Introduction

Electrostatic capacitors-based dielectrics are ubiquitous components in modern electronic devices owing to their high power density1,2,3,4,5,6,7,8. As power electronics converter technology toward high frequency and miniaturization, the need for capacitors has greatly increased in applications at elevated temperatures, such as oil and gas extraction (170~250 °C), electrified aircraft (180–300 °C), and space exploration where environmental temperatures can reach 400 °C9,10,11,12,13,14,15. Unfortunately, the upper working temperature of the best commercially available high-temperature dielectric capacitors (biaxially oriented polypropylene) is <~105 °C, far from the critical requirement of 100–400 °C service by modern industrial development16. Given existing limitations, the development of next-generation dielectric capacitors that have high-energy storage characteristics and stable performance over a broad temperature range is crucial for a variety of industries.

The discharged-energy density Ue of a dielectric is defined as \({U}_{{\rm {e}}}={\int }_{{P}_{{\rm {r}}}}^{{P}_{{\rm {m}}}}E{\rm {d}}p\), where Pm and Pr are the maximum polarization and remnant polarization, respectively (Supplementary Fig. 1)17. From the formula and P–E loop, large Pr not only limits the achievable Ue but also generates significant energy loss Uloss and degrades the energy storage efficiency η6. Massive energy loss will transform into Joule heat and lead to irreversible thermal breakdown of the device9. Hence, integration of high Pm, low Pr, and large breakdown strength Eb, is highly desired to realize a high energy storage density and efficiency at high temperature. However, leakage current and conduction loss significantly increase at elevated temperatures and highly applied electric fields and cause a sharp deteriorating energy storage performance and lifetime15,18.

Lead-based ferroelectric thin films with high curie temperatures Tc are attractive for high-temperature applications, but they generally suffer from substantially reduced Eb, decreased Ue, and η at high temperatures, even if the temperature is below Tc11. Furthermore, the toxic nature of lead and environmental policy limitations are another concern for their practical usage. Recently, lead-free relaxor ferroelectric (RFE) films have been considered to be the best potential stocks for high-temperature dielectrics capacitors among dielectric materials due to their high electric-field-induced polarization, dramatically reduced Pr, and more importantly, a dispersed dielectric constant over a broad temperature range6. Considerable efforts have been devoted to improving the energy storage performance of RFEs through designing the domain structure3,6,19, defects types4,20, strain and interface state of the film21,22,23,24,25, or selecting suitable material to construct composite dielectrics10,26. However, the current results have shown very limited success. A major challenge is the Joule heat arising from conduction loss that causes thermal breakdown and capacitor failure8,9. Until now, this challenge has still not been effectively addressed.

We departed from the traditional high-temperature dielectric capacitors design strategy by focusing on metadielectrics (MDs) for superior energy storage properties and exceptional thermal stability. The basis of this approach is the strong anisotropic thermal conductivity properties in MDs (Supplementary Fig. 3), which can efficiently dissipate Joule heat energy and avoid thermal runaway27,28. Furthermore, electric and magnetic multipolar Mie-type resonances theory shows that MD nanostructures exhibit low dielectric loss, which can greatly reduce Joule heat generation29. The integration of high thermal conductivity and low dielectric loss is a benefit for high-temperature energy storage capacitors. The MDs are an emerging new composite material designed and manufactured artificially with unexpected properties30,31. Till now, however, MDs for high-temperature energy storage applications are still unexplored.

Results

To demonstrate the effectiveness of the MD design for improving high-temperature energy storage performance, we first conducted phase-field simulations (as described in the “Methods” section) to study the polarization response and dielectric breakdown process at high temperatures. We choose RFE BaHf0.17Ti0.83O3 as the matrix because of their stable dielectric constant in a wide temperature range (Supplementary Fig. 4)32, and HfO2 with a large band gap (5.3–5.7 eV) as the second phase to construct composite MDs33. We design three representative composite nanostructures (Fig. 1a): single-phase dielectric, 0–3 composite dielectric (The first number indicates the connectivity of the active phase while the second number indicates the connectivity of the passive second phase.)34 and the MD. The simulated dynamic evolution of the breakdown path at 573 K is shown in Fig. 1a. It is found that tortuous paths (black region) in 0–3 composite dielectric is more than that in single-phase dielectric. The tortuous paths start to split and diffuse in the MD, making the breakdown strength of the MD film increase to 2.6 times that of the single-phase dielectric. On the basis of ultrahigh voltage resistance of the MD, we simulated corresponding P–E loops and local polarization distribution at the maximal acceptable applied electric field, as shown in Fig. 1b and Supplementary Fig. 5. It can be noted that the 0–3 composite dielectric shows slimmer P–E loop with small Pr, compared with visible hysteresis loss and high Pr in single-phase dielectric. The P–E loop of the MD film is slimmed to a notable degree, resulting in a minimal hysteresis. Thus, high energy storage density (area of shaded part) and efficiency can be achieved simultaneously in the MD. The great improvement of energy storage performance at high-temperature benefits from the blocking effect of the ordered second phases on delaying and hindering the propagation of the breakdown path35. The blocking effect could be further understood from the electron avalanche theory and heat dissipation, as shown in Fig. 1c. The dielectric breakdown is typically induced by the avalanche multiplication of free charge carriers at a high electric field or high temperature since sufficient kinetic energy acquired by charge carriers will give a high probability of collision and ionization. However, for the MD, the initial impact ionization could be delayed and the chain reaction is restricted in the x–y plane due to insufficient kinetic energy and high insulation perpendicular to the direction of electrostatic force. Thus, the electron avalanche would be restricted to form more tortuous breakdown paths. At the same time, the high thermal conductivity and low electrical conductivity of the HfO2-branch reduce the Joule heating and act as the exit for thermal runway, which is beneficial for high-temperature stability. From an energy perspective, the electrostatic energy and Joule heat energy under high electric field and high temperature can be accounted for the performance improvement of the MD design (Fig. 1d, Supplementary Fig. 6). For single-phase dielectric, the distributions of electrostatic energy and Joule heat energy are located at high-value position with high peak intensity. In 0–3 composite dielectric, two peak positions shift toward lower values, and their peak intensities weaken. While for the MD, the intensity of peaks is smallest and the peaks become diffuse, which is beneficial for high breakdown field strength at high temperatures. Based on the above discussions, the phase-field simulations show that the MD design can greatly enhance energy storage performance at high temperatures.

Fig. 1: Phase field simulations of the metadielectric(MD).
figure 1

a Breakdown phase evolutions of three dielectric structures at 573 K. b Corresponding polarization–electric field hysteresis loops of three dielectric structures under their breakdown electric fields. c Illustrations of possible electron avalanche and heat source modes of the MD at high temperatures. d The statistical distributions of local electric energy and local Joule energy of three dielectric structures under 3 MV·m−1 and 573 K.

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To realize MD design, a series of BaHf0.17Ti0.83O3(BHTO)–xHfO2 composite films (x = 0, 0.06, 0.12, 0.18, 0.25, 0.32 and 0.38) with thicknesses of ~300 nm were grown on (001) Nb-doped SrTiO3 (Nb:STO) substrates by using radio frequency magnetron sputtering. A BHTO single-phase film was obtained for x = 0, validated by HAADF-STEM image (Fig. 2a) and X-ray diffraction (Fig. 2h). Energy-dispersive X-ray spectroscopy (EDS) mapping shows homogeneous distributions of chemical element components for x = 0 (Fig. 2f). Secondary phases HfO2 appeared (Fig. 2h) as nanoprecipitates with a high density and uniform intragranular distribution for x ≥ 0.18, bright particles in Fig. 2b. Combined the HAADF-STEM image (Fig. 2c) and EDS mapping (Fig. 2g and Supplementary Fig. 9), it can be concluded the desired film structure for the MD is realized as x increased to 0.25. The shape of the phase-separated metadielectric be explained in Supplementary Fig. 10. Further increasing x to 0.32, the sample evolves into 0–3 composite dielectric (Fig. 2d and h), and for x = 0.38 amorphous BHTO (A-BHTO) and HfO2 phase coexist in the film (Fig. 2h, Supplementary Fig. 11). The evolution of the structure in the BHTO–xHfO2 films is summarized in Fig. 2i.

Fig. 2: Structure and characterization.
figure 2

a–e HAADF-STEM image of BHTO–xHfO2 (x = 0, 0.18, 0.25, 0.32 and 0.38). Energy-dispersive X-ray spectroscopy (EDS) mapping of BHTO–xHfO2 with x = 0 (f) and 0.25 (g). h Enlarged XRD patterns of the BHTO–xHfO2 films. i Schematic illustration of the evolution of the structure in BHTO-xHfO2 films.

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As expected, the breakdown strength is highly determined by the designed microstructure of the composited film. It can be seen from Fig. 3a and Fig. 3b, the breakdown strength determined by a two-parameter Weibull statistical analysis first increases as x and then decreases, and goes to the maximum value 12.4 MV·cm−1 at x = 0.25, namely MD structure, which is much higher than that of the traditional BaTiO3 films (~0.79 MV·cm−1)36. Furthermore, the β value also increases from 17 for the film with x = 0 to 32 for the MD film, indicating the MD film has better reliability. The superior insulation properties of the MD film over the single-phase BHTO dielectric stems from its substantially reduced leakage current, as shown in Fig. 3c. For example, at a DC field of 4.66 MV·cm−1, a current density of 1.3 × 10−7 A·cm−2 is found for the MD film, which is two orders of magnitude lower than that of the single-phase BHTO film (1.7 × 10−5 A·cm−2). The analysis of the conductive mechanism reveals that the MD structure can greatly prolong the ohmic conduction from 1.99 to 4.30 MV·cm−1 indicated by the red arrows in Fig. 3c, and significantly delays the witch-on of the Fowler–Nordheim tunneling mechanism at high fields indicated by the blue arrows in Supplementary Fig. 12c. Bipolar P–E hysteresis loops of the films measured at an electric field of 4 MV·cm−1 and 1 kHz are shown in Fig. 3d. The single-phase (x = 0) film possesses the largest Pm and Pr with remarkable hysteresis, but the MD film (x = 0.25) exhibits much narrowed P–E loop and substantial reduction of Pr. What’s more important, the ratio Pm/Pr obtained from the maximum applied electric field goes to the maximum value at the MD structure (Fig. 3e), indicating that the energy storage density and efficiency will be improved. Further analyzing the P–E curves, we can find that the MD structure can greatly enhance the proportion of the linear dielectric response to the energy storage density (Uln/Ue) as shown in Supplementary Fig. 13, which induces the energy storage efficiency \(\eta\) increases from 80.3% (Single-phase film) to 90.3% (MD film). The energy storage performance of the BHTO–xHfO2 films with the applied electric field are summarized in Fig. 3f and Supplementary Fig. 14 derived from their unipolar P–E loops (Supplementary Fig. 15). It is noted that the energy storage density Ue of the MD film with x = 0.25 is ~177 J·cm−3, which is 119% higher than that of the single-phase film (x = 0). A significant enhancement of \(\eta\) > 83% was also achieved at 11 MV·cm−1 in the MD film due to the enhanced relaxation behavior

Fig. 3: Dielectric breakdown and energy storage performance.
figure 3

a Weibull distribution analysis of the breakdown strength Eb and Weibull parameter β. b Comparison of the Eb of the BHTO-xHfO2 films. c Logarithmic plot of leakage current density versus electric field. d P–E loops at an electric field of 4.0 MV·cm−1 and 1 kHz. e The Pm, Pr, and Pm/Pr are determined from ten points. The error bars in e present the standard deviation of the corresponding data. f Energy density and efficiency at the electric fields up to Eb.

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(Supplementary Figs. 16 and 17) and lowered domain reversal potential (Supplementary Figs. 18 and 19)3. Such superior behavior (Ue of 177 J·cm−3 and η of 83%) of the MD film is even better than that of the best-reported lead-based RFE films (Ue of 133 J·cm−3 and η of 75%)4, indicating that our MD strategy has a great advantage in achieving high performance of energy storage.

To evaluate the effect of the MD structure on thermal stability, we investigated the temperature-dependent energy density and efficiency over the temperature range from –150 to 400 °C. As shown in Fig. 4a, the energy storage efficiency of traditional single-phase film (x = 0) drops sharply above 100 °C with enormous hysteresis loss (Supplementary Fig. 20), and only <60% remains at 400 °C even at a low electric field of 3 MV·cm−1 (Supplementary Fig. 21). Compared to single-phase film, the 0–3 composite films (x = 0.18 and 0.32) show relative higher energy density, but the energy storage efficiency also degrade above 100 °C. However, the variation of energy storage density and efficiency of MD film is less than 5% and 8%, respectively, indicating the energy storage properties are stable in the tested temperature range. More importantly, a high energy density of 85 J·cm−3 with an efficiency exceeding 81% is achieved in MD film, which is 300% higher than that of the single-phase film with x = 0. The excellent thermal stability implies that the MD film is directly applicable in the extreme-environment. The ultrahigh temperature stability of our MD film stems from three aspects: one is the highest Fowler–Nordheim barrier height (ϕB), as shown in Fig. 4b and Supplementary Fig. 23, derived from their leakage current densities at various temperatures (Supplementary Fig. 22). It is noted that the ϕB (1.92 eV) of the MD film is much higher than that of the 0–3 composite films (1.62, 1.73 eV) and the single phase (1.28 eV) at room temperature. The ϕB of the MD film is still as high as 1.33 eV, even at 200 °C, which is higher than the ϕB of a single phase at room temperature. The high barrier height will produce a strong barrier against charge injection assisted by thermal and electrical energy9. The second one is the temperature and electric field insensitive dielectric constant further suppresses dielectric loss at high temperatures4. As shown in Fig. 4c, the maximum change of the normalized dielectric constant of the MD film (x = 0.25) is as low as 1.1 over the temperature range of 25–400 °C. The corresponding value of the single-phase film (x = 0) reaches 1.656 in the same condition, which is 1.5 times as high as that of the MD film. This suggested the MD film (x = 0.25) exhibits a suppression of the extrinsic temperature and electric field contributions to the dielectric constant over an ultra-wide temperature range. The third one is the suppressed conduction loss and ferroelectric loss. To clearly visualize the suppression of energy loss for the MD structure at high temperatures, we calculated the conduction loss and ferroelectric loss (Fig. 4d) of the samples at 400 °C under the same electric field of 6 MV·cm−1 from the P–E loop (Supplementary Fig. 20). The conduction loss of the MD film (x = 0.25) is as low as 1.53%, that is much <25.8% of the single-phase film (x = 0). The MD film also shows the lowest ferroelectric loss (embedded image).

Fig. 4: Reliability and cycling stability.
figure 4

a Temperature-dependent energy storage performance. b Temperature-dependent Fowler–Nordheim barrier height. c AC-field (From 0 to 2 MV·cm−1) and temperature (from 25 to 400 °C) dependence of dielectric constant. All data are normalized to the dielectric constant. d Conduction loss and ferroelectric losses at 400 °C. The error bars in d present the standard deviation of the corresponding data. e Charging–discharging reliability of different films at 400 °C. f Comparison of the energy storage performances of our films with representative dielectrics.

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Cycling reliability and lifetime are other critical prerequisites for dielectric capacitors. The fatigue behaviors of the films are tested at 400 °C. It can be seen from Fig. 4e derived from Supplementary Fig. 24 that the energy storage efficiency decreases sharply for the single-phase (x = 0) and 0–3 composite films (x = 0.18 and 0.32) after 2 × 104 cycles, and the single-phase film becomes breakdown after 2.15 × 105 cycles. However, the MD film (x = 0.25) survives over 1 × 106 cycles, and the degradations of Ue and \(\eta\) is <2%, which meets the commercial standard. At the same time, the MD film can discharge the storage energy ultra-fast in ~4 μs and induce a power density is larger than 100 MW·cm−3 (Supplementary Fig. 25). The superior energy storage and lifetime over a wide temperature range from −150 to 400 °C can meet almost all the urgent need for extreme conditions from the low temperature at the South Pole –90 °C to extremely high-temperature circumstances, for example, oil and gas extraction and space explore, and it is much better than the current-state-of-the-art high-temperature lead-based4,24,37,38,39 and lead-free6,7,10,19,23,36,40,41,42,43,44,45,46 dielectric capacitors, as shown in Fig. 4f.

Discussion

In summary, we proposed the metadielectrics strategy to solve the long-standing problem of capacitors with severe deterioration of electrical and dielectric properties at high temperatures and realize thermal-stable thin film capacitors at ultra-high temperatures. The designed metadielectrics capacitors can effectively block the charge carriers from transporting in the film and make the breakdown strength go up to 12.4 MV·cm−1. Besides, the strong anisotropic thermal conductivity properties in MDs can aid the thermal energy to release and reduce the energy loss, making the up-limited working temperature go up to 400 °C and exhibit excellent thermal stability in the temperature range from −150 to 400 °C, which are expected to be used directly in extreme environments without the need for additional cooling devices. The proposed approach should be applicable to many other material systems, especially composite dielectric materials.

Methods

Phase-field simulation (dielectric breakdown simulations)

In this work, based on the phase field theory, an electro-thermal breakdown model was developed to dynamically simulate the dielectric breakdown process. Here, a continuous phase field variable η (r, t) is introduced to describe the breakdown path in dielectrics, where η (r, t) = 1 represents the breakdown phase and η (r, t) = 0 represents the non-breakdown phase.

The free energy in an inhomogeneous system contributes from the phase separation fsep, the gradient energy fgrad, the electrostatic energy density felec, and the Joule heating energy fJoule, which can be written as

$$F=\int\limits_{\!\!\!{V}} \left[\,{f}_{{{{\rm{sep}}}}} \left(\eta \left({{{\bf{r}}}}\right)\right)+{f}_{{{{\rm{grad}}}}} \left(\eta \left({{{\bf{r}}}}\right)\right)+{f}_{{{{\rm{elec}}}}} \left(\eta \left({{{\bf{r}}}}\right)\right)+{f}_{{{{\rm{Joule}}}}} \left(\eta \left({{{\bf{r}}}}\right)\right)\right]{\rm {d}}V$$
(1)

In this simulation, a double-well function is used to describe the phase separation energy as follows:

$${f}_{{{{\rm{sep}}}}}(\eta ({{{\bf{r}}}}))=\alpha {\eta }^{2}{(1-\eta )}^{2}$$
(2)

where α is a positive coefficient describing the energy barrier of the phase separation with a value of 108 J/m3 in this work. The gradient energy is expressed as

$${f}_{{{{\rm{grad}}}}} \left(\eta ({{{\bf{r}}}})\right)=\frac{1}{2}\gamma {\left|\nabla \eta ({{{\bf{r}}}})\right|}^{2}$$
(3)

where γ is the gradient energy coefficient with the value of 10−10 J/m. The electrostatic energy can be calculated by

$${f}_{{{{\rm{elec}}}}}\left(\eta ({{{\bf{r}}}})\right)=\frac{1}{2}{\varepsilon }_{0}{\varepsilon }_{ij}({{{\bf{r}}}}){E}_{i}({{{\bf{r}}}}){E}_{j}({{{\bf{r}}}})$$
(4)

where ε0 is the vacuum dielectric constant, εij is the relative dielectric constant, and Ei(r) is the electric field. The local electric field is determined by solving Ohm’s law \(\nabla J=\nabla (\sigma E)=0\) due to the relatively high electrical conductivity at high temperatures.

The Joule heating energy is expressed by

$${f}_{{{{\rm{Joule}}}}} \left(\eta ({{{\bf{r}}}})\right)=\sigma ({{{\bf{r}}}}){E}_{i}({{{\bf{r}}}}){E}_{j}({{{\bf{r}}}}){\rm {d}}t$$
(5)

where σ(r) is the electrical conductivity tensor, and dt is the operating period of the applied electric field. In order to describe the matrix and the second phase in the nanocomposite, a non-evolving field variable ρ(r) is introduced, which takes the value of 1 in the second phase and 0 in the matrix. Then, the spatially dependent permittivity and electrical conductivity could be written as

$${\varepsilon }_{ij}({{{\bf{r}}}})= {\eta }^{3} \left(10-15\eta+6{\eta }^{2}\right){\varepsilon }_{ij}^{{{{\rm{B}}}}}+\left[1-{\eta }^{3}\left(10-15\eta+6{\eta }^{2}\right)\right] \\ \times \left\{{\rho }^{3} \left(10-15\rho+6{\rho }^{2}\right){\varepsilon }_{ij}^{{{{\rm{S}}}}}+\left[1-{\rho }^{3} \left(10-15\rho+6{\rho }^{2}\right)\right]{\varepsilon }_{ij}^{{{{\rm{M}}}}}\right\}$$
(6)
$${\sigma }_{ij}({{{\bf{r}}}})= {\eta }^{3} \left(10-15\eta+6{\eta }^{2}\right){\sigma }_{ij}^{{{{\rm{B}}}}}+\left[1-{\eta }^{3} \left(10-15\eta+6{\eta }^{2}\right)\right] \\ \times \left\{{\rho }^{3} \left(10-15\rho+6{\rho }^{2}\right) {\sigma }_{ij}^{{{{\rm{S}}}}}+\left[1-{\rho }^{3} \left(10-15\rho+6{\rho }^{2}\right)\right]{\sigma }_{ij}^{{{{\rm{M}}}}}\right\}$$
(7)

where \({\varepsilon }_{ij}^{{{{\rm{B}}}}}\), \({\varepsilon }_{ij}^{{{{\rm{S}}}}}\) and \({\varepsilon }_{ij}^{{{{\rm{M}}}}}\) indicate the permittivity of the breakdown phase, second phase and the matrix, \({{\sigma }}_{ij}^{{{{\rm{B}}}}}\), \({{\sigma }}_{ij}^{{{{\rm{S}}}}}\) and \({{\sigma }}_{ij}^{{{{\rm{M}}}}}\) indicate the electrical conductivity of the breakdown phase, second phase and the matrix, respectively.

A modified Allen–Cahn equation is developed to govern the dynamic process of dielectric breakdown,

$$\frac{\partial \eta ({{{\bf{r}}}},t)}{\partial t}= -{L}_{0}H \left({f}_{{{{\rm{elec}}}}}+{f}_{{{{\rm{Joule}}}}}-{f}_{{{{\rm{critical}}}}}\right)\\ \left[\frac{\partial {f}_{{{{\rm{sep}}}}}(\eta )}{\partial \eta ({{{\bf{r}}}},t)}-\gamma {\nabla }^{2}\eta ({{{\bf{r}}}},t)+\frac{\partial {f}_{{{{\rm{elec}}}}}({{{\bf{r}}}})}{\partial \eta ({{{\bf{r}}}},t)}+\frac{\partial {f}_{{{{\rm{Joule}}}}}({{{\bf{r}}}})}{\partial \eta ({{{\bf{r}}}},t)}\right]$$
(8)

where L0 is the kinetic coefficient determining the interface mobility, H(felec + fJoule−fcritical) is the Heaviside unit step function H((felec + fJoule)< fcritical)= 0 and H((felec + fJoule) > fcritical)= 1), and fcritical is the maximal energy density calculated by \({f}_{{{{\rm{critical}}}}}=1/2{\varepsilon }_{0}{\varepsilon }_{{{{\rm{r}}}}}{E}_{{{{\rm{b}}}}}^{2}+{\sigma }_{{{{\rm{r}}}}}{E}_{{{{\rm{b}}}}}^{2}{{{\rm{d}}}}t\). The purpose of introducing the Heaviside function into the Allen–Cahn equation is to ensure that the breakdown phase can grow only if the electric energy of a local point is greater than its maximal energy endurance. The detailed parameters for each material are listed in supplementary Information.

Phase-field simulation (polarization response simulations)

Here, the spontaneous polarization Pi(r, t) is selected to describe the temporal evolution of the polarization field and the domain structure to solve the three-dimensional time-dependent Ginzburg–Landau equation in the phase field simulation,

$$\frac{\partial {P}_{i}({{{\bf{r}}}},t)}{\partial t}=-L\frac{\delta F}{\delta {P}_{i}({{{\bf{r}}}},t)}+{\xi }_{i}({{{\bf{r}}}},t)$$
(9)

where L is the kinetic coefficient related to the domain wall migration rate, F is the total energy, r is the spatial position, t is time, Pi(r, t) is the polarization intensity at a certain space position and a certain time, ξi(r, t) is the impact of thermal noise, which conforms to a random Gaussian distribution.

The total energy F includes the Landau bulk free-energy density, gradient energy density, elastic energy density, and electrostatic energy density, as follows,

$$F=\int\limits_{\!\!\!{V}} \left[\,{f}_{{{{\rm{bulk}}}}}+{f}_{{{{\rm{grad}}}}}+{f}_{{{{\rm{elas}}}}}+{f}_{{{{\rm{elec}}}}}\right]{{{\rm{d}}}}V$$
(10)

where fbulk, fgrad, felas and felec represent the Landau bulk free-energy density, gradient energy density, elastic energy density and electrostatic energy density, respectively. The bulk energy density fbulk can be described as a six-order polynomial in terms of polarization, which is expressed as

$${f}_{{{{\rm{bulk}}}}}= {\alpha }_{1}({P}_{1}^{2}+{P}_{2}^{2}+{P}_{3}^{2})+{\alpha }_{11}({P}_{1}^{4}+{P}_{2}^{4}+{P}_{3}^{4})\\ +{\alpha }_{12}({P}_{1}^{2}{P}_{2}^{2}+{P}_{1}^{2}{P}_{3}^{2}+{P}_{2}^{2}{P}_{3}^{2})+{\alpha }_{112}[{P}_{1}^{4}({P}_{2}^{2}+{P}_{3}^{2})\\ +{P}_{2}^{4}({P}_{1}^{2}+{P}_{3}^{2})+{P}_{3}^{4}({P}_{1}^{2}+{P}_{2}^{2})]+{\alpha }_{111}({P}_{1}^{6}+{P}_{2}^{6}+{P}_{3}^{6})\\ +{\alpha }_{123}{P}_{1}^{2}{P}_{2}^{2}{P}_{3}^{2}$$
(11)

where P1, P2, and P3 are polarization components. α1, α11, α12, α111, α112 and α123 are Landau energy coefficients.

Owing to the contribution of domain walls, the gradient energy density fgrad is expressed as follows:

$${f}_{{{{\rm{grad}}}}}= \frac{1}{2}{G}_{11}({P}_{1,1}^{2}+{P}_{2,2}^{2}+{P}_{3,3}^{2})+{G}_{12}({P}_{1,1}{P}_{2,2}+{P}_{2,2}{P}_{3,3}+{P}_{1,1}{P}_{3,3}) \\ +\frac{1}{2}{G}_{44} \left[{({P}_{1,2}+{P}_{2,1})}^{2}+{({P}_{2,3}+{P}_{3,2})}^{2}+{({P}_{1,3}+{P}_{3,1})}^{2}\right] \\ +\frac{1}{2}{G}_{44} ^{\prime} \left[{({P}_{1,2}-{P}_{2,1})}^{2}+{({P}_{2,3}-{P}_{3,2})}^{2}+{({P}_{1,3}-{P}_{3,1})}^{2}\right]$$
(12)

where Gij is gradient energy coefficient, and Pi, j is \(\frac{\partial {P}_{{i}}}{\partial {r}_{{j}}}\).

The elastic energy density felas has the following expression:

$${f}_{{{{\rm{elas}}}}}=\frac{1}{2}{C}_{ijkl}{e}_{ij}{e}_{ikl}=\frac{1}{2}{C}_{ijkl}({\varepsilon }_{ij}-{\varepsilon }_{ij}^{0})({\varepsilon }_{kl}-{\varepsilon }_{kl}^{0})$$
(13)

where Cijkl is the elastic stiffness tensor, eij is the elastic strain, εij is the total strain and \({\varepsilon }_{ij}^{0}\) is electrostrictive stress-free strain.

For a given domain structure, the electrostatic energy density felec consists of applied external electric field and electric field, which is formulated as follows:

$${f}_{{{{\rm{elec}}}}}=-{P}_{i}{E}_{ij}^{{{{\rm{ex}}}}}-\frac{1}{2}{E}_{ij}^{{{{\rm{in}}}}}{P}_{i}$$
(14)

where \({E}_{ij}^{{{{\rm{ex}}}}}\) is the applied external electric field, \({E}_{ij}^{{{{\rm{in}}}}}\) is the electric field induced by the dipole moment in the sample. The detailed parameters for each material are listed in supplementary Information.

Phase-field simulation (thermal conduction simulations)

Here, to investigate the strong thermal conductivity anisotropy of metadielectric, the heat transport process is described by the steady-state heat conduction equation

$$\kappa \left({{{\bf{r}}}}\right) {\nabla }^{2}T \left({{{\bf{r}}}},{t}\right)=0$$
(15)

where κ and T are the thermal conductivity and temperature, respectively. When solving Eq. (15), the temperature difference is set at both ends without considering the heat exchange with the ambient environment. The detailed parameters for each material are listed in supplementary Information.

Sample preparation

BaHf0.17Ti0.83O3–xHfO2(BHTO–xHfO2) ceramic targets with x = 0, 0.06, 0.12, 0.18, 0.25, 0.32, and 0.38 were sintered by a conventional solid-state reaction method23. High-purity BaCO3, TiO2, and HfO2 powders were mixed at designed stoichiometric ratios and then ball-milled for 4 h in ethano. After drying, the mixture was calcined at 850 °C for 4 h. The as-synthesized powders were further ball milled to reduce the particle size. Alcohol solution was added to the dried powder as a binder, and the powder was pressed into pellets with diameters of 50 mm at 15 MPa for 15 min. The pellets were heated to 550 °C for burning out the binder and then sintered at 1050 °C for 2 h in a sealed alumina crucible.

Before formal growth, the targets were sanded, cleaned, and pre-sputtered to ensure that the target surface had reached steady state prior to growth. 300 nm BHTO-xHfO2 films were grown on 0.7% Nb-doped SrTiO3 (001) single-crystalline substrates via the radio-frequency magnetron sputtering system. The BHTO–xHfO2 films were grown at 700 °C under a total Ar/O2 mixture pressure of 0.2 mbar with the mixed ratio of 1:1. Following the growth, the films were annealed at 700 °C for 15 min under the Ar/O2 mixture pressure of 200 mbar, then cooled down to room temperature at 10 °C·min−1.

Characterizations

The phase and crystal structure of the films were acquired with a high-resolution X-ray diffractometer (HRXRD; PANalytical X’Pert MRD) with Cu Kα radiation (λ = 1.540598 Å). HAADF-STEM imaging and EDS mapping were acquired on a scanning transmission electron microscope (JEOL ARM 200F), equipped with a probe aberration corrector, operated at 200 keV. For measuring the dielectric and ferroelectric properties, the samples are patterned with 50-nm-thick Pt square top electrodes with 200 μm sides. The structural configuration of the film and electrodes is shown in Supplementary Fig. 2. Measurements of the dielectric spectra over broad frequency (102–106 Hz) and temperature ranges (25–500 °C) were conducted on a precision impedance analyzer (HP-4980A, Agilent) with a Quatro-Cryosystem temperature control system. Polarization–electric field hysteresis loops (P-E loop) tests were performed by using a ferroelectric tester (TF2000 analyzer; axi ACCT, Aachen, Germany) and the high-voltage amplifier is Trek PA05035. The maximum output voltage is 400 V. Lakeshore cryocooled probe station (CRX-6.5K, Lake Shore Cryotronics, Inc., USA) provides different temperature environments for the P–E test. The leakage currents were measured with a delay time of 2 s using an Agilent B2901A precision source. The cyclic charge–discharge tests were carried out in a PK-CPE1901 test system (PolyK Technologies). The first-order reversal curve (FORC) loops were recorded by using biased half-triangular electric field pulses of Emax = 3 MV/cm and ∆Er = ∆E = 0.1 MV/cm, and 60 FORC loops were measured with 40 Hz. For the fatigue behavior for charge–discharge cycling test, successive bipolar triangular voltage waves were applied on the films; after that, a unipolar field was applied to characterize the P–E loops and energy storage performance.

Reporting summary

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