Thermal and Quantum critical Properties of overdoped La_{2-x}Sr_{x}CuO_{4}

We analyze the extended superfluid density data of Bo\v{z}ovi'{c} \textit{ et al}. taken on homogenous thin films to explore the critical properties of the thermal (TSM) and quantum superconductor to metal (QSM) transitions in overdoped La${}_{2-x}$Sr${}_x$CuO${}_4$. The temperature dependence reveals remarkable agreement with d-wave BCS behavior, where $\sigma \propto 1/\lambda ^{2}$. No sign of the expected KTB transition is observable. We show that the critical amplitude ${\rho }_{s0}={\rho }_s\left(T\right)\mathrm{/}(1-T\mathrm{/}{T}_c)$ scales as $T_c\propto {\rho }_{s0}^{1\mathrm{/}2}$ and with that as $T_{c}\propto \rho_{s}\left( 0\right) $, empirically verified by Bo\v{z}ovi'{c} \textit{et al}. Together with additional evidence for BCS behavior in overdoped La${}_{2-x}$Sr${}_x$CuO${}_4$ we show that the scaling relation $T_{c}\propto \lambda \left( 0\right) ^{-1}\propto \Delta \left( 0\right) \propto \xi \left( 0\right) ^{-1}\propto H_{c2}\left( 0\right) ^{-1/2}\propto \lambda_{0}^{-1}\propto \Delta_{0}\propto \xi_{0}^{-1}\propto H_{c20}^{-1/2}$ applies, by approaching the QSM and TSM transitions. This differs from the dirty limit behavior $\lambda \left( 0\right) \propto \xi \left( 0\right) ^{1/2}$,maintainsthe$T_{c}$independenceoftheGinzburg-Landauparameter$\kappa =\lambda_{o}/\xi_{0}$, and puts a stringent constraint on the effect at work. We notice that potential candidates appear to be the nonlocal effects on the penetration depth of clean d-wave superconductors explored by Kosztin and Legget.