Charge transfer and oxygen ordering in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$

The electronic structure of the ${\mathrm{CuO}}_{\mathit{x}}$ planes is studied using a generalized Hubbard model including Cu-O repulsion ${\mathit{U}}_{\mathit{p}\mathit{d}}$, for each value of x and two different assumptions on the oxygen (O) ordering. The result explains qualitatively the experimentally observed hole count in the ${\mathrm{CuO}}_2$ planes, the amount of ${\mathrm{Cu}}^{+}$ and the metal-insulator transition near x=0.5. For large enough ${\mathit{U}}_{\mathit{p}\mathit{d}}$, the energy ΔE favoring ordering in chains is positive. A simple explanation of this and the relation between charge transfer and O ordering is given. The screening length λ is calculated using Thomas-Fermi theory, an effective one-band model for the ${\mathrm{CuO}}_2$ planes and experimental data. This information is used to construct an effective lattice-gas model for the O ordering, based on O-O screened repulsions in which ΔE is the only parameter. The superstructures predicted by this model provide an explanation of almost all observed diffraction patterns and of recently observed photoinduced changes in the transport properties. The electronic and structural results are consistent with the observed dependence of the superconducting ${\mathit{T}}_{\mathit{c}}$ vs x.