Introduction

Dielectric ceramic capacitors, celebrated for the exceptional power density and swift charge/discharge prowess, are increasingly earning a place as pivotal constituents within the realm of cutting-edge electronic devices and power electronics systems1,2,3,4,5,6,7. The predominant challenge confronting ceramic capacitors is their relatively low recoverable energy storage density (Wrec). According to the theory of dielectric energy storage, optimizing the energy storage performance (ESP) can be achieved by attaining a large polarization difference and high breakdown strength (Eb), with specific methods including grain refinement, control of grain orientation, domain structure regulation, enhancement of electrical uniformity, and defect engineering8,9,10,11,12,13,14. These strategies have indeed facilitated a gradual improvement in Wrec over the past decade. However, with the burgeoning growth of emerging high-tech fields such as electric vehicles, aerospace crafts, and space exploration, low energy loss and broad temperature range stability and reliability have also become critical performance indicators that cannot be overlooked15,16,17,18,19. Therefore, achieving superior comprehensive ESP has become an urgent focal point that demands immediate and heightened attention in the current stage.

High-entropy perovskite oxides represent a novel class of functional ceramics engineered by modulating the configurational entropy (Sconfig) of the system20,21,22,23,24. Materials designed through entropy engineering for dielectric energy storage often exhibit the characteristics of relaxor ferroelectrics (RFE)25,26,27,28,29,30. This is attributed to the increased atomic-level disorder arising from Sconfig, which in turn enhances the disorder in polarization configurations. Consequently, these materials are endowed with a suite of desirable properties, including ultra-low loss, high Eb, and large saturation polarization (Ps).

BaTiO3-BiMO3 (BT-BM) system stands as one of the most classic RFE systems for ESP, where M is composed of no more than three elements31,32,33,34. The function of M is to induce disorder, thereby disrupting the long-range B-O bond coupling. In conjunction with Bi, which facilitates the formation of new A-O bond couplings, the induction of BiMO3 enhances the Eb and concurrently preserves the Ps. Enhancing the disorder within the system to the extent of achieving a high-entropy state, it remains to be explored whether such extreme polarization disorder can lead to enhanced comprehensive ESP.

In this work, we initiate with the 0.8BT-0.2Bi(Mg0.5Zr0.5)O3 system and elevate its Sconfig in a two-step process. Firstly, we substitute (Mg0.5Zr0.5) with (Mg0.2Zn0.2Al0.2Sn0.2Zr0.2), denoted the ceramics as BT-5, introducing a variety of B-site cations with different ionic radii to further disrupt long-range order. Secondly, we introduce NaTaO3 (NT) to form the high-entropy system abbreviated as BT-H. On one hand, the presence of three distinct valences of perovskite structures ( + 1 + 5 type, +2 + 4 type, and +3 + 3 type) and high Sconfig promotes atomic-scale polar inhomogeneity. On the other hand, the introduction of NT, which exhibits an O phase (Pbnm) below 720 K, facilitates the coexistence of local multiphases, thereby smoothing the energy barriers between phases and diminishing hysteresis35. Delightedly, the BT-H ceramics achieve an exceptional Wrec of 8.9 J cm−1 with an ultrahigh efficiency of 97.8%, and demonstrate superior temperature stability across an exceedingly broad temperature range (ΔWrec ~±9 % and Δη ~ ±4.8%, −85 °C–220 °C).

Results and discussion

In pursuit of superior comprehensive energy storage capabilities, outstanding Eb is an essential virtue. Figures 1a, b and S2c shows the SEM micrographs of the BT-2, BT-5, and BT-H ceramics. All compositions exhibit a compact structure, characterized by distinct grain boundaries and an absence of discernible pores. From the grain size distribution diagrams in Fig. S2e–g, it is directly observed that the grain size was diminished from 0.31 to 0.23 μm with the increase of the Sconfig, which is beneficial for realizing high Eb3. Introducing multiple elements into the A and B sublattices lead to a reduction in grain boundary migration rates due to the sluggish diffusion effect, consequently restraining the driving force for grain growth. EDS mappings for all ten elements in BT-H proves that the ceramics still keep macroscopic homogeneity, as shown in Fig. S4. In addition, the ultraviolet-visible (UV-vis) absorption spectra in Fig. S2d show that the fitted Eg increased to 3.38 eV for the BT-H ceramics, indicating a potentially higher intrinsic breakdown strength36. Finite element analysis was used to simulate the distributions of local electric field based on the microstructure (Fig. 1) and dielectric (Fig. S7) characteristics of the BT-2 and BT-H ceramics. The reduction in grain size and its more uniform distribution enhances the breakdown resistance of the BT-H ceramics, suppressing the formation of localized high electric fields13,37. As a result, under the breakdown electric field of the BT-2 sample, the electrical treeing in BT-H ceramics propagates by only approximately 65%.

Fig. 1: The systematic comparison of breakdown strength was conducted between BT-2 and BT-H ceramics.
figure 1

SEM images of polished and thermally etched ceramics and simulation of breakdown path distribution of a BT-2 and b BT-H ceramics. Imaginary part of complex impedance and complex modulus plots at 550 °C and Arrhenius fit of the grain and the grain boundary for c BT-2 and d BT-H ceramics.

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Another perspective for assessing the enhancement of Eb is through the intrinsic resistivity. Complex impedance spectra of the BT-2 and BT-H ceramics from 475 °C to 575 °C are shown in Fig. S7a, b. The larger Z’ and Z” and the shift of the M” and Z” peaks Fig. 1c, d towards lower frequency at the same temperature illustrate the increase of resistivity from BT-2 to BT-H. The bridged gap between the M” and Z” peak (ΔfBT-2 = 1.6 × 104 Hz, ΔfBT-H = 6 × 103 Hz) indicates the elevated electrical homogeneity11. Arrhenius plots of the conductivity for the two samples’ grain and grain boundary (simulated with the equivalent circuit of two RC connected in series) are also shown in Fig. 1c, d. For the BT-H ceramics, the activation energy of grain boundary (EGB = 1.47 eV) and the activation energy of grain (EG = 1.224 eV) are both increased and closer, which is consistent with the Z&M-f plots, benefiting for high Eb38. The dielectric constant (ε′) and loss tangent (tanδ) of BT-2, BT-5, and BT-H ceramics as a function of temperature are shown in Fig. S7c–e. All the samples exhibit a frequency dispersion phenomenon, which is consistent with the XRD patterns shown in Fig. S2a, b and indicate a relaxor behavior. And the slope (γ) of the line fitted by modified Curie–Weiss law is higher than 1.5 also confirms the relaxor behavior of the three ceramics. Notably, the Tm (281.15 K) of the BT-H ceramics has distinctly shifted below room temperature, indicating that it become the superparaelectric (SPE) state at room temperature24. In the SPE regime, the polar nanoregions (PNRs) exhibit significantly reduced dimensions and weakened dipolar interactions compared to conventional RFEs. The unique microstructure leads to substantially lowered energy barriers for polarization reorientation, which facilitates near-zero hysteresis loss and enhanced energy storage efficiency (η). Besides, the high-entropy BT-H ceramics also possess higher temperature stability of dielectric performance, which is confirmed by the temperature coefficient of capacitance (TCC) shown in Fig. S7f. Compared with the original composition BT-2, the dielectric stability of the BT-H ceramics in the low-temperature region was elevated, widening the value of the TCC < 15% to exceed 280 °C (−66 °C–222 °C). The subtle ε′ that spans across both high and low-temperature regions also contributes to achieving excellent temperature stability in ESP.

Figure 2a shows the unipolar polarization hysteresis (P-E) loops for the three components at 50 kV mm−1, which all exhibit slender features and RFE behaviors. Despite this, the transition from BT-2 to BT-H is still characterized by a process from a band of canonical ergodic relaxation state to the SPE state, accompanied by a metamorphosis of the loop shape from a crescent (η ~ 92%) to a needle-like form (η ~ 99%) with negligible polarization hysteresis, which also indicates the increase of free-switching PNRs and the evanescent long-range polar region24. The unipolar P-E loops for the three components at the Eb are shown in Fig. S8a. The initial component BT-2 exhibits a middle Wrec of 6 J/cm3 with a high η of 92.1%. Due to the reduced polarization and no particularly large increase in Eb, the Wrec (5.89 J/cm3) of the BT-5 decreased slightly, while the η increased to 96% as shown in Fig. S8b. Promisingly, the high-entropy BT-H ceramics can maintain the near-zero hysteresis and needle-like shape with the elevated electric field, thus the giant Wrec of 8.9 J/cm3 with ultrahigh η of 97.8% are simultaneously achieved under the highest electric field of 83.3 kV mm−1, as shown in Fig. 2b. Compared to other RFE ceramics, the Pm in the BT-H system increases almost linearly with the rise in electric field, while the remanent polarization (Pr) remains at an exceedingly low value (<0.4 μC cm−3). This confirms that the increased configurational entropy (from 1.14 to 1.64 R) promotes atomic-scale chemical heterogeneity and lattice distortion, resulting in local compositional fluctuations and spatially heterogeneous lattice strain fields, which essentially eliminate long-range ordering at the atomic level and impede the cooperative alignment of macroscopic polar domains, as shown in Fig. 2c. The Weibull distribution plot in Fig. S8c confirms the reliability of the measurement results of Eb for all the shape parameters (β, the slope of the fitted lines) are higher than 10. The great value of η (97.8%) obtained here is one of the highest values among all lead-free ceramics compositions reported ever, thus leading to the excellent comprehensive ESP as shown in Fig. 2d10,11,12,13,14,26,27,28,31,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75.

Fig. 2: The energy storage performance and charge/discharge characteristics of BT-H ceramics at room temperature.
figure 2

a Unipolar P-E hysteresis loops of BT-2, BT-5, and BT-H ceramics under 50 kV/mm at 10 Hz; b Unipolar P-E hysteresis loops; c Discharge energy density, maximum polarization, energy efficiency, and remnant polarization of the BT-H ceramics with different electric fields at 10 Hz. Error bars represent standard deviation. d Comparison of discharge energy density and energy efficiency between BT-H and reported lead-free ceramics. e discharge energy density-time curves and f discharge current-time curves after up 1200 cycles of the BT-H ceramics.

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The charge/discharge performance is another crucial criterion to evaluate the energy storage potential as the applied capacitor. Figure S8d exhibits the regular overdamped oscillation waveforms of the BT-H ceramics using a RLC load circuit with a 300 Ω resistance from 12.5 to 57.5 kV mm−1. Moreover, calculated by the discharge current here, the high discharge energy density (Wd~4.8 J cm−3) and the fast discharge rate (t0.9 ~ 250 ns) was achieved for the BT-H ceramics, as shown in Figs. 2e and S8e. The underdamped discharge waveform plot of the BT-H ceramics was shown in Fig. S8f. Both high power density (PD ~ 550 MW cm−3) and current density (CD ~ 774.55 A cm−2) was obtained under 55 kV mm−1. Besides, the excellent cycling discharging stability further confirmed the reliability during the long-term discharge progress. The Wd and η of BT-H ceramics show minimal variation after 4 million cycles and 1440 h of aging, as illustrated in the Figs. S9 and S10. The Wd and the t0.9 of the BT-H ceramics changed slightly (±3.8% and ±1.9%) after up to 1200 cycles (shown in Figs. 2f and S8h), which may be attributed to the near-zero energy hysteresis observed in the P-E loops. Such subtle variations indicate the excellent long-term reliability and fatigue durability of the BT-H ceramics.

In addition to the ESP at room temperature, the ESP under extreme temperature conditions is also of great importance due to the exploration of the extended realms. Figure 3a shows the P-E loops of BT-H in the temperature range of −85–240 °C under an electric field of 50 kV mm−1. The Pmax is well maintained (ΔPmax ~ 5.5%) from −85 to 170 °C and the Pr remains below 1 μC cm−3 up to 220 °C, which demonstrates that the localized strain fields generated by multi-component ionic size mismatch effectively inhibit domain wall motion, thereby preventing significant growth of the nanodomains across an exceptionally broad temperature range, with the atomic-scale chemical disorder caused by high Sconfig being maintained, effectively suppressing the macroscopically polarization degradation. Moreover, the pronounced compositional disorder in BT-H ceramics leads to distorted oxygen vacancy migration pathways and increased activation energy, which effectively restricts oxygen vacancy mobility and their pinning effect on domain walls. As a result, the BT-H ceramics maintain slender polarization-electric field hysteresis loops even at elevated temperatures, thereby preserving high energy storage efficiency. Remarkably, the value of Wrec varies only ±9% and the η varies ±4.8% in the extra-wide temperature range (−85 °C–220 °C), as shown in Fig. 3b. The in-situ temperature-variable Raman spectra of the BT-H ceramics are provided in Fig. 3c. As the temperature increases, the Raman peak around 300 cm−1 for B-site mode remains diffuse and only slightly shifting towards lower wavenumbers, while the peak around 500 cm−1 for BO6 octahedrons persists with a subtle broadening. Considering that the polarization origin of the BT-H ceramic system stems from the B-site Ti-O bonds, the results suggest that the highly disordered atomic-scale structure and polarization configurations exhibit strong thermal stability, as evidenced by their broad energy barrier distribution for cooperative polarization switching. This enables retention of significant polarization contribution even at elevated temperatures, resulting in relatively stable polar structures17,70. The full-spectrum Raman spectrum of the BT-H from 100 to 1000 cm−1 is presented in the Fig. S14a, b. Figure 3d presents a comparative analysis of energy storage temperature stability for lead-free ceramics reported recently across two dimensions. It can be clearly seen that the both small variation (<10%) and wide temperature range (ΔT > 300 °C) for Wrec are only simultaneously obtained in this work, indicating promising application for extreme-temperature and high-precision devices10,12,13,14,19,26,27,28,29,31,36,37,38,39,40,42,44,45,46,47,48,51,52,53,55,56,57,58,60,61,62,63,64,65,66,67,68,69,70,72,73,74,75,76,77,78,79,80,81,82,83.

Fig. 3: The energy storage performance of BT-H ceramics from −85 °C to 240 °C.
figure 3

a Unipolar P-E hysteresis loops and b Storable energy density, discharge energy density and energy efficiency of the BT-H ceramics from −85 °C to 240 °C under the 50 kV/mm at 10 Hz, error bars represent standard deviation; c Raman spectra of BT-H ceramics magnified curves at ~300 and 510 cm−1 from −70 °C to 230 °C; d Comparison of the variation and temperature range between this work and others; e Unipolar P-E hysteresis loops of the BT-H ceramics at 200 °C, error bars represent standard deviation.

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Figure 4a displays the bright-field, high-resolution, selected-area electron diffraction (SAED) and inverse-fast Fourier transform (IFFT) TEM image along [110]c for the BT-H ceramics. The SAED pattern of BT-H sample shows a near-cubic phase (Pm-3m) symmetry and no other diffraction spots are present. In the high-resolution image, no discernible domain morphology but weak contrast can be found, illustrating that the disordered distribution of multi-component cations leads to the predominance of short-range disordered polar structures within the crystalline grains. Furthermore, in the noise-reduced inverse fast Fourier transform (IFFT) TEM images, the stochastic distribution of multi-cation species leads to pronounced intensity fluctuations in diffraction spots, while concurrently yielding enhanced clarity in the morphological visualization of PNRs. To gain a deeper understanding of the impact of high Sconfig at the atomic resolution on the system and the origin of polarization configurations in the BT-H ceramic system, a high-angle annular dark-field (HAADF) STEM was conducted. The original HAADF-STEM images captured along the [110]c zone axis is presented in Fig. S15b. Then, the off-centering displacement vector mapping and the polarization angle distribution mapping for [110]c direction were obtained using 2D Gaussian peak fitting, as shown in Fig. 4b, c. As evident from the figure, the oxygen octahedral tilt angles in BT-H ceramics exhibit a spatially disordered distribution, which effectively disrupts the cooperative alignment of long-range polarization. Stipulating that the horizontal right direction is defined as 0°, the ±180° vectors represent the O (Orthogonal) phase, the ±45° and ±135° vectors represent the R (Rhombohedral) phase, the ±90° vectors represent the T (Teragonal) phase, and vectors with almost no displacement represent the C (Cubic) phase84. Thus, the extreme atomic-scale disorder and the coexistence of R-O-T-C multiphases were discerned through the disarray of vector arrows. The multiphase PNRs regions, observable in Fig. 4c, exhibit a staggered and discontinuous distribution, which suggests weak coupling between dipoles and benefits for achieving near-zero hysteresis. Excluding the C phase, the specific statistical results of the individual PNRs proportions are shown in Fig. 4f, with the T phase being predominant at 61.7%, and the R and O phases at 30.9% and 7.4%, respectively. If vectors with displacements less than 0.02 Å represent the C phase, then the C phase accounts for approximately 17%, while the relative proportions of the T, R, and O phases remain unchanged. The relative atomic intensity of A-site sublattice and B-site sublattice are shown in Fig. 4d, e and a region of 7 × 10 atoms have been extracted and magnified, respectively. Upon examining the entire image, there is no periodic arrangement or cluster aggregation of cations, indicating that the 10 types of cations are uniformly distributed on a large scale, which is further corroborated by the atomic-resolution energy-dispersive X-ray spectroscopy (EDS) maps along with [100]c zone axis shown in Fig. S16. However, upon close examination of a specific local atomic area, it is observed that there are persistent and pronounced intensity fluctuations in the cations of both the A-site and B-site sublattices, with the fluctuating regions not overlapping. This phenomenon visually illustrates the atomic-scale polarization inhomogeneity caused by high Sconfig, leading to a majority of dipoles being independently oriented without hysteresis85. Similar domain morphologies, polarization vector distributions, and sublattice atomic intensity distributions can be observed from the HAADF-STEM images of the [100]c zone axis, as shown in Fig. S16. These results strongly confirm the significant role of entropy engineering in regulating polarization configurations and achieving comprehensive ESP.

Fig. 4: Crystal structure analysis and local atom structure analysis of the of BT-H ceramics.
figure 4

a Morphology, high-resolution, inverse fast Fourier transform STEM images and SAED pattern; b, c HAADF-STEM polarization vector images; d, e Relative integrated intensity of A-site sublattices and B-site sublattices with magnified extraction images; f statistic of calculated polarization angle distribution of the BT-H ceramics along [110]c.

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Finally, phase-field simulations were employed to theoretically verify the exceptional ESP of the designed BT-H high-entropy ceramics. Both physical and calculation parameters are grounded in the BT-H high-entropy ceramics. The calculated P-E hysteresis loops and the corresponding evolution of domain structures at various temperatures are depicted in Fig. 5a–e. The domain structure diagrams, each with a side length of 100 nm, utilize four distinct colors to represent four different domain configurations. For each temperature, domain structures under three distinct states are presented: (I) at zero electric field; (II) under the maximum applied electric field, and (III) after the removal of the electric field. At room temperature, it is evident that the long-range ferroelectric order within the constructed high-entropy ceramics is entirely disrupted, resulting in a disordered domain structure with domain sizes predominantly below 5 nm. Notably, even under the applied electric field, the domain orientations remain non-uniform, indicating that the random field is sufficiently strong to prevent the alignment of all polar regions in a unified direction despite the external electric field. The extremely small domain sizes in PNRs facilitate the reduction of the energy barrier for domain switching and thereby achieving ultra-low hysteresis, which is reflected in the nearly zero Pr in the P-E hysteresis loop, as depicted in Fig. 3c. Furthermore, it is notably remarkable that, under the condition of specified physical parameters at varying temperatures and an invariant random field, the domain evolution of BT-H high-entropy ceramics demonstrates strong temperature independence within the range of −65–200 °C. For BT-H ceramics, the dynamic and reversible nature of local PNRs (with response times on the picosecond scale) makes it difficult for oxygen vacancies to achieve stable pinning. Additionally, the nanoscale domain size (<5 nm) significantly reduces the interaction probability between oxygen vacancies and domain walls. As a result, the domain structure within BT-H ceramic grains remains well-preserved at elevated temperatures, preventing significant hysteresis loop broadening. This indicates that temperature exerts a negligible impact on local energy barriers, enabling the preservation of the room-temperature polarization structure over a wide temperature range. Consequently, the shape of the corresponding P-E hysteresis loop remains virtually unaltered, thereby achieving outstanding temperature stability in ESP.

Fig. 5: The phase-field simulation results of BT-H.
figure 5

Calculated P-E hysteresis loops and calculated domain structure evolutions of the BT-H ceramic at various temperatures using phase-field simulation method: a −65 °C, b −5 °C, c 25 °C, d 100 °C, e 200 °C.

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In summary, through entropy engineering optimization, we demonstrate enhanced ESP in the BT-Bi(Mg0.5Zr0.5)O3 RFEs. Microstructurally, the incorporation of ten distinct ionic species has enhanced the configurational entropy of the system, thereby fostered local chemical heterogeneity and introduced PNRs with diverse symmetries. The structural modulation effectively suppressed grain boundary migration and increase the bulk resistivity. Macroscopically, the entropy-engineered BT-H ceramic exhibits near-zero hysteresis, retarded polarization saturation, and exceptional breakdown strength, which leads to an outstanding Wrec of 8.9 J cm−3 under 83.3 kV mm−1, with an ultra-high η of 97.8%. Notably, owing to the preservation of polarization configurations over an ultra-wide temperature range, the BT-H ceramic demonstrated unparalleled temperature stability across an extended operational range (ΔWrec ~±9%, η ~ ±4.8%, −85 °C–220 °C). This study exemplifies the cocktail effect of high-entropy materials in the field of dielectric energy storage, demonstrating that the design of dielectrics with superior integrated ESP through entropy engineering is a viable approach.

Methods

Sample Preparation: (Ba0.8Bi0.2)(Ti0.8Mg0.1Zr0.1)O3 (abbreviated as BT-2), (Ba0.8Bi0.2)(Ti0.8Mg0.04Zn0.04Al0.04Sn0.04Zr0.04)O3 (abbreviated as BT-5) and (Ba0.76Na0.04Bi0.2)(Ti0.76Ta0.04Mg0.04Zn0.04Al0.04Sn0.04Zr0.04)O3 (abbreviated as BT-H) ceramics were fabricated via a conventional solid-state reaction method with high purity raw chemicals of Ba2CO3 (≥99.95%), Na2CO3 (≥99.7%), Bi2O3 (≥99.9%), TiO2 (≥99%), MgO (≥99.99%), ZnO (≥99.99%), ZrO2 (≥99.7%), Al2O3 (≥99.99%), SnO2 (≥99.5%), Ta2O5 (≥99.99%). All the chemicals were dried at 200 °C, weighted according to the stoichiometric amounts and then milled for 18 h in the zirconia jars with ethanol and zirconia balls. The well-mixed powders were dried and calcined at 950 °C for 5 h and then ball milled again for 12 h. Subsequently, the dried powders (40.5 wt%) were mixed with butanone (50.7 wt%), glycerol trioleate (0.7 wt%), dibutyl phthalate (0.7 wt%), polyvinyl butyral (6.7 wt%), and polyethylene glycol (0.7 wt%) and ball milled for 12 h to obtain a uniform ceramic slurry. The tapes were fabricated by employing a tape-casting method. After drying and cutting, the square tapes were stacked and hot-pressed at 65 °C. Subsequently, four layers of the green tapes were stacked and laminated using a parallel-plate hot press under a pressure of 20 MPa. After sintering, the resulting ceramic samples achieved a final thickness of about 50 μm. Finally, the green ceramic samples were maintained at 600 °C for 8 h to remove the organics and sintered at 1120–1200 °C for 1.5–2 h in air.

Structure characterization: all samples were examined by XRD (Cu-Kα source) on a Bruker D8 Advance diffractometer. XRD patterns were collected over a range of 2θ = 10°–90°, with 0.02° per scanning step. The grain morphologies and element distribution of the polished and thermal etched samples were detected by a field-emission scanning electron microscope (FE-SEM, Quanta 200FEG, FEI Company, USA). The Raman spectra were collected using a Raman scattering spectrometer (LabRAM HR Evolution, Horiba, France) with a heating stage (Linkam, THM600, UK) at 532 nm laser excitation. Optical absorption properties of the grinded ceramic particle powder were measured from 200 to 800 nm using a UV-vis spectrophotometer (Shimadzu UV-3600i Plus, Japan). The ceramic specimens were processed into TEM thin foils through standard procedures involving sectioning, mechanical thinning, precision polishing, and Ar-ion milling, followed by deposition of a conductive carbon layer. Microstructural characterization was performed using a JEM-F200 transmission electron microscope to obtain dark-field images and SAED patterns. Atomic-resolution high-angle annular dark-field (HAADF) imaging was conducted on a Cs-corrected Spectra300 microscope operated with convergence and collection semi-angles of 30 mrad/50–200 mrad, respectively. The polarization vectors were quantified by measuring the displacement of B-site cations relative to the centroid of their surrounding A-site cation tetrahedra. The atomic column positions at picometer-precision fitting were performed using MATLAB software.

Electric property measurements: The P-E loops measured at room temperature were derived from the ferroelectric measurement system (Radiant Technologies Inc., Albuquerque, USA). For the test of temperature-dependent P-E loops, another ferroelectric measurement system (TF analyzer 2000, Aachen, Germany) was used with a low-temperature probing stage (Janis, USA) and a high-temperature probing stage (TFA 370-7, aixACCT, Germany). A dielectric charge-discharge measurement system (CFD-003, Tongguo Technology, China) was used to measure the energy release properties. The storage energy was discharged under a load resistance RL of 300 Ω. The electrodes were deposited by magnetron sputtering at a current of 10 mA for 5 min and For energy-related measurements, the ceramic samples were fabricated into electrodes with dimensions of 1.0–2.0 mm in diameter and 0.05–0.10 mm in thickness. For dielectric and impedance spectroscopy, the ceramics were polished to ~0.7–1 mm in thickness and ~8 mm in diameter to ensure proper electrical contact and measurement accuracy. Dielectric and impedance measurements are performed using a precision LCR meter (E4980A, Agilent Technologies, USA) combined with a temperature control system (DMS-2000, Balab Technologies, China). The heating rate was set at 3 °C min–1, and measurements were performed within a test frequency range of 1 kHz to 1 MHz.

Finite element simulation: we apply a mechanically coupled phase-field model with reference to86,87,88. The ferroelectric system’s total free energy density for the single crystal in the simulation model is expressed as

$$H={H}_{{\mbox{bulk}}}+\frac{1}{2}{c}_{{ijkl}}{{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{e}}}{{{\rm{\varepsilon }}}}_{{kl}}^{{\mbox{e}}}-\frac{1}{2}{K}_{{ij}}{E}_{i}{E}_{j}-{P}_{i}{E}_{i}+\frac{1}{2}{g}_{{ijkl}}{P}_{i,j}{P}_{k,l}$$
(1)

The first term \({H}_{{\mbox{bulk}}}\) of Eq. (1) represents the bulk free energy density,

$${H}_{{\mbox{bulk}}}= {a}_{{ij}}{P}_{i}{P}_{j}+{a}_{{ijkl}}{P}_{i}{P}_{j}{P}_{k}{P}_{l}+{a}_{{ijklmn}}{P}_{i}{P}_{j}{P}_{k}{P}_{l}{P}_{m}{P}_{n} \\ +{a}_{{ijklmnpq}}{P}_{i}{P}_{j}{P}_{k}{P}_{l}{P}_{m}{P}_{n}{P}_{p}{P}_{q}$$
(2)

where \({a}_{{ij}}\), \({a}_{{ijkl}}\), \({a}_{{ijklmn}}\), and \({a}_{{ijklmnpq}}\) are the Landau-Devonshire potential coefficients. To be specific82,

$${H}_{{\mbox{bulk}}}= \,{a}_{1}({P}_{1}^{2}+{P}_{2}^{2}+{P}_{3}^{2})+{a}_{11}\left({P}_{1}^{4}+{P}_{2}^{4}+{P}_{3}^{4}\right) \\ +{a}_{12}\left({P}_{1}^{2}{P}_{2}^{2}+{P}_{2}^{2}{P}_{3}^{2}+{P}_{3}^{2}{P}_{1}^{2}\right)+{a}_{111}\left({P}_{1}^{6}+{P}_{2}^{6}+{P}_{3}^{6}\right) \\ +{a}_{112}\left[{P}_{1}^{2}\left({P}_{2}^{4}+{P}_{3}^{4}\right)+{P}_{2}^{2}\left({P}_{1}^{4}+{P}_{3}^{4}\right)+{P}_{3}^{2}\left({P}_{1}^{4}+{P}_{2}^{4}\right)\right] \\ +{a}_{123}{P}_{1}^{2}{P}_{2}^{2}{P}_{3}^{2}+{a}_{1111}\left({P}_{1}^{8}+{P}_{2}^{8}+{P}_{3}^{8}\right) \\ +{a}_{1112}\,\left[{P}_{1}^{6}\left({P}_{2}^{2}+{P}_{3}^{2}\right)+{P}_{2}^{6}\left({P}_{1}^{2}+{P}_{3}^{2}\right)+{P}_{3}^{6}\left({P}_{1}^{2}+{P}_{2}^{2}\right)\right] \\ +{a}_{1122}\left({P}_{1}^{4}{P}_{2}^{4}+{P}_{2}^{4}{P}_{3}^{4}+{P}_{3}^{4}{P}_{1}^{4}\right)+{a}_{1123}\left({P}_{1}^{4}{P}_{2}^{2}{P}_{3}^{2}+{P}_{2}^{4}{P}_{3}^{2}{P}_{1}^{2}+{P}_{3}^{4}{P}_{1}^{2}{P}_{2}^{2}\right)$$
(3)

Among these coefficients, all are assumed temperature-independent except for \({a}_{1}\), which follows a linear temperature dependence based on the Curie-Weiss law. The values of these coefficients, summarized in Table 1, were determined through fitting to different ferroelectric phase transitions. Specifically, \({a}_{11}\), \({a}_{111}\), and \({a}_{1111}\) were derived considering the transition from the paraelectric cubic phase to the ferroelectric tetragonal phase, along with the spontaneous polarization and dielectric constant of the tetragonal phase. The coefficients \({a}_{12}\), \({a}_{112}\), \({a}_{1112}\), and \({a}_{1122}\) were determined based on the properties of the orthorhombic phase, while \({a}_{123}\) and \({a}_{1123}\) were extracted from the characteristics of the rhombohedral phase.

Table 1 Coefficients of Landau–Devonshire potential where T is temperature in °C
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The second term in Eq. (1) describes the elastic energy density, where \({c}_{{ijkl}}\) is the elastic stiffness tensor and \({{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{e}}}\) represents the elastic strain. The elastic strain \({{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{e}}}\) can be further decomposed into the total strain and the electrostrictive eigenstrain as:

$${{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{e}}}={{{\rm{\varepsilon }}}}_{{ij}}-{{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{P}}},$$
(4)

where the eigenstrain induced by polarization follows:

$${{{\rm{\varepsilon }}}}_{{ij}}^{{\mbox{P}}}={Q}_{{ijkl}}{P}_{k}{P}_{l},$$
(5)

with \({Q}_{{ijkl}}\) being the electrostrictive coefficient.

The third and fourth terms of Eq. (1) account for the electrostatic energy density, with the dielectric tensor

$${K}_{{ij}}={{{\rm{\varepsilon }}}}_{0}{{\rm{\kappa }}}{{{\rm{\delta }}}}_{{ij}},$$
(6)

where \({{{\rm{\varepsilon }}}}_{0}\) is the permittivity of free space, and \({{\rm{\kappa }}}\) denotes the relative permittivity of the material. The final term captures the energy contribution from polarization gradients, with \({g}_{{ijkl}}\) representing the gradient energy coefficient.

The total electric field in the RFEs model consists of three distinct components, expressed as83

$${E}_{i}={E}_{i}^{{{\rm{d}}}}+{E}_{i}^{{{\rm{ext}}}}+{E}_{i}^{{{\rm{rand}}}},$$
(7)

where \({E}_{i}^{{\mbox{d}}}\) represents the depolarization field, \({E}_{i}^{{\mbox{ext}}}\) corresponds to the external electric field, and \({E}_{i}^{{\mbox{rand}}}\) denotes the random field arising from chemical inhomogeneity follows a Gaussian distribution \(N\),

$${E}_{i}^{{{\rm{rand}}}} \sim N(\mu,\Delta ).$$
(8)

where its expectation \(\mu\) is set to be zero in the simulation, and the standard deviation is denoted as \(\Delta\).

The mechanical equilibrium equation for the Cauchy stress tensor \({{{\rm{\sigma }}}}_{{ij}}\) in Cartesian coordinates is given by

$${{{\rm{\sigma }}}}_{{ij},j}=0.$$
(9)

And the corresponding boundary conditions can be either displacement-controlled or traction-controlled,

$${u}_{i}={\bar{u}}_{i},\, {{{\rm{\sigma }}}}_{{ij}}{n}_{j}={\bar{t}}_{i},$$
(10)

where \({n}_{j}\) is the unit normal vector on the boundary, \({\bar{u}}_{i}\) represents a prescribed displacement, and \({\bar{t}}_{i}\) denotes the applied traction force.

By differentiating the total free energy \(H\) with respect to strain, the constitutive law is obtained as:

$${{{\rm{\sigma }}}}_{{ij}}=\frac{\partial H}{\partial {{{\rm{\varepsilon }}}}_{{ij}}}={c}_{{ijkl}}\left({{{\rm{\varepsilon }}}}_{{kl}}-{{{\rm{\varepsilon }}}}_{{kl}}^{{{\rm{P}}}}\right).$$
(11)

In electrostatics, the charge conservation law dictates that the electric displacement field \({D}_{i}\) satisfies

$${D}_{i,i}=q,$$
(12)

where \(q\) is the volume charge density. The associated boundary conditions can be specified through electric potential or surface charge:

$$\varphi=\bar{\varphi },\, {D}_{i}{n}_{i}=-\bar{{{\rm{\omega }}}},$$
(13)

where \(\bar{\varphi }\) and \(\bar{{{\rm{\omega }}}}\) correspond to prescribed electric potential and surface charge density, respectively.

The electric displacement vector gives by differentiating the total free energy \(H\) with respect to electric field,

$${D}_{i}=-\frac{\partial H}{\partial {E}_{i}}={K}_{{ij}}{E}_{j}+{P}_{i}.$$
(14)

The evolution of polarization (order parameter) follows a Ginzburg-Landau type equation given by:

$${\dot{P}}_{i}= -{{\rm{\alpha }}}\frac{\delta H}{\delta {P}_{i}}=-{{\rm{\alpha }}}\left[\frac{\partial H}{\partial {P}_{i}}-{(\frac{\partial H}{\partial {P}_{i,j}})}_{,j}\right] \\ =-{{\rm{\alpha }}}\left[\frac{\partial {H}_{{\mbox{bulk}}}}{\partial {P}_{i}}-\frac{\partial {\varepsilon }_{{jk}}^{{{\rm{P}}}}}{\partial {P}_{i}}{{{\rm{\sigma }}}}_{{jk}}-{E}_{i}-{g}_{{ijkl}}{P}_{k,{lj}}\right],$$
(15)

where the over dot denotes a time derivative, and \({{\rm{\alpha }}}\) represents the mobility parameter governing the polarization dynamics.