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In this work, we study the characteristics I-V and P-V of a silicon solar cell as well as its fill factor, its electrical power from the optimum thickness obtained in the base under variation of the irradiation energy flow of charged particles. The recombination velocity at the junction corresponding to the maximum power point was also deduced.

Authors have studied the electrical parameters of the solar cell namely the fill factor, the conversion efficiency, the power, the I-V and P-V characteristics under the Influence of Irradiation [

Our study is to determinate these electrical parameters from the optimal base thickness of the solar cell under variation of the irradiation energy flow and extracting the values of the recombination velocity at the junction, corresponding to the maximum power.

^{+}-p-p^{+} silicon solar cell [^{+}) zone covered by the front grids. Then comes the space charge region (SCR), which is formed by migration of the majority charges coming from the two semiconductors (n^{+} and p), according to the principle of Helmotz or compensation law. They are formed as fixed charges that delimit a space, where there is an intense electric field, which will allow the dissociation of photogenerated electron-hole pairs, and their acceleration to the deficit areas in corresponding charge. The (p) zone is doped with boron atoms and represents the larger thickness base (170 - 300 µm). It is the zone of pair creation (electron-hole), the most important, which justifies, the interest of its study in a solar cell. The (p^{+}) zone over doped in boron atoms, allows the creation of another rear space charge region, where the created electric field (Back surface Field) will allow the minority carriers of the (p) zone to be pushed back to the junction, to be then collected and participated in the photocurrent.

The solar cell thus achieved (

G ( x ) = n × ∑ i = 1 3 a i × e − b i ⋅ x (1)

δ ( x , ϕ p , k l ) follows the charge transport equation given by (Equation (2)):

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

It is accompanied by the Equation (3) and Equation (4) specifying the conditions at the (p) base boundaries in the 1D model, which define, the Sf [^{+}/p) at x = 0, and in the rear (p/p), at x = H.

D ( k l , ϕ p ) ∂ δ ( x , k l , ϕ p ) ∂ x | x = 0 = S f × δ ( 0 , k l , ϕ p ) (3)

D ( k l , ϕ p ) ∂ δ ( x , k l , ϕ p ) ∂ x | x = H = − S b × δ ( H , k l , ϕ p ) (4)

The diffusion term influenced by the irradiation conditions of the solar cell, is given by the following empirical relationship [

L ( k l , ϕ p ) = 1 ( 1 L 0 2 + k l × ϕ p ) 1 / 2 (5)

where:

L 0 is the diffusion length of the excess minority carriers in absence of irradiation energy flux ( ϕ p ). L ( k l , ϕ p ) is 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) and which may be related to Einstein’s relationship by:

[ L ( k l , ϕ p ) ] 2 = τ × D ( k l , ϕ p ) (6)

D ( k l , ϕ p ) and τ are respectively the diffusion coefficient and lifetime of the electrons in the base of the solar cell under irradiation.

The continuity Equation (2) solution is provided by:

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, coefficients A and B, will be obtained by use of Equation (3) and Equation (4).

1) Fick’s charged particle law establishes the photocurrent density of minority carriers, derived from the base by the following relationship:

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 = 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 (8)

2) At the high values of recombination velocity at the junction, it is established that [

[ ∂ J p h ( S f , k l , ϕ p ) ∂ S f ] S f ≻ 5 × 10 5 cm ⋅ s − 1 = 0 (9)

Derived from this equation, it gives the following relations:

i) S b 0 ( H , k l , ϕ p ) L ( k l , ϕ p ) D ( k l , ϕ p ) = − tanh ( H L ( k l , ϕ p ) ) (10)

Equation (10), gives rise the definition of intrinsic velocity and becomes a diffusion velocity [

ii) S b 1 ( H , k l , ϕ p ) L ( k l , ϕ p ) D ( 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 (11)

Equation (11) is marked by a term of absorption (b_{i}) and leads to generation velocity, when H is small compared to L [

3) By Boltzmann’s law, the photovoltage at the junction is written as:

V p h ( S f , H , k l , ϕ p ) = K b × T q ⋅ ln ( N b n i 2 ⋅ δ ( 0 , H , k l , ϕ p ) + 1 ) (12)

where, Kb is the Boltzmann constant, q is the elementary charge of the electron and T is the temperature. Nb is the solar cell base doping rate, and n_{i} is the intrinsic density of minority charge carriers.

The expressions (Equation (10) and Equation (11)) are represented by the

The profile of the J p h ( S f , ϕ p , H o p t ) - V ( S f , ϕ p , H o p t ) characteristic for different values of the irradiation energy flow and base optimum thickness is shown in

ϕ p (MeV) | 60 | 80 | 100 | 120 | 140 |
---|---|---|---|---|---|

Hopt (µm) | 127 | 123 | 120 | 117 | 114 |

with the increase of the irradiation energy flow. And the photovoltage increases slightly.

The Ohm law applied to the circuit in

P ( S f , k l , ϕ p ) = V p h ( S f , k l , ϕ p ) × J ( S f , k l , ϕ p ) (13)

with:

J ( S f , k l , ϕ p ) = J p h ( S f , k l , ϕ p ) − J d ( S f , k l , ϕ p ) (14)

where J_{d} is the solar cell density of current under dark expressed as:

J d ( S f , k l , ϕ p ) = q × S f 0 × n i 2 N b × exp ( V p h ( S f , k l , ϕ p ) V T − 1 ) (15)

Sf_{0} is the excess minority carrier recombination velocity associated with shunt resistance-induced charge carrier losses [

S f 0 ( k l , ϕ p ) = ∑ i = 1 3 K i × D ( k l , ϕ p ) × [ b i × L ( k l , ϕ p ) − e − b i × H × ( s h ( H L ( k l , ϕ p ) ) + b i × L ( k l , ϕ p ) × c h ( H L ( k l , ϕ p ) ) ) ] L ( k l , ϕ p ) × [ e − b i × H × ( c h ( H L ( k l , ϕ p ) ) + b i × L ( k l , ϕ p ) × s h ( H L ( k l , ϕ p ) ) ) − 1 ] (16)

We note on

On

The fill factor is an important parameter for a solar cell. It shows the physical quality of the solar cell for a conversion efficiency and indicates the performance of a perfect cell. The expression of the fill factor is given [

F F = P max V O C × J S C (17)

P_{max} is the maximum power. Voc is the open circuit photovoltage.

J_{sc} is the short circuit current density

The conversion efficiency of a solar cell is the ratio between the maximum power supplied provided by the solar cell and the power of absorbed illumination. It is written as follows:

η = J S C max × V max P i n c i d e n t (18)

P i n c i d e n t is the incident light power absorbed by the solar cell.

With: P i n c i d e n t = 100 mW ⋅ cm − 2 in the standard conditions (Air Mass 1.5).

The obtained solar cell fill factor and efficiency for different irradiation energy flow values corresponding to the optimum thickness are shown in

ϕ p (MeV) | 60 | 80 | 100 | 120 | 140 |
---|---|---|---|---|---|

Hopt (cm) | 0.0127 | 0.0123 | 0.0120 | 0.0117 | 0.0114 |

J_{sc}_{max} (A/cm^{2}) | 0.0295 | 0.0263 | 0.0242 | 0.0224 | 0.0212 |

J_{d} (Sf_{max}) (A/cm^{2}) | 0.1374 × 10^{−3} | 0.1355 × 10^{−3} | 0.1342 × 10^{−3} | 0.1330 × 10^{−3} | 0.1318 × 10^{−3} |

V_{max} (V) | 0.5931 | 0.5940 | 0.5946 | 0.5951 | 0.5955 |

P_{max} (W/cm^{2}) | 0.01741 | 0.01554 | 0.01431 | 0.01325 | 0.01254 |

FF | 0.9951 | 0.9947 | 0.9945 | 0.9939 | 0.9933 |

η_{max} (%) | 17.41 | 15.54 | 14.31 | 13.25 | 12.55 |

The equation obtained from the power versus the irradiation energy is given by the following relation:

P max = u × ϕ p 2 − w × ϕ p + z (19)

with: u = 4 × 10 − 7 W ⋅ cm − 2 / MeV , w = 10 − 4 W ⋅ cm − 2 / MeV , z = 0.0247 W / cm 2

The equation obtained from the power versus the base thickness is given by the following relation:

P max = α × H 2 − β × H + ν (20)

with: α = 1140.6 W / cm 4 , β = 23.709 W / cm 3 , ν = 0.1346 W / cm 2

The fit equation obtained from the efficiency versus the base thickness is given by the following relation:

η max = m × H 2 − n × H + k (21)

with: m = 10 6 cm − 2 , n = 24038 cm − 1 , k = 136.58

We note on

Sf_{max}, the excess minority carrier recombination velocity at the junction corresponding to the maximum power point is point out by solving the following equation [

∂ P ∂ S f = 0 (22)

From this Equation (22), the transcendental equation depending on recombination velocity Sf and the irradiation energy is obtained, for each Hopt. It is given by the following expressions:

M ( S F max , k l , ϕ p ) = 1 S F max × L ( k l , ϕ p ) × [ 1 − S F max × L ( k l , ϕ p ) Y 1 × D ( k l , ϕ p ) + S F max × L ( k l , ϕ p ) ] (23)

N ( S F max , k l , ϕ p ) = [ Γ max ( 0 , k l , ϕ p ) ( Γ max ( 0 , k l , ϕ p ) + n i 2 N b ) × ( S F max × L ( k l , ϕ p ) + Y 1 × D ( k l , ϕ p ) ) ] × [ 1 log ( N b × Γ max ( 0 , k l , ϕ p ) n i 2 + 1 ) ] (24)

Γ max ( 0 , k l , ϕ p ) is the density of the minority excess minority carrier at the point of maximum power, its expression is given by the following relations:

Γ max ( 0 , k l , ϕ p ) = β × D ( k l , ϕ p ) × [ Y 2 + Y 1 − b i × L ( k l , ϕ p ) S F max × L ( k l , ϕ p ) + Y 1 × D ( k l , ϕ p ) ] (25)

with:

β = − a i × L ( k l , ϕ p ) 2 × n D ( k l , ϕ p ) × ( L ( k l , ϕ p ) 2 × b i 2 − 1 ) (26)

Y 1 = D ( k l , ϕ p ) L ( k l , ϕ p ) × sinh ( H L ( k l , ϕ p ) ) + S b ( k l , ϕ p ) × cosh ( H L ( k l , ϕ p ) ) D ( k l , ϕ p ) L ( k l , ϕ p ) × cosh ( H L ( k l , ϕ p ) ) + S b ( k l , ϕ p ) × sinh ( H L ( k l , ϕ p ) ) (27)

Y 2 = ( D ( k l , ϕ p ) × b i − S b ( k l , ϕ p ) ) × exp ( − b i ⋅ H ) D ( k l , ϕ p ) L ( k l , ϕ p ) × cosh ( H L ( k l , ϕ p ) ) + S b ( k l , ϕ p ) × sinh ( H L ( k l , ϕ p ) ) (28)

The graphical resolution of this transcendental equation as a function of the excess minority carrier recombination velocity Sf at the junction [_{max} values by the intercept point of the two curves represented by

The results obtained from _{max}, are given in

The recombination velocity Sf_{max} of the excess minority carrier at the junction yielding P_{max}, decreases while irradiation energy flow increases.

_{max}(H

_{opt})

_{max} of the excess minority carrier at the junction yielding P_{max}, as function of optimum base thickness.

Irradiation energy (MeV) | Base thickness (cm) | Intercept points curves (p) | Sfmax (p∙10^{p} cm/s) |
---|---|---|---|

60 | 0.0127 | 1.9683 | 182.975 |

80 | 0.0123 | 1.9638 | 180.675 |

100 | 0.0120 | 1.9600 | 178.754 |

120 | 0.0117 | 1.9564 | 176.953 |

140 | 0.0114 | 1.9534 | 175.465 |

The equation obtained from the best fit of Sf_{max} versus base optimum thickness is given by the following relation:

S f max = a × H O p t + b (29)

with: a = 5859.6 s − 1 , b = 108.53 cm ⋅ s − 1

The recombination velocity Sf_{max} of the excess minority carrier at the junction increases with the optimum base thickness.

In this work, a technique for obtaining the optimum thickness of the solar cell under variation of the irradiation energy flow has been presented. It is also deduced from this optimal thickness, the fill factor, the electrical power, the efficiency of the solar cell as well as the recombination velocity at the junction through a transcendental equation, leading to maximum power. We found that the electrical parameters of the solar cell decrease with the increasing of the irradiation energy flow. Then we have plotted and fitted the curves of the power, the efficiency and the recombination velocity (at the maximum power) versus the optimum base thickness.

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

Sow, O., Ba, M.L., El Moujtaba, M.A.O., Traore, Y., Sow, El.H., Sarr, C.T., Diop, M.S. and Sissoko, G. (2020) Electrical Parameters Determination from Base Thickness Optimization in a Silicon Solar Cell under Influence of the Irradiation Energy Flow of Charged Particles. Energy and Power Engineering, 12, 1-15. https://doi.org/10.4236/epe.2020.121001