1. Introduction

Perovskite-silicon tandem solar cells provide the opportunity to surpass the efficiency limits governing single junction solar cells at a competitive price [1] and low resource and energy consumption [2]. Progressing rapidly, they evolved from an emerging technology [3] to a real contender for mass production within a single decade, exhibiting efficiencies of above 30% in the laboratory [4]. Approaching the theoretical efficiency limit, detailed minimization of all loss channels becomes increasingly important to further advance the technology. An important group of loss channels originates from the optical properties of the tandem solar cell system. A comparative look at the total generated current (sum of current absorbed in perovskite and silicon) allows for an estimation of the remaining potential for optical improvements: While in the most efficient perovskite-silicon tandem solar cells published so far, a total current of around 40.0 mA/cm² is obtained [5], values of up to 42.9 mA/cm² have been measured in silicon single junction solar cells [6]. Assuming no change in the other parameters, a short circuit current improvement of this magnitude would result in an absolute efficiency improvement of around 5%_rel by reducing optical loss channels. Perovskite-silicon tandem solar cells provide the opportunity to surpass the efficiency limits governing single junction solar cells at a competitive price [1] and low resource and energy consumption [2]. Progressing rapidly, they evolved from an emerging technology [3] to a real contender for mass production within a single decade, exhibiting efficiencies of above 30% in the laboratory [4]. Approaching the theoretical efficiency limit, detailed minimization of all loss channels becomes increasingly important to further advance the technology. An important group of loss channels originates from the optical properties of the tandem solar cell system. A comparative look at the total generated current (sum of current absorbed in perovskite and silicon) allows for an estimation of the remaining potential for optical improvements: While in the most efficient perovskite-silicon tandem solar cells published so far, a total current of around 40.0 mA/cm² is obtained [5], values of up to 42.9 mA/cm² have been measured in silicon single junction solar cells [6]. Assuming no change in the other parameters, a short circuit current improvement of this magnitude would result in an absolute efficiency improvement of around 5%_rel by reducing optical loss channels.

There are three such optical loss channels: (i) Reflectance losses, where light is reflected from any of the interfaces within the tandem cell stack and leaves the cell without being absorbed; (ii) Thermalization losses where light that could potentially be absorbed in the top cell is transmitted to and absorbed by the bottom cell, resulting in less current generated at the high top cell voltage, and (iii) Parasitic absorption. Multiple studies introduced modifications aiming at reducing the impact of one or multiple of these loss channels. These include antireflective coatings [7], intermediate reflectors. There are three such optical loss channels: (i) Reflectance losses, where light is reflected from any of the interfaces within the tandem cell stack and leaves the cell without being absorbed; (ii) Thermalization losses where light that could potentially be absorbed in the top cell is transmitted to and absorbed by the bottom cell, resulting in less current generated at the high top cell voltage, and (iii) Parasitic absorption. Multiple studies introduced modifications aiming at reducing the impact of one or multiple of these loss channels. These include antireflective coatings [7], intermediate reflectors [8], adjustment of layer thicknesses [9] and structuring of one or multiple interfaces within the tandem solar cell stack [5,10–14].

While these studies showed that structured interfaces have a beneficial effect, a broader overview comparing the impact of structure geometry, position in the layer stack and combination of multiple, individually optimized structures for a single reference system is lacking. Within the presented work, such a comparison is provided for structures at the two interfaces (i) air-perovskite and (ii) perovskite-silicon. To this end, a state-of-the-art tandem solar cell is simulated using a combined model of rigorous coupled wave analysis (RCWA) [15–17] for the thin film top cell and a simple bottom cell model including Lambertian scattering at the rear side [18]. Nanostructures at the two interfaces are introduced in the model. Their geometry is varied across a broad parameter range in period and structure height as wells as in shape, while all material volumes are kept constant. The spectrally dependent effect of this variation on the reflective behavior and the locally resolved absorption distribution is evaluated. In addition, the interaction effect arising from different structure combinations is investigated. A fully textured tandem solar cell deposited on random pyramids serves as reference system. This is modeled by ray tracing.While these studies showed that structured interfaces have a beneficial effect, a broader overview comparing the impact of structure geometry, position in the layer stack and combination of multiple, individually optimized structures for a single reference system is lacking. Within the presented work, such a comparison is provided for structures at the two interfaces (i) air-perovskite and (ii) perovskite-silicon. To this end, a state-of-the-art tandem solar cell is simulated using a combined model of rigorous coupled wave analysis (RCWA) [15–17] for the thin film top cell and a simple bottom cell model including Lambertian scattering at the rear side [18]. Nanostructures at the two interfaces are introduced in the model. Their geometry is varied across a broad parameter range in period and structure height as wells as in shape, while all material volumes are kept constant. The spectrally dependent effect of this variation on the reflective behavior and the locally resolved absorption distribution is evaluated. In addition, the interaction effect arising from different structure combinations is investigated. A fully textured tandem solar cell deposited on random pyramids serves as reference system. This is modeled by ray tracing.

2. Methods

2.1 Solar cell and structure configurations

The different configurations analyzed in this work are shown in Fig. 1(a)-(e). The flat reference structure (a) features a thin film top cell stack placed upon a crystalline silicon absorber representing the bottom cell. From bottom to top, the top cell stack consists of an ITO recombination layer, a PTAA/PFN hole transport layer (for the optical simulation, this is treated as one layer), the perovskite absorber, C₆₀/SnO_x/ITO_top electron transport layers and a MgF₂ anti-reflection coating. This configuration is based on Ref. [19] and [9]., but different optical datasets for both MgF₂ [20,21] and perovskite [20,22] were used. The MgF₂ dataset takes dispersion into account in both n and k. The perovskite dataset chosen for this work exhibits a higher bandgap of 1.72 eV (see Supplement 1 section S3 for bandgap calculation). This was done as the optical structuring investigated within this paper is expected to benefit current matching, omitting the need for a bandgap decrease to cater this property. Subfigures b-e depict the different structure combinations discussed in this study, namely a structure at the air-perovskite interface (b), the perovskite-silicon interface (c), the same structure at both interfaces (d) and separately optimized structures at the two interfaces (e). For all structured system configurations, the volume of all materials was kept constant relative to the reference system. The flat reference structure (a) features a thin-film top cell stack placed upon a crystalline silicon absorber representing the bottom cell. From bottom to top, the top cell stack consists of an ITO recombination layer, a PTAA/PFN hole transport layer (for the optical simulation, this is treated as one layer), the perovskite absorber, C₆₀/SnO_x/ITO_top electron transport layers and a MgF₂ anti-reflection coating. This configuration is based on Ref. [19] and [9]., but different optical datasets for both MgF₂ [20,21] and perovskite [20,22] were used. The MgF₂ dataset takes dispersion into account in both n and k. The perovskite dataset chosen for this work exhibits a higher bandgap of 1.72 eV (see Supplement 1 section S3 for bandgap calculation). This was done as the optical structuring investigated within this paper is expected to benefit current matching, omitting the need for a bandgap decrease to cater this property. Subfigures b-e depict the different structure combinations discussed in this study, namely a structure at the air-perovskite interface (b), the perovskite-silicon interface (c), the same structure at both interfaces (d) and separately optimized structures at the two interfaces (e). For all structured system configurations, the volume of all materials was kept constant relative to the reference system unless explicitely mentioned otherwise.

 

Fig. 1. The reference perovskite solar cell stack (a) and configurations with a structure at the air-perovskite interface (b), the perovskite-silicon interface (c), the same structure at both interfaces (d) and optimized structures at the two interfaces(e).

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2.2 Wave-optical modelling of nanostructured interfaces

To correctly represent the effect of a structure on the optical behavior, and especially to understand the interaction effects between multiple structures, the system is described using a coherent wave-optical model. To this end, the Rigorous Coupled Wave Analysis (RCWA) method in the “reticolo” implementation as provided by Lalanne, Hugonin et al. [15–17] was used. For all simulated systems, reflectance as well as transmittance towards the bottom cell are calculated. For selected system configurations, additionally the spatially resolved absorptance was calculated. This allows for the quantification of parasitic absorption as well as for visualization of the absorption process. Due to the complexity of the presented systems, the near field calculation was not computationally feasible for the entire parameter variation ranges. As parasitic absorption did not change significantly for any of the evaluated systems, this does not impact the conclusions drawn from this optimization. A careful convergence analysis for all simulation parameters was performed, as described in section S2 in the Supplement 1. As parasitic absorption did not change significantly for any of the evaluated systems, this does not impact the conclusions drawn from this optimization.

2.3 Model for the silicon bottom cell

The target of the presented simulation study is to compare different structure configurations at the interfaces of the top solar cell under ideal conditions. To exclude the influence of a specific bottom solar cell concept, the bottom cell is represented by a crystalline silicon absorber located directly underneath the top cell. Lambertian scattering at the rear side was assumed, and absorptance and reflectance at the rear side were calculated according to Green [18], with an absorber thickness of 250 µm. For the rear surface reflection in the model, a best fit to a reflectance measurement of a silicon wafer with silver back side metallization and random pyramid texture on both interfaces was conducted. A comparison of the model described in this and the previous section with literature (both experimental and simulation) is given in section S5 of the Supplement 1. The target of the presented simulation study is to compare different structure configurations at the interfaces of the top solar cell under ideal conditions. To exclude the influence of a specific bottom solar cell concept, the bottom cell is represented by a crystalline silicon absorber located directly underneath the top cell. Lambertian scattering at the rear side was assumed, and absorptance and reflectance at the rear side were calculated according to Green [18], with an absorber thickness of 250 µm. For the rear surface reflection in the model, a best fit to a reflectance measurement of a silicon wafer with silver back side metallization and random pyramid texture on both interfaces was conducted. A comparison of the model described in this and the previous section with literature (both experimental and simulation) is given in section S5 of the Supplement 1.

2.4 Random pyramid model

The random pyramid configuration is described using the ray tracing software Raytrace3D [20], which was previously used to successfully describe experimental reflection data of structured silicon in combination with a thin film system in [23]. In analogy to [21] our simulation model consists of a single pyramid and boundary conditions which include a random jump in beam propagation. All thin film layers forming the top solar cell are included by setting a transfer matrix formalism as surface on top of the silicon pyramid.

2.5 Calculation of the photo current

To compare the performance of the different configurations, the generated photo current was calculated. For this, absorptance and reflectance were calculated as described above in the wavelength range between 300 nm and 1200 nm in steps of 15 nm. This result was interpolated to the wavelength resolution of the AM1.5 g spectrum given by [24]. The current was then calculated as the integral of the product of absorptance A and the photon flux given by the spectral data ${\mathrm{\Phi }_{\textrm{AM}1.5\textrm{g}}}$:

The integral has been approximated using the trapezoid rule. As the quantification of parasitic absorption in the top solar cell was only possible for a limited amount of system configurations, for parameter variations the generated photo current in the entire top solar cell was calculated approximately from the reflectance and transmittance data. This simplification does not affect the position of the extrema within the parameter variations. To distinguish the different calculation the approximated current is labelled as top solar cell current, while the more precise values are referred to as current absorbed in the perovskite. To distinguish between the actual reflectance and absorptance values and the values weighted by the AM1.5 g spectrum as described in this section, the latter are referred to as reflected and absorbed equivalent current throughout this work.

3. Results

3.1 Effect of a structure at the air-perovskite interface

First, the configuration depicted in Fig. 1 is considered: The perovskite top solar cell is deposited on top of a planar silicon interface, with the top of the perovskite absorber being structured. All layers above follow the structure of the absorber. For the evaluation of the basic effects of structure height and period, a square unit cell with sinusoidal height variation in x- as well as y-direction was chosen. The geometry is described in more detail in Fig. 3 and section S1 of the Supplement 1.

Both parameters were varied, and the resulting equivalent currents are depicted in Fig. 2(a) for the reflectance of the tandem stack and Fig. 2(b) for absorptance in the top cell. A structure height of 0 nm corresponds to the reference. Three effects can be observed for the configuration: First, independent of structure period, a higher structure height results in lower reflectance and higher absorptance. Second, there is an effect of the period on the magnitude of this effect and therefore an ideal period for both minimum reflectance and maximum top cell absorptance. Last, the extrema for reflected and absorbed equivalent current do not coincide.

 

Fig. 2. Equivalent current reflected (a) and absorbed in the top cell (b) for different combinations of structure height and period. The structure is located at the air-perovskite interface as in Fig. 1 (b) and made up of a quadratic unit cell with a sinusoidal height profile as illustrated in Fig. 3(a)-(b). For this and the following parameter variations, a point density of at least 100 nm in both directions was used, with a higher density of 10 nm around the maxima.

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Overall, this means that depending on the chosen parameters, a structure at this interface can reduce both reflection and thermalization losses. The latter is achieved by an increase in the total share of photons absorbed in the perovskite top cell (for constant reflectance and constant parasitic absorption, this is directly proportional to the equivalent current absorbed in the top cell). The parameter ranges across which the observed effects reach their maximum are rather broad, which makes an experimental realization more feasible.

3.2 Influence of structure geometry

In the previous subsection, the effect of period and structure height was shown for one structure geometry. In the following, this consideration is expanded to three different periodic geometries, which are depicted in Fig. 3. All these geometries share a square unit cell (crossed grating configuration). Different cross section profiles, namely sinusoidal, parabolic and a pyramidal are approximated by quadratic slices as displayed in the top view in Fig. 3(a)-(d). The structures are employed at the top interface and the reflective behavior is compared. As monotonically decreasing reflectance towards higher structure heights was observed before, it was considered sufficient to conduct this comparison at two constant aspect ratios (ratio structure height/period) of 1.0 and 0.5. The resulting reflectance values are depicted in Fig. 3(e). In general, a similar dependence of the reflected equivalent current on period and height is observed for all three configurations. For the aspect ratio of 1.0, all structures perform comparably. At the lower aspect ratio of 0.5, the sinusoidal structure slightly outperforms both other configurations by around 0.3 mA/cm². While this is an indication that for some structures, the antireflective effect might be more stable at parameters away from the optimum, overall different unit cell geometries enable similarly good antireflective properties. Note that only a selection of continuous geometries has been evaluated here. For all further calculations, the sinusoidal concept is chosen because of the expected higher stability under parameter variation. The calculation presented in this and the previous section suggests an optimal period of 650 nm for a sinusoidal structure with aspect ratio 1.0.

 

Fig. 3. a-d: The different unit cell geometries considered within this work. (a) depicts a cut in the x-y-plane, where all geometries share a square layout. (b-d) show cuts along the x-z-plane, with the different geometries: sinusoidal (b), quadratic (c) and pyramidal (d). Supplement 1 section S1 gives a mathematical description of the geometries illustrated here. (e) shows the equivalent current reflected by a system with a structure at the air-perovskite interface at different structure periods. Results are shown for the three different unit cell geometries illustrated in b-d and for two different feature aspect ratios (feature size equal to period and equal to half the period).

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3.3 Effect of a structure at the perovskite-silicon interface

Next, the sinusoidal structure is inserted at the interface between perovskite and silicon (Fig. 1 (c)). The silicon absorber is structured, as are the contacting layers towards the perovskite absorber. The perovskite absorber then fills the structure and has a planar top interface. For this case, the reflectance for different structure heights and periods is depicted in Fig. 4. An optimum region can be identified around a structure height of 400 nm and a period range from 250 nm to above 600 nm. This region is very broad, and near-optimal results can already be reached with much smaller structure heights: At a period of 350 nm, the reflected equivalent current is increased by less than 0.2 mA/cm² by decreasing the structure height to 150 nm. Overall, a reduced reflected equivalent current of around 1.0 mA/cm² can be achieved with a structure at this position, while 3.1 mA/cm² were reflected in the flat configuration.

 

Fig. 4. Equivalent current reflected by a system with a sinusoidal structure with varied height and period at the perovskite-silicon interface.

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3.4 Same geometry at both interfaces

The improvement observed for a nanostructure at the lower interface in the previous section is smaller than the one achieved with the structure at the upper interface. It should not be neglected, however: This gain is achieved by reducing reflection at a different interface, so this approach can be combined with the previous structure to achieve an even larger improvement. This is done in the following by combining two structures of similar geometry at both interfaces, as illustrated in Fig. 1(d). The silicon substrate is structured, and all top cell layers follow that structure. The results of the parameter variation for both structure height and period are depicted in Fig. 5(a). For the reflective properties, a similar pattern as before is observed: Higher features lead to lower reflectance, with a slight dependence on feature period for this effect. For structure heights above around 350 nm, no relevant improvements are observed. For the absorbed equivalent current in the top solar cell, a different result compared to the concepts above is achieved: In both structure height and period, there is an optimum resulting in minimal thermalization losses at 350 nm.

 

Fig. 5. Equivalent current reflected (a) and absorbed in the top cell (b) for the configuration with the same structure at both interfaces (compare Fig. 1 (d)). A sinusoidal structure is employed and height and period are varied.

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For this optimum, the distribution of the current between the two sub-cells and useful/parasitic absorptance has been calculated wavelength-resolved and is depicted in Fig. 6. Under ideal conditions, almost no reflectance occurs except for the region close to the silicon bandgap. Up until 1000 nm, a total of just 0.02 mA/cm² is reflected. This explains the saturation of the reflectance plot in Fig. 5(a) towards higher structure heights.

 

Fig. 6. Wavelength-resolved reflectance and absorptance for the optimized structure extracted from Fig. 5 for the case of the same structure at both interfaces (compare Fig. 1(d)). As a comparison, the top and bottom cell absorption (including parasitic) for a conformally coated random pyramid structure with ideal, infinitely large pyramids as calculated by ray tracing are displayed. The equivalent currents are given in brackets and are calculated according to section 2.5.

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The configuration discussed in this subsection bears close resemblance to conformally coated random pyramids as presented in the work of [10]. The major difference is the feature scale: While so far, nanostructures have been investigated and their wave-optical effects had to be accounted for, random pyramids with a typical feature size of several micrometers can be well described by ray optics. To explore the effect of this scale, random pyramids with the same perovskite top solar cell were modeled using ray tracing. For the pyramids, an ideal facet angle of 54.7 degrees [25] was assumed. For the absorber thickness, again a constant perovskite absorber volume relative to the planar stack was assumed, resulting in a thickness of 277 nm normal to the pyramid surface. Regarding reflectance, both nanostructure and random pyramids perform equally well. It is reduced with a similar quality for both the random pyramid configuration as when using the nanostructure. Absorptance close to the perovskite bandgap is significantly lower in the random pyramid case. Ultimately, this results in larger thermalization losses because of two aspects: (i) more high-energy photons are transmitted to the bottom cell and absorbed at the low bandgap and (ii) to achieve current matching a lower perovskite bandgap has to be chosen, leading to more thermalization losses in the top cell. To give an estimation of the possible thermalization loss reduction, absorptance in the top cell close to the bandgap at 700 nm is compared in the following. A complete thermalization loss analysis including bandgap adjustment is given at the end of the next section for selected systems. For the random pyramid case, absorption in perovskite at 700 nm is reduced to 65%, relative to 85% for the optimized nanostructure and 82% for the flat reference case (not included in the plot). This means that to achieve a similar reduction in thermalization losses, a bigger perovskite material volume has to be employed when following the random pyramid architecture. This has been quantified in section S6 in the Supplement 1: a similar reduction is achieved with a perovskite thickness of 665 nm normal to the surface, which is equal to the material volume of a planar layer of 1150 nm thickness. For a planar cell design, only an absorber thickness of 700 nm would be necessary, but with the drawback of increased reflection relative to the two structured approaches.

3.5 Wavelength-resolved absorptance and reflectance and structure combination optimization

Fig. 7 shows the wavelength-resolved absorptance and reflectance of the structure configurations discussed so far. For each configuration, parameter combinations have been chosen from the region of minimal reflection in the parameter variations discussed above. As there are no clear minima, but larger regions with a change in reflected equivalent current smaller than 0.5 mA/cm², maximum equivalent current absorbed in the top cell was taken as a secondary selection criterium. The champion parameters chosen are listed in Table 1.

 

Fig. 7. Reflectance (top) and absorptance in perovskite and silicon (bottom) for the different configurations illustrated in Fig. 1. For each combination, optimized parameters have been used, which are listed in Table 1.

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Table 1. Equivalent currents calculated for the different configurations shown in Fig. 7.

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For the flat reference configuration depicted in grey, the reflection losses are mainly originated in the wavelength regions 300 nm – 400 nm and 800 nm – 1200 nm. A significant amount of thermalization loss is present as well, as close to the bandgap at 700 nm, just 79% of the total absorbed equivalent current is absorbed in the perovskite.

Introducing a structure at the interface between air and perovskite firstly reduces the reflectance over the entire wavelength range by 1.8 mA/cm² relative to the 3.1 mA/cm² observed for the flat configuration. Only the non-absorbed light in the bottom cell between 1000 nm and 1200 nm still contributes a significant amount of reflection, making up 1.1 mA/cm² of the remaining 1.3 mA/cm². Secondly, thermalization losses are significantly reduced by improved absorptance in the perovskite close to the bandgap (+10% at 700 nm).

The structure at the interface between perovskite and silicon also induces a slight improvement in the absorptance close to the bandgap (+4% at 700 nm), but mainly reduces reflectance by a total of 1.2 mA/cm². The reduction is solely achieved at the lower interface. This is strongly indicated by the perfect agreement with the reflective properties of the planar stack at small wavelengths, where the lower interface only has marginal interaction with the light due to the strong perovskite absorption.

The result for geometrically identical structures at both interfaces is also aligned with this hypothesis, as here, the reflectance is reduced to 0.02 mA/cm² across the wavelength range until around 1000 nm. Compared to the air-perovskite configuration, the second structure eliminated the remaining reflectance below 1000 nm. The same comparison also shows that this structure combination only yields small improvements regarding thermalization losses, with an improvement of perovskite absorption of just 3% at 700 nm. This is less than what is achieved with just a structure at the air-perovskite interface. This indicates that with regards to thermalization losses, the structure at the lower interface can impede the positive effect of the one at the upper interface.

With the aim of reducing this negative effect of the structure at the perovskite-silicon interface, but still retain the reduction in reflectance achieved by it, a fourth concept is evaluated: At the air-perovskite interface the same structure as in the air-perovskite configuration is employed, while at the bottom interface a much smaller structure height is employed. As visible in Fig. 4, with the right choice of period at 325 nm, this should still lead to almost optimal antireflective properties at this interface. The results in Fig. 7 and Table 1 show that with this configuration, a similar reduction in reflectance as in the previous case is achieved, with a total reduction of 2.0 mA/cm² in reflection losses across the entire wavelength range. At the same time, this configuration provides the largest reduction in thermalization losses, increasing the absorption in the perovskite at 700 nm by 12%.

To set the overall current gain and reduction in thermalization losses in relation to a possible efficiency gain, the flat system configuration was recalculated with the bandgap adjusted to obtain current matching. For this, the refractive index dataset closest to the bandgap desired for this application was chosen from Manzoor et al. [22] and shifted linearly along the energy axis by a further 10 meV, until current matching was obtained at a bandgap of 1.68 eV (calculated according to section S3 in the Supplement 1) at a matched current of 20.7 mA/cm². Assuming a fill factor of 80%, bottom cell V_OC of 700 mV and top cell V_OC of 980 mV for the 1.72 eV and 940 mV for the 1.68 eV bandgap absorber respectively [19], the optical improvements result in an efficiency improvement of 1.3%_abs or 4.9%_rel.To set the overall current gain and reduction in thermalization losses in relation to a possible efficiency gain, the flat system configuration was recalculated with the bandgap adjusted to obtain current matching. For this, the refractive index dataset closest to the bandgap desired for this application was chosen from Manzoor et al. [22] and shifted linearly along the energy axis by a further 10 meV, until current matching was obtained at a bandgap of 1.68 eV (calculated according to section S3 in the Supplement 1) at a matched current of 20.7 mA/cm². Assuming a fill factor of 80%, bottom cell V_OC of 700 mV and top cell V_OC of 980 mV for the 1.72 eV and 940 mV for the 1.68 eV bandgap absorber respectively [19], the optical improvements together with the bandgap adaption result in an efficiency improvement of 1.3%_abs or 4.9%_rel.

3.6 Spatial distribution of absorbed intensity

To gain a deeper understanding of the way the structures discussed above impact the absorption in perovskite close to the bandgap, Fig. 8 displays the spatial distribution of the absorption intensity for the different optimized configurations in the wavelength range 500 nm – 700 nm.

 

Fig. 8. Spatially resolved absorbed intensity in the bandgap region 500 nm – 700 nm for a cut along the path depicted in subfigure (a), for the different configurations: (b) Flat system, (c) structure at lower interface, (d) structure at top interface, (e) same structure at both interfaces, (f) optimized structures at each interface. The interfaces between each layer is depicted by a white line, silicon and air half space are depicted in grey.

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Throughout the figure, the absorbed intensity is depicted along a cut of the structure as depicted in subfigure (a). Subfigure (b) shows the absorption behavior of the flat system configuration, with no variation of the absorptance intensity in the x-y-plane and a decay of the same in the negative z-direction. The structure concept with the least impact on the absorption behavior is the structuring of the lower interface depicted in subfigure (c). The equivalent current analysis in Table 1 already showed that this concept mainly improves antireflective behavior but does not significantly impact the balance of the sub-cell currents. This is confirmed by the absorption analysis conducted here, with the spatial distribution of light absorptance in perovskite essentially like the one in the flat configuration. Opposite to this, placing the structure at the top interface in subfigure (d) massively changes the absorption distribution. Within the peaks of the structure, absorption maxima are generated, while underneath the valleys, almost no absorption is visible. As shown in the previous section, this leads to improved absorption in the perovskite close to its bandgap, ultimately reducing thermalization losses. Correlating this increase in current with the absorption maxima within the structure peaks also allows for an explanation of the reduced absorptance increase in the case depicted in subfigure (e): Here, peaks of the bottom structure reach into the areas of high intensity/generation, as they are of the same height as the top texture. So, while using a higher top structure enhances the overall generation, the similarly higher bottom structure reaches further into the maxima, hindering further absorptance increase in the perovskite for higher structures. Therefore, the system performing best regarding both reflectance and thermalization is found at a lower structure height. Figure S4 in the Supplement 1 shows the same configuration with the feature height and period of 650 nm, where the negative impact of the higher features at the perovskite-silicon Interface becomes visible. This problem is solved with the usage of two structures of different height, of which the absorption characteristics are depicted in subfigure (f). Here, the higher top structure creates the absorptance maximum already observed in (d), while the lower bottom texture does not protrude into it. Nevertheless, Table 1 shows that this lower bottom texture is still able to effectively reduce reflectance at that interface.

While the combination of two structures of different height and width promises the best optical performance according to the simulations conducted in this work, it can also be expected to pose the greatest challenges for experimental realization. Nevertheless, there is a reasonable pathway for the fabrication of such a solar cell: The inclusion of a nanostructure at the interface between perovskite and silicon, small enough to result in a planarized perovskite layer, is currently included in most highest-efficiency perovskite-silicon solar cells [5,13]. Direct nanopatterning of perovskite layers has also been demonstrated by multiple groups [14,26,27]. As the resulting structures do not interact geometrically and they are introduced at different points in the solar cell processing chain, it should be possible to integrate them alongside each other and create any desired structure combination.

3.7 Interaction with module cover glass

While currently a lot of effort is put into achieving highest cell efficiencies for perovskite-silicon tandem solar cell, ultimately high performance must be achieved with a perovskite-silicon solar module. Therefore, in the following the transferability of the results to the module application is discussed. For this, in the configurations from Fig. 1 (a), (d) and (e) with the parameters from Table 1, we replace the incident medium air (n = 1) by module glass/EVA laminate (as approximation set to n = 1.5) and remove the MgF₂ ARC layer which is only efficient against air. We omit the interface air-module glass to not favor any concept by a specific surface structure or antireflective coating at that interface. In the resulting absorption and reflection data visible in Fig. 9, a beneficial effect similar to the one observed in the previous sections is visible. To quantify this, the equivalent current reflected (excluding the air-module glass interface) is reduced from 3.4 mA/cm² for the flat case to 1.1 mA/cm² for the case of different optimized geometries and 1.0 mA/cm² for the same geometry at both interfaces. Note that all structure parameters and layer thicknesses are optimized for the application against air. Therefore, an even bigger benefit in the module application might be achieved through optimization, which is out of the scope of this paper.

 

Fig. 9. (a) Reflectance and Absorptance of the configurations from Fig. 1 (a), (d) and (e) with the parameters from Table 1 with the incident medium set to module glass/EVA and the MgF₂ layer removed. (b) Illustration of the planar stack.

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4. Conclusion

Within this work, the beneficial effect of nanostructured interfaces in perovskite-silicon tandem solar cells has been shown. It has been shown that these can lead to a reduction in reflection as well as thermalization losses.

Regarding reflectance, structures at both the air-perovskite as well as the perovskite-silicon interface have been shown to be beneficial across a large variation of structure parameters. The largest gain has been observed with structures at both interfaces with a reduction in reflectance of 2.1 mA/cm² relative to the planar reference stack, which exhibits a total reflection of 3.1 mA/cm².

To reduce thermalization losses, a structure at the air-perovskite interface has been shown to be most beneficial, which can be complemented by a flatter structure at the second interface. Here, the best result yielded an increase of 12% absorption in perovskite close to the bandgap at 700 nm. The reflectance reduction and the increase in bandgap enabled by the reduction in thermalization losses has been calculated to be equivalent to an efficiency gain of 4.9%_rel.

In comparison to perovskite-silicon tandem solar cells deposited on random pyramids, the nanostructured concepts evaluated in this work exhibit a similar reduction in reflectance while leading to lower thermalization losses. Therefore, they are a promising alternative tandem solar cell design concept and should be evaluated experimentally.