## 3 Theoretical Results and Discussion  
In this section we theoretically derive the SBH dependence on the bias applied to the junction in order to show how the graphene Schottky PDs based on different semiconductors could be optimized. It is well-known that the SBH \(\Phi_{B}\) of Schottky PDs can be determined by the two following equations (i.e., the Schottky-Mott relations) [33]:  
\[q\Phi_{B}^{(n)}(V_{R}) = q\Phi_{\rm gr}^{0}-\Delta E_{\rm F}(V_{R})-q\chi_{\rm sm}\ \ \ \ \ \ \ \ \ \  
In Table 2 we report the bandgap energy and the electron affinity for various semiconductors and the SBH at zero bias \(\Phi_{\text{B0}}\), calculated through Equation (15) or (16) when \(V_{R}=0\). The values of \(\Phi_{\text{B0}}\) were evaluated by considering a graphene work function \(\Phi_{\text{gr}}^{0}=4.6\) eV [35; 36], a built-in potential \(V_{\text{bi}}=0.6\) V, an initial SLG extrinsic p-doping \(n_{0}=10^{12}\) cm\({}^{-2}\), and a low doping of the semiconductors \(N=10^{16}\) cm\({}^{-3}\). Figure 3a shows the intersections between these values of SBH \(\Phi_{\text{B0}}\) for different semiconductors and the curve of the optimized \(\Phi_{B}^{*}(\lambda)\) at room temperature (given by Equation (14)), suggesting the working wavelength to achieve the highest \(R/NEP\) ratio for each material. In the range of wavelengths where \(\Phi_{\text{B0}}>\Phi_{\text{B}}^{*}(\lambda)\), the SBH can be lowered down to its optimal value as in Equation (14) by simply applying a specific reverse bias \(V_{R}\) to the junction. \begin{table}
\begin{tabular}{l l l l l} \hline \hline \(\mathbf{Semiconductor}\) & \(\mathbf{E_{g}}\) (eV) & \(\mathbf{\chi_{\text{sm}}}\) (eV) & \(\mathbf{\Phi_{\text{B0}}^{(n)}}\) (eV) & \(\mathbf{\Phi_{\text{B0}}^{(p)}}\) (eV) \\ \hline \(Si\) & 1.12 & 4.00 & 0.73 & 0.39 \\ \(GaAs\) & 1.43 & 4.07 & 0.66 & 0.77 \\ \(Al_{0.3}Ga_{0.7}As\) & 1.77 & 3.77 & 0.96 & 0.84 \\ \(Ge\) & 0.66 & 4.13 & 0.60 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bandgap \(E_{g}\) and electron affinity \(\chi_{\text{sm}}\) of various semiconductors together with values of SBH when the Schottky junction is formed, calculated thanks to Equations (15)–(17) by taking into account an initial extrinsic p-doping \(n_{0}=10^{12}\) cm\({}^{-2}\) of the single-layer graphene (SLG) and the thermal equilibrium contact with the substrate. For the calculations we considered low-doped semiconductors (i.e., \(N=10^{16}\) cm\({}^{-3}\)). Figure 2: Band diagrams of (**a**) graphene/n-semiconductor and (**b**) graphene/p-semiconductor junctions at the thermal equilibrium and when a reverse bias \(V_{R}\) is applied. At the thermal equilibrium, graphene has an initial carrier density \(n(V_{R}=0)\). After a reverse bias this charge density becomes \(n(V_{R})\). \(E_{0}\) represents the vacuum energy level while \(E_{\text{F}}^{0}\) is the Dirac point. \(\Phi_{\text{gr}}^{0},\chi_{\text{sm}}\), \(E_{g}\), \(E_{C}\), and \(E_{V}\) are respectively the intrinsic graphene work function, electron affinity, conduction band, bandgap, and valence band. \(E_{F}^{\text{sm}}\) is the Fermi energy level in the semiconductor and \(q\Phi_{\text{B0}}\) the Schottky barrier at zero bias. The values of the Schottky barrier \(q\Phi_{\text{B}}\) depend on the graphene Fermi energy level \(E_{F}\) that shifts when a voltage is applied. By inverting Equation (18) and using Equation (17) and Equation (15) or (16) it is possible to calculate, for each wavelength and for each semiconductor, the values of the reverse voltage \(V_{R}\) such that \(\Phi_{B}=\Phi_{B}^{*}\). We report this plot in Figure 3b by considering a maximum reverse bias of 20 V. It is interesting to observe that within this limit, graphene Schottky PDs based on p-Al\({}_{0.3}\)Ga\({}_{0.7}\)As and p-GaAs can be optimized only in a narrow window of the NIR spectrum, whereas n-Si can be optimized in a broader range, including at 1.55 \(\upmu\)m where only a small reverse voltage \(V_{R}=0.66\) V for maximizing the \(R/NEP\) ratio is required.