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

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).