First-principles study of phonon-mediated superconductivity beyond the Migdal approximation*

The Migdal-Eliashberg theory has been remarkably successful in explaining phonon-mediated superconductivity in materials under the adiabatic limit, where the ratio of the phonon frequency to the electronic Fermi energy is much smaller than unity. The breakdown of the adiabatic limit has been pointed out in various systems such as high-T_c, flat band, and low-carrier density superconductors. To address the impact of nonadiabatic effects in this class of materials, we recently implemented first-order vertex corrections for electron-phonon interactions within the Eliashberg formalism in the EPW code. In this talk, we highlight the significance of nonadiabaticity in predicting the transition temperature of representative high-T_c superconductors.