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" (ordinary) 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 (including vacuum, which is topologically trivial) is forced to support conducting edge states. Since this results from a global property of the topological insulator's band structure, local (symmetry-preserving) 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(1) 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 Z 2 {\displaystyle \mathbb {Z} _{2}} or Z {\displaystyle \mathbb {Z} } , 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 (spin-momentum locking). 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 (also known as the quantum spin Hall insulators) were proposed by Charles L. Kane and Eugene J. Mele in 2005, and also by B. Andrei Bernevig and Shoucheng Zhang in 2006. The Z 2 {\displaystyle \mathbb {Z} _{2}} 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 (1D) helical edge states would be realized in quantum wells (very thin layers) 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 were 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. (Note that Majorana zero-mode can also appear without topological insulators.) 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 (i.e., not topological), while the gapless surface states in quantum Hall effect are topological (i.e., robust against any local perturbations that can break all the symmetries). The Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the Z 2 {\displaystyle \mathbb {Z} _{2}} invariants. An experimental method to measure Z 2 {\displaystyle \mathbb {Z} _{2}} topological invariants was demonstrated which provide a measure of the Z 2 {\displaystyle \mathbb {Z} _{2}} topological order. (Note that the term Z 2 {\displaystyle \mathbb {Z} _{2}} topological order has also been used to describe the topological order with emergent Z 2 {\displaystyle \mathbb {Z} _{2}} gauge theory discovered in 1991.) More generally (in what is known as the ten-fold way) for each spatial dimensionality, each of the ten Altland—Zirnbauer symmetry classes of random Hamiltonians labelled by the type of discrete symmetry (time-reversal symmetry, particle-hole symmetry, and chiral symmetry) has a corresponding group of topological invariants (either Z {\displaystyle \mathbb {Z} } , Z 2 {\displaystyle \mathbb {Z} _{2}} or trivial) 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 (n-type thermoelectrics) and Sb2Te3 (p-type thermoelectrics). High thermoelectric power conversion efficiency is realized in materials with low thermal conductivity, high electrical conductivity, and high Seebeck coefficient (i.e., the incremental change in voltage due to an incremental change in temperature). 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 (i.e., high electrical conductivity and Seebeck coefficient) 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 (i.e., the number of electronic bands that are contributing to charge transport). 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 (or quasimomenta) and are usually written as k = ( k 1 , k 2 , . . . , k d ) {\displaystyle \mathbf {k} =(k_{1},k_{2},...,k_{d})} (where d is the number of dimensions of the system). 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: k {\displaystyle \mathbf {k} } and k + G {\displaystyle \mathbf {k} +\mathbf {G} } , where G {\displaystyle \mathbf {G} } is a reciprocal wavevector, describe the same state (or sets of states). 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, T d {\displaystyle T^{d}} .
Two-band Hamiltonians The topological insulators that are easiest to describe are periodic crystalline insulators with two degrees of freedom (e.g. two orbitals) 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 (describing polyacetylene). By separating the degrees of freedom from inside and outside the cell, one can write the wavefunction as | ψ k ⟩ = 1 N ∑ m e i k ⋅ R m | m ⟩ ⊗ | α ⟩ ≡ | k ⟩ ⊗ | α ⟩ {\displaystyle |\psi _{\mathbf {k} }\rangle ={\frac {1}{\sqrt {N}}}\sum _{m}e^{i\mathbf {k} \cdot \mathbf {R} _{m}}|m\rangle \otimes |\alpha \rangle \equiv |\mathbf {k} \rangle \otimes |\alpha \rangle } where m {\displaystyle m} labels the unit cell, R m {\displaystyle \mathbf {R} _{m}} are lattice vectors, N {\displaystyle N} is the number of unit cells (which may or may not be infinite) and | α ⟩ {\displaystyle |\alpha \rangle } is the internal state of a unit cell, which in our case can be generically written | α ⟩ = a | A ⟩ + b | B ⟩ {\displaystyle |\alpha \rangle =a|A\rangle +b|B\rangle } (where A and B are the internal degrees of freedom). Due to Bloch's theorem, this basis block-diagonalizes the Hamiltonian. One such block, h k ≡ ⟨ k | H | k ⟩ {\displaystyle h_{\mathbf {k} }\equiv \langle \mathbf {k} |H|\mathbf {k} \rangle } corresponding to the quasimomentum k {\displaystyle \mathbf {k} } , can be represented by a 2 × 2 {\displaystyle 2\times 2} hermitian matrix for the two-band case (this matrix is often called the Bloch Hamiltonian). Any 2 × 2 {\displaystyle 2\times 2} hermitian matrix can be written as a linear combination of Pauli matrices, σ 1 , σ 2 {\displaystyle \sigma ^{1},\sigma ^{2}} and σ 3 {\displaystyle \sigma ^{3}} and the identity matrix. Hence any two-band periodic Hamiltonian can be written as h k = d i ( k ) σ i + d 0 ( k ) I {\displaystyle h_{\mathbf {k} }=d_{i}(\mathbf {k} )\sigma ^{i}+d_{0}(\mathbf {k} )I} where summation over repeated indices is assumed, d i {\displaystyle d_{i}} is a three dimensional real valued vector, and I {\displaystyle I} is the identity matrix. The advantage of this formulation is that the Hamiltonian can now be viewed as a vector in R 3 {\displaystyle \mathbb {R} ^{3}} , and n-dimensional subsets of the Brillouin zone now parametrize a n-dimensional surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} . In this sense, we can think of d = ( d 1 , d 2 , d 3 ) {\displaystyle \mathbf {d} =(d_{1},d_{2},d_{3})} as a mapping between two spaces: d : T d → R 3 {\displaystyle \mathbf {d} :T^{d}\to \mathbb {R} ^{3}}
T d {\displaystyle T^{d}} corresponds to k-space and R 3 {\displaystyle \mathbb {R} ^{3}} in our case corresponds to the space of 2 × 2 {\displaystyle 2\times 2} hermitian matrices. Moreover, this basis of the Hamiltonian allows to perform scalar products and rotations of vectors d {\displaystyle \mathbf {d} } . The latter is of particular interest since σ 3 {\displaystyle \sigma ^{3}} is diagonal, meaning h k {\displaystyle h_{\mathbf {k} }} can be diagonalized by rotating d {\displaystyle \mathbf {d} } to be parallel to the z {\displaystyle z} axis. Therefore, there exists a unitary matrix of the form R k = exp [ i θ ( k ) ⋅ σ 2 ] {\displaystyle R_{\mathbf {k} }=\exp[i\mathbf {\theta (\mathbf {k} } )\cdot {\frac {\mathbf {\sigma } }{2}}]} such that R k h k R k † = ( | d ( k ) | + d 0 0 0 − | d ( k ) | + d 0 ) {\displaystyle R_{\mathbf {k} }h_{\mathbf {k} }R_{\mathbf {k} }^{\dagger }={\begin{pmatrix}|\mathbf {d(\mathbf {k} )} |+d_{0}&0\\0&-|\mathbf {d(\mathbf {k} )} |+d_{0}\end{pmatrix}}} where θ ( k ) {\displaystyle \mathbf {\theta } (\mathbf {k} )} is the axis of rotation, and its magnitude determines the angle of rotation. The energies can be deduced from the fact that ( h k − d 0 I ) 2 = d i d i {\displaystyle (h_{\mathbf {k} }-d_{0}I)^{2}=d_{i}d^{i}} . Most importantly, the rotation R k {\displaystyle R_{\mathbf {k} }} only involves the direction of d {\displaystyle \mathbf {d} } , thus the eigenstates of h k {\displaystyle h_{\mathbf {k} }} do not depend on its magnitude, and so does any quantity that only involves eigenstates.
Connection to topology An important examples of such a quantity is the Berry connection: A α , β ( k ) ≡ i ⟨ ψ α ( k ) | ∇ k | ψ β ( k ) ⟩ {\displaystyle {\mathcal {A}}_{\alpha ,\beta }(\mathbf {k} )\equiv i\langle \psi _{\alpha }(\mathbf {k} )|\nabla _{\mathbf {k} }|\psi _{\beta }(\mathbf {k} )\rangle } where α {\displaystyle \alpha } and β {\displaystyle \beta } are band indices. Its integral over a path gives the geometric phase, which corresponds to the phase gained by a state during adiabatic evolution that is not due to the change in energy. In the context of band structure, such a geometric phase is called a Zak phase. In two dimensional materials, an integral of A {\displaystyle {\mathcal {A}}} over the Brillouin zone gives the Hall conductance and its integral can also be used to compute the polarization of a crystal. Notice that the Berry connection is not gauge invariant, but its integral over a closed path is, therefore only the integral of A {\displaystyle {\mathcal {A}}} over a closed domain can correspond to an observable quantity. A {\displaystyle {\mathcal {A}}} depends only on the direction of d ( k ) {\displaystyle \mathbf {d} (\mathbf {k} )} , d ^ ( k ) {\displaystyle {\hat {\mathbf {d} }}(\mathbf {k} )} , which can be thought of as a mapping from d-torus T d {\displaystyle T^{d}} to the 2-sphere S 2 {\displaystyle S^{2}} . d ^ : T d → S 2 {\displaystyle {\hat {\mathbf {d} }}:T^{d}\to S^{2}} Let O ∈ R {\displaystyle {\mathcal {O}}\in \mathbb {R} } be a physically observable quantity obtained by integrating the Berry phase over T d {\displaystyle T^{d}} , the Brillouin zone (for example the Hall conductance). This can be seen as the composite map O ( d ^ ( T d ) ) {\displaystyle {\mathcal {O}}({\hat {\mathbf {d} }}(T^{d}))}
O : T d → S 2 → R {\displaystyle {\mathcal {O}}:T^{d}\to S^{2}\to \mathbb {R} } Thus such an observable must depend on d ^ ( T d ) {\displaystyle {\hat {\mathbf {d} }}(T^{d})} . Let's consider d ^ ( T d ) {\displaystyle {\hat {\mathbf {d} }}(T^{d})} and d ^ ′ ( T d ) {\displaystyle {\hat {\mathbf {d} }}'(T^{d})} , two continuous mappings from T d {\displaystyle T^{d}} to S 2 {\displaystyle S^{2}} . If d ^ ( T d ) {\displaystyle {\hat {\mathbf {d} }}(T^{d})} can be smoothly deformed into d ^ ′ ( T d ) {\displaystyle {\hat {\mathbf {d} }}'(T^{d})} , they are said to be homotopically equivalent. For example a loop on a sphere can be thought as such a mapping where T d = S 1 {\displaystyle T^{d}=S^{1}} is the unit circle and all loops on a sphere are homotopically equivalent since they can all be smoothly deformed into a single point. If instead d ^ : T d → S 1 {\displaystyle {\hat {\mathbf {d} }}:T^{d}\to S^{1}} , which is the case when d {\displaystyle \mathbf {d} } lies on a plane (and thus the Bloch Hamiltonian is only composed of two Pauli matrices), this is no longer the case: not all loops on a circle can be contracted to a point. A set of homotopically equivalent mappings is called an equivalence class. Given a manifold M {\displaystyle M} , the set of equivalence classes of mappings between M {\displaystyle M} and S n {\displaystyle S^{n}} , with the operation of composition of the mappings, is called the n-th homotopy group of M {\displaystyle M} , noted π n ( M ) {\displaystyle \pi _{n}(M)} . The relevant homotopy groups for two-band systems are π 2 ( T d ) {\displaystyle \pi _{2}(T^{d})} and π 1 ( T d ) {\displaystyle \pi _{1}(T^{d})} since we can only consider mappings from closed subsets of the Brillouin zone ( T d {\displaystyle T^{d}} ) to the 2-sphere S 2 {\displaystyle S^{2}} or the circle S 1 {\displaystyle S^{1}} (which is a subset of S 2 {\displaystyle S^{2}} ). For example if we take S 1 {\displaystyle S^{1}} , the unit circle, then π 1 ( S 1 ) = Z {\displaystyle \pi _{1}(S^{1})=\mathbb {Z} } : when we map a circle to a circle, we get a loop that lives on the perimeter of the circle, such loops can wind an integer number of times around the circle and loops with a different winding number cannot be deformed into one another. Another example is π 2 ( S 1 ) = { I } {\displaystyle \pi _{2}(S^{1})=\{I\}} since every loop on a sphere can be smoothly deformed into a single point, hence all mappings from the circle to the 2-sphere are homotopically equivalent. A number that labels an element of the homotopy group is called a topological invariant. An example of such a topological invariant for the first homotopy group of a circle is the winding number mentioned before. In this case d ^ ( k ) = cos ( ϕ ( k ) ) x ^ + sin ( ϕ ( k ) ) y ^ {\displaystyle {\hat {\mathbf {d} }}(\mathbf {k} )=\cos(\phi (\mathbf {k} )){\hat {\mathbf {x} }}+\sin(\phi (\mathbf {k} )){\hat {\mathbf {y} }}} and one way to compute the winding number ν {\displaystyle \nu } is ν = 1 2 π ∫ Σ d ϕ ( k ) {\displaystyle \nu ={\frac {1}{2\pi }}\int _{\Sigma }d\phi (\mathbf {k} )}
Insulating phases as elements of the homotopy group The unit vector d ^ ( k ) {\displaystyle {\hat {\mathbf {d} }}(\mathbf {k} )} can be defined as d ( k ) | d ( k ) | {\displaystyle {\frac {\mathbf {d} (\mathbf {k} )}{|\mathbf {d} (\mathbf {k} )|}}} as long as | d ( k ) | ≠ 0 {\displaystyle |\mathbf {d} (\mathbf {k} )|\neq 0} , therefore, curves or surfaces passing through the origin cannot be continuously mapped to the 2-sphere. Two elements of the homotopy group are separated by a discontinuity and hence they must describe curves or surfaces d ( k ) {\displaystyle \mathbf {d} (\mathbf {k} )} that can only be deformed into one another by passing through the origin in ( d 1 , d 2 , d 3 ) {\displaystyle (d_{1},d_{2},d_{3})} space. Looking at the form of two-band Bloch Hamiltonian given above, one can see that if | d ( k ) | = 0 {\displaystyle |\mathbf {d} (\mathbf {k} )|=0} , the Hamiltonian becomes degenerate, and thus two bands must be touching. Therefore, according to band theory, a two-band Hamiltonian describes a conductor if d ( k ) {\displaystyle \mathbf {d} (\mathbf {k} )} passes through the origin at some point in the Brillouin zone. One can deduce from this that two elements of the homotopy group correspond to two insulating Hamiltonians separated by a conducting phase, or alternatively, those two Hamiltonians are not adiabatically connected. Pairs of Hamiltonians representing different elements of the homotopy group are said to have a different topology. Integrals of the Berry connection A {\displaystyle {\mathcal {A}}} over the Brillouin zone cannot vary smoothly between Hamiltonians of different topology since they depend on d ^ ( B Z ) {\displaystyle {\hat {\mathbf {d} }}({\mathcal {BZ}})} which cannot vary smoothly between those two Hamiltonians. It turns out that this integral is a topological invariant. For example, in the case where d {\displaystyle \mathbf {d} } lies on a plane, it can be shown, using the unitary matrix that diagonalises the Hamiltonian, R k = exp [ i θ ( k ) ⋅ σ 2 ] {\displaystyle R_{\mathbf {k} }=\exp[i\mathbf {\theta (\mathbf {k} } )\cdot {\frac {\mathbf {\sigma } }{2}}]} , that A ( k ) = ∇ k ϕ ( k ) {\displaystyle {\mathcal {A}}(\mathbf {k} )=\mathbf {\nabla } _{\mathbf {k} }\phi (\mathbf {k} )} (where ϕ {\displaystyle \phi } is the angle on the circle), hence ν = 1 2 π ∫ B Z d Σ ⋅