Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

Abstract

We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form Omega _t:= Omega times [0,t] for Omega Subset mathbb {R}^2 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}$$\\begin{aligned} \\mathcal {E}[M]= \\int _{\\Omega _t} d^2 |\ abla M|^2+ Q \\int _{\\Omega _t} (M_1^2+M_2^2) + \\int _{\\mathbb {R}^3} |\\mathcal {H}(M)|^2 - \\int _{\\mathbb {R}^3} |\\mathcal {H}(e_3 \\chi _{\\Omega _t})|^2, \\end{aligned}$$\\end{document}describing the energy difference for a given magnetization M: mathbb {R}^3 rightarrow mathbb {R}^3 with |M| = chi _{Omega _t} and the uniform magnetization e_3 chi _{Omega _t}. For t ll d, 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 |Omega |^{{frac{1}{2}}} sim (Q-1)^{{-frac{1}{2}}} d e^{frac{2pi d}{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.