Superconductivity in an extended Hubbard model with attractive   interaction
  In this work, a two-dimensional one-band Hubbard model is investigated within a two-pole approximation. The model presents a non-local attractive potential $U (U<0)$ that allows the study of d-wave superconductivity and also includes hopping up to second-nearest-neighbors. The two-pole scheme has been proposed to improve the Hubbard-I approximation. The analytical results show a more complex form for the gap $\Delta(T)$, when compared to the one obtained in the latter approximation. Indeed, new anomalous correlation functions associated with the superconductivity are involved in the calculation of $\Delta(T)$. Numerical results in a range of temperatures are presented. Moreover, the structure of the quasiparticle bands and the topology of the Fermi surface are studied in detail in the normal state. Connections with some experimental results are also included. 
E. J. Calegari, S. G. Magalhaes, C. M. Chaves and A. Troper
cond-mat.str-el