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 ${\tilde{\Omega }}_t:=\tilde{\Omega }\times [0,t]$ for $\tilde{\Omega }⋐{\mathbb{R}}^2$ 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 $|M|={\chi }_{{\tilde{\Omega }}_t}$ and the uniform magnetization $e_3{\chi }_{{\tilde{\Omega }}_t}$. 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 $|\tilde{\Omega }|\sim (Q-1{)}^{1/2}d{e}^{\frac{2\pi d}{t}\sqrt{Q-1}}$. We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.

27 pages