# Design and Fabrication of Capacitive Silicon Nanomechanical Resonators with Selective Vibration of a High-Order Mode

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## Abstract

**:**

## 1. Introduction

## 2. Device Description

_{AC}together with DC bias voltage V

_{DC}are applied to a driving electrode, which results in an electrostatic force that acts on the resonant body vibration. This motion results in the changes of the motional capacitance of the resonators owing to the changes in the size of the capacitive gaps. Based on monitoring in a time-varying electrostatic force, the resonant frequency of the resonators can be observed.

_{el}is the area of the electrode plate, ε

_{r}is the dielectric constant of the material between the plates (for an air environment, ε

_{r}≈ 1), ε

_{0}is the electric constant (ε

_{0}≈ 8.854 × 10

^{−12}F·m

^{−1}), and g is the distance between two plates called the capacitive gap.

_{n}are determined by the formula of the effective spring constant k

_{eff}and the effective mass m

_{eff}, as follows:

_{n}is the frequency coefficient for each resonance mode, E is the Young’s modulus of the resonator material, I

_{z}is the area moment of inertia, L is the length of the resonant body, and m

_{0}is the mass of the resonators.

_{n}is the corresponding constant value for each resonance mode and ρ is the density of the structure material. The k

_{n}values for the first, second, third, and fourth resonance modes are k

_{1}= 1.027, k

_{2}= 2.833, k

_{3}= 5.54, and k

_{4}= 9.182, respectively.

_{m}, motional inductance L

_{m}, motional capacitance C

_{m}, and feed-through capacitance C

_{f}.

_{dB}is the insertion loss of the transmission and its unit is in decibels (dB).

## 3. Experiments

#### 3.1. Experimental Methodology

_{2}layer is formed on the entire surface of the SOI wafer via wet thermal oxidation (Figure 3b). Then, a 400-nm-thick EB resist (ZEP 502A) is patterned on the above SiO

_{2}layer (on the device layer side). The reactive ion etching (RIE) method is employed to etch SiO

_{2}with the EB resist as a mask, using a gas mixture of CHF

_{3}and Ar with a power of 120 W and a chamber pressure of 5 Pa. Narrow gaps with smooth and vertical etched shapes were achieved, as shown in Figure 4a. After removing the EB resist, the nano trenches on the top silicon layer are then formed by the deep RIE with Bosch process using SF

_{6}(etching cycles of 2.5 s) and C

_{4}F

_{8}(passivation cycles of 2.5 s) gases. Figure 4b shows the nano trenches formed using the above process. The resonant body and capacitive gaps are 500 nm and 300 nm, respectively. Following this, the resonator structures are created by employing photolithography following the deep RIE of silicon (Figure 3c).

_{6}gas. After the buried SiO

_{2}layer is etched out by buffered hydrofluoric acid (BHF) solution and devices are dried by a supercritical CO

_{2}process to avoid sticking issues, the electrode pads using Cr-Au are formed by a sputtering process via a shadow mask. Finally, the Au wire bonding process is conducted, as shown in Figure 3e.

#### 3.2. Measurement Setup

_{DC}, electrical components including capacitors and resistors, and coaxial cables. The resonators are set in a vacuum chamber at a pressure chamber of 0.01 Pa.

#### 3.3. Measurement Results

_{DC}= 15 V, V

_{AC}= 0 dBm and a vacuum chamber of 0.01 Pa. Similar resonant frequency values are observed for all fabricated devices, although their resonant lengths are significantly different (Table 1). Thus, by placing the driving electrodes along the resonant body, the high-order mode capacitive resonators can be demonstrated.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Fixed-fixed beam capacitive silicon resonators. (

**a**) First mode vibration structure; (

**b**) third mode vibration structure; (

**c**) cross-sectional structure.

**Figure 2.**Finite element method (FEM) simulation. (

**a**) First mode vibration; (

**b**) third mode vibration.

**Figure 3.**Fabrication process. (

**a**) Silicon on insulator (SOI) wafer (7 μm/ 1 μm/ 300 μm); (

**b**) thermal oxidation; (

**c**) combination of electron beam (EB) lithography, photolithography, and deep reactive ion etching (RIE) process; (

**d**) anodic bonding; (

**e**) backside silicon etching, SiO

_{2}removal, and metal contact pads.

**Figure 4.**Fabricated results. (

**a**) SiO

_{2}patterning with EB resist and using the RIE technique; (

**b**) resonant body and narrow trenches formed by deep RIE; (

**c**) first mode vibration structure; (

**d**) second mode vibration structure; (

**e**) third mode vibration structure; (

**f**) fourth mode vibration structure.

**Figure 5.**Measurement setups. (

**a**) First mode vibration structure: (

**b**) high-order mode vibration structure.

**Figure 6.**Frequency responses. (

**a**) First mode vibration structure; (

**b**) second mode vibration structure; (

**c**) third mode vibration structure; (

**d**) fourth mode vibration structure.

Resonator Structures | Vibration Modes | First Mode | Second Mode | Third Mode | Fourth Mode |
---|---|---|---|---|---|

Parameters | Resonant length | 21.3 μm | 35.5 μm | 49.3 μm | 63.3 μm |

Resonant width | 0.5 μm | 0.5 μm | 0.5 μm | 0.5 μm | |

Resonant thickness | 7 μm | 7 μm | 7 μm | 7 μm | |

Capacitive gap | 0.3 μm | 0.3 μm | 0.3 μm | 0.3 μm | |

Number of driving electrodes | 1 | 2 | 3 | 4 | |

Calculation | Frequency | 9.66 MHz | 9.60 MHz | 9.73 MHz | 9.79 MHz |

Finite element method (FEM) Simulation | Frequency | 9.71 MHz | 9.68 MHz | 9.73 MHz | 9.78 MHz |

Vibration mode (resonant body only) |

**Table 2.**Summary of measurement conditions and evaluation results of the first, second, third, and fourth mode capacitive resonators.

Resonator Structures | Vibration Modes | First Mode | Second Mode | Third Mode | Fourth Mode |
---|---|---|---|---|---|

Measurement conditions | V_{AC} | 0 dBm | 0 dBm | 0 dBm | 0 dBm |

V_{DC} | 15 V | 15 V | 15 V | 15 V | |

Pressure level | 0.01 Pa | 0.01 Pa | 0.01 Pa | 0.01 Pa | |

Experimental results | Resonant frequency | 10.15 MHz | 10.85 MHz | 10.85 MHz | 10.36 MHz |

Quality factor | 10078 | 8768 | 4255 | 844 | |

Insertion loss | −75 dB | −71 dB | −65.6 dB | −51.5 dB | |

Motional resistance | 281 kΩ | 181 kΩ | 95 kΩ | 18.7 kΩ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Toan, N.V.; Shimazaki, T.; Inomata, N.; Song, Y.; Ono, T.
Design and Fabrication of Capacitive Silicon Nanomechanical Resonators with Selective Vibration of a High-Order Mode. *Micromachines* **2017**, *8*, 312.
https://doi.org/10.3390/mi8100312

**AMA Style**

Toan NV, Shimazaki T, Inomata N, Song Y, Ono T.
Design and Fabrication of Capacitive Silicon Nanomechanical Resonators with Selective Vibration of a High-Order Mode. *Micromachines*. 2017; 8(10):312.
https://doi.org/10.3390/mi8100312

**Chicago/Turabian Style**

Toan, Nguyen Van, Tsuyoshi Shimazaki, Naoki Inomata, Yunheub Song, and Takahito Ono.
2017. "Design and Fabrication of Capacitive Silicon Nanomechanical Resonators with Selective Vibration of a High-Order Mode" *Micromachines* 8, no. 10: 312.
https://doi.org/10.3390/mi8100312