Continuum elastic modeling of graphene resonators

Ultrahigh-quality graphene resonators by liquid-based strain-engineering

Two-dimensional (2D) material-based nanoelectromechanical (NEM) resonators are expected to be enabling components in hybrid qubits that couple mechanical and electromagnetic degrees of freedom. However, challenges in their sensitivity and coherence time have to be overcome to realize such mechanohybrid quantum systems. We here demonstrate the potential of strain engineering to realize 2D material-based resonators with unprecedented performance. A liquid-based tension process was shown to enhance the resonance frequency and quality factor of graphene resonators six-fold. Spectroscopic and microscopic characterization reveals a surface-energy enhanced wall interaction as the origin of this effect. The response of our tensioned resonators is not limited by external loss factors and exhibits near-ideal internal losses, yielding superior resonance frequencies and quality factors to all previously reported 2D material devices. Our approach represents a powerful method of enhancing 2D NEM resonators for future quantum systems.

Research Progress of Graphene Nano-Electromechanical Resonant Sensors—A Review

Graphene nano-electromechanical resonant sensors have wide application in areas such as seawater desalination, new energy, biotechnology, and aerospace due to their small size, light weight, and high sensitivity and resolution. This review first introduces the physical and chemical properties of graphene and the research progress of four preparation processes of graphene. Next, the principle prototype of graphene resonators is analyzed, and three main methods for analyzing the vibration characteristics of a graphene resonant sheet are described: molecular structural mechanics, non-local elastic theory and molecular dynamics. Then, this paper reviews research on graphene resonator preparation, discussing the working mechanism and research status of the development of graphene resonant mass sensors, pressure sensors and inertial sensors. Finally, the difficulties in developing graphene nano-electromechanical resonant sensors are outlined and the future trend of these sensors is described.

Discrete breathers and energy localization in a nonlinear honeycomb lattice

Discrete breathers (DBs) in nonlinear lattices have attracted much attention in the past decades. In this work, we focus on the formation of DBs and their induced energy localization in the nonlinear honeycomb lattice derived from graphene. The key step is to construct a reduced system (RS) with only a few degrees of freedom, which contains one central site and its three nearest neighbors. The fixed points and periodic orbits of the RS can be obtained from the Poincaré section of the dynamics. Our main finding is that the long-running DB solution of the full honeycomb system corresponds to the periodic orbit given by one of the fixed points of RS, where the central site and its nearest neighbors vibrate inversely. When the initial condition deviates from this fixed point, the local vibration is attracted to it after a short transient process. When the initial condition is assigned to other fixed points of the RS, the initial excitation energy flows to the other part of the full system quickly, resulting in a delocalized wave propagation. Another main finding is that the long-lived DB solutions emerge only when the initial excitation energy is larger than a threshold value, above which the frequency of the DB exceeds the phonon band edge. The excitation energy generally dissipates from the local region due to the interactions between the DB and phonons near the Γ point in the dispersion relation. These results provide a holistic physical picture for the DB solutions in two-dimensional nonlinear lattices with complex potentials, which will be crucial to the understanding of energy localization in the realistic two-dimensional materials.

An Ultrahigh-Sensitivity Graphene Resonant Gyroscope

In this study, a graphene beam was selected as a sensing element and used to form a graphene resonant gyroscope structure with direct frequency output and ultrahigh sensitivity. The structure of the graphene resonator gyroscope was simulated using the ANSYS finite element software, and the influence of the length, width, and thickness of the graphene resonant beam on the angular velocity sensitivity was studied. The simulation results show that the resonant frequency of the graphene resonant beam decreased with increasing the beam length and thickness, while the width had a negligible effect. The fundamental frequency of the designed graphene resonator gyroscope was more than 20 MHz, and the sensitivity of the angular velocity was able to reach 22,990 Hz/°/h. This work is of great significance for applications in environments that require high sensitivity to extremely weak angular velocity variation.

Modeling and Observation of Nonlinear Damping in Dissipation-Diluted Nanomechanical Resonators

Dissipation dilution enables extremely low linear loss in stressed, high aspect ratio nanomechanical resonators, such as strings or membranes. Here, we report on the observation and theoretical modeling of nonlinear dissipation in such structures. We introduce an analytical model based on von Kármán theory, which can be numerically evaluated using finite-element models for arbitrary geometries. We use this approach to predict nonlinear loss and (Duffing) frequency shift in ultracoherent phononic membrane resonators. A set of systematic measurements with silicon nitride membranes shows good agreement with the model for low-order soft-clamped modes. Our analysis also reveals quantitative connections between these nonlinearities and dissipation dilution. This is of interest for future device design and can provide important insight when diagnosing the performance of dissipation dilution in an experimental setting.

The Effect of Edge Mode on Mass Sensing for Strained Graphene Resonators

Edge mode could disturb the ultra-subtle mass detection for graphene resonators. Herein, classical molecular dynamics simulations are performed to investigate the effect of edge mode on mass sensing for a doubly clamped strained graphene resonator. Compared with the fundamental mode, the localized vibration of edge mode shows a lower frequency with a constant frequency gap of 32.6 GHz, despite the mutable inner stress ranging from 10 to 50 GPa. Furthermore, the resonant frequency of edge mode is found to be insensitive to centrally located adsorbed mass, while the frequency of the fundamental mode decreases linearly with increasing adsorbates. Thus, a mass determination method using the difference of these two modes is proposed to reduce interferences for robust mass measurement. Moreover, molecular dynamics simulations demonstrate that a stronger prestress or a higher width–length ratio of about 0.8 could increase the low-quality factor induced by edge mode, thus improving the performance in mass sensing for graphene resonators.

Strain engineering of 2D semiconductors and graphene: from strain fields to band-structure tuning and photonic applications

Two-dimensional (2D) transition metal dichalcogenides (TMDCs) and graphene compose a new family of crystalline materials with atomic thicknesses and exotic mechanical, electronic, and optical properties. Due to their inherent exceptional mechanical flexibility and strength, these 2D materials provide an ideal platform for strain engineering, enabling versatile modulation and significant enhancement of their optical properties. For instance, recent theoretical and experimental investigations have demonstrated flexible control over their electronic states via application of external strains, such as uniaxial strain and biaxial strain. Meanwhile, many nondestructive optical measurement methods, typically including absorption, reflectance, photoluminescence, and Raman spectroscopies, can be readily exploited to quantitatively determine strain-engineered optical properties. This review begins with an introduction to the macroscopic theory of crystal elasticity and microscopic effective low-energy Hamiltonians coupled with strain fields, and then summarizes recent advances in strain-induced optical responses of 2D TMDCs and graphene, followed by the strain engineering techniques. It concludes with exciting applications associated with strained 2D materials, discussions on existing open questions, and an outlook on this intriguing emerging field.

Dynamically-enhanced strain in atomically thin resonators

Graphene and related two-dimensional (2D) materials associate remarkable mechanical, electronic, optical and phononic properties. As such, 2D materials are promising for hybrid systems that couple their elementary excitations (excitons, phonons) to their macroscopic mechanical modes. These built-in systems may yield enhanced strain-mediated coupling compared to bulkier architectures, e.g., comprising a single quantum emitter coupled to a nano-mechanical resonator. Here, using micro-Raman spectroscopy on pristine monolayer graphene drums, we demonstrate that the macroscopic flexural vibrations of graphene induce dynamical optical phonon softening. This softening is an unambiguous fingerprint of dynamically-induced tensile strain that reaches values up to ≈4 × 10⁻⁴ under strong non-linear driving. Such non-linearly enhanced strain exceeds the values predicted for harmonic vibrations with the same root mean square (RMS) amplitude by more than one order of magnitude. Our work holds promise for dynamical strain engineering and dynamical strain-mediated control of light-matter interactions in 2D materials and related heterostructures.

Here, the authors use Raman spectroscopy on circular graphene drums to demonstrate dynamical softening of optical phonons induced by the macroscopic flexural motion of graphene, and find evidence that the strain in graphene is enhanced under non-linear driving.

On-chip Heaters for Tension Tuning of Graphene Nanodrums

For the study and application of graphene membranes, it is essential to have means to control their resonance frequency and temperature. Here, we present an on-chip heater platform for local tuning of in-plane tension in graphene mechanical resonators. By Joule heating of a metallic suspension ring we show thermomechanical resonance frequency tuning in a few-layer (FL) graphene nanodrum, which is accompanied by an increase in its quality factor, which we attribute to the increase of the in-plane tension. The in situ control of temperature, in-plane tension, resonance frequency, and quality factor of suspended two-dimensional (2D) nanodrums makes this device a unique platform for investigating the origin of dissipation in these ultrathin structures and can be of fundamental importance for studying the thermal properties of 2D materials. Moreover, by simultaneously controlling the heater and the backgate voltage, we can independently control the resonance frequency and quality factor, which is of great importance for applications in sensors and resonant mechanical filters.

Metastable states and energy flow pathway in square graphene resonators

Nonlinear interaction between flexural modes is critical to heat conductivity and mechanical vibration of two-dimensional materials such as graphene. Much effort has been devoted to understand the underlying mechanism. In this paper, we examine solely the out-of-plane flexural modes and identify their energy flow pathway during thermalization process. The key is the development of a universal scheme that numerically characterizes the strength of nonlinear interactions between normal modes. In particular, for our square graphene system, the modes are grouped into four classes by their distinct symmetries. The couplings are significantly larger within a class than between classes. As a result, the equations for the normal modes in the same class as the initially excited one can be approximated by driven harmonic oscillators, therefore, they get energy almost instantaneously. Because of the hierarchical organization of the mode coupling, the energy distribution among the modes will arrive at a stable profile, where most of the energy is localized on a few modes, leading to the formation of "natural package" and metastable states. The dynamics for modes in other symmetry classes follows a Mathieu type of equation, thus, interclass energy flow, when the initial excitation energy is small, starts typically when there is a mode that lies in the unstable region in the parameter space of Mathieu equation. Due to strong coupling of the modes inside the class, the whole class will get energy and be lifted up by the unstable mode. This characterizes the energy flow pathway of the system. These results bring fundamental understandings to the Fermi-Pasta-Ulam problem in two-dimensional systems with complex potentials, and reveal clearly the physical picture of dynamical interactions between the flexural modes, which will be crucial to the understanding of their abnormal contribution to heat conduction and nonlinear mechanical vibrations.

Accurate and Precise Determination of Mechanical Properties of Silicon Nitride Beam Nanoelectromechanical Devices

Accurate and precise determination of mechanical properties of nanoscale materials is mandatory since device performances of nanoelectromechanical systems (NEMS) are closely related to the flexural properties of the materials. In this study, the intrinsic mechanical properties of highly stressed silicon nitride (SiN) beams of varying lengths are investigated using two different techniques: Dynamic flexural measurement using optical interferometry and quasi-static flexural measurement using atomic force microscopy. The resonance frequencies of the doubly clamped, highly stressed beams are found to be inversely proportional to their length, which is not usually observed from a beam but is expected from a string-like structure. The mass density of the SiN beams can be precisely determined from the dynamic flexural measurements by using the values for internal stress and Young's modulus determined from the quasi-static measurements. As a result, the mass resolution of the SiN beam resonators was predicted to be a few attograms, which was found to be in excellent agreement with the experimental results. This work suggests that accurate and precise determination of mechanical properties can be achieved through combined flexural measurement techniques, which is a crucial key for designing practical NEMS applications such as biomolecular sensors and gas detectors.

Grain boundaries guided vibration wave propagation in polycrystalline graphene

Propagation of mechanical stransverse wave in polycrystalline graphene sheet.

Nanoscale Mechanics of Graphene and Graphene Oxide in Composites: A Scientific and Technological Perspective

Graphene shows considerable promise in structural composite applications thanks to its unique combination of high tensile strength, Young's modulus and structural flexibility which arise due to its maximal chemical bond strength and minimal atomic thickness. However, the ultimate performance of graphene composites will depend, in addition to the properties of the matrix and interface, on the morphology of the graphene used, including the size and shape of the sheets and the number of chemical defects present. For example, whilst oxidized sp(3) carbon atoms and vacancies in a graphene sheet can degrade its mechanical strength, they can also increase its interaction with other materials such as the polymer matrix of a composite, thus maximizing stress transfer and leading to more efficient mechanical reinforcement. Herein, we present an overview of some recently published work on graphene mechanical properties and discuss a list of challenges that need to be overcome (notwithstanding the strong hype existing on this material) for the development of graphene-based materials into a successful technology.

Finite-size effect on the dynamic and sensing performances of graphene resonators: the role of edge stress

We have studied the finite-size effect on the dynamic behavior of graphene resonators and their applications in atomic mass detection using a continuum elastic model such as modified plate theory. In particular, we developed a model based on von Karman plate theory with including the edge stress, which arises from the imbalance between the coordination numbers of bulk atoms and edge atoms of graphene. It is shown that as the size of a graphene resonator decreases, the edge stress depending on the edge structure of a graphene resonator plays a critical role on both its dynamic and sensing performances. We found that the resonance behavior of graphene can be tuned not only through edge stress but also through nonlinear vibration, and that the detection sensitivity of a graphene resonator can be controlled by using the edge stress. Our study sheds light on the important role of the finite-size effect in the effective design of graphene resonators for their mass sensing applications.

Localized surface plasmons in vibrating graphene nanodisks

Localized surface plasmons are confined collective oscillations of electrons in metallic nanoparticles. When driven by light, the optical response is dictated by geometrical parameters and the dielectric environment and plasmons are therefore extremely important for sensing applications. Plasmons in graphene disks have the additional benefit of being highly tunable via electrical stimulation. Mechanical vibrations create structural deformations in ways where the excitation of localized surface plasmons can be strongly modulated. We show that the spectral shift in such a scenario is determined by a complex interplay between the symmetry and shape of the modal vibrations and the plasmonic mode pattern. Tuning confined modes of light in graphene via acoustic excitations, paves new avenues in shaping the sensitivity of plasmonic detectors, and in the enhancement of the interaction with optical emitters, such as molecules, for future nanophotonic devices.

Temperature-Dependent Adhesion of Graphene Suspended on a Trench

Graphene deposited over a trench has been studied in the context of nanomechanical resonators, where experiments indicate adhesion of the graphene sheet to the trench boundary and sidewalls leads to self-tensioning; however, this adhesion is not well understood. We use molecular dynamics to simulate graphene deposited on a trench and study how adhesion to the sidewalls depends on substrate interaction, temperature, and curvature of the edge of the trench. Over the range of parameters we study, the depth at the center of the sheet is approximately linear in substrate interaction strength and temperature but not trench width, and we explain this using a one-dimensional model for the sheet configuration.

Self-assembly of water molecules using graphene nanoresonators

Inspired by macroscale self-assembly using the higher order resonant modes of Chladni plates, we use classical molecular dynamics to investigate the self-assembly of water molecules using graphene nanoresonators.

A review on the flexural mode of graphene: lattice dynamics, thermal conduction, thermal expansion, elasticity and nanomechanical resonance

Single-layer graphene is so flexible that its flexural mode (also called the ZA mode, bending mode, or out-of-plane transverse acoustic mode) is important for its thermal and mechanical properties. Accordingly, this review focuses on exploring the relationship between the flexural mode and thermal and mechanical properties of graphene. We first survey the lattice dynamic properties of the flexural mode, where the rigid translational and rotational invariances play a crucial role. After that, we outline contributions from the flexural mode in four different physical properties or phenomena of graphene-its thermal conductivity, thermal expansion, Young's modulus and nanomechanical resonance. We explain how graphene's superior thermal conductivity is mainly due to its three acoustic phonon modes at room temperature, including the flexural mode. Its coefficient of thermal expansion is negative in a wide temperature range resulting from the particular vibration morphology of the flexural mode. We then describe how the Young's modulus of graphene can be extracted from its thermal fluctuations, which are dominated by the flexural mode. Finally, we discuss the effects of the flexural mode on graphene nanomechanical resonators, while also discussing how the essential properties of the resonators, including mass sensitivity and quality factor, can be enhanced.

Detecting the mass and position of an adsorbate on a drum resonator

The resonant frequency shifts of a circular membrane caused by an adsorbate are the sensing mechanism for a drum resonator. The adsorbate mass and position are the two major (unknown) parameters determining the resonant frequency shifts. There are infinite combinations of mass and position which can cause the same shift of one resonant frequency. Finding the mass and position of an adsorbate from the experimentally measured resonant frequencies forms an inverse problem. This study presents a straightforward method to determine the adsorbate mass and position by using the changes of two resonant frequencies. Because detecting the position of an adsorbate can be extremely difficult, especially when the adsorbate is as small as an atom or a molecule, this new inverse problem-solving method should be of some help to the mass resonator sensor application of detecting a single adsorbate. How to apply this method to the case of multiple adsorbates is also discussed.

Graphene optomechanics realized at microwave frequencies

Cavity optomechanics has served as a platform for studying the interaction between light and micromechanical motion via radiation pressure. Here we observe such phenomena with a graphene mechanical resonator coupled to an electromagnetic mode. We measure thermal motion and backaction cooling in a bilayer graphene resonator coupled to a microwave on-chip cavity. We detect the lowest flexural mode at 24 MHz down to 60 mK, corresponding to 50±6 mechanical quanta, which represents a phonon occupation that is nearly 3 orders of magnitude lower than that which has been recorded to date with graphene resonators.

Piezoelectric monolayers as nonlinear energy harvesters

We study the dynamics of h-BN monolayers by first performing ab-initio calculations of the deformation potential energy and then solving numerically a Langevine-type equation to explore their use in nonlinear vibration energy harvesting devices. An applied compressive strain is used to drive the system into a nonlinear bistable regime, where quasi-harmonic vibrations are combined with low-frequency swings between the minima of a double-well potential. Due to its intrinsic piezoelectric response, the nonlinear mechanical harvester naturally provides an electrical power that is readily available or can be stored by simply contacting the monolayer at its ends. Engineering the induced nonlinearity, a 20 nm2 device is predicted to harvest an electrical power of up to 0.18 pW for a noisy vibration of 5 pN.

Removal of cobalt ions from aqueous solution by an amination graphene oxide nanocomposite

A newly designed amination graphene oxide (GO-NH2), a superior adsorption capability to that of activated carbon, was fabricated by graphene oxide (GO) combining with aromatic diazonium salt. The resultant GO-NH2 maintained a high surface area of 320 m(2)/g. When used as an adsorbent, the GO-NH2 demonstrated a very quick adsorption property for the removal of Co(II) ions, more than 90% of Co(II) ions could be removed within 5 min for dilute solutions at 0.3g/L adsorbent dose. The adsorption capability approaches 116.35 mg/g, which is one of the highest capabilities of today's materials. The thermodynamic parameters calculated from temperature-dependent adsorption isotherms suggested that the Co(II) ions adsorption on GO-NH2 was a spontaneous process. Considering the superior adsorption capability, the GO-NH2 filter membrane was fabricated for the removal of Co(II) ions. Membrane filtration experiments revealed that the removal capabilities of the materials for cobalt ions depended on the membrane's thickness, flow rate and initial concentration of Co(II) ions. The highest percentage removal of Co(II) exceeds 98%, indicating that the GO-NH2 is one of the very suitable membrane materials in environmental pollution management.

Fermi-Pasta-Ulam physics with nanomechanical graphene resonators: intrinsic relaxation and thermalization from flexural mode coupling

Thermalization in nonlinear systems is a central concept in statistical mechanics and has been extensively studied theoretically since the seminal work of Fermi, Pasta, and Ulam. Using molecular dynamics and continuum modeling of a ring-down setup, we show that thermalization due to nonlinear mode coupling intrinsically limits the quality factor of nanomechanical graphene drums and turns them into potential test beds for Fermi-Pasta-Ulam physics. We find the thermalization rate Γ to be independent of radius and scaling as Γ∼T*/εpre2, where T* and εpre are effective resonator temperature and prestrain.

MoS2 nanoresonators: intrinsically better than graphene?

We perform classical molecular dynamics simulations to examine the intrinsic energy dissipation in single-layer MoS2 nanoresonators, where the point of emphasis is to compare their dissipation characteristics with those of single-layer graphene. Our key finding is that MoS2 nanoresonators exhibit significantly lower energy dissipation, and thus higher quality (Q)-factors by at least a factor of four below room temperature, than graphene. Furthermore, this high Q-factor endows MoS2 nanoresonators with a higher figure of merit, defined as frequency times Q-factor, despite a resonant frequency that is 50% smaller than that of graphene of the same size. By utilizing arguments from phonon-phonon scattering theory, we show that this reduced energy dissipation is enabled by the large energy gap in the phonon dispersion of MoS2, which separates the acoustic phonon branches from the optical phonon branches, leading to a preserving mechanism for the resonant oscillation of MoS2 nanoresonators. We further investigate the effects of tensile mechanical strain and nonlinear actuation on the Q-factors, where the tensile strain is found to counteract the reductions in Q-factor that occur with higher actuation amplitudes. Overall, our simulations illustrate the potential utility of MoS2 for high frequency sensing and actuation applications.

Adsorbate migration effects on continuous and discontinuous temperature-dependent transitions in the quality factors of graphene nanoresonators

We perform classical molecular dynamics simulation to investigate the mechanisms underpinning the unresolved, experimentally observed temperature-dependent scaling transition in the quality factors of graphene nanomechanical resonators (GNMRs). Our simulations reveal that the mechanism underlying this temperature scaling phenomenon is the out-of-plane migration of adsorbates on GNMRs. Specifically, the migrating adsorbate undergoes frequent collisions with the GNMR, which strongly influences the resulting mechanical oscillation, and thus the quality factors. We also predict a discontinuous transition in the quality factor at a lower critical temperature, which results from the in-plane migration of the adsorbate. Overall, our work clearly demonstrates the strong effect of adsorbate migration on the quality factors of GNMRs.

Boron-nitride nanotube triggered self-assembly of hexagonal boron-nitride nanostructure

Molecular mechanics results show that a hexagonal boron nitride (h-BN) membrane can spontaneously assemble on the single-walled boron nitride nanotube (BNNT) in a scroll or helical manner, showing an interesting dependence on h-BN width.

Preserving the Q-factors of ZnO nanoresonators via polar surface reconstruction

We perform molecular dynamics simulations to investigate the effect of polar surfaces on the quality (Q)-factors of zinc oxide (ZnO) nanowire-based nanoresonators. We find that the Q-factors in ZnO nanoresonators with free polar (0001) surfaces are about one order of magnitude higher than in nanoresonators that have been stabilized with reduced charges on the polar (0001) surfaces. From normal mode analysis, we show that the higher Q-factor is due to a shell-like reconstruction that occurs for the free polar surfaces. This shell-like reconstruction suppresses twisting motion in the nanowires such that the mixing of other modes with the resonant mode of oscillation is minimized, and leads to substantially higher Q-factors in ZnO nanoresonators with free polar surfaces.

Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators

We study circular nanomechanical graphene resonators by means of continuum elasticity theory, treating them as membranes. We derive dynamic equations for the flexural mode amplitudes. Due to the geometrical nonlinearity the mode dynamics can be modeled by coupled Duffing equations. By solving the Airy stress problem we obtain analytic expressions for the eigenfrequencies and nonlinear coefficients as functions of the radius, suspension height, initial tension, back-gate voltage and elastic constants, which we compare with finite element simulations. Using perturbation theory, we show that it is necessary to include the effects of the non-uniform stress distribution for finite deflections. This correctly reproduces the spectrum and frequency tuning of the resonator, including frequency crossings.

Intrinsic loss due to unstable modes in graphene

The mechanism of dissipation operative at the nanoscale remains poorly understood for most cases. In this work, using molecular dynamics simulations, we show that the unstable out-of-plane mode leads to the absorption of energy from the in-plane motion in graphene. The in-plane vibration modulates the potential energy profile for the out-of-plane modes. For the fundamental out-of-plane mode in the loading direction, the minimum of the potential energy shifts because of in-plane compressive strain. The structure takes a finite amount of time to relax to the new potential energy configuration. A hysteresis in the out-of-plane dynamics is observed when the time period of in-plane excitation becomes comparable to the time required for this relaxation. Increasing the stiffness of the out-of-plane modes by giving an initial tensile strain leads to a considerable decrease in dissipation rate.

Observation of pull-in instability in graphene membranes under interfacial forces

We present a unique experimental configuration that allows us to determine the interfacial forces on nearly parallel plates made from the thinnest possible mechanical structures, single and few layer graphene membranes. Our approach consists of using a pressure difference across a graphene membrane to bring the membrane to within ~10-20 nm above a circular post covered with SiOx or Au until a critical point is reached whereby the membrane snaps into adhesive contact with the post. Continuous measurements of the deforming membrane with an AFM coupled with a theoretical model allow us to deduce the magnitude of the interfacial forces between graphene and SiOx and graphene and Au. The nature of the interfacial forces at ~10-20 nm separation is consistent with an inverse fourth power distance dependence, implying that the interfacial forces are dominated by van der Waals interactions. Furthermore, the strength of the interactions is found to increase linearly with the number of graphene layers. The experimental approach can be used to measure the strength of the interfacial forces for other atomically thin two-dimensional materials and help guide the development of nanomechanical devices such as switches, resonators, and sensors.

Slip corona surrounding bilayer graphene nanopore

The electronic and magnetic properties of bilayer graphene (BLG) depend on the stacking order between the two layers. We introduce a new conceptual structure of "slip corona" on BLG, which is a transition region between A-A stacking close to a nanopore composed of bilayer edges (BLEs) and A-B stacking far away. For an extremely small nanopore (diameter D(pore) < ~5 nm), both atomistic simulations and a continuum model reach consistent descriptions on the shape and size of this "corona" (diameter ~50 nm), which is much larger than the width of the typical dislocation core (~1 nm) in 3D metals or the nanopore itself, due to the weak van der Waals interactions and low interlayer shear resistance between two adjacent layers of graphene. The continuum model also suggests that the width of this "corona" from the BLE to the A-B stacking area would increase as D(pore) increases and converge to ~40 nm when D(pore) is more than ~80 nm. This large stacking transition region provides a new avenue for tailoring BLG properties.

Nonlinear vibration behavior of graphene resonators and their applications in sensitive mass detection

Graphene has received significant attention due to its excellent mechanical properties, which has resulted in the emergence of graphene-based nano-electro-mechanical system such as nanoresonators. The nonlinear vibration of a graphene resonator and its application to mass sensing (based on nonlinear oscillation) have been poorly studied, although a graphene resonator is able to easily reach the nonlinear vibration. In this work, we have studied the nonlinear vibration of a graphene resonator driven by a geometric nonlinear effect due to an edge-clamped boundary condition using a continuum elastic model such as a plate model. We have shown that an in-plane tension can play a role in modulating the nonlinearity of a resonance for a graphene. It has been found that the detection sensitivity of a graphene resonator can be improved by using nonlinear vibration induced by an actuation force-driven geometric nonlinear effect. It is also shown that an in-plane tension can control the detection sensitivity of a graphene resonator that operates both harmonic and nonlinear oscillation regimes. Our study suggests the design principles of a graphene resonator as a mass sensor for developing a novel detection scheme using graphene-based nonlinear oscillators.

Determination of the bending rigidity of graphene via electrostatic actuation of buckled membranes

Classical continuum mechanics is used extensively to predict the properties of nanoscale materials such as graphene. The bending rigidity, κ, is an important parameter that is used, for example, to predict the performance of graphene nanoelectromechanical devices and also ripple formation. Despite its importance, there is a large spread in the theoretical predictions of κ for few-layer graphene. We have used the snap-through behavior of convex buckled graphene membranes under the application of electrostatic pressure to determine experimentally values of κ for double-layer graphene membranes. We demonstrate how to prepare convex-buckled suspended graphene ribbons and fully clamped suspended membranes and show how the determination of the curvature of the membranes and the critical snap-through voltage, using AFM, allows us to extract κ. The bending rigidity of bilayer graphene membranes under ambient conditions was determined to be 35.5−15.0 +20.0 eV. Monolayers are shown to have significantly lower κ than bilayers.

Interaction between Eu(III) and graphene oxide nanosheets investigated by batch and extended X-ray absorption fine structure spectroscopy and by modeling techniques

The interaction mechanism between Eu(III) and graphene oxide nanosheets (GONS) was investigated by batch and extended X-ray absorption fine structure (EXAFS) spectroscopy and by modeling techniques. The effects of pH, ionic strength, and temperature on Eu(III) adsorption on GONS were evaluated. The results indicated that ionic strength had no effect on Eu(III) adsorption on GONS. The maximum adsorption capacity of Eu(III) on GONS at pH 6.0 and T = 298 K was calculated to be 175.44 mg·g(-1), much higher than any currently reported. The thermodynamic parameters calculated from temperature-dependent adsorption isotherms suggested that Eu(III) adsorption on GONS was an endothermic and spontaneous process. Results of EXAFS spectral analysis indicated that Eu(III) was bound to ∼6-7 O atoms at a bond distance of ∼2.44 Å in the first coordination shell. The value of Eu-C bond distance confirmed the formation of inner-sphere surface complexes on GONS. Surface complexation modeling gave an excellent fit with the predominant mononuclear monodentate >SOEu(2+) and binuclear bidentate (>SO)(2)Eu(2)(OH)(2)(2+) complexes. This paper highlights the application of GONS as a suitable material for the preconcentration and removal of trivalent lanthanides and actinides from aqueous solutions in environmental pollution management.

Ripples in a string coupled to Glauber spins

Each oscillator in a linear chain (a string) interacts with a local Ising spin in contact with a thermal bath. These spins evolve according to Glauber dynamics. Below a critical temperature, there appears an equilibrium, time-independent, rippled state in the string that is accompanied by a nonzero spin polarization. On the other hand, the system is shown to form "metastable," nonequilibrium long-lived ripples in the string for slow spin relaxation. The system vibrates rapidly about these quasistationary states, which can be described as snapshots of a coarse-grained stroboscopic map. For moderate observation times, ripples are observed irrespective of the final thermodynamically stable state (rippled or not). Interestingly, the system can be considered as a "minimal" model to understand rippling in clamped graphene sheets.

Stamp transferred suspended graphene mechanical resonators for radio frequency electrical readout

We present a simple micromanipulation technique to transfer suspended graphene flakes onto any substrate and to assemble them with small localized gates into mechanical resonators. The mechanical motion of the graphene is detected using an electrical, radio frequency (RF) reflection readout scheme where the time-varying graphene capacitor reflects a RF carrier at f = 5-6 GHz producing modulation sidebands at f ± f(m). A mechanical resonance frequency up to f(m) = 178 MHz is demonstrated. We find both hardening/softening Duffing effects on different samples and obtain a critical amplitude of ~40 pm for the onset of nonlinearity in graphene mechanical resonators. Measurements of the quality factor of the mechanical resonance as a function of dc bias voltage V(dc) indicates that dissipation due to motion-induced displacement currents in graphene electrode is important at high frequencies and large V(dc).

Carbon nanotube field effect transistors with suspended graphene gates

Novel field effect transistors with suspended graphene gates are demonstrated. By incorporating mechanical motion of the gate electrode, it is possible to improve the switching characteristics compared to a static gate, as shown by a combination of experimental measurements and numerical simulations. The mechanical motion of the graphene gate is confirmed by using atomic force microscopy to directly measure the electrostatic deflection. The device geometry investigated here can also provide a sensitive measurement technique for detecting high-frequency motion of suspended membranes as required, e.g., for mass sensing.

Buckling of a single-layered graphene sheet on an initially strained InGaAs thin plate

The elastic buckling behavior of a defect-free single-layered graphene sheet deposited on a strained InGaAs substrate is investigated. Such a buckled sandwich structure can be formed by local etching of an initially strained InGaAs substrate. We numerically investigated the necessary buckling conditions for a single-layered graphene sheet of circular geometry on an initially strained InGaAs thin plate. A criterion for buckling for various axisymmetric buckling shapes was obtained. It is shown that for a thin circular InGaAs plate with a monolayer graphene sheet of radius 80 nm and thickness 4 nm three axisymmetric buckling shapes can be obtained. For an initial value of the elastic deformation of the plate of 3%, the in-plane strain in graphene can reach a value of 1%. This deformation is shown to be distributed inhomogeneously along the radius of the graphene monolayer.

Self-assembly of graphene nanostructures on nanotubes

We demonstrate by molecular dynamics simulations that carbon nanotubes can activate and guide on their surfaces and in their interiors the self-assembly of planar graphene nanostructures of various sizes and shapes. Nanotubes can induce bending, folding, sliding, and rolling of the nanostructures in vacuum and in the presence of solvent, leading to stable graphene rings, helices, and knots. We investigate the self-assembly conditions and analyze the stability of the formed nanosystems, with numerous possible applications.

Compression behavior of single-layer graphenes

Central to most applications involving monolayer graphenes is its mechanical response under various stress states. To date most of the work reported is of theoretical nature and refers to tension and compression loading of model graphenes. Most of the experimental work is indeed limited to the bending of single flakes in air and the stretching of flakes up to typically approximately 1% using plastic substrates. Recently we have shown that by employing a cantilever beam we can subject single graphenes to various degrees of axial compression. Here we extend this work much further by measuring in detail both stress uptake and compression buckling strain in single flakes of different geometries. In all cases the mechanical response is monitored by simultaneous Raman measurements through the shift of either the G or 2D phonons of graphene. Despite the infinitely small thickness of the monolayers, the results show that graphenes embedded in plastic beams exhibit remarkable compression buckling strains. For large length (l)-to-width (w) ratios (> or =0.2) the buckling strain is of the order of -0.5% to -0.6%. However, for l/w < 0.2 no failure is observed for strains even higher than -1%. Calculations based on classical Euler analysis show that the buckling strain enhancement provided by the polymer lateral support is more than 6 orders of magnitude compared to that of suspended graphene in air.

Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators

We use suspended graphene electromechanical resonators to study the variation of resonant frequency as a function of temperature. Measuring the change in frequency resulting from a change in tension, from 300 to 30 K, allows us to extract information about the thermal expansion of monolayer graphene as a function of temperature, which is critical for strain engineering applications. We find that thermal expansion of graphene is negative for all temperatures between 300 and 30 K. We also study the dispersion, the variation of resonant frequency with DC gate voltage, of the electromechanical modes and find considerable tunability of resonant frequency, desirable for applications like mass sensing and RF signal processing at room temperature. With a lowering of temperature, we find that the positively dispersing electromechanical modes evolve into negatively dispersing ones. We quantitatively explain this crossover and discuss optimal electromechanical properties that are desirable for temperature-compensated sensors.

Nonlinear vibrational analysis of single-layer graphene sheets

Recent experiments have shown the applicability of single-layer graphene sheets (SLGSs) as electromechanical resonators. Existing theoretical models, based on linear continuum or atomistic methods, are limited to the study of linear vibrations of SLGSs. Here we introduce a hybrid atomistic-structural element which is capable of modelling nonlinear behaviour of graphene sheets. This hybrid element is based on an empirical inter-atomic potential function and can model the nonlinear dynamic response of SLGSs. Using this element, nonlinear vibrational analysis of SLGSs is performed. It is shown that the nonlinear vibrational analysis of SLGSs predicts significantly higher fundamental frequencies. Also, the effects of vibration amplitude as well as the geometry of the SLGSs on the fundamental frequency are studied and predictive relations between the fundamental frequency, the SLGS length and the non-dimensional vibration amplitude are presented. The results are verified with experimental observations and are in remarkable agreement.