Resonant nonlinear response of a nanomechanical system with broken symmetry

We study the response of a weakly damped vibrational mode of a nanostring resonator to a moderately strong resonant driving force. Because of the geometry of the experiment, the studied flexural vibrations lack inversion symmetry. As we show, this leads to a nontrivial dependence of the vibration amplitude on the force parameters. For a comparatively weak force, the response has the familiar Duffing form, but for a somewhat stronger force, it becomes significantly different. Concurrently there emerge vibrations at twice the drive frequency, a signature of the broken symmetry. Their amplitude and phase allow us to establish the cubic nonlinearity of the potential of the mode as the mechanism responsible for both observations. The developed theory goes beyond the standard rotating-wave approximation. It quantitatively describes the experiment and allows us to determine the nonlinearity parameters.

Figure 1

Scanning electron micrograph of the doubly clamped silicon nitride sting resonator (green) and two adjacent gold electrodes (yellow) for dielectric drive and detection. Depicted is one of the clamping pads (right), the onset of the string, and the two control electrodes.

Figure 2

Duffing response and non-Duffing response. Measured nonlinear response curves at drive voltages of $V_d=9$ (gray) and $40\mathrm{mV}$ (black). Both traces display unprocessed raw data; the noise level remains below the size of the dots. A fit to the Duffing model at $V_d=9\mathrm{mV}$ as well as a theory curve using the obtained ${\gamma }_{\mathrm{eff}}^{\left(\mathrm{V}\right)}/{\left(2\pi \right)}^2=2.48\times {10}^{15}{\mathrm{V}}^{-2}{\mathrm{s}}^{-2}$ for $V_d=40\mathrm{mV}$ are included (red lines). The deviation of the data from the Duffing model is clearly visible for the $V_d=40\mathrm{mV}$ data. A better agreement is achieved by taking the influence of the cubic nonlinearity of the potential $\beta$ into account (green line). The curve is calculated for ${\beta }^{\left(\mathrm{V}\right)}/{\left(2\pi \right)}^2=2.08\times {10}^{15}{\mathrm{V}}^{-1}{\mathrm{s}}^{-2}$ and is truncated at a detuning of 2 kHz.

Figure 3

Spectrum of the resonantly driven fundamental mode ($M$). For a resonant drive at $f_d=f_0=6.528\mathrm{MHz}$ with $V_d=100\mathrm{mV}$, the overtone ($O$) at $13.056\mathrm{MHz}$ is clearly visible. Higher-order overtones are barely discerned in this representation. A constant noise background has been subtracted from the data.

Figure 4

(a) Drive voltage dependence of the resonant amplitude of the fundamental mode ($M$, black) as well as the amplitudes of the vibrations at twice ($O$, blue) and three times the drive frequency (dark blue); the amplitudes of the vibrations at the overtones have been rescaled to account for the different frequency-dependent displacement-to-voltage conversion factor. The drive frequency $f_d$ is fixed at the resonance frequency of the fundamental mode $f_0$. (b) The amplitude of the vibrations at $2f_d$ [marked by $O$ in (a)] depends quadratically on the amplitude of the fundamental mode. Data shown in blue. The red line is a quadratic fit.

Figure 5

The dependence of the vibration frequency as a function of the action variable of the mode $\omega \left(I\right)$ on the amplitude $A$ of the first harmonic. The squared amplitude is scaled by its characteristic value $\gamma /2{\omega }_0^2$ for a Duffing oscillator, whereas $\omega \left(I\right)-{\omega }_0$ is scaled by the eigenfrequency ${\omega }_0\equiv \omega \left(0\right)$ of small-amplitude vibrations. (a) The brown, petrol, and light green curves refer, respectively, to $\beta /{\omega }_0\sqrt{\gamma }=0,0.6$, and $\sqrt{0.9}$; the latter value corresponds to the critical cubic nonlinearity for which ${\gamma }_{\mathrm{eff}}=0$. The dark green curve is computed for the value of $\beta /{\omega }_0\sqrt{\gamma }$ used in the comparison with the experiment in Fig. 2. (b) Closeup showing $\omega \left(I\right)$ for the same $\beta$ as in Fig. 2 (dark green) compared with the result of the RWA for the Duffing model with the effective Duffing parameter ${\gamma }_{\mathrm{eff}}$ calculated for the same $\beta$ (light gray). The deviation of the green curve from the straight line demonstrates the inapplicability of the conventional RWA. Remarkably, for the considered case where $\gamma /{\gamma }_{\mathrm{eff}}\gg 1$ this deviation is pronounced in the range where the amplitude is still comparatively small, $A^2\ll 2{\omega }_0^2/\gamma$, but $A^2\sim \left|{\gamma }_{\mathrm{eff}}\right|{\omega }_0^2/{\gamma }^2$.

Figure 6

Resonant excitation of the overtone for a drive at half the eigenfrequency. (a) Amplitude of the resonantly excited vibrations at $2f_d$ as a function of the drive frequency $f_d$ swept around $f_0/2$ with $V_d=100\mathrm{mV}$. The red line displays a Lorentzian fit with half-width $\mathrm{\Gamma }$. (b) Peak amplitude of the resonantly excited vibrations at $f_0$ for $f_d=f_0/2$ as a function of the drive voltage. The red line corresponds to a quadratic fit.

Figure 7

Linear response of the fundamental out-of-plane mode ($M$) at a drive voltage of $V_d=1$ mV. A Lorentzian fit (red line) yields an eigenfrequency of 6.528 MHz, a linewidth of $2\mathrm{\Gamma }/\left(2\pi \right)=20\mathrm{Hz}$, and a quality factor of $Q\approx 325000$.

Figure 8

Response of the second spatial harmonic out-of-plane eigenmode $M_2$ appearing 200 kHz higher in frequency than the overtone $O$ of the fundamental flexural out-of-plane eigenmode $M$.

Figure 9

Influence of the nonlinear coupling to the driving force on the signal at $2f_d$. The green line shows the amplitude of vibrations at $2f_d$, which result from the nonlinear component of the drive and are described by Eq. (B2). The black and dashed blue lines are taken from Fig. 4 and show, respectively, the measured amplitudes of the fundamental mode $M$ and the signal at $2f_d$ (the amplitude of the signal at $2f_d$ has been rescaled to account for the different displacement-to-voltage conversion factor at this frequency). The results demonstrate a negligible influence of nonlinear driving.

Figure 10

DC voltage dependence of the nonlinearity parameters. Both ${\beta }^{\left(V\right)}$ and ${\gamma }_{\mathrm{eff}}^{\left(V\right)}$ are plotted against the square of the applied DC voltage. The sign of ${\gamma }_{\mathrm{eff}}^{\left(V\right)}$ changes from positive to negative near $V_{\mathrm{DC}}=6\mathrm{V}$. The cubic nonlinearity ${\beta }^{\left(V\right)}$ has been determined using the procedure described in Sec. 5. It is only accessible in the region of not too small $|{\gamma }_{\mathrm{eff}}^{\left(V\right)}|$.

Figure 11

Amplitudes $A_{nE}$ of the overtones at frequencies $n{\omega }_E$ with $n=2,3,4$. The amplitudes are calculated from Eq. (D9) by varying the energy $E$ and, respectively, the amplitude $A_{1E}$ of the main tone. For convenience of the comparison with the experiment, the amplitudes are scaled by the drive-dependent factor, but in fact they are calculated from Eqs. (D8) and (D9) in the absence of driving. Shown are the amplitudes $A_{2E}=2\left|a_2\right|$ (blue), $A_{3E}=2\left|a_3\right|$ (mid blue), and $A_{4E}=2\left|a_4\right|$ (dark blue).