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Magnetic Ginzburg-Landau energy with a periodic rapidly oscillating and diluted pinning term. (English. French summary) Zbl 1489.35265

This interesting work is devoted to the pinning phenomenon in type-II superconducting composites. The superconductivity of physical point of view is characterized by a total absence of electrical resistance and a perfect diamagnetism. If the imposed conditions are very intense, then superconductivity is destroyed in certain areas of the material called vorticity defects. In the case when the vorticity defects first appear in small areas, then there exist so called type II superconductors. The behavior of a superconductor is modeled by minimizers of a Ginzburg-Landau type energy.
Here it is investigated the presence of traps for the vorticity defects. To this end the author considers an energy including a pinning term that models impurities in the superconductor. These impurities play the role of traps for the vorticity defects. Thus the author has focused this study on the type-II superconducting composites with impurities. The mathematical model in the case of an infinite long homogenous type II superconducting cylinder was intensively studied during last three decades [E. Sandier and S. Serfaty, Calc. Var. Partial Differ. Equ. 17, No. 1, 17–28 (2003; Zbl 1037.49001); Vortices in the magnetic Ginzburg-Landau model. Basel: Birkhäuser (2007; Zbl 1112.35002); S. Serfaty, Commun. Contemp. Math. 1, No. 2, 213–254 (1999; Zbl 0944.49007)]. The main task in the Ginzburg-Landau theory is to find the minimizers of the functional \(\mathcal{E}_{\varepsilon,h_{ex}} :\mathcal{H}\to \mathbb{R}^{+}\), that is, \[ (u,A)\longmapsto (1/2)\int\limits_{\Omega } |\nabla u-\iota Au|^2+ (2^{-1}\varepsilon^{-2}) (a_{\varepsilon}^2-|u|^2)^2+ |\textrm{curl}(A)-h_{ex}|^2, \] where \(\Omega \subset\mathbb{R}^2\) is a smooth bounded simply connected open set, \(\mathcal{H}= H^1(\Omega ,\mathbb{C}) \times H^1(\Omega ,\mathbb{R}^2)\), \(a_{\varepsilon }: \Omega\to\{1, b\}\) (\(b\in (0,1)\) independent of \(\varepsilon \)) is a periodic diluted pinning term. In the definition of \(a_{\varepsilon } \) the quantity \( \delta =\delta (\varepsilon )\to 0\) as \(\varepsilon\to 0\) is the parameter of period, \(\lambda = \lambda (\varepsilon ) \to 0\) is the parameter of dilution, and \(\omega\subset\mathbb{R}^2\) (\(0 \in\omega \)) is a smooth bounded simply connected open set which gives the form of the impurities. The author considers a strongly diluted case \(\lambda^{1/4}| \ln{\varepsilon }|\to 0\) with not too small connected components of \(\omega_{\varepsilon }\) to trap the vorticity defects \(|\ln{\lambda \delta }|= \mathcal{O}(\ln{| \ln{\varepsilon }|})\) but with sufficiently small parameter of the period. Thus if \((u_{\varepsilon}, A_{\varepsilon})\) minimizes \(\mathcal{E}_{\varepsilon ,h_{ex}}\), then the vorticity defects can be interpreted as the set \(M_{\varepsilon}= \{|u_{\varepsilon }|<b/2\}\). It is excepted that the connected components of \(M_{\varepsilon}\) are close to disks with radii of order \(\varepsilon \).
This study is focused on the extreme type II case \(\varepsilon\to 0\) and it is also assumed a divergent upper bound for \(h_{ex}\). Vorticity defects appear for the minimizers greater than a critical \(H_{c_1}\). To calculate the critical value \(H_{c_1}\) it is used so called London solution \(\xi_0\in H_0^1\cap H^2\) which is the unique solution of the London equation:
\( -\triangle^2\xi_0+\triangle \xi_0=0\) in \(\Omega , \ \ \) \(\triangle \xi_0=1\) on \(\partial\Omega , \ \ \) \(\xi_0=0\) on \(\partial\Omega \) .
Note that the London solution has several minima. The value \(H_{c_1}\) is calculated by a standard balance of the energetic costs of a configuration without vorticity defects, i.e. \(|u| \geq b/2\) with well prepared competitors having an arbitrary number of quantized vorticity defects. The quantization problem is interpreted by the degree of \(u\) around a vorticity defect. It is an observable quantity related with the circulation of the superconducting currents. The pinning effect is explicitly established. The asymptotic location of the vorticity defects with various scales is discussed as well. The macroscopic location of the vorticity defects is considered in the frame of the Bethuel-Brézis-Hélein renormalized energy restricted to the minima of the London solution coupled with a renormalized energy obtained by Sandier-Serfaty. It turns out that the arrangement of the vorticity defects around the minima of the London solution can be described, as in the homogenous case, by a renormalized energy obtained by Sandier-Serfaty. It is shown that the microscopic location is the same as it is in the heterogeneous case without magnetic field.

MSC:

35Q56 Ginzburg-Landau equations
35J20 Variational methods for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
82D55 Statistical mechanics of superconductors
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References:

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