Polarons in organic conjugated polymers

Solid State Communications, Printed in Great Britain.

Vol.52,No.2,

pp.99-I02,

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

]984.

POLARONS IN ORGANIC CONJUGATED POLYMERS M.C. dos Santos Departamento de Qu[mica Fundamental Universidade Federal de Pernambuco 50000 R e c i f e - PE - BRAZIL c.P. ~ Z o

Departamento de F[sica Universidade Federal de Pernambuco 50000 Recife - PE - BRAZIL u.s.

B,~ndi

Departamento de F[sica Pontificia Universidade Catolica Caixa Postal 38011 - Rio de Janeiro - R J - BRAZIL ( R E C E I V E D ON J U N E 2 6 t h 1 9 8 4 by R . C . C . L E I T E )

An unified treatment of the electronic structure of organic conjugated polymers based on a renormalization approach is presented• The changes in the electronic structures of trans-polyacetylene, poly-(p-phenylene), polypyrrole and polythiophene, brought about by the presence of polarons in these systems, are studied. For all systems, two localized states, which can be interpreted as the symmetric and antisyrmmetric combinations of soliton wavefunctions, are created at the central gap.

As a consequence of the fact that the electrical conductivity of trans-polyacetylene, trans-PA, c a n b e varied, b y effect of doping, over thirteen orders of magnitude [i], particular attention has been given to the study of conjugated organic polymers. Although a mechanism of solltens [2,3] is consistent with the observed transport properties of trans-PA [4,5], the discovery that other conjugated polymers, as poly (p-phenylene), PPP, could be doped to conductivity levels comparable to trans-PA [6] has made clear that a more general mechanism had to be found t o respond for the conductivity of these materials. Only in trans-PA a soliton will be free to move isoenergetically along the chain; all other systems do not present degenerated ground states and, hence, cannot house solitons. However, once a neutral and a charged soliton interact, they can form a bound polaron-like structure. Due to their local nature, polarons can exist in all conjugated polymers. In fact, polarons, either single or double charged, could respond for the observed properties of PA, PPP, polypyrrole (PPY) and polythiophene (PTP) [7,8]. In this letter we present preliminary results of an unified treatment of defects in conjugated polymers, based on a renormalization approach. This technique has been applied before in condensed matter physics to treat the electronic spectrum of disordered chains [9-11]. Here we use the renormalization idea to study the electronic stru~ ture of localized states associated to the presence of polarons in infinite conjugated chains. Starting from the model hamiltonian associated to trans-PA, •

H - 1%cic. i

÷

÷ % (vii'i% ÷

el)

Zl

tonians of the conjugated polymers to the same form given by Eq.l. Using Dyson's equation and the Transfer-Matrix approach, the expression for the Green's function for trans-PA has been derived previously [12] as g-(x G

= oo

99

_ (S+1)~][(c_~)2

(2)

_ (B_l)2]

where ~ and 8 are, respectively, the self-energy of a given site and the hopping parameters, defined in units of V I (See Fig.la). Formally, it is possible to substitute the structures presented in Figures Ib, Ic and Id hy the simpler "renormalized polymer" represented in Fig.le. For this is only necessary to define renormalized self-energies and bopping integrals after following a treatment similar to that of Reference 12 to relate the carbon atoms connecting rings. In Table i we present the analytical expressions for the renormalized parameters for the polymers of interest. The numerical values for these parameters which best fit an Extended-H~ckel band-structure calculation [15] are also presented in the same Table. In Fig.2 we present the simplest structures associated to the existence of polarons in the systems represented in Fig.l. Since we are interested i n the qualitative changes in the electronic structure of the polymers brought about by the presence of the polaron, we have confined the defect to only two rings. The parameters appropriate for each one of the "renormalized" polymers are shown in Table 2. In this manner, we reduce the other polymers to a structure formally equivalent to trans-PA, and therefore may use the transfer matrices T and T 2 derlved in prevlous works [12-14]. The set of equations for the Green's functions associated to the defect sites are •

where the parameters are defined as usual [12-14], we have the purpose of reducing all the model hamil-

, ¢[(c_~)2

.

,

1

lO0

POLARONS

IN ORGANIC CONJUGATED

Vol.

POLYMERS

52, No.

2

(a)

2 -I

0

I

2

3

~

VI

•

VI

8

(b)

V

2 e

~

I

0

I

2

I

2

4

~

5

e,

el -2

-I

0

0Cx

-2 '~=;~-I

0 v--J

OtX

(d )

I

2 ""--" 5

el

x

e3/x--

5 02

e_ A

el/"/'--~

82

---/X~

Va

-2

-I

0

I

2

5

4

5

{hi

-2

(el -2

4

O~x

,'d=:zz~e2

_A_Y,

Figure

3

eI

-I

0

I

2

3

-I

0

I

2

5

4

5

Figure 2 : (a) Polaron type defect structures for trans-PA, PPP and PPY or PTP, and (b) their equivalent renormalized chain.

1 : Schematic structures for the polymers: (a) trans-PA, (b) PPP-benzenoid, (c) PPP-quinonoid, (d) PPY and PTP. The renormalized structure is shown in (e). Hopping parameters are indicated over the bonds. Sites to he renormalized are numbered.

TABLE i. Self-energy (d) and hopping (B) renormalized parameters, e 81 e 8% are the hopping parameters defined in Fig.l. ~x is the self-energy of the heteroatom X. All parameters are gzven in unlts of V1•

Polymer trans-PA

~

B

0

1.32

PPP-benzenoid

2e2t/(e2-e 2)

2e3/(E2-82)

ppP-quinonoid

2 2 2 2e2~/(~ -61)

2 2 2 28182/(g -01)

fitted values

e = 1.087 81 = 1.000 82 = 0.830

PPY(PTP)

@21g/(2_e22)

+

e3~2/(E_~x)

82e21/(2_e22)

+ 8~/(c-~ x)

81 = 1.387 (1.411) 82 = 1.139 (1.188) 83 = 1.071 (0.940) ~

(g-e-Tl)Goo- V~G10 = i (E-~)c10

- V~Coo

(c-~)G20

- V~G

- V~G20

= 0

0 - V~C30

= 0

(3)

(g-~-T)G30-

V~G20 = 0

From the we obtain as usual the local den1 sities of states given by pi.(~)= -~- ImGi.(E) . The localized states are assgciated to theJpoles of Gij and, for all cases, two poles are found in the

x

= 1.791 (0.608)

central gap as presented in Fig.3. The corresponding wavefunctions for trans-PA are shown in Fig.4. They can be regarded as the symmetric and antisymmetric linear combinations of the soliton wavefunctions for these systems. As expected, we have found that charge is redistributed along the chain. The results presented above indicate the convenience of using renormalization ideas to treat the whole class of conjugated polymers from an unified point of view. A more complete version of this work will be presented elsewhere. We acknowledge the financial support of the Brazilian agencies FINEP and CNPq.

Vol. 52, No. 2

TABLE Z.

POLARONS

IN ORGANIC CONJUGATED POLYMERS

]0!

Self-energy and hopping renormalized parameters for the defect region, as stated in equations Fig.2b. All values are given in units of V I.

!

!

a

VI

V2

fitted values

0

i

1,32

-

2 0 22E I ( c 2 - 0 I2)

01

Polymer trans-PA PPP

(3) and

2010~/(E2_0~)__

01 = 1 . 1 9 6 O2 = 0 . 9 9 1

2 2 -O 2I) + e21(C-~x)3 O2c/(g

PPY(PTP)

(*)

01

(*)

Z 2 -Ol2 ) + 0 )2 / ( g3 - f i x 01021(s

See fitted values in Table i.

(c) (o) I

,

9

!

I

I

-2

-I

r

EF

2

E(V,)

-3

-2

-I

EF

E(VI)

I

'p (h

(d)

i I I I I

I i

I I I [ I I

it I'

I

1I

j

, i

-3

-I

E

-3 I

2

E

2

~1

VI }

Figure 3 : Local densities of states (arbitrary units) for the perfect chain Poo (---), and in the presence of the polaron for Poo (-) ; (a) trans-PA; (b) PPP; (c) PPY; (d) PTP. Energies in units of V I. Two poles splitt-off above and below the Fermi level gF"

EF

I

I 2

e(~.)

102

POLARONS IN ORGANIC CONJUGATED POLYMERS

W

W

0.4

.0.4

I i

!

I /

,-3 -2

II

Vol. 52, No. 2

1

'!

~J

I

3

5

6/

4 I I

/

I

I

C

l

"L 7

8 sites

/1 -5/

ii

I

-4

I ~ I II

(a)

/ ' -31 ,

. O. 2

/

IL 1"

-2

-I

2

/ o

!,

3 /]

|

I ~6

7

/I

l .

;

(b)

Figure 4 : Defect wavefunctions for trans-PA. (a) bonding (lower energy) and (b) anti-bonding (higher energy) localized states.

REFERENCES

I. 2. 3.

4.

5. 6.

7.

A.J. Heeger, Comm. Solid State Phys. IO, 53 (1981). J.A. Pople and S.H. Walmsley, Mol. Phys. 5, 15 (1962). H.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979) and Phys. Rev. B22, 2099 (1980). D. Moses, A. Denenstein, J. Chen, A.J. Heeger, P. McAndrew, T. Woerner, A.G. MacDiarmid , and Y.W. Park, Phys. Rev. B25, 7652 (1982). S. Etemad, A.J. Heeger and A.G. MacDiarmid, Ann. Rev. Phys. Chem. 33, 443 (1982). D.M. Ivory, G.G. M~ller, J.M. Sowa, L.W. Shacklette, R.R. Chance and R.H. Baughman, J. Chem. Phys. 71, 1506 (1979). R.R. Chance, D.S. Boudreaux, H. Eckhardt, R.L. Elsenbaumer, J.E. Frommer, J.L. Bredas and R. Silbey, Proc. NATO Adv. Study Inst. on "Quantum Chemistry of Polymers: Solid-State Aspects", July 1983.

8. 9. i0.

II. 12. 13. 14. 15.

G. Crecelius, M. Stamm, J. Fink and J.J. Ritsko, Phys. Rev. Letters 50, 1498 (1983). C.E.T. GonGalves da Silva and B. Koiller, Solid State Cor~num. 40, 215 (1981). B. Koiller, M.O. Robbins, M. Davidovieh and C.E.T. GonCalves da Silva, Solid State Commun. 45, 959 (1983). M.O. Robbins and B. Koiller, Phys. Rev. B27, 7703 (1983). H.S. Brandi and C.P. de Melo, Solid State C o ~ u n 44, 37 (1982). C.P. de Melo, H.S. Brandi and A.A.S. da Gama, Theoret. Chim. Acta (Berl.) 63, 1-8 (1983). C.P. de Melo, H.S. Brandi and A.A.S. da Gama, accepted by Theoret. Chim. Aeta (Berl.) M.C. dos Santos and C.P. de Melo, Solid State Commun. 50, 389 (1984).