A universal scaling relation in high-temperature superconductors

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).