Observation of multi-order polar radial vortices and their topological transition

Topological states have garnered enormous interest in both magnetic and ferroelectric materials for promising candidates of next-generation information carriers. Especially, multi-order topological structures with modulative topological charges are promising for multi-state storage. Here, by engineering boundary conditions, we directly observe the self-assembly two-order ferroelectric radial vortices in high-density BiFeO₃ nanostructures. The as-observed two-order radial vortex features a doughnut-like out-of-plane polarization distribution and four-quadrant in-plane distribution, with the topological charge of Q = 0. Systematic dimensional control of the BiFeO₃ nanostructures reveals size-dependent stabilization of distinct topological states, from elementary one-order to complex three-order radial vortices, which is further rationalized by phase-field simulations. The transition between different topological states with various topological charges is also realized under an external electric field. This study opens up an avenue for generating configurable polar topological states, offering potential advancements in designing high-performance multi-state memory devices.