The model developed in this work is based on the structure shown in Figure 1. The device is a monofacial silicon solar cell consisting of a thin, heavily doped n⁺ emitter, a p-type base of thickness H, and a rear p⁺ region acting as a back-surface field (BSF). The n⁺/p junction separates the photogenerated electron-hole pairs; the p/p⁺ transition produces an internal electric field that drives minority electrons toward the collecting junction and thereby reduces their recombination at the rear surface [17]. The originx = 0 is located at the edge of the space-charge region on the base side, and x = H at the p/p⁺ interface. Polychromatic illumination is incident on the front surface (n⁺ side). The uniform static magnetic field B forms an angle θ with the main carrier-diffusion direction (x axis). The adopted model is one-dimensional and assumes low-level injection, a homogeneous quasi-neutral base, and physical parameters (mobility, lifetime, and diffusion coefficient) that are uniform across the base thickness.

Figure 1. Structure of the monofacial n⁺-p-p⁺ silicon solar cell.

This representation brings together the three coupled physical mechanisms in the base: electron-hole pair generation under polychromatic illumination, diffusion toward the collecting junction, and bulk (Shockley-Read-Hall, radiative, and Auger) [18] and surface recombination. The pair (B, T) acts directly on diffusive transport, whereas the p⁺ back-surface field affects the recombination boundary condition at x = H. This structure provides the basis for the mathematical model developed in the remainder of this section.

In the frequency-domain dynamic regime, the continuity equation coupling the diffusion, generation, and recombination of minority carriers (electrons) in the p-type base is written as:

where δ n ( x , t ) and G ( x , t ) respectively represent the excess minority-electron density in the base and the overall generation rate of carriers under polychromatic illumination. They are given by the following relations:

where δ ( x ) and g ( x ) denote the spatial component and e i ω t the temporal component.

For this type of solar cell, the spatial component of the minority-carrier generation rate is given by Equation (3):

where a i and b i are tabulated coefficients of solar radiation that depend on the wavelength-dependent absorption coefficient of silicon under AM 1.5.

n : represents the illumination level expressed in suns. It relates the experimental illumination level to the AM 1.5 reference illumination level; x: denotes depth in the base, measured from the emitter junction (x = 0) to the rear surface (x = H). This generation model provides the source term in Equation (1); the diffusion coefficient D_n, which governs transport, must now be characterized as a function of magnetic field, temperature, and modulation frequency.

2.3.1. Thermomagnetic Diffusion Coefficient

The temperature-dependent diffusion coefficient is given by the Einstein relation:

where k_B is the Boltzmann constant, q is the elementary charge, and µ_n(T) is the electron mobility in the base. Over the temperature range studied, its thermal dependence is represented by Equation (5), in accordance with the parameterization of silicon transport properties proposed by Jacoboni et al. [19]. At the reference temperature T_ref = 300 K, µ_n (300 K) = 1350 cm²/V·s is adopted for lightly doped silicon [19][20].

Under a static magnetic field B, the Lorentz force deflects minority-carrier trajectories and reduces their longitudinal diffusion coefficient. Under steady conditions, this reduction is expressed as:

This relation shows that the relative magnetic suppression of diffusion increases with the dimensionless parameter μ ( T ) B . Here, the magnetic field is perpendicular to the main diffusion direction, corresponding to θ = π/2, so that B_⟂ = B.

Magnetic-field effects are represented by a one-dimensional scalar approximation for diffusion along the carrier-collection direction. This model captures the reduction of longitudinal transport but neglects off-diagonal Hall terms, transverse carrier redistribution, Hall electric fields, and finite-width edge effects. The results should therefore be interpreted within this restricted approximation rather than as a complete tensorial description of magnetotransport [7][19][20].

2.3.2. Coupled Thermomagnetic Diffusion Length

The complex diffusion length under magnetic field B and temperature T is given by:

This expression highlights the coupling among the thermal, magnetic, and frequency effects considered in the present model. Temperature acts on electron mobility and the diffusion coefficient, the magnetic field reduces longitudinal transport, and the modulation frequency produces attenuation and phase lag through the factor ωτ_n. In all simulations, the minority-carrier lifetime is fixed at τ n = 1.0   μ s for every temperature and magnetic-field strength; consequently, f c = [ 2 π τ n ] − 1 = 159.15   kHz remains constant over 280 - 360 K. No temperature- or magnetic-field-dependent lifetime law is assumed in this work.

2.4.1. General Solution of the Diffusion Equation

The general solution of continuity Equation (1) for the excess minority-carrier density in the base as a function of depth x is:

where the coefficients βi are given by:

A and B are two integration constants determined by the boundary conditions at the n⁺/p junction and the rear p/p⁺ interface.

2.4.2. Boundary Conditions

At the n⁺/p junction, continuity between the diffusion current and the current crossing the junction imposes:

where S_f is the junction recombination velocity [21][22]. This velocity equals the sum of the recombination velocity associated with the external load ( S f j = 10 j ) and the intrinsic recombination velocity S_f0, an effective emitter-base interface recombination velocity. It accounts for both interfacial recombination mechanisms and electron extraction by the junction.

At the rear p/p⁺interface, the recombination boundary condition is:

where S_b is the back-surface recombination velocity. A low S_b value corresponds to effective passivation and efficient action of the p⁺ back-surface field [23].

The thermomagnetically coupled frequency-dependent photocurrent density is obtained from the diffusion current evaluated at the junction:

Its magnitude and phase are respectively given by:

These two quantities, the magnitude and phase of the complex photocurrent, are the principal observables of the parametric study presented in the following section. The effects of magnetic field, temperature, and modulation frequency on minority-carrier transport and collection are examined successively through numerical simulations of the model established above.

To quantify the influence of the field on diffusive transport, the relative reduction in the diffusion coefficient is defined as:

At 300 K, the Einstein relation gives a zero-field diffusion coefficient D_n(300 K) = 34.90 cm²∙s⁻¹ for an electron mobility of 1350 cm²∙V⁻¹∙s⁻¹. Substitution into D n ( B , T ) = D ( T ) / ( 1 + [ μ ( T ) B ] 2 ) shows that fields below 100 mT modify electron diffusion only very weakly (Table 1). At 50 mT, for example, D_n decreases only from 34.9002 to 34.8986 cm²∙s⁻¹, corresponding to a reduction of 0.00456%. The conventional 1% threshold is reached at B threshold = 0.74447 T. The decrease then becomes more noticeable, reaching 1.79% at 1 T, 6.79% at 2 T, and 31.30% at 5 T.

Table 1. Values of D_n and R_D at different magnetic-field strengths at T = 300 K.

The value D_n(0) = 34.90 cm²·s⁻¹ obtained at 300 K differs by only 0.285% from the value of 35 cm²·s⁻¹ adopted by Combari et al. [26], indicating satisfactory zero-field agreement. Under a magnetic field, however, the comparison reveals a substantial quantitative divergence. At 50 mT, the present model predicts a relative diffusion-coefficient reduction of only R_D = 4.56 × 10⁻³%, whereas some theoretical and experimental studies report a much stronger decrease in short-circuit current within this field range [6][26]. This divergence cannot be explained by the small difference between 34.90 and 35 cm²·s⁻¹. It may arise from additional mechanisms not represented by the sole factor 1/[1+ (µ_nB)²], differences in whether the applied field is static or alternating, experimental conditions, or the treatment of mobility and magnetic-field units. The present results should therefore be interpreted within the validity domain of the classical magnetodiffusion model. Their extrapolation to the overall electrical response of a photovoltaic module requires experimental validation under a controlled field.

Irrespective of this divergence from some electrical responses reported in the literature, three magnetodiffusion regimes can be distinguished from the R_D values listed in Table 1.

The range 0 ≤ B < 0.745 T corresponds to a change in D_n below 1% according to the adopted criterion; it broadly includes the geomagnetic field, approximately 25 - 65 µT [27], and local fields produced by many permanent magnets. The range 0.745 ≤ B < 2.469 T corresponds to a to a model-defined transport modification between the adopted 1% and 10% criteria; such fields can be generated by powerful electromagnets or superconducting magnets, including the static fields of 1.5 and 3 T commonly used in magnetic resonance imaging (MRI) systems [28]. Finally, 2.469 ≤ B ≤ 5 T represents a strong-perturbation regime in which the reduction in D_n reaches approximately 31.3% at 5 T. This range corresponds to specialized laboratory characterization rather than ordinary photovoltaic operation; its technical feasibility is demonstrated by laboratory superconducting magnets capable of producing continuous fields well above 5 T and exceeding 45 T in the most powerful research facilities [24][25].

Figure 2 shows the minority-electron diffusion coefficient and its relative reduction R_D as functions of magnetic field for five temperatures between 280 and 360 K.

Figure 2. Minority-electron diffusion coefficient and relative reduction R_D as functions of magnetic field for 280 ≤ T ≤ 360 K.

For all temperatures considered, D_n decreases as B increases, in agreement with D n ( B , T ) = D 0 ( T ) / [ 1 + ( μ n ( T ) B ) 2 ] . This decrease reflects the greater deflection of minority carriers by the Lorentz force and the resulting reduction in their longitudinal diffusive transport. Within the adopted model, the diffusion coefficient also decreases with temperature because the thermal decrease in mobility, described by μₙ(T) ∝ T⁻²·⁴², dominates the increase in the thermal factor k_BT/q.

At the reference temperature of 300 K, the three regimes previously defined from R_D are clearly visible in Figure 2(b). For 0 ≤ B < 0.745 T, R_D remains below 1%, so the magnetic influence may be considered negligible under the adopted criterion. Between 0.745 and 2.469 T, the reduction ranges from 1% to 10%, corresponding to a model-defined modification of minority-carrier transport exceeding the adopted 1% criterion. Finally, for 2.469 ≤ B ≤ 5 T, the reduction exceeds 10% and reaches approximately 31.3% at 5 T, characterizing a strong magnetic-perturbation regime.

The thresholds are nevertheless temperature dependent. Their shift toward higher fields as temperature increases results from the decrease in electron mobility. Thus, the 10% threshold rises from approximately 2.09 T at 280 K to 3.84 T at 360 K. Conversely, at a given field, the relative reduction is greater at lower temperatures. The convergence of the curves at high fields results from competition between the higher initial D₀ at low temperatures and the increased magnetic sensitivity associated with greater mobility.

This first analysis identifies the conditions under which the internal parameter D_n is affected. To determine how this modification propagates to the dynamic response, the complex diffusion length, which combines D_n with carrier lifetime and modulation frequency, is examined next.

Figure 3 presents the frequency dependence of the magnitude and phase of the complex diffusion length for different magnetic-field strengths at T = 300 K and τ_n = 1 µs. The characteristic frequency associated with this lifetime is f_c = 159.15 kHz.

Figure 3. Magnitude and phase of the complex diffusion length as functions of frequency for different B values at T = 300 K and τ_n = 1 µs.

In the low-frequency regime, the magnitude approaches its quasi-static value D n ( B , T )   τ n . Its progressive decrease with increasing magnetic field results from the field-induced reduction in the diffusion coefficient of minority carriers. At B = 5 T, the magnitude is approximately 17.1% lower than its zero-field value. Near the characteristic frequency f_c = 159.15 kHz, the magnitude begins to decrease significantly and reaches 2^−1/⁴, or approximately 84.1% of its quasi-static value. At higher frequencies, it follows the asymptotic behavior | L ω * | ∝ f − 1 / 2 , indicating a progressive reduction in the dynamic penetration depth of minority carriers.

The phase evolves from nearly 0˚ in the quasi-static regime to −22.5˚ at f = f_c and then approaches −45˚ in the high-frequency asymptotic regime. Assuming that the minority-carrier lifetime is independent of magnetic field, the phase has no explicit dependence on B; the phase curves associated with the different magnetic-field strengths therefore overlap. Within the limits of the present model, the magnetic field modifies the magnitude of the complex diffusion length without changing its characteristic frequency or phase response. The diffusion length describes a transport scale but does not, by itself, specify the actual carrier distribution in the base. The next step therefore incorporates polychromatic generation and the boundary conditions to analyze the spatial density profiles.

The combined analysis of the magnitude and phase of the complex minority-carrier density (Figure 4) fully characterizes its dynamic response. The magnitude describes attenuation of the photogenerated population through diffusion and recombination, whereas the phase quantifies the carrier delay relative to the modulated optical excitation. This analysis identifies the transition between the quasi-static and dynamic regimes and assesses the influence of temperature and magnetic field on carrier transport and collection at T= 300 K and B= 1 T.

Figure 4. Magnitude and phase of the complex minority-electron density at different modulation frequencies, for T = 300 K and B = 1 T.

Under front-side polychromatic illumination, the density magnitude rises rapidly from the junction, reaches a maximum at a depth of approximately 40 - 50 µm, and then decreases toward the rear surface. The low density near the junction results from the high recombination-velocity value, which limits carrier accumulation in this region.

In the quasi-static regime, the profiles at 10 Hz and 0.1fc nearly coincide, showing that the carriers can follow the illumination modulation. When the frequency reaches f_c = 159.15 kHz, the amplitude begins to decrease noticeably and a substantial spatial phase lag appears. For f ≫ f_c, the density is strongly attenuated and becomes more localized near the generation region because the carriers no longer have sufficient time to diffuse deeply into the base.

The phase, which is nearly zero at low frequency, becomes progressively more negative as frequency and depth increase. This behavior reflects the delay of the carrier response relative to the optical excitation. At high frequencies, nonmonotonic phase variations result from interference among the complex contributions of the homogeneous solution, polychromatic generation, and the boundary conditions.

These profiles reveal the internal dynamics of the base. To relate these dynamics to a directly usable electrical quantity, the remainder of the analysis focuses on the junction photocurrent and its normalization relative to the zero-field case.

To quantify the specific influence of the magnetic field on the dynamic response of the cell independently of its absolute photocurrent level, a relative magnetic-response factor is defined as:

This relative factor is interpreted as follows:

C_TM = 1: no magnetic effect on the amplitudeC_TM < 1: photocurrent attenuationC_TM > 1: photocurrent enhancement

A value of unity corresponds to a photocurrent that is insensitive to the field, whereas C_TM < 1 denotes attenuation and C_TM > 1 denotes enhancement.

This factor is related to the signed sensitivity by:

S_B > 0 indicates photocurrent attenuation under magnetic field, whereas S_B < 0 indicates an increase.

Because these two quantities are redundant, C_TM is used to represent the normalized response, whereas S_B is used only to define the operational thresholds B_J,1% and B_J,10%. These thresholds apply to the photocurrent and may differ substantially from those derived from Dₙ.

The electrical thresholds B_J,1% and B_J,10% are operational, model-defined criteria obtained from |S_B(T, B, f)| = 1% and |S_B(T, B, f)| = 10%, respectively. They are not experimental detection limits unless a measurement uncertainty and instrumental resolution are explicitly specified.

Figure 5 and Figure 6 respectively show the magnetic-field and frequency dependence of the normalized magnetic response.

Figure 5 shows that C_TM starts at unity for B = 0 and progressively decreases with field strength; the field therefore slightly attenuates the photocurrent magnitude in the present model. The departure from unity is more pronounced at low temperature, consistent with the higher electron mobility and increased sensitivity of D_n. It also increases in the dynamic regime, thereby lowering the field required to reach a given level of electrical variation.

Figure 5. Normalized magnetic response C_TM as a function of B at 0.1f_c, f_c, and 10f_c for different temperatures.

A key result is that B_1%,D = 0.745 T, defined by a 1% reduction in D_n at 300 K, produces only a 0.0858% reduction in |J_ph| at f = f_c. At 300 K, the electrical threshold |S_B| = 1% is approximately 2.61 T at f_c. Geometry, generation, and collection conditions therefore filter out part of the change in the diffusion coefficient.

Using the numerical parameter set stated above, the electrical threshold B_J,1% is obtained by solving |S_B(300 K, B, f_c)| = 1%, which yields approximately 2.61 T. This value is specific to the adopted base thickness, lifetime, optical-generation model, recombination velocities, illumination level, and operating frequency; it is not a universal magnetic threshold for silicon solar cells.

Figure 6. Frequency dependence of CTM for B = 0.745 T and B = 1 T at different temperatures.

Figure 6 confirms that, for B = 0.745 and 1 T, C_TM remains very close to unity throughout the studied frequency range. The effect is nearly negligible in the quasi-static regime and becomes slightly more visible when f exceeds fc, although it does not reach 1% over this magnetic-field interval. The separation of the curves with temperature shows that normalization by J_ph(T, 0, f) does not remove the thermal dependence of magnetic sensitivity. This residual dependence motivates the introduction of I_TM.

To isolate the thermal modification of the relative magnetic response, the following indicator is defined with respect to T_ref = 300 K:

At identical magnetic field and frequency, this indicator compares the relative field effect at temperature T with that obtained at 300 K.

The sign of I_TM indicates whether the relative magnetic attenuation is weaker or stronger than in the reference case.

By construction, I_TM is zero at the reference temperature and in the absence of a magnetic field. A nonzero value indicates that temperature modifies the relative effect of the magnetic field on the photocurrent. Joint consideration of C_TM and I_TM therefore distinguishes a simple magnetic variation in photocurrent from a genuine thermomagnetic interaction in the dynamic response of the cell.

Thus, C_TM describes the normalized magnetic response, S_B provides a directly interpretable threshold criterion, and I_TM reveals the intrinsic thermal dependence of this effect.

Figure 7. Thermomagnetic interaction I_TM as a function of B relative to T_ref = 300 K for three frequency regimes.

Figure 7 verifies the normalization properties: I_TM is zero at B = 0 for every temperature and identically zero at 300 K. For T < 300 K, I_TM is negative, meaning that C_TM is below its reference value and the relative magnetic attenuation is stronger. For T > 300 K, I_TM becomes positive: the thermal decrease in mobility makes the photocurrent relatively less sensitive to the field. The magnitude of the interaction increases with B and becomes more noticeable in the dynamic regime.

To summarize these observations in a single representation, Figure 8 maps the thermomagnetic interaction indicator I_TM in the (B, T) plane at the characteristic frequency f_c = 159.15 kHz.

Figure 8. Map of the thermomagnetic interaction I_TM in the (B, T) plane at f = f_c = 159.15 kHz.

This map concisely confirms the trends identified in Figure 7: I_TM is zero along the T = 300 K isotherm (central green band), increases with B above this isotherm (yellow-orange to red shades, indicating weaker relative photocurrent attenuation), and decreases below it (blue shades, indicating stronger relative attenuation). The steepest gradient appears in the upper-right corner of the map (high T and high B), where the mobility difference between T and T_ref is greatest and its effect on the term µ_n(T)B is most pronounced. This two-dimensional representation therefore provides a rapid means of anticipating, at a given frequency, the direction and magnitude of the expected thermomagnetic interaction for a specified pair (B, T), before consulting the detailed curves in Figures 5-7.

Table 2 summarizes the numerical values of R_D, C_TM, S_B, and I_TM, together with the magnitude and phase of the photocurrent, for three representative magnetic fields (B ≈ 0 T, represented here by 0.001 T as a quasi-zero reference; B = 0.745 T, corresponding to the B_1% threshold; and B = 1 T) and for the five temperatures considered in the preceding figures.

The table quantitatively confirms all the qualitative trends identified in the preceding subsections. At a quasi-zero field (B = 0.001 T), R_D is negligible (of the order of 10⁻⁶%), C_TM is numerically indistinguishable from unity at the displayed precision, and I_TM remains of the order of 10⁻⁷ - 10⁻⁸%, confirming the internal numerical consistency of the model. For the two nonzero fields (0.745 and 1 T), both R_D and S_B decrease monotonically with increasing temperature. At B = 1 T, for example, R_D decreases from 2.48% at 280 K to 0.75% at 360 K, while S_B decreases from 0.212% to 0.066% over the same interval. This confirms that the magnetic sensitivity of transport decreases as temperature rises, in agreement

Table 2. Summary of the thermomagnetic indicators (R_D, C_TM, S_B, and I_TM) and the complex photocurrent (magnitude and phase) as functions of temperatureT for three magnetic-field strengths (≈ 0, 0.745, and 1 T) at the characteristic frequency f_c = 159.15 kHz.

with Figure 2 and Figure 5. Correspondingly, C_TM approaches unity as T increases, indicating weaker magnetic attenuation of the photocurrent at higher temperatures.

The I_TM indicator faithfully reproduces the behavior observed in Figure 7: it is zero by construction at 300 K, negative at 280 K, and positive at 320, 340, and 360 K, with a magnitude that increases with magnetic field. The ranges are −0.03% ≤ I_TM ≤ 0.06% at B = 0.745 T and −0.06% ≤ I_TM ≤ 0.09% at B = 1 T. The photocurrent magnitude |J_ph| decreases only slightly as both B and T increase (from 26.655 to 25.830 mA·cm⁻² over the entire table), confirming that most of the photocurrent variation originates from temperature rather than magnetic field. Indeed, the field effect remains below 0.22% over the investigated range B ≤ 1 T. Finally, the photocurrent phase becomes systematically more negative as temperature increases (from −3.91˚ at 280 K to −4.06˚ at 360 K at fixed field), indicating a slight additional delay in the dynamic response of the cell at higher temperatures, consistent with the decrease in D_n(B, T) shown in Figure 2. The table thus provides a compact numerical reference, complementary to the graphical results, for validating or parameterizing future experimental and simulation studies.