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

TL;DR

This work analyzes Homes scaling through a unified treatment of elastic and inelastic scattering in superconductors. It shows that elastic scattering in dirty $d$-wave superconductors yields $\rho_{s0} \propto (\sigma_{\mathrm{dc}} T_c)^2$, in contrast to the linear Homes law for dirty $s$-wave and optimally doped materials; incorporating inelastic scattering via Migdal–Eliashberg theory restores Homes scaling and reveals a Planckian regime. By normalizing both axes of the Homes plot with the Drude weight $\omega_{p,D}^2$, the authors uncover a universal four-regime scaling (Planckian, superclean, dirty $s$-wave, dirty $d$-wave) with a central Planckian point, supported by data across 45 materials. The revised superfluid scaling provides a practical framework for classifying superconductors by order-parameter symmetry and scattering characteristics and connects observed universality to Planckian physics within a largely Fermi-liquid description.

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, $ρ_{s0}$, is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature, $σ_\mathrm{dc} T_c$. 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 $ρ_{s0} \propto \left(σ_\mathrm{dc} T_c \right)^2$ 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, $ω_{p,D}^2$, 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.

Figures (5)

• Figure 1: Homes scaling (gray line) shown alongside theory curves for pointlike elastic scattering disorder, revealing different dirty-limit power-laws for $s$-wave and $d$-wave superconductors. Elastic scattering curves have been plotted for two choices of Drude plasma frequency: $\omega_{p,D} = 10^4$ cm$^{-1}$ (solid lines) characteristic of typical cuprates; and $\omega_{p,D} = 7 \times 10^4$ cm$^{-1}$ (dashed lines) characteristic of an elemental metal such as Nb. In the $d$-wave case, results depend on the scattering phase shift of the impurities and are bounded by the Born and unitarity limits. Also plotted is a selection of data for various superconductors, used throughout the paper and summarized in Table \ref{['tab:materials']}.
• Figure 2: The revised superfluid scaling plot factors out the material-specific Drude weight, $\omega_{p,D}^2$, to reveal the underlying universal behavior, expressed in terms of the dimensionless quantities $f_s \equiv \omega_{p,s}^2/\omega_{p,D}^2$ and $k_B T_c/\hbar \Gamma_\mathrm{tr}$. Rescaled data from Fig. \ref{['fig:standard_Homes_scaling']} are plotted, along with the $s$-wave and $d$-wave curves for elastic scattering disorder. The rescaled Homes line is shown for reference.
• Figure 3: Revised superfluid scaling, including the effects of inelastic scattering and disorder. For the $d$-wave case, three regimes of inelastic scattering are shown: strong coupling ($\Omega_B = 3.5 T_{c0}$); intermediate coupling ($\Omega_B = 8 T_{c0}$); and weak coupling (BCS limit). The $d$-wave behavior is delimited in each case by the Born and unitarity limits for point scatterers. In the $s$-wave case, only strong coupling and BCS limits are shown. The near optimally cuprates and the heavy fermion superconductor CeCoIn$_5$ are dominated by inelastic scattering and cluster tightly in the Planckian regime. Radiating out along one line are the dirty $s$-wave superconductors. Dirty $d$-wave superconductors follow a different scaling line, as typified by overdoped La$_{2-x}$Sr$_x$CuO$_4$ and underdoped YBa$_2$Cu$_3$O$_{6.333}$. Superclean Sr$_2$RuO$_4$ exists in a class of its own.
• Figure 4: Model normal-state spectrum containing Drude and Lorentz components, showing real and imaginary parts of the normal-state conductivity. (a) Drude--Lorentz fits to $\sigma_1(\nu)$ can be used to isolate the Drude contribution to the integrated weight, shown shaded in blue. (b) The Drude component dominates $\sigma_2(\nu)$ at low frequencies, making complex conductivity, measured by techniques such as THz-TDS, a powerful and unambiguous means of determining the Drude component of the spectrum, and therefore $\omega_{p,D}^2$.
• Figure 5: Optical conductivity spectra, $\sigma_1(\nu)$, for various superconductors, illustrating the formation of the zero-temperature superfluid from the normal-state Drude weight. Each plot contains a normal-state spectrum ($T > T_c$) and a superconducting spectrum in the $T \to 0$ limit, with the shaded area in between indicating the spectral weight that condenses to form the zero-temperature superfluid density. (a) The optical conductivity of optimally doped YBa$_2$Cu$_3$O$_{7-\delta}$, based on optical data from Ref. Rotter1991 and dc conductivity from Ref. Ando2004, showing Planckian-limit inelastic scattering ($\Gamma_\mathrm{tr} \approx 2 T_c$) in the normal state. (b) and (c) Optical conductivity of a dirty $s$-wave superconductor, for normal-state elastic scattering rates $\Gamma_\mathrm{tr} = 2 T_c$ and $9 T_c$, based on the Zimmermann's calculation Zimmermann1991. (d) to (f) Optical conductivity of a dirty $d$-wave superconductor, calculated for normal-state elastic scattering rates $\Gamma_\mathrm{tr} = 2 T_c$, $9 T_c$ and $20 T_c$, using an impurity model and formalism taken from ab initio calculations for overdoped La$_{2-x}$Sr$_x$CuO$_4$Ozdemir:2022Broun.2024.