## 2 Methods
### Numerical Simulation
From these equations, through numerical simulation, and by performing a parametric sweep of the light wavelength (that enters Equation (11), further affecting Equations (9) and (10)), the current that each photodiode produces is obtained, thus obtaining its responsivity and quantum efficiency, as presented in Equations (1) and (2). The photodiodes were modeled and simulated in COMSOL Multiphysics software, in a 2D geometry, with a cross-section as shown in Figure 2:1 The simulations only took into account the p-n junction of the photodetectors, without the oxide layers (which are usually included in the microfabrication processes). Furthermore, an \"out-of-plane\" parameter (100 \(\upmu\)m) that represents the virtual depth of the structure was added to the software, allowing to take into account the microstructure as a volume instead of a plane, but without the computational cost of a 3D simulation. In these bidimensional simulations, the semiconductor and wave optics as well as the frequency domain modules were used to represent all the physical phenomena involved in the photodiode operation. The first simulated photodiode junction was the n+/p-substrate photodiode, for all the considered CMOS technologies (0.18, 0.35 and 0.7 \(\upmu\)m), due, theoretically, to its higher quantum efficiency in the visible spectral range (as in Figure 2) [7,11]. The photodiode was designed according to the Figure 3 representation, with two p+ and one n+ regions on top of a p-type silicon substrate (with 1.14 \(\times\) 10\({}^{15}\)/cm\({}^{3}\) acceptor concentration in the 0.18 \(\upmu\)m technology). Two p+ regions were doped in the top corners of the photodiode, with a n+ doped region in the center of the photodiode, which will be the active area, i.e., the photosensitive region. This active area has a width of 100 \(\upmu\)m. The distance \(d_{0}\) corresponds to the substrate thickness. In this work, the value of 500 \(\upmu\)m was considered. Two ports were considered in the domain: one on the top of the photodiode, in the photosensitive region, where the wave excitation is defined (defined by the blue line and the red arrow in Figure 3a), and another on the bottom of the photodiode (with no wave excitation), that works as a radiation outlet. On the top of the photodiode, four regions with ideal ohmic contacts were defined (through the option: metal contact in COMSOL Multiphysics). All the other exterior boundaries of the photodiode geometry were defined as electric insulation. The boundary interface regions between the p+ and n+ doping and the p-type substrate were defined as heterojunction boundaries, through a continuous quasi-Fermi levels model. Figure 2:2 Illustrative example of a n+/p-substrate photodiode simulation: (**a**) complete geometry and (**b**) detail of the top of the photodiode. The dark blue area represents the p+ doping on the p-type silicon substrate. In the center, the colored area represents the n+ doping region, representing the active area of the photodiode, with a 100 \(\upmu\)m width (represented by the dimension _a_). \(d_{0}\) represents the substrate thickness; the value of 500 \(\upmu\)m was used. The length of the p + doping zones (represented in the figure by the dimension _b_) considered was 2 \(\upmu\)m for all technologies. Photodiodes manufactured using the 0.18 \(\upmu\)m technology have this same value. The total width of the photodiode is 110 \(\upmu\)m.  
According to the different technologies, the junction's depth (Table 1) and the concentrations of dopants (Table 2) are the main varying parameters. These junction depths enter Equation (12), which defines the decay length \(l_{i}\). The decay length represents the distance where the decay occurs between dopant concentration \(N_{0}\) and background doping \(N_{b}\). Furthermore, \(D_{j,i}\) represents the junction depth in the \(i\)-direction, \(N_{b}\) is background doping and \(N_{0}\) is the dopant concentration within the uniformly doped region [15].