We study circular nanomechanical graphene resonators by means of continuum elasticity theory, treating them as membranes. We derive dynamic equations for the flexural mode amplitudes. Due to the geometrical nonlinearity the mode dynamics can be modeled by coupled Duffing equations. By solving the Airy stress problem we obtain analytic expressions for the eigenfrequencies and nonlinear coefficients as functions of the radius, suspension height, initial tension, back-gate voltage and elastic constants, which we compare with finite element simulations. Using perturbation theory, we show that it is necessary to include the effects of the non-uniform stress distribution for finite deflections. This correctly reproduces the spectrum and frequency tuning of the resonator, including frequency crossings.

https://iopscience.iop.org/article/10.1088/0957-4484/24/39/395702/pdf

Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators.

We study circular nanomechanical graphene resonators by means of continuum elasticity theory, treating them as membranes. We derive dynamic equations for the flexural mode amplitudes. Due to geometrical nonlinearity these can be modeled by coupled Duffing equations. By solving the Airy stress problem we obtain analytic expressions for eigenfrequencies and nonlinear coefficients as functions of radius, suspension height, initial tension, back-gate voltage and elastic constants, which we compare with finite element simulations. Using perturbation theory, we show that it is necessary to include the effects of the non-uniform stress distribution for finite deflections. This correctly reproduces the spectrum and frequency tuning of the resonator, including frequency crossings.

Frequency tuning, nonlinearities and mode coupling in circular graphene resonators

We use a scalable optical technique to characterize more than 21000 circular nanomechanical devices made out of suspended single- and double-layer graphene on cavities with different diameters ($D$) and depths ($g$). To maximize the contrast between suspended and broken membranes we used a model for selecting the optimal color filter. The method enables parallel and automatized image processing for yield statistics. We find the survival probability to be correlated to a structural mechanics scaling parameter given by $D^4/g^3$. Moreover, we extract a median adhesion energy of $\Gamma =$ 0.9 J/m$^2$ between the membrane and the native SiO$_2$ at the bottom of the cavities.

Very large scale characterization of graphene mechanical devices using a colorimetry technique

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Freestanding 2D Materials and their Applications: From Lab to Fab

Since the introduction of the iPhone and Wii gaming console in 2007 the market for mobile sensor applications has exploded. A continuous effort is made to make these sensors smaller, better and cheaper. This allowed for new applications that caused the market to grow exponentially in recent years. Pressure sensors have recently found implementation in the mobile sensor market, however they require lots of fabrication steps making them relatively expensive. Due to its exceptional properties, graphene can be used as both an electrical and mechanical element reducing the amount of fabrication steps needed to build a pressure sensor. Using this principle, concepts for an absolute (atmospheric) pressure sensor and pressure difference (gage) sensor were developed. Novel analytical solutions were developed and compared to existingmeasurements of similar devices or to numerical simulations. It is found that these devices will not only be smaller and cheaper, but also have a larger bandwidth and less sensitivity to mechanical disturbances. The thermal noise of these devices is orders of magnitude lower than similar silicon based devices, therefore it is expected that graphene based sensors will meet the demands on accuracy, resolution and signal-to-noise ratio of current high-precision pressure sensors. It is concluded that graphene based resonant pressure sensors have the potential to meet the future mobile market demands.

Graphene Based Pressure Sensors: Resonant Pressure Transduction using Atomically Thin Materials

Free axisymmetric vibrations of a stretched circular membrane are studied using a membrane theory consisting of a pair of non-linear partial differential equations coupled between the transverse and radial displacements of the membrane. A systematic perturbation method, in which the amplitude of the transverse displacement is taken as the perturbation parameter, is used to obtain periodic solutions of the non-linear equations. The initial membrane strain enters the problem as a parameter which is allowed to vary over a range of values. A case of self-resonance is encountered when the initial membrane strain approaches some critical values. This self-resonance case is also treated through an appropriate modification of the perturbation method.

T.W. Lee

Papers

On the non-linear vibrations of a circular membrane☆