Effect of free carriers on the elastic constants ofp-type silicon and germanium

Numerical calculations of the effect of free holes on the elastic constant $C_{44}$ of Ge and Si are presented. The energy of the three top valence bands is obtaired by diagonalizing a 6 × 6 deformation-potential Hamiltonian, exact to second order in the components of $\overrightarrow{\mathrm{k}}$. The numerical calculations underline the shortcomings of previous analytical work performed by angular averaging of the energy bands. Our calculations are extended to obtain the effect of a large [001] stress on the hole contribution to $C_{44}$ mentioned above. This effect is anisotropic: The elastic constants $C_{44}$ and $C_{66}$ become inequivalent. The calculations are shown to be in good agreement with experiment in the low-temperature, low-carrier concentration region. This limitation is a direct consequence of having assumed a Hamiltonian quadratic in $\overrightarrow{\mathrm{k}}$.