Indian Journal of Pure & Appli ed Physi cs Vol. 40, May 2002, pp. 355-360

Mobile charge density wave model for third order non-linearity of 1t- conjugate polymers

Aranya B Bhattac herjee

Department of Physics, Atma Ram Sanatan Dharma College, University of Delhi , Dhaula Kuan, New Delhi I 10021

(EMai l: aranyab @bol.net. in . sibsagar200 1@yahoo.com)

Recei ved 23 October 2000; revi sed 2 1 February 2002; accepted 8 Marc h 2002

A theoretical one dimensional model hased on Huckel ' s molecular orbital theory appli ed to one electron de-localized n electron Hamil toni an in a conjugate chain of regul ar po lyene (ignoring electron-electron repulsion) is developed. It demonstrates the dependence of third order non-li nearity on the fo ll owin g: (a) the chain lengt h. (b) linear correlati on plot of v (C=C) versus IIAmax (parameters measured fro m linear spectroscopic technique of Raman and absorption spectroscopy). (c) ratio of the electron velocity to the sound velocity in the chain and (d) deformation potent ial coupling of the phonons to the on-site electronic charge density.

] Introduction

The n -conj ugate po lymeri c structures are known to have large va lues of non-resonant third orde r non-linearity. T hi s large third order hyper­polari zability has been shown to re fl ect the effecti ve

n-e lectron conj ugati on '·x. Because of the ir re lati ve ly large non-resonant non-I inear optical properti es, organic po lymers show promi se for applications in optical switching and logic operators for optical computing and signal process ingY

·11I When an

ex te rn a l uniform e lec tri c fi e ld of strength F ac ts

upon a mo lecule of polari zabili ty a, the induced dipole moment dev iates from a simple linear dependence on the fie ld and can be written as:

... ( I )

T he coeffic ients ~ and y (known respect ive ly as the second and third order hyper-po lari zab ility) are assoc iated with the strength of the non- linear response to applied f ie lds in the zero frequency

limit. S ince ~ is zero fo r systems with in version sy mmetry , non-linear optica l experiments for organic c rysta ls and po lymers have usua lly been

devoted to the de te rminati on of y.

The fi rst syste matic in vesti gation of y of conjugate systems was carri ed out by Hermann e f

ar'.This work generated considerable interes t in conj ugated structures and spurred theoretical analys is by Rustag i el 0/." who recogn ised the

importance of n-conjugati on in determining y. Since then severa l theore tical studies have been reported:1.5 ·7 . Inte rest centres on these theoretical

studies on enhanc ing y and hence it is of extreme importance to understand the re lati onship between

mo lecul ar structure and y in order to pred ict structures w ith large third order process. T he inte rest in the e lec tronic structure of po lyenes has grown tremendous ly after the suggesti on that conformationa l defec ts such as so li tons 12 and po larons " ·'.J should account for the unusua l transport properties .

Tn thi s paper, a theoreti cal one d imensional mode l based on Hucke l' s mo lecula r orbita l theory

applied to one e lec tron de-locali zed n -electron Hamiltoni an in a conjug:lte cha in of regul ar pol yene (ignoring e lectron-electron repul sion) is developed that demonstrates the dependence of third o rde r non-linearity on (a) the c ha in length , (b) linear

corre lati on pl ot of v (C=C) versus I IA,",,, (paramete rs measured from linear spectroscopic techn ique of Raman and absorpt ion spectroscopy), (c) ratio of the e lectron ve locity to the sound ve loci ty in the cha in , and (d) deformati on potenti a l coupling of the phonons to the on-si te e lectron ic charge dens ity.

Earl ie r attempts to pt ..... d ict y of organi c molecules fro m linear spectroscopic properties were made b a number of workersX.'5.'h.

356 INDIAN J PURE & APPL PHYS , VOL 40, MAY 2002

T

E ~

u

1560

1540

;,.~ 1520

1500

148 18,000 22POO 26,000

I/~r~x (em-I)

Fig. I - Correlati on of v c=c with I~ / i\ll1ax

2 The Theoretical Modlel

One can begin with an idea l case of a chain of N carbon atoms evenl y spaced a di stance a apart. Here a is the average C-C di stance. In a one-d imensional chain , it is suffic ient to assume interacti on between nearest ne ighbours onl y. In th is approx imati on, the

state of a single de- locali zed 1t-e lec tron in a conjugate cha in of carbon atoms is characterized by the Huckel type (n glecting e lectron-electron

interacti on) Hamiltoni an.

1-1 = 1-1 C' + 1-1 ,. + 1-1 ,.,. ... (2)

Here, 1-1 ,. is the e lectronic part, 1-1 ,. the vibrational

and 1-1 (' 1' the e lectron ie-vibrational coupl ing part. [n

the presence of an externa l e lec tri c fi e ld of strength

F , 1-1 e + 1-1 ev is written as:

" . (3)

Here, E'k (x;) = [Ek( x;) - eF x; ] , where Ek ( x;) is the

on-s ite molecular orbita l energy of the e" molecular orbital assumed to be identical for all carbon atoms .

-J ; (x;) is the e lectron transfer integral of the

'1" l C-C bond . at (a;) are the Fermion creati on and

annihilation operators. X; is the di spl acement of the

'1" I

displacement coordinate of the i l" carbon atom. The

vibrati onal Hamiltoni an can be wri tten in harmonic approx imation as:

... (4)

where M is the reduced mass of the carbon-carbon molecule, k the longitudinal e lasticity coeffi c ient and P; the momentum operator canoni call y

conjugate to the di splacement operator V; . The

Hamiltonian for the e lec tron-phonon inte raction.

( 1-1 ev ) may be written in the simples t case by

rewriting both J ; (x;) and E~ (x;) take n as funct ions

of the di spl acement x;. These can be ex panded in

Taylor Series as :

I " dEk(X;) Ek (x;) = Ek + LY ) - cFl .. =ox) + ......

. . dX,. I.)

" .(5)

" dEk ( X;) f . I ' ' bl F The te rm fi ( dX) ) or i =t= j IS neg Igl e. rom

Eg . (5), the transfer integral depends upon the separation between adjacent s ites i

::.i 7 For a

dimeri zed chain (i.e. broken sym met ry), the s ites move a lternately c loser together and furthe r apart. Thi s leads to two types of transfer integra ls:

BHATIACHERJEE; MODEL FOR 1t-CONJUGATE POLYMERS 357

1'1 1) f 0 J 1 = J II -- or V i+1 -Vi> 2

... (6)

For organic compounds, the bandwidth is given by 4J

II '" O.2IeV « 21'111 (the band gap '" 2.5 eV~6

eV). Further, C=C is known to be a true measure of conjugation , consequentl y the transfer integral is effectively taken as that for V i+1 -Vi < O. Under the

. 1'1 above assumptions, J '" _ II and

2

N

/-l eI = L U)aO;a i(V i+1 -Vi) } i=1

... (7)

where 8 = (X-eF), X = aCk IS an non-linear aXi

parameter specifyin g the strength of the deformation potential coupling of the phonons to the on-site electronic charge density and is identical for all carbon-carbon units. One can minimize the average value of H with respect to the wave function:

1111)= LL Ckie CT (t )ar o > ... (8) k

where Cki IS the k '" molecular orbital

(MO)coeffici ent at the i'" carbon atom and

sati sfy L lck;j2 = I ... (9) i.k

aCt) is an operator of the form:

where

The n-electron density fJ i on the i'" carbon and

bond order Pi of the / ' C-C bond are defined as:

.. . ( 12)

where summation is made over occupieu orbital s. Minimization procedure yields the foll owing rc.' ult:

... ( 13)

Eq . ( 13) in the continuum approximation along with the Hamilton equation of motion is reduced to the nonlinear Schrodinger equation (NLS):

a 1'1 a2 0

ih-- A +_11 -+CIC(e; ,'fC(e; ,,) = 0 al 2 ae

... ( 14)

. (x - VI ) whele A = ck - tl

ll, e; = , V IS the electron

a

I' S=-,

Va

o a 2 k v - =­

II M IS the

longitudinal sound velocity. Soliton so lutions of the NLS Eq. ( 14) ex ists for s < I, normali zed by the condition over the length L of the chain L

jIC(e; ,t)1 2 de; = I . The mobile charge density wave o

along the conjugate chain is determined by:

... ( 15)

... ( 16)

Now, only the electroni c and e lectron-phonon coupling part of the total Hamiltonian contributes to the non-linear process. Transforming Eq. ( 13) into continuum approximation for I = 0 and

L aO

(1111 /-1 .. +/-1".1111)= IRe (c*rA- 1'1;, ac;-1

1C-20ICI2 p }dC;

.. . ( 17)

One can obtain the tota l electroni c and e lectronic-vi hronic coup ling energy as:

2 2AN .1 0 1 • 6 E = ( I N - (--) - K - N ' l( X - er ) .

k ·1 A 2 (1_ .2), 3,1 0 n '\ II

.. . ( 18)

358 INDIAN J PURE & APPL PHYS, VOL 40, MAY 2002

where K = mv%, m is the electron mass. In

deriving Eq. (18), the following assumptions can be made: for ~ < 1, sech2 [1lN] "" 1 and tanh [1lNJ "" (IlN + 1l3pf/3). The energy can be rescaled and set Ek

= 0, also K2 is a small quantity and can be neglected . Now according to Hellman-Feynman theorem lx

:

J1=--= -HllfI)+(lfIl-llfI)+(lfIIH- 11 oE (Olfll oH IOlfl) of of of of

.. . (19)

The first and the third terms on the right hand side of Eq. (19) are zero if", is the true wave function. Consequently:

12e 2N' 1 J1 = ( 1 2 2 1 )[N + --leX - eF) ·

k · Ll,,(I-s ) . 3 ... (20)

Now, expanding the right side of Eq. (20) and comparing the coefficient of P with the right side of Eq. (1), one can obtain the third order hyper­polarizability y.

Irl=( 432e4x2 )[N+ 2N'] k' Ll ~, (I - S 2)' 3

. . . (21)

To convert Eq. (21) into esu units, the following conversion can be used:

y (esu) = y (S.I)x7 .16x I o I3/E" 3 !

The dependence of the longitudinal elasticity coefficient k on the conjugation length can be reproduced by first writing k in terms of the wave number of the C=C Raman mode, since C=C stiffness is a genuine measure of the degree of conjugation through the 1t-electron chain l9

.

with v<;< = 1459 cm l + (nO/N') cm- I

... (22)

... (23)

From v<;< versus A·lm. , correlation plot, where N'

= NI2 is the number of double bonds, M is the reduced mass of C=C (9.96xlO·27 kg), c is the velocity of light. The absorption maximum (A· l

m. ,) is taken as the band gap for 1t~1t* transition. Before proceeding to fit the experimental data to the author's model (Eq. 21), a rough estimate of X would be helpful. In order to obtain the order of magnitude estimate for the level of anharmonicity to

be expected in C=C vibrations. The C=C bond is assumed to have the anharmonic potential (about the minimum at xo ).

... (24)

Non-linearity enters our model as:

Hev = nxx . . . (25)

where n gives the number of quanta In the C=C bond .

kx2

Also nnw =­" 2

... (26)

where nwo = E" is the energy of C=C vibration.

From Eqs (24)-(26) X = 2n2ba" / at the

/EoM

minimum of V(x) , ao being the average C=C bond

length.

dV I k From Eq. (24) - .r=a = 0, thus Ibl =-dx" 4 2 ao

... (27)

Finally, one can have X = nk 2a"EoM

... (28)

Thus from Eq. (28), one can obtain the order estimate for any N as X = 10. 10 Newtons.

3 Results and Discussion

The best agreement between this model and the experimental results for y is obtained using

S2 = 0.95 and X = 1.14 X 10.10 Newtons. Table 1 shows experimentally obtained and calculated y values as a function of the number of repeat units. One can see that the agreement between experimental values and those obtained from this model is very good. The present results suggest that, onset of saturation of y of linear polyenes should occur for N > 24.The effect of the coupling constant on y is evident from Eq. (21). On increasing the value of X , the increase of y as a function of chain length is faster and saturation is observed only for large N. The influence of s on y is not yet clear and requires further investigation. In the present analysis of the power-law behaviour, the author has

used the general equation r = aN h . The parameter

BHATTACHERJEE; MODEL FOR 1t- CONJUGATE POLYMERS 359

a was found to be 5 .89 x 10.37 esu, while b was

est imated to be 3 .175 [compared to that predicted by free e lectron mode l( b = 5.0 [Ref. I I], simpl e Huckel theory (b = 5.257 [Ref.20)) , (b = 4.25 [Ref.5)) and (b = 3.2 -3.4[Ref.7)). The rati o of the

bandwidth 410 to the bandgap 2~o pl ays the rol e of

coherence length 21• <;(= 2aJo{) determines the

jt." range of nonlinear excitations. For linear pol yenes <; « ({ , consequently detai Is on the scale of the

lattice spac ings are important and diffe rent members of the po lyene series will ex hibit different

Tabl e I - Comparison between the experimentall y measured

and calcu lated values( from Eq. 2 1) of y as a fun ction o f the repeat units (N)

N k (N/m) ~,,(eV ) Yoxp(esu) Yl1lo,,(esu) 2 1680 4.13 1.32x 10 .. 37 1.32x 10.37

4 11 71 2.57 7.00x 10.3(, 6.3 1 X 1O.3fi

6 102 1 2.06 5.0 1x 10 .. 35 5.0 1x 10.35

8 95 1 1. 8 1.86x 10 .. 34 1.82x 10 ·3~

10 909 1.64 5.01x 10·3~ 5.0 1x 1 0·3~

12 882 1. 54 1.26x 10.33 1.00x 10·n

14 863 1.47 2.00x 10.33 2.00x 10·)3

16 849 1.41 3.09x 10.3.3 3. 16X 10.33

18 838 1.37 5.01 x 10.33 5.0 1x 10 .. 33

20 830 1.33 7.00x 1O .. n 7.94x 10.3.3

22 822 1.30 1.1 2x 10 .. 32 1. ll x 10.32

24 8 17 1.28 1.5 1x 10 .. 31 1.51 x 10.32

26 8 12 1.26 2.00x 10 .. 31 2.02x 10.32

features. Contrary to thi s, the author has found a s ingle value of the coupling constant, which can be

used to represent the IT-electron de-localizati on effect throughout the po lyene series. Thi s implies that each monome r unit bears the same structural correlation with its nei ghbouring units. Thi s justification may be true for linear po lyenes but not for other seri es such as benzene o ligomers . In thi s mode l the equality of C=C and C-C bonds has been assumed, but , stri ctly speaking, bond alteration

results in a reduced de-localizati on of IT-e lec tron s.

Thi s implies that va lues of X and s found from the least square fit are less than the true va lues.

4 Conclusions

In thi s paper, a simple mode l has been presented wh ich can be llsed to inte rpret the non-linear optica l behaviour of conjugated structures . The close connection that ex ists between linear and non-linear properties of conjugated organic molecules has been

demonstrated . Such an approach to predict non­linear parameters from linear spectroscopic data will enable the se lection of potenti a l conjugate systems from the a lready ex isting large amount o f linear spectroscopic data for use in non-linear applications. The method can very readil y be used

to ex plain the dependence of y on the c hain length by adjusting two parameters: the coupling constant

(X) and the rati o of the e lectron ve loc ity to the

velocity of sound (s). The dependence of y on s is

a new feature of thi s mode l. Despite the gross over simplification, the author is indeed surpri sed with the excellent agreement found be tween the va lues calculated usmg this mode l and those ex peri mentally observed.

5 Acknowledgement

This work was done within the framework of the Associateship scheme o f the Abdus Salam International Centre for Theore tical Phys ics, Trieste, Ital y.

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