Mathematical analysis of magnetic domain patterns in thin ferromagnetic films
Final Report Abstract
Ferromagnetic materials play a key role in data storage technologies, based on their property to form complex magnetization patterns [8]. A major focus in recent research is devoted to ultrathin ferromagnetic films with perpendicular anisotropy. Typical experimentally observed structures in such films are stripe and bubble domain patterns. Other patterns which appear in thin ferromagnetic films are so called zigzag domain walls. While these patterns have been explored on the basis of specific ansatz configurations in the physical literature, a comprehensive (ansatz-free) mathematical theory which explains these patterns is still missing. The aim of this project hence was to contribute to the development of such a theory and to understand the structure of these domain patterns from the underlying micromagnetic energy. The difficulty in the treatment of the underlying variational model lies in its non-convexity, its nonlocality and its vectorial character. Another peculiarity of the magnetic system is their dipolar character, i.e. the spatial interrelation between positive and negative charges. While there are some tools available for the analysis of such models, no general theory to solve such problems exists. This requires to use different methods from the field of calculus of variations and asymptoptic analysis such as Γ-convergence, the derivation of sharp interpolation estimates and tools from geometric measure theory. In the first part of the project we have derived the scaling law for the ground state energy and the scaling law for the typical domain width for stripe and bubble domain patterns in a thin ferromagnetic film of critical size. As a result we could give a criterion for the size of the film for the onset of pattern formation. The difficulty in this structures is that at leading order we have a cancellation of the interfacial part of the energy and the nonlocal magnetostatic energy. For a corresponding sharp interface model, we were, however, still able to derive a Γ–limit towards some nonlocal perimeter type energy. We note that the Γ–limit can also be seen as a second order variant of the classical nonlocal approximation of the BV-functional. This result has been possible by a novel use of the autocorrelation function for the minimization of these types of functionals and we were able to substantially expand previous results in this direction. The advantage of this approach is the linearization of the functional although with the cost of a substantially more complex (non–convex) underlying function space. The resulting functional is naturally stated in terms of the autocorrelation function and still demands much further research such as basic reguarity results for weak (and strong) solutions. We have also derived the first Γ– limit results for the so called zigzag walls by a delicate use of a suitable test function together with some geometric considerations. We also have further investigated the Ohta-Kawasaki energy where we have derived a macroscopic limit for the three–dimensional model. The result shows in particular that the energy prefers a uniform distribution on large scales in the considered small (but critical) volume fraction limit. We have furthermore investigated the structure of isolated ferromagnetic domains which are relevant in phase tranformation and nucleation processes. Finally, we considered optimal structures in lattice models in the presence of charges of different sign. These investigations are in parts motivated by the dipolar nature of the magnetic models and in fact can explain the domain wall structure of one–dimensional stripe domain wall patterns in thin ferromagnetic films.
Publications
- “Optimal shape of isolated ferromagnetic domains”. In: SIAM J. Math. Anal. 50.6 (2018), pp. 5857–5886
H. Knüpfer and F. Nolte
(See online at https://doi.org/10.1137/18M1175719) - “Emergence of nontrivial minimizers for the three-dimensional Ohta-Kawasaki energy”. In: Pure Appl. Anal. 2.1 (2020), pp. 1–21
H. Knüpfer, C. B. Muratov, and M. Novaga
(See online at https://doi.org/10.2140/paa.2020.2.1) - “On the optimality of the rock-salt structure among lattices with charge distributions”. In: Math. Models Methods Appl. Sci. 31.2 (2021), pp. 293–325
L. Bétermin, M. Faulhuber, and H. Knüpfer
(See online at https://doi.org/10.1142/S021820252150007X) - “Γ-limit for two-dimensional charged magnetic zigzag domain walls”. In: Arch. Ration. Mech. Anal. 239.3 (2021), pp. 1875–1923
H. Knüpfer and W. Shi
(See online at https://doi.org/10.1007/s00205-021-01606-x) - “Asymptotic shape of isolated magnetic domains” (2022)
H. Knüpfer and D. Stantejsky
(See online at https://doi.org/10.48550/arXiv.2110.02384) - “Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy” (2022)
B. Brietzke and H. Knüpfer
(See online at https://doi.org/10.48550/arXiv.2205.10061)
