Intrinsic and Extrinsic Size Effects

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Intrinsic and Extrinsic Size Effects in Fine-Grained

Morphotropic-Phase-Boundary Lead Zirconate Titanate Ceramics

Clive A. Randall,* Namchul Kim, John-Paul Kucera,* Wenwu Cao, and Thomas R. Shrout*

Materials Research Laboratory, Pennsylvania State University, University Park, Pennsylvania 16802

The processing, electromechanical properties, and micro-structure of lead zirconate titanate (PZT) ceramics over thegrain-size range of 0.110 m were studied. Using measure-ments over a large temperature range (15600 K), the rela-tive role of extrinsic contribution (i.e., domain-wall motion)was deduced to be influenced strongly by the grain size,particularly for donor-doped PZT. Analytical transmissionelectron microscopy studies were conducted to investigatethe trend in domain configurations with the reduction ofgrain size. The correlations between domain density, do-main variants, domain configurations (before and after pol-ing), spontaneous deformation, and the elastodielectricproperties were qualitatively discussed, leading to new in-

sights into the intrinsic and extrinsic effects and relevantsize effects in ferroelectric polycrystalline materials.

I. Introduction

THE piezoelectric activity in the solid solution PbZrO3PbTiO3 (PZT) system is optimum for compositions nearthe morphotropic phase boundary (MPB).1,2 The MPB is analmost-temperature-independent phase boundary that separatestwo ferroelectric phases: a tetragonal crystal structure (P4mm)and a rhombohedral structure (R3c), as shown in Fig. 1. Com-positions near the MPB may have both of these phases coex-isting to give a total of fourteen possible polarization directions(six tetragonal 001 and eight rhombohedral 111in refer-ence to the cubic prototype cell axes). The large number of

polarization directions enables optimized crystallographic ori-entations to be established from grain to grain in the polingprocess and, in turn, results in anomalously high piezoelectricproperties. Further improvements are made to the electrome-chanical properties through compositional modifications. Do-nor and acceptor doping are used to obtain so-called softand hard PZT materials, respectively.313 Properties con-trasting the elastodielectric properties in hard and soft PZT aregiven in Table I.

There have been many earlier attempts to investigate theinfluence of grain size on the piezoelectric properties in PZTceramics. In general, grain-size studies were limited in a rangeof 1.010 m in bulk ceramics. The variations in propertieswith grain size showed little consistency, as summarized inTable II. Reasons for the inconsistencies are believed to be

related to processing and the control of grain-boundary resis-tivities that lead to space-charge accumulation, which masksthe size-effect response in the ferroelectric grains.

In a polycrystalline system, dielectric and piezoelectric prop-

erties are dependent on both intrinsic and extrinsic mecha-nisms. Intrinsic contributions are from the relative ion/cationshift that preserves the ferroelectric crystal structure. Intrinsiccontributions in soft PZTs at room temperature are


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agreement with the direct scanning and transmission electronmicroscopy (SEM and TEM, respectively) studies. Particle-size distributions of the powders were determined using a sedi-mentation apparatus (Sedigraph Model 5000, Micromeritics In-strument Group, Norcross, GA). Prior to the measurement, ahomogeneous suspension was obtained using an ultrasonic bathand Sedisperse, which is a highly purified nonpolar alphatichydrocarbon surfactant (Micromeritic P-12).

(2) Sintering Ceramics

These powders densified at temperatures in the range of9501000C, as shown in Fig. 4(a). The sintering tempera-tures and times were varied to obtain grain-size variations. The

firings were performed in a lead atmosphere that was main-tained by a PbZrO3 source in a double closed Al2O3 crucible.The submicrometer-sized grains were produced via pressure-assisted densification, hot uniaxial pressing (HUP), and hotisostatic pressing (HIP) techniques. HUP was performed oncalcined powders that were placed in a disk-shaped Al2O3 dieand hot pressed at 800C for 3 h under a uniaxial pressure of17 MPa. Oxygen flow was maintained only during the heatingportion of the cycle, to aid the organic binder burnout. HIP wasperformed (Model 3000, Leci Co System). Sintered ceramicpellets were placed on a platinum foil inside a chamber andisostatically pressed in air at 21 MPa at a temperature 100200C below the conventional sintering temperature. Thesamples could then be taken to higher densities (98%) withoutincreasing the grain growth. This densification process then

enabled better electric characterization of the ceramics. Themean grain size varies systematically with the sintering tem-perature, as shown in Figs. 4(b) and (c), for undoped and dopedcompositions, respectively. After sintering, the samples wereannealed in an oxygen atmosphere furnace. This treatment wasperformed to compensate oxygen vacancies that formed during

the sintering and to also remove excess PbO from the grainboundaries. Samples were placed on a platinum plate in anopen Al2O3 crucible and annealed at 800900C for 6 h andcooled very slowly for 24 h in a flowing oxygen atmosphere.

(3) Electrical Characterization

Electrical measurements were made on ceramics with den-sities that were >95% of the theoretical value. Prior to mea-surements, the ceramics were ground parallel and polished us-ing a final grit of 3.0 m Al2O3 abrasive. After an ultrasoniccleaning in an acetone bath, samples were sputtered with goldand/or platinum. In addition, an air-dry silver paste was appliedto the sputtered electrode, to ensure good electrical contact.

Dielectric properties were obtained by measuring the capaci-tance with an automated multiple-frequency impedancecapacitanceresistance (LCR) meter (Model HP 4172, Hew-lett-Packard, Palo Alto, CA) in a kanthal-wound resistancefurnace. Piezoelectric coefficients and electromechanical cou-pling factors were determined with different grain-sized PZTceramics, the samples that were prepared in shapes based onIEEE standards,16 and after poling under a field level of 3050kV/cm for 1060 min at 120C. The piezoelectric constant, d33,was measured on poled and then one-day-aged samples usinga d33 meter (Berlincourt). Electromechanical properties weremeasured using an impedance analyzer (Model HP 4194A,Hewlett-Packard). The low-temperature dielectric and piezo-electric-related properties were measured using a cryogenicrefrigeration system (Model MC-2-APD, Cryogenics). The po-

larization-versus-electric-field hysteresis curves were obtainedusing a modified SawyerTower circuit.

(4) Structural Characterization

X-ray diffractometry (XRD) studies were performed onpowders and sintered ceramics (Model XRG-3000, Philips

Table I. Dielectric and Piezoelectric Properties of Undoped and Commercial PZT Ceramics

PZT ceramicCurie point,

Tc(C)Dielectric

constant, KDielectric

loss, tan Piezoelectric coefficient (pC/N) Piezoelectric coupling factor

d33 d31 d15 kp k33 k31 k15

Undoped Pb(Zr0.52Ti0.48)O3 386 730 0.004 220 93 494 0.52 0.67 0.31 0.69Hard PZT-4 328 1300 0.004 289 123 496 0.58 0.70 0.33 0.71Soft PZT-5 365 1700 0.02 374 171 584 0.60 0.60 0.34 0.68

PZT-4 is (Pb0.94Sr0.06)(Zr0.53Ti0.47)O3, and PZT-5 is Pb0.988(Zr0.52Ti0.48)0.976Nb0.024O3.At 1 kHz.

Fig. 1. PbZrO3PbTiO3 (PZT) subsolidus phase diagram (from Jaffe et al.1).

678 Journal of the American Ceramic SocietyRandall et al. Vol. 81, No. 3


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Table II. Grain-Size Effects on PZT Systems

Reference YearGrain size

(m)Dielectric constant,

K, at 25CPiezoelectric property

at 25C Transition parameters Model Comments

Haertling3 1964 25 Increases As Ec increases,Pr decreases

Hot-pressedceramics

Gesemann4 1965 37 As kp decreases,Pr decreases

Webster andWeston5

1968 1.76 Increases As kp and Qmdecrease, Prdecreases

Internal-stressmodel

Clamping ofdomainmotion

Okazaki andNagata6

1971 1.76 Decreases kij decreases As Tc increases,Kmax decreases

Porosity effectsare alsoimportant

Ouchi et al.7 1965 14 Decreases As dij andkp decrease,

Ec increases

As Tc increases,Kmax decreases

Space-chargemodel

Lanthanum-dopedPZT

Martirena andBurfoot8

1974 1.84.6 Decreases dij decreases As Tc increases,Kmax decreases

Gaussiandistribution

Hot-pressedniobium-dopedPZT

Keizer andcoworkers9,10

1973,1975

0.610 As Tc increases,Kmax decreases

Internal-stressmodel

Propose hydrostaticeffect; lanthanum-doped PZT

Pisarski11 1988,

1989

3 As Tc increases,

Kmax decreases

Width of MPB

region changeRossetti12 1991 2D Tc increases Two- or three-

dimensionalstress model

Devonshire theory;thin-film PbTiO3

Yamamoto13 1992 0.610 As Ec increases,Pr and kpdecrease

As Tc decreases,Kmax decreases

PbO grainboundary

Processingdependence

For Haertling,3 Okazaki and Nagata,6 Ouchi et al., 7 and Martirena and Burfoot8 data, the indication is for an unpoled dielectric constant. For Webster and Weston5 data, theindication is for a poled dielectric constant.

Fig. 2. Dielectric constant (K) as a function of temperature (4.2300 K) and the theoretical curve of an average of the single-crystal values,calculated assuming a tetragonal solution to the energy function (from Zhang et al.14).

March 1998 Intrinsic and Extrinsic Size Effects in Fine-Grained MPB PZT Ceramics 679


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Electronic Instruments, Mahwah, NJ). All data analyses wereperformed using APD 1700 software (Philips). Precise latticeparameters were determined by the Cohenss least-squares ana-lytical method using fifteen to thirty of the tetragonal andrhombohedral reflections that appeared in the scan range of1090, with a step scan of 0.01 and scan rate of 0.01/s. Thepeak positions were determined using a variable-tip-widthpeak-locating algorithm that was based on a second derivative

test. The results of the algorithm were visually inspected byexpanding the peak region on the monitor, and manual correc-tions to the peak positions were made if necessary. The peakpositions so determined were then corrected by applying asecond-order polynomial to the known peak positions of ahigh-purity external reference standard of quartz. Using thisprocedure, the absolute angular precisions of the least-squaresfit seldom exceeded 0.005.

SEM (Model DS 130, International Scientific Instruments(ISI), Milpitas, CA) was used to determine grain-size distribu-

tions on polished and etched samples. The samples were pol-ished systematically with 20, 12, 5, and 3 m SiC powders.The surface was brought to a final polish with diamond paste(1 and 0.5 m). Thermal etching of the polished surfaces wasperformed using a fast-fire furnace at a temperature that was200C below the sintering temperature. SEM samples werecoated with gold to reduce the charging effects before obser-vations were made. TEM was performed on carefully polishedand ion-beam-thinned samples of both poled and unpoled ce-ramics. Ion-beam thinning was made on a dual-beam ion mill(Gatan, Pleasanton, CA) with argon-ion beams that were ac-celerated by a 4 kV potential and at a current of 1 mA. Theangle of beam incidence was 12. A TEM microscope (Model420 STEM, Philips) was used in conjunction with a single-tilthot stage (Gatan) for high-temperature observations.

Table III. Characteristics ofComponent and Reacted Powders

Material

Specificsurface

area(m2/g)

Equivalentparticle

size (m)

MedianSedigraph

particlesize (m)

Averageagglomerationnumber, AAN

PbCO3 0.8 1.0 3.0 27TiO2 8.4 0.2 0.4 17ZrO2 21.4 0.05 0.9 5800Nb2O5 2.0 0.7 1.4 9

Precursor ZrxTi1xO2 2.2 0.65 2.5 56

PZT (56/44) 16.3 0.05 0.6 2200

Niobium-dopedPZT (52/48) 17.6 0.42 0.8 6900

Fig. 3. Processing flow chart for the undoped and niobium-doped PZT ceramics.



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III. Results

(1) Electrical Measurements

The temperature dielectric constant and loss, measured at 1kHz, are compared in poled and unpoled ceramics, in dopedand undoped PZT, respectively, as a function of grain size. Thedielectric constant decrease as the grain size increases. Plots ofthe dielectric constant versus temperature are shown in Figs.5(a) and (b), for undoped and doped PZT, respectively. Thedielectric constant maximum at the transition temperature sys-tematically decreases as the grain size decreases for both dopedand undoped PZT. Close inspection of the transition tempera-

ture (Tc) reveals that there is no change in Tc with grain size forthe niobium-doped PZT. However, there is an increase in Tcwith decreasing grain size for undoped PZT, which is consis-tent with the observations made earlier by Okazaki and Na-gata.6 This result was related to a space-charge effect and wasnot a true ferroelectric size-effect phenomenon; this can beverified by the application of the CurieWeiss law to the di-electric temperature dependence in the paraelectric region. Fig-ure 6(a) shows inconsistent Curie constants for undoped PZT,whereas Fig. 6(b) shows the Curie constant to be constant overthe entire grain range for the niobium-doped samples.

The ferroelectric switching behavior of the doped and un-doped is investigated in terms of the polarization-versus-electric-field hysteresis loops. Figure 7 shows typical loops forthe doped PZT, with increasing coercive field and with de-

creasing grain size. The remnant and saturated polarizationvalues decreased as the grain size decreased. In the case of theundoped PZT, the loops demonstrated a propeller shape, whichis typical of an aged sample. The aging in undoped PZT istypical of a hard PZT. The samples could be de-aged by cy-cling through many loops (3500 cycles). The coercive fieldalso increased as the grain size decreased in the undoped PZT;however, the remnant polarization did not show a systematictrend. The hot-pressed sample had an increase in the coercivefield; however, it is believed that the grain-boundary resistivitywas very sensitive to the processing in undoped PZT. With theinability to limit the space-charge contribution in the undopedPZT, only the niobium-doped PZTs are considered for detailed

electromechanical characterization.In general, the piezoelectric properties at room temperature

decreased consistently with grain-size reduction. The grain-size dependence of piezoelectric coefficients d33, d31, and dh(the hydrostatic constant) are shown for niobium-doped PZT atroom temperature in Fig. 8. The temperature dependence ofpiezoelectric coefficient d31 is compared in niobium-dopedPZT at 300, 100, and 15 K in Fig. 9; the temperature sensitivityat room temperature is the highest. Similarly, the trend is alsoobserved with the electromechanical coupling coefficients (kpand k31, Figs. 10(a) and (b), respectively).

(2) Structural Aspects

SEM studies on polished and etched surfaces were used toobtain statistical data of the grain-size distributions. Typical

Fig. 4. (a) Densities for undoped and niobium-doped PZT ceramics and mean grain sizes for (b) undoped PZT (52/48) and (c) niobium-doped PZT(52/48), all as a function of sintering temperature.



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symmetrical distributions of the niobium-doped PZT grain-sizedistributions are shown in Figs. 11(a) and (b).

Lattice parameters for the niobium-doped PZT ceramics areplotted in Fig. 12(a) as a function of grain size. The latticedistortion along the tetragonal c-axis is strongly reduced as thegrain size is reduced to 1.0 m, whereas the a-axis dimensionis not as strongly influenced. The crystal tetragonality, c/a,decreases as the grain size decreases, which also translates to areduction in the volume of the unit cell (Figs. 12(b) and (c)).

These dimensional changes are the result of an increase in theinternal stress in the ceramic when the transition from the para-electric to the ferroelectric phase occurs.17,18 Similar resultsand trends have been observed and reported by Arlt19 in Ba-TiO3 polycrystals. The spontaneous unit-cell distortion is mu-tually clamped in the fine-grained materials, which reduces theintrinsic polarization and also increases the internal stress in theceramic; this is consistent with the observation that the distor-tion of the tetragonality of fine-grained ceramics is reduced incomparison to that of free powder of similar crystal size.17

TEM investigations have been conducted to ascertain thedomain configurations and domains sizes as the grain size var-ies. Some caution must be taken in regard to the validity of theobserved domain sizes. There is a possibility that domains canredistribute in the thinned TEM foils to obtain a new equilib-rium configuration, respective to the thin foil, that does not

represent the desired bulk ceramic sample. The TEM samplesrequire that the metastable bulk domain configuration and sizeremain locked into the grain during the preparation andanalysis. Two experiments have been performed to validate thestate of the domains in the thin foil:

Fig. 7. Grain-size dependence of electric-fieldpolarization hyster-esis loop for niobium-doped PZT (52/48) samples.

Fig. 5. Dielectric constant as a function of temperature and grain sizefor (a) undoped PZT (52/48) ceramics at 100 kHz and (b) niobium-doped PZT (52/48) ceramics at 1 kHz.

Fig. 6. Grain-size dependence of Curie constants for (a) undopedPZT compositions at 100 kHz and (b) niobium-doped ceramics at 100kHz.



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(1) Thinned samples were analyzed to examine the do-main-size distribution; then, these same samples were heatedabove the transition temperature and cooled slowly at a rate of4C/min back to room temperature, where a second domain-size distribution was observed.

(2) Poled ceramics were analyzed after ion-beam thinningto confirm that the poled domain structure was preserved dur-ing the thinning process.18

Using the same preparation routes, we ion-beam thinned PZT

ceramics that had a wide range of grain sizes (both doped andundoped). Typical domain-size distributions all show skeweddistributions; therefore, in terms of describing a typical domainsize, the mode of the distribution was selected. We note that,for grain sizes >1.0 m, there is a multiple number of domainvariants in each grain. However, for the submicrometer grainsizes, the domain configurations are simply striated across thegrain, with the domains apparently pinned to the grain bound-ary, as shown in Fig. 13. Additional details of the domainstructures and the grain-size relationships are discussed else-where.20

IV. Discussion

All the above-mentioned elastodielectric properties in nio-bium-doped PZT ceramics demonstrated pure size effects (in-

dependent of space-charge polarization mechanisms). The gen-eral trends in the properties are summarized in Table IV forniobiumPZT.

(1) Intrinsic and Extrinsic Contributions

The general trends of the PZT ceramics that have similarcomposition and density indicate a reduction in the magnitude

Fig. 8. Grain-size dependence of piezoelectric coefficients (a) d33and d31 and (b) the hydrostatic coefficient dh at room temperature forniobium-doped PZT (52/48) ceramics.

Fig. 9. Grain-size dependence of piezoelectric coefficient d31 at dis-crete temperatures (300, 100, and 15 K) for niobium-doped PZT (52/48) ceramics.

Fig. 10. Grain-size dependence of electromechanical coupling fac-tors (a) kp and (b) k31 at discrete temperatures (300, 100, and 15 K) forniobium-doped PZT (52/48) ceramics.



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of the elastodielectric properties with decreasing grain size inthe range of 100.1 m. Because the magnitude of the elasto-dielectric properties is dependent on both intrinsic and extrinsicproperties, it is expected that both intrinsic and extrinsic con-tributions are influenced by the reduction in grain size in thePZT ceramics. In principle, one may distinguish the intrinsicand extrinsic portions by simply expressing the physical prop-erties in terms of a sum:21

= ex + in (1)

d= dex + din (2)

and

s = sex + sin (3)

where is the dielectric permittivity, d the piezoelectric coef-ficient, and s the elastic compliance.

There has been some debate on how to separate the intrinsicand extrinsic portions from the apparent measured quantities.Herbiet et al.21 argued that the extrinsic contribution is themain source of dielectric loss, because the domain wall motioncan induce mechanical friction. Using this argument, they de-rived that the extrinsic contribution to the dielectric propertycan reach 70% of the total at room temperature. Damjanovicand Demartin22 approached the problem from the use of pi-

ezoelectric measurements under a periodic stress perturbation.They assumed that the extrinsic contribution could be dividedinto two portions: the in-phase component and the out-of-phasecomponent, where only the out-of-phase component is associ-ated with loss. From their conclusion, the extrinsic contributionwas 30%40%.

Recently, Zhang et al.23 assumed that the extrinsic domainwall motion had no contribution to the hydrostatic piezoelectricconstant dh, considering that the domain redistribution pro-duced no volumetric change. With this assumption, they cal-

culated the temperature dependence of the dielectric and pi-ezoelectric constants and attributed most of the increase ofthese properties as being extrinsic in nature. If this assumptionis valid, the change of dh with grain size will be a good indi-cation of the suppression of the intrinsic properties in the fine-grained system. As shown in Fig. 14, the variation of dh hastwo distinct regions. In the region of 0.12 m, the decrease ofthe dh value with the reduction of the grain size is fairly linearand remains practically unchanged for grain sizes >3 m. Us-ing the assumption of Zhang et al.,23 this indicates a criticalsize of 2 m.

On the other hand, as shown in the same figure, the value ofthe remnant polarization does not decrease until a grain size of0.8 m is attained. Considering that the remnant polarizationdirectly reflects the degree of lattice distortion, it is a much-better indicator for the intrinsic contributions. In other words,

only when the grain size decreases to


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Fig. 12. Grain-size dependence of (a) lattice parameters (aT and cT), (b) crystal tetragonality (c/a), and (c) unit-cell volume for PZT (52/48)ceramics.



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Table IV. Property Trends for PZT Ceramics

Elastodielectri c pr operty Grain-size (GS) eff ect , 0.110 m Comments

Transition temperature Independent Provided processing eliminates space-chargeeffects that perturb the fundamentaldielectric anomaly

Maximum dielectric constant Decreases as GS decreases Peak broadening also occurs with decreasinggrain size

Room-temperature dielectricconstant and dielectric losstan

Under poled conditions, and tan decreaseas GS decreases; under unpoled conditions, and grain growth decrease as GS decreases

The origin of dielectric loss seems to bedifferent in poled and unpoled PZT

Hysteresis of PE loops Ec increases as GS decreases; Pr decreases asGS decreases

Higher switching fields are required;saturation polarization Ps also decreasesas GS decreases

Piezoelectric coefficients d33and d31

Decrease as GS decreases At low temperatures, domain size is basicallyindependent of the grain

Electromechanical couplingcoefficients kp and k31

Decrease as GS decreases At temperatures of 100 K, the couplingcoefficients are independent of the grainsize

Fig. 14. Grain-size dependence of remnant polarization Pr and hydrostatic coefficient dh for niobium-doped PZT (52/48).

Fig. 13. TEM image showing striated twin domain configurations indoped PZT (52/48) 0.2-m-grain-size ceramics.

Fig. 15. Grain-size dependence of d33 and d31 for niobium-dopedPZT (52/48).



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change), it will affect the redistribution of differently orientateddomains, which causes the induced charge response. Such aneffect will be more substantial for systems that have a smallerpolarization than for systems that have a larger polarization.

With such an hypothesis, we can proceed to analyze thegrain-size-induced changes in the physical quantities. There aretwo critical grain sizes in the niobium-doped PZT system: 2.0and 0.8 m (Fig. 16).20 The former marks the beginning of adrastic reduction of the extrinsic contributions to the piezoelec-tric and dielectric properties, mainly through the reduction of

orientational variants and the direct coupling of the domainwalls to the grain boundaries. This point will be clearer fromthe discussion in the next section. The 0.8 m grain size sig-nifies the beginning of the intrinsic property suppression,which, in turn, is related to the fact that the grains are elasti-cally clamped in the dense ceramic. The spontaneous strain,measured via the c/a ratio in the ceramic, shows a sharp re-duction in magnitude below the 0.8 m grain size. X-ray mea-surements on powder that is of similar size to the grains showno reduction in the spontaneous strain; from this difference, itis assumed that an internal stress exists that reduces the unit-cell polarization, and, therefore, we call it an intrinsic effect.

(2) Domain Configurations and Grain Sizes

The basic variation of crystal size influences the domainsize. From the balance of the depolarization energy and the

domain wall energy,25 one can show that the relationship be-tween the domain size and the grain size is

d= *Po

2t

12

(5)

where is the energy density of the domain wall, Po thespontaneous polarization, and * the effective dielectric con-stant.

In the grain-size range of 310 m, the grain-sizedomain-size relationship obeyed the expected parabolic relationship;however, for the submicrometer grain sizes, there is a departureto smaller domain widths than expected by the parabolic rela-tionship.20 This trend was also consistent with BaTiO3 andPZT thin films, as reported by Arlt19 and Warren et al.,26

respectively. We believe the deviation is due to changes in theboundary conditions. Clamping of domain walls at the grainboundaries from its neighbors will cause the relative increase indomain density at smaller grain sizes.18,21

Another noticeable fact is the reduction in the number ofdomain variants for grain sizes


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Relation in Morphotropic Phase Boundary, Pb(Zr0.52Ti0.48)O3, U.S.Jpn. Se-min. Dielectr. Piezoelectr. Ceram., 7th, 14548 (1995).

19G. Arlt, The Influence of Microstructure on the Properties of FerroelectricCeramics, Ferroelectrics, 104, 21727 (1990).

20W. Cao and C. A. Randall, Grain Size and Domain Size Relations in BulkCeramic Ferroelectric Materials, J. Phys. Chem. Solids, 57 [10] 1499505(1996).

21R. Herbiet, U. Robels, H. Dederichs, and G. Arlt, Domain Wall andVolume Contributions to Material Properties of PZT Ceramics, Ferroelectrics,98, 10721 (1989).

22D. Damjanovic and M. Demartin, Dependence of the Direct PiezoelectricEffect in Coarse and Fine Grain Barium Titanate Ceramics on Dynamic StaticPressure, Appl. Phys. Lett., 68 [21] 304648 (1996).

23

Q. M. Zhang, H. Wang, N. Kim, and L. E. Cross, Direct Evaluation of

Domain-Wall and Intrinsic Contributions to the Dielectric and PiezoelectricResponse and Their Temperature Dependence on Lead Zirconate-Titanate Ce-ramics, J. Appl. Phys., 75 [1] 45459 (1994).

24M. J. Haun, Thermodynamic Theory of the Lead Zirconate-Titanate SolidSolution System; Ph.D. Thesis. The Pennsylvania State University, UniversityPark, PA, 1988.

25C. Kittel, Ferroelectric Domain Wall Thermodynamics, Phys. Rev., 70,965 (1946).

26W. L. Warren, D. Dimos, B. A. Tuttle, R. D. Nasby, and G. E. Pike, Elec-tronic Domain Pinning in Pb(Zr,Ti)O3 Thin-Films and Its Role in Fatigue,

Appl. Phys. Lett., 65 [8] 101820 (1994).27G. Arlt and N. A. Pertsev, Force Constant and Effective Mass of 90

Domain Walls in Ferroelectric Ceramics, J. Appl. Phys., 70 [4] 228389

(1991).


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