Lamella Silicon Solar Cell under Both Temperature and Magnetic Field: Width Optimum Determination

This work deals with determining the optimum thickness of the lamella wafer of silicon solar cell. The (p) base region makes up the bulk of the thickness of the wafer. This thickness has always been a factor limiting the performance of the solar cell, as it produces the maximum amount of electrical charges, con-tributing to the photocurrent. Determining the thickness of the wafer cannot be only mechanical. It takes into account the internal physical mechanisms of generation-diffusion-recombination of excess minority carriers. They are also influenced by external factors such as temperature and magnetic field. Under these conditions, magneto transport equation is required to be applied on excess minority carrier in lamella base silicon solar cell. It yields maximum diffusion coefficient which result on Lorentz law and Umklapp process. Then from photocurrent, back surface recombination velocity expressions are de-rived, both maximum diffusion coefficient and thickness dependent. The plot of the back surface recombination calibration curves as function of lamella width, leads to its maximum values, trough intercept points. Lamella optimum width is then obtained, both temperature and magnetic field dependent and expressed in relationships to show the required base thickness in the elaboration process.

However solar cell base thickness (H) is a geometric parameter to consider, compared to minority carriers diffusion length, to ensure a high probability collection of photocreated carriers [47] [48] [49] [50] [51].
The vertical multi-junction silicon solar cells (VMJ) [52] [53] [54] [55], use materials having charge carriers with short diffusion length, but its architecture gives the advantage of excess minority carriers to be collected, without traveling great distances. Indeed the low thickness base can be combined with two emitter allowing the collection of minority carrier (PVMJ) [56] [57], or by existence a rear field (junction p/p + ) who drives them back, thus reducing the distance to be covered (SVMJ) [58]. This rear field induces a recombination velocity minority charge carriers (Sb) that characterize the back surface of solar cell (BSF or ohmic contact) and then gives the rate of charge carrier loss [27] [28] [29] [30] [48] [49].
Our study is interested in the lamella thickness determination, through the new expression of recombination velocity at the back side. This allows extending the life of minority charge carriers in lamella and promotes the solar cell performance, under the effect of both external magnetic field and temperature. Figure 1 shows the structure of vertical multi-junction silicon solar cells connected in series [52] [53] [58]. It is composed a succession of junctions (n + -p-p + ) joined together with metallic (Al) contacts. Incidental illumination occurs parallel to junctions i.e. space charge region plane (SCR) [59] [60]. The elaboration of junction (p-p + ) produces the back field effect, that induces excess minority carriers back surface recombination velocity (Sb), that straugths back them towards the junction (SCR) and thus avoids their recombination [28]. Figure 2 shows a section of vertical junction silicon solar cell unit, with the different regions (emitter, junction, base, rear field area). The axis (Ox), to origin

Magneto Transport Equation
Excess minority carrier's density δ(x), generated on the abscissa x and at depth z, in the base of solar cell in steady regime, undergo the law magneto-transport, presented through the following continuity equation [61]: , With µ(T) is the minority carriers mobility temperature dependent in the base and expresses as [38]: q is the electron elementary charge. k b is Boltzmann's constant given as: The generation rate of minority charge carriers generated at depth z in the base is modeled and expressed by the following relation [65]: The coefficients a i and b i are obtained from the tabulated values of the radiation.

Solution
The solution of magneto transport equation is given by the following expression of the minority charge carrier density as: With ( ) ( ) ( )

Boundary Conditions
The previous relationship is fully defined, by determining the coefficients A 1 and A 2 , using base boundary conditions, what are junction (SCR) and back side: 1) At the junction (n + /p), x = 0, it is given by [66] ( ) ( ) ( ) 2) At back surface (p/p + ), x = H, it is given by [28] [67]: Sf is excess minority carrier junction recombination velocity. It has two components, one defines the operating point, thus, it is imposed by the external load resistor, and the other is the intrinsic recombination velocity, which is related to the solar cell shunt resistance in electric equivalent model [66] [68].

D. Faye et al.
Sb is back surface recombination velocity (x = H), where there is an electric field (p/p + ), allowing repel the minority charge carriers towards junction (n + /p) and avoid their back side recombination [28]. Thus the collection rate of minority carries participating in the photocurrent increases.

Photocurrent Density
The excess minority carriers collected through junction give photocurrent density Jph obtained from the following Fick relation:

Back Surface Recombination Velocity
Solving Equation (12) , , The maximum values of diffusion coefficient as a function of optimum temperature for different values of magnetic field were determined by comparisons of two different methods according to relationship [69]: Other authors, using the same approach, proposed in 3D study or in frequency modulation the following expressions:  Optimum temperature depending magnetic field [70] These relationships show that the choice of values of parameters like the temperature, the magnetic field and the frequency must obey certain conditions for obtaining solar cell good performance.
In Figure 3, we represent the profiles of two back surface recombination velocity of excess minority carriers depending on thickness base solar cell for different diffusion coefficient maximum values as a function of optimum temperature and magnetic field.
For each value of maximum diffusion coefficient, the optimum thickness H op of base is determined by projection on abscissa-axis of the intercept point of Sb 1 and Sb 2 curves. Thus the different values are presented in Table 1. Figure 4 shows the lamella optimum width (H op ) as function of maximum diffusion coefficient.
We note that lamella optimum thickness increases linearly according to maximum diffusion coefficient. Considering the best fit, we can write the following relation: The constants a and b are respectively the slope and the ordinate at origin of line. We get the following equation: The base optimum thickness decreases depending on the applied magnetic field. Indeed, when the magnetic field increases, mobility and diffusion of minority carriers decrease with the increase in the intensity of Lorentz force slowing down the movement of charge carriers [21]. There is thus a decrease in the diffusion coefficient resulting in the decrease of base optimum thickness. Figure 6 shows the lamella optimum thickness H op as a function optimum temperature.

Conclusions
This thickness optimization technique plays an important role in the case of vertical solar cell junction, which uses low quality materials, whose minority carriers have low diffusion lengths. It makes the back surface recombination velocity at (p-p + ) more efficient by a judicious choice of lamella thickness. That's why, the two expressions of back surface recombination of excess minority carriers are required to determine the lamella optimum thickness for different values of diffusion coefficient as a function of optimum temperature for different magnetic field values. So the different relationships found justify the choice of the lamella optimal thickness either as a function of temperature or magnetic field. Consequently these results can be used as a tool for selecting lamella elaboration process.