Calculated photoinduced dynamics in trans-polyacetylene

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Synthetic Metals 76 (1996) 31-33

Calculated photoinduced dynamics in tram-polyacetylene S. Block, H.W. Streitwolf Mar-Planck-Arbeitsgruppe

Halbieitertheorie,

Hausvogteiplutz

5-7, D-lOI

Berlin,

Germany

Abstract The generation and dynamics of self-localized excitationsin polyacetylene following a resonantfs pulse after introduction of some nucleation centresare calculated using the Su-Schrieffer-Heeger model. We obtain and investigate soliton pairs with an oscillating dipole moment and localized breathers. Furthermore, the linear photoinduced absorption is calculated and discussedin random phase approximation using the instantaneouseigenstates. Keywords:

Polyacetylene; Photoinduced absorption; Soliton; Mean field approximation

1. Introduction A strong optical pump pulse applied to a polymer chain leads to characteristic changes in the optical absorption [ 11 due to localized excitations like solitons [ 21 and soliton pairs [ 31 as well as breather-like [ 41 oscillations of the chain. In order to discuss these photoinduced absorptions we use the Su-Schrieffer-Heeger (SSH) Hamiltonian [ 21 extended to include an external electric field in the dipole approximation as a model for truns-polyacetylene (TPA) . Contrary to most investigations in the literature which rely on the adiabatic dynamics (e.g. [5] ), where the occupation numbers of the instantaneous eigenstates are not allowed to change, we solved the time-dependent Schrodinger equation for the electrons in mean field approximation [ 61 and calculated the nonlinear response to the electric field numerically. Furthermore, earlier investigations exclusively used some excited state of the system (e.g., electron-hole pair, soliton, polaron [ 51) as the starting point of their simulation. Because of the electron-phonon coupling in ‘PA the optically excited electrons strongly distort the bond length alternation already during the absorption process. An electron-hole pair on a dimerized chain, therefore, does not seem to be an adequate initial condition. We, therefore, for the first time started from a dimerized chain in the ground state and solved the equations of motion for the SSH model [2] under the influence of a strong resonant electric pulse of some 30 fs length. We followed the dynamics during a time span of 1500 fs. In a perfectly uniform chain self-localized excitations do not form since the absorbed energy tends to form extended excitations like optical phonons even if the chain has absorbed sufficient energy. However, we found small pertur0379-6779/96/$15.00

0 1996 Elsevier Science S.A. All rights reserved

bations of the dynamics sufficient to serve as nucleation centres, which aid the formation of spatially localized excitations. One possible way is to modify the Hamiltonian by a small defect at some site [7]. Another way used in this contribution is to introduce small chain fluctuations given by au,(t) =w,(t)Au with - 1
2. The model We have used the SSH Hamiltonian Xc(t) +Z’Jt) +ZnE(t) where Se(t) = ~c;tshnn~wnr,= m’s

=

-

&I(

&I,+

1+

f%,nr-

with tight-binding hopping inte-

1)

and a linear electron-phonon coupling Pft,l=~(s,,-s,,,,>(s,,“,+l-s,,~,-,>

Z’=

CC;;;,(~n’+CPfinlUl(t>>Cnr, m’s 1

is the electronic Hamiltonian grals &

[2]:

32

S. Block, H.W. Streitwolf/Synthedc Metals 76 (1996) 31-33

The lattice is described in harmonic approximation: q&L

1 -I- -p&up 1 2M 2 [I’

with Dll, = K( 26,,, - &, + 1 - &,- *), and the electric field E( t) is introduced in dipole approximation: cTE(t) =e~x,(t)(c~,c,,ns

1/2)E(t)

where x,(t) = na + u,(t) are the monomer positions. The equations of motion for the mean values of the displacements uI( t) read

1ooc ul w ; 800 2 vi u 6OC 400 200 C 3

s

k

is the density matrix with fk the initial Fermi distribution function, and $,,.,kobeys the time-dependent Schrtjdinger equation:

20

40

60

80

100 120 140 site n Fig. 1. Staggered order parameter r,,( t) for a chain of 146-monomer length. Maximum (white) and minimum (black) values are about 8X 10-l’ and about - 8 X IO-“’ m, respectively.

n’

where h,,,(t) is the electronic Hamiltonian in mean field approximation and explicitly depends on time t through the lattice displacements uI( t) . We use the parameters from [ 81 generally accepted for TPA: cr=4.1 eV A-‘, K=21 eV Am2, t,=2.5 eV, and M= 31 14fi2 eV-’ A-‘.

w n

t

-0.5

3. Dynamics We discuss the dynamics of our system in terms of the distortions of the monomer positions (the staggered order parameter) r,,= -(l/4)(-1)“(2u,-24,-,-u,+,), which displays the time development of localized optical excitations, and the instantaneous electronic eigenvalue spectrum. The system being deterministically chaotic nevertheless shows characteristic features (‘scenarios’) in its time development. In particular, we have found [ 71 charged bound and free soliton pairs and charge neutral localized lattice oscillations (‘breathers’) moving with lower velocities than the solitons. These excitations remain stable for some 100 fs until they locally interact with each other. The bound soliton pairs are similar to polaronic lattice distortions [9] but, contrary to polarons, have an electric dipole moment. As an example of various calculations with different fluctuations, electric field strengths and chain lengths, we discuss the results obtained for an electric field of the laser pulse with frequency fiw = 2.0 eV and amplitude E, = 2.5 X 10’ V m- ’ at 150 fs for a chain of 146 monomers. The fluctuation parameters are given by AU= lo-l6 m.

.I i-l I

.Y

600 800 1000 1200 1400 time t (fs) Fig. 2. Instantaneous energy spectrum I for a chain of 146.monomer length showing the gap states (k=73, 74). The shaded regions below the statek= 72 and above the statek = 75 represent the valence and conduction bands, respectively. 0

200

400

A density plot of the staggered order parameter shown in Fig. 1 reveals a bound soliton pair created at about 300 fs moving to the left. It interacts with breathers and is reflected from the chain edge at about 480 fs. At about 520 fs the bound soliton pair separates into an unbound pair, while the gap states in the instantaneous eigenvalue spectrum (Fig, 2) coalesce at 0 eV. 4. Induced absorption In order to compare the induced optical properties of our excited chain model with the measured photoinduced absorp-

S. Block,

0

0.2

0.4

0.6

0.6

1.0

1.2

1.4

H.W. Streitwolf/Synthetic

1.6

1

hv (eV) Fig. 3. Photoinduced absorption, Y Im E(V), averaged over time (greater than 500 fs) and different chains for an ensemble of chains developing separated solitons (curve c) and those with no solitons (curve b), compared to the absorption without pump pulse (curve a).

Metals

76 (1996)

31-33

33

sitions between the valence band and the gap states as well as the gap states and the conduction band. All chains show a distinct decrease of the gap, which is responsible for a high energy peak at about 1.2 eV. This peak is lower for chains with solitons since oscillator strength is transferred to their low energy peak. Similarly, the bleaching of the transitions near the former absorption edge is due to the shift of oscillator strength to lower transitions. We furthermore average over some chains of each type with different chain lengths ranging about 120 monomers. To compare these results with the linear response we have also drawn the absorption without pump pulse in Fig. 3. We have not included Coulomb interaction of the r-electrons in our model, so neutral solitons have states at the centre of the gap. Without Coulomb interaction neutral solitons cannot be related to the high energy feature in our calculations.

5. Conclusions tion change [ 11 we notice that the induced current density on the average over chains with different lengths has vanished some 30 fs after the pulse. We have, therefore, calculated the instantaneous dielectric function of the excited chain in random phase and rotating wave approximation:

using the energy spectrum &k and the dipole matrix elements dkkjdetermined from the instantaneous eigenfunctions (Pnk(t) of the electronic Hamiltonian: ~hn&)

%Sk(t)

= Ek(t)

(Pnk(t)

at each instant of time (r= 0.02 eV) . The occupation numbers nk are determined from the decomposition of the wavefunctions $,+(t) in terms of the instantaneous eigenfunctions: tClnk(t)

=

x%k’(tbk’k(t)

kr

by nk = Ck’fk, 1cu,, 1’. To eliminate the particular evolutionary details of each chain we finally average the dielectric function of each chain over time after some initial evolution of 500 fs has passed. We distinguish chains which develop free solitons from those which do not. The former type shows a photoinduced absorption peak at about 0.5 eV (Fig. 3) which results from tran-

We have investigated the time evolution of the dimerized chain excited by a 30 fs electric pulse in the SSH model. We have found free and bound soliton pairs and breathers. Using the instantaneous electronic spectrum and eigenfunctions we have calculated the induced dielectric function in random phase approximation and averaged over time and two typical ensembles of chains of different length. The result compares qualitatively with the experimental induced absorption [ 11.

References [l] J. Orenstein, in T.A. Skotheim (ed.), Handbook of Conducting Polymers, Marcel Dekker, New York, 1986. [2] W.P. Su, J.R. SchriefferandA.J. Heeger, Phys. Rev. B, 22 (1980) 2099. [3] S.A. Brazovskii and N.N. Kirova, Pis’ma Zh. Eksp. Teor. Fiz., 33 (1981) 6. [4] A.R. Bishop, D.K. Campbell, P.S. Lomdahl, B. Horovitz and S.R. Phillpot, Phys. Rev. Left., 52 (1984) 671. [5] W.P. Su and J.R. Schrieffer, Proc. Nutl. Acad. Sci. USA, 77 (1980) 5626. [6] A. Terai, in T. Kobayashi (ed.), Relaxation in Polymers, World Scientific, Singapore, 1993. [7] S. Block and H.W. Streitwolf, to be published. [8] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Left., 42 (1979) 1698. [9] D.K. Campbell, A.R. Bishop and K. Fesser, Phys. Rev. B, 26 (1982) 6862.