Strong interplay between polar and structural topologies

Domain: physics
Published: 2026-03-13
Editorial outcome: accept

Abstract
Topological structures in condensed matter systems unlock new possibilities for the development of nanoelectronic devices. However, the potential of antiferroelectrics to host topological features remains largely unexplored, constrained by significant energy barriers from antiparallel dipole coupling that suppress polarization rotation and challenges in high-quality film fabrication. Here for the first time, we find that, dislocations, the most common one-dimensional topological structures in crystals, exhibit unexpectedly strong couplings with polar topologies and induce ordered polar antihedgehog lattices in antiferroelectric PbZrO3 driven by the interplay of electrostrictive effect and the flexoelectric field. Combined atomic-resolution transmission electron microscopy and phase-field simulations, it is revealed that the polarizations converging at dislocation cores and diverging between dislocations define lattices characterized by checkerboard-like antihedgehogs, respectively. Unexpected interplay between polar and structural topologies establishes a new paradigm for topology design.

Introduction
For centuries, geometry has provided a precise framework for describing the physical world. The advent of topology, however, has fundamentally transformed condensed matter physics, revealing that phase classification extends beyond the symmetry-based Landau–Ginzburg–Wilson paradigm1. By focusing on global invariants rather than local symmetries, topology has introduced a new dimension to phase characterization, enabling the discovery of novel phases with distinct and robust topological signatures. Topological structures in condensed matter systems arise from the intricate interplay among various degrees of freedom, including lattice distortions, charge ordering, spin alignment, and orbital interactions. In ferroelectric materials, lattice and charge degrees of freedom dominate, unlike in magnetic systems where spin degrees of freedom play the primary role. The coupling of elastic, electrostatic, and gradient energies facilitates the stabilization of exotic topological configurations, including flux-closures2, vortices3, skyrmions4, polar waves5, merons6 and polar Bloch points7. Remarkably, the spatial dimensions of ferroelectric topological domains are often orders of magnitude smaller than their magnetic counterparts, positioning them as ideal candidates for next-generation functional electronic devices, particularly in information storage and sensor-driven devices. The discovery of an ever-growing family of ferroelectric topological states continues to motivate the exploration of new material platforms, further expanding the landscape of polar topologies and their potential applications.

Research on complex topological structures has predominantly focused on ferroelectric systems for decades. In contrast, antiferroelectrics, as counterparts to ferroelectrics, are distinguished by their inherently zero remnant polarization and field-driven antiferroelectric-ferroelectric phase transitions. Such characteristics render antiferroelectrics highly desirable for high-performance applications, including high-energy storage capacitors8,9, memories10, cooling devices11,12, thermal switches13, and electrostrain devices14,15. However, the exploration of topological structures in antiferroelectric materials remains largely unexplored, mainly due to two formidable challenges. First, the strong antiparallel dipole coupling in antiferroelectrics, characterized by high switching energy barriers, hinders the control of local polarization rotation and limits effective long-range manipulation under homogeneous external fields. Second, the pronounced mismatch between typical antiferroelectrics, e.g., lead zirconate-based antiferroelectrics, and conventional substrates pose formidable challenges to the fabrication of ultrathin films and superlattices, which serve as critical platforms for topology stabilization and exploration. Nevertheless, recent studies have shown that the unique sensitivity of antiferroelectric order parameters to boundary conditions, strain, electric fields, and temperature enables favorable conditions for polarization rotation14,16-20, a key precursor for the stabilization of topological states. The potential realization of topological structures in antiferroelectric materials would complement those observed in ferroelectrics, enriching the understanding of polarization phenomena, and expanding the design possibilities for functional materials. Such topological states could enable functionalities like electric field-driven transformations, offering unprecedented design flexibility and tunability for next-generation high-performance electronic devices and sensors. Despite the exciting prospect of topologies in antiferroelectric materials, the experimental observation of topological textures remains an unresolved challenge.

Here, by using the most common one-dimensional topological structures in crystals, those are, dislocations, we have induced novel polar topological arrays in a classical antiferroelectric material, marking a significant step forward in polar topology research. Specifically, using aberration-corrected scanning transmission electron microscopy (AC-STEM) and phase-field simulations, we report that a[010] high-density interfacial dislocation arrays form within the PbZrO3/KTaO3 (PZO/KTO) heteroepitaxial system to accommodate lattice mismatch. The electrostrictive effect and the flexoelectric field induced by the dislocation network drive the emergence of polar antihedgehog arrays. In these arrays, the combination of polarizations converging at dislocation cores and diverging between dislocations defines a lattice characterized by checkerboard-like antihedgehog. These findings reveal unexpectedly strong couplings between polar and structural topologies, expand the range of materials capable of hosting such structures from ferroelectric to antiferroelectric systems, and provide innovative strategies for designing functional materials.

Results
High-density dislocation arrays at heterointerface

High-quality epitaxial PZO thin films were successfully fabricated on (001)-oriented KTO substrates by pulsed laser deposition (PLD) technique. High-resolution X-ray diffraction (HRXRD) confirms the high crystalline quality of the films (Supplementary Fig. 1). The broadening of HRXRD peaks and elongation of the (002) Bragg reflection along the Q direction in reciprocal space maps (RSMs) indicate a continuous changing in the out-of-plane lattice constant (Supplementary Fig. 1 and Supplementary Fig. 2a–2c). Moreover, the (103) reflection reveals the coexistence of strained (S) and relaxed (R) regions, reflecting a distribution of local strain states within the film (Supplementary Fig. 2d–2f). This spatial variation in strain is further corroborated by selected area electron diffraction (SAED) patterns, which exhibit diffuse and elongated diffraction spots (Supplementary Fig. 3), consistent with the presence of multiple strain states in the film.

The cross-sectional TEM dark-field image (Fig. 1a) reveals an array of high-density interfacial dislocations at the PZO/KTO interface, effectively accommodating the large lattice mismatch of 4.2% between the film and the substrate (a=3.989 nm for KTO substrate, a ≈b =4.160 nm, and c =4.120 nm for PZO21, where the subscript ‘pc’ represents pseudo cubic). To further investigate the characteristics of the interfacial dislocations, HAADF-STEM imaging was performed (Fig. 1b) and the Burgers vector of the dislocations was determined as b=a[010], as shown in the dashed box in Fig. 1b. The in-plane strain (ε ) distribution, extracted via geometric phase analysis (GPA) and corresponding to Fig. 1b, is shown in Fig. 1c, demonstrating that the dislocations induce a periodic strain field within the film. Further insights into the arrangement of dislocation arrays were obtained from planar TEM observations of a 5 nm PZO film. The planar-view bight field-STEM (BF-STEM) image (Fig. 1d) provides a clear visualization of the dislocation array from an in-plane perspective. Statistical analysis of the dislocation spacing, based on extensive data collection (Fig. 1e), indicates an average spacing of 9.7±1.2 nm.

To gain a deeper understanding of the dislocation array, Supplementary Fig. 4 shows the atomic arrangement schematics in PZO/KTO heterostructures from both planar and cross-sectional perspectives. The interfacial dislocation lines are oriented along the [100] and [010] directions at the interface. The two sets of interfacial dislocation lines intersect to form a two-dimensional interfacial dislocation array, as shown in Supplementary Fig. 4.

Experimental observation of polar antihedgehog domains

Fig. 2 shows the atomic-resolved HAADF-STEM image and resolves the strain field and polar topological structure induced by the interfacial dislocations. Detailed analysis of the region above the dislocation core (highlighted by the dashed box in Fig. 2a) was performed from the perspective of lattice distribution. Supplementary Fig. 5 provides definitions for lattice rotation and Pb2+ displacements. Using quantitative analysis based on two-dimensional Gaussian peak fitting, the distributions of the lattice rotation in the horizontal direction (R ) and vertical direction (R ) are obtained in Fig. 2b and 2c, respectively, displaying significant lattice distortions around the high-density dislocations. The R map in Fig. 2c illustrates that lattice rotation distortion decreases gradually from the dislocation core at the interface toward the film surface. To quantitatively visualize this effect, spacing profiles along the [001] direction on both sides of the dislocation core were extracted from the R map (Fig. 2d), revealing a strain gradient exceeding 108 m-1 along the [001] direction. Additionally, a similar trend was observed by analyzing the R values over four unit cells above the dislocation core along the [010] direction (Fig. 2e), further confirming the periodic strain distribution and its gradient near the dislocations.

Functional oxides exhibit strong coupling between lattice and charge, where significant lattice distortions near dislocations will strongly affect polarization at the nanoscale22-29. To explore the coupling relationship among dislocations, lattice, and polarization, further analysis was conducted. The in-plane (Shift X) and out-of-plane (Shift Y) components of δ , extracted from the atomic-resolution HAADF-STEM image (highlighted in the dashed box in Fig. 2a), were shown in Supplementary Fig. 6a and 6b, respectively. The Shift X map reveals a periodic positive and negative distribution, with maximum and minimum values adjacent to the dislocation core and an in-plane component that diminishes with increasing distance away from the core, indicating the presence of a polar structure that converges or diverges around the core. In addition, there are periodic distortions between two adjacent dislocations, and the corresponding Shift Y map (Supplementary Fig. 6b) indicates that the displacement directions near the interface and surface are opposite in those regions. The combined observations from the lattice rotation map and Pb2+ displacement map point to the existence of an ordered distributed polar structure strongly coupled to the dislocation array in the 5 nm PZO film.

To reveal the polar structure in the PZO film, polar mapping analysis is performed on atomically resolved HAADF-STEM images (the dashed box in Fig. 2a). The positions of Pb2+ and Zr4+ columns are marked by green and red circles, respectively. SLight orange arrows denote the directions of δ vectors, representing the displacement of Pb2+ cation from the center of its four surrounding Zr4+ columns, and thus indicate the local spontaneous polarization. We performed additional analysis using integrated differential phase contrast STEM (iDPC-STEM), a technique highly sensitive to light elements. This allowed us to directly measure the displacement of Pb cations relative to the geometric center of the oxygen octahedra. The strong agreement between the polarization vectors derived from both methods confirms that δ is a reasonable method for determining local polarization in our system (Supplementary Note 1 and Supplementary Fig. 7).

The polarization is observed to converge at the core of the dislocations, revealing the presence of ordered polar arrays, as visualized more intuitively in the polarization mapping shown in Fig. 2f. The polarization structure can be decomposed into two basic units, represented by the black and blue dashed boxes (Supplementary Fig. 8). The two units are intertwined rather than spatially separated, with unit at the dislocation core features polarization converging toward the core, reminiscent of center-type domains30-33. In contrast, the derived unit situated between adjacent dislocations exhibits a polarization configuration with head-to-head alignment along the out-of-plane [001] direction and tail-to-tail alignment along the in-plane [010] direction, forming a cross-like topology characteristic of antivortex domains34-36. Notably, the centers of the two basic units are not coplanar, a distribution influenced by the strain field generated by the dislocation array. These results provide direct evidence of the strong coupling between the dislocation array and the polar topological structure.

To gain deeper insights into the ordered polar topology from different perspectives, planar samples of 5 nm PZO films were extensively analyzed to complement the cross-sectional observations. An ordered, dot-like contrast in the topography TEM image (Fig. 3a) indicates an ordered structural arrangement in the 5 nm PZO film, which is further supported by the satellite spots in the SAED pattern (Fig. 3b). The spacing between the main and first-order satellite peaks corresponds to a real-space period of ~10 nm, this value shows excellent agreement with the average dislocation spacing of ~9.7 nm (Fig. 1e). Polarization in dielectric materials is often accompanied with a local polarization field, which can be effectively visualized using differential phase-contrast (DPC) imaging37. In the DPC-STEM analysis (Fig. 3c), the [100] and [010] directions of the PZO unit cells are aligned with the A-C and B-D segments of the detectors, respectively, as illustrated in the upper-left inset of Fig. 3c. High-resolution DPC-STEM mapping reveals regular modulation of the polarization field across the PZO film, further confirming the ordered nature of the polar topology.

Atomic-resolution HAADF-STEM imaging of the 5 nm PZO film was performed under in-focus conditions to avoid spurious contrast from Moiré patterns, which can arise due to lattice mismatch between the film and substrate (example in Supplementary Fig. 9). The resulting image (Fig. 3d) provides the intrinsic polarization structure. This interpretation is further supported by control experiments on a sample flipped to position the substrate on top (Supplementary Fig. 10). The polarization distribution within one cycle, highlighted in the top-left inset of Fig. 3d, is represented by light orange arrows, which denote the δ vectors. The convergence centers of the topological structures, identified in the polarization mapping of Fig. 3e (derived from Fig. 3d), are marked with blue circles and reveal an ordered topological lattice. A quantitative analysis of the spacing distribution between these cores indicates a clear ordered arrangement overall, despite local fluctuations (Supplementary Fig. 11). Combining the analysis of planar and cross-sectional TEM samples, we speculate that the polarization configuration at the core of the dislocation corresponds to an antihedgehog structure, as illustrated in the three-dimensional schematic in Fig. 3f. Notably, the polar topological structure exhibits adaptability to slight irregular spacing of the dislocation array (Fig. 1e), and can adapt to these local perturbations without losing its fundamental order (Supplementary Fig. 12).

Phase-field simulations and theoretical rationalization

To elucidate the specific role of dislocations on the formation of the polar antihedgehog observed experimentally, systematic phase-field simulations were performed (details provided in the Methods section). As shown in Fig. 4a, the topological domain structure exhibits a checkerboard-like pattern composed of divergent (orange circles) and convergent (blue circles) polarization centers. Notably, the convergent centers are precisely aligned with the dislocation cores (Fig. 4b and Supplementary Fig. 13), underscoring the essential role of dislocations in shaping the observed ordered structure. To further elucidate the polarization characteristics within the ordered arrays, the local polarization distribution in the highlighted region I in Fig. 4a was extracted and presented in Fig. 4c–4h. As shown in Fig. 4c–4e, the analysis identifies two distinct polarization behaviors: in the upper region, the polarization converges downward toward the dislocation, while in the lower part, the polarization converges upward toward the dislocation. Near the dislocation core, the region of upward-oriented polarization decreases from approximately 3 nm to about 1 nm along the [001] direction, with upward components concentrated only near the interface (Fig. 4d and 4e). Away from the interface, the polarization predominantly remains downward-oriented. Planar projections across different thicknesses in Fig. 4f–4h reveal a distinct trend of polarization convergence. As the polarization converges at the dislocation cores in the upper region, a divergent polarization pattern emerges in the region enclosed by four adjacent dislocation, as highlighted as the region II in Fig. 4a and shown in Supplementary Fig. 14.

These configurations are reminiscent of the hedgehog and antihedgehog domains, with the former possessing a positive topological charge (Fig. 4i) and the latter a negative one (Fig. 4j). The topological charge of the regions can be calculated using the formula7,38, 𝑄 = 1 ∫𝑑𝐴 𝜀 𝐩̂(r)⋅ [∂ 𝐩̂(r)×∂ 𝐩̂(r)], where 𝐩̂ is the normalized polarization vector, 𝐴 is the surface enclosing the volume containing the topological region, and 𝜀 is the Levi-Civita symbol. The calculated topological charge is −1 for the polarization convergence unit in region I and −1 for the polarization divergence unit in region II, corresponding to two distinct types of topological configurations. The first represents a convergent-type antihedgehog structure, characterized by the inward orientation of polarization vectors toward a central singularity. In contrast, the second configuration exhibits a polarization field that converges vertically toward the core, forming a head-to-tail alignment along the out-of-plane direction, while diverging laterally within the plane toward the opposite edges. This combination of vertical convergence and in-plane divergence gives rise to a composite topological texture, closely resembling the divergent–divergent–convergent–type Bloch point reported in our previous work7. The combination of polarizations converging at dislocation cores and diverging between dislocations defines a lattice characterized by checkerboard-like antihedgehog (Fig. 4k).

The formation of the novel polar antihedgehog structure is attributed to the interplay between the electrostrictive effect and the flexoelectric field induced by the dislocation network. First, the periodic strain field of the dislocation network (Fig. 4b and Supplementary Fig. 13) significantly reconstructs the local polarization distribution by altering the local lattice environment. In addition, the strain gradient associated with the dislocation field cannot be neglected. As illustrated in Fig. 5a–5d and Supplementary Fig. 15, strong strain gradients emerge near the dislocation cores, particularly along the out-of-plane direction. The gradients of the normal strains (𝜀 , 𝜀 ) and the shear strain (𝜀 ) along [001] are much larger than those in-plane, with the maximum 𝜀 reaching 4.8×108 m-1. The resultant flexoelectric field shown in Fig. 5e–5j is highly localized at the interface. The calculated in-plane flexoelectric field 𝐸 and 𝐸 (Fig. 5i and Supplementary Fig. 16) exhibits opposite directions on either side of the dislocation line. Specifically, 𝐸 is positive on the left of the dislocation line and negative on the right with an absolute peak magnitude of 6.8 MV/cm, which drives the polarization between adjacent dislocation cores near the interface to converge toward the dislocation core. In addition, the pronounced out-of-plane strain gradient amplifies 𝐸 , which reaches a positive value of 34 MV/cm near the interface and drives the polarization to reverse from downward to upward. This strong, localized flexoelectric field drives polarization in the interface region to flip upward and converge toward the dislocation core. Hence, the synergistic interaction between the electrostrictive effect and the highly localized flexoelectric field is essential for stabilizing the polar antihedgehog structure. In contrast, as illustrated in Supplementary Fig. 17, in the absence of the flexoelectric field, the out-of-plane polarizations show a convergence pattern. For the in-plane polarizations, those in the upper region exhibit pronounced convergence toward the dislocation core, while those in the lower region show a divergent pattern, different from the experimental observation.

Chemical analysis of dislocation cores and emergent properties

To assess potential chemical changes induced by dislocation strain fields39, we performed atomic-resolution electron energy loss spectroscopy (EELS) near the dislocation cores. Atomic-scale EELS elemental mapping (Fig. 6a–6h) shows no significant chemical inhomogeneity. This reveals the elemental distribution above the interface are not affected by the dislocations, showing no clear sign of chemical aggregation or deficiency in the PZO films. These combined results suggest that the observed polar topological structures originate primarily from strain gradients and flexoelectric effects, rather than from chemical inhomogeneity.

Recent work has established that local negative capacitance can emerge within a range of ferroelectric topological textures, including skyrmions, and Bloch points7,40. Here, we resolve the spatial arrangement of the antihedgehog lattice in the PZO film (Fig. 6i). Remarkably, both the polarization-converging cores (orange arrows) and the polarization-diverging cores (blue arrows) exhibit regions of negative dielectric permittivity, as captured by the  = −500 isosurface. The manifestation of negative capacitance at these distinct topological centers emphasizes the functional role of the antihedgehog lattice in shaping the local electrostatic landscape and reveals a direct link between polar topology and dielectric response.

Moreover, the influence of the dislocation strain field is spatially limited. As thickness increases, the strain gradient relaxes with distance from the interface (Supplementary Note 2). Both experimental and computational results indicate that the polarization rotation is primarily confined to a thickness of approximately 5 nm (Supplementary Figs. 18–19). Beyond this range, the effect of the dislocation strain field diminishes significantly, leading to a minimal influence on the polarization, with the in-plane component becoming effectively absent near the surface. To quantitatively reveal the piezoresponse behaviors of the above topological structure, local butterfly loops of the PZO films with different thicknesses are measured using an atomic force microscope equipped with an interferometric displacement sensor (IDS). Representative piezoresponse amplitude loops for 5 nm, 7 nm, and 9 nm PZO films are shown in Fig. 6j, with the corresponding effective d peak values summarized in Fig. 6k. The 5 nm PZO film hosting the antihedgehog lattice exhibits higher d value of ~10.6 pm/V, exceeding those of the 7 nm (~5.8 pm/V) and 9 nm (~8.4 pm/V) films. This trend is different from the strong thickness dependence typically observed in ferroelectric d for systems such as BaTiO 41, and highlights the role of the ordered topological structure in enhancing the electromechanical response. The magnitude of this response is comparable to that reported for 8 nm-thick BiFeO films containing polar skyrmion superlattices (d ~9.0 pm/V)42, underscoring the potential of topology-engineered phases in functional oxide devices.

To further demonstrate the field-tunable nature of the dislocation-stabilized antihedgehog lattice, we performed phase-field simulations under an applied out-of-plane electric field. The results, presented in Supplementary Fig. 20, show that the local polarization within the antihedgehog domains undergoes a reversible switching process in response to the external field. Upon application of a sufficiently strong bias of 2.0 MV/cm, the polarization at both the converging and diverging topological centers is reoriented, leading to a controlled reconfiguration of the lattice. Notably, once the electric field is removed, the system spontaneously relaxes back to its initial topological state, highlighting the recoverable and dynamically reconfigurable character of the antihedgehog lattice.

Methods
Film deposition details

A series of epitaxial PbZrO3 (PZO) ultrathin films were grown on single-crystal KTaO3 (001) substrates by PLD, using a Coherent ComPexPRO 201KrF excimer laser (λ 248 nm). The KTaO3 (001) substrates used here are commercial substrates without extra chemical or heat treatment. Before film deposition, the substrate was heated up to 800 °C with a heating rate of 25 °C/min stayed for 5 minutes to clean the substrate surface and the laser was focused on a ceramic PZO target for 10 minutes pre-sputtering to clean the target surface. Then cooled down to 700 °C with a cooling rate of 15 °C/min to grow the PZO films. Throughout the deposition process, an oxygen pressure of 6.6 Pa, a laser repetition rate of 4 Hz, and a laser energy of 200 mJ were used. After deposition, these samples were stabilized at 700 °C for 5 minutes and then cooled down to room temperature at a cooling rate of 5 °C/min in an oxygen pressure of 2.6×104 Pa.

Structural analysis

Crystal structure study was performed by X-ray θ-2θ scans and reciprocal space mappings (RSMs) using a high-resolution X-ray diffractometer (Bruker D8 Discover X-ray diffractometer, λ=1.5406 Å).

TEM, STEM and determining the positions of atom columns

Cross-sectional samples for TEM and STEM imaging were prepared by slicing, gluing, grinding, dimpling, and finally ion milling by using Precision Ion Polishing System 691 of Gatan company. Before ion milling, the samples were dimpled down to 10 μm by Gatan 657 Dimple Grinder. During the ion milling, a voltage of 4.5 kV and an incident angle of 7° were used at first, and then the voltage and incident angle were gradually reduced to 4.0 kV and 5°, respectively. The final ion milling voltage was 0.1 kV for 5 min to reduce the amorphous layer produced in the milling process. Unlike the cross-sectional samples, the plane-view sample was milled only from the substrate side to protect the film from damage.

Dark-field images and selected area electron diffraction (SAED) patterns were acquired by JEOL F200 high-resolution TEM. High-angle annular dark-field (HAADF) STEM images were acquired using ThermoFisher Spectra 300 microscope with a high-brightness field-emission gun, double aberration (Cs) correctors from CEOS operating at 300 kV. The beam convergence angle is 25 mrad, and the collection angle ranges from 36 mrad to 200 mrad. Each of the atomic-scale HAADF-STEM images was acquired by adding up 20 original images, recorded from the same area. Image acquisition and processing were performed using the Velox software (FEI). Acquiring images in this way can reduce the influence of sample drift and scanning noises. (i)DPC-STEM imaging was performed with a convergence angle of 25 mrad and a panther detector. The thickness ranges of planar-view and cross-sectional samples from ~20 nm to 30 nm as measured by electron energy loss spectroscopy (EELS). The EELS elemental mapping data were acquired using K3 camera in Gatan Digital Micrograph software, with a dispersion of 0.9 eV/Ch. The large-scale strain fields were extracted by using geometry phase analysis (GPA)46, carried out by Gatan Digital Micrograph software.

By using a two-dimensional Gaussian peaks fitting47 in Matlab software, the atom positions can be accurately determined48, thus making it possible to acquire information of the lattice spacing, lattice rotation, and ionic displacement. The lattice rotation is defined as the deviation angle of each lattice unit relative to the ideal crystal axes (horizontal x and vertical y directions), with clockwise and counterclockwise rotations assigned as negative and positive, respectively. The resulting parameters, R and R , not only provide a quantitative measure of lattice rotation along each axis but, more importantly, collectively reflect the local lattice distortions induced by dislocation strain fields. In perovskite-structured PZO materials, the polarization primarily originates from the displacement of Pb2+ cations. Based on this, the present study characterizes the displacement vector of the Pb2+ cations by calculating the displacement of Pb atom column relative to the geometric center of the surrounding four Zr atomic columns, thereby approximating the polarization direction equivalently.

Phase-field simulation

In the three-dimensional phase-field modeling of the PZO system, the spontaneous polarization components 𝑃 (𝑖 = 1,2,3) are selected as the order parameters. The corresponding temporal evolution follows the time-dependent Ginzburg-Landau (TDGL) equation49:

𝜕𝑃(𝒓,𝑡) 𝛿𝐹
𝑖 = −𝐿 ,(𝑖 = 1,2,3) (1)
𝜕𝑡 𝑖
𝛿𝑃(𝒓,𝑡)
𝑖

where 𝐿 represents the kinetic coefficient associated with the domain wall mobility and 𝛿𝐹⁄𝛿𝑃(𝐫,𝑡) signifies the driving force governing the spatial and temporal behavior of 𝑃(𝐫,𝑡). The total free energy 𝐹 accounts for contributions from bulk, gradient, elastic, electrostatic, and flexoelectric energies, formulated as:

𝐹 = ∭(𝑓 +𝑓 +𝑓 +𝑓 +𝑓 )𝑑𝑉 (2)
𝑏𝑢𝑙𝑘 𝑔𝑟𝑎𝑑 𝑒𝑙𝑎𝑠 𝑒𝑙𝑒𝑐 𝑓𝑙𝑒𝑥𝑜

where 𝑉 is the volume of the film.

The expression of bulk free energy density is as follows:

𝑓 = 𝛼 (𝑃2 +𝑃2 +𝑃2)+𝛼 (𝑃4 +𝑃4 +𝑃4)+𝛼 (𝑃2𝑃2 +𝑃2𝑃2 +𝑃2𝑃2)
𝑏𝑢𝑙𝑘 1 1 2 3 11 1 2 3 12 1 2 2 3 1 3
+𝛼 (𝑃6 +𝑃6 +𝑃6)+𝛼 [𝑃4(𝑃2 +𝑃2)+𝑃4(𝑃2 +𝑃2)+𝑃4(𝑃2 +𝑃2)] (3)
111 1 2 3 112 1 2 3 2 1 3 3 1 2
+𝛼 𝑃2𝑃2𝑃2
123 1 2 3

where 𝛼 , 𝛼 , and 𝛼 are local potential coefficients representing the stiffness with respect to polarization changes.

The gradient energy density can be expressed as50,51:

𝜕𝑃 2 𝜕𝑃 2 𝜕𝑃 2
𝑖 𝑖 𝑖
𝑓 = ∑ {𝜆 [( ) +( ) +( ) ]
𝑔𝑟𝑎𝑑 0 𝜕𝑥 𝜕𝑦 𝜕𝑧
𝑖 (4)
𝜕2𝑃 2 𝜕2𝑃 2 𝜕2𝑃 2
𝑖 𝑖 𝑖
+𝑔 [( ) +( ) +( ) ]}
0 𝜕𝑥2 𝜕𝑦2 𝜕𝑧2

Here, 𝜆 is a negative constant which stabilizes the AFE phase by accounting for the coupling between polarization and oxygen octahedral tilts. In contrast, 𝑔 is a positive constant representing the next-nearest-neighbor interaction among polarizations, driving the phase transition from AFE to FE. In this work, the renormalized values of 𝜆 = −0.1 and 𝑔 = 0.3 are adopted to simulate the FE phase observed in the ultrathin PZSO film.

The elastic energy density can be written as:

1
𝑓 = 𝐶 (𝜀 −𝜀0)(𝜀 −𝜀0 ) (5)
𝑒𝑙𝑎𝑠 2 𝑖𝑗𝑘𝑙 𝑖𝑗 𝑖𝑗 𝑘𝑙 𝑘𝑙

in which 𝜀 is the total strain and 𝜀0 represents the eigenstrain. The difference 𝜀 −𝜀0 corresponds to the elastic strain. In this work, the eigenstrain 𝜀0 is decomposed into two components: the spontaneous polarization-induced strain 𝜖0𝑃 and the dislocation-induced strain 𝜖0𝐷: 𝜖0 = 𝜖0𝑃 +𝜖0𝐷. The spontaneous strain, 𝜀0𝑃 = 𝑄 𝑃 𝑃, is related to be the polarization through electrostrictive coefficients 𝑄 . For a dislocation loop located on a slip plane s with a Burgers vector 𝑏 = (𝑏 ,𝑏 ,𝑏 ), the induced strain is given by 𝜀0𝐷 = 1 [(𝑏 𝑛 +𝑏 𝑛 )]𝛿(𝑥 − 2𝑑0
𝑥𝐷) where 𝑛 = (𝑛 ,𝑛 ,𝑛 ) is the unit vector normal to the slip plane, 𝑑 is the interplanar distance of the slip plane, 𝛿(𝑥 −𝑥𝐷) is the Dirac delta function, and 𝑥𝐷 is a point inside the dislocation loop. To model the experimentally observed dislocation networks, the simulation incorporates an ordered dislocation array, which consists of dislocation lines with Burgers vectors 𝒃 = 𝑎[010] along the [010] direction and 𝒃 = 𝑎[100] along the [100] direction. The spacing between adjacent dislocation lines is set as approximately 9 nm, consistent with experimentally measured dislocation spacings.

The electrostatic energy density can be expressed as:

1
𝑓 = − 𝜀 𝜀 𝐸2 −𝐸 𝑃 (6)
𝑒𝑙𝑒𝑐 2 0 𝑏 𝑖 𝑖 𝑖

where 𝐸 is the electric filed component, 𝜀 is the permittivity of vacuum and 𝜀 is the background dielectric constant.

The last term is the flexoelectric energy density:

𝑓 𝜕𝑃 𝜀
𝑖𝑗𝑘𝑙 𝑘 𝑖𝑗
𝑓 = (𝜀 −𝑃 ) (7)
𝑓𝑙𝑒𝑥𝑜 2 𝑖𝑗 𝜕𝑥 𝑘𝜕𝑥
𝑙 𝑙

where 𝑓 represents the flexocoupling coefficient tensor52,53. In Voigt notation, the components are simplified as 𝑓 =𝑓 , 𝑓 = 𝑓 , and 𝑓 = 2𝑓 . Given the uncertainties in the exact magnitude of 𝑓 , we use 𝑓 = 𝑓 = 1 V and assume 𝑓 = 0.5 V for simplicity based on the reasonable estimation from previous literatures54,55. The driving force from the flexoelectric effect yields the flexoelectric field 𝐸 𝑓𝑙𝑒𝑥𝑜 :

𝛿𝑓 𝑓𝑙𝑒𝑥𝑜 𝜕𝑓 𝑓𝑙𝑒𝑥𝑜 𝜕 𝜕𝑓 𝑓𝑙𝑒𝑥𝑜 𝜕𝜀 𝑖𝑗 𝑓𝑙𝑒𝑥𝑜
= − = −𝑓 = −𝐸 (8)
𝛿𝑃 𝜕𝑃 𝜕𝑥 (𝜕𝑃 /𝜕𝑥 ) 𝑖𝑗𝑘𝑙 𝜕𝑥 𝑘
𝑘 𝑘 𝑙 𝑘 𝑙 𝑙

Mechanical stress and electric field equilibrium is assumed to arrive much faster than the evolution of domain structures. Thus, we solve the mechanical equilibrium equations and Maxwell equation:

𝜎 = 0
𝑖𝑗,𝑗
(9)
𝐷 = 0
𝑖,𝑖

to obtain the corresponding driving forces for each polarization configuration.

The three-dimensional computational domain is discretized into a 64𝛥𝑥×64𝛥𝑦× 24𝛥𝑧 mesh. The grid spaces are 𝛥𝑥 = 𝛥𝑦 = 𝑎 and 𝛥𝑧 = 𝑎 /2, where 𝑎 = 𝑎0 = 0.416 nm, √2
setting up a 5 nm-thick PZO film. Beneath the PZO film, a substrate layer of 20 grids thickness is included to emulate a deformable substrate. Above the PZO film, a vacuum layer with a thickness of 12 grid points is incorporated. The material parameters of PZO are sourced from prior literature51 and summarized in Supplementary Table 1. Periodic boundary condition is applied along the in-plane directions. The mechanical boundary conditions are defined as a traction-free top surface for the PZO film and a fully constrained bottom surface for the deformable substrate. Additionally, short-circuit electrical boundary conditions are imposed on the top and bottom surfaces of the PZO layer. The initial domain configuration is set as a single-domain state, with the polarization oriented entirely downward. Considering that dislocations relieve most of the substrate-induced strain, only a -1% compressive strain is applied to stabilize the out-of-plane domain configuration.

We evaluate the local out-of-plane dielectric constant  , defined as the functional derivative of the electric displacement D with respect to the total electric field E : 𝜀 = ∂𝐷 /∂𝐸 . The electric displacement is given by 𝐷 = 𝜀 𝜀 𝐸 +𝑃, where is the vacuum permittivity, is the background dielectric constant, and P is the local polarization. To compute  , a small probing electric field of 400 kV/cm is applied along the out-of-plane direction. The resulting variations in D and E are evaluated at each grid point, from which the local dielectric response is obtained.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this revised manuscript, Jiang and collaborators have comprehensively addressed all of the critical concerns I previously\nraised. In particular, the justification of the relationship between Zr displacement and local polarization is now well\nsupported, and the definition of the topological charges has become reasonable and convincing. I am also comfortable with\nthe authors’ claims regarding the antiferroelectricity-related novelty. The comments raised by the other reviewers also\nappear to have been adequately addressed.\nOverall, I now believe that this manuscript provides important insights into the interplay between local topological polar\nordering and dislocation arrays, revealed with sufficient scientific rigor. I therefore support publication of this manuscript.\nOne minor point: I remain uncomfortable with the authors’ use of the term “periodic structure” to describe the dislocation\narray. In my view, the dislocation arrays shown in Fig. 1c, Fig. 1d, and Fig. 3a are far from exhibiting long-range periodicity.\nIn fact, the distribution resembles a highly distorted lattice and does not appear periodic even over short ranges of a few\ndislocation length scales. To me, a “periodic structure” would imply something closer to the atomic arrangement of the\nKTO substrate shown in Fig. 2a. I believe many readers will share this concern. I therefore strongly suggest that the authors\nreplace the term “periodic” with wording that more accurately reflects the actual distribution of the dislocations.", "reviewer_2": "Authors clearly discussed and addressed the issues which I raised, then I recommend this manuscript for publication in its\ncurrent form."}, "author_response": "Reply to reviewers’ comments:\nRef number: NCOMMS-26-005794-T\nTitle: Strong interplay between polar and structural topologies\nAuthors: Ru-Jian Jiang, Mei-Xiong Zhu, Su-Zhen Liu, Yu-Ting Chen, Desheng Ma,\nYan-Peng Feng, Min-Jie Zou, Meng-Jiao Han, Chi Hou Lei, Yu-Jia Wang, Yun-Long\nTang, Yin-Lian Zhu, Xiu-Liang Ma\nResponse to reviewer #1 (R1):\nWe appreciate the positive evaluations by the reviewer that “In this revised\nmanuscript, Jiang and collaborators have comprehensively addressed all of the critical\nconcerns I previously raised. In particular, the justification of the relationship between\nZr displacement and local polarization is now well supported, and the definition of the\ntopological charges has become reasonable and convincing. I am also comfortable\nwith the authors’ claims regarding the antiferroelectricity-related novelty. The\ncomments raised by the other reviewers also appear to have been adequately\naddressed. Overall, I now believe that this manuscript provides important insights into\nthe interplay between local topological polar ordering and dislocation arrays, revealed\nwith sufficient scientific rigor. I therefore support publication of this manuscript”.\nIn the meanwhile, the reviewer also raised one concern. We fully understand this\npoint and have addressed it specifically below. The key revision is highlighted in RED\nin the manuscript.\nQuestion 1): One minor point: I remain uncomfortable with the authors’ use of the\nterm “periodic structure” to describe the dislocation array. In my view, the dislocation\narrays shown in Fig. 1c, Fig. 1d, and Fig. 3a are far from exhibiting long-range\nperiodicity. In fact, the distribution resembles a highly distorted lattice and does not\nappear periodic even over short ranges of a few dislocation length scales. To me, a\n“periodic structure” would imply something closer to the atomic arrangement of the\nKTO substrate shown in Fig. 2a. I believe many readers will share this concern. I\ntherefore strongly suggest that the authors replace the term “periodic” with wording\nthat more accurately reflects the actual distribution of the dislocations.\nResponse to Question 1):\nWe thank the reviewer for this insightful and valuable comment. We fully agree\nthat the term “periodic” can imply a mathematically strict long-range order, which\ndoes not precisely describe the dislocation arrays or the associated polar structures in\nour system, as they exhibit a certain degree of spatial fluctuation.\nOur intention was to convey their distinctly non-random and correlated spatial\narrangement, in contrast to a completely disordered state. To accurately reflect this\nnature and to eliminate any potential confusion regarding strict periodicity, we have\nrevised the terminology throughout the manuscript as follows:\n1.The description of dislocation arrangements has been simplified by removing\nthe prefix “periodic”. They are now referred to simply as “dislocation networks” or\n“dislocation arrays,” as their specific configuration is sufficiently characterized in the\ncontext.\n2.The terms “periodic polar topology” and “periodic polar arrays” have been\nreplaced with “ordered polar topology” and “ordered polar arrays,” respectively. This\nterminology more precisely captures their spatially correlated and regular, yet not\nstrictly periodic configuration.\nWe believe these revisions enhance the descriptive accuracy of our work. The\nmanuscript has been updated accordingly.\nReply to reviewers’ comments:\nRef number: NCOMMS-26-005794-T\nTitle: Strong interplay between polar and structural topologies\nAuthors: Ru-Jian Jiang, Mei-Xiong Zhu, Su-Zhen Liu, Yu-Ting Chen, Desheng Ma,\nYan-Peng Feng, Min-Jie Zou, Meng-Jiao Han, Chi Hou Lei, Yu-Jia Wang, Yun-Long\nTang, Yin-Lian Zhu, Xiu-Liang Ma\nResponse to reviewer #2 (R2):\nWe sincerely thank the reviewer for the constructive comments and positive\nrecommendation for our work's publication in Nature Communications."}]

Decision letter