Vertex corrections to the theory of superconductivity

Abstract

With the eventual goal of understanding vertex corrections in the doped fullerenes and the bismuthates, assuming that they may be electron-phonon superconductors, we have constructed a weak coupling perturbation theory that includes the lowest-order vertex corrections (and reduces to Migdal-Eliashberg theory in the absence of the vertex corrections). Vertex corrections are studied in the local approximation which neglects the momentum dependence of the self energy. This approximation is exact in the limit of infinite dimensions and appears to be accurate in three dimensions. We present results for the superconducting transition temperature as a function of ω ϱ/E ƒ and λ and make comparison with similar results calculated in the infinite-dimensional Holstein model in the weak coupling limit. We find that vertex corrections reduce T c and the energy gap Δ o; however, the ratio 2 Δ 0/ k B T c remains unaffected. We also find that the isotope effect is reduced in the presence of vertex corrections.