Theoretical Aspect of Physical Phenomena in Inorganic Photovoltaic Cells. Electrical Modeling and Numerical Simulation

This work is based on the development of a theoretical model describing the drift and diffusion transport of photogenerated charge carriers and the impact of space charge on this transport in relation to the different physical phenomena characterizing the photovoltaic conversion in an inorganic silicon-based cell. In a second step, we used a numerical solution of the transport differential equations based on the Runge-Kutta algorithm in the framework of the finite difference method, This led us to an electrical model of the photovoltaic cell and of the photo-generated currents by RLC circuit equipped with a diode modeling the direction of electron and hole transport and allowed us to study the relations between the optical and electrical properties of the cell, as well as the influence of the different concentrations of impurities used for the n-type and p-type doping of the silicon on the properties of absorption of the light photons, the spectral response as well as the conductivity, the open-circuit potential and the short-circuit current.


Introduction
The absorption of light photons by a semiconductor leads to the photo-generation of charge carriers, electrons and holes, through the formation of excitons which are considered as bound states of an electron and a hole, the photovoltaic effect [1][2]. The dissociation of these excitons under the influence of the electric field of the p-n junction leads to the generation of electrons and holes which diffuse from the p-n junction towards the electrode-semiconductor contact zones and thus their collection on the cathode for electrons and on the anode for holes. This transport phenomena are due to the concentration and junction potential gradients, the transport equations describing the main physical phenomena related to photovoltaic conversion and charge carrier diffusion are derived from the differential equations related to the electron and hole current densities by calculating the divergence of the current density vectors and using the kinetic evolution of charge densities [5][6].The effect of the space charge zone on the generation and recombination phenomena as well as on the transport of the load carriers is also considered. The study of the hope charge zone is based on the Nernst-Planck equations, the transport equations and the Poisson equation. [7] These equations allowed us to *Corresponding author:mohammedazza81@gmail.com establish the theoretical model on which the numerical simulation of the cell is based, using the Runge-Kutta algorithm derived from the finite difference method for solving the differential equations obtained. On the other hand, we proceeded to model the silicon-based inorganic photovoltaic cell by an equivalent circuit focusing on the generated photo-currents and their relationship with the densities of the donor and acceptor impurities, the wavelength, the open circuit potential and the luminous flux, In a second row, we have also approached the resistances equivalent to the series, source and shunt resistances by trying to elucidate with maximum clarity, the links between the structural properties of the material at the microscopic scale with the optical and electrical properties of the photovoltaic cell. [8][9]].

Diffusion and drift of electrons and holes
For the electron: For the holes:

Continuity equations relating to electrons and holes
Let us calculate the one-dimensional divergence of the two vector fields J ⃗ A and J ⃗ B in order to arrive at the diffusion and drift equations relating to the charge carriers: [10].
For J = et J = The continuity equation reduces to: The same for the holes gives: The charge carriers diffuse from the (p-n) junction to the cathode for electrons and to the anode for holes.

Gauss -Poisson Equation
The Gauss-Poisson equation is an ecliptic equation deduced from the laws of electrostatics, for intrinsic one-dimensional semiconductors this equation takes the form [11][12][13][14]: For extrinsic semiconductors provided with ionized sites (impurities) and structural defects, we can write [12].

Determination of the expression of the junction electric field
By integrating the Nernst -Planck equations relating to the densities of electronic currents and those of holes we find as an expression for the junction electric field In the case of an open circuit It is the junction electric field which appeared during the irradiation of the material whose electric neutrality is no longer respected; this field is due to the diffusion currents which appeared under the influence of the concentration gradients of the carriers of load out of equilibrium when irradiation is not uniform.
This permits to write the Poisson equation: By solving the Poisson equation we find: Therefore: [ a − J + e f + c d + g ] Where N is the total density: N= n A − n B + N z f + N { d + n | The absorption of the radiation gives rise to a process of photoionization and causes an increase in the energy of the electrons and holes which remain bound and form an exciton when the concentration of the ND donor centers of low ionization energy is large enough ;here is overlap of the orbitals of the valence band and the conduction band which leads to a transfer of electrons by tunnel effect when the donor material is subjected to the action of an external electric field Ẽ ⃗ B| , these transitions electronics give rise to conduction by jumps, this is the phenomenon of photoconductivity.

The space charge
In the case of unipolar conductivity (nh = 0) we write the electric field by virtue of the Poisson equation: It's about Nernst potential For the holes we have a ( ) = Ö exp ( Çà(p) âÅ ) Densities follow the Maxwell-Boltzmann distribution law by substituting the coordinate x by the spherical coordinate. The electric charge density will then be: and by virtue of the expression of ϕ (r) ρ(r) = qn è exp − q² 4πεrkT (14) Since the electrostatic interactions are relegated to thermal agitation therefore we have qϕ≪kT, the expansion of the exponential gives as an expression of n (r) by virtue of the make that ne = nh j ( ) = − ² j 4 (15) And the nonlinear equation of ϕ (r) in spherical coordinates is linearized to take the form This admits for the solution Let's pose: From the laws of thermodynamics, the dissociation energy of the excitons is related to the dissociation equilibrium constant That is It is therefore the dissociation constant of the excitons at equilibrium taking into account the space charge zone; the electric double layer reigning in this zone favors even more the dissociation of the excitons; the potential drop due to the space charge.
The drop in potential becomes negligible the current limit by the space charge is given by the Mott-Gurney law: k o + k q = 0 With Jn = 0 at the acceptor-cathode interface and Jp = 0 at the anode-donor interface, because p> n and because μe>μh we then have V è≤≤Ω ≠ V ø¿¡ and this is due to the space charge, there is therefore accumulation of charge 3 Method for numerical resolution of differential equations (numerical modeling) The equations obtained are non-linear and strongly coupled equations so they do not admit analytical solutions except in very simplified cases. The method used for the numerical solution approximation analysis of these differential equations of diffusion and drift for electrons and holes is the Runge-Kutta method because of his simplicity, its ease and its adequacy for the numerical resolution of conservation equations for physical systems with simple geometry. [11] The adopted meshes are: These are steps of discretization; i and j are the indices of discretization.

Results and discussion
A dimensional writing of the diffusion equation in the form of a Boltzmann kinetic equation

Discretization of the exciton diffusion equation in space and time
The application of the discretization operator So differential equation (26) can be written as the numerical difference equation as follows: In stationary regime we can write The method followed here is that of Runge-Kutta of the second order which requires two iteration steps of the system of implicit equations: [12][13] = 0 → = + 1 2 Based on equation 27 we find:

Discretization of the current density equation
In adimensional writing the equations take below the form: * = − * * * + * * * This equation can be written as a system of two equations: The application of the discretization operator results in the following differences equations: So :

Photo-currents generated
The generated photo-current is the sum of three currents. The variation of the total current as a function of the ND impurities is represented by the following graph: The total current increases rapidly with the density of the donor impurities and this is due to the decrease in dark current with the doping this results in the strong dependence of the total current mainly on the photocurrent. the doping This type increases the photoinduced electron donor sites and further decreases the dark current density. This is the equation of the curve J (V) of the photovoltaic cell .At short-circuit that is to say at V = 0, the current density equals the density of the short-circuit current Jcc which is expressed as a function of luminous flux Φ in the form: ˇˇ= Φ( ) With K the photovoltaic sensitivity and r (λ) the spectral reflection coefficient:

Energy efficiency of the cell:
This efficiency is dependent on the quantum efficiency by Where QE is the quantum efficiency of the photovoltaic conversion the spectral wavelength response of the photovoltaic cell depends more often on the associated electrical circuit has the form By virtue of the dependency of photo-current on Voc input, the variation of this spectral response in relation to the open circuit potential is illustrated by the graph below This graph illustrates the variation of the polarization inversion current density with the wavelength of the incident light. It can be seen that this current density is an increasing function of the wavelength, that is to say that the polarization inversion current density is a function of the wavelength of the incident light, that is to say that the polarization inversion current density is an increasing function of the wavelength of the incident light. that is to say that the optimal wavelengths for a current density of the photo induced charge carriers is in the visible range of the electromagnetic spectrum and more precisely the long wavelengths this could appear paradoxical but it is not the case of the fact that the polarization inversion current density depends more on the luminous flux than on the wavelength and this means that the current density depends on the photon numbers much more than on the individual energy and this means that the photo current in turn then increases with the photon number with the luminous flux. According to the above graph we see that the density of the polarization inversion current decreases rapidly with the density of the donor impurity and this is due to the decrease of the saturation current density, which leads to the increase of the photo-current density with the density of the donor impurity. This can be explained by the fact that the width of the forbidden band decreases, thus the increase of the mobility of the photo induced charge carriers that constitute the photo-current, especially the part that is due to the transition of the electrons from the valence band to the conduction band.
If we account for the parallel variation of the polarization inversion current as a function of the density of the donor impurities and the luminous flux, we obtain the following 3D graph In this 3D graph it is constant that the polarization inversion current density decreases strongly with the density of the donor impurities which determines the saturation current density which becomes negligible in front of the photo current, that is to say that the polarization inversion current density depends in this case essentially on luminous flux and increases with its increase, and since j inv decreases with the donor doppage it leads to an increase of the photo current.

Space charge modeling condenser:
The capacity of SCZ is expressed by modeling it with a condenser as follows: The thickness of the space charge zone depends strongly on the potential drop as well as the density of the donor and acceptor impurities this dependence is given by the equation And it is illustrated by the following 3D graph We can see from this 3d graph that the thickness of the space charge zone decreases with type P doping and increases with type n dropping. This leads to an increase in the dissociation of the excitons by the increase of the electric field generated by the pn junction, which leads to an increase in the current generated by the space charge zone, which is due to the separation of the charges generated by the dissociation of the excitons.

Solar cell modelling
This cell can be assimilated to an R.L.C circuit where C represents the capacitance of the space charge zone the resistances R = R ¡ + R ; + R ¡A and the coil is assimilated to the polarization current this circuit can be described by the differential equation The total impedance is: And the phase of the current has the form:

Conclusion
The modulation of the electronic properties of semiconductors, especially silicon, allows the optical and electrical characteristics of silicon-based photovoltaic cells to be optimized. This modulation is only possible by an optimal doping as well as by the control of the morphology and the thickness of the p-n junction, which requires the implementation of laborious and expensive physico-chemical characterization techniques [15][16]. This is the reason why we have resorted to the electrical modeling of the photovoltaic cell by an RLC circuit as well as to the numerical simulation of some of its optical and structural characteristics (density of impurities) (input functions) in relation to the electrical characteristics of the cell (output function). In future research, since we used numerical simulations to solve our problem, we can make use of the work of Daaif et al. [18] who programmed computer solutions to solve different manipulations of crystallography, thus by the development of virtual laboratories [19][20]. A multitude of mathematical and numerical techniques are possible, but we have opted for the finite difference method and the Runge-Kutta algorithm, which allows easy numerical resolution of drift diffusion and charge distribution equations that describe the essential transport properties of photo-induced charge carriers as well as the influence of space charge zone on the electrical properties of the cell, especially in relation to exciton dissociation. In this work, we have also drawn attention to the theoretical description of the physical phenomena characterizing the photovoltaic conversion, which allowed us to highlight the links between the electronic structure of silicon and the optical and electrical properties of the cell [17]. The electrical modeling allowed us to represent the cell by an equivalent electrical circuit modeling the different layers as well as the interfaces of the cell through three resistors, a capacitor as well as a diode and a coil; it also allowed us to model the physical processes on which the operation of the photovoltaic cell is based. By numerical simulation we tried to understand the relationship between the total current, the inversion current, and the optical parameters of the incident light such as luminous flux and wavelength, we also found a strong dependence of the polarization inversion current, and photocurrent as well as the thickness of the space charge zone and the density of impurities due to the donor doping which influences the current due to electrons and the acceptor doping which influences the current due to holes, the influence of the open circuit potential on the spectral response was also studied.