The Formation of the Concertina Pattern

Abstract

The concertina is a magnetization pattern observed in elongated thin-film elements of a soft-magnetic material during the switching-process by an external magnetic field. The almost periodic pattern consists of stripe-like quadrangular and triangular domains of uniform, in-plane magnetization. The domains are separated by so-called walls in which the magnetization quickly turns. The main goal of this thesis consists in the analytical and numerical investigation of the formation and coarsening -- which is closely related to the hysteresis of the switching-process -- of the pattern starting from the micromagnetic energy functional. <br /> Experimental observations of very elongated samples suggest that the concertina pattern bifurcates from an oscillatory buckling mode simultaneously all over the sample. The existence of a corresponding parameter regime was confirmed on the level of a linear stability analysis based on the micromagnetic energy. <br /> On the basis of a reduced model derived in that particular parameter regime, we analytically (interpolation estimates, stability and Bloch-wave analysis, etc.) and numerically (numerical bifurcation, path-following, branch-switching, etc.) investigate the formation and the coarsening of the pattern. We quantitatively predict the average period of the concertina pattern and qualitatively predict the coarsening and the hysteresis. We show that on a mesoscopic scale the pattern corresponds to a weak solution of Burgers' equation where walls can be interpreted as shocks. <br /> On the theoretical side, the main ingredients are given by non-linear interpolation estimates derived from the analysis of the inhomogeneous Burgers equation and a Bloch wave analysis linking the concavity of the minimal energy per period to a modulation instability (Eckhaus instability) of the pattern that is at the origin of the coarsening. With the help of a numerical bifurcation analysis, we are able to systematically compute all metastable states. <br /> Finally, we also link the concertina pattern to the magnetization ripple and discuss the effect of a weak (crystalline or induced) anisotropy.