## 3 Results and Discussion
### Simulation of the SERS Substrate  
Simulation of the real-world of silicon wire covered with silver and gold and emerged in a medium that contains S and RBD proteins were performed using an effective and accurate tool: COMSOL Multiphysics software. The software's main feature is that it is based on the most advanced numerical method, the finite element method. We used an electromagnetic module in the frequency domain to understand how the bio-medium may influence the silicon covered with Ag/Au. Accordingly, two main equations are used. The first is the Maxwell Equation (1):  
\[\Delta\times\mu_{r}^{-1}\bigg{(}\
abla\times\overset{\rightarrow}{E}\bigg{)}- k_{0}^{2}\bigg{(}\varepsilon-\frac{j\sigma}{\omega\varepsilon_{0}}\bigg{)} \overset{\rightarrow}{E}=0 \tag{1}\]  
where \(\omega\), \(\mu_{r}\), \(\varepsilon\), and \(\sigma\) are the excitation frequency, the relative permeability (fixed to 1), the relative permittivity, and the electrical conductivity, respectively. \(\varepsilon_{0}\) and \(k_{0}\) represent the permittivity and wave number in free space, being \(k_{0}=\omega/c_{0}\) (with \(c_{0}\) the speed of light in a vacuum). The approximate relative permittivity inside the RBD protein is 4, the relative permittivity of water is 90 and the average relative permittivity in the protein-water interface is 20-30 based on an explicit model with atomic details [41,42]. The second, Equation (2), is used to calculate the electromagnetic enhanced Raman scattered light denominated g from the structure of silicon wire surrounded by silver and gold. \[g=\bigg{(}\frac{I_{SERS}}{I_{Raman}}\times\frac{N_{Raman}}{N_{SERS}}\bigg{)}^{2} \tag{2}\]  
The field enhancement can be created close to the Ag nanoparticles where the polaritons are constructively interfered. Then, the local field is further enhanced and a dipole is produced, leading to the amplification of the Raman scattering in the hot-spots and a formation of plasmons [11,43]. An interactive excitation from the Ag and Au nanoparticles system at a resonant frequency (plasmon resonance) generates and enhances apparent Raman polarizability. The designed structure is a three-dimensional (3D) model of silicon wire with a 12 nm radius and 300 nm length, surrounded with 108 silver and gold nanospheres (silver and gold were randomly selected on the geometry with 95% of silver nanospheres and 5% of gold nanospheres) with a 7 nm radius, as depicted in Figure 12a. We used a sequence  
Figure 11: Principal component analysis (PCA) of the collected spectra for SL and SL-BSA. Red dots represent SL. Black dots represent SL-BSA. (300 nm length of silicon wire) of the real image in Figure 2a since, with more length and number of Ag/Au nanospheres, the time of calculation will be increased and surpass the memory of the software. However, this sequence helps to understand the physical mechanism at the basis of the enhancement in the SERS system used for detection. We used the same experimental conditions, 632.8 nm laser excitation, and involved the entire geometry in a medium with the same optical properties of the bio-medium that contains S and RBD proteins. Figure 12b depicts the absorption cross-section of the silicon wire surrounded with silver and gold in two mediums: air and bio-medium (proteins). Three resonant plasmon modes were observed in both air (365 nm, 508.7 nm, 754 nm) and bio medium (394.1 nm, 520 nm, 775.4 nm). An apparent redshift of surface plasmon of about 34 nm, 12 nm, and 21.4 nm of plasmon bands, located initially in the air medium, is observed when we place the structure in a bio-medium. Those shifts go along with a slight change in band spectrum. Plasmon shift confirms that the entire 3D geometry behavior is sensitive to the optical properties of the corresponding medium. Hence, SERS frequency shift sensing is a powerful tool for protein detection. We also calculated the quantified enhancement factor G of the structure in a bio-medium, see Figure 12c.