Recent experiments revealed unusual dielectric behaviors of Pb(Mg 1/3Nb 2/3)O 3 originated from polar nanoregions (PNRs). Thus, Pauli's master equation is adopted to investigate the distinct dielectric responses, correlation strength, and volume evolutions of static and dynamic PNRs. Our findings have not only validated the percolation theory but also ascertained Lorentzian distribution of the rate of thermal change of PNR volume. Finally, based on Maxwell's equation the observed dielectric deviations of bulk permittivity from Curie-Weiss law are attributed to the thermal effects on static volume fraction and polarization rotation. © 2011 American Institute of Physics.published_or_final_versio

Shi, YP

Soh, AK

Applied Physics Letters

Y. P. Shi

A. K. Soh

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Effects of volume evolution of static and dynamic polar nanoregions on the dielectric behavior of relaxors

Recent experiments revealed unusual dielectric behaviors of Pb(Mg1/3Nb2/3)O3 originated from polar nanoregions (PNRs). Thus, Pauli’s master equation is adopted to investigate the distinct dielectric responses, correlation strength, and volume evolutions of static and dynamic PNRs. Our findings have not only validated the percolation theory but also ascertained Lorentzian distribution of the rate of thermal change of PNR volume. Finally, based on Maxwell’s equation the observed dielectric deviations of bulk permittivity from Curie-Weiss law are attributed to the thermal effects on static volume fraction and polarization rotation

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Title Effects of volume evolution of static and dynamic polarnanoregions on the dielectric behavior of relaxorsAuthor(s) Shi, YP; Soh, AKCitation Applied Physics Letters, 2011, v. 99 n. 9, article no. 092908Issued Date 2011URL http://hdl.handle.net/10722/157139Rights Applied Physics Letters. Copyright © American Institute ofPhysics.Effects of volume evolution of static and dynamic polar nanoregionson the dielectric behavior of relaxorsY. P. Shi and A. K. Soha)Department of Mechanical Engineering, The University of Hong Kong, Hong Kong(Received 23 July 2011; accepted 10 August 2011; published online 2 September 2011)Recent experiments revealed unusual dielectric behaviors of Pb(Mg1/3Nb2/3)O3 originated from polarnanoregions (PNRs). Thus, Pauli’s master equation is adopted to investigate the distinct dielectricresponses, correlation strength, and volume evolutions of static and dynamic PNRs. Our findings havenot only validated the percolation theory but also ascertained Lorentzian distribution of the rate ofthermal change of PNR volume. Finally, based on Maxwell’s equation the observed dielectricdeviations of bulk permittivity from Curie-Weiss law are attributed to the thermal effects on staticvolume fraction and polarization rotation.VC 2011 American Institute of Physics.[doi:10.1063/1.3632082]The physical nature underlying colossal piezoelectricresponse and broad temperature (T) dielectric dispersion ofrelaxor ferroelectrics (relaxors) still remains debatable inview of a few controversies between the existing experimen-tal and theoretical results. In contrast to wide acceptability ofCurie-Weiss (C-W) relation for the sharp phase transition innormal ferroelectrics, apparent deviations from C-W lawhave been frequently observed in prototypical relaxors1,2Pirc et al.3 found that the deviation that peaked near theCurie-temperature (TC) was even more significant when ahigher electric field (E) was imposed. More and more experi-ments suggested that unusual relaxor behaviors were closelyrelated to the coexistence and competition between the long-rang order and polar nanoregions (PNRs) that appearedbelow the Burns temperature (Td). To-date, it is believed thatthe typical PNRs in relaxors are mutually coupled via inter-actions with polarizable matrix. In addition to the influenceof local PNR polarization,1 the correlation strength of a“connected” network4 composed of high-density PNRs waseasily modulated by random fields stemming from structuraldisorder in multivalent relaxors.3 Recently, Xu et al.5 haverevealed that T- and E-driven redistribution of PNR polariza-tions played a key role in governing the vagarious dielectricresponses of relaxors. Although PNR dynamics in relaxorshave been widely studied, some of the observations clearlycontradicted the related theories. First, adoption of differentnanostructure was unavoidable to interpret the recentlyobserved E-independent source of diffuse scatteringintensity.5 Second, the latest neutron spin-echo (NSE) meas-urements6 exhibited distinct T-dependence of static anddynamic scattering intensity in Pb(Mg1/3Nb2/3)O3 (PMN)crystals. Furthermore, the following essential problemsencountered for clarification of typical relaxor behaviorshave yet to be resolved: (i) The methodology for distinguish-ing different dielectric features of static and dynamic evolu-tion of PNRs and, hence, correlating them with bulkdielectric dispersion and (ii) explanation of the three lengthscales,7 i.e., 5-25 nm sized PNRs, chemical ordering region(COR) of <5 nm, and ferroelastic domain walls (DWs) of100 nm in length, which are required for a PMN crystal tostabilize inhomogeneities?In this letter, these essential issues are addressed to bet-ter understand the dielectric behaviors and related propertiesof relaxors. The total volume of a PNR, Vtotal, can be dividedinto two components, i.e., a T-dependent static volume (Vsta)and a dynamic volume (Vdyn) initiated by adequate correla-tion among PNRs. Vdyn is sensitive to T and E in terms ofmodulation for the interaction strength.PNRs in PMN have rhombohedral symmetry and [111]-type polarization (with eight easy directions). In general,time (t)-dependent probability (pn) for PNRs to occupy thenth ground state is governed by Pauli’s master equation:8,9sddtpnðtÞ ¼Xm 6¼nhpmðtÞ  eUapnðtÞi; (1)where integer n, m 2 ½1; 8 and s is relaxation time. To fur-ther simplify Eq. (1), we restrict it to a two-state system. Ageneric activation parameter Ua that denotes the energy dif-ference between two ground states is established by combin-ing Ne´el’s proposal4 and our recent findings,8 i.e., Ua¼QV/kT, where k is Boltzmann’s constant and V is either Vsta orVdyn; Q is the energy density given by 2EjPjm, in which jPjmdenotes the PNR polarization maxima. Based on the pro-posed solution9 of Eq. (1), we obtain the bulk PMN polariza-tion magnitudehPi ¼ jPjm 1  eUa1 þ eUa ð1  et=sÞ: (2)In order to distinguish the dielectric contributions from Vstaand Vdyn, it is assumed that Vsta has a T-independent polar-ization maxima, whereas, that of Vdyn is the sum of a con-stant jPjm and e(T)E, where e(T) is T-dependent dielectricpermittivity. In the asymptotic limit, substituting Eq. (2) intoe(T)¼ @hPi/@E yields the contribution of Vsta and Vdyn to e,i.e., esta and edynesta ¼ jPjmE2Ua2 þ eUa þ eUa edyn ¼ jPjmEUa1  Ua þ eUa  :8><>: (3)a)Author to whom all correspondence should be addressed: Electronic mail:aksoh@hkucc.hku.hk. Tel.: þ852-28598061. FAX: þ852-28585415.0003-6951/2011/99(9)/092908/3/$30.00 VC 2011 American Institute of Physics99, 092908-1APPLIED PHYSICS LETTERS 99, 092908 (2011)Downloaded 29 Aug 2012 to 147.8.230.79. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsThe dissimilarity in Ua-dependence of esta and edyn shown inFig. 1(a), represents the distinct dielectric features of Vstaand Vdyn. esta disperses over a wide Ua range, whereas edynonly shows a substantial change near Ua¼ 1.278, at which itdiverges in a manner similar to normal ferroelectrics.Furthermore, since the inverse Curie constant (1/C) reflects@e1/@T, Eq. (3) indicates that the QV/k ratio in relaxorsis comparable with the C value. For PMN crystal Q 2.3 103 eV/(nm)3 (Ref. 4) and C  2.05 105 K(Ref. 7), Eq. (3) predicts a mean volume of 103 nm3, whichtallies with the PNR size of 5-25 nm observed by SEM.7In view of a latest finding7 that the C-W relation vali-dates even if T is near TC, it is postulated that Eq. (3) forlocal PNRs requires the C-W law to be satisfied. The inset ofFig. 1(a) shows that the 1/esta given by Eq. (3) is in excellentagreement with the C-W law over a broad T-range, while theminor deviation between them due to the absence of estadivergence can be eliminated by an appropriate 1/edyn.The C-W behavior of esta actually arises from the evolu-tion of Vtotal and Vdyn pursuant to the thick and thin curve inFig. 1(b), respectively. Fig. 1(b) exhibits an outstandingagreement of our predicted Vtotal with the calculated volumefraction of PNRs.10 Equation (3) underlines the existence ofa peak in both the Vtotal and Vdyn curve. This suggests thatthe PNR correlation has both attractive and repulsive feature,and it finally becomes negligible compared with thermalfluctuations. Accordingly, the histogram of PNR evolution isdivided into three regions, as depicted in Figs. 1(b) and 1(c).In region I, elliptical PNRs composed of spherical Vsta andheterogeneous Vdyn, increase gradually while cooling till Tpwhere all the PNRs percolate through PMN crystal. Vstaexpands at the expense of Vdyn in a narrower region II and itbecomes immobile at the freezing temperature (Tf). Mean-while, a sudden Vtotal decrease and the formation of ellipticalVsta would induce profound stresses7 due to a higher symme-try of a newly formed Vsta than that of Vdyn. Finally, inregion III, a similar percolation process will take place in theremaining medium, leading to the rebounding of Vtotal[dashed curve in Fig. 1(b)] and formation of smaller CORs.7Another explanation is that the interfacial energies andresultant stresses drive the CORs and Vsta self-assembly11and stabilize long-range order at low temperatures. This mayhelp to explain the observation12 that PNR/COR was presentover a much wider T range in a PMN crystal than in aSr1-xBaxNb2O6 where the stress factor was insignificant. Asmaller Vsta near Td should be ineffective for self-stress-relaxation and, thus, formation of a sufficiently large Vdyn,e.g., Vdyn/Vsta  1 (Ref. 6) appears inevitable for maintain-ing the overall cubic symmetry via elimination of stresses.This dynamics is able to redistribute bulk polarization from ainitial direction close to [111] easy direction.13 Similar tonormal ferroelectrics where ferroelastic DWs are involved inminimization of mechanical energy, ferroelastic DWs suchas 90 DWs in Fig. 1(c) could also be stabilized to relax thestresses of PNRs, which may be used to explain the meas-ured perpendicularity13 of PNR polarization to that of thematrix. Besides, once the dimension of remaining medium isless than a critical size,11,14 both high-aspect-ratio DWs andirregular stress field7 will appear and persist at low Ts.For explaining the unconventional NSE results,6 Fig.2(a) is plotted, which shows that the Vsta curve tallies withthe static intensity (SI) when T is decreased from 500 K to200 K. Besides, before reaching a sudden freezing at 190 K,the Vsta curve could be divided into a nearly straight and anonlinear portion, which agrees with the calculated curve1 oflocal order parameter q, as shown in the inset of Fig. 2(a). AsFIG. 1. (Color online) (a) Ua-dependence of sta and dyn; the inset compares 1/sta and Curie-Weiss law. (b) Comparison of Vtotal and Vdyn with the calcu-lated PNR volume fraction (Ref. 10). (c) Schematics for thermal evolutions of Vsta and Vdyn and rotation of overall polarization ( P!); Pdyn!denotes the polar-ization direction in Vdyn.FIG. 2. (Color online) Comparison of (a) the predicted Vsta with measured NSR SI data (Ref. 6). The inset displays q vs T calculated in Ref. 1. (b) Variation of@Vsta/@(T) with the NSE DI (solid circle). (c) Our predicted Vdyn with the renormalized DI data; Tdyn is the critical temperature to stabilize Vdyn.092908-2 Y. P. Shi and A. K. Soh Appl. Phys. Lett. 99, 092908 (2011)Downloaded 29 Aug 2012 to 147.8.230.79. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsfor the unusual dynamic intensity (DI), we would attributethem to two causes: @Vsta/@(T) and variation of Vdyn. Intui-tively, a larger @Vsta/@(T) is bound to augment the meas-ured DI. Hence, @Vsta/@(T) is plotted using the Vsta data inFig. 2(a), as shown in Fig. 2(b), in which another two fittingsare also displayed. In a mean field case,4 percolation theorypredicts the PNR mean volume asV ¼ V0=ð1  T0=TÞ: (4)The dashed-curve in Fig. 2(b) is a dV/d(T) fitting based onEq. (4) with V0  0.1075 and T0¼ 243.5 K, which is closeto 224 K given in Ref. 15. Although Pirc and Blinc4 assumedthat Eq. (4) was valid as long as T>T0, Fig. 2(b) shows thatEq. (4) is only applicable for T> 315 K. However, a Lorent-zian fitting based on@Vsta=@ Tð Þ ¼ mm T$=p T  Tstam 2 þ T$2h i (5)displays an excellent agreement with @Vsta/@(T) data downto 250 K. mm¼ 0.369 and T$ ¼ 59:12 K and Tstam indicate thatthe rate maxima is at 290 K, which is quite close to the cal-culated1 300 K [refer to Fig. 1(a)]. In general, the relaxordynamic response is supposed to increase with the growth ofVdyn. Thus, the Vdyn predicated is also plotted in Fig. 2(c), inwhich shows perfect agreement with the observed DI data.Since diffuseness scattering in a cooled relaxor is charac-terized by a gradual increase in static volume fraction a,6 twomechanisms are proposed to predict the thermal evolution ofa. Mechanism I estimates a based on the known Vtotal and Vsta[see Figs. 1(b) and 2(a)] as aI¼Vsta/Vtotal, which yields thesolid curve in Fig. 3(a). It can be seen that the decreasing trendof aI is similar to that of the experimental data (aExp).6 Instead,mechanism II is based on a thermodynamic stability analysisfor a representative PNR. Basically, PNRs can be stabilized bycooling the relaxor from Td down to a sufficiently low temper-ature. This is equivalent to that adding Vdyn to the already sta-ble Vsta requires further cooling the whole PNR to a muchlower temperature (Tdyn) than Tsta to ensure the stability ofVsta mechanism II estimates a asaII ¼ ðTd  TstaÞ=ðTd  TdynÞ; (6)where Tsta and Tdyn are the known temperatures of SI and DIdata determined in Figs. 2(a) and 2(c). Td is the only adjusta-ble parameter to our fitting of Eq. (6) for the eight-pair (SI-DI) data. For PMN crystals, the best estimation of Td is 580K, which lies in the measured range of 550 K (Ref. 6) to 600K (Ref. 10).Lastly, we would correlate the T-dependent a with thethermal dispersion of average permittivity (ePNR) among allthe PNRs. For a two-phase dielectric composite,16 Max-well’s equation predicts that the ePNR satisfiesa13þ edynePNR  edyn ¼ 13þ edynesta  edyn : (7)The symmetric form of Eq. (7) implies that the observed devi-ation of 1/ePNR from the C-W law is solely originated fromthe thermal change of a. Fig. 3(b) shows that @(1/aI)/@T and@(1/aExp)/@T have similar trend in a broad T-range. Equation(7) infers that the nearly constant aExp for T< 249.6 K andT> 458.6 K ensure the obedience of crystal permittivity withthe C-W relation, and that dielectric deviation phenomenoncan be observed in the middle T-range.In conclusion, the distinct dielectric responses static anddynamic PNRs are studied based on Pauli’s master equation.The PNR correlation is found to peak near the Curie temper-ature. Two mechanisms have also been proposed to estimatethe thermal change of a. Finally, based on Maxwell’s equa-tion, the observed deviations of bulk permittivity from the C-W law are attributed to thermal effects on and polarizationrotations that occur from Td to Tf. Our findings are in excel-lent agreement with the existing theoretical and experimentalresults.Research Grants Council of the Hong Kong SpecialAdministrative Region, under Project Nos. HKU 716508Eand HKU 717011E, is acknowledged. We thank Prof. G.J.Weng at Rutgers University for helpful discussions.1D. Viehland, S. J. Jang, L. E. Cross, and M. Wuttig, Phys. Rev. B 46, 8003(1992).2W. Kleemann, J. Dec, V. V. Shvartsman, Z. Kutnjak, and T. Braun, Phys.Rev. Lett. 97, 065702 (2006).3R. Pirc, R. Blinc, and V. S. Vikhnin, Phys. Rev. B 74, 054108 (2006).4R. Pirc and R. Blinc, Phys. Rev. B 76, 020101(R) (2007).5Z. J. Xu, J. S. Wen, G. Y. Xu, C. Stock, J. S. Gardner, and P. M. Gehring,Phys. Rev. B 82, 134124 (2010).6C. Stock, L. V. Eijck, P. Fouquet, M. Maccarini, P. M. Gehring, G. Y. Xu,H. Luo, X. Zhao, J.-F. Li, and D. Viehland, Phys. Rev. B 81, 144127 (2010).7D. S. Fu, H. Taniguchi, M. Itoh, S.-Y. Koshihara, N. Yamamoto, and S.Mori, Phys. Rev. Lett. 103, 207601 (2009).8Y. P. Shi and A. K. Soh, Acta Mater. 59, 5574 (2011).9M. Vopsaroiu, J. Blackburn, M. G. Cain, and P. M. Weaver, Phys. Rev. B82, 024109 (2010).10I.-K. Jeong, T. W. Darling, J. K. Lee, Th. Proffen, R. H. Heffner, J. S.Park, K. S. Hong, W. Dmowski, and T. Egami, Phys. Rev. Lett. 94,147602 (2005).11F. M. Bai, J. F. Li, and D. Viehland, J. Appl. Phys. 97, 054103 (2005).12V. V. Shvartsman, J. Dec, T. Lukasiewicz, A. L. Kholkin, and W. Klee-mann, Ferroelectrics 373, 77 (2008).13G. Y. Xu, Z. Zhong, Y. Bing, Z.-G. Ye, and G. Shirane, Nature Mater. 5,134 (2006).14Y. P. Shi, A. K. Soh, and G. J. Weng, J. Appl. Phys. 109, 024102 (2011);Y. P. Shi and A. K. Soh, EPL 92, 57006 (2010).15A. Levstik, Z. Kutnjak, C. Filipicˇ, and R. Pirc, Phys. Rev. B 57, 11204(1998).16G. J. Weng, Mech. Mater. 42, 886 (2010).FIG. 3. (Color online) (a) Thermal-evolution of a; aII is determined fromthe best fitting of Eq. (6) for aExp (Ref. 6). (b) Plots of @(1/aI)/@T and @(1/aExp)/@T; the dashed line in (a) and solid curve in (b) is a visual guide.092908-3 Y. P. Shi and A. K. Soh Appl. Phys. Lett. 99, 092908 (2011)Downloaded 29 Aug 2012 to 147.8.230.79. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

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Effects of volume evolution of static and dynamic polar nanoregions on the dielectric behavior of relaxors

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