# An Optimized Graphene-Based Surface Plasmon Resonance Biosensor for Detecting SARS-CoV-2
## 2 Materials and Methods
### Modeling Approach  
We have carried out a numerical analysis to calculate the reflectance curve using the Transfer Matrix Method (TMM) and Fresnel equation; details about the mathematical model can be seen in [20; 25]. The proposed biosensors are constructed by sequentially stacking multiple layers in parallel (see Figure 1). The thickness of each layer is measured along the perpendicular axis (\(z\)-axis). The interface boundary conditions of the first and final layers are represented as \(Z=Z_{1}=0\) and \(Z=Z_{n-1}\), respectively.  
Figure 1: Proposed SPR sensors for SARS-CoV-2 sensing.  
The transfer matrix describes the relationship between the tangential components of the electric and magnetic fields as  
\[\begin{bmatrix}\text{E}_{1}\\ \text{H}_{1}\end{bmatrix}=\text{M}\begin{bmatrix}\text{E}_{\text{N}-1}\\ \text{H}_{\text{N}-1}\end{bmatrix} \tag{1}\]  
where E\({}_{1}\), H\({}_{1}\), E\({}_{\text{N}-1}\), and H\({}_{\text{N}-1}\) correspond to the tangential components of electric and magnetic fields at the first and last layer interfaces, respectively. M represents the characteristic matrix of the \(N\)-layer structure with elements \(M_{ij}\):  
\[\text{M}=\prod_{k=2}^{N-1}M_{k}=\begin{bmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{bmatrix} \tag{2}\]  
And \(M_{k}\) is expressed as  
\[M_{k}=\begin{bmatrix}\cos\beta_{k}&-i\sin\beta_{k}/q_{k}\\ -iq_{k}\sin\beta_{k}&\cos\beta_{k}\end{bmatrix} \tag{3}\]  
where \(k\) is an integer number. Additionally, \(\beta_{k}\) is the phase thickness and \(q_{k}\) is the refractive index of the corresponding layer:  
\[\beta_{k}=\frac{2\pi d_{k}}{\lambda_{0}}\sqrt{\varepsilon_{k}-n_{1}^{2}\sin^{ 2}\theta} \tag{4}\]  
and  
\[q_{k}=\frac{\sqrt{\varepsilon_{k}-n_{1}^{2}\sin^{2}\theta}}{\varepsilon_{k}} \tag{5}\]  
Here, \(\theta\) is the angle of incidence, \(\lambda_{0}\) is the incident wavelength light, \(n_{1}\) is the refractive index of the prism, \(d_{k}\) is the thickness layer, and the local dielectric function \(\varepsilon(\lambda_{0})\) can be adopted as \(n(\lambda_{0})\). In this work, we use the data for the He-Ne laser with \(\lambda_{0}=633\) nm. Hence, the total reflection analysis of the N-layer system is obtained as  
\[R=\left|\frac{(M_{11}+M_{12}q_{N})q_{1}-(M_{21}+M_{22}q_{N})}{(M_{11}+M_{12}q_ {N})q_{1}+(M_{21}+M_{22}q_{N})}\right|^{2} \tag{6}\]  
By using Equation (6), the SPR curve as a function of the angle of incidence is computed. It is noted that for each SPR curve, surface plasmon excitation is identified as a dip in the reflected intensity \(R\), corresponding to the minimum in attenuated total reflection (ATR). The angle of incidence at the ATR minimum is called the SPR angle.  
Moving forward, to analyze the performance of the biosensor, it is necessary to consider the following metrics. The sensitivity of the biosensors (\(S^{P}\)) is defined as the multiplication of the sensitivity to the refractive index change (\(S^{N}_{RI}\)) and the adsorption efficiency of the target analyte (\(E\)):  
\[S^{P}=S^{P}_{RI}\cdot E \tag{7}\]  
To optimize the biosensor, we mainly focus on the sensitivity enhancement concerning conventional biosensors (i.e., P/Ag/M) (here, \(P\) is the parameter to be optimized) expressed as  
\[\Delta S^{P}_{RI}=\left(S^{P}_{RI}-S^{0}_{RI}\right)/S^{0}_{RI} \tag{8}\]  
The sensitivity to the refractive index change can be expressed as  
\[S^{P}_{RI}=\Delta\theta/\Delta n \tag{9}\]  
The parameter \(\Delta\theta\) represents the angle variation.  
The detection accuracy (DA) (i.e., signal-to-noise ratio) can be expressed in terms of \(\Delta\theta\) and the full width at half maximum (FWHM) as  
\[\text{DA}=\Delta\theta/\text{FWHM} \tag{10}\]  
Finally, quality factor (QF) (or figure of merit, FoM) can be expressed in terms of \(S\) and FWHM as  
\[\text{QF}=S_{RI}^{\text{P}}/\text{FWHM} \tag{11}\]