Compressible spherical dipolar glass model of relaxor ferroelectrics
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t Compressible spherical dipolar glass model of relaxor ferroelectrics
R. Pirc, ∗ Z. Kutnjak, and N. Novak
Joˇzef Stefan Institute, P.O. Box 3000, 1001 Ljubljana, Slovenia (Dated: October 15, 2018)The interactions between the dielectric polarization and the fluctuations of the strain (stress)tensor in relaxor ferroelectrics are shown to give rise to the anisotropy of the anharmonic P -termin the Landau-type free energy, however, the harmonic P -term is still properly described by therigid spherical random bond–random field model. These are the essential features of the compressiblespherical dipolar glass model, which is used to calculate the singularities of the specific heat nearfield-induced critical points. The results agree with recent high-resolution calorimetric experimentsin PMN [110]. PACS numbers: 77.80.Jk,77.84.-s,64.70.Q-,77.80.B-
I. INTRODUCTION
Relaxor ferroelectrics (relaxors) exhibit a variety ofphysical properties which are interesting for numerouspractical applications, such as tunable capacitors, ul-trasonic transducers, actuators, and pyroelectric detec-tors [1]. Sometimes relaxors are regarded as a subgroupof incipient ferroelectrics in view of the fact that theydo not possess a polarized long-range ordered phase inzero applied electric field. However, in contrast to nor-mal incipient ferroelectrics, relaxors undergo a freezingtransition into a nonergodic glass-like phase below theso-called freezing temperature. If the relaxor is slowlycooled in a nonzero electric field E , it will pass througha sequence of quasi-stationary states. Thus, in order tostay close to thermal equilibrium, the experimental scaleshould increase steadily as the temperature is lowered.The corresponding E - T phase diagram is shown in Fig. 1for the case of PbMg / Nb / O (PMN) in a field alongthe [110] direction [2]. The solid line in Fig. 1 separatesthe field-cooled dipolar glass phase from the field-inducedlong-range correlated ferroelectric phase. Similarly, thedotted line represents the boundary between the ergodicparaelectric phase and the frozen-in nonergodic dipolarglass phase. On approaching this line from the right,the longest dielectric relaxation time τ ( E, T ) divergesaccording to the Vogel-Fulcher law [3, 4], reflecting therandom character of the relaxor state.Dielectric experiments in PMN [111] [5, 6] have shownthat there is no frequency dispersion of the dielectric sus-ceptibility in the region above the solid line, indicatingthat the relaxation times are finite and the system is er-godic. The transitions across the solid line, indicatedby the arrows, are all first order and are characterizedby a jump in the polarization P = P ( E, T ). As onemoves towards higher temperatures, the size of the jump ∗ Electronic address: [email protected]
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NonergodicDG ErgodicDGFE CP
T (K) E ( k V / c m ) Widom lineT f FIG. 1: Phase diagram for a relaxor with b <
0. The solidline separates the dipolar glass (DG) phase from the field-induced ferroelectric (FE) phase. The arrows indicate thedirection of the field-cooling (-heating) process. Dashed line:Supercritical regime. Dotted line: Freezing line separating theergodic dipolar glass phase from the nonergodic one. Opencircles: Data from Ref. [2]. becomes smaller and finally disappears at a liquid-vapor-type critical point T CP , E CP , where the transition is sec-ond order [7]. Beyond this point, the relaxor is in a su-percritical state characterized by a smooth evolution of P ( E, T ) and of the field-dependent dielectric susceptibil-ity χ ( E, T ) = ( ∂P/∂E ) T . The dashed line marks thepositions of the maxima of χ ( E, T ) (Widom line).
II. POLARIZATION-STRESS COUPLING INLANDAU FREE ENERGY
It had been suggested earlier [8] that when dealing withquasi-equilibrium states as in Fig. 1, the relaxor can bedescribed in terms of a Landau theory based on the freeenergy density F = F + 12 a P + 14 b P + 16 c P + · · · − EP. (1)For simplicity we are dealing with a scalar order param-eter P = P ( E, T ), corresponding to the polarization vec-tor along one of the symmetry directions in the crystal,i.e., [100], [110], or [111] in a system with average cubicsymmetry. For an oblique direction of the electric field ~E the quartic term should be written as F = 14 b ijkl P i P j P k P l , (2)where the summation over all Cartesian indices is im-plied. Thus for a given symmetry direction, b will be afunction of b ijkl , which are components of a fourth ranktensor b .The first term F in Eq. (1) contains the contribu-tion of all other degrees of freedom such as electrons,phonons, etc. The coefficient a = a ( T ) is related to theinverse quasistatic field-cooled susceptibility χ , namely, a = ( ε χ ) − . The susceptibility χ can be calculatedfrom the static spherical random bond–random field (SR-BRF) model of relaxor ferroelectrics [9, 10] and has thegeneral form, χ = Θ(1 − q ) T − T (1 − q ) , (3)which is well known from the theory of spin glasses andwas found empirically to hold in the case of relaxors [11].Here, Θ is the Curie constant and T a measure of theaverage interaction between the elementary dipolar enti-ties in the system. In relaxors, these are known to be thepolar nanoregions (PNRs), which are formed below theBurns temperature [12]. Finally, q = q ( T ) is the dipolarglass order parameter, which is nonzero at all temper-atures due to the presence of quenched random electricfields [13]. In zero applied field, the order parameter q isdetermined by the real solution of the following algebraicequation [9, 10]: q = ( J/kT ) ( q + ∆ /J )(1 − q ) . (4)The parameter J is defined in terms of the variance J /N of the infinitely ranged random interactions of a spin-glass type, and ∆ the variance of local random fields.For PMN, the estimated values are J/k ∼
217 K and∆ /J = 0 . T in Eq. (3) is of the order kT ≡ J ∼ . J [14].The parameters b, c, ... in Eq. (1) are related to thenonlinear susceptibilities χ , χ ,..., which are defined asusual by the expansion P/ε = P s + χ E + χ E + χ P + · · · . In relaxors, the spontaneous polarization vanishes,thus by definition P s ≡
0, and the Landau coefficient b isgiven by b = − χ / ( ε χ ). It should be emphasized thatin general both b and χ , as well as higher order Landaucoefficients in Eq. (1), depend on the direction of the field ~E due to the anisotropy term (2).The SRBRF model was originally introduced for anideal isotropic relaxor system [9] in a rigid environment.Thus the nonlinear susceptibility χ derived from it is independent of the orientation and χ rigid <
0. Conse-quently, b rigid >
0, i.e., b rigid is a positive scalar. Exper-iments on various relaxors have shown that χ can eitherbe positive or negative, depending on the particular sys-tem studied and on the field orientation [6, 15].In order to derive a more general version of the modelcapable of reproducing the observed anisotropy of thecoefficient b , we introduce a coupling between the polar-ization P and the strain tensor u ij (or the internal stresstensor X ij ). This suggests that we should consider thestress dependence of the Landau coefficients in Eq. (1).Focusing on the P -term, we first introduce a generalizedLandau coefficient a kl = ε − ( χ − ) kl . Next, by expanding a ( X ij , T ) kl to linear order in X ij , we replace the P -termin F by12 ε " χ − δ kl + (cid:18) ∂ ( χ − ) kl ∂X ij (cid:19) E,T X ij + · · · P k P l . (5)The partial derivative is related to the electrostrictiontensor Q ijkl , namely, [16] Q ijkl = 12 ǫ (cid:18) ∂ ( χ − ) kl ∂X ij (cid:19) E,T . (6)By adding the elastic energy, the free energy F acquiresan additional term, which can be written as F X = X · Q · P + 12 X · C − · X . (7)Here, C is the elastic constant tensor and ( P ) ij = P i P j .Minimizing F X with respect to X ij at constant tem-perature and field, we formally recover the free energy(1), however, the quartic term is now replaced by thegeneral expression (2) with b ijkl = b rigid δ ij δ kl + B ijkl , (8)where the fourth rank tensor B is given by B = − Q · C · Q . (9)In relaxors, the electrostriction effect is usually large andthe magnitude of the tensor components B ijkl may ex-ceed the value of b rigid . Obviously, the sign of B ijkl willin general depend on the balance between the individualcomponents of Q ijkl and C ijkl . Thus the resulting valueof b for a symmetry direction can either be positive ornegative.The generalized Landau free energy (1) with a ( T )given by the SRBRF model and the P -term havingthe form (2), and with quartic coefficients (8) given byEq. (8), will be referred to as the Compressible SphericalDipolar Glass (CSDG) Model.We can evaluate the coefficients B ijkl for the caseswhere values of Q ijkl and C ijkl are explicitly known.In Table I, these are listed for the PMN crystal us-ing the Voigt notation, i.e., B = B , etc. The TABLE I: Values of C ij and Q ij used to calculate B ij from Eq. (9) and B [ p ] from Eq. (10). C a,b C a C a,b [GPa] Q c Q -0.96 c Q d [10 − m C ] B B B -1.654 [10 Vm C ] B [100] B [110] -0.84 B [111] -1.836 [10 Vm C ] a Reference [18]; b Reference [19]; c References [16, 17]; d estimated [17] value of Q can be estimated from the relation [17] Q = ( Q − Q ) /
2. From Eq. (9) we can then cal-culate the coefficients B , B , and B (see Table I).For a symmetry direction p , where p refers to [100], [110],or [111], the Landau coefficient b = b [ p ] in Eq. (1) can beexpressed in terms of b ij as follows: b [100] = b ; (10a) b [110] = ( b + b + 2 b ); (10b) b [111] = ( b + 2 b + 4 b ) . (10c)We can now write b [ p ] = b [ p ] rigid + B [ p ] , where b rigid > b [ p ] , however, the new term B [ p ] is ingeneral anisotropic. The corresponding values of B [ p ] forPMN are listed in Table I. While B [100] >
0, we can seethat B [110] and B [111] are both negative. Thus, b [100] > χ [100]3 <
0. On the other hand, if | B [ p ] | > b rigid for p = [110] and [111], the values of b [110] and b [111] will benegative, implying χ [110]3 > χ [111]3 >
0, respectively.It is interesting to note that Tagantsev and Glazounov[15] observed χ [111]3 >
0, but χ [100]3 < b [ p ] has important consequences for theexistence of field-induced critical points for fields alongthe direction [ p ]. III. FIELD INDUCED CRITICAL POINTS
The temperature and field dependence of the dielec-tric polarization during a field-cooled (or field-heated)quasi-stationary process is calculated by minimizing nu-merically the free energy (1). This procedure automallyselects the correct solution of the minimization condition( ∂F/∂P ) E,T = 0. The parameter a = a ( E, T ) is cal-culated from the SRBRF model [8], while b and c aretreated as free parameters. To ensure stability, we as-sume that c > b > b < b > P ( E, T ) is found to increase monotonicallywith E and decrease with T , and no critical singularitiesof the susceptibility can be expected. For b <
0, how-ever, P ( E ) makes a discontinuos jump at some value of E at low temperatures. As T increases, the jump be-comes smaller and finally disappears at the critical point P / P E/E CP T/T CP =10.8 0.9 0.95 1.1 FIG. 2: Field dependence of P ( E ) for a relaxor with b < T close to the critical tempera-ture T CP , obtained by minimizing the free energy (1). Notethat these calculations are only valid in the ergodic regionabove the freezing line shown in Fig. 1. E CP , T CP , where the slope of P ( E ) is infinite. This isillustrated in Fig. 2 for b = − . c = 0 .
08, corre-sponding to PMN [110]. A similar behavior of P ( T ) hadbeen obtained earlier for PMN [111], where b = − / c = | b | [8].The coordinates of the critical point are determinedfrom the relations [8, 20] a ( T CP ) = 9 b c ; E CP = 6 b c P CP , (11)where P CP = p − b/ (10 c ) is the polarization at the crit-ical point. The critical exponents at the field-inducedcritical point differ from the usual mean field exponentsfor ferroelectrics in zero field [8].The E - T phase diagram for PMN corresponding tocooling in a field along the [110] direction is plotted inFig. 1 using the data points from Ref. [2]. Similar phasediagrams were obtained earlier for PMN [100] and [111][6, 21]. For E k [100] no critical point was found, but thephase diagram for the [111] direction was shown to beanalogous to the [110] case, in agreement with the aboveestimates for b [ p ] . A similar conclusion had been reachedearlier by Zhao et al. [22].The existence of field-induced phase transitions andcritical points has recently been confirmed in PMN [110] T/T CP CP =10.750.50.25 ( E , T ) ( a r b . un it s ) (a) (b) C s i ng E ( a r b . un it s ) T/T CP CP =1 1.251.51.752 FIG. 3: (a) Calculated temperature dependence of the sucep-tibility χ ( E, T ) for a set of field values
E/E CP , as indicated.(b) Same, but for the singular part of the specific heat fromEq. (16). by measuring the specific heat using high-resolutioncalorimetry [2]. The excess specific heat ∆ C E ( T ), whichis due to the contribution of the dipolar degrees of free-dom, namely, PNRs can be derived from the free energyEq. (1) by applying the thermodynamic relation for theentropy S = − ( ∂F/∂T ) E . We can write S = S + S dip ,where S = − ( ∂F /∂T ) E and the dipolar part S dip isdefined as the contribution of all P -dependent terms inEq. (1), S dip = − (cid:18) a P + 14 b P + 16 c P + · · · (cid:19) , (12)where a ≡ da/dT , b ≡ db/dT , etc., and the condition( ∂F/∂P ) E = 0 has been applied [8]. The dipolar excessspecific heat capacity at constant field is given by ∆ C E = T ( ∂S dip /∂T ) E , and at constant polarization similarly by∆ C P = T ( ∂S dip /∂T ) P . These two quantities are relatedby the standard thermodynamic relation∆ C E = ∆ C P + T χ ( E, T )[( ∂E/∂T ) P ] , (13)where χ ( E, T ) = ( ∂P ( E, T ) /∂E ) T is the field-dependentsusceptibility. The partial derivative in Eq. (13) can beevaluated from the equation of state E = aP + bP + cP + · · · , i.e.,( ∂E/∂T ) P = a P + b P + c P + · · · . (14)To calculate the expression on the right hand side wewould, therefore, need to know the temperature depen-dence of the coefficients a, b, c, etc. In practice, a ( T )
220 240 260 2800.0000.0010.0020.0030.004 Experiment C E ( J / g K ) T (K)(a)
220 240 260 2800102030
T (K)
Theory (b) C s i ng E ( a r b . un it s ) FIG. 4: (a) Experimental data showing the specific heatanomalies occurring in PMN [110] for four selected values ofthe electric field, obtained by high-resolution calorimetry [2].(b) Calculated temperature dependence of the singular part ofthe excess specific heat C singE for a relaxor with b = − . c = 0 .
08 and the same field values as used in the experiment.The remaining parameters are taken from Ref. [8]. is known from the SRBRF model through the relation a = ( ε χ ) − , and a ( T ) follows from the temperaturederivative of χ ( T ). On the other hand, b ( T ) and c ( T )could, in principle, be estimated from the nonlinear sus-ceptibilities as given by the SRBRF model and the Q and C tensors. In the following we will simply assumethat b and c are effectively constant in the temperaturerange of interest, and thus b and c will be neglected.The calculated temperature dependence of χ ( E, T ) isshown in Fig. 3a for a set of values 0 ≤ E ≤ E CP . Herewe used the parameter values of J , J , and ∆ determinedearlier from the dielectric data [14], and the remainingparameters were chosen as b = − . c = 0 .
08. At E = 0, the zero-field cooled susceptibility χ is recov-ered. For 0 < E ≤ E CP , χ ( E, T ) exhibits a jump atthe first order transitions and diverges at the criticalpoint. For
E > E CP , however, χ ( E, T ) is character-ized by rounded maxima, in accordance with the smoothbehavior of P ( E, T ) in the supercritical regime.The first term in Eq. (13) is readily shown to be∆ C P = − T (cid:18) a P + 14 b P + 14 c P + · · · (cid:19) , (15)where a ≡ da /dT etc. Again, the corresponding deriva-tives of b, c, ... are not known, and we will neglect them.The quantity a can, however, be calculated from a ( T ).It shows a sharp peak around the static ”freezing” tem-perature T f = ( J + ∆) / /k , but is rather small else-where. We may conclude that ∆ C P will be nonsingularfor all values of E, T , however, its precise behavior couldonly be determined if the values of b ( T ) and c ( T ) wereknown.The temperature dependence of the singular part of∆ C E ( T ) can be calculated from Eqs. (13) and (14),namely, ∆ C singE ∼ = T χ ( E, T )( a P ) , (16)and is displayed in Fig. 3b for the same set of parame-ters as in Fig. 3a. The positions of the singularities of∆ C singE ( T ) coincide with those of χ ( E, T ), however, thedirection of the jumps is reversed due to the last factorin Eq. (16).The experimental data for ∆ C E ( T ) in PMN [110] ob-tained by high-resolution calorimetry [2] are shown inFig. 4a for four discrete values of the electric field E . Forcomparison, the theoretical prediction for the singularpart of ∆ C E ( T ), calculated from Eq. (16) at the samefield values, is plotted in Fig. 4b. The experimental val-ues for the critical field and the critical temperature aregiven by E CP ∼ = 8 kV/cm and T CP ∼ = 240 K, respectively[2].The predicted behavior of ∆ C singE ( T ) qualitativelyagrees with the experimental values of ∆ C E ( T ), how-ever, the experimental anomalies appear to be broader than the calculated ones. There are several reasons forthis broadening, for example, finite size effects, structuralinhomogeneities, and slow relaxation.It should be stressed that at zero field ( E = 0), noanomalies in ∆ C E ( T ) were found in the entire tempera-ture range studied, in accordance with the CSDG model.This contrasts with the so-called random field scenario,according to which PMN [110] is assumed to undergo aferroelectric phase transition at E = 0. This point hasbeen discussed in more detail in Ref. [2]. IV. CONCLUSIONS
We have shown that the electrostrictive coupling be-tween the dielectric polarization P and the strain orstress tensor fluctuations in a relaxor ferroelectric givesrise to an anisotropy of the P -term in the Landau freeenergy. For a given symmetry direction of the appliedfield the effective Landau coefficient b may become nega-tive, thus leading to the field-induced critical points. Thecompressible spherical dipolar glass model predicts singu-larities of the dipolar specific heat near critical points, inagreement with high-resolution calorimetry experimentsin PMN [110] [2]. Acknowledgment.
This work was supported by theSlovenian Research Agency through Grants P1-0044, P1-0125, J1-0155, and J1-2015. [1] K. Uchino,
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