University of Groningen
Light intensity dependence of open-circuit voltage of polymer
Koster, L. J. A.; Mihailetchi, V. D.; Blom, P. W. M.
Published in:
Applied Physics Letters
DOI:
10.1063/1.1889240
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Citation for published version (APA):
Koster, L. J. A., Mihailetchi, V. D., & Blom, P. W. M. (2005). Light intensity dependence of open-circuit
voltage of polymer: fullerene solar cells.
Applied Physics Letters
,
86
(12), [123509].
https://doi.org/10.1063/1.1889240
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Light intensity dependence of open-circuit voltage of polymer:fullerene
solar cells
L. J. A. Koster,
a兲
V. D. Mihailetchi, R. Ramaker, and P. W. M. Blom
Materials Science Centre
Plus
, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands
共Received 21 July 2004; accepted 3 February 2005; published online 17 March 2005兲
The open-circuit voltage V
oc
of polymer:fullerene bulk heterojunction solar cells is investigated as
a function of light intensity for different temperatures. Devices consisted of a blend of a poly
共p-phenylene vinylene兲 derivative as the hole conductor and 6,6-phenyl C
61
-butyric acid methyl
ester as the electron conductor. The observed photogenerated current and V
oc
are at variance with
classical p–n junction-based models. The influence of light intensity and recombination strength on
V
oc
is consistently explained by a model based on the notion that the quasi-Fermi levels are constant
throughout the device, including both drift and diffusion of charge carriers. © 2005 American
Institute of Physics. 关DOI: 10.1063/1.1889240兴
Organic photovoltaic elements are a promising alterna-
tive to conventional inorganic solar cells because of their
low-cost fabrication of large areas. The best performance is
currently obtained with polymer:fullerene bulk heterojunc-
tion solar cells,
1
yielding power conversion efficiencies of
typically 2.5% under AM1.5 illumination. One of the key
parameters of photovoltaic devices is the open-circuit volt-
age 共V
oc
兲, which is the voltage for which the current in the
external circuit equals zero. In polymer:fullerene solar cells
limitations of the open-circuit voltage have been attributed to
Fermi level pinning
2
and to band bending at the contact due
to the injection of charges.
3
For further optimization of solar
cell performance fundamental understanding of the mecha-
nisms governing the photovoltaic performance is indispens-
able.
For a conventional 共Si兲 p–n junction solar cell the cur-
rent density under illumination J
L
is given by
4
J
L
= J
s
共e
qV/nkT
−1兲 − J
ph
, 共1兲
where J
s
is the 共reverse bias兲 saturation current density, V is
the applied voltage, q is the elementary charge, k is Boltz-
mann’s constant, T is temperature, and n is the ideality fac-
tor. The photogenerated current density is denoted by J
ph
.
Subsequently, the open-circuit voltage is given by 共J
L
=0兲
V
oc
= 共nkT/q兲ln共J
sc
/J
s
+1兲, 共2兲
where J
sc
is the short-circuit current density. It should be
noted that Eq. 共2兲 is only valid for an ideal solar cell since it
has been assumed that the photogenerated current density is
voltage independent, meaning that J
ph
=J
sc
at any applied
voltage. Recently, Eq. 共2兲 has also been applied to explain
the temperature dependence of V
oc
of polymer:fullerene bulk
heterojunction solar cells.
5,6
However, it is not clear whether
such an analysis in terms of an ideal solar cell is justified for
the case of polymer:fullerene bulk heterojunctions. More-
over, the fact that both J
sc
and J
s
are also temperature depen-
dent further complicates the applicability of Eq. 共2兲 to the
effects of temperature on the device characteristics of or-
ganic bulk heterojunction devices. A more direct way of test-
ing the applicability of Eq. 共2兲 toward organic solar cells is to
investigate the dependence of V
oc
on light intensity at differ-
ent temperatures. Since it has been demonstrated that J
sc
is
nearly linearly dependent on light intensity,
7,8
it follows from
Eq. 共2兲 that V
oc
should exhibit a slope of nkT/q, when plot-
ted as a function of the logarithm of light intensity. In this
study we demonstrate that the light intensity dependence of
V
oc
of polymer:fullerene bulk heterojunction solar cells is in
contradiction with the predictions of the conventional p–n
junction based model 关Eq. 共2兲兴. An alternative expression for
V
oc
is presented that is based on the fact that at zero current
the quasi-Fermi levels are constant throughout the device,
which incorporates both drift and diffusion of charge carri-
ers. This expression consistently explains the experimental
dependence of V
oc
on light intensity for bulk heterojunction
devices.
The solar cells addressed in this study are bulk hetero-
junctions consisting of a blend of poly关2-methoxy-5-共3
⬘
,7
⬘
-dimethyloctyloxy兲-p-phenylenevinylene兴共MDMO-PPV兲 as
electron donor and 6,6-phenyl C
61
-butyric acid methyl ester
共PCBM兲 as electron acceptor in a 1:4 weight ratio. This
blend is sandwiched between a hole-conducting layer
of poly共3,4-ethylenedioxythiophene兲/poly共styrenesulfonate兲
共PEDOT:PSS兲, and an evaporated lithium fluoride 共LiF兲
共1nm兲/aluminum 共100 nm兲 top electrode. After fabrication
the current–voltage characteristics of these devices were
measured in a nitrogen atmosphere, both in dark and under
illumination. A white light halogen lamp set at 800 W/m
2
共spectral range 450–750 nm兲 was used to illuminate the de-
vices. Incident light power dependent measurements were
performed by using a set of six neutral density filters with a
constant optical density in the involved spectral range. The
generation rate of electrons and holes is assumed to be pro-
portional to the intensity.
In Fig. 1 the dark current density J
dark
is shown as a
function of voltage V for a MDMO-PPV:PCBM based solar
cell at different temperatures. From the slope of the exponen-
tial part of the J–V characteristics the ideality factors are
determined. The results are summerized in Table I. At room
temperature the ideality factor n typically amounts to 1.4 and
then further increases to 2.0 at 210 K, in agreement with
other observations.
9
Subsequently, the current–voltage char-
acteristic 共J
L
–V兲 of an illuminated 共800 W/m
2
兲 device at
room temperature is shown in Fig. 2, together with the cur-
rent predicted by Eq. 共1兲. It is clear that there is a large
a兲
Electronic mail: l.j.a.koster@rug.nl
APPLIED PHYSICS LETTERS 86, 123509 共2005兲
0003-6951/2005/86共12兲/123509/3/$22.50 © 2005 American Institute of Physics86, 123509-1
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discrepancy between the predictions of the model and the
experimental data: near V
oc
the predicted current is much too
high. This already strongly indicates that the p–n junction
model is not applicable to polymer:fullerene bulk heterojunc-
tion devices. Figure 3 shows V
oc
as a function of the loga-
rithm of light intensity at various temperatures, the highest
intensity corresponds to 800 W/m
2
共no filter兲. The experi-
mental data are fitted with a linear function with slope S
which is given in Table I in units of kT/q. Suprisingly, the
experimental slopes are within experimental error equal to
kT/q instead of nkT/q 关Eq. 共2兲兴 for all temperatures. Thus,
next to the photocurrent 共Fig. 2兲 the light intensity depen-
dence of V
oc
is also not in agreement with the classical
model. It should be mentioned that we have also verified this
for other PPV derivatives.
The main reason for this disagreement is that, as stated
previously, Eq. 共2兲 is based on the assumption of a voltage-
independent photogenerated current J
ph
. Recently, it has been
shown by Mihailetchi et al.
10
that the photogenerated current
of MDMO-PPV:PCBM devices shows a very different be-
havior: In the inset of Fig. 2 the photogenerated current of
such a device is plotted as a function of effective applied
voltage, V
oc
−V, where V
oc
has been corrected for dark cur-
rent. Near the open-circuit voltage, a linear dependence of
the photogenerated current upon applied voltage is observed.
This behavior is caused by the opposite effect of drift and
diffusion of charge carriers.
11
At V
oc
drift and diffusion bal-
ance and the current is zero. At higher effective voltage
V
oc
−V⬎ 0.1 V the drift contribution is dominant and the
photogenerated current tends to saturate. However, due to an
increased dissociation efficiency of photogenerated bound
electron–hole pairs, the photocurrent further increases before
it reaches full saturation at V
oc
−V⬎ 10 V.
10
Consequently,
the assumption of a constant photogenerated current is not
valid. When the photocurrent near the open-circuit voltage is
equated to J
sc
共inset of Fig. 2, line兲 the photocurrent is
strongly overestimated, hence Eq. 共2兲 cannot be expected to
meticulously reproduce the experimental data. We note that
the fit of Eq. 共1兲 to experimental photocurrent data is often
improved by including series and shunt resitivities.
9
How-
ever, the physical meaning of these quantities is not clear
though.
We suggest an alternative expression for the open-circuit
voltage, based on the metal–insulator–metal picture.
12
In this
approach, the device is described as one semiconducting ma-
terial with the highest occupied molecular orbital 共HOMO兲
of the polymer functioning as the valence band and the low-
est unoccupied molecular orbital 共LUMO兲 of PCBM acting
as conduction band. The energy difference between the
HOMO and LUMO levels will be denoted by the band gap
E
gap
. As a first step, the quasi-Fermi levels
n,p
are intro-
duced as
13
n共p兲 = n
int
exp关共− 兲q共V −
n共p兲
兲/共kT兲兴, 共3兲
where n共p兲 is the electron 共hole兲 concentration under illumi-
nation and n
int
is the intrinsic concentration of both electrons
and holes. The intrinsic carrier concentration n
int
is given by
n
int
= N
c
exp共− E
gap
/共2kT兲兲, 共4兲
where N
c
is the effective density of states, which is equal to
2.5⫻ 10
25
m
−3
.
14
The product np is known to statisfy np
=n
int
2
in equilibrium,
15
however,
np = n
int
2
exp关q共
p
−
n
兲/共kT兲兴, 共5兲
when the system is not in equilibrium. The familiar expres-
sion for the electron 共hole兲 current density, including both
drift and diffusion, is,
6
J
n共p兲
= q
n共p兲
n共p兲E + 共− 兲kT
n共p兲
x
n共p兲, 共6兲
where
n共p兲
is the electron 共hole兲 mobility and E is the elec-
tric field strength. Equation 共6兲 can be rewritten in terms of
the quasi-Fermi levels as
6
TABLE I. Overview of ideality factors n obtained from Fig. 1 and slopes S
obtained from Fig. 3.
295 K 250 K 210 K
n 1.34 1.62 1.98
S共kT/q兲 1.03 1.01 0.90
FIG. 2. Experimental current under illumination of an MDMO-PPV:PCBM
device at 295 K 共symbols兲 and the current density predicted by Eq. 共2兲
共line兲. Inset: The photogenerated current density J
ph
of an MDMO-
PPV:PCBM device 共symbols兲 as a function V
oc
−V. The line denotes the
short-circuit current density corresponding to the assumption of J
ph
being
constant.
FIG. 3. V
oc
of a MDMO-PPV:PCBM device 共symbols兲 as a function of light
intensity, the solid lines denote linear fits to the experimental data, and the
dotted line represents the prediction at room temperature of Eq. 共12兲.The
inset shows the spectrum of the white light halogen lamp used to illuminate
the devices.
FIG. 1. Experimental dark current of a MDMO-PPV:PCBM device 共sym-
bols兲 and fit to the exponential part 共lines兲 at various temperatures.
123509-2 Koster
et al.
Appl. Phys. Lett. 86, 123509 共2005兲
Downloaded 07 Apr 2005 to 129.125.25.39. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
J
n共p兲
=−q
n共p兲
n共p兲
x
n共p兲
. 共7兲
At open circuit the current densities are 共virtually兲 zero, con-
sequently, the quasi-Fermi levels are constant. Since the
共ohmic兲 contacts are in thermal equilibrium, the quasi-Fermi
levels have to be equal to the potential at the contacts. This
implies that the difference
p
−
n
is constant throughout the
device and equal to the applied voltage at open-circuit, there-
fore
np = n
int
2
exp关qV
oc
/共kT兲兴. 共8兲
We have recently developed a numerical model succes-
fully describing the current–voltage characteristics of poly-
mer:fullerene solar cells which includes drift and diffusion of
charge carriers, bimolecular recombination, and the effect of
field- and temperature-dependent generation of free charge
carriers.
16
In this model the continuity equation for electrons
is given by
1
q
x
J
n
共x兲 = PG − 共1−P兲R, 共9兲
where P is the dissociation probability of a bound electron–
hole pair into free charge carriers, G is the generation rate of
bound electron-hole pairs, and R the Langevin recombination
rate of free electrons and holes given by
R =
␥
共np − n
int
2
兲, 共10兲
where
␥
is the Langevin recombination constant. The gen-
eration rate of free charge carriers is then represented by PG.
The recombination rate can be written as, to a very good
approximation, R=
␥
np, since the photogenerated charges
outnumber the thermally excited charge carriers by many
orders of magnitude 关see Eq. 共8兲兴. Since the current densities
are zero, so are their derivatives and hence recombination
and generation cancel everywhere in the device. Subse-
quently, it follows from Eq. 共9兲 that,
G =
␥
np共1−P兲/P. 共11兲
Therefore, using Eq. 共8兲 and solving for V
oc
one has
V
oc
=
E
gap
q
−
kT
q
ln
冉
共1−P兲
␥
N
c
2
PG
冊
. 共12兲
This formula predicts the right slope S of V
oc
versus light
intensity, viz., kT/q, since P and
␥
do not depend on inten-
sity. Further, Eq. 共12兲 is consistent with the notion of a field-
dependent photogenerated current, in contrast to Eq. 共2兲,
since both drift and diffusion have been taken into account
through the use of Eqs. 共6兲 and 共7兲. Using the appropriate
values for the electron and hole mobility for an MDMO-
PPV:PCBM device,
16
E
gap
=1.3 eV 共corresponding to an en-
ergy difference between the HOMO of MDMO-PPV and the
LUMO of PCBM of 1.3 eV
16
兲, P=0.474,
16
N
c
=2.5
⫻ 10
25
m
−3
,
16
and G=2.7⫻ 10
27
m
−3
s
−1
for the generation
rate, corresponding to illumination by a white light halogen
lamp 共spectrum shown in inset of Fig. 3兲 set to 800 W/m
2
,
9
Eq. 共12兲 predits V
oc
=0.8 V at room temperature. This is in
good agreement with the corresponding experimental value
of 0.77 V as shown in Fig. 3. Figure 3 also shows the pre-
dicted light intensity dependence of V
oc
as predicted by Eq.
共12兲共dotted line兲. It should be noted that the analysis of the
temperature dependence of V
oc
of polymer:fullerene solar
cells by using Eq. 共12兲 is strongly complicated by the ab-
sence of a sharply defined band gap. Due to the presence of
energetic disorder in both materials, their HOMO and
LUMO levels exhibit a Gaussian broadening
of typically
0.1 eV.
17,18
Since the exact distribution of energy levels in-
side the PPV:PCBM blend is not known, the uncertainty in
E
gap
is of the same order of magnitude as the variation of V
oc
with temperature, thereby prohibiting an exact quantitative
analysis. For further analysis temperature-dependent charge
transport measurements performed on blends are necessary.
In summary, we have investigated the open-circuit volt-
age at various temperatures and demonstrated that the open-
circuit voltage, when plotted as a function of light intensity,
has a slope equal to kT/q. This cannot be explained by using
a formula derived from p–n junction-based models for
current–voltage characteristics in dark and under illumina-
tion. The main cause of this discrepancy lies in the fact that
the strong voltage dependence of the photogenerated current
is not taken into account. An alternative model for the open-
circuit voltage has been presented, based on the notion that
the quasi-Fermi levels are constant throughout the device.
This model consistently explains the light intensity depen-
dence of the open-circuit voltage of polymer:fullerene bulk
heterojunction devices.
The work of one of the authors 共L.J.A.K.兲 forms part of
the research program of the Dutch Polymer Institute 共#323兲.
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