Two-orbital analysis on the material dependence of $T_c$ in the single-layered cuprates

The significant material dependence of $T_c$ in the cuprates remains an important puzzle, even within the single-layer family. A recent paper[1] has demonstrated, with a two-orbital model, that, while the usual wisdom is to considered the cuprate as a one-band ($d_{x^2-y^2}$) system, a hybridization with the second ($d_{z^2}$) one around the Fermi energy significantly affects $T_c$ in the spin-fluctuation mediated pairing. There, the energy offset ($\Delta E$) between the two orbitals has been shown to govern the extent of the $d_{z^2}$ mixture, hence $T_c$. Here we further extend this line of approach to identify the key factors that determine $\Delta E$ in the cuprates, focusing on the structural difference among broader (La, Hg, Bi, and Tl) single-layer cuprates. We have revealed that the apical oxygen height ($h_{\rm{O}}$) above the CuO$_2$ plane and the separation ($d$) between the CuO$_2$ planes are the important parameters that determine $\Delta E$, thereby causing the material dependence of $T_c$. This picture enables us to capture the $T_c$ variation among the single-layered cuprates in a simple lattice-parameter space. [1]H. Sakakibara {\it et al.} Phys. Rev. Lett. {\bf 105}, 057003 (2010)