Calculus of Variations and Geometric Measure Theory

E. Davoli - L. D'Elia - J. Ingmanns

Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions

created by davoli on 08 Jun 2023
modified on 29 Dec 2023


Accepted Paper

Inserted: 8 jun 2023
Last Updated: 29 dec 2023

Journal: Journal of Nonlinear Science
Year: 2023


We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere, and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.

Keywords: Micromagnetics, stochastic homogenization, chiral magnetic materials, Dzyaloshinskii-Moriya interaction