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 nD. 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 ρ∝exp(ε₃kT). For small K, ε₃=(e²κ₀)(4πnD3)1/3(1-1.35K1/3), where κ₀ is the dielectric constant. At higher K, ε₃ and ρ attain a minimum near $K=0.5$. The dependence on nD is extracted; the agreement of the latter and of ε₃ with experiment is satisfactory. The magnitude of ρ is in fair agreement with experiment. The influence of excited donor states on ρ is discussed.