Critical Fields in Ferromagnetic Thin Films: Identification of Four Regimes

We are interested in critical fields for ferromagnetic elements: At which strength of the external field does a branch of stationary magnetizations become unstable, and what is the unstable mode? We consider samples which are infinite in the direction of the external field and have a rectangular cross-section, of much smaller thickness than width, as an idealization of a thin film element. For this geometry Aharoni [Phys. Stat. Sol. 16, 3-42 (1966)] claims that there are only three different regimes: The unstable mode is either of coherent rotation type, of buckling type, or of curling type. We discover a large fourth parameter regime with an unstable mode displaying an oscillation in the infinite direction. We prove the existence of exactly four regimes by rigorously analyzing the scaling of the Rayleigh quotient of the Hessian of the energy functional. The parameters are the film width, the film thickness, and the exchange length.