Unified Description of Charge-Carrier Mobilities in Disordered Semiconducting Polymers
W.F. Pasveer,1J. Cottaar,1C. Tanase,2R. Coehoorn,3P.A. Bobbert,1,*P.W.M. Blom,2
D.M. de Leeuw,3and M.A.J. Michels1
1Group Polymer Physics, Eindhoven Polymer Laboratories and Dutch Polymer Institute, Technische Universiteit Eindhoven,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2Materials Science Centre and Dutch Polymer Institute, Nijenborgh 4, 9747 AG Groningen, The Netherlands
3Philips Research Laboratories, Professor Holstlaan 4, 5656 AA Eindhoven, The Netherlands
(Received 8 December 2004; published 23 May 2005)
From a numerical solution of the master equation for hopping transport in a disordered energy
landscape with a Gaussian density of states, we determine the dependence of the charge-carrier mobility
on temperature, carrier density, and electric field. Experimental current-voltage characteristics in devices
based on semiconducting polymers are excellently reproduced with this unified description of the
mobility. At room temperature it is mainly the dependence on carrier density that plays an important
role, whereas at low temperatures and high fields the electric field dependence becomes important.
Omission in the past of the carrier-density dependence has led to an underestimation of the hopping
distance and the width of the density of states in these polymers.
DOI: 10.1103/PhysRevLett.94.206601 PACS numbers: 72.20.Ee, 72.80.Le, 73.61.Ph
Introduction.—The use of conjugated semiconducting
polymers in light-emitting diodes (PLEDs)  and field-
effect transistors (FETs)  has triggered intensive re-
search into the optoelectronic and electrical transport prop-
erties of these materials [3,4]. Understanding their charge-
carrier transport is of crucial importance to design and
synthesize better materials and further improve device
performances. One of the most important parameters de-
termining performance is the mobility ? of the charge
carriers. In particular, the dependence of ? on temperature
T and electric field E has been extensively addressed in
literature [5–8]. Chargetransport in disordered polymers is
regarded as a hopping process between localized sites,
which are thought to consist of conjugated polymer chain
segments. The variations in the on-site energies due to
disorder are usually assumed to be Gaussian. Very recent
direct measurements of the energy distribution in an elec-
trochemically gated polymer transistor, indeed, showed a
nearly Gaussian shape . Monte Carlo simulations of
hopping transport were performed by Ba ¨ssler et al. for
the case of a Gaussian disorder model (GDM), showing a
non-Arrhenius temperature dependence ? / exp??c ^ ?2?,
with c ? 0:44, ^ ? ? ?=kBT, and ? the width of the
Gaussian, while a Poole-Frenkel ? / exp??
was found, in a limited field range, for the dependence on
the electric field [5,6]. However, as pointed out by
Gartstein and Conwell , a spatially correlated potential
for the charge carriers is needed to explain the Poole-
Frenkel behavior in a wide region of field strengths.
Several suggestions were put forward as a cause for this
correlation, such as charge-dipole interactions [11,12] or
thermal fluctuations in molecular geometries .
Recently, it was realized that the importance of another
parameter had been overlooked: the charge-carrier density
p [14,15]. Experiments on hole-only diodes and FETs,
with the same polymer as active material, showed that ?
can differ up to 3 orders of magnitude between the diode
and the FET . It was demonstrated that the only way to
explain this huge difference is by taking a strong depen-
dence of the mobility on p into account. More recent
experiments suggested that at room temperature it is suffi-
cient to take only the p dependence of ? into account, but
that at low temperatures it is still necessary to assume an E
dependence . Experimentally, it is hard to separate the
influences of E and p on ?, since under a higher electric
field more carriers will be injected. In Ref.  the p
dependence of ? was taken equal to an empirical form,
consisting of an algebraic contribution pT0=T?1, derived by
Vissenberg and Matters for the high-density regime ,
plus a density-independent contribution, determined from
the current-voltage characteristics of hole-only diodes at
low voltages. The failure of this empirical form to repro-
duce the experimental current-voltage characteristics at
low temperatures and high voltages was attributed to an
unknown dependence of ? on E. The purpose of this Letter
is to establish a unified theoretical description of the full T,
p, and E dependence of ?, and to critically compare the
results with experiments.
Theory.—We determine the mobility ? unambiguously
from a numerical solution ofthe master equation represent-
ing hopping of charge carriers on a lattice of sites:
Here piis the probability that site i is occupied by a charge
and Wijis the transition rate for hopping from site i to j.
The factors 1 ? piaccount, in a mean-field approximation,
for the fact that only one carrier can occupy a site, due to
the high Coulomb penalty for the presence of two or more
carriers. We consider hopping as a thermally assisted tun-
?Wijpi?1 ? pj? ? Wjipj?1 ? pi?? ? 0:
PRL 94, 206601 (2005)
27 MAY 2005
2005 The American Physical Society
neling process and assume coupling to a system of acous-
tical phonons, which, in line with earlier work, leads to
transition rates of the form 
??0exp??2?Rij? ??"j? "i??;"j? "i;
where ? ? 1=kBT, ?0is an intrinsic rate, Rij? jRj? Rij
is the distance between sites i and j, ? is the inverse
localization length of the localized wave functions under
consideration, and "iis the on-site energy of site i. The
energy differences in Eq. (2) are supposed to contain a
contribution ?eERijxdue to an electric field E in the x
direction (e is the charge of the carriers). In the absence of
an electric field the occupational probabilities are given by
the Fermi-Dirac distribution.
We solve Eq. (1) by an iteration procedure similar to that
suggested byYuet al. .Forsimplicity wetake a regular
cubic lattice with lattice constant a. For the inverse local-
ization length we take ? ? 10=a, a typical value for the
relevant polymers  (we checked that varying ? pre-
dominantly changes the prefactor of the mobility). Our
method allows for variable-range hopping , but for
the parameter range studied here it was sufficient to con-
sider hopping to a maximum distance of
periodic boundary conditions and the site energies without
electric field are drawn randomly from a Gaussian distri-
butionofwidth ?.From the solution of Eq.(1)the mobility
? is obtained as ? ?P
lattices are taken large enough such that finite size effects
are negligible (sizes up to 1503for the lowest tempera-
tures). Averages over a number of different disorder con-
figurations are taken until an accuracy better than 10% is
obtained for ?.
a. We apply
i;jWijpi?1 ? pj?Rijx=pEV, with
p ? hpii=a3and V the system volume. The sizes of the
Results.—In Fig.1wedisplay the pdependence of?for
different temperatures. This dependence shows a striking
similarity to the p dependence found by Tanase et al.
[15,16]. The numerical data of Fig. 1 can be rather well
described by the following parametrization scheme :
??T;p? ? ?0?T?exp?1
?0?T? ? ?0c1exp??c2^ ?2?;
? ? 2ln? ^ ?2? ^ ?? ? ln?ln4?
with c1? 1:8 ? 10?9and c2? 0:42. This parametrization
is, in particular, satisfactory for not too high densities.
Because of particle-hole symmetry ??T;p? ? ??T;a?3?
p?, so the parametrization obviously fails at densities
approaching a?3=2. In the limit of vanishing densities,
we recover the ? / exp??c ^ ?2? temperature dependence
found by Ba ¨ssler et al. [5,6].
In Fig. 2 we display the E dependence of ? for different
temperatures at a low and a high density, typical for LEDs
and FETs, respectively. We find that the E dependence can
be approximately modeled by a density-independent pre-
??T;p;E? ? ??T;p?f?T;E?:
The prefactor can be rather well parametrized by
2? ^ ?2? ^ ???2pa3???;
f?T;E? ? exp
0:44? ^ ?3=2? 2:2?
1 ? 0:8
This parametrization is optimized for the low-density re-
gion, but it is also rather accurate in the high-density region
(see inset of Fig. 2). The parametrization Eq. (5) correctly
yields an E2dependence at low fields, which should hold
because of field-reversal symmetry E ! ?E, and de-
FIG. 1 (color online).
bility at various temperatures for a vanishing electric field, in
units of ?0[see Eq. (3c)]. Symbols: numerical results. Lines: fits
using the parametrization scheme given in Eq. (3).
Carrier-density dependence of the mo-FIG. 2 (color online).
various temperatures, for a typical density in LEDs (main panel)
and FETs (inset). Symbols: numerical results. Lines: Eqs. (3)–
Field dependence of the mobility at
PRL 94, 206601 (2005)
27 MAY 2005
scribes the approximately linear field dependence of
ln??=?0? found when a critical field strength of about
?=ea is passed. At very high fields, where ? saturates
and eventually decreases as a function of E, the parame-
trization breaks down. Although a field range could be
givenin which anapproximate
Roichman and Tessler  determined a p and E de-
pendence of ? for the GDM with a very simple model in
which local fluctuations of the electrochemical potential
are neglected. We checked that in this model the mobilities
are overestimated by several orders of magnitude at low
temperatures, making this model very unrealistic.
Application to hole-only devices.—Using the above re-
sults, we now proceed to calculate the current-voltage
characteristics of a polymer layer of thickness L, sand-
wiched between two electrodes. The carrier density at the
injecting electrode is assumed to be so high that compli-
cations related to injection need not be considered. The
relation between the space-charge limited current (SCLC)
density J and voltage V then follows from the solution of
? holds, the parametrization Eq. (5) is a more
J ? p?x?e??T;p?x?;E?x??E?x?;
where x is the distance from the injecting electrode, ?0
is the vacuum permeability, and ?ris the relative dielec-
tric constant of the polymer. We take ?r? 3, a typical
value for the relevant polymers. From calculations for
a constant mobility including diffusion we conclude
that diffusion effects, causing a significant increase of
the current only at low voltages and a change of the
density and field only close to the electrodes, can be
J ? V
done on hole-only devices of poly[40-(3,7-dimethylocty-
loxy)-1,10-biphenylene-2,5-vinylene] (NRS-PPV), with
L ? 560 nm,
(OC1C10-PPV), with L ? 275 nm, as well as with an in-
dium tin oxide bottom electrode as anode and an evapo-
rated gold contact as top electrode. The results are
displayed in Fig. 3. We also display in Fig. 3 the solution
of Eq. (6) with the T-, p-, and E-dependent mobility from
the parametrization given above. The parameters ?0, ?,
and a are determined in such a way that an optimal overall
fit is obtained.
A clear observation is that the agreement between ex-
periment and theory is excellent. For the NRS-PPV device
the E dependence at lower temperatures and high voltages
is clearly important in explaining the measurements,
whereas under these conditions the E dependence for the
OC1C10-PPV device is on the edge of becoming important.
In the inset of Fig. 3 the dotted line gives the current
density in the absence of the p and E dependence of ?,
as expressed by the Mott-Gurney law J ?9
, for T ? 235 K. The huge effect of the density depen-
dence of ? is clearly visible. We checked that the calcu-
lated carrier-density and electric field distribution in the
device are very different for the three different situations
(no p or E dependence, only p dependence, and p and E
dependence of the mobility). Hence, for a correct descrip-
tion of the charge-carrier injection, transport, and recom-
bination in PLEDs, the effects of both p and E need to be
accounted for. The strong density dependence and the
weaker field dependence, as compared to previous studies,
also has important consequences for the interpretation of
time-of-flight measurements .
The fits lead to values of ? for the two polymers that are
essentially the same (0:14 ? 0:01 eV) and to values for the
lattice constants that are slightly different (1:8 ? 0:1 nm
and 1:6 ? 0:1 nm). An important result is that the values
for ? and a found by us are markedly larger than reported
in previous work , where the deviation from the Mott-
Gurney law was attributed to an E dependence of the
mobility. In the case of OC1C10-PPV, values of ? ?
0:112 eV and a ? 1:2 nm were reported as optimal fitting
parameters  (these values are also typical for the present
samples). The lower value of ? can be mainly attributed to
the omission of the p dependence, whereas the lower value
of a can be mainly attributed to the overestimation of the E
FIG. 3 (color online).
(lines) current-density vs voltage results for polymer layers of
NRS-PPV (main panel) and OC1C10-PPV (inset). Full line:
solution of SCLC Eq. (6) with the T-, p-, and E-dependent
mobility from the present work. Dashed line: SCLC result
without E dependence. Dotted line: SCLC result without p or
E dependence (shown only in the inset, for T ? 235 K). The
values used for ?0are 72 and 6400 m2=Vs, respectively.
Experimental (symbols) and theoretical
PRL 94, 206601 (2005)
27 MAY 2005
Discussion and conclusions.—The excellent fits we ob- Download full-text
tain with experimental current-voltage data of two semi-
conducting polymers strongly suggest that the underlying
dependencies of the mobility on temperature, carrier den-
sity, and electric field in these materials are correctly
described by this Letter, at least in the regime of carrier
densities considered here. Because in the present work the
site energies are uncorrelated, an important conclusion is
that, for the materials investigated here, our analysis does
not indicate that there is a need to assume a certain spatial
energy correlation, as suggested before [10–13]. Another
important conclusion is that a satisfactory description can
be given with a Gaussian site energy distribution. Hence, a
simple uncorrelated Gaussian disorder model can be used.
Considering the excellent agreement with experiment,
there is no indication for a need to go beyond the simple
starting point Eq. (2) forthe transition rates, by considering
multiphonon hop processes . We confirm recent claims
that the carrier-density dependence of the mobility in
PLEDs is more important than the field dependence, but
show that a field dependence is still required to describe
current-voltage characteristics at low temperatures and
high fields. These conclusions have consequences for the
internal field distribution and density of charge carriers of
both signs in devices like LEDs and FETs, which should be
different than assumed thus far. Moreover, markedly larger
values of the width of the Gaussian energy distribution and
the typical hopping distance are found than obtained
We should, however, point at the drastic approximation
made in replacing the hopping problem in a polymer by an
isotropic hopping model on a regular lattice. The fitted
values of 1.6–1.8 nm for the lattice constants are consid-
erably smaller than the typical length of 6–7 nm of a
conjugated segment of derivatives of PPV (about 10 repeat
units) and considerably larger than the typical distance of
0.3–0.4 nm between neighboring polymer chains. Also,
intrachain and interchain hopping processes are expected
to have very different characteristics. Therefore, our model
parameters should be considered as the effective result of
an average of intrachain and interchain hopping. This
complicates an identification with microscopic parameters
of the polymer. Finally, we mention that the description
presented here should also be applicable to small-molecule
based materials, for which such an identification should be
easier to make.
This work forms part of the research program of the
Dutch Polymer Institute (Projects No. 274 and No. 276).
The computations were made possible by support from
NCF (Dutch National Computer Facilities).
Electronic address: P.A.Bobbert@tue.nl
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