Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics

Abstract

We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.

Homogenization of Chiral Magnetic Materials: A Mathematical Evidence of Dzyaloshinskii’s Predictions on Helical Structures

“…The homogenization analysis performed in our paper is motivated by recent technological advances in the field of spintronics, first and foremost, by the observation, in magnetic systems lacking inversion symmetry, of chiral spin textures known as magnetic skyrmions (Ferriani et al 2007;Bogdanov and Hubert 1999), whose origin is ascribed to the Dzyaloshinskii-Moriya interaction (DMI) (Dzyaloshinsky 1958;Fields 1956). We refer to Melcher (2014), Muratov and Slastikov (2017), as well as Li and Melcher (2018) and the references therein, for a mathematical analysis of micromagnetic models including DMI [see also (Cicalese and Solombrino 2015;Cicalese et al 2016;Cicalese 2019) for a study of effective theories and chirality transitions in the discrete-to-continuous setting, and Hrkac et al (2019) for recent results on the numerics of chiral magnets]. More precisely, our work builds on Dzyaloshinskii's observations in Dzyaloshinskii (1964), Dzyaloshinskii (1965) where, based on the Landau theory of second-order phase transitions, the emergence of helicoidal structures is predicted (see also Bak and Jensen 1980).…”

Numerical analysis of the Landau-Lifshitz-Gilbert equation with inertial effects

“…For ease of presentation, in the micromagnetic energy functional (4) below, we consider only the leading-order exchange contribution. The numerical treatment of standard lower-order energy contributions (e.g., magnetocrystalline anisotropy, Zeeman energy, magnetostatic energy, Dzyaloshinskii-Moriya interaction) is well understood; see, e.g., [14,25,17,20].…”

Computational micromagnetics with Commics

“…The original tangent plane scheme from [12] is formally first-order in time and was analyzed for the energy being only the exchange contribution. The scheme was extended to general lower-order effective field contributions [14,33], DMI [52], and the coupling with other partial differential equations, e.g., various forms of Maxwell's equations [60,19,59,42], spin diffusion [10], and magnetostriction [20]. A projection-free version of the method was analyzed in [10,67].…”