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 $\ensuremath{\sigma}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}{(\frac{\ensuremath{\lambda}{\ensuremath{\alpha}}^{3}}{{\ensuremath{\rho}}_{0}\mathrm{kT}})}^{\frac{1}{4}}]$. Here $T$ is the temperature of the system, ${\ensuremath{\rho}}_{0}$ is the density of states at the Fermi level, $\ensuremath{\lambda}$ is a dimensionless constant, and ${\ensuremath{\alpha}}^{\ensuremath{-}1}$ is the distance for exponential decay of the wave functions. We rederive these results, relating $\ensuremath{\lambda}$ to the critical density of a certain dimensionless percolation problem, and we estimate $\ensuremath{\lambda}$ to be approximately 16. The applicability of the model to experimental observations on amorphous Ge, Si, and C is discussed.