Crossover from superfluidity to superconductivity in a system with doping dependent attraction

Zero temperature crossover from superfluidity to superconductivity with carrier density increasing is studied for a two-dimensional system in the s-wave and d-wave pairing channels. It was assumed that the particle attraction correlation length depends on carrier density $n_f$ as $r_0\sim1/\sqrt{n_f}$. Such a dependence was found experimentally for the radius of magnetic correlations in $La_{2-x}Sr_xCuO_4$. The short range Coulomb repulsion was also taken into account. It is shown that the behaviour of the system with doping is fundamentally different from the case with $r_0(n_f)=const$. In particular, similarly to the d-wave case, the crossover in the s-channel takes place only if the coupling is larger of some minimal value, otherwise the Cooper pairing scenario takes place at any small carrier density. The relevance of the model to high-temperature superconductors is discussed.