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

Abstract

In this paper, we investigate the influence of the bulk Dzyaloshinskii-Moriya interaction on the magnetic properties of composite ferromagnetic materials with highly oscillating heterogeneities, in the framework of-convergence and 2-scale convergence. The homogeneous energy functional resulting from our analysis provides an effective description of most of the magnetic composites of interest nowadays. Although our study covers more general scenarios than the micromagnetic one, it builds on the phenomenological considerations of Dzyaloshinskii on the existence of helicoidal textures, as a result of possible instabilities of ferromagnetic structures under small relativistic spin-lattice or spin-spin interactions. In particular, we provide the first quantitative counterpart to Dzyaloshinskii's predictions on helical structures.

Lattice Solutions in a Ginzburg–Landau Model for a Chiral Magnet

“…The occurrence of helical phases is common to other mathematically related theories for condensed matter such as the Oseen-Frank model for chiral liquid crystals (see, e.g., Virga 1995) or the Gross-Pitaevskii model for spin-orbit coupled Bose-Einstein condensates (see, e.g., Aftalion and Rodiac 2019). Helical structures in chiral ferromagnets are also discussed in Davoli and Di Fratta (2020) and Muratov and Slastikov (2017).…”

Linearized von Kármán theory for incompressible magnetoelastic plates

“…As already mentioned in Section 1, in the present work we neglect other contributions in the magnetostatic energy, like the anisotropy energy [1,41] or the DzyaloshinskiiMoriya interaction energy [14]. For simplicity, we also do not consider applied forces or external magnetic fields.…”

“…We mention that the magnetostatic energy usually involves additional terms such as the anisotropy energy [1,41] and the DzyaloshinskiiMoriya interaction energy [14] that, on first approximation, we are neglecting. Also, we are not considering the effect of body or surface forces applied on the body, as well as the one given by the presence of an external magnetic field through the so-called Zeeman energy.…”

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

“…Embedding planar structures in the three-dimensional space permits altering their magnetic properties by tailoring their local curvature. The interplay between geometry, topology, and Dzyaloshinskii-Moriya interaction (DMI) leads to the formation of novel magnetic spin textures, e.g., chiral domain walls and skyrmions [9,13,32]. The curvature effects have been shown to play a crucial role in stabilizing these chiral spin-textures.…”