Dielectric-screening matrix and lattice dynamics of Si

The dynamical matrix of a semiconductor is set up by evaluating the full dielectric matrix $\epsilon (\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{G}},\overrightarrow{\mathrm{q}}+{\overrightarrow{\mathrm{G}}}^{\prime })$ to the lowest order in a Brillouin-Wigner perturbation scheme. The introduction in the dynamical matrix of the nondiagonal elements of the dielectric matrix, which account for the lattice effects on the electron screening, gives rise to noncentral forces between the atoms and restores the correct $q\to 0$ limit for the longitudinal acoustic frequencies through the fulfillment of the acoustic sum rule. A calculation of the phonon frequencies of Si is presented. The results, which are in reasonable agreement with experimental data, show that the off-diagonal elements are responsible for the stability of the crystal against shear and give real values for transverse acoustic frequencies. It is also shown that a third-order Rayleigh-Schrödinger perturbation theory, with some correction to include higher-order effects, can provide a good description of phonon spectrum of Si, but poorly reproduces the experimental elastic constants.