# Spider Web-Like Phononic Crystals for Piezoelectric MEMS Resonators to Reduce Acoustic Energy Dissipation

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## Abstract

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## 1. Introduction

^{2}

_{eff}) [10]. Consequently, the AlN-on-Silicon MEMS resonator is a satisfactory technology for single-chip multi-band wireless communications [11].

## 2. Phononic Crystals Design

^{3}and 8500 m/s, respectively.

_{c}) and lattice constant (a) of the circle hole PnC (C-PnC) unit cell were designed as 16.5 and 34 μm, respectively. Moreover, the radius (r

_{1}) of the circle, width of the narrow beams, and width of the rings (w

_{1}, w

_{2}) of the spider web-like PnC (SW-PnC) unit cell were defined as 24, 1, 0.5, 2, and 2 μm, respectively. In general, in comparison with the 3 × 3 C-PnC array (102 μm × 102 μm), the proposed 4 × 4 SW-PnC array (96 μm × 96 μm) has an approximate size but offers a lighter weight and greater ability of acoustic wave isolation (i.e., which will be demonstrated later). In addition, the proposed SW-PnC works around the frequency range of 76 MHz. For other frequency ranges, the in-plane dimension size of the lattice constant and radii of the circles for the PnC unit cell can be reduced or enlarged proportionally. Changing the width of the beams is not recommended, as this width would be likely set at the minimum width imposed by the fabrication process. Besides, the SW-PnC is difficult to fabricate due to its large thicknesses/aspect ratios in the current Bosch process. So, the SW-PnC is at present suggested to enlarge the in-plane size to reduce the thicknesses/aspect ratios, and then use it in some lower frequency applications.

_{out}and P

_{in}are the values of the output and input power in the delay line and solid line, respectively. The initial P

_{in}was 0 dBm. Moreover, S21 is the S-parameter of transmitted waves, and represents the power transmission coefficient from the input port to the output port. Consequently, the proposed multiphysics finite-element analysis (FEA) simulation models of the delay line and solid line can be used to map the mechanical quantities to electrical ones, and further prove the transmission properties of PnCs.

_{up}and f

_{down}are the bounding frequencies of the acoustic bandgap. In this research, the frequency range of the complete bandgap for C-PnC is from 67.7 to 83.5 MHz and the frequency range of the largest complete bandgap for SW-PnC is from 68.0 to 84.5 MHz. Apparently, the mid-gap frequency of C-PnC is approximately equal as 76 MHz to that of SW-PnC. The BG% of C-PnC and SW-PnC is 20.9% and 21.6%, respectively. In general, the proposed SW-PnC possesses a smaller lattice constant (70.6%) and much lighter weight (44.2%) in a similar bandgap frequency range compared with C-PnC.

## 3. Resonators Design

_{p}) of resonators was set as 56 µm, and designed to be transduced at the fifth-order symmetric lamb mode, which resonates at the frequency of 76 MHz. The resonant frequency of resonators for any given harmonic mode is given by [34]:

_{r}is the width of the resonator and n is the number of harmonic modes. For a fifth-order symmetric mode of resonators, the W

_{r}is equal to the five-fold W

_{p}(i.e., W

_{r}= 5 W

_{p}).

_{anc}) of resonators was calculated by [35]:

## 4. Results and Discussion

_{anc}of 5870, indicating a large part of the acoustic wave dissipated through supporting tethers. The resonator RCP shows a relatively higher Q

_{anc}of 52,500, and the device RSWP with the SW-PnC array plate shows the most significant displacement isolation in undercut regions, as well as the anchoring substrate, with a Q

_{anc}of up to 64,800. From the above discussion, the simulated Q

_{anc}of resonators agrees well with the transmission properties of the delay line and solid line.

_{s}), loaded Q, motional resistance (R

_{m}), and the effective electromechanical coupling coefficient (k

^{2}

_{eff}). The relationships among the f

_{s}, Q, R

_{m}, and k

^{2}

_{eff}were defined as [36,37,38]:

_{-3dB}is the −3dB bandwidth around the series resonant frequency. The max {Re(Y11)} and f

_{p}refer to the maximum value of the real part for the admittance Y11 and the parallel resonant frequency of resonators, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Three-dimensional (3D) illustration of the 2D phononic crystals (PnC) plate with circle holes, and its unit cell with the definition of geometry parameters. The thickness of PnC structures was fixed as 10 μm in this research. The radius (r

_{c}) and lattice constant (a) of the circle hole PnC (C-PnC) are 16.5 and 34 μm, respectively. (

**b**) Illustration of the spider web-like PnC (SW-PnC) plate and its unit cell. (

**c**) Schematic top-view of the geometry parameters and irreducible brillouin zone for the SW-PnC unit cell. The lattice constant, radius (r

_{1}) of circle, and width of rings (w

_{1}, w

_{2}) of SW-PnC are 24, 1, 2, and 2 μm, respectively. Therefore, the geometry designs ensured the minimum width of the SW-PnC unit cell and C-PnC unit cell was the same as 0.5 μm.

**Figure 2.**(

**a**) Schematic set-up for the computation of the transmission property of the acoustic delay line with a finite C-PnC strip. Furthermore, (

**b**) the delay line and solid line with the PnC strip, and with the solid silicon strip as the transmission medium between interdigital transducers (IDTs) were performed, verifying the existence of acoustic bandgaps formed by the associated PnC. In this research, periodic boundary conditions were applied to surfaces along the y-direction of the delay line and solid line to form the 2D PnC slab and reference solid slab, while improving the calculation efficiency.

**Figure 3.**(

**a**) 3D Illustration of the Floquet boundary conditions applied to a C-PnC unit cell. The boundary condition applied on the orange surfaces means an infinite number of repetitions in the x-direction, and the condition applied on the purple surfaces means an infinite number of repetitions in the y-direction. (

**b**) Corresponding eigenmode shapes of the first 10 frequency band structures for the proposed C-PnC, with ka/2π = 0.25. (

**c**) Band structures of an infinite PnC are consistent with the transmission characteristics through a finite number of PnC unit cells for C-PnC. The C-PnC shows one complete acoustic bandgap in the frequency range from 67.7 to 83.5 MHz.

**Figure 4.**(

**a**) 3D Illustration of the Floquet boundary conditions applied to a SW-PnC unit cell. Moreover, the boundary condition applied on the orange surfaces means an infinite number of repetitions in the x-direction, and the condition applied on the purple surfaces means an infinite number of repetitions in the y-direction. (

**b**) Corresponding eigenmode shapes of the first 16 frequency band structures for the proposed SW-PnC with ka/2π = 0.25. (

**c**) Band structures of an infinite PnC are consistent with the transmission characteristics through a finite number of PnC unit cells for SW-PnC. The SW-PnC shows three complete bandgaps, and the widest frequency range of them is from 68.0 to 84.5 MHz.

**Figure 5.**Illustration of the displacement distribution in the z-direction of two types of delay line and solid line working in the frequency of 76 MHz, and a more significant isolation of the acoustic wave in the delay line where a smaller-sized SW-PnC strip is observed. In general, due to the large circle hole, the C-PnC is already a very lightweight PnC structure compared with previously reported PnCs. Nevertheless, the proposed SW-PnC possesses a smaller lattice constant (70.6%), much lighter weight (44.2%), and greater energy loss suppression (two-fold) in a similar bandgap frequency range compared with C-PnC.

**Figure 6.**Dispersion relations of the C-PnC unit cell with a minimum width of (

**a**) 0.5 μm and (

**b**) 0.25 μm, respectively. Meanwhile, the mid gap frequency of these two kinds of C-PnC are approximately equal as 76 MHz.

**Figure 7.**(

**a**) 3D Illustration of the fifth-order symmetric lamb mode aluminum nitride (AlN)-on- silicon-on-insulator (SOI) MEMS resonator with C-PnC array plates in each of the undercut regions. (

**b**) Three types of MEMS resonators, including the conventional resonator (C), the resonator with 3 × 8 C-PnC array plates (RCP), and with 4 × 10 SW-PnC array plates (RSWP) were designed in each of undercut regions to further verify the effectiveness of a finite PnC structure in reducing the anchor loss and indicating that the resonator RSWP could realize the optimal Q. In this research, (

**c**) the width of perfectly matched layer (PML) and the width of the undercut region in the x-direction of resonators had a three-fold wavelength (3λ) and were identical to one λ, respectively. In addition, finite element analysis was only performed for a quarter section of the resonator due to the symmetric width-extensional resonant mode (a symmetric boundary condition was applied to the symmetric surfaces).

**Figure 8.**Illustration of displacement distributions of the fifth-order width-extensional resonant mode of the conventional resonator (C), resonator with C-PnC array plates (RCP), and resonator with SW-PnC array plates (RSWP). It is noteworthy that the left side and right side are total displacement and displacement in the z-direction distributions, respectively. The simulated anchor loss (Q

_{anc}) of resonators agrees well with the transmission property of the delay line and solid line.

**Figure 9.**Illustration of the admittance spectra for resonators C, RCP, and RSWP in the same frequency range. The simulated results of admittance curves are well consistent with the transmission characteristics, verifying that the SW-PnC can sufficiently reduce the energy dissipation of resonators, thereby achieving the optimal Q.

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**MDPI and ACS Style**

Bao, F.-H.; Wu, X.-Q.; Zhou, X.; Wu, Q.-D.; Zhang, X.-S.; Bao, J.-F.
Spider Web-Like Phononic Crystals for Piezoelectric MEMS Resonators to Reduce Acoustic Energy Dissipation. *Micromachines* **2019**, *10*, 626.
https://doi.org/10.3390/mi10090626

**AMA Style**

Bao F-H, Wu X-Q, Zhou X, Wu Q-D, Zhang X-S, Bao J-F.
Spider Web-Like Phononic Crystals for Piezoelectric MEMS Resonators to Reduce Acoustic Energy Dissipation. *Micromachines*. 2019; 10(9):626.
https://doi.org/10.3390/mi10090626

**Chicago/Turabian Style**

Bao, Fei-Hong, Xue-Qian Wu, Xin Zhou, Qi-Die Wu, Xiao-Sheng Zhang, and Jing-Fu Bao.
2019. "Spider Web-Like Phononic Crystals for Piezoelectric MEMS Resonators to Reduce Acoustic Energy Dissipation" *Micromachines* 10, no. 9: 626.
https://doi.org/10.3390/mi10090626