Superconductor-to-Metal Quantum Phase Transition in Overdoped $La_{2-x}Sr_xCuO_4$

We investigate $T_c$ and magnetic penetration depth $\lambda (T)$ near the superconductor-metal quantum phase transition in overdoped $La_{2-x}Sr_{x}CuO_{4}$ films. Both $T_c$ and superfluid density $n_s$, $\propto \lambda^{-2}$, decrease with overdoping. They obey the scaling relation $T_c \propto [\lambda^{-2}(0)]^{\alpha}$ with $\alpha \approx 1/2$. We discuss this result in the frameworks of disordered d-wave superconductors and of scaling near quantum critical points. Our result, and the linear scaling ($\alpha \approx 1$) found for the more anisotropic $Tl_2Ba_2CuO_{6+\delta}$, can both be understood in terms of quantum critical scaling, with different dimensionalities for fluctuations.