# Aluminum-Based Plasmonic Photodetector for Sensing Applications
## 3 Simulation of the Proposed Structure and Parameter Optimization
### Theoretical Framework for Parameter Optimization Problem  
Figure 4 illustrates the main geometrical features of the structure and depicts its main parameters, namely the grating period \(a\), the groove depth (etching depth) \(D\), the intermediated metallic layer depth \(d\), and the trapezoidal grating profile bases \(w_{1}\) and \(w_{2}\). The latter can be expressed through dimensionless parameters, such as the filling factor \(f=w_{2}/a\) and the base ratio \(w_{1}/w_{2}\). The optimal value for the groove depth was determined from a previous experimental investigation of SPP excitation in aluminum gratings with different modulation patterns [33]. These experiments revealed the optimal ratio of the groove depth to the grating period \(D/a\approx 0.086\). Therefore, the optimal groove depth is \(D\approx 0.086\)\(\times\)\(0.63\)\(\upmu\)m \(\approx 0.05\)\(\upmu\)m = \(50\) nm. The optimal values for the rest of the geometrical parameters were estimated using numerical simulations. In order to better estimate the photoresponse, the amount of electromagnetic energy absorbed in each layer was estimated separately. Such an approach allowed for separating the parasitic absorption in the metal and the useful absorption in the Si substrate. By knowing the electromagnetic energy distribution inside each layer, we can integrate over the spatial coordinates to find a partial absorption \(A_{j}(\omega)\) inside the j-th layer. \[A_{j}(\omega)=\frac{2\pi}{\lambda}\int_{0}^{a}\frac{\mathrm{dx}}{a}\int_{Z_{ \mathrm{j}}}^{Z_{\mathrm{j+1}}}\mathrm{d}z\,\epsilon^{\prime\prime}(r,\omega) \frac{\left|E(r)\right|^{2}}{\left|E_{0}\right|^{2}} \tag{4}\]  
where \(\epsilon^{\prime\prime}(r,\omega)\) is the local dielectric constant, \(E_{0}\) is the tangential component of the amplitude of an incident wave, \(E(r)\) is the local electric field, and \(r=(x,y,z)\) are the spatial coordinates. Figure 5 shows the calculated angular dependencies of the \"useful\" absorption in the Si substrate for the PPPDs with different values of the intermediate layer thickness (\(d\)) of the Al grating. A clearly pronounced plasmonic maximum is observed around \(\theta_{\mathrm{res}}=15^{\circ}\), corresponding to SPP excitation. The shape of the resonance curve for such structures is closer to a Fano shape, as shown in [20] for gold gratings with a similar groove shape. Figure 6a (Curve 1) shows the dependence of the maximum light absorption in the silicon substrate (max \(A(\theta)\)) on \(d\) for the simulated structure. A monotonic decrease in the maximum useful absorption was observed with an increase in the thickness of the intermediate Al layer, in agreement with the decrease in its transparency. However, besides maximizing absorption, another parameter had to be optimized to improve the sensitivity, namely the full width at half-maximum (FWHM), because narrower absorption peaks correspond to a steeper change in the detection signal caused by the resonance frequency shift. Figure 6a (Curve 2) shows the dependence of the FWHM on \(d\) for the same structure. It is evident that the increase in the thickness of the intermediate Al layer led to a decrease in the FWHM, which is a positive effect. Therefore, it was reasonable to select the goal function taking into account the FWHM; for example, we used the ratio of the maximal absorption to the FWHM, \(\max A(\theta)/\mathrm{FWHM}\) (Figure 6b), in this work. The optimal thickness \(d\) was found from the analysis of the goal function \(\max A(\theta)/\mathrm{FWHM}\). As can be seen in Figure 6b, the goal function has two maxima, namely at a minimum thickness of 5 nm and at around 14 nm. Thermally deposited ultra-thin Al films with an effective thickness of 5 nm practically lose their conductivity after oxidation in the air. At the same time, layers with a thickness of 14 nm are well-passivated and provide efficient excitation of SPPs. Therefore, for the considered structure, \(d=14\) nm is the optimal thickness of the intermediate Al layer.