Matheon-C15 Pattern formation in magnetic thin films

Background.

Magnetic materials display a complex variety of patterns and singular structures on many different length scales. The way such patterns form and evolve in time is vitally important for many key technologies related to sensors, memory and recording devices where particularly magnetic thin films offer themselves in practice. Very recently strong interest in magnetic films stems from their combination with semiconductor elements in view of hybrid devices for future spin electronics, a rapidly developing new technology. A widely accepted theoretical framework for ferromagnetism is based on a continuum model due to Landau and Lifshitz. Within their theory, a universal energy functional, the micromagnetic energy, that captures in principle magnetic phenomena on every relevant length scale, including the coarse domain structures as well as fine substructures of domain walls and their mutual interaction. Mathematically, the formation of these patterns and structures is driven by the interplay of non-convexity and non-locality due to the coupling with Maxwell's equations. In practice, however, the general model is by far to complex in order to extract interesting predictions or to effectively perform suitable computations in high resolution. Hence there is a need for suitable reduced or effective models that capture specific features in parameter regimes of specific interest.

Whereas in physics community micromagnetic theory is mainly used on the basis of computation, our main focus concerns rigorous model reductions and the investigation and derivation of qualitative features of patterns and microstructure. Mathematically, thin films distinguish themselves from bulk materials by a different form of induced stray-field interaction that becomes singular in dimensionally reduced theories and constitutes the main analytical challenge in such models. In recent time important analytic contribution have been made that were mostly focused on energy minimizing configuration. Particularly typical domain wall structures have been elucidated, cf.[Mel1, Mel2,Mel2, DKMO1] , and reduced models for soft magnetic films were rigorously derived on the basis of Gamma-convergence cf. [DKMO2],. To a much lesser extend, however, dynamic questions were addressed so far. This will be one of the main direction of research within the research group. On the other hand we aim to provide analytical background for phenomena that emerge in the production of new magnetic materials for which an adequate theoretical framework is still to be developed:

Research.

The treatment of dynamic micromagnetic phenomena and relaxation processes is clearly a multiscale problem and by far to complex in order to draw refined qualitative properties from the global picture of Landau-Lifshitz dynamics. Thus, as in the static case, a promising strategy consists in the separate treatment of magnetic structure elements. In this context we investigate the motion of domain walls driven by applied fields, a mechanism common to most memory and recording devices. The goal is to give a rather complete analysis of the dynamics of diffuse domain walls including dynamic scaling laws in terms of all involved parameters. The main challenge is then the modeling and analysis of possible dynamic interactions of several domain walls in complex networks.

An efficient dynamic description of magnetic patterns, however, requires in the first place the derivation of the effective dynamics of micromagnetic singularities like sharp domain walls and vortices. The latter program has successfully been carried out in the context of certain Ginzburg-Landau models for superconductivity, but it remains a challenge in the context of reduced models for magnetic thin films: If, for a soft material, the dimensions of the specimen are much larger than the single-domain limit, closed-flux magnetizations with vanishing stray-field, somewhat comparable with velocity fields in hydro-dynamics, are favored. We will investigate the gradient flow associated to a divergence constrained model from the gradient theory of phase transitions. We focus on asymptotic features and (geometric) evolution laws for vortices and interfaces.

In the context of equilibrium patterns, novel magnetic devices, such as magnetic multilayers or semiconducting hybrid structures as intended for future spin electronic applications, exhibit surprising new structural and magnetic properties. An essential task to make such new materials available for technological applications is an adequate qualitative understanding in order to optimize their fabrication. Important examples are epitaxially grown ferromagnetic films, such as manganese arsenide films on semiconducting substrates, like e.g. gallium arsenide. One of the most striking features of such films is the formation of self-organized striped patterns of co-existing ferro- and paramagnetic phases rising from anisotropic stresses. At the same time, complex domain structures develop within the ferromagnetic phase, cf. [PDI1], reflecting a subtle interplay of demagnetization and magneto-elastic effects. In collaboration with the Paul-Drude-Institute for solid state electronics, a systematic understanding of the micromagnetic properties of such manganese arsenide films is to be developed within this project.

Literature.