Introduction

Inorganic dielectric capacitors, benefitting from ultrafast charge/discharge rate (at a time scale of microseconds), large voltage endurance (up to several kilovolts or higher), and excellent stability and reliability (operating temperature typically ranging from −55 to 150 °C, with a cycling life exceeding 106 cycles)1,2,3, are essential components in high-power energy storage systems. They can be used in applications such as pulsed ignition units, high-voltage DC-AC inverters, pulsed laser generators, and electromagnetic power sources4,5,6. Dielectric capacitors utilize electric field-induced polarization and accumulated charge to store energy—the charged energy density is given by \({U}_{{{\rm{c}}}}={\int }_{-{P}_{{{\rm{r}}}}}^{{P}_{{{\rm{m}}}}}E{{\rm{d}}}P\) (here electric displacement D ~ polarization P as permittivity » 1) and recoverable energy density is given by \({U}_{{{\rm{e}}}}=-{\int }_{{P}_{{{\rm{m}}}}}^{{P}_{{{\rm{r}}}}}E{{\rm{d}}}P\). The efficiency η is therefore defined as the ratio of Ue and Uc. It is easy to find that a high recoverable energy density and a high efficiency require a small remnant polarization Pr, a large maximum polarization Pm, and a high electric field. In other words, a high “active” polarization Pm - Pr is necessary. Therefore, it is pivotal to tune the polarization-electric field (PE) behavior to achieve high energy storage performance. Ferroelectrics (FEs) with high polarization are materials of interest; however, in most pristine FEs, the energy density is typically <2 J cm−3 for ceramic bulks and <20 J cm−3 for thin films due to large loss7,8. There have been extensive investigations to tackle this issue. Among all efforts, modifying normal FEs into relaxor ferroelectrics (RFEs) perhaps is the most widely used and the most often effective one9,10,11,12,13,14,15,16.

In RFEs, it is well recognized that local inhomogeneity plays a critical role in their superior properties to conventional FEs, which produces diffused phase transition arising from the unsynchronized kinetics of different regions in RFEs17,18. Microscopically, local inhomogeneity can induce polar nanoregions (PNRs) or polar slushes19,20,21, which contributes to higher domain dynamics and susceptibility as well as reduced domain switching barriers, making RFEs promising for high energy density and efficiency9,14,22,23. Guided by this principle, chemical modification has been employed to break long-range ordered domains into nanodomains. This strategy, commonly termed domain engineering, is exemplified in various systems, e.g., BiFeO3-SrTiO3 (BFO-STO)9,24, BiFeO3-BaTiO3-SrTiO310, Sm doped BiFeO3-BaTiO311 and BiMg0.5Ti0.5O3-SrTiO314. More recently, an entropy tactic, i.e., increasing configurational entropy of FEs, has been proposed. This approach also utilizes increased local inhomogeneity to manipulate RFEs and thus achieved an ultrahigh energy density13,25.

Although there have been numerous experimental pieces of research elucidating the role of local inhomogeneity by advanced scanning transmission electron microscopy characterizations9,26, the current techniques are still beset by several inherent limitations: (1) in situ visualization of local switching dynamics of domains with high temporospatial resolution is still challenging albeit some constructive progress27,28,29, (2) local inhomogeneity cannot be precisely assessed hindered by atomic column projections in microscopy, and (3) processing defects can blur the interpretation of characterizations. These can be tackled by theoretical methods, of which phase-field simulations are the most successful practice for FEs and RFEs10,30,31, because the conventional domain scale is typically of hundreds of nanometers to several microns (i.e., mesoscopic scale), which is beyond the limit of ab initio methods32. Phase-field simulations have unveiled how the energy storage performance can be boosted with various domain setups, such as nanodomains, polymorphic nanodomains, superparaelectric polar clusters and isolated polar domains10,11,14,33,34. However, the current work mostly serves as a screening for experiments; the underlying mechanism for substantially improving energy storage performance from the perspective of local inhomogeneity is not sufficiently studied. Moreover, the simulation setups for PNRs, polar slushes and other highly inhomogeneous RFEs rely too much on preset domain structures. Driven by this motivation, this work aims to offer an understanding of the improved energy storage performance in RFEs with local inhomogeneity, and attempts to develop a model directly incorporating compositional fluctuations to demonstrate the linkage between local inhomogeneity and dielectric properties.

Results

The phenomenological PNR model

It is recognized that PNRs exist in many RFEs due to local inhomogeneity9,35,36,37. In general, compounding paraelectric (PE) end members can most effectively disrupt long-range polar ordering and thus break large domains into nanosized ones9,10,38. In phase-field simulations, a common way to simulate this structure characteristic is to composite nanosized polar regions (with FE features) and a non-polar matrix (with PE features or linear dielectric (LD) features)10,23,33,34 (See Methods). This composite-like model is reasonable because experimental studies revealed that RFEs are somewhat composite-like given the PNR structure39. To evaluate the role of PNRs, we first studied the effect of domain size and domain concentration in the typical system xBFO-(1-x)STO, where x is treated as domain concentration. Figure 1a involves the domain size evolution with a fixed domain concentration of 30%, while Fig. 1b involves the domain concentration evolution with a fixed domain size of ~3 nm in radius. With the decrease in domain size, the P–E hysteresis loops become slightly less hysteretic, i.e., with smaller coercive fields, contributing to a higher energy efficiency (from 43% to 61%), but the enhancement in energy density is rather limited (from 11.8 to 12.7 J cm−3). With the decrease in domain concentration, the polarization is sacrificed but the hysteresis loop is apparently slimmed with increased energy density (from 10.0 to 12.9 J cm−3) and efficiency (from 14% to 85%). The result indicates that, given the property of PNRs intact, domain concentration (i.e., doping concentration of FE components) is more influential in improving the overall energy storage performance; small domains together with a relatively small domain concentration can achieve the best overall energy storage performance. This result resonates with many experimental and theoretical observations10,11,33.

Fig. 1: Effect of PNRs on improving energy storage performance.
figure 1

a PNR configurations of xBFO-(1-x)STO solid solutions with a fixed domain concentration of 30% and an increased domain size from left to right, and the corresponding P–E hysteresis loops. b PNR configurations of xBFO-(1-x)STO solid solutions with a fixed domain size of 3 nm in radius and an increased domain concentration from left to right, and the corresponding P–E hysteresis loops. ce energy density Ue (c), efficiency η (d) and system permittivity (e) as a function of domain size, domain concentration and matrix permittivity of xBFO-(1-x)LD solid solutions.

Apart from PNRs, the matrix can have an impact on the energy storage performance. Intuitively, matrices with higher permittivity should contribute to a larger polarization and thus better energy storage performance. To investigate the point, we then investigated xFE-(1-x)LD with the permittivity of the LD matrix as the third attribute, and calculated the energy density, efficiency, permittivity of the entire “PNR + matrix” system and characteristic polarizations (Fig. 1c–e and Supplementary Fig. 1 for BFO-LD and Supplementary Fig. 2 for PbTiO3-LD). For energy density, the increased LD permittivity has two effects: increasing the energy density by elevating the maximum polarization (Supplementary Fig. 1a) and shifting the peak values to small domain size and concentration regions. Higher matrix permittivity can compensate for the loss of polarization, so that high energy density can be sustained even with small domain size and concentration. For efficiency in Fig. 1d, although high efficiency occurs in the region of small domain size and concentration, higher matrix permittivity can compress the area of this region, indicating that high matrix permittivity does not favor efficiency. Therefore, to achieve optimal energy storage, a permittivity of ~102 is desired, which is exactly the case for the classical PE STO40. In Fig. 1e, the permittivity for the entire “PNR + matrix” system is calculated, with a similar evolution of energy density, implying that an increase in permittivity of the system at zero field is a good indicator for energy storage, because the permittivity enhancement originates in the increased Pm - Pr (higher than pure FE because of reduced Pr, and higher than LD because of considerably maintained Pm), being the same requirement for energy density. A similar trend is also observed in PTO-based systems (Supplementary Fig. 2), manifesting the universal effects of PNRs on improving energy storage performance.

The key to improving energy storage with PNRs is the slimmed hysteresis loop. This change is not trivial if we compare the P–E loops of PNRs and those with a simple average of PNRs and matrices by concentration (Supplementary Fig. 3). Previous literature attributes this phenomenon to reduced Landau energy barrier33—this explanation holds true but is insufficient. A simple question is: since our simulation setup does not change the parameter of polar regions, how is the Landau energy barrier lowered with the decrease in domain size? Thus, to further understand the mechanism of PNRs, we mapped energy and field distributions for BFO-STO systems in Fig. 2. In Fig. 2a–d, as the domain size increases (left upper panels), the Landau energy decreases (right upper panels, equivalent to the increase in Landau energy barrier) until it reaches the thermodynamic minimum when the domain size is large enough. The change in Landau energy is rational, because Fig. 1a shows that polarization is suppressed in smaller domains, and thus the Landau energy in such PNRs has not reached the minimum of the double well potential (Supplementary Fig. 4). Electric energy (left lower panels) and local electric field mappings (right lower panels) unveil that polarization discontinuity of non-polar and polar regions can cause electric field redistribution that enhances the field in non-polar matrices and offsets the field in PNRs to achieve bounded charge neutrality as much as possible. Such an internal field distribution can minimize the electric energy of matrices at the cost of elevating that of PNRs. Systems with small domain size and concentration benefit from this effect most because energy penalty in PNRs is more tolerable. The redistribution of the electric field on the one hand delays the polarization of polar regions upon applying the electric field, and on the other hand minimizes hysteresis as non-polar matrices tend to force back PNRs to the low-polarization state upon removing the field via electrostatic interactions, thus contributing to the slanted and slimmed P–E loops that are essential in high-performance dielectrics1,41. The smaller domain size and concentration will amplify such a control and the overall hysteresis loop is more matrix-like (i.e., PE- or LD-like). Upon applying an electric field, we find that polarization direction inhomogeneity is successfully induced by the electrostatic interactions between PNRs and the matrix, which is key to an enhanced energy storage (Supplementary Fig. 5). The effect is also confirmed by elevating the temperature which weakens polarization, where a more dispersed polarization direction distribution contributes to higher energy density and efficiency (Supplementary Fig. 6).

Fig. 2: Electrostatic view to understand how PNRs improve energy storage.
figure 2

ad, Structure, Landau energy, electric energy and electric field distributions with the field Ey is applied. The domain size increases from (a) to (d). e Local P–E loops along the arrow marked in (b) from bottom to top. The two ends represent the matrix and the middle three represent the PNR. Note that the fourth loop is very near the PNR/matrix interface, showing the most controlled behavior. f Local susceptibility mapping as the domain size increases from top to bottom.

We also observed that the redistribution is more significant in regions where the matrix and PNRs are in series connected (i.e., interfaces perpendicular to the electric field Ey). Figure 2e plots local P–E loops along the arrows marked in Fig. 2b. In the matrix, the P–E loops are as normal PE-like (with slight hysteresis due to the impact of adjacent PNRs), but in PNRs S-shaped loops occur, demonstrating the stabilization of local negative capacitance which is a unique feature for FEs42,43,44. Only FE/PE connected in series can stabilize negative capacitance, because the P–E loop is no longer electric field-controlled (the case that PNRs are directly biased), but charge-controlled (the case that PNRs and matrix are in series connected)45,46,47. Once PNRs are charge-controlled, the first derivative of Landau energy (i.e., the intrinsic electric field-polarization relationship, Supplementary Fig. 7) becomes a single-valued mapping, and therefore an S-shaped loop is allowed. The local P–E loops are further confirmed by the mapping of local dielectric susceptibility (proportional to 1/ε22) where negative regions coincide with PNRs. Such analysis demonstrates that a non-polar matrix can control the behavior of PNRs with less hysteretic and more reversibly switchable features, contributing to an enhanced energy efficiency. The electrostatic explanation for PNRs can also answer why higher matrix permittivity does not favor high efficiency—the polarization incompatibility is weakened between the matrix and PNRs and the matrix-controlled behavior is hence weakened.

To validate the proposed electrostatic mechanism, we performed more simulations incorporating PNRs with different shapes. It is inferred that more in-series connection topologies contribute to better matrix controllability, and thus higher energy storage performance. Therefore, a set of PNRs were constructed in Fig. 3a, with different aspect ratios (b/a, where b is the length along x and a is along y) but a fixed domain concentration of 30% and a fixed equivalent domain radius of 8 nm (geometrically averaged as \(\frac{\sqrt{{ab}}}{2}=8\) nm). As the aspect ratio increases, the maximum polarization is slightly suppressed (Fig. 3a, the last panel, from 0.39 to 0.29 C m−2), consistent with previous reports48,49,50, but the remnant polarization is reduced dramatically (Fig. 3a, the last second panel, from 0.22 C m−2 to 0.03 C m−2). This leads to a remarkable increase in energy density Ue from 10.2 to 14 J cm−3, efficiency η from 27% to 91%, and the system permittivity from 222 to 558 (Fig. 3a, first three panels). Note that this performance improvement is more significant than simply tuning domain size and concentration as demonstrated in Fig. 1 (29% vs. 37% improvement). The efficiency is increased with limited sacrifice of polarization, demonstrating great promise in boosting the overall energy storage performance. When the PNR is in-series connected to the non-polar matrix and the electric field is applied, the ferroelectric/paraelectric interface vertical to the electric field must maintain charge neutrality as soon as possible to lower the electric energy, which can only be done by polarizing the matrix more and polarizing the PNR less, which we term as an interfacial charge-controlled behavior. This is actually a sense of depolarization in PNRs. This effect can lead to the delayed occurrence of domain switching and therefore an overall delayed polarization, together with significantly lowered remnant polarization and slightly reduced maximum polarization. Thus, the energy storage performance is boosted. If, by contrast, the PNR is more in-parallel connected to the matrix (i.e., large radius along the electric field), although the interface vertical to the electric field still requires charge neutrality, the primary part of the PNR is less impacted by the interface, leading to a more ferroelectric behavior—in other words, they are less depolarized by the non-polar matrix.

Fig. 3: Connection topology of PNRs and the matrix.
figure 3

a PNRs with different aspect ratios. From top to bottom are morphology, P–E loops and energy storage parameters. The color marks polarization (P). b PNRs with a 32:2 aspect ratio rotate from parallel (0.0 deg) to vertical (90.0 deg) to the electric field. c PNRs with plate, fiber and sphere shapes either vertical (v) or parallel (p) to the electric field. The electric field is along y direction.

Figure 3b, c show another two cases of increasing in-series connectivity. In Fig. 3b, the PNRs with a 32:2 aspect ratio rotate from parallel to vertical to the electric field, and the energy performance is boosted analogous to the case in Fig. 3a (the case for PTO-STO can be found in Supplementary Fig. 8). In Fig. 3c, a three-dimensional model is created with PNR plates (the largest in-series/in-parallel connectivity), PNR fibers (middle in-series/in-parallel connectivity) and PNR spheres (the smallest in-series/in-parallel connectivity) vertical or parallel to the electric field. Plate PNRs vertical to the electric field exhibit the largest energy density (~20 J cm−3) among the five cases, again echoing the proposed electrostatic mechanism. It is even more interesting that in an antiferroelectric (AFE)–PE system that in-series connectivity can substantially increase the AFE–FE transition field EA and FE–AFE transition field EF, resulting in a simultaneous improvement in energy density and efficiency (Supplementary Fig. 9). The polarization in non-polar regions is typically smaller than antipolar regions (either before or after their polarization), which renders bound charges, and consequently depolarizes antipolar regions as well as delaying EA. When the field is removed, the non-polar regions return to their non-polar state, and the bound charges therefore induce recovery of polarized antipolar regions, leading to a reduced hysteresis45. It is, therefore, important to emphasize that electrostatic interactions that are usually neglected are actually crucial factors to improve energy storage performance in polarization inhomogeneous systems induced by local inhomogeneity.

The polar slush model with strong random fields

The PNR model has a series of successful practice in RFEs with FE and PE regions; however, if a solid solution contains two FE end members, the microscopic morphologies can be connected to polar domains with very few PE matrices. In such a case, the rendered P–E loops are usually rectangular with large loss and poor energy storage performance (Supplementary Figs. 10, 11). In experiments, although FE-FE solid solutions can be FE-like, it is however often observed that a complex FE system can be relaxor-like with substantially improved energy density51. This behavior can be described by the random field theory. In the random field theory framework, uncompensated charged defects arising from lattice distortion, off-centered ions, point defects, and dipoles act as the source of random fields38,52. A more complex system (with e.g., complex stoichiometry, ion valence and ion radii, etc.) tends to possess larger random fields. In phase-field simulations, such random fields are introduced by generating a field following a normal distribution \({E}_{{{\rm{rand}}}} \sim N(0,{E}_{{{\rm{mag}}}})\) independent of external electric fields (See Methods)53. With the increase in the magnitude of random fields, Fig. 4a displays progressively broken domains from macro-sized striped domains into nanosized domain slushes of BFO without any PE matrices (the case for KNN is in Supplementary Fig. 12). Note that this process still does not involve any change in Landau coefficients; the driving force is purely the random field (local inhomogeneity), which at low fields can make polarization more disordered, resulting in a small remnant polarization and minimized hysteresis (Fig. 4b, Pr reduced from 0.62 C m−2 to 0.19 C m−2). This brings in significant boost in energy storage density from 9 to 25 J cm-3 and efficiency from 8% to 70% (Fig. 4c, d), with roughly a threefold increase in both energy density and efficiency for various systems, such as BFO, K0.5Na0.5NbO3 (KNN) and BiFeO3-PbTiO3 (BFPT) solid solutions. The polar slush model should also be applied universally in complicated ferroelectric systems and ferroelectric-paraelectric systems. We also note that in polar slushes, the polarization is significantly higher than in PNRs, so ferroelectric solid solution with high local inhomogeneity should be better candidates for energy storage. Along with the increased energy storage performance, increased permittivity peaked at a certain random field magnitude is found (Fig. 4e), indicating a maximal active polarization. The change in both domain size and P–E loops demonstrates the decrease in long-range ordered ferroelectricity—this is well reflected in decreased Tm (at this temperature the permittivity reaches the maximum, which indicates roughly the FE-PE transition). It should be noted that the critical magnitude of random fields that can considerably alter the performance of FEs is highly related to their intrinsic coercive fields, which is ~2 MV cm−1 for BFO and 0.1 MV cm−1 for KNN. If we consider random fields are mainly induced by extrinsic factors, ferroelectrics with lower coercive fields are likely to be more sensitive.

Fig. 4: Random field-induced nanodomain slushes and boosted energy storage.
figure 4

a Domain patterns of BFO with the increased random fields. The colors represent different domain variants, including 8 rhombohedral (R), 12 orthorhombic (O) and 6 tetragonal (T). b P–E loops of BFO with the increased random fields. cf, energy density Ue (c), energy efficiency η (d), permittivity (e) and Tm (f) as a function of random field for BFO, KNN and BFPT.

Our results imply that minimized domains in polar slush models are spontaneous results of increased local inhomogeneity, and such an effect can suppress the macroscopic ferroelectricity, activate domain responses to electric fields, and boost energy storage performance. In this regard, increasing local inhomogeneity is equivalent to increasing the temperature (Supplementary Fig. 13). It is also important to point out that nanodomains are actually a result of local inhomogeneity and intense random fields. The conventional notion that small domains are accountable for enhanced energy density may be superficial; enhanced energy storage performance and small domains are both results of local inhomogeneity.

Commonality of PNR and polar slush models

The above sections have revealed how local inhomogeneity enhances energy storage performances in the framework of two models. Admittedly, the two models differ in their simulation settings: (1) the Landau energy of the PNR model is composed of ferroelectric and paraelectric, and that of the polar slush model is fully composed of ferroelectric; (2) depolarization mechanism at the field-free state is different, i.e., the PNR model is from the electrostatic interaction between ferroelectric and paraelectric regions, and the polar slush model is from the random fields caused by inhomogeneity that act as a extrinsic structural parameter to depolarize the monodomain to the multidomain. However, both effectively incorporate inhomogeneity into the system. To further explore the commonalities in how the two models impose inhomogeneity, we conducted frequency-dependent permittivity versus temperature calculations (see Methods). Since we are not focusing on very high frequencies, we still adopt the first-order time-dependent evolution equation54. It is seen from Fig. 5 that both BFO-STO with PNRs (polarization inhomogeneity, Fig. 5a, b) and BiFeO3 with polar slushes (polarization disordering induced by random fields, Fig. 5c, d) exhibit conspicuous frequency dispersion, indicating the unsynchronized domain dynamics across the temperature and frequency range39,55,56. With the increase in non-polar end members in xBFO-(1-x)STO solid solutions, a decreased Tm along with an increased ΔT from ~0 to over 200 K is observed, as weakened ferroelectricity and significant local inhomogeneity are accompanied by the decrease in x. This implies that regions with polarization inhomogeneity have different dynamics. It is also found in Supplementary Fig. 14 that smaller domain sizes can also induce decreased Tm and increased ΔT, demonstrating that larger polarization inhomogeneity as imposed by surrounding matrices in smaller-sized PNRs. Similarly, in BFO with polar slushes, the same trend occurs: Tm decreases as the random field is intensified, along with the ΔT increased from ~0 to 250 K, implying local inhomogeneity can alter domain dynamics from collective to non-synchronized. In both models, the largest frequency dispersion always occurs in systems with the most disordered domain configurations, which are also systems with significantly improved energy storage performances (Figs. 1 and 4). The dynamical dielectric properties validate that strong RFE characteristics are induced by intense local inhomogeneity, and excellent energy storage performance is necessitated by strong RFE features. In addition, the similar results from two different models manifest their ability to describe local inhomogeneity although it is differently introduced in two models, and demonstrate that local inhomogeneity is a general factor in influencing dielectric properties and capacitive energy storage.

Fig. 5: Dynamical dielectric properties.
figure 5

a Permittivity versus temperature near Tm of xBFO-(1-x)STO at different frequencies. b Frequency dispersion as a function of x. c Permittivity versus temperature near Tm of BFO with different random fields. d Frequency dispersion as a function of the random field.

Modeling energy storage performance in the framework of local inhomogeneity

Having investigated the two commonly used models of RFEs and their success in explaining the improvement in energy storage performance, we still see weaknesses in them: (1) the PNR model relies too much on preset domain structures; (2) the random field magnitude in the polar slush model is weakly related to the information of the material system; and (3) both models contain very little information of the chemical composition. In most cases, RFEs are solid solutions or chemically complex systems, so it is necessary to incorporate compositional fluctuations for accurate representation, as it is the origin of all kinds of inhomogeneity in dielectrics. Herein, combining the advantages of both models and the consideration of chemical inhomogeneity, we develop a model that involves the following:

  1. 1)

    Chemical inhomogeneity. The entire system is composed of solid solutions of various end members A, B, C, … with nominal concentrations cA0, cB0, cC0, … and

    $${\sum}_{i=A,\,B,C,\ldots }{c}_{i}^{0}=1$$
    (1)

    We use a Dirichlet distribution to generate real concentrations

    $${c}_{i}\left({{\bf{r}}}\right) \sim D\left({{k}_{1}S}_{{{\rm{config}}}}^{0};{c}_{i}^{0}\right)$$
    (2)

    where \({c}_{i}^{0}\) is the nominal concentration and \({S}_{{{\rm{config}}}}^{0}\) is the calculated configurational entropy from nominal concentrations that indexes the local inhomogeneity25,57. k1 is set to make configurational entropy dimensionless and more meaningful to generate concentrations. Such a generation contains local inhomogeneity as ci is a dispersed distribution rather than a certain value, and includes information of both original nominal concentration and the complexness of the entire system.

  2. 2)

    Dielectric inhomogeneity. Dielectric inhomogeneity (i.e., different Landau coefficients at different regions) is induced by compositional inhomogeneity. Here for the sake of simplicity, we use a linear combination of Landau coefficients αmnl with respect to real concentrations

    $${\alpha }_{{mnl}}\left({{\bf{r}}}\right)={\sum}_{\begin{array}{c}i=A,\,B,C,\ldots \\ {mnl}\; {{\rm{are}}}\; {{\rm{Tensor}}}\; {{\rm{indices}}}\end{array}}{c}_{i}{\alpha }_{i,{mnl}}$$
    (3)
  3. 3)

    Other local inhomogeneity. Other local inhomogeneity depends on chemical fluctuations and the ensuing inhomogeneity from charge, lattice, dipole, etc. We simply use random field to characterize it. We define the random field to follow

$${E}_{{{\rm{rand}}}}\left({{\bf{r}}}\right)={{k}_{2}S}_{{{\rm{config}}}}^{0}{\sum}_{i=A,B,C}\nabla {c}_{i}$$
(4)

where k2 represents the characteristic random field of the system. The magnitude of random fields is dependent of the concentration gradient, i.e., local chemical fluctuations.

The three setups integrate the key considerations of both models—dielectric inhomogeneity in the PNR model and random fields in the polar slush model—while addressing current limitations by linking them to local chemical composition. Detailed parameters can be found in Methods. Using this method, we first conducted simulations for the same classical system xBFO-(1-x)STO. Figure 6a displays x distributions (content of BFO) from x = 0.9 to 0.1. It is clear that real concentration of the BFO end member is inhomogeneously distributed in the system. Figure 6b and Supplementary Fig. 15 are the corresponding polarization distribution—those with more BFO, regardless of inhomogeneity, contain large domains, while those with more STO contain small, disordered domains owing to increased inhomogeneity. Statistics show that the incorporation of PE components and accompanied local inhomogeneity induce the decrease in domain size from 25 nm to 2.5 nm and the decrease in domain concentration from 95% to 10% (Fig. 6d). This result reveals that local inhomogeneity modulates domain configuration by simultaneously decreasing the domain concentration and size; PNRs can effectively form with the consideration of local compositional fluctuations. Correspondingly, the P–E loops are slimmer (Fig. 6c). Benefitted from reduced hysteresis, the energy density peaks at x ~ 0.4, with a nearly double-fold improvement, along with a high efficiency exceeding 90%. Our simulations are in good alignment with previous simulations and experimental observations (Supplementary Fig. 16)9,33, demonstrating the validity of the model in describing of local inhomogeneity and the reasonability of the linkage between compositional fluctuations and domain structures/dielectric properties. We also extended the method to the (1-x)AFE-xPE solid solution and the results are consistent with the PNR model (Supplementary Figs. 9 and 17).

Fig. 6: Simulating solid solutions xBFO-(1-x)STO using the inhomogeneity model.
figure 6

a Real concentration cBFO distribution as a function of nominal concentration x. b Polarization magnitude distribution as a function of x. c P–E loops of different x. d Statistics of simulated domain size and domain concentrations as a function of x. e Calculated energy density and efficiency as a function of x.

Notice the introduced configurational index in Dirichlet distribution, we could unify entropy-modulated dielectrics into this framework of local inhomogeneity, instead of the integrated attempts of (1) random fields and (2) change in Curie temperature25, which is although effective but lacks rationale because the chosen parameters highly rely on existing data to fit experiments. High-entropy compositions like Ax(B1,B2,B3…)yOz can be regarded as solid solutions AxB1yOz-AxB2yOz-AxB3yOz-… Usually, only limited end members of the solid solution are ferroelectrics, and others can be taken as PEs or LDs. Random fields are assigned according to the total entropy of the system and the local inhomogeneity in concentration. Therefore, high-entropy compositions should have lower Curie temperature (assured by linear combination of Landau parameters with PEs and LDs) and higher random fields (assured by higher entropy and higher fluctuation in concentration), in accordance with previous simulation practices. We exemplified this model in the high-entropy system (Bi,A,B,C,D)Ti3O12, where A, B, C, and D are ions with equimolar substitution to original Bi sites. With the addition of A, B, C and D, entropy increases from 0.69 R to 1.61 R, and random field is therefore enhanced and local compositional fluctuations are more intensified (Fig. 7a, c, and Supplementary Fig. 18). Δc/c rather than Δc is used here to normalize the chemical fluctuations. Accordingly, the P–E loops transit from FE into RFE and eventually PE as entropy increases (Fig. 7b). Although spontaneous polarization decreases monotonously due to the introduction of non-polar end members, polarization fluctuations are enhanced as a consequence of the local inhomogeneity (Fig. 7d). This results in an enhancement in energy density at around medium-entropy to high-entropy compositions (Sconfig ~1.5 R) (Fig. 7e), in good consistency with previous phase-field simulations25.

Fig. 7: Simulating high-entropy dielectrics using the inhomogeneity model.
figure 7

a Random field magnitude distribution with increased configurational entropy. b P–E loops for different configuration entropy. ce Normalized compositional fluctuations Δc/c (c), spontaneous polarization Ps and normalized polarization fluctuations ΔP/P (d), and energy density and efficiency (e) as a function of configurational entropy.

If we further incorporate valence and radius fluctuations of doped sites, by calculating

$${\delta }_{{radius}}=\sqrt{{\sum}_{i}{c}_{i}^{0}{\left(1-\frac{{r}_{i}}{\bar{r}}\right)}^{2}}$$
(5)
$${\delta }_{{valence}}=\sqrt{{\sum}_{i}{c}_{i}^{0}{\left(1-\frac{{v}_{i}}{\bar{v}}\right)}^{2}}$$
(6)

where \({r}_{i}\), \(\bar{r}\), \({v}_{i}\) and \(\bar{v}\) are radius of each cation, average radius of all cations on the site, valence of each cation and average valence of all cations on the site, respectively, we rewrite the random field as

$${E}_{{{\rm{rand}}}}\left({{\bf{r}}}\right)={\left({k}_{3}{\delta }_{{radius}}+{k}_{4}{\delta }_{{valence}}\right)S}_{{{\rm{config}}}}^{0}{\sum}_{i=A,B,C}\nabla {c}_{i}$$
(7)

The two terms represent contributions from ion radius fluctuations and valence fluctuations. Employing this modified model, we re-conducted simulations for high-entropy (Bi,A,B,C,D)Ti3O12 given, e.g., (1) A = La, B = Pr, C = Nd, D = Sm, and (2) A = La, B = Pr, C = Ca, D = Ce. The results are shown in Supplementary Fig. 19. It is seen that case (1) has larger radius fluctuations and case (2) has larger valence fluctuations (Supplementary Fig. 19a, b). According to the \({k}_{3}\) and \({k}_{4}\) settings, valence fluctuations have a larger impact on random fields (which is reasonable as valence fluctuations directly impose local fields). Thus, the evolution of P–E loops is more rapid for case (2) (Supplementary Figs. 19c, d), which also causes an earlier peak in energy density for case (2) as shown in Supplementary Figs. 19e. This case of entropy-modulated dielectrics again demonstrates the effectiveness and applicability of our model incorporating local inhomogeneity.

Discussions

We systematically investigated the role of local inhomogeneity in enhancing energy storage performance using the PNR model and the polar slush model, both incorporating local inhomogeneity in different ways. The PNR model introduces regionally inhomogeneous polarization, while the polar slush model considers intense random fields. Despite the common observation that smaller domains improve energy storage, it is local inhomogeneity that plays a pivotal role. In PNR-based materials, electrostatic interactions between inhomogeneous regions enhance energy storage, with an optimal in-series PNR-matrix configuration. In polar slushes, random fields disrupt long-range ferroelectricity and induce polarization disorder, boosting energy storage. The two models share common foundations in terms of describing local inhomogeneity and correlating it to dielectric properties, manifesting the universal role of local inhomogeneity in high-performance RFEs.

To unify these models, we developed an approach directly incorporating local inhomogeneity, successfully simulating solid solutions and high-entropy dielectrics. This model straightforwardly emphasizes the central role of local inhomogeneity in forming PNRs and polar slushes, ultimately enhancing performance. We hope these insights will inspire better material design and contribute to the development of outperforming dielectrics.

Methods

Phase-field simulations

In the basic model, the polarization vector \({{\bf{P}}}=\left({P}_{1},{P}_{2},{P}_{3}\right)\) is chosen as the only order parameter in all the models31,58,59. The free energy distribution and domain structure are obtained by solving the time-dependent Ginzburg–Landau (TDGL) equation

$$\frac{\partial {{\bf{P}}}}{\partial t}=-L\frac{\partial F}{\partial {{\bf{P}}}}$$
(8)

where L is the kinetic coefficient, t is the time, and F is the free energy, which is expressed as the integral of the individual free energy densities:

$$F=\int\int\int \left({f}_{{{\rm{Landau}}}}+{f}_{{{\rm{gradient}}}}+{f}_{{{\rm{elastic}}}}+{f}_{{{\rm{electric}}}}\right){{\rm{d}}}V$$
(9)

The free energy terms are given by

$${f}_{{{\rm{Landau}}}}={\alpha }_{i}{P}_{i}^{2}+{\alpha }_{{ij}}{P}_{i}^{2}{P}_{j}^{2}+{\alpha }_{{ijk}}{P}_{i}^{2}{{P}_{j}^{2}P}_{k}^{2}+{\alpha }_{{ijkl}}{P}_{i}^{2}{{P}_{j}^{2}P}_{k}^{2}{P}_{l}^{2}$$
(10)
$${f}_{{{\rm{gradient}}}}={\frac{1}{2}G}_{11}{P}_{i,i}^{2}+\frac{1}{2}{G}_{44}{\left({P}_{i,j}+{P}_{j,i}\right)}^{2}+\frac{1}{2}{G}_{44}^{{\prime} }{\left({P}_{i,j}-{P}_{j,i}\right)}^{2}$$
(11)
$${f}_{{{\rm{elastic}}}}=\frac{1}{2}{C}_{{ijkl}}\left({\varepsilon }_{{ij}}-{\varepsilon }_{{ij}}^{0}\right)\left({\varepsilon }_{{kl}}-{\varepsilon }_{{kl}}^{0}\right)$$
(12)
$${f}_{{{\rm{electric}}}}=-\frac{1}{2}{K}_{{ij}}{\varepsilon }_{0}{E}_{i}{E}_{j}-{E}_{i}{P}_{i}$$
(13)

where \(\alpha\), G, C, \({\varepsilon }_{{ij}}\), \({\varepsilon }_{{ij}}^{0}\), K, \({\varepsilon }_{0}\), and E are the Landau coefficients, gradient coefficients, elastic stiffness, total strain, spontaneous strain, background dielectric constant, dielectric permittivity of vacuum, and local electric field, respectively. The spontaneous strain, which is also called the eigenstrain, is calculated using the electrostrictive coefficients Q as follows:

$${\varepsilon }_{{ii}}^{0}={Q}_{11}{P}_{i}^{2}+{Q}_{12}\left({P}_{j}^{2}+{P}_{k}^{2}\right)$$
(14)
$${\varepsilon }_{{ij}}^{0}={Q}_{44}{P}_{i}{P}_{j}$$
(15)

The local field can be obtained by calculating the gradient of the electric potential \({{\bf{E}}}=-\nabla \varphi\), which is solved using the Poisson equation:

$${K}_{{ij}}\frac{{\partial }^{2}\varphi }{\partial {x}_{i}\partial {x}_{j}}=\frac{\partial {P}_{i}}{{\varepsilon }_{0}\partial {x}_{i}}$$
(16)

In the PNR model, elliptic PNRs are randomly generated in a grid with axis a and b, and a rotation angle, until the desired domain concentration is reached. Except for the case with different aspect ratios, all PNRs have a = b and a rotation angle of zero. For PNRs and matrices, strictly speaking, \(\alpha\), C, Q and K should all be different; however, to accelerate the computation process, only regionally different Landau energy coefficients \(\alpha\) that are pivotal to dielectric characteristics are employed. Related coefficients are listed in Supplementary Table 160,61,62,63,64,65.

In the random field model, the electric energy is modified as

$${f}_{{{\rm{electric}}}}=-\frac{1}{2}{K}_{{ij}}{\varepsilon }_{0}{E}_{i}{E}_{j}-{E}_{i}{P}_{i}-{E}_{{{\rm{rand}}},i}{P}_{i}$$
(17)

where Erand is the random field66,67, which is generated as a structure parameter and does not change in iterations53. Erand follows

$${E}_{{{\rm{rand}}},i}\left({{\bf{r}}}\right) \sim N\left(0,{E}_{{{\rm{mag}}}}\right)$$
(18)

where N represents a normal distribution, and Emag is the standard deviation of the distribution (i.e., magnitude of a single component of random field). Related coefficients are also included in Supplementary Table 1.

In the developed inhomogeneity model, following the procedure in the main text, a Dirichlet distribution should be generated. \({k}_{1}=\frac{30}{R}\) is set for all cases where R is the ideal gas constant. Then, random field is obtained according to the gradient of ci and a customized coefficient k2, which is \(\frac{{5\times 10}^{6}{l}_{0}}{R}\) for BFO, \(\frac{{1\times 10}^{6}{l}_{0}}{R}\) for Bi4Ti3O12 and AFE. l0 is the length normalizer of the system (10-9 m). In the random field incorporating valence and radius fluctuations, we set \({k}_{3}=\frac{{2.8\times 10}^{8}{l}_{0}}{R}\) and \({k}_{4}=\frac{{1.1\times 10}^{9}{l}_{0}}{R}\), by estimating the magnitude of coefficient following

$${k}_{3} \sim \frac{e\times {10}^{-10}}{{a}^{3}{\varepsilon }_{0}{\varepsilon }_{r}}\frac{{l}_{0}}{R}$$
(19)
$${k}_{4} \sim \frac{e}{{a}^{2}{\varepsilon }_{0}{\varepsilon }_{r}}\frac{{l}_{0}}{R}$$
(20)

Expect for the case in Fig. 3c, the simulation grid has dimensions of \(128\Delta \times 128\Delta \times 1\Delta\) in the x, y and z directions, where \(\Delta \,=1\,{{\rm{nm}}}\) is the grid spacing. In the 3D simulations, a \(32\Delta \times 32\Delta \times 32\Delta\) grid is adopted. For elastic and TDGL equations, periodic boundary condition is used; for electric equations, short-circuit boundary condition is used. The initial polarization nuclei consisted of randomized noise within 0.001 C m−2 in magnitude. All PE loops and permittivity were calculated under the periodic sweeping of a [010] electric field. The kinetic coefficient L and time step dt are 0.1 and 0.01, respectively. For dynamical simulations, time step T of each cycle of the hysteresis loop satisfies \(T=\frac{1}{f\Delta t}\), where f is the frequency of interest. If \(f\Delta t\) is larger than 1, then enlarge \(\Delta t\) to satisfy that T is an integer.

The phase-field simulations were conducted using the commercial Mu-Pro package (http://mupro.co/).