Nonlinear damping in graphene resonators

Based on a continuum mechanical model for single-layer graphene, we propose and analyze a microscopic mechanism for dissipation in nanoelectromechanical graphene resonators. We find that coupling between flexural modes and in-plane phonons leads to linear and nonlinear damping of out-of-plane vibrations. By tuning external parameters such as bias and ac voltages, one can cross over from a linear- to a nonlinear-damping dominated regime. We discuss the behavior of the effective quality factor in this context.

Figure 1

Schematic view of a suspended graphene membrane over a trench in an insulating substrate. A metallic gate is used for actuating the resonator. In-plane phonons are created in the suspended region and dissipate energy as they propagate away.

Figure 2

Resonance frequency ${\Omega }_0$ vs bias voltage. Symbols denote results of numerical calculation. The dashed (red) and dashed-dotted (blue) lines show the contributions of mechanical stiffening and electrostatic softening for $T_0={10}^{-3}{T}_1$, respectively. Parameters are given in Table .

Figure 3

Ratio $\tilde{\delta }$ of nonlinear (NLD) and linear (LD) damping terms according to Eq. (9); (a) bias voltage and (b) ac voltage dependence. The thin dashed and dashed-dotted lines show the asymptotic behavior for strong LD and NLD. A crossover between the two regimes is achieved by changing the bias voltage. Parameters are given in Table .

Figure 4

Quality factor $Q$ vs bias voltage. (a) $Q$ calculated from Eq. (20) and (b) with additional voltage-independent damping $Q_{\mathrm{eff}}^{-1}=Q^{-1}+Q_0^{-1}$ with $Q_0={10}^5$. The gray area indicates the region of attainable $Q$ factors. The dashed lines correspond to the behavior in (a). Parameters are given in Table .

Figure 5

Static deflection $q_0$ vs bias voltage for three values of initial tension. The linear approximation [Eq. (B10)] is shown as dashed lines in the figure, while the squares and triangles correspond to the full numerical solution of the static problem.

Figure 6

Quality factor $Q$ vs bias voltage calculated from Eq. (20) for $V_{\mathrm{ac}}={10}^{-4}\text{V}$. The full and dashed lines show the result for a voltage dependent ${\Omega }_0\left(V_{\mathrm{dc}}\right)$ and constant ${\Omega }_0={\Omega }_0\left(0\right)$, respectively. Parameters are given in Table .