PHYSICAL REVIEW B VOLUME 46, NUMBER 9 1 SEPTEMBER 1992-I

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Magnetic-field-induced kinetics of ferroelectric phase transitions

A. Gordon* and P. WyderMax Plane-k Institut fur Festkorperforschung, Hochfeld Magne-tlabor, Borte Postal 166X, F 38042, -Grenoble CEDEX, France

(Received 24 April 1992)

The dynamics of ferroelectric interphase boundaries is studied in external magnetic fields. On thebasis of the exact solution of the time-dependent Ginzburg-Landau equation the velocity of the inter-phase boundary and its width are calculated as functions of the applied magnetic field.

In recent years the kinetics of first-order phase transi-tions has been studied on the basis of the time-dependentGinzburg-Landau model. ' In particular, the experi-mental' and theoretical work ' ' for ferroelectric phasetransitions showed that the thermo-induced motion ofthe interphase boundary between paraelectric and fer-roelectric phases may be realized as a motion of a solitarywave or a kink. Although the temperature-dependent dy-namics of the interphase boundaries at ferroelectric phasetransitions has been extensively investigated, their field-induced motion has not been considered so far.Meanwhile, magnetic field effects in ferroelectrics are wellknown: the phase transition temperature is shifted by theexternal magnetic field due to the magnetoelectriceffect. All the experiments on the influence of exter-nal magnetic fields were carried out in materials undergo-ing first-order phase transitions, in which the interphaseboundary between the paraelectric and ferroelectricphases could exist. Nevertheless, neither the staticfeatures of the first-order phase transitions, nor their ki-netics under the influence of magnetic fields have beenstudied.

In this paper we develop a theory of the influence ofapplied magnetic fields on the motion of the ferroelectricinterphase boundary.

In the presence of an external magnetic field H the freeenergy density f can be described by

f i ap2 & bp4~ I cp6+dX2+eXP2+ i gp2H2

+ ,'hP H +iXH +D—(VP)

where P is the polarization, coefficient a=a(T), con-stants b, c, d, e, g, h, and i are positive, X is the strain,and D is the positive coefficient at the inhomogeneityterm. The microscopic analysis for perovskites givesthat the direct coupling of the magnetic field to the polar-ization dominates the influence of magnetostriction. Themagnetostriction term is therefore negligible. We pay at-tention to the case of the perovskites for two reasons.

(l) The thermal dynamics of the ferroelectric inter-

phase boundaries was experimentally studied in BaTi03and PbTi03. "

(2) The data on the external magnetic influence on thephase transition temperature are available for these sub-stances. '

The strain effect on the dynamics of the ferroelectricphase boundary can be taken into account by renormal-ization of the coefficient a. ' We restrict ourselves withterms H and H since the experiments ' show that mag-netic field-induced shift of the ferroelectric phase transi-tion temperature is determined by terms with H and H .

We suppose here that the interphase boundary dynam-ics is governed by the time evolution of the order parame-ter, the polarization. The time-dependent Ginzburg-Landau equation for the evolution of the polarization isused here:

ap 5FBt 5P ' (2)

where I is the Landau-Khalatnikov transport coefficient.5F /5P is the variational derivative, tending to restore thevalue of P to its thermal equilibrium value. The model as-sumes that the relaxation of the system to equilibriumprovides the propagation rate of the interphase boundary.Usually the kinetics of relaxation of the order parameteris only considered if the conduction of heat is sufficientlyfast. It implies that the latent heat of the phase transitionis conducted very fast to the external medium or is quick-ly absorbed by the crystal maintaining isothermal condi-tions. Then the temperature may be treated as a constantand so no kinetic equation for it is necessary. We are notdealing with substances which conduct heat well. How-ever, the ferroelectric first-order phase transitions areclose to second-order ones. For this reason their latentheat is very small. Then the interphase boundary is not aheat source during its motion and we can neglect thethermal gradient at its surface.

Substituting (2) into (l) we obtain the following equa-tion for the uniaxial case:

46 5777 1992 The American Physical Society

A. GORDON AND P. WYDER

P= Po

[1+exp( —s/b, ) ]'/2 (5)

+I (aP b—P +cP.+gPH +hPH ) 2—I D =0 .at Bx

(3)

Substituting s =x Ut—into (3) we have an ordinarydi6'erential equation in the variable s instead of the par-tial differential equation (3) in the independent variables xand t:

2I D +U —I (.aP —bP +eP +gPHdP dP 3 5 2

ds2 ds

+hPH )=0. (4)

Equation (4) was sr)lved and studied in detail in Refs. 2,3, and 17. The solution of (4) for the boundary condi-tions of an interphase, boundary type has the kink form

0.74

0.72—

0.70—

0.68—

0.660

I I

5 10 15 20 25H (Tj

30

FIG. 1. The magnetic-field dependence of the interphaseboundary width for BaTi03. The width is given in units of(6Dc)' /b. The strength of the applied magnetic field is

presented in tesla (T).

where Po is the equilibrium value of polarization

Po = ')/(b /2c) I 1+[1—(4Ac /b ~) ]'/2J

and 5 is the width of the interphase boundary' 1/2

3D

2(bPO —A )

which moves with th(: velocity v

(4A —bPO)U = I (2D/3)'

"i/ bP —A

(6)

(7)

5=(T To)/(T—, —To) (13)

is a dimensionless temperature. The solution (8) was also

derived for the nonlinear lattices with dissipation. '

Consider the case of the magnetic-Geld influence. Tak-

ing into account (7)—(9) and (11)—(13) we obtain the fol-

lowing expressions for the interphase boundary width 5and its velocity v as functions of the applied magneticfield H at T=T,(5=1):

b, = [(6Dc)' /b ][[—,'(5 —B)+(1—B)'/ ]]'/

andHere

A =a+gH +h3V (9)

[1+B—(1—B) ]U = I"b(D/6c)'

[—,'(5 —B)+(1—BB 1/2 1/2(15)

The solution (5) is stable under small perturbations.In the absence of tk[e external magnetic field the inter-phase boundary exists at the temperature rangeTo& T(T*, where To and T* are limits of stability ofparaelectric and ferroelectric phases correspondinglygiven by

T =To+(b /4a'c), (10

where a =a'( T —To ); T, is the phase transition tempera-ture. Equation (5) describes the propagation of the inter-phase boundary leading to the phase transition. We canrewrite (7) and (8) as

where

B=(16c/b )(gH'+ hH )

=(16a'c/b )(aH +PH ),

0.6

0.5—

0.4—

(16)

=3 D4 a '( T, —To)[1—(35/8)+ &1—(35/4) ]

' 1/2 0.3—

0.2—

0.1—

and

v =2I [Da'(T, —To)]'

00 10 15 20

H (T)25 30

where

[5—(2/3)[l+(1 —35/4)' ]][1—(35/8)+[1 —(35/4)' ']j' ' (12)

FIG. 2. The magnetic-field dependence of the interphase

boundary velocity U for BaTi03 ~ The velocity is presented in

units of 1 b(D/3c)' . The strength of the applied magnetic

field is presented in tesla (T).

MAGNETIC-FIELD-INDUCED KINETICS OF FERROELECTRIC. . . 5779

where a and p are coefficients measured in Ref. 7 fromthe magnetic field-induced shift of the phase transitiontemperature. We assume here that coefficients D and Ido not depend on the magnetic field strength. The argu-ments are the following ones. According to Ref. 19,D ~ l, where l is the lattice parameter. Since the latticeparameter l does not depend on the strength of the ap-plied magnetic field, D is independent of H. The trans-port coefficient I does not include the dependence of thephase transition temperature. For this reason we as-sume that the magnetic-field dependence of I is negligi-ble. We see from (14) that the width of the interphaseboundary increases when the magnetic field increases.The dependence of the width 6 on the magnetic fieldstrength H is shown in Fig. 1. In Fig. 2 the interphaseboundary velocity v is presented as a function of theexternal magnetic field H. Figures 1 and 2 are plottedwith the help of (14) and (15) for BaTi03. a'=6. 7X10mks b =9.7X1Q mks c =3.9X1P mksa=6.27X10 KT, P=6.28X10 KT . The

magnetic field strength is measured in tesla (T). In a widerange of applied magnetic fields 8 (& 1 and 5 and v are

and

b, =(2/b)(2Dc/3)'i [1+(B/6)]

u =(I b/3)(D/6c)' 8 [I+(8/6)] .

(17)

(18)

It is seen from Figs. 1 and 2 that the maximal increaseof the interphase boundary width for BaTi03 is about10%, while there is a very considerable effect of the mag-netic field on its velocity. Since the movement of the in-

terphase boundary is related to the growth of ferroelec-tric crystals, the magnetic-field effect can be used togovern the growth processes.

The support of the Minerva foundation is acknowl-edged. This research was supported by the German-Israel Foundation for Scientific Research and Develop-ment, Grant No. G-112-279.7/88.

'On sabbatical leave from the Department of Mathematics andPhysics, Oranim, Haifa University, 36910Tivon, Israel.

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