Systematic Analysis of Carbon-based Microdisk Resonators

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Please cite this article as: M.T. Chorsi, H.T. Chorsi, Systematic Analysis of Carbon-based Microdisk Resonators, FlatChem (2020), doi: https://doi.org/10.1016/j.flatc.2020.100159

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Systematic Analysis of Carbon-based Microdisk Resonators Meysam T. Chorsi1* and Hamid T. Chorsi2

1

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA

2

Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA

*

Corresponding author: E-mail: [email protected]

ABSTRACT

Unique mechanical, electrical, and chemical properties of carbon-based nanomaterials make them particularly suited for the modern class of micro-electromechanical resonators. Among several micro/nano-electromechanical system (MEMS/NEMS) resonator components, microdisk profile resonators have recently gained significant attention due to their ultra-high-quality factors and facile system-on-chip integration. Here, we apply an analytical model for the dynamics of graphite microdisk resonators encompassing the effects of electrostatic, van der Waals and Casimir forces. We expound on the electrical and mechanical characteristics of the graphite disk resonator and compare the results to those of a conventional disk resonator made of Silicon. Our studies show that graphite has larger capacitance changes and hence larger electrical sensitivity. The first and second resonant peaks are determined to be around 38 MHz and 104 MHz. The larger amplitude response of graphite disk resonator can lead to a larger signal to noise ratio and a quality factor as high as 1.88×106, a four-fold increase compared to silicon. We applied the developed model to investigate the emerging class of 2-dimentional (2D) materials. We have shown that a one-atom thick graphene resonator can exhibit sixteen times larger resonant frequency than that of a graphite resonator. The proposed method can find applications in 1

modeling the electromechanical behavior of a variety of resonating systems based on 3D/2D carbonbased nanomaterials.

Keywords: carbon-based nanomaterials; graphene; microdisk resonators; intermolecular forces; thermoelastic damping

1. Introduction

Carbon-based nanomaterials (CBNs) such as thin film graphite, one-atom thick graphene, and diamondlike carbon are establishing themselves as useful tools within micro/nano-electromechanical system (MEMS/NEMS) technologies [1-10]. CBNs’s unique properties make them instrumental materials to so many current systems and devices and hence they have enticed the attention of researchers. CBNs are unique in that they can have properties of both metals and non-metals. They are flexible but not elastic, have high thermal and electrical conductivity (in fact graphene has the largest thermal and electrical conductivity of any known material), and are highly refractory and chemically inert. In addition, the thickness of CBNs can be precisely controlled through mechanical exfoliation or chemical vapor deposition (CVD), enabling even micron-sized graphene sheets with thicknesses down to a single atomic layer. Graphene is mechanically very robust, transparent, elastic, impermeable and is the best-known conductor of electricity. CBNs can be fabricated using several methods, each of which results in different physical properties of the final material. Because of these fascinating properties at nanoscale dimensions, CBNs have the potential to be used in the next generation of efficient, miniaturized opto-electromechanical devices. CBNs have recently garnered significant attention in MEMS/NEMS technology [11-16]. Typically, MEMS/NEMS resonators are based on three structures: microbeams [17-24], microdisks [25-28] or 2

microrings [29-32]. Microbeam resonators suffer from large thermoelastic losses because of bending motion which prevents their utilization in commercial devices. Radial-contour mode disk resonators experience no such losses and can reach extremely high resonant frequencies while retaining relatively larger dimensions.

MEMS disk resonators have been widely used as gyroscope in aerospace and

automotive applications. Examples of recent advances include a record high quality-factor MEMS disk resonator with decaying time constant of 74.9 s [33], a cobweb-like disk resonator gyroscope with the lowest relative frequency split between two degenerate modes [34], and a spoke length distribution optimization method based high-quality disk resonator [35]. In biosensing, microdisk resonators can be implemented to detect biological molecules and nanoparticles [36-41]. High frequency disk resonators also are essential components in wireless communication systems [42-45]. Our research was motivated by the need to understand the dynamic response of an electromechanical model for CBN microdisk resonators. Considering the low accuracy reported for most “lumped parameter” models around the natural frequency, we built a “continuous mathematical” model of the resonator. The results show superior performance in quality factor and electrical sensitivity of CBNs microdisks compared to conventional resonators. This paper is organized as follows. The problem formulation, mathematical model and the material properties are briefly introduced in Section 2. For thorough and in-depth derivation of the analytical method and equations readers can refer to our previous reports [27, 28, 32]. In Section 3, the Galerkin method and the discretized equation of motion are introduced. Then the shooting method is applied to capture the periodic solutions corresponding to each excitation frequency. Section 4 reports the frequency response curves of the microdisk near the fundamental resonance frequency and discusses its behavior for different materials. Finally, Section 5 summarizes the results and gives concluding remarks.

2. Operation Principle and Mathematical Model 3

We consider a disk micro-resonator resting on a polysilicon stem as shown in Figure 1. The resonator is actuated by a combination of a DC-bias voltage, VDC, and an AC excitation voltage with amplitude VAC and frequency Ω, which are applied through the electrode-to-resonator gap. The electric load exerts a force on the plate and excites a resonance in the radial-contour mode shape, where the plate undergoes inplane vibration about its static equilibrium position. We start off our analysis with a straightforward case study consisting of a graphite micro-resonator. Table 1 summarizes the geometrical and material properties of the microdisk resonator.

Table 1: Parameters used in the numerical calculation. Class Physical Constants

The Graphite Disk

The Gap

Parameter

Symbol

Values

Units

Planck’s Constant Divided by 2π

ħ

1.055 × 10−34

Js

Speed of Light

c

2.988 × 108

m/s

Permittivity of Vacuum

ε0

8.85 × 10−12

C2.N-1.m-2

Disk Radius

R

12

μm

Disk Thickness

h

2

μm

Electrode-to-Disk Gap

d0

100

nm

Considering the displacement field in the cylindrical coordinate system as u=urer+uθeθ+uzez and using the extended Hamilton’s principle, the non-dimensional governing equation of motion for the resonator can be written as [27]:

4

 2u r t 2

C

u r t



 2u r r 2



1 u r r r



ur r2

 1

 ( r  1) V DC V AC Cos(t )  r 1  u r 

2

2

 Fn

  r  1  , n 1 3  Fn   2 r 1  u r    F     r  1 , n  2 3 4  n r 1  u r  

(1)

and the ensuing non-dimensional boundary condition holds:

ur r

 ur

(2)

r 1

where in Eq. (1) C is the non-dimensional damping coefficient and,

1 

R 2 1  2   0 2Ed 03

, 2 

R 2 1  2  A 6 Ed 04

, 3 

R 2 1  2   2 ћc 240Ed 05

.

(3)

The terms v, ε0, u, ζ, A, δ, and E are Poisson’s ratio, permittivity of vacuum, radial displacement, a dummy parameter, Hamaker constant, Dirac delta function and Young’s modulus, respectively. For graphite A = 4.7 × 10−19 J [46]. Notice that ħ = 1.055 × 10−34 Js is Planck’s constant divided by 2π and c = 2.998 × 108 m/s is the speed of light [47]. The index, n, is 1 and 2 for the van der Waals and Casimir forces, respectively. To obtain the frequency response of the system we need to solve the equation of motion and its boundary condition with an appropriate method. The results are applied to three different materials to investigate their electromechanical behavior and understand the advantages and disadvantages of using graphite . For the case studies presented in this paper, we have used the obtained experimental parameters summarized in Table 2 [48-50].

5

Table 2: Properties of three different materials used in fabrication of the disk resonator. Property

Young’s

Poisson’s

Acoustic

Electrical

Thermal

Hamaker

Velocity (vp)

Conductivity

Conductivity

Constant (A)

Density (ρ) Material

Modulus (E)

Ratio (ν)

Graphite

4.1 [GPa]

0.17

1950 [Kg/m3]

1450 [m/s]

3 × 103 [S/m]

150 [W/(m.K)]

4.7 × 10−19 [J]

Diamond

1050 [GPa]

0.1

3515 [Kg/m3]

17284 [m/s]

1 × 10-14 [S/m]

990 [W/(m.K)]

2.8 × 10−19 [J]

Silicon

170 [GPa]

0.28

2329 [Kg/m3]

8544 [m/s]

1 × 10-12 [S/m]

130 [W/(m.K)]

2.5 × 10−19 [J]

3. Dynamic Analysis

Using the separation-of-variables approach and solving the frequency equation for the in-plane vibration of a disk, natural frequencies and the appertaining mode shapes for each excited mode of the disk are obtained as [27]: f 0i 

i 2 R

E

 1  2 

(Hz)

(4)

where λi is the ith non-dimensional frequency of the resonator. Figure 2(a) illustrates mode shapes of the graphite disk.

Figure 2(b) depicts contour plots for a disk vibrating in the radial-contour mode up to fourth order. In the first mode, the entire disk is moving in-phase along the radius, with maximum displacement at the edges and a stationary nodal point in the center. In addition to the central node, the second mode adds a nodal circumference at which the resonator is also stationary and the phase of vibration reverses. The third mode adds yet another nodal circumference, creating three discrete vibrating regions. Further ith-order overtones are likely, with each mode having a central node and i-1 nodal circumferences [51]. To obtain the modal ordinary differential equations, the orthogonality of mode shapes is applied for Eq. (4), resulting in:

6

M

 q (t )M i

i 1

M

ij



 q (t )C i

i 1

ij



 q (t )K i

i 1

1 ( r  1) V DC V AC Cos(t )   j ( r ) 2

M

ij





1

 1  M q (t ) ( r )    i  i  i 1 

0

 2  r  1  j ( r ) 1  dr , n  1 3   n  0 M     1   q i (t )i ( r )   i 1      1  3  r  1  j ( r ) dr , n  2 4  n 0  M  1 ( ) ( ) q t r      i  i  i 1  

2

dr   n

(5)

where, 1

M ij   i (r ) j (r )rdr 0

1

C ij  C  i (r ) j (r )rdr

(6)

0

1

1

1

0

0

0

K ij    i(r ) j (r )rdr   i (r ) j (r )dr  

1 i (r ) j (r )dr r

The prime means differentiation with respect to r and the overdot denotes differentiation with respect to the time t. Considering a single-mode approximation, the shooting method is applied to capture the periodic solutions of Eq. (5). Unlike the approximate analytical methods of multiple time scales and the Lindstedt-Poincaré scheme [52], the shooting method imposes no restrictions on the response amplitude or the quality factor. It is computationally efficient compared to direct numerical integration. Once a periodic solution is captured, its stability is investigated via the eigenvalues of the monodromy matrix, called Floquet multipliers. If the absolute values of Floquet multipliers are less than unity, the periodic solution is stable, otherwise it is considered unstable.

4. Results and Discussion

We begin our analysis with the effect of intrinsic material properties of each material of interest on the fundamental mode shape of the disk resonator. Figure 3 presents the total material dependent 7

displacement attained via the finite element method (FEM) in COMSOL software. As it can be seen, graphite indicates the largest tip displacement; about two orders of magnitude larger than silicon and diamond. This can be contributed to the low modulus of elasticity observed in CBNs. The size of the gap between the resonator and the electrode can impact the effect of electrodes on the disk displacement. Detailed analysis of such effect can be found in [53, 54].

We obtain the frequency response curve for the graphite resonator using Eq. (5); results are shown in Figure 4(a) and 4(b). Curves obtained via our analytical method (blue) are compared to FEM simulation results for the first and the second mode resonant frequencies with the fundamental resonant peaks around 38 MHz and 104 MHz, respectively. Very close accord between the two models can be witnessed. Approximately a 50 Hz difference in the resonant frequency and approximately 1.3 pm in displacement were obtained. We attribute these differences to the intermolecular forces that have been considered in the proposed analytical model in this paper. The first resonance peak of graphite is around 38 MHz compared to 461 MHz for a polydiamond disk resonator. Although the frequency is smaller in this case, the three orders of magnitude higher amplitude considerably increases the signal-to-noise ratio.

Electrostatic force causes frequency softening in the system, i.e., as the DC voltage increases, the linear natural frequency decreases. Figure 4(c) shows the variation of the frequency response curve near the first mode of vibration as the DC voltage is increased. As required, there is a direct relation between the DC voltage and the amplitude of the periodic orbits shown in Figure 4 (d). The decrease is about 60 Hz as the DC voltage increases from 0 to 40 V.

We then investigated the effect of AC voltage amplitude on the response. Figure 5(a) presents the radial displacement as the frequency of excitation varies. It is noted that as the amplitude of the AC voltage

8

increases, the amplitude of the periodic solutions also increases. Responses are linear even with very small damping ratios in a vacuum environment. The results indicate that nonlinearities are not significant in case of disk resonators because of their small displacements.

An interesting characteristic of graphite is its large capacitance compared to diamond and silicon. This can be attributed to the large electrical conductivity of graphite which is orders of magnitude higher than silicon [55]. Figure 5(b) presents the change of the electrode-to-resonator capacitance as DC voltage varies for silicon, diamond and graphite resonators. Here, we have set the gap size to be 40nm. This can give us capacitance values in the femtofarad (fF) range which can be measured using methods such as differential-mode charge-based capacitance [56]. Due to linear relation between the gap size and capacitance, we expect to see similar curves with about an order of magnitude less capacitance for the 100 nm gap size case. The capacitance value can be modified by changing the dimensions or the material of the disk resonator. Results show that the electrode-to-resonator capacitance for the graphite resonator is considerably higher than the other two resonators and is a nonlinear function of the disk radial displacement. This property indicates a high electrical sensitivity that is desirable for capacitive sensors.

We further study the influence of van der Waals and Casimir forces on the frequency response of the model for CBN microresonators. The radial displacement at the edge of the disk resonator is shown in Figure 5(c) with and without considering the van der Waals and Casimir effects. Such intermolecular forces tend to increase the edge displacement; the difference is shown for an electrode gap of 100 nm. Because of the relatively large gap, the effect of intermolecular forces is small, but it can be increased drastically for distances smaller than 20 nm [57].

9

Figure 6 shows the mode shape and the corresponding temperature distribution within the disk resonator. The performance of microresonators is largely determined via the quality factor of the resonance and the stability of the resonant frequency. Higher quality factor resonators have a sharper peak in their frequency spectrum at the resonant frequency and have a larger signal-to-noise ratio resulting in better sensitivity. For many resonant modes the limit to the achievable quality factor is determined by thermoelastic damping. The resonator quality factor is calculated considering the thermoelastic damping as the dominant damping mechanism. As can be seen from Figure 6, graphite disk resonators can achieve a very high-quality factor compared to silicon and diamond resonators. The calculated graphite quality factor is about 1.88×106. This quality factor is much higher than the reported values based on conventional silicon and aluminium nitride resonators [58, 59].

Finally, to study the effect of thickness on the frequency of CBN electromechanical system, we modeled graphene, a one-atom thick layer of graphite. We discovered that graphite and graphene present distinct resonant frequencies. Figure 7 shows the frequency response of the resonators for frequencies from 0 to 700 MHz. Graphite has the resonant frequency around 38 MHz, while graphene has the resonant frequency around 618 MHz. Although, as shown, the peak amplitude of graphene is about 300 times smaller than that of graphite, and this poses an immense challenge on the detection of the frequency peak for graphene sheets using many other techniques, the developed model is capable of capturing this feature.

Conclusion

This work presented a generalized theoretical framework to model the dynamics of CBN microdisk resonators and compared their electromechanical behavior with silicon resonators. The effects of 10

electrostatic, van der Waals, and Casimir forces were included in the analytical model. The validity of the results was investigated via FEM-based numerical simulations. Results show that CBN disk resonators in general have higher electrical sensitivity than silicon, mainly because of their larger slope in capacitance as the DC voltage varies. Disk resonators made of graphite show a four-fold increase in the quality factor compared to their silicon counterparts. Our results further show that the graphite disk resonator exhibits a high-quality factor, low actuation voltage, and high capacitance compared to conventional materials. The distinct frequencies of graphite and graphene obtained from our mathematical model can be used to distinguish and detect the type of each resonator. The proposed mathematical model is capable of simulating the electromechanical behavior of 3D/2D microdisks for general operating conditions as well as for a wider range of applied electric loads.

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Figures:

Figure 1: Schematic of a contour mode disk resonator with one fully surrounding electrode shown in layout view.

Figure 2: (a) Radial displacement for the center-supported disk resonator. (b) The first three radial resonance mode plots for a microdisk with colorbar representing the relative magnitude of displacement for each mode.

Figure 3: Calculated total displacement in the fundamental resonance mode of the microdisk resonator. Colorbars are in micrometer scale. About two orders of magnitude enhancement of the radial displacement with graphite was obtained.

Figure 4: (a, b) Determined frequency response for the graphite disk resonator from the proposed method. FEM simulation results are provided for comparison. Modeling parameters are VDC = 20 V, VAC = 0.1 V and d0 = 100 nm (a) fundamental mode, (b) second mode. (c, d) Fundamental mode calculated for VAC = 0.1 V and d0 = 100 nm. (c) The effect of DC voltage on the radial displacement. (d) Effect of the applied DC voltage on the resonance frequency.

Figure 5: (a) AC voltage variation effects on the frequency response curves of the fundamental mode for VDC = 20 V and d0 = 100 nm. (b) Change in the electrode-to-resonator capacitance for VDC = 20 V, VAC = 1 V and d0 = 40 nm. The inset shows a zoom for Silicon. (c) Fundamental modes for VDC = 5 V, VAC = 0.1 V and d0 = 40 nm considering the effect of van der Waals and Casimir forces on the frequency response curve.

Figure 6: Thermoelastic damping in the MEMS Resonator (a) Fundamental mode shape and corresponding temperature distribution within the disk. (b) Comparison of results for three different materials.

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Figure 7: Difference in the frequency response of the graphite and the graphene disk resonators at the fundamental mode for VDC = 10 V, VAC = 0.1 V and d0 = 100 nm.

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GRAPHICAL ABSTRACT

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