Superconducting state in an oxygen hole metal

We study within BCS theory the properties of an effective Hamiltonian to describe conduction by holes through an oxygen anion network. The Hamiltonian contains an on-site repulsive interaction $U_p$ and a modulated hopping interaction $\Delta t$ that yields a larger hopping amplitude between sites when other holes are present on those sites. The superconducting state is found to be $s$ wave with an energy-dependent gap. Superconductivity is restricted to low hole densities and the critical temperature increases with the hopping amplitude. The particular form of the interaction allows for superconductivity even in the presence of large Coulomb repulsion, up to $\frac{U_p}{\Delta t}\approx 30$. We discuss the behavior of the tunneling density of states, specific heat, gap ratio, and coherence length as a function of hole density and parameters in the Hamiltonian, and the relationship between our results and existing, as well as possible future, experimental results on high-$T_c$ oxides. Our model provides a natural explanation for the spread in gap values observed in different experiments, for the observed broadening of the resistive transition in a field, and for the observed superconducting glass behavior.