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 . Thus it limits the electric output current and reduces the efficiency    ; it then influence solar cells industrial manufacturing processes   . Therefore, its determination is the subject of several measurement techniques, especially from electrical current and voltage  . Some characterization techniques   ; generally maintain, the solar cell in steady state, under darkness or polychromatic illumination   and depending to the model to be used  . Others use an electrical or optical signal in modulated frequency, to extract impedance and deduce its components (series  and shunt resistances  , capacitance  ).
In addition to the experimental conditions of illumination constant or variable , there are additional: temperature variations , the presence of electromagnetic field  , its exposure to an irradiation flow of charged particles  . These conditions impose a variation in the diffusion coefficient of minority carriers in the base of the solar cell . Manufacturing conditions set the material doping rate (Nb)   , which itself influences the diffusion coefficient of minority carriers D (Nb).
Theoretically, the investigation is done by considering the crystallographic aspect i.e., in a one-dimensional (1D)  or three-Dimensional (3D) model of space . The latter case takes into account the grain size (g) and the grains boundaries recombination velocity (Sbg), which influence the shunt resistance  .
Regardless of the model used , the recombination rate at the junction (x = 0) and in the rear (x = H) in the base of the solar cell, is taken into account through, respectively the surfaces recombination velocity Sf and Sb   . These phenomenological parameters (recombination in the bulk and surfaces), have studied and extracted, with theoretical and experimental methods, keeping the solar cell under steady   or dynamic state   .
In this work, we determine the shunt resistance, of a n+-p-p+ silicon solar cell, whose base has undergone irradiation of charged particles  . Its thickness   is taken into account and optimized through the study of the theoretical expression of excess minority carrier recombination velocity in the rear face . Then, using the concept of recombination velocity at the junction initiating the short circuit of the solar cell   under steady state, we determine the shunt resistance, for each thickness imposed by the intensity of the irradiation flow of charged particles.
2. Theoretical Study
2.1. Presentation of the Solar Cell
Figure 1 represents n+-p-p+ silicon solar cell  illuminated by the polychromatic light and under irradiation. The cell consists of a emitter doped in (n+), 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 is the irradiation energy flow.
When the solar cell is illuminated by a constant polychromatic light, all the
Figure 1. Structure of the silicon solar cell (n+-p-p+) under irradiation.
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)):
represents the excess minority carrier density in the base of the solar cell at the x-position, dependent of the irradiation energy.
and 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:
with the diffusion length of the excess minority carriers in the base as a function of the irradiation energy flux ( ) 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  :
is diffusion length of the excess minority carriers in the base before irradiation,
is the irradiation energy flow,
is the damage coefficient intensity.
➢ G(x) is the excess minority carrier generation rate , given by:
n is the number of sun or level of illumination, indicating the concentration of incident light.
The coefficients ai and bi 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:
The expressions of, A and B are determined from the following boundary conditions:
a) At the junction emitter-base (x = 0)
Sf 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)
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 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:
3. Results and Discussions
3.1. Photocurrent Density
Figure 2 represents the profiles of the photocurrent density as a function of the recombination velocity at the junction for different values of the irradiation energy and the optimum base thickness .
3.2. Back Surface Recombination Velocity
The Sb expression is obtained from the derivative of the photocurrent density for large Sf values  .
Figure 2. Photocurrent density as function the recombination velocity at junction for different irradiation energy flow and optimum base thickness with kl = 5 cm−2/MeV.
The resolution of this equation yields to establish the following expressions of the excess minority carrier recombination velocity at the rear face, i.e. and :
It represents the intrinsic minority carrier’s recombination velocity at the p-p+ junction.
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 (bi) and leads to a generation rate, for .
Figure 3 represents the profile of the two expressions of the recombination velocity at the rear face Sb0 and Sb1 as function of the base thickness .
Figure 3. Back surface recombination velocity versus solar cell base thickness = 140 MeV.
3.3. Expression of Recombination Velocity Initiating the Short Circuit
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  .
Equation (12) becomes replacing the photocurrent density by its expression:
The resolution of Equation (13) yields to the mathematical expression of the minority carrier recombination velocity initiating the short circuit, given as:
3.4. Shunt Resistance
The expression of the phototension is obtained using the well-known Boltzmann relation as:
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, ni is the intrinsic density of minority charge carriers.
3.4.2. Experimental Shunt Resistance
Figure 4 represents the well-known (Jsc(Sf) – Vph(Sf)) electric characteristic of the solar cell under illumination.
By applying the law of nodes and that of the meshes to the circuit of the Figure 5 we obtain the expression of the shunt resistance expressed as:
Figure 6 gives the profile of the calibrated curve of the expression of the shunt resistance as a function of excess minority carrier recombination velocity at the junction, for different irradiation energy.
3.4.3. Experimental Determination Technique of Shunt Resistance
Figure 7 gives the technique of experimental determination of the shunt resistor, from the surface recombination velocity at the junction initiating the short circuit (Sfsc) in the solar cell, which intercepts the calibration curve to give an ordinate equal to the experimental shunt resistance (Rshexp)  .
Table 1 represents the values of the recombination velocity Sfsc initiating the short circuit in the solar cell, obtained for different irradiation energy, damage coefficient and yield to the shunt resistors Rshexp.
Figure 8 shows a decreasing plot of the shunt resistance reverse Rshexp versus irradiation energy for given intensity.
The equation obtained from the shunt resistance as a function of the irradiation energy is given by the following relation:
Figure 4. Photocurrent density as a function of photovoltage.
Figure 5. Equivalent electrical circuit of the short circuit.
Figure 6. Calibrated shunt resistance curve versus junction surface recombination velocity for different irradiation energy corresponding the optimum base thickness with kl = 5 cm−2/MeV.
Figure 7. Calibrated curve for the determination of the experimental shunt resistance.
Figure 8. Experimental shunt resistance reverse versus irradiation energy.
Table 1. Rshexp values for different irradiation energy with kl = 5 cm−2/MeV.
With: , and
Among previous work on solar cells that have undergone, irradiation of charged particles and their effects on physical mechanisms influencing performance  , our work proposes a mathematical modeling of the results of shunt resistance based on the optimum thickness specific to each flow and intensity of irradiation. This result allows the choice of the thickness of the solar cell in the process of its manufacture, in anticipation of the conditions of its operation.
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.
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