topological insulator

History

In 2005, scientists Charles Kane and Eugene Mele theoretically predicted that graphene could be a topological insulator. However, due to the weak effect, this did not happen. In 2007, Laurens Molenkamp and his group first created a topological insulator based on mercury telluride. Since then, many such materials have been discovered.

How it works

The secret lies in topology — a branch of mathematics that studies properties that do not change under smooth deformations. The quantum states of electrons in such a crystal are 'twisted,' and this twisting is preserved. Therefore, on the surface, electrons are forced to move in a certain direction, not changing course even when colliding with irregularities of the crystal. This is similar to how water in a river flows around a rock without stopping.

History

In 2005, scientists Charles Kane and Eugene Mele theoretically predicted that graphene could be a topological insulator. However, due to the weak effect, this did not happen. In 2007, Laurens Molenkamp and his group first created a topological insulator based on mercury telluride. Since then, many such materials have been discovered.

How it works

The secret lies in topology — a branch of mathematics that studies properties that do not change under smooth deformations. The quantum states of electrons in such a crystal are 'twisted,' and this twisting is preserved. Therefore, on the surface, electrons are forced to move in a certain direction, not changing course even when colliding with irregularities of the crystal. This is similar to how water in a river flows around a rock without stopping.

History

The foundations were laid by work on the quantum Hall effect (1980s), when David Thouless, F. Duncan Haldane, and Michael Kosterlitz used topology to explain the robustness of Hall conductivity. In 2005, Charles Kane and Eugene Mele proposed that graphene, when spin-orbit coupling is considered, could be a two-dimensional topological insulator (quantum spin Hall phase), but the effect in graphene turned out to be negligibly small. In 2006, Bernevig, Hughes, and Zhang predicted the quantum spin Hall effect in HgTe/CdTe quantum wells. Experimentally, a two-dimensional topological insulator was realized by Laurens Molenkamp's group in 2007. Almost simultaneously (2008-2009), three-dimensional topological insulators — tetradymites (Bi₂Se₃, Bi₂Te₃) — with surface Dirac fermions were discovered.

How it works

A topological insulator forms due to band inversion driven by strong spin-orbit coupling. The bulk Hamiltonian can be described in the k·p approximation, and under certain parameters, a condition for a nonzero Z₂ invariant arises. Surface states are solutions of the two-dimensional Dirac equation H = v_F (σ_x p_x + σ_y p_y), where v_F is the Fermi velocity. Owing to spin polarization and chirality (spin direction is rigidly tied to momentum), backscattering is suppressed, and conductivity is quantized. Topological protection is destroyed only by perturbations capable of closing the bulk band gap, such as a strong magnetic field or breaking of time-reversal symmetry.

Open problems & fun facts

Search for materials with a large bulk band gap (>0.3 eV) for room-temperature applications; Understanding the role of correlations and interactions in topologically ordered phases; Realization of topological superconductivity and detection of Majorana fermions; Fabrication of ultra-clean heterostructures for defect-free transport

💡 The surface states of a three-dimensional topological insulator resemble two-dimensional graphene: they obey the Dirac equation and have zero effective mass. However, unlike graphene, a topological insulator has only one Dirac cone, not two, making it 'lonesome graphene'.