Extrinsic effects in twinned ferroelectric polycrystalsRafel Pérez, Jose E. García, Alfons Albareda, and Diego A. Ochoa Citation: J. Appl. Phys. 102, 044117 (2007); doi: 10.1063/1.2769339 View online: http://dx.doi.org/10.1063/1.2769339 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v102/i4 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Extrinsic effects in twinned ferroelectric polycrystalsRafel Pérez,a� Jose E. García, Alfons Albareda, and Diego A. OchoaDepartament de Física Aplicada, Universitat Politècnica de Catalunya, c/Jordi Girona 1-3, Mòdul B4,08034 Barcelona, Spain

�Received 30 March 2007; accepted 30 June 2007; published online 30 August 2007�

The structure of medium-grain piezoelectric ceramics often consists of alternate lamellae ofnon-180° domains. In this work, the extrinsic effects of the electric and elastic fields on suchstructures are studied. A description of the extrinsic behavior of a single grain is given, and it isshown that the relations between piezoelectric and dielectric or elastic constants must beindependent of the wall mobility, being solely dependent on the relation between spontaneouspolarization and strain. By using an appropriate coordinate system, the conditions under which theintrinsic and extrinsic effects can be added are analyzed. The linear global behavior of a grain canthen be described as a function of its orientation and of two additional parameters: the lack ofequilibrium between domains � and its mean thickness d. The basis is established to describe thestate of a ceramic through a distribution function that accounts for domain orientation and whichdepends on the poling, fatigue, and ageing of the sample. Finally, the goodness of the model isanalyzed, although some aspects must be still modified in order to describe the overall behavior ofthe ceramic. © 2007 American Institute of Physics. �DOI: 10.1063/1.2769339�

I. INTRODUCTION

The discovery of perovskite-type piezoelectric ceramics,and especially lead zirconate titanate �PZT�, has lead to agreat advance in the field of piezoelectric materials1 becauseof their very high piezoelectric coefficient and high electro-mechanical coupling factor. They have become unsurpass-able in many applications such as transducers andactuators.2–4 The best materials are attained by an appropri-ate composition in the vicinity of the morphotropic phaseboundary, while some improvements of specific propertiescan be obtained by addition of dopants.5 Depending on itscomposition, PZT has a tetragonal or rhombohedral struc-ture, which determines the direction toward which it sponta-neously polarizes.

The internal structure of such ceramics is rather com-plex. The material is formed by randomly oriented grains,each one consisting of a single monocrystal finely divided inmultiple ferroelectric domains, which have a definite sponta-neous polarization and deformation. In order to avoid inter-nal stress, these domains or twins can only be arranged oneagainst each other in certain ways to form domain walls,which can be of 180° or non-180°, giving rise to well definedstructures.6,7 As a result of the stress that appears near thegrain boundaries due to the existence of these domains, theymust have a thickness proportional to the square root of thegrain size. This size then determines the level of complexityof the inner domain structure.8–10

The application of an external electric field produces anelectrical polarization in the ceramic as well as a mechanicalstrain �dielectric and reverse piezoelectric effects�, which canalso be produced by applying an external stress �direct piezo-electric effect and elasticity�.11

These effects are produced in two different ways: by

deforming the unit cell �intrinsic effect� or by displacing do-main walls �extrinsic effect�. In the latter case, this is accom-plished by changing the direction of spontaneous polariza-tion and strain of a thin slice of material that moves from onedomain to another. It is assumed that wall mobility decreaseswhen the temperature falls. Thus the only mechanism that isstill expected to work in the vicinity of 0 K is the intrinsicone,12 while at room temperature the extrinsic effect can bedominant, depending on the impurities.13 It can also beshown that the extrinsic effect is related to the strong non-linear behavior of the ceramics.14,15

Direct observation of the ceramic by means of scanningelectronic microscopy1 or by atomic force microscopy16

shows that one of the preferred configurations of themedium-grain ceramics consists of a lamellar structureformed by only two types of domain separated by non-180°walls.17 Although this is only a possibility, it appears to bethe most probable for a certain range of grain sizes. Thisstructure can appear simultaneously with a more complexstructure, in which lamellae formed by regular non-180°walls are combined with irregular 180° walls, while in largergrains lamellar structures appear containing other thinnerlamellae. For grains smaller than those considered, internalstress acquires greater importance, so a discrepant behavioris expected, and domain thickness begins to lose the propor-tionality to the square root of the grain size. Finally, in thesmallest grains, the domain walls disappear, so there is nolonger any extrinsic effect, and even the intrinsic effectceases because the ferroelectric phase cannot be held.10,18

The present work is based on the hypothesis that the materialis constituted entirely by a lamellar structure formed by twotypes of domain separated by non-180° walls, so discrepan-cies between theory and experience may be attributed to thefact that other structures may be present in real ferroelectricceramics.

The behavior of a grain with such a structure is analyzeda�Electronic mail: rafel@fa.upc.edu

JOURNAL OF APPLIED PHYSICS 102, 044117 �2007�

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by assuming that the type of domain, the intrinsic behavior,and the wall mobility are known. It is necessary to find asuitable coordinate system for studying the microcrystal, aswell as the conditions that allows us to add up both effects.The aim of this work mainly concerns the study of non-180°wall domain movement within such a structure and how thisfact influences the properties of the grain considered as awhole.

Subsequently, it is also established how the whole be-havior of the ceramic can be predicted by considering twoaspects: the fact that a grain can have any orientation andthat the sample can be described by a distribution functiondescribing how many grains are oriented in each of the pos-sible directions.

II. STRUCTURE MORPHOLOGY

The symmetry groups of ferroelectric and correspondingparaelectric phases give the different configurations of a mi-crocrystal. The number of orientations of the spontaneouspolarization vector is equal to the relation between the re-spective orders of the groups. Then, in a perovskite such asPZT, whose paraelectric phase belongs to the m3m group,there are six or eight different domains, depending onwhether the ferroelectric phase belongs to the tetragonal4mm or the rhombohedral 3m group. In each case, the cell isdeformed �spontaneous strain� in three or four differentways, respectively, since the strain does not change when thepolarization is turned to the opposite direction.

Polarization vectors P point to the directions �1 0 0�,�0 1 0�, and �0 0 1� in the tetragonal case or to the directions

�1 1 1�, �1̄ 1 1�, �1 1̄ 1�, and �1̄ 1̄ 1� in the rhombohedralcase. In all cases, the opposite directions are also possible.Each type of domain can be symbolized by a point at wherethe direction of P intersects with the surface of a cube rep-resentative of the unit cell �Fig. 1�.

There is a type of domain wall for each pair of polariza-tion vectors P. Then, each type of wall can be represented bya segment joining both points. Taking into account the con-ditions that must be fulfilled in a wall in order to maintainthe periodicity along the wall and its electrical neutrality,20

the surface must be perpendicular to the bisectrix of thespontaneous polarization directions PI and PII, which auto-matically guarantees both mechanical and electrical condi-tions �Fig. 2�. When PI and PII form a 180° angle, this con-dition does not univocally determine the orientation of thewalls. They, therefore, may have a cylindrical shape with an

irregular base; however, when they form any other angle thewall orientation is well defined, so walls must be flat, andthey can form a lamellar structure where both domains alter-nate.

In the tetragonal case, there is only one kind of walldifferent from 180°, which is that defined by two vectors Ppointing toward adjacent faces and forming a 90° angle.Therefore, there are as many equivalent possibilities as edgesin a cube �12�. In the rhombohedral case, on the other hand,there are two kinds of walls:19 the vectors can point towardtwo adjacent vertices, forming a 71° angle, or to the verticesat the ends of a face diagonal, whose angle is 109°. Thenumber of walls of the first type is equal to the number ofdiagonals �also 12�, while there are as many walls of thesecond type as face diagonals �12 more�, as can be seen inFig. 2.

A coordinate system is defined in order to describe suchobjects. The vectors of its basis are uP, which has the direc-tion of the sum of vectors PI and PII, uA, which has thedirection of their difference, and uN, which is perpendicularto both. The symbols P, A, and N have been selected torepresent the polarization, active, and neutral directions, aswill be explained later, in reference to the dielectric behaviorof such a structure. In this paper, the planes are denoted bytwo symbols �i.e., PA is the plane containing the vectors uP

and uA�. These symbols are also used as a subindex of thetensors when these are expressed in the local reference alliedto lamellar structures �i.e., SAP denotes the shear componenton the plane AP of the strain tensor�. Due to the spontaneousstrain, the angles differ slightly from the values indicated, ascan be seen in Fig. 3, where this effect has been exaggerated.Although the domains tend to have the same thickness, thisis not always the case due to the interaction of electricalfields and stress, whether they are external or produced byneighboring grains.15 The lack of equilibrium between the

FIG. 1. �Color online� Domain orientations. Each orientation is representedby a point on the surface of a cube, oriented according to the unit cell, andshowing the direction of the respective polarization vector. �a� Tetragonalcase. �b� Rhombohedral case.

FIG. 2. �Color online� Polarization direction and domain wall orientation.�a� The average polarization vector is a linear combination of PI and PII, soits end belongs to the segment joining both ends. The domain wall is per-pendicular to the bisectrix of both vectors: �b� 90° walls �tetragonal case�and ��c� and �d�� 71° and 109° walls �rhombohedral case�.

FIG. 3. �Color online� Domain arrangement in a lamellar twinned structure.Both domain types alternate between parallel planes, separated by a meanthickness d. In the inset one may see that, in order to maintain the sameperiodicity on the wall plane, the cells must be slanted with the same anglewith respect to such a plane. Moreover, the normal component of the polar-ization vector P of the cell is continuous. Domains of one type or anothermay have different thicknesses, the fraction � describes their lack of equi-librium. The figure shows the coordinate system used.

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volumes occupied by both domains is described by a variable� whose value belongs to �−1, +1� range. In this model,since inequalities between the thicknesses of domains of thesame type have no incidence, they are considered to beequal. Mean domain thickness d is defined as the mean dis-tance between adjacent walls. Due to the stress produced bythe domain strain near the grain boundary, d is expected tobe proportional to the square root of the domain size.21

For a suitable microcrystal size,21 such a structure usu-ally fills the grain, so the ceramic can be considered as a setof elements of this type, oriented at random in all directionsin an unpoled ceramic. Each grain is characterized by itsorientation, the mean area of its domains, its mean thicknessd, and the value of �.

In the proposed system of coordinates, spontaneous po-larization vectors of both domains are

PI = �P1, + P2,0� ,

PII = �P1,− P2,0� , �1�

since PA plane is a symmetry plane for both domains andspontaneous polarizations are symmetrical with respect toPN plane, while the spontaneous strain tensor can be writtenas

SI = �S1,S2,S3,0,0, + S6� ,

SII = �S1,S2,S3,0,0,− S6� , �2�

since spontaneous strains are symmetrical with respect to PNplane.

As the volumes of each type of domain are proportionalto �1/2��1+�� and �1/2��1−��, the equivalent value of thegrain polarization is equal to

P = P0 + � · �P0, �3�

where

P0 = 12 �PI + PII� = �P1,0,0� ,

�P0 = 12 �PI − PII� = �0,P2,0� . �4�

It is worth pointing out that PI and PII have the samemodulus, so the vectors P0 and �P0 are perpendicular anddirected, respectively, toward the directions of uP and uA

�Fig. 4�.Similarly

S = S0 + ��S0, �5�

where

S0 = �S1,S2,S3,0,0,0� ,

�S0 = �0,0,0,0,0,S6� . �6�

In general, if ��0, the macroscopic symmetry of thegrain is monoclinic �m group�, where PA is the symmetryplane. In the case that �=0, the system is orthorhombic�mm2 group�, where P is an axis of binary symmetry and PAand PN are symmetry planes.

III. DYNAMICAL ANALYSIS

The structure under consideration has certain dielectric,piezoelectric, and elastic behaviors, due either to the dis-placement of the atoms inside the cell �intrinsic effect� or tothe displacement of domain walls �extrinsic effect�. The in-trinsic behavior is characteristic of each domain, so the over-all behavior depends on the rate between the domains �vari-able ��. However, the extrinsic behavior is determined by thearea and number of walls, and therefore depends on the meanthickness d, although not on �.

If a displacement of a domain wall � is produced byapplying a field, so that one type of domain expands whilethe other retracts, then changes in the mean polarization andstrain are produced. These changes are an exclusive functionof ��, which not only depends on � but also on the numberof walls per unit of length, which is inversely proportional tod. The values of such increments are �� �P0 and �� �S0,respectively, so the polarization increment is a vector thatwill point toward uA, while the strain increment will consistexclusively of a shear strain over the plane PA. As can beobserved in Fig. 3, the movement of the wall toward uP

produces a change in a thin layer of cells, whose nodes moveexclusively parallel to uA.

Assuming that a grain is a cube of side a, each of its nwalls has a surface a2. Since the total volume is V=a3, thevolume gained by the first type of domain is

�V = �a2n = �a3/d �7�

because a=nd but

�V/V = �/d = ��/2, �8�

then,

�� = 2�/d . �9�

In this case, the extrinsic contribution can be describedvery simply. The wall displacement �, which points towarduP, only produces a change of electrical displacement DA anda shear strain SPA. Otherwise, since spontaneous polariza-tions of both domains are equal in direction uP and null indirection uN, when an electric field is applied in any of suchdirections there is no preference for one domain over an-other, so these fields are not expected to move the domainwall. Something similar occurs with the stress in which casethe only component that can interact with the wall is TPA.From this, one may deduce that the entire extrinsic effect isreduced to three coefficients: one dielectric �AA

ext, another pi-ezoelectric dAPA

ext , and a third elastic sPAPAext .

If only the extrinsic effect is considered, and assumingits behavior to be linear, under the application of an electricfield EA or a stress TPA, the walls move on average a distance� from its position of equilibrium, in proportion with themobilities �E and �T,

FIG. 4. �Color online� The average polarization can be described in terms offraction �.

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� = �EEA + �TTPA = ��T �E��TPA

EA� . �10�

Such a displacement produces an alteration on �, whichin turn produces increments of DA and SPA,

�DA = ��P2,

�SPA = ��S6, �11�

where P2 and S6 represent half the spontaneous polarizationand strain differences between both domains,

��SPA

�DA� = �S6

P2� · �� = �S6

P2�2/d · �

= 2/d · �S6

P2���T �E��TPA

EA�

= ��2/d��TS6 �2/d��ES6

�2/d��TP2 �2/d��EP2��TPA

EA� . �12�

However, by definition,

��SPA

�DA� = �sPAPA

ext dAPAext

dAPAext �AA

ext ��TPA

EA� . �13�

Then,

�AAext =

2

d�EP2, dAPA

ext =2

d�TP2,

dAPAext =

2

d�ES6, sPAPA

ext =2

d�TS6. �14�

According to thermodynamics, if there are neither lossesnor nonlinearity, direct and reverse piezoelectric coefficientsmust be equal. Thus, the ratio between both mobilities �E

and �T must be equal to the ratio between spontaneous strainand polarization,

�E/�T = P2/S6. �15�

In a similar form, we have

dAPAext /�AA

ext = S6/P2,

dAPAext /sPAPA

ext = P2/S6. �16�

Thus, we must expect the ratios between dielectric, pi-ezoelectric, and elastic coefficients to depend on the sponta-neous strain and polarization, while remaining independentof their respective mobilities.

From these relations, the extrinsic electromechanicalcoupling factor,

k2 = dAPAext2 /��AA

extsPAPAext � , �17�

is expected to be equal to 1, so in the absence of other effectsthe electromechanical conversion should be perfect.

The value of � is altered by the wall movement, andtherefore some of the intrinsic coefficients are also changed.

This phenomenon, similar to that described by Trolier-McKinstry et al.22 referring to the 180° wall movement,gives rise to some nonlinear behavior. In this case, the effectmust be proportional to the square of the applied instanta-neous field or stress.

In order to analyze the overall behavior of this structure,the respective tensors must be expressed in the coordinatesystem �uP uA uN�. In order to avoid local stress and fields inthe domain wall, it must be pointed out that the uniaxialstrains parallel to the wall must be the same on both sides, sothe strains SN, SN, and SAN must be continuous throughoutthe wall. Furthermore, stress applied on the wall must beidentical, so the stresses TP, TPA, and TPN must also be con-tinuous. From the electrical point of view, electric field isconservative, so EA and EN must be continuous. Finally, theelectric displacement DP, perpendicular to the wall, mustalso be continuous, assuming that there are no electric spacecharges. Thus it is possible to see that the set of variables

F = �TP,SA,SN, SAN,TPN,TPA, DP,EA,EN� �18�

is common to both domains, while the set

X = �SP,TA,TN, TAN,SPN,SPA, EP,DA,DN� �19�

depends on each one.15

Inside every domain, the intrinsic effect can be describedby a 9�9 matrix, Mint

I or MintII, which relates both sets of

variables, such that

XI = MintIF and XII = Mint

IIF . �20�

These matrices can be obtained from the tensors s, d,and � by applying the adequate matrix transformations.23

The matrix Mint0 is constructed by all the coefficients that are

common to both domains, while the matrix �Mint is made bythose that change their sign according to the domain underconsideration. Since the set F is common to both domains,the matrix can be averaged by weighing with the volume ofthe respective domains. Thus, the equivalent matrix Mint willlinearly depend on �. It is worth pointing out that the extrin-sic effect can be represented by a matrix Mext, similar topreceding ones, which relates �TPA ,EA� belonging to F, with�SPA ,DA�, which forms part of X. No other variables arerelated by Mext. Therefore, the overall effect can be de-scribed by

M = M0int + ��Mint + Mext. �21�

Then, the matrix M has the following format:

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M = � � � � � �

� � � � � �

� � � � � �

� � �

� � �

� � � � � �

� � � � � �

� � � � � �

� � �

Tp

SA

SN

SAN

TNP

TAP

DP

EA

EN

= Sp

TA

TN

TAN

SNP

SAP

EP

DA

DN

,

�22�

where coefficients * and � are fixed �they do not depend on��, while coefficients � are proportional to �. The terms �

also describe the extrinsic behavior.The fact that the extrinsic effect can be described as a

function of the variables common to both domains is a criti-cal fact that enables us to consider the overall effect on atwinned structure as a sum of the intrinsic and extrinsic ef-fects. Thus, extrinsic effect can only be treated separately ifthe sets F and X are used.

Unlike the intrinsic effect, a part of which is proportionalto �, extrinsic effect is proportional to its increment ��,which depends on the displacement � and on the inverse ofthe mean domain thickness d. For a given field or stress thatproduces a certain displacement �, the extrinsic effect isgreater when the crystal is divided into fine domains, whilethe intrinsic effect is completely unaffected by that fact.

There is a collateral effect whose importance is neces-sary to consider. Although in the proposed model the wallmotion does not produce any space charge or mechanicalstress inside the grain, it does produce a charge redistributionand stress at the grain boundaries. Thus, wall motion willincrement both electrical and mechanical energies. This factis more important where small grains are concerned.

IV. EFFECT OF THE GRAIN ORIENTATION ON THEEXTRINSIC EFFECT

Let us now take a ceramic sample with a plane-parallelcapacitor shape and with electrodes on its parallel faces �Fig.5�a��. The external electric field is applied toward a directionZ, normal to electrodes, which coincides with the direction ofthe macroscopic poling. Up to now, we have not taken intoaccount that the ceramic consists of a large number of grains,each of them with its own orientation with regard to Z. Thus,it is necessary to consider which angles can be used to de-

scribe the grain orientation, how the properties depend onsuch orientation, and how the device can be described asexhaustively as possible by one or more distribution func-tions.

Given that the electric field or the uniaxial stress is oftenapplied in direction Z, it is necessary to establish the orien-tation of the grain with respect to this direction. However,the directions perpendicular to Z are usually equivalent,which does not occur with the different orientations relatedto the grain. For this reason, it is preferable to orient thedirection Z with respect to the grain rather than in the oppo-site way.

The direction Z of the external field is determined by theangles � and �, by taking as a reference the local coordinatesallied to each grain �uP uA uN� �Fig. 5�b��. Moreover, in or-der to study the lateral effects such as �11, a third angle must be introduced, which measures the rotation around theZ axis.

If there is no intrinsic effect, the system is only sensitiveto EA and TPA, so these fields must be described as a functionof an external field EZ or of a stress TZ applied in the direc-tion Z, which is defined by the angles � and �,

EA = cos � sin � EZ,

TAP = cos � sin � cos � TZ. �23�

Furthermore, in a similar way, DA and SPA are expressedin the poling direction as

DZ = cos � sin � DA,

SZ = cos � sin � cos � SAP. �24�

Thus the extrinsic effect appears differently according tothe dielectric, piezoelectric, and elastic coefficients, sincetheir respective tensors have different orders. Then, by as-suming that the relation is purely linear and that there is nointrinsic effect, the relations

�ZZ = cos2 � sin2 � �AA,

dZZ = cos2 � sin2 � cos � dAPA,

sZZ = cos2 � sin2 �cos2 � sPAPA, �25�

must apply, as shown in Fig. 6.When there is only the extrinsic effect, dielectric, piezo-

electric, or elastic behavior can be deduced directly by view-ing such images. As can be observed, all coefficients are nullat �=0 and also at =90°. This means that in a ceramic

FIG. 5. �Color online� Grain orientation with respect to the sample andangles that define the orientation of the poling direction with respect to thegrain.

FIG. 6. �Color online� Extrinsic response of a grain as a function of direc-tion Z: �a� dielectric coefficient �zz, �b� piezoelectric coefficient dzz, and �c�elastic coefficient szz.

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where there are only fully oriented elements of this type,extrinsic effects cannot appear. The effects then becomelarger when the grains move away from the ideal orientation.Moreover, the orientation in which the effect is maximum isnot the same for all three effects. While the maximum dielec-tric effect is produced at 90°, the maximum elastic effectoccurs at 45°, and the maximum piezoelectric effect occursat an intermediate orientation, 55°. All coefficients are nullfor any other direction perpendicular to uA. While the per-mittivity or elasticity lobes are all of the same sign, the signis opposite for the piezoelectricity higher lobes ���90° �than for the lower ones.

For the dielectric effect, the maximum response can beobtained when measured along the direction uA and a nullvalue along the direction uP. This result is consistent with thesimulation carried out by Alhuwalia et al.24 By assuming thatin a material poled in direction Z the angle � is small formost of the material, �11 is the average for the permittivityvalues in all directions perpendicular to uP, and is thereforeexpected to be higher than in direction Z, whose value is zerofor a perfectly oriented material. Thus, the extrinsic contri-bution to �11 is expected to be greater than to �33.

Something similar to the last case occurs for the piezo-electric effect d33. Its value only moves from zero when thedirection Z moves away from P. The direct evaluation ofextrinsic d31 is rather difficult because the third angle, whichhas yet to be considered, must be used and averaged accord-ingly. However, this difficulty can be overcome if we ob-serve that when the wall moves, only a shear strain is pro-duced, which does not alter its volume. The strain obtainedby a field toward direction Z is expected to modify the lateraldimensions without changing the volume, so d31 must beequal to −�1/2�d33. The qualitative evaluation of d15 is easysince it can be seen that this is the only extrinsic coefficientwhen the system is fully oriented. Indeed, the application ofa field in the direction uA �corresponding to E1� produces ashear over the plane AP �given by S13=S5�. Then, d15 isexpected to be the highest of all the piezoelectric coefficientswhen they are due to the extrinsic coefficient.

No noteworthy extrinsic contributions are expected to befound in the compliance s, neither in the direction Z�s33� norperpendicular to it �s11�, because the grain will only contrib-ute when it moves away from the ideal orientation. However,an important contribution to the coefficient s55 is expectedbecause the system may undergo a shear displacement overthe plane PA.

V. MACROSCOPIC BEHAVIOR AND DOMAINDISTRIBUTION

In order to obtain the overall behavior of the ceramic,the grain distribution must be considered. A single grain ischaracterized by its volume V, its orientation �� ,��, the de-gree of polarization �, and by the mean domain thickness d.In order to find its response to an electrical or mechanicalexcitation applied in the direction Z, the 9�9 matrices rep-resenting the behavior of each grain must be evaluated: thefixed term M0, the part �M variable on � and the extrinsicpart Mext, which depends on 1/d. In order to make the addi-

tion linear, those matrices must be described by taking the setF as a basis, so once M is computed, it may be transformedto the basis �TP ,TA ,TN ,TAN ,TNP ,TAP ,EP ,EA ,EN�, fromwhich it is possible to change the coordinate system in orderto adapt it to the device reference. Once the grain behavior isdetermined in terms of �, �, �, and d, statistical methodsmust be applied by evaluating how many grains there are ineach orientation, �, and d, with the aim of predicting thecollective behavior.

A reasonable simplification of the problem consists inconsidering that all equally oriented grains have the samethickness d and the same �. Since d depends on the grainsize, this approximation is only valid for uniform grain sizeceramics, which is a commonly occurring case. Based on thisassumption, the ceramic can be characterized by three distri-bution functions, which all depend on the two orientationangles � and �: P�� ,�� �volume of the ceramic with suchorientation�, ��� ,��, and d�� ,��.

In order to evaluate the intrinsic behavior, it is necessaryto give the volume of the grains having an orientation de-fined by � and �, as well as a degree of polarization �. Sincethe extrinsic behavior depends on the total surface related toeach orientation, it is proportional to 1/d, irrespective of thevalue of �.

The probability that a grain is oriented within a solidangle defined by �� ,�+d�� and �� ,�+d�� is

P��,��sin �d�d� . �26�

Thus, the mean contribution of any variable can be writ-ten as

�x� = 0

�

sin �d� −�/2

�/2

P��,��x��,�,���,���d� , �27�

where only angles � belonging to the interval �−� /2 ,� /2�are considered. Since domains I and II are interchangeable�which implies a change of sign in ��, only one of the twopossibilities must be taken into account.

It is reasonable to assume that P does not depend on �because without poling all directions are equivalent. How-ever, we must assume that � does indeed depend on �.Therefore, the evaluation of the mean value is straightfor-ward when the variable x does not depend on �, especially ifthe dependence on � is of the form cos2 �.

If

x��,�,���,��� = x���cos2 � , �28�

then

�x� =�

2

0

�

sin � P���x���d� . �29�

This relation can be applied for evaluating the extrinsiccontribution �in the absence of an intrinsic effect�. As can beseen, each coefficient has a specific dependence on �, whichcan be expressed by the functions denoted as f��� �Fig. 7�,

�zz��� = 12 sin2 � �AA

ext = 12�AA

ext f���� ,

dzz��� = 12 sin2 � cos � dAPA

ext = 12dAPA

ext fd��� ,

044117-6 Perez et al. J. Appl. Phys. 102, 044117 �2007�

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szz��� = 12 sin2 � cos2 � sPAPA

ext = 12sPAPA

ext fs��� , �30�

��zz� = �AAext �

2

0

�

sin � f����P���d� = �AAextg�,

�dzz� = dAPAext �

2

0

�

sin � fd���P���d� = dAPAext gd,

�szz� = sPAPAext �

2

0

�

sin � fs���P���d� = sPAPAext gs, �31�

where P must fulfill

1 =�

2

0

�

P���sin �d� . �32�

If a Gaussian distribution P��� with a width w is con-sidered, the values of the respective integrals can be com-puted. Therefore, the average values of the three extrinsiccoefficients can be obtained, expressed by the coefficients g,which depend on the distribution function and on the respec-tive functions f���.

The three coefficients as a function of the width w arerepresented in Fig. 8. Although �s� and ��� increase when thedistribution widens, this is not the case of �d� since in thiscase the effect can be positive or negative depending on thedirection.

Moreover, the relations gd /g� and gd /gs can be com-puted, as well as the extrinsic electromechanical coefficientkext, which is equal to the product gd /g� ·gd /gs �see Fig.8�b��. One may observe that such a coefficient tends to 1 inthe absence of dispersion, while it tends to 0 if there is abroad distribution, as occurs in unpoled ceramics.

In an unpoled ceramic, all directions are expected to beequally probable, so the distribution function must be flat.This gives rise to elastic and dielectric isotropies but a nullpiezoelectricity. However, when the ceramic is poled, mostof the domains and domain walls are expected to be orientedaccording to a small angle � producing a small piezoelectriceffect. Taking into account the real extrinsic contribution to

permittivity and piezoelectricity in the poling direction, it isevident that such a model, as we have conceived, cannotfully explain the behavior in this respect.

In the rhombohedral case, there are two different typesof domain walls, of 71° and 109°, so two types of twinnedstructures can be formed. Assuming that both are present in aconsiderable amount, the description of the ceramic can onlybe possible if two different distributions P71��� and P109���are used. It is worth pointing out that the sum of both prob-abilities, P71��� and P109���, extended to all orientations,must be normalized to one.

VI. CORRELATION BETWEEN THEORY ANDBEHAVIOR

If a material has a behavior similar to that predicted bythe model, its domain structure can be expected to be similarto that previously described. Agreements and divergences inthe properties of some common commercial ceramics arecompared in Table I.

First of all, one may see that the extrinsic effect willprobably be much greater in soft than in hard ceramics, andthis is likely to be the cause of the high differences betweentheir �33

T values, a fact that is also apparent in the rest ofcoefficients. However, this may not be the only reason forthese differences, given that the differences in compositionmay cause the crystallographic phase to change.25 In additionto these characteristics, concerning poled ceramics, we musttake into account that in these ceramics the dielectric con-stant is less than in unpoled ones. This fact is in agreementwith the present model, as can be seen in Fig. 8, given thatthe distribution width diminishes in the poling process.

The present model predicts that extrinsic �11 should benotably higher than �33. It must be taken in consideration thatin monocrystals a similar anisotropy is produced by the in-trinsic effect. However, nonlinear measurement of the per-mittivity shows that the increment in �11 due to the fieldamplitude is significant and of the same order of magnitude

FIG. 8. �Color online� �a� g�, gd, and gs as a function of the width w of thedistribution. �b� Ratios gd /g� and gd /gs and electromechanical coupling fac-tor kext as a function of w.

FIG. 7. �Color online� Functions f����, fd���, and fs���.

TABLE I. Relevant linear characteristics of some commercial ceramics �Ref. 31�.

Material �33T �11

T d33 d31 d15 s33E s11

E s55E k33

l

PZT8 �very hard� 1000 1290 225 −97 330 13.5 11.5 31.9 0.64PZT4 �hard� 1300 1475 289 −123 496 15.5 12.3 39.0 0.70PZT5A �soft� 1700 1730 374 −171 584 18.8 16.4 47.5 0.705

Barium Titanate 1700 1450 190 −78 260 9.5 9.1 22.8 0.50

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both in hard and in soft materials,26 which is in agreementwith our model. It is necessary to point out that the highvalue of �33 can be explained if the coexistence of 180°domain walls is taken into account.

The relation d33=−2d31 is sufficiently fulfilled in allcases. The value of −d31/d33 ranges from 0.43 to 0.46, some-what less than the theoretical value of 0.5. This causes thehydrostatic piezoelectric coefficient to be small but not null.As expected, the coefficient d15 is the highest of all, althoughonly moderately so. However, the electromechanical cou-pling factor is somewhat less than 1, showing that an appre-ciable dispersion of grain orientation exists. In textured ce-ramics, where such dispersion is reduced to a minimum, thevalue of k reaches its highest value, as predicted by themodel.

The application of high electric fields gives rise to non-linear phenomena, attributable to extrinsic mechanisms,which are revealed by an increase in the dielectric constantand in the reverse piezoelectric coefficient. The non-180°wall displacement produces an increment in the electriccharge and in deformation, both proportional to such dis-placement, so that they must be mutually proportional. In theproposed model, if the dispersion of P is small, the extrinsiccontribution to d33 and to �33 must be proportional to thespontaneous strain and polarization, respectively. Thus, theplot of d33

inv vs �33 for different applied field amplitudes notonly shows this proportionality but also that the slope mustbe the same for all reasonably poled PZT ceramics. Thisoccurs experimentally with all PZT ceramic types, whetherthe nonlinear effect is linear or quadratic dependent on thefield amplitude.27,28

VII. DISCUSSION AND CONCLUSIONS

The proposed model enables the dielectric, piezoelectric,and elastic behaviors of ceramics to be analyzed taking intoaccount their domain structure. A grain is taken as an el-emental unit containing a lamellar structure exclusivelyformed by two domain types that alternate in layers andseparated by non-180° walls, which are subjected neither tostress nor space electric charge. The advantage of using aparticular coordinate system that simplifies the description isshown. The model can be applied either to tetragonal or torhombohedral ceramics, although such structures are morefrequent in the first type than in the second one.

Assuming that the state of a grain is defined by the meandomain thickness d and the fraction �, it is shown that itsbehavior can be described when electric field or mechanicalstress is applied, taking into account both intrinsic and ex-trinsic effects. It can be shown that the matrix describing thelinear electromechanical behavior must be the sum of bothcontributions, extrinsic and intrinsic, which has a fixed partand a part that is dependent on �. In order to add up theseeffects correctly, the set of variables that must be taken asindependent is also shown.

The basis enabling us to compute the response of a grainas a function of its orientation is likewise established. The

limiting case in which only the extrinsic effect is present isalso studied by evaluating the effect it produces on the valuesof the different coefficients of the material.

The study of the response as a function of the orientationallows us to describe the ceramic as a whole, in terms of acertain distribution function, whose independent variablesare the two angles describing its orientation. It is observedthat the distribution of the volume as a function of the anglesmust be complemented by the domain thickness �extrinsic�and the fraction � �intrinsic�. To this effect, the electrome-chanical coupling factor, which value must be one �kext=1�in a twinned structure with an exclusively extrinsic contribu-tion, reduces its value due to the broadening of the distribu-tion and to the intrinsic contribution.

Finally, a review of the characteristics of commercialceramics shows to what extent the theoretical behavior re-flects reality. Soft ceramics are closer to the model than hardones but only in certain respects. Some nonlinear behaviors,such as the close relation between the increments of piezo-electric d33 and dielectric �T

11 coefficients or the incrementof �T

11 observed when the amplitude is increased, corrobo-rate the validity of the model.

The presence of impurities and defects, which has notbeen included in the model, may modify the results in twodifferent ways: by altering the mobility, which does notchange the results obtained, except if an anisotropic mobilityis considered and by delivering nucleation centers that en-hance the rise of more complex structures.

Such validity is expected to be in agreement with thefulfillment of the following three conditions. First, the de-scribed structure represents a high percent of the whole vol-ume of the ceramic. Second, the ceramic has been suitablypoled, so the described distribution differs little from the realone. Finally, the disturbing effects are not particularly impor-tant, e.g., nonlinear wall mobility due to the impurities, thecoexistence of both 180° and non-180° domain walls, thecreation of four-domain complex structures, and the electri-cal and mechanical interactions between neighboring grains.

Although, in the interests of limiting complexity, the in-teraction between grains is not included in the present model;this interaction must nevertheless be taken into account sinceany intrinsic or extrinsic modification of a grain will produceelectric fields and stresses in its surroundings, which willaffect the neighboring grains. Thus, the hypothesis thatgrains are independent is far from being true. Under the ac-tion of a favorable field, the grains cannot all expand withoutrestraint. Only those that are best oriented will expand, at theexpense of the rest. Due to those effects, the domain struc-ture is not broken at the grain boundary, so domains canextend over multiple grains. This effect is especially impor-tant for the smallest grains.29,30

The possibility that mobility may have a nonlinear be-havior, which would depend on the nature of defects, hasalso not been dealt with here, neither has the possibility thatcomplex structures may be formed, such as those that com-bine this type of structure with 180° domain walls. An accu-rate prediction of the whole behavior can only be expectedwhen such factors, not dealt with in this approximation, aretaken into account.

044117-8 Perez et al. J. Appl. Phys. 102, 044117 �2007�

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ACKNOWLEDGMENTS

This work is supported by the Spanish MEC �Project No.MAT2004-01341� and the European Network POLECER�No. G5RT.CT-2001-05024�.

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