## 2 Design and Modeling  
The layered structure for the planar perovskite/c-Si tandem solar cell shown in Figure 1a is one of the commonly used in other works [8, 28, 29]. Our nanostructured layer arrangement and design are shown in Figure 1b,c. The role, material and thickness of each layer are listed in Table 1, where they are ordered from top to bottom. When moving from the planar tandem cell to our design, we consider that the grating constitutes the upper cell. The materials of the core of the perovskite cell are the same for both designs. Table 1 also contains the references used to retrieve the optical constants of the materials. The 5 nm-thick hydrogenated amorphous silicon layers are to facilitate the charge carrier transport in the c-Si subcell by generating the required built-in potential in the junction. Due to their small thickness, their optical effects are negligible and they will be ignored in our model [7, 39]. In the proposed design we replaced the ITO thin film top contact with a nanostructured metallic electrode. The perovskite multilayer cell structure is distributed as a 1D rectangular grating on top of the cSi cell, which includes a gold thin film 20 nm-thick on top of it (refractive index from [32]). Figure 1b,c show the top perovskite cell layers arranged as a multilayer grating with a period \(P=800\) nm. As we see in Figure 1c, the width of the perovskite cell layers and the gold contact is the same, \(W\). Both the width and the thickness of the perovskite layer, \(W\) and \(H\), are the geometrical parameters that we optimized. Finally, we used a thin dielectric layer of aluminum nitride (AlN) as a protective coating. This material is being included in a wide variety of optoelectronic sensing devices because of its good biocompatibility and mechanic-electrical properties and is typically fabricated by sputtering technique [40]. The wavelength-dependent refractive index data for AlN was extracted from a standardized table [30]. Our design incorporates this material as a thin film, having a thickness of 50 nm. We have checked that the addition of this layer does not compromise the performance of the cell. We calculated the optical performance of both the reference and proposed designs with the finite element method implemented in COMSOL Multiphysics. The nanostructured array is infinite along the grating direction and can be computationally reduced to a unit \begin{table}
\begin{tabular}{c c c} \hline
**Layer** & **Material** & **Thickness** \\  & **(Planar Cell \(\rightarrow\) Nanostructured)** & **(Planar Cell \(\rightarrow\) Nanostructured)** \\ \hline Protective & None \(\rightarrow\) AlN [30] & 0 \(\rightarrow\) 50 nm \\ Top contact & ITO [31] \(\rightarrow\) Au grating [32] & 100 nm \(\rightarrow\) 20 nm \\ Hole transport layer (HTL) & SPIRO-OMeIAD [33] & 160 nm \\ Active layer & Perovskite (CH\({}_{3}\)NH\({}_{3}\)PbI\({}_{3}\)) [34] & 260 nm \(\rightarrow\) 750 nm \\ Electron transport layer (ETL) & TiO\({}_{2}\)[35] & 30 nm \\ Tunnel junction & ITO [31] & 44 nm \\ Hole transport & p-type hydrogenated amorphous silicon (paSi-H) [36] & 10 nm \\ Buffer layer & intrinsic hydrogenated amorphous silicon (iaSi-H) [36] & 5 nm \\ Active layer & Crystalline Si (c-Si) [37] & 200 \(\upmu\)m \\ Buffer layer & Intrinsic hydrogenated amorphous silicon (iaSi-H) [36] & 5 nm \\ Electron transport layer & n-type hydrogenated amorphous silicon(naSi-H) [36] & 10 nm \\ Botton contact & Aluminum (Al) [38] & 200 nm \\ \hline \end{tabular}
\end{table} 
Table 1: Tandem perovskite/c-Si solar cell layer structure. Both the materials and thicknesses of the layers are included for the planar cell and the one presented in this paper (nanostructured). We have also included the references where the optical constant are taken from. Figure 1: (**a**) Layered structure of the reference planar tandem perovskite/c-Si solar cell (the active c-Si is much thicker than the rest of the layer). (**b**) Schematic design of a nanostructured perovskite solar cell located on top of a crystalline silicon cell.