Origin of the material dependence of Tc in the single-layered cuprates

In order to understand the material dependence of Tc within the single-layered cuprates, we study a two-orbital model that considers both dx2-y2 and dz2 orbitals. We reveal that a hybridization of dz2 on the Fermi surface substantially affects Tc in the cuprates, where the energy difference ΔE between the dx2-y2 and the dz2 orbitals is identified to be the key parameter that governs both the hybridization and the shape of the Fermi surface. A smaller ΔE tends to suppress Tc through a larger hybridization, whose effect supersedes the effect of diamond-shaped (better-nested) Fermi surface. The mechanism of the suppression of d-wave superconductivity due to dz2 orbital mixture is clarified from the viewpoint of the ingredients involved in the Eliashberg equation, that is, the Green's functions and the form of the pairing interaction described in the orbital representation. The conclusion remains qualitatively the same if we take a three-orbital model that incorporates the Cu 4s orbital explicitly, where the 4s orbital is shown to have an important effect of making the Fermi surface rounded. We have then identified the origin of the material and lattice-structure dependence of ΔE, which is shown to be determined by the energy difference ΔEd between the two Cu 3d orbitals (primarily governed by the apical oxygen height) and the energy difference ΔEp between the in-plane and apical oxygens (primarily governed by the interlayer separation d).