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

Abstract

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.

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Acknowledgements

We are grateful to the referee for carefully reading the file and his useful comments. We also thank J. Fabiszisky for carefully proofreading and drawing some of the pictures. Hans Knüpfer was partially supported by the German Research Foundation (DFG) by the project #392124319 and under Germany’s Excellence Strategy—EXC-2181/1—390900948. Wenhui Shi was partially supported by the German Research Foundation (DFG) by the Project SH 1403/1-1.

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Communicated by A. Braides

Appendix A: Real Space Representation of the Stray Field

Appendix A: Real Space Representation of the Stray Field

We first recall the following standard representation of the homogeneous \(H^{\frac{1}{2}}\)-norm. We give the short proof since the constant in front of the identity is essential in our arguments:

Lemma A.1

(Finite difference representation of \(H^{\frac{1}{2}}\)-norm) For \(m \in H^{\frac{1}{2}}(Q_\ell )\) we have

$$\begin{aligned} \int _{Q_\ell } \left| |\nabla |^{\frac{1}{2}} m\right| ^2 \ \mathrm{d}x \ = \ \frac{1}{4\pi } \int _{Q_\ell } \int _{\mathbb {R}^2} \frac{|m(x+h)-m(x)|^2}{|h|^3} \ \mathrm{d}h \mathrm{d}x. \end{aligned}$$
(97)

Proof

Using Plancherels identity (8) and Fubini’s theorem, we obtain

$$\begin{aligned}&\int _{Q_\ell } \int _{\mathbb {R}^2} \frac{|m(x+h)-m(x)|^2}{|h|^3} \ \mathrm{d}x \mathrm{d}h = \ \int _{Q_\ell } |{\widehat{m}}(\xi )|^2 \int _{\mathbb {R}^2} \frac{|1-e^{i\xi \cdot h}|^2}{|h|^3} \ \mathrm{d}h \mathrm{d}\xi \\&\quad = \ \int _{Q_\ell } |\xi ||{\widehat{m}}(\xi )|^2 \int _{\mathbb {R}^2} \frac{|1-e^{i\xi \cdot h}|^2}{|h|^3|\xi |} \ \mathrm{d}h \mathrm{d}\xi = \ 4\pi \int _{Q_\ell } |\xi | |{\widehat{m}}(\xi )|^2 \ \mathrm{d}\xi . \end{aligned}$$

The last identity follows with the change of variables \(h\mapsto \frac{h}{|\xi |}\) and since \(\int _{\mathbb {R}^2} \frac{|1-e^{ih_1}|^2}{|h|^3} \ \mathrm{d}h = 4\pi \) (cf. [13, (39)]). \(\square \)

The next lemma yields another representation for the \(H^{\frac{1}{2}}\)-norm when \(j \rightarrow - \infty \) and \(k \rightarrow \infty \).

Lemma A.2

(\(H^{\frac{1}{2}}\)-norm vs. \(H^{-\frac{1}{2}}\)-norm) For \(f \in C^\infty _c(Q_\ell ;\mathbb {R}^2)\) we have

$$\begin{aligned}&\frac{1}{2} \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{|f(y)-f(x)|^2}{|x-y|^3} \ \mathrm{d}x\mathrm{d}y \nonumber \\&\quad = \ \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{(\nabla \cdot f)(x)(\nabla \cdot f)(y)}{|x-y|}\ \mathrm{d}x\mathrm{d}y \nonumber \\&\qquad + \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{(\nabla \times f)(x)(\nabla \times f)(y)}{|x-y|}\ \mathrm{d}x\mathrm{d}y \nonumber \\&\quad \geqq \ \frac{1}{2} \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{(\nabla \cdot f)(x)(\nabla \cdot f)(y)}{|x-y|}\ \mathrm{d}x\mathrm{d}y, \end{aligned}$$
(98)

where \(\tilde{Q}_\ell = \mathbb {R}\times [0, \ell )\) and \(\tilde{P}_\ell = \mathbb {R}\times [j \ell , k \ell )\) for any \(j < k \in \mathbb {Z}\).

Proof

Let I (resp. J) be first (second) integral on the second line of (98). We recall the identity \(D (\frac{z}{|z|^3})\) = \(D^t (\frac{z}{|z|^3})\) \(=\) \(\frac{1}{|z|^5}(z^\perp \otimes z^\perp \)\(2 z \otimes z)\) where \(z^\perp := (-z_2,z_1)\). Integrating by parts in x and y, since \(\nabla _z (\frac{1}{|z|}) = - \frac{z}{|z|^3}\) and since \(\nabla (f \cdot g) = (D^t g) f + (D^tf) g\) then yields

$$\begin{aligned} I&= - \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } f(x) \cdot \frac{y-x}{|x-y|^3} \left[ \nabla _y \cdot (f(y) - f(x))\right] \\&= - \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{2 f(x)\cdot (y-x) (f(y)-f(x))\cdot (y-x)}{|x-y|^5} \\&\quad + \frac{f(x) \cdot (y-x)^\perp (f(y)-f(x))\cdot (y-x)^\perp }{|x-y|^5}. \end{aligned}$$

First, integrating in y then in x similarly yields that

$$\begin{aligned} I&= \ \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{ 2 f(y)\cdot (y-x) (f(y)-f(x))\cdot (y-x)}{|x-y|^5} \\&- \frac{f(y) \cdot (y-x)^\perp (f(y)-f(x))\cdot (y-x)^\perp }{|x-y|^5}. \end{aligned}$$

Taking the sum of these two expressions, one gets

$$\begin{aligned} I&= \ \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{1}{|x-y|^3} \left( \left| (f(y)-f(x))\cdot \frac{y-x}{|y-x|}\right| ^2 - \frac{1}{2} \left| (f(y)-f(x))\cdot \left( \frac{y-x}{|y-x|} \right) ^\perp \right| ^2 \right) . \end{aligned}$$
(99)

Since \(\nabla \times f = \nabla \cdot f^\perp \), the same calculation as before, replacing f by \(f^\perp \), yields

$$\begin{aligned} J&= \ \int \!\!\!\!\int _{\tilde{Q}_\ell \times \tilde{P}_\ell } \frac{1}{|x-y|^3} \left( \left| (f(y)-f(x))\cdot \left( \frac{y-x}{|y-x|} \right) ^\perp \right| ^2\right. \\&\quad \left. - \frac{1}{2} \left| (f(y)-f(x))\cdot \frac{y-x}{|y-x|} \right| ^2 \right) \ \mathrm{d}x\mathrm{d}y. \end{aligned}$$

The identity (98) follows by taking the sum of the last two equations. The inequality in (98) follows from (99). \(\square \)

We have the following singular integral characterization for the magnetostatic energy:

Lemma A.3

(Integral representations of magnetostatic energy) Let \(\sigma \in L^2(Q_\ell )\) with

$$\begin{aligned} \quad \text {with } {{\,\mathrm{spt}\,}}\sigma \Subset Q_\ell \quad \text { and } \quad \int _{Q_\ell } \sigma \ \mathrm{d}x \ = \ 0. \end{aligned}$$
(100)

Then there is a unique \(q \in H^1(Q_\ell ;\mathbb {R}^2)\) with \(\nabla \cdot q = \sigma \) and \(\nabla \times q = 0\) such that

$$\begin{aligned} \int _{Q_\ell } \left| |\nabla |^{-\frac{1}{2}}\sigma \right| ^2 \ \mathrm{d}x&= \ \frac{1}{4\pi } \int _{Q_\ell } \int _{\mathbb {R}^2} \frac{|q(x+h) - q(x)|^2}{|h|^3} \ \mathrm{d}h \mathrm{d}x \nonumber \\&= \ \frac{1}{2\pi } \lim _{N \rightarrow \infty , N \in \mathbb {N}} \int _{Q_\ell } \int _{\mathbb {R}\times [-N\ell ,N\ell ]}\frac{\sigma (x+h)\sigma (x)}{|h|} \ \mathrm{d}h \mathrm{d}x. \end{aligned}$$
(101)

Proof

By assumption (100) we have \({\widehat{\sigma }}(0) = 0\) and \(\nabla {\widehat{\sigma }} \in L^\infty (\mathbb {R}\times \frac{2\pi }{\ell } \mathbb {Z})\). This implies that \(q \in H^1(Q_\ell ;\mathbb {R}^2)\), where q is defined by its Fourier transform \(\widehat{q} := -i \frac{\xi }{|\xi |^2} {\widehat{\sigma }}\). By construction q satisfies \(\nabla \cdot q = \sigma \) and \(\nabla \times q = 0\). This solution is unique by the uniqueness of the Helmholtz decomposition. By (9), since \(|{\widehat{\sigma }}| = |\xi | |{\widehat{q}}|\) and by (97) we then get

$$\begin{aligned} \int _{Q_\ell } ||\nabla |^{-\frac{1}{2}} \sigma |^2 \ \mathrm{d}\xi \ {\mathop {=}\limits ^{(9)}} \ \int _{\mathbb {R}\times \frac{2\pi }{\ell } \mathbb {Z}} |\xi | |\widehat{q}|^2 \ \mathrm{d}\xi \ {\mathop {=}\limits ^{(97)}} \ \frac{1}{4\pi } \int _{Q_\ell } \int _{\mathbb {R}^2} \frac{|q(x+h)-q(x)|^2}{|h|^3} \ \mathrm{d}h \mathrm{d}x. \end{aligned}$$

Together with Lemma A.2 this yields (101). \(\square \)

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Knüpfer, H., Shi, W. \(\Gamma \)-Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls. Arch Rational Mech Anal 239, 1875–1923 (2021). https://doi.org/10.1007/s00205-021-01606-x

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