electric field scaling in Ferroelectrics

ARTICLE IN PRESS Physica B 405 (2010) 2757–2761

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Review

Temperature/electric field scaling in Ferroelectrics Abdelowahed Hajjaji a,n, Daniel Guyomar a, Sebastien Pruvost a, Samira Touhtouh b, Kaori Yuse a, Yahia Boughaleb b a b

Laboratoire de Ge´nie Electrique et Ferroe´lectricite´, LGEF, INSA LYON, Bat. Gustave Ferrie, 69621 Villeurbanne Cedex, France Laboratoire de Physique de la Matiere Condensee, LPMC, De´partement de Physique, Faculte´ des Sciences, 24000 El-Jadida, Maroc

a r t i c l e in fo

abstract

Article history: Received 2 September 2009 Received in revised form 21 January 2010 Accepted 9 March 2010

The effects of the field amplitude (E) and temperature on the polarization and their scaling relations were investigated on rhombohedral PMN-xPT ceramics. The scaling law was based on the physical symmetries of the problem and rendered it possible to express the temperature variation (Dy) as an electric field equivalent DEeq ¼ (a + 2b  P(E,y0))  Dy. Consequently, this was also the case for the relationship between the entropy (G) and polarization (P). Rhombohedral Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics were used for the verification. It was found that such an approach permitted the prediction of the maximal working temperature, using only purely electrical measurements. It indicates that the working temperature should not exceed 333 K. This value corresponds to the temperature maximum before the dramatic decrease of piezoelectric properties. Reciprocally, the polarization behavior under electrical field can be predicted, using only purely thermal measurements. The scaling law enabled a prediction of the piezoelectric properties (for example, d31) under an electrical field replacing the temperature variation (Dy) by DE/(a + 2b  p(E,y0)). Inversely, predictions of the piezoelectric properties (d31) as a function of temperature were permitted using purely only electrical measurements. & 2010 Elsevier B.V. All rights reserved.

Keywords: Ceramics Ferroelectrics materials Scaling law MPN-PT Temperature Polarization Model

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . Scaling law . . . . . . . . . . . Experimental procedure Results and discussion. . Conclusions . . . . . . . . . . References . . . . . . . . . . .

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1. Introduction The solid solution Pb(Mg1/3Nb2/3)1  xTixO3 has been extensively studied for its ferroelectric properties [1–3]. Due to their electromechanical properties, piezoelectric materials are widely used as medical ultrasound imaging probes, sonars for underwater communications, high-sensitivity sensors and actuators [4–6]. Some of the applications, such as those in space, involve environments, where the temperature or/and electric field is/are subject to large variations. It is therefore necessary to characterize the behavior of these ceramics over a wide range of possible operating temperatures or electric fields. Variations in

n

Corresponding author: Tel.: + 33 660474477. E-mail address: [email protected] (A. Hajjaji).

0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.03.023

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temperature or/and electric field would results/result in a significant nonlinear behavior of the material coefficients; thus, leading to the overall performance of the material being affected [7–11]. The occurrence of such a nonlinearity is dependent on the material composition, dopants and internal defects as well as on the magnitude of the temperature variations. Aside from the hysteresis and nonlinearity discussed above, ceramics are also useful in many applications for enhancing the pyroelectric and electrocaloric effects of piezoelectric materials employed for energy harvesting and refrigeration [12–16]. The resultant nonlinear and hysteretic nature of piezoelectric materials induce a power limitation for heavy duty transducers or a lack of controllability for positioners. Consequently, a nonlinear modeling, including a hysteresis appears to be a key issue in order to obtain a proper understanding of transducer behavior and to determine the boundary conditions of use. Several models have

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been proposed in the literature for comprehending the hysteretic behavior of various materials [17–19]. However, a majority of these phenomenological models are purely electric, and it is consequently difficult to interpret the results as a function of the temperature in order to obtain a clear physical understanding. In order to complement these models, a previous study has proposed a scaling law between stress and the electric field [20]. As a continuation of this study, the present paper first puts forward a simple scaling law linking the electric field (E) and the temperature (y). This law was based on the physical symmetries of the problem and rendered it possible to express the temperature (y) as an electrical field equivalent (DEeq  (a +2b  P(E,y0))  Dy, with a and b two constants), and consequently the relationship between the entropy (G) and the polarization (P) also. Rhombohedral Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics were used for the verification, and it was shown that such an approach permitted the prediction of the maximal temperature, using only purely electrical measurements (i.e., of G(E) and P(E)). The maximum temperature for application is the temperature that can be applied to materials without losing their piezoelectric properties (depolarization temperature). The second part of the study describes the use of a scaling law, giving rise to the electric field effect from the temperature effect, in order to predict the behavior of the ferroelectric properties under an electric field.

In order to determine a scaling law between the electric field and the temperature, one should start by following the piezoelectric constrictive equations, restricting them in one dimension. These equations can be formulated with the temperature and the electric field as independent variables; thus, giving dy

y

þ p:dE

GðPÞ ¼ a9PðE, y0 Þ9 þ bPðE, y0 Þ2 ,

ð7Þ

with a and b as constant. Value 9P(E,y0)9 is the absolute value of the polarization. Differentiating the entropy versus P(E,y0) leads to G(E,y0)/ p(E,y0)¼ a  sign(p(E,y0))+ 2b  p(E,y0) In our case, as P(E,y0) remains positive, the expression (7) becomes

GðPÞ ¼ aPðE, y0 Þ þ bPðE, y0 Þ2

ð8Þ

The derivatives of the entropy can be written as: dGðE, y0 Þ ¼ a þ2b  PðE, y0 Þ dPðE, y0 Þ

ð9Þ

Introducing Eq. (9) in the previous calculations leads to ða þ 2b  PðE, y0 ÞÞ 

dPðE, y0 Þ dPðE0 , yÞ ¼p ¼ dE dy

ð10Þ

dPðE, y0 Þ dPðE0 , yÞ ¼p ¼ dE ða þ 2b  PðE, y0 ÞÞdy

ð11Þ

The function a + 2b  P(E,y0) does not depend on temperature. Thus, Eq. 11 can be written as dPðE, y0 Þ dPðE0 , yÞ ¼p ¼ dE dðða þ 2b  PðE, y0 ÞÞyÞ

ð12Þ

According to Fig. 1, for a given value of polarization (P), we can write the following equality P¼dP(E,y0)¼dP(E0,y). Thus,

2. Scaling law

dG ¼ c

the entropy remains positive, leading to

and

dD ¼ eT33 dE þp dy

D ¼ e0 E þP

ð1Þ ð2Þ

where D, P, E, y and G represent the electric displacement, the polarization, the electric field, the temperature and the entropy, respectively, and where c and p, respectively, correspond to the heat capacity and the pyroelectric coefficient. Here, the superscripts signify the variable that is held constant, and the subscript 3 indicates the poling direction. Since the polarization is large enough compared to e0E, Pce0E, then DEP. The coefficients are defined as: dGðE, y0 Þ ¼p dE

ð3Þ

dDðE0 , yÞ dPðE0 , yÞ ¼ ¼p dy dy

ð4Þ

DE  ða þ2b  PðE, y0 ÞÞ  Dy and Dy 

DE , ða þ2b  PðE, y0 ÞÞ

with DE  E E0 ¼E and Dy ¼ y  y0 The term (a + 2b  P(E,y0))  Dy can thereby be considered to play an equivalent role as that of the electric field (DE). Such a statement is fraught with a consequence, since this equivalence must be preserved for all cycles (P, G or coefficients). Moreover (a +2b  P(E,y0))  Dy is equal to aDyC (a  DyC ¼EC, P¼0) when the temperature tends to the Curie temperature (yC). The   E ¼ EC is equivalence thus precisely implies that the couple P¼0   y ¼ yC equivalent to the couple . Hence, P¼0 lim ða þ 2b  PðE, y0 ÞÞ  Dy ¼ EC ) lim ða þ 2b  PðE, y0 ÞÞ ¼

y-yC

y-yC

ð5Þ

ð6Þ

here, y0 and E0 correspond to room temperature (298 K) and the initial electric field (0 kV/mm), respectively. From a physical point of view, the entropy cannot depend on the polarization orientation in the ferroelectrics material. It means that the entropy must be an even function of polarization. Limiting the entropy expansion to the second order and ensuring

¼a

ð14Þ

which can also be expressed as dGðE, y0 Þ dPðE, y0 Þ dPðE0 , yÞ ¼p ¼ dPðE, y0 Þ dE dy

EC

DyC

As illustrated in Fig. 1, the scaling law can be used to derive the behavior of the polarization as a function of the temperature P(y) from P(E) cycle, or reciprocally to drive the polarization behavior versus the electrical field, once the P(E) cycle is known.

For a given P dGðE, y0 Þ dPðE0 , yÞ ¼ p, ¼ dE dy

ð13Þ

Fig. 1. Schematic illustration of the scaling law.

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3. Experimental procedure Rhombohedral Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics were synthesized using the solid-state two-stage process described by Swartz and Shrout [21]. The reagents MgO and Nb2O5 were mixed in ethanol, dried, and reacted at 1200 1C to form the precursor columbite, MgNb2O6. This precursor was blended with TiO2 and PbO, after which a calcination was performed at 850 1C to form a pure perovskite solid solution. Upon milling, polyvinyl alcohol (PVA) was added as a binder. The mixture was cold-pressed into rods and burnt out at 600 1C. The resultant green pellets were sintered at 1250 1C for 4 h, using PbZrO3 as a lead source. The sintering was carried out in sealed alumina crucibles to prevent the evaporation of PbO during densification. During the procedure, a pyrochlore phase, demonstrating poor dielectric properties, may appear [22–24]. All the steps of the process were controlled by means of an X-ray diffraction and it was confirmed that the synthesized ceramics were made up of pure perovskite. For the stabilization of the perovskite phase, excess amounts of MgO and PbO were added, which led to the suppression of the pyrochlore phase formation. The density of all sintered ceramics was above 95% of the theoretical value. The ceramics were silver paste electroded on both sides, and when necessary, poling was performed in a silicone oil bath at 50 1C for 5 min under an electric field applied along the axial direction. The field value was 3.5 kV/mm for the disk samples. The dielectric constant (K) and the dielectric loss factor (tan d) were measured 24 h after poling using a low-capacitance resonator meter (HP 4284 LCR) at 1 kHz. Moreover the piezoelectric coefficient (d33) was determined with the Berlincourt meter. At room temperature, the dielectric constant (K) of the samples was equal to 2100e33, the dissipation factor (tan d) was 1%, and the piezoelectric coefficient (d33) was 250 pC/N. These values were similar to those given by Choi et al. [25]. The temperature dependence of the polarization of the ceramics was obtained through integration of the delivered current using a current amplifier (Keithley model 428). For electric field measurements, the polarization was measured on disk-shaped samples placed in an oil bath at room temperature by applying the electric field in opposite direction to the polarization vector.

4. Results and discussion The effects of various electric fields and temperatures on the polarization profile are illustrated in Fig. 2, where Fig. 2(a)

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represents the polarization variation as a function of the temperature for an electric field E¼ 0 V/mm. It was shown by Hajjaji et al. [26] that the depolarization as a function of the temperature was mainly due to the decrease in the dipole moment, and the fact that the variation in this dipole moment was reversible. In the vicinity of the ferroelectric to paraelectric transition, the temperature depolarization of the ceramics was the result of a 0–901 domain switching, whereas a 0–1801 domain switching did not occur with temperature. At a fixed y (cf. Fig. 2(b)), the polarization variation was minor for low applied electric fields. It then began to increase as E increased gradually from 350 V/mm (a value close to Ec). For the electric field, the depolarization of the ceramic was governed by the domain wall motion. As demonstrated by Pruvost et al. [27], the depolarization process under an electric field was more complicated than its counterpart under a compressed stress or temperature, in the sense that the electric field depolarization involved more than one mechanism. For electric field experiments, there existed three possibilities for domain switching: 0–901, 90–1801 and 0–1801. It should be pointed out that the focus of the present study was to investigate the characteristics of the polarization variation, when the sample was in a stable state. For this, the employed fields (E) were below 450 V/mm (EoEc) and the temperature dependence took place below 373 K. Despite the difference between the mechanisms of depolarization as a function of electric field and temperature, we have try determining a law that links the two (electric field E and temperature y) and to identify one from another. In order to obtain a suitable scaling relation for the ceramic, one can first follow the suggested scaling law given in Eq. (13). This enables a direct determination of the proportionality coefficients a and b from the experimental data. The coefficient a can be determined from the following Eq. (14) (Ec/Dyc ¼ a ¼4300). According to Fig. 2(a) and (b), a plot of the eclectic field (DE) as a function of Dy renders it possible to obtain the coefficient b (b ¼3000). Based on the plot in Fig. 3, it was revealed that the experimental data could be fitted (with R2 ¼0.99), within the measured uncertainty, by DE¼ (a + 2b  P(E,y0))  Dy. In addition, the viability of the proposed scaling law was explored by a way of two distinct experiments on PMN-25PT. Starting from the experimental depolarization under temperature P(y), the depolarization was plotted as a function of (a +2b  P(E,y0))  Dy (giving P(a +2b  P(E,y0)  Dy) and was compared to the direct measurement of P(E). The experimental result under an electric field, P(E), was plotted as a function of E/(a +2b  p(E,y0)) (giving

0.22

0.2 Polarization (C/m2)

Polarization (C/m2)

0.2 0.18 0.16 0.14

0.16 0.14 0.12

0.12 0.1 280

0.18

300

320 340 Temperature (°K)

360

380

-3

-2

-1

Electric field (V/m)

0 x 105

Fig. 2. (a) Polarization versus temperature on Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramic. (b) Polarization versus electric field on Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramic.

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p(E/(a +2b  p(E,y0)))) and was compared to the direct measurement of P(y). This is depicted in Fig. 4. The second comparison was helpful in determining the appropriateness of the scaling law for fields close to the coercive field (Ec). In this area, a small portion of curve P(E) produced a wide range of temperatures on the line P(y), due to y-yC, when E-Ec (cf. Figs. 3 and 4). In a general manner, the experimental and reconstructed cycles were in reasonably good agreement, with regard to both increasing and decreasing paths, for the rhombohedral ceramics (PMN-25PT). This decent correlation for both the P(E) and P(y) cycles thus confirmed the viability of the scaling law. It is interesting to note that for purely electrical measurements, the presented law rendered it possible to determine the maximum temperature for practical use (cf. Fig. 4). Small variations in polarization were observed for an applied electric field lower than EM (here, 150 V/mm), leading to the conclusion that the polarizations underwent a rapid change. Based on the obtained EM value, one can determine the equivalent temperature (yM), corresponding to the maximum temperature used. The relationship Dy ¼ E/(a + 2b  p(E,y0)) leads to both a negative, i.e., Dymin ¼Emin/(a + 2b  p(Emin,y0)), and a positive, i.e., Dymax ¼Emax/(a +2b  p(Emax,y0)), bound. The absolute value of Dymin can thus be considered to be much larger than Dymax.

Fig. 3. Scaling of an electric field against (Dy) for Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramic.

Fig. 5. Piezoelectric constant d31 as a function of temperature for PMN-25PT ceramics.

Consequently, a symmetric electrical field cycle would give rise to a dissymmetric cycle in terms of temperature. Reciprocally, a symmetric temperature cycle would result in an asymmetric cycle in terms of an electrical field. The solid solution Pb(Mg1/3Nb2/3)1  xTixO3 has been extensively studied for its piezoelectric properties, and particularly for its crystallinity. These materials are of interest due to their use in piezoelectric transducers and actuators [4,5]. The stability of the piezoelectric properties under external excitation plays an important role in the choice of material for an application, but it remains difficult to measure these properties under an electric field. The proposed scaling law (DE¼(a + 2b  P(E,y0))  Dy) can thus be used to predict the piezoelectric properties under an electric field using only measures of temperature. Fig. 5 depicts the piezoelectric constant (d31) as a function of temperature for PMN-25PT ceramics. Starting for this Fig. 5 and Fig. 2(a), we can determine the piezoelectric constant (d31) under an electric field. In this case, the temperature variation (Dy) has been replaced by DE¼ (a + 2b  P(E,y0)). Reciprocally, predictions of the piezoelectric constant (d31) as a function of the temperature were permitted using only purely electrical measurements (the electric field E has been replaced by (a + 2b  P(E,y0))  Dy). The proposed scaling law, in this paper, can be used for several models in order to comprehend the hysteretic behavior of materials. The temperature can be added in the electrics models as an equivalent electric field. However, a majority of these phenomenological models are purely electric, and it is consequently difficult to interpret the results as a function of the temperature, in order to obtain a clear physical understanding. The behavior of ferroelectric materials under a combined electric field (E) and temperature (y) can thus be determined, which will help in the identification and understanding of the effect of the simultaneous action of temperature and electric fields on ceramics.

5. Conclusions

Fig. 4. Experimental validation of the scaling law for PMN-25PT ceramic.

In summary, the present paper has proposed a simple scaling law linking the electric field (E) and the temperature (y) (DE ¼(a +2b  P(E,y0))  Dy). The nonlinear behavior was considered and compared to that predicted by a linear reversible constitutive law, in order to demonstrate the range of validity of the linear assumptions. The ease of conversion between P(E) and P(y) cycles by such a simple law provided numerous opportunities regarding the use of piezoelectric materials. The proposed scaling law (DE ¼(a +2b  P(E,y0))  Dy) was employed for predicting the piezoelectric properties under an electrical field using only

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measures of temperature. Reciprocally, predictions of the piezoelectric properties as a function of temperature were permitted using purely only electrical measurements. Such an approach thus rendered it possible to predict the maximal temperature with only purely electrical measurements. References [1] X. Wan, H. Luo, J. Wang, H.L.W. Chan, Solid State Commun. 129 (2004) 401. [2] G. Sebald, L. Lebrun, D. Guyomar, IEEE Trans. UFFC 51 (2004) 1491. [3] S.E. Park, T.R. Shrout, IEEE Trans. Ultrason. Ferroelectrics Freq. Control 44 (1997) 1140. [4] L.E. Cross, 76 (1987) 241. [5] G.H. Haertling, J. Am. Ceram. Soc. 82 (1999) 797. [6] S.E. Park, T.R. Shrout, IEEE Trans. UFFC 44 (1997) 1140. [7] M. Davis, D. Damjanovic, N. Setter, J. Appl. Phys. 96 (2004). [8] R.G. Sabata, B.K. Mukherjee, J. Appl. Phys. 101 (2007) 064111. [9] R. Yimnirun, R. Wongmaneerung, S. Wongsaenmai, A. Ngamjarurojana, S Ananta, Y. Laosiritaworn, Appl. Phys. Lett. 90 (2007) 112906. [10] N. Aurelle, D. Guyomar, C. Richard, P. Gonnard, Ultrasonics 34 (1996) 187. [11] A. Albareda, P. Gonnard, V. Perrin, R. Briot, D. Guyomar, IEEE Trans. Ultrason. Ferroelectrics Freq. Control 47 (2000) 844.

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[12] R. Radebaugh, W.N. Lawless, J.D. Siegwarth, A.J. Morrow, Cryogenics 19 (1979) 187. [13] A.S. Mischenko, Q. Zhang, R.W. Whatmore, J.F. Scott, N.D. Mathur, Appl. Phys. Lett. 89 (2006) 242912. [14] G. Sebald, L. Seveyrat, D. Guyomar, L. Lebrun, B. Guiffard, S. Pruvost, J. Appl. Phys. 100 (2006) 124112. [15] D. Guyomar, S. Pruvost, G. Sebald, IEEE Trans. Ultrason. Ferroelectrics Freq. Control 55 (2008) 279. [16] X.L. Zhang, Z.X. Chen, L.E. Cross, W.A. Schulze, J. Mater. Sci. 18 (1983) 968. [17] S.C. Hwang, J.E. Huber, R.M. McMeeking, N.A. Fleck, J. Appl. Phys. 84 (1998) 1530. [18] T. Steinkopff, J. Eur. Ceram. Soc. 19 (1999) 1247. [19] B. Ducharne, D. Guyomar, G. Sebald, J. Phys. D: Appl. Phys. 40 (2007) 551. [20] D. Guyomar, A. Hajjaji, G. Sebald, K. Yuse, J. Appl. Phys. 105 (2009) 124103. [21] S.L. Swartz, T.R. Shrout, Mater. Res. Bull. 17 (1982) 1245. [22] M. Lejeune, J. Boilot, Mater. Res. Bull. 20 (1985) 493. [23] P.A. Joy, K. Sreedhar, J. Am. Ceram. Soc. 80 (1997) 770. [24] J.P. Guha, J. Eur. Ceram. Soc. 23 (2003) 133. [25] S.W. Choi, R.T.R. Shrout, S.J. Jang, A.S. Bhalla, Ferroelectrics 100 (1989) 29. [26] A Hajjaji, S Pruvost, G Sebald, L Lebrun, D Guyomar, K Benkhouja, Acta Mater. 57 (2009) 2243. [27] S. Pruvost, G. Sebald, L. Lebrun, D. Guyomar, L. Severat, Acta Mater. 56 (2008) 215.