Topological Insulators

TL;DR

The paper frames topological insulators as band-theory realizations of topological order, unified with the quantum Hall paradigm via bulk invariants and bulk–boundary correspondence. It develops the theoretical toolkit (Chern and Z2 invariants, BdG formalism, and the periodic table of topological phases) and surveys experimental milestones in 2D QSHI (HgTe/CdTe) and 3D TIs (Bi1−xSbx, Bi2Se3 family). It then explores how symmetry breaking at surfaces (magnetism, superconductivity) can yield exotic states such as the surface quantum Hall effect, axion electrodynamics, and Majorana fermions with potential for topological quantum computation. The discussion culminates in prospects for material improvements, heterostructures, and device concepts that could harness TI surface states for new technologies and quantum information processing.

Abstract

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. The 2D topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A 3D topological insulator supports novel spin polarized 2D Dirac fermions on its surface. In this Colloquium article we will review the theoretical foundation for these electronic states and describe recent experiments in which their signatures have been observed. We will describe transport experiments on HgCdTe quantum wells that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. We will then discuss experiments on Bi_{1-x}Sb_x, Bi_2 Se_3, Bi_2 Te_3 and Sb_2 Te_3 that establish these materials as 3D topological insulators and directly probe the topology of their surface states. We will then describe exotic states 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. We will close by discussing prospects for observing these exotic states, a well as other potential device applications of topological insulators.

Figures (20)

• Figure 1: (a, b, c) The insulating state. (a) depicts an atomic insulator, while (b) shows a simple model insulating band structure. (d, e, f) The quantum Hall state. (d) depicts the cyclotron motion of electrons, and (e) shows the Landau levels, which may be viewed as a band structure. (c) and (f) show two surfaces which differ in their genus, $g$. $g=0$ for the sphere (c) and $g=1$ for the donut (f). The Chern number $n$ that distinguishes the two states is a topological invariant similar to the genus.
• Figure 2: The interface between a quantum Hall state and an insulator has chiral edge mode. (a) depicts the skipping cyclotron orbits. (b) shows the electronic structure of a semi infinite strip described by the Haldane model. A single edge state connects the valence band to the conduction band.
• Figure 3: Electronic dispersion between two boundary Kramers degenerate points $\Gamma_a=0$ and $\Gamma_b=\pi/a$. In (a) the number of surface states crossing the Fermi energy $E_F$ is even, whereas in (b) it is odd. An odd number of crossings leads to topologically protected metallic boundary states.
• Figure 4: Boundary states for a topological superconductor (T-SC). (a) shows a 1D superconductor with bound states at its ends. (b,c) show the end state spectrum for an ordinary 1D superconductor (b) and a 1D topological superconductor (c). (d) shows a topological 2D superconductor with a chiral Majorana edge mode (e). A vortex with flux $\Phi = h/2e$ is associated with a zero mode (c).
• Figure 5: Edge states in the quantum spin Hall insulator. (a) shows the interface between a QSHI and an ordinary insulator, and (b) shows the edge state dispersion in the graphene model, in which up and down spins propagate in opposite directions.