Effect of interlayer single-particle hoppings on the superconducting transition temperature

It is well known that the superconducting transition temperature of high-$T_c$ cuprates depends on the number of CuO$_2$ planes in the unit cell. The multilayer structure implies the possibility of interlayer hopping. Under the assumption that the interlayer hopping can be specified by the parameter $t_{\bot}({\mathbf k}) = t_{\bot}({\cos(k_x)-\cos(k_y)})^2$, the quasiparticle excitation spectrum for the bilayer cuprate in the superconducting state has been determined in the framework of the $t-t'-t''-t_\bot-j^*$ model using the generalized mean-field approximation. It turns out that the interlayer hoppings does not create any additional mechanism of the cooper paring and does not lead to an increase in $T_c$. The splitting of the upper hubbard quasiparticle band attributed to the interlayer hoppings is manifested as two peaks in the doping dependence of the superconducting transition temperature at temperatures below the maximum $T_c$ value for a single-layer cuprate. It has been found that antiferromagnetic interlayer correlations suppress the interlayer splitting. This probably leads to the common doping dependence of $T_c$ for both single-layer and bilayer cuprates.