Disorder and density of states in conjugated polymers

Synthetic Metals, 41-43 (1991) 3425-3430

3425

DISORDER AND DENSITY OF STATES IN CONJUGATED POLYMERS

K. FESSER Physikalisches

Institut, Universit~t Bayreuth,

D-8580 Bayreuth

(F.R.G.)

ABSTRACT Electronic analyzed,

states

Different

approaches

for

in

conjugated

polymers

containing

impurities

types of disorder,

bond and site are considered.

average

impurity

the

over

the

Possible mechanisms for the metal-insulator

distribution

are

are

Various

presented.

transition as function of doping

are discussed.

INTRODUCTION The enormous arisen state

from

interest

the fact

in conjugated polymers

that upon doping

to a highly conducting

this metal-lnsulator

(M-I)

attempts

a change

over from

state can be achieved.

transition have

been

is,

however,

made

in

the last ten years has a pristine

insulating

The detailed

nature

of

not yet fully understood

although

various

behavior.

In this paper some of these approaches are reviewed,

the

past

to

explain

this

the main focus

will be on the role of disorder as introduced during the doping process. In the first section the model will be introduced and some general remarks on

the

relation

to

real

systems

will

be

made.

Then

the

results

for

the

density of states will be presented in case the impurities are treated within Born-approximation. the

coherent

simulations.

In a subsequent

potential Finally,

we

chapter we will

approximation want

and

to discuss

compare different

discuss these

the results with

approaches

of

numerical to

the

M-I

transition problem and address some open questions.

0379-6779/91/$3.50

© Elsevier Sequoia/Printed in The Netherlands

3426

MODEL In

a

simplified

conjugated predictions its

of

merits

this

many-particle

ground state

This

is

and e l e c t r o n i c

Justified

oscillating

impurities.

factor

The n u m e r i c a l

The d i s o r d e r impurities:

bond d i s o r d e r

neighboring

whereas

site

diagonal

and o f f - d i a g o n a l

sites

disorder

scaled

for all

the is

during

the

doping

the value

is

in

ingredients

mind

(Fermi

of

that

discussed

a

later

(TLM) [3] v e r s i o n of

even this

lattice

true simple

should

in

the

up t o a

presence

creates

the

on-site

~-orbital

two

continuum energy

of

is valid.

the hopping matrix in

of this

dimerization

lattice

approximation

process of

as

aspects

conducting polymers.

off-diagonal

the

experiments

c o m p r i s e s t h e wide c l a s s

variation

show t h a t

types

of

element version)

leading

to

terms in the continuum limit.

t h e model can be w r i t t e n

, dx @+(x)(hTL M + h lmp)~s (x) + ~1

H = ~

some

The r e s u l t s

component)

modulates

modulates

keep

conjugated,

spatial

the

physical

slower than the underlying

(optical

(this

to

use the continuua

simulations

introduced

between

Properly

if

wave f u n c t i o n s

want

already

and K i r o v a a l s o

polymers.

p a p e r we s h a l l

with

of

A l t h o u g h some

h a v e t o be i n t e r p r e t e d

include

We a l s o

description

is widely used.

quantitatively

parameters

be r e g a r d e d a s g e n e r i c

tight-binding

[1]

t h e most r e l e v a n t

which

[2] due t o B r a z o v s k i i

Throughout this

rapid

sense its

parameters

of non-degenerate

model.

agree

model w i t h

interactions.

generalization

u-electron SSH model

model do n o t

In this

effective

therefore

particle

as a tractable

are unchanged. liquid)

one

polymers the familiar

in the form

dx A2(x)

S

w i t h @ (x) a t w o - c o m p o n e n t s p i n o r of spin

s,

A(x)

the dimerized

coupling constant.

h

TLM

describing

lattice

The e l e c t r o n i c

right

distortion,

and l e f t

moving e l e c t r o n s

and A t h e

electron-phonon

Hamiltonians are

= - i~ a + o" A(X} 3 x I

for the free part,

and the i n t e r a c t i o n w i t h the impurities is

hlmp = Z 6(x - Xl)[Ub~ 1 + Us(1 ± ~2)] [

with U

b,s

random, odd s i t e .

b o n d and site impurity strength, and t h e

± sign

in

the

last

part

respectively.

depends

The p o s i t i o n s x

on w h e t h e r

x

l

is

l

are

an e v e n o r

3427 The

dimerization

amplitude

has

A(x)

to

be

determined

selfconsistently

through ~E O = ~

tot

= ~



.

BORN APPROXIMATION Using

the quasi-classical

differential

equations

can

Green's function approach be

derived

for

the

[4] a coupled set of

impurity

averaged

(matrix

component) Green's functions

0

3

0

x

b

= 2i A(x) b

1

b

= 20

x 4

x

b

b

-

5

= - 2ob

5

where

only

4

21A(x)

b I

4

b

5

+ 2i/T

b

b 1 5

4,

bond

concentration),

disorder

has

bee n

taken

into

account

(I/T

~

c U2 b'

C

In this formulation the selfconsistency equation reads E

c

= - ~/2=i y

A(X)

- 2i/~ b

do b s ( O , x )

with a cut-off E A

(i.e.

being the full band width. With the assumption of a uniform c relaxation of the lattice around an impurity) the problem of

no

nonmagnetic

impurities

impurities connected

in a polymer can be mapped onto the case of magnetic

in a superconductor to

left-

and

[4]. (The broken

right-moving

electrons

time in

reversal

the

symmetry

continuum

is

model.)

Consequently the dimerization amplitude and the electronic gap are no longer related as in the pure system,

one finds that upon doping the electronic gap

closes whereas the dimerization as the relevant order parameter decreases and stays finite within a certain concentration range (gapless Peierls phase).

T-MATRIX AND CPA It is well known that the Born treatment this

problem,

electronic

rather

state

one

expects

that

is only a poor approximation

around

in the gap will be formed.

an

impurity

to

a

localized

In the case of many

impurities

this will finally lead to an impurity band within the gap. In order

to go

beyond

the Born

approximation

a single

considered with the help of the correspondin E t-matrix. follow

impurity

The localized

immediately from the energy poles of the t-matrix.

can

be

levels

Since all Green)s

3428 functions

are

only

2x2 m a t r i c e s

can be computed e x a c t l y V 1 t - N I - EV

t h e y c a n be i n v e r t e d

1 g = N ~ G°[k'

'

easily

and the

t-matrix

[5]

iEn )

k

with V the impurity potential bond

and

site

sufficiently The

impurity

averaged

potential

concentration

through

c(V

-

-

and

the

g(V

-

Green's

reduces

~)]-1

-

for

states

in the gap carries

order

the

single

parameter

to

(1

function

before

the

Green's

account

function

approximation.

this

determined

Z)[1

into

we

Taking now both

find

that

only

for

within

the

strong site strength such localized gap states exist.

impurity

coherent

and N the size of the system.

scattering

follows

is now

low

Born

C)~(1

+ g

I) -1

=

been

treated

as

and/or

self-energy

The

finding

Z.

concerning

the

o v e r t o t h e many i m p u r i t y has

strength

The

Z

is

0

o

case

determined

impurity

treatment.

f r o m ~-1 = G-I

impurity

h

For

the

-

G

a

existence

case.

space

discussed

of

localized

Note that

also

independent

here

(uniform)

function. Physically than

in

one

the

concerned

an

this

and f i n a l l y

that

free

impurity

concentration

the

finds

impurity

the

band

the electronic

by r e c e n t

parameter far

it

as

(dimerization)

the

in

the

joins

the

gap closes.

transition.

have been confirmed

as

appears

band b r o a d e n s ,

metal-lnsulator

order

system;

electronic gap;

upon

conduction

numerical

of

this

simulations

smaller

increasing (or

are the

valence)

Thus o n e h a s a d e t a i l e d

The p r e d i c t i o n s

is

properties

band

description

analytical

of

approach

[6].

OTHER APPROACHES In an early have

used

attempt

procedure,

however,

selfconststently. concentration and

broadening

applicability nondegenerate consequently lattice phase.

to understand

variational

band.

trial does

not

By c o u p l i n g they

have

of of

a dopant model

ground

state

has

two b a n d s

They

also

for

treat

the

described

this

the M-I-transition

functions

induced since

the

form the

of

sollton the

in

the the

gap

functions

This

nonlinear

polarons; from

existence

the

dopant

the

might

formation limit

excitations a

of

[7] This

interaction to

through

lattice.

different

possible

and Rice

distortion.

electron-lattice these

generic are

Mele

lattice

M-I-transition

system

found

the

polaron

the in

lattice

single

soliton

a gapless

Peierls

the

a

3429 The

influence

properties numerically also

of

single impurity

a

on

both

by Philpott

predict

the

et al.

of

"ultragap"

respectively).

of such a state requires

electronic

structure

contains

than

Just

conduction

more than one impurity

over

the

impurity

valence

distribution

distribution

to those obtained with more conventional

investigations The

results,

impurity

studies

by Xu and

technique.

to treating

the effects

the results are therefore methods.

a result which as Ziegler Such a pseudogap,

The

similar

They obtain a pseudogap

at

[iO] claims is related to the use

however,

does

not

appear

in other

using the continuum version of the SSH model.

analogy

amorphous

a real

path integral

is equivalent

the Born approximation,

Since

has been performed

of a Gaussian

model.

and

of the many impurity case.

assumption

the Fermi energy,

level they

valence

bands.

on a chain single

[9] with the help of a supersymmetric

of a continuum

(beyond

to include more of the true

and

Trullinger

of disorder within

state

In the spirit of the original SSH model

are only a first step towards an understanding The average

localized

states

a reliable description

ground

has been studied in detail

[8]. Besides an intragap

existence

conduction band edges,

material

(homogeneous)

as well as kink and polaron excitations

drawn

by

Bryant

semiconductors namely

the strong

kink excitations

[7,8]

and

appears

Glick

to

be

[II]

too

interaction

of

and the possible

of

conjugated

simple

impurity

in

the

states

suppression

polymers

light

of

with polaron

of such

to

these

impurity

and

levels

[5]. FUTURE WORK As

stressed

earlier

obtained by assuming

most

results

that around an impurity the original et

al.

such

in

the

effects

single

kink)

the gap from

excitations

impurity/many

to

transfer

by Mele

One advantage

the Fermi

impurity

other

(kink)

hand

case

from

and Rice

other than numerical

wavefunctions

the

It

for

the dopant

[7] and

paper

have

One expects,

been

however,

The

important

interaction

of

on one hand

and

has

studied

a

to

complete

however, to

recently

be

the

to

the

localized

from polaron in

the

understanding

to include polymer

by Conwell

include

of

Coulomb

chain

as

(and many the

effects has

and Jeyadev

been

[12] by

studies.

of numerical

calculations

can be extracted

energy

this

is therefore

treatment.

the

in

parameter.

lattice is deformed as shown by Philpott

case.

It will be complicated,

charge

proposed

on

polaron

M-I-transition. due

impurity

in an analytical

levels within

presented

a uniform dimerization

quite

(top of valence

are much more sensitive

is that the spatial

easily.

and bottom

to localization

We have

found

of conduction

extent

of the

[6] that

around

band)

than the states deep

the states

in both bands.

3430 In addition valence and conduction regions: is

conceivable

models

band states tend to localize

around the impurities and in impurity free regions,

for

that

the

using

this

conductivity

information

in

these

connections

materials

such

in different

respectively.

can as

be

made

variable

It

with range

hopping.

ACKNOWLEDGEMENTS I

am

very

much

collaboration. (TOPOMAK,

indebted

This

Bayreuth)

work

to

was

Y.

Wada

performed

and

K.

Harigaya

wlthin

the

for

auspices

a of

fruitful SFB

213

of the Deutsche Forschungsgemeinschaft.

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