Nanoelectromechanical systems were fabricated from single- and multilayer graphene sheets by mechanically exfoliating thin sheets from graphite over trenches in silicon oxide. Vibrations with fundamental resonant frequencies in the megahertz range are actuated either optically or electrically and detected optically by interferometry. We demonstrate room-temperature charge sensitivities down to 8 × 10 –4 electrons per root hertz. The thinnest resonator consists of a single suspended layer of atoms and represents the ultimate limit of two-dimensional nanoelectromechanical systems.

https://science.sciencemag.org/content/sci/315/5811/490.full.pdf

Electromechanical Resonators from Graphene Sheets

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Nanomechanical resonators enable the measurement of mass with extraordinary sensitivity. Previously, samples as light as 7 zeptograms (1 zg = 10(-21) g) have been weighed in vacuum, and proton-level resolution seems to be within reach. Resolving small mass changes requires the resonator to be light and to ring at a very pure tone-that is, with a high quality factor. In solution, viscosity severely degrades both of these characteristics, thus preventing many applications in nanotechnology and the life sciences where fluid is required. Although the resonant structure can be designed to minimize viscous loss, resolution is still substantially degraded when compared to measurements made in air or vacuum. An entirely different approach eliminates viscous damping by placing the solution inside a hollow resonator that is surrounded by vacuum. Here we demonstrate that suspended microchannel resonators can weigh single nanoparticles, single bacterial cells and sub-monolayers of adsorbed proteins in water with sub-femtogram resolution (1 Hz bandwidth). Central to these results is our observation that viscous loss due to the fluid is negligible compared to the intrinsic damping of our silicon crystal resonator. The combination of the low resonator mass (100 ng) and high quality factor (15,000) enables an improvement in mass resolution of six orders of magnitude over a high-end commercial quartz crystal microbalance. This gives access to intriguing applications, such as mass-based flow cytometry, the direct detection of pathogens, or the non-optical sizing and mass density measurement of colloidal particles.

/pdf/weighing-of-biomolecules-single-cells-and-single-35bbp7ev6j.pdf

Weighing of biomolecules, single cells and single nanoparticles in fluid

We show that the Raman scattering technique can give complete structural information for one-dimensional systems, such as carbon nanotubes. Resonant confocal micro-Raman spectroscopy of an (n,m) individual single-wall nanotube makes it possible to assign its chirality uniquely by measuring one radial breathing mode frequency omega(RBM) and using the theory of resonant transitions. A unique chirality assignment can be made for both metallic and semiconducting nanotubes of diameter d(t), using the parameters gamma(0) = 2.9 eV and omega(RBM) = 248/d(t). For example, the strong RBM intensity observed at 156 cm(-1) for 785 nm laser excitation is assigned to the (13,10) metallic chiral nanotube on a Si/SiO2 surface.

Structural (n,m) determination of isolated single wall carbon nanotubes by resonant Raman scattering

Using an atomic force microscope, we measured effective spring constants of stacks of graphene sheets (less than 5) suspended over photolithographically defined trenches in silicon dioxide. Measurements were made on layered graphene sheets of thicknesses between 2 and 8nm, with measured spring constants scaling as expected with the dimensions of the suspended section, ranging from 1to5N∕m. When our data are fitted to a model for doubly clamped beams under tension, we extract a Young’s modulus of 0.5TPa, compared to 1TPa for bulk graphite along the basal plane, and tensions on the order of 10−7N.

/pdf/mechanical-properties-of-suspended-graphene-sheets-4mrc05pwn9.pdf

Mechanical properties of suspended graphene sheets

We report on the fabrication of nanometer-scale mass sensors with subattogram sensitivity. Surface micromachined polycrystalline silicon and silicon nitride nanomechanical oscillators were used to detect the presence of well-defined mass loading. Controlled deposition of thiolate self-assembled monolayers on lithographically defined gold dots were used for calibrated mass loading. We used a dinitrophenyl poly(ethylene glycol) undecanthiol-based molecule (DNP-PEG4-C11thiol) as a model ligand for this study. Due to the fact that the gold mass is attached at the distance l0 from the end x=l of the cantilever beam, an additional moment evolves in the boundary condition of the oscillator, which was taken into consideration through the rotational inertia of the attached mass. We showed that the corresponding correction of the frequency is on the order of γ(l0/l), where γ is the attached mass normalized to the mass of the beam. The rotational inertia correction to the frequency is on the order of γ(l0/l)2. The a...

Attogram detection using nanoelectromechanical oscillators

The results of theoretical and experimental investigation of an initially curved clamped–clamped microbeam actuated by a distributed electrostatic force are presented. Reduced-order Galerkin and consistently constructed lumped models of the shallow Euler–Bernoulli arch were built and verified by numerical analysis, and the influence of various parameters on the stability was investigated. Due to the unique combination of generic mechanical and electrostatic nonlinearities, the voltage–deflection characteristic of the device may have two maxima implying the existence of sequential snap-through buckling and pull-in instability and of bistability of the beam. The first critical voltage can be higher or lower than the second one, while the stable deflections are significantly larger than in a straight beam. The minimal initial elevation required for the appearance of the snap-through in the electrostatically actuated beam is smaller than in the case of uniform deflection-independent loading; a closed-form approximation of this elevation was evaluated. The devices were fabricated from silicon on insulator (SOI) wafer using deep reactive ion etching and in-plane responses were characterized by means of optical and scanning electron microscopy. Model results obtained for the actual dimensions of the device were in good agreement with the experimental data.

https://iopscience.iop.org/article/10.1088/0960-1317/18/5/055026/pdf

The pull-in behavior of electrostatically actuated bistable microstructures

Resonant nanoelectromechanical systems (NEMS) are being actively investigated as sensitive mass detectors for applications such as chemical and biological sensing. We demonstrate that highly uniform arrays of nanomechanical resonators can be used to detect the binding of individual DNA molecules through resonant frequency shifts resulting from the added mass of bound analyte. Localized binding sites created with gold nanodots create a calibrated response with sufficient sensitivity and accuracy to count small numbers of bound molecules. The amount of nonspecifically bound material from solution, a fundamental issue in any ultra-sensitive assay, was measured to be less than the mass of one DNA molecule, allowing us to detect a single 1587 bp DNA molecule.

Enumeration of DNA molecules bound to a nanomechanical oscillator.

A detailed study of the transient nonlinear dynamics of an electrically actuated micron scale beam is presented. A model developed using the Galerkin procedure with normal modes as a basis accounts for the distributed nonlinear electrostatic forces, nonlinear squeezed film damping, and rotational inertia of a mass carried by the beam. Special attention is paid to the dynamics of the beam near instability points. Results generated by the model and confirmed experimentally show that nonlinear damping leads to shrinkage of the spatial region where stable motion is realizable. The voltage that causes dynamic instability, in turn, approaches the static pull-in value.

Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.

Slava Krylov

Papers

Attogram detection using nanoelectromechanical oscillators

The pull-in behavior of electrostatically actuated bistable microstructures

Enumeration of DNA molecules bound to a nanomechanical oscillator.

Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures