The influence of magnetic steps on bulk superconductivity

Abstract

We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.

The breakdown of superconductivity in the presence of magnetic steps

“…• In the intensity-regime H < κ/|a|: [AK16] establishes the existence of superconductivity in the whole bulk of Ω, and the results of our SDSF case are similar to those of the SDUF and CDUF cases (see e.g. [SS03] [CG17] proved that this uniform distribution is not affected (to leading order) by the presence of corners.…”

“…(1.11) This paper is an integral part of a research that started in [AK16,AKPS19]. Throughout these papers, we present tools for studying the distribution of superconductivity in a smooth domain submitted to a step magnetic field satisfying Assumption 1.1 (the SDSF case), when κ is large, considering various regimes of the intensity of this magnetic field.…”

“…Since the functional in (1.1) is gauge invariant 1 , one may restrict its minimization with respect to (ψ, A) (originally done in H 1 (Ω; C) × H 1 (Ω; R 2 )) to the space H 1 (Ω; C) × H 1 div (Ω), where H 1 div (Ω) = A ∈ H 1 (Ω; R 2 ) : div A = 0 in Ω, A · ν = 0 on ∂Ω (1.2) and ν is a unit normal vector of ∂Ω. This restriction is beneficial due to the nice regularity properties of the space H 1 div (Ω) (see [AK16,Appendix B]). Hence, we introduce the following ground-state energy:…”

“…).• In the intensity-regime κ/|a| < H ≤ κ/(|a|Θ 0 ):[AK16] shows the disappearance of superconductivity from the bulk of Ω 1 and Ω…”

The Distribution of Superconductivity Near a Magnetic Barrier

“…At some stage, the expected behaviour of the superconductor in question deviates from the one submitted to a uniform magnetic field. Recently, this case was considered in [AK16] and the following was obtained. Suppose that H = bκ and κ is large.…”