Concentric Split Aluminum with Silicon-Aluminum Nitride Annular Rings Resonators

This paper presents a novel approach of annular concentric split rings microelectromechanical resonators with tether configuration to reduce anchor loss and gives very high-quality factor (Q) 2.97 Million based on FEA (Finite Element Analysis) simulation. The operating frequencies of these resonators are 188.55 MHz to 188.62 MHz. When the proposed SR (square rectangle) hole shaped one dimensional phononic crystal (1D PnC), and two dimensional phononic crystal (2D PnC) structure consist of very wide and complete band gaps is applied to novel design rings MEMS resonators, the quality factor (Q) further improved to 19.7 Million and 1750 Million, respectively, by using the finite element method. It is also observed that band gaps become closer by reducing the value of filling fraction, and proposed SR PnC gives extensive peak attenuation. Moreover, harmonic response of ring resonator is verified by the perfect match layers (PML) technique surrounded by resonators with varying width 1.5λ, and 3λ effectively reduce the vibration displacement.

Several researchers [8][9][10][11][12][13][14][15][16] have demonstrated the ring resonators or (disc resonator with hole is developed in center). It is noted that operating frequency, and design of resonator play an important role in the performance and improving the quality factor of resonators. Moreover, the phononic crystal (PnC) having wide acoustic band gap is the key factor that can give the high-quality Q of microelectromechanical resonators.
This work describes a design of novel SR (Square-Rectangle) hole shaped Phononic crystal, and this phononic crystal applied to novel annular concentric rings shaped Aluminium Nitride piezoelectric MEMS resonator.
At first, we have designed and simulated the unique silicon phononic crystal (PnC) structure of width 10 µm. we have also observed that reducing the value of filling fraction closer the band gaps.

Analysis of Phononic Band Gaps
The band gap in Phononic crystal (PnC) depends upon the properties of materials from which PnC made, density, and speed of sound in materials. Simply, we can say that band gap's position with respect to frequency is the ratio of sound velocity to the size of periodic unit cell, and filling factor [45,46]. We have investigated the wave propagation in a novel air SR (Square-Rectangle) shaped phononic crystal unit cell consist of Si (silicon, anisotropic) through FE parametric solver which can sweep wave vector k. The air SR shape means the holey square and holey rectangles in PnC unit cell. The geometrical dimension of unit cell is 18.6 µm × 18.6 µm with centered horizontal and centred vertical air rectangles 18 µm × 4 µm (rectangles hole) and centered air square of 9.2 µm (square hole) each side, as shown in Figure 1b. The dimensions of centered rectangles hole and centered square hole, as shown in Figure 1b. The 3D view of PnC structure as shown in Figure 1c. We analyzed the above periodic unit cell to evaluate the frequency response for 1D phononic crystal. The k parametric sweep fluctuates from 0 to 1. Where (0-1) indicate wave number of irreducible Brillouin zone (Γ-Χ). More generally, k swept from Γ to Χ (0 to π/a).
Moreover, the dispersion occurs when different wavelengths' waves have different velocities of propagation. The dispersion relation associated with wave vector k to its frequency as follows (1) Where 'ꞷ' and 'c' are the angular frequency and wave velocity respectively. Particularly for phononic crystal the dispersion relation is complex and is represent by frequency band curves schematic as shown in Figure 2.   We analyzed the above periodic unit cell to evaluate the frequency response for 1D phononic crystal. The k parametric sweep fluctuates from 0 to 1. Where (0-1) indicate wave number of irreducible Brillouin zone (Γ-X). More generally, k swept from Γ to X (0 to π/a).
Moreover, the dispersion occurs when different wavelengths' waves have different velocities of propagation. The dispersion relation associated with wave vector k to its frequency as follows where 'ω' and 'c' are the angular frequency and wave velocity respectively. Particularly for phononic crystal the dispersion relation is complex and is represent by frequency band curves schematic as shown in Figure 2. We analyzed the above periodic unit cell to evaluate the frequency response for 1D phononic crystal. The k parametric sweep fluctuates from 0 to 1. Where (0-1) indicate wave number of irreducible Brillouin zone (Γ-Χ). More generally, k swept from Γ to Χ (0 to π/a).
Moreover, the dispersion occurs when different wavelengths' waves have different velocities of propagation. The dispersion relation associated with wave vector k to its frequency as follows (1) Where 'ꞷ' and 'c' are the angular frequency and wave velocity respectively. Particularly for phononic crystal the dispersion relation is complex and is represent by frequency band curves schematic as shown in Figure 2.    The band gaps generated by the 1D phononic crystal in Brillouin zone (Γ-X) is depicted in Figure 2a.
It is worth mention that bandgap also depend upon the geometrical dimension, and inclusion filling factor in PnC by using the following relation Air is used as high acoustic impedance material, and Aluminum Nitride (AIN) is low acoustic impedance material. The filling fraction of this SR (square rectangles) hole PnC is 0.542.
If the horizontal and vertical air rectangle is taken as 17 µm × 5 µm in size, and centered air square of 9 µm of each side, the band gap has reduced. We have observed that reducing the value of filling fraction closer the band gaps. The band gaps created into higher order modes due to smaller air inclusion area. The filling fractions and the band gap of phononic crystal obtained from different geometrical dimension, as shown in Table 1  We have also analyzed the above periodic unit cell to evaluate the frequency response for 2D phononic crystal. In 2D PnC structure the k parametric sweep fluctuates from 0 to 3. Where (0-1), (1-2), and (2-3) indicate wave number of irreducible Brillouin zone (Γ-X), (X-M), and (M-Γ) respectively. More generally k swept from Γ to X (0 to π/a), X to M (0 to π/a), and then M to Γ (π/a to 0). Figure 3 obtained from FEA (Finite Element Analysis)) simulation, blue region shows that there is no wave propagation or no resonant mode in structure, which covers a wide and complete bandgap. It is worth mention that bandgap also depend upon the geometrical dimension, and inclusion filling factor in PnC by using the following relation area of inclusion area of unit cell (2) Air is used as high acoustic impedance material, and Aluminum Nitride (AIN) is low acoustic impedance material. The filling fraction of this SR (square rectangles) hole PnC is 0.542.
If the horizontal and vertical air rectangle is taken as 17 μm × 5 μm in size, and centered air square of 9 μm of each side, the band gap has reduced. We have observed that reducing the value of filling fraction closer the band gaps. The band gaps created into higher order modes due to smaller air inclusion area. The filling fractions and the band gap of phononic crystal obtained from different geometrical dimension, as shown in Table 1  We have also analyzed the above periodic unit cell to evaluate the frequency response for 2D phononic crystal. In 2D PnC structure the k parametric sweep fluctuates from 0 to 3. Where (0-1), (1)(2), and (2-3) indicate wave number of irreducible Brillouin zone (Γ-Χ), (Χ-Μ), and (Μ-Γ) respectively. More generally k swept from Γ to Χ (0 to π/a), Χ to Μ (0 to π/a), and then Μ to Γ (π/a to 0). Figure 3 obtained from FEA (Finite Element Analysis)) simulation, blue region shows that there is no wave propagation or no resonant mode in structure, which covers a wide and complete bandgap.
The red shaded, green shaded, and purple shaded area represents the deaf acoustic mode or deaf bandgap [47] [48], because only resonant modes are allow to propagate in these range of frequencies. Deaf bands are the uncoupled band in dispersion relation. The frequency band structure of 3D plate model and eigenmode shapes of frequency bands with ka/2π = 0.25 is shown in Figure 3. The red shaded, green shaded, and purple shaded area represents the deaf acoustic mode or deaf bandgap [47,48], because only resonant modes are allow to propagate in these range of frequencies.
Deaf bands are the uncoupled band in dispersion relation. The frequency band structure of 3D plate model and eigenmode shapes of frequency bands with ka/2π = 0.25 is shown in Figure 3.
As this PnC structure employed to resonator for further reduction of energy leakage, so we have calculated transmission spectra through PnC for analyzing the leakage of energy, because transmission of acoustic waves is forbidden in a bandgap. The transmission is defined as the ratio of transmitted waves to incident waves, and calculated as the difference of probe displacement and source displacement.
As the phononic crystal is employed to resonator with supporting tether length 12 µm, and width 4 µm on both sides of resonator. The motion of resonator is in y-direction and movement of Poisson coupling in x-, and z-directions. The x, y, and z excitations, as shown in Figure 4. As this PnC structure employed to resonator for further reduction of energy leakage, so we have calculated transmission spectra through PnC for analyzing the leakage of energy, because transmission of acoustic waves is forbidden in a bandgap. The transmission is defined as the ratio of transmitted waves to incident waves, and calculated as the difference of probe displacement and source displacement.

log
(3) As the phononic crystal is employed to resonator with supporting tether length 12 μm, and width 4 μm on both sides of resonator. The motion of resonator is in y-direction and movement of Poisson coupling in x-, and z-directions. The x, y, and z excitations, as shown in Figure 4. We have calculated the transmission wave spectra through finite thickness of 10 μm Phononic crystals as shown in Figure 4. Here we see that the attenuation peak is wider for x and y excitation as well as for z excitation, and no dip in transmission as compared in band diagram of Figure 5. A very minor transmission dip is observed only at 120 MHz in z excitation. Form Figure 5 we see that the band gap, wide peak attenuation, and operational frequency of resonator lie in the same frequency region. Thus, wider peak of attenuation makes a broader choice of desire operational frequencies of resonators. Thus, it is clear from Figure 5 that the transmission is very low and acoustic wave is banned in the yellow highlighted area (band gaps area). We have calculated the transmission wave spectra through finite thickness of 10 µm Phononic crystals as shown in Figure 4. Here we see that the attenuation peak is wider for x and y excitation as well as for z excitation, and no dip in transmission as compared in band diagram of Figure 5. A very minor transmission dip is observed only at 120 MHz in z excitation. Form Figure 5 we see that the band gap, wide peak attenuation, and operational frequency of resonator lie in the same frequency region. Thus, wider peak of attenuation makes a broader choice of desire operational frequencies of resonators. Thus, it is clear from Figure 5 that the transmission is very low and acoustic wave is banned in the yellow highlighted area (band gaps area). Micromachines 2018, 9, x FOR PEER REVIEW 6 of 15

Device Analysis
Annular concentric rings shaped Aluminum Nitride piezoelectric MEMS resonator is designed in this work, as shown in Figure 6. The resonator is described with width of tether (Tw = 4 μm) without phononic crystal (PnC), and also with one dimensional phononic crystal (1D PnC), and two dimensional phononic crystal (2D PnC).

Device Analysis
Annular concentric rings shaped Aluminum Nitride piezoelectric MEMS resonator is designed in this work, as shown in Figure 6.

Device Analysis
Annular concentric rings shaped Aluminum Nitride piezoelectric MEMS resonator is designed in this work, as shown in Figure 6. The resonator is described with width of tether (Tw = 4 μm) without phononic crystal (PnC), and also with one dimensional phononic crystal (1D PnC), and two dimensional phononic crystal (2D PnC).
where (J, and Y) are the first, and second kind of Bessel functions, and (M, and N) are modified form of Bessel functions, R 0 , and R i are the outer and inner radius.
The Aluminum Nitride (AlN) as piezoelectric material is taken in ring shaped resonator, so the resonant frequency [9,52] can be governed in the following way where λ = 2W, E p , ρ, and σ are width of resonator mode (wavelength), Young's Modulus, Poisson's ratio, and effective mass density of piezoelectric material, respectively. We have set the primary alignment of resonators structure to <110> frame of reference for silicon elasticity matrix [53] in (COMSOL Multiphysics 5.4) as below Some other important parameters of medium/materials used in this work are listed in Table 2.  When the potential is applied between aluminum electrodes, the electric field is developed the induced mechanical strain along aluminum nitride piezoelectric material, this is due to inverse effect of piezoelectric, as a result mode of vibration appears. The mode shape vibration of resonator is depicted in Figures 7 and 8  In this study we have also obtained high Q by employing PnCs to resonators through FEA (Finite Element Analysis) simulation. There are few energy loss mechanisms comprise the total Q-factor of the device [54] . Their losses are given as Support (anchor) loss, Viscous damping, Electrical damping, Loss due to structural asymmetry, and Thermo-Elastic Damping (TED).  In this study we have also obtained high Q by employing PnCs to resonators through FEA (Finite Element Analysis) simulation. There are few energy loss mechanisms comprise the total Q-factor of the device [54] . Their losses are given as Support (anchor) loss, Viscous damping, Electrical damping, Loss due to structural asymmetry, and Thermo-Elastic Damping (TED). In this study we have also obtained high Q by employing PnCs to resonators through FEA (Finite Element Analysis) simulation. There are few energy loss mechanisms comprise the total Q-factor of the device [54]. Their losses are given as Support (anchor) loss, Viscous damping, Electrical damping, Loss due to structural asymmetry, and Thermo-Elastic Damping (TED).

Medium/Materials Parameters
In this study only anchor loss is considered (through FEA simulation), and the other losses are completely omitted. Due to nature of motion resonator it is not affected by air damping, and the material losses are difficult to predict, so the major loss is anchor loss [33,55,56]. The energy loss mechanism in MEM resonator is also demonstrated by Akhieser effect (AKE) [57]. Moreover Frangi et al. [58,59] reported that all thermoelastic losses (Akhieser losses) are included due to sources of dissipation. These Akhieser losses are generally negligible because piezoelectric MEM resonators operate in one of its bulk modes, and only the anchor loss deserve attention.
We have analyzed the Quality Q of proposed resonator without PnC, employed with 1D PnC, and with 2D PnC by the following summarized relation [60,61].
where E s and E l are the energy stored (vibration energy) and energy loss (energy dissipation per cycle of vibration) in resonator. Figure 9 show that the tether (width 4 µm) is attached with 2D PnC, and resonator is bounded with PML (perfect match layer). The width of PML is set as varying 'aλ' (a = 1.5 and 3, and λ = 45 µm) ( Table 3). Elastic waves are produced by resonator through tether, this is the major loss of energy. Specific dimension of PML can prevent resonator from reflecting elastic waves, because the elastic waves can be absorbed by PML [57,62,63]. Thus, we have used the perfect match layer through FEA (Finite Element Analysis) simulation to reduce the energy leakage through tethers. Reducing the energy leakage means increasing the quality factor. So Q anchor is obtained by resonant frequency f 0 over band width −3 dB [33,56,58] where 'ω' is the desired mode's eigen frequency of the resonator in FEA simulation. In this study only anchor loss is considered (through FEA simulation), and the other losses are completely omitted. Due to nature of motion resonator it is not affected by air damping, and the material losses are difficult to predict, so the major loss is anchor loss [33][55] [56] . The energy loss mechanism in MEM resonator is also demonstrated by Akhieser effect (AKE) [57]. Moreover Frangi et al. [58], [59] reported that all thermoelastic losses (Akhieser losses) are included due to sources of dissipation. These Akhieser losses are generally negligible because piezoelectric MEM resonators operate in one of its bulk modes, and only the anchor loss deserve attention.
We have analyzed the Quality Q of proposed resonator without PnC, employed with 1D PnC, and with 2D PnC by the following summarized relation [60], [61].

2π
(7) Where Es and El are the energy stored (vibration energy) and energy loss (energy dissipation per cycle of vibration) in resonator. Figure 9 show that the tether (width 4 μm) is attached with 2D PnC, and resonator is bounded with PML (perfect match layer). The width of PML is set as varying 'aλ' (a = 1.5 and 3, and λ = 45 μm). Elastic waves are produced by resonator through tether, this is the major loss of energy. Specific dimension of PML can prevent resonator from reflecting elastic waves, because the elastic waves can be absorbed by PML [56][60] [62]. Thus, we have used the perfect match layer through FEA (Finite Element Analysis) simulation to reduce the energy leakage through tethers. Reducing the energy leakage means increasing the quality factor. So Qanchor is obtained by resonant frequency f0 over band width −3 dB [56][58] [33] 3 dB ∆ 2 Where 'ꞷ' is the desired mode's eigen frequency of the resonator in FEA simulation.    The response of annular rings resonators is obtained by suppling the voltage source of 1 V to electrodes. Figures 10 and 11 show the displacement of resonators along x, y, and z directions without, and with pnC structure through frequency domain FEA simulations.
Micromachines 2018, 9, x FOR PEER REVIEW 10 of 15 The response of annular rings resonators is obtained by suppling the voltage source of 1 V to electrodes. Figures 10 and 11 show the displacement of resonators along x, y, and z directions without, and with pnC structure through frequency domain FEA simulations. These results declare that resonators with perfect match layer (PML) width approximately equal to 3λ ≈ 135 μm reduced the displacement of vibration, as shown in Figure 11. The Perfect match layer (artificial absorbing layers) technique is firstly described by Pierre [63]. Moreover, wide PML curtail the frequency domain and acts as an isotropic absorber or simply we can say it acts like an artificial boundary.   Figure 12 differentiate the Quality (obtained from FEA simulations) between resonators without PnC structure, with 1D PnC structure, with 2D PnC (with perfect match layer (PML) width = 1.5λ, 3λ, respectively). These results show that Quality Q of resonator is the highest when 2D Phononic crystal structure is employed to resonators. These results declare that resonators with perfect match layer (PML) width approximately equal to 3λ ≈ 135 µm reduced the displacement of vibration, as shown in Figure 11. The Perfect match layer (artificial absorbing layers) technique is firstly described by Pierre [64]. Moreover, wide PML curtail the frequency domain and acts as an isotropic absorber or simply we can say it acts like an artificial boundary. Figure 12 differentiate the Quality (obtained from FEA simulations) between resonators without PnC structure, with 1D PnC structure, with 2D PnC (with perfect match layer (PML) width = 1.5λ, 3λ, respectively). These results show that Quality Q of resonator is the highest when 2D Phononic crystal structure is employed to resonators.
We see that, this unique design of resonators gives high quality, and when PnC is employed to resonator, the energy isolate from the resonator to substrate (reducing the loss of energy) as a result Q further increased. Furthermore, band gap of designed PnC covers the operating frequency of designed resonators, and the design of this SR PnC will help to increase the quality of further designing of resonators in future. We see that, this unique design of resonators gives high quality, and when PnC is employed to resonator, the energy isolate from the resonator to substrate (reducing the loss of energy) as a result Q further increased. Furthermore, band gap of designed PnC covers the operating frequency of designed resonators, and the design of this SR PnC will help to increase the quality of further designing of resonators in future.

Conclusion
In this work, we have presented a new approach of annular concentric split rings microelectromechanical resonators with tether configuration to reduce anchor loss and gives very high-quality factor (Q) 2.9 Million. These resonators were operating between the frequency 188.55 MHz and 188.62 MHz. In this works we have also proposed SR (square rectangle) hole shaped phononic crystal structure, which gives very wide and complete band gaps. When SR (square rectangle) shaped one dimensional phononic crystal (1D PnC), and two dimensional phononic crystal (2D PnC) is employed to novel design rings MEMS resonators the quality factor (Q) further improved to 19.7 Million and 1750 Million, respectively. We have concluded that value of filling fraction is reduced then band gaps become closer. It is observed that wider peak of attenuation makes broader desired operational frequencies of resonators.
We have also verified the harmonic response of annular rings resonators by perfect match layers (PML) technique surrounded to resonators with varying width 1.5λ, and 3λ effectively reduce the vibration displacement.

Conclusions
In this work, we have presented a new approach of annular concentric split rings microelectromechanical resonators with tether configuration to reduce anchor loss and gives very high-quality factor (Q) 2.9 Million. These resonators were operating between the frequency 188.55 MHz and 188.62 MHz. In this works we have also proposed SR (square rectangle) hole shaped phononic crystal structure, which gives very wide and complete band gaps. When SR (square rectangle) shaped one dimensional phononic crystal (1D PnC), and two dimensional phononic crystal (2D PnC) is employed to novel design rings MEMS resonators the quality factor (Q) further improved to 19.7 Million and 1750 Million, respectively. We have concluded that value of filling fraction is reduced then band gaps become closer. It is observed that wider peak of attenuation makes broader desired operational frequencies of resonators.
We have also verified the harmonic response of annular rings resonators by perfect match layers (PML) technique surrounded to resonators with varying width 1.5λ, and 3λ effectively reduce the vibration displacement.