Nanomechanical Resonators Research Articles

Quality factor comparison between graphene and SiN with triangular lattice phononic crystal structure based on dissipation dilution theory

Nanomechanical resonators attract interests both in fundamental and applied fields, offering outstanding mass and force sensitivity, relying on both the low mass m and high quality factor Q. However, the quality factor typically does not exceed the inverse of the material loss angle φ, related to resonant energy dissipation. Recently, some exceptions breaking the 1/φ limiting rule in a great extent have been reported in use of highly stressed silicon nitride material. Up to now it is interpreted that the pre-stress seems to ‘dilute’ the intrinsic energy dissipation according to Zener model. Dissipation dilution effect could be better if the device dimension is reduced, which implies that the smaller the resonator, the higher the quality factor. Single-layer graphene's thickness is 0.335 nm, which is one hundredth of silicon nitride’ typical thickness about 30 nm, indicating that the dissipation dilution effect of graphene could be much better than that of silicon nitride. Through theoretical calculation, the quality factor amplification coefficient defining the dissipation dilution effect quantitatively of graphene is 2272 times that of SiN. By COMSOL finite element simulation, we further confirm this conclusion. Take the first defect mode for example, the amplification coefficient of graphene and SiN is about 3.02 × 105 and 151 respectively, with the division result about 2004 times. As the overall size increases, the division results gradually converge to a stable value about 2260.

Electrically Tunable MXene Nanomechanical Resonators Vibrating at Very High Frequencies

As an emerging class of two-dimensional (2D) layered nanomaterial, MXene exhibits a number of intriguing properties, such as good electrical conductivity and high elastic modulus, and has witnessed continued growth in related device research. However, nanoscale MXene devices which leverage both the intrinsic electrical and mechanical properties of these 2D crystals have not been experimentally studied. Here we demonstrate nanoelectromechanical resonators based on 2D MXene crystals, where Ti3C2Tx drumheads with a wide range of thickness, from more than 50 layers all the way down to a monolayer, exhibit robust nanomechanical vibrations with fundamental-mode frequency f0 up to >70 MHz in the very high frequency (VHF) band, a displacement noise density down to 52 fm/Hz1/2, and a fundamental-mode frequency-quality factor product up to f0 × Q ≈ 6.85 × 109 Hz. By combining experimental results with theoretical calculations, we independently derive the Young's modulus of 2D Ti3C2Tx crystals to be 270-360 GPa, in excellent agreement with nanoindentation measurements, based on which we elucidate frequency scaling pathways toward microwave frequencies. We further demonstrate electrical tuning of resonance frequency in MXene resonators and frequency-shift-based MXene vacuum gauges with responsivity of 736%/Torr and detection range down to 10-4 Torr. Our study can lead to the design and creation of nanoscale vibratory devices that exploit the intrinsic electrical and mechanical properties in 2D MXene crystals.

Direct Observation of Thermal Vibration Modes Using Frequency-Selective Electron Microscopy

Thermal vibration properties of nanometer-scale objects are critical for their application in devices such as nanomechanical resonators. An imaging method has been developed which allows the direct visualization of higher-order thermal vibration modes at room temperature, which have so far been inaccessible to observation due to their subangstrom amplitudes and the much stronger overlapped first mode. This technique, combining aberration-corrected scanning transmission electron microscopy with broad-band signal acquisition in the time domain, can display the amplitude distribution of several thermal vibration modes simultaneously by selecting specific frequency windows. This is showcased by mapping the first six thermal vibration modes of a singly clamped nanowire and comparing them to natural vibration mode profiles obtained by finite element calculations. This implementation furthers our understanding of the collective Brownian motion of nanostructures and extends the analysis capabilities of electron microscopy.

Nonlinear Stiffness and Nonlinear Damping in Atomically Thin MoS2 Nanomechanical Resonators

We report on experimental measurements and quantitative analyses of nonlinear dynamic characteristics in ultimately thin nanomechanical resonators built upon single-layer, bilayer, and trilayer (1L, 2L, and 3L) molybdenum disulfide (MoS2) vibrating drumhead membranes. This synergistic study with calibrated measurements and analytical modeling on observed nonlinear responses has led to the determination of nonlinear damping and stiffness coefficients at cubic and quintic orders for these two-dimensional (2D) resonators operating in the very high frequency (VHF) band (up to ∼90 MHz). We find that the quintic force can be ∼20% of the Duffing force at larger amplitudes, and thus, it generally cannot be ignored in a nonlinear dynamics analysis. This study provides the first quantification of nonlinear damping and frequency detuning characteristics in 2D semiconductor nanomechanical resonators and elucidates their origins and dependency on engineerable parameters, setting a foundation for future exploration and utilization of the rich nonlinear dynamics in 2D nanomechanical systems.

Persistent Nonlinear Phase-Locking and Nonmonotonic Energy Dissipation in Micromechanical Resonators.

Many nonlinear systems are described by eigenmodes with amplitude-dependent frequencies, interacting strongly whenever the frequencies become commensurate at internal resonances. Fast energy exchange via the resonances holds the key to rich dynamical behavior, such as time-varying relaxation rates and signatures of nonergodicity in thermal equilibrium, revealed in the recent experimental and theoretical studies of micro- and nanomechanical resonators. However, a universal yet intuitive physical description for these diverse and sometimes contradictory experimental observations remains elusive. Here we experimentally reveal persistent nonlinear phase-locked states occurring at internal resonances and demonstrate that they are essential for understanding the transient dynamics of nonlinear systems with coupled eigenmodes. The measured dynamics of a fully observable micromechanical resonator system are quantitatively described by the lower-frequency mode entering, maintaining, and exiting a persistent phase-locked period-tripling state generated by the nonlinear driving force exerted by the higher-frequency mode. This model describes the observed phase-locked coherence times, the direction and magnitude of the energy exchange, and the resulting nonmonotonic mode energy evolution. Depending on the initial relative phase, the system selects distinct relaxation pathways, either entering or bypassing the locked state. The described persistent phase locking is not limited to particular frequency fractions or types of nonlinearities and may advance nonlinear resonator systems engineering across physical domains, including photonics as well as nanomechanics.

Phase Transition of MoTe2 Controlled in van der Waals Heterostructure Nanoelectromechanical Systems.

This work reports experimental demonstrations of reversible crystalline phase transition in ultrathin molybdenum ditelluride (MoTe2 ) controlled by thermal and mechanical mechanisms on the van der Waals (vdW) nanoelectromechanical systems (NEMS) platform, with hexagonal boron nitride encapsulated MoTe2 structure residing on top of graphene layer. Benefiting from very efficient electrothermal heating and straining effects in the suspended vdW heterostructures, MoTe2 phase transition is triggered by rising temperature and strain level. Raman spectroscopy monitors the MoTe2 crystalline phase signatures in situ and clearly records reversible phase transitions between hexagonal 2H (semiconducting) and monoclinic 1T' (metallic) phases. Combined with Raman thermometry, precisely measured nanomechanical resonances of the vdW devices enable the determination and monitoring of the strain variations as temperature is being regulated by electrothermal control. These results not only deepen the understanding of MoTe2 phase transition, but also demonstrate a novel platform for engineering MoTe2 phase transition and multiphysical devices.

Ultra-high-Q nanomechanical resonators for force sensing

Nanomechanical resonators with ultra-high quality factors have become a central element in fundamental research, enabling measurements below the standard quantum limit and the preparation of long-lived quantum states. Here, I propose that such resonators will allow the detection of electron and nuclear spins with high spatial resolution, paving the way to future nanoscale magnetic resonance imaging instruments. The article lists the challenges that must be overcome before this vision can become reality, and indicates potential solutions.

Nano-Optomechanical Resonators Based on Suspended Graphene for Thermal Stress Sensing

Nanomechanical resonators made from suspended graphene combine the properties of ultracompactness and ultrahigh detection sensitivity, making them interesting devices for sensing applications. However, nanomechanical systems can be affected by membrane stress. The present work developed an optomechanical resonator for thermal stress sensing. The proposed resonator consists of a section of hollow core fiber (HCF) and a trampoline graphene-Au membrane. An all-optical system that integrated optical excitation and optical detection was applied. Then, the resonance frequency of the resonator was obtained through this all-optical system. In addition, this system and the resonator were used to detect the membrane's built-in stress, which depended on the ambient temperature, by monitoring the resonance frequency shift. The results verified that the temperature-induced thermal effect had a significant impact on membrane stress. Temperature sensitivities of 2.2646 kHz/°C and 2.3212 kHz/°C were obtained when the temperature rose and fell, respectively. As such, we believe that this device will be beneficial for the quality monitoring of graphene mechanical resonators.

Hemolysis-Inspired, Highly Sensitive, Label-Free IgM Detection Using Erythrocyte Membrane-Functionalized Nanomechanical Resonators

Immunoglobulin detection is important for immunoassays, such as diagnosing infectious diseases, evaluating immune status, and determining neutralizing antibody concentrations. However, since most immunoassays rely on labeling methods, there are limitations on determining the limit of detection (LOD) of biosensors. In addition, although the antigen must be immobilized via complex chemical treatment, it is difficult to precisely control the immobilization concentration. This reduces the reproducibility of the biosensor. In this study, we propose a label-free method for antibody detection using microcantilever-based nanomechanical resonators functionalized with erythrocyte membrane (EM). This label-free method focuses on the phenomenon of antibody binding to oligosaccharides (blood type antigen) on the surface of the erythrocyte. We established a method for extracting the EM from erythrocytes and fabricated an EM-functionalized microcantilever (MC), termed EMMC, by surface-coating EM layers on the MC. When the EMMC was treated with immunoglobulin M (IgM), the bioassay was successfully performed in the linear range from 2.2 pM to 22 nM, and the LOD was 2.0 pM. The EMMC also exhibited excellent selectivity compared to other biomolecules such as serum albumin, γ-globulin, and IgM with different paratopes. These results demonstrate that EMMC-based nanotechnology may be utilized in criminal investigations to identify blood types with minimal amounts of blood or to evaluate individual immunity through virus-neutralizing antibody detection.

Sliding nanomechanical resonators

The motion of a vibrating object is determined by the way it is held. This simple observation has long inspired string instrument makers to create new sounds by devising elegant string clamping mechanisms, whereby the distance between the clamping points is modulated as the string vibrates. At the nanoscale, the simplest way to emulate this principle would be to controllably make nanoresonators slide across their clamping points, which would effectively modulate their vibrating length. Here, we report measurements of flexural vibrations in nanomechanical resonators that reveal such a sliding motion. Surprisingly, the resonant frequency of vibrations draws a loop as a tuning gate voltage is cycled. This behavior indicates that sliding is accompanied by a delayed frequency response of the resonators, making their dynamics richer than that of resonators with fixed clamping points. Our work elucidates the dynamics of nanomechanical resonators with unconventional boundary conditions, and offers opportunities for studying friction at the nanoscale from resonant frequency measurements.

Classifiable Limiting Mass Change Detection in a Graphene Resonator Using Applied Machine Learning

Nanomechanical resonator devices are widely used as ultrasensitive mass detectors for fundamental studies and practical applications. The resonance frequency of the resonators shifts when a mass is loaded, which is used to estimate the mass. However, the shift signal is often blurred by the thermal noise, which interferes with accurate mass detection. Here, we demonstrate the reduction of the noise interference in mass detection in suspended graphene-based nanomechanical resonators, by using applied machine learning. Featurization is divided into image and sequential datasets, and those datasets are trained and classified using 2D and 1D convolutional neural networks (CNNs). The 2D CNN learning-based classification shows a performance with f1-score over 99% when the resonance frequency shift is more than 2.5% of the amplitude of the thermal noise range.

Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams

Small-sized structural elements such as beams, plates, and shells are usually used as nanomechanical resonators, nanoscale mass sensors, nanoelectromechanical actuators, and nanoenergy harvesters. At the nanoscale, the structures usually possess a high surface area-to-bulk volume ratio, leading to the free energy related to surface atoms becoming considerable compared to that of the bulk part. Earlier reports indicated several physical reasons for size-dependent phenomena, e.g., nonlocal stress, surface energy, and couple stress. To provide an in-depth insight into the mechanical behavior of small-scale structures, size-dependent continuum models including two or more physical factors have attracted the attention of the academic community. This research analyzes the thermal buckling and postbuckling characteristics of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams with a tri-parameter, nonlinear elastic foundation and subjected to a uniform temperature rise. Chen-Yao’s surface energy theory and Yang’s symmetrical couple stress theory are combined to capture two types of size effects in nanobeams. The postbuckling model is formulated based on the Euler–Bernoulli deformation hypothesis and Euler–Lagrange equation. Using a two-step perturbation technique, the related postbuckling equilibrium path is determined. In numerical analysis, the impacts of surface energy, couple stress, elastic foundation, boundary conditions, geometric factor, layout type, and volume fraction of CNTs on the thermal buckling and postbuckling behaviors of nanobeams are revealed. It is indicated that considering couple stress or surface energy can lead to a significant increase in the postbuckling stability of nanobeams compared to the case in which it is not considered. In addition, there is a reverse competition between couple stress or surface energy effects on the thermal buckling responses of nanobeams. As the temperature rise will cause the material elastic moduli softening, the thermal buckling load–deflection curves of nanobeams with the temperature-independent case are much higher than those with the temperature-dependent cases.

Nanomechanical Spectroscopy of 2D Materials.

We introduce a nanomechanical platform for fast and sensitive measurements of the spectrally resolved optical dielectric function of 2D materials. At the heart of our approach is a suspended 2D material integrated into a high Q silicon nitride nanomechanical resonator illuminated by a wavelength-tunable laser source. From the heating-related frequency shift of the resonator as well as its optical reflection measured as a function of photon energy, we obtain the real and imaginary parts of the dielectric function. Our measurements are unaffected by substrate-related screening and do not require any assumptions on the underling optical constants. This fast (τrise ∼ 135 ns), sensitive (noise-equivalent power = ), and broadband (1.2-3.1 eV, extendable to UV-THz) method provides an attractive alternative to spectroscopic or ellipsometric characterization techniques.

Snap-through of graphene nanowrinkles under out-of-plane compression

Nanowrinkles (i.e. the buckled nanoribbons) are widely observed in nano-devices assembled by two-dimensional (2D) materials. The existence of nanowrinkles significantly affects the physical (such as mechanical, electrical and thermal) properties of 2D materials, and thus further, impedes the applications of those devices. In this paper, we take the nanowrinkle formed in a monolayer graphene as a model system to study its deformation behaviours, especially the configuration evolution and the snap-through buckling instabilities, when subjected to the out-of-plane compression. By performing molecular dynamics simulation, the graphene nanowrinkles with or without self-adhesion (which are notated as ‘clipped’ state or ‘bump’ state, respectively) are obtained depending on the geometric size and the applied axial compressive pre-strain. The elastica theory is employed to quantify the shape of ‘bump’ nanowrinkles, as well as the critical condition of the transition between ‘clipped’ and ‘bump’ states. By applying out-of-plane compression to the generated graphene nanowrinkle, it flips to an opposite configuration via snap-through buckling. We identify four different buckling modes according to the configuration evolution. An unified phase diagram is constructed to describe those buckling modes. For the cases with negligible van der Waals interaction getting involved in the snap-buckling process, i.e. without self-adhesion, the force–displacement curves for nanowrinkles with same axial pre-strain but different sizes can be scaled to collapse. Moreover, the critical buckling loads can also be scaled and predicted by the extended elastica theory. Otherwise, for the cases with self-adhesion, which corresponds to the greater axial pre-strain, the van der Waals interaction makes the scaling collapse break down. It is expected that the analysis about the snap-through buckling of graphene nanowrinkles reported in this work will advance the understanding of the mechanical behaviours of wrinkled 2D materials and promote the design of functional nanodevices, such as nanomechanical resonators and capacitors.

Simulating and studying the topological properties of the photon-phonon coupled modes in a one-dimensional superconducting circuit lattice

We theoretically study the topological properties in a one-dimensional superconducting circuit lattice, where each unit cell is composed of a superconducting microwave cavity, a nanomechanical resonator (NAMR) and a superconducting qubit. By using diagonalization and dimensional reduction methods, the quantum spin Hall system of the photon-phonon coupled modes can be achieved. We analyze the energy spectrum and edge states of different lattice sizes. It is found that the number of lattices affects the topological characteristics of the system, and the inversion of the edge state distribution can be realized. In addition, considering the effects of defects, dissipation and disorder on the system, we find that they will respectively expand the region of the edge state, change the maximum value of the energy spectrum and cause small fluctuation of the energy spectrum. But the edge state is still very stable, which is due to the topological protection of the edge state. This scheme opens up a new way to study the application of topological matter in quantum information processing, meanwhile, the research results can be used to design novel quantum devices.

Bidirectional Optical Non‐Reciprocity in a Multi‐Mode Cavity Optomechanical System

AbstractOptical non‐reciprocity that allows unidirectional flow of optical field is pivoted on time reversal symmetry breaking, which originates from radiation pressure because of light–matter interaction in cavity optomechanical systems. Here, the non‐reciprocal transport of optical signals across two ports via three optical modes optomechanically coupled to the mechanical excitations of two nanomechanical resonators (NMRs) is studied under the influence of strong classical drive fields and weak probe fields. It is found that there exists the conversion of reciprocal to non‐reciprocal signal transmission via tuning the drive fields and perfect non‐reciprocal transmission of output fields is realized when the effective cavity detuning parameters are near resonant to the NMRs' frequencies. The unidirectional non‐reciprocal transport is robust to the optomechanical couplings around resonance conditions. Moreover, the loss rates of cavities play an inevitable role in the unidirectional flow of signal across the two ports. Bidirectional transmission can also be controlled by the phase changes associated with the probe and drive fields along with their relative phase. This scheme may provide a foundation for the compact non‐reciprocal communication and quantum information processing, thus enabling novel devices that route photons in unconventional ways such as all‐optical diodes, optical transistors, and optical switches.

Nanomechanical Resonators: Toward Atomic Scale.

The quest for realizing and manipulating ever smaller man-made movable structures and dynamical machines has spurred tremendous endeavors, led to important discoveries, and inspired researchers to venture to previously unexplored grounds. Scientific feats and technological milestones of miniaturization of mechanical structures have been widely accomplished by advances in machining and sculpturing ever shrinking features out of bulk materials such as silicon. With the flourishing multidisciplinary field of low-dimensional nanomaterials, including one-dimensional (1D) nanowires/nanotubes and two-dimensional (2D) atomic layers such as graphene/phosphorene, growing interests and sustained effort have been devoted to creating mechanical devices toward the ultimate limit of miniaturization─genuinely down to the molecular or even atomic scale. These ultrasmall movable structures, particularly nanomechanical resonators that exploit the vibratory motion in these 1D and 2D nano-to-atomic-scale structures, offer exceptional device-level attributes, such as ultralow mass, ultrawide frequency tuning range, broad dynamic range, and ultralow power consumption, thus holding strong promises for both fundamental studies and engineering applications. In this Review, we offer a comprehensive overview and summary of this vibrant field, present the state-of-the-art devices and evaluate their specifications and performance, outline important achievements, and postulate future directions for studying these miniscule yet intriguing molecular-scale machines.

Determining Young's modulus via the eigenmode spectrum of a nanomechanical string resonator

We present a method for the in situ determination of Young's modulus of a nanomechanical string resonator subjected to tensile stress. It relies on measuring a large number of harmonic eigenmodes and allows us to access Young's modulus even for the case of a stress-dominated frequency response. We use the proposed framework to obtain Young's modulus of four different wafer materials, comprising three different material platforms amorphous silicon nitride, crystalline silicon carbide, and crystalline indium gallium phosphide. The resulting values are compared with theoretical and literature values where available, revealing the need to measure Young's modulus on the sample material under investigation for precise device characterization.

Structural Optimization of Graphene Triangular Lattice Phononic Crystal Based on Dissipation Dilution Theory.

Nanomechanical resonators offer brilliant mass and force sensitivity applied in many fields, owing to a low mass m and high-quality factor Q. However, in vibrating process, resonant energy is inevitably dissipated. Typically, quality factor does not surpass the inverse of the material loss angle φ. Recently, some exceptions emerged in the use of highly stressed silicon nitride material. As yet, it is interpreted that the pre-stress seems to “dilute” the intrinsic energy dissipation according to the Zener model. Is there any other material that could further break the 1/φ limit and achieve higher quality factors? In our previous research, through theoretical calculation and finite element simulation, we have proved that graphene’s quality factor is two orders of magnitude larger than silicon nitride, on account of the extremely thin thickness of graphene. Based on this, we further optimize the structure of phononic crystals to achieve higher quality factors, in terms of duty cycle and cell size. Through simulation analysis, the quality factor could improve with a larger duty cycle and bigger cell size of triangular lattice phononic crystal. Unexpectedly, the Q amplification coefficient of the 3 × 5-cell structure, which is the least number to compose a phononic crystal with a central defect area, is the highest. In contrast, the minimal cell-number structure in hexagonal lattice could not achieve the brilliant dissipation dilution effect as well as the triangular one. Then we consider how overall size and stress influence quality factor and, furthermore, compare theoretical calculation and finite simulation. Lastly, we start from the primitive 3 × 5 cells, constantly adding cells to the periphery. Through simulation, to our surprise, the largest Q amplification coefficient does not belong to the largest structure, instead originating from the moderate one consisting of 7 × 13 cells.

Photothermal Responsivity of van der Waals Material-Based Nanomechanical Resonators.

Nanomechanical resonators made from van der Waals materials (vdW NMRs) provide a new tool for sensing absorbed laser power. The photothermal response of vdW NMRs, quantified from the resonant frequency shifts induced by optical absorption, is enhanced when incorporated in a Fabry–Pérot (FP) interferometer. Along with the enhancement comes the dependence of the photothermal response on NMR displacement, which lacks investigation. Here, we address the knowledge gap by studying electromotively driven niobium diselenide drumheads fabricated on highly reflective substrates. We use a FP-mediated absorptive heating model to explain the measured variations of the photothermal response. The model predicts a higher magnitude and tuning range of photothermal responses on few-layer and monolayer NbSe2 drumheads, which outperform other clamped vdW drum-type NMRs at a laser wavelength of 532 nm. Further analysis of the model shows that both the magnitude and tuning range of NbSe2 drumheads scale with thickness, establishing a displacement-based framework for building bolometers using FP-mediated vdW NMRs.