Minority Carrier Diffusion Coefficient D * ( B , T ) : Study in Temperature on a Silicon Solar Cell under Magnetic Field

This work deals with minority carrier diffusion coefficient study in silicon solar cell, under both temperature and applied magnetic field. New expressions of diffusion coefficient are pointed out, which gives attention to thermal behavior of minority carrier that is better understood with Umklapp process. This study allowed to determine an optimum temperature which led to maximum diffusion coefficient value while magnetic field remained constant.


Introduction
The photovoltaic conversion efficiency depends on the nature and structure of the semiconductor, its manufacturing processes and the operating conditions.In order to improve solar cell performance, several characterization techniques of semiconductor material have been proposed.Among the most important parameters in the different characterization techniques, it can be noted the diffusion coefficient [1] [2] of the minority carrier (D).Thus, the diffusion coefficient was the grain size (g) [10], the temperature (T) [11] [12] and the electric field (E) [13] [14] [15].
It then affects the determination of the recombination parameters in the bulk i.e. lifetime (τ) and on the surfaces, specially, the back surface recombination velocity (Sb) and junction surface recombination velocity (Sf)) [24] [25].It then affects the determination of the recombination parameters in the bulk i.e. lifetime (τ) and on the surfaces, specially, the back surface recombination velocity (Sb) and junction surface recombination velocity (Sf)) [24] [25] [26] according to the operating conditions [11] [12] [27] [28] (steady state, dynamic frequency and transient) and according to the space dimensional model [29] under study i.e. (1D) or (3D) of the solar cell, diffusion coefficient gets new expressions [30]- [35].Taking into account the emitter, the ambipolar diffusion coefficient is then derived [20] [21].
In static regime, the photocurrent Iph is studied versus absorption coefficient wavelength dependent (λ) and leads to spectral response [36] [37] [38].The well known current-voltage (I-V) characteristic (under dark or illumination) allows the determination of the electrical parameters such as series (Rs) and shunt resistances (Rsh), and junction transition capacitance (Cz) [39] [40].
In frequency regime, we note the studies of both Sb and Sf, excess minority carrier recombination velocity respectively at the junction and at the back side surfaces, by the help of Bode and Nyquist diagrams, leading to electrical equivalent models, with effect of both external (B, E, Φ, kl) and internal (g, Sg, (λ)) parameters [3] [41] [42] [43].
In this article, the study focuses on the minority carriers diffusion coefficient in silicon solar cell under both temperature and applied magnetic field.

Presentation of the Solar Cell
We consider a back surface field (B.S.F) silicon solar cell (n + -p-p + type) under influence of temperature and applied magnetic field (Figure 1).

Diffusion Coefficient
When the solar cell is illuminated, the phenomena of generation, diffusion and recombination of the minority carriers in the solar cell base are considered.
The minority carrier diffusion coefficient D × (B) in the base under the influence of applied magnetic field B [3], is extended with applied temperature T and then gives the following equation: where D 0 (T) is the diffusion coefficient versus temperature T, in the solar cell without magnetic field.It is given by the Einstein-Smoluchowski relation [44] [45]: With μ(T) is the minority carriers mobility temperature [46] [47] dependent in the base and expresses as: ( ) q is the electron elementary charge and kb is Boltzmann's constant given as kb = 1.38 × 10 −23 m 2 •kgs −2 •K −1 .

Magnetic Field Effect on the Diffusion Coefficient
Figure 2 shows the minority carrier diffusion coefficient versus magnetic field logarithm for different temperature values.For a given temperature, the diffusion coefficient is maximum and almost constant when the magnetic field is weak.Indeed, for low magnetic field values, the carrier mobility is not strongly influenced by magnetic field variation and this explains the bearing observed.On the other hand, when the magnetic field is  greater than 10 −3 T, mobility and minority carrier diffusion decrease with the magnetic field [48] [49].The diffusion coefficient is more sensitive to temperature for weak magnetic field values.However an inversion is observed when B is greater than 10 −3 T, where the diffusion coefficient increases with temperature.For lower magnetic field values (<10 −3 T), the diffusion coefficient increases with temperature and reaches a maximum value corresponding to a temperature called optimum temperature Topt (B) then decreases.Indeed, when the temperature is below Topt (B), the Umklapp process [50] does not limit the thermal conductivity which varies with T3 [51], so the thermal resistance decreases according to the temperature which leads to an increase of the diffusion coefficient [51] [52].High thermal resistance induced by high temperatures is due to the exponential establishing of Umklapp process which provides 1/T thermal conductivity dependent [51] [52] [53].Thermal agitation reduces minority charge carrier mobility of and causes the diffusion coefficient decreasing [48].

2. Temperature Effect on Diffusion Coefficient
On the other hand, when the magnetic field is greater than 10 −3 T, the diffusion coefficient increases with temperature.
Moreover, it may be noted that the optimum temperature increases according to the magnetic field intensity

Magnitude of the Diffusion Coefficient as a Function of the Optimum Temperature for Different Magnetic Field Values
The optimum temperature Topt (B) for maximum diffusion is determined using two methods: From the curves in Figure 3, the maximum diffusion coefficient values are determined according to the optimum temperature for different magnetic field values.
Figure 3. Diffusion coefficient versus temperature for different magnetic field values.
From Table 1, we represent in Figure 4 the profile on log-log scale, diffusion coefficient versus optimum temperature.
Considering the average right, the following relationship is obtained: The constants a and b are determined from the curve, the following equations is obtained: The resolution of the equations constituted by relations ( 6) and (7) gives: Hence the relationship Topt: • Analytical method The diffusion coefficient is maximum when the temperature is equal to Topt for a given magnetic value B which remained constant.Thus, by annulling its derivative versus temperature, we can determine Topt while keeping B constant value.
Ln(Dmax) The derivative of the diffusion coefficient at T = Topt is given by the relation as: We then deduce the relationship: Using the relation (10), the optimum temperature can be calculated for different magnetic field values.Results are presented in Table 2.
For a comparative study of the two methods, we represent in Figure 5, on loglog scale, profiles of the amplitude of diffusion coefficient versus the optimum temperature.The results for the two methods are identical to one decimal place.
The two curves are almost confused.So for the rest of this work, we can justify the choice of temperatures set in the study of various parameters of the solar cell.
For a given value of the magnetic field, the temperature to be used must obey the relation (8) in order to obtain an optimal response of the solar cell under magnetic field.

Conclusions
The minority carrier diffusion coefficient *( , ) D B T study has shown much more sensitivity to temperature for weak applied magnetic field.For low magnetic field value, the minority carrier diffusion decreases with temperature which reduces the solar cell performance.
Otherwise, the diffusion coefficient increases with temperature, reaches a maximum value corresponding to a temperature called optimum temperature.For a fixed magnetic field value, the diffusion coefficient decreases with the optimum temperature.The relation obtained between the maximum value of the diffusion coefficient and the optimum temperature allows justifying the selection of the temperature values for the study of the solar cell parameters.

Figure 3
Figure 3 shows the profile of the diffusion coefficient versus temperature for different magnetic field values, obtained by plotting combined Equations (1) (2) and (3).

Figure 5 .
Figure 5. Log-log maximum diffusion coefficient versus optimum temperature for both methods.

Table 1 .
Diffusion coefficient with optimum temperature.

Table 2 .
Optimum temperature with magnetic field.