Smooth crossover from variable-range hopping with Coulomb gap to nearest-neighbor interchain
hopping in conducting polymers
Sanjib Maji,
1
Soumik Mukhopadhyay,
1
R. Gangopadhyay,
2
and A. De
2
1
ECMP Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700 064, India
2
CS Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700 064, India
Received 7 June 2006; revised manuscript received 20 November 2006; published 8 February 2007
The temperature T dependence of the conductivity of a set of poly-3,4 ethylenedioxythiophene
samples synthesized by varying the oxidizing agent and its molar ratio with the monomer is in excellent
agreement with that in the model prescribed by Aharony et al. Phys. Rev. Lett. 68, 3900 1992, which
predicts a smooth crossover from the Coulomb-gap dominated variable-range hopping VRH at low tempera-
tures to the Mott three-dimensional-VRH at high-enough temperatures. In the temperature regime of such
crossover, it takes the form log T
-1/3
, which is attributed to the nearest-neighbor interchain hopping char-
acteristic of short average conjugation length.
DOI: 10.1103/PhysRevB.75.073202 PACS numbers: 72.80.Le
I. INTRODUCTION
Variable-range hopping VRH of charge carriers is one of
the prime mechanisms of electrical conduction in disordered
solids such as amorphous semiconductors, conducting poly-
mers, etc.
1–7
The shape of the density of states DOS in
these materials plays a key role in governing the kinds of
VRH exhibited. Temperature, magnetic field, and other pa-
rameters can influence the DOS and, in turn, can result in a
crossover from one kind of VRH to another even in a single
disordered material.
8,9
Extensive studies, both theoretical and
experimental, have been performed on the temperature de-
pendent crossover of VRH in disordered inorganic as well as
organic e.g., conducting polymers solids.
4–11
Mott and Davis pointed out that the phonon-assisted hop-
ping of charge carriers in disordered solids is not necessarily
restricted to nearest-neighboring sites only. Rather, a tem-
perature driven optimization of hopping distance with re-
spect to hopping energy takes place.
1
At relatively higher
temperatures, the effect of electron-electron interaction being
negligible, Mott and Davis assumed a constant DOS near the
Fermi level while arriving at the relation log T
-1/d+1
,
where d is the dimensionality of the hopping process. There
are several instances of three-dimensional 3D-VRH, where
the logarithmic conductivity varies with T
-1/4
, in both inor-
ganic and organic disordered solids.
4,5,7,8,12
The temperature
dependence of conductivity of the form log T
-1/3
result-
ing from two-dimensional 2D-VRH has also been observed
in various systems.
6,13,14
Again, ln varies as T
-1/2
for quasi-
one-dimensional VRH Refs. 3 and 15, as well as for Cou-
lomb charging between isolated conducting islands occurring
in granular metals
16–18
or in systems consisting of polaronic
clusters,
19
as observed for doped polypyrrole,
20
poly3,4 eth-
ylenedioxythiophene PEDOT/polystyrenesulfonate,
21
etc.
Knotek and Pollak observed that the nearest-neighbor in-
teraction between localized electrons gives rise to a mini-
mum in DOS near the Fermi level.
22
In the strongly localized
regime, the long-range electron-electron e-e Coulomb in-
teraction opens up a soft gap in DOS at the Fermi level,
which governs the VRH mechanism as described by Efros
and Shklovskii ES.
2
There exist a number of experimental
evidences in which hopping is indeed influenced by the pres-
ence of Coulomb gap at sufficiently low temperatures, hop-
ping energy being less than or comparable to the gap
energy.
4,5,7,8,23
In such cases, the temperature dependence of
conductivity appears in the form log T
-1/2
in the lower-
temperature range and exhibits a crossover to the Mott-VRH
at higher temperatures. According to the phenomenological
analysis by Aharony et al., the temperature dependent cross-
over in resistivity can actually be interpreted with the help of
a universal scaling function which asymptotically reduces to
the Mott 3D-VRH at higher temperatures and to the ES-VRH
in the lower-temperature range.
10
This scaling function pre-
dicts a smooth crossover, whereas Meir described a rather
abrupt crossover introducing a scaling function based on a
percolation picture of transport in strongly localized
systems.
11
In the present study, a set of PEDOT samples exhibits a
smooth crossover from the Coulomb-gap dominated VRH to
a kind of hopping mechanism which arises out of the struc-
tural complexity of conducting polymer samples having a
short conjugation length.
II. EXPERIMENT
Doped-PEDOT, an easily processable and highly stable
intrinsically conducting polymer with appreciably high
conductivity,
24,25
has been chemically synthesized. Follow-
ing a standard method, ethylenedioxythiophene EDOT was
treated separately with ferric chloride FeCl
3
, 6H
2
O in mo-
lar ratios of 1.167:1 and 1:2 and with ammonium peroxodis-
ulfate APS in 1.286:1 and 2.25:1. In order to synthesize a
sample, EDOT was dissolved in water with 0.2M dodecyl-
benzenesulphonic acid solution. Subsequently, an aqueous
solution of one of the oxidizing agents FeCl
3
or APS was
added dropwise on constant stirring under inert atmosphere.
After continuation of the reaction overnight, ethanol was
added for complete precipitation of a dark blue solid, pre-
sumably the doped-PEDOT. The precipitate was thoroughly
washed with ethanol, dried, and pelletized. Two samples S1
and S2 using FeCl
3
and two samples S3 and S4 using APS
PHYSICAL REVIEW B 75, 073202 2007
1098-0121/2007/757/0732024 ©2007 The American Physical Society 073202-1