Symmetry-adaptive ferroelectric mesostates in orientedPb(BI1/3BII2/3)O3–PbTiO3 crystalsDwight Viehland Citation: J. Appl. Phys. 88, 4794 (2000); doi: 10.1063/1.1289789 View online: http://dx.doi.org/10.1063/1.1289789 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v88/i8 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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JOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 8 15 OCTOBER 2000

Symmetry-adaptive ferroelectric mesostates in orientedPb„BI1Õ3BII2Õ3…O3–PbTiO3 crystals

Dwight Viehlanda)

Naval Seacommand, Division Newport, Newport, Rhode Island 02841-1708

~Received 29 February 2000; accepted for publication 30 June 2000!

An intermediate orthorhombic ferroelectric phase has been found in oriented crystals of(0.92)Pb~Zn1/3Nb2/3!O3–0.08PbTiO3 ~PZN-PT 92/8!. Investigations have been performed byelectrically induced polarization and strain methods, and reciprocal phase space mapping. Thelattice parameters of this intermediate ferroelectric state have been shown to be equal to thosepredicted by the adaptive martensite theory. A hierarchy of symmetries is believed to exist on themesoscale which are due to symmetry reductions by domain averaging. ©2000 American Instituteof Physics.@S0021-8979~00!07219-4#

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I. INTRODUCTION

^001& oriented single crystals of (12x)Pb~Zn1/3Nb2/3!O3–xPbTiO3 @PZN-PT (12x)/x# and (12x)Pb~Mg1/3Nb2/3!O3–xPbTiO3 @PMN-PT (12x)/x# havebeen reported to possess high electromechanical couplingefficients of 0.94,1–6 high longitudonal piezoelectric constants of between 1500 and 2400 pC/N,1–6 and high electri-cally induced strains of up to 1.7%.3–6 The origins of thehigh electrically induced strain have been attributed toinduced rhombohedral to tetragonal ferroelectric phtransition.3 The high values of the total induced strainsaturation are consistent with thec/a-ratio change of a ferro-electric tetragonal state.7

An understanding of the mechanism of the electricainduced phase transformation in these crystals has notdeveloped. However, a considerable amount of workgone into understanding the phase transformation menism in mixedB-site cation perovskites~crystals and poly-crystalline! under zero field. MixedB-site cation ferroelectricperovskites are well known to have unusual phase transmational mechanisms.8,9 Both random bond~or spin-glasslike!10–12 and random field~or pinning!13,14 models havebeen proposed. Both have validity and have been suggeto be equally important in mixedB-site cation perovskites.12

Recently, Blinc has proposed a random-field induced sphcal bond model,15 which incorporates both previous modein a self-consistent manner. In this model, the dynamicsthe polarization are controlled by the random/spherical bcharacteristics, which are induced by random fields. Thebrid random-field random-bond model allows for frustratidue to the near spherical nature of the free energy. Multcompeting orderings of the polarization along multiple diretions is allowed. Such an approach allows sophisticatedderstanding of competing interactions on a mesoscale.

Obtaining an atomic level understanding of the comping interactions has proven more difficult. Ferroelectric trasitions are displacive in nature and consequently martensNormally, ferroelectric transformations are weakly martentic as the elastic deformations are small. However, in

a!Electronic mail: ddviehland@aol.com

4790021-8979/2000/88(8)/4794/13/$17.00

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electrically induced rhombohedral to tetragonal transitiothe transformation may be more strongly martensitic. Tpurpose of this investigation was to study the phase trational mechanism of oriented Pb~B1B2!O3–PbTiO3 crystals.Electrically induced strain and polarization, and reciprophase space mapping investigations have been performe

II. EXPERIMENTAL PROCEDURE

Single crystals of PZN-PT 92/8 were obtained from TRCeramics, State College, PA. Crystals of^001&, ^111&, and^110& orientations were obtained. These crystals were groby a flux method. Plate-like specimens were cut into dimsions of ;0.5 cm30.5 cm30.05 cm. The specimens werelectroded with gold.

Reciprocal phase space mapping was performed^110&-oriented PZN-PT 92/8 crystals. A PhillipsX8 Pert sys-tem equipped with a PW 3040 goniometer and a PW 30open Euleran cradle was used. Measurements focused o~330! reflection. The goniometer was swept through a narrrange ofu–2u space focused upon the~330! reflection whilerotating the Euleran cradle in order to obtain twdimensional mapping of the diffracted intensity within a ploptimized for a third angle~f!.

P–E measurements were made using a modifiSawyer–Tower bridge. In addition,e –E measurements wersimultaneously performed using an inductance method.measurements were performed using a drive frequencyHz. Electrically induced strain (e –E) and polarization(P–E) measurements were performed under various cotions. Measurements were taken on specimens that weresentially ‘‘free’’ to deform, and that were slightly stressed bthe sample holder. The difference between the stress statthe measurements was controlled by the mechanical spconstant of the sample holder.

III. ELECTROSTRICTIVE NATURE OF INDUCEDSTRAIN

A. e – E investigations of Š001‹ orientation

Figure 1~a! shows the bipolare –E response of an001&-oriented crystal. This measurement was performed on a c

4 © 2000 American Institute of Physics

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4795J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

tal in a mechanically ‘‘free’’ condition. Figure 1~a! clearlyillustrates an electrically induced phase transformation invicinity of 15 kV/cm. Strong hysteretic effects are evident,can be seen by the difference between the forward andverse switching fields. The double-loop nature of this cuis qualitatively similar to that observed in the electricainduced antiferroelectric orthorhombic (AFEo) to ferroelec-tric tetragonal (FEt) phase transformation in PbZrO3.

16 Themagnitude of the electrically induced shape change;1.231022 under an electric field of 20 kV/cm.

Figure 1~b! shows the unipolare –E response of an^001&-oriented crystal. The measurement was performed ocrystal in a slightly stressed condition. In Figure 1~b! thee –E response can be seen to be nearly anhysteretic.magnitude of the electrically induced strain at 120 kV/cwas 1.2%. Comparisons of the data in Figs. 1~a! and 1~b!will demonstrate that the total shape change~the sum of con-tractive and expansive contributions! for the free bipolar

FIG. 1. Comparative« –E data for^001&-oriented PZN-PT 92/8 crystals thaillustrate a strong stress/clamping sensitivity in the phase transformatcharacteristics.~a! ‘‘Free’’ condition under bipolar drive, and~b! slightly‘‘stressed’’ condition under unipolar drive.

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drive was close in magnitude to the maximum value ofe forthe slightly stressed unipolar drive. Clearly, small changethe mechanical boundary conditions make dramatic chanin thee –E characteristics. Such changes have previouslybeen reported for any piezoelectric crystal or ceramic.

Figures 2~a! and 2~b! shows theP–E ande –E charac-teristics taken in the slightly stressed condition at variodrive fields. These data were taken simultaneously. Thesults show quadraticP–E behavior, with little energy lossThe results are similar to that expected for electrostrictPMN-PT ceramics. In order to obtain better insights into F2, the e data were analyzed as a function ofP2. Figure 3shows a plot ofe –P2. Figure 3 illustrates a near linear dependence ofe on P2. This clearly demonstrates that the eletrically induced strain is electrostrictive in nature.

A comparison of the results in Figs. 1~a! and 1~b! willreveal that the total strain near saturation is equivalentboth responses. This is an important point to distinguiClearly, the strain is electrostrictive in nature in each ca~compare with Fig. 3!. However, the mechanism by whicthe strain is recovered under field changes significantly wsmall changes in mechanical boundary conditions. The stcan be recovered abruptly at low fields by an electricastimulated phase transformation, or it can be recoveslowly by an electrostrictive process. The total shape chais constant for both cases, but the transformational pathby which the strain is recovered is different. There is nothunusual about either case, since both transformatiostrains17 in normal ferroelectric transitions and electricalinduced strains in electrostrictive PMN-PT relaxferroelectrics18 are electrostrictive in nature. However, whis unusual is that one can observe either response insingle crystal depending upon how the measurement isformed.

In electrostrictive relaxor ferroelectrics, the symmetrythe low temperature state is broken by quenched disorRather than undergoing a phase transformation into thetemperature state, the system freezes into a state with cluof the low temperature state embedded within the aversymmetry of the high temperature state. In the relaxor sttransmission electron microscopy~TEM! investigations haverevealed the presence of nanometric scale polar clusters19,20

Under high electric field, the polar clusters are drivenwards a long-range ordered ferroelectric state. Thus, the etrostrictive strain, which is the phase transformational strais recovered gradually with increasing field in a reversibmanner. However, in a normal ferroelectric, long-range poorder develops discontinuously at a phase transformatThus, the electrostrictive strain is recovered abruptly acritical field level in an irreversible manner.

Clearly, before application of an electric field, the sytem is in a low temperature polarized state. The electrosttive process must then be due to a transition betweenlow temperature polarized states which have significandifferent deformations. The transition between the two ferelectric states can occur discontinuously as in a normal feelectric transition@Fig. 1~a!#, or continuously as in an electrostrictive relaxor ferroelectric@Fig. 1~b!#. One couldconjecture a continuous process that involved the rotation

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4796 J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

the polarization vector in a homogeneous polarization stIn this case, a transition between rhombohedral and tetranal ferroelectric states would occur through a monocliferroelectric state where the monoclinic angle varsignificantly.21 However, x-ray and TEM data to be presented in Sec. V clearly demonstrate the importance of sttural irregularities. Such irregularities are consistent withcontinuous transformational process in the presencequenched random fields, as occurs in electrostrictive relaferroelectrics. Monoclinic symmetry may exist with a vaable angle, however it would be a simple cell and not the ucell. In the random-field case, polarization switching wouoccur by a domain breakdown and realignment process.main breakdown under moderate ac drives in ‘‘soft’’ ferrelectrics near morphotropic phase boundaries have prously been reported byex situTEM.22 Furthermore, receninvestigations of polarization relaxational dynamicsPMN-PT 65/35 ceramics have also demonstrated evidencpolarization switching by domain breakdown.23

FIG. 2. ~Color! P–E and« –E data for^001&-oriented PZN-PT 92/8 crystaunder unipolar ac electrical drive at various drive amplitudes.~a! P–E dataand ~b! « –E data. These measurements were made in a slightly strecondition.

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B. Electrostrictive equations of state

In order to obtain more insights, it is necessary to qutify the electrostrictive response,17 the electrostrictive strainin a tetragonal ferroelectric state given in Eq.~1!, wheree33

is the longitudonal electrostrictive strain along the~001!,e11, ande22,

e335Q33P~001!2 ~1a!

and

e115e225Q13P~001!2 ~1b!

are the transverse electrostrictive strains along the~010! and~100!, P3 is the polarization along the~001!, P1 is the po-larization along the~100! and~010!, Q33 is the longitudonalelectrostrictive coefficient, andQ31 is the transverse electrostrictive coefficient. Haun has previously reported valuesQ33 and Q31 for MPB compositions of PZT of 9.6631022

and 24.631022, respectively.17 Recently, Cross and coworkers have found similar values for001&-orientedPZN-PT 92/8 crystals.21

These equations can now be used to estimate thepected magnitudes of the electrostrictive strains. Under hac electrical drive, close to saturation in thee –E curves,P(001) was found to be equal toPs , and obviouslyP(100) andP(010) are zero. Placing these values into Eqs.~1!, the maxi-mum magnitudes ofe33 and e11 can be estimated as 1.631022 and 27.7331023. The c-lattice constant of the tetragonal ferroelectric phase (ct) of PZN-PT 92/8 has previ-ously been shown to be 4.09 Å by Durbinet al.,24,25 whichwas close to that previously reported by Kuwataet al. forPZN-PT 88/12.1,2 The lattice parameter of the cubic sta(ac) is related to those of the tetragonal (at ,ct) by

ct5ac~11e33!, ~2a!

at5ac~11e11!. ~2b!

Using Eq.~2a!, the value ofac can be estimated as 4.028 ÅThe value ofat can then be estimated using Eq.~2b! as3.995 Å.

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FIG. 3. ~Color! Plot of e as a function ofP2. Data are shown for variousfields.

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4797J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

Interestingly, this value forac is in significant disagreement with that previously reported by Kuwataet al.1,2 and byDurbin et al.24,25Each of these investigators reported a vafor the lattice constant of the rhombohedral ferroelectric s(ar), which is equal to that ofac , of 4.055 Å for PZN-PT92/8. However, if one would take the difference between tvalue for ac and that forct , the maximum value ofe33

would be 8.6331023, which is significantly lower than thashown in Figs. 1~a! and 1~b! and lower than that previouslreported by other investigators. One could attribute thisference in total strain to domain switching effects. Howevthere are several reasons why care must be taken. Firststrain shown in Fig. 1~b! was electrostrictive, whereas strafrom domain switching is not. Only strain from a phaswitching process is electrostrictive. Second, this value oar

is significantly higher than that previously reported fromvestigations of MPB compositions of PMN-PT.26 These in-vestigations have reported values of;4.02 Å, which iscloser to that determined above using Eqs.~1! and ~2!.

Recently, Guo and Bhalla27 have investigated PZN-PT4.5/95.5 and 92/8 by neutron diffraction. In the as-grocondition of the crystal, the value ofar was determined to be;4.05 Å. However, in annealed specimens, the value oar

was found to be;4.02Å. These results demonstrate impotant changes in the structure with specimen history. Cleathe system can be trapped into a metastable intermedstate during either processing or potentially electrical cyclof the field.

Amin et al. have previously phenomenologically moeled MPB compositions of PZT.28 They observed that anorthorhombic ferroelectric phase was always present ametastable state. On the rhombohedral side of the MPBwas higher in free energy (DG) than the rhombohedraphase, but lower than the tetragonal one. Whereas ontetragonal side of the MPB, it was higher inDG than thetetragonal phase, but lower than the rhombohedral one.thermore, in the BaTiO3 system, a stable orthorhombic ferroelectric phase is well known to exist between ferroelecrhombohedral and tetragonal ones.29–31 The unit cell in thisphase is double that of the simple unit cell. Interestingly,simple cell has monoclinic symmetry,30 having the samepoint group as that recently suggested by Cross for PZNcrystals.21

Considerating the facts mentioned in the proceedparagraph, it is reasonable to anticipate the metastable imediate state in PZN-PT to be orthorhombic ferroelectricorder to obtain more insights, it is necessary to quantifyelectrostrictive response of an orthorhombic ferroelecstate. The equations of state are given in Eqs.~3!,17 whereQ44 an electrostrictive shear coefficient.

«335«115~Q331Q13!P~001!2 , ~3a!

«2252Q13P~001!2 , ~3b!

«135Q44P~001!2 . ~3c!

Using Eqs.~3a! and ~3b!, the values of«33 and «22 can beestimated as 7.931023 and 4.531023, respectively. Con-sidering that the values ofQ11 andQ13 were equal to those

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reported for MPB compositions of PZT, the value ofQ44 wastaken to be equal to that determined by Haun,17 which was8.1931022 m4/C2. By placing this value into Eq. 3~c!, «13

can be estimated as 6.8931023. The lattice constants of theorthorhombic ferroelectric phase referenced to its monoclsimple cell (am , am , cm) can thus be calculated using.

am5bm5ac~11«33!, ~4a!

cm5ac~11«11!. ~4b!

Using these equations, the values ofam , bm , cm can be es-timated as 4.043, 4.043, and 3.995 Å, respectively. Thetational angle of the monoclinic structure from the cubic ois illustrated in Fig. 4~a! asf,29–31and can readily be calculated from the triangle in Figure 4~a! by

f5sin21~am«13/am!5sin21~«13!. ~5!

Using this equation,f can be estimated as 0.395°.The orthorhombic cell is obtained from the monoclin

shown in Fig. 4~b! in a manner similar to that for orthorhombic BaTiO3.

29–31 The orthorhombic lattice paramete(ao , bo , co) can be obtained from the monoclini(am , am , cm ;f),31 given in

ao52am sin@~901f!/2#, ~6a!

bo52am cos@~901f!/2#, ~6b!

co5cm . ~6c!

Using Eqs.~6! and referencing the parameters to the^001&,the values of the lattice parameters (221/2ao,221/2bo ,co) canbe determined to be 4.024, 4.057, and 3.995 Å, respectivAn important point to note is thatao is equal toac , thatco

is equal toat , and thatbo is equal to the lattice parameter othe intermediate ferroelectric state. The importance of twill be discussed further in Sec. VI.

Recently, Cross21 has determined the electrostrictive cefficients of PZN-PT crystals. Different values were foudepending upon how the coefficients were calculated. Tvalues ofQ33 andQ31 were determined to be 9.431022 and24.731022 m4/C2 by estimations of slope changes in thstrain~which are close to the values used in this article!, and

FIG. 4. Diagrams illustrating the structure of the orthorhombic ferroelecstate.~a! The simple cell which has monoclinic symmetry and~b! the unitcell which consists of a doubled monoclinic cell.

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5.331022 and22.631022 m4/C2 by estimations from totastrain changes. Naturally, the question arises as to wvalues ofQ33 andQ31 are the correct values. The answerboth. Considering the extreme stress sensitivity of the mrial @compare Figs. 1~a! and 1~b!#, the values which one determines depend significantly upon how determinationsmade. To illustrate this better, calculations can be redusing values ofQ11 and Q12 equal to 0.053 and20.026m4/C2, respectively, with the assumption that the systeminitially in an intermediate, metastable orthorhombic ferrelectric state. The values («33,«11) can be estimated as 8.31023 and 24.431023 using Eqs.~3!. The values of thelattice parameters (ao , bo) can then be estimated as 4.00 a4.093 Å, which are consistent with those forat , ct .

The lattice parameter results of Sec. III are summariin Table I. The results indicate that the electrically inductransformation@Fig. 1~b!# may proceed by gradual symmetadaptations from one low temperature polarized statewards another that is much more deformable via an interdiate orthorhombic ferroelectric state which contains a mture of the lattice constants of the other two states. In thecondition @Fig. 1~a!#, the system may be metastably trappinto the orthorhombic ferroelectric state.

IV. DIRECT OBSERVATION OF THE FERROELECTRICORTHORHOMBIC STATE

The results presented in Sec. III indicate that the fitransitional step involves an electrically induced^111& to^110& polarized state. However, no evidence has previoubeen reported which demonstrates or even indicates thferroelectric orthorhombic state might exist in PZN-PTPMN-PT.

A. A Fully saturated polarizable deformable Š110‹state

Figure 5~a! shows the« –E behavior for the110& orien-tation. These data were taken in the free condition. Norbutterfly-like hysteresis looks were observed. The magnitof the total electrically induced shape change was;0.6%.The magnitude of the contractive and expansive strains w;0.4% and 0.2%, respectively. Figure 5~b! showsP–E datafor an ^110&-oriented crystal of PZN-PT 92/8 in the free codition. These data clearly show the presence of a ferroeleP–E response. Furthermore, the values ofPs and Pr were0.41 and 0.38 C/m2, respectively. These results unambi

TABLE I. Summary of lattice constants~determined by electrostrictionequations!.

PhaseaÅ

bÅ

cÅ

Cubic 4.028 4.028 4.028Ferroelectricrhombohedral

4.028 4.028 4.028

Ferroelectricorthorhombic

4.028 4.058 3.995

Ferroelectrictetragonal

3.995 3.995 4.090

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iously demonstrate the presence of a fully polarizable, fudeformabable orthorhombic ferroelectric state.

Figures 6~a!–6~c! show theP–E data for^001&, ^110&,and^111& orientations, respectively. Inspection of these dwill reveal that the saturation polarization (Ps) along all ofthese directions is equivalent. However, significant vartions were observed in the area of the hysteresis loops anthe coercive field (Ec). The coercive field was found to increase from 5 kV/cm for the001&, to 10.5 kV/cm for the^110&, to 12.5 kV/cm for the111& orientations. Corresponding changes in the area of the hysteresis loops wereserved. These results are consistent with the phenomenocal predictions of Amin et al.28 that the orthorhombicferroelectric phase is metastable with respect to the tetranal and rhombohedral ferroelectric ones. Clearly; a fusaturated orthorhombic ferroelectric phase is stablePZN-PT that is higher inDG than the tetragonal ferroelectristate and that is slightly lower inDG than the rhombohedraone.

B. Evidence of an intermediate FE o state in the phasetransformational sequence for Š001‹-orientedcrystals

P–E data are presented in Fig. 7 for a freshly annea^001&-oriented crystal for various magnitudes of ac electridrive. These measurements were made in the free condi

FIG. 5. « –E and P–E data for^110&-oriented PZN-PT 92/8 crystal undebipolar ac electrical drive.~a! « –E data and~b! P–E data. These measurements were made in a free condition.

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4799J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

For Eac510 kV/cm, which is below that required to inducthe secondary transition in the free condition, the value ofPr

was equal toPs/31/2(0.24 C/m2). This is consistent with a

rhombohedral projection of the polarization onto the^001&.However, it should be noticed that under this drive levelpolarization tended towards a saturation value of 0.29 C/2,which is equal toPs/2

1/2. This is consistent with an orthorhombic projection of the polarization onto the^001&.

It is interesting to note some of the changes that ocwith increasingEac. For Eac520 kV/cm, a secondary phastransition occurred near 15 kV. In this case,Ps was equal to0.41 C/m2, which is consistent with a fully polarized tetragonal ferroelectric state. However, on removal of the field,Pr

FIG. 6. P–E data for various crystal orientations.~a! ^001&, ~b! ^110&, and~c! ^111&.

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changed to 0.29 C/m2. This demonstrates that the saturatipolarization of the first transitional state is equal to thePr ofthe secondary transitional state.

The results of Sec. IV clearly demonstrate two trantional steps with a ferroelectric orthorhombic phase asintermediate state. Furthermore, the results also demonsthat the ferroelectric orthorhombic state can be metastatrapped in^001&- oriented crystals.

V. STRUCTURAL INVESTIGATIONS OF THEINTERMEDIATE FEo STATE

v scans are made by rocking the sample parallel toplane of diffraction. Consequently, Ewald’s circle is shifteallowing determination of the scattering intensity along tki direction, i.e., as a function of depth in the specimen.specimen can also be rocked in an Eularian cradle perpdicular to the plane of diffraction~i.e., kx–ky scans!. How-ever, in the current investigation, only a constant anguoffset (b I) alongky was found necessary for optimization othe diffraction intensity with a cradle. The use of triple axdiffractometers that havev scan capabilities offers severaadvantages.32 First, in thekx–ky plane, homogeneous strainand strain gradients can be separated from structural imfections such as tilts and mosacitiy. Second, in thev scan,the mosaic spread and tilting can be determined along thki

direction. This diffraction method is often used to investigaepitaxial layers and heterostructures. The lattice constanthe layers, the epitaxial strain, and the coherency of thetaxial layers can all be determined.

Figure 8 shows a~330! diffraction phase space map foan^110&- oriented PZN-PT 92/8 crystal in the as-grown crytal. They axis of Fig. 8 is thev scan~i.e., ki!, and thex axisis thev–2u scan. The corresponding~330! reciprocal spacemap is shown in Fig. 9. Figures 8 and 9 reveal importfeatures, which will be discussed sequentially in more dein the following paragraphs. The principal structural featuof interest in this figure include

~1! a ~330! peak splitting which resulted in a principapeak centered at 2u5107.675°(0.8073 r.l.u.) and a satallitpeak centered at 2u5106.835°(0.8030 r.l.u.);

~2! a rotation anglev of 0.35° between the two peakand a rotation angleb1 of 0.63° for both peaks about th^110& which is required for optimization of the diffractionintensity;

~3! strong diffuse scattering alongki in the satallite peak,with the full width at half maximum~FWHM! extendingbetween 0.10° and 0.35°, and significant diffuse scatterbetween the two~330! peaks.

Assuming that the splitting of the~330! peak is due tomultidomain formation, the peak splitting~D2u! can be re-lated to an angle between domains (b II), given by24

D2u52~u12u2!50.84°52lb II /@a cos~2u/2!#, ~7!

wherea is the lattice spacing~;4 Å!, l is the wavelength ofthe CuKa radiation~1.542 Å!, and ~2u/2! is the peak posi-tion of the principal peak~53.84°!. By rearranging Eq.~7!,b II can be estimated to be 0.65°. Clearly, the rotatio

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4800 J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

anglesb I andb II are equivalent, and result in homogeneostrain of the specimen when multiple domains are arrangeometrically. The first domain would be rotated 0.65° wrespect to the axis of a Cartesian coordinate, and thedomain would be rotated 0.65° with respect to the first

FIG. 7. ~Color! P–E for a freshly annealed001&-oriented PZN-PT 92/8crystal under bipolar drive. The measurement atE510 kV/cm was madefrom the virgin state, and the measurement atE520 kV/cm was made sub-sequently.

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sd

xtr

1.3° with respect to the Cartesian coordinate. The importaof this angular rotation will be discussed in more detailSec. VI.

A second angular rotation was observed in thev scan. InFigs. 8 and 9, the satallite peak can be seen to be displby 0.34°~0.057 r.l.u.! from kx . This rotation does not resulin homogeneous strain between domains, and is close topredicted by Eq.~5! for the monoclinic simple cell of theorthorhombic ferroelectric phase~f!. The value ofd^110& canbe determined from the principal peak using Bragg’s lawbe 2.863 Å. Both sides of the monoclinic simple cell aequal, as can be seen in Fig. 4~b!.29–31Consequently,am canbe approximated as 21/2d^110& which can then be estimated a4.048 Å. Interestingly, this is close to the value ofam pre-dicted for the simple monoclinic cell of the orthorhombunit cell in Sec. III B. Using Eqs. 6~a! and 6~b! and referenc-ing the parameters to the001&, the values of the lattice parameters of the orthorhombic ferroelectric phase (ao , bo)can be estimated to be 4.030 and 4.059 Å, respectivThese values are in good agreement with those predicteSec. III B.

The line broadening along theki direction in reciprocalmapping yields the lateral correlation lengthj. The Scherrerequation33 can be used to determinej, as given in

j5l/@2W cos~2u/2!#, ~8!

FIG. 8. ~Color! ~330! diffraction space map for a110&-oriented PZN-PT 92/8 crystal.

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4801J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

FIG. 9. ~Color! ~330! reciprocal space map for a110&-oriented PZN-PT 92/8 crystal.

e

p-rmM

Te

,

ii

ntit

eninac

civehs

si-lae

srgy

-

.near

where W is the width of the satellite peak in radians. Thvalue ofW alongki was 0.24°~;0.016 rad!, as can be de-termined from Fig. 8 or 9. The value ofj can then be esti-mated as;80 Å. This value is a correlation length that reresents the distance over which the domains are unifoViehland and co-workers have previously performed TEinvestigations of MPB compositions of PMN-Pceramics.20,34,35 They reported tweed-like structures, raththan normal micron-sized domains. Domains of;103 Å inlength and;102 Å in width were reported. Furthermorethese tweed-like structures were oriented along the^110&.

The results presented in Sec. V demonstrate that antermediate orthorhombic ferroelectric phase existsPZN-PT 92/8. This orthorhombic phase can be represeeither by a simple cell of monoclinic symmetry, or by a uncell of orthorhombic symmetry. Orthorhombic domains thexist which are long and thin. These orthorhombic domahave a small rotational angle of 0.65° with respect to eother along the110&.

VI. ADAPATIVE SYMMETRY MODEL FOR THEINTERMEDIATE FEo STATE

Khachaturyan et al.36 developed a thermodynamitheory of martensite transformations which allows adaptsymmetries. The adaptive phase was believed to be a mstable alternative to normal nucleation and growth. Ttransformation path is characterized by a sequence of me

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.

r

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cale coherent structures formed by atomically thin martentic plates consisting of quasiperiodic alternating lamelwith periodl, as shown in Fig. 10.

The condition for the minimization of the twin thicknesis controlled by a balance of elastic and surface eneterms, as given in Eq.~9!,36 whereg tw is the twin surface

l;~g twD/m«^001&2 !1/2 ~9!

energy,m is the shear modulus, and«^001& is the elastic strainalong the^001&. Twin thickness minimization to atomic lev

FIG. 10. Illustration of mesothin domain configurations~see Ref. 36!. d1

and d2 are the domain thicknesses, andD is the average domain lengthSymmetry reduction occurs due to effective domain averaging on aatomic scale.

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4802 J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

els will occur under simultaneous conditions of a lowg tw , asoft elastic shear modulus, and high transformational stin the cubic to tetragonal transformation. The symmetrystriction upon the adaptive phase is that it must have losymmetry than that of the normal~untwinned! martensitephase and that this lower symmetry point group must bsubgroup of that of the parent phase. Consequently, aquirement of an adaptive phase is that it is related to thathe parent phase through an invariant-plane strain, whmust also be parallel to the twinning plane of the prodphase.

In a cubic-to-tetragonal martensitic transformation,adaptive phase must have a average orthorhombic symmwith symmetry elements of~or lower than! mm2. Khachatu-ryan et al.36 have also shown that the crystal lattice paraeters of the adaptive martensite (aad ,bad ,cad) are related tothose of the normal martensite (at , ct) and the parent cubicphase (ac) as given in

aad5ac , ~10a!

bad5at1ct2ac , ~10b!

cad5at . ~10c!

The relations in Eqs.~10! among the crystal-lattice parameters of the orthorhombic ferroelectric phase, the tetrago~normal martensite! ferroelectric phase, and the parent cubphase are the defining fingerprints of an adaptive phase36

For PZN-PT, this theory can be applied to the rhombhedral to tetragonal ferroelectric transformation by assumthat the rhombohedral phase is pseudocubic. This is a coassumption for the most part, as the lattice constant chain the cubic to rhombohedral transition are small, particulacompared to those of the rhombohedral to tetragonal tration. The lattice constants of the cubic, tetragonal ferroetric, rhombohedral ferroelectric, and orthorhombic ferroeltrics states of PZN-PT 92/8 were summarized in TableEquations~10! can now be applied to these data to determthe feasibility of understanding the intermediate ferroelecphase as an adaptive state. The applicability of Eqs.~10a!and ~10c! are clear, i.e.ac , ct are equal toaad , cad whichare equal 4.028 and 3.995 Å, respectively. The value ofbad

can be estimated as 3.99514.09024.028 Å from Eq.~10b!,which is equal to 4.057 Å. Thus, the adaptive martenstheory predicts the presence of an intermediate ferroelephase of average orthorhombic symmetry having latticerametersaad , bad , and cad of 4.028, 4.057, and 3.995 Årespectively. These values are in remarkable agreementthose of the intermediate ferroelectric orthorhombic phasPZN-PT listed in Table I, whereao , bo , andco are equal to4.028, 4.058, and 3.995 Å, respectively.

Further confirmation of the validity of this model can bfound by considering structural details predicted by Kachaturyan et al.36 The adaptive phase transformationmodel predicts a rotation of the average orthorhombic strture along the (110)c in order for the boundary between twatomically thin domain variants to be stress free. This resin a monoclinic texture of the average orthorhombic strture. The average rotation angle~b! between two atomicallythin domain variants is given as

Downloaded 25 Sep 2013 to 147.26.11.80. This article is copyrighted as indicated in the abstract.

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By placing the values forct and at determined from thelattice parameters of the average orthorhombic structurePZN-PT 92/8 above into Eq.~11!, b can be estimated to b1.35°. This estimate ofb is in remarkable agreement with thexperimental value of the rotation angle between two ortrhombic ferroelectric domains~b I or b II!, which was deter-mined from the x-ray data given in Figs. 8 and 9, and wequal to 1.3°. These results show that the PZN-PT 92/8an orthorhombic unit cell of 4.0275, 4.0575, and 3.995with a monoclinic texture due to a coherent adjustmenttwo orientational variants by an angle of 1.3°.

Following the concepts of Khachaturyanet al.,36 theadaptive ferroelectric phase results from a mixture of msothick layers of tetragonal ferroelectric domains, as illutrated in Fig. 10. The thickness of these domains approacthat of atomic distances. Consequently, the thickness ofdomains becomes on the order of the thickness of the domboundaries. Due to the high volume fraction of surface arthe lattice ~and consequently polarization! becomes effec-tively microheterogeneous. Thus, the adaptive orthorhomferroelectric phase must have lower symmetry than the nmal ~untwinned! tetragonal ferroelectric phase (4mm), andthis lower symmetry group must be a subgroup of that ofparent rhombohedral ferroelectric phase (3m) from whichthe tetragonal one is electrically induced. Thus, by symmerestrictions, the symmetry of the simple cell of the adaptphase must be the monoclinic point group~m!, which is alsothe symmetry of the twin structure separating two tetragoferroelectric variants. This is consistent with the x-ray dataFigs. 8 and 9. An orthorhombic unit cell with two formulunits can then be obtained, as previously shown in Fig. 4~b!.A monoclinic texture symmetry subsequently results whthe two orthorhombic variants are coherently adjusted althe ^110&c .

Accordingly,36 in the adaptive ferroelectric phase, msothin ferroelectric domains will be stacked parallel to t^110&c , similar to what was previously shown bTEM.20,34,35 The stacking will occur in such a manner ththe transformation strain mismatch is accommodated althe ^110&c . This requirement in conjunction with the mesothinness of the domains is the origin of the adaptive symetry. In this manner, the lattice constants of the polarizatvariants can be considered as being symmetry reducehaving the distortions of the domain boundary structure. Tintrinsic microinhomogeneous lattice of the adaptive phasnot random~in the compositionally uniform approximation!,rather the patterns of the microinhomogeneity are symmrestricted according to the geometrical martensitic theoryWechsler, Lieberman, and Read37 and of Bowles andMackenzie.38 In this manner, the elastic strain energy of ttetragonal ferroelectric transformation is stored in atomlevel lattice shuffles, not in conventional domain boundariresulting in a lowering of the average point group symmeof the lattice to that of the boundaries. Geometrical and symetry restrictions force the system to preserve the trantional invariance of the symmetry, which is achievedstacking mesothin domains in periodic arrangements al

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4803J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

the ^110&c . Various stacking sequences may occur alo^110&c .

The structural results presented in Sec. VI provide stroevidence of the presence of a symmetry-adaptive intermate orthorhombic ferroelectric phase in oriented single crtals of PZN-PT 92/8. In this intermediate state, a hierarchysymmetries exists. On an atomic level, ferroelectric disttions having tetragonal (4mm) symmetry are thermodynamcally stable. Tetragonal ferroelectric domain statesmixed/averaged on an atomic/mesoscale level resultingsymmetry reduction to a monoclinic simple cell. Monoclinsimple cells are then arranged into an orthorhombic unit cas previously shown in Fig. 4~b!. A texture monoclinic~m!symmetry results due to coherent rotations of mesothvariants of the average orthorhombic ferroelectric staround the^110&c to achieve complete strain accommodtion.

A. Diffuse scattering and disorder of sequentiallayers

Diffuse scattering arises due to structural microinhomgeneities. In Sec. V, the strong diffuse scattering alongki

was shown. A correlation length of;80 Å along the 110&was determined, which was in agreement with the widthstweed-like structures previously reported by TEM.20,34,35

The reciprocal mapping data in Fig. 8 also reveastrong diffuse scattering of near constant intensity ovelarge area of reciprocal space. Significant diffuse scattewas observed along110& between~330! peaks, and alongkx . The net effect of the combination of these contributioin addition to that alongki , is that significant diffuse scattering occurs over a large portion of reciprocal space. Anitropic diffuse scattering~streaking! is well known in marten-sitic transformations along110&c ,39–42and is due to tweedlike precursor structures, which have similar morphologythose reported for relaxor ferroelectrics near the boundbetween rhombohedral and tetragonal states.20,34,35 Accord-ingly, diffuseness results from spatial variations~faults! inthe stacking sequence and variations in the periodicitymesothin variants along the110&c .36 In Fig. 8, the nearconstant intensity of the diffuse scattering along the^110&c

indicates that mesothin ferroelectric domains~or tweeds!have significant variability in stacking sequences.

The adaptive symmetry model was developed forideal case of a compositionally uniform system. Inherstructural heterogeneity was predicted in the homogenecase. Symmetry restrictions were imposed upon the micrhomogeneous nature of the lattice by the geometrical mtensite theory. However, in a system containing significrandom fields43 due to quenched disorder, the effect of symetry restrictions will be increasingly reduced with an icreasing concentration of quenched disorder. This is becrandom fields will break the translational invariance, relaing the elastic energy by incoherency. Significant variatioin stacking sequences and in domain thicknesses can reschanges in the lattice constants along the^001&, and may bethe origin of the electrostrictive nature of the electrically iduced strain~Fig. 1!, as will be discussed in more detail iSec. VII C.

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The strong near isotropic diffuse scattering in the dataFig. 8 is due to irregularities introduced into the stackisequence ~and ordering within layers! introduced byquenched random fields. Stacking of mesothin ferroelecdomains occurs in sequences which minimize the elasticergy, but which are limited in order and spatial length parlel to the^110&c by restrictions imposed by the random-fieconditions.

B. Modulated structure of the adaptive phase

The adaptive symmetry model predicts a long wavlength modulated structure, which results from stackingquences of domain variants. The modulation wavelength~l!can be obtained from elastic energy considerations, sincevolume fraction of twin-related domains~v! must be keptconstant during domain thickness reduction in order to matain invariant plane transformational strain. This conditiongiven in the following equations:36

v5«~ tet!^100& /@«~ tet!^100&2«~ tet!^001&#5d1 /l, ~12a!

«~ tet!^100&5~at2ac!/ac , ~12b!

«~ tet!^001&~ct2ac!/ac , ~12c!

l5d11d2 . ~12d!

By placing the values ofct , at , andac into Eq.~7!, v can bedetermined to be 0.34, and the ratio ofd1 /d2 is ;2/5. Thus,the adaptive symmetry will result in a 7R superstructure.

The unit cell of the intermediate orthorhombic ferroeletric state was a doubled cell. The value ofd^110& was;5.75Å. The width of the domains was determined by x-ray recrocal phase space mapping to be;80 Å along the^110&.Each mesothin domain can then be approximated as b14d^110& in thickness. Thus, the modulated structure resufrom a ferroelectric tetragonal domain 4 unit cells thick this twinned along the110&c against a ferroelectric tetragonadomain 10 unit cells thick. This mixing of tetragonal domastates results in an orthorhombic ferroelectric state. The pposed modulated structure is illustrated in Fig. 11.

One of the ferroelectric layers is 4d^110& thick. 1/4 110&structural modulations are well known to occur in antiferrelectric orthorhombic (AFEo) PbZrO3.

16,44–47Structural in-vestigations of PZ~Ref. 22! have revealed mesoscale thiclayers of polar regions oriented along the^110&c which are4d^110& thick. Interestingly, Egami and co-workers have i

FIG. 11. Diagram illustrating the atomically thin ferroelectric domain stacing sequence in the 7R superstructure~see Ref. 36!. Each of the layers in the7R structure is two unit cells thick along the^110&.

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4804 J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

vestigated the Pb(Zr12xTix)O3 and PMN-PT crystalline solutions and found weak Pb displacements along^110&c ,structurally consistent with those in AFEo .48

C. Applicability of the adaptive symmetry model tothe electrically induced transformationalsequence

In order to begin developing a better understandingthe role of the adaptive symmetry on the phase transfortional mechanism under electrical field,« –E data need to beconsidered. The macroscopic shape change of the spec(«I i j ) due to two alternating twin-related tetragonal domais given in Eqs. 13~a!–13~c!,

«I i j 5v«~1! i j 1~12v!«~2! i j , ~13a!

«~1! i j 5F «33 0 0

0 «11 0

0 0 «11

G ~13b!

«~2! i j 5F «11 0 0

0 «33 0

0 0 «11

G ~13c!

where«(1)i j and «(2)i j are the elastic strain tensors of ttragonal variant 1 and tetragonal variant 2, respectively.adaptive symmetry is a result of Eq. 12~a! being placed intoEq. 13~a!. This restricts the volume fraction of the twinrelated domains to being a constant during reduction of tthicknesses to the mesoscale. However, the restrictionposed by Eq.~12a! applies only in the stress free conditiounder zero field, i.e., it predicts the metastability of thetermediate adaptive phase.

The volume fractions of the domains can be changedresponse to either mechanical or electrical ordering fieAccordingly,v can be changed by varyingl, as given in.

vn5d1 /l5d1 /~d11d2!52/~21n!. ~14!

The value ofd1 can be chosen as being fixed@i.e., 2 or l51/(n12)#, allowing d2 to vary in multiples of the unit cell~n!. Equation 13 can then be written as.

« i j 5$@2/~21n!#«~1! i j 1@n/~21n!#«~2! i j %. ~15!

This equation gives the macroscopic shape change for vous stacking sequences of domains.

Figure 12~a! shows«33 and«11 as a function ofn. In themechanically free condition, the periodicity of the stackisequence (l521n) can be considered as increasing wincreasing electric field~i.e., of n!. In this case, the value on may increase from 5 to infinity. The resulting macroscostrain is 8.431023. However, under mechanical stress, tproduct 1

2s« will be minimized. Thus, the stacking of thlayers is changed. In the partially stressed/clamped cotion, n may be decreased to 2. In this case, the value onmay increase from 2 to 5 with electric field, and subsquently from 5 to infinity. Accordingly, the resulting macrostrain is 1.231022. Correspondingly, Fig. 12~b! showsmacroscopic lattice constants~c, a! as a function ofn. Forn52, the values ofc anda are equivalent, and are close to

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value of 4.045 Å. Forn55, the values ofc anda are equal to4.06 and 4.028 Å, respectively. And, forn5102, the valuesof c anda are equal to 4.09 and 3.995 Å.

Figure 12 predicts a maximum electrically induced straof 1.231022 for ^001&-oriented crystals. This is in agreement with the« –E data presented in Fig. 1. In Fig. 12, thmaximum induced strain was equal to 1.231022, whereasthat predicted by electrostriction in Eq.~1! was 1.631022.These results demonstrate that polarization switching^001&-oriented crystals is by changes in a meso structureresult from symmetry reductions by domain averaging.near continuous sequence of mesostates seemingly e~which have slightly different mesoscale symmetries! be-tween the rhombohedral ferroelectric and the tetragoferroelectric states. Applied ordering fields~s and E! canthus readily change the stability of the system.

VII. DISCUSSION AND SUMMARY

The data support a model in which the free energy lascape is multivallied with a near continuous sequence ofgenerate states. The continuous sequence of degenerate

FIG. 12. ~Color! Macroscopic lattice strains and lattice parameters as pdicted by Eq.~15! as a function of the number of unit cells in the modulatstructure.~a! The lattice strains and~b! the lattice parameters.

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4805J. Appl. Phys., Vol. 88, No. 8, 15 October 2000 Dwight Viehland

occurs due to a combination of the variability in stackingmesothin ferroelectric domains and limitation of their ordand spatial lengths by random fields.

Random fields partially relax the effects of the symmerestrictions imposed by the Wechsler–Liebermann–Rtheory37 of martensitic transitions. The advantage of tadaptive symmetry model in the presence of quenchedorder is that the transformational pathway between twodered states is by a broken symmetry mechanism. Incase, polarization reversal does not occur by conventionucleation and growth. Rather, a metastable alternativfound in mesoscale spatial regions by the breakdownlong-range polar order, rearrangement of lattice shuffles,restacking of mesothin ferroelectric domains. Thus, the stem can pass through a nearly continuously degeneratequence of symmetry-related states within the transitiopathway.

Karthaet al.49 developed a ‘‘glassy’’ mesophase modfor metastable tweed structures observed in premartenttransformations. They modeled precursor phenomena unonlinear nonlocal elastic free energy coupled to quenclocal compositional inhomogeneites. Viehland and cworkers have previously discussed the similarities of twelike structures in MPB compositions of PMN-PT with thoof premartenstic states, based upon this glassy mesopmodel.34,35 They further discussed the importance of thetweed-like structures on the polarization switching mecnism. However, at that time, an atomic level model of tglassy mesophase~or random-field random-bond state! wasnot developed for PMN-PT. The adaptive symmetry mode36

in conjunction with quenched random fields43 can describesome of the complex aspects of frustrated atomic-levelplacement~polar and antipolar! responsible for this glassmesophase.

Due to the degeneracy of free energy along many dirtions in reciprocal space, many different types of variantsbe created/destroyed by the application of electrical anstress fields. For oriented single crystals of mixedB-site cat-ion perovskites such as PZN-PT, a theory of domain enneering is not necessary,3 as there are many degenerate stawhich the system can naturally trap a domain state into unan ordering field. To understand the unique propertiesoriented crystals, one needs to understand the nature oglassy state~both the random-bond and random-field charteristics! in relaxor ferroelectrics.20,34,35 All the importantcomponents that are necessary to describe the complexisotropic electromechanical response of oriented crystalspresent in this state. That is, except for the symmetrygeometrical restrictions imposed on the transformatiopathway37,38 between rhombohedral and tetragonal phaswhich can be accounted for by the inclusion of adaptive mtensite theory.36

ACKNOWLEDGMENTS

This work was supported by the Office of Naval Rsearch. Recognition must be given to A. G. Khachaturyanhis reiterated suggestion over a number of years to look

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adaptive symmetry states in ferroelectrics. Recognition malso be given Ruyan Guo for sharing her work before pucation, such kindness is a hallmark of professionalThanks must also be given to Lou Carriero for help in tx-ray measurements.

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