## 2 Theoretical Background  
IPE theory was first developed by [PERSON] in 1931, and it was focused on the injection of electrons from a metal into vacuum [25]. Several authors have extended [PERSON]'s theory to the emission of carriers into semiconductors, conceiving the modified [PERSON] theory [26; 27; 28] and providing the following expression for the internal quantum efficiency (IQE) \(\eta_{\rm int}\) of IPE-based PDs, defined as the number of charge carriers \(N_{e}\) produced per absorbed photons \(N_{\rm ass}\)[26]:\[\eta_{\rm int}=\frac{N_{e}}{N_{\rm ass}}=\frac{1}{8E_{F}}\cdot\frac{(h\
u-q\Phi_{B} )^{2}}{h\
u} \tag{1}\]  
where \(E_{F}\) represents the Fermi level, \(h\
u=hc/\lambda\) is the energy of the incident photon (\(\lambda\) is the wavelength and \(c\) the speed of light in a vacuum), \(q\) is the electron charge, and \(\Phi_{B}\) is the potential barrier at the interface between the metal and the semiconductor. This expression is derived by taking into account the ratio of charge carriers having kinetic energy _normal_ to the surface of the junction, necessary to overcome the potential barrier. This mechanism usually leads to poor efficiency (about 1%) [29; 30]; however, it has been demonstrated that two-dimensional materials replacing metals in the Schottky junctions provide an IQE enhancement [18]. In particular, in single-layer graphene (SLG)/semiconductor junctions a still higher ratio of photon conversion in charge carriers is observed. Regarding this, [PERSON] et al. [18] explain how this can be ascribed to the molecular structure of the graphene. Indeed, the \(\pi\) orbitals are normal to the interface with the semiconductor, and the charge carriers' momentum can be directed only towards the semiconductor or in the opposite direction, leading to an enhancement of the emission probability up to \(\frac{1}{2}\). When SLG is used as active medium in an IPE-based PD, Equation (1) can no longer be applied due to the linearity of the dispersion relation near the Dirac point [31], different density of states, and probability of emission. However, the IQE of Schottky PDs based on SLG has been derived as [18]:  
\[\eta_{\rm int}^{\rm SLG}=\frac{1}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h \
u)^{2}}. \tag{2}\]  
The responsivity \(R\) is related to \(\eta_{\rm int}^{\rm SLG}\) by the following relation:  
\[R=\frac{I_{ph}}{P_{inc}}=S\cdot\frac{1}{h\
u}\cdot\eta_{\rm int}^{\rm SLG}= \frac{S}{2}\cdot\frac{(h\
u)^{2}-(q\Phi_{B})^{2}}{(h\
u)^{3}} \tag{3}\]  
where \(I_{ph}\) is the photogenerated current, \(P_{inc}\) is the incident optical power, and \(S\) is the graphene optical absorbance. It is worth mentioning that in Equation (3) the charge carrier \(q\) is been considered in order to express the responsivity in A/W. Graphene has an optical absorption related to the universal fine-structure constant \(\alpha=e^{2}/(\pi e\theta hc)\)[32] and independent of the frequency, \(A_{G}=\pi\alpha\approx 2.3\%\). Here we focus our attention on devices that provide the complete absorption of the incident radiation such as long waveguides and resonant structures, thus we consider \(S=1\). As the Schottky barrier \(\Phi_{B}\) decreases, more electrons can pass into the semiconductor, giving rise to higher responsivities, as shown in Equation (2). Unfortunately, the dark current \(I_{d}\) of the junction also increases as \(\Phi_{B}\) diminishes due to thermal effects [33]:  
\[I_{d}=A_{j}A^{*}T^{2}\cdot e^{-\frac{\Phi_{B}}{kT}} \tag{4}\]  
where \(A_{j}\) is the area of the Schottky junction, \(A^{*}\) is the Richardson constant, \(T\) is the absolute temperature and \(k\) is the Boltzmann constant. Furthermore, there is a component of noise intrinsic to the photodetection mechanism: due to the quantized nature of the light, the current is constituted by a succession of random impulses, which cause fluctuations of the measured current (shot noise).