Performance Analysis of a Radial N+/P Silicon Solar Cell in Steady State and Monochromatic Illumination ()

Aboubacar Savadogo^{}, Bernard Zouma^{}, Bruno Korgo^{}, Ramatou Konaté^{}, Sie Kam^{}

Departement de Physique, Laboratoire d’Energies Thermiques Renouvelables, Unité de Formation et de Recherche en Sciences exactes et appliquées, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso.

**DOI: **10.4236/ampc.2023.1312015
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Departement de Physique, Laboratoire d’Energies Thermiques Renouvelables, Unité de Formation et de Recherche en Sciences exactes et appliquées, Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso.

In this paper, we investigate theoretically a radial
n^{+}/p silicon solar cell in steady state and monochromatic
illumination. The purpose of this work is to analyze the effect of the cell
base radius on its electrical parameters. The continuity equation in
cylindrical coordinates is established and solved based on Bessel functions and
boundaries conditions; this led us to the photovoltage and photocurrent density
in the cell. The open circuit voltage and the short circuit current density are
then deduced and analyzed considering the base radius. Based on J-V and P-V
curves, series and shunt resistances, fill factor and maximum power point are
derived and the conversion efficiency of the cell is deduced. We showed that
short circuit current density, maximum power, conversion efficiency and shunt
resistance decrease with increasing base radius contrary to the open circuit
voltage, the fill factor and the series resistance.

Share and Cite:

Savadogo, A. , Zouma, B. , Korgo, B. , Konaté, R. and Kam, S. (2023) Performance Analysis of a Radial N+/P Silicon Solar Cell in Steady State and Monochromatic Illumination. *Advances in Materials Physics and Chemistry*, **13**, 207-217. doi: 10.4236/ampc.2023.1312015.

1. Introduction

To improve the performance and reduce the cost of solar cells or to overcome the limitations exhibited by Shockley and Queisser [1] , many and diverse studies on n-p junction solar cells are conducted. There are thus several configurations: we encounter parallelepiped or cubic configuration with the classic horizontal junction between the base and emitter assuming that the generated carriers move from the base to the emitter parallel to the illumination [2] [3] [4] and vertical junction for the purpose to reduce the path of the carriers to the junction to decrease recombination in the base of the cell [5] [6] [7] . Also, we can find in the literature cylindrical configurations with planar [8] [9] [10] and radial [11] [12] [13] [14] p-n junction as depicted in Figure 1 which is deeply investigated in this paper.

Trabelsi *et al.* [9] and Mbodji *et al.* [8] carried out work on a cylindrical grain N^{+}/P polycrystalline solar cell. The study [9] reveals that the recombination at the grain boundaries becomes non-negligible and the photocurrent decreases for a photovoltaic cell whose thickness H is greater than the diffusion length for a small cylindrical grain radius. And the optimum thickness to obtain more than 95% of the available current density is around 50 μm. And on the other hand Mbodji *et al.* [9] showed that photocurrent density and photovoltage increase with grain radius. And the shunt resistance also increases with the radius of the grain for large values of the dynamic speed at the junction.

Leye *et al.* [10] studied the temperature effect on the performance of an N/P solar cell considering the columnar cylindrical grain. They showed that when the temperature increases the short circuit increases however, the open circuit decreases. Some parameters, such as the short-circuit photocurrent density, the open circuit photovoltage, the fill factor, and the efficiency are linearly dependent on the temperature. These results mentioned above were obtained by considering a planar junction.

As for the radial junction, it should be noted that it is essentially perceived in nanotechnologies such as nanowires offering many advantages. Failing to increase the maximum efficiency beyond the standard limits, radial geometry allows to reduce both the quantity and quality of the material necessary to approach these limits, thus allowing a substantial reduction in costs [12] [13] .

The geometric configuration of solar cells therefore seems to be a key element. Especially since studies carried out on cubic and cylindrical solar cells have shown a slight improvement in photovoltaic performance in favor of the cylindrical model [9] . Simulation work has highlighted the advantage of the radial junction over a planar junction [14] .

Figure 1. Radial p-n junction solar cell.

Sam *et al.* [11] were able, through a 2D approach, to study the effect of the base radius, the thickness and the wavelength on the photocurrent density and the quantum efficiency of a radial junction solar cell. However their study doesn’t give any information on the effect of the base radius on the other electrical parameters such as photovoltage, electric power, fill factor, shunt and series resistances.

The purpose of this article is to broaden the field of study in this field by studying the influence of the base radius of a radial junction cell on its electrical parameters.

2. Theoretical Background

Our model is illustrated in Figure 1. It consists of three parts including two coaxial cylinders:

The first solid cylinder forms the base of the solar cell. It is a p-type region (10^{15} to 10^{17} cm^{−3}) of radius *R _{b}* and depth

The second hollow cylinder, which circles the first one, represents the n^{+}-type emitter of thickness d and is heavily doped (10^{18} to 10^{19} cm^{−3}).

The third part is what is called the space charge region (SCR). Located and created between the emitter and the base, it is a transition region where there is an intense electric field which separates electron-hole pairs arriving at the junction.

2.1. Assumptions

To simplify our study, we assume that:

The contribution of the emitter is not considered [8] [11] also implying the non-taking into account of the grain limits.

The penetration of light is along the z- axis parallel to the junction and the generation of excess minority charge carriers.

The transport of excess minority charge carriers is according to the position r in the base [14] .

The base doping is uniform, so no field exists outside the space charge regions: the base is quasi-neutral [9] [15] .

The Earth’s magnetic field is not considered because that magnetic fields of values smaller than 10^{−5} T have no significant effect on a solar cell [2] .

2.2. Continuity Equation

When the solar cell is illuminated, there are three major phenomena that occur inside: carrier generation, recombination, and drift/diffusion. These phenomena are described by the continuity equation. Assuming that carriers generation is uniform according to *θ* and taking into account the symmetry of the model, carriers density should be invariant by any *θ* angle rotation [8] [10] [13] [16] . The continuity equation can then be written as:

$\begin{array}{l}\frac{{\partial}^{2}{\delta}_{n}\left(r,z,\lambda \right)}{\partial {r}^{2}}+\frac{1}{r}\cdot \frac{\partial {\delta}_{n}\left(r,z,\lambda \right)}{\partial r}+\frac{{\partial}^{2}{\delta}_{n}\left(r,z,\lambda \right)}{\partial {z}^{2}}-\frac{{\delta}_{n}\left(r,z,\lambda \right)}{{L}_{n}^{2}}\\ =-\frac{\alpha \left(\lambda \right)\left(1-R\right){\varphi}_{0}{\text{e}}^{-\alpha \left(\lambda \right)z}}{{D}_{n}}\end{array}$ (1)

*δ _{n}* is the excess minority carrier density,

Equation (1) is a partial differential equation whose general solution can be given in the following form [9] [11] :

${\delta}_{n}\left(r,z,\lambda \right)={\displaystyle \underset{k\ge 0}{\sum}f\left(r,\lambda \right)\mathrm{sin}\left({C}_{k}z\right)}$ (2)

*C _{k}* and

At the emitter base interface (*r *= *R _{b}*):

${\frac{\partial {\delta}_{n}\left(r,z,\lambda \right)}{\partial r}|}_{r={R}_{b}}=-\frac{{S}_{f}}{2{D}_{n}}\cdot {\delta}_{n}\left({R}_{b},z,\lambda \right)$ (3)

At the backside (*z* = *H*):

${\frac{\partial {\delta}_{n}\left(r,z,\lambda \right)}{\partial z}|}_{z=H}=-\frac{{S}_{b}}{{D}_{n}}\cdot {\delta}_{n}\left(r,H,\lambda \right)$ (4)

With *S _{f}* and

At the front side (*z* = 0):

${\delta}_{n}\left(r,z=0,\lambda \right)=0$ (5)

3. Electrical Parameters

3.1. Photocurrent Density

The photocurrent density is determined by using Fick’s law:

${J}_{ph}=\frac{q{D}_{n}}{\pi {R}_{b}^{2}}{\displaystyle \underset{0}{\overset{H}{\int}}{\frac{-\partial {\delta}_{n}\left(r,z,\lambda \right)}{\partial r}|}_{r={R}_{b}}\left(2\pi {R}_{b}\right)\text{d}z}$ (6)

*q* is the electron charge.

Inserting Equation (3) into Equation (6) and doing the integration, we obtain Equation (7):

${J}_{ph}={\displaystyle \underset{k\ge 0}{\sum}\frac{-q{S}_{f}}{{R}_{b}{C}_{k}}\left[{A}_{k}\left(\lambda \right){I}_{0}\left(\frac{{R}_{b}}{{L}_{nk}}\right)+{M}_{k}\left(\lambda \right)\right]\left[\mathrm{cos}\left({C}_{k}H\right)-1\right]}$ (7)

In Equation (7) *I*_{0} represent the Bessel function of modified 1^{st} species of order 0, the expressions of *M _{k}* and

${M}_{k}\left(\lambda \right)=\frac{2\alpha \left(\lambda \right)\left(1-R\left(\lambda \right)\right){\varphi}_{0}{L}_{nk}^{2}{C}_{k}^{2}\left[1+{\left(-1\right)}^{k+1}\mathrm{exp}\left(-\alpha H\right)\right]}{{D}_{n}\left({C}_{k}^{2}+{\alpha}^{2}\right)\left({C}_{k}H-\mathrm{sin}\left({C}_{k}H\right)\mathrm{cos}\left({C}_{k}H\right)\right)}$ (8)

${A}_{k}\left(\lambda \right)=-\frac{{M}_{k}\left(\lambda \right)}{\frac{2{D}_{n}}{{S}_{f}{L}_{nk}}{{I}^{\prime}}_{0}\left(\frac{{R}_{b}}{{L}_{nk}}\right)+{I}_{0}\left(\frac{{R}_{b}}{{L}_{nk}}\right)}$ (9)

3.2. Photovoltage

The photovoltage is obtained from Boltzmann relation in the form:

${V}_{ph}={V}_{T}\mathrm{ln}\left[1+\frac{1}{{n}_{0}}{\displaystyle \underset{0}{\overset{H}{\int}}{\delta}_{n}\left(r={R}_{b},z,\lambda \right)\left(2\pi {R}_{b}\right)\text{d}z}\right]$ (10)

By using Equation (3), Equation (10) can be rewritten as:

${V}_{ph}={V}_{T}\mathrm{ln}\left[1+\frac{4\pi {R}_{b}{D}_{n}}{{n}_{0}{S}_{f}}{\displaystyle \underset{k\ge 0}{\sum}\frac{{A}_{k}\left(\lambda \right){{I}^{\prime}}_{0}\left(\frac{{R}_{b}}{{L}_{nk}}\right)}{{C}_{k}{L}_{nk}}\left[\mathrm{cos}\left({C}_{k}H\right)-1\right]}\right]$ (11)

${V}_{T}=\frac{{K}_{B}T}{q}$ is the thermal voltage,
${n}_{0}=\frac{{n}_{i}^{2}}{{N}_{B}}$ is the electron density at equilibrium, *n _{i}* the intrinsic carrier’s density,

4. Results and Discussion

Figure 2 presents the photocurrent density versus junction dynamic velocity for various base radius.

For lower *S _{f}* (≤10

In Figure 3, we plotted the photovoltage versus junction dynamic velocity for various base radius.

As noted on Figure 3, when the cell radius increases there is more carrier’s accumulation across the junction. The more the cell radius is, the more the n-p junction is far for charge carrier. For those carriers not recombined, there is an accumulation and then an increase of the photovoltage.

Those observations can be seen on Figure 4 with the Photocurrent density versus Photovoltage curves.

Figure 2. Photocurrent density versus junction dynamic velocity for various base’s radius. *L _{n}* = 50 µm,

Figure 3. Photovoltage versus junction dynamic velocity for various base’s radius. *L _{n}* = 50 µm,

Figure 4. Photocurrent density versus photovoltage for various base’s radius. *S _{b}* = 2 × 10

The short circuit current density decreases with increasing base radius while the open circuit voltage increases. Figure 4 also shows that short circuit current density is more sensitive to the base radius contrary to the open circuit voltage.

For a given Photocurrent density—Photovoltage curve, the solar cell behaves like a real current generator near short circuit and like a real voltage generator near open circuit. The solar cell can then be represented by the following electrical equivalent circuits [6] [7] .

From the electrical equivalent circuits depicted in Figure 5 and Figure 6, expressions of shunt and series resistances are derived as follow:

${R}_{s}=\frac{{V}_{phco}-{V}_{ph}\left({S}_{f}\right)}{{J}_{ph}\left({S}_{f}\right)}$ (12)

${R}_{sh}=\frac{{V}_{ph}\left({S}_{f}\right)}{{J}_{phcc}-{J}_{ph}\left({S}_{f}\right)}$ (13)

Series and shunt resistances curves are presented respectively on Figure 7 and Figure 8 versus junction dynamic velocity for various base radiuses.

Figure 5. Solar cell electrical equivalent circuit near open circuit.

Figure 6. Solar cell electrical equivalent circuit near short circuit.

Figure 7. Series resistance versus junction dynamic velocity for various base’s radius. *S _{b}* = 2 × 10

Figure 8. Shunt resistance versus junction dynamic velocity for various base’s radius *R _{b}*;

Beyond a slight increase in the series resistance *R _{s}* with increasing junction dynamic velocity, it can be noticed through Figure 7, that when

Also, it is noted through Figure 8 that the shunt resistance (*R _{sh}*) increases as the dynamic velocity at the

Figure 9 presents the output power versus photovoltage for various base’s radius Rb.

The output power increases with the photovoltage up to a maximum value, called the maximum power point or peak power, before decreasing and canceling out at the open circuit. Increasing the radius of the base leads to a decrease in maximum power (*P _{m}*) and an increase in circuit voltage (

Figure 9. Output power versus photovoltage for various base’s radius *R _{b}*.

Table 1. Electrical parameters of the solar cell for various base radius.

5. Conclusion

In this paper, we studied the effect of the base radius on the electrical parameters of a radial n-p junction solar cell. After establishing and solving the continuity equation in cylindrical coordinates, new expressions for the photovoltage and the photocurrent density were deduced. This allowed the deduction of the series and shunt resistances, the fill factor (FF) and conversion efficiency. The study reveals that when the radius of the base increases, the short-circuit current density, the maximum power, shunt resistance and the conversion efficiency (*η*) decrease. However, there is an increase in the open circuit photovoltage, the fill factor and the series resistance.

Acknowledgment

The authors wish to express their gratitude to the international Scientific Program (ISP), University of Uppsala, Sweden for financial support through the project BUF01.

Conflicts of Interest

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

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