An effective model for boundary vortices in thin-film micromagnetics

Kurzke, Matthias; Ignat, Radu

Abstract

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of "global Jacobian"detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a Γ-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity 1. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

Citation

Kurzke, M., & Ignat, R. (2023). An effective model for boundary vortices in thin-film micromagnetics. Mathematical Models and Methods in Applied Sciences, 33(9), 1929-1973. https://doi.org/10.1142/S021820252350046X

• Journal Article Type: Article
• Acceptance Date: Apr 8, 2023
• Online Publication Date: Jun 21, 2023
• Publication Date: Aug 1, 2023
• Deposit Date: Apr 4, 2023
• Publicly Available Date: Jun 22, 2024
• Journal: Mathematical Models and Methods in Applied Sciences
• Print ISSN: 0218-2025
• Electronic ISSN: 1793-6314
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 9
• Pages: 1929-1973
• DOI: https://doi.org/10.1142/S021820252350046X
• Keywords: Applied Mathematics; Modeling and Simulation
• Public URL: https://nottingham-repository.worktribe.com/output/19270567
• Publisher URL: https://www.worldscientific.com/doi/10.1142/S021820252350046X
• Additional Information: This is a pre-copyedited, author-produced version of an article accepted for publication in Mathematical Models and Methods in Applied Sciences. The published version of record Radu Ignat and Matthias Kurzke, An effective model for boundary vortices in thin-film micromagnetics, Mathematical Models and Methods in Applied Sciences 2023 33:09, 1929-1973 is available online at: https://doi.org/10.1142/S021820252350046X