Topological transformation of magnetic hopfion in confined geometries

Three-dimensional (3D) topological magnetic structures have attracted enormous interest due to their exceptional spatial structures and intriguing physics. Hopfions, characterized by the Hopf index, are 3D spin textures that emerged as closed twisted skyrmion strings. A comprehensive understanding of magnetic structural transitions within nanostructures is crucial for their applications in spintronics devices. Despite the demonstration of stabilization and current-driven dynamics of hopfion, their behavior in geometric confinement has remained unexplored. Here, we investigate the transformation between hopfions and torons in various nanostructures using micromagnetic simulations. By tailoring the axial ratio of elliptical nanodisks, the elliptical hopfion is found to be transformed into a toron structure. Moreover, the current-driven topological transformation between hopfion and toron has also been realized in finite-sized nanostripes and stepped nanostructures. This deformation and transformation arise from the repulsive potential of the boundaries or edges. To connect real-space observations and 3D topological spin configurations, we simulate the Lorentz transmission electron microscope images of the aforementioned magnetic structures. This study, uncovering the dynamics and transformation of hopfions, will invigorate 3D magnetic structures-based memory and logic devices.