Topological insulator

A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material.
A topological insulator is an insulator for the same reason a "trivial" insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. The topological insulator cannot be continuously transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state. Thus, due to the continuity of the underlying field, the border of a topological insulator with a trivial insulator is forced to support conducting edge states.
Since this results from a global property of the topological insulator's band structure, local perturbations cannot damage this surface state. This is unique to topological insulators: while ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property.
This leads to a more formal definition of a topological insulator: an insulator which cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state. In other words, topological insulators and trivial insulators are separate regions in the phase diagram, connected only by conducting phases. In this way, topological insulators provide an example of a state of matter not described by the Landau symmetry-breaking theory that defines ordinary states of matter.
The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries, and can be classified using the so-called periodic table of topological insulators. Some combinations of dimension and symmetries forbid topological insulators completely. All topological insulators have at least U symmetry from particle number conservation, and often have time-reversal symmetry from the absence of a magnetic field. In this way, topological insulators are an example of symmetry-protected topological order. So-called "topological invariants", taking values in or, allow classification of insulators as trivial or topological, and can be computed by various methods.
The surface states of topological insulators can have exotic properties. For example, in time-reversal symmetric 3D topological insulators, surface states have their spin locked at a right-angle to their momentum. At a given energy the only other available electronic states have different spin, so "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic.
Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media. There exist photonic, magnetic, and acoustic topological insulators, among others.

Prediction

The first models of 3D topological insulators were proposed by B. A. Volkov and O. A. Pankratov in 1985,
and subsequently by Pankratov, S. V. Pakhomov, and Volkov in 1987. Gapless 2D Dirac states were shown to exist at the band inversion contact in PbTe/SnTe and HgTe/CdTe heterostructures.
Existence of interface Dirac states in HgTe/CdTe was experimentally verified by Laurens W. Molenkamp's group in 2D topological insulators in 2007.
Later sets of theoretical models for the 2D topological insulator were proposed by Charles L. Kane and Eugene J. Mele in 2005, and also by B. Andrei Bernevig and Shoucheng Zhang in 2006. The topological invariant was constructed and the importance of the time reversal symmetry was clarified in the work by Kane and Mele. Subsequently, Bernevig, Taylor L. Hughes and Zhang made a theoretical prediction that 2D topological insulator with one-dimensional helical edge states would be realized in quantum wells of mercury telluride sandwiched between cadmium telluride. The transport due to 1D helical edge states was indeed observed in the experiments by Molenkamp's group in 2007.
Although the topological classification and the importance of time-reversal symmetry was pointed in the 2000s, all the necessary ingredients and physics of topological insulators were already understood in the works from the 1980s.
In 2007, it was predicted that 3D topological insulators might be found in binary compounds involving bismuth, and in particular "strong topological insulators" exist that cannot be reduced to multiple copies of the quantum spin Hall state.

Properties and applications

Spin-momentum locking in the topological insulator allows symmetry-protected surface states to host Majorana particles if superconductivity is induced on the surface of 3D topological insulators via proximity effects. The non-trivialness of topological insulators is encoded in the existence of a gas of helical Dirac fermions. Dirac particles which behave like massless relativistic fermions have been observed in 3D topological insulators. Note that the gapless surface states of topological insulators differ from those in the quantum Hall effect: the gapless surface states of topological insulators are symmetry-protected, while the gapless surface states in quantum Hall effect are topological. The topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the invariants. An experimental method to measure topological invariants was demonstrated which provide a measure of the topological order. More generally for each spatial dimensionality, each of the ten Altland—Zirnbauer symmetry classes of random Hamiltonians labelled by the type of discrete symmetry has a corresponding group of topological invariants as described by the periodic table of topological invariants.
The most promising applications of topological insulators are spintronic devices and dissipationless transistors for quantum computers based on the quantum Hall effect and quantum anomalous Hall effect. In addition, topological insulator materials have also found practical applications in advanced magnetoelectronic and optoelectronic devices.

Thermoelectrics

Some of the most well-known topological insulators are also thermoelectric materials, such as Bi₂Te₃ and its alloys with Bi₂Se₃ and Sb₂Te₃. High thermoelectric power conversion efficiency is realized in materials with low thermal conductivity, high electrical conductivity, and high Seebeck coefficient. Topological insulators are often composed of heavy atoms, which tends to lower thermal conductivity and are therefore beneficial for thermoelectrics. A recent study also showed that good electrical characteristics can arise in topological insulators due to warping of the bulk band structure, which is driven by band inversion. Often, the electrical conductivity and Seebeck coefficient are conflicting properties of thermoelectrics and difficult to optimize simultaneously. Band warping, induced by band inversion in a topological insulator, can mediate the two properties by reducing the effective mass of electrons/holes and increasing the valley degeneracy. As a result, topological insulators are generally interesting candidates for thermoelectric applications.

Theoretical background

Topology of the Brillouin zone

A periodic system is, by definition, invariant under some translations, allowing us to label the eigenstates of a periodic Hamiltonian by the eigenvalues of the translation operators that leave the system unchanged. The eigenvalues of the translation operator are called crystal momenta and are usually written as . Those quantum numbers play a role analogous to the wavevector of a free particle. The periodicity of the crystal induces a periodicity in k-space: and, where is a reciprocal wavevector, describe the same state. The set of all quasimomenta that are not equivalent up to a reciprocal wavevector is called the Brillouin zone. As a consequence of the periodicity of the Brillouin zone, its opposite edges or faces should be identified with each other, therefore the Brillouin zone describes a d-dimensional torus,.

Two-band Hamiltonians

The topological insulators that are easiest to describe are periodic crystalline insulators with two degrees of freedom per unit cell. Although a lot of the following treatment is only true for those systems, it is easily generalizable to any crystalline topological insulator. Some notable examples of two-band systems are graphene and the Su-Shrieffer-Heeger model. By separating the degrees of freedom from inside and outside the cell, one can write the wavefunction aswhere labels the unit cell, are lattice vectors, is the number of unit cells and is the internal state of a unit cell, which in our case can be generically written . Due to Bloch's theorem, this basis block-diagonalizes the Hamiltonian. One such block, corresponding to the quasimomentum, can be represented by a hermitian matrix for the two-band case. Any hermitian matrix can be written as a linear combination of Pauli matrices, and and the identity matrix. Hence any two-band periodic Hamiltonian can be written aswhere summation over repeated indices is assumed, is a three dimensional real valued vector, and is the identity matrix. The advantage of this formulation is that the Hamiltonian can now be viewed as a vector in, and n-dimensional subsets of the Brillouin zone now parametrize a n-dimensional surfaces in. In this sense, we can think of as a mapping between two spaces: corresponds to k-space and in our case corresponds to the space of hermitian matrices. Moreover, this basis of the Hamiltonian allows to perform scalar products and rotations of vectors. The latter is of particular interest since is diagonal, meaning can be diagonalized by rotating to be parallel to the axis. Therefore, there exists a unitary matrix of the form such that where is the axis of rotation, and its magnitude determines the angle of rotation. The energies can be deduced from the fact that. Most importantly, the rotation only involves the direction of, thus the eigenstates of do not depend on its magnitude, and so does any quantity that only involves eigenstates.