Scaling relation for the superfluid density in cuprate superconductors

Abstract

A universal scaling relation, $ρ_s \propto σ(T_c)\times T_c$ has been reported by Homes $et$ $al$. (Nature (London) {\bf 430}, 539 (2004)) where $ρ_s$ is the superfluid density and $σ(T)$ is the DC conductivity. The relation was shown to apply to both c-axis and in-plane dynamics for high-$T_c$ superconductors as well as to the more conventional superconductors Nb and Pb, suggesting common physics in these systems. We show quantitatively that the scaling behavior has several possible origins including, marginal Fermi-liquid behavior, Josephson coupling, dirty-limit superconductivity and unitary impurity scattering for a d-wave order parameter. However, the relation breaks down seriously in overdoped cuprates, and possibly even at lower doping.