Electrical Parameters Determination from Base Thickness Optimization in a Silicon Solar Cell under Influence of the Irradiation Energy Flow of Charged Particles

In this work, we study the characteristics I-V and P-V of a silicon solar cell as well as its fill factor, its electrical power from the optimum thickness obtained in the base under variation of the irradiation energy flow of charged particles. The recombination velocity at the junction corresponding to the maximum power point was also deduced.


Introduction
Authors have studied the electrical parameters of the solar cell namely the fill factor, the conversion efficiency, the power, the I-V and P-V characteristics under the Influence of Irradiation [1], from the back surface recombination velocity modeling in white biased [2], under temperature with the junction surface recombination concept [3], by acquisition automatic of I-V properties and temperature [4] [5], under influence of incidence angle on a vertical Silicon Solar [6], by illumination wavelength effect on a parallel vertical junction silicon solar cell and under irradiation [7] and illumination level effects on macroscopic parameters of a bifacial solar cell [8]. Our study is to determinate these electrical parameters from the optimal base thickness of the solar cell under variation of the irradiation energy flow and extracting the values of the recombination velocity at the junction, corresponding to the maximum power. Figure 1 represents a n + -p-p + silicon solar cell [9] [10]. The emitter is the thin (n + ) zone covered by the front grids. Then comes the space charge region (SCR), which is formed by migration of the majority charges coming from the two semiconductors (n + and p), according to the principle of Helmotz or compensation law. They are formed as fixed charges that delimit a space, where there is an intense electric field, which will allow the dissociation of photogenerated electron-hole pairs, and their acceleration to the deficit areas in corresponding charge. The (p) zone is doped with boron atoms and represents the larger thickness base (170 -300 µm). It is the zone of pair creation (electron-hole), the most important, which justifies, the interest of its study in a solar cell. The (p + ) zone over doped in boron atoms, allows the creation of another rear space charge region, where the created electric field (Back surface Field) will allow the minority carriers of the (p) zone to be pushed back to the junction, to be then collected and participated in the photocurrent.

Theory
The solar cell thus achieved (Figure 1) is previously subjected to a flow of charged particles ( p φ ) and intensity (kl) [11], allowing to simulate the conditions of operation outside the atmosphere, in the supply of satellites. Under polychromatic illumination, the density of charge carriers ( ) , , x p kl δ φ generated at point of abscissa x in the base, according to the law is defined by equation (Equation (1)), describing the generation rate [12] [13] and expressed as: follows the charge transport equation given by (Equation (2)): The diffusion term influenced by the irradiation conditions of the solar cell, is given by the following empirical relationship [11]: where: 0 L is the diffusion length of the excess minority carriers in absence of irradiation energy flux ( p φ ). ( ) , p L kl φ is the diffusion length of the excess minority carrier in the base as a function of the irradiation energy flux ( p φ ) and the damage coefficient intensity (kl) and which may be related to Einstein's relationship by: and τ are respectively the diffusion coefficient and lifetime of the electrons in the base of the solar cell under irradiation.
The continuity Equation (2) solution is provided by: where, coefficients A and B, will be obtained by use of Equation (3) and Equation (4).

Expressions of Photocurrent, Recombination Velocity (Sb) and Phototension
1) Fick's charged particle law establishes the photocurrent density of minority carriers, derived from the base by the following relationship: 2) At the high values of recombination velocity at the junction, it is established that [ Derived from this equation, it gives the following relations: Equation (10), gives rise the definition of intrinsic velocity and becomes a diffusion velocity [19], when H is very large compared to L. ii) Equation (11) is marked by a term of absorption (b i ) and leads to generation velocity, when H is small compared to L [19].
3) By Boltzmann's law, the photovoltage at the junction is written as: where, Kb is the Boltzmann constant, q is the elementary charge of the electron and T is the temperature. Nb is the solar cell base doping rate, and n i is the intrinsic density of minority charge carriers.

Optimum Thickness of the Base, Deduced from Each Case of Irradiation Flow
The expressions (Equation (10) and Equation (11)     with the increase of the irradiation energy flow. And the photovoltage increases slightly. Figure 4 represents the equivalent electric circuit of an illuminated solar cell [9].  The Ohm law applied to the circuit in Figure 4 yields the electric power delivered by the base of the solar cell to an external load as follows:

Electrical Power of the Solar Cell
with: where J d is the solar cell density of current under dark expressed as: Sf 0 is the excess minority carrier recombination velocity associated with shunt resistance-induced charge carrier losses [24] [25], which characterizes the quality of the solar cell [15] [16] [20]. The expression intrinsic recombination velocity at the junction is given in static regime [26] [27] and under polychromatic illumination, by [28] and applied her for solar cell under irradiation as:

( ) ( )
, , -, , P Sf p Hopt V Sf p Hopt φ φ Characteristic Figure 5 and Figure 6 show the variations in electrical power as function of both, the excess minority carrier recombination velocity at the junction and the photovoltage for different irradiation energy flow and optimum base thickness. We note on Figure 5, the decrease of maximum power amplitude with the irradiation energy flow corresponding the base thickness.
On Figure 6, it is also observed a decrease in power with the increase of irradiation energy flow corresponding the optimum base thickness.

Fill Factor
The fill factor is an important parameter for a solar cell. It shows the physical quality of the solar cell for a conversion efficiency and indicates the performance of a perfect cell. The expression of the fill factor is given [29] as:

Efficiency
The conversion efficiency of a solar cell is the ratio between the maximum power supplied provided by the solar cell and the power of absorbed illumination. It is written as follows:  Figure 7 represents the profil of the power versus irradiation energy, Figure 8 and Figure 9 represent the profils of the power and the efficiency versus the optimum base thickness.    The equation obtained from the power versus the irradiation energy is given by the following relation:

Curves of Power and Efficiency
with:  We note on Figure 8 and Figure 9 that the power and the efficiency increase with the increasing base thickness.

Recombination Velocity Sfmax at the Junction
Sf max , the excess minority carrier recombination velocity at the junction corresponding to the maximum power point is point out by solving the following equation [3] [30].
From this Equation (22) is the density of the minority excess minority carrier at the point of maximum power, its expression is given by the following relations: with:   The graphical resolution of this transcendental equation as a function of the excess minority carrier recombination velocity Sf at the junction [30], for different irradiation energy flow corresponding the optimum base thickness, gives the Sf max values by the intercept point of the two curves represented by Figure   10.
The results obtained from Figure 10 corresponding to the numerical values of Sf max , are given in Table 3.
The recombination velocity Sf max of the excess minority carrier at the junction yielding P max , decreases while irradiation energy flow increases.
Curve Sf max (H opt ) Figure 11 represents the profil of the recombination velocity Sf max of the excess minority carrier at the junction yielding P max , as function of optimum base thickness.   The recombination velocity Sf max of the excess minority carrier at the junction increases with the optimum base thickness.

Conclusion
In this work, a technique for obtaining the optimum thickness of the solar cell under variation of the irradiation energy flow has been presented. It is also deduced from this optimal thickness, the fill factor, the electrical power, the efficiency of the solar cell as well as the recombination velocity at the junction through a transcendental equation, leading to maximum power. We found that the electrical parameters of the solar cell decrease with the increasing of the irradiation energy flow. Then we have plotted and fitted the curves of the power, the efficiency and the recombination velocity (at the maximum power) versus the optimum base thickness.