Reduced theory of symmetric and antisymmetric exchange interactions in nanowires

¹ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy
² Department of Physics and Astronomy, Uppsala University, Box-516, Uppsala SE-751 20, Sweden
³ School of Mathematics, University of Bristol, University Walk, Bristol BS8 1UG, UK

^* Corresponding author: valeriy.slastikov@bristol.ac.uk

Received: 22 July 2024
Accepted: 30 November 2024

Abstract

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve γ and defined in the space of 𝕊²-valued functions. The perturbation K is represented by a matrix-valued function defined on 𝕊² with values in ℝ^3×3. Under natural regularity conditions on K, we show that the family of perturbed Dirichlet energies converges, in the sense of Γ-convergence, to a simplified energy functional on γ. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of γ. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii–Moriya interaction (DMI) is present.