The influence of nonlocality of quasi-particle response to the nonlinear Meissner effect in d-wave superconductors

A nonlinear correction to the magnetic penetration depth into a (d_{x^2 - y^2 } )-wave superconductor is found that takes into account nonlocal effects. The calculation is carried out for the self-consistent distribution of the screening supercurrent. An analytical expression for the nonlinear correction is obtained in the limit of the pure superconductor for orientations that do not admit the formation of surface states. It is shown that the presence of unitary impurities makes the nonlinear Meissner effect unobservable in the nonlocal regime for almost all orientations of the crystal either due to the suppression of the nonlocal response of quasi-particles by impurities or due to a considerable contribution of Andreev low-energy surface states to the penetration depth. In the Born scattering limit, there exists a broad interval of orientations and mean free paths in which the non-locality of the response must be taken into account in the calculation of the nonlinear correction to the penetration depth at rather low temperatures.