Superconducting critical temperature in quasi-two-dimensional systems

We calculate the superconducting critical temperature for systems consisting of weakly coupled planes. We consider an attractive-U Hubbard model in a tetragonal structure. The anisotropy is characterized by the ratio between the hopping matrix elements within the planes (t) and between them (t’). We calculate the critical temperature ${\mathit{T}}_{\mathit{c}}$ for the onset of off-diagonal long-range order by calculating fluctuations around the mean-field solution. For independent planes (t’=0) ${\mathit{T}}_{\mathit{c}}$ is zero and each plane has a Kosterlitz-Thouless transition at a temperature ${\mathit{T}}_{\mathrm{KT}}$. As the interplane coupling increases ${\mathit{T}}_{\mathit{c}}$ increases very rapidly and for t’/t≃0.03 we obtain ${\mathit{T}}_{\mathit{c}}$≃${\mathit{T}}_{\mathrm{KT}}$. For larger values of t’ the superconducting critical temperature increases almost linearly with the interplane coupling.