Introduction

With the advent of the 5G information era, electronic devices are gradually evolving towards miniaturization, high precision, and high performance, leading to an increasingly prominent issue of heat dissipation in components such as chips. Traditional vapor compression refrigeration systems suffer from low efficiency and huge sizes, thus limiting the upgrade of electronic devices. The new generation of refrigeration technologies, represented by electrocaloric refrigeration, has garnered widespread attention from both the scientific and industrial communities due to their advantages of high efficiency, controllable size, and environmental friendliness1,2,3,4,5,6. However, the electrocaloric refrigeration materials based on PbTiO3 (PTO) and BaTiO3 (BTO) systems are mostly constrained by issues such as small adiabatic temperature change (ΔT) and narrow operating temperature ranges7,8,9,10. Research has shown that the operating temperature range and ΔT can be enhanced through doping and materials structure design11,12,13,14,15,16. In recent years, several research teams have proposed design concepts that combine positive and negative electrocaloric effects (ECE) to develop electrocaloric refrigeration materials with broad operating temperature ranges and huge ΔT17,18,19,20,21.

For the positive ECE in ferroelectric (FE) materials, there is a clear understanding and abundant experimental results. Current studies show that the ultra-high peaks of the positive ECE usually appear near the Curie temperature (TC), where a large external electric field leads to FE-paraelectric (PE) phase transition, accompanied by a large polarization entropy change22,23,24. However, in the AFE materials represented by the PbZrO3 (PZO), there is a negative ECE that distinguishes it from the positive ECE. This means that the AFE-FE phase transition caused by an electric field leads to a decrease in temperature. The cause of this anomalous physical phenomenon is still debated, and it has been suggested that it may be caused by a field-induced first-order endothermal transition between AFE and FE25 or antiparallel polarization instability26,27. The coexistence of AFE-FE phases and the evolution of the phase interface during the electric field-induced phase transition have been captured by high-resolution electron microscopy28,29. However, it is still difficult to observe the polarization switching process in field-induced AFE-FE phase transition in real time under high-temperature conditions. Therefore, studying the electric field-induced AFE phase nucleation and interface motion at the polarization scale has become the key to revealing the mechanism and improving the performance of the negative ECE.

Thermodynamic calculations have been employed to explain the negative ECE of PZO-based ceramics. Simulation results show that the change in the negative ECE effect is related to the free energy barrier of the AFE-FE phase transition30. However, effects of temperature, external electric field, and doping on the AFE-FE phase transition at the polarization scale still need further study. Phase-field simulations have emerged as powerful tools for the understanding of positive ECE/negative ECE at the polarization scale6,14,31,32,33,34,35. In this work, we investigate the AFE-FE phase transition domain switching process of prototype PZO bulk, including AFE stripe domain and polymorphic domain, AFE nanodomain produced by the PZO-based doping ceramics at different temperatures and electric field conditions via phase-field simulations. Calculate the negative ECE properties resulting from the phase transitions according to Maxwell relation. The results show that the polar boundaries generated by the local incommensurate antiphase boundaries, the AFE domain boundaries, and the AFE nanodomains, etc. provide nucleation sites for the AFE-FE phase transition process, which reduces the temperature and the electric field strength required for the phase transition. Polar boundaries realize a stepwise rather than transient phase transition process of AFE-FE. Indirect method calculations show that high-density polar boundaries can significantly increase the operating temperature range of the negative ECE, though it reduces the peak ΔT. The peak value of ΔT up to −13 K is realized in the temperature range of 400–450 K. A wide temperature range of about 75 K can be realized at 42 kV/cm. In addition, we further validate the conclusions in the PTO bulk system. The results show that the enhancement of the operating temperature range by the introduction of polar boundaries is also applicable in the positive ECE in FE materials.

Results

Antiferroelectric phase-field model

Firstly, to calculate the negative ECE of PZO-based AFE materials, we need to improve the current FE phase-field models. Xue et al. 36 have defined two order parameters of p and q to describe the antiparallel polarization in two sublattices in Sm-doped BiFeO3. Liu et al. 37 have constructed a high-order gradient energy-dependent model to describe the next-nearest-neighbor interaction between antiparallel polarizations. In this work, it is appropriate to simulate incommensurate AFE domain structures by using the second model. However, we still need to improve the phase-field model by introducing temperature and electric field factors quantitatively. In this work, AFE materials are taken as an example by solving the time-dependent Ginzburg-Landau (TDGL) equation for the temporal evolution of the polarization vector field38,39,

$$\frac{\partial {P}_{i}({\bf{r}},t)}{\partial t}=-L\frac{\delta F}{\delta {P}_{i}({\bf{r}},t)},(i=1,2,3)$$
(1)

where \({P}_{i}({\bf{r}},t)\) is polarization, L is the kinetic coefficient, and F is the total free energy of the system, which is expressed as,

$$F={\iiint} ({f}_{Land}+{f}_{Elas}+{f}_{Elec}+{f}_{Grad})dV$$
(2)

where V is the system volume. The gradient energy density can be expressed as37,

$$\begin{array}{c}{F}_{Grad}={\beta }_{11}\mathop{\sum}\limits_{i}{P}_{i}^{2}{P}_{i,i}^{2}+{\beta }_{12}\mathop{\sum}\limits_{i\ne j\ne k}{P}_{i}^{2}\left({P}_{j,k}^{2}+{P}_{k,j}^{2}\right)-{\gamma }_{11}\mathop{\sum}\limits_{i}{\theta }_{i}^{2}{P}_{i,i}^{2}-{\gamma }_{12}\mathop{\sum}\limits_{i\ne j\ne k}{\theta }_{i}^{2}({P}_{j,k}^{2}+{P}_{k,j}^{2})\\ +{\varphi }_{11}\mathop{\sum}\limits_{i}{\left(\frac{{\partial }^{2}{P}_{i}}{\partial {x}_{i}^{2}}\right)}^{2}+{\varphi }_{12}\mathop{\sum}\limits_{i\ne j}{\left(\frac{{\partial }^{2}{P}_{i}}{\partial {x}_{j}^{2}}\right)}^{2}\end{array}$$
(3)

θi is the oxygen tilt. β11 and β12 are the positive constants, which describe polar-polar interactions. γ11 and γ12 are the positive constants, which describe the coupling between oxygen tilt and polarizations. φ11 and φ12 are the positive constants of high-order gradient energy describing next-nearest-neighbor interaction between polarizations, which can drive the phase transition from AFE to FE. For the convergence of the numerical solution, we use Eq. (4) to normalize the gradient energy coefficients,

$${g}_{11}=\frac{{\gamma }_{11}\cdot {\theta }_{1}^{2}\cdot {P}_{0}}{{({a}_{c})}^{2}\cdot {a}_{0}}$$
(4)

where 0.757 C·m−2 for P0 and 5.54 × 107 J·m·C−2 for a0. The oxygen tilt can be correlated with temperature by Eq. (5),

$$\theta =\sqrt{-\frac{{k}_{0}}{{b}_{0}}(T-{T}_{\theta })}$$
(5)

Tθ is the transition temperature of the oxygen tilt, and k0 and b0 are the positive material constants. We can conclude that the first-order of the gradient energy is always smaller than zero in AFE materials40. Thus, Eq. (6) could be simplified as follows,

$${F}_{Grad}=-{\gamma }_{11}\mathop{\sum}\limits_{i}{\theta }_{i}^{2}{P}_{i,i}^{2}-{\gamma }_{12}\mathop{\sum}\limits_{i\ne j\ne k}{\theta }_{i}^{2}\left({P}_{j,k}^{2}+{P}_{k,j}^{2}\right)+{\varphi }_{11}\mathop{\sum}\limits_{i}{\left(\frac{{\partial }^{2}{P}_{i}}{\partial {x}_{i}^{2}}\right)}^{2}+{\varphi }_{12}\mathop{\sum}\limits_{i\ne j}{\left(\frac{{\partial }^{2}{P}_{i}}{\partial {x}_{j}^{2}}\right)}^{2}$$
(6)

then, a linear relation between temperature and coefficients of the first-order gradient energy can be obtained, such as γ11θi2 (i = 1, 3) = γ11m(T - Tθ), where m = k0/b0. Moreover, we have fitted the linear relation between temperature and coefficients of the first-order gradient energy in PZO materials. In this work, we use γ11θi2(i = 1, 3) = 8.64 × ac2 × 107 J·m·C−2 for 480 K and 7.78 × ac2 × 107 J·m·C−2 for 450 K, ac = 0.5 × 10−9 m.

Mechanism of polar boundaries enhancing negative ECE

To clearly explain the mechanism of polar boundaries modulating the negative ECE performance of PZO-based AFE materials, a schematic diagram is drawn, as shown in Fig. 1. Figure 1c shows the stripe domains of the prototype PZO, in which the polarization strictly follows the 2 × 2 antiparallel configurations. The interface contains only typical antiphase boundaries. Under an applied electric field, the polarizations possessing a component opposite to the direction of the electric field are gradually destabilized and switched. Upon reaching the phase transition point, the intermediate state hardly stabilizes and the whole system rapidly transforms to a FE single domain, which corresponds to the jumping point of the polarization-temperature (P-T) curve in Fig. 1a, the huge ΔT and the narrow temperature range in adiabatic temperature change-temperature (ΔT-T) curve in Fig. 1b. Figure 1d corresponds to the bulk ceramic domain structures, polymorphic domain, where multiple Orthorhombic-phase (O-phase) variants and local incommensurate antiphase boundaries (shown by white lines and white arrows) appear due to the defect dipoles or other lattice distortions. Under the applied electric field, the AFE-FE phase transition will preferentially occur at the local incommensurate antiphase boundaries and the AFE domain boundaries. Boundary transition process occurs, which broads the negative ECE operating temperature range. Based on this, our work further increases the inhomogeneity of the AFE domain structure by using elements doping which leads to random rise and fall of the thermodynamic potentials. The decrease in nanodomain size is accompanied by a very high density of polar boundaries. Under the same strength of the external electric field, the AFE domains are made to undergo a local transition at lower temperatures, so that the P-T curves cross the phase transition point in the form of a gradual rather than a sudden change, which broads the working temperature range, as shown in Fig. 1e. Moreover, four different types of polar boundaries have been found in PZO-based AFE materials via phase-field methods, as shown in Supplementary Fig. 1.

Fig. 1: Schematic diagrams of polar boundaries modulating the negative ECE properties of PZO-based AFE materials.
figure 1

a P-T curve, b ΔT-T curve. The phase transition paths of: c stripe domains, d polymorphic domains, e AFE nanodomains under the applied electric field. (AFE phase polarization, black arrows. FE phase polarization, red arrows. Polar boundaries, white arrows, and lines.).

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In this work, the temperature-electric field revision was applied to the PZO AFE phase-field model to simulate the temperature-electric field-induced AFE-FE phase transition phase diagram of the prototype PZO polymorphic domains, as shown in Fig. 2a. The temperature ranges from 298 K to 510 K, and the electric field strength ranges from 0 to 84 kV/cm. As the same as the experimentally obtained phase diagram of the prototype PZO41, the temperature-induced AFE-FE phase transition region is not found without electric field applied. Further increase in temperature will induce the direct transition from AFE into PE. Besides, the stabilized existence of the mixed phase near the phase boundary between AFE (Fig. 2b) and FE (Fig. 2d) has been found. The simulation results indicate that there exists a transition region in which the two phases stably coexist in the AFE-FE phase transition (Fig. 2c). This phenomenon is hardly observed in the AFE stripe domains40. Therefore, we believe that the appearance of AFE domain boundaries in polymorphic domains and local incommensurate antiphase boundaries are responsible for the generation of the local FE phase at conditions below the phase transition temperature or electric field.

Fig. 2: Temperature-electric field phase diagram of PZO polymorphic domain.
figure 2

a Phase diagram of prototype PZO polymorphic domain as a function of temperature-electric field, b AFE phase domain, c Mixed phase domain, d FE phase domain. (Domain structures below the phase diagram are the local results in the 256Δx × 256Δy × 1Δz. Green points are AFE-FE phase transition points from PZO ceramics experiments25).

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Next, P-E loops under different temperatures and electric field strengths have been simulated in Supplementary Fig. 2. To verify that polar boundaries promote the stability of mixed phases, we select the temperature and electric field conditions where the mixed phase could stably exist, and simulate the dynamic domain switching process. As shown in Fig. 3a, a P-E loop is obtained by a gradual electric field applied up to 42 kV/cm to the stable polymorphic domains at 474 K. The corresponding domain structures have been output at four representative moments. Before point A, the polarization and electric field exhibit a basic linear relationship because the average polarization change at this stage is mainly caused by the gradual switching of polarizations with antiparallel components under the applied electric field. The polarization increased significantly at point A, and phase-field simulations indicate that local nucleation sites of AFE-FE phase transition appear near the AFE domain boundaries, and yet most areas remain in the AFE phase, as shown in Fig. 3b. Point B, where the slope of the P-E curve is the steepest, is usually considered as the phase transition point. At this moment, the already nucleated FE phase regions near the AFE domain boundaries exhibit significant expansion, and preferential phase transition also occurs at the local incommensurate antiphase boundaries (2 × 3) in Fig. 3c. At point C, the whole domain stabilizes in the FE phase (Fig. 3d), and a significant hysteresis phenomenon has been found. As the electric field continues to decrease and approach zero, nucleation sites of the AFE phase in the FE begin to occur at point D, with reverse phase transformation also occurring more likely at the FE domain walls (Fig. 3e). The field-induced phase transition and domain-switching process in the polymorphic mixed phase of PZO demonstrate that the introduction of polar boundaries provides nucleation sites for phase transition. This gradual diffusion rather than instantaneous phase transition feature may promote the increase in the operating temperature range of negative ECE. We have reason to believe that to obtain AFE materials with a wider negative ECE operating temperature range, it is necessary to introduce inhomogeneous domain structures, namely higher-density polar boundaries, through doping or other means.

Fig. 3: P-E loop and corresponding domain structures of PZO.
figure 3

a P-E loop of prototype PZO polymorphic domains under 474 K temperature with 42 kV/cm electric field strength is applied. Domain structures: b FE phase nucleation at AFE domain boundaries at point A, c AFE-FE phase transition at point B, d FE phase at point C, e AFE phase nucleation at FE domain boundaries at point D. (local magnifications are in the red lines.).

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Therefore, in this work, we use the spatial rise and fall of the potential function to approximate the distribution of inhomogeneous composition due to doping in experiments. The randomly rising and falling potential function distribution in the phase-field model implies that the domain structures with lower TC possess lower AFE-FE phase transition barriers and can undergo AFE-FE phase transition at lower temperatures and electric field strengths. As shown in Fig. 4a, the P-T curves of the prototype PZO stripe domains (Fig. 4b, c), polymorphic domains (Fig. 4d), and doping-induced AFE nanodomains (Fig. 4e) are compared. Domain structures at the characteristic temperatures are selected for the analysis. The P-T curve slope of the AFE-nanodomains is much lower than that of the stripe domains and polymorphic domains, and the AFE-FE phase transition has already begun to occur at 453 K. Typical AFE-nanodomains appear at D point, under 468 K, 42 kV/cm, most of the domain structures have completed the AFE-FE phase transition, but there still exists a certain percentage of nanodomains with extremely high phase transition barriers due to the inhomogeneity of doping, which results in their isolation and stabilization in AFE phases, in Fig. 4e.

Fig. 4: P-T curves and corresponding domain structures of PZO.
figure 4

a P-T curves of PZO AFE stripe domains, polymorphic domains, and AFE nanodomains at 42 kV/cm electric field strength, b transient domain structure at point A of phase transition of the stripe domains, c domain structure of the stripe domains after the phase transition at point B (stripe domain are the local results in the 32Δx×32Δy×1Δz), d mixed phase domain structure of the polymorphic domains during the phase transition at point C, e mixed phase domain structure of the AFE nanodomains during the phase transition at point D. (local magnifications are in the red lines.).

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Negative ECE calculations

Finally, we simulate the P-T curves of polymorphic domains and AFE nanodomains at different external electric field strengths in Fig. 5a, c. The ΔT-T curves of them are obtained by the indirect method based on the Maxwell relation, as shown in Fig. 5b, d. Entropy change (ΔS) has been simulated in Supplementary Fig. 3. The polymorphic domains possess a high negative ΔT at the low electric field. As the electric field strength is increased from 21 kV/cm to 42 kV/cm, the peak ΔT is increased from −5.77 K to −9.58 K in Fig. 5b. However, the negative ECE strength shows a decreasing trend from 2.75 to 2.28 K·m·MV−1 in Fig. 5e. Meanwhile, the operating temperature range stays around 460–480 K. The maximum ΔT of the AFE nanodomains is somewhat decreased than that of the polymorphic domains. The peak ΔT is enhanced from −1.40 K to −13.05 K, corresponding to the electric field strength from 21 kV/cm to 84 kV/cm, accompanied by the negative ECE strength from 0.67 to 1.55 K·m·MV−1, as shown in Fig. 5d, e. In addition, the widths of the operating temperature ranges are all more significantly improved than those of the polymorphic domains. Especially under the electric field of 42.0 kV/cm, a wide width of about 75 K (402–477 K) operating temperature region has been realized with a ΔT larger than 0.1 K, as shown in Fig. 5e. In Fig. 5f, we compare the results of phase-field simulations with negative ECE strength and operating temperature range of PZO-based ceramics measured in experiments. AFE nanodomains have advantages in both negative ECE strength and operating temperature range.

Fig. 5: Negative ECE properties of PZO-based AFE materials by phase-field simulations and Maxwell relation.
figure 5

P-T curves: a polymorphic domains, c AFE nanodomains. ΔT-T curves: b polymorphic domains, d AFE nanodomains. e The maximum negative ECE strength versus operating temperature range, f the comparison of electrocaloric performance |ΔT/ΔE | - |ΔT| for PZO-based polymorphic domain and nanodomain in this work, including (Pb0.97La0.02)(Zr0.66Sn0.27Ti0.07)O3 single crystals46, (Pb0.97La0.02)(ZrxSn0.94-xTi0.06)O3 ceramics30, (Pb0.97La0.02)(Zr0.95Ti0.05)O3 ceramics47, PbMg0.5W0.5O3 ceramics17, Pb0.99Nb0.02(Zr0.85Sn0.13Ti0.02)0.98O3 ceramics24, PbZrO3 ceramics25.

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Discussion

In this work, we have simulated the most representative prototype PZO AFE materials by phase-field method and constructed a temperature-electric field phase diagram. Unlike previous studies, the incommensurate AFE is shown to be stabilized near TC42 in Fig. 6b. Although room temperature up to ~450 K remains a 2 × 2 commensurate AFE phase, as shown in Fig. 6a. Above TC, the domain exhibits a PE phase structure, as shown in Fig. 6c. Theoretically, after considering the AFE-FE-PE phase transition beyond TC, the negative and positive ECE can be realized successively in PZO-based AFE materials. In our subsequent studies, the coupled temperature-electric field AFE phase-field model can provide theoretical support for the design of ECE cooling materials combining positive and negative ECE.

Fig. 6: Temperature-induced phase transition of prototype PZO domain structures by phase-field simulations.
figure 6

a commensurate AFE, b incommensurate AFE, c PE.

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Moreover, the mechanism by which polar boundaries modulate the operating temperature range of ECE cooling materials can also be verified in FE materials. The PTO system is used in this work to briefly analyze the positive ECE properties. The P-T curves have been obtained by applying an electric field of 100 kV/cm to the PTO bulk domains and nanodomains, respectively, in the temperature range of 300–700 °C, as shown in Fig. 7a. For PTO nanodomains, 10% volume fraction of defect dipoles have been introduced into the simulations. Based on the initial state obtained from the phase-field simulation, it can be found that the PTO nanodomains not only possess many small domains in the in-plane Tetragonal phase (T-phase), but also defects, doping, and other factors-induced polar nanoregions in Fig. 7c. The nanodomains introduce a high density of polar boundaries, which promotes the domain switching of the materials at lower temperatures at B point (Fig. 7e), effectively increasing the operating temperature range of the positive ECE. In contrast, PTO bulk domains have almost no polar boundaries and require higher temperatures to realize domain switching, as shown in Fig. 7b. However, in the magnified image of the domain structure at point A, before switching, the polarization preferred switching near the 180° domain walls of the T-phase can still be found in Fig. 7d. This also indirectly indicates that the polar boundaries are not only the preferential nucleation sites for the AFE-FE phase transition but also easily become the sites for the domain switching in the FE phase. Besides, we further simulate effect of the concentration of defect dipoles on P-T curve, as shown in Supplementary Fig. 4. Results show that defect dipoles increase from 0 to 10%, the peak polarization of the PTO bulk appears at lower temperature. It indicates that the system transforms from FE to relaxor-ferroelectric, which broadens the operating temperature range of the ECE.

Fig. 7: P-T curves and corresponding domain structures of PTO.
figure 7

a P-T curves of PTO bulk domains and nanodomains under temperature 300 ~ 700 °C, b bulk domain structure at initial state, c nanodomain at initial state, d bulk domain at A point e nanodomain at B point under 100 kV/cm and 400 °C. (Local magnified polarizations are in the red lines).

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In short, we refine the temperature-electric field phase-field model for PZO-based AFE materials. It is found that the AFE-FE phase transition occurs preferentially at polar boundaries and transforms the transient phase transition into a form of regional nucleation and diffusion. Moreover, modulation of polar boundaries’ density can be achieved in the polymorphic and AFE nanodomains. We have found that polar boundaries promote the operating temperature range of negative ECE. Finally, a wide width of about 75 K (402–477 K) operating temperature range with maximum ΔT = −4.57 K has been realized under the electric field of 42.0 kV/cm. Based on our theoretical analysis, the operating temperature range of the ECE can be improved by introducing polar boundaries for AFE/FE materials. We envision this work will attract broad interest from the community of ECE materials.

Methods

Thermodynamic methods

To verify the accuracy of the PZO potential function of the coupled temperature term, we performed a three-dimensional free energy surface plotting of the Landau free energy only without applying external stresses and electric fields, as shown in Supplementary Fig. 5. Thermodynamic free energy density (ΔGpolar) is as follows15,

$$\begin{array}{rcl} {\Delta}G_{polor}&=& \alpha_{1} (P_{1}^{2}+P_{2}^{2}+P_{3}^{2})+\alpha_{11} (P_{1}^{4}+P_{2}^{4}+P_{3}^{4})+\alpha_{12}(P_{1}^{2}P_{2}^{2}+P_{1}^{2}P_{3}^{2}+P_{2}^{2}P_{3}^{2})\\ &+& \alpha_{112}\left[P_{1}^{4}(P_{2}^{2}+P_{3}^{2})+P_{2}^{4}(P_{1}^{2}+P_{3}^{2})+P_{3}^{4}(P_{1}^{2}+P_{2}^{2})\right]\\ &-&\alpha_{111}(P_{1}^{6}+P_{2}^{6}+P_{3}^{6})+\alpha_{123}P_{1}^{2}P_{2}^{2}P_{3}^{2}\\ &-&\frac{1}{2}s_{11}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}\right)-\frac{1}{2}s_{44}\left(\sigma_{4}^{2}+\sigma_{5}^{2}+\sigma_{6}^{2}\right)-s_{12}\left(\sigma_{1}\sigma_{2}+\sigma_{1}\sigma_{3}+\sigma_{2}\sigma_{3}\right)\\ &-&Q_{11}\left(\sigma_{1}P_{1}^{2}+\sigma_{2}P_{2}^{2}+\sigma_{3}P_{3}^{2}\right)-Q_{44}\left(\sigma_{2}P_{2}P_{3}+\sigma_{5}P_{1}P_{3}+\sigma_{6}P_{1}P_{2}\right)\\ &-&\mu_{1}\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)\left(T-T_{0}\right)\\ &-&\lambda_{eff}\left[\left(P_{1}^{2}-P_{2}^{2}\right)P_{3}\sigma_{6}+\left(P_{3}^{2}-P_{1}^{2}\right)P_{2}\sigma_{5}+\left(P_{2}^{2}-P_{3}^{2}\right)P_{1}\sigma_{4}\right]\\ &-&\tau_{eff}\left[\left(\sigma_{1}-\sigma_{2}\right)P_{3}\sigma_{6}+\left(\sigma_{3}-\sigma_{1}\right)P_{2}\sigma_{5}+\left(\sigma_{2}-\sigma_{3}\right)P_{1}\sigma_{4}\right]\\ &-&E_{1}P_{1}-E_{2}P_{2}-E_{3}P_{3} \end{array}$$
(7)

where P1, P2, P3 are polarization components. α1, α11, α12, α111, α112 and α123 are Landau coefficients. s11, s12 and s44 are elastic compliance. σi (i = 1, 6) are the stress, λeff and τeff represent the high-order electromechanical term. Ei is the applied electric field. μ1 is the linear thermal expansion coefficient. Q11, Q12 and Q44 are the electrostrictive tensors. α1 is given by the Curie-Weis law,

$${\alpha }_{1}={\alpha }_{0}\left(T-{T}_{C}\right)/(2{\varepsilon }_{0}C)$$
(8)

where T is the temperature and TC is the Curie temperature, ε0 is the vacuum permittivity and C is the Curie constant. In this work, TC of prototype PZO is set as 510 K41.

For simplification, we have not considered the stress and electric field applied in the thermodynamic models. The results show that PZO is stabilized in the O-phase at room temperature, while the Rhombohedral-phase is slightly higher in energy than the O-phase, and the T-phase has the highest energy. Above the TC, PZO polarization should be isotropic and become Cubic-phase. However, the thermodynamic potential function of the polarization is not completed to describe the antiparallel configuration of the polarization. Therefore, it is necessary to introduce a higher-order gradient energy term through the phase-field model to construct the energy for the antiparallel configuration of the AFE polarization.

Phase-field model

The Landau free energy density fLand can be calculated by,

$$\begin{array}{l}{f}_{Land}={\alpha }_{1}({P}_{1}^{2}+{P}_{2}^{2}+{P}_{3}^{2})+{\alpha }_{11}({P}_{1}^{4}+{P}_{2}^{4}+{P}_{3}^{4})+{\alpha }_{12}({P}_{1}^{2}{P}_{2}^{2}+{P}_{1}^{2}{P}_{3}^{2}+{P}_{2}^{2}{P}_{3}^{2})\\ \qquad\qquad+\,{\alpha }_{112}\left[{P}_{1}^{4}({P}_{2}^{2}+{P}_{3}^{2})+{P}_{2}^{4}({P}_{1}^{2}+{P}_{3}^{2})+{P}_{3}^{4}({P}_{1}^{2}+{P}_{2}^{2})\right]\\ \qquad\qquad+\,{\alpha }_{111}({P}_{1}^{6}+{P}_{2}^{6}+{P}_{3}^{6})+{\alpha }_{123}{P}_{1}^{2}{P}_{2}^{2}{P}_{3}^{2}\end{array}$$
(9)

however, to simulate the spatial rise and fall caused by elements doping, α1(r) is related to the location. TC(r) has a spatial rise and fall of plus or minus 70 K.

$${\alpha }_{1}({\bf{r}})={\alpha }_{0}(T-{T}_{C}({\bf{r}}))/(2{\varepsilon }_{0}C)$$
(10)

where ε0 is the vacuum permittivity and C is the Curie constant. The elastic energy density can be expressed as,

$${f}_{Elas}=\frac{1}{2}{C}_{ijkl}({\varepsilon }_{ij}-{\varepsilon }_{ij}^{0})({\varepsilon }_{kl}-{\varepsilon }_{kl}^{0})$$
(11)

where Cijkl is the elastic stiffness tensor, \({\varepsilon }_{ij}\) and \({\varepsilon }_{ij}^{0}\) are the total local strain, and the spontaneous strain, respectively. The electrostatic energy density fElec is given by,

$${f}_{Elec}=-{P}_{i}{E}_{i}-\frac{1}{2}{\varepsilon }_{0}{\kappa }_{ij}{E}_{i}{E}_{j}$$
(12)

where E is the total electric field, which can be described as, E = Eappl + Edip + Erf, where Eappl is the applied electric field, Edip is the dipole-dipole interaction field, and Erf is the local electric field caused by the random defect dipoles, which has been used in the PTO nanodomain simulations43. κij are the relative dielectric constants. The electric displacement satisfied the electrostatic equilibrium equation44:

$$\frac{\partial {D}_{i}}{\partial {x}_{i}}={\varepsilon }_{0}\frac{\partial \left({\kappa }_{ij}{E}_{j}\right)}{\partial {x}_{i}}+\frac{\partial {P}_{i}}{\partial {x}_{i}}=0$$
(13)

where Di (i = 1, 3) is the electric displacement.

The equation was solved by a semi-implicit Fourier spectral method and the simulation size is 256Δx × 256Δy × 1Δz. (Δx is the number of grid points and equals 0.5 nm in this work) To obtain AFE stripe domain without polar boundaries, we use 32Δx × 32Δy × 1Δz in the phase-field model. The mechanical boundary conditions of PZO-based ceramics are periodic. We use Landau coefficients of the Pb(Zr1-xTix)O3 (x ≤ 0.1) system modified for the calculation45. α0 = 5.54 × 107 J·m·C−2, α11 = 5.60 × 108 J·m5·C−4, α12 = 2.89 × 108 J·m5·C−4, α111 = 1.65 × 109 J·m9·C−6, α112 = − 8.66 × 108 J·m9·C−6, α123 = 3.19×1010 J·m9·C−6, C11 = 15.6 × 1010 N·m−2, C12 = 9.6×1010 N·m−2, C44 = 12.7 × 1010 N·m−2, Q11 = 0.048 m4·C−2, Q12 = −0.015 m4·C−2, Q44 = 0.047 m4·C−2.

Calculations of the electrocaloric effect

In this work, the isothermal entropy change (ΔS) and ΔT of PZO is calculated by the indirect method with Maxwell relation for the negative ECE as follows,

$$\Delta S=-\frac{1}{\rho }{{\int }_{{E}_{1}}^{{E}_{2}}\left(\frac{\partial P}{\partial T}\right)}_{E}dE$$
(14)
$$\Delta T=-\frac{T}{{C}_{p}\rho }{\int }_{{E}_{1}}^{{E}_{2}}{\left(\frac{\partial P}{\partial T}\right)}_{E}dE$$
(15)

The density ρ and heat capacity Cp of PZO used in the work are 8.2 g·m−3 and 330 J·kg−1·K−1 46.