Apical oxygen ions and the electronic structure of the high-Tccuprates

We analyze a five-band extended Hubbard model involving four orbitals on the ${\mathrm{CuO}}_2$ planes of the high-temperature superconducting oxides (${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ and ${\mathit{d}}_{3\mathit{z}}^2$-${\mathit{r}}^2$ for the copper, and 2${\mathit{p}}_{\mathit{x}}$ and 2${\mathit{p}}_{\mathit{y}}$ for the oxygen) and the 2${\mathit{p}}_{\mathit{z}}$ orbital(s) for the out-of-plane apical oxygen ion(s). The strong local repulsion between holes on copper is treated by means of a slave-boson approach in mean-field approximation, whereas the nearest-neighbor Cu-O Coulombic repulsion is treated within a Hartree decoupling scheme. We systematically investigate the variation of the resulting band structure with doping and with varying model parameters, and examine its stability with respect to lattice deformations. The results are compared with experimental data on photoemission, polarized x-ray-absorption spectroscopy, electron-energy-loss spectroscopy, and optical absorption. We analyze in particular the effects of the apical oxygen(s) on the electronic structure, and we identify the amount ${\mathit{n}}_{\mathit{a}1}$ of holes in states with local ${\mathit{a}}_1$ symmetry as the quantity most directly affected by their presence. This quantity differentiates between the various high-${\mathit{T}}_{\mathit{c}}$ cuprates, which are otherwise very similar as far as the planar structure is concerned. An analysis of ten different classes of compounds reveals a correlation between the maximum critical temperature ${\mathit{T}}_{\mathit{c}}^{\max }$ attained within each class at the optimum doping ${\mathrm{\delta }}^{\max }$ and the excess of ${\mathit{a}}_1$ holes with respect to the doping itself ${\mathit{n}}_{\mathit{a}1}$(${\mathrm{\delta }}^{\max }$)-${\mathrm{\delta }}^{\max }$. This correlation indicates that the highest critical temperature can be reached in the compounds where the interaction between apical oxygen(s) and ${\mathrm{CuO}}_2$ plane is weakest. We discuss some implications of our results in the light of various theoretical models.