Doping Dependence of the Magnetic Penetration Depth and Coherence Length in YBa$_{2}$Cu$_{3}$O$_{y}$

While linear scaling of $T_{c}$ with superfluid density has been inferred from early muon depolarization rate ($\sigma )$ measurements on high-temperature superconductors, this interpretation of $\mu $SR data assumes an invalid relation between $\sigma $ and the in-plane magnetic penetration depth $\lambda _{ab}$. From detailed measurements and analysis of the $\mu $SR line shapes in YBa$_{2}$Cu$_{3}$O$_{y}$ single crystals, we have accurately determined the doping dependence of $\lambda _{ab}$ and the superconducting coherence length $\xi _{ab}$. We find that $T_{c}$ exhibits a sublinear dependence on 1/$\lambda _{ab}^{2}$, and that $\xi _{ab}$ increases with decreasing hole doping. In addition, the $\mu $SR line shapes for $y$ = 6.50 are found to be consistent with $y$ = 0 antiferromagnetism in and around the vortex cores, compatible with static stripes. We find that the antiferromagnetic order occurs only where superconductivity is suppressed, indicating that the ordering of copper spins competes with high-temperature superconductivity.