Hopping Conductivity in Disordered Systems

Abstract

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 $\sigma \propto \exp [-{\left(\frac{\lambda {\alpha }^3}{{\rho }_0\mathrm{kT}}\right)}^{\frac{1}{4}}]$. Here $T$ is the temperature of the system, ${\rho }_0$ is the density of states at the Fermi level, $\lambda$ is a dimensionless constant, and ${\alpha }^{-1\mathrm{}}$ is the distance for exponential decay of the wave functions. We rederive these results, relating $\lambda$ to the critical density of a certain dimensionless percolation problem, and we estimate $\lambda$ to be approximately 16. The applicability of the model to experimental observations on amorphous Ge, Si, and C is discussed.