The solar cell under study is composed of silicon layers forming a structure known as N⁺/P/P⁺. It operates under polychromatic light (composed of multiple wavelengths) and is analyzed under time-varying operating conditions, referred to as frequency-domain dynamic operation. The key physical phenomena occur in the central layer, known as the base (P-type), where electrons are generated by light, diffuse through the material, and may recombine.

Diagram 1.Structure of the silicon solar cell (N⁺/P/P⁺).

2.2.1. Diffusion Equation (Minority Carriers in the P-Type Base)

The minority carrier density equation is given as follows.

Where D n ( ω ) , is the effective diffusion coefficient in the frequency domain and is given by the following relation:

τ n is the minority carrier lifetime, δ n ( x , t ) the excess electron density, and G ( x , t ) the carrier generation rate due to illumination.

The expressions for δ n ( x , t ) and G ( x , t ) are given by the following relations [9]:

Where δ ( x ) and g ( x ) represent the spatial components, and e j ω t represents the temporal component. For this type of solar cell, the spatial component of the minority carrier generation rate is given by expression (4):

Where L ( ω ) is the complex diffusion length as a function of the modulation frequency, given by:

The constants A and B are determined from the boundary conditions at the junction and the back surface [10].

2.2.2. Boundary Conditions

At the junction: x = 0

At the rear face: x = H

Where, Sf is the recombination velocity at the junction of excess minority carriers, and also determines the operating point of the photovoltaic cell [11]-[13]. Sb is the recombination velocity at the back surface of excess minority carriers [13][14].

The expression for the recombination velocity at the rear surface (Sb) is determined from the derivative of the photocurrent density (Jph) with respect to the recombination velocity at the junction (Sf). For large values of Sf (Sf ≥ 10⁶ cm/s), corresponding to a region where the photocurrent density curves become asymptotic with a zero gradient, we have:

Solving this equation leads to two distinct expressions for the recombination rate at the back surface, denoted Sb₁ and Sb₂ [6].

In our study, and in order to simplify the calculation, we used the expression for Sb₂ to determine the explicit relationship between mobility (equation 11) and the geometric (H), magnetic (B), thermal (T), and optical (Φ) parameters.

As part of the study on charge carrier transport in a photovoltaic cell subjected to a magnetic field, it is crucial to understand how thermal and magnetic parameters influence diffusion mechanisms. The diffusion coefficient, denoted D, is a fundamental parameter that describes the mobility of minority carriers, especially in dynamic regimes where operating conditions deviate from equilibrium. To properly calibrate the values of the magnetic field B and the temperature T used in the modeling, a preliminary analysis was conducted to evaluate their combined effect

Figure 1. Diffusion coefficient as a function of temperature.

on the diffusion coefficient. Figure 1 shows the variation of the minority carrier diffusion coefficient D as a function of temperature, for different magnetic field intensities ranging from 3 × 10⁻⁴T to 10⁻³T.

This modeling makes it possible to identify the optimal temperature range for each magnetic field intensity, which is crucial for maximizing the efficiency of solar cells in varying environments.

The performance of a solar cell is strongly influenced by the recombination of minority carriers, particularly at the back surface, where passivation is generally more challenging. In this context, the present modeling focuses on evaluating the combined impact of temperature T and magnetic field B on the back surface recombination velocity Sb, as a function of the charge carrier mobility μ. Figure 2 presents the simulated results of the magnitude of the recombination velocity Sb as a function of mobility μ for different temperature and magnetic field values. Each curve shows an increasing linear trend, indicating a proportional relationship between μ and Sb, which means that as mobility increases, surface recombination becomes more significant.

Figure 2. Recombination velocity Sb as a function of mobility µ for different temperature and magnetic field values.

The curves also reveal that, for the same mobility μ, the recombination velocity Sb increases with both temperature and magnetic field. This indicates that at high temperatures, thermal agitation enhances carrier recombination. Furthermore, under stronger magnetic fields, the Lorentz force alters the trajectories of charge carriers, which can increase recombination losses especially in regions with a strong field gradient.

Figure 3 illustrates the combined effect of the magnetic field B and temperature T on two key electrical parameters of solar cells: the short-circuit current density Jsc and the open-circuit voltage Voc.

(a) (b)

(c) (d)

Figure 3. Variation of Jsc and voc as a function of magnetic field (a) and (b) and temperature (c) and (d).

The analysis of the variations in short-circuit current density (Jsc) and open-circuit voltage (Voc) reveals the following: the short-circuit current density (Jsc) decreases with increasing magnetic field due to the Lorentz force acting on carrier trajectories. Similarly, the open-circuit voltage (Voc) also decreases with the magnetic field B, albeit more moderately. Temperature has a dual effect: it initially increases Jsc due to enhanced thermal generation, but ultimately degrades it beyond a certain threshold as recombination processes become dominant. In contrast, Voc continuously decreases with temperature T, in line with the classical theory of PN junctions. Overall, this modeling highlights the importance of accounting for the coupled effects of magnetic field (B) and temperature (T) in the design and performance prediction of solar cells intended for extreme environments (e.g., space, industrial settings, integrated magnetic sensors). It also emphasizes the need to optimize surface passivation, thermal management, and the selection of materials with high magneto-thermal robustness, as well as the design of more resilient structures (e.g., magnetic shielding, passive cooling, and low-recombination materials).

his section presents the frequency-domain study of the dynamic behavior of a silicon solar cell subjected to varying magnetic fields. The analysis is based on the evolution of the phase of the complex impedance Z(ω) (Figure 4(a)) and its representation in the complex plane (Nyquist diagram, Figure 4(b)). It highlights delayed transport mechanisms as well as capacitive and resistive effects induced by the physical parameters of the system.

3.4.1. Effect of the Magnetic Field

Figure 4 analyzes the frequency-dependent behavior of the complex impedance Z(ω) of a silicon solar cell under different magnetic field strengths.

Figure 4(a) shows that the impedance phase exhibits a minimum located in the range log ( ω ) ∈ [ 6 , 7 ] . This range corresponds to the system’s relaxation resonance frequency, where capacitive and resistive effects temporarily balance each other. As the magnetic field B increases, a more pronounced phase shift (i.e., a more negative phase) is observed, indicating slower carrier dynamics, charge accumulation, or increased interfacial resistance.

(a) (b)

Figure 4. Frequency response of the impedance under magnetic field. (a) Phase analysis, (b) Complex impedance analysis—Nyquist diagram.

In Figure 4(b) we also observe semi-circular curves typical of RC type response systems. As the magnetic field (B) increases, the real part of the impedance (Z) shifts toward higher values, indicating an increase in series resistance (Rs) or recombination resistance (Rp). Similarly, the imaginary part of (Z) reaches more negative minimum values, suggesting enhanced junction or diffusion capacitance. The widening of the arcs further reveals a dispersion in time constants, linked to the non-uniformity of recombination processes under the magnetic field.

These results demonstrate the influence of the magnetic field on the global capacitive and resistive behavior of the cell, reflecting a degradation in carrier collection dynamics and a modification of the junction properties.

3.4.2. Effect of Temperature

Figure 5 shows the evolution of the phase (Figure 5(a)) and the complex impedance (Figure 5(b)) as a function of frequency for different temperatures. The phase exhibits a minimum around log(ω) ≈ 6, indicating capacitive behavior, slightly modulated by temperature.

(a) (b)

Figure 5. Variation of the phase (a) and of the real and imaginary parts of the impedance (b) as a function of frequency for different temperatures.

The Nyquist diagrams (Figure 5(b)) reveal depressed arcs, indicating non-ideal behavior typical of a distributed RC circuit. As temperature increases, the real part of the impedance decreases, suggesting a reduction in internal resistance due to improved carrier mobility or a lowering of recombination barriers. These results highlight the thermal impact on transport mechanisms within the solar cell.

3.4.3. Combined Effect of Magnetic Field and Temperature

The Nyquist diagram presented illustrates the evolution of the complex impedance as a function of both temperature and applied magnetic field (Figure 6). The typical semi-circular behavior can be modeled by an equivalent electrical circuit consisting of a series resistance Rs in series with a parallel branch composed of a resistance Rp and capacitance C. He is equivalent to (Rs + (Rp // C).

Figure 6. Nyquist diagrams of the measured complex impedance for different temperatures and applied magnetic field intensities.

An increase in the series resistance (Rs) and a slight decrease in the parallel resistance (Rp) are observed with rising temperature and magnetic field intensity.

The increase in Rs with temperature and magnetic field indicates a reduction in carrier mobility. Simultaneously, the decrease in Rp reflects an increase in recombination or internal losses, while the capacitance C varies depending on charge accumulation at the interface. This model enables a detailed analysis of transport phenomena under thermo-magnetic effects.

These results confirm that thermo-magnetic coupling significantly affects transport and recombination mechanisms in photovoltaic structures and highlight the relevance of frequency analysis as a diagnostic tool for characterizing internal phenomena under multiple external stresses.