Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form Ωt:=Ω×[0,t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _t:= \Omega \times [0,t]$$\end{document} for Ω⋐R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \Subset \mathbb {R}^2$$\end{document} with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by E[M]=∫Ωtd2|∇M|2+Q∫Ωt(M12+M22)+∫R3|H(M)|2-∫R3|H(e3χΩt)|2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$\end{document}describing the energy difference for a given magnetization M:R3→R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$\end{document} with |M|=χΩt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|M| = \chi _{\Omega _t}$$\end{document} and the uniform magnetization e3χΩt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_3 \chi _{\Omega _t}$$\end{document}. For t≪d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ll d$$\end{document}, we derive the scaling of the minimal energy and a BV-bound in the critical regime, where the base area of the film has size of order |Ω|12∼(Q-1)-12de2πdtQ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$\end{document}. We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.