Ferroelectric incommensurate spin crystals

Ferroics, especially ferromagnets, can form complex topological spin structures such as vortices¹ and skyrmions^2,3 when subjected to particular electrical and mechanical boundary conditions. Simple vortex-like, electric-dipole-based topological structures have been observed in dedicated ferroelectric systems, especially ferroelectric-insulator superlattices such as PbTiO₃/SrTiO₃, which was later shown to be a model system owing to its high depolarizing field^4-8. To date, the electric dipole equivalent of ordered magnetic spin lattices driven by the Dzyaloshinskii-Moriya interaction (DMi)^9,10 has not been experimentally observed. Here we examine a domain structure in a single PbTiO₃ epitaxial layer sandwiched between SrRuO₃ electrodes. We observe periodic clockwise and anticlockwise ferroelectric vortices that are modulated by a second ordering along their toroidal core. The resulting topology, supported by calculations, is a labyrinth-like pattern with two orthogonal periodic modulations that form an incommensurate polar crystal that provides a ferroelectric analogue to the recently discovered incommensurate spin crystals in ferromagnetic materials^11-13. These findings further blur the border between emergent ferromagnetic and ferroelectric topologies, clearing the way for experimental realization of further electric counterparts of magnetic DMi-driven phases.