Nanofiber-like polar configurations enable ultrahigh energy storage in relaxor ferroelectrics via high-entropy design

Abstract

Dielectric capacitors are highly promising for advanced power electronics due to their superior power density. However, their widespread application has been limited by inadequate energy storage performance. In this work, we propose a distinct strategy, centered on creating a nanofiber-like polar configuration, to achieve ultrahigh energy storage performance in relaxor ferroelectrics. This is realized through a high-entropy composition design in NaNbO₃-based perovskites, where multiple cations with significant ferroelectric activity and ionic radius differences are introduced at both A- and B-sites. This approach effectively disrupts the long-range ferroelectric and antiferrodistortive orders simultaneously. The resulting nanofiber-like domain structure, which features several nanometers in length but only a few unit cells in width, exhibits a low-hysteresis and nearly linear polarization response to applied electric field (E). Consequently, an exceptional energy storage density (W_rec) of ~19.3 J cm⁻³, a high efficiency (η) of ~95.2%, and an impressive W_rec/E coefficient of 208 J kV mm⁻³ are simultaneously achieved in the NaNbO₃-(Bi_0.5Li_0.5)(Ti_0.8Zr_0.1Sn_0.1)O₃ multilayer ceramic capacitors. These results underscore the immense potential of the nanofiber-like polar configuration strategy for developing high-performance dielectric energy storage materials.

Authors create nanofiber-like polar configurations in relaxor ferroelectrics via high-entropy design, achieving ultrahigh energy storage density (~19.3 J cm⁻³) and efficiency (~95.2%) in multilayer ceramic capacitors.

Introduction

The relentless drive toward miniaturized energy-autonomous systems, coupled with the extreme demands of high-power electronics in applications such as ballistic missile defense, oil well drilling, and next-generation electric vehicles, has placed unprecedented requirements on energy storage components. These applications demand not only ultrahigh power density and rapid charge-discharge capabilities but also exceptional reliability under harsh mechanical and thermal conditions^1–5. Dielectric capacitors, owing to their intrinsic fast charging/discharging kinetics and high voltage tolerance, have emerged as critical enablers in such scenarios. Among them, ceramic-based capacitors, particularly multilayer ceramic capacitors (MLCCs), stand out due to their excellent stability, large volumetric capacitance enabled by their multilayer structure, and reliable operation under high voltages^6–8. This volumetric advantage allows MLCCs to achieve higher overall capacitance within a given footprint compared to ceramic-based thin film capacitors. The performance of MLCCs is largely governed by the properties of their dielectric ceramic layers. Consequently, considerable research efforts in recent years have been directed toward developing ceramic dielectric materials with superior energy storage characteristics^2,6–13.

The energy storage performance of dielectric materials is primarily determined by their polarization-electric field (P-E) response. Based on the nature of P-E behavior, dielectric materials can be categorized into several types, among which antiferroelectrics (AFEs) and relaxor ferroelectrics (FEs) have shown great promise for energy storage applications due to their distinctive P-E characteristics^3–5. However, each type faces inherent limitations. AFEs exhibit high maximum polarization (P_max) driven by field-induced AFE-FE phase transitions, yet the associated large hysteresis from the first-order transition leads to low energy storage efficiency (η). In contrast, relaxor FEs display low hysteresis and thus high η owing to the dynamics of highly active polar nanoregions (PNRs), but their relatively low P_max often restricts the recoverable energy density (W_rec). Therefore, achieving simultaneous optimization of W_rec and η remains challenging, and most current studies report a compromise between the two. In recent years, various strategies and concepts, such as multi-phase nanodomain coexistence^2,6, relaxor AFE^14,15, superparaelectric state¹⁶, high-entropy design¹⁷, supercritical state¹⁸, and dipole glass¹², have been proposed to address the W_rec-η trade-off, leading to remarkable progress. Nevertheless, attaining an ultrahigh W_rec (>15 J cm⁻³) along with η exceeding 95% remains a bottleneck^3–18. Moreover, many reported high-performance values are achieved under extremely high electric fields, which hinder practical device integration. In this context, achieving efficient energy storage per unit volume under low electric fields is a critical consideration for applications that demand miniaturization and system-level integration¹⁹. In parallel, the figure of merit W_rec/E—representing the energy storage capacity per unit applied field—has also been widely adopted in the literature to assess the field efficiency of dielectric materials^20,21, as it provides complementary insight into material performance across different field ranges. Addressing these field-dependent performance metrics is therefore essential for translating material-level innovations into viable device platforms, where MLCCs offer a particularly compelling architecture. Their multilayer structure enables high volumetric energy storage without relying solely on extreme operating fields, directly tackling the integration challenges that often limit practical deployment.

Conventional design strategies for relaxor FEs, which involve disrupting long-range ferroelectric order to generate PNRs, inevitably face a trade-off between hysteresis and polarization. Furthermore, the high dynamics of PNRs can induce premature polarization saturation. A recent phase-field simulation study on relaxor FE energy storage materials suggests that the formation of nano-columnar domains can simultaneously yield low hysteresis and high polarization¹⁹; however, this prediction has not yet been experimentally verified. In this work, we propose a new strategy to resolve the conflict between W_rec and η by constructing a nanofiber-like polar configuration within a relaxor FE matrix, as shown in Fig. 1. We selected NaNbO₃ (NN), known for its coexisting AFE and FE phases^21,22, as the base matrix. The incorporation of Bi_0.5Li_0.5TiO₃ (BLT) stabilizes the FE phase, while its small tolerance factor enhances antiferrodistortive behavior, promoting stripe-like domains, and the mixed valence and ionic radii of Bi³⁺, Li⁺, and Ti⁴⁺ effectively disrupt long-range FE order. Further substitution of Zr⁴⁺ and Sn⁴⁺ at the B-site increases configurational entropy, refining the domains into a nanofiber-like configuration that differs from conventional PNRs. These nanofiber-like domains exhibit low hysteresis and a nearly linear, high polarization response under an applied electric field, ultimately enabling outstanding energy storage performance in MLCCs (Fig. 1b, c)

Fig. 1. Strategy for nanofiber-like polar configuration engineering in energy-storage relaxor FEs.

a Phase-field calculated domain configurations and corresponding polarization-electric field (P-E) loops in NaNbO₃-based antiferroelectric system. a1, b4. Pure NaNbO₃ with striped long-range-ordered domains, showing square P-E loop; a2, a5. NaNbO₃ with mediate entropy; a3, a6. NaNbO₃ with elaborate high-entropy design, showing nanofiber-like nanodomains. b schematic diagram of the MLCC structure. c a comparison of energy storage properties between this work and some recently-reported works.

Results and discussion

To gain theoretical insight into the impact of configurational entropy on domain morphology and polarization behavior, we first performed phase-field simulations on the NaNbO₃-based system with increasing entropy levels. As shown in Fig. 1a, pure NaNbO₃ exhibits long-range ordered striped antiferroelectric domains, characteristic of its pristine state, which yields a square polarization–electric field (P–E) loop with large hysteresis²². With moderate entropy increase (0.7NN-0.3(Bi_0.5Li_0.5)TiO₃, abbreviated as NN-BLT), the long-range order is partially disrupted, resulting in a more complex and fragmented domain structure accompanied by a slimmer P–E loop. Remarkably, at the highest entropy level (0.7NN-0.3(Bi_0.5Li_0.5)(Ti_0.8Zr_0.1Sn_0.1)O₃, abbreviated as NN-BLTSZ), the system spontaneously reorganizes into anisotropic, nanofiber-like polar domains. This configuration exhibits a nearly hysteresis-free, linear P–E response while maintaining high polarization, suggesting an optimal balance between local disorder and cooperative polarization.

The emergence of such nanofiber-like domains can be understood in terms of the unique energy landscape of a high-entropy state. In this regime, the gradient energy term becomes dominant due to the extreme compositional fluctuations. The polarization vectors along the short axis of the fibers are highly constrained, forming quasi-one-dimensional polarization channels that enable continuous polarization rotation with a low energy barrier, a fundamentally different mechanism from the nucleation-and-growth mechanism in conventional polar nanoregions. However, the atomic-scale structural features underpinning this simulated behavior, including the precise cation arrangements, oxygen octahedral tilting, and the real-space morphology of these domains, require direct experimental validation. In the following sections, we employ multiscale characterization techniques, from X-ray diffraction to atomically resolved scanning transmission electron microscopy (STEM), to experimentally verify the existence of nanofiber-like domains and elucidate their structural origin in our high-entropy NN-BLTSZ system.

The ceramics with compositions of NN, 0.7NN-0.3(Bi_0.5Li_0.5)TiO₃ (NN-BLT), 0.7NN-0.3(Bi_0.5Li_0.5)(Ti_0.8Zr_0.1Sn_0.1)O₃ (NN-BLTSZ) were designed and synthesized via a conventional solid-state reaction route. X-ray diffraction analysis (Fig. S1) confirms the formation of a single-phase perovskite structure without any secondary phases for all compositions. Scanning electron microscopy (SEM) images (Fig. S2) reveal densely packed microstructures. For the pure NN sample, clearly defined grain boundaries are observed, whereas the NN-BLT sample exhibits relatively blurred grain boundaries, likely due to the incorporation of BLT components that introduce local compositional heterogeneity and disrupt grain boundary contrast. The domain morphology evolution was investigated using transmission electron microscopy (TEM) and piezoresponse force microscopy (PFM), as illustrated in Fig. 2. In NN ceramics, the coexistence of AFE orthorhombic P and FE orthorhombic Q phases is confirmed by striped macroscopic domains, along with 1/4 and 1/2 superlattice spots in selected-area electron diffraction (SAED) patterns (Fig. 2a), consistent with earlier reports^20–22. In contrast, NN-BLT exhibits only the FE orthorhombic Q phase, characterized by nanoscale stripe-like domains with 1/2 superlattice spots (Fig. 2b), indicating that BLT incorporation stabilizes the FE phase²¹. Further substitution of Sn⁴⁺ and Zr⁴⁺ at the B-site enhances local structural inhomogeneity, leading to additional domain refinement and the emergence of nanofiber-like domain configurations in NN-BLTSZ (Fig. 2c). Moreover, TEM and SAED analyses also confirm that the NN-BLTSZ composition maintains a single-phase perovskite structure without any detectable secondary phases or local phase separation, indicating that the high-entropy design results in a homogeneous solid solution. Domain dynamics were probed via in-situ PFM (Fig. 2d-f). Under a low electric field (10 V), NN shows pronounced amplitude signals corresponding to FE macrodomains, whereas NN-BLT and NN-BLTSZ exhibit a featureless response indicative of polar nanodomains, corroborating the relaxor FE character. Temperature-dependent dielectric spectroscopy (Fig. S3) further supports their relaxor FE behaviors. At higher voltages (50 V), NN-BLT develops micron-scale FE domains with strong piezoelectric activity, while NN-BLTSZ maintains finer domain structures. This suggests a progressive suppression of the electric-field-induced transition from short-range to long-range polar order from NN-BLT to NN-BLTSZ compositions. Correspondingly, the P-E loops become progressively slimmer from NN to NN-BLT to NN-BLTSZ, as shown in Fig. S4. To ensure a fair comparison, NN-BLT was measured up to its maximum applicable field of 35 kV mm⁻¹, under which its η remains below 80% and decreases with increasing field. In contrast, NN-BLTSZ maintains an efficiency above 90% even at fields up to 60 kV mm⁻¹, clearly demonstrating the benefit of the high-entropy design in reducing hysteresis. We plot the calculated configurational entropy (ΔS_conf) against the measured W_rec and η across our composition series, as shown in the Table S1. A clear positive correlation is observed, where the compositions with higher ΔS_conf consistently trend toward superior W_rec and η. Concurrently, the current density-electric field (J-E) curves exhibit broadened and suppressed current peaks. As a result, the NN-BLTSZ composition achieves an outstanding energy storage performance, with a recoverable energy density W_rec = 8.2 J cm⁻³ and an η = 93.1%, demonstrating significant potential for high-performance dielectric capacitors^23–25.

Fig. 2. Domain Morphology of NN, NN-BLT, and NN-BLTSZ bulk ceramics.

a bright-field and SEAD images of NN ceramics. b bright-field and SEAD images of NN-BLT ceramics. c bright-field and SEAD images of NN-BLTSZ ceramics. In-situ PFM results of d NN, e NN-BLT, and f NN-BLTSZ ceramics. The subscripts 1 and 2 represent the test results at 10 V and 50 V, respectively.

To elucidate the unique polar structure in the NN-BLTSZ ceramics, atomic-scale elemental mapping was performed, as presented in Fig. 3a. The principal constituent elements, including Na, Nb, Ti, and Bi, all exhibit a homogeneous spatial distribution. In contrast, the minority substituents Sn and Zr display discernible local agglomeration, indicating the presence of pronounced local chemical heterogeneity^14,26. This chemical fluctuation is further corroborated by the line-scan elemental profiles in Fig. 3b. The atomic-resolution high-angle annular dark-field (HAADF) image in Fig. 3c. The atomic displacements of the B-site cations are extracted with respect to their four nearest A-site cations as polarizations^27,28, revealing a commensurately modulated polar structure with a width of only approximately two unit cells. That is, the magnified local atomic structure shows a near “up-down” polarization configuration, which is consistent with the 1/2 (010) superlattice spots in the FFT image. The larger view (Fig.3d) presents the distinctive nanofiber-like domains assembled by the localized “up-down” polar regions, which can be more directly visualized in the enlarged area 1 and area 2. This nanofiber-like feature is not only reflected in the angle modulation but also in the magnitude modulation (Fig. S5), which clearly illustrates the highly anisotropic in statistical distributions (Fig. S6). Specifically, the polarization magnitudes fluctuate from high polar to nonpolar, where the average value is ~13.25 pm, and the polarization angles present a wide-angle distribution characteristic ranging from -180° to 180°. The polarization angle as a function of distance along [0-10]_c direction once again verifies the local “up-down” polarization configuration (Fig. 3e).

Fig. 3. Atomic-scale domain structures of the x = 0.06 ceramic.

a,b Atomic element distribution and the corresponding line-scan elemental profiles. c Atomic-resolution HAADF images along [100]_c direction. The inset is fast Fourier transform (FFT) pattern where the 1/2 (010) superlattice spots suggest the 2-fold modulation structures based on local “up-down” polarization configuration from the magnified localized atomic structure. d Atomic displacement mapping. Color scales in the polarization denote the polarization angle. e Polarization angle as a function of distance along [0-10]_c derived from the white line 1-4 in (d). f Atomic-resolution iDPC image along [100]_c direction. The magnified localized atomic structure shows the oxygen octahedral tilt. g Calculated oxygen octahedral tilt mapping. Color scales denote the oxygen octahedral tilt angle. h Oxygen octahedral tilt angle as a function of distance along [0-10]_c derived from the white line 1-4 in (g).

This fiber-like nature of the nanodomains is as opposed to the commonly observed equiaxed PNRs. Typically, the disruption of long-range FE order favors the formation of spherically shaped PNRs to minimize the system’s energy^29,30. The formation of the observed fiber-like domains, which are only several unit cells in width and extend for tens of nanometers, is intimately linked to the patterns of oxygen octahedral tilting. The oxygen octahedral tilts are calculated from the iDPC image and the corresponding tilt angle is shown in the schematic diagram (Fig. 3f). The oxygen octahedral tilt mapping (Fig. 3e) reveals a short-range-ordered tilting pattern with a modulation period nearly twice (showed in the enlarged aera 1 and aera 2) that of the polar modulation, yet without developing long-range correlation, which is also visibly shown in the oxygen octahedral tilt angle as a function of distance along [0-10]_c direction (Fig. 3h). This specific structural manifestation is attributed to the low tolerance factor (τ) inherent to the high-entropy composition design. As listed in Table S1, the τ decreases from 0.967 for NN to 0.955 for NN-BLT and further to 0.953 for NN-BLTSZ. The reduced τfavors the preservation and enhancement of oxygen octahedral tilting^31,32, while the high-entropy effect subsequently disrupts the coherence of these tilts, preventing long-range ordering²⁶. This relaxor FE system, characterized by its nanofiber-like domains, is fundamentally distinct from conventional relaxors with quasi-spherical PNRs. It concurrently maintains high spontaneous polarization and short-range ferroic order, which is instrumental in achieving the concurrent high polarization and low hysteresis observed in the macroscopic P-E loops. Furthermore, the local high orientational alignment of these fiber-like domains is believed to be responsible for the delayed polarization saturation under increasing electric fields.

The MLCCs fabricated using the NN-BLTSZ composition demonstrate outstanding structural integrity and high functional performance. Cross-sectional SEM imaging (Fig. 4a) reveals a well-defined, homogeneous laminated structure consisting of four densely packed dielectric layers, each with an average thickness of approximately 13 μm, alternated with continuous and coherent internal electrodes. Energy-dispersive X-ray spectroscopy (EDS) elemental mapping confirms minimal interdiffusion across the dielectric-electrode interfaces, indicating excellent chemical compatibility and interfacial stability. Furthermore, the dielectric layers exhibit very low porosity and are free from macroscopic defects such as cracks or delamination, with interlayer thickness variation tightly controlled within ±0.7 μm. P-E loop measurements (Fig. 4b) underscore the superior energy storage characteristics of the NN-BLTSZ MLCCs. The materials exhibit slim hysteresis loops with strongly enhanced P_max of ~52.4 μC cm^-2 at 93 kV mm^-1 and ultralow remnant polarization (P_r < 0.5 μC cm^-2), resulting in hysteresis losses accounting for less than 5% of the total energy input. This behavior can be attributed to the presence of localized nanofiber-like domains, which effectively restrict long-range ferroelectric ordering under high electric fields while facilitating rapid and reversible polarization switching. As a result, the NN-BLTSZ MLCCs achieve a remarkable W_rec of 19.3 J cm^-3 and a high η of 95.2% under an applied electric field of 93 kV mm^-1 (Fig. 4d). These values significantly surpass those of most state-of-the-art lead-free and even lead-based MLCCs (Fig. 4f, g), highlighting the efficacy of the nanofiber-like polar configurations in enabling ultrahigh energy storage performance^{4–15,17,18,23–26,33–43}.

Fig. 4. Energy-storage performances of the NN-BLTZS MLCCs.

a The cross-section image, b SEM image on surface, and c EDS images of representative elemental distributions of the MLCCs. d P-E loops and e field-dependent W_rec and η values of the MLCCs. f A comparison of W_rec/E etween the NN-BLTZS MLCC and other recently-reported ceramic capacitors^{4–15,17,18,23–26,33–43}. g A comparison of W_rec and corresponding testing electric field between the NN-BLTZS MLCC and other recently-reported MLCCs^{4–15,17,18,23–26,33–43}.

The stability of energy storage performance is essential for ensuring the reliable operation of NN-BLTZS MLCCs^41–44. As illustrated in Fig. 5a, P-E hysteresis loops were measured over a frequency range of 1-100 Hz. The MLCCs exhibit remarkable frequency stability, with a W_rec of 7.7 J cm^-3 varying within only ±1%, and η maintained above 96.7% with less than 2% fluctuation across the tested frequencies (Fig. 5c). In addition, the P-E loops show minimal temperature dependence over a broad temperature span from 25 ^oC to 200 ^oC, highlighting excellent thermal stability. Both hysteresis loss and P_max are well retained, leading to negligible variations in energy storage parameters (ΔW_rec < 3%, Δη < 2%), as summarized in Fig. 5b and e. The fatigue endurance was also evaluated by subjecting the MLCCs to repeated unipolar cycling. As shown in Fig. 5c, the P-E loop remains nearly unchanged even after 10³ fatigue cycles. After 10⁴ cycles, a slight decrease in P_max is observed, resulting in less than 5% reduction in W_rec, which is within an acceptable range for practical applications. This indicates that the nanofiber-like polar configuration maintains its structural integrity under repeated electrical cycling. The charge-discharge performance further underscores the application potential of these capacitors. Under external electric fields ranging from 30 to 50 kV mm^-1, the NN-BLTZS MLCCs display well-defined underdamped discharge waveforms (Fig. 5e). The power density (P_D) increases with applied electric field, reaching an ultrahigh value of ≈167.5 MW cm^-3 at 50 kV mm^-1 (inset of Fig. 5e). In the overdamped discharge mode with a fixed load resistance of 150 Ω, stable oscillatory waveforms are observed across the same field range. As shown in Fig. 5f, a discharge energy density (W_D) of ≈4.7 J cm^-3 is achieved at 50 kV mm^-1, with 90% of the stored energy released within an extremely short duration of ≈0.34 μs. This rapid discharge behavior is attributed to the fast polarization-field response⁴², consistent with the frequency-insensitive P-E loops observed in Fig. 5a. In view of these results, the NN-BLTZS MLCCs demonstrate outstanding overall energy storage characteristics, combining high stability under varying operational conditions with superior pulse discharge capability, showing great promise for use in advanced pulsed-power electronic systems.

Fig. 5. Stability and charge/discharge performances of the NN-BLTSZ MLCCs.

a P-E loops under different temperatures and b W_rec and η values as a function of temperature at 50 kV mm⁻¹. c P-E loops under different frequencies and d W_rec and η values as a function of frequency at 50 kV mm⁻¹. e cycling-dependent P-E loops and f corresponding W_rec and η values at 50 kV mm⁻¹. g Overdamped discharge waveforms. h W_D as a function of electric fields. i Field-dependent P_D and W_D values.

In summary, we have demonstrated an effective strategy to overcome the long-standing trade-off between W_rec and η in dielectrics by constructing a nanofiber-like polar configuration via a rationally designed high-entropy system. Multiscale structural analyses reveal that the high-entropy composition disrupts the long-range and antiferrodistortive orders of the NaNbO₃ matrix, spontaneously forming anisotropic nanodomains with lengths of several nanometers and widths of merely a few unit cells, where a morphology fundamentally distinct from conventional polar nanoregions. This unique, constrained geometry enables a nearly anhysteretic and linear polarization response, yielding record-breaking MLCC performance: an ultrahigh W_rec of ~19.3 J cm^-3, a superior η of ~95.2%, and an impressive W_rec/E coefficient of 208 J kV·mm^-3 in MLCCs, outperforming most reported lead-free dielectrics. Beyond these metrics, our work establishes a new paradigm of entropy-mediated nanofiber-like texture engineering, offering a viable pathway to tailor dielectrics for demanding real-time applications such as extreme-environment power electronics, high-frequency traction inverters, and next-generation grid infrastructure, while potentially extending to other functional material systems.

Methods

Fabrication of bulk ceramics and MLCCs

The perovskite powders (NN, NN-BLT, and NN-BLTSZ) were prepared through a conventional solid-state reaction using high-purity precursors (>99%): Na₂CO₃, Nb₂O₅, Li₂CO₃, Bi₂O₃, TiO₂, ZrO₂ and SnO₂. After weighing, the raw materials were thoroughly mixed and ball-milled in ethanol with zirconia media for 6 h. The dried mixtures were then calcined at 900°C for 8 h in covered alumina crucibles. To produce bulk ceramics, the calcined powders were pressed into pellets. To minimize sodium volatilization during sintering, a layer of powder with the same composition as the pellet was placed on top of each sample. The pellets were then sintered in air at optimized temperatures: 1350 °C for pure NN, 1100 °C for NN-BLT, and 1150 °C for NN-BLTSZ, all for 2 h in sealed crucibles.

The fabrication process for MLCCs began with dispersing the calcined NN-BLTSZ powder in a solvent containing a dispersant, followed by ball milling for 12 h. Subsequently, binders and plasticizers were introduced, and the mixture was milled for another 12 h. After filtration and de-airing, the slurry was tape-cast into green sheets. Ag/Pd (70:30) electrode paste was screen-printed onto the sheets, which were then stacked, laminated at 50 °C for 5 min, and warm isostatically pressed at 50 MPa and 55 °C to improve green density. Thermal debinding was performed at 600 °C for 2 h with a ramping rate of 0.5 °C/min, followed by sintering at 1210 °C for 2 h to obtain the final MLCCs.

Phase-field simulations

The total free energy for the NaNbO₃-based antiferroelectric system can be expressed as following⁴⁴:

$$F=\int \int \int \left(f_{lan}+f_{grad}+f_{elas}+f_{elec}+f_{localelec}\right)dV$$ 1

where V is the volume of the system.

The landau free energy f_lan can be calculated by

$$f_{lan}=a_1\left(P_1^2+P_2^2+P_3^2\right)+a_{11}\left(P_1^4+P_2^4+P_3^4\right)+a_{12}\left(P_1^2{P}_2^2+P_2^2{P}_3^2+P_1^2{P}_3^2\right)+a_{112}\left[P_1^4\left(P_2^2+P_3^2\right)+P_2^4\left(P_1^2+P_3^2\right)+P_3^4\left(P_1^2+P_2^2\right)\left)\right)\right]+a_{111}\left(P_1^6+P_2^6+P_3^6\right)+a_{113}\left(P_1^2{P}_2^2{P}_3^2\right)$$ 2

where a₁, a₁₁, a_12, a₁₁₂, a₁₁₁ and a₁₁₃ are Landau coefficients.

The gradient free energy f_grad representing the energy from polarization inhomogeneity is described as⁴⁵:

$$f_{grad}={\sum }_i\left\{\begin{array}{c}\alpha \left[{\left(\frac{\partial P_i}{\partial x}\right)}^2+{\left(\frac{\partial P_i}{\partial y}\right)}^2+{\left(\frac{\partial P_i}{\partial z}\right)}^2\right]\hfill \\ +\beta \left[{\left(\frac{{\partial }^2{P}_i}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^2{P}_i}{\partial y^2}\right)}^2+{\left(\frac{{\partial }^2{P}_i}{\partial z^2}\right)}^2\right]\hfill \end{array}\right\},\left(i=1,2,3\right)$$ 3

where α is a negative coefficient representing the stability of AFE phase, β is a positive coefficient representing the stability of FE phase.

The long-range elastic interaction energy f_elas includes strain gradient field and external stress field energy, which can be expressed as:

$$f_{elas}=\frac{1}{2}\left\{C_{ijkl}\left({\epsilon }_{ij}-{\epsilon }_{ij}^0\right)\left({\epsilon }_{kl}-{\epsilon }_{kl}^0\right)+C_{ijkl}{e}_{ij}^{\prime }\left({\epsilon }_{kl}-{\epsilon }_{kl}^0\right)\right\}$$ 4

where C_ijkl is the elastic stiffness tensor, ε_ij, ε⁰_ij and e’_ij are the total strain, electrostrictive stress-free strain and local strain of ion radius difference.

The electric energy f_elec can be expressed as:

$$f_{elec}=-\frac{1}{2}{E}_i{P}_i-\frac{1}{2}{E}_{i,depol}\overline{P_i}$$ 5

where E_i represents the inhomogeneous electric field caused by dipole-dipole interactions, E_i,depol is the average depolarization field. $\overline{P_i}$ is the average polarization.

The local random distribution electric field energy f_localelec related with dopant valence and concentration can be expressed as⁴⁶:

$$f_{localelec}=E_{i,localelec}\overline{P_i}$$ 6

where E_i,localelec is the random distribution electric field.

The temporal evolution of the polarization field can be calculated by solving the time-dependent Ginzburg-Landau (TDGL) equation:

$$\frac{d{P}_i\left(x,t\right)}{dt}=-M\frac{\delta F}{\delta P_i\left(X,t\right)},i=1,2,3$$ 7

where M is the kinetic coefficient related to the domain mobility, t is time and i is the variant of polarization. The simulations were carried out in two dimensions with cell sizes 128Δx×128Δy (Δx and Δy are the number of grid points, which are equal to 2 nm) with periodic boundary conditions.

Structure characterizations

XRD data were acquired using a diffractometer (D/Max-RB, Rigaku, Japan). Grain morphology of the ceramics was examined by field-emission SEM (SU8020, JEOL, Japan). Domain evolution under bias voltage was studied via PFM (Asylum Research MFP-3D Origin + , Oxford Instruments, UK). Raman spectra were recorded on well-polished samples with a Raman spectrometer (LabRam HR Evolution, HORIBA JOBIN YVON, France). A probe-corrected Thermo Scientific™ Spectra 300 S/TEM, equipped with a Super-X Detector, was used for aberration-corrected scanning transmission electron microscope (STEM) imaging. The operating conditions were as follows: accelerating voltage=300 kV, convergence semi-angle=30 mrad. HAADF images were obtained at the atomic scale using a detector semi-angle range of 64−200 mrad. The integrated differential phase contrast (iDPC) images were collected with a segmented four‑quadrant DF4 detector. The atomic column positions at picometer precision fitting were operated by MATLAB code^47,48. Additional domain-morphology observations and SAED were conducted on a JEOL JEM-F200 microscope at 200 kV. Specimens for (S)TEM analysis were prepared by mechanical thinning followed by final Ar⁺ ion-milling using a Gatan PIPS II system.

Electric property measurements

Dielectric properties as functions of temperature and frequency were measured with an LCR meter (E4980A, Agilent Technologies, USA) coupled with a high/low-temperature probing stage (HCT1821, Tongguo Technology, China). P-E hysteresis loops were obtained using a FE test system (TF Analyzer 2000E, aixACCT Systems GmbH, Germany). Charge-discharge characteristics were evaluated on a dedicated energy-storage testing platform (CFD-003, Tongguo Technology, China).

Author contributions

N.L. and A.X. conceived the experiments. A.X., N.L., A.T., X.W., Z.L., X.S., W.W., and X.F. synthesized the samples and measured macroscopic properties. L.H. performed the calculation. N.L. and F.Q. performed the SEM, TEM, and STEM experiments. N.L., A.X., L.H., R.Z., Z.Y., Z.W., and S.D. analyzed the data and wrote and edited the manuscript. All authors contributed to the discussion and manuscript preparation.

Peer review

Peer review information

Nature Communications thanks Ampattu R. Jayakrishnan, Eunmi Lee and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

N. L. discloses support for the research of this work from National Natural Science Foundation of China (grant number 52502108) and Natural Science Foundation of Zhejiang Province (LQN25E020002). A. X. discloses support for the research of this work from National Natural Science Foundation of China (52302131) and Natural Science Foundation of Anhui Province (2308085QE140). L. H. discloses support for the research of this work from National Natural Science Foundation of China (grant number 12504114).

Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information files.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-73375-8.