Shallow core levels, or how to determine the doping and $T_c$ of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ and Bi${}_2$Sr${}_2$CuO${}_{6+\delta }$ without cooling

Determining the doping level in high-temperature cuprate superconductors is crucial for understanding the origin of superconductivity in these materials and for unlocking their full potential. However, accurately determining the doping level remains a significant challenge due to a complex interplay of factors and limitations in various measurement techniques. In particular, in Bi${}_2$Sr${}_2$CuO${}_{6+\delta }$ and Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$, where the mobile carriers are introduced by non-stoichiometric oxygen $\delta$, the determination has been extremely problematic. Here, we study the doping dependence of the electronic structure of these materials in angle-resolved photoemission and find that both the doping level, $p$, and the superconducting transition temeprature, $T_c$ can be precisely determined from the binding energy of the Bi $5d$ core-levels. The measurements can be performed at room temperature, enabling the determination of $p$ and $T_c$ without cooling the samples. This should be very helpful for further studies of these materials.