The n⁺-p-p⁺ structure of the PV cell is assumed. This study is carried out in the quasi-neutral base (QNB) assumption [18] [19] which assumes that intrinsic electric field in the cell’s base is neglected. First, the PV cell is submitted to a moderate light concentration (C = 50 Suns) as showed by Figure 1 and then to a normal light illumination (C = 1 Sun).

The continuity equation of minority charge carriers in steady state is given by:

∂ 2 δ β ( x , N b ) ∂ x 2 − δ β ( x , N b ) ( L β ( N b ) ) 2 = − G β ( x ) D β ( N b ) (1)

In Equation (1), δ^β (x, N_b) represents the excess minority carrier density which is a function of the both position x in the base and doping level N_b. D^β (N_b) and L^β (N_b) respectively represent the diffusion coefficient and the diffusion lenght. G^β (x) is the carrier generation rate at the base deph x. The index β is linked to the illumination mode. So, under moderate light illumination C = 50 Suns and β = con. And under normal illumination mode C = 1 Sun and β = nor.

· Under moderate concentration (C = 50 Suns and β = con)

We take into account the electric field of electrons concentration gradient given by Equation (2) below [5] [6] [18]:

E ( x ) = D p − D n μ n + μ p ⋅ 1 δ c o n ( x ) ⋅ ∂ δ c o n ( x ) ∂ x (2)

The diffusion coefficient is given by Equation (3):

D c o n ( N b ) = μ n ( N b ) [ 2 D n ( N b ) − D p ] + μ p D n ( N b ) μ n ( N b ) + μ p (3)

In Equation (3), D_n (N_b) And D_p Respectively represent diffusion coefficients of electrons and holes. m_n (N_b) and m_p respectively represent mobility coefficients of electrons and holes.

Equation (3) shows that under moderate concentration (C = 50 Suns), carriers diffusion depends on diffusion and mobility coefficients of electrons and holes.

The carrier generation rate G^con (x) at the base depth x is given by [5] [6] [18] [20]:

G ( x ) = C ⋅ ∑ i = 1 3 a i e − b i x (4)

· Under normal illumination mode (C = 1 Sun and β = nor)

The electric field of electrons concentration gradient is no longer taken into account. The diffusion coefficient in the base is then reduces to that of the electrons.

D n o r ( N b ) = D n ( N b ) (5)

The electron generation rate becomes:

G n o r ( x ) = ∑ i = 1 3 a i e − b i x (6)

As shown by Equations (7) to (9) below, electrons diffusion coefficient D_n (N_b), their mobility coefficient m_n (N_b) and their lifetimeτ (N_b) depend on doping level (N_b) [17].

D n ( N b ) = 1350 ⋅ V T 1 + 81 ⋅ N b N b + 3 , 2.10 8 (7)

μ n ( N b ) = q k T ⋅ D n ( N b ) (8)

τ ( N b ) = 12 1 + N b 5 × 10 16 ( μ s ) (9)

Then, whatever the illumination mode the diffusion coefficient in the base D^β (N_b) is a function of doping level (N_b) as well as carriers the diffusion length L^β (N_b) given by:

L β ( N b ) = D β ( N b ) τ ( N b ) (10)

By solving the continuity Equation (1), excess minority carriers density is determined by:

δ β ( x , N b ) = A β c h ( α β ( N b ) x ) + B β s h ( α β ( N b ) x ) + ∑ i = 1 3 K i β ⋅ e − b i x (11)

Coefficients A^β and B^β are determined by using following boundary conditions [5] [6] [18] [20]:

· at the junction (x = 0)

D β ( N b ) ⋅ ∂ δ β ( x , N b ) ∂ x | x = 0 = S f ⋅ δ β ( x = 0 , N b ) (12)

Sf represents the junction dynamic velocity and is the sum of two contributions:

S f = S f 0 + S f j (13)

Sf₀ is the junction intrinsic recombination velocity and is related to carriers losses at the junction. Sf_j is the junction dynamic velocity and defines the carriers flow imposed by an external load.

Thus, near the open circuit, the carriers flow through the junction is practically null and therefore Sf_j ≈ 0 and Sf ≈ Sf₀. In the short-circuit situation, the carriers flow through the junction is very important and therefore Sf_j → +∞ and Sf→ +∞.

· at the cell’s back (x = H)

D β ( N b ) ⋅ ∂ δ β ( x , N b ) ∂ x | x = H = − S b ⋅ δ β ( x = H , N b ) (14)

S_b corresponds to the back surface recombination velocity and quantifies carriers losses at the PV cell rear side.

The ratio of doping level by carriers density is given by the following Equation (15).

Q β ( x , N b ) = N b δ β ( x , N b ) (15)

This ratio allows to compare carriers density evolution versus the doping level.

· The current density

The current density given by Equation (16) Is Found By Applying The Fick second law [5] [6] [18] [20] [21]:

J p h β ( S f , N b ) = q ⋅ D β ( N b ) ⋅ ∂ δ β ( x , S f , N b ) ∂ x | x = 0 (16)

The short-circuit current density is then defined by:

J s c β = lim S f → + ∞ J p h β (17)

· The photo-voltage

Using Boltzmann relation, we determine the expression of the voltage across the junction, given by Equation (18) [5] [6] [18] [20] [21]:

V p h β ( S f , N b ) = V T ln [ δ β ( x = 0 , S f , N b ) n 0 + 1 ] (18)

Then the cell’s open-circuit voltage is given by:

V o c β = lim S f → 0 V p h β (19)

V_T represents the thermal voltage and is given by V_T = kT/q, n₀ represents electrons density in thermodynamic equilibrium and is given by n₀ = (n_i)²/N_b. At T = 300 K, the both electrons intrinsic concentration n_i and the silicon energy gap Eg (T) are calculated by following Equations (20) and (21) [6]:

n i = A n ⋅ T 3 2 ⋅ exp ( − E g ( T ) 2 k T ) (20)

E g ( T ) = 1.1557 − 7.021 × 10 − 4 T 2 T + 1108 (21)

A_n represents the materiel specific constant. For the silicon A_n = 3.87 × 10¹⁶ and k the Boltzmann constant.

The electric power The electric power delivered by the PV cell is given by Equation (22) below [5] [6] [20] [21]:

P β ( S f , N b ) = V p h β ( S f , N b ) ⋅ J p h β ( S f , N b ) (22)

· The conversion efficiency

The solar cell’s conversion efficiency is calculated by using the following Equation (23) [5] [6] [20]:

η β ( N b ) = ( P β ) max ( S f , N b ) P i n c (23)

(P^β)_max is the maximum power at the illumination mode β and P_inc represents the power of the incident light under Air Mass 1.5 standard conditions.

The variation rate of a parameter X is calculated as a percentage. It is defined by:

| X f − X i X i | × 100 (24)

where X_i and X_f represent respectively the initial and final values of X.

Figure 2 and Figure 3 below give respectively variations of diffusion coefficient and diffusion length with doping level (N_b) increase, under moderate light concentration (C = 50 Suns) and normal light illumination mode (C = 1 Sun).

Figure 2 and Figure 3 show that for a given doping level (N_b), diffusion parameters are higher under moderate light concentration (C = 50 Suns) compared to the normal illumination mode (C = 1 Sun).

It also appears that whatever the illumination mode (C = 50 Suns or C = 1 Sun), the diffusion parameters decrease with the doping level increase. This decrease of the diffusion parameters results in a decrease of carriers flux through the junction. This allows predicting a decrease of the current density with the base doping level increase. The decrease of the diffusion parameters also results in carriers storage near the pn junction. This also allows us to predict an increase of the photo voltage with the base doping level increase.

Table 1 gives respectively under moderate light concentration and normal light illumination, doping level values and corresponding diffusion parameters values as well as their variation rates.

Table 1 shows that under moderate light concentration (C = 50 Suns), doping

level increase from 10¹³ cm⁻³ to 10¹⁶ cm⁻³ leads to diffusion coefficient decrease from 62.68 cm²∙s⁻¹ to 55.34 cm²∙s⁻¹. This is equivalent to 11.71% decrease in the diffusion coefficient. At the same time, under normal light illumination the diffusion coefficient decreases from 34.93 cm²∙s⁻¹ to 31.21 cm²∙s⁻¹. That is equivalent to 10.65% decrease in the diffusion coefficient.

It also appears from Table 1 that under moderate light concentration, the diffusion length decreases by 97.00% while under normal light illumination it decreases by 96.98%.

These results show that doping level impact on diffusion parameters is more felt under the moderate light concentration mode compared to normal light illumination.

The current density profile is plotted under a moderate light concentration (C =

50 Suns) for different doping level as showed by the following Figure 4.

Figure 4 shows that for a given doping level, the current density is null near the open circuit where the dynamic velocity Sf≈ 0. The current density then increases with the dynamic velocity to reach its maximum in the short-circuit situation (Sf→ +∞).

Figure 4 also shows that doping level increasing leads to current density decrease. This result agrees with the study of doping level influence on diffusion parameters carried out above. Indeed, this study had predicted a decrease in current density with doping level increase. These results are also consistent with those of Ould Cheikh et al. [14] and Samoura et al. [15] who worked under normal illumination.

Table 2 gives for the two illumination modes and for different doping level, the short-circuit current as well as its variation rate.

Table 2 shows that under moderate light concentration (C = 50 Suns), the short-circuit current density decreases from 1262.82 mA∙cm⁻² to 379.36 mA∙cm⁻², when the doping level increases from 10¹³ cm⁻³ to 10¹⁶ cm⁻³. This is equivalent to 69.96% decrease in short-circuit current density.

At the same time, under normal ligh illumination, the current density decreases from 23.88 mA∙cm⁻² to 6.32 mA∙cm⁻². This is equivalent to 73.53% decrease in short-circuit current density.

These results show that doping concentration effect on short-circuit current density is greater under normal illumination compared to moderate illumination mode. These results are in contradiction with diffusion parameters study. Indeed, this study had shown a greater effect of the doping level under moderate concentration compared to the normal illumination mode. This contradictory result is linked to the fact that under moderate light concentration carriers generation is strong. The carriers density is very high and close to doping level and consequently less sensitive to the doping rate variations. The PV cell is then said

to be in high injection condition [18] [22].

As show by following Figure 5, the PV cell’s voltage profile versus dynamic velocity was also plotted for different doping level (N_b) and under moderate light concentration.

For a given doping level, Figure 5 shows that near the open-circuit the voltage is maximum and constant. It then decreases with the dynamic velocity and becomes null in short-circuit situation.

It also emerges from Figure 5 that the doping level increase leads to photo voltage increase. This result also is also agrees with diffusion parameters study which had predicted an increase in voltage with doping level increase. Working under non-concentrated illumination Ould Cheikh et al. [14] and Trukhanov et al. [16] also found similar results.

The following Table 3 gives for the two illumination modes and for different doping levels, the open-circuit voltage as well as its variation rate.

Table 3 shows that under moderate light concentration, the open-circuit voltage increases from 995.74 mV to 1055.89 mV when the doping level increases from 10¹³ cm⁻³ to 10¹⁶ cm⁻³. This is equivalent to 6.04% increase in open-circuit voltage.

At the same time, under normal illumination mode the open-circuit voltage increases from 900.63 mV to 957.32 mV. This is equivalent to 6.29% increase in open-circuit voltage.

This result shows that the doping level effect on the open-circuit voltage is also greater for the normal illumination mode compared to the moderate light concentration.

This is in contradiction with the diffusion parameters study carried out earlier in this article. But is explained by importance of carriers density under moderate light concentration. The carriers density is close to the doping level and less sensitive to variations in this doping level. This implies open-circuit voltage low variation under moderate light concentration compared to normal illumination mode.

The electric power profile is plotted under moderate light concentration and for different values of the doping level (N_b) as shows by following Figure 6.

For a given doping level, Figure 6 shows that the electric power increases with the dynamic velocity to reach it maximum before decreases. The electric power is null near open-circuit and in short-circuit situation. Figure 6 also shows that the maximum power decreases when the doping level increases from 10¹³ cm⁻³ to 10¹⁶ cm⁻³.

Table 4 gives for the two illumination modes, the maximum power for different doping level and its variation rate.

Under moderate light concentration the maximum power decreases from 1108.50 mW∙cm⁻² to 355.22 mW∙cm⁻² when the doping level increases from 10¹³ cm⁻³ to 10¹⁶ cm⁻³. This is equivalent to 67.95% decrease in the maximum power. At the same time under normal illumination mode the maximum power decreases from 18.76 mW∙cm⁻² to 5.31 mW∙cm⁻². This corresponds to 71.70% decrease in the maximum power.

This result also shows that doping level effect on the maximum power is greater for normal illumination mode compared to moderate light concentration.

Figure 7 below gives, for the two illumination modes, the variations of the conversion efficiency with the doping level.

Figure 7 shows that whatever the illumination mode, the conversion efficiency decreases when the doping level increases.

The following Table 5 gives under moderate light concentration and under normal illumination, the conversion efficiency for different doping level and its variation rate.

Table 5 shows that under moderate light concentration, the conversion efficiency decreases from 30.79% to 9.87% with a variation rate of 67.94%. Under normal illumination, the conversion efficiency decreases from 18.76% to 5.31% with a variation rate of 71.70%. These results show a significant effect of doping level under normal illumination mode compared to moderate light concentration mode.

As we have already explained, this fact is due to carriers density importance under moderate light concentration compared to normal illumination. Thus, under moderate light concentration, the carriers density is significant and close to the doping level and therefore not very sensitive to variations in the doping level. However, under normal illumination, the carriers density is significantly lower and then more sensitive to variations in the doping level.

We plotted for the two illumination modes, the variations of the ratio of the doping level by the carriers density as show by Figure 8. The carriers density is calculated near the open-circuit operating point and at the base depth x = 0.0001 cm, very close to the junction.

Figure 8 shows that the quotient of doping level by carriers density increases very rapidly in normal illumination mode compared to moderate light concentration mode. This reflects the fact that under moderate light concentration, the carrier density is very close to the doping level.

Pelanchon et al. [18] also showed that under light concentration carriers density is close to doping level. This is due to carriers strong generation related to the illumination level. The cell then is in high injection condition [18] [23].

However, in normal illumination mode, the carrier density is very low compared to the doping rate.

As shown by Figure 9, the variations of the ratio of the doping level by carriers density were also plotted, in an intermediate operating point and for the two illumination modes at the same depth x = 0.0001 cm.

It emerges from Figure 9 that the ratio of the doping level by carriers density also increases rapidly in normal illumination mode in comparison with the

moderate concentration illumination mode. This result proves that in an intermediate operating point and under moderate light concentration, the doping level is close to carriers density. However, under normal illumination mode, carriers density is significantly low compared to the doping level.

Under the two illumination modes, the variations of the ratio of the doping level by carriers density were also plotted in the short-circuit situation as shown by Figure 10 below.

The results also show that in the short-circuit situation and under moderate light concentration, carrier density is close to doping level. However under normal illumination mode, carriers density is low compared to the doping level. These facts result in a greater effect of doping rate on carriers density under normal illumination mode compared to the moderate light concentration mode.