## 3 Transduction Principles of MEMS AE Sensors
### Piezoelectric MEMS AE Sensors  
Since the development of thin piezoelectric film deposition on a substrate, piezoelectric MEMS sensors have been designed for sensing and actuating [32]. The thin-film piezoelectric layers have size constraints, stress from substrate, epitaxial growth affecting dependence on crystal orientation and substrate preparation to be compatible with piezoelectric thin-film [33]. The most common piezoelectric MEMS are resonating devices made of a piezoelectric layer deposited on a membrane operating at its resonance frequency. The piezoelectric AE sensor operates in the direct effect, which means that a mechanical strain or stress causes an electrical displacement. The coupled mechanical and electrical domains of bulk piezoelectric materials apply to thin piezoelectric films as  
\[S_{j}=s_{ij}^{E}T_{j}+d_{ij}E_{i} \tag{11}\]  
\[D_{i}=d_{ij}T_{j}+\varepsilon_{ij}^{T}E_{i} \tag{12}\]  
where \(S_{j}\) and \(T_{j}\) are strain and stress tensors, \(E_{i}\) and \(D_{i}\) are electric field and displacement, \(s_{ij}\) represents elastic compliance and \(d_{ij}\) are piezoelectric coefficients. For thin-film piezoelectric MEMS devices, the most common coefficients are \(d_{33}\) and \(d_{31}\), which are related to the electric field in 3 direction to strain, either in the same direction (3) or the orthogonal direction (1 or 2 due to symmetry) [34; 35]. The directions 1, 2 and 3 are shown in Figure 5. The most important materials and geometric properties for sensing are piezoelectric coefficients, electromechanical coupling coefficient, mechanical quality factor, resonant frequency and residual stress.  
* Piezoelectric coefficient and doping direction depend on ferroelectric (e.g., PZT) or nonferroelectric (e.g., AlN) materials. Ferroelectric materials are doped after deposition, while the doping orientation of nonferroelectric materials is controlled during the deposition [34]. Piezoelectric coefficients of common piezoelectric thin films can be found in [35].
* Electromechanical coupling factor is the ratio of the mechanical energy converted to the input electrical energy of the piezoelectric material. It can be calculated by motional capacitance and passive capacitance.
* Quality factor measures the mechanical damping of a microstructure where a high quality factor is needed for MEMS resonators [35]. Piezoelectric design has a higher quality factor than capacitive  
Figure 5: Three configurations to tune the resonant frequency, (**a**) diaphragm anchored around circumference to tune at MHz range, (**b**) diaphragm connected with flexible beams to tune the 100–500 kHz range, (**c**) silicon mass connected with flexible beams to tune less than 100 kHz. The color scale indicates the total displacement in micrometers.  
design, caused by squeeze film damping. The quality factor can be extracted from the impedance curve and half-power bandwidth formula as \[Q\approx\frac{f_{res}}{\Delta f}\] (13) where \(f_{res}\) is the resonance frequency and \(\Delta f\) is the difference of two frequencies with \(\frac{1}{\sqrt{2}}\) magnitude of \(f_{res}\)[36].
* Resonant frequency is controlled by mass and stiffness of microstructure. Further discussion is presented in the next section.
* The residual stress causing pretention to circular plate design can influence the resonant frequency [37]. If a thin diaphragm is selected for designing a low-frequency sensor (less than 200 kHz), geometric nonlinearities should be considered due to high aspect ratio of diameter to thickness [38]. To reduce the aspect ratio for designing low frequency piezoelectric resonators, [PERSON] et al. [39] designed a membrane connected with four beam elements and a mass made of an entire silicon substrate to reduce the resonant frequency of the microstructure as shown in Figure 5.