Electrical Modeling of Dye-Sensitized Solar Cells for Improving the Overall Photoelectric Conversion Efficiency

Swati Sahu*

S.O.S in Electronics & Photonics, Pt. Ravishankar Shukla University, Raipur (C.G.)

*Corresponding author: swati.luck05@gmail.com

[Received: 18 January 2019; Revised version: 18 April 2019; Accepted: 22 April 2019]

Abstract. An electrical model of dye-sensitized solar cell (DSSC) is derived on continuity and transport equations for all the four charged species i.e. electrons, iodide ions (I−), triiodide ions (I₃−) and cations. The device model comprises of a pseudo-homogeneous active layer, where solar photovoltaic effect including both diffusion of electrons in nanoporous TiO₂ layer as well as ions in electrolyte occur, and a bulk electrolyte layer, where only ions diffuse take place. The distribution of the electrons, iodide and tri-iodide ions as function of the pseudo-homogeneous active layer thickness of the DSSC under both the open-circuit and short-circuit operation conditions were performed. Parametric studies were conducted to analyze (J–V) characteristic of the DSSC with three different sets of porosity and also for different sets of TiO2 layer thicknesses.

Key words: dye-sensitized solar cell; photovoltatic effect; pseudo-homogeneous active layer; porosity.

Introduction

DSSCs are suitable photovoltaic systems for energy harvesting in low as well as diffuse light conditions for instance those created in cloudy climates and indoors [Boschloo et al., 2009]. DSSCs have many advantages compared to conventional solar cells for inexpensive manufacturing technology, flexibility and different color availability [Belarbi et al., 2015, Ferber et al., 1998]. DSSC is assembled using many different components such as photoanode, dye, electrolyte and a counter electrode [Ferber et al., 2001].

       Södergren et al. [2002] illustrated an analytical model based on electrons diffusion in mesoporous TiO2 film. Papageorgiou et al. [1999] explained diffusion of tri-iodide ions in redox couple and influences of porosity of TiO2 layer. Ferber et al. [2001] presented a numerical model based on charge transport processes through quasi-homogeneous medium and later integrated in 2D model of TiO2 structure [Manoucheri et al., 2018]. Belarbi et al. [2015] illustrated an electrical model based on pseudo-homogeneous effective medium. Dye sensitized solar cells comprise a variety of different components such as photoanode, dye, electrolyte and a counter electrode and many different possible combinations. It is more promising to optimize their overall performance in assembled devices by selecting the thickness of TiO2 film, selection of sensitizers and redox mediator and also identify ideal conditions. The development of model for the photovoltaic response of the DSSC is an important topic for improving the operation and extract information concerning the internal mechanisms [Oda et al., 2006].

        In this paper, electrical model of DSSC will be presented. Their input parameters will be examined via simplified structures and the simulation results will be demonstrated on DSSCs. The present model aims to be vigorous and complete to execute prediction of new DSSCs as well as their performance in the future.

Materials and Mehtod

The DSSC is a nanocrystalline photoelectrochemical device, as illustrated in Figure 1. The photon is absorbed by a monolayer of photoactive dye molecules covered to a thin nanocrystalline TiO2 layer. The photoexcited dye molecule shifts an electron into the conduction band of TiO₂ semiconductor, resulting in the oxidation of the dye and is rapidly regenerated by the liquid electrolyte generally containing the redox couple i.e., iodide/triiodide, which permeates the pores. The injected electron diffuses through the mesoporous TiO₂ semiconductor film to a FTO (fluorine-doped tin oxide) layer [Papageorgiou et al., 1999]. Finally, the oxidized mediator is regenerated at the counter electrode and moves through an external electric circuit.

(I) P + hυ → P* Photo-excitation

(II) P* → P+ + (𝑇𝑖𝑂2)− Charge Injection

(III) P+ +2I− → P + I2∗− and 2I2∗− → I3− + I− Dye Regeneration

(IV) I3− + 2e− →3I− Electrolyte Regeneration

(V) P* → P Photosensitizer Relaxation

(VI) P+ + 𝑒𝑇𝑖𝑂2− → P Recombination through dye

(VII) 2𝑒𝑇𝑖𝑂2− + I3− → 3I− Recombination through Electrolyte

Figure 1.The working principle of a Dye-Sensitized Solar Cell.

Equations of continuity

The governing equations for electrical model of device is derived from a continuity equation for the electron number density in the conduction band of the mesoporous TiO2 layer and is written as 1                   

qedje(x)dx=G(x)−Ψ(x) (1)

Where G(x) and Ψ(x) are denoted as photo-generations term and recombination rate qe and je are the elementary charge and current density of electron. G(x) is the photo-generation rate and expressed as a unit quantum efficiency of particular photon spectrum [Soedergren et al., 1994]; such as  G(x)=∫α(λ)φ(λ)λmaxλmine−α(λ)xdλ (2) Here G(x) is integrated in the wavelength range from (λmin = 300 nm) to (λmax = 800 nm) that limit the absorption band of cell. α is the absorptivity of the dye molecules at wavelength λ, φ(λ) is the spectral incident photon flux density at AM 1.5 global solar spectrum.  When the DSSC operate at steady-state conditions with irradiance, assuming that only electrons recombine in conduction band with (I3−) specie in the electrolyte, so the recombination rate are given as [Usami et al., 2001]

                           Ψ(x)= ke{ne(x).√nI3 −,1(x)nI− ,1 (x)−ne0.√nI3 −,10(nI−,10)3 .nI−,1(x)} (3)

In the domain 1, G(x) and Ψ(x) are present in active layer only. ne and 𝑛𝑒 0 are electron concentrations and initial electron under irradiance and in equilibrium. Relaxation rate constant of electron is denoted by ke. 𝑛𝐼− ,1,𝑛𝐼3 −,1,𝑛𝐼−,10 and 𝑛𝐼3 −,10 are the concentration of (I3−I−⁄) under irradiance and initial (I3−I−⁄) with no irradiance.  The electron diffusion coefficient (De) depend on the porosity value of the TiO2; such as

                                                                                   𝐷𝑒=𝛹.|𝑃−𝑃𝑐𝑝|𝜇 (4)

Where, the value of constant set (𝛹) equal to 4.10-8 m-1s-1, Pcp is the critical porosity and equal to 0.76. The power law exponent term (𝜇) is equal to 0.82. For porosity (P) values such that P < 0:41; the electron diffusion coefficient (De) is expressed as

                                                      𝐷𝑒=1.69×10−8(−17.48 𝑃3+7.39 𝑃2−2.89 𝑃+2.15) (5)

Using Einstein relation in this electrical model 𝐷𝑒=𝜇𝑒𝑉𝑡ℎ (6)

The generation of two electrons from the redox reaction at the front electrode and back electrode is equal to the generation of one I3− ions and the total recombination of three I− ions. [Katoh et al., 2014, Usami et al., 2001]    

                                                           12(𝐺𝑒−Ψ𝑒)=(𝐺𝐼3−−Ψ𝐼3−)=13(Ψ𝐼−−𝐺𝐼−) (7)

Now the continuity equations for all three ionic species (I−, I3− and cation) is linked with the continuity equations for electrons in the domain 1; such as

                         1qe.𝑑𝐽𝐼−(𝑥)𝑑𝑥=−32.qe.𝑑𝐽𝑒(𝑥)𝑑𝑥 (8) 1qe𝑑𝐽𝐼3−(𝑥)𝑑𝑥=12qe𝑑𝐽𝑒(𝑥)𝑑𝑥 (9)  1qe(𝑥)=0 (10)

Where 𝐽𝐼−,3− and 𝐽𝑐 are iodide, tri-iodide and cation current densities. There are no cations present in charge transfer in domain 2, so the generations or recombinations do not occur and the current density of cation is also zero. Similarly the continuity equations for I− and I3− also equals to zero.

Transport equations

The transport equations in DSSC describe the movement of all the four charged species (electron, I−, I3− and cation) in both the domains i.e. the active layer and bulk layer. Due to its negligible role, electric field is ignored in the numerical model and the current densities of all the four species are associated to their concentration (ne, nI−, nI3− and nc), so the charged species are able to move in diffusion only in all the four transport equations (Eqs.(11) to (14)). 1qe𝐽𝑒(𝑥)=𝐷𝑒.𝑑𝑛𝑒(𝑥)𝑑𝑥 (11) 1qe𝐽𝐼−(𝑥)=𝐷𝐼−,𝑟.𝑑𝑛𝐼−,𝑦(𝑥)𝑑𝑥 (12) 1qe𝐽𝐼3−(𝑥)=𝐷𝐼3−,𝑟.𝑑𝑛𝐼3−,𝑦(𝑥)𝑑𝑥 (13) 1qe𝐽𝑐(𝑥)=−𝐷𝑐,𝑟.𝑑𝑛𝑐,𝑦(𝑥)𝑑𝑥 (14)  De , DI−,y ,DI3−,y and Dc,y are the diffusion coefficients of the electron, I−, I3−and cation in the domain 1 and 2 (denoted by domain y).

Boundary conditions

A set of boundary conditions are required in order to get a unique solution of the differential equations [Vignati et al., 2009].

At x = 0 only the electrons involve to the net current, whereas at x = A or x = B since. TiO2 is not present so the contribution from electron current density is zero, Eq. (15) and (16). At x = A, ionic densities of ions in the active layer corresponds to the porosity p of the semiconductor TiO2 and change discontinuously at boundary of active layer (in domain 1) and bulk electrolyte (in domain 2) and follows Eq. (17).

𝐽𝑒(0)=𝐽𝑒𝑥𝑡 (15) 𝐽𝑐(0)=𝐽𝐼−(0)=𝐽𝐼𝑒−(0)=𝐽𝑒(A)=𝐽𝑒(B)=0 (16)

                                                                            𝑛𝑖,1(𝐴)=𝑝.𝑛𝑖,2(𝐴) (17)

The conservation charge of the iodine nuclei and electrons are given in the integral form in the equations (18)–(21). The total number of cations present within the device remains constant is given by:

                                                   ∫𝑛𝑐,1(𝑥)𝑑𝑥𝐴0+∫𝑛𝑐,2(𝑥)𝑑𝑥=𝑛𝑐,10.𝐴+𝑛𝑐,20.(𝐴−𝐵)𝐵𝐴 (18)

The next boundary condition (Eq. (19)):

                                     ∫𝑛𝐼3−,1𝐴0(𝑥)+1/3.𝑛𝐼−,1(𝑥))𝑑𝑥+∫(𝑛𝐼3−,2𝐵𝐴(𝑥)+1/3.𝑛𝐼−,2(𝑥))𝑑𝑥=(𝑛𝐼3−,10+1/3      

                                                      𝑛𝐼−,10).𝐴+(𝑛𝐼3−,20+1/3.𝑛𝐼−,20).(𝐵−𝐴) (19)

In Eq.(20), every three iodide ions generates two conduction band (CB) electrons. ∫(12.𝑛𝑒(𝑥)+13.𝑛𝐼−,1(𝑥))𝑑𝑥+∫13.𝑛𝐼−,2𝐵𝐴𝐴0(𝑥)𝑑𝑥=(12.𝑛��0+13.𝑛𝐼−,10).𝐴+13.𝑛𝐼−,20.(𝐵−𝐴) (20)

Eq. (21) describes the fact that the total number of charge neutrality within the cell does not change in the dark configuration, hence the total number of iodide ions and tri-iodide ions equals to total number of cations inside the cell. (𝑛𝐼−,10+𝑛𝐼3−,10).𝐷+(𝑛𝐼−,20+𝑛𝐼3−,20).(𝐵−𝐴)=𝑛𝑐,20.𝐷+𝑛𝑐,20.(𝐵−𝐴) (21)

The external current density and the cell’s voltage (VOC) is calculated by Eq. (22),

                                                                    𝑉𝑂𝐶=1qe.[𝐸𝐹𝑛(0)−𝐸𝑅(𝐵)] (22)

Where ER(d) is the redox potential at counter electrode, 𝐸𝐹𝑛(0) is the Fermi level of TiO2 semiconductor at the TCO/TiO2 interface (x=0) and is verified from the concentration of free electrons at TCO/TiO2 interface, ne(0), Eq. (23).

                                            (0)=𝐸𝑐+𝑘.𝑇.(0)𝑁𝐶 (23) 𝑁𝑐=2.(2.𝜋.𝑚𝑒∗.𝑘.𝑇ℎ2)32⁄ (24)

NC is effective density of states in the mesoporous TiO2 conduction band (Eq. (24)), where 𝑚𝑒∗ denoted as effective electron mass and Planck’s constant is denoted as h.

Results and Discussion

Firstly, the validation of the improvements amended in the presented mathematical model. For the suitable assessment with the results, we employed the same parameters extracted from literature as reported in Table 1. The set of simulations is aimed to analyze the effect of parameters on the performance of DSSC using our validated model.

Under the open-circuit and short-circuit operation conditions, the distribution of the electrons, iodide and tri-iodide ions as function of the distance x within the 12μm thick active layer of cell are illustrated in fig. 2-4. The concentration of electrons decreases with increasing value of distance x under open circuit condition as illustrated in Fig. 2.The generation of photons near the surface is more intense and they recombine with I− ions when they diffuse in the direction of back electrode. However, at short-circuit condition, some micrometers of the nanoporous TiO2 layer involve to the total short circuit current, whereas generated electrons in the active layer recombine with I3− ions. Since most of the electrons move to the front contact, so the concentration of electrons almost equal to zero with monotonically increases in distance x.

Figure 2. The distribution of the electrons as function of the distance (x) within 12μm thick active layer of the cell under short-circuit and open-circuit conditions.

The concentration of I− ions in the DSSC increases with increasing value of distance x (μm) because I−ions are normally generated at back electrode as illustrated in Fig. 3. The purpose of I− is to bring back the positively charged photo-exited sensitizer into its ground state. Most of the photons are generated because the concentration of I− ions is the least close to front electrode under short-circuit condition. Moreover, under open-circuit conditions, the concentration of I− ions is increased near the front electrode since I− can also be formed from recombination process. On the other hand, the I3− ions monotonically decrease with increasing the value of distance x (μm) as illustrated in fig. 4. The I3− ions are formed by the recombination process in active electrode and then they diffuse in the direction of the counter electrode.

The porosity of the TiO2 layer also affects the performance of the DSSC [ni et al., 2006]. As the porosity increases, the current density decreases as illustrated in Fig. 5. Due to this the effective diffusion coefficient and diffusion length decreases, so that lesser electrons are extracted. Therefore the current density increases as the porosity decreases.

The (J–V) characteristic of the DSSC with three different TiO2 thicknesses as illustrated in fig. 6. As the thickness of TiO2 layer increase photovoltage is slightly decreases because the electrons are easily recombined with I3− ions and also the electrolyte by reason of the expanded electron transport path. However, the current density increases with the increase of TiO2 thickness.  

Conclusions

In this investigation, we presented an implementation of an electrical model to simulate the DSSC and optimize their parameters using the calculation of the current density–voltage characteristic. The model based on continuity and transport equations in the pseudo-homogeneous active layer as well as the bulk electrolyte layer is demonstrated. The improved model has facilitated us to analyze the effect of different parameters on the performance of the device. This model will enable in future to predict optimized parameters for highly efficient device.

Acknowledgements

The authors would like to thank Photonics Research Laboratory, School of Studies in Electronics & Photonics, Pt. Ravishankar Shukla University, Raipur (C.G.), India for supporting this work.

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