Ferroelectric incommensurate spin crystals

Ferroics, especially ferromagnets, can form complex topological spin structures such as vortices1 and skyrmions2,3 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 field4–8. 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 materials11–13. 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.