In Figure 1, an n+-p-p+ type of a bifacial silicon solar cell [17] [18], in 1D, is represented:

With d and H respectively the emitter thickness and the solar cell base thickness

We consider a constant magnetic field vector B → [19] [20] [21], that is applied following the y-axis perpendicularily to the solar cell plane.

So, the expression of the photoconductivity is given by [22]:

σ p h = q μ o δ n 1 + μ o 2 B 2 (1)

where q is the elementary charge of electron and μ o the quasi-static mobility of the minority carriers without magnetic field.

One can see, in Equation (1) that the expression of the photoconductivity depends on the minority carriers’ density δ n and magnetic field. Thus, in steady-state, the minority carriers’ density can be determined by resolving the following continuity equation:

D n ∂ 2 δ n ∂ x 2 − δ n τ = − G (2)

where τ is the average lifetime of the minority carriers in the base; D_n is the minority carriers’ diffusion coefficient which expression is given by:

D n = D o 1 + μ o 2 B 2 (3)

With D_o being the minority carriers diffusion coefficient without applied magnetic field; G the minority carriers generation rate [23] at the position x.

The expression of the minority carriers’ generation rate, can be given as:

G = η ∑ i = 1 3     a i e − b i x (4)

with η being the solar number, a_i and b_i are coefficients deduced from modelling of the generation rate considered over all solar radiation spectrum under AM 1.5.

The solution of Equation (2) can be written as:

δ n = A e − x L n + C e x L n − ∑ i = 1 3 a i L n 2 e − b i x D n ( b i 2 L n 2 − 1 ) (5)

with:

L n = D n τ (6)

and:

( b i 2 L n 2 − 1 ) ≠ 0 (7)

where L_n is the minority carriers diffusion length in the base.

The coefficients A and C are determinated by the following boundary conditions [24] [25]:

- at the junction (x = 0)

D n ⋅ ∂ δ n ∂ x | x = 0 = S f ⋅ δ n | x = 0 (8)

- at the rear side (x = H)

D n ⋅ ∂ δ n ∂ x | x = H = − S b ⋅ δ n | x = H (9)

where Sf and Sb are respectively the junction recombination velocity that defines the solar cell operating point and the back surface recombination velocity.

- The photocurrent density is related to the gradient of the photogenerated minority carriers at the junction and is given by:

J p h = q D n ∂ δ n ∂ x | x = 0 (10)

- The storage of the photogenerated minority carriers to the vicinity of the junction creates a photovoltage whose expression is given by the following Boltzmann equation:

V p h = V T ln ( 1 + N B n i 2 δ n ( 0 ) ) (11)

With:

V T = K T / q (12)

where V_T is the thermal voltage at temperature T = 300 K; N_B is the doping acceptor atoms in the base; n_i is the intrinsic concentration of the minority carriers at temperature T and K is the Boltzmann constant.

- The solar cell output power is the energy supplied, at a time t, by the solar cell. Its expression is given by:

P p h = J p h ⋅ V p h (13)

In Figure 2, the photoconductivity versus the junction recombination velocity, is represented for different values of the magnetic field:

In Figure 2, we obtain the same behavior for the four curves of the photoconductivity versus junction recombination velocity. Considering curve 1, without an applied magnetic field, in the interval of the junction recombination velocity [0 cm·s⁻¹; 2 × 10² cm·s⁻¹] where the solar cell operates in an open circuit situation, we note a high value of the photoconductivity that stays constant. This is related to a great photogeneration of the minority carriers around the junction where they cannot cross since they have not enough kinetic energy. However, in the interval [2 × 10² cm·s⁻¹; 5 × 10⁵ cm·s⁻¹] that corresponds to a variable operating point of the solar cell, the photoconductivity decreases because the minority carriers gradually gain a kinetic energy to across the junction. When the solar cell operates in a short-circuit situation, for a junction recombination velocity S f ≥ 5 × 10 5   cm ⋅ s − 1 , the photoconductivity is lesser than of in open circuit one. In this situation, the minority carriers have enough kinetic energy to across the junction and there is no more new photogenerated carriers.

The I-V characteristics, for different values of the magnetic field, are given in Figure 3:

The four curves in Figure 3, exhibit the same behaviour. For a low photovoltage, the photocurrent is maximum and corresponds to the short-circuit photocurrent: there is no significant storage of the photogenerated minority carriers in the vicinity of the junction since the majority of these carriers have acrossed the junction. This situation occurs when the junction recombination velocity is very high while the load is low. However, in the open circuit situation of the solar cell, there is a significant storage of the minority carriers in the vicinity of the junction; this implies a high photovoltage which corresponds to the open circuit photovoltage and where the photocurrent is closed to zero. With a magnetic field, the amplitude of the photocurrent decreases while the photovoltage increases. We note that the magnetic field slows down the photogenerated minority carriers in the vicinity of the junction or deviates them from their initial direction. We note that the operating point of the solar cell depends on the load and the magnetic field. That is why this operating point will impact on the output power delivered by the solar cell.

In Figure 4, the output power versus the photovoltage, for different values of the magnetic field, is represented:

The curves of the P-V characteristics, in Figure 4, show similar profiles. For a given curve and its corresponding value of magnetic field, we note three zones:

1) a first zone which corresponds to the short-circuit situation of the solar cell and where the power increases linearly with the photovoltage. The fit of the curve give positive slope that corresponds to a short-circuit photocurrent;

2) a second zone corresponding to the open circuit situation of the solar cell where the power decreases with the photovoltage: it is like a solar cell that loses an energy and cannot work correctly since the load is high;

3) a third zone which is curvated and located between the two zones mentioned above, where the solar cell operates around the optimal power point.

The increase of the magnetic field decreases the power delivered by the solar cell on the load.

From I-V and P-V characteristics, we propose a method for determining the optimal power operating point and the corresponding junction electric field.

We give, at first, a relationship between the photocurrent density and the electric field, as followed:

J p h = σ p h ⋅ E (14)

where E is the electric field.

In Figure 5, a graphical determination method of the optimal operating point, is presented [26].

The I-V and P-V characteristics, in Figure 5, are given without an applied magnetic field and for an operating point that varies from the open circuit situation to the short-circuit one. The operating point is as function of the junction recombination velocity which is linked to the load. A high load corresponds to a low junction recombination velocity while a feeble load corresponds to a high junction recombination velocity. Therefore, there is a load that is linked to an operating point in which the solar cell delivers a maximum power P m a x . This operating point of the solar cell, which corresponds to the maximum power, is called the optimal operating point and is showed in Figure 5 by the point P. The P point projection on the photovoltage axis, gives us the optimal photovoltage V p h o p . Its extension on the I-V characteristic, gives a point whose projection on the photocurrent axis allows us to obtain the optimal photocurrent J p h o p . With the obtained values of V p h o p and J p h o p , we deduce the optimal junction recombination velocity S f o p which corresponds to the optimal operating point of the solar cell.

In Figure 6, the optimal junction recombination velocity S f o p and the optimal photoconductivity σ p h o p , are determined:

The photoconductivity and the photocurrent density are represented on both sides, in Figure 6, according to the junction recombination velocity which defines the operating point of the solar cell. From the curve of the photocurrent density versus the junction recombination velocity, the optimal photocurrent J p h o p gives a point whose projection on the junction recombination velocity axis, allows us to obtain the value of S f o p . With the curve of the photoconductivity versus the junction recombination velocity, we determine the optimal photoconductivity σ p h o p . From the values of J p h o p and σ p h o p that are obtained, we deduce the value of the junction electric field E for a given optimal operating point of the solar cell.

These steps of determining all of the optimal values corresponding to an optimal operating point, are considered for different magnetic field values. What permits us to give Table 1 below:

Both the optimal photovoltage and the junction electric field increase with the magnetic field because the magnetic field slows down and stores the photocreated minority carriers in the vicinity of the junction. The optimal photocurrent density, the maximum power, the optimal junction recombination velocity and the optimal photoconductivity decrease with the magnetic field. Therefore, we note that, a high photoconductivity corresponds to an ohmic solar cell in which the electric field is weak since the recombination of the minority carriers in the bulk of the solar cell becomes weak. However, a low photoconductivity corresponds to a nonohmic solar cell in which the electric field and the minority carriers’ recombination in bulk are high. From these results we note a degradation of the intrinsic parameters of the solar cell when a magnetic field is applied.