Short abstract
A recently published1 one-dimensional electrical model of the nanocrystalline TiO₂ dye-sensitized solar cell (DSC) was improved, using now a two-dimensional approach. Based on material parameters, the model permits the determination of steady-state charge-carrier distributions, the calculation of I-V curves, dark characteristics and the spectral response of a DSC. In the one-dimensional model, the treatment of the electric field was questionable. The two-dimensional model under high spatial resolution allows to answer the question of the distribution of both the electric and the electrochemical potential in the cell. Thus, a deeper insight into the operation mechanism of a DSC is obtained. It is shown quantitatively that the electric potential drops mainly at the TCO / TiO₂ interface and not at a Helmholtz - layer.

Description of the two-dimensional model
A schematic of a DSC and steps in the developement of a not too complicated model are shown in fig. 1. The inner cell consists of nanosized TiO₂ colloids and of the iodide / triiodide redox electrolyte. The TiO₂ colloids are sintered together and coated with suitable light-absorbing, charge-transfer dye molecules. The electrolyte fills the pores in the nanoporous TiO₂ layer. This compound is sandwiched between two glass substrates, which are coated with transparent conductive oxide (TCO) layers. The electrolyte / platinized TCO - contact is modelled as a redox electrode via the Nernst equation and a charge-transfer resistance. The TiO₂ / TCO - contact is modelled as an ohmic metal-semiconductor contact. The electrolyte is also in contact with the unplatinized TCO. However, in the absence of platinum, which acts as a catalyst for the redox reaction, the charge-transfer resistance is very high. Therefore, the current at the electrolyte / unplatinized TCO - contact is very small and was neglected for simplicity.
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. It is assumed that the transport of all these charge carriers can be described by effective diffusion constants or mobilities. If the assumed ordering of the TiO₂ colloids into columns, as shown in fig. 1, were real, strongly improved effective diffusion constants or mobilities would be obtained. Apart from that, cell properties should not change. The assumed geometrical structure is essential for the realization of the model calculations.
The electric potential j in the cell, and therefore the electric field , is calculated using Poisson's equation both for the TiO₂ semiconductor and for the redox electrolyte. These two materials are described by the two dielectric constants eTiO₂ and eel. Beside the mobile charge carriers involved, also the positive charge of the ionised donor sites of the TiO₂ is taken into account (nanocrystalline TiO₂ is weakly n-doped due to a non-perfect crystal structure). At the TiO₂ / electrolyte phase boundary, the conditions of continuity for the electric potential j and for the dielectric displacement vector are applied.
Within the cell, only one electron loss mechanism is considered: The capture of conduction band electrons by the oxidized species (triiodide) of the electrolyte. This process is described by an electron relaxation current density jR across the phase boundary TiO₂ / electrolyte. The absorption of each photon in the dye is assumed to be coupled with the injection of one electron into the TiO₂ conduction band and subsequent oxidation of the electrolyte. This process is described by an electron generation current density jG across the phase boundary TiO₂ / electrolyte.
The electric double layer at the phase boundary betweeen the electrolyte and the TiO₂ or the TCO is modelled by Stern's modification of the Gouy-Chapman theory2, which assumes that the ions in the diffuse layer at the electrolyte side of the electric double layer cannot approach the surface any closer than the ionic radius. There is a plane of closest approach for the centers of the ions at some distance xH, which is called the outer Helmholtz plane (fig. 2). The influence of the dye molecules, which partially block the TiO₂ / electrolyte interface, was neglected. The dye centers the model via its content in the cell and its spectral absorption coefficient only.

Fig. 1: Different models for the dye-sensitized solar cell. Three steps of simplification are shown.

The laminar structure at the right was used for the calculations. The TiO₂ - colloids are sintered together, coated with light-absorbing dye molecules and wetted with an iodide / triiodide redox electrolyte. This compound is sandwiched between two glass substrates, which are coated with transparent conductive oxide (TCO) layers. The TCO layer on the electrolyte side is covered with some platinum (Pt), which acts as a catalyst for the redox reaction. At the lower right side, coordinates x and y are indicated. aTiO₂ = 10 nm, ael = 10 nm.

Fig. 2: Stern's model of the distribution of charge and of the electric potential j in the electric double layer at a solid / electrolyte interface.

Results

Fig. 3: Electric potential distribution at the electrolyte / TiO2 interface in x - direction (y ³ 1 mm, see fig. 1 for coordinates and dimensions). TiO2 colloid radius aTiO2 = 10 nm. Two-dimensional modelling of the DSC in thermal equilibrium (non-illuminated cell).

Fig. 4: Electric potential distribution in y - direction within the electrolyte (x = ael = 10 nm), within the TiO2 (x = aTiO2= 10 nm) and at the electrolyte / TiO2 interface (x = 0). Two-dimensional modelling of the DSC in equilibrium.

In equilibrium, i.e. in the dark and no bias voltage applied, there is only one Fermi level. Only a very small electric potential drop in the order of 0.1 mV appears at the TiO₂ / electrolyte interface (fig. 3). Due to the small size of the TiO₂ colloids, band bending within a colloid is negligible. In this case, the electric potential also cannot drop in the Helmholtz layer of the electrolyte, due to electrostatic reasons. The Fermi level at the nanosized TiO₂ / electrolyte interface mainly adjusts by a change in the chemical potential of the TiO₂ semiconductor, i.e. by a change in the density of its conduction-band electrons. The model calculations show that the electric potential drops mainly at the TCO / TiO₂ contact (fig. 4).
Under operation, significant electric fields can appear within the TiO₂ semiconductor. Thus, beside diffusion, there are also significant drift currents. Nethertheless, the current-voltage behaviour of the one-dimensional modelled DSC could be confirmed. In the two-dimensional modelling, higher drift currents are compensated by lower diffusion currents, thus the modelled overall currents do not change significantly. The magnitude of the current density is determined by the gradient in the quasi-Fermi-level only, i.e. the gradient in the electrochemical potential of the electrons in the TiO₂ conduction band. The distribution of the quasi-Fermi-level within the modelled DSC under operation is shown in fig. 6.

Fig. 5: Electric potential distribution in y - direction within the electrolyte (x = ael = 10 nm) and within the TiO2 (x = aTiO2= 10 nm), see fig. 1 for the coordinates. Two-dimensional modelling of the DSC under open-circuit (OC) and short-circuit (SC) conditions. At the TCO / TiO2 contact at y =0, large potential drops occur (not completely shown here).

Fig. 6: Distribution of the electrochemical potential in y - direction within the TiO2 (x = aTiO2= 10 nm). Two-dimensional modelling of the DSC under open-circuit (OC), maximum-power-point (MPP) and short-circuit (SC) conditions.

References

[1]	J. Ferber, R. Stangl, J. Luther, "An electrical model of the dye-sensitized solar cell", Sol. En. Mat. 53 (1998) 29-54.
[2]	A. L. Bard, L. R. Faulkner, "Electrochemical methods", John Wiley & Sons 1980.

[BACK] [MAIN] [FURTHER]

---

emails: ferber@fmf.uni-freiburg.de

---

Last updated April 9, 1999