## 2 Materials and Methods
### Optical Properties of Graphene  
This paper studies the use of graphene as a transparent conductive oxide on the front layer of the solar cell. To assess the impact of the proposal on the performance of the cell, the optical properties resulting from considering different graphene layers (from one to ten layers) are modelled. The optical properties calculated for graphene are refractive index (\(n\)), reflectance (\(R\)), and absorbance (\(A\)). The thickness of each layer of graphene is 0.34 nm [42]. Figure 2a shows the diagram of graphene layers on the substrate. Figure 2b shows the spectral transmittance, \(T(\lambda)\), of the considered graphene layers, and the values are obtained from refractiveindex.info database [43]. Absorbance \(A(\lambda)\) is calculated with Equation (1), where the wavelength is in nm [36]:  
\[A(\lambda)=\text{log}\bigg{(}\frac{1}{T(\lambda)}\bigg{)}, \tag{1}\]  
Extinction coefficient \(k(\lambda)\) is calculated in terms of absorption, according to Equation (2) [44]:  
\[k(\lambda)=2.303\frac{A(\lambda)}{4\pi d}, \tag{2}\]  
Figure 1: The black line shows the standard G173 reference spectrum with a 1.5 air mass (AM1.5). The red line shows the spectrum calculated for the Atacama Desert. Both are normalized to 1000 W m\({}^{-2}\)[34]. Figure 2: (**a**) Graphene layers on the glass substrate. (**b**) Transmittance graph from refractiveindex.info database. where \(d\) is the sample thickness of graphene in nm. The complex refractive index of graphene is calculated by optimizing the transmittance expressed as \(T_{cal}=T(s(\lambda),\ n(\lambda),\ k(\lambda),d)\), which depends on the substrate refraction index \(s(\lambda)\), the graphene refraction index \(n(\lambda)\), the graphene extinction coefficient \(k(\lambda)\), and the thickness \(d\). To do this, the approximate transmittance \(T_{apx}(\lambda,d)\) and \(T_{cal}\) are minimized [44]:  
\[ \begin{array}{c}\text{{minimized}}\sum_{\lambda}[T_{apx}(\lambda,d)-T_{cal} (s(\lambda),\ n(\lambda),\ k(\lambda),d)]^{2}\end{array}  \tag{3}\]  
where \(T_{apx}(\lambda,d)=T(\lambda)-\Delta T\) is detailed in Equation (4) according to the refractiveindex.info database, with \(T(\lambda)\) as shown in Figure 2b and the correction factor, \(\Delta T\), being the approximation function that changes according to the wavelength range obtained by approximating the experimental values of \(n(\lambda)\)[42]:  
\[ \begin{array}{c}T_{apx}\left(\lambda,d\right)=\left\{\begin{array}{ll}T( \lambda)-\left(0.0003\lambda-0.0578\right),&210<\lambda<270\\ T(\lambda)+\left(0.0002\lambda-0.0701\right),&271<\lambda<348\\ T(\lambda)+\left(5*10^{-6}\lambda-0.0039\right),&349<\lambda<532\\ T(\lambda)+\left(9*10^{-7}\lambda-0.0020\right),&533<\lambda<1200\\ \end{array}\right.\end{array}  \tag{4}\]  
To calculate the refractive index, non-linear equations are solved numerically using Equation (5) [44; 45]:  
\[ \begin{array}{c}T_{cal}\left(s(\lambda),\ n(\lambda),\ k(\lambda),d\right)= \frac{Ax}{B-Cx+Dx^{2}}\end{array}  \tag{5}\]  
where  
\[ \begin{array}{c}A=16s\left(n^{2}+k^{2}\right)\\ B=\left[\left(n+1\right)^{2}+k^{2}\right]\left[\left(n+1\right)\left(n+s^{2} \right)+k^{2}\right]\\ C=\left[\left(n^{2}-1+k^{2}\right)\left(n^{2}-s^{2}+k^{2}\right)-2k^{2}\left( s^{2}+1\right)\right]2 cos\varphi\\ -k\left[2\left(n^{2}-s^{2}+k^{2}\right)+\left(n^{2}+1\right)\left(n^{2}-1+k^{ 2}\right)\right]2 sin\varphi\\ D=\left[\left(n-1\right)^{2}+k^{2}\right]\left[\left(n-1\right)\left(n-s^{2 }\right)+k^{2}\right]\\ \varphi=\frac{4\pi dN}{\lambda},\qquad x=e^{-ad},\qquad\alpha=\frac{4\pi k}{\lambda}\end{array} \]  
The substrate corresponds to glass, so the substrate refraction index, \(s(\lambda)\), is given by Equation (6) [36; 37]:  
\[s(\lambda)=\sqrt{1+\left(0.7568-7930/\lambda^{2}\right)^{-1}} \tag{6}\]  
Equations (1)-(6) were used to calculate the refractive index and the extinction coefficient, obtaining values similar to those reported by [PERSON] [42]. Figure 3 shows \(n(\lambda)\) and \(k(\lambda)\) for the graphene from the monolayer up to ten layers.