Theoretical analysis of the concentration and distribution of carriers in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$

An ionic model and the total-energy method are used to calculate the density and distribution of carriers in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ systems and, in particular, to understand the relationship between the density of carriers and the density of oxygen vacancies in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$. It is shown that the density of carriers is normally 1-x per unit cell and the carriers are holes on the oxygen ions in the ${\mathrm{CuO}}_2$ planes. In addition, for the 1+ valence copper ions ${\mathrm{Cu}}^{+}$ present in the CuO□ plane, the density of ${\mathrm{Cu}}^{+}$ is normally x per unit cell.