2D Modeling of Solar Cell p-n Radial Junction: Study of Photocurrent Density and Quantum Efficiency in Static Mode under Monochromatic Illumination

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1. Introduction

The importance that has taken renewable energies in the search for alternative sources of energy to fossil traditional energies because of their rarefaction and harmful impact on the environment is translated specially to photovoltaic energy by many studies on the solar cells. We can obtain energy by converting the solar energy to electrical energy with the semiconductor optoelectronic device such as the vertical junction solar cell or planar junction [1]. Solar cell geometry seems to be very important for solar high performance. Especially since study conducted on cubic or cylindrical solar cell [2] showed a slight improvement in solar cell parameters in favor of a cylindrical model notably the radial junction solar cell [3]. The quantum efficiency is a fundamental parameter of the solar cell. Its knowledge allows determining of recombination parameters of solar cell as effective diffusion length, the effective lifetime and recombination velocity because it depends on recombination and generation phenomena in the solar cell. Several studies [4] - [9] have led to find these above parameters in order to characterize the solar cell. The originality of this paper lays its innovative and efficient aspect. The aim of this study is to investigate photocurrent density and quantum efficiency respectively in the base of a radial junction polycrystalline solar cell.

2. Materials and Methods

In this study based on a 2D modeling of a polycrystalline silicon radial junction solar cell, we made the following hypothesis:

- The grains have identical structures and electrical properties. Therefore, the study is restricted to a grain whose model is schematized in Figure 1;

- The solar cell is subjected to monochromatic front side illumination;

- The electrical field of crystal lattice and the thickness of space charge region are neglected.

Figure 1 presents a radial junction solar cell geometry:

The operation of radial junction solar cell is based on optical absorption in vertical direction (*z*). The photogenerated carriers’ charge collection is made in radial direction.

Figure 1. Radial junction model.

2.1. Study of Photocurrent Density and Quantum Efficiency

The transfer phenomenas of solar cell are model by the following equation of continuity numbered Equation (1) in cylindrical coordinates [10].

$\begin{array}{l}\frac{{\partial}^{2}\delta \left(r,\theta ,z\right)}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial \delta \left(r,\theta ,z\right)}{\partial r}+\frac{1}{r}\frac{{\partial}^{2}\delta \left(r,\theta ,z\right)}{\partial {\theta}^{2}}\\ \text{\hspace{0.05em}}+\frac{{\partial}^{2}\delta \left(r,\theta ,z\right)}{\partial {z}^{2}}-\frac{\delta \left(r,\theta ,z\right)}{{L}^{2}}=-\frac{g\left(\lambda ,z\right)}{D}\end{array}$ (1)

where:

$g\left(\lambda ,z\right)$ is carriers generation rate [10] [11]:

$g\left(\lambda ,z\right)=\alpha \left(\lambda \right){\varphi}_{0}\left[1-R\left(\lambda \right)\right]{\text{e}}^{-\alpha \left(\lambda \right)z}$ (2)

- $\delta \left(r,z\right)$ : minority carriers’ density;

- *D*: carriers’ diffusion coefficient;

- *L*: carriers’ diffusion length.

As, we have an azimuthal symmetry, the angle $\theta $ is not processed [10] [11]. Therefore the continuity equation becomes:

● In the base region, the minority carriers’ movement is governed by:

$\frac{{\partial}^{2}{\delta}_{n}\left(r,z\right)}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {\delta}_{n}\left(r,z\right)}{\partial r}+\frac{{\partial}^{2}{\delta}_{n}\left(r,z\right)}{\partial {z}^{2}}-\frac{{\delta}_{n}\left(r,z\right)}{{L}_{n}^{2}}=-\frac{g\left(\lambda ,z\right)}{{D}_{n}}$ (3)

2.1.1. Photocurrent Density Expression in the Base

Referring to Figure 1, the boundaries conditions are:

– At the junction (*r *= *R _{b}*):

${\frac{\partial {\delta}_{n}\left(r,z\right)}{\partial r}|}_{r={R}_{b}}=-\frac{{S}_{f}}{2{D}_{n}}\delta \left(r={R}_{b},z,\lambda \right)$ (5)

– At the rear side (*z *= *H*):

${\frac{\partial {\delta}_{n}\left(r,z\right)}{\partial z}|}_{z=H}=-\frac{{S}_{b}}{{D}_{n}}\delta \left(r,z=H,\lambda \right)$ (6)

A solution of equation is given by:

${\delta}_{n}\left(r,z\right)={\displaystyle \underset{k\ge 0}{\overset{\infty}{\sum}}f\left(r,\lambda \right)}\mathrm{cos}\left({c}_{k}z\right)$ (7)

The eigenvalues *C _{k}* which satisfy the boundary condition (6) are expressed by:

${c}_{k}\mathrm{tan}\left({c}_{k}H\right)=\frac{{\delta}_{b}}{{D}_{n}}$ (8)

The expression of photocurrent density is given by the following equation:

${J}_{ph}^{b}=-q{D}_{n}{\displaystyle {\int}_{0}^{H}{\frac{\partial {\delta}_{n}\left(r,z\right)}{\partial r}|}_{r={R}_{b}}\text{d}z}$ (9)

Substituting ${\delta}_{n}\left(r,z\right)$ in Equation (9); this equation becomes:

${J}_{ph}^{b}=q\frac{{S}_{f}}{2{c}_{k}}{\displaystyle \underset{k\ge 0}{\overset{\infty}{\sum}}\left({A}_{k}\left(\lambda \right){I}_{0}\left(\frac{{R}_{b}}{{L}_{nk}}\right)+{M}_{k}\left(\lambda \right)\right)}\mathrm{sin}\left({c}_{k}H\right)$ (10)

2.1.2. Quantum Efficiency in the Base

1) Internal quantum efficiency in the base

The expression of internal quantum efficiency (IQE) is giving by the following expression:

$\text{IQE}\left(\lambda \right)=\frac{{J}_{ph}^{B}}{q{\varphi}_{0}\left[1-R\left(\lambda \right)\right]}$ (11)

By replacing ${J}_{ph}^{B}$ expression (10) in Equation (11), we get Equation (12):

$\text{IQE}\left(\lambda \right)={\displaystyle \underset{k\ge 0}{\sum}\frac{-{S}_{f}}{2{\varphi}_{0}{C}_{k}\left[1-R\left(\lambda \right)\right]}}\mathrm{sin}\left({C}_{k}\cdot H\right)\left[{A}_{k}\left(\lambda \right){I}_{0}\left(\frac{{r}_{b}}{{L}_{nk}}\right)+{M}_{k}\left(\lambda \right)\right]$ (12)

2) External quantum efficiency in the base

The expression of external quantum efficiency (EQE) is giving by the following expression:

$\text{EQE}\left(\lambda \right)=\frac{{J}_{ph}^{B}}{q{\varphi}_{0}}$ (13)

By replacing ${J}_{ph}^{B}$ expression (10) in Equation (13), we obtain

$\text{EQE}\left(\lambda \right)={\displaystyle \underset{k\ge 0}{\sum}\frac{-{S}_{f}}{2{\varphi}_{0}{C}_{k}}}\mathrm{sin}\left({C}_{k}\cdot H\right)\left[{A}_{k}\left(\lambda \right){I}_{0}\left(\frac{{r}_{b}}{{L}_{nk}}\right)+{M}_{k}\left(\lambda \right)\right]$ (14)

${I}_{0}\left(x\right)$ is the modified Bessel function of the first kind of zero order and ${{I}^{\prime}}_{0}\left(x\right)$ is its first derivative with respect to its argument.

3. Results and Discussions

We present here the simulation results obtained from the previous modeling equations by using Mathcad software.

3.1. Photocurrent Density in the Base

The effects of wavelength, base radius and base thickness photocurrent density are presented and analyzed.

3.1.1. Wavelength Effects on Photocurrent Density

Figure 2 shows the profile of photocurrent density versus wavelength for various values of junction recombination velocity*S _{f}*

We remark a low photocurrent density for low and big wavelength. The absorption peak is notified around 700 nm.

These phenomena are explained by electrons density variations. In additional, photocurrent density increase for ${10}^{2}\text{cm}/\text{s}\le {S}_{f}\le {10}^{6}\text{cm}/\text{s}$ because more electrons cross the junction.

For ${S}_{f}\prec {10}^{2}\text{cm}/\text{s}$, photocurrent density is almost zero because there is a low displacement and strong accumulation of electrons near the junction. Solar cell is then an open circuit state [1] [12].

3.1.2. Base Radius Effects on Photocurrent Density

The following figures present variations of photocurrent density as a function

Figure 2. Profile of the photocurrent density as a function of the wavelength for different values of the recombination velocity of the radius *R _{b}* of the base with,

of base radius for different values of the depth creating more electrons because the emitter thickness increase.

On Figure 3, the profile of photocurrent density as a function of dynamic velocity to the junction for various values of base radius *R _{b}.*

The photocurrent increase with the increase of base radius *R _{b}*. The base radius increase creates a large incident light absorption, thus leading to a large of carriers photogenerations in solar cell. Also, this is due to the increasing of base radius minimizes the recombinations which lead to a storage of charge carriers in the base. Results are in good with those obtained by [3] [11] and presented and that lower base radius corresponds to an increase of activities in recombination centers leading to decrease the photocurrent and hence the solar cell efficiency.

3.2. Internal Quantum Efficiency in the Base

3.2.1. Base Radius Effect on Internal Quantum Efficiency

In Figure 4, we present the IQE as a function of wavelength for different values base radius.

The internal quantum efficiency decrease when the base radius increase. This is a consequence of the increase of carriers’ density with base radius because much carriers are collected. Also, the electron maximum decrease in the base depth explaining these variations.

3.2.2. Base Thickness Effect on Internal Quantum Efficiency

In Figure 5, we present the variations of internal quantum efficiency as a function of wavelength with various base thickness values.

Figure 3. Profile of the photocurrent density as a function of the recombination velocity at the junction for different values of the base radius with: *S _{b}* = 10

Figure 4. Internal quantum efficiency versus wavelength with various base radius values, *S _{b}* = 10

Figure 5. Internal quantum efficiency versus wavelentgth with various base thickness values, *S _{b}* = 10

The observing of curves indicate the decrease of internal quantum efficiency with base thickness increase. This increase is relatively greater than the increase with base radius. This increase is explained also by the silicon polycrystalline which grains are small thickness.

3.2.3. Wavelength Effect on Internal Quantum Efficiency

In figures, the variations of internal quantum efficiency with wavelength for various values of base thickness and base radius are presented.

Whether Figure 4 or Figure 5, the curves of internal quantum efficiency present three parts:

- $300\text{\hspace{0.17em}}\mu \text{m}\le \lambda \le 600\text{\hspace{0.17em}}\mu \text{m}$, the internal quantum efficiency increase with wavelength for various values of base thickness or base radius because the absorption increase with wavelength up to reach the peak (silicon energy gap 1.12 eV). The generation of electrons is great for small values of wavelength. This low increase is

- The internal quantum efficiency maximum value is reached to 600 µm. The generation of electron is very important for this wavelength value.

- $600\text{\hspace{0.17em}}\mu \text{m}\le \lambda \le 900\text{\hspace{0.17em}}\mu \text{m}$, the internal quantum efficiency decrease with wavelength for various values of base thickness or base radius. This decrease can be explained by the reduction of electrons density in the base. Also, for big values of wavelength, there is no electron generation in the base.

The different variations of internal quantum efficiency are explained by the photocurrent variations. Also, the internal quantum is as a function of photocurrent.

3.3. External Quantum Efficiency in the Base

3.3.1. Base Radius Effects on External Quantum Efficiency

The influences of base radius on external quantum efficiency are illustrated in Figure 6.

Figure 6. External quantum efficiency versus wavelentgth with various base radius values, *S _{b}* = 10

Figure 7. External quantum efficiency versus wavelentgth with various base radius values, *S _{b}* = 10

The curves analysis indicates that the external quantum efficiency increase crease with base radius. For $\lambda \ge 700\text{\hspace{0.17em}}\mu \text{m}$, the decrease is asymptotic as if there is none incident absorption so none photocurrent.

3.3.2. Base Radius Effects on External Quantum Efficiency

The impacts of base thickness on external quantum efficiency are illustrated in Figure 7.

The external quantum efficiency decrease with the increase of base thickness. So the reduction of base thickness traduce the decrease of the carriers surface generation.

3.4. Wavelength Effects on External Quantum Efficiency

In figures, the curves of external quantum efficiency with wavelength for various values of base thickness and base radius are plotted.

Whether the figure, the curves of internal quantum efficiency present two parts:

- $300\text{\hspace{0.17em}}\mu \text{m}\le \lambda \le 700\text{\hspace{0.17em}}\mu \text{m}$, the decrease of external quantum efficiency is slow.

- $\lambda \ge 700\text{\hspace{0.17em}}\mu \text{m}$, the decrease of external quantum efficiency is asymptotic.

For short wavelengths, only a small fraction of the light is converted into electron –holes pairs. Most photons are already absorbed in the layers that the light traverses prior to the absorber layer; this is called parasitic absorption.

4. Conclusions

This paper carried out 2D modelling of radial junction of solar cell. The effects of base radius, base thickness and wavelength on carriers’ photocurrent density and quantum efficiency are investigated.

Also, the influences of base radius, base thickness and charges carriers’ recombination velocity on photocurrent density and quantum efficiency of solar cell are presented and analyzed. The results of this study which are in good agreement with other research works [3] [13] validate thus our results.

The 2D modeling of solar cell p-n radial junction quantum efficiency is studied in first time in cylindrical coordinates according to our research. So this work will permit to study better the radial junction of solar cell. In the future, we will try to make an experimental study to confirm theoretical results with experimental results.

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