Beyond Homes Scaling: Disorder, the Planckian Bound, and a New Universality

Beginning with high-Tc cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density ρs0 is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature σdcTc. For conventional, s-wave superconductors, such scaling has been shown to be a natural consequence of elastic-scattering disorder, not only in the extreme dirty limit, but across a broad range of scattering parameters. Here we show that when an analogous calculation is carried out for elastic scattering in d-wave superconductors, a stark contrast emerges, with ρs0∝(σdcTc)2 in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal-Eliashberg treatment of inelastic scattering, we show how the observed Homes scaling is recovered. The normal-state behavior of near-optimally-doped cuprates is dominated by inelastic scattering, but significant deviations from Homes scaling occur for disorder-dominated cuprate systems, such as underdoped YBa2Cu3O6.333 and overdoped La2−xSrxCuO4, and in very clean materials with little inelastic scattering, such as Sr2RuO4. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight ωp,D2 and show that a new universal scaling emerges, in which the superfluid fractions of dirty s-wave and dirty d-wave superconductors coalesce to a single point at which normal-state scattering is occurring at the Planckian bound. The combined result is a new tool for classifying superconductors in terms of order parameter symmetry, as well as scattering strength and character. Although our model starts from a Fermi-liquid assumption, it describes underdoped cuprates surprisingly well.