Analysis and Modeling of Magnetic Skyrmions

Ferromagnetic materials possess a spontaneous magnetization. Sufficiently small volumes of a ferromagnet have an intensity of magnetization controllable through external magnetic fields. The giant magnetoresistance, for which Fert and Grünberg were awarded the Nobel prize in 2007, allows for a massive change in the resistance of a conductor in response to an applied magnetic field. The ability to read magnetization states through resistance measurements and write magnetic domains through electrical currents has inaugurated the field of spintronics and led to the design of new magnetic recording devices. However, several challenges must be addressed before spintronics can be turned into a competitive technology. Magnetic skyrmions are one of the promises of the young science of spintronics and might help tackle most issues. This is due to their ultimate small size and stability and because they can be moved by weak spin-polarized currents. Skyrmions are a class of solitons that are topologically stable and have quasiparticle properties. Skyrmions may behave like particles, but they are inherently more complex structures due to their collective nature. Magnetic skyrmions emerge as topological defects in the magnetization texture that carry a specific topological charge, referred to as the skyrmion winding number. Their mathematical understanding, which has gained signicant attention these days, is still in its infancy. The proposed research plans to contribute to our theoretical understanding of the emergence of magnetic skyrmions in curved geometries. Also, it intends to elucidate the origin of antisymmetric exchange interaction through a refined analysis of the Heisenberg model in the short -range regime. We briefly describe the main project aims. First, the derivation of reduced higher-order variational problems that capture the curvature`s influence on the observable magnetization states. Second, the analysis of topologically protected states in spherical thin films. Indeed, spherical thin films can support magnetic skyrmions stabilized by curvature effects only, and the main objective here is towards geometric characterizations of ground states with prescribed skyrmion winding numbers. Finally, the analysis of symmetric and antisymmetric exchange energies as short-range limits of non-local Heisenberg energies. Micromagnetic models of symmetric and antisymmetric exchange contributions account only for local interactions. In contrast, the Heisenberg model allows, more generally, for non-local interactions, and the interest here is in showing that exchange energies are short-range limits of non-local energies of the Heisenberg type. The comprehensive understanding of the mentioned phenomena is a fundamental issue that relies on a combination of ideas from various elds, including solid mechanics, topology, partial dierential equations, and the calculus of variations. The methods involved will contribute to the mathematical theory of partial differential equations and will propel applications in related fields. Indeed, similar open questions permeate additional condensed and soft matter physics (such as superconductivity and nematic liquid crystals), and their answers often demand new mathematical machinery.

This project successfully established a new mathematical foundation for predicting the behavior of complex, microscopic magnetic structures. By developing rigorous formulas to describe how magnetic forces interact over distances and within curved, ultra-thin materials, we have provided the theoretical tools necessary to design the next generation of highly efficient data storage and computing devices. As technology shrinks, the materials used in computer memory and sensors become incredibly thin and often take on curved nano-shapes. At this microscopic scale, classical physics models are no longer sufficient. Magnets do not just interact with their immediate neighbors; they experience nonlocal interactions, where magnetic forces influence each other across distances. Furthermore, tiny magnetic whirls called skyrmions can form, which hold massive potential for data storage but are notoriously difficult to stabilize and predict mathematically. Until now, the scientific community lacked the precise mathematical rules needed to fully understand and control these complex phenomena. Over the course of this project, our team developed new mathematical frameworks to solve these problems. First, we created accurate mathematical models for curved, ultra-thin magnetic films, specifically accounting for the unique forces that create magnetic whirls. Second, we solved a major puzzle regarding nonlocal magnetic interactions. By extending and proving complex mathematical formulas, we provided exact conditions for how these long-distance magnetic forces operate in both symmetric and asymmetric scenarios. Finally, we successfully validated classical theorems for microscopic ferromagnetic particles under these new, complex conditions. These mathematical advances have direct, profound implications for technology and the economy. By understanding how tiny magnetic whirls behave in confined spaces, engineers can develop new spintronic devices. These devices promise to store vastly more data than current hard drives while consuming a fraction of the electricity. Ultimately, our mathematical blueprints will help drive the development of faster, smaller, and more energy-efficient electronics, contributing to a more sustainable technological future.