Nonlinear vibration behavior of graphene resonators and their applications in sensitive mass detection
© Dai et al.; licensee Springer. 2012
Received: 27 June 2012
Accepted: 6 August 2012
Published: 4 September 2012
Graphene has received significant attention due to its excellent mechanical properties, which has resulted in the emergence of graphene-based nano-electro-mechanical system such as nanoresonators. The nonlinear vibration of a graphene resonator and its application to mass sensing (based on nonlinear oscillation) have been poorly studied, although a graphene resonator is able to easily reach the nonlinear vibration. In this work, we have studied the nonlinear vibration of a graphene resonator driven by a geometric nonlinear effect due to an edge-clamped boundary condition using a continuum elastic model such as a plate model. We have shown that an in-plane tension can play a role in modulating the nonlinearity of a resonance for a graphene. It has been found that the detection sensitivity of a graphene resonator can be improved by using nonlinear vibration induced by an actuation force-driven geometric nonlinear effect. It is also shown that an in-plane tension can control the detection sensitivity of a graphene resonator that operates both harmonic and nonlinear oscillation regimes. Our study suggests the design principles of a graphene resonator as a mass sensor for developing a novel detection scheme using graphene-based nonlinear oscillators.
Graphene has recently been attracting the scientific community due to its excellent electrical [1–3] and/or mechanical properties [4–8]; these remarkable properties have enabled the exploitation of graphene for the development of nano-electro-mechanical system (NEMS) such as nanoresonators [9, 10]. Specifically, since a pioneering work by researchers at Cornell , graphene has recently been extensively taken into account for designing nanoresonators that can exhibit high-frequency dynamic range [11–13] with favorable high Q factors [13–17]. The high-frequency dynamics of graphene is attributed to its excellent mechanical properties such as Young's modulus of approximately 1 TPa [4–8, 18]; it is noted that a resonant frequency is linearly proportional to the square root of Young's modulus when a device operates in harmonic oscillation [9, 10]. Until recently, most research works [11–15] (except a work by Eichler et al. ) have focused on the harmonic oscillation of a graphene resonator. However, the nonlinear vibration of a graphene resonator has not been well studied yet, albeit a recent study  reports an experimental observation of the nonlinear vibration of a graphene resonator. The nonlinear elastic deformation of a graphene is ubiquitous due to the fact that a monolayer graphene is an atomically thin sheet so that the out-of-plane deflection of a graphene is much larger than its thickness , which indicates that a graphene can easily undergo a nonlinear elastic deflection. Moreover, as discussed in our previous study [9, 20, 21], the nonlinear vibration is a useful route to the development of novel sensitive detection scheme based on nanoresonators made of nanomaterials such as carbon nanotubes.
To gain a detailed insight into the underlying mechanism of the vibration of a graphene resonator, an atomistic simulation such as molecular dynamics (MD) simulation has been widely utilized. For instance, Park and coworkers [22, 23] have studied various effects such as edge effect and/or internal friction effect on the vibrational behavior of graphene resonators using MD simulation. Furthermore, Park and coworkers  have investigated the energy dissipation mechanism of vibrating polycrystalline graphenes fabricated from chemical vapor deposition method by using MD simulation. Despite the ability of MD simulation to provide detailed characteristics of the vibrational behavior of graphene resonators, MD simulation is computationally restricted to studying the vibrational behavior of a graphene resonator whose length scale is <10 nm (e.g., see refs. [22, 24]). On the other hand, most experimental studies have considered a graphene resonator whose length scale is >1 μm (e.g., see refs. [11–15]). This clearly indicates that a current atomistic simulation is unable to be utilized to analyze an experimentally observed vibrational behavior of a graphene resonator whose length scale is in the order of micrometer.
The computational limitation of atomistic simulations in depicting the underpinning principles of experimentally observed mechanics of graphene resonators has led researchers [19, 25–27] to consider a continuum elastic model, particularly a plate model, for unveiling the vibrational characteristics of a graphene resonator. In order for a continuum elastic model to dictate the atomistic feature of the mechanics of a graphene, the elastic constants of a continuum elastic model (e.g., plate model) for a graphene have to be determined from an atomistic simulation such as MD simulation as it was taken into account for deciding the elastic constants of atomic structures (e.g., lattice) [7, 8]. Recently, a plate model with its elastic constants obtained from MD simulation has allowed revealing the mechanisms of the mechanics of a graphene. More remarkably, in a recent study by Isacsson and coworkers , a plate model has been utilized for studying the vibrational behavior of a graphene resonator; it is shown that the vibrational behavior of a graphene resonator predicted from a plate model, whose elastic constants were determined from atomistic model, is consistent with an experimentally observed vibration of a graphene resonator. However, a recent study by Isacsson et al.  has only concentrated on the harmonic oscillation of a graphene resonator, even though a graphene resonator can easily reach the nonlinear vibration regime. To the best of our knowledge, despite recent studies [28, 29] theoretically reporting the nonlinear vibration of a graphene resonator, the nonlinear oscillation of a graphene resonator (particularly, nonlinearity tuning), as well as atomic mass detection using graphene-based nonlinear oscillators, has not been well studied based on a continuum elastic model and/or MD simulation.
In this work, we have studied the nonlinear vibration of a graphene resonator using a continuum elastic model, i.e., plate model. We have found that nonlinear oscillation is a useful avenue for improving the detection sensitivity of a graphene resonator and that the detection sensitivity of a graphene-based nonlinear oscillator is governed by both the actuation force (which determines the nonlinearity of vibration) and the size of a graphene resonator. It is shown that the nonlinearity of vibration for a graphene resonator can be tuned by an in-plane tension and that such in-plane tension can modulate the detection sensitivity of a graphene resonator that operates in both harmonic and nonlinear oscillations. In particular, an in-plane tension improves the dynamic frequency range and detection sensitivity of a graphene resonator that operated in harmonic oscillation, while an in-plane tension deteriorates the dynamic frequency range and sensing performance of a graphene-based nonlinear oscillator. Our study sheds light on a continuum elastic model for gaining insight into not only the underlying mechanisms of nonlinear vibration-based enhancement of the dynamic frequencies and sensing performance of a graphene resonator, but also the role of an in-plane tension in modulating the nonlinearity of a graphene resonator.
Theory and model
Here, is a potential field prescribed to an i-th carbon atom for a graphene, r is the atomic coordinates of a graphene, and N is the total number of carbon atoms for a graphene. Here, it should be noted that when the elastic constants of a graphene (i.e. κ and ES) are determined from Equation 2, an axial tension N0 is assumed to be zero (i.e., pre-stress is not applied to a graphene). As described in a literature , the force field parameters provide the elastic constants of a graphene such as κ = 1.5 eV and ESh = 2,000 eV/nm2.
The vibrational behavior of a graphene resonator can be numerically described by solving the Duffing equation given by Equation 8.
Here, Δm0 is the mass of atoms adsorbed onto the unit area of a graphene, while Δm is the total mass of adsorbed atoms onto the entire surface of a graphene. Based on inertia terms for both mass-adsorbed graphene (as described in Equation 10) and a bare graphene (represented in Equation 9.a), it is straightforward to compute the resonant frequency shift, Δω, of a graphene resonator due to the mass adsorption; Δω = ω(m + Δm) – ω(m), where ω(m) is the resonant frequency of a bare graphene resonator whose effective mass is given as m (with m = ρab), and ω(m + Δm) is the resonant frequency of a graphene resonator onto which the mass adsorption with the amount of Δm occurs. This frequency shift due to mass adsorption is typically negative since mass adsorption increases the overall mass of a resonator and, consequently, reduces the resonant frequency of a resonator with respect to that of a bare resonator.
Results and discussion
Resonance behavior of graphene
As described above, the resonant frequency of a graphene resonator predicted from a continuum elastic model is overestimated in comparison with that measured from experiment. This may be attributed to the boundary condition that we used in our simulation; in our simulation, we have utilized the fully clamped boundary conditions such as ψ(x y) = ∇ψ(x y) = 0 along all edges; this boundary condition is referred to as “fully clamped” boundary condition. In order to understand the effect of boundary condition on the resonant frequency of a graphene resonator, we have taken into account the deflection eigenmode in the form of ψ(x y) = sin(πx/a)⋅sin(πy/b), which satisfies the boundary condition as follows: ψ(x y) = 0 at all edges, but ∇ψ(x y) ≠ 0 along all edges. This boundary condition is referred to as “weakly clamped” boundary condition. As shown in Figure 2b, the resonant frequency of a graphene resonator that is weakly clamped is almost close to the theoretical predictions obtained from the membrane model . Moreover, it is shown that the resonant frequency of a graphene resonator is critically dependent on the boundary condition; the weak clamping of a graphene resonator reduces its resonant frequency. In this work, we have considered a fully clamped graphene resonator, otherwise specified. The resonant frequencies of graphene resonators predicted from our continuum elastic model (i.e., plate model) are consistent with experimentally measured frequencies of graphene resonators . In addition, it is shown that pre-strain increases the resonant frequency of a graphene (Figure 2c).
Resonant response of graphene resonators to mass adsorption
Even though graphene resonator has recently been extensively considered as a NEMS device for its applications in actuation, it has been barely employed as a mass sensor that enables highly sensitive atomic detection (e.g., measurement of atomic weight). The high detection sensitivity of a graphene resonator is attributed to the high-frequency dynamic range that is achieved due to high elastic stiffness and low mass density of a graphene resonator. In this study, we have scrutinized the resonant responses of graphene resonators, which undergo not only harmonic oscillation but also nonlinear vibration, to atomic adsorption onto the surface of a graphene resonator. Our study on the nonlinear response of a graphene resonator to mass adsorption is ascribed to our previous finding [9, 20, 21] that nonlinear oscillation is useful in improving the detection sensitivity of a nanoresonator.
We have also studied the effect of graphene size (D) on the frequency shift of a graphene resonator due to mass adsorption, Δω, with respect to the actuation amplitude (p0). It is found that for a graphene resonator whose size is D ≥ 150 nm, the increase of actuation amplitude enhances the frequency shift of a graphene resonator due to mass adsorption, which is consistent with our conjecture that nonlinear vibration improves the detection sensitivity of a graphene resonator. On the other hand, for a graphene resonator whose size is D ≤ 100 nm, an increase in the actuation amplitude does not significantly amplify the frequency shift of a graphene resonator due to mass adsorption in comparison to that of a large-scale graphene resonator (i.e., D > 200 nm). This indicates that even though large actuation amplitude induces the nonlinear oscillation of a small-scale graphene resonator (e.g., D < 100 nm), the nonlinear vibration does not remarkably increase the frequency shift due to mass adsorption. This result suggests that the length scale of a graphene resonator plays a central role in not only the dynamic frequency range of a graphene resonator but also in the sensing performance of graphene-based nonlinear oscillators.
Effect of pre-strain applied to graphene resonators on their resonance behaviors and sensing performances
As described in previous studies [9, 20, 21], a mechanical tension (due to pre-strain or pre-stress) applied to a resonator leads to the increase of the dynamic frequency range of a resonator as well as its sensing performance. In this study, we have investigated how a pre-strain applied to a graphene resonator improves not only the dynamic behavior of a graphene resonator but also the detection sensitivity of a graphene resonator that operates in both harmonic and nonlinear oscillations.
In order to validate our conjecture that an in-plane tension can play a role in the transition from nonlinear vibration to harmonic oscillation, as shown in Figure 5b, we have plotted the dimensionless parameter θ (representing the degree of nonlinearity) as a function of actuation amplitude and in-plane tension. When the actuation amplitude is in the order of 0.01 fN, the vibration behavior of a graphene becomes almost harmonic oscillation regardless of in-plane tension, as anticipated. For a graphene resonator actuated by an actuation amplitude of 1 fN, the dimensionless parameter for a bare graphene resonator (i.e. N0 = 0) is in the order of 10, which indicates that the vibration behavior of a bare graphene is almost nonlinear oscillation. On the other hand, when an in-plane tension with the amount of 10 pN/nm is applied to a graphene resonator actuated by the amplitude of 1 fN, the dimensionless parameter is in the order of 0.5, which indicates that an in-plane tension reduces the nonlinearity of the resonance of a graphene. As in-plane tension increases, the nonlinearity is significantly reduced even up to the order of 10−1, indicating that the resonance behavior is almost harmonic oscillation. Our result suggests that an in-plane tension plays a central role not only in increasing the dynamic frequency range of a graphene resonator but also in inducing the transition from nonlinear vibration to harmonic oscillation.
Figure 5c depicts the critical in-plane tension that is responsible for the transition from nonlinear resonance to harmonic oscillation. Here, the critical in-plane tension is defined as an in-plane tension at which nonlinear oscillation is transitioned to harmonic vibration. It is found that the role of in-plane tension in such transition is highly correlated with the size of a graphene resonator. For instance, for a graphene resonator whose size is D = 200 nm, the resonance behavior of a graphene actuated by an amplitude of ≤0.3 fN is almost close to harmonic oscillation. On the other hand, the vibrational behavior of a 350-nm graphene resonator driven by even an amplitude of 0.1 fN becomes nonlinear oscillation. This indicates that the size of a graphene resonator determines the actuation amplitude that is required to induce the nonlinear vibration of a graphene resonator. Moreover, it is shown that the smaller the graphene resonator is, the smaller is the amount of an in-plane tension that can induce the transition from nonlinear oscillation to harmonic resonance. This suggests that an in-plane tension-driven transition from nonlinear vibration to harmonic oscillation is determined by the size of a graphene resonator.
In this work, we have studied the vibrational behaviors of graphene resonators as well as their sensing performance based on continuum elastic model such as plate model. It is shown that nonlinear vibration is useful in improving the detection sensitivity of a graphene resonator and that an in-plane tension is able to tune both the nonlinearity of the vibrating graphene resonators and their detection sensitivity. It should be noted that, in this work, our continuum model is only applicable to a monolayer graphene resonator. For modeling the multilayered graphene resonator, the interactions between graphene sheets have to be considered in the continuum modeling , which will be studied for our future work. Moreover, our continuum elastic model discards the finite size effect (i.e., edge effect) on the dynamic behavior (and also sensing performance) of a monolayer graphene resonator; here, edge (stress) effect arises from the imbalance between coordination numbers for edge atoms and bulk atoms, respectively [25, 44]. This edge stress effect on a monolayer graphene is conceptually identical to the surface stress effect on a nanowire resonator . Such edge effect on the frequency behavior of a graphene resonator and its sensing performance will be studied for our future work.
This work was supported by the National Research Foundation (NRF) of Korea under grant numbers NRF-2010-0026223 and 2012R1A1A2008616 (to KE). One of the authors (C-WK) gratefully acknowledges the support from NRF under grant numbers 2012R1A2A2A04047240 and 2012R1A1A2008870.
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