## 2 Methods
### Numerical Simulation
The dopants' profiles, away from the boundary, are defined in terms of the distance \(D\) from the selected boundaries, as described by (13). In the equation, \(N_{a,d}\) are the concentrations of the acceptors or donors and \(N_{0}\) is the concentration of dopants at the selected boundaries [15]. Figure 2:3  
Figure 2:4  
Regarding the concentrations of p and n type dopants, they also vary from technology to technology. Therefore, the values presented in Table 2 were used. Here, the acceptor concentration in p-substrate corresponds to \(N_{b}\) and the donor concentration or acceptor concentration at the boundary is represented by \(N_{0}\), as in (12). Additionally, more parameters were needed to model and perform the numerical simulations (Table 3). Particularly, in COMSOL Multiphysics, the lifetime of electrons and holes is already predefined, by default, in the silicon material properties. Finally, an incident optical power of 0.1 \(\upmu\)W was used in Equations (1) and (2). The initial temperature of the domain was set to 300 K.  
Regarding the mesh, each 2D domain was refined with a small grid, comprised by triangular elements, as shown in Figure 2:5 From this figure, it is possible to see a greater refinement at the top of the photodiode, where the type n and type p diffusers are located, \begin{table}
\begin{tabular}{c c c c} \hline
**CMOS Technology** & **Junction Depth n+ (\(d_{n}\)) (\(\upmu\))** & **Junction Depth p+ (\(d_{p}\)) (\(\upmu\))** & **n-Well Depth (\(d_{nw}\)) (\(\upmu\))** \\ \hline
0.18 \(\upmu\)m & 0.16 & 0.18 & 1.8 \\
0.35 \(\upmu\)m & 0.20 & 0.20 & 2.0 \\
0.7 \(\upmu\)m & 0.35 & 0.35 & 2.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Junction depths for CMOS technologies 0.18 \(\upmu\)m, 0.35 \(\upmu\)m and 0.7 \(\upmu\)m [17; 18; 19]. \begin{table}
\begin{tabular}{c c c c} \hline
**CMOS Technology** & **Acceptor Concentration in p-Substrate (\(N_{b}\)) (\(l/cm^{3}\))** & **D Donor Concentration n+ (\(N_{0}\)) (1/cm\({}^{3}\))** & **Acceptor Concentration p+ (\(N_{0}\)) (1/cm\({}^{3}\))** & ** Donor Concentration n-Well (\(N_{0}\)) (1/cm\({}^{3}\))** \\ \hline
0.18 \(\upmu\)m & \(1.14\times 10^{15}\) & \(4.38\times 10^{19}\) & \(4.30\times 10^{19}\) & \(1.19\times 10^{17}\) \\
0.35 \(\upmu\)m & \(1.25\times 10^{15}\) & \(5.00\times 10^{19}\) & \(5.00\times 10^{19}\) & \(1.28\times 10^{17}\) \\
0.7 \(\upmu\)m * & \(1.42\times 10^{15}\) & \(6.80\times 10^{19}\) & \(6.80\times 10^{19}\) & \(1.35\times 10^{17}\) \\ \hline \end{tabular}
* Values were calculated through a linear regression, since they were not found in literature.  
\end{table} 
Table 2: Acceptor and donor concentrations (1/cm\({}^{3}\)) for technologies 0.18 \(\upmu\)m, 0.35 \(\upmu\)m and 0.7 \(\upmu\)m [20]. \begin{table}
\begin{tabular}{c c c} \hline Hole lifetime & 10 \(\upmu\)s \\ Electron lifetime & 10 \(\upmu\)s \\ Incident optical power & 0.1 \(\upmu\)W \\ Substrate thickness (\(d_{0}\)) & 500 \(\upmu\)m \\ \hline \end{tabular}
\end{table} 
Table 3: Additional relevant parameters included in the numerical simulations. since it is the domain area where a more accurate simulation is required. Table 4 shows the number of elements and quality of the mesh of the n+/p-substrate photodiodes, for each of the technologies under study. As reported in Table 4, the nine generated meshes have high quality, with an average quality above 0.9 (mesh quality varies from 0 to 1) for all of them. Two different solvers were considered: (1) a frequency-stationary study was used when the illuminated photodiodes were simulated (without polarization); (2) without lighting, when only the semiconductor physics was implemented, a stationary study was used. In the frequency-stationary solver, a parametric sweep of the wavelength was considered, so the spectral response could be obtained.