# Reduced models for ferromagnetic thin films with periodic surface roughness

## TL;DR

This paper derives a simplified two-dimensional model to understand how periodic surface roughness affects magnetic anisotropy and magnetization in thin ferromagnetic films, revealing mechanisms behind perpendicular and uniaxial anisotropy.

## Contribution

The paper introduces a novel reduced model for ferromagnetic thin films with periodic surface roughness using homogenization and $ ext{Gamma}$-convergence techniques.

## Key findings

- Insights into the formation of perpendicular magnetic anisotropy.
- Understanding of uniaxial anisotropy with arbitrary magnetization direction.
- A new two-dimensional local model for thin ferromagnetic films.

## Abstract

We investigate the influence of periodic surface roughness in thin ferromagnetic films on shape anisotropy and magnetization behavior inside the ferromagnet. Starting from the full micromagnetic energy and using methods of homogenization and $\Gamma$-convergence we derive a two dimensional local reduced model. Investigation of this model provides an insight on the formation mechanism of {\it perpendicular magnetic anisotropy} and uniaxial anisotropy with an {\it arbitrary} preferred direction of magnetization.