Speaker
Description
In the Ginzburg-Landau nodel superconductors are characterized by a parameter $\kappa$ called the Ginzburg-Landau parameter. If $\kappa<\frac{1}{\sqrt{2}}$ the superconductors are classified as type-I, if $\kappa>\frac{1}{\sqrt{2}}$ they are classified as type-II. While in type-II superconductors vortices appear, in type-I superconductors normal and superconducting regions are formed, separated by interfaces. In particular by the Meissner Effect, if $\rho$ is the density of superconducting electrons and $B$ is the magnetic field, it is observed that $\rho B\simeq 0$. Considering a 3D sample, it is experimentally observed that complex patterns appear at the surface. It is believed that these patterns are a manifestation of branching patterns inside the sample.
In a first work Conti Otto and Serfaty derive two regimes of parameters for the 3D type-I model, corresponding to uniform and non uniform branching patterns. Moreover, in a subsequent work Conti Otto Goldman and Serfaty prove a $\Gamma$-convergence result for the full 3D model in the case of uniform branching patterns.
In this talk I present a $\Gamma$-convergence result for the 2D type-I Ginzburg-Landau model in the crossover of the two regimes found in the former work. This is a first step in understanding how to extend the results of the latter work to the second regime.
With these hypothesis on the parameters the energy functional shares similarities with a Modica-Mortola type functional and in the limit $\Gamma$-converges to the area functional. To prove this result, it is necessary to carefully treat the global interaction between the phase of the complex order parameter $u$ and the vector potential $A$, taking into account the gauge invariance satisfied by the functional.
This talk is based on an ongoing work with Michael Goldman and Alessandro Zilio.
