Investigation of the Behavior of a Photovoltaic Cell under Concentration as a Function of the Temperature of the Base and a Variable External Magnetic Field in 3D Approximation

The photovoltaic (PV) cell performances are connected to the base photoge-nerated carriers charge. Some studies showed that the quantity of the photo-generated carriers charge increases with the increase of the solar illumination. This situation explains the choice of concentration PV cell (


Introduction
The performances of a PV cell are dependent on external factors, such as solar radiation, temperature, gamma radiation, etc.Indeed, in the presence of gamma radiation, the performances of a PV cell could degrade [1].The same is true for operating a PV cell in a high heat environment, resulting in a rise in its temperature.The rise of temperature can come from the activities in the cell or from exposure under the sun, through the radiant light.Whatever the source of temperature rise, it contributes to reducing the proper functioning of a PV cell [2].
On the other hand, the increase in illumination produces an improvement in the performances of a PV cell.Note that the photovoltaic market is dominated by silicon PV cells [3].However, the laboratory conversion efficiency of a silicon PV cell single junction is only about 26.8%, under non illumination concentration [3].This efficiency seems low given the growth in energy demand.Consequently, research has turned to other materials such as thin films and organic semiconductors.Other research work has also focused on concentrated polycrystalline silicon PV cells.For a concentrated PV cell, given the illumination mode, the number of photogenerated charge carriers in the base is very high.
The concentrated PV cell has the advantage of having better efficiency than the non-concentrated one, up to 27.6% in the laboratory [2], using less material thanks to its reduced size and occupying less space.The inconvenience of concentrating PV cells is that they produce a lot of heat during operation, causing high temperature.Indeed, the variations of the photocurrent and the photovoltage are coming from internal factors related to the movements of the charge carriers and which affect the temperature of the base [4] [5].In addition, the multiplicity of magnetic field sources in our living environment (telecommunication antennas, transformers, TV and radio antennas, etc.) causes a magnetic influence on the PV cells during their operation.Indeed, the magnetic field, through the Lorentz force, acts on the movement of charge carriers by modifying their trajectory.Consequently, the presence of magnetic field influences the performance of PV cells [6] [7].It is therefore instructive to investigate the cumulative influence of residual heat due to the movement of charge carriers and the magnetic field on the behaviour of a concentrated PV cell in order to contribute to the deployment of this technology in Sahel region.Thus, this study presents the combined effects of base heating and a variable external magnetic field on the operation of a concentration PV cell.After the introduction, the methods and theories, the hypotheses and the mathematical model used will be discussed afterwards.Finally, the results and discussion on the simultaneous influence of heat and magnetic field on the electrical parameters of a concentrated PV cell will be presented.This work will end with a conclusion and recommen-

Mathematical Modulization and Assumptions
In this study, the polycrystalline silicon PV cell used is formed by a lot of grains with different sizes.All the grains are assumed cubic [8] [9].That leads to finding the real physical situation with the consideration of the grain boundaries recombination velocity.Figure 1 illustrates the cubic grain.
Each grain presents the sizes g x following (ox) axis and g y following (oy) axis.
In cubic grain x y g g = and H gives the PV cell base depth following (oz) axis.
Recombination planes are the adjacent g x surfaces of two grains and are located at positions 2 x g x = ± and 2 y g y = ± respectively perpendicular.These planes are perpendicular to (ox) and (oy) axes of the coordinate system [9] [10].In this model, the magnetic field is applied along the direction of the (oy) axis, i.e. parallel to the junction and by this assumption the base doping rate is uniform.This leads to an almost zero crystalline electric field.
Moreover, the rate of absorption of light being a function of the depth z ( 0 z H ≤ ≤ ) of the base, there is thus a non-uniform generation of the charge carriers in the volume of the base.The concentration of charge carriers is then high in the regions close to the illuminated surface and then decreases as moving away from this surface.This variation in charge carrier density along the oz axis is the source of an internal electric field ( ) E z called the concentration gra- dient electric field of excess minority carriers [6] [11].This electric field will have an influence on the mobility of the charge carriers in the volume of the base.Light enters the PV cell along the oz axis.This leads to a generation of charge carriers along this same axis.By assumption, the charge carrier density is uniform along the ox and oy axes.The concentration gradient electric field is therefore zero along these two axes: The grain is also subjected to a variable magnetic field B oriented along the (oy) axis as shown in Figure 1 above.By assumption, the PV cell is under a static illumination regime.

Mathematical Modulization
The magneto-transport equation in the base of the PV cell under an intense multi-spectral illumination and under a variable magnetic field, is given by the following relationship [12].( ) and ( ) With As the diffusion coefficient depends on the mobility of the carriers which causes the heating of the base, the diffusion coefficients of electrons and holes consequently depend on the temperature T of the base of the PV cell and are expressed as follows [5]: B k is the Boltzmann constant and q the elementary electric charge.The illu- mination being intense, the expression of the crystalline electric field due to the concentration gradient is given by [11]: The charge carrier conservation equation at the p-n junction is materialized by the following continuity equation [13] ( ) ( ) ( ) ( ) , , , , 1 The multi-spectral illumination under concentration is given by [9] B.  According to the study model and in accordance with Equations ( 1) and ( 7), the differential equation of continuity of the charge carriers of the PV cell under intense illumination and under variable magnetic field is equal to: ,  given by: where ( ) ( ) and jk B coefficients are determined using the boundary conditions at the junction ( 0 z = ) and at the rear face of the base ( z H = ) of the PV cell: x y z B T D B T S x y H B T z

Density of Photocurrent
When the concentrating PV cell is under light incidence, some photogenerated charge carriers cross the junction and produce photocurrent in the external circuit.Thus, from Fick's first law we derive the expression for the photocurrent density, given by the following expression [4]:

Photovoltage
The photovoltage characterizes the rate of charge carriers accumulated at the junction of the PV cell.The photovoltage supplied by the solar cell under concentration 50 suns C = and under an external magnetic field, in 3D, is given by the Boltzmann relation according to the following expression [9]: , ln 1 , , 0, , d d where is the thermal voltage; i n is the intrinsic electron concen- tration; B N is the base doping rate.

Electric Power
The electrical power delivered by the PV cell is the product of the photovoltage and the photocurrent depending on the magnetic field and temperature of the base.The expression of the electrical power is given by the following equation [4] [9]:

Conversion Efficiency
The conversion efficiency is the ratio of the maximum power supplied by the PV cell to the incident light power.In this work, the conversion efficiency is a function of the intensity of the magnetic field and the temperature of the base.Thus, the expression of the conversion efficiency is given by [4]: In the case of a concentrated PV cell, the incident power .From the mathematical model and the hypotheses formulated, the expressions of the electrical parameters such as the photocurrent, the photovoltage, the electrical power delivered and the conversion efficiency were established.In the following, it will be a question of presenting the results of the study of the simultaneous influence of the temperature of the base and the magnetic field on the operation.It appears in Figure 2 that the increase in the intensity of the magnetic field leads to decrease in the photocurrent density.This is explained by the fact that the magnetic field causes a deflection of the charge carriers at due to Lorentz force whose intensity increases with the intensity of the magnetic field.This increase in magnetic field strength the recombination rate of charge carriers in the base and hence a decrease in photocurrent with increasing magnetic field strength [7].It also appears in this Figure 2 that, for values of the magnetic field B such that 0 0.4 mT B < <

Photocurrent
, the increase in the photocurrent density is accompanied by an increase in the temperature of the base.On the other hand, for values of the magnetic field 0.4 mT B > , there is an inversion of the curves, characterized by a drop in the photocurrent with the increase in temperature.
Indeed, for 0 0.4 mT B < < , the charge carriers diffuse towards the junction and the three phenomena which are thermalization, collisions between charge carriers .Thus, for a given operating temperature, the photocurrent density produced by the PV cell is optimal for low values of the magnetic field intensity.

Photovoltage
The followings Figure 3 shows the behavior of the photovoltage supplied by the PV cell, as a function of magnetic field strength, for different values of the temperature of the base.The energy released in the base decreases and the temperature decreases.On the other hand, for values of the magnetic field 0.4 mT B > , there is an inversion of the curves, characterized by an increase in the photovoltage at high temperatures.This result is explained by the fact that at large values of the magnetic field, the deviations become significant and this tends to block the charge carriers in the base.The possibilities of collision between load and braking carriers increase; therefore, the increase in photovoltage is accompanied by an increase in temperature.In Figure 4, it appears that the electric power decreases when the intensity of the magnetic field increases.This result is the consequence of charge carrier deflections with increasing magnetic field strength, which reduces the number of charge carriers that arrive at the junction and the number of charge carriers that diffuse across the junction.As a result, the electrical power decreases with the increase of the magnetic field strength.From below Figure 4, it also emerges that for some given values of the magnetic field induced between 0 and 4 × 10 −4 T, the value of the power decreases with the increase in the temperature of the base.In this range of magnetic field values, charge carriers are always oriented towards the junction (but their number decreases with the increase of the magnetic field strength).Then, the energy released in the base by collisions and by

Figure 1 .
Figure 1.PV cell cubic grain under concentrated illumination and under magnetic field.

nD
and p D being respectively the diffusion coefficients of electrons and holes in the base of the PV cell; n µ and p µ are respectively the mobilities of electrons and holes in the base of the silicon PV cell.The mobilities of electrons and holes depend on the absolute temperature T of the base of the PV cell and are given by the following expressions [5]: the recombination rate of the base excess minority carriers is found by[9] are respectively the diffusion coefficient and the diffusion length of the PV cell depending on the variable magnetic field and the temperature: The general solution of the Equation (8) is provided by J. Dugas et al. as: the transcendental equations makes it possible to obtain the values of the coefficients j C and k C along the axes (Ox) and (Oy).The ex- pression of the function is obtained by injecting the expression of Equation (9) into the charge carrier continuity equation indicated by Equation(8); this leads to a second order differential equation in , i k Z whose solution is

Figure 2
Figure 2 below shows the behavior of the photocurrent density supplied by the PV cell in an intermediate operating situation.

Figure 2 .
Figure 2. Evolution of the photocurrent versus the temperature of the base and the intensity of the external magnetic field ( 50 suns C = , 3 3 10 cm x y g g

Figure 3 Figure 3 .
Figure3shows that the photovoltage decreases with the increase in the intensity of the magnetic field, when the PV cell is in an intermediate operating situation.This result is explained by the fact that the increase in the magnetic field leads to increase in the deviations of the charge carriers.Carriers that are deflected to the junction diffuse and those that are deflected to the side surfaces recombine.These two phenomena contribute to a reduction in the density of the carriers in the volume of the base and consequently to a reduction in the photovoltage.It also appears in Figure3that, for magnetic field values such as 0 < < 0.4mT B , the increase in photovoltage is accompanied by a decrease in temperature.This result is the consequence of the increase in the rate of charge

Figure 4
Figure 4 presents the variations in the electrical power delivered as a function of the intensity of the magnetic field for different values of the temperature of the concentration PV cell base.

Figure 4 .
Figure 4. Variations of the electric power according to the temperature of the base and the intensity of the external magnetic field ( 50 suns C = , 3 3 10 cm x y g g