Electron-phonon interaction and longitudinal-transverse phonon splitting in doped semiconductors

We study the effect of doping on the electron-phonon interaction and on the phonon frequencies in doped semiconductors, taking into account the screening in presence of free carriers at finite temperature. We study the impact of screening on the Fr\"ohlich-like vertex and on the long-range components of the dynamical matrix, going beyond the state-of-the-art description for undoped crystals, thanks to the development of a computational method based on maximally localized Wannier functions. We apply our approach to cubic silicon carbide, where in presence of doping the Fr\"ohlich coupling and the longitudinal-transverse phonon splitting are strongly reduced, thereby influencing observable properties such as the electronic lifetime.