Determination of the Series Resistance of a Series Vertical-Junction Silicon (N+/P/P+) Solar Cell under Polychromatic Illumination and Magnetic Field: Effect of Optimum Thickness

Dibor Faye^*, Babou Dione, Mountaga Boiro, Pape Diop

Semiconductor and Solar Energy Laboratory, Department of Physics, Cheikh Anta Diop University, Dakar, Senegal.

DOI: 10.4236/jmp.2024.1510064                

Abstract

By solving the magneto-transport equation for excess minority charge carriers in the base of the series vertical-junction silicon cell, the phenomenological parameters of the cell can be determined from the boundary conditions. Photocurrent density and photovoltage are determined for each value of applied magnetic field and corresponding optimum thickness, to establish the current-voltage characteristic (J_ph(Sf, Sb, z, B, Hop)-V_ph(Sf, Sb, z, B, Hop) of the silicon cell under polychromatic illumination. This study will make it possible to reduce the material used (by reducing the optimum thickness), which will help to lower prices. It will also enable us to reduce betting effects (lower series resistance), thereby boosting solar cell efficiency.

Keywords

Series Vertical Junction Silicon Cell, Static Regime, Magnetic Field, Optimum Thickness, Series Resistance

1. Introduction

To improve the quality of solar cells, several characterization techniques are used [1], depending on the operating regime established (static or dynamic regime) [2]-[7] in order to derive phenomenological parameters such as lifetime, mobility of minority carriers, diffusion length and coefficient, recombination velocities [1] [6]-[11] of minority carriers, as well as the electrical parameters of the solar cell, such as series and shunt resistances, capacitance and power [12]-[17].

In this work, we propose the determination of the series resistance of the N+/P/P+ silicon solar cell with vertical junctions [18]-[20] under polychromatic illumination in static regime placed in a magnetic field [3] [6] for different values of the optimum thickness of the base of the solar cell imposed by the magnetic field [21]-[24].

2. Theory

The structure of the series vertical-junction solar cell [18] [20] [25] [26] under polychromatic illumination and magnetic field is shown in Figure 1 below.

Figure 1. Series vertical-junction solar cell under polychromatic illumination and magnetic field.

Figure 2 shows a schematic diagram of a series of vertical junction solar cell units under polychromatic illumination and a magnetic field.

Figure 2. Series vertical junction solar cell unit under polychromatic illumination and magnetic field.

These parts are shown in following sections.

3. The n+ Type Transmitter

The thickness is small (0.5 to 1 μm), it is heavily doped with donor atoms (10¹⁷ to 10¹⁹ atoms per cm³) and covered with a metallic contact which allows the photocreated electrical charges to be collected.

4. The p-Type Base

This part is relatively lightly doped (10¹⁵ to 10¹⁷ atoms per cm³) in acceptor atoms. Its thickness is much greater than that of the transmitter. Being p-type (doped with acceptor atoms), this part of the structure presents a deficit of electrons (minority charge carriers in the base).

5. The Transmitter-Base Junction (Space Charge Zone)

Between the two zones of the two differently doped semiconductors (n-type emitter and the p-type base), there is a junction where a very intense electric field reigns allowing the separation of the electron-hole pairs photogenerated in the base arriving at this junction.

6. Continuity Equation

The phenomena of generation, diffusion and recombination of minority excess charge carriers in the base under magnetic field and polychromatic illumination are described by a so-called continuity equation. It is defined by the expression below.

$D\left(B\right)\cdot \frac{{\partial }^2\delta \left(x\right)}{\partial x^2}-\frac{\delta \left(x\right)}{\tau }=-G\left(z\right)$ (1)

$\delta \left(x\right)$ is the density of excess minority charge carriers photogenerated in the cell base.

$D\left(B\right)$ is the diffusion coefficient of electrons in the base in the presence of the magnetic field.

$\tau$ is the lifetime of the minority carriers in the base and is defined by the following Einstein relation:

$\tau =\frac{L^2\left(B\right)}{D\left(B\right)}$ (2)

$L\left(B\right)$ is the diffusion length of excess minority charge carriers in the base.

$D\left(B\right)$ is the diffusion coefficient of minority carriers in the base [3] [6] [23], it is defined by:

$D\left(B\right)=\frac{D_0}{1+{\left(\mu \cdot B\right)}^2}$ (3)

The expression for the generation ratio of minority carriers at depth z in the base is given by the following relation:

$G\left(z\right)=n\times {\displaystyle {\sum }_{i=1}^3\left(a_i\times \exp \left(-b_i\times z\right)\right)}$ (4)

With $a_i$ et $b_i$ tabulated coefficients of solar radiation and depend on the absorption coefficient of silicon with the wavelength [27]. It makes it possible to correlate the experimental illuminance level with the reference illuminance level taken under AM 1.5.

7. Solving the Continuity Equation

The expression for the continuity equation becomes:

$\frac{{\partial }^2\delta \left(x,z,B\right)}{\partial x^2}-\frac{\delta \left(x,z,B\right)}{L^2\left(B\right)}+\frac{1}{D\left(B\right)}\times n\times {\displaystyle {\sum }_{i=1}^n\left(a_i\times \exp \left(-b_i\times z\right)\right)}=0$ (5)

The solution to this equation is in the following form:

$\begin{array}{c}\delta \left(x,z,B\right)=A\left(Sf,Sb,z,B\right)\cdot cosh\left(\frac{x}{L\left(B\right)}\right)+C\left(Sf,Sb,z,B\right)\cdot sinh\left(\frac{x}{L\left(B\right)}\right)+\\ {\displaystyle {\sum }_{i=1}^3\left(\frac{a_i\times L^2(B)}{D(B)}\times exp\left(-b_i(z)\right)\right)}\end{array}$ (6)

The coefficients $A\left(Sf,Sb,z,B\right)$ et $C\left(Sf,Sb,z,B\right)$ are determined from the boundary conditions.

8. Boundary Conditions

At the junction (x = 0):

${\frac{\partial \delta \left(x,z,B\right)}{\partial x}|}_{x=0}={\frac{Sf\cdot \delta \left(x,z,B\right)}{D\left(B\right)}|}_{x=0}$ (7)

Sf is the recombination velocity of minority charge carriers at the junction, imposed by the external charge, and reflects the flow of minority carriers through the junction [6] [9] [28] [29].

At the back surface (x = H):

${\frac{\partial \delta \left(x,z,B\right)}{\partial x}|}_{x=H}=-{\frac{Sb\cdot \delta \left(x,z,B\right)}{D\left(B\right)}|}_{x=H}$ (8)

Sb is the recombination rate of excess minority charge carriers on the back of the cell [8]-[11] [29]. It characterizes the loss of carriers at the back of the cell. The existence of the backside electric field (BSF) enables photogenerated minority carriers to be returned from the backside (p/p+ junction) to the emitter-base junction to participate in the photocurrent.

9. Photocurrent Density and Photovoltage

The expression for photocurrent density is determined from the density of minority charge carriers using Fick’s law. It is given by the following expression:

$J_{ph}\left(Sf,Sb,z,B\right)=q\cdot D\left(B\right)\cdot {\frac{\partial \delta \left(x,z,B\right)}{\partial x}|}_{x=0}$ (9)

With q the elementary electric charge.

The expression for the photovoltage across the cell under illumination is given by Boltzmann’s relation below.

$V_{ph}\left(Sf,Sb,z,B\right)=\frac{Kb\cdot T}{q}\times \ln \left[\frac{Nb}{{\left(n_i\right)}^2}\cdot \delta \left(0,Sf,Sb,z,B,Hop\right)+1\right]$ (10)

Kb is Boltzmann’s constant, T is the absolute temperature, Nb is the doping rate in the base and n_i is the intrinsic electron concentration.

10. Technique for Determining the Recombination Velocity at the Junction Limiting the Open Circuit

Figure 3 shows the technique for determining the recombination velocity of minority carriers at the junction limiting the open circuit.

At low recombination velocities of minority carriers at the junction (Sf), the photovoltage is maximum and constant: this is the value of the open-circuit photovoltage. Referring to this part of the curve, the orthogonal projection of the point limiting the open circuit situation onto the x-axis gives the value of the minority carrier recombination velocity at the junction limiting the open circuit Sfco.

Figure 3. Photovoltage profile as a function of recombination rate at the junction (τ = 10⁻⁵ s; μ = 1350 cm²∙V⁻¹ s⁻¹).

11. Effect of the Magnetic Field and the Optimum Thickness of the Base on the Characteristic J_ph(Sf, Sb, z, B, Hop)-V_ph(Sf, Sb, z, B, Hop)

Figure 4 represents the profile of the J_ph-V_ph characteristic of the silicon solar cell under polychromatic illumination for different values of the magnetic field and different values of the optimum thickness of the base imposed by the magnetic field.

Figure 4. J_ph-V_ph characteristic of the solar cell for different values of the magnetic field and optimum thickness (τ = 10⁻⁵ s; μ = 1350 cm²∙V⁻¹ s⁻¹).

Figure 4 shows that the photocurrent density is maximum and constant at low values of the photovoltage, it corresponds to the short-circuit photocurrent density. When the photocurrent density decreases and tends towards a zero value, the photovoltage increases until it reaches a maximum value: this is the open circuit photovoltage.

We also note that the increase in the magnetic field and the optimum thickness imposed by the magnetic field lead to a reduction in the short-circuit photocurrent density, on the other hand, the open circuit photovoltage increases slightly.

12. Study of Series Resistance

Series resistance is a fundamental parameter which depends on the nature of the substrate, the temperature, the technology used and is very important for the quality of a solar cell. It should ideally be as low as possible to limit its influence on the current of the solar cell [12]-[14].

13. Electrical Model of the Open Circuit Solar Cell

To determine the series resistance R_s, we propose the equivalent electrical model of the silicon solar cell in open circuit (low values of Sf which correspond to the maximum value of the photovoltage). In Figure 5, in an open circuit situation (circled area), the photocurrent density characteristic as a function of the photovoltage is assumed to be an oblique line allowing the modeling of the solar cell as a voltage generator.

Figure 5. J_ph-V_ph characteristic.

The equivalent electrical diagram of the solar cell operating in an open circuit situation is shown in the figure below.

14. Expression of Series Resistance

The expression for the series resistance is obtained by applying the mesh law to the electrical circuit in Figure 6 [30] [31]. It is given by the equation below.

Figure 6. Equivalent electrical circuit of the solar cell in open circuit.

$R_s\left(Sf,B,z\right)=\frac{V_{co}\left(B,z\right)-V_{ph}\left(Sf,B,z\right)}{J_{ph}\left(Sf,B,z\right)}$ (11)

$V_{co}\left(B,z\right)$ is the open circuit voltage.

Figure 7 shows the evolution of the series resistance as a function of the recombination velocity of minority charge carriers at the junction for different values of the magnetic field and the corresponding optimum thickness.

Figure 7. Profile of series resistance as a function of recombination velocity at the junction for different values of magnetic field and optimum thickness (τ = 10⁻⁵ s; μ = 1350 cm²∙V⁻¹ s⁻¹).

Figure 7 shows that whatever the value of the magnetic field, the series resistance increases with the velocity of recombination of the minority carriers at the Sf junction. As Sf increases, the quantity of excess minority charge carriers passing through the junction increases, causing the metal grid to heat up and the series resistance to increase with the recombination velocity at the junction.

Furthermore, we also see that when the magnetic field and the optimum thickness increase, the series resistance increases, whatever the value of the recombination velocity at the junction. The increase in series resistance with the magnetic field is due to the magnetoresistance effect.

(τ = 10⁻⁵ s; μ = 1350 cm²∙V⁻¹ s⁻¹) gives some values of the series resistance obtained from the recombination velocity of the minority carriers at the junction limiting the open circuit for different values of the magnetic field and optimum thickness imposed on the magnetic field.

Table 1. Series resistance values for different values of magnetic field and optimum thickness.

B (10−4T)	0	2	4	6	8
Hop (cm)	0.0086	0.0085	0.0082	0.0079	0.0076
Sfco (cm/s)	400	350	200	150	130
Rs (Ω/cm2)	22.5	22.8	23	24.15	25.78

Figure 8 shows the change in series resistance as a function of magnetic field. It can be seen that the series resistance increases with the magnetic field.

Figure 8. Profile of the series resistance as a function of the magnetic field.

The values in Table 1 were used to plot the series resistance profile as a function of the optimum thickness of the cell base (Figure 9).

Figure 9. Profile of series resistance as a function of optimum base thickness.

Figure 9 shows that the series resistance decreases as the optimum thickness of the solar cell base increases. The mathematical correlation equation between series resistance and optimum thickness obtained from the curve in Figure 9 is also shown in this figure.

5. Conclusion

The series resistance of the series vertical junction silicon solar cell in the static regime under polychromatic illumination and magnetic field, taking into account the optimum thickness of the base of the solar cell imposed by this magnetic field, is studied in this work. Using the boundary conditions, we first solve the diffusion equation for the minority carriers in the base, and then determine the photocurrent density and photovoltage. From the expressions for photocurrent density and photovoltage, the current-voltage characteristic of the solar cell under illumination and magnetic field conditions is presented, enabling the series resistance in the equivalent electrical circuit to be determined in an open-circuit situation. The profile of the series resistance as a function of the recombination velocity of the minority carriers at the junction for different values of the magnetic field and the corresponding optimum thickness is plotted to study the evolution of the series resistance as a function of the optimum thickness.

Notation List of All the Variables

Symbol	Physical size	Unit
x	Width of cell base	cm
z	Depth of cell base	cm
G(z)	Carrier generation ratio in the cell base	cm−3/s
Nb	Base impurity doping ratio	cm−3
ni	Intrinsic density of minority carriers with ni = 1010	cm−3
τ	Life of minority holders	s
B	Magnetic field	T
δ(x)	Charge carrier density in the cell base	cm−3
ai	Tabulated solar radiation coefficient	cm−3/s
bi	Tabulated solar radiation coefficient	cm−1
D	Diffusion coefficient	cm2/s-1
H	Thickness of the P-doped base	cm
L	Minority carrier diffusion length	cm
Sf	Recombination velocity of minority carriers at the junction	cm/s
T	Absolute temperature	K
Sb	Recombination velocity of minority carriers in the rear zone	cm/s
Vph	Photovoltage	V
Vco	Open circuit voltage	V
VT	Thermal voltage	V
Kb	Boltzmann coefficient	Sans unité
q	Elementary charge	C
Jph	Photocurrent density	A/cm2
Rs	Series resistance	Ω.cm2
Sfco	minority carrier recombination velocity at the junction limiting the open circuit Sfco.	cm/s

Conflicts of Interest

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

References

[1]	Dhariwal, S.R. and Vasu, N.K. (1981) A Generalised Approach to Lifetime Measurement in PN Junction Solar Cells. Solid-State Electronics, 24, 915-927.[CrossRef]
[2]	Diallo, H.L., Seïdou Maiga, A., Wereme, A. and Sissoko, G. (2008) New Approach of Both Junction and Back Surface Recombination Velocities in a 3D Modelling Study of a Polycrystalline Silicon Solar Cell. The European Physical Journal Applied Physics, 42, 203-211.[CrossRef]
[3]	Betser, Y., Ritter, D., Bahir, G., Cohen, S. and Sperling, J. (1995) Measurement of the Minority Carrier Mobility in the Base of Heterojunction Bipolar Transistors Using a Magnetotransport Method. Applied Physics Letters, 67, 1883-1884.[CrossRef]
[4]	Stokes, E.D. and Chu, T.L. (1977) Diffusion Lengths in Solar Cells from Short-Circuit Current Measurements. Applied Physics Letters, 30, 425-426.[CrossRef]
[5]	Sissoko, G., Nanéma, E., Corréa, A., Biteye, P.M., Adj, M. and N’Diaye, A.L. (1998) Silicon Solar Cell Recombination Parameters Determination Using the Illuminated I-V Characteristic. World Renewable Energy Congress, Florence, 20-25 September 1998, 1847-1851.
[6]	Vardayan, R.R., Kerst, U., Wawer, P., Nell, M.N. and Wagemann, H.G (1998). Method of Measurement of All Recombination Parameters in the Base Region of Solar Cells. Proceedings of 2nd Conference and Exhibition on Photovoltaic Solar Energy Conversion, Vienna, 6-10 July 1998, 191-193.
[7]	Jung, T.-W., Lindholm, F.A. and Neugroschel, A. (1984) Unifying View of Transient Responses for Determining Lifetime and Surface Recombination Velocity in Silicon Diodes and Back-Surface-Field Solar Cells, with Application to Experimental Short-Circuit-Current Decay. IEEE Transactions on Electron Devices, 31, 588-595.[CrossRef]
[8]	Sissoko, G., Museruka, C., Corréa, A., Gaye, I. and Ndiaye, A.L. (1996) Light Spectral Effect on Recombination Parameters of Silicon Solar Cell. World Renewable Energy Congress, Pergamon, Part III, 1487-1490.
[9]	Joardar, K., Dondero, R.C. and Schroder, D.K. (1989) A Critical Analysis of the Small-Signal Voltage-Decay Technique for Minority-Carrier Lifetime Measurement in Solar Cells. Solid-State Electronics, 32, 479-483.[CrossRef]
[10]	Rose, B.H. and Weaver, H.T. (1983) Determination of Effective Surface Recombination Velocity and Minority-Carrier Lifetime in High-Efficiency Si Solar Cells. Journal of Applied Physics, 54, 238-247.[CrossRef]
[11]	Fossum, J.G. (1977) Physical Operation of Back-Surface-Field Silicon Solar Cells. IEEE Transactions on Electron Devices, 24, 322-325.[CrossRef]
[12]	Bouzidi, K., Chegaar, M. and Bouhemadou, A. (2007) Solar Cells Parameters Evaluation Considering the Series and Shunt Resistance. Solar Energy Materials and Solar Cells, 91, 1647-1651.[CrossRef]
[13]	Bashahu, M. and Habyarimana, A. (1995) Review and Test of Methods for Determination of the Solar Cell Series Resistance. Renewable Energy, 6, 129-138.[CrossRef]
[14]	El-Adawi, M.K. and Al-Nuaim, I.A. (2001) A Method to Determine the Solar Cell Series Resistance from a Single I-V. Characteristic Curve Considering Its Shunt Resistance—New Approach. Vacuum, 64, 33-36.[CrossRef]
[15]	Diallo, H.L., Dieng, B., Ly, I., Dione, M.M., Ndiaye, M., Lemrabott, O.H., Bako, Z.N., Wereme, A. and Sissoko, G (2012) Determination of the Recombination and Electrical Parameters of a Vertical Multijunction Silicon Solar Cell. Research Journal of Applied Sciences, Engineering and Technology, 4, 2626-2631.
[16]	Kumar, R.A., Suresh, M.S. and Nagaraju, J. (2001) Measurement of AC Parameters of Gallium Arsenide (GaAs/Ge) Solar Cell by Impedance Spectroscopy. IEEE Transactions on Electron Devices, 48, 2177-2179.[CrossRef]
[17]	Thongpron, J., Kirtikara, K. and Jivacate, C. (2006) A Method for the Determination of Dynamic Resistance of Photovoltaic Modules under Illumination. Solar Energy Materials and Solar Cells, 90, 3078-3084.[CrossRef]
[18]	Gover, A. and Stella, P. (1974) Vertical Multijunction Solar-Cell One-Dimensional Analysis. IEEE Transactions on Electron Devices, 21, 351-356.[CrossRef]
[19]	Wise, J.F. (1970) Vertical Junction Hardened Solar Cell. US Patent 3, 690-953.
[20]	Terheiden, B., Hahn, G., Fath, P. and Bucher, E. (2000) The Lamella Silicon Solar Cell. 16th European Photovoltaic Solar Energy Conference, Glasgow, 1-5 May 2000, 1377-1380.
[21]	Ndiaye, A.M., Gueye, S., Mbaye Fall, M.F., Diop, G., Ba, A.M., Ba, M.L., et al. (2020) Diffusion Coefficient at Resonance Frequency as Applied to N+/p/p+ Silicon Solar Cell Optimum Base Thickness Determination. Journal of Electromagnetic Analysis and Applications, 12, 145-158.[CrossRef]
[22]	Faye, D., Gueye, S., Ndiaye, M., Ba, M.L., Diatta, I., Traore, Y., et al. (2020) Lamella Silicon Solar Cell under both Temperature and Magnetic Field: Width Optimum Determination. Journal of Electromagnetic Analysis and Applications, 12, 43-55.[CrossRef]
[23]	Flohr, T. and Helbig, R. (1989) Determination of Minority-Carrier Lifetime and Surface Recombination Velocity by Optical-Beam-Induced-Current Measurements at Different Light Wavelengths. Journal of Applied Physics, 66, 3060-3065.[CrossRef]
[24]	Diop, M.S., Ba, H.Y., Thiam, N., Diatta, I., Traore, Y., Ba, M.L., et al. (2019) Surface Recombination Concept as Applied to Determinate Silicon Solar Cell Base Optimum Thickness with Doping Level Effect. World Journal of Condensed Matter Physics, 9, 102-111.[CrossRef]
[25]	Hu, C., Carney, J.K. and Frank, R.I. (1977) New Analysis of a High-Voltage Vertical Multijunction Solar Cell. Journal of Applied Physics, 48, 442-444.[CrossRef]
[26]	Sarfaty, R., Cherkun, A., Pozner, R., Segev, G., Zeierman, E., Flitsanov, Y., Kribus, A. and Rosenwaks, Y. (2011). Vertical Junction Si Micro-Cells for Concentrating Photovoltaics. Proceedings of the 26th European Photovoltaic Solar Energy Conference and Exhibition, Hamburg, 5-6 September 2011, 145-147.
[27]	Furlan, J. and Amon, S. (1985) Approximation of the Carrier Generation Rate in Illuminated Silicon. Solid-State Electronics, 28, 1241-1243.[CrossRef]
[28]	Sissoko, G., Sivoththanam, S., Rodot, M. and Mialhe, P (1992) Constant Illumination-Induced Open Circuit Voltage Decay (CIOCVD) Method, as Applied to High Efficiency Si Solar Cells for Bulk and Back Surface Characterization. 11th European Photovoltaic Solar Energy Conference and Exhibition, Montreux, 12-16 October 1992, 352-354.
[29]	Diasse, O., Diao, A., Wade, M., Diouf, M.S., Diatta, I., Mane, R., et al. (2018) Back Surface Recombination Velocity Modeling in White Biased Silicon Solar Cell under Steady State. Journal of Modern Physics, 9, 189-201.[CrossRef]
[30]	Ly, I., Lemrabott, O.H., Dieng, B., Gaye, I., Gueye, S., Diouf, M.S., et al. (2023) Techniques de détermination des paramètres de recombinaison et le domaine de leur validité d’une photopile bifaciale au silicium polycristallin sous éclairement multi spectral constant en régime statique. Journal of Renewable Energies, 15, 187-206.[CrossRef]
[31]	Diouf, M.S., Gohan, S., Thiam, A., Faye, K., Ngom, M.I., Gaye, D. and Sissoko, G. (2015) Determination of the Junction Surface Recombination Velocity Limiting the Open Circuit (SFOC) for a Bifacial. International Journal of Innovative Science, Engineering & Technology, 2, 931-938.