Since the discovery of superconductivity at elevated temperatures in the copper oxide materials 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 relation, and it works reasonably well for the underdoped materials. It does not, however, describe optimally doped (where Tc is a maximum) or overdoped materials. Similarly, an attempt to scale the superfluid density with the d.c. conductivity (σdc) was only partially successful. 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 disorder, and direction (parallel or perpendicular to the copper-oxygen planes).
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Acknowledgements 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 HgBa2CuO4þd 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.
Acknowledgements This work was supported by NASA, the UK Particle Physics Research Council (PPARC) and NSF Hungary. The Swedish Solar Telescope is operated on the island of La Palma by the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. R.E. thanks M. Kéray for encouragement. We thank C.J. Schrijver, T. Tarbell, M. DeRosa and A. Title for discussions, and M. Carlsson for pointing out the importance of 3 min oscillations.