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Springer Series in Solid-State Sciences 187

Shun-Qing Shen

Topological

Insulators

Dirac Equation in Condensed Matter

Second Edition

Springer Series in Solid-State Sciences

Volume 187

Series editors

Bernhard Keimer, Stuttgart, Germany Roberto Merlin, Ann Arbor, MI, USA Hans-Joachim Queisser, Stuttgart, Germany Klaus von Klitzing, Stuttgart, Germany

Shun-Qing Shen

Topological Insulators

Dirac Equation in Condensed Matter

Second Edition

123

Shun-Qing Shen Department of Physics The University of Hong Kong Hong Kong China

ISSN 0171-1873 ISSN 2197-4179 (electronic) Springer Series in Solid-State Sciences ISBN 978-981-10-4605-6 ISBN 978-981-10-4606-3 (eBook) DOI 10.1007/978-981-10-4606-

Library of Congress Control Number: 2017946646

1st edition: © Springer-Verlag Berlin Heidelberg 2012 2nd edition: © Springer Nature Singapore Pte Ltd. 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Preface to the First Edition

Recent years, we have seen rapid emergence of topological insulators and super- conductors. The field is an important advance of the well-developed band theory in solids since its birth in 1920s. The band theory or Fermi liquid theory and Landau’s theory of spontaneously broken symmetry are two themes for most collective phenomena in many-body systems, such as semiconductors and superconductors. Discovery of the integer and fractional quantum Hall effects in 1980s opens a new window to explore the mystery of condensed matters: Topological order has to be introduced to characterize a large class of quantum phenomena. Topological insulator is a triumph of topological order in condensed matter physics. The book grew out of a series of lectures I delivered in an international school on “Topology in Quantum Matter” at Bangalore, India, in July 2011. The aim of this book is to provide an introduction for a large family of topological insulators and superconductors based on the solutions of the Dirac equation. I believe that the Dirac equation is a key to the door of topological insulators. It is a line that could thread all relevant topological phases from one to three dimensions, and from insulators to superconductors or superfluids. This idea actually defines the scope of this book on topological insulators. For this reason, a lot of topics in topological insulators are actually not covered in this book, for example, the interacting systems and topological field theory. Also I have no ambition to review rapid developments of the whole field and consequently no intention to introduce all topics in this introductory book. I would like to express my gratitude to my current and former group members, and various parts of the manuscript benefited from the contributions of Rui-Lin Chu, Huai-Ming Guo, Jian Li, Hai-Zhou Lu, Jie Lu, Wen-Yu Shan, Yan-Yang Zhang, An Zhao, Yuan-Yuan Zhao, Rui Yu, and Bin Zhou. Especially, I would like to thank Hai-Zhou Lu for critical reading the manuscript and replotting all figures. I benefited from numerous discussions and collaborations with Qian Niu,

vii

Jainendra K. Jain, Jun-Ren Shi, Zhong Fang, and Xin Wang on the relevant topics. I am grateful for the support and suggestions from Lu Yu while writing this book. Some of the results in this book were obtained in my research projects funded by Research Grants Council of Hong Kong.

Hong Kong, China Shun-Qing Shen June 2012

viii Preface to the First Edition

• 1 Introduction
  • 1.1 From the Hall Effect to the Quantum Spin Hall Effect
    • Spin Hall Systems 1.2 Topological Insulators as a Generalization of the Quantum
  • 1.3 Beyond Band Insulators: Disorder and Interaction
  • 1.4 Topological Phases in Superconductors and Superfluids
  • 1.5 Topological Dirac and Weyl Semimetals
  • 1.6 Dirac Equation and Topological Insulators
    • of Phase Transition 1.7 Topological Insulators and Landau Theory
  • 1.8 Summary
  • 1.9 Further Reading
  • References
• 2 Starting from the Dirac Equation
  • 2.1 Dirac Equation
  • 2.2 Solutions of Bound States - 2.2.1 Jackiw-Rebbi Solution in One Dimension - 2.2.2 Two Dimensions - 2.2.3 Three and Higher Dimensions
  • 2.3 Why not the Dirac Equation?
  • 2.4 Quadratic Correction to the Dirac Equation
  • 2.5 Bound State Solutions of the Modified Dirac Equation - 2.5.1 One Dimension: End States - 2.5.2 Two Dimensions: Helical Edge States - 2.5.3 Three Dimensions: Surface States - Insulators 2.5.4 Generalization to Higher-Dimensional Topological
  • 2.6 Summary
  • 2.7 Further Reading
  • References
• 3 Minimal Lattice Model for Topological Insulators
  • 3.1 Tight Binding Approximation
  • 3.2 Mapping from a Continuous Model to a Lattice Model
  • 3.3 One-Dimensional Lattice Model
  • 3.4 Two-Dimensional Lattice Model
    • 3.4.1 Integer Quantum Hall Effect
    • 3.4.2 Quantum Spin Hall Effect
  • 3.5 Three-Dimensional Lattice Model
  • 3.6 Parity at the Time Reversal Invariant Momenta
    • 3.6.1 One-Dimensional Lattice Model
    • 3.6.2 Two-Dimensional Lattice Model
    • 3.6.3 Three-Dimensional Lattice Model
  • 3.7 Summary
  • References
• 4 Topological Invariants
  • 4.1 Bloch’s Theorem and Band Theory
  • 4.2 Berry Phase
  • 4.3 Quantum Hall Conductance and the Chern Number
  • 4.4 Electric Polarization in a Cyclic Adiabatic Evolution
  • 4.5 Thouless Charge Pump
  • 4.6 Fu–Kane Spin Pump
  • 4.7 Integer Quantum Hall Effect: The Laughlin Argument
  • 4.8 Time Reversal Symmetry and the Z 2 Index
  • 4.9 Generalization to Two and Three Dimensions
  • 4.10 Phase Diagram of the Modified Dirac Equation
  • 4.11 Further Reading
  • References
• 5 Topological Phases in One Dimension
  • 5.1 Su–Schrieffer–Heeger Model for Polyacetylene
  • 5.2 Topological Ferromagnet
  • 5.3 p-Wave Pairing Superconductor
  • 5.4 Ising Model in a Transverse Field
  • 5.5 One-Dimensional Maxwell’s Equations in Media
  • 5.6 Summary
  • References
  • Spin Hall Effect 6 Quantum Anomalous Hall Effect and Quantum
  • 6.1 Quantum Anomalous Hall Effect - Number 6.1.1 Two-Dimensional Dirac Model and the Chern
    • 6.1.2 Haldane Model
    • 6.1.3 Experimental Realization
  • 6.2 From the Haldane Model to the Kane-Mele Model
  • 6.3 Transport of Edge States - 6.3.1 Landauer-Büttiker Formalism - 6.3.2 Transport of Edge States
  • 6.4 Stability of Edge States
    • Quantum Wells 6.5 Realization of the Quantum Spin Hall Effect in HgTe/CdTe
      • 6.5.1 Band Structure of HgTe/CdTe Quantum Wells
      • 6.5.2 Exact Solution of Edge States
      • 6.5.3 Experimental Measurement
    • Quantum Well 6.6 Quantized Conductance in InAs/GaAs Bilayer
    • A Case Study 6.7 Quantum Hall Effect and Quantum Spin Hall Effect:
      • 6.7.1 Quantum Hall Effect (m ¼ 2)
      • 6.7.2 Quantum Spin Hall Effect
  • 6.8 Coherent Oscillation Due to the Edge States
  • 6.9 Further Reading
  • References
• 7 Three-Dimensional Topological Insulators. - Topological Insulators 7.1 Family Members of Three-Dimensional - 7.1.1 Weak Topological Insulators: PbxSn 1 xTe - 7.1.2 Strong Topological Insulators: Bi 1 xSbx - Bi 2 Se 3 and Bi 2 Te 7.1.3 Topological Insulators with a Single Dirac Cone: - 7.1.4 Strained HgTe
  • 7.2 Electronic Model for Bi 2 Se
  • 7.3 Effective Model for Surface States
  • 7.4 Physical Properties of Topological Insulators - 7.4.1 Absence of Backscattering - 7.4.2 Weak Antilocalization - 7.4.3 Shubnikov-de Haas Oscillation
  • 7.5 Surface Quantum Hall Effect
  • 7.6 Surface States in a Strong Magnetic Field
  • 7.7 Topological Insulator Thin Film - 7.7.1 Effective Model for Thin Film - 7.7.2 Structural Inversion Asymmetry - 7.7.3 Experimental Data of ARPES
  • 7.8 HgTe Thin Film
  • 7.9 Further Reading
  • References
• 8 Impurities and Defects in Topological Insulators
  • 8.1 One Dimension
  • 8.2 Integral Equation for Bound State Energies - 8.2.1 dPotential
  • 8.3 Bound States in Two Dimensions
  • 8.4 Topological Defects - 8.4.1 Magnetic Flux and Zero Energy Mode - 8.4.2 Wormhole Effect - 8.4.3 Witten Effect
  • 8.5 Disorder Effect on Transport
  • 8.6 Further Reading
  • References
• 9 Topological Superconductors and Superfluids. - or Spin Polarized Fermions 9.1 Complex (p þ ip)-Wave Superconductor for Spinless
  • 9.2 Spin Triplet Pairing Superfluidity: 3 He-A and -B Phases - 9.2.1 3 He: Normal Liquid Phase - 9.2.2 3 He-B Phase - 9.2.3 3 He-A Phase: Equal Spin Pairing
  • 9.3 Spin-Triplet Superconductor: Sr 2 RuO
  • 9.4 Superconductivity in Doped Topological Insulators
  • 9.5 Further Reading
  • References
• 10 Majorana Fermions in Topological Insulators
  • 10.1 What Is a Majorana Fermion?
  • 10.2 Majorana Fermions in p-Wave Superconductors - 10.2.1 Zero Energy Mode Around a Quantum Vortex - 10.2.2 Majorana Fermions in Kitaev’s Toy Model - 10.2.3 Quasi-One-Dimensional Superconductors
  • 10.3 Majorana Fermions in Topological Insulators
    • Superconductors 10.4 Sau–Lutchyn–Tewari–Das Sarma Model for Topological
  • 10.5 4 p-Josephson Effect
    • Quantum Computing 10.6 Non-Abelion Statistics and Topological
  • 10.7 Further Reading
  • References
• 11 Topological Dirac and Weyl Semimetals
  • 11.1 Weyl Equations and Weyl Fermions - 11.1.1 Weyl Equations - 11.1.2 A Single Node and Magnetic Monopole
  • 11.2 Emergent Dirac and Weyl Semimetals - 11.2.1 Dirac Semimetal - 11.2.2 Topological Dirac Semimetal - 11.2.3 Topological Weyl Semimetal
  • 11.3 Graphene: A Topological Dirac Semimetal
  • 11.4 Two-Node Model - 11.4.1 Model - 11.4.2 The Chern Number and Fermi Arc - 11.4.3 Quantum Anomalous Hall Effect
  • 11.5 Tight-Binding Model and Topological Phase Transition
  • 11.6 Chiral Anomaly
  • 11.7 Exotic Magnetotransport - 11.7.1 Three-Dimensional Weak Antilocalization - 11.7.2 Negative Magnetoresistance - 11.7.3 Linear Magnetoconductance Near the Weyl Nodes - 11.7.4 High Mobility and Large Magnetoresistance
  • 11.8 Further Reading
  • References
• 12 Topological Anderson Insulator
  • 12.1 Band Structure and Edge States
  • 12.2 Quantized Anomalous Hall Effect
  • 12.3 Topological Anderson Insulator
    • Anderson Insulator 12.4 Effective Medium Theory for Topological
  • 12.5 Band Gap or Mobility Gap
  • 12.6 Summary
  • 12.7 Further Reading
  • References
• 13 Summary: Symmetries and Topological Classification - Fermion Systems 13.1 Ten Symmetry Classes for Non-interacting
  • 13.2 Physical Systems and the Symmetry Classes - 13.2.1 Standard (Wigner–Dyson) Classes - 13.2.2 Chiral Classes - 13.2.3 Bogoliubov-de Gennes (BdG) Classes
  • 13.3 Characterization in the Bulk States
  • 13.4 Five Types in Each Dimension
  • 13.5 Conclusion
  • 13.6 Further Reading
  • References
• Appendix A: Derivation of Two Formulae
• Appendix B: Time Reversal Symmetry
• Appendix C: The Dirac Matrices and the Dirac Gamma Matrices
• Index

Chapter 1

Introduction

Abstract The discovery of topological insulators and superconductors is an impor- tant advance in condensed matter physics. Topological phases reflect global prop- erties of the quantum states in materials, and the boundary states are characteristic of the materials. Such phases constitute a new branch in condensed matter physics. Here a historic development is briefly introduced, and the known family of phases in condensed matter are summarized.

1.1 From the Hall Effect to the Quantum Spin Hall Effect

In 1879, Edwin H. Hall observed an effect that now bears his name; he measured the voltage that arises from the deflected motion of charged particles in solids under externally applied electric and magnetic fields [1]. Consider a two-dimensional sam- ple subjected to a perpendicular magnetic field B. Charged particles passing through the sample are deflected by the Lorentz force and accumulate near the boundary. As a result, the charge accumulation along the boundary produces an electric field E. When electric and magnetic forces are balanced, the Lorentz force on a moving charged particle is zero,

F = q ( E + v × B ) = 0 , (1.1)

where v is the velocity of the particle and q is the charge of particle. The voltage difference between the two boundaries is V (^) H = E W ( W is the width of the sample) and the electric current through the sample is I = q ρ e v W (ρ e is the density of the charge carriers). The ratio of the voltage to the electric current is known as the Hall resistance

R H =

V H

I

B

q ρ e

© Springer Nature Singapore Pte Ltd. 2017 S.-Q. Shen, Topological Insulators , Springer Series in Solid-State Sciences 187, DOI 10.1007/978-981-10-4606-3_

1

1.1 From the Hall Effect to the Quantum Spin Hall Effect 3

Initially, theorists argued that the spin accumulation was caused through asymmet- ric scattering of the spin-up and spin-down electrons within the impurity potentials, hence termed the extrinsic spin Hall effect [10]. In 2003, two independent groups demonstrated that the spin-orbit coupling in the electron band structure can produce a transverse spin current even without impurity scattering, hence called the intrin- sic spin Hall effect [11, 12]. In the quantum Hall regime, the competition between Zeeman splitting and spin-orbit coupling leads to the resonant spin Hall effect, in which a small current induces a finite spin current and spin polarization [13]. The spin Hall effect has been observed experimentally in a GaAs and InGaAs thin film [14] and in the spin light-emitting diode of a p-n junction [15]. The discovery of the integer quantum Hall effect opened a new phase in the study of the various forms of the Hall effect. In 1980, von Klitzing, Dorda, and Pepper discovered experimentally that, in a two-dimensional electron gas produced at a semiconductor hetero-junction subjected to a strong magnetic field, the longitudinal conductance vanishes while quantum plateaus appear in the Hall conductance at values ν e^2 / h [16]. The prefactor is an integer (ν = 1 , 2 ,.. .), known as the filling factor. The quantum Hall effect is a quantum mechanical version of the Hall effect in two dimensions. This effect is now very well understood, and can be simply explained in terms of the single-particle orbitals of an electron in a magnetic field [17]. It is known that the motion of a charged particle in a uniform magnetic field is equivalent to that of a simple harmonic oscillator in quantum mechanics, in which the energy levels are quantized with energy ( n + 12 )ℏω c , and ω c = eB / m is the cyclotron frequency. The energy levels, called Landau levels, are highly degenerate. When one Landau level is fully filled, the filling factor is ν = 1 and the corresponding Hall conductance is e^2 / h. It is realized now that the integer ν is actually a topological invariant, i.e, Thouless- Kohmoto-Nightingale-Njis (TKNN) invariant, that is insensitive to the geometry of the system and the interaction of electrons [18]. For clarity, physicists like to use a semi-classical picture to explain the quantization of the Hall conductance. For a charged particle in a uniform magnetic field, the particle cycles rapidly around the magnetic flux because of the Lorentz force. The

cyclotron radius is given by the magnetic field R (^) n =

ℏ eB (^2 n^ +^1 ).^ When the particle is close to the boundary, the particle bounces off the rigid boundary, and thus skips forward along the boundary. As a result, it forms a conducting channel called edge state (see Fig. 1.1). The group velocity of the particle in the bulk is much slower than the cyclotron velocity, and hence the particles in the bulk are pinned or localized by impurities or disorders. However, the rapid-moving particles along the edge channel are not affected by the impurities or disorders and thus form a perfect one-dimensional conducting channel with a quantum conductance e^2 / h. Because the Landau levels are discrete, each Landau level will generate one edge channel. Consequently, the number of filled Landau levels, i.e., the filling factor, determines the quantized Hall conductance. Thus the key feature of the quantum Hall effect is that all electrons in the bulk are localized, whereas the electrons near the edge form a series of edge conducting channels [19], which is characteristic of a topological phase.

4 1 Introduction

Fig. 1.1 Schematic of the formation of the chiral edge channel in the quantum Hall effect under the Lorentz force on charge carriers. In the quantum anomalous and spin Hall effects, the driving force is replaced by the spin transverse force

In 1982, Tsui, Stormer, and Gosard observed that, in a sample with higher mobility,

the quantum plateaus appear at filling factors ν with rational fractions (ν = 13 , 2 3 ,^

1 5 ,^

2 5 ,^

3 5 ,^

12 5 ,^ · · ·^ ). Known as the fractional quantum Hall effect [20], this effect relies fundamentally on the electron–electron interaction as well as the Landau level quantization. Laughlin proposed that the ν = 1 /3 state is a new type of many- body condensate, which can be described by the Laughlin wave function [21]. The quasi-particles in the condensate carry the fractional charge e /3 because of their strong Coulomb interaction. The observed Hall conductance plateaus arise from the localization of the fractionally charged quasi-particles in the condensate. Thus, the fractional quantum Hall effect can be regarded as the integer quantum Hall effect of these quasi-particles. In 1988, Jainendra K. Jain proposed that the quasi-particles, called composite fermions, can be regarded a combination of an electron charge and quantum magnetic flux [22]. This picture is applicable to all the quantum plateaus observed in the fractional quantum Hall effect, which is now well-accepted in terms of a topological quantum phase of composite fermions that breaks time-reversal symmetry. In 1988, Duncan Haldane proposed that the integer quantum Hall effect can be realized in a lattice system of spinless electrons in a periodic magnetic flux [23]. Although the total magnetic flux is zero, electrons are driven to form a conducting edge channel by the periodic magnetic flux. As there is no pure magnetic field, the quantum Hall conductance originates from the electron band structure for the lattice not from the discrete Landau levels with a strong magnetic field. Thus this is a version of the quantized anomalous Hall effect in the absence of an external field or Landau levels. Furthermore, it was found that the role of the periodic magnetic flux can be replaced by spin-orbit coupling. The quantized anomalous Hall effect can be realized in a ferromagnetic insulator with strong spin-orbit coupling. The anomalous Hall effect persists in an insulating regime. The anomalous Hall conductance can be expressed in terms of the integral of the Berry curvature over the momentum space or the Chern number for fully-filled bands [24]. The Haldane model produces a non- zero Chern numbers for an electron band without the presence of a magnetic field or Landau levels. According to the bulk-edge correspondence, the quantized Hall conductance originates from the dissipationless transport of topologically protected edge states. There have been extensive investigations on this topic [25–29]. One of the promising schemes is based on a magnetically doped topological insulator thin

6 1 Introduction

Fig. 1.2 Evolution from the ordinary Hall effect to the quantum spin Hall effect or two-dimensional topological insulator. Here B signifies the strength of the magnetic field, and M the magnetization for the ferromagnet. The year indicates when the effect was discovered experimentally. σ H is the Hall conductance and σ S is the spin Hall conductance

1.2 Topological Insulators as a Generalization

of the Quantum Spin Hall Systems

There is no Hall effect in three dimensions. However, the generalization of the quan- tum spin Hall effect to three dimensions is one of the milestones in the develop- ment of topological insulators [40–43]. It is not a simple generalization from two dimensions to three dimensions of the transverse transport of an electron charge or spin, or the Hall effect. Instead, bound states evolve near the system boundary based on the intrinsic band structure; the one-dimensional helical edge states in the two-dimensional quantum spin Hall system can evolve into two-dimensional surface states surrounding the three-dimensional topological insulator. A topological insu- lator is a material in a state of quantum matter that behaves as an insulator in its interior but as a metal on its boundary. In the bulk of a topological insulator, the elec- tronic band structure resembles an ordinary insulator, with separated conduction and valence bands. Near the boundary, the surface states within the bulk energy gap allow electron conduction. Electron spins in these states are locked to their momenta. A topological insulator preserves the time-reversal symmetry. Because of the Kramers degeneracy, for a given energy, there always exists a pair of states that have oppo- site spins and momenta; the backscattering between these states is forbidden. These states are characterized by a topological index. Kane and Mele proposed a Z 2 index to classify materials with time-reversal invariance as strong and weak topological insulators [44]. For a weak topological insulator, the resultant surface states are not robust against disorder or impurities, although its physical properties are very similar to those of two-dimensional states. The relationship of a strong topological insulator with the quantum spin Hall system is more subtle. It is possible to classify conven- tional and topological insulators by time-reversal symmetry. The surface states in a strong topological insulator is protected by time-reversal symmetry. Bi (^1) − x Sb x was the first candidate as a three-dimensional topological insulator predicted [45] and verified experimentally [46] by Fu and Kane. Zhang et al. [47] and Xia et al. [48] pointed out that Bi 2 Te 3 and Bi 2 Se 3 are topological insulators with

1.2 Topological Insulators as a Generalization of the Quantum Spin Hall Systems 7

a single Dirac cone of surface states. Angle-resolved photoemission spectroscopy data showed clearly the existence of a single Dirac cone in Bi 2 Se 3 [48] and Bi 2 Te 3 [49]. Electrons in the surface states possess a quantum spin texture, and the electron momenta are strongly coupled with the electron spins. These can produce many exotic magneto-electric properties. Qi et al. [50] proposed an unconventional magneto- electric effect for the surface states, in which electric and magnetic fields are coupled together and are governed by the so-called axion equation instead of by Maxwell’s equations. This is now regarded as one of the characteristic features of topological insulators [51, 52]. Reducing the system to one dimension brings some new insights with respect to topological properties. The boundary of one-dimensional system is simply an end point. A one-dimensional topological insulator is an insulator with two end states

Fig. 1.3 The boundary states and their energy dispersions of topological materials. A d -dimensional material has a ( d − 1)-dimensional boundary