Minimizers of the Lawrence-Doniach Functional with Oblique Magnetic Fields

TL;DR

This paper analyzes the minimizers of the Lawrence-Doniach energy in layered superconductors under oblique magnetic fields, revealing flux lock-in transitions and staircase vortices through sharp energy bounds and convex minimization.

Contribution

It provides the first rigorous derivation of energy bounds and flux lattice orientation for Lawrence-Doniach models with oblique fields in the thin-layer limit.

Findings

Identifies the lower critical field and flux lattice orientation to leading order.

Discovers a flux lock-in transition for small inclination fields.

Suggests the existence of staircase vortices in the energy profile.

Abstract

We study minimizers of the Lawrence--Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing $s\to 0$ and the Ginzburg--Landau parameter $\kappa=\eps^{-1}\to\infty$, under the hypotheses that $s=\eps^\alpha$ with $0<\alpha<1$. By deriving sharp matching upper and lower bounds on the energy of minimizers, we determine the lower critical field and the orientation of the flux lattice, to leading order in the parameter $\eps$. To leading order, the induced field is characterized by a convex minimization problem in $\RR^3$. We observe a ``flux lock-in transition'', in which flux lines are pinned to the horizontal direction for applied fields of small inclination, and which is not present in minimizers of the anisotropic Ginzburg--Landau model. The energy profile we obtain suggests the presence of ``staircase vortices'', which have been described qualitatively in the physics literature.