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

Abstract

We solve the Eliashberg equations for the case of an explicit dependence of the interactions, and of the resulting self-energies XI (I?, a), X2(% w ) . We consider a strong energy-dependence of the electron-electron scattering-rate r;', 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 tl, which is of the order of the phonon frequency Up),. 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 Aph, and small values of the cutoff C1. Specifically, the Coulomb interaction is found not to depress T,; thejsotope effect is strongly reduced when c1 <Up),. We find solutions in which the gap function A ( k , w ) 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-renormaliition X, depend very strongly on the superconducting gap A, and thus display a very strong temperature-dependence between T, 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. IntroductionHigh-T, cuprates, as well as several other systems, are characterized by a very large ionic polarizability of the background non-metallic BaO, Y203 layers. The ionic dielectric constant at low frequencies is in the range eion(w) = 30-60 [l]. This dielectric constant is obtained after substracting the Lindhard-like dielectric polarizability of the conduction electrons. We believe that this huge ionic polarizability plays a key role in the superconducting process, and is the primary factor responsible for the high superconducting transition temperature. The conventional pairing theory of superconductivity is based on the Eliashberg Eqs. [2]. These are based on the Migdal scenario, in which

s- and d-wave symmetries of the solutions of the Eliashberg equations

“…As we already pointed out 11 , it is possible to explain some typical high T c properties from a model where the electron-phonon interaction is severely cut off along the ξ k direction. Indeed, the dielectric constant of the ionic background ǫ ion is anomalously big at small energies 6 , i.e.…”

“…Instead of using (k x , k y ) as momentum variables, we will use the more convenient "polar-like" κ ≡ (θ, ξ k ) where θ is the angle and ξ k ≡ ǫ k − ǫ F . As we already did in previous publications [9][10][11]13 , we do not integrate out the dependence on the momentum ξ k . The Eliashberg equations can then be written in the Matsubara frequencies formalism (ω n = πT (2n + 1)):…”

“…The strong ξ k -dependence of the interaction has very interesting effects on the gap parameter Φ and on the renormalization functions X and Z that we studied in some length elsewhere [9][10][11] . In the context of the study of the symmetries of the gap, these effects are not very important because they are somewhat hidden by the effects of the anisotropy of the coupling.…”

“…In the context of the study of the symmetries of the gap, these effects are not very important because they are somewhat hidden by the effects of the anisotropy of the coupling. Nevertheless, we wanted to keep the same formalism in order to continue to explore the richness of the solutions it can bring, and in order to be consistent with our earlier approach [9][10][11] .…”

“…We therefore consider a strong electron-phonon interaction through our generalized Eliashberg scheme [9][10][11] . In this scheme we solve the Eliashberg equations 12 for an electron-phonon interaction having a very strong k-dependence of its matrix elements g(k, k ′ ) that enters in the phonon propagator D(k, k ′ ; ω n − ω n ′ ) through eq.…”