interface motion in ferroelectrics

Physica 138B (1986) 239-243 North-Holland, Amsterdam

INTERFACE MOTION IN FERROELECTRICS

A. G O R D O N Department of Physics, Technion- Israel Institute of Technology, Haifa 32000, Israel and Department of Physics and Mathematics, Oranim-School of Education of the Kibbutz Movement, Haifa University, Tivon 36910, Israel

Received 31 July 1985

The phase plane method is used for solution of the time-dependent Ginzburg-Landau equation. It is shown that the obtained kink solution is a moving interface. The stability of the kink solution under small perturbations is proved. The estimate of the interface velocity at ferroelectric phase transitions in BaTiO3 and PbTiO3 crystals shows a satisfactory agreement with experiments.

1. Introduction

Kinetics of first order phase transitions was investigated on a basis of the time-dependent Ginzburg-Landau equation [1-3]. It has been shown that the interface motion may be presented as a soliton or kink motion. The results obtained in [1, 2] may be applicable for liquids, liquid crystals and "divertible ferroelectrics" [4, 5], in which the free energy expansion in an order parameter has a cubic term. However, for the majority of ferroelectrics results of [1, 2] are not applicable. The kink solutions presented in [1, 2] have been obtained by Montroll [6] using the direct substitution.

We use here the phase plane method [7] to solve the time-dependent Ginzburg-Landau equation for kinetics in reversible ferroelectrics. The kink solution to the time-dependent Ginz- burg-Landau equation describes the interface between the paraelectric and ferroelectric phases in uniaxial ferroelectrics. We investigate stability of this kink solution under small perturbations thus complementing the results [1-3]. Com- parison with experiments is carried out for ferroelectric BaTiO 3 and PbTiO 3 crystals.

2. Interface as a kink solution of the time-dependent Ginzburg-Landau equation

The time-dependent Ginzburg-Landau equa-

tion for evolution of the order parameter 77 is given by [8, 9]

o+1 6 F - r (1 )

a t 8 ~ '

where F is the Landau-Khalatnikov damping coefficient, which is assumed to depend non- critically on a temperature [9], F is the free energy, which is given by the first-order phase transition by [10, 11]

F = F o + l 2 1 4 1 6 (t917) 2 ~AT/ -71B17 +gCT/ + D \-~-x/ ' (2)

where A, B, C > 0 and D is the coefficient of the inhomogeneity term [10], 8F/&7 is the variational derivative of F in respect with ~7 [11, 12]. The expression (2) describes a first-order phase transition in uniaxial proper ferroelectrics. In the vicinity of the coexistence line the free energy F(rl) has three minima. One of them cor- responds to the paraelectric phase and the others correspond to domains of the ferroelectric phase. At the coexistence line all the minima have the same depth. As a result of a temperature change, the minima corresponding to the metastable phase become higher than those of the stable phase.

After the substitution of (2) into (1) we obtain

O__ff_~ + F ( A ~ - BTI 3 + Cr/s) - 2 F D 02r/= 0. (3) Ot Ox 2

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240 A. Gordon I Interface motion in ferroelectrics

The original partial differential equation in the independent variables of coordinate x and time t (3) can be reduced to ordinary differential equation in the variable s by rewriting x - vt, where v is a velocity in the direction x. After this transformation in (3) we obtain

dn 2 F D d2n + v - - - F ( A n - B n 3 + Cn 5) = 0. (4) ds z ds

The solutions, which we seek, are stationary in the moving coordinate system and they are of wave front type. We assume that these solutions satisfy the following boundary conditions; dn /ds --> O, where s ---> ---~ and r / ~ nl for s --> +0% n --> 0 for s --> -o% where 71 and 0 are minima of the free energy as a function of n-

We can present eq. (4) as a system of the following equations:

dn ds Y' (5a)

2 F D dy+ vy- (AT - Bn 3 + Cr/5) = 0 . ds

(Sb)

Using eqs. (5a) and (5b) we obtain a differential equation in the (y, n) phase plane:

dy v 1 + - - dn 2 F D 2 D y

- - (AT - Bn3+ C17s) = O. (6)

The functions y(s ) and n(s), corresponding to a solution of (4), give a trajectory in the phase plane. The trajectory has the slope

dy v Cn(n z - n~ ) (n : - n~) . . . . + , (7) dn 2 F D 2Dy

where

4 n~ ~ 1 + 1 - B2 ] , (8)

2_ B ~ 1 4 A C ~ n2 - ~ (1 - B2 j (9)

are extrema of the free energy F.

Using the phase plane method [7] we obtain that eq. (7) can be satisfied by a trajectory of the form

y =/3n (n 2 (10) -

where fl is a constant which is determined upon substitution of (10) into (7). From (5a) and (10) we have, after the integration, the following kink solution:

= n~ (11) n [ l+ex p ( - s / z l ) ] m '

where

1 / 3D X / 2 "

(12) Brim - A

The kink solution (11) describes the interface between the ferroelectric phase (7/# 0) and the paraelectric phase (n = 0), where A is the interface width. This solution has been also obtained in our paper [3]. There it has been checked by direct substitution that eq. (4) has the solution (11).

After the substitution (10) into (7) we obtain the following expression for the interface velocity

v \ X / B n ~ - A / " ( 1 3 )

where

A = A ' ( T - To), (14)

where T O is the stability temperature of the paraelectric phase, which is connected with the phase transition temperature T c by the equation (see, e.g. [5]):

3 B 2 T~= To4 16 A ' C " (15)

The interface exists in the temperature interval T * < T < T O , where T* is the stability tem-

A. Gordon / Interface motion in [erroelectrics 241

perature of the ferroelectric phase determined by the equation (see, e.g., [5]):

B 2

T * = To+ 4 A ' C " (16)

The temperature dependences of the interface width and velocity may be written in the form

A = ~DV2[2A'(T~ - To)(1 - 36 + ~/1 _3:~6)]-1/2 ,

3(1 + , # 1 - -6)1 v = F [ 2 D A ' ( T ~ - To) ] 1/2 [ 6 - 2 ( 1 - 3 6 + ~ / 1 - ~ 6 ) 3 1/2'

where

T - T o 6 - - - -

L-To

(17)

(18)

(19)

It is seen that at the phase transition temperature T c, 6 = 1 and the interface velocity is equal to zero. The interface does not move, because of the equality of free energies of coexisting phases. The sign of the interface velocity depends on the direction in which the interface propagates thus leading to formation of the paraelectric or ferroelectric phase.

3. Stability analysis of the kink solution

We can write 7 ' as the infinite sum

ao

7' = ~] e x p ( - ~iT)71i(s), (23) i=1

where n~ are eigenfunctions of (22). They are also solutions of the following equation:

dEn, dn, [ (d2F2) ] 2 F D ~ + v ds + ~r~ - F ~=~ "0~ = 0

(24)

with corresponding eigenvalues cry. We define new eigenfunctions

ni = exp(-4--~D s) f/. (25)

Then we obtain the following equation for f/:

ds -----~ 2 F D 16F2D 2 2--D \dn2][ n=no fi = O.

(26)

Eq. (26) has the form of a Schr6dinger's equation with an energy equal to ~ i / 2 F D and a potential energy

v 1 (d2F l 16---f2-~ + ~-/~ \ d.q2 / i •

In order to prove the stability of the kink solution (11) under small perturbations, we transform eq. (3) by following transformations:

x ~ s = x - v t and t--* ~'. (20)

We also let

n(s, r) = no(S)+ n'(s, (21)

where ]n'(s, ~')] "~ In0(s)l and n0(s) is the solution (11). As a result of the transformation we obtain

, r(a F l 0 n °71' °Tl' n ' 2 F D --~-52 + v Os 0~" \ C917 2 ] 7/='0o

= O. (22)

For stability of the solution (11) a perturbation must not grow with time. A necessary condition for stability is therefore that none of the tr i in (23) can be negative. A sufficient condition for stability is that all the values o- i be positive. Since the values o'i form a sequence bounded from below [13] it is necessary only to evaluate the ground state eigenvalue ~ r If T0(s) is a solution of (4), the ground state eigenvalue tr 1 is equal to zero. Indeed, we suppose that o- 1 - -0 in (26). Then we obtain*

2 F D d27/-----~1 d n l _ F (d2F~[ d s 2 + t~ ds \ ~ 2 ] 1 n l = 0. (27)

~/= ~0

* 71 is here not equal to ~h given by (8).

242 A. Gordon / Interface motion in ferroelectrics

After the differentiation of (4) with respect to s we have

d (dr/0 d (dr/0 2FD ~ +

\ d s / V ds \ d s /

_r(d2F l (dn03 \dr/2]l .=no \ ds / = 0 . (28)

Thus dr/o/ds is an eigenfunction of (22) with a zero eigenvalue. It is seen f rom (11) that the derivative dr/rids has no zero crossings (nodeless function). Thus it is the ground state eigenfunction in a Schr6dinger 's equation and

dr/° and 81 = 0 . (29) '171 = ds

All the eigenvalues are therefore positive. Con- sequently, all solutions (23) of the linearized equation decay exponentially with time and the kink solution (11) is stable.

4. Comparison with experiments

It is clear that the nonlinearity in kinetics of first order phase transitions appearing as the kink solution (11) arises from the nonlinearity in the free energy expansion leading to a phase transition.

Above results are suitable to uniaxial ferroelectrics. However , known measurements of the interface motion were carried out in BaTiO 3 and PbTiO 3 crystals [14, 151, which do not belong to this category of ferroelectrics. Thus these calculations serve as an approximate estimate, not pretending to exact description of the phenomenon. We also do not take into account the dependence of the interface velocity on the crystal thickness, the cooling rate and the thermal gradient in samples.

The first qualitative evidence in favour of this theory is the observable interface motion which occurs with conservation of the interface profile. In ferroelectric crystals KTa0.65Nb0.asO 3 [16], BaTiO 3 [17] and PbTiO 3 [18] the phase transition

proceeds by movement of a well-defined planar phase boundary. Consequently, the interface motion can be presented as a soliton motion or a wave of permanent profile given by (11).

To compare the theoretical results with experiments on the measurement of the interface velocity in the ferroelectric BaTiO 3 [14], we estimate the coefficient D in (2) according to [10]: D = 1r/15.d2; D = 3.35x 1 0 - 6 c m 2 for the lattice paramete r d = 4 x 10-8cm [19]. We take A ' = 6.66 × 10 -6 ( l /K) [201, T~- T O = 15 K [20]. The damping coefficient F = 0.6 x l0 s s -1 at T = T¢ + 2 K [21]. We suppose that the slow thermal motion of the interface contributes to the central peak width [22]. The data for the coefficient F were taken for SrTiO 3 [21], because the analo- gous measurements in BaTiO 3 and PbTiO 3 crystals have not yet been carried out. Calculations in (18) give the interface velocity v = 3.2 × 10 -3 cm/s. In [14] the experimental value is equal to v = (0.6-9.5)x 10 -3 cm/s. The interface motion velocity depends on the cooling rate, the thermal gradient in the crystal and the thickness of samples.

We compare between theoretical and experimental results for PbTiO 3 crystals. For parameters D = 3.3 x 10 -16 cm 2 (d = 4.0 x 10 -s cm [19]), A ' = 9.09 x 10 -6 ( l /K) [19], T c - T o = 22 K and for the same F we obtain v = 7.0 x 10 -3 cm/s. In [15] the experimental value of the interface velocity v = (4.3 × 10-3-7 x 10 -2) cm/s. This velocity also depends on the cooling rate, the thermal gradient and the thickness of samples. The interface motion in PbTiO 3 crystals is jerky, and consists of alternating motions and stops [15]. The interface velocity was defined as a ratio of the distance passed by a phase boundary to the whole period of the passage.

Comparison between experimental results in BaTiO 3 and PbTiO 3 shows that the interface velocity grows as the difference T c - T O increases. It is also seen from the theoretical expression (18).

Therefore, there is a satisfactory agreement between the theory and experiments both in the value of an interface velocity and in its dependence on the tempera ture difference T c - T 0.

In [16] the motion of the sharply defined in-

A. Gordon / Interface motion in ferroelectrics 243

terrace was observed at the cubic-tetragonal phase transition in the ferroelectric KTa0.65Nb0.350 3 crystals. This planar interface moves from one end of the crystal to the other during the phase transition, leaving behind a single domain ferroelectric crystal. Such diffusionless phase boundaries are well known in metallurgy and have been studied extensively in a number of metal alloys, most notably the austenite-martensite transformation in steel [23]. In [24] it has been demonstrated that the free energy (2) with the shear strain serving as an order parameter 7/ describes the observed static phenomena in shape-memory alloys. We can suppose that eqs. (1), (2), (11), (13) describe the kinetics of the austenite-martensite transformation in the above-mentioned case.

5. Summary

The interface motion at first-order ferroelectric phase transitions with the reversible spon- taneous polarization is described by the kink solution of the time-dependent Ginzburg-Lan- dau equation. The expressions for the velocity and width of an interface are derived. The stability of the kink solution under small perturbations is proved. The satisfactory agreement between calculated and measured values of the interface velocity in ferroelectric BaTiO 3 and PbTiO 3 crystals is obtained.

Acknowledgements

The author is indebted to Profs. J. Genossar and L. Benguigui for useful discussions.

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interface motion in ferroelectrics

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