A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

Abstract

We study the Landau-Lifshitz model for the energy of multi-scale transition layers -called "domain walls" -in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors m ± α ∈ S 2 that differ by an angle 2α. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in S 2 by an angle 2θ, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle θ to α in S 1 . The angle θ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7].Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).

Stability of point defects of degree $$\pm \frac{1}{2}$$ ± 1 2 in a two-dimensional nematic liquid crystal model

“…Two-dimensional point defects in the Landau-de Gennes framework have been studied for quite some time in the literature; see e.g. [5,9,11,15,17,20,22,25,30,31] (and also [16,27] in micromagnetics). Our motivation came from the paper [15] which concerns the extreme low-temperature regime (b 2 = 0).…”

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

“…Proof. The proof follows the same arguments as in the second proof of Theorem 2.3 based on Lemma 3.9 (see [23,Lemma 1.]). Take a minimizing sequence (γ n ) n≥1 ⊂Ḣ 1 (R, R d a ) such that γ n (±∞) = u ± and E(γ n ) → geod a W (u − , u + ) as n → ∞.…”

“…We point out a second proof, based on the following compactness result [23, Lemma 1. ], which can be seen as a generalization of Lemma 3.8 in terms of the average sequence {ū n }: Lemma 3.9 (L. Döring, R. Ignat, F. Otto [23]). Let (v n ) n≥1 be a sequence of scalar functions v n : R → R uniformly bounded inḢ 1 (R), i.e., sup n v n L 2 (R) < ∞, and such that…”

“…[19,38,35]), and on the other hand, the domain walls involving microstructures, such as the so-called cross-tie walls (see e.g., [2,45]), the zigzag walls (see e.g., [37,43]) or the asymmetric Néel / Bloch walls (see e.g. [23,22]). Our paper aims to give a general approach in identifying the anisotropy potentials W for which the domain walls are one-dimensional in the system (1.1).…”

“…The proof is based on the following general compactness result which is somehow reminiscent from [23,Lemma 1. ] and [33,Lemma 4.4.].…”