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The Formation and Coarsening of the Concertina Pattern

Joint work with F. Otto, R. Schaefer, and H. Wiczoreck.

The concertina is a magnetization pattern observed in elongated thin-film elements of a soft-magnetic material. It is a ubiquitous domain pattern that occurs in the switching process of the uniform magnetization due to the reversal of an applied magnetic field in the direction of the long axis of the small element. The almost periodic pattern consists of stripe-like quadrangular and triangular domains of uniform, in-plane magnetization. The domains are separated by sharp transition layers namely walls in which the magnetization quickly turns.

Experimental observations suggest that the concertina pattern bifurcates from an oscillatory buckling mode simultaneously all over the sample. The existence of a corresponding parameter regime, i.e., thin, wide samples, was confirmed by Cantero & Otto on the level of a linear stability analysis based on the micromagnetic energy — a non-convex and due to Maxwell’s equations non-local variational model.

On the basis of a reduced model derived in the particular parameter regime by Gamma-convergence, we investigate the formation and the coarsening of the pattern using various tools from (asymptotic) analysis — non-linear interpolation estimates, Bloch-wave analysis, bifurcation analysis and amplitude functionals — and numerical simulations — path-following, branch-switching. Exploring the energy landscape with the help of these methods, we quantitatively predict the average period of the concertina pattern and qualitatively predict its hysteresis. In particular, we argue that the experimentally observed coarsening of the concertina pattern is due to secondary bifurcations related to an Eckhaus instability. The latter is a non-linear instability that is known to occur in convective systems.

We finally discuss the effect of a weak (crystalline or induced) anisotropy and contrast this instance of a quenched disorder to thermal fluctuations in the Landau-Lifschitz equations.

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