$$\Gamma $$-Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls

Abstract

Charged domain walls are a type of domain walls in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization $m \in \mathbb{S}^1$ is given by \begin{align*} E_\epsilon[m] \ = \ \epsilon\|\nabla m\|_{L^2}^2 + \frac {1}{\epsilon} \|m \cdot e_2\|_{L^2}^2 + \frac{\pi\lambda}{2|\ln\epsilon|} \|\nabla \cdot (m-M)\|_{\dot H^{-\frac{1}{2}}}^2, \end{align*} where magnetization in $e_1$-direction is globally preferred and where $M$ is an arbitrary fixed background field to ensure global neutrality of magnetic charges. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on $m$. In the limit $\epsilon \to 0$ and for fixed $\lambda> 0$, corresponding to the macroscopic limit, we show that the energy $\Gamma$-converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime $\lambda \leq 1$ one-dimensional charged domain walls are favorable, in the supercritical regime $\lambda > 1$ the limit model allows for zigzaging two-dimensional domain walls.