Solution of the Eliashberg equations for a very strong electron‐phonon coupling with a low‐energy cutoff

We solve the Eliashberg equations for the case of an explicit k dependence of the interactions, and of the resulting self‐energies Σ₁(k,ω), Σ₂(k,ω). We consider a strong energy‐dependence of the electron‐electron scattering‐rate τ ee−1, which is associated with a strong energy‐dependence of the electron‐phonon matrix element g(k,k′). We characterize this energy‐dependence by a cutoff ζ₁, which is of the order of the phonon frequency ω_ph. We find that we can account for a large number of unexpected features of the superconductivity of the cuprates by the BCS electron‐phonon theory, if we consider very large values of the McMillan coupling constant λ_ph, and small values of the cutoff ζ₁. Specifically, the Coulomb interaction is found not to depress T_c; the isotope effect is strongly reduced when ζ₁ < ω_ph. We find solutions in which the gap function Δ(k, ω) has extended s‐wave symmetry but is very anisotropic. These large anisotropies are in good agreement with various experiments. We suggest that the underlying cause of the strong energy‐dependence is a very small electronic screening parameter at the Fermi surface; the electron‐phonon matrix element g is abnormally large, and this accounts for the high transition temperatures of the cuprates. An order of magnitude estimate suggests that the electron‐phonon mechanism can account for transition temperatures up to about 200 K. We thus propose a very‐strong‐coupling theory, in which the renormalization functions, in particular the energy‐renormalization X, depend very strongly on the superconducting gap Δ, and thus display a very strong temperature‐dependence between T_c and T = 0. An experimental manifestation of the very strong coupling with a small cutoff is a zero bias anomaly sometimes observed in tunneling experiments.