Optical design of spectrally selective interlayers for perovskite/silicon heterojunction tandem solar cells

: Monolithic perovskite/c-Si tandem solar cells have the potential to exceed the Shockley-Queisser limit for single junction solar cells. However, reflection losses at internal interfaces play a crucial role for the overall efficiency of the tandem devices. Significant reflection losses are caused by the charge selective contacts which have a significantly lower refractive index compared to the absorber materials. Here, we present an approach to overcome a significant part of these reflection losses by introducing a multilayer stack between the top and bottom cell which shows spectrally selective transmission/reflection behavior. The layer stack is designed and optimized by optical simulations using transfer matrix method and a genetic algorithm. The incident sun light is split into a direct part and an isotropic diffuse part. The tandem solar cell with interlayer shows an absolute improvement of short-circuit current density of 0.82 mA/cm 2 .


Introduction
The rapid increase in power conversion efficiency of metal halide perovskite solar cells (PSC) [1][2][3][4] and their high optical band-gap make these devices to attractive candidates for top cells for crystalline silicon (c-Si) solar cell technology [5][6][7][8][9]. Silicon heterojunction (SHJ) solar cells show very high efficiencies, even higher than silicon homojunction solar cells [10,11]. Recently, many different process technologies and layer combinations were investigated for PSC as this technology still needs much work in order to optimize the device performance [12][13][14]. In particular, a lot of research is currently ongoing to identify optimal electron and hole transport layers [15][16][17].
When looking for a tandem cell application, one has to distinguish between monolithic two-terminal devices and four-terminal devices [17][18][19][20][21][22][23]. The big advantage of the twoterminal device is that only one matching box is necessary for the module application. In contrast, the four-terminal device allows that both cells can be individually driven at their respective maximum power point, whereas current matching has to be taken into account for the two-terminal concept. From an optical point of view, one has to take care about any layer between the top cell absorber and the bottom cell absorber for both tandem device concepts to achieve a good optical matching. In particular, layers with refractive indices significantly lower than those of the absorber layers cause reflection losses for light with photon energies lower than the band gap of the top cell absorber [15,23,24].
In case of the four-terminal architecture, two contact layers with sufficiently low sheet resistance need to be deposited between the top cell and bottom cell absorber in order to collect the charge carriers from both cells. Such layers typically show significant absorption losses. Additionally, the mechanical stacking demands for a transparent medium like glass, air or an index matching liquid which separates these contact layers. Therefore, an optically thick region with low refractive index and high absorption can hardly be avoided here.  [25,26]. e PSC still e cular in the cas the optical ligh oupling [27]. T flection losses nt the optical d rocrystalline si ced dielectric B ill be shown t tricted spectra n losses. We w C/SHJ tandem e is 0. 18 [36], although lower process temperatures were reported [37,38]. Besides other possible materials that do not require such high temperatures, Poly (3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) is used as hole transport layer for our simulation study as the processing is simple and flexible, resulting in reasonable efficiencies [39]. As electron transport layer, we use [6,6]-phenyl-C 61 -butyric acid methyl ester (PCBM). The whole layer stack is illustrated in Fig. 1.
As back reflector, an 80 nm thick aluminum doped zinc oxide (ZnO:Al) and a 500 nm thick silver (Ag) layer stack is assumed. The c-Si absorber is passivated at both sides with 5 nm thick a-Si:H. 30 nm thick p-doped and n-doped nc-SiO x :H was assumed at the back and front side of the 250 µm thick silicon wafer as carrier selective contact layers for the bottom cell, respectively. The bottom cell is complemented with an ITO layer. The IL consists of three µc-SiO x :H layers with high, low and high refractive index, respectively.
For the top cell, PEDOT:PSS is assumed as hole transport layer, followed by a quadruple cation perovskite layer with a bandgap energy of 1.65 eV, which can be prepared similar to a previously published recipe [40] with added guanidinium iodide. Details about this process will be published elsewhere. The perovskite layer is coated with a 20 nm thick PCBM as electron transport layer, capped with a buffer layer consisting of ZnO nanoparticles (ZnO NP) and a 700 µm thick glass cover sheet.
During the study, the thicknesses of the ITO, the individual layers of the IL, the PEDOT:PSS, the perovskite and the ZnO NP were varied in order to achieve the highest possible J SC . All interfaces were assumed to be flat.

Simulation
The optical design of the layer stack is done based on the transfer matrix method (TMM) [41]. The complex refractive indices for Ag and c-Si were taken from literature [42,43]. The values for all other materials were experimentally determined by a combination of reflection/transmission measurement, spectroscopic ellipsometry and photo-thermal deflection spectroscopy on reference samples. All layers with silicon and silicon-based alloys were deposited by plasma enhanced chemical vapor deposition, ITO and back contact by sputtering and the layers of the top solar cell by spin coating from liquid phase. For the most important layers for the optical matching condition between top cell and bottom cell, the wavelength dependent refractive indices n (a) and extinction coefficients k (b) were shown in Fig. 2. The wavelength of 750 nm is depicted as dotted vertical line. This wavelength almost equals the optical bandgap of the perovskite layer and, therefore, represents the spectral region where the optical matching between top and bottom cell is mostly relevant.
At this wavelength, the two absorber layers have the highest refractive indices n (3.71 for c-Si and 2.44 for perovskite) with respect to the other materials, whereas the two layers that extract the charge carriers, PEDOT:PSS and ITO, have the lowest refractive indices (1.54 and 1.71, respectively). As interlayers, the high refractive silicon oxide alloy with lower oxygen content and a refractive index of 2.17 at a wavelength of 750 nm was assumed. The second component of the IL consists of a µc-SiO x :H with higher oxygen content and, therefore, lower refractive index of 1.77 at a wavelength of 750 nm. For the TMM simulation, the 700 µm thick glass sheet as well as the 250 µm thick c-Si were assumed as incoherent layers, whereas all other layers were assumed as coherent layers. As the illumination conditions are of major importance for tandem solar cells, the incident light is assumed to consist of a direct and a diffusive component. For the direct light, the AM1.5d spectrum is assumed to impinge along the surface normal. The diffuse part of the sun light (AM1.5g-AM1.5d) is spread over nine angles of incidence, 0°-80° in steps of 10°. The intensities for each angle were weighted in a way to assume an isotropic illumination characteristic from the upper half space. This illumination and the glass cover sheet were chosen to represent a much more realistic condition for real application compared to common approaches. More advanced sky models split the sky into a circumsolar region, a near-horizon region and the remaining region. Although those models describe the solar irradiance more accurately compared to the isotropic assumption [44,45], the deviations are small compared to regional variations of the sun spectrum and the impact of mounting angles on the incident solid angle range. The absorptance in each layer was extracted and the corresponding J SC of the tandem solar cell calculated from the minimum of the charge carrier generation in both absorber layers. By applying a genetic algorithm [46], the layer thicknesses were optimized to achieve the maximal J SC . The advantages of a genetic algorithm are its ability to deal with multiple variables and its applicability to parallel processing. Furthermore, a genetic algorithm does not need derivative information for optimization. The population size, selection fraction, mutation rate and number of iterations were chosen to be 450, 0.65, 0.08 and 400, respectively, and the generated J SC is taken as fitness value. For the ZnO NP, PEDOT:PSS and ITO layers, lower boundaries for the thickness were defined as 35 nm, 20 nm and 35 nm, respectively.

Results and discussions
This section is structured as follows. First, we present an optical loss analysis for the optimized reference tandem solar cell without IL stack. We will demonstrate that reflection losses in the spectral range from 600 nm -900 nm play a major role in the limitation of the device. Next, we will introduce the three-layer stack as IL between the top and bottom cell. We will demonstrate that this IL3 causes a significant reduction of reflection losses which increases the absorptance in both, top cell and bottom cell absorber. This will be complemented by a direct comparison of absorptance and solar cell reflectance for the layer stacks with and without IL. For comparison, we also designed a tandem solar cell with a single IL1 [47] consisting of the µc-SiO x :H layer with higher refractive index. Our study will be finalized by a quantification of the effect of the IL3 on the J SC as a function of the angle of incidence.  Table 1 summarizes the layer thicknesses that were determined by the optimization procedure of the genetic algorithm for the tandem solar cell without and with IL. The order of the columns follows the order of the layers in the tandem solar cell stack in the direction from the bottom cell to the front contact. IL1 and IL3 represent the high refractive µc-SiO x :H, IL2 shows the thickness of the low refractive µc-SiO x :H layer. Note that the optimized thicknesses for the ITO, PEDOT:PSS and ZnO NP layers were very close to the lower boundaries of 35 nm, 20 nm and 35 nm, respectively. Figure 3(a) shows the optical loss analysis for the tandem solar cell stack without IL. Any colored area corresponds to the amount of incident light that is absorbed in the related layer. The top and bottom cell absorber, perovskite and c-Si, were represented by the black and red color, respectively. The absorptance in the ZnO NP and PCBM layers at the front side of the tandem solar cell were shown in magenta and green, respectively. PEDOT:PSS is shown as dark blue and ITO as light blue area. The optical losses within the passivation and charge carrier selective contacts of the SHJ bottom cell were represented together in orange. The absorptance in the back reflector stack (ZnO:Al/Ag) is shown in dark yellow. The white area corresponds to light that is reflected by the whole tandem solar cell stack. Only the black and red areas correspond to the J SC . Any other colored area contributes to parasitic absorption losses.
The parasitic absorption in the short-wavelength range (wavelengths λ<600 nm) is dominantly determined by the absorptance in the ZnO NP and PCBM layers, which correspond to losses in J SC of 0.52 mA/cm 2 and 0.77 mA/cm 2 , respectively. For light with λ>600 nm, the main contribution to the parasitic absorption is given by the absorption losses in the PEDOT:PSS layer (1.13 mA/cm 2 ) followed by ITO (0.67 mA/cm 2 ). Absorption losses originating from other layers are negligible. A huge contribution to optical losses arises from solar cell reflectance. Reflection losses in the long wavelength range (λ>1000 nm) can hardly be avoided with the assumed flat interfaces due to the weak absorption in both absorber layers. A remarkable reflection loss occurs in the spectral range 600 nm<λ<900 nm with a well-pronounced dip. This reflection loss originates from the significant steps in the refractive indices along the transition from the top cell to the bottom cell, mainly the perovskite/PEDOT:PSS interface and the ITO/Si interface. Another issue can be seen in the spectral range from 500 nm<λ<750 nm. Here, the perovskite layer can still absorb photons but obviously not fully. Instead, a significant amount of incident light is absorbed in the c-Si wafer. Due to the higher open-circuit voltage of the perovskite top cell, a photon that is absorbed in the c-Si causes higher thermalization losses compared to a photon that is absorbed in the perovskite. Therefore, it would be advantageous to improve the absorptance in the perovskite layer in this spectral range. In this case, it would mean that a good optical interlayer should (i) lead to a decrease of the absorptance in the c-Si in the spectral range 500 nm<λ<750 nm by increasing the absorptance in the top cell, (ii) increase the absorptance in the c-Si wafer in the spectral range 750 nm<λ<900 nm by a higher transmittance of impinging light to the bottom cell and (iii) reduce the overall solar cell reflectance in the spectral range 600 nm<λ<900 nm.
The optical loss analysis for the tandem solar cell stack with IL3 is shown in Fig. 3(b) with the same color code as in Fig. 3(a). The additional interlayers were represented in yellow. Compared to the tandem cell without IL, the reflection losses in the spectral range 600 nm<λ<900 nm appears to be reduced when the IL3 is implemented into the layer stack and layer thicknesses optimized. It can also be seen that parasitic absorption losses still only play a minor role in this spectral range suggesting that the absorptance, and thereby the EQE, of the c-Si wafer is increased. In order to compare the layer stacks better, the overall solar cell reflectance R for the layer stack without IL (solid line), with IL1 (dotted lines) and with IL3 (dashed line) is shown in Fig. 4(a). The absorptance of the top cell (black lines) and bottom cell (red lines) are illustrated in Fig. 4(b) for the tandem solar cell without IL (solid lines), with IL1 (dotted lines) and with IL3 (dashed lines).
Obviously, a significant reduction of the solar cell reflectance in the range 600 nm<λ<900 nm is achieved by introducing an IL between the top and bottom cell. Outside this spectral range, both reflectance curves appear very similar. This demonstrates a significant improvement of the utilization of sun light. In order to evaluate how this reduced solar cell reflectance is affecting the performance of the tandem solar cell, the absorptance of top and bottom cell were plotted together in Fig. 4(b) for the tandem solar cells with and without IL. The absorptance in the bottom cell is significantly reduced in the spectral range 500 nm<λ<750 nm. Simultaneously, the absorptance in the top cell is significantly increased in the same spectral range. This is caused by either the reflectance of impinging light at the IL or the larger thickness of the perovskite layer in the case of the tandem solar cell with IL (256 nm instead of 220 nm, cf. Table 1). In the spectral range 750 nm<λ<850 nm, a strong increase in the absorptance of the bottom cell is found. This increase is directly related to the decrease in solar cell reflectance and demonstrates the strong spectral selectivity of the IL. Impinging light with 500 nm<λ<750 nm is efficiently reflected, whereas impinging light with 750 nm<λ<850 nm is efficiently transmitted. Thus, the designed IL behaves as desired. This behavior leads to an increase in the J SC of the tandem solar cell with IL compared to the tandem solar cell without IL. The corresponding values are summarized in Table 2. The J SC of the tandem solar cell without IL was determined to be 17.06 mA/cm 2 . The integrated reflection losses in the spectral range 600 nm<λ<900 nm corresponds to a shortcircuit current density of 3.25 mA/cm 2 . The tandem solar cell with IL1 shows a J SC and reflection losses of 17.70 mA/cm 2 and 1.67 mA/cm 2 , respectively. For the tandem solar cell with IL, the J SC and the reflection losses were 17.88 mA/cm 2 and 1.70 mA/cm 2 , respectively. This means that the IL3 leads to a reduction of reflection losses by 1.55 mA/cm 2 and an increase in J SC by 0.82 mA/cm 2 compared to the tandem cell without IL. Assuming that the reduction of reflection losses is distributed equally to top and bottom cell absorber, remaining the current matching condition, the J SC is expected to be increased by half of the value that the reflection losses are reduced. This condition is almost fulfilled. For the tandem solar cell with IR1, the reflection losses are slightly lower compared to the tandem solar cell with IR3 but the achieved J SC is also lower. This means that the spectral selectivity of the IR1 is lower than that of the IR3.
Another issue that needs to be addressed is the dependency of the device performance on the angle of incidence. On the one hand, elongated path lengths of impinging light in the top cell absorber due to oblique angles impact the amount of absorbed photons therein. Therefore, the spectral density that reaches the bottom cell is influenced by the angle of incidence. This probably disturbs the current matching condition and induces significant losses in the device [48]. On the other hand, a multilayered IL, as introduced in our study, makes the situation even more relevant. The discussed effects of the IL on the solar cell reflectance and EQE of top and bottom cell depend on the angle at which light impinges the IL. Figure 5 presents the angular dependency of the J SC of the tandem solar cells without (black lines) and with (red lines) IL3 for diffuse irradiance (solid) and specular irradiance (dotted). Note, that in case of the diffuse irradiance the incident solar spectrum is split into the direct part and the diffuse part. Only the direct part of incident solar irradiance is affected by the angle of incidence, whereas the diffuse part is always assumed to illuminate the solar cells isotropically from the upper half space. In case of the specular irradiance, the same solar cell stacks were illuminated with the full AM1.5g spectrum at the angle of incidence. It can be seen that, although the decrease in J SC with the angle of incidence is slightly stronger for the tandem cell with IL3, the absolute value of the J SC remains higher over the whole angular regime compared to the tandem cell without IL. This means that the introduction of an IL3 between the top and bottom cell leads to an improvement in J SC at any angle of incidence making this concept highly relevant for application. Furthermore, it can be seen that the angular dependency is stronger for the situation with specular irradiance. In other words, the tandem solar cells are more robust against the angle of incidence when the diffuse part of solar irradiance is taken into account. Future work could implement a back side texture to the c-Si bottom cell to increase the absorptance in the long-wavelength region [49]. Furthermore, electrical properties of the involved materials and interfaces should be better understood to find the optimal layer combination, in particular for the top cell.

Conclusion
We have shown that reflection losses in the spectral range 600 nm<λ<900 nm play a crucial role for the performance of PSC/SHJ tandem solar cells. These losses originate at the optical transition from the top cell to the bottom cell and are mainly caused by the low refractive indices of PEDOT:PSS and ITO. In order to compensate these losses, we designed a multilayer stack which is sandwiched between top and bottom solar cell and consists of an alternating series of high, low and high refracting µc-SiO x :H. Applying this interlayer, the EQE of both, top and bottom solar cell, is significantly improved while reducing the solar cell reflectance. The IL acts as a spectrally selective photonic component that reflects light with 500 nm<λ<750 nm and transmits light with 750 nm<λ<850 nm. The improved optical properties by the IL lead to an increase in J SC by 0.82 mA/cm 2 . We have demonstrated that the designed tandem solar cell with IL shows a higher J SC compared to the reference tandem cell independent from the angle of incident light.