This study was carried out on an n⁺-p-p⁺ type polycrystalline silicon bifacial solar cell grain. It should be noted that the solar cell grain has the same electrical properties (doping rate, mobility of minority carriers, lifetime and diffusion length) as a polycrystalline silicon PV cell. This is because studies have shown that the PV cell is made up of an association of pieces of very small single crystals separated by transition zones called grain boundaries [13] - [15] . These transition zones are therefore major centres of recombination. For modelling purposes, we assume that the grain boundaries are perpendicular to the front and back surfaces, and therefore; at the junction. Also, the perturbation (injection of carriers) applied to the base of the grain is not very intense so as to modify the mobility of the free electrons, so the crystalline field in the base is not taken into account.

The value of the emitter of the photopile grain is H = 0.03 cm and those of the diffusion length and diffusion coefficient are: Ln = 0.015 cm; Dn = 26 cm²/s. In order to simplify the expressions for minority carrier density and photocurrent density, photovoltage, series and shunt resistance, we have assumed a square cross-section for the grains. The analysis can then be extended to a parallelepiped grain shape [16] .

When the photopile grain, placed in a region where the magnetic field is constant (B = 7.5 mT) and makes an angle θ with the Oy axis, is simultaneously illuminated from both sides with multispectral light ( Figure 1 ) and the light penetrates along the (Oz) axis as well as generating excess minority charge carriers, the transport phenomena are governed by the continuity equation [11] [12] .

$\begin{array}{l}\frac{{\partial }^2\delta \left(x,y,z\right)}{\partial x^2}+C_y\cdot \frac{{\partial }^2\delta \left(x,y,z\right)}{\partial y^2}+\left[1+{\left({\mu }_n{B}_0\sin \theta \right)}^2\right]\frac{{\partial }^2\delta \left(x,y,z\right)}{\partial z^2}-\frac{\delta \left(x,y,z\right)}{L_n^{\ast 2}}\\ =-\frac{G\left(z\right)}{D_n^{\ast }}\end{array}$ (1)

$C_y=\left[1+{\left({\mu }_n{B}_0\sin \theta \right)}^2\right]$, $D_n^{\ast }=\frac{D_n}{1+{\left({\mu }_0{B}_0\right)}^2}$ and $L_n^{\ast }=\frac{L_n}{1+{\left({\mu }_0{B}_0\right)}^2}$

The general solution to the continuity equation is [11] :

$\delta \left(x,y,z\right)={\displaystyle \underset{j}{\overset{\infty }{\sum }}{\displaystyle \underset{k}{\overset{\infty }{\sum }}\left[A_{j,k}\cosh \left(\frac{z}{L_{j,k}}\right)+B_{j,k}\sinh \left(\frac{z}{L_{j,k}}\right)-{\displaystyle \underset{i=1}{\overset{3}{\sum }}{K}_i}\left[{\text{e}}^{-b_{i}z}+{\text{e}}^{-b_i\left(H-z\right)}\right]\right]}}$(2)

$K_i=\frac{n\cdot a_i\cdot L_{j,k}^2}{D_{j,k}\cdot \left[{\left(b_i\cdot L_{j,k}\right)}^2-1\right]}$ (3)

The coefficients $A_{j,k}$ and $B_{j,k}$ are fully determined using the boundary conditions at the junction and at the back of the solar cell [11] [15] [16] . In this way, the expression for the density of minority charge carriers will be completely determined.

By applying Fick’s law to the junction of the bifacial cell grain, we obtain the expression for the photocurrent density given by Equation (3):

$J_{Ph}\left(Sf,B,\theta \right)=e\cdot D\left(\theta \right){\displaystyle \underset{j}{\sum }{\displaystyle \underset{k}{\sum }{R}_{jk}}}\left[\frac{B_{jk}}{L_{jk}}+{\displaystyle \underset{i=1}{\overset{3}{\sum }}{K}_i\cdot b_i\left(1+{\text{e}}^{-b_i\cdot H}\right)}\right]$ (4)

This expression has two significant variables: the dynamic velocity at the junction $Sf$ and the angle of incidence of the magnetic field intensity.

The expression for the photovoltage (Equation (4)) across the bifacial cell junction is obtained using the Boltzmann approximation.

$V_{ph}\left(Sf,B,\theta \right)=V_T\cdot \ln \left[1+\frac{1}{n_0}{\displaystyle \underset{j}{\sum }{\displaystyle \underset{k}{\sum }{R}_{jk}}}\cdot \left[A_{jk}-{\displaystyle \underset{i=1}{\overset{3}{\sum }}{K}_i\left(1+{\text{e}}^{-b_{i}H}\right)}\right]\right]$ (5)

with V_T the thermal voltage with value V_T = 26 mV; n_o is the electron density at thermodynamic equilibrium.

The expression for series resistance is given by Equation (5).

$R_s\left(Sf,B,\theta \right)=\frac{V_{co}\left(B,\theta \right)-V_{ph}\left(Sf,B,\theta \right)}{J_{ph}\left(Sf,B,\theta \right)}$ (6)

The expression for the shunt resistance is given in Equation (6).

$R_{sh}\left(Sf,B,\theta \right)=\frac{V_{ph}\left(Sf,B,\theta \right)}{J_{phcc}-J_{ph}\left(Sf,B,\theta \right)}$ (7)

Figure 2 shows the profiles of the density of excess minority charge carriers as a function of depth for different values of the angle of incidence of the magnetic field.

Simulations are carried out for two types of operation of the solar cell: open-circuit operation (low values of the dynamic speed at the junction) and short-circuit operation (high values of the dynamic speed at the junction).

Figure 2 shows negative slopes at the junction and a maximum at the back when the magnetic field is intense, i.e. for θ = 0 rad. In addition, the carrier density decreases as the angle of incidence increases. Figure 2 also shows that after a depth of 0.020 cm the curves are curved. These curvatures are due to the contribution of the rear face of the solar cell grain. The decrease in the carrier density with the increase in the angle of incidence of the magnetic field leads to a drop in the open circuit voltage and therefore a drop in the series resistance.

Figure 3 below shows the carrier density profiles for different values of the angle of incidence of the magnetic field. We also observe a decrease in carrier density as the angle of incidence of the magnetic field increases. The profiles in Figure 3 show peaks justifying the presence of carriers on the front and back of the solar cell. This is justified by the Lorentz force, which reflects the photocarriers back towards the junction and onto the photovoltaic cell interfaces.

Then we see in this Figure 3 that as the angle of incidence of the magnetic field increases, the maxima of the carrier densities between 0.01 cm and 0.02 cm decrease and become linear, thus justifying that the charge carriers photocreated in the base are less present because they have recombined [11] [12] .

Figure 4 shows the profiles of photocurrent density versus photovoltage for different values of the angle of incidence of the magnetic field intensity.

Analysis of the curves in Figure 4 shows that when the angle of incidence varies from 0 to π/2 rad the short-circuit photocurrent density increases and the open-circuit voltage decreases.

Phenomenon observed in Figure 4 . The increase in photocurrent density and the decrease in open circuit voltage can be explained by the fact that when the angle of incidence increases, the Lorentz force that sends the carriers back towards the lateral faces decreases until it is cancelled, causing the migration of the carriers towards the junction that cross it without being accumulated too much [12] .

The series resistance is associated with the manufacture of the solar cell. It characterises the loss of carriers at the junction [7] . The higher the resistance, the fewer carriers pass through the junction. As a result, the junction heats up. It is therefore important to minimise it.

Figure 5 shows the variation in series resistance as a function of the angle of incidence of the magnetic field. This variation is periodic with period π.

Figure 6 shows the variation in shunt resistance as a function of the angle of incidence of the magnetic field.

We can see that the shunt resistance increases from 0 to π/2 rad and π rad of 3π/2 rad then decreases π/2 rad of π rad and 3π/2 rad of 2π rad. For values of the angle of incidence equal to π/2 rad of 3π/2 rad the shunt resistance reaches its peak. This is because at these values the Lorentz force has no effect on the movement of the carriers because the magnetic field vector is collinear with the electron velocity vector.

In short, we have shown that when the angle of incidence increases from 0 to π/2 rad of

π rad to 3π/2 rad, the magnetic field tends to cancel out, as does the Lorentz force, resulting in little deflection of the carriers towards the side surfaces of the cell. As a result, there is less loss of carriers at the junction. The increase in the shunt resistance curves with the angle of incidence of the magnetic field from 0 to π/2 rad and π rad of 3π/2 rad is due to the disturbing effect caused by the magnetic field. This is because we expected a decrease in the shunt resistance for the angle of incidence varying from 0 to π/2 rad and π rad of 3π/2 rad.

Table 1 shows that as the angle of incidence of the magnetic field intensity increases, the values of the series resistance decrease while those of the shunt resistance increase. This is the result of the decrease in voltage at the junction due to the Lorentz force, which tends to cancel out. Hence the strong migration of carriers across the junction.