# Frequency Tuning of Graphene Nanoelectromechanical Resonators via Electrostatic Gating

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}/(V⋅s) [2]), graphene also exhibits remarkable mechanical properties such as ultrahigh Young’s modulus E

_{Y}= 1 TPa, and a large breaking strain limit up to 25% [3,4]. These attributes have made graphene an attractive and promising candidate for highly miniaturized and aggressively scaled resonant-mode nanoelectromechanical systems (NEMS) with unprecedented device performance. To date, mainly doubly clamped [5,6,7,8] membranes and ribbons, and circumference-clamped circular drumhead graphene resonators [9,10,11,12,13,14,15,16] have been prototyped, and fundamental device physics and basic device characteristics have been studied. Further, potential applications of these graphene 2D NEMS resonators in sensing of external stimuli and perturbations [17], and components for radio frequency (RF) signal processing and communications (e.g., oscillators [11]), have been attempted. For these applications, continuous and wide frequency tuning controlled by moderate level electrical signals (i.e., voltage or current) is highly desirable, to render these systems tunable, flexible, or even programmable and reconfigurable. Indeed, strong frequency tunability has been quite heavily pursued in more conventional and state-of-the-art microelectromechanical systems (MEMS) based resonators and oscillators; however, tuning range is often limited up to 5% due to their high stiffness [18]. Thanks to ultra-strong yet highly stretchable crystals and their related material properties, graphene NEMS resonators can exhibit remarkably broad tunability of resonance frequency, with Δf

_{res}/f

_{res}> 300% [12].

## 2. Analytical Model and Computational Methods

#### 2.1. Development of Frequency Tuning Model and Analysis Procedure

_{e}) on the center of the devices is solved by minimizing the total energy (sum of elastic energy U

_{el}and electrostatic energy U

_{es}) upon the application of electrostatic force. To make the model accurate, we perform several iterative calculations of the capacitance between the gate and suspended membrane C

_{g}, equilibrium displacement z

_{e}, and total strain ε after deflection. Substituting the modified strain ε and C

_{g}into the effective spring constant k

_{eff}, the resonance frequency f

_{res}can then be obtained using a relation of f

_{res}= (k

_{eff}/m

_{eff})

^{1/2}/2π, with the calculated effective mass of resonance mode, m

_{eff}.

#### 2.2. Frequency Tuning of Doubly Clamped Graphene Resonator

_{d}(x) = 4z

_{e}(Lx − x

^{2})/L

^{2}, (0 < x < L), where L is the length of the suspended graphene membrane (or ribbon), and it has the maximum static deflection of z

_{e}at the midpoint (x = L/2) due to the center of the symmetric structure. Under uniformly distributed force, we neglect the Poisson’s ratio in doubly clamped case since the resonance frequency is independent on width of the membrane. For the doubly clamped device structure shown in Figure 1b, the elastic energy stored in the stretched membrane U

_{el,d}can be described as [24]

_{Y}, t, and ε

_{0}are the width, Young’s modulus, thickness, and built-in strain of membrane, respectively. The total tension after deflection is γ = E

_{Y}tε

_{d}, where ε

_{d}is the total strain that is composed of both the built-in strain and the added strain according to the deflection. By introducing deflection profile u

_{d}(x) into Equation (1), the elastic energy is then

_{g,d}is

_{0}is the depth of the air gap. The electrostatic energy stored in the device structure is then

_{g}is the gate voltage. The equilibrium displacement z

_{e}can be obtained by minimizing the total energy thus finding the deflection position where the elastic and electrostatic forces are equal:

_{g,d}= ${\u0454}_{0}$wL/z

_{0}+ 2${\u0454}_{0}$wLz

_{e}/3${z}_{0}^{2}$ + 8є

_{0}wLz

_{e}

^{2}/15${z}_{0}^{3}$ + …, and the initial three terms of the series can be used for calculating z

_{e}using Equation (5), this calculation produces an approximated value of z

_{e}and it may give quite large error, especially when we calculate the exponent of z

_{e}. Instead, here we perform several iterations to find accurate device C

_{g}by repeating a process of plugging z

_{e}obtained using Equation (5) back into Equation (3) till it is convergent. The total strain ε

_{d}in the graphene includes built-in strain and the added strain upon deformation. The added strain after deflection is obtained by comparing the extended length from the deformed graphene sheet and the initial graphene length. The total strain is

_{f,d}(x) = δzsin(πx/L), where δz is the dynamic displacement at the midpoint. We assume that at the small static defection z

_{e}of the graphene membrane, the vibration mode shape remains to be in the sinusoidal form. The elastic energy δU

_{el}for the fundamental mode resonance of the strained graphene resonator is

_{eff,d}can be given by the second order differentiation of the total energy:

_{res}= (1/2π)(k

_{eff}/m

_{eff})

^{1/2}, where m

_{eff}is effective mass which can be calculated from kinetic energy of membrane at resonance. The peak kinetic energy E

_{kin,d}is

_{eff,d}= 0.5ρtwL in doubly clamped case, where ρ is the mass density. The frequency tuning for the doubly clamped resonator can be given by

#### 2.3. Circumference-Clamped Circular Membrane

_{el,c}can be obtained by [19,25]

_{0,r}, ε

_{0,θ}are radius, Poisson’s ratio, initial radial strain, initial tangential strain, respectively. Similar to the doubly clamped case, we assume that the curvature of the static deflection forms the parabolic shape, and it has the maximum static deflection at its center due to symmetry. The solution of deflection can also be u

_{c}(r) = z

_{e}(R

^{2}− r

^{2})/R

^{2}, where z

_{e}is the static defection at the center of the drumhead. The elastic energy induced by electrostatic deformation is

_{e}can be obtained by minimizing the total energy

_{e}, we conduct iterations using Equations (12)–(14) till it is convergent. Based on the deflection curvature and estimated z

_{e}, the total radial strain from stretching of the drumhead is estimated by calculating the radial elongation of the membrane:

_{0}. Considering the circumference-clamped boundary conditions, the mode shape of the fundamental resonance is u

_{f}

_{,c}(r) = δzJ

_{0}(2.405r/R). The elastic energy of the strained circular drumhead at resonance is then modified by substituting the mode shape u

_{f}

_{,c}(r) into Equation (11):

_{eff}= 0.271ρtπR

^{2}for the fundamental mode resonance in the circular drumhead case. Thus, the resonance frequency for the circular drumhead under electrostatic gating is given by

## 3. Results and Discussions

_{g}= 0 V as a function of device characteristic dimension (length for the doubly clamped structures, and diameter for the circular drumhead devices) with different built-in strain levels of 0.002%, 0.01%, 0.05%, and 0.25%, respectively. Due to possible photoresist residue on the device during fabrication and surface adsorbates, we assume the mass is higher than the intrinsic device mass estimated by device dimensions and mass density, and we take an effective mass ratio of 2 (i.e., m

_{device}= 2 × m

_{graphene}) as a typical value. With strain levels from 0.002% to 0.25%, the resonance frequency depends on the length for the doubly clamped case and the diameter for the circular drumhead case, with f

_{res}~L

^{−1}, and f

_{res}~D

^{−1}power laws, at zero bias condition. It is clearly shown that in both doubly clamped and circular drumhead cases, resonance frequency increases as the characteristic device dimension decreases and as built-in strain increases. For the same characteristic length and built-in strain, the circular device gives higher resonance frequency than the doubly clamped device does, since its circumference-clamped structure provides higher spring constant and smaller effective mass. These correlations for both cases agree with the results from previous models [6,9,10,19,20,21,22,23,24] at gate voltage V

_{g}= 0 V. To enable resonance frequency above gigahertz, f

_{res}> 1 GHz, built-in strain ≈0.25% with length smaller than 0.4 μm or built-in strain ≈0.05% with length smaller than 0.18 μm are required for doubly clamped graphene resonators. For circular drumhead resonators to achieve f

_{res}> 1 GHz, it needs a built-in strain ≈0.05% with a diameter smaller than 0.22 μm or a built-in strain ≈0.25% with a diameter smaller than 0.6 μm. Accordingly, fabricating smaller size membranes, larger built-in strain, and circular drumhead structure are preferred to attain higher resonance frequency.

_{g}. At different built-in strain levels, we find three frequency tuning behaviors: resonance frequency increases monotonically with |V

_{g}|, it decreases monotonically with |V

_{g}|, and it first decreases, then increases with increasing |V

_{g}|. All these cases are observed in existing experiments [5,6,7,8,9,10,11,12,13,14,15,16] for both of doubly clamped and circular drumhead graphene resonators. Figure 3 shows simulated frequency tuning of the doubly clamped and circular drumhead single-layer (1L) graphene resonators, with typical effective mass ratio = 2, air gap z

_{0}= 300 nm, length L = 1 μm and diameter D = 1 μm, respectively. The results show that a big difference between our model and the previous modeling. For the device with relatively small built-in strain (ε

_{0}= 0.01% in Figure 3a,d), applying |V

_{g}| leading to elastic stiffening which is much bigger than that of capacitive softening, causing frequency elevation (“U” shape). Whereas for the device with large built-in strain (ε

_{0}= 0.25% in Figure 3c,f), the capacitive softening dominates as |V

_{g}| increases, which keeps reducing the resonance frequency. For intermediate built-in strain (ε

_{0}= 0.05% in Figure 3b,e), initially the capacitive softening dominates, then the spring constant stiffening dominates as |V

_{g}| increases, showing “W” shape frequency tuning. These tuning behaviors also affect frequency tuning range of the graphene resonators. For smaller built-in strain (ε

_{0}= 0.01%), the resonance frequency of the doubly clamped device (L = 1 μm, z

_{0}= 300 nm) shifts from 75 MHz to 98 MHz when |V

_{g}|= 0 to 10 V (Figure 3a), showing the frequency tuning range of Δf

_{res}/f

_{res}≈ 31%. For the device with the higher built-in strain (ε

_{0}= 0.25%), although it leads to higher initial resonance frequency, the frequency shift is very small (378.9 MHz to 378.3 MHz), offering very limited frequency tunability, Δf

_{res}/f

_{res}≈ 0.15%. These results suggest that there is an important trade-off between achieving high initial resonance frequency and wide frequency tuning range, which should be considered for designing the desired device performance for the specific applications. For example, devices with large initial strain are preferred for high frequency signal processing applications, and small initial strain facilitates realizing voltage controlled tunable devices that could be useful for making systems with tunable or programmable functions.

_{0}= 0.01%) with the smaller air gap can provide much larger frequency tuning range for the doubly clamped and circular drumhead resonators (e.g., Δf

_{res}/f

_{res}≈ 24.4% for z

_{0}= 350 nm, and Δf

_{res}/f

_{res}≈ 57.8% for z

_{0}= 150 nm, for doubly clamped devices). Interestingly, with the same device parameters, such as the characteristic length, initial strain, depth of air gap, doubly clamped devices show larger tuning ranges when the gate voltage is changed (|V

_{g}| = 0–10 V) (Δf

_{res}/f

_{res}≈ 57.8% for L = 1 μm, z

_{0}= 150 nm, ε

_{0}= 0.01%), compared to those of the circular drumhead resonators (Δf

_{res}/f

_{res}≈ 48.2% for D = 2R = 1 μm, z

_{0}= 150 nm, ε

_{0}= 0.01%). This is attributed to the fact that the fully clamped circular drumhead devices have larger stiffness at the beginning, making it less responsive to incremental external electrostatic force.

_{0}= 0.01%, z

_{0}= 300 nm) without an applied gate voltage, the resonance frequencies are independent of the number of layers. With the gate voltage, the resonance frequency for the 1L graphene resonators exhibits a much higher frequency tuning capability, compared to those in the few-layer graphene resonators (Δf

_{res}/f

_{res}≈ 30.2% for 1L, Δf

_{res}/f

_{res}≈ 17.5% for 2L, and Δf

_{res}/f

_{res}≈ 12.0% for 3L doubly clamped graphene resonators), and we also find a similar trend in the circular drumhead resonators. According to Equations (9) and (19), the elastic stiffening is independent of the thickness t, while the capacitance softening increases with the increasing thickness of the membrane. For the built-in strain ≈0.01% in these cases, the softening effects of the few-layer are stronger compared to the softening in the single layer. Therefore, 1L graphene resonators give a much higher frequency at the same voltage and thus a wider tuning capability, compared to those in the few-layer graphene resonators.

_{g}= 0 V to 10V as a function of the characteristic dimension and air gap with high built-in strain (0.2%) and low built-in strain (0.002%), for both doubly clamped and circular drumhead graphene resonators, respectively. With an air gap from 200 nm to 350 nm, length or diameter from 0.2 μm to 2 μm, and lower built-in strain (0.002%), this gives a larger frequency tuning range, while a higher built-in strain (0.2%) leads to a smaller frequency tuning range for both types of geometries for the graphene resonators. We find that a smaller depth of air gap, smaller initial strain, and longer characteristic dimension help achieve a wide frequency tuning range.

_{e}, which is a positive correlate to the strain, depends on the DC gate voltage V

_{g}, with z

_{e}~${V}_{\mathrm{g}}^{2}$ power laws [6,10]. This is the only correct form for sufficiently small V

_{g}; while we expect z

_{e}~${V}_{\mathrm{g}}^{2/3}$ for large V

_{g}. This is the reason that the estimated resonance frequency from the previous modeling is much larger than the experimental results. In our model, we obtain z

_{e}by performing several iterative calculations, to make the model accurate at a wide voltage range. Based on the iterative calculations, we solve the effective spring constant by considering both static deflection and the fundamental mode shape, which are largely unexplored in the previous model.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Device structures and modeling procedure of electrostatic tuning of resonance frequency in graphene nanoelectromechanical systems (NEMS). (

**a**) Deflection of graphene resonator under electrostatic force. Without gate voltage, the membrane is flat (thin magenta dashed line) and suspended over the trench at equilibrium. With applied DC gate voltage, the membrane deflects to new equilibrium displacement z

_{e}(thick black solid line). With AC actuation, the membrane vibrates with amplitude δz (thick black dashed lines). (

**b**,

**c**) Schematics of doubly clamped and circular drumhead graphene resonators, respectively. (

**d**) Flowchart for computational implementation of the frequency tuning model.

**Figure 2.**Calculated resonance frequencies of single-layer (1L) graphene resonators without applying gate voltage. Due to photoresist residue in fabrication and surface absorption, we assume the mass is higher than the intrinsic device mass (of carbon atoms only), and we take an effective mass ratio of 2 as the typical value. Frequency scaling of (

**a**) doubly clamped membranes and (

**b**) circular drumhead graphene resonators. The labels represent built-in strain levels in the devices.

**Figure 3.**Three typical frequency tuning behaviors for both a doubly clamped graphene resonator (upper row) and a circular drumhead graphene resonator (lower row) with varying built-in strain of (

**a**,

**d**) 0.01%, (

**b**,

**e**) 0.05%, and (

**c**,

**f**) 0.25%. The red dashed lines show previous modeling and blue solid lines show this model by assuming that effective mass ratio = 2, air gap z

_{0}= 300 nm, length L = 1 μm and diameter D = 1 μm, respectively.

**Figure 4.**Dependence of resonance characteristics on depth of air gap and number of layers. Frequency tuning of doubly clamped graphene resonators with varying (

**a**) air gap and (

**c**) number of layers. Simulated resonance frequencies and tuning characteristics for circular drumhead resonators with (

**b**) different depth of air gap and (

**d**) number of layers.

**Figure 5.**3D plots of computed frequency tuning with varying characteristic dimension and air gap with high built-in strain (0.2%) and low built-in strain (0.002%). (

**a**) Frequency tunability of doubly clamped single-layer (1L) graphene resonators, and (

**b**) circular drumhead 1L graphene resonators.

**Figure 6.**Fitting frequency tuning model to experimental data. The measured data from [6] and fitted data with our modeling (blue solid lines) and previous modeling (red dashed lines) [24] by assuming that built-in strain ε

_{0}, effective mass ratio, length L, air gap z

_{0}are (

**a**) 0.003%, 3.5, 1.1 μm 250 nm and (

**b**) 0.23%, 5.7, 1.8 μm 250 nm, respectively.

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## Share and Cite

**MDPI and ACS Style**

Mei, T.; Lee, J.; Xu, Y.; Feng, P.X.-L.
Frequency Tuning of Graphene Nanoelectromechanical Resonators via Electrostatic Gating. *Micromachines* **2018**, *9*, 312.
https://doi.org/10.3390/mi9060312

**AMA Style**

Mei T, Lee J, Xu Y, Feng PX-L.
Frequency Tuning of Graphene Nanoelectromechanical Resonators via Electrostatic Gating. *Micromachines*. 2018; 9(6):312.
https://doi.org/10.3390/mi9060312

**Chicago/Turabian Style**

Mei, Tengda, Jaesung Lee, Yuehang Xu, and Philip X.-L. Feng.
2018. "Frequency Tuning of Graphene Nanoelectromechanical Resonators via Electrostatic Gating" *Micromachines* 9, no. 6: 312.
https://doi.org/10.3390/mi9060312