Introduction to topological insulators

Journal Club Talk by Ting Pong Choy on 13th May 2010
Summary by Fazel Fallah Tafti

References:
Nature 452, 970-974 (2008)
Phys. Rev. Lett. 98, 106803 (2007)
Phys. Rev. B 74, 195312 (2006)
Phys. Rev. Lett. 95, 146802 (2005)


Ting Pong introduced two model topological insulators. The first model was a simple two band honey comb lattice with a Hamiltonian consisting of a NN hopping term and an "effective" spin-orbit NNN hopping term. This Hamiltonian can be decoupled in k space and the eigenvalues may be studied in two cases; (a) if the spin-orbit coupling is zero it simply describes the tight binding Graphene system with two Dirac points related through the inversion symmetry (b) if the spin-orbit coupling is non-zero a gap opens at both Dirac points with each band being doubly degenerate with spin up and spin down states. However due to very small spin-orbit coupling in C atoms, this is mostly a theoretical toy model with no experimental realization.

The second model is a Cd(Hg)Te quantum well system composed of a thin HgTe slab sandwiched between two thick CdTe layers. The band structure of the two compounds are inverted with respect to each other. The band structure of the quantum well is similar to CdTe in the thin regime but once the thickness of the HgTe is raised above some critical value (6 nm) the bands are inverted and at d=dc the gap must close. This material has been experimentally tested by measuring the hall resistivity of the quantum well. Ting Pong showed that the Hamiltonian of this system is identical to the toy Graphene model mentioned above in small momentum limit.

In the second half of the discussion, a robust definition of the topological insulator was given based on the spin Chern number assuming spin being conserved. The spin Chern number is always an integer and it is calculated by integrating the curl of a Berry phase term in k-space. This number can be calculated for each occupied state and the sum over all the occupied states gives the total spin Chern number which is proved to always be an integer. This number is even for a conventional band insulator and odd for a topological band insulator. A model calculation was done to show how one can derive the Chern number using the model Hamiltonian which preserves the time reversal symmetry. Time reversal symmetry was introduced as iSy.K where K is the complex conjugate operator. The calculation was done through a mapping from kx,ky space into theta,phi space using a Jacobian. The "topology" comes from the way we define this mapping.