The solar cell under study is a bifacial polycrystalline silicon n⁺-p-p⁺ solar cell, whose structure is shown in Figure 1. It consists of four distinct parts:

The n⁺type emitter (front face): the most heavily doped region, which receives incident light and collects electrons.The p-type base: weakly doped with acceptor atoms; electrons are minority carriers.The rear panel of type p⁺:p⁺ overdoping generating a back surface field (BSF) that repels minority carriers generated near the backside toward the junction.The emitter-base junction (space charge region):the internal electric field separates the electron-hole pairs generated by the absorption of photons.

The solar cell is subjected successively to three illumination modes. front face (α = 1), back face (α = 2), and simultaneous illumination of both faces (α = 3). The study is conducted in a static regime, which means that all parameters are independent of time.

Figure 1. Structure of an n⁺-p-p⁺ silicon bifacial solar cell [4].

The model is one-dimensional, with the junction (x = 0) and the base extending to x = H as its origin [5]. The contribution of the emitter is neglected given its small thickness (about 1 μm). In the static regime under multispectral illumination, the density of excess minority carriers in the base obeys the continuity equation.

where:

δ α ( x ) : Denotes the density of excess minority carriers as a function of depth x in the base according to the different illumination modes;L: Diffusion length of minority charge carriers in the base. It represents the average distance traveled by minority carriers before their recombination;D: Diffusion coefficient of minority carriers in the base; G α ( x ) : Rate of generation of minority carriers at a distance x from the solar cell junction.

The generation rate G_α(x) expressed as a function of the light absorption depth x in the base can be written as follows [6]:

where n is the illumination level. The absorption coefficients for the AM1.5 spectrum are:

a₁ = 6.13 × 10²⁰ cm⁻³s⁻¹; a₂ = 0.54 × 10²⁰ cm⁻³s⁻¹; a₃ = 0.0991 × 10²⁰ cm⁻³s⁻¹

b₁ = 6630 cm⁻¹; b₂ = 10³ cm⁻¹; b₃ = 130 cm⁻¹

The ξ α and χ α parameters are defined according to the illumination mode (Table 1).

Table 1.Values of the parameters ξ_α and χ_α depending on the illumination mode [7][8].

Boundary conditions are written: at the junction (x = 0): ∂δ_α/∂_x = (Sf_α/D)·δ_α(0); to the rear face (x = H). ∂δ_α/∂_x = −(Sb_α/D)·δ_α(H), où Sf_α et Sb_α are the recombination speeds at the junction and on the rear face, respectively.

The parameter values used in the simulation are. L = 0.015 cm; D = 26 cm²/s; H = 0.03 cm; Nᵇ = 10¹⁶ cm⁻³; n_i = 10¹⁰ cm⁻³; T = 300 K. These values are consistent with the reported experimental data for polycrystalline silicon under AM1.5.

Photocurrent density

The photocurrent density is deduced from Fick’s law applied to the junction.

For recombination speeds Sf_α → ∞, J_phα reaches a constant value corresponding to the short-circuit current J p h α .

A_α denotes the illuminated active area of the solar cell for illumination mode α (the front-face area for α = 1, the rear-face area for α = 2, and the sum of the front- and rear-face areas for the simultaneous mode α = 3), used to convert the photocurrent density J_phα (mA/cm²) into the photocurrent I_phα = A_α·J_phα (mA), and the power density P_α (mW/cm²) obtained from Equation (5) into the electrical power P_α (W) = A_α·P_α (mW), for a cell of a given physical surface area.

Photovoltage

The photovoltage is deduced from the carrier density at the junction via the Boltzmann relation.

where V T = K ⋅ T q is the thermal voltage (~0.026 V à 300 K).

Electrical power

Taking into account the diode current J p h α , the power per unit area is written.

All these expressions have been implemented in MATHCAD and evaluated as a function of the recombination rate S f α at the junction, for the different illumination levels n and the three illumination modes.

The illumination level n used throughout Equation (2) is a dimensionless multiplicative factor applied to the AM1.5 reference spectral irradiance from which the absorption coefficients ai and bi were derived; n = 1 corresponds to one standard sun (AM1.5G, total irradiance ≈ 100 mW/cm²), and the sub-unity values used in this study (n = 0.065, 0.175 and 0.375) correspond to fractions of this reference irradiance, consistent with the illumination conditions reported by Bernard (2006) and with the experimental levels used for validation against Ly et al.(2012).

For the simultaneous illumination mode (α = 3), the same illumination level n is applied to the front and rear faces simultaneously (ξ₃ = χ₃ = 1 in Equation (2)) rather than a fixed total irradiance being split between the two faces. Consequently, for a given n the total incident energy under simultaneous illumination is twice that received under single-face illumination, and part of the higher photocurrent, photovoltage and power obtained for this mode reflects this larger total input rather than a purely bifacial gain. A strictly energy-normalized comparison between modes would require halving n on each face for the simultaneous case, which is not done in the results presented here and is left for a dedicated bifacial-gain analysis in future work.

In Equation (5), Sf₀ denotes the recombination velocity associated with the diode dark (saturation) current; it is taken as a fixed, mode-independent parameter and is used, together with the thermal voltage V_T, to represent the reversedark current q S f 0 ( 1 + e V p h α V T ) that is subtracted from the illuminated photocurrent J_phα to obtain the net current used to compute the powerP_α for each illumination mode.

All expressions for δ_α(x), J_phα, V_α and P_α were implemented in Mathcad by solving the linear second-order ordinary differential Equation (1), together with its boundary conditions at x = 0 and x= H, in closed analytical form (a homogeneous cosh/sinh solution plus a particular solution driven by the exponential generation term G_α(x)); the resulting closed-form expressions were then evaluated numerically over a logarithmic sweep of the junction recombination velocity, Sf_α = 10^p cm/s with p varied from 0 to 6, for each illumination level n and each mode α. Unless stated otherwise in a figure caption, all other parameters (L, D, H, Nb, n_i, T and the back-surface recombination velocity Sb_α) were held fixed at the values given above.

For each illumination mode and level n, the current-voltage (I-V) characteristic shown in Section 3.4 was constructed parametrically. J_phα and V_α were both computed as functions of the same junction recombination velocity Sf_α over the sweep described above, and paired point-by-point at each value of Sf_α to trace the J_phα-V_α curve, using the dark-current assumption described for Equation (5) for all three illumination modes.

3.1.1. Illumination from the Front Face

The substitution of integration coefficients in the continuity equation makes it possible to obtain the expression of the density of excess minority carriers for an illumination by the front face of the solar cell. The density profiles are illustrated in Figure 2 for different levels of illumination.

In the vicinity of the junction (x = 0), a carrier storage is observed, which corresponds to a maximum density of minority carriers when the recombination rate is low (open circuit). The carrier density decreases as one moves away from the junction towards the rear face. The increase in the level of illumination leads to an increase in the density of minority carriers, especially at the junction. This effect becomes negligible for a depth greater than 0.025 cm.

Figure 2. Charge carrier density as a function of depth x in the base and for different levels of illumination from the front face with Sf₁ = 1000 cm/s; D = 26 cm²/s; L = 0.015 cm et Sb₁ = 1000 cm/s.

When the recombination rate at the junction increases significantly, the carrier density δ(x) decreases. This behavior illustrates the transition from open circuit mode (Sf → 0, maximum density) to short-circuit mode (high Sf, low density at the junction). Figure 3 shows the influence of the recombination rate at the junction on the carrier density for a fixed illumination level n = 0.2.

According to the curve, the recombination rate at the junction increases, which leads to a decrease in the minority charge carrier density.

3.1.2. Illumination from the Rear Face

For backside illumination, the carrier density increases from the junction to a maximum located at the backside of the base (positive gradient in this area). This behavior is the opposite of what was observed for front side illumination. Figure 4 shows the profiles for different levels of illumination n.

Figure 3. Carrier density as a function of depth x in the base for illumination from the front face and for different recombination speeds at the junction with D = 26 cm²/s; n = 0.2; Sf = 300 cm/s L = 0.015 cm; H = 0.03 cm; Sb₁ = 4000 cm/s.

Figure 4. Charge carrier density as a function of depth x in the base and for different levels n of illumination through the rear face with Sf₂ = 1000 cm/s; D = 26 cm²/s; L = 0.015 cm; H = 0.03 cm etSb₂ = 1000 cm/s.

Figure 5 illustrates the influence of the recombination rate at junction S f 2 for different values and for an illumination level n = 0.2. Whatever the recombination rate at the S f 2 junction, the carrier density increases with depth x up to a maximum, then decreases. In the decreasing zone, carriers can no longer cross the junction and recombine. When S f 2 is very high (short circuit), the junction density is almost zero.

Figure 5. Carrier density as a function of depth x in the base for illumination from the rear face and for different recombination speeds at the junction with D = 26 cm²/s; n = 0.2; Sf₂ = 300 cm/s; L = 0.015 cm; H = 0.03 cm; Sb₂ = 40,000 cm/s.

3.1.3. Simultaneous Illumination of Both Sides

For simultaneous illumination, the carrier density has two maxima. one at the junction (influence of the front face) and the other at the rear face (influence of the rear face). Figure 6 illustrates these profiles for different levels of illumination.

Figure 6. Density of charge carriers as a function of depth x in the base and for different levels n of simultaneous illumination with Sf₃ = 1000 cm/s; D = 26 cm²/s; L = 0.015 cm; H = 0.03 cm etSb₃ = 1000 cm/s.

Figure 7 shows the influence of the recombination rate at the S f 3 junction for simultaneous illumination. The density decreases with the increase of S f 3 ; this effect attenuates as we move away from the junction, to become almost zero at the end of the base (x between 0.028 and 0.03 cm).

Figure 7. Carrier density as a function of depth x in the base for simultaneous illumination and for different recombination speeds at the junction Sf₃ with D = 26 cm²/s; n = 0.2; L = 0.015 cm; H = 0.03 cm etSb₃ = 40,000 cm/s.

3.2.1. Current Density for Illumination from the Front Face

Figure 8 illustrates the variation of the photocurrent density as a function of the recombination rate at junction S f 1 for different illumination levels n, for illumination from the front face.

Figure 8. Current density as a function of the recombination rate at the junction Sf₁ for different values of n for illumination by the front face of the solar cell with D = 26 cm²/s; L = 0.015 cm et Sb₁ = 1000 cm/s.

For low values of S f 1 (open circuit), the photocurrent density is almost zero because the carriers are blocked at the junction. When S f 1 increases, the current gradually increases until it reaches a constant value corresponding to the short-circuit current. The photocurrent density increases with the level of illumination n, which is explained by the increase in the density of minority carriers under more intense illumination [8].

3.2.2. Current Density for Backlighting

Figure 9 illustrates the variation of current density as a function of recombination rate at junction S f 2 for different illumination levels n, for backside illumination.

Figure 9. Current density as a function of the recombination rate at the Sf₂ for different values of n for illumination by the rear face of the solar cell with D = 26 cm²/s; L = 0.015 cm et Sb₂ = 1000 cm/s.

The current density profiles present a similar pattern to those obtained for the front side illumination. However, the short-circuit photocurrent density for a front face illumination is higher than that obtained by the rear face [8]. The level of illumination leads to an increase in photocurrent density in both cases.

3.2.3. Current Density for Simultaneous Illumination of Both Sides

Figure 10 shows the variation profiles of photocurrent density J p h 3 as a function of recombination rate at junction S f 3 for different illumination levels n.

The photocurrent density for simultaneous illumination is higher than those obtained for front or rear face illuminations alone. The density of the currents increases according to the level of lighting.

3.2.4. Comparative Study of Current Densities for Different Illumination Modes

Figure 11 presents a comparison of the photocurrent curves for the three illumination modes (front face, rear face, simultaneous) at an illumination level n = 1.

Figure 10. Current density as a function of the recombination rate at the junction Sf₃ for different values of n for simultaneous illumination of both sides with D = 26 cm²/s; L = 0.015 cm et Sb₃ = 1000 cm/s.

Figure 11. Current density as a function of the recombination rate Sf_α for different illumination modes with n = 1; H = 0.03 cm; Sb₃ = 1000 cm/s; D = 26 cm²/s; L= 0.015 cm.

For a given illumination level (n = 1), the theoretical short-circuit current of the front face is greater than that of the rear face, in accordance with the experimental results presented in Table 2. The short-circuit current of simultaneous illumination is the highest of the three modes.

3.3.1. Phototension for Illumination from the Front Face

Figure 12 shows the variation of the photovoltage as a function of the recombination rate at junction S f 1 for front side illumination and for different illumination levels n.

Figure 12. Photovoltage as a function of the recombination rate Sf₁ for different values of n for illumination from the front face with D = 26 cm²/s; Sb₁ = 1000 cm/s; H = 0.03 cm etSf₁ = 10^P cm/s.

The photovoltage is maximum and constant for low values of S f 1 (open circuit, storage of carriers at the junction). When S f 1 reaches high values, the voltage decreases and tends to zero (short circuit). The photovoltage increases with the level of illumination, because the density of minority carriers, on which the photovoltage directly depends, is higher under intense illumination.

3.3.2. Photovoltage for Illumination from the Back Side

Figure 13 illustrates the profile of photovoltage V₂ as a function of recombination rate at the junction S f 2 for different illumination levels n.

The rear side photovoltage profiles have the same overall appearance as those on the front side. However, for the same level of illumination, the photovoltage for a front side illumination is higher than that obtained by the rear side.

3.3.3. Photovoltage for Simultaneous Illumination of Both Sides

Figure 14 shows the profile of the photovoltage V₃ as a function of the recombination rate at the junction S f 3 for different illumination levels n and for simultaneous illumination.

The photovoltage V₃ remains constant and maximum when S f 3 is low (open circuit), then rapidly decreases to zero for high values of S f 3 (short circuit). The photovoltage increases with the level of illumination, in accordance with the trends observed for the other modes.

Figure 13. Photovoltage as a function of the recombination rate Sf₂ for different values of n for illumination from the rear front withD = 26 cm²/s; Sb₂ = 1000 cm/s; H = 0.03 cm et Sf₂ = 10^P cm/s.

Figure 14. Photovoltage as a function of the recombination rate Sf₃ for different values of n for simultaneous illumination with D = 26 cm²/s; Sb₃ = 1000 cm/s; H = 0.03 cm et Sf₃ = 10^P cm/s.

3.3.4. Comparative Study of Photovoltages for Different Types of Illumination

Figure 15compares the photovoltage as a function of the recombination rate at the junction for the three illumination modes at n = 1.

For a given illumination level (n = 1), the open-circuit voltage obtained by simultaneous illumination of both sides is higher than that of the front and rear faces, in accordance with the experimental result from Table 3.

Figure 15. Photovoltage as a function of the recombination rate Sf_α for different types of illumination and for n = 1 with D = 26 cm²/s; L = 0.015 cm; H = 0.03 cm; Sf_α = 1000 cm/s.

3.4.1. Characteristic I-V for Different Levels of Illumination from the Front Face

Figure 16 shows the current-voltage characteristic I-V obtained for illumination from the front face and for different levels of illumination n.

Figure 16. Characteristic I-V for different values of n for a front side illumination with Sb₁ = 1000 cm/s; D = 26 cm²/s; H = 0.03 cm; L = 0.015 cm.

For a voltage below 0.5 V, the current density remains practically constant, close to the short-circuit current. Above 0.5 V, it rapidly drops towards zero in the vicinity of the open circuit. The open-circuit voltage and the short-circuit current increase with the level of illumination, the variation in current being much greater than that in voltage.

3.4.2. Characteristic I-V for Different Levels of Backlighting

Figure 17 shows the characteristic I-V for backlighting and for different levels of illumination n.

Figure 17. Characteristic I-V for different values of n for a rear side illumination with Sb₂ = 1000 cm/s; D = 26 cm²/s; H = 0.03 cm et L = 0.015 cm.

For a voltage below 0.45 V, the current density is practically constant and equal to the back side short-circuit current. The current density and open-circuit voltage increase with the level of illumination, the increase in current being much greater than that in voltage.

3.4.3. Characteristic I-V for Simultaneous Illumination

Figure 18 shows the variation in current density as a function of voltage for simultaneous illumination of both sides and different levels n.

Figure 18. Characteristic I-V for different values of n for simultaneous illumination of both sides with Sb₃ = 1000 cm/s; D = 26 cm²/s; H = 0.03 cm etL = 0.015 cm.

For a voltage lower than 0.5 V, the current density is constant (short-circuit). It then decreases quickly to cancel itself at the open circuit point. The effect of the illuminance level is more pronounced in the vicinity of the short circuit.

3.4.4. Comparative Study of I-V Characteristics for Different Illumination Modes

Figure 19 compares the I-V characteristics of the three illumination modes for n = 1.

Figure 19. I-V characteristic for different illumination modes and for n = 1; Sb_α = 1000 cm/s; D = 26 cm²/s; H = 0.03 cm et L= 0.015 cm.

Simultaneous illumination of both sides produces the highest current density and open circuit voltage. The back side alone yields the lowest values. These results are consistent with experimental results from the literature [8].

3.5.1. Power for Illumination from the Front Face

Figure 20 shows the variations of the electric power P1 as a function of the recombination rate at junction Sf₁ for different levels of illumination n.

There are four operating zones. 1) open circuit (low Sf, P ≈ 0); 2) zone between open circuit and maximum power; 3) zone between maximum power and short circuit; 4) short circuit (high Sf, P ≈ 0). The maximum power point is around p ≈ 3.85. This point moves to the open circuit when the illumination level increases.

3.5.2. Power for Rear-Side Illumination

Figure 21 shows the variations in P₂ power as a function of Sf₂ for backside illumination.

Figure 20. Power of the solar cell illuminated by its front face as a function of the recombination rate Sf₁ at the junction, for different illumination levels n of the solar cell, Sb₁ = 3∙10³ cm/s; et Sf₁ = 10^P cm/s.

Figure 21. Power of the solar cell illuminated by its back surface as a function of the recombination rate Sf₂ at the junction, for different illumination levels n of the solar cell, Sb₂ = 3∙10³ cm/s; et Sf₂ = 10^P cm/s.

The same four areas of variation are observed. For the same level of illumination, the power obtained by backlighting is lower than that of the front side, but cannot be neglected.

3.5.3. Power for Simultaneous Illumination of Both Sides

Figure 22 shows the variations in P3 power as a function of Sf₃ for simultaneous illumination.

The influence of the recombination rate at the junction on the power is similar to the observations of previous modes. The maximum power is the highest for simultaneous illumination.

Figure 22. Power of the solar cell for simultaneous illumination of both sides as a function of the recombination rate Sf₃ at the junction, for different illumination levels n of the solar cell with Sb₃ = 3∙10³ cm/s; et Sf₃ = 10^P cm/s.

3.5.4. Comparative Study of Power Characteristics for Different Illumination Modes

Figure 23compares the power characteristics for the three illumination modes at n = 1.

Figure 23. Power characteristics for different illumination modes and for n= 1; Sb_α = 3∙10³ cm/s; etSf_α = 10^P cm/s.

For n = 1, the simultaneous illumination of both sides provides the highest power, followed by the front side and then the rear side. This result is consistent with the photocurrents and photovoltages obtained for each mode.

Table 2 presents a comparison of theoretical and experimental short-circuit currents for different levels of illumination, while Table 3 compares open-circuit voltages.

Table 2. Comparison of the short-circuit currents J_cc1 and J_cc2 (front and rear face) for different levels of illumination—Theoretical and experimental results.

Table 3. Comparison of open circuit voltages V_co1, V_co2 and V_co3 for different levels of illumination—Theoretical and experimental results [8].

Experimental J_cc values are taken from Table 1 and experimental V_co values from Tab. 4 of reference [8]. The theoretical trends obtained (short-circuit current and open-circuit voltage both increasing with illumination level, and simultaneous illumination outperforming either single-face mode) are consistent with the experimental observations reported in reference [8]. Quantitatively, the theoretical short-circuit current densities deviate from the experimental values in reference [8] by about 4.5% to 6.4% for J_cc1 (Table 2), and the theoretical open-circuit voltages deviate by about 1.6% to 4.5% (Table 3), which supports the concordance claimed above.