Question:

Band structure of topological insulators

Nolan: 3 days ago

The above figure is Rashba-split free electron surface state. The below on is TI surface state figure.

The above figure is Rashba-split free electron-like surface state in a projected bulk band gap. The below figure is the band structure of a topological insulators. x axis is the wave vector, y axis is the energy. red and blue lines are surface states with one spin up one spin down. What is the difference between the two figures why TI has this linear one but the above is parabolic? They all have Rashba spin orbit interaction.

Answer:
Everly: 3 days ago

The difference is obvious: In the second figure, the blue and red line connect the valence and conduction bands. These are actually surface states. So regardless of where your chemical potential lies, there will be low-energy excitations on the surface, i.e. the surface is conducting. This is what happens in a topological insulator. In the first figure, you can put your Fermi energy between the valence band and the surface states (there is a whole energy window where nothing happens), and the whole system, both bulk and surface included, is insulating.

You should also notice that the red/blue lines have opposite spins. Imagine there are impurities on the surface. Normally electrons just backscatter by the impurities, causing dissipation of electric current. But in topological insulators, backscattering means you have to go from the red to the blue, since one is right-moving and the other is left-moving. But in the process the spin has to flip, which breaks time-reversal symmetry. So if the impurities do not break time-reversal symmetry (i.e. non-magnetic), they do not cause backscattering (to the leading order). This is why the surface states are robust: they are protected by time-reversal symmetry.