On the Ginzburg--Landau Functional in the Surface Superconductivity Regime

Abstract

We present new estimates on the two-dimensional Ginzburg-Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg-Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg-Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg-Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, related to a conjecture by Xing-Bin Pan.

Revision of the version published in Communications in Mathematical Physics 332 (3), 1297-1343 (2014). Minor mistakes corrected and Theorem 2.4 reformulated accordingly

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Cited by in corpus (14)

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• Multi-component Ginzburg-Landau theory: microscopic derivation and examples
• The Ginzburg-Landau functional with vanishing magnetic field
• The breakdown of superconductivity in the presence of magnetic steps
• Effects of boundary curvature on surface superconductivity
• Universal and shape dependent features of surface superconductivity
• On the Third Critical Speed for Rotating Bose-Einstein Condensates
• Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field
• Magnetic steps on the threshold of the normal state
• Concentration Behavior and Lattice Structure of Surface Superconductivity
• Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in General External Fields
• On the Ginzburg-Landau Energy of Corners
• Almost Flat Angles in Surface Superconductivity
• Differential Equations of Quantum Mechanics