Beyond Homes scaling: disorder, the Planckian bound and a new universality

Abstract

Beginning with high-$T_c$ cuprate materials, it has been observed that many superconductors exhibit so-called "Homes scaling", in which the zero-temperature superfluid density, ${\rho }_{s0}$, is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature, ${\sigma }_{\mathrm{dc}}{T}_c$. 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 ${\rho }_{s0}\propto {\left({\sigma }_{\mathrm{dc}}{T}_c\right)}^2$ in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal--Eliashberg treatment of inelastic scattering, we show how 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 YBCO and overdoped LSCO, and in very clean materials with little inelastic scattering, such as Sr${}_2$RuO${}_4$. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight, ${\omega }_{p,D}^2$, and show that 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.

15 pages, 5 figures

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