Abstract
We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the $\Gamma$-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes.
Highlights
Vortices determine electromagnetic properties of superconductors that are important for practical applications
The sample is subjected to a uniform magnetic field, which is weak so that the vortices do not appear in the bulk of the superconducting sample and they may exist only in the holes
In this work we study a homogenization problem for a large number of vortices and large number of pinning holes that are described by a perforated domain Ωε
Summary
Vortices determine electromagnetic properties of superconductors that are important for practical applications (e.g., resistance). A key practical issue is to decrease the energy dissipation in superconductors, which occurs due to the motion of vortices. This dissipation can be suppressed by the pinning of vortices. Problems of pinning in superconducting thin films lead to the analysis of a two-dimensional Ginzburg-Landau (GL) energy functional in a domain with periodic or random arrays of holes called antidots in physical literature (see e.g., [6] and references therein). In this work we consider a two-dimensional mathematical model of pinning of vortices by many holes in relatively small superconducting samples (comparable to the London depth).
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