Introduction

Antiferroelectrics possess a compensated antiparallel arrangement of electric dipoles, resulting in a zero net polarization1. Another critical feature is that electric field should convert this antiferroelectric phase into a ferroelectric phase, and this transformation is reversible when turning off the field2. This volatile phase transition results in characteristic double hysteresis of polarization as a function of the electric field. As the antiferroelectric-to-ferroelectric phase transition encompasses large variations in charge, volume and temperature, antiferroelectrics are very attractive for applications such as high-density energy storage3, electromechanical actuators4, and electrocaloric refrigeration5.

PbZrO3 was the first material to be considered as antiferroelectric6. Still, it remains intensively investigated both theoretically and experimentally. Below its critical temperature of about 500 K, PbZrO3 transitions between the high-temperature Pm\(\bar{3}\)m cubic symmetry and the low-temperature Pbam orthorhombic symmetry with the characteristic up-up-down-down antipolar pattern of Pb ions. This phase transition is associated with antiferrodistortive oxygen octahedra tilts. Theoretical studies demonstrated the critical role of these tilts in the stabilization of antiferroelectric ordering in PbZrO37,8. Recently, first-principles calculations revealed that another polymorph of PbZrO3 with Ima2 space group, showing the same antiferrodistortive tilts but with an uncompensated up-up-down ferrielectric pattern, may actually be the ground state9. On the experimental side, scanning transmission electron microscopy (STEM) investigations in PbZrO3 single crystals concluded that both the antiferroelectric and ferrielectric phases can coexist10. This ferrielectric phase seems to develop via the condensation of translational boundaries11. In thick PbZrO3 films prepared by chemical solution deposition, a stripe-like pattern with alternated antiferroelectric and ferrielectric phases was interpreted as due to residual compressive strain12. As the film thickness is reduced in epitaxial PbZrO3, the situation becomes even more complex. Initial studies showed signatures of ferroelectric phase stabilization via electric measurements13,14, and recent STEM investigations revealed complex phase transitions from the classical antipolar Pbam to ferrielectric Ima2, orthorhombic or rhombohedral ferroelectric phases, as the thickness of the film is reduced to the 45-5 nm range15,16,17,18. Indeed, while ferroelectrics tend to break into nanodomains looking like antiferroelectrics when grown as ultrathin films, antiferroelectric PbZrO3 seems to follow an inverse trend by transiting to a non-zero polar state19. Overall, the modern studies on ultrathin films of PbZrO3 are typically carried out on relaxed layers with possible residual compressive strain. On the other hand, theory suggested that ferroelectricity would be favored under compressive strain, while antiferroelectricity would persist if PbZrO3 was grown under tensile strain20.

A long-standing issue for the growth of high-quality epitaxial thin films of PbZrO3 is the difficulty in finding single-crystal oxide substrates with cell parameters that are well matched with those of PbZrO3. Indeed, in its bulk antiferroelectric phase, PbZrO3 displays a Pbam orthorhombic symmetry with ao = 5.882 Å, bo = 11.783 Å and co = 8.228 Å21 (o stands for orthorhombic), resulting in pseudo-cubic parameters of 4.163 Å in the ao–bo plane and 4.114 Å along the co axis. However, PbZrO3 epitaxial thin films are usually grown on standard perovskite substrates such as SrTiO312,13,15,17,18,22,23, DyScO34,24 or GdScO325 with unit-cell parameters ranging from 3.905 Å to 3.971 Å, resulting in large compressive strain. This large lattice mismatch causes the PbZrO3 thin films to relax during growth, forming defects such as edge dislocations12,14,18. A related issue is that studying the electrical properties of PbZrO3, in order to exploit its antiferroelectric properties, becomes challenging when the film thickness is decreased below 50 nm.

In order to reduce the number of defects with epitaxially-strained PbZrO3 films, we employed a buffer layer of LaLuO3, a perovskite with large lattice parameters. Single crystals of LaLuO3 prepared by the Czochralski method were shown to crystallize in a Pbnm orthorhombic structure26,27,28 with ao = 5.810 Å, bo = 6.013 Å, and co = 8.373 Å28, corresponding to a pseudo-cubic lattice parameter of 4.181 Å in the ao–bo plane and 4.186 Å along the co axis. As there are no single-crystal substrates of LaLuO3 commercially available, we developed the growth of LaLuO3 epitaxial thin films as artificial substrates. To our knowledge, Schubert et al. were the only ones to report the growth of epitaxial LaLuO3 by pulsed laser deposition29. Following their work, we were able to stabilize fully-relaxed thick films of LaLuO3 on commercially-available DyScO3(110)o single crystals (“Methods”, Supplementary Fig. 1). Despite the large lattice mismatch, the LaLuO3 orthorhombic layer, with typical thicknesses of 100 nm, shows good crystalline quality and grows with the same (110)o orientation as the DyScO3 substrate. We then used such LaLuO3-buffered DyScO3(110)o crystals for the growth of PbZrO3.

PbZrO3 thin films with thicknesses ranging from 200 nm down to 9 nm were grown by pulsed laser deposition using a KrF excimer laser. In order to perform electrical measurements, a bottom electrode of SrPbO3 with typical thicknesses of 15 nm was grown on LaLuO3 prior to the growth of PbZrO3 (Methods). SrPbO3 also crystallizes in a Pbnm orthorhombic structure with ao = 5.852 Å, bo = 5.969 Å, and co = 8.324 Å30,31, corresponding to a pseudo-cubic lattice parameter of 4.179 Å in the ao–bo plane and 4.162 Å along the co axis. Hence, if the orthorhombic symmetry is preserved and given the lattice constants of SrPbO3 and LaLuO3, we expect PbZrO3 to grow under an anisotropic epitaxial tensile strain of +0.4% and +1.2–1.8% in the two in-plane directions (Supplementary Note 1). Our experimental strategy is motivated by first-principles calculations, which suggested that such tensile strain would stabilize antiferroelectricity20.

Results

Epitaxial arrangement of the films

We first examined the structural properties of these heterostructures using STEM. The cross-section high-angle annular dark field (HAADF) STEM image of the whole epitaxial stack (Fig. 1a) shows well-defined layers with thicknesses of 30 nm, 19 nm, and 108 nm for PbZrO3, SrPbO3, and LaLuO3, respectively. Using local FFTs, we could identify the epitaxial orientation of each layer with respect to their FFT diffraction peaks (color-filtered in Fig. 1b). The blue color of the DyScO3 substrate and its associated FFT shows that the zone axis of the image is parallel to the co axis of the substrate with characteristic ½{110} pseudo-cubic reflections (yellow circles in the FFT of Fig. 1b) corresponding to the ao and bo orthorhombic axes and the (110)o surface orientation. As previously mentioned, the orthorhombic layer of LaLuO3 follows the same (110)o orientation with an overall homothety of the unit-cell (see the FFT with yellow color). SrPbO3 appears sensitive to either the sample thinning process or the electron beam, and its crystallinity is not preserved throughout the whole cross-section specimen. Nevertheless, local observations confirm that SrPbO3 grows with the same (110)o orientation as the LaLuO3 and DyScO3 (Supplementary Fig. 2). Hence, the SrPbO3/LaLuO3/DyScO3 stack harbors a single crystal-like (110)o orientation with all the co axes parallel with one another. Regarding the PbZrO3 film, FFTs show two diffraction patterns (green and red) with ¼{110} superlattice reflections (green and red circles in the FFT of Fig. 1b). Indeed, the antiferroelectric Pbam phase of PbZrO3 consists of antiparallel up-up-down-down Pb displacements, forming stripes with a periodicity of four layers in the {110} pseudo-cubic plane, often coined as commensurate modulations32 of ¼{110}. Consequently, these two FFTs correspond to a co axis of the film parallel to the zone axis and with (120)o and (1\(\bar{2}\)0)o orientations, as sketched in Fig. 1b. The antiferroelectric domains can extend over typical lateral sizes of 150 nm, significantly larger than the 30-nm PbZrO3 thickness. Thus, the number of possible variants is reduced from six25 possible to only two, due to the preserved orthorhombic symmetry throughout the epitaxial stack. The presence of dark lines in the PbZrO3 domains (Fig. 1b) is attributed to antiphase boundaries, but the film remains purely in the antipolar Pbam phase (Supplementary Fig. 3). From the geometrical phase analysis (Fig. 1c, d and Supplementary Fig. 4), we observe that the large lattice mismatch strain (>5%) between LaLuO3 and DyScO3 is relieved within the first 10 nm of the layer. Additional strain profile analyses taken on different areas with weakly (Supplementary Fig. 5) or strongly amorphized SrPbO3 layers (Supplementary Fig. 6) show that LaLuO3, SrPbO3, and PbZrO3 display a constant in-plane lattice parameter value, suggesting that the epitaxial stack is coherently strained to the LaLuO3 buffer layer.

Fig. 1: Structural analysis of the complex epitaxial stack using scanning transmission electron microscopy.
figure 1

a HAADF-STEM image of the carbon-covered PbZrO3/SrPbO3/LaLuO3 orthorhombic perovskite layers grown on the DyScO3(110)o orthorhombic substrate. The layer thicknesses are 30, 19, and 108 nm for PbZrO3, SrPbO3, and LaLuO3, respectively. The SrPbO3 electrode reacts under the electron beam of the microscope. b Colored images obtained by filtering the FFT contributions of the different layers (as displayed on the right panel), showing DyScO3 (DSO), LaLuO3 (LLO), and the two orientations of PbZrO3 (PZO) corresponding to 90° rotation around the co axis. The ½{110} superlattice reflections of DyScO3 and LaLuO3 are emphasized by the yellow circles. The ¼{110} superlattice reflections of PbZrO3 are underlined by the red and green circles. The zone axis is parallel to the co axes of the four orthorhombic layers, giving rise to the following epitaxial relationship: PbZrO3(120)o,(1-20)o \(\parallel\) SrPbO3(110)o \(\parallel\) LaLuO3(110)o \(\parallel\) DyScO3(110)o with PbZrO3[001]o \(\parallel\) SrPbO3[001]o \(\parallel\) LaLuO3[001]o \(\parallel\) DyScO3[001]o. c, d Out-of-plane unit-cell deformation (εzz) (c) and in-plane unit-cell deformation (εxx) (d) obtained from geometrical phase analysis of the bright field STEM image (Supplementary Fig. 4). The uniform lattice expansion observed for the in-plane unit-cell deformation shows that the epitaxial stack is coherently strained to the LaLuO3 buffer layer.

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Evaluation of the epitaxial strain

We used X-ray diffraction to get more quantitative insights into the structure of the epitaxial heterostructure. A typical 2θ-ω X-ray diffraction pattern is displayed in Fig. 2a, showing high crystalline quality for the LaLuO3, SrPbO3 and PbZrO3 thin films and no secondary phases (other film thicknesses are displayed in Supplementary Fig. 7). Wide range reciprocal space maps taken with a 2D detector (Supplementary Fig. 8) confirm the overall epitaxy with PbZrO3(120)o \(\parallel\) SrPbO3(110)o \(\parallel\) LaLuO3(110)o \(\parallel\) DyScO3(110)o and PbZrO3[001]o \(\parallel\) SrPbO3[001]o \(\parallel\) LaLuO3[001]o \(\parallel\) DyScO3[001]o. The asymmetric high-resolution reciprocal space map around DyScO3(332)o (Fig. 2b) shows that LaLuO3, SrPbO3 and PbZrO3 share a common in-plane lattice constant, and this is the case for all thicknesses between 200 nm and 9 nm (Supplementary Fig. 9). We also performed reciprocal space maps around the (420)o, (332)o, (240)o, and (33\(\bar{2}\))o of DyScO3 (Supplementary Fig. 10 and Supplementary Fig. 11) for the 200, 80, 40, 30, 20, and 9 nm thicknesses of PbZrO3 to calculate the lattice parameters of PbZrO3 (Fig. 2c). For all film thicknesses, the in-plane pseudo-cubic lattice parameters along co (Fig. 2c, green dots) and perpendicular to co (Fig. 2c, red dots) are larger than the bulk parameters (Fig. 2c, dashed gray lines). By contrast, the out-of-plane parameters (Fig. 2c, blue dots) are smaller than the bulk values, in line with a biaxial epitaxial tensile strain. Considering this (120)o epitaxy of PbZrO3, we estimated the strain for the two in-plane and the out-of-plane directions (Fig. 2d and Supplementary Note 1). The films display a relatively constant anisotropic in-plane tensile strain as a function of thickness, reaching +1.7 ± 0.1% along co and +0.3 ± 0.1% perpendicular to co, in agreement with fully-strained layers on the artificial substrate of LaLuO3. The out-of-plane strain is estimated to be −1.3 ± 0.1 %. Hence, growing PbZrO3 on SrPbO3/LaLuO3-buffered DyScO3 substrates, we were able to successfully impose a coherent tensile strain that is robust over a wide range of film thicknesses.

Fig. 2: Epitaxial PbZrO3 thin films under tensile strain.
figure 2

a 2θ-ω X-ray diffraction pattern showing the (hh0)o symmetric peaks of the DyScO3, LaLuO3 and SrPbO3 as well as the (h,2h,0)o peak for PbZrO3, with h = 1, 2, 3, 4 for layer thicknesses of 30, 19, and 108 nm for PbZrO3, SrPbO3, and LaLuO3, respectively. The small peak at around 40° corresponds to the Pt(111) of the capacitors. b Reciprocal space map around DyScO3(332)o for a similar sample as in (a) with a PbZrO3 thickness of 40 nm. The three epitaxial layers of LaLuO3, SrPbO3, PbZrO3 share a common in-plane lattice parameter. c Pseudo-cubic in-plane (green and red) and out-of-plane (blue) lattice parameters of PbZrO3 as a function of the film thickness estimated from reciprocal space maps around DyScO3(220)o, (420)o, (332)o, (240)o, (33-2)o (Supplementary Fig. 10 and Supplementary Fig. 11, Supplementary Note 1). The bulk parameters apc and cpc of PbZrO3 are shown as dashed lines. d Epitaxial strain for the two in-plane axes and the out-of-plane axis deduced from c (Supplementary Note 1). A large in-plane tensile strain is measured along the PbZrO3 co axis (green) and a more moderate one perpendicularly to the co axis (red). This results in a strong out-of-plane compressive strain (blue). Overall, the tensile epitaxial strain does not vary significantly with the PbZrO3 film thickness from 9 to 200 nm. The error bars in (c) and (d) represent the standard deviation; if not visible, they are smaller than the symbol size.

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Electrical properties of the films

We now explore the influence of such an epitaxial tensile strain on the antiferroelectric properties of PbZrO3. Capacitors were fabricated on the PbZrO3/SrPbO3/LaLuO3 stack by a combination of optical lithography and lift-off of sputtered Pt top square electrodes, with lateral sizes varying from 5 to 120 μm. The electrical properties of typical 30 × 30 μm2 capacitors are displayed in Fig. 3 for PbZrO3 thicknesses of 200, 40, 30, 20, and 9 nm, under a triangular voltage excitation with a frequency of 2 kHz. Current vs. voltage loops (Fig. 3a) for thicknesses of 20-200 nm show four sharp current peaks characteristic of antiferroelectric-to-ferroelectric field-induced phase transitions. By integrating the displacement current densities as a function of time, we obtain the corresponding polarization vs. voltage (Fig. 3b), which shows clear double hysteresis loops. Despite the low 20–40 nm film thickness, no remanent polarization is detected, with no evidence of ferroelectric phases, attesting to the high quality of these epitaxial PbZrO3 films. Remarkably, these displacement current peaks are still clearly detectable for 9-nm-thick PbZrO3 films, though a large capacitive background impedes reliable polarization integration. The critical fields for the antiferroelectric-to-ferroelectric phase transitions increase from 0.3 MV/cm to 2.4 MV/cm when the film thicknesses decrease from 200 nm to 9 nm (Fig. 3c). Such large critical fields are consistent with measurements in pure PbZrO3 single crystals33, while incommensurate phases stabilized by doping these crystals show reduced critical fields32. A possible scenario is that the ferroelectric phase needs to nucleate first in a pure antiferroelectric matrix, while local ferrielectric or ferroelectric coexisting phases act as nucleation points in doped PbZrO3. Another striking feature of these double hysteresis loops is the horizontal voltage shift that tends to increase as thickness decreases (Fig. 3b). This imprint field, reaching 0.3 MV/cm for the lowest PbZrO3 thickness (Fig. 3c), could be related to the asymmetry between the Pt/PbZrO3 and PbZrO3/SrPbO3 interfaces. Overall, the capacitors based on epitaxial PbZrO3 grown on SrPbO3/LaLuO3 display exceptional electrical properties with sharp double hysteresis loops down to 20 nm and antiferroelectric signatures down to 9 nm thicknesses.

Fig. 3: Thickness dependence of the electrical properties of Pt/PbZrO3/SrPbO3 capacitors grown on LaLuO3-buffered DyScO3 substrates.
figure 3

a Current vs. voltage loops for a triangular waveform of voltage at 2 kHz for film thicknesses of 200, 40, 30, 20, and 9 nm (from left to right). Four displacement current peaks are observed for all the film thicknesses. b Corresponding polarization vs. voltage loops show clear double hysteresis, characteristic of antiferroelectric capacitors. Capacitive contributions for the lowest film thickness of 9 nm impede the integration of polarization vs voltage. c Critical electric field of the antiferroelectric-to-ferroelectric (AFE-FE) phase transition as a function of the film thickness (black, in log-log scale). The AFE-to-FE critical electric field is calculated by averaging the positive and negative critical voltages in (b), and dividing this average voltage by the film thickness. The imprint field, corresponding to the shift of the loops towards positive voltage, is also plotted (in red) as a function of the PbZrO3 thickness. Voltage is applied to the SrPbO3 electrode while the Pt top electrode is grounded.

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Visualization of the atomic-scale polar displacements

To confirm that this electrical behavior is associated with the antiferroelectric nature of PbZrO3, we used high-resolution HAADF-STEM imaging to map the local polar displacements of the Pb atoms (Fig. 4 and Supplementary Fig. 12) for the 30, 20, and 9 nm thick films. The 30 nm PbZrO3 film displays stripes with polar displacements alternating along <110> pseudo-cubic directions for every two {110} planes (inset of Fig. 4a). This characteristic up-up-down-down antipolar pattern is expected for the antiferroelectric phase of PbZrO3 with Pbam symmetry. Two antipolar domains are observed in Fig. 4a with a 90-degree domain wall aligned along the {100} pseudo-cubic plane. The dipoles locally form head-to-tail boundaries in order to minimize the depolarization energy, as also observed for PbZrO3 ceramics32. Interestingly, this characteristic antipolar pattern is preserved for PbZrO3 film thicknesses of 20 nm (Fig. 4b) and 9 nm (Fig. 4c), though the interfaces may introduce some local polar disorder (e.g., bottom right corner of Fig. 4b). The antipolar nature of these 30 nm to 9 nm-thick PbZrO3 films is strikingly different from multiple recent results on relaxed epitaxial thin films of PbZrO3 with similar thicknesses, reporting homogeneous polar displacements characteristic of a ferroelectric phase15,17,18. Overall, we have demonstrated that the excellent electrical properties of the PbZrO3 films grown under tensile epitaxial strain are associated with their true antipolar nature down to 9 nm.

Fig. 4: Antipolar texture of PbZrO3 thin films using scanning transmission electron microscopy.
figure 4

Polar displacements were obtained by comparing the location of the Pb with respect to the barycenter of the Zr square lattice, in high-resolution HAADF-STEM images (raw images are displayed in Supplementary Fig. 12). In all the images, the zone axis is parallel to the co axis of DyScO3. The resulting dipoles are represented as colored arrows. a Polar textures of the 30-nm-thick PbZrO3 film, showing two large antipolar domains with a 90-degree domain wall. The bottom inset is a zoom emphasizing the characteristic up-up-down-down antipolar pattern of the Pbam phase. b Polar textures in the 20-nm-thick PbZrO3 film. c Polar textures in the 9-nm-thick PbZrO3 film. Horizontal scale bars are 5 nm.

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Regarding this thinnest PbZrO3 film, while a clear antipolar pattern is observed in its pristine state, we discovered instabilities of this pattern under the STEM electron beam. In PbZrO3 single crystals, similar time-dependent STEM experiments revealed a transition from the antipolar to a cycloidal polar state under the electron beam34. In our case, these instabilities were not detected in thicker PbZrO3 films. Sequential HAADF-STEM images taken every two seconds allowed us to visualize changes in the antipolar pattern (Fig. 5). Indeed, while the up-up-down-down antipolar pattern is initially very well ordered over the whole region of interest (Fig. 5a, t = 0–38 s), local defects start to nucleate after several scans (Fig. 5a, t = 40 s). We first observe the nucleation of −π/2 translational boundaries10,11 characterized by a phase shift of the up-up-down-down square (4 unit-cell) wave by 3 unit cells (Fig. 5b, yellow). The nucleation of such a topological defect gives rise to a local uncompensated up-up-down-down-down pattern. This local defect promotes the subsequent shift of the whole antipolar pattern from left to right by one unit cell (Fig. 5b, t = 42 s). This cascade-like movement is enabled by the nucleation and annihilation of +π/2 (phase shift by 1 unit cell) translational boundaries10,11 in other areas, resulting in transient up-down-down (Fig. 5b, black) polar patterns. Consequently, the instability of the antipolar order under the STEM electron beam suggests competing ferrielectric ordering for such low PbZrO3 thicknesses. Nevertheless, we did not detect the aggregation of translational boundaries in the form of ferrielectric phases10,12,17 throughout the different areas of the cross-section specimen. Further investigations are required to identify the possible roles of the interfaces and surfaces in the nucleation of the translation boundaries.

Fig. 5: Antipolar instabilities in the 9-nm-thick PbZrO3 film: translation boundary nucleation and movement of the antipolar pattern under the electron beam.
figure 5

a Sequential STEM-HAADF images taken with 2 s integration time on the same area, displayed at 0, 38, 40 and 42 s. b Zooms taken from the left images for the regions in yellow and black, emphasizing local modification of the antipolar pattern. The horizontal scale bar is 5 nm.

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Discussion

To summarize, we have fabricated pure PbZrO3 antiferroelectric thin films by pulsed laser deposition. The thin films were successfully grown under tensile strain on LaLuO3 artificial substrates and did not show any strain relaxation from 9 to 200 nm. Local atomic map investigations by scanning transmission electron microscopy show the characteristic up-up-down-down dipole pattern of the orthorhombic Pbam phase, with no indication for ferroelectric or ferrielectric phase coexistence, down to 9 nm. Capacitors based on these PbZrO3 films display sharp double hysteresis of polarization as a function of the electric field with no remanent polarization, evidencing purely antiferroelectric behavior. Our results give clear evidence that the proposed epitaxy stabilizes the antiferroelectric phase against competing ferroelectric or ferrielectric phases down to thicknesses of 9 nm. These findings suggest that epitaxial tensile strain can stabilize antiferroelectricity in PbZrO3 down to nanometer thicknesses, opening the way for antiferroelectric-based nanodevices.

Methods

Sample preparation

All the films were grown by pulsed laser deposition using a KrF excimer laser (Coherent Compex Pro 110, 248 nm wavelength), with sintered ceramic targets (Toshima Manufacturing Co.) of stoichiometric composition for LaLuO3 and SrPbO3, and with 10% Pb excess for PbZrO3. The DyScO3 substrates were preliminarily annealed at 1000 °C for 3 h under pure O2 flow, and single termination was systematically checked with atomic force microscopy (Bruker Innova). The 100 nm-thick LaLuO3 layer was grown at 730 °C under a dynamic O2 pressure of 0.4 mTorr, at 5 Hz laser repetition rate and 2.5 J/cm2 fluence on the target. The sample temperature was then reduced to 560 °C, while O2 was replaced by N2O with dynamic pressures of 40 mTorr and 120 mTorr for SrPbO3 and PbZrO3, respectively. For both materials, higher growth temperatures result in high lead deficiency and increased surface roughness. The 15 nm-thick SrPbO3 layer was grown at a laser repetition rate of 2 Hz and 1 J/cm2 fluence, and these values were then increased to 5 Hz and 2.5 J/cm2 for the PbZrO3 layer. Lower fluence gives rise to non-stoichiometric precipitates on the film surface. Following the growth, the sample was cooled down to room temperature under 300 Torr of static O2 pressure. The top electrodes, composed of 20 nm of Pt, were deposited by RF magnetron sputtering.

Electrical measurements

P-E hysteresis loops were recorded at room temperature with a Radiant Multiferroic tester.

X-ray diffraction

Conventional 2θ-ω scans and reciprocal space maps were performed using Cu Kα–1 radiation in a 9-kW rotating anode Rigaku SmartLab diffractometer.

Scanning transmission electron microscopy

STEM images were acquired using a Cs-corrected USTEM Nion microscope at 100 and 200 keV. Polar displacements were evaluated using two-dimensional Gaussian fitting of the atomic positions in the HAADF images, as implemented in the Atomap software35. The off-center displacements of Pb cations were calculated with respect to the geometric center of the four surrounding Zr-site columns, and the arrows representing the off-center intensities and directions were overlaid on the HAADF images at the Pb column positions.