Abstract

Analytical model is developed to give insight into the influence of the incident wavelength of the incoming light illumination on optical and electrical properties mesostructured hybrid perovskite-based solar cell for different photoactive layer thickness. Bruggeman theory and T-matrix are used to compute the analytical expressions of optical parameters such as absorption coefficient and generation rate. Then, the short-circuit current density is deduced from the continuity equation. The absorption coefficient of the photoactive layer decreases with high incident wavelength of the incoming light of the visible spectrum (400-800 nm) while the absorbance and generation rate grow up with the phototactive layer thickness. The current density increases with illuminated wavelength ranging 400-520 nm while it decreases with higher active medium (TiO₂ +Perovskite) layer thickness ranging from 500 nm to 1.7 µm. Beyond to 520 nm of the wavelength of the incoming light, the current density decreases to vanish at 800 nm. As finding in the literature, it can be noticed high absorption, charge carriers generation and current density for low wavelength in the visible spectrum which correspond to high energies.

1. Introduction

Recently, the investigation of hybrid perovskite-based solar cells attract the attention of scientific community around the world because of their important and remarkable luminous characteristics ¹. The first paper on perovskite solar cells was published in 2009 by Akihiro Kojima and al., with low conversion efficiencies of 3.13% and 3.81% ². Several investigations have been developed to improve their efficiency ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}. Due to their rapid evolution, perovskite are very competitive compared to the third generation of photovoltaic devices (organic solar cells) and conventional (silicon) technologies which have been studied in the last decades. A record conversion efficiency of about 22.1% has been certified in 2016 ²¹. There is a series of organic-inorganic perovskite materials (CH₃NH₃AX₃) where M is a metal (M = Pb, Sn) and X is a halogen (X = I, Br, Cl). Meanwhile, perovskite materials CH₃NH₃PbI_3-xcl_x (x=0;1 ;2 ;3) are much more widely used, with very good efficiencies ^{22, 23, 24, 25, 26, 27, 28}. These perovskite-based organic-inorganic hybrid cells are derived from the so-called third-generation dye-like cells commonly referred to as "Grätzel solar cell conception".

To date two different architectures exist: mesostructure and plane structures.

Mesostructure consisting of titanium dioxide (TiO₂) and perovskite (CH₃NH₃PbI₃) layer sandwiched between an anode transparent electrode and a metal cathode electrode as depicting in Figure 1. Many experimental investigations have been devoted to the mesostructure perovskite solar cells in the last years ^{2, 3, 4, 5, 6, 7}. However, theoretical models to predict and optimize parameters affected the devices efficiency are almost unexisted.

Our present analytical and numerical model is developed to give insight into the dependence effects of wavelength of the incoming light and the medium (TiO₂+Perovskite) layer thickness on the absorption, the charge carrier generation and the current density.

2. Computational Model

Computational results consisting of determining in one hand optical parameters such as the absorption and the charge carriers generated within the photoactive layer (TiO₂+ CH₃NH₃PbI₃) and in other hand the electrical parameters as short-circuit current are presented in this section.

Some assumptions are considered to determine the absorption of the photoactive layer which consisting of a blending of nanoparticles of TiO₂ and methylammonium lead iodide perovskite. In this part, only the photoactive layer is considered and represented as below in Figure 2.

One dimensional Cartesian system is associated to the active layer. The origin is taken at the anode/photoactive interface, z the depth and d the thickness.

This part is devoted to the determination of the optical parameters of the composite photoactive layer (TiO₂+CH₃NH₃PbI₃) such as the absorption coefficient, the absorbance and the generation rate. These parameters depend on the refractive index of the composite material. However, the individual elements refractive indexes of titanium dioxide (TiO₂) and CH₃NH₃PbI₃ for the visible spectrum wavelength ranging from 400 nm to 800 nm are obtained from the literature ²⁹.

Consequently, the refractive index of the photoactive layer is calculated by using Bruggeman theory (effective medium theory) which takes in spherical shape the nanoparticles ^{30, 31, 32, 33, 34}. This theory leads to a second-degree equation given by relation (1):

(1)

Where is the average dielectric constant of the composite medium, and are the dielectric permittivity of TiO₂ and perovskite respectively, the porosity of TiO₂ ³⁵.

The resolution of equation(1) leads to the average dielectric constant of the composite medium which allows to deduce the refractive index complex () given by the equation (2):

(2)

with

(3)

and being respectively the refractive index and extinction coefficient of the composite medium.

Finally, the analytical expression of the absorption coefficient of the photoactive layer is given by equation (4):

(4)

To determine the absorbance of the solar cell, the previous consider a photoactive layer is deposited on flat glass. The stacking thin layers is as shown in Figure 3.

The stacking thin films method so-called T-matrix is used to determine the electrical field component which depend on the parameters of the matrix given by equation (5) as below ³⁰:

(5)

and are the amplitudes of the fields and , at the input () and the output of the stack. being the thickness of the stack. is the equivalent matrix of the layer given by equation(6):

(6)

and are the wave admittance in the layer and the phase shift undergone by the wave when it passes through the layer respectively. By simplifying, equation (5) leads to:

(7)

being the admittance wave in the substrate. Parameters and are defined by the following relations:

(8)

(9)

The optical parameters such as the reflectance (), transmittance () and absorbance of a stack of thin layers, are given below and depend on B and C the parameters matrix ³⁶.

(10)

(11)

(12)

is the admittance of the wave in a vacuum.

Using calculations based on electromagnetism and considering the static energy conservation equation ³⁶ defined by equation (13), one can easily obtain the electron generation rate in term of the stacking layers depth for the wavelength of the monochromatic incident light as in equation(14) considering equation (13).

(13)

with total energy density per unit of time and , Poynting Vector.

(14)

being the photons injection rate.

The photon absorption rate is given by equation (15):

(15)

where,

(16)

and c are Planck constant and the velocity of light in vacuum respectively.and being the electric field amplitude and the wave admittance in the photoactive layer respectively.

The analytical expression of the short-circuit current density is computed in this section. The perovskite material is the main responsible of incoming light absorption within the photoactive layer. Actually, all the mechanism such as the charge carriers transport, recombination are carried out within the photoactive layer identically in dye sensitized solar cells (DSSCs). It can be noticed that electrons conduction predominance in charge transport of conduction through mesoporous titanium dioxide (TiO₂) ³⁷.

It has also been suggested that the high dielectric constant of perovskite material allows photo-generated excitons to dissociate immediately into free carriers (electrons and holes) ^{5, 38}. Under illumination, electrons photogenerated diffuse through the photoactive layer to be collected at the anode electrode.

Therefore, it is important to develop electrical model to solve the steady-state equation of electron density across the photoactive layer ³⁹ defined as below:

(17)

where, is electrons density in term of the depth . and are the diffusion coefficient and length respectively. represents the electron generation rate in various wavelength of the light.

In order to resolve equation (17), charge carriers recombinations within the photoactive region are negligible because their diffusion length is very important compared to the layer thickness in the perovskite material ^{40, 41}.

To ensure the unicity of the solution of equation (17), the following boundary conditions at the anode/active layer () and active layer/cathode () interfaces respectively:

(18)

(19)

where and denote the recombination rates of the electrons at the interfaces () and () respectively.

Finally, the short-circuit current density is deduced from the electrons density photogenerated within the photoactive layer and shown in equation (20):

(20)

where is the electron charge.

3. Results and Discussions

The analytical expressions of the optical and electrical parameters are depicted by establishing a numeric based-code implemented by the commercial software Matlab. The numerical results are presented in this section.

The composite layer (TiO₂+ CH₃NH₃PbI₃) absorption coefficient, the absorbance, charge carrier generation rate and the short-circuit current density as a function of the incoming light wavelength for various thickness are presented and discussed.

The Figure 4 shows the absorption coefficient of the photoactive layer as a function of the incident light wavelength of the spectrum of the visible.

The absorption coefficient decreases the high incoming light wavelength while it increases with high energies as summarize in N.G. Park studies ²⁰.

It can be predicted form this result, the variation of others optical parameters such as , and depending on the parameters of the matrix.

In Figure 5 is depicted the absorbance versus the incident light wavelength for different thickness of the absorber layer.

The absorbance shows various pics in different magnitudes which decreases very quickly with high incident light wavelength of the spectrum of the visible to vanish at 800 nm while it increases for high thickness ranging from 500 nm to 1.7µm. This decrease of the absorbance can be noticed with low thickness around 500 nm for the wavelength ranging from 500 nm to 725 nm.

These results are in good agree with G. Sfyrie & al. for any fixed thickness ⁴². The results give insight into the dependence effects of the both the incident light wavelength in the spectrum of the visible and the medium (TiO₂+perovskite) region thickness on the absorbance of the absorber material. It can be predicted that the blend of TiO₂ and CH₃NH₃PbI₃ perovskite has a large band of absorption in the spectrum of the visible ranging from 500nm to 775 nm and the limited value of thickness to deposit on substrate in order to obtain high magnitude of absorbance for photovoltaic conversion and efficiency.

Charge carriers are photogenerated within the medium region subsequent to the absorption of the incoming light excitation. The absorption coefficient and the absorbance are used to investigate the charge carriers generation rate.

Figure 6 shows the charge carriers generation rate versus the incident light wavelength for different thickness of the photoactive layer (TiO2+ CH₃NH₃PbI₃).

The charge carrier generation rate decreases with the incident light wavelength by exhibiting two pics at 500 nm and 725 nm. The former presents high magnitude (3.4x10²⁰cm^-3) compared to the latter (7.5x10¹⁹cm^-3). It vanishes with highest wavelengths at 775 nm. Compared to the absorbance, the charge carriers generation increases very slowly with high thickness ranging from 500 nm to 1.7 µm.

Subsequently to the charge carriers generation within the photoactive layer consisting of blending of TiO₂ and CH₃NH₃PbI₃ absorber material, electrons diffuse towards the electrodes to be collected as current. Consequently, one can be said that the short-circuit current of the cell is related to the active region absorption. The short-circuit current density versus the incident wavelength for different thickness is depicted in Figure 7.

Figure 7 depicts the short-circuit current density as a function of the wavelength for different thicknesses of photoactive composite layer.

The current density increases with illuminated wavelength ranging 400-520 nm while it decreases with higher active layer thickness ranging from 500-1700 nm. Beyond to 520 nm of the wavelength of the incoming light, the current density decreases slowly then quickly to vanish at 800nm. In term of the active layer thickness, the current density grows weakly due to the limited charge carriers diffusion length ranging 1200-1500 nm ³⁷. Basically, charge collection is as well as higher the active layer thickness being less than the charge carriers diffusion length for the illumination wavelength ranging 400-775 nm while no effect could not be affect beyond this region. Consequently, the short-circuit current density of about 24 mA.cm^-2 is recorded for 670 nm thickness under 400 nm illumination. These results are in good agreement with G.Sfyrie and al. ⁴².

4. Conclusion

From our model we have been able to investigate the influence of the wavelength of the incident light for different thicknesses of the composite photoactive layer, on the optical and electrical properties of a hybrid solar cell based on perovskite material CH₃NH₃PbI₃ mesostructured. This model allowed us to know the efficiency of the solar cell considered on the absorption of light, the generation of electrons and currents in the visible wavelength range. We have also found that in the visible domain, the absorption of light and the generation of charge carriers and currents are much greater for the low wavelengths corresponding to the large energies. However, it would be interesting to investigation the internal and external quantum yields of the whole cell for better optimization.

References