# Magnetic domains in thin ferromagnetic films with strong perpendicular anisotropy

## TL;DR

This paper analyzes the energy scaling and domain patterns in thin ferromagnetic films with strong perpendicular anisotropy, identifying the transition from single to multi-domain states and deriving related energy laws.

## Contribution

It introduces a detailed analysis of the transition between single and multi-domain states in ferromagnetic films, including asymptotic energy behavior and domain size scaling laws.

## Key findings

- Transition point between single and multi-domain states identified.
- Energy converges to a simpler 2D local model in the single domain regime.
- Scaling laws for minimal energy and domain size derived.

## Abstract

We investigate the scaling of the ground state energy and optimal domain patterns in thin ferromagnetic films with strong uniaxial anisotropy and the easy axis perpendicular to the film plane. Starting from the full three-dimensional micromagnetic model, we identify the critical scaling where the transition from single domain to multidomain ground states such as bubble or maze patterns occurs. Furthermore, we analyze the asymptotic behavior of the energy in two regimes separated by a transition. In the single domain regime, the energy $\Gamma$-converges towards a much simpler two-dimensional and local model. In the second regime, we derive the scaling of the minimal energy and deduce a scaling law for the typical domain size.