Scaling of the superfluid density in severely underdoped \boldmath YBa$_2$Cu$_3$O$_{6+y}$\unboldmath

Microwave impedance techniques have been used to determine the critical temperature ($T_c$), the in-plane superfluid density ($\rho_{s0}$), and the dc conductivity ($\sigma_{dc}$) just above $T_c$ in a highly-underdoped sample of YBa$_2$Cu$_3$O$_{6+y}$ for $y\simeq 0.333$. In this state the sample may be annealed to yield different levels of chain oxygen order and electronic doping in the copper-oxygen planes, resulting in a range of $T_c \simeq 2 - 17$~K.\footnote{D. M. Broun {\it et al.}, Phys. Rev. Lett. {\bf 99}, 237003 (2007).} The linear relation between $\rho_{s0}$ and $T_c$ is not observed, instead $\rho_{s0}\propto T_c^2$. However, the results do follow the more general scaling relation $\rho_{s0}/8 \simeq 4.4\,\sigma_{dc}T_c$,\footnote{C. C. Homes {\it et al.}, Phys. Rev. B {\bf 72}, 134517 (2005).} extending the validity of this relation for the in-plane data by an order of magnitude. In addition, these new results now provide a region of overlap between the scaling observed in the copper-oxygen planes, and perpendicular to the planes along the poorly-conducting {\it c} axis.