Topological insulators

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Shoucheng Zhang (2015), Scholarpedia, 10(7):30275. doi:10.4249/scholarpedia.30275 revision #151969 [link to/cite this article]
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Metals and insulators

The electronic band structure determines the conductivity of metals and insulators. The band theory of solid describes the electronic structure of such states, which exploits the 'discrete' translational symmetry of the crystal to classify electronic states in terms of their crystal momentum $\mathbf{k}$, defined in a periodic Brillouin zone. As stated by the Bloch theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves \begin{equation} H|\psi_{n}(\mathbf{k})\rangle=E_{n}(\mathbf{k})|\psi_{n}(\mathbf{k})\rangle \end{equation} with \begin{equation} |\psi_{n}(\mathbf{k})\rangle = e^{i\mathbf{k}\cdot\mathbf{r}}|u_n(\mathbf{k})\rangle, \end{equation} where $|u_n({\mathbf{k})}\rangle$ is defined in a single unit cell of the crystal, $\mathbf{r}$ is the position operator. They are eigenstates of the Bloch Hamiltonian $H(\mathbf{k})$ with $H(\mathbf{k})|u_n({\mathbf{k})}\rangle=E_n(\mathbf{k})|u_n({\mathbf{k})}\rangle$, with $H=\sum_{\mathbf{k}}\psi^{†}(\mathbf{k})H(\mathbf{k})\psi(\mathbf{k})$, where $\psi(\mathbf{k})$ is a basis. In a crystal with two sublattices, $\psi(\mathbf{k})$ would be a two component vector of annihilation operator $(c_{A\mathbf{k}}, c_{B\mathbf{k}})^T$. The eigenvalues $E_n(\mathbf{k})$ define energy bands that collectively form the band structure, where $n$ is the band index. Fig. 1 illustrates the simplest case of two energy bands (valence and conduction band) and the band gap. Energy bands and the gaps between them determine the conductivity and other properties of solids.

  • Insulators have a fully occupied valence band and unoccupied conduction band with an energy gap (sometimes a few eV). There are no gapless electronic states available for electrical conduction.
  • Metals have a partially filled band. Gapless electronic states in the partially filled band are responsible for electrical conduction, even at $T=0$ K.s electronic states in the partially filled band are responsible for electrical conduction, even at $T=0$ K.
Figure 1: The energy bands of metals and insulators. For the insulators, the filled valence band is separated from the conduction band by an energy gap. On the other hand, the lower energy band in metals is partially filled with electrons.

Conventional surface states of insulators

Surface states are electronic states found at the surface of materials. The termination of a crystal obviously causes deviation from perfect periodicity, in this case, $\mathbf{k}$ becomes complex, leading to the formation of a surface state. The wave function of a surface state decays monotonically in the direction of vacuum and damps in an oscillatory fashion in the direction of the crystal, and it is localized close to the crystal surface.

There are two kinds of surface states, conventionally called Shockley states and Tamm states. However, there is no real physical distinction between the two terms, only the mathematical approach in describing surface states is different. Shockley states are states that arise due to the change in the electron potential associated solely with the crystal termination, this approach is suited to describe normal metals and some narrow gap semiconductors. While surface states that are calculated in the framework of a tight-binding model are often called Tamm states, they are suitable to describe also transition metals and insulators. These surface states are called topologically trivial in the modern sense, since they originate from one bulk band and return to the same bulk band, say the conduction band, as illustrated in Figure 2a. Potential disorder and oxidation can easily remove these trivial surface states. There also exist surface states on metal surfaces, such as Cu (100) and Ag (110) surface.

Figure 2: Schematic representation of the surface energy levels of a crystal in either 2D or 3D, as a function of surface crystal momentum. a, conventional insulator; b, topological insulator. The shaded region shows the bulk continuum states, and the lines show discrete surface (or edge) bands localized near one of the surfaces. b occurs in topological insulators and guarantees that the surface bands cross any Fermi level inside the bulk gap.

Topological insulators

A topological insulator is a material with time reversal symmetry and topologically protected surface states. These surface states continuously connect bulk conduction and valence bands, as illustrated in Figure 2b. Topological insulator has an energy gap in the bulk interior, just as in an ordinary insulator, but it contains conducting states localized on its surface. Although ordinary band insulators can also support conductive surface states, the surface states of topological insulators are special since they are topologically protected by the time reversal symmetry, and no non-magnetic disorder can destroy them.

In the bulk of a non-interacting topological insulator, the electronic band structure resembles an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow metallic conduction on the surface. An odd number of surface states connects conduction band with valence bands, and crossings at time-reversal-invariant points in Brillouin zone (as shown in Figure 2b), which leads to topologically protected metallic boundary states (Zhang et al.(2011),Hasan et al.(2010)).

Time-reversal symmetry is represented by an antiunitary operator $\mathcal{T}=\exp(i\pi S_y/\hbar)K$, where $S_y$ is the spin operator and $K$ is complex conjugation. For spin $1/2$ electrons, $\mathcal{T}$ has the property $\mathcal{T}^2=-1$. This leads to an important constraint, known as Kramers' theorem, which states that all eigenenergies of a time-reversal invariant Hamiltonian are at least twofold degenerate. This follows because if a nondegenerate state $|u\rangle$ existed then $\mathcal{T}|u\rangle=c|u\rangle$ for some constant $c$. This would mean $\mathcal{T}^2|u\rangle=|c|^2|u\rangle$, which is not allowed because $|c^2|\neq-1$. If we apply the Kramers' theorem to the Bloch wave in an insulator, we find that for any Bloch state $\psi_{n}(\mathbf{k},\sigma)$, there is another state $\mathcal{T}\psi_{n}(\mathbf{k},\sigma)$ which has the same energy, \begin{equation} \mathcal{T}\psi_{n}(\mathbf{k},\sigma)=\psi'_n(-\mathbf{k},-\sigma). \end{equation} where $\sigma$ is the spin index. In general, the Kramers doublet are located at different momentum point $\mathbf{k}$ and $-\mathbf{k}$. However, if $\mathbf{k}=-\mathbf{k}+\mathbf{G}$, where $\mathbf{G}$ is reciprocal lattice vector, i.e. at the time-reversal-invariant points, every single energy level is double degenerate. This guarantees the surface states must cross at time-reversal-invariant points in Brillouin zone.

Two-dimensional topological insulator

In 2005, Kane and Mele identified a new type of topological property characterizing two dimensional insulators, different from the property of the integer quantum Hall effect (Kane et al.(2005)). Two groups independently introduced the models for two-dimensional topological insulators (also synonymously called the quantum spin Hall insulator). Adapting an earlier model for graphene by F. Duncan M. Haldane (Haldane(1988)) which exhibits an integer quantum Hall effect, Kane and Mele proposed a quantum spin Hall model in graphene (Kane et al.(2005)), where the spin up electron exhibits a chiral integer quantum Hall effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect. Independently, a quantum spin Hall model was proposed by Bernevig and Zhang (Bernevig et al.(2005)) in strained GaAs, with an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient of both proposals is the existence of spin-orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron. This theoretical prediction was soon confirmed experimentally, leading to the first discovery of the topological insulator in nature.

Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state under realistic conductions. In 2006, Bernevig, Hughes and Zhang (Bernevig et al.(2006)) predicted that mercury telluride quantum wells are topological insulators beyond a critical thickness $d_c=6.5$~nm. The general mechanism for topological insulators is band inversion, in which the usual ordering of the conduction band and valence band is inverted by spin-orbit coupling.

In most common semiconductors, the conduction band is formed from electrons in $s$ orbitals and the valence band is formed from electrons in $p$ orbitals. In HgTe, however, the relativistic effect including spin-orbit coupling is so large that the bands are inverted-that is, the $p$-orbital dominated valence band is pushed above the $s$-orbital dominated conduction band. HgTe quantum wells are grown by sandwiching the material between CdTe, which has a similar lattice constant but much weaker spin-orbit coupling with normal band ordering. Therefore, increasing the thickness $d$ of the HgTe layer increases the strength of the spin-orbit coupling for the entire quantum well. For a thin quantum well, as shown in the left column of Figure 3a, the CdTe has the dominant effect and the bands have a normal ordering: The $s$-like conduction subband E1 is located above the $p$-like valence subband H1. In a thick quantum well, as shown in the right column, the opposite ordering occurs due to increased thickness $d$ of the HgTe layer.

Figure 3: HgTe quantum wells are two-dimensional topological insulators. (a) The behavior of a HgTe/CdTe quantum well depends on the thickness $d$ of the HgTe layer. Here the blue curve shows the potential-energy well experienced by electrons in the conduction band; the red curve is the barrier for holes in the valence band. Electrons and holes are trapped laterally by those potentials but are free in the other two dimensions. For quantum wells thinner than a critical thickness $d_c\simeq 6.5 {\rm nm}$, the energy of the lowest energy conduction subband, labeled E1, is higher than that of the highest-energy valence band, labeled H1. But for $d>d_c$, those electron and hole bands are inverted. (b) The energy spectra of the quantum wells. The thin quantum well has an insulating energy gap, but inside the gap in the thick quantum well are edge states, shown by red and blue lines. (c) Experimentally measured resistance of thin and thick quantum wells, plotted against the voltage applied to a gate electrode to change the chemical potential. The thin quantum well has a nearly infinite resistance within the gap, whereas the thick quantum well has a quantized resistance plateau at $R = h/2e^2$, due to the perfectly conducting edge states. Moreover, the resistance plateau is the same for samples with different widths, from 0.5 $\mu$m (red) to 1.0 $\mu$m (blue), proof that only the edges are conducting.

The QSH state in HgTe can be described by a simple model for the E1 and H1 subbands (Bernevig et al.(2006)) (in appendix). Explicit solution of that model gives one pair of edge states for $d>d_c$ in the inverted regime and no edge states in the $d<d_c$, as shown in Figure 3b. The pair of edge states (called helical states) carry opposite spins and disperse all the way from valence band to conduction band, and backscattering by nonmagnetic impurities is forbidden. A simple semiclassical picture illustrates why single-particle backscattering is forbidden for degrees of freedom with half-odd-integer angular momentum (Figure 4). The mechanism is analogous to the way anti-reflective coatings on eyeglasses and camera lenses work. In such a system, reflected light from the top and bottom surfaces of the antireflective coating interfere destructively, suppressing the overall amount of reflected light (Figure 4a). This effect is, however, not robust, as it requires precise matching of the coating thickness to the wavelength of the light. Just as photons can be reflected from an interface between two dielectrics, so can electrons be backscattered by an impurity, and different backscattering paths will interfere with each other (Figure 4b). On a quantum spin Hall edge, the two paths correspond to the electron going around the impurity in either a clockwise or counterclockwise fashion, with the spin rotating by an angle of $\pi$ or $-\pi$, respectively. Consequently, the phase difference between the two paths is a full $\pi-(-\pi)=2\pi$ rotation of the electron spin. However, the wave function of a spin-$1/2$ particle picks up a minus sign under a full $2\pi$ rotation. Therefore, the two backscattering paths related by time-reversal always interfere destructively, leading to perfect transmission.

Figure 4: (a) On a lens with antireflection coating, light waves reflected by the top (blue line) and the bottom (red line) surfaces interfere destructively, which leads to suppressed reflection. (b) A quantum spin Hall edge state can be scattered in two directions by a nonmagnetic impurity. Going clockwise along the blue curve, the spin rotates by $\pi$; counterclockwise along the red curve, by $-\pi$. A quantum mechanical phase factor of $-1$ associated with that difference of $2\pi$ leads to destructive interference of the two paths---the backscattering of electrons is suppressed in a way similar to that of photons off the antireflection coating.

As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is $G_{xx}=2e^2/h$ in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to the width of the sample. In 2007, a team at the University of Würzburg led by Laurens Molenkamp observed the quantum spin Hall effect in HgTe quantum wells grown by molecular-beam epitaxy (Köonig et al.(2007)). The transport experiment shows finite conductance of $2e^2/h$ for quantum spin Hall insulator indicating the contribution from helical edge states. In contrast, a trivial insulator phase is indeed insulating, with vanishing conductance [Figure 3c].

In 2008, Zhang's group predicted that the type II quantum well GaSb/InAs is a two-dimensional topological insulator (Liu et al.(2008)). The conduction and valence band are spatially separated in two materials with inverted order and weakly coupled in this broken gap quantum well. Soon after the theoretical prediction, Rui-Rui Du's group in Rice University observed the quantum spin Hall effect in GaSb/InAs quantum well (Knez et al.(2011)).

Three dimensional topological insulator

The two-dimensional topological insulator with a one-dimensional helical edge state can be simply generalized to a three-dimensional topological insulator (Fu et al. (2007),Moore et al. (2007),Roy (2009)), for which the surface state consists of a single two-dimensional massless Dirac fermion and the dispersion forms a so-called Dirac cone, as illustrated in Figure 5.

Three dimensional topological insulator was first realized in the alloy ${\rm Bi_{1-x}Sb_x}$ with a special range of $x$ (Liang et al.(2007),Hsieh et al. (2008)). However, the surface states and the underlying mechanism turn out to be extremely complex. Soon afterwards, bulk crystals of ${\rm Bi_2Te_3}$, ${\rm Bi_2Se_3}$, and ${\rm Sb_2Te_3}$ are predicted to be three dimensional topological insulators with bulk energy gap as large as $0.3$~eV, with the topological surface states consisting of a single Dirac cone (Zhang et al. (2009),Xia et al.(2009)). The single Dirac-cone surface state of ${\rm Bi_2Se_3}$ has been observed in the Angle-Resolved PhotoEmission Spectroscopy (ARPES) experiments by a Princeton based group (Xia et al.(2009)). Furthermore, the group's spin-resolved ARPES measurements showed that the electron spin lies in the plane of the surface and is always perpendicular to the momentum. A pure topological insulator phase without bulk carriers was first observed in ${\rm Bi_2Te_3}$ by a Stanford based group in ARPES experiments (Chen et al.(2009)). As shown in Figure 5c, the observed surface states indeed disperse linearly, crossing at the point with zero momentum. By mapping all of momentum space, the ARPES experiments show convincingly that the surface states of ${\rm Bi_2Te_3}$ and ${\rm Bi_2Se_3}$ consist of a single Dirac cone.

Similar to HgTe, the nontrivial topology of the ${\rm Bi_2Te_3}$ family is due to band inversion between two orbitals with opposite parity, driven by the strong spin-orbit coupling of Bi and Te. Due to such similarity, that family of 3D topological insulators can be described by a 3D version of the HgTe model (see the appendix). First-principle calculations show that the materials have a single Dirac cone on the surface. The spin of the surface state lies in the surface plane and is always perpendicular to the momentum, as shown in Figure 2b.

Further generalization of topological insulators can be made to higher dimensions with different symmetries, and the complete classifications are first given by Schnyder et al. (2008) and Kitaev (2009).

Figure 5: (a) The crystal structure of the 3D topological insulator ${\rm Bi_2Te_3}$ consists of stacked quasi-2D layers of Te-Bi-Te-Bi-Te. The arrows indicate the lattice basis vectors. The surface state is predicted to consist of a single Dirac cone (Zhang et al.(2009)). (b) This ARPES plot of energy versus wavenumber in ${\rm Bi_2Te_3}$ shows the linearly dispersing surface-state band (SSB) above the bulk valence band (BVB). The top two dashed green lines denote the bulk insulating gap; the bottom line marks the point of the Dirac cone (Chen et al.(2009)). (c) Angle-resolved photoemission spectroscopy maps the energy states in momentum space. Spin dependent ARPES of the related compound ${\rm Bi_2Se_3}$ reveals that the spins (red) of the surface states lie in the surface plane and are perpendicular to the momentum (Xia et al.(2009))


Topology studies the shapes of manifolds or spaces. The topology of a space is preserved under continuous deformations (e.g. stretching and bending, but not tearing or gluing), and is usually characterized by topological numbers. For example, closed oriented two-dimensional surfaces (spaces) are characterized by genus $g$, the number of holes of the closed surface. A sphere has $g=0$, and a torus has $g=1$ as shown in Figure 6. The genus can be calculated by integration of the two-dimensional curvature $R$ over the space, given by: \begin{equation} 2-2g=\frac{1}{2\pi}\int\limits_{M}RdS \end{equation}

Figure 6: Sphere, $g=0$; and torus, $g=1$

Instead of the Gaussian curvature defined above, one can define the Berry curvature in the Brillouin zone, which can be viewed as a torus for a two dimensional insulator. The Berry curvature $F$ can be defined as \begin{equation} F = (\nabla_{\mathbf{k}}\times\mathbf{A})_z, \end{equation} where $\mathbf{A}$ is the Berry connection of Bloch states defined as \begin{equation} \mathbf{A} = i\langle u(\mathbf{k})|\nabla_{\mathbf{k}}|u(\mathbf{k})\rangle. \end{equation}

This enables us to define a (topological) Chern number $N$ for the insulator \begin{equation} N=\frac{1}{2\pi}\int F dk_xk_y=\frac{1}{2\pi}\oint_C\mathbf{A}\cdot d\mathbf{k}, \end{equation} where the integration is over the Brillouin Zone. In the absence of magnetic fields, an insulator with $N\neq0$ is called a quantum anomalous Hall insulator. As a consequence, a quantum anomalous Hall insulator has $|N|$ gapless chiral edge states on the edge. A quantum anomalous Hall insulator breaks the time reversal symmetry. To make the physical picture of Chern number clearer, the simplest case of a two-band model can be studied as an example. The Hamiltonian of a two-band model can be generally written as $h(\mathbf{k})=\sum\limits_{a=1}^3d_a(\mathbf{k})\sigma^a+\epsilon(\mathbf{k})1$. The energy spectrum is easily obtained: $E_{\pm}(\mathbf{k})=\epsilon(\mathbf{k})\pm\sqrt{\sum_ad_a^2(\mathbf{k})}$. When $\sum_ad_a^2(\mathbf{k})>0$ for all $\mathbf{k}$ in the Brillouin zone, the two bands never touch each other. In the single particle Hamiltonian $h(\mathbf{k})$, the vector $d(\mathbf{k})$ acts as a Zeeman field applied to a pseudospin $\sigma_i$ of a two level system. The occupied band satisfies $[\mathbf{d}(\mathbf{k})\cdot\boldsymbol{\sigma}]|-,\mathbf{k}\rangle=-|\mathbf{d}(\mathbf{k})||-,\mathbf{k}\rangle$ which thus corresponds to the spinor with spin polarization in the $-\mathbf{d}(\mathbf{k})$ direction. Thus Berry's phase gained by $|-,\mathbf{k}\rangle$ during an adiabatic evolution along some path $C$ in $\mathbf{k}$ space is equal to Berry's phase a spin-1/2 particle gains during the adiabatic rotation of the magnetic field along the path $\mathbf{d}(C)$. This is known to be half of the solid angle subtended by $\mathbf{d}(C)$. Consequently, the first Chern number $N$ is determined by the winding number of $\mathbf{d}(\mathbf{k})$ around the origin.

For time reversal invariant two-dimensional insulators, the Chern number $N$ is always 0. However, with the time reversal invariant points in the Brillouin Zone identified, a $Z_2$ topological number $(-1)^{N_2}$ can be defined Kane et al. (2005). An insulator is trivial if $N_2=0$, and is called a quantum spin Hall if $N_2=1$. Consequentially, a trivial insulator has gapped edge, while a quantum spin Hall has a pair of gapless helical edge states carrying opposite spins.

Topological field theory

To describe the novel topological properties of a single Dirac cone on the surface of a topological insulator. One can use the topological field theory in terms of elementary concepts in electromagnetism (Qi et al.(2008)).

Inside an insulator, the electric field ${\bf E}$ and the magnetic field ${\bf B}$ are both well defined. In a Lagrangian-based field theory, the insulator's electromagnetic response can be described by the effective action $S_0=1/8\pi\int d^3xdt(\epsilon E^2-B^2/\mu)$, with $\epsilon$ the electric permittivity and $\mu$ the magnetic permeability, from which Maxwell's equations can be derived. The integrand depends on geometry, though, so it is not topological. To see that dependence, one can write the action in terms of the field strength tensor $F_{\mu\nu}$, the relativistic electromagnetic field tensor: $S_0=1/16\pi \int d^3xdtF_{\mu\nu}F^{\mu\nu}$. The implied summation over the repeated indices $\mu$ and $\nu$ depends on the metric tensor---that is, on geometry. (Indeed, it is that dependence that leads to the gravitational lensing of light.) There is, however, another possible term called axion term Wilczek (1987) in the action of the electromagnetic field: \begin{eqnarray} S_\theta&=&\frac{\theta\alpha}{4\pi^2}\int d^3xdt{\bf E\cdot B}\equiv\frac{\theta\alpha}{32\pi^2} \int d^3xdt \epsilon_{\mu\nu\rho\tau}F^{\mu\nu}F^{\rho\tau}\nonumber\\ &=&\frac{\theta}{2\pi}\frac{\alpha}{4\pi}\int d^3xdt \partial^\mu (\epsilon_{\mu\nu\rho\tau} A^\nu\partial^\rho A^\tau) \tag{1} \end{eqnarray} where $\alpha=e^2/\hbar c\simeq 1/137$ is the fine-structure constant, $\theta$ is an angular variable which is defined modulo $2\pi$, and $\epsilon_{\mu\nu\rho\tau}$ is the fully antisymmetric 4D Levi-Civita tensor. Unlike the Maxwell action, $S_\theta$ is a topological term---it depends only on the topology of the underlying space, not on the geometry.

The axion term in general breaks time-reversal symmetry and inversion symmetry. However, for an insulator with time reversal symmetry, the parameter $\theta$ can only take values of $0$ and $\pi$. $\theta=0$ corresponds to a conventional insulator, and $\theta=\pi$ corresponds to a topological insulator. The definition is generally valid for interacting topological insulators as well.

Solving Maxwell's equations with the topological term included leads to predictions of novel physical properties. It should be mentioned that such topological term is an integral of a total derivative, so it only has an effect on the boundary where $\theta$ jumps from $\theta=0$ to $\theta=\pi$. Otherwise, it does not enter into the Maxwell equations. A point charge above the surface of a 3D topological insulator is predicted to induce not only an image electric charge but also an image magnetic monopole below the surface (Qi et al.(2009)). If we uniformly cover the surface with a thin ferromagnetic film, an insulating gap also opens up at the Dirac point giving rise to $\frac{1}{2}e^2/h$ Hall quantum Hall conductance (Qi et al.(2008)).

Possible applications

The topological protection of the surface state or edge state of topological insulators made them promising candidate for potential applications. One possible application is to use the edge channels of the quantum anomalous Hall insulator as interconnects for integrated circuits Zhang et al. (2012). For these edge channels propagate along the insulator edges without any dissipation, they could greatly reduce heat dissipation in integrated circuits. Especially, the quantum anomalous Hall insulator with multi-channels lowers the contact resistance, significantly improving the performance of the interconnect devices (Wang et al. (2013)).

Topological insulators are interesting for thermoelectrics due to their unique electronic structure. In fact, many topological insulators were known as excellent thermoelectric materials, for topological insulator and thermoelectric compounds usually favor the same material features, such as heavy elements and the smaller energy gap. Bi$_2$Te$_3$ and Bi$_2$Se$_3$ comprise some of the best performing room temperature thermoelectrics with a temperature-independent thermoelectric effect, $ZT$, between $0.8$ and $1.0$, where the carrier density is between a semiconductor and a metal. Zhang's group predicted in 2014 that $ZT$ is strongly size dependent in topological insulators ( Xu et al. (2014)), and the size parameter can be tuned to enhance $ZT$ to be significantly greater than 1.0, making topological insulator a promising material for thermoelectric science and technology.


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Models of topological insulators

The essence of the quantum spin Hall effect in real materials can be captured in explicit models that are particularly simple to solve. The two-dimensional topological insulator HgTe can be described by an effective Hamiltonian that is essentially a Taylor expansion in the wave vector ${\bf k}$ of the interactions between the lowest conduction band and the highest valence band (Bernevig et al.(2006)): \begin{eqnarray}\tag{2} H(k)&=&\epsilon({\bf k})\mathbb{I}+\left(\begin{array}{cccc}M({\bf k})&A(k_x+ik_y)&0&0\\A(k_x-ik_y)&-M({\bf k})&0&0\\ 0&0&M({\bf k})&-A(k_x-ik_y)\\0&0&-A(k_x+ik_y)&-M({\bf k})\end{array}\right)\nonumber\\ \epsilon({\bf k})&=&C+D{\bf k}^2,~M({\bf k})=M-B{\bf k}^2 \end{eqnarray}

where the upper $2\times 2$ block describes spin-up electrons in the $s$-like E1 conduction and the $p$-like H1 valence bands, and the lower block describes the spin-down electrons in those bands. The term $\epsilon({\bf k})\mathbb{I}$ is an unimportant bending of all the bands ($\mathbb{I}$ is the identity matrix). The energy gap between the bands is $2M$, and $B$, typically negative, describes the curvature of the bands; $A$ incorporates interband coupling to lowest order. For $M/B < 0$, the eigenstates of the model describe a trivial insulator. But for thick quantum wells, the bands are inverted, $M$ becomes negative, and the solution yields the edge states of a quantum spin Hall insulator.

The three-dimensional topological insulator in the ${\rm Bi_2Te_3}$ family can be described by a similar model (Zhang et al. (2009)): \begin{eqnarray}\tag{3} H(k)&=&\epsilon({\bf k})\mathbb{I}+\left(\begin{array}{cccc}M({\bf k})&A_2(k_x+ik_y)&0&A_1 k_z\\A_2(k_x-ik_y)&-M({\bf k})&A_1 k_z&0\\ 0&A_1 k_z&M({\bf k})&-A_2(k_x-ik_y)\\A_1 k_z&0&-A_2(k_x+ik_y)&-M({\bf k})\end{array}\right)\nonumber\\ \epsilon({\bf k})&=&C+D_1k_z^2+D_2k_\perp^2,~M({\bf k})=M-B_1k_z^2-B_2k_\perp^2 \end{eqnarray} in the basis of the Bi and Te bonding and antibonding $p_z$ orbitals with both spins. The curvature parameters $B_1$ and $B_2$ have the same sign. As in the two-dimensional model, the solution for $M/B_1 < 0$ describes a trivial insulator, but for $M/B_1 > 0$, the bands are inverted and the system is a topological insulator.

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