Real-Time Measurement of Nanotube Resonator Fluctuations in an Electron Microscope

Imaging Nanomechanical Vibrations and Manipulating Parametric Mode Coupling via Scanning Microwave Microscopy

In this study, we present a novel platform based on scanning microwave microscopy for manipulating and detecting tiny vibrations of nanoelectromechanical resonators using a single metallic tip. The tip is placed on the top of a grounded silicon nitride membrane, acting as a movable top gate of the coupled resonator. We demonstrate its ability to map mechanical modes and investigate mechanical damping effects in a capacitive coupling scheme, based on its spatial resolution. We also manipulate the energy transfer coherently between the mode of the scanning tip and the underlying silicon nitride membrane, via parametric coupling. Typical features of optomechanics, such as anti-damping and electromechanically induced transparency, have been observed. Since the microwave optomechanical technology is fully compatible with quantum electronics and very low temperature conditions, it should provide a powerful tool for studying phonon tunnelling between two spatially separated vibrating elements, which could potentially be applied to quantum sensing.

Analysis of the vibrational characteristics of diamane nanosheet based on the Kirchhoff plate model and atomistic simulations

Single layer diamond-diamane, has been reported with excellent mechanical properties. In this work, molecular dynamics (MD) simulation and Kirchhoff plate model are utilized to investigate the vibrational characteristics of diamane sheets. The mechanical parameters of diamane sheets, including bending stiffness, Young's modulus, Poisson's ratio and coefficient of thermal expansion, are calibrated by using MD simulations. The natural frequencies and corresponding modal shapes of the diamane sheets predicted by the Kirchhoff plate model agree well with that obtained from the MD simulations. It is found that the edges exert marginal effect on the modal shapes when free boundary conditions are applied. Additionally, the Kirchhoff plate model considering the thermal expansion provides reasonable prediction for the natural frequencies of the diamane sheets with all boundary clamped under varying temperatures. This study offers valuable insights into the vibrational properties of diamane sheets, from both a simulation and theoretical standpoint. The findings would be beneficial for the design of nanoscale mechanical resonators utilizing these novel carbon materials.

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.

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.

A molecular dynamics simulation on the atomic mass sensor made of monolayer diamond

The recently synthesized monolayer diamond-diamane has proved to possess excellent mechanical and electrical properties, and it holds great potential in the field of nano-mass sensors. Herein, a molecular dynamics (MD) simulation is employed to systematically investigate the vibration response of the diamane nanoribbon (DNR) for the mass inspection. The results show that under different attached masses, the natural frequency of DNR is about three times of that of the bilayer graphene nanoribbon (BGNR) with the same size. The edge flatness of the DNR can be maintained during the vibration process, while the edge of the BGNR will warp in the initial state. Increasing the pre-strain can significantly increase the natural frequency of the DNR, leading to a higher response sensitivity of the DNR. In addition, the DNR has a higher mass resolution than the BGNR, and can detect smaller attached mass. The position of the attached mass in the resonator has a significant effect on the detection response. When the attached mass is near the center of the resonator, the frequency shift reaches the maximum value, and then it rapidly decreases to zero when the attached mass is close to the edge of DNR. Finally, the attached mass has no obvious effect on the quality factor of the DNR, and its value is stable between 10⁵and 10⁶orders of magnitude. The theoretical results demonstrate the accuracy of the MD results. The MD simulations reveal that the DNR has important implications as a resonant material for nano-mass sensor in the future.

Interrelation of Elasticity and Thermal Bath in Nanotube Cantilevers

We report the first study on the thermal behavior of the stiffness of individual carbon nanotubes, which is achieved by measuring the resonance frequency of their fundamental mechanical bending modes. We observe a reduction of the Young's modulus over a large temperature range with a slope -(173±65)  ppm/K in its relative shift. These findings are reproduced by two different theoretical models based on the thermal dynamics of the lattice. These results reveal how the measured fundamental bending modes depend on the phonons in the nanotube via the Young's modulus. An alternative description based on the coupling between the measured mechanical modes and the phonon thermal bath in the Akhiezer limit is discussed.

Heat diffusion-related damping process in a highly precise coarse-grained model for nonlinear motion of SWCNT

Second sound and heat diffusion in single-walled carbon nanotubes (SWCNT) are well-known phenomena which is related to the high thermal conductivity of this material. In this paper, we have shown that the heat diffusion along the tube axis affects the macroscopic motion of SWCNT and adapting this phenomena to coarse-grained (CG) model can improve the precision of the coarse-grained molecular dynamics (CGMD) exceptionally. The nonlinear macroscopic motion of SWCNT in the free thermal vibration condition in adiabatic environment is demonstrated in the most simplified version of CG modeling as maintaining finite temperature and total energy with suggested dissipation process derived from internal heat diffusion. The internal heat diffusion related to the cross correlated momentum from different potential energy functions is considered, and it can reproduce the nonlinear dynamic nature of SWCNTs without external thermostatting in CG model. Memory effect and thermostat with random noise distribution are not included, and the effect of heat diffusion on memory effect is quantified through Mori-Zwanzig formalism. This diffusion shows perfect syncronization of the motion between that of CGMD and MD simulation, which is started with initial conditions from the molecular dynamics (MD) simulation. The heat diffusion related to this process has shown the same dispersive characteristics to second wave in SWCNT. This replication with good precision indicates that the internal heat diffusion process is the essential cause of the nonlinearity of the tube. The nonlinear dynamic characteristics from the various scale of simple beads systems are examined with expanding its time step and node length.

A coherent nanomechanical oscillator driven by single-electron tunnelling

A single-electron transistor embedded in a nanomechanical resonator represents an extreme limit of electron-phonon coupling. While it allows fast and sensitive electromechanical measurements, it also introduces backaction forces from electron tunnelling that randomly perturb the mechanical state. Despite the stochastic nature of this backaction, it has been predicted to create self-sustaining coherent mechanical oscillations under strong coupling conditions. Here, we verify this prediction using real-time measurements of a vibrating carbon nanotube transistor. This electromechanical oscillator has some similarities with a laser. The single-electron transistor pumped by an electrical bias acts as a gain medium and the resonator acts as a phonon cavity. Although the operating principle is unconventional because it does not involve stimulated emission, we confirm that the output is coherent. We demonstrate other analogues of laser behaviour, including injection locking, classical squeezing through anharmonicity, and frequency narrowing through feedback.

Turing-complete mechanical processor via automated nonlinear system design

Nanomechanical computers promise a greatly improved energetic efficiency compared to their electrical counterparts. However, progress towards this goal is hindered by a lack of modular components, such as logic gates or transistors, and systematic design strategies. This article describes a universal logic gate implemented as a nonlinear mass-spring-damper model, followed by an automated method to translate computations, expressed as source code of arbitrary complexity, into combinations of this basic building block. The proposed approach is validated numerically in two steps: First, a set of discrete models are generated from code. The models implement computations with increasing complexity, starting by a simple adder and ending in a eight-bit Turing complete mechanical processor. Then the models are forward integrated to demonstrate their computing performance. The processor is validated by executing the Erathostenes' sieve algorithm to mechanically compute prime numbers.

Mass Sensing for the Advanced Fabrication of Nanomechanical Resonators

We report on a nanomechanical engineering method to monitor matter growth in real time via e-beam electromechanical coupling. This method relies on the exceptional mass sensing capabilities of nanomechanical resonators. Focused electron beam-induced deposition (FEBID) is employed to selectively grow platinum particles at the free end of singly clamped nanotube cantilevers. The electron beam has two functions: it allows both to grow material on the nanotube and to track in real time the deposited mass by probing the noise-driven mechanical resonance of the nanotube. On the one hand, this detection method is highly effective as it can resolve mass deposition with a resolution in the zeptogram range; on the other hand, this method is simple to use and readily available to a wide range of potential users because it can be operated in existing commercial FEBID systems without making any modification. The presented method allows one to engineer hybrid nanomechanical resonators with precisely tailored functionalities. It also appears as a new tool for studying the growth dynamics of ultrathin nanostructures, opening new opportunities for investigating so far out-of-reach physics of FEBID and related methods.

Flexural Wave Propagation in Mass Chain-Filled Carbon Nanotubes

The propagation characteristics of terahertz (THz) flexural waves in mass chain-filled single-walled carbon nanotubes (MCSCs) are studied using a continuum mechanics approach and molecular dynamics (MD) simulations, where each single-walled carbon nanotube (SWCNT) is modeled as a nonlocal Timoshenko beam based on the nonlocal strain gradient theory. The effect of the surrounding elastic medium and the van der Waals (vdW) interactions between the mass chain and the SWCNT on the wave propagation is quantitatively considered in governing equations, respectively. The analytical expressions of two flexural wave branches and the bandgap between the two branches are derived. When combining our MD simulations of the carbon-atom chain-filled SWCNT, the wave within the bandgap disperses rapidly, and the mass chain has a significant influence on the phase velocity of the flexural wave. The present theoretical solution has a high accuracy in a wide frequency range up to the THz region. In particular, the surrounding elastic medium of the MCSCs remarkably affects the phase velocity for low frequencies, but not for high frequencies. The present study indicates that the wave propagation of a SWCNT could be modulated by changing the filled mass chain and the surrounding elastic medium.

Force sensing with nanowire cantilevers

Nanometer-scale structures with high aspect ratios such as nanowires and nanotubes combine low mechanical dissipation with high resonance frequencies, making them ideal force transducers and scanning probes in applications requiring the highest sensitivity. Such structures promise record force sensitivities combined with ease of use in scanning probe microscopes. A wide variety of possible material compositions and functionalizations is available, allowing for the sensing of various kinds of forces. In addition, nanowires possess quasi-degenerate mechanical mode doublets, which allow for sensitive vectorial force and mass detection. These developments have driven researchers to use nanowire cantilevers in various force sensing applications, which include imaging of sample surface topography, detection of optomechanical, electrical, and magnetic forces, and magnetic resonance force microscopy. In this review, we discuss the motivation behind using nanowires as force transducers, explain the methods of force sensing with nanowire cantilevers, and give an overview of the experimental progress so far and future prospects of the field.

Shot-Noise-Limited Nanomechanical Detection and Radiation Pressure Backaction from an Electron Beam

Detecting nanomechanical motion has become an important challenge in science and technology. Recently, electromechanical coupling to focused electron beams has emerged as a promising method adapted to ultralow scale systems. However the fundamental measurement processes associated with such complex interaction remain to be explored. Here we report a highly sensitive detection of the Brownian motion of μm-long semiconductor nanowires (InAs). The measurement imprecision is found to be set by the shot noise of the secondary electrons generated along the electromechanical interaction. By carefully analyzing the nanoelectromechanical dynamics, we demonstrate the existence of a radial backaction process that we identify as originating from the momentum exchange between the electron beam and the nanomechanical device, which is also known as radiation pressure.

Real-time vibrations of a carbon nanotube

The field of miniature mechanical oscillators is rapidly evolving, with emerging applications including signal processing, biological detection¹ and fundamental tests of quantum mechanics². As the dimensions of a mechanical oscillator shrink to the molecular scale, such as in a carbon nanotube resonator^3-7, their vibrations become increasingly coupled and strongly interacting^8,9 until even weak thermal fluctuations could make the oscillator nonlinear^10-13. The mechanics at this scale possesses rich dynamics, unexplored because an efficient way of detecting the motion in real time is lacking. Here we directly measure the thermal vibrations of a carbon nanotube in real time using a high-finesse micrometre-scale silicon nitride optical cavity as a sensitive photonic microscope. With the high displacement sensitivity of 700 fm Hz^-1/2 and the fine time resolution of this technique, we were able to discover a realm of dynamics undetected by previous time-averaged measurements and a room-temperature coherence that is nearly three orders of magnitude longer than previously reported. We find that the discrepancy in the coherence stems from long-time non-equilibrium dynamics, analogous to the Fermi-Pasta-Ulam-Tsingou recurrence seen in nonlinear systems¹⁴. Our data unveil the emergence of a weakly chaotic mechanical breather¹⁵, in which vibrational energy is recurrently shared among several resonance modes-dynamics that we are able to reproduce using a simple numerical model. These experiments open up the study of nonlinear mechanical systems in the Brownian limit (that is, when a system is driven solely by thermal fluctuations) and present an integrated, sensitive, high-bandwidth nanophotonic interface for carbon nanotube resonators.

Quasi-periodic magnetization reversal of ferromagnetic nanoparticles induced by torsional oscillations in static magnetic fields

In order to reverse the magnetization of small ferromagnetic particles it is necessary to overcome an energy barrier, which is mainly defined by the magnetic anisotropy. Usual reversal stimuli include the application of static or time-dependent external magnetic fields, thermal activation, spin transfer torque, or combinations thereof. Here, we report on repeated, quasi-periodic magnetization reversal in single-domain particles that are exposed to a constant magnetic field perpendicular to the magnet's easy axis. The continuous sequence of reversals is induced by torsional oscillations of the magnet's anisotropy landscape, which are caused by angular oscillations of the magnet's body. In our experiments, a nickel nanowire constitutes both a mechanical resonator and a nanomagnetic sample with uniaxial anisotropy. We measure the transient flexural vibration behavior by electron beam based methods and find strong signatures of periodic magnetization switching between two magnetic states of the nanowire. Our system can be modeled as a driven damped harmonic oscillator under the influence of switchable magnetostatic interactions.

MicroMegascope

Atomic force microscopy (AFM) allows us to reconstruct the topography of surfaces with resolution in the nanometer range. The exceptional resolution attainable with the AFM makes this instrument a key tool in nanoscience and technology. The core of a standard AFM set-up relies on the detection of the change of the mechanical motion of a micro-oscillator when approaching the sample to image. This is despite the fact that AFM is nowadays a very common instrument for both fundamental and applied research. The fabrication of the micrometric scale mechanical oscillator is still a very complicated and expensive task requiring dedicated platforms. Being able to perform AFM with a macroscopic oscillator would make the instrument more versatile and accessible for an even larger spectrum of applications and audience. Here, we present atomic force imaging with a centimetric oscillator, an aluminum tuning fork of centimeter size as a sensor on which an accelerometer is glued on one prong to measure the oscillations. We show that it is possible to perform topographic images of nanometric resolution with a gram tuning fork. In addition to the stunning sensitivity, we show the high versatility of such an oscillator by imaging both in air and liquid. The set-up proposed here can be extended to numerous experiments where the probe has to be heavy and/or very complex, and so too the environment.

Ultrasensitive Displacement Noise Measurement of Carbon Nanotube Mechanical Resonators

Mechanical resonators based on a single carbon nanotube are exceptional sensors of mass and force. The force sensitivity in these ultralight resonators is often limited by the noise in the detection of the vibrations. Here, we report on an ultrasensitive scheme based on a RLC resonator and a low-temperature amplifier to detect nanotube vibrations. We also show a new fabrication process of electromechanical nanotube resonators to reduce the separation between the suspended nanotube and the gate electrode down to ∼150 nm. These advances in detection and fabrication allow us to reach [Formula: see text] displacement sensitivity. Thermal vibrations cooled cryogenically at 300 mK are detected with a signal-to-noise ratio as high as 17 dB. We demonstrate [Formula: see text] force sensitivity, which is the best force sensitivity achieved thus far with a mechanical resonator. Our work is an important step toward imaging individual nuclear spins and studying the coupling between mechanical vibrations and electrons in different quantum electron transport regimes.

Optomechanics with a hybrid carbon nanotube resonator

In just 20 years of history, the field of optomechanics has achieved impressive progress, stepping into the quantum regime just 5 years ago. Such remarkable advance relies on the technological revolution of nano-optomechanical systems, whose sensitivity towards thermal decoherence is strongly limited due to their ultra-low mass. Here we report a hybrid approach pushing nano-optomechanics to even lower scales. The concept relies on synthesising an efficient optical scatterer at the tip of singly clamped carbon nanotube resonators. We demonstrate high signal-to-noise motion readout and record force sensitivity, two orders of magnitude below the state of the art. Our work opens the perspective to extend quantum experiments and applications at room temperature.

Electron beam detection of a Nanotube Scanning Force Microscope

Atomic Force Microscopy (AFM) allows to probe matter at atomic scale by measuring the perturbation of a nanomechanical oscillator induced by near-field interaction forces. The quest to improve sensitivity and resolution of AFM forced the introduction of a new class of resonators with dimensions at the nanometer scale. In this context, nanotubes are the ultimate mechanical oscillators because of their one dimensional nature, small mass and almost perfect crystallinity. Coupled to the possibility of functionalisation, these properties make them the perfect candidates as ultra sensitive, on-demand force sensors. However their dimensions make the measurement of the mechanical properties a challenging task in particular when working in cavity free geometry at ambient temperature. By using a focused electron beam, we show that the mechanical response of nanotubes can be quantitatively measured while approaching to a surface sample. By coupling electron beam detection of individual nanotubes with a custom AFM we image the surface topography of a sample by continuously measuring the mechanical properties of the nanoresonators. The combination of very small size and mass together with the high resolution of the electron beam detection method offers unprecedented opportunities for the development of a new class of nanotube-based scanning force microscopy.