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In this work, we propose a method to determinate the optimum thickness of a monofacial silicon solar cell under irradiation. The expressions of back surface recombination velocity depending the damage coefficient (
*kl*) and irradiation energy (
ø_{p}) are established. From their plots, base optimum thickness is deduced from the intercept points of the curves. The short-circuit currents
*Jsc*0 and
*Jsc*1 corresponding to the recombination velocity
*Sb*0 and
*Sb*1 are determinated and a correlation between the irradiation energy, the damage coefficient and optimum thickness of the base is established.

Studies of the effect irradiation of charged particles on photovoltaic solar cells have always preoccupied scientific minds around different aspects and parameters. The study of the displacement of atoms during irradiation was first presented in 1956 [

On the other side, authors have studied the effect of irradiation of these particles on the solar cells (mono or bifacial [

Also, on determining phenomenological parameters i.e. excess minority carrier recombination velocity [

The incident illumination wavelength under steady [

This work deals with a method, to determinate the optimum thickness of a silicon solar cell under the effect of irradiation charged particles. Then, from the excess minority carrier density continuity equation in the base, expressions of photocurrent density [

^{+}-p-p^{+}) [

When the solar cell is properly illuminated by a static polychromatic light, all the processes for generation, recombination in the bulk and surfaces and diffusion of excess minority carrier in the base are governed by the following continuity equation:

D ( k l , ϕ p ) ∂ 2 δ ( x , k l , ϕ p ) ∂ x 2 − δ ( x , k l , ϕ p ) τ + G ( x ) = 0 . (1)

δ ( x , k l , ϕ p ) represents the excess minority carrier density in the base of the solar cell at the x-position, dependent of the irradiation energy.

D ( k l , ϕ p ) and τ are respectively the diffusion coefficient of the electrons in the base under irradiation and the lifetime of the excess minority carrier in the base of the solar cell linked by the following Einstein relationship:

[ L ( k l , ϕ p ) ] 2 = τ × D ( k l , ϕ p ) , (2)

with L ( k l , ϕ p ) the diffusion length of the excess minority carrier in the base as a function of the irradiation energy flux ( ϕ p ) and the damage coefficient intensity (kl). It also represents the average distance traveled by the minority carrier before their recombination in the base under irradiation. It is related to the diffusion length before irradiation by the following empirical relation [

L ( k l , ϕ p ) = 1 ( 1 L 0 2 + k l ⋅ ϕ p ) 1 / 2 , (3)

where:

L 0 is the diffusion length of the excess minority carriers in the base before irradiation.

ϕ p is the irradiation energy flux.

kl is the damage coefficient intensity.

➢ G (x) is the excess minority carrier generation rate [

K l G ( x ) = n ⋅ ∑ i = 1 3 a i e − b i ⋅ x . (4)

• n is the number of sun or illumination concentration [

• The coefficients a_{i} and b_{i} take into account the tabulated values of solar radiation and the dependence of the absorption coefficient of silicon with the wavelength [

The carrier density is subjugated to the following boundary conditions:

1) At the junction: emitter-base (x = 0)

D ( k l , ϕ p ) ∂ δ ( x , k l , ϕ p ) ∂ x | x = 0 = S f ⋅ δ ( 0 , k l , ϕ p ) . (5)

2) At the back side (x = H)

D ( k l , ϕ p ) ∂ δ ( x , k l , ϕ p ) ∂ x | x = H = − S b ⋅ δ ( H , k l , ϕ p ) . (6)

Sf is the excess minority carrier recombination velocity at the junction and also indicates the solar cell operating point [

Sb is the excess minority carrier recombination velocity on the back side surface [

It is the consequence of the electric field produced by the p-p^{+} junction and characterizes the behavior of the density of the excess carrier at this interface. It yields to send back to the emitter-base interface the minority carriers generated near the rear face.

The resolution of the differential Equation (1) gives the expression of the excess minority carrier density in the base as:

δ ( x , k l , ϕ p ) = A ⋅ cosh [ x L ( k l , ϕ p ) ] + B ⋅ sinh [ x L ( k l , ϕ p ) ] − ∑ K i ⋅ e − b i ⋅ x , (7)

where:

K i = − n × [ L ( k l , ϕ p ) ] 2 × a i D ( k l , ϕ p ) ( b i 2 × L ( k l , ϕ p ) 2 − 1 ) . (8)

The expressions of, A and B are determined from the following boundary conditions and are given by:

A = L ( k l , ϕ p ) × K i [ D ( k l , ϕ p ) × S b ( k l , ϕ p ) − D 2 ( k l , ϕ p ) × b i ] e − b i ⋅ H + χ ( k l , ϕ p ) Y × sinh ( H L ( k l , ϕ p ) ) + X × cosh ( H L ( k l , ϕ p ) ) , (9)

χ ( k l , ϕ p ) = ( D ( k l , ϕ p ) × cosh ( H L ( k l , ϕ p ) ) + L ( k l , ϕ p ) × S b ( k l , ϕ p ) × sinh ( H L ( k l , ϕ p ) ) ) × [ S f + D ( k l , ϕ p ) × b i ] , (10)

Y = [ L 2 ( k l , ϕ p ) × S b ( k l , ϕ p ) × S f + D 2 ( k l , ϕ p ) ] , (11)

X = D ( k l , ϕ p ) × L ( k l , ϕ p ) × [ S f + S b ( k l , ϕ p ) ] , (12)

B = L ( k l , ϕ p ) × K i L ( k l , ϕ p ) × S f × [ S b ( k l , ϕ p ) − D 2 ( k l , ϕ p ) × b i ] e − b i ⋅ H + ζ ( k l , ϕ p ) Y × sinh ( H L ( k l , ϕ p ) ) + X × cosh ( H L ( k l , ϕ p ) ) , (13)

ζ ( k l , ϕ p ) = ( D ( k l , ϕ p ) × sinh ( H L ( k l , ϕ p ) ) + L ( k l , ϕ p ) × S b ( k l , ϕ p ) × cosh ( H L ( k l , ϕ p ) ) ) × [ S f + D ( k l , ϕ p ) × b i ] . (14)

The expression of the photocurrent density is given by the relation:

J p h ( S f , H , k l , ϕ p ) = q ⋅ D ( k l , ϕ p ) ⋅ [ ∂ δ ( S f , x , H , k l , ϕ p ) ∂ x ] | x = 0 . (15)

For polychromatic illumination we obtain:

J p h ( S f , H , k l , ϕ p ) = q ⋅ D ( k l , ϕ p ) [ B ( S f , H , k l , ϕ p ) L ( k l , ϕ p ) + ∑ i = 1 3 K i ⋅ b i ] . (16)

This photocurrent density is constant for the large values of the carrier recombination rate at the junction between 3 × 10^{3} ≤ Sf ≤ 6 × 10^{6} cm/s [

The Sb expression is obtained from the derivative of the photocurrent density for large Sf values [

[ ∂ J p h ( S f , k l , ϕ p ) ∂ S f ] S f ≻ 4 × 10 4 cm ⋅ s − 1 = 0 . (17)

The resolution of this equation yields to establish the following expressions of the excess minority carrier recombination velocity at the rear face, S b 0 ( H , k l , ϕ p ) and S b 1 ( H , k l , ϕ p , b i ) :

S b 0 ( H , k l , ϕ p ) = − D ( k l , ϕ p ) L ( k l , ϕ p ) × tanh ( H L ( k l , ϕ p ) ) . (18)

It represents the intrinsic recombination velocity at the p-p^{+} junction of the minority carrier.

S b 1 ( H , k l , ϕ p ) = D ( k l , ϕ p ) L ( k l , ϕ p ) ⋅ ∑ i = 1 3 L ( k l , ϕ p ) ⋅ b i ( e b i ⋅ H − cosh ( H L ( k l , ϕ p ) ) ) − sinh ( H L ( k l , ϕ p ) ) − L ( k l , ϕ p ) ⋅ b i ⋅ sinh ( H L ( k l , ϕ p ) ) + cosh ( H L ( k l , ϕ p ) ) − e b i ⋅ H . (19)

It represents the recombination rate at the rear face influenced by the effect of the absorption of light in the material through the coefficients (b_{i}) and leads to a generation rate.

From the expression (16), we represent in Figures 2-4 the profiles of the photocurrent density as a function of excess minority carrier recombination velocity at the junction for different values of the irradiation energy, the damage coefficient and the base thickness.

On this figure, we note three different parts on the profile of the photocurrent density:

• The photocurrent density is almost zero for low values of the recombination velocity (Sf < 200 cm/s), the solar cell operates then in open circuit.

• Then for 200 cm/s < Sf < 4 × 10^{4} cm/s, the photocurrent density increases with the recombination velocity to reach a maximum of amplitude. This shows that the excess minority carrier has acquired sufficient energy to cross the junction.

• For Sf > 4 × 10^{4} cm/s, the photocurrent density is maximum and constant, corresponding to the short-circuit photocurrent. The figure also shows that as the irradiation energy and damage coefficient increases, the maximum amplitude of the photocurrent density decreases. This phenomenon can be explained by the interaction of the irradiating particles with the silicon material which increases and reduces the excess minority carrier density.

In

ϕ p (MeV) | 100 | 130 | 160 | 190 | 220 |
---|---|---|---|---|---|

D (cm^{2}/s) | 30.063 | 29.204 | 28.375 | 27.546 | 26.688 |

H (cm) | 0.0127 | 0.0123 | 0.0119 | 0.0116 | 0.0113 |

Sb0 (cm/s) | 5377.5 | 5287.5 | 5202.5 | 5127.5 | 5042.5 |

Sb1 (10^{5} cm/s) | 2.222 | 2.150 | 2.082 | 2.022 | 1.954 |

Jsc0 (A/cm^{2}) | 0.0353 | 0.0353 | 0.0353 | 0.0354 | 0.0354 |

Jsc1 (A/cm^{2}) | 0.0266 | 0.0266 | 0.0265 | 0.0265 | 0.0264 |

The correlation between the irradiation energy and optimum thickness of the base is established:

H ( cm ) = a × ϕ p 2 − b × ϕ p + c . (20)

With: a = 2 × 10 − 8 cm / MeV , b = 2 × 10 − 5 cm / MeV , c = 0.014 cm .

In

kl (cm^{−2}/MeV) | 5 | 7 | 9 | 10 | 11 |
---|---|---|---|---|---|

D (cm^{2}/s) | 30.079 | 28.895 | 27.895 | 27.368 | 26.789 |

H (cm) | 0.0126 | 0.0121 | 0.0117 | 0.0115 | 0.0113 |

Sb0 (cm/s) | 5377.5 | 5267.5 | 5165 | 5107.5 | 5052.5 |

Sb1 (10^{5} cm/s) | 2.222 | 2.134 | 2.052 | 2.006 | 1.962 |

Jsc0 (A/cm^{2}) | 0.0353 | 0.0353 | 0.0353 | 0.0354 | 0.0354 |

Jsc1 (A/cm^{2}) | 0.0266 | 0.0265 | 0.0265 | 0.0264 | 0.0264 |

The correlation between the damage coefficient and optimum thickness of the base is established:

H ( cm ) = a × ϕ p 2 − b × ϕ p + c . (21)

With: a = 6 × 10 − 6 MeV ⋅ cm / cm − 2 , b = 3 × 10 − 4 MeV ⋅ cm / cm − 2 , c = 0.014 cm .

In this work, we have proposed a method for determining the optimal thickness of a monofacial solar cell subjected to the effect of the irradiation of charged particles. The expressions of the excess minority carrier in the base and the photocurrent density have been proposed. Calibration curves of the photocurrent density were plotted versus the junction recombination rate for different values of the irradiation energy and the damage coefficient.

The expressions of the excess minority carrier recombination rates at the back face have been deduced from the derivative of the photocurrent density with respect to the excess minority carrier recombination velocity at the junction, when this tends to large values corresponding to the short-circuit situation of the solar cell.

The graphical resolution of the equations of recombination rates at the rear face yields to obtain at the points of intersection of the curves the value of the optimum thickness for a given irradiation energy and a given damage coefficient in the vicinity of the short-circuit current.

Finally, a correlation between the irradiation energy, the damage coefficient and the optimal thickness of the solar cell has been established.

The authors declare no conflicts of interest regarding the publication of this paper.

Ba, M.L., Thiam, N., Thiame, M., Traore, Y., Diop, M.S., Ba, M., Sarr, C.T., Wade, M. and Sissoko, G. (2019) Base Thickness Optimization of a (n^{+}-p-p^{+}) Silicon Solar Cell in Static Mode under Irradiation of Charged Particles. Journal of Electromagnetic Analysis and Applications, 11, 173-185. https://doi.org/10.4236/jemaa.2019.1110012