Nonlinear optical properties of conjugated polymers

Synthetic Metals, 41-43 (1991) 3759-3762

3759

NONLINEAR OPTICAL PROPERTIES OF CONJUGATED POLYMERS

K.A.PRONIN and YU.I.DAKHNOVSKII Institute of Chemical Physics, Acad. Sci.USSR,

Moscow

(USSR)

ABSTRACT With

the

help

of

the

nonequilibrium

diagram

technique

we

rigorously calculate the nonlinear optical susceptibility for third harmonic

generation

carriers

is

taken

frequencies smeared

in

a

into

account.

corresponding

out

by

Peierls

dielectric. The

square

to one-third

damping.

Theoretical

Slow root

relaxation

of

divergencies

at

and the total results

are

in

band gap are qualitative

agreement with experiments on polyacetylene.

INTRODUCTION Conjugated polymers are promising materials for nonlinear optics (NLO) because of their high hyperpolarizabilities analytical been

expression

rigorously

for

found

the

third

without

order

taking

correlations and relaxation processes. soliton

contribution

to ts~

have

account

generation the

slow

in

a

been

of

(3)

Coulomb

presented

account in

[5].

for the process

one-dimensional

relaxation

susceptibility into

has

Computer calculations of the

paper we calculate the NLO susceptibility harmonic

[1-3]. In [4] the

carriers

dielectric that

In

this

of third

taking

into

originates

from

electron scattering on thermal phonons and impurities

[6].

RESULTS We use the continuum model of a Peierls dielectric Lin-Liu-Maki

type

contains

solitons.

no

0379-6779~1/$3.50

[7].

The polymer

ground

The

gap

complex

state

A=A s +A e i~ p

of Takayama-

is dimerised in

the

and

general

© Else~erSequoia~rintedin The Netherlands

3760 case

consists

dielectrics structure

two

for

the

of the

component

A

molecule)

P

dimerization

diagram

technique

=

and the function

is left. obtain

With the

generation

for

the

expression

the

trans-(CH)x

interaction

f is defined

only

with

from

the

resonance

weak damping

[4]. Drastic

square

root

because

of the

nonequilibrium

of

susceptibility factors

e :

divergences

by (2)

c3) = B 3 s

For brevity

inequality

are

now

smoothed

~ [~(l+i)

the resonance

divergence the

to ~-I/2>>i.

frequency

real

given real Note

in the resonance

damping,

can

but

is still

into part

the of

and

the

area

term

imaginary

z<~ where

b ] +...}

= 1/2

values

.

asymptotics

that damping

inversely

6~(I-i)

_5/2.1

the

by the first

values

of

the

latter

312

for

parts

produces for c=0

formula

+ ...} the

exhibits

in Eq.(4).

(3)

constants:

~/3 ~< ~1 _ z ~< 6-3~2a -2 the

- a -

63/2(~_Z)1/2

smoothly The

by

structure

finite

+

we give only the approximate

Below for ~(3)

damping.

takes

a -(z-o ) 8 =312

{

out

~e<
~(l-i)

Z--8

==he/IA l we

in the main term the result i arise in the vicinity of z=l,~. Here the

{ 12= I/2

, b = 81.63.

transforms

Ii-nzl>>=,

(Fig.l).

a = 2.41

= B

I/2] -2!in2}

frequences

For Iz-~l<<= the susceptibility proportional to small =i/2: XTHG

(i)

in f(z) and obtain

changes

highly pronounced

the

of the the

s

frequency,B=3-22-Te4 (~vf) lA l-~

= z-l(l-z2)-I/2{in[(l-z)*/2-i(l+z)

neglect

A

rigid

3

g(z)

of both

while

with

valid for small damping

= g(z-ic~IAl -I) - g(-z)

(3)

Peierls

state the component

electron

the help

f(z)

~THG

the

account

Here z--~/2 IA I is the dimensionless

of Wu

ground

of

into

'combined'

z -8 {3(l-8z2)f(3z)-8(l-4z2)f(2z)+(5-8z2)f(z)}

B

Far

For

is nonzero,

takes

we

for third harmonic (3)

interaction

that

Peierls

~THG

contributions.

that have a nondegenerate

(accounting the

of

Now

case a

of

zero

square

root

the peak

of X ~3~ are

a nonzero

(4)

values

proportional

tail

for

it was vanishing.

Im (3~

It means

that the nonlinear resonant behaviour is accompanied by absorption. Above the resonance z-!>>u, X (3) also matches the ==0 result: 3

3761 (3) = B z -8

~THG

Again that

the

even

peak

value

without

but reaches

- a -

65/2(Z--~) 3/2

{

of

Im

relaxation

a constant

63/2(Z--~)1 1/2 }

(3)

Re

is

proportional

X (3) for

z ~ !+0 3

, q ~ -83.18

For e=0 t h e p e a k

d(z)

= -2(2z-i) [ i + i ~ 2 - 3 / 2 ( 2 z - i ) - I / 2 ) ,

(l_2z) I/2,

! = 8~2(y-iA)

d(z)

does

not

Eq.(8)

The in

(6)

0 ~ 2! _ z ~ 1

(7a)

0 ~ z - i ~ 1 2

(7b)

damping { [ l + y 2 1 1 / 2 - y } l / 2 - i { [ l + y 2 1 1 / a + y } I/2

2z-i , y=

slightly

~i,

shifts

curves

of t h e The

while

its

within

in

and the

mechanism latter

of

show

dependence

of

parts

square to

is small

and

model

numerical gap

root

identical

do

Analysis with

of weak

for

damping.

for

e=0

Fig.l(b)

the

the

the

of

final

the

of with

lying Peierls

result

so

the

systems with a linearized

intensity

damping

both

difference

specifics

enter

coefficient

dependence

polymers

states, The

not

o n l y the

frequency

to all id d i e l e c t r i c

in s y s t e m s

divergences

nonzero

the

ground

prefactor.

formation

spectrum.

refraction

is

of X(3)are p r e s e n t e d

IX(3)1.

Consequently

nondegenerate

is a p p l i c a b l e

electron

imaginary

correspond

modulus.

this

degenerate only

Its m a g n i t u d e

w e n o t e t h a t the b a n d g a p A e n t e r s

through

X (3)

z=~ peak.

and

lines

curves

shows the frequency

B

real

fine

bold

Finally

the

(8)

2~

to i / 2

Fig.l(a).

[4],

diverge

in X(3):

[l+y2] I/2

proportional

Note

- i 19.11

= -4 2 I/2 ~

for n o n z e r o

-i/2

is d e f i n e d by

d(z)

while

to

n e g a t i v e v a l u e - B 38a.

T h e z =! ' r e s o n a n c e ' p r o d u c e s no d i v e r g e n c e 2 (3) 28 XTH c = B { p + q (2Z-I) + d(z) } p ~ 8.98 + i 2.81

(5)

will

dependent be g i v e n

index

of

in a f u t u r e

publication. After discrete Our

are

present

work

was

Su-Schrieffer-Heeger

curves

there and

the

agree some

with

First,

presented

in the

interval

z =~ r e s o n a n c e s the c u r v e of R e f . 8 2 the z e r o d a m p i n g result, w h i l e the

latter

and

is

learned

those

lower with

we

m o d e l was c a l c u l a t e d

qualitatively

differences.

completed

closer

to

for ours the

that

XTH c(3) in

recently in

[8],

between

IX(3)i g o e s

in

a

[8].

though i the z=~

considerably

practically experimental

coincides data

for

3762 (3) ~ T HG

(3) ZTH G I

v-

.j

I

i

I

3

2

I

Fig. l. The nonlinear susceptibility:

a) the real

(solid line)

imaginary

~ G(3) ("Z");

the

(dashed

line)

parts

of

b)

modulus

and of

(3) "z) XTHG(

•

polyacetylene

[3]. And second:

here Im~¢3)takes positive

the interval 0.35szs0.52 as in the c=0 regime,

while

values

in

in [8] it is

small and negative. REFERENCES 1

P.A.Chollet Nonlinear

et

al,

in

P.N.Prasad

Ovtic~

and

~lectroactive

and

D.R.Ulrich

Polymers,

(eds.),

Plenum,

N.Y.,

1988, p.121. 2

M.Sinclair, D.McBranch, (1989) D645.

3

W.S.Fann,

4

F.Kajzar, Phys.Rev.Lett.,62 (1989) 1492. W.Wu, Phvs.Rev.Lett.,61 (1988) 1119.

5

D.M.Mackie, R.J.Cohen and A.J.Glick, Phys.Rev. B,39 (1989) 3442.

6

Yu.I.Dakhnovskii

S.Benson,

and

D.Moses and A.J.Heeger,

J.M.J.Madey,

K.A.Pronin,

S.Etemad,

Svnth.Met.,~8

G.L.Baker

ProG.Symp. on Correlation

Effects in Low Dimensional Conductors and Superconductors, Kiev,May 1990, Springer, Berlin, in press. 7 8

and

H.Takayama, Y.R.Lin-Liu, K.Maki, Phvs.Rev.B,21 C.Wu and X.Sun, Phvs.Rev.B,41 (1990) 12845.

(1980) 2388.