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.
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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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