S. Scandolo, ICTP

Lesson 17 (Electronic transport in crystals)

In the previous lecture, the rate of change of for an electron in a crystal in constant electric field was derived: . The goal is calculate the induced current: , with . We can see that if then the integral of gives zero because the occupied states (the ground state in this case) are distributed symmetrically around . The current is also zero if the band is filled completely as in this model when an electron exits at one end of the BZ it enters at the opposite side. This means that the same states are occupied at any given time. In this model, the current will be periodic with . We can also notice that the mean of the current is zero. (This is obviously wrong.) In order to understand electronic transport, we have to incorporate scattering into the model. (E.g. scattering of electrons on impurities, on other electrons, etc.) The time scale of scattering, , is a lot smaller than , so while the electric field moves the electrons away from the ground state, scattering holds them back. From these two opposite effects a qualitative picture of a new equilibrium of occupied states can be drawn: the ground state is shifted by . This gives a non-zero current as is not symmetric anymore and it goes as