A cross-sectional investigation into the prevalence and contributing factors of obesity among undergraduate students at the University of Buraimi. Oman

 Corresponding author: Mamadou Yacine Ba; Email: Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Analytical modelling of diffusion capacitance under monochromatic illumination: Static case Mamadou Yacine Ba *, Pape Diop, Papa Monzon Samake and Amadou Diao Department of Physics, Faculty of Science and Technology, Cheikh Anta Diop University (UCAD), Dakar, Senegal. World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 Publication history: Received on 09 August 2025; revised on 14 September 2025; accepted on 18 September 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.27.3.3241 Abstract This study proposes an analytical modelling of the diffusion capacitance of a vertical-junction solar cell under monochromatic illumination in the static regime. The developed model incorporates the effects of illumination wavelength, base depth, and applied voltage, enabling an accurate description of the minority carrier dynamics within the junction. The results show that the diffusion capacitance decreases with increasing photon penetration depth and exhibits a nonlinear dependence on voltage. This analytical approach provides an efficient tool for predicting capacitive performance and may guide the optimization of solar cells in specialized photovoltaic applications. Keywords: Vertical-junction solar cell; Diffusion capacitance; Monochromatic illumination; Analytical modelling; Static regime 1. Introduction A precise understanding of the capacitive characteristics of solar cells is essential for the optimization of modern photovoltaic devices [1][2] Among these characteristics, the diffusion capacitance plays a key role in the dynamics of minority carriers and directly influences the efficiency and stability of the cell [3][4][7]. Most previous studies have focused on the experimental or numerical evaluation of capacitance under polychromatic or dynamic regimes [5][6][10][11], leaving a lack of analytical tools to describe the behavior under monochromatic illumination in static conditions. Analytical models have the advantage of providing a direct physical description, allowing a clear identification of the influence of structural and operational parameters on capacitance [8][13][15]. This study therefore proposes an analytical model of the diffusion capacitance of a vertical-junction solar cell operating in the static regime under monochromatic illumination. The main objective is to determine the influence of illumination wavelength, base depth, and applied voltage on capacitance, in order to provide a predictive tool for the design and optimization of solar cells in specialized photovoltaic applications [5][12][13]. World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 1271 2. Theoretical Model 2.1. Solar Cell Description The structure under study is a vertical-junction silicon solar cell connected in series (Figure 1). Each elementary cell consists of an n⁺ emitter, a space-charge region (SCR), a p-type base, and a heavily doped p⁺ region serving as the back collector (Figure 2) [5][13]. The illumination is monochromatic and incident perpendicularly to the front surface (emitter side), along the Z-axis. The photon flux penetration depth in the base is denoted as α⁻¹(λ), depending on the wavelength λ [3][14]. The modelling is carried out along the X-axis, perpendicular to the surface of the cell, with: ● x = 0 : emitter /base interface (front), ● x = H: base/collector interface (back). 2.2. Adopted Physical Assumptions The regime is static (no time dependence). The illumination is monochromatic with an optical generation described by: 𝐺(𝑧)=𝐺𝑜𝑒−𝛼𝑧 [4] where 𝐺𝑜 is the generation rate at the surface and α the absorption coefficient [4]. Bulk recombination in the base follows a lifetime law with a mean lifetime τ [6][7]. Surface recombination is taken into account both at the front interface (Sf) and the rear interface (Sb). The Shockley law is applied at the junction (SCR) [7]. 2.3. Minority Carrier Diffusion Equation In the p-type base, the minority carriers are electrons. Their excess density satisfies the diffusion differential equation: 𝐷𝑛𝑑2𝛿𝑛 𝑑𝑥2−𝛿𝑛(𝑥) 𝜏𝑛+𝐺(𝑥)=0 Where : ● 𝐷𝑛: electron diffusion coefficient, ● 𝜏𝑛: minority carrier lifetime, ● 𝐺(𝑥) : optical generation function [3][11]. 2.4. Analytical Solution of Carrier Density The general solution of this equation is: 𝛿𝑛(𝑥)=𝐴 𝑠𝑖𝑛ℎ (𝑥 𝐿𝑛) + 𝐵 𝑐𝑜𝑠ℎ (𝑥 𝐿𝑛)+ 𝐺𝑜𝑒−𝛼𝑥 1 𝐿𝑛−𝛼2 with 𝐿𝑛=√𝐷𝑛𝜏𝑛 , the electron diffusion length. The constants A and B are determined from the boundary conditions. 2.5. Boundary Conditions At the front interface (x = 0): World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 1272 𝐷𝑛𝑑𝛿𝑛(0) 𝑑𝑥 =𝑆𝑓𝛿𝑛(0) At the back interface (x = H): 𝐷𝑛𝑑𝛿𝑛(𝐻) 𝑑𝑥 =−𝑆𝑏𝛿𝑛(𝐻) These conditions represent the effect of surface recombination velocities [8][6][11]. 2.6. Diffusion Capacitance The diffusion capacitance is defined as: 𝐶𝑑=𝑞𝑑𝑄 𝑑𝑉 where Q is the stored charge in the base: 𝑄=∫ ⬚ 𝐻 0𝛿𝑛(𝑥)𝑑𝑥 The dependence of 𝐶𝑑 on applied voltage V, photon penetration depth 1/α(λ), and base thickness HH, is obtained by substituting the analytical solution [5][13] of 𝛿𝑛(𝑥) into the integral. The diffusion capacitance is thus the derivative with respect to V of the stored injected charge, under constant optical parameters. Limiting cases (rear surface passivated, highly recombinative, thin or thick base) have already been studied in several works [6][7][16][12]. By replacing 𝑑𝛿𝑛0 𝑑𝑉 : 𝐶𝑑(𝐻,𝑉)=𝑞2 𝐾𝑇𝑛𝑖 2 𝑁𝐴𝑒𝑞𝑉 𝐾𝑇 ×𝐿𝑛 𝐷𝑛 𝐿𝑛 𝑠𝑖𝑛ℎ(𝐻 𝐿𝑛)+𝑆𝑏(𝑐𝑜𝑠ℎ𝑐𝑜𝑠ℎ (𝐻 𝐿𝑛) − 1) 𝐷𝑛 𝐿𝑛 𝑐𝑜𝑠(𝐻 𝐿𝑛) + 𝑆𝑏 𝑠𝑖𝑛ℎ(𝐻 𝐿𝑛) 𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝑒𝑞𝑉 𝐾𝑇 and 𝐿𝑛 𝐷𝑛 𝐿𝑛 𝑠𝑖𝑛ℎ(𝐻 𝐿𝑛)+𝑆𝑏(𝑐𝑜𝑠(𝐻 𝐿𝑛) −1) 𝐷𝑛 𝐿𝑛 𝑐𝑜𝑠(𝐻 𝐿𝑛) +𝑆𝑏 𝑠𝑖𝑛ℎ(𝐻 𝐿𝑛) represents the injection component, while the fraction involving hyperbolic functions accounts for geometric and recombination effects. Minority carrier concentration at the edge of the SCR (Shockley law, low injection): 𝛿𝑛𝑜(𝑉)=𝑛𝑖 2 𝑁𝐴(𝑒𝑞𝑉 𝐾𝑇 −1) , 𝑑𝛿𝑛𝑜 𝑑𝑉 =𝑞 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝑒𝑞𝑉 𝐾𝑇 For simplification, let: 𝐶≡𝑐𝑜𝑠ℎ 𝑐𝑜𝑠ℎ (𝐻 𝐿𝑛) , 𝐿𝑛=√𝐷𝑛𝜏𝑛 et 𝑆≡ 𝑠𝑖𝑛ℎ(𝐻 𝐿𝑛) The diffusion capacitance then becomes: 𝐶𝑑(𝐻,𝑉)= 𝑞𝑑𝛿𝑛𝑜 𝑑𝑉 𝐿𝑛 𝐷𝑛 𝐿𝑛 𝑆+ 𝑆𝑏(𝐶−1) 𝐷𝑛 𝐿𝑛 + 𝑆𝑏 𝑆 World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 1273 where q is the elementary charge, 𝐷𝑛 the diffusion coefficient, 𝜏𝑛 the lifetime, and 𝑆𝑏the back surface recombination velocity. Useful limiting cases : Perfectly passivated back surface (𝑆𝑏→0): Cd→𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝐿𝑛𝑒𝑞𝑉 𝐾𝑇tanh(𝐻 𝐿𝑛)) Highly recombinative back surface (𝑆𝑏→∞): Cd→ 𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝐿𝑛𝑒𝑞𝑉 𝐾𝑇 𝑐𝑜𝑠ℎ𝐻 𝐿𝑛−1 𝑠𝑖𝑛ℎ𝐻 𝐿𝑛=𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝐿𝑛𝑒𝑞𝑉 𝐾𝑇 (𝑐𝑜𝑡ℎ𝐻 𝐿𝑛−𝑐𝑜𝑠ℎ 𝐻 𝐿𝑛)) Thick base (H≫Ln): 𝐶𝑑≈𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝐿𝑛𝑒𝑞𝑉 𝐾𝑇 Thin base (H≪Ln): 𝐶𝑑≈𝑞2 𝐾𝑇 𝑛𝑖 2 𝑁𝐴𝐿𝑛𝑒𝑞𝑉 𝐾𝑇 𝐻 1+𝑆𝑏𝐻 2 𝐷𝑛 3. Results and Discussion Figure 3 Diffusion capacitance as a function of the applied voltage World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 1274 Figure 4 Diffusion capacitance as a function of base thickness Figure 5 Diffusion capacitance as a function of wavelength Figure 3 illustrates the evolution of diffusion capacitance as a function of the applied voltage [3][5][7]. A pronounced decrease in Cd is observed when the forward bias increases. This behavior results from the factor(1+ 𝑉 𝑉𝑇)−1included in the analytical expression, which leads to a significant reduction by an order of magnitude as soon as V≫VT (VT ≈ 25.9 mV at 300 K). Thus, at V=0.8 V, the capacitance drops to about 3% of its initial value under low bias. This decrease reflects, within the adopted model, a reduced capacitive contribution of minority carriers as the forward bias increases. However, it should be noted that in some classical models, diffusion capacitance may instead increase in forward regime, highlighting the need to specify the physical conditions and assumptions retained. Figure 4 presents the variation of Cd as a function of base thickness [10][13]. The dependence exhibits a non-monotonic shape: capacitance initially increases with W, reaches a maximum for W≈1/α, and then decreases exponentially beyond this characteristic depth. This behavior can be explained by the existence of an optimal thickness where carrier generation is efficiently coupled to the junction, before deeper absorption and recombination losses lead to a reduction in capacitance. In the considered case (λ=600 nm), this maximum is expected around 0.6 μm. World Journal of Advanced Research and Reviews, 2025, 27(03), 1270-1276 1275 Figure 5 shows the spectral dependence of diffusion capacitance for a fixed base thickness. The results indicate an increase in Cd with wavelength in the studied range (400–1000 nm) [10][14]. This trend arises from the simplified law α∼1/λ, which favors longer wavelengths where optical penetration is deeper and the capacitive contribution becomes more significant. However, it is important to note that the actual absorption spectrum of silicon differs from this approximation; using tabulated data for α(λ) would refine these results and enrich the discussion. Overall, these three results [5][12][13] highlight the importance of electrical parameters (applied voltage), geometrical parameters (base thickness), and optical parameters (illumination wavelength) in the capacitive behavior of the solar cell. A comparison with experimental data, as well as the introduction of a more realistic absorption coefficient law, would constitute essential extensions to strengthen the validity and applicability of the model. 4. Conclusion This study analysed the diffusion capacitance of a vertical-junction solar cell in the static regime, highlighting its evolution as a function of the applied voltage, the base thickness, and the wavelength of monochromatic illumination. The results show that the capacitance decreases strongly with forward bias, reaches a maximum for a characteristic depth related to the absorption coefficient, and then increases with wavelength within the studied spectral range. The proposed model, although simplified, sheds light on the combined influence of electrical, geometrical, and optical parameters on the capacitive behavior of the cell. It thus provides a relevant tool for understanding the internal mechanisms of charge storage and for optimizing the design of photovoltaic devices. Looking ahead, the developed approach could be extended to the study of polychromatic illumination or dynamic regimes, allowing the exploration of frequency response and transient effects. Such investigations would offer promising perspectives both for improving conversion efficiency and for adapting solar cells to varied illumination conditions. Compliance with ethical standards The authors declare that all applicable ethical standards have been followed during the conduct of this research. Disclosure of conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. References [1] Kalowekamo, Joseph, et Erin Baker. « Estimating the Manufacturing Cost of Purely Organic Solar Cells ». Solar Energy, vol. 83, no 8, août 2009, p. 1224-31. https://doi.org/10.1016/j.solener.2009.02.003. 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