Frequency tuning, nonlinearities and mode coupling in circular graphene resonators

TL;DR

This paper models circular graphene resonators using continuum elasticity, deriving equations for their vibrational modes, analyzing nonlinear effects, and providing analytic expressions for eigenfrequencies and nonlinear coefficients influenced by various physical parameters.

Contribution

It introduces a continuum elasticity model for circular graphene resonators, deriving analytic expressions for eigenfrequencies and nonlinear coefficients considering stress distribution effects.

Findings

Analytic expressions for eigenfrequencies as functions of physical parameters.

Non-uniform stress distribution significantly affects frequency tuning.

Frequency crossings are explained through the model.

Abstract

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.