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 $\mathrm{ka}\ll 1$ (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, $\mu$, 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 ${(\pm \frac{2m^{*}{q}^{2}a}{{\hslash }^{2}D})}^{\frac{1}{2}}$, 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 $\mu \propto T^{-\frac{1}{2}}{N_I}^{-\frac{1}{3}}$ for repulsive centers, while for attractive centers there are three possible limiting cases: ${\mu }_1\propto T^{-\frac{1}{2}}{N_I}^{-\frac{1}{3}}$, ${\mu }_2\propto T^{\frac{1}{2}}{N_I}^{-1\mathrm{}}$ and ${\mu }_3\to \infty$. 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 $\mathrm{ka}\gg 1$ which yield $\mu \propto T^{\frac{3}{2}}{N_I}^{-1\mathrm{}}$ for both attractive and repulsive centers.