Intrasublattice Hopping in the Extended $\mathit{t}$- $\mathit{J}$ Model and ${\mathit{T}}_{\mathit{c}}^{\max }$ in the Cuprates

It is shown that BCS-like scenarios for high- $T_c$ superconductivity based on the extended $t$- $J$ model yield a characteristic prediction for $T_c^{\max }$, the critical temperature at optimum doping. A pivotal role is played by $t_{-}\equiv t^{\prime }{-2t}^{\prime \prime }$ ( $t^{\prime }$ is the next-nearest-neighbor hopping, $t^{\prime \prime }$ is the third-neighbor hopping). Because $t_{-}$ determines the shape of the Fermi surface and the nature of saddle points, it strongly affects $T_c^{\max }$. Owing to symmetry, structural differences outside the ${\mathrm{CuO}}_2$ planes are described exclusively by $t_{-}$. This explains semiquantitatively the actual variation of $T_c^{\max }$ among the various cuprates, with estimated parameters that are consistent with the observed Fermi surfaces.