*3.1. Thermoelastic Losses*

Thermoelastic damping (TED) is a loss mechanism due to the irreversible heat flow in vibrating structures. A temperature gradient occurs between regions under tension (where the temperature drops) and regions under compression (where the temperature rises).

We use an analytical model proposed by Lifshitz [26,27], where the thermoelastic quality factor is given by:

$$Q\_{thermo} = \frac{C\_p}{E a\_T^2 T} \left(\frac{6}{\xi^2} - \frac{6}{\xi^3} \left(\frac{\sinh(\frac{x}{\xi}) + \sin(\frac{x}{\xi})}{\cosh(\frac{x}{\xi}) + \cos(\frac{x}{\xi})}\right)\right)^{-1} \tag{10}$$

*ω*, *Cp*, *αT*, *T*, *E* are the pulsation, specific heat capacity, linear thermal expansion coefficient, temperature, Silicon Young's modulus, respectively. *ξ* = *h ωρbCp* <sup>2</sup>*<sup>K</sup>* represents a dimensionless number where *K* is the thermal conductivity. The values of all these parameters can be found in Table 3. The maximum of thermoelastic damping [26] occurs for *ξ* = 2.225. This value corresponds to a transition frequency *ft* = *<sup>π</sup>* <sup>2</sup> *<sup>K</sup> <sup>ρ</sup>bCph*<sup>2</sup> . For a cantilever frequency *fn* lower than the transition frequency *ft* (*fn* < *ft*), the beam is permanently in thermal equilibrium. In this case the vibration is called isothermal. On the other hand, when *fn* > *ft* the cantilever frequency is higher than the transition frequency, the beam does not have enough time to thermally equilibrate and this vibration is called adiabatic. In both cases, the energy dissipation is low. However, the *Qthermo* quality factor is higher in isothermal than in adiabatic regime [28]. In case of constant-frequency regime, one needs to calculate the thickness that gives the maximal damping. Based on the *ft* expression, the isothermal zone corresponds to the thin cantilever thickness and the adiabatic zone to the large thickness. For *fn* = 11 kHz, the maximal thermoelastic damping, i.e., the lowest *Qthermo* = 12, 500, corresponds to a cantilever with thickness *h* = 90 μm. Therefore, for frequency of 11 kHz, thermoelastic damping is not a limiting factor.


**Table 3.** Parameters used to describe the damping mechanism.

## *3.2. Acoustic Losses*

Acoustic losses refer to losses caused by a vibrating structure being a source of acoustic wave radiation. A good approximation of these losses can be expressed with an analytical model for cantilever with elliptical cross-section [29–31]. In this approach the quality factor related to acoustic losses is given by the following equation:

$$Q\_{acuistic} = \frac{256}{\pi} \frac{\rho\_b}{\rho\_f} \frac{1}{(k\_b b)^3} \frac{h \int\_0^L \phi\_n^2(\mathbf{x}) \, d\mathbf{x}}{\int\_{\varphi=0}^\pi \sin^3 \theta \left| \int\_0^L \phi\_n(\mathbf{x}) \exp(-ik\_b \mathbf{x} \cos(\varphi)) \, d\mathbf{x} \right|^2 d\varphi} \tag{11}$$

where *ρ<sup>f</sup>* is a fluid density, *ks* = *ω*/*cs* the acoustic wave number and *cs* is the speed of sound. Numerical calculations using Equation (11) show that the losses due to acoustic radiation become important when the cantilever length is comparable to the acoustic wavelength *λa*. It is less significant at low frequencies. Moreover, acoustic losses increase quickly as the width increases and the thickness decreases (for constant-frequency regime). For instance, for *b* = 5000 μm and *h* = 1 μm *Qacoustic* 605.
