In the following we will describe a simple application of tiberCAD for the 1D simulation of a Dye Sensitized Solar Cell (DSC).

The basic structure of a DSC is the following:

1. A photoanode made of a semiconductor mesoporous medium (TiO2), plunged into a liquid electrolyte.
2. A region of liquid electrolyte in contact with a counter-electrode covered by a thin platinum layer.

The surface of the semiconductor is covered by a mono-layer of dye molecules which make the oxide photoactive. The mesoporous material is obtained by sintering together nanoparticles of TiO2 of 15-20 nm in diameter. The diameter of the nanoparticles is of fundamental importance to obtain an adequate porosity of the material. In fact, in order to have a high light harvesting from the dye molecules it is necessary to have a large effective area that can be obtained only with a sponge-like material. Moreover, the porosity of the semiconductor lets the electrolyte be in contact with the dye molecules.

Here is a scheme of a DSC (in the inset: a typical Dye molecule)

The functioning of the DSC device is the following: when a photon is absorbed by a dye, it excites an electron which is transferred into the conduction band of the porous material. The charge transfer is extremely fast, in the order of tens of fs. Then, the electron percolates inside the porous material until it reaches the anode contact: a transparent conductive oxide (TCO).
The ionized dye is regenerated by the electrolyte by a redox process where iodide is oxidized and triiodide is formed. This regeneration has two important consequences: on one hand the dye is now ready to absorb another photon and on the other hand the equilibrium of the redox pair concentration in the electrolyte is broken. In particular, the concentration of triiodide is increased at the expense of iodide concentration.
Thus, a concentration gradient is formed that moves triiodide ions towards the counter-electrode and iodide ions towards the mesoporous material. Finally, the triiodide ions reach the platinum where they are reduced and transformed back in iodide ions. The cycle of charge carriers is completed and a current has been established inside the cell exploiting the energy of the photon absorbed.

Two important facts must be noted:

1. In the cell there is no permanent chemical transformation;
2. The split of the electron-hole pair created by the photon occurs directly at the dye level, the electron percolates inside the TiO2 while the hole is handled by the electrolyte. This means that charge carriers in DSCs are majority carriers and the recombination is suppressed. This is the reason why DSCs allow reasonable efficiencies despite they are made of very poor quality materials.

DEVICE MODEL

Here we will see a simple 1D model of a DSC, made of two physical regions:

1. TiO2 region, which is the mesoporous medium where the light absorption and recombination take place
2. Electrolyte region

Two boundary condition points represent the photoanode (left) and the counter-electrode (right)

Two pysical Regions, TiO2 and Electrolyte are defined, which are distinguished by the different value of the parameter porosity.
Porosity, in general, may assume a value comprised between 0 and 1 and models the rate of porosity of the nanocrystalline material. In particular, a value of porosity = 1 associated to a region defines it as a pure electrolyte region, where no semiconductor is present. On the other hand, a porosity = 0 indicates a pure semiconductor region.
A value of 0.5 is used for the TiO2 region, to indicate that both electrolyte and semiconductor are present in equal ratio. In the region electrolyte, instead, porosity=1 (default value) indicates that only electrolyte liquid is present in this region. The value of porosity parameter affects the value of the initial densities of electrons and ions. Namely, a lower porosity imply a higher value of initial electron density, since higher is the volume occupied by the semiconductor. On the other hand, a value 1 for porosity means that no initial electron density is present in the electrolyte region.

DSC SIMULATION MODEL

The numerical model of DSCs includes the dynamic and continuity equations for four charge carriers: the electrons in TiO2, the iodide/triiodide ions in the electrolyte and the cation. The inclusion of the cation is needed although it does not contribute directly to the current, but it is fundamental for the charge neutrality of the whole cell. We assume a Boltzmann distribution for the concentration of the charged species, both for the ions in the electrolyte and the electrons in the semiconductor.
The propagation of the different charged particles is assumed to be described by drift-diffusion equations.

The boundary conditions are defined as an ohmic contact (anode) with an electron collection rate (kinetic rate) and an electrochemical (Pt) contact (the counter-electrode cathode). The cathode is a contact of the special type Pt (platinum): it is modeled as a Butler-Volmer equation, the only needed parameter is the exchange current.

PHOTOGENERATION

For a device under illumination we have to define a generation term. The generation term is related to the flux of photons which reaches the active TiO2 regions and the dye present in the cell. We assume a simple Lambert-Beer exponential decay for charge generation.

The parameters for the generation include the vector which fixes the direction of light, the light intensity (in units of Sun) the absorption spectrum of the used dye, the spectrum of the light source (the default is the standard 1.5 AM sun spectrum).

I-V CHARACTERISTIC UNDER ILLUMINATION

Two sweeps are needed for the calculation of the entire I-V characteristic under illumination.

The first sweep is needed to perform the transition from dark condition to full short-circuit condition under illumination.
Thus, the generation rate (due to light irradiation) is increased step by step and at each step the current is calculated, until the desired illumination is reached.

The second sweep is on the bias at the anode and computes the I-V characteristic of the DSC under illumination.

In case of dark simulation (application of an external bias without illumination) the first sweep is not needed.

Here are a picture of the two sweeps performed in the calculation.

For every working point, nodal and elemental quantities are calculated and saved. The nodal quantities include electrochemical potential, electrostatic potential, density and recombination concentration profiles inside the cell. The elemental quantities include the currents components (remind that in a DSC the total current is the sum of three currents for the electrons, iodide and triiodide ions respectively) and the electrostatic field.

Note that in a DSC the current is fundamentally diffusion driven, while the electric field and the consequent drift current are rather small.

The figure below shows the potential profiles for the ionic species and the electrostatic potential in short-circuit condition, along the cell.

Here are current contributions, that is the values of the iodide, triiodide and electron current components along the cell (at short-circuit condition).

Finally, this is the calculated IV characteristic of the cell.