Mechanistic Origin of a Stable Magnetic Vortex in Three-Dimensional Pyramid Fe Thin Films

A magnetic vortex, characterized by curling in-plane magnetization, is generally unstable in two-dimensional (2D) ferromagnetic thin films. Here, we demonstrated that this vortex could be stable in three-dimensional (3D) pyramid-shaped Fe thin films and elucidated mechanistic origin of the stable vortex. Magnetization measurements reveal characteristic M−H hysteresis loops with a pronounced bending and a gradual slope near zero magnetization, contrasting strongly with the abrupt switching seen in 2D films. By decomposing the magnetization processes on each facet in pyramid, we identify the sequence of vortex formation, stabilization, and annihilation. The key factor is the 3D geometry: non-coplanar facet junctions at the ridge lines act as structural singularities that naturally pin domain walls (DWs). These ridge lines restrict DW motion, confine local magnetic structures, and mediate inter-facet interactions, creating geometrical constraints enhancing vortex stability. Vortex formation is driven by magnetostatic energy minimization, as in 2D films. However, ridge-induced weakening of inter-facet exchange becomes the dominant factor in the 3D pyramidal structure. Overall, the interplay of shape anisotropy, magnetostatic, exchange, and Zeeman energies under 3D constraints provides a clear framework for vortex stability, offering the first mechanistic insight into stable vortices in 3D ferromagnetic films and supporting future 3D magnetic devices.