s-wave superconductivity in the presence of local Coulomb correlations

We derive the superconductiong mean-field equations for an attractive interaction, V, in the s-wave channel when local Coulomb interactions are taken into account for any value of U. Our results show that the Coulomb repulsion is detrimental to the critical temperature, $T_c$, and the order parameter, $\mathrm{\Delta }(T)$, for values of $U\ge \mathrm{\mid }V\mathrm{\mid }$. Furthermore, our results depend on band filling in a sensible way. In the presence of local correlations, $2\mathrm{\Delta }(0)\mathrm{/}{T}_c$ differs from the BCS ratio, since Coulomb interactions affect much more the superconducting critical temperature, $T_c$, than the superconducting order parameter, $\mathrm{\Delta }(T)$. We conclude that the presence of Coulomb interactions play an additional role in the analysis of experimental data, specially in narrow band systems.