Study of Nonlinear Coupling in Atomically Thin MoS₂ Nanoelectromechanical Resonators

Academic Background

With the rapid development of nanotechnology, Nanoelectromechanical Systems (NEMS) have shown great potential in fields such as sensors, signal processing, and quantum computing. Particularly, two-dimensional (2D) materials like molybdenum disulfide (MoS₂) have become ideal materials for constructing NEMS due to their atomic thickness, excellent mechanical properties, and electrical characteristics. At the nanoscale, MoS₂ and other 2D materials exhibit multimode resonances and nonlinear dynamic behaviors, providing a unique platform for studying novel device physics.

In NEMS resonators, nonlinear mode coupling is an important research topic. When a resonator is driven into the nonlinear regime, energy exchange occurs between different vibrational modes, leading to resonance frequency shifts and other complex dynamic phenomena. Understanding these nonlinear coupling mechanisms is crucial for designing high-performance NEMS devices. However, existing research has mostly focused on mode coupling with integer frequency relationships, while studies on nonlinear coupling between closely spaced modes without integer relationships are limited.

This study aims to explore the nonlinear coupling mechanisms between closely spaced modes in a bilayer (2L) MoS₂ nanoelectromechanical resonator and to quantify this coupling effect through experiments and theoretical models. The research not only provides new insights into understanding nonlinear dynamics in NEMS but also lays the foundation for future designs of multimode resonators and phononic frequency combs.

Paper Source

This paper was co-authored by S. M. Enamul Hoque Yousuf, Steven W. Shaw, and Philip X.-L. Feng, from the Department of Electrical and Computer Engineering at the University of Florida, the Department of Mechanical and Civil Engineering at the Florida Institute of Technology, and the Department of Mechanical Engineering at Michigan State University, respectively. The paper was published in 2024 in the journal Microsystems & Nanoengineering, titled Nonlinear coupling of closely spaced modes in atomically thin MoS₂ nanoelectromechanical resonators.

Research Process

1. Resonator Design and Fabrication

The research team used an all-dry transfer technique to transfer a bilayer MoS₂ film onto a pre-fabricated array of microcavities and electrodes, constructing a drumhead resonator with a diameter of 4 micrometers. The microcavity, with a depth of approximately 290 nanometers, ensured high responsivity between optical reflectance and displacement. The electrode array included a set of four-point contact leads and a local gate at the bottom of the microcavity for electrostatic excitation of the resonator.

2. Resonator Characterization

The resonator was characterized using Raman spectroscopy and an optical interferometry system. Raman spectroscopy confirmed that the suspended MoS₂ film was bilayer with a thickness of about 1.3 nanometers. The optical interferometry system was used to measure the thermomechanical noise, driven resonance, and nonlinear mode coupling of the resonator. By scanning the x and y coordinates of the sample stage, the team also mapped the mode shapes of the resonator.

3. Nonlinear Mode Coupling Experiments

The team studied the nonlinear coupling between the second mode (f₂=20.45 MHz) and the first mode (f₁=18.41 MHz) by driving the resonator near f₂ and measuring the thermomechanical noise spectrum of f₁. A function generator was used to drive the resonator at frequencies close to f₂, and the resonance frequency shift (f₁s) of f₁ was measured using a spectrum analyzer. The nonlinear coupling coefficient λ was extracted by fitting the experimental data.

4. Theoretical Model and Data Analysis

To describe the nonlinear coupling between the two modes, the team developed a resonator model that included a dispersive coupling term. The coupled mode equations were solved using the method of averaging, yielding a closed-form expression for the nonlinear coupling coefficient λ. The fitting of experimental data to the theoretical model showed that λ=0.027 ± 0.005 pm⁻²·μs⁻².

Main Results

1. Extraction of the Nonlinear Coupling Coefficient: By driving the second mode and measuring the thermomechanical noise spectrum of the first mode, the team successfully extracted the nonlinear coupling coefficient λ. The experimental results showed that λ=0.027 ± 0.005 pm⁻²·μs⁻².

2. Observation of Anomalous Frequency Shift: When the driving frequency was close to the resonance frequency of the second mode, an anomalous frequency shift was observed in the first mode. This shift was caused by dynamic tension changes induced by the large-amplitude vibrations of the second mode.

3. Mapping of Mode Shapes: Using the optical interferometry system, the team successfully mapped the mode shapes of the resonator, confirming the system’s spatial resolution and sensitivity.

Conclusion

This study successfully quantified the nonlinear coupling effect between closely spaced modes in a bilayer MoS₂ nanoelectromechanical resonator through experiments and theoretical models. The results demonstrated that nonlinear mode coupling can be directly quantified by measuring thermomechanical noise spectra and that anomalous frequency shifts caused by dynamic tension can be captured. This method is not only applicable to MoS₂ resonators but can also be extended to other materials and MEMS/NEMS resonators.

Research Highlights

1. Novel Experimental Method: This study is the first to quantify the nonlinear mode coupling coefficient in NEMS resonators by measuring thermomechanical noise spectra, avoiding potential errors introduced by traditional methods.

2. Discovery of Anomalous Frequency Shift: The team observed for the first time an anomalous frequency shift caused by dynamic tension, providing new insights into understanding nonlinear dynamics in NEMS.

3. Broad Application Prospects: The study of nonlinear mode coupling provides a theoretical foundation for designing multimode resonators, phononic frequency combs, and other devices, with broad application prospects.

Research Significance

This study not only provides new experimental and theoretical tools for understanding nonlinear dynamics in NEMS but also lays the foundation for designing high-performance nanoelectromechanical devices in the future. By quantifying the nonlinear mode coupling coefficient, the team offers important references for developing new sensors, signal processors, and quantum computing devices. Additionally, the methods and models used in this study are universal and can be extended to other materials and systems, further advancing the field of nanoelectromechanical systems.