Einstein's relationship for hopping electrons

Abstract

Photoconductivity in amorphous semiconductors at very low temperatures is temperature-independent, being determined by the energy-loss hopping of carriers through localized band-tail states. In such a hopping relaxation, neither diffusion coefficient, D, nor mobility of carriers, μ, depend on temperature and the conventional form of the Einstein's relationship μ = eD kT is not valid. The relationship between μ and D for the hopping relaxation of electrons in the exponential band tail is calculated and it is shown that it has the form μ ∝ eD ϵ 0 , where ϵ 0 is the energy scale of the exponential band tail.