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

Dos Santos, Mickaël

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 705-799.

À l’aide d’un argument perturbatif, on étudie une énergie de type Ginzburg–Landau bidimensionnelle avec un champ magnétique et présentant un terme de chevillage périodique rapidement oscillant, discontinu et (fortement) dilué. Cette énergie modélise l’état d’un supraconducteur hétérogène de type II soumis à un champ magnétique. On calcule la valeur du premier champ critique à partir duquel les défauts de vorticité apparaissent. Ensuite on démontre une dépendance classique reliant les défauts de vorticité quantifiés avec l’intensité du champ appliqué. Notre étude traite aussi le cas où la solution de London admet plusieurs point de minimum. L’effet d’ancrage des défauts de vorticité est clairement établi et on précise suivant différentes échelles l’emplacement asymptotique des défauts de vorticité. La position macroscopique des défauts de vorticité est donnée par la célèbre énergie renormalisée de Bethuel–Brézis–Hélein restreinte au points de minimum de la solution de London couplée avec une énergie renormalisée obtenue par Sandier–Serfaty. La position mesoscopique, i.e., l’arrangement des défauts de vorticité autour des points de minimum de la solution de London, est décrite, comme dans le cas homogène, par une énergie renormalisée obtenue par Sandier–Serfaty. La position microscopique est exactement la même que dans le cas sans champ magnétique. On calcule aussi des champs critiques secondaires qui incrémentent la vorticité quantifiée.

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 dependence of the quantized vorticity defects with the intensity of the applied field. Our study includes the case of a London solution having several 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–Brézis–Hélein renormalized energy restricted to the minima of the London solution coupled with a renormalized energy obtained by Sandier–Serfaty. The mesoscopic location, i.e., the arrangement of the vorticity defects around the 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.

Classification : 35Q56, 35J20, 35B27
Mots clés : Superconductivity, Ginzburg–Landau, pinning

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