Question

Explain in detail the difference between extended state conduction and localized state conduction while clearly explaining the mechanism of each and under what conditions do each of them work.

Answer 1

Let us start by thinking of a three dimensional metal with a low density of impurities and hence weak disorder. In this limit, electrons moving through the system are only scattered by impurities every now and then but otherwise they move ballistically between collisions. This picture is known as the Drude model and yields a conductivity given by,

  

where is the electron density, is the election mass and is the mean free time between collisions. In this limit, the election wave functions are extended throughout the bulk of the system, much like the plane waves of a clean crystal.

The other extreme is the strong disorder limit, in which there is a strongly fluctuating random potential in the material, for instance due to a high density of impurities. This potential has minima at random positions in the material where electrons can be trapped. The corresponding wave functions are localised exponentially close to the minima. If the density of impurities is high in the material, all electronic states are of this type. In this situation, the only mechanism for conduction in the material is tunneling between different potential minima. However, the minima may be far apart from each other which leads to a suppression of conductance and to an insulating state.

Let's discuss qualitatively the simplest effect of strong disorder on topological states, which are gapped. If we imagine a disorder potential with a Gaussian distribution, there are always "rare regions"where the disorder potential is very deep. Such potentials may thus yield energy levels in the middle of the gap. If disorder is strong, there might be no gap at all. However the electrons are trapped in these rare region are localised and do not participate in any of the transport properties of the material.

Thus even weak disorder can close the energy gap which is what protects topology as discussed so far. However, since these localized states are isolated from each other and most importantly from the edges, they do not affect any of the topological properties.

The considerations above make it clear that the presence or absence of energy gap is not a good criterion to tell whether the system is insulating or not in the presence of strong disorder and that we need to understand the nature of the electronic states in the system.