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Even a relatively weak drive force is enough to push a typical nanomechanical resonator into the nonlinear regime. Consequently, nonlinearities are widespread in nanomechanics and determine the critical characteristics of nanoelectromechanical systems (NEMS) resonators. A thorough understanding of the nonlinear dynamics of higher eigenmodes of NEMS resonators would be beneficial for progress, given their use in applications and fundamental studies. Here, we characterize the nonlinearity and the linear dynamic range (LDR) of each eigenmode of two nanomechanical beam resonators with different intrinsic tension values up to eigenmode $n=11$. We find that the modal Duffing constant increases as $n^4$, while the critical amplitude for the onset of nonlinearity decreases as $1/n$. The LDR, determined from the ratio of the critical amplitude to the thermal noise amplitude, increases weakly with $n$. Our findings are consistent with our theory treating the beam as a string, with the nonlinearity emerging from stretching at high amplitudes. These scaling laws, observed in experiments and validated theoretically, can be leveraged for pushing the limits of NEMS-based sensing even further., Comment: 7 pages, 3 figures

Brownian motion imposes a hard limit on the overall precision of a nanomechanical measurement. Here, we present a combined experimental and theoretical study of the Brownian dynamics of a quintessential nanomechanical system, a doubly-clamped nanomechanical beam resonator, in a viscous fluid. Our theoretical approach is based on the fluctuation-dissipation theorem of statistical mechanics: We determine the dissipation from fluid dynamics; we incorporate this dissipation into the proper elastic equation to obtain the equation of motion; the fluctuation-dissipation theorem then directly provides an analytical expression for the position-dependent power spectral density (PSD) of the displacement fluctuations of the beam. We compare our theory to experiments on nanomechanical beams immersed in air and water, and obtain excellent agreement. Within our experimental parameter range, the Brownian force noise driving the nanomechanical beam has a colored PSD due to the ``memory" of the fluid; the force noise remains mode-independent and uncorrelated in space. These conclusions are not only important for nanomechanical sensing but also provide insight into the fluctuations of elastic systems at any length scale.

The oscillatory dynamics of nanoelectromechanical systems (NEMS) is at the heart of many emerging applications in nanotechnology. For common NEMS, such as beams and strings, the oscillatory dynamics is formulated using a dissipationless wave equation derived from elasticity. Under a harmonic ansatz, the wave equation gives an undamped free vibration equation; solving this equation with the proper boundary conditions provides the undamped eigenfunctions with the familiar standing wave patterns. Any harmonically driven solution is expressible in terms of these undamped eigenfunctions. Here, we show that this formalism becomes inconvenient as dissipation increases. To this end, we experimentally map out the position- and frequency-dependent oscillatory motion of a NEMS string resonator driven linearly by a non-symmetric force on one end at different dissipation limits. At low dissipation (high Q factor), we observe sharp resonances with standing wave patterns that closely match the eigenfunctions of an undamped string. With a slight increase in dissipation, the standing wave patterns become lost and waves begin to propagate along the nanostructure. At large dissipation (low Q factor), these propagating waves become strongly attenuated and display little, if any, resemblance to the undamped string eigenfunctions. A more efficient and intuitive description of the oscillatory dynamics of a NEMS resonator can be obtained by superposition of waves propagating along the nanostructure., Comment: 8 pages, 5 figures

Piezoresistive strain gauges allow for electronic readout of mechanical deformations with high fidelity. As piezoresistive strain gauges are aggressively being scaled down for applications in nanotechnology, it has become critical to investigate their physical attributes at different limits. Here, we describe an experimental approach for studying the piezoresistive gauge factor of a gold thin-film nanoresistor as a function of frequency. The nanoresistor is fabricated lithographically near the anchor of a nanomechanical doubly-clamped beam resonator. As the resonator is driven to resonance in one of its normal modes, the nanoresistor is exposed to frequency-dependent strains of {$\varepsilon \lesssim 10^{-5}$} in the $4-36~\rm MHz$ range. We calibrate the strain using optical interferometry and measure the resistance changes using a radio-frequency mix-down technique. The piezoresistive gauge factor $\gamma$ of our lithographic gold nanoresistors is $\gamma \approx 3.6$ at 4 MHz, in agreement with comparable macroscopic thin metal film resistors in previous works. However, our $\gamma$ values increase monotonically with frequency and reach $\gamma \approx 15$ at 36 MHz. We discuss possible physics that may give rise to this unexpected frequency dependence., Comment: 26 pages, 30 figures

Single-crystal diamond nanomechanical resonators are being developed for countless applications. A number of these applications require that the resonator be operated in a fluid, i.e., a gas or a liquid. Here, we investigate the fluid dynamics of single-crystal diamond nanomechanical resonators in the form of nanocantilevers. First, we measure the pressure-dependent dissipation of diamond nanocantilevers with different linear dimensions and frequencies in three gases, He, N$_2$, and Ar. We observe that a subtle interplay between the length scale and the frequency governs the scaling of the fluidic dissipation. Second, we obtain a comparison of the surface accommodation of different gases on the diamond surface by analyzing the dissipation in the molecular flow regime. Finally, we measure the thermal fluctuations of the nanocantilevers in water, and compare the observed dissipation and frequency shifts with theoretical predictions. These findings set the stage for developing diamond nanomechanical resonators operable in fluids., Comment: Nano Lett., Article ASAP (2015)

Here, we study noisy transitional flows in imperfect millimeter-scale channels. For probing the flows, we use microcantilever sensors embedded in the channel walls. We perform experiments in two nominally identical channels. The different set of imperfections in the two channels result in two random flows in which high-order moments of near-wall fluctuations differ by orders of magnitude. Surprisingly however, the lowest order statistics in both cases appear qualitatively similar and can be described by a proposed noisy Landau equation for a slow mode. The noise, regardless of its origin, regularizes the Landau singularity of the relaxation time and makes transitions driven by different noise sources appear similar.

Direct and noninvasive measurement of the pressure distribution in test sections of a micro-channel is a challenging, if not an impossible, task. Here, we present an analytical method for extracting the pressure distribution in a deformable micro-channel under flow. Our method is based on a measurement of the channel deflection profile as a function of applied \emph{hydrostatic} pressure; this initial measurement generates "constitutive curves" for the deformable channel. The deflection profile under flow is then matched to the constitutive curves, providing the \emph{hydrodynamic} pressure distribution. The method is validated by measurements on planar micro-fluidic channels against analytic and numerical models. The accuracy here is independent of the nature of the wall deformations and is not degraded even in the limit of large deflections, $\zeta_{\rm{max}}/2h_{0}= {\cal{O}}(1)$, with $\zeta_{\rm{max}}$ and $2h_0$ being the maximum deflection and the unperturbed height of the channel, respectively. We discuss possible applications of the method in characterizing micro-flows, including those in biological systems.

Dynamic interactions between two oscillating micromechanical cantilevers are studied. In the experiment, the tip of a high-frequency cantilever is positioned near the surface of a second low-frequency cantilever. Due to the highly nonlinear interaction forces between the two surfaces, thermal oscillations of the low-frequency cantilever modulate the driven oscillations of the high-frequency cantilever. The dissipations and the frequencies of the two cantilevers are shown to be coupled, and a simple model for the interactions is presented. The interactions studied here may be useful for the design of future micro and nanoelectromechanical systems for mechanical signal processing; they may also help realize coupled mechanical modes for experiments in non-linear dynamics.

We present an experimental study of a confined nanoflow, which is generated by a sphere oscillating in the proximity of a flat solid wall in a simple fluid. Varying the oscillation frequency, the confining length scale and the fluid mean free path over a broad range provides a detailed map of the flow. We use this experimental map to construct a scaling function, which describes the nanoflow in the entire parameter space, including both the hydrodynamic and the kinetic regimes. Our scaling function unifies previous theories based on the slip boundary condition and the effective viscosity.

We show that oscillating flow of a simple fluid in both the Newtonian and the non-Newtonian regime can be described by a universal function of a single dimensionless scaling parameter $\omega\tau$, where $\omega$ is the oscillation (angular) frequency and $\tau$ is the fluid relaxation-time; geometry and linear dimension bear no effect on the flow. Experimental energy dissipation data of mechanical resonators in a rarefied gas follow this universality closely in a broad linear dimension ($10^{-6}$ m$< L < 10^{-2}$ m) and frequency ($10^5$ Hz $< \omega/2\pi < 10^8$ Hz) range. Our results suggest a deep connection between flows of simple and complex fluids., Comment: To be published in Physical Review Letters

Here we apply nanomechanical resonators to the study of oscillatory fluid dynamics. A high-resonance-frequency nanomechanical resonator generates a rapidly oscillating flow in a surrounding gaseous environment; the nature of the flow is studied through the flow-resonator interaction. Over the broad frequency and pressure range explored, we observe signs of a transition from Newtonian to non-Newtonian flow at $\omega\tau\approx 1$, where $\tau$ is a properly defined fluid relaxation time. The obtained experimental data appears to be in close quantitative agreement with a theory that predicts purely elastic fluid response as $\omega\tau\to \infty$.

In a recent letter, Gaidarzhy et al. claim to have observed evidence for quantized displacements of a nanomechanical oscillator. We contend that the evidence, analysis, claims, and conclusions presented are contrary to expectations from fundamentals of quantum mechanics and elasticity theory, and that the method used by the authors is unsuitable in principle to observe the quantized energy states of a nanomechanical structure.

We describe the application of nanoelectromechanical systems (NEMS) to ultrasensitive mass detection. In these experiments, a modulated flux of atoms was adsorbed upon the surface of a 32.8 MHz NEMS resonator within an ultrahigh vacuum environment. The mass-induced resonance frequency shifts by these adsorbates were then measured to ascertain a mass sensitivity of 2.53x10^-18 g. In these initial measurements, this sensitivity is limited by the noise in the NEMS displacement transducer; the ultimate, limits of the technique are set by fundamental phase noise processes. Our results and analysis indicate that mass sensing of individual molecules will be realizable with optimized NEMS devices., Comment: Submitted to Appl. Phys. Lett. on December 22, 2003

Nanomechanical resonators can now be realized that achieve fundamental resonance frequencies exceeding 1 GHz, with quality factors (Q) in the range 1,000 - 100,000. The minuscule active masses of these devices, in conjunction with their high Qs, translate into unprecedented inertial mass sensitivities. This makes them natural candidates for a variety of mass sensing applications. Here we evaluate the ultimate mass sensitivity limits for nanomechanical resonators operating in vacuo, which are imposed by a number of fundamental physical noise processes. Our analyses indicate that nanomechanical resonators offer immense potential for mass sensing - ultimately with resolution at the level of individual molecules., Comment: preprint, subm. to J. Appl. Phys. - 32 pages, 6 figures, 2 tables