Theory of Impurity Scattering in Semiconductors

TLDR

Experiments on germanium show that lattice scattering alone cannot explain the temperature dependence of resistivity, suggesting ionized impurity scattering as a significant contributor. The study seeks to quantify the resistivity due to ionized impurity scattering. Using the Rutherford scattering formula within the Lorentz‑Boltzmann equation, the authors assume independent scattering by individual ions to evaluate collision terms. They obtain ρ = 2.11×10^2 κ^−2 T^−3/2 ln[1+36 κ^2 d^2 (kT)^2 e^−4], with d the half average impurity spacing and κ the dielectric constant.

Abstract

Experiments by Lark-Horovitz and collaborators on the Hall effect and resistivity of germanium semiconductors have shown that the simple theory of lattice scattering alone cannot explain the temperature dependence of the resistivity. Another probable source of resistance is scattering by ionized impurity centers. This resistance can be calculated by using the Rutherford scattering formula. Evaluation of the collision terms in the Lorentz-Boltzmann equation of state is made by assuming that scattering of an electron by one ion is approximately independent of all other ions. This results in a resistivity given by (in ohm cm): $\ensuremath{\rho}=2.11\ifmmode\times\else\texttimes\fi{}{10}^{2}{\ensuremath{\kappa}}^{\ensuremath{-}2}{T}^{\ensuremath{-}\frac{3}{2}}\mathrm{ln}{1+36{\ensuremath{\kappa}}^{2}{d}^{2}{(\mathrm{kT})}^{2}{e}^{\ensuremath{-}4}}$ where $d$ is half the average distance between impurity ions and $\ensuremath{\kappa}$ the dielectric constant of the semiconductor.