Scaling of the superfluid density in high-temperature superconductors

Abstract

A scaling relation $N_c\simeq 4.4{\sigma }_{\mathit{dc}}{T}_c$ has been observed in the copper-oxide superconductors, where $N_c$ is the spectral weight associated with the formation of the superconducting condensate ${\rho }_s=8N_c$, $T_c$ is the critical temperature, and ${\sigma }_{\mathit{dc}}$ 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 $2\Delta$ and a normal-state scattering rate $1∕\tau$; in the clean limit $1∕\tau ⪡2\Delta$, and in the dirty limit $1∕\tau >2\Delta$. The dirty limit may also be defined operationally as the regime where ${\rho }_s$ varies with $1∕\tau$. It is shown that the scaling relation $N_c$ or ${\rho }_s\propto {\sigma }_{\mathit{dc}}{T}_c$, 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 ${\Delta }_0$, if $1∕\tau >2{\Delta }_0$ 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 $a\text{−}b$ 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.