Scaling of the superfluid density in high-temperature superconductors

A scaling relation Nc or ρs ασdcΤc has been observed in the copper-oxide superconductors, where ρs is the superconducting condensate and the spectral weight is Nc=ρs/8, Tc is the critical temperature, and σdc is the normal-state dc conductivity close to Tc. This scaling relation is examined within the context of clean and dirty-limit BCS superconductors. It is shown that the scaling relation Nc ≃ 4.4σdcTc, which follows directly from the Ferrell-Glover-Tinkham sum rule, is the hallmark of a BCS system in the dirty-limit. The scaling relation implies that the copper-oxide superconductors are likely to be in the dirty limit, and that as a result the energy scale associated with the formation of the condensate scales linearly with Tc. The a-b planes and the c axis also follow the same scaling relation. It is observed that the scaling behavior for the dirty limit and the Josephson effect (assuming a BCS formalism) are essentially identical, suggesting that in some regime these two pictures may be viewed as equivalent.