Impurity Conduction at Low Concentrations

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 $\ensuremath{\rho}\ensuremath{\propto}\mathrm{exp}(\frac{{\ensuremath{\epsilon}}_{3}}{\mathrm{kT}})$. For small $K$, ${\ensuremath{\epsilon}}_{3}=(\frac{{e}^{2}}{{\ensuremath{\kappa}}_{0}}){(\frac{4\ensuremath{\pi}{n}_{D}}{3})}^{\frac{1}{3}}(1\ensuremath{-}1.35{K}^{\frac{1}{3}})$, where ${\ensuremath{\kappa}}_{0}$ is the dielectric constant. At higher $K$, ${\ensuremath{\epsilon}}_{3}$ and $\ensuremath{\rho}$ attain a minimum near $K=0.5$. The dependence on ${n}_{D}$ is extracted; the agreement of the latter and of ${\ensuremath{\epsilon}}_{3}$ with experiment is satisfactory. The magnitude of $\ensuremath{\rho}$ is in fair agreement with experiment. The influence of excited donor states on $\ensuremath{\rho}$ is discussed.