Ionized Impurity Scattering in Nondegenerate Semiconductors

Abstract

The treatment for the scattering by ionized impurities in a semiconductor using the partial wave technique is set up and applied using a square well for the attractive impurity and a square barrier to represent the repulsive impurity. For the case $\mathrm{ka}\ensuremath{\ll}1$ (where $k$ is the wave number of the charge carriers and $a$ is the range of the impurity potential), analytical solutions are obtained for the mobility, $\ensuremath{\mu}$, and Hall coefficients applicable to nondegenerate semiconductors at low temperatures with high impurity content. The mean free time for scattering is a function of the parameters $\mathrm{ka}$ and ${(\ifmmode\pm\else\textpm\fi{}\frac{2{m}^{*}{q}^{2}a}{{\ensuremath{\hbar}}^{2}D})}^{\frac{1}{2}}$, where $q$ is the charge of the carrier and $D$ is the dielectric constant. The present treatment for reasonable choices of $a$ yields limiting laws of $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}\frac{1}{2}}{{N}_{I}}^{\ensuremath{-}\frac{1}{3}}$ for repulsive centers, while for attractive centers there are three possible limiting cases: ${\ensuremath{\mu}}_{1}\ensuremath{\propto}{T}^{\ensuremath{-}\frac{1}{2}}{{N}_{I}}^{\ensuremath{-}\frac{1}{3}}$, ${\ensuremath{\mu}}_{2}\ensuremath{\propto}{T}^{\frac{1}{2}}{{N}_{I}}^{\ensuremath{-}1}$ and ${\ensuremath{\mu}}_{3}\ensuremath{\rightarrow}\ensuremath{\infty}$. Case 2 refers to a form of resonance scattering and case 3 is the Ramsauer effect in semiconductors. These results are markedly different from previous formulas valid for $\mathrm{ka}\ensuremath{\gg}1$ which yield $\ensuremath{\mu}\ensuremath{\propto}{T}^{\frac{3}{2}}{{N}_{I}}^{\ensuremath{-}1}$ for both attractive and repulsive centers.