## 2 Theoretical Background  
In order to quantify the performance of IPE-based PDs, a very important parameter is the internal quantum efficiency \(\eta_{int}\), i.e., the ratio of the number of charge carriers generated by the PD to the number of absorbed photons. On the other hand, the external quantum efficiency \(\eta_{ext}\) (the ratio of the number of charge carriers generated by the PD to the number of incident photons) is linked to the internal by \(\eta_{ext}=A\)\(\eta_{int}\), being A the active material absorption. A macroscopic magnitude is the responsivity \(R\), i.e., the ratio of the photogenerated current (\(I_{ph}\)) to the incident optical power (\(P_{inc}\)). The responsivity R is linked to the external quantum efficiency \(\eta_{ext}\) by the following:  
\[R=\frac{I_{ph}}{P_{inc}}=\frac{\lambda[nm]}{1242}\cdot\eta_{ext} \tag{1}\]  
In 1931, [PERSON] derived the first model of carrier photoemission from metal into vacuum [25]. In the 1960s, [PERSON], [PERSON] and [PERSON] [26] modified [PERSON]'s theory in order to extend the carrier photoemission into semiconductors. Subsequently, [PERSON] and [PERSON] reviewed the previous model and, under the zero temperature approximation, they obtained that the device internal quantum efficiency \(\eta^{3D}_{int}\) (number of carriers generated per absorbed photons) can be written as [20]:  
\[\eta^{3D}_{int}=\frac{1}{8q\phi_{B0}}\cdot\frac{\left(lnv-q\phi_{B0}\right)^{2 }}{lnv} \tag{2}\]  
where the factor \(1/8q\Phi_{B0}\)[35] is very often replaced by a generic factor C (named quantum efficiency coefficient) used to put in agreement the theory with the experimental data, \(lnv=1242/\lambda_{0}\) (nm) is the photon energy being \(\lambda_{0}\) the vacuum wavelength, \(\Phi_{B0}\) is the Schottky barrier height (SBH) under zero bias, and q is the electron charge. The 3D apex is used to indicate that the formula refers to junctions involving 3D materials, i.e., metals. Equation (2) was derived for metal-based junctions, i.e., on three-dimensional materials, however, it has been proved that it fails to correctly describe carrier photoemission involving Schottky junctions based on two-dimensional materials [27; 28], thus IPE theory must be properly modified in order to be applied to the Graphene/Si junctions whose band diagram is reported in Figure 2. It could be worth mentioning that the band structure of graphene is characterized by a valence and conduction bands touch each other at six-points in the first Brillouin zone. These particular points, where the Fermi energy is set to 0 eV, are named Dirac points. The band structure close to one of the Dirac points, responsible for the electronic properties of graphene, represents the conic surface shown in Figure 2[29]. By following [PERSON]'s approach [20] the number of excited holes \(N_{T}\) is:  
\[N_{T}=\int_{0}^{h\
u}D(E)\cdot dE \tag{3}\]  
where \(D(E)\) is the graphene density of states (DOS) that can be written as \(D(E)=\frac{2E}{\pi\hbar^{2}\
u_{F}^{2}}\), [30]. \(E\) is the hole energy referred to the Fermi level, \(\hbar\) is the reduced Planck constant, \(\
u_{F}\) is the Fermi velocity, and \(h\
u\) is the photon energy. On the other hand, the number of states \(N\) occupied by holes having a certain probability to be emitted into silicon is:  
\[N=\int_{q\phi_{\text{B0}}}^{h\
u}D(E)\cdot P(E)\cdot dE \tag{4}\]  
where \(P(E)\) is the carrier emission probability. In three-dimensional materials, \(P(E)\) can be expressed as \(P(E)=(1-\cos\mathfrak{I})/2\)[20], where \(\mathfrak{I}\) is the so defined carrier escape angle [20]. On the contrary, \(P(E)\) in graphene assumes a very simple value of \(\frac{1}{2}\) because the \(\pi\) orbitals are always normal to the graphene/Si interface, thus the photo-excited hole momenta can have two directions: one pointing towards Si and the other in the opposite direction [27].