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.