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Colloquium: Topological insulators M. Z. Hasan* Joseph Henry Laboratories, Department of Physics, Princeton University, Princeton, New Jersey 08544, USA C. L. Kane† Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA �Published 8 November 2010� Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional �2D� topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional �3D� topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on Bi1−xSbx, Bi2Se3, Bi2Te3, and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators. DOI: 10.1103/RevModPhys.82.3045 PACS number�s�: 73.20.�r, 73.43.�f, 85.75.�d, 74.90.�n CONTENTS I. Introduction 3045 II. Topological Band Theory 3046 A. The insulating state 3046 B. The quantum Hall state 3047 1. The TKNN invariant 3047 2. Graphene, Dirac electrons, and Haldane model 3047 3. Edge states and the bulk-boundary correspondence 3048 C. Z2 topological insulator 3049 D. Topological superconductor, Majorana fermions 3050 1. Bogoliubov–de Gennes theory 3050 2. Majorana fermion boundary states 3051 3. Periodic Table 3051 III. Quantum Spin Hall Insulator 3052 A. Model system: Graphene 3052 B. HgTe/CdTe quantum well structures 3053 IV. 3D Topological Insulators 3054 A. Strong and weak topological insulators 3054 B. The first 3D topological insulator: Bi1−xSbx 3055 C. Second generation materials: Bi2Se3, Bi2Te3, and Sb2Te3 3058 V. Exotic Broken Symmetry Surface Phases 3061 A. Quantum Hall effect and topological magnetoelectric effect 3061 1. Surface quantum Hall effect 3061 2. Topological magnetoelectric effect and axion electrodynamics 3061 B. Superconducting proximity effect 3062 1. Majorana fermions and topological quantum computing 3062 2. Majorana fermions on topological insulators 3063 VI. Conclusion and Outlook 3064 Acknowledgments 3064 References 3065 I. INTRODUCTION A recurring theme in condensed-matter physics has been the discovery and classification of distinctive phases of matter. Often, phases can be understood using Landau’s approach, which characterizes states in terms of underlying symmetries that are spontaneously bro- ken. Over the past 30 years, the study of the quantum Hall effect has led to a different classification paradigm based on the notion of topological order �Thouless et al., 1982; Wen, 1995�. The state responsible for the quantum Hall effect does not break any symmetries, but it defines a topological phase in the sense that certain fundamen- tal properties �such as the quantized value of the Hall *mzhasan@princeton.edu †kane@physics.upenn.edu REVIEWS OF MODERN PHYSICS, VOLUME 82, OCTOBER–DECEMBER 2010 0034-6861//82�4�/3045�23� © 2010 The American Physical Society3045 conductance and the number of gapless boundary modes� are insensitive to smooth changes in material parameters and cannot change unless the system passes through a quantum phase transition. In the past five years a new field has emerged in condensed-matter physics based on the realization that the spin-orbit interaction can lead to topological insulat- ing electronic phases �Kane and Mele, 2005a, 2005b; Fu, Kane, and Mele, 2007; Moore and Balents, 2007; Roy, 2009b� and on the prediction and observation of these phases in real materials �Bernevig, Hughes, and Zhang, 2006; Fu and Kane, 2007; König et al., 2007; Hsieh et al., 2008; Xia, Qian, Hsieh, Wray, et al., 2009; Zhang, Liu, et al., 2009�. A topological insulator, like an ordinary insu- lator, has a bulk energy gap separating the highest occu- pied electronic band from the lowest empty band. The surface �or edge in two dimensions� of a topological in- sulator, however, necessarily has gapless states that are protected by time-reversal symmetry. The topological in- sulator is closely related to the two-dimensional �2D� integer quantum Hall state, which also has unique edge states. The surface �or edge� states of a topological insu- lator lead to a conducting state with properties unlike any other known one-dimensional �1D� or 2D electronic systems. In addition to their fundamental interest, these states are predicted to have special properties that could be useful for applications ranging from spintronics to quantum computation. The concept of topological order �Wen, 1995� is often used to characterize the intricately correlated fractional quantum Hall states �Tsui, Stormer, and Gossard, 1982�, which require an inherently many-body approach to un- derstand �Laughlin, 1983�. However, topological consid- erations also apply to the simpler integer quantum Hall states �Thouless et al., 1982�, for which an adequate de- scription can be formulated in terms of single-particle quantum mechanics. In this regard, topological insula- tors are similar to the integer quantum Hall effect. Due to the presence of a single-particle energy gap, electron- electron interactions do not modify the state in an essen- tial way. Topological insulators can be understood within the framework of the band theory of solids �Bloch, 1929�. It is remarkable that after more than 80 years, there are still treasures to be uncovered within band theory. In this Colloquium, we review the theoretical and ex- perimental foundations of this rapidly developing field. We begin in Sec. II with an introduction to topological band theory, in which we explain the topological order in the quantum Hall effect and in topological insulators. We also give a short introduction to topological super- conductors, which can be understood within a similar framework. A unifying feature of these states is the bulk-boundary correspondence, which relates the topo- logical structure of bulk crystal to the presence of gap- less boundary modes. Section III describes the 2D topo- logical insulator, also known as a quantum spin Hall insulator, and discusses the discovery of this phase in HgCdTe quantum wells. Section IV is devoted to three- dimensional �3D� topological insulators. We review their experimental discovery in Bi1−xSbx, as well as more re- cent work on “second-generation” materials Bi2Se3 and Bi2Te3. Section V focuses on exotic states that can occur at the surface of a topological insulator due to an in- duced energy gap. An energy gap induced by a magnetic field or proximity to a magnetic material leads to a novel quantum Hall state along with a topological magneto- electric effect. An energy gap due to proximity with a superconductor leads to a state that supports Majorana fermions and may provide a new venue for realizing pro- posals for topological quantum computation. In Sec. VI we conclude with a discussion of new materials, new ex- periments, and open problems. Some aspects of this subject have been described in other reviews, including the review of the quantum spin Hall effect by König et al. �2008� and surveys by Moore �2010� and Qi and Zhang �2010�. II. TOPOLOGICAL BAND THEORY A. The insulating state The insulating state is the most basic state of matter. The simplest insulator is an atomic insulator, with elec- trons bound to atoms in closed shells. Such a material is electrically inert because it takes a finite energy to dis- lodge an electron. Stronger interaction between atoms in a crystal leads to covalent bonding. One of the tri- umphs of quantum mechanics in the 20th century was the development of the band theory of solids, which pro- vides a language for describing the electronic structure of such states. This theory exploits the translational sym- metry of the crystal to classify electronic states in terms of their crystal momentum k, defined in a periodic Bril- louin zone. The Bloch states �um�k��, defined in a single unit cell of the crystal, are eigenstates of the Bloch Hamiltonian H�k�. The eigenvalues Em�k� define energy bands that collectively form the band structure. In an insulator an energy gap separates the occupied valence- band states from the empty conduction-band states. Though the gap in an atomic insulator, such as solid ar- gon, is much larger than that of a semiconductor, there is a sense in which both belong to the same phase. One can imagine tuning the Hamiltonian so as to interpolate con- tinuously between the two without closing the energy gap. Such a process defines a topological equivalence between different insulating states. If one adopts a slightly coarser “stable” topological classification scheme, which equates states with different numbers of trivial core bands, then all conventional insulators are equivalent. Indeed, such insulators are equivalent to the vacuum, which according to Dirac’s relativistic quantum theory also has an energy gap �for pair production�, a conduction band �electrons�, and a valence band �posi- trons�. Are all electronic states with an energy gap topologi- cally equivalent to the vacuum? The answer is no, and the counterexamples are fascinating states of matter. 3046 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 线条 B. The quantum Hall state The simplest counterexample is the integer quantum Hall state �von Klitzing, Dorda, and Pepper, 1980; Prange and Girvin, 1987�, which occurs when electrons confined to two dimensions are placed in a strong mag- netic field. The quantization of the electrons’ circular orbits with cyclotron frequency �c leads to quantized Landau levels with energy �m=��c�m+1/2�. If N Lan- dau levels are filled and the rest are empty, then an en- ergy gap separates the occupied and empty states just as in an insulator. Unlike an insulator, though, an electric field causes the cyclotron orbits to drift, leading to a Hall current characterized by the quantized Hall conductivity, �xy = Ne 2/h . �1� The quantization of �xy has been measured to 1 part in 109 �von Klitzing, 2005�. This precision is a manifestation of the topological nature of �xy. Landau levels can be viewed as a “band structure.” Since the generators of translations do not commute with one another in a magnetic field, electronic states cannot be labeled with momentum. However, if a unit cell with area 2��c /eB enclosing a flux quantum is de- fined, then lattice translations do commute, so Bloch’s theorem allows states to be labeled by 2D crystal mo- mentum k. In the absence of a periodic potential, the energy levels are simply the k independent Landau lev- els Em�k�=�m. In the presence of a periodic potential with the same lattice periodicity, the energy levels will disperse with k. This leads to a band structure that looks identical to that of an ordinary insulator. 1. The TKNN invariant What is the difference between a quantum Hall state characterized by Eq. �1� and an ordinary insulator? The answer, explained by Thouless, Kohmoto, Nightingale, and den Nijs �1982� �TKNN�, is a matter of topology. A 2D band structure consists of a mapping from the crystal momentum k �defined on a torus� to the Bloch Hamil- tonian H�k�. Gapped band structures can be classified topologically by considering the equivalence classes of H�k� that can be continuously deformed into one an- other without closing the energy gap. These classes are distinguished by a topological invariant n�Z �Z denotes the integers� called the Chern invariant. The Chern invariant is rooted in the mathematical theory of fiber bundles �Nakahara, 1990�, but it can be understood physically in terms of the Berry phase �Berry, 1984� associated with the Bloch wave functions �um�k��. Provided there are no accidental degeneracies when k is transported around a closed loop, �um�k�� ac- quires a well defined Berry phase given by the line inte- gral of Am= i�um��k�um�. This may be expressed as a sur- face integral of the Berry flux Fm=��Am. The Chern invariant is the total Berry flux in the Brillouin zone, nm = 1 2� � d2k Fm. �2� nm is integer quantized for reasons analogous to the quantization of the Dirac magnetic monopole. The total Chern number, summed over all occupied bands, n =�m=1 N nm is invariant even if there are degeneracies be- tween occupied bands, provided the gap separating oc- cupied and empty bands remains finite. TKNN showed that �xy, computed using the Kubo formula, has the same form, so that N in Eq. �1� is identical to n. The Chern number n is a topological invariant in the sense that it cannot change when the Hamiltonian varies smoothly. This helps to explain the robust quantization of �xy. The meaning of Eq. �2� can be clarified by a simple analogy. Rather than maps from the Brillouin zone to a Hilbert space, consider simpler maps from two to three dimensions, which describe surfaces. 2D surfaces can be topologically classified by their genus g, which counts the number of holes. For instance, a sphere �Fig. 1�c� has g=0, while a donut �Fig. 1�f� has g=1. A theorem in mathematics due to Gauss and Bonnet �Nakahara, 1990� states that the integral of the Gaussian curvature over a closed surface is a quantized topological invariant, and its value is related to g. The Chern number is an integral of a related curvature. 2. Graphene, Dirac electrons, and Haldane model A simple example of the quantum Hall effect in a band theory is provided by a model of graphene in a periodic magnetic field introduced by Haldane �1988�. We briefly digress here to introduce graphene because it will provide insight into the conception of the 2D quan- tum spin Hall insulator and because the physics of Dirac electrons present in graphene has important parallels at the surface of a 3D topological insulator. B Insulating State Quantum Hall State E k 0 E k EG (a) (b) (c) (d) (e) (f) /a−π/a−π 0 /a−π/a−π hωc FIG. 1. �Color online� States of matter. �a�–�c� The insulating state. �a� An atomic insulator. �b� A simple model insulating band structure. �d�–�f� The quantum Hall state. �d� The cyclo- tron motion of electrons. �e� The Landau levels, which may be viewed as a band structure. �c� and �f� Two surfaces which differ in their genus, g. �c� g=0 for the sphere and �f� g=1 for the donut. The Chern number n that distinguishes the two states is a topological invariant similar to the genus. 3047M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 线条 Graphene is a 2D form of carbon that is of current interest �Novoselov et al., 2005; Zhang et al., 2005; Geim and Novoselov, 2007; Castro Neto et al., 2009�. What makes graphene interesting electronically is the fact that the conduction band and valence band touch each other at two distinct points in the Brillouin zone. Near those points the electronic dispersion resembles the linear dis- persion of massless relativistic particles, described by the Dirac equation �DiVincenzo and Mele, 1984; Semenoff, 1984�. The simplest description of graphene employs a two band model for the pz orbitals on the two equivalent atoms in the unit cell of graphene’s honeycomb lattice. The Bloch Hamiltonian is then a 2�2 matrix, H�k� = h�k� · �� , �3� where �� = ��x ,�y ,�z� are Pauli matrices and h�k� = „hx�k� ,hy�k� ,0…. The combination of inversion �P� and time-reversal �T� symmetry requires hz�k�=0 because P takes hz�k� to −hz�−k�, while T takes hz�k� to +hz�−k�. The Dirac points occur because the two components h�k� can have point zeros in two dimensions. In graphene they occur at two points, K and K�=−K, whose locations at the Brillouin-zone corners are fixed by graphene’s rotational symmetry. For small q k−K, h�q�=�vFq, where vF is a velocity, so H�q�=�vFq ·�� has the form of a 2D massless Dirac Hamiltonian. The degeneracy at the Dirac point is protected by P and T symmetry. By breaking these symmetries the de- generacy can be lifted. For instance, P symmetry is vio- lated if the two atoms in the unit cell are inequivalent. This allows hz�k� to be nonzero. If hz�k� is small, then near K �Eq. �3� becomes a massive Dirac Hamiltonian, H�q� = �vFq · �� + m�z, �4� where m=hz�K�. The dispersion E�q�= ±���vFq�2+m2 has an energy gap 2�m�. Note that T symmetry requires the Dirac point at K� to have a mass m�=hz�K�� with the same magnitude and sign, m�=m. This state describes an ordinary insulator. Haldane �1988� imagined lifting the degeneracy by breaking T symmetry with a magnetic field that is zero on the average but has the full symmetry on the lattice. This perturbation allows nonzero hz�k� and introduces a mass to the Dirac points. However, P symmetry requires the masses at K and K� to have opposite signs, m�=−m. Haldane showed that this gapped state is not an insula- tor but rather a quantum Hall state with �xy=e2 /h. This nonzero Hall conductivity can be understood in terms of Eq. �2�. For a two level Hamiltonian of the form of Eq. �3� it is well known that the Berry flux �Berry, 1984� is related to the solid angle subtended by the unit vector hˆ�k�=h�k� / �h�k��, so that Eq. �2� takes the form n = 1 4� � d2k��kxhˆ� �kyhˆ� · hˆ . �5� This simply counts the number of times hˆ�k� wraps around the unit sphere as a function of k. When the masses m=m�=0, hˆ�k� is confined to the equator hz=0, with a unit �and opposite� winding around each of the Dirac points where �h�=0. For small but finite m, �h� �0 everywhere, and hˆ�K� visits the north or south pole, depending on the sign of m. It follows that each Dirac point contributes ±e2 /2h to �xy. In the insulating state with m=m� the two cancel, so �xy=0. In the quantum Hall state they add. It is essential that there were an even number of Dirac points since otherwise the Hall conductivity would be quantized to a half integer. This is in fact guaranteed by the fermion doubling theorem �Nielssen and Ninomiya, 1983�, which states that for a T invariant system Dirac points must come in pairs. We return to this issue in Sec. IV, where the surface of a topological insulator provides a loophole for this theorem. 3. Edge states and the bulk-boundary correspondence A fundamental consequence of the topological classi- fication of gapped band structures is the existence of gapless conducting states at interfaces where the topo- logical invariant changes. Such edge states are well known at the interface between the integer quantum Hall state and vacuum �Halperin, 1982�. They may be understood in terms of the skipping motion electrons execute as their cyclotron orbits bounce off the edge �Fig. 2�a� . Importantly, the electronic states responsible for this motion are chiral in the sense that they propa- gate in one direction only al