Low-Temperature Magnetic Penetration Depth in d-Wave Superconductors:   Zero-Energy Bound State and Impurity Effects
  We report a theoretical study on the deviations of the Meissner penetration depth $\lambda(T)$ from its London value in d-wave superconductors at low temperatures. The difference arises from low-energy surface Andreev bound states. The temperature dependent penetration depth is shown to go through a minimum at the temperature $T_{m0}\sim \sqrt{\xi_0/\lambda_0}T_c$ if the broadening of the bound states is small. The minimum will straighten out when the broadening reaches $T_{m0}$. The impurity scattering sets up the low-temperature anomalies of the penetration depth and destroys them when the mean free path is not sufficiently large. A phase transition to a state with spontaneous surface supercurrent is investigated and its critical temperature determined in the absence of a subdominant channel activated at low temperatures near the surface. Nonlinear corrections from Andreev low-energy bound states to the penetration length are obtained and shown, on account of their broadening, to be small in the Meissner state of strong type II superconductors. 
Yu. S. Barash, M. S. Kalenkov, and J. Kurkijarvi