Invited
Ferroelectric incommensurate spin crystal

Ferroics can form complex topological spin structures such as vortices and skyrmions, especially when subjected to particular boundary conditions. In ferroelectrics vortex-like electric dipole-based topological structures have been observed in dedicated ferroelectric systems, especially PbTiO3/SrTiO3 ferroelectric/insulator superlattices, which have proven to be an ideal model system due to their high depolarising field. The large electrostatic energy is minimised by local rotations of surface dipoles, similar to ferromagnetic Kittel domains, in the which local dipoles rotate in such a way to reduce both the depolarization and stray fields, avoiding the suppression of the ferroelectricity in the thin films.
In single PbTiO3 epitaxial layers sandwiched between SrRuO3 electrodes we observe a more complex domain structure analogue of the double-Q magnetic spin crystal phase. This comprises of periodic clockwise and anti-clockwise ferroelectric vortices which are modulated by a second cycloidal ordering along their toroidal core. Thus, one Q vector determines the periodicity of vortices and a second Q vector breaks the uniformity of the domains in the perpendicular direction, leading to a state described by a double- Q modulation. The interplay of only two orthogonal periodic modulations results in the so-called incommensurate spin crystal. [1]
The presence of such a double-Q structure, mediated by incommensurate interactions, would require an electric counterpart of the magnetic Dzyaloshinskii-Moriya interaction (DMi). Such an electric DMi could provide the phenomenological explanation of the emergence of magnetic-like phases in ferroelectric systems.

References
[1] D. Rusu et al., Nature 602, 240 (2022)