On the Einstein relation for hopping electrons

Transport properties of disordered semiconductors at low temperatures are determined by hopping of electrons via localized band tail states. Much attention has been paid recently in both experimental and theoretical studies to the relation between the diffusion coefficient of carriers, D, and their mobility, mu, in the hopping regime. Rather controversial results have been reported. In particular, some computer simulations have been claimed to show that the conventional Einstein relation mu = eD/kT is violated in the hopping regime even in the case of thermal equilibrium. We study the relation between mu and D by a computer simulation and show that such statements were based on a wrong interpretation of the simulation results. In thermal equilibrium, the Einstein relation between mu and D must hold, although in a nonequilibrium system this relation can be violated. (C) 1998 Elsevier Science B.V. All rights reserved.