Order parameter and scaling behavior in BaZrxTi1−xO3 (0.3 < x < 0.6) relaxor ferroelectrics

Order parameter and scaling behavior in BaZr x Ti1 x O3 (0.3 0.6) relaxor ferroelectricsMuhammad Usman, Arif Mumtaz, Sobia Raoof, and S. K. Hasanain Citation: Applied Physics Letters 103, 262905 (2013); doi: 10.1063/1.4860967 View online: http://dx.doi.org/10.1063/1.4860967 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/26?ver=pdfcov Published by the AIP Publishing

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Order parameter and scaling behavior in BaZrxTi12xO3 (0.3 < x < 0.6) relaxorferroelectrics

Muhammad Usman, Arif Mumtaz,a) Sobia Raoof, and S. K. HasanainDepartment of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan

(Received 8 November 2013; accepted 15 December 2013; published online 31 December 2013)

We report the relaxor behavior of the zirconium doped barium titanate BaZrxTi1�xO3 solid solutions

and discuss the temperature, frequency, and concentration dependence in terms of correlations

among the polar nanoregions. The relaxor behavior is analyzed within the mean field theory by

estimating the Edward-Anderson order parameter qEA. Additionally, we find that qEA calculated for

the different concentrations obeys a scaling behavior qEA ¼ 1� T=Tmð Þn, where Tm are the

respective dielectric maxima temperatures and n¼ 2.0 6 0.1. The frequency dependence of the qEA

also shows results consistent with the above mentioned picture. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4860967]

Ferroelectric relaxor systems and the understanding of

their complex behavior have recently attracted a lot of inter-

est. In part, this is due to their tunable dielectric properties

and potential applications in microwave devices, ferroelec-

tric random access memories (FERAMs), and multi layered

capacitors, to name a few.1 Apart from technological appli-

cations there is also a strong interest in these materials as

typifying a complex form of polar order in a system with

manifest randomness.2,3 Relaxor ferroelectrics are character-

ized, in particular, by a diffuse and frequency dependent

dielectric peak temperature. Perovskites, represented as

ABO3, are common ferroelectric materials and have been

extensively studied. In these perovskite structures relaxor

behavior sets in for certain substitutions at either A or B or

both A and B sites. Typical examples are PbMg1/3Nb2/3O3

(PMN),4,5 PbZn1/3Nb2/3O3 (PZN),5 and the lead (Pb) free

systems BaZrxTi1�xO3 (BZTx) and BaTi1�xSnxO3 (BSnT).6

Generally speaking relaxor ferroelectrics may be divided

into two broad categories, pertaining to heterovalent or

homovalent substitutions. The systems with heterovalent

compositions such as PMN and their solid solutions, e.g.,

PbMg1/3Nb2/3O3-PbTiO3 (PMN-PT)7 show a well-defined

dielectric peak that exhibits relaxor features. The relaxor

behavior in these heterovalent systems is understood as

occurring mainly due to two reasons: the charge imbalance

arising due to the differing dopant valence and second from

the presence of ordered polar regions of nanosize.8,9 Both

these effects produce random quenched fields, an essential

ingredient in canonical relaxor materials. A lot of work has

been done in such systems, and they are relatively well

understood. On the other hand, the homovalent substitution

of Tiþ4, for example, in BaTiO3, with Zrþ4 does not produce

a significant charge imbalance and therefore is not expected

to produce quenched random fields.2 In contrast to the heter-

ovalent relaxors the relaxor behavior in these latter materials

appears only after a significantly higher level substitution.

For example, in the case of BaZrxTi1�xO3 the relaxor behav-

ior starts for x> 0.27.10 The origin of relaxor behavior in

such homovalent titanate material has recently attracted

attention for both experimental and theoretical studies.

It has been argued in the past and also in recent works

that ferroelectric relaxors have many features that resemble

those of magnetic spin glass systems and are describable gen-

erally by the models developed for those systems.8

Akbarzadeh et al. in their numerical studies have described

the behavior of the dielectric susceptibility for the

BaZr0.5Ti0.5O3,11 using the Edward-Anderson order parame-

ter.12 Their work also illustrates the presence of polarization

clusters that grow in size and number with decreasing temper-

atures and begin to interact below a certain characteristic

temperature. These polarization clusters are identical to the

polar nanoregions (PNRs) that are understood to be central to

the relaxor behavior, with the addition that they are now con-

sidered to be dynamic entities, i.e., the center of the cluster

may shift from one Ti ion to another with time. The predic-

tions of this model were found to be in good agreement with

the experiments of Bhalla et al.10 for the temperature and fre-

quency dependence of the dielectric constant of BZT materi-

als. This is consistent with the generally accepted picture of

the development of polar nanoregions, their growth, and the

development of interactions between them. However, it

remains to be seen how (and if) these features manifest them-

selves experimentally in the order parameter, both with

increasing levels of substitution and increasing frequency. In

the current work we have studied the BaZrxTi1�xO3 relaxor

ferroelectrics (0.3� x� 0.6) and present our results for the

dielectric susceptibility as a function of concentration, tem-

perature, and frequency (x, T, f) and analyze the variations of

the obtained Edward-Anderson (EA) order parameter for

these variables (x, T, f). We further explore the important

question whether the order parameter for different concentra-

tions can be described by a universal behavior, qEAðT=TmÞ, as

is typically seen in classical spin glasses. In a similar vein the

variation of qEA(T, f) is studied to determine how the dynami-

cal response at different temperatures manifests itself in the

order parameter.

The BaZrxTi1�xO3 bulk ceramic samples with different

Zr concentration (0.3� x� 0.6) were prepared by conven-

tional solid state reaction with a two-step sintering process,

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

0003-6951/2013/103(26)/262905/5/$30.00 VC 2013 AIP Publishing LLC103, 262905-1

APPLIED PHYSICS LETTERS 103, 262905 (2013)

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and the preparation and characterization of the obtained sin-

gle phase structures have been detailed elsewhere.13

Dielectric measurements were carried out over a frequency

range 0.2–500 kHz using Wayne Kerr LCR meter

(WK-4275) in the temperature range of 10–300 K using a he-

lium closed cycle system (Janis CCS-350 Cryostat).

In Fig. 1, both real and imaginary parts of the dielectric

susceptibility measured at various frequencies as a function

of temperature are shown for different Zr concentrations. For

all concentrations studied, clear frequency dispersion can be

seen. It is also evident from the figure that the dielectric peak

shifts to low temperature as the Zr concentration increases.

Furthermore peak broadening with increasing Zr concentra-

tion is also evident. These observations are consistent with

many reported studies and are usually associated with com-

positional heterogeneity of the PNRs. The random distribu-

tion of B-site cations creates compositional variations of

local polar nanoregions which in turn lead to locally varying

ferroelectric transition temperatures.14 Note also that the fre-

quency dependence of the temperature of the dielectric per-

mittivity maximum, Tm, follows the Vogel-Fulcher law (not

shown here) as reported by us previously.3

The dielectric permittivity e0 of the relaxor ferroelectric

can be described by the Curie Weiss law (Eq. (1)) in the high

temperature paraelectric phase

1

e0¼ T � h

C: (1)

To determine the critical parameters C (Curie constant)

and the transition temperature h, the higher temperature data

for the dielectric constant e0 was fitted to Eq. (1) for all the

BZTx compositions. The fitted data are shown in Fig. 2 that

yield the values of Curie constant C and Curie temperature

h. For example, for the composition BZT0.5 the Curie con-

stant C is 3.33� 104 K while h is 91 K, consistent with val-

ues reported previously.10 Note that the deviation of the data

from the Curie-Weiss behavior initiates at a temperatures

Tdev higher than the peak temperature Tm, as marked in

Fig. 2 for BZT0.6.

It has been argued8,14 that the relaxor ferroelectrics may

be considered as electric dipole glass analog of magnetic

spin glass systems, and therefore relaxor ferroelectrics may

be treated and analyzed using the well-established models

such as the Edwards-Anderson model developed for the spin

glass systems. Hence to discuss the development of the

relaxor state we take into account the EA-order parameter

which in the present context describes the average correla-

tions between the different PNRs. The order parameter qEA

can be written as qEA � hPiPji,3,15,16 where Pi and Pj are the

dipole moments corresponding to the ith and jth polar nano-

regions, respectively. Sherrington and Kirkpatrick (SK)

developed17 an infinite range model for the spin glass to cal-

culate the EA-parameter. This model relates the temperature

dependence of the susceptibility (v) to the local order param-

eter qEA (Refs. 16–19)

v ¼ C 1� qEAð ÞT � h 1� qEAð Þ : (2)

FIG. 1. The real and imaginary parts

of dielectric permittivity measured at

different frequencies as a function of

temperature for various compositions

BZTx (0.3� x� 0.6).

FIG. 2. The inverse of the dielectric permittivity (1/e0) as function of temper-

ature for various concentrations, for f¼ 500 kHz. The lines are the fit to the

Curie-Weiss behavior. The deviation temperature Tdev and the x-intercept To

are marked for the BZT0.6.

262905-2 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)

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This relation has been used extensively16,19 to extract

the EA-order parameter from the experimental data for relax-

ors. For this purpose the values of C and h as determined

from the Curie Weiss fit of the data and the value of dielec-

tric susceptibility (at a particular frequency) for varying tem-

perature are inserted in Eq. (2).

Using the data of Figs. 1 and 2 the value of the EA-order

parameter was calculated for different concentrations of Zrat a fixed frequency of 500 kHz. The results are shown in

Fig. 3. It is evident from the figure that for all concentrations

of Zr, qEA Tð Þ starts smoothly and slowly at high tempera-

tures and then rises rapidly and appears to merge at low

enough temperatures. For each composition the value of qEA

becomes nonzero at a temperature closely coinciding with

the temperature Tdev defined earlier as the temperature where

the 1/e0 vs. T data begin to deviate from the Currie–Weiss fit

(see Fig. 2). In general, we observe that the onset point for

nonzero qEA value shifts to lower temperatures with increas-

ing x. Note that at fixed temperature, e.g., at 100 K, qEA Tð Þfor higher concentrations is lower than qEA Tð Þ at low con-

centrations. All these observations are consistent with the

PNR or polar clusters picture.11 For example, at low Zr con-

centration paraelectric BaZrO3 forms a dilute solution in fer-

roelectric BaTiO3. At this stage the polar clusters that are

centered at Tiþ4 sites are in close proximity to each other

and are able to develop strong correlations even at relatively

higher temperatures. On the other hand increasing Zr con-

centration results in a lower number of Ti-centers in the BZT

solid solution and hence to poor correlations among the

PNRs at higher temperatures. These correlations and the

EA-order parameter qEA Tð Þ therefore grow rapidly as the

temperature is reduced since more and more PNRs forms

and smaller PNRs merge to form even bigger clusters at low

temperatures. Finally, at low enough temperatures the num-

ber of PNRs begins to saturate and the order parameters

qEA Tð Þ for the respective concentrations merge at low

enough T.

In order to determine if the behavior of the order parame-

ter satisfies any universal relation, we plotted the qEA values

for different concentrations vs. their respective scaled temper-

atures T/Tm. The data are shown in Fig. 4. Interestingly we

found that the qEA T=Tmð Þ curves for different concentrations

(but the same frequency) follow a universal pattern. This is

evident from Fig. 4 where the curves for the four different

concentrations overlap to a very high degree up to

T� 0.85 Tm. Above this temperature range all the curves

show a clear tail with qEA vanishing at T significantly higher

than Tm. The non-vanishing of qEA at T¼Tm and the devia-

tion from the extrapolated fit at higher temperatures is not

surprising since the ordered regions or PNRs are understood

to exist at temperature well beyond the Curie point, up to the

Burns temperature.10 However their numbers are too few, or

the interactions between them are too weak to develop a cor-

related behavior.

Typically spin glass systems have been shown to follow

a temperature dependence of universal sort given as

1� T=Tg

� �n, where the value of n can vary depending on

the dimensionality of the system, the assumed form of the

random field distribution, and the proximity to the critical

temperature.19,20 In our case we found that the qEA T=Tmð Þdependence that most closely describes the universal behav-

ior as shown by the solid line in Fig. 4 is given by the de-

pendence qEA T=Tmð Þ ¼ 1� T=Tmð Þn with n¼ 2.0 6 0.1.

While such a dependence is not typical for spin glass sys-

tems, it has been observed in an unconventional spin glass

Mg1þtTitFe2�2tO4.21 At this point the scaling behavior that

we observe can only be justified as an empirical fit.

The observed behavior of qEA has also been compared to

the predictions of the spherical random bond-random field

(SRBRF) model which has been applied to heterovalent

relaxors.22 First this model predicts a linear T dependence

for qEA at low temperatures, if one is to ignore random fields.

We however do not find linear qEA(T/Tm) dependence for any

appreciable temperature range. Second, if random fields are

considered as playing a significant role, then the SRBRF

model predicts a systematic change in the qEA(T/Tm) behav-

ior with increasing random field strength, i.e., the absence of

FIG. 3. The EA order parameter qEA as a function of temperature for various

BZTx compositions and f¼ 500 kHz determined using Eq. (2). Note that the

onset temperature Tdev of qEA is the same as the deviation temperature deter-

mined from the Curie-Weiss behavior.

FIG. 4. EA-order parameter qEA as a function of scaled temperature T/Tm for

various BZTx compositions and f¼ 500 kHz. The degree of overlap of the

curves below T¼ 0.85 Tm and the deviations at higher T are evident. The solid

curve indicates the scaling function qEA ¼ 1� T=Tmð Þn with n¼ 2.05 6 0.1.

262905-3 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)

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a universal scaled behavior. We, on the other hand, find an

overlapping or universal qEA(T/Tm) behavior for the various

compositions. We understand the higher x compositions as

corresponding to higher level of random field strengths D.22

Thus if random fields were to be playing a significant role in

determining the order in these relaxors than the qEA(T/Tm)

curves for different x should not overlap. Therefore we

emphasize that our data show that in case of homovalent sub-

stituted relaxors, the role of random fields is not substantial.

This is also in agreement with the results of Refs. 11, where

they found that the relaxor behavior continues in BZT0.5

even if they completely switched off the random fields in

their simulations. They have argued that small antiferroelec-

tric contributions present in this system may initiate the

observed relaxor behavior.

Finally we have addressed the question of the dynamical

behavior of the order parameter by comparing the qEA(T/Tm)

behavior over a wide range of frequencies. This behavior has

been studied for all the four compositions (Fig. 5). It is appa-

rent from Fig. 5(a) that qEA(T/Tm, f) shows a frequency inde-

pendent behavior for x¼ 0.3. We note that this is the

composition where the relaxor behavior is known to just set

in for the BZT system. This may also be understood as the

concentration where polar cluster sizes grow large enough

and correlations set in between them. However as x increases

we note that the scaled qEA(T/Tm, f) demonstrates significant

frequency dependence, as evident from Fig. 5. We see that

the high frequency curves lie systematically below the low

frequency curves. In other words, at the same scaled temper-

ature smaller order parameter qEA(T/Tm, f) values are associ-

ated with the higher frequencies suggesting obviously that

the degree of correlation decreases with increasing fre-

quency. Note also that as we move to higher concentrations

(Figs. 5(b)–5(d)) and thereby deeper into the relaxor state,

the gap between the low frequency and the high frequency

curves increases. This can be interpreted in the sense that for

larger x compositions there is a broader distribution of relax-

ation times which is reflected in a greater level of disper-

sion.23 We also note that for a fixed x the onset point, where

qEA(T/Tm, f) initially assumes a non-zero value, shifts to

lower scaled temperatures with increasing frequencies. This

indicates that with increasing frequency less time is available

for the correlations to develop and therefore low values of

qEA(T/Tm, f).In conclusion we have presented a detailed study of BZT

relaxor ferroelectrics with the view to relate the dielectric

response of this system with the picture of a dipolar glassy

system. The behavior is seen to be well described by the de-

velopment of an order parameter via mean field theory. For

fixed frequency a universal behavior of the order parameter

is reported for a range of relaxor concentrations. This univer-

sal behavior also supports the argument that in these isova-

lent substituted relaxor systems the random fields may not

play a significant role. The dynamic behavior of the order pa-

rameter is also consistent with a weakening of the correla-

tions between the ordered regions with increasing frequency.

This work was financially supported by the Higher

Education Commission of Pakistan under the grant for the

project Development and Study of Magnetic Nanostructures.

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FIG. 5. EA-order parameter qEA for

various compositions BZTx as function

of reduced temperature T/Tm measured

at frequencies of 0.2–500 kHz.

262905-4 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)

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262905-5 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)

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