The model of this study is a three-dimensional (3D) grain extracted from a bifacial polycrystalline silicon PV cell. It mainly includes three parts: the emitter, the base and the p⁺ overdoped zone equipped with an active surface for albedo collection.

Figure 1 illustrates the three-dimensional structure of a bifacial photovoltaic cell at the grain scale. The device is composed of three successive regions: the emitter (−W ≤ z ≤ 0), the base (0 ≤ z ≤ H), and the heavily doped p⁺ layer (H ≤ z ≤ H + W_bsf).

To simplify the formulation and solution of the governing equations, the following assumptions are made: The excess minority carrier generation is assumed to occur only along the (OZ) direction, which is perpendicular to the junction. The electric field is confined within the space charge region (SCR) [4] . The parameters W, H, and W_bsf represent the thicknesses of the emitter, the base, and the overdoped layer, respectively. Finally, g_x and g_y denote the grain dimensions along the x and y axes, which define the flat surface exposed to incident light.

The three-dimensional continuity equations in the steady-state regime, describing the base, the p⁺ region, and the emitter, are derived from the magnetotransport equations under multispectral illumination [5] .

$\frac{{\partial }^2{\delta }_n}{\partial x^2}+\frac{{\partial }^2{\delta }_n}{\partial y^2}+\frac{{\partial }^2{\delta }_n}{\partial z^2}-\frac{{\delta }_n}{{\tau }_n{D}_n^{*}}+\frac{G\left(z\right)}{D_n^{*}}=0$ (1)

$\frac{{\partial }^2{\delta }_{n^{+}}}{\partial x^2}+\frac{{\partial }^2{\delta }_{n^{+}}}{\partial y^2}+\frac{{\partial }^2{\delta }_{n^{+}}}{\partial z^2}-\frac{{\partial }_{n^{+}}}{L_{n^{+}}^{*}}+\frac{G\left(z\right)}{D_{n^{+}}^{*}}=0$(2)

$\frac{{\partial }^2{\delta }_p}{\partial x^2}+\frac{{\partial }^2{\delta }_p}{\partial y^2}+\frac{{\partial }^2{\delta }_p}{\partial z^2}-\frac{{\delta }_p}{{\tau }_p{D}_p^{*}}+\frac{G\left(z\right)}{D_p^{*}}=0$(3)

The expression of the generation rate, G(z), is given in [6] .

$G\left(z\right)=n{\displaystyle \underset{m=1}{\overset{3}{\sum }}{a}_m\left[{\text{e}}^{-b_m\left[W+z\right]}+{\text{e}}^{-b_m\left[H+W_{bsf}-z\right]}\right]}$(4)

The coefficients a_m and b_m are determined from tabulated solar irradiance data [7] [8] . These coefficients are given for an AM 1.5 solar spectrum by:

The parameter n, commonly referred to as the number of suns, represents the concentration factor of the incident solar radiation.

Define The electron and hole photocurrent densities depend on the gradients of the excess minority carriers [13] . Under simultaneous illumination on both sides, the photocurrent density in the emitter is expressed as follows:

$J_p\left(S_f,N_b\right)=-e{D}_p{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{S}_{pjk}\left[\frac{B_{pjk}}{L_{pjk}^{*}}+{\displaystyle \underset{m=1}{\overset{3}{\sum }}{b}_m{T}_{pjk}\left[{\text{e}}^{-b_{m}W}-{\text{e}}^{-b_m\left[H+W_{bsf}\right]}\right]}\right]}}$(12)

Under simultaneous illumination on both faces, the photocurrent density in the base is given by:

$J_n\left(S_f,N_b\right)=e{D}_n{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{S}_{njk}\left[\frac{B_{njk}}{L_{njk}^{*}}+{\displaystyle \underset{m=1}{\overset{3}{\sum }}{b}_m{T}_{njk}\left[{\text{e}}^{-b_{m}W}-{\text{e}}^{-b_m\left[H+W_{bsf}\right]}\right]}\right]}}$(13)

Under simultaneous illumination on both faces, the photocurrent density in the overdoped (p⁺) region is expressed as:

$\begin{array}{l}{J}_{n^{+}}\left(S_f,N_{bsf}\right)\\ =e{D}_{n^{+}}^{*}{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{S}_{n^{+}jk}\left[\frac{A_{n^{+}jk}+B_{n^{+}jk}}{L_{n^{+}jk}}+{\displaystyle \underset{m=1}{\overset{3}{\sum }}{b}_m{T}_{n^{+}jk}\left[{\text{e}}^{-b_m\left[W+H\right]}-{\text{e}}^{-b_m\left[W_{bsf}\right]}\right]}\right]}}\end{array}$(14)

The voltage at the terminals of the photovoltaic cell under illumination is calculated using the Boltzmann relation [14] .

For simultaneous illumination on both faces, the photovoltage in the base is given by the following equation:

$V_{php}\left(S_f,N_e\right)=V_T\ln \left(1+\frac{1}{n_0}\ast {\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{S}_{p^{\prime }jk}\left(A_{pjk}-{\displaystyle \underset{m=1}{\overset{3}{\sum }}{T}_{pjk}\left[{\text{e}}^{-b_{m}W}+{\text{e}}^{-b_m\left(H+W_{bsf}\right)}\right]}\right)}}\right)$ (15)

Under simultaneous illumination on both sides, the photovoltage in the base is expressed as follows:

$V_{phn}\left(S_f,N_b\right)=V_T\ln \left(1+\frac{1}{n_0}\ast {\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{S}_{n^{\prime }jk}\left(A_{njk}-{\displaystyle \underset{m=1}{\overset{3}{\sum }}{T}_{njk}\left[{\text{e}}^{-b_{m}W}+{\text{e}}^{-b_m\left(H+W_{bsf}\right)}\right]}\right)}}\right)$ (16)

For simultaneous illumination on both faces, the photovoltage in the overdoped zone is evaluated using Equation (17).

$V_{ph{n}^{+}}\left(S_f,N_{bsf}\right)=V_T\ln \left(1+\frac{1}{n_0}\ast {\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\left(Z_{n^{+}jk}\left(H\right)\right)S_{n^{+}{}^{\prime }jk}}}\right)$(17)

The electrical power produced by the PV cell is given in emitter, base and the overdoped zone by the Equations (18), (19) and (20)

$P_p\left(S_f,N_e\right)=J_{php}\ast V_{php}$(18)

$P_n\left(S_f,N_b\right)=J_{phn}\ast V_{phn}$(19)

$P_{n^{+}}\left(S_f,N_{bsf}\right)=J_{ph{n}^{+}}\ast V_{ph{n}^{+}}$(20)

The fill factor defines the efficiency of the PV cell; it can also provide information on the aging of the PV cell. It is the ratio between the maximum power delivered and the ideal power [15] . For simultaneous illumination, the fill factor is given the electrical power produced by the PV cell is given in emitter, base and the overdoped zone by the Equations (21), (22) and (23).

$F{F}_p\left(S_f,N_e\right)=\frac{J_{php\max }\ast V_{php\max }}{J_{phcc}\ast V_{phco}}$(21)

$F{F}_n\left(S_f,N_b\right)=\frac{J_{phn\max }\ast V_{phn\max }}{J_{phcc}\ast V_{phco}}$(22)

$F{F}_{n^{+}}\left(S_f,N_{bsf}\right)=\frac{J_{ph{n}^{+}\max }\ast V_{ph{n}^{+}\max }}{J_{phcc}\ast V_{phco}}$(23)

The determination of intrinsic parameters, namely diffusion coefficients, lifetimes and densities of excess minority charge carriers in addition to the evaluation of extrinsic parameters such as photovoltage and photocurrent density, electrical power and fill factor have been carried out in this section. In the following section the results and discussions of the influence of thickness on extrinsic parameters will be given.

In this section, Figure 2 presents the evolution of the photocurrent density as a function of the dynamic velocity at the junction for different Ne doping rates under simultaneous illumination of both faces.

In Figure 2 , we observe a decrease in the photocurrent density for the values of S_f ≻ 10² cm∙s⁻¹ when the doping rate of the emitter increases. But for a doping rate varying between 10¹⁷ cm⁻³ and 10¹⁹ cm⁻³, this is very low. In short circuit, we observe a reduction in the photocurrent density of about 20.68% when the doping rate of the emitter goes from 10¹⁷ cm⁻³ to 10²¹ cm⁻³. The increase in the doping rate of the emitter leads to an increase in the electrons which are the majority carriers. the holes will be recombined and this will lead to a drop in the photocurrent density of the holes at the emitter. Indeed, the diffusion coefficient and the diffusion length both decrease with increasing N_e doping rate [2] . These two diffusion parameters are characteristic of the movements of charge carriers. The diffusion length is the distance traveled by the photogenerated charge carriers before recombining and the diffusion coefficient is related to the mobility of charge carriers. Hence the reduction in photocurrent density.

The evolution of the photovoltage as a function of the dynamic velocity at the junction for different doping rates of the emitter under simultaneous illumination of both faces is presented in Figure 3 .

We notice that in open circuit, when the doping rate of the emitter increases from 10¹⁷ cm⁻³ to 10²¹ cm⁻³, it appears an increase in the open circuit photovoltage of about 10%. Indeed, increasing.

The doping rate of the emitter leads to an increase in positive ions near the junction. the photovoltage being a difference of potentials will increase at the junction on the emitter side. This will cause an increase in the open circuit photovoltage. In the following part, we will determine the behavior of the electrical power as a function of the doping rate of the transmitter.

The figure below ( Figure 4 ) represents the evolution of the electrical power as a function of the dynamic velocity at the junction for different doping rates of the N_e emitter under simultaneous illumination of both faces.

We notice in this figure the impact of the doping rate of the transmitter on the maximum electrical power. We note that the maximum electrical power increases for the values of the doping rate 10¹⁸ cm⁻³ then decreases from this value. This state of affairs confirms the effect of the doping rate of the emitter N_e observed on the photocurrent density and the photovoltage. According to the results recorded in Table 1 , we also notice that the very high doping rate acts negatively on the fill factor.

As we can see, the fill factor increases when the doping rate N_e increases from 10¹⁷ cm⁻³ to 10¹⁹ cm⁻³ then decreases from 10¹⁹ cm⁻³. The reduction in the fill factor is due to the fact that for the values of the doping rate N_e ≤ 10¹⁹ cm⁻³ the increase in the photovoltage com pensates for the short-circuit photocurrent losses. On the other hand, for N_e ≻ 10¹⁹ cm⁻³ the short circuit photocurrent loss becomes significant and can no longer be compensated by the open circuit photovoltage. Therefore, the doping rate N_e should be within the interval [10¹⁸ cm⁻³ - 10¹⁹ cm⁻³]. Subsequently, we will study the impact of the base doping rate.

Figure 5 represents the evolution of the photocurrent density as a function of the dynamic velocity at the junction for different doping rates of the base under simultaneous illumination of the both faces.

We also notice that for the values S_f ≤ 10² cm∙s⁻¹ the doping rate of the base has no impact on the photocurrent density. On the other hand, for the values S_f ≻ 10² cm∙s⁻¹ the short-circuit photocurrent density decreases when the doping rate of the base increases. In short circuit, this decrease in photocurrent density is about 24.67% when the base doping rate increases from 10¹⁵ cm⁻³ to 10¹⁹ cm⁻³. Indeed, when the doping rate of the base is high, the holes which are the majority carriers increase. Consequently, very few electrons will be able to cross the junction to be collected and participate in the current of the short-circuited external circuit. Indeed, increasing the doping of the base leads to a reduction in the diffusion length as well as the diffusion coefficient [2] . The electrons in the base of the PV cell will undergo significant recombination, hence the reduction in the short circuit photocurrent density. In the following section, we will study the impact of the doping rate of the base on the photovoltage.

The study of the evolution of the photovoltage as a function of the dynamic velocity at the junction under simultaneous illumination of both faces, the curves obtained are presented in Figure 6 .

In short circuit, the photovoltage is very insensitive to variations in the doping rate of the base. On the other hand, in an open circuit the photovoltage increases by 5.23% when the doping rate of the base goes from 10¹⁵ cm⁻³ to 10¹⁶ cm⁻³ then remains constant from this base doping rate value.

Indeed, increasing the doping rate of the base leads to an increase in negative ions at the junction on the base side. Thus, the open circuit photovoltage which is a potential difference will increase. The electrons are then blocked in the base of the PV cell, resulting in an increase in the open circuit photovoltage despite the volume recombinations which increase with the doping rate of the base. Reason why the open circuit photovoltage increases when the doping rate of the base increases. In the following part, the evolution of the electrical power as a function of the doping rate of the base will be analyzed.

The maximum electrical power increases for the values of the doping rate N_b ≤ 10¹⁶ cm⁻³ then decreases for the values of the doping rate dopage N_b ≻ 10¹⁶ cm⁻³. As the doping rate increases, the electrons in the base become less and less mobile, which favors their recombinations. This fact is linked to the decrease in the photocurrent density of the electrons observed previously. Which clearly justifies the reduction in electrical power when the doping rate of the base increases. We present, in Table 2 , the values of the fill factor as a function of the doping rate of the base.

We see that the fill factor undergoes a slight increase when the doping rate N_b goes from 10¹⁵ cm⁻³ to 10¹⁶ cm⁻³ then remains constant for the values of the doping rate between 10¹⁶ cm⁻³ and 10¹⁷ cm⁻³. Then for values of N_b greater than 10¹⁷ cm⁻³ the fill factor decreases sharply. However, the optimal doping rate of the base must be between 10¹⁶ cm⁻³ and 10¹⁷ cm⁻³. In the following part, the impact of the doping rate of the overdoped zone will be studied.

Figure 8 shows the evolution of the photocurrent density as a function of the dynamic velocity at the junction for different doping rates of the overdoped zone under simultaneous illumination on both faces.

For values of S_f ≤ 10² cm∙s⁻¹, the doping rate of the zone p⁺ has no impact on the photocurrent density. On the other hand, for values of S_f ≻ 10² cm∙s⁻¹ we have an increase in the photocurrent density. In short circuit, when the doping rate of the overdoped zone p⁺ increases from 10¹⁷ cm⁻³ to 10²¹ cm⁻³, an increase in the photocurrent density of approximately 24.52% is observed. This increase is due to the presence of the electric field on the rear surface of the PV cell. Consequently, the minority carriers (electrons) generated near the surface escape the recombination process at the back face [16] . Indeed, the presence of the back electric field of the PV cell makes it possible to minimize recombinations although the doping rate of this zone is high. we can say that the increase in impurities in this area leads to an increase in the photocurrent density at the back face of the PV cell. Now, we will show the impact of the doping rate of the overdoped zone on the photovoltage.

In Figure 9 , we present the photovoltage curve as a function of the dynamic velocity at the junction for different doping rates of the overdoped zone under simultaneous illumination on both faces.

We see an increase in photovoltage when the doping rate increases in the open circuit. Indeed, we see an increase in the photovoltage up to N_bsf = 10²⁰ cm⁻³ then it remains constant from this value. When the doping rate of the zone increases from 10¹⁷ cm⁻³ to 10²¹ cm⁻³, it appears an increase of about 25% in the open circuit photovoltage. Indeed, the doping rate of the overdoped zone will impact the back electric field (Back Surface Field) which is directed from the base towards this overdoped zone. The potential barrier induced by the difference in doping level between the base and the overdoped zone therefore tends to confine the minority carriers in the base [15] . However, very high doping of the overdoped zone reduces this potential barrier due to the BGN band gap narrowing phenomenon [17] . So the dopant atoms become negative ions at the base-zone interface increasing. Hence an increase in photovoltage. The evolution of the electrical power as a function of the doping rate of the overdoped zone is analyzed in the following part.

To determine the behavior of the electrical power as a function of the doping rate of the overdoped zone p⁺, we represent the evolution of the electrical power as a function of the doping rate of the overdoped zone p⁺. The evolution obtained for simultaneous illumination is presented in Figure 10 .

We observe in Figure 10 that the electrical power by the PV cell is almost zero in the vicinity of the open circuit and the short circuit; it reaches its maximum at an intermediate operating point. For this mode of operation, when the doping rate increases from 10¹⁷ cm⁻³ to 10²⁰ cm⁻³, the electrical power increases and then remains constant from 10²⁰ cm⁻³. This increase is explained by the increase in the intensity of the eclectic field. This prevents recombination at the back side of the PV cell. This observation was made at the level of the photovoltage density. In Table 3 the values of the fill factor are recorded for different values of the doping rate of the overdoped zone.

We notice that increasing the doping rate of the overdoped zone p⁺ gives a significant improvement in all the parameters of the PV cell until reaching a constant value, but the fill factor remains constant from this value. Indeed, recombination in the overdoped zone p⁺ is very weak given its location on the back face of the cell. In addition, the heavily doped overdoped zone p⁺ makes it possible to reduce recombination at the semiconductor metal contact. Thus, the optimal doping rate value N_bsf should be within the interval [10²⁰ cm⁻³ - 10²¹ cm⁻³].