Nonadiabatic pairing effects for tight-binding electrons interacting with phonons

The nonadiabatic electron-phonon corrections for the superconducting pairing are investigated for a specific tight-binding model corresponding to a 2d square lattice. This permits us to investigate the role of various specific properties like the band filling, nesting effects, and a realistic van Hove singularity on the superconducting effective pairing beyond Migdal’s limit. The main results are the following: (i) Starting from a momentum-independent electron-phonon coupling, the nonadiabatic effects lead to an effective pairing that is strongly dependent on frequency and momentum. (ii) If instead the electron-phonon coupling is mainly forward (as due to correlation effects), the resulting pairing becomes strongly enhanced. These results confirm but also extend the simplified scheme used up to now to compute these properties. In this respect our results link the nonadiabatic effects to the specific properties of realistic materials.