Introduction

With the continuous booming of high-power electronics in recent years, lead-free energy-storage dielectric has garnered increasing research attention towards green power devices. Eco-friendly dielectric ceramics with fast discharge speed and performance stability, in comparison to supercapacitors, flywheels and batteries, exhibit huge advantages in energy storage applications1,2,3. Driven by the demands for high power density and extended cycling lifespan in advanced electronic systems, lead-free multilayer ceramic capacitors (MLCCs) have gained much research interest4,5,6. In general, the total energy density (W), the recoverable energy density (Wrec) and energy efficiency (\(\eta\)) of a dielectric can be calculated according to the equations: \(W={\int }_{o}^{{P}_{\max }}{EdP},\,{W}_{{rec}}={\int }_{{P}_{r}}^{{P}_{\max }}{EdP},\,\eta=\frac{{W}_{{rec}}}{W}\times 100\%\), where E, Pmax and Pr are the applied electric field, the maximum and remanent polarization, respectively7,8. Therefore, simultaneously achieving large Pmax, low hysteresis loss and high breakdown electric field (Eb) is highly desired to realize high energy storage performance (ESP). In the past decade, various lead-free matrix materials have been explored to suit the needs of MLCCs, including BaTiO3 (BT)4, NaNbO3 (NN)5, AgNbO3 (AN)9, BiFeO3 (BF)10, Bi0.5Na0.5TiO3 (BNT)11. Manufacturing textured ceramics and weakening polarization-strain coupling are regarded as useful approaches to enhance the performance of MLCCs6,12. For instance, textured ceramic was a proposed solution to enhance the breakdown strength of MLCCs by lowering the strains, but the low energy efficiency (η ~ 80%) did not own practical worth. Besides, Wrec of 14.6 J cm-3 and η of 93% were achieved in MLCCs with ultraweak polarization-strain coupling. One of the main issues for energy storage performance in MLCCs is finding the best balance between energy density (Wrec over 20 J cm-3) and efficiency (\(\eta\) over 90%), as optimizing one parameter is typically achieved at the expense of sacrificing the other. This issue significantly hinders MLCCs development towards real applications for energy storage, restricting their potential for miniaturization and integration in next-generation device system.

The energy storage performance of MLCCs is highly dependent on dielectric types with distinct polarization behavior. The polarization response in classical ferroelectrics is closely associated with the long-range-order polarization configuration, which typically results in large hysteresis loss and unsatisfactory ESP13,14,15. Hence, relaxor ferroelectrics have been developed to realize high ESP via domain engineering. In relaxors, long-range ferroelectric order is broken and replaced by an amount of scattered polar nanoregions (PNRs), which suppress long-range dipole ordering under high electric field, resulting in field-insensitive dielectric constants and delayed polarization saturation16,17,18. Meanwhile, PNRs lower the domain-switching barrier and minimize hysteresis loss from a thermodynamic perspective19,20. Therefore, employing sophisticated strategies to transform ferroelectric domains into highly dynamic PNRs is widely regarded as key to simultaneously achieving high Wrec and η.

High-entropy strategy is recently emerging as a delicate approach for realizing relaxor ferroelectric that affords excellent compositional flexibility for improving energy storage performance owing to its unique effects21,22,23,24. Configurational entropy (Sconfig) is defined as \(-R[\left({\sum }_{i=1}^{N}{y}_{i}{\mathrm{ln}}{y}_{i}\right)+\left({\sum }_{j=1}^{M}{y}_{j}{\mathrm{ln}}{y}_{j}\right)]\), where R, N(M) and yi(j) are the ideal gas constant, atomic species and contents at the equivalent positions, respectively25. In most studies, high-entropy design is achieved by the incorporation of diverse cations into equivalent sites, facilitating the formation of a single perovskite structure within a stabilized lattice framework as Sconfig increases26,27,28. In general, mixing size- and charge-mismatched cations tends to generate heterogeneous nanoscale phases whose formation is difficult to control precisely, so that identifying useful compositions often relies on extensive trial-and-error. Moreover, this approach also tends to promote the excessive formation of weakly polar phases. As a result, superparaelectric or even paraelectric states appear at room temperature, accompanied by a dramatic decrease in dielectric constant (εr)29,30. However, low εr usually corresponds to small polarization saturation and thus relatively low ESP under practical operating fields. It is therefore highly desirable to develop novel high-entropy design strategies that can induce heterogeneous nanoscale phases while maintaining a high εr.

We proposed a high-entropy design strategy that engineers competing ferroic orders to achieve high energy storage performance in BNT-based MLCCs. Guided by phase-field simulation, we proportionally introduced various oxides with different symmetry, including BaTiO3 (tetragonal phase, T)31, CaTiO3 (cubic phase, C), SrTiO3 (C), AgNbO3 (orthorhombic phase, O)32 and KNbO3 (O)33, into a high spontaneous-polarization BNT (rhombohedral phase, R) matrix for tuning Sconfig (Fig. S1) of (Bi0.5Na0.5)(1-x)(Sr0.2Ca0.2Ba0.2K0.2Ag0.2)xTi1-0.4xNb0.4xO3 (abbreviated as BNT-CE, named by their Sconfig value) system. Introducing symmetry-distinct oxide components into the same lattice targetedly gives rise to locally competing polar states. Consequently, it could lead to pronounced structural distortion and heterogeneity (Fig. 1a, b), thereby possibly enabling the formation of a crossover region between relaxor ferroelectric and superparaelectric states. This strategy is expected to concurrently enhance relaxor behavior and maintain large Pmax in lead-free high-entropy MLCCs and therefore serves as an effective approach to simultaneously realize large Wrec and high η.

Fig. 1: Phase-field simulation results for an entropy-driven perovskite structure along [001].
Fig. 1: Phase-field simulation results for an entropy-driven perovskite structure along [001].The alternative text for this image may have been generated using AI.
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Schematic diagram of (a) Low-entropy model and (b) high-entropy model. c Three-dimensional domain structures of the 0.85R and 2.17R models under different electric fields by phase-field simulation. Simulated P-E loops of (d) the 0.85R and (e) 2.17R models. The insets illustrate domain states before and after electric field application. f Simulated evolution of Pr and Eb as functions of Sconfig. g Simulated evolution of Wrec and η as functions of Sconfig.

Results and discussion

Phase-field simulation of competing orders with high-entropy design

Herein, we employed phase-field simulation based on oxide-symmetry considerations to rationalize our compositional design and visualize the competing process of different polar orders17,29,34. To directly visualize the designed competing ferroic orders, we simulated the evolution of domains in both two- and three-dimensional space as a function of Sconfig. Distinct phase states and domain structures emerge upon the incorporation of various oxides with different phases into the matrix as displayed in Fig. 1c and Fig. S2. The distribution of simulated polarization vectors becomes gradually random with the increase of Sconfig, generating diverse polar regions. Based on the result of the competition among ferroic orders of R/T/O/C phases represented by different vectors and colorful regions, the long-range order of the host phase is totally broken and transforms to nanoregions. Therefore, the high-entropy model manifests a heightened level of disorder. As the insets of Fig. 1d, e show, the evolution of the polar structure under an electric field reveals that the high-entropy model is much easier to revert to their initial state and maintains fast domain-switching speed with small hysteresis. In addition, corresponding simulated P-E loops are elucidated with increasing entropy (Fig. S3), revealing a remarkable transition from classic ferroelectric to relaxor ferroelectric and eventually to linear dielectric response. It also demonstrates that minimal Pr and high breakdown strength can be obtained as entropy increases (Fig. 1f). Besides, based on the simulated P-E loops, the Wrec and η are calculated in Fig. 1g as dependence on Sconfig. It shows the η gradually raises with increasing entropy while the highest Wrec is achieved at the Sconfig of 2.17R, beyond which Wrec begins to decline as the polarization response becomes weak and linear (Fig. S3). Meanwhile, the 2.17R model exhibits negligible variation in simulated P-E loops across a practical operating temperature range from −50 to 150 oC, demonstrating excellent thermal stability for application (Fig. S4). Therefore, these results highlight that optimizing Sconfig within an appropriate window is critical for forming competing orders and achieving superior energy storage performance.

Under the guidance of phase-field simulation, we prepared a series of (Bi0.5Na0.5)(1-x)(Sr0.2Ca0.2Ba0.2K0.2Ag0.2)xTi1-0.4xNb0.4xO3 ceramics (BNT-CE, x = 0.02/0.85R, 0.1/1.27R, 0.2/1.65R, 0.3/1.94R, 0.4/2.17R, 0.5/2.34R). BNT-CE ceramics exhibit a pure perovskite structure with diffraction peaks regularly shifting towards low angle as Sconfig increases, indicative of lattice expansion (Fig. S5)35,36. However, for samples with entropy higher than 2.17R, Sr/Ca-rich secondary phase emerges, indicating that the solid solution limitation is reached37. Meanwhile, Rietveld refinements (Fig. S6) indicate that raising Sconfig reduces the R polar fraction, weakening the overall polarity of the BNT-CE ceramics. As revealed by Raman spectra (Fig. S7), the peak ν2 representing the A-site vibration mode exhibits a red shift and reduced intensity with the increase of entropy, indicating enhanced random fields arising from A-site chemical complexity. The B-O (200-400 cm−1) modes gradually broaden while BO6-related modes (450-800 cm−1) become narrower and more intense with the increase of Sconfig, which means short-range chemical disorder and dispersed octahedra distribution38,39,40. In order to evaluate the overall ESP, unipolar P-E loops of BNT-CE ceramics were measured at their characteristic Eb (Fig. 2a). Consistent with the simulation results, the transition from classical ferroelectric to relaxor ferroelectric is observed with increasing Sconfig, characterized by delayed polarization saturation. Pr is abruptly decreased from 44.7 to 1.1 μC cm−2 while Pmax exhibits only a slight reduction from 60.9 to 60.5 μC cm-2. Such negligible Pr and large Pmax in high-entropy ceramic contribute to the improved ESP. Consequently, the 2.17R ceramic realizes high Wrec of 10.51 J cm−3 and high η of 88.2%, which are over 7- and 4- folds higher than those of the low-entropy 0.85R ceramic (Fig. 2b), respectively. In Fig. S8, the statistical Eb of BNT-CE ceramics were calculated by Weibull distribution fitting based on ten samples, with a good reliability supported by all the slopes β higher than 1041. The result shows that the Eb increases steadily from 243 kV cm-1 to 535 kV cm-1 with the increase of entropy. The average grain size of the BNT-CE ceramics decreases markedly from 1.42 μm to 0.63 μm with increasing Sconfig while the relative density of all samples increases only slightly from 96.8% to 97.4% (Fig. S9), owing to enhanced lattice strain energy and reduced driving force for grain coarsening42. Refined and dense microstructure helps to depress leakage current (Fig. S10), which induces enhanced barriers in carrier transport and diminishes the conduction loss at high electric field, thus substantially improving Eb21. To clarify the evolution of Eb in BNT-CE ceramics, we carried out finite-element simulations based on the SEM images (Fig. S9) to visualize the electric-field distribution and the propagation of electrical trees at 400 kV cm−1, as shown in Fig. 2c43. The higher grain-boundary density in 2.17R ceramic provides more blocking and dissipation sites, which slows the propagation of electrical trees and therefore contributes to the decrease of leakage current and the higher breakdown strength.

Fig. 2: Energy storage performance and relaxor behavior of BNT-CE ceramic.
Fig. 2: Energy storage performance and relaxor behavior of BNT-CE ceramic.The alternative text for this image may have been generated using AI.
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a Unipolar P-E loops of BNT-CE ceramics with different Sconfig. b Comparison of Wrec and η with different Sconfig. Evolution of electric field and the growth of breakdown paths at the same electric field of 400 kV cm−1 for (c) the 0.85R and (d) 2.17R cerami. e Temperature-dependent dielectric constant and loss tangent of the 0.85R and 2.17R ceramics under different frequencies. f Simulated domain evolution of the 2.17R ceramic under various temperatures and electric fields. Mapping of unit cell c/a ratio of (g) the 0.85R and (h) 2.17R ceramics along the [001] direction. i Statistical c/a ratio distributions for the 0.85R and 2.17R ceramics.

The temperature-dependent dielectric properties of BNT-CE ceramics were then measured to evaluate their relaxor behavior (Fig. 2e and Fig. S11). A shoulder-like flat peak emerges as Sconfig reaches 1.65R and subsequently disappears for Sconfig higher than 1.65R, accompanied by a shift of the Curie temperature (Tm) towards lower temperature, which facilitates the coexistence of multiple phases and promotes domain switching44,45. Notably, by tuning Sconfig, the 2.17R ceramic is placed in the crossover region between relaxor ferroelectric and superparaelectric states with a high dielectric constant (~ 1800) at room temperature, rather than falling into a superparaelectric state with low εr as in many reported high-entropy systems27,29,46. In this crossover region, highly dynamic polar nanoregions and strong competition among ferroic orders allow a large polarization with a small remanent polarization and a high breakdown field7,47. Meanwhile, 2.17R ceramic exhibits excellent temperature coefficient of εr (within ± 15%) in −30 to 125 oC (Fig. S12), which meets the Y7R capacitor standard for application. Illustrated by the simulated domain evolution results of 2.17R model during a wide temperature span of −100 to 300 oC (Fig. 2f), the fraction of C phase (gray region) gradually increases with rising temperature, while the polarization configuration under maximum electric field keeps unvaried across all temperatures. To further elucidate the variation of relaxor behavior, the diffuseness coefficient γ is fitted by the modified Curie-Weiss equation: \(\frac{1}{\varepsilon }-\frac{1}{{\varepsilon }_{m}}=\frac{{(T-{T}_{m})}^{\gamma }}{C}\)48. The magnitude of γ increases from 1.03 to 1.91 as Sconfig increases (Fig. S13), supporting the enhanced relaxor behavior in 2.17R high-entropy ceramic17. The dynamic domain responses of the 0.85R and 2.17R ceramics were also measured by piezo-response force microscopy (PFM) to monitor the evolution of ferroelectric domains (Fig. S14). With increasing entropy, the switched domains induced by a 20 V electric field fade more rapidly and are more readily reverted to their initial configuration upon field removal, indicating shorter relaxation time and low hysteresis loss in 2.17R high-entropy ceramic46,49. In addition, we calculated the distribution of the unit cell c/a ratio to clarify the structural origin of this crossover and the enhanced breakdown strength through HAADF-STEM analysis (Fig. 2g, h). The average c/a ratio significantly decreases from 1.027 for the 0.85R ceramic to 1.009 for the 2.17R ceramic, as shown in Fig. 2i, which indicates weak long-range ferroelectric anisotropy and polarization-strain coupling in the 2.17R ceramic50. The reduced average c/a helps to diminish local field amplification and thermal loss under high electric field, which is beneficial for improving breakdown strength39. Moreover, the 2.17R ceramic exhibits a dispersed distribution of c/a ratio, in contrast to the more clustered distribution observed in the 0.85R ceramic, with the standard deviation increasing from 0.04 (0.85R) to 0.06 (2.17R) as entropy rises, which indicates stronger spatial heterogeneity of lattice distortion.

Local structural evolution of entropy-driven ceramics

As revealed in phase-field simulations, competing orders generated by chaotic chemical environments and distorted local structure serve as the structural foundation for achieving high energy storage performance. Thus, the evolutions of polarization configuration in both high-entropy and low-entropy ceramics were systematically investigated to elucidate the underlying polarization response mechanism through high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM). The atomic-resolution images of the 0.85R (Fig. 3a) and 2.17R (Fig. 3b) ceramics along [001] are marked by colorful vectors, which represent the polarization angle of the B-site cations relative to the lattice centers of the four nearest neighboring A-site cations8. The 0.85R ceramic exhibits pronounced polarization displacement (Fig. 3c) and large domain sizes, with the R phase as the dominant phase. With increasing Sconfig, the notable reduction of R/O phase, combined with the existing tiny amount of T phase and increased fraction of C phase induce the competition of diverse ferroic orders within local regions. Compared to the low-entropy ceramic, the remarkable distinction in polarization orientation of high-entropy ceramic reflects substantial structural distortion and diverse heterogeneous polarization configuration, wherein strongly polar phases are intricately embedded within a weakly polar cubic matrix. Figure 3c stresses a contrast in the displacement distribution that the average displacement of ~ 9.4 pm in the 2.17R ceramic is much smaller than that of ~ 18.4 pm in the 0.85R ceramic. This trend is consistent with the grain-size evolution (Fig. S9), where smaller grains are expected to confine the polarization correlation length and suppresses domain coalescence51. This reduced polarization displacement promotes complex interactions among randomly distributed ions and competing ferroic orders, giving rise to highly dynamic polar nanoregions with weak domain coupling22,49,52. Therefore, tuning configurational entropy to induce local compositional inhomogeneity provides an effective way to regulate the formation of competing orders and to modulate the associated structural distortions. To comprehensively clarify structural features in local polar fluctuations of BNT-CE ceramic, we discerned the local E-field distribution by differential phase contrast (DPC) analysis12. The DPC-STEM (Fig. 3d) image of the 2.17R ceramic shows rapidly varying hues in the local electric-field distribution, with color indicating the field direction and brightness representing its magnitude. The pronounced fluctuations in both color and brightness highlight the polar heterogeneity of the local fields. Three representative types of E-field distribution are observed, as displayed in Fig. 3e, which can be associated with polar regions (region 1), disordered polar nanoregions (region 2) and weakly polar matrix (region 3), respectively. These distinct E-field distributions reflect the presence of different local polar states within the crossover region. It directly visualizes the coexisting polar states that arise from competing ferroic orders in the high-entropy ceramic, which is favorable for achieving high saturation polarization and low polarization hysteresis.

Fig. 3: Local structural evolution in BNT-CE ceramics and the BO6 octahedral distortion in high-entropy ceramic.
Fig. 3: Local structural evolution in BNT-CE ceramics and the BO6 octahedral distortion in high-entropy ceramic.The alternative text for this image may have been generated using AI.
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Atomic-resolution HAADF-STEM polarization vector image along [001] direction of (a) the 0.85R and (b) 2.17R ceramics. c Polarization displacement of the 0.85R and 2.17R ceramics. d DPC image with diverse characteristics of E-field distribution and (e) enlarged images of the regions in (d). f iDPC image of A/B sites and oxygen ions along [001] direction of the 2.17R high-entropy ceramic, and the colorful squares represent the BO6 with different rotation angle. g Distance and relative position of adjacent O atoms of the enclosed region in (f). h Statistical result of the rotation angle distribution of the BO6 octahedral of the 2.17R ceramic.

We also utilized integrated differential phase contrast (iDPC) to particularly capture the otherwise imperceptible light O atoms, in order to directly analyze the rotation behavior of BO653. The O atoms can be discernible in the iDPC images because this technique offers high sensitivity to light elements, and the relatively small contrast difference with neighboring cations enhances their visibility54. Figure S15 gives the high-resolution iDPC image of 2.17R high-entropy ceramic along [001] direction and illustrates the atom occupancy of the A/B site and O position. The squares with various colors in Fig. 3f denote different BO6 rotation behaviors, revealing irregular alternations between clockwise and anticlockwise orientation. As seen in Fig. 3g, O atoms within the same horizontal layer display varying displacement amplitudes, accompanying by noticeable differences in adjacent O-O distance. Then, the O-O distances (Fig. S16) extracted from the iDPC image are statistically analyzed, which demonstrates that most O atoms deviate from their average positions. This reflects a random and diffuse distribution of oxygen displacements. Quantitatively, octahedral rotation angles range from −4.3 to +4.3o, indicating relatively small rotation with a non-collective distribution. The statistical histogram in Fig. 3h presents a normal distribution of the octahedra, further confirming the dispersed and random nature of octahedral rotations in high-entropy ceramic. This heterogeneity of octahedral rotation caused by competing orders locally disrupts long-range symmetry and shifts the positive/negative charge center, generating spatially dispersed local dipoles and reduced switching barriers55. Such a configuration delays polarization saturation and facilitates reversible domain switching, thereby sustaining a large Pmax. Furthermore, local distortions of the oxygen octahedra will consume a portion of the electrical energy required for establishing relatively large domains under an applied field23, thus impeding domain growth and delaying saturation polarization. The structural variations by increasing Sconfig promote the formation of a weakly polar region, thereby lowering the energy barrier for reorienting the polarization configuration. Thus, the designed competing orders induced by high-entropy strategy modulate oxygen octahedral rotations, which delays polarization saturation and enhances breakdown strength, thereby maintaining a large Pmax while reducing Pr and enhancing ESP.

Energy Storage Performance of high-entropy MLCCs

To validate the efficacy of the entropy-induced strategy for actual pulse power application, we further fabricated 1210-type MLCCs comprising four active ceramic layers, based on the optimal composition of Sconfig = 2.17R. Figure 4a, b and Fig. S17 present the SEM image and energy-dispersive X-ray spectroscopy (EDS) element mapping of the high-entropy multilayer capacitor, which shows a uniform thickness of ~ 5.5 μm in the dielectric layer and 1.5 μm in the electrode layer, respectively. Both dielectric layers and electrode layers exhibit a dense microstructure, and the elements in each layer are uniformly distributed. The selected area electron diffraction (SAED) was conducted in the interface region between the two layers (Fig. 4(c1-c2)), which exhibits a well-defined and sharp boundary. No obvious element diffusion can be found between the dielectric layer and the electrode layer according to the result of linear scanning (Fig. S18), and both regions exhibit sharp diffraction spots (Fig. 4(d1-d2)). The inset of Fig. 4e and Fig. S19 show the unipolar P-E loops of high-entropy MLCCs as a function of E field. High Wrec ~ 20.64 J cm−3 and η ~ 94.2% are achieved in high-entropy MLCCs at a high Eb of 1472 kV cm−1. As Fig. 4f reveals, our work represents superior energy storage performance compared to other MLCCs in recent reports. To quantitatively evaluate the trade-off between Wrec and η, we used a figure of merit UF = Wrec/(1-η) to reflect the overall energy storage performance2, as shown in Fig. S20. UF of our work reaches a high value of 355.7 owing to the combined high Wrec and η. Hence, these results demonstrate the effectiveness of regulating competing orders via a high-entropy strategy in realizing superior energy-storage-performance MLCCs.

Fig. 4: Microstructures and energy storage performance of high-entropy 1210-type MLCCs.
Fig. 4: Microstructures and energy storage performance of high-entropy 1210-type MLCCs.The alternative text for this image may have been generated using AI.
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a Optical photo of high-entropy MLCCs. b SEM image of high-entropy MLCCs. TEM images for (c1) low resolution and (c2) high resolution, (d1-d2) SAED pattern acquired from the electrode area marked in (c2). e Wrec and η of the MLCCs under different electric fields. The inset is the corresponding P-E loops. f Comparison of energy storage performance in this work with other recently reported MLCCs. g Thermal stability of Wrec and η as temperature increases. h Anti-fatigue stability of Wrec and η as cycling number increases. i Time-dependent discharge energy density under various electric fields for high-entropy MLCCs.

From a practical application viewpoint, we also evaluated the thermal stability and cycle reliability of the high-entropy MLCCs. In Fig. 4g and Fig. S21, we tested the temperature stability of energy storage performance at the electric field of 900 kV cm−1 in the range from 20 to 140 oC, which shows a high Wrec of ~ 12.21 J cm–3 and η of over 93.3%, respectively, and displays a slight degradation below 5% for both Wrec and η. Besides, the anti-fatigue reliability tests were conducted to further assess the practical application. As shown in Fig. 4h, the MLCCs survive after 107 cycles with only below 2% attenuation, keeping a Wrec of ~ 12.15 J cm−3 and η of over 96.2%. In addition, as a key parameter, the charge-discharge performance of high-entropy MLCCs was measured at different electric field. Figure 4i and Fig. S22 illustrate the overdamped discharge curves of the high-entropy MLCCs within the field range of 200 kV cm−1 to 900 kV cm−1, where MLCCs achieve a large discharge energy density (Wdis) of 10.92 J cm−3 with a fast t0.9 ~ 0.81 μs (the time to release 90% of Wdis). Meanwhile, high-entropy MLCCs exhibit high current density (CD) of 424 A cm−2 and large power density (PD) of 191 MW cm−3 (Fig. S23). The charge-discharge performance exceeds that of most reported MLCCs, as shown in Fig. S24. The excellent temperature, fatigue stability and charge-discharge property fully show wide application scenarios and huge application potential for the next-generation MLCC devices.

In summary, we demonstrated a high-entropy design that directly harnessed the symmetry of different oxides, together with phase-field simulations, to engineer competing polar orders and form the crossover region. At the local scale, the competition among diverse orders yields more dispersed and random polar configuration with the increase of Sconfig. In composition with optimized Sconfig value, distinct E-field distributions, oxygen displacements and dispersed distribution of BO6 rotations are observed, and these features together contribute to broad switching barrier and enhanced relaxor behavior. Ultimately, superior performance of Wrec ~ 20.64 J cm−3 and η ~ 94.2% is achieved in high-entropy MLCCs, along with excellent temperature and fatigue stability, and a large charge-discharge capacity of Wdis ~ 10.92 J cm−3. Our work provides a predictive and transferable solution to the longstanding challenge of simultaneous improvement of Wrec and η in pulse capacitors and highlights a broader technological potential for high-entropy multilayer devices in next-generation energy-storage applications.

Methods

Phase-field simulation and finite-element simulation

The total free energy of the high-entropy BNT-based system can be expressed as the sum of the following contributions56,57: \(F=\int {f}_{{landau}}{dV}+{f}_{{grad}}{dV}+{f}_{{elas}}{dV}+{f}_{{elec}}{dV}\). The Landau free energy flan can be presented as function of Pi as: \({f}_{{landau}}={\alpha }_{1}\left({P}_{1}^{2}+{P}_{2}^{2}+{P}_{3}^{2}\right)+{\alpha }_{11}\left({P}_{1}^{4}+{P}_{2}^{4}+{P}_{3}^{4}\right)+{\alpha }_{12}\left({P}_{1}^{2}{P}_{2}^{2}+{P}_{2}^{2}{P}_{3}^{2}+{P}_{1}^{2}{P}_{3}^{2}\right)+{\alpha }_{111}\left({P}_{1}^{6}+{P}_{2}^{6}+{P}_{3}^{6}\right)+{\alpha }_{112}\left({P}_{1}^{4}{P}_{2}^{2}+{P}_{2}^{4}{P}_{3}^{2}+{P}_{1}^{4}{P}_{3}^{2}+{P}_{1}^{2}{P}_{2}^{4}+{P}_{2}^{2}{P}_{3}^{4}+{P}_{1}^{2}{P}_{3}^{4}\right)+{\alpha }_{113}({P}_{1}^{2}{P}_{2}^{2}{P}_{3}^{2})\), where α is the Landau coefficients. \({f}_{{grad}}\) is the gradient energy, representing the domain boundary energy, which can be defined as: \({f}_{{grad}}=\frac{1}{2}G{\sum }_{i,j}^{3}{({{{\bf{P}}}}_{{{\bf{ij}}}})}^{2}\), where G is the gradient energy coefficient. The long-range elastic interactions energy felas, including strain-gradient and external-stress fields, is expressed as: \({f}_{{elas}}=\frac{1}{2}\left\{{C}_{{ijkl}}\right.\left.\left({\varepsilon }_{{ij}}-{\varepsilon }_{{ij}}^{0}\right)\left({\varepsilon }_{{kl}}-{\varepsilon }_{{kl}}^{0}\right)+{C}_{{ijkl}}{e}_{{ij}}^{{\prime} }\left({\varepsilon }_{{kl}}-{\varepsilon }_{{kl}}^{0}\right)\right\}\), where Cijkl is the elastic stiffness tensor, εij, ε0ij and \({e}_{{ij}}^{{\prime} }\) are the total strain. \({f}_{{elec}}=-\frac{1}{2}{E}_{i}{P}_{i}-\frac{1}{2}{E}_{i,{depol}}\overline{{P}_{i}}\), where Ei represents the inhomogeneous electric field caused by dipole-dipole interactions, Ei,depol is the average depolarization field. \(\overline{{P}_{i}}\) is the average polarization. The temporal evolution of the polarization field can be calculated by solving the time-dependent Ginzburg-Landau (TDGL) equation: \(\frac{d{P}_{i}(x,t)}{{dt}}=-M\frac{\delta F}{\delta {P}_{i}\left(X,t\right)},i={\mathrm{1,2,3}},\) where M is the kinetic coefficient related to the domain mobility, t is time, and i denotes polarization variants. Phase-field simulations were performed in a three-dimensional grid of size 64 × 64 × 64 and in the length scale l0 (the numerical grid size) of ~ 2.0 nm with periodic boundary conditions. The probability of the electrical tree channels is defined using the following equation: \(p\left({i}^{{\prime }},{j}^{{\prime}} \to i,j\right)=\frac{{({\aleph }_{{i}^{{\prime}},{j}^{{\prime}}}-{\aleph }_{i,j}-\aleph )}^{m}}{\sum {({\aleph }_{{i}^{{\prime}},{j}^{{\prime} }}-{\aleph }_{i,j}-\aleph )}^{m}}+{({\aleph }_{{i}^{{\prime} },{j}^{{\prime} }}-{\aleph }_{{i}^{{\prime \prime}},{j}^{{\prime \prime} }}-\aleph )}^{m}-{loss}\), where \({\aleph }_{i,j}\), \({\aleph }_{{i}^{{\prime} },{j}^{{\prime} }}\) and \({\aleph }_{{i}^{{\prime} {\prime} },{j}^{{\prime} {\prime} }}\) represent the electrical potential of the discharged point, probable point, and linked point, respectively. \(\aleph\) is the threshold electrical potential of the ceramic for the grain and grain boundary. Loss and \(m\) denote the evolution loss of the tip electrical tree channels and the fractal dimension, respectively.

Ceramic preparation

A conventional solid sintering was used to prepare (Bi0.5Na0.5)(1-x)(Sr0.2Ca0.2Ba0.2K0.2Ag0.2)xTi1-0.4xNb0.4xO3 ceramics (x = 0.02, 0.1, 0.2, 0.3, 0.4, 0.5). The raw powders of Bi2O3 (99.99%), Na2CO3 (99.8%), BaCO3 (99.95%), SrCO3 (99.99%), CaCO3 (99.99%), Ag2O (99.7%), K2CO3 (99%), TiO2 (99.9%), Nb2O5 (99.99%) were mixed and milled for 4 hours by using ZrO2 balls in ethanol. Subsequently, the dried powder was then calcined in an alumina crucible at 850 oC for 2 h. Powders were milled again using smaller size ZrO2 balls, and 0.3 wt.% MnCO3 was added as the sintering aid. Then, the calcined powders were mixed with 6 wt.% polyvinyl alcohol. The green pellet was first burn out binder at 600 oC for 2 h and finally sintered at 1120-1180 oC for 2 h in an oxygen atmosphere.

MLCC preparation

BNT-based high-entropy powder (x = 0.4/2.17R) was utilized in the fabrication of 1210-type MLCCs via tape casting with an MTI MSK-AFA-II system. A 70 Ag/30 Pd electrode was printed via a DEK 247 screen printer. The stacked layers were then subjected to hot-pressing at 80 oC for 20 min. The resulted MLCCs were sintered at 1080 oC for 2 h in an oxygen atmosphere.

Structure characterizations

Powder X-ray diffraction patterns were collected using a powder X-ray diffractometer (XRD, Rigaku, D/max-2550 V) with Cu Kα radiation (λ = 1.5406 Å). Raman spectra were obtained with a Raman spectrometer (Renishaw, inVia). Field-emission scanning electron microscopy (FE-SEM, Hitachi, S-4800, Tokyo, Japan) was used to examine the microstructures and density of the ceramics. Apparent density of all ceramic samples was measured by the vacuum-saturated Archimedes method. The theoretical density was determined from the unit-cell parameters obtained by Rietveld refinement of the diffraction peaks. The relative density is the ratio of the apparent density and the theoretical density. The STEM and DPC/iDPC tests were carried out on a 300 kV Thermofisher Spectra 300 electron microscope equipped with double aberration correctors. During the process of testing, the probe convergence angle was 18 mrad, and the collection angle was 50-200 mrad to obtain the HAADF images. For determining the B-site cation displacement vectors and rotation of BO6 octahedra, the noise in the obtained HAADF-STEM and DPC/iDPC images was reduced by Gaussian blur of Velox (Thermofisher) software with sigma of 1.0-1.8. The projected atom positions of A sites, B sites, and O were determined via OpenCV using the Python software, thus, the projected B-site cation displacement was deduced as a vector between each B-site cation and the center of mass of its four nearest corner A-site cations. The rotation of BO6 octahedra was deduced as an angle from the minimum inscribed square of each B-site cation and its four nearest O ions with [001] direction.

Ferroelectric and dielectric measurements

P-E loops were measured using ceramic samples with an electrode size of 1 mm diameter and a thickness of ~ 60 μm under an alternating current voltage with a frequency of 10 Hz by using a ferroelectric measuring system (aixACCT TF Analyzer 2000E). A broad frequency dielectric spectrometer (Novocontrol GmbH, Concept 80) was used to measure temperature- and frequency-dependent dielectric properties. Underdamped and overdamped charge-discharge tests were conducted on a commercial charge-discharge platform (CFD-001, Gogo Instruments Technology, Shanghai, China) using an RLC load circuit under a direct current voltage, operated at a 1 kHz. We calculated charge-discharge properties by employing the following formula: \({W}_{{dis}}=\frac{R\int {I}^{2}(t){dt}}{{V}^{{\prime} }}\), \({C}_{D}=\frac{{I}_{\max }}{S}\) and \({P}_{D}=\frac{E\times {I}_{\max }}{2S}\), where S represents the effective electrode area and \({V}^{{\prime} }\) is the effective volume of the tested samples.