Impurity Conduction at Low Concentrations

Abstract

The conductivity of an $n$-type semiconductor has been calculated in the region of low-temperature $T$ and low impurity concentration $n_D$. The model is that of phonon-induced electron hopping from donor site to donor site where a fraction $K$ of the sites is vacant due to compensation. To first order in the electric field, the solution to the steady-state and current equations is shown to be equivalent to the solution of a linear resistance network. The network resistance is evaluated and the result shows that the $T$ dependence of the resistivity is $\rho \propto \exp \left(\frac{{\epsilon }_3}{\mathrm{kT}}\right)$. For small $K$, ${\epsilon }_3=\left(\frac{e^2}{{\kappa }_0}\right){\left(\frac{4\pi n_D}{3}\right)}^{\frac{1}{3}}(1-1.35\mathrm{}{K}^{\frac{1}{3}})$, where ${\kappa }_0$ is the dielectric constant. At higher $K$, ${\epsilon }_3$ and $\rho$ attain a minimum near $K=0.5\mathrm{}$. The dependence on $n_D$ is extracted; the agreement of the latter and of ${\epsilon }_3$ with experiment is satisfactory. The magnitude of $\rho$ is in fair agreement with experiment. The influence of excited donor states on $\rho$ is discussed.