*Discussion about Signal-to-Noise Ratio*

The optimum for the SNR presented in Figure 10 does not match with the highest output signal amplitude presented in Figure 9. The highest output signal corresponds to the highest mechanical displacement *wn* (Equation (25)). The Brownian noise can

be considered as a force acting on the cantilever. The optimal way to improve SNR is maximization of the photoacoustic force which comes down to increasing the surface area for photoacoustic pressure collection. Therefore, the optimum SNR is shifted to greater widths and thicknesses, where the acoustic force is greater (Figure 4), and the collected energy increases. As in the previous section, the increase of the surface collecting the photoacoustic energy will be limited to the larger widths by the squeeze film damping, and by the acceleration term for the larger thicknesses.

**Figure 10.** Signal-to noise ratio for a cantilever of fundamental resonance frequency equal to 11 kHz (**a**) and 60 kHz (**b**), a gap between support and cantilever equal to *d* = 10 μm.

The second significant parameter in Equation (27) is the pulsation *ωn*. Unlike the previous result, *ω<sup>n</sup>* reduces the difference between low and high frequencies. This is due to the fact that SNR is inversely proportional to the square root of frequency *SNR* ∝ <sup>√</sup><sup>1</sup> *ωn* , while the amplitude of displacement *wn*(*ωn*) is inversely proportional to the frequency square power *wn*(*ωn*) ∝ <sup>1</sup> *ω*2 *n* . For instance, the ratio between maximal value of SNR at 11 kHz and 60 kHz *max*(*SNR*(11*kHz*)) *max*(*SNR*(60*kHz*)) <sup>1</sup> 22 *max*(*wn*(11*kHz*)) *max*(*wn*(60*kHz*)) is around 22 times lower than the ratio of displacement. This indicates that for SNR the frequency term is less significant than in terms of displacement (voltage output).

#### **7. Study of the Gap Effect**

This section focuses on the effect of the distance between electrodes *d* (Figure 1) on the general sensor performance for a constant resonance frequency of 11 kHz. The analytic solutions previously presented (Equation (27)) were implemented in a Python programming environment to estimate an optimal value of signal-to-noise ratio for each value of the gap *d*. Subsequently, for each optimal SNR value, we get the geometrical parameters of the cantilever (width, length, thickness, Figure 11a) and their corresponding output voltages (Figure 11b).

When *d* increases, the signal-to-noise ratio increases as the displacement increases due to the decrease of squeeze film damping. At the same time, the signal output voltage decreases due to the increase of the distance between electrodes. This improvement on the SNR can also be explained by the optimized width, which increases with the gap *d* and which allows more energy collection. This increase in width is made possible by the decrease in the squeeze film damping for large gap *d*. Above *d* 200 μm the acoustic damping becomes dominant and some saturation appears on the width curve. The variations on the thickness curve are less important than on the width. The length curve follows the thickness rise to satisfy the constant frequency condition. As for the curve of the width, we can identify on the curves for length and thickness two different regimes that are probably due to the transition where acoustic damping becomes more important than squeeze film damping.

By taking into account the fabrication process issues, a cantilever with a gap *d* = 10 μm can be realized on a silicon-on-oxide (SOI) wafer. In this case, the SNR ratio will reach 150 and the amplitude of the output signal should reach 0.9 μV. Depending on the possibilities of the fabrication process, size of the final design, and required performance of the device (SNR), Figure 11 can be used as a reference to create a cantilever for optimal photoacoustic gas detection with capacitive transduction mechanisms.

**Figure 11.** (**a**) Geometrical values giving the highest signal-to-noise ratio as a function of the gap *d* between support and cantilever. (**b**) Highest signal-to-noise ratio and its corresponding output amplitude signal as a function of the gap *d* between support and cantilever. For these simulations, the cantilever fundamental resonance frequency is equal to 11 kHz.
