First-order vertex corrections to phonon-mediated superconductivity in H₃S and Pb

The Migdal-Eliashberg (ME) theory can breakdown in phonon-mediated superconductors within the adiabatic regime where the Fermi energy is comparable to the phonon energy. In this study, we incorporated first-order vertex corrections for electron-phonon interactions within the Eliashberg formalism and investigated their effect on the superconducting properties of H₃S and Pb. In the case of H₃S, where the adiabatic assumption remains questionable, we observed a significant reduction in the critical temperature (T_c) and superconducting energy gap compared to the adiabatic predictions. On the other hand, in Pb, where the adiabatic limit holds, vertex-corrected T_c and energy gap closely match the ME results.