2.1.

Two-Dimensional Simulation Conditions and Process

Though 3-D simulations are ideal, many 2-D simulations have been performed in the past due to constraints of computing power.^12–16 Sometimes, the applicability of the results was understood to be limited to the qualitative aspects. However, in this work, we analyze the behavior of a device whose orthogonal dimension is significantly larger than its lateral dimensions, so that a 2-D model is more than sufficient to simulate its behavior. This detector has a large extent in the $z$-direction, orthogonal to the plane of the simulation. Its aperture is thus a slit—subwavelength in the plane of the simulation, unconfined in the orthogonal direction. Accordingly, the impinging radiation is incident in the plane of the radiation; the Gaussian beams employed are 2-D Gaussian beam.

The simulation was performed with the COMSOL Multiphysics, which uses the FEM. The simulation couples the semiconductor and the wave optics modules using the optoelectronics interface.

The semiconductor module computes the electronic transport and charge distribution within the silicon using the drift-diffusion equations for the electron and hole densities in the semiconductor coupled to Poisson’s equation for the electric “band-bending” potential. These are augmented by statistical mechanics to describe the effect of doping on the carrier charge population. Light–matter interactions, described ahead, occur primarily within the Schottky junction, which develops between the semiconductor bulk and metal surface. The junction is a depletion layer, which develops in the semiconductor due to the difference in Fermi levels between the two materials. The work function of the metal, the electron affinity of the semiconductor, and the doping level are input parameters. In addition, the phenomenon of thermionic emission from the metal to the semiconductor is built as a boundary condition on the drift-diffusion current, which determines the boundary conditions on the charge distribution.

The material chosen for the interior of the device was silicon, n-doped to a donor concentration $N_d={10}^{17}{\mathrm{cm}}^{-3}$ using analytic doping. Trap-assisted recombination was activated, with default values. The sides of the device serve as the Schottky junction. This is realized using the metal contact boundary condition with the rectifying junction option selected, together with “thermionic emission.” The material chosen for the model was aluminum, which sets the work function, and the external bias was set to the appropriate voltage. Together these determine the boundary condition for the electric potential $\varphi$

The wave optics module simulates the propagation of the electromagnetic wave using Maxwell’s equations. On the entire outer surface of the device (excluding the ends), a perfect electric conduction boundary condition was used, which imposes a Dirichlet condition on the component of the electric field parallel to the surface. This affects a perfectly reflecting boundary, i.e., a perfect metal. This is a good approximation, as the entering electromagnetic wave decays entirely in the aluminum before penetrating through to the silicon in the interior. In order to simulate photodetection, “optical transitions” were activated in the interior. This adds a photogeneration source term to the convection–diffusion equations, which is proportional to the intensity of the electromagnetic field. The interaction between the radiation and the silicon—photodetection—involves indirect and direct interband transitions. The precise form of this term is determined by Fermi’s golden rule, together with an empirical model for indirect absorption in silicon, based on an empirical model of Green and Keevers.¹⁷

The incoming optical excitation is affected at the upper surface using a port boundary condition with the relevant value of the electromagnetic intensity. The lower surface is given an open boundary condition using the scattering boundary condition selection. Finally, with regards to the electric potential, the upper surface is given an insulating boundary condition, whereas the lower surface is defined as a metal contact boundary condition with the ohmic contact option.

2.2.

Polarization Aspects

This work reports on an incident transverse electric (TE) wave linearly polarized in the out-of-plane direction. This restriction from the longitudinal polarization is a consequence of the large extent of the device in this direction—absorption of this polarization is significantly stronger than of waves polarized in the direction transverse to the slit. This is a familiar result from the interaction of EM waves with wires and nanorods,¹⁸ where it serves as the basis for design of many polarizers.

2.3.

Metallic Boundaries and Surface Polariton-Plasmon Resonances

The exterior walls of the device are composed of a layer of aluminum (Fig. 1). In the upper section of the device, this layer serves as the anode of the Schottky junction. In the simulation, a perfect metal was assumed, i.e., a perfect electric conduction boundary condition was used, which imposes a Dirichlet condition on the component of the electric field parallel to the surface. Regarding the semiconductor component, the metal–semiconductor junction is modeled as a boundary condition. The metal component itself has no thickness and does not model internal bulk currents.

Fig. 1

COMSOL simulation of the NSOM photodetector structure. (a) The depletion layer boundaries are marked by red dashed lines for a given bias of $-0.5\mathrm{V}$ applied at the Schottky contact. The TE wave components are indicated in the inset. (b) COMSOL simulation with the mesh structure.

Regarding the possibility of surface polariton-plasmon (SPP) resonances at the insulator to metal interface, we can argue that only transverse magnetic (TM) electromagnetic waves can excite such resonances. Assuming that the metal boundaries are modeled as perfect conductors, this translates into the Dirichlet boundary condition on the electric field: $E_{\parallel }=0$. This in turn corresponds to zero impedance and vanishing skin depth, so that the electric field is entirely excluded from the interior of the metal. In this limit, the conductance is infinite so that the complex dielectric function is also infinite according the Drude equation ¹⁹

The SPP wave number¹⁸ is then reaching the total wavenumber $k$ according

3.1.

Two-Dimensional Scanning of Laser Beam Illumination

The NSOM detector is simulated in 2-D as a trapezoidal structure of $1.54\text{-}\mu \mathrm{m}$ height, $2\text{-}\mu \mathrm{m}$ bottom width, and 150-nm top aperture diameter. As shown in Figs. 2 and 3, the device—identifiably by its trapezoid silhouette—is shifted horizontally across the vertical projected Gaussian laser beam of 550-nm wavelength having a $0.69\text{-}\mu \mathrm{m}$ full width at half maximum (FWHM). The color scale indicates the beam intensity as the electric field value (V/m). In Fig. 2, the device is presented at its starting point, on the left side of the beam. The scanning proceeds when the device is shifting to the right direction. In Fig. 3, the device’s position approaches the center of the beam where it absorbs part of the intensity.

Fig. 2

Initial conditions of the scanning process, with the device located on the left side of the laser beam, incoming from above. The Gaussian laser beam has an FWHM of $1.1\mu \mathrm{m}$. The distance from the top center of the photodetector to the beam center is $2\mu \mathrm{m}$.

Fig. 3

Simulation result of the beam scanning, using a shifted trapezoid moving horizontally under the vertical illumination. The distance from the top center of the photodetector to the beam center is $0.5\mu \mathrm{m}$.

Our analysis consists in varying several parameters in order to optimize the photocurrent response. The metal contact is $1\text{-}\mu \mathrm{m}$ height from the top aperture (both sides of the trapezoid). The ground electrode is at the bottom of the structure. The first class of parameters we will consider is the device intrinsic specifications such as the silicon doping level and the metal work function. Then, we will consider the influence of external parameters such as the bias voltage, the radiant intensity, and the laser wavelength.

3.2.

Photocurrent Detection Resolution

As discussed above, the FWHM of the impinging beam is $0.69\mu \mathrm{m}$. In comparison, Fig. 4 shows that the FWHM of the signal detected by the device is $\sim 0.77\mu \mathrm{m}$. This reflects geometric averaging due to the finite width of the aperture—the FWHM is increased by an amount 1 to 2 times the width, which is 150 nm in this case. It is to be stressed that the increased FWHM of the signal is geometric and is less than that would be expected due to diffraction.

Fig. 4

Current as a function of the photodetector’s position ($\mu \mathrm{m}$), relative to the beam center, for several doping values. The work function is 4.72 eV (Al), $\lambda$ is 550 nm, the voltage bias is 0 V, and the radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

3.3.

Photocurrent Scan as a Function of Silicon Doping

Figure 4 shows the simulation of the current measured at the metal electrode by scanning the detector across the laser beam as described above. The varying parameter is the doping level of the n-type silicon. The simulation was performed under the following conditions: the metal work function is 4.72 eV (Al), wavelength $\lambda$ is 550 nm, the voltage bias is $-0.5\mathrm{V}$, and the electric field is ${10}^3\mathrm{V}/\mathrm{m}$ equivalents to a radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$. The doping is reduced here from ${10}^{18}$ to ${10}^{16}{\mathrm{cm}}^{-3}$. An enhancement of the simulated current is obtained.

3.4.

Photocurrent Scan as a Function of Metal Work Function

In order to enhance the applicability of the study, the simulations were performed for several contact metals. The considered elements are: aluminum (Al), tungsten (W), and gold (Au) with respective work function values of 4.72, 4.97, and 5.82 eV. The scanning of the laser beam in these conditions is presented in Fig. 5.

Fig. 5

Current as a function of the photodetector’s position ($\mu \mathrm{m}$), relative to the beam center, for several work function values. The simulation was performed under the following conditions: doping is ${10}^{17}{\mathrm{cm}}^{-3}$, $\lambda$ is 550 nm, voltage bias is $-0.5\mathrm{V}$, and radiant intensity in this case is $1\mathrm{W}/{\mathrm{m}}^2$.

3.5.

Photocurrent Scan as a Function of Bias Voltage

In Fig. 6, the bias voltage has been changed from 0 to $-5\mathrm{V}$ in steps of $-0.5\mathrm{V}$. The simulation was performed under the following conditions: silicon doping is ${10}^{17}{\mathrm{cm}}^{-3}$, metal work function is 4.72 eV (Al), $\lambda$ is 550 nm, and the radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

Fig. 6

Current as a function of the photodetector’s position ($\mu \mathrm{m}$), relative to the beam center, for several bias voltage values. Doping is ${10}^{17}{\mathrm{cm}}^{-3}$, work function is 4.72 eV (Al), $\lambda$ is 550 nm, and radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

We note here two contributions for the electric current, which are varying with the bias voltage: the minimum value attributed to the dark current $I_{\text{dark}}$ (taken for example at $d=-2\mu \mathrm{m}$), and the maximum value $I_{\max }$ corresponding to the laser beam center ($d=0\mu \mathrm{m}$). In order to extract the detector’s selectivity to the photocurrent, we define the detector’s contrast factor (CF) as

In Fig. 7, $I_{\text{dark}}$, $I_{\max }$, and the CF are plotted as function of the bias voltage. The less is the bias voltage the more is the CF. An explanation will be developed later in the discussion.

Fig. 7

$I_{\max }$, $I_{\text{dark}}$, and CF are all presented a function of the applied $V_{\text{bias}}$, based on Fig. 6.

3.6.

Photocurrent Scan as a Function of Radiant Intensity

In order to evaluate the detector responsivity of the current, it was important to evaluate the influence of the laser beam parameters namely the radiant intensity and wavelength. In Fig. 8, the photocurrent scan was simulated as a function of several orders of magnitude of the radiant intensity (0.15 to $15\mathrm{W}/{\mathrm{m}}^2$). Surprisingly, the “dark” current (located at $\pm 2\mu \mathrm{m}$) is also sensitive to the radiant intensity. The beam seems to be larger at higher power. It is conceivable that a larger beam illuminates the trapezoid from the sides below the metal contact, and this contributes to a higher “dark” current.

Fig. 8

Semilog plot of the photocurrent as a function of the photodetector’s position ($\mu \mathrm{m}$), relative to the beam center, for several radiant intensity values. The simulation was performed under the following conditions: silicon doping is ${10}^{17}{\mathrm{cm}}^{-3}$, work function is 4.72 eV (Al), $\lambda$ is 550 nm, and the voltage bias is $-0.5\mathrm{V}$.

As expected, the increase of the radiant intensity implies a linear increasing of the current. Also, the photocurrent response should be consistent with realistic laser intensities as encountered in AFM systems. Of course, additional factors should be taken in account, including different types of decay, absorption, and quantum efficiency.

3.7.

Photocurrent Scan as a Function of Wavelength

Now for a given radiant intensity of $0.15\mathrm{W}/{\mathrm{m}}^2$, the photocurrent response was studied for wavelengths ranging between 400 and 1100 nm, by steps of 100 nm. Several interesting characteristics are notable in Fig. 9. From 400 to 600 nm, the current values are seen to steadily increase with the wavelength. However, jumps of the current are observed at 800 and 1000 nm revealing an oscillating phenomenon. The study was limited to 1100 nm, corresponding to nominal range, in which the absorption of silicon is significant. A further analysis will be presented below in order to elucidate this point.

Fig. 9

Current as a function of the photodetector’s position ($\mu \mathrm{m}$), relative to the beam center, for several wavelength values. The simulation was performed under the following conditions: doping is ${10}^{17}{\mathrm{cm}}^{-3}$, work function is 4.72 (Al), the voltage bias is $-0.5\mathrm{V}$, and the radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

4.1.

Semiquantitative Model of the Photocurrent

Assuming the photocurrent is due primarily to the drift of the photogenerated carriers in the depleted layer and assuming a one-dimensional model for simplicity sake, the steady state photocurrent density $J_{\mathrm{ph}}$ (per unit area of the electrode area) is expected to increase with the depletion layer width $w$ like²⁰

With ${\epsilon }_s$ is the silicon relative permittivity ($=11.7$) and $N_D$ is the silicon doping (${10}^{17}{\mathrm{cm}}^{-3}$). $V_{\mathrm{bi}}$ is the built-in voltage of the Schottky diode defined as the difference between the Fermi levels of the metal and the silicon. However, the $w$ value is limited here by the detector aperture of 150 nm. Above this limit, the depletion areas from both sides of the metal contact are reaching each other and no further expansion is possible. This may explain why the CF of the photocurrent is decreasing while increasing $V_{\mathrm{rev}}$ above 1 V as shown in Fig. 7.

If the energy level reference is taken at the vacuum level, $V_{\mathrm{bi}}$ can be expressed as the difference between the silicon Fermi level $E_{F,n}$ and the metal Fermi level $E_{F,\text{metal}}=-{\mathrm{\varphi }}_m$ according to

The silicon Fermi level is given by

Thus we get

Since the dark current is zero at zero voltage (the thermal generation current is negligible), it turns out that the CF is maximum (100%) at zero voltage and decay to a constant value as shown in Fig. 7. This semiquantitative model is helpful to interpret the increase of the simulated current with the decreasing of the doping level as shown in Fig. 4. Moreover, according to Eqs. (3) and (5), the depletion width $w$ increases for metals with larger work function, with a square root behavior at low voltage bias. This indeed is borne out in Fig. 5, where it can be observed that the three elements show values of current, which increase with the value of their respective work function.

4.2.

Responsivity

As suggested from Fig. 8, one of the most important results expected from our simulation is a pure linear relation between the maximum value of the photocurrent ($d=0\mu \mathrm{m}$) and the irradiance, i.e., the radiant intensity per unit area of the detector surface. When developing electro-optical devices and in particular nanoscale-based sensors, such a linear relation is of considerable importance since it allows easy prediction of the current produced for any desired irradiance. In Fig. 10, the plot of this relation is clearly displayed.

Fig. 10

Photocurrent as a function of the irradiance. The simulation was performed under the following conditions: Silicon doping is ${10}^{17}{\mathrm{cm}}^{-3}$, work function is 4.72 eV (Al), $\lambda$ is 550 nm, and the voltage bias is $-0.5\mathrm{V}$.

In order to evaluate the responsivity of the photodetector, it was important to quantify the electrical field from the Gaussian laser beam, outside and inside the photodetector, through the trapezoid aperture. The responsivity is the ratio between the current and the irradiance. We need to estimate it from the slope of Fig. 10 and get around $0.66\mathrm{pA}/\mathrm{W}/{\mathrm{m}}^2$.

Figure 11 shows the vertical electrical field on top of the upper surface of the trapezoid. Figure 12 shows its simulated values of the electric field all along the vertical line centered at $d=0\mu \mathrm{m}$ (no lateral shift). The horizontal red line represents the height ($h=-0.5\mu \mathrm{m}$), in which the trapezoid’s top aperture stands and defined as an insulating boundary condition. Between the field’s upper value of $1480\mathrm{V}/\mathrm{m}$ (just above the aperture) and the field’s lower value of $600\mathrm{V}/\mathrm{m}$ (just under the aperture), there is a ratio of 40%, which is a significant value. A clear distinction appears between the propagating field and the evanescent field at the detector aperture (150-nm diameter).

Fig. 11

Electrical field outside and inside the photodetector. The border at the aperture’s height is represented by the horizontal red line. The vertical line is the path, in which the field values have been measured in Fig. 12.

Fig. 12

Laser beam electrical field outside and inside the photodetector. The border at the aperture’s height is represented by the horizontal red line. Aperture diameter is 150 nm and the radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

4.3.

Spectral Response

The last interesting check was to evaluate the behavior of the photocurrent as a function of the wavelength, this time for several aperture diameters ($A=150$, 300, 500 nm). As observed in Fig. 13, the smaller the aperture is, the lower the current is. Also, there are two distinct areas: the left one, till $\lambda =600\mathrm{nm}$, where the current keeps the same shape of oscillations, and decreasing with smaller aperture diameters. However, the right area shows some interference pattern but the current is less sensitive to the aperture diameter. The attenuation of the current at very short values of $\lambda$ ($<450\mathrm{nm}$) is due to strong absorption of light at silicon surface. We can see that, in addition to this diffraction phenomenon, there is also an interference additional phenomenon, shown in the right side of this figure; these oscillations may be due to the fact that part of the electrical field is reflected by the detector, and interfering with the original laser beam as simulated in Fig. 11.

Fig. 13

Current as a function of the wavelength for several top aperture diameters. The simulation was performed under the following conditions: silicon doping is ${10}^{17}{\mathrm{cm}}^{-3}$, work function is 4.72 eV (Al), the voltage bias is $-0.5\mathrm{V}$, and the radiant intensity is $0.15\mathrm{W}/{\mathrm{m}}^2$.

4.4.

Optimization and Recommended Setup Values

Series of above simulations and analyses enable to forecast and optimize the behavior and the accuracy of the NSOM photodetector for two different series of parameters: the device’s specifications (doping, work function, etc.) and the setup parameters (external electrical field, wavelength, etc.). This is why it was important to summarize all the recommended values in Table 1. Summarizing above analyses, it appears that the highest measured current was obtained for a doping of ${10}^{16}{\mathrm{cm}}^{-3}$ (Fig. 4), for a work function of gold (Fig. 5) and a wavelength above 600 nm (Fig. 13). Regarding the work function’s preference, if the gold enables a higher current, the aluminum is more suitable to work with silicon and is usually preferable from an evaporation process point of view since less contaminating. This is why we kept the choice of aluminum in Table 1. A low reverse bias ($-0.5\mathrm{V}$) is recommended to get a high CF of the photocurrent (Fig. 7).

Table 1

Recommended parameters of the photodiode and of the setup.