Mathematical analysis of magnetic domain patterns in thin ferromagnetic films

• Applicant: Professor Dr. Hans Knüpfer
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 392124319

Ferromagnetic materials play a key role in data storage technologies, based on their property to form complex magnetization patterns. Of particular interest are ultra-thin ferromagnetic films with perpendicular anisotropy. Commonly experimentally observed structures in such films are stripe and bubble domain phases. While these patterns have been explored on the basis of specific ansatz configurations in the physical literature, a comprehensive mathematical theory which explains these patterns from the underlying micromagnetic energy is still missing. The aim of this project is hence to contribute to the development of such a theory and to derive the structure of these domain patterns from the underlying energy functional. One main tool is the rigorous derivation of relevant asymptotic models in the framework of Gamma-convergence.In particular, we will address the following questions: We first investigate the formation of domain patterns under application of an external field and derive related macroscopic models. Furthermore, we will derive an asymptotic model related to the limit of vanishing film thickness and analyze ground states for this model. We will also consider discrete lattice energies related to one-dimensional configurations. The main goal is to show periodicity, based on reflection positivity principles. In another part of the project, we will derive an effective thin-film energy which takes the effect of the sample boundary into account. Finally, we will address further questions as e.g. the analysis of thicker films and the effect of Dzyaloshinskii-Moriya interaction on the domain pattern formation.The analytical challenge in this project stems from the non-convexity, nonlocality and vectorial character of the underlying micromagnetic energy. While there are some tools available, a general theory to solve such complex problems does not yet exist. We use different methods from the fields of calculus of variations and asymptoptic analysis such as Gamma-convergence, interpolation estimates and tools from geometric measure theory. From a broader perspective, the micromagnetic model can be seen as a prototype model for other non-convex, nonlocal structure forming systems. In particular, we believe that the tools developed in the course of this project will be applicable also in the the study of related pattern forming systems such as e.g. superconductor type I+II models, models for nematic crystals and models from elastoplasticity.

• DFG Programme: Research Grants