Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 5 / a / Solution

A parabolic isotropic band has group velocity $\mathbf{v}=\hslash \mathbf{k}/m^{\ast }$ and semiclassical force $\hslash \dot{\mathbf{k}}=-e(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ for electrons with charge $-e$, where $e>0$. In the relaxation-time approximation the nonequilibrium drift velocity loses momentum at rate $1/\tau$. Combining these givesThe relaxation term represents the average effect of scattering rather than a force on every carrier between collisions. Linear response, an approximately energy-independent $m^{\ast }$ and $\tau$ near the Fermi surface, and wavelengths long enough for a local bulk response are implicit. The equilibrium random velocities average to zero and do not themselves contribute a drift current.