Visualizing nonlinear resonance in nanomechanical systems via single-electron tunneling
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nghua University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Received: 12 August 2020 / Revised: 8 October 2020 / Accepted: 8 October 2020
ABSTRACT 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.
KEYWORDS carbon nanotube, mechanical resonator, quantum dot, nonlinear, coupling
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Introduction
Carbon nanotube (CNT) mechanical systems have been the model systems to elucidating the physics of nonlinear dynamics [1–6] and the electromechanics of coupling individual electrons with nanometer-scale vibrations [7–13]. To date, based on the quantum-dot-embedded CNT mechanical resonator, the investigates of the modulated mechanical vibrations by single-electron transport have led to insightful discoveries, which include softening of the restoring force [6, 7, 14], driving CNTs into resonance spontaneously [8], and inducing a coherent oscillation (which has been achieved recently) [15]. These effects can be exploited to control the CNT vibration in the quantum limit [16, 17] or to detect the charge from the consequent mechanical clues [11]. In contrast, how the mechanical motion modulating single-electron transport hasn’t been explored much yet [12], especially for driving a large amplitude into a nonlinear regime.
In this study, we construct a quantum dot-mechanical resonator hybrid system based on the doubly clamped CNT, whose flexural modes exhibit an intrinsic geometric nonlinearity that is caused by nonlinear stretching or a large curvature [18, 19]. When working in a large-amplitude regime, the mechanical oscillation modulates the single-electron tunneling, resulting in the single-electron chopper mode. At the same time, the vibration amplitude of the CNT versus its frequency can be directly displ
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