ARTICLE IN PRESS
Solar Energy Materials & Solar Cells 87 (2005) 117–131
www.elsevier.com/locate/solmat
Influence of electrolyte in transport and
recombination in dye-sensitized solar cells studied
by impedance spectroscopy
Francisco Fabregat-Santiagoa,, Juan Bisquerta,
Germà Garcia-Belmontea, Gerrit Boschloob, Anders Hagfeldtb
a
Department de Ciències Experimentals, Universitat Jaume I, Avda. V. Sos Baynat s/n,
Castelló 12071, Spain
b
Department of Physical Chemistry, Uppsala University, Husarg 3, P.O. Box 579, Uppsala 75123, Sweden
Received 18 May 2004; received in revised form 22 July 2004; accepted 26 July 2004
Available online 23 November 2004
Abstract
The main features of the characteristic impedance spectra of dye-sensitized solar cells are
described in a wide range of potential conditions: from open to short circuit. An equivalent
circuit model has been proposed to describe the parameters of electron transport,
recombination, accumulation and other interfacial effects separately. These parameters were
determined in the presence of three different electrolytes, both in the dark and under
illumination. Shift in the conduction band edge due to the electrolyte composition was
monitored in terms of the changes in transport resistance and charge accumulation in TiO2.
The interpretation of the current–potential curve characteristics, fill factor, open-circuit
photopotential and efficiency in the different conditions, was correlated with this shift and the
features of the recombination resistance.
r 2004 Elsevier B.V. All rights reserved.
Keywords: Dye-sensitized solar cell; Transport; Electrochemical impedance spectroscopy; Band shift
Corresponding author. Tel.: +34 964 728 094; fax: +34 964 728 066.
E-mail address: [email protected] (F. Fabregat-Santiago).
0927-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.solmat.2004.07.017
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1. Introduction
The optimization of dye-sensitized solar cells for their commercial use as an
alternative power source of energy is the origin of research dedicated to the
identification and description of the physical processes, such as injection, transport,
accumulation and recombination of charge, that determine and limit the
performance of the device [1].
Impedance spectroscopy is a well-known technique used for the study of
electrochemical systems. The method is easy to measure but the correct interpretation
of results needs the use of a suitable model. In the case of dye-sensitized solar cells the
structure is a complex network of interconnected titanium dioxide (TiO2) nanosized
colloids deposited on a transparent conducting oxide (TCO) and permeated with a
redox electrolyte. This is the reason why impedance models published so far, to
describe the behavior of the dye-sensitized solar cell, are restricted to certain working
conditions [2,3] or apply only to separate parts of the cell [4,5].
In this work we aim to describe the general features of impedance spectroscopy of
complete dye-sensitized solar cells in a broad range of experimental conditions, both
in the dark and under illumination, for the full range of operating conditions, i.e.
from open circuit to short circuit passing through the different possible loads
attainable by the solar cell under illumination. The results are interpreted using
impedance models based on transmission lines [6] that describe the transport,
accumulation and recombination of electrons in the semiconductor phase of the
cells. Furthermore, the effect that electrolytes with different ion composition have on
these properties is also analyzed in order to monitor the displacement of the
conduction band of the semiconductor [7] and to describe differences found in
parameters such as the open-circuit potential, fill factor, short-circuit current and
efficiency of the different samples.
2. Experimental
Dye-sensitized solar cells were made from 12 nm diameter colloids deposited onto
bare TCO (F:SnO2, Hartford 8 O/&), sintered at 450 1C for half an hour and sensitized
with cis-(NCS)2(2,20 -bipyridyl-4,40 dicarboxylate)2Ru(II) (N3 dye from Solaronix)
overnight. Three different electrolytes were used in this study as shown in Table 1,
Table 1
Characteristics of the samples. Performance data were taken at 0.1 sun
Sample name
Electrolyte composition
#noMBI
#Li
#Na
#TCO
0.5 M
0.5 M
0.5 M
0.5 M
LiI, 0.05 M I2 in 3-MPN
LiI, 0.05 M I2, 0.5 MBI in 3-MPN
NaI, 0.05 M I2, 0.5 MBI in 3-MPN
LiI, 0.05 M I2, 0.5 MBI in 3-MPN
Z (%)
Voc (V)
Isc(mA)
FF
2.0
4.7
4.5
0.31
0.58
0.70
Bare TCO
1.26
1.23
0.96
blank
0.52
0.66
0.67
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119
20
-Z'' (kΩ)
-0.2 V
-0.35 V
4
-0.4 V
2
10
1
2
0
(a)
0
0
0
10
20
30
(b)
0
2
4
6
(c)
0
1
2
3
4
Z' (kΩ)
20
300
200
1000
-Z'' (Ω)
-0.45 V
-0.6 V
0
500
0
0
(d)
10
100
100
100
200
300
0
50
160
170
180
0
0
500
1000
1500
(e)
Z' (Ω)
0
50
100
150
Fig. 1. Impedance spectra of a dye-sensitized solar cell, #Li (Table 1), at different applied potentials:
(J) experimental data; (—) fit result. Insets represent enlargement of the area marked with the circle.
where 3-MPN stands for 3-methoxypropionitrile, and MBI for 1-methylbenzimidazole,
which is expected to have a similar effect as 4-tert-butylpyridine (4-TBP). Cells were
sealed with Surlyn (Du Pont) and a piece of lightly platinized TCO was used as counter
electrode. A sample of bare TCO was also prepared to be used as blank.
Electrochemical measurements were done using a CHI-660 electrochemical work
station with impedance analyzer in a two-electrode configuration. Bias potentials
ranged between 0.2 and 0.8 V depending on the open-circuit photopotential of the
cell under illumination at 0.1 sun. To enable comparisons, the same potentials were
applied in the dark. For impedance measurements, a 10 mV AC perturbation was
applied ranging between 10 kHz and 10 mHz. 0.1 sun illumination was obtained
from a 50 W halogen lamp.
3. Results and modeling
Typical impedance spectra of a dye-sensitized solar cell at different applied
potentials are shown in Fig. 1. Experimental data (J) have been fitted (—) to the
model represented by the equivalent circuit shown in Fig. 2(a). In this figure, the
representation of the network of colloids of TiO2 has been simplified to a columnar
model. The equivalent circuit elements have the following meaning [6]:
C m (=cmL) is the chemical capacitance that stands for the change of electron
density as a function of the Fermi level [8].
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RS
rt
RTCOrr
rt
rt
rr
cµ
rt
TCO+Pt
rr
rr
cµ
solution
rt
cµ
cµ
CTCO
CPt
(a) TCO
TiO2
solution
RS
TCO+Pt
R TCO
CTCO
CPt
(b) TCO
TiO2
RS
solution
TCO+Pt
Rr
Cµ
(c)
TCO
CPt
TiO2
Fig. 2. (a) Equivalent circuit for a complete solar cell. (b) Simplified circuit for insulating TiO2 (potentials
around 0 V) as currents are low, Zd may be skipped. (c) Simplified circuit for TiO2 in the conductive state.
Rt (¼ rt L) is the electron transport resistance.
Rr (¼ rr =L) is a charge-transfer resistance related to recombination of electrons at
the TiO2/electrolyte interface.
Rs is a series resistance accounting for the transport resistance of the TCO.
RTCO is a charge-transfer resistance for electron recombination from the
uncovered layer of the TCO to the electrolyte.
CTCO is the capacitance at the triple contact TCO/TiO2/electrolyte interface [9].
Zd(sol) is the impedance of diffusion of redox species in the electrolyte [4,5].
RPt is the charge-transfer resistance at the counter electrode/electrolyte
interface [2,3].
CPt is the interfacial capacitance at the counter electrode/electrolyte interface [2,3].
The first three mentioned elements are denoted in lowercase letters in Fig. 2(a)
meaning the element per unit length for a film of thickness L, because they are
distributed in a repetitive arrangement of a transmission line. The physical meaning
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121
of this network corresponds to the impedance of diffusion and reaction that is
discussed in previous papers [6,10–12]. The equivalent circuit indicates the internal
distribution of electrochemical potential (Fermi level for electrons) in response to the
modulated small perturbation of the external electrical potential at a steady state
[12]. The impedance function of the diffusion–reaction model is
Z¼
Rt Rr
1 þ io=ok
1=2
coth½ðok =od Þ1=2 ð1 þ io=ok Þ1=2 ;
(1)
where od ¼ Dn =L2 ¼ 1=Rt C m is the characteristic frequency of diffusion in a finite
layer (Dn being the electron chemical diffusion coefficient [13]), ok ¼
1=R
pffiffiffiffiffiffi
ffi r C m is the
rate constant for recombination, o is the angular frequency and i ¼ 1: Note that
od and ok are related, respectively, with the inverse values of the more commonly
used parameters of transit (or transport) time and lifetime [10].
The model of Eq. (1) is strictly valid for a homogeneous distribution of electrons
in the semiconductor, which yields to constant elements in the transmission line. It
has been shown that even in the case of strong inhomogeneity this model is
approximately valid [12].
Fig. 3 illustrates the theoretical shapes of impedance spectra corresponding to
Eq. (1), i.e. the transmission line model of Fig. 2(a) without considering the effects
of the uncovered TCO, the Pt electrode or the diffusion in the electrolyte. The
different shapes are obtained by changing only one parameter, the charge-transfer
resistance Rr.
It is interesting to describe here some cases of Eq. (1) corresponding to particular
physical situations, see Refs. [10,11] for a more extended discussion. In the
conditions of low recombination, Rt oRr (or ok ood ), Eq. (1) reduces at low
frequencies to the expression of a semicircle
Z¼
1
Rr
Rt þ
3
1 þ io=ok
(2a)
Fig. 3. Simulations of the impedance spectra of Eq. (1) varying Rr from a very high value, curve 1, to a
low value, curve 8, while keeping Rt constant. In curves 1–5, Rt oRr while in curves 7–8, Rt 4Rr :
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and to the expression of a straight line of slope 1, the conventional Warburg
impedance
o 1=2
Z ¼ Rt i
(2b)
od
at high frequencies. od indicates the frequency of transition from one behavior to the
other. Theoretical curves 1–5 in Fig. 3 are well described by these equations. Note
that for very large values of Rr (ok very small), curves 1–3, the 451 line of diffusion
given by Eq. (2b) is a minor feature at high frequencies. The impedance is largely
dominated by the reaction arc, the second term in Eq. (2a), with the characteristic
frequency ok.
On the other hand, in conditions of large recombination, Rt bRr ; the general
impedance of Eq. (1) becomes the Gerischer impedance
1=2
Rt Rr
Z¼
(3)
1 þ io=ok
that corresponds to curve 8 in Fig. 3. This behavior is similar to diffusion and
reaction in semi-infinite space [11]. As in the preceding case, at high frequency the
impedance shows a diffusion line of slope 1, corresponding to Eq. (2b), which is
implied by Eq. (1) at obok :
Note in Fig. 2(a) that the impedance of the TCO/electrolyte interface is connected
as a terminal (boundary) element to the diffusion–reaction impedance of electrons in
TiO2. The analytical model is an extension of Eq. (1) that is discussed in Ref. [14].
The different impedance elements in the equivalent circuit are determined by the
stationary distribution of carriers that will change strongly depending on the steadystate illumination and potential. This is the reason for the strong variations that can
be observed in the shape of the impedance spectra of Fig. 1. We will see that the
major factor controlling the impedance spectra is the state of the semiconductor
TiO2 at the different potentials [6], and this will allow us to extract important
information on the processes of the dye-sensitized solar cell.
The dye molecules adsorbed in the semiconductor surface control the rate of
injected electrons in the colloids and, in addition, decrease the area that is active for
recombination, as it is usually thought that recombination of electrons in the
semiconductor with the electrolyte dominates over charge losses through oxidized
dye molecules [15,16]. However, the dye molecules are not directly acted upon by the
modulation of the Fermi level that is applied in impedance spectroscopy, and are
therefore not represented distinctly as a separate element in the impedance model.
We discuss now the interpretation of the measured spectra according to the
previous model. At the lower applied potentials, Fig. 1(a), the main characteristic is a
simple semicircle displaced from the origin by a quantity RS. As will be explained
below, in this case the TiO2 is an insulator and the equivalent circuit of Fig. 2(a) may
be reduced to the parallel association of the resistance RTCO and the capacitor CTCO,
in series with RS, Fig 2(b). At higher frequencies, a small deformation of this single
arc occurs (not shown). This effect arises from the contribution of the platinized
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123
counter electrode and may be modeled by the parallel combination of the platinum
charge transfer resistance, RPt, and double layer capacitance CPt. In this case
currents flowing through the complete cell are very small, so that diffusion effects of
triiodide are negligible and Zd(sol) may be ignored.
The spectrum shown at intermediate potentials, Fig. 1(d), corresponds to the
behavior of the whole transmission line [6] as described above in relation to Eq. (1)
and in Fig. 3 for Rt oRr : The diffusion behavior is clearly appreciated in the highfrequency wing, with a slope close to 1, see the inset of Fig. 1(d). The low-frequency
semicircle is the result of the parallel association of the electron chemical capacitance
C m with the charge-transfer resistance, Rr, along the TiO2, as given by Eq. (2a).
Zd(sol) is also negligible in this case.
As mentioned before, Fig. 1(d) appears in the case in which the charge-transfer
resistance is considerably larger than the transport resistance (lifetime4transit time).
The opposite case, Rt 4Rr ; is the Gerischer impedance that implies a strong
recombination rate relative to conductivity observed under illumination where, as
will be shown later, Rr decreases significantly with respect to its value in the dark,
Fig. 4. In these conditions, rising up the Fermi level position with the applied
potential turns into the case of Rt oRr that yields again the shape of Fig. 1(d).
We remark that in the case represented in Fig. 1(d) RTCO bRr : This fact has
allowed to clearly identify the characteristic spectrum of diffusion–reaction given by
the condition Rt oRr and as consequence, the model of Fig. 2(a) can be simplified by
eliminating RTCO. In the potentials that range between Figs. 1(a) and (d), the
complete model of Fig. 2(a) is needed to fit the data, and the spectra have an
intermediate shape between both, Figs. 1(b) and (c). In these cases, the responses of
the TCO, the TiO2 and even the counter electrode overlap in the frequency domain
and cannot be separated without using the model.
At more negative potentials the Fermi level in TiO2 approaches the lower edge of
the conduction band, and both Rt and Rr become smaller due to the increasing
electron density. At a certain potential the TiO2 becomes sufficiently conductive so
that Rt is negligible and C m bC TCO : In this case the circuit of Fig. 2(a) reduces to the
one in Fig. 2(c). It consists of the parallel connection of Rr and Cm in series with RS
plus the effects of the counter electrode and diffusion in the electrolyte. At these
-Z'' (Ω)
200
100
0
100
200
300
400
Z' (Ω)
Fig. 4. Impedance spectra of the solar cell #Na (Table 1) under 0.1 sun illumination at 0.55 V: (J)
experimental data; (—) fit result. This set of data is well described by the Gerischer impedance.
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Rt / Ω
10
10
10
10
5
10-2
4
10
Cfilm / F
10
3
2
10
10
1
10
-0.8
(a)
-0.6
-0.4
-0.2
10 5
-3
-4
10 4
10 3
10 2
-5
10 1
-6
10 0
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
0.0
Potential / V
10 6
Rct / Ω
124
Potential / V
(b)
Potential / V
(c)
Fig. 5. Results from the impedance data for the three samples: #Na (r), #Li (J) and #noMBI (&) in the
dark at the different applied potentials. (a) Evolution of transport resistance in TiO2. (b) Capacitance of
the cell without the contribution of the platinum capacitance. (c) Charge transfer resistance of the cell,
after subtracting platinum and series resistance.
10
3
10 6
10-2
10 5
2
10
10
10
(a)
1
-0.8
-0.6
-0.4
-0.2
Potential / V
10
0.0
(b)
-3
Rct / Ω
10
Cfilm / F
Rt / Ω
10
-4
10 4
10 3
10 2
-5
10 1
10 0
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
-6
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
Potential / V
(c)
Potential / V
Fig. 6. Same as Fig. 5 under 0.1 sun illumination: #Na (r), #Li (J) and #noMBI (&).
potentials we found typical spectra as that shown in Fig. 1(e). The high-frequency
semicircle is the result of the evolution with the potential of the small deformation
mentioned for lower potentials, related to the parallel connection of RPt and CPt
from the counter electrode. The intermediate semicircle is the combination of Rr and
C m : Finally, the small feature at the lowest measured frequencies, see inset, is the
effect of the diffusion of the ions in the electrolyte [3–5]. In this work we are not
considering a detailed description of this effect.
Impedance data of three cells with different electrolytes indicated in Table 1 were
measured within a range of bias potentials in the dark and under illumination. The
main physical parameters obtained from the impedance model by fitting to the
equivalent circuits are plotted in Figs. 5 and 6.
4. Discussion
The observation of the impedance spectrum of Fig. 1(d) has two important
implications on the description of the behavior of the dye-sensitized solar cell. The
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125
first one is related with the
Rt oRr that yields to a value of the normalized
ffiffiffiffiffiffiffiffiffiffiffiffiffi
pcondition
diffusion length Ld =L ¼ Rr =Rt [10] greater than 1. A diffusion length of electrons
larger than the thickness of the film means that the transit time is shorter than the
lifetime, and this is a necessary condition to efficiently collect the charge injected by
the dye when illuminating the solar cell. Therefore, the impedance shape of Fig. 1(d)
is an important characteristic feature of an efficient nanocomposite solar cell device.
Furthermore, the condition Ld =L41 implies nearly homogeneous distribution of
the Fermi level along the nanoporous structure, at least in the vicinity of open
circuit [12].
The second implication is that at this potential and at the more negative ones,
electron losses are mainly produced at the TiO2/electrolyte interface provided that
RTCO 4Rr : Therefore, to improve the performance of the cell the recombination
through this interface needs to be diminished as remarked by many authors [17,18].
In the following, we discuss the trends and interpretation of the main impedance
parameters as a function of steady-state conditions for the different electrolyte
compositions.
4.1. The transport resistance
Figs. 5(a) and 6(a) show the change of transport resistance of electrons in TiO2,
from a very high to a very low value when the applied potential is displaced toward
negative values. The resulting Rt–potential curves are dependent on the electrolyte
composition. Both observations may be explained as follows. The conductivity of a
semiconductor is
s ¼ emncb
(4)
with e the electron charge, m the mobility (that is proportional to the free electrons
diffusion coefficient) and ncb the density of electrons in the conduction band (cb)
given by
E F E cb
ncb ¼ N cb exp
;
(5)
kT
where N cb is the effective density of states in the conduction band, k the Boltzmann
constant, T the temperature, E cb the position of the lower edge of the cb and E F the
position of the Fermi level in the semiconductor that is governed by the applied
potential, V a ¼ ðE F E F0 Þ=e:
If we consider that the mobility is constant, from Eqs. (4) and (5) we obtain that
E F E cb
Rt ¼ R0 exp ;
(6)
kT
with R0 constant for all the samples provided that their geometrical dimensions
are similar. Eq. (6) presents the exponential behavior shown in Figs. 5(a) and 6(a),
with a theoretical slope of 60 mV per decade that agrees with the values obtained in
the dark (66 mV per decade). On the other hand for the data taken under
illumination it is obtained as 130 mV per decade. This difference is related to the
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presence of the injected charge in the sample, that yields smaller values of Rt at the
same potentials, but this feature has not been successfully modeled yet.
Focusing the attention on the shift of the transition potentials now, it follows from
Eq. (6) that it can only be attributed to a displacement of the conduction band
energy levels. For samples #Na and #Li, the shift in Ecb, that was attributed to the
difference in the cation size in a previous work [7], agrees quite well with the
measured open-circuit photopotential difference, see Table 1. On the other hand, the
relative displacement in Ecb between samples #Li and #noMBI, is associated with the
surface adsorption of this basic compound that also shifts upwards the conduction
band in TiO2. However, this shift is only partially responsible for the total VOC
difference. The other part has its origin in the recombination process as will be
explained below.
4.2. The capacitance
Fig. 5(b) shows the capacitance of film in the dark which, according to Fig. 2(a) is
the sum of the exposed substrate and the TiO2 capacitances: C film ¼ C TCO þ C m : In
good agreement with data from transport of Fig. 5(a), the capacitance of the film
starts to rise exponentially only when the TiO2 has a measurable conductivity. This
exponential rise is a behavior typical of the chemical capacitance of TiO2 that is
related to the total density of electrons, n, by the expression [8,19]
e2
n;
(7)
kT
where n / nbcb and 14b40 indicate an exponential distribution of traps below the
conduction band edge ðb 2 ½0:2; 0:3Þ as found in our results here and in previous
work [20]. The case b=1 means that all the electrons belong to the conduction band.
Eqs. (6) and (7) imply that both Cfilm and Rt should switch to exponential behavior
at the same potential. This fact is confirmed by Figs. 5(a) and (b). Before this
happens, Cfilm is due exclusively to CTCO, as can be directly deduced from the
comparison of the capacities of the #Li complete cell and the #TCO blank, Fig. 7(a).
Under illumination, due to the electron injection from the dye to TiO2, transport
resistance and consequently charge accumulation in TiO2 become dominant at
slightly lower externally controlled potentials, Figs. 6(a) and (b).
Cm ¼
4.3. Recombination resistance
The overall charge transfer resistance of the film, Rct, that is the parallel
combination of RTCO and Rr, is represented in Figs. 5(c) and 6(c). Rct has also two
well-defined regions of behavior: When the TiO2 is an insulator, electron
recombination takes place mainly at the uncovered TCO that presents a transfer
factor a 0:25; see below. When the TiO2 becomes electronically conducting,
Rct changes its slope indicating that recombination occurs through the colloids
surface, yielding a0 0:5: To avoid the recombination through the TCO, it can be
covered by a thin layer of compact TiO2 or polymer, but from these figures we can
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106
10-2
#TCO
#Li
105
10-3
Rct / Ω cm2
Cfilm / F cm-2
127
10-4
104
103
102
10-5
101
-0.9
(a)
-0.6
-0.3
Potential / V
-0.9
0.0
(b)
-0.6
-0.3
0.0
Potential / V
Fig. 7. Comparison between the results of Cfilm (a) and Rct (b) obtained for bare TCO and a complete dyesensitized solar cell.
conclude that a large difference could not be expected for these samples as most of
the recombination at the potentials of maximum efficiency occur at the TiO2
interface.
The values of Rct under illumination, Fig. 6(b) are lower than those obtained in
the dark. Once discarded, recombination losses through the unregenerated dye
[15,16], this decrease may be related to the higher concentration of I
3 ions near the
surface of the colloids produced during the regeneration of the dye and to the
recombination of the photogenerated electrons, specially at the more positive bias.
In order to show graphically the potential region in which TCO response
dominates the behavior of Rct and its transition to the region dominated by the TiO2
nanocrystalline network, in Fig. 7(b) the results for the sample of bare TCO and the
complete sample #Li, both with the same electrolyte composition, are presented. To
make a better comparison, the obtained Rct have been normalized to the geometric
area and the exposed factor of TCO, e ¼ Rct ð#LiÞ=Rct ð#TCOÞ ¼ 0:90ð0:05Þ
calculated at the most anodic potentials in the dark. Thus, Fig. 7(b) shows clearly
that electron recombination in the region of potentials more positive than 0.4 V is
given mainly at the uncovered TCO. The change in the slope of Rct in the complete
dye-sensitized solar cell for potentials more negative than 0.4 V indicates that
electron recombination takes place through the TiO2. Note that the low coverage of
the bottom TCO, 10% (=(1e) 100), obtained here agrees very well with
previously obtained values that were attributed to the mismatch between the crystal
lattices of TiO2 (anatase) and SnO2 in the TCO (cassiterite) [9].
Analytical description of charge transfer resistance for the TCO follows: Fluorinedoped SnO2 is a high conductivity degenerated semiconductor that behaves very
similarly to a metal. Current of charge transfer in a semiconductor/electrolyte phase
is thus governed by the Buttler–Volmer relation [21]
ha i
ð1 aÞ
E j 0 exp E ;
(8)
j BV ¼ j 0 exp
kT
kT
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j0 being the exchange current density, a the transfer factor and E ¼ E F E redox the
energy difference between the Fermi level in the semiconductor and the redox level.
In a two-electrode configuration, for small potential drops in the counter
electrode, the E is governed by the applied potential, V a ¼ E=e; and it is possible to
calculate the charge transfer resistance as
1
di
RTCO ¼ e
;
(9)
dE
i being the current flux. Applying Eq. (9) to Eq. (8) we obtain the resistances
h ae
i
Va
RC ¼ R0C exp
(10)
kT
and
ð1 aÞe
Va ;
RA ¼ R0A exp (11)
kT
for cathodic and anodic applied potentials, respectively. From the cathodic and
anodic slopes of the data of the bare TCO sample, we obtain very good agreement
with theory as a ¼ 0:25 and 1 a ¼ 0:69:
On the other side, for sample #Li, in the region of TiO2 recombination, Eq. (10) is
followed with a0 ¼ 0:5; as commented before. This change in the transfer factor
thereby provides a way to distinguish between the two recombination processes.
However, the description of recombination from TiO2 through Eq. (10) can be
considered only a phenomenological expression, indeed, note that the dependence of
Rct in Fig. 6(c) is not exponential but tends to a Gaussian shape. Electrons may
recombine not only from the conduction band but also from localized intraband
states and from an exponential distribution of surface states in the bandgap [13]. In
addition, the different surface states differently match the fluctuating energy levels of
the acceptors in solution, depending on the reorganization energy. Henceforth, the
charge-transfer resistance for recombination from the nanostructured TiO2 network
necessitates a more complete model that will be presented in future work.
Note that in a previously reported analysis [22] the value obtained for the transfer
factor for the TCO was 0.5. We attributed this difference to the use of acetonitrile as
solvent and methylhexylimidazolium iodide, MHII, as salt in that case. Preparation
of cells with similar electrolytes, yielded the same value of a as theirs. This suggests
that an electrolyte with these components induces a recombination process involving
the double number of electrons (most likely 2 instead of 1) than in the case of its
absence, that yields to a value of a00 =2a.
4.4. Effect of bandshift and steady-state characteristics
A comment has to be made about the origin of the change in slope observed at the
more negative potentials of Figs. 6(b) and (c), for C film and Rct : Under illumination
at these potentials Rct and RPt have values within the same order of magnitude.
Therefore, a significant drop of potential takes place in the counter electrode. The
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129
pertinent correction in these figures (not shown) yields the same tendency as that in
the dark.
Taking into account this last comment, let us discuss the effect of bandshift on
charge transfer resistance under illumination shown in Fig. 6(c). In the region of the
more negative potentials, where the recombination takes place through the TiO2, we
observe that the shift in the potential at which Rct reaches the same values for the
samples #Na and #Li, is nearly the same as the conduction band change observed in
Fig. 6(a). However, if we compare this shift for samples #Li and #noMBI, we find
that it is larger than the conduction band displacement. At the same time it agrees
very well with the difference obtained in the open-circuit voltage, Table 1. A
reasonable explanation for this fact may be partial passivation of TiO2 by MBI
adsorbed at the surface of the colloids that reduces the effective area available for
recombination. The consequent rise in Rct ; that can be observed in Figs. 5(c) and 6(c)
for the samples with MBI, produces an increment in Ld =L in the region of transition
from insulating to the conducting state of TiO2. Thus, Ld =L passes from slightly
minor to greater than 1 and is the element responsible for the increment in the fill
factor.
Thus, the addition of MBI in the electrolyte, similar to what happens with 4-TBP,
has two effects on the performance of the cell: it rises up the conduction band
position and reduces recombination yielding higher V OC ; fill factor and efficiency.
Differences in I SC between #Li and #Na are mainly due to the different
thicknesses of the sample, around 17% in average, but there are also two other
minor contributions due to the rise of the conduction band of the semiconductor:
The first, a lower matching/driving force for the injection of the electrons from the
excited levels in the dye towards the TiO2 [23]. The second, a higher driving force for
recombination that would lead to a decrease in Rct : Our data do not give sufficient
resolution to evaluate these corrections.
Finally, note that Eq. (10) follows the behavior of a diode for V a 4e=kT: Thus,
diode models typically used to fit the I–V curves data may be applied, and indeed the
origin of the diode equation is the recombination rate dependence on electron
concentration [8]. We remark that in the present case, in a first approximation
(considering the peculiarities of electron recombination from the TiO2 nanoparticles
and the particular physical meaning of the parameters obtained), the proper model
would need two diodes of different exponents to fit the data.
4.5. Chemical diffusion coefficient
From the data of transport and the chemical capacitance, the effective (chemical)
diffusion coefficient, Dn ¼ L2 =Rt C m ; was calculated and results in the dark and
under illumination are shown in Fig. 8. Together with the same shifts described
before, two new features that affect Dn under illumination should be remarked: (i) It
is much larger than the value in the dark, which is consistent with the electron
injection in the semiconductor network that yields the better conductivity shown in
Fig. 6(a). (ii) Its exponential behavior flattens as could happen if it approached the
conduction band diffusion coefficient [13]. However, the values of Rt and Cm from
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-3
10
-4
D / cm2 s-1
10
-5
10
-6
10
-7
10
-0.8
-0.6
-0.4
-0.2
0.0
Potential / V
Fig. 8. Chemical diffusion coefficient for samples in the dark #Na (.), #Li (K) and #noMBI (’) and
under illumination #Na (r), #Li (J) and #noMBI (&).
which it is calculated do not present the expected tendency for a cb mechanism.
Thus, this last point remains open for further research.
5. Conclusions
A transmission line-based model was successfully used to describe the electrochemical behavior of dye-sensitized solar cells with different electrolytes in the dark
and under illumination from the results of impedance spectroscopy. The
characteristic impedance spectra have been discussed and the conditions needed to
obtain an efficient dye-sensitized solar cell were commented on.
The parameters for transport, accumulation and recombination processes have
been separated and determined using the impedance model. The range of potentials
at which recombination of electrons is given through the TCO or the TiO2 is
distinguished and related to the conducting/insulating state of TiO2.
Conduction band shifts were clearly monitored when varying the electrolyte
composition. In combination with recombination data these displacements of the
band were utilized to describe the changes observed in the changes in photopotential,
fill factor and short-circuit current that govern the efficiency of the cell.
Acknowledgments
The authors want to acknowledge Eric Lewin for the preparation of the cells. This
work was supported by projects BFM2001-3604 from MCyT, 02G014.31/1 from
Fundació Bancaixa and JMB/JG/AP from Generalitat Valenciana, and by the
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131
Ångstrom Solar Center program of the Swedish Energy Agency (Swedish
Foundation for Strategic Environmental Research).
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