Scaling of the superfluid density in high-temperature superconductors

A scaling relation N-c similar or equal to 4.4 sigma T-dc(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(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)proportional to sigma T-dc(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-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.