Effects of External Magnetic Field and Air Mass on Space Charge Region Width Extension of a Bifacial Solar Cell Front Side Illumination

The environmental and economical merits of converting solar energy into electricity via photovoltaic cells have caused an ever increasing interest among developed and developing countries to allocate more budget on photovoltaic systems in order to boost up their efficiency in recent years. Besides the material and design parameters, there are several external factors such as magnetic field, air mass, intense light, external electric field, solar spectrum.... that can influence the PV cell’s performance. There have been a handful of studies conducted on the effect of various influential parameters on the efficiency and performance of photovoltaic cells; however none has taken these two parameters (magnetic field and air mass) into account simultaneously. In this 3D study the effects of magnetic field and the air mass illumination on space charge region width extension of a bifacial polycristalline solar cell front side illumination will be elaborated. Based on the columnar model of the grain and the quasi-neutral base, the continuity equation is established and the boundaries conditions are defined in order to use Green’s functions to solve this equation. New analytical expression of charge carriers’ density is found and the diffusion capacitance to the junction is calculated. The normalized carriers’ density plot versus base depth and magnetic field with various air mass illumination are presented and analyzed. The effects of magnetic field and air mass illumination on space charge region width extension are then deducted. The influences of magnetic field and air mass illumination on the junction capacitance and on the reverse of junction capacitance are also shown and analyzed.


Introduction
The characterization of the performance of the solar cell do not take into account the effects of such environmental factors as insulation level, solar spectrum, electric field, magnetic field and other meteorological conditions. However, some external factors as magnetic field, external electric field, intense light, air mass and internal factors as grain size, grains boundaries recombination velocity, electric field in the bulk due to carrier concentration [1][2][3][4][5][6][7][8][9][10][11][12] can modify the solar cells operating whatever the solar cell quality.
Due to natural spectral sensitivity of solar cell devices, the solar spectrum (air mass) and magnetic field are two of those factors which may strongly influence the solar cell performance. Whatever is solar cell quality, it is important at its installation moment to take into account an outside factors susceptible to lower efficiency: the magnetic field.
Since the parameters as charge carriers density, junction capacitance and reverse of junction capacitance of the solar cell are closely dependent of carriers' distribution in the cell, the magnetic field and air mass illumination will influence these parameters.
In this work, we present the effects of magnetic field and air mass illumination in 3D study of bifacial solar on space charge Extension of a Bifacial Solar Cell Front Side Illumination width extension and junction capacity.

Study Assumptions
This study is based on a 3D modeling of a bifacial polycrystalline silicon solar cell; we made the following assumptions: a) Considering the weak thickness of the emitter and the space charge region, we neglected their contributions to the photocurrent, so the quasi-totality of the current is provided therefore by the base [13,14]; b) The grains are under shape parallelepipedic (2a; 2b; H) and the joints of grains are perpendicular to the junction; 1. The surfaces between two adjacent grains and perpendicular to the junction are characterized by the same carriers recombination processes evaluated by a grain boundary recombination velocity Sgx Sgy Sg = = ; 2. The electric field of crystal lattice is negligible [15]

Diffusion Equation Resolution
In the base of the solar cell, the minority carriers are electrons, and their density satisfies to the equation below: where ( , ) ( , , , ) r t x y z t δ δ = r D * is the diffusion coefficient of excess minority carriers in the base of the bifacial cell in presence of magnetic field [18,19]; it can be expressed as: θ is a coefficient which depend on magnetic field intensity.
The carriers' generation rate under multispectral light at the depth z in the base can be written by the following expression: In this expression of ( , ) g z t , cci cco I n I = indicates the illumination level (sun number) and 1 n = for our study. The coefficients m a and m b are the modeling coefficients of illumination air mass values indicated on the table 1 [17,18]. Equation (1) is solved with the following boundaries conditions: At the junction z = 0 At the rear side z = H At surfaces limited by x = ±a and y = ±b Sf , Sb and Sg are the recombination velocity of minority carriers respectively at surfaces z = 0, z = H and x = ± a (or y = ±b).
a, b and H are the grain sizes as indicated on figure 1.
A solution of equation (1) is given by expression (11) according to the author of the reference [16]. where: β * and j l * are expressed by: The expression (11) can be simply write for front side illumination: where: The quantities

Results and Discussions
We deal here the simulation results obtained from the previous modeling equations from software Mathcad 15 and origin pro8. On this figure, we notice tree types of region whatever the curve:

Effects of Air Mass on Charge Carriers Density
1. Firstly, a region where the charge carriers density gradient is positive. These explain the passage of a flow of electrons through the junction; 2. Secondly, a region where the charge carriers density gradient is negative. These is translated by a gradual decrease in the incident light flux in the base depth thus reducing the carriers charge generation which contribute to the photocurrent; 3. Thirdly, a region where the charge carriers density gradient is zero. This region was limited by the two regions to the depth There is a storage of electrons which induce a capacity with a space charge region establishing itself to the junction ( 0 z = ) to the base depth 0 z . In additional, we also note that the charge carriers density increase when the air mass of solar cell illumination decreases: we are therefore witnessing a reduction in charge carriers photogenerated in the base when the flow of incident light deviates from the perpendicular to the junction. Hence an increase in junction capacitance when the air mass of solar cell illumination increase.
So the space charge width extension has affected by the solar illumination air mass variations.

Effects of Magnetic Field on Charge Carriers Density
The figure 3 shows the effects of magnetic field on charge carriers density. We remark tree forms of gradients whatever the curve: 1. The positive gradient 0 z z p : in this region the charge carriers cross the junction because they have more excess energy to cross junction and participate to current production 2. The zero gradients in 0 z , at this position, carrier density is maximum and electrons are shifted, this situation explain an open circuit and the junction width extend to 0 z , 3. The negative gradient corresponding to 0 z z f where electrons are recombined in volume in the base. This decay is been explained by Beer Lambert law [19]; We also note that the maximum of carriers density increase with the magnetic field increase. This is explained by the storage of charge carriers across the junction.
This behavior of maximum of charge carriers were not only explained by the decrease of diffusion length and diffusion coefficient but also by the number of electrons which through the junction with the increase of magnetic field.
So the space charge width extension has affected by the magnetic field variations.

Effects Magnetic Field and Air Mass on Junction Capacitance
The solar cell junction behaves like a capacitor with capacitance C [2,20] depending on the space charge region width. However, the expression of junction capacity has been given by [21]: . .exp Through the figure 4, junction capacitance is plotted versus magnetic field for different illumination air mass values. The curves of figure 4 shows the decay of junction capacitance with the magnetic field but this decay is slow for small values of magnetic field. So the junction capacitance seems not to depend on magnetic field for values less than 0 B B ≤ by presenting a small tray. But for more intense magnetic field 0 B B f , the junction capacitance decreases very rapidly. This behavior of the junction capacitance is directly due to the space charge region widening with magnetic field.
Analyzing the influence of illumination air mass on the solar cell, capacitance behaviors poorly once illumination air mass is increasing, this in case of open circuit mode.
In fact increasing the illumination air mass means that intensity of light is reduced: hence the amount of charge carriers decreases also. But once in short circuit mode, capacitance is very poor and also influence of illumination air mass is slow.

Effects of Magnetic Field and Air Mass on the Reverse of Junction Capacitance
The reverse of junction capacitance is proportional to the space charge region width. Then we can also deduce the effects of magnetic field and air mass on the space charge region width extension.
The figure 5 present the profiles of the reverse of junction capacitance with magnetic field and air mass illumination. The reverse of junction capacitance increase when magnetic field whatever the illumination air mass. So this increase is slow for 0 B B ≤ but for more intense values of magnetic field 0 B B f this increase is faster. Then space charge width is reduced when the magnetic field increase The decrease of illumination air mass number translate the increase of photogeneration charge carriers, so the reverse of maximum of carriers density decrease reducing the reverse of junction capacitance and the space charge region width. The space charge region width extension is affected by the magnetic field and air mass illumination.

Conclusion
In this paper, a three dimensional approach of electrons diffusion in the p region of a bifacial polycrystalline silicon solar cell is presented. The solar cell is front side illuminated by a pulsed light under external magnetic field versus various air mass illumination values.
This manuscript shows the effects of air mass illumination and magnetic field on carriers' density, junction capacitance decay and on junction capacitance reverse. The influences of magnetic field and air mass illumination on space charge region extension are pointed out.
In summary, the limiting effects for larges values of magnetic field on the quality of junction capacitance and air mass illumination impacts of a solar cell are pronounced. The air mass illumination and large values of magnetic field affect the solar cell space charge region width extension.