COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational Materials Science
I I ( 1998) 227232
Thermally activated motion of domain wall in a crystal with a small degree of discreteness S.V. Dmitriev
*, T. Shigenari, K. Abe
Department of Applied Physics and Chemistry, University of NectroCommunications, Chofushi. Tokyo 182, Japan Received 18 January 1998; accepted 9 February 1998
Abstract In the frame of the onedimensional elastically hinged molecule (EHM) model of crystal the thermally activated motion of a rather wide domain wall (DW) was studied by numerical simulation at temperatures much higher than PeierlsNabarro barrier but much lower than kinkpair nucleation temperature. It was found that DW undergo Brownianlike motion and at some moments the drift velocity of DW can be changed by obtaining (losing) some energy from (to) thermal fluctuations. 0 1998 Elsevier Science B.V. All rights reserved. PACS: 64.70.Rb; 64.70.Kb; 61.72.C~ Keywords: Heated chain; Domain wall; Kinkphonon interaction; Computer simulation
1. Introduction The modulational instability of lattice was experimentally found for a great variety of crystals [1,2]. If the period of modulation is in an irrational ratio with the lattice period then the modulated phase appears as an incommensurate phase. Near the critical point modulation has the sinusoidal form but further evolution changes the modulation to a rectangular form which can be considered as an array of periodically arranged, commensurate domains separated by the discommensurations (domain walls). The process of nucleation and growth of discommensurations was extensively studied by moleculardynamics tech
* Corresponding author. General Physics Dept., Barnaul State Technical University, 46 Lenin St., 656099, Bamaul, Russia.
nique [351. As a rule, there exists the second critical point, socalled lockin transition, when the walls disappear and a homogeneous commensurate structure with a new symmetry is formed. The adequate description of the lockin transition has been given in the frame of the EHM model, recently proposed by the present authors [6]. The transition occurs when the mutual repulsion between DWs is changed to attraction, then the walls start to move and annihilate [7,8]. The law of increasing of the average domain width during such a process has been studied by Ikeda [9] with the using of the neutron scattering technique in a magnetic system and has been measured in ferroelectric NaNO, by Hamano et al. [lo] and by Zhang et al. [I 11. When analysing the kinetics of a lockin transition, due attention should be given to the behaviour
O9270256/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. P/I SO9270256(98)00008l
228
S.V. Dmitrier~ et ul. / Computational
of DWs in a heated crystal. As it will be shown later, the drift velocity of DWs can be changed not only due to their mutual attractive interaction, as, for example, in calculations by Nagai and Kawasaki [12], but also due to the interaction with phonons. If the width of DW is much smaller than the average distance between them then they can be considered as onedimensional gas of particles interacting with phonons. It was shown by Hasenfratz and Klein [13] and by Wada and Schrieffer [14] that nonlinear effects can give rise to the Brownianlike motion of the kink in the continuum ‘p4 equation. The kink is accelerated only when the gradient of the amplitude of the phonon wave packet does not vanish [ 131. The diffusion constant at low temperature is proportional to T* and does not depend on the wall thickness [14]. In a discrete lattice the PeierlsNabarro potential prohibits a wall from moving freely and can trap a wall. In this case a DW at low temperatures performs jump diffusion between wells of the PeierlsNabarro potential [ 1% 171. The interaction between kink and phonons in the displacive regime is in marked contrast to that of the orderdisorder regime [ 13,16,17]. In the present paper the effect was studied for the displacive case. 2. Domain wall in twoperiodic
EHM crystal
In the limiting case of undeformable molecules and small displacements, each hinge of EHM chain has one degree of freedom, namely, the transversal displacement y,. The chain is compressed by the force along its axis and the hinges are in the anharmanic, onewell fourth order polynomial background potential. In dimensionless form the equations of motion are given by [6] d2y, ~+F(Y~~~Y~,
+6~,4~,+,
+Y~+~)
+P(y,_,2yn+yn+,)+yn+Yn3=0, (2.1) where coefficients F, P are proportional to the stiffness of a hinge and the compression force, respectively. The twoperiodic stationary solution to Eq. (2.1) is Y2n = y,,_,
=
+fi,
(2.2)
Muterids
Scierrce II f 1998) 227232
where A = 4P  l6F1, which is possible if A > 0. In order to describe a solitonlike solution in the twoperiodic structure, let us consider two functions u,(n) and o,(n) instead of y(n), defined on the hinges with the even and odd numbers, respectively, moreover, u0 =  U, = u. Eq. (2.1) gives d2 ~1, +F(u,_~+~u,_, dt*
+6u,+4u,+,
+LJ~+~)
P(u,_,+2u,+u,+,)+u,+u~=0. (2.3) In the longwave Eq. (2.3)
approximation,
V,,(P8F)V,,AV+V’=O,
one obtains
from
(2.4)
where V(x,t) is the unknown slowly varying continuous function and we have chosen the length of a molecule as a unite of length. This equation has the solution in the form of a kink (antikink): V= ifitanh(
g(t+&)).
where 5=xct, B=(P8Fc2)‘. The width of the kink \/2/(
(2.5) increases as the
magnitude of AB tends to 0. For \/~/(AB) 1 one can use Eq. (2.5) to write the solution to Eq. (2.1) in the form of a moving DW in the twoperiodic structure: ?‘”DW= +(1)“fitanh
c(t+t,)]
,
(2.6)
i where n, defines the position of DW at a time t = to. The solution exists if A > 0 and AB > 0. The first condition is condition for the existence of the twoperiodic solution (2.2). From the second condition it is clear that solution (2.6) exists not in the whole region where the twoperiodic structure exists, but only in the region P > 8 F. In the vicinity of the line P = 8 F the solution (2.6) also cannot be used, because the term V,, vanishes in Eq. (2.4) and the higher order gradient term should be retained in this equation.
S.V. Dmitrieu et al. / Computational
Let us define a wave packet
rjsin(GwfiX) ocosh(Gq&iY)
(2.7) where
S=(l
v=dD,
d*)‘I*, (n  n&Y
= d(r
X=ttr,  To) + dt, +
\I1/(P8F)( n  n,), parameters 0 _
2 A3/* sin $ rr
= 0.
Note, that Eqs. (2.8) and (2.4) coincide earized with respect to the V= + fi.
(2.8) when lin
ture T * was estimated by Koehler et al. [20] as, roughly, h/2, where h is the height of the doublewell potential. In our model h = A*/4 [see Eq. (2.4)] and T * = 2. 10e6. We studied the temperatures much higher than the height of the PeierlsNabarro potential and much smaller than the kinkpair nucleation temperature. The initial conditions have been chosen as to describe a heated crystal with a DW: y,(t)
of numerical
experiments
The equations of motion Eq. (2.1) were integrated numerically using the Stiirmer method of order six [19] with a time step size of At = 0.002. This is sufficient to ensure that the total energy of the system is conserved with the accuracy 0.05% over an interval of t = 10’. The chain consisted of N = 151 molecules and the periodic boundary conditions y, = yn+v were used. Temperature is given by T = (dy,/dr*) and ( . > denotes the ensemble and/or time average. An important point is the choice of the parameters. First of all, the condition of stability of the twoperiodic solution must be satisfied. Second, width of DW should be much smaller than the length of simulated crystallite, but should not be very small, otherwise the solution (2.6) would not be a good approximation and the PeierlsNabarro barrier would be very high. We have chosen F = 0.06, P = 0.491 for which the width of DW is equal to 2.35 (approximately as in [13]) and the height of the PeierlsNabarro barrier, as was estimated from Eq. (2.61, is 5 . lo“. The kinkpair nucleation tempera
=y,“”
+Y,‘(t)v
(3.1)
where yfW is given by Eq. (2.6) at c = 0, no = N/2 and y:(r) represents the thermal fluctuation. According to the equipartition theorem, in the statistical equilibrium state the energy of the system is shared equally among linear modes and there is no phase correlation between the Fourier amplitudes of the harmonics. To satisfy these conditions let us represent the Nperiodic grid function y:(r) (second term in the righthand side of Eq. (3.1)) in the form
El1 y:(r)
3. Conditions
229
Materials Science II (1998) 227232
1 = Nb,cOs(o,r+ 2 (N1)/2 +N c k=l
pO)
b,cos
2kn.n ~~r+cp,, N
(3.2) where n = 0,. . , N  1, N is an odd number. The fundamental frequencies o k can be estimated under the assumption that the DW is absent by substitution of Eq. (3.2) into the Eq. (2.1) linearized with respect to the deviation from the solution (2.2) as follows w*k = 16F sin4 
N
 4P sin* ;+1+3A
(3.3)
with 0 I: k I (N  1)/2. The N amplitudes b,,b,,& k = 1,. . . ,(N  1)/2 were chosen in a way that each mode has energy E/N, where E is total energy of the system and the initial phases p,,,tpk,, are the random variables uniformly distributed in the domain (0, 27r).
230
S. V. Dmitrieo et al. / Computational
Muteriuls Science II (1998) 227232
4. Results and discussions The typical example of the time variation of the DW position is presented in Fig. la for T= lo’ and the absolute value of velocity of the wall during this process is shown in Fig. 1b. The DW moved like a Brownian particle and its drift velocity was changed at some time moments. At these moments DW obtained (lost) some kinetic energy from (to) thermal fluctuations. In the course of observation the DW was many times accelerated and decelerated. As Fig. lb suggests, the highest velocity peaks have approximately the same heights at given temperature. The temperature dependence of the maximum drift velocity is presented in Fig. 2. In Fig. 3 Fourier coefficients of the function v!(t) are shown or T= 10e7 at t = 0: 105: 106: 10’. I The modes b,,b,,b ,,..., b~,v,j,2,bCN_,j,2 are numbered by index n as 0, 1,. . . , N  1, respectively. Recall that Fourier coefficients at I = 0 were obtained from the condition that all the modes have the same energy. Due to the presence of DW and nonlinearity of system there is an energy exchange between the modes. In the initial stage of the process (prior to t= 105) the modes with big wave vectors are involved in the energy exchange. By the time t = lo6 all the modes are involved in the energy exchange and the Fourier coefficients slowly vary inside some
,,I

Fig. 2. Temperature dependence of the maximum drift velocity.
belt about the initial magnitudes. The evolution of the energy exchange between modes have no noticeable effect on the motion of DW and hence one can tell that Eqs. (3.1) and (3.2) give reasonably good initial conditions in the case being considered. Let us now discuss the possible mechanisms of the domain wall acceleration effect. The linear continuum theory of kinkphonon interaction shows that phonons do not affect on the kink. The travelling fluctuations simply suffer a phase shift upon passing through the cp4kink (see, for example, Refs. [22,23]). The nonlinear continuum theory, as mentioned above, predicts the acceleration of the kink by phonon wave packet as long as there is an energy flux through the kink. In a heated crystal there are thermal fluctua
l.0~0.0 a* 0.
t=O
0.0 _i;:
t Fig. 1. The timedependence DW at T=
IO‘.
of (a) position and (b) velocity of
n
n
[X106] Fig.
y:(t)
3. Timeevolution at T=
IO‘.
of
Fourier
coefficients
The modes b,,b,,q
are numbered by index n as 0, 1,.
of the function
,..., b~N_,J,2.b~N
, N  1, respectively.
LJ,z
S. V. Dmitrieu et al. /Computational
tions of phonon amplitude, hence this theory explains the Brownianlike motion of DW but the abrupt changes of the drift velocity of DW are not predicted. Probably, the effects of discreteness play here an important role. The discreteness can be treated as a perturbation of a continuum equation. For perturbed continuum equations the effects of inelastic collision between solitons were found [24]. One can assume that the nonlinearity and the discreteness of the considered system give rise to the formation of the breatherlike modes with a small amplitude, which are a special kind of wave packets. The possibility of the energy exchange between a kink and a highamplitude breather during their collision in a crystal with a small degree of discreteness has been reported [25]. The acceleration of the kink by collision with a smallamplitude breather can also be expected. The result of collision between a DW Eq. (2.6) and a smallamplitude wave packet Eq. (2.7) is shown in Figs. 4 and 5 for a chain consisted of N = 601 molecules. At t = 0 DW was at rest (c = 0) and wave packet with w = 0.995 moved from left to right. The only difference between Figs. 4 and 5 is the velocity of wave packet was equal to d = 0.5 and d = 0.1, respectively. As a result of the collision the fast wave packet passed through DW and the slow one was reflected by DW. In the first case the DW is scarcely affected by the wave packet but collision
t=15OO
Materials
Science I I (1998) 227232
1
0
c
I
,
100
231
I
200
m
I
300 n
400
k
500
I
600
Fig. 5. The same as in Fig. 4 but for d = 0.1. The slowly moving wave packed is reflected by DW
and the collision involves an
energy and momentum exchange between quasiparticles.
with the slowly moving wave packet involved the energy and momentum exchange between quasiparticles (see Fig. 5). The dynamics of the twoperiodic EHMmodel in continuum limit is described by Eq. (2.4) which is, at the considered magnitudes of parameters P,F, the (p4 equation. This causes us to anticipate that the effect described in this paper may occur in such a popular models as a (p4 and a sineGordon field.
5. Conclusions It has been demonstrated by numerical experiments that at law temperature the drift velocity of a rather wide DW can be changed due to the interaction with phonons. The mechanism of this effect is not well understood. It is likely that the effect of discreteness cannot be neglected even if the degree of discreteness was small in present calculations. It was shown that DW can be accelerated by collision with a slowly moving small amplitude wave packet.
t=3000 wave packet c L
0

1
100
I
I
200
I
300 n
11
400
500
Acknowledgements 600
Fig. 4. The collision between a DW and a wave packet. At r = 0 DW
is at rest and wave packet moves from left to right with a
fairly high velocity d = 0.5. The wave packet passes through DW and the DW is scarcely affected by it.
The authors would like to thank T. Kumata for the help during the course of the study. One of the authors (S.V.D.) wishes to thank the Ministry of Education, Science, Sports and Culture of Japan for their financial support.
S.V.
232
Dmitrieti
et 01. /
Computationul Mute&Is
References [I] R. Blinc, A.P. Levanyuk (Eds.), Incommensurate Phases in Dielectrics, Vols. 1 and 2, North Holland, Amsterdam, 1986. [2] [3] [4] [5] (61 [7] [8] (91 [IO] [I I]
H.Z. Cummins, Phys. Rep. 185 (1990) 211. K. Parlinski, F. D&toyer, Phys. Rev. B 41 (1990) 11428. K. Parlinski, Ferroelectrics I04 (1990) 73. K. Parlinski, Y. Watanabe, K. Ohno, Y. Kawazoe, Phys. Rev. B 50 (1994) 16173. S.V. Dmitriev, K. Abe, T. Shigenari, J. Phys. Sot. Jpn. 65 (1996) 3938. S.V. Dmitriev, T. Shigenari. A.A. Vasiliev, K. Abe, Phys. Rev. B 55 (1997) 8155. A.A. Ovcharov, S.V. Dmitriev, M.D. Starostenkov, Comp. Mater. Sci. 9 (1998) 325. H. Ikeda, J. Phys. C I6 (1983) 3563. K. Hamano, J. Zhang, K. Abe, T. Mitsui, H. Sakata. K. Ema, J. Phys. Sot. Jpn. 65 (1995) 142. J. Zhang, K. Hamano, K. Abe, T. Mitsui, H. Sakata, K. Ema. J. Phys. Sot. Jpn. S9 (1995) 149.
[ 121 [13] [14] [l5] [16] [I71 [lx]
[ 191 [20] [2 I] [22] [23] [24] 1251
Science
II
(1998)
227232
T. Nagai, K. Kawasaki, Physica l20A (1983) 587. W. Hasenfratz, R. Klein, Physica 89A (1977) 191. Y. Wada, J.R. Schrieffer, Phys. Rev. B I8 (1978) 3897. T. Munakata, Phys. Rev. A 45 (1991) 1230. J.A. Combs, S. Yip, Phys. Rev. B 28 (1983) 6873. J.A. Combs, S. Yip, Phys. Rev. B 29 (1984) 438. V.E. Zakharov, L.A. Takhtadjan, L.D. Faddeev, Dokl. Acad. Nauk SSSR 219 (1974) 1334. E. Hairer, S.P. Nlirsett, G. Wanner, Solving Ordinary Differential Equations. Nonstiff Problems, Springer, Berlin, 1993. T.R. Koehler, A.R. Bishop, J.A. Krumhansl, J.R. Schrieffer, Solid State Commun. I7 (1975) 1515. A.A. Samarskii, ES. Nikolaev, Numerical Methods for Grid Equations, Vol. I, BirkhLser Verlag, Base], 1989. J. Goldstone, R. Jackiw, Phys. Rev. D 1I (1975) 1486. CM. Varma, Phys. Rev. B 14 (1976) 244. YuS. Kivshar, B.A. Malomed, Rev. Mod. Phys. 61 (1989) 763. S.V. Dmitriev, L.V. Nauman, A.M. WusatowskaSarnek, D. Starostenkov, Phys. Status Solidi B 201 (1997) 89.