A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

TL;DR

This paper derives a simplified energy model for domain walls in soft ferromagnetic films, revealing the transition from symmetric to asymmetric wall types and enabling better analysis of their structure and bifurcations.

Contribution

The paper rigorously derives a reduced energy model for domain walls, capturing asymmetric core and tail contributions, and facilitates analysis of bifurcation phenomena in domain wall structures.

Findings

Energy splits into asymmetric core and symmetric tails

Reduced model confirms experimental and numerical predictions

Enables analysis of asymmetric domain walls and bifurcations

Abstract

We study the Landau-Lifshitz model for the energy of multi-scale transition layers -- called "domain walls" -- in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors $m^\pm \in \mathbb{S}^2$ that differ by an angle $2\alpha$. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in $\mathbb{S}^2$ by an angle $2\theta$, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle $\theta$ to $\alpha$ in $\mathbb{S}^1$. The angle $\theta$ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7]. Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).