Tailoring the Resonance of Bilayer Graphene Sheets by Interlayer sp3

H. F. Zhan†, Y. Y. Zhang‡, J. M. Bell†, B. C. Zhang§, and Y. T. Gu*† ... If you have an individual subscription, please log in using your ACS...
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Tailoring the Resonance of Bilayer Graphene Sheets by Interlayer sp3 Bonds H. F. Zhan,† Y. Y. Zhang,‡ J. M. Bell,† B. C. Zhang,§ and Y. T. Gu*,† †

School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, 2 George Street, Brisbane, QLD 4001, Australia ‡ School of Computing, Engineering & Mathematics, University of Western Sydney, Locked Bag 1797, Sydney, NSW 2751, Australia § School of Mechanical and Power Engineering, North University of China, Xueyuan Road 3, Taiyuan Shanxi 030051, China ABSTRACT: Graphene-based resonators are envisioned to build the ultimate limit of 2-D nanoelectromechanical system due to their ultrasensitive detection of mass, force, pressure, and charge. However, such application has been greatly impeded by their extremely low quality factor. In the present work, we explore, using the large-scale molecular dynamics simulation, the possibility of tailoring the resonance properties of a bilayer graphene sheet (GS) with interlayer sp3 bonds. For the bilayer GS resonator with interlayer sp3 bonds, we discovered that the sp3 bonds can either degrade or enhance the resonance properties of the resonator depending on their density and location. It is found that the distribution of sp3 bonds only along the edges of either pristine or hydrogenated bilayer GS leads to a greatly enhanced quality factor. A quality factor of ∼1.18 × 105 is observed for a 3.07 × 15.31 nm2 bilayer GS resonator with sp3 bonds, which is more than 30 times larger comparing with that of its pristine counterpart. The present study demonstrates that the resonance properties of a bilayer GS resonator can be tuned by introducing sp3 bonds. This finding provides a useful guideline for the synthesis of the bilayer GS for its application as a resonator component.

1. INTRODUCTION

The low Q-value originates from different extrinsic and intrinsic loss mechanisms. The extrinsic loss mechanisms include gas damping, clamping or support losses, and coupling losses mediated by transducers.8,11 For instance, by introducing a tensile strain to the resonator, the inherent clamping losses can be significantly decreased.12 Experiments have demonstrated that the Q-value can exceed 7000 for graphene resonators when they are subjected to a tensile strain induced by annealing.13 Numerical studies report that the spurious edge modes arising from the free edges of the graphene GS are the dominant intrinsic loss mechanisms underlying the low Qfactor.14,15 This result has been further affirmed by an experiment. In the experiment, a circular monolayer graphene GS with all sides fixed is found to possess a Q-factor around an order of magnitude larger than that in the previous experimental measurements.16 Recent theoretical studies also revealed that the coupling between flexural modes and in-plane phonons will lead to the linear and nonlinear damping of the out-of-plane vibrations, which consequently affect the Q-factor of the graphene resonator.17,18 The damping is strongly dependent on the amplitude of the motion. When the driving force decreases until the damping motion becomes barely detectable, an extremely high Q-factor (about 100 000) is observed for a graphene resonator at 90 mK.19 In addition, the thermoelastic damping losses is also an important factor leading

Graphene has been an intense focus of research as well as applications ever since its discovery in 2004.1 Graphene is a flat monolayer arranged in a honeycomb crystal structure containing sp2-bonded carbon atoms. This endows it with enticing properties such as very high mechanical stiffness, electron mobility, electrical and thermal conductivity, and many others.2,3 These record-breaking properties of graphene open up huge potential applications in the areas of electronics, photonics, composite materials, energy generation and storage, and sensors.3−6 Being a single layer of carbon atoms, graphene resonators are envisioned to build the ultimate limit of the 2-D nanoelectromechanical system (NEMS) due to their ultrasensitive detection of mass, force, and charge.7 To attain a NEMS with such high sensitivity and reliability, it is crucial to keep a lowenergy dissipation, which is inversely proportional to the quality (Q)-factor within a resonant mechanical element.8 However, graphene-based resonators have been found to possess an extremely low Q. This has greatly impeded its application as a sensing component or detector. In situ testings have shown that for a 20 nm thick multilayer graphene sheet (GS), the Q-factor ranges from 100 to 1800 as the temperate decreases from 300 to 50 K.7 By using scanning force microscopy, multilayer GSs suspended over SiO2 trenches have been reported with an extremely low Q between 2 and 30.9 A recent measurement on monolayer graphene resonator showed a Q around 1500 at the temperature of 7 K.10 © 2013 American Chemical Society

Received: November 6, 2013 Revised: December 15, 2013 Published: December 16, 2013 732

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to the low Q-factor. Experimental results have demonstrated a strong relationship between Q-factor and the temperature for a monolayer graphene resonator, with the Q approaching 10 000 at 5 K20 and 9000 at 10 K.21 Attempts continue to find an effective means to enhance the Q-factor of graphene-based resonators. Robinson et al.22 found that the multilayer graphene oxide films exhibit a high Q-factor (up to 4000) due to the existence of covalent cross-linking in the neighboring platelets. Inspired by this, we have looked for the opportunities of tailoring the resonance properties of a bilayer GS resonator through the modification of sp2 bonds to a hybrid sp2-sp3 bonds. Experiments have shown that electron irradiation23 and femto second-laser excitation24 can create interwall bonds for graphite and pristine multiwall carbon nanotubes (MWCNTs), which in turn enhance interwall coupling and sliding resistance. It is also verified from experiments and from molecular dynamics (MD) simulations that the mechanical properties of MWCNTs can be improved dramatically by connecting the concentric layers via the formation of interwall sp3 bonds25 or other cross-linkings.23 Previous works have also proven that the edges of graphene can be effectively reconstructed by creating bilayer and even triplelayer looped edges through Joule heating treatment.26−28 In particular, recent quantum MD calculations have evidenced the existence of sp3-linked bilayer graphene, which is induced by the vacancies (created by electron irradiation) on adjacent GSs.29

that the oscillations occur in the linear regime. As illustrated in Figure 1, both ends of the bilayer GS were fully fixed in the three directions so as to mimic a doubly clamped configuration. The zigzag and armchair edges are oriented along the x axis (width direction) and y axis (length direction), respectively. To capture the impacts on the spurious edge modes14 from the presence of sp3 bonds, no periodic boundary conditions were utilized in three dimensions during the simulation process, which provides a close mimic to the real working condition of a nanoresonator. The overall simulation methodology to study the oscillatory properties of the bilayer GS is identical to that used previously for metal nanowires.12,14,35−37

2. NUMERICAL IMPLEMENTATION The resonance properties of the bilayer GS coupled by sp3 interlayer bonds have been investigated by a series of MD simulations, carried out using the open-source LAMMPS code.30 Following the previous definition,25,31 the sp3 interlayer bonds were created in the bilayer GS by gradually moving two atoms with the same planar coordinates (one in the upper layer and the other in the lower layer) to the sp3 bond distance (∼1.54 Å). The density of the sp3 bonds is defined as the number of sp3 bonds divided by the effective number of atoms in the system. The pristine bilayer GS contains 1728 carbon atoms with a length of ∼10 nm and a width of ∼2 nm. The interlayer distance between the pristine bilayer GS is assumed to be 3.35 Å.31 The densities of sp3 bonds range from 0.45 to 12.84%. To ensure a complete sp3 bond, edge bonds will not be modified and the largest density of sp3 bonds is ∼21.97% for an AB stacking bilayer GS studied here. In the MD simulations, the interaction between the carbon atoms in GS is described by the reactive empirical bond order (REBO) potential,32 which has been widely used to investigate mechanical and thermal properties of carbon-based nanomaterials. During the simulation, the initial equilibrium configuration of bilayer GS was accomplished by using the conjugate gradient algorithm. The bilayer GS was then equilibrated at 10 K (NVT ensemble) for 400 ps by the Nosé−Hoover thermostat33,34 with a time step of 0.5 fs while keeping both ends fixed. Finally, the bilayer GS was actuated by applying a sinusoidal velocity excitation ν(z) = A sin(ky) along the z axis, where A is the actuation amplitude (equal to 1.5 Å/ps), and k is π/L (here L is the effective length of the bilayer GS, which excludes the two fixed edges; see Figure 1). Thereafter, the bilayer GS undergoes free vibration in the NVE ensemble. The applied velocity field increased the total potential energy by merely 0.1% (corresponding to an initial transverse displacement less than 2% of the bilayer GS’s height), which ensures

3. RESULTS AND DISCUSSION To elucidate the influence of the interlayer sp3 bonds on the resonance properties of bilayer GS, we define the Q-factor as the ratio between the total system energy and the average energy loss in one radian at resonant frequency,38 that is, Q = 2πE/ΔE, where E is the total energy of the vibration system and ΔE is the energy dissipated by damping in one cycle of vibration. Assuming that Q-factor is constant during vibration, then after n vibration cycles, the maximum energy En is related to the initial maximum energy E0 by En = E0(1 − 2π/Q)n.39 Because the energy-preserving (NVE) ensemble was adopted during the vibration, the loss of potential energy (PE) must be converted to kinetic energy (KE). Therefore, the time history of the external energy (EE) will be tracked for the calculation of Q-factor. EE is defined as the difference of PE before and after the initial excitation is applied to the GS.12 It is worth noting that the extrinsic loss mechanisms that reduce the Q-factor, such as the gas damping and clamping losses, are not considered in this work. A discussion of the vibration frequency from the discrete Fourier transform40 will be given in the following section. First of all, the resonance properties of a pristine bilayer GS are furnished as a reference for the effect of sp3 bonds. As shown in Figure 2a, the amplitude of the external energy decreases quickly with the increase in time, which indicates a low Q-factor (estimated as 2310). Figure 2b gives the corresponding frequency spectrum of the external energy, which clearly shows the natural frequency of the external energy as 236 GHz. As the energy is a square function of the velocity; that is, the natural frequency of the bilayer GS is half of the external energy frequency, or 118 GHz. The atomic configurations of the bilayer GS at 495 and 510 ps are plotted in Figure 2c, from which the edge atoms are found to vibrate differently from that of a bulk atom (with a normal coordination number). As previously reported by Kim and

Figure 1. Simulation model of a doubly clamped bilayer GS with randomly distributed interlayer sp3 bonds: (a) top view and (b) front view. Inset shows a sp3 bond. (c) Profile of a sinusoidal velocity actuation.

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Figure 2. Simulation result for a pristine bilayer GS: (a) Time history of the external energy; the red line describes the energy dissipation trend. (b) Frequency spectrum. (c) Atomic configurations of the bilayer GS at 495 and 510 ps. Inset shows the edge mode of the edge atoms. Atoms are colored based on their z coordinates.

Figure 3. Simulation results for bilayer GS with randomly distributed sp3 interlayer bonds: (a) Time history of the external energy for the GS with 0.45% sp3 bonds; inset 1 shows the dispersion locations of the sp3 bonds, and insets 2 and 3 show the atomic configurations of the bilayer GS at 15 ps. Atoms are colored according to their z coordinates. (b) Frequency spectrum. Time history of the external energy for the GS with: (c) 3.96% sp3 bonds and (d) 7.98% sp3 bonds. The red line describes the energy dissipation trend.

Park14 for a monolayer resonator, the spurious edge modes of vibration are the dominant intrinsic loss mechanism behind the low Q-factor for a bilayer GS resonator. 3.1. Randomly Distributed sp3 Interlayer Bonds. To eliminate the edge effect, one can virtually fix all edges of the bilayer GS during the simulation. As previously reported for a monolayer GS resonator,14 a significantly higher Q-factor can be obtained through such a scheme, but fixing all edges is not the usual working condition for a real resonator. With this respect, we explore the possibility of using sp3 interlayer bonds to improve the Q-factor. At the beginning, we investigate a bilayer GS with randomly distributed sp3 bonds. The density of sp3 bonds ranges from 0.45 to 12.84%. To guarantee a reasonable comparison, we restricted the generation of the random distribution locations with the same starting value; that is, the locations of the sp3 bonds in a lower density case will also be contained in a larger density case. In Figure 3, the time history of the external energy for the defective bilayer GS with different densities of sp3 bonds is compared with that of their pristine counterpart. As is seen in Figure 3, the GS with sp3 bonds exhibits greatly changed vibration behavior. For the case with 0.45% sp3 bonds, the amplitude of the external energy initially decreases sharply and then arrives at a relatively stable state but shows fluctuation in the external energy with a period of ∼500 ps. Besides of the edge modes of vibration, this defective GS is also found to exhibit two vibration modes, as revealed by the frequency spectrum in Figure 3b; for example, two frequency peaks are detected. According to the atomic configurations of the GS at 15 ps, these two resonance frequencies are actually related to the longitudinal and transverse vibration of the GS (see the insets 2 and 3 in Figure 3a). It can be seen in the first inset of

Figure 3a that the random dispersion of the sp3 bonds has broken the homogeneity of the GS along the x axis (transverse direction), and this is likely to make the transverse vibration mode easier to activate, as seen in the simulation. Similar profiles of the external energy are observed for the bilayer GS with 0.96−3.96% sp3 bonds, which indicate that the proportion of sp3 bonds is not the only factor determining the energy dissipation during vibration. One example of such profile is presented in Figure 3c for the GS with 3.96% sp3 bonds. The amplitude of the external energy initially decreases rapidly with the increasing time and then stabilizes after 1000 ps. This implies more rapid energy dissipation and a lower Qfactor when compared with the results in Figure 2 for the pristine GS. The bilayer GS with 5.94−12.84% sp3 bonds exhibits an apparently enhanced Q-factor. As seen in Figure 3d, the amplitude of the external energy for the GS with 7.98% sp3 bonds decreases slowly with the increasing time. The amplitude is as high as 1.0 eV after 1500 ps. Such trajectory has led to a high Q-factor (estimated as 25 370), more than ten times larger than that of the pristine GS. This observation signifies that the presence of sp3 bonds can greatly enhance the Q-factor when its proportion is above a certain value. Next, we turn our attention to the effects of the sp3 bonds on other properties of the bilayer GS. Figure 4a compares the vibration density of states (VDOS) among five different cases (computed based on the autocorrelation function of the atomic velocities41). From Figure 4a, it is readily seen that the VDOS of the bilayer GS with sp3 bonds has clearly changed over the entire frequency range when compared with the pristine counterpart. The most significant changes are found in the lowand high-frequency modes. This result indicates that the 734

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bonds in the bilayer GS will increase its resonance frequency as well as VDOS. 3.2. Controlled Dispersion of sp3 Interlayer Bonds. From the previous discussion, the Q-factor of the bilayer GS may either decrease (in Figure 3a,c) or increase (in Figure 3d), while the proportion of sp3 bonds is not the only factor that determines this influence. Such observation agrees well with the previous study,42 which reported that the influence of mythyl (CH3) on the tensile properties of a monolayer GS heavily relies on the location, distribution, and coverage of CH3. In the following, we investigate the location of sp3 bonds by allocating sp3 bonds to different specific locations. Initially, 0.96% sp3 bonds were randomly distributed along either one or two available horizontal locations (the edge locations), as shown in the insets of Figure 5a,b. It is clearly observed that the GS with sp3 bonds locating along these two edges (see Figure 5b) possesses much better Q-factor than that of the GS with sp3 bonds distributed along one edge only (in Figure 5a). This result suggests that a symmetric dispersion of sp3 bonds along the two transverse edges is more effective in reducing the edge modes of vibration. Following this finding, we considered the GS with constrained allocation of sp3 bonds along both edges, with varying density of sp3 bonds. Figures 5c,d shows the results for the GS with the edges being partially and fully occupied by sp3 bonds, respectively. Evidently, the latter case shows an excellent vibration performance than the former one, and its Q-factor is up to 13 360, over five times larger than that of the pristine GS in Figure 2a. In general, if there are more sp3 bonds along the two edges, the bilayer GS resonator will have a higher Q-factor. Other configurations that contain sp3 bonds along transverse lines, as schematically illustrated in the insets of Figure 5e,f, were also investigated. As expected, the GS with evenly distributed sp3 bonds exhibits better resonance performance (than the one with concentrated sp3 bonds), although it is not as good as the GS with a similar density (i.e., 1.47%) of sp3 bonds locating along the two outmost transverse lines (not shown here, which shows a Q around 2740). Further testing has shown that the bilayer GS with a higher density of vertically

Figure 4. (a) Vibration density of states (VDOS) of the bilayer GS with different densities of sp3 bonds. (b) Relative frequency of the bilayer GS as a function of the density of sp3 bonds. Black rectangular marker represents the transverse resonance frequency that is detected during vibration for the bilayer GS with 0.45% sp3 bonds.

introduction of sp3 bonds will also affect the phonon−phonon scattering mechanisms and the associated thermal property of the bilayer GS. A comparison of the natural frequency is presented in Figure 4b. Interestingly, we found a quasiparabolic relationship of the relative frequency with respect to the percentage of sp3 bonds (relative frequency is the ratio of the natural frequency between the bilayer GS with and without sp3 bonds). The relative natural frequency of all the cases is greater than unity for all samples, with sp3 bonds density lower than 10%, implying an increased resonant frequency due to the presence of sp3 bonds. Simulation results have shown that higher density of sp3 bonds will significantly decrease the resonance frequency of the bilayer GS, and the higher vibration modes can be easily excited, which is attributed to the reduced flexural rigidity of the bilayer GS when more sp2 bonds are replaced by sp3 bonds. Therefore, the existence of certain sp3

Figure 5. MD results from the bilayer GS with controlled distribution of sp3 bonds. Time history of the external energy for the bilayer GS with: (a) Randomly distributed 0.96% sp3 bonds along one edge, (b) randomly distributed 0.96% sp3 bonds along both edges, (c) both edges partially occupied by sp3 bonds, (d) both edges fully occupied by sp3 bonds, (e) sp3 bonds along three middle adjacent transverse lines, and (f) sp3 bonds along three equally spaced transverse lines. The insets in all Figures represent the corresponding models. 735

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aligned sp3 bonds does not perform better than that the configuration presented in Figure 5f. From Figure 5, we conclude that the impacts of the sp3 bonds on the resonance properties of the bilayer GS are determined by both their density and location, and the case with the two edges being fully occupied by sp3 bonds exhibits the highest Q-factor and excellent resonant properties. 3.3. Effect of Ordered sp3 Interlayer Bonds. In this section, the findings above were further assessed by moving the sp3 bonds from the outermost edge line of atoms successively to the next line of atoms of the bilayer GS. For the bilayer GS with a width of 2 nm (16 atoms in the width direction excluding two edge rows), eight different configurations can be constructed (as schematically shown in Figure 6a). More

to weaken the edge vibration modes. It is clearly shown in Figure 6b that the other seven samples also show a higher Qfactor than the pristine one due to the presence of interlayer sp3 bonds, although such enhancement is not as significant as the second sample. The natural frequency as a function of the locations of sp3 bonds (indexed as above) is shown in the inset of Figure 6b. Similarly to the results presented in Figure 4b, a quasi-parabolic relationship between the relative resonance frequency and the location is observed, and all of the relative resonance frequencies are larger than unity. We also tested the bilayer GS with all dangling bonds being passivated by hydrogen. As illustrated in Figure 7a, without the sp3 bonds, the pristine bilayer GS is under a mixed vibration state that includes both transverse and longitudinal vibrations. Such vibration state can be easily distinguished from the front and side views of the GS at 190 and 195 ps (Figure 7b). However, only longitudinal vibration is observed for the bilayer GS with sp3 bonds (see insets 1−3 in Figure 7c), which leads to a high Q-factor of 1.31 × 105. Same as the previous case, the hydrogen-passivated bilayer GS with sp3 bonds being aligned along the second outmost edges exhibits the most superior resonance property. The bilayer GS resonators with larger size were also examined, including the size of about 3.07 × 15.31 × 0.335 and 4.05 × 20.42 × 0.335 nm3. Similar phenomena were observed from these simulations; that is, the bilayer GS with sp3 bonds locating along the second outmost lines of atoms possesses a greatly enhanced Q-factor. The profiles of the external energy obtained from the bilayer GS resonator (size around 3.07 × 15.31 × 0.335 nm3) without and with sp3 bonds are presented in Figure 8a,b, respectively. It is found that the Q-

Figure 6. (a) Schematic view of different locations of sp3 bonds. (b) Relative Q-factor as a function of the locations of sp3 bonds. The solid red line represents the unity. Inset shows the relationship between the relative frequency and the locations of sp3 bonds.

specifically, the sp3 bonds in the first configuration were aligned along the two outmost lines of atoms on the edge of the GS, as shown in inset of Figure 5d, while for the last configuration these sp3 bonds were constrained in two adjacent lines that are located in the middle of the bilayer GS. Figure 6b shows the Qfactor determined for each configuration as a function of the locations of these sp3 bonds, that is, indexed by the number of the lines of atoms from the edge, with the edge atoms being indexed as 1, and the second line of atoms from the edge indexed 2, and so on. Strikingly, the bilayer GS with the sp3 bonds located at the second outmost lines (index 2) is found to possess the excellent resonance properties with the Q-factor as high as 7.83 × 104, which is nearly 34 times that of the pristine bilayer GS. Such results suggest that locations along the second outmost lines are the most efficient locations for the sp3 bonds

Figure 8. Time history of the external energy for: (a) A pristine bilayer GS with the size of 3.07 × 15.31 × 0.335 nm3 and (b) the bilayer GS with sp3 bonds being aligned along the two second outmost lines.

Figure 7. (a) Time history of the external energy for the bilayer GS being fully passivated by hydrogen. (b) The top shows the front view of the corresponding atomic configurations at 190 ps (inset shows the enlarged view of the structure), and the bottom 1 and 2 illustrate the side view of the sample at 190 and 195 ps, respectively. (c) The bilayer GS with both hydrogen-passivation and sp3 bonds (that are aligned along the two second outmost lines); Bottom 1 shows the distribution of the sp3 bonds, and insets 2 and 3 show the front view of the sample at 190 and 200 ps, respectively. 736

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Figure 9. (a) Time history of the external energy for a bilayer GS with V12(ββ) divacancy bridging. (b) The top shows the top view of the corresponding model (inset shows a divacancy bridging), and the bottom shows the atomic configuration of the sample at 385 ps (inset shows an enlarged view of the structure). (c) Time history of the external energy for a bilayer GS with ‘spiro’ interstitial bridging. (d) The top shows the top view of the corresponding model (inset shows a ‘spiro’ interstitial bridging), bottom 1 shows the atomic configuration of the sample at 50 ps, and bottom 2 shows the corresponding sinusoidal schematic profile. (e) Time history of the external energy for a bilayer GS with Frenkel pair defect. (f) The top shows the top view of the corresponding model (inset shows a Frenkel pair defect), and the bottom shows the atomic configuration of the sample at 100 ps. For these three cases, a smaller velocity actuation amplitude is applied to avoid the activation of higher vibration modes (namely, 1.0 Å/ps).

factor for the bilayer GS with sp3 bonds is approaching 1.18 × 105, which is more than 30 times larger than that of its pristine counterpart, as illustrated in Figure 8a. 3.4. Other Interlayer Cross-Linkings. Finally, we extended the present study of sp3 bonds to other different types of interlayer cross-linkings that are formed due to the existence of primary vacancy defects:23,43 a V12(ββ) divacancy bridging (which has two two-fold coordinated C atoms surrounding each vacancy), a ‘spiro’ interstitial bridging (which has a four-fold coordinated C atom inserted between the two graphene layers), and a Frenkel pair defect (i.e., a threefold coordinated C atom sits below the vacancy in the upper layer). Similarly, we aligned these three different cross-linkings along the two edges of the bilayer GS. For the bilayer GS with bivacancy bridging, a surprisingly high Q is estimated (∼8.82 × 108). As shown in lower of Figure 9b, all edge atoms are in a buckled configuration after energy minimization, and this configuration appears intact during the simulation. This observation indicates that no edge modes of vibration have been triggered, despite the fact that the bivacancy bridging introduces two-fold coordinated C atoms to the bilayer GS. A similar phenomenon is also found for the bilayer GS with ‘spiro’ interstitial bridging (see bottom 2 of Figure 9d). It is worth mentioning that for the bilayer GS with ‘spiro’ interstitial bridging, two resonance frequencies are detected, but the amplitude of the second resonance frequency is much smaller than the first one. (The frequency spectrum is not shown here.) Such results signify the excitation of the second vibration mode of the bilayer GS, which is further affirmed from the sinusoidal profile of the bilayer GS presented in the bottom 2 of Figure 9d. Unlike these two cases, a much smaller Q-factor (∼2740) is obtained for the bilayer GS with Frenkel pair defects, although the bottom of Figure 9f shows that there is also no edge mode of vibration in this case. We should note that according to our results obtained from the presence of sp3 bonds, it is reasonable to observe such a lower Q for the case with Frenkel pair defects because those defects are not located at the outmost edges of the bilayer GS (considering the chemical stability of the structure, as compared in the top of Figure 9b,d,f).

Meanwhile, it is found that different bilayer cross-linkings exert different influence on the first natural frequency. The first frequencies for the bilayer GS with bivacancy bridging and Frenkel pair defect are 110.62 and 110.67 GHz, respectively, lower than that of the pristine GS. For the bilayer GS with “spiro” interstitial bridging, a relative higher frequency (∼119.67 GHz) is observed. According to the above discussion, it can be concluded that the resonance properties of the bilayer GS can be effectively manipulated through the specific allocation of interlayer bridging or cross-linkings.

4. CONCLUSIONS We have employed large-scale MD simulation to investigate the effect of interlayer sp3 bonds on the resonant properties of bilayer GS. By introducing randomly distributed or controlled configuration of sp3 bonds to the bilayer GS resonator, we found that the sp3 bonds can either degrade or improve the resonance properties of the resonator depending on their density and location. It is found that distribution of sp3 bonds along the entire two second outermost lines of the edge of the bilayer GS can enhance the Q-factor significantly. A Q-factor of ∼1.18 × 105 is observed for a 3.07 × 15.31 nm2 bilayer GS resonator with sp3 bonds, which is more than 30 times larger than that of a pristine bilayer GS. In addition, the presence of sp3 bonds can boost the first natural frequency of the bilayer GS. Like sp3 bonds, other cross-linkings such as divacancy bridging, ‘spiro’ interstitial bridging, and Frenkel pair defect are found to play a positive effect on the Q-factor. Our present study suggests that the bilayer GS holds great potential application as a resonator component with improved Q-factor via the introduction of controlled interlayer sp3 bonds, although the precise control of the locations of sp3 bonds remain a challenge in the current experiments.

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

Support from the ARC Discovery Project (DP130102120), the High Performance Computer resources provided by the Queensland University of Technology, and the National Natural Science Foundation of China (51275487) is gratefully acknowledged.

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