A probabilistic approach to the problem of electron localization in disordered systems and sharpness of the mobility edge

Abstract

By applying the theory of the asymptotic distribution of extremes and a certain stability criterion to the question of the domain of convergence in the probability sense, of the renormalized perturbation expansion (RPE) for the site self-energy in a cellularly disordered system, an expression has been obtained in closed form for the probability of nonconvergence of the RPE on the real-energy axis. Hence, the intrinsic mobility mu (E) as a function of the carrier energy E is deduced to be given by mu (E)= mu ₀exp(-exp( mod E mod -E_c) Delta ), where E_c is a nominal 'mobility edge' and Delta is the width of the random site-energy distribution. Thus mobility falls off sharply but continuously for mod E mod >E_c, in contradistinction with the notion of an abrupt 'mobility edge' proposed by Cohen et al. and Mott. Also, the calculated electrical conductivity shows a temperature dependence in qualitative agreement with experiments on disordered semiconductors.