NMR and the spherical random bond–random field model of relaxor ferroelectrics

Journal of Physics and Chemistry of Solids 61 (2000) 177–183 www.elsevier.nl/locate/jpcs

NMR and the spherical random bond–random field model of relaxor ferroelectrics R. Blinc*, J. Dolinsˇek, A. Gregorovicˇ, B. Zalar, C. Filipicˇ, Z. Kutnjak, A. Levstik, R. Pirc Jozef Stefan Institute, P.O. Box 3000, 1001 Ljubljana, Slovenia Received 1 March 1999; accepted 21 April 1999

Abstract To determine the nature of the relaxor state and, in particular, to discriminate between a ferroelectric state broken up into nanodomains under the constraint of quenched random fields and a glassy state we have measured the temperature dependences of the nonlinear dielectric susceptibility in zero field as well as the temperature dependence of the local polarization distribution function and the Edwards–Anderson order parameter in PMN single crystal via 93Nb NMR. The experimental results can be quantitatively described by the newly proposed Spherical Random Bond–Random Field model of relaxor ferroelectrics. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Glasses; D. Dielectric properties; D. Ferroelectricity; D. Nuclear magnetic resonance; D. Phase transitions

1. Introduction In spite of intensive investigations, the nature of the diffuse phase transition [1] in relaxors has remained the subject of some controversy. For example, it has been suggested some time ago that relaxors could be described as a dipolar glass due to the reorientation of large superparaelectric clusters [2–6]. The existence of nanometer sized polar domains in lead magnesium niobate (PMN) has in fact been demonstrated by X-ray and neutron scattering experiments [7,8]. However, subsequent experiments have indicated that a dipolar glass should be ruled out and that one is rather dealing with a random-field induced freezing of ferroelectric domains on a nanometric length scale [9]. A more recently established electric field-temperature (E,T) phase diagram [10] shows that by cooling PMN in a d.c. electric field higher than EC , 1:7 kV=cm a diffuse phase transition into a long-range ferroelectric phase takes place. In contrast, in zero electric field no macroscopic symmetry change has been observed down to the lowest temperatures.

* Corresponding author. Tel.: 1386-61-177-3281; fax: 1386-61126-3269. E-mail address: [email protected] (R. Blinc)

The low temperature state, which is widely known as the relaxor phase, is characterized by a broad frequency dispersion in the complex dielectric constant, slowing dynamics, and logarithmic polarization decay [2,3,11]. At present it is not clear whether the relaxor glass phase in PMN and related systems in zero field is either (i) a ferroelectric state broken up into nanodomains under the constraint of quenched random electric fields, or (ii) a glass state of high pseudo-spin dimensionality with randomly interacting polar micro-regions in the presence of random fields. Here we discuss two sets of experiments which allow one to discriminate between the ferroelectric and glass behavior in zero field …E ˆ 0† as well as between different glass states. Specifically we will focus on (A) the temperature dependence of nonlinear anomaly a3 ˆ x3 =x41 in PMN, where x1 is the linear and x3 the third order nonlinear dielectric susceptibility, and (B) the temperature behavior of the Edwards–Anderson dipolar glass order parameter qEA and the local polarization distribution function W(p) obtained by 93 Nb NMR in PMN. These experiments will be analyzed in terms of the spherical random bond–random field (RBRF) model of relaxor ferroelectrics, which is an extension of the Ising RBRF of dipolar glasses [12], and is capable of reproducing the observed features.

0022-3697/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00279-6

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2. The spherical random bond–random field model The symmetry of PMN is pseudo-cubic and the Pb and Nb atoms are randomly disordered in a multi-site potential around their high symmetry perovskite lattice positions. The Pb displacements in particular according to Vakhrushev [13] are of the spherical shell type and thus quasi-continuous. The Nb/Mg atom shifts are smaller than the Pb ones but as well seem quasi-isotropic. The number of allowed sites is thus so large that the two-site pseudo-spin “Ising” model developed for proton dipolar glasses [12] or the three-, four- and six-site models developed for cyanide type quadrupolar glasses [14] are not appropriate. This is even more true if short range ordered clusters or nanodomains are present. We will therefore adopt the physical picture of interacting polar clusters [2–5,15] and assume that there are ni Nb-type unit cells in a cluster Ci, where i ˆ 1; 2; …; N and N is the total number of reorientable clusters. The main contribution to the dipole moment p~0 …i† of the l-th cell in Ci is due to the relative displacements of the Nb 51 and Pb 21 ions. The clus~ i ˆ ni p~0 …i†: ter dipole moment is thus P ~ i =p0 …i† as a dimensionless Introducing S~i ˆ …‰n2i Šav =3†21=2 P order parameter field, we find that its components satisfy the closure relation 3 X N X

mˆ1 iˆ1

S2im

ˆ 3N

…m ˆ x; y; z†;

…1†

P where ‰n2i Šav ˆ …1=N† i n2i : The Edwards–Anderson order parameter can then be defined as qEA m ˆ

N 1 X kS l2 ˆ ‰kSim l2 Šav : N iˆ1 im

…2†

In zero field, due to the average cubic symmetry qEA m will not depend on m and can be simply written as qEA. Considering the order parameter field S~i we note that physically the displacements of the Nb and Pb ions are restricted to a discrete but large set of directions. The number of allowed orientations of S~i is thus so high that we can consider S~i as a continuous vector 2∞ , Sim , ∞

…3†

subject only to the spherical constraint (1). This together with the existence of randomly competing ferroelectric (FE) and antiferroelectric (AFE) interactions leads to the SRBRF model of relaxor ferroelectrics [15] Hˆ2

X X 1X ~ ~ ~ S~i ; Jij Si ·Sj 2 h~i ·S~i 2 E· 2 ij i i

…4†

where Jij are random interactions and h~i local random fields. Further we have in analogy with dipolar glasses [12] 2

‰Jij2 Šcav ˆ

J ; N

‰Jij Šcav ˆ

J0 N

…5†

Fig. 1. Temperature dependence of the Edwards–Anderson glass order parameter for various values of the random field D~ according to the SRBRF model.

and ‰him ; hjn Šcav ˆ dij dmn J 2 D~

…6†

with D~ ; D=J 2 : We further assume in analogy with dipolar glasses that the randomly frustrated interactions are infinitely ranged. This last assumption is supported by the long range nature of the acoustic and optic phonon mediated strain-polarization inter-cluster interactions, which seem to play a dominant role in relaxors. The average free energy can be now calculated in a standard manner by applying the replica trick and imposing the spherical constraint (1) [16]. The equations for the polarization P ˆ Pm and the glass order parameter P ˆ Pm in a field Ei‰111Š are now obtained as q ˆ b2 J 2 …q 1 D~ †…1 2 q†2 1 P2

…7†

and P ˆ b…1 2 q†…J0 P 1 E†

…8†

where b ˆ 1=kT: The temperature dependence of the glass order parameter q in the case E ˆ 0 and J0 ˆ 0 for various values of the random field D~ is shown in Fig. 1. The average probability distribution of local polarization p~i ˆ kS~i l is defined as W…~p† ˆ

1 X d…p 2 p~i †: N i

…9†

~ and the The first moment of W…~p† is the polarization P second moment the glass order parameter q. For a glass phase P ˆ 0 but q is nonzero. For this spherical model

R. Blinc et al. / Journal of Physics and Chemistry of Solids 61 (2000) 177–183

Fig. 2. Local polarization distribution function W(p) for (i) a twosite dipolar glass, (ii) a two-site inhomogeneous ferroelectric, (iii) a three-site quadrupolar glass and (iv) a relaxor described by the SRBRF glass model. The parameters are: (i) J0 ˆ 0:2J; D~ ˆ 1; and T=J ˆ 10 (a), 3 (b), 1.5 (c), 0.85 (d); (ii) J0 ˆ 5J; D~ ˆ 1 and T=J ˆ 2 (a), 4.8 (b), 4.55 (c), 2.8 (d); (iii) J0 ˆ 0; D~ ˆ 0:1 and T=J ˆ 2 (a), 1.25 (b), 0.75 (c); (iv) J0 ˆ 0; D~ ˆ 1; and T=J ˆ 8 (a), 3 (b), 0 (c).

W…~p† is found to be Gaussian over the whole temperature interval. p For E ˆ 0 and J0 , J0c ; where J0c ˆ J 2 1 D2 ; i.e. for a spherical glass we find " # 1 1 p2 W…~p† ˆ exp 2 …10† ; p ˆ u~pi u: 2 q 2pq3=2 For J0 . J0c or E ± 0; i.e. for an inhomogeneous “spherical” ferroelectric, we have, on the other hand W…~p† ˆ W0 …px †W0 …py †W0 …pz †

…11†

Fig. 3. Phase diagram of a relaxor according to the SRBRF model.

a long range ordered ferroelectric state will appear for E . EC : The corresponding phase diagram for a spherical glass type relaxor ferroelectric is shown in Fig. 3.

3. The nonlinear susceptibility anomaly A straightforward way to discriminate between a ferroelectric state broken up into nanodomains under the constraint of quenched random fields or a (dipolar) glass state is to check the temperature dependence of the nonlinear dielectric anomaly. In a system with average cubic symmetry the relation between the electric field Em …m ˆ 1; 2; 3† and polarization Pm is given by [19,20] P1 ˆ x1 E1 2 x122 E1 …E22 1 E32 † 2 x111 E13 1 …

with " # 2 1 1 …pm 2 P† W0 …pm † ˆ p exp 2 ; 2 q 2 P2 2p…q 2 P2 †

…12†

E1 ˆ a1 P1 1 a122 P1 …P22 1 P23 † 1 a3 P31 1 … where

It should be noted that the above local polarization distribution function W(p) for a spherical glass is significantly different from the forms of W(p) previously obtained [17,18] for two-site dipolar glasses, two-site ferroelectrics or three-site quadrupolar glasses (Fig. 2). A determination of the form of W(p) in relaxors should thus allow one to discriminate between the different possible glassy or ferroelectric states in these systems. Assuming

a1 ˆ

g.0

…14†

or

m ˆ x; y; z:

J0 ˆ J0 …E† ˆ J00 1 gE2 ;

179

…13†

1 ; x1

a122 ˆ

…15†

x122 x and a3 ˆ 111 : x41 x41

According to the scaling theory one has for a ferroelectric in zero electric field 

g22b aFE : 3 / …T 2 TC †

…16†

For a system with cubic symmetry in d ˆ 3 spatial dimensionalities one has g 2 2b . 0 where the mean field value is g 2 2b ˆ 0: For a random field frustrated ferroelectric a3 thus decreases on approaching TC from above. For a dipolar

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Fig. 4. T-dependence of the third order nonlinear dielectric susceptibility x 3 for various values of the random field D~ :

glass, on the other hand, x 1 remains finite at Tf whereas

x3 / …T 2 Tf †2g3

…17†

with g3 ˆ 1: Thus the nonlinear coefficient 2g3 aDG 3 / …T 2 Tf †

…18†

should—in contrast to the ferroelectric case—increase with decreasing T on approaching Tf. For a relaxor described by the spherical RBRF model with D~ ± 0 one finds for E ! 0

bx21 …J0 x1 1 1† x3 ˆ ‰1 2 b2 J 2 …1 2 q†…1 2 3q 2 2D~ †Š‰1 2 bJ0 …1 2 q†Š …19† whereas the linear susceptibility is given by

x1 ˆ

b…1 2 q† : 1 2 bJ0 …1 2 q†

…20†

The temperature dependence of x 3 is shown in Fig. 4 for various values of D~ : For J0 , J0c and small D~ ; we have a significant peak around T ˆ J=k though x 3 remains finite at all temperatures. For a large D~ ; the peak in x 3 disappears. The corresponding expression for a3 for the glassy phase without long range order …J0 , J0c † is a3 ˆ

…1 2

q†2 ‰1

2

T : 2 q†…1 2 3q 2 2D~ †Š

b2 J 2 …1

…21†

For D~ ± 0 this quantity never diverges but has for D~ p 1 a sharp peak near T < J=k: For J0 . J0c ; corresponding to the case of an inhomogeneous random field smeared ferroelectric, a3 is not peaked but rather drops to a constant value for T ! TC : For T . TC and J ! 0 with J0 ± 0 and D ± 0; i.e. for the

Fig. 5. Temperature dependence of the quasi-static nonlinear dielectric susceptibility coefficient a3 ˆ x3 =x41 in PMN for E ˆ 0 and fit to the SRBRF model. The fit parameters are J=k ˆ 220 K and D~ ˆ 2 × 1024 : The T-dependence of a3 evaluated for E ± 0 is also shown.

pure case of a ferroelectric in a random field we have q 1 T . TC …22† …1 1 2b2 D 2 1 1 4b2 D†; q. ˆ 2 2b D and q, ˆ 1 2

T ; J0

T , TC

…23†

so that a3 ˆ

1 : …1 2 q †‰1 1 2b2 D…1 2 q†Š 2

…24†

Here the nonlinear dielectric susceptibility does not diverge but in contrast to the case of a glass drops on approaching TC. The quasi-static a3(T) measured in PMN [15,19,20] in zero d.c. electric field E ˆ 0 is shown in Fig. 5. It increases sharply with decreasing T between 320 and 220 K. This is incompatible with the case of a ferroelectric in a random field but quantitatively agrees with the predictions of the SRBRF model for J0 , J0c : In the presence of an electric field the behavior completely changes and the peak in a3 disappears (Fig. 5) in agreement with the SRBRF model predictions. 4. The 93Nb NMR spectra of PMN 4.1. Theory The quadrupole coupling of 93Nb …I ˆ 9=2† in perovskite crystals vanishes as long as the Nb nucleus is located in the cubic perovskite site in the center of the oxygen octahedron. The quadrupole coupling is however extremely sensitive to small displacements of the nuclei from the high symmetry

R. Blinc et al. / Journal of Physics and Chemistry of Solids 61 (2000) 177–183

and given by frequency distribution f …n† : 1 X 1 X f …n† ˆ d…n 2 nil †: N i ni l

181

…26†

If we expand relation (25) up to second order terms, we find:

ni ˆ n0;i 1 a~ ·~pi 1 p~i ·b·~pi 1 …

…27†

where the coefficients a~ and b depend on the orientation of the external magnetic field with respect to the crystal axes. The frequency distribution f …n† is now related to the local polarization distribution function W…~p† via: Z f …n† ˆ d3 pW…~p†d…n 2 n0 2 a~ ·~p 2 p~·b·~p†: …28† In the linear case, where ua~ u q ibi; we have for a given orientation

nil ˆ n…0† ~ ·~p ) v ˆ n0 1 ap il 1 a

Fig. 6. Comparison between experimental and theoretical SRBRF 93 Nb 12 ! 2 12 NMR lineshape in PMN for the “quadratic” orientation.

sites. In KNbO3, for instance, the quadrupole coupling constant e2 qQ=h is 25.8 MHz in the tetragonal 93Nb phase and 23.4 MHz in the orthorhombic phase, where it is of course zero in the cubic phase. NMR investigations of glasses and inhomogeneous ferroelectrics are based on the fact that in the fast motion limit the quadrupole perturbed NMR frequency of a given nucleus depends on the local value of the order parameter pim …m ˆ x; y; z† which measures the displacement of the nucleus from the high symmetry sites:

ni ˆ n…pi †:

…25†

It should be noted that in Pb(Mg1/3Nb2/3)O3 —abbreviated as PMN—the distribution of electric field gradient (EFG) tensors at the Nb-sites which leads to a distribution of NMR frequencies, is due to two different effects: 1. The static compositional (i.e. substitutional) disorder Nb $ Mg; 2. The positional disorder of a given kind of ions which are in the high temperature phase dynamically disordered between a large number of equilibrium positions around the high symmetry perovskite site in a multiple well potential. The random freeze out of this positional disorder is related to the formation of the relaxor phase. Because of the inhomogeneous local polarization distribution W(p) the NMR lineshape will be also inhomogeneous

and the NMR lineshape f …n† is using expression (10) for W(p) obtained as: " # 1 …n 2 n0 †2 f …n† ˆ p exp 2 : …29† 2qa2 2pqa2 The lineshape for the spherical model is thus for the linear case Gaussian and the width is determined by the glass order parameter q. By measuring f …n† we can determine W(p). The second moment of the frequency distribution f …n† Z 1 X 2 2 M2 ˆ f …n†…n 2 n0 †2 dn ˆ a p ˆ a2 q …30† N i;m m im is proportional to the Edwards–Anderson glass order parameter q [15]. In the quadratic case, ibi q ua~ u; the situation is different. Here we have, e.g. for a given orientation 1 0 1 0 0 C B C …31† b ˆ b0 B @0 0 0A 0

0 0

thus

nil ˆ n0il 1 p~·b·~p ) n ˆ n0 1 b0 p2x

…32†

and we find with the help of expression (10) for W(p) " # 1 1 …n 2 n0 †2 f …n† ˆ Q…n† p exp 2 ; 2 qb0 2pqb0 …n 2 n0 † …33†

b0 . 0; where Q…n† is a step function. Obviously f …n† has here a singularity at n 2 n0 ˆ 0; i.e. at p ˆ 0: f …n† decreases with increasing n 2 n0 : If b0 , 0; f …n† is mirrored around the vertical axis. The lineshape f …n† is now strongly asymmetric (Fig. 6) and the glass order parameter is related to the first moment M1 of f …n†:

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Fig. 7. Comparison between the theoretical SRBRF and the experimental 93Nb 12 ! 2 12 NMR lineshape in PMN for a “linear” orientation. The lineshape can be decomposed into two Gaussians.

Fig. 9. Local polarization distribution function W(p) derived for the 93 Nb 12 ! 2 12 NMR lineshape in PMN at two different temperatures. The agreement with the predictions of the SRBRF model is excellent.

4.2. Experiment

Fig. 8. T-dependence of the second moment of the “narrow” 93Nb 1 1 2 ! 2 2 NMR component of PMN. Also shown is the T-dependence of the Edwards–Anderson glass order parameter q. The solid line represents a fit to the SRBRF model with J=k ˆ 20 K and D~ ˆ 0:002:

Since the macroscopic cubic symmetry of PMN is locally broken, the NMR lineshape is anisotropic. By varying the orientation of the crystal in the magnetic field we can thus go from the “linear” to the “quadratic” case. The 93Nb 12 ! 2 12 NMR spectra of a PMN single crystal have been measured in a magnetic field of B ˆ 9:3 T corresponding to a Larmor frequency nL ˆ 92:9 MHz: Spin– lattice (T1) relaxation time measurements have shown that we are indeed in the fast motion limit and that expressions (27)–(33) apply. ~ The experimental The “quadratic” case is found for Bic: and theoretical lineshapes for this orientation are compared in Fig. 6. The predicted singularity at n 2 n0 (i.e. p ˆ 0) is clearly seen and the overall agreement between theory—Eq. (33)—and experiment is rather good. ~ The “linear” case is found for Bi‰111Š (Fig. 7). The two dimensional (2D) “separation of interactions” NMR spectra clearly show that the lineshape is indeed inhomogeneous and that we deal with a frequency distribution f …n†: The NMR lineshape can be decomposed into a “narrow” Gaussian component and a “broad” Gaussian component (Fig. 7). Both components are inhomogeneously broadened. The apparent width of the “broad” component is T-independent between 400 and 30 K, whereas the width of the

R. Blinc et al. / Journal of Physics and Chemistry of Solids 61 (2000) 177–183

“narrow” component strongly increases with decreasing temperature. The ratio between the intensities of these two components does not change in the investigated Tinterval. At 200 K the “inhomogeneous” linewidth of the “narrow” component is 16 kHz whereas the homogeneous one is only 720 Hz. The inhomogeneous width increases to 53 kHz at 130 K whereas the homogeneous one still equals < 700 Hz. We believe that the T-independent “broad” component is connected with unresolved satellite transitions whereas the T-dependent “narrow” component is due to the central 12 ! 2 12 NMR transition [21]. It is this last component which should reflect the motion of the “reorientable” polar clusters and indicate the “glassy” freeze-out. The T-dependence of the second moment of the “narrow” component and the corresponding T-dependence of the Edwards–Anderson glass order parameter are shown in Fig. 8. The break in the T-dependence of M2 and q around the freezing temperature is much more pronounced than in “dipolar” and “quadrupolar” glasses. Thus the random field contribution to the glass transition is here much weaker. The observed T-dependence of q can be fitted to the SRBRF model. The fit yields J=k ˆ 265 K and D~ ˆ 0:002 as well as J0 , J: This agrees relatively well with the values derived from the T-dependence of a3 if the difference in time scales is taken into account. It is important to note that not only the T-dependence of the second moment of the lineshape but also the lineshape itself and its T-dependence can be quantitatively described by the SRBRF model. The corresponding local polarization distribution function W…~p† is Gaussian at all temperatures (Fig. 9). This shows that the relaxor state in PMN can be described for E ˆ 0 as a “spherical” SRBRF glass and not as a dipolar or quadrupolar glass or as an inhomogeneous ferroelectric.

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