TopologicalInsulator

The concept is illustrated pictorially by chemists when describing the Molecular Orbital, e.g. in a biatomic molecule. Two isolated atoms with one valence electron each: these electrons occupy states degenerate in energy. When close together the degeneracy is removed by additional interaction terms to yield bonding and antibonding states

This means that in zero-th approximation the two electron obey a 2-electron Hamiltonian {${\cal H}_0$} without interaction terms. A small interaction {$\cal{V}\ll{\cal H}_0$} may be considered as a perturbation for the two higher energy degenerate bound states. This leads to the diagonalization of a 2x2 matrix with equal diagonal {$H_0$} and small non diagonal {$V$} terms.

The same thing applies to Bloch states at the Brillouin Zone (BZ) boundaries, e.g. in the tight binding scheme. Since they are connected by a reciprocal lattice vector {$\mathbf{G}$}, they represent the same state (periodicity of the lattice), i.e. they are degenerate in energy. The fact that a lattice vector {$\mathbf{G}$} joins equivalent states indicates a nesting. If there is a neglected non diagonal interaction (e.g. the crystal potential) the well known result is the opening of a band gap. The energies in between the crystal equivalent of the bonding and antibonding state are not available to electrons.

By this mechanism a generic non interacting gas of valence electrons (thought as the first approximation to a quantum solid), may give rise to an insulator. When this happens one may think of the bonding and antibonding states as the two stationary quantum waves that are produced by the superposition of progressive and regressive plane waves at wavevector {$\mathbf{G}/2$} (in some sense by the scattering of these plane waves on the lattice).

A related crystal phenomenon is the Peierls mechanism, by which a linear metallic chain of atoms will almost inevitably dimerize and produce an insulator (Jahn-Teller distortion cames into this as well). If spin is present one may have an analogous spin Peierls mechanism, by which the metallic chain turns into an antiferromagnetic insulator chain.

We must take for granted that for certain insulators (the topological insulators) the vicinity of the crystal boundary (the surface) produces additional states in the gap, where the Fermi level lies. This gives rise to a peculiar two dimensional (2-d) Fermi surface, in the shape of circle of linearly energy-dependent radius, i.e. a cone in {$E$} vs. {$(k_x,k_y)$}. This corresponds to the so called Dirac cone, which appears also in 2-d graphene.

The Fermi surface indicates that a surface metal exists, but surface states are generally prone to the nesting mechanism: any disorder, impurity etc. may provide additional interaction terms, destroying the metal (the so-called Anderson localisation for strong disorder). However the 2-d metal may be topologically protected if a strong spin-orbit interaction dictates that the spin of the electron must be correlated to the wave vector direction. Then the circular energy surface implies that the spin must turns by {$2 \pi$} going around it, leading to possible analogies of the spin Peierls mechanism to yield antiferromagnetic insulating states.

It is well known, though, that a spin {$1/2$} wave function requires a rotation of {$4\pi$} to be brought back into itself. This topological feature prevents the nesting to take place in the case described above. The prototypic topological insulators are Bi₂Se₃, Bi₂Te₃. Their relative Bi₂S₃ is a normal insulator. The spin-orbit interaction is the magnetic field acting on the spin, due to the apparent motion of the nucleus around it (like the apparent motion of the Sun around the Earth). This field becomes large for heavier atoms where the orbiting velocities are larger, approaching the relativistic limit. So heavier Se, Te bring in a stronger spin-orbit interaction and do the trick.