Scaling of the superfluid density in high-temperature superconductors

A scaling relation Nc 4.4 dcTc has been observed in the copper-oxide superconductors, where Nc is the spectral weight associated with the formation of the superconducting condensate s=8Nc, Tc is the critical temperature, and dc 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 2 and a normal-state scattering rate 1 / ; in the clean limit 1 / 2 , and in the dirty limit 1 / 2 . The dirty limit may also be defined operationally as the regime where s varies with 1/ . It is shown that the scaling relation Nc or s dcTc, 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 0, if 1 / 2 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 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.