Résumé
Electron-phonon interaction and phonon frequencies of doped polar semiconductors are sensitive to long-range Coulomb forces and can be strongly affected by screening effects of free carriers, the latter changing significantly when approaching the two-dimensional limit. We tackle this problem within a linear-response dielectric-matrix formalism, where screening effects can be properly taken into account by generalized effective charge functions and the inverse scalar dielectric function, allowing for controlled approximations in relevant limits. We propose complementary computational methods to evaluate from first principles both effective charges -- encompassing all multipolar components beyond dynamical dipoles and quadrupoles -- and the static dielectric function of doped two-dimensional semiconductors, and provide analytical expressions for the long-range part of the dynamical matrix and the electron-phonon interaction in the long-wavelength limit. As a representative example, we apply our approach to study the impact of doping in disproportionated graphene, showing that optical Fr\"ohlich and acoustic piezoelectric couplings, as well as the slope of optical longitudinal modes, are strongly reduced, with a potential impact on the electronic/intrinsic scattering rates and related transport properties.