Numerical Study of the Effect of Temperature on the Performance of a Silicon Heterojunction Solar Cell (HIT) in the Presence of Excitons

In this article, a detailed study of the physical phenomena in the base of a silicon heterojunction solar cell (HIT) is elaborated. To carry out this work we have established a mathematical model which is in the form of a system of two continuity equations. The latter are subjected to physical conditions of nature to define our field of study. This system of continuity equations is solved using a computational program in a digital programming language. Numerical analysis is used in this study because the mathematical system describing the transport phenomena of load carriers (electrons and excitons) in a silicon heterojunction photovoltaic cell is very complex. Thus, to facilitate numerical resolution, the dimensional parameters of the physical system are rendered dimensionless. The resulting dimensionless equations are discretized by the finite volume method. They are then implemented in a calculation program by an iterative line-by-line relaxation method of the Gauss-Siedel type. In addition, with a low density coupling coefficient b=10 -16 cm 3 s -1 that depends on the material’s properties, the influence of temperature on the diffusion lengths, on the carrier and photocurrent densities, and on the internal quantum yield is studied. This study is carried out using polychromatic illumination with ultraviolet, visible and infrared wavelengths.


Introduction
The study of the effect of temperature on the solar cell has always aroused interest of some authors. Several researchers have conducted research in this area, including: M. Faye, M. Niane, S. Ndiaye, O. Ngom, C. Mbow, B. Ba [1] have shown by considering a non-linear problem and voluminal coupling coefficient b which depends on the temperature, the reduction of the effects of the high values wavelength on the total density of the photocurrent by those of the heat factor and the Fourier number.
M. Burgelman Minnaer et B. [2] which consider that the base is the sector that produces the majority of the photocurrent density.
Zh. Karazhanov [3], carried out his study in the space load area neglecting the electric field. He noticed that excitons have a diffusion length about 3 to 6 times smaller than electrons. Therefore, taking into account the excitons, the author found a significant increase in the current under dark and a decrease in the short-circuit current, the open-circuit voltage, the form factor and the efficiency.
As for M. Faye, C. Mbow, B. Ba [4][5], they also made a numerical study on the influence of the heating factor, the conversion rate and the density coupling coefficient of excitons on the photocurrent density in the base, including the space load area of a silicon homojunction cell. They used monochromatic light. They took into account the presence of a temperature-dependent electric field.
In the continuity of all this work, we studied the influence of temperature on the density of photocourant, on the internal quantum efficiency of a solar cell with silicon heterojunction (HIT) in the presence of excitons, under polychromatic illumination.
To do this, the following plan is drawn up: First, we have developed a mathematical and numerical model with which we have highlighted a calculation program.
The latter leads to the results that will be commented on.

Position of the Problem
The study of a physical problem enabled us to establish a mathematical model, equations. These are governed by the phenomena of generation, recombination, diffusion and dissociation in a silicon heterojunction photovoltaic cell. We consider a silicon heterojunction cell of type (n + p), L length, illuminated by a mono (poly) chromatic light in the presence of excitons.
Our photocell consists of three parts a) The transmitter: area highly doped in donor atoms (1, 5.10 19 cm -3 ); b) Space Load Area (SLA): located between the transmitter and the base of an electric field. This allows the separation and acceleration of electron-hole pairs that arrive at the junction; c) The base: almost neutral zone, doped in accepting atoms (10 17 cm -3 ). The mechanisms of generation-recombination, diffusion and dissociation govern the equations of continuity.

Transport of Electrons in the Base
In the doped type base (p), the minority carriers are charged electrons (-q). These electrons come from free electron-hole pairs and excitons. The density of electrons obeys the law of storage of charges whose system of equations is given by: We describe the photo generation transport of the minority electrons in the base (n e ) and electrons from the linked electron hole pairs: excitons (n x ), by the system of differential equations (1) closed by initial conditions and boundaries.

Boundary Conditions
The evolution of a physical system depends on these boundary conditions. In the theoretical field of physics, we very often use mathematical models to solve physical phenomena. But mathematical equations applied to physics are often difficult or impossible to solve analytically or digitally without resorting to the physical conditions of nature often called boundary conditions. a) For electrons from free electron-hole pairs: b) For electrons from excitons: Excitons being more present in organic semiconductors [6][7][8] where efficiency of organic photovoltaic cells rarely exceeds 5% [9]. The highest occupied molecular orbital (energy) can be compared to the top of the valence band in inorganic semiconductors. The lowest unoccupied molecular orbital (energy) can be compared to the bottom of the conduction band in inorganic semiconductors.
In contrast, in inorganic semiconductors where we have bands with direct or indirect gap, the free electron-hole pairs are more numerous than excitons which are often neglected. Here the electron-hole pairs are weakly bound (excitons of Wanier Mott) [10] and can be easily separated.
Since silicon is an inorganic compound, therefore by hypothesis, the flux of photon generating electron-hole pairs linked can be expressed as a function of the flux of photon generating free electron-hole pairs by the following

Diffusion Equations and Boundary Conditions Without Dimensions
To facilitate the numerical resolution of these equations, it is often necessary to render them dimensionless in order to have homogeneous solutions. a) The electron transport equations and the adimensional boundary conditions of the base: c) For electrons from excitons: With: Some parameters such as electron scatter coefficients and excitons of these equations depend on temperature. Since the equations are non-linear, the result is that the calculation must be self-consistent and not present an analytical solution. Therefore, a numerical resolution of these equations is required.

Digital Modelling
To obtain a numerical solution to our problem, we must transform the differential equations of the mathematical model into a system of algebraic equations obtained after discretization. We have chosen the law of power as the method of discretization. This method involves integrating equations into a discrete set called control volume.
Then we will rewrite the system in the following conservative form:

Meshing and Discretization of the Domain
The adimensionalization allowed us to transform our field of study between 0 z = and  We will therefore use a trigonometric type mesh in the different fields: the space loading zone consists essentially of crystalline silicon of thickness  The discretization steps are then given by: The segment centred around the i z node called the "control volume" and delimited by the nodes (w) and (r) is defined by If we consider i z abscissa knot, we note:

Discretizations of the Equations
To discreet the equations of the mathematical model that describes our physical system, we will use the finite volume method.
The finite volume method is based on a control volume approach and the formulation of an equation that often includes the following terms: a) Unsteady term b) Convective term c) Diffusif term d) Source term In the case of this section, the unstable term will not be considered. In addition, some time-dependent optoelectric phenomena will be studied with time parameters fixed to appropriate values that correspond to our field of study (around s µ and ns ). Let us now integrate the system into the control volume i dυ it comes: To evaluate the gradients of the k n densities delimiting the control volume, we choose an interpolation between neighbouring nodes. , We can rewrite Equation (11) by taking into account the expressions (13) and (14) . Then we get: There are different schematics to approximate the values of the k n functions: centred differences, upwind, hybrid, power law, 2nd order upwind, QUICK etc. In our work, we chose the law of power because of its stability and precision.
By posing the expressions of new coefficients represented in the following Table 2.
Finally, a discrete equation is obtained for the variable k n of the carrier density in the general form of equation (16) . .
a n a n a n S The source term is uniform and linear: in the control volume. We can therefore express it by equation (17) : .   In the same way as in the previous resolution of the diffusion equations of electrons and excitons, we will rewrite the conditions at the initial and final limits respectively in the form of two conservative equations; then he comes:

Discretization of Initial and Boundary Conditions
To solve the first and last terms of these equations, we need a Taylor serial development of the k n derivative as a function of z in the first order.
Let us consider the point of abscissa i z as the point of origin of discrete space then: We deduce from this the approximate expressions of the partial derivatives of k n as a function of the thickness , 1 ,  Finally, we will establish the complete matrix system that approaches our problem.  Expressions of these coefficients are given in the studies [11,12].

The Photocurrent Density of the Carriers
The photocurrent density consists of the diffusion current due to the diffusion of the carriers via the photons and the conduction current due to the effect of the electric field of the space load zone.

Current of Conduction
In a material, the charged electrons collide with the atoms of the crystalline network. Each shock corresponds to a loss of energy. Between two shocks, the electrons are accelerated uniformly in the opposite direction of the electric field ( )

Diffusion Current
Fick's law, which is an experimental law established in 1856 by Adolphe Fick, states that: the number of particles per unit of time and volume ( )

Spectral Response and Quantum Performance
The spectral response is similarly the ratio of the intensity ( ) One distinguishes external quantum yield, from internal quantum yield in which the flux of photon absorbed by the active layer of the cell is taken into account.
The expression presented here takes into account optical losses such as reflection or transmission through the cell: this is the external quantum yield.
Optical losses can be corrected to give the internal quantum yield that takes into account the characteristics specific to the cell (diffusion, surface and volume recombinations and/or absorption of the material under consideration).
By definition, A is the ratio of the number of photon photons generated to the number of photon absorbed. It can also be defined as the ratio between the external quantum yield and the absorption coefficient of the thickness layer z.
With here: ( ) A λ being the absorption coefficient of the thickness layer z .

Results and Discussions: The Influence of Temperature on the Optoelectric Parameters of Doped Crystalline Silicon, Type (p)
Temperature is a very important parameter and its behaviour cannot be neglected in solar cells. The performance of a silicon solar cell is highly temperature sensitive. The latter acts on the molecular structure (the crystalline network) of silicon by exciting electrons and reducing the energy of the absorption gap.
In this article, we set out to study the effect of temperature in the base of a silicon heterojunction solar cell (HIT), often referred to as a high-efficiency solar cell. The basis of the HIT cell is the c-Si (p) with a thickness , we were able to assess the intrinsic parameters on which the mass depends. Indeed, excitons in inorganic semiconductors being pseudo particles whose electron-hole pairs are weakly bound; therefore, we can choose a very low-density coupling coefficient 16 3 in the c-Si (p). In addition to the physical parameters related to the study and the intrinsic parameters, we can give the values of the physical constants involved in this work: The maximum diffusion length of electrons is about six times greater than that of excitons ( Figure 6). At temperatures below 173 K, excitons diffuse more than electrons. While at temperatures ranging from 173 K to 500 K, electrons are more diffusible than excitons ( Figure 5). In addition, we noted in Figure 5b that the temperature has an antagonistic effect on the diffusion lengths of the two types of load carriers: electrons and excitons. That is, the increase of one leads to a decrease of the other and vice versa. Since excitons are in an excited state and are weakly related: Wannier-Mott excitons. As a result, the lower the temperature, the more stable the excitons are and can easily spread. On the other hand, the higher the temperature, the more unstable the molecules in the cell, especially excitons that are already on excited levels.
In addition, at low temperatures, only those electrons that have enough energy to be allowed to rise above the forbidden band are scattered. But as the temperature increases, the electrons propagate deep in the cell, because in addition to the energy of sufficiently charged electrons, the temperature causes a reduction of the gap energy thus favoring the movement of the carriers free charge (electrons). The evolution of the diffusion lengths of electrons and excitons as a function of temperature allowed us to understand the movements of the carriers in depth in the cell, but does not edify us on the real effect of temperature on the photovoltaic cell. To better understand the influence of temperature, we will evaluate it on the density of the load carriers.   In addition, at low temperatures (200 K), molecules in the cell are less subject to external influences such as thermal excitation. As the temperature rises, the molecules in the cell become more and more agitated. This agitation of thermal origin causes some electrons to be in excited positions. As a result, the density of excitons increases to the detriment of that of electrons.

The Density of the Carriers of Charge
In summary, we were able to assess the amount of carriers in a part of the cell made up of c-Si (P). But it is not enough to have a very large number of carriers in the cell to predict its ability to produce energy because in addition to the diffusion and the number of carriers in the cell, the movement of free and bound electrons must be ordered, hence the importance of studying the photocurrent density of electrons and excitons.

Photocurrent Density of Charge Carriers
Figures 9-20 provide information on the photocurrent density of electrons, that of excitons and on the total density as a function of the depth of crystalline silicon for different temperature values.   In this part, the cell is illuminated by a polychromatic light with wavelengths between 0,5 m µ and 2,5 m µ . When we evaluate the photocurrent density as a function of the wavelength, we observe a gradual decrease in the photocurrent density to a value of 1, 6 m λ µ = . Beyond this value, the photocurrent density of the electrons, that of the excitons and the total density remain practically constant. We have also noticed that crystalline silicon only responds to wavelengths between 0, 6 m µ and 1, 6 m µ . Outside of this wavelength range, the photocurrent density is virtually unchanged (see Figures 9-20). Contrary to what is often explained in the literature, crystalline silicon has a range of wavelengths encompassing its gap energy . do not have enough energy, but since they have an energy close to the energy of the gap, there are excited levels between the bands of conduction and valence. They are able to attract electrons that have less energy than the gap. In addition to the phenomena that make it possible for the carriers to pass from one level of energy to the other, the charged electrons interact with each other during their transitions, for example they repel each other, exchange energies among themselves. There are forces like the electric field that deflects them from their trajectories.  For the movement of the load carriers to be orderly, it is necessary that electrons that are in stable positions or that have sufficient energy to reach permissible or intermediate energy levels (excitonic levels) have no influence. In addition, a thermal excitation creates disorder in crystalline silicon by bringing stable electrons out of the valence band of their orbits: that is to say, they are in interstitial positions and this only occurs at very high temperatures. This disorder creates recombinant sites that attract excitons and trap them. On the other hand, an increase in temperature also results in a decrease in the absorption gap, facilitating the orderly movement of free electrons.   On the other hand, at temperature values ranging from 100 K à 500 K , there is an increase in the photocurrent density of the electrons and a decrease in that of the excitons (Figures 12 and 16). The total photocurrent density (electrons and excitons) varies in the same way as that of electrons, but with a difference of 2

2, 22
. mA cm − (Figure 20).  Thus, we can say that as long as we do not reach temperatures that exceed the critical value for the material used, the temperature can be considered favourable to the photocurrent density.
But here too, we cannot say with certainty that the temperature is favourable to the efficiency of our photovoltaic cell.
To do this, we will use the calculation of internal quantum efficiency to measure the efficiency of the photovoltaic cell illuminated by a polychromatic light with wavelengths between 0, 6 m µ et 2,5 m µ .
In this section, we were able to define a very precise wavelength range between 0, 6 m µ et 1, 6 m µ , in which we have variations in photocurrent density.

The Internal Quantum Efficiency of the Photovoltaic Cell
In the case of quantum solar cell yields, transparent and conductive oxides (TCO), considered as window layers, are highlighted. Indium oxide (ITO) is used in our solar cell as transparent electrodes. It has a high optical transmission due to its high gap, good electrical conductivity which allows it to obtain the least transport losses of photogenerated charges and low reflection. This allows it to efficiently transport the photons to the active layer. The photocurrent density study allowed us to define a wavelength range between 0, 6 m µ et 1, 6 m µ in which the c-Si (p) is more active. Photons passing through the active layer by photogeneration undergo recombination and dissociation phenomena that are at the origin of losses in the base.
In Figures 21-23 we reproduce the internal quantum yield of electrons, that of excitons and the total internal quantum yield as a function of the wavelength for different temperature values.
The curves in Figures 21-23 show, for wavelengths between 0, 6 m   In other words, an increase in temperature leads to an increase in the internal quantum yield of electrons and a decrease in the yield of excitons (Figures 21 and 22).
Silicon absorbs photons with wavelengths between 0, 6 m  At a wavelength 1,5 m λ µ = value that corresponds to the plasma wavelength, most photons have low energy that does not allow them to pass through the transparent indium oxide (ITO) window layer.
Two very interesting effects of temperature lead to antagonistic results on the internal quantum performance of electrons and excitons. The increase in temperature leads to a decrease in the energy of the gap which is favourable to the internal quantum efficiency of electrons while a thermal excitation is unfavourable to the internal quantum efficiency of excitons.
The curves in Figure 23 show an increase in the total internal quantum efficiency of the photovoltaic cell of 10% for the different temperature values. This increase is due to the intake of excitons.

Conclusion
Our study shows that excitons are more stable and diffusible than electrons at very low temperatures. This is confirmed by researchers in this field [13][14][15]. In addition, a disorderly increase in carrier density was observed. Thus, the results obtained on the densities of, photocurrent allowed us to define a wavelength range between 600 nm and 1600 nm in which the c-Si (p) is more active to generate the load carriers. Finally, we obtained quantum yields of around 80% with a 10% participation of excitons for the different temperature values.