(1) Freiburger Materialforschungszentrum
Stefan-Meier-Str. 21, 79104 Freiburg, Germany
(2) Fraunhofer Institut für Solare Energiesysteme
Oltmannsstr. 5, 79100 Freiburg, Germany
An electrical model of the dye-sensitized solar cell (DSC) is presented. Based on
material parameters, the model permits the determination of charge-carrier
distributions, the
effective electric field in the cell, the calculation of I-V curves, dark characteristics and
the
spectral response of a DSC.
A sketch of a DSC is shown in Fig. 1. The cell is
modelled as a
pseudohomogeneous effective medium, consisting of the nanoporous
TiO2
semiconductor, the light-absorbing dye and the redox electrolyte, which are
intermixed. The
front cell boundary (TCO/TiO2) is modelled as an ohmic
metal-semiconductor
contact. The back cell boundary (electrolyte/ platinized TCO) is modelled as a redox
electrode via a current-overpotential equation. Continuity and transport equations are
applied to all the mobile charge carriers involved: the electrons in the
TiO2
conduction band, and the iodide, the triiodide and the cations in the electrolyte. The
macroscopic effective electric field, resulting from the unbalanced charge-carrier
distribution
under illumination, is calculated using Poisson's equation. The internal cell voltage is
determined from the difference between the quasi-Fermi level of the electrons in the
TiO2 conduction band and the redox potential of the electrolyte. The
external
voltage depends on the external load and is calculated using an appropriate
equivalent
circuit, which includes the series resistance due to the TCO layers and contacts, and
also
shunt resistances due to internal leakages.
Within the cell, only one electron loss mechanism is considered: The capture of
conduction band electrons by the oxidized species (triiodide) of the electrolyte
(relaxation Re
of electrons). In a first approach, potential-independent rate constants are used to
describe
this reaction.
The absorption of each photon is assumed to be coupled with the injection of
one
electron into the TiO2 conduction band and subsequent oxidation of
the
electrolyte (generation Ge of electrons, assumed to be independent of the electric
potential). The characteristics of the dye enter only via its spectral absorptivity and its
concentration into the model. Thus, the following total reaction occurs everywhere
within
the effective medium of the modelled DSC:
Fig.1: Schematic diagram of the modelled DSC. The inner cell is
modelled as a
pseudo-homogeneous effective medium, consisting of the interconnected
TiO2
semiconductor particles (grey), the light-absorbing dye (small black dots) and the
electrolyte (filling the pores). The coordinates x = 0 and x = d indicate the
TCO/TiO2 interface or the electrolyte/platinum interface respectively.
The
coordinates x = 0- and x = d+ indicate positions close to the interfaces, but within
the
TCO or platinized TCO, respectively. The quasi-Fermi level EF of the
electrons near the TiO2/TCO interface (x = 0-) and the redox level
ERedox at the electrolyte/ platinized TCO interface (x =
d+) are
used to calculate the internal cell voltage Uint. UPt is
the
overvoltage and EOCRedox is the redox level of the
platinum
electrode in open circuit. EF depends implicitly on the intensity of
incident
radiation as well as on the net current density j. These reactions are assumed to be
always
in equilibrium, so the mass action law can be applied.
The model in its present state is based on the following additional assumptions:
Parameter | typical numerical value |
electron relaxation rate constant ke | 3 x102 s-1 |
electron mobility ue | 10-2 cm2/Vs |
electron effective mass me* | 5.6 me |
I- and I3- diffusion constant DI | 8.5 x10-6 cm2/s |
initial I- concentration C0I- | 0.45 M |
initial I3- concentration C0I3- | 0.05 M |
exchange current density of the platinum electrode j0 | 0.1 A /cm2 |
symmetry parameter beta | 0.78 |
effective dielectric constant epsilon | 65 |
conduction band potential ECB | - 0.81 V vs. SCE |
standard electrolyte redox potential E0 | + 0.11 V vs. SCE |
TCO resistance RTCO | 6 |
shunt resistance RP | 10 k |
incident radiation phi(lambda) | = AM1.5, 1 kW/m2 |
thickness of inner cell d | 15 um |
cell area A | 1 cm2 |
porosity p | 0.5 |
dye content | 1000 monolayers |
Fig. 2 I-V curve of the modelled DSC, calculated with the parameters of Tab. 1. |