## 3 Theoretical Results and Discussion
Indeed, at a reverse voltage of 0.66 V the \(\Phi_{\text{B0}}=0.73\) eV of the graphene/n-Si junction can be reduced to its optimum value of \(\Phi_{B}^{*}(1.55\upmu\text{m})=0.71\) eV. In contrast, p-GaAs requires a higher reverse voltage of 12 V to maximize \(R/NEP\). Finally, n-Ge stands out among the analyzed semiconductors in view of the possibility to be employed over a region of the NIR spectrum above 2 \(\upmu\)m. The range of wavelengths where \(R/NEP\) can be optimized for various semiconductors, by applying a reverse bias up to 20 V, is summarized in Table 3.  
In Figure 4a,b we report the values of the quantities of interest in this work--the \(R/NEP\) ratio and the optimized \(NEP\)--for all the examined semiconductors by considering a graphene circular area with radius of 500 \(\upmu\)m and a PD closed on a load resistance of 10 M\(\Omega\). We compute these optimized quantities through Equation (14) substituted into Equations (3) and (12). Recall that the results shown in Figure 4 are valid when the condition in Equation (8) is fulfilled. In order to verify it, we consider the dark current \(I_{d}\) one order of magnitude higher than \(\frac{2V_{\text{th}}}{R_{\text{L}}}\) (\(I_{d}=10\frac{2V_{\text{th}}}{R_{\text{L}}}\)), and we calculate both optimized \(R/NEP\) and \(NEP\) by Equation (12). The solid black lines drawn in Figure 4a,b represent  
Figure 3: (**a**) Intersection between the curve \(\Phi_{B}^{*}(\lambda)\) at 300 K and the values of SBH \(\Phi_{\text{B0}}\) at the interface between graphene and several semiconductors in conditions of thermal contact (no voltage applied to the junction); (**b**) reverse voltage \(V_{R}\) to apply to the graphene/semiconductor junction as function of the wavelength for maximizing the signal-to-noise ratio (\(SNR\)) (\(\Phi_{B}=\Phi_{B}^{*}\)) for various semiconductors. The values of \(\Phi_{\text{B0}}=\Phi_{\text{B}}(V_{R}=0)\) were calculated through Equations (15)–(18) by considering an initial graphene p-doping of \(n_{0}=10^{12}\) cm\({}^{-2}\) and a doping of \(N=10^{16}\) cm\({}^{-3}\) for all the semiconductors reported in Table 2. \begin{table}
\begin{tabular}{l l l} \hline \hline \(Semiconductor\) & \(\lambda_{\text{min}}\) (nm) & \(\lambda_{\text{max}}\) (nm) \\ \hline \(n-Si\) & 1541 & 2099 \\ \(p-GaAs\) & 1459 & 1582 \\ \(n-GaAs\) & 1692 & 2417 \\ \(p-Al_{0.3}Ga_{0.7}As\) & 1346 & 1447 \\ \(n-Al_{0.3}Ga_{0.7}As\) & 1197 & 1508 \\ \(n-Ge\) & 1852 & 2843 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Range of wavelengths in which the \(R/NEP\) ratio of the Schottky photodetectors (PDs) can be maximized by applying a reverse bias up to 20 V.  
the validity thresholds of our discussion: graphene Schottky PDs can be optimized in terms of \(R/NEP\) ratio at a given wavelength by means the use of semiconductors placed below and above the solid black lines drawn in Figure 4a,b, respectively. These thresholds depend on the load resistance \(R_{L}\), the SBH \(\Phi_{\text{B}}\), and the graphene active area \(A_{j}\), as is clearly shown by Equation (8). We discover that in the case analyzed here, only graphene/n-Si, graphene/n-Ge, and graphene/n-GaAs Schottky PDs can be suitable for this optimization procedure. Although Si is typically used for visible detection, analysis shows that graphene/n-Si Schottky PDs with a maximized \(R/NEP\) ratio could be adopted for detecting sub-bandgap NIR wavelengths with responsivity and \(NEP\) of 133 mA/W and 500 fW/\(\sqrt{\text{Hz}}\) at 1.55 \(\upmu\)m, respectively. These devices provide low NEP, enabling their employment for power monitoring and lab-on-chip applications. Note that the predicted responsivity of graphene/n-Si PDs is higher than that reported for NIR Si PDs based on bulk-defect-mediated absorption. Indeed, taking advantage of mid-gap defects introduced into Si ring and disk resonators, [PERSON] et al. reported a responsivity of only 23 mA/W at \(-\)5 V [37] and 45 mA/W at \(-\)3 V [38] at 1560 nm, respectively.