Angular dependence of hole—acoustic-phonon transition rates in silicon

Hole—acoustic-phonon transition rates in silicon, calculated by us previously, are expanded in a double set of cubic harmonics of Von der Lage and Bethe. The expansion allows us to quantify the degree of anisotropy in the scattering rates for all valence-band pairs and as a function hole energy. The rates are shown to exhibit a high degree of anisotropy requiring for a good fit angular functions with combined angular momentum indices $L+L^{\prime }\le 8$. Just as important, owing to the nonparabolic nature of silicon's valence bands, the expansion coefficients are markedly energy dependent. Therefore, calculations of transport coefficients for $p$-type silicon which neglect the anisotropy and nonparabolicity of the valence-band structure are not expected to be quantitatively correct. The parabolic limit will result in an incorrect $T^{-\frac{3}{2}}$ temperature dependence for acoustic-phonon limited mobility due to the temperature dependence of the carrier concentrations. It will also fail to predict the correct magnitude of the conductivity since the distribution functions for each band will not reflect the anisotropy and energy dependence of the transition rates.