Electromechanical coupling and temperature-dependent polarization reversal in piezoelectric ceramics

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0885–3010/$25.00 © 2011 IEEE

Electromechanical Coupling and Temperature-Dependent Polarization Reversal

in Piezoelectric Ceramicspaul m. Weaver, markys G. cain, Tatiana m. correia, and mark stewart

Abstract—Electrostriction plays a central role in describ-ing the electromechanical properties of ferroelectric materials, including widely used piezoelectric ceramics. The piezoelectric properties are closely related to the underlying electrostric-tion. Small-field piezoelectric properties can be described as electrostriction offset by the remanent polarization which characterizes the ferroelectric state. Indeed, even large-field piezoelectric effects are accurately accounted for by quadratic electrostriction. However, the electromechanical properties de-viate from this simple electrostrictive description at electric fields near the coercive field. This is particularly important for actuator applications, for which very high electromechanical coupling can be obtained in this region. This paper presents the results of an experimental study of electromechanical cou-pling in piezoelectric ceramics at electric field strengths close to the coercive field, and the effects of temperature on elec-tromechanical processes during polarization reversal. The roles of intrinsic ferroelectric strain coupling and extrinsic domain processes and their temperature dependence in determining the electromechanical response are discussed.

I. Introduction

Electrostriction plays a central role in describing the electromechanical properties of ferroelectric ma-

terials [1], including widely used piezoelectric ceramics. Electrostrictive strain has a quadratic relationship to the polarization [1]. In simplified form,

S QP= 2, (1)

where S is the strain, P is the polarization, and Q is the electrostrictive coefficient. In the case of a ferroelectric below the curie temperature, the polarization and strain may be considered as comprising both spontaneous and field-induced components. The piezoelectric properties are closely related to the underlying electrostriction. small-field piezoelectric properties can be described as elec-trostriction offset by the remanent polarization which characterizes the ferroelectric state [2]. For small-field piezoelectric effects in a poled ceramic at room tempera-ture, the total polarization is not significantly affected by the applied field. In this case, the total remanent polar-ization is unchanged by relatively small changes in the

induced polarization. It is straightforward to show [1] that the small-field piezoelectric coefficients can be obtained from the electrostriction coefficients by the following rela-tions [3]:

d Q P31 12 0= 2 rε ε (2)

d Q P33 11 0= 2 rε ε. (3)

However, in many actuator applications, soft pZT materi-als are used at high electric field. They therefore show a high degree of induced polarization which is comparable to the remanent polarization. many applications also require operation across a wide temperature range. at elevated temperatures, the remanent polarization decreases and the induced polarization increases [4], [5], so the assump-tions on which (2) and (3) are based are no longer valid. This means that electrostriction dominates the thermal expansion and high-field electromechanical response, and it must be taken into account directly to fully describe high-field electromechanical coupling in soft ferroelectrics or at high temperatures.

For unipolar actuator operation with the electric field applied in the direction of poling, the electrostrictive re-lationship of (1) provides a very accurate description of the high-field electromechanical coupling across a wide temperature range [4]. This is remarkable, considering the complexity of the extrinsic domain processes [2] taking place. However, when the electric field is reversed and bi-polar operation is considered, strong deviations from the electrostrictive behavior are observed in the region of the coercive field [5]. The behavior of the ferroelectric material in this regime is important in two respects. First, many actuator applications employ bipolar electric fields to ex-tend the actuation range [6]. a better understanding of high-field electromechanical coupling in this regime will lead to improved design of actuators and their control sys-tems. second, measurement of electromechanical coupling under bipolar electric fields can provide a unique insight into the complex temperature dependency of domain pro-cesses in ferroelectrics. This is an area that is not fully un-derstood and is relevant to a wide range of technological applications of ferroelectrics and piezoelectrics.

This paper presents the results of an experimental study of electromechanical coupling at electric field strengths close to the coercive field of piezoelectric ceramics, and the effects of temperature on electromechanical processes during polarization reversal. The roles of intrinsic ferro-

manuscript received January 4, 2011; accepted april 22, 2011. The authors acknowledge the financial support of the Technology strategy board and the UK national measurement office.

The authors are with the national physical laboratory, Teddington, UK (e-mail: [email protected]).

digital object Identifier 10.1109/TUFFc.2011.2010

weaver et al.: electromechanical coupling and temperature-dependent polarization reversal 1731

electric strain coupling and extrinsic domain processes and their temperature dependence in determining the electromechanical response are discussed. It is shown that measurement of mechanical as well as electrical properties provides a rich source of new information on domain pro-cesses and actuator response.

II. Experimental details

a commercial soft pZT-based ceramic (c91, Fuji ce-ramics, Tokyo, Japan) was used for the experiments de-scribed here. datasheet values for d33, d31, and the curie point are 640 × 10−12 m·V−1, −330× 10−12 m·V−1, and 165°c, respectively. samples were in the form of sheets 25 × 5 × 0.3 mm with approximately 2 μm of ni electrode on both 25 × 5 mm faces. structural characterization of the ceramic was performed by X-ray diffraction using cuKα radiation (Fig. 1). The highly visible (200)/(002) and (112)/(211) doublet peaks are indicative that the in-vestigated ceramic crystallized in a tetragonal structure.

Electrically and thermally induced strain of the ceramic were measured using a vertical pushrod dilatometer. cor-rections were made for expansion of the pushrod by cali-bration against a reference material of known expansivity. strain was measured in the 25 mm length for an electric field applied across the 0.3 mm thickness (this was also the direction of poling), i.e., transverse (d31) expansion or contraction. a positive strain refers to an expansion. note that results in this paper are mostly plotted as negative strain (noted on the graph labels) to present the familiar direction of the strain loops.

Electric field was applied using a high-voltage ampli-fier under computer control. simultaneous measurements were made of the applied voltage, charge, extension, and temperature. The bipolar loop sequence comprised a series of linear ramps 0 to +400 V to −400 to 0 V applied over a period of 40 s (Fig. 2). Two loops were performed for each temperature point (approximately every 10°c) for a heating/cooling rate of 20°c per hour.

polarization was measured by means of a capacitor (C = 17.2 μF) placed in series with the piezoelectric element

(approximately 17 nF at room temperature). The volt-age (V) across the capacitor was measured by means of a high-input-impedance unity-gain instrumentation ampli-fier. The capacitor was discharged only at the start of the temperature cycle and not between successive loops. This is important, because the calculation of electrostric-tive strain requires knowledge of the absolute polariza-tion, which varies with temperature, and the definition of an experimentally realizable polarization zero requires some care. changes in polarization were computed from the relation

∆∆

PC VA= , (4)

where ΔV is the change in voltage from the start of the temperature cycle and A is the ceramic electrode area. The small amount of drift caused by input bias currents was measured and compensated for in the results. For each loop, the loop average of the polarization was calcu-lated. It has been shown previously under similar experi-mental conditions [5] that the loop average polarization does not change significantly with temperature, and that the low-temperature value was the same as that obtained in the paraelectric phase above the curie temperature. The loop average polarization was then averaged over the whole temperature range to calculate a single zero point for the polarization measurement, ΔP0. all polarization values presented here were obtained using

P P P= 0∆ ∆− . (5)

III. deviation from Electrostriction

Eq. (1) represents an idealized quadratic electrostric-tion, which is closely adhered to in the paraelectric phase above the curie temperature [5] and for unipolar electri-cal cycles below the curie temperature [4]. For bipolar loops below the curie temperature, the strain deviates from that predicted by (1). Fig. 3 shows a typical polar-

Fig. 1. X-ray diffraction pattern for Fuji c91 ceramic. Fig. 2. applied electric field waveform.

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ization-electric field (pE) loop at room temperature. The coercive field is defined as the electric field at the point in the pE loop where the polarization becomes equal to zero [1]. There are two such points (coercive points) in Fig. 3: one for the field changing from a negative extremum, through the coercive field, to a positive extremum (posi-tive-going electric field), and a corresponding point for a negative-going electric field. The part of the pE loop (and corresponding strain-electric field loop) where the electric field is close to, and passes through, the coercive field is referred to, in this paper, as the coercive region.

It is possible to apply (1) to the polarization data to obtain the strain that would be expected if a purely qua-dratic electrostriction was obtained. The square of the polarization is shown in Fig. 4 and has the form of the

familiar butterfly loop typical of strain in ferroelectric ma-terials.

To convert the P 2 values to strain, the electrostric-tive coefficient must be known. because the strain ori-gin is arbitrary, an offset figure is also required. both of these figures were obtained by applying a quadratic fit to the strain-polarization curve in the region of decreas-ing electric field between maximum field and zero field as described previously [5]. a value for the electrostriction coefficient of Q12 = 0.0105 c−2·m4 was used for the data shown in Fig. 5, where the strain calculated from the po-larization is compared with the directly measured strain. (a slight asymmetry in both the quadratic strain loop and the measured strain loop was removed by averaging the polarization amplitude, field amplitude, and strain over positive and negative field values.)

The two loops match very closely for the region of de-creasing electrical field from the maximum strain to zero field, as expected from the fitting procedure used for the calculation of the electrostrictive coefficient. The close-ness of the fit over this region confirms that the strain very closely follows the electrostrictive quadratic function. However, as the field is reversed, the P 2 strain starts to deviate from the measured value, particularly around the coercive field. There is a whole region of strain amplitude predicted from the P 2 function that is not observed in the strain measurement. To examine this more closely, the difference between the measured strain and the P 2 strain is plotted in Fig. 6.

This difference between the electrostrictive strain and the measured strain represents large changes in polariza-tion that are not accompanied by a corresponding strain. It is interesting to note that the peak in this strain dis-crepancy occurs at a field slightly higher than the coercive field. When the electric field is decreasing from maximum, there is no strain discrepancy. The widely accepted view of the piezoelectric response of poled ceramics is [2], [3]

Fig. 3. pE loop at 19°c. Transitions through the coercive region are marked in gray for (a) positive-going electric field and (b) negative-going electric field (the direction of change is marked by the arrows).

Fig. 4. P 2 variation with electric field at 19°c (from the pE loop shown in Fig. 3).

Fig. 5. strain field loops directly measured (circles) and calculated from polarization data (dashed line).

weaver et al.: electromechanical coupling and temperature-dependent polarization reversal 1733

that the strain response is formed of two contributions 1) distortion of the unit cells within a single domain, re-ferred to as intrinsic; and 2) deformation caused by the motion of non-180° domain walls. In soft pZT ceramic, the contribution from domain wall movement is dominant [7]. Fig. 6 shows that as the coercive region is approached, there is an increasing contribution from non-strain induc-ing polarization changes that must, therefore, be caused by increased mobility of 180° domain walls. conversely, the close adherence to the electrostrictive description in the unipolar region implies that 180° domain walls are largely inactive in this region, confirming earlier obser-vations made by comparing dielectric and piezoelectric coefficient variation with temperature [8]. comparison of the electrostrictive and measured strain, therefore, per-mits discrimination of the different domain wall processes within the material.

We can now examine how the strain discrepancy is af-fected by temperature. Fig. 7 shows the strain discrepancy across a range of temperatures. It is clear that as the temper-ature increases, the strain discrepancy decreases, until, above the curie temperature, a precisely quadratic electrostriction is followed [5] and the strain discrepancy disappears.

This can be seen clearly in Fig. 8, which plots the maxi-mum strain discrepancy as a function of temperature. It follows an almost linear decrease up to 140°c, then rap-idly decreases to zero. This closely reflects the behavior of the remanent strain (the strain at zero electric field) after compensation for thermal expansion, which is also plot-ted. (Thermal expansion was compensated by subtracting the value of the strain zero obtained from the quadratic fit, as described previously [5].) Fig. 9 shows that the strain discrepancy as a fraction of the remanent strain hardly varies at all across the temperature range with a

value ranging from 0.58 in the low-temperature limit to 0.67 at a temperature just below the curie temperature. The slight rise in this ratio would suggest a small increase in the relative contribution of 180° domain wall movement as the temperature increases. This correlates with a small decrease in the electrostriction coefficient with increasing temperature reported previously [5].

IV. strain rate in the coercive region

as discussed previously, the dominant contribution to the piezoelectric strain is from non-180° domain wall

Fig. 6. difference between the directly measured strain and the strain calculated from polarization data. Increasing E refers to a field applied opposite to the polarization direction and increasing in magnitude from zero to maximum. decreasing E refers to the field decreasing from maxi-mum amplitude back to zero. The vertical line represents the value of the coercive field.

Fig. 7. Temperature variation of the difference between the directly measured strain and the strain calculated from polarization data. The temperature interval between plots is approximately 10°c. The lowest temperature plotted is 19°c and the highest is 186°c. plots at 19°c and 137°c are marked (above 137°c the peaks become indistinct).

Fig. 8. Temperature variation of the peak strain discrepancy, and the remanent strain after compensation for thermal expansion.

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movement. If this were the only contribution, then, in the tetragonal system, the strain rate and current density could be estimated from the tetragonal distortion and the spontaneous polarization by the relationship [9]

34 =0 0

0 0

3c aa P

ej

− �, (6)

where a0 and c0 are the lattice constants, P0 is the spon-taneous polarization, �e3 is the strain rate in the direction of the applied field,

�edSdt3

3= , (7)

and j is the current density:

jdPdt= 3 . (8)

The physical reason for this relationship can be un-derstood by considering a small homogeneous region of a tetragonal crystal polarized at 90° to the direction of the applied field. If the polarization direction in this region switches by 90° to align with the field, the polarization component in the direction of the field will change from zero to P0, i.e., δP3 = P0. The unit cell dimension in the direction of the electric field changes from a0 to c0, so a strain in this direction of δS3 = (c0 − a0)/a0 is also cre-ated. dividing δS3 and δP3 by a small time interval δt and taking their ratio leads to (6):

dSdP

dSdt

dPdt

ej

c aa P

3 3 3 0 0

0 0= = =

34/

� −. (9)

The factor of 3/4 accounts for the fact that the crystal-lographic axes within each grain in a ceramic will, in gen-

eral, not be aligned with the electric field direction. This relation would still be approximately true even in the presence of some rhombohedral structure because all the possible twinning strains would average to nearly 90° [9].

measured values for the current density and strain rate through the bipolar cycle (see Fig. 2 for the electrical cy-cle) are shown in Fig. 10 and Fig. 11, respectively. note that at lower temperatures we observe a double peak in the polarization response that is not seen in the strain response. The extra peak in the polarization response is most likely due to the 180° domain wall processes shown previously to make a large contribution to the polarization in the coercive region, without contributing to the strain.

To estimate S3 from the measured S1 strain a factor of 2 was used, i.e., S3(estimated) ≈ 2 × S1. This reflects the fact that the d33 piezoelectric coefficient is approxi-mately double the d31 coefficient. results for the full bipo-

Fig. 9. Temperature variation of the strain discrepancy as a fraction of the remanent strain.

Fig. 10. current density response at 19 and 125°c.

Fig. 11. strain rate response at 19 and 125°c.

weaver et al.: electromechanical coupling and temperature-dependent polarization reversal 1735

lar loop at 19°c are shown in Fig. 12. room-temperature values cited in [9] for the parameters used in (9) are a0 = 0.4051 nm, c0 = 0.4086 nm, P0 = 0.32 c·m−2, which gives a value for dS3/dP of 0.0202 c−1·m2. This is shown as horizontal lines in Fig. 12 for comparison with the experi-mental results. To aid clarity, Fig. 13 shows only the pos-itive-going transition through the coercive field at 19°c. The measured value of dS3/dP is the same order of mag-nitude as, but is still significantly smaller (approximately 3.5 times) than, that predicted by (9).

The smaller measured value of dS3/dP indicates that the polarization term is changing faster than the strain term, i.e., it is an indication of the relative contribution of 180° domain wall processes. It was demonstrated in

section III that the relative contribution of 180° domain wall processes varies only slightly with temperature. We can therefore rearrange (9) to estimate an effective lattice distortion parameter from the experimental data:

α = =43

43

20 0

00

30

1c aa P

ej P

ej

−≈

� �. (10)

results are shown in Fig. 14. The value of the slope in Fig. 14 is −13.12 ppm·°c−1. note that in contrast to the remanent strain and polarization, there is no transition at the curie temperature, but the effective lattice constant varies almost linearly with temperature, reaching zero at the curie temperature.

V. conclusion

comparison of the electrostrictive and measured strain permits discrimination of the different domain processes within ferroelectric materials, and provides valuable infor-mation on the temperature dependency of these domain processes. strain and polarization follow a closely elec-trostrictive relation in the stable region of the free energy diagram, implying that the same mechanism is respon-sible for both strain and polarization changes, i.e., that non-180° domain wall processes are responsible for both strain and polarization response in this region. However, strong deviation from electrostriction is associated with the electric fields in the region of the coercive field because of a large contribution from 180° domain wall movement. results presented here show that the relative contribution of 180° domain wall processes is approximately constant below the curie temperature. a comparison of strain and polarization rates in the coercive region was used to pro-vide an estimate of the effective lattice distortion and its temperature dependency.

Fig. 12. dS3/dP as a function of electrical field at 19°c. The light gray markers show the data points in the coercive regions identified in Fig. 3. The vertical lines are the positive and negative coercive field values. The horizontal lines represent the value calculated from the data of [9] using (9).

Fig. 13. dS3/dP as a function of electrical field at 19°c for the positive-going coercive region. The vertical line marks the coercive field value.

Fig. 14. Unit cell distortion parameter α as a function of temperature.

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acknowledgments

The authors acknowledge the financial support of the Technology strategy board and the UK national mea-surement office. We also acknowledge the support of the apaHoE project partners servocell ltd., diameter ltd., and University of southampton.

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[4] p. m. Weaver, m. G. cain, and m. stewart, “Temperature depen-dence of strain–polarization coupling in ferroelectric ceramics,” Appl. Phys. Lett., vol. 96, no. 14, art. no. 142905, 2010.

[5] p. m. Weaver, m. G. cain, and m. stewart, “Temperature depen-dence of high field electromechanical coupling in ferroelectric ceram-ics,” J. Phys. D, vol. 43, no. 16, art. no. 165404, 2010.

[6] p. m. Weaver, “a sensorless drive system for controlling tempera-ture dependent hysteresis in piezoelectric actuators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 4, pp. 704–710, 2011.

[7] s. li, W. cao, and l. cross, “The extrinsic nature of nonlinear be-havior observed in lead zirconate titanate ferroelectric ceramic,” J. Appl. Phys., vol. 69, no. 10, pp. 7219–7224, 1991.

[8] q. Zhang, H. Wang, n. Kim, and l. cross, “direct evaluation of do-main-wall and intrinsic contributions to the dielectric and piezoelec-tric response and their temperature dependence on lead zirconate-titanate ceramics,” J. Appl. Phys., vol. 75, no. 1, p. 454, 1994.

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Paul Weaver is principal research scientist with the multi-functional materials research group of the UK’s national physical laboratory. His re-search interests include the development of novel electromechanical devices and the application of piezoelectrics and ferroelectrics for sensing and actuation. He is a visiting reader at southampton University. He holds an m.a. degree in natural science from cambridge University, and a ph.d. degree from southampton University. He is a chartered engineer and member of the IET.

Markys Cain graduated with his ph.d. degree from Warwick University in 1990 and spent the next 2 years in the materials department of the University of california, santa barbara, studying thin film epitaxial science. subsequent research in ceramic composite materials technology in the UK utilized many of the principles learned at santa barbara in the deployment of new interfacial fiber coatings for advanced high-temperature ceramic matrix composites for gas turbine applications. research with an oxford-based company led to

the development of a prototype sEm-based instrumented indentation system, and he joined the national physical laboratory (ndl) in 1997 to lead the Functional materials research group. His research activity includes the development of measurement methods to elucidate materi-als behavior in ferroelectric and piezoelectric ceramics and thin film ma-terials, and more recently in multiferroic materials and materials metrol-ogy for spintronics and energy harvesting. The focus of his research is materials metrology, and he has published more than 80 peer-reviewed scientific papers in the field. He chairs the Iom3 smart materials and systems committee. He is Knowledge leader for the materials division at npl and principal research scientist for the multifunctional materi-als technical area, and he is also a member of the Institute of physics. In 2009, he was awarded the Institute of materials’ Verulam medal for out-standing contributions to ceramic science, and is a visiting professor at queen mary, University of london.

Tatiana Correia is currently a Higher research scientist at the national physical laboratory. she graduated in physics from the University of porto, having later received an m.sc. degree in material science and engineering from aveiro University. she obtained her ph.d. degree at cranfield Uni-versity, where she studied the electrocaloric effect in thin films. Her research interests include fabri-cation and characterization of piezoelectric, pyro-electric, and ferroelectric materials and applica-tions for energy harvesting, energy storage, and refrigeration.

Mark Stewart received a b.Eng. degree in met-allurgy and materials science from the University of liverpool in 1982, followed by a ph.d. degree in 1985. He is currently a senior research scientist with the multi-functional materials research group of the UK’s national physical laboratory, focus-ing on the development of characterization tech-niques for functional materials, in particular piezoelectric materials.