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Energy measurement in nonlinearly coupled nanomechanical modes
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) The coupled-element flexural micro-resonator with mode shapes and resonance measurements. (a) Diagram of the experimental setup involving a high frequency collective mode and a low frequency fundamental, with nonlinear elastic coupling between them. The insets show the simulated mode shapes. (b) The fundamental transverse mechanical resonance of the resonator was measured at 37.3 MHz with quality factor 13 500, using magneto-motive actuation and detection. (c) The collective resonance mode at 837 MHz with quality factor 1075. (d) SEM micrograph of a typical ultra-nano-crystalline diamond resonator device.

Image of FIG. 2.
FIG. 2.

(Color online) Two methods for inducing modal frequency shift. (a) Plot of the fundamental mode resonance illustrating center frequency shift at large amplitudes and the onset of nonlinear response regime. The peaks are normalized to the signal transmission background. The fitted curves are perturbation solutions to the Duffing equation that describes the response close to the resonance peak. (b) The frequency shift on the fundamental mode can also be induced by actuating it in the linear regime with constant forcing amplitude and applying a second excitation on the collective mode resonance. The plot shows the initial and the shifted peaks. The smooth curves are fitted analytical solutions, see Ref. 13. The diagram on the plot illustrates the standard AFM-type slope detection scheme for measuring the frequency shift. The inset shows amplitude modulation of the carrier frequency, which is proportional to the frequency shift at small drives.

Image of FIG. 3.
FIG. 3.

(Color online) Monitoring of the collective mode excitation. (a) Plot of the fundamental mode frequency shift as a function of the collective mode driving frequency, applied near the collective mode resonance. The smooth curve is a numerical fit to the data. (b) By varying the excitation of the collective mode, the frequency shift signal on the fundamental mode is monitored down to the noise floor of the experiment. The collective mode excitation force (bottom abscissa axis) is converted to equivalent average mechanical energy (top abscissa axis) and normalized by the energy quantum for the collective mode.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Energy measurement in nonlinearly coupled nanomechanical modes