# Concepts and Key Technologies of Microelectromechanical Systems Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MEMS Resonator Operation Principle

#### 2.1. Equivalent Model

#### 2.2. Vibration Modes

#### 2.3. Transduction Mechanisms

## 3. Performance and Optimization

#### 3.1. Quality Factor

#### 3.1.1. Air Damping Loss

_{air}, the predictions sometimes vary widely because air damping is affected by resonator size, gap size, ambient pressure, vibration modes, and non-ideal fluid motion. Several estimation methods are summarized below.

#### 3.1.2. Anchor Loss

#### 3.1.3. Thermoelastic Loss

^{19}quality factor was obtained by optimizing the size and distribution of holes. That material, orientation, doping level, and slot location all affect the temperature stability of tuning fork MEMS resonators was experimentally demonstrated by [51]. Therefore, it is proposed that intelligent optimization algorithms, such as Covariance-Matrix-Adaptation-Evolution-Strategy can be used to determine the geometry of MEMS resonators to maximize quality factor and temperature stability. It is the belief of [60] that the traditional TED model is not suitable for partially coated resonators, so an analytical TED model of a partially covered cantilever with a silicon oxide coating was developed. The results show that the proposed TED model matches well with the finite element method. It is also pointed out that the length of the metal coating should be less than 70%, and the influence of TED will be reduced to 25% when the position of the metal coating is far from the clamping end.

#### 3.1.4. Other Losses

#### 3.2. Motional Resistance

#### 3.3. Frequency Accuracy

#### 3.3.1. Mechanical Trimming

#### 3.3.2. Electrical Tuning

#### 3.4. Temperature Stability

#### 3.4.1. Passive Compensation

_{2}coating [92] has a second order flip temperature similar to that of a quartz crystal. The flip temperature point is controlled by varying the ratio of the resonator’s SiO

_{2}to silicon content. This means that a resonator with temperature stability at room temperature can be obtained by designing the inversion point. An extended mode MEMS resonator based on an oxide refilling process was proposed and reported in [93]. The first order TCF of the extended mode MEMS resonator can be compensated more effectively by placing silicon dioxide islands in high strain regions, resulting in a TCF of 4 ppm/K. AlN lamb wave resonators with negative TCF can also be temperature compensated by a composite structure [94]. The addition of a layer of SiO

_{2}underneath the AlN achieved 250 ppm from −55 °C to 125 °C. A MEMS resonant gas sensor with oxide trenches on the edge of the cantilever, which has little degradation to the quality factor, was proposed by [95]. Experimental results show that the proposed design reduces the frequency temperature coefficient to 1.7 ppm/°C and the quality factor can reach 4700.

^{19}cm

^{−3}. The results also show that the greater the stress, the greater the frequency change of the MEMS resonator. a 105 MHz concave silicon bulk acoustic resonator (CBAR) with a linear TCF of −6.3 ppm/°C by boron doping was fabricated by [103]. Some highly doped silicon DETFs were fabricated and the effect of different alignment orientations on temperature stability were tested by [104]. The results show that a highly doped silicon resonator with proper alignment can achieve a 200 ppm frequency change from −35 °C to 85 °C.

#### 3.4.2. Active Compensation

_{e}is the effective capacitive area, k

_{n}is the dynamic stiffness, and d is the initial capacitive gap.

## 4. Summary and Future Perspective

_{3}[137] and multimode resonators [138]; (4) Development of new MEMS devices that are highly immune to environmental drift, such as amplitude ratio output [139,140] and multi-mode excitation resonant sensor; (5) Flexible MEMS resonators [141] that can be used in the human body; (6) Monolithic integrated units for multiple sensors or multiple frequency outputs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Mass damping spring system; (

**b**) Equivalent electrical model including feedthrough capacitor and parasitic capacitance.

**Figure 2.**Diagrams of flexural modes (

**a**–

**d**) and bulk modes (

**e**–

**i**): (

**a**) Single-ended fixed out-of-plane mode; (

**b**) Double-ended fixed out-of-plane mode; (

**c**) Double ended tuning forks (DETF) in-plane mode; (

**d**) Above-membrane out-of-plane mode; (

**e**) Length extension (LE) mode; (

**f**) Width expansion (WE) mode; (

**g**) Radial breathing mode; (

**h**) Lamé mode; (

**i**) Wine-glass mode, or face-shear (FS) mode when the resonator is rectangular.

Structure and Mode | Resonant Frequency | Parameter |
---|---|---|

Cantilever beam | ${f}_{0}={C}_{n}\sqrt{\frac{{E}_{eq}}{{\rho}_{eq}}}\frac{t}{{L}^{2}}$ | ${f}_{0}$ resonant frequency ${C}_{n}$ mode coefficient ${E}_{eq}$ modulus of elasticity ${\rho}_{eq}$ density $t$ device thickness $L$ device length $n$ mode number $W$ device width $R$ device radius $\sigma $ Poisson’s ratio $D$ feature size $D=L$ for LE mode $D=W$ for WE mode and $D=R$ for radial extension ${G}_{eq}$ shear modulus ${C}_{0}$ constant parameter ${\kappa}_{n}$ frequency parameter. |

Double-clamped tuning fork | ${f}_{0}=\frac{{[(1+2n)\pi /2]}^{2}}{2\pi \sqrt{12}}\sqrt{\frac{{E}_{eq}}{{\rho}_{eq}}}\frac{W}{{L}^{2}}$ | |

Circular membrane | ${f}_{0}=\frac{10.22}{2\pi {R}^{2}\sqrt{12{\rho}_{eq}\left(1-{\sigma}^{2}\right)/{E}_{eq}{t}^{2}}}$ | |

Extension mode | ${f}_{0}=\frac{n}{2D}\sqrt{\frac{{E}_{eq}}{{\rho}_{eq}}}$ | |

FBAR | ${f}_{0}=\frac{1}{2t}\sqrt{\frac{{E}_{eq}}{{\rho}_{eq}}}$ | |

Lamé | ${f}_{0}=\frac{1}{\sqrt{2}L}\sqrt{\frac{{G}_{eq}}{{\rho}_{eq}}}$ | |

Face shear | ${f}_{0}=\frac{{C}_{0}}{L}\sqrt{\frac{{G}_{eq}}{{\rho}_{eq}}}$ | |

Wineglass | ${f}_{0}=\frac{{\kappa}_{n}}{2\pi R}\sqrt{\frac{{E}_{eq}}{{\rho}_{eq}\left(1-{\sigma}^{2}\right)}}$ |

Type | Frequency | Q | Pressure (mTorr) | Transmission (dB) | Vibration Modes | Reference | Schematic Illustration |
---|---|---|---|---|---|---|---|

Capacitive | 32.768 kHz | ~15,000 | 50 | −74 | Flexural | [22] | |

Capacitive | 6.35 MHz | 1,700,000 | 0.15 | −17 | Lamé | [26] | |

Capacitive | 51.3 MHz | 128,400 | 0.08 | −90 | Lamé | [27] | |

Capacitive | 107.3 MHz | 11,000 | Standard atmosphere | −80 | Whispering gallery | [28] | |

Capacitive | 150.9 MHz | 18,000 | 0.225 | −72 | Radial-contour | [24] | |

Piezoelectric | 10 MHz | 4682 | Standard atmosphere | −20 | Width expansion | [29] | |

Piezoelectric | 14.02 MHz | 5000 | ~mTorr | −24 | Length extension | [23] | |

Piezoelectric | 48.14 MHz | 10,000 | ~mTorr | −8 | Width expansion | [23] | |

Piezoelectric | 52 MHz | 4743 | Standard atmosphere | −25 | Lateral-extension | [30] | |

Piezoelectric | 882 MHz | 220 | Standard atmosphere | −46 | Contour mode | [31] |

Frequency | Mode | Type | Original Q | Enhanced Q | Methods | Reference | Schematic Illustration |
---|---|---|---|---|---|---|---|

52 MHz | Lateral-extension | Anchor loss | 606 | 4743 | Frame structure with PnC | [30] | |

51.3 MHz | Lamé | Anchor loss | 56,400 | 128,400 | The beam with root slots | [27] | |

10.03 MHz | Lateral mode | Anchor loss | 2618 | 3945 | Reflective structures | [41] | |

10.03 MHz | Lateral mode | Anchor loss | 2618 | 4522 | PnC | [41] | |

10 MHz | Width expansion | Anchor loss | 1570 | 4682 | PnC + Reflector | [29] | |

610 kHz | Flexural mode | TED | 13,000 | 16,000 | Slots | [50] | |

400 kHz | Flexural mode | TED | 15,000 | 40,000 | Slots | [51] | |

20 kHz | Flexural mode | Coating loss | 3000 | 8000 | Coating coverage | [52] |

Frequency (MHz) | Type | Methods | Reference | Stability |
---|---|---|---|---|

0.39 | In-plane flexural | SiO_{2} | [95] | 1.7 ppm/°C [10 °C to 90 °C] |

1 | DETF | SiO_{2} | [89] | −0.02 ppm/°C^{2}[−55 °C to 125 °C] |

1.024 | DETF | SiO_{2} | [92] | −0.02 ppm/°C^{2}[−40 °C to 120 °C] |

711 | Lamb Wave | SiO_{2} | [94] | −0.021 [−55 °C to 125 °C] |

0.47 | DETF | Doping | [104] | 190 ppm [5 °C to 85 °C] |

9 | Lateral extensional | Doping | [101] | ±20 ppm [−40 °C to 85 °C] |

10 | square extensional | Doping | [101] | ±16 ppm [−40 °C to 85 °C] |

23 | Extensional mode | Doping | [90] | 10 ppm [−40 °C to 85 °C] |

25.09 | Lateral extensional | Doping | [88] | 245 ppm [−40 °C to 85 °C] |

24.44 | Width extensional | Doping and SiO_{2} | [112] | ±21.5 ppm [−40 °C to 85 °C] |

Frequency (MHz) | Type | Methods | Reference | Stability |
---|---|---|---|---|

2.92 | Free-free beam | Electrostatic | [118] | 0.44 ppm/°C 25 °C to 55 |

5.5 | I-shaped bulk | Electrostatic | [11] | 39 ppm 25 °C to 125 °C |

1.126 | DETF | SiO_{2}+ electrostatic | [117] | ±2.5 ppm −10 °C to 80 °C |

0.54 | In-plane flexural | Doping and Single-Temperature Calibration | [123] | ±8 ppm 5 °C to 85 °C |

77.7 | Lamé mode | Oven control | [127] | ±0.3 ppm −25 °C to 85 °C |

1.2 | DETF | Oven control | [124] | ±1 ppm −20 °C to 80 °C |

1.2 | DETF | Calibration and control | [124] | ±0.05 ppm −20 °C to 80 °C |

1.2 | Plate Bending | Doping and control | [133] | ±25 ppb −40 °C to 40 °C |

10 | Length-extensional | Doping and control | [128] | ±0.5 ppm −35 °C to 85 °C |

13 | Lamé | Doping and control | [133] | ±5 ppb −40 °C to 40 °C |

42.7 | Shear mode | Doping and control | [130] | ±0.4 ppm −40 °C to 80 °C |

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**MDPI and ACS Style**

Feng, T.; Yuan, Q.; Yu, D.; Wu, B.; Wang, H.
Concepts and Key Technologies of Microelectromechanical Systems Resonators. *Micromachines* **2022**, *13*, 2195.
https://doi.org/10.3390/mi13122195

**AMA Style**

Feng T, Yuan Q, Yu D, Wu B, Wang H.
Concepts and Key Technologies of Microelectromechanical Systems Resonators. *Micromachines*. 2022; 13(12):2195.
https://doi.org/10.3390/mi13122195

**Chicago/Turabian Style**

Feng, Tianren, Quan Yuan, Duli Yu, Bo Wu, and Hui Wang.
2022. "Concepts and Key Technologies of Microelectromechanical Systems Resonators" *Micromachines* 13, no. 12: 2195.
https://doi.org/10.3390/mi13122195