*3.5. Results and Discussion*

Despite the lack of a general trend for the quality factor optimization in terms of all losses, it is possible to find the optimal value of the quality factor in terms of geometry for a given frequency. This optimum is presented in Figure 6. It takes into account all damping mechanisms presented in the previous subsections. The values of parameters used in this numerical simulation are presented in Table 3. Simulations have been realized at 11 kHz where the acoustic force is maximal, and for comparison at 60 kHz. An optimum has been found for laser modulation frequency at 11 kHz. For other physical mechanisms like damping mechanism, other optimums in modulation frequency can be expected. Simulation at higher frequency illustrates the evolution of these physical parameters with the frequency. As it will be seen in Section 6, although the different frequency dependency of physical mechanisms, the optimum frequency for the gas sensor is the same as the one for the acoustic force, here 11 kHz. A complete frequency study is presented in Appendix A.4. For the present simulations, a gap value of *d* = 10 μm has been chosen as a good compromise between fabrication constraint and sensor performances. The extensive study of the gap is presented in Section 7. Simulations show that the maximum value is slightly larger at high frequencies, and the most significant effect is the shift of the optimum to the lowest thickness when the frequency increases. As we will detail below, this effect is due to the losses of the mechanical supports.

In the figure, we identified the areas which correspond to the main limiting mechanisms. As it was shown *Qsupport* ∝ √ <sup>1</sup> *ω*3 *nh*<sup>3</sup> , therefore the limitation for the quality factor with high thickness originates from the damping of the support. The term in *ω*<sup>3</sup> in this equation explains the shift of the optimum to the lowest thickness, when the frequency increases.

The effect of the squeeze film damping appears for the largest width when the gas is trapped under the cantilever. For the smallest width, where the inertial forces are smaller than the viscous forces, the total quality factor is limited by the viscous damping. This area corresponds to the lowest Reynolds number.

In this model, neither thermoelastic nor acoustic damping are limiting factors. For all geometries and frequencies, the two associated quality factors are at least one order of

magnitude higher than the other damping mechanisms. For more detail on the individual limits of each quality factor, the reader can refer to Appendix A.2, Figures A2 and A3.

**Figure 6.** Total quality factor as a function of width and thickness for a cantilever of fundamental resonance frequencies equal to 11 kHz (**a**) and 60 kHz (**b**), for a gap between support and cantilever equal to *d* = 10 μm.
