The electric field is created using the planar capacitor principle. Two aluminum conductors are used to be connected to the first silicon solar cells as presented on the Figure 1.

The both aluminum conductors connected to the first solar cells become the cathode and the anode of the planar capacitor. When the external solar cells are illuminated, they give the open circuit photovoltage expressed by the Equation (1) [16] .

V C O = lim S f → 0 V p h ( S f ) (1)

where V p h ( S f ) in Volt (V) depending of the junction dynamic velocity ( S f ) in cm∙s⁻¹, is the photovoltage given by the first solar cells. S f will be defined in the next. By this open circuit photovoltage, the electric field is created in the vacuum (air) presents between both aluminum conductors. The first solar cells are kept only in open circuit state. It has not contribution in the photocurrent and photovoltage of the final system. The main role of the fist solar cells is to provide the electrical field. The expression of this electric field is:

E ( d , n ) = n × V C O d (2)

where d(cm) is the distance between the both aluminum conductors, n the numbers of the solar cells used to create the electric field and E(d, n) in V.m⁻¹ the electrical field depending of the distance d(cm), of the number of the solar cells and of the different intrinsic parameters of the solar cell. The Figure 2 illustrates a schematic representation of the installation of the silicon solar cells under the uniform electrical field created between the both aluminum conductors.

The silicon solar cells are influenced by the uniform electrical field which is generated between the both aluminum conductors. That causes the apparition of the conduction current. The orientations of the solar cells between the two

aluminum conductors is chosen for the kind where the internal electric field between p-n junction and external electrical field will give a resulting component. This assumption is based on the study developed by Zerbo et al. [10] about electromagnetic field. One solar cell will be placed between the two aluminum conductors for one solar cell chosen for the creation of the external electric field. To solve the illumination problem of the solar cell and to avoid the perpendicularity between internal electrical field of the space charge region and external electrical field created, the height of the anode will be greater than the height of the cathode. It will be added a tilt of the solar cell between anode and cathode. These all means will put an angle between the both electrical fields. The variation of this angle allows to choose a better position giving an optimum output of the second solar cell. Some other work will study the real quantity of external electrical field which will influence the second solar cell. In the present work it will be assumed that the external electrical field which will be provided to the solar cells between anode and cathode will be optimum. The following subsection is going to provide the calculations of the density of the photocurrent, the photovoltage and the electric power.

The electric parameters of the silicon solar cell installed between the both aluminum conductors are obtained by solving of the transport equation of the exceeding minority carriers. It is given by the Equation (3) [17]

J n = e D n ⋅ g r a d δ ( x ) + e ⋅ μ n ⋅ E δ ( x ) (3)

D n ( cm ⋅ s − 2 ) is the electron diffusion coefficient, μ n ( cm 2 ⋅ V − 1 ⋅ s − 1 ) is the electron mobility coefficient caused by the electrical field E , e is the electronic charge and δ ( x ) in cm⁻³ which is the density of the exceeding minority carriers charge. This equation is solved in 1D approach. The assumptions of the 1D theory are described by Zerbo et al. [18] , Zerbo et al. (2014) [8] and Zoungrana et al. (2012) [12] . The resolution of this equation allows to find the different components of the Equation (3) [8] [18] . The equation 3 leads to the calculation of the equation of conservation of exceeding minority carriers charge called continuity equation presented in the Equation (4) [7] . This equation is calculated for the structure of the polycrystalline silicon solar cell considered as n + − p − p + [8] .

∂ δ ( x ) ∂ t = 1 e d i v ( J n ) + G n ( x ) − R n ( x ) (4)

With G n ( x ) in carriers∙s⁻¹∙cm⁻³ giving the exceeding minority carriers charge generation rate. To analyze the influence of the different light, this study will be done under the monochromatic illumination. R n ( x ) in carriers∙s⁻¹∙cm⁻³ is the exceeding minority carriers charge recombination rate after the generation.

G n ( x ) − R n ( x ) provides the quantity of the exceeding minority carriers charge which are conserved during the cross of the solar cell junction without being recombined [7] . The expression of the recombination rate is given by

R n ( x ) = δ ( x ) τ [19] .

This study is led under monochromatic illumination because this illumination allows to evaluate the effect of any light wavelength where photovoltaic conversion takes place only between 0.4 μm - 1.1 μm. The expression of G n ( x ) for monochromatic illumination is given by Mathieu H. [7] and Moliton A. [6] .

G n ( x ) = α ( λ ) ϕ 0 [ 1 − ρ ( λ ) ] e − α ( λ ) x (5)

α ( λ ) and ρ ( λ ) are respectively absorption and reflection coefficient at the wavelength λ ( μ m ) and ϕ 0 ( W ⋅ m − 2 ) is the incident photon flux. The monochromatic illumination parameters are provided by Green M. A. (2008) [20] . This work is realized in steady state conditions, the carriers charge density is not

function of the time. Then, ∂ δ ( x ) ∂ t = 0 . Defining δ(x) as the exceeding minority

carriers charge at x position which across the p − n + junction of the solar cell, the relevant differential equation is simple.

∂ 2 δ ( x ) ∂ x 2 + μ n E D n ∂ δ ( x ) ∂ x − δ ( x ) L n 2 = − G n ( x ) D n (6)

With L_n the minority carrier length. The solution of Equation (6) is the sum of two members as shown by the Equation (7)

δ ( x ) = δ O ( x ) + δ I ( x ) (7)

where δ O ( x ) is the density of minority carriers charge in excess for a solar cell at obscurity because it is calculated without illumination and its expression is δ O ( x ) = e β x [ A cosh ( γ x ) + B sinh ( γ x ) ] . With A and B are found by the solar cell boundaries conditions given by Zerbo et al. (2011, 2014) [8] [18] :

- At the solar cell p-n junction

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

- At the rear face

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

where S b provides the rear face recombination velocity. S f gives the expression of the junction dynamic velocity. It is the sum of two contributions: S f 0 which is the intrinsic junction recombination velocity related to the losses of carriers at the junction interface and S f j which is the junction dynamic velocity imposed by an external load resistance and defining the operating point of the cell [21] [22] [23] , therefore S f = S f 0 + S f j . To carry out this study, the ideal case will be assumed. Hence, S f 0 = 0 .

The second member of the Equation (7) is the contribution under solar illumination. Its expression is

δ I ( x ) = k e − α ( λ ) x with k = − α ( λ ) ϕ 0 [ 1 − R ( λ ) ] L n 2 D n [ α 2 ( λ ) L n 2 − 1 ] . The density of the carrier’s

minority in excess will be depending of the S f , S b , λ , x , d and the other intrinsic parameters of the silicon semi-conductor. The calculation of the exceeding minority carrier charge allows to find the electrical parameters. These electric parameters of the solar cells are presented in the next subsection.

The first electrical parameter of the silicon solar cell is the density of photocurrent. By application of the Fick law, it is expressed as [8] :

J p h ( S f , S b , λ , d ) = e [ D n ∂ δ ( x , S f , S b , λ , d ) ∂ x | x = 0 + μ n ⋅ E ⋅ δ ( 0 , S f , S b , λ , d ) ] (10)

The second electric parameter is the photovoltage which is expressed in Equation (11) by application of the Boltzmann law [8] .

V p h ( S f , S b , λ , d ) = V T ln ( δ ( 0 , S f , S b , λ , d ) n 0 + 1 ) (11)

where V_T is the thermal photovoltage. At T = 300 K, V_T = 0.026 V, n 0 = N B n i 2

with N B = 10 17   cm − 3 , the doping level and n i = 1.45 × 10 10   cm − 3 , the intrinsic carriers’ density at thermal equilibrium. This subsection will finish by the electric power which is the arithmetical multiplication between the density of photocurrent and the photovoltage.

P p h ( S f , S b , λ , d ) = V p h ( S f , S b , λ , d ) × J p h ( S f , S b , λ , d ) (12)

The maximum electric power is found by using the curves of the electric power in function of the junction dynamic velocity obtained with Mathcad 15 software. The conversion efficiency is the ratio of the maximum electric power ( P M P P ( W ⋅ m − 2 ) ) by the incident power ( P a b ( W ⋅ m − 2 ) ) from solar illumination absorbed in the base of the solar cell. The conversion efficiency for monochromatic is expressed by the Equation (13) [7]

η = P M P P P a b (13)

With P a b = α ( λ ) ϕ 0 [ 1 − R ( λ ) ] h c λ where h = 6.632 × 10 − 34   J ⋅ s is the Planck

constant and c = 3 × 10 10   cm ⋅ s − 1 is the vacuum light speed. The next section will be concerned to the results and discussions of this work.

The electrical field created between the both aluminum conductors is depending of solar wavelength, the distance between the cathode and the anode, the number of the silicon solar cells which are using to create the open circuit voltage and the intrinsic parameters of the solar cell. The Figure 3 presents the behavior of the electrical field in function of the number of the silicon solar cells.

The electrical field increases for the increase of the solar cells number. In fact, the increase of the solar cells number causes the increase of the open circuit photovoltage. One of the purposes of this work is to provide an improvement of the conversion efficiency by reducing of the silicon matter. Consequently, the creation of the electrical field must use less solar cells reducing the semiconductor matter used for the manufacture of the solar cells. The Figure 4 gives the evolution of the electrical field in function of the distance between the both aluminum conductors.

The electrical field is very strong for the short distances and it is weak for the greater distances between anode and cathode of the planar capacitor. The physics signification is that the short distances are causing higher electrical field. So, it will be better to have short distance between the aluminum conductors in order to obtain higher electrical field. The evolution of the electric field in function of the solar illumination wavelengths is given by the Figure 5.

The evolution of the electric field in function of the solar illumination wavelength has the solar spectrum characteristic because of the solar illumination atmospheric absorption. The short wavelength causes more storage of the generated electronic near the junction i.e. p side. Moreover, the electrical field is decreasing with the increase of wavelength. Hence, to simulate the electrical field effect on the silicon solar cells installed into influence area of this electrical field, one solar cell is chosen for the illumination wavelength of 0.70 μm with the variation of the distances. The following subsection will present the influence of the electrical field on one silicon solar cell’s electrical parameters in 1D approach.

The electric parameters treated in this point concern the photocurrent, the photovoltage, the electric power and the conversion efficiency. The Figure 6 is the

graphic representation of the density of photocurrent in function of the dynamic velocity and the variation of the distances.

From low values of the junction dynamic velocity to high values of this velocity, the solar cell operating point is going from open circuit state to short circuit situation. It appears the increase of the photocurrent from open circuit to short circuit situation. For the intermediate level between open circuit and short circuit situations, the photocurrent presents an improvement. This state is the real operating point of the solar cell. The photocurrent at this state is improving also for the short distance i.e. for the high value of the electrical field. But the important value obtained at the open circuit state can cause the heating by Joule effect of the junction of the solar cell. It will be better to minimize this open circuit photocurrent which presents only the loss of the generated exceeding minority carriers at the junction. The photovoltage is represented on the Figure 7 in function of the junction dynamic velocity for different distances.

The photovoltage is greatest in the open circuit state. It decreases when the solar cell is operating in short circuit. The photovoltage is slightly insensitive to the electric field. But the values of the photovoltage is decreasing with the short distances between the both aluminum conductors. This reduction is the consequence of the crossing of the p - n junction by the surplus minority carriers. We observe that the increase of the photocurrent is more important than the decrease of the photovoltage. The electric power is presented on the Figure 8 for different distances when the junction dynamic velocity varies.

For the short circuit situation, the solar cell does not deliver the electric power. But for the open circuit situation there is an electric power. This power increases for the short distances between the both aluminum conductors. The electric power in open circuit causes the heating of the junction p-n of the solar cell by Joule effect. So, for this dispositive of improvement of the solar cell efficiency, the air free convection can be used to evacuate the heat coming from this open circuit electric power. Then, the maximum electric power (P_MPP) is improving for the short distances. The Table 1 presents the P_MPP collected from the Figure 8 and the conversion efficiency calculated using the Equation (11).

From 3.5 cm to 0.5 cm i.e. the electric field passes from 0.198 V∙cm⁻¹ to 1.387 V∙cm⁻¹, the electric power increases from 20.834 mW∙cm⁻² to 21.477 mW∙cm⁻² and the conversion efficiency increases from 24.5% to 25.2%. In experimental study, E. Serafettin [13] finds in red color wavelength of a monochromatic illumination that the open circuit voltage decreases with the increase of the electric field but the short circuit current does not change. However, we observe an increase of 0.7% about conversion efficiency. This efficiency is better improved compared at efficiency of the silicon polycrystalline solar cell which does not influence by the electric field [7] . This increase can improve with the reduction of distance between the both aluminum conductors. It can also improve by the increase of the number of the solar cells used for the creation of the electrical field and choosing the good angle between internal and external electric fields. Moreover, the short distances values bring the operating point of the solar cell in vicinity of the open circuit state.