## 1. Introduction

Many IoT (internet of things) devices can be connected to the internet via a wireless network [

1]. Increasing the amount of transmitted and received information and requiring the accurate data transmissions are necessaries. Satisfying these issues, micro clock generators for transmitters and receivers with a smaller size and higher performance are required.

Quartz crystal resonators are usually employed for the above applications, as they exhibit high-quality factor [

2,

3], good power handling [

4], and excellent temperature stability [

5,

6]. However, their vibration is on a small scale owing to the direct physical contact between the electrodes and the resonant body. Their working frequencies, dependent on the thickness of a piezoelectric film, are in a low range frequency. Also, their fabrication process is not compatible with complementary metal-oxide-semiconductor (CMOS) fabrication. Capacitive silicon resonators, on the other hand, are expected to overcome the above problems, as has been presented in many works [

7,

8,

9,

10,

11,

12,

13]. An ultra-high

Q factor can be achieved by capacitive silicon resonators, as reported in References [

7,

8,

9]. In addition, they are capable of integration with a CMOS chip [

10] and they exhibit excellent long-term stability [

7,

8]. Their resonant frequency (fundamental mode) depends on the geometric dimensions of the resonators [

7,

13]. For instance, resonant frequencies of bar-type [

13], square-type [

12], and disk-type [

9] capacitive silicon resonators are decided by the width of the resonant body. A fixed-fixed beam capacitive silicon resonator, such as that presented in Reference [

7], contains the resonant beam body, as well as driving and sensing electrodes. Its resonant frequency is designed by its length and width. In attaining high-frequency capacitive silicon resonators, downscaling (reducing the length and width of the resonant body) is a common solution, although this induces problems such a large motional resistance and high insertion loss. The motional resistance of resonators is always desired to be as low as possible for an impedance that matches the CMOS chip. Hence, the downscaling method makes capacitive silicon resonators hard to apply to practical applications (such as integration with electrical circuits). Pursuant to an increase of the operating frequency without downscaling the resonant structures, this work focuses on capacitive silicon nanomechanical resonators that are able to vibrate at a higher mode selectively based on placing the driving electrodes along the resonant body. The first, second, third, and fourth mode fixed-fixed beam capacitive silicon resonators are produced and examined.

## 2. Device Description

Figure 1a,b present a perspective-view schematic of the fixed-fixed beam capacitive silicon resonators with first and third mode vibration, respectively. The basic components of resonator structures are the resonant body, capacitive gaps, the driving electrode, and the sensing electrode. The resonant body is suspended by the two anchors at each end of the resonant body on the patterned glass substrate. The cross-sectional view of the resonator structures is shown in

Figure 1c. For the first mode vibration structure, the driving electrode is placed along the side of the resonant body and the sensing electrode is placed in another side of the resonant body, as shown in

Figure 1a. They are separated from the resonant body by narrow capacitive gaps. In turn, for the third mode vibration, the driving electrodes are designed and placed on the both sides of the resonant body and the sensing electrode, as the motional detection is on the resonant body electrode (

Figure 1b). The high-order mode vibration structures are decided by the number of the driving electrodes along the resonant body. The number of driving electrodes for the second, third, and fourth mode vibration is 2, 3, and 4, respectively.

To operate the resonators, an AC input signal V_{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.

The resonant body is actuated by a delta deviance electrostatic force, which is generated by the combined effects of DC voltage and AC voltage, given as:

where

A_{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.

The resonant frequencies

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

k_{eff} and the effective mass

m_{eff}, as follows:

The effective spring constant and effective mass of the resonators are given by:

where

λ_{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.

Equations (2)–(4) can be combined into the equation below [

14]:

where

k_{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.

The equivalent circuit model of the capacitive resonators is reported in many works [

10,

11,

12], and consisted of the motional resistance

R_{m}, motional inductance

L_{m}, motional capacitance

C_{m}, and feed-through capacitance

C_{f}.

where

Q is the quality factor of the resonator,

t is the thickness of the resonator, and

η is an electromechanical transduction factor.

The larger the transduction factor η in the resonator, the more electrical energy will be converted into the mechanical domain, and consequently the bigger the force difference that can be gained for the vibration in the capacitive resonators. In this work, the high-order mode vibration has a longer resonant body compared to that of other resonators. This means that its transduction factor is larger, which results in a greater chance of capacitance compared to others. Thus, a high vibration peak (low insertion loss) can be achieved. Also, the small motional resistance can be expected (the motional resistance is proportional to the second order of the transduction factor (Equation (8)).

The motional resistance of resonators can be calculated by Equation (3); however, this equation becomes complex when considered at the high-order mode vibration. Another way to estimate the motional resistance of resonators is based on the insertion loss, which is not dependent on the vibration mode of structures, as follows [

15]:

where

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

The fixed-fixed beam resonators presented in this work are designed for lateral vibration as a flexural mode. A finite element method model is built by COMSOL (Version 5.2a, Keisoku Engineering System Co., Ltd, Tokyo, Japan) for a prediction of the vibration shape and the resonant frequency.

Figure 2a,b show the vibration shapes of the first and third modes, respectively. The colors correspond to total in-plane displacement, where red denotes the maximum displacement and blue represents no displacement. Other vibration mode shapes can be found in

Table 1.

In this work, all resonators are designed to have the same resonant frequency for performance comparison. The fixed-fixed beam resonators with the first, second, third, and fourth mode vibrations are designed and fabricated. The resonator parameters, their theoretical calculations, and finite element method (FEM) simulations are shown in

Table 1.