Singly-Ionized-Impurity Scattering in Degenerate Material

The work of Blatt has been extended to include ionized-impurity scattering in degenerate $n$-type material having a single-valley conduction band and a spherical effective mass. The transport cross section $Q$ has been calculated for a potential of the Thomas-Fermi type where the inverse screening length is fixed by the Friedel sum rule. By writing the Schrödinger equation in terms of dimensionless variables, we have found that, as a consequence of the sum rule, $Q{k}^2$ is a universal function of $a_{0}k$, where $a_0$ is the first Bohr radius in the material and $k$ is the Fermi wave number. A comparison of the computed $Q{k}^2$ with $Q_B{k}^2$ (the Born-approximation result for Thomas-Fermi screening, which is also a universal function of $a_{0}k$) shows that for $a_{0}k<\mathrm{}0.3\mathrm{}$. the Born approximation overestimates the resistivity, a result which is typical for metals. However, for larger $a_{0}k$, which corresponds to many degenerately doped semiconductors, the Born approximation understimates the resistivity, the discrepancy being as much as 51% at $a_{0}k=0.8\mathrm{}$. For $a_{0}k>\mathrm{}10\mathrm{}$, the Born approximation is in error by no more than 3%.