Systematic variation of magnetic-field penetration depth in high-Tcsuperconductors studied by muon-spin relaxation

The muon-spin relaxation rate $\sigma$ has been measured in the high-$T_c$ superconductors $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_x$ for $x=6.66, 6.95, 7.0$, and ${\mathrm{La}}_{1.85}$ ${\mathrm{Sr}}_{0.15}$Cu${\mathrm{O}}_4$ in transverse external magnetic fields ∼1-4 kG. We find a simple relation which connects the transition temperature $T_c$, the magnetic-field penetration depth ${\lambda }_L$, the carrier concentration $n_s$, and the effective mass $m^{*}$ as $T_c\propto \sigma \propto \frac{1}{{\lambda }_L^2}\propto \frac{n_s}{m^{*}}$. The linear dependence $T_c\propto \frac{n_s}{m^{*}}$ suggests a high-energy scale for the coupling between superconducting carriers.