Warner J. Venstra

Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators

A theoretical and experimental investigation is presented on the intermodal coupling between the flexural vibration modes of a single clamped-clamped beam. Nonlinear coupling allows an arbitrary flexural mode to be used as a self-detector for the amplitude of another mode, presenting a method to measure the energy stored in a specific resonance mode. The observed complex nonlinear dynamics are quantitatively captured by a model based on coupling of the modes via the beam extension; the same mechanism is responsible for the well-known Duffing nonlinearity in clamped-clamped beams.

Size-dependent effective Young’s modulus of silicon nitride cantilevers

The effective Young’s modulus of silicon nitride cantilevers is determined for thicknesses in the range of 20–684 nm by measuring resonance frequencies from thermal noise spectra. A significant deviation from the bulk value is observed for cantilevers thinner than 150 nm. To explain the observations we have compared the thickness dependence of the effective Young’s modulus for the first and second flexural resonance mode and measured the static curvature profiles of the cantilevers. We conclude that surface stress cannot explain the observed behavior. A surface elasticity model fits the experimental data consistently.

Buckling beam micromechanical memory with on-chip readout

We have used double clamped beams to implement a mechanical memory. Compressive stress is generated by resistive heating of the beams and beyond the buckling limit the bistable regime is accessed. Bits are written by applying lateral electrostatic forces. The state of the beam is read out by measuring the capacitance between beam and electrodes. Two ways to implement a mechanical memory are discussed: compensation of initial beam imperfections and snap through of the postbuckled beam. Although significant relaxation effects are observed, both methods prove reliable over thousands of write cycles.

Mechanical stiffening, bistability, and bit operations in a microcantilever

We investigate the nonlinear dynamics of microcantilevers. We demonstrate mechanical stiffening of the frequency response at large amplitudes, originating from the geometric nonlinearity. At strong driving the cantilever amplitude is bistable. We map the bistable regime as a function of drive frequency and amplitude, and suggest several applications for the bistable microcantilever, of which a mechanical memory is demonstrated.

Effect of undercut on the resonant behaviour of silicon nitride cantilevers

We present a systematic investigation of the dynamic properties of silicon nitride cantilevers in air. The thermal noise spectra of cantilevers have been measured using a home-made optical deflection setup. Torsional and flexural resonances up to the seventh mode are observed. The dependence of resonance frequencies on the dimensions and mode number is studied in detail. It is found that undercut increases the effective length of the cantilever by a value ?L, which depends on the undercut distance and the resonance mode shape, but not on the cantilever length. Finite element modelling confirms these experimental findings. A simple model is suggested for the shape of the undercut region, which agrees well with experimental findings. Using this model, the undercut cantilever can be approximated by a stepped beam, where the clamp distance depends on the underetch duration and the mode shape.

Q-factor control of a microcantilever by mechanical sideband excitation

We demonstrate the coupling between the fundamental and second flexural modes of a microcantilever. A mechanical analogue of cavity-optomechanics is then employed, where the mechanical cavity is formed by the second vibrational mode of the same cantilever, coupled to the fundamental mode via the geometric nonlinearity. By exciting the cantilever at the sum and difference frequencies between fundamental and second flexural modes, the motion of the fundamental mode of the cantilever is damped and amplified. This concept makes it possible to enhance or suppress the Q-factor over a wide range.

Observation of decoherence in a carbon nanotube mechanical resonator

In physical systems, decoherence can arise from both dissipative and dephasing processes. In mechanical resonators, the driven frequency response measures a combination of both, whereas time-domain techniques such as ringdown measurements can separate the two. Here we report the first observation of the mechanical ringdown of a carbon nanotube mechanical resonator. Comparing the mechanical quality factor obtained from frequency- and time-domain measurements, we find a spectral quality factor four times smaller than that measured in ringdown, demonstrating dephasing-induced decoherence of the nanomechanical motion. This decoherence is seen to arise at high driving amplitudes, pointing to a nonlinear dephasing mechanism. Our results highlight the importance of time-domain techniques for understanding dissipation in nanomechanical resonators, and the relevance of decoherence mechanisms in nanotube mechanics.

Strongly coupled modes in a weakly driven micromechanical resonator

We demonstrate a strong coupling between the flexural vibration modes of a clamped-clamped micromechanical resonator vibrating at low amplitudes. This coupling enables the direct measurement of the frequency response via amplitude- and phase modulation schemes using the fundamental mode as a mechanical detector. In the linear regime, a frequency shift of 0.8 Hz is observed for a mode with a line width of 5.8 Hz in vacuum. The measured response is well-described by the analytical model based on the Euler-Bernoulli beam including tension. Calculations predict an upper limit for the room-temperature Q-factor of 4.5×105 for our top-down fabricated micromechanical beam resonators.

Interactions between directly- and parametrically-driven vibration modes in a micromechanical resonator

The interactions between parametrically- and directly-driven vibration modes of a clamped-clamped beam resonator are studied. An integrated piezoelectric transducer is used for direct and parametric excitation. First, the parametric amplification and oscillation of a single mode are analyzed by the power and phase dependence below and above the threshold for parametric oscillation. Then, the motion of a parametrically-driven mode is detected by the induced change in resonance frequency in another mode of the same resonator. The resonance frequency shift is the result of the nonlinear coupling between the modes by the displacement-induced tension in the beam. These nonlinear modal interactions result in the quadratic relation between the resonance frequency of one mode and the amplitude of another mode. The amplitude of a parametrically-oscillating mode depends on the square root of the pump frequency. Combining these dependencies yields a linear relation between the resonance frequency of the directly-driven mode and the frequency of the parametrically-oscillating mode.

Time-domain response of atomically thin MoS2 nanomechanical resonators

We measure the energy relaxation rate of single- and few-layer molybdenum disulphide (