Prior to the actual fabrication process, it is beneficial to conduct simulations of the proposed solar-cell structure to obtain an estimation of its expected performance and perform reliability analyses. This proves beneficial given the complex and time-consuming process involved in manufacturing solar cells, while also reducing both the associated risks and costs. Several software programs are available, including SILVACO ATLAS, COMSOL, SETFOS, AMPS, Wx-AMPS, and SCAPS-1D, which enable the simulation of solar cell characteristics based on input device parameters.

The SCAPS-1D software for numerical simulation was developed by the Department of Electronics and Information Systems, Ghent University, Belgium. Its primary function is to simulate the properties of the device by solving three key equations: Poisson's equation, electron/hole transport equation, and continuity equation. This facilitates the simulation of various device configurations and offers the capability of performing multiple batch runs for consistent data collection. In addition, it includes powerful tools for data analysis and visualization via user-friendly graphical interfaces.

The SCAPS-1D software is capable of solving continuity equations for charge carriers, including the equation for hole continuity and electron continuity. Furthermore, it handles the optical absorption coefficient, overall charge transport, electron transport, and hole transport equations. ⁵ Figure 1 and Figure 2 illustrate the solar cell definition panel and design of solar cell structures for D1 and D2 device structures respectively. Architecture of both the designed models are shown in Figure 3 followed by the energy band diagrams for both the device structures in Figure 4.

In this study, a simulation was conducted for two different device structures, namely D1 and D2. Device D1 encompassing a specific HTL that is Graphene Oxide, which was chosen to assess its compatibility with titanium dioxide (TiO ₂) as an ETL, along with CH ₃NH ₃SnI ₃ as the absorber layer, was optimized by varying the thickness, acceptor density, defect density of both the absorber layer and HTL, and defect density at the interface between the HTL and the perovskite layer. Another model, D2, encompassing CH ₃NH ₃SnI ₃ as the absorber layer without including any HTL was optimized by varying the thickness, acceptor density, and defect density of the absorber layers. Furthermore, adjustments were made to other PV parameters of the cell for both device models, such as series and shunt resistance and operating temperature. The aim is to identify the optimal conditions for achieving the highest possible device performance for both designed models through a simulation study. The properties of each layer were referenced from existing literature. The input parameters for the different layers are listed in detail in Tables 1 and 2.

3.1.1 Thickness

The basic role of the HTL in PSCs is to facilitate the transfer of holes from the perovskite layer to the electrode while also acting as a barrier between the perovskite layer and the electrode. The thickness of the HTL is crucial for optimal performance, because if the HTL is excessively thick, it increases the series resistance, impeding the efficient transfer of holes to the electrode. This can lead to a decrease in the overall cell performance. However, an HTL that is too thin may not provide sufficient separation between the perovskite layer and electrode. ³ Consequently, this proximity can promote the recombination of charge carriers at the interface between the perovskite layer and the electrode, reducing the efficiency of the device. Therefore, finding the right balance in the thickness of the HTL is important to ensure efficient hole transfer and minimize recombination at the interface between the perovskite layer and the electrode in the cells.

Figure 5 illustrates the impact of adjusting the HTL thickness, ranging from 50 to 500 nm, on the resulting output parameters. The variation in the thickness of the HTL had a minimal impact on the photovoltaic parameters of the cell. Short-circuit current (J _SC), efficiency (ɳ), open-circuit voltage (V _OC) remained nearly unchanged, indicating that they were not significantly affected by the HTL thickness. Although there was a slight increase observed in the Fill Factor (FF) initially from 54.56 % to 54.57 % as HTL thickness was increased from 50 nm to 300 nm, the FF reverted to its initial value of 54.56 % within the range of HTL thickness from 350 nm to 500 nm. However, this small variation is considered negligible in practical terms, and does not have a meaningful impact on the overall performance of the cell.

In summary, changes in photovoltaic parameters due to variations in the HTL thickness were minimal, indicating that the effect was insignificant at such levels of variation. Hence, 300 nm was taken as the optimal thickness for the HTL, resulting in V _oc = 1.17 V, J _sc = 31.95 mA/cm ², FF = 54.57 %, and Efficiency = 20.47 %.

3.1.2 Acceptor density

To achieve the ideal acceptor density for doping the HTL, device simulations were conducted with varying HTL acceptor densities ranging from 10 ¹⁰ cm ⁻³ to 10 ²¹ cm ⁻³, while the acceptor density value of the absorber layer was fixed at 1×10 ¹⁴ cm ⁻³ and the donor density value of the ETL was fixed at 1×10 ¹⁸ cm ⁻³ respectively. The objective is to determine the doping density that would optimize the performance of PSC by striking a balance between the enhanced PV characteristics and minimizing any adverse effects on the device. Figure 6 provides a visual representation of the correlation between the doping concentration of GO and cell performance parameters. By increasing doping concentration from 10 ¹⁰ cm ⁻³ to 10 ²¹ cm ⁻³, notable improvement in PCE was observed, with PCE increasing from approximately 18 % to 22 %. This was due to the increase in the GO doping concentration, and the disparity between the energy bands of GO and the absorber layer became more pronounced. This leads to an enhanced electric potential, resulting in a stronger electric field that facilitates efficient separation of photogenerated carriers. Consequently, the recombination rate decreased, ultimately improving the overall performance of the cell. ⁵ However, achieving doping concentrations higher than 10 ²⁰ cm ⁻³ for the hole transport layer is challenging. Hence, the optimal value was considered for an HTL doping concentration of 10 ²⁰ cm ⁻³.

Moreover, a high density of acceptor dopants can create deep Coulomb traps, which can negatively affect the mobility of holes within the material. ⁷ With a doping concentration of the hole transport layer of 4×10 ²⁰ cm ⁻³, the solar cell performance was coming a PCE of 22.15 %, FF of 58.70 %, V _oc of 1.18 V, and J _sc of 31.95 mA/cm ².

3.1.3 Defect density

The defect density in the HTL of perovskite solar cells is an important parameter that affects the overall device performance. Defect density in HTL can arise from various sources, including impurities, lattice imperfections, and grain boundaries. ⁸ To examine how the performance of the PSC is affected by changes in the defect values for the HTL, the defect values were varied in the range of 10 ¹⁰ cm ⁻³ to 10 ²⁰ cm ⁻³. Figure 7 shows a plot of the fluctuation of the defect density in the HTL. However, as the defect density increased, there was a noticeable decrease in J _SC. Defects in HTL can act as trap states for charge carriers, leading to increased recombination rates. ⁸ Recombination refers to the loss of photogenerated electrons and holes before they can contribute to the photocurrent. Defects in the HTL hinder the efficient transport of charge carriers, specifically holes, from the perovskite layer to the electrode. ⁹ This can result in lower hole mobility and decreased charge extraction, leading to a reduced photocurrent and overall device performance. Furthermore, defects in the HTL can introduce additional resistance into the device, particularly in the series resistance pathway. This increased series resistance can impede the flow of charge carriers, leading to a reduction in the photocurrent and J _sc. Therefore, increasing the defect density in the HTL of a cell is generally expected to decrease J _sc owing to the increased recombination, hindered charge transport, and increased series resistance.

Hence, optimizing the HTL and minimizing the defect densities are critical for improving the efficiency and stability of perovskite solar cells. According to present simulation studies, the cell exhibits optimal performance at a defect density of N _t = 10 ¹² cm ⁻³, resulting in J _SC = 31.95 mA/cm ², V _OC = 1.18 V, FF = 58.70 %, PCE = 22.15 %.

3.1.4 Interface defect density of HTL/Active layer

The interface plays a vital role in the performance of photovoltaic devices, as it directly affects the recombination of electrons and holes. ³ To examine the impact of defects at the interface between GO and CH ₃NH ₃SnI ₃, the performance of the cell was analysed by varying the defect densities from 10 ⁵ cm ⁻³ to 10 ¹² cm ⁻³. As depicted in Figure 8, the J _SC remained constant, while the V _OC experienced a significant decrease from 1.18 V at 10 ⁵ cm ⁻³ to 0.64 V at 10 ¹² cm ⁻³. As the defect density at the interface increased, the FF of the photovoltaic device exhibited an upward trend until it reached a threshold of approximately 10 ⁹ cm ⁻³. However, the PCE of the device continued to decrease from 22.15 % at 10 ⁵ cm ⁻³ to 15.49 % at 10 ¹² cm ⁻³. An elevated level of interface defects at the interface leads to an increase in trapping and recombination sites, thereby diminishing device performance. ³ The occurrence of excess carrier density at the HTL/perovskite interface leads to an amplified charge recombination process. ³ Thus, prioritizing the improvement of the GO/perovskite interface quality by minimizing the interface defects in the region requires significant attention. Based on the simulation study, for GO/perovskite interface, the density of interface defects needs to be 10 ⁹ cm ⁻³ to achieve enhanced results, with an FF of 78.07 % and a PCE of 20.76 %.

3.2.1 Thickness

The choice of absorber layer thickness is a trade-off between light absorption and charge collection efficiency. A thicker absorber layer can absorb more light because of its higher photon absorption, resulting in increased photocurrent generation. However, it can also lead to a longer path for charge carriers to travel, potentially increasing the charge recombination losses. ¹⁰ However, a thinner absorber layer reduces the path length for charge carriers, thus enhancing the charge collection efficiency. However, this may limit the amount of light absorbed and decrease the overall photocurrent. Optimizing the absorber layer thickness involves finding a balance between maximizing the light absorption and minimizing the charge carrier transport losses. ¹⁰

To determine the optimal thickness of the absorber layer, simulations were conducted over a range of 100–1000 nm for both device structures D1 and D2 while keeping the other parameters constant. The dependence of the performance parameters on the thickness of the absorber layer is shown in Figure 9.

As shown in Figure 9(b), increasing the absorber layer thickness leads to increasing values of J _SC for D1 and D2. The increase in J _SC can be attributed to the greater absorption of photons as the absorber layer thickness increases. With a thicker absorber layer, more photons are absorbed, resulting in an increase in the excess carrier concentration, which consequently leads to an increase in J _SC. ¹¹

Conversely, when the absorber layer thickness increased, the series resistance within the cell and its internal power depletion also increased. This, in turn, causes a continuous decrease in FF, which is evident in both D1 and D2. ¹¹

V _oc can be determined using the following equation: V OC = nkT q ln ( I L I O + 1 ) (1) where I O = dark saturation current, I L = light generated current, n is a factor, nkT q is the thermal voltage.

As the absorber layer thickness decreased, the recombination of electrons and holes decreased, leading to a decrease in the reverse saturation current (I _o). Meanwhile, as the thickness of the absorber layer increased, the concentration of excess carriers also increased, resulting in an increase in the photocurrent (I _L) and subsequent enhancement of the open-circuit voltage (V _oc). ¹¹ Figure 9(c) clearly shows that for D2, the values of V _OC continued to increase with increasing absorber layer thickness. For D1, the V _OC increases to an optimal value of 0.83 V when the thickness is 450 nm. After reaching this optimal value, Voc reached a saturation point with minimal decrease. However, the decline in Voc was not significant. ¹²

The PCE of the cell can be determined by the combined effects of the V _oc, J _sc, and FF. PCE = J SC V OC FF P S (2) where, P _S = irradiance from Sun to Earth. ¹¹

By extracting the values of J _sc, V _oc, and FF from the results depicted in the Figures, it is evident that as the thickness increased, the PCE also increased for both D1 and D2. This observation can be deduced by substituting the obtained values into (2). ¹² Thus, 450 nm can be considered as the optimal thickness for D1 and D2, yielding the following set of PV parameters V _oc = 0.83 V, J _sc =31.95 mA/cm ², FF=78.07 %, Efficiency=20.76 % (for D1 structure) and V _oc = 0.72 V, J _sc =31.94 mA/cm ², FF=76.79 %, Efficiency=17.71 % (for D2 structure) respectively.

3.2.2 Acceptor density

The acceptor concentration in a tin-based perovskite absorber layer refers to the doping of perovskites with acceptor-type dopants. In doping, impurities are intentionally introduced into the crystal lattice of a material to modify its properties. In the context of perovskite solar cells, acceptor doping typically involves introducing cations that are electron-deficient, leading to p-type conductivity. Acceptor doping can be used to tune the electrical properties of perovskite materials, such as their conductivity and charge carrier concentration, which in turn affect the overall performance of the solar cell. The acceptor densities of the absorber layers of the two structures were investigated by varying them within the range of 10 ¹⁰ cm ⁻³ to 10 ¹⁹ cm ⁻³ as illustrated in Figure 10. With an acceptor density of 10 ¹⁰ cm ⁻³, the open-circuit voltage is 0.83 V, while at 10 ¹⁹ cm ⁻³, it rises to 0.91 V for D1. This increase in V _OC can be attributed to the declining Fermi level of holes with increasing acceptor density. ⁹ For D2, at an acceptor density of 10 ¹⁰ cm ⁻³, open-circuit voltage is 0.48 V and it keeps on rising with increase in acceptor density value, reaches maximum that is 2.45 V at 10 ¹⁸ cm ⁻³. However, beyond this value, the plot of V _oc started declining. The graph for J _SC exhibits a declining pathway with the current density decreasing from 31.95 mA/cm ² at 10 ¹⁰ cm ⁻³ to 22.08 mA/cm ² at 10 ¹⁹ cm ⁻³ for D1 and from 31.94 mA/cm ² at 10 ¹⁰ cm ⁻³ to 29.61 mA/cm ² at 10 ¹⁹ cm ⁻³ for D2. This behaviour was attributed to the increased recombination of charge carriers at higher acceptor densities. ⁹ As acceptor density increased from 10 ¹⁰ cm ⁻³ to 10 ¹⁹ cm ⁻³, graph for Fill Factor exhibited declining trend, dropping from approximately 78 % to nearly 22 % for D1 and from approximately 68 % to nearly 33 % for D2 respectively. This decrease in the Fill Factor is associated with an increase in the charge-carrier recombination rate within the absorber layer. ⁹ Increased acceptor concentrations can lead to detrimental effects on charge carrier transport, recombination rates, and other optoelectronic properties, thereby reducing the efficiency of the solar cell, as is evident from the efficiency plots of the two devices. ⁹ The analysis reveals that the simulated solar cell demonstrates the finest performance outcomes with V _oc = 0.84 V, J _sc = 31.77 mA/cm ², FF = 79.04 %, Efficiency = 21.04 %, when acceptor density of absorber layer is set to 7×10 ¹⁷ cm ⁻³ for D1 and with V _oc = 0.84 V, J _sc = 31.84 mA/cm ², FF = 80.56 %, Efficiency = 21.78 %, when the acceptor density of absorber layer is set to 2×10 ¹⁶ cm ⁻³ for D2.

3.2.3 Defect density

The defects present in an absorber layer, both in terms of their type and quantity, lead to nonradiative recombination, which significantly affects the performance of the device. Figure 11 shows the performance parameter curves of the two cells for different defect densities ranging from 10 ⁸ cm ⁻³ to 10 ¹⁶ cm ⁻³. This Figure clearly illustrates a significant reduction in the performance parameters as the defect density (N _t) increases. For D1, at N _t = 10 ⁸ cm ⁻³, J _sc = 31.78 mA/cm ², V _oc = 0.84 V, PCE = 21.06 %, FF = 79.09 % and when N _t = 10 ¹⁶ cm ⁻³, J _sc = 31.34 mA/cm ², V _oc = 0.83 V, PCE = 19.73 %, FF = 76.16 %. For D2, at N _t = 10 ⁸ cm ⁻³, J _sc = 31.84 mA/cm ², V _oc = 0.84 V, PCE = 21.79 %, FF = 80.57 % and when N _t = 10 ¹⁶ cm ⁻³, J _sc = 31.84 mA/cm ², V _oc = 0.84 V, PCE = 21.29 %, FF = 79.30 %. Observing this data, it becomes evident that as the defect density increases, there is a noticeable decrease in all the above parameters for both cells. The decrease in performance with increasing defect density is primarily attributed to the recombination process, resulting in the annihilation of charge carriers. ¹⁰ When the defect density was low, the carrier diffusion length was greater, leading to a reduced recombination process. ¹⁰ Consequently, this condition contributes to improved PV performance. The most favourable outcome is achieved at a N _t of 10 ¹⁰ cm ⁻³ with J _sc = 31.78 mA/cm ², V _oc = 0.84 V, PCE = 21.06 %, FF = 79.09 % for D1 and N _t of 10 ¹¹ cm ⁻³ with J _sc = 31.84 mA/cm ², V _oc = 0.84 V, PCE = 21.79 %, FF = 80.57 % for D2.

The components of a PSC, namely the HTL, perovskite layer, ETL, and front contact (FTO), introduce electrical resistance to the device. To investigate the impact of the series resistance on the PSCs performance, simulations were conducted by varying the series resistance from 0 Ω cm ² to 10 Ω cm ² for both device structures. The results, as shown in Figure 12, reveal that the open-circuit voltage remained unchanged with the variation in series resistance for D1 and D2, while other parameters such as short-circuit current, fill factor, and power conversion efficiency decreased with higher series resistance for both D1 and D2. Hence, it is evident that a lower series resistance value is preferable for achieving better overall PSC performance. ^{13, 14} Consequently, a series resistance value of 1 Ω cm ² is considered the optimal choice for both the structures, resulting in a fill factor of 75.60 % (D1), 77.15 % (D2) and an efficiency of 20.13 % (D1), 20.86 % (D2).

The presence of various charge recombination paths in PSC leads to a shunt resistance. ¹³ To investigate the impact of shunt resistance on the device performance, simulations were conducted by varying the shunt resistance from 1×10 ¹ Ω cm ² to 5×10 ⁵ Ω cm ². Figure 13 shows the variation in the device parameters with different shunt resistance values for the D1 and D2 devices. Notably, the efficiency and fill factor of both PSCs exhibit significant improvements with increasing shunt resistance. ¹⁴ At a shunt resistance of 5×10 ⁵ Ω cm ², both the PSCs demonstrated an impressive power conversion efficiency of 21.06 % (D1); 21.79 % (D2), and a high fill factors of 79.09 % (D1); 80.57 % (D2). Thus, 5×10 ⁵ Ω cm ² is taken as the optimal shunt resistance for the D1 and D2 structures; at this value, the cell delivers enhanced performance in terms of efficiency and fill factor.

Solar cells tend to operate more efficiently at 300 K, which corresponds to room temperature, than at higher temperatures. To investigate the impact of the operating temperature on the performance of the proposed cells, variations in temperature ranging from 296 to 400 K were employed, as illustrated in Figure 14. With increasing temperature, the plots of V _oc, J _sc and efficiency show an upward trend for the D1 structure. However, the plot for FF shows an upward trend until 306 K; thereafter, the plot shows degradation with a further increase in temperature. This can be attributed to the fact that, with an increase in temperature, electrons experience heightened instability owing to the influx of greater energy. Consequently, their recombination with holes becomes more challenging when they enter the charging zone, resulting in lower FF. ⁹

For D2, with an increase in temperature, the plot of V _oc and PCE followed a downward trend. However, the plot for FF shows an upward trend until 360 K, and thereafter, the plot degrades with a further increase in temperature. The decrease in the output values observed with increasing temperature occurs because the electrons receive additional energy, rendering them highly unstable. Consequently, they encounter difficulties recombining with holes as they reach the charging zone. ¹⁵ Although the value of J _sc diminishes to some extent, it subsequently increases with increasing temperature. This phenomenon is attributed to the decrease in the band gap energy, which facilitates the generation of a greater number of hole-electron pairs from high-energy photons. ¹⁵

The simulation indicates that the cell D1 achieves its most optimal performance characteristics at a temperature of 306 K with V _oc = 0.85 V, J _sc = 31.78 mA/cm ², FF = 79.14 %, Efficiency = 21.28 % and the cell D2 attains its peak performance at a temperature of 357 K with V _oc = 0.78 V, J _sc = 31.91 mA/cm ², FF = 81.73 %, Efficiency = 20.52 %.