Uncooled PV cell under variable light concentration: Determination of profiles of the temperature, the intrinsic properties and the carrier density

Studies on concentrated light influence do not take into account the effect of the heating and this proves to be harmful on photovoltaic parameters. The main purpose of this work is to study the effects of light concentration and the heating caused by this concentration on intrinsic properties and carrier density profile. A thermal model of the PV cell is proposed. By applying the power balance at the steady-state, the PV cell thermal equation is determined. The resolution of this equation leads to temperature profile which shows a rapid increase with light concentration. The mobility n  and diffusion n D coefficients of electrons increase to reach their maxima, respectively


INTRODUCTION
Since the development of the first photovoltaic cell, the field of photovoltaic energy has obtained encouraging results. However, photovoltaic energy still remains uncompetitive on the market compared to traditional energy sources (Royne et al., 2005). This is an obstacle to its large-scale adoption.
To win this challenge of the market, the photovoltaic sectors have been diversified. Among these, highefficiency photovoltaic sector can be cited for light concentration. This technique consists of sending on the PV cell an illumination whose intensity can go up to 1000 times that of sunlight. This concentration of the intensity of the incident light on the PV cell is done by using a system of parabolic mirrors or Fresnel lenses (El Chaar et al., 2011).
It is also well-known that most of the solar radiation absorbed by a photovoltaic cell is not converted to electricity but contributes to increase the temperature (Mattei et al., 2006;Cui et al., 2009). It has been shown that this increase of the PV cell's temperature becomes particularly rapid under light concentration (Cui et al., 2009;Swapnil et al., 2013;Wang et al., 2015).
In addition, many works have shown that temperature increase has strong influences on electronic and intrinsic properties of semiconductor materials and photovoltaic cells (Reggiani et al., 2000;Souza and Sousa, 2019;Dhar et al., 2005;Alkuhayli et al., 2021;Amar et al., 2021;Kabbani and Honnurvali, 2021;Medekhel et al., 2022;Santos et al., 2022). Reggiani et al. (2000) showed the strong decrease of the mobility of the electrons and the holes with temperature increase in silicon devices. In the same direction, Souza and Sousa (2019) have shown that temperature increase leads to a decrease of the mobility of the electrons and the holes. They also showed that the intrinsic carrier density and the reverse saturation current density increase with temperature increase. Ravindra and Srivastava (1979) have demonstrated the decrease of the gap energy with the increase of temperature in semiconductor materials such as germanium (Ge), indium arsenide (InAs), phosphide of indium (InP), gallium arsenide (GaAs), gallium phosphide (GaP), silicon carbide (6H-SiC), and silicon (Si).
Thus, a study of light concentration effect on photovoltaic parameters which does not consider the temperature influence cannot lead to realistic results.
The main purpose of this work is to study the effects of light concentration and the heating caused by this concentration on the intrinsic properties and carrier's density in the base of a silicon PV cell under variable light concentration. Savadogo et al. 97 A thermal model was proposed and which allows to determine the cell temperature profile versus concentration ratio. Thereafter, an electrical model is proposed. On the basis of this electrical model, the influence of concentration ratio on intrinsic properties and carrier's density in the cell base are studied. This study takes into account the increase in temperature linked to the heating caused by light concentration.

Thermal model
The solar incident light is concentrated by using an optical system which can be a lens or a reflecting mirror. As shown in Figure 1, the PV cell receives light power which is converted into electrical power. A large part of the concentrated light received by the PV cell is dissipated in thermal forms. In this model, the following assumptions were made: (1) Heat dissipation is assumed to take place by radiation and natural convection.
(2) Heat exchanges are assumed to take place at the front and back surfaces of PV cell. (3) The thermal convection coefficient is assumed to be the same at the front and at the back of the PV cell. (4) The emissivity is also assumed to be the same on the front side and the back side. (5). Because of the cell small thickness, its temperature is assumed to be uniform and the heat exchanges on the lateral surfaces are neglected. Thus, the powers exchanged by the PV cell are: (1) The absorbed concentrated light power: In Equation 1, 0  represents the concentrator transmissivity and 0  the PV cell surface absorption coefficient.

0
A represents the PV cell surface, C is the concentration ratio and in P the solar illumination power density which is assumed to be 2 1000 / in P W m  under standard air mass conditions AM 1.5 (Ravindra and Srivastava, 1979;Savadogo et al., 2020).
(2) The PV cell electrical power output: where  represents the photo-conversion efficiency and is given by Cui et al. (2009): (1 ) with a=0.425 and b=0.00176.
Author ( where ray A represents the radiative surface and  its emissivity. (4) The power dissipated by natural convection is: where c A represents the convective surface and h its convection coefficient: The PV cell reaches its steady state temperature when the absorbed concentrated light power is equal to the sum of the electrical power output and the powers dissipated in heat forms. lum el ray con Introduction of the above equations into Equation 6 leads to Equation 7: This equation is a polynomial equation of degree four whose numerical resolution by the Newton method leads to the cell temperature profile versus light concentration ratio.

The silicon intrinsic properties
Equations 8 to 15 present the silicon intrinsic electronic parameters.
(1) The electrons and the holes mobility ( , The index m is linked to the type of the doping material (type n or type p). In this work the n-type doping concentration of N d =10 18 cm −3 and p-type of N a =10 16 cm -3 are considered (Souza and Sousa, 2019;Dhar et al., 2005;Savadogo et al., 2020).  (2) The electrons and the holes diffusion ( (4) The intrinsic concentration of electrons in the silicon (Ouedraogo et al., 2021): with n A is a specific constant of the material and for the silicon 3.87 10 .
It appears through these equations that the silicon intrinsic parameters are temperature dependent. As the temperature is function of light concentration ratio, the intrinsic parameters and then the PV cell electronic parameters will vary with light concentration.

The continuity equation
This study is based on a n + -p-p + silicon PV cell operating under an increasing concentrated light C. The following assumptions are used in this study: (1) the contribution of the emitter is negligible (2) we only take into account the base contribution (Savadogo et al., 2020Ouedraogo et al., 2021;Saria et al., 2020;Barandja et al., 2021;Soro et al., 2017).
Because of light concentration, carrier's distribution within the base is non-uniform. This leads us to take into account the electric field of electrons concentration gradient given by Equation 16 (Pelanchon et al., 1992;Savadogo et al., 2020Savadogo et al., , 2021Soro et al., 2017).  ( In Equation 19,  represents the electron lifetime.

Carriers density
The generation rate of carriers n G is equal to the sum of two contributions as show by Equation 20 (Savadogo et al., 2020;Dieme et al., 2015;Gökhan, 2016): (1) The photo generation rate

Influence of light concentration on temperature profile
The numerical resolution of Equation 7 using Matlab software provides data. These data are used to plot with the OriginPro 8 software the temperature profile given in Figure 3. Figure 3 shows a rapid increase in cell temperature with increasing light concentration ratio. Indeed, concentration ratio increase leads to increase of heat received by the PV cell and therefore its temperature increases.
The temperature increases from 300 K to more than 1710 K when light concentration increases from C= 1 Sun to C=666.60 Suns. 1710 K represents the melting temperature of the silicon (Talyzin et al., 2019). This result shows that, a concentrated PV system which is not associated to a system of cooling cannot achieve a concentration ratio of C=666.60 Suns.
According to Reggiani et al. (2000), Equations 8 to 13 which give the diffusion and mobility coefficients in silicon remain valid for temperatures below 650K. Also Ravindra and Srivastava (1979)  In continuation of the present work, we will take as maximum value of concentration ratio, C=50 Suns which corresponds according to the temperature profile to a temperature of approximately T=600K. Figure 4 gives the variations of the mobility of electrons and holes versus light concentration ratio in the silicon. Figure 4 shows that electrons mobility decreases when light concentration goes from 1 Sun to 3 Suns before increase slightly to reach its maximum

Influence of concentration ratio on the electrons and the holes coefficients of mobility and diffusion
The decrease of the electrons and the holes mobility with light concentration increase can be explained by the temperature increase with concentration ratio shown earlier.
These results agree with authors such as Reggiani et al. (2000), Souza and Sousa (2019) and Soro et al. (2017) who showed that the electrons and the holes mobility decrease with temperature increase.
The variations of the electrons and the holes diffusion coefficients versus concentration ratio are also plotted as shown in Figure 5. Figure 5 shows that the electrons diffusion coefficient  Influence of concentration ratio on silicon gap energy Figure 6 gives the variations of the silicon gap energy with light concentration increase. From Figure 6 it appears that a decrease of the silicon energy gap with light concentration increases. This decrease of the silicon gap energy with light concentration increase can be explained by: (1) its decrease with the temperature and which has been shown by authors such as Reggiani et al. (2000). (2) The temperature increase with concentration  Figure 7 gives variations of electrons intrinsic density with light concentration ratio. Figure 7 shows that for concentration values lower than 12 Suns 12 C Suns  , the electrons intrinsic density is constant and practically null. Beyond C=12 Suns where the cell temperature is T= 501.32 K, this intrinsic density increases exponentially with light concentration.

Influence of concentration ration on electrons intrinsic density
The electrons intrinsic density increase can also be explained by two facts: (1) its increase with temperature which has been shown by authors such as Ravindra and Srivastava (1979). (2) The temperature increase with light concentration shown by temperature profile earlier. The curves in Figure 8 show that light concentration increase leads to an increase of carrier density. This result agrees with Pelanchon et al. (1992) and Zoungrana et al. (2012) who did not take into account the effect of light concentration on the temperature.

Influence of concentration ratio on minority charge carrier's density profile
It was noted that for concentrations C=1 Sun to C=9 Suns, carrier density curves present mainly two (02) parts: (1) In the zone near the junction (x≤ 0.01 cm): the carrier density increases slowly with the depth x. This variation of the density can be explained by passage through the cell junction of carriers located in this zone.
(2) Beyond x=0.01 cm, carrier density is practically invariable with the position x in the base. This invariance of the density with the position x is linked to the consideration of carriers thermal generation which is done uniformly in the volume of the base. The photogeneration being however preponderant in the zone close to junction and decreases with the position x in the base.
Results therefore show that carrier's density is greater in rear side compared to the zone near the junction. This result is in contradiction with Pelanchon et al. (1992) and Zoungrana et al. (2012) who showed that carriers density is greater at the illuminated face. This contradiction is due to the fact that Pelanchon et al. (1992) and Zoungrana et al. (2012) did not take into account temperature effect. Indeed, as shown earlier, beyond C=12 Suns where the cell temperature is T= 501.32 K, the carriers intrinsic density increases exponentially with light concentration.
This contradiction is also explained by variations of the gap energy, the mobility and the diffusion of electrons and holes as shown in this study.
The positive slope presented by the carrier density curve at C=12 Suns, shows that all the generated carriers will cross the cell junction to participate to the current. We must therefore expect from C=12 Suns very high values for the current density.
The carrier density profile was also plotted in the shortcircuit situation and for different concentrations as shown in Figure 9. Carrier density and slopes of the curves at junction increase with concentration ratio increase. Opposite to Pelanchon et al. (1992) and Zoungrana et al. (2012), the carrier's density is greater in rear side compared to the The cell being in the short-circuit situation, all the carriers cross the cell junction. There is no carrier storage near the cell junction: at the cell junction, the minority carrier's density is practically null. Also, the augmentation of the slope of curves of carriers density reflects the short-circuit current density increase with the concentration ratio. Near the open-circuit, the carrier density profile was also plotted for different light concentrations as shown in Figure 10.
For a given concentration, we note that carrier density is maximum at the cell junction and decreases with the position x in the base. It was also noted that carrier density curves present negative slopes at the cell junction.
These negative slopes presented by density curves at the cell junction show that the carriers do not cross the cell's junction; we have carrier's storage at the cell's junction.

Conclusion
In this study, a thermal model of the PV cell under variable light concentration was proposed. In steady state, the PV cell thermal equation was obtained. The resolution of this equation led to the temperature profile.
The results show that the cell temperature goes from 300 K to more than 1710 K when light concentration goes from C= 1 Sun to C=666.60 Suns. However, the silicon melting temperature is 1710 K and this result shows that a concentration of C=666.60 Suns cannot be reached without the association of cooling system to the concentrated PV system.
The mobility and diffusion coefficients of the electrons first increase with the concentration to reach their maximum (at C=6.77 Suns for the mobility coefficient and at C= 12.59 Suns for the diffusion coefficient, respectively) before decrease. However, the mobility and diffusion coefficients of the holes decrease with light concentration increase.
The silicon gap energy decreases while electrons intrinsic density increases with light concentration increase. These results are explained on one hand by dependence these parameters on temperature and on the other hand by temperature increase with light concentration ratio.
An electrical model of the PV cell under concentration is also proposed. This model is used to determine the carrier density.
It emerges that whatever the operating point, carrier's density increases significantly with light concentration.
The results also show that carrier's density is greater in the rear side compared to the zone near the junction. This result is linked to carrier's intrinsic density exponential increase with light concentration. This result is in contradiction with Pelanchon et al. (1992) and Zoungrana et al. (2012) who did not take into account temperature effect and who showed that carrier's density is greater at the illuminated face.
This contradiction is also explained by variations of the gap energy, the mobility and the diffusion of electrons and holes as shown in this study. These intrinsic parameters were assumed to be constant by Pelanchon et al. (1992) and Zoungrana et al. (2012).