Galvanomagnetic Theory forn-Type Germanium and Silicon: Hall Theory and General Behavior of Magnetoresistance

Abstract

A general expression for the resistivity tensor, appropriate to $n$-type germanium and silicon, is deduced from which the magnetoresistance $\frac{\Delta \rho }{\rho }$ and Hall coefficient $R_H$ relations are evaluated. The angular dependence of $\frac{\Delta \rho }{\rho }$ in germanium shows precisely the qualitative features noted in the experiments of Pearson and Suhl. Additional details emerge, however, for silicon that were not detected by Pearson and Herring—presumably because of the restricted range of $\omega \tau$ they employed. The field dependence of $\frac{\Delta \rho }{\rho }$ for both germanium and silicon is examined for a number of high-symmetry orientations of the current J and magnetic B vectors with the finding of a departure from the square law at high fields. A detailed study of the $R_H$ field dependence is made for the combinations $J_{100}$, $B_{010}$ and $J_{110}$, $B_{1\overline{1}0}$. A minimum is observed in germanium for the latter case and in silicon for the former case. The minima occur between the limiting values for $b\to \infty$, $R_H=\frac{1}{\mathrm{nqc}}$, and $b\to 0$, $R_H=\left(\frac{1}{\mathrm{nqc}}\right)\left[\frac{3K(K+2\mathrm{})\mathrm{}}{{\mathrm{}(2K+1)\mathrm{}}^2}\right]$; these limiting values are invariant for all alignments of J and B.