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 E_F of the electrons near the TiO₂/TCO interface (x = 0-) and the redox level E_Redox at the electrolyte/ platinized TCO interface (x = d⁺) are used to calculate the internal cell voltage U_int. U_Pt is the overvoltage and E^OC_Redox is the redox level of the platinum electrode in open circuit. E_F 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:

• Band bending within a TiO₂ colloid is negligible due to the small size of the colloids. A detailed investigation shows that the drop of the electrical potential in a TiO₂ colloid is less than 1 mV.
• The steady-state electron transport is described by an effective diffusion constant or mobility.
• The model is a macroscopic one. Microscopic properties such as Helmholtz and diffuse layers, microscopic electric fields, screening effects etc. are neglected.

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

Tab. I: Parameters involved in the electric model of the DSC. AM1.5 radiation is incident from the TiO₂ side (front) and is corrected for 11 % loss by reflection and absorption of the conducting glass substrate. The spectral absorption coefficient of the dye alpha(lambda) was determined from the dye content and a measured absorption spectrum of a RuL₂(NCS)₂, L=2,2'-bipyridyl-4,4'-dicarboxylate dye. Potentials are indicated versus a standard calomel electrode (SCE).

Fig. 2 I-V curve of the modelled DSC, calculated with the parameters of Tab. 1.

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