Introduction

The robust structure of perovskites grants them exceptional tolerance to many instabilities and distortions, which further generates a variety of extraordinary functionalities1,2,3. When different structural instabilities possess similar free energies, the interaction between them determines the manifested properties4,5,6. More remarkably, some instabilities can even combine to interact with the electronic degrees of freedom or other stable lattice modes to create new phases with extended functionalities7,8,9. A particularly intriguing scenario is the cooperative coupling between soft antiferrodistortive (AFD) modes and the hard polar/antipolar modes, which gives rise to hybrid “improper” dipole order that is fundamentally different from conventional “proper” ferroelectrics8,10. Such an improper mechanism not only opens up avenues to merge electric-field-controllable (anti)ferroelectricity with octahedral-mediated magnetism among a wide spectrum of layered perovskites such as (Ca, Sr)3Mn2O7 and NaLaMnWO68,11,12,13,14, but also accounts for the unusual twin-wall ferroelectricity in many simple perovskites like CaTiO3 and SrTiO315,16,17. With the dipole order as a secondary effect, the triggered polar states are geometrically protected12,18, which can overcome the tremendous energy penalty when dipoles curling periodically19. In this context, exploring polar structures induced by AFD can also offer an alternative scheme for more complex polar textures that are energetically disfavored in proper ferroelectric crystals12,19,20. In this work, we report the experimental observation of an unconventional antiferroelectric (AFE) order generated by AFD modulation in NaNbO3-based perovskites, revealing such capability.

Acknowledged as “the most complex ferroelectric perovskite”21, NaNbO3 has been the subject of extensive crystallographic studies for over half a century6,22,23,24,25,26. This complexity originates from the instability competition among multiple octahedral rotations and off-center cation displacement23,27, resulting in a series of symmetry-lowering phase transitions (Supplementary Fig. 1). Some phases such as R-NaNbO3 (R stands for an orthorhombic phase named by H D Megaw et al.22,23) exhibit AFD modulation with quadrupled or sextupled periodicities28,29,30, or even incommensurate modulation (when doped)31,32,33, whose detailed structure is still under debate with discrepancies from various characterization techniques30,32,33,34. In particular, the exact octahedral rotation pattern associated with the AFD modulation still remains unresolved30,33. The challenge to resolve such structure ambiguity lies in the difficulty in solving large superstructures containing irregular structure fluctuations, a task traditional diffraction analysis struggling with35,36,37,38. This difficulty is particularly pronounced in R-NaNbO3, existing in a high temperature range, whose diffraction signals can be compromised by mixed phases and varying modulation periods34,39,40. Here, we stabilize the R phase down to room temperature by composition engineering (see Supplementary Figs. 2–8), which allows us to access the nature of its modulated structure via robust room-temperature characterization. Applying advanced scanning transmission electron microscopy (STEM) on compositionally engineered NaNbO3-based compounds, we map both the cation displacement and octahedral rotation across multiple dimensions. This comprehensive approach enables us to unveil in R phase not only an unusual AFE order but also its improper origin as AFD modulation, demonstrating a new design avenue for intriguing ferroelectric structures in perovskites.

Results

Composition engineering enables robust room-temperature R-NaNbO3

The characteristics of R-NaNbO3 include the presence of both \(\frac{1}{2}\) oo0p (o = odd, and the subscript p stands for the prototype perovskite unit cell) reflections from in-phase octahedral rotation (c+ in Glazer notation), which vanish in the room-temperature P phase, and the satellite reflections indexed by the modulation wavevector q = \(\frac{1}{\lambda }\) q010 (λ ≥ 4) (Supplementary Fig. 1b)28,29. The transition between R and P phases is also characterized by the dielectric anomaly at ~360 °C in pure NaNbO3 (Supplementary Fig. 1a). To explicitly resolve the structure of R-NaNbO3 using STEM, we face two peculiar challenges: (i) its stable temperature range of 360−480 °C requires high-temperature STEM imaging, whose image resolution may degrade due to in situ heating; (ii) its incommensurate structure modulation with illy-defined and potentially fluctuating periodicities. Composition engineering is thus exploited to overcome these issues. Firstly, we stabilize R phase down to room temperature by alloying a paraelectric material such as CaTiO331, which simplifies our experimental characterization. Secondly, we tune the modulation wavevector from incommensurate ~\(\frac{1}{12.3}\) q010 to commensurate \(\frac{1}{4}\) q010 by 1.5 mol% Mn doping (Supplementary Fig. 3), which further erases the complexity associated with the incommensurate modulation. The obtained Mn-doped (NaNbO3)0.85(CaTiO3)0.15 exhibits all the characteristic diffraction signals of R phase, which persist from room temperature up to 450 °C (Supplementary Fig. 4). At last, due to the low Z-contrast (Z is the atomic number) of Na (Z = 11) compared to Nb (Z = 41) in annular dark-field (ADF) STEM imaging, we partially substitute Na with Ag (Z = 47) to enhance the contrast of A cations (Supplementary Fig. 5), enabling more reliable determination of atomic positions and displacement. Despite the subtle ionic size difference between Na+ and Ag+, a proper amount of Ag substitution does not change the characteristic structure of R phase (both with and without Mn doping, see Supplementary Fig. 6). The obtained ceramics clearly exhibit multidomain structure with a ___domain size of hundreds of nanometers, as revealed by TEM analysis in Supplementary Fig. 7. Atomic-scale elemental mapping in STEM does not reveal any cation segregation in our samples (Supplementary Fig. 8), indicating that the modulation is purely displacive without chemical separation. All the steps involved in the synthesis of Mn-doped (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3 allow us to perform room-temperature characterization on the commensurately modulated structure of R phase.

Mapping the antiferroelectric distortion in R-NaNbO3

The enhanced Z contrast of A sites enables us to visualize both the A and B columns by ADF-STEM, as shown in Fig. 1a–c. We adopt the modulation-free lattices as the references (see Supplementary Note 1 and Supplementary Fig. 9), which allows us to elucidate the displacement of both A and B cations responsible for the supercell multiplicity along the three principal projections (namely, [100]p, [010]p and [001]p). As presented in Fig. 1d–f, all three projections exhibit antipolar displacement, revealing the three-dimensional (3D) AFE distortion with the periods of 4ap (ap—lattice parameter of the pseudocubic cell) along [010]p and 2ap along both [100]p and [001]p, consistent with the supercell identified by electron diffraction (Supplementary Fig. 2d). Particularly, Fig. 1d shows the largest displacement along [100]p, which forms a transverse wave along [010]p, manifesting the charge separation and AFE ordering akin to prototypical AFE PbZrO341,42. Since the B cations are much less displaced (as indicated by cyan arrows in Fig. 1d–f), we use the positions of B cations to represent the centers of the corresponding BO6 octahedra and then the polarization field can be reflected by calculating the relative displacement of A columns against the four neighboring B columns, as presented in Fig. 1g–i. Such polar displacement is also confirmed by integrated differential phase contrast (iDPC) STEM imaging with all the A, B, and O atomic columns being considered (Supplementary Figs. 10 and 11). Despite the long-standing belief that the R phase possesses AFE22,27,34,38,43, our study, to the best of our knowledge, presents the first experimental evidence revealing its AFE order, with the detailed lattice displacement mapped out. Furthermore, Fig. 1d shows that the displacement of B cations is non-collinear (see the averaged displacement profiles in Supplementary Fig. 12), which also contradicts with previous prediction22,34 and suggests that it is not generated solely from the soft antipolar mode connecting to a Brillouin zone center instability44. This, together with the absence of characteristic dielectric anomaly during the paraelectric-to-AFE transition to form the R phase (at ~480 °C, see Supplementary Fig. 1a), signifies an unconventional origin of its AFE.

Fig. 1: Imaging the AFE modes in Mn-doped (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3 by ADF-STEM.
figure 1

ADF-STEM images projected along [001]p (a), [100]p (b), and [010]p (c) zone axes. The mirror planes (or “tilt-twin boundaries”) are denoted by vertical dashed lines in (a, b). d–f Cation displacement maps with respect to the reference lattices derived from (a–c). Arrows in yellow and cyan show the displacement of A and B cations, respectively, with length scaling with the magnitude. The eigenmodes dominating the cation displacement fields in (d–f) according to group theoretical analysis are overlaid in (a–c). g–i Relative displacement maps of A cations with respect to the four B-cation neighbors derived from (a–c).

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Resolving the novel octahedral rotation pattern

The capability to visualize both heavy and light elements with high sensitivity of iDPC-STEM also enables us to explicitly correlate the lattice displacements with the BO6 octahedral rotations. As illustrated in Supplementary Fig. 13, the identified octahedral rotation involves ~8° c+-rotation around [001]p (Fig. 2a) and ~11° rotation around [110]p (Fig. 2b), which resembles the classic a−a−c+ rotation in Glazer notation45. Indeed, the evident elongation of O columns in [100]p and [010]p projections (suggested by dashed outlines in Fig. 2c, d) clearly indicates the misalignment of O due to out-of-phase octahedral rotations about these two axes (a−b−). The a−b−c+ rotation, akin to a−a−c+ in Pnma perovskites, is consistently accompanied by antipolar displacement of A cations to optimize its coordination environment46,47. However, the periods of antipolar displacement should be 2ap along [100]p and [010]p directions47, rather than the complex displacement pattern mapped out in Fig. 1e, f. This implies the presence of inhomogeneous octahedral rotations around [010]p axis deviating from conventional a−a−c+ in Glazer notation30. A closer inspection of the displacement maps (Fig. 1 and Supplementary Fig. 10) identifies the existence of (010)p mirror planes recurring every two ap along [010]p, as indicated by vertical dashed lines in Fig. 1a, b and Supplementary Fig. 10a, b. Across the mirror planes, the out-of-phase b−-rotation should switch to the local in-phase b+-rotation (Supplementary Fig. 14), giving rise to an unusual AFD modulation following the sequence of a−b−c+/a−b+c+/a−b−c+/a−b+c+ along [010]p. The simulated iDPC-STEM images based on the derived AFD modulation can reproduce the observed O column distortions, including both the zigzag O-B-O chain (Fig. 2b) and the O column elongation (Fig. 2c, d). While such mirror planes have been reported as “twin walls” in CaTiO3 and “tilt-twin boundaries” in complex double perovskites15,17,35,48,50,50, the detailed octahedral rotation sequence across them has never been directly imaged before, owing to the overlap of the a−b+c+ and a−b−c+ heterostructures along the rotational axis [010]p (Supplementary Fig. 14).

Fig. 2: Imaging the AFD modes in Mn-doped (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3 by iDPC-STEM.
figure 2

Magnified iDPC-STEM images projected along [001]p (a), [110]p (b), [100]p (c), and [010]p (d) zone axes. Displacement or elongation of O columns, as indicated by arrows in (a, b) or dashed outlines in (b–d), manifesting octahedral rotations. The zigzag O-B-O chain in (b) indicates out-of-phase octahedral rotation commonly seen in a−b−c+/− Glazer rotation. Structure models aside and the simulated images with derived octahedral rotation show consistency with observations. The A/B/O atoms are shown in green/blue/red.

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Instead of directly imaging along the rotational axis [010]p, our approach involves examining projections along the orthogonal axes ([001]p and [100]p) to identify the modulated rotation based on the shift of apical O atoms within BO6 octahedra51. As illustrated in Fig. 3a, b, combined octahedral rotations around [100]p and [010]p (a−b− or a−b+) cause shift of apical O atoms approximately along [110]p/[\(1\bar{1}0\)]p (Fig. 3b), which would otherwise coincide with the position of B cations when no rotation occurs (Fig. 3a). This shift leads to the diagonal elongation of B-O columns that can be detected in the [001]p-projected ADF-STEM image (see bottom panel of Fig. 3b). Furthermore, across the (010)p mirror plane with a−b+ rotation, elongation of B-O columns switches to the orthogonal direction (i.e., from [110]p to [\(1\bar{1}\)0]p elongation as indicated by cyan box in Fig. 3b), in contrast to the a−b− rotation that only relates unidirectional elongation (as indicated by magenta box in Fig. 3b). Consequently, mapping the elongation variation enables us to efficiently distinguish a−b+ from a−b−, as demonstrated in Fig. 3c–f. To carry out this differentiation, principal component analysis (PCA) (see ref. 52 and Methods) was applied to the [001]p-projected ADF-STEM image of Fig. 1a. As shown in Fig. 3c, d, the first PCA component (PCA1) reflects the average column shape which is nearly round, while the following PCA component (PCA2) signifies the diagonal elongation of B-O columns along [110]p or [\(1\bar{1}\)0]p. The exact elongation direction can be determined by the sign of PCA2 weight factor (Fig. 3d): the positive/negative weight factor corresponds to [110]p/[\(1\bar{1}\)0]p elongation, respectively (indicated by dashed ovals in Fig. 3d), while the mirror planes are located where the elongation direction (sign of weight factor) switches, representing the local a−b+ rotation. Combining with the c+ rotation imaged from the equatorial O atoms in Fig. 2a, we can reconstruct the full octahedral rotation pattern as Fig. 3b. It unequivocally reveals the periodic modulation following the sequence of a−b−c+/a−b+c+/a−b−c+/a−b+c+ along [010]p, with the largest cation displacement located at the “tilt-twin boundaries” (Fig. 1d). Column shape analysis on the [100]p-projected image and STEM imaging on Mn-doped (Na0.85Ca0.15)(Nb0.85Ti0.15)O3 provide consistent results, as presented in Supplementary Figs. 15–17.

Fig. 3: AFD modes spatially resolved via principal component analysis (PCA) in Mn-doped (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3.
figure 3

a Model of the parent structure with non-rotated octahedron (left) and the simulated ADF-STEM image (right) showing round shape of the B-O column, as indicated by dashed outlines. b Model (top) of the fully relaxed structure from first-principles calculations with AFD modulation along b axis and the simulated ADF-STEM image (bottom) showing unidirectional B-O column elongation for a−b−c+ rotation and switching elongation at the mirror plane (a−b+c+ “tilt-twin boundary”). Column shape elongation in the ADF-STEM image is indicated by dashed ovals. c PCA1 revealing the average round shape of B-O columns in the ADF-STEM image of Fig. 1a. d PCA2 showing the second eigenshape with positive (top left) and negative (top right) weight factors. The corresponding reconstructed shapes considering PCA1 and PCA2 are given at the bottom. e Weight factor map of PCA2. f Magnified experimental ADF-STEM image from the region in dashed box in (e), matching the simulated ADF-STEM image at the bottom of (b).

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Coupled antiferroelectric and antiferrodistortive orders

We emphasize that fluctuation in modulation periods does not alter the essential characteristics of the AFE ordering and modulated octahedral rotation observed above. As illustrated in Fig. 4 and Supplementary Fig. 18, similar AFE ordering with (a−b−c+)m/a−b+c+/(a−b−c+)n/a−b+c+ (m, n = integer) modulation is identified in (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3 samples without Mn doping. The only variation observed pertains to the periodicity, which is in line with its incommensurate modulation revealed by electron diffraction (Supplementary Fig. 2c). This finding not only excludes the possibility that Mn doping changes the nature of the structure modulation in R-NaNbO3, but also demonstrates an effective way to tune the octahedral modulation and the underpinned AFE via chemical doping. We also note that even though modulated octahedral rotation in R phase has been previously documented in literature, with many contradictory rotation sequences claimed27,30,33,34,38,52, our STEM observation here has explicitly unveiled its nature as a−b+c+ (or mirror plane) modulated a−b−c+ rotation, as generally described by (a−b−c+)m/a−b+c+/(a−b−c+)n/a−b+c+. With a larger modulation wavelength, it presents a solid validation of the AFD-polar coupling mechanism for ferroelectricity: polarization arises from the spatial variation of the AFD mode53,54. As revealed in Fig. 4b–e and Supplementary Fig. 18c–e, polar displacements develop across the “tilt-twin boundaries” with their magnitude reaching a maximum right at these boundaries, where AFD variation (gradient) is the most pronounced, rather than within the coherent a−b−c+ nanodomains.

Fig. 4: Antiferroelectric order in incommensurately modulated (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3.
figure 4

a ADF-STEM image projected along [001]p zone axis. The mirror planes (or “tilt-twin boundaries”) are denoted by vertical dashed lines. b Cation displacement map with respect to the reference lattice derived from (a). Arrows in yellow and cyan show displacement of A and B cations, respectively. c Weight factor map of PCA2 derived from (a) showing incommensurate AFD modulation. The AFD modulation wave can be manifested by the spatial variation of the PCA2 weight factors. d Vertically averaged displacement profiles for A (dA) and B (dB) cations along [100]p extracted from (b). e Vertically averaged weight factor profile extracted from (c). The displacive modulation wave shown in (d) and the AFD modulation revealed in (e) are in quadrature (90° out of phase), signifying the AFD-polar coupling. Error bars indicate ±1 SD.

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In addition to the cation displacement and modulated rotation of the BO6 octahedra, we have also identified the additional deformation of the octahedra due to the shift of O atoms along [100]p. As revealed by iDPC-STEM imaging in Supplementary Fig. 18d, the [100]p shift of O occurs primarily at the (010)p mirror planes. Interestingly, the largest A-cation displacement also happens at these mirror planes (Supplementary Fig. 18c), which is along [100]p but opposite to the aforementioned O shift (Supplementary Fig. 18d). This leads to the largest charge separation and the associated polarity at the mirror planes, resembling the ferroelectricity reported at the twin walls of CaTiO315,17. Indeed, the twin walls in CaTiO3 are also mirror planes across which one of the out-of-phase rotation components (a−) in a−a−c+ switches to the local in-phase rotation, similar to the a−b+c+ rotation identified in this study. We can then interpret the observed (a−b−c+)m/a−b+c+/(a−b−c+)n/a−b+c+ modulation in R-NaNbO3 as a−b−c+ nanodomains being interrupted by a−b+c+ twin walls, a feature that has been proposed in numerous other perovskite oxides beyond CaTiO3 as “tilt-twin boundaries”49,50,55. It is ubiquitous in all Pnma perovskite oxides that the a−a−c+ rotation induces collective A-cation displacement along [110]p/[\(1\bar{1}0\)]p, but in an antipolar manner, resulting in polarization cancellation and paraelectricity46,47,56. At the (010)p twin walls, however, the local mirror symmetry dictates the displacement perpendicular to the walls (i.e., [010]p components) to vanish, which leaves the dominant [100]p displacement at the walls. A detailed analysis by Bellaiche and Iniguez47 has explicitly proposed that a−b+c+ rotation induces pure [100]p A-cation displacement in perovskites. While such displacement has led to local ferroelectricity at the CaTiO3 twin walls15,17, it is distinctively manifested as AFE here in R-NaNbO3. The analysis by Bellaiche and Iniguez has also been extended to modulated octahedral rotation a−bkc+, which indeed predicts the emergence of the AFE ordering along [010]p47. However, such a modulated tilt scheme has not yet been experimentally validated before and the associated improper AFE has thus remained unsettled. Here by resolving the R phase structure in real space, we have not only directly mapped out the predicted tilt scheme, but also explicitly unearthed the underlying AFE order, demonstrating a prototypical example of hybrid improper AFE in simple perovskites. It is also worth noting that most previous structure models of R phase, proposed based on diffraction analysis, emphasized the AFE displacement of B cations22,27,34. Despite the larger Born effective charge of Nb (+9.54e) compared to Na (+1.13e) and O (−1.67e), which could contribute more to the local polarization of R-NaNbO3, our real-space imaging here unambiguously unravels the dominant role of A-cation and O displacement, rather than the B cations, in establishing the AFE order.

Multimode interaction triggers the unique antiferroelectric order

So, what is the driving force for such an unusual modulation in octahedral rotation? First-principles calculations on the prototype cubic NaNbO3 (space group Pm\(\bar{3}\)m) reveal structural instabilities represented by phonons with negative frequencies, which tend to condense at lower temperature to trigger symmetry-lowering phase transitions accompanied by atomic displacement. As shown in Fig. 5a, the lowest phonon branch reveals pronounced AFD instabilities forming a flat band along the M-T-R path39,57. This flat band indicates comparable energies for exhibiting pure in-phase or out-of-phase rotation patterns, or some intermediate/combined configurations, interpreting the presence of rich competing phases in NaNbO3 (Supplementary Fig. 1a)6,33,39. It is also consistent with the experimentally identified octahedral rotations with tunable modulation periods in R-NaNbO3, where the in-phase rotation (c+) corresponds to the irreducible representation (irrep.) M3+ of Pm\(\bar{3}\)m, the out-of-phase rotation (a−) corresponds to the irrep. R4+ 58, while the intermediate configuration (bk) corresponds to the irrep. T4(½, ½ − 1/λ, ½) with λ varying according to the position along this flat branch (between 0 and ½). We note that the modulated octahedral rotation in R-NaNbO3 appears between the high-temperature a−b+c+ rotation24,46,52 and the low-temperature a−b−c+ rotation59 (Supplementary Fig. 1a), which can be interpreted as an intermediate state due to the competition between M3+ and R4+ modes.

Fig. 5: AFD modulation triggered AFE order in R-NaNbO3.
figure 5

a Phonon dispersions of cubic NaNbO3 along the M-T-R path exhibiting a flat soft phonon band (in thick) involving AFD modulation (irrep. T4). b Structure model of 2 × 4 × 2 superstructure. c Local AFD mode amplitudes resolved along [010]p. d Cation displacements of AFE Δ5 mode. Experimental value derived from Fig. 1d is shown in dashed lines. e Electric polarization resulted from the AFE Δ5 mode. AFD amplitude and polarization are determined based on A-site-centered perovskite cells. f Energy profiles in order-parameter (eigenmode) space (M3+,R4+,T4,Δ5,Z1) along selected paths bridging (M3+,R4+,T4,Δ5,Z1) = (1, 1, 1, 0, 0) and (M3+,R4+,T4, Δ5,Z1) = (1,1,1,1,1) states. The amplitudes of the order parameters (horizontal axis) are normalized to 1. The vertical coordinate indicates the energy levels when gradually incorporating Δ5 and Z1 modes into the reference structure with only the M3+, R4+, and T4 modes [namely, (M3+,R4+,T4,Δ5,Z1) = (1,1,1,0,0)] along different paths.

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Of particular significance is that by condensing the experimentally identified rotational modes, namely M3+(½,½,0), R4+(½,½,½), and T4(½,¼,½), the relaxed supercell structure successfully reproduces all the structural characteristics observed by our STEM imaging. This includes not only the a−b−c+/a−b+c+/a−b−c+/a−b+c+ (λ = ¼) AFD modulation (Fig. 5b, c), but also the AFE displacement (Fig. 5d) that matches the experimental maps in Fig. 1d, with the A-cation dominant displacement occurring at the mirror planes. The obtained structure also fits well with X-ray diffraction data (Supplementary Fig. 19), inferring the pure R phase of the compositionally engineered samples. It is worth emphasizing that the condensed modes pertain exclusively to octahedral rotations with no requirement of an antipolar instability. These rotational instabilities can spontaneously generate a transverse polarization wave (Fig. 5e), thus demonstrating the existence of improper AFE driven by octahedral rotation. Indeed, rotational modes such as R4+(½,½,½) break the A-site symmetry and can activate the X5+ mode that manifests as collective [110]p/[\(1\bar{1}0\)]p shift of A cations for a−a−c+ rotation and [100]p shift for a−b+c+ rotation, as illustrated by numerous theoretical studies15,47,53 and also unveiled by our STEM analysis. Such octahedral-rotation-driven A-cation displacement has been identified as the origin of hybrid improper (anti)ferroelectricity in layered perovskites8,11,12,13,14, while here it gives rise to hybrid improper AFE in the non-layered R-NaNbO3.

To disclose the origin of the unique non-collinear AFE displacement (Fig. 1d), group theoretical analysis is adopted to decompose all the lattice modes contributing to R-NaNbO3 (see Supplementary Note 2). It suggests a multimode scenario relating to the parent Pm\(\bar{3}\)m structure: three primary AFD modes [M3+(½,½,0), R4+(½,½,½) and T4(½,½ − 1/λ,½), λ ≥ 4] establishing the Pnmm space group and multiple secondary AFE modes induced by the hybrid AFD mode as byproducts8 (see Supplementary Fig. 20, Supplementary Table 1, and Supplementary Note 3). This multimode picture is indeed identified by our observations, with all the significant lattice distortions mapped out, as overlaid in Fig. 1 (also see Supplementary Fig. 20 and Supplementary Tables 2 and 3). More remarkably, it reveals two types of Lifshitz invariants to trigger the unusual AFE order in Fig. 1d. One involves the interaction between primary AFD and secondary AFE modes which takes the form60,61:

$${E}_{{coupling}}({{{\bf{u}}}},{{{\mathbf{\Phi }}}})=\frac{{{{\rm{\xi }}}}}{2}\left[{{{\mathbf{\Phi }}}}\frac{\partial {{{\bf{u}}}}}{\partial y}-{{{\bf{u}}}}\frac{\partial {{{\mathbf{\Phi }}}}}{\partial y}\right]$$

where \({{{\mathbf{\Phi }}}}\) and \({{{\bf{u}}}}\) represent the AFD and AFE distortions, respectively, and ξ is the coupling coefficient. In R-NaNbO3, it entails three AFD modes with octahedral rotation around the three cubic axes [\({{{\mathbf{\Phi }}}}={{{\mathbf{\Phi }}}} ({\Phi}_{x},{\Phi}_{y},{\Phi}_z)\)], which together couple with an AFE mode (\({{{\bf{u}}}}\)), manifesting a quadrilinear flexo-AFD coupling mechanism. For example, considering the AFE mode Δ5, which is directly responsible for the polarization wave \({P}_{x}\), this coupling contributes to an energy decrease of ~4.7 meV/atom, as indicated by the energy profiles (purple) in Fig. 5f54,60,61. The other type of Lifshitz invariants describes the higher-order interaction between two secondary AFE modes with the presence of the AFD modes, which leads to the 3D displacement configuration (see Supplementary Fig. 20). Such interaction is exemplified by the Δ5 and Z1 pair modes, where the Z1 mode can be further activated if the Δ5 mode is incorporated into the three AFD modes, corresponding to an additional energy decrease of ~1.4 meV/atom (green profile). It reveals an asymmetric coupling between these two AFE modes (Δ5⊕Z1), mimicking the so-called electric Dzyaloshinskii-Moriya interaction proposed by Zhao et al.62, where two AFE modes are coupled, capable of establishing a non-collinear cation displacement configuration due to the presence of the AFD modes (see ref. 63). The left-right canting behavior of B-cation displacement as mapped in Fig. 1d can therefore be explained by these multimode interactions. Such a physical picture accounting for the unique AFE order in NaNbO3 demonstrates the critical role of AFD modulation in establishing AFE rather than the previously recognized flexoelectric effect, therefore broadening the understanding of the origin of AFE in perovskites and related materials.

Discussion

We note that NaNbO3-based perovskites have been recognized as viable lead-free AFE materials for dielectric energy storage26,38,43. Their superb storage capacity and efficiency are largely attributed to the incommensurately modulated R phase. Conventional diffraction methods are difficult to tackle such incommensurate modulations due to the streaky superlattice reflections pertaining to the highly fluctuated nature. While atomic-scale ADF-STEM imaging provides a robust avenue to image the local structure in real space, it primarily presents a projected structure. Due to the lack of depth resolution and the obscure Z-contrast of light O atoms, directly visualizing the 3D octahedral rotation in perovskite oxides has been challenging. As such, the exact octahedral rotation pattern behind R phase is much underexplored. Here, we simplify this structure problem by stabilizing a commensurate phase at room temperature and adopt PCA methodology to extract the subtle contrast of O columns along orthogonal zone axes so that the 3D octahedral rotation pattern could be unambiguously reconstructed. This methodology is further consolidated by picometer precision cation displacement measurement and first-principles calculations, eventually allowing us to unlock the intricate interplay between the AFD and AFE modes underpinning the complicated 3D structure. We have also measured the temperature-dependent dielectric response (Supplementary Fig. 21) and room-temperature polarization-electric field (P-E) loop (Supplementary Fig. 22) of Mn-doped (Na1 − xCax)(Nb1 − xTix)O3 ceramics (x = 0.06, 0.08, 0.10, 0.11, 0.13, and 0.15) crossing the P-R phase boundary, to clarify the effect of the identified AFE structure on the physical properties. Upon increasing CaTiO3 content to x ≥ 0.11, the dielectric anomaly signifying the transition to conventional AFE P phase is heavily damped (Supplementary Fig. 21d–f), indicating the suppression of polar instability and stabilization of AFE R phase down to room temperature. As a result, the electric switching to a polar state is dispersed and significantly delayed, and the square-shaped double P-E loop then turns rather slimmer (Supplementary Fig. 22e, f) with minimized hysteresis. Yet, a large polarization can still be maintained upon applying an electric field (Supplementary Fig. 22f). This thus manifests a substantial enhancement of efficiency (>80%) without loss of capacity, heralding the superior figure of merit of R phase for dielectric energy storage applications among NaNbO3-based compounds.

To summarize, by comprehensively mapping all structural distortions using STEM in conjunction with first-principles calculations, we have revealed a previously unobserved AFD modulation described by (a−b−c+)m/a−b+c+/(a−b−c+)n/a−b+c+ in R phase of NaNbO3-based perovskites. This unique rotation triggers A-cation-dominated AFE polarization along [100]p, especially at the periodic (010)p mirror planes with local a−b+c+ rotation, resembling the twin-wall ferroelectricity in CaTiO315,17. The tunable wavelength via chemical modification offers additional opportunity to regulate the density of the dipole states. Such improper AFE is fundamentally different from the conventional AFE in P phase and other perovskite AFEs such as PbZrO363, where an antipolar instability condenses and plays an important role in establishing AFE, as evidenced by the Curie–Weiss type dielectric anomaly23,64,65. It also challenges the earlier proposed chemical ordering32 or B-cation displacement22,34 as the origin of AFE in R-NaNbO3. Our results also suggest that the octahedral rotational phonon mode responsible for the AFE ordering is highly localized, as evidenced by the flat phonon band with zero group velocity and the highly fluctuated periodicity. Such a localized mode implies localization of the observed electric dipoles, reminiscent of the scale-free ferroelectricity in HfO266, indicating the potential for ferroelectric devices via electric control at the unit-cell level. Having all the ion displacement mapped out, we have also identified an abnormal non-collinear B-cation displacement pattern with a considerable component along the modulation wavevector (q[010]p). Such a dipolar configuration resembles the non-collinear spin structures in magnets and would further guide the design for non-collinear/non-coplanar ground-state polar textures that are rare but highly appreciated in ferroelectric materials. In this regard, our study opens up an exceptional venue to explore complex dipole topologies in perovskite materials through AFD engineering.

Methods

Ceramics fabrication and XRD/electrical measurements

Ceramics with the compositions of NaNbO3, (Na0.85Ca0.15)(Nb0.85Ti0.15)O3, (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3, and 1.5 mol% Mn-doped (Na0.85Ca0.15)(Nb0.85Ti0.15)O3 and (Na0.65Ag0.20Ca0.15)(Nb0.85Ti0.15)O3 were fabricated using the conventional solid-state method. Stoichiometric amounts of reagents [Na2CO3 (99.8%), CaCO3 (99.8%), Ag2O (99.7%), Nb2O5 (99.99%), and TiO2 (99.99%)] were first mixed by ball milling and then calcined at 900–980 °C. The obtained powders were milled again with the addition of proper amounts of MnO2 (98.8%), and then pressed into pellets and sintered at 1190–1320 °C. Ceramics with varying Mn, Ca, or Ag content were fabricated by the same process. Phase purity and crystal structure of the as-prepared samples were characterized using a powder X-ray diffractometer (XRD, SmartLab-3kW, Rigaku Ltd, Tokyo, Japan). The Rietveld refinement was performed using GSAS. For dielectric measurement, the sintered pellets were polished and coated with silver electrodes, and the dielectric permittivity was measured as a function of temperature using an impendence analyzer. Polarization versus electric field (P-E) loops were measured using a TF Analyzer 3000 ferroelectric tester.

S/TEM characterization and image simulation

Electron diffraction patterns were taken using a JEOL JEM-2100F transmission electron microscope (TEM) operated at 200 kV. In situ TEM heating was performed using a Protochips Fusion holder with sample loaded on the E-chips. To reduce the sample damage, the beam was blanked during the heating process. STEM was carried out on a probe corrected Spectra 300 S/TEM (ThermoFisher Scientific) equipped with an X-FEG source operated at 300 kV. A probe current of 40 pA and a convergence semi-angle of 29.9 mrad were adopted. The ADF-STEM images were collected with a semi-angle range of 72–200 mrad, while the iDPC-STEM images were collected using a four-segment detector with a semi-angle range of 5–21 mrad. The atomic column positions were determined via 2D Gaussian fitting of the ADF-STEM images using StatSTEM package67 or via the center of mass for iDPC-STEM images considering the elongated O column shape deviating from a 2D Gaussian peak. Energy dispersive X-ray spectroscopy was performed using the super-X detection system equipped in the Spectra 300 S/TEM. The probe currents used for atomic-scale elemental mapping and quantitative composition analysis were 50 pA and 80 pA, respectively. The multislice image simulation was performed using the Dr. Probe package68 under the conditions in accordance with experiments. The sample thickness was set to ~32 nm with 1/4 Na replaced by Ag to mimic the experimental A-site contrast.

Principal component analysis

A column-shaped analysis procedure based on the PCA following ref. 52 was adopted to analyze the B-column shape in the ADF-STEM images. In short, the exact positions of the atomic columns are firstly determined via 2D Gaussian fitting. This gives the coordinates (xij, yij) of the atomic column centers in the i-th column and j-th row of the B sublattice. To locate the column center with subpixel resolution, the original image is linearly interpolated, and a square segment covering the full shape is selected around each position. Subsequently, the set of image segments is analyzed via principal component decomposition, i.e., each column shape is represented as an average shape plus a linear superposition of orthonormal eigenvectors with position-dependent weight factors

$${S}_{{i}{j}}=A+{\sum }_{k=2}^{\infty }{w}_{{ij}}(k){x}_{{ij}}(k)$$

where A is the first PCA component (PCA1) reflecting the average column shape, k (≥2) is the order of the following PCA component (PCAk), xij(k) are the k-th order eigenvectors that are orthonormal, and wij(k) are the corresponding position-dependent weight factors. The k-th PCA component can be described by PCAk =  wij(k)xij(k).

First-principles calculations

First-principles calculations were performed on pure NaNbO3 with the Perdew-Burke-Ernzerhof form and the generalized gradient approximation for exchange-correlation potential. Plane-wave basis sets with the projector augmented plane-wave method were used as implemented in the Vienna ab initio simulation package (VASP). The valence states used for the calculations were 2p63s1 for Na, 4s24p64d45s1 for Nb, and 2s22p4 for O, respectively. An energy cut-off of 600 eV was used for the plane-wave basis set. The structure was considered as relaxed when the maximum component of the Hellmann-Feynman force acting on each ion was less than 0.005 eV/Å. The phonon dispersions of cubic NaNbO3 were calculated based on the fully relaxed structure with a 2 × 2 × 2 supercell using the density functional perturbation theory method implemented in the PHONOPY code interfaced with VASP. The atomic structure of R phase was calculated by freezing the unstable phonon modes with a subsequent relaxation. Gamma-centered k-point meshes of 8 × 8 × 8 and 4 × 2 × 4 were used to sample the Brillouin zones for structure relaxation of cubic and R-NaNbO3, respectively, while a Gamma-centered k-point mesh of 4 × 4 × 4 was used for phonon calculation. To obtain the ferroelectric polarization of R-NaNbO3, we first computed the Born effective charges Z*, and then the unit-cell electric polarization was calculated by evaluating P ≈ \(\frac{1}{\varOmega }\)ΣZ*u, where u is the ionic displacement in R phase relative to the ideal cubic lattice of NaNbO3 and Ω is perovskite unit-cell volume.