Hole concentrations, Hall number, and $T_c$ relationships in substituted $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$

One of the fundamental questions involving the high-temperature copper oxide containing superconductors is to understand the relationship between hole concentration and the superconducting transition temperature, $T_c$. We help resolve this question by a study of Nd-, La-, Ga-, Ca-, and Zn-substituted $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$ with controlled oxygen stoichiometry. The total hole concentration is determined by a chemical method and trends in the mobile hole concentration by Hall measurements. By considering $T_c$ to be a function of the mobile sheet holes only, there is a maximum in $T_c$ (92 K) at a sheet-hole concentration of about 0.20-0.25 per Cu site.