Influence of Both Magnetic Field and Temperature on Silicon Solar Cell Base Optimum Thickness Determination

The minority carrier’s recombination velocity at the junction and at the back surface is used for the modeling and determination of the optimum thickness of the base of a silicon solar cell in the static regime, under magnetic field and temperature influence. This study takes into account the Umklapp process and the Lorentz effect on the minority carriers photogenerated in the base.


Introduction
The parameters of the electric equivalent model of the solar cell, under darkness or illumination lead to its characterization, through the measurement of electrical current and voltage (I-V) [1]. These measurements are made by maintaining the solar cell under static [2] [3] or dynamic (transient [4] [5] [6] or frequency [7] [8] [9]) regimes and thus defining the different characterization technics.
However, these parameters are not expressed as function of solar cell geometrical parameters, especially the thicknesses of emitter, space charge region, base, and grain size [16].
In this work, the phenomenological parameters, such as minority carrier recombination velocity [17] [18] in the bulk (τ) [19], at the emitter-base junction (Sf) and at the back surface (Sb) [20] [21] of the base with thickness (H), are studied to extract the optimum thickness H, of the base of the silicon solar cell, placed under magnetic field B and with variation of the temperature T. This optimum thickness is obtained from the curves of the back surface recombination velocity, expressed as dependent of both the diffusion coefficient D(T, B) [22] [23] revealing the Umklapp process [24] and the base thickness. Figure 1 represents a n + -p-p + silicon solar cell under polychromatic illumination, by the emitter (n + ), through the collected grids. The space charge region (SCR), in x = 0, constitutes the junction (n + -p), allowing the separation of photogenerated electron-hole pairs. The rear face (p + ), in x = H, is a zone where an electric field exists (Back surface Field), which allows the return of the minority charge carriers towards the junction [25] [26].

Theory
When the solar cell is under illumination, the density δ(x, T, B) of photogenerated carriers in the base under magnetic field B at temperature T, is governed by the following magneto transport equation. Under magnetic field, the diffusion coefficient is given by the following relation [27]: Figure 1. The solar cell structure to the n + -p-p + type, under both magnetic field and temperature.

Polychromatic illumination
Metal grid And the mobility coefficient is given as [28]: , .

L T B D T B
= L represents the diffusion length of minority carriers in excess. Plot of expression (2) allows to extract maximum diffusion coefficient which is related to optimum temperature by following relation [23]: and 11.87 β ′ = the density of photogenerated carriers in the base, is produced by the generation rate, expressed by [29]: where n is the number of sun or level of illumination.
The parameters a i et b i stem from the modeling of the incident illumination as defined under A.M1.5. The expression of the excess minority carrier density in the base is given by the resolution of the continuity equation and is written as:

Photocurrent Density
Fick's law allows us to obtain the expression of the photocurrent density. This expression is given by the following equation: For Sf beyond 10 5 cm/s, the photocurrent is constant with Sf and corresponds to the short-circuit current density Jphsc, which is a plateau that increases with the temperature and decreases with magnetic field (Lorentz's law).

Back Surface Recombination Velocity Sb (T, B)
where in Sb1 expression, appears the effect of light absorption (b i coefficient) in the material and leads to a generation rate ( [20] [21], [23]. Given Fick's law on the back surface, the Sb2 (<0) expression is the intrinsic minority carriers recombination velocity at the back surface [22].
Sb1 and Sb2 lead asymptotically to the quotient D/L, which is the diffusion velocity (for L << H) [20], [21].   The intersection of the curves Sb1 and Sb2, has for abscissa, the optimum thickness of the base of the solar cell for each diffusion coefficient D max (B, Topt) [23]. Table 1 summarizes the variation of the solar cell base thickness for each diffusion coefficient (D max (B, Topt)) and the respective short-circuit currents Jsc1 and Jsc2 which remain maximum and constant. Figure 4 and Figure 5, give the representation of the thickness of the base of the solar cell necessary for each case of the diffusion coefficient. The correlation between the diffusion coefficient D max (T, B) and the optimum base thickness is established.
The current-voltage characteristics, under constant illumination of the solar cell having different base thicknesses, are simulated and the efficiency is obtained according to the thickness and under the influence of the surface recombination velocity [31]. The influence of thickness is highlighted in dynamic regime [16] [30] [32] through the constant decay time, as well as in studies of the solar cell in 3D model [24] [33] [34] where the electrical (D, Sf, Sb) [35] [36] and geometry (grain size) [37] parameters are involved.
Thus the results we propose in this work, constitute a contribution for the modelling and manufacturing of the solar cell thickness, for optimum efficiency under specific operating conditions [36].

Conclusion
The proposed study on determining the optimum thickness of the base of a silicon solar cell under temperature and magnetic field, takes into account the behaviour of minority carriers in physical processes of thermal agitation (Umklapp) and deflection (Lorentz). These physical mechanisms are quantified through the diffusion coefficient and recombination velocity of minority carriers

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