2.3.3 Nonlinear and Nonlocal Effects in the Vortex State

Predicting the behaviour of $\lambda$ in the vortex state is far more complicated. This is because, compared to the Meissner state, the problem must be solved in two dimensions, rather than one. A second complication is that the magnitude of the order parameter is spatially inhomogeneous due to the presence of vortices. Very recently, some progress has been made in understanding the nature of the field dependence of $\lambda$ in the vortex state of a d_x²-y²-wave superconductor [40]. The nonlinear effects which were discussed in the previous section also affect the supercurrent response in the vortex state. The mechanism for these nonlinear effects is identical to that in the Meissner phase. In a d_x²-y²-wave superconductor, the supercurrent response is also nonlocal due to the nodes on the Fermi surface. Near the nodes $\xi \! \gg \! \lambda$,which was discussed earlier in connection with Eq. (2.37). There it was noted that these nonlocal effects will affect the temperature dependence of $\lambda$at very low T.

Nonlocal effects will not affect the measured field dependence of $\lambda$ in the Meissner state. This can be realized, for instance, by incorporating a ${\bf k}$-dependence into the London kernel $Q_L ({\bf k})$. The terms in Q_L (k) which contain k will affect the precise way the field decays into the superconductor. However, since k itself does not depend on the magnetic field, this will not create a field-dependence for $\lambda$.

The situation is quite different in the vortex state. Since the theory is nontrivial and the predicted behaviour of $\lambda (H)$ in Ref. [40] is given as a numerical result, the discussion here will be qualitative only. The authors of Ref. [40] introduce an appropriate ${\bf k}$-dependent kernel into the London model to account for the nonlocal effects arising at the nodes. The higher-order k terms are more important for large values of k. Since large k values correspond to small values of r in real space, the nonlocal effects are most important near the vortex cores. At fields just above H_c1 where the vortex cores are isolated from each other, the measured penetration depth is virtually unaffected. However, with increasing applied field, the vortex cores move closer together and the nonlocal regions overlap. This in turn leads to significant changes in the distribution of magnetic field between the vortices and a field dependence for the corresponding effective penetration depth. In particular, the effective $\lambda$ is found to increase with increasing magnetic field. At fields in which the vortices begin to interact, this field dependence is very strong. The strength of the field dependence weakens somewhat at higher magnetic fields. In the same study, the nonlinear corrections are found to have a small effect on the field dependence of $\lambda$.However, it should be noted that the calculations in Ref. [40] were performed assuming that the size of the vortex cores are small, which is not always the case.