A nanoscale parametric feedback oscillator

Parametric Frequency Divider Based Ising Machines

We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.

Dynamical response and noise limit of a parametrically pumped microcantilever sensor in a Phase-Locked Loop

We investigate the response of a digitally controlled and parametrically pumped microcantilever used for sensing in a Phase-Locked Loop (PLL). We develop an analytical model for its dynamical response and obtain an explicit dependence on the rheological parameters of the surrounding viscous medium. Linearization of this model allows to find improved responsivity to density variations in the case of parametric suppression. Experiments with a commercial microcantilever validate the model, but also reveal an increase of frequency noise in the PLL associated with the parametric gain and phase, which, in most cases, restricts the attainable limit of detection. The noise in open-loop is studied by measuring the random fluctuations of the noise-driven deflection of the microcantilever, and a model for the power spectral density of amplitude, phase and frequency noises is discussed and used to explain the frequency fluctuations in the closed-loop PLL. This work concludes that parametric pumping in a PLL does not improve the sensing performance in applications requiring detecting frequency shifts.

A Precise Closed-Loop Controlled ZnO Nanowire Resonator Operating at Room Temperature

To realize the real-time measurement of masses of nanoparticles, virus molecules, organic macromolecules, and gas molecules, and to analyze their physical and chemical properties, a ZnO nanowire (NW) resonator operating at room temperature with an ultrahigh resonant frequency, real-time detection, and high precision was designed and developed in this study. The machining method is simple and easy to integrate into an integrated circuit. A closed-loop detection system based on a phase-locked loop (PLL) and frequency modulation technology (FM) was used to perform closed-loop testing of electromagnetically excited ZnO NW. The first-order resonance frequency of the resonator was 10.358 MHz, the quality factor Q value was about 600, the frequency fluctuation value f_RMS was about 300 Hz, and the FM range could reach 200 kHz. The equivalent circuit model of the resonator was established, the parasitic parameters during the test were obtained, and the frequency accuracy and phase noise of the resonator were analyzed and tested. The experimental results show that the closed-loop system can automatically control the resonator in a wide range of frequency bands, with good tracking performance of the resonant frequency, small frequency fluctuation, and low phase noise level.

Reaching silicon-based NEMS performances with 3D printed nanomechanical resonators

The extreme miniaturization in NEMS resonators offers the possibility to reach an unprecedented resolution in high-performance mass sensing. These very low limits of detection are related to the combination of two factors: a small resonator mass and a high quality factor. The main drawback of NEMS is represented by the highly complex, multi-steps, and expensive fabrication processes. Several alternatives fabrication processes have been exploited, but they are still limited to MEMS range and very low-quality factor. Here we report the fabrication of rigid NEMS resonators with high-quality factors by a 3D printing approach. After a thermal step, we reach complex geometry printed devices composed of ceramic structures with high Young’s modulus and low damping showing performances in line with silicon-based NEMS resonators ones. We demonstrate the possibility of rapid fabrication of NEMS devices that present an effective alternative to semiconducting resonators as highly sensitive mass and force sensors.

NEMS devices, nano-electro-mechanical systems, by virtue of their minute size, offer ultra-high sensitivity, though at the expense of manufacturing complexity. Here, Stassi et al succeed in manufacturing high quality factor NEMS devices using high resolution 3D printing.

High Dynamic Range Nanowire Resonators

Dynamic range quantifies the linear operation regime available in nanomechanical resonators. Nonlinearities dominate the response of flexural beams in the limit of very high aspect ratio and very small diameter, which leads to expectation of low dynamic range for nanowire resonators in general. However, the highest achievable dynamic range for nanowire resonators with practical dimensions remains to be determined. We report dynamic range measurements on singly clamped silicon nanowire resonators reaching remarkably high values of up to 90 dB obtained with a simple harmonic actuation scheme. We explain these measurements by a comprehensive theoretical examination of dynamic range in singly clamped flexural beams including the effect of tapering, a usual feature of semiconductor nanowires. Our analysis reveals the nanowire characteristics required for broad linear operation, and given the relationship between dynamic range and mass sensing performance, it also enables analytical determination of mass detection limits, reaching atomic-scale resolution for feasible nanowires.

Strong internal resonance in a nonlinear, asymmetric microbeam resonator

Exploiting nonlinear characteristics in micro/nanosystems has been a subject of increasing interest in the last decade. Among others, vigorous intermodal coupling through internal resonance (IR) has drawn much attention because it can suggest new strategies to steer energy within a micro/nanomechanical resonator. However, a challenge in utilizing IR in practical applications is imposing the required frequency commensurability between vibrational modes of a nonlinear micro/nanoresonator. Here, we experimentally and analytically investigate the 1:2 and 2:1 IR in a clamped-clamped beam resonator to provide insights into the detailed mechanism of IR. It is demonstrated that the intermodal coupling between the second and third flexural modes in an asymmetric structure (e.g., nonprismatic beam) provides an optimal condition to easily implement a strong IR with high energy transfer to the internally resonated mode. In this case, the quadratic coupling between these flexural modes, originating from the stretching effect, is the dominant nonlinear mechanism over other types of geometric nonlinearity. The design strategies proposed in this paper can be integrated into a typical micro/nanoelectromechanical system (M/NEMS) via a simple modification of the geometric parameters of resonators, and thus, we expect this study to stimulate further research and boost paradigm-shifting applications exploring the various benefits of IR in micro/nanosystems.

Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization

A microcantilever is a suspended micro-scale beam structure supported at one end which can bend and/or vibrate when subjected to a load. Microcantilevers are one of the most fundamental miniaturized devices used in microelectromechanical systems and are ubiquitous in sensing, imaging, time reference, and biological/biomedical applications. They are typically built using micro and nanofabrication techniques derived from the microelectronics industry and can involve microelectronics-related materials, polymeric materials, and biological materials. This work presents a comprehensive review of the rich dynamical response of a microcantilever and how it has been used for measuring the mass and rheological properties of Newtonian/non-Newtonian fluids in real time, in ever-decreasing space and time scales, and with unprecedented resolution.

Nonlinear parametric oscillator: A tool for probing quantum fluctuations

Nanomechanical oscillators have, over the last few years, started probing regimes where quantum fluctuations are important. Here we consider a nonlinear parametric oscillator in the quantum domain. We show that in the classical subharmonic resonance zone, the quantum fluctuations are finite but greatly magnified depending on the strength of the nonlinear coupling. This should make such oscillators useful in probing quantum fluctuations.

Rapid Flipping of Parametric Phase States

We experimentally demonstrate flipping the phase state of a parametron within a single period of its oscillation. A parametron is a binary logic element based on a driven nonlinear resonator. It features two stable phase states that define an artificial spin. The most basic operation performed on a parametron is a bit flip between these two states. Thus far, this operation involved changing the energetic population of the resonator and therefore required a number of oscillations on the order of the quality factor Q. Our technique takes a radically different approach and relies on rapid control of the underlying potential. Our work represents a paradigm shift for phase-encoded logic operations by boosting the speed of a parametron bit flip to its ultimate limit.

Experimental Verification of a Bigrating Beam Rider

An optical beam rider making use of a light sail comprising two opposing diffraction gratings is experimentally demonstrated for the first time. We verify that the illuminated space-variant grating structure provides an optical restoring force, exhibiting stable oscillations when the bigrating is displaced from equilibrium. We further demonstrate parametric cooling by illuminating the sail with synchronized light pulses. This experiment enhances the technical feasibility of a laser-driven light sail based on diffractive radiation pressure.

Gate-controlled phase switching in a parametron

The parametron, a resonator-based logic device, is a promising physical platform for emerging computational paradigms. When the parametron is subject to both parametric pumping and external driving, complex phenomena arise that can be harvested for applications. In this paper, we experimentally demonstrate deterministic phase switching of a parametron by applying frequency tuning pulses. To our surprise, we find different regimes of phase switching due to the interplay between a parametric pump and an external drive. We provide full modeling of our device with numerical simulations and find excellent agreement between model and measurements. Our result opens up new possibilities for fast and robust logic operations within large-scale parametron architectures.

Resonance tracking in a micromechanical device using phononic frequency combs

Micro and nanomechanical resonators have been extensively researched in recent decades for applications to time and frequency references, as well as highly sensitive sensors. Conventionally, the operation of these resonant sensors is implemented using a feedback oscillator to dynamically track variations in the resonant frequency. However, this approach places limitations on the frequency stability of the output response, particularly owing to near-carrier noise shaping, limiting measurement stabilities at short-to-moderate integration times. Here, in this paper, utilizing the recent experimental demonstration of phononic frequency combs, we demonstrate an alternative resonance tracking approach with the potential to provide significant improvements in near-carrier phase noise and long-term stability. In addition, we also showcase comb dynamics mediated resonant frequency modulation which indirectly points to the possible control of inevitable noise processes including thermomechanical fluctuations. This resonant tracking approach may also have general applicability to a number of other physical oscillators.

A Nonlinear Rate Microsensor utilising Internal Resonance

Micro- and nano-resonators have been studied extensively both for the scientific viewpoint to understand basic interactions at small scales as well as for applied research to build sensors and mechanical signal processors. Majority of the resonant microsystems, particularly those manufactured at a large scale, have employed simple mechanical structures with one dominant resonant mode, such as in timing resonators, or linearly coupled resonant modes, as in vibratory gyroscopes. There is an increasing interest in the development of models and methods to better understand the nonlinear interactions at micro- and nano-scales and also to potentially improve the performance of the existing devices in the market beyond limits permissible by the linear effects. Internal resonance is a phenomenon that allows for nonlinear coupling and energy transfer between different vibration modes of a properly designed system. Herein, for the first time, we describe and experimentally demonstrate the potential for employing internal resonance for detection of angular rate signals, where the Coriolis effect modifies the energy coupling between the distinct drive and sense vibration modes. In doing so, in addition to providing a robust method of exciting the desired mode, the proposed approach further alleviates the mode-matching requirements and reduces instabilities due to the cross-coupling between the modes in current linear vibratory gyroscopes.

Exotic states in a simple network of nanoelectromechanical oscillators

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

A Temperature-Compensated Single-Crystal Silicon-on-Insulator (SOI) MEMS Oscillator with a CMOS Amplifier Chip

Self-sustained feedback oscillators referenced to MEMS/NEMS resonators have the potential for a wide range of applications in timing and sensing systems. In this paper, we describe a real-time temperature compensation approach to improving the long-term stability of such MEMS-referenced oscillators. This approach is implemented on a ~26.8 kHz self-sustained MEMS oscillator that integrates the fundamental in-plane mode resonance of a single-crystal silicon-on-insulator (SOI) resonator with a programmable and reconfigurable single-chip CMOS sustaining amplifier. Temperature compensation using a linear equation fit and look-up table (LUT) is used to obtain the near-zero closed-loop temperature coefficient of frequency (TCf) at around room temperature (~25 °C). When subject to small temperature fluctuations in an indoor environment, the temperature-compensated oscillator shows a >2-fold improvement in Allan deviation over the uncompensated counterpart on relatively long time scales (averaging time τ > 10,000 s), as well as overall enhanced stability throughout the averaging time range from τ = 1 to 20,000 s. The proposed temperature compensation algorithm has low computational complexity and memory requirement, making it suitable for implementation on energy-constrained platforms such as Internet of Things (IoT) sensor nodes.

Electrically tunable single- and few-layer MoS2 nanoelectromechanical systems with broad dynamic range

Atomically thin semiconducting crystals [such as molybdenum disulfide (MoS₂)] have outstanding electrical, optical, and mechanical properties, thus making them excellent constitutive materials for innovating new two-dimensional (2D) nanoelectromechanical systems (NEMS). Although prototype structures have recently been demonstrated toward functional devices such as ultralow-power, high-frequency tunable oscillators and ultrasensitive resonant transducers, both electrical tunability and large dynamic range (DR) are critical and desirable. We report the first experimental demonstration of clearly defined single-, bi-, and trilayer MoS₂ 2D resonant NEMS operating in the very high frequency band (up to ~120 MHz) with outstanding electrical tunability and DR. Through deterministic measurement and calibration, we discover that these 2D atomic layer devices have remarkably broad DR (up to ~70 to 110 dB), in contrast to their 1D NEMS counterparts that are expected to have limited DR. These 2D devices, therefore, open avenues for efficiently tuning and strongly coupling the electronic, mechanical, and optical properties in atomic layer semiconducting devices and systems.

Optically driven ultra-stable nanomechanical rotor

Nanomechanical devices have attracted the interest of a growing interdisciplinary research community, since they can be used as highly sensitive transducers for various physical quantities. Exquisite control over these systems facilitates experiments on the foundations of physics. Here, we demonstrate that an optically trapped silicon nanorod, set into rotation at MHz frequencies, can be locked to an external clock, transducing the properties of the time standard to the rod’s motion with a remarkable frequency stability f_r/Δf_r of 7.7 × 10¹¹. While the dynamics of this periodically driven rotor generally can be chaotic, we derive and verify that stable limit cycles exist over a surprisingly wide parameter range. This robustness should enable, in principle, measurements of external torques with sensitivities better than 0.25 zNm, even at room temperature. We show that in a dilute gas, real-time phase measurements on the locked nanorod transduce pressure values with a sensitivity of 0.3%.

Nanomechanical sensors that rely on intrinsic resonance frequencies usually present a tradeoff between sensitivity and bandwidth. In this work, the authors realise an optically driven nanorotor featuring high frequency stability and tunability over a large frequency range.

Probing the chaotic boundary of a membrane resonator with nanowire arrays

Mechanically induced nonlinearities in nano-electromechanical systems (NEMSs) are typically avoided in design due to their unpredictable nature; however, by incorporating these normally unwanted nonlinear and chaotic phenomena, the performance of NEMS devices displays substantially different characteristics opening a broad new range of potential applications for their use. In this work, experiments have been conducted for probing the chaotic boundary of a circular membrane mechanical resonator with and without a silicone nanowire array (Si NWA). The NWA resonator can transition from linear to nonlinear quasi-periodic behaviour, and further transition into a chaotic state at resonance. Moreover, the NWA resonator demonstrated a high level of complex nonlinear behaviours, as the device expands the power spectral response from a single frequency at a linear regime to a wideband continuous frequency spectrum when chaotic behaviour was initiated; the threshold power of this transition decreased with a smaller NWA diameter. It was also observed that the NWA resonator had higher damping compared to the resonator without a NWA; however, as the vibration velocity of the NWA resonator increased, complex air damping and thin squeeze film damping lowered the threshold for probing the chaotic boundary condition of the NWA resonator.

Complex Dynamical Networks Constructed with Fully Controllable Nonlinear Nanomechanical Oscillators

Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network's full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization.

Low-Power Photothermal Self-Oscillation of Bimetallic Nanowires

We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in the presence of incident laser light and a reflecting Si mirror. Situated in a standing wave of optical intensity and subject to photothermal forces, the nanowire undergoes self-induced oscillations at low incident light thresholds of <1 μW due to engineered strong temperature-position (T-z) coupling. Along with inducing self-oscillation, laser light causes large changes to the mechanical resonant frequency ω₀ and equilibrium position z₀ that cannot be neglected. We present experimental results and a theoretical model for the motion under laser illumination. In the model, we solve the governing nonlinear differential equations by perturbative means to show that self-oscillation amplitude is set by the competing effects of direct T-z coupling and 2ω₀ parametric excitation due to T-ω₀ coupling. We then study the linearized equations of motion to show that the optimal thermal time constant τ for photothermal feedback is τ → ∞ rather than the previously reported ω₀ τ = 1. Lastly, we demonstrate photothermal quality factor (Q) enhancement of driven motion as a means to counteract air damping. Understanding photothermal effects on nano- and micromechanical devices, as well as nonlinear aspects of optics-based motion detection, can enable new device applications as oscillators or other electronic elements with smaller device footprints and less stringent ambient vacuum requirements.

Self-tuning bistable parametric feedback oscillator: Near-optimal amplitude maximization without model information

Theory predicts that parametrically excited oscillators, tuned to operate under resonant condition, are capable of large-amplitude oscillation useful in diverse applications, such as signal amplification, communication, and analog computation. However, due to amplitude saturation caused by nonlinearity, lack of robustness to model uncertainty, and limited sensitivity to parameter modulation, these oscillators require fine-tuning and strong modulation to generate robust large-amplitude oscillation. Here we present a principle of self-tuning parametric feedback excitation that alleviates the above-mentioned limitations. This is achieved using a minimalistic control implementation that performs (i) self-tuning (slow parameter adaptation) and (ii) feedback pumping (fast parameter modulation), without sophisticated signal processing past observations. The proposed approach provides near-optimal amplitude maximization without requiring model-based control computation, previously perceived inevitable to implement optimal control principles in practical application. Experimental implementation of the theory shows that the oscillator self-tunes itself near to the onset of dynamic bifurcation to achieve extreme sensitivity to small resonant parametric perturbations. As a result, it achieves large-amplitude oscillations by capitalizing on the effect of nonlinearity, despite substantial model uncertainties and strong unforeseen external perturbations. We envision the present finding to provide an effective and robust approach to parametric excitation when it comes to real-world application.

Parametric Symmetry Breaking in a Nonlinear Resonator

Much of the physical world around us can be described in terms of harmonic oscillators in thermodynamic equilibrium. At the same time, the far-from-equilibrium behavior of oscillators is important in many aspects of modern physics. Here, we investigate a resonating system subject to a fundamental interplay between intrinsic nonlinearities and a combination of several driving forces. We have constructed a controllable and robust realization of such a system using a macroscopic doubly clamped string. We experimentally observe a hitherto unseen double hysteresis in both the amplitude and the phase of the resonator's response function and present a theoretical model that is in excellent agreement with the experiment. Our work unveils that the double hysteresis is a manifestation of an out-of-equilibrium symmetry breaking between parametric phase states. Such a fundamental phenomenon, in the most ubiquitous building block of nature, paves the way for the investigation of new dynamical phases of matter in parametrically driven many-body systems and motivates applications ranging from ultrasensitive force detection to low-energy computing memory units.

Micromachined Resonators: A Review

This paper is a review of the remarkable progress that has been made during the past few decades in design, modeling, and fabrication of micromachined resonators. Although micro-resonators have come a long way since their early days of development, they are yet to fulfill the rightful vision of their pervasive use across a wide variety of applications. This is partially due to the complexities associated with the physics that limit their performance, the intricacies involved in the processes that are used in their manufacturing, and the trade-offs in using different transduction mechanisms for their implementation. This work is intended to offer a brief introduction to all such details with references to the most influential contributions in the field for those interested in a deeper understanding of the material.

Self-Sustained Micromechanical Oscillator with Linear Feedback

Autonomous oscillators, such as clocks and lasers, produce periodic signals without any external frequency reference. In order to sustain stable periodic motion, there needs to be an external energy supply as well as nonlinearity built into the oscillator to regulate the amplitude. Usually, nonlinearity is provided by the sustaining feedback mechanism, which also supplies energy, whereas the constituent resonator that determines the output frequency stays linear. Here, we propose a new self-sustaining scheme that relies on the nonlinearity originating from the resonator itself to limit the oscillation amplitude, while the feedback remains linear. We introduce a model for describing the working principle of the self-sustained oscillations and validate it with experiments performed on a nonlinear microelectromechanical oscillator.

Frequency fluctuations in silicon nanoresonators

Frequency stability is key to the performance of nanoresonators. This stability is thought to reach a limit with the resonator's ability to resolve thermally induced vibrations. Although measurements and predictions of resonator stability usually disregard fluctuations in the mechanical frequency response, these fluctuations have recently attracted considerable theoretical interest. However, their existence is very difficult to demonstrate experimentally. Here, through a literature review, we show that all studies of frequency stability report values several orders of magnitude larger than the limit imposed by thermomechanical noise. We studied a monocrystalline silicon nanoresonator at room temperature and found a similar discrepancy. We propose a new method to show that this was due to the presence of frequency fluctuations, of unexpected level. The fluctuations were not due to the instrumentation system, or to any other of the known sources investigated. These results challenge our current understanding of frequency fluctuations and call for a change in practices.

Optimal Parametric Feedback Excitation of Nonlinear Oscillators

An optimal parametric feedback excitation principle is sought, found, and investigated. The principle is shown to provide an adaptive resonance condition that enables unprecedentedly robust movement generation in a large class of oscillatory dynamical systems. Experimental demonstration of the theory is provided by a nonlinear electronic circuit that realizes self-adaptive parametric excitation without model information, signal processing, and control computation. The observed behavior dramatically differs from the one achievable using classical parametric modulation, which is fundamentally limited by uncertainties in model information and nonlinear effects inevitably present in real world applications.

Modelling the Size Effects on the Mechanical Properties of Micro/Nano Structures

Experiments on micro- and nano-mechanical systems (M/NEMS) have shown that their behavior under bending loads departs in many cases from the classical predictions using Euler-Bernoulli theory and Hooke's law. This anomalous response has usually been seen as a dependence of the material properties on the size of the structure, in particular thickness. A theoretical model that allows for quantitative understanding and prediction of this size effect is important for the design of M/NEMS. In this paper, we summarize and analyze the five theories that can be found in the literature: Grain Boundary Theory (GBT), Surface Stress Theory (SST), Residual Stress Theory (RST), Couple Stress Theory (CST) and Surface Elasticity Theory (SET). By comparing these theories with experimental data we propose a simplified model combination of CST and SET that properly fits all considered cases, therefore delivering a simple (two parameters) model that can be used to predict the mechanical properties at the nanoscale.

A self-stabilized coherent phonon source driven by optical forces

We report a novel injection scheme that allows for “phonon lasing” in a one-dimensional opto-mechanical photonic crystal, in a sideband unresolved regime and with cooperativity values as low as 10⁻². It extracts energy from a cw infrared laser source and is based on the triggering of a thermo-optical/free-carrier-dispersion self-pulsing limit-cycle, which anharmonically modulates the radiation pressure force. The large amplitude of the coherent mechanical motion acts as a feedback that stabilizes and entrains the self-pulsing oscillations to simple fractions of the mechanical frequency. A manifold of frequency-entrained regions with two different mechanical modes (at 54 and 122 MHz) are observed as a result of the wide tuneability of the natural frequency of the self-pulsing. The system operates at ambient conditions of pressure and temperature in a silicon platform, which enables its exploitation in sensing, intra-chip metrology or time-keeping applications.

Low Temperature Reactive Sputtering of Thin Aluminum Nitride Films on Metallic Nanocomposites

Piezoelectric aluminum nitride thin films were deposited on aluminum-molybdenum (AlMo) metallic nanocomposites using reactive DC sputtering at room temperature. The effect of sputtering parameters on film properties was assessed. A comparative study between AlN grown on AlMo and pure aluminum showed an equivalent (002) crystallographic texture. The piezoelectric coefficients were measured to be 0.5±0.1 C m^-2 and 0.9±0.1 C m^-2, for AlN deposited on Al/0.32Mo and pure Al, respectively. Films grown onto Al/0.32Mo however featured improved surface roughness. Roughness values were measured to be 1.3nm and 5.4 nm for AlN films grown on AlMo and on Al, respectively. In turn, the dielectric constant was measured to be 8.9±0.7 for AlN deposited on Al/0.32Mo seed layer, and 8.7±0.7 for AlN deposited on aluminum; thus, equivalent within experimental error. Compatibility of this room temperature process with the lift-off patterning of the deposited AlN is also reported.

Low phase-noise autonomous parametric oscillator based on a 226.7 MHz AlN contour-mode resonator

We present the first parametric oscillator based on the use of a 226.7 MHz aluminum nitride contour-mode resonator. This topology enables an improvement in the phase noise of 16 dB at 1 kHz offset with respect to a conventional feedback-loop oscillator based on the same device. The recorded phase noise is -106 dBc/Hz at 1 kHz offset.

Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

Stabilization of a linear nanomechanical oscillator to its thermodynamic limit

The rapid development of micro- and nanomechanical oscillators in the past decade has led to the emergence of novel devices and sensors that are opening new frontiers in both applied and fundamental science. The potential of these devices is however affected by their increased sensitivity to external perturbations. Here we report a non-perturbative optomechanical stabilization technique and apply the method to stabilize a linear nanomechanical beam at its thermodynamic limit at room temperature. The reported ability to stabilize a nanomechanical oscillator to the thermodynamic limit can be extended to a variety of systems and increases the sensitivity range of nanomechanical sensors in both fundamental and applied studies.

Phase synchronization of two anharmonic nanomechanical oscillators

We investigate the synchronization of oscillators based on anharmonic nanoelectromechanical resonators. Our experimental implementation allows unprecedented observation and control of parameters governing the dynamics of synchronization. We find close quantitative agreement between experimental data and theory describing reactively coupled Duffing resonators with fully saturated feedback gain. In the synchronized state we demonstrate a significant reduction in the phase noise of the oscillators, which is key for sensor and clock applications. Our work establishes that oscillator networks constructed from nanomechanical resonators form an ideal laboratory to study synchronization--given their high-quality factors, small footprint, and ease of cointegration with modern electronic signal processing technologies.

Graphene mechanical oscillators with tunable frequency

Oscillators, which produce continuous periodic signals from direct current power, are central to modern communications systems, with versatile applications including timing references and frequency modulators. However, conventional oscillators typically consist of macroscopic mechanical resonators such as quartz crystals, which require excessive off-chip space. Here, we report oscillators built on micrometre-size, atomically thin graphene nanomechanical resonators, whose frequencies can be electrostatically tuned by as much as 14%. Self-sustaining mechanical motion is generated and transduced at room temperature in these oscillators using simple electrical circuitry. The prototype graphene voltage-controlled oscillators exhibit frequency stability and a modulation bandwidth sufficient for the modulation of radiofrequency carrier signals. As a demonstration, we use a graphene oscillator as the active element for frequency-modulated signal generation and achieve efficient audio signal transmission.

Surpassing fundamental limits of oscillators using nonlinear resonators

In its most basic form an oscillator consists of a resonator driven on resonance, through feedback, to create a periodic signal sustained by a static energy source. The generation of a stable frequency, the basic function of oscillators, is typically achieved by increasing the amplitude of motion of the resonator while remaining within its linear, harmonic regime. Contrary to this conventional paradigm, in this Letter we show that by operating the oscillator at special points in the resonator's anharmonic regime we can overcome fundamental limitations of oscillator performance due to thermodynamic noise as well as practical limitations due to noise from the sustaining circuit. We develop a comprehensive model that accounts for the major contributions to the phase noise of the nonlinear oscillator. Using a nanoelectromechanical system based oscillator, we experimentally verify the existence of a special region in the operational parameter space that enables suppressing the most significant contributions to the oscillator's phase noise, as predicted by our model.

Ultra low power consumption for self-oscillating nanoelectromechanical systems constructed by contacting two nanowires

We report here the observation of a new self-oscillation mechanism in nanoelectromechanical systems (NEMS). A highly resistive nanowire was positioned to form a point-contact at a chosen vibration node of a silicon carbide nanowire resonator. Spontaneous and robust mechanical oscillations arise when a sufficient DC voltage is applied between the two nanowires. An original model predicting the threshold voltage is used to estimate the piezoresistivity of the point-contact in agreement with the observations. The measured input power is in the pW-range which is the lowest reported value for such systems. The simplicity of the contacting procedure and the low power consumption open a new route for integrable and low-loss self-excited NEMS devices.

Nonlinear mode-coupling in nanomechanical systems

Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced nanomechanical devices; it has important implications for applications ranging from quantitative sensing to fundamental research. However, achieving accurate experimental characterization of nonlinearities in nanomechanical systems (NEMS) is problematic. Currently employed detection and actuation schemes themselves tend to be highly nonlinear, and this unrelated nonlinear response has been inadvertently convolved into many previous measurements. In this Letter we describe an experimental protocol and a highly linear transduction scheme, specifically designed for NEMS, that enables accurate, in situ characterization of device nonlinearities. By comparing predictions from Euler-Bernoulli theory for the intra- and intermodal nonlinearities of a doubly clamped beam, we assess the validity of our approach and find excellent agreement.

Optimal operating points of oscillators using nonlinear resonators

We demonstrate an analytical method for calculating the phase sensitivity of a class of oscillators whose phase does not affect the time evolution of the other dynamic variables. We show that such oscillators possess the possibility for complete phase noise elimination. We apply the method to a feedback oscillator which employs a high Q weakly nonlinear resonator and provide explicit parameter values for which the feedback phase noise is completely eliminated and others for which there is no amplitude-phase noise conversion. We then establish an operational mode of the oscillator which optimizes its performance by diminishing the feedback noise in both quadratures, thermal noise, and quality factor fluctuations. We also study the spectrum of the oscillator and provide specific results for the case of 1/f noise sources.

Subkelvin parametric feedback cooling of a laser-trapped nanoparticle

We optically trap a single nanoparticle in high vacuum and cool its three spatial degrees of freedom by means of active parametric feedback. Using a single laser beam for both trapping and cooling we demonstrate a temperature compression ratio of four orders of magnitude. The absence of a clamping mechanism provides robust decoupling from the heat bath and eliminates the requirement of cryogenic precooling. The small size and mass of the nanoparticle yield high resonance frequencies and high quality factors along with low recoil heating, which are essential conditions for ground state cooling and for low decoherence. The trapping and cooling scheme presented here opens new routes for testing quantum mechanics with mesoscopic objects and for ultrasensitive metrology and sensing.

Challenges for nanomechanical sensors in biological detection

Nanomechanical biosensing relies on changes in the movement and deformation of micro- and nanoscale objects when they interact with biomolecules and other biological targets. This field of research has provided ever-increasing records in the sensitivity of label-free detection but it has not yet been established as a practical alternative for biological detection. We analyze here the latest advancements in the field, along with the challenges remaining for nanomechanical biosensors to become a commonly used tool in biology and biochemistry laboratories.

Ultra-low power hydrogen sensing based on a palladium-coated nanomechanical beam resonator

Hydrogen sensing is essential to ensure safety in near-future zero-emission fuel cell powered vehicles. Here, we present a novel hydrogen sensor based on the resonant frequency change of a nanoelectromechanical clamped-clamped beam. The beam is coated with a Pd layer, which expands in the presence of H(2), therefore generating a stress build-up that causes the frequency of the device to drop. The devices are able to detect H(2) concentrations below 0.5% within 1 s of the onset of the exposure using only a few hundreds of pW of power, matching the industry requirements for H(2) safety sensors. In addition, we investigate the strongly detrimental effect that relative humidity (RH) has on the Pd responsivity to H(2), showing that the response is almost nullified at about 70% RH. As a remedy for this intrinsic limitation, we applied a mild heating current through the beam, generating a few μW of power, whereby the responsivity of the sensors is fully restored and the chemo-mechanical process is accelerated, significantly decreasing response times. The sensors are fabricated using standard processes, facilitating their eventual mass-production.

Passive phase noise cancellation scheme

We introduce a new method for reducing phase noise in oscillators, thereby improving their frequency precision. The noise reduction is realized by a passive device consisting of a pair of coupled nonlinear resonating elements that are driven parametrically by the output of a conventional oscillator at a frequency close to the sum of the linear mode frequencies. Above the threshold for parametric instability, the coupled resonators exhibit self-oscillations which arise as a response to the parametric driving, rather than by application of active feedback. We find operating points of the device for which this periodic signal is immune to frequency noise in the driving oscillator, providing a way to clean its phase noise. We present results for the effect of thermal noise to advance a broader understanding of the overall noise sensitivity and the fundamental operating limits.

Stress-induced variations in the stiffness of micro- and nanocantilever beams

The effect of surface stress on the stiffness of cantilever beams remains an outstanding problem in the physical sciences. While numerous experimental studies report significant stiffness change due to surface stress, theoretical predictions are unable to rigorously and quantitatively reconcile these observations. In this Letter, we present the first controlled measurements of stress-induced change in cantilever stiffness with commensurate theoretical quantification. Simultaneous measurements are also performed on equivalent clamped-clamped beams. All experimental results are quantitatively and accurately predicted using elasticity theory. We also present conclusive experimental evidence for invalidity of the long-standing and unphysical axial force model, which has been widely applied to interpret measurements using cantilever beams. Our findings will be of value in the development of micro- and nanoscale resonant mechanical sensors.