Sliding nanomechanical resonators

Abstract

The motion of a vibrating object is determined by the way it is held. This simple observation has long inspired string instrument makers to create new sounds by devising elegant string clamping mechanisms, whereby the distance between the clamping points is modulated as the string vibrates. At the nanoscale, the simplest way to emulate this principle would be to controllably make nanoresonators slide across their clamping points, which would effectively modulate their vibrating length. Here, we report measurements of flexural vibrations in nanomechanical resonators that reveal such a sliding motion. Surprisingly, the resonant frequency of vibrations draws a loop as a tuning gate voltage is cycled. This behavior indicates that sliding is accompanied by a delayed frequency response of the resonators, making their dynamics richer than that of resonators with fixed clamping points. Our work elucidates the dynamics of nanomechanical resonators with unconventional boundary conditions, and offers opportunities for studying friction at the nanoscale from resonant frequency measurements.

Fig. 1. Unconventional resonant frequency tuning spectra in few-layer graphene (FLG) resonators.

a Colorized scanning electron microscope (SEM) image of the device. FLG (blue shaded stripe) is connected to source (S) and drain (D) electrodes and is suspended over a gate electrode (G). Scale bar: 1 micrometer. b Schematic of the device. A frequency-modulated voltage $V_{\mathrm{SD}}=V_0\cos \left[2\pi f_{\mathrm{d}}t+\left(f_{\mathrm{\Delta }}/f_{\mathrm{m}}\right)\sin 2\pi f_{\mathrm{m}}t\right]$ is applied between S and D, where $f_{\mathrm{d}}$ is the drive frequency, $f_{\mathrm{m}}=1.37$ kHz is the modulation frequency, and $f_{\mathrm{\Delta }}/f_{\mathrm{m}}\approx 75$ (see Methods). A dc voltage $V_{\mathrm{G}}$ is applied to G. A drain current $I$ at frequency $f_{\mathrm{m}}$ is measured. c Resonant frequency $f$ of the first and second vibrational modes of the resonator shown in a as a function of $V_{\mathrm{G}}$. d $f$ as a function of increasing (upper panel) and decreasing $V_{\mathrm{G}}$ (lower panel). Arrows indicate the stepping direction of $V_{\mathrm{G}}$. The drive power is −39 dBm in c, d.

Fig. 2. Resonant frequency loops and their dependence on V_G range.

a Frequency response of the mode whose resonant frequency is shown in Fig. 1d as a function of $f_{\mathrm{d}}$ and $V_{\mathrm{G}}$. The response is extracted from the spectrum of current $I$ which peaks at the resonant frequency. In stage 1, $V_{\mathrm{G}}$ is stepped from $20$ V up to ≈$30$ V. In stage 2, $V_{\mathrm{G}}$ is stepped from ≈$30$ V down to $25$ V. In stage 3, $V_{\mathrm{G}}$ is stepped from $25$ V up to ≈$30$ V again. Stages 1–3 are done sequentially and in one go. The left panel shows the superimposed spectra from stages 1–3. b Frequency response measured upon increasing and decreasing $V_{\mathrm{G}}$ over various $V_{\mathrm{G}}$ ranges. The panels are made by superimposing data measured with increasing $V_{\mathrm{G}}$ and data measured with decreasing $V_{\mathrm{G}}$. c Superimposed dependences of the resonant frequency $f$ on $V_{\mathrm{G}}$ extracted from the four panels in b.

Fig. 3. Resonant frequency loops and their dependence on stepping rate dV_G/dt.

a Response of the mode shown in Fig. 1d as a function of $f_{\mathrm{d}}$ and $V_{\mathrm{G}}$ for four different stepping rates. b Width of frequency loops $△V_{\mathrm{G}}$ as a function of $\mathrm{d}{V}_{\mathrm{G}}/\mathrm{d}t$. Marker shape identifies the data set in a from which $△V_{\mathrm{G}}$ is extracted.

Fig. 4. Mechanical model reproducing frequency loops.

a We assume that the supported membrane slides over the substrate controllably (double headed arrows in the inset) as a varying electrostatic pressure (vertical arrows) modulates its vertical displacement. b We model this sliding by assuming that the clamping area behaves as a spring and dashpot system. c Schematic of the sliding model. Extending spring feeds extra membrane into the suspended area, making the resonator longer and lowering the resonant frequency $f$. Contracting spring pulls the membrane away from the suspended area, shortening the resonator and increasing $f$. d Calculated $f$ as a function of $V_{\mathrm{G}}$ for different $V_{\mathrm{G}}$ ranges. Arrows indicate the path followed by $f\left(V_{\mathrm{G}}\right)$. The rightmost panel in d displays the superimposed frequency loops shown in the four leftmost panels. e Calculated width of frequency loop $△V_{\mathrm{G}}$ as a function of $\mathrm{d}{V}_{\mathrm{G}}/\mathrm{d}t$. Marker shape identifies the data set (d and insets to e) from which $△V_{\mathrm{G}}$ is extracted.

Fig. 5. Physics of the reversible sliding.

a SEM image obtained by zooming out of the area shown in Fig. 1a, revealing three mechanically connected resonators, $R_{\mathrm{L}}$, $R_{\mathrm{M}}$, and $R_{\mathrm{R}}$. The resonator investigated in the text is the middle one, labelled as $R_{\mathrm{M}}$. Scale bar: 1 micrometer. b Schematic of the reversible sliding. As extra membrane length is fed into the middle trench in response to electrostatic pulling (state (1) to state (2)), strain within $R_{\mathrm{L}}$ and $R_{\mathrm{R}}$ increases. The concomitant in-plane sliding and out-of-plane displacement of $R_{\mathrm{L}}$ and $R_{\mathrm{R}}$ give rise to a tension within the resonators, which acts as a restoring force (state (2) to state (3)). c Schematic plot showing the sliding distance, the out-of-plane displacement and the tension within the resonators upon increasing and decreasing $V_{\mathrm{G}}$.