Introduction

The escalating demand for devices characterized by superior energy and power densities is catalyzing breakthroughs in the development of materials for energy storage applications1,2. While batteries and supercapacitors possess high energy densities, their power densities (batteries ~10–102 W kg−1, supercapacitors ~ 102–106 W kg−1) fail to meet the requirements for pulse power applications. Owing to the energy storage mechanisms facilitated by dipole orientation, dielectric capacitors achieve exceptionally high-power densities (~107–108 W kg−1). However, their comparatively low energy densities present significant obstacles to the miniaturization and integration of advanced devices. The energy storage performance of dielectric capacitors is primarily determined by two critical metrics of the recoverable energy density (Wrec) and efficiency (η), which are closely associated with the saturation polarization (Pmax), remnant polarization (Pr), and breakdown strength (Eb) of their materials3,4.

In pursuit of alternatives to lead-based materials, a variety of eco-friendly ceramics have been rigorously explored, including BaTiO3 (BT), Bi0.5Na0.5TiO3(BNT), K0.5Na0.5NbO3 (KNN), BiFeO3 (BF), NaNbO3 (NN), and AgNbO3 (AN). Notably, BiFeO3 stands out due to its substantial spontaneous polarization (~100 μC cm−2) and high Curie temperature (TC) of around 830 °C, showcasing potential for achieving outstanding energy storage performance. However, the square P-E loop of pure BiFeO3, characterized by large Pr, is suboptimal for energy storage applications. Additionally, the volatility of Bi3+ ions and the valence fluctuations of Fe ions between Fe3+ and Fe2+ ions lead to a relatively high electrical leakage current, thereby inhibiting the Eb5,6. To address the aforementioned challenges, various methods have been employed to enhance the energy storage properties of BF-based ceramics, such as high-entropy design7, introduction of aliovalent ions and liquid phases as sintering aids8, defect engineering9. For example, the substitution of NN for BF-BT ceramic unexpectedly offers several advantages, including increased bandgaps, reduced grain size, and enhanced resistivity, all of which collectively contribute to a significant enhancement in dielectric breakdown strength10. Incorporating non-isovalent Zn2+ and Nb5+ ions into BF-based ceramics effectively suppresses polarization hysteresis caused by ergodic relaxor zones and random fields, while also inducing a highly delayed polarization saturation, with the polarization magnitude continuously increasing under the electric field of critical evolution, thereby enhancing energy storage performances11. Through entropy engineering, long-range ferroic orders are disrupted into local polymorphic distortions featuring diverse BO6 tilt types and heterogeneous polarization configurations in Bi0.47Na0.47Ba0.06TiO3-Sr0.7La0.2Ta0.2Ti0.75O3 ceramic, resulting in a high polarization response, negligible remnant polarization, delayed polarization saturation, and enhanced Eb12. Despite ongoing efforts, the energy storage performances of BF-based ceramics still fall short of anticipated for application (Wrec < 10 J cm−3). Therefore, an in-depth exploration of the internal structure of BF-based ceramics is imperative to significantly enhance their energy storage capabilities.

Regulating the characteristics of microscopic nanoregions offers a promising approach for developing high-performance relaxor ferroelectric (RFE) BF-based ceramics. In particular, modulating chemical compositions and sintering temperatures facilitates the creation of diverse PNR configurations. However, the Eb still shows a prematurely reached at the critical condition, especially single-phase distribution. Additionally, the pronounced relaxation behavior in RFE ceramics leads to decreased polarization, posing persistent challenges in achieving high Wrec13. Consequently, the design of polar nanoregions with multi-oriented is proposed to alleviate anisotropy and rotational barriers between dipoles, which is expected to achieve high polarization, enhanced breakdown strength, and reduced hysteresis comprehensively, thereby optimizing the comprehensive performance.

Along this way, the (0.7-x)BiFeO3-0.3BaHf0.05Ti0.95O3-xSr(Al0.5Nb0.5)O3 (BF-BHT-xSAN) ceramics are prepared via solid-phase sintering. Phase-field simulations and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) demonstrate an improved polarization restoration and a distinctly non-periodic phase structure arrangement. This unique phase configuration with multiple disordered phases significantly minimizes hysteresis loss. Additionally, grain refinement and introduction of high-bandgap ions synergistically improves the breakdown electric field. As a result, the designed lead-free BF-based ceramics simultaneously achieve an ultra-high Wrec of 12.2 J cm−3 and η of 90.1% at 830 kV cm−1. Additionally, the BF-BHT-0.15SAN ceramic exhibits minimal fluctuations in temperature stability (ΔWrec < 7.7%, η < 13.8%) and insensitivity in frequency stability (Wrec < 2.8%, η < 2%, from 1 to 200 Hz), and anti-fatigue characteristics (Wrec < 1%, η < 2%, over 10,000 st cycles). The design concept of the phase structure configuration in this system also provides a reference for further development of advanced ceramic dielectric capacitors.

Results and discussion

Phase field simulation is a powerful computational method for characterizing physical properties. As shown in Figs. 1a, b and S1, when E = 0, the ‘initial’ model is set up with a relatively homogeneous dipole arrangement, including a large ferroelectric domain with essentially the same direction and amplitude of polarization over the observed range. Subsequently, more cubic phases with undirected and smaller polarization amplitudes are added to the simulation as the ‘optimal’ model, which lead to a more multiplicative and dispersed distribution of arrows in the simulation region. Figure 1c illustrates the amplified plots of the various phases that set up in the simulation, including tetragonal (T), rhombohedral (R) and cubic (C) phases with different macroscopic symmetries. From the theoretical point of view, the disorder of the local composition is derived from the increase of elemental species, the mismatch of atomic size, mass and electronegativity of the dopant atoms, which is able to reduce the polarization anisotropy and enhance the relaxor behavior effectively14,15,16. Therefore, we performed a 2D phase field simulation with the ‘initial’ and ‘optimal’ models as matrices and reproduced the evolution of the polarization in the microstructure with respect to the electric field. When the electric field is changed, it can be explicitly observed in the ‘initial’ model that the dipole direction is aligned with the applied electric field and maintains a certain degree of polarization distribution after the electric field is withdrawn, which exhibits the characteristics of a conventional ferroelectric with a large residual polarization in the P-E hysteresis loop (Fig. 1d). On the contrary, the ‘optimal’ model shows a disordered arrangement of phases with narrower polar regions, and this correspondingly smaller domain structure provides higher activity, which is conducive to the rapid recovery from the long ordered state to the initial state after the electric field is withdrawn17. Therefore, the simulated polar region exhibits a slender P-E curve as shown in Fig. 1e. In addition, from the line shape of the Landau-Devonshire free energy curve (Fig. 1f), the ‘optimal’ model is flatter than the ‘initial’, which proves that the domain switching barriers in the multiphase and disordered mode can be further weakened, and the hysteresis loss will be suppressed to the maximum extent18. Therefore, based on theoretical simulations and calculations, the highly disordered modulation of multiple phases becomes a feasible strategy to achieve excellent energy storage performance.

Fig. 1: Phase field simulation.
figure 1

a, b The associated evolution of polar structures of “initial” and “optimal” models under a applied electric field of E0 at points A,B,C. c Magnified images of different phases setting in the simulation. d, e Calculated P-E hysteresis loops under a given field of E0. f The landau energy curves.

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Figure S2a displays the X-ray diffraction (XRD) patterns of BF-BHT-xSAN ceramics at room temperature, revealing that all compositions maintain a pure perovskite structure without any detectable impurity phases. According to the magnified XRD patterns, (111) diffraction peaks gradually shift towards higher angles as SAN content increases, indicating lattice contraction due to the replacement of larger radius Bi3+ and Fe3+ ions by smaller Sr2+ and (Al0.5Nb0.5)4+ ions. To further elucidate the crystal structures of the ceramics, Rietveld refinement analyses of XRD patterns are performed, as presented in Figs. 2a and S2. The incorporation of SAN induces the phase transition from R3c (R) to Pm-3m (C), with the C phase content increasing from 30.1% to 75.8%, the details are displayed in Table S1.

Fig. 2: The microstructural researches of the BF-BHT-xSAN ceramics.
figure 2

a Rietveld refinement analysis of BF-BHT-0.15SAN ceramics. b Phase content variations diagram. c The Raman spectra of BF-BHT-xSAN ceramics at room temperature. d The SAED patterns along the [110] direction. The HAADF-STEM image of (e) x = 0.10 and (f) x = 0.15. The polarization angle variation of (g) x = 0.10 and (h) x = 0.15. The polarization magnitude variation of (i) x = 0.10 and (j) x = 0.15. k Typical dipole orientation of each phase. l, m Enlarged plots of HAADF for x = 0.15.

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The vibrational dynamics of the internal atomic structure of BF-BHT-xSAN ceramics are characterized using Raman spectroscopy, as depicted in Fig. 2c. In view of the difficulty in directly observing the width variations of Raman spectral peaks between 150 and 400 nm−1, this study calculated the Full Width at Half Maximum (FWHM) within this wavelength range, as shown in Fig. S3. The FWHM values increase with x, indicating that the Raman peaks become flatter and broader. This spectral behavior is indicative of reduced hybridization between the O2− 2p orbitals and the vacant orbitals of Ti4+ d0 and Fe3+ d5, which is closely associated with the formation of nanoscale domains19. The peaks at 500 cm−1 and 640 cm−1 exhibit shifts to lower and higher wavenumbers with increment of x content, signifying that increased local structural disorder within the BO6 octahedra induced by the co-doping of Sr2+ and (Al0.5Nb0.5)4+, which generates localized random electric fields and promotes the formation of polar nanoregions (PNRs)20.

Figures 2d and S4 display the TEM and SAED images of the BF-BHT-0.15SAN ceramic. Notably, no superlattice diffraction spots can be observed from the three zone axes of [110]c, [111]c, and [100]c, indicating that the polarization configuration is influenced by cation displacements rather than oxygen octahedral tilting. To gain deeper understanding of the intrinsic atomic-scale structure, the atomic arrangement of BF-BHT-0.15SAN ceramic is observed using a high-angle annular dark-field transmission electron microscope (HAADF-TEM), and a variety of polarization configurations are analyzed in terms of the cation vector motions, as shown in Fig. 2e, f. The positions of atomic columns at the strongly contrasting A sites and the weaker B sites are estimated through 2D Gaussian fitting. The arrows indicate the atomic displacement vectors, encompassing both direction and magnitude, quantified as the deviation of A (B) site atoms relative to the center of their four nearest B (A) site neighbors. In Fig. 2g, h, the polarization angle along the [001]c direction broadens from a range of 0°–125° (x = 0.10) to 0°–360° (x = 0.15), suggesting the local structural distortions caused by differences in ionic radii and ferroelectric activity between various A-site and B-site cations, hindering the formation of long-range ordered polarization alignment. At the same time, the internal dipole shifts to a smaller scale, and the corresponding average polarization amplitude decreases from 13.72 pm (Fig. 2ix = 0.10) to 5.72 pm (Fig. 2jx = 0.15) (x = 0.12 of 8.9 pm and x = 0.20 of 5.23 pm, Fig. S5). Magnified HAADF images show that these clusters of atoms consisting of co-oriented dipoles are not directly connected, but rather are connected together by “bridges”, which behave as weakly displaced phases or transitions between two phases states, as shown by the circled vector arrows in Fig. 2l and the dashed lines in Fig. 2m. This implies that the long-range ferroelectric order of BiFeO3 is effectively disrupted, and the interspersed C phase within the matrix can hinder polarization rotation and reduce internal stress under an applied electric field. Therefore, the design of the polarization models in the theoretical simulation are justified by the experimental observations.

The statistical Eb of the BF-BHT-xSAN ceramics using the two-parameter Weibull distribution fitting by eight samples for each composition are illustrated in Fig. 3a. We found that the Eb increases steadily with increasing content of SAN from 530 kV cm−1 for x = 0.10 to 860 kV cm−1 for x = 0.20. Meanwhile, the Weibull modulus β of all the samples exceeds 18.4, indicating an excellent reliability and uniformity of the ceramics. The substantial improvement of Eb can be ascribed to reduced grain size and insulated performance. The grain size becomes smaller as the SAN increases from 5.8 μm (x = 0.1) to 3.1 μm (x = 0.2) (Fig. S6). According to the relationship between the grain size and Eb (Eb ∝ (Ga)−a)21,22,23, the composition of x = 0.2 attain the highest Eb as expected. The reduced grain size, coupled with the presence of high-density grain boundaries from 5.86% (x = 0.10) to 14.63% (x = 0.20) (Table S3), establishes substantial barriers that inhibit the formation of breakdown paths.

Fig. 3: The Weibull distribution, Ea, leakage current density and breakdown simulation of BF-BHT-xSAN ceramics.
figure 3

a Linear fitting of Weibull distribution, β and Eb for the BF-BHT-xSAN ceramics. b Arrhenius fitting of Inf (InHz)/Tm of x = 0.10, x = 0.12, x = 0.15, x = 0.20 ceramics according to the Vogel-Fulcher model. c The leakage current density of BF-BHT-xSAN ceramics. d The nominal electric field of x = 0.10, 0.12, 0.15, and 0.20 ceramics. e Breakdown paths and (f) local electric field distribution of BF-BHT-xSAN ceramics under phase field simulation.

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In order to get a deeper understanding of the insulating mechanism at high Eb, we introduce the Vogel-Fulcher (V-F) model based on a temperature-dependent dielectric as follows24:

$$f={f}_{0}\exp \left(\frac{-{E}_{a}}{{k}_{B}({T}_{m}-{T}_{f})}\right)\,$$
(1)

where f0, Ea, kB, Tm, and Tf represent the Debye frequency, activation energy, Boltzmann constant, maximum dielectric constant temperature (Tm comes from Fig. S7), and freezing temperature (Tf)25,26. The experimental results, R2 > 0.99, show the reliability of the nonlinear fitting of the V-F model in Fig. 3b and Table S2. The activation energy Ea shows an increasing trend with higher x, indicating a suppression of dipole coupling under multiphase coexistence. Moreover, during the accumulation process, the increase in Ea effectively reduces the migration rate of oxygen vacancies, thereby improving the Eb. Additionally, the ultra-small polar nanoregions (size about 1–3 nm) and the reduction in leakage current, for example, 2.4 × 10−4 A cm−2 for x = 0.10 and 1.2 × 10−4 A cm−2 for x = 0.20 (Fig. 3c), will be able to make the BF-BHT-0.15SAN ceramic capacitors generate the minimized heat loss under high electric fields from polarization rotation and carrier migration27.

The breakdown paths and electric field distributions are simulated using 2D finite element simulations, as shown in Fig. 3e and Table S3. Under the same conditions, the critical breakdown time increases significantly from 22.81 ns for x = 0.10 to 27.79 ns for x = 0.20, suggesting that the ceramic is capable of withstanding a higher applied electric field. Grain boundaries (GB) deplete free charge carriers in adjacent grains, serving as critical barriers to electric tree propagation and improving breakdown strength20,23. When the breakdown paths cross the whole of the samples, the ceramics of x = 0.2 with the smallest grain sizes have more high-density grain boundaries, and the high-energy region of the electric field around the interface (the red portion in the GB) is obviously suppressed (Fig. 3f)28,29,30, leading to a higher breakdown strength. Finally, the nominal electric field of the BF-BHT-xSAN ceramics is enhanced as expected with the increase of the components (Fig. 3d), indicative of higher dielectric breakdown resistance as the same above.

The bipolar P-E loops and corresponding current-electric field (I-E) curves of BF-BHT-xSAN ceramics are measured at 360 kV cm−1 (Fig. S8). With the increase of SAN content, the P-E curves of BF-BHT-xSAN ceramics becomes significantly slimmer, and no obvious current peaks can be detected in the compositions with high x content. The transformation from a bimodal to a rectangular state signifies the transition of ceramics from strong ferroelectricity to pronounced relaxor behavior, aligning well with the trends of PFM results (Fig. S9). To elucidate the mechanisms behind the exceptional energy density and efficiency of the “strong relaxor P-E loop,” we performed a detailed analysis by calculating the curvature based on P-E loops (Fig. 4a)31. The curvature (K) for each component is computed using the following equation32:

$$K={\rm{d}}\theta /{\rm{ds}}={\rm{y}}({\rm{x}})^{\prime\prime} /{\{1+{[y^{\prime} (x)]}^{2}\}}^{3/2}$$
(2)

the K value gradually decrease (from 0.037 at x = 0.10 to 0.022 at x = 0.20, Fig. 4b) along with a notable straightening of the loop, which is closely associated with the rapid polarization response induced by the formation of nano-domains featuring diverse dipole orientations within the ceramic, effectively mitigating the Pr. Consequently, as the SAN content increases from 0.10 to 0.20 under the critical electric field, Pr exhibits a monotonic variation from 8.83 μC cm−2 to 2.275 μC cm−2. According to P-E loops, the Wrec and η for each component are calculated, as shown in Fig. 4c. Notably, the Wrec reaches the maximum value of 12.2 J cm−3 and efficiency of up to 90.1% at x = 0.15, which are superior to other reported BF-based ceramics10,33,34,35,36,37. Meanwhile, under the condition of η > 90%, only a few systems attain the performance frontier of Wrec ~ 10 J cm−3, in contrast, the Wrec of BF-BHT-xSAN surpasses the current upper limits for lead-free bulk ceramics, as illustrated in Fig. 4d. To intuitively compare the trade-off between Wrec and η across systems, the WF is defined as WF = Wrec /(1-η) to represent the comprehensive energy storage characteristics of the samples38,39. Benefiting from high Wrec and η, the WF reached an unattainable value of 123, which surpasses the maximum values reported for other ceramics (Figs. 4e and S10), validating the feasibility of this novel strategy.

Fig. 4: Excellent energy storage performance of BF-BHT-xSAN ceramics.
figure 4

a P-E loops of BF-BHT-xSAN ceramics up to breakdown. b The maximum curvatures extracted from the fitting line. c The Wrec and η parameters of the BF-BHT-xSAN ceramics. d Comparisons of Wrec (η ≥ 90%) between x = 0.15 and x = 0.20 ceramics in this work and other reported bulk ceramics. e Comparisons of WF between x = 0.15 and x = 0.20 ceramics in this work and other reported BF-based ceramics. f Wrec and η dependent temperature for BF-BHT-0.15SAN ceramics at 450 kV cm−1. g Comparison of Tmax–Tmin/ΔWrec(%) (Tmax ≥ 100 °C) between BF-BHT-0.15SAN and other reported ceramics. h Curves of discharge energy density as a function of temperature.

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Outstanding temperature, frequency, and cyclic stability are essential for the stable operation of energy storage devices in harsh environments. Temperature-variation P-E curves for BF-BHT-0.15SAN ceramic at 450 kV cm−1 between 25 and 170 °C are shown in Figs. 4f and S11a. Pm remains almost the same (<7%), implying the transition from PNR to long-distance ordered structures is unaffected by temperature within the tested range40. Attributed to the activation of conductive mechanisms under high electric fields, the efficiency decreases with increasing temperature, especially in the BF-BT system, the high electric fields above 100 °C always promote the generation of oxygen vacancies41, which is usually accompanied by a decrease in the Wrec. However, the variation in Wrec of BF-BHT-0.15SAN ceramic is less than 7.7%, which is better than most of lead-free ceramics (Fig. S11b, c), particularly vital for devices that struggle with effective heat dissipation. As shown in the Fig. 4g, we compared the ceramics of Tmax–Tmin/ΔWrec(%) (Tmax ≥ 100 °C, a trade-off factor for temperature stability) between BF-BHT-0.15SAN and other reported ceramics42,43,44,45,46,47,48,49. As a result, BF-BHT-0.15SAN ceramic shows a high value of 18.8, and the larger value shows the greater ability to maintain stable energy storage performance even under extreme conditions, ensuring the material’s suitability for high-temperature applications. In addition, the BF-BHT-0.15SAN ceramic also exhibits frequency-independent (Wrec < 2.8%, η < 2%, from 1 to 200 Hz) and anti-fatigue characteristics (Wrec < 1%, η < 2%, over 10,000 st cycles) under 450 kV cm−1 (Fig. S12).

The underdamped and overdamped tests conducted at varying electric fields of BF-BHT-0.15SAN ceramic are illustrated in Fig. S13. As the electric field increases, the power density (PD = EImax/2S) and current density (Imax/S) rise from 10.96 MW cm−3 and 273.89 A cm−2 at 80 kV cm−1 to 182.17 MW cm−3 and 828.23 A cm−2 at 440 kV cm−1, respectively. Compared to other ceramics, BF-BHT-0.15SAN exhibits superior performance in power density (Fig. S13c)49,50,51,52,53,54,55,56,57,58. Subsequently, Fig. S13d–f shows the overdamped tests under different electric fields with a load resistance of 150 Ω. When the electric field reaches 650 kV cm−1, BF-BHT-0.15SAN achieves a high discharged energy density (Wd) of approximately 6.6 J cm−3 with a rapid charging/discharging rate of 107 ns. In addition, the BF-BHT-0.15SAN ceramic also exhibit exceptional temperature stability with Wd < 8.3% (from 6.06 to 6.6 J cm−3) over the temperature variation range of 40–170 °C (Fig. 4h).

The study shows an exceptional breakdown field and negligible hysteresis loss by leveraging the characteristics of high disorder of polarization angle and low magnitude that arise from constructing various polarization orientations at the nanoscale. An ultrahigh recoverable energy storage density of 12.2 J cm−3 and efficiency of 90.1% under 830 kV cm−1 are realized in BF-BHT-0.15SAN ceramic that outperforms previously most of studied lead-free bulk ceramics. In addition, BF-BHT-0.15SAN ceramic shows excellent temperature (from 25 °C to 170 °C), frequency (from 1 Hz to 200 Hz), cycling stability (over 10,000 st cycles) and charging/discharging properties, making a breakthrough in the field of ceramic capacitors. This work demonstrates that introducing a diversity of polarization orientations in bulk ceramics is an effective strategy for designing materials with outstanding energy storage properties.

Methods

Material preparation

Lead-free (0.7-x)BiFeO3-0.3BaHf0.05Ti0.95O3-xSr(Al0.5Nb0.5)O3 (x = 0.1, 0.12, 0.15, and 0.20) ceramics were synthesized by solid-phase sintering. The raw materials, including Bi2O3 (99%), Fe2O3 (99%), BaCO3 (99.8%), HfO2 (99.9%), TiO2 (99%), SrCO3 (99.99%), Al2O3 (99.99%), Nb2O5 (99.5%), and MnO2 (99.95%). All powders were weighed according to their chemical ratios and mixed with ethanol, then ball-milled for 12 h with ZrO2 balls. Following complete dry, the mixed powders were calcined at 780 °C for 3 h with sintering aids MnO2. The final powder was cast into discs with a radius of 1 cm and sintered at 1040 °C–1100 °C for 3 h after removing PVA at 500 °C. The synthesized ceramics were polished down to a thickness of 40–50 μm and then plated with gold electrodes with radius of 1 mm for measurements.

The internal crystallographic structure and phase content of the ceramic samples were analyzed by X-ray powder diffraction and Rietveld refinement in GSAS-II. Surface morphology of the ceramics was characterized using Scanning Electron Microscopy. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images were taken by a double spherical aberration corrected transmission electron microscope (Spectra 300). The response of domain structures to DC voltage was characterized by using piezoresponse force microscopy (PFM, Cypher S). The P-E hysteresis curves were measured at room temperature using a ferroelectric tester (Poly K, USA). Charge/discharge behavior of the samples were evaluated using a high voltage charge/discharge measurement platform at high range from 120 kV cm−1 to 650 kV cm−1, with a resistive load of RL = 150 Ω during charging.

Phase-field simulations

Phase-field simulations were performed for the BiFeO3 based bulk ceramic with different degrees of phase structure, with the spontaneous polarization \(\overrightarrow{P}\) as the order parameter. The polarization evolution can be obtained by solving the time-dependent Ginzburg-Landau equations59,60:

$$\frac{\partial \overrightarrow{P}}{\partial t}=-L\frac{{\rm{\delta }}{\rm{F}}}{{\rm{\delta }}\overrightarrow{P}}$$
(3)

where t is the evolution timestep, L is the kinetic coefficient related to the domain wall mobility. The total free energy F can be calculated by the volume integration of each energy densities, e.g., the elastic, electric, Landau, and gradient energy densities:

$$F=\int ({f}_{elas}+{f}_{elec}+{f}_{Land}+{f}_{grad})dV$$
(4)

The Landau free energy fLand can be calculated by:

$${f}_{Land}= {\alpha }_{ij}{P}_{i}{P}_{j}+{\alpha }_{ijkl}{P}_{i}{P}_{j}{P}_{k}{P}_{l}+{\alpha }_{ijklmn}{P}_{i}{P}_{j}{P}_{k}{P}_{l}{P}_{m}{P}_{n} \\ +{\alpha }_{ijklmnop}{P}_{i}{P}_{j}{P}_{k}{P}_{l}{P}_{m}{P}_{n}{P}_{o}{P}_{p}+\ldots$$
(5)

For the sake of simplicity, the Landau polynomial is expanded to the eighth order, with the Landau parameters set as a linear combination of the different materials61. To model the concentration inhomogeneity, a local fluctuation of the disordered concentration with magnitude of 0.1 (“initial” model) and 0.15 (“optimal” model) are added. The electric energy density can be expressed as:

$${f}_{elec}=-{E}_{i}{P}_{i}-\frac{1}{2}{k}_{ij}{\varepsilon }_{0}{E}_{i}{E}_{j}$$
(6)

Where kij represents the background dielectric constant, which is set as 40 in this study62,63,64. An incremental out-of-plane electric field of 100 kV cm−1 (and 60 kV cm−1) is applied to the bulk crystal with “optimal” model (and “initial” model) until reaching 1000 kV cm−1 (and 600 kV cm−1).

The numerical methods of the phase-field model are reported in previous literature60. A quasi-two-dimensional mesh of 200 × 2 × 200 was used, with each grid representing 1 nm. A three-dimensional periodic boundary condition is applied to model the bulk material. The normalized timestep for this study was set at 0.01.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.