Research

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Introduction
Due to limited fossil energy reserves, increasing energy consumption in the world and increasing environmental pollution, we can no longer rely on existing energy sources. Due to these issues and challenges such as increasing efciency, reducing prices, sustainability, and storage of energy facing renewable energy sources, so many researches have been done. So far, various layered solar cells based on semiconductors have been studied. Recently, perovskite solar cells have been extensively studied theoretically and experimentally [1][2][3][4][5][6][7]. Such as the work of Yoo and his colleagues in 2021, the work of Wu and his colleagues in 2020, and other works mentioned in references 1 to 7. Silicon solar cells have also been considered for their high efciency and stable and nontoxic materials. However, silicon is a material with an indirect energy gap that has a low light absorption coefcient [8]. In contrast, cadmium telluride solar cells have a direct band gap that corresponds to the spectrum of the sun and has excellent performance and a high absorption coefcient [9]. In addition, these cells can be produced at a lower cost than silicon solar cells [10]. All of these properties have attracted scientists' attention to cadmium telluride-based solar cells. In 1982, these cells were examined by a group of researchers [11], they reported a yield of 10% for these cells, but the results of another group [12] showed a yield of 15.8%, and another American laboratory [13] reported the yield to be 16.5%. Recently, returns of 18% and 22% [12,14], and even returns of 44% have been reported [15]. In cadmium telluride-based solar cells, this layer is often used as an adsorbent layer. Tis adsorbent layer needs a window layer. Te most suitable layer that is used as a window layer next to the cadmium telluride adsorbent layer is cadmium sulfde [16], which forms a PN bond next to cadmium telluride. Cadmium sulfde has a suitable energy gap in the range of 2.42 electron volts. Because cadmium sulfde is a transparent substance, light can penetrate through the cadmium telluride layer, resulting in a photovoltaic efect. Recently, scientists have shown that in addition to experimental approaches, in which optimization is sometimes done experimentally and fails, attempts to analyze device performance through simulation models have shown particular success [17]. Te study of solar cell simulations will be very useful for predicting the output performance of solar cells before the actual construction of the cell, and will signifcantly reduce production costs and avoid excessive costs. Various tools are readily available for simulating solar cells, including wxAMPS, Silvaco ATLAS software, SCAPS-1D software, GPVDM software, and so on. So far, a lot of research has been done and published using these simulation tools [18][19][20][21][22][23][24][25]. But the remarkable point in this feld is that with the use of this ready-made software, it is not possible to study the physical equations governing the problem accurately and tangibly. Also, the errors that occur in numerical calculations are not visible. Results are presented only by giving values and constants to the software, which include the absorption coefcient, energy gap and etc., and the software automatically solves equations and does simulation. So far, no study has been done on the simulation of solar cells without the use of ready-made simulation software. In fact, an important feature that distinguishes our work from other studies that have been done so far is avoiding the use of ready-made simulation software. Because optimization requires a proper understanding of physical processes, we frst coded in MATLAB and Maple software. Our main goal in this research is to simulate and optimize the diferent layers used in solar cells. Terefore, in terms of all mathematical and physical parameters, and by simultaneously numerically solving the transport equations, continuity equations, and Poisson equations, we optimized the performance of a multilayer solar cell by calculating unrestricted parameters in terms of independent and limited variables. Layers and their impact on open-circuit voltage, short-circuit current, flling factor, and efciency have been carefully studied [26].

The Governing Physical Equations of the Problem
First, the transport equations, the electron ̶ hole equation, and the Poisson equation must be solved simultaneously. In some previous studies, a quasi-drift difusion model has been proposed in which only the electron and hole continuity equations were solved, without considering the Poisson equation. Te cavity and Poisson equation was, the extended one-dimensionaldrift-difusion model which can design, optimize, and describe the performance of various cell parameters [27]. Te purpose of this paper is to increase the accuracy of a solar cell using the extended drift-difusion model, adding a layer for optical trapping and thus compensating for the reduction of current, and fnally engineering and optimizing the layers to achieve maximum physical accuracy in the parameters of the solar cell [26]. Te changes in the electric potential of a semiconductor can be determined from the answer to the Poisson equation.
where Ψ is the electrostatic potential, n, p are the densities of the electron and the hole, N − A is the acceptor impurity, and N + D is the impurity. Te densities of electrons and holes in heat balance obey these relations.
where J n and J p are the current density of electrons and holes.

Engineering and Simulated Structure
We frst solved the physical equations governing the problem simultaneously, then discretized the equations for coding in MATLAB and Maple software. We simulated important conditions and variables reported in other work [28,29] for solar cells. We performed the simulation in the conditions of sunlight (100 mW·cm −2 ) and mass of world air (AM) 1.5 at a temperature of 300 K. Figure 1 shows the structure of a simulated CdTe-based solar cell. Te structure consists of a transparent TCO conductive oxide layer, a SnO 2 bufer layer, and a CdS and CdTe layer. Te n-type CdS layer is used as the window layer, and a p-type CdTe layer is used as the absorbent layer.
Te optical band gap, carrier concentration, and other parameters used in simulating the structure of the solar cells were collected from experimental data and other research [9]. Te parameters used in this simulation to describe the structure of the simulated CdTe-based solar cells are listed in Table 1.

Layer Tickness Engineering.
In order to engineer the layer thickness in 4-layer solar cells, including TCO/SnO 2 / CdS/CdTe layers, frst the mathematical and physical equations governing the problem were solved accurately, then discretizing the equations, in MATLAB and Maple software were done to be performed for programming equations. To ensure the accuracy of the results, equations were solved in both MATLAB and Maple software separately and then the results obtained from each software were compared. Additionally, we compare our results with the work done by other researchers in other similar software. Te preparation can also be used to analyze and evaluate the results.
Te thickness of the layers is one of the important and infuential parameters on the electrical properties of solar cells, so frst, by considering the data in Table 1, the thickness of the frst layer of TCO has been engineered. In Figure 2, the efciency results in terms of the thickness of the TCO layer are shown. In the range where the thickness of TCO is less than 0.2 μm, the yield is almost constant and from 0.2 to 0.4 μm the yield starts to increase, and at 0.4 μm it reaches its maximum value. For thicknesses greater than 0.4 μm, the frst layer of yield starts to reduce again. Terefore, when the thickness of the frst layer is 0.4 micrometers, the efciency of the solar cell reaches its maximum.
Ten, considering the thickness of the TCO layer, which is the most efcient for solar cells, the thickness of the second layer of SnO 2 is engineered in terms of efciency. Te thickness of the second layer in terms of efciency is shown in Figure 3. As shown in Figure 3, at a thickness of 0.1 of the second layer, the efciency of the solar cell reaches its maximum value, and for thicknesses greater than 0.1 micrometers, the efciency begins to decrease.
Te third layer that we used in the structure of the solar cell is the CdS layer. Tis layer has a suitable energy gap in the range (2.42-3.1 ev). Due to its suitable chemical properties, it is often used as a window layer in the structure of solar cells. Figure 4 shows the efciency of the solar cell in terms of the thickness of the window layer. As shown in this fgure, when the window layer thickness is greater than 0.1 micrometers, the efciency of the solar cell begins to decrease, and at a thickness of 0.1 micrometers for the window layer, the efciency reaches its maximum value. Table 2 shows the thickness of layers in Figures 5-7. Other physical parameters used are listed in Table 1.  International Journal of Optics 3 We considered the thickness of the frst, second, and fourth layers as 0.4 micrometers, 0.1 micrometers, and 3 micrometers, respectively, and we changed the thickness of the third layer from 0.2 micrometers to 0.8 micrometers, and the changes in the flling and fow coefcients We showed the short circuit and open-circuit voltage according to the thickness change of the third layer in Figures 5-7, respectively.
Te fourth layer that is engineered in the structure of our solar cells is the CdTe layer. Tis layer is used as the absorber layer. CdTe plays the role of the adsorbent layer in the structure of solar cells due to its high and suitable absorption coefcient. Figure 8 shows the efciency diagram in terms of the thickness of the adsorbent layer. Te absorbent layer thickness was carefully engineered in           the range of zero to 5 micrometers. From a thickness of 0 to 3.9 μm for the absorber layer, the efciency increases, and at a thickness of 3.9 μm CdTe, the efciency of the solar cell reaches its maximum. Te efciency is then reduced for thicknesses greater than 3.9 μm for the adsorbent layer. Table 3 shows the thickness of the layers in Figures 9-11. Other physical parameters used are listed in Table 1.
We considered the thickness of the frst, second, and third layers as 0.4 micrometers, 0.1 micrometers, and 0.1 micrometers, respectively, and we changed the thickness of the fourth layer from 0.2 micrometers to 4 micrometers, and the changes in the flling and fow coefcients We showed the short circuit and open-circuit voltage according to the thickness change of the fourth layer in Figures 9-11, respectively.     Diferent values were reported [30,31] for the CdTe adsorption coefcient, in previous studies. Terefore, we examined the reports in the previous works one by one, and after comparing them, we chose the value that would give the maximum return in our work. Figure 12 shows the efciency diagram in terms of the absorption coefcient of the adsorbent layer. Figure 13 shows the efciency of a solar cell in terms of the impurity of the window layer. Te impurity of the cadmium sulfde layer is examined as the window layer from     International Journal of Optics (10 21 to 10 25 )1/m 3 . As shown in Figure 13, the efciency is increased from 10 21 to 10 23 1/m 3 and from 10 23 to 10 25 1/m 3 is almost constant.
Carrier lifespan is another important parameter that has a signifcant impact on the efciency of solar cells. In Figure 14, we showed the lifespan of carriers in the absorber layer in terms of solar cell performance.
Another important parameter that afects the efciency of solar cells is the zero resistance current or in other words the short-circuit current. First, at a zero constant voltage, we examined the short-circuit current in terms of the thickness of the absorber layer, the results of which are shown in Figure 15. Ten, we studied the short-circuit current in terms of the thickness of the absorber layer, at a constant voltage of 0.7 volts. We compare the short-circuit current in terms of the thickness of the absorber layer at two constant voltages of 0 and 0.7 volts and combined both results in Figure 16.
Dark current is another important parameter that is usually less considered in articles. In Figure 17, at a constant voltage of 0.7 volts, we studied the dark current in terms of the thickness of the absorber layer.

Conclusion
So far, many studies have been done on the numerical simulation and optimization of solar cells, most of which have been prepared using simulation software. For the frst time, in a completely diferent way and with accurate consideration of mathematical and physical equations, we simulate these equations ourselves. Rule without using ready-made software packages for simulation, 4-layer TCO/ SnO 2 /CdS/CdTe solar cells were simulated by solving equations and then discretization and programming in MATLAB and Maple software. Ten, for optimization, various parameters such as the thickness of each of the four layers were studied carefully. Te results showed that a 4layer solar cell with TCO/SnO 2 /CdS/CdTe layers with a total thickness of 4.5 microns can be efcient by more than 20%.

Data Availability
Data are available upon request to the corresponding author.

Conflicts of Interest
Te authors declare that they have no conficts of interest.