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Device simulation of low-band gap polymer solar cells: Influence of
electron-hole pair dissociation and decay rates on open-circuit voltage
Yuan Shang,1Qikai Li,1Lingyi Meng,1Dong Wang,2and Zhigang Shuai1,2,a兲
1Key Laboratory of Organic Solids, Beijing National Laboratory for Molecular Science (BNLMS),
Institute of Chemistry, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China
2Department of Chemistry, MOE Key Laboratory of Organic Opto-Electronics and Molecular Engineering,
Tsinghua University, 100084 Beijing, People’s Republic of China
共Received 17 July 2010; accepted 7 September 2010; published online 5 October 2010兲
We simulated the performance of recently developed highly efficient bulk heterojunction
photovoltaic cells with poly 关N-9⬙-hepta-decanyl-2,7-carbazole-alt-5,5-共4⬘,7⬘-di-2-thienyl-
2⬘,1⬘,3⬘-benzothiadiazole兲兴 as the donor and 关6,6兴-phenyl C70-butyric acid methyl ester as the
acceptor, using a device model. The simulated current-voltage curve is in excellent agreement with
the experiment. This enables us to analyze how microscopic processes of excitons and charges
govern the device performance. The influence of dissociation rate and decay rate of photoinduced
electron-hole pairs on the open-circuit voltage VOC is investigated. It is shown that a high
dissociation rate relative to decay rate will lead to enhanced VOC.©2010 American Institute of
Physics.关doi:10.1063/1.3494527兴
Polymer solar cells have attracted tremendous attention
for their potential in low-cost fabrication as well as applica-
tions in flexible, lightweight, and large-area devices. Since
the proposal of bulk heterojunction 共BHJ兲structure consist-
ing of electron donating and accepting moieties,1there has
been a dramatic improvement in the efficiency of polymer
solar cells. Very recently, power conversion efficiencies
above 6% under air mass 1.5 global 共AM 1.5 G兲illumination
were reported.2,3In these devices, low band gap polymers
with deeper highest occupied molecular orbital 共HOMO兲en-
ergies were fabricated, leading to a high short-circuit current
density JSC and an increased open-circuit voltage VOC.
VOC is one of the key parameters in solar cell devices. In
order to optimize the performance of organic solar cells, it is
essential to understand the fundamental processes of excitons
and carriers governing photovoltaic conversion.4–6In a
polymer/fullerene solar cell, the photogenerated excitons re-
sult in bound electron-hole pairs via an ultrafast electron
transfer5from the donor to the acceptor.6The bound
electron-hole pairs either decay to the ground state with a
decay rate kfor dissociate into free carriers with a rate kd.4In
this work, we show that both dissociation rate and decay rate
are of vital importance to the performance of solar cells,
especially to VOC.
In order to disentangle the effect of kdand kfon VOC,we
performed device simulations of BHJ solar cells based on
poly 关N-9⬙-hepta-decanyl-2,7-carbazole-alt-5,5-共4⬘,7⬘-di-
2-thienyl-2⬘,1⬘,3⬘-benzothiadiazole兲兴 共PCDTBT兲and 关6,6兴-
phenyl C70-butyric acid methyl ester 共PC70BM兲, using a nu-
merical model first developed by Koster et al.7incorporating
exciton diffusion, bimolecular recombination 共the Langevin
type, governed by the slowest charge carrier8兲, space-charge
effect, and charge dissociation or decay of bound electron-
hole pairs.4The net generation of free charge carriers de-
pends on exciton generation and its subsequent dissociation
as well as nongeminate recombination. The net generation
rate Uis then written as U=PG−共1−P兲R, where Gis the
exciton generation rate, Ris the recombination rate and P
=kd/共kd+kf兲.
The space-dependent Uis related to the gradient of cur-
rent density Jn共p兲through the continuity equations
xJn共x兲=qU共x兲and
xJp共x兲=−qU共x兲,共1兲
with qthe elementary charge. The current density has two
contributions: the drift current due to the electrostatic poten-
tial gradient and the diffusion current due to the charge den-
sity gradient
Jn=−qn
n
x
+qDn
xn
and Jp=−qp
p
x
−qDp
xp,共2兲
where Dn共p兲=
n共p兲kBT/qis the carrier diffusion coefficient
and
n共p兲is the carrier mobility. The electrostatic potential
and the charge density satisfy the Poisson equation
2
x2
共x兲=q
关n共x兲−p共x兲兴,共3兲
where is the dielectric constant. With appropriate boundary
conditions, the Poisson and continuity equations can be
solved iteratively based on the scheme of Gummel.9Finally,
the current-voltage curve and carrier densities can be ob-
tained. The device structure and the flow chart of the simu-
lation program are depicted in Fig. 1.
We choose physical parameters based on the BHJ solar
cells consisting of PCDTBT as the hole conductor and the
fullerene derivative PC70BM as the electron conductor.3The
PCDTBT/PC70BM solar cells exhibit one of the best perfor-
mance of polymer solar cells studied to date, with JSC
=10.6 mA cm−2,VOC =0.88 V, FF=0.66, and
e=6.1%.
The thickness of the active layer is 80 nm.
a兲Author to whom correspondence should be addressed. Electronic mail:
zgshuai@tsinghua.edu.cn.
APPLIED PHYSICS LETTERS 97, 143511 共2010兲
0003-6951/2010/97共14兲/143511/3/$30.00 © 2010 American Institute of Physics97, 143511-1
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
First, the exciton generation rate Gunder AM 1.5 G
irradiation is calculated10 by the experimental optical absorp-
tion spectrum for the blend3and the AM 1.5 G solar spec-
trum through the following equation:
G=
冕
0
800 nm
兵Ni共兲⫻关1–10
−A共兲⫻L兴其d/L,共4兲
where Ais the normalized absorption coefficient, is the
photon wavelength, Niis the incident photon number per unit
area, and Lis the thickness of the active layer of the solar
cell.
In the above, the exciton generation is assumed to be
uniform, nonspace-dependent. This is a reasonable assump-
tion since the active layer of the device is very thin 共80 nm兲.
Previous studies also show that it does not give rise to seri-
ous inconsistencies.7,11 Through Eq. 共4兲, we obtain the rate as
G=1⫻1028 m−3 s−1 under AM 1.5 G irradiation. The charge
carrier mobilities are taken from the experimental measure-
ments as
n=3.5⫻10−3 cm2V−1 s−1 共Ref. 12兲and
p=1.0
⫻10−3 cm2V−1 s−1.13 The dielectric constant for conju-
gated polymers is typically between 3 and 4, here we set it to
be 3.5. The effective density of states for electrons and holes
at respective electrode is chosen to be 2.5⫻1025 m−3, which
gives the boundary condition for carrier densities.7The en-
ergy gap Egap between the lowest unoccupied molecular or-
bital 共LUMO兲of the PC70BM and the HOMO of the
PCDTBT is 1.3 eV,14,15 which sets the boundary condition
for solving the Poisson equation as
共0兲−
共L兲= 1.3 − Va,共5兲
where L=80 nm is the position of anode and Vais the ap-
plied external voltage.
Our particular interest in this work is to investigate the
influence of the electron-hole pair dissociation rate kdand
decay rate kfon VOC. It has been shown that kddepends on
the electron-hole pair separation distance aas well as the
built-in field and temperature as4,7
kd=3R
4
a3e−Eb/kT
冉
1+b+b2
3
冊
,共6兲
where Ris the Langevin bimolecular recombination rate, EB
is the electron-hole pair binding energy q2/共4
a兲, and
b=q3F/共8
kB
2T2兲where Fis the field strength.
Here, we vary the electron-hole pair distance afrom 1 to
2.2 nm, which results in a range of kdfrom 105to 107s−1.
The decay rate kfis varied accordingly from 105to 107s−1
as a parameter. These values cover the practically accessible
organic materials useful in photovoltaic cell
applications.16–18 The simulated VOC versus kdand kfis
shown in Fig. 2. For VOC ⬎0.9 V, which corresponds to the
regime of kd⬎4kf, 79% of the bound electron-hole pairs dis-
sociate into free charge carriers without significant decay to
the ground state. In this case, a large number of free charge
carriers can participate in the transport and reach the elec-
trodes. We show in Fig. 3the electron and hole density dis-
tributions for various kdat fixed kf=5⫻105s−1. It can be
seen that the amount of electron-hole pairs which decay in
the bulk reduces dramatically as kdincreases.
In BHJ organic solar cells, VOC is equal to the splitting
of quasi-Fermi levels between the contacts. Based on the
concept of zero gradient of the electrochemical potential,
VOC can be defined from the quasi-Fermi level splitting as19
FIG. 1. 共Color online兲共a兲Device structure of the BHJ solar cell. 共b兲Flow
chart of the simulation process. It starts with an initial guess for the potential
and carrier densities. The steady state is obtained by solving the Poisson and
continuity equations iteratively with a convergence criterion of 10−7.
FIG. 2. 共Color online兲Influence of the dissociate rate kdand the decay rate
kfon the open-circuit voltage VOC of the PCDTBT/PC70BM solar cells.
FIG. 3. Electron and hole density distributions for various dissociation rate
kdat fixed kf=5⫻105s−1 under open-circuit condition.
143511-2 Shang et al. Appl. Phys. Lett. 97, 143511 共2010兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
共7兲
where NLand NHare the densities of states in the LUMO of
the acceptor and HOMO of the donor, respectively,20 that
cannot be exceeded by nand p. The high kdand low kf
values will lead to high values of nand p, and as a result
increased VOC according to Eq. 共7兲. This is fully in line with
the simulation results shown in Fig. 3.
In another extreme case, for kd⬍0.071kf, Fig. 2shows
that VOC ⬍0.8 V. Under such situation, only 6.6% of
electron-hole pairs dissociate into free charge carriers, most
of them decay to the ground state. This means that the elec-
tron 共hole兲density within the device is low. The open-circuit
voltage is consequently low according to Eq. 共7兲.
The experimental and simulated J-Vcurves under the
illumination of both monochromatic green light 共532 nm兲
and AM 1.5 G irradiation are shown in Fig. 4for compari-
son. The best fit is obtained by choosing kf=1.5⫻106s−1
and kd=1.8⫻106s−1 共the value given is for the open-circuit
condition because kdis field dependent兲. The good agree-
ment between the experiment and the simulation justifies the
model adopted in our investigation.
To conclude, we have investigated the influence of dis-
sociation rate and decay rate on the open-circuit voltage by a
numerical simulation model. We demonstrated that to
achieve a high open-circuit voltage, the photogenerated
electron-hole pairs must dissociate with a rate much faster
than their decay rate. The experimental current-voltage curve
for the BHJ PCDTBT/PC70BM device can be perfectly re-
produced by a set of reasonable parameters, either predeter-
mined by measurements or by fit.
This work was supported by the National Natural Sci-
ence Foundation of China 共Grant Nos. 20833004, 20773145,
20733006, 20920102031, and 20903060兲and the Ministry of
Science and Technology of China through 973 program.
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FIG. 4. Comparison between the simulated and experimental J-Vcurves for
the PCDTBT/PC70BM device. Two types of illumination condition are con-
sidered, AM 1.5 G and monochrome at 532 nm. Both manifest very nice fit
with a single set of parameters.
143511-3 Shang et al. Appl. Phys. Lett. 97, 143511 共2010兲
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
