Origin of large electrostrain in Sn4+ doped Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 ceramics

Accepted Manuscript 4+ Origin of large electrostrain in Sn doped Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 ceramics Zhen Liu, Ruihao Yuan, Dezhen Xue, Wenwu Cao, Turab Lookman PII:

S1359-6454(18)30528-7

DOI:

10.1016/j.actamat.2018.07.004

Reference:

AM 14681

To appear in:

Acta Materialia

Received Date: 6 February 2018 Revised Date:

1 June 2018

Accepted Date: 2 July 2018

Please cite this article as: Z. Liu, R. Yuan, D. Xue, W. Cao, T. Lookman, Origin of large electrostrain 4+ in Sn doped Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 ceramics, Acta Materialia (2018), doi: 10.1016/ j.actamat.2018.07.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Origin of large electrostrain in Sn4+ doped Ba(Zr0.2 Ti0.8 )O3 -x(Ba0.7 Ca0.3 )TiO3 ceramics Zhen Liua,b , Ruihao Yuanc , Dezhen Xuec , Wenwu Caob,d,∗ , Turab Lookmana,∗∗ a Theoretical

Matter Science and Technology Institute and Department of Physics, School of Science, Harbin Institute of Technology, Harbin 150080, China c State

Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

Research Institute and Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

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Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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Abstract

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A relatively large bipolar strain of 0.23% was recently reported in 3% Sn4+ doped (BaTi0.8 Zr0.2 )O3 -x(Ba0.7 Ca0.3 )TiO3 (BZT-xBCT) ceramics under a field of 20 kV/cm [1]. This strain is 53% higher than that in BZT-50BCT (0.15% at 20 kV/cm). To systematically study the mechanism and origin of the large electrostrain enhancement in Sn4+ doped BZT-53.3BCT, we develop a parameterized Landau free energy model and perform phase field simulations. Our results indicate that the softening of the elastic modulus C 0 = C11 − C12 accompanied by the reduction of anisotropy energy due to the 3% Sn4+ doping is primarily responsible for the large electrostrain enhancement. Keywords: high electrostrain, Sn4+ doped, BZT-x BCT, Landau parameters, phase field simulation

1. Introduction

electric-induced bipolar strain of 0.23% (unipolar strain 0.19%) at a field of 20 kV/cm (See fig.1(d)) in 3%Sn4+ The lead based solid solution ceramics Pb(Ti1−x Zrx )O3 doped BZT-53.3BCT ceramics[1]. So far it is the largest (PZT) are the most commonly used piezoelectric materials electrostrain reported in the BZT-xBCT system at electric for applications as sensors and actuators because of their fields below 30kV/cm. large piezoelectric coefficient d33 and high Curie temperature[2]. A question we address in this work is why a relatively However, lead-free piezoelectrics have attracted attention 35 small percentage of Sn4+ doped in BZT-53.3BCT can lead in recent years as alternatives to lead based systems. One to such a large electrostrain enhancement (53.3%) comsuch family of alternatives is the Barium Titanate based pared to BZT-xBCT. It has been suggested[3] that the ceramic system (BaTi0.8 Zr0.2 )O3 -x(Ba0.7 Ca0.3 )TiO3 (BZT- large electromechanical response at the MPB composition xBCT) with a d33 of 620 pC/N at the morphotropic phase in BZT-xBCT is related to the free energy flattening and boundary (MPB) composition of x = 0.5[3, 4]. A large di- 40 weak anisotropy near the tricritical point, which leads to electric response was also reported in BZT-xBCT at coman almost negligible energy barrier to polarization rotapositions near the triple point[5] where the energy landtion. However, the existence of a single tricritical point scape is expected to be more favorable in terms of reduced does not necessarily enhance the piezoelectric response, barrier energies. The piezoelectric response can be furand the reduction in the anisotropy energy near the phase ther optimized by ion doping so that it is compositionally 45 boundaries is not the only reason for the large piezoelectric modified[6]. It has been reported that 0.08 wt% CeO2 response [10, 11, 12]. The BZT-xBCT compositions with doped BZT-xBCT has an excellent piezoelectric coeffilarge piezoelectric d33 at the orthorhombic (O) to tetragocient d33 = 673 pC/N and higher Tc ∼ 110 ℃[7] comnal (T) phase boundary also feature a large elastic complipared to the Tc ∼ 93 ℃[3] for BZT-50BCT. The 2% Fe3+ ance S33 value[11, 13] to suggest that the elastic softening doped BZT-50BCT ceramic has an almost hysteresis-free 50 of the system also contributes to a large d33 . An enhanced electric-field-induced strain with an electrostrictive coeffishear elastic compliance S44 has been reported at the O-T cient Q33 = 0.04 m4 /C2 and unipolar strain of 0.08%, and phase boundary [14, 15]. Moreover, the experimental evithus shows promise as an alternative in hard PZT actuadence also shows that the piezoelectric response from the tor applications [8]. However, in BZT-xBCT, the largest tetragonal phase PZT is higher than that for the monoreported bipolar electrostrain is 0.17% at a field 30 kV/cm 55 clinic phase, which is linked to an anomalous softening of E in BZT-45BCT due to tetragonal and rhombohedral phase the elastic modulus 1/S11 of the tetragonal compositions coexistence[9]. We have recently demonstrated a large

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(a)BZT-50BCT,

(b) (Ba0.84 Ca0.16 ) (Ti0.90 Zr0.10 )O3 i.e. (BZT-53.3BCT) without Sn4+ (BZCT-0Sn), and

(c) 3% Sn4+ doped (Ba0.84 Ca0.16 )(Ti0.90 Zr0.07 Sn0.03 )O3 135 (BZCT-3Sn) ceramic.

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The intrinsic and extrinsic piezoelectric properties of the three compounds are calculated and we study the influence of elastic modulus on the electrostrain through phase140 field simulations. Our finding reveals that the softening of elastic modulus C 0 = C11 − C12 along with a flattening of anisotropic free energy contribute to the large electrostrain. 2. Experimental

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2.1. Specimen preparation

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2.2. Experimental results Figure 1(a, b) show the dielectric permittivity and dielectric loss versus temperature properties of the three compounds. Note that there is an offset between the dielectric permittivity peaks and the dielectric loss peaks. As the peaks in the dielectric loss show the temperatures at which the transformation rates are maximum, we use the location of the dielectric loss peaks as a measure of the transformation temperature. The loss tangent data plot is shown as Figure S1 in the Supplementary Material. The heating/cooling rate was 2 K/min (usually employed in literature) controlled by a temperature chamber from the SIGMA company. We have measured the permittivity and dielectric loss versus temperature curves for unpoled and poled samples, and also compared the influence of the heating rate on the T-O phase transition temperature, as shown in Fig. S3. The dielectric loss curves (red and blue) of Fig. S3 show that there is little change of T-O transition temperature(less than 5℃). The Curie temperatures of BZT-50BCT, BZCT-0Sn and BZCT-3Sn are estimated as 91.2 ℃, 89.5 ℃, 75.1 ℃, respectively. The BZCT-3Sn compound has a much lower Curie temperature compared to the other two compounds. Below the Curie temperature, the BZCT-3Sn ceramic has a larger dielectric constant (See Fig.1(a)) and dielectric loss (See Fig.1(b)). The three compounds undergo a cubic (C)-tetragonal (T)-orthorhombic (O)-rhombohedral (R) phase transition with decreasing temperature. The T to O phase transition temperatures, TT O , are estimated as 24.6 ℃, 23.0 ℃, 17.6 ℃, respectively. Similarly, the O to R phase transformation temperatures, TOR , are -6.8℃, -8.3 ℃, -10.3 ℃, respectively. The P-E loop curves in Fig.1(c) show that the coercive fields for the three compounds are lower (∼ 2.5 kV/cm) than those of BaTiO3 (∼ 10 kV/cm) ceramics[23]. The magnitudes of the spontaneous polarization of the three compounds are close, with the BZT-50BCT compound showing a slightly smaller polarization than the other two compounds. The BZCT-3Sn compound shows a very large electrostrain response under an external electric field in Fig.1(d), with an electrostrain enhancement as high as 53.3% compared to BZT-50BCT and BZCT-0Sn at 20 kV/cm.

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temperature. The air conditioning temperature of the lab was set to be 25 ℃. The frequency used for the P-E loops and strain curves was 1 Hz, the uncertainty associated with the measurement of the phase transition temperature is estimated as ±0.5 K. For the electrostrain, we repeated the measurements several times, and estimate the absolute error in the strain as 1%

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closest to the MPB[16]. We perform thermodynamic calculations and phase field110 simulations to study the mechanisms underlying the large electrostrain behavior of Sn4+ doped BZT-xBCT. This is enabled by first parameterizing a Landau phenomenological model for the different dopants and compositions.[17, 18, 19]. An elegant and simple model has been proposed to115 calculate the Landau parameters for solid solutions based on a phase diagram using two special points, namely, the triple and tricritical points,[19, 20]. However, the anisotropic free energy of the sextic terms is neglected in the model. In addition, recent experiments on the BZT-xBCT system indicate that the triple point also appears to be a120 tricritical point[21], in which case, the parameters of the quartic terms may not be fully determined. To date, few studies have carried out and reported a systematic parameterization of the Landau coefficients for BZT-xBCT compounds extracted from experiments, although there125 have been studies on the piezoelectric response at the O-T phase boundary using Landau parameters based on those for BaTiO3 [22]. In this work, we perform a comparative analysis using Landau free energy parameters from experiments for the following: 130

The ceramics were fabricated by a conventional solidstate method with the raw materials BaCO3 (99.8%), CaCO1503 (99.9%), SrCO3 (99.9%), BaZrO3 (99.9%), SnO2 and TiO2 (99.9%). The calcination was performed at 1350 ℃ for 3 h and sintering was done at 1450 ℃ for 3 h in air. All the samples were synthesized under the same conditions to reduce the influence of variable processing on the targeted155 property. The sintered samples for ferroelectric measurements were polished to obtain parallel sides and painted with silver electrodes. The polarization hysteresis loops 3. Landau coefficients from experimental data (P-E loops) were measured by a ferroelectric tester by Radiant Technologies (Precision LCII) and the electrostrain According to Landau-Devonshire theory, the free enwas measured by an MTI 2000 photonic sensor under an ergy G for a ferroelectric can be expressed as an expansion electric field of 20kV/cm, all with disk-shaped samples. of symmetry allowed terms in terms of Pi (i = 1, 2, 3), the The loops were measured at 25 ℃, which was the room 2

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order parameter, to sixth order[14, 24]: θ between the grain’s orientation and the field direction, 175 as shown in Fig.2(b). Thus, the measured polarization X X X X G =α1 Pi2 + α11 Pi4 + α12 Pi2 Pj2 + α111 Pi6 with field is an average of different polarization orientai i i>j i tions. Using the modified equation, P = hPs i + 0 r E, X Y 4 2 2 4 2 where hPs i is the average spontaneous polarization, hPs i + α112 (Pi Pj + Pi Pj ) + α123 Pi , can be extracted by fitting the P-E loop data in Fig.2(c). i>j i 180 This gives hPs i ≈ 0.17 C/m2 for BZCT-3Sn. In bulk (1) ferroelectric ceramics with a large number of grains, we where Pi is the component of spontaneous polarization assume that all grain orientations are equally probable along the three axial directions and α1 , α11 , α12 , α111 , over the whole solid angle before poling. As shown in α112 , and α123 are the coefficients of the free energy denFig. S2, after poling by a [001] electric field, the polarsity. 185 ization probability distribution becomes conical in the poIn principle all the six Landau parameters are temperlarization sphere. For the three compounds BZT-50BCT, ature and composition dependent. However, extracting BZCT-0Sn, and BZCT-3Sn which have stable T phase and all the parameters from experimental data is a difficult metastable O phase at room temperature, after poling the task. Most work in this field has centered on using only range of the angle between the spontaneous polarization two or three temperature dependent parameters, with the190 and z-axis is [0, π/4] according to the symmetry of the assumption that the other parameters are temperature intwo phases. At the two-phase coexisting temperature, dependent or very weakly temperature dependent so that the magnitudes of the polarizations of the T phase and the temperature dependence is ignored. For example, for O phase are close (as estimated in Fig. 3, PO ≈ 1.13PT , BaTiO3 parameters, Li et al. [25], and Heitmann and where PO , PT are the spontaneous polarizations at the Rossetti [20] only allow α1 to be temperature dependent,195 T-O transition), and some metastable O phase domains whereas Pertsev et al. [26] assume that α1 , α11 and α123 could transform to T phase after poling. Thus, after polare temperature dependent. However, the most widely ing by an external field, the intrinsic spontaneous polarused Landau parameters for BaTiO3 reported in Ref. [27] ization Ps is related to hPs i by hPs i = Ps hcos(θ)i, where R π/4 R with α1 , α11 and α123 are temperature dependent, α1 = cos(θ) sin(θ)dθ cos(θ)dA 0 R R π/4 = ≈ 0.85, where dA hcos(θ)i = dA sin(θ)dθ 3.34 × 105 (T − 381) C−2 m2 N, α11 = 4.69 × 106 (T − 393) − 0 2.02 × 108 C−4 m6 N , and α111 = −5.52 × 107 (T − 393) +200 is the unit area of the spherical surface. Thus, the intrinsic spontaneous polarization Ps of the T phase is estimated 2.76 × 109 C−6 m10 N. We see that if T > 443K, then as 0.2 C/m2 for BZCT-3Sn at room temperature. α111 < 0 and the parameters become invalid at temperaThe free energy densities, GT , GO and GR of the tetragtures above 443K since in the Landau model, the coeffional (T), orthorhombic (O) and rhombohedral (R) phases, cient of sixth order should be positive. In our model, we 205 respectively, can be written as follows: choose α1 and α12 to be temperature dependent to avoid this issue. However, by ignoring the temperature dependence in the higher order terms, our parameters will give GT = α0 (T − T0 )PT2 + α11 PT4 + α111 PT6 (3) rise to error in the spontaneous polarization at tempera1 tures far below room temperature so that the pure R phase GO = α0 (T − T0 )PO2 + (2α11 + α12 )PO4 4 polarization might not be adequately captured. The coef1 ficient α1 is related to temperature T by α1 = α0 (T − T0 ), + (α111 + α112 )PO6 (4) 4 where T0 is the Curie-Weiss temperature, and α12 is as1 sumed to be linear with α12 = a(T − T0 ) + b, where a, b GR = α0 (T − T0 )PR2 + (α11 + α12 )PR4 3 ∂2G 1 are constants. Using the relationship εr ε0 = ∂P 2 [28], we 1 obtain: + (3α111 + 6α112 + α123 )PR6 (5) 1 27 = 2α0 (T − T0 ), (2) εr ε0 where PT , PO , PR are the magnitudes of the polarizawhere εr is the relative permittivity of the material, ε0 is tion at the three phases. At the cubic to tetragonal phase the permittivity of vacuum, respectively, and α0 and T0 transition temperature, we have that GT T =Tc = 0, and can be extracted from fitting the dielectric-temperature dGT 1 dPT T =Tc = 0, giving the well known relationship data. Above the curie temperature Tc , εr ε0 is linear in temperature with a slope given by 2α0 . For BZCT-3Sn, 2 α11 Tc = T0 + , (6) Fig.2(a) illustrates the fitting of coefficient α0 = 2.4 × 105 4α0 α111 VmC−1 K−1 . Above the saturation field, the polarization along the between Tc and T0 [28]. For all the three compounds with T T field in a single crystal can be written as P = Ps + 0 r E phase at room temperature, using the condition dG dPT T =25 = [29]. In ferroelectric ceramics, due to the different orienta0, we obtain: tions of different gains, the polarization cannot be poled to α0 (25 − T0 ) + 2α11 Ps2 + 3α111 Ps4 = 0 (7) the field direction at large electric fields, but with an angle

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The spontaneous polarization at a given temperature270 is generally measured from the electric hysteresis loop. At the ferroelectric-ferroelectric phase transformation temperatures TT O and TOR where the two phases coexist, it’s difficult to measure the values of the polarizations of the two different phases. The spontaneous polarizations of275 the coexisting phases are quite close to each other at the transformation temperatures, and they can be related such that PO = γ1 PT , and PR = γ2 PO0 , where γ1 , γ2 are scale factors that are ≈ 1, and PT , PO , PO0 , PR are the spontaneous polarizations at the T-O and O-R phase transi-280 tion temperatures. We estimate the scale factors γ1 , γ2 of the coexisting phases as follows: (1) Measure the spontaneous polarization at several temperatures where the phases are pure T phase, O phase or R phase. (2) Fit the spontaneous polarization function as a function of tem-285 perature for each phase. (3) Calculate γ1 , γ2 by extrapolating the spontaneous polarization value of each phase to the transformation temperature. Taking BZCT-3Sn as an example, Fig.4 shows the normalized polarization P/PRT for different phases at temperatures away from290 the transformation temperatures, where PRT is the polarization value at room temperature. At the T phase temperature,pthe polarization is related to temperature by P 2 = λ[1 + 1 − φ(T − T0 )] according to Landau theory, where λ and φ are two constants, and T0 is the CurieWeiss temperature. By fitting the measured data and ex-295 trapolating to the T-O phase transformation temperature TT O = 17.6℃, we obtain PT /PRT = 1.02 and PO /PRT = 1.15, thus γ1 = PO /PT ≈ 1.13. Also at the O-R phase transformation temperature TOR = −10.3℃, we obtain PO0 /PRT = 1.25 and PR /PRT = 1.32, thus γ2 = PR /PO0 ≈300 1.06. By substituting the values for T0 , TO , TR , α11 , α111 and the polarization scale factors into Eq.(8-13), the interaction parameters α12 , α112 , α123 can be obtained. Using this approach, we list in Table 1 the calculated Landau parameters from the experimental data for the three305

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compounds BZT-50BCT, BZCT-0Sn and BZCT-3Sn. The limitation of this parameterized model is that it’s only applicable to the compositions undergoing a C-T-O-R phase transformation. In the BZT-xBCT system, this approach is valid for compositions 0.35 < x < 0.6 near the morphotropic phase boundary. In order to estimate the changes in the parameters due to the proximity to room temperature of the T-O phase transition temperature, we evaluated the parameters for BZT-50BCT at 25 ℃ and 35 ℃, and BZCT-3Sn at 25 ℃ and 15 ℃, that is, approximately a ten degree difference from their transition temperatures. As shown in Tables S1 and S2, the maximum change in the parameters is about 5%, not sufficient to affect the salient physics that we study here. In our parameterized model, the variation of the spontaneous polarization Ps from the P-E loops at room temperature and the scale factors γ1 and γ2 are the major contributors to the errors of the results. In Fig. S6(a), we calculated the spontaneous polarization versus temperature by varying Ps from 0.19 C/m2 to 0.20 C/m2 , keeping the scale factors γ1 = 1.13 and γ2 = 1.06. We can see that the two polarization curves have the same transformation behavior with decreasing temperature. However, the Ps =0.20C/m2 curve has a higher spontaneous polarization. The difference in the polarizations between the two curves slightly increases as the temperature decreases. At the R phase temperature with T = −35℃, the polarization difference increases to 0.015C/m2 . Therefore, a 5% variation of Ps at room temperature will lead to an error of 7.5% in the polarization at T = −35℃, and the error could be even larger if the temperature is far below -35℃, which means that the low temperature R phase polarization might not be adequately captured. In Fig. S6(b) we vary the scale factors, keeping Ps = 0.2 C/m2 . Varying γ1 and γ2 will influence the magnitude of the spontaneous polarization of O phase and R phase, with larger values of γ1 and γ2 leading to even higher spontaneous polarizations of O phase and R phase. The relationship between the spontaneous strain and polarization is given by 0ij = Qijkl Pk Pl (i, j, k, l = 1, 2, 3)[30], where Qijkl is the electrostrictive coefficient. For BZT50BCT, the electrostrictive coefficients Q11 = 0.066, Q12 = −0.023, and Q44 = 0.055 using the Voigt notation. Followa −a0 ing the same method as in Ref.[31], ε011 = ε022 = Ta0 C ,

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By substituting the values for Tc , T0 , Ps into Eq.(6)(7),250 the solved parameters a11 , a111 for BZCT-3Sn are −1.63 × 108 C−4 m6 N and 4.94 × 109 C−6 m10 N, respectively. At the T to O transition temperature TT O , both T and O stable phases coexist. Hence, 255 GT T =T = GO T =T (8) TO TO dGT = 0 (9) dPT T =TT O dGO = 0 (10)260 dPO T =TT O

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ε033 = Ta0 C , where a0C is the cubic cell length extrapC olated into the tetragonal region. Using our parameters in Table 1, the lattice parameters as a function of temperature can be estimated. In Fig.4, we use the available experimental data for BZT-50BCT [32] and compare the measured and predicted values for the lattice parameters. The agreement is reasonable and suggests that the parameterized model captures the salient aspects. In order to understand the larger electrostrain enhancement associated with the 3% Sn4+ doped ceramic BZCT3Sn compared to BZT-50BCT and BZCT-0Sn, we carried out several theoretical calculations by using the fitted Lan-

ACCEPTED MANUSCRIPT Parameters α1 (105 C−2 m2 N) α11 (108 C−4 m6 N) α12 (106 C−4 m6 N) α111 (109 C−6 m10 N) α112 (109 C−6 m10 N) α123 (109 C−6 m10 N)

BZT-50BCT 2.4 × (T − 86.1) -1.88 −0.093 × (T − 86.1) + 464 7.22 0.64 -3.89

BZCT-0Sn 3.1 × (T − 80.3) -3.18 −1.48 × (T − 80.3) + 369 8.87 2.37 -4.50

BZCT-3Sn 2.4 × (T − 69.5) -1.63 −0.59 × (T − 69.5) + 299 4.94 0.78 -2.71

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4. Thermodynamic discussion of the intrinsic properties of the compounds

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In order to investigate the domain structure and evolution for the BZT-50BCT, BZCT-0Sn and BZCT-3Sn compounds, we calculated the behavior of the minimized365 Landau free energy density versus polarization with decreasing temperature, as shown in Figure 5. At the room temperature of 25 ℃, all three compounds show a stable T and metastable O phase. However, the free energy of the metastable O phase in the BZCT-3Sn compound is370 much higher compared to BZT-50BCT and BZCT-0Sn, indicating that the O phase is more unstable in BZCT3Sn. On decreasing the temperature to 15 ℃, BZT-50BCT possess a stable O phase (See Fig.5(a)), whereas BZCT0Sn and BZCT-3Sn have coexisting T and O phases (See375 Fig.5(b)(c)). At a temperature -10 ℃, which is close to the O-R transformation temperature, all the three compounds show coexisting O and R phases. The calculated Landau energy density from the fitted parameters is thus consistent with the phase transformation process as per380 experimental observations. The calculated predictions from the model for all three compounds are shown in Fig.6. The peaks of the rela2 tive permittivity εr calculated by εr1ε0 = ∂∂PG2 shown in Fig. 6(a) for the three compounds occur at the Curie385 temperature and decrease to below 600 at room temperature. Fig.6(b) shows the spontaneous polarization versus temperature. For the compounds undergoing a first order phase transformation, the spontaneous polarization is discontinuous at the Curie temperature. The spontaneous390 polarization increases as the temperature decreases below Tc . The magnitude of the polarizations for BZCT-0Sn and BZCT-3Sn is about 0.20 C/m2 , which is slightly larger

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than that for BZT-50BCT (∼ 0.19 C/m2 ). Fig.6(c) shows the intrinsic P-E curves for the three compounds, the coercive field for BZCT-3Sn is close to that for BZT-50BCT (∼ 45 kV/cm), but smaller than ∼ 60 kV/cm for BZCT0Sn. The intrinsic piezoelectric coefficients can be calculated using dij = λik ηkj , where λik = (∂ 2 F/∂Pi ∂Pk )−1 is the dielectric stiffness and ηkj = ∂ 2 F/∂Pk ∂σj defines the piezoelectric coefficients[31, 33]. Thus, the intrinsic piezoelectric coefficient d33 can be written as d33 = 2r 0 Q11 Ps so that large values of intrinsic permittivity, spontaneous polarization Ps and electrostrictive coefficient Q11 can lead to a large intrinsic d33 . The electrostrictive coefficient Q11 for BZT-50BCT is 0.066 m4 /C2 , smaller than 0.08 m4 /C2 for the other two compounds. The calculated intrinsic d33 values for BZT-50BCT, BZCT-0Sn and BZCT-3Sn are 146 pC/N, 134 pC/N and 232 pC/N, respectively. Note that Gao et al. evaluated an intrinsic d33 ≈ 196 pC/N for BZT-50BCT ceramic under an electric field of 35 kV/cm in experiments[34]. However, the actual value of intrinsic d33 should be smaller than 196 pC/N since their applied maximum electric field is smaller than the saturation field (over 40 kV/cm). On the other hand, our calculated intrinsic d33 of BZT-50BCT is much lower than that for the total d33 (∼ 600 pC/N)[3]. The intrinsic contribution to the total d33 is only about 25%, indicating that the extrinsic piezoelectric activity is the major contributor to the observed d33 . Experimental studies show that the extrinsic contribution to the total d33 is as high as 67% for BZT50BCT ceramic and over 60% for BZT-45BCT [34, 35], in agreement with our calculation. We can consider the enhanced piezoelectric response to be related to the reduction in the anisotropy energy leading to a lowering of energy barriers to facilitate ease of domain switching or polarization rotation[3, 36, 37]. The energy variation with polarization angle in the (1¯10) plane starting from [001] direction is calculated in Fig.6(d). Compared to BZCT-0Sn, one can clearly see that the 3% Sn4+ doped compound has a reduced anisotropy energy and lower energy barrier between the T and O phases. However, the BZCT-3Sn compound has similar anisotropy energy and energy barrier compared to BZT-50BCT, indicating that the reduced anisotropy energy is not the only reason for the large electrostrain response. Therefore, the extrinsic contribution to the large electrostrain needs to be further studied.

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dau parameters in Table 1. The contributions to the dielectric and piezoelectric properties can be essentially di-350 vided into two aspects: the intrinsic contributors related to the lattice property and the extrinsic contribution from the domain switching behavior. Hence, we combine thermodynamic calculations of the intrinsic properties with phase field simulations of domain switching behavior to355 provide insight into the origin of the larger electrostrain enhancement in BZCT-3Sn compared to BZT-50BCT and BZCT-0Sn.

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5. Phase field simulations of domain structure and430 phase is determined by the annealing time to relieve interswitching behavior nal stresses. The volume fraction of the metastable phase generally decreases with the annealing time. This is due The thermodynamic discussions of the intrinsic properto the quenching and residual stress; not all metastable O ties of the three compounds above indicate that the BZCTphase is relaxed to T phase and the actual domain struc3Sn compound has higher intrinsic d33 (∼ 230 pC/N)435 ture is a mixture of O phase and T phases. This is similar than the other two compounds (∼ 140 pC/N). Here, the to the PZT ceramics at the MPB composition in which T extrinsic effect on total piezoelectric response connected and R phases coexist, and the system tends to increase the to domain switching and polarization rotation has been volume fraction of R phase with annealing time accompastudied via phase field simulations. In these studies, benied by the relief of internal stress[40]. sides the Landau free energy G, the elastic energy Fela , electrostatic energy Fe and gradient energy Fg contribute440 5.2. Study of domain switching behavior to the total free energy F (Details of the total free enWe applied an external field along [10] direction in the ergy functional forms can be found in Ref.[38]), such that simulations to study in domain switching. As shown in F = G + Fela + Fe + Fg and the temporal evolution of the Fig.8, the three compounds have similar domain morpholpolarization field can be obtained by solving the dependent ogy without an external electric field. The {01} domains Ginzburg-Landau equation [38, 39]: 445 will switch to {10} domains as the field is increased and there is movement of 90◦ domain walls accompanying the δF ∂Pi = −L , (i = 1, 2, 3) (14) domain switching. At E[10] = 20 kV/cm, the average {01} ∂t δPi domain size in BZCT-3Sn is smaller than that in BZTHere L is the kinetic coefficient, t is evolution time and 50BCT and BZCT-0Sn, indicating that the domains are Pi is the polarization component along the three axis di-450 easier to rotate in BZCT-3Sn compared to the other two rections. We used a 128×128 mesh with periodic condicompounds. The unipolar electrostrain versus electric field tion to simulate the domain structures and switching bedependence is calculated in Fig.9. The maximum value havior. The gradient energy density is written as fg = of the calculated electrostrains are slightly smaller than 1 2 2 g[(∇P ) + (∇P ) ], where g is the gradient coefficient. the experimental results because the rotation of {001} dox y 2 We use g ≈ 17 × 10−11 C2 m4 N for the three compounds455 mains are neglected in our two dimensional simulations. in the simulation. The elastic parameters used for BZTThe BZCT-3Sn compound has a larger strain than the 50BCT, BZCT-0Sn and BZCT-3Sn are C11 = 13.0 × 1010 other two compounds due to domain switching, consistent N/m2 , C12 = 9.0 × 1010 N/m2 , C44 = 3.44 × 1010 N/m2 ; with the experimental results in Fig.1(d). C11 = 14.0 × 1010 N/m2 , C12 = 11.0 × 1010 N/m2 , C44 = As emphasized in Ref [13], a drastic reduction in anisotropy 3.44 × 1010 N/m2 and C11 = 13.5 × 1010 N/m2 , C12 =460 energy in the absence of enhanced elastic softening will not 11.4 × 1010 N/m2 , C44 = 3.44 × 1010 N/m2 , respectively. result in a larger piezoelectric response. It has been sugSimilarly, the electrostrictive coefficients for BZT-50BCT gested that a large piezoelectric coefficient is connected are Q11 = 0.066 m4 /C2 , Q12 = −0.023 m4 /C2 with Q44 = with a larger elastic compliance S33 [11]. Regarding this 0.055 m4 /C2 . For both BZCT-0Sn and BZCT-3Sn we used point, the measured elastic compliance S33 and elastic Q11 = 0.08 m4 /C2 , Q12 = −0.034 m4 /C2 with Q44 = 0.06465 stiffness C33 of the three compounds are given in Fig.10. E D m4 /C2 . The elastic compliance parameters S33 and S33 decrease E D as the elastic stiffness parameters C33 and C33 increase. 5.1. Domain structure evolution with time Thus, no elastic softening related to enhanced S33 occurs in BZCT-3Sn. This suggests that other elastic parameters The evolution of the domain structure in BZCT-3Sn is shown in Fig.7. The domain patterns are visualized by470 may also affect the polarization rotation or domain switching. Therefore, we carried out several phase field simuladifferent colors, and the red vectors in Fig.7(b - d) repretions by adjusting the elastic parameters C11 , C12 , and C44 sent the polarization distribution within the domains. For and find that the electrostrain is significantly influenced by the T phase domains, the polarization directions are along the shear C 0 = C11 − C12 . As shown in Fig.10(b), a small [10] (light green), [¯ 10], [01], and [0¯ 1]. For the metastable O 0 phase domains, the polarization directions are along [11],475 reduction in C gives rise to a larger electrostrain. If the ferroelectric system has weak anisotropy, the shear modu[¯11], [1¯ 1], and [¯ 1¯ 1]. The initial condition at the start of the lus 2C44 ≈ C11 − C12 , namely, softening of elastic modusimulation takes the form of a small fluctuation in polarlus C 0 is softening of shear C44 , accompanied by enhanced ization (Fig.7(a)) and after iterating for 3000 time steps elastic compliance S44 . However, for a ferroelectric syswith the normalized time t∗ = 30, the T phase and O phase 480 tem with strong anisotropy, C 0 is no longer related to C44 structures are seen to coexist as shown in Fig.7(b). The and C 0 is then a typical constant. Thus, we suggest that metastable O phase is suppressed with the relaxation time ∗ ∗ the doping of 3% Sn4+ in BZT-53.3BCT leads to a small in Fig.7(c) at t = 60. After a time interval of t = 500 in reduction in C 0 , which gives rise to ∼ 50% electrostrain Fig.7(d), most O phase domains are seen to relax to staincrease. ble T phase. Experimentally, the fraction of metastable O 6

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We have proposed a parameterized model of the BZTBCT system to fit the Landau parameters to the compounds BZT-50BCT, BZCT-0Sn and BZCT-3Sn. Using550 these parameters, we carried out thermodynamic calculations and phase field simulations to investigate the mechanisms underlying the large electrostrain enhancement due to a relatively small 3% Sn4+ fraction. Our results indicate555 that a small reduction in C 0 = C11 − C12 accompanied by a decrease in the anisotropy energy leads to a large electrostrain. This finding can potentially serve as a guide for experiments to optimize the electrostrains in piezoelectrics560 by ion doping.

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[17] The work was supported by the Center for Nonlinear Science and LDRD at Los Alamos National Laboratory and by the National Key Basic Research Program of China (Grant No.2013CB632900), and the State Scholar-570 ship Fund of the China Scholarship Council (No.201506120195).[18]

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Figure 1: Experimental dielectric and piezoelectric properties for the three Barium Titanate compounds BZT-50BCT, BZCT-0Sn and BZCT3Sn. (a) The corresponding dielectric-temperature curves. (b) The dielectric loss curves. (c) The electric hysteresis loops and (c)The electrostrain vs. external field dependence showing a large strain at 20 kV/cm.

Figure 2: Schematic of fitting the α0 and spontaneous polarization Ps from permittivity curve and P-E loop. (a) Over the Curie temperature, 1/εr is linear with the temperature, the coefficient α0 can be calculated from Eq.(2). (b) Schematic of local polarization direction in different oriented grains. (c) Schematic of fitting average spontaneous polarization in the ceramic.

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Figure 3: Polarization versus temperature for the BZT-3Sn compound. The polarization P is normalized by PRT , where PRT is the polarization for BZCT-3Sn at room temperature. The black points are measured data at temperatures away from the transformation temperature, the polarizations of different phases at the coexisting temperatures can be extrapolated by fitting to the measured data

Figure 4: Measured and computed crystal lattice parameters of BZT50BCT. The experimental data points shown are taken from Ref.[32].

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Figure 5: The free energy density pattern with polarization at 25℃, 15℃and -10℃, respectively in the three compounds. The colorbar from blue to red denotes the free energy density from low to high. At room temperature, (a)(b)(c) indicate that the three compound have stable T phase and metastable O phase. With the temperature decreases, the three compounds transform from T to O to R phase.

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Figure 7: Domain evolution for bulk BZCT-3Sn, the red vectors in (b)(c)(d) represent the direction of the polarization vectors. For the T phase domains, the polarization directions are along [10] (light green), [¯ 10], [01], and [0¯ 1]. For the metastable O phase domains, the polarization directions are along [11], [¯ 11], [1¯ 1], and [¯ 1¯ 1]. The domain structures in the dashed rectangular regions depict the metastable O phases. The metastable O phase domains relax to T phase domains as the evolution time t∗ increases. (a) t∗ = 0 (b)t∗ = 30 (c) t∗ = 90 and (d) t∗ = 500 (t∗ = t/t0 and t0 = 50/L ns).

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Figure 8: Phase field simulations of domain switching behavior under [10] electric field. The white arrows denote the directions of the polarization in the domains. At E[10] = 20 kV/cm, the average {01} domain size in BZCT-3Sn is smaller than that in BZT-50BCT and BZCT-0Sn, indicating that the domains are easier to rotate in BZCT-3Sn compared to the other two compounds.

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Figure 9: Calculated electrostrain vs. electric field of the three compounds. The BZCT-3Sn compound has a larger electrostrain compared to the BZT-50BCT and BZCT-0Sn compounds.

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E and S D denote the short and open circuit elastic compliances parameters. Figure 10: (a) The measured elastic compliance and stiffness. S33 33 E and C D represent the short and open circuit elastic stiffness parameters . (b) The influence of the elastic parameter C 0 (C 0 = And C33 33 C11 − C12 ) on the electrostrain of BZCT-3Sn compound.

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