Universal optimal hole-doping concentration in single-layer high-temperature cuprate superconductors

We argue that in cuprate physics there are two types, hole content per CuO${}_2$ plane ($P_{pl}$) and the corresponding hole content per unit volume ($P_{3D}$), of hole-doping concentrations for addressing physical properties that are two-dimensional (2D) and three-dimensional (3D) in nature, respectively. We find that superconducting transition temperature ($T_c$) varies systematically with $P_{3D}$ as a superconducting \textquotedblleft $dome$\textquotedblright with a universal optimal hole-doping concentration $P_{3D}^{opt\mathrm{.}}$ = 1.6 $\times$ 10${}^{21}$ cm${}^{-3}$ for single-layer high temperature superconductors. We suggest that $P_{3D}^{opt\mathrm{.}}$ determines the upper bound of the electronic energy of underdoped single-layer high-$T_c$ cuprates.