Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

TL;DR

This paper analyzes the conditions under which pattern formation begins in ultrathin ferromagnetic films with perpendicular anisotropy, focusing on energy scaling and the critical regime for pattern emergence.

Contribution

It provides a mathematical analysis of the onset of pattern formation in thin ferromagnetic films, including energy scaling laws and bounds in the critical regime, which was not previously characterized.

Findings

Energy scales with film size and anisotropy parameter

BV-bound established in the critical regime

Pattern formation onset depends on rescaled film size

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 $\tilde{ \Omega} \Subset \mathbb{R}^2$ with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by \begin{align} \mathcal{E}[M] &= \int_{\tilde{\Omega}_t} d^2 |\nabla M|^2+ Q \int_{\tilde{\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_{\tilde{ \Omega}})|^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.