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Vortices and antivortices in antiferroelectric PbZrO3

Nature Materials volume 24, pages 1359–1363 (2025)Cite this article

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Abstract

Ferroelectric materials are characterized by a parallel arrangement of electric dipoles, but at the nanoscale they can present vortices and other non-trivial topological structures1,2,3,4,5,6,7,8,9 that combine small size and topological protection, rendering them functionally attractive10,11,12,13. The driving force for the appearance of vortices in ferroelectrics is the need to minimize the depolarizing fields at interfaces3,4,5,14; by making the polarization rotate, depolarization fields vanish4,5,8,9. Antiferroelectrics, by contrast, are defined by an antiparallel arrangement of electric dipoles15. A priori, therefore, they lack the depolarization fields that drive the appearance of non-trivial topologies in ferroelectrics. At the atomic scale of the dipoles, however, we find that polar discontinuities can still happen, driving the appearance of topological singularities at ferroelastic domain walls.

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Fig. 1: STEM-HAADF image acquired from twinning boundaries and overlaid by corresponding Pb displacement maps.
Fig. 2: Four typical twin-boundary configurations obtained from second-principles relaxation.
Fig. 3: Distribution of polarization P(x, y) obtained in the continuum-field study of twin boundaries in PbZrO3.
Fig. 4: Dynamics of antivortex ferroelastic twin boundary.

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Data availability

All the TEM data are available via the public Sydney eScholarship Repository at https://ses.library.usyd.edu.au/handle/2123/33747. The structure files related to the second-principles simulations in the paper (provided in the POSCAR format), including both structures from relaxations and from 300-K molecular dynamics simulations, are available via the ULiège Open Data Repository at https://doi.org/10.58119/ULG/K6YK6Y.

Code availability

All the first- and second-principles calculations in this study rely on the open-source software ABINIT, DeePMD and LAMMPS (Supplementary Section 3). Pb displacement maps were extracted using the open-access, Python-based Atomap library.

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Acknowledgements

We thank J. Hlinka for insights about domain-wall motion. This work was funded by the FET-Open programme of the EU, project Topological Solitons in Antiferroics (TSAR). Y.L. acknowledges the BIST Postdoctoral Fellowship Programme (PROBIST) funded by the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement number 754510. ICN2 is supported by the Severo Ochoa Centres of Excellence programme, grant number CEX2021-001214-S. G.C. acknowledges financial support from the Catalan government (grant number 2021 SGR 0129), and from the National Research Agency (Agencia Estatal de Investigación), project number PID2023-148673NB-I00. K.R. was supported by the National Science Centre, Poland, grant number 2022/47/B/ST3/02778. P.G. and K.S. acknowledges support from F.R.S. FNRS Belgium under PDR grant T.0107.20 (PROMOSPAN). H.Z. acknowledges the Research IPD STEMA Program of University of Liège. Simulations were performed at the CECI supercomputer facilities funded by the F.R.S. FNRS (grant number 2.5020.1) and also made use of the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the Walloon Region (grant number 1117545). We are grateful for the scientific and technical support from the Australian Centre for Microscopy and Microanalysis (ACMM) as well as the Microscopy Australia node at the University of Sydney.

Author information

Author notes

  1. These authors contributed equally: Ying Liu, Huazhang Zhang, Konstantin Shapovalov.

Authors and Affiliations

  1. Catalan Institute of Nanoscience and Nanotechnology (ICN2), Campus Universitat Autonoma de Barcelona, Bellaterra, Spain

    Ying Liu, Jordi Arbiol & Gustau Catalan

  2. School of Aerospace, Mechanical and Mechatronic Engineering, and Australian Centre for Microscopy and Microanalysis, The University of Sydney, Sydney, New South Wales, Australia

    Ying Liu, Ranming Niu, Julie M. Cairney & Xiaozhou Liao

  3. Theoretical Materials Physics, Q-MAT, Université de Liège, Sart-Tilman, Belgium

    Huazhang Zhang, Konstantin Shapovalov & Philippe Ghosez

  4. School of Physics and Mechanics, Wuhan University of Technology, Wuhan, China

    Huazhang Zhang

  5. Institute of Physics, University of Silesia, Chorzów, Poland

    Krystian Roleder

  6. Institute of Applied Physics, Military University of Technology, Warsaw, Poland

    Andrzej Majchrowski

  7. Institut Català de Recerca i Estudis Avançats (ICREA), Barcelona, Spain

    Jordi Arbiol & Gustau Catalan

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Contributions

Y.L., R.N., X.L., J.M.C. and J.A. acquired and analysed the TEM data. K.R. and A.M. made the antiferroelectric single-crystal sample of PbZrO3. H.Z. performed the second-principles atomistic simulations and K.S. performed the continuum simulations of the twin structures: both worked together on the data analysis. Both types of simulation and related interpretation of the results were conducted under the supervision and guidance of P.G. G.C. and Y.L. conceived the study and wrote the first draft. G.C., Y.L., H.Z., K.S. and P.G. revised all subsequent versions of the text and converged the paper.

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Correspondence to Gustau Catalan.

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Nature Materials thanks Fangfang Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Low-magnification bright-field image showing the ferroelastic domain wall in PZO.

The ferroelastic wall is indicated by a white arrow. The stripes along the pseudocubic <110> directions correspond to translational boundaries.

Extended Data Fig. 2 An STEM-HAADF image showing a twin boundary with the center at PbO plane.

The HAADF image is overlaid by corresponding Pb displacement maps. Head-to-tail Pb displacement configurations are evident.

Extended Data Fig. 3 Oxygen octahedral rotation configuration at a head-to-tail ferroelastic twin boundary.

(a)STEM-HAADF image overlaid with a Pb displacement map (yellow arrows). (b) A STEM-iDPC image showing positions of Pb, Zr, and O atoms. Red lines connect O positions situated between Zr columns in ZrO2 planes along the horizontal direction. The fluctuation magnitude and directional trend of the O-chain indicate both the magnitude and direction of oxygen octahedral rotation. (c) O chain characteristic in PZO single crystal lamella without a twin boundary. The oxygen octahedral rotation angle is defined in the inset as the angle between the line connecting two neighbouring oxygen ions (red line) and the horizontal reference line (yellow dashed line). (d) A rotation map of oxygen octahedra extracted from (b). The sign of O octahedra rotation angle is defined in the inset.

Extended Data Fig. 4 Oxygen octahedral rotation configuration at a head-to-head and tail-to-tail ferroelastic twin boundary.

(a) A STEM-HAADF image overlaid with a Pb displacement map. (b) A STEM-iDPC image showing positions of Pb, Zr, and O ion columns. Red lines connect O positions situated between Zr columns in ZrO2 planes along the horizontal direction. (c) A rotation map of oxygen octahedra.

Extended Data Fig. 5 Second-principles simulation predicted twin boundaries in PZO and the corresponding domain wall energies (EDW).

Each row illustrates one type of twinning boundary: (a) Type-1, (b) Type-2, (c) Type-3, (d) Type-4, (e) Type-5, (f) Type-6, (g) Type-7, and (h) Type-8. The arrows represent local Pb displacements (δPb), with colors indicating their directions. The colors of the squares in the first, second, and third columns correspond to the amplitudes of \({\phi }_{x}={\omega }_{x}{\left(-1\right)}^{{i}_{x}+{i}_{y}+{i}_{z}}\), \({\phi }_{y}={\omega }_{y}{\left(-1\right)}^{{i}_{x}+{i}_{y}+{i}_{z}}\), and \({\omega }_{z}\), respectively. Here, ωx, ωy, ωz are the local oxygen octahedral rotations around the three Cartesian axes, while ix, iy, iz are integers locating the oxygen octahedra.

Extended Data Fig. 6 Averaged twin boundary structures from the finite-temperature simulations at 300K.

The twin boundary types are: (a) Type-1, (b) Type-2, (c) Type-3, (d) Type-4, (e) Type-5, and (f) Type-6. The simulations were performed using the twinning boundary Types 1-6 shown in Figs. S2.2–S2.5 as the initial structures. Types 7, 8 evolved into Type 6 prior to reaching 300 K.

Extended Data Fig. 7 Distribution of polarisation P(x,y) obtained in the continuum field study of ‘hard’-type twin boundaries.

(a) Head-to-tail and (b) head-to-head dipole configurations. Colour hue and intensity correspond to, respectively, local orientation and magnitude of polarisation. The highest colour intensity corresponds to ~30% of maximum polarisation obtained in the calculations. Inset: colour hue-intensity wheel.

Extended Data Fig. 8 Distribution of tilt ϕ(x,y) obtained from the continuum field study.

(a) “Easy”-type head-to-tail (HtT) twin boundary, (b) “Easy”-type head-to-head (HtH) twin boundary, (c) “Hard”-type HtT twin boundary, and (d) “Hard”-type HtH twin boundary.

Extended Data Fig. 9 Intermediate distribution of polarisation P(x,y) obtained in the continuum-field damped relaxation study based only on polarisation-dependent energy terms.

The highest colour intensity corresponds to ∼30% of maximum polarisation obtained in the calculations. Inset: colour hue-intensity wheel.

Extended Data Fig. 10 Comparative effect of reduced correlation length on continuum-field simulations, and comparison for the cases of head-to-tail wall (Fig. 3(a) of the manuscript) and head-to-head wall (Fig. 3(b) of the manuscript).

Reduced correlation lengths reduce the core size, but not the topological winding number.

Supplementary information

Supplementary Information

Supplementary Sections 1–3, Table 1.1 and Figs. 1–7 and discussions.

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Liu, Y., Zhang, H., Shapovalov, K. et al. Vortices and antivortices in antiferroelectric PbZrO3. Nat. Mater. 24, 1359–1363 (2025). https://doi.org/10.1038/s41563-025-02245-3

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