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Numerous reports have elucidated the importance of mechanical resonators comprising quantum-dot-embedded carbon nanotubes (CNTs) for studying the effects of single-electron transport. However, there is a need to investigate the single-electron transport that drives a large amplitude into a nonlinear regime. Herein, a CNT hybrid device has been investigated, which comprises a gate-defined quantum dot that is embedded into a mechanical resonator under strong actuation conditions. The Coulomb peak positions synchronously oscillate with the mechanical vibrations, enabling a single-electron “chopper” mode. Conversely, the vibration amplitude of the CNT versus its frequency can be directly visualized via detecting the time-averaged single-electron tunneling current. To understand this phenomenon, a general formula is derived for this time-averaged single-electron tunneling current, which agrees well with the experimental results. By using this visualization method, a variety of nonlinear motions of a CNT mechanical oscillator have been directly recorded, such as Duffing nonlinearity, parametric resonance, and double-, fractional-, mixed- frequency excitations. This approach opens up burgeoning opportunities for investigating and understanding the nonlinear motion of a nanomechanical system and its interactions with electron transport in quantum regimes.

The carbon nanotube (CNT) hybrid device composed of a gate-defined quantum dot embedded into a mechanical resonator is investigated under strong actuation condition. The Coulomb peaks are driven to the synchronous and large oscillation by the nonlinear mechanical vibration, and can be coherently modulated to the single-electron shuttle mode. Conversely, the mediated single-electron tunnelling current provides an in situ and sensitive characterization strategy for the multi-order mechanical nonlinear resonance. We show that, under the strong modulation applied via gate voltage, the swing and trampoline modes exhibit different nonlinearity behaviour, depending on both the quadratic and cubic coefficients. The nonlinearities in this hybrid system also give rise to coupling modes of quadratic and cubic resonances at combined frequencies and even fractional frequencies. Especially, the subharmonic resonance due to quadratic nonlinearity is predicted to enable the excitation of the coherent coupling state between two modes. Our theoretical mode of the intra- and intermodal nonlinearities shows a good agreement of our experimental observations quantitatively.