3.1. The Radiation Diffraction on the Structure
Figure 2 presents the calculation results of the irradiating light diffraction over the analyzed structure (AS). We consider a monochromatic plane electromagnetic wave, a normal incident on the AS in the negative direction of the
y axis. This wave with the complex vector of the electric field
E0 is linearly polarized along the
x axis. Here are the normalized distributions of the squared absolute value of the electric vector |
E|
2/|
E0|
2, where
E =
Es +
E0 is the complex vector of the total electric field, that is, the sum of the complex vectors of the scattered
Es and the given irradiating
E0 fields. These distributions, calculated at a wavelength of the illuminating beam of 500 nm and 700 nm, are presented in the fragments of
Figure 2a,b, respectively.
It can be seen that the field distributions have the form of a series of bands with the minimum and maximum values alternating in the vertical direction with the period close to λ/2. These distributions have a pronounced interference structure characteristic of the standing waves with the alternating nodes and antinodes. Thus, the field distribution is largely determined by the interference of the waves, which are incident and reflected from the blade and the gate. With a change in λ, the location of the interference fringes changes. As a result, the tip of the blade may be in the region of low or high field values. This leads to migration of the areas of increased field absorption in the area of the tip of the blade-emitter, which is analyzed in the next section.
3.2. Spectral and Spatial Dependences of Amplification of the Optical and Electrostatic Fields on the Blade of the Structure
The efficiency of the photosensor is largely determined by the absorption intensity of radiation by its cathode. It is desirable to maximize this intensity and, consequently, the number of photoexcited (hot) electrons [
21,
22,
23,
24]. The local intensity value of radiation absorption is determined by the following equation [
24]:
where ν is the radiation frequency and ε
″ is the imaginary part of the dielectric function of the absorbing material. The calculation of the local intensity distribution of radiation absorption is actually reduced to the calculation of the squared modulus distribution of the electric field vector |
E|
2 in the absorbing material. The results of the given calculation are presented in
Figure 3.
The calculation results of the two-dimensional distribution of the function |E|2/|E0|2 inside the nanoscale tip of the molybdenum blade and outside in the vicinity of the tip, that is, in the vacuum, are presented in this figure. We can see a high degree of field localization in molybdenum in the near-surface layer, with the depth of several nanometers. In addition, there is a heterogeneity of the near-surface field in the orthogonal direction along the perimeter of the forming blade. Moreover, the location of these field inhomogeneities varies depending on the wavelength of the incident radiation. A similar pattern is observed near the tip of the emitter in the vacuum. The maxima of the field change their position along the perimeter of the forming blade with a change in the wavelength of the incident radiation. Thus, the spectral dependence of the areas of increased field absorption location in the emitter blade is obvious, which should be taken into account when choosing the operation modes of the given type of photosensors with increased efficiency.
The qualitative proximity of the field distribution in
Figure 3a,c is explained by the fact that a node of a standing wave is located in the center of the end face of the blade. Field distribution in
Figure 3b differs significantly from the previous ones owing to the fact that the antinode of the standing wave is located approximately in the center of the end face of the blade. We emphasize that near the nanosized edge of the blade, there is a significant deformation of the field distribution. As a result of this, in particular, the period of following nodes and antinodes decreases.
Formula (2) allows to quantify the specific power level of the absorbed photons and identify the patterns of their dependence on the absorption depth based on the analysis of results of spectral distribution estimation of the function presented in
Figure 4. The calculation is carried out in the vicinity of point 2 (the designation is entered in
Figure 3), which is located at the interface of the emitter tip rounding with its end surface. The curves in
Figure 4 correspond to the specific absorption rate determined at different depths from the surface, starting from zero and ending at the depth of 20 nm. Additionally, we take into account that the data relating the specific absorbed power level are important in terms of estimating potential efficiency of the photosensor, as it is proportional to the concentration of hot electrons. Then, the top four curves in
Figure 4 give a quantitative characteristic of the absorbed photon power, which is spent on excitation of the electrons, which can transport to the surface of the emitter (the depth is less than the free path of the hot electron in molybdenum
Lτ = 10 nm).
An unexpected result referred not to the maxima or minima of the spectral absorption function, but to their significant difference in the magnitude (by 5–8 times depending on the wavelength of the radiation). The concentration of the hot electrons changes accordingly. It is also seen that the attenuation degree of the optical signal as the function of depth has spectral dependence. The observation point over the surface of the emitter tip was chosen arbitrarily during the computational experiment. However, an additional investigation demonstrated the existence of similar patterns in the other points across the emitter parameter.
In particular, this evidence is provided by the results of numerical simulation of the function |
E|
2/|
E0|
2 built along the perimeter of the blade generator inside molybdenum (
Figure 5a), and within the vacuum (
Figure 5b) when exposed to light with different wavelengths. Here,
Figure 5c,d show distribution of the modulus of electrostatic field intensity when applied to the gate of the potential U
g = 1 V (along the perimeter of the blade generator outside the emitter and a two-dimensional distribution near the tip of the emitter, respectively). The electrostatic field is also characterized by a certain degree of localization. However, the actual localization zone does not change the position compared with the spectral dependence of the position of optical field intensity extrema.
It is obvious that efficiency of the analyzed structure of the photosensor will improve if an additional adjustment element is introduced that ensures implementation of the matching principle to the localization zones of the absorbed optical and electrostatic fields.
3.3. A Model for the Tunnel Photocurrent within the Structure
In contrast to traditional metal photosensors with a flat radiation receiver, the surface of the sensitive element being investigated has a two-dimensional structure, where the depth of topological irregularities can be compared with the radiation wavelength. Therefore, as a result of the direct and reflected wave interference, we find a complex pattern of standing waves, particularly near the tip of the emitter blade (see
Figure 2 and
Figure 3). The calculations also showed that, when the emitter is irradiated by electromagnetic radiation of the visible or near IR range, the penetration depth of electromagnetic waves into the molybdenum blade (the skin depth) can be about 30 nm (distribution of the normalized specific absorption power at the depths up to 20 nm is shown in
Figure 4). In this case, the diffraction effects of the fields in the vacuum form significant spatial inhomogeneity of the optical field and inside the emitter along the blade perimeter, as shown in
Figure 4. This leads to substantial inhomogeneity of the specific absorption power not only along the depth of the skin layer, but also along the perimeter of the tip of the blade.
In the external strong electrostatic field, the height of the potential barrier over the metal surface decreases by
where
e is the absolute value of electron charge.
Then, the tunneling probability of the hot electrons from the emitter surface when exposed to optical radiation in both the visible and near-IR ranges increases exponentially. This happens despite the low photon energy, which is noticeably less than the work function
φ0. The results provided in the recent works [
5,
12,
13,
25,
26] demonstrate the experimental observation of the tunnel photoemission.
As follows from the analysis of strength distribution in the electrostatic field presented in
Figure 5c,d, the localization zone in the emitter is limited to the part of the segment designated by the points 3 and 4. According to (3), it is only in this zone that the Schottky barrier decreases to the maximum, whereas the permeability increases, respectively, and conditions can arise for the tunneling of photoelectrons into the vacuum.
A generalized scheme and mechanisms for sequential irradiation of the tip of the blade, generation of hot electrons at the volume of the molybdenum emitter skin layer, the transport of hot electrons from the layer with
Lτ thickness to the surface of the emitter, and their tunneling into the vacuum in the localization zone of the electrostatic field are given in
Figure 6.
The probability of tunneling of nonequilibrium photoelectrons generated in the surface layer of molybdenum with the thickness
Lτ with the energy
EF +
hν and the momentum directed normally to the surface of the emitter is determined by the following relationship [
13,
21]:
Here, hν is the energy of irradiating photons, y ≡ (e3F)1/2/(φ − hν) is the relative decrease in the potential barrier height for nonequilibrium photoelectrons, is the Nordheim elliptic function, the range of the argument variation is , m is the effective mass electron, and F is intensity of the electrostatic field.
Accounting of the temperature effects carried out in the work of [
12] allowed us to obtain the relations needed to determine the tunneling probability of hot electrons that reach the surface of the emitter
Pλ, and the tunneling photocurrent density
IPh from the metal emitter depending on the electrostatic field
F and the monochromatic light intensity
Iλ.
where
βT = 1/(
kT);
G is the Fowler function;
Rλ = |(1 −
nr)/(1 +
nr)|
2, where (
nr)
2 = ε
r (the relative permittivity of the metal);
fλ is the coefficient taking into account the ratio of the free path
Lτ of hot electrons to the absorption length of electromagnetic radiation at the wavelength
λ;
αλ is the absorption coefficient, and
L is the molybdenum blade thickness.
It follows from relations (4)–(6) that the photocurrent of hot electrons depends exponentially on the electrostatic field. This pattern is similar to the dependence of the field emission on the strength of the local field. At the same time, the photocurrent of the hot electrons increases linearly with an increase in intensity of the detected optical radiation.
A new design of the photosensor modernized by introducing an additional electrode from the opposite side of the gate will broaden the possibilities to influence the configuration and position of the electrostatic field localization zone (the second type). To make the influence of the potential Ua on the additional control electrode significant, it is necessary to ensure a relatively small gap between the electrode and the plane of the emitter. There is experience in manufacturing the structures of the given type [
27,
28]. To make it clear, let us set the gap at 5 nm when simulating a transformed electrostatic field.
As can be seen in
Figure 7a,b, application of a unit potential to an additional electrode leads to the appearance of the localization zone of the electrostatic field on the surface of the upper curve of the emitter blade and its end portion. However, the amplification level of the local field is noticeably lower than when the same unit potential is applied to the gate. This conclusion follows from comparing the data relating the field strength in
Figure 5c,d (on the fragment of the boundary surface between the points 3 and 4) on the lower rounding with relevant data relating the field strength in
Figure 7a,b on the upper rounding (between the points 1 and 2). Therefore, approximately equivalent enhancement of the field strength with control of both electrodes is achieved when the potential ratio U
a/U
g is equal to 4. The results of the calculation of the field strength distribution for the given potential ratio are shown in
Figure 7c,d, where both the expansion of the area of the localization zone and the increase in the maximum field strength are demonstrated simultaneously. The calculation examples provided in
Figure 6 and
Figure 8 demonstrate the possibilities for the changes within a wide range of topology and position of electrostatic localization zones, as well as the maximum field level within the zones.
3.5. Results of the Experimental Study and Their Interpretation
Spectral dependence of the photo-sensor response was recovered using the experimental data obtained in the accordance with the above subsection. The switch-on mode of the device corresponded to that shown in
Figure 5, when a single electrode, a gate, was used for controlling. Analysis of the measurement results in
Figure 8a, performed at various levels of the gate potential, shows that an increase in the electrostatic field strength leads to a noticeable increase in the photon emission with the wavelength ranging between 520–580 nm. A two-fold increase in the photocurrent is achieved with a slight increase (about 3%) in electrostatic field strength. Regarding the rest of the investigated range of the wavelength of irradiating laser, a growth in the photocurrent is also observed, though it is less evident. It is necessary to notice an increased irregularity of the spectral curves in
Figure 8a. The laser with a tunable working wavelength used in the experiment has a spectral dependence of the pulse energy, as shown in
Figure 8b, and a gap ranging within 700–800 nm. During the tests in the available range of tuning of the near-infrared radiation (from 800 to 1064 nm), 0.2–0.8 µA photocurrent of the vacuum sensor under laser pulse irradiation of 0.2–0.5 mJ was also detected.
In our view, the observed regularities of the far-field diffraction effect of the optical wave at the submicron scale stage (emitter—gate) on significant phase dispersion in the amplitude and direction of the Poiting vector over the molybdenum emitter surface (see simulation results in
Figure 5) provide an adequate interpretation of the experimentally observed processes. The position of localization zones of the specific optical signal absorption power, on the one hand, and unchanging position of the electrostatic field localization zone, on the other hand, which rapidly change with the frequency of the irradiating beam, lead to their asynchronous behavior. A lack of coordination of the two types of localization zones (optical absorption and electrostatic external field) relating the mutual position results in the decrease of the photo-sensor response within a relatively narrow frequency range. With a further change in the radiation wavelength, there conditions are favorable for both the absorption and tunneling of hot electrons into the vacuum.
Under such conditions, an emerging possibility for an additional adjustment of the system by varying positions of electrostatic field localization zones and changing the potential of control electrodes, as shown in
Figure 7, ensures the prospect for improving the vacuum photosensor parameters.
The illustrated possibility that ensures the performance of the vacuum photosensor in a wide wavelength range (ranging from the visible up to near infrared radiation) is explained by a single photon effect of the tunnel photoemission process. To prove the performance reliability, we provide the measurements of the photocurrent as the function of the laser pulse energy. Linearity of the dependence designed in
Figure 9 indicates the absence of nonlinear processes characteristic of the multiphoton photoemission. This allows us to predict a possibility to further upgrade the efficiency of the tunnel photoemission.
Note the structure of a 1 µm cell period with the localization of the electrostatic field [
5], the measured photo current rates for the fixed bias voltages showed a significant non-linear increase with an increase in the optical power. In our view, the reason for such photo current behavior could be extremely high levels of the form-factor and small inter-tip distances. As a result of intensified near-field effects (localization of the optical field on the nanoscale tips), redistribution of the concentration of hot electrons can lead to the nonlinear effects of electron–electron and electron–phonon interactions. Linearity of the photo current-optical power characteristics of the proposed and investigated photosensor is estimated as preferable for the majority of optical-electronic applications.
Known devices based on arrays of gold plasmon nanosized elements can be mentioned as similar photosensors: in the form of “mushrooms” [
6] and in the form of a triangular unit-cell on adjacent rows [
5]. In both cases, the effect of local surface plasmon resonance of gold nanostructures is used. On the basis of a theoretical analysis, the above studies show that the electric field enhancement can reach 100–120 only at certain resonant frequencies. The experimental data on the study of the photoeffect in a vacuum diode are presented for two wavelengths of 633 and 785 nm in the work of [
6] and for one value
λ = 785 nm in the work of [
5]. A feature of the nanostructure [
5] is a very small vacuum gap between adjacent opposite points, which is 50 nm. The photoeffect is observed while ensuring the electric field in the gap at a level of up to 1 GV/m. When the field was varied, a photocurrent of 0.3–8 μA was detected upon irradiation with a laser beam with an intensity of 0.2 kW/cm
2. The analysis of the results of the study of the photosensor in this article allows us to determine the intensity of the working field. The photoeffect is detected at a field of 0.22–0.24 GV/m. The magnitude of the photocurrent in this case varies in the range 0.3–5.8 μA (see
Figure 8a), which approximately corresponds to the level of the photoresponse of the analog [
5].
Along with the broadband of the studied photosensor, it is necessary to note the following additional advantages in comparison with analogues:
- (i)
The simplicity and low cost of manufacturing technology (traditional photolithography compared with e-beam lithography, molybdenum instead of gold).
- (ii)
The “open” structure of the photosensor electrodes allows the formation of a directed electron flow from the tip of the blade into the free space. This is important for creating current sources for vacuum devices (X-ray tubes, UHF (Ultra high frequency) and THz (Terahertz) generators, and amplifiers) with ultrafast optical signal modulation. In known analogues with nanoscale vacuum gap, the possibility of forming an electron flow with a controlled trajectory is associated with great difficulties.