Skinny Polarization Vortices from Frustrated Dipoles*

Topological textures, such as vortices and skyrmions, have been observed in both magnetic and electric dipolar systems. Despite their similarities, the origin of these textures has been discussed differently in the two systems. In magnetic systems, long-range ordering of magnetic dipoles consists of smooth and continuous rotation of the magnetic moments, and the textures are stabilized by exchange interactions. In dipolar systems, however, non-collinear orderings are hindered by the preference of neighboring dipoles to be aligned either parallel or antiparallel. As such, topological polar textures have been demonstrated in oxide heterostructures via boundary-condition engineering of the electric, elastic, gradient energies and their couplings. Here, we report the existence of skinny polar vortices spanning only ~1 nm in bulk crystals of a quasi-1D chalcogenide, BaTiS₃, by combining the synchrotron measurements, first-principles calculations and phenomenological modeling. Analogous to the magnetic systems, these polar vortices emerge in the bulk state due to geometrical frustration of dipoles on a triangular lattice, which is removed by the superposition of three antipolar waves (triple-Q) along both in-plane and out-of-plane directions. In particular, antipolar interactions between nearest-neighbor sites enables confinement of the vortices to the atomic scale. We will end with a discussion of general phase diagrams for the formation of such textures, and the effect of temperature, strain and electric fields on their stability.