Scaling of the superfluid density in high-temperature superconductors

Abstract

A scaling relation ${N}_{c}\ensuremath{\simeq}4.4{\ensuremath{\sigma}}_{\mathit{dc}}{T}_{c}$ has been observed in the copper-oxide superconductors, where ${N}_{c}$ is the spectral weight associated with the formation of the superconducting condensate ${\ensuremath{\rho}}_{s}=8{N}_{c}$, ${T}_{c}$ is the critical temperature, and ${\ensuremath{\sigma}}_{\mathit{dc}}$ 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 $2\ensuremath{\Delta}$ and a normal-state scattering rate $1∕\ensuremath{\tau}$; in the clean limit $1∕\ensuremath{\tau}⪡2\ensuremath{\Delta}$, and in the dirty limit $1∕\ensuremath{\tau}g2\ensuremath{\Delta}$. The dirty limit may also be defined operationally as the regime where ${\ensuremath{\rho}}_{s}$ varies with $1∕\ensuremath{\tau}$. It is shown that the scaling relation ${N}_{c}$ or ${\ensuremath{\rho}}_{s}\ensuremath{\propto}{\ensuremath{\sigma}}_{\mathit{dc}}{T}_{c}$, 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 ${\ensuremath{\Delta}}_{0}$, if $1∕\ensuremath{\tau}g2{\ensuremath{\Delta}}_{0}$ 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, as a result, that the energy scale associated with the formation of the condensate scales linearly with ${T}_{c}$. The $a\text{\ensuremath{-}}b$ 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.