This paper is focused on electrode design for piezoelectric tuning fork resonators. The relationship between the performance and electrode pattern of aluminum nitride piezoelectric tuning fork resonators vibrating in the in-plane flexural mode is investigated based on a set of resonators with different electrode lengths, widths, and ratios. Experimental and simulation results show that the electrode design impacts greatly the multimode effect induced from torsional modes but has little influence on other loss mechanisms. Optimizing the electrode design suppresses the torsional mode successfully, thereby increasing the ratio of impedance at parallel and series resonant frequencies (*R*_{p}/*R*_{s}) by more than 80% and achieving a quality factor (*Q*) of 7753, an effective electromechanical coupling coefficient (*kt*_{eff}^{2}) of 0.066%, and an impedance at series resonant frequency (*R*_{m}) of 23.6 kΩ. The proposed approach shows great potential for high-performance piezoelectric resonators, which are likely to be fundamental building blocks for sensors with high sensitivity and low noise and power consumption.

## HIGHLIGHTS

A new strategy is explored for suppressing the multimode effect by tuning the electrode pattern of piezoelectric tuning fork resonators.

Comprehensive resonator performance improvement is achieved by effective multimode suppression.

Crucial parameters are improved simultaneously with

*Q*= 7753,*kt*_{eff}^{2}= 0.066%,*R*_{m}= 23.6 kΩ, and an 86% higher*R*_{p}/*R*_{s}.

## I. INTRODUCTION

Microelectromechanical systems (MEMS) resonators have attracted much research and industry interest because of their unique features and their advantages of small size, high stability, low cost, and integration ability with integrated circuits. A major performance metric of MEMS resonators is the quality factor *Q*, with a higher total quality factor indicating that more energy is trapped in the resonator rather than dissipated, which contributes to high-resolution sensors and low-phase-noise oscillators. Besides *Q*, the effective electromechanical coupling coefficient *kt*_{eff}^{2} is used to evaluate the conversion rate between mechanical energy and electrical energy, with higher *kt*_{eff}^{2} implying a greater potential for a filter with a broader bandwidth and an oscillator with a larger frequency pulling range.^{1} In a sustained oscillation loop, the motional impedance *R*_{m} dominates the power consumption and phase noise.

Tuning forks (TFs) are among the most common types of MEMS resonators and have been used widely in applications such as timing references,^{2,3} inertial sensors,^{4,5} pressure sensors,^{6,7} and atomic force microscopy.^{8,9} In these applications, TF resonators (TFRs) are used in symmetric in-plane flexural modes to reduce the impact of anchor loss, and their performance is crucial for the resonant sensing capabilities of sensing elements. Compared to electrostatic MEMS TFRs, piezoelectric ones generally have higher *kt*_{eff}^{2} and smaller *R*_{m} but are more likely to suffer from the multimode effect,^{10} which not only reduces *Q* but also leads to deteriorations in *kt*_{eff}^{2} and *R*_{m}. The closer the frequencies of the unwanted modes to that of the target mode or the stronger the resonances of the unwanted modes, the more significant the multimode effect. Among the few previous publications on suppressing the multimode effect in piezoelectric TFRs, Gil *et al.* demonstrated its existence,^{11} Vigevani *et al.* simulated the trends of the frequency difference between the unwanted and target modes by adjusting the anchor length,^{12} and Liu *et al.* proposed a two-port actuation design for enhanced symmetric in-plane motion and decreased out-of-plane motion.^{13}

Herein, we explore a new strategy for suppressing the multimode effect. By tuning the electrode pattern of a piezoelectric TFR vibrating in the in-plane flexural mode, comprehensive resonator performance improvement is achieved via effective multimode suppression. Devices with different electrode designs including electrode length, gap, and ratio are fabricated and tested, and their impacts on *Q* and *R*_{m} are analyzed, the aim being to propose a set of design guidelines for electrode patterns. By following those guidelines, the multimode effect is suppressed, and simultaneous improvement of the crucial resonator parameters *Q*, *kt*_{eff}^{2}, and *R*_{m} is achieved. The proposed design has *Q* = 7753, *kt*_{eff}^{2} = 0.066%, *R*_{m} = 23.6 kΩ, and a value of *R*_{p}/*R*_{s} (ratio of impedance at parallel and series resonant frequencies) that is 86% higher than that of a reference design.

## II. DESIGN

*d*

_{31}piezoelectric coefficient. Under proper electrical actuation, the device vibrates in the in-plane flexural mode, as shown in Fig. 1(c). Although the base alters the boundary conditions of the arms, for the dimensional dependence of the frequency of the in-plane flexural mode of the TF, that of a cantilever can be referenced,

^{14}i.e.,

*ω*is the angular frequency of the cantilever,

*E*and

*ρ*are the effective Young’s modulus and density of the material, respectively, and

*W*and

*L*are the cantilever width and length, respectively.

To investigate the effects of the electrode pattern, we designed several resonators with identical structural dimensions as listed in Table I but different electrode layouts as listed in Table II. For each resonator, only one of the electrode length, gap, and ratio was set as a variable, while the other two parameters remained constant at the values highlighted in green in Table II. The device with an electrode length of 222 μm, an electrode gap of 4 μm, and an electrode ratio of 1 is the basic device. In each resonator, the variables are applied to both the top and bottom electrodes, meaning that the bottom ones always have the same dimensions as those of the top ones.

. | . | . | Unit: μm . |
---|---|---|---|

Anchor length | 20 | Top electrodes thickness | 0.15 |

Anchor width | 35 | ||

Base length | 30 | Piezoelectric layer thickness | 1 |

Arm length | 224 | Bottom electrodes thickness | 0.15 |

Arm width | 50 | ||

Arm gap | 32.5 | Passive layer thickness | 2.5 |

. | . | . | Unit: μm . |
---|---|---|---|

Anchor length | 20 | Top electrodes thickness | 0.15 |

Anchor width | 35 | ||

Base length | 30 | Piezoelectric layer thickness | 1 |

Arm length | 224 | Bottom electrodes thickness | 0.15 |

Arm width | 50 | ||

Arm gap | 32.5 | Passive layer thickness | 2.5 |

. | . | . | Electrode ratio . | |
---|---|---|---|---|

. | Electrode length . | Electrode gap . | (widths of outer/inner . | |

Variables . | (μm) . | (μm) . | electrodes [μm]) . | |

50 | 4 | 0.125 | (4.4/35.6) | |

Value | 100 | 10 | 0.25 | (8/32) |

15 | 0.5 | (13.3/26.7) | ||

150 | 1 | (20/20) | ||

200 | 20 | 2 | (26.7/13.3) | |

25 | 4 | (32/8) | ||

222 | 30 | 8 | (35.6/4.4) |

. | . | . | Electrode ratio . | |
---|---|---|---|---|

. | Electrode length . | Electrode gap . | (widths of outer/inner . | |

Variables . | (μm) . | (μm) . | electrodes [μm]) . | |

50 | 4 | 0.125 | (4.4/35.6) | |

Value | 100 | 10 | 0.25 | (8/32) |

15 | 0.5 | (13.3/26.7) | ||

150 | 1 | (20/20) | ||

200 | 20 | 2 | (26.7/13.3) | |

25 | 4 | (32/8) | ||

222 | 30 | 8 | (35.6/4.4) |

With varying electrode length, the electrodes extend from the fixed ends of the arms toward the free ends for differing lengths, as shown in Fig. 2(a); at the maximum electrode length of 222 μm, the electrodes cover almost the whole arms, as shown in Fig. 1(a). With varying electrode gap, the inner and outer electrodes broaden from the inner and outer surfaces of the arms toward the middle, respectively, as shown in Fig. 2(b). As the electrode ratio is increased from 0.125 to 8, the gap with a fixed width shifts from the outer boundary toward the inner boundary, altering the width ratio of the outer to inner electrodes, as shown in Fig. 2(c).

## III. FABRICATION AND TEST METHODS

To investigate experimentally how the performance of AlN-based TFRs depends on the electrode pattern, a complementary metal–oxide–semiconductor (CMOS)-compatible MEMS manufacturing process was used to fabricate AlN-based TF devices. The microfabrication process began with a high-resistivity silicon wafer. First, an air cavity was etched on the substrate, then phosphosilicate glass (PSG) was deposited as a sacrificial layer, and the surface was smoothed with chemical mechanical polarization [Fig. 3(a)]. A 2.5-μm-thick AlN passive layer and a 150-nm-thick molybdenum layer were deposited in sequence, and the Mo layer was patterned to form bottom electrodes [Fig. 3(b)]. Then, 1-μm-thick AlN and another 150-nm-thick Mo layer were deposited sequentially to form the piezoelectric layer and top electrode layer [Fig. 3(c)]. Afterward, the top electrode layer was partly removed to define the top electrode pattern [Fig. 3(d)], and the piezoelectric layer was patterned to uncover the testing pads [Fig. 3(e)]. Next, the passive layer and piezoelectric layer were partly etched together to form the shape of the TF [Fig. 3(f)]. Finally, gold was sputtered and patterned to serve as electrical connections [Fig. 3(g)], and the sacrificial layer was removed by hydrofluoric acid to suspend the device [Fig. 3(h)].

*f*

_{s},

*Q*, and

*kt*

_{eff}

^{2}, we also derived and included

*C*

_{0},

*R*

_{p}/

*R*

_{s}, and

*R*

_{s}in the evaluation system, where

*R*

_{p}/

*R*

_{s}is a form of a figure of merit (FoM) defined as

^{15}

## IV. EXPERIMENTAL RESULTS

### A. Basic device

The directly measured impedance curves—including magnitude and phase curves—of a fabricated basic device are shown in Fig. 4(a). The basic device has a series resonant frequency (*f*_{s}) of 795.96 kHz with *kt*_{eff}^{2} = 0.046% and *R*_{s} = 26.19 kΩ. The quality factor at *f*_{s} (*Q*_{s}) is calculated to be 6738 by the 3-dB method, and *R*_{p}/*R*_{s} is calculated to be 6.56. The measured impedance curves are fitted with a modified Butterworth–Van Dyke (mBVD) circuit model, as shown in the inset of Fig. 4(a), which models the electrical behavior of a resonator near its resonance frequency.^{16} The values of the equivalent circuit components were extracted by fitting the impedance predicted based on the equivalent circuit parameters to the measured impedance data, and the extracted parameters are listed in Table III.

Parameter . | Value . | |
---|---|---|

Before de-embedding . | After de-embedding . | |

R_{m} | 30.70 kΩ | 30.70 kΩ |

L_{m} | 52.95 H | 54.03 H |

C_{m} | 0.76 fF | 0.74 fF |

R_{0} | 6 kΩ | 100 Ω |

C_{0} | 2.72 pF | 1.85 pF |

R_{se} | 0.8 kΩ | 0 kΩ |

Parameter . | Value . | |
---|---|---|

Before de-embedding . | After de-embedding . | |

R_{m} | 30.70 kΩ | 30.70 kΩ |

L_{m} | 52.95 H | 54.03 H |

C_{m} | 0.76 fF | 0.74 fF |

R_{0} | 6 kΩ | 100 Ω |

C_{0} | 2.72 pF | 1.85 pF |

R_{se} | 0.8 kΩ | 0 kΩ |

The directly measured impedance includes not only the electrical response of the resonator itself but also that of the testing pads and connecting lines.^{17} To estimate the parasitic impedance from the testing pads and connecting lines, de-embedding was carried out in the Advanced Design System 2014 software. The impedance curve and corresponding mBVD parameters after de-embedding are shown in Fig. 4(b) and Table III. The value of *f*_{s} for the de-embedded impedance is still 795.96 kHz, with *kt*_{eff}^{2} = 0.058%, *R*_{s} = 28.85 kΩ, *Q*_{s} = 7903.69, and *R*_{p}/*R*_{s} = 14.79. Comparing the parameters before and after de-embedding indicates the quite notable impact of the testing pads and connecting lines. However, because all the devices in this work share the same testing pads and connecting lines, which therefore do not impact the performance comparison among devices with different electrode designs, the directly measured data are adopted in the following sections.

The value of *R*_{m} extracted from the impedance both before and after de-embedding is 30.70 kΩ, which is the main source of *R*_{s}. Thus, *R*_{s} is often used as a substitute for *R*_{m} because it can be read directly from the impedance curve.

### B. Electrode length

Figure 5 shows the parameters derived from the measured electrical impedance as functions of the electrode length. The quality factor at *f*_{s} increases as the electrode coverage in terms of length increases from 23% to 45% (50 μm to 100 μm), reaching a maximum value of 7752. As the electrode length is extended further, *Q*_{s} starts to decrease until it reaches 6424 when the electrode edge reaches the boundary of the arms. *Q*_{s} is improved by 21% by an appropriate electrode length design compared to the basic device. *kt*_{eff}^{2} shares the same trends as *Q*_{s}. The maximum value of *kt*_{eff}^{2} of 0.066% arises under the same condition that the electrodes cover 45% of the whole arms (corresponding to 100 μm). As the electrode coverage is increased to 1, *kt*_{eff}^{2} decays to 0.047%, which is only 71% of the maximum value. As the electrode length is increased, *f*_{s} presents a declining trend in general. More specifically, *f*_{s} at an electrode length coverage ratio of 45% (100 μm) is slightly larger than that at 23% (50 μm), and as the electrodes are lengthened further, *f*_{s} decreases from 847 kHz to 794 kHz. Finally, when the electrode length is increased, *C*_{0} grows linearly.

The effect of *Q*_{s} combines with that of *kt*_{eff}^{2}, resulting in an *R*_{p}/*R*_{s} peak at a 45% electrode length coverage ratio (100 μm), and the improvement in *R*_{p}/*R*_{s} caused by reducing the electrode length coverage ratio from 100% to 45% (100 μm) is greater than 85%. As the combined result of *f*_{s}, *C*_{0}, *kt*_{eff}^{2}, and *Q*_{s}, as the electrode length increases, *R*_{s} decreases rapidly by 33% from 34 kΩ to 22 kΩ and then increases to 26 kΩ, and the minimum value appears when the electrodes cover 67% (150 μm) of the arms.

### C. Electrode gap

Figure 6 shows the measured electrical parameters of devices with different electrode gaps. On the whole, *Q*_{s} increases and then decreases with increasing electrode gap. As the electrode gap is increased from 4 μm to 20 μm, *Q*_{s} increases significantly with a gradually decreasing growth rate. The maximum *Q*_{s} is achieved when the electrode gap is 20 μm, corresponding to an electrode coverage ratio of 55% in width. When the electrode gap exceeds 20 μm, *Q*_{s} begins to decrease with an increased rate, reaching 5925 when the electrode gap reaches 30 μm. *kt*_{eff}^{2} shows a similar trend. As the electrode gap is increased from 4 μm to 10 μm, *kt*_{eff}^{2} increases rapidly by 13%, and as the electrode gap is increased further to 25 μm, *kt*_{eff}^{2} fluctuates between 0.053% and 0.054%. Then, *kt*_{eff}^{2} begins to decrease as the electrode gap is increased further to 35 μm, which corresponds to a single-electrode width of only 7 μm. Under this condition, the electrode coverage ratio in width is too low to induce significant electromechanical coupling. *f*_{s} grows approximately linearly as the electrode gap is increased, whereas *C*_{0} decreases.

Because *Q*_{s} and *kt*_{eff}^{2} exhibit similar tendencies, the end result is that *R*_{p}/*R*_{s} also increases and then decreases with increasing electrode gap. The peak value of *R*_{p}/*R*_{s} = 8.1 appears at an electrode gap of 20 μm, and the growth rate in reaching that peak is greater than that for either *Q*_{s} or *kt*_{eff}^{2}. As the electrode gap is increased from 4 μm to 15 μm, *R*_{s} decreases slightly from 26.4 kΩ to 24.8 kΩ, and with further increase of the electrode gap, *R*_{s} increases rapidly to 37 kΩ at an electrode gap of 30 μm.

### D. Electrode ratio

Figure 7 shows the relationships between the measured electrical parameters and the electrode ratio. Overall, a larger electrode ratio gives a higher *Q*_{s}. When the electrode ratio is 1/8 or 1/4, *Q*_{s} is too low to be derived by the 3-dB method, which is why the corresponding points are missing. When the electrode ratio is 1/2 (i.e., the width of the outer electrodes is only half that of the inner electrodes), *Q*_{s} is reduced by 24% compared to that of devices with equal-width electrodes. When the electrode ratio exceeds 1, the outer electrodes are wider than the inner ones, and a conspicuous increase in *Q*_{s} can be observed. *Q*_{s} reaches its maximum value of 6995 when the electrode ratio is 2, after which it decreases. The curve of *kt*_{eff}^{2} seems to have 180° rotational symmetry about the point corresponding to an electrode ratio of 1. As a whole, wider outer electrodes increase *kt*_{eff}^{2}, while narrower ones decrease *kt*_{eff}^{2}. The maximum *kt*_{eff}^{2} of 0.054% is realized when the electrode ratio is 4. For *f*_{s} and *C*_{0}, as the electrode ratio is increased, these two parameters basically remain unchanged, with volatility in small ranges; the fluctuations in *f*_{s} and *C*_{0} are within only 0.66% and 0.76%, respectively.

When the width of the outer electrodes is one-eighth of that of the inner electrodes, *R*_{p}/*R*_{s} is only 1.2, meaning that the impedance curve around the resonant frequency is quite flat, which is why the corresponding *Q*_{s} cannot be derived by the 3-dB method. The very small values of both *R*_{p}/*R*_{s} and *Q*_{s} indicate rather weak resonance under such an actuation. As the electrode ratio is increased from 1/8 to 4, *R*_{p}/*R*_{s} increases remarkably by a factor of 6.1, reaching 7.2 when the electrode ratio is 4. Then, as the electrode ratio is increased further, *R*_{p}/*R*_{s} decreases to 6.57. Generally, increasing the electrode ratio reduces *R*_{s}. As the electrode ratio is increased from 1/8 to 4, *R*_{s} decreases by 67%, and the minimum value of *R*_{s} = 22 kΩ is achieved when the electrode ratio is 4. Comparing the results suggests that low-electrode-ratio patterns degrade the performance more than high-electrode-ratio patterns improve it.

## V. DISCUSSION

### A. Air damping and anchor loss

Air damping is among the best-known energy loss mechanisms, especially for resonators working at low and medium frequencies. As described in Sec. III, all the devices were tested under a pressure below 0.01 Pa, and this vacuum condition ensures that air damping can be ignored, as demonstrated in our previous work.^{17}

Anchor loss occurs when part of the elastic energy stored in a structure is transmitted through an anchor into the surrounding substrate during resonant microstructure vibration. In Ref. 21, the quality factor *Q*_{anchor} related to the anchor loss of a similar double-ended TFR working in the in-plane flexural mode was measured as being ∼9 × 10^{5}. This large value of *Q*_{anchor} is two orders of magnitude larger than the total quality factor measured in the present work, so the role of anchor loss in changing the quality factor is also negligible.

### B. Thermoelastic damping

TED is an intrinsic energy loss mechanism induced by the coupling of strain gradients and temperature gradients. When the resonator is vibrating, a compressed region with a positive coefficient of thermal expansion undergoes a temperature increase, while a stretched region undergoes a temperature decrease. This temperature difference causes heat flow from warmer to cooler regions, and thus elastic energy is dissipated irreversibly as heat.

To simulate the *Q*_{TED} of devices with different electrode patterns, we performed a finite-element analysis. The simulated temperature distribution is shown in Fig. 8(a), where yellow represents hotter and red represents cooler, and the temperature field is obviously coincident with the strain field. The simulated *Q*_{TED} values for devices with different electrode lengths, gaps, and ratios are shown in Figs. 8(b)–8(d), respectively. The simulated *Q*_{TED} is larger than but of the same order of magnitude as the measured total *Q*, according to Eq. (3), which shows that TED indeed contributes strongly to the overall energy dissipation. Nevertheless, although the values of the simulated *Q*_{TED} vary with the different patterns, the variances are quite limited; with increasing electrode length, gap, and ratio, the fluctuations are no more than 1.9%, and 0.4%, respectively, which are significantly less than those of the measured values. Also, the tendencies of the simulated data are inconsistent with those of the measured data. These disagreements in both the variation range and tendency between the simulated and measured quality factors reveal that TED is not the major contributor to the variation in the measured *Q*.

### C. Multimode effect

The correlation between quality factor and electrode pattern is considered to be related to the multimode effect, which is an energy loss mechanism related to the interaction of adjacent resonance modes.^{10}

In this work, both the structure and electrode pattern are designed to stimulate the in-plane flexural mode, but in practice other unwanted resonance modes may also be excited. If an adjacent mode has a frequency close to the target resonance, then the energy of the target mode will be consumed, leading to target resonance attenuation.^{13} Moreover, if the frequency of a spurious mode approaches that of the target mode or if the resonance of the spurious mode strengthens, then the multimode effect will be more significant.

For the in-plane flexural mode investigated here, the nearest resonance mode is the torsional mode,^{22} as shown in Fig. 9. The frequencies and resonance strengths of devices with different electrode designs are compared in Fig. 10, where *R*_{p}/*R*_{s} is used to evaluate the resonance strength. As shown in Fig. 10(a), with increasing electrode length, the frequencies of both the flexural and torsional modes decrease, but their difference is almost unchanged. However, the trend in Fig. 10(b) is obviously the opposite: as the electrode coverage in terms of length is decreased from 100% to 45%, the resonance of the torsional mode decays gradually, leading to a resonance enhancement of the flexural mode. As the electrode coverage in terms of length is decreased further from 45% to 23%, although the resonance strength of the torsional mode continues to decrease, the electrodes are now too short to offer adequate actuation, resulting in a decrease in *R*_{p}/*R*_{s} of the flexural mode.

The situation for the resonators with different electrode gaps is similar, with the frequency difference remaining basically unchanged with increasing electrode gap. As the electrode gap is increased from 4 μm to 20 μm, the resonance of the torsional mode weakens, thereby suppressing the multimode effect and strengthening the resonance of the flexural mode. With an electrode gap of 25 μm or 30 μm, the electrode coverage in terms of width of a single electrode is less than 20%, and both the torsional and flexural modes are weakened as a result of a weak driving force.

For the resonators with different electrode ratios, the frequencies of both the flexural and torsional modes remain virtually unchanged as the electrode ratio is varied, and the two curves for the resonance strength intersect. As the electrode ratio is decreased from 1, the torsional mode strengthens gradually, as does the multimode effect, whereas the trend for the strength of the in-plane flexural mode is downward. In contrast, *R*_{p}/*R*_{s} of the torsional mode decreases as the electrode ratio is increased from 1, thereby strengthening the flexural mode. When the electrode ratio is 1/8 or 8, the wider electrode almost covers the whole arm, so it is difficult to drive the resonator to vibrate in both the flexural and torsional modes.

Because the effective electromechanical coupling coefficient is defined as the ratio of mechanical energy and electrical energy, as the energy loss mechanism of the resonators is suppressed, so the restored mechanical energy increases, leading to the rise in *kt*_{eff}^{2}. That is why the trend of *kt*_{eff}^{2} is always similar to that of *Q*, and as a product of *Q* and *kt*_{eff}^{2}, the performance of the resonator vibrating in the in-plane flexural mode shares a similar trend.

In general, resonators are used in a specific application, such as a filter with a given pass band^{23} or an energy harvester working at a low frequency.^{24} Therefore, the frequency is an important characteristic that requires particular attention during the resonator design process. As shown in Figs. 5 and 6, as the electrode coverage is decreased, although the performance is optimized, the frequency increases because of the mass loading effect, and so steps must be taken to maintain the frequency. For the TFRs investigated herein, adjusting the arm length is one of the most efficient methods, as presented in Eq. (1). Here, we take the set of resonators with different electrode lengths as an example. If we set the resonator with fully covered electrodes as a reference, then the maximum frequency change induced by varying the electrode coverage in terms of length is 6.4%, which can be calibrated by lengthening the arms by only 3.7%. This limited extension of the arms has no impact on the anchor loss or TED in principle and has a negligible influence on the multimode effect.

Table IV compares the performances of the present resonators with those of other one-port piezoelectric TFRs working in the in-plane flexural mode reported recently elsewhere. As can be seen, with proper electrode design, we have achieved the joint-lowest *R*_{m} and high *f* × *Q* and FoM; *f* × *Q* is considered an upper limit of the FoM due to material-limited intrinsic dissipation mechanisms,^{28} and the FoM is the product of *Q* and *kt*_{eff}^{2}. Moreover, the performance could be enhanced further by synergizing these three electrode designs and via proper structural design.

Frequency(kHz) . | Q value
. | f × Q (10^{8} Hz)
. | R_{m}(kΩ)
. | kt_{eff}^{2}(%)
. | FoM . | Reference . |
---|---|---|---|---|---|---|

204.5 | 3 632 | 7.4 | 23.6 | 0.055 | 2.0 | 13 |

375 | 3 028 | 11.3 | 175 | ⋯ | ⋯ | 12 |

522.8 | 4 800 | 25.1 | 250 | 0.099 | 4.8 | 25 |

175 | 25 000 | 43.7 | ⋯ | ⋯ | ⋯ | 26 |

430 | 18 000 | 77.4 | 70 | ⋯ | 6.0 | 27 |

794.4 | 6 424 | 51.0 | 26.4 | 0.047 | 3.0 | This work (basic device) |

845.3 | 7 753 | 65.6 | 23.6 | 0.066 | 5.1 | (100-μm-length electrodes) |

Frequency(kHz) . | Q value
. | f × Q (10^{8} Hz)
. | R_{m}(kΩ)
. | kt_{eff}^{2}(%)
. | FoM . | Reference . |
---|---|---|---|---|---|---|

204.5 | 3 632 | 7.4 | 23.6 | 0.055 | 2.0 | 13 |

375 | 3 028 | 11.3 | 175 | ⋯ | ⋯ | 12 |

522.8 | 4 800 | 25.1 | 250 | 0.099 | 4.8 | 25 |

175 | 25 000 | 43.7 | ⋯ | ⋯ | ⋯ | 26 |

430 | 18 000 | 77.4 | 70 | ⋯ | 6.0 | 27 |

794.4 | 6 424 | 51.0 | 26.4 | 0.047 | 3.0 | This work (basic device) |

845.3 | 7 753 | 65.6 | 23.6 | 0.066 | 5.1 | (100-μm-length electrodes) |

In summary, by accounting jointly for *Q*, *kt*_{eff}^{2}, *R*_{p}/*R*_{s}, and *R*_{m}, we offer the following advice on electrode design for better performance of piezoelectric TFRs: a range of electrode coverage in terms of length of 40% to 80%, a range of electrode coverage in terms of width of 40% to 65%, and a range of electrode ratios of 3 to 6 are conducive to obtaining better-performing piezoelectric TFRs.

## VI. CONCLUSIONS

In this work, the effects of electrode design on the performance of AlN TFRs vibrating in the in-plane flexural mode were investigated. Resonators with different electrode coverages in terms of length and width and electrode ratios were designed, fabricated, and tested, and the analysis indicated that the multimode effect influences performance strongly but fortunately can be suppressed via optimized electrode design. In terms of *R*_{p}/*R*_{s}, properly designed electrode length, gap, and ratio can increase its value by 86%, 54%, and 36%, respectively, which provides some guidelines for designing AlN-based TFRs for high performance the in-plane flexural mode. TF resonators with outstanding performance would find applications in sensing fields such as in gyroscopes, accelerometers, and pressure sensors, and the enhanced performance in terms of *Q*, *R*_{m}, and *kt*_{eff}^{2} (for example) will contribute to high signal-to-noise ratio, great sensitivity, and low power consumption in such application scenarios.

## ACKNOWLEDGMENTS

This work was supported in part by the National Key Research and Development Program of China (Grant No. 2020YFB2008800) and the Nanchang Institute for Microtechnology of Tianjin University. The authors thank engineers Quanning Li, Xuejiao Chen, Chen Sun, and Wenlan Guo of the M/NEMS laboratory at Tianjin University for their help in fabricating our devices.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

**Yi Yuan** received a B.S. degree in Instrument Science and Technology from Tianjin University in China in 2018 and is currently pursuing a Ph.D. degree at the School of Precision Instruments and Optoelectronics Engineering at Tianjin University. His current research interests include the design and fabrication of MEMS piezoelectric resonators.

**Qingrui Yang** received B.S. and Ph.D. degrees in Instrument Science and Technology from Tianjin University in 2011 and 2018, respectively, and is currently an Assistant Professor in the State Key Laboratory of Precision Measurement Technology & Instruments at Tianjin University. Her research interests focus on the design, fabrication, and application of piezoelectric MEMS resonators, filters, and sensors.

**Haolin Li** received a B.S. degree in Mechanical Design, Manufacturing, and Automation from the China University of Petroleum in 2019 and is currently pursuing a Ph.D. in Instrument Science and Technology at Tianjin University. His current research interests include MEMS resonators and sensors.

**Shi Shuai** received a B.S. degree from the China University of Mining and Technology in 2020 and is currently a student at the State Key Laboratory of Precision Measuring Technology and Instruments. His research field is the design and fabrication of MEMS resonators.

**Pengfei Niu** is currently an Associate Professor at the State Key Laboratory of Precision Measuring Technology and Instruments at Tianjin University. He is working on acoustic MEMS platforms, wearable electrochemical sensing devices, and their coupling for health diagnosis applications.

**Chongling Sun** received a B.S. degree from Tianjin University and is currently an Engineer at the School of Precision Instruments and Opto-electronics Engineering at Tianjin University. Her research is focused on MEMS processes.

**Bohua Liu** received a B.S. degree from Hebei University of Technology in China and is currently an Engineer at the School of Precision Instruments and Opto-electronics Engineering at Tianjin University. He is working on MEMS devices and processes.

**Menglun Zhang** received B.S. and Ph.D. degrees from Tianjin University. Since 2016, he has been an Assistant Professor in the School of Precision Instruments and Optoelectronics Engineering and the State Key Laboratory of Precision Measurement Technology & Instruments at Tianjin University. He has also been a Visiting Scholar in the Bioelectronics Group at the University of Cambridge. His research is focused on piezoelectric MEMS sensors and actuators.

**Wei Pang** received a B.S. degree from Tsinghua University in China in 2001 and M.S. and Ph.D. degrees in Electrical Engineering from the University of Southern California in 2005 and 2006, respectively. He has been a full professor at Tianjin University (TJU) since 2009, where he has built the TJU MEMS Laboratory, a 15 000-ft^{2} facility completely dedicated to MEMS devices. This facility has a clean room (ca. 7000 sq. ft), a CAD lab, and a measurement/test/metrology lab. Over the past 12 years at TJU, he has launched a major research effort in a variety of areas, including resonators/filters/oscillators, acoustofluidics, ultrasound imaging and TOF sensors, pressure sensors, gas and particle sensors, uncooled infrared sensors, sweat sensing, droplet ejectors, microphones, and microspeakers. He has authored or coauthored more than 110 SCI journal papers, 70 international conference papers, and 350 granted and pending patents in the MEMS field. He has also pioneered the commercialization of piezoelectric MEMS in China, developed over 400 BAW filter products (in frequency ranges between 600 MHz and 10 GHz, covering fractional bandwidth ranges between 0.01% and >20%), and shipped hundreds of millions of aluminum nitride MEMS devices to the world’s respected OEMs.