Correlation between Superfluid Density and T_C of Underdoped YBa₂Cu₃O_6+x Near the Superconductor-Insulator Transition

We report measurements of the ab-plane superfluid density n_s (magnetic penetration depth λ) of heavily underdoped films of YBa₂Cu₃O_6+x, with T_C’s from 6 to 50 K. We find the characteristic length for vortex unbinding transition equal to the film thickness, suggesting strongly coupled CuO₂ layers. At the lowest dopings, T_C is as much as 5 times larger than the upper limit set by the 2D Kosterlitz-Thouless-Berezinskii transition temperature calculated for individual CuO₂ bilayers. Our main finding is that T_C is not proportional to n_s(0); instead, we find T_C∝n_s^1/2.3±0.4. This conflicts with a popular point of view that quasi-2D thermal phase fluctuations determine the transition temperature.