n + -p-p + Silicon Solar Cell Base Optimum Thickness Determination under Magnetic Field

Base optimum thickness is determined for a front illuminated bifacial silicon solar cell n + -p-p + under magnetic field. From the magneto transport equation relative to excess minority carriers in the base, with specific boundary conditions, the photocurrent is obtained. From this result the expressions of the carrier’s recombination velocity at the back surface are deducted. These new expressions of recombination velocity are plotted according to the depth of the base, to deduce the optimum thickness, which will allow the production, of a high short-circuit photocurrent. Calibration relationships of optimum thickness versus magnetic field were presented according to study ranges. It is found that, applied magnetic field imposes a weak thickness material for solar cell manufacturing leading to high short-circuit current.


Introduction
One major problem of silicon solar cells is the small collection of minority charge carriers which may be due among others at short diffusion lengths and carrier's mobility and surfaces recombination velocity issues. In order to improve its performance, several characterization techniques relating to minority carriers deflection under magnetic field, were presented [1]- [6]. Thus, the structure solar cell studied can be with: 1) horizontal junction (monofacial, bifacial or double side surface field) [7] [8] [9].
In this work, we present a method to determinate the optimum thickness (Hopt) of silicon solar cell under external conditions i.e. magnetic field (B) and polychromatic illumination.

Monofacial Solar Cell Presentation
Silicon solar cell type n + /p/p + [5] subjected to multi spectral illumination and a constant magnetic field (perpendicular to Ox axis), is presented in Figure 1.

Magnetotransport Equation
The B magnetic field influences the movement of minority charge carriers. In this condition the distribution equation relative to minority charge carriers in the base is given as follows [4] [6] [14] [31] [32].
G(x) is the minority carrier's generation rate [33] ( ) L is the minority carrier's diffusion length B depending and τ their lifetime.

Solution
The Magnetotransport equation solution is given by following expression ( ) , x B δ for front illumination: The previous relationship is fully defined, by determining the coefficients E and F, using base boundary conditions, what are junction (i.e. space charge region) and back side (p/p + surface).

Boundary Conditions
At the junction x = 0: Sf is excess minority carrier junction recombination velocity and describes the solar cell operating point [35] [36].
At back side x = H: Sb is back surface recombination velocity induced by the back surface field for low high junction (p/p + ) and thus minority carriers are pushed back to the junction. The space charge region's (n + /p) electrical field allows them to be collected and to contribute to the photocurrent [37] [38] [39].

Photocurrent Density for Different Magnetic Field Values
The excess minority charge carriers collected through junction give photocurrent density ( ) , Jph Sf B obtained from the following Fick relation. Figure 2 gives the plot of photocurrent density versus minority carrier's recombination velocity at the junction (Sf).
Regardless of the magnetic field values, the photocurrent increases with the junction recombination (Sf). When junction recombination velocity is high, short circuit photocurrent is obtained, and then, magnetic field reduced by deflection the electric charges due to increased Lorentz force intensity. Thus two study intervals will be defined according to magnetic field B value. Journal of Electromagnetic Analysis and Applications

Back Surface Recombination Velocity and Optimum Thickness Determination
Otherwise, we note that at high values recombination velocity Sf, photocurrent remains constant and becomes short circuit current Jsc(B, H). So its derivative with respect to Sf, is therefore zero [39].
Sb1 electronic parameters dependent (D and L), is designed as intrinsic back surface recombination, while Sb2 also depending of average (composite) absorption coefficient (b i ) [33] is considered as extrinsic one. Journal of Electromagnetic Analysis and Applications

The Base Optimum Thickness Determination
The optimum thickness determination technique, already used on solar cells maintained under other conditions [ Table 1, and represented on Figure 4, as Hopt versus B.
The correlation between optimum thickness and magnetic field is established below:      The correlation between optimum thickness and magnetic field is established below:  The results obtained by the application of the optimum thickness determination technique, show here, a thickness decrease with the magnetic field, for the two magnetic field ranges. This means that Lorentz's strength increases with the magnetic field imposes lower thicknesses to recover minority carriers, for a maximum photocurrent delivered by the solar cell. Both lowest and highest magnetic field values give respectively 176 μm and 117 μm solar cell base optimum thickness. This appears as a compromise between the different physical mechanisms of generation-diffusion-recombination-deflection, which take place in the base of the solar cell. This allows us to conclude that a front illuminated silicon solar to operate under magnetic field requires less material for its manufacturing.
It should be noted that previous work, using the same technique or other [44], has produced very interesting results, maintaining the solar cell (horizontal or vertical junction [41] [45]) under variation of: absorption coefficient [45], doping rate (hence the lifetime, the diffusion coefficient) [26] and irradiation flux by nuclear particles [40].
Modelling studies by combination of two to two or three of the previous conditions [29] [30] [42] have revealed the important economy of matter in the manufacture of solar cell, for these specific uses. The mathematical relationships between the optimum thickness of the base of the solar cell and the parameters of these specific conditions have been established It appears from the analysis of these results that the deflection of minority carriers due to the magnetic field, leads to lower optimum thicknesses than in other cases, such as thermal agitation (Umklap process), or the use of monochromatic absorption coefficient radiation (short wavelengths).

Conclusion
The calibrating silicon solar cell base thickness under polychromatic illumina- tion operating and applied magnetic field, was realized. The optimal thickness (Hopt) decreases significantly with the external applied magnetic field. This yield makes a judicious and optimal choice of the thickness of the base solar cell during its manufacture for an application of this kind.