## Abstract

Barium zirconate titanate [Ba(Zr_{x}Ti_{1−x})O_{3}] solid solutions are promising lead-free ferroelectric materials that have received substantial interest. Thermodynamic analysis based on phenomenological Landau–Devonshire theory is a powerful method for theoretical investigation of ferroelectric materials, but cannot be applied to Ba(Zr_{x}Ti_{1−x})O_{3} because there is no thermodynamic potential. In this paper, a thermodynamic potential for Ba(Zr_{x}Ti_{1−x})O_{3} (0 ≤ *x* ≤ 0.3) solid solutions is constructed, and then a thermodynamic analysis carried out. The results accurately reproduce known phase structures and their transition temperatures, with good agreement with experimentally measured polarization, dielectric, and piezoelectric constants. It is found that Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions at room temperature have three phase boundaries, including a tetragonal–orthorhombic phase boundary at *x* = 0.013, an orthorhombic–rhombohedral phase boundary at *x* = 0.0798, and a rhombohedral–paraelectric phase boundary at *x* = 0.2135. The results also indicate that the chemical composition-induced ferroelectric–paraelectric phase boundary has superior electromechanical properties, suggesting a new way to enhance electromechanical coupling in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions.

## Introduction

Ferroelectrics with switchable polarization, excellent dielectric, piezoelectric, and pyroelectric properties have a wide range of applications in memories, capacitors, actuators, sensors, and electrocaloric cooling devices.^{1,2,3,4,5,6,7,8,9,10,11,12} As a typical ferroelectric material, barium titanate (BaTiO_{3}) possesses a rich variety of phase transitions.^{13} Above its Curie temperature (*T*_{C} ~ 120 °C), the BaTiO_{3} single crystal is paraelectric phase with cubic symmetry.^{13} When the BaTiO_{3} single crystal is cooled through Curie temperature, it undergoes a paraelectric to ferroelectric phase transition, and has a tetragonal phase structure with polarization along <100> direction.^{13} When the cooling is continued, the BaTiO_{3} single crystal exhibits a tetragonal ferroelectric to orthorhombic ferroelectric phase transition at about *T*_{T↔O} ~ 0 °C with polarization along <110> direction.^{13} Upon further cooling, the BaTiO_{3} single crystal shows an orthorhombic ferroelectric to rhombohedral ferroelectric phase transition around *T*_{O↔R} ~ −80 °C with polarization along <111> direction.^{13} Thus, BaTiO_{3} has three types of phase transitions, including one ferroelectric–paraelectric (FE-PE) phase transition and two ferroelectric–ferroelectric (FE-FE) phase transitions. The corresponding phase transition temperatures can be tuned by the substitution of Ti^{4+} with Zr^{4+},^{14,15} which forms barium zirconate titanate [Ba(Zr_{x}Ti_{1−x})O_{3}] solid solutions. In Ba(Zr_{x}Ti_{1−x})O_{3} system, the FE-FE phase transition temperatures *T*_{T↔O} and *T*_{O↔R} increase with the increase in composition of Zr, whereas the FE-PE phase transition temperature *T*_{C} decreases with increasing composition *x*, and all three phase transitions are pinched into a single phase transition around *x* = 0.15,^{15,16} suggesting a pinched phase transition feature. With further increase in composition *x*, Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions only have FE-PE phase transitions between rhombohedral and cubic phase structures, and the corresponding phase transition temperature *T*_{C} decreases.^{15}

Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions have been studied widely on the experimental aspects. For example, the phase structures of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions at different temperatures were investigated by Yu^{16,17}et al., and the phase transition temperatures were identified for the low-composition range (*x* = 0, 0.05, 0.08, 0.1, 0.15, 0.2, 0.25, 0.3),^{13,16,17,18,19,20,21,22} revealing that a pinched phase transition occurs near the composition *x* = 0.15.^{15,16} Also, previous experimental observations displayed that Ba(Zr_{x}Ti_{1−x})O_{3} with composition *x* = 0.08 has a coexistence of orthorhombic and rhombohedral phases at a temperature of 30 °C, and possesses an enhancement of piezoelectric response near this polymorphic phase boundary.^{22} In addition, Yu et al. studied the dielectric properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, and found that Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions have stronger dielectric response than that of pure BaTiO_{3}.^{21} Recently, Qian et al. discovered that Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions show a giant electrocaloric effect over a broad temperature range.^{23} Thus, the previous experimental results indicate that Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions have great potential applications. However, the phase diagram and electromechanical properties with continuous variation in composition have not been accomplished.

Although the experimental advances in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions are encouraging, the corresponding theoretical studies are rare. Thermodynamic analysis based on phenomenological Landau–Devonshire theory is a powerful method for the theoretical investigation of ferroelectric materials.^{3,7,24,25,26,27,28,29,30,31,32,33,34,35} In order to implement thermodynamic analysis, the thermodynamic potential of a particular ferroelectric system should be established first. Haun et al. have established the thermodynamic potential for lead zirconate titanate (PZT) solid solution based on its phase structures and electromechanical properties in single crystals and bulk ceramics.^{36} Similarly, Bell and Cross^{37} constructed a sixth-order thermodynamic potential for BaTiO_{3}, which was revised as an eighth-order thermodynamic potential by Li et al.^{26} After that, Wang et al.^{38} reported another eighth-order thermodynamic potential, in which the second and higher-order coefficients are dependent on temperature. In order to consider the quantum mechanical effects at low temperature, Wang et al.^{39} proposed a modified thermodynamic potential for BaTiO_{3}, based on which the calculated phase transitions in temperature–pressure phase diagram show excellent agreement with experimental data. With the established thermodynamic potentials, the thermodynamic analysis on PZT solid solution and BaTiO_{3} have been implemented widely for explaining and predicting new physical phenomena.^{40,41,42} On the other hand, the known thermodynamic potential is a necessary condition for simulating ferroelectric domain structures via the conventional phase-field method.^{43,44,45,46} Thus, thermodynamic potential plays an important role in the theoretical analysis of ferroelectrics. However, the thermodynamic potential of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions is unestablished and lacking. This is also one reason for rare theoretical studies on Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. Thus, we seek to construct thermodynamic potential for Ba(Zr_{x}Ti_{1−x})O_{3} (0 ≤ *x* ≤ 0.3) solid solutions, and then employ it to study and predict the phase diagram as well as electromechanical properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions via theoretical analysis.

## Results and Discussion

### Constructing thermodynamic potential

Thermodynamic potential is the key factor for the thermodynamic theory of ferroelectrics, which is lacking for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. Thus, we seek to construct it for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. Based on the phenomenological Landau–Devonshire theory, the thermodynamic potential always uses polarization as an order parameter to describe the phase transitions in ferroelectrics.^{36} With the free energy of paraelectric phase at Curie temperature as a reference, the thermodynamic potential energy of a ferroelectric crystal under isothermal condition can be expanded in powers of polarization *P*_{i}(*i* = 1,2,3) and stress *σ*_{I}(*I* = 1,2,...,6),^{47,48,49}

in which *E*_{i} is the electric field, *Q*_{IJ}, *S*_{IJ} are the electrostrictive coefficient and elastic compliance constant under constant polarization in Voigt notation, respectively. It should be noted that *α*_{1}, *α*_{ij}, *α*_{ijk}, and *α*_{ijkl} are the thermodynamic potential coefficients, all of which are unknown for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. Estimating the values of these coefficients can then construct the thermodynamic potential of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions.

According to the thermodynamic theory, phase structures and electromechanical properties of ferroelectrics can be calculated using thermodynamic analysis from thermodynamic potential.^{36,49} The polarization of ferroelectrics can be determined by minimizing thermodynamic potential, that is, by solving the derivatives of thermodynamic potential with respect to polarization components for zeros,

The phase structures are classified by the solutions of the equilibrium polarization states, which include four sets of solutions corresponding to four different phase structures: (i) the paraelectric (PE) phase with *P*_{1} = *P*_{2} = *P*_{3} = 0; (ii) the tetragonal (T) phase with *P*_{1} = *P*_{2} = 0,*P*_{3} ≠ 0 (or *P*_{1} = *P*_{3} = 0, *P*_{2} ≠ 0; *P*_{2} = *P*_{3} = 0, *P*_{1} ≠ 0), (iii) the orthorhombic (O) phase with *P*_{1} = 0,|*P*_{2}| = |*P*_{3}| ≠ 0 (or *P*_{2} = 0,|*P*_{1}| = |*P*_{3}| ≠ 0; *P*_{3} = 0,|*P*_{1}| = |*P*_{2}| ≠ 0); and (iv) the rhombohedral (R) phase with |*P*_{1}| = |*P*_{2}| = |*P*_{3}| ≠ 0. With the determination of polarization, the relative dielectric constants can be evaluated by

with *ε*_{0} being the permittivity of free space and

The piezoelectric coefficients *d*_{in} (*n* = 1,2,...,6)

where the strain *ε*_{n} can be obtained by the relationship

The above derivation of thermodynamic analysis reveals that phase structures and electromechanical properties of ferroelectrics are correlated with thermodynamic potential directly. As such, the inverse determination of thermodynamic potential coefficients of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions can be estimated via fitting the available experimental data of phase structures, phase transition temperatures, polarization, and dielectric constants reported in literatures.^{13,16,17,18,19,20,21,22,26,50,51,52,53,54,55,56,57} The experimental data used for fitting are listed in Supplementary Table S1. The details of determining thermodynamic potential coefficients for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions are given in section METHODS and schematically illustrated in Supplementary Fig. S1. A similar procedure has been applied successfully to construct the thermodynamic potentials for (K_{1−x}Na_{x})NbO_{3} and (Ba_{1−x}Sr_{x})TiO_{3}.^{58,59} The determined thermodynamic potential coefficients of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions are tabulated in Table 1.

It is noticed that electromechanical coupling terms are also included in the thermodynamic potential energy [Eq. (1)], and can be determined if elastic compliance constant *S*_{IJ} and electrostrictive coefficient *Q*_{IJ} are known. Since the elastic compliance constants and electrostrictive coefficients data of Ba(Zr_{x}Ti_{1−x})O_{3} are unavailable in literature, both of them are evaluated from a linear interpolation of the corresponding data of BaTiO_{3} and BaZrO_{3}, given by \(S_{IJ} = (1 - x)S_{IJ}^{{\mathrm{BaTiO}}_{\mathrm{3}}} + xS_{IJ}^{{\mathrm{BaZrO}}_{\mathrm{3}}}\) and \(Q_{IJ} = (1 - x)Q_{IJ}^{{\mathrm{BaTiO}}_{\mathrm{3}}} + xQ_{IJ}^{{\mathrm{BaZrO}}_{\mathrm{3}}}\). A similar method was adopted by Huang et al. to obtain the electrostrictive coefficients of (Ba_{1−x}Sr_{x})TiO_{3}.^{59} Notice that \(S_{11}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = 8.3 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(S_{12}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = - 2.7 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(S_{44}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = 9.24 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(S_{11}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = 3.43 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(S_{12}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = - 0.675 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(S_{44}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = 14.29 \times 10^{ - 12}{\mathrm{m}}^{\mathrm{2}}{\mathrm{/N}}\), \(Q_{11}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = 0.1{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\), \(Q_{12}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = - 0.034{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\), and \(Q_{44}^{{\mathrm{BaTiO}}_{\mathrm{3}}} = 0.029{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\).^{29,60,61} Due to the lack of available data, we adopt the similar method described by Wang et al.,^{62} and obtain the electrostrictive coefficients of BaZrO_{3} (\(Q_{11}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = 0.096{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\), \(Q_{12}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = - 0.039{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\), and \(Q_{44}^{{\mathrm{BaZrO}}_{\mathrm{3}}} = 0.031{\mathrm{m}}^{\mathrm{4}}{\mathrm{/C}}^{\mathrm{2}}\)) using combined first-principles calculations and Berry’s phase approach (details in the Supporting Information). Since experimental data of electrostrictive coefficients of BaZrO_{3} are not available, we compare the hydrostatic electrostrictive coefficient *Q*_{h} = *Q*_{11} + 2*Q*_{12} between the calculation and experiment, and good agreement is observed, as details in the Supporting Information. As such, the thermodynamic potential of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions has been constructed, as tabulated in Table 1. The known thermodynamic potential allows us to carry out thermodynamic analysis for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, and also makes it possible to realize phase-field simulations for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. At low temperature, phase transitions and electromechanical properties of ferroelectrics are significantly affected by low-temperature quantum mechanical effects,^{39,63} which is not considered in this work, due to the lack of corresponding low-temperature experimental data of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions for fitting thermodynamic potential coefficients.

### Thermodynamic analysis

We now adopt the constructed thermodynamic potential to predict the phase structures and electromechanical properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, including the variations of properties in both specific compositions and continuous variations of compositions, which are also compared with some reported experimental data for verifying the reliability of the established thermodynamic potential.

### Phase structures and electromechanical properties in specific compositions

We first examine phase structures and electromechanical properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions in specific compositions, such as Ba(Zr_{x}Ti_{1−x})O_{3} with composition *x* = 0.05, 0.1, 0.15, 0.2, 0.25, and 0.3. The free energy of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions under stress-free boundary condition is plotted in Supplementary Fig. S3, according to which the energy minimizing phase structures are identified. Supplementary Fig. S3(a) indicates that Ba(Zr_{x}Ti_{1−x})O_{3} with the composition *x* = 0.05 undergoes three phase transitions, and has a rhombohedral phase, an orthorhombic phase, a tetragonal phase, and a paraelectric phase in order of increasing temperature. A similar characteristic has been observed in Ba(Zr_{x}Ti_{1−x})O_{3} with the composition *x* = 0.1 [Supplementary Fig. S3(b)], while the temperature range of rhombohedral phase extends, and the temperature ranges for orthorhombic and tetragonal phases shrink. For the composition *x* = 0.15 [Supplementary Fig. S3(c)], all the phases have the same energy at *T*_{C}, suggesting a quadruple point in Ba(Zr_{x}Ti_{1−x})O_{3}, which has been confirmed by experimental observations.^{15,16} For the composition *x* = 0.2, 0.25, 0.3 below the Curie temperature *T*_{C} [Supplementary Figs. S3(d)–(f)], the rhombohedral phase has the lowest energy though the contrast between the three ferroelectric phases is not obvious. The phase transition temperatures can also be identified, which are marked in Supplementary Fig. S3. We compare phase transition temperatures between calculations and experiments^{13,16,17,18,19,20,21,22,26,50} in Supplementary Table S2, and a good agreement between them is observed, suggesting that the constructed thermodynamic potential reproduces phase transitions in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions correctly.

Based on the constructed thermodynamic potential, we also calculate the electromechanical properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions in specific compositions as a function of temperature, as shown in Figs. 1–3, in which some available experimental data from literatures are referred for comparison. Figure 1 reveals that the modulus of polarization decreases with an increase in temperature as expected, while the polarization components show discontinuous variations across phase boundaries, suggesting Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions having first-order phase transition. At the same temperature, the magnitude of polarization decreases with an increase in composition *x*. The experimental data for polarization of Ba(Zr_{x}Ti_{1−x})O_{3} single-crystalline at some compositions is unavailable. For this case, we evaluate the single-crystalline polarization values *P*_{3} by using its counterpart ceramic values \(P_3^c\) measured in experiments for comparison. The used relationships between them are \(P_3 = 3P_3^c\) in tetragonal phase and \(P_3 = \sqrt 2 P_3^c\) in rhombohedral phase^{64} (details in the Supporting Information). The values of polarization calculated based on thermodynamic theory are consistent with those measured in experiments,^{16,20,51,52,53,54} as evidenced in Fig. 1. The dielectric and piezoelectric responses show peaks at phase transition temperatures, as seen in Figs. 2 and 3, especially a high electromechanical response existing at ferroelectric–paraelectric phase transition temperature, implying that superior electromechanical properties always appear near phase boundaries. The experimental data of *ε*_{33}^{19,21,55,56,57} and *d*_{33}^{17,20,22,55} are also plotted in Figs. 2 and 3, showing that our thermodynamic calculation results agree with experimental data. For some composition *x*, the obvious differences between the calculation values and experimental measurements can be observed, since the counterpart ceramic values are used for comparison due to the lack of experimental data for the corresponding single crystals. All the above comparisons between thermodynamic calculations and experimental data, including the comparisons of phase structures, phase transition temperatures, and electromechanical properties, suggest that the constructed thermodynamic potential of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions is reliable.

### Phase diagram and electromechanical properties of continuous variations of compositions

In order to study the compositional dependence of phase structures and electromechanical properties, we fit the correlations between thermodynamic potential coefficients (Table 1) and composition *x* of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, as shown in Supplementary Fig. S4, the corresponding expressions of the fitting curves are listed in Supplementary Table S3. Based on these expressions of the thermodynamic potential coefficients, the temperature-composition phase diagram of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions is constructed in Fig. 4a by searching for the lowest energy state between different phases. It exhibits that a paraelectric phase locates at high temperature, and a rhombohedral ferroelectric phase occupies in low temperature range, while tetragonal and orthorhombic ferroelectric phases appear only between paraelectric and rhombohedral phases at low composition *x*. Also, it can be found that the Curie temperature *T*_{C} decreases with an increase in composition *x*, while phase transition temperatures *T*_{T↔O} and *T*_{O↔R} show an opposite trend, and both of them increase with increasing composition *x*, leading to a pinched phase transition and a quadruple point near *x* = 0.15, as shown in Fig. 4a. The experimental phase transition temperatures reported in literature^{13,16,17,18,19,20,21,22} are marked in Fig. 4a for comparison, indicating that our calculation results are consistent with experimental results, which again confirms the reliability of thermodynamic potential constructed in this work.

The contour plot of polarization components in terms of composition and temperature is shown in Figs. 4b–d. It can be observed that Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions with a low composition *x* have high polarization, and the polarization components are discontinuous at phase boundaries, which implies first-order phase transitions in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. The contour plots of dielectric constants and piezoelectric coefficients are also presented in Figs. 5a–c and Figs. 6a-b, respectively, where the anomalies of dielectric and piezoelectric responses are observed around the phase boundaries, and superior dielectric and piezoelectric properties appear near the ferroelectric–paraelectric phase boundary, especially around quadruple point. To see this more clearly, the corresponding polarization, dielectric constants and piezoelectric coefficients at room temperature are given in Figs. 4e, 5d, and 6c. It is noticed that the tetragonal–orthorhombic (ferroelectric–ferroelectric) phase boundary is at *x* = 0.013, the orthorhombic–rhombohedral (ferroelectric–ferroelectric) phase boundary is located at *x* = 0.0798, and the rhombohedral–paraelectric (ferroelectric–paraelectric) phase boundary is at *x* = 0.2135. The dielectric and piezoelectric properties show peaks near the tetragonal–orthorhombic (ferroelectric–ferroelectric) and rhombohedral–paraelectric (ferroelectric–paraelectric) phase boundaries. Comparing the electromechanical properties among the three phase boundaries, it is evident that the ferroelectric–paraelectric phase boundary has more superior electromechanical properties, suggesting a new way to enhance electromechanical coupling in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. In addition, good agreements between calculated results and experimental data are observed in Figs. 4e, 5d, and 6c. We also examine the effect of electric field on piezoelectric coefficients at room temperature in Figs. 6d–f, where electric field induced the enhancement of piezoelectric response is obvious, especially at high composition *x* (for example, *x* = 0.225) near the ferroelectric–paraelectric phase boundary.

In summary, we have constructed a thermodynamic potential for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, and carried out the thermodynamic analysis of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions using the constructed thermodynamic potential. It is found that the phase structures, phase transition temperatures, polarization, dielectric, and piezoelectric properties of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions are reproduced well. The results also reveal that three phase boundaries, including tetragonal–orthorhombic, orthorhombic–rhombohedral, and rhombohedral–paraelectric phase boundaries, can be formed at different chemical compositions. The chemical composition-induced ferroelectric–paraelectric phase boundary has superior electromechanical properties, which suggests a new way to enhance electromechanical coupling in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions. Notice that stress and strain constraints are always used to tune the phase structures and electromechanical properties in ferroelectrics.^{39,68} The constructed thermodynamic potential can be also used to predict the changes in phase structures and electromechanical properties under mechanical constraints.

## Methods

The process of determining thermodynamic potential coefficients for Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions is schematically illustrated in Supplementary Fig. S1. Among all thermodynamic potential coefficients, only *α*_{1} is assumed to be dependent on temperature *T*, following Curie–Weiss law

where *C*, *T*_{0} are the Curie constant and Curie–Weiss temperature, respectively. *α*_{1} can be determined by fitting Curie temperature *T*_{C} and dielectric constants of paraelectric phase in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions, as shown in Supplementary Fig. S1. The thermodynamic potential coefficients *α*_{11}, *α*_{111}, and *α*_{1111} need to be fitted to the phase transition temperatures *T*_{C} and *T*_{T↔O}, as well as dielectric constant and polarization of tetragonal phase. The coefficients *α*_{12}, *α*_{112}, *α*_{1112}, and *α*_{1122} are fitted to the phase transition temperatures *T*_{T↔O} and *T*_{O↔R}, and dielectric constant and polarization of orthorhombic phase. Similarly, the coefficients *α*_{123} and *α*_{1123} can estimated via fitting the phase transition temperature *T*_{O↔R} and electrical properties in rhombohedral phase. After that, all the thermodynamic potential coefficients of Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions can be obtained, as tabulated in Table 1. It is noted that the experimental observations^{17,20,21} indicated only paraelectric and rhombohedral two phases existing in Ba(Zr_{x}Ti_{1−x})O_{3} solid solutions with high concentration of Zr (*x* > 0.15). For this reason, the experimental data of paraelectric and rhombohedral phases is used to determine thermodynamic potential coefficients at high concentration *x*, as well as the eighth-order thermodynamic potential coefficients *α*_{1111}, *α*_{1112}, *α*_{1122}, and *α*_{1123} are set to be zero for this case.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was partially supported by the National Key Research and Development Program of China (2016YFA0201001), NSFC (Approval Nos. 11572276, 11772286, 11502225, 11627801, and 11402221), Project supported by Hunan Provincial Natural Science Foundation of China (2017JJ3312 and 2016JJ6146), Scientific Research Fund of Hunan Provincial Education Department (16A202 and 17B251), and Hunan Provincial Innovation Foundation for Postgraduate (CX2017B279). Lei acknowledges startup fund supported jointly by the Dean of Parks College and the Provost of the Saint Louis University.

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Y.L. conceived the idea and designed the calculations. Y.L. and J.P. implemented constructing thermodynamic potential. Q.Y. calculated electrostrictive coefficients. All authors performed thermodynamic analysis. Y.L. and J.P. analyzed the results and wrote the paper. All authors reviewed and contributed to the final paper preparation.

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Peng, J., Shan, D., Liu, Y. *et al.* A thermodynamic potential for barium zirconate titanate solid solutions.
*npj Comput Mater* **4, **66 (2018). https://doi.org/10.1038/s41524-018-0126-3

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DOI: https://doi.org/10.1038/s41524-018-0126-3