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**Electrical Parameters Determination from Base Thickness Optimization in a Silicon Solar Cell under Influence of the Irradiation Energy Flow of Charged Particles** ()

^{1}, Mamadou Lamine Ba

^{2}, Mohamed Abderrahim Ould El Moujtaba

^{2}, Youssou Traore

^{3}, El Hadji Sow

^{2}, Cheikh Tidiane Sarr

^{2}, Masse Samba Diop

^{2}, Grégoire Sissoko

^{2}

Keywords

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*Energy and Power Engineering*,

**12**, 1-15. doi: 10.4236/epe.2020.121001.

1. Introduction

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 [1], from the back surface recombination velocity modeling in white biased [2], under temperature with the junction surface recombination concept [3], by acquisition automatic of I-V properties and temperature [4] [5], under influence of incidence angle on a vertical Silicon Solar [6], by illumination wavelength effect on a parallel vertical junction silicon solar cell and under irradiation [7] and illumination level effects on macroscopic parameters of a bifacial solar cell [8].

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.

2. Presentation of the Solar Cell

Figure 1 represents a n^{+}-p-p^{+} silicon solar cell [9] [10]. The emitter is the thin (n^{+}) 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.

3. Theory

The solar cell thus achieved (Figure 1) is previously subjected to a flow of charged particles ( $\varphi p$ ) and intensity (kl) [11], allowing to simulate the conditions of operation outside the atmosphere, in the supply of satellites. Under polychromatic illumination, the density of charge carriers $\delta \left(x,\varphi p,kl\right)$ generated at point of abscissa x in the base, according to the law is defined by equation (Equation (1)), describing the generation rate [12] [13] and expressed as:

$G\left(x\right)=n\times {\displaystyle \underset{i=1}{\overset{3}{\sum}}{a}_{i}\times {e}^{-{b}_{i}\cdot x}}$ (1)

$\delta \left(x,\varphi p,kl\right)$ follows the charge transport equation given by (Equation (2)):

$D\left(kl,{\varphi}_{p}\right)\frac{{\partial}^{2}\delta \left(x,kl,{\varphi}_{p}\right)}{\partial {x}^{2}}-\frac{\delta \left(x,kl,{\varphi}_{p}\right)}{\tau}+G\left(x\right)=0$ (2)

Figure 1. Structure of the monofacial silicon solar cell (n^{+}-p-p^{+}) under irradiation.

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 [14] [15] [16] and Sb [17] [18] [19] recombination velocity rates, respectively at the junction (n^{+}/p) at x = 0, and in the rear (p/p), at x = H.

${D\left(kl,\varphi p\right)\frac{\partial \delta \left(x,kl,\varphi p\right)}{\partial x}|}_{x=0}={S}_{f}\times \delta \left(0,kl,\varphi p\right)$ (3)

${D\left(kl,\varphi p\right)\frac{\partial \delta \left(x,kl,\varphi p\right)}{\partial x}|}_{x=H}=-{S}_{b}\times \delta \left(H,kl,\varphi p\right)$ (4)

The diffusion term influenced by the irradiation conditions of the solar cell, is given by the following empirical relationship [11]:

$L\left(kl,{\varphi}_{p}\right)=\frac{1}{{\left(\frac{1}{{L}_{0}^{2}}+kl\times {\varphi}_{p}\right)}^{1/2}}$ (5)

where:

${L}_{0}$ is the diffusion length of the excess minority carriers in absence of irradiation energy flux ( ${\varphi}_{p}$ ). $L\left(kl,{\varphi}_{p}\right)$ is the diffusion length of the excess minority carrier in the base as a function of the irradiation energy flux ( $\varphi p$ ) and the damage coefficient intensity (kl) and which may be related to Einstein’s relationship by:

${\left[L\left(kl,{\varphi}_{p}\right)\right]}^{2}=\tau \times D\left(kl,{\varphi}_{p}\right)$ (6)

$D\left(kl,{\varphi}_{p}\right)$ and $\tau $ 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:

$\delta \left(x,kl,\varphi p\right)=A\times \mathrm{cosh}\left[\frac{x}{L\left(kl,\varphi p\right)}\right]+B\times \mathrm{sinh}\left[\frac{x}{L\left(kl,\varphi p\right)}\right]+{\displaystyle \sum {K}_{i}\times {e}^{-{b}_{i}\cdot x}}$ (7)

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

4. Results and Discussion

4.1. Expressions of Photocurrent, Recombination Velocity (Sb) and Phototension

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

$\begin{array}{c}Jph\left(Sf,H,kl,{\varphi}_{p}\right)=q\cdot D\left(kl,{\varphi}_{p}\right)\cdot {\left[\frac{\partial \delta \left(Sf,x,H,kl,{\varphi}_{p}\right)}{\partial x}\right]}_{x=0}\\ =q\cdot D\left(kl,{\varphi}_{p}\right)\frac{B\left(Sf,H,kl,{\varphi}_{p}\right)}{L\left(kl,{\varphi}_{p}\right)}+{\displaystyle \underset{i=1}{\overset{3}{\sum}}{K}_{i}\cdot {b}_{i}}\end{array}$ (8)

2) At the high values of recombination velocity at the junction, it is established that [16] [19] [20] [21]:

${\left[\frac{\partial Jph\left(Sf,kl,{\varphi}_{p}\right)}{\partial Sf}\right]}_{Sf\succ 5\times {10}^{5}\text{cm}\cdot {\text{s}}^{-1}}=0$ (9)

Derived from this equation, it gives the following relations:

i) $Sb0\left(H,kl,\varphi p\right)\frac{L\left(kl,\varphi p\right)}{D\left(kl,\varphi p\right)}=-\mathrm{tanh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)$ (10)

Equation (10), gives rise the definition of intrinsic velocity and becomes a diffusion velocity [19], when H is very large compared to L.

ii) $\begin{array}{l}Sb1\left(H,kl,\varphi p\right)\frac{L\left(kl,\varphi p\right)}{D\left(kl,\varphi p\right)}\\ ={\displaystyle \underset{i=1}{\overset{3}{\sum}}\frac{L\left(kl,\varphi p\right)\times {b}_{i}\times \left({e}^{{b}_{i}\cdot H}-\mathrm{cosh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)\right)-\mathrm{sinh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)}{-L\left(kl,\varphi p\right)\times {b}_{i}\times \mathrm{sinh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+\mathrm{cosh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)-{e}^{{b}_{i}\cdot H}}}\end{array}$ (11)

Equation (11) is marked by a term of absorption (b_{i}) and leads to generation velocity, when H is small compared to L [19].

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

$Vph\left(Sf,H,kl,{\varphi}_{p}\right)=\frac{Kb\times T}{q}\cdot \mathrm{ln}\left(\frac{Nb}{{n}_{i}^{2}}\cdot \delta \left(0,H,kl,{\varphi}_{p}\right)+1\right)$ (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.

4.2. Optimum Thickness of the Base, Deduced from Each Case of Irradiation Flow

The expressions (Equation (10) and Equation (11)) are represented by the Figure 2, allowing to obtain the optimum thickness (abscissa of intercept point) of the base [22] [23] for different cases of irradiation of the solar cell, and presented by Table 1. These results are decisive for the characterization of the solar cell under irradiation, through the calculation of photocurrent density, photovoltage and conversion efficiency.

4.3. $Jph\left(Sf,\varphi p,Hopt\right)-V\left(Sf,\varphi p,Hopt\right)$ Characteristic

The profile of the $Jph\left(Sf,\varphi p,Hopt\right)\text{-}V\left(Sf,\varphi p,Hopt\right)$ characteristic for different values of the irradiation energy flow and base optimum thickness is shown in Figure 3. We note that the photocurrent density $\left(J\left(Sf,\varphi p,Hopt\right)\right)$ decreases

Table 1. Base optimum thickness values obtained for different irradiation energy flow.

Figure 2. Recombination velocity versus solar cell base thickness with kl = 5 cm^{−2}/MeV, τ = 10^{−}^{6} s.

Figure 3. Characteristic photocurrent density versus photovoltage for different irradiation energy flow and base thickness with kl = 5 cm^{−2}/MeV.

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

4.4. Electrical Power of the Solar Cell

Figure 4 represents the equivalent electric circuit of an illuminated solar cell [9].

Figure 4. Equivalent electrical circuit of an illuminated solar cell.

The Ohm law applied to the circuit in Figure 4 yields the electric power delivered by the base of the solar cell to an external load as follows:

$P\left(Sf,kl,\varphi p\right)=Vph\left(Sf,kl,\varphi p\right)\times J\left(Sf,kl,\varphi p\right)$ (13)

with:

$J\left(Sf,kl,\varphi p\right)=Jph\left(Sf,kl,\varphi p\right)-{J}_{d}\left(Sf,kl,\varphi p\right)$ (14)

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

${J}_{d}\left(Sf,kl,\varphi p\right)=q\times S{f}_{0}\times \frac{{n}_{i}^{2}}{Nb}\times \mathrm{exp}\left(\frac{Vph\left(Sf,kl,{\varphi}_{p}\right)}{{V}_{T}}-1\right)$ (15)

Sf_{0} is the excess minority carrier recombination velocity associated with shunt resistance-induced charge carrier losses [24] [25], which characterizes the quality of the solar cell [15] [16] [20]. The expression intrinsic recombination velocity at the junction is given in static regime [26] [27] and under polychromatic illumination, by [28] and applied her for solar cell under irradiation as:

$S{f}_{0}\left(kl,\varphi p\right)={\displaystyle \underset{i=1}{\overset{3}{\sum}}{K}_{i}}\times \frac{D\left(kl,\varphi p\right)\times \left[{b}_{i}\times L\left(kl,\varphi p\right)-{e}^{-{b}_{i}\times H}\times \left(sh\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+{b}_{i}\times L\left(kl,\varphi p\right)\times ch\left(\frac{H}{L\left(kl,\varphi p\right)}\right)\right)\right]}{L\left(kl,\varphi p\right)\times \left[{e}^{-{b}_{i}\times H}\times \left(ch\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+{b}_{i}\times L\left(kl,\varphi p\right)\times sh\left(\frac{H}{L\left(kl,\varphi p\right)}\right)\right)-1\right]}$ (16)

4.5. $P\left(Sf,\varphi p,Hopt\right)-V\left(Sf,\varphi p,Hopt\right)$ Characteristic

Figure 5 and Figure 6 show the variations in electrical power as function of both, the excess minority carrier recombination velocity at the junction and the photovoltage for different irradiation energy flow and optimum base thickness.

We note on Figure 5, the decrease of maximum power amplitude with the irradiation energy flow corresponding the base thickness.

On Figure 6, it is also observed a decrease in power with the increase of irradiation energy flow corresponding the optimum base thickness.

4.6. Fill Factor and Efficiency

4.6.1. Fill Factor

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 [29] as:

Figure 5. Solar cell power versus junction recombination velocity for different irradiation energy flow and base thickness with kl = 5 cm^{−2}/MeV.

Figure 6. Solar cell power versus photovoltage for different irradiation energy flow and base thickness with kl = 5 cm^{−2}/MeV.

$FF=\frac{{P}_{\mathrm{max}}}{{V}_{OC}\times {J}_{SC}}$ (17)

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

J_{sc} is the short circuit current density

4.6.2. Efficiency

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:

$\eta =\frac{{J}_{SC}{}_{\mathrm{max}}\times {V}_{\mathrm{max}}}{{P}_{incident}}$ (18)

${P}_{incident}$ is the incident light power absorbed by the solar cell.

With: ${P}_{incident}=100\text{\hspace{0.17em}}\text{mW}\cdot {\text{cm}}^{-2}$ 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 Table 2.

4.6.3. Curves of Power and Efficiency

Figure 7 represents the profil of the power versus irradiation energy, Figure 8 and Figure 9 represent the profils of the power and the efficiency versus the optimum base thickness.

Table 2. Table of parameters leading to the fill factor and efficiency corresponding to the optimum base thickness for different irradiation energy flow.

Figure 7. Power versus irradiation energy.

Figure 8. Power versus base thickness.

Figure 9. Efficiency versus optimum base thickness.

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

${P}_{\mathrm{max}}=u\times \varphi {p}^{2}-w\times \varphi p+z$ (19)

with: $u=4\times {10}^{-7}\text{W}\cdot {\text{cm}}^{-\text{2}}/\text{MeV}$, $w={10}^{-4}\text{W}\cdot {\text{cm}}^{-\text{2}}/\text{MeV}$, $z=0.0247\text{\hspace{0.17em}}\text{W}/{\text{cm}}^{2}$

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

${P}_{\mathrm{max}}=\alpha \times {H}^{2}-\beta \times H+\nu $ (20)

with: $\alpha =1140.6\text{\hspace{0.17em}}\text{W}/{\text{cm}}^{4}$, $\beta =23.709\text{\hspace{0.17em}}\text{W}/{\text{cm}}^{3}$, $\nu =0.1346\text{\hspace{0.17em}}\text{W}/{\text{cm}}^{2}$

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

${\eta}_{\mathrm{max}}=m\times {H}^{2}-n\times H+k$ (21)

with: $m={10}^{6}{\text{cm}}^{-\text{2}}$, $n=24038\text{\hspace{0.17em}}{\text{cm}}^{-1}$, $k=136.58$

We note on Figure 8 and Figure 9 that the power and the efficiency increase with the increasing base thickness.

4.7. Recombination Velocity Sfmax at the Junction

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 [3] [30].

$\frac{\partial P}{\partial Sf}=0$ (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:

$\begin{array}{l}M\left(S{F}_{\mathrm{max}},kl,\varphi p\right)\\ =\frac{1}{S{F}_{\mathrm{max}}\times L\left(kl,\varphi p\right)}\times \left[1-\frac{S{F}_{\mathrm{max}}\times L\left(kl,\varphi p\right)}{Y1\times D\left(kl,\varphi p\right)+S{F}_{\mathrm{max}}\times L\left(kl,\varphi p\right)}\right]\end{array}$ (23)

$\begin{array}{l}N\left(S{F}_{\mathrm{max}},kl,\varphi p\right)\\ =\left[\frac{{\Gamma}_{\mathrm{max}}\left(0,kl,\varphi p\right)}{\left({\Gamma}_{\mathrm{max}}\left(0,kl,\varphi p\right)+\frac{n{i}^{2}}{Nb}\right)\times \left(S{F}_{\mathrm{max}}\times L\left(kl,\varphi p\right)+Y1\times D\left(kl,\varphi p\right)\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\times \left[\frac{1}{\mathrm{log}\left(\frac{Nb\times {\Gamma}_{\mathrm{max}}\left(0,kl,\varphi p\right)}{n{i}^{2}}+1\right)}\right]\end{array}$ (24)

${\Gamma}_{\mathrm{max}}\left(0,kl,\varphi p\right)$ is the density of the minority excess minority carrier at the point of maximum power, its expression is given by the following relations:

${\Gamma}_{\mathrm{max}}\left(0,kl,\varphi p\right)=\beta \times D\left(kl,\varphi p\right)\times \left[\frac{Y2+Y1-{b}_{i}\times L\left(kl,\varphi p\right)}{S{F}_{\mathrm{max}}\times L\left(kl,\varphi p\right)+Y1\times D\left(kl,\varphi p\right)}\right]$ (25)

with:

$\beta =-\frac{{a}_{i}\times L{\left(kl,\varphi p\right)}^{2}\times n}{D\left(kl,\varphi p\right)\times \left(L{\left(kl,\varphi p\right)}^{2}\times {b}_{i}^{2}-1\right)}$ (26)

$Y1=\frac{\frac{D\left(kl,\varphi p\right)}{L\left(kl,\varphi p\right)}\times \mathrm{sinh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+Sb\left(kl,\varphi p\right)\times \mathrm{cosh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)}{\frac{D\left(kl,\varphi p\right)}{L\left(kl,\varphi p\right)}\times \mathrm{cosh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+Sb\left(kl,\varphi p\right)\times \mathrm{sinh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)}$ (27)

$Y2=\frac{\left(D\left(kl,\varphi p\right)\times {b}_{i}-Sb\left(kl,\varphi p\right)\right)\times \mathrm{exp}\left(-{b}_{i}\cdot H\right)}{\frac{D\left(kl,\varphi p\right)}{L\left(kl,\varphi p\right)}\times \mathrm{cosh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)+Sb\left(kl,\varphi p\right)\times \mathrm{sinh}\left(\frac{H}{L\left(kl,\varphi p\right)}\right)}$ (28)

The graphical resolution of this transcendental equation as a function of the excess minority carrier recombination velocity Sf at the junction [30], for different irradiation energy flow corresponding the optimum base thickness, gives the Sf_{max} values by the intercept point of the two curves represented by Figure 10.

The results obtained from Figure 10 corresponding to the numerical values of Sf_{max}, are given in Table 3.

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

Curve Sf_{max}(H_{opt})

Figure 11 represents the profil of the recombination velocity Sf_{max} of the excess minority carrier at the junction yielding P_{max}, as function of optimum base thickness.

Table 3. The numerical values of Sf_{max} for different irradiation energy flow and the optimum base thickness.

Figure 10. Representation of transcendental equation versus.

Figure 11. Sf_{max} versus optimum base thickness.

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

$S{f}_{\mathrm{max}}=a\times {H}_{Opt}+b$ (29)

with: $a=5859.6\text{\hspace{0.17em}}{\text{s}}^{-1}$, $b=108.53\text{\hspace{0.17em}}\text{cm}\cdot {\text{s}}^{-\text{1}}$

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

5. Conclusion

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.

Conflicts of Interest

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

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