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 CuO₂ 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 the superconducting transition temperature (T_c) varies systematically with P_3D as a superconducting 'dome' with a universal optimal hole-doping concentration of P_3D^opt = 1.6 × 10²¹ cm^-3 for single-layer high-temperature superconductors. We suggest that P_3D^opt determines the upper bound of the electronic energy of underdoped single-layer high-T_c cuprates.