Influence of Temperature and Frequency on Minority Carrier Diffusion Coefficient in a Silicon Solar Cell under Magnetic Field

Seydina Diouf^{1},
Mor Ndiaye^{1},
Ndeye Thiam^{2},
Youssou Traore^{1},
Mamadou Lamine Ba^{2},
Ibrahima Diatta^{1},
Marcel Sitor Diouf^{1},
Oulimata Mballo^{1},
Amary Thiam^{1},
Ibrahima Ly^{2},
Grégoire Sissoko^{1}

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1. Introduction

Minority carrier diffusion coefficient is a recombination parameter which has a big impact on photovoltaic conversion efficiency. His determination is fundamental for different techniques characterization of solar cells. Many studies have been conducted on the minority carrier diffusion coefficient under the influence of temperature, damage coefficient, irradiation flux and magnetic field in static regime [1] [2] [3] or dynamic frequency regime [4] [5] [6] [7].

In this work, the maximum diffusion coefficient in silicon solar cell is determined according to the optimum temperature for different frequency and magnetic field values.

2. Study of the Diffusion Coefficient

The expression of minority carrier diffusion coefficient in solar cell under dynamic frequency regime versus the magnetic field and the temperature is given by the following relation [4] [5] :

$D\left(\omega ,B,T\right)=D\left(B,T\right)\times \frac{\left[\left(1+{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}+{\omega}^{2}\right)\right)\right]+j\omega \tau \left[{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}-{\omega}^{2}\right)-1\right]}{{\left[1+{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}-{\omega}^{2}\right)\right]}^{2}+4{\omega}^{2}{\tau}^{2}}$ (1)

With

$D\left(B,T\right)=\frac{D\left(T\right)}{1+{\left[\mu \left(T\right)\times B\right]}^{2}}$ (2)

$D\left(B,T\right)$ is the minority carrier diffusion coefficient under influence temperature T and applied magnetic field B [2].

$D\left(T\right)=\mu \left(T\right)\times \frac{Kb\times T}{q}$ (3)

D(T) is the diffusion coefficient versus temperature T, in the solar cell without magnetic field [3] [8].

$\mu \left(T\right)$ is the minority carriers mobility temperature [9] [10], dependent in the base and expresses as:

$\mu \left(T\right)=1.43\times {10}^{9}{T}^{-2.42}\text{\hspace{0.17em}}{\text{cm}}^{2}\cdot {\text{V}}^{-1}\cdot {\text{s}}^{-1}$ (4)

q is the electron elementary charge

Kb is Boltzmann’s constant given as Kb = 1.38 × 10^{−23} m^{2}∙kg∙s^{−2}∙K^{−1 }

${\omega}_{c}\left(B\right)=\frac{qB}{{m}_{e}}$ (5)

${\omega}_{c}\left(B\right)$ is cyclotronic frequency of electron [11] [12].

Figure 1 represents the minority carrier diffusion coefficient according to frequency for different values magnetic field at the temperature T = 300 K.

The maximum minority carrier diffusion coefficient in solar cell is obtained when the modulation frequency is equal to cyclotronic frequency. Thus, the

Figure 1. Diffusion coefficient versus frequency for different values magnetic field at the temperature T = 300 K.

curves of this figure make it possible to obtain the cyclotronic frequency for different values magnetic field.

For the rest of the work, Table 1 will allow us to set the value magnetic field for each cyclotronic frequency.

2.1. Determination of the Optimal Temperature by Graphic Method

Figure 2 represents the minority carrier diffusion coefficient according to temperature for different pairs’ values cyclotronic frequency and the magnetic field.

The minority carrier diffusion coefficient increases with temperature up to a maximum value Dn_{max}(ω, B) corresponding to temperature called optimum temperature T_{opt}(ω, B) for a given cyclotronic frequency and magnetic field. Indeed, when the temperature is lower than the optimal temperature T_{opt}(ω, B). Indeed, when the temperature is lower than the optimal temperature T_{opt}(ω, B), the number phonons [13] [14] of high energy varies as an exponential form, according to Boltzmann’s law. The Umklapp processes [13] [14] no longer limit thermal conductivity, which varies in T^{3}. There is not too much thermal agitation hence the increase in the minority carrier diffusion coefficient.

On the other hand for temperatures higher than T_{opt}(ω, B) the minority carrier diffusion coefficient decreases. Phonons are excited for temperatures above the optimum temperature. There is thermal agitation hence the decrease in minority

Table 1. Cyclotronic frequency for different magnetic field values.

Figure 2. Diffusion coefficient versus temperature for different pairs values cyclotronic frequency and the magnetic field.

carrier diffusion coefficient.

From the curves in Figure 3, the values of optimum temperature and maximum diffusion coefficient for each pair cyclotronic frequency and magnetic field are determined and presented in the following Table 2.

Table 3 allowed to represent the following figures.

The curve obtained can be assimilated to an affine function of equation:

$\mathrm{ln}{D}_{\mathrm{max}}\left(\omega ,B\right)=a\mathrm{ln}{T}_{opt}\left(\omega ,B\right)+b$ (6)

${D}_{\mathrm{max}}\left(\omega ,B\right)={\text{e}}^{b}\times {\left[{T}_{opt}\left(\omega ,B\right)\right]}^{a}$ (7)

The constants a and b are determined from the curve, the following equations is obtained:

$2.645=5.67a+b$ (8)

$2.296=5.838a+b$ (9)

${D}_{\mathrm{max}}\left(\omega ,B\right)=1.717\times {10}^{6}{\left[{T}_{opt}\left(\omega ,B\right)\right]}^{-2.065}$ (10)

These results obtained by the graphical method can be verified by an analytical method.

2.2. Determination of Optimal Temperature by Analytical Method

The minority carrier diffusion coefficient curve versus temperature admits a

Table 2. Values optimum temperature and maximum diffusion coefficient for each pair cyclotronic frequency and magnetic field.

Figure 3. Log-log maximum diffusion coefficient versus optimum temperature.

maximum corresponding to optimum temperature T_{opt}(ω, B). This optimum temperature can be obtained by solving the following equation:

$\frac{\text{d}D\left(\omega ,B,T\right)}{\text{d}T}=0$ (11)

Finally we get:

$\begin{array}{l}{T}_{opt}\left(\omega ,B\right)\\ =\sqrt[-1.84]{\frac{2.272\times {10}^{-19}}{1.184\times {B}^{2}}}\frac{\left[1+{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}+\omega \right)\right]+j\omega \tau \left[{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}-{\omega}^{2}\right)-1\right]}{j\omega {\tau}^{3}{\omega}_{c}{\left(B\right)}^{2}-j{\omega}^{3}{\tau}^{3}-j\omega \tau +\left[1+{\tau}^{2}\left({\omega}_{c}{\left(B\right)}^{2}+{\omega}^{2}\right)\right]}\end{array}$ (12)

The relation allows deducing the values of optimum temperature T_{opt}(ω, B) for different values cyclotronic frequency and magnetic field.

Log-log maximum diffusion coefficient versus optimum temperature is presented by Figure 4.

For a comparative study of two methods, we represent in Figure 5, on log-log scale, profiles of amplitude diffusion coefficient versus optimum temperature.

Note that two curves are almost confused. Thus, the relation obtained will make it possible to justify the choice the values temperature, magnetic field and

Table 3. Values optimum temperature and maximum diffusion coefficient for each pair cyclotronic frequency and magnetic field.

Figure 4. Log-log maximum diffusion coefficient versus optimum temperature.

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

frequency in the study of different parameters a silicon solar cell.

3. Conclusion

The study of minority carrier diffusion coefficient in silicon solar cell has shown that the choice of parameter values such as temperature, magnetic field and frequency must obey certain conditions for a good performance of solar cells. Thus, the optimum temperature T_{opt}(ω, B) for a maximum minority carrier diffusion coefficient is obtained using the pairs of cyclotronic frequency and magnetic field values presented in Table 2 and Table 3.

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