Nanomechanical Beam Resonator Research Articles

Nanomechanical Measurement of the Brownian Force Noise in a Viscous Liquid.

We study the frequency spectrum of the thermal force giving rise to Brownian motion of a nanomechanical beam resonator in a viscous liquid. In the first set of experiments, we measure the power spectral density (PSD) of the position fluctuations of the resonator around its fundamental mode at its center. Then, we measure the frequency-dependent linear response of the resonator, again at its center, by driving it with a harmonic force that couples well to the fundamental mode. These two measurements allow us to determine the PSD of the Brownian force noise acting on the structure in its fundamental mode. The PSD of the force noise from multiple resonators spanning a broad frequency range displays a "colored spectrum" and follows the dissipation of a blade oscillating in a viscous liquid-by virtue of the fluctuation-dissipation theorem of statistical mechanics.

Size-dependent nonlinear dynamic modeling and vibration analysis of piezo-laminated nanomechanical resonators using perturbation method

This article investigates size-dependent nonlinear dynamic modeling and vibration analysis of nanomechanical resonators equipped with piezoelectric layers employing perturbation methods. For nonlinear dynamic modeling, the nanomechanical beam resonator is modeled based on the nonlocal elasticity theory and von-karman type of nonlinear formulation. Then, the Hamilton’s principle is employed to derive the nonlinear vibration equation of the piezoelectric laminated beam resonator. The Galerkin separation approach is employed to develop the governing equation as a time-dependent nonlinear differential one which is solved by the multiple scales perturbation method. Obtaining an analytical formulation, a detailed study is conducted to investigate the size effects on nonlinear free vibration of the beam at several modes of vibration. The obtained results indicate that the nonlocal parameter causes a reduction effect on the nonlinear frequencies particularly for its higher values and higher modes of vibration. Also, it is observed that the nonlocal effect on the natural frequencies diminishes with an increase in the vibration amplitude.

Numerical Analysis of Multidomain Systems: Coupled Nonlinear PDEs and DAEs With Noise

We present a numerical modeling and simulation paradigm for multidomain, multiphysics systems with components modeled both in a lumped and distributed manner. The lumped components are modeled with a system of differential-algebraic equations (DAEs), whereas the possibly nonlinear distributed components that may belong to different physical domains are modeled using partial differential equations (PDEs) with associated boundary conditions. We address a comprehensive suite of problems for nonlinear coupled DAE–PDE systems including 1) transient simulation; 2) periodic steady-state (PSS) analysis formulated as a mixed boundary value problem that is solved with a hierarchical spectral collocation technique based on a joint Fourier–Chebyshev representation, for both forced and autonomous systems; 3) Floquet theory and analysis for coupled linear periodically time-varying DAE–PDE systems; 4) phase noise analysis for multidomain oscillators; and 5) efficient parameter sweeps for PSS and noise analyses based on first-order and pseudo-arclength continuation schemes. All of these techniques, implemented in a prototype simulator, are applied to a substantial case study: a multidomain feedback oscillator composed of distributed and lumped components in two physical domains, namely, a nano-mechanical beam resonator operating in the nonlinear regime, an electrical delay line, an electronic amplifier and a sensor-actuator for the transduction between the two physical domains.

Thermoelastic damping in micro- and nanomechanical beam resonators considering size effects

In this paper, we describe governing equations for modified coupled thermoelasticity in micro- and nanomechanical beam resonators, which can treat the effects of size by taking the relaxation time, the phonon mean-free path and the material length scale parameter into account. An analytical model of thermoelastic damping is derived using the complex-frequency approach. Numerical results of thermoelastic damping calculated using the proposed model are presented and compared to those calculated using the models proposed by Zener and Lifshitz and Roukes for a silicon thin beam. The results show that the nonlocal effect characterized by the length scale parameter is negligible even at sizes down to the phonon mean-free path. The size effects characterized by the relaxation time and the phonon mean-free path are significant for a micron-scale beam. Device miniaturization beyond the submicron scale will lead to increased energy dissipation due to thermoelastic damping considering size effects. The influence of size effects on thermoelastic damping can be suppressed by increasing aspect ratios. Finally, we present the range of geometry of a silicon beam resonator where the effects of size can be neglected by taking 10% as the permitted error bound.

Size modulated transition in the fluid–structure interaction losses in nano mechanical beam resonators

An understanding of the dominant dissipative mechanisms is crucial for the design of a high-Q doubly clamped nanobeam resonator to be operated in air. We focus on quantifying analytically the viscous losses—the squeeze film damping and drag force damping—that limit the net quality factor of a beam resonator, vibrating in its flexural fundamental mode with the surrounding fluid as air at atmospheric pressure. Specifically, drag force damping dominates at smaller beam widths and squeeze film losses dominate at larger beam widths, with no significant contribution from structural losses and acoustic radiation losses. The combined viscous losses agree well with the experimentally measured Q of the resonator over a large range of beam widths, within the limits of thin beam theory. We propose an empirical relation between the maximum quality factor and the ratio of maximum beam width to the squeeze film air gap thickness.

Air damping models for micro- and nano-mechanical beam resonators in molecular-flow regime

Researches on energy dissipation of micro- and nano-mechanical beam resonators into the surrounding air are classified into two models, air drag force damping of the resonant beam vibrating in free space and squeeze film air damping of the resonant beam vibrating nearby walls. Appropriate application of these two models are crucial to the design of high-Q micro- and nano-mechanical beam resonators. In this paper, we will present a detail analysis on appropriate application range of these two models for evaluating air damping in micro- and nano-mechanical beam resonators. The two models are applied for investigating damping in the cantilevered and bridged beam resonators. The obtained analysis results show the significance difference between the two models for estimating the quality factor of the micro- and nano-mechanical beam resonators. The criterion for the distinction between the two models is derived by using the ratio factor of damping coefficients of micro- and nano-mechanical beam resonators.

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators.

Highly sensitive conversion of motion into readable electrical signals is a crucial and challenging issue for nanomechanical resonators. Efficient transduction is particularly difficult to realize in devices of low dimensionality, such as beam resonators based on carbon nanotubes or silicon nanowires, where mechanical vibrations combine very high frequencies with miniscule amplitudes. Here we describe an enhanced piezoresistive transduction mechanism based on the asymmetry of the beam shape at rest. We show that this mechanism enables highly sensitive linear detection of the vibration of low-resistivity silicon beams without the need of exceptionally large piezoresistive coefficients. The general application of this effect is demonstrated by detecting multiple-order modes of silicon nanowire resonators made by either top-down or bottom-up fabrication methods. These results reveal a promising approach for practical applications of the simplest mechanical resonators, facilitating its manufacturability by very large-scale integration technologies.

Nonadiabatic Dynamics of Two Strongly Coupled Nanomechanical Resonator Modes

The Landau-Zener transition is a fundamental concept for dynamical quantum systems and has been studied in numerous fields of physics. Here, we present a classical mechanical model system exhibiting analogous behavior using two inversely tunable, strongly coupled modes of the same nanomechanical beam resonator. In the adiabatic limit, the anticrossing between the two modes is observed and the coupling strength extracted. Sweeping an initialized mode across the coupling region allows mapping of the progression from diabatic to adiabatic transitions as a function of the sweep rate.

Monolithic integration of a nanomechanical resonator to an optical microdisk cavity

We report a Silicon nano-opto-mechanical device in which a nanomechanical doubly-clamped beam resonator is integrated to an optical microdisk cavity. Small flexural oscillations of the beam cause intensity modulations in the circulating optical field in the nearby microdisk cavity. By monitoring the corresponding fluctuations in the cavity transmission via a fiber-taper, one can detect these oscillations with a displacement sensitivity approaching 10 fm·Hz-1/2 at an input power level of 50 μW. Both the in-plane and out-of-plane fundamental flexural resonances of the beam can be read out by this approach - the latter being detectable due to broken planar symmetry in the system. Access to multiple mechanical modes of the same resonator may be useful in some applications and may enable interesting fundamental studies.

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.

Effects of tensile stress on the resonant response of Al thin-film and Al-CNT nanolaminate nanomechanical beam resonators

We report on the dependence of resonant response to applied tensile stress in Al and Al-CNT (carbon nanotube) nanolaminate thin-film nanomechanical beam resonators. We optically measure the dynamic flexural response, while a chip-bending method is utilized to apply a static tension along its axis. For Al thin-film resonators, large and reversible increases in the resonant frequencies are observed along with near doubling of the resonator quality factor; while, for Al-CNT, largely unaffected with a slight decrease in resonant frequencies. Such behavior gives an insight on the role of CNT as a structural reinforcement in the Al thin-film based nanolaminate.

Damping and decoherence of a nanomechanical resonator due to a few two-level systems

We consider a quantum model of a nanomechanical flexing beam resonator interacting with a bath comprising a few damped tunneling two-level systems. In contrast with a resonator interacting bilinearly with an ohmic free oscillator bath (modeling clamping loss, for example), the mechanical resonator damping is amplitude dependent, while the decoherence of quantum superpositions of mechanical position states depends only weakly on their spatial separation.

Optomechanical coupling in photonic crystal supported nanomechanical waveguides

We report improved optomechanical properties of a nano-mechanical beam resonator embedded in an optical race-track resonator. Precise control of the mechanical resonator is achieved by clamping the beam between two low-loss photonic crystal waveguide couplers. The low insertion loss and the rigid mechanical support provided by the couplers yield both good mechanical and optical Q-factors for improved signal quality.

Nonlinear dynamics of nanomechanical beam resonators: improving the performance ofNEMS-based sensors

In order to compensate for the loss of performance when scaling resonant sensors down toNEMS, it proves extremely useful to study the behavior of resonators up to very highdisplacements and hence high nonlinearities. This work describes a comprehensivenonlinear multiphysics model based on the Euler–Bernoulli equation which includes bothmechanical and electrostatic nonlinearities valid up to displacements comparable to the gapin the case of an electrostatically actuated doubly clamped beam. Moreover, the modeltakes into account the fringing field effects, significant for thin resonators. The model hasbeen compared to both numerical integrations and electrical measurements of devicesfabricated on 200 mm SOI wafers; it shows very good agreement with both. Animportant contribution of this work is the provision for closed-form expressions ofthe critical amplitude and the pull-in domain initiation amplitude including allnonlinearities. This model allows designers to cancel out nonlinearities by tuning somedesign parameters and thus gives the possibility to drive the resonator beyondits critical amplitude. Consequently, the sensor performance can be enhancedto the maximum below the pull-in instability, while keeping a linear behavior.

Macroscopic Tuning of Nanomechanics: Substrate Bending for Reversible Control of Frequency and Quality Factor of Nanostring Resonators

We have employed a chip-bending method to exert continuous and reversible control over the tensile stress in doubly clamped nanomechanical beam resonators. Tensile stress is shown to increase the quality factor of both silicon nitride and single-crystal silicon resonators, implying that added tension can be used as a general, material-independent route to increased quality factor. With this direct stretching technique, we demonstrate beam resonators with unprecedented tunability of both frequency and quality factor. Devices can be tuned back and forth between a high and low stress state, with frequency tunability as large as several hundred percent demonstrated. Over this wide range of frequency, quality factor is also tuned by as much as several hundred percent, providing insights into the loss mechanisms in these materials and this class of nanoresonator. Devices with frequencies in the 1-100 MHz range are studied, with quality factor as high as 390,000 achieved at room temperature, for a silicon nitride device with cross-sectional dimensions below 1 microm, operating in a high stress state. This direct stretching technique may prove useful for the identification of loss mechanisms that contribute to the energy balance in nanomechanical resonators, allowing for the development of new designs that would display higher quality factors. Such devices would have the ability to resolve smaller addendum masses and thus allow more sensitive detection and offer the potential for providing access to previously inaccessible dissipation regimes at low temperatures. This technique provides the ability to dramatically tune both frequency and quality factor, enabling future mechanical resonators to be used as variable frequency references as well as variable band-pass filters in signal-processing applications.

Detection of coherent acoustic oscillations in a quantum electromechanical resonator

Coherent control of occupation numbers in quantum mechanical multilevel systems is widely studied driven by its application in lasers and its prospects for quantum computational elements. Here the authors present a nanoelectromechanical resonator equivalent to the coherent control of a quantum mechanical two level system. The distinct eigenmodes of a nanomechanical beam resonator represent the two levels whose amplitude mode occupation numbers are controlled by a frequency matched acoustic excitation, mediated by a pulsed surface acoustic wave. They show that similar to quantum mechanical systems it is possible to transfer occupation numbers from one mode to another by matched acoustic pulses.

Tunable response nano-mechanical beam resonator

A theoretical proposal is made to dynamically tune the frequency response function of a nano-mechanical beam resonator. The resonator is excited by a magneto-motive drive mechanism. The resonator features multiple drive electrodes with the ability to control the phase and magnitude of the excitation force at each electrode. This nano-resonator behaviour is desirable in the application of MEMS filters. It provides tunable null responses and can increase the relative stability of higher order resonant modes enabling higher operating frequencies. The proposal is supported with the technique of modal analysis and finite element results.

Nanomechanical hydrogen sensing

A nanomechanical beam resonator is used as a sensitive, specific hydrogen sensor. The beam is fabricated from AuPd alloy and tested by magnetomotive transduction at room temperature. The fundamental resonance frequency decreases significantly and reversibly at hydrogen pressures above 10−5Torr, whereas the frequency shifts observed for other gases are significantly smaller. The large frequency shift is likely due to the formation of interstitial hydrogen in the metal alloy lattice, which relieves the built-in tensile stress in the resonator beam. The uptake of hydrogen as measured by frequency shift is consistent with previous studies.

Standing acoustic waves imaged by a nanomechanical resonator

A nanomechanical beam resonator is used to image the standing wave pattern formed by two counter propagating surface acoustic waves of the Rayleigh type. As the intrinsic properties of the resonator are affected by a fast off-resonant mechanical excitation, it is possible to scan the wave pattern when shifting the positions of the wave's knots through the resonator's position. Due to the small size of the beam the wave is hardly disturbed by the nanoresonator, and the observed shifts in the resonator's eigenfrequency follow the expected form of a standing wave. At high acoustic amplitudes deviation from the linear behavior is observed while the dependence of the effect on the beam's magneto-impedance probe power elucidates the physical origin of the resonator tuning.

Tunable coupled nanomechanical resonatorsfor single-electron transport

Nano-electromechanical systems (NEMS) are ideal for sensorapplications and ultra-sensitive force detection, since theirmechanical degree of freedom at the nanometre scale can becombined with semiconductor nano-electronics. We present asystem of coupled nanomechanical beam resonators in siliconwhich is mechanically fully Q-tunable ~700-6000. This kind ofresonator can also be employed as a mechanical charge shuttlevia an insulated metallic island at the tip of an oscillatingcantilever. Application of our NEMS as an electromechanicalsingle-electron transistor (emSET) is introduced andexperimental results are discussed. Three animation clipsdemonstrate the manufacturing process of the NEMS, the Q-tuningexperiment and the concept of the emSET.