Fate of solitons, polarons and bipolarons in conjugated polymers: The role of interchain coupling

Synthetic Metals, 28 (1989) D507-D512

D507

FATE OF SOLITONS, POLARONS AND BIPOLARONS IN CONJUGATED POLYMERS: THE ROLE OF INTERCHAIN COUPLING

DIONYS BAERISWYL* IBM Research Division, Zurich Research Laboratory, 8803 Ri~schlikon, Switzerland KAZUMI MAKI Department of Physics, University of Southern California, Los Angeles, CA 90089-0484, U.S.A.

ABSTRACT The effect of interchain coupling on the relative order of dimerization on neighboring chains and on the local states in conjugated polymers is studied within the framework of the SSH model including interchain electron transfer. Parallel ordering is preferred for alternating

interchain

hopping integrals.

In this case neutral solitons on neighboring

chains are strongly bound, whereas polarons are destabilized even for weak interchain coupling. INTRODUCTION It has been widely proclaimed that conjugated polymers offer a unique playground for various

nonlinear states

recognized

that

the

like solitons

arrangement

and

of the

polarons. polymer

At the

chains

same time

within

it has been

a three-dimensional

periodic lattice can affect the translational freedom or even the very existence of these states. This is quite obvious for topological solitons which connect two configurationaHy different but energetically degenerate phases in the case of a single chain but not so in the crystal where one of the two phases has lower energy. Thus solitons have to occur in pairs, close to chain ends or within amorphous regions. We have discussed

in an earlier paper both the interchain order and the soliton

confinement in terms of the SSH model [1] supplemenled by interchain hopping between adjacent chains [2].

Assuming that the hopping integral did not change sign along the

chains, we obtained anti-parallel ordering for the dimerization patterns of neighboring chains.

On

the

other

hand,

recent

X-ray

diffraction

experiments

on

well-oriented

trans-polyacetylene [3] not only indicate that the chains are rotated with respect to each other but also that the bond alternation on neighboring chains is in phase (parallel).

We

have subsequently realized that the interchain-hopping integrals not only alternate in size *Permanent address: Swiss Federal Institute of Technology, Physics, ETH-HSnggerberg, 8093 Zurich, Switzerland.

0379-6779/89/$3.50

Institute

of Theoretical

© Elsevier Sequoia/Printed in The Netherlands

D 508

but can also alternate in sign between even and odd sites [4]. indeed it is easy to show that

hopping

integrals

with

alternating

signs

lead to

parallel

ordering,

as

required

experimentally. Recently it has been argued on the basis of scaling arguments that interchain coupling can readily destabilize polarons [5]. this conclusion.

More detailed calculations [6] have substantiated

In this paper we briefly explain the effect of parallel ordering on the

confinement of solitons and explore the consequences for tile stability of polarons and bipolarons. Tile two-chain problem can be mapped on the solvable model of the AB p o l y m e r chain introduced by Rice and Mele [7]. We find that the polaron is unstable for very small interchain-hopping integrals, in agreement with the earlier work of Gartstein and Zakhidov [6]. RELATIVE ORDER OF NEIGHBORING CHAINS We consider two neighboring polyacetylene chains described by the Hamiltonian H = HI+H2+H'

1

H, =

,

(1)

+ KE. U,n- °,o+,/2- Et,,.(C,nC,n. n

+ he). j = '2

-Xtin(Cto c2° + h c )

.'=

(2)

n

~3/

n

where Ujn and Cjn, respectively, are the displacement coordinate of a CH group and the annihilation operator of a ~-electron on the j-th chain at the n-th lattice site. Spin indices are omitted for simplicity.

The transfer integrals tll n and t j_ n are assumed to have the

form

tll n = t o + -~--( -1)nAe + cc(Ujn -- Ujn+l ) ,

(4)

tin

(5)

= t 1 +(--1)nt2

,

where a finite external field &e represents the two inequivalent bonds of cis-polyacetylene (or similar situations in other conjugated polymers like polythiophene). The form of the transverse hopping is deduced on the basis of geometrical considerations [4]. For parallel ordering (Uln = U2n = (-1)nu0) the energy gain per site due to interchain tunneling is AE (p) ~ - (t22 / ~t0) In(8t 0 / 4)

where

,

(6)

D509

(7)

A = A e + 4c~u0 is the (single-chain) gap p a r a m e t e r , w h e r e a s for anti-parallel o r d e r i n g we find

(8)

AE (a) ~ z&E(p) 4- (t 2 - t ~ ) / ( ~ t o ) .

This i m m e d i a t e l y we

limit

implies that parallel o r d e r i n g

ourselves

to this

case.

Moreover

is preferred for t 2 > t 1. we

consider

situations

In the following where

the

order

parameters

Ajn = A e 4- 20~( -1)n(ujn - Ujn.Fl )

(9)

are the s a m e on both chains, I.e. Aln = A2n = An.

It is then helpful to introduce bonding

operators

b n = 2 - ' / ' ( c l n 4- C2n)

(10)

and anti-bonding o p e r a t o r s

a n = 2 - % ( C l n - C2n)

(11)

in t e r m s of which Eq.(1) reads

H

-~K~(ujn nj

- Ujn+l) 2 + tl(N a - Nb) + (12]

,7___d[tin(an+an

--

b n + b n )-t-

iIn(an+ an.4_1 nL bn+ bn4_1 ~l- h c ) -

,

n

where

N a and N b are the

number

of electrons

in lhe

a- and

b-bands,

respectively.

B e si de s the extra e n e r g y p r o p o r t i o n a l to t 1 this H a m i l t o n i a n c o r r e s p o n d s to t w o u n c o u p l e d diatonlic p o l y m e r s introduced by Rice and M e l e [7]. For the h o m o g e n e o u s

ground state

(An = A0) the quasi-particle s p e c t r u m is

Ek~ ) =

+tl+[(2t

0 cos k ) 2 + & 2 + ( A 0

sin k ) 2 + t ~ ] v~

(13)

SOLITON MOLECULE We s h o w n o w that, in the case of parallel ordering, the interchain co u p l i n g leads to a strong binding of neutral solitons on n e i g h b o r i n g

chains. C o n s i d e r t w o kinks on adjacent

chains, both kinks having the s a m e shape and the s a m e center site. The H a m i l t o n i a n can then

be written

in the form

of Eq. (12) which

admits

exact

soliton

solutions.

In the

D510

t e r m i n o l o g y of Rice and Mele [7] the soliton pair oil the two chains is transformed to an A-soliton on the a-chain and a B-soliton on the b-chain or vice versa. In the continuum limit [4,7] the energy of this complex relative to the ground state is E(2S 0) = 2E0(S 0) - (4/=)(t 2 _+ t l ) t a n - l ( A / t 2 ]

where E0(S 0 ) = ( 2 / = ) A from

the

charge

,

(14)

is the soliton energy [6] of an isolated chain.

irnbalance

between

the

a-

and

b-chains

for

The + sign arises the

two

possible

configurations. The binding energy Eb ~ 2 ( t 2 + t l ) , t

2
,

(15)

is simply the bonding energy provided by the splitting of the midgap levels. Therefore a pair of neutral solitons on adjacent chains forms a tightly bound molecule.

For a pair of

(positively or negatively) charged solitons there is no bonding. In this case the binding is provided by the confinement potential [3] Ec(C) ~ 2~ coth (~'/~) (t 2 - t 2)/(~VF)

(16)

,

where # is the distance between the solitons and ¢ = vF/A is the coherence length. Since this energy becomes appreciable only at a certain distance the relative motion of charged solitons is less restricted than that of neutral solitons.

On the other hand, a charged and

a neutral soliton on adjacent chains are again expected to be strongly bound, their binding energy being about half that of two neutral solitons. This suggests that a transport mechanism

involving the separation

of neutral

and charged

solitons

on

neighboring

chains [9] is unlikely to be relevant in crystalline samples. POLARON INSTABILITY The polaron energy [10,11] is only weakly affected by interchain coupling as long as the inhomogeneous lattice distortion does not leak out to adjacent chains [2].

We show

now that, for parallel ordering, the extension of the inhomogeneity in the transverse direction can lower the energy due to the splitting of the gap states. We consider again a system

of two chains

and assume that they support tile same polaronic

distortion.

Generalizing the polaron solution of the AB polymer [12] to the case of a two-chain structure we find Ep = (8/~)A 0 I(yA/Ao) t a n h - l [ ( A o / A ) sin ,9] +

(I

-- 7) sin ,9 +

[0z/6)(4 4- n+ --n_) - ,9] cos ,9} + (N a - Nb)t , ,

(17)

where n+ and n_ are the total occupation numbers of the upper and lower gap states, respectively, A 0 = ( A 2 + t22)'/~ is the gap parameter and 7 ~ Ae/(2)-A)(with ~ = 2cc2/~t0K) is the confinement parameter of Brazovskii and Kirova [10].

The variational parameter $

DSII

determines

the e x t e n s i o n

x 0 of the s t r u c t u r e

in chain

direction

(in units of the

lattice

constant) t h r o u g h (18)

A 0 sirl ,9 = A t a n h [(A0/t0)x 0 sin ,9] .

Eq. (17) can also be used for d i s c u s s i n g ),-+ 0 , solitons. assurne y -

We

neglect

o t h e r states like b i p o l a r o n s or, taking the limit

n o w the w e a k d e p e n d e n c e

O. We c o n s i d e r a d d i t i o n a l e l e c t r o n s (v =

oil t 2 and. for simplicity,

also

1,2) which are d i s t r i b u t e d o v e r m

c h a i n s s u p p o r t i n g identical p o l a r o n i c d i s t o r t i o n s . The gap levels split into two bands. The additional

e l e c t r o n s are a d d e d at the b o t t o m of the u p p e r band, i.e. at A c o s , 9 - / T m t 1

w h e r e /7m i n c r e a s e s from 1 for m = 1 to 4 for m -> ,,-~. Thc~ e n e r g y of such a structure is E(m, v, ,9) = ( 4 m / ~ ) A ( sin ,9 - ,9 cos ,el] + v(A cos ,9

flint1)

(19)

.

M i n i m i z i n g with r e s p e c t to ,9 y i e l d s D = (Trvl4m) and E(m, ,,)

(4m/~)A sin(nv/4m)-

=

v/Trot 1 .

(20)

This function d e c r e a s e s m o n o t o n i c a l l y with i n c r e a s i n g m and a s s u m e s its m i n i m u m E(v) = v ( A -

for m ~

(21)

o~. C o r r e s p o n d i n g l y ,

conduction with

4tl)

the

the gap states

bands. This e n e r g y for c o m p l e t e l y polaron

(bipolaron)

or b i p o l a r o n

energy

for

m o v e to tile e d g e s delocalized

a single

chain.

charges

of the

valence

and

has to be c o m p a r e d

It f o l l o w s

that

the

polaron

is s t a b l e as long as the binding e n e r g y is larger than the e n e r g y g a i n e d by

the t r a n s v e r s e d e l o c a l i z a t i o n of the a d d e d e l e c t r o n s . This yields the stability criteria

(22)

for the case of a p o l a r o n , and

tl < ~-{1 - (2/,,)b = 0.0gt A

(23)

for the case of a b i p o l a r o n . T h e s e two r e l a t i o n s a g r e e with the r e s u l t s of G a r t s t e i n and Z a k h i d o v [6] and with Emin's s c a l i n g a r g u m e n t s [5]. DISCUSSION We have s h o w n that i n t e r c h a i n t r a n s f e r integrals of a l t e r n a t i n g signs lead to a parallel o r d e r i n g of d i m e r i z a t i o n , as o b s e r v e d e x p e r i m e n t a l l y . for

the

localized

excitations

of c o n j u g a t e d

This has f a r - r e a c h i n g c o n s e q u e n c e s

polymers.

Solitons,

which

experience

the

D512 confinement potential of the misaligned segment separating them, can in addition be strongly bound due to the splitting of midgap levels. Thus two neutral solitons (or a neutral and a charged soliton) on adjacent chains form a "soliton molecule". Similarly, polarons are easily destabilized by interchain coupling. In facl, for a gap of the order of 2 eV the non-alternating component of the transverse hopping integrals has only to be larger than about 25 meV to delocalize completely the additional charge. On the other hand, for bipolarons the transverse coupling has to be larger by nearly a factor of 4 to destabilize the local state. Various groups have found evidence for gap levels in polythiophene on the basis of photo-induced and dopant-induced optical absorption experiments and interpreted their data in terms of polarons or bipolarons, or both [13-16]. It is not yet clear to what extent the idealized model used in this paper is appropriate for- discussing the experimental findings. The potential of dopant ions or of other impurities could play an important role in stabilizing polarons and bipotarons. Oil the other hand Coulomb interactions between the added charges could be detrimental to bipolarons. REFERENCES 1

W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev~B , 22 (1980) 2099; Erratum Phys. Rev. B, 28 (1983) 1138.

2

D. Baeriswyl and K. Maki, Phys. Rev. B, 28 (1983) 2068.

3

H. Kahlert, O. Leitner and G. Leising, _Synth. Met.± 17 (1987) 467.

4 5 6

D. Baeriswyl and K. Maki, Phyhs. Rev. B._, to be published. D. Emin, Phys. Rev. B, 33 (1986) 3973. Yu. N. Gartstein and A.A. Zakhidov, Sol. St. Cgmm___un.,__62(1987) 213; Erratum ibidem,

7

65 (1987) it. M.J. Rice and E.J. Mele, Phys. Rev. L ett~ 4_9 (1982) 1455.

8

H. Takayama, Y.R. Lin-Liu and K. Maki, _Phys. Rev.__BL2_!. (1980) 2388.

9

S. Kivelson, Phys._ Rev. B, 25 (1982) 3798.

10 S.A. Brazovskii and N.N. Kirova, Pis'ma Zh. Eksp_. Te_o[..Fi_z~,33 (1981) 6 [_Soy. Phys. JETP Lett., 33 (1981) 4 ]. 11 D.K. Campbell and A.R. Bishop, ~ R _ e v _ . BB~24 (1981)4859. 12 D.K. Campbell, Phys. Rev. Lett., 50 (1983) 865; see also T. Martin and D.K. Campbell, Phys. Rev. B, 35 (1987) 7732. 13 T.-C. Chung, J.H. Kaufmann, A.J. Heeger and F. Wudl, P__I~s. Rev. B, 30 (1984) 702. 14 T. Hattori, W. Hayes, K. Wong, K. Kaneto and K. Yoshino, J. Phys. C, 17 (1984) L803. 15 G. Harbeke, E. Meier, W. Kobel, M. Egli, I-I. Kiess and E. TosaLti, Sol. St. Commun., 55 (1985) 419; G. Harbeke, D. Baeriswyl, H. Kiess and W. Kobel, Physica Scripta, T13 (1986) 302. 16 Z. Vardeny, E. Ehrenfreund, O. Brafman, M. Nowak, H. Schaffer, A.J. F. Wudl, Phys. Rev. Left., 56 (1986) 671.

Heeger and