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Concentration Behavior and Lattice Structure of 3D Surface Superconductivity in the Half Space

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  • Søren Fournais
  • Jean Philippe Miqueu
  • ,
  • Xing Bin Pan, East China Normal University, NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai

We study a 3D Ginzburg-Landau model in a half-space which is expected to capture the key features of surface superconductivity for applied magnetic fields between the second critical field HC2 and the third critical field HC3. For the magnetic field in this regime, it is known from physics that superconductivity should be essentially restricted to a thin layer along the boundary of the sample. This leads to the introduction of a Ginzburg-Landau model on a half-space. We prove that the non-linear Ginzburg-Landau energy on the half-space with constant magnetic field is a decreasing function of the angle ν that the magnetic field makes with the boundary. In the case when the magnetic field is tangent to the boundary (ν = 0), we show that the energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. For non-parallel applied fields, we also construct a periodic problem with vortex lattice minimizers reproducing the effective energy, which suggests that the order parameter of the full Ginzburg-Landau model will exhibit 3 dimensional vortex structure near the surface of the sample.

Original languageEnglish
Article number12
JournalMathematical Physics Analysis and Geometry
Number of pages33
Publication statusPublished - 2019

    Research areas

  • 3D vortices, Calculus of variation, Estimate of eigenvalue, Ginzburg-Landau equations, Magnetic Schrodinger operator, Partial differential equations, Superconductivity, Surface concentration, Surface superconductivity

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