## 3 Results and Discussion  
The duty cycle of the grating--the portion of the period occupied by the perovskite cell--determines the amount of short-wavelength irradiation that is absorbed by the perovskite cell and the energy transferred to the c-Si cell. It considers the radiation both directly through the gap of the grating, and after being transmitted through the perovskite cell. We optimized the design to maximize \(J_{SC}\) from both junctions by considering the multilayered grating's width, \(W\), and the perovskite cell thickness, \(H\). For a two-terminal tandem cell to operate properly, an additional condition must be fulfilled: currents from both device junctions need to be equal. This condition ensures that both junctions electrically matched, so that \(J_{SC}\)(perovskite) = \(J_{SC}\)(c-Si). Considering this electrical matching condition, the term \(J_{SC}\) will refer to the short-circuit current of the individual cells. We performed calculations with values of \(W\) between 50 nm and 650 nm (grating duty cycle ranging from 6.25% to 81.25%). The height ranges of the perovskite active layer went from 100 nm to 1000 nm. The diffusion length of the photo-generated charge carriers in perovskites is about 1 \(\upmu\)m, so we limited the maximum height of the perovskite layer in our calculations to this value [42]. The results for the optimization calculation are shown in Figure 2. They show that increasing the width, \(W\), generates a lower response from the c-Si cell (see Figure 2a). The same trend is observed when enlarging the thickness of the perovskite layer, \(H\). The increase in both \(W\) and \(H\) leads to a larger volume of the perovskite multilayer and a higher absorption inside it, which leads to a lower generation of power at the c-Si cell. This result is expected because the perovskite multilayer extracts power from the incoming radiation, reducing the amount to be absorbed by the c-Si cell. The behavior of the perovskite cell is a little more complex (see Figure 2b). The delivered short-circuit current, \(J_{SC}\) increases with thickness, \(H\) (more active medium involved), but it shows an optimum value for the width of the grating (\(\simeq\)50% duty cycle). The black lines in the maps of Figure 2a,b, define the geometrical parameters that yield the same \(J_{SC}\) value as the reference planar device (16.2 mA/cm\({}^{2}\)) [7]. These lines divide the maps into two regions of interest: I and II. Region I corresponds to a \(J_{SC}\) value lower than the value of \(J_{SC}\) for the reference device with ITO and region II yields a \(J_{SC}\) value higher than the one for the reference planar device. Therefore, the optimum geometry is in region II, where the design delivers a short-circuit current higher than the reference planar tandem cell. When choosing the geometrical arrangement of the grating lying in region II of both Figure 2a,b, we also need to fulfill the matching current condition. This situation is marked by the black star on the maps of Figure 2. It corresponds to a width \(W=250\) nm and a thickness \(H=750\) nm of the perovskite layer. The \(J_{SC}\) value at this point is 16.7 mA/cm\({}^{2}\), which is 1.03 times the value of the reference planar device that uses an ITO top contact. When analyzing the electromagnetic behavior of the proposed nanostructure, we observed how the top metallic contact efficiently redirects light towards the active layers of both devices. At the same time, the structure responds differently in terms of the polarization state of the incoming radiation. The results of this analysis is shown in Figure 3a,b for the TE and TM linear polarizations, respectively. Here, transversal means that the field vector is aligned along the grooves of the grating (perpendicular to the cross section presented in Figure 1c). The short wavelength portion is mostly absorbed by the active layer at the perovskite cell (shown in gray) and the long wavelength portion is absorbed by the active layer at the c-Si cell (shown in light blue).