Reduced theory of symmetric and antisymmetric exchange interactions in nanowires

TL;DR

This paper derives a simplified energy model for ferromagnetic nanowires considering symmetric and antisymmetric exchange interactions, highlighting how Dzyaloshinskii-Moriya interaction influences magnetic properties.

Contribution

It introduces a reduced energy functional for nanowires that captures the effects of exchange interactions and DMI, based on $ ext{Gamma}$-convergence analysis.

Findings

Reduced energy functional explicitly includes anisotropic terms influenced by curvature.

Antisymmetric exchange interactions contribute to shape-dependent anisotropy.

Results have implications for understanding magnetic behavior in nanowires with DMI.

Abstract

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve $\gamma$ and defined in the space of $\mathbb{S}^2$-valued functions. The perturbation $K$ is represented by a matrix-valued function defined on $\mathbb{S}^2$ with values in $\mathbb{R}^{3 \times 3}$. Under natural regularity conditions on $K$, we show that the family of perturbed Dirichlet energies converges, in the sense of $\Gamma$-convergence, to a simplified energy functional on $\gamma$. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of $\gamma$. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.