Universal Correlations between $T_c$ and $\frac{n_s}{m^{*}}$ (Carrier Density over Effective Mass) in High-$T_c$ Cuprate Superconductors

The muon-spin-relaxation rate $\sigma$ has been measured in sixteen specimens of high-$T_c$ cuprate superconductors (the 2:1:4, 1:2:3, 2:2:1:2, and 2:2:2:3 series). This has allowed us to study the magnetic field penetration depth $\lambda$ and thus the superconducting carrier density $n_s$ divided by the effective mass $m^{*}\left(\sigma \propto \frac{1}{{\lambda }^2}\propto \frac{n_s}{m^{*}}\right)$. A universal linear relation between $T_c$ and $\sigma (T\to 0)\propto \frac{n_s}{m^{*}}$ has been found with increasing carrier doping. In heavily doped samples, however, $T_c$ shows saturation and suppression with increasing $\frac{n_s}{m^{*}}$. This saturation starts at different values of $\frac{n_s}{m^{*}}$ for materials with different multiplicities of CuO planes.