Nonlocal Effects on the Magnetic Penetration Depth ind-Wave Superconductors

We show that, under certain conditions, the low temperature behavior of the magnetic penetration depth $\lambda \left(T\right)$ of a pure $d$-wave superconductor is determined by nonlocal electrodynamics and, contrary to the general belief, the deviation $\Delta \lambda \left(T\right)=\lambda \left(T\right)-\lambda \left(0\right)$ is proportional to $T^2$ and not $T$. We predict that the $\Delta \lambda \left(T\right)\propto T^2$ dependence, due to nonlocality, should be observable experimentally in nominally clean high- $T_c$ superconductors below a crossover temperature $T^{*}=\left({\xi }_0/{\lambda }_0\right){\Delta }_0\sim 1$ K. Possible complications due to impurities, surface quality, and crystal axes orientation are discussed.