Gamma-limit for zigzag walls in thin ferromagnetic films

Abstract: In the continuum theory, the magnetization of a ferromagnetic sample $\Omega \subset \R^3$ is described by a unit vector field $m \in H^1(\Omega,S^2)$. The minimization of the underlying micromagnetic energy leads to the formation of extended magnetic domains with uniform magnetization, separated by thin transition layers. One type of such transition layers, observed in thin ferromagnetic films are the so called zigzag walls. We consider the family of energies $$E_\varepsilon[m] \ = \ \frac{\epsilon}{2}\|\nabla m\|_{L^2(\Omega)}^2 + \frac 1{2\varepsilon} \|m \cdot e_2\|_{L^2(\Omega)}^2 % + \frac{\pi\lambda}{2|\ln \varepsilon|} \|\nabla \cdot (m-M)\|_{\dot H^{-\frac 12}}^2, $$ valid for thin ferromagnetic films. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on $m$. Here, $M$ is an arbitrary fixed background field to ensure global neutrality of magnetic charges. In the limit $\varepsilon \to 0$ 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.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic

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