Since the discovery of superconductivity at elevated temperatures in the copper oxide materials1 there has been a considerable effort to find universal trends and correlations amongst physical quantities, as a clue to the origin of the superconductivity. One of the earliest patterns that emerged was the linear scaling of the superfluid density (ρs) with the superconducting transition temperature (Tc), which marks the onset of phase coherence. This is referred to as the Uemura relation2, and it works reasonably well for the underdoped materials. It does not, however, describe optimally doped (where Tc is a maximum) or overdoped materials3. Similarly, an attempt to scale the superfluid density with the d.c. conductivity (σdc) was only partially successful4. Here we report a simple scaling relation (ρs∝σdcTc, with σdc measured at approximately Tc) that holds for all tested high-Tc materials. It holds regardless of doping level, nature of dopant (electrons versus holes), crystal structure and type of disorder5, and direction (parallel or perpendicular to the copper–oxygen planes).
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We thank A. Chubukov, P. D. Johnson, S. A. Kivelson, P. A. Lee, D. B. Tanner, J. J. Tu, Y. Uemura and T. Valla for discussions. Work in Canada was supported by the Natural Sciences and Engineering Research Council of Canada, and the Canadian Institute for Advanced Research. The HgBa2CuO4+δ crystal growth work at Stanford University was supported by the Department of Energy's Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Work at the University of California at San Diego was supported by the National Science Foundation and the Department of Energy. Work at Brookhaven was supported by the Department of Energy.
The authors declare that they have no competing financial interests.
Supplementary Table 1: The values for a variety of high-Tc cuprate superconductors for the critical temperature, dc conductivity close to the critical temperature, superfluid density and the penetration depth, in the a-b planes. Supplementary Table 2: The same quantities described in Supplementary Table 1, except along the c axis. (DOC 36 kb)
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Homes, C., Dordevic, S., Strongin, M. et al. A universal scaling relation in high-temperature superconductors. Nature 430, 539–541 (2004). https://doi.org/10.1038/nature02673
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