Figure 1

$ab$-plane surface impedance at 2.64 GHz for the ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$ ellipsoid, at two dopings. $R_s(T)$ is measured directly in the experiment. Absolute reactance is obtained by offsetting $\Delta X_s(T)$ so that $R_s$ and $X_s$ match in the normal state. Inset: a fluctuation peak in ${\sigma }_1(T)$ at one doping. $\Delta T_c$ is set to the difference between inflection points in ${\sigma }_1(T)$ on opposite sides of the transition.Reuse & Permissions

Figure 2

$ab$-plane superfluid density ${\rho }_s(T)=1/{\lambda }^2(T)$ shown at 20 of the 37 dopings measured in this study. Measurements were made starting in the most ordered state ($T_c\approx 17\mathrm{K}$) followed by controlled oxygen annealing in small steps down to $T_c\approx 3\mathrm{K}$. At the end of the experiment the sample was reordered and measured again to verify reproducibility. Lines mark where the vortex-unbinding transition should occur for a 2D superconductor. The dashed line corresponds to ${\rho }_s^{2\mathrm{D}}\equiv {\hslash }^{2}d/4k_B{e}^2{\mu }_0{\lambda }^2=(2/\pi )T$ in each ${\mathrm{CuO}}_2$ plane. The solid line shows ${\rho }_s^{2\mathrm{D}}=(2/\pi )T$ in each ${\mathrm{CuO}}_2$ bilayer. ${\rho }_s(T)$ instead passes smoothly through this region. While mean-field-like over most of the doping range, ${\rho }_s(T)$ develops downwards curvature near $T_c$ at the highest dopings, a possible indication of the onset of the 3D-$XY$ critical fluctuations.Reuse & Permissions

Figure 3

Our ${\rho }_{s0}(p)$ data (solid diamonds, right-hand scale) and $T_c(p)$ data from Ref. 13 (solid circles, left-hand scale). ${\rho }_{s0}\equiv 1/{\lambda }^2(T\to 0)$ is obtained by linear extrapolation to $T=0$. Hole doping for the superfluid density data is determined from our values of $T_c$, using a linear fit and extrapolation of the $T_c(p)$ data from Ref. 13 (solid line). To the extent that this extrapolation is valid, ${\rho }_{s0}(p)\sim (p-p_c{)}^2$ at the onset of superconductivity, before crossing over to a linear doping dependence. The dashed line is a linear fit to ${\rho }_{s0}(p)$ at higher doping.Reuse & Permissions

Figure 4

The results of assuming a linear mapping between doping and ${\rho }_{s0}$ (solid diamonds). $T_c$ (open squares) varies as $(p-p_c{)}^{1/2}$ as $p\to p_c$, with the solid line a square-root $T_c(p)$ curve for comparison. $T_c(p)$ data from Ref. 13 (solid circles) are sparse in this range. Vertical bars on the $T_c$ data show transition widths estimated from rounding of ${\sigma }_1(T)$ fluctuation peaks. The corresponding doping spreads (horizontal bars) show little variation with doping.Reuse & Permissions

Figure 5

Fluctuation peaks in ${\sigma }_1(T)$ at 2.64 GHz as doping is varied. Peak width is approximately constant in the higher doping range and then broadens considerably below $T_c\approx 8\mathrm{K}$, consistent with a steepening of $T_c(p)$ in that range.Reuse & Permissions