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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