Deformation Potentials in Silicon. II. Hydrostatic Strain and the Electron-Phonon Interaction

The hydrostatic pressure dependence of the energy of ${\Gamma }_{{25}^{\prime }}$, the top of the valence band in Si, is calculated and expressed in terms of the deformation potential constant ${D_d}^v$. There are three types of terms which contribute to ${D_d}^v$: those involving the valence wave functions explicitly, a term involving the over-all zero of the crystal potential and an exchange and correlation term depending on the average valence charge density. It is shown that to obtain the correct change in the zero of energy with strain one may take the usually arbitrary zero of energy appearing in the Hartree-Fock self-consistent potential to be equal to the total energy of the ions in their equilibrium position divided by the number of electrons. The first term is determined by applying perturbation theory to the wave functions calculated previously by Kleinman and Phillips. The second term is calculated using the Ewald summing technique and the third term is determined from the Bohm-Pines approximation. Our calculated value of the pressure dependence of the ${\Gamma }_{{25}^{\prime }}-L_1$ energy gap in Si is similar to the experimental value obtained in Ge. Our absolute energy shifts lie between those estimated by Herring from transport data, and those of Bagguley et al., who determined the relaxation times appearing in Herring's theory from cyclotron resonance.