1. Introduction

JMP

Journal of Modern Physics

2153-1196

Scientific Research Publishing

10.4236/jmp.2018.912135

JMP-87766

Articles

Physics&Mathematics

Irradiation Energy Effect on a Silicon Solar Cell: Maximum Power Point Determination

Mamadou

Lamine Ba

¹Hawa

Ly Diallo

²Hamet

Yoro Ba

²Youssou

Traore

²Ibrahima

Diatta

²Marcel

Sitor Diouf

²Mamadou

Wade

¹Gregoire

Sissoko

²^*

Laboratory of Sciences and Techniques of Water and Environment, Polytechnic School of Thiès, Thiès, Senegal

Laboratory of Semiconductors and Solar Energy, Physics Department, Faculty of Science and Technology, University Cheikh Anta Diop, Dakar, Senegal

29092018

09122141215529, August 20188, October 2018 11, October 2018

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Silicon Solar Cell Irradiation Electrical Parameters Maximum Power Point

1. Introduction

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 [1] [2] [3] . These same concerns are studied at the terrestrial level in order to investigate the relationship between the solar cell parameters and those of the irradiation [4] .

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 [5] [6] [7] [8] [9] .

This work gives the expressions the excess minority carrier density continuity equation in the base. Then, the expressions [10] [11] of the photocurrent density, the photovoltage, the excess minority carrier excess minority carrier recombination velocity at the back side Sb and the electrical power, all depending on the irradiation energy are deduced. These parameters are also represented graphically as a function of the excess minority carrier recombination velocity at the junction.

The characteristic curve of the photocurrent density as a function of the photovoltage I(Sf)-V(Sf) [12] [13] [14] , is produced as irradiation parameters dependent. The power [15] [16] [17] , as a function of both the recombination velocity of the excess minority carrier at the junction and the photovoltage, is also represented graphically.

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.

2. Theory

Consider a crystalline silicon solar cell (n⁺-p-p⁺) [18] . Its structure is illustrated in Figure 1. Where:

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 [19] [20] :

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 [21] [22] .

S_b is the excess minority carrier recombination velocity on the back side surface [23] [24] [25] . Its expression is obtained from the derivative of the photocurrent density for large Sf values [21] [26] [27] that can be seen on Figure 2.

[ ∂ J p h ∂ S f ] = 0 (8)

From the relation (Equation (8)), the calculation gives the recombination velocity S_b [10] [28] of the excess minority carrier at the back side of the solar cell, depending on parameters influenced by irradiation energy, such as, L(kl, φp) and D(kl, φp):

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)

3. Photocurrent Density

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)

Figure 2 shows the profile of the photocurrent density as a function of the excess minority carrier recombination velocity at the junction for different given values of the irradiation energy.

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 × 10³ 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 × 10³ 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.

4. Photovoltage

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

Figure 3 shows the profile of the photovoltage as a function of the excess minority recombination velocity at the junction for different values of the irradiation energy.

We note in this figure that the photovoltage is maximum and constant for recombination velocity lower than 2 × 10² 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.

5. Illuminated Solar Cell I(Sf)-V(Sf) Characteristic Study

The profile of the illuminated solar cell I(Sf)-V(Sf) characteristic for different values of the irradiation energy is shown in Figure 4.

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

6. Study of the Power and the Maximum Power Point6.1. Electrical Power of the Solar Cell

The equivalent electric circuit of a real solar cell under illumination is shown in Figure 5. This circuit gives the solar cell as an ideal current generator that outputs an illumination depending photocurrent density Iph, connected in parallel with a diode and a shunt resistor Rsh and in series with a series resistor Rs [29] .

The ohm law applied to the circuit in Figure 5 yields the electric power delivered by the base of the solar cell to an external load as follows:

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 Figure 5 the current delivered by an illuminated solar cell to an external load, is given by the following relationship:

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)

Sf₀ is the excess minority carrier recombination velocity associated with shunt resistance-induced charge carrier losses [30] [31] , which characterizes the good quality of the solar cell [21] [22] [24] .

Figure 6 and Figure 7 show the variations in electrical power as a function of both, the excess minority carrier recombination velocity at the junction and the photovoltage for different values of the irradiation energy.

We note that the power grows from the high recombination velocity at the junction (Figure 6) and reaches a maximum amplitude where it decreases to tend to zero value. The maximum of this power amplitude decreases with the

increase of the irradiation energy. There are three zones of variation of the power as a function of the photovoltage (Figure 7):

➢ 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.

6.2. Maximum Power Point and Efficiency

The maximum power point of a photovoltaic generator corresponds to the photocurrent density-photovoltage couple generating the maximum electrical power [17] . The product of the maximum photocurrent density Jph_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 [17] .

∂ 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 Figure 8. At each Sf_max operating point, the electrical power delivered by the photovoltaic generator, is maximum.

Figure 8 shows a decrease in the Sf_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 Figure 8 corresponding to the numerical values of Sf_max for each maximum power point, are given in Table 1.

The influence of the irradiation energy on the Sf_max is represented by Figure 9.

Table 1 The numerical values of Sf<sub>max</sub> corresponding to the maximal power point for different irradiation energy values

Irradiation energy (MeV)	Intercept points for given irradiation energy values (p)	Sf_max (p∙10^pcm/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.

7. The Efficiency

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 (Figure 4). The graphical values corresponding to the maximum power point, leading to both, the maximum photocurrent and the maximum photovoltage, allowed to obtain the photovoltaic efficiency conversion for different values of the irradiation energy. These results are noted in Table 2.

Table 2 <xref ref-type="table" rid="table">Table </xref>of parameters leading to the efficiency η<sub>max</sub> corresponding to the maximal power point for different irradiation energy values

Irradiation energy (MeV)	50	60	70	80	90
I_max (A/cm²)	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/cm²)	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.

8. Conclusions

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_maxand η_max versus the applied irradiation energy.

Conflicts of Interest

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

Cite 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

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