Microscopic theory of electron-phonon interaction in insulators or semiconductors

A unified treatment of the electron-phonon interaction matrix $M$ in insulators (or semiconductors at low temperatures) is given in terms of the complete linear electron response within the adiabatic approximation. The effective electron-one-phonon (el-ph) potential $M$ which is constructed, analogously to metals, as a screened vertex-corrected renormalized (bare) ion potential is defined, including all many-body corrections. We derive the small-wave-vector ($\overrightarrow{\mathrm{q}}$) limit of $M$ leading to multipole as well as short-range contributions by separating out the electron response for fixed macroscopic electric field and using the analytical properties of the inverse dielectric tensor ${\epsilon }^{-1\mathrm{}}$. We obtain explicit expressions for all standard types of el-ph couplings in terms of microscopic quantities in principle calculable from band structure. The main results of Lawaetz are rederived and new—indirectly induced—multipole terms are obtained. A new quadrupole sum rule for $\epsilon$ is proved which guarantees, together with the already known acoustic sum rule, the well-known behavior of the acoustic scattering potential as a function of $q$ in the elastic limit. An effective el-two-ph scattering is investigated; further sum rules are proved representing constraints on a nonlinear response function.