On random walks and the AC conductivity of hopping systems

Abstract

The continuous-time random walk theory developed by Scher and Lax (1973) to describe the AC conductivity of disordered solids is extended. An exact formal expression, which is in a form suitable for either analytical or direct numerical evaluation, is derived for the conductivity. This expression is evaluated analytically for the special case of a system of randomly distributed hopping sites with randomly distributed energy levels. Simple exact formulae are obtained for the conductivity for a general disordered system at high frequencies and for an ordered system at all frequencies.