As part of this three-dimensional modeling, the study focuses on a bifacial solar cell composed of polycrystalline silicon, consisting of an assembly of grains with varied morphologies and dimensions, yet exhibiting homogeneous electrical properties [14]. These grains, characterized by square cross-sections and a given thickness H, are separated by intergranular boundaries [9], which constitute preferential sites for recombination phenomena. This structural configuration justifies the adoption of a Cartesian coordinate system for the analytical treatment. Figure 1 illustrates a representative segment of a grain, where the recombination planes are defined as the adjacent surfaces between two perpendicular grains, located at positions x = ± a 2 and y = ± a 2 . Interface recombination is modeled using a uniform surface recombination velocity (S_gx = S_gy = S_g) [7], assumed to be independent of solar irradiance. This assumption allows the imposition of linear boundary conditions in the continuity equations. The diffusion of minority carriers is thus governed by temperature and the intensity of the magnetic field. For the sake of simplification, the contributions of the emitter and the space-charge region are neglected, with the analysis focusing exclusively on the base of the cell. It is subjected to uniform, non-concentrated front-side illumination and to a variable external magnetic field oriented perpendicularly to the depth of light penetration [9][15]. Light enters the junction at the plane (z = 0).

In the present model, the magnetic field is applied perpendicularly to the (xOz) plane, that is, parallel to the junction, so that the base doping level is considered uniform. This leads to a quasi-zero crystalline electric field.

Temperature is correlated with solar irradiance through the following relation [16]-[18]:

where NOCT is the cell’s nominal operating temperature, specified by the manufacturer. It is determined under standardized test conditions of 800 W/m² irradiance, and a wind speed of 1 m/s, Tis the real-time temperature of the solar cell in Kelvin (K), T_a is the ambient temperature in Kelvin (K), G is the solar irradiance in W.m⁻² or in suns with 1 C = 1000   W ⋅ m − 2 .

Figure 1. Theoretical model of a grain segment in a polycrystalline silicon photovoltaic cell.

2.2.1. Continuity Equation

Under conditions of light intensity and external magnetic field, the excess minority carrier density δ ( x , y , z ) generated within the base is governed by the following continuity equation:

where

with

g ( z ) denotes the generation rate [19] of minority carriers; G represents the number of suns or the solar irradiance [3][4][20][21], suns; a_i and b_i are constants [18][22] tabulated in the spectral modeling of the generation rate under AM 1.5 illumination.

The parameters D ( T , B ) and L ( T , B ) respectively denote the diffusion coefficient and the diffusion length in the presence of a magnetic field and temperature coupled with irradiance with T = f ( G ) .

The values ofa_i and b_i, proposed by Noor S. Mohammad in the three-term approximation of the generation rate in silicon under AM 1.5 illumination, are defined as follows (Table 1).

Table 1. The coefficients a_i and b_i under AM 1.5 illumination.

2.2.2. Solution of the Continuity Equation

The solution to the continuity equation takes the following form [22]:

C j and C k are the eigenvalues of the transcendental equations, determined respectively from the boundary conditions provided:

The nonlinear relationships C j and C k are given by:

The transcendental coefficients C j and C k are explicitly determined using the Newton-Raphson numerical method, based on successive approximations of their roots, according to the following iterative formula:

By applying the orthogonality condition of the cosine terms, the function Z j , k ( z ) is defined as follows:

with

and

The constants A j , k ( T , B , G ) and B j , k ( T , B , G ) are determined based on the boundary conditions at the junction and the rear surface:

At the junction z = 0 ,

At the Back side z = H ,

This resolution leads to:

and

with

The expression for the photocurrent [14][22] is given by:

After computation, the photocurrent density is defined as follows:

When the dynamic recombination velocity Sf at the junction becomes sufficiently large, a simplified expression for the short-circuit photocurrent is obtained, defined as follows:

As for the expression of the photovoltage, it is determined by the Boltzmann equation [14][22]:

After computation, the resulting expression is given by:

with

The intrinsic carrier concentration, which depends on both the temperature and the electronic band structure of the material, is defined by the relation [23]:

where A is a constant given by A = 3.87 × 10 16   cm − 3 ⋅ K − 3 / 2 , and E g = 1.12 × 1.6 × 10 − 19   J denotes the energy bandgap of silicon.

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The numerical modeling of photovoltaic devices relies on a rigorous characterization of their fundamental electronic parameters. Within the scope of this simulation, these parameters are incorporated into the transport and generation equations of the photocurrent, enabling precise evaluation of the solar cell’s electrical response under various climatic and environmental conditions. The corresponding values are summarized in Table 2.

Table 2. Fundamental electronic parameter values.

To better understand the combined effect of solar irradiance and temperature on the excess minority carrier density, two distinct simulation profiles have been established

The first profile (Figure 2 with T = 293 K) illustrates the variation of carrier density as a function of irradiance, while maintaining a constant temperature. This configuration isolates the direct impact of irradiance on carrier generation.The second profile examines the carrier density as a function of irradiance correlated with temperature, thereby highlighting the interdependent effects of these two physical parameters. Table 3 illustrates the variation of cell temperature as a function of irradiance with NOCT = 47˚C.

Table 3. Solar cell temperature as a function of irradiance.

This dual representation aims to distinguish the respective and combined contributions of irradiance and temperature on the behavior of minority carriers, with the ultimate goal of optimizing the operating conditions of solar cells exposed to varying thermal and radiative environments.

Figure 2.Carrier density profile of a polycrystalline silicon solar cell as a function of magnetic field and irradiance coupled with temperature.

The analysis of Figure 2 demonstrates that, for each irradiance level, the carrier density progressively decreases as the magnetic field intensity increases, approaching zero around 1.2 mT when temperature is held constant, and slightly beyond this threshold when temperature varies jointly with irradiance. Conversely, irradiance consistently enhances carrier density, thereby confirming the stimulating effect of incident energy on carrier generation. Nonetheless, the density remains systematically higher when temperature is not coupled with irradiance, underscoring the attenuating influence of thermal coupling on excess electron density. These findings highlight that the interaction between temperature and irradiance induces a complex dynamic, potentially constraining carrier generation under variable thermal conditions. Such behavior must be accounted for in the optimization of photovoltaic device performance within fluctuating environmental contexts.

In order to deepen the understanding of the mechanisms influencing the photocurrent density in a solar cell, Figure 3 presents a series of profiles illustrating its evolution as a function of magnetic field, irradiance, and irradiance coupled with temperature.

Figure 3. Photocurrent density profile of a polycrystalline silicon solar cell as a function of magnetic field under irradiance coupled with temperature.

The examination of Figure 3 highlights the evolution of the photocurrent profile as a function of magnetic field intensity, ranging from 0 to 1.4 mT. This behaviour is analysed both under constant temperature conditions (T = 293 K) and when temperature is considered as dependent on solar irradiance (T = f(G)), for irradiance levels varying in increments of 0.2 between 1 sun and 0.2 sun. The results reveal that an intensification of the magnetic field induces a spatial redistribution of charge carriers, thereby altering their trajectories under the effect of the Lorentz force and reducing their collection efficiency. In fact, the Lorentz force, orthogonal to both the magnetic field and the direction of carrier diffusion, causes a deflection of the electrons toward the lateral faces along the (Ox) axis. This deflection leads to a localized decrease in the photocurrent. In parallel, irradiance enhances carrier generation and increases photocurrent density, although this effect is mitigated by the influence of the magnetic field. When temperature is coupled with irradiance, thermal recombination further reduces carrier lifetime as well as collection efficiency, resulting in a photocurrent lower than that observed under constant temperature conditions.

These findings underscore the importance of distinguishing thermal effects from those related to irradiance in the analysis of photovoltaic performance, as their interaction can generate significant energy losses that must be taken into account in the design and optimization of devices.

The photovoltage of a photovoltaic cell is significantly influenced by temperature and solar irradiance. Figure 4 presents the photovoltage profile as a function of magnetic field for various irradiance levels, both under constant temperature and when the two climatic factors are coupled.

Figure 4. Photovoltage density profile of a polycrystalline silicon solar cell as a function of magnetic field under irradiance coupled with temperature.

The analysis of Figure 4 highlights the evolution of the photovoltage profile as a function of magnetic field intensity, ranging from 0 to 1.4 mT, under both constant temperature conditions (T  = 293 K) and when temperature is considered as dependent on solar irradiance (T = f(G)), for irradiance values varying in steps of 0.2 between 1 sun and 0.2 sun. The comparative examination reveals a progressive decrease in photovoltage with increasing magnetic field intensity, consistent with the results reported by Soro et al. [6]. Under constant temperature, photovoltage decreases markedly as irradiance is reduced, whereas under thermal coupling (T = f(G)) it remains slightly higher, reflecting the dominant role of irradiance in carrier generation. This indicates that the beneficial effect of enhanced light intensity outweighs the adverse influence of temperature rise, as corroborated by previous studies [24]. In fact, solar irradiance constitutes the primary driver of photovoltaic conversion, while temperature exerts an inhibitory influence. When irradiance is sufficiently high, its beneficial effect outweighs the thermal constraints, thereby sustaining a comparatively elevated photovoltage. Finally, the decline in photovoltage under strong magnetic fields and low irradiance is explained by the combined action of Lorentzforce deviation of carriers and reduced photon flux, which together intensify recombination and limit carrier density, thereby compromising overall efficiency.

The study underscores the decisive role of solar irradiance in the electrical performance of photovoltaic generators, particularly under the combined influence of temperature and magnetic field. Indeed, exposure to a magnetic field reduces the density of excess electrons and consequently the photocurrent, an effect further amplified by thermal and irradiance variations. These observations are consistent with the findings of Combari et al. [25], who demonstrated that the application of a magnetic field to silicon photovoltaic modules leads to a marked decrease in carrier density and overall efficiency. Similarly, the work of Afonso et al. [26] confirms the joint influence of irradiance and temperature, with the former stimulating carrier generation while the latter enhances recombination. Thus, the convergence of these results with prior studies validates the proposed model and highlights that the magnetic field, in conjunction with thermal and radiative variations, constitutes a critical factor in the assessment and optimization of photovoltaic performance.