# Magnetic Ginzburg-Landau energy with a periodic rapidly oscillating and diluted pinning term

Abstract : We study the $2D$ full Ginzburg-Landau energy with a periodic rapidly oscillating, discontinuous and [strongly] diluted pinning term using a perturbative argument. This energy models the state of an heterogeneous type II superconductor submitted to a magnetic field. We calculate the value of the first critical field which links the presence of vorticity defects with the intensity of the applied magnetic field. Then we prove a standard dependance of the quantized vorticity defects with the intensity of the applied field. Our study includes the case of a London solution having several {\it minima}. The pinning effect is explicitly established and we give the asymptotic location of the vorticity defects with various scales. The macroscopic location of the vorticity defects is understood with the famous Bethuel-Brezis-Hélein renormalized energy restricted to the {\it minima} of the London solution coupled with a renormalized energy obtained by Sandier-Serfaty. The mesoscopic location, {\it i.e.}, the arrangement of the vorticity defects around the {\it minima} of the London solution, is described, as in the homogenous case, by a renormalized energy obtained by Sandier-Serfaty. The microscopic location is exactly the same than in the heterogeneous case without magnetic field. We also compute the value of secondary critical fields that increment the quantized vorticity.
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https://hal.archives-ouvertes.fr/hal-02089571
Contributor : Mickaël Dos Santos <>
Submitted on : Tuesday, April 16, 2019 - 12:51:36 AM
Last modification on : Friday, April 19, 2019 - 1:14:16 AM

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PinnedMagneticGL.pdf
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### Identifiers

• HAL Id : hal-02089571, version 2
• ARXIV : 1904.02381

### Citation

Mickaël Dos Santos. Magnetic Ginzburg-Landau energy with a periodic rapidly oscillating and diluted pinning term. 2019. ⟨hal-02089571v2⟩

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