Kinks in disordered coupled Peierls chains

ELSEVIER

Synthetic Metals 101 (1999) 341-342

Kinks in disordered coupled Peierls chains J. Mertsching Institute of Physics, Humboldt University, Hausvogteiplatz 5-7, D-.lO_ll7 Berlin, Germany

Abstract The theory of sharp kinks in a disordered Peierls chain is extended to broad kinks and kinks in coupled chains. Both the finite kink width and the chain coupling reduce the expected kink density. The estimated disorder strength does not considerably contribute to the tail in the optical absorption. Keywords:

Semi-empirical models and model calculations; Ising models; Computer simulations; Polyacetylene and derivatives

1. Introduction As recently discussedby Mostovoy, Figge, and Knoester [1,2], uncorrelated disorder should induce neutral kinks in Peierls chains. Experimentally, only few kinks with a density n N l/3000 are detected in polyacetylene [3], corresponding to a very small disorder according to [1,2]. This result, however, was obtained assuming the kinks to be sharp and neglecting any interaction of the Peierls chains. Both these assumptions lead to some overestimation of the kink density, becausebroad kinks gain lessenergy from the disorder than sharp ones, and additional energy is required to put parts of neighbouring chains in a mismatched configuration [4]. In this paper, the disorder-induced kink density is calculated for broad kinks and interacting Peierls chains. As a result, the disorder strength corresponding to the observed small kink density becomes-larger than according to [1,2], but still gives only a small contribution to the tail in the optical absorption. 2. Model

of disordered

coupled chains

rpV are uncorrelated random numbers with (rlrv) = 0, (r&j = 1, and ti > to chain coupling parameters. The ground state for 7 = 0 is the dimerized matched configuration ufiU = u, where long and short bonds on neighbouring chains are in phase. An elementary excitation of a single uncoupled chain is a kink ZL~,, = ‘1Lupu, g/.Lv = i tanh[(p - ,~a)/<] with the formation energy (2/7r)A , A = 2au. In coupled chains an additional energy (t: - $)/(nT) per bond pair in a mismatched configuration (see Fig. 1) is required [4].

T

mismatched kink

3. Transformation

Fig. 1. Mismatched bond configuration on one side of a single kink.

to a random-field

Ising model

In an approximation for small disorder, i.e. linear in T, the model becomes [l]

We start from a bond-disordered Su-Schrieffer-Heeger model including the chain coupling where 2, is the number of kinks and Z,, the number of mismatched bond pairs. For sharp kinks with opV M il, the model becomesequivalent to an anisotropit random-field Ising model

p(vv’)u

where (-l)%lrv is the change of the bond length between the sites FLUand ,D+ IV, 7 the disorder strength, 0379-6779/99/$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)007991

J, = a

7r ’

Jyz = $

0” , h=-, KAr

2cY2

J. Mertsching

342

i Synthetic

Metals

1Oi (1999)

341-342

The ground state of a sample is calculated exactly by transformation to a maximum flow problem in a network [5] which is solved by the Ford-Fulkerson algorithm [6]. The random numbers rpv are chosen with a box distribution, and an average over m 2 50 samples is taken.

dimensional space. For weak chain coupling, the kink density only slightly decreases with increasing coupling strength and increasing number NY+ of coupled chains, but asymptotically decreases by a factor l/NyZ for C,s > 1 and large chain coupling when no mismatched region occurs.

4. Results

5. Comparison

The kink density n = 2,/N depends on two constants C, = XT/r and C,, = x(tT - ti)/(2Ar), where N = N,N,, is the number of CH units and X = 4cr”/(nkT) the electron-phonon interaction constant.

Spin resonance experiments with polyacetylene yi&d ng;/=“;O;; fG ,y,‘ie4,= 55 , r = XTIC, = 0.022 iv

1.2

*ox c

0.6 0.4 0.2

o.ov 0

'

'

10

20

30

CX

40

Fig. 2. Averaged normalized kink density for sharp kinks on uncoupled chains, rings, and tanhkinks on chains of length N, = 10000.

Asymptotically, n + l/2 for C, < 1, and n 25 l/C,” - r2 for weak disorder C, >> 1 in agreement with [1,2], probably nC,” + 1- for N --t 00 and C, -+ co (r + 0). As follows from Fig. 2, open chains contain somewhat less kinks than rings since one or two kinks near the ends may be missing. For broad tanh-kinks as in polyacetylene, the kinks in a long disordered chain can only be approximately determined. We start from a configuration with kinks at every sign change of ~~1 and successively eliminate kink pairs or single kinks near the chain ends with maximum energy gain. This method works very well for sharp kinks and, hopefully, also for broad kinks. The density of broad kinks with width [ = 7 is also included in Fig. 2. Indeed, broad kinks arise less easily than sharp kinks since in the central region of a kink (gpr (< 1) little energy can be gained from the disorder. For coupled chains, the calculation is done for sharp kinks only. Fig. 3 shows the averaged normalized kink density for 2 chains in a plane, and 4 chains in a 3-

'\

.\

--_ -----____

___--

,-J$

-

d=3 --------_-____v

Fig. 3. Averaged normalized kink density for sharp kinks OIi Nyz = 2d-1 coupled chains of length N, = lo-00 with C, = 10.

with

experiment

and conclusion

Without Collomb interaction, the kinks would be neutral and spinless [7]. With Coulomb interaction, neutral kinks have quasi-free spins l/2 and should be detectable by spin resonance (a strong coupling of distant spins seems improbable). The tail of the linear optical absorption, when assumed to be due to static disorder [8], would yi$d T = 0.21 eV [9], corresponding to_C, = 5.6 and a large kink density n 5 l/25 never observed. However, the tail is due to kink-pair production and large lattice fluctuations [lo1 which do not oroduce static kinks. A chain coupling dm M 0.1 eV yields C,, M 0.12 for a half gap - - A = 0.83 eV, and a reduction of the kink density by x 12 %. The observed small density of neutral kinks n M 0.00033 in polyacelene is compatible with a small uncorrelated disorder T M 0.03 eV, which does not considerably contribute to the tail in the optical absorption. L

1

Acknowledgement I am indebted to Dr. H.-W. Streitwolf for interesting discussions and a direct numerical check of the-kink formation within the disordered Su-Schrieffer-Heeger model. References [l] M. V. Mostovoy, M. T. Figge, and J. Knoester,-Europhys. Lett. 38, 687_(1997) [2] M. V. Mostovoy, M. T. Figge, and J. Knoester, Phys. Rev. B 57, 2861 (1998) [3] A. J. Heeger, S. Kivelson, J. R. Sckieffer, and W. P. Su, Rev. Mod. Phys. 60, 781 (1988) and references therein [4] D. Baeriswyl and K. Maki, Phys. Rev. B 38, 8135 (1988) [5] A. K. Hartmann and K. D. Usadel, Physica A 214, 141 (1995) [S] R. E. Tarjan, Data Structures and Network Algorithms, Philadelphia 1983 [7] K. Iwano,Y. Ono, A. Terai, Y. Ohfuti, and Y. Wada, J. Phys. Sot. Japan 58, 49048 (1989) [8] K. Kim, R. H. McKenzie, and J. W. Wilkins, Phys. Rev. Lett. 71, 4015 (1993) [9] J. Mertsching and B. Starke, Solid State Corn=. 101, 249 (1997) IlO] M. V. Mostovoy and J. Knoester, Phys. Rev. B 53,12057 (1996) and references therein