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$$\Gamma $$ Γ -Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls
Archive for Rational Mechanics and Analysis ( IF 2.6 ) Pub Date : 2021-02-04 , DOI: 10.1007/s00205-021-01606-x
Hans Knüpfer , Wenhui Shi

Charged domain walls are a type of domain wall 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

$$\begin{aligned} E_\varepsilon [m] \ = \ \varepsilon \Vert \nabla m\Vert _{L^2}^2 + \frac{1}{\varepsilon } \Vert m \cdot e_2\Vert _{L^2}^2 + \frac{\pi \lambda }{2|\ln \varepsilon |} \Vert \nabla \cdot (m-M)\Vert _{\dot{H}^{-\frac{1}{2}}}^2, \end{aligned}$$

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 \(\varepsilon \rightarrow 0\) 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 \(\lambda \leqq 1\), one-dimensional charged domain walls are favorable, in the supercritical regime \(\lambda > 1\), the limit model allows for zigzaging two-dimensional domain walls.



中文翻译:

$$ \ Gamma $$Γ-二维带电磁之字形畴壁的限制

带电畴壁是铁磁薄膜中的一种畴壁,由于整体拓扑约束而出现。具有面内磁化\(m \ in {\ mathbb {S}} ^ 1 \)的单轴铁磁薄膜的无量纲微磁能由下式给出

$$ \ begin {aligned} E_ \ varepsilon [m] \ = \ \ varepsilon \ Vert \ nabla m \ Vert _ {L ^ 2} ^ 2 + \ frac {1} {\ varepsilon} \ Vert m \ cdot e_2 \垂直_ {L ^ 2} ^ 2 + \ frac {\ pi \ lambda} {2 | \ ln \ varepsilon |} \ Vert \ nabla \ cdot(mM)\ Vert _ {\ dot {H} ^ {-\ frac {1} {2}}} ^ 2,\ end {aligned} $$

其中M是任意固定的背景场,以确保磁性的全局中性。我们认为材料是细条状,并通过在m上的适当边界条件来强制带电畴壁。在极限\(\ varepsilon \ rightarrow 0 \)中,对于固定的\(\ lambda> 0 \),对应于宏观极限,我们显示出能量\(\ Gamma \)-收敛到极限能量,其中跳跃不连续磁化强度各向异性地受到惩罚。特别地,在亚临界状态\(\ lambda \ leqq 1 \)中,一维带电畴壁是有利的,在超临界状态\(\ lambda> 1 \),极限模型可以使二维畴壁成Z字形。

更新日期:2021-02-04
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