Topological Insulators

University of Pennsylvania ScholarlyCommons

Department of Physics Papers Department of Physics

11-8-2010

Colloquium: Topological Insulators

M. Zahid Hasan Princeton University

Charles L. Kane University of Pennsylvania, [email protected]

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Recommended Citation Hasan, M. Z., & Kane, C. L. (2010). Colloquium: Topological Insulators. Retrieved from https://repository.upenn.edu/physics_papers/39

Suggested Citation: Hasan, M.Z. and C.L. Kane (2010). "Colloquium: Topological insulators," Reviews of Modern Physics. vol. 82, pp. 3045-3067.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/39 For more information, please contact [email protected]. Colloquium: Topological Insulators

Abstract Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted 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 teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, 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.

Disciplines Physical Sciences and Mathematics | Physics

Comments Suggested Citation: Hasan, M.Z. and C.L. Kane (2010). "Colloquium: Topological insulators," Reviews of Modern Physics. vol. 82, pp. 3045-3067.

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/39 REVIEWS OF MODERN PHYSICS, VOLUME 82, OCTOBER–DECEMBER 2010

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 2010͒

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 ͑2D͒ topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional ͑3D͒ 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 number͑s͒: 73.20.Ϫr, 73.43.Ϫf, 85.75.Ϫd, 74.90.ϩn

CONTENTS V. Exotic Broken Symmetry Surface Phases 3061 A. Quantum Hall effect and topological magnetoelectric effect 3061 I. Introduction 3045 1. Surface quantum Hall effect 3061 II. Topological Band Theory 3046 2. Topological magnetoelectric effect and A. The insulating state 3046 axion electrodynamics 3061 B. The quantum Hall state 3047 B. Superconducting proximity effect 3062 1. The TKNN invariant 3047 1. Majorana fermions and topological 2. Graphene, Dirac electrons, and Haldane quantum computing 3062 model 3047 2. Majorana fermions on topological insulators 3063 3. Edge states and the bulk-boundary VI. Conclusion and Outlook 3064 correspondence 3048 Acknowledgments 3064 C. Z topological insulator 3049 2 References 3065 D. Topological superconductor, Majorana fermions 3050 1. Bogoliubov–de Gennes theory 3050 2. Majorana fermion boundary states 3051 I. INTRODUCTION 3. Periodic Table 3051 III. Quantum Spin Hall Insulator 3052 A recurring theme in condensed-matter physics has A. Model system: Graphene 3052 been the discovery and classification of distinctive B. HgTe/CdTe quantum well structures 3053 phases of matter. Often, phases can be understood using IV. 3D Topological Insulators 3054 Landau’s approach, which characterizes states in terms A. Strong and weak topological insulators 3054 of underlying symmetries that are spontaneously bro- B. The first 3D topological insulator: Bi Sb 3055 1−x x ken. Over the past 30 years, the study of the quantum C. Second generation materials: Bi Se ,BiTe ,and 2 3 2 3 Hall effect has led to a different classification paradigm Sb Te 3058 2 3 based on the notion of topological order ͑Thouless et al., 1982; Wen, 1995͒. The state responsible for the quantum Hall effect does not break any symmetries, but it defines *[email protected] a topological phase in the sense that certain fundamen- †[email protected] tal properties ͑such as the quantized value of the Hall

conductance and the number of gapless boundary experimental discovery in Bi1−xSbx, as well as more re- ͒ modes are insensitive to smooth changes in material cent work on “second-generation” materials Bi2Se3 and parameters and cannot change unless the system passes Bi2Te3. Section V focuses on exotic states that can occur through a quantum phase transition. at the surface of a topological insulator due to an in- In the past five years a new field has emerged in duced energy gap. An energy gap induced by a magnetic condensed-matter physics based on the realization that field or proximity to a magnetic material leads to a novel the spin-orbit interaction can lead to topological insulat- quantum Hall state along with a topological magneto- ing electronic phases ͑Kane and Mele, 2005a, 2005b; Fu, electric effect. An energy gap due to proximity with a Kane, and Mele, 2007; Moore and Balents, 2007; Roy, superconductor leads to a state that supports Majorana 2009b͒ and on the prediction and observation of these fermions and may provide a new venue for realizing pro- phases in real materials ͑Bernevig, Hughes, and Zhang, posals for topological quantum computation. In Sec. VI 2006; Fu and Kane, 2007; König et al., 2007; Hsieh et al., we conclude with a discussion of new materials, new ex- 2008; Xia, Qian, Hsieh, Wray, et al., 2009; Zhang, Liu, et periments, and open problems. al., 2009͒. A topological insulator, like an ordinary insu- Some aspects of this subject have been described in lator, has a bulk energy gap separating the highest occu- other reviews, including the review of the quantum spin pied electronic band from the lowest empty band. The Hall effect by König et al. ͑2008͒ and surveys by Moore surface ͑or edge in two dimensions͒ of a topological in- ͑2010͒ and Qi and Zhang ͑2010͒. sulator, however, necessarily has gapless states that are protected by time-reversal symmetry. The topological insulator is closely related to the two-dimensional ͑2D͒ integer quantum Hall state, which also has unique edge II. TOPOLOGICAL BAND THEORY states. The surface ͑or edge͒ states of a topological insulator lead to a conducting state with properties unlike A. The insulating state any other known one-dimensional ͑1D͒ or 2D electronic systems. In addition to their fundamental interest, these The insulating state is the most basic state of matter. states are predicted to have special properties that could The simplest insulator is an atomic insulator, with elec- be useful for applications ranging from spintronics to trons bound to atoms in closed shells. Such a material is quantum computation. electrically inert because it takes a finite energy to dis- The concept of topological order ͑Wen, 1995͒ is often lodge an electron. Stronger interaction between atoms used to characterize the intricately correlated fractional in a crystal leads to covalent bonding. One of the tri- quantum Hall states ͑Tsui, Stormer, and Gossard, 1982͒, umphs of quantum mechanics in the 20th century was which require an inherently many-body approach to un- the development of the band theory of solids, which pro- derstand ͑Laughlin, 1983͒. However, topological consid- vides a language for describing the electronic structure erations also apply to the simpler integer quantum Hall of such states. This theory exploits the translational sym- states ͑Thouless et al., 1982͒, for which an adequate de- metry of the crystal to classify electronic states in terms scription can be formulated in terms of single-particle of their crystal momentum k, defined in a periodic Bril- ͉ ͑ ͒͘ quantum mechanics. In this regard, topological insula- louin zone. The Bloch states um k , defined in a single tors are similar to the integer quantum Hall effect. Due unit cell of the crystal, are eigenstates of the Bloch H͑ ͒ ͑ ͒ to the presence of a single-particle energy gap, electron- Hamiltonian k . The eigenvalues Em k define energy electron interactions do not modify the state in an essen- bands that collectively form the band structure. In an tial way. Topological insulators can be understood within insulator an energy gap separates the occupied valence- the framework of the band theory of solids ͑Bloch, band states from the empty conduction-band states. 1929͒. It is remarkable that after more than 80 years, Though the gap in an atomic insulator, such as solid ar- there are still treasures to be uncovered within band gon, is much larger than that of a semiconductor, there is theory. a sense in which both belong to the same phase. One can In this Colloquium, we review the theoretical and ex- imagine tuning the Hamiltonian so as to interpolate con- perimental foundations of this rapidly developing field. tinuously between the two without closing the energy We begin in Sec. II with an introduction to topological gap. Such a process defines a topological equivalence band theory, in which we explain the topological order between different insulating states. If one adopts a in the quantum Hall effect and in topological insulators. slightly coarser “stable” topological classification We also give a short introduction to topological super- scheme, which equates states with different numbers of conductors, which can be understood within a similar trivial core bands, then all conventional insulators are framework. A unifying feature of these states is the equivalent. Indeed, such insulators are equivalent to the bulk-boundary correspondence, which relates the topo- vacuum, which according to Dirac’s relativistic quantum logical structure of bulk crystal to the presence of gap- theory also has an energy gap ͑for pair production͒,a less boundary modes. Section III describes the 2D topo- conduction band ͑electrons͒, and a valence band ͑posi- logical insulator, also known as a quantum spin Hall trons͒. insulator, and discusses the discovery of this phase in Are all electronic states with an energy gap topologi- HgCdTe quantum wells. Section IV is devoted to three- cally equivalent to the vacuum? The answer is no, and dimensional ͑3D͒ topological insulators. We review their the counterexamples are fascinating states of matter.

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3047

B. The quantum Hall state (a)Insulating State (b) (c)

E E The simplest counterexample is the integer quantum G Hall state ͑von Klitzing, Dorda, and Pepper, 1980; Prange and Girvin, 1987͒, which occurs when electrons confined to two dimensions are placed in a strong mag- −π/a 0 k −π/a netic field. The quantization of the electrons’ circular (d)Quantum Hall State (e) (f) ␻ orbits with cyclotron frequency c leads to quantized E ⑀ ប␻ ͑ ͒ hω Landau levels with energy m = c m+1/2 .IfN Lan- B c dau levels are filled and the rest are empty, then an energy gap separates the occupied and empty states just as in an insulator. Unlike an insulator, though, an electric −π/a 0 k −π/a field causes the cyclotron orbits to drift, leading to a Hall FIG. 1. ͑Color online͒ States of matter. ͑a͒–͑c͒ The insulating current characterized by the quantized Hall conductivity, state. ͑a͒ An atomic insulator. ͑b͒ A simple model insulating band structure. ͑d͒–͑f͒ The quantum Hall state. ͑d͒ The cyclo- ␴ = Ne2/h. ͑1͒ xy tron motion of electrons. ͑e͒ The Landau levels, which may be viewed as a band structure. ͑c͒ and ͑f͒ Two surfaces which ␴ differ in their genus, g. ͑c͒ g=0 for the sphere and ͑f͒ g=1 for The quantization of xy has been measured to 1 part in 109 ͑von Klitzing, 2005͒. This precision is a manifestation the donut. The Chern number n that distinguishes the two ␴ states is a topological invariant similar to the genus. of the topological nature of xy. Landau levels can be viewed as a “band structure.” Since the generators of translations do not commute 1 with one another in a magnetic field, electronic states ͵ 2 F ͑ ͒ nm = ␲ d k m. 2 cannot be labeled with momentum. However, if a unit 2 cell with area 2␲បc/eB enclosing a flux quantum is de- nm is integer quantized for reasons analogous to the fined, then lattice translations do commute, so Bloch’s quantization of the Dirac magnetic monopole. The total theorem allows states to be labeled by 2D crystal mo- Chern number, summed over all occupied bands, n mentum k. In the absence of a periodic potential, the ͚N = m=1nm is invariant even if there are degeneracies be- energy levels are simply the k independent Landau lev- tween occupied bands, provided the gap separating oc- ͑ ͒ ⑀ els Em k = m. In the presence of a periodic potential cupied and empty bands remains finite. TKNN showed with the same lattice periodicity, the energy levels will ␴ that xy, computed using the Kubo formula, has the disperse with k. This leads to a band structure that looks same form, so that N in Eq. ͑1͒ is identical to n. The identical to that of an ordinary insulator. 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 ␴ 1. The TKNN invariant of xy. The meaning of Eq. ͑2͒ can be clarified by a simple What is the difference between a quantum Hall state analogy. Rather than maps from the Brillouin zone to a characterized by Eq. ͑1͒ and an ordinary insulator? The Hilbert space, consider simpler maps from two to three answer, explained by Thouless, Kohmoto, Nightingale, dimensions, which describe surfaces. 2D surfaces can be and den Nijs ͑1982͒ ͑TKNN͒, is a matter of topology. A topologically classified by their genus g, which counts 2D band structure consists of a mapping from the crystal the number of holes. For instance, a sphere ͓Fig. 1͑c͔͒ momentum k ͑defined on a torus͒ to the Bloch Hamil- has g=0, while a donut ͓Fig. 1͑f͔͒ has g=1. A theorem in tonian H͑k͒. Gapped band structures can be classified mathematics due to Gauss and Bonnet ͑Nakahara, 1990͒ topologically by considering the equivalence classes of states that the integral of the Gaussian curvature over a H͑k͒ that can be continuously deformed into one an- closed surface is a quantized topological invariant, and other without closing the energy gap. These classes are its value is related to g. The Chern number is an integral distinguished by a topological invariant nʦZ ͑Z denotes of a related curvature. the integers͒ called the Chern invariant. The Chern invariant is rooted in the mathematical theory of fiber bundles ͑Nakahara, 1990͒, but it can be 2. Graphene, Dirac electrons, and Haldane model understood physically in terms of the Berry phase A simple example of the quantum Hall effect in a ͑Berry, 1984͒ associated with the Bloch wave functions band theory is provided by a model of graphene in a ͉ ͑ ͒͘ um k . Provided there are no accidental degeneracies periodic magnetic field introduced by Haldane ͑1988͒. ͉ ͑ ͒͘ when k is transported around a closed loop, um k ac- We briefly digress here to introduce graphene because it quires a well defined Berry phase given by the line inte- will provide insight into the conception of the 2D quan- A ͗ ٌ͉ ͉ ͘ gral of m =i um k um . This may be expressed as a sur- tum spin Hall insulator and because the physics of Dirac F ٌϫA face integral of the Berry flux m = m. The Chern electrons present in graphene has important parallels at invariant is the total Berry flux in the Brillouin zone, the surface of a 3D topological insulator.

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3048 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

Graphene is a 2D form of carbon that is of current Conduction Band ͑ E interest Novoselov et al., 2005; Zhang et al., 2005; Geim Insulator n=0 and Novoselov, 2007; Castro Neto et al., 2009͒. What makes graphene interesting electronically is the fact that (a) (b) E the conduction band and valence band touch each other F at two distinct points in the Brillouin zone. Near those Quantum Hall points the electronic dispersion resembles the linear dis- State n=1 Valence Band persion of massless relativistic particles, described by the −π −π Dirac equation ͑DiVincenzo and Mele, 1984; Semenoff, /a 0 k /a ͒ 1984 . The simplest description of graphene employs a FIG. 2. ͑Color online͒ The interface between a quantum Hall two band model for the pz orbitals on the two equivalent state and an insulator has chiral edge mode. ͑a͒ The skipping atoms in the unit cell of graphene’s honeycomb lattice. cyclotron orbits. ͑b͒ The electronic structure of a semi-infinite The Bloch Hamiltonian is then a 2ϫ2 matrix, strip described by the Haldane model. A single edge state connects the valence band to the conduction band. H͑k͒ = h͑k͒ · ␴ជ , ͑3͒ ␴ជ ͑␴ ␴ ␴ ͒ ͑ ͒ where = x , y , z are Pauli matrices and h k Ј ˆ ͑ ͒ ͑ ͒ ͑ ͒ ͑P͒ masses m=m =0, h k is confined to the equator hz =0, =„hx k ,hy k ,0…. The combination of inversion and ͑ ͒ ͑T͒ ͑ ͒ P with a unit and opposite winding around each of the time-reversal symmetry requires hz k =0 because Dirac points where ͉h͉=0. For small but finite m, ͉h͉ takes h ͑k͒ to −h ͑−k͒, while T takes h ͑k͒ to +h ͑−k͒. z z z z ˆ ͑ ͒ The Dirac points occur because the two components 0 everywhere, and h K visits the north or south pole, h͑k͒ can have point zeros in two dimensions. In depending on the sign of m. It follows that each Dirac 2 ␴ graphene they occur at two points, K and KЈ=−K, point contributes ±e /2h to xy. In the insulating state Ј ␴ whose locations at the Brillouin-zone corners are fixed with m=m the two cancel, so xy=0. In the quantum by graphene’s rotational symmetry. For small qϵk−K, Hall state they add. ͑ ͒ ប H͑ ͒ ប ␴ It is essential that there were an even number of Dirac h q = vFq, where vF is a velocity, so q = vFq· ជ has the form of a 2D massless Dirac Hamiltonian. points since otherwise the Hall conductivity would be The degeneracy at the Dirac point is protected by P quantized to a half integer. This is in fact guaranteed by ͑ and T symmetry. By breaking these symmetries the de- the fermion doubling theorem Nielssen and Ninomiya, ͒ T generacy can be lifted. For instance, P symmetry is vio- 1983 , which states that for a invariant system Dirac lated if the two atoms in the unit cell are inequivalent. points must come in pairs. We return to this issue in Sec. ͑ ͒ ͑ ͒ IV, where the surface of a topological insulator provides This allows hz k to be nonzero. If hz k is small, then near K ͓Eq. ͑3͔͒ becomes a massive Dirac Hamiltonian, a loophole for this theorem. H͑q͒ = បv q · ␴ជ + m␴ , ͑4͒ F z 3. Edge states and the bulk-boundary correspondence ͑ ͒ ͑ ͒ ͱ͉ប ͉2 2 where m=hz K . The dispersion E q =± vFq +m A fundamental consequence of the topological classi- ͉ ͉ T has an energy gap 2 m . Note that symmetry requires fication of gapped band structures is the existence of Ј Ј ͑ Ј͒ the Dirac point at K to have a mass m =hz K with the gapless conducting states at interfaces where the topo- same magnitude and sign, mЈ=m. This state describes an logical invariant changes. Such edge states are well ordinary insulator. known at the interface between the integer quantum Haldane ͑1988͒ imagined lifting the degeneracy by Hall state and vacuum ͑Halperin, 1982͒. They may be breaking T symmetry with a magnetic field that is zero understood in terms of the skipping motion electrons on the average but has the full symmetry on the lattice. execute as their cyclotron orbits bounce off the edge ͑ ͒ This perturbation allows nonzero hz k and introduces a ͓Fig. 2͑a͔͒. Importantly, the electronic states responsible mass to the Dirac points. However, P symmetry requires for this motion are chiral in the sense that they propa- the masses at K and KЈ to have opposite signs, mЈ=−m. gate in one direction only along the edge. These states Haldane showed that this gapped state is not an insula- are insensitive to disorder because there are no states ␴ 2 tor but rather a quantum Hall state with xy=e /h. available for backscattering—a fact that underlies the This nonzero Hall conductivity can be understood in perfectly quantized electronic transport in the quantum terms of Eq. ͑2͒. For a two level Hamiltonian of the Hall effect. form of Eq. ͑3͒ it is well known that the Berry flux The existence of such “one way” edge states is deeply ͑Berry, 1984͒ is related to the solid angle subtended by related to the topology of the bulk quantum Hall state. the unit vector hˆ ͑k͒=h͑k͒/͉h͑k͉͒, so that Eq. ͑2͒ takes Imagine an interface where a crystal slowly interpolates the form as a function of distance y between a quantum Hall state ͑n=1͒ and a trivial insulator ͑n=0͒. Somewhere along 1 the way the energy gap has to vanish because otherwise hˆ ͒ · hˆ . ͑5͒ ץ hˆ ϫ ץn = ͵ d2k͑ 4␲ kx ky it is impossible for the topological invariant to change. There will therefore be low energy electronic states This simply counts the number of times hˆ ͑k͒ wraps bound to the region where the energy gap passes around the unit sphere as a function of k. When the through zero. This interplay between topology and gap-

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3049 less modes is ubiquitous in physics and has appeared in E (a)Conduction Band E (b) Conduction Band many contexts. It was originally found by Jackiw and Rebbi ͑1976͒ in their analysis of a 1D ﬁeld theory. Simi- lar ideas were used by Su, Schrieffer, and Heeger ͑1979͒ EF EF to describe soliton states in polyacetalene. A simple theory of the chiral edge states based on Valence Band Valence Band Jackiw and Rebbi ͑1976͒ can be developed using the two Γ k Γ Γ k Γ band Dirac model ͑4͒. Consider an interface where the a b a b mass m at one of the Dirac points changes sign as a FIG. 3. ͑Color online͒ Electronic dispersion between two → ͑ ͒ ͑ ͒Ͼ function of y. We thus let m m y , where m y 0 boundary Kramers degenerate points ⌫ =0 and ⌫ =␲/a.In͑a͒ Ͼ ͑ ͒Ͻ a b gives the insulator for y 0 and m y 0 gives the quan- the number of surface states crossing the Fermi energy EF is tum Hall state for yϽ0. Assume mЈϾ0 is ﬁxed. The even, whereas in ͑b͒ it is odd. An odd number of crossings .Schrödinger equation, obtained by replacing q by −iٌជ in leads to topologically protected metallic boundary states Eq. ͑4͒, has a simple and elegant exact solution, ⌬ ͑ ͒ NR − NL = n, 7 y 1 ⌬ ␺ ͑x,y͒ ϰ eiqxx expͩ− ͵ dyЈm͑yЈ͒dyЈ/v ͪͩ ͪ, ͑6͒ where n is the difference in the Chern number across qx F 0 1 the interface.

͑ ͒ ប with E qx = vFqx. This band of states intersects the C. Z2 topological insulator Fermi energy EF with a positive group velocity dE/dqx ប Since the Hall conductivity is odd under T, the topo- = vF and defines a right moving chiral edge mode. In the 1980s related ideas were applied to narrow gap logically nontrivial states described in Sec. II.B.3 can T semiconductors, which can be modeled using a 3D mas- only occur when symmetry is broken. However, the sive Dirac Hamiltonian ͑Volkov and Pankratov, 1985; spin-orbit interaction allows a different topological class T Fradkin, Dagotto, and Boyanovsky, 1986͒. An interface of insulating band structures when symmetry is unbro- ͑ ͒ where the Dirac mass changes sign is associated with ken Kane and Mele, 2005a . The key to understanding T gapless 2D Dirac fermion states. These share some simi- this new topological class is to examine the role of larities with the surface states of a 3D topological insu- symmetry for spin 1/2 particles. T lator, but as we shall see in Sec. IV.A there is a funda- symmetry is represented by an antiunitary operator ⌰ ͑ ␲ ប͒ mental difference. In a separate development, Kaplan =exp i Sy / K, where Sy is the spin operator and K is ͑1992͒ showed that in lattice quantum chromodynamics complex conjugation. For spin 1/2 electrons, ⌰ has the four-dimensional ͑4D͒ chiral fermions could be simu- property ⌰2 =−1. This leads to an important constraint, lated on a five-dimensional lattice by introducing a simi- known as Kramers’ theorem, which all eigenstates of a T lar domain wall. This provided a method for circumvent- invariant Hamiltonian are at least twofold degenerate. ing the doubling theorem ͑Nielssen and Ninomiya, This follows because if a nondegenerate state ͉␹͘ existed 1983͒, which prevented the simulation of chiral fermions then ⌰͉␹͘=c͉␹͘ for some constant c. This would mean on a 4D lattice. Quantum Hall edge states and surface ⌰2͉␹͘=͉c͉2͉␹͘, which is not allowed because ͉c͉2 states of a topological insulator evade similar doubling −1. In the absence of spin-orbit interactions, Kramers’ theorems. degeneracy is simply the degeneracy between up and The chiral edge states in the quantum Hall effect can down spins. In the presence of spin-orbit interactions, be seen explicitly by solving the Haldane model in a however, it has nontrivial consequences. semi-infinite geometry with an edge at y=0. Figure 2͑b͒ A T invariant Bloch Hamiltonian must satisfy shows the energy levels as a function of the momentum ⌰H͑k͒⌰−1 = H͑− k͒. ͑8͒ kx along the edge. The solid regions show the bulk conduction and valence bands, which form continuum states One can classify the equivalence classes of Hamiltonians and show the energy gap near K and KЈ. A single band, satisfying this constraint that can be smoothly deformed describing states bound to the edge, connects the va- without closing the energy gap. The TKNN invariant is lence band to the conduction band with a positive group n=0, but there is an additional invariant with two pos- velocity. sible values, ␯=0 or 1 ͑Kane and Mele, 2005b͒. The fact By changing the Hamiltonian near the surface the dis- that there are two topological classes can be understood persion of the edge states can be modified. For instance, by appealing to the bulk-boundary correspondence. ͑ ͒ E qx could develop a kink so that the edge states inter- In Fig. 3 we plot analogous to Fig. 2 the electronic T sect EF three times—twice with a positive group velocity states associated with the edge of a invariant 2D insu- and once with a negative group velocity. The difference, lator as a function of the crystal momentum along the Ͻ Ͻ␲ NR −NL, between the number of right and left moving edge. Only half of the Brillouin zone 0 kx /a is modes, however, cannot change and is determined by shown because T symmetry requires that the other half the topological structure of the bulk states. This is sum- −␲/aϽkϽ0 is a mirror image. As in Fig. 2, the shaded marized by the bulk-boundary correspondence, regions depict the bulk conduction and valence bands

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3050 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators separated by an energy gap. Depending on the details of n↑ +n↓ =0, but the difference n␴ =͑n↑ −n↓͒/2 deﬁnes a the Hamiltonian near the edge there may or may not be quantized spin Hall conductivity ͑Sheng et al., 2006͒. The states bound to the edge inside the gap. If they are Z2 invariant is then simply present, however, then Kramers’ theorem requires they ␯ ͑ ͒ T = n␴ mod 2. 11 are twofold degenerate at the invariant momenta kx ␲ ͑ ␲ ͒ =0 and /a which is the same as − /a . Away from While n↑, n↓ lose their meaning when Sz nonconserving ⌫ these special points, labeled a,b in Fig. 3, a spin-orbit terms ͑which are inevitably present͒ are added, ␯ retains interaction will split the degeneracy. There are two ways its identity. ␲ ͑ ͒ the states at kx =0 and /a can connect. In Fig. 3 a they If the crystal has inversion symmetry, there is another connect pairwise. In this case the edge states can be shortcut to computing ␯ ͑Fu and Kane, 2007͒.Atthe ⌳ ͑⌳ ͒ eliminated by pushing all of the bound states out of the special points a the Bloch states um a are also parity ␲ ␰ ͑⌳ ͒ gap. Between kx =0 and /a, the bands intersect EF an eigenstates with eigenvalue m a = ±1. The Z2 invari- even number of times. In contrast, in Fig. 3͑b͒ the edge ant then simply follows from Eq. ͑10͒ with states cannot be eliminated. The bands intersect E an F ␦ ␰ ͑⌳ ͒ ͑ ͒ odd number of times. a = ͟ m a , 12 Which of these alternatives occurred depend on the m topological class of the bulk band structure? Since each where the product is over the Kramers pairs of occupied band intersecting EF at kx has a Kramers partner at −kx, bands. This has proven useful for identifying topological the bulk-boundary correspondence relates the number insulators from band-structure calculations ͑Fu and NK of Kramers pairs of edge modes intersecting EF to Kane, 2007; Teo, Fu, and Kane, 2008; Guo and Franz, ͒ the change in the Z2 invariants across the interface, 2009; Zhang, Liu, et al., 2009; Pesin and Balents, 2010 .

⌬␯ ͑ ͒ NK = mod 2. 9 D. Topological superconductor, Majorana fermions

We conclude that a 2D topological insulator has topo- Considerations of topological band theory can also be logically protected edge states. These form a unique 1D used to topologically classify superconductors. This is a conductor, whose properties will be discussed in Sec. III. subject that has seen fascinating recent theoretical de- The above considerations can be generalized to 3D to- velopments ͑Roy, 2008; Schnyder et al., 2008; Kitaev, pological insulators, discussed in Sec. IV, which have 2009; Qi, Hughes, et al., 2009͒. We give an introduction protected surface states. that focuses on the simplest model superconductors. The There are several mathematical formulations of the Z2 more general case will be touched on at the end. This invariant ␯ ͑Kane and Mele, 2005b; Fu and Kane, 2006, section will provide the conceptual basis for topological 2007; Fukui and Hatsugai, 2007; Moore and Balents, superconductors and explain the emergence of Majo- 2007; Fukui, Fujiwara, and Hatsugai, 2008; Qi, Hughes, rana fermions in superconducting systems. It will also and Zhang, 2008; Roy, 2009a; Wang, Qi, and Zhang, provide background for Sec. V. B , where we discuss Ma- 2010͒. One approach ͑Fu and Kane, 2006͒ is to define a jorana states in superconductor-topological insulator ͑ ͒ ͗ ͑ ͉͒⌰͉ ͑ ͒͘ unitary matrix wmn k = um k un −k built from the structures along with possible applications to topological ͉ ͑ ͒͘ ⌰ occupied Bloch functions um k . Since is antiunitary quantum computing. and ⌰2 =−1, wT͑k͒=−w͑−k͒. There are four special ⌳ points a in the bulk 2D Brillouin zone where k and −k 1. Bogoliubov–de Gennes theory coincide, so w͑⌳ ͒ is antisymmetric. The determinant of a In the BCS mean-field theory of a superconductor the an antisymmetric matrix is the square of its Pfaffian, ␦ ͓ ͑⌳ ͔͒ ͱ ͓ ͑⌳ ͔͒ Hamiltonian for a system of spinless electrons may be which allows us to define a =Pf w a / Det w a ͑ ͒ ͉ ͑ ͒͘ written in the form de Gennes, 1966 , = ±1. Provided um k is chosen continuously through- ͑ ͒ 1 c out the Brillouin zone which is always possible , the ␮ ͚ ͑ † ͒H ͑ ͒ͩ k ͪ ͑ ͒ branch of the square root can be specified globally, and H − N = ckc−k BdG k † , 13 2 k c−k the Z2 invariant is † H where ck is an electron creation operator and BdG is a 4 2ϫ2 block matrix, which in Nambu’s notation may be ͑ ͒␯ ␦ ͑ ͒ ␶ជ −1 = ͟ a. 10 written in terms of Pauli matrices as a=1 H ͑ ͒ ͓H ͑ ͒ ␮͔␶ ⌬ ͑ ͒␶ ⌬ ͑ ͒␶ ͑ ͒ BdG k = 0 k − z + 1 k x + 2 k y. 14 H ͑ ͒ This formulation can be generalized to 3D topological Here, 0 k is the Bloch Hamiltonian in the absence of ⌬ ⌬ ⌬ insulators and involves the eight special points in the 3D superconductivity and = 1 +i 2 is the BCS mean-field Brillouin zone. pairing potential, which for spinless particles must have The calculation of ␯ is simpler if the crystal has extra odd parity, ⌬͑−k͒=−⌬͑k͒. For a uniform system the ex- symmetry. For instance, if the 2D system conserves the citation spectrum of a superconductor is given by the H perpendicular spin Sz, then the up and down spins have eigenvalues of BdG, which exhibit a superconducting T H independent Chern integers n↑, n↓. symmetry requires energy gap. More generally, for spatially dependent 0

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3051

(a) 1DT-SC(d) 2DT-SC Read and Green, 2000; Ivanov, 2001; Stern, von Oppen, ͒ Φ and Mariani, 2004; Nayak et al., 2008 . Due to the (b) (c) particle-hole redundancy the quasiparticle operators sat- ∆ ∆ ⌫ ⌫† Γ isfy 0 = 0. Thus, a quasiparticle is its own antiparticle— E Γ =Γ† (e) the defining feature of a Majorana fermion. A Majorana 0 0 0 † 0 Γ =Γ Γ fermion is essentially half of an ordinary Dirac fermion. −E E E 0 Γ =Γ † Due to the particle-hole redundancy, a single fermionic −∆ −∆ −E E state is associated with each pair of ±E energy levels. 0 k The presence or absence of a fermion in this state de- ͑ ͒ fines a two-level system with energy splitting E. Majo- FIG. 4. Color online Boundary states for a topological super- ͑ ͑ ͒ ͑ ͒ rana zero modes must always come in pairs for in- conductor TSC . a A 1D superconductor with bound states ͒ at its ends. The end state spectrum for ͑b͒ an ordinary 1D stance, a 1D superconductor has two ends , and a well superconductor and ͑c͒ a 1D topological superconductor. ͑d͒ A separated pair defines a degenerate two-level system, topological 2D superconductor with ͑e͒ a chiral Majorana edge whose quantum state is stored nonlocally. This has pro- mode. ͑c͒ A vortex with flux ⌽=h/2e is associated with a zero found implications, which we return to in Sec. V. B when mode. discussing the proposal of Kitaev ͑2003͒ to use these properties for quantum information processing. In two dimensions the integer classification gives the ⌬ H Z and the Schrödinger equation associated with BdG is number of chiral Majorana edge modes ͓Figs. 4͑d͒ and ͑ ͒ known as the Bogoliubov–de Gennes BdG equation. 4͑e͔͒, which resemble chiral modes in the quantum Hall ͑ ͒ † Since Eq. 13 has c and c on both sides there is an effect but for the particle-hole redundancy. A spinless inherent redundancy built into the BdG Hamiltonian. superconductor with p +ip symmetry is the simplest ⌬ H H x y For =0, BdG includes two copies of 0 with opposite model 2D topological superconductor. Such supercon- H sign. More generally, BdG has an intrinsic particle-hole ductors will also exhibit Majorana bound states at the symmetry expressed by core of vortices ͑Caroli, de Gennes, and Matricon, 1964; ⌶H ͑ ͒⌶−1 H ͑ ͒ ͑ ͒ Volovik, 1999; Read and Green, 2000͒. This may be un- BdG k =− BdG − k , 15 derstood by considering the vortex to be a hole in the ⌶ ␶ ⌶2 where the particle-hole operator = xK satisfies superconductor circled by an edge mode ͓Fig. 4͑d͔͒. ء͒ ͑ H ͑ ͒ H ͒ ͑ =+1. Equation 15 follows from 0 −k = 0 k and When the flux in the hole is h/2e the edge modes are the odd parity of the real ⌬͑k͒. It follows that every quantized such that one state is exactly at E=0. H eigenstate of BdG with energy E has a partner at −E. Majorana fermions have been studied in particle phys- These two states are redundant because the Bogoliubov ics for decades but have not been definitively observed ⌫† ͑ ͒ quasiparticle operators associated with them satisfy E Majorana, 1937; Wilczek, 2009 . A neutrino might be a ⌫ = −E. Thus, creating a quasiparticle in state E has the Majorana fermion. Efforts to observe certain lepton same effect as removing one from state −E. number violating neutrinoless double ␤ decay processes The particle-hole symmetry constraint ͑15͒ has a simi- may resolve that issue ͑Avignone, Elliott, and Engel, lar structure to the time-reversal constraint in Eq. ͑8͒,so 2008͒. In condensed-matter physics, Majorana fermions it is natural to consider the classes of BdG Hamiltonians can arise due to a paired condensate that allows a pair of that can be continuously deformed into one another fermionic quasiparticles to “disappear” into the conden- without closing the energy gap. In the simplest case, sate. They have been predicted in a number of physical spinless fermions, the classification can be shown to be systems related to the spinless px +ipy superconductor, Z2 in one dimension and Z in two dimensions. As in Sec. including the Moore-Read state of the ␯=5/2 quantum II.C, this can be most easily understood by appealing to Hall effect ͑Moore and Read, 1991; Greiter, Wen, and ͒ ͑ the bulk-boundary correspondence. Wilczek, 1992; Read and Green, 2000 ,Sr2RuO4 Das Sarma, Nayak, and Tewari, 2006͒, cold fermionic atoms ͑ 2. Majorana fermion boundary states near a Feshbach resonance Gurarie, Radzihovsky, and Andreev, 2005; Tewari et al., 2007͒, and 2D structures At the end of a 1D superconductor ͑Kitaev, 2000͒ that combine superconductivity, magnetism, and strong there may or may not be discrete states within the en- spin-orbit coupling ͑Lee, 2009; Sato and Fujimoto, 2009; ergy gap that are bound to the end ͓Figs. 4͑a͒–4͑c͔͒.If Sau et al., 2010͒. In Sec. V. B we discuss the prospect for they are present, then every state at +E has a partner at creating Majorana fermion states at interfaces between −E. Such finite-energy pairs are not topologically pro- topological insulators and ordinary superconductors ͑Fu tected because they can simply be pushed out of the and Kane, 2008͒. energy gap. However, a single unpaired bound state at E=0 is protected because it cannot move away from E 3. Periodic Table =0. The presence or absence of such a zero mode is determined by the Z2 topological class of the bulk 1D Topological insulators and superconductors fit to- superconductor. gether into a rich and elegant mathematical structure The Bogoliubov quasiparticle states associated with that generalizes the notions of topological band theory the zero modes are fascinating objects ͑Kitaev, 2000; described above ͑Schnyder et al., 2008; Kitaev, 2009;

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3052 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

TABLE I. Periodic table of topological insulators and superconductors. The ten symmetry classes are labeled using the notation of Altland and Zirnbauer ͑1997͒ ͑AZ͒ and are speciﬁed by presence or absence of T symmetry ⌰, particle-hole symmetry ⌶, and chiral symmetry ⌸=⌶⌰. ±1 and 0 denote the presence and absence of symmetry, with ±1 specifying the value of ⌰2 and ⌶2.As ͑ ͒ a function of symmetry and space dimensionality d, the topological classiﬁcations Z, Z2, and 0 show a regular pattern that repeats when d→d+8.

Symmetry d

AZ ⌰⌶⌸12345678

A0000Z 0 Z 0 Z 0 Z AIII 0 0 1 Z 0 Z 0 Z 0 Z 0

AI1 00000Z 0 Z2 Z2 Z BDI 1 1 1 Z 000Z 0 Z2 Z2 D010Z2 Z 000Z 0 Z2 DIII −1 1 1 Z2 Z2 Z 000Z 0 AII −1 0 0 0 Z2 Z2 Z 000Z CII −1 −1 1 Z 0 Z2 Z2 Z 000 C0−100Z 0 Z2 Z2 Z 00 CI 1 −1 1 0 0 Z 0 Z2 Z2 Z 0

Schnyder et al., 2009; Ryu et al., 2010͒. The classes of exist in graphene ͑Kane and Mele, 2005a͒ and in 2D equivalent Hamiltonians are determined by specifying semiconductor systems with a uniform strain gradient the symmetry class and the dimensionality. The symme- ͑Bernevig and Zhang, 2006͒. It was subsequently pre- try class depends on the presence or absence of T sym- dicted to exist ͑Bernevig, Hughes, and Zhang, 2006͒ and metry ͑8͒ with ⌰2 = ±1 and/or particle-hole symmetry was then observed ͑König et al., 2007͒ in HgCdTe quan- ͑15͒ with ⌶2 = ±1. There are ten distinct classes, which tum well structures. In Sec. III.A we introduce the phys- are closely related to the classification of random matri- ics of this state in the model graphene system and de- ces of Altland and Zirnbauer ͑1997͒. The topological scribe its novel edge states. Section III.B reviews the classifications, given by Z, Z2, or 0, show a regular pat- experiments, which have also been the subject of the tern as a function of symmetry class and dimensionality review article by König et al. ͑2008͒. and can be arranged into the Periodic Table of topological insulators and superconductors shown in Table I. The quantum Hall state ͑class A, no symmetry; d=2͒, ͑ ⌰2 ͒ A. Model system: Graphene the Z2 topological insulators class AII, =−1; d=2,3 , and the Z and Z topological superconductors ͑class D, 2 In Sec. II.B.2 we argued that the degeneracy at the ⌶2 =1; d=1,2͒ described above are entries in the Peri- Dirac point in graphene is protected by inversion and T odic Table. There are also other nontrivial entries de- symmetry. That argument ignored the spin of the elec- scribing different topological superconducting and su- trons. The spin-orbit interaction allows a new mass term perfluid phases. Each nontrivial phase is predicted via in Eq. ͑3͒ with respect to all of graphene’s symmetries. the bulk-boundary correspondence to have gapless In the simplest picture, the intrinsic spin-orbit interac- boundary states. One notable example is superfluid tion commutes with the electron spin S , so the Hamil- 3HeB ͑Volovik, 2003; Roy, 2008; Schnyder et al., 2008; z tonian decouples into two independent Hamiltonians for Nagato, Higashitani, and Nagai, 2009; Qi, Hughes, et al., 2 2 the up and down spins. The resulting theory is simply 2009; Volovik, 2009͒,in͑class DIII, ⌰ =−1, ⌶ =+1; d ͑ ͒ ͒ two copies of the model of Haldane 1988 with opposite =3 which has a Z classification along with gapless 2D signs of the Hall conductivity for up and down spins. Majorana fermion modes on its surface. A generaliza- This does not violate T symmetry because time reversal tion of the quantum Hall state introduced by Zhang and ␴ ͑ ͒ flips both the spin and xy. In an applied electric field, Hu 2001 corresponds to the d=4 entry in class A or the up and down spins have Hall currents that flow in AII. There are also other entries in physical dimensions opposite directions. The Hall conductivity is thus zero, that have yet to be filled by realistic systems. The search but there is a quantized spin Hall conductivity, defined is on to discover such phases. ↑ ↓ ␴s ␴s ␲ by Jx −Jx = xyEy with xy=e/2 —a quantum spin Hall effect. Related ideas were mentioned in earlier work on 3 III. QUANTUM SPIN HALL INSULATOR the planar state of He films ͑Volovik and Yakovenko, 1989͒. Since it is two copies of a quantum Hall state, the The 2D topological insulator is known as a quantum quantum spin Hall state must have gapless edge states spin Hall insulator. This state was originally theorized to ͑Fig. 5͒.

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3053

The edge states are similarly protected from the ef- E Conduction Band Conventional ν=0 fects of weak electron interactions, though for strong Insulator interactions the Luttinger liquid effects lead to a mag- (a) (b) netic instability ͑Wu, Bernevig, and Zhang, 2006; Xu and EF Moore, 2006͒. This strong interacting phase is interesting

Quantum spin because it will exhibit charge e/2 quasiparticles similar ν=1 to solitons in the model of Su, Schrieffer, and Heeger Hall insulator Valence Band ͑1979͒. For sufﬁciently strong interactions similar frac- −π −π /a 0 k /a tionalization could be observed by measuring shot noise FIG. 5. ͑Color online͒ Edge states in the quantum spin Hall in the presence of magnetic impurities ͑Maciejko et al., insulator ͑QSHI͒. ͑a͒ The interface between a QSHI and an 2009͒ or at a quantum point contact ͑Teo and Kane, ordinary insulator. ͑b͒ The edge state dispersion in the 2009͒. graphene model in which up and down spins propagate in opposite directions. B. HgTeÕCdTe quantum well structures

The above discussion was predicated on the conserva- Graphene is made out of carbon—a light element tion of spin Sz. This is not a fundamental symmetry, with a weak spin-orbit interaction. Though there is dis- though, and spin nonconserving processes—present in agreement on its absolute magnitude ͑Huertas- ␴s any real system—invalidate the meaning of xy. This Hernando, Guinea, and Brataas, 2006; Min et al., 2006; brings into question theories that relied on spin conser- Boettger and Trickey, 2007; Yao et al., 2007; Gmitra et ␴s ͑ ͒ vation to predict an integer quantized xy Volovik and al., 2009 , the energy gap in graphene is likely to be Yakovenko, 1989; Bernevig and Zhang, 2006; Qi, Wu, small. Clearly, a better place to look for this physics and Zhang, 2006͒, as well as the influential theory of the would be in materials with strong spin-orbit interactions, ͑nonquantized͒ spin Hall insulator ͑Murakami, Nagaosa, made from heavy elements near the bottom of the Peri- and Zhang, 2004͒. Kane and Mele ͑2005a͒ showed that odic Table. To this end, Bernevig, Hughes, and Zhang due to T symmetry the edge states in the quantum spin ͑2006͒ ͑BHZ͒ had the idea to consider quantum well Hall insulator are robust even when spin conservation is structures of HgCdTe. This paved the way to the experi- violated because their crossing at k=0 is protected by mental discovery of the quantum spin Hall insulator the Kramers degeneracy discussed in Sec. II.C. This es- phase. tablished the quantum spin Hall insulator as a topologi- Hg1−xCdxTe is a family of semiconductors with strong cal phase. spin-orbit interactions ͑Dornhaus and Nimtz, 1983͒; The quantum spin Hall edge states have the important CdTe has a band structure similar to other semiconduc- “spin filtered” property that the up spins propagate in tors. The conduction-band-edge states have an s-like one direction, while the down spins propagate in the symmetry, while the valence-band-edge states have a other. Such edge states were later termed “helical” ͑Wu, p-like symmetry. In HgTe, the p levels rise above the s Bernevig, and Zhang, 2006͒, in analogy with the corre- levels, leading to an inverted band structure. BHZ con- lation between spin and momentum of a particle known sidered a quantum well structure where HgTe is sand- as helicity. They form a unique 1D conductor that is wiched between layers of CdTe. When the thickness of Ͻ essentially half of an ordinary 1D conductor. Ordinary the HgTe layer is d dc =6.3 nm the 2D electronic states conductors, which have up and down spins propagating bound to the quantum well have the normal band order. Ͼ in both directions, are fragile because the electronic For d dc, however, the 2D bands invert. BHZ showed states are susceptible to Anderson localization in the that the inversion of the bands as a function of increas- presence of weak disorder ͑Anderson, 1958; Lee and ing d signals a quantum phase transition between the Ramakrishnan, 1985͒. By contrast, the quantum spin trivial insulator and the quantum spin Hall insulator. Hall edge states cannot be localized even for strong dis- This can be understood simply in the approximation that order. To see this, imagine an edge that is disordered in the system has inversion symmetry. In this case, since the a finite region and perfectly clean outside that region. s and p states have opposite parity the bands will cross The exact eigenstates can be determined by solving the each other at dc without an avoided crossing. Thus, the ͑ ͒ scattering problem relating incoming waves to those re- energy gap at d=dc vanishes. From Eq. 12 , the change flected from and transmitted through the disordered re- in the parity of the valence-band-edge state signals a ͑ ͒ ␯ gion. Kane and Mele 2005a showed that the reflection phase transition in which the Z2 invariant changes. amplitude is odd under T—roughly because it involves Within a year of the theoretical proposal the flipping the spin. It follows that unless T symmetry is Würzburg group, led by Laurens Molenkamp, made the broken, an incident electron is transmitted perfectly devices and performed transport experiments that across the disordered region. Thus, eigenstates at any showed the first signature of the quantum spin Hall in- energy are extended, and at temperature T=0 the edge sulator. König et al. ͑2007͒ measured the electrical con- state transport is ballistic. For TϾ0 inelastic back- ductance due to the edge states. The low-temperature scattering processes are allowed, which will, in general, ballistic edge state transport can be understood within a lead to a finite conductivity. simple Landauer-Büttiker ͑Büttiker, 1988͒ framework in

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3054 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

Normal Inverted ky ky (a) (b) Γ Γ Γ Γ EF sp 3 4 3 4 CdTe d kx kx Γ Γ Γ Γ E HgTe E 1 2 1 2 ps6.3 nm (a) (b) (c) CdTe d FIG. 7. ͑Color online͒ Fermi circles in the surface Brillouin E (d) zone for ͑a͒ a weak topological insulator and ͑b͒ a strong to- ͑ ͒ (c) eV pological insulator. c In the simplest strong topological insulator the Fermi circle encloses a single Dirac point.

k (Ω)

L 2 14,23 G=.3e /h R ducting surface states. Soon after, this phase was pre- G=2e2/h dicted in several real materials ͑Fu and Kane, 2007͒, in- w ␣ V0QSHI cluding Bi1−xSbx as well as strained HgTe and -Sn. In 2008, Hsieh et al. ͑2008͒ reported the experimental discovery of the first 3D topological insulator in Bi1−xSbx. V -V (V) g thr In 2009 second-generation topological insulators, includ- FIG. 6. ͑Color online͒ Experiments on HgTe/CdTe quantum ing Bi2Se3, which has numerous desirable properties, ͑ wells. ͑a͒ Quantum well structure. ͑b͒ As a function of layer were identified experimentally Xia, Qian, Hsieh, Wray, thickness d the 2D quantum well states cross at a band inver- et al., 2009͒ and theoretically ͑Xia, Qian, Hsieh, Wray, et sion transition. The inverted state is the QSHI, which has he- al., 2009; Zhang, Liu, et al., 2009͒. In this section we lical edge states ͑c͒ that have a nonequilibrium population de- review these developments. termined by the leads. ͑d͒ Experimental two terminal conductance as a function of a gate voltage that tunes EF Ͻ through the bulk gap. Sample I, with d dc, shows insulating A. Strong and weak topological insulators behavior, while samples III and IV show quantized transport associated with edge states. Adapted from König et al., 2007. A 3D topological insulator is characterized by four Z2 ͑␯ ␯ ␯ ␯ ͒͑ topological invariants 0 ; 1 2 3 Fu, Kane, and Mele, 2007; Moore and Balents, 2007; Roy, 2009b͒. They can which the edge states are populated according to the be most easily understood by appealing to the bulk- chemical potential of the lead that they emanate from. 2 boundary correspondence, discussed in Sec. II.C. The This leads to a quantized conductance e /h associated surface states of a 3D crystal can be labeled with a 2D ͑ ͒ with each set of edge states. Figure 6 d shows the resis- crystal momentum. There are four T invariant points tance measurements for a series of samples as a function ⌫ 1,2,3,4 in the surface Brillouin zone, where surface of a gate voltage which tunes the Fermi energy through states, if present, must be Kramers degenerate ͓Figs. 7͑a͒ the bulk energy gap. Sample I is a narrow quantum well and 7͑b͔͒. Away from these special points, the spin-orbit that has a large resistance in the gap. Samples II–IV are interaction will lift the degeneracy. These Kramers de- wider wells in the inverted regime. Samples III and IV 2 generate points therefore form 2D Dirac points in the exhibit a conductance 2e /h associated with the top and surface band structure ͓Fig. 7͑c͔͒. The interesting ques- bottom edges. Samples III and IV have the same length tion is how the Dirac points at the different T invariant L=1 ␮m but different widths w=0.5 and 1 ␮m, indicat- ⌫ points connect to each other. Between any pair a and ing that transport is at the edge. Sample II ͑L=20 ␮m͒ ⌫ b, the surface-state structure will resemble either Fig. showed finite temperature scattering effects. These ex- 3͑a͒ or 3͑b͒. This determines whether the surface Fermi periments convincingly demonstrate the existence of the ⌫ ⌫ surface intersects a line joining a to b an even or an edge states of the quantum spin Hall insulator. Subse- odd number of times. If it is odd, then the surface states quent experiments have established the inherently non- are topologically protected. Which of these two alterna- ͑ local electronic transport in the edge states Roth et al., tives occurred is determined by the four bulk invari- ͒ Z2 2009 . ants. The simplest nontrivial 3D topological insulators may IV. 3D TOPOLOGICAL INSULATORS be constructed by stacking layers of the 2D quantum spin Hall insulator. This is analogous to a similar con- In the summer of 2006 three theoretical groups inde- struction for 3D integer quantum Hall states ͑Kohmoto, pendently discovered that the topological characteriza- Halperin, and Wu, 1992͒. The helical edge states of the tion of the quantum spin Hall insulator state has a natu- layers then become anisotropic surface states. A pos- ral generalization in three dimensions ͑Fu, Kane, and sible surface Fermi surface for weakly coupled layers Mele, 2007; Moore and Balents, 2007; Roy, 2009b͒. stacked along the y direction is shown in Fig. 7͑a͒. In this Moore and Balents ͑2007͒ coined the term “topological figure a single surface band intersects the Fermi energy ⌫ ⌫ ⌫ ⌫ insulator” to describe this electronic phase. Fu, Kane, between 1 and 2 and between 3 and 4, leading to the and Mele ͑2007͒ established the connection between the nontrivial connectivity in Fig. 3͑b͒. This layered state is ␯ bulk topological order and the presence of unique con- referred to as a weak topological insulator and has 0

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3055

͑␯ ␯ ␯ ͒ =0. The indices 1 2 3 can be interpreted as Miller in- (a) Pure Bismuth (x=0)E (b) Bi1-xSbx, 0.07

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3056 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

T ⌳ ␦ TABLE II. Symmetry labels for the Bloch states at the 8 invariant momenta a for the ﬁve valence bands of Bi and Sb. a’s are ͑ ͒ ͑␯ ␯ ␯ ␯ ͒ ͑ ͒ given by Eq. 12 and determine the topological class 0 ; 1 2 3 by relations similar to Eq. 10 . The difference between Bi and ϳ Sb is due to the inversion of the Ls and La bands that occurs at x 0.04.

Bi: Class ͑0;000͒ Sb: Class ͑1;111͒ ⌳ ␦ ⌳ ␦ a Symmetry label a a Symmetry label a ⌫ ⌫+ ⌫− ⌫+ ⌫+ ⌫+ ⌫ ⌫+ ⌫− ⌫+ ⌫+ ⌫+ 1 6 6 6 6 45 −1 1 6 6 6 6 45 −1 3LLs La Ls La La −1 3LLs La Ls La Ls +1 3XXa Xs Xs Xa Xa −1 3XXa Xs Xs Xa Xa −1 − + − + − − + − + − 1T T6 T6 T6 T6 T45 −1 1T T6 T6 T6 T6 T45 −1 states near L have a nearly linear dispersion that is well direction perpendicular to the surface whereas the bulk described by a ͑3+1͒-dimensional Dirac equation states do. Moreover, unlike in a transport experiment, ͑Wolff, 1964͒ with a small mass. These facts have been ARPES carried out in a spin resolution mode can, in used to explain many peculiar properties of bismuth. addition, measure the distribution of spin orientations Substituting bismuth with antimony changes the criti- on the Fermi surface which can be used to estimate the cal energies of the band structure ͓Fig. 8͑b͔͒.AtanSb Berry phase on the surface. Spin sensitivity is critically Ϸ ⌬ concentration of x 0.04, the gap between La and Ls important for probing the existence of spin-momentum closes and a truly massless 3D Dirac point is realized. As locking on the surface expected as a consequence of x is further increased this gap reopens with an inverted bulk topological order. ordering. For xϾ0.07 the top of the valence band at T Experiments by Hsieh et al. ͑2008͒ probed both the moves below the bottom of the conduction band at L, bulk and surface electronic structures of Bi0.09Sb0.91 with and the material becomes an insulator. Once the band at ARPES. Figure 9͑a͒ shows the ARPES spectrum, which T drops below the valence band at L,atxϳ0.09, the can be interpreted as a map of the energy of the occu- system is a direct-gap insulator with a massive Dirac-like pied electronic states as a function of momentum along bulk bands. As x is increased further, the conduction and the line connecting ⌫¯ to M¯ in the projected surface Bril- տ valence bands remain separated, and for x 0.22 the va- louin zone ͓Fig. 9͑b͔͒. Bulk energy bands associated with lence band at a different point rises above the conduc- the L point are observed, which reflect the nearly linear tion band, restoring the semimetallic state. 3D Dirac-like dispersion. The same experiments ob- Since pure bismuth and pure antimony both have a served several surface states that span the bulk gap. finite direct band gap, their valence bands can be topo- The observed surface-state structure of Bi1−xSbx has logically classified. Moreover, since they have inversion similarities with the surface states in pure Bi, which have symmetry, Eq. ͑12͒ can be used to determine the topo- been studied previously ͑Patthey, Schneider, and Mick- logical indices. Table II shows the symmetry labels that litz, 1994; Agergaard et al., 2001; Ast and Höchst, 2001; specify the parity of the Bloch states for the occupied Hirahara et al., 2006; Hofmann, 2006͒. In pure Bi, two bands at the 8 T invariant points in the bulk Brillouin ¯ zone ͑Liu and Allen, 1995͒. Fu and Kane ͑2007͒ used bands emerge from the bulk band continuum near ⌫ to this information to deduce that bismuth is in the trivial form a central electron pocket and an adjacent hole ͑0;000͒ class while antimony is in the ͑1;111͒ class. Since lobe. These two bands result from the spin splitting of a the semiconducting alloy is on the antimony side of the surface state and are thus expected to be singly degen- band inversion transition, it is predicted to inherit the ¯ erate. In Bi1−xSbx, there are additional states near M, ͑1;111͒ class from antimony. which play a crucial role. Charge transport experiments, which were successful As explained in Sec. IV.A, Kramers’ theorem requires ͑ for identifying the 2D topological insulator König et al., surface states to be doubly degenerate at the T invariant 2007͒, are problematic in 3D materials because the sig- points ⌫¯ and each of the three equivalent M¯ points. Such nature in the conductivity of the topological character of the surface states is more subtle in three dimensions. a Kramers point is indeed observed at M¯ approximately Moreover, it is difficult to separate the surface contribu- 15±5 meV below EF. As expected for a system with tion to the conductivity from that of the bulk. ARPES is strong spin-orbit interactions, the degeneracy is lifted an ideal tool for probing the topological character of the away from M¯ . The observed surface bands cross the surface states. ARPES uses a photon to eject an electron Fermi energy five times between ⌫¯ and M¯ . This odd from a crystal and then determines the surface or bulk ͑ ͒ electronic structure from an analysis of the momentum number of crossings is analogous to Fig. 3 b and indi- of the emitted electron. High-resolution ARPES per- cates that these surface states are topologically pro- formed with modulated photon energy allows for a clear tected. Accounting for the threefold rotational symme- ͑ ͒ isolation of surface states from that of the bulk 3D band try and mirror symmetry of the 111 surface, these data structure because surface states do not disperse along a show that the surface Fermi surface encloses ⌫¯ an odd

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3057

0.1 3D Topo. Insulator (Bi1-x Sb x) bulk

) 12 345, gap 0.0 hn Mott spin detector

( e B 40 kV E Au foil e beam -0.1 accelerating optics z’ q z y’ 0.0 0.2 0.4 0.6 0.8 1.0 (a) x’ (a) sample x G (KP) - k (Å-1 ) M (KP) y X low high Spin Polarization Distribution z 0.20.2 1 1 (b) k (c) M G y 1 M - K x=0 (Å ) 0.0 P P Deg 8 y 0.0 y 0 0 z kx k T x=0.1 -0.2-0.2 Berry’s Phase π -1 -1 X cm) 6 L -1 -0.2-0.2 0.00.0 0.20.2 0.40.6 0.8 1.0 1.01.2 (c) 0 1 01

E W (b) -1 P P kx (Å ) x in plane T L 2 4 1 (m ´ 80 X r FIG. 10. ͑Color online͒ Topological spin textures: Spin re- L 2 S solved photoemission directly probes the nontrivial spin tex- ͑ ͒ La 0 tures of the topological insulator surface. a A schematic of x 0 100 200 300 Bi 4% 7% 8% spin-ARPES measurement setup that was used to measure the T (K) spin distribution on the ͑111͒ surface Fermi surface of ͑ ͒ FIG. 9. ͑Color online͒ Topological surface states in Bi Sb : Bi0.91Sb0.09. b Spin orientations on the surface create a vor- 1−x x ⌫ ␲ ͑a͒ ARPES data on the ͑111͒ surface of Bi Sb which probes texlike pattern around point. A net Berry phase is ex- 0.9 0.1 ͑ ͒ the occupied surface states as a function of momentum on the tracted from the full Fermi-surface data. c Net polarizations along x, y, and z directions are shown. P ϳ0 suggests that line connecting the T invariant points ⌫¯ and M¯ in the surface z spins lie mostly within the surface plane. Adapted from Hsieh, Brillouin zone. Only the surface bands cross the Fermi energy Xia, Wray, Qian, et al., 2009a, 2009b and Hsieh et al., 2010. five times. This, along with further detailed ARPES results ͑Hsieh et al., 2008͒, establishes that the semiconducting alloy ͑ ͒ Bi1−xSbx is a strong topological insulator in the 1;111 class. provided even more information about the topological ͑b͒ A schematic of the 3D Brillouin zone and its ͑111͒ surface ͑ ͒ structure by probing a mirror Chern number, which projection. c The resistivity of semimetallic pure Bi con- ͑ ͒ trasted with the semiconducting alloy. Adapted from Hsieh et agreed favorably with theory Teo, Fu, and Kane, 2008 . al., 2008. Spin polarized ARPES also enables a similar charac- terization of surface states in the metallic regime of the Bi1−xSbx series. Pure Sb is predicted to have a topologi- number of times, while it encloses the three equivalent cally nontrivial valence band despite the semimetallic M¯ points an even number of times. This establishes band overlap. Hsieh, Xia, Wray, Qian, et al. ͑2009b͒ ␯ Bi1−xSbx as a strong topological insulator, with 0 =1. found that the surface states of Sb carry a Berry phase The data are consistent with the predicted ͑1;111͒ topo- and chirality property predicted by theory ͑Teo, Fu, and logical class. Kane, 2008͒ that is unlike the conventional spin-orbit A distinguishing feature of topological insulator sur- metals such as gold, which has zero net Berry phase and face states is the intimate correlation between spin and no net chirality. Additional compositions of the Bi1−xSbx momentum they exhibit, which underlies the ␲ Berry series provided further evidence for the topological phase associated with the Fermi surface. Spin resolved character of the surface states ͑Nishide et al., 2010͒. ARPES, shown schematically in Fig. 10͑a͒, is ideally These results demonstrated that ARPES and spin suited to probe this physics. Experiments by Hsieh, Xia, ARPES are powerful probes of topological order. Wray, Qian, et al. ͑2009b͒ measured the spin polarization As discussed in Sec. IV.A the topological surface of the surface states. These experiments proved that the states are expected to be robust in the presence of non- surface states are indeed nondegenerate and strongly magnetic disorder and immune from Anderson localiza- spin polarized ͓Fig. 10͑b͔͒, providing even more decisive tion. The origin of this is the fact that T symmetry for- evidence for their topological classification. In addition, bids the backscattering between Kramers pairs at k and the spin-polarization data also established the connectiv- −k. Random alloying in Bi1−xSbx, which is not present in ͑ ity of the surface state bands above EF which is inacces- other material families of topological insulators found to sible to ARPES͒, showing that bands labeled 2 and 3 in date, makes this material system an ideal candidate in Fig. 9͑a͒ connect to form a hole pocket. Finally, they which to examine the impact of disorder or random po- directly mapped the spin texture of the Fermi surface, tential on topological surface states. The fact that the 2D providing the first direct evidence for the ␲ Berry phase states are indeed protected from spin-independent scat- by showing that the spin polarization rotates by 360° tering was established by Roushan et al. ͑2009͒ by com- around the central Fermi surface, shown in Fig. 10͑c͒. bining results from scanning tunneling spectroscopy and The measurement of the handedness of this rotation spin ARPES. Figure 11 shows the analysis of the inter-

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3058 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

Spin-ARPES C. Second generation materials: Bi2Se3,Bi2Te3, and Sb2Te3

The surface structure of Bi1−xSbx was rather compli- cated and the band gap was rather small. This motivated a search for topological insulators with a larger band gap and simpler surface spectrum. A second generation of 3D topological insulator materials ͑Moore, 2009͒, especially Bi2Se3, offers the potential for topologically protected behavior in ordinary crystals at room temperature and zero magnetic field. In 2008, work by the Princeton group used ARPES and first-principles calculations to study the surface band structure of Bi2Se3 and observe the characteristic signature of a topological insulator in the form of a single Dirac cone ͑Xia, Qian, Hsieh, Wray, et al., 2009͒. Concurrent theoretical work by Zhang, Liu, et al. ͑2009͒ used electronic structure methods to show that Bi2Se3 is just one of several new large band-gap topological insulators. Zhang, Liu, et al. ͑2009͒ also provided a simple tight-binding model to capture the single Dirac cone observed in these materials. Detailed and ͑ systematic surface investigations of Bi2Se3 Hor et al., 2009; Hsieh, Xia, Qian, Wray, et al., 2009a; Park et al., ͒ ͑ 2010 ,Bi2Te3 Chen et al., 2009; Hsieh, Xia, Qian, Wray, et al., 2009a, 2009b; Xia, Qian, Hsieh, Shankar, et al., ͒ ͑ ͒ 2009 , and Sb2Te3 Hsieh, Xia, Qian, Wray, et al., 2009b confirmed the topological band structure of these three materials. This also explained earlier puzzling observa- ͑ ͒ tions on Bi2Te3 Noh et al., 2008 . These works showed that the topological insulator behavior in these materials FIG. 11. ͑Color online͒ Absence of backscattering: Quasipar- is associated with a band inversion at k=0, leading to the ͑ ͒ ͑ ͒ ticle interference observed at the surface of Bi0.92Sb0.08 exhibits 1;000 topological class. The 1;000 phase observed in ͑ ͒ ͑ ͒ the absence of elastic backscattering. a Spatially resolved the Bi2Se3 series differs from the 1;111 phase in conductance maps of the ͑111͒ surface obtained at 0 mV over a Bi1−xSbx due to its weak topological invariant, which has 1000 Åϫ1000 Å. ͑b͒ Spin-ARPES map of the surface state implications for the behavior of dislocations ͑Ran, measured at the Fermi level. The spin textures from spin- Zhang, and Vishwanath, 2009͒. ͑ ͒ ARPES measurements are shown with arrows. c Fourier Though the phase observed in the Bi Se class has the ͑ ͒ ͑ ͒ 2 3 transform scanning tunneling spectroscopy FT-STS at EF. d ␯ ͑ ͒ ͑ ͒ same strong topological invariant 0 =1 as Bi1−xSbx, The joint density of states JDOS at EF. e The spin- ͑ ͒ ͑ ͒ there are three crucial differences that suggest that this dependent scattering probability SSP at EF. f Closeup of ⌫ series may become the reference material for future ex- the JDOS, FT-STS, and SSP at EF, along the -M direction. Adapted from Hsieh, Xia, Wray, Qian, et al., 2009b and Rous- periments. The Bi2Se3 surface state is found from han et al., 2009. ARPES and theory to be a nearly idealized single Dirac cone as seen from the experimental data in Figs. 12, 13, ͑ and 16. Second, Bi2Se3 is stoichiometric i.e., a pure ͒ ference pattern due to scattering at the surface. Figure compound rather than an alloy such as Bi1−xSbx and 11͑c͒ shows the Fourier transform of the observed pat- hence can be prepared, in principle, at higher purity. tern ͓Fig. 11͑a͔͒, while Figs. 11͑d͒ and 11͑e͒ show the While the topological insulator phase is predicted to be joint density of states computed from the Fermi surface quite robust to disorder, many experimental probes of ͓Fig. 11͑b͔͒ with and without a suppression of k to −k the phase, including ARPES of the surface band struc- backscattering. The similarity between Figs. 11͑c͒ and ture, are clearer in high-purity samples. Finally and per- haps most important for applications, Bi Se has a large 11͑e͒ shows that despite strong atomic scale disorder, k 2 3 band gap of ϳ0.3 eV ͑3600 K͒. This indicates that in its to −k backscattering is absent. Similar conclusions have high-purity form Bi Se can exhibit topological insulator emerged from studies of the electronic interference pat- 2 3 behavior at room temperature ͑Fig. 13͒ and greatly in- terns near defects or steps on the surface in other topo- creases the potential for applications. To understand the ͑ logical insulators Urazhdin et al., 2004; Zhang, Cheng, likely impact of these new topological insulators, an ͒ et al., 2009; Alpichshev et al., 2010 . In graphene there is analogy can be drawn with the early days of high- an approximate version of this protection if the disorder temperature cuprate superconductivity: the original cu- has a smooth potential which does not mix the valleys at prate superconductor LBCO was quickly superseded by K and KЈ, but real graphene will become localized with second-generation materials such as YBCO and BSCCO strong disorder ͑Castro Neto et al., 2009͒. for most applied and scientific purposes.

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators 3059

Half Dirac Gas TopologicalInsulator

2 1 DOS 2 Θaxion =π node E Dirac Γ k y 1 node

background k x

ARPES DOS0 (arb. units) -0.15 -0.1 -0.05 0.0 (b) Bi230 Se (v =1=Θπ / ) (a) E - ED (eV) Room-Temperature Topological Insulators 300K 10K 0

-0.2

E (eV) -0.4

(c)-0.1 00.1 (d) -0.1 00.1(e) -0.1 00.1 k(Å)-1 k(Å)-1 k(Å)-1 FIG. 12. ͑Color online͒ Helical fermions: Spin-momentum x x x locked helical surface Dirac fermions are hallmark signatures FIG. 13. ͑Color online͒ Room temperature topological order of topological insulators. ͑a͒ ARPES data for Bi Se reveal ͑ ͒ 2 3 in Bi2Se3: a Crystal momentum integrated ARPES data near surface electronic states with a single spin-polarized Dirac Fermi level exhibit linear falloff of density of states, which ͑ ͒ cone. b The surface Fermi surface exhibits a chiral left- combined with the spin-resolved nature of the states suggest ͑ ͒ handed spin texture. c Surface electronic structure of Bi2Se3 that a half Fermi gas is realized on the topological surfaces. ͑b͒ computed in the local-density approximation. The shaded re- Spin-texture map based on spin-ARPES data suggest that the ͑ ͒ gions describe bulk states, and the lines are surface states. d spin chirality changes sign across the Dirac point. ͑c͒ The Dirac Schematic of the spin-polarized surface-state dispersion in node remains well defined up a temperature of 300 K suggest- ͑ ͒ Bi2X3 1;000 topological insulators. Adapted from Xia et al., ing the stability of topological effects up to the room tempera- 2008, Hsieh, Xia, Qian, Wray, et al., 2009a, and Xia, Qian, ture. ͑d͒ The Dirac cone measured at a temperature of 10 K. Hsieh, Wray, et al., 2009. ͑e͒ Full Dirac cone. Adapted from Hsieh, Xia, Qian, Wray, et al., 2009a. All key properties of topological states have been demonstrated for Bi2Se3 which has the simplest Dirac ing disorder potential introduced on the surface. Non- cone surface spectrum and the largest band gap. In magnetic disorder created via molecular absorbent NO2 Bi2Te3 the surface states exhibit large deviations from a or alkali atom adsorption ͑KorNa͒ on the surface ͑ ͒ simple Dirac cone Fig. 14 due to a combination of leaves the Dirac node intact ͓Figs. 15͑c͒ and 15͑d͔͒ in ͑ ͒ smaller band gap 0.15 eV and a strong trigonal poten- both Bi Se and Bi Te ͑Hsieh, Xia, Qian, Wray, et al., ͑ ͒ 2 3 2 3 tial Chen et al., 2009 , which can be utilized to explore 2009a; Xia, Qian, Hsieh, Shankar, et al., 2009͒. These ͑ some aspects of its surface properties Fu, 2009; Hasan, results are consistent with the fact that the topological Lin, and Bansil, 2009͒. The hexagonal deformation of the surface states is confirmed by scanning tunneling mi- croscopy ͑STM͒ measurements ͑Alpichshev et al., 2010͒; Fig. 14. Speaking of applications within this class of materials, Bi2Te3 is already well known to materials scien- tists working on thermoelectricity. It is a commonly used thermoelectric material in the crucial engineering regime near room temperature. Two defining properties of topological insulators— spin-momentum locking of surface states and ␲ Berry phase—can be clearly demonstrated in the Bi2Se3 series. The surface states are expected to be protected by T symmetry which implies that the surface Dirac node should be robust in the presence of nonmagnetic disorder but open a gap in the presence of T breaking pertur- bations. Magnetic impurities such as Fe or Mn on the ͓ surface of Bi2Se3 open a gap at the Dirac point Figs. FIG. 14. ͑Color online͒ Hexagonal warping of surface states in ͑ ͒ ͑ ͔͒͑ 15 a and 15 b Xia et al., 2008; Hsieh, Xia, Qian, Wray, Bi2Te3: ARPES and STM studies of Bi2Te3 reveal a hexagonal et al., 2009a; Hor, Roushan, et al., 2010; Wray et al., deformation of surface states. Fermi-surface evolution with in- 2010͒. The magnitude of the gap is likely set by the increasing n-type doping as observed in ARPES measurements. teraction of Fe ions with the Se surface and the T break- Adapted from Alpichshev et al., 2010.

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3060 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

0.2

-1

(Å0.0 )

-0.2 (a) -0.2 0.0 0.2-0.2 0.0 0.2-0.2 0.0 0.2

-0.2

E (eV) -0.4

(b) -0.1 0.0 0.1 -0.1 0.0 0.1 -0.1 0.0 0.1 -1 kx (Å )

FIG. 16. ͑Color online͒ Chemical gating a topological surface FIG. 15. ͑Color online͒ Protection by time-reversal symmetry: to the spin-degenerate point: Topological insulator surfaces are Topological surface states are robust in the presence of strong most interesting if the chemical potential can be placed at the nonmagnetic disorder but open a gap in the presence of T Dirac node without intercepting any bulk band. This can be breaking magnetic impurities and disorder. ͑a͒ Magnetic impu- achieved in Bi Se via the chemical tailoring of the surface or rity such as Fe on the surface of Bi Se opens a gap at the 2 3 2 3 using electrical gating methods. ͑a͒ Evolution of surface Fermi Dirac point. The magnitude of the gap is set by the interaction surface with increasing NO adsorption on the surface. NO of Fe ions with the Se surface and the T breaking disorder 2 2 extracts electrons from the Bi Se surface leading to an effec- potential introduced on the surface. ͑b͒ A comparison of sur- 2 3 tive hole doping of the material. ͑b͒ Chemical gating of the face band dispersion with and without Fe doping. ͑c͒ and ͑d͒ surface can be used to place the chemical potential at the spin- Nonmagnetic disorder created via molecular absorbent NO or 2 degenerate Dirac point. Adapted from Hsieh, Xia, Qian, Wray, alkali atom adsorption ͑KorNa͒ on the surface leaves the et al., 2009a. Dirac node intact in both Bi2Se3 and Bi2Te3. Adapted from Hsieh, Xia, Qian, Wray, et al., 2009a, Xia, Qian, Hsieh, Shan- kar, et al., 2009, and Wray et al., 2010. tion in the bulk from impurity or self-doping states. Electrical transport measurements on Bi2Se3 show that surface states are protected by T symmetry. doping with a small concentration of Ca leads to insulat- Many of the interesting theoretical proposals that uti- ing behavior. Figure 17͑a͒ shows the resistivity of several lize topological insulator surfaces require the chemical samples with varying Ca concentrations. For 0.002Ͻx potential to lie at or near the surface Dirac point. This is Ͻ0.025, the resistivity shows a sharp upturn below similar to the case in graphene, where the chemistry of 100 K before saturating. The low-temperature resistivity carbon atoms naturally locates the Fermi level at the Dirac point. This makes its density of carriers highly tunable by an applied electrical ﬁeld and enables applications of graphene to both basic science and microelec- (a) tronics. The surface Fermi level of a topological insula- CB SS tor depends on the detailed electrostatics of the surface E and is not necessarily at the Dirac point. Moreover, for k VB naturally grown Bi2Se3 the bulk Fermi energy is not even in the gap. The observed n-type behavior is be- lieved to be caused Se vacancies. By appropriate chemical modiﬁcations, however, the Fermi energy of both the bulk and the surface can be controlled. This allowed Hsieh, Xia, Qian, Wray, et al. ͑2009a͒ to reach the sweet spot in which the surface Fermi energy is tuned to the Dirac point ͑Fig. 16͒. This was achieved by doping bulk FIG. 17. ͑Color online͒ Electrical transport in Bi Se . ͑a͒ Re- with a small concentration of Ca, which compensates the 2 3 sistivity for samples of pure Bi2Se3 doped with a small concen- Se vacancies, to place the Fermi level within the bulk tration of Ca. Increasing the Ca concentration moves the band gap. The surface was hole doped by exposing the Fermi level from the conduction band into the gap and then to surface to NO2 gas to place the Fermi level at the Dirac the valence band. Samples with 0.002ϽxϽ0.0025, labeled G, point. show insulating behavior below 100 K ͑Checkelsky et al., ͒ ͑ ͒ The main remaining complication with these materi- 2009 . b Bi2Se3 doped with Cu shows superconducting behav- als, especially for experimental techniques that ͑unlike ior below 3.8 K for x=.12. The inset shows the magnetic sus- ARPES͒ do not distinguish directly between bulk and ceptibility which exhibits the Meissner effect. Adapted from surface states, is that they have some residual conduc- Hor, Williams, et al., 2010 and Wray et al., 2010.

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is still too small to be explained by the surface states B alone. However, the low-temperature transport exhibits 2 (b) interesting 2D mesoscopic effects that are not com- 1 (a) E 0 σ =e2/2h pletely understood ͑Checkelsky et al., 2009͒. Doping xy -1 Bi2Se3 with copper leads to a metallic state that shows TI ͓ ͑ ͔͒ -2 superconducting behavior Fig. 17 b below 3.8 K σ =e2/2h ͑Wray et al., 2009; Hor, Williams, et al., 2010͒. This has xy important ramiﬁcations for some of the devices pro- (d) posed in Sec. IV. (c) E M M

TI V. EXOTIC BROKEN SYMMETRY SURFACE PHASES FIG. 18. ͑Color online͒ Surface quantum Hall effect. ͑a͒ The Now that the basic properties of topological insulators Dirac spectrum is replaced by Landau levels in an orbital mag- have been established, we may ask what can be done netic field. ͑b͒ The top and bottom surfaces share a single chi- with them. In this section we argue that the unique prop- ral fermion edge mode. ͑c͒ A thin magnetic film can induce an erties of topological insulator surface and edge states are energy gap at the surface. ͑d͒ A domain wall in the surface most dramatic if an energy gap can be induced in them. magnetization exhibits a chiral fermion mode. This can be done by breaking T symmetry with an exter- nal magnetic field ͑Fu and Kane, 2007͒ or proximity to a magnetic material ͑Qi, Hughes, and Zhang, 2008͒,by can share a single chiral edge state, which carries the breaking gauge symmetry due to proximity to a super- integer quantized Hall current. conductor ͑Fu and Kane, 2008͒, or by an excitonic insta- A related surface quantum Hall effect, called the bility of two coupled surfaces ͑Seradjeh, Moore, and anomalous quantum Hall effect, can be induced with the ͒ Franz, 2009 . In this section we review the magnetic and proximity to a magnetic insulator. A thin magnetic film superconducting surface phases. on the surface of a topological insulator will give rise to a local exchange field that lifts the Kramers degeneracy at the surface Dirac points. This introduces a mass term A. Quantum Hall effect and topological magnetoelectric effect m into the Dirac equation ͓Eq. ͑16͔͒,asinEq.͑4͒.Ifthe 1. Surface quantum Hall effect EF is in this energy gap, there is a half integer quantized ␴ 2 ͑ ͒ Hall conductivity xy=e /2h Pankratov, 1987 , as dis- A perpendicular magnetic field will lead to Landau cussed in Sec. II.B.2. This can be probed in a transport levels in the surface electronic spectrum and the quan- experiment by introducing a domain wall into the mag- tum Hall effect. The Landau levels for Dirac electrons net. The sign of m depends on the direction of the mag- are special, however, because a Landau level is guaran- ͓ teed to exist at exactly zero energy ͑Jackiw, 1984͒. This netization. At an interface where m changes sign Fig. ͑ ͔͒ zero Landau level is particle-hole symmetric in the sense 18 d there will be a 1D chiral edge state, analogous to ͑ ͒ that the Hall conductivity is equal and opposite when unfolding the surface in Fig. 18 b . the Landau level is full or empty. Since the Hall conductivity increases by e2 /h when the Fermi energy crosses a Landau level the Hall conductivity is half integer quan- 2. Topological magnetoelectric effect and axion electrodynamics tized ͑Zheng and Ando, 2002͒, ␴ ͑ ͒ 2 ͑ ͒ The surface Hall conductivity can also be probed xy = n + 1/2 e /h. 17 without the edge states either by optical methods or by This physics has been demonstrated in experiments on measuring the magnetic field produced by surface cur- ͑ ͒ graphene Novoselov et al., 2005; Zhang et al., 2005 . rents. This leads to an intriguing topological magneto- However, there is an important difference. In graphene electric effect ͑Qi, Hughes, and Zhang, 2008; Essin, Eq. ͑17͒ is multiplied by 4 due to the spin and valley Moore, and Vanderbilt, 2009͒. Imagine a cylindrical to- degeneracy of graphene’s Dirac points, so the observed pological insulator with magnetically gapped surface Hall conductivity is still integer quantized. At the sur- states and an electric field E along its axis. The azi- face of the topological insulator there is only a single ͑ 2 ͉͒ ͉ Dirac point. Such a “fractional” integer quantized Hall muthal surface Hall current e /2h E leads to a effect should be a cause for concern because the integer magnetic-dipole moment associated with a magnetiza- ␣ quantized Hall effect is always associated with chiral tion M= E, where the magnetoelectric polarizability is 2 edge states, which can only be integer quantized. The given by ␣=e /2h. resolution is the mathematical fact that a surface cannot A field theory for this magnetoelectric effect can be have a boundary. In a slab geometry shown in Fig. 18͑a͒, developed by including a ␪ term in the electromagnetic the top and bottom surfaces are necessarily connected to Lagrangian, which has a form analogous to the theory of each other and will always be measured in parallel ͑Fu axion electrodynamics that has been studied in particle and Kane, 2007͒, doubling the 1/2. The top and bottom physics contexts ͑Wilczek, 1987͒,

Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010 3062 M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators

2 ⌬L = ␪͑e /2␲h͒E · B. ͑18͒ Measure (|012034>+|112134>)/ 2 The field ␪, which is a dynamical variable in the axion theory, is a constant ␲ in the topological insulator. Im- 1234 portantly, when expressed in terms of the vector poten- t ␪ tial E·B is a total derivative, so a constant has no Braid effect on the electrodynamics. However, a gapped interface, across which ␪ changes by ⌬␪, is associated with a 1234 surface Hall conductivity ␴ =⌬␪e2 /͑2␲h͒. xy Create |012034> As in the axion theory, the action corresponding to Eq. ͑18͒ is invariant under ␪→␪+2␲. Physically, this re- FIG. 19. ͑Color online͒ A simple operation in which two vor- flects the fact that an integer quantum Hall state with tices are exchanged. The vortex pairs 12 and 34 are created in ␴ 2 ͑ ͒ xy=ne /h can exist at the surface without changing the the vacuum zero quasiparticle state. When they are brought bulk properties ͑Essin, Moore, and Vanderbilt, 2009͒. back together they are in an entangled superposition of 0 and This resembles a similar ambiguity in the electric polar- 1 quasiparticle states. ization. Qi, Hughes, and Zhang ͑2008͒ showed that since T ␪ ␲ T E·B is odd under , only =0 or are consistent with making a quantum computer is preventing the system ␪ symmetry, so is quantized. By computing the magne- from accidentally measuring itself. The 2N Majorana ␪ toelectric response perturbatively, can be computed in bound states define N qubits a quantum memory. ␴ a manner similar to the Kubo formula calculation of xy. Adiabatically interchanging the vortices or more gen- ␪ ␲ ␯ / is identical to 0, the invariant characterizing a erally braiding them leads to the phenomenon of non- strong topological insulator. Abelian statistics ͑Moore and Read, 1991͒. Such pro- Observation of the surface currents associated with cesses implement unitary operations on the state vector this magnetoelectric effect will be an important comple- ͉␺ ͘→ ͉␺ ͘ a Uab b that generalize the usual notion of Fermi ment to the ARPES experiments. It should be empha- and Bose quantum statistics ͑Nayak and Wilczek, 1996; sized, however, that despite the topologically quantized Ivanov, 2001͒. These operations are precisely what a status of ␪, the surface currents are not quantized the quantum computer is supposed to do. A quantum com- way edge state transport currents are quantized in the putation will consist of three steps, depicted in Fig. 19. quantum Hall effect. The surface currents are bound ͑ ͒ currents, which must be distinguished from other bound i Create: If a pair i,j of vortices is created, they will ͉ ͘ currents that may be present. Nonetheless, it may be be in the ground state 0ij with no extra quasipar- possible to account for such effects, and signatures of ␪ ticle excitations. Creating N pairs initializes the will be interesting to observe. Qi, Li, et al. ͑2009͒ pointed system. ␴ out that a consequence of a nonzero surface xy is that ͑ii͒ Braid: Adiabatically rearranging the vortices an electric charge outside the surface gives rise to a pat- modifies the state and performs a quantum com- tern of surface currents that produces a magnetic field putation. the same as that of an image magnetic monopole. ͑iii͒ Measure: Bringing vortices i and j back together allows the quantum state associated with each ͉ ͘ ͉ ͘ B. Superconducting proximity effect pair to be measured. 1ij and 0ij will be distinguished by the presence or absence of an extra Combining topological insulators with ordinary super- fermionic quasiparticle associated with the pair. conductors leads to a correlated interface state that, such as a topological superconductor, is predicted to Though the quantum operations allowed by manipu- host Majorana fermion excitations. In this section we lating the Majorana states do not have sufficient struc- ͑ begin by reviewing the properties of Majorana fermion ture to construct a universal quantum computer Freed- ͒ excitations and the proposal by Kitaev ͑2003͒ to use man, Larsen, and Wang, 2002 , the topological those properties for fault tolerant quantum information protection of the quantum information makes the ex- processing. We then describe methods for engineering perimental observation of Majorana fermions and non- Majorana fermions in superconductor-topological insu- Abelian statistics a high priority in condensed-matter ͑ ͒ lator devices and prospects for their experimental obser- physics Nayak et al., 2008 . Current experimental ef- ␯ vation. forts have focused on the =5/2 quantum Hall state, where interferometry experiments ͑Das Sarma, Freed- man, and Nayak, 2005; Stern and Halperin, 2006͒ can, in 1. Majorana fermions and topological quantum computing principle, detect the non-Abelian statistics predicted for As discussed in Sec. II.D.2, a well-separated pair of the quasiparticles. Though recent experiments on the Majorana bound states defines a degenerate two-level quantum Hall effect have shown encouraging indirect system—a qubit. Importantly, the quantum information evidence for these states ͑Dolev et al., 2008; Radu et al., in the qubit is stored nonlocally. The state cannot be 2008; Willett, Pfeiffer, and West, 2009͒, definitive obser- measured with a local measurement on one of the bound vation of the Majorana states has remained elusive. In states. This is crucial because the main difficulty with Sec. V.B.2 we describe the possibility of realizing these

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states in topological insulator-superconductor structures. (a)Φ (b) The large energy scale associated with the energy gap in =h/2e S S φ φ Bi2Se3 may provide an advantage, so the required tem- 1 2 perature scale will be limited only by the supercon- S TI φ ductor. S 3 (c) (d) SSM 2. Majorana fermions on topological insulators M Φ eh=h/2e Consider an interface between a topological insulator S and an s wave superconductor. Due to the superconduct- M QSHI ing proximity effect, Cooper pairs may tunnel from the superconductor to the surface, leading to an induced su- FIG. 20. ͑Color online͒ Majorana fermions on topological in- perconducting energy gap in the surface states. The re- sulators. ͑a͒ A superconducting vortex or antidot with flux sulting 2D superconducting state is different from an or- h/2e on a topological insulator is associated with a Majorana dinary superconductor because the surface states are not zero mode. ͑b͒ A superconducting trijunction on a topological spin degenerate and contain only half the degrees of insulator. Majorana modes at the junction can be controlled by ␾ freedom of a normal metal. The superconducting state adjusting the phases 1,2,3. 1D chiral Majorana modes exist at resembles the spinless p +ip topological supercon- a superconductor-magnet interface on a topological insulator. x y ͑ ͒ ductor discussed in Sec. II.D, which is also based on a c A 1D chiral Dirac mode on a magnetic domain wall that splits into two chiral Majorana modes around a superconduct- spin nondegenerate Fermi surface. Unlike the p +ip x y ing island. When ⌽=h/2e interference of the Majorana modes superconductor, the surface superconductor does not ͑ ͒ T converts an electron into a hole. d Majorana modes at a violate symmetry, and its Cooper pairs have even par- superconductor-magnet junction on a 2D QSHI. ity. The minus sign required by Fermi statistics is sup- plied by the ␲ Berry phase of the surface states. Like the px +ipy superconductor, the surface superconductor will ingenuity—what makes them good for quantum com- have a zero energy Majorana state bound to a vortex puting makes them hard to measure. A first step would ͑Fu and Kane, 2008͒. Similar zero modes were later be to detect the Majorana state at a vortex, antidot, or found for superconducting graphene ͑Ghaemi and Wilc- trijunction by tunneling into it from a normal metal. A zek, 2007; Bergman and Le Hur, 2009͒, though those signature of the zero mode would be a zero-bias modes were intrinsically doubled. Undoubled Majorana anomaly, which would have a characteristic current- bound states were found earlier by Jackiw and Rossi voltage relation ͑Bolech and Demler, 2007; Law, Lee, ͑1981͒ in a related field theory model that had an extra and Ng, 2009͒. chiral symmetry. Interestingly, the Majorana states on a Another venue for Majorana fermions on a topologi- topological insulator emerge as solutions to a 3D BdG cal insulator surface is a linear interface between super- theory, so there is a sense in which their non-Abelian conducting and magnetically gapped regions ͑Fu and statistics is inherently three dimensional ͑Teo and Kane, Kane, 2008; Tanaka, Yokoyama, and Nagaosa, 2009; 2010͒. Linder et al., 2010͒. This leads to a 1D chiral Majorana Majorana states can, in principle, be engineered and mode, analogous to the edge state of a 2D topological manipulated using junctions of superconductors on the superconductor ͓Fig. 4͑e͔͒. This can be used to construct surface of a topological insulator ͑Fu and Kane, 2008͒.If a novel interferometer for Majorana fermions ͑Akh- the phases on three superconductors that meet at a tri- merov, Nilsson, and Beenakker, 2009; Fu and Kane, ͓ ͑ ͔͒ ͑␾ ␾ ␾ ͒ junction Fig. 20 b are arranged such that 1 , 2 , 3 2009b͒. Figure 20͑c͒ shows a superconducting island sur- =͑0,2␲/3,4␲/3͒, then a vortex is simulated, and a zero rounded by magnetic regions with a magnetic domain mode will be bound to the junction. If the phases are wall. The chiral Dirac fermions on the magnetic domain changed, the zero mode cannot disappear until the en- wall incident from the left split into two chiral Majorana ergy gap along one of the three linear junctions goes to fermions on opposite sides of the superconductor and zero. This occurs when the phase difference across the then recombine. If the superconductor encloses a flux junction is ␲. At this point the Majorana bound state ⌽=h/2e, then the Majorana fermions pick up a relative moves to the other end of the linear junction. Combin- minus sign analogous to the Aharonov-Bohm effect. ing these trijunctions into circuits connected by linear This has the effect of converting an incident electron junctions could then allow for the create-braid-measure into an outgoing hole, with a Cooper pair of electrons protocol discussed in Sec. V.B.1 to be implemented. The absorbed by the superconductor. This could be observed state of two Majorana modes brought together on a lin- in a three terminal transport setup. ear junction can be probed by measuring the supercur- Majorana bound states can also be engineered at the rent across that junction. edge of a 2D quantum spin Hall insulator utilizing mag- There are many hurtles to overcome before this am- netic and superconducting energy gaps ͓Fig. 20͑d͔͒͑Nils- bitious proposal can be realized. The first step is finding son, Akhmerov, and Beenakker, 2008; Fu and Kane, a suitable superconductor that makes good contact with 2009a͒. This and other geometries can, in principle, be a topological insulator. Probing the signatures of Majo- used to test the inherent nonlocality of Majorana fer- rana fermions and non-Abelian statistics will require mion states ͑Fu, 2010͒

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VI. CONCLUSION AND OUTLOOK insulators to take advantage of the special spin properties of the surface states. Though the basic properties of topological insulators Topological insulating behavior is likely to arise in have been established, the field is at an early stage in its other classes of materials in addition to the binary com- development. There is much work to be done to realize pounds Bi1−xSbx,Bi2Te3,Bi2Se3, and Sb2Te3. If one ex- the potential of these new and fascinating materials. In pands ones horizon to ternary compounds ͑or beyond͒ this concluding section we discuss some recent develop- the possibilities for exotic materials multiply. Candidate ments and look toward the future. materials will be narrow gap semiconductors which in- In the history of condensed-matter physics, the single clude heavy elements. One intriguing class of materials most important ingredient in the emergence of a new is transition-metal oxides involving iridium. Shitade et al. ͑ ͒ field is the perfection of the techniques for producing 2009 predicted that Na2IrO3 is a weak topological in- ͑ ͒ high quality materials. For example, the intricate physics sulator. Pesin and Balents 2010 suggested that certain of the fractional quantum Hall effect would never have iridium based pyrochlore compounds may be strong to- emerged without ultrahigh mobility GaAs. Topological pological insulators. Since these materials involve d elec- insulator materials need to be perfected so that they ac- trons, a crucial issue will be to understand the interplay tually insulate. There has been substantial progress in between strong electron-electron interactions and the this direction. For instance, transport experiments on spin-orbit interaction. Recent theoretical work predicts topological insulator behavior in ternary Heusler com- Ca-doped crystals of Bi Se show clear insulating behav- 2 3 pounds ͑Chadov et al., 2010; Lin, Wray, Xia, Jia, et al., ior below around 100 K ͑Checkelsky et al., 2009͒. How- 2010͒ and other materials ͑Lin, Markiewicz, et al., 2010; ever, the electrical resistance saturates at low tempera- Lin, Wray, Xia, Xu, et al., 2010; Yan et al., 2010͒. These ture and the surface currents appear to be overwhelmed are exciting new directions where further theoretical and by either bulk currents or currents in a layer near the experimental work is called for. surface. This is a challenging problem because narrow Topological superconductors present another exciting gap semiconductors are very sensitive to doping. None- frontier direction. In addition to observing the surface theless, it seems clear that there is ample room for im- Majorana modes predicted for 3HeB ͑Chung and Zhang, provement. Thin films produced, for instance, by me- 2009͒, it will be interesting to predict and observe elec- chanical exfoliation ͑as in graphene͒, or catalytically ͑ ͒ tronic topological superconductors, to characterize their generated Bi2Se3 ribbons and wires Peng et al., 2010 surface modes, and to explore their potential utility. To may be helpful in this regard. A particularly promising this end there has been recent progress in developing new direction is the growth of epitaxial films of Bi2Se3 methods to theoretically identify topological supercon- ͑ ͒ ͑ ͒ Zhang et al., 2009, 2010 and Bi2Te3 Li et al., 2009 . ductors based on their band structure ͑Fu and Berg, This has led to the recent observation of Landau quan- 2010; Qi, Hughes, and Zhang, 2010͒. ͑ tization in the Dirac surface states Cheng et al., 2010; More generally there may be other interesting corre- ͒ Hanaguri et al., 2010 . It will be interesting to observe lated states related to topological insulators and super- such quantization in a transport study. Present materials conductors. For example, Levin and Stern ͑2009͒ pose a challenge due to competition between surface showed that a T invariant fractional quantum spin Hall and bulk states ͑Checkelsky et al., 2009; Taskin and state can be topologically stable. This points to a general Ando, 2009͒. The detailed study of the electronic and theoretical problem associated with the topological clas- spin transport properties of the surface states is called sification of interacting systems: How do we unify the for, complemented by a host of other probes, ranging seemingly different notions of topological order epito- from optics to tunneling spectroscopy. mized by Thouless et al. ͑1982͒ and by Wen ͑1995͒? The Another direction for future innovation will be the topological field theories studied by Qi, Hughes, and study of heterostructures involving topological insulators Zhang ͑2008͒, as well as connections with string theory and other materials. In addition to providing a means ͑Ryu and Takayanagi, 2010͒, provide steps in this direc- for protecting and controlling the population of the sur- tion, but a complete theory remains to be developed. face states, such structures could provide a step toward To conclude, recent advances in the physics of topo- the longer term goal of engineering exotic states, such as logical insulators have been driven by a rich interplay Majorana fermions, with the surface states. There are between theoretical insight and experimental discover- many material problems to be solved in order to find ies. There is reason for optimism that this field will con- appropriate magnetic and superconducting materials tinue to develop in exciting new directions. which exhibit the appropriate proximity effects with the surface states, and detailed experiments will be neces- ACKNOWLEDGMENTS sary to characterize those states. A recent development along these lines is the discovery of superconductivity in C.L.K. thanks L. Fu, E. J. Mele, and J. C. Y. Teo for ͑ ͒ Cu-doped Bi2Se3 Hor, Williams, et al., 2010 . One can collaboration and NSF Grant No. DMR 0906175 for imagine devices fabricated with techniques of modula- support. M.Z.H. thanks L. A. Wray for assistance pre- tion doping that have proved extremely powerful in paring figures, A. Bansil, R. J. Cava, J. G. Checkelsky, J. semiconductor physics. One can also imagine other de- H. Dil, A. V. Fedorov, Y. S. Hor, D. Hsieh, H. Lin, F. vices that integrate magnetic materials with topological Meier, N. P. Ong, J. Osterwalder, L. Patthey, D. Qian, P.

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