Numerical simulations were performed using two in-house developed optical simulators, both used for specific optical analysis. To determine complex refractive indices of fabricated rGO layers, a simulator Nika (Čampa et al., 2005 and Čampa, 2012) was used, which was developed to determine real and imaginary parts of the complex refractive index of unknown material from reflectance–reflectance and/or reflectance–transmittance or transmittance–transmittance measurements at different incident angles and light polarization. The advantage of the tool is that the wavelength dispersion function of the material does not need to be known in advance. This is important, especially when dealing with novel materials, such as rGO. All possible mathematical solutions are calculated that are based on known relations between complex refractive indices and measured reflectance and/or transmittance characteristics from which the physical solution is extracted. It also enables detailed determination of the thickness of the characterized layer based on wavelength continuation of solutions.

For simulation of a single-junction and tandem thin-film solar cells, presented by a multilayer stack of layers, the simulator SunShine was used (Krč et al., 2002a; 2002b, and 2003). The simulator was developed specifically for simulations of thin-film solar cells with flat and randomly nanotextured interfaces. It is based on a one-dimensional, semicoherent optical model, enabling the analysis of coherent (thin-layers) and incoherent (thick layers) propagation of light in terms of electromagnetic waves and incoherent rays, respectively, in solar cell structures. Light scattering, if applied, is described by the scalar scattering theory including calibration functions for specific textures. For the angular distribution of light, predefined functions can be used or external values are imported as calculated by an appropriate external model or deduced by measurements (Krc et al., 2005, 2006, 2016, and 2017). In the presented analysis of PK/CIGS tandem devices, the native nanotexture of the CIGS absorber was not considered as the prior test simulations of the structures including the nanotextures show no visible changes in the trends investigated here. The SunShine simulator has been applied before to various concepts of single and multijunction thin-film solar cells (Krc et al., 2005; Zeman and Krc, 2008; Schmid et al., 2010; and Krč et al., 2012).

Using presented optical models, we simulated wavelength-dependent reflectance and transmittance of the entire structures and determined absorptances of individual layers. To determine external solar cell parameters which are directly linked to optical behavior, i.e., external quantum efficiency, EQE, and short-circuit current density, Jsc, we considered the following electrical simplifications which are close to state-of-the-art-devices, ideal extraction of charge carriers from the absorber (PK and CIGS) layers, and neglecting the contribution of the generated carriers from other supporting thin layers. Considering these assumptions, each absorbed photon results in an electron–hole pair, and all generated electrons and holes are expected to be extracted to the contacts; thus, the ratio between charge-carriers flux and incident photon flux, defining the EQE, can be equalized with the absorptance curve of the absorber layer (we denote such obtained EQE as EQE _opt). Applying the AM1.5 g solar spectrum, we calculated J _SCopt from EQE _opt. Such an approach turns out to be effective and sufficiently accurate when studying effects related to optical properties, as shown in our previous research (Krc et al., 2006, 2017; Kovacic et al., 2019; Kovačič et al., 2019).

From simulated J _SCopt values, it is possible to predict the power conversion efficiency of solar cells by accessing (from electrical simulations or measurements) the two other external electrical parameters, i.e., open-circuit voltage (V _OC) and fill factor (FF). However, in this article, we focused on optical effects only by expecting that the main benefits related to rGO layers are expected from the electrical improvements of devices. In any case, rGO layers and the entire structure have to be optimized from both optical and electrical point of view. A detailed electrical simulation study of the structures is underway.

In this section, we present the single-junction PK and CIGS solar cell structures and the complete tandem PK/CIGS solar cell. The single-junction structures were used for model calibration and validation, whereas the tandem structure was considered in the main simulation analysis.

In Figure 1A, a schematic representation of the fabricated PK solar cell is given. This cell was designed already in the direction of a top cell in the tandem device, thus as a semitransparent PK cell. The cell layers are deposited on the top of a glass substrate, where the PK active layer (with energy band-gap E _G = 1.6 eV) is sandwiched between the hole- (NiO) and the electron- (C60&BCP) transport layers, while indium tin oxide (ITO) transparent conductive oxide (TCO) electrode is used on both, top and bottom side of the device. Presented antireflection (AR) layers, ∼100 nm of MgF₂, deposited on top of ITO and on the glass substrate, are used only for single-junction PK cell, while in tandem structure, these are not used. In a tandem device, a bit thinner front ITO contact (70 nm) is used than in single-junction PK cell (100 nm) to minimize near-infrared absorption and assure higher transparency of long-wavelength light to be absorbed in the bottom of the CIGS solar cell. The illumination is considered from the top side in both cases, for single-junction and for tandem devices.

A single-junction CIGS cell, already designed for usage as a bottom cell in tandem structure, was also fabricated and used for calibration and verification of the optical model. Its schematic structure is presented in Figure 1B . An Mo back contact was deposited on a soda-lime glass (SLG) substrate, followed by an absorbing CIGS layer (with E _G = 1.07 eV) as p-type material and thin CdS as an n-type material, forming the p–n solar cell junction. On the top side, the cell was contacted with thin ZnO followed by an ITO electrode in our case. A complete PK/CIGS tandem structure, as used in simulations, and the corresponding layer thicknesses are presented in a schematic cross-section in Figure 1C. The incorporation of thin rGO layers at positions (1) and (2) in the PK/CIGS structure is indicated.

The first step was to determine the optical properties of fabricated GO-based layers. Here, results of wavelength-dependent complex refractive indices of a GO film and a rGO film, both deposited on glass substrates, are presented. The rGO film was reduced upon irradiation with 50 UV laser pulses of 248 nm laser wavelength, 20 ns pulse duration, and laser fluence of 20 mJ/cm². The thicknesses of the GO and the rGO layers were determined to be 70 and 57 nm, respectively. Both layers were deposited on a 1 mm thick glass substrate. Complex refractive indices were determined by the reflectance–reflectance method under different incident angles of the applied beam.

Real part (n) and the imaginary part extinction coefficient (k) of the extracted complex refractive indices for the initial GO and the reduced rGO are shown in Figures 2A,B, respectively. Results show that, in this case, the initial GO exhibits a bit higher refractive index, compared to the rGO layer. On the other hand, the extinction coefficient shows higher values for the characterized rGO, indicating higher absorption in this layer. Increased absorption due to the reduction process may be linked to increased carbon density as a result of the rGO volume reduction. The gradual increase of the absorption also suggests that electronic conjugation within the graphene sheets is restored while irradiating and reducing GO (Huang et al., 2011). In addition, during the reduction process, removal of oxygen-containing groups occurs. There is evidence that the oxygen-containing group is removed through the change of the GO layer transparency (coloration) after laser treatment. This can be attributed to a higher carbon density in the reduced material, due to oxygen group removal. The gradual increase in the absorption also suggests that electronic conjugation within the graphene sheets is restored while irradiating and reducing GO (Huang et al., 2011). Also, during the reduction process, removal of oxygen-containing groups occurs, which leads to the recovery of the π-network. The alteration of the refractive index value during the reduction process could also be attributed to the significant increase in the free carrier density. However, the laser reduction process allows further optimization of the optical and electrical properties of layers. In this article, we used the as-determined optical properties of the rGO layer when simulating the tandem device. Parasitic optical losses were analyzed and quantified inside the solar cell in order to indicate directions and need for further rGO optimizations.

Complex refractive indices of other materials used in the tandem PK/CIGS solar cell have been also determined for realistic layers in CIGS cell by either reflectance–reflectance, reflectance–transmittance, or ellipsometry measurements, as published in (Kovacic et al., 2019). For PK cells, optical constants and thicknesses of the NiOx, PK, C60, BCP, and ITO layers were extracted by spectroscopic ellipsometry measurements using the J.A. Woollam RC2^® ellipsometer.

Before starting with the loss analysis and optimization of structures in relation to rGO in tandem devices, it is important to check the exact status of simulations with respect to obtained experimental results of the available current state solar cells. Here, the validation of optical simulator SunShine was done on single-junction cells used later as subcells in simulated tandem configuration, namely, PK cell and CIGS cell. Selected structure configurations for this investigation are given in Tables 1, 2 for the case of top semitransparent PK and bottom CIGS cells, respectively. Solar cell parameters of both fabricated cells can be found in Table 3. Relatively low-performance values, especially for the CIGS cell, can be attributed to, so far, the nonoptimized electrical and optical properties of devices. The fabricated cells are still from the early stage of optimization processes.

A comparison of measured and simulated EQEs is given in Figures 3A,B for the PK and CIGS cell, respectively. In general, relatively good matching between measurements and simulations can be observed for both cells. In simulations, only minor variation of layer thicknesses, well in the range of fabrication uncertainties, both for perovskite solar cell as well as CIGS solar cell, had to be performed to match the positions of interference fringes in wavelength-dependent characteristics. Detailed layer thicknesses as specified for fabrication and as determined by simulations are given in Tables 1 and 2 for PK and CIGS cells, respectively. In this case, the CIGS cell was simulated by considering the front random native roughness of ∼1500 nm thick CIGS film, with a value of σ _rms = 52 nm (Krc et al., 2005). This value was applied to all interfaces of layers on top of the CIGS absorber. Additional simulations reveal that roughness, in this case, has only minimal effect and is thus not considered in further simulations.

For the PK cell, simulated EQE _opt shows higher values than the measured EQE. We attribute this variation to limited charge carrier extraction, as the developed PK recipe for semitransparent cell and the layers stack was not yet fully optimized in this case. Some deviations can be also observed in reflectance and transmittance curves, which can be contributed to the deviations of structures regarding the antireflection coatings. The EQE was measured for cells without added antireflection coatings (no MgF₂ at back or front), while R and T were measured on devices with 100 nm MgF₂ antireflection coatings at both, front and backside of the device.

For the CIGS cell, significantly reduced back reflection (reduced by 80% compared to the value calculated from refractive indices) had to be considered in simulations, indicating optically nonoptimized CIGS/Mo interface and Mo back contact in the case of these test cells. In case of better back reflection, an increase in long-wavelength reflectance and EQE _opt is expected (Krc et al., 2017). In further simulations, reflectance of the CIGS/Mo interface was considered as expected from our previous experiments (including a thin MoSe₂ interfacial layer in between, resulting in a ∼25% decrease in reflectance characteristics) (Lin et al., 2016 and Kovačič et al., 2019). Besides, nonperfect extraction of charge carriers from the CIGS absorber may be attributed to somewhat lower measured EQE compared to simulated EQE _opt.

The results of validation and previous model applications (Kovacic et al., 2019 and Kovačič et al., 2019) present in our opinion a sufficient level of confidence in the models and input data to study the trends in single-junction and tandem PK/CIGS devices. In our optical study of tandem devices, we still consider an ideal extraction of charge carriers from absorber layers; thus, EQE _opt can be compared to EQE.

Tandem devices can be contacted via two terminals (2T), at the most front and most rear contact or intermediate terminal(s) can be used in 3- or 4-terminal (3T and 4T) devices, where each subcell in stacked configuration can be contacted individually. In the case of a 2T device, the subcell with minimal photocurrent determines the total current of the tandem cell due to a series of electrical connections. Thus, there exists a requirement of current matching to avoid the current losses. In the case of 3T or 4T, the photocurrents and voltages can be accessed individually and the restriction on current matching is not important anymore.

In the scope of further optical simulations, we detect short-circuit current densities, J _SCopts, of the top and bottom cell individually; however, we consider the 2T configuration, where only the initial tandem cell without the rGO layers was approximately current matched. In further modifications of the cell (introduction of rGO layers), current matching has not been performed in this study (by tunning the PK layer thickness and bandgap) since our intention was to decouple and clearly indicate the effect of introduced rGOs on the performance of the individual subcell. For 2T device configuration, thus, the minimal J _SC (corresponding either to the top or bottom subcell) has to be considered as J _SC of the entire 2T tandem cell. In the final device, a further optimization process of current matching is required as well.

Initially, a detailed optical analysis of reflectance, transmittance, and absorptance in individual layers for tandem cells without introduced rGO layers was performed. The results of wavelength-dependent R, T, and A in individual layers are presented in Figure 4A. Absorbances of PK and CIGS active layers, corresponding to the EQE _opt of the top and the bottom subcell, lead to the photocurrents, whereas A of other supporting layers is considered as parasitic losses with neglected contributions to the photocurrent. J _SCopt values calculated from As by considering AM1.5g solar spectrum in the range from 300 to 1200 nm are presented in Figure 4B. Results show that absorbed light in active PK and CIGS layers contributes to J _SCopt of 17.7 mA/cm² and 17.0 mA/cm² for the top and bottom subcell, respectively. Together, around 75% of incident light photons in the given wavelength range is converted to useful photocurrent, leaving 25% of light photons lost due to different optical loss channels.

Results point towards relatively high reflectance as one of the main optical loss factors for both PK and CIGS cells in tandem configuration, contributing up to 12.3% of lost J _SCopt. This loss can be decreased by applying antireflection films (e.g., MgF₂) or additional microtextured light management foils to the device to trap long-wavelength light in the cell and increase the probability to be absorbed in active PK and CIGS layers (Jošt et al., 2017; Kovacic et al., 2019; and Kovačič et al., 2019).

For the top PK cell, the main optical losses are due to absorption in the top ITO and C60 layer, together contributing up to 3.8% of J _SCopt losses in tandem.

For the bottom CIGS cell, main optical losses seem to be attributed to the poorly reflecting Mo back reflector, contributing up to 5.5% current loss. However, it should be noted that these losses occur in the wavelength range where photon energies are close or lower with respect to the selected CIGS bandgap. Therefore, by improving the back reflector of the cell with a 1500 nm thick CIGS absorber, only a part of reflected light can be absorbed in the CIGS layer, being rather reflected out of the device. Additional optical simulations of the hypothetical tandem device with ideally reflecting flat back contact would increase the J _SCopt of the bottom cell by 0.7 mA/cm² which is 4.1% relatively. Further analysis of the results shown in Figure 4 indicates that nonnegligible optical loss is also due to the ITO layer in the middle of the device, presenting up to 3.2% of J _SCopt. Here, losses could be decreased by thinning down the layer or using an alternative solution at this recombination junction between subcells.

The introduction of rGO layers at the two possible positions (see Figure 1C) is analyzed from the optical point of view. As mentioned, electrical simulations are not carried out in this article. The optical properties of the rGO layers as presented in Figure 2A and B are used in simulations. The thickness of rGO layers was selected as a variable parameter in simulations and was varied from 0 to 70 nm. Although in some applications very thin rGO films are used (less than 3 layers of material), we extend the research here to a much broader range of thicknesses (up to 70 nm) to achieve the best compromise between optical (transparency and low parasitic absorption) and electrical properties (conductivity and carrier recombination) of the layer in the PK/CIGS tandem devices.

Results of optical simulations of tandem PK/CIGS device with the rGO layer in the middle (1), at the top (2), and at both positions are presented in plots in Figure 5. In addition, rGO with the optical properties as determined in Figure 2, and a hypothetical case for optically idealized rGO with zero parasitic absorption (zero extinction coefficient) was included in simulations to quantify the theoretical optical potential of such interlayers. Optimization of the laser reduction process allows for the improvement of optical and electrical properties of layers. With the further tuning of the reduction process much lower absorption of rGO material should be achievable. In Figure 5, graphs on the left present the EQE _opt and parasitic absorptance in the rGO layer. 1-R curves are added as well. Graphs on the right present the corresponding J _SCopts as a function of rGO thickness.

Results show that the tandem cell with rGO at the middle position (1), has an impact on EQE _opt of the bottom (CIGS) cell predominantly, while the top PK cell is almost not affected optically, see Figures 5A and B. Such a small effect on the PK top cell is expected in this case since the amount of incident light that passes the top PK cell is not affected and a negligible change in absorptance is associated with slightly changed reflectivity properties at the backside of the PK cell due to the introduced rGO layer (see discussion section). However, by the introduction of rGO and by increasing its thickness from 0 to 70 nm, EQE _opt of the CIGS cell noticeably falls. This decrease is mainly attributed to the increased parasitic absorption in the thicker rGO layer. Focusing on 1-R curves, in the wavelength range λ = 800–900 nm, a decrease with the increased thickness or rGO layer is observed, whereas in the long-wavelength range (λ > 900 nm), the opposite trend occurs. The decrease in 1-R curves (increase in R) is due to increased parasitic absorption in the thicker rGO layer in this wavelength range, letting less light to leave the structure. The increase in 1-R curves at longer wavelengths (decrease in R) is attributed to interference effects due to introduced rGO as it will be addressed in the discussion by partial structure simulations.

The same trend as observed for absorptances of the active CIGS layer in Figure 5A is reflected in the J _SCopt values of the CIGS cell (Figure 5B). For example, in a 5 nm thin rGO, a decrease in the J _SCopt of the CIGS cell is around −1.3% and increases linearly with the increased thickness, according to simulations. For the idealized rGO with zero absorption, the J _SCopt values stay almost unchanged as indicated by dashed lines in the plot.

Next, we analyze the results of simulations of the tandem cell with the rGO layer at top position 2), i.e., between the top ITO and BCP layer. Both EQE _opt for PK and CIGS subcell shown in Figure 5C decrease with an increased rGO layer thickness in this case. This is since the rGO located at the top position absorbs short- and long-wavelength incident light prior to reaching both active layers (PK and CIGS). The corresponding J _SCopt dependencies are plotted in Figure 5D. Again, almost linear decreases in J _SCopts are observed for increased thickness of rGO. A higher rate of decrease is present in J _SCopt of the PK cell. This is because rGO has a higher extinction coefficient and thus parasitic absorption in the short-wavelength range of light where the PK subcell is sensitive to light absorption. For a 5 nm thick rGO layer, J _SCopts of the top and bottom cells are reduced by 3.5% and 0.6%, respectively. In this case, the requirement for using either very thin or low-absorbing rGO material (at short and long wavelengths) is even more important in the final optimization of the device. Simulation results of a hypothetical rGO layer without absorption losses show shallow interference effects in J _SCopt dependency on rGO thickness, resulting in even slight improvement in J _SCopt of the CIGS cell around the thickness of 20 nm.

The last study case includes rGO at both positions in the middle and at the top of the tandem cell [(1) and (2)]. Again, reduced EQE _opt in both, PK and CIGS cells is observed ( Figure 5E) and the corresponding J _SCopts (Figure 5F ). In both graphs, the values at the horizontal axis correspond to both rGOs (thickness variation of both rGOs simultaneously). As indicated before, the main restriction are parasitic optical losses, now in both rGOs, resulting in a cumulative reduction of J _SCopt in the CIGS subcell. However, due to the higher extinction coefficient of rGO at shorter wavelengths, the losses in the PK subcell are higher. If both rGO layers are set to 5 nm, the corresponding decreases in J _SCopts are 3.5% for the top PK and 1.9% for the bottom CIGS cell. Simulation results with hypothetical rGOs with zero absorption losses show a very similar trend as in the case study before.