Phase Transitions under the Electric Field in Ternary Ferroelectric Solid Solutions of Pb(In 1/2 Nb 1/2 )O 3 –Pb(Mg 1/3 Nb 2/3 )O 3 –PbTiO 3 near the Morphotropic Phase Boundary: Electric Approach

: Temperature–field phase diagrams in the [001] c and [011] c directions in the cubic coordinate in 24%Pb(In 1/2 Nb 1/2 )O 3 –46%Pb(Mg 1/3 Nb 2/3 )O 3 –30%PbTiO 3 (24PIN–46PMN–30PT) and 31PIN– 43PMN–26PT near the morphotropic phase boundary have been clarified by measuring the temperature dependences of permittivity under an electric field. Field-induced intermediate orthorhombic and tetragonal phases have been newly found in 24PIN–46PMN–30PT and 31PIN–43PMN–26PT, respectively. The temperature dependences of the remanent polarization have also been determined by polarization–electric field (P–E) hysteresis loop evaluation. On the basis of our experimental results, the phase transition and dielectric anisotropy in PIN–PMN–PT have been discussed.

To understand such properties near MPB, a simple theoretical model based on the Landau-Devonshire free energy was reported, where the permittivity perpendicular to the spontaneous polarization becomes extremely high, since the anisotropy of the free energy becomes small in the parameter space [6].A similar mechanism underlying such a giant response was also found in BaTiO 3 on the basis of the first principles studies [7].In any case, it is certain that the anisotropic energy of the polarization near MPB in the relaxor ferroelectrics plays an essential role in their colossal dielectric and piezoelectric responses.
For PMN-xPT solid solution systems, the temperature-concentration phase diagram near MPB has been reported, where the rhombohedral, monoclinic, and tetragonal phases appear in ferroelectric phases [4,8].An electromechanical coupling coefficient k * 33 = 94% was reported for PMN-33%PT, which is the highest reported among all piezoelectric materials [9].However, the operating temperature range in PMN-33%PT is narrow, because the transition temperature between the tetragonal and rhombohedral phases is about 60 • C [4].
The temperature-field phase diagrams under various directions of an electric field in PMN-xPT were reported to clarify the average symmetry in the ferroelectric phase, where the ferroelectric critical endpoint (CEP) was found in the phase diagram [10][11][12][13][14]. On the basis of such temperature-field phase diagrams, we showed that relaxor ferroelectric Crystals 2024, 14, 121 2 of 16 crystals almost behave similarly to a normal ferroelectric material under a DC biasing field [15].Indeed, these field-induced phase transitions in the vicinity of MPB can be well reproduced on the basis of the Landau-Devonshire free energy [16].The nonlinear dielectric susceptibility in PMN-xPT was also found to be well-analyzed within the Landau theory [17].Recently, we have also found the aging effect on PMN-xPT [18].
On the other hand, in PIN, the chemical ordering of B-site cations (In and Nb) was clarified to be controlled by appropriate thermal treatment [19][20][21][22].PIN crystals with different chemical orderings formed by different thermal treatments can be classified into three groups: the "ordered PIN", "disordered PIN", and "partly disordered PIN".An asgrown single crystal is the partly disordered PIN, where the partly disordered PIN shows a broad peak of the dielectric constant without dielectric dispersion at about 90 • C [22].
Hosono et al. proposed the ternary ferroelectric solid solution system PIN-PMN-PT as a candidate material that realizes both a large electromechanical coupling coefficient (PMN-PT) and a high transition temperature (PIN-PT), and they reported that 16%PIN-51%PMN-33%PT (16PIN-51PMN-33PT) with a high transition temperature of 187 • C shows a large piezoelectric constant of 2200 pC/N [1].To improve the performance of ternary ferroelectric solid solutions of PIN-PMN-PT, their physical properties with respect to the phase transition and MPB were extensively investigated using ceramic and single crystal samples of this system [28][29][30][31][32][33][34][35].The temperature-field phase diagrams for 33PIN-35PMN-32PT and 23PIN-52PMN-25PT were studied to clarify the structural phase transition and stability of these materials under a biasing field [36][37][38].A phenomenological approach to analyzing PIN-PMN-PT near MPB based on the Landau-Devonshire energy function with 10th-order terms in the polarization was discussed to explain qualitatively the engineered domain mechanism [39].It seems that further experimental data are needed to determine the expansion coefficients taking into account the anisotropy of thermodynamic potential, which is the most important factor to explain the large dielectric and piezoelectric responses near MPB [6,7].We pointed out in our previous paper that the data of a temperature-field phase diagram in various electric field directions are useful for evaluating the anisotropy [16].
Under these circumstances, in this paper, dielectric permittivities under the biasing field and the polarization-electric field (P-E) hysteresis loops in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT near MPB were investigated.The temperature-field phase diagram in the [001] c and [011] c directions in the cubic coordinates and the spontaneous polarization as a function of temperature were clarified.On the basis of our experimental results, the phase transition and dielectric anisotropy in PIN-PMN-PT are discussed.

Experimental Procedure
Single crystal wafers in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT near MPB were grown by the Bridgman technique [40].Figure 1 shows the phase diagram of the ternary system of PIN-PMN-PT at room temperature, where 24PIN-46PMN-30PT and 31PIN-43PMN-26PT are shown by the solid and open circles, respectively.The straight dasheddotted line connects the triple points in PIN-PT and PMN-PT, and the straight dotted line connects MPB at room temperature in PIN-PT and PMN-PT [4,5].It is conjectured from Figure 1 that both materials are located near MPB and show phase sequences of cubic-rhombohedral and cubic-tetragonal-rhombohedral in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT, respectively.dashed-dotted line connects the triple points in PIN-PT and PMN-PT, and the straight dotted line connects MPB at room temperature in PIN-PT and PMN-PT [4,5].It is conjectured from Figure 1 that both materials are located near MPB and show phase sequences of cubic-rhombohedral and cubic-tetragonal-rhombohedral in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT, respectively.Sample plates with thicknesses of about 250-500 µm were used in our experiments after annealing treatment for 3 h at 500 °C.For the dielectric measurement, the parallelplate capacitor of a sample with Au electrodes deposited on its face was prepared.Permittivity measurements under a DC biasing field were performed using an impedance/gain phase analyzer (NF ZGA5900) and a high-voltage amplifier (Trek 609E-6).In our measurement system, the AC probe voltage applied to measure dielectric permittivity is about 0.1-0.2V, and the maximum DC biasing voltage applied to a sample during the measurement is about 800 V. Complex dielectric permittivity, ˆi ε ε ε ′ ′′ = − , was obtained in the range from 1 to 100 kHz after carefully removing the effects of the stray capacitance and residual impedance from the system.
A Sawyer-Tower circuit was used with a standard capacitor of 10 µF to evaluate P-E hysteresis loops, where a sinusoidal field in the frequency of 1 Hz and the amplitude of 14 kV/cm was applied to the sample.No correction of the phase lag using the phase compensation circuit was performed because of the low conductivity in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT samples.

Permittivity under Biasing Electric Field in 24PIN-46PMN-30PT
Figure 2a-c show typical examples of the temperature dependence of permittivity under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm in the [001]c direction in 24PIN-46PMN-30PT.Three dielectric anomalies at Tm, TCT, and TTR are observed in each figure.It is seen that the temperatures TCT and TTR strongly depend on the electric field strength, and the temperature interval between TCT and TTR widens with increasing field Sample plates with thicknesses of about 250-500 µm were used in our experiments after annealing treatment for 3 h at 500 • C. For the dielectric measurement, the parallel-plate capacitor of a sample with Au electrodes deposited on its face was prepared.Permittivity measurements under a DC biasing field were performed using an impedance/gain phase analyzer (NF ZGA5900) and a high-voltage amplifier (Trek 609E-6).In our measurement system, the AC probe voltage applied to measure dielectric permittivity is about 0.1-0.2V, and the maximum DC biasing voltage applied to a sample during the measurement is about 800 V. Complex dielectric permittivity, ε = ε ′ − iε ′′ , was obtained in the range from 1 to 100 kHz after carefully removing the effects of the stray capacitance and residual impedance from the system.
A Sawyer-Tower circuit was used with a standard capacitor of 10 µF to evaluate P-E hysteresis loops, where a sinusoidal field in the frequency of 1 Hz and the amplitude of 14 kV/cm was applied to the sample.No correction of the phase lag using the phase compensation circuit was performed because of the low conductivity in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT samples.

Permittivity under Biasing Electric Field in 24PIN-46PMN-30PT
Figure 2a-c show typical examples of the temperature dependence of permittivity under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm in the [001] c direction in 24PIN-46PMN-30PT.Three dielectric anomalies at T m , T CT , and T TR are observed in each figure .It is seen that the temperatures T CT and T TR strongly depend on the electric field strength, and the temperature interval between T CT and T TR widens with increasing field strength along the [001] c direction, whereas the temperature T m does not depend on the field strength within an experimental error.We conclude that T CT and T TR are the transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases, respectively, whereas at least T m does not indicate a ferroelectric phase transition.The details on T m will be discussed in Section 4.1.
strength along the [001]c direction, whereas the temperature Tm does not depend on the field strength within an experimental error.We conclude that TCT and TTR are the transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases, respectively, whereas at least Tm does not indicate a ferroelectric phase transition.The details on Tm will be discussed in Section 4.1.In Figure 3a-c -c and 3a-c in 24PIN-46PMN-30PT are summarized in Table 1.In Figure 3a-c Table 1.Phase transition temperatures measured on heating in 24PIN-45PMN-30PT.Tm is determined from permittivity peak at 1 kHz.Table 1.Phase transition temperatures measured on heating in 24PIN-45PMN-30PT.T m is determined from permittivity peak at 1 kHz.

Permittivity under Biasing Electric Field in 31PIN-43PMN-26PT
The temperature dependences of permittivity under the DC biasing fields of 0, 1.0, and 2.0 kV/cm along the [001]c direction in 31PIN-43PMN-26PT are respectively shown in Figure 5a-c as typical examples.It is seen that only the temperature Tm showing the maximum permittivity is found in Figure 5a, whereas three dielectric anomalies at Tm, TCT, and TTR appear in Figure 5b,c.The temperatures TCT and TTR strongly depend on the electric field strength, and the temperature interval between TCT and TTR widens with increasing field strength, whereas the temperature Tm does not depend on the field strength within an experimental error.Consequently, TCT and TTR are determined to be the transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases, respectively, and Tm does not indicate a ferroelectric phase transition.

Permittivity under Biasing Electric Field in 31PIN-43PMN-26PT
The temperature dependences of permittivity under the DC biasing fields of 0, 1.0, and 2.0 kV/cm along the [001] c direction in 31PIN-43PMN-26PT are respectively shown in Figure 5a-c as typical examples.It is seen that only the temperature T m showing the maximum permittivity is found in Figure 5a, whereas three dielectric anomalies at T m , T CT , and T TR appear in Figure 5b,c.The temperatures T CT and T TR strongly depend on the electric field strength, and the temperature interval between T CT and T TR widens with increasing field strength, whereas the temperature T m does not depend on the field strength within an experimental error.Consequently, T CT and T TR are determined to be the transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases, respectively, and T m does not indicate a ferroelectric phase transition.
Figure 6a-c also show typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing field of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [011] c direction in 31PIN-43PMN-26PT.One or two dielectric anomalies appear in each figure along the [011] c direction.The temperature T CR depends on the electric field strength, whereas the temperature T m does not within an experimental error.We conclude that T CR is the transition temperature from cubic to rhombohedral phases, and at least T m does not indicate a ferroelectric phase transition.The transition temperatures obtained from Figures 5a-c and 6a-c in 31PIN-43PMN-26PT are summarized in Table 2.
Figure 7a,b show the temperature-field phase diagrams along the [001] c and [011] c directions in 31PIN-43PMN-26PT, respectively.Circles and squares show the transition temperatures determined from the permittivity measured during the heating and cooling processes, respectively.The letters C, T, O, R, M A , and M B indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, and monoclinic B symmetries, respectively [8].The letters in parentheses show the rigorous symmetry under the electric field along each direction.To confirm sample dependence, the results of the transition temperature for two samples are shown in Figure 7b.It is seen that phase transition temperatures below 1 kV/cm are not consistent with those above 1 kV/cm owing to the relaxor nature of the diffuse phase transition.The assignment of the symmetry in the ferroelectric phases will be discussed in Section 4.1.Table 2. Phase transition temperatures measured on heating in 31PIN-45PMN-30PT.T m is determined from permittivity peak at 1 kHz.

P-E Hysteresis Loops
Figure 8a-d show typical examples of the P-E hysteresis loops in different electric fields along the [001]c direction in 24PIN-46PMN-30PT, where the frequency of the electric fields is 1 Hz.The P-E hysteresis loops were measured in the temperature range from 180 to 30 °C during the cooling process.It is considered that in Figure 8a, the imperfect   The temperature dependence of remanent polarization determined by the P-E hysteresis loop measurement is shown in Figure 9.The dotted lines indicate the transition temperature determined from Figure 4a, where TCT = 171 °C and TTR = 92 °C (see Section 3.1).At the transition temperature between the tetragonal and rhombohedral phases, an anomaly of the remanent polarization is found, although no jump of the polarization appears.This implies the coexistence of the tetragonal, orthorhombic, and rhombohedral phases.10a-d, where the frequency of the electric fields is 1 Hz, and the temperature range measured is from 180 to 30 • C during the cooling process.It is guessed that in Figure 10a, the imperfect triple-loop pattern basically indicates the field-induced transition in the paraelectric phase.Figure 10b-d show typical P-E hysteresis loops revealing the polarization reversal in the ferroelectric phase.

P-E Hysteresis Loops
The temperature dependence of remanent polarization determined by the P-E hysteresis loop measurement is shown in Figure 9.The dotted lines indicate the transition temperature determined from Figure 4a, where T CT = 171 • C and T TR = 92 • C (see Section 3.1).At the transition temperature between the tetragonal and rhombohedral phases, an anomaly of the remanent polarization is found, although no jump of the polarization appears.This implies the coexistence of the tetragonal, orthorhombic, and rhombohedral phases.
teresis loop measurement is shown in Figure 9.The dotted lines indicate the transition temperature determined from Figure 4a, where TCT = 171 °C and TTR = 92 °C (see Section 3.1).At the transition temperature between the tetragonal and rhombohedral phases, an anomaly of the remanent polarization is found, although no jump of the polarization appears.This implies the coexistence of the tetragonal, orthorhombic, and rhombohedral phases.Typical examples of the P-E hysteresis loops in 31PIN-43PMN-26PT under different electric fields along the [001]c direction are shown in Figure 10a-d, where the frequency of the electric fields is 1 Hz, and the temperature range measured is from 180 to 30 °C during the cooling process.It is guessed that in Figure 10a, the imperfect triple-loop pattern basically indicates the field-induced transition in the paraelectric phase.Figures 10bd show typical P-E hysteresis loops revealing the polarization reversal in the ferroelectric phase.Figure 11 shows the temperature dependence of the remanent polarization obtained by the P-E hysteresis loop measurement, where the dotted line indicates the transition temperature determined from Figure 7b, where TCR = 127 °C (see Section 3.2).Note that in Figure 11, the true polarization value in the rhombohedral phase must be multiplied by 3 .

Assignment of the Symmetry in 24PIN-46PMN-30PT
Let us start with the assignment of the ferroelectric phase in the phase diagram along the [001]c direction in 24PIN-46PMN-30PT.It is seen in Figure 4a that the temperature interval of the tetragonal phase extends as the electric field along the [001]c direction increases, which is consistent with the stability of the polarization in the tetragonal phase under the field in the [001]c direction.Therefore, by extrapolating the phase boundary from the electric field above 1 kV/cm to zero field, we conclude that the phase transition sequence under zero biasing field is determined to be the C-T-R phases.The phase transitions at TCT and TTR under zero biasing field are of the first order, and the transition temperatures are at TCT = 171 °C and TTR = 92 °C.We were unable to determine CEP in the C-T phase transition because of the diffuseness of this phase transition.By extrapolating the phase boundary above 1 kV/cm to E = 0, we also estimated the slopes of the C-T and T-R phase boundaries to be dE/dTCT|E=0 = 0.11 kV/cmK and dE/dTTR|E=0 = −0.16kV/cmK under an electric field along the [001]c direction, respectively.
Next, we assign the ferroelectric phases in the phase diagram along the [011]c direction shown in Figure 4b.The temperature interval of the orthorhombic phase extends as the electric field along the [011]c direction increases, which is consistent with the stability of the polarization in the orthorhombic phase under the field in the [011]c direction.We conclude that the orthorhombic phase appears only under the field along the [011]c direction.The field-induced orthorhombic phase is determined to be a metastable phase under zero biasing field, because no orthorhombic phase appears in the electric field along the [001]c direction.
The transition temperature TCT is determined to be 175 °C by extrapolating the phase boundary from the electric field above 1 kV/cm to zero field.The slope of the C-T phase boundary is obtained to be dE/dTCT|E=0 = 0.17 kV/cmK under the electric field along the [011]c direction.The reason for the difference of 4 °C in the phase transition temperature TCT along the [001]c and [011]c directions is guessed to be the sample dependence.
In general, the ferroelectric transition temperature depends on the direction and strength of the biasing field, because the electric field is the conjugate force to the polarization.Indeed, TCT and TTR were confirmed to depend on the biasing field, as shown in Figure 4a,b.However, Tm does not completely depend on the electric field within an experimental error.We considered that at least the temperature Tm at which the permittivity is maximum is not a ferroelectric transition temperature.

Assignment of the Symmetry in 31PIN-43PMN-26PT
Let us assign the ferroelectric phases in the phase diagram along the [001]c direction in 31PIN-43PMN-26PT.In Figure 7a, the temperature interval of the tetragonal phase  In general, the ferroelectric transition temperature depends on the direction and strength of the biasing field, because the electric field is the conjugate force to the polarization.Indeed, T CT and T TR were confirmed to depend on the biasing field, as shown in Figure 4a,b.However, T m does not completely depend on the electric field within an experimental error.We considered that at least the temperature T m at which the permittivity is maximum is not a ferroelectric transition temperature.

Assignment of the Symmetry in 31PIN-43PMN-26PT
Let us assign the ferroelectric phases in the phase diagram along the [001] c direction in 31PIN-43PMN-26PT.In Figure 7a, the temperature interval of the tetragonal phase extends with increasing electric field along the [001] c direction, which is consistent with the stability of the polarization in the tetragonal phase under the field along the [001] c direction.The intermediate ferroelectric phase newly found is determined to be the tetragonal phase.With respect to the C-T phase transition, we find that the thermal hysteresis of the transition temperature decreases with increasing electric field.By extrapolating with straight lines (thin dotted line in Figure 7a), the critical endpoint is determined to be 173 • C and 2.4 kV/cm.By extrapolating the phase boundary from the field above 1 kV/cm to zero field, we determined the transition temperatures to be T CT = 136 • C and T TR = 126 • C, and estimated the slopes of the boundaries C-T and T-R to be dE/dT CT | E=0 = 6.5 × 10 −2 kV/cmK and dE/dT TR | E=0 = −0.24kV/cmK in the electric field along the [001] c direction, respectively.
In the phase diagram only along the [001] c direction shown in Figure 7a, we were unable to determine whether a stable tetragonal phase exists under zero electric field.In the phase diagram along the [011] c direction shown in Figure 7b, no intermediate tetragonal phase was found.This indicates that the tetragonal phase is not stable under zero electric field.As for the C-T phase transition, it is found that the thermal hysteresis of the transition temperature decreases as the electric field increases.By extrapolating with straight lines (thin dotted line in Figure 7b, the temperature at which the phase transition changes from first to second order is determined to be 160 • C and 2.5 kV/cm, indicating the tricritical point.Dul'kin et al. showed the existence of a tricritical point in the C-R phase transition of 26PIN-46PMN-28PT with different compositions under an electric field along [011] c direction [35].From the point of view of symmetry, these are presumed to be critical points of the same kind.Further detailed study of such tricritical points is needed.
By extrapolating the phase boundary above 1 kV/cm to E = 0, we determined the transition temperature T CR to be 127 • C, and the slope of the C-R phase boundary is obtained to be dE/dT CR | E=0 = 8.3 × 10 −2 kV/cmK in the electric field along the [011] c direction.From the above, we conclude that the phase transition sequence under zero biasing field is considered to be the C-R phases, and the tetragonal phase is a metastable phase under zero biasing field.

Evaluation of the Phase Boundary Based on the Clausius-Clapeyron Equation
Let us focus on the slope of the phase boundary in the temperature-field phase diagram of the perovskite-type ferroelectrics on the basis of the Clausius-Clapeyron equation.We start with the Landau-Ginzburg-Devonshire free energy function f expressed in terms of the polarization components p i (i = 1-3) as where α is temperature-dependent, as shown by α = a(T − T 0 ), a > 0, T 0 > 0. The parameters β 1 , β 2 , γ 1 , γ 2 , and γ 3 are constants, E = (E 1 , E 2 , E 3 ) is the external electric field, and p = (p 1 , p 2 , p 3 ) the polarization.We truncated the free energy function at the sixth order of the polarization for simplicity.At this truncated free energy, the cubic (C), tetragonal (T), orthorhombic (O), and rhombohedral (R) phases are stable under zero external field, where the stable spontaneous polarizations in the C, T, O, R phases are defined as (0, 0, 0), (0, 0, p), (0, q, q), and (r, r, r), respectively.We consider the slope of the boundary between the A and B phases at zero field in the temperature-field phase diagram based on the free energy in Equation (1), where the A and B phases are the C, T, O, and R phases.According to the Clausius-Clapeyron equation, the slope of the phase boundary is obtained as [41] where e E is the directional vector of the electric field E/|E|, and ∆p = p B − p A and ∆S = S B − S A = −a(p 2 B − p 2 A )/2 are the jumps of the polarization and the entropy at the phase boundary between the A and B phases, respectively.The derivation of the extended Clausius-Clapeyron equation in ferroelectrics is given in Appendix A. All the slopes of the phase boundary at zero field in the temperature-field phase diagram are summarized in Table 3.The slope of the phase boundary can be determined if the polarizations at the phase boundary in the A and B phases are known.Note that the Clausius-Clapeyron equation presented in Equation ( 2) is also applicable to the free energy expanded to the 10th-order term of the polarization recently proposed by Lv et al. [39].√ 2a(3r 2 −2q 2 ) 4(r−q) √ 3a(3r 2 −2q 2 ) 2(3r−2q) Since no jump of the spontaneous polarization at the transition point can be observed in our experimental result, we only evaluate the slope of the boundary between the cubic and tetragonal phases in 24PIN-46PMN-30PT, where the slopes of the C-T boundaries along the [001] c and [011] c directions are 0.11 and 0.17 kV/cmK, respectively.From Table 3, the ratio of the slopes is √ 2. It is seen that the ratio 0.17/0.11 is 1.5 ∼ = √ 2 within an experimental error, which is consistent with our experimental results.

Conclusions
In this study, we have clarified the temperature-field phase diagrams along the [001] c and [011] c directions in the cubic coordinate in 24PIN-46PMN-30PT and 31PIN-43PMN-26PT near MPB.The temperature dependences of the remanent polarization have also been determined by P-E hysteresis loop observation.
In 24PIN-46PMN-30PT, we conclude that the phase transition sequence without an external field is the C-T-R phases, where the phase transition temperatures are 171 and 92 • C. The field-induced transition to the ferroelectric orthorhombic phase appears only under the electric field along the [011] c direction.This indicates that the orthorhombic phase observed in the electric field is a metastable phase under zero field.We analyzed the slope of the phase boundary at zero field in the temperature-field phase diagram on the basis of the Clausius-Clapeyron equation, and consequently, we confirmed that the phase diagrams along the [001] c and [011] c directions are consistent within an experimental error.
In 31PIN-43PMN-26PT, the phase transition sequence without an external field is the C-R phases, as determined by extrapolating the phase boundary above 1 kV/cm to E = 0, where the transition temperature is 127 • C. The field-induced transition to the tetragonal phase appears only under the electric field along the [001] c direction, indicating a metastable phase under zero field.
We experimentally found that many ferroelectric phases including metastable orthorhombic and tetragonal phases exist in PIN-PMN-PT.This implies that the local minima of the free energy as a function of polarization in various directions compete with each other, and then the anisotropy of the Landau-Ginzburg-Devonshire free energy in the polarization space is small.Therefore, we conclude that the large dielectric and piezoelectric responses in these materials near MPB come from the transversal instability [6].Further investigations from the viewpoint of the anisotropy in the thermodynamic potential are required to clarify the physical properties in PIN-PMN-PT solid solution systems.

Figure 1 .
Figure 1.Ternary phase diagram for PIN-PMN-PT at room temperature, where 24PIN-46PMN-30PT and 31PIN-43PMN-26PT are shown by the solid and open circles, respectively.The straight dashed-dotted line connects with the triple points in PIN-PT and PMN-PT, and the straight dotted line connects with MPBs at room temperature in PIN-PT and PMN-PT.

Figure 1 .
Figure 1.Ternary phase diagram for PIN-PMN-PT at room temperature, where 24PIN-46PMN-30PT and 31PIN-43PMN-26PT are shown by the solid and open circles, respectively.The straight dashed-dotted line connects with the triple points in PIN-PT and PMN-PT, and the straight dotted line connects with MPBs at room temperature in PIN-PT and PMN-PT.

Figure 2 .
Figure 2. Typical examples of the permittivity as a function of temperature under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [001]c direction in 24PIN-46PMN-30PT.The temperatures TCT, TTR, and Tm are the phase transition temperatures between the cubic and tetragonal phases and between tetragonal and rhombohedral phases and the temperature showing a peak of permittivity (not ferroelectric transition temperature), respectively.
, we also show typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing fields of (a) 0, (b) 2.0, and (c) 3.0 kV/cm along the [011]c direction in 24PIN-46PMN-30PT.At a permittivity along the [011]c, three or four dielectric anomalies appear in each figure.All the temperatures showing the dielectric anomalies, except for Tm, depend on the electric field strength.We conclude that at least Tm does not indicate a ferroelectric phase transition.The subscripts of the temperatures indicating the dielectric anomalies and the assignment of the symmetry of the ferroelectric phases will be discussed in Section 4.1.The transition temperatures obtained from Figures 2a

Figure 2 .
Figure 2. Typical examples of the permittivity as a function of temperature under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [001] c direction in 24PIN-46PMN-30PT.The temperatures T CT , T TR , and T m are the phase transition temperatures between the cubic and tetragonal phases and between tetragonal and rhombohedral phases and the temperature showing a peak of permittivity (not ferroelectric transition temperature), respectively.

18 Figure 3 .
Figure 3.Typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing fields of (a) 0, (b) 2.0, and (c) 3.0 kV/cm along the [011]c direction in 24PIN-46PMN-30PT.The temperatures TCT, TTO, TOR, and Tm are the phase transition temperatures between the cubic and tetragonal phases, between the tetragonal and orthorhombic phases, and between the orthorhombic and rhombohedral phases, and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.
Figure 4a,b show the temperature-field phase diagrams along the [001]c and [011]c directions in 24PIN-46PMN-30PT, respectively.Circles and squares show the transition temperature determined from the permittivity measured on heating and cooling processes, respectively.The letters C, T, O, R, MA, MB, and MC indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, monoclinic B, and monoclinic C symmetries, respectively [8].The letters in parentheses show the rigorous symmetry under the electric

Figure 3 .
Figure 3.Typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing fields of (a) 0, (b) 2.0, and (c) 3.0 kV/cm along the [011] c direction in 24PIN-46PMN-30PT.The temperatures T CT , T TO , T OR , and T m are the phase transition temperatures between the cubic and tetragonal phases, between the tetragonal and orthorhombic phases, and between the orthorhombic and rhombohedral phases, and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.
along each direction.Experimental results for two samples are shown in Figure4a,b to confirm sample dependence.It is seen that phase transition temperatures below 1 kV/cm are not consistent with those above 1 kV/cm owing to the relaxor nature of the diffuse phase transition.The assignment of the symmetry in the ferroelectric phases will be discussed in Section 4.1.

Figure 4 .
Figure 4. Temperature-field phase diagrams along (a) [001]c and (b) [011]c directions in 24PIN-46PMN-30PT.Circles and squares show transition temperatures determined from the permittivity measured during heating and cooling processes, respectively.Solid lines are the phase boundary, and dotted lines show the eye guide extrapolating the phase boundary.The letters C, T, O, R, MA, MB, and MC indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, monoclinic B, and monoclinic C symmetries, respectively [8].The letters in the parentheses show the rigorous symmetries under the electric field along each direction.

Figure 4 .
Figure 4. Temperature-field phase diagrams along (a) [001] c and (b) [011] c directions in 24PIN-46PMN-30PT.Circles and squares show transition temperatures determined from the permittivity measured during heating and cooling processes, respectively.Solid lines are the phase boundary, and dotted lines show the eye guide extrapolating the phase boundary.The letters C,T, O, R, M A , M B , and MC indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, monoclinic B, and monoclinic C symmetries, respectively[8].The letters in the parentheses show the rigorous symmetries under the electric field along each direction.

Figure 5 .
Figure 5.Typical examples of the temperature dependence of permittivity under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [001]c direction in 31PIN-43PMN-26PT.The temperatures TCT, TTR, and Tm are the phase transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.

Figure
Figure 6a-c also show typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing field of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [011]c direction in 31PIN-43PMN-26PT.One or two dielectric anomalies appear in each figure along the [011]c direction.The temperature TCR depends on the electric field strength, whereas the temperature Tm does not within an experimental error.We conclude that TCR is the transition temperature from cubic to rhombohedral phases, and at least Tm does not indicate a ferroelectric phase transition.The transition temperatures obtained from Figures 5a-c and 6a-c in 31PIN-43PMN-26PT are summarized in Table2.

Figure 5 .
Figure 5.Typical examples of the temperature dependence of permittivity under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [001] c direction in 31PIN-43PMN-26PT.The temperatures T CT , T TR , and T m are the phase transition temperatures between the cubic and tetragonal phases and between the tetragonal and rhombohedral phases and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.

Figure 6 .
Figure 6.Typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [011]c direction in 31PIN-43PMN-26PT.The temperatures TCR and Tm are the phase transition temperature between the cubic and rhombohedral phases and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.
Figure 7a,b show the temperature-field phase diagrams along the [001]c and [011]c directions in 31PIN-43PMN-26PT, respectively.Circles and squares show the transition temperatures determined from the permittivity measured during the heating and cooling processes, respectively.The letters C, T, O, R, MA, and MB indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, and monoclinic B symmetries, respectively [8].The letters in parentheses show the rigorous symmetry under the electric field along each direction.To confirm sample dependence, the results of the transition temperature for two

Figure 6 .
Figure 6.Typical examples of the temperature dependence of permittivity at 1 kHz under the DC biasing fields of (a) 0, (b) 1.0, and (c) 2.0 kV/cm along the [011] c direction in 31PIN-43PMN-26PT.The temperatures T CR and T m are the phase transition temperature between the cubic and rhombohedral phases and the temperature showing a peak of the permittivity (not ferroelectric transition temperature), respectively.

Figure 7 .
Figure 7. Temperature-field phase diagrams along (a) [001]c and (b) [011]c directions in 31PIN-43PMN-26PT.Circles and squares show transition temperatures determined from the permittivity measured during heating and cooling processes, respectively.Solid lines are the phase boundary, and dotted lines show the eye guide extrapolating the phase boundary.The letters C, T, O, R, MA, and MB indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, and monoclinic B symmetries, respectively [8].The letters in the parentheses show the rigorous symmetries under the electric field along each direction.

Figure 7 .
Figure 7. Temperature-field phase diagrams along (a) [001] c and (b) [011] c directions in 31PIN-43PMN-26PT.Circles and squares show transition temperatures determined from the permittivity measured during heating and cooling processes, respectively.Solid lines are the phase boundary, and dotted lines show the eye guide extrapolating the phase boundary.The letters C, T, O, R, M A , and M B indicate cubic, tetragonal, orthorhombic, rhombohedral, monoclinic A, and monoclinic B symmetries, respectively [8].The letters in the parentheses show the rigorous symmetries under the electric field along each direction.

Figure
Figure 8a-d show typical examples of the P-E hysteresis loops in different electric fields along the [001] c direction in 24PIN-46PMN-30PT, where the frequency of the electric fields is 1 Hz.The P-E hysteresis loops were measured in the temperature range from 180 to 30 • C during the cooling process.It is considered that in Figure 8a, the imperfect triple-loop pattern basically indicates the field-induced transition in the paraelectric phase.Figure 8b-d show typical P-E hysteresis loops revealing the polarization reversal in the ferroelectric phase.

18 Figure 8 .
Figure 8a-d show typical examples of the P-E hysteresis loops in different electric fields along the [001] c direction in 24PIN-46PMN-30PT, where the frequency of the electric fields is 1 Hz.The P-E hysteresis loops were measured in the temperature range from 180 to 30 • C during the cooling process.It is considered that in Figure 8a, the imperfect triple-loop pattern basically indicates the field-induced transition in the paraelectric phase.Figure 8b-d show typical P-E hysteresis loops revealing the polarization reversal in the ferroelectric phase.Crystals 2024, 14, x FOR PEER REVIEW 10 of 18

Figure 8 .
Figure 8. P-E hysteresis loops along the [001] c direction in 24PIN-46PMN-30PT.The frequency of the electric field applied is 1 Hz.The measurement temperatures are (a) 180 • C, (b) 130 • C, (c) 80 • C, and (d) 30 • C.Typical examples of the P-E hysteresis loops in 31PIN-43PMN-26PT under different electric fields along the [001] c direction are shown in Figure10a-d, where the frequency of the electric fields is 1 Hz, and the temperature range measured is from 180 to 30 • C during the cooling process.It is guessed that in Figure10a, the imperfect triple-loop pattern basically indicates the field-induced transition in the paraelectric phase.Figure10b-dshow typical P-E hysteresis loops revealing the polarization reversal in the ferroelectric phase.The temperature dependence of remanent polarization determined by the P-E hysteresis loop measurement is shown in Figure9.The dotted lines indicate the transition temperature determined from Figure4a, where T CT = 171 • C and T TR = 92 • C (see Section 3.1).At the transition temperature between the tetragonal and rhombohedral phases, an anomaly of the remanent polarization is found, although no jump of the polarization appears.This implies the coexistence of the tetragonal, orthorhombic, and rhombohedral phases.

Figure 11
Figure11shows the temperature dependence of the remanent polarization obtained by the P-E hysteresis loop measurement, where the dotted line indicates the transition temperature determined from Figure7b, where T CR = 127 • C (see Section 3.2).Note that in Figure11, the true polarization value in the rhombohedral phase must be multiplied by √ 3.

1 .
Assignment of the Symmetry in 24PIN-46PMN-30PT Let us start with the assignment of the ferroelectric phase in the phase diagram along the [001] c direction in 24PIN-46PMN-30PT.It is seen in Figure 4a that the temperature interval of the tetragonal phase extends as the electric field along the [001] c direction increases, which is consistent with the stability of the polarization in the tetragonal phase under the field in the [001] c direction.Therefore, by extrapolating the phase boundary from the electric field above 1 kV/cm to zero field, we conclude that the phase transition sequence under zero biasing field is determined to be the C-T-R phases.The phase transitions at T CT and T TR under zero biasing field are of the first order, and the transition temperatures are at T CT = 171 • C and T TR = 92 • C. We were unable to determine CEP in the C-T phase transition because of the diffuseness of this phase transition.By extrapolating the phase boundary above 1 kV/cm to E = 0, we also estimated the slopes of the C-T and T-R phase boundaries to be dE/dT CT | E=0 = 0.11 kV/cmK and dE/dT TR | E=0 = −0.16kV/cmK under an electric field along the [001] c direction, respectively.Next, we assign the ferroelectric phases in the phase diagram along the [011] c direction shown in Figure 4b.The temperature interval of the orthorhombic phase extends as the electric field along the [011] c direction increases, which is consistent with the stability of the polarization in the orthorhombic phase under the field in the [011] c direction.We conclude that the orthorhombic phase appears only under the field along the [011] c direction.The field-induced orthorhombic phase is determined to be a metastable phase under zero biasing field, because no orthorhombic phase appears in the electric field along the [001] c direction.The transition temperature T CT is determined to be 175 • C by extrapolating the phase boundary from the electric field above 1 kV/cm to zero field.The slope of the C-T phase boundary is obtained to be dE/dT CT | E=0 = 0.17 kV/cmK under the electric field along the [011] c direction.The reason for the difference of 4 • C in the phase transition temperature T CT along the [001] c and [011] c directions is guessed to be the sample dependence.

Table 3 .
Slope of the phase boundary between A and B phases at zero electric field in the temperaturefield phase diagram.∆p = p B − p A and ∆S = S B − S A .