1. Introduction

Organo-metallic halide perovskite materials have made extremely rapid advances in solar cell applications; progressing from 3.5% efficiency in 2009 [1] to 22.1% in 2016 [2]. This rapid improvement can be attributed to their ability to be solution-processed at low temperatures, high material quality, and simple architectures that allow for flexibility with regards to the design of perovskite solar cells in both stand-alone and tandem applications. Development of tandem perovskite solar cells has progressed steadily for both monolithic 2-terminal [3–6] and stacked 4-terminal [6–8] architectures, with the most promising results occurring when utilized with a silicon bottom cell due to its low bandgap (1.1 eV) allowing for higher potential current. The stacked 4-terminal tandem structure allows for the independent development of the perovskite and silicon cells, as the cells are only optically coupled, not electronically. This is not the case for monolithic 2-terminal tandem cells, where the perovskite and silicon sub-cells are optically and electronically coupled. As a result, the performance of each cell depends on the parameters of the other since the current is limited by the lower of the individual cell currents.

To achieve high performance, careful material selection for the charge transport and extraction layers is crucial as the materials must be both transparent and conductive to allow sufficient light harvesting and charge carrier extraction for both cells in the tandem. However, there are no clear criteria in the literature for selecting these materials in terms of tandem performance. The current high performance perovskite/silicon 2-terminal tandem cells by Albrecht [3], Werner [4] and Mailoa [5] utilize different material combinations. It is therefore necessary to identify which layers have the greatest impact on tandem cell performance and which of the available material options offer the best properties. One method to do this is to utilize optical modelling to analyze how different material choices affect the light distribution within the cell. Optical modelling of perovskite/silicon 2-terminal tandems already exists in the literature [9, 10], where a strong focus has been placed on optimizing the thickness of the layers in a cell to maximize efficiency. Although this approach is effective for reducing the parasitic absorption, it has only limited value for reducing reflection loss, much of which originates from internal interfaces within the cell. Tackling this internal reflection (as opposed to front-surface reflection that can be reduced with standard anti-reflection coatings) requires a study to identify, first the key layers involved, and then the ideal properties for these layers to guide material selection for future high efficiency tandem cell development.

In this paper, we analyze monolithic 2-terminal perovskite/silicon solar cells from an optical perspective by developing a complete optical model for a monolithic stack; systematically identifying sources of optical loss, and proposing solutions to minimize these losses. We use EMUstack [11], an open-source, simulation package that utilizes a combination of scattering matrices and Finite Element Method (FEM) to calculate light propagation through multilayered structures. From the simulation results, we establish material selection criteria to minimize optical losses within the tandem cell. We reinforce the importance of high transparency front-side conductive oxide and charge transport materials to minimize parasitic absorption in the cell, taking note of the material and processing constraints imposed by the monolithic fabrication process. We then demonstrate the importance of considering reflection at the charge transport interfaces, and show that current gains of up to 3 mA/cm² are possible through appropriate selection of currently available charge transport materials. Finally, we quantify the potential performance gains of light trapping in the silicon cell using a modified Lambertian scattering model. We show that the combination of effective light trapping in the silicon, and optimization of the perovskite active layer thickness can achieve a matched current of more than 20 mA/cm², without the need for front-side texturing [12]. In doing so, we also find that the refractive index of the tunnel junction has a surprisingly low impact on the performance of the tandem cell. By systematically improving the performance of the tandem cell through the aforementioned steps, we identify the ideal optical properties for the charge transport and transparent conductive oxide layers, providing guidance in material selection for the development of monolithic tandem cells of >30% efficiency.

2. Tandem cell architecture

Monolithic perovskite/silicon tandem solar cells are fabricated by depositing the perovskite cell, layer by layer, onto the silicon cell. This allows for two possible perovskite cell orientations depending on the whether the Electron Transport Layer (ETL) or Hole Transport Layer (HTL) is deposited first. Schematics of these two tandem orientations are shown in Fig. 1, where for the p-i-n orientation (from top to bottom) the ETL is deposited first; while for the n-i-p orientation the HTL is deposited first. It is important to note that from an optical standpoint, there is no distinction between the two perovskite cell orientations outside of the varying optical properties of different materials.

 

Fig. 1 Schematics of device structures for simulation in the (a) p-i-n or (b) n-i-p orientations. The arrows indicate the direction of incident light. TCO: Transparent Conductive Oxide. ETL: Electron Transport Layer. HTL: Hole Transport Layer.

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Naturally, the doping of the bulk crystalline silicon (c-Si) must be chosen such that charge carriers generated within the c-Si bulk can be extracted to the contacts. As doping c-Si has a relatively small effect on its optical properties (compared to variations in the perovskite cell), we treat the c-Si optical properties as intrinsic [13] regardless of the cell orientation. The perovskite material that was chosen was Methylammonium Lead Iodide (MAPbI₃), which has a band gap of 1.55 eV. According to theoretical studies [14], the ideal top cell bandgap for a 2-terminal tandem with a silicon bottom cell (1.1 eV) is 1.7 eV. However, the non-unity absorption of the top cell allows for flexibility in tuning the MAPbI₃ layer thickness to achieve optimal current-matching despite the bandgap being lower than the theoretical optimum.

Two types of silicon cells have been used in the highest performing 2-terminal perovskite/silicon tandem cells in the literature: a heterojunction with intrinsic thin layer (HIT) silicon cell as used by Werner [4] and Albrecht [3], or a homojunction silicon cell as used by Mailoa [5]. As the doped silicon transport layers in these cells have a relatively small optical impact, and to avoid unnecessary complexity, we ignore these layers and treat the silicon cell as a c-Si bulk layer with a silver back contact. The type of silicon cell does, however, influence the choice of tunnel junction due to temperature constraints of the doped silicon transport layers. A TCO (typically Indium Zinc Oxide, IZO or Indium Tin Oxide, ITO) tunnel junction is compatible with both types of silicon cell, whereas a polycrystalline silicon (poly-Si) tunnel junction [5] is only compatible with the homojunction silicon cell as the annealing step for the poly-Si layer would compromise the electronic properties of the amorphous silicon layers in the HIT cell. The poly-Si tunnel junction has lower parasitic absorption and is a better index match to c-Si than a TCO tunnel junction, which allows for more light to enter the c-Si layer. Despite these apparent advantages, we demonstrate here that poly-Si only provides a minor optical benefit over ITO as a tunnel junction material, even in the presence of light trapping within the c-Si sub-cell. As the ITO tunnel junction is compatible with both types of silicon cell and is only slightly worse than the poly-Si tunnel junction, we focus our numerical studies on a perovskite/silicon tandem with an annealed ITO tunnel junction (optical data for annealed ITO from Duong et al. [7]) to retain generality of the silicon cell. We note, however, that the fabrication of the perovskite cell may impose additional requirements that render the use of the HIT cell structure impractical. For example, the use of a TiO₂ ETL typically requires a high temperature (~500°C) anneal.

To understand how different charge transport materials affect the optical properties of a perovskite/silicon 2-terminal tandem solar cell, we first analyze a p-i-n orientation cell with layer composition MgF₂/ unannealed IZO/ spiro-OMeTAD/ MAPbI₃/ compact (cp)-TiO₂/ annealed ITO/ c-Si/ Ag with layer thicknesses 105 nm/ 44 nm/ 160 nm/ 300 nm/ 44 nm/ 30 nm/ 200 μm/ 200 nm, respectively. These materials are represented in our numerical model with dispersive experimental data from the literature [7, 9, 13, 15–18], where this data has previously provided good agreement with experimental cell measurements [7]. This preliminary perovskite/silicon tandem structure was derived from our previous work on semi-transparent mesoporous perovskite solar cells for use in a 4-terminal tandem with silicon [7]. A buffer layer (such as MoO_x or WO_x) between the TCO and the HTL (as used in several works to protect the HTL from sputtering damage [3, 4, 6, 7]) was not included in the model as we consider this to be an artefact of the current cell fabrication process rather than an integral layer of the cell. The impact of this omission is small: including a 10 nm MoO_x buffer layer (using optical data from [7]) in the model increases total parasitic absorption by ~0.4 mA/cm², which reduces the overall tandem cell current by <0.2 mA/cm². Therefore, our overall results are not significantly affected by the absence of a MoO_x buffer layer.Alternative buffer layers such as CO₂ plasma treated WO_x promise even lower parasitic losses [19] and future material combinations may render such a buffer layer obsolete.

One key distinction between the 2-terminal cell studied in this paper and our previous work on semi-transparent perovskite cells for mechanically-stacked 4-terminal tandems arises from the layer deposition sequence imposed by the monolithic 2-terminal cell structure.

In a monolithic tandem, the front-side TCO is one of the final layers to be deposited, and as a result it cannot be annealed due to the low thermal budget of the underlying perovskite cell layers. As this layer is at the front of the cell, it must have high transparency across the entire visible-to-near infrared solar spectrum. In contrast, for a 4-terminal cell, the final TCO layer is typically on the rear of the perovskite cell, and thus only requires high transparency for the near infrared wavelengths. This additional constraint for monolithic tandems necessitates a less absorptive TCO than the unannealed ITO used in our previous 4-terminal cell work [7]. An alternative to ITO that has been demonstrated in perovskite-silicon tandems is IZO [18]. Unannealed IZO has been shown to be less absorptive and more conductive than unannealed ITO [7, 18], allowing further thickness reduction provided film uniformity and quality can be maintained. The IZO thickness was set to 44 nm as it has been shown that through the application of an appropriately-optimized metal grid, the TCO layer thickness can be reduced while maintaining an adequately low sheet resistance [20]. Selecting IZO rather than ITO as the front-side TCO in the 2-terminal cell structure of Fig. 1(a), we calculate a total parasitic absorption in the TCO of only 0.45 mA/cm², compared with 2.65 mA/cm² if unannealed ITO was used. This significant reduction in loss highlights the importance of low absorption front-side TCOs in achieving high performance 2-terminal tandem cells.

For our starting cell geometry, we also optimized the thickness of the spiro-OMeTAD HTL within an experimentally-constrained range of 100-200 nm, finding a maximum perovskite current of 17.52 mA/cm² with a thickness of 160 nm. The lower bound of this range is constrained by pinhole formation in spiro-OMeTAD, which based on our experimental experience occurs for film thicknesses below 100 nm. It is worth noting that the spiro-OMeTAD layer thickness strongly influences thin film resonance effects in the cell, and consequently a thinner layer is not necessarily beneficial. This is in contrast to the 4-terminal configuration studied previously [7], where the HTL was at the rear of the perovskite cell, and optical losses were minimized by using the thinnest pinhole-free spiro-OMeTAD layer. The influence of spiro-OMeTAD thickness on tandem cell performance has been previously studied and discussed in the literature [4, 5, 21, 22].

The optical loss breakdown of this initial perovskite/silicon 2-terminal tandem structure is shown in Fig. 2, where the absorptance spectra for each layer and total reflectance spectra of the cell are shown along with the equivalent current loss (parasitic or reflective mechanisms) or generation (perovskite or c-Si absorption) for each layer as calculated by integrating the photon flux of the AM1.5G solar spectrum between wavelengths 350 and 1200 nm.

 

Fig. 2 Light distribution for the initial p-i-n perovskite/silicon tandem structure shown in Fig. 1(a), with layer thicknesses as specified in the text.

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From Fig. 2, the largest optical losses are parasitic absorption in the spiro-OMeTAD layer and reflection loss out the top of the cell with current losses of 1.85 mA/cm² and 10.12 mA/cm², respectively. As the parasitic absorption within the spiro-OMeTAD layer cannot be further reduced by decreasing the thickness, it is imperative that it is replaced by an alternative material with lower absorption, which we later demonstrate also has a large impact on the reflection loss of the cell. These results show that parasitic absorption can be substantially reduced by selecting materials with low absorptivity for all layers that do not contribute to the current, while thickness optimization only results in small reductions in parasitic loss. This is especially important for layers on top of the perovskite, namely the TCO and top charge transport layer.

The 2-terminal cell modelled in Fig. 2 has a silicon-limited current of 15.51 mA/cm², where the low performance of the silicon cell can be attributed to reflection from the multiple material interfaces above the Si layer. Therefore, the reflection at these interfaces must be reduced to improve the silicon cell current. To this end, we next explore the impact of charge transport refractive indices on the reflection loss.

3. Minimizing reflection loss via appropriate charge transport material selection

The largest light loss mechanism for the 2-terminal perovskite/silicon tandem cell is reflection. The majority of this reflection can be attributed to two sets of interfaces, the IZO/spiro-OMeTAD/Perovskite layer interfaces and the cp-TiO₂/ITO/c-Si layer interfaces, as the large index contrasts between the adjacent layers results in high Fresnel reflection. The refractive indices of these interface sets at λ = 1000 nm are 1.9/1.6/2.3 and 2.4/1.8/3.5 for the layers adjacent to the HTL and ITO tunnel junction, respectively. As the HTL is on the front-side of the perovskite cell, reflection from the HTL interfaces affects the entire light spectrum, and thus impacts both the perovskite and silicon cell currents. Hence, reducing the index contrast at this interface set is the most critical.

3.1 Minimizing reflection at the HTL interfaces

To investigate this interface set, we replace the spiro-OMeTAD layer in the numerical model with an idealized (lossless) HTL of constant refractive index, and calculate the integrated reflection loss as a function of the refractive index and thickness of this layer. The dependence of reflection loss on the refractive index and thickness of the HTL is shown in Fig. 3(a), where the refractive index of the MAPbI₃ layer and commonly chosen hole transport materials at λ = 1000 nm are represented by a dashed line and arrows, respectively. The organic HTLs, spiro-OMeTAD and PEDOT:PSS, result in high reflection loss at most thicknesses; demonstrating the importance of developing high index transport materials. The yellow stars, labelled A and B, denote locations of highly non-ideal (n = 1.5) and ideal (n = 2.8) refractive indices for 120 nm layer thickness. The reflectance spectra and integrated reflection loss at these points are shown in Fig. 3(b), with the higher index (B) demonstrating broadband reflectance suppression crucial for high performance.

 

Fig. 3 (a) Reflection loss of the monolithic tandem cell as a function of HTL refractive index and thickness. Refractive indices of MAPbI₃ and common hole transport materials at λ = 1000 nm are denoted by a dashed line and arrows, respectively. The white line depicts analytic thin film interference minima for the half-wavelength optical path length denoted by inset schematics. The yellow stars, A and B, denote the locations of the reflectance spectra shown in (b). (b) Reflectance spectra of highly non-ideal (A) and ideal (B) refractive indices and their corresponding reflective loss.

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From Fig. 3(a), the dependence of reflection on the thickness and refractive index of the HTL can be separated into two distinct regions. In region I (n < n_MAPbI3), low reflection only occurs for very low HTL thicknesses of a few tens of nanometres, while in region II (n > n_MAPbI3), reflection is strongly dependent on HTL thickness. The features in these regions can be described by thin film interference of the HTL layer with other layers in the perovskite top cell. According to the theory of thin film interference, reflection from a film is minimized when destructive and constructive interference occurs simultaneously for reflected and transmitted light, respectively. The interference conditions depend on the optical path length (OPL) between two highly reflective interfaces with minimum reflection occurring at optical path lengths of either half or quarter of the suppressed wavelength. In multilayered structures, interference effects are usually dominated by the interfaces with the most Fresnel reflection, while intermediate interfaces with lower reflection can often be ignored. In Fig. 3(a), the low reflectance conditions in both regions I and II can be explained by a half-wavelength OPL, as either a High-Low-High (region I) or a Low-High-Low (region II) refractive index sequence is formed by the layers adjacent to the highly reflective interfaces. In both cases, minimum reflection occurs when:

In region I, the relevant OPL corresponds to the HTL since the most reflective interfaces are the IZO/HTL and HTL/MAPbI₃ interfaces. In this region, only the zeroth order (m = 0) minimum is visible as the spacing of the reflection fringes is larger than 250 nm due to the low refractive index of the HTL. For HTL thicknesses greater than about 30 nm, the reflection decreases with increasing HTL refractive index as a result of improved index matching between the HTL and MAPbI₃ layers which increases absorption in the perovskite and silicon. When these refractive indices match, the relevant OPL increases to include the MAPbI₃ and cp-TiO₂ layers as the HTL/MAPbI₃ interface is no longer highly reflective.

In region II, the dominant reflective interfaces are IZO/HTL and cp-TiO₂/ITO, with the relevant OPL being comprised of the HTL, MAPbI₃ and cp-TiO₂ layers as shown in the region II schematic in Fig. 3(a). The higher refractive index of the HTL reduces the reflection fringe spacing, increasing the sensitivity to the HTL thickness. The reflection minimum at n = 2.8 for a 120 nm thick HTL coincides with the analytic expression:

One material that fits these constraints is NiO_x, which has a refractive index of ~2.3 and a typical deposition thickness of 80 nm [23]. The light loss breakdown for the tandem cell with an 80 nm NiO_x HTL is shown in Fig. 4, where a significant decrease in reflection loss from 10 to 7 mA/cm² and HTL absorption from 1.85 to 0.34 mA/cm² can be seen. These reductions in optical loss allow significantly more light to reach the perovskite and silicon layers, increasing the J_sc values for the perovskite and silicon cells to 19.5 and 18.2 mA/cm², respectively. It should be noted that the optical data used for NiO_x in the simulation is for unannealed NiO_x, deposited by atomic layer deposition (ALD) at a temperature of 300°C [24]. While this process is not compatible with direct deposition onto the perovskite layer, we assume that the optical properties are indicative of NiO_x deposited by lower temperature deposition methods, such as solution-processing or pulsed laser deposition (PLD) as listed in Table 1.

 

Fig. 4 Light distribution for the perovskite/silicon tandem with 80 nm NiOx HTL.

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Table 1. Electron (Blue) and Hole (Red) Transport Material properties

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3.2 Minimizing reflection at the ETL interfaces

After replacing the front-side HTL with NiO_x, we next investigate the dependence of reflection on the refractive index and thickness of an idealized ETL at the rear of the perovskite cell as shown in Fig. 5(a). Similar to Fig. 3(b), the reflectance spectra in Fig. 5(b) also demonstrate the importance of high index transport layers for low reflectance; however, the reflectance suppression is localized to long wavelengths due to the ETL being located behind the perovskite layer. The same thin film interference phenomena observed for the HTL in Fig. 3(a) can also be observed in Fig. 5(a), where two distinct regions can be identified. In region I, the high reflection interfaces are MAPbI₃/ETL and ITO/c-Si with an OPL comprised of the ETL and ITO layers; while in region II, the high reflection interfaces are IZO/NiO_x and ETL/ITO with an OPL comprised of the ETL, MAPbI₃ and NiO_x layers. The inclusion of the ITO layer in the OPL for region I shifts the reflection features to thinner ETLs, introducing an additional low reflection feature compared to Fig. 3(a). Similar to the HTL case, the locations of the low reflection features in Fig. 5(a) for both regions I and II can be described by Eq. (1) for a suppressed wavelength of λ = 930 nm for integers m = 1 for region I and m = 2, 3 for region II. One caveat of this analytic expression is the assumption that the suppressed wavelength is constant in both regions, for all ETL thicknesses and refractive indices, resulting in the imperfect fit seen for the m = 2 curve in region II. This expression does however, give a strong indication of the locations of the primary features of the figure.

 

Fig. 5 (a) Reflection loss of the monolithic tandem cell as a function of ETL refractive index and thickness. Refractive index of MAPbI₃ and common electron transport materials at λ = 1000 nm are denoted by a dashed line and arrows, respectively. The white lines depict analytic thin film interference minima for half-wavelength optical path length denoted by inset schematics. The yellow stars, A (n = 1.5, thickness 120 nm) and B (n = 3.6, thickness 20 nm), denote the locations of the reflectance spectra shown in (b). (b) Reflectance spectra of highly non-ideal (A) and ideal (B) refractive indices and their corresponding reflective loss.

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The reflection minimum in Fig. 5(a) occurs at a refractive index of n = 3.6 for 20 nm thick ETL, however there is no ETL with such a high refractive index and minimal parasitic absorption. A more practical option is therefore to use an ETL with a refractive index near that of MAPbI₃, where the reflection loss is only slightly larger than at the minimum and is also less sensitive to the ETL thickness. An ETL with a refractive index similar to that of MAPbI₃ is cp-TiO₂; hence there is no benefit to changing our ETL for this cell orientation.

From these results, it is clear that while the theoretical reflection minimum is not achievable with currently available charge transport materials, low reflection is possible by selecting materials with refractive indices close to that of MAPbI₃, with the added benefit that the thickness of the charge transport layers has little effect on the overall optics of the cell. By decoupling the optics from the charge transport layer thickness in this way, optimization of the electronic properties of these layers can proceed with fewer optical constraints. We also note that since both charge transport layers have been individually optimized in a manner that is independent of the perovskite layer orientation, these results apply to both perovskite top cell orientations with deposition constraints being the only limiting factor in material selection.

An alternative method to reduce internal reflection may be to use front-side texturing of the Si sub-cell [12], however our results show that low reflectance is possible for a planar structure without any such texturing. Not only does an all-planar front surface simplify the processing of the cell, but it is also more compatible with achieving the high-quality solution-processed films necessary for high-efficiency perovskite top cells.

Now that the reflection losses due to the HTL/ETL layer interfaces have been mitigated, the remaining dominant reflection peak in Fig. 4 occurs at wavelengths beyond 1000 nm. Reducing this reflection component requires light trapping within the silicon cell to increase the absorption of light with wavelengths close to the silicon bandgap.

4. Light trapping and optimized current-matching between top/bottom cells

To increase absorption within the silicon layer at long wavelengths, we next introduce light trapping into our optical model in the form of a rear Lambertian scattering surface, as illustrated in Fig. 6(a). A number of experimental textured surfaces can result in approximately Lambertian performance, such as random pyramids and random spherical caps, and can be formed by a number of processes [25]. In Fig. 6(a), the dielectric/metal back surface is assumed to scatter light (red arrows) in a Lambertian distribution, increasing the optical path length of the light within the silicon, as well as allowing for total internal reflection of scattered light from interfaces above the silicon layer. The methodology followed for the light trapping was inspired by Green [26], where angle-averaged propagation length, front-side reflection and back-side reflection for the c-Si layer are calculated numerically. Unlike many Lambertian scattering models, which typically consider the average front-side reflectance of the c-Si layer as a constant value, we take into account the internal reflectance from all layers above the silicon cell to calculate the average front-side reflectance (R_F) back into the c-Si layer. An in-depth description of how the Lambertian model was implemented in the model is contained in Appendix A.

 

Fig. 6 (a) Schematic of the multilayer tandem cell structure with a rear dielectric/metal Lambertian scattering surface, where the splayed red arrows represent Lambertian scattering. The internal front-side reflectance, R_F, denotes the average reflectance of scattered light back into c-Si layer, including light that enters and returns from the layers above. (b) Reflectance vs scattering angle θ at a wavelength of 1000 nm for light incident from the silicon cell onto the layers above for ITO (blue) and poly-Si (red) tunnel junctions. The vertical dashed lines denote the critical angles for the interfaces of c-Si/Air and c-Si/cp-TiO₂ at 16.5° and 42°, respectively. The inset shows the angle averaged front-side reflectance, R_F, vs wavelength for the tandem cell with ITO (blue) and poly-Si (red) tunnel junctions and an ideal Si/Air interface (magenta).

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The figure of merit for light trapping in the silicon cell is the angle-averaged front-side reflectance of light incident from the silicon assuming a Lambertian distribution,

Where R(λ,θ) is the reflectance of light back into the silicon cell from the layers above (colored arrows in Fig. 6(a)) at wavelength λ, and incident angle θ. Calculating R_F in this manner takes into account any parasitic absorption in the upper layers of the cell before the light returns to the c-Si layer, which could have a significant impact on the light trapping efficiency. The internal reflectance R(λ,θ) at a wavelength of 1000 nm is plotted in Fig. 6(b) as a function of incident angle for an ITO (blue) and a poly-Si (red) tunnel junction, where the vertical dashed lines represent the critical angles of c-Si/Air and c-Si/cp-TiO₂ at angles of 16.5° and 42°, respectively. These critical angles separate the reflectance spectra into three distinct regimes: 0≤θ≤θ_c,Si/Air, θ_c,Si/Air≤θ≤θ_c,Si/TiO2, and θ_c,Si/TiO2≤θ≤π/2. In the first regime, 0≤θ≤θ_c,Si/Air, a large amount of light escapes through the front of the cell. In the second regime, θ_c,Si/Air≤θ≤θ_c,Si/TiO2, total internal reflection at the top (air) interface prevents light from escaping through the front of the cell, so any reduction in reflection from unity is a result of parasitic absorption. In the third regime, θ_c,Si/TiO2≤θ≤π/2, light is reflected back into the silicon before reaching the perovskite top cell, with only a small reduction in reflection due to evanescent tunnelling through the thin ITO/cp-TiO₂ layers. The internal reflectance for a poly-Si tunnel junction is slightly larger than an ITO tunnel junction due to lower parasitic absorption.

The inset in Fig. 6(b) shows the internal angle-averaged reflection, R_F, plotted against wavelength for an ITO tunnel junction (blue), poly-Si tunnel junction (red), and an ideal c-Si/Air interface (magenta). From this inset, it is clear that despite the parasitic absorption in the layers above the silicon layer, the internal reflectance is only ~5% less than for an ideal c-Si/Air interface for both tunnel junctions, with the poly-Si slightly outperforming the ITO. In either case, the absorption in the layers above silicon has a surprisingly small impact on the light trapping in the c-Si layer.

The loss breakdown for the tandem cell incorporating Lambertian light trapping is shown in Fig. 7, where the back reflector of the silicon cell comprised a SiO₂ (50 nm)/Ag (300 nm) bilayer and the perovskite layer was adjusted to an optimum thickness of 355 nm for a matched J_SC of 20.09 mA/cm². The calculated silicon current is over 2 mA/cm² greater than the best experimental monolithic tandem cell with rear-side light trapping [6]. The thicker perovskite layer was made necessary by the light trapping, which increased the silicon current without reducing the perovskite current. The remaining reflection peak at 900 nm is a result of the index mismatch between the ITO tunnel junction and the c-Si layer. This could be mitigated through the use of an anti-reflection coating between the layers; however the resultant current gain of 0.09 mA/cm² does not warrant the added fabrication complexity. The same optimization was also applied for light trapping in a tandem with a poly-Si tunnel junction, with an optimized perovskite layer thickness of 375 nm resulting in a matched J_SC of 20.36 mA/cm². Despite the increased light trapping and lower index mismatch at the c-Si interface provided by the poly-Si tunnel junction, the current gain was only 0.27 mA/cm² compared to the cell with the ITO tunnel junction, confirming the relatively small impact of the tunnel junction optics on the tandem cell performance. For completeness, the same optimization was also applied for light trapping in a n-i-p tandem with a poly-Si tunnel junction with the same charge transport layers as in the p-i-n orientation. The optimized perovskite layer thickness in this case is 355 nm, resulting in a matched J_SC of 20.32 mA/cm². This result demonstrates that the top cell orientation has little effect on the overall tandem performance for materials with refractive indices close to that of perovskite.

 

Fig. 7 Light distribution in a perovskite/silicon tandem cell including Lambertian light trapping in the c-Si cell with ITO tunnel junction. The perovskite thickness was optimized to 355 nm for maximum current-matching.

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From these light trapping results, it is clear that rear-side light trapping can provide increased silicon absorption, which in turn allows for thicker perovskite cells for current matching, and thus significant tandem cell efficiency gains.

5. Charge transport material compatibility

In the preceding sections, we showed that charge transport materials with a high refractive index and low absorption are critical for achieving high performance monolithic tandems. However, we have so far ignored the experimental constraints of the various charge transport layers, such as deposition and annealing temperatures, and chemical compatibility with perovskite materials.

A range of charge transport materials that are commonly used in high performance perovskite and tandem solar cells are shown in Table 1, along with typical deposition and annealing temperatures, deposition methods, average long wavelength refractive index and the relative amount of parasitic absorption. Parasitic absorption was determined by calculating the wavelength-averaged single-pass absorptance of a 50 nm thick film and grouping materials according to this value as high (average absorptance >2%), medium (1%-2%) and low (1%). It should be noted that this method underestimates the absorption exhibited by a spiro-OMeTAD HTL, as the 50 nm film considered is significantly thinner than that of a typical spiro-OMeTAD layer (~200 nm).

From Table 1, it is clear that the materials that have both a high refractive index and low absorption are the inorganic charge transport materials, such as cp-TiO₂ and NiO_x. However, to produce films with sufficient charge mobility and film quality, these materials are typically deposited/annealed at temperatures above what the perovskite layer can withstand (~150°C). Therefore if these deposition methods are chosen, the material must be deposited before the perovskite layer, enforcing the orientation of the tandem cell. Additionally, a number of deposition processes that may physically damage the perovskite or use aqueous solutions (e.g. PEDOT:PSS) are also not applicable for deposition on top of the perovskite layer. Reactivity between perovskite and the charge transport layers must also be considered as some organic [35] and inorganic [34] charge transport materials have been shown to react with perovskite and its precursors, such that many of the currently used materials may be unsuitable for use in long-term stable cells. Taking these restrictions into account, the best high index charge transport materials for depositing on top of perovskite are solution-processed/sputtered NiO_x for the HTL and np-ZnO or np-TiO₂ for the ETL. These materials do not require a post-annealing step after deposition.

It is worth noting that the recent development of Cs-containing perovskite materials may allow for a larger variety of compatible charge transport materials due to their greater temperature and moisture stability [36, 37]. However, the high performance mixed-halide perovskite cells only appear to provide a slight increase in thermal stability, which does not significantly increase the range of compatible materials. All inorganic perovskite materials [38, 39] (such as CsPbBr₃) have higher thermal stability; however current cell performance is insufficient to justify use in tandem with c-Si. Until more stable, high performing perovskite compositions become available, development of new transport materials should focus on inorganic materials with low absorption and high refractive indices that can be deposited at low temperatures.

6. Summary and conclusion

By investigating and mitigating the dominant loss mechanisms in monolithic perovskite/silicon tandem cells, we have identified practical steps to substantially improve the performance of both the perovskite and silicon sub-cells by reducing parasitic and reflective losses. A summary of the improvement of J_SC for both the perovskite and silicon cells in the 2-terminal tandem after each refinement is shown in Fig. 8, where currents generated in both the perovskite (blue) and silicon (red) cells and currents lost by parasitic absorption (green) and reflection (purple) for each stage are shown.

 

Fig. 8 Progression of generated and lost J_SC after each adjustment to the tandem architecture. The additional cases of light-trapping within the c-Si layer for a poly-Si tunnel junction for both the p-i-n and n-i-p orientations are also included, with optimized perovskite thicknesses of 375 nm and 355 nm for a J_SC = 20.36 mA/cm² and 20.32 mA/cm², respectively.

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From Fig. 8, the major improvements in the currents of the two cells are a result of three main refinements. First, low parasitic absorption was achieved by replacing the front-side layers of the perovskite cell by materials with low absorption across the solar spectrum. Second, reflection was reduced by 3 mA/cm² by carefully managing the internal reflection within the cell by selecting materials for the ETL and HTL such that the refractive index mismatch between these layers and the adjacent layers are minimized. Third, by incorporating light trapping within the silicon layer via a rear Lambertian scattering surface, the current generated by the silicon cell was greatly enhanced; which allowed for a thicker perovskite layer to achieve a matched-current greater than 20 mA/cm². Through the application of these refinements, in conjunction with the best V_OC and fill factor for individual perovskite [36] (V_OC = 1.15 V, FF = 0.79), and silicon [40] (V_OC = 0.74 V, FF = 0.827) cells, we calculate possible tandem efficiencies of 27.7% and 31.0% without and with light trapping, respectively. Therefore, with further improvement to the open-circuit voltage and fill factor of the tandem cell, achieving tandem efficiencies greater than 30% is within reach with appropriate material selection and light trapping mechanisms.

Appendix A Light Trapping Methodology

We incorporate light trapping using a Lambertian scattering model to estimate the maximum performance gains achievable with light trapping. To fully simulate the perovskite/silicon tandem with a rear Lambertian scattering surface, three distinct scenarios are individually simulated: 1. Light normally incident onto the front side of the tandem cell with a semi-infinite c-Si substrate, 2. Light incident onto the perovskite top cell from below (i.e. from the c-Si layer), and 3. Light incident from c-Si onto a planar dielectric/metal back surface. Scenarios 2 and 3 are calculated for incident angles 0≤θ≤90°. The three cases are shown in Figs. 9(a), 9(b) and 9(c), respectively. In these sub-figures, the reflectance (R) and transmittance (T) at each interface are shown, in addition to the absorptance within a layer (A) and the amplitude of the incident light (I). The complete Lambertian light trapping model is depicted in Fig. 9(d), with the A, R, T values calculated in the three previous scenarios being incorporated into the front (F subscript), back (B subscript), propagation (P and P`) and normally incident (Perov subscript) terms used to calculate the light trapping within the c-Si layer of thickness W. The Front, Back and propagation trapping terms are averaged over the scattering angles produced by a Lambertian distribution.

 

Fig. 9 Schematics for Lambertian light trapping within the silicon layer via a rear Lambertian scattering surface. (a) Light enters the tandem structure at normal incidence from the front side of the tandem cell (b) Light approaches top cell layers from the c-Si substrate over a range of incident angles. (c) Light approaches the dielectric/metal back reflector from the c-Si substrate over a range of incident angles. (d) Complete schematic for Lambertian light trapping in the c-Si layer of thickness W.

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The angle-averaged absorption A_F, reflection R_F, and transmission T_F from the top cell layers are represented by the following equations:

(4)$$R_F\left(\lambda \right)=2\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}R\left(\lambda ,\theta \right)\cos \left(\theta \right)\sin \left(\theta \right)d\theta$$

(5)$$T_F\left(\lambda \right)=2\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}T\left(\lambda ,\theta \right)\cos \left(\theta \right)\sin \left(\theta \right)d\theta$$

where θ is the scattering angle, λ is the wavelength of the light and R(λ,θ) / T(λ,θ) are the angle-resolved reflectance/transmittance from the top cell layers when light approaches from the c-Si substrate. Therefore the angle-averaged absorption is A_F(λ) = 1− R_F(λ) − T_F(λ).

Similarly, the angle-averaged reflection and absorption at the c-Si back surface are:

(6)$$A_B\left(\lambda \right)=2\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}{A}_{Metal}\left(\lambda ,\theta \right)\cos \left(\theta \right)\sin \left(\theta \right)d\theta .$$

where A_Metal(λ,θ) is wavelength and angle-dependent absorption within the rear dielectric/metal layers. Rear transmittance is assumed zero as thick metals (>100 nm) are fully opaque. Therefore the angle-averaged back surface reflection is R_B(λ) = 1− A_B(λ).

To define the c-Si absorption enhancement, the average propagation length in the silicon must be calculated. We define two different propagation terms, P and P`, where P`=exp(−α(λ)W) is the single-pass transmittance through the Si at normal and P is the angle-average single-pass transmittance of scattered light with a Lambertian distribution:

(7)$$P\left(\lambda \right)=2\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\cos \left(\theta \right)\sin \left(\theta \right)\text{exp}\left(-\frac{\alpha \left(\lambda \right)W}{\cos \left(\theta \right)}\right)d\theta$$

(8)$$P^2\left(\lambda \right)=2\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}\cos \left(\theta \right)\sin \left(\theta \right)\text{exp}\left(-\frac{2\alpha \left(\lambda \right)W}{\cos \left(\theta \right)}\right)d\theta ,$$

where W and α(λ) are the thickness and absorption co-efficient of the c-Si layer, respectively. The c-Si layer thickness W is set to 200 μm in these calculations. P² is the double-pass angle-averaged transmittance. We can now define the total reflectance of the cell and the total absorptance in the top cell, c-Si and rear metal layers.

(9)$$R_{Tot}\left(\lambda \right)=R_{Perov}\left(\lambda \right)+\frac{T_{Perov}{P}^{\prime }\left(\lambda \right)R_B\left(\lambda \right)P\left(\lambda \right)T_F\left(\lambda \right)}{1-P^2\left(\lambda \right)R_B\left(\lambda \right)R_F\left(\lambda \right)}$$

(10)$$A_{Top,Tot}\left(\lambda \right)=A_{Perov}\left(\lambda \right)+\frac{T_{Perov}{P}^{\prime }\left(\lambda \right)R_B\left(\lambda \right)P\left(\lambda \right)A_F\left(\lambda \right)}{1-P^2\left(\lambda \right)R_B\left(\lambda \right)R_F\left(\lambda \right)}$$

(11)$$A_{Metal,Tot}\left(\lambda \right)=\frac{T_{Perov}{P}^{\prime }\left(\lambda \right)A_B\left(\lambda \right) }{1-P^2\left(\lambda \right)R_B\left(\lambda \right)R_F\left(\lambda \right)}$$

(12)$$A_{Si,Tot}\left(\lambda \right)=1-R_{Tot}\left(\lambda \right)-T_{Tot}\left(\lambda \right)-\left[A_{Metal,Tot}\left(\lambda \right)+A_{Top,Tot}\left(\lambda \right)\right]$$

Funding

This work has been supported by the Australian Government through the Australian Renewable Energy Agency (ARENA). Responsibility for the views, information or advice expressed herein is not accepted by the Australian Government. KC acknowledges support from the Australian Research Council Future Fellowship Scheme.

Acknowledgements

Computational resources were provided by the National Computational Infrastructure, Australia and the NeCTAR Research Cloud, Australia.

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