Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields

Abstract

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors with Josephson constant λ and Ginzburg-Landau parameter 1/ǫ in a bounded generalizedwhere Ω is a bounded simply connected Lipschitz domain in R 2 . Our main result is that in an applied magnetic field H ex = h ex e 3 which is perpendicular to the layers with |ln ǫ| ≪ h ex ≪ ǫ −2 , the minimum Lawrence-Doniach energy is given by (1)) as ǫ and the interlayer distance s tend to zero. We also prove estimates on the behavior of the order parameters, induced magnetic field, and vorticity in this regime. Finally, we observe that as a consequence of our results, the same asymptotic formula holds for the minimum anisotropic three-dimensional Ginzburg-Landau energy in D with anisotropic parameter λ and o ǫ,s (1) replaced by o ǫ (1).

Convergence of the Lawrence--Doniach Energy for Layered Superconductors with Magnetic Fields Near $H_c_1$

“…Note that the constructed magnetic potential˜ A ǫ in the recovery sequence does not depend on s. The regime for h ex under consideration is a lot more subtle than the higher regime studied in [10], since for the regime h ex ∼ | ln ǫ|, the superconductor undergoes a phase transition from the superconducting state to the mixed state. The same regime for the Lawrence-Doniach model was considered in the work of Alama et al [2], in which the energy is minimized among configurations whose gauge-invariant quantities are periodic with respect to a given parallelepiped.…”

“…It is crucial to understand how the vector field A relates to its traces in this particular context. In [10], some powerful a priori estimates for the magnetic potential were established based on the results in [9]. Those estimates turned out to be very useful for the analysis of the Lawrence-Doniach model.…”

“…They also studied the cases when the magnetic fields are parallel to the layers or oblique in [2] and [3]. Without the additional periodicity assumptions, Bauman and the author [10] proved an asymptotic formula for the minimum Lawrence-Doniach energy with | ln ǫ| ≪ h ex ≪ ǫ −2 in the limiting case as (ǫ, s) → (0, 0), and obtained vortex structure information in this case. In the last regime, h ex ≥ C ǫ 2 , it was shown by Bauman-Ko [9] that if C is sufficiently large, all minimizers of the Lawrence-Doniach energy are in the normal phase.…”

A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

“…Other models are considered in [2], where the authors treat the structure of symmetric vortices in a Ginzburg-Landau model of high-temperature superconductivity and antiferromagnetism. In [4], the authors give an analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields.…”

First Critical Field of Highly Anisotropic Three-Dimensional Superconductors via a Vortex Density Model

“…Without the periodicity assumptions, a great simplification to a mean field model in the form of a Γ-convergence result with h ex ∼ | ln ǫ| is achieved by the second author in [28]. In a higher regime, an asymptotic formula for the minimum Lawrence-Doniach energy with | ln ǫ| ≪ h ex ≪ ǫ −2 in the limit as (ǫ, s) → (0, 0) is obtained in [8] together with information of the vortex structure. In the regime h ex ≥ C ǫ 2 , it was shown by Bauman-Ko [7] that if C is sufficiently large, all minimizers of the Lawrence-Doniach energy are in the normal phase.…”