## 2 Materials and Methods
### The Free Path Length of Hot Electrons in Molybdenum  
When a conduction electron interacts with a photon, it transits to a non-equilibrium (hot) state. Its energy can be estimated as \(E_{F}+h\
u\pm kT\), where \(E_{F}\) is the Fermi energy, \(h\
u\) is the photon energy, \(k\) is the Boltzmann constant, and \(T\) is the temperature. Typically, the characteristic time \(\tau\) of non-equilibrium electron relaxation in metals is between 1 fs and 10 fs. In particular, the relaxation time for molybdenum is equal to 6 fs [15].  
With the known characteristic time, the mean free path of hot electrons can be estimated as \(L_{\tau}=\tau\cdot\upsilon\)[16]. Here, \(\upsilon\) is the maximal electron velocity, which is defined as \(\upsilon=\{2(E_{F}+h\
u)/\mathrm{m}\}^{1/2}\) (m is the mass of the electron). Rough estimations of \(\upsilon\) for molybdenum give the value with the order of \(\upsilon=1.6\times 10^{6}\) m/s. Accordingly, the mean free path of hot electrons in molybdenum is approximately equal to 10 nm.  
The parameter \(L_{\tau}\) is directly related to the maximal depth of the subsurface layer characterized by the non-zero probability of hot electron tunneling into the vacuum in the case when the electron  
Figure 1: A photosensor (**a**) in the vacuum housing with a planar glass molybdenum blade structure; (**b**) the top view of the parallel blade system; (**c**) the computational domain (with a half-period of 16.5 \(\upmu\)m) for constructing a finite element model of a section of a blade emitter—(1) molybdenum emitter, with the thickness of 300 nm and potential of 0 V; (2) dielectric layer of SiO\({}_{2}\), with the thickness of 700 nm; (3) molybdenum gate with the potential Ug; (4) vacuum.  
momentum is directed perpendicularly to the emitter surface. That is why the mean free path of hot electrons is a fundamental parameter controlling the quantum yield of the sensor with the tunneling photoemission.  
The classical photoelectric effect in the metals is observed if the resulting energy \(E_{F}+h\
u\) of the nonequilibrium photon-interacting electron that was initially located in the conduction band near the Fermi level is larger than the work function \(\varphi_{0}\)[17]. In this case, the probability for a hot electron to escape from the metal is equal to the unity. Correspondingly, a long-wavelength cutoff for photoemission is defined as  
\[\lambda_{r}(\text{nm})=1240/\varphi_{0}\ (\text{eV}). \tag{1}\]  
The features of the photoelectric effect, taking into account the effect of the temperature T and irradiation with the wavelength near the red border in the ultraviolet area at \(h\
u_{r}+kT\leq\varphi_{0}\), were considered in terms of the quantum mechanics in [PERSON]'s work [18].  
For molybdenum, \(\lambda_{r}\) = 288 nm, that is, it belongs to the ultraviolet (UV) range. Therefore, to ensure photosensitivity of the molybdenum emitter during irradiation in the visible and NIR ranges is a nontrivial task. Its solution is connected with the use of special impact methods in order to deform the potential Schottky barrier on the emitter surface and create conditions for realization of the tunnel photoemission mode.