$Γ$-limit for two-dimensional charged magnetic zigzag domain walls

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_ε[m] \ = \ ε\|\nabla m\|_{L^2}^2 + \frac {1}ε \|m \cdot e_2\|_{L^2}^2 + \frac{πλ}{2|\lnε|} \|\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 $ε\to 0$ and for fixed $λ> 0$, corresponding to the macroscopic limit, we show that the energy $Γ$-converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime $λ\leq 1$ one-dimensional charged domain walls are favorable, in the supercritical regime $λ> 1$ the limit model allows for zigzaging two-dimensional domain walls.

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