Two orbital anaysis on the correlation between $T_c$ and the Fermi surface shape in the cuprate superconductors

Correlation between the Fermi surface shape and $T_c$ in the cuprates has been an issue of great interest. Experimentally, materials with more warped Fermi surfaces tend to have higher $T_c$. In our recent studies(PRL 105, 057003(2010)), we have given an explanation to this by considering a two-orbital model that explicitly takes account of the $d_{z^2}$ orbital on top of the $d_{x^2-y^2}$ orbital. Namely, when the $d_{z^2}$ orbital component mixes on the Fermi surface, $d$-wave pairing is degraded, while the Fermi surface becomes better nested. In our previous study, however, we had only one example of actual materials in which the $d_{z^2}$ mixture is strong, i.e., La214. In order to show that $T_c$ is indeed systematically correlated with the $d_{z^2}$ mixture, we investigate further examples, namely, Pb$_2$Sr$_2$Cu$_2$O$_6$, Pb$_2$Sr$_2$YCu$_3$O$_8$ and La$_2$CaCu$_2$O$_6$, which have relatively low $T_c$ and Fermi surfaces that are not strongly warped. Applying the fluctuation exchange approximation to the two-orbital model obtained for these materials, we show that the $d_{z^2}$ mixture does indeed reduce $T_c$. Present result endorses our conclusion that the $d_{z^2}$ orbital mixture is an important key factor for the material dependence of $T_c$ in the cuprates.