## 2 Optics of Microcrystalline Silicon Solar Cells
### Solar Cells with Textured Silicon Absorbers  
The optical model used in this study is based on a periodic arrangement of silicon nanowires. The array of nanowires represents a solar cell with a patterned absorber. The dimensions of the nanowire solar cell are described by the period of the unit cell, \(P\), and the diameter of the nanowire, \(D\) (Figure 1b). The nanowire is square-based, so the term \"diameter\" is equivalent to the edge-lengthof the nanowire. The area filling factor of the nanowire is equal to \((D/P)^{2}\). A constant unit cell period of 900 nm is chosen for these simulations as previous studies have shown that the optimal period of the surface texture is approximately equal to the thickness of the solar cell [9]. The diameter of the nanowire is varied from 50 nm to 900 nm corresponding to area fill factors ranging from 0.3% to 100%. In other words, the area filled with air is varied from 0% to 99.7%. The simulated power loss profiles for a solar cell with textured absorber are shown in Figure 2a-c for incident wavelengths of 400 nm, 600 nm and 800 nm, respectively. These power loss profiles are shown for a diameter of 640 nm and period of 900 nm. For short wavelengths, absorption of light is most prominent along the front surface and edges of the absorber layer. The edges of the silicon nanowires show strong absorption since the light propagates in the gaps between nanowires. The light that is absorbed by the edges lead to an increased quantum efficiency in the spectral range from 300 to 380 nm. The quantum efficiency in this spectral region is increased because the sidewalls are not covered with a \(p\)- or \(n\)-layer. For long wavelengths, the absorption and quantum efficiency of light in the solar cell is enhanced (Figure 3a). The long wavelength light is diffracted by the silicon nanowire. It is important to note that the simulations are carried out in 3D and hence, the light is diffracted in all three dimensions. In the case of 2D simulations the nanowire array would be replaced by a line grating and the light would only by diffracted in 2D. The 3D diffraction grating (nanowire array) leads to a higher quantum efficiency and short circuit current density compared to 2D diffraction gratings (line grating). For nanowire diameters larger than 640 nm a drop of the quantum efficiency for long wavelengths is observed (Figure 3a). Figure 4a shows the red, green, blue and total short circuit current densities for nanowires with a constant period of 900 nm and varying diameter. The dashed line exhibits the short circuit current density generated by a solar cell on a flat substrate. For the flat substrate (Figure 1a), a total short circuit current density of 12.4 mA/cm\({}^{2}\) is calculated. For the silicon nanowire solar cell (Figure 1b) with a nanowire diameter between 400 and 900 nm, an improvement of short circuit current density over the flat substrate is achieved, with optimal performance at nanowire diameter of 640 nm. At this optimal diameter, the short circuit current density is found to be 16.8 mA/cm\({}^{2}\) (Figure 4a). For diameters of the nanowire smaller than 400 nm, the total short circuit current density drops below the short circuit current density of a solar cell on a flat substrate (Figure 4a). With decreasing diameter,  
Figure 2: Power loss profile of an axial nanowire solar cell for incident wavelengths of 400 nm (**a**), 600 nm (**b**) and 800 nm (**c**). The edge-length (\(D\)) of the nanowire is 640 nm, while the period (\(P\)) of the unit cell is 900 nm. long wavelengths light cannot couple in the nanowires anymore (Figure 3b). The cut-off wavelength for coupling light in the wire is given by  
\[\lambda_{cut}=2\times n_{si}\times D, \tag{6}\]  
where \(n_{Si}\) is the refractive index of the silicon wire. For nanowires with diameters smaller than 135 nm, the quantum efficiency is limited by the incoupling of long wavelength light in the nanowire. Hence, the short circuit current density drops with decreasing diameter of the nanowire.