Conductivity in Anisotropic Semiconductors: Application to Longitudinal Resistivity and Hall Effect in Saturation-Stressed Degenerately Dopedn-Type Germanium

A variational iteration technique is developed for calculating the distribution function and electrical resistivity for semiconductors with anisotropic band structure. When only a single iteration is performed the result for the resistivity is identical to that obtained by an application of Kohler's principle with a relaxation-time approximation and the calculated distribution function includes anisotropic corrections to the zeroth-order energy-dependent relaxation time. The accuracy of the technique is demonstrated by comparing the results of a first-order calculation with the exact solution of the Boltzmann equation for the transverse resistivity and the distribution function for Brooks-Herring scattering by current carriers moving on an ellipsoidal constant-energy surface in the limit of infinite-mass anisotropy. The technique is applied to the calculation of the longitudinal resistivity and Hall $r$ coefficient of saturation-stressed degenerately doped $n$-type Ge at $T=0\mathrm{}°$K. We find that $r$ is within 5% of unity, thus justifying Katz's determination of the number of carriers from his Hall data but the theoretical upper bound to the resistivity is found to significantly underestimate the experimental results. However, when the calculation is approximately corrected for the known inaccuracies of the Born approximation together with contributions from multiple scattering and dressing effects, the theory is found to be in good agreement with experiment.