Two-square-well model for layered superconductors: Effect of intralayer and interlayer Coulomb interactions on thed-wave gap

Abstract

On the basis of Fermi liquid theory, we study the effect of intralayer and interlayer Coulomb interactions on the energy gap of layered superconductors assuming the BCS d-wave pairing model. The gap equation is derived for the two-square-well model, assuming a boson-mediated attraction and a long-range Coulomb repulsion that leads to intralayer and interlayer Coulomb scattering of Cooper pairs. The d-wave gap amplitude ${\ensuremath{\Delta}}_{1}$ at $T=0\mathrm{K}$ is evaluated as a function of the attractive pairing strength and of the screened Coulomb interactions. For optimally doped ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ we find that intralyer Coulomb scattering reduces ${\ensuremath{\Delta}}_{1}$ by a factor between 1.5 and 2. The additional effect of interlayer Coulomb scattering depends in a crucial manner on the relative phases of the gap functions in adjacent layers. If Josephson and boson-assisted interlayer pair tunneling (ILT) are responsible, the three-dimensional phase coherence, that is, the phase difference between adjacent layers $\ensuremath{\Delta}\ensuremath{\varphi}=0,$ interlayer Coulomb scattering leads to a further reduction of the gap that can be comparable with or larger than the effect of intralyer scattering. If, however, the phase of the gap function alternates, that is $\ensuremath{\Delta}\ensuremath{\varphi}=\ensuremath{\pi},$ because the interlayer Coulomb scattering dominates the ILT effect, the interlayer Coulomb scattering of Cooper pairs enhances ${\ensuremath{\Delta}}_{1}.$ This effect is studied as a function of the strength of the Coulomb interaction and it is shown that, with increasing strength, the pseudo-Coulomb parameter ${\mathrm{\ensuremath{\mu}}}^{\mathrm{*}}$ decreases and may change sign when the Coulomb interaction is sufficiently strong. The possible role of the interlayer Coulomb effect is discussed in relation to scanning tunneling microscopy experiments.