1. Introduction
Reference oscillators are essential components that are widely employed in various electronics systems to provide frequency control and fulfill timing requirements [
1]. With the development of real-time clocks and the Internet of Things (IoT), the application of miniaturized ultra-low-power oscillators is essential [
2]. In the design of oscillators, low-frequency resonators are the preferred choice because the use of higher-frequency oscillators results in a significant increase in current consumption by several orders of magnitude [
3,
4]. Quartz tuning fork resonators vibrating at 32,768 Hz are the main technology used in low-frequency resonators due to their high quality factor (
Q) and intrinsic level of stability [
5]. However, it is difficult to further reduce the size of quartz tuning fork resonators, which results in the dramatic degradation of
Q and an increase in the motional impedance (
Rm) [
6,
7]. In the case of one-port resonators operating within a sustained oscillation loop,
Rm and Q influence the power consumption, close-to-carrier phase noise, and zero-phase oscillation conditions.
By contrast, MEMS resonators have intrinsic advantages with regard to their capacity to be miniaturized [
8]. Compared with capacitive MEMS resonators, piezoelectric resonators have a higher electromechanical coupling factor [
9,
10] and thus lower motional impedance [
11,
12], which is beneficial to the development of ultra-low-power oscillators. However, their conventional stand-alone cantilever structure is an obstacle to the attainment of further reductions in the motional impedance via an increase in the resonator area. Once the resonant frequency of the cantilever resonator is determined, the length-to-thickness ratio of the resonator is held constant [
5]. Consequently, elongating the length of the resonator to increase its area would lead to a power series expansion in thickness, thereby exerting pressure on the fabrication process. On the other hand, widening the cantilever resonator would result in an increase in the thermoelastic damping loss and anchor loss [
13,
14], which in turn would diminish the resonator’s
Q.
In this paper, the low motional impedance of mechanically coupled, double-sided actuating multi-cantilever piezoelectric MEMS resonators is investigated. While using multiple cantilevers is straightforward in principle, the coupled system exhibits complex behavior that modulates key parameters, such as the effective electromechanical coupling coefficient (Kteff2) and spurious modes, which will be discussed later. Microcantilevers of varying numbers and with different vibrational arm widths are designed, fabricated, and characterized. The quadruple cantilever proves to be a resonator structure that can achieve a low Rm, high Kteff2, and compact dimensions.
2. Device Design
The
Rm of the resonator can be expressed as follows [
15]:
In Equation (1), fs, C0, Kteff2, and Q represent the series resonance frequency, static capacitance, effective electromechanical coupling coefficient, and quality factor, respectively. It is evident that the performance of Rm is not isolated but influenced by other performance parameters. The primary method used to reduce Rm involves increasing the resonator’s C0. For multi-cantilever resonators, the area of the resonator is N times that of a single cantilever, where N denotes the number of arms. Consequently, the C0 of the resonator increases by a factor of N, ideally reducing the Rm of a multi-cantilever beam resonator to 1/N of that for a single cantilever. Equation (1) also indicates that Kteff2 and Q should not be compromised with the multi-cantilever beam resonator design for reduced Rm.
The multi-cantilever resonator comprises a base and arms, with all arms connected to the base, and the base affixed to the structure. As shown in
Figure 1a, the triple-cantilever resonator is featured as an example, and the other multi-cantilever resonators have arms with identical dimensions and gaps. The resonator stack, from bottom to top, includes a passive layer (PL), a bottom electrode (BE), a piezoelectric layer (PZ), and a top electrode (TE).
Figure 1b illustrates the method used to connect the electrode in the triple-cantilever resonator; the BE and TE of each arm are connected to signal or ground terminals, respectively, and electrodes carrying the same signal are connected to identical test electrode pads. As shown in
Figure 2, compared to the single-sided actuation configuration with a floating bottom electrode, the double-sided actuation configuration serves to increase the resonator’s
C0, thereby reducing its
Rm.
The Rm of cantilever-type resonators typically lies in the kiloohm range, similar to the impedance of the silicon substrate. This precludes treating the silicon substrate as an insulating substrate, and thus a consideration of its impedance properties is necessitated. Therefore, an insulating layer (passive layer) between the BE and the substrate is required to ensure isolation. In addition, to ensure compatibility with the fabrication process, this study employs an aluminum nitride (AlN) thin film as both the piezoelectric layer and passive layer to achieve an out-of-plane flexural resonant mode.
Due to the presence of the AlN passive layer in the resonator, there is a parasitic capacitance (
CFt) that exists in parallel with the resonator’s test electrode pads, as illustrated in
Figure 3a. The thickness of the passive layer inversely affects this parasitic capacitance; thinner layers result in higher capacitance, which severely degrades the resonator’s performance. This degradation is evidenced by a reduction in the parallel impedance (
Rp) and a decrease in the
Kteff2, along with phase distortion.
Figure 3b demonstrates how varying thicknesses of the AlN passive layer influence the resonator’s performance. Considering that using an AlN with an excessive thickness can complicate the fabrication process, this study employs passive and piezoelectric layers of AlN with thicknesses of 2.5 μm and 1 μm, respectively. At this thickness, the
Kteff2 of the resonator achieves 88.5% of the performance that is possible with an infinitely thick passive layer. The calculation formula for the
Kteff2 is as follows:
In Equation (2),
fs and
fp represent the series and parallel resonance frequencies, respectively. The table in
Figure 3 presents the dimensional parameters for the stack of the resonator, where both the top and bottom electrode materials are molybdenum (Mo).
To balance the torque at the base, the electrical polarity of each arm is symmetrically arranged along the resonator centerline, with adjacent arms in half of the resonator displaying inverse electrical polarity. When arms with the opposite polarity vibrate in the opposite direction, the main mode is excited. Conversely, if the arms with opposite electrical polarities vibrate in the same direction and the amplitudes of the arms differ, the partial cancellation of the piezoelectrically induced charge results in the generation of a spurious resonant response.
Figure 4 displays the simulated frequency response curves obtained for single- to sextuple-cantilever resonators with a width of 30 μm, alongside 3D finite element analysis (FEA) images illustrating the main mode and the spurious mode of vibration. These figures also show the arrangement of signals for the top and bottom electrodes in each image, where “+” and “−” represent the signal and ground terminals, respectively. Due to this arrangement, when two adjacent arms vibrate in the same direction, the piezoelectrically induced charge on the electrode surface provides counterbalance. It can be seen that the spurious mode is observed from the triple-cantilever resonator to the sextuple-cantilever resonator.
In comparison with single-cantilever resonators, the vibrations of each arm in multi-cantilever resonators are mechanically coupled through the base. Diagrams of the stress distribution on the xy plane of the upper surface for the main modes of the resonators with double- to sextuple-cantilever resonators are exhibited in
Figure 5a. It is evident that double-, quadruple- and sextuple-cantilever resonators experience greater stress at the base. The
Kteff2 for multi-cantilever resonators, which is calculated using the amplitude–frequency characteristic curves obtained from finite element simulations, is presented in
Figure 5b. The results show that the double-, quadruple- and sextuple-cantilever resonators have a higher
Kteff2. Based on the foregoing FEA, it is observable that, under identical actuating voltages, the higher the stress at the resonator’s base, the greater the
Kteff2 of the resonator. We speculate that the stress at the base modulates the coupling between the electrical and mechanical domains of the multi-cantilever resonators, which will be further investigated in our future research. Due to the varying number of arms, the
Kteff2 differs among multi-cantilever resonators. Consequently, the
Rm in Equation (1) does not decrease in strict accordance with the multiple relations.
3. Device Fabrication
Figure 6 shows a simplified fabrication process. First, an air cavity is etched directly onto a silicon wafer using reactive-ion etching and then filled with phosphosilicate glass (PSG) as a sacrificial layer, as shown in
Figure 6a. A total of 2.5 μm of AlN is deposited and used as the passive layer via RF sputtering. Then, 150 nm of Mo is also deposited and patterned as the bottom electrode via plasma etching, as shown in
Figure 6b. Next, 1 μm of AlN is deposited and used as the piezoelectric layer. Then, 150 nm of Mo is deposited and patterned as the top electrode, as shown in
Figure 6c. Next, 1 μm of AlN is etched via Cl
2-based plasma etching and potassium hydroxide (KOH) wet etching; this exposes the bottom electrode, as shown in
Figure 6d. Afterward, 3.5 μm of AlN is etched via Cl
2-based plasma etching; this forms the shape of the resonant cavity, as shown in
Figure 6e. To connect the bottom and top electrodes, 600 nm of Au is deposited via physical vapor deposition and is patterned based on the lift-off, as shown in
Figure 6f. Finally, the PSG is released in the HF solution, as shown in
Figure 6g.
Figure 7 shows scanning electron microscopy (SEM) images of the fabricated multi-cantilever resonators with a width of 30 μm.
4. Measurement Results
The frequency response measurements for the single- to sextuple-cantilever resonators with a width of 30 μm in a vacuum environment are presented in
Figure 8. For one-port measurements, probes were connected to test pads and an impedance analyzer (E4990A, Keysight Technologies, USA). The spurious and main modes observed in the measured frequency response curves are consistent with the finite element simulation results presented in
Figure 4. It should be noted that the curve near the resonance of each parallel resonator appears unsmooth. This is an artifact that arises when the scanning frequency range is broad. The artifacts come from the impedance analyzer we used when spanning frequency is large. When the scanning frequency range is reduced, the frequency interval is accordingly reduced and these oscillations around the
fp disappear, as shown in Figure 10a.
Figure 9 presents a scatter distribution of
Q and
Kteff2, along with a fitting curve that depicts the variations in
C0 and the series impedance (
Rs, the impedance at the series resonant frequency, which directly reflects the level of
Rm), as measured when the W_Arm value of the single- to sextuple-cantilever resonators is set to 30 μm, 40 μm, and 50 μm. As shown in
Figure 9a,
Q is randomly distributed at approximately 10,000, indicating that neither the number of arms nor the variation in width established in this study significantly affect the Q value of the resonator. However, both the number and width of the arms influence
Kteff2, as illustrated in
Figure 9b. In the trends in
Kteff2 that can be observed, the measurements align closely with the simulation trends depicted in
Figure 5b. Notably, the
Kteff2 values measured for the sextuple-cantilever resonators are lower than those predicted via the finite element simulations. This discrepancy can be attributed to the increased complexity of the external electrode leads, which is caused by the greater number of arms; in turn, this deteriorates the
Kteff2.
When W_Arm remains constant,
C0 exhibits a linear relationship with the number of arms, as shown in
Figure 9c. In addition,
Rs is inversely proportional to the number of arms, as depicted in
Figure 9d. Nevertheless, the
Rs value of quintuple-cantilever resonators is slightly higher than that of quadruple-cantilever resonators. The reason for this is that the
Kteff2 value of quintuple-cantilever resonators is significantly smaller than that of other resonators. When the number of cantilevers exceeds four, the downward trend observed in
Rs becomes less pronounced; meanwhile, there is a significant increase in area, which is disadvantageous regarding the miniaturization of MEMS devices. Additionally, quadruple-cantilever resonators exhibit a
Kteff2 comparable to that of single- and double- cantilever resonators. Therefore, quadruple-cantilever resonators represent the best multi-cantilever resonator design, combining high electromechanical coupling and low motional impedance to achieve superior resonator performance.
5. Discussion
Table 1 compares the performance of the quadruple-cantilever resonator proposed in this study with resonators that have previously been developed and feature piezoelectric AlN out-of-plane flexural modes; the comparison parameters include
Q,
Rs,
Kteff2,
f ×
Q, and the figure of merit (FoM). Resonators with a high Q enhance the frequency stability and phase noise of oscillators and contribute to an improved resolution and signal-to-noise ratio. A low
Rs improves the signal-to-noise ratio and reduces the energy consumption of oscillators. Additionally, a large
Kteff2 provides a broader range of excitation frequencies for oscillators. Furthermore, for miniaturized MEMS devices, the size of the components is significant; hence, the area of the resonators has also been compared. The resonator introduced in this research exhibits an enhanced overall performance regarding the
f ×
Q product and FoM compared to prior studies. Its performance is nearly comparable to that of devices developed by Murata Manufacturing Co., Ltd. [
16], but it offers added advantages; these include a significantly lower
Rs and a resonator size that is only one-fifth of that of Murata’s corresponding chip dimensions.
Figure 10a shows the electrical properties of the main mode for the smallest quadruple-cantilever resonator designed in this study, which features a W_Arm value of 30 μm, measured at a pressure of 0.02 Pa. The measured 3 dB quality factor (
QMea) is 10,300, with an
Rs of 28.6 kΩ at an
fs of 55.8 kHz. The corresponding parameters of the modified Butterworth–van Dyke (mBVD) model are displayed in
Figure 10b, which shows a fitted quality factor (
QmBVD) of 10,175.
Figure 10c presents the measured pressure dependence of both
QMea and
Rs for the main mode of the resonator. When the pressure is below 5 Pa, the
QMea of the resonator is able to maintain over 83% of its maximum value; this indicates that the resonator operates within the “intrinsic region” [
7]. At this pressure range,
Rs does not decrease significantly.