Recent progress in relaxor ferroelectrics withperovskite structure

FRONTIERS OF FERROELECTRICITY

J O U R N A L O F M A T E R I A L S S C I E N C E 41(2006)31–52

Recent progress in relaxor ferroelectrics

with perovskite structure

A.A.BOKOV,Z.-G.YE

Department of Chemistry,Simon Fraser University,8888University Drive,Burnaby,BC,V5A 1S6,Canada

Relaxor ferroelectrics were discovered almost 50years ago among the complex oxides with perovskite structure.In recent years this ?eld of research has experienced a revival of interest.In this paper we review the progress achieved.We consider the crystal structure including

quenched compositional disorder and polar nanoregions (PNR),the phase transitions including compositional order-disorder transition,transition to nonergodic (probably spherical cluster glass)state and to ferroelectric phase.We discuss the lattice dynamics and the peculiar

(especially dielectric)relaxation in relaxors.Modern theoretical models for the mechanisms of PNR formation and freezing into nonergodic glassy state are also presented.

C 2006Springer Science +Business Media,Inc.

1.Introduction Relaxor ferroelectrics or relaxors are a class of disordered crystals possessing peculiar structure and properties.At high temperature they exist in a non-polar paraelectric (PE)phase,which is similar in many respects to the PE phase of normal ferroelectrics.Upon cooling they trans-form into the ergodic relaxor (ER)state in which polar regions of nanometer scale with randomly distributed di-rections of dipole moments appear.This transformation which occurs at the so-called Burns temperature (T B )can-not be considered a structural phase transition because it is not accompanied by any change of crystal structure on the macroscopic or mesoscopic scale.Nevertheless,the polar nanoregions (PNRs)affect the behaviour of the crystal dramatically,giving rise to unique physical prop-erties.For this reason the state of crystal at T T f )shifts with frequency due to the dielectric dispersion.Because of the diffuseness of the dielectric anomaly and the anomalies in the temperature dependences of some other properties,relaxors are often called (especially in early literature)the “ferroelectrics with diffuse phase transition,”even though no transition into FE phase really occurs.

The nonergodic relaxor (NR)state existing below T f can be irreversibly transformed into a FE state by a strong enough external electric ?eld.This is an important charac-teristic of relaxors which distinguishes them from typical dipole glasses.Upon heating the FE phase transforms to the ER one at the temperature T C which is very close to T f .In many relaxors the spontaneous (i.e.without the ap-plication of an electric ?eld)phase transition from the ER into a low-temperature FE phase still occurs at T C and thus the NR state does not exist.

Compositional disorder,i.e.the disorder in the ar-rangement of different ions on the crystallographically equivalent sites,is the common feature of relaxors.The relaxor behaviour was ?rst observed in the perovskites with disorder of non-isovalent ions,including the stoichiometric complex perovskite compounds, e.g.0022-2461C 2006Springer Science +Business Media,Inc.

DOI:10.1007/s10853-005-5915-731

FRONTIERS OF

FERROELECTRICITY

Figure1Temperature dependences of the real and imaginary parts of

the relative dielectric permittivity measured at different frequencies in a

crystal of the prototypical relaxor Pb(Mg1/3Nb2/3)O3.Enlarged view in the

insert shows the universal relaxor dispersion(URD)(after Bokov and Ye,

unpublished).

Pb(Mg1/3Nb2/3)O3(PMN)[1]or Pb(Sc1/2Ta1/2)O3

(PST)[2](in which Mg2+,Sc3+,Ta5+and Nb5+ions

are fully or partially disordered in the B-sublattice of

the perovskite ABO3structure)and nonstoichiometric

solid solutions,e.g.Pb1?x La x(Zr1?y Ti y)1?x/4O3(PLZT)

[3,4]where the substitution of La3+for Pb2+ions

necessarily leads to vacancies on the A-sites.Recently

an increasing amount of data reported has shown that

many homovalent solid solutions,e.g.Ba(Ti1?x Zr x)O3

(BTZ)[5,6]and Ba(Ti1?x Sn x)O3[7]can also exhibit

relaxor behaviour.Other examples of relaxor ferro-

electrics are complex perovskites Pb(Zn1/3Nb2/3)O3

(PZN)Pb(Mg1/3Ta2/3)O3(PMT),Pb(Sc1/2Nb1/2)O3

(PSN),Pb(In1/2Nb1/2)O3(PIN),Pb(Fe1/2Nb1/2)O3

(PFN),Pb(Fe2/3W1/3)O3(PFW)and the solid solutions:

(1?x)Pb(Mg1/3Nb2/3)O3?x PbTiO3(PMN-PT)and

(1?x)Pb(Zn1/3Nb2/3)O3?x PbTiO3(PZN-PT).

Although relaxor ferroelectrics were?rst reported

nearly half a century ago,this?eld of research has experi-

enced a revival of interest in recent years.In this paper,we

try to provide an overview of the current understanding

of the various issues of relaxors.Emphasis is put on the

latest developments.For a review of the earlier studies,

readers can refer to Refs.[8–11].

f7b23503ce2f0066f433228epositional order-disorder phase

transitions and quenched disorder in

complex perovskites

As mentioned above,the disordered distribution of differ-

ent ions on the equivalent lattice sites(f7b23503ce2f0066f433228epositional

disorder,also called chemical,ionic or substitutional dis-

order)is the essential structural characteristic of relax-

ors.The ground state of the complex perovskites should

be compositionally ordered,e.g.in the A(B 1/2B 1/2)O3

compounds each type of the cations,B or B ,should

be located in its own sublattice,creating a superstructure

with complete translational symmetry.This is because

the electrostatic and elastic energies of the structure are

minimized in the ordered state due to the difference in

both the charge and the size of B and B ions.Ther-

mal motion is capable of destroying the order at a certain

nonzero temperature(T t).This occurs in the form of struc-

tural phase transition,the order parameter(compositional

long-range order,s)of which can be measured by the

X-ray or other diffraction methods.Such kind of phase

transitions had been known long ago(e.g.in many metal-

lic alloys)and was also discovered comparatively recently

at T t～1500K in PST,PSN[12]and several other complex

perovskites.Ordering implies the site exchange between

B and B cations via diffusion.It is a relaxation process

with a nearly in?nite characteristic time at low tempera-

tures,but at1500K it can be quite fast.As a result,in

some perovskites(e.g.in PST,PSN,PIN),by annealing

at temperatures around T t and subsequent quenching,one

can obtain the metastable states with different s at low

temperatures.In some other materials(e.g.in PFN and

PMN)the compositional disorder cannot be changed by

any heat treatment because the relaxation time of order-

ing is too long.However in all known relaxors,at T B and

below,the compositional order is frozen(quenched),i.e.

cannot vary during practically reasonable time.

In the real complex perovskite crystals and ceramics

the quenched compositional disorder is often inhomoge-

neous,e.g.small regions of the ordered state are embedded

in a disordered matrix.These regions can be regarded as a

result of incomplete compositional order-disorder phase

transformation or as quenched phase?uctuations.In the

prototypical relaxor PMN this kind of inhomogeneous

structure always exists and cannot be changed by any

heat treatment.

In Pb(B 1/2B 1/2)O3perovskites the ordering of

B-site ions converts the disordered PE Pmˉ3m structure

into the ordered Fmˉ3m structure in which B ions

alternate with B ions along the 100 directions(1:1

ordering).In the ordered phase of many non-ferroelectric

A(B2+1/3B5+2/3)O3perovskites,B2+ions alternate

with two B5+ions along the 100 directions(1:2

ordering).The type of ordering in lead-containing relaxor

perovskites,Pb(B2+1/3B5+2/3)O3,has been the subject of

debates.In the early works only inhomogeneous ordering

(ordered regions within disordered surroundings)was

found in the samples studied.High-resolution electron

microscopy of PMN revealed nano-size(～2–5nm)

regions in which the ordering of1:1type(Fmˉ3m)

was observed(see e.g.Refs.[13,14]).These chemical 32

FRONTIERS OF FERROELECTRICITY

nanoregions (CNR)give rise to weak superlattice re?ections (the so-called F-spots).The results of anoma-lous X-ray scattering measurements [15]showed that the CNRs in PMN exhibit an isotropic shape and a temperature-independent (as expected for the quenched order)size in the temperature interval of 15–800K.Alternating Mg 2+and Nb 5+ions,i.e the same type of ordering as in the ordered Pb(B 1/2B 1/2)O 3perovskites,were initially supposed to exist in these regions.This structural model was called “space charge model”because it implies the existence of the negatively charged compositionally ordered non-stoichiometric nanoregions,and the positively charged disordered non-stoichiometric f7b23503ce2f0066f433228eter,by means of appropriate high temperature treatments,Davies and Akbas [16]were able to increase the size of CNRs and obtained highly 1:1ordered samples without the disordered matrix in the PMT and modi?ed PMN ceramics.The existence of such ordering in overall stoichiometric samples is obviously inconsistent with the space charge model.The results of X-ray energy dispersive spectroscopy with a nanometer probing size revealed that the Mg/Nb ratio is the same in the CNRs as in the disordered regions of PMN [17],which also dis-agrees with the space charge model.A charge-balanced “random-site”model has been suggested in which one of the B-sublattices is occupied exclusively by B 5+ions while the other one contains a random distribution of B 2+and B 5+ions in a 2:1ratio so that the local stoichiometry is preserved [16].The inhomogeneous compositional disorder,characteristic of the Pb(B 2+1/3B 5+2/3)O 3perovskites,are shown schematically in Fig.2.

The degree of compositional disorder can greatly in-?uence the FE properties.For example,the disordered PIN crystals are relaxor ferroelectrics,but in the ordered state,they are antiferroelectrics with a sharp phase transi-tion [18,19],con?rming the general rule that the relaxor

Figure 2Schematic representation of the ordered chemical nanoregion,CNR (the area delimited by the solid line)within the disordered matrix in Pb(B 2+1/3B 5+2/3)O 3perovskites according to the random-site model.One of the two sublattices inside CNR (shown by dashed lines)is formed by B 5+ions only.Pb and O ions are not shown.

behaviour can only be observed in disordered crystals.The possibility for real perovskite samples to have differ-ent states of compositional disorder,depending on crystal growth or ceramic sintering conditions,should be taken into account in research work.More detailed discussion on the compositional ordering and its impact on FE prop-erties can be found in Refs.[20–22],and the literature therein.

3.Relaxors in the ergodic state 3.1.Paraelectric structure

The PE phase of all perovskite ferroelectrics has the cubic m ˉ3

m average symmetry,but locally the ion con?guration can be distorted,i.e.the ions are not located in the special crystallographic sites of the ideal perovskite structure.For example,in the classical ferroelectrics BaTiO 3,the random displacements of Ti cations along the 111 di-rections caused by the multiple-well structure of potential surface were found [23,24].Such kind of displacements is due to the hybridization between electronic states of cations and the 2p states of oxygen (and should not exist in the case of purely ionic bonds).This effect is an impor-tant factor in the FE instability [25]and is also expected to occur in perovskite relaxors.Moreover,owing to the dif-ferent sizes of the compositionally disordered cations and the random electric ?elds created because of the different charges of these cations in relaxors,all ions are expected to be displaced from special positions.These shifts should exist in the PE phase and also at lower temperatures.Permanent uncorrelated displacements of ions from the high-symmetry positions of the (fully or partially compositionally disordered)cubic perovskite-type struc-ture were indeed found in relaxors at temperatures much higher,as well as lower,than T B .They are shown schemat-ically in Fig.3.The displacements of Pb 2+were detected by X-ray and neutron diffraction in PMN [26–28],PZN,PSN,PST,PIN,PFN,PZN-PT and PMN-PT with small x (see Refs.[29–31],and references therein).To describe the Pb distribution,a spherical layer model has been pro-posed [28]according to which the shifts of ions are ran-dom both in length and direction so that they are dis-tributed isotropically within the spherical layer centred on the special Pb site.1The typical radius of the sphere is

～0.3A ?

.It decreases slightly with increasing temperature.The off-symmetry displacements of Pb ions in PMN were found to vanish at T >925K [26](for other relaxors no data up to so high temperatures are available).The spher-ical layer model for Pb displacements in PMN was con-?rmed by the NMR investigations [32]and by the pulsed neutron atomic pair-distribution function (PDF)analy-sis [33].Note that the signi?cant random off-centring of Pb ions in perovskites is not the result of compositional

1For PZN the shifts of Pb from the ideal positions along the eight equivalent

111 directions were reported,instead of spherical layer distribution [29].

33

FRONTIERS OF

FERROELECTRICITY

= Pb= O= B5+

Figure3Typical uncorrelated ion displacements(shown by small arrows)

in the unit cell of the lead-containing complex perovskite relaxor.Thick

arrows show the direction of the local spontaneous polarisation P caused by

the correlated displacements of ions inside PNRs.

disorder.It was also found in the PE phase of the ordi-

nary perovskite PbZrO3[34].On the other hand,in the

PMN-PT solid solution with x=0.4which is still com-

positionally disordered,the Pb displacements from the

special perovskite positions were not observed at T>T C

[31].

According to neutron diffraction data[27,29,35]the

shifts of oxygen ions in the planes parallel to the corre-

sponding faces of the perovskite cubic cell are isotropic

(in PMN the shifts are close to0.2A?).The oxygen ions are

also shifted(by about0.06A?in PMN)in the perpendic-

ular direction so that the distribution of shifts forms two

rings parallel to the face of the cube.The displacements

of B-site ions(Nb5+,Mg2+,Zn2+etc.)from the ideal

positions were not noticed in diffraction experiments[29,

35](some authors found small seemingly isotropic dis-

placements of about0.1A?in PMN[27]).Nevertheless

the investigations of the extended X-ray absorption?ne

structure(EXAFS)and the pre-edge regions of absorption

spectra revealed the off-centre random displacements of

Nb in the direction close to 110 in PMN,PZN,PSN

and PIN[36].These displacements are not sensitive to

the change of temperature(in the range of290–570K),

nor to the degree of compositional disorder(in PSN and

PIN).The pulsed neutron PDF studies con?rm[33]that

the Nb displacements(in PMN at room temperature)are

comparatively small(much smaller than in KNbO3).

In the canonical relaxors such as PMN,the average

crystal symmetry seems to remain cubic with decreas-

ing temperature(see,however,the discussion in Sections

7.1.1and7.2),but the local structure changes.In addition

to the uncorrelated local distortions described above,the

clusters of FE order(i.e.PNRs)appear at T

≈620K in PMN).Due to their extremely small(nano-

metric)size,these clusters cannot be detected from the

pro?les of the X-ray and neutron Bragg diffraction peaks.

Other experiments are needed to validate their existence.

3.2.Experimental evidence for PNRs

The?rst experimental(although indirect)evidence for the

PNRs came from the temperature dependences of the op-

tic index of refraction(n)which appear to be linear at T>

T B,as shown in Fig.4[37].At lower temperatures a de-

viation from linearity was observed which was attributed

to the variation of n induced(via quadratic electrooptic

effect)by local spontaneous polarization inside the PNRs.

The existence of PNRs was later con?rmed by elastic dif-

fuse neutron and X-ray scattering around the reciprocal

lattice points[38–41].In the PMN crystals,signi?cant

diffuse scattering appears at T

increasing with decreasing temperature.This effect re-

sembles the scattering caused by FE critical?uctuations,

but an important difference(found in synchrotron X-ray

experiments[41])is that the shape of wavevector depen-

dence of scattering intensity at large distances from the

reciprocal lattice point deviates from the Lorentzian.This

means[41]that the PNRs are more compact than the

usual FE critical?uctuations and have better de?ned bor-

ders.The correlation length(ξ)of the atomic displace-

ments contributing to the diffuse scattering,which is a

measure of the size of PNR,can be derived from the ex-

periment:it is inversely proportional to the width of the

T

T

B

T

f

Figure4Schematical typical temperature dependences of the refractive

index,n,unit cell volume,V,reciprocal dielectric permittivity,1/ε,and

birefringence, n,in the canonical relaxor.

34

FRONTIERS OF FERROELECTRICITY

diffuse (Lorentzian)peak.According to the recent high-resolution neutron elastic diffuse scattering study of PMN [42],the size of the emerging PNRs is very small (ξis around 1.5nm)and practically temperature independent at high temperatures (Fig.5).The perovskite unit cell pa-rameter being ～0.4nm,each PNR is composed of only a few unit cells.Below about 300K,ξbegins to grow on cooling,reaching ～7nm at 10K.The most signif-icant growth is found around T f .Qualitatively the same behaviour was observed in the bulk of PZN crystals (the structure of PZN surface layers is different,see Section 7.2)but the size of PNRs is larger:they grow from ～7nm at high temperatures to ～18nm at 300K [43,44].From the analysis of the relation between ξand the integrated intensity of scattering,it was concluded [42]that the num-ber of PNRs also increases on cooling,but in contrast to the temperature evolution of ξ,the increase begins right from T B and at T ≈T f a sharp decrease of this number occurs (presumably due to the merging of smaller PNRs into larger ones).Below T f the number of PNRs remains practically the same at any temperature.

Emergence of PNRs below T B was also observed in the PMN crystal by means of transmission electron mi-croscopy (TEM)[14],but their size was an order of mag-nitude larger than that determined from the neutron diffuse scattering,probably because of the in?uence of electron beam irradiation.

The directions of ionic displacements responsible for the spontaneous dipole moment of PNRs were investi-gated in several works.By means of dynamic structural analysis of diffuse neutron scattering in PMN crystals it was found that the B -site cations (Nb and Mg)and the O anions are displaced with respect to the Pb cations in the opposite directions along the body diagonal (i.e.the [111]direction)of the perovskite unit cell,forming a rhombohedral polar structure [45].The rhombohedral R 3m symmetry was also derived from the analysis

of

Figure 5Average size of PNRs in the Pb(Mg 1/3Nb 2/3)O 3crystal (deter-mined from diffuse neutron scattering)as a function of temperature.Vertical dashed line corresponds to T f .(after Xu et al .[42]).

ion-pair displacement correlations obtained by an X-ray diffuse scattering technique [46],but according to this study,O displacements deviate from the body diagonal and remain parallel to the 110 direction.The shape of PNR was found to be ellipsoidal [46].The same shape was revealed by TEM [14].

Besides the structural features,many properties of re-laxors can be adequately explained on the basis of the idea of PNRs.For example,in contrast to ordinary ferro-electrics,where a sharp anomaly of speci?c heat is known to appear at phase transition,in relaxors such anomaly is smeared over a wide temperature range and thus is hardly distinguishable from the background of the lattice contri-bution.The excess speci?c heat (total minus lattice con-tribution)has been determined in PMN and PMT crystals using precise adiabatic and thermal relaxation techniques [47].It appears as a diffuse symmetric maximum located within the same temperature interval where PNRs nucle-ate and grow (between 150and 500K in PMN).Therefore the anomaly is likely to be caused by the formation of PNRs and/or by dipolar interactions among them.

Brillouin spectra of PMN-PT at T T C revealed signif-icant relaxation mode (central peak)which was attributed to the thermally activated fast (10–100GHz)relaxation of PNRs [48].The intensity and the width of the peak in-crease with decreasing temperature,indicating an increase of the number of PNRs and a slowing-down of their dy-namics,respectively.The hypersonic damping was also observed.It increases upon cooling,and is attributed to the scattering of acoustic mode by PNRs [48].

PNRs can be thought as unusually large dipoles whose direction and/or magnitude are dependent on an exter-nal electric ?eld.Therefore the related properties are expected to be unusual.Indeed,at those temperatures where PNRs exist,relaxors are characterized by giant electrostriction [49–51],remarkable electrooptic effect [50]and extremely large dielectric constant (see Fig.1and Section 6).

Even though no unambiguous structural con?rmations for the phase transition at T B are known,the anoma-lies of properties at this temperature were reported.The frequency-independent maximum of the dielectric loss tangent was found at this temperature in PMN [52].In the course of thermal cycling of PMN and PMN-PT crystals unannealed after growth,a narrow maximum of the acous-tic emission activity is observed (and decreases with the increase of number of cycles)in the vicinity of T B [53].Not only the temperature dependence of the index of re-fraction deviates from linearity at T

Little is known about the relation between the CNRs and the PNRs in relaxors,although such relation can a

35

FRONTIERS OF FERROELECTRICITY

priori be expected.Based on the TEM data,it was con-

cluded that the PNRs in PMN may contain CNRs inside

and in this case the regions in which PNRs and CNRs

overlap remain non-polar[14].In the framework of the

theoretical models discussed in Sections5and7.3,the

CNRs can be considered as one of the factors in?uencing

the formation and behaviour of PNRs,but not necessarily

the determining factor.

f7b23503ce2f0066f433228ettice dynamics in relaxors

Phase transitions in displacive ferroelectrics(including

perovskites)are known to be caused by softening and

condensation of transverse optic(TO)phonon mode at the

Brillouin zone centre.Since the frequency of this mode

(ω0)is connected to the static lattice dielectric constant

(through the Lyddane-Sachs-Teller relation,1/ε∝ω02),

the divergence ofεin ferroelectrics at T CW according

to the Curie-Weiss law(1/ε∝T?T CW)implies that

the mode condenses(its frequency tends to zero)at T CW,

too.In relaxors,the Curie-Weiss law also holds in the PE

phase,i.e.at T>T B,(see Fig.4and Section6),but the

corresponding softening of phonon modes had not been

detected until recently.During the last few years neutron

inelastic scattering technique was applied to investigate

the lattice dynamics in relaxor crystals.In the PE phase

of PMN far above T B,the dispersion of the transverse

acoustic(TA)and low-energy TO phonons were found to

be very similar to that existing in the PE phase of classical

displacive ferroelectrics PbTiO3[39,54](see the curves

for1100K in Fig.6).On cooling down to T≈T B,the

optic branch softens in the same manner as in displacive

ferroelectrics,i.e.,the frequency of the mode at wave

vector q=0(zone centre mode)follows the Cochran law

[54,55]:

ω20=A(T?T0),(1)

with A>0,as shown in Fig.7.At T

de?ned TO modes are also observed[55](see the curve

for150K in Fig.6).Once again,the temperature evolu-

tion is consistent with the typical behaviour of a FE soft

mode below the Curie temperature(Fig.7),i.e.Equation1

holds with A<0(note that T0is close to T B in this case).

The same dependence is observed at low temperatures in

PZN crystal[44].But in the temperature range between

T f and T B,the lattice dynamics is different.The prop-

agating TO modes are observed here only for the wave

vectors larger than q wf.For q

damped.The TO phonon branch drops sharply into the

TA branch at q wf,resembling a waterfall(as shown in

Fig.6for500K),and for this reason,the phenomenon is

called“waterfall.”It has been observed not only in PMN

but also in other relaxor materials for which neutron in-

elastic experiments were performed,i.e.in PZN[56]and

PZNT[57].In the same temperature range where the

TO

Figure6Phonon dispersion curves in PMN crystal for the TA branch at

1100K(solid circles)and the lowest-energy TO branch at three different

temperatures:T=1100K>T B(dashed line),T=500K

circles),and T=150K

]).

Figure7Temperature dependence of TO-phonon energy squared measured

at(200)in PMN crystal.Vertical dashed lines correspond to T f and T B.The

temperature range in which the waterfall feature appears is indicated by the

thick horizontal line.The other dashed and solid lines are guides to the eye.

(after Wakimoto et al.[55]).

mode is overdamped(between T f and T B),the damping

of TA phonons is also enhanced(e.g.neutron inelastic

experiments in PMN revealed a large maximum of the

TA-phonon linewidth at～400K[55]).

It was initially proposed that the large damping of TO

modes is due to the presence of PNRs which prevent the

propagation of phonons with wavelength larger than the

size of PNR,and thus q wf is the measure of the average 36

FRONTIERS OF FERROELECTRICITY

size of the PNRs [54,57].Later it was shown [58]that q wf depends on the choice of the Brillouin zone and the relation of q wf to the size of PNRs is improbable.The waterfall effect was then explained by the interactions of acoustic and optic branches.It was also noticed that damping of the soft mode near phase transition is not the unique feature of relaxors;similar effect can be found in some ordinary ferroelectrics [58].

An important question arises as to whether the low-energy TO mode found in relaxors is really the FE soft mode,i.e.whether the frozen mode displacements are re-sponsible for the spontaneous dipole moment of PNRs and for the spontaneous polarization of the low-temperature FE phase.Initially the answer to the question was neg-ative.Nabereznov et al .[39]compared the correlated atomic displacements in PNRs (found by elastic diffuse scattering experiments)and the TO-mode atomic dis-placements and concluded that they are incompatible.Besides,Vakhrushev and Shapiro [59]noticed that the value of T 0derived from Equation 1does not coincide with T CW found in dielectric measurements and there-fore this mode could not be the FE soft mode.They also identi?ed an additional “quasioptic”branch lying signif-icantly lower in energy than the TO branch and having the temperature variation consistent with the Curie-Weiss behaviour of the dielectric constant.The quasioptic mode was proposed to be the true soft mode.This opinion was later disputed.The apparent con?ict between diffuse and soft mode scattering experiments was reconciled with the help of the “phase-shifted condensed soft mode”model proposed by Hirota et al .[40].According to this model the total displacement of atoms inside a PNR consists of two components.The ?rst component is created by the TO soft-mode condensation and gives rise to the sponta-neous polarisation of PNR.The second one results from a uniform displacement of all atoms leading to the shift of the PNR along their polar direction relative to the surrounding non-polar matrix.Wakimoto et al .[60]de-scribed the lattice dynamics by a coupling between TA and TO modes without the need for considering any ad-ditional quasioptic modes.The concept of soft coupled optic mode was introduced.Being condensed,this mode has the optic component responsible for the dipole mo-ment of PNR and the acoustic component giving rise to the uniform displacement of PNR as a whole.The mi-croscopic origin of the uniform component is not yet clear.

Infrared (IR)spectroscopy of PMN between 20and 300K [61]and of PLZT between 10and 530K [62]revealed three main zone-centre TO modes as typical of cubic perovskites.High-energy TO modes do not show any pronounced temperature dependence both in PMN and PLZT.The lowest-energy mode (which was also re-solved by time-domain THz spectrometry)was found to soften following Equation 1with A <0and T 0≈T B ,i.e.in agreement with the above-discussed neutron scat-tering data.Nevertheless,in contrast to the neutron data,the mode remains underdamped in the whole tempera-ture range.This discrepancy is related to the fact that phonons with different wave-vectors are probed in IR (q ≈10?5?–1)and neutron (q ≥10?2??1)experiments.Raman scattering,which is known to be an effective tool for studying the optic soft modes in the crystals with normal phase transitions,was also widely applied to re-laxors (see Refs.[63–65],for a review),but no soft mode has been found here.Light scattering spectra in relaxors appear to be quite complex and their interpretation is not straightforward.In particular,it is not clear what kind of disorder gives rise to the observed ?rst-order lines that

are forbidden by the Pm 3m (O I

h

)average symmetry of relaxors.

Recently,vibration spectra in PMN were determined theoretically from the ?rst principles [66].Although the computations were performed for the case of composi-tionally ordered structure (the real structure is disordered with the inclusions of ordered CNRs,see Section 2),it appeared to be possible to assign the calculated phonons to the main peculiarities of Raman and IR spectra.

5.Origin and evolution of PNRs:Models and theories

Although the very existence of PNRs in relaxors seems to be doubtless,the cause and mechanisms of their formation are not conclusively understood.At temperatures higher than T B the structure and properties of relaxors closely resemble those of normal displacive ferroelectrics.When a relaxor becomes compositionally ordered after high-temperature annealing (without changing the chemical composition),a sharp ferro-or antiferroelectric (AFE)phase transition is observed (see Section 2).These facts seemingly suggest that the relaxor crystal tends to be ferro-or antiferroelectric at low temperatures,but the quenched compositional disorder somehow prevents the normal transition into the phase with macroscopic FE or AFE order from happening.Instead,the PNRs appear.There exist different approaches to explain the formation of PNRs.All of them can be schematically subdivided into two categories.The models of the ?rst category [8,67–70]consider the PNRs as a result of local “phase tran-sitions”or phase ?uctuations so that the crystal consists of nanosize polar islands embedded into a cubic matrix in which the symmetry remains unchanged (as shown in Fig.8a).The models of the second category assume the transition to occur in all regions of the crystal and the crystal consists of low-symmetry nanodomains separated by the domain walls but not by the regions of cubic sym-metry [71,72](the example is shown in Fig.8b).Note that these two situations can hardly be distinguished ex-perimentally by structural examinations [73]because the local symmetry of cubic matrix is not expected to be cubic

37

FRONTIERS OF

FERROELECTRICITY

-

polar nanoregions-

regions of cubic symmetry

(a)(b)

Figure8Schematic representation of PNRs in relaxors according to the

different models.

and the thickness of domain walls(i.e.the regions where

polarization is not well-de?ned)is comparable with the

size of nanodomains.

The second category is represented by the random-?eld

model proposed by Westphal,Kleemann and Glinchuk

(WKG model)[71,74],who applied the results of a the-

oretical work by Imry and Ma[75]to the relaxors.It was

shown in Ref.[75]that in the systems with a continuous

symmetry of order parameter,a second-order phase transi-

tion should be destroyed by quenched random local?elds

conjugate to the order parameter.Below the Curie temper-

ature the system becomes broken into small-size domains

(analogy of PNRs)instead of forming a long-range or-

dered state.It should be emphasized that this model does

not consider the trivial case of the local spontaneous po-

larization which is directed parallel to the quenched?eld

when the?eld is strong enough.Instead the situation is

determined by the interplay of the surface energy of do-

main walls and the bulk energy of domains in the presence

of arbitrary weak random?elds[75].

For displacive transitions,continuous symmetry means

that the spontaneous deformation is incommensurate

with the PE lattice.However,this is not the case for

the perovskite ferroelectrics in which the spontaneous

deformation and the polarization(order parameter)are

aligned along de?nite crystallographic directions(e.g.

the 111 directions for the rhombohedral phase).Never-

theless,when the number of allowed directions is large

(e.g.eight for the rhombohedral phase),the symmetry of

order parameter can be considered quasi-continuous and

the approach appears to be applicable.The disordered

distribution of the heterovalent ions inherent to the

compositionally disordered structure(e.g.Nb5+and

Mg2+ions in PMN)provides the source for quenched

random electric?elds.

Ishchuk[72]analysed the thermodynamic potential in

the framework of Landau phenomenological theory for

the systems in which the energies of the FE and AFE

phases are close to each other.It was shown that the state

with coexisting FE and AFE domains may have lower

thermodynamic potential than the homogeneous(FE or

AFE)state.This effect is due to the interactions(electro-

static and elastic)between the FE and AFE domains.It

was suggested that relaxors are just the crystals in which

this effect occurs.In other words,the nonpolar regions,

coexisting with PNRs(FE domains),are the domains of

AFE structure.

The best-known model of the?rst category was devel-

oped in the early works by Isupov and Smolenskii[8,

22].Due to the compositional disorder the concentrations

of different ions(e.g.Mg2+and Nb5+in PMN)are sub-

ject to quenched spatial?uctuations.As the FE Curie

temperature(T C)depends on the concentration,spatial

?uctuations of local T C are expected.It was suggested

that upon cooling,local FE phase transitions occur?rst in

those regions where T C is higher,whereas the other parts

of the crystal remain in the PE phase.Therefore,PNRs

are simply the regions with elevated Curie temperature.

Several other models use the microscopic approach and

consider the structural evolution and formation of PNRs in

terms of interatomic interactions.The FE lattice distortion

in the ordinary perovskites is known to be determined by a

delicate balance between the electrostatic(dipole-dipole)

interactions and the short-range repulsions.Hybridiza-

tion between the oxygen2p states and electronic states of

cations(covalent bonding)is able to change this balance,

in?uencing thereby the phase transition temperature[25].

In the compositionally ordered(translationally symmet-

ric)crystals,exactly the same forces affect all the atoms

of a certain type because they have the same coordination

neighbourhood.In the case of compositional disorder,the

ions of different types may be found in the neighbour-

ing unit cells on the same crystallographic positions(e.g.

in the B-sublattice of PMN,both Mg and Nb ions are

the nearest neighbours of Nb ions).The interatomic in-

teractions which would cause ferro-or antiferroelectric

order in the compositionally ordered state become ran-

dom in this case,and as a result,the long-range polar

order is disturbed.The models described below empha-

size the importance of different interactions:the interac-

tions under random local electric?elds only(including

dipole-dipole interactions)[67],and the dipole-dipole in-

teractions together with random short-range repulsions[9,

68]or random covalent bonding[33].

In the random?eld theory developed for relaxors by

Glinchuk and Farhi(GF model)[67](see also Ref.[76]

in which the related papers are reviewed),the transition

is regarded as an order-disorder one,i.e.at high tempera-

ture the crystal is represented by a system of reorientable

dipoles(dipoles caused by the shifts of ferroactive ions

from their ideal perovskite positions,see Section3.1).

These random-site dipoles are embedded in highly polar-

izable“host lattice”(the high polarizability is due to the

transverse optic soft mode existing in relaxors,see Section

4).The dipole-dipole interactions are indirect(they occur

via the host lattice)and random.Nevertheless,according 38

FRONTIERS OF FERROELECTRICITY

to the theory,they should lead to uniformly directed local ?elds and thus to FE ordering at low temperature (in con-trast to direct dipole-dipole interactions which can lead to a dipole glass state).Thus to explain the absence of macro-scopic FE order in relaxors,additional sources of random local electric ?elds are considered.These additional ?elds can be static (coming from quenched compositional disor-der,lattice vacancies,impurities and other imperfections)or dynamic (associated with shifts of non-ferroactive ions from the special positions).In contras to the ?elds consid-ered in the WKG model,these ?elds should be rather large (larger than critical value)to destroy the long-range FE order.The FE order parameter,phase transition temper-ature T C ,linear and nonlinear dielectric susceptibilities are calculated within the framework of statistical theory using the distribution function for local ?elds.It is found that depending on the model parameters (concentration of dipoles,other ?eld sources and the host lattice correla-tion length),the low-temperature phase can be FE,dipole glass or mixed ferroglass.In the temperature interval be-tween T C and T B ,the short-range clusters may appear,in which the reorientable dipoles are ferroelectrically cor-related (i.e.PNRs).In the ferroglass state these clusters coexist with the macroscopic regions in which the dipoles are coherently ordered.

Note that the GF model for relaxors is the extension of the analogous theory for incipient ferroelectrics with off-centre impurities (e.g.KTaO 3:Li,Nb,or Na).In the later case the off-centre impurities are the interacting dipoles.Due to their small concentration the crystal can be con-sidered as a system of identical dipoles with random long-range interactions.In the case of complex perovskite relaxors,the dipole concentration cannot be considered small.The random interactions of different (short-range)nature are also involved and thus the dipoles are not iden-tical.It was ?rst recognized in the model proposed by one of the co-authors of the present review [9,68].In this model the PNRs are the result of local condensation of the soft phonon mode (which exists in relaxors as discussed in Section 4).The consideration is based on the model of coupled anharmonic oscillators which is often applied to ordinary ferroelectrics.The effective Hamiltonian is given by the sum of Hamiltonians of the individual unit cells:

H =

l

0.5 2l +A l ξ2l +B l ξ4l ? l

νll ξl ξl

,where l and ξl are the generalized momentum and co-ordinate of the soft mode displacements,A l and B l are pa-rameters of one-particle potential,which are determined

by the interactions (mainly short-range repulsive)between ions of the l th unit cell,and υll are parameters character-izing the interactions (long-range dipole-dipole)between the different cells.In the translationally invariant crystal,

all the parameters,A l ,B l ,and υll ,would be the same.In

the case of compositional disorder they are different.The distribution function for these parameters is introduced in the model.This distribution gives rise to the spatial distribution of local “Curie temperature”T C .PNRs ap-pear in the regions with enhanced local T C .The model parameters are linked to the parameters of real structure (in particular,the size of ions).Based on the crystal com-position,this model is able to predict quantitatively the degree of “diffusion”of the transition,i.e.the extent of temperature interval in which the PNRs develop before the crystal transform into the low-temperature nonergodic phase.In particular,the degree of diffusion increases with increasing difference in the radii of ions in the ferroactive sublattice (A or B perovskite sublattice)or with increas-ing compositional disorder in this sublattice.2On the other hand,the diffusion is much less sensitive to the disorder in the non-ferroactive sublattice.The in?uence of the degree of compositional disorder on T C is also explained.Based on the arguments similar to those used in the original model [68]it was recently suggested [77]that,because of the randomness of microscopic forces responsible for the onset of spontaneous polarization,each PNR can consist of unit cells polarized in different directions.This model of “soft nanoregions”also implies that,due to thermally activated reorientations of some unit cells inside PNR,not only the direction (as believed before),but also the mag-nitude of the spontaneous dipole moment of individual PNR can strongly change with time (due to ?uctuations or under the external ?eld),while the size of PNR remains the same.

The Hamiltonian considered in the model by Egami [33]consists of two terms,

H =H 1+H 2.

(2)

The ?rst term is written in a standard form

H 1=?

i j

J i j S i · S j ,(3)

where S

i is the local polarization caused by the displace-ment of i -th Pb ion from its special position (as discussed in Section 3.1),J ij describes the random interaction be-tween local polarizations mediated by oxygen and B-site ions.It is explained that in PMN the Pb ions cannot form the covalent bonds with those O ions which are bonded to Nb.On the other hand,Mg ions create purely ionic bonds and do not prevent the Pb–O bonding.Consequently the direction towards Mg is an “easy”direction for Pb dis-placement.This directional dependence of the energy of Pb displacements resembles the crystalline anisotropy in

2In lead-containing complex perovskites the Nb and Ta cations are supposed

to be ferroactive.

39

FRONTIERS OF FERROELECTRICITY

magnetic systems.It is random in compositionally disor-

dered crystal and can be described by model Hamiltonian

H2.This model was established to account for the re-

laxor properties in ER as wall as in NR phases,but the

appearance of PNRs was not derived.

Timonin[69]suggested that the ergodic phase in relax-

ors is an antilog of Grif?ths phase theoretically predicted

long ago(but not yet experimentally found)for dilute fer-

romagnetics.Ferroelectric clusters of various sizes(i.e.

PNRs)appear in this model at T

Curie temperature for non-dilute crystal)and speci?c non-

exponential relaxation is predicted.

Speci?c temperature evolution of PNRs can be ex-

plained in terms of the phenomenological kinetic theory

of phase transitions in compositionally disordered crys-

tals[70].The emergence of PNR,i.e.the region of polar

crystal symmetry within the cubic surrounding,should

be accompanied by the creation of electric and elastic

?elds around PNR,which increase the total energy of the

system.Due to the similar effects in the compositionally

ordered crystals undergoing a?rst-order phase transition,

the regions of the new phase(nuclei)are not stable.They

tend to grow if their size is larger than the critical one

or disappear otherwise.As follows from the theory[70],

in disordered crystals the nuclei of the new phase can be

stable and the equilibrium size of newly formed nuclei

can be arbitrary small.The PNRs in relaxors which are

really small(contain several unit cells)and stable can

be regarded as such kind of nuclei.The theory predicts

that PNRs begin to appear in the PE phase at T B as a

result of local“phase transitions”(e.g.condensation of

phonon soft mode).Upon cooling,the number of PNRs

increases but the equilibrium size of each PNR remains

unchanged within a certain temperature interval just be-

low the temperature at which it appears.Upon further

cooling,the PNR grows slowly with decreasing tempera-

ture while remaining in a stable equilibrium,and?nally

at T=T C,becomes metastable so that the size of PNR in-

creases steeply due to phase instability.In other words,the

behaviour predicted by this model is the same as experi-

mentally observed in PMN(see Fig.5).But this theory is

unable to describe quantitatively the real behaviour at T<

T C,because it does not take into account the interactions

between different PNRs,which are obviously signi?cant

at low temperatures.It was further explained[70,78]

that depending on the model parameters(in particular,the

mean T C and the width of the distribution of local tran-

sition temperatures),a sharp phase transition can occur,

resulting in large FE domains at T

small width and a comparatively high T C)or the transi-

tion is diffuse and the low-temperature polar regions are

of nanometer size.The dipole-dipole interactions between

them can lead to the formation of a glass-type phase at

a certain temperature T f.The intermediate situations are

also possible with moderately diffuse transition and meso-

scopic polar regions(domains).Note that these different

types of behaviour have indeed been observed experimen-

tally in different perovskite materials(see Section7.2).

6.Dielectric response in relaxors

Small-signal dielectric response has been intensively stud-

ied in a large number of relaxor materials,but most in-

vestigations were restricted within the frequency range

of10–109Hz or narrower.In the past few years,mod-

ern measurement facilities with enlarged frequency range

have been applied to relaxors.It has been found that sig-

ni?cant dielectric dispersion exists in the whole spectrum

starting from the frequency of lattice vibrations down to

the lowest practically measurable frequency of f～10?5

Hz.The present section will focus on these works.

The?eld-induced polarization in relaxors can be di-

vided into several qualitatively different parts so that the

total relative permittivity in the temperature range of per-

mittivity maximum can be written as

ε=1+χe+χPh+χR+χU+χLF,(4)

whereχe,χPh,χR,χU andχLF are the susceptibilities

(complex numbers)describing the electronic,phonon,

“conventional relaxor”(CR),“universal relaxor”(UR)

and“low-frequency”contributions,respectively.All the

contributions are frequency dependent[in Equation4the

susceptibilities are ranked in the order of increasing typ-

ical characteristic time].As in any materials,electronic

contribution persists in relaxors at all temperatures and at

frequencies up to1015–1017Hz,but at lower frequencies,

the value ofχ e=(n2?1)～10is small as compared with

other susceptibilities.

The phonon(lattice)susceptibility(caused by the mu-

tual displacements of cation and anion sublattices)is ac-

tive up to the frequencies of1012–1014Hz.To separateχPh

from other contributions,measurements at these frequen-

cies are necessary.In relaxorsχPh has been determined

from IR re?ectivity spectra at temperatures lower than T B

only.In PMN crystals,χPh increases from～40at20K to

～100at300K[61].In other words,it constitutes less than

1%of the total low-frequency permittivity measured at T m

(see Fig.1).This is an important difference of relaxors

from ordinary displacive ferroelectrics in which phonon

polarization totally accounts for the permittivity peak at

the phase transition.

The susceptibilitiesχR andχU related to the relaxation-

type polarizations are the main contributions giving rise to

the peculiar relaxor peaks in the temperature dependences

of permittivity(shown in Figs1and9).The real part of

χR is constant at low enough frequencies and decreases

to zero when the frequency reaches the(temperature-

dependent)characteristic value.This decrease is accom-

panied by the peak in the frequency dependence of the 40

FRONTIERS OF FERROELECTRICITY

imaginary part.3Both real and imaginary parts of the UR susceptibility continuously (without any loss peak)de-crease in the whole frequency range practically available for measurements according to the power law

χ U =tan(n π/2)χ U ∝f n ?1,

(5)

where n is close to but smaller than unity.Note that the

same empiric classi?cation (i.e.the monotonic frequency variation versus the variation with loss maximum)applies not only to relaxors,but also to the relaxation processes found in many other solids [79].Nevertheless,the val-ues of χR and χU in relaxors are extraordinary large as compared to other dielectrics.The CR dispersion (CRD)is observed at the low-temperature slope of permittivity peak giving rise to the frequency shift of T m ,while the UR dispersion (URD)exists at temperatures lower,as well as higher,than T m (as shown in Figs 1and 9).

The last term χLF in Equation 4combines all possible relaxation contributions not related to the relaxor ferro-electricity,which may include the polarization of hopping charge carriers [79,80],Maxwell-Wagner-type polarisa-tion,etc.Typically,these contributions become signi?cant in good-quality samples of relaxor perovskites at compar-atively high temperatures and/or low frequencies.In the PMN crystal presented in Fig.1the contribution of χLF at lowest frequency f =10?2leads to the noticeable increase of ε at temperatures above ～300K.

In the radio-and audio-frequency ranges,where the dielectric properties of relaxors are most often studied,the

value of χ R is much larger than χ

U .That is probably the reason why the UR contribution has been discovered only recently with the help of the frequency response analyser that is able to work at ultra-low frequencies [as one can see from Equation 5,χ U increases with decreasing f ].The χU component was separated from the χR one by means of the analysis of dielectric spectra at T >T C (or T >T f )?rst in PMN-PT [80–82]and then in PMN [77],PSN [83]and BTZ [84].Since CR is the dominant

contribution giving rise to the diffuse ε (T )peak,χ

R

≈ε in the vicinity of T m (at least for the frequencies that are not very low or very high).Therefore,most of the dielectric investigations of relaxors dealt in fact with the CR contribution,even though it was not identi?ed by the authors.

As shown recently for many relaxors [85,86],the high-temperature slope of the diffuse ε (T )≈χ

R

(T )peak can be scaled with the empirical Lorenz-type relation,

εA

ε ?1=(T ?T A )22δ2

,(6)

3The

well-known Debye relaxation is an example of such kind of the be-

haviour.

Figure 9Different possibilities for the temperature evolution of struc-ture and dielectric properties in compositionally disordered perovskites:(a)

canonical relaxor;(b)crystal with a diffuse relaxor-to-ferroelectric phase transition at T C

where T A (εm )are the ?tting parame-ters de?ning the temperature and magnitude of the Lorenz peak (6),and δis as a measure of the degree of diffuse-ness of the peak.This formula gives a more adequate description of the experimental data than the previously used relation,εm /ε ?1∝(T ?T m )γ(where 1<γ<2,and εm is the value of ε at T m ).Equation 6holds from temperature T 1,which is typically several degrees higher than T m ,to temperature T 2,which is a few dozens of de-41

FRONTIERS OF FERROELECTRICITY

grees lower than T B(see Fig.9).The diffuse peak ofε (T)

≈χ R(T)can be scaled with more complex relations(see

Refs.[19,87]for details).

At T>T B,the dielectric constant is described by the

Curie-Weiss law,

ε =C/(T?T CW),

where the Curie constant,C,has the same order of mag-

nitude(～105K)as in ordinary displacive ferroelectrics

and T CW is typically higher than the low-frequency value

of T m(as shown in Fig.4)but at high frequencies(e.g.in

PMN at f>20GHz)T m can become larger[52].

The CR contribution can consist of several compo-

nents in itself involving different polarization mecha-

nisms.Each of the mechanisms gives rise to the corre-

sponding dispersion and can be seen in theε (f)curve

as an individual maximum(or an anomaly if neighbour-

ing maxima overlap each other).For example,in PMN

crystals three components(dispersion regions)have been

found at T

neously between210K(≈T f)and290K.The?rst compo-

nent appears at the(temperature-independent)frequency

of～1THz and gives rise to a comparatively small input

to the static dielectric constant(about130).The two other

dispersion regions become broadened on cooling and their

mean relaxation time increases[i.e.the frequency of the

correspondingε (f)peak decreases],so that at T

the low-frequency component shifts out of the measure-

ment frequency range and the higher-frequency compo-

nent develops into a constant(frequency-independent in

the range of102–1011Hz)loss.This effect of constant

loss is a noticeable property of the low-temperature non-

ergodic phase in relaxors.It can also be observed in other

relaxors,e.g.in the PLZT ceramics[89]and the com-

positionally disordered PIN crystals[19].The magnitude

of constant loss decreases exponentially on cooling,but

still remains measurable at liquid-helium temperature.At

extremely low frequenciesε in NR phase is no longer

constant and slightly increases with decreasing frequency

(see Fig.1).

To describe the CR dielectric spectra,the same empiri-

cal expressions as used for other dielectrics were applied.

The Kohlrausch-Williams-Watts[80,90],the Havriliak-

Negami[91]and the simpler Cole-Cole[19,89]formulae

have been employed by different authors to?t the ex-

perimentalε(f)data.The alternative way to analyse the

dispersion is to?nd the appropriate function for the distri-

bution of relaxation times.For example,Rychetsky et al.

[89]?tted the relaxation in PLZT to a uniform distribution

that broadens upon cooling.

A remarkable feature which was observed?rst in PMN

[92]and then in many other relaxor ferroelectrics is the

V ogel-Fulcher(VF)law connecting the temperature and

the frequency of theε (T)peak:

f=(2πτ0)?1exp[?E a/(T m?T VF)],(7)

where f is the measurement frequency,τ0,E a and T VF are

the?tting parameters.The same relation but with slightly

different parameters has also been reported for the peak

temperature in theε (T)dependences(T mi).Investigations

of PIN crystals showed that the parameters of Equation7

can be different in different frequency intervals[19].The

VF law was known in structural and spin glasses.When

revealed in relaxors,it became one of the main reasons to

postulate the existence of a dipole glass phase at T

Equation7might(but not necessarily,see below)signify

the similar VF relation for the characteristic relaxation

timeτof the corresponding relaxation process:

τ=τ∞exp E b/(T?T f) ,(8)

whereτ∞,E b are the parameters and T f is the freezing tem-

perature(i.e.the temperature below which the relaxation

time becomes in?nite).This divergence ofτindicates that

the thermally activated reorientations of dipoles respon-

sible for polarization slow down with decreasing tem-

perature and become impossible(consequently dipoles

cannot respond to the electric?eld)at T=T f,but not at T

=0as prescribed by the Arrhenius law for the dynamics

of independent dipoles.In dipole glasses the interactions

among the dipoles are the cause for such kind of freez-

ing.These interactions(bonds)are frustrated(i.e.can be

either FE or AFE but cannot be satis?ed simultaneously)

and thus favour the con?gurations with random directions

of dipoles,in contrast to the ferroelectrics and antiferro-

electrics in which the dipole directions are parallel and

antiparallel,respectively.

The relations between Equations7and8in relaxors

were studied by several authors.In the case of Debye re-

laxation(which can be expected in the system of identical

non-interacting dipoles)this relation should be simple:τ

follows the Arrhenius law[which is the same as the VF law

(8),but with T f=0 and this automatically means that

law(7)also holds for T mi with the same parameters,i.e.

τ0=τ∞,E a=E b and T VF=0.The relaxation in relaxors

is much more complex and can be characterized by a wide

spectrum for the distribution of the relaxation times.Sim-

ple relations between the parameters of Equations7and

8are not evident.Furthermore,different relaxation times

from the spectrum may have different freezing tempera-

tures.It was also shown theoretically that the situations

are possible in which Equation7holds with T VF=0,but

no freezing at a non-zero temperature really takes place

[93],i.e.the VF relationships for T m and T mi do not nec-

essarily imply glass-type dipole dynamics.The problem

seems to be solved in some relaxors by means of a special

analysis of the frequency-temperature dependences of the 42

FRONTIERS OF FERROELECTRICITY

real part of permittivity.It was shown that the longest re-laxation time in the spectra of PMN [94,95],PST [94]and PLZT [96]diverges according to the relationship (8)with T f =T VF (≈220K in PMN),while the bulk of the distribution of relaxation times remains ?nite even below T f [95,96].The divergence of the longest relaxation time means that,at least empirically,the behaviour of relaxors in small-signal electric ?eld is similar to the behaviour of dipole glasses.However,a microscopic interpretation of this fact is not so clear.In contrast to ordinary dipole (or spin)glasses in which the susceptibility can by un-ambiguously attributed to the reorientation of certain per-manent dipoles (spins),the structure of relaxors is more complex and the polarization mechanisms responsible for the large and diffuse ε(T )peak have not been de?nitely identi?ed.

Most of the existing explanations relate the dielectric re-laxation in relaxors to the PNRs.The PNRs are very small and can be considered as individual thermally activated dipoles giving rise to the orientational polarization.Thus,the dominant contribution to the measured ε(T )relaxor peak (i.e.the CR contribution according to the classi?ca-tion described above)may be attributed to the thermally activated reorientation of dipole moments of PNRs (lo-cal spontaneous polarization vectors).Many authors pro-ceeded upon this assumption when analysing the dielec-tric data (see e.g.Refs.[97–99]).The dipole moments of PNRs are considered in many models as interacting (di-rectly or via surrounding matrix)entities constituting a glassy system (see Sections 5and 7.3).The reorientations may be affected by the random anisotropy and (in contrast to magnetic spin glasses)by an environment of random electric and elastic ?elds.

The second possible mechanism associated with PNRs is the side-way motion of their boundaries without the change of the orientation.In the course of such motion,the volume (and thereby the dipole moment)of the polar re-gion changes,giving rise to the characteristic polarization response.This looks like breathing of PNRs and there-fore,the corresponding model developed by Glazounov and Tagantsev is called “breathing”model [100].The model considers the vibrating PNR boundaries in terms of the theory of randomly pinned interface,which was developed earlier for magnetic materials.In the case of relaxors the internal random local ?elds induced by charge disorder act as the pinning centres.Another approach was used by Rychetsky et al .who proposed a thermo-dynamic model for the polarization reversal near the PNR boundary,which is equivalent to the displacement of the boundary [89].In particular,this model describes well the constant loss effect at low temperatures.From the analy-sis of the behaviour of PMN crystals in large dc and ac electric ?elds,it was suggested [101]that the dielectric re-sponse in the ergodic phase [i.e.in the vicinity of the ε(T )maximum]is controlled by the vibration of PNR bound-aries,rather than by the thermally activated reorientations of PNRs.

Note that in PMN and some other relaxors two main components determine the CR dielectric response in the ergodic phase (see above in this section).The low-frequency component may result from the reversal of the spontaneous dipole moments of PNRs and the high-frequency one may originate from the PNR boundary mo-tion [88].

The value of the universal susceptibility [i.e.the sus-ceptibility whose dispersion is described by Equation 5at all frequencies]in relaxors is several orders of magni-tude larger than in non-relaxor materials with the same n [80].Thus it is reasonable to suggest that the UR po-larization mechanism is also connected with PNRs which are inherent only in relaxors.Within the scope of the soft nanoregions model (see Section 5),the UR response has been attributed to the thermally activated reorientations of dipole moments of individual unit cells inside PNRs [77].The Curie-Weiss law in relaxors can be treated in two different ways,depending on the polarization mechanism which is supposed to be valid in the temperature range of the law (i.e.at T >T B ).The ?rst way (see e.g.Refs.[52,59,60,88,102])implies that,as in the case of normal per-ovskite (displacive)ferroelectrics,the ?eld-induced po-larization is due to the phonon contribution.The second approach suggests that the polarization mechanism in the temperature range of the Curie-Weiss law is qualitatively the same as at T m (by analogy to order-disorder ferro-electrics and spin glasses)and involves the relaxation of individual dipoles (see e.g.Ref.[103]in which the Sherrington-Kirkpatrick model was used to analyze the susceptibility in PMN).To determine which way is ad-equate,the experimental investigation of high-frequency (IR)dispersion at T >T B is needed.

7.Relaxors at low temperatures:A glassy state or a ferroelectric phase

In the previous sections we have mainly considered the relaxors at comparatively high temperatures,i.e.in the PE and ER phases.We have also discussed some basic aspects of the low-temperature behaviour of canonical relaxors,i.e.those in which the structure remains macro-scopically cubic at all temperatures and the FE phase can be achieved only by poling (e.g.by applying an external electric ?eld).In the canonical relaxors (e.g.PMN,PMT,PLZT with large x ),a nonergodic (glassy)state appears at low temperatures.In many other materials (e.g.PSN,PST,PLZT with small x,and PMN-PT with large x )that exhibit relaxor properties and related structural features (e.g.PNRs)at high temperatures,a spontaneous (i.e.with-out poling)structural phase transition into the FE phase occurs.These two different paths of temperature evolu-tion are shown schematically in Fig.9.In this Section,

43

FRONTIERS OF FERROELECTRICITY

we describe the low-temperature behaviour of relaxors in

more detail.

7.1.Glassy nonergodic relaxor phase

7.1.1.Structure

As mentioned in Section4the soft mode in the prototypi-

cal relaxor PMN recovers below T f so that the temperature

dependence of the mode frequency shows the behaviour

characteristic of a normal ferroelectric phase[i.e.follows

Equation1with A<0].A sharp peak of hypersonic

dumping was observed at T f[104].However,no other

evidence of the structural phase transition at T f has been

detected.The average cubic symmetry of PMN at low

temperatures was con?rmed in many structural studies by

the absence of any splitting of X-ray and neutron Bragg

re?ections(which means that the shape of unit cell is cu-

bic)as well as by the analysis of the intensities of the

re?ections(which are sensitive to the positions of atoms

in the cell).For instance,in Refs.[27,73],the unit cell

was determined to be cubic by X-ray and neutron powder

diffraction experiments performed down to5K,but due to

the limited number of re?ections analysed,the positions

of atoms and the thermal parameters could not be re?ned

simultaneously.In Refs.[105,106],the analysis of a large

number of re?ections obtained from X-ray diffraction of

PMN single crystals con?rmed the Pm3m space group

in the range of100–300K.The cubic structure is also

con?rmed by the absence of birefringence[107,108].4

Even though the structural phase transition in PMN

is not de?nitely observed,some important structural

changes not affecting the average symmetry are still

found.With decreasing temperature,the average size

of PNRs increases signi?cantly around T f(Fig.5).The

synchrotron X-ray scattering revealed the emergence of

very weak and wide1/2(hk0)superlattice re?ections(α

spots)in the vicinity of T f[110].These re?ections were

attributed to the antiferroelectric nanoregions(AFNR)

formed by the correlated anti-parallel(static or dynamic)

displacements of Pb ions along the 110 directions with

a magnitude of～0.2A?.Signi?cant enhancement of the

intensity ofαspots below T f is believed to arise from an

increase in the total number of the AFNRs,whose average

size of～30A?(determined from the width of re?ections)

remains constant down to the lowest measured tempera-

ture of10K[110].AFNRs appear to be different from

PNRs and CNRs,and unrelated to either of them[110].

7.1.2.Broken ergodicity in relaxors

Relaxors show nonergodic behaviour resembling the be-

haviour of spin(or dipole)glasses.In the high temperature

4While most researchers agree that the average structure of PMN is cubic,

the rhombohedral structure was also reported[109].The possible reason

for this discrepancy will be discussed in Section7.2.

(ergodic)phase of glasses,the spins(or dipoles,which can

be considered as pseudospins)are weakly correlated and

free to rotate,so that after any excitation(e.g.after appli-

cation and removal of an external?eld)the system quickly

comes back to the state with the lowest free energy,i.e.the

state with zero total magnetization.It is always the same

state regardless of the initial conditions(i.e.the strength

and direction of the?eld in our example).At lower tem-

peratures,due to the correlations between spins,the free

energy surface has very many minima of almost the same

depth separated by energy barriers of different heights

(each minimum corresponds to a speci?c con?guration of

spins).In the glass phase,some of these barriers are so

high that the time needed to overcome them is larger than

any practically reasonable observation time.Therefore,

during this time the system cannot reach all the con?gu-

ration states,and consequently,the usual thermodynamic

averaging and the time averaging give different results,i.e.

the system is in a nonergodic state.On its way to a new

state of minimum free energy required by the changed ex-

ternal conditions,the system should pass many barriers of

different heights.This leads to a process with a wide distri-

bution of relaxation times.The maximum relaxation time

from this distribution may be so large(in?nite for an in-

?nite crystal)that the system cannot effectively reach the

equilibrium.As a result,the state and the physical proper-

ties of the material depend on the history(i.e.the external

?eld applied,the temperature variations,the observation

time,etc.).In particular,substantial ageing effects should

be observed,i.e.the change of properties with time spent

by the sample at certain?xed external thermodynamic

parameters(temperature,?eld,etc.).

All the main(mutually related)characteristics of noner-

godic behaviour typical of spin glasses,i.e.anomalously

wide relaxation spectrum,ageing,dependence of the ther-

modynamic state on the thermal and?eld history of a

sample,are observed in relaxors at temperatures around

and below T f.The slowing-down of dipole dynamics was

already discussed in terms of small-signal dielectric re-

sponse in Section6.Slow relaxation manifests itself also

in other properties related to the local and/or macroscopic

polarization.In particular,the relaxation of optical linear

birefringence induced in PMN by a weak(E

ternal electric?eld was studied[107](E cr is the critical

?eld needed to induce the transition to the FE phase).

The Kohlrausch-Williams-Watts-type and the Curie-von

Schweidler-type relaxations were found in the tempera-

ture intervals of180

respectively.The results were successfully described in

terms of Chamberlin’s approach to dynamic heterogene-

ity[111],implying a broad relaxation spectrum.Applica-

tion of a strong(E>E cr)d.c.?eld to the PMN crystal at

T

permittivity[112]and a slow evolution of X-ray Bragg 44

FRONTIERS OF

FERROELECTRICITY

Figure 10Linear birefringence measured subsequently as a function of temperature on zero-?eld cooling (ZFC),?eld heating with E =1.2kV/cm

peaks re?ecting the change of crystal symmetry [113].5The effects of ageing of susceptibility in the NR phase of PMN and in the typical spin glass phase were found to be very similar (and much stronger than in typical dipole glasses)[114].The example of the dependence of prop-erties on the thermal and electrical history of sample is shown in Fig.10.The other examples are the splitting in the temperature dependences of the ?eld-cooled and zero-?eld-cooled quasistatic dielectric constants in PMN and PLZT [95,96]and the P (E )hysteresis loops (see Section 7.1.3),

The ergodicity is clearly broken in relaxors at low tem-peratures,but this does not necessarily mean that relax-ors are really dipole glass systems.Many other systems may also be nonergodic [115].In particular,an ordinary FE phase is also nonergodic,but its potential landscape contains only a few minima (which are symmetric and correspond to the different directions of spontaneous po-larization).As a result,the properties are easily distin-guishable from those of nonergodic spin glass (or relaxor)phase.Wide relaxation spectrum and ageing phenomena are absent in the ideal FE crystal.But in the composi-tionally disordered perovskite crystal the situation is very different and different explanations for the nonergodic be-haviour are possible.For instance,the above-mentioned Kohlrausch-Williams-Watts-type relaxation of birefrin-gence was explained by domain wall displacements,rather than by the reorientations of dipoles [107].Furthermore,some peculiarities of the relaxor behaviour have never been observed in spin and dipole glasses.In particular,the Barkhausen jumps during poling process (detected optically in PMN)are not compatible with the glassy reorientation of dipoles,which takes place on a micro-5After

a long (several hours)waiting time the entire crystal suddenly trans-forms to the FE phase via a ?rst-order transition.

scopic length scale and hence should be continuous and monotonic [71].Field-induced FE phase and FE hystere-sis loops have not been observed in typical dipole glasses.Thus,the nature of the nonergodic phase in relaxors re-mains the subject of intensive discussion.In particular,the WKG model suggests that the low-temperature phase of canonical relaxors is a ferroelectric state,but broken into nanodomains by quenched random ?elds.We will discuss the origin of nonergodic phase in more detail in Section 7.3.

Note also that in terms of compositional disorder,relax-ors are frozen in a metastable state,as discussed in Section 2.The degree of compositional disorder can depend on thermal prehistory.This is also an effect of nonergodic-ity.6However,at temperatures around T B and below,the compositional disorder remains unchanged on the exper-imental time scales (i.e.frozen),and at the same time,the motion of dipoles (at T >T f )is fast.Thus,when consid-ering the subsystem of dipoles at T >T f ,one can believe that the crystal reaches the equilibrium 7and the phase is effectively ergodic.On the other hand,if the sample has been annealed during experiment at high temperatures (～700K or higher)the possible effects of nonergodicity related to the compositional disorder should be taken into account.

7.1.3.Electric-?eld-induced ferroelectric

phase in relaxors

An important feature of the NR state is that,it can be irreversibly transformed to the phase with the FE dipole order when poling by an electric ?eld larger than the crit-ical strength (in PMN the minimal E cr is about 1.7kV/cm at T C 210[108]).This feature points to the common na-ture of relaxor and normal ferroelectrics.The FE hystere-sis loops,which are known to be the determinative char-acteristic of FE phase,are observed in relaxors with the values of remnant polarization and coercive ?eld typical of normal ferroelectrics.Pyro-and piezoelectric effects are also observed after poling.X-ray diffraction [113,117]and optical [108]investigations of poled PMN crystals showed that the ?eld-induced phase has the rhombohe-dral 3m symmetry,i.e.the same symmetry as in several normal perovskite ferroelectrics.On the other hand,lo-cally the structure is inhomogeneous,i.e.different from normal ferroelectric structure.The traces of cubic phase were observed at low temperature by X-ray diffraction experiments in poled PMN crystal [113].The NMR in-vestigations of PMN crystal poled by a ?eld almost two

6In contrast to spin (dipole)or FE state where the relaxation time is expected

to become in?nite in in?nite crystal,the rate of compositional ordering does not depend on a crystal size.The corresponding relaxation time at nonzero temperature can be very large,but not in?nite.

7According to Feynman [116],a system is in equilibrium if “all the fast things have happened and all the slow things have not”.

45

FRONTIERS OF FERROELECTRICITY

times as large as E cr,revealed that only about50%of Pb

ions are displaced parallel to the[111]poling direction

in a FE manner,while the other50%exhibit spherical

layer-type displacements characteristic of PE phase[32].

The size and number of AFNRs found in PMN in the un-

poled state(see Section7.1.1)remain unchanged in the

FE phase[110].

Upon heating,the FE phase transforms to the cubic

(ER)phase at a well-de?ned temperature,T C(≈210K

in PMN).This?rst-order phase transition is accompa-

nied by a step-like drop of spontaneous polarization(as

determined from pyroelectric current),sudden vanishing

of birefringence,and sharp peak of dielectric constant.A

more detailed description of the?eld-induced transition

and FE phase in relaxors is given in Ref.11.

7.2.Spontaneous relaxor-to-ferroelectric

phase transition

The transition from the ER to the FE phase typically

takes place at temperature T C,which is several degrees

or several dozens of degrees lower than T m,as schemat-

ically shown in Fig.9b and f7b23503ce2f0066f433228eually the transition is

observed in those relaxors where theε (T)peak is not

very diffuse(i.e.the diffuseness parameterδis relatively

small).X-ray and neutron diffraction experiments unam-

biguously indicate the change of symmetry at T C from

the high-temperature cubic to a low-temperature tetrag-

onal or rhombohedral(in most cases)one[118,119].

The symmetry breaking is also con?rmed by the appear-

ance of Brillouin scattering peaks which are forbidden

in cubic phase[48],the appearance of optical birefrin-

gence[120],and the formation of FE domains which are

clearly observed by optical polarizing microscopy[121,

122],electron microscopy[123]and scanning force mi-

croscopy[124–126].The FE phase in relaxors exhibits

typical FE properties,namely the large dielectric con-

stant,the FE hysteresis loops[118,121,127],the pyro-

[128]and piezoelectric(see below)effects,etc.At tem-

peratures slightly above T C,double hysteresis loops can

be observed[118],as is typical of normal ferroelectrics.

The spontaneous relaxor-to-ferroelectric phase transi-

tion can be accompanied by signi?cant anomalies in the

temperature dependences of structural parameters[118,

119,128],dielectric[4,82,118],optical[4,120,128],

thermal[118,128]and other properties.The transition

can be very sharp:e.g.in PFN crystals,the related jump

ofε(T)occurs in a temperature interval smaller than0.1K

[130].In many other cases,it is smeared for different rea-

sons.The change ofε(T)in ceramics is usually not as

sharp as in single crystals,probably because of the inho-

mogeneity related to the existence of grains and bound-

aries[131].Another type of inhomogeneity that can smear

the transition in the relaxor-based solid solution crystals

(e.g.PMN-PT,PZN-PT)is the macroscopic variation of

x across the sample[120,132].Besides,some“intrinsic”

causes for the smearing of phase transition also exist so

that a clear boundary between the canonical relaxors(in

which the anomalies of structure and properties are dif-

fuse or absent)and the relaxors with a sharp FE transition

cannot be de?ned.The intermediate behaviour can appear

in different ways.X-ray diffraction studies of PZN crys-

tals revealed the coexistence of the mesoscopic domains

of FE phase and the regions of cubic(relaxor)phase in a

temperature range of about70K around the mean T C[78,

133].The concentration of the cubic phase gradually de-

creases on cooling,i.e.the transition is highly diffuse.The

size of FE domains in PZN(40–200nm)[78,133]and

in disordered PST(25–75nm)[123]is smaller than in a

normal FE phase,but larger than the size of typical PNRs.

The average domain size(at room temperature)in the

(1?x)PMN?x PT solid solutions was found,by scanning

force microscopy,to gradually increase from～40nm in

the rhombohedral phase with x=0.1,to～2μm(which is

comparable to the domain size in ordinary ferroelectrics)

in the tetragonal phase with x=0.4[126].

Because of the similarity between the FE phase in or-

dinary ferroelectrics and the FE phase in relaxor ferro-

electrics,the transition at T C was initially called“sponta-

neous relaxor-to-normal ferroelectric transition,”but later

investigations showed that the low-temperature phase in

relaxors is not exactly a“normal”ferroelectric phase,even

in those relaxors where the FE transition is relatively sharp

and FE domains are large.In particular,in the PMN-PT

crystals with x=0.35,the central peak in the Brillouin

spectra,which is related to the relaxation of PNRs,was

observed not only at T>T C(see Section3.2),but also at

T

[48].Furthermore,macroscopic(1–2μm)areas of aver-

age cubic symmetry were found alongside with the areas

of FE phase[48].In PMN-PT crystals with x=0.20,

the piezoresponse force microscopy revealed a continu-

ous distribution of the sizes of polar regions starting from

～5nm(resolution limit).The complex structure of the

micron-size FE domains with the PNRs of the opposite

polarity embedded in them was observed[125].A mix-

ture of the rhombohedral domains and the domains of a

different low-symmetry(presumably monoclinic)phase

was observed by synchrotron X-ray diffraction in the

FE phase of PZN crystal[78].Unlike the plane walls

in ordinary ferroelectrics,the domain walls in relaxor fer-

roelectrics are usually diffuse and irregular[122,124].

The IR spectroscopy of PZN-PT and PMN-PT crystals

(with x=0.08and x=0.29,respectively)did not re-

veal any phonon softening that was expected for normal

FE phase transitions at T C and below[134].Accordingly,

the phonon contribution to the dielectric constant at these

temperatures is small(～100),i.e.much less than the low-

frequency value that reaches～5×104at T=T C and～5

×103at T<

FRONTIERS OF FERROELECTRICITY

canonical relaxors (and in contrast to ordinary displacive ferroelectrics),the dielectric response is determined by the relaxation polarization at all temperatures around and below the dielectric peak.It was concluded that the tran-sition into the FE phase consists in a stepwise increase in the size of PNRs which transform into FE domains [134].Another phenomenon,which is unusual for ordinary ferroelectrics,is the speci?c macroscopic phase inhomo-geneities discovered recently in good-quality crystals of some relaxor ferroelectrics.Diffraction experiments per-formed in PZN with X-ray of different energies (and thus different penetration lengths)revealed that the outer layer (an estimated thickness of ～10–50μm)undergoes a struc-tural phase transition into the FE phase while the lattice inside the crystal maintains the cubic unit cell [135,136].8Another interesting point is that at all temperatures the lattice parameter of the outer layers is slightly (～0.2%)smaller than that of the bulk (inside)[136].The same feature,i.e.a FE “skin”(observed by low-energy X-ray diffraction)and a cubic phase in the bulk (observed by high-energy X-ray or neutron diffraction),was also found in PZN-PT [137,138]and PMN-PT [139]crystals with small x .It was suggested that this cubic phase (named X-phase)is similar to the average cubic phase in pure PMN [140],i.e.it is a NR phase.As a typical NR phase,the X-phase can be irreversibly transformed into the FE phase by poling [137].

Using spatially resolved neutron diffraction technique,it was found that even in PMN crystals,where the rhom-bohedral phase has not been detected,the near-surface layer (of ～100μm thick)has the lattice constant notice-ably smaller than the bulk structure [141].It was supposed [140]that a very thin rhombohedral skin possibly exists in PMN also,with a thickness much smaller than the pen-etration length of X-rays,so that the skin could not be detected in the usual diffraction experiments.Note in this connection that,as reported in Ref.[109],the Rietveld re?nement of neutron diffraction data collected on PMN powder revealed a rhombohedral macroscopic symmetry,namely R 3m at 300K (i.e.above T f )and R 3c at 10K.Second harmonic generation signal was also detected,in-dicating a non-centric symmetry.In these experiments,very ?ne powder (of 4–5μm particle size)synthesized by a special route was used,so the whole material can be considered a near-surface region.This is the possible reason why the rhombohedral phase was found instead of the cubic phase usually observed by other authors in crystals and large-grain powders of PMN.

The behaviour unusual for relaxor as well as for ordi-nary ferroelectrics was recently found in PMN-PT crystals with large x (～0.5)[132].The dielectric properties typical of the ER phase are observed at T >T m ,namely the

8A

tetragonal unit cell was initially reported [135],but more elaborate investigations [136]later showed that it is indeed cubic.

deviation from the Curie-Weiss law at T

While an external electric ?eld transforms a NR state to a FE one,the hydrostatic pressure is able to induce the reverse transformation.In the crystals exhibiting the relaxor-to-FE phase transition,the FE phase does not ap-pear if the sample is cooled under a high enough pressure and the behaviour typical of the canonical relaxors is ob-served.The pressure-induced crossover from a ferroelec-tric to a relaxor state was discussed in detail in the recent reviews by Samara [142,143].

Excellent piezoelectric properties were found near the morphotropic phase boundary in PMN-PT,PZN-PT and some other solid solutions of complex perovskite relaxors with PbTiO 3.The transition from FE to ER phase is ob-served in these crystals at temperatures much higher than room temperature.Thus,they are considered and inves-tigated as promising materials for practical applications.This ?eld of research,currently very active,has been re-viewed in a number of recent papers [144–148].

7.3.Theoretical description of nonergodic

phase in relaxors

Early works on relaxors (e.g.the composition ?uctuations model by Smolenskii and Isupov [8,97]and the super-paraelectric model by Cross [10])considered the PNRs to be relatively independent noninteracting entities.It was later understood that the speci?c nonergodic behaviour of relaxors at low temperatures cannot be explained without taking into account the interactions among PNRs and/or quenched random local ?elds existing in the composition-ally disordered structure.The interactions among PNRs may lead to anomalous slowing-down of their dynam-ics (nonergodicity effects)or,when becoming frustrated,even to the formation of the glass state in which the dipole moments of individual PNRs are randomly ?xed in differ-ent directions.Note that these interactions are of dipole-dipole nature and can be considered as dynamic local ?elds.Additionally PNRs can be in?uenced (or probably even ?xed)by quenched local random ?elds stemming from the compositional disorder or other types of lattice defects.

In Section 5we have already discussed the modern the-ories explaining the formation of PNRs.Some of these

47

FRONTIERS OF FERROELECTRICITY

theories can also explain the transition from the ergodic

to nonergodic relaxor state.In particular,in the GF model,

the PNRs naturally appear in the temperature interval be-

tween the PE and the low-temperature dipole glass or

mixed ferroglass phase.In the WKG model,the forma-

tion of PNRs as well as the transition to the NR state is

ascribed to the quenched random?elds exclusively.

However,the mechanisms leading to the formation of

PNRs at high temperatures are not necessarily responsi-

ble for their freezing and for the development of the low-

temperature nonergodic state.The formation and freez-

ing of PNRs are possibly two distinct phenomena requir-

ing different approaches.The“semi-microscopic”mod-

els[98,149,150]of glass state in relaxors describe only

the latter phenomenon,while PNRs are believed to be

already-existing objects and the mechanisms of their for-

mation are not examined.

In the spherical random-bond-random-?eld(SRBRF)

model proposed by Pirc and Blinc[149,150],the Hamil-

tonian is formally written with Equations2and3,but

the meanings of the parameters are different from those

discussed in Section5.Pseudospins S i proportional to

the dipole moments of PNRs are introduced so that the

relation

i

( S i)2=3N(9)

is satis?ed(N is the number of pseudospins in the crystal).

It is assumed that each component of S i can?uctuate

continuously and take any value,9i.e.

?∞

J ij in Equation3are the random interactions(bonds)be-

tween PNRs which are assumed to be in?nitely ranged.

The second term in the Hamiltonian in Equation2de-

scribes the interaction of pseudospins with quenched ran-

dom electric?elds h i,

H2=?

i

h

i·

S

i.

Both random bonds J ij and random?elds h i obey the(un-

correlated)Gaussian probability distributions with an rms

variance of J/

√

N and ,respectively.The mean value

of the distributions equals J0/N(for random bonds)and

zero(for random?elds).In the absence of random?elds

( =0),if J

the PE phase(in the model this phase is equivalent to

the ER phase)into an inhomogeneous FE phase with a

9Models in which the order parameter satis?es conditions(9),(10)are called

“spherical”models.Due to these conditions the model is exactly solvable

by the replica method.

nonzero spontaneous polarization;if J>J0,the system

transforms,at a well-de?ned temperature T=J,from the

PE to a spherical glass phase without long range order,

and the glass order parameter(which is equivalent in this

model to the well-known Edwards-Anderson order pa-

rameter,q EA)decreases linearly from1at T=0to zero at

T=J.The presence of random?elds( =0)destroys the

phase transition so that q EA remains nonzero at T=J,and

approaches zero when the temperature further increases.

Fig.11a shows the temperature dependence of q EA deter-

mined experimentally from the NMR data of PMN(q EA is

shown to be proportional to the second moment,M2,of the

frequency distribution corresponding to the narrow93Nb

NMR line)[150].The solid line represents the?t with

the parameters J/k=20K and /J2=0.002,con?rming

the applicability of the model.The local polarization dis-

tribution function W( p)(where p=< S>)predicted by

the model and determined experimentally from the NMR

lineshape also appears to be the same as shown in the inset

of Fig.11a[150].The W( p)shape observed in dipolar

and quadrupolar glasses look very different,as shown in

Fig.11b.These results suggest that the NR phase in PMN

cannot be described as a dipolar or quadrupolar glass.It is

a new type of glass which can be called“spherical cluster

glass”[151].The SRBRF model is also able to explain the

dielectric non-linearity in PMN.The dynamic version of

SRBRF model describing the dispersion of liner and non-

linear dielectric susceptibility has been developed[152].

In the coupled SRBRF-phonon model[99],the coupling

of PNRs with soft TO phonons leads to the modi?cation

of interactions among PNRs.The effect of pressure on

the relative stability of different phases in relaxors are

explained.

Vugmeister and Rabitz[98,153]considered in their

model the hopping of PNRs in multi-well potentials.The

PNRs exist in a highly polarizable PE host lattice with

a displacive dielectric response.The theory takes into

account the broad distribution of the potential barriers

controlling PNR dynamics and the effect of interactions

between PNRs mediated by highly polarizable host.These

two aspects are described in terms of the local?eld dis-

tribution function.In this model,the dipole glass freezing

is believed to be accompanied by the critical FE slowing-

down.It is shown that the true glass state in which all

dipoles(PNRs)are frozen is not achieved in relaxors:the

degree of the local freezing is rather small even at low

temperatures.10The role of the critical slowing-down is

shown to be signi?cant in the dynamics of the system due

to the closeness of FE instability.In other words,relaxors

can be considered incipient ferroelectrics.This explains

their very large dielectric constant.In the framework of

this model,the shape of the frequency-dependent permit-

10This is in agreement with the experimental?nding that in relaxors only

the longest relaxation time diverges at T f,while the bulk of the relaxation

spectrum remains active at low temperatures(see Section6).

48

FRONTIERS OF

FERROELECTRICITY

Figure 11(a)Temperature dependence of the Edwards-Anderson glass order parameter q EA in PMN.The solid line is the ?t to the “spherical random bond random ?eld”(SRBRF)model.The inset shows the local polarization distribution function W ( p )along the p x axis according to the SRBRF model.(b)Examples of the W (p )functions for dipolar and quadrupolar glasses are shown for comparison (after Blinc et al .[150]).

tivity as a function of temperature in typical relaxors is explained qualitatively.The glasslike freezing of the dy-namics of PNRs is characterized by the non-equilibrium spin-glass order parameter,the temperature behaviour of which is consistent with the NMR experiments (shown in Fig.11).The kinetics of the electric ?eld induced tran-sition from the NR to FE phase was also successfully reproduced [154](while the glass models experience dif-?culties in explaining this transition).

The behaviour of PNRs can be in?uenced by the elec-tronic subsystem.In particular,the thermo-localization of charge carriers on the defects in the temperature range of phase transition can change the relaxation dynamics.The direction of spontaneous polarization of PNRs can be pinned by localized charge carriers,preventing the align-ment of PNRs in the external electric ?eld.The related phenomena are studied in Refs.[155,156].

As mentioned above,the models so far discussed in this Section consider PNRs (pseudospins)to be already-existing entities.In order to describe the process of their formation and development (which begins from T B T f ),other models are needed.Recently,it has been proposed that quenched random ?elds give rise to the formation of PNRs in the PE phase,as prescribed by the WKG model,and then,upon further cooling,the crystal un-dergoes a transition into the spherical cluster glass state due to random interactions between PNRs [151].Alter-natively,some other models can be used to describe the formation of SRBRF pseudospins,in particular,the soft nanoregions model [77][which justi?es the ful?lment of condition (10)]together with the kinetic model [70](as discussed in Section 5).

Let us now discuss the mechanisms of the spontaneous relaxor-to-ferroelectric phase transition.There are two ways to explain the formation of FE phase at T C from the system of disordered PNRs in relaxors.The ?rst one suggests that the dipole-dipole interactions between PNRs (or individual ions)lead to their FE-type arrangement (as,e.g.,in the SRBRF model discussed above).The second mechanism arises from the kinetic model of phase tran-sitions in compositionally disordered crystals [70](see Section 5)and suggests the thermally activated growth of PNRs at T C .It is not easy to discriminate these two mech-anisms from each other.In fact,it is also possible that both mechanisms contribute to the process of the formation of FE phase.

8.Conclusions

In this paper we have analyzed the peculiar behaviour of relaxor ferroelectrics that occurs in compositionally disor-dered perovskites.The quenched compositional disorder in these compounds gives rise to another type of disorder,i.e.the glassy nonergodic state that can be observed at low temperatures,instead of a FE or AFE ordering that exists in many simple perovskites.The research in this ?eld has undergone such a tremendous growth that it was not pos-sible to review all the important works in this short paper.Some subjects were discussed only brie?y just to give the examples characterizing the peculiarities of the behaviour.Some other important topics have been left out,in partic-ular,the materials technology of crystals,ceramics and thin ?lms and applications of relaxor ferroelectrics.

Despite the remarkable progress achieved in the re-cent years,fundamental physics of the relaxors remains a fascinating puzzle.Some key questions,such as what the origin of relaxor behaviour is,still have no de?nite answers.Several theoretical models have been proposed;some of them contradict each other.Further experiments need to be performed in order to prove or reject these models,while new and more satisfactory theories are yet to be worked out.With their complex structures and in-triguing properties,relaxors represent truly a frontier of research in ferroelectrics and related materials,offering great opportunities both for fundamental research and for technological applications.

Acknowledgements

This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC)and the U.S.Of?ce of Naval Research (Grant#N00014-99-1-0738).

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