Superconductivity in a strongly correlated anisotropic three-dimensional system

An effective Hamiltonian resulting from a Hubbard model with strong on-site Coulomb interaction and in the presence of local electron-phonon coupling is studied with use of the Gutzwiller approximation (GA) and the X-operator technique for extended s- and d-wave superconductivity in an anisotropic three-dimensional system. A comparison of the results obtained by the GA and the X-operator technique demonstrates that the latter approximation, which preserves the local constraint, is more appropriate for studying superconductivity in strongly correlated systems. In a two-dimensional system, this method predicts extended s-wave superconductivity for both the low-filling and almost-half-filled systems, whereas d-wave superconductivity is stable for intermediate carrier concentrations. The anisotropy in the hopping and the pairing interactions has a stronger effect on the s-wave than on the d-wave superconductivity and decreases the transition temperature ${\mathit{T}}_{\mathit{c}}$ for s-wave pairing with increasing interplanar coupling. A possible interpretation of the doping dependence of ${\mathit{T}}_{\mathit{c}}$ in high-temperature superconductors is discussed.