# The Effect of Edge Mode on Mass Sensing for Strained Graphene Resonators

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## Abstract

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## 1. Introduction

^{−24}g) with a mass responsivity of 0.34 GHz/yg. Then, Jiang et al. [29] investigated the feasibility of the graphene nanomechanical resonant mass sensor and showed that the mass sensitivity would triple if the actuation energy was about 2.5 times the initial kinetic energy of the nanoresonator. In addition, some efforts, including decreasing geometric sizes [30], utilizing vacancies [31], and adjusting capacitive force [32], have been made to improve the performance of graphene-based resonant mass sensors. Though ultrahigh mass resolution of up to 1 yg has been reported, the impact of edge modes, which always occur at the free edges of graphene sheets and even break the coherence of fundamental oscillation [3,33], has rarely been discussed.

## 2. Simulation Structures and Methods

## 3. Results and Discussions

#### 3.1. Two Distinct Vibration Eigenmodes

^{3}for pure grapheme [2,5].

#### 3.2. Effect of Edge Mode on Mass Sensing

#### 3.2.1. Distinct Response to Centrally Distributed Adsorbates

_{0}is the density of pure graphene and ∆ρ is the increment of equivalent density, assuming the absorbed mass is evenly distributed. The factor K is used to adjust the frequency shifts caused by distinct distributions of adsorbates and equals 1 when adsorbates are evenly distributed. Since the volume of graphene sheet can be regarded to be unchanged, Equation (2) is then rewritten as

_{0}is the mass of pure graphene sheets and ∆m represents the mass of adsorbates. In contrast, due to the unpredicted results when using standard elastic beam theory, the frequency of edge modes shows a considerably different response to central adsorbates.

^{0.5}according to Equation (1), and it matches with the MD results considerably well, with an R-square of more than 0.99 and a maximum relative error of about 6%. Moreover, this relative error decreases to about 2% if a stronger tension (>15 GPa) is exerted, which can be explained by the fact that the stretched graphene sheet is closer to the ideal tensioned membrane structure. On the other hand, the two curves in Figure 4b reveal that the difference between fundamental frequency and edge frequency seems to remain steady despite the variational stress. Thus, the resonant frequency of edge modes can be described as

_{bias}is a constant. Equation (4) is then used to fit the MD results, showing an excellent R-square value up to 0.99, as displayed in Figure 4b, and the frequency gap between the fundamental modes and edge modes is about 32.6 GHz. From a different perspective, unlike the surface of fundamental modes warping along the axis of absorbate mass in the chart in Figure 4a, the edge modes’ fitting surface remained flat, thus revealing that edge modes are quite unsensitive to the centrally distributed adsorbates. To exhibit the relations between frequencies and adsorbed mass clearly, the sections of 10, 15, 20, and 25 GPa from Figure 4a were chosen and are individually presented in Figure 4c. It is apparent that the fundamental frequencies show a clear linear relation with the mass of attached gold atoms. Moreover, the further fitting results show the slopes of −60.9, −85.9, −96.0, and −107 with R-squares of 0.98, 0.98, 0.99, and 0.99, respectively. In contrast, the frequencies of edge modes show low variances of less than 4% in response to the absorbates. The corresponding R-squares are 0.43, 0.30, 0.36, and 0.17, respectively, which indicates weak linear relationships. As a result, the frequency of the edge mode still conforms to Equation (4) with the centrally distributed adsorbates.

#### 3.2.2. Determination of Centrally Distributed Mass in the Two Modes

_{0}is the mass of the graphene sheet (excluding fixed areas); C

_{bias}is a constant as illustrated before, which is irrelevant to the stress in graphene. Note that the above derivation is under the premise that the mass is centrally distributed; nevertheless these adsorbates cannot be located at the exact middle of graphene sheets. Thus, the acceptable range of the adsorbates where the mass determination method proposed above is applicable is worth discussing.

#### 3.2.3. Constraints for Evenly Distributed Adsorbates

_{edge}≈ f

_{fundamental}− 30. In this case, we cannot obtain two independent equations regarding absorbate mass and stress and solve them together like in the case of central distribution. That is, the frequency of edge modes cannot provide more independent and valid information for mass determination, so the mass determination method proposed above is not applicable for evenly distributed adsorbates.

#### 3.3. Effects of Edge Mode on the Q Factor

_{0}and A

_{t}are the amplitudes of fundamental oscillation at the beginning and end of simulation, respectively. In terms of Equation (6), the Q factors of these four samples are then calculated to be 1463, 2520, 2321 and 4405, respectively. It is apparent from these four charts that the graphene sheet with smallest width and stress has much more energy loss in oscillation, while that with the largest width and stress has the least energy loss, and its Q factor is three times bigger than that of the former. The intermediate two also clearly show less amplitude diminishing, with Q factors of 2520 and 2321, respectively. Incidentally, the ripples on the margins of these amplitude curves occur due to the incoherent mixing of vibrations of two modes with different frequencies, which is called the beating phenomenon. Comparing the four results, it can be seen that enlargement of the width and stress could effectively improve the Q factor of graphene resonators.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Atomic schematic of doubly clamped monolayer graphene resonator for classic molecular dynamics (MD) simulation.

**Figure 2.**Normalized fundamental spectrum of kinetic energy for a doubly clamped graphene sheet with a length of 100 Å and a width of 50 Å.

**Figure 3.**(

**a**) Shape of edge eigenmodes. Lengths of fixed and free edges are 50 and 100 Å, respectively. Maximum out-of-plane displacement of the edge mode is 3.8 Å. (

**b**) Normalized spectrums of average out-of-plane displacements of center group and edge group of carbon atoms, respectively.

**Figure 4.**Resonant frequencies of fundamental and edge modes versus (

**a**) centrally distributed absorbed mass and axial stress, (

**b**) axial stress with no adsorbates, and (

**c**) absorbed mass the axial stress in graphene sheet ranging from 10 to 25 GPa. Length and width of the doubly clamped graphene sheet are 100 and 50 Å, respectively.

**Figure 5.**Resonant frequencies of the fundamental modes and edge modes versus the adsorbate positions along the width of graphene sheets. (

**a**) Width of the graphene sheet is 50 Å. (

**b**) Width of the graphene sheet is 80 Å.

**Figure 6.**(

**a**) Resonant frequencies of fundamental modes and edge modes versus evenly distributed absorbed mass and axial stress. Length and width of the considered doubly clamped graphene sheet are 100 and 50 Å, respectively. (

**b**) Resonant frequency gap between fundamental modes and edge modes versus the absorbed mass with axial stress of 10 GPa.

**Figure 7.**(

**a**–

**d**) Diminishing vibration amplitude with time for four graphene sheets with different widths and stress. (

**a**) 50 Å, 10 GPa. (

**b**) 50 Å, 25 GPa. (

**c**) 80 Å, 10 GPa. (

**d**) 80 Å, 25 GPa. (

**e**) Q factor versus axial stress. (

**f**) Q factor versus width.

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**MDPI and ACS Style**

Xiao, X.; Fan, S.-C.; Li, C.
The Effect of Edge Mode on Mass Sensing for Strained Graphene Resonators. *Micromachines* **2021**, *12*, 189.
https://doi.org/10.3390/mi12020189

**AMA Style**

Xiao X, Fan S-C, Li C.
The Effect of Edge Mode on Mass Sensing for Strained Graphene Resonators. *Micromachines*. 2021; 12(2):189.
https://doi.org/10.3390/mi12020189

**Chicago/Turabian Style**

Xiao, Xing, Shang-Chun Fan, and Cheng Li.
2021. "The Effect of Edge Mode on Mass Sensing for Strained Graphene Resonators" *Micromachines* 12, no. 2: 189.
https://doi.org/10.3390/mi12020189