Misfit-Strain Phase Diagram, Electromechanical and Electrocaloric Responses in Epitaxial PIN–PMN–PT Thin Films

xPb(In1/2Nb1/2)O3-(1−x−y)Pb(Mg1/3Nb2/3)O3−yPbTiO3 (PIN–PMN–PT) bulks possess excellent electromechanical coupling and dielectric properties, but the corresponding epitaxial PIN–PMN–PT thin films have not yet been explored. This paper adopts a nonlinear thermodynamics analysis to investigate the influences of misfit strains on the phase structures, electromechanical properties, and electrocaloric responses in epitaxial PIN–PMN–PT thin films. The misfit strain–temperature phase diagram was constructed. The results reveal that the PIN–PMN–PT thin films may exist in tetragonal c-, orthorhombic aa-, monoclinic M-, and paraelectric PE phases. It is also found that the c-M and aa-PE phase boundaries exhibit a superior dielectric constant ε11 which reached 1.979 × 106 with um = −0.494%, as well as the c-M phase boundary showing a large piezoelectric response d15 which reached 1.64 × 105 pm/V. In comparison, the c-PE and M-aa phase boundaries exhibit a superior dielectric constant ε33 over 1 × 105 around um = 0.316% and the piezoelectric response d33 reached 7235 pm/V. The large electrocaloric responses appear near the paraelectric- ferroelectric phase boundary. These insights offer a guidance for experiments in epitaxial PIN–PMN–PT thin films.

There are more studies on PIN-PMN-PT bulk. For instance, in the experimental aspect, Li et al. [11] investigated the ferroelectric, dielectric, elastic, piezoelectric, and electromechanical properties of tetragonal PIN-PMN-PT crystals. The electromechanical coupling exhibited a high dc bias electric field stability compared to its rhombohedral counterpart, and the single domain piezoelectric coefficients d 33 and d 15 were found to be 530 and 2350 pC/N, respectively. Lin et al. [18] studied the piezoelectric thermal stability of PIN-PMN-PT ternary ceramics near the morphotropic phase boundary. The resulting temperature-dependent piezoelectric effects in the PIN-PMN-PT ceramics indicate that this ternary ceramics system within the MPB region shows a better temperature stability and increased usable temperature range compared with the PMN-32PT single crystals. In the theoretical aspect, Lv et al. developed a Landau-Devonshire energy functional for PIN-PMN-PT to investigate the phase transformation, phase diagrams, and the electromechanical properties of the PIN-PMN-PT single crystal [19], which match well with the experiments. On the contrary, research on PIN-PMN-PT thin films has been rare. Compared to bulk films, thin films are grown on a substrate, which impose a strain constraint due to a substrate lattice mismatch [9,[20][21][22][23]. It is known that the misfit strain can, in general, influence the phase structures and electromechanical properties in thin films [24]. The films have wider applications and a better adjustment than the bulks [8]. However, the influences of the misfit strain on the phase structures, electromechanical properties, and electrocaloric response in the epitaxial PIN-PMN-PT thin films have been lacking, which hinders the corresponding experimental studies. Notice that the misfit strain in the films is caused by the mismatch of lattice between the substrate and the film, which can be relaxed by a defect [25][26][27][28][29], such as a dislocation in the thicker films [30]. Thus, this article employs a nonlinear thermodynamic analysis to establish the misfit strain-temperature phase diagram for PIN-PMN-PT (26PIN-42PMN-32PT) thin films, which the stoichiometric composition are described by atomic%, from which the influences of the misfit strain on the phase structures, electromechanical properties, and electrocaloric responses are studied, offering a guidance for PIN-PMN-PT thin film experiments.
The structure of this article is as follows: the theory of nonlinear thermodynamics for ferroelectric thin films is outlined in Section 2, including the calculation methods for the electromechanical properties and electrocaloric response. The influences of the misfit strain on the phase structures, electromechanical properties, and electrocaloric responses of PIN-PMN-PT thin films are investigated in Section 3. The important discoveries and conclusions are summarized in Section 4.
According to the principle of the minimum energy, the polarization components of the thin films at stable configurations can be computed by solving the system of the equation: With the computed polarization components (P 1 , P 2 , P 3 ), the relative dielectric constant of the thin film can be calculated [32]: For (001)-oriented thin films, the piezoelectric coefficients d in can be computed [22]: where the strain S i is obtained from the stress-strain relation [30]. The main focus has been placed on the significant piezoelectric coefficients d 15 and d 33 . The strain components used for calculating the piezoelectric coefficients d 15 and d 33 are given as [32]:  [11,19,33].

Adiabatic Temperature Change in Electrocaloric Response
The electrocaloric (EC) effect is a phenomenon in a dielectric material that shows an adiabatic temperature change ∆T under an applied electric field change, or the entropy change induced by the isothermal conditions [34]. Following the method developed by Liu et al. in the previous work [35,36], we use an entropy-based analysis to calculate the EC adiabatic temperature change ∆T for the epitaxial PMN-PT-PIN thin films. In the literature [35][36][37], the total entropy S total of a ferroelectric thin film can be written as the sum of the dipolar entropy S dip and the lattice entropy S latt , In Equation (12), S dip (E, T) is the contribution from the dipolar degree of freedom, which is a function of polarization, depending on the working temperature T, the external electric field E, and the misfit strain. S latt (T) is assumed to be only correlated to the lattice contribution. Under the adiabatic condition, the total entropy change of the thin film is zero, leading to where subscripts i and f correspond to the initial and final states, respectively. The change in y S latt can be approximated by Note that C latt is the lattice heat capacity per unit volume. Combining Equations (13) and (14), the final state temperature T f can be calculated by Thus, the adiabatic temperature change in the electrocaloric response is given by where the dipolar entropy S dip is associated with the dipolar free energy ofthe ferroelectric thin films and can be determined by [35][36][37].

Results and Discussion
The above nonlinear thermodynamics method is adopted to analyze the influences of the misfit strain on the phase structures, electromechanical properties, and electrocaloric response in PIN-PMN-PT thin films. The material parameters used in the calculations are listed in Table 1 [11,19,33], which accurately reproduce the phase diagrams and electromechanical properties in the PIN-PMN-PT bulk, indicating the reliability of the material parameters.
We first investigate the influence of the misfit strain on the phase structures of the PIN-PMN-PT thin films, as shown in Figure 1. Figure 1a,b show that under the in-plane biaxial misfit strain, PIN-PMN-PT thin films may exhibit the four phase structures, and their polarization characteristics are summarized in Table 2. It can be seen that the tetragonal c phase is easily formed under a compressive strain, and the orthorhombic aa phase is more easily formed under a tensile strain. The tetragonal phase c has a polarization along the [001]-direction. In contrast, the monoclinic M phase can exist under both a compressive strain and a tensile strain. At high temperatures, the paraelectric PE phase is formed. To more clearly observe the variation of polarization with the mismatch strain more clearly, the change in the polarization components with the misfit strain at room temperature is plotted in Figure 1c. It can be seen that with the misfit strain change from compressive to tensile, the PIN-PMN-PT films exhibit a tetragonal c phase, monoclinic M phase, and orthorhombic aa phase in turn. The out-of-plane component P 3 decreases within the monoclinic M phase. In contrast, the in-plane component P 1 exhibits the opposite trend. At room temperature, the c-M phase boundary is around u m = −0.49%, while the M-aa phase boundary is around u m = 0.315%. Table 2. The polarization components of the epitaxial PIN-PMN-PT thin films in the absence of an external electric field.

Phase Polarization
Paraelectric PE P 1 = P 2 = P 3 = 0 Tetragonal c P 1 = P 2 = 0, P 3 = 0 Orthorhombic aa P 3 = 0, |P 1 | = |P 2 | = 0 Monoclinic M |P 1 | = |P 2 | = 0, P 3 = 0 Next, we investigate the influence of the misfit strain on the electromechanical properties of the PIN-PMN-PT thin films, including the dielectric and piezoelectric responses. Figure 2 presents the dielectric constants ε 11 , ε 22 , and ε 33 of the PIN-PMN-PT films at different temperatures and misfit strains. Due to the in-plane biaxial misfit strain on the thin films, it is expected that ε 11 = ε 22 . Figure 2a shows the excellent transverse permittivity in the vicinity of the c-M phase boundary and the aa-PE phase boundary. Figure 2b shows an excellent longitudinal permittivity in the vicinity of the c-PE phase boundary and the M-aa phase boundary. Figure 2c shows the trend of the dielectric constant at room temperature with respect to the misfit strain. Similarly, the dielectric response enhancement at the phase boundary is also observed in the BaTiO 3 [22] and BiFeO 3 [24,38] thin films due to the abrupt change in the polarization slope near the phase boundary. The sudden change in the polarization slope also causes the PIN-PMN-PT film to exhibit an excellent piezoelectric response near the phase boundary, as shown in Figure 3a,b, where the c-M phase boundary has an excellent transverse piezoelectric response d 15 , the c-PE phase boundary and M-aa phase boundary exhibit an excellent longitudinal piezoelectric response d 33 . The piezoelectric response of the PIN-PMN-PT thin film at room temperature is shown in Figure 3c, which reaches a peak at the c-M phase boundary, and a peak at the M-aa phase boundary.
within the monoclinic M phase. In contrast, the in-plane component 1 P exhibits the opposite trend. At room temperature, the c-M phase boundary is around um = −0.49%, while the M-aa phase boundary is around um = 0.315%.    = . Figure 2a shows the excellent transverse permittivity in the vicinity of the c-M phase boundary and the aa-PE phase boundary. Figure 2b shows an excellent longitudinal permittivity in the vicinity of the c-PE phase boundary and the M-aa phase boundary. Figure 2c shows the trend of the dielectric constant at room Finally, we investigate the influence of the misfit strain on the adiabatic temperature change ∆T in the electrocaloric response in PIN-PMN-PT thin films, as shown in Figure 4, where the electrical field is applied along the [001] direction with a variation (∆E) of 10 MV/m. The results show that large electrocaloric responses ∆T appear near the ferroelectric-paraelectric phase boundary at high working temperature because the dipoles in the paraelectric PE phase are easier to reorient when the external electrical field is changed. The corresponding ∆T at a fixed temperature and under a fixed misfit strain are shown in Figure 4b,c, where the peaks of the EC responses ∆T near the phase boundaries can be observed more clearly, suggesting that an appropriate misfit strain can enhance the EC response of the PIN-PMN-PT thin films. temperature with respect to the misfit strain. Similarly, the dielectric response enhancement at the phase boundary is also observed in the BaTiO3 [22] and BiFeO3 [24,38] thin films due to the abrupt change in the polarization slope near the phase boundary. The sudden change in the polarization slope also causes the PIN-PMN-PT film to exhibit an excellent piezoelectric response near the phase boundary, as shown in Figure 3a   Finally, we investigate the influence of the misfit strain on the adiabatic temperature change ∆T in the electrocaloric response in PIN-PMN-PT thin films, as shown in Figure   4, where the electrical field is applied along the [001] direction with a variation ( E ) of 10 MV/m. The results show that large electrocaloric responses ∆T appear near the ferroelectric-paraelectric phase boundary at high working temperature because the dipoles in the paraelectric PE phase are easier to reorient when the external electrical field is changed. The corresponding ∆T at a fixed temperature and under a fixed misfit strain are shown in Figure 4b,c, where the peaks of the EC responses ∆T near the phase boundaries can be observed more clearly, suggesting that an appropriate misfit strain can enhance the EC response of the PIN-PMN-PT thin films.

Conclusions
In summary, we adopt a nonlinear thermodynamics analysis to study the influences of misfit strains on the phase structures, electromechanical properties, and electrocaloric responses of epitaxial PIN-PMN-PT thin films. It is found that the PIN-PMN-PT thin films may appear in tetragonal c-, orthorhombic aa-, monoclinic M-, and paraelectric PE phases. We also found that the c-M and aa-PE phase boundaries show a superior dielectric

Conclusions
In summary, we adopt a nonlinear thermodynamics analysis to study the influences of misfit strains on the phase structures, electromechanical properties, and electrocaloric responses of epitaxial PIN-PMN-PT thin films. It is found that the PIN-PMN-PT thin films may appear in tetragonal c-, orthorhombic aa-, monoclinic M-, and paraelectric PE phases. We also found that the c-M and aa-PE phase boundaries show a superior dielectric constant, ε 11 , as well as the c-M phase boundary having a large piezoelectric response, d 15 , while the c-PE and M-aa phase boundaries show a superior dielectric constant, ε 33 , and the piezoelectric response d 33 . The adiabatic temperature change ∆T indicates that the paraelectric-ferroelectric phase boundary shows a large electrocaloric response. The findings offer guidance for PIN-PMN-PT thin film experiments.