Planar hole-doping concentration and effective three-dimensional hole-doping concentration for single-layer high-T_c superconductors

We propose that physical properties for the high temperature superconductors can be addressed by either a two-dimensional planar hole-doping concentration ($P_{pl}$) or an effective three-dimentional hole-doping concentration ($P_{3D}$). We find that superconducting transition temperature ($T_c$) exhibits a universal dome-shaped behavior in the $T_c$ $vs.$ $P_{3D}$ plot with a universal optimal doping concentration at $P_{3D}$ $\sim$ 1.6 $\times$ 10$^{21}$ cm$^{-3}$ for the single-layer high temperature superconductors.