ZHEW YE: Vibrational Modes around a Moving Soliton in Trans-Polyacetylene 393

phys. stat. sol. (b) 174, 393 (1992)

Subject classification: 63.10; 63.20; S12. I

Department of Physics, University of Ottawa ')

Vibrational Modes around a Moving Soliton in Trans-Polyacetylene (CH),

BY ZHEN YE^)

The vibrational modes of a soliton in trans-polyacetylene are studied. Using a quasi-realistic continuum model which generalizes the Takayama-Lin-Liu- Maki model by including acoustic-phonon effects, a self-consistent mean-field eigenequation for the vibrational modes of the soliton is derived. Then it is shown that these vibrational modes are damped due to the acoustic-phonon effects, which contributes to the linewidths in the lineshape observation. Die Schwingungsmoden eines Solitons in trans-Polyacetylen werden untersucht. Mittels eines quasi- realistischen Kontinuumsmodells, das das Takayama-Lin-Liu-Maki-Model1 durch Einbeziehung der Effekte akustischer Phononen verallgemeinert, wird eine selbstkonsistente ,,mean-field"-Eigen- gleichung des Solitons abgeleitet. Diese Schwingungsmoden werden durch die Effekte der akustischen Phononen gedampft, was zur Linienbreite beitragt.

1. Introduction Since the ~O 'S , many quasi-one-dimensional conductors have been synthesised, such as KCP, TTF-TCNQ, (CH),. With the development of experimental techniques, a number of new physical phenomena have been discovered, namely the soliton conduction mechanism in polyacetylene, nonlinear optical properties, and narrow band noise in TaS. To explain these phenomena, many theories have been proposed. One of the most successful theories is the SSH theory [l] for polyacetylene. Soon after the SSH model, a continuum version was proposed by Takayama, Lin-Liu, and Maki (TLM) [2]. Later Leblanc et al. [3] extended the TLM model by including the acoustic-phonon effects. They pointed out that due to the acoustic phonons, the soliton in polyacetylene is moving along the chain with a constant velocity. In the present paper, we would investigate the local vibrational modes around the moving soliton in trans-polyacetylene. We will derive an eigenequation for these vibrational modes, which will clearly show that acoustic phonons give rise to the intrinsic damping of these local modes. The damping is related to the photon absorption linewidth.

A number of reviews on the progress in the area of conducting polymers have appeared in recent years. For a good review, readers should refer to [4]. In this paper, first I would like to review briefly the general properties and the Peierls transition [5], then start from a realistic model to calculate the soliton vibrational modes in polyacetylene.

2. Polyacetylene and Peierls Phase Transition Since the polyacetylene was first synthesised by Shirakawa et al. in 1974, it has been studied extensively both experimentally and theoretically due to its simple molecular structure. It

I ) 34 George Glinski, Ottawa K1N 6N5, Canada. ') Address after Jan. 93: Institute of Ocean Sciences, P.O. Box 6000, Government of Canada, Sidney,

B.C., V8L 482, Canada.

394 ZHEN YE

is considered as a model material for the study of quasi-one-dimensional systems. Polyacetylene is a linear polymer consisting of weakly coupled chains of carbon-hydrogen units that form a quasi-one-dimensional lattice. If all the bond lengths were equal, the pure trans-(CH), would be a quasi-one-dimensional metal with a half-filled band. However, such a system is unstable with respect to a dimerization distortion, in which adjacent CH groups move toward each other forming alternately short bonds and long bonds.

We assume N atoms are placed along a chain with spacing a. Under the periodicity, the wave function in the chain will have wave vectors

2nn N a

k , = - I -

with n = 1,2,3 ... . If there is one valence electron per atom, then the total number of the states in the first Brillouin zone k , = z/a is

2 4 a 2nlNa

2- = 2 N .

The Fermi vector k , is determined by

i.e., 71

k , = - 2a

Therefore the band is half-filled. However this state is unstable. Assume now that the odd-numbered atoms move to the right with a displacement u, the even-numbered atoms move to the left with a displacement -u. The new lattice will have spacing 2a. Then the first Brillouin zone will be k, = 4 2 a which will coincide with the Fermi surface. Then the system is fully filled and an isolator. However when (CH), is doped by, e.g., AsF,, it may become a conductor or semiconductor. In the following we will prove that the distorted state has lower energy and therefore is a stable state.

We use the tight-bonding model with nearest-neighbor interaction (SSH model). The Hamiltonian is

H = - c [ t o - 4 & + 1 - %)I (C;f+,Cn + C,tC,+l) n

K M 2 n 2 n

+ - c ( % + 1 - u n ) 2 + - c u , 2 , (2.5)

where u, is the displacement for the n-th atom, Ci the creation operator of an electron on the n-th atom, to describes the hopping of electron along the chain. M is the mass of the CH group. Typically, to = 2.5 eV, CI = 4.1 eV/nm, and K = 2 1 4 0 ~ eV/nm2.

Now we study two static cases: 1. No Peierls distortion. In this case u, = 0, Hamiltonian becomes

Vibrational Modes around a Moving Soliton in Trans-Polyacetylene (CH),

Performing Fourier transformation

C, = ( ~ 1 - 1 ' 2 C e-iknaCk k

we have

where ll 71

E(k) = 2t0 cos (ka) ; - - 2 k 5 - . a a

So the energy of the ground state is

or the energy per site is

(2.10)

(2.1 1)

2. After distortion. Now, if the n-th atom has a displacement of un = (- 1)" u, the Hamiltonian becomes

(2.12)

Because the even-numbered and the odd-numbered atoms have different displacements, we use the follwing unitary transformation:

for the odd atoms,

for the even atoms. Then H becomes

H = ~ E ~ ( C ; I + C ; I - CitCi) k

+ i C d ( k ) (C;ItCi - CitC;) + 2NKu2 , k

(2.13)

(2.14)

(2.15)

where A ( k ) = 471 sin ka. Now we make a Bogolyubov transformation to diagonalize the Hamiltonian,

a; = -iakC;I + pkCg,

a; = a,*C; + ipzCi .

(2.16)

(2.17)

396 ZHEN YE

We obtain

where E ( k ) = I-, H = E ( k ) (aitaB - u;~u;) + 2NKu2

k

(2.18)

(2.19)

Therefore, after distortion two energy branches appear. The ground state energy can be calculated from the above equation,

U ( U ) = - 2 C E ( k ) + 2 N K d k j k F

4Nt0 8Na2u2 ( -__-___ (2.20) 71 ntO [ au 2 /

or the energy per site,

4t, 8a2u2 ( 2t: l.) ln--- + 2 K u 2 . (2.21) U(U) -

N n nt, _ _ _ - _ - _ _

The second term is the decrease of the energy, while the third term is the energy gain from the distortion. The displacement u is determined by minimizing this energy, i.e. (aU(u)/au)(,, = 0. Using the known parameters for polyacetylene, we find u , = 0.003 nm and 24(0) = 1.4eV. The condensation energy per site is E , = ( U ( u ) / N ) - (U(0) jN) = -0.015eV. These data agree with the experiments. It should be pointed out that the energy is degenerate for u = f u , .

3. Moving Soliton and Vibrational Modes

In the last section, we have seen that there are two degenerate states of distortion. It can be proved that when these two states appear in one chain, the boundary between these two is the soliton excitation. In this section we study the soliton vibration using the quasi-realistic polyacetylene kink-dynamics model with acoustic-phonon effects [3].

Based on the SSH model, Leblanc et al. [3] proposed a continuum model of polyacetylene by keeping terms up to the second order of lattice spacing a*. The Lagrangian density is

where @ is the optical phonon field, 5 the acoustic phonon field, and y the electron field,

(3.2) w = (;;) >

Vibrational Modes around a Moving Soliton in Trans-Polyacetylene (CH), 397

where yll and w2 correspond to conducting and valence bands, respectively,

r i a

vl(x) = ($”’ 1 dkeik”Cc(k), - n/a

4 0

y 2 ( x ) = ( ~ ) ” ’ 1 dkeikx(-i) C’(k) - nla

The parameters vF, us, wQ, and g are the Fermi velocity, the acoustic-phonon velocity, the bare optical phonon mass, and the bare electron-phonon interaction constant. They are related to the parameters of the discrete model by the following relations vF = 2at,,

uf = ~ a’, 06 = - , g = 4 a ( ~ / M ) ~ / ’ . In order to facilitate the analytical study by ignoring

the deviation of the gap equation from the BCS equation, the last term of the above Lagrangian can be put as follows after integration by parts:

K 4K M M

The dimensionless constant i is introduced to correct roughly the change of effective coupling.

We define the following order parameters:

s(0l @ 10) = d(x, t ) = 4,&, 0 9

(01 f lo> = Y(X, 4 2

(3.4)

(3.5)

where 24, is the Peierls gap. Then in the mean-field approximation, we have the following equations of motion:

(3.6)

(3.7)

This set of mean-field equations has been solved using perturbation theory in a boosted frame. Here we simply give the solutions. The static soliton results are

(3.9)

(3.10)

29 physicd (b) 174/2

398 ZHEN YE

where (po(X) = tanh (1/2X),

c~i(X) = Q ( v o - 9:) 9

y o ( X ) = Rq, + const and

The boosted coordinate is

The soliton velocity v, is

The constants R and Q are defined as

A 0 R = - - (1 - g

(3.1 1)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

Now we study the vibration around these solutions in the boosted frame. In the new frame, the coordinates are X,( = (Xo, XI)). The transformation is

x, = A E X " , where

cosh6 s inhe sinhI9 cosh I9

and

(3.18)

(3.19)

(3.20)

In the new frame the mean-field equations of motion become

(3.22)

Vibrational Modes around a Moving Soliton in Trans-Polyacetylene (CH), 399

and

(3.23) u,"

WQ = -g (01 [y+z3y" - ~ p " f t ~ ~ p ] 10) .

In this frame we have a static soliton. Now we follow the linear analysis approach proposed by Ito et al. [6], we study the

optical-phonon vibrational modes in the presence of the soliton. Assume a small vibration around the static solutions,

(3.24)

(3.25)

Ip(X,,X,) = (1C)p '"FXO + 6y0) + u;(y;ia'"F=J + 61pl),

A(Xi, X o ) = (Ao(X1) + 6Ao(Xi, Xo)) - u:(Ai(Xi) + (6A,(Xo, X - I ) ) ,

Y(X1, XO) = YO(X1) + 6YO(X,, XO) ' (3.26) We denote

+ 6wo7 (3.27)

(3.28)

(3.29)

(3.30)

Substituting (3.24) to (3.26) into the mean-field equations of motion we obtain by matching u: terms

- ia/vpXo Yb = wo

w; = w1 - i E l w X o + 6Y19

Ab = do + 6 A o ,

A; = A , + 6 A l .

(3.32)

a2 - [&& + 041 A; + (1 + 0 2 ) - A b ax:

(3.35)

29 li

400

If we consider only up to v, order, the soliton equations are reduced to

and

Equation (3.37) leads to

Now we make a Fourier transformation

6wo = 6y,(o, X,) e-iwi”FxO d o ,

6Ao = 6do(o, X,) e-iwi”FxO d o .

Then (3.39) becomes

a 8x1

o 6wo(% X , ) + &73 ~ b o ( o , X,) + b o w o ( o , Xl) Ao(X1) Z 1

+ 6d,(w - E,X1)71pO(X - 1) = 0 .

If we expand 6wo(w, XI) in eigenfunctions of the operator - then we have

ZHEN YE

(3.37)

(3.38)

(3.39)

(3.40)

(3.41)

(3.42) \

(3.44)

where

From (3.38), we have

where 2 means summation over all occupied states. Notice n

/ ~ & I , ( X , ) = g 2 2 Cwt’ e-i&n/WXO)? z , ( y l ~ ) e-iEn/WXO)

n

(3.46)

(3.47)

Vibrational Modes around a Moving Soliton in Trans-Polyacetylene (CH), 401

Finally we obtain the eigenequation which determines the frequencies of the local vibrational modes together with its intrinsic damping effects due to the presence of acoustic phonons,

(1 - $ - 2i ") 6do(X,, w ) = dy K ( X 1 , y) Fd,(y, w ) , (3.48) co; ax,

- m

where

From (3.48) we can study the dynamical processes of the vibrational modes around the moving soliton in trans-plyacetylene. The third term on the left-hand side arising from the acoustic-phonon effects gives rise to the imaginary part of the frequencies of the vibrational modes determined by this eigenequation. Without the third term on the left-hand side, the equation becomes the one obtained by Ito et al. 161. Using the explicit wave function in the presence of soliton, the kernel K(x, y) can be simplified as [6]

e l U - k ' ) ( x - Y )

sech (x) sech ( y ) + - d k K ( x , y ) = [dk 17 2n ' S Ek + E,.

x [(k + z) - i(i - &)tanhx]

(3.50)

in which Ek = v m . Three local eigenmodes wi can be found from (3.48), namely a Goldston mode, an

infrared-active mode, and a Raman-active mode [6]. In the presence of an acoustic phonon, the eigenfrequencies take the form: w = wi + ivJ+ The imaginary parts correspond to the broadening of local modes or the linewidth in the photon absorption. In the present paper, we do not attempt to solve numerically the broadening of these three modes due to the acoustic-phonon effects.

To conclude, we discussed the Peierls distortion in trans-(CH), in a simple approach. In the presence of an acoustic phonon, the soliton in trans-polyacetylene is moving along the chain. Using the quasi-realistic continuum mode by including the acoustic-phonon effects, we derived the eigenequation for the vibrational modes around the moving soliton. Our result shows that the acoustic phonon gives rise to the intrinsiclinewidths of these modes.

Acknowledgements

I am grateful to Prof. H. Umezawa for his guidance during the course of this work. I also thank Dr. Y. Leblanc for his stimulating discussion.

402 ZHEN YE: Vibrational Modes around a Moving Soliton in Trans-Polyacetylene

References

[l] W.-P. Su, J. R. SCHRIEFFER, and A. J. HEEGER, Phys. Rev. Letters 42, 1698 (1979); Phys. Rev. B

[2] M. TAKAYAMA, Y. R. LIN-LIU, and K. MAKI, Phys. Rev. B 21, 2388 (1980). [3] Y. LEBLANC, H. MATSUMOTO, H. UMEZAWA, and F. MANCINI, Phys. Rev. B 30, 5958 (1984). [4] A. 1. HEEGER, S. KIVELSON, J. R. SCHR~EFFER, and W.-P. Su, Rev. mod. Phys. 60, 781 (1988). [5] R. E. PEIERLS, Quantum Theory of Solids, Clarendon, Press Oxford 1955. [6] H. ITO, A. TERAI, Y. ONO, and Y. WADA, J. Phys. SOC. Japan 53, 3520 (1984).

22, 2099 (1980).

(Received July 31, 1992)