Influences of relative humidity on the quality factors of MEMS cantilever resonators in gas rarefaction

Abstract

In this paper, the effect of relative humidity of moist air is discussed on the quality factor (Q factor) of micro-electro-mechanical systems (MEMS) cantilever resonators in wide range of gas rarefaction (ambient pressure and accommodation coefficients, ACs). The modified molecular gas lubrication (MMGL) equation is used to model the squeeze film damping problem of MEMS cantilever resonators. Dynamic viscosity and Poiseuille flow rate are used to modify the MMGL equation to consider the coupled effects of relative humidity and gas rarefaction. Thermoelastic damping and anchor loss, which are dominant damping mechanisms of MEMS cantilever resonators, are also included to calculate total Q factor. Thus, the influences of relative humidity are discussed on the Q factors of MEMS cantilever resonators in wide range of gas rarefaction and dimension of cantilever. The results showed that the Q factor decreases as relative humidity increases in wide range of gas rarefaction (pressure, and ACs) and dimension of cantilever (length, width, and thickness). The influences of relative humidity on the Q factor become more significantly in larger length, larger width, smaller thickness of cantilever, and higher gas rarefaction (lower pressure and ACs). Whereas, the influences of relative humidity on the Q factor reduce or are neglected in smaller length, larger thickness of cantilever and lower gas rarefaction (higher pressure and ACs).

Appendices

Appendix 1

In this result, Q factor of TED (Q_TED) is calculated and plotted as function of length (L_b) and thickness (T_b) of cantilever, respectively. The material properties of silicon cantilever are implemented as the thermal conductivity $\kappa$ = 90 W/(m K), the specific heat capacity $C_{P}$ = 700 J/(kg K). In Fig. 11, the result shown that Q_TED increases as L_b increases. In Fig. 12, the result shown that Q_TED decreases down to a minimum value and then increases as T_b increases. Furthermore, the obtained results of Q_TED from the FEM model (COMSOL 2018) shown good agreement with those obtained results from Zener (1937, 1938) model and LR (Lifshitz and Roukes (2000)) model in wide range of length (L_b) and thickness (T_b) of cantilever. Then, the obtained results of Q_TED from Zener (1937, 1938) model are applied in the present analysis. Thus, it can be noted that smaller length and larger thickness of cantilever results in higher TED and then lower Q_TED produced. The obtained results of Q_TED can be used to calculate the total Q factor (Q_total) in wide range of length (L_b) and thickness (T_b) of cantilever.

Fig. 11

The Q factor of TED (Q_TED) plotted with length of cantilever (L_b) at first resonator mode

Fig. 12

The Q factor of TED (Q_TED) plotted with thickness of cantilever (T_b) at first resonator mode

Appendix 2

In this result, the Q factor of anchor loss ($Q_{anch}$), which is evaluated by Hao et al. (2003) model in Eq. (21), is calculated for various length (L_b) (Table 3) and thickness (T_b) (Table 4) of cantilever. The assumption for the correctness in Eq. (21) is accepted as the transverse elastic wavelength ($\lambda_{T}$) is much larger than the width ($W_{b}$) of the micro-plate, $\lambda_{T} /W_{b} \gg 1$ as shown in Tables 3 and 4. In Table 3, the results shown that $Q_{anch}$ increases L_b increases. Whereas, Q_anch decreases as T_b increases as showed in Table 4. Thus, it can be noted that smaller length and larger thickness of cantilever results in higher anchor loss and lower Q_anch. The obtained results of Q_anch can be used to calculate the total Q factor (Q_total) in wide range of length (L_b) and thickness (T_b) of cantilever.

Table 4 The Q factor of anchor loss ($Q_{anch}$) is calculated for different thickness of cantilever ($T_{b}$)