Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

Abstract

AbstractWe consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $$\Omega _t:= \Omega \times [0,t]$$ Ω t : = Ω × [ 0 , t ] for $$\Omega \Subset \mathbb {R}^2$$ Ω ⋐ R 2 with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by $$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$ E [ M ] = ∫ Ω t d 2 | ∇ M | 2 + Q ∫ Ω t ( M 1 2 + M 2 2 ) + ∫ R 3 | H ( M ) | 2 - ∫ R 3 | H ( e 3 χ Ω t ) | 2 , describing the energy difference for a given magnetization $$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$ M : R 3 → R 3 with $$|M| = \chi _{\Omega _t}$$ | M | = χ Ω t and the uniform magnetization $$e_3 \chi _{\Omega _t}$$ e 3 χ Ω t . For $$t \ll d$$ t ≪ 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 $$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$ | Ω | 1 2 ∼ ( Q - 1 ) - 1 2 d e 2 π d t Q - 1 . We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.