Scaling of the superfluid density in high-temperature superconductors

A scaling relation Nc ≃ 4.4σdcTc has been observed parallel and perpendicular to the copper-oxygen planes in the high-temperature superconductors; Nc is the spectral weight and σdc is the dc conductivity just above the critical temperature Tc. In addition, Nb and Pb also fall close to this scaling line. The application of the Ferrell-Glover-Tinkham sum rule to the BCS optical properties of Nb above and below Tc yields Nc ≃ 8.1σdcTc when the normal-state scattering rate is much greater than the superconducting energy gap (1/τ > 2Δ, the “dirty” limit). This result suggests that the high-temperature superconductors may be in the dirty limit. The superconductivity perpendicular to the planes is explained by the Josephson effect, which again yields Nc ≃ 8.1σdcTc in the BCS formalism. The similar forms for the scaling relation in these two directions suggests that in some regime the dirty limit and the Josephson effect may be viewed as equivalent.