Numerical Procedure for Optimizing Dye-Sensitized Solar Cells

We propose a numerical procedure consisting of a simplified physical model and a numerical method with the aim of optimizing the performance parameters of dye-sensitized solar cells (DSSCs). We calculate the real rate of absorbed photons (in the dye spectral range) Greal(x) by introducing a factor β < 1 in order to simplify the light absorption and reflection on TCO electrode. We consider the electrical transport to be purely diffusive and the recombination process only to occur between electrons from the TiO 2 conduction band and anions from the electrolyte. The used numerical method permits solving the system of differential equations resulting from the physical model. We apply the proposed numerical procedure on a classical DSSC based on Ruthenium dye in order to validate it. For this, we simulate the J-V characteristics and calculate the main parameters: short-circuit current density Jsc, open circuit voltage Voc, fill factor FF, and power conversion efficiency η. We analyze the influence of the nature of semiconductor (TiO 2 ) and dye and also the influence of different technological parameters on the performance parameters of DSSCs.The obtained results show that the proposed numerical procedure is suitable for developing a numerical simulation platform for improving the DSSCs performance by choosing the optimal parameters.


Introduction
The technology and materials used for the third generation solar cells give the opportunity to obtain cells with high efficiency [1][2][3][4][5].The solar cells based on dye-sensitized nanostructure with mesoporous metal oxides (DSSCs) have attracted considerable attention since the work of O'Regan and Grätzel [6], their manufacturing being environmentfriendly and energy-efficient [1,7].Up to now, certified efficiencies over 10% under standard conditions or even higher (12,4%) at the laboratory scale were reported [7][8][9][10].Based on the low cost of materials and the simplicity of fabrication process, DSSC can have lower fabrication costs than conventional silicon-based solar cells.Taking into account this advantage, the improvement of DSSC parameters for making them widely used appears as a strong necessity.
Further optimization of the DSSC parameters requires a better correlation between interrelated processes of transport and accumulation of electrons in the mesoporous oxide phase and recombination of electrons with electron acceptors [11].In order to understand the different processes governing the DSSC's mode of operation and to enhance the DSSCs performance, modeling of processes and numerical simulation of the cells were carried out [12][13][14][15][16][17][18][19][20][21][22][23].The thickness, morphology (particle diameter and porosity) for the TiO 2 layer, electron mobility and recombination rate (electron lifetime), absorption spectrum of the dye, thickness and material quality of transparent conductive oxide (TCO) layers and the used electrolyte determine the - characteristics [17,18,[20][21][22]24].Consequently, the extracted parameters from simulation, the short-circuit current density  sc , the open circuit voltage  oc , the fill factor FF, and the power conversion efficiency  were analyzed.
In the present paper, we propose a numerical procedure for optimizing the DSSCs consisting of a simplified physical model and a numerical method capable of solving the system of differential equations resulted from the physical model.We applied this proposed procedure on a DSSC with Ru dye in order to show its validity.For this, the current densityvoltage (-) characteristics were simulated, and the main parameters  sc ,  oc , FF, and  were obtained and discussed.The influence of the thickness of TiO 2 film and electron lifetime on the cell performance parameters was analyzed aiming to be used in the cell manufacturing.

Physical Model and Numerical Method
We use a classical DSSC which is presented in Figure 1.This device contains two electrodes made of TCO glass, commonly used being fluorine-doped tin oxide (FTO).The illuminated electrode is coated with a nanoporous TiO 2 layer which is also coated with a monolayer of the Ru-complex dye.On the counter electrode of TCO glass, a thin layer of Pt (∼5 nm) which acts as a catalyst for the redox reaction is deposited [25].The space between the two electrodes is filled with an electrolyte (propyl methyl imidazolium iodide) containing an iodide/triiodide ( − / − 3 ) redox couple.In a DSSC under visible light, an electron from the dye molecule is excited, then injected into the conduction band of TiO 2 nanostructured semiconductor, and finally collected by the TCO glass electrode and transported in the external load.The positive charge (dye cations) is reduced by receiving an electron from the iodide ion ( − ) which will be subsequently regenerated by reducing triiodide ions ( − 3 ) at the platinized counter electrode.
By using the nanostructured TiO 2 , the depletion process is less pronounced, so that under reverse bias the electric field is negligible [14,26].The origin of the photovoltage is explained by a built-in potential barrier between TiO 2 semiconductor and the TCO electrode and/or by changing the position of Fermi level due to the electron injection [26].The most probable recombination process takes place between electrons from TiO 2 and  − 3 ions [18,19].In the ideal case, the rate of the absorbed photons in the volume unit can be written as where  ∈ [0, ] describes the position inside the TiO 2 film (including the dye) with a thickness ,  is the wavelength of the radiation, limits  1 and  2 are imposed by the absorption characteristics of the dye through the absorption coefficient (), and  ideal () is the spectral incident photon flux density.
In the real case, the rate of the absorbed photons  ideal () in the volume unit is reduced to  real () due to the light absorption in TCO glass electrodes and to the reflection onto [16].We describe the contributions of these processes by a factor  < 1 aiming to propose a simplified model to be used for designing and optimizing any type of DSSCs.Therefore,  real () is given by The rate of electron injection  inj () should take into account the injection efficiency  inj [11], so that In our simplified model we consider the electrical transport to be purely diffusive, and therefore it is described by a diffusive transport equation for the electrical current density  [17,19].We neglect the internal electric field generated by unbalanced local charge because of its very low value [12,17].The electron density () in the conduction band of the TiO 2 layer is described by the continuity equation in quasistationary regime.In this case, the system of equations has the following form: where  is the elementary charge,  describes the position inside the TiO 2 film  ∈ [0, ], () is the recombination rate, and  is the electron diffusion coefficient.We neglect the trapping/detrapping processes, so that in the continuity equation (see ( 4)) the corresponding terms do not appear [15].Also we assume that the recombination process takes place only between electrons from conduction band of TiO 2 layer and anions present in the electrolyte [18].Consequently, the recombination rate is proportional to () −  0 , so that where  0 is the electron density at equilibrium (in the dark) and  is the electron lifetime.
The boundary conditions are the following.
(1) Assuming that electrons are extracted by application of an external voltage , the boundary condition at where   is the density of states in the conduction band of TiO 2 semiconductor, given by the formula where  *  is the effective mass of the electron,   is Boltzmann constant, ℎ is Planck constant,  is the cell temperature, and  =   −   is the energy between the conduction band edge   and the Fermi quasilevel   .
(2) Assuming that all the electrons are collected at  =  (there is no recombination process inside TiO 2 film meaning that TiO 2 film has a good quality from the technological point of view), the boundary condition at  =  is The open circuit voltage  oc is derived from the condition  = 0 and is given by the expression where  inj (0) is the rate of electron injection at  = 0, while the short-circuit current density  sc is obtained for the condition  = 0.
The system of differential equations ( 4) and ( 5) together with boundary conditions (see (7) and ( 9)) has no enough conditions at  = 0 but has the boundary condition at  =  (see (9)).This problem is known as "the boundary value problem, " being defined by differential equations in which some conditions are specified at the initial point while the others are specified at the end point.In order to solve this system, we used a numerical method based on the "shooting method" [27] which generates different values for the initial condition at  = 0 until the final condition at  =  is satisfied.We used this method in a program developed in Mathcad.
Consequently, the numerical procedure we propose for designing and optimizing any type of DSSCs considers the essential processes which occur in a real DSSC and its technological parameters and simplifies the others, for example, the contribution of the light absorption and reflection on the TCO electrodes by introducing  < 1 for obtaining  real ().

Results and Discussion
In order to validate our numerical procedure (simplified model and numerical method) we apply it on a DSSC based on Ru dye and simulate the - characteristics.For this, we took into account the absorption coefficient () of the Ru dye between the spectral limits  1 = 300 nm and  2 = 800 nm where its absorption is very good (∼10 3 M −1 cm −1 ) [28].The numerical procedure can be applied on a DSSC based on a different dye by changing the spectral limits and  the absorption coefficient.For the incident flux density we considered the AM1.5 G spectrum [29].The other constants and parameters used in the simulation were taken from the literature [11,12,21] and are presented in Table 1.
Figure 2 shows the simulated -characteristics obtained for  = 20 m and  = 0.9 and different values for the electron lifetime  in the conduction band.
The values obtained for  sc ,  oc , FF, and  using the parameters  = 20 ms and  = 20 m and different values for  are listed in Table 2.One can see that an increase of  with 20% (by using suitable TCO electrodes) produces an increase of both  sc and  with about 24%, while  oc and FF remain practically constant.
The simulated values for  sc ,  oc , FF, and  as a function of the TiO 2 film thickness, using  = 20 ms, are presented in Table 3.According to this, Figures 3(a   thickness dependence of the short-circuit current density  sc and the power conversion efficiency , the TiO 2 film thickness being an important technological parameter for designing and optimizing the cell.One can see that both  sc and  increase with the film thickness up to maximum values of  sc = 12.601 mA/cm 2 and  = 8.589%, respectively, reached for  = 15m.By further increasing the film thickness up to 50 m, they insignificantly decrease (2%-3%).Consequently, a thicker layer will not bring any improvement to  sc or  (helpful information for design and technological processes).Also, one can observe that the increase of film thickness from 2 to 10 m does not influence the open circuit voltage  oc (remains constant) and the fill factor FF (decreases with about 2%).
In Table 4 the values for  sc ,  oc , FF, and  calculated for different values of the lifetime  using  = 20 m are given.Figures 4(a) and 4(b) present the corresponding curves, that is, the lifetime dependence of the short-circuit current  sc and the power conversion efficiency .They show that both  sc and  significantly increase when  increases up to 50 ms.This means that if other materials instead of TiO 2 ( takes values up to 50 ms) are used, the performance parameters of DSSC can be controlled and consequently optimized.
The values for all parameters we obtained from simulation are in good agreement with those reported in the literature for this type of DSSC [14,19,21,25,[30][31][32] which demonstrate that our proposed numerical procedure is valid and versatile.Therefore, it can be used for designing and optimizing DSSCs.

Conclusions
In this paper we propose a numerical procedure for optimizing the parameters of any type of DSSC, consisting of a simplified physical model and a numerical method capable of solving the system of differential equations resulted from the model.
We applied this procedure on a classical DSSC based on Ru dye by simulating the - characteristic and calculating all main parameters,  sc ,  oc , FF, and , and the obtained results are in good agreement with those reported in the literature, which validate our proposed procedure.For this, we analyzed the influence of the nature of semiconductor and dye and also the influence of different technological parameters on the performance parameters of DSSCs.Thus, for obtaining the real rate of the absorbed photons  real () we simplified the light absorption and reflection of TCO by introducing the factor  and we considered the nature of dye by using the specific absorption coefficient ().We analyzed the influence of TiO 2 /semiconductor thickness and electron lifetime in the conduction band on the main cell parameters.We showed that the increase of TiO 2 semiconductor thickness over the optimal thickness of 15 m does not influence the  sc and  values ( sc = 12.601 mA/cm 2 and  = 8.589%).We obtained a similar result for the electron lifetime higher than 40-50 ms.The obtained results in good agreement with the literature validate our numerical procedure.The analysis made on this classical DSSC based on TiO 2 with Ru dye can be extended to any other DSSCs.
The proposed numerical procedure allows a comprehensive analysis of the performance parameters of DSSCs and permits the development of a numerical simulation platform for designing and optimizing any type of DSSCs.

Figure 1 :
Figure 1: Sketch of a classical DSSC.Nanoporous TiO 2 with dye has  thickness.
) and 3(b) present the

Table 2 :
The values of  sc ,  oc , FF, and  obtained for different .

Table 3 :
The values of  sc ,  oc , FF, and  obtained for different TiO 2 thicknesses.

Table 4 :
The values of  sc ,  oc , FF, and  obtained for different electron lifetimes .