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The aim of this study is to determinate the electrical parameters of a white biased silicon solar cell submitted to an irradiation energy of particles (protons, helium, electrons and heavy ions). A theoretical study of the influence of irradiation energy on the photocurrent density, the photovoltage, the maximum power, as well as the maximum efficiency of the solar cell is presented through a resolution of the continuity equation relative to excess minority carrier. Then the expressions of the photocurrent density Jph, the photovoltage Vph, and the excess minority carrier recombination velocity at the back side
*Sb* are established dependent of irradiation parameters
*∅*
*p*,
*Kl* respectively irradiation flux and intensity. In this work, we propose a method for determining the recombination velocity of the excess minority carrier at the junction
*Sf*
_{max} corresponding to the maximum power point delivered by the photovoltaic generator under the influence of the irradiation. It is then obtained by calculating the derivative of the power with respect to the excess minority carrier recombination velocity
*Sf* at the junction emitter-base. A transcendental equation solution is deduced as eigenvalue, leading to the junction recombination velocity of excess minority carrier and also yields the solar cell maximum conversion efficiency.

The study of the effect of radiation on solar cells designed for space applications has long occupied research fields, in order to understand the stakes in their performance opposite the high-energy radiating particles of the space environment [

The aim of this study is to show the influence of irradiation energy on the electrical parameters of a silicon solar cell: photocurrent density, photovoltage, I-V characteristic, electric power and efficiency.

This work deals with a method, to determinate the maximum power point of the solar cell under the effect of the irradiation energy. Maximum Power Point Trackers (MPPT) is a well-known technique allowing the solar cell to operate at the maximum power point under varying illumination and temperature [

This work gives the expressions the excess minority carrier density continuity equation in the base. Then, the expressions [

The characteristic curve of the photocurrent density as a function of the photovoltage I(Sf)-V(Sf) [

A transcendental equation giving the velocity of recombination of the excess minority carrier at the points of maximum power Sf_{max} is determined and the numerical values of Sf_{max} are extracted graphically. We then calculated the fill factor FF of the solar cell for different values of the irradiation energy. Finally, the profiles of Sf_{max}, V_{max}, I_{max} and η_{max} versus irradiation energy are shown graphically.

Consider a crystalline silicon solar cell (n^{+}-p-p^{+}) [

x is the depth in the base of the solar cell measured from the emitter-base junction, called space charge region (SCR) (x = 0) to the back side face (x = H). H is the base thickness, where a back surface field (BSF) is created by help of the p^{+} zone.

Kl is the damage coefficient while ϕp is the irradiation energy.

The set of different processes taking place in the base can result in the so called continuity equation:

∂ 2 δ ( x , k l , ϕ p ) ∂ x 2 − δ ( x , k l , ϕ p ) [ L ( k l , ϕ p ) ] 2 = − 1 D ( k l , ϕ p ) ⋅ G ( x ) (1)

with:

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

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

D ( k l , ϕ p ) is the diffusion coefficient of the electrons in the base under irradiation.

L ( k l , ϕ p ) is the diffusion length of the excess minority carriers 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 carriers before their recombination in the base under irradiation.

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.

G(x) is the excess minority carrier generation rate, given by [

G ( x ) = ∑ i = 1 3 a i e − b i ⋅ x (4)

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 resolution of the differential equation 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 (5)

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

2-4-a: 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 ) (6)

2-4-b: 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 ) (7)

S_{f} is the excess minority carrier recombination velocity at the junction and also indicates the operating point of the solar cell [

S_{b} is the excess minority carrier recombination velocity on the back side surface [

[ ∂ J p h ∂ S f ] = 0 (8)

From the relation (Equation (8)), the calculation gives the recombination velocity S_{b} [

S b ( 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 (9)

The expression of the photocurrent density is deduced from the excess minority carrier density in the base. It is given by the following relation:

J p h ( S f , k l , ϕ p ) = q ⋅ D ( k l , ϕ p ) ⋅ [ ∂ δ ( x , k l , ϕ p ) ∂ x ] x = 0 (10)

We note in this figure that the photocurrent density is almost zero for recombination velocity lower than 10 cm/s (solar cell operating in open circuit). Then for 10 cm/s < Sf < 3 × 10^{3} 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 some energy to cross the junction.

Indeed, for recombination velocity greater than 3 × 10^{3} cm/s, the photocurrent density is maximum and constant, corresponding to the short-circuit photocurrent.

The figure also shows that as the irradiation energy 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 density of the excess minority carrier.

The illuminated solar cell photovoltage expression, is obtained by the Boltzmann relation.

V p h ( S f , k l , ϕ p ) = V T ⋅ ln ( N b n i 2 ⋅ δ ( 0 , k l , ϕ p ) + 1 ) (11)

V_{T} is the thermal voltage, defined by:

V T = K b q T (12)

• T is the absolute temperature = 300 K

• Nb is the doping rate in acceptor atoms in the base

• n_{i} is the intrinsic concentration

• K_{b} is the constant of Boltzmann

• q is the elementary charge of the electron

We note in this figure that the photovoltage is maximum and constant for recombination velocity lower than 2 × 10^{2} cm/s; thus corresponding to solar in open circuit condition. Beyond this recombination velocity, the photovoltage linearly decreases very rapidly to reach almost zero value in the vicinity of the short-circuit and consequently, yields the crossing of almost all excess minority carrier at the junction.

Then, we note in this figure that the increase of the irradiation energy influences weakly the photovoltage.

The profile of the illuminated solar cell I(Sf)-V(Sf) characteristic for different values of the irradiation energy is shown in

We note that the photocurrent density decreases with the increase of the irradiation energy. And the photovoltage increases slightly.

The equivalent electric circuit of a real solar cell under illumination is shown in

The ohm law applied to the circuit in

P ( S f , k l , ϕ p ) = V p h ( S f , k l , ϕ p ) ⋅ I ( S f , k l , ϕ p ) (13)

Applying the first Kirchhoff law to the circuit of

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

I_{d} is the diode current, its expression is given by the following relation:

I 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 [

We note that the power grows from the high recombination velocity at the junction (

increase of the irradiation energy. There are three zones of variation of the power as a function of the photovoltage (

➢ A short-circuit zone with a maximum photocurrent density (where the photovoltage is zero) gives a power equally zero;

➢ A region in the vicinity of the open circuit with a maximum photovoltage (with a zero photocurrent density) provides such a zero power;

➢ And an intermediate zone with increasing and decreasing power passing through a maximum which is located at an intermediate operating point.

It is also observed a decrease in power with the increase of the irradiation energy.

The maximum power point of a photovoltaic generator corresponds to the photocurrent density-photovoltage couple generating the maximum electrical power [_{max} and the maximum photovoltage Vph_{max} gives a maximum power as P_{max} = Jph_{max} × Vph_{max}.

The recombination velocity Sfmax of the excess minority carrier at the junction corresponding to the maximum power point is bring out by solving the following equation [

∂ P ∂ S f = 0 (16)

Let Sf_{max} denote the recombination velocity of the excess minority carrier at the junction corresponding to the maximum power point. It depends on both the phenomenological and geometrical parameters of the solar cell, respectively,

L(kl, φp), D(kl, φp), Sf(kl, φp), Sb(kl, φp), μ, τ, n_{i}, Nb, b_{i} and H in the one dimensional model.

From Equation (16), the transcendental equation depending on recombination velocity Sf and the irradiation energy is obtained. It is given by the following expressions:

M ( S f , 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 ) ] (17)

And:

N ( S f , 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 ) ] (8)

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 relation:

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 ) ] (19)

with:

β = n ⋅ a i ⋅ L ( k l , ϕ p ) 2 D ( k l , ϕ p ) ⋅ ( L ( k l , ϕ p ) 2 ⋅ b i 2 − 1 ) (20)

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 ) ) (21)

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 ) ) (22)

The graphical resolution of this transcendental equation as a function of the excess minority carrier recombination velocity Sf at the junction, for different values of the irradiation energy, gives the Sf_{max} values by the intercept point of the two curves represented by _{max} operating point, the electrical power delivered by the photovoltaic generator, is maximum.

_{max} values as the irradiation energy increases. This reflects the decrease in maximum power as the irradiation energy increases. We observe the intercept points in the figure, corresponding to the values of Sf_{max}. These values of Sf_{max} correspond to a condition of solar cell operating at its maximum power point.

The results obtained from _{max} for each maximum power point, are given in

The influence of the irradiation energy on the Sf_{max} is represented by

Irradiation energy (MeV) | Intercept points for given irradiation energy values (p) | Sf_{max} (p∙10^{p}cm/s) |
---|---|---|

50 | 1.506 | 47.624 |

60 | 1.432 | 35.970 |

70 | 1.349 | 26.924 |

80 | 1.251 | 24.967 |

90 | 1.139 | 15.728 |

The recombination velocity Sf_{max} of the excess minority carrier at the junction decreases with the irradiation energy.

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

η = I max ⋅ V max P incident (23)

P incident is the incident light power absorbed by the solar cell, with P incident = 100 mW / cm 2 in the standard conditions Air Mass 1.5.

The representation of the efficiency is deduced from the I-V characteristic curve (

Irradiation energy (MeV) | 50 | 60 | 70 | 80 | 90 |
---|---|---|---|---|---|

I_{max} (A/cm^{2}) | 0.02802 | 0.02619 | 0.02470 | 0.02346 | 0.02241 |

I_{d} (Sf_{max}) (A) | 0.19578 × 10^{−3} | 0.19616 × 10^{−3} | 0.19655 × 10^{−3} | 0.19692 × 10^{−3} | 0.19729 × 10^{−3} |

V_{max} (V) | 0.59549 | 0.59606 | 0.59652 | 0.59692 | 0.59726 |

P_{max} (W/cm^{2}) | 0.016569 | 0.015494 | 0.014617 | 0.013886 | 0.013267 |

FF | 0.016682 | 0.015605 | 0.014731 | 0.014003 | 0.013381 |

Ƞ_{max} (%) | 16,569 | 15.494 | 14.617 | 13.886 | 13.267 |

Figures 10-12 show the maximum photocurrent I_{max}, the maximum photovoltage V_{max} and the maximum conversion efficiency η_{max} of the solar cell as a function of the irradiation energy.

These results show that, the recombination velocity of the excess minority carrier Sf_{max} at the junction yielding, the maximum photocurrent as well as the photovoltaic conversion efficiency, decrease when the irradiation energy increases, contrary to the maximum photovoltage which increases with the energy of irradiation. This results in a variation of the peak power point as the irradiation energy increases.

In this work, from the expression of the excess minority carrier density in the base, the photocurrent density and the photovoltage, lead to the illuminated I-V characteristic of the solar cell under irradiation. This study showed us a decrease in short-circuit photocurrent and an increase in open-circuit photovoltage as irradiation energy increases. The decrease of the short-circuit photocurrent is manifested by a decrease in the excess minority carrier density that crosses the junction as the irradiation energy increases.

From the dark and illuminated I-V characteristics, we have studied the electric power delivered by the base of the solar cell as a function of the excess minority carrier recombination velocity Sf at the junction. The power increases with the recombination velocity Sf as well as the photovoltage up to a maximum value which represents the maximum power point. Then it decreases to cancel at a value corresponding to the open circuit voltage (very low Sf).

A transcendental equation graphically resolved, gives eigenvalue, represented by the recombination velocity of the excess minority carrier Sf_{max} corresponding to the maximum power point of the solar cell under different values of the irradiation energy.

Finally we calculated and plotted solar cell electrical parameters I_{max}, V_{max}_{ }and η_{max} versus the applied irradiation energy.

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

Ba, M.L., Diallo, H.L., Ba, H.Y., Traore, Y., Diatta, I., Diouf, M.S., Wade, M. and Sissoko, G. (2018) Irradiation Energy Effect on a Silicon Solar Cell: Maximum Power Point Determination. Journal of Modern Physics, 9, 2141-2155. https://doi.org/10.4236/jmp.2018.912135