A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types.
Lukas Döring, Radu Ignat, and Felix Otto
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Submission date: 13. Jun. 2012 (revised version: September 2013)
published in: Journal of the European Mathematical Society, 16 (2014) 7, p. 1377-1422
DOI number (of the published article): 10.4171/JEMS/464
MSC-Numbers: 49S05, 49J45, 78A30, 35B32, 35B36
Keywords and phrases: thin-film micromagnetics, domain walls, gamma convergence, reduced model
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We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called “domain walls” – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors m±∈ S2 that differ by an angle 2α. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which performs a partial rotation in S2 by an angle 2θ, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angle θ to α in S1. The angle θ is chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed asymptotically and numerically in . Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asymmetric domain walls including their extended tails, which were previously inaccessible to brute-force numerical simulation.