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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 field4,5,6,7,8. 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 materials11,12,13. 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.

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Fig. 1: Unit-cell-scale electric dipole topologies.
Fig. 2: Macroscopic ordering.
Fig. 3: Plan-view TEM imaging.
Fig. 4: Double modulation.

Data availability

The data that support the findings of this study are available at the University of Warwick open access research repository ( or from the corresponding author on reasonable request.


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This work was partly supported by the EPSRC (UK) through grant nos. EP/P031544/1 and EP/P025803/1. M.A. acknowledges the Theo Murphy Blue Skies Award of the Royal Society. This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science user facility, operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357. We also acknowledge the technical support from M. Crosbie. We would like to acknowledge the University of Warwick Research Technology Platform for assistance in the research described in this paper.

Author information

Authors and Affiliations



M.A. conceived the idea. D.R., M.A. and A.M.S. designed the experiments. D.R. prepared the samples, performed DFT and DF-TEM experiments and analysed the data. J.J.P.P. and J.A.G. performed the STEM experiments and analysis. G.A.A.N., J.S., D.H. and D.R. collected the synchrotron data. T.P.A.H. and D.R. analysed the XRD data. R.B. performed the two-beam diffraction contrast simulations. All authors contributed to the discussions. All authors wrote the manuscript.

Corresponding author

Correspondence to Marin Alexe.

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Extended data figures and tables

Extended Data Fig. 1 Structure characterization.

a, AFM topography of the surface of our (SrRuO3)11/(PbTiO3)13/(SrRuO3)11 trilayer sample. b, Low-magnification cross-sectional STEM of the sample. The scale bar is 10 nm.

Extended Data Fig. 2 RSM data.

a, 3D reciprocal space map around the DSO 002pc Bragg peak. The side panels show the 2D projected intensity of the recorded scatter. b, RSM around the asymmetric reflection (103)pc. c, Reciprocal space map Qz versus Qx. d, Comparison plot of the extracted Qz scan at Qx = 0 and Qx = 0.07 Å−1. e, Integrated (boxed area) line profile showing the first-order and second-order satellite peaks and their widths. f, Reciprocal space map Qz versus Qy. g, The integrated (boxed area) line profile shows weak first-order satellite peaks corresponding to a periodicity of about 8.05 nm. h, Plan-view projection into a Qx versus Qy RSM map with extracted line scans showing the in-plane distribution of the satellite peaks.

Extended Data Fig. 3 Cross-sectional DF-TEM.

a, Image of a (100)pc cross section taken under the g = 020pc excitation condition. b, Image of the same (100)pc cross section taken under the g = 002pc excitation. c, Image of a (010)pc cross section (that is, cut at 90° from a and b) taken using g = 002pc. The scale bars are 20 nm.

Extended Data Fig. 4 Plan-view DF-TEM.

a, Low-magnification plan view of the complex domain pattern take under the g1 = 110pc condition. The figure inset shows the enlarged boxed area. b, Plan-view dark-field image taken under g2 = 100pc. c, Plan-view diffraction contrast taken under g3 = 010pc excitation. The scale bars are 100 nm for a and 30 nm for b and c.

Extended Data Fig. 5 Noise filter.

a, 2D Fourier transform of the plan-view image taken under g1= 110 excitation. b, Bandpass filter that removes the noise and retains the signal for |Q| < 0.16 nm−1. c, Bandpass filter that also removes the central spot and the signal for |Q| < 0.015 nm−1. The scale bar is 30 nm.

Extended Data Fig. 6 Filtered/unfiltered plan-view DF-TEM.

a, The bandpass filter improves the signal-to-noise ratio without introducing artefacts. Both the filtered and unfiltered images show, apart from the labyrinth pattern, a periodic modulation in the contrast along the individual domains. b, The second modulation permeates the labyrinth pattern.

Extended Data Fig. 7 Diffraction contrast simulations.

Left, experimental plan-view DF-TEM images of the vortex array. Note that the g = 110pc image is at higher magnification and Bragg filtered. Right, two-beam Howie–Whelan diffraction contrast simulations of the contrast arising from the strain fields of a 2D array of vortices as described in the main text (deviation parameter s = 0.01 nm). The scale bars are 30 nm for g = 010, 30 nm for g = 100 and 10 nm for g = 110.

Extended Data Fig. 8 Tilt map.

Left, oxygen tilt behaviour along a row of unit cells. Right, tilt map throughout the PTO layer.

Extended Data Fig. 9 Cross-sectional polar maps.

a, Polarization maps along the ordered [010]pc direction. The projection of a cycloidal and helical modulated vortex array into the [001]pc–[010]pc plane shows that the domain topology is retained. b, The projection of the modulated vortex array into the [001]pc–[100]pc plane shows that the cycloidal modulation allows the polar vector to rotate in plane, similar to the experimental polar map, whereas the helical modulation does not.

Extended Data Fig. 10 X-ray CD data.

a, [100]pc//[001]o in the scattering plane. b, [010]pc//[−110]o in the scattering plane. The first row presents the sum of the dichroic signal, (I+ + I)/2, in which I± refers to the measured intensity for opposite helicities of the incoming light and the second row shows the CD signal, (I+ − I)/(I+ + I), and its behaviour on 180° rotation of the sample. The third panel shows the dichroic signal, (CDϕ1 − CDϕ2)/2, associated with a rotation of the sample by 180°, demonstrating a weak signal at the ± satellites in a, which is absent in b. In c and d, we show the dichroism under sample rotation when [100]pc//[001]o is in the scattering plane (c) and when the [010]pc//[−110]o direction is in the scattering plane (d). The upper panels of c and d plot the data from the two satellites onto a common axis, with the lower panels showing their average.

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Rusu, D., Peters, J.J.P., Hase, T.P.A. et al. Ferroelectric incommensurate spin crystals. Nature 602, 240–244 (2022).

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