Graphene has recently been attracting the scientific community due to its excellent electrical [1
] and/or mechanical properties [4
]; these remarkable properties have enabled the exploitation of graphene for the development of nano-electro-mechanical system (NEMS) such as nanoresonators [9
]. Specifically, since a pioneering work by researchers at Cornell [11
], graphene has recently been extensively taken into account for designing nanoresonators that can exhibit high-frequency dynamic range [11
] with favorable high Q factors [13
]. The high-frequency dynamics of graphene is attributed to its excellent mechanical properties such as Young's modulus of approximately 1 TPa [4
]; 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
]. Until recently, most research works [11
] (except a work by Eichler et al. [17
]) 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 [17
] 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 [19
], which indicates that a graphene can easily undergo a nonlinear elastic deflection. Moreover, as discussed in our previous study [9
], 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
] 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 [24
] 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
]). On the other hand, most experimental studies have considered a graphene resonator whose length scale is >1
μm (e.g., see refs. [11
]). 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
] 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
]. 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 [19
], 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. [19
] 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
] 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.