Effect of anisotropy on the energy gap and the critical temperature in a strongly coupled layered superconductor

Abstract

The Eliashberg theory is applied to study the critical temperature and a gap anisotropy of strongly coupled layered superconductors. The electron-ion pseudopotential, the electron, and the phonon energy spectra are proposed to be anisotropic in layered crystals with an open Fermi surface. The anisotropic electron-phonon spectral density ${[{\ensuremath{\alpha}}^{2}F(z)]}_{{p}_{z}}$ is found to be expanded in $cos(n{p}_{z}d)$ functions. Such an expression for ${[{\ensuremath{\alpha}}^{2}F(z)]}_{{p}_{z}}$ demands the energy gap $\ensuremath{\Delta}({p}_{z}, \ensuremath{\omega})$ to present also as an expansion in the harmonic functions. Therefore, the value of the energy gap along the $c$ axis should display ${p}_{z}$ dependence. The value of $\ensuremath{\Delta}$ differs from that obtained by in-plane measurements. By using McMillan's method we present each amplitude ${\ensuremath{\Delta}}_{n}(\ensuremath{\omega})$ in the harmonic expansion of the energy gap $\ensuremath{\Delta}({p}_{z}, \ensuremath{\omega})$ by trial function. The infinite set of coupled homogeneous equations for ${\ensuremath{\Delta}}_{0}, {\ensuremath{\Delta}}_{1}, {\ensuremath{\Delta}}_{2}, \dots{}$ is obtained, the solution of which should yield the critical temperature. We estimate only those equations which include the three amplitudes, ${\ensuremath{\Delta}}_{0}$, ${\ensuremath{\Delta}}_{1}$, and ${\ensuremath{\Delta}}_{2}$, in the harmonic expansion of the energy gap. An approximate calculation of ${T}_{c}$ shows that the critical temperature must be enhanced due to the influence of anisotropy.