2D Modeling of Solar Cell Radial Junction: Study of Carriers Charge Density and Photocurrent Density in Static Mode under Monochromatic Illumination

A theoretical study of a polysilicon solar cell with a radial junction in static regime under monochromatic illumination is presented in this paper. The junction radial solar cell geometry is illustrated and described. The carriers’ diffusion equation is established and solved under quasi-neutral base as-sumption with boundaries conditions and Bessel equations. New analytical expressions of electrons and holes density and photocurrent are found. The wavelength and structural parameters (base radius, emitter thickness) influences on charge carriers density and photocurrent are shown and examined.


Introduction
Solar energy is an inexhaustible energy source. Solar cell technology has been pioneered in space industry, essentially because solar energy is one of the main power sources for satellites. However, space environment is a very harsh environment for electronic devices, such as silicon solar cells and other semiconductor based detectors. We can obtain energy by converting the solar energy in electrical energy with the semiconductor optoelectronic device, such as the vertical junction solar cell or planar junction [1]. Solar cell geometry seems to be very important for solar high performance. Especially since study conducted on cubic or cylindrical solar cell [2] showed a slight improvement in solar cell parameters in favor of a cylindrical model, notably the radial junction solar cell [3].
The originality of this work lies on its innovative and efficient aspect. The aim of this study is to investigate on charge carriers' density and photocurrent density respectively in the base and in the emitter of a radial junction polycrystalline solar cell.

Materials and Methods
In this study based on a 2D modeling of a polycrystalline silicon radial junction solar cell, we made the following hypothesis: -The grains have identical structures and electrical properties. Therefore, the study is restricted to a grain whose model is schematized in Figure 1.

Study of Carriers' Charge Density
The transfer phenomena of solar cell are model by the following equation of continuity numbered Equation (1) in cylindrical coordinates [4].
Therefore the continuity equation becomes: • In the base region, the minority carriers' movement is governed by: • In the emitter region, the transport equation is given by: The eigenvalues c k which satisfy the boundary condition (6) are expressed by: Replacing the Equation (7) in Equation (5) where the following expressions gives Equation (10):

I x ′
is its first derivative with respect to its argument [6].

b) Electrons photocurrent density expression
The equation of photocurrent density is given by: (12); this equation becomes: 2) Holes density and photocurrent in the emitter a) Holes density expression Referring to Figure 1, the boundary condition in the emitter is: ─ At the junction: ─ At the rear side: ─ The continuity Equation (4) general solution is: The eigenvalues k a which satisfy the boundary condition Equation (15) are given by: The function ( ) , p f r λ can be shown to have the form:

Results and Discussions
We present here the simulation results obtained from the previous modeling equations by using Mathcad software.

Electrons Density and Photocurrent in the Base
The effects of wavelength and base radius on electrons density and photocurrent density are presented and analyzed.

Radius Rb Effects on Electrons Density
In Figure 2, we present the variations of electrons density versus base radius for different values of base depth. We observe that the electrons density and base radius of solar cell vary in the same order. The generation of electrons in the base is important because base radius increase. Also electrons density decrease versus base depth.

Wavelength Effects on Electrons Density
In Figure 3, we present the variations of electrons density versus wavelength for different values of solar cell thickness.    (1.12 eV). Also, the weak absorption for low and big values explain these electrons density variations. The absorption peak is observed around 700 nm. Figure 4 shows the profile of photocurrent density versus wavelength for various values of junction recombination velocity Sf.

Wavelength Effects on Photocurrent Density
We remark a low photocurrent density for low and big wavelength. The absorption peak is notified around 700 nm.
These phenomena are explained by electrons density variations. In additional, photocurrent density increase for

Study of Holes Density and Holes Photocurrent Density in the Emitter
The following figure presents variations of holes density as a function of emitter thickness for different values of the depth creating more electrons because the emitter thickness increases.

Emitter Thickness Effects on Holes Photocurrent Density
In Figure 5, below we study the evolution of photocurrent density versus holes recombination velocity It is observed that the holes density increase with the emitter thickness increase. This behavior is explained by the increase of incident light absorption   due to the emitter thickness increase. Also, the holes density decrease with the base depth the holes photogeneration decrease.

Emitter Thickness Effects on Holes Photocurrent Density
In Figure 6, the variations of holes photocurrent density versus holes recombination velocity for various values of emitter thickness.
The increase of holes photocurrent density is observed with the emitter thickness increase. This is due to the fact that more the emitter thickness increases, more there is a large absorption of incident photons. These is consequently generates more carriers in the emitter region.

Solar Cell Photocurrent Density versus Base Radius and Emitter Thickness
The photocurrent density of solar cell is linked to the electrons and holes recombination governed respectively by electrons recombination velocity and holes recombination velocity.
In Figure 7, the profile of photocurrent density as a function of dynamic velocity to the junction for various values of base radius R b .

Conclusions
This paper carried out 2D modelling of radial junction of solar cell. The effects of base radius, emitter thickness and wavelength on carriers' charge density and carriers' photocurrent density are investigated.
Also, the influences of base radius, emitter thickness and charges carriers' recombination velocity on photocurrent density of solar cell are presented and analyzed. The results of this study which are in good agreement with others research works [3] validate thus ours results.
The 2D modeling of solar cell radial junction is studied in first time in cylin- ter the radial junction of solar cell. In the future, we will try to do an experimental study to confirm theoretical results with experimental results.