Deviation from the Normal Mode Expansion in a Coupled Graphene-Nanomechanical System

A significant deviation from the normal mode expansion is observed in the optomechanically measured thermal noise of a graphene membrane suspended on a silicon nitride nanoresonator. This deviation is due to the heterogeneous character of mechanical dissipation over the spatial extent of coupled eigenmodes, which is tuned through an avoided anticrossing. We demonstrate that the fluctuation-dissipation theorem permits a proper evaluation of the thermal noise of the coupled nanomechanical system. Since a good spatial homogeneity is delicate to ensure at the nanoscale, this approach is fundamental to correctly describing the thermal noise of nanomechanical systems which ultimately impact their sensing capacity.

Figure 1

The experimental setup. (a) SEM of a $20\times 20\mu {\mathrm{m}}^2$ suspended CVD-grown graphene monolayer supported on a 300-nm-thick SiN nanomembrane, as sketched in (b). (c) Experimental setup. A balanced homodyne detection measures the phase fluctuations of the probe-laser field reflected by the sample and monitors its position fluctuations. A second counterpropagating pump-laser beam can be intensity modulated to optomechanically drive the coupled nanoresonators. The experiment is performed at pressures below ${10}^{-3}\mathrm{mbar}$. (d) Model describing the inertially coupled nanoresonators. (e) Thermal noise of a graphene membrane. The sharp peaks on each side are weakly coupled SiN eigenmodes.

Figure 2

Thermal noise of the hybridized eigenmodes. (a) Thermal noise of the coupled nanomechanical system measured in the middle of the graphene membrane ($S_{{\delta x}_G}[\mathrm{\Omega }]$) when tuned to an anticrossing region by adjusting the pump intensity ($400\mu \mathrm{W}$). Lower traces are obtained after numerical background substraction. The solid lines are the best fits derived employing the normal mode expansion. The dashed green lines are fits using expression (4). (b) Spectra measured through the anticrossing for increasing tuning laser powers. The dashed lines are fits using Eq. (4) with the fitting parameters ${\mathrm{\Omega }}_{S,G}$, ${\mathrm{\Gamma }}_{S,G}$ reported in (c),(d) using $\mu =0.002$. The purple disks represent the measured coupled eigenfrequencies ${\mathrm{\Omega }}_{\pm }/2\pi$, and the solid lines are deduced from Eq. (3). (d) A similar analysis for damping rates ${\mathrm{\Gamma }}_{\pm }/2\pi$. (e) Relative Brownian temperature variation deduced from the fits.

Figure 3

Role of damping heterogeneity. (a) Numerical simulations of thermal-noise spectra deduced from the model (see the SM [20]) for varying uncoupled graphene frequency ${\mathrm{\Omega }}_G$ and different graphene damping rates. From left to right, ${\mathrm{\Gamma }}_G/2\pi =5$, 50, and 500 kHz (${\mathrm{\Omega }}_S/2\pi =2\mathrm{MHz}$, $\mu =0.1$, ${\mathrm{\Gamma }}_S/2\pi =5\mathrm{kHz}$). (Bottom panels) Thermal-noise spectra calculated with ${\mathrm{\Omega }}_G/2\pi =1.5\mathrm{MHz}$ for different graphene damping rates [5, 10, 20, 50, 100, 200, 300, and 500 kHz from (i) to (viii)] and respective coupling strength of $\mu =0.1$ and 0.01 for (b) and (c), respectively. The peak asymmetry and the noise reduction are absent in the case of homogeneous damping (i), and the simulations are well described by a modal expansion (the dashed lines).

Figure 4

Optomechanical response of the hybridized nanomechanical system. (a) Optomechanical response obtained by modulating the pump intensity for increasing modulation depths $\delta P$ with a fixed average tuning power ($P_0=60\mu \mathrm{W}$). (b) Maximum driven displacement reported as a function of $\delta P$. The solid line has a slope of $17\mu \mathrm{m}/\mathrm{W}$. (c) Optomechanical responses obtained for an increasing optical pump power $P_0$ (30% modulation strength). (d) Amplitude and phase of the mechanical susceptibility ${\chi }_{GG}$ derived for $400\mu \mathrm{W}$ of tuning power. The corresponding thermal-noise spectrum expected using Eq. (4) is reported in (e)(i) and presents a very good agreement with the measured spectrum (ii). The detection noise is included in traces (iii) and (iv). A 30% correction is used here on the optical to the force conversion factor determined in Fig. 4 to account for a slight modification of the actuation efficiency between both measurements.

Figure 5

Probing fluctuation-dissipation theorem validity at different tuning powers while following the same conventions as in Fig. 4. Response measurements of Fig. 4 are used to compute the local effective mechanical susceptibility subsequently injected in Eq. (4) and compared to the experimental spectra. For the entire data set, the only free parameter is an overall correction factor (from 0.8 to 1.2) to account for slight drifts in the readout and actuation efficiencies.