Polaron transport in a long phonon waves approximation

One seeks to understand directly the smooth electron distribution functions long ago observed in semiconductors by Haynes and Westphal, and found to satisfy the Einstein relation for diffusion.A zero-order approximation to the distribution function analogous to Chandrashekhar's solution of the Brownian motion problem is established. This is done on the basis of the expression of Feynman et al., for the density matrix of the electron with phonon coordinates eliminated. Justification of the underlying approximation of expansion in long phonon waves, as well as a discussion of the relation of the scheme to the Boltzmann or rate equation approach, is given.