# A Review on Coupled Bulk Acoustic Wave MEMS Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of BAW Resonators

#### 2.1. BAW Resonators

#### 2.1.1. BAW Propagation

#### 2.1.2. Bulk Vibration Modes

#### 2.1.3. BAW Resonators with Different Suspension Structures

#### 2.2. Multi Degrees-of-Freedom (Multi-DoF) Coupled BAW Resonators

#### 2.2.1. Mass-Spring-Damper System for a Single Degree of Freedom (1-DoF) BAW Resonator

#### 2.2.2. Coupling Mechanisms

#### 2.2.3. Mass-Spring-Damper System for 2-DoF Weakly Coupled BAW Resonators

_{1}and M

_{2}are the mass of two resonators, K

_{1}and K

_{2}are the suspension spring stiffness, K

_{c}is the stiffness of the coupling spring, c

_{1}and c

_{2}are the damping coefficient. Similarly, the displacements of the two resonators are X

_{1}and X

_{2}, and the external forces applied to the two resonators are F

_{1}and F

_{2}.

_{1}= F

_{2}= 0 and c = 0, the resonant frequency of this system can be obtained, expressed as [48]:

#### 2.2.4. Mass-Spring-Damper System for 3-DoF Weakly Coupled BAW Resonators

_{3}is the mass of resonator 3, K

_{3}is the supporting spring stiffness, Kc

_{1}and Kc

_{2}are the stiffness of the coupling spring, and c

_{3}is the damping. Similarly, the displacement of resonator 3 is X

_{3}, and the force applied to the three resonators is F

_{1}, F

_{2}, and F

_{3}, respectively.

_{1}= F

_{2}= F

_{3}= 0 and c = 0, the resonant frequency of this system can be obtained, expressed as [79]:

#### 2.2.5. Mode Aliasing

_{3dB}is the 3 dB bandwidth of each mode, f

_{1}and f

_{2}are the resonant frequency of two adjacent modes.

#### 2.3. Weakly Coupled BAW Resonators

#### 2.3.1. Mode Localization

_{1}/X

_{2}| of two resonators as an output metric is several orders of magnitude higher than that using the conventional frequency shift as an output metric, which provides a new sensing mechanism with high sensitivity. As a result of the increase of the parametric sensitivity, the resolution limit of the mode-localized sensors can also be improved by orders of magnitude especially when more than 2-DoF resonators are coupled together, which has first been theoretically validated in [81].

_{1}/X

_{2}| of the resonator amplitude at the same resonant frequency is then utilized for sensing applications. In Figure 13, the relationship between the relative mass change on resonator 2 and the amplitude ratio |X

_{1}/X

_{2}| is presented. When there is no mass perturbation in this system, the amplitude ratio is equal to 1, indicating both resonators have the same deformation. in the presence of a mass perturbation, the amplitude ratio is no longer equal to 1. Besides, the in-phase and out-of-phase modes have different mode shape changes with the same mass perturbation, which could be used to detect the position of the perturbation (for details see Section 2.3.4).

#### 2.3.2. Sensitivity Characterization

_{2}= M + ΔM or K

_{2}= K + ΔK) is used to characterize the normalized sensitivity which is obtained as:

_{2}= M + ΔM or K

_{2}= K + ΔK) represented by the amplitude change (X) is obtained as [48]:

_{c}) times larger than that of the 1-DoF resonator. However, when the amplitude ratio (AR) of the two resonators is adopted as an index to characterize the normalized sensitivity, the normalized sensitivity represented by the AR is obtained as:

_{1}/X

_{2}|.

_{3}= M + ΔM or K

_{3}= K + ΔK) is obtained as [48,49,50,51]:

_{1}/X

_{3}|.

_{2}is designed to be two times or larger than K (Kc < K/10 < K

_{2}/20, K

_{1}= K

_{3}= K) [48,49,50,51], the normalized sensitivity can be expressed as:

#### 2.3.3. Common Mode Rejection

_{T}, which is applied to both resonators as a common-mode signal. Now let’s consider a situation in which a small external stiffness perturbation (ΔK) is applied to only one of the two resonators (e.g., resonator 2 here) in the presence of ΔK

_{T}:

_{T}on the sensitivity is canceled out as a common-mode signal. However, in reality, the two resonators are not fully identical due to fabrication errors, and the influence of ΔK

_{T}on the sensitivity (the frequency shift or the AR) can be reduced dramatically but not completely [83], as shown in Figure 14.

_{T}and ΔK are both introduced to the resonator at the same time, the system stiffness K

_{s}is:

_{T}or caused by the ΔK, making the sensors based on the 1-DoF resonator vulnerable to temperature fluctuation if no extra temperature compensation method is used.

#### 2.3.4. Detection of the Position of Perturbations

## 3. Coupled BAW Resonators with Different Transduction Methods

#### 3.1. Capacitive Actuation and Sensing

**Table 8.**Description of several classic transduction mechanisms applied to coupled BAW MEMS resonators.

Transduction Methods | Capacitive Actuation and Sensing | Piezoresistive Sensing | Piezoelectric Actuation and Sensing |
---|---|---|---|

Actuation force (F) | $F\approx \left(\frac{1}{2}{V}_{\mathrm{d}}^{2}+2\left|{V}_{\mathrm{ac}}\right|{V}_{\mathrm{d}}\right)\frac{\mathrm{d}C}{\mathrm{d}x}$ [76] | Not applicable yet | $F=\frac{{e}_{33}{A}_{3}{V}_{ac}}{h}-\mathrm{Longitudinal}$ $F=\frac{{e}_{31}{A}_{1}{V}_{ac}}{h}$- Transverse [92] |

Sensed motional current (i_{mot}) | ${i}_{\mathrm{mot}}={V}_{\mathrm{d}}\frac{\partial C}{\partial x}\frac{\partial x}{\partial t}$ [76] | ${i}_{\mathrm{mot}}\approx \frac{{V}_{\mathrm{d}}}{R}\left(\frac{\Delta {R}_{\mathrm{r}}}{{R}_{\mathrm{r}}}\right)$ [95] | $\left|{i}_{\mathrm{mot}}\right|=\left|\omega \cdot {e}_{33}{A}_{3}{S}_{3}\right|\mathrm{Longitudinal}$ $\left|{i}_{\mathrm{mot}}\right|=\left|\omega \cdot {e}_{31}{A}_{3}{S}_{1}\right|$ Transverse [92] |

Cons | Parasitic capacitance; Nonlinearity; complex circuit design; thin film damping; limitation in liquids [96] | Temperature dependency; high power consumption with using high DC voltages; noisy | Processing difficulty in piezoelectric materials; not possible static measurement |

Pros | Low power consumption; good noise performance; easy to fabricate | Simple setup; inherent shielding; applicable to liquids [97] | Eliminating frequency drifts caused by DC voltage variations due to no DC voltages needed here [13]; Applicable to liquids [98,99,100,101] |

#### 3.1.1. One-Port Configuration

#### 3.1.2. Two-Port Configuration

#### 3.2. Capacitive Actuation and Piezoresisitive Sensing

_{d}) is applied to the resonator at the anchors to detect the change of resistance, and then the motional current can be obtained [39,40].

_{d}rather than shrinking the transduction gap or increasing the applied actuation voltage, but it is limited to the piezoresistive coefficient of the single crystal silicon and the dissipation of the electric power of the resonator.

#### 3.3. Piezoelectric Actuation and Piezoelectric Sensing

_{3}), and the transverse actuation configuration (see Figure 21b), where the force (F) is perpendicular to the applied electric field (ε

_{3}). The electrodes (marked in grey in Figure 21) on the top and bottom surface of the piezoelectric material form a capacitor and the current through the capacitor will have a piezoelectric component in addition to the regular capacitance current.

#### 3.4. Capacitive-Piezo Transduction

## 4. Applications of Coupled BAW Resonators

#### 4.1. Sensors Based on Weakly Coupled BAW Resonators

^{2}) [116] and C-FBARs (1.78 ng/cm

^{2}) [117]), the detection limit of this device operating in the air (1.46 ng/cm

^{2}or 36 ng/cm

^{2}) is not superior. To improve the sensitivity, further miniaturization of the proof mass or a secondary label of mass may be needed.

#### 4.2. Oscillators Based on Coupled BAW Resonators

#### 4.3. Filters Based on Coupled BAW Resonators

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Finite element model (FEM) simulation of a BAW resonator with square plate structure in COMSOL. (

**a**) WG mode; (

**b**) Extensional mode.

**Figure 2.**FEM simulation of a BAW resonator with disk structure in COMSOL. (

**a**) WG mode; (

**b**) Extensional mode.

**Figure 3.**Other FEM simulated bulk modes in COMSOL. (

**a**) BL mode (Adapted with permission from Ref. [63]); (

**b**) Face-shear mode; (

**c**) Butterfly mode.

**Figure 4.**Schematic diagrams of suspensions. (

**a**) Straight-beam suspension; (

**b**) T-shaped suspension. Adapted with permission from Ref. [26].

**Figure 7.**2-DoF coupled BAW resonators with different coupling methods. (

**a**) Fixed-fixed straight coupling beam; (

**b**) Folded coupling beam; (

**c**) Electrostatic coupling. Note: d is the lateral transduction gap between the resonator R1 and resonator R2, and the dotted arrow lines indicate the electric field direction.

**Figure 8.**WG modes of a 2-DoF coupled BAW square plate resonator system with the same mechanical coupling beam located at different positions (represented by y) simulated in COMSOL. (

**a**) Out-of-phase WG mode, y = 0 μm; (

**b**) In-phase WG mode, y = 0 μm; (

**c**) Out-of-phase WG mode, y = −100 μm; (

**d**) In-phase WG mode, y = −100 μm; (

**e**) Out-of-phase WG mode, y = −200 μm; (

**f**) In-phase WG mode, y = −200 μm; (

**g**) Out-of-phase WG mode, y = −300 μm; (

**h**) In-phase WG mode, y = −300 μm. Note: y represents the y-axis value of the center location of the coupling beam, and the origin is at the center of the first square-plate proof mass. Note: There is a coordinate system in (

**a**) with the original point set at the center of the left resonator. The y-axis value shows the position of the coupling beam. When the coupling beam is in the center position of both two resonators, y = 0. Then the beam moves down to the edge part, the y axis value is −100 μm, −200 μm, and −300 μm, respectively.

**Figure 9.**A 2-DoF electrostatically coupled BAW disk resonator system operating in different modes simulated in COMSOL. (

**a**) Out-of-phase mode 1; (

**b**) In-phase mode 1; (

**c**) Out-of-phase mode 2 (WG mode); (

**d**) In-phase mode 2 (WG mode). The deformation of resonators is magnified. In reality, the frequencies of the two modes are prone to be aliased due to the limited Q factor.

**Figure 12.**A 2-DoF weakly coupled BAW disk resonator system with mode localization simulated in COMSOL. (

**a**) Out-of-phase mode without perturbation; (

**b**) Out-of-phase mode with a small mass perturbation on the right resonator; (

**c**) In-phase mode without perturbation; (

**d**) In-phase mode with a small mass perturbation on the right resonator.

**Figure 13.**Relationship between the amplitude ratio change and the relative mass change on resonator 2.

**Figure 14.**Effect of temperature change on the normalized sensitivity of resonators(Reprinted with permission from Ref. [83]). (

**a**) 1-DoF resonators; (

**b**) 2-DoF resonators.

**Figure 17.**Measurement setup for a coupled BAW resonator system using capacitive actuation and piezoresistive sensing (Reprinted with permission from Ref. [40]).

**Figure 18.**The frequency response measured for a BAW resonator with square extensional (SE) mode in the air (Adapted with permission from Ref. [95]). (

**a**) One-port capacitive actuation and sensing method; (

**b**) One-port capacitive actuation and piezoresistive sensing method.

**Figure 19.**SEM image of the fabricated BAW rectangular-plate resonators with capacitive actuation and piezoresistive sensing setup (Adapted with permission from Ref. [37]). (

**a**) A 2-DoF coupled BAW resonator device; (

**b**) A 3-DoF coupled BAW resonator device.

**Figure 20.**(

**a**) FEM simulation of 2-DoF BAW disk resonators with in-phase extensional mode shown; (

**b**) SEM image of the fabricated 2-DoF BAW disk resonators with capacitive actuation and piezoresistive sensing. Reprinted with permission from Ref. [38].

**Figure 21.**Piezoelectric transduction configuration. (

**a**) Longitudinal configuration; (

**b**) Transverse configuration. Note: t is the thickness of the piezoelectric layer, A

_{1}is the cross-sectional area of the piezoelectric layer, A

_{3}is the electrode area, ε

_{3}is the intensity of the electric field, and F is the generated force.

**Figure 22.**Schematic of the piezoelectric transduction mechanism applied to a 2-DoF weakly coupled BAW square-plate resonator device vibrating at in-phase and out-of-phase WG modes. The mass perturbation is conducted by applying particles onto the bottom surface of the resonators. Reprinted with permission from Ref. [42].

**Figure 23.**Working principle of a conventional piezoelectric resonator, capacitive resonator and novel capacitive-piezo resonator (Reprinted with permission from Ref. [114]).

**Figure 24.**(

**a**) Schematic of a 1-DoF capacitive-piezo disk resonator operated at the radial contour mode with the anchor at the center (Reprinted with permission from Ref. [115]); (

**b**) Schematic of a 2-DoF capacitive-piezo disk resonator device operated at the WG mode with the anchor connected to the resonator body by a suspension beam (Reprinted with permission from Ref. [114]).

**Figure 25.**(

**a**) Optical micrograph of the 2-DoF BAW resonator system; (

**b**) Comparison between the analytical frequency shift and the measured one with respect to the deposited Cr film thickness. Reprinted with permission from Ref. [39].

**Figure 26.**Optical micrograph of attached analytes. (

**a**) SCPMs with diameters of 5.61 μm and 15.68 μm; (

**b**) High Five insect cells. Reprinted with permission from Ref. [40].

**Figure 27.**Relationship between the frequency shift and the number of analytes attached to the resonator from. (

**a**) SCPMs; (

**b**) High Five insect cells. Reprinted with permission from Ref. [40].

**Figure 29.**(

**a**) Comparison of Q factors among the single, 2-DoF and 3-DoF square-plate BAW resonators as the drain current (Reprinted with permission from Ref. [37]); (

**b**) Comparison of Q factors between the single and 2-DoF disk BAW resonators with bias current increasing (Reprinted with permission from Ref. [38]).

**Figure 30.**(

**a**) SEM image of a 7-DoF strongly coupled BAW resonator system with anchors at the center; (

**b**) FEM simulated out-of-plane motion of the coupled resonator systems; (

**c**) Frequency response of several multi-DoF coupled resonators. Reprinted with permission from Ref. [37].

**Figure 31.**(

**a**) Schematic of a 3-DoF strongly coupled disk resonator system with the equivalent electric circuit; (

**b**) Measured frequency response for several multi-DoF disk resonator systems; (

**c**) Relationship between the offset frequency and phase noise. Adapted with permission from Ref. [118].

**Figure 32.**(

**a**) Schematic and SEM image of a 2-DoF and 96-DoF coupled BAW disk resonator filter system; (

**b**) Revised schematic of the out-of-phase radial contour mode shape; (

**c**) Revised schematic of the in-phase radial contour mode shape. Adapted with permission from Ref. [80].

**Figure 33.**A comparison of frequency spectra between a 30-DoF disk system and a single disk system. Adapted with permission from Ref. [80].

Type of Waves | Characteristics | Schematic of Wave Propagation |
---|---|---|

Longitudinal wave | Propagation through the medium in the same or opposite direction of particle vibration or oscillation. It is also called pressure waves. | Adapted with permission from Ref. [25]. |

Transverse wave | Propagation through the medium in the direction perpendicular to particle oscillation direction. It is also called shear-mode waves. | Adapted with permission from Ref. [4]. |

Lamb wave | It exists in thin plates, also called the plate wave, and the particle motion lies in the plane that is perpendicular to the plate and the propagation direction of the wave [25]. It is divided into symmetric waves and antisymmetric waves [24]. | Adapted with permission from Ref. [24]. |

**Table 2.**Dimensions and materials of a 2-DoF coupled BAW square plate resonator system modeled in COMSOL (Adapted with permission from Ref. [26]).

Parameters | Value |
---|---|

Length of the square plate (μm) | 800 |

Thickness of the device (μm) | 25 |

Length and width of the stem (μm) | 67.2 and 14.1 |

Length and width of the cap (μm) | 70.7 and 10.6 |

Length and width of the coupling beam (μm) | 400 and 14 |

Materials selection for FEM simulation | Single crystal silicon (SCS) |

**Table 3.**The in-phase and out-of-phase mode resonant frequencies of a 2-DoF coupled BAW square plate resonator system with the same mechanical coupling beam located at different positions obtained by FEM simulation in COMSOL.

Location of Coupling Beam | In-Phase Mode (Hz) | Out-of-Phase Mode (Hz) | Frequency Difference |
---|---|---|---|

y = 0 μm | 4,702,389 | 4,769,160.74 | 1.42% |

y = −100 μm | 4,704,931.73 | 4,762,091 | 1.21% |

y = −200 μm | 4,711,448.66 | 4,746,444.54 | 0.74% |

y = −300 μm | 4,717,826.99 | 4,730,406.99 | 0.27% |

**Table 4.**Dimensions and materials of a 2-DoF electrostatically coupled BAW square resonator system modeled in COMSOL.

Parameters | Value |
---|---|

Diameter of the disk plate (μm) | 1500 |

Thickness of the device (μm) | 30 |

Lateral transduction gap (μm) | 2 |

Length and width of the stem (μm) | 225 and 37.5 |

Length and width of the convex plate (μm) | 5.5 and 450 |

Materials selection for FEM simulation | Single crystal silicon (SCS) |

**Table 5.**Differences in resonant frequencies of a 2-DoF electrostatically coupled BAW disk resonator system operated at different vibrational modes obtained by FEM simulation in COMSOL.

Vibration Modes | Out-of-Phase Mode (Hz) | In-Phase Mode (Hz) | Frequency Difference |
---|---|---|---|

Mode 1 | 365,455.475 | 365,537.439 | 0.02% |

Mode 2 (WG mode) | 2,654,149.09 | 2,654,150.75 | 0.00006% |

**Table 6.**Differences in resonant frequencies of a 2-DoF mechanically coupled BAW square resonator system operated at different vibrational modes obtained by FEM simulation in COMSOL.

Vibrational Modes | In-Phase Mode (Hz) | Out-of-Phase Mode (Hz) | Frequency Difference |
---|---|---|---|

Mode 1 | 381,093.019 | 411,584.939 | 8% |

Mode 2 (WG mode) | 4,711,448.66 | 4,746,444.54 | 0.74% |

**Table 7.**Dimensions and materials of a 2-DoF weakly coupled BAW disk resonator system modeled in COMSOL.

Parameters | Value |
---|---|

Diameter of the disk plate (μm) | 200 |

Thickness of the device (μm) | 30 |

Lateral transduction gap (μm) | 2 |

Length and width of the stem (μm) | 50 and 10 |

Length and width of coupling beams (μm) | 125 and 10 |

Materials selection for FEM simulation | Single crystal silicon (SCS) |

References | [39] | [40] | [41,42] |
---|---|---|---|

Vibration mode | Extensional mode | Extensional mode | WG mode |

Actuation way | Capacitive | Capacitive | Piezoelectric |

Sensing way | Piezoresistive | Piezoresistive | Piezoelectric |

Frequency (MHz) | 5.492, 5.423 | 5.492; 3.145 | ~2.5 |

Q factors | 5139, 8505 | 5139; 8676 | 1773.8 |

Mass perturbation | Cr film | Microbeads and cells | Particles |

Characterize sensitivity | Frequency shift | Frequency shift | Frequency shift; AR shift |

Sensitivity | 34 Hz/ng | −12 Hz/cell | 0.0735; 148.22 |

Detection limit | Not mentioned | 1.46/36 ng/cm^{2};~1.68 dried cells | Not mentioned |

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**MDPI and ACS Style**

Wang, L.; Wang, C.; Wang, Y.; Quan, A.; Keshavarz, M.; Madeira, B.P.; Zhang, H.; Wang, C.; Kraft, M.
A Review on Coupled Bulk Acoustic Wave MEMS Resonators. *Sensors* **2022**, *22*, 3857.
https://doi.org/10.3390/s22103857

**AMA Style**

Wang L, Wang C, Wang Y, Quan A, Keshavarz M, Madeira BP, Zhang H, Wang C, Kraft M.
A Review on Coupled Bulk Acoustic Wave MEMS Resonators. *Sensors*. 2022; 22(10):3857.
https://doi.org/10.3390/s22103857

**Chicago/Turabian Style**

Wang, Linlin, Chen Wang, Yuan Wang, Aojie Quan, Masoumeh Keshavarz, Bernardo Pereira Madeira, Hemin Zhang, Chenxi Wang, and Michael Kraft.
2022. "A Review on Coupled Bulk Acoustic Wave MEMS Resonators" *Sensors* 22, no. 10: 3857.
https://doi.org/10.3390/s22103857