, 118 – constants, 107, 112–115, 120, 134 – filter, 118 – properties, 107 optoelectronic materials, 135 orbital, 193 – angular momentum, 159 oscillation frequency, 108 penetration depth, 110, 134 perfect – electric conductor (PEC), 86 – optical lenses, 107 permeability, 108, 116–118 perovskite oxide, 192 perovskite oxide films, 200 phase – angle, 118 – change, 64, 66 – propagation, 127 – transition, 162 – velocity, 127 photo-generated carrier, 168 photo-induced current, 170 photo-induced – magnetization reversal, 174 – phase transition, 196 – spin switching, 175 photoelectric effect, 191 photonic crystals, 95 photovoltaic effect, 205

273

plasma – energy, 128 – frequency, 108, 109, 132 – oscillation, 110 – resonance, 109, 110, 130, 131 polarization modulation, 148 positive – refraction, 120 – refractive index, 120, 121 power splitters, 118 Poynting vector, 116 precession, 140 propagation characteristics, 107 pulsed – laser deposition, 200 – magnetic fields, 175 quasi-free electrons, 107 quasi-free s electrons, 108 radio frequency (RF), 121 real refractive angle, 114, 115 reduce, 121 reflection, 120 – high-energy electron diffraction, 201 reflective index, 119 reflectivity, 110 refraction – angle, 116, 123, 126 – law, 112 refractive – angle, 112, 113, 115, 120, 134 – index, 110, 112, 114, 119–121, 127, 128, 130, 131 relaxation time, 128 reversed – Cerenkov radiation, 117 – Doppler effect, 117 RF (radio frequency), 121 right-hand rule, 116 right-handed material, 117 scattering, 120 SEM measurements, 121 slit-hole resonators, 242 Snell’s law, 112, 113, 120, 121 Snell’s law of refraction, 112 space resolution, 123 spectroscopic ellipsometer, 121

274 | Preface spin – angular momentum, 156 – exchange field, 164 – orbital coupling, 158 – polarized density of states, 161 – reversal, 176 – wave, 162 split-ring resonator, 232 splitters, 92 SPP (surface plasmon polariton), 130 standing spin wave, 164 stealth – devices, 107 – materials, 118 stereometamaterials, 237 Stoner excitation, 159 strong dispersion, 109 super-resolution imaging, 107 superdiffusive spin transport, 160 superexchange, 194 surface – plasmon resonance, 133 – plasma, 107, 108 – plasma excitation, 107 – plasma resonance, 107, 131, 132, 135 – plasmon, 130, 131, 134 – plasmon polariton (SPP), 107, 130, 134 – plasmon resonance, 107, 131,132–135 – plasmon vector, 134 TE mode, 120 temporal coherence, 53, 68, 72 time – delay, 62, 66

– resolved magneto-optical Kerr effect (TR-MOKE), 146 time-dependent – drift-diffusion model, 212 – scattering cross-section, 77 time-resolved magnetic second-harmonic-generation (TR-MSHG), 151 TM mode, 120 tolerance factor, 192 TR-MOKE, 146 TR-MSHG, 151 transmission, 120 transparent materials, 120 two dimensional drift-diffusion model, 219 ultrafast – demagnetization, 154 – laser, 139 – magnetization reversal, 179 – photoelectric response, 206 ultraviolet photodetector, 198 uniform spin precession, 162 unusual lateral photoelectric effect, 217 vacuum wave vector, 131 wave vector, 116, 131, 135 wavelength, 110 x-ray, 156, 185 – magnetic circular dichroism (XMCD), 156 XMCD, 156

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