Electrical Modeling of DyeSensitized 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 dyesensitized solar cell (DSSC) is derived on
continuity and transport equations for all the four charged species i.e.
electrons, iodide ions (I−), triiodide ions (I_{3}−) and cations. The
device model comprises of a pseudohomogeneous active layer, where solar
photovoltaic effect including both diffusion of electrons in nanoporous TiO_{2}
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
triiodide ions as function of the pseudohomogeneous active layer thickness of
the DSSC under both the opencircuit and shortcircuit 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: dyesensitized solar cell;
photovoltatic effect; pseudohomogeneous 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 triiodide 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 quasihomogeneous medium and later
integrated in 2D model of TiO2 structure [Manoucheri et al., 2018]. Belarbi et
al. [2015] illustrated an electrical model based on pseudohomogeneous
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 TiO_{2} 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 TiO_{2}
semiconductor film to a FTO (fluorinedoped 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* Photoexcitation
(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 DyeSensitized 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
photogenerations term and recombination rate qe and je are the elementary
charge and current density of electron. G(x) is the photogeneration 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 steadystate
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−𝑃𝑐𝑝𝜇 (4)
Where, the value of constant set (𝛹) equal to 4.108 m1s1, 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, triiodide 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 triiodide 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 opencircuit and shortcircuit operation
conditions, the distribution of the electrons, iodide and triiodide ions as
function of the distance x within the 12μm thick active layer of cell
are illustrated in fig. 24. 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 shortcircuit 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 shortcircuit and opencircuit 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 photoexited 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 shortcircuit condition. Moreover, under opencircuit
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.


Figure 3. The distribution of
the iodide ions as function of the distance (x) within 12μm thick active
layer of the cell under shortcircuit and opencircuit conditions

Figure
4. The distribution of the triiodide ions as function of the distance
(x) within 12μm thick active layer of the cell under shortcircuit and
opencircuit conditions

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.


Figure 5. Representation of
voltage as a function of the current density for the three different porosity
(p)

Figure 6. Representation of
voltage as a function of the current density for the three different TiO2
thicknesses




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
pseudohomogeneous 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.
References
Boschloo, G. and Hagfeldt, A. (2009). Characteristics of the
iodide/triiodide redox mediator in dyesensitized solar cells. Accounts
of chemical research, 42(11), pp.18191826..
Belarbi, M., Benyoucef, B., Benyoucef, A., Benouaz, T. and
GoumriSaid, S. (2015). Enhanced electrical model for dyesensitized solar cell
characterization. Solar Energy, 122, pp.700711.
Ferber, J., Stangl, R. and Luther, J. (1998). An electrical
model of the dyesensitized solar cell. Solar Energy Materials and
Solar Cells, 53(12), pp.2954.
Ferber, J. and Luther, J., (2001). Modeling of photovoltage
and photocurrent in dyesensitized titanium dioxide solar cells. The
Journal of Physical Chemistry B, 105(21), pp.48954903.
Hagfeldt, A. and Graetzel, M. (1995). Lightinduced redox
reactions in nanocrystalline systems. Chemical Reviews, 95(1),
pp.4968.
Ito, S., Murakami, T.N., Comte, P., Liska, P., Grätzel, C.,
Nazeeruddin, M.K. and Grätzel, M., (2008). Fabrication of thin film dye
sensitized solar cells with solar to electric power conversion efficiency over
10%. Thin solid films, 516(14), pp.46134619.
Katoh, R. and Furube, A., (2014). Electron injection
efficiency in dyesensitized solar cells. Journal of Photochemistry and
Photobiology C: Photochemistry Reviews, 20, pp.116.
Manouchehri, S., Zahmatkesh, J. and Yousefi, M.H. (2018).
Twodimensional optical fiberbased dyesensitized solar cell simulation: the
effect of different electrodes and dyes. Journal of Computational
Electronics, 17(1), pp.329336.
Ni, M., Leung, M.K., Leung, D.Y. and Sumathy, K. (2006). An
analytical study of the porosity effect on dyesensitized solar cell
performance. Solar Energy Materials and Solar Cells, 90(9),
pp.13311344.
Oda, T., Tanaka, S. and Hayase, S. (2006). Differences in
characteristics of dyesensitized solar cells containing acetonitrile and ionic
liquidbased electrolytes studied using a novel model. Solar energy
materials and solar cells, 90(16), pp.26962709..
Papageorgiou, N., Liska, P., Kay, A. and Grätzel, M. (1999).
Mediator transport in multilayer nanocrystalline photoelectrochemical cell
configurations. Journal of the Electrochemical Society, 146(3),
pp.898907.
Soedergren, Soedergren, S.,
Hagfeldt, A., Olsson, J. and Lindquist, S.E. (1994). Theoretical models for the
action spectrum and the currentvoltage characteristics of microporous
semiconductor films in photoelectrochemical cells. The Journal of
Physical Chemistry, 98(21), pp.55525556.
Usami, A. and Ozaki, H. (2001).
Computer simulations of charge transport in dyesensitized nanocrystalline
photovoltaic cells. The Journal of Physical Chemistry B, 105(20),
pp.45774583.
Vignati, S., 2012. Solutions
for indoor light energy harvesting.
Villanueva, J., Anta, J.A.,
Guillén, E. and Oskam, G. (2009). Numerical simulation of the current− voltage
curve in dyesensitized solar cells. The Journal of Physical Chemistry
C, 113(45), pp.1972219731.