Download A comparative analysis of photovoltaic principles

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A comparative analysis of photovoltaic principles governing
dye-sensitized solar cells and p-n junctions
Juan Bisquert*a, Jorge García-Cañadasa, Iván Mora-Seró,a Emilio Palomaresb
a
Departament de Ciències Experimentals, Universitat Jaume I, 12080 Castelló, Spain
b
Department of Chemistry, Imperial College of Science, Technology, and Medicine, Exhibition
Road, London SW7 2AY, United Kingdom
*e-mail: [email protected]
ABSTRACT
We discuss a generalized model for a solar cell, and the realization with heterogeneous photochemical photovoltaic
converters such as the dye-sensitized solar cell. The different steps involved in the conversion of photon energy to
electrical energy, indicate that a key point to consider is maintaining the separation of Fermi levels in the selective
contacts to the absorber. In order to understand the irreversible processes limiting the efficient operation of the solar
cell, it is necessary to obtain a precise description of the internal distribution of Fermi levels. We suggest the equiva lent
circuit as a central tool for obtaining such description, in relation with small perturbation measurement techniques. The
fundamental steps of excitation and charge separation, and the losses by transport and charge transfer, can be
represented by suitable circuit elements, and the overall circuit configuration indicates the operation of the selective
contacts. The comparison of the equivalent circuits for heterogeneous dye solar cells and solid-state p-n junctions,
shows the significant difference in the mechanisms of the selective contacts of these solar cells.
Keywords: solar cell, TiO 2, photovoltaic principles, equivalent circuit
1. INTRODUCTION
Because of the large practical success of solid-state junction photovoltaic (PV) devices, there exists a tendency to
view properties of rectifying semiconductor p-n junctions as the fundamental basics of PV action . Junction space charge
regions with an associated built-in potential in the dark are often regarded as a major factor for obtaining PV action.
Presently, we are interested in the investigation of a range of opportunities for new devices, with distinct
characteristics, employing liquid of glassy materials (photoelectrochemical and organic cells), and heterogeneous
configurations in which the materials are spatially mixed down to the nanoscale.1-3 Especially the dye-sensitized solar
cell (DSSC)4 based on nanoporous titanium dioxide shows high conversion efficiency and stands out as a prototype of
the new heterogeneous solar cells.1 Important advantages of heterogeneous devices like the DSSC or polymer/fullerene
blend solar cells, are the low materials cost, ease of fabrication and low process temperatures that should permit
preparing cheap and versatile devices in a large scale. DSSC are distinct from conventional p-n junction devices by the
distribution of the functions of light absorption, electron transport and hole transport to different materials. These new
features have caused a scientific controversy, in respect to whether the origin of the open-circuit voltage in DSSCs
should be interpreted in terms of built-in potentials, which is suggested by some authors and denied by others.5-7 Similar
questions arise in another area with promising future development, that of organic (or plastic) solar cells.2
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Any solar cell is a device that causes a voltage difference between the outer contacts at open circuit under incident
photon irradiation. The voltage difference that is measured is a difference of the electrochemical potential of electrons in
the metal leads, i.e., it is a difference of Fermi levels, E F . Supposing metal contacts M of identical composition, no
difference of chemical potentials can exist between them, so that the difference of Fermi levels between left and right
contacts corresponds to a difference of electrostatic (Galvani) potentials ( φ ), and the open-circuit potential is
Voc = φ M (left ) − φ M (right ) . However, concerning the materials in the solar cell, there is no restriction as to which
factor causes the separation of electrochemical potentials under photon irradiation: it may be chemical or electrostatic
potential variations, or a combination of the two. Under incident irradiation Voc = −[E F (left ) − E F (right )]/ e , e being
the absolute electron charge.8,9
These basic remarks set the stage for the general approach adopted in this paper, based on a recent report on the
physical chemical principles of PV conversion.10 These principles indicate the limitation of an analysis that focuses on
electrostatic potential differences inside the cell, especially for describing the operation of heterogeneous solar cells. It
becomes necessary to analyze the electrochemical potential (quasi-Fermi level) distribution in the solar cell. How can
we access this internal distribution? One answer is given by reciprocity relationships. If the device produces the
separation of Fermi levels when it absorbs light, conversely it must emit light when the separation is imposed externally
(by forward bias).8 This leads to the study of conversion efficiency from electroluminiscence data.11,12
Our strategy to obtain a detailed picture of the internal distribution of dissipative processes, in terms of spatial
variations of Fermi levels, is based in obtaining the equivalent circuit (EC) that represents the response of the solar cell
to a given stimulus, such as a perturbation of incident illumination or bias potential applied to the solar cell. This work
aims to cover this description, from the general principles of solar energy conversion applied to the heterogeneous
photochemical PV converter,10 to the EC providing a detailed and realistic representation of DSSC operation.
It is known that ECs for DSSCs can be obtained starting from a diffusion model,13 for instance. This approach has
been developed for nanostructured semiconductor electrodes,14,15 giving the transmission line ECs that are
characteristic of diffusion.16 In this paper we adopt a different route, which constructs the circuit of the solar cell
heuristically, by considering the different steps of PV action. These fundamental steps will be described in Section 2,
summarizing the original discussion given in Ref. 10. In heterogeneous devices like DSSCs it is important to
characterize and reduce a number of non-radiative dissipative processes (carrier transport and interfacial charge transfer
between separated phases) that limit seriously the conversion efficiency. The associated impedances are indicated in
Section 3, and the special configuration imposed by the necessity of selective contacts to the absorber gives rise to the
full EC that is discussed in Section 4. This approach provides similar results as those obtained before starting from the
transport model,14,15 but the new approach shows much more clearly the interpretation of the EC in terms of PV
principles, and indicates how the ECs should be applied in the characterization of heterogeneous solar cells. The EC for
DSSCs will be validated by impedance measurements at forward bias in dark, and in Section 5 it will be compared with
the EC of a p-n junction.
2. PHOTOVOLTAIC PRINCIPLES
The simple two-level system represented in Fig. 1 illustrates the main general features of PV devices.10 Energy
conversion is realized physically by the combination of light absorption and charge separation.17 The absorber material
can be a single molecule, a semiconductor crystal network, or an organic polymer. The excitation can be an electron
hole pair in a semiconductor, an electronic excitation of a molecule, the production of excitons, etc. The excitation in the
absorber promotes electrons to the high energy level, H, and leaves an electron deficiency behind, in the low level, L,
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with the correspondent separation of the chemical potentials of the two states, ∆µ = µ H − µ L . In the case of the DSSC,
µ D and µ D* are the chemical (redox) potentials of the unexcited and excited dye molecule, respectively, and
∆µ = µ D* − µ D .10, 12
Figure 1: Scheme of an idealized solar cell converter. It
consists on an absorber A where the photons excite carriers
from a low energy (L) to a high energy (H) state, as indicated
by the arrow. Selective contacts B and C to the upper and
lower state, respectively, maintain the splitting of Fermi levels
of the carriers in the absorber up to the outer metal contacts.
The only recombination in the cell is the radiative emission
process from H to L, indicated by the dotted arrow.
The second step requires filters to each of the carriers (or “membranes”18), B and C (Figure 1), that will enable to
contact the Fermi levels of high and low states of the absorber, separately to the two metals M at the external leads. That
is, B and C are selective contacts to the absorber that realize the step of charge separation.19 Selectivity of contacts to
different carriers may be based on potential barriers at interfaces, but also on the kinetic conditions of interfacial charge
transfer. Selectivity to the dye molecules in the DSSC is obtained by preferential injection of electrons in excited state to
the TiO2 conduction band, and hole transfer from the dye ground state to the redox species.
According to our definition each selective contact extends from the absorber (dye) to the respective metal electrode.
Thus the function of B (or C) is to bring the electron from the place of excitation by transport to the electrodes through
which the PV effect is measured. In the DSSC the contact to the high-energy state of the dye is formed by the
combination of nanoporous TiO2 and transparent conducting oxide (TCO), and the redox couple in the electrolyte forms
the selective contact to the ground state of the dye. A more complete account is given in Ref. 10. In a heterogeneous
device such as the DSSC, the existence of a large internal area where both the excitation in the absorber and charge
separation by the selective contacts can be realized implies that the selective contacts to the absorber, B and C, are
closely merged spatially, typically in the nanoscale range. In Fig. 1 the two contacts have been separated at both sides of
the absorber for conceptual clarity, but the real configuration with coexistence of spatially separated contacting phases is
shown in Fig. 2. A high conductivity of the electronic species, in the phase B, is desirable in order to avoid the waste of
a substantial part of the original driving force, ∆µ , in driving the transport of the species by migration or diffusion. The
transport resistances in the EC represent this condition below.
If the device is essentially electroneutral, the excitation produced by the light can be considered as producing an
increase in the population of electrons in a set of states of fixed number, with a corresponding decrease in the population
of another set of states of fixed number.20 These devices operate by the modification of chemical potential of the
carriers and can be denominated photochemical PV converters.10 In the DSSC the original chemical potential difference
in the absorber, ∆µ = µ D* − µ D , produces a change of the chemical potential in the α phase (TiO2) of the contact B
(Fig. 2), i.e. an increase of the electron concentration. Assuming completely selective contacts means that both phase α
(TiO2) and phase γ (electrolyte) can only exchange electrons with the dye, and not with each other. In these conditions
the Fermi level in the semiconductor, E Fn * , equilibrates to the chemical potential of the excited electrons in the dye,
µ D* . E Fn * rises above E redox = µ D and
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−eVoc = ∆µ = µ D* − µ D ,
(1)
as shown in Ref. 12.
Figure 2: Scheme of the structure of a DSSC, showing the
spatially separated phases (α, β, and γ) and the dye
molecules adsorbed on the surface. Size of particles and
voids is typically in the 10 nm range. The lower part shows
the EC for an impedance measurement, containing the
transport resistances rt in phase α, the charge transfer
resistance between phases α → γ , and the chemical
capacitance c ch between TiO2 nanoparticles and the redox
system (I/I3-). This chemical capacitance corresponds to the
*
variation of Fermi level, E Fn
, with respect to the
conduction band energy, E c , and redox potential, E redox ,
as indicated in the upper part.
3. IMPEDANCES IN THE SOLAR CELL
In brief, any solar cell can be described as a pump of electrons from lower to higher Fermi level regions (the
selective contacts), and an internal resistance that shorts partially the pumping process. For example, in DSSC light
absorption at the dye molecules attached to the surface of TiO2 nanoparticles has the effect of raising an electron from a
redox particle in solution to TiO 2 conduction band.
The solar cell as a whole must have a finite impedance. The “pumping effect” in PV action, requires that the phases
with separated Fermi levels (selective contacts) cannot be shorted electronically, neither disconnected electronically.
These two conditions may be expressed in terms of impedance, respectively:
(a) The total impedance cannot be zero, because the Fermi levels would never separate.
(b) The total impedance can neither be infinite, because microscopic reversibility requires that the converse of the
excitation process (recombination with emission of a photon) proceed at a finite rate.
Therefore, the internal resistance is not a failure of device building; it is a necessary component of any solar cell. But
the magnitude of the recombination resistance is a crucial aspect for the conversion efficiency. Ideally, the only
impedance (dissipative process) of the solar cell should be the process of radiative emission by decay of the
photoexcited species, as assumed in thermodynamic studies of solar cells. 8,17 This process alone implies that the contact
phases B and C are electronically connected through the absorber and reach a thermodynamic equilibrium in the dark,
with a constant Fermi level throughout the device. Therefore, direct electron transfer between B and C is not necessary
in a heterogeneous solar cell. Consequently, in the idealized model of DSSC, indicated formerly, the only possible route
for electron exchange between phases α (TiO2) and γ (electrolyte), is the absorber phase (dye molecules).
In reality the TiO2 is exposed to the electrolyte, so recombination can occur both via oxidized dye molecules and
iodide ions. Since the iodide concentration in standard DSSCs electrolytes is rather high (typically 500 mM), the
primary recombination loss mechanism is between the injected electrons and the oxidized ions in solution. A decrease of
selectivity of contacts occurs because phase γ accepts electrons from phase α.
Current efforts to reduce the avoidable α/γ recombination use the core-shell electrode, consisting of an inner
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semiconductor matrix coated by a thin shell with a more negative conduction band potential. Results with these
electrodes show that the efficiency of the DSSC can be increased significantly via improvement of all cell
parameters.21,22
4. EQUIVALENT CIRCUITS OF DYE-SENSITIZED SOLAR CELLS
We indicate here the equivalent circuits that are valid for small perturbation measurements over a steady state, either
in frequency or time domain. These circuits are more general than the equivalent circuits usually used for describing i-V
curves in solar cells (containing resistances, diodes and current generators), because the latter circuits adopt a restriction
to zero frequency.
4.1. Equivalent circuit for the idealized photochemical PV converter
We start with the EC of the idealized converter indicated in Fig. 1. The circuit, shown in Fig. 3(a), contains a
photocurrent generator i ph for the current of excitation from ground state to high energy level of Fig. 1. The resistance
rrad indicates the radiative recombination process. From first principles, this resistance is irreducible, as discussed in
Section 3. The EC of the absorber molecules is completed with a chemical capacitance c ch that reflects the capability of
the different energy states of the absorber to accept or release additional carriers due to a change in their chemical
potentials.16
The equivalent circuits discussed here do not imply physical electrical potential differences. Rather, the general
convention adopted here is that local potentials in the EC have the meaning of local electrochemical potentials in the
solar cell. Therefore the potential difference across the chemical capacitor in Fig. 3(a) has the meaning of a difference of
Fermi levels; this is the splitting of Fermi levels in the absorber. The primary effect of PV action is to charge the
chemical capacitor in the absorber.
Figure 3: EC for an ideal solar cell. (a) The transverse elements represent the three relevant processes occurring in the light
absorber: A photocurrent generator i ph stands for excitation from ground to high energy level; the chemical capacitance c ch
accounts for the change of Fermi levels; the resistance rrad indicates the radiative recombination. The upper and lower lines
represent the ideal, resistanceless selective contacts from the high and low energy states of the absorber to the metal electrodes. (b)
The impedance of the circuit under forward bias.
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As remarked in the foregoing, another key point for the solar cell operation is the functioning of selective contacts.
Ideal selective contacts are represented in Fig. 3(a) by the horizontal resistanceless wires. Note that each wire unites one
side of the absorber (i.e. either of the carriers with seperated Fermi levels) to one and only one metal electrode. The
overall configuration of the circuit, with the different lines of selective contacts from the absorber to each metal lead,
converts the excess, photo-induced charge in the chemical capacitor to an output (photo)voltage and (photo)current.
We remark also the distributed representation of the circuit. This is because the light absorber is spatially distributed
through the volume. In principle we can attach a transverse circuit element of Fig. 3(a) to each dye molecule of the
DSSC. However, it is much more convenient to consider the transverse circuit elements as the result of a suitable
averaging procedure over a small volume element.
It is important to point out the fundamental role of the chemical capacitors of Fig. 3(a) in solar cells, because these
capacitors are not usually included in the ECs for steady-state conditions. By explicitly including this capacitor in the
equivalent circuit, we are able to describe kinetic measurements like open-circuit voltage decay (OCVD)23 or intensitymodulated photovoltage spectroscopy (IMVS).24 For example consider open circuit conditions: The capacitors of Fig.
3(a) will be charged. In physical terms this means an excess of carriers both in the high and low energy states, with
respect to the dark equilibrium values, while the potential across the capacitor represents the photovoltage, as already
mentioned. Release of this charge is measured in OCVD,23 giving the lifetime at each steady state. This transient
measurement is therefore correctly represented by our EC. Indeed it can be shown that τ = 1 / rrad c ch is the time
constant for recombination. Without c ch in the EC we can describe only i-V curves, because at frequency zero the
capacitor becomes an infinite impedance that can be removed.
The EC shown in Fig. 3(a) represents the transfer function to a modulated perturbation of the light intensity. For
instance when the potential between the outer contacts is measured, the circuit represents the response of the solar cell in
IMVS. If the contacts are short-circuited and the output current is measured, then the circuit represents the response in
intensity-modulated photocurrent spectroscopy (IMPS).24
On another hand, if we measure the relationship of modulated potential to current we obtain the electrical impedance
of the solar cell, which is represented by the circuit shown in Fig. 3(b). The current generators have been removed
because the excitation process, E L → E H , has no associated electrical impedance. The voltage applied to the leads
cannot induce this process; it requires the excitation of the absorber by light. On another hand the converse of excitation,
the recombination, can be stimulated by voltage-injection of carriers to the upper state, for example, thus it has
associated a finite impedance (resistance rrad ).
We note that the EC for impedance, Fig. 3(b) may be used both for conditions of dark and steady state illumination.
The changes of steady-state condition usually affect the parameters of the EC measured in impedance or other frequency
techniques.
4.2. Realistic equivalent circuits for the photochemical PV converter
In Section 3 we indicated that there are different kinds of dissipative processes in a solar cell, which give internal
resistances representing interfacial, transport and recombination processes. The more realistic EC for a heterogeneous
PV converter such as a DSSC is shown in Fig. 4. With respect to the idealized model in Fig. 3(a), in reality the
recombination resistance will be rct << rrad , due to parallel mechanisms of charge transfer of the species in the high
Fermi level phase to that with a lower Fermi level,25 and the recombination via non-radiative processes will be
dominant. Interfacial and transport impedances will exist also.
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Figure 4: EC for a heterogeneous PV converter such as a DSSC. The transverse elements represent the three main processes
occurring in the solar cell: A photocurrent generator i ph stands for excitation of the dye from ground to high energy level; the
chemical capacitance c ch accounts for the change of Fermi levels; the resistance rct indicates the charge transfer events from the
phase of high to the phase of low Fermi level. The upper line shows a diffusion resistance along the selective contact to the highenergy state (nanoporous TiO2), and the lower line represents the resistanceless (ideal) selective contact from the low energy state of
the absorber to the Pt (redox electrolyte).
The main difference of Fig. 4 with respect to the idealized EC of Fig. 3(a) is the transport resistance rt across the
selective contact to the high energy state. This resistance indicates a hindrance for electron transport (diffusion).
Furthermore the chemical capacitance is predominantly associated to the TiO2 phase instead of the absorber, though the
latter may contribute also. When we reduce the EC of Fig. 4 to that representing exclusively the impedance, we obtain
the circuit shown in the bottom of Fig. 2. The transmission line arrangement of rt and c ch in Fig. 2 corresponds to the
standard diffusion impedance in an extended layer.26 With the inclusion of the recombination resistance, we obtain in
Fig. 2 the impedance of diffusion coupled with reaction with the blocking (reflecting) boundary condition in a layer of
thickness L , that has been described elsewhere. 14 The impedance function has the form
1/ 2
[
]
 R R 
(2)
Z =  W k  coth (ω k / ω d )1 / 2 (1 + iω / ω k )1 / 2
 1 + iω / ω k 
where RW = Lrt is the diffusion resistance, Rk = rct / L is the recombination resistance, ω d = 1 /(c ch rt L2 ) = Dn / L2 is
the characteristic frequency of diffusion in a finite layer ( Dn being the electron diffusion coefficient) and ω k is the rate
constant for recombination ( rct = ω k / c ch ). In the conditions of low recombination, RW << Rk , Eq. (2) reduces at low
frequencies to the expression
Rk
1
(3)
Z = RW +
3
1 + iω / ω k
which traces a semi arc in the complex plane plot, shifted RW / 3 in the real axis. On another hand Eq. (2) provides at
high frequencies the characteristic pattern of diffusion consisting on a line with slope of 1 in the complex plot.
In Fig. 5 we show impedance measurements of a DSSC in the dark. The results reported here were obtained with a
DSSC in the standard sandwich-type configuration in which a dyed nanoporous TiO2 electrode is facing a Pt counter
electrode. A 4 µm film of nanostructured TiO2 was deposited over the substrate and sensitized with a Ru-complex dye.
A platinized F:SnO2 sheet was used as a counterelectrode. The cell was filled with organic electrolyte (acetonitrile, (Bu)
4NI 0.6M, LiI 0.1 M, I2 0.1 M, 0.5 M 4-tertbutylpyridine) and sealed. The conversion efficiency of the solar cell is 3.5%
(without scatter layer).
The EC of Fig. 2 implies rather characteristic impedance spectra, as that observed in Fig. 5. Both the large arc at low
frequency, and the Warburg (diffusion) 45º line are appreciated in Fig. 5. The results in Fig. 5 allow identifying the
lifetime (frequency at the maximum of the arc, τ = ω k −1 ) and the transit time for diffusion ( ≈ ω d −1 , frequency of
turnover from the 45º line to the arc). Fig. 5 shows that the electron lifetime is considerably larger than the transit time.
This enables efficient extraction of electrons, i.e. an efficient functioning of the selective contacts, implying that
conversion of solar energy to electricity is indeed possible with this DSSC.
The present analysis shows that the diffusion-reaction impedance with a blocking boundary obtained previously14 is
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the fundamental characteristic of heterogeneous photochemical PV converters at forward bias. This impedance model
can be used for obtaining the main kinetic constants of the DSSC.
Figure 5: Impedance measurement on a DSSC in the dark at
0.4 V forward bias. The lower part shows an enlargement of
the high frequency side of the spectrum.
5. EQUIVALENT CIRCUITS OF P-N JUNCTION DEVICES
In a p-n junction solar cell the functions of light absorption and charge separation are realized in the same material.
In the p side of the junction (Fig. 6) the photogenerated electron-hole pairs are transported to the junction where the
charges are separated by the potential barrier: The barrier accepts electrons and rejects partially the holes incoming from
the p-side. In the n-side the converse mechanism occurs. This has the effect of modifying the electrical energy of the
majority carriers at both sides of the junction. As emphasized recently,19 the field in the junction region is largely
incidental to the operation of the solar cell. Green19 indicates that the p-n junction solar cell works well because the ntype region forms a selective contact for the high energy state of Fig. 1 (conduction band), while the p-type region
allows selective contact to the valence band (low energy state of Fig. 1).
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Figure 6: Scheme of a p-n junction solar cell,
showing the direction of electron and hole
current, and the electron and hole quasi-Fermi
levels E Fn and E Fp . The splitting of the quasiFermi levels in the junction gives the external
voltage, V. The lower part shows the EC for
electron diffusion and recombination into the pdoped region, corresponding to the impedance at
forward bias.
The ideal EC model for the p-n junction solar cells should contain essentially the same elements as those shown in
Fig. 3(a). A current generator describes the excitation of a carrier from valence to conduction band (with energy
difference ∆E = EC − EV ). The splitting Fermi levels, i.e. the difference between the quasi-Fermi levels for electrons
and holes ( ∆µ = E Fn − E Fp ) implies a chemical capacitor. Photon-radiative recombination of electrons and holes gives
the resistance rrad .
The realistic EC for the p-n junction is shown in Fig. 6. The recombination resistance is usually rrec > rrad . For
example silicon is an indirect gap semiconductor in which band-to-band recombination is low with respect to crystal
defect-assisted Shockley-Read-Hall recombination (although it is possible to select silicon samples with very low bulk
defect assisted Shockley-Read-Hall recombination, giving a long carrier lifetime27). Included in the EC of Fig. 6 is also
the transport resistance rt for minority electron carriers diffusion in the p-doped region.
We now emphasize the main differences between the ECs of heterogeneous PV devices (DSSC) and p-n junction
devices, and the physical origin of such differences. In Fig. 6 we note that the transmission line arrangement is
terminated in a short circuit between the high and low longitudinal lines, in contrast to Fig. 2, where such shortcircuiting does not occur. This is because the metal contact to the p-doped region equalizes necessarily the Fermi levels
of both electron and hole carriers. It follows that if we reduce the diffusion resistance rt to an arbitrarily low value, as in
the ideal EC of Fig. 3(b), the junction device will be short-circuited and will not operate as a solar cell. The physical
reason for this is that the mechanism of selectivity in p-n junction is formed by different conductivities of holes and
electrons in the p-doped region.28 The p-doped region is a membrane that conducts poorly electrons and shows very
good conduction for holes. The resistanceless wire in the lower part of the EC of Fig. 6 represents this last feature. If the
conductivity is similar for both species, the selectivity is lost and the device ceases to be a solar cell. The lower electron
conductivity in the p-region (indicated rt in the EC of Fig. 6) permits to contain the electrons in the n-type side that
spill over the barrier towards the p-type side. As is well known, the leak of majority carrier electrons through the
selective contact gives rise to the ideal diode response at forward bias (Shockley equation). This model can be obtained
from the EC of Fig. 6 at frequency zero.
In contrast to this, in a DSSC the selectivity mechanism consists on different rates of charge transfer from the
absorber (dye) to the contacting phases (TiO2, electrolyte), so that it is possible to improve the conductivity in these
contacting phases as much as desired, in principle, without loosing the necessary property of contacts selectivity for the
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device being a solar cell.
6. CONCLUSIONS
We have analyzed a generalized model for a solar cell with ideal selective contacts, and the realization with
heterogeneous photochemical photovoltaic converters such as the dye-sensitized solar cell. The processes that oppose
the Fermi level separation in the selective contacts to the absorber, i.e. irreversibilities from the absorber to the metal,
can be suitably characterized by equivalent circuit elements. The overall connection of the elements also provides
essential information on the operation of the solar cell, in particular the functioning of the selective contacts. The main
difference between heterogeneous dye solar cells and solid-state p-n junctions was found in the physical mechanisms of
the selective contacts, which implies a significant structural contrast of the respective equivalent circuit configurations.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
M. Grätzel, "Photoelectrochemical cells", Nature, 414, 338, 2001.
C. J. Brabec, N. S. Sariciftci and J. C. Hummelen, "Plastic solar cells", Adv. Mat., 11, 15, 2001.
B. A. Gregg, "Excitonic solar cells", J. Phys. Chem. B, 107, 4688, 2003.
B. O' Regan and M. Grätzel, "A low cost, high efficiency solar cell based on dye sensitized colloidal TiO2 films",
Nature, 353, 737, 1991.
K. Schwarzburg and F. Willig, "Origin of photovoltage and photocurrent in the nanoporous dye-sensitized
electrochemical solar cell", J. Phys. Chem. B, 28, 5743, 1999.
F. Pichot and B. A. Gregg, "The photovoltage-determining mechanism in dye-sensitized solar cells", J. Phys.
Chem. B, 104, 6, 2000.
D. Cahen, G. Hodes, M. Grätzel, J. F. Guillemoles and I. Riess, "Nature of photovoltaic action in dye -sensitized
solar cells", J. Phys. Chem. B, 104, 2053, 2000.
W. Shockley and H. J. Queisser, "Detailed balance limit of efficiency of p-n junction solar cells", J. Appl. Phys.,
32, 510, 1961.
P. T. Landsberg and T. Markvart, "The Carnot factor in solar -cell theory", Solid-St. Electron., 42, 657, 1998.
J. Bisquert, S. Rühle, D. Cahen, G. Hodes and A. Zaban, "Physical chemical principles of photovoltaic
conversion with nanoparticulate, mesoporous dye-sensitized solar cells." J. Phys. Chem. B, submitted, 2003.
G. Smestad, "Testing of dye sensitized TiO2 solar cells II. Theoretical voltage output and photoluminescence
efficiencies", Sol. Energy Mater. Sol. Cells, 32, 273, 1994.
T. Trupke, P. Würfel, I. Uhlendorf and I. Lauermann, "Electroluminiscence of the dye-sensitized solar cell", J.
Phys. Chem. B, 103, 1905, 1999.
S. Södergren, A. Hagfeldt, J. Olsson and S. E. Lindquist, "Theoretical models for the action spectrum and the
current-voltage characteristics of microporous semiconductor films in photoelectrochemical cells", J. Phys.
Chem., 98, 5552, 1994.
J. Bisquert, "Theory of the impedance of electron diffusion and recombination in a thin layer", J. Phys. Chem. B,
106, 325, 2002.
F. Fabregat-Santiago, G. Garcia-Belmonte, J. Bisquert, A. Zaban and P. Salvador, "Decoupling of transport,
charge-storage and interfacial charge-transfer in the nanocrystalline TiO2/electrolyte system by impedance
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16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
methods", J. Phys. Chem. B, 106, 334, 2002.
J. Jamnik and J. Maier, "Generalised equivalent circuits for mass and charge transport: chemical capacitance and
its implications", Phys. Chem. Chem. Phys., 3, 1668, 2001.
T. Markvart and P. T. Landsberg, "Thermodynamics and reciprocity of solar energy conversion", Physica E, 14,
71, 2002.
P. Würfel, "Solar energy conversion with hot electrons from impact ionisation", Sol. Energy Mater. Sol. Cells,
46, 43, 1997.
M. A. Green, "Photovoltaic principles", Physica E, 14, 11, 2002.
R. T. Ross, "Some thermodynamics of photochemical systems", J. Chem. Phys., 46, 4590, 1967.
Y. Diamant, S. G. Chen, O. Melamed and A. Zaban, "Core-shell nanoporous electrode for dye sensitized solar
cells: the effect of the SrTiO3 shell on the electronic properties of the TiO2 core", J. Phys. Chem. B, 107, 1977,
2003.
E. Palomares, J. N. Cliffort, S. A. Haque, T. Lutz and J. R. Durrant, "Control of charge recombination dynamics
in dye sensitized solar cells by the use of conformally deposited metal oxide blocking layers", J. Am. Chem. Soc.,
125, 475, 2003.
A. Zaban, M. Greenshtein and J. Bisquert, "Determination of the electron lifetime in nanocrystalline dye solar
cells by photovoltage decay measurements", ChemPhysChem, 4, 859, 2003.
A. C. Fisher, L. M. Peter, E. A. Ponomarev, A. B. Walker and K. G. U. Wijayantha, "Intensity dependence of the
back reaction and transport of electrons in dye-sensitized nanocrystalline TiO2 solar cells", J. Phys. Chem. B,
104, 949, 2000.
J. Bisquert, A. Zaban and P. Salvador, "Analysis of the mechanism of electron recombination in nanoporous
TiO2 dye-sensitized solar cells. Nonequilibrium steady state statistics and transfer rate of electrons in surface
states.", J. Phys. Chem. B, 106, 8774, 2002.
J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago and P. R. Bueno, "Theoretical models for ac impedance of
diffusion layers exhibiting low frequency dispersion", J. Electroanal. Chem., 475, 152, 1999.
E. Yablonovitch and T. Gmitter, "Auger recombination in silicon at low carrier densities", J. Appl. Phys., 49,
587, 1986.
P. Würfel, "Thermodynamic limitations to solar energy conversion", Physica E, 14, 18, 2002.
SPIE USE, V. 6 5215-12 (p.11 of 11) / Color: No / Format: A4/ AF: A4 / Date: 2003-10-27 13:59:04