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 $\tildeΩ_t := \tildeΩ \times [0,t]$ 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.

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