论文标题
薄铁磁膜中具有垂直方向性的模式形成的发作
Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy
论文作者
论文摘要
我们考虑$ \tildeΩ_t的超薄铁磁膜中的模式形成的开始：= \tildeΩ相对微磁能由\ begin {align} \ mathcal {e} [m]＆= \ int _ {\tildeΩ_t} d^2 | \ nabla m |^2+ q \ q \ inb \ int _ {\ Mathbb {r}^3} | \ Mathcal {h}（m）|^2 - \ int _ {\ Mathbb {\ Mathbb {r}^3} | \ Mathcal {h}磁化$ M：\ Mathbb {r}^3 \ to \ Mathbb {r}^3 $ with $ | m | =χ_ {\ tilde {ω} _t} $和统一磁化$e_3χ_ {\ tilde {ω} _t} $。对于$ t \ ll d $，我们在关键体制中建立了能量的缩放和bv限制的胶片的基本区域$ | \ tilde {ω} | \ sim（q-1）^{1/2} d E^{\ frac {2πd} t \ sqrt {q-1}} $。我们此外，根据重新制定的膜的大小，研究了临界状态中非平凡模式形成的发作。
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $\tildeΩ_t := \tildeΩ \times [0,t]$ for $\tilde{ Ω} \Subset \mathbb{R}^2$ with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by \begin{align} \mathcal{E}[M] &= \int_{\tildeΩ_t} d^2 |\nabla M|^2+ Q \int_{\tildeΩ_t} (M_1^2+M_2^2) + \int_{\mathbb{R}^3} |\mathcal{H}(M)|^2 - \int_{\mathbb{R}^3} |\mathcal{H}(e_3 χ_{\tilde{ Ω}})|^2, \end{align} describing the energy difference for a given magnetization $M : \mathbb{R}^3 \to \mathbb{R}^3$ with $|M| = χ_{\tilde{ Ω}_t}$ and the uniform magnetization $e_3 χ_{\tilde{ Ω}_t}$. For $t \ll d$, we establish the scaling of the energy and a BV-bound in the critical regime here the base area of the film is of order $|\tilde{ Ω}| \sim (Q-1)^{1/2} d e^{\frac{2πd}t \sqrt{Q-1}}$. We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.
