Microscopic theory of dielectric screening and lattice dynamics. II. Phonon spectra and effective charges

The factorization scheme for treating dielectric screening in solids, introduced in a previous paper, is here used to calculate the dynamical matrix for the phonon frequencies and eigenvectors. By considering the limit as $\overrightarrow{\mathrm{q}}\to 0$ we derive explicit microscopic expressions for the effective-charge tensor, the optic modes, the low-frequency dielectric constant $\epsilon \left(\omega \right)$, the acoustic sum rules, and the elastic and piezoelectric constants. The Lyddane-Sachs-Teller relation is obeyed, and the deviation of the Sziegeti charge from the ionic charge is discussed. We discuss the relationship to other models for the dynamics of insulators and semiconductors, and finally present results of explicit calculations for the phonon spectra of silicon and germanium.