[Paper Review] Tunable parametric amplification of a graphene nanomechanical resonator in the nonlinear regime

Zi-Jia Su, Yue Ying+7 more|arXiv (Cornell University)|Sep 29, 2019

Mechanical and Optical Resonators47 references9 citations

TL;DR

This study demonstrates electrically tunable parametric amplification in a doubly clamped graphene nanomechanical resonator by applying microwave pumping at twice the resonant frequency. Despite significant higher-order nonlinearity beyond Duffing and van der Pol types, the system achieves a maximum parametric gain of 10.2 dB at 19 V gate voltage, with gain tunability and saturation behavior under increasing pump power.

ABSTRACT

Parametric amplification is widely used in nanoelectro-mechanical systems to enhance the transduced mechanical signals. Although parametric amplification has been studied in different mechanical resonator systems, the nonlinear dynamics involved receives less attention. Taking advantage of the excellent electrical and mechanical properties of graphene, we demonstrate electrical tunable parametric amplification using a doubly clamped graphene nanomechanical resonator. By applying external microwave pumping with twice the resonant frequency, we investigate parametric amplification in the nonlinear regime. We experimentally show that the extracted coefficient of the nonlinear Duffing force {\alpha} and the nonlinear damping coefficient {\eta} vary as a function of external pumping power, indicating the influence of higher-order nonlinearity beyond the Duffing (~x^3) and van der Pol (~x^2 dx/dt) types in our device. Even when the higher-order nonlinearity is involved, parametric amplification still can be achieved in the nonlinear regime. The parametric gain increases and shows a tendency of saturation with increasing external pumping power. Further, the parametric gain can be electrically tuned by the gate voltage with a maximum gain of 10.2 dB achieved at the gate voltage of 19 V. Our results will benefit studies on nonlinear dynamics, especially nonlinear damping in graphene nanomechanical resonators that has been debated in the community over past decade.

Motivation & Objective

• To investigate parametric amplification in graphene nanomechanical resonators under nonlinear dynamics.
• To examine the influence of higher-order nonlinearity beyond Duffing (~𝑥³) and van der Pol (~𝑥²𝑑𝑥/𝑑𝑡) types on amplification.
• To achieve electrical tuning of parametric gain via gate voltage in the nonlinear regime.
• To quantify the evolution of nonlinear Duffing and damping coefficients under varying external pumping power.

Proposed method

• Employed a doubly clamped few-layer graphene flake suspended over a prepatterned Si₃N₄ trench to form a nanomechanical resonator.
• Used frequency-modulated microwave signals (FM) at frequency 𝑓 to drive the resonator, with mixing current detected at the drain electrode.
• Applied a continuous microwave pump at 2𝑓 via a bias-tee to the gate electrode to induce parametric amplification.
• Measured the resonant frequency and mixing current as functions of gate voltage and pump power to extract nonlinear coefficients.
• Fitted experimental data to a nonlinear Langevin equation to extract the Duffing coefficient 𝛼, nonlinear damping coefficient 𝜂, and effective driving force.
• Used Eq. (7) to model parametric gain as a function of pump power, with fitting validated at low nonlinearity.

Experimental results

Research questions

• RQ1 How does higher-order nonlinearity beyond Duffing and van der Pol types affect parametric amplification in graphene nanomechanical resonators?
• RQ2 Can parametric amplification be achieved and tuned in the nonlinear regime when higher-order nonlinearities are significant?
• RQ3 How do the nonlinear Duffing coefficient 𝛼 and nonlinear damping coefficient 𝜂 depend on external pumping power?
• RQ4 To what extent can the parametric gain be electrically tuned via gate voltage in the nonlinear regime?

Key findings

• The absolute values of the Duffing coefficient |𝛼| and nonlinear damping coefficient 𝜂 increase with increasing external pumping power, indicating the presence of higher-order nonlinearity beyond standard Duffing and van der Pol terms.
• Parametric gain increases with pump power and shows a saturation trend, consistent with nonlinear saturation effects in the system.
• The maximum parametric gain of 10.2 dB is achieved at a gate voltage of 19 V, demonstrating effective electrical tuning of amplification.
• The nonlinear damping coefficient 𝜂 increases linearly with vibrational amplitude 𝑥₀ for 𝑥₀ < 0.53 nm, indicating amplitude-dependent energy dissipation.
• The Duffing coefficient |𝛼| exhibits a linear dependence on 𝑥₀, confirming the dominance of cubic nonlinearity at moderate amplitudes.
• The system remains operable under strong nonlinearities, confirming robustness of parametric amplification in the nonlinear regime.