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Journal of Non-Crystalline Solids 137&138 (1991) 927-930 North-Holland
IOURNA L OF
NON-CRYSTALLINE SOLIDS
THE ORIGIN OF EXTENDED STATES IN CONDUCTING POLYMERS
Philip PHILLIPS and H.-L. WU
Rm. 6-223, Department of Chemistry, MIT, Cambridge, Mass. 02139
Abstract: We show in this lecture that the metal transition in polyacetylene and polyaniline can be explained by a single disordered model that possesses a set of conducting states in 1-dimension.
1. STATEMENTOF PROBLEM In this lecture, I am going to focus on the
insulator-metal transition in a class of amorphous
semiconductors that are perceived as somewhat ancillary to the principal concerns of this conference. I will focus on trans-polyacetylene and poly-aniline and show how the metal transition in these systems
is described by a disordered model that supports a set of current-carrying states in 1-dimension. Let us start by reviewing the transport properties of trans-
polyacetylene. 1 In its undoped form, polyacetylene is a semiconductor with a band gap of 1.2eV. High
conductivities on the order of 1000 S/cm are acheived when the polymer is either n or p-type doped. In its doped state, however, polyacetylene is different from other kinds of semiconductors in that solitonic excitations (which constitue a moving domain wall) are favoured above electron or hole- like excitations because they cost less energy to create. Such excitations populate the mid gap states. Because charged soliton excitations are
spinless, they are consistent with the lack of any observable spin susceptibility when the polymer is
lightly-doped. 2 Although the soliton model has had
much success in describing the electronic properties of lightly-doped polyacetylene, many fundamental issues associated with the metallic state in the
heavily-doped polymer remain unresolved.3 Probably the most dramatic change observed in the heavily-doped state is the sudden emergence of a temperature-independent spin susceptibility at a doping level of 5-6%. The sudden appearance of a
metal-like Pauli susceptibility has been interpreted
as a first-order phase transition. 3 More importantly,
however, the onset of a Pauli magnetic susceptibility indicates that in the heavily-doped material, the
band gap closes thereby giving rise to a non-zero density of states at the Fermi level.
An early proposal to explain the metallic state in heavily-doped polyacetylene is the polaron model of
Kivelson and Heeger.4 Although this model would account for the Pauli susceptibility, it is inconsistent with the intense IRAV modes (the signature of soliton excitations) observed in the experiments of Kim and
Heeger.3 Additional models have attempted to
address the problem of the apparent closing of the band gap as the dopant level increases. Numerous
authors have pointed out qualitatively that the soliton band merges with the valence band as the doping
level approaches 10%.5, 6 Inclusion of interchain
coupling is generally thought to decrease the dopant concentration (to the experimentally-observed value of 5%) at which the midgap states become populated.5,6 Such heuristic arguments, however,
cannot by themselves explain the origin of the Pauli susceptibility. The abrupt onset of a Pauli susceptibility will occur naturally upon doping only if 1) increasing the dopant level closes the band gap thereby creating a finite density of states at the Fermi level, and most importantly 2) the electronic states at
the Fermi level are extended. Mele and Rice 7 have
shown that the disorder inherent in the random distribution of dopant ions is capable of partially closing the band gap. They found, however, that the
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
928 P. Philips, Z-L. Wu / Origin of extended states in conducting polymers
states at the Fermi level were Iocalised and hence
incapable of accounting for the observed Pauli
susceptibility.
The role of disorder, however, cannot be so easily
dismissed because the work of Saldi8 demon-
strates clearly that the distribution of dopant ions can
be incommensurate with the lattice. This suggests
that randomness in the distribution of dopant ions
gives rise to a random distribution of solitons. In an
attempt to understand the role of such disorder,
Lavarda, et al.9 considered single strands of trans-
polyacetylene with 3000 repeat units containing
various concentrations of randomly-placed solitons.
For the case of randomly-placed neutral or charged
solitons, they concluded that the soliton band merges
'with the valence band at a doping level of ~5-6% 9.
It is the merging of the soliton with the valence band
that gives rise to a non-zero density of states at the
Fermi level. The results of Lavarda, et. al.9are
significant because 1)the Pauli susceptibilty appears
experimentally at a dopant concentration of 5-6%
and 2) their results represent the first time it has been
shown that a single strand model for polyacetylene
can account for the closing of the band gap upon
doping. Lavarda, et al.9 conclude that disorder in the
distribution of solitons could play a role in the
insulator-metal transition if the states at the Fermi
level are now extended. It appears then that an
explanation of the metallic state in polyacetylene
requires a model which is of low dimension,
disordered but nonetheless supports a set of
extended, current-carrying states. Such a model
would necessarily be in contrast to the standard
view that disorder precludes the presence of
extended states in 1-dimension. 10
Another polymer whose metallic state appears to
require such a model is polyaniline. Polyaniline
refers to the general class of nitrogen-connected
aromatic rings of benzeneoid and quinoid character.
Upon electrochemical oxidation of the leucoemeraldine or acidification of the emeraldine
parent forms of the polymer, polyaniline undergoes
an insulator-metal transition. 11 Experimentally, it is
clear that the metallic phase is characterized by a
conductivity that is on the order of 10 S/cm, a factor of
1011 greater than the conductivity in the insulating
forms of the polymer. 12 Though complete consesus
has not been reached on the nature of the metallic
state, it seems fairly well-established that bi-polarons
are the stable excitation. 13 The role of disorder in
the distribution of bi-polarons in the metallic state of
this polymer has recently been examined by Galvao,
et a1.14 They conclude that 1) the peak in the
absorption spectrum at 1.5eV is intrinsically tied to
disorder and 2) upon protonation of the emeraldine
form of the polymer, the Fermi level moves within the
band to a region where extended states are
located.14 They conclude then that the insulator-
metal transition in polyaniline arises from the
movement of the Fermi level (upon protonation or
oxidation) to conducting states located in the vicinity
of the band edge. The results of Galvao et al. 14
provide another clear example that somehow the
presence of disorder does not preclude the
existence of extended states in one dimension.
2. RESOLUTION
We want to show that the origin of the extended
states in a disordered soliton or bi-polaron lattice can
be explained by a simple model which is a slight
twist on the standard binary alloy. In the standard
binary alloy, two site energies e a and e b are
assigned at random to the lattice sites. A constant
nearest-neighbour matrix element V connects the
lattice sites. It is well-known that none of the states in
this model carry current. 10 Consider the following
trivial change in the binary alloy in which pairs of
lattice sites ( that is, two sites in succession) are
assigned ~a and eb, respectively. We refer to a lattice
in which at least one of the site energies is assigned
at random to pairs of lattice sites as the random
dimer model. 15 We have shown 15 that this simple
twist on the binary alloy leads to a set of extended states which ultimately lead to near ballistic transport
P. Philips, J.-L. Wu l Ofigin of extended states in conducting polymers 929
of an initially-localised particle. To show how the
extended states arise, we compute the reflection
coefficient
IR[ 2 = W2(W+cosk) 2 W2(W+cosk)2+W2sin2 k (1)
through a dimer defect, where W=(ea-Eb)/2V. As is
evident the reflection coefficient vanishes when
W=cosk or equivalently the electronic state with
wavevector ko=cos - l w will be unscattered by the
defect provided that -2V<ea-Eb<2V. The vanishing of
the reflection coefficient obtains because at the wave
vector ko the scattering from the secondsite in the
dimer is 180 degrees out of phase from the scattering
from the first. It is straightforward 15 to show that the
single dimer resonance survives when the dimers
are randomly placed. In fact in the random system,
the total number of states that are unscattered by the
disorder scales as ~ , where N is the length of the
sample. The random dimer model, then, represents
a clear example of a 1-dimensional model which
sports a set of conducting states in the presence of disorder.
The relationship between the random dimer
model and conducting polymers can be established
as follows. We first note that the results of the
random dimer model are easily generalized and
shown to be applicable to any lattice in which the
defects possess internal structure and a plane of
symmetry.16 That is, any random lattice in which the
defects are symmetrical and span at least two sites
will contain a set of current-carrying electronic states.
Soliton defects in polyacetylene and bipolarons in
polyaniline both possess symmetric internal
structure. Consequently, random lattices containing
solitons or bipolarons will necessarily possess a set
of conducting states at a particular energy in the
band. The location of the extended states is easily
established from the reflection coefficient. 17 Shown
in Fig. 1 is the reflection coefficient for both positive
and negative solitons. Figures l a and lb contain
1.0
0.8 Positive Soliton
IR~ 0"6
0.4
0.2
0.0 L._ - 6 - 4 - 2 0 2
E(eV)
(a)
i
4
1.0
0.8 j Negative 0 . 6 Soliton
IR~20. 4 ,
0 . 2
o o - J - 6 - 4 - 2 0
(b)
2 4 E(eV)
6
FIGURE 1 Reflection coefficient for a) positive soliton and b) negative soliton in polyacetylene.
the results for positive and negative solitens spread
over 40 (dashed line) and 32 sites, respectively,
the equilibrium spatial extent of positive and
negative solitons18. As Fig. 1 indicates, the
reflection coefficient vanishes indentically below
-.7eV (in the valence band) for a positive soliton and
between .7 and 1.1eV (in the conduction band) for a
negative soliton. It was vedfied numerically that
decreasing the size of the soliton decreased the
width of extended states. The lower bound is of
course "~/N-. We conclude then that there is a wide
band of extended states in the vicinity of the band
edge for either n or p-type doping: Because the
extended states lie so close to the band edge, a
doping level of 5% is sufficient to move the Fermi level from the band edge where the states are
strongly Iocalised to either within the valence band
(p-doping) or the conduction band (n-doping) where
the conducting states lie. We argue that it is the
930 R Philips, J.-L. Wu / Origin of extended states in conducting polymers
electrons lying in these states that are responsible for
the experimentally-observed Pauli susceptibilty in
heavily-doped trans- polyacetylene. Analogously, the reflection coefficient for a
quinoid defect in the highest occupied band for polyaniline is shown in Fig. 2. As is evident from Fig. 2, we find that whether or not the quinoid defect is
protonated, the reflection coefficient vanishes ~-.3213
to -.2913, with I]=2.5eV, the resonance integral for
benzene. Hence, the states at -.3213 to -.29[3 are
completely unscattered by the quinoid defect and thus are conducting states. When the polymer is
protonated, Galvao, et al. 14 showed that the Fermi
level moves continuously from -.2113 to approximately
-.3513 at a protonation level of 50%. At this level the polymer exhibits its maximum conductivity. Hence, the energy of the current-carrying states in the random dimer model of protonated polyaniline is in
excellent agreement with the location of the extended states as calculated in ref. 14 and is in
close proximity to the location of the Fermi level when the polymer exhibits its maximum conductivity.
We conclude then that the extended states observed
in the calculations of ref. 14 can be explained with
the random dimer model.
IRI 2
1.0
0.8 Unprotonated 0,6
0.4
0.2
0.0
. ¢/~otonated
.0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 E(IB)
FIGURE 2 Reflection coefficient through a protonated and unprotonated bipolaron in polyaniline.
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