· AC carrier generation rate G ( x , t ) is given by the relationship:

G ( x , t ) = g ( x ) ⋅ e − j ω t (3)

The steady state generation rate is expressed as:

g ( x ) = α ( λ ) ⋅ I 0 ( λ ) ⋅ ( 1 − R ( λ ) ) ⋅ e − α ⋅ ( λ ) . x (4)

I₀ is the incident flux, while R(λ) and α(λ) are respectively, the reflection and absorption coefficients in the silicon material

By replacing Equations (2) and (3) in Equation (1), the continuity equation for the excess minority carriers’ density in the base is reduced to the following relationship:

∂ 2 δ ( x , ω ) ∂ x 2 − δ ( x , ω ) L 2 ( ω ; B ) = − g ( x ) D ( ω , B ) (5)

L ( ω , B ) is the complex diffusion length, under magnetic field and frequency modulation, of excess minority carriers in the base given by:

L ( ω , B ) = D ( ω , B ) τ 1 + j ω τ (6)

τ is the excess minority carriers lifetime in the base.

The solution of Equation (5) is:

δ ( x , ω , B , λ ) = A ⋅ cosh [ x L ( ω , B ) ] + B ⋅ sinh [ x L ( ω , B ) ] + K ⋅ e − α ⋅ x (7)

With

K = α ( λ ) ⋅ I 0 ⋅ ( 1 − R ( λ ) ) ⋅ [ L ( ω , B ) ] 2 D ( ω , B ) [ L ( ω , B ) 2 ⋅ α ( λ ) 2 − 1 ] (8)

and

( L ( ω , B ) 2 ⋅ α ( λ ) 2 ≠ 1 ) (9)

Coefficients A and B are determined through the boundary conditions:

· At the junction (x = 0)

D ( ω , B ) ∂ δ ( x , ω , B , λ ) ∂ x | x = 0 = S f ⋅ δ ( x , ω , B , λ ) D ( ω , B ) | x = 0 (10)

· On the back side in the base (x = H)

D ( ω , B ) ∂ δ ( x , ω , B , λ ) ∂ x | x = H = − S b ⋅ δ ( x , ω , B , λ ) D ( ω , B ) | x = H (11)

Sf and Sb are respectively the recombination velocities of the excess minority carriers at the junction and at the back surface. The recombination velocity Sf reflects the charge carrier velocity of passage at the junction, in order to participate in the photocurrent. It is then imposed by the external load which fixes the solar cell operating point [15]. It has an intrinsic component which represents the carrier losses associated with the shunt resistor in the solar cell electrical equivalent model [20] [44] [45] [46]. The excess minority carrier recombination velocity Sb on the back surface is associated with the presence of the p⁺ layer which generates an electric field for throwing back the charge carrier toward the junction [8] [17] [18].

The representation of AC photocurrent density according to the junction recombination velocity of minority carriers shows that, for very large Sf, a bearing sets up and corresponds to the AC short-circuit current density (Jphsc) [44] [45] [46]. So, in this junction recombination velocity interval, we can write [16]:

∂ J p h ( S f , S b , ω , B , λ ) ∂ S f | S f ≥ 10 5 cm ⋅ s − 1 = 0 (12)

The solution of Equation (12) leads to the expressions of AC recombination velocity in the rear surface [41] [44] [45] given by Equations (13) and (14):

S b 1 ( ω , B ) = − D ( ω , B ) L ( ω , B ) ⋅ tanh ( H L ( ω , B ) ) (13)

S b 2 ( ω , B , λ ) = D ( ω , B ) L ( ω , B ) ⋅ [ α ( λ ) ⋅ L ( ω , B ) ⋅ ( exp ( − α ( λ ) ⋅ H ) − cosh ( H L ( ω , B ) ) + sinh ( H L ( ω , B ) ) ) exp ( − α ( λ ) ⋅ H ) − cosh ( H L ( ω , B ) ) + α ( λ ) ⋅ L ( ω , B ) ⋅ sinh ( H L ( ω , B ) ) ] (14)

Minority carrier diffusion coefficient is influenced by external conditions applied to solar cell. It expression is dependent of parameter such as, temperature [52], magnetic field [10], charged particles fluence and intensity of irradiation [53], and others combined external conditions [42] [54] [55]. For this case under study D ( ω , B ) is the complex diffusion coefficient of excess minority carrier in the base under magnetic field and frequency modulation. Its expression is given by the relationship [40] [41] [43]:

D ( ω , B ) = D ⋅ [ ( 1 + τ 2 ⋅ ( ω c 2 + ω 2 ) ) 4 ⋅ ω 2 ⋅ τ 2 + [ 1 + τ 2 ( ω c 2 − ω 2 ) ] 2 − j ⋅ ω ⋅ τ ⋅ ( 1 − τ 2 ( ω c 2 − ω ) 2 ) 4 ⋅ ω 2 ⋅ τ 2 + [ 1 + τ 2 ( ω c 2 − ω 2 ) ] 2 ] (15)

With

ω c = q ⋅ B m e ∗ (16)

is the cyclotron frequency, m e ∗ the electron mass, and q the elementary charge.

The diffusion coefficient is plotted in Figure 2. It decreases with the modulation frequency for different magnetic field values. We observe the appearance of resonance peaks [41] [43] for intensities of the upper magnetic field 10⁻⁷ Tesla, giving the maximum diffusion coefficient (Dmax) value.

Two areas of very weak magnetic fields emerge in Figure 2, with semi-logarithmic scale of frequency:

· a first zone (ω < 10⁴ rad/s) where the complex diffusion coefficient is virtually constant corresponding to the quasi-static regime, which ends at the point of the cut-off frequency [24];

· a second zone (from 10⁴ rad/s to 10⁸ rad/s) where the complex diffusion coefficient decreases drastically with pulsation, marking the frequency dynamic regime, or excess minority carriers appear frozen, because of their low relaxation time.

The application of a magnetic field to the solar cell in this frequency region, reveals a third area where the amplitude of the complex diffusion coefficient produces a peak, at a given frequency called resonance frequency [43].

This resonance phenomenon is obtained when the modulation frequency is equal to the cyclotron frequency [41] [43] (i.e. frequency of the electron in its orbit in the presence of a magnetic field).

Thus, the frequency of modulation with or without the application of a magnetic field to the solar cell, causes the reduction of the effective diffusion of excess minority carriers.

Also the study of the solar cell in frequency modulation, requires a choice of the values of the magnetic field (Table 1) for an optimal response through the diffusion coefficient (Dmax) to produce large photocurrent. Then such data will be used for the next calculations.

Figure 3 gives the profile of the two expressions of AC back surface recombination velocity for different ringing frequencies inducing Dmax values, versus thickness of the base of the solar cell, under long wavelength (α(λ) = 6. 2 cm⁻¹). The technique [47] [48] [56] of the intercept point of the curves yields the optimum thickness of the base, and allow the establishment of data in Table 1.

As obtained in previous study [56] [57] [58] [59], the optimum thickness is an increasing straight line versus diffusion coefficient (Figure 4). Thus solar cell with large diffusion coefficient can be used for thick base solar cell manufacturing.

The relationship obtained is expressed as:

H o p ( cm ) = 1.8 × 10 − 4 × D max ( cm 2 /s ) + 0.0063 (17)

From obtained diffusion length in Table 1, base optimum thickness is represented in Figure 5.

The representation is also an increase strait line [50] expresses as:

H o p ( cm ) = 0.39 × L max ( cm ) + 0.0042 (18)

Figure 6 and Figure 7, give the representation of the optimum thickness versus (ωr) and magnetic field (B) respectively. The plots of Hopt, are similar, because of the ratio ( q / m e ∗ ) in Equation (6), then the effects of both the resonance frequency and the magnetic field impose thin thicknesses base.