Nonadiabatic superconductivity. II. Generalized Eliashberg equations beyond Migdal’s theorem

Abstract

We generalize the Eliashberg equations to include the first nonadiabatic effects beyond Migdal's theorem. The resulting theory is nonperturbative with respect to \ensuremath{\lambda} and perturbative with respect to (\ensuremath{\lambda}${\mathrm{\ensuremath{\omega}}}_{\mathit{D}}$/${\mathit{E}}_{\mathit{F}}$). The main effects are due to the vertex corrections and the cross diagram that show a complex behavior with respect to the exchanged momentum (q) and frequency (\ensuremath{\omega}). Positive corrections and a corresponding enhancement of ${\mathit{T}}_{\mathit{c}}$ arise naturally if the electron phonon scattering is characterized mainly by small q values. For this reason we discuss our results in terms of an upper cutoff ${\mathit{q}}_{\mathit{c}}$ for the scattering. The generalized Eliashberg equations are solved numerically and analytically and we also provide a generalization of the McMillan equation that includes the nonadiabatic effects. For relatively small values of ${\mathit{q}}_{\mathit{c}}$, normal values of the coupling (\ensuremath{\lambda}\ensuremath{\simeq}0.5--1.0) can lead to strong enhancements of ${\mathit{T}}_{\mathit{c}}$ in the range of the observed values. This situation leads also to a complex behavior of the isotope effect that can be anomalously large (\ensuremath{\alpha}>1/2) in some region of parameters but it can also vanish for ${\mathrm{\ensuremath{\omega}}}_{\mathit{D}}$>${\mathit{E}}_{\mathit{F}}$. It is therefore important to identify which features of more realistic models can lead to a situation in which the nonadiabatic effects are mainly positive. One way to achieve this is to have an upper cutoff ${\mathit{q}}_{\mathit{c}}$ for the electron-phonon scattering and electronic correlations appear to be a natural candidate to produce this effect. The nonadiabatic effects are expected to play an important role also for the normal properties of the system that can deviate appreciably from a normal Fermi liquid. It is important to study these effects in the future because they should lead to specific predictions of new effects that can be tested experimentally.