Effect of impurity scattering on the low temperature magnetic penetration depth of a nonlocal and nonlinear d-wave superconductor

The magnetic penetration depth of a quasi-two dimensional d-wave superconductor in the presence of nonlineary, nonlocality, and impurity effects is investigated by using Green’s function method. It is shown that a d-wave superconductor would inevitably avoid the violation of the Nernst theorem by creating a T² term in its penetration depth through a competition of nonlinear, nonlocal, and impurity effects and this system may be stable at low temperatures. I also show that in the impure sample at low temperatures, T < T^* ∝ γ the impurity effect determines the temperature dependence of the penetration depth, i.e., nonlocal and nonlinear effects are completely masked by impurities.