论文标题
$γ$ - 二维电荷磁锯齿形域墙壁的限制
$Γ$-limit for two-dimensional charged magnetic zigzag domain walls
论文作者
论文摘要
带电域壁是薄铁磁膜中的一种域壁，由于全球拓扑限制而出现。单轴薄铁磁性膜的非二量化微磁能，带有面内磁化$ m \ in \ mathbb {s}^1 $由 \ begin {align*} e_ε[m] \ = \ 这些 + \ \ frac {πλ} {2 | \lnε|} \ | \ nabla \ cdot（m-m）\ | _ {\ dot h^{ - \ frac {1} {2}}}}}}}}}}}}^2， \ end {align*}，其中$ e_1 $ - 方向是全球优先的，而$ m $是任意固定的背景字段，以确保磁性电荷的全局中立性。我们认为形式的材料是薄带，并通过$ M $的合适边界条件强制实施带电的域墙。在限制$ε\至0 $的情况下，对于固定$λ> 0 $，对应于宏观限制，我们表明能量$γ$ - 连接到极限能量，而磁化磁化的跳跃不连续性受到各向异性的惩罚。特别是，在亚临界体制中，$λ\ leq 1 $一维电荷域墙是有利的，在超临界状态$λ> 1 $中，极限模型允许在二维域壁上锯切。
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.
