## 2 Theoretical Background
The quadratic mean value of the fluctuations linked to both photocurrent \(I_{ph}\) and dark current \(I_{d}\) is the following:  
\[i_{s}^{2}(\Phi_{B})=2 qB(I_{d}(\Phi_{B})+I_{ph}(\Phi_{B})) \tag{5}\]  
where \(B\) is the device bandwidth. In addition to the shot noise, there is a thermal noise (Johnson noise) with quadratic mean value:  
\[i_{R}^{2}=\frac{4 TB}{R_{L}}, \tag{6}\]where \(R_{L}\) is the load resistance of the PD. Since the two contributions of the noise current are statistically independent, the total noise \(i_{n}\) is given by their squared sum:  
\[i_{n}=\sqrt{2 qB(I_{d}(\Phi_{B})+I_{ph}(\Phi_{B}))+\frac{4 kTB}{R_{L}}}. \tag{7}\]  
At low signal levels \(I_{ph}<<I_{d}\), the condition to make the thermal noise negligible compared to the shot noise in Equation (7) is:  
\[I_{d}>>2V_{\text{th}}/R_{L}, \tag{8}\]  
where the thermal voltage \(V_{\text{th}}=kT/q\). At room temperature, Equation (8) mainly depends on both SBH and \(R_{L}\). Of course, if the thermal noise dominates the shot noise, \(i_{n}\) does not depend on the SBH and the optimization procedure reported here can no longer be adopted. Compared to the absolute value of \(i_{n}\), its magnitude compared to the generated signal \(I_{ph}\), defined as the signal-to-noise ratio \(SNR=I_{ph}/i_{n}\), is even more important. In order to find the value of photogenerated current \(I_{ph}\) that brings \(SNR=1\), we can take advantage of the definition of the SNR and considering Equations (7) and (8), we obtain:  
\[SNR=\frac{I_{ph}}{\sqrt{2 qB(I_{d}(\Phi_{B})+I_{ph}(\Phi_{B}))}}=1. \tag{9}\]  
The square of the previous equation gives a quadratic form in the unknown \(I_{ph}\); by solving it we find:  
\[I_{ph}=qB\Bigg{(}1\pm\sqrt{1+\frac{2I_{d}}{qB}}\Bigg{)}. \tag{10}\]  
This expression makes it possible to obtain the minimum incident optical power \(P_{inc}\) necessary to get \(SNR=1\) for a PD characterized by a responsivity \(R\). Since the NEP is defined as the incident optical power \(P_{inc}\) necessary to get \(SNR=1\) divided by the square root of the bandwidth (\(NEP=P_{inc}/\sqrt{B}\)), we numerically obtain NEP by considering \(B=1\) Hz in Equation (10) and dividing it by the responsivity \(R\):  
\[\eta_{\rm int}^{\rm SLG}=\frac{1}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h \
u)^{2}}. \tag{2}\]0  
which reduces to the very well-known formula:  
\[\eta_{\rm int}^{\rm SLG}=\frac{1}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h \
u)^{2}}. \tag{2}\]1  
where \(2I_{d}/q\) is much larger than 1 in typical PDs. It is worth noting that in Equation (12) the sign of \(R\) follows the sign of \(I_{ph}\), as is clear when looking at Equation (3). Optimized PDs are characterized by high responsivity and low NEP. However, by looking at Equations (3) and (12) it is clear that by increasing the SBH, the NEP improves at the expense of the responsivity. On the other hand, an SBH decrease is beneficial in terms of responsivity but it degrades the NEP. Hence, we sought investigate the Schottky barrier \(\Phi_{B}\) that maximizes the \(R\) to \(NEP\) ratio. Toward this aim we introduce the function \(G(\Phi_{B})=\sqrt{\frac{R}{NEP}}\) using Equations (2)-(4), and (12):  
\[\eta_{\rm int}^{\rm SLG}=\frac{1}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h \
u)^{2}}. \tag{2}\]2where \(C=1/(2\sqrt[4]{2 qA_{j}A^{*}})\) depends on the geometry through the junction area \(A_{j}\) and on the semiconductor through the Richardson constant \(A^{*}\). Figure 1a displays the behavior of \(g(\Phi_{B})\) at 300 K for three different wavelengths, 1.3 \(\upmu\)m, 1.55 \(\upmu\)m, and 2 \(\upmu\)m, showing the presence of a peak. By calculating the first and second derivatives of \(G(\Phi_{B})\) we can find the value \(\Phi_{B}^{*}\) of SBH corresponding to this peak:  
\[\eta_{\rm int}^{\rm SLG}=\frac{1}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h \
u)^{2}}. \tag{2}\]3  
We define \(\Phi_{B}^{*}\) as the optimized SBH because it is the value at which the \(R\)-to-\(NEP\) ratio is maximized.