2015 Enhanced Electrical Model For Dye-Sensitized Solar Cell Characterization
2015 Enhanced Electrical Model For Dye-Sensitized Solar Cell Characterization
2015 Enhanced Electrical Model For Dye-Sensitized Solar Cell Characterization
com
ScienceDirect
Solar Energy 122 (2015) 700–711
www.elsevier.com/locate/solener
Received 10 March 2015; received in revised form 16 July 2015; accepted 31 August 2015
Abstract
The dye-sensitized solar cells (DSSC) have aroused in recent decades a growing interest from researchers in photovoltaic area, and
those for their low cost and their performance very respectable (11.2%). The physical and chemical phenomena that take place inside
DSSC cells are complex compared to the conventional one. They are related to disordered and tangled nature of materials that made the
recipe of these DSSC, such as TiO2, electrolyte, and dye. In the present paper, we deliver a detailed theoretical model based on electrical
considerations to study the impact of physical parameter of DSSC cell on the J –V characteristic, performance and photovoltaic
efficiency. The DSSC cell is modeled as a ‘‘pseudo-homogeneous effective medium” consisting of a TiO2 semi-conductor, dye absorber
of light and electrolyte in order to study the transport and electrochemical phenomena. The model is resolved numerically using the
Broyden–Fletcher–Goldfarb–Shanno (BFGS) approach and allow access to several physical parameters and their impact on the perfor-
mance of the cell. The main target is to control theses parameters to get an optimized DSSC cell for a performing photovoltaic device.
Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Dye-sensitized solar cell (DSSC); TiO2; Electrical model; Parameters impact; J–V characteristics
1. Introduction the best thin-film cells (Han et al., 2004). Most recently, due
to their competitive price/performance ratio, DSSC have
Dye-sensitized solar cells (DSSC) are thin film solar cells gained an important place among solar technologies. Their
based on semiconductor grown between a photo-sensitized efficiency is more than 11.2% (Nazeeruddin et al., 1993,
anode, photo-electrochemical material and an electrolyte 2001; O’Regan and Grätzel, 1991; Han et al., 2004) with
(Nazeeruddin et al., 1993, 2001). They are called also Grätzel costs of production significantly lower compared to the con-
cell due to the name of their co-inventor (with O’Regan) ventional solar cells. However, not only technological prob-
O’Regan and Grätzel (1991). They became attractive due to lems (long-term stability) must be solved, but also physics of
their low cost and simple to make compared to other solar devices of this type of cells is not yet augur well in detail.
cell devices although their conversion efficiency is less than It is important to optimize the performance of DSSC
and joint (theoretical and experimental) efforts were made
and continue to progress nowadays. In theoretical side,
⇑ Corresponding author. quantitative modeling of the photovoltaic response of the
E-mail addresses: souraya.goumri-said@chemistry.gatech.edu, DSSC is an important topic for improving the operation
sosaid@alfaisal.edu (S. Goumri-Said).
http://dx.doi.org/10.1016/j.solener.2015.08.037
0038-092X/Ó 2015 Elsevier Ltd. All rights reserved.
M. Belarbi et al. / Solar Energy 122 (2015) 700–711 701
Table 1
Nomenclature used to characterize the analytical model for modelling the
DSSC cell.
Symbols Nomenclature with unit
C 0I Initial concentration of iodide (M)
C 0I Initial concentration of tri-iodide (M)
3
tTiO2 Thickness of TiO2 (lm)
G Generation constant (m3 s1 )
w Recombination rate (m3 s1 )
N CB Effective density of states in the TiO2 conduction band (m3 )
ECB Conduction band energy (J)
ERedox Redox energy (J)
KB Boltzmann constant (J K1 )
T Temperature (K)
j Current density (A m2 )
ninj Electron injection efficiency
Fig. 2. The working principle of a DSSC. Band energy and Quasi-Fermi D Diffusion coefficient (m2 s1 )
energy of the conduction band of the semiconductor SC as well as the E Electric field (V=m)
redox potentials of the dye and of the electrolyte are indicated. Electron q Elementary charge (C)
injection (reactions #1 and #2) combined with electrolyte oxidation p Porosity
(reaction #3) and electron loss (reactions #5 and #6) can take place within nstref Standard reference concentration (m3 )
the whole cell. S Cell Surface (m2 )
V ext External voltage of the cell (V)
a Absorptivity of the molecules dye (m1 )
u AM 1.5 global solar spectrum (m3 s1 )
mediator is reduced to the counter electrode (CE). The se Electron lifetime (s)
k e Electron recombination constant (s1 )
maximum voltage is the difference between the redox
l Coefficient of mobility (m2 /V s)
potential of the mediator and the Quasi-Fermi level of n Symmetry parameter (A s/V m)
the semi-conductor. The positive charge is transferred from e Dielectric constant
the dye (S þ ) to a mediator (iodide I ) present in the solu-
tion. This mediator, then oxidized in tri-iodide (I
3 ), diffuses
through the solution. Thus, the cycle of redox reactions is conduction band of the TiO2 layer (Wenger et al., 2009).
looped by transformation of solar energy which is So, the equations of continuity have the form:
absorbed by an electric current. We introduce the nomen-
clature related to different mathematical symbols used in 1 dje ðxÞ
¼ GðxÞ wðxÞ ð7Þ
the following mathematical modeling in Table 1. q dx
The operating cycle can be summarized by the following
chemical reactions (Papageorgiou et al., 1996): where GðxÞ is the generation term, je is the electrical current
density, and wðxÞ is the recombination rate.
S þ ht ! S The electrons generation term is given by the Beer–Lambert
½Absorption; electron excitation in the dye molecule ð1Þ law, such as:
S ! S þ þ e Z kmax
½Injection; e injection into the conduction band of the TiO2 layer ð2Þ Ge ðxÞ ¼ ninj aðkÞuðkÞeaðkÞx dk ð8Þ
kmin
3 1
Sþ þ I ! I þS ½Dye regeneration ð3Þ
2 2 3 The Generation rates are integrated in the wavelength
I þ 2e ! 3I range from (kmin = 300 nm) to (kmax = 800 nm), where the
3 ðCathodeÞ ðCEÞ ðCathodeÞ
DSSC is active (Topič et al., 2010; Gacemi et al., 2013).
½Charge transfer reaction at the counter electrode ð4Þ ninj is the electron injection efficiency, aðkÞ is the absorptiv-
S þ þ e ity of the molecules dye (Ferber et al., 1998), uðkÞ is the
ðCBÞ ! S ½Dye regeneration by injected electron
AM 1.5 global solar spectrum, which is reduced by the
ð5Þ reflectance and the absorption of the front TCO glass.
I x denotes the location within cell, where x ¼ 0 indicates
3 ðAnodeÞ þ 2eðCBÞ ! 3 I ðAnodeÞ
TCO/TiO2 interface at the front electrode (Fig. 2), x ¼ tTiO2
½Recombination of injected e with I
3 ð6Þ indicates the interface between electrolyte and Pt–TCO.
We assume that only electron from the conduction band
recombines with tri-iodide in the electrolyte and that the
2.2. Equations of continuity
recombination rate is linear in ðne neq Þ, from where:
The electrical model of the DSSC is based on the conti- ne neq
w¼ ð9Þ
nuity equation of the electron number density nðxÞ in the se
M. Belarbi et al. / Solar Energy 122 (2015) 700–711 703
ne is the electron density; se is the electron lifetime; To solve all differential equations, boundary conditions
1 are necessary (Nithyanandam and Pitchumani, 2010):
such as se ¼
k e
At x ¼ 0, only the electron contributes to the net
k e denotes electron recombination rate constant;
current:
neq is the dark equilibrium density: ð10Þ
je ð0Þ ¼ jext ¼ jcell ; jc ð0Þ ¼ jI ð0Þ ¼ jI ð0Þ ð19Þ
3
Taking into account the stoichiometry of reaction (3)
and (4), the terms of generation and recombination of At x ¼ tTiO2 (TiO2–Pt interface), the contribution from
tri-iodide and iodide species
must
3 be affected by the corre- electron current density is zero and the charge carriers
1
sponding coefficient 2 and 2 . In the other hand, the are only the ionic species
cations are neither generated nor lost.
Following reference Topič et al. (2010), the continuity je ðtTiO2 Þ ¼ 0 ð20Þ
equations of all the three species (iodide, tri-iodide and Assuming that all the electrons are collected at x ¼ tTiO2 ,
cation) could be linked with continuity equations for the the boundary condition at
electrons and they can be written as:
dnðxÞ
x ¼ tTiO2 is ¼0 ð21Þ
1 djI ðxÞ 3 dje ðxÞ dx
¼ ð11Þ
q dx 2q dx
The metal semi-conductor contact is assumed to be
1 djI 3 ðxÞ 1 dje ðxÞ Ohmic, for which:
¼ ð12Þ
q dx 2q dx E¼0 ð22Þ
1 djc ðxÞ
¼0 ð13Þ At x ¼ 0, the net current is carried by the electron which
q dx
is governed by Nernst potential coupled with the Butler–
where jI ; jI , jc indicates iodide, tri-iodide and cation Volmer equation (Villanueva et al., 2009; Cameron
3
current densities. et al., 2005) expressed as:
2vu
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
unI ðtTiO2 Þ nOC I ðt TiO2 Þ q 1 n
2.3. Transport equations jcell ¼ jPt 4t OC 3
exp ð1 nÞ E ð0Þ EOC V int
nI ðtTiO2 Þ nI ðtTiO2 Þ KbT q F Redox
3
nI ðtTiO2 Þ q 1 n
The movement of all four charged species [electron, OC exp n ðE ð0Þ EOC
Redox Þ V int ð23Þ
nI ðtTiO2 Þ KbT q F
iodide, tri-iodide and cation] could be described by
transport equations: where jPt is the exchange current density at Pt electrode,
1 dne ðxÞ n is the symmetry parameter, nOC I is the concentrations
j ðxÞ ¼ le ne ðxÞEðxÞ þ De ð14Þ of iodide in open-circuit, nOC is the concentrations of
q e dx I 3
where E0Redox stands for the standard potentiel of the where if is the ideality factor, De is the diffusion coeffi-
I =I
3 redox couple and nstref is the standard reference
cient of electron, tTiO2 is the thickness of TiO2, and Le
concentration = 1 mol/l. is the electron diffusion length given by:
pffiffiffiffiffiffiffiffiffi
Le ¼ D e s e ð34Þ
2.5. Improvement of the model se is the electron lifetime.
The short-circuit current is given by:
Due to the conservation of particle numbers, the integral 0 1
of the concentration of the charge carriers is always quLe a @ tTiO2 Le a expðtTiO2 aÞA
J sc ¼ Le a þ tanh þ t
Table 2
Parameters used for modelling the dye-sensitized solar cell.
Parameter Value Reference
2
S 1 cm Ferber et al. (1998)
tTiO2 10 lm Wang et al. (2005b)
n 0.78 Ferber et al. (1998)
se 23.6 ms Gacemi et al. (2013)
N CB 1 1021 cm3 Bisquert and Mora-Seró (2010)
me 5.6 me Filipi et al. (2012)
Fig. 3. Equivalent circuit used for modeling DSSC. Rsh is the shunt p 0.41 Ni et al. (2006)
resistance due to internal leakages in the cell, Rs represents series resistance aðkÞ 1000 cm1 Andrade et al. (2011)
due to TCO layers, and Rext is the external load. T 298 K /
De 1.10 104 cm2 s1 Wang et al. (2005b)
DI 4.91 106 cm2 s1 Andrade et al. (2011)
The internal voltage of the cell is calculated as follow: DI 3 4.91 106 cm2 s1 Andrade et al. (2011)
C 0I 1.1 M Andrade et al. (2011)
V int ¼ I ext ðRext þ Rs Þ ð42Þ C 0I 0.1 M Andrade et al. (2011)
3
The external voltage of the cell is calculated as follow: ECB ERedox 0.93 eV Andrade et al. (2011)
uðkÞ AM 1.5 Global Ferber et al. (1998)
Rsh le 0.3 cm2/V s Ferber et al. (1998)
V ext ¼ I ext Rext ¼ Rext Sjcell ð43Þ Rs 8.4 X Andrade et al. (2011)
Rext þ Rsh þ Rs
Rsh 10 k X Ferber et al. (1998)
By varying Rext , the characteristic J –V can be obtained. k e 104 Ferber et al. (1998)
e 0.63 Wang et al. (2005b)
jPt 6.81 102 Andrade et al. (2011)
3. Method of resolution ninj 0.90 Andrade et al. (2011)
Our Model
Causon and Mingham, 2010) on an appropriate mesh of 12
J. Ferber et al
points which extends over an interval [0, tTiO2 ] (the finite 10
element methods approximates the solution within each
8
element by using some elementary shape function that
can be constant, linear or higher order). The set of equa- 6
tions are solved using the Broyden–Fletcher–Goldfarb–
4
Shanno (BFGS) method (Nazareth, 1979; Shanno, 1970;
Dai, 2013), the function BFGS has the following parame- 2
x 10
16
open-circuit conditions, the concentration of I ions
4.5
Short-Circuit increases near the front electrode. The iodide ions are
4 Open-Circuit
formed by the recombination process, while tri-iodide ions
3.5
are formed in active electrode and afterwards, they diffuse
3
toward the counter electrode.
ne (cm )
-3
2.5
In DSSC technology, it is important to control different
2
process such as the charge injection from the dye to the
1.5
semiconductor (TiO2), losses induced by the charge trans-
1
port and recombination. In our calculations, the electron
0.5
recombination constant impact on the characteristic
0
0 1 2 3 4 5 6 7 8 9 10 (J –V ) is displayed in Fig. 7. The electron recombination
x (µm) constant k e determines the dark current and therefore influ-
ences the behavior of the open-circuit voltage (Filipi et al.,
Fig. 5. Electron particle density as function of the location within cell (x).
2012). The more constant relaxation of electrons is larger,
the more concentration of electrons in the band of
electron particle density as displayed in Fig. 5 decreases conduction of TiO2 is higher (Ferber et al., 1998). A higher
with increasing x. When photons are generated, they dif- concentration of electrons means a high level of
fuse toward counter electrode and after that they recom- Quasi-Fermi, which gives us a higher voltage. In addition,
bine with iodide ions. At short-circuit condition, the first lower values of k e means less recombination and therefore
few micrometers of TiO2 layer contributes mostly to the higher yields.
external circuit. The concentration of electrons practically We display the influence of electron mobility le on the
equals to zero at the front contact and increases on func- characteristic (J –V ) in Fig. 8a It is obvious to see that
tion with position x (lm) in the cell. higher values of le lead to a shorter transit times str . Con-
Fig. 6a and b shows the concentration of I ions sequently, less electron loss in the TiO2 electrode inducing
increases with increasing of position x (lm), in the cell. higher efficiencies. For low values of the electron mobility,
However, the I 3 ions decrease with decreasing x (lm). At it results in higher open-circuit voltages. In practice, it is
short-circuit condition, the concentration of iodide ions is often observed that cells with higher open-circuit voltage
the smallest near the front electrode. Furthermore, under have smaller short-circuited currents, that will limit the
conversion efficiency. In the other hand, the in Fig. 8b we
illustrate the dependence of the transient time str on the
19 electron recombination constant. As it can be observed
x 10
a 30 from the curve, the transit time, str , is decreasing when k e
Short-Circuit
28
Open-Circuit increases. For a higher value of k e , the transient time is
nearly independent from the recombination constant.
26
The porosity of the semiconductor layer play an impor-
(cm )
-3
20
that the current density of short-circuit decreases as the
n
x 10
19 x 10 3
4.5 0.016
b Open-Circuit
Current Density (mA / cm )
0.014
2
Short-Circuit
4
0.012
(cm )
-3
0.01
3.5
0.008
tri-iodide
3 0.006
n
2 0
0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
x (µm) Voltage ( V )
Fig. 6. (a, b) Concentration of I ions increases with increasing of Fig. 7. The impact of the electron recombination constant k e on J –V
position x (lm), in the cell. characterization.
M. Belarbi et al. / Solar Energy 122 (2015) 700–711 707
14 20
18
12
Current Density (mA / cm )
2
8 12
10
6
8
4 2 6
µe1=0.0053 (Cm /Vs)
µe2=0.06 (Cm2/Vs) 4
2 d = 0.013 (µm)
Diameter of TiO 2
µe3=6.25 (Cm2/Vs) d = 0.019 (µm)
2
d = 0.027 (µm)
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Voltage (V)
Voltage (V)
Fig. 8a. The influence of electron mobility le on the characteristic (J –V ). Fig. 10. Impact of TiO2 nanoparticles diameter on J –V characterization.
0.032 14
0.03
12
8
0.024
6
0.022
0.02 4
3 µm
0.018 2 9 µm
Thickness of TiO 2
1 x 10
2
2 x 10 3 3 x 10 4 4 x 10 5 5 x 10 6 6 x 10 7 7 x 10 8 8 x 10 9 9 x 1010 12 µm
k e (s -1 ) 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Voltage (V)
Fig. 8b. The dependence of the transient time str on the electron
recombination constant. Fig. 11. Impact of TiO2 nanoparticle thickness on J –V characterization.
3.5 0.65
(eV)
0.6
2.5
Fn
µm -1)
1.5
0.5
1 Short-Circuit
Open-Circuit
0.45
0.5
0 0.4
0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 12
x (µm) x (µm)
Fig. 12. The evolution of electron conductivity in TiO2 electrode with the Fig. 14a. Short-circuit and open-circuit Quasi-Fermi levels behaviors in
position in the cell x (lm). the cell.
x 10 -7 0.8
9
0.75
(eV)
8 0.7
Fn
Effective Diffusion ( m 2 / s)
0.6
6
0.55 t TiO = 3 µm
2
5 t TiO = 9 µm
0.5 2
t T iO 2 = 12 µm
4
0.45 t TiO = 14 µm
2
3
0.4
0 2 4 6 8 10 12 14
2 Thickness of TiO 2 (µm)
0 5 10 15 20 25
Thickness of TiO 2 (µm)
Fig. 14b. The changes in electron Quasi-Fermi level as function of the
TiO2 film thickness.
Fig. 13a. The dependence of the effective diffusion coefficient on the TiO2
film thickness.
0.9
Fig. 14a shows the simulated short-circuit and open-
circuit Quasi-Fermi levels. It may be detected in the figure
Effective Electron Life Time (s)
0.5
0 0
0 10 20 30 0 10 20 30
Thickness (µm) Thickness (µm)
0.8 15
c d
0.6
10
0.4
5
0.2
0 0
0 10 20 30 0 10 20 30
Thickness (µm) Thickness (µm)
Fig. 15. (a, b, c and d) Variations of V oc , V mp , J sc and J mp respectively as function of TiO2 film thickness.
a b c
0.85 8 8
0.8
7 7
0.75
6 6
Maximum Power (mW / cm 2)
0.7
5 5
Efficiency (%)
0.65
Fill Factor
0.6 4 4
0.55
3 3
0.5
2 2
0.45
1 1
0.4
0.35 0 0
0 20 40 0 20 40 0 20 40
Thickness (µm) Thickness (µm) Thickness (µm)
Fig. 16. (a, b, and c) The variations of fill factor, efficiency and maximum power respectively as function of TiO2 film thickness.
thickness. In fact, when the light is transmitted by the TiO2 a thicker electrode can absorb more photons and leading
electrode, the intensity decreases gradually (El Tayyan, to higher J sc , and J mp .
2011). Consequently, when the thickness increases, the den- More importantly for DSSC cell characterization.
sity of excessive electrons becomes lower, resulting in a Fig. 16 represents the variations of fill factor, efficiency
decrease in V oc and V mp . On the other hand, the J sc ; J mp and maximum power with TiO2 film thickness. The fill fac-
increase suddenly with increasing thickness, then reach a tor decreases with the increasing of electrode thickness. As
peak and decrease gradually after that. The electron the figure show, it is clearly that the efficiency and the
photo-generation can easily explain the variation of maximum power follow the same variation since the energy
J sc ; J mp with thickness. Any increase of the thickness conversion of DSSC is the ratio between the maximum
of electrode directly increases the internal surface area of output power and the incident illumination power
the semi-conductor, ending in a higher load of dye. Thus, (Meng et al., 2008).
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