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 $\rho_{s0}=\rho_{s}\left( T\right) /\left( 1-T/T_{c}\right)$ scales as $T_{c}\propto \rho_{s0}^{1/2}$ 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 $\lambda \left( 0\right) \propto \xi \left( 0\right) ^{1/2}$, maintains the $T_{c}$ \ independence of the Ginzburg-Landau parameter $\kappa =\lambda_{o}/\xi_{0}$, 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.