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The concept of the recombination of the minority carrier’s recombination velocity at the junction and in the rear, is used for determination, optimum thickness and then shunt resistance in the base of the silicon solar cell, maintained in steady state and under energy from the irradiation flow of charged particles. Resistance shunt is obtained and modeled through a relationship expressed according to the flow and energy of irradiation imposed on the solar cell.

Shunt resistance is a very important parameter of the equivalent electric model of the solar cell, which induces the leaking current of electrical charges when it is of low value [

In addition to the experimental conditions of illumination constant or variable [

Theoretically, the investigation is done by considering the crystallographic aspect i.e., in a one-dimensional (1D) [

Regardless of the model used [

In this work, we determine the shunt resistance, of a n^{+}-p-p^{+} silicon solar cell, whose base has undergone irradiation of charged particles [

^{+}-p-p^{+} silicon solar cell [^{+}), a base doped in (p), a space charge zone where there is an intense field to separate the pairs of electron holes that arrives there, 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 ϕ p is the irradiation energy flow.

When the solar cell is illuminated by a constant polychromatic light, all the

physical processes that governed the excess minority charge carriers in the base are: photogeneration, diffusion, recombination in the bulk and surfaces. The minority carriers obey to the continuity equation and boundary conditions, expressed by the following Equations ((1), (6) and (7)):

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 carriers in 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 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. It is related to the minority carrier’s 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 diffusion length of the excess minority carriers in the base before irradiation,

ϕ p is the irradiation energy flow,

k l is the damage coefficient intensity.

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

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

n is the number of sun or level of illumination, indicating the concentration of incident light.

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 (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 (5)

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

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)

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

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_{b} is the excess minority carrier recombination velocity on the back side surface [^{+} junction and characterizes the behavior of the density of the charge carriers at this surface. It yields to send back towards the emitter-base junction, the minority carriers generated near the rear face, to then contribute to the photocurrent.

c) Photocurrent density

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 = 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)

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 ≻ 5 × 10 5 cm ⋅ s − 1 = 0 (9)

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

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 ) ) (10)

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

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 (11)

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

The recombination velocity of the charge carriers at the junction initiating the Sfcc short-circuit can be obtained from the different expressions of the photocurrent and the value of the short-circuit photocurrent [

J p h ( S f , H , k l , ϕ p ) − J p h S C ( H , k l , ϕ p ) = 0 (12)

Equation (12) becomes replacing the photocurrent density by its expression:

q ⋅ D ( k l , ϕ p ) ⋅ S f D ( k l , ϕ p ) [ A ( S f , H , k l , ϕ p ) + ∑ i = 1 3 K i ] − J p h S C ( H , k l , ϕ p ) = 0 (13)

The resolution of Equation (13) yields to the mathematical expression of the minority carrier recombination velocity S f c c ( H , k l , ϕ p ) initiating the short circuit, given as:

S f c c ( H , k l , ϕ p ) = [ ϕ ( k l , ϕ p ) ⋅ D 2 ( k l , ϕ p ) ⋅ sinh ( H L ( k l , ϕ p ) ) + ϕ ( k l , ϕ p ) ⋅ S b ( k l , ϕ p ) ⋅ L ( k l , ϕ p ) ⋅ D ( k l , ϕ p ) ⋅ cosh ( H L ( k l , ϕ p ) ) − ψ ( k l , ϕ p ) − ( ( ϕ ( k l , ϕ p ) ⋅ L 2 ( k l , ϕ p ) ⋅ S b ( k l , ϕ p ) ⋅ sinh ( H L ( k l , ϕ p ) ) ) − ϕ ( k l , ϕ p ) ⋅ L ( k l , ϕ p ) ⋅ D ( k l , ϕ p ) ⋅ cosh ( H L ( k l , ϕ p ) ) + γ ( k l , ϕ p ) ) ] (14)

ψ ( k l , ϕ p ) = L ( k l , ϕ p ) ⋅ D 2 ( k l , ϕ p ) ⋅ ∑ i = 1 3 ( K 1 ⋅ b i ) (15)

γ ( k l , ϕ p ) = L ( k l , ϕ p ) ⋅ D ( k l , ϕ p ) ⋅ X ( H , k l , ϕ p ) ⋅ ∑ i = 1 3 ( K i ) − L ( k l , ϕ p ) ⋅ ∑ i = 1 3 ( K i ⋅ α ( k l , ϕ p ) ) (16)

α ( k l , ϕ p ) = L ( k l , ϕ p ) [ S b ( k l , ϕ p ) − D ( k l , ϕ p ) b i ] ( e − b i H ) (17)

ϕ ( k l , ϕ p ) = L ( k l , ϕ p ) [ J p h S C ( H , k l , ϕ p ) q ⋅ D ( k l , ϕ p ) − ∑ i = 1 3 ( K i ⋅ b i ) ] (18)

X ( H , 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 ) ) (19)

The expression of the phototension is obtained using the well-known Boltzmann relation 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 ) (20)

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, n_{i} is the intrinsic density of minority charge carriers.

By applying the law of nodes and that of the meshes to the circuit of the

R s h ( S f , H , k l , ϕ p ) = V p h ( S f , H , k l , ϕ p ) J p h s c ( H , k l , ϕ p ) − J p h ( S f , H , k l , ϕ p ) (21)

The equation obtained from the shunt resistance as a function of the irradiation energy is given by the following relation:

Irradiation energy (MeV) | Rsh_{exp} (Ω/cm^{2}) |
---|---|

60 | 2321.4 |

80 | 3392.9 |

100 | 4285.7 |

120 | 5000 |

140 | 6071.4 |

1 R s h e x p = a × ϕ p 2 − b × ϕ p + c (22)

With: a = 4 × 10 − 8 Ω − 1 ⋅ cm 2 , b = 10 − 5 Ω − 1 ⋅ cm 2 and c = 0.001 Ω − 1 ⋅ cm 2 .

Among previous work on solar cells that have undergone, irradiation of charged particles and their effects on physical mechanisms influencing performance [

In this work, a technique has been proposed for determining the shunt resistance from the calibration curves as a function of the recombination velocity at the junction initiating the short circuit under variation the irradiation energy flow and the optimum base thickness of silicon solar cell.

The graphical resolution of the rear faces recombination velocity at the intrinsic or electronic and that dependent on the absorption coefficient bi, yields to determinate the optimum base thickness under irradiation energy flow. The shunt resistance increases with the irradiation energy flow and is modeled through a mathematical relationship, taking into account the useful optimum thickness adapted to the conditions of irradiation by charged particles.

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

Sow, O., Ba, M.L., Ba, H.Y., El Moujtaba, M.A.O., Traore, Y., Diop, M.S., Lemrabott, H., Wade, M. and Sissoko, G. (2019) Shunt Resistance Determination in a Silicon Solar Cell: Effect of Flow Irradiation Energy and Base Thickness. Journal of Electromagnetic Analysis and Applications, 11, 203-216. https://doi.org/10.4236/jemaa.2019.1112014