Hopping Conductivity in Disordered Systems

By considering a model in which charge is transported via phonon-induced tunneling of electrons between localized states which are randomly distributed in energy and position, Mott has obtained an electrical conductivity of the form - ({{^3}{₀kT}) }^1{4}. Here T is the temperature of the system, ₀ is the density of states at the Fermi level, is a dimensionless constant, and ^-1 is the distance for exponential decay of the wave functions. We rederive these results, relating to the critical density of a certain dimensionless percolation problem, and we estimate to be approximately 16. The applicability of the model to experimental observations on amorphous Ge, Si, and C is discussed.