A zigzag pattern in micromagnetics

AbstractWe study a simplified model for the micromagnetic energy functional in a specific asymptotic regime. The analysis includes a construction of domain walls with an internal zigzag pattern and a lower bound for the energy of a domain wall based on an “entropy method”. Under certain conditions, the two results yield matching upper and lower estimates for the asymptotic energy. The combination of these then gives a Γ-convergence result

Renormalised energy between boundary vortices in thin-film micromagnetics with Dzyaloshinskii-Moriya interaction

We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg-Landau type model in terms of the averaged magnetization in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a $\Gamma$-convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.Comment: Comments are very welcom

Separation of domain walls with nonlocal interaction and their renormalised energy by Γ-convergence in thin ferromagnetic films

We analyse two variants of a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter ϵ&gt;0. The model gives rise to transition layers, called Néel walls, and we study their behaviour in the limit ϵ→0. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between Néel walls to leading order. But while Ginzburg-Landau vortices show attraction for degrees of the same sign and repulsion for degrees of opposite signs, the pattern is reversed in this model. In a previous paper, we determined the renormalised energy for one of the models studied here under the assumption that the Néel walls stay separated from each other. In this paper, we present a deeper analysis that in particular removes this assumption. The theory gives rise to an effective variational problem for the positions of the walls, encapsulated in a Γ-convergence result. In the second part of the paper, we turn our attention to another, more physical model, including an anisotropy term. We show that it permits a similar theory, but the anisotropy changes the renormalised energy in unexpected ways and requires different methods to find it.</p

Uniqueness result for a weighted pendulum equation modeling domain walls in notched ferromagnetic nanowires

We prove an existence and uniqueness result for solutions $\varphi$ to a weighted pendulum equation in $\mathbb{R}$ where the weight is non-smooth and coercive. We also establish (in)stability results for $\varphi$ according to the monotonicity of the weight. These results are applied in a reduced model for thin ferromagnetic nanowires with notches to obtain existence, uniqueness and stability of domain walls connecting two opposite directions of the magnetization