Scaling of the superfluid density in high-temperature superconductors

A scaling relation Nc4.4σdcTc has been observed in the copper-oxide superconductors, where Nc is the spectral weight associated with the formation of the superconducting condensate ρₛ=8Nc, 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 a clean and dirty-limit BCS superconductor. These limits are well established for an isotropic BCS gap 2Δ and a normal-state scattering rate 1∕τ; in the clean limit 1∕τ⪡2Δ, and in the dirty limit 1∕τ>2Δ. The dirty limit may also be defined operationally as the regime where ρₛ varies with 1∕τ. It is shown that the scaling relation Nc or ρₛ∝σdcTc, which follows directly from the Ferell-Glover-Tinkham sum rule, is the hallmark of a BCS system in the dirty-limit. While the gap in the copper-oxide superconductors is considered to be d wave with nodes and a gap maximum Δ₀, if 1∕τ>2Δ₀ then the dirty-limit case is preserved. The scaling relation implies that the copper-oxide superconductors are likely to be in the dirty limit and, as a result, that 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.