Universal Correlations between<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:mfrac></mml:math>(Carrier Density over Effective Mass) in High-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>Cuprate Superconductors

The muon-spin-relaxation rate $\ensuremath{\sigma}$ 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 $\ensuremath{\lambda}$ and thus the superconducting carrier density ${n}_{s}$ divided by the effective mass ${m}^{*}(\ensuremath{\sigma}\ensuremath{\propto}\frac{1}{{\ensuremath{\lambda}}^{2}}\ensuremath{\propto}\frac{{n}_{s}}{{m}^{*}})$. A universal linear relation between ${T}_{c}$ and $\ensuremath{\sigma}(T\ensuremath{\rightarrow}0)\ensuremath{\propto}\frac{{n}_{s}}{{m}^{*}}$ 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.