Scaling of the superfluid density in high-temperature superconductors

Abstract

A scaling relation ρ_s \simeq 35σ_{dc}T_c has been observed in the copper-oxide superconductors, where ρ_s is the strength of the superconducting condensate, T_c is the critical temperature, and σ_{dc} is the normal-state dc conductivity close to T_c. 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 2Δand a normal-state scattering rate 1/τ; in the clean limit 1/τ\ll 2Δ, and in the dirty limit 1/τ> 2Δ. The dirty limit may also be defined operationally as the regime where ρ_s varies with 1/τ. It is shown that the scaling relation ρ_s \propto σ_{dc}T_c 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 Δ_0, if 1/τ> 2Δ_0 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 that as a result the energy scale associated with the formation of the condensate is scaling linearly with T_c. 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 effects may be viewed as equivalent. This raises the possibility that electronic inhomogeneities in the copper-oxygen planes may play an important role in the nature of the superconductivity in the copper-oxide materials.