Optical simulations and optimization of perovskite/CI(G)S tandem solar cells using the transfer matrix method

Optical simulations were carried out using the transfer matrix formalism [17] with a MATLAB code built in-house (further details will be published elsewhere). For this we used complex refractive indices measured experimentally by spectroscopic ellipsometry prior to this study, namely PVK (E_g = 1.59 eV), C₆₀, tin oxide (SnO₂), hydrogenated indium oxide (IO:H), indium tin oxide (ITO), indium zinc oxide (IZO), nickel oxide (NiO_x ), and extracted from the literature, namely CuInSe₂ (CIS) (E_g 1.01 eV), CI(G)S (E_g 1.10 eV), molybdenum (Mo); molybdenum selenide (MoSe), aluminum-doped zinc oxide (AZO), intrinsic zinc oxide (i-ZnO), cadmium selenide (CdS) [18] and PVKs with bandgaps other than 1.59 eV [19], considered only in the last section of the results, i.e. bandgap variation. The code calculates wavelength-dependent external and internal optical properties, such as reflectance, transmittance and absorptivity of each layer. From the absorptivity in the active layer A_abs, we can calculate the maximal optical equivalent short-circuit current density (J_SC) which is an estimate of the device's real J_SC (assuming all photogenerated electron–hole pairs are extracted) by using (1) [20, 21]:

$$\begin{equation}{J_{{\text{SC}}}} = q\mathop {{{\mathop \int \nolimits }}}\nolimits_0^\infty {A_{{\text{abs}}}}\left( \lambda \right){{{\Phi }}_{{\text{AM1}}{\text{.5G}}}}\left( \lambda \right){\text{d}}\lambda \end{equation} \tag{ 1 }$$

in which q is the elementary charge and ${{{\Phi }}_{{\text{AM1}}{\text{.5G}}}}$ is the AM1.5G spectrum photon flux. To write equation (1) we consider that the external quantum efficiency (EQE) is equal to the absorptivity of the active layer, i.e. that the internal quantum efficiency is 100% in the active layer, and that the other layers do not contribute to current generation (the light absorbed in other layers is considered parasitic absorption). All the layers are treated as optically flat and no interface roughness is taken into account. Therefore, the light is coherent within the structure and optical interference effects will be always present in the simulations. Although models considering incoherent or partial incoherent conditions exist [22, 23] they add complexity and are not needed in the present study. Nonetheless, our approach still provides accurate results when compared to experiments, as shown later.

The optimization process was carried out step-by-step by first considering the single junction cells and then the tandem device with the simulated structures depicted in figure 1. The tandem and sub-cells have already been described in earlier publications and further information on their fabrication can be found in [24].

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The structures from figure 1 only correspond to the optical part of the devices. The structure shown in figure 1(c) combines the structure of figure 1(b) which serves as the bottom sub-cell with the PVK structure of figure 1(a) (with minor variations) which serves as the top sub-cell. Similar structures were fabricated and characterized to compare the results from the simulation with experimental data. The formula of the PVK used here is Cs_0.05MA_0.10FA_0.85Pb(I_0.9Br_0.1)₃, with band gap of E_g = 1.59 eV. The 2PACz interlayer, present in the fabricated PVK device between the PVK and NiO_x , is omitted in the simulation as we have observed that it does not contribute relevantly to the optical properties due to its very small thickness of only a few molecular monolayers. The simulations were carried out in the wavelength range of 300–1300 nm, corresponding to an energy range of 0.95–4.1 eV. The direction of illumination is indicated by the arrows in figure 1, and we consider normal incidence. The thin-film structures are built on a semi-infinite glass substrate (n = 1.5) and are surrounded by air.

The set of refractive indices and the thicknesses of the structures shown in figure 1 were refined using an iterative process of simulation and comparison with experimental data. The importance of the exact refractive indices of the materials to use in simulations has been underlined as early as in 2014 [25], and it is still relevant. The complete list of refractive indices used in this work can be found in the SI.

The simulated data is relevant only if it can be supported by the experimental results of the same device structures. For this, we measured the samples of the structures shown on the figure 1 by two complementary techniques: EQE and reflectance (R), described in SI. The structure from the figure 1(b) is not a working solar cell, since it lacks a transparent conductive oxide (TCO) top electrode, for which it has been characterized only using reflectance measurements.

First, we simulated the structures presented in figure 1 and compared the resulting EQE and reflectance to the experimental results, to verify the model (i.e. the selection of refractive indices and thicknesses). Within an iterative process we found the thicknesses and refractive indices of the layers, so that the simulated results match well the experimental data (see figure 2). The complete list of the complex refractive indices used in this work can be found in the SI.

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We show the final results of the simulated models compared with experimental EQE and R in figure 2. In figure 2(a) we observe a good match for the EQE of the PSC, where the shape follows well the experimental points. However, the reflectance spectra are deviating from the experimental one at near-infrared (NIR) wavelengths. This is caused here by the strong reflection from the inadvertently illuminated silver contacts, that were present in the real sample which significantly increased measured reflectance. Nevertheless, the spectral region from 300 to 800 nm is well reproduced by the simulated curve. In case of the CI(G)S bottom sub-cell, the results of simulations are much closer to the experimental ones. We can see in figure S1(a) that the simulated reflectance matches the experiment, with only small deviations in the NIR region. As a reminder, the code used for simulations does not consider incoherent effects nor the roughness, which can cause the reflectance minima and maxima to be much more pronounced in the simulated curves, while in the experimental data the amplitude of these features is less distinct [18]. As long as the wavelength positions of minima and maxima of the interference fringes in the reflectance spectrum matches to those in the experimental spectrum, together with a good match of the average values of R and EQE, we consider the result sufficient. We note that since the CI(G)S bottom sub-cell does not contain the top TCO layer, the EQE has been not measured. Verification was based only on the reflectance spectrum.

After the individual assessment of the PVK semitransparent solar cell and CIS substrate, the two models were combined to simulate the tandem device. For the single solar cell, we used commercial substrate of ITO-covered glass slides, whereas in case of the tandem architecture, the recombination layer was IZO. For this reason, in the model we also substituted the TCO material, i.e. the ITO of the semitransparent PSC (figure 1(a)) with an IZO film (figure 1(c)) and adjusted the thicknesses of the other layers. The optimization process consisted of simulating the R and EQE spectra for different thicknesses of one layer at a time, and selecting the result that best fitted the experimental data. The thicknesses were kept always in the range of experimentally acceptable values. That is, the thicknesses were in accordance to the production method limitations as for instance it would not be possible to produce very thin layers of some materials, because resulting thin films do not provide homogeneous coverage, or simply to assure good enough electrical properties as is the case especially for TCOs. Since in this work we were not simulating the electrical properties, these thicknesses were selected based on our calibrated nominal TCO layer thickness. The optical response of the optimized tandem model is depicted in figure 2(b), showing a good match between model and experiment for the wavelength range from 300 to 800 nm. In the NIR region (800–1300 nm), the fit is perturbed by interference effects in the R and EQE spectra. This is caused by the fact, that the real samples have some roughness which is not considered in the simulation, and this leads to some partial incoherence and deviation between experiments and simulations, especially for longer wavelengths.

We then calculated J_sc by integrating the experimental and simulated EQE spectra, to further constrain the model, by comparing the values of the current density generated in the absorbers. In the experimental PSC, the generated J_sc from EQE spectra equals 19.4 mA cm⁻², while from the simulated model we find a J_sc of 20.2 mA cm⁻². For the CI(G)S bottom sub-cell, the J_sc generated in the absorber was 39.6 mA cm⁻², which is in range of J_sc values obtained for CIS cells [26]. However, we note that in the presence of the top TCO electrode this value would probably be lower due to increased parasitic absorption in the UV-range (300–400 nm), and possible free carrier absorption in the NIR. For the tandem solar cell, the current density generated in the PVK sub-cell is 20.3 mA cm⁻², both in the experimental and simulated data with an excellent match to the integrated EQE of this sub-cell. The current density generated in the CIS sub-cell was 21.2 mA cm⁻² determined experimentally and 21.3 mA cm⁻² derived from the simulation, proving that average values of the interference-impacted regions are resulting in a J_sc comparable to the measured one. This also indicates that the current limiting sub-cell in the tandem is the PSC, for which we can attribute a loss of 1 mA cm⁻², due to the current mismatch between the sub-cells. The precise values of the current density calculations can be found in the SI (table S1 and figure S2).

In the following we use the models described above as a starting point for the optimization of the tandem structure to reach higher efficiencies beyond the reported record of 24.9% [24]. In this regard, a detailed optical loss analysis of the simulated structures, as shown in figure 3, allows us to identify layers which are the sources of strong parasitic absorption—that is, the light that has been absorbed by this layer does not contribute to the J_sc of the device.

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In case of the semitransparent PSC (figure 3(a)), the largest loss is due to the thick top TCO layer (170 nm of IZO), which absorbs almost 10% of the incident illumination (see table S2 in the SI). Also, C₆₀ is a significant source of parasitic absorption. In the NIR part of the graph (780–1300 nm) we observe significant free carrier absorption in the sub-bandgap region in both TCOs, resulting in a decrease in transmittance of the PSC top cell. TCOs and C₆₀ have been already identified as a main parasitic absorption sources in PVK-based solar cells [15, 27–30], underlining the need to optimize these layers.

Analyzing the optical losses of the CIS sub-cell (figure S1(b) and table S2), we observe very strong parasitic absorption in i-ZnO and CdS layers. For tandems, however, this does not account for a major loss, because photons in the wavelength range absorbed in these layers (300–600 nm) will be already fully absorbed by the PVK film in the top sub-cell. Their thicknesses, however, can play a role in the total reflectance spectrum of the tandem device, thus, we also considered them in the optimization of the tandem device. In this work, the rest of the CIS sub-cell will not be optimized, however we refer the interested reader to further publications related to this task [18, 31, 32].

Last, the optical loss analysis of the tandem shown in figure 3(b) suggests that the part for which we can reach the largest improvement is the PVK top sub-cell, as it is the one where the highest share of parasitic absorption occurs. Similarly, to the semitransparent solar cell (figure 3(a)), the IZO top electrode is responsible for the main losses in the UV–VIS range.

These analyses allowed us to formulate clear goals for the optimization procedure. First, the semitransparent PSC (figure 1(a)) needs to be optimized, to (i) increase sub-bandgap transmittance and (ii) maximize the generated current density by decreasing parasitic absorption in the functional layers and minimizing the reflectance. The threshold value of the sub-bandgap transmittance (T_sub-BG—below the band gap of PVK E_g = 1.59 eV, corresponding to a wavelength of 780 nm) is more than 90%. T_sub-BG is defined as the ratio of the integrated area below the transmittance curve in the range 780–1300 nm to the sum of the A+ R+ T in the same wavelength range. Similarly, any percentage values of R, A or T given later are defined in the same way, but in the full range of wavelengths (300–1300 nm). Finally, the optimized PVK sub-cell will be integrated into the tandem model as a starting point with subsequent adjustments of all layers, in order to maximize the current generated in both absorbers (PVK and CIS) and achieve current matching between the two sub-cells.

In the semitransparent PVK solar cell optimization, we first optimized the top TCO thickness (170 nm IZO), which has been identified as the main source of parasitic absorption in the UV-region. Several TCO materials are considered as suited for top electrodes in PVK-based solar cells, including IZO, ITO, and IO:H [8–10, 15, 29]. We have simulated the J_sc and sub-bandgap transmittance of the PVK solar cell with IZO and IO:H of various thicknesses as the top TCO layer, shown in the supplementary information in figure S3(a). Devices with ITO for the top electrode have also been simulated in this work, however, this material exhibits inferior optical properties compared to IZO and IO:H, and hence we have excluded it from further consideration. In figure S3(a) we can see a result of this optimization. By enabling more light to reach the PVK absorber, the solar cell with IO:H as the top TCO achieves higher current densities than the solar cell employing IZO. Also, sub-bandgap transmittance is higher in this case. In figure S3(a) we observe that for both TCOs the J_sc values exhibit minima for the thickness values between 50 and 120 nm.

Since there was no particular current density established as a goal of this optimization, the T_sub-BG > 90% was first considered when selecting the appropriate TCO thickness. Any of the thicknesses of both materials allows to reach 90% transmittance; moreover, the maximum transmittance corresponded to the thicknesses with the lowest achievable J_sc of the device. From an optical point of view, a thinner TCO will result in better transmittance, however if the TCO layer is too thin it will not provide sufficient electrical conductivity. For this reason, we decided to select IO:H of the thickness of 130 nm as the top contact, which will provide high NIR transmittance of the PVK top sub-cell, while being thick enough to exhibit good electrical conductivity.

We applied a similar procedure for the optimization of the rear TCO in the PSC device. Here, to exchange the ITO, which was resulting in very high NIR parasitic absorption due to free carrier absorption, we considered two other materials: IZO and AZO. All these materials are often used in state-of-the-art PVK/CI(G)S tandem devices [14, 24, 33–35]. In this optimization step, we observe almost no change to the current density, which is as expected, since this TCO layer is located below the PVK absorber. The main goal was to further increase the sub-BG transmittance. In figure S3(b) we can see that AZO layer is resulting in much better transmittance than rear IZO (both layers result in similar current densities of the device). For AZO thicknesses of 70–120 nm, the transmittance is between 89% and 90%. While it can appear counterintuitive at first glance, that the transmittances increase with increasing TCO thickness, the improvement is explained by the decrease in the reflectance spectrum. Since A+ T+ R= 1 (or 100%), we attribute the decrease of reflectance as the source for improved transmittance, even if the absorption in the TCO is increased due to its increased thickness. This relation is shown in figure S4. We can conclude, that AZO as the rear TCO can act as antireflective coating, as its refractive index is in-between that of NiO_x and glass, and hence increase the photo-generated current with increasing thickness.

Thus, the AZO layer with a thickness of 110 nm has been selected as a replacement of the 100 nm thick ITO layer. Although, the optimum may further change later in case of the tandem stack, since here we consider the AZO/glass interface and will need to include the full CIS sub-cell in between them, which results in best improvement of the optical performance for the semitransparent solar cell. Some other minor changes allowed us to eventual reach a J_sc of 21.8 mA cm⁻² generated in the PVK absorber, while a sub-bandgap transmittance of 92% is maintained. These adjustments include decreasing the thickness of C₆₀ and SnO₂ from 20 to 12 nm. The lower the thickness of these layers, the lower the parasitic absorption. However, layers thinner than 10 nm can be challenging to deposit and exhibit inferior electrical properties. Hence we did not decrease their thickness any further. Furthermore, the thickness of the antireflection coating (MgF₂) has been increased from 80 to 100 nm. The final spectra with the concomitant optical loss analysis and the final structure are shown in figure 4. Detailed absorption analysis is listed in the table S2.

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The new PVK sub-cell from figure 4 has been directly applied to the tandem stack, which already enabled us to decrease the parasitic absorption in the UV and blue spectral region with a concomitant increase of absorption in the PVK absorber. However, the following additional adjustments were necessary for the tandem configuration, to obtain higher current density in both sub-cells and the corresponding current matching. We decreased the AZO thickness to 80 nm and changed the thickness of the i-ZnO and CdS layers, which do not absorb any significant amount of light in the tandem configuration (see table S2 in the SI), to obtain a more favorable reflectance. As has been already noted by Jost et al [14], it is possible to find many thickness pairs of these two materials that allow to increase the absorption in the CIS sub-cell. Here, for the tandem configuration, the optimal results were obtained for an increase of the i-ZnO thickness from 30 nm to 50 nm, and a decrease in the CdS thickness from 100 nm to 80 nm. Finally, we adjusted the PVK layer thickness to an optimal value of 640 nm (figure S5), which had the highest impact on the current matching.

With such optimized tandem, we obtained almost ideally matched J_sc of more than 21.7 mA cm⁻² (21.75 in PVK sub-cell and 21.76 in CIS sub-cell). This improves the initial J_sc of the tandem device by almost 1.5 mA cm⁻². The final structure and the corresponding R and EQE spectra with optical loss analysis are shown in figure 5.

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The results of the optical loss analysis of the initial structures from figure 1 ('start') and final optimized structures from figures 4 and 5 ('optimized') are presented in figure 6. The comparison shows the distribution of the optical properties in the devices (that is, the fraction of incident light that is absorbed in, reflected from, and transmitted through the device), distinguishing the absorption in the active layer (absorbers: PVK and CIS) in light green, parasitic absorption A_par in dark green, reflectance R in blue, and transmittance T in grey. All distributions fulfill the condition: 1 = (A_act + A_par) + R + T and clearly indicate that the key to increasing the absorption in the active layers to finally increase J_sc is to decrease the parasitic absorption by selecting more adequate materials with adjusted thickness values.

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Going beyond optical simulations, we use the results of the optical optimization to predict achievable efficiencies of such cells (figure 7). Sha et al for example, calculated the efficiency limit of PVK single solar cells of about 31% (using detailed balance model) [36]. For tandems, Chen et al, predicted 29% efficiency for PVK/CI(G)S tandems [37] if the tandem has an optimized design. Since our work is strictly focused on optical simulations, the values for open circuit voltageV_oc and fill factor (FF) that are required for the calculation of the PCE have been selected by revising the results for the state-of-the-art PVK/CI(G)S tandem devices.

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First, as a general observation we find that the best-performing PVK/CI(G)S tandems have very similar device configuration [14, 24, 33, 35]: IZO as a top electrode, SnO₂/C₆₀ (or PCBM) as an electron transport layer, ETL, NiO_x with self-assembled monolayers (SAMs), or SAMs alone, as a hole transport layer, HTL. The most-used TCO for the recombination layer is AZO, while the CI(G)S bottom-cell configuration remains almost the same. The optimized structure in this work is also very similar to those, with the main difference being the top TCO IO:H instead of IZO. We can thus assume that such structure would reach similar values of the FF and V_oc. For the V_oc, there is a spread of different values that are reached by tandem cells in literature ranging from 1.59 V to 1.77 V (current record devices [14, 24]). Since this parameter depends strongly on the bandgaps of the absorbers, film quality and non-radiative recombination, we decided to calculate the PCE using V_oc for the broad range of values from 1.5 V to 1.9 V. Considering best performing small bandgap CIS (E_g = 1.01 eV) and PVK (1.59 eV) single solar cells, these devices can reach V_oc on the order of 0.6 V [38, 39] and 1.1 V respectively. Regarding the FF, we selected two values to also provide a range of possible combinations of V_oc and FF. An FF = 75% is currently achieved in the state-of the-art devices, and FF = 80% is assumed to be achievable in practice [14]. With this, we can see that the optimized tandem device of this work would be able to reach PCE > 30% for the V_oc = 1.75 V, and FF of 79%. These values might be quite challenging to achieve for such bandgap selection for the absorbers (1.01 eV for CI(G)S and 1.59 eV for PVK), especially assuming quite high idealized current from these simulations of 21.7 mA cm⁻² and high FF of almost 80%. Nevertheless, with this new optimized stack (figure 5), 30% efficiency of CIS/PVK 2T tandem should be within our reach. These improvements could include: passivation to increase V_oc by decreasing non-radiative recombination or trying different bandgap combination of the absorbers. To increase fill factor, management of parasitic resistances (shunt and sheet resistances) would need to be achieved by improving solar cell manufacturing processes.

Our work thus gives the following perspectives for the current PVK/CI(G)S tandem development direction:

• Exchanging top TCO to IO:H would allow to reach J_sc higher than 21 mA cm⁻² (current highest certified J_sc from mentioned tandem structures from the literature is 19.9 mA cm⁻² [34]).
• With no more significant material changes in this tandem stack, the >30% efficiencies will by possible by slightly improving V_oc to 1.75 V and FF to 79%.

This would allow PVK/CI(G)S tandems to catch up with PVK/c-Si technology.

Furthermore, we simulated the J_sc of the tandem structure from figure 5 using PVKs with different bandgaps (1.56 eV, 1.62 eV, 1.67 eV, 1.72 eV and 1.78 eV) and also bottom sub-cell absorbers with two different bandgaps: CI(G)S with bandgap 1.10 eV, typically used in PVK/CI(G)S tandems in literature, and small bandgap CI(G)S, here referred as CIS of a bandgap 1.01 eV used for our cells. Different bandgaps will not only change the theoretically attainable V_oc of the tandem, but will also change the amount of absorbed light by each sub-cell and thus generated J_sc. Refractive indices of the PVK absorbers of different band gaps mentioned in this section has been found in the literature [19].

In our simulations using the optimized structure from figure 5, we simulated J_sc for the different bandgaps of the absorbers. The thickness of the PVK absorber was systematically varied to ensure current matching. When it was possible—we then selected this thickness and the corresponding J_sc of the tandem. However, achieving current matching conditions for most of the bandgap combinations in this structure was impossible. In such cases, the current of the limiting sub-cell has been selected as the tandem J_sc, for the PVK thickness that resulted in the 'closest' current matching conditions possible.

Having simulated J_sc of the tandem devices with different absorber bandgaps, we then estimated attainable V_oc based on the bandgaps. Following Jost et al [14], we assumed a V_oc loss of 400 mV for the PVK, and for CI(G)S cases, we assumed the max V_oc currently achievable in stand-alone solar cells. For CIS (1.01 eV) it was 600 mV and for CI(G)S (1.10 eV) it was 730 mV. Again, for the predictions we used two fill factors: 75% being the currently achievable in the literature for 2T PVK/CI(G)S tandems and a theoretically achievable FF = 80% for the reference.

The results are shown in figure 8. We can see that for larger bandgap absorbers (both PVK and CI(G)S), the attainable PCE is higher compared to the cases with smaller bandgaps. The maximum corresponds to the PVK (1.67 eV)/CI(G)S (1.10 eV) combination, achieving almost 32% efficiency for FF = 80%. For small bandgap CIS, the best bandgap of PVK appears to be the one used in our work (1.59 eV), leading to 30% efficiency as discussed before (figure 7).

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