Nonadiabatic and anharmonic effects in high-pressure H3S and D3S superconductors

Superconductivity in compressed H3S arises from the interplay between high-frequency phonons and a pronounced van Hove singularity near the Fermi level. Using first-principles calculations, we investigate the superconducting properties of H3S and D3S at 160 and 200 GPa, explicitly incorporating anharmonic lattice dynamics and first-order vertex corrections to electron-phonon (e-ph) interactions, thereby going beyond the Migdal approximation underlying conventional Migdal-Eliashberg theory. We find that both anharmonicity and nonadiabatic vertex corrections suppress the effective e-ph coupling and reduce the superconducting critical temperature (Tc). Calculations performed within the energy-dependent full-bandwidth Eliashberg formalism, including both anharmonic and vertex effects, yield Tc values in close agreement with experimental measurements for D3S at both pressures and for H3S at 200 GPa.