Band-Structure Trend in Hole-Doped Cuprates and Correlation with ${\mathit{T}}_{\mathit{c}\max }$

By calculation and analysis of the bare conduction bands in a large number of hole-doped high-temperature superconductors, we have identified the range of the intralayer hopping as the essential, material-dependent parameter. It is controlled by the energy of the axial orbital, a hybrid between $\mathrm{Cu}4s$, apical-oxygen ${2p}_z$, and farther orbitals. Materials with higher $T_{c\max }$ have larger hopping ranges and axial orbitals more localized in the ${\mathrm{CuO}}_2$ layers.