# Topological insulator

An idealized band structure for a topological insulator. The Fermi level falls within the bulk band gap which is traversed by topologically-protected surface states.

A topological insulator is a material with non-trivial symmetry-protected topological order that behaves as an insulator in its interior but whose surface contains conducting states,[1] meaning that electrons can only move along the surface of the material. However, having a conducting surface is not unique to topological insulators, since ordinary band insulators can also support conductive surface states. What is special about topological insulators is that their surface states are symmetry-protected[2][3][4][5] by particle number conservation and time-reversal symmetry.

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 surface metallic conduction. Carriers in these 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 the "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic. Non-interacting topological insulators are characterized by an index (known as ${\displaystyle \mathbb {Z} _{2}}$ topological invariants) similar to the genus in topology.[1]

As long as time-reversal symmetry is preserved (i.e., there is no magnetism), the ${\displaystyle \mathbb {Z} _{2}}$ index cannot change by small perturbations and the conducting states at the surface are symmetry-protected. On the other hand, in the presence of magnetic impurities, the surface states will generically become insulating. Nevertheless, if certain crystalline symmetries like inversion are present, the ${\displaystyle \mathbb {Z} _{2}}$ index is still well defined. These materials are known as magnetic topological insulators and their insulating surfaces exhibit a half-quantized surface anomalous Hall conductivity.

## Prediction and discovery

Time-reversal symmetry-protected two-dimensional edge states were predicted in 1987 by Oleg Pankratov[6] to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride, and were observed in 2007.[7]

In 2006, it was predicted that similar topological insulators might be found in binary compounds involving bismuth,[8][9][10][11] and in particular "strong topological insulators" exist that cannot be reduced to multiple copies of the quantum spin Hall state.[12]

The first experimentally-realized 3D topological insulator state (symmetry-protected surface states) was discovered in bismuth-antimony in 2008.[13] Shortly thereafter symmetry-protected surface states were also observed in pure antimony, bismuth selenide, bismuth telluride and antimony telluride using angle-resolved photoemission spectroscopy (ARPES).[14] Many semiconductors within the large family of Heusler materials are now believed to exhibit topological surface states.[15][16] In some of these materials, the Fermi level actually falls in either the conduction or valence bands due to naturally-occurring defects, and must be pushed into the bulk gap by doping or gating.[17][18] In 2012, topological Kondo insulators were discovered in samarium hexaboride, which is a bulk insulator at low temperatures.[19][20] The surface states of a 3D topological insulator is a new type of two-dimensional electron gas (2DEG) where the electron's spin is locked to its linear momentum.[21]

Fully bulk-insulating or intrinsic 3D topological insulator states exist in Bi-based materials.[22]

In 2014, it was shown that magnetic components, like the ones in spin-torque computer memory, can be manipulated by topological insulators.[23][24]

The effect is related to metal–insulator transitions (Bose–Hubbard model).[citation needed]

## Properties and applications

Spin-momentum locking[21] 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.[25] (Note that Majorana zero-mode can also appear without topological insulators.[26]) 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 ${\displaystyle \mathbb {Z} _{2}}$  topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the ${\displaystyle \mathbb {Z} _{2}}$  invariants. An experimental method to measure ${\displaystyle \mathbb {Z} _{2}}$  topological invariants was demonstrated which provide a measure of the ${\displaystyle \mathbb {Z} _{2}}$  topological order.[27] (Note that the term ${\displaystyle \mathbb {Z} _{2}}$  topological order has also been used to describe the topological order with emergent ${\displaystyle \mathbb {Z} _{2}}$  gauge theory discovered in 1991.[28][29]) 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 ${\displaystyle \mathbb {Z} }$ , ${\displaystyle \mathbb {Z} _{2}}$  or trivial) as described by the periodic table of topological invariants.[30]

The most promising applications of topological insulators are spintronic devices and dissipationless transistors for quantum computers based on the quantum spin Hall effect[7] and quantum anomalous Hall effect.[31] In addition, topological insulator materials have also found practical applications in advanced magnetoelectronic and optoelectronic devices.[32][33]

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