Steady one-dimensional domain wall motion in biaxial ferromagnets: mapping of the Landau-Lifshitz equation to the sine-Gordon equation

TL;DR

This paper demonstrates that the Landau-Lifshitz equation for a one-dimensional ferromagnetic domain wall can be mapped to the sine-Gordon equation under certain conditions, revealing a relativistic-like behavior of domain wall dynamics.

Contribution

It shows the formal reduction of the Landau-Lifshitz equation to the sine-Gordon equation for steady-state motion in biaxial ferromagnets, highlighting velocity-dependent energy and width.

Findings

Mapping is valid only below a critical velocity.

Domain wall energy and width follow relativistic-like formulas.

Beyond steady motion, the mapping is no longer applicable.

Abstract

Motivated by the difference between the dynamics of magnetization textures in ferromagnets and antiferromagnets, the Landau-Lifshitz equation of motion is explored. A typical one-dimensional domain wall in a bulk ferromagnet with biaxial magnetic anisotropy is considered. In the framework of Walker-type of solutions of steady-state ferromagnetic domain wall motion, the reduction of the non-linear Landau-Lifshitz equation to a Lorentz-invariant sine-Gordon equation typical for antiferromagnets is formally possible for velocities lower than a critical velocity of the topological soliton. The velocity dependence of the domain wall energy and the domain wall width are expressed in the relativistic-like form in the limit of large ratio of the easy-plane/easy-axis anisotropy constants. It is shown that the mapping of the Landau-Lifshitz equation of motion to the sine-Gordon equation can be performed only by going beyond the steady-motion Walker-type of solutions.