Nonlinear optical spectra of conjugated polymers with dephasing

Prog. Crystal Growth and Charact. Vol. 33, pp. 101-104, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-8974/96 $32.00

Pergamon

NONLINEAR OPTICAL SPECTRA OF CONJUGATED POLYMERS WITH DEPHASING V. A. Shakin 1,2, S. Abe 1 and T. Kobayashi2 1 Electrochemical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan 2 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

Abstract

We have theoretically studied X(3) spectra of degenerated four wave mixing and electroabsorption for conjugatedpolymerswithin a treatment of double-excitationconfiguration interaction for the Pariser-Parr-Pople model. Comparison between the generally accepted OnWard formula and the density matrix approach in deriving X(3) reveals in the latter case some new features of the nonlinear spectra, for example, additional resonances, and demonstrates an importance of the density matrix approach for analyzing correctly the regions of the two-photon and saturated absorptions in conjugated polymers. Delocalizedn-electrons in conjugated polymers are responsible for relatively large optical nonlinearity of this class of materials. Here we present calculation data for the third order optical nonlinearities in conjugated polymers, especially for the degenerate four wave mixing and electroabsorption. Eigenstates of the system of the n-electrons are found in the Pariser-Parr-Pople model by use of double-excitation configuration interaction method [1]. To calculate X(3) spectra one can use the generally accepted Orr-Ward formula [2] or apply the density matrix formalism (see, for example, [3]). There is a principal difference between these two approaches, because the former assumes a validity of a wave function to describe a behavior of the quantum system and hence cannot treat rigorously its interaction with a reservoir, while the latter is adequate to take correctly into account the energy absorption process. As a result, the Orr-Ward formula for X(3) satisfactorily treats two-photon resonances, where excited state population is relatively small [2], but fails to describe correctly the saturation absorption. Figure 1 shows a comparison between X(3)(-oXo~,-03,c0) spectra calculated using the Orr-Ward formula and the density matrix approach for a polymer ring of N=30 sites with modulation of transfer integral 5t=0.1t and on-site and off-site coulomb potentials of the electron-electron interaction (we use Pople potential) U=2t 101

102

V. A. Shakin et aL

and V=t, where t stands for the 5 x104 (a) Orr-Ward averaged magnitude of the transfer E integral. The broadening constant in -5 x10 4 the Orr-Ward formula, F, and I -1 xl05 longitudinal and transverse d a m p i n g constants in the density matrix ~ 1 . 5 x l 0 ~ formalism, F 1 and 1-'2, all are here the -2 xl 0 s . . . . . . . . . ' ......... I ......... 0.5 1 1.5 same and equal 0.2t/h.x(l) and X(3) Photon energy (units of t) presented in this report are those per site and normalized respectively by o

Z~) 1) =

e2a 2/t,

Z~~) = e 4 a 4 / t 3,

5 x104

(b) Density matrix

J 2

~.

0 $ -5 x104 I

-1 xl05 £ JL,1.5 xl05 .o

where e is electron charge and a is an averaged distance between adjacent sites. Figures l(a) and (b) demonstrate a failure of the Orr-Ward formula to describe the saturation absorption, while the regions of the two-photon absorption resolved in Figures l(c) and (d) are very alike for both approaches. However, a careful examination of ( c ) a n d ( d ) b r i n g s to light some additional peaks in the latter picture. These additional peaks correspond to the additional resonances, which represent a specific feature of the density matrix approach and cannot be obtained from the OrrWard formula, The additional resonances can be observed if the photon energy equals the energy difference between one-photon and two-photon excited states. It is possible to show from the general expression of X(3)(-o);o),-co,6o) obtained in the density matrix

-2 xl05 0.5

i

1200 -

~°

1000

-

600 400 200 0 -200 -400 ' 0.5 800

I

? ~

~" g a

i

i

i

i

I

i

i

i

I

i

i

i

i

i

i

i

i

I p

I

i

I

I

I

I

1 1.5 Photon energy (units of t)

(c) Orr-Ward

i!i ,,i,,-,,,,,i:,,,1,Iiil L

1200 (d) 1000 800 6OO 4OO

I

I

|

2

::

De~n~ia 1 1.5 Photon energy (units of t)

2

1

2000 ; -200 r-400 0.5

55 ;i 1

1.5

C

2

Photon energy (units of t)

Fig.1 ComparisonbetweenOrr-War and density matrix approaches. The dashed ;rodst)lid lines ct)rrespond to the real and imaginaryparts respectively.

103

Nonlinear Optical Spectra of Conjugated Polymers

formalism that the intensity of the 100 (b) additional peaks is proportional to the c° hFl=0.002t ratio Fz/F1, while the two-photon ~ 80 resonances do not vary if F2 doesn't ~ 60 AR change [4]. ~ 40 Figure 2 shows X(3)(-o~;m,-co,co) ~ 20 spectra for the polymer ring of N=6 ~ 0 ,ib I I I II I I I II I I II I sites with dt=0.2t, hF~=0.020t. The 3.5 3 1 1.5 2 2.5 0.5 Photon energy (units of t) vertical arrow points the lowest onephoton resonance. Plot (a) is for hF~=0.020t, (b) for hF~=0.002t, and (c) 100 (a) II for hF~ 0.001t. Here we intentionally I 80 - hFl=0.020t consider a small molecule to facilitate ~£ 60 an assignment of the peaks. TPA 40 i marks one of the resonances, which TPA l s 20 corresponds to the two-photon 0 absorption, i.e. the coherent transitions , , , .... , .... 1 1.5 2 2.5 3 3.5 0.5 from the ground state to the twoPhoton energy (units of t) photon excited states absorbing two photons. Doubled photon energy at 100 such resonance equals the excitation ~o 80 energy of the corresponding twohF]=0.O01t photon excited state. The additional ~g 6o resonances (one of them is marked as L 4o s AR) result from the incoherent s 20 transitions to the two-photon excited ~ 0 I I I II I I I II i I I I I I]ll I'lLI states, that is why its intensities are 1 1.5 2 2.5 3 3.5 0.5 Photon energy (units of t) inversely proportional to the longitudinal damping constant F~. In Fig.2 The real (dashed line) and imaginary (solid line) this case the photon energy at the parts of X(3)(-(o;~,-o),o)) as functions of photon energy for various FI, while F 2 is fixed. resonance equals one of the possible energy difference between one-photon excited and two-photon excited states. It is clearly demonstrated by Figure 2(c) that if F2>>F~ (it is a usual case for conjugated polymers) the additional resonances may be considerably stronger than ordinary two-photon absorption resonances, one should be very careful making an assignment of experimentally observed resonances. The density matrix formalism enables us to investigate a spectral region around the lowest one-photon resonance. Figure 3 demonstrates a relationship between IX(3)(-co;~o,-co,os)l and Im[x(1)(co)] - - s o - c a l l e d scaling law [51. The solid

(c)

AR

i

! TPA

i

V . A . S h a k i n et al.

104

lines correspond to the low energy side of the linear absorption peak, and the dashed ones, to the high energy side. As a whole these lines indicate a power-law behavior

Zt3)(_w;co,_og, co)

=c{,+

(l)

107

t,

"'1

'

.......

I

......

hFz=0.02t The exponent p is close to one (p-l.2) on the low energy side, while it is much larger than one (p~l.7) on the high energy side. What should be noted here is that the coefficient c rapidly decreases upon increasing the transverse damping constant F 2. Due to relative simplicity of measuring the absolute value of X(3)(-o);co,-o),eo), this fact can be useful for an estimation of F2. In cases when the transverse damping constant is about two orders of magnitude larger than the longitudinal one, that is when additional resonances certainly dominate over two-photon resonances, we numerically observe in the very wide range of photon energies, which embrace important regions of the two-photon and saturated absorptions, the following relation between Z(3)(-co;eo,-~,co) and X(3)(-co;co,0,0) spectra

5

c~

lOe

hF2=0.04t~. hF z = y ;

s" I,L

105

104

,;:.,..," 1000 0.1

1

10

Im (Z(l)) / Z(l~o Fig.4 The absolute value of X(3)(-o~;o~,-~,o3) as a function of the imaginary part of z(l)(o~) for the polymer ring of N=20 sites with ~t=O.2t, hFl=().OOlt and wtrious F2. The solid lines correspond to the low-energy side of the resonance peak and the dashed lines correspond to the high-energy side.

N Z(3)(-eo;co,-co,eo)XfJ)o --- C ZO)(-eo;eo,0,0) hn(z< l)(co)), with a real factor C independent on N and co. This relation can help in the evaluation of g(3)(-o3;co,-co,m) for materials with Fz>>FI oil the basis of linear and electromodulation spectra, which are relatively easier to measure.

1. 2. 3. 4. 5.

References V.A.Shakin and S.Abe, Phys.Rev. B50, 4306 (1994). B.J. Orr and J,F. Ward, Mol. Phys. 20, 513 (1971). R.W. Boyd, Nonlinear optics, Academic Press, San Diego (1992) V.A.Shakin, S.Abe and T.Kobayashi, submitted to Phys.Rev. C. Bubeck, A. Kaltbeitzel, A. Grund and M. LeClerc, Chem. Phys. 154, 343 (1991).