On approximation of Ginzburg–Landau minimizers by S1-valued maps in domains with vanishingly small holes

Abstract

We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently discovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of S 1 -valued and C-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.1 have been studied, in particular, in [16,18] where the existence of two critical magnetic fields, H c1 and H c2 , was established rigorously for simply-connected domain when ε > 0 is small. When the external magnetic field is weak (h ext < H c1 ) it is completely expelled from the bulk semiconductor (Meissner effect) and there are no vortices. When the field strength is ramped up from H c1 to H c2 , the magnetic field penetrates the superconductor through an increasing number of isolated vortices while the superconductivity is destroyed everywhere, once the field exceeds H c2 .The pinning phenomenon that we consider in this paper is observed in non-simply-connected domains with holes that may or may not contain another material. If a hole "pins" a vortex, the order parameter u has a nonzero winding number on the boundary of the hole. We refer to this object as a hole vortex. Note that degrees of the hole vortices increase along with the strength of the external magnetic field. This situation is in contrast with the regular bulk vortices that have degree ±1 and increase in number as the field becomes stronger.An alternative way to model the impurities is to consider a potential term (a(x) − |u| 2 ) 2 where a(x) varies throughout the sample. It was proven in [9] that the impurities corresponding to the weakest superconductivity (where a(x) is minimal) pin the vortices first. This model was studied further in [1] and [4] to demonstrate the existence of nontrivial pinning patterns and in [2] to investigate the breakdown of pinning in an increasing external magnetic field, among other issues. A composite consisting of two superconducting samples with different critical temperatures was considered in [5,14] where nucleation of vortices near the interface was shown to occur.In our model we consider a superconductor with holes, similar to the setup in [3]. In that work, the authors considered the asymptotic limits of minimizers of GL ε as ε → 0 and determined that holes act as pinning sites gaining nonzero degree for moderate but bounded magnetic fields. For magnetic fields below the threshold of order | ln ε| the degree of the order parameter on the holes continues to grow without bound, however beyond the critical field strength, the pinning breaks down and vortices appear in the interior of the superconductor. Since...