Diffusive motion of solitons in trans-polyacetylene

Synthetic Metals, 17 (1987) 27-32

27

DIFFUSIVE MOTIONOF SOLITONS IN TRANS-POLYACETYLENE

M. OGATA*, A. TERAI and Y. WADA Department of Physics, University of Tokyo, Hongo 7-3-I, Bunkyo-ku, Tokyo 113 (Japan)

ABSTRACT Diffusive motion of a soliton in trans-polyacetylene is studied theoretically within

the adiabatic approximation of the Takayama, Lin-Liu,

The collective

coordinate

method is applied to

between the soliton and classical phonons. to

the

diffusive motion.

investigate

and Maki model. the

interaction

There are two mechanisms which lead

Their contributions can be distinguished when the

dynamical diffusion coefficient D(m) and the f r i c t i o n function F(m) are studied. They are

calculated in the low temperature region.

Quantum corrections

and

possible other effects are also discussed.

INTRODUCTION Recently i t has become apparent experimentally that there are mobile spins in trans-polyacetylene, which are considered to be solitons

[I].

Analyses of ESR

and NMR experiments have suggested that the diffusion rate of the solitons would be proportional to T2 at low temperatures [2,3].

Our purpose is to study the

soliton dynamics theoretically in trans-(CH) x to find out possible mechanisms of the diffusive motion [4,5]. I t has recently been shown that the soliton gives a reflectionless potential for

phonons with large wave numbers, while phonons with

suffer reflection

[6].

small wave numbers

In Ref. [4], we have found that the c o l l i s i o n with the

phonon gives rise to a s h i f t of the location of the soliton in the second order perturbation approximation with respect to the phonon amplitude.

When phonons

*Present address: Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan. 0379-6779/87/$3.50

© Elsevier Sequoia/Printed in The Netherlands

28 with

small wave numbers collide with the soliton,

momentum in the second order processes.

t h e y also transfer

the

It is thus expected that the diffusive

motion is induced by two mechanisms. One is a random walk whose basic steps are the shifts of the soliton due to collisions with thermally excited phonons. The other is an ordinary Brownian motion, where friction is caused by the momentum transfer. The interplay of the two mechanisms is to be studied, using collective coordinate method.

COLLECTIVE COORDINATEMETHODFOR trans-POLYACETYLENE We consider the soliton motion in the TLM model [7]

H = - - l I dx {g4p(x t)2+ ~Q2A (x t ) 2} 2g2 ' ,

+ Xs' I dx ~ s t ( X ' t ) { - i v F ° 3 B x + ° I A ( x ' t ) }

~s(X't)

(I)

where A(x,t) is the order parameter proportional to the continuum limit of the staggered lattice displacement, its conjugate momentum field being defined by P(x,t)=A(x,t)/g 2, g the coupling constant, WQ the bare optical phonon frequency, t vF the Fermi velocity, o I and 03 the Pauli matrices, and the electron fields ~s and ~s with the spin index s have two components which represent the right- and left-going waves. According to the collective coordinate method [8],

the soliton coordinate is

introduced by a relation A ( x , t ) = As(X-Qo(t)) + X ( X - Q o ( t ) , t )

(2)

with a constraint

I×(x,t)A (x)dx = o

(3)

where a prime indicates the spatial derivative and As(X) represents the solution As(X ) = Aotanh(x/~),

C = VF/&0

soliton

(4)

A0 being the magnitude of the order parameter in the perfectly dimerized states. Since linear modes around the soliton contain the Goldstone mode go(x)

29 (proportional to As(X)), the field modes; {gn(x)}, X(x,t) :

X(x,t) can be expanded in terms of the other

~ Qn(t)gn(X) nm0

(5)

Explicit forms of gn(x) were obtained numerically

[6,9].

The adiabatic approximation for the electronic motions means that the

Fermi

distribution holds even when A(x,t) varies. Thenthe average <~(x,t)~s(x,t)> , in (I), becomes a functional of A(x,t), giving rise to effective unharmonic terms HQ among Qn's (nmO). The Hamiltonian (I) is thus reduced to

g2(Po+ f~x'dx) 2 H = E

+ s

2Mo( 1 + ~ / M o ) 2

2 g . l +-~ P P +-~ fln2Qn*Qn + H2 nmO n n 2g2 n#O

M0 : l{As(X)}2dx

(6)

where Es and M0 are s o l i t o n energy and mass, r e s p e c t i v e l y ,

and

= I X ' ( x , t ) A sw (X)dx

11" =

nmO

P (t)gn(X) n

(7)

Here, Pn is the momentum conjugate to Qn [ 5 ] . QO :

I t gives

g2(Po+ / ~ x ' d x )

(8)

MO( I+ ~/Mo)2

FRICTION FUNCTION AND DIFFUSION CONSTANT OF THE SOLITON Using a simple approximation to the Mori formula F(m) is estimated correlation

by

the

Fourier-Laplace

[ 1 0 ] , the f r i c t i o n

transform

of

the

total

function force

30

I <<00'(~0 o(t,jo >e x p ( - i m t ) >

£(w) =

where < , >

dt

(9)

is the canonical ensemble average.

At low temperatures and 1ow

frequencies, i t is estimated by perturbation method to be

Re£(w) = {O.OOl + 5O(m----)2} kBT mO AO WO + O(T2)

(10)

where m 0

is the renormalized optical phonon frequency [5,11]• The ~ independent term is obtained from the momentum-transfer processes and t h e w 2

term from the reflectionless collision

processes.

The dynamical diffusion coefficient D(m) is given by the formula •

D(w) -

°

~Qo'Qo> i~ + r(w)

(11)

In the static l i m i t e+ O, i t gives the Einstein relation

D(O)

g2kBT :

~

(12)

Mot(O) which

is

temperature independent.

On the other

hand, when the

frequency

satisfies w>£(~), we get

ReD(w) =

g2kBT Re£(w) 2 MOW

= { 0 • 0 0 1 --~-,

~ A0 1

where we have used relations g2=~VFm02/2 and MO=4A02/3~.

(13

Equation (13)

proportional to T2 in the low temperature region.

DISCUSSION We have found that the soliton-phonon interactions can give rise to the diffusive motion of the soliton through the two mechanisms. For polyacetylene, however, the quantum effect should be drastic for the optical phonons since wO

31

10 -3

z/ //f 10 -4

3o

o

//////

L..

I 0 -5

io-6 i

0.1

0.2

O~

1

2

kBT/~0 Fig. I. Log-log plot of r(O) as a function of temperature. The dashed line represents the classical value given by (10), which is proportional to T.

is

of the order of I03K [12].

diminishes given

by

at low temperatures, (10).

Since the number of thermally excited

phonons

r(O) is much reduced from the classical

The temperature dependence is estimated and shown in

value Fig. I.

Then,

i t becomes highly possible that the acoustic phonons are playing relevant

roles

for

the

electrons.

soliton

There s t i l l

motion,

in spite

of

their

weak interactions

with

is a long way to go to understand the diffusive motion

of soliton in polyacetylene.

ACKNOWLEDGEMENTS

The authors are very thankful to Prof.

H.

Shiba and Dr. Y. Ono for f r u i t f u l

discussions.

REFERENCES I See, for instance, the various reports in J. de Physique(colloq. C3) 44 (1983), Mol. Cryst. Liq. Cryst.,117 (1985), and Synthetic Metals~13 (1986).

32 2

M. Nechtschein, F. Devreux, F. Genoud, M. Guglielmi and K. Holczer, Phys.

3

Rev. B, 27 (1983) 61. K. Mizoguchi, K. Kume and H. Shirakawa, Solid State Commun.,50 (1984) 213

4

A. Terai, M. Ogata and Y. Wada, J. Phys. Soc, Jpn.,55 (1986) no. 7.

5

M. Ogata, A. Terai and Y. Wada, J. Phys. Soc, Jpn.,55 (1986) no. 7.

See also their contribution in these Proceedings.

6

Y. Ono, A. Terai and Y. Wada, J. Phys. Soc. Jpn.~ 55 (1986) no. 5.

7

H. Takayama, Y.R. Lin-Liu and K. Maki, Phys. Rev. B, 21 (1980) 2388.

8

For a review see, R. Jackiw, Rev. Mod. Phys., 49 (1977) 681.

9

H. Ito, A. Terai, Y. Ono and Y. Wada, J. Phys. Soc. Jpn.,53 (1984) 3520.

I0

H. Mori, Pro9. Theor. Phys., 33 (1965) 423.

II

M. Ogata, A. Terai and Y. Wada, in preparation.

12

for @4 case, see Y. Wada, J. Phys. Soc. Jpn.,51 (1982) 2735.