London model allows. The higher the applied field, the more strongly the order parameter is suppressed and the more flux is accumulated in the near-edge layer. On the other hand, when the vortex jumps inside, the screening is restored to a significant extent and, accordingly, there is less flux in the near-edge layer (Fig. ). The competition between the flux expelled from this layer and the vortex's flux determines the sign and amplitude of flux jumps. Despite the deceptive simplicity of this explanation, there is no simple way to explain why the vortex entry restores screening while the field at the edge remains the same. This is a nonlinear property of superconductors.We now turn to the question of why the flux jumps are not quantized, even when the surface barrier is suppressed by edge roughness, and why the distance between the curves with and without a vortex is less than f 0 (see Fig. ). The latter implies that the vortex's flux even in equilibrium (that is, not only the corresponding flux jumps) is considerably less than f 0 . This observation can be explained by the changes in the structure of near-edge vortices predicted in refs 1-4. Figure plots the measured amount of flux f carried by a vortex versus its distance from the disk's edge. We can see that all our data for different samples and temperatures fall on a single curve, if plotted in units of the effective penetration length, โฆ 2 8 % l eff = โฆ 2 8 % l 2 /h. There is also excellent agreement with the corresponding theoretical dependence. We note that, for a typical experimental situation, h and โฆ 2 8 % l are about 0.1 m, and it is very unlikely that a vortex can jump farther than 1 m from the edge before being stopped by pinning, even in samples with low pinning. According to Fig. , in such a case the flux carried by vortices is reduced to about 0.5f 0 . Only vortices located as far as 100 m away from the film edge have their flux quantized with an accuracy better than 1%.We have shown that there are two independent effects that lead to non-quantized penetration of magnetic field in type II superconductors. The first (theoretically established a long time ago, but never observed and often perceived as small) arises due to changes in the structure of near-edge vortices. This should be important in thin films and, in our opinion, may account for a number of unexplained observations. The second, unexpected, effect is more general, and appears owing to the inevitable presence of barriers for flux motion through a superconducting boundary (for example, Bean-Livingston barriers). If such barriers are sufficiently high, nonlinear screening can lead to the extreme situation, causing 'negative vortices'; but if this is not the case, surface barriers can still prevent the quantized penetration. One or both of the above effects can be expected in many-if not most-relevant experimental situations. โ
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