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Nonlinear dynamic simulations of mechanical resonators have been facilitated by the advent of computational techniques that generate nonlinear reduced order models (ROMs) using the finite element (FE) method. However, designing devices with specific nonlinear characteristics remains inefficient since it requires manual adjustment of the design parameters and can result in suboptimal designs. Here, we integrate an FE-based nonlinear ROM technique with a derivative-free optimization algorithm to enable the design of nonlinear mechanical resonators. The resulting methodology is used to optimize the support design of high-stress nanomechanical Si3N4 string resonators, in the presence of conflicting objectives such as simultaneous enhancement of Q-factor and nonlinear Duffing constant. To that end, we generate Pareto frontiers that highlight the trade-offs between optimization objectives and validate the results both numerically and experimentally. To further demonstrate the capability of multi-objective optimization for practical design challenges, we simultaneously optimize the design of nanoresonators for three key figure-of-merits in resonant sensing: power consumption, sensitivity and response time. The presented methodology can facilitate and accelerate designing (nano)mechanical resonators with optimized performance for a wide variety of applications.

Heat transport in two-dimensions is fundamentally different from that in three dimensions. As a consequence, the thermal properties of 2D materials are of great interest, both from scientific and application point of view. However, few techniques are available for accurate determination of these properties in ultrathin suspended membranes. Here, we present an optomechanical methodology for extracting the thermal expansion coefficient, specific heat and thermal conductivity of ultrathin membranes made of 2H-TaS2, FePS3, polycrystalline silicon, MoS2 and WSe2. The obtained thermal properties are in good agreement with values reported in the literature for the same materials. Our work provides an optomechanical method for determining thermal properties of ultrathin suspended membranes, that are difficult to measure otherwise. It can does provide a route towards improving our understanding of heat transport in the 2D limit and facilitates engineering of 2D structures with dedicated thermal performance., Comment: 14 pages, 10 figures

Suspended drums made of 2D materials hold potential for sensing applications. However, the industrialization of these applications is hindered by significant device-to-device variations presumably caused by non-uniform stress distributions induced by the fabrication process. Here we introduce a new methodology to determine the stress distribution from their mechanical resonance frequencies and corresponding mode shapes as measured by a laser Doppler vibrometer (LDV). To avoid limitations posed by the optical resolution of the LDV, we leverage a unique manufacturing process to create ultra-large graphene drums with diameters of up to 1000 um. We solve the inverse problem of a F\"oppl--von K\'arm\'an plate model by an iterative procedure to obtain the stress distribution within the drums from the experimental data. Our results show that the generally used uniform pre-tension assumption overestimates the pre-stress value, exceeding the averaged stress obtained by more than 47%. Moreover, it is is found that the reconstructed stress distributions are bi-axial, which likely originates from the transfer process. The introduced metholodogy allows one to estimate the tension distribution in drum resonators from their mechanical response and thereby paves the way for linking the used fabrication processes to the resulting device performance.

Nanomechanical resonances of two-dimensional (2D) materials are sensitive probes for condensed-matter physics, offering new insights into magnetic and electronic phase transitions. Despite extensive research, the influence of the spin dynamics near a second-order phase transition on the nonlinear dynamics of 2D membranes has remained largely unexplored. Here, we investigate nonlinear magneto-mechanical coupling to antiferromagnetic order in suspended FePS$_3$-based heterostructure membranes. By monitoring the motion of these membranes as a function of temperature, we observe characteristic features in both nonlinear stiffness and damping close to the N\'{e}el temperature $T_{\rm{N}}$. We account for these experimental observations with an analytical magnetostriction model in which these nonlinearities emerge from a coupling between mechanical and magnetic oscillations, demonstrating that magneto-elasticity can lead to nonlinear damping. Our findings thus provide insights into the thermodynamics and magneto-mechanical energy dissipation mechanisms in nanomechanical resonators due to the material's phase change and magnetic order relaxation.

High-aspect-ratio mechanical resonators are pivotal in precision sensing, from macroscopic gravitational wave detectors to nanoscale acoustics. However, fabrication challenges and high computational costs have limited the length-to-thickness ratio of these devices, leaving a largely unexplored regime in nano-engineering. We present nanomechanical resonators that extend centimeters in length yet retain nanometer thickness. We explore this expanded design space using an optimization approach which judiciously employs fast millimeter-scale simulations to steer the more computationally intensive centimeter-scale design optimization. By employing delicate nanofabrication techniques, our approach ensures high-yield realization, experimentally confirming room-temperature quality factors close to theoretical predictions. The synergy between nanofabrication, design optimization guided by machine learning, and precision engineering opens a solid-state path to room-temperature quality factors approaching 10 billion at kilohertz mechanical frequencies -- comparable to the performance of leading cryogenic resonators and levitated nanospheres, even under significantly less stringent temperature and vacuum conditions.

The high susceptibility of ultrathin two-dimensional (2D) material resonators to force and temperature makes them ideal systems for sensing applications and exploring thermomechanical coupling. Although the dynamics of these systems at high stress has been thoroughly investigated, their behavior near the buckling transition has received less attention. Here, we demonstrate that the force sensitivity and frequency tunability of 2D material resonators are significantly enhanced near the buckling bifurcation. This bifurcation is triggered by compressive displacement that we induce via thermal expansion of the devices, while measuring their dynamics via an optomechanical technique. We understand the frequency tuning of the devices through a mechanical buckling model, which allows to extract the central deflection and boundary compressive displacement of the membrane. Surprisingly, we obtain a remarkable enhancement of up to 14x the vibration amplitude attributed to a very low stiffness of the membrane at the buckling transition, as well as a high frequency tunability by temperature of more than 4.02 %/K. The presented results provide insights into the effects of buckling on the dynamics of free-standing 2D materials and thereby open up opportunities for the realization of 2D resonant sensors with buckling-enhanced sensitivity., Comment: 20 pages and 4 figures

Mechanical nonlinearities dominate the motion of nanoresonators already at relatively small oscillation amplitudes. Although single and coupled two-degrees-of-freedom models have been used to account for experimentally observed nonlinear effects, it is shown that these models quickly deviate from experimental findings when multiple modes influence the nonlinear response. Here, we present a nonlinear reduced-order modelling methodology based on FEM simulations for capturing the global nonlinear dynamics of nanomechanical resonators. Our physics-based approach obtains the quadratic and cubic nonlinearities of resonators over a wide frequency range that spans 70 MHz. To qualitatively validate our approach, we perform experiments on a graphene nanodrum driven opto-thermally and show that the model can replicate diverse ranges of nonlinear phenomena, including multi-stability, parametric resonance, and different internal resonances without considering any empirical nonlinear fitting parameters. By providing a direct link between microscopic geometry, material parameters, and nonlinear dynamic response, we clarify the physical significance of nonlinear parameters that are obtained from fitting the dynamics of nanomechanical systems, and provide a route for designing devices with desired nonlinear behaviour.

The temperature dependent order parameter provides important information on the nature of magnetism. Using traditional methods to study this parameter in two-dimensional (2D) magnets remains difficult, however, particularly for insulating antiferromagnetic (AF) compounds. Here, we show that its temperature dependence in AF MPS$_{3}$ (M(II) = Fe, Co, Ni) can be probed via the anisotropy in the resonance frequency of rectangular membranes, mediated by a combination of anisotropic magnetostriction and spontaneous staggered magnetization. Density functional calculations followed by a derived orbital-resolved magnetic exchange analysis confirm and unravel the microscopic origin of this magnetization inducing anistropic strain. We further show that the temperature and thickness dependent order parameter allows to deduce the material's critical exponents characterising magnetic order. Nanomechanical sensing of magnetic order thus provides a future platform to investigate 2D magnetism down to the single-layer limit., Comment: 9 pages, 4 figures, 1 table. Appended SI

Atomic layer deposition (ALD), a layer-by-layer controlled method to synthesize ultrathin materials, provides various merits over other techniques such as precise thickness control, large area scalability and excellent conformality. Here we demonstrate the possibility of using ALD growth on top of suspended 2D materials to fabricate nanomechanical resonators. We fabricate ALD nanomechanical resonators consisting of a graphene/MoS$_2$ heterostructure. Using AFM indentation and optothermal drive, we measure their mechanical properties including Young's modulus, resonance frequency and quality factor, showing similar values as their exfoliated and chemical vapor deposited counterparts. We also demonstrate the fabrication of nanomechanical resonators by exfoliating an ALD grown NbS$_2$ layer. This study exemplifies the potential of ALD techniques to produce high-quality suspended nanomechanical membranes, providing a promising route towards high-volume fabrication of future multilayer nanodevices and nanoelectromechanical systems., Comment: 8 pages, 6 figures

The resonance frequency of ultra-thin layered two-dimensional (2D) materials changes nonlinearly with the tension induced by the pressure from the surrounding gas. Although the dynamics of pressurized 2D material membranes have been extensively explored, recent experimental observations show significant deviations from analytical predictions. Here, we present a multi-mode continuum approach to capture the nonlinear pressure-frequency response of pre-tensioned membranes undergoing large deflections. We validate the model using experiments conducted on polysilicon drums excited opto-thermally and subjected to pressure changes in the surrounding medium. We demonstrate that considering the effect of pressure on the membrane tension is not sufficient for determining the resonance frequencies. In fact, it is essential to also account for the change in the membrane's shape in the pressurized configuration, the mid-plane stretching, and the contributions of higher modes to the mode shapes. Finally, we show how the presented high-frequency mechanical characterization method can be used to determine Young's modulus of ultra-thin membranes.

Mechanical resonators that possess coupled modes with harmonic frequency relations have recently sparked interest due to their suitability for controllable energy transfer and non-Hermitian dynamics. Here, we show coupling between high Q-factor ($>\!\!10^4$) resonances with a nearly 1:1 frequency relation in spatially-symmetric microresonators. We develop and demonstrate a method to analyze their dynamical behavior based on the simultaneous and resonant detection of both spectral peaks, and validate this with experimental results. The frequency difference between the peaks modulates their ringdown, and creates a beat pattern in the linear decay. This method applies both to the externally driven and the Brownian motion (thermal) regime, and allows characterization of both linear and nonlinear parameters. The mechanism behind this method renders it broadly applicable to both optical and electrical readout, as well as to different mechanical systems. This will aid studies using near-degenerate mechanical modes, for e.g., optomechanical energy transfer, synchronization and gyroscopic sensors.

Microphones exploit the motion of suspended membranes to detect sound waves. Since the microphone performance can be improved by reducing the thickness and mass of its sensing membrane, graphene-based microphones are expected to outperform state-of-the-art microelectromechanical (MEMS) microphones and allow further miniaturization of the device. Here, we present a laser vibrometry study of the acoustic response of suspended multilayer graphene membranes for microphone applications. We address performance parameters relevant for acoustic sensing, including mechanical sensitivity, limit of detection and nonlinear distortion, and discuss the trade-offs and limitations in the design of graphene microphones. We demonstrate superior mechanical sensitivities of the graphene membranes, reaching more than 2 orders of magnitude higher compliances than commercial MEMS devices, and report a limit of detection as low as 15 dBSPL, which is 10 - 15 dB lower than that featured by current MEMS microphones., Comment: 34 pages, 6 figures, 7 supplementary figures

Levitation offers extreme isolation of mechanical systems from their environment, while enabling unconstrained high-precision translation and rotation of objects. Diamagnetic levitation is one of the most attractive levitation schemes, because it allows stable levitation at room temperature without the need for a continuous power supply. However, dissipation by eddy currents in conventional diamagnetic materials significantly limits the application potential of diamagnetically levitating systems. Here, we present a route towards high $Q$ macroscopic levitating resonators by substantially reducing eddy current damping using graphite particle based diamagnetic composites. We demonstrate resonators that feature quality factors $Q$ above 450,000 and vibration lifetimes beyond one hour, while levitating above permanent magnets in high vacuum at room temperature. The composite resonators have a $Q$ that is more than 400 times higher than that of diamagnetic graphite plates. By tuning the composite particle size and density, we investigate the dissipation reduction mechanism and enhance the $Q$ of the levitating resonators. Since their estimated acceleration noise is as low as some of the best superconducting levitating accelerometers at cryogenic temperatures, the high $Q$ and large mass of the presented composite resonators positions them as one of the most promising technologies for next generation ultra-sensitive room temperature accelerometers.

State-of-the-art nanomechanical resonators are heralded as a central component for next-generation clocks, filters, resonant sensors and quantum technologies. To practically build these technologies will require monolithic integration of microchips, resonators and readout systems. While it is widely seen that mounting microchip substrates into a system can greatly impact the performance of high-Q resonators, a systematic study has remained elusive, owing to the variety of physical processes and factors that influence the dissipation. Here, we analytically analyze a mechanism by which substrates couple to resonators manufactured on them, and experimentally demonstrate that this coupling can increase the mechanical dissipation of nanomechanical resonators when the resonance frequencies of resonator and substrate coincide. More generally, we then show a similar coupling mechanism can exist between two adjacent resonators. Since the substrate-mode coupling mechanism strongly depends on both the resonator position on the substrate and the mounting of the substrate, this work provides key design guidelines for high-precision nanomechanical technologies.

The resolution of resonant sensors is fundamentally limited by the presence of noise. Thermomechanical noise, intrinsic to the resonator, sets the ultimate sensor performance when all other noise sources have been eliminated. For linear resonators, the sensing resolution can always be further improved by increasing the driving power. However, this trend cannot continue indefinitely, since at sufficiently high driving powers non-linear effects emerge and influence the noise performance. As a consequence, the resonator's non-linear characteristics play an inextricable role in determining its ultimate resolution limits. Recently, several works have studied the characteristic performance of non-linear resonators as sensors, with the counter intuitive conclusion that increasing the quality factor of a resonator does not improve its sensing resolution at the thermomechanical limit. In this work we further analyze the ultimate resolution limits, and describe different regimes of performance at integration times below and above the resonator's decay time. We provide an analytical model to elucidate the effects of Duffing non-linearity on the resolution of closed-loop sensors, and validate it using numerical simulations. In contrast to previous works, our model and simulations show that under certain conditions the ultimate sensing resolution of a Duffing resonator can be improved by maximizing its quality factor. With measurements on a nanomechanical membrane resonator, we experimentally verify the model and demonstrate that frequency resolutions can be achieved that surpass the previously known limits., Comment: 15 pages, 5 figures

Nonlinearities are inherent to the dynamics of two-dimensional materials. Phenomena like intermodal coupling already arise at amplitudes of only a few nanometers, and a range of unexplored effects still awaits to be harnessed. Here, we demonstrate a route for generating mechanical frequency combs in graphene resonators undergoing symmetry-breaking forces. We use electrostatic force to break the membrane's out-of-plane symmetry and tune its resonance frequency towards a two-to-one internal resonance, thus achieving strong coupling between two of its mechanical modes. When increasing the drive level, we observe splitting of the fundamental resonance peak, followed by the emergence of a frequency comb regime. We attribute the observed physics to a non-symmetric restoring potential, and show that the frequency comb regime is mediated by a Neimark bifurcation of the periodic solution. These results demonstrate that mechanical frequency combs and chaotic dynamics in 2D material resonators can emerge near internal resonances due to symmetry-breaking.

Although 2D materials hold great potential for next-generation pressure sensors, recent studies revealed that gases permeate along the membrane-surface interface that is only weakly bound by van der Waals interactions, necessitating additional sealing procedures. In this work, we demonstrate the use of free-standing complex oxides as self-sealing membranes that allow the reference cavity of pressure sensors to be sealed by a simple anneal. To test the hermeticity, we study the gas permeation time constants in nano-mechanical resonators made from SrRuO3 and SrTiO3 membranes suspended over SiO2/Si cavities which show an improvement up to 4 orders of magnitude in the permeation time constant after annealing the devices for 15 minutes. Similar devices fabricated on Si3N4/Si do not show such improvements, suggesting that the adhesion increase over SiO2 is mediated by oxygen bonds that are formed at the SiO2/complex oxide interface during the self-sealing anneal. We confirm the enhancement of adhesion by picosecond ultrasonics measurements which show an increase in the interfacial stiffness by 70% after annealing. Since it is straigthforward to apply, the presented self-sealing method is thus a promising route toward realizing ultrathin hermetic pressure sensors., Comment: 7 pages, 5 figures

From ultra-sensitive detectors of fundamental forces to quantum networks and sensors, mechanical resonators are enabling next-generation technologies to operate in room temperature environments. Currently, silicon nitride nanoresonators stand as a leading microchip platform in these advances by allowing for mechanical resonators whose motion is remarkably isolated from ambient thermal noise. However, to date, human intuition has remained the driving force behind design processes. Here, inspired by nature and guided by machine learning, a spiderweb nanomechanical resonator is developed that exhibits vibration modes which are isolated from ambient thermal environments via a novel "torsional soft-clamping" mechanism discovered by the data-driven optimization algorithm. This bio-inspired resonator is then fabricated; experimentally confirming a new paradigm in mechanics with quality factors above 1 billion in room temperature environments. In contrast to other state-of-the-art resonators, this milestone is achieved with a compact design which does not require sub-micron lithographic features or complex phononic bandgaps, making it significantly easier and cheaper to manufacture at large scales. Here we demonstrate the ability of machine learning to work in tandem with human intuition to augment creative possibilities and uncover new strategies in computing and nanotechnology.

Through rational chemical design, and thanks to the hybrid nature of metal-organic frameworks (MOFs), it is possible to prepare molecule-based 2D magnetic materials stable at ambient conditions. Here, we illustrate the versatility of this approach by changing both the metallic nodes and the ligands in a family of layered MOFs that allows the tuning of their magnetic properties. Specifically, the reaction of benzimidazole-type ligands with different metal centres (MII = Fe, Co, Mn, Zn) in a solvent-free synthesis produces a family of crystalline materials, denoted as MUV-1(M), which order antiferromagnetically with critical temperatures that depend on M. Furthermore, the incorporation of additional substituents in the ligand results in a novel system, denoted as MUV-8, formed by covalently bound magnetic double-layers interconnected by van der Waals interactions, a topology that is very rare in the field of 2D materials and unprecedented for 2D magnets. These layered materials are robust enough to be mechanically exfoliated down to a few layers with large lateral dimensions. Finally, the robustness and crystallinity of these layered MOFs allow the fabrication of nanomechanical resonators that can be used to detect -- through laser interferometry -- the magnetic order in thin layers of these 2D molecule-based antiferromagnets., Comment: Main Text + Supplementary Information

Coupled nanomechanical resonators made of two-dimensional materials are promising for processing information with mechanical modes. However, the challenge for these types of systems is to control the coupling. Here, we demonstrate strong coupling of motion between two suspended membranes of the magnetic 2D material FePS$_3$. We describe a tunable electromechanical mechanism for control over both the resonance frequency and the coupling strength using a gate voltage electrode under each membrane. We show that the coupling can be utilized for transferring data from one drum to the other by amplitude modulation. Finally, we also study the temperature dependence of the coupling, and in particular how it is affected by the antiferromagnetic phase transition characteristic of this material. The presented electrical coupling of resonant magnetic 2D membranes holds promise of transferring mechanical energy over a distance at low electrical power, thus enabling novel data readout and information processing technologies.

The resonance frequency of membranes depends on the gas pressure due to the squeeze-film effect, induced by the compression of a thin gas film that is trapped underneath the resonator by the high frequency motion. This effect is particularly large in low-mass graphene membranes, which makes them promising candidates for pressure sensing applications. Here, we study the squeeze-film effect in single layer graphene resonators and find that their resonance frequency is lower than expected from models assuming ideal compression. To understand this deviation, we perform Boltzmann and continuum finite-element simulations, and propose an improved model that includes the effects of gas leakage and can account for the observed pressure dependence of the resonance frequency. Thus, this work provides further understanding of the squeeze-film effect and provides further directions into optimizing the design of squeeze-film pressure sensors from 2D materials.

Diamagnetic levitation offers stable confinement of an object from its environment at zero power, and thus is a promising technique for developing next generation unclamped resonant sensors. In this work, we realize a resonant weighing scale using a graphite plate that is diamagnetically levitating over a checkerboard arrangement of permanent magnets. We characterize the bending vibrations of the levitating object using laser Doppler vibrometry and use microgram glass beads to calibrate the responsivity of the sensor's resonance frequency to mass changes. The sensor is used for real-time measurement of the evaporation rate of nano-litre droplets with high-accuracy. By analyzing the resonator's frequency stability, we show that the millimeter graphite sensor can reach mass resolutions down to 4.0ng, relevant to biological and chemical sensing concepts., Comment: 16 pages, 8 figures

The dynamics of suspended two-dimensional (2D) materials has received increasing attention during the last decade, yielding new techniques to study and interpret the physics that governs the motion of atomically thin layers. This has led to insights into the role of thermodynamic and nonlinear effects as well as the mechanisms that govern dissipation and stiffness in these resonators. In this review, we present the current state-of-the-art in the experimental study of the dynamics of 2D membranes. The focus will be both on the experimental measurement techniques and on the interpretation of the physical phenomena exhibited by atomically thin membranes in the linear and nonlinear regimes. We will show that resonant 2D membranes have emerged both as sensitive probes of condensed matter physics in ultrathin layers, and as sensitive elements to monitor small external forces or other changes in the environment. New directions for utilizing suspended 2D membranes for material characterization, thermal transport, and gas interactions will be discussed and we conclude by outlining the challenges and opportunities in this upcoming field., Comment: 39 pages, 16 figures

The charge density wave (CDW) state in van der Waals systems shows interesting scaling phenomena as the number of layers can significantly affect the CDW transition temperature, $T_{CDW}$. However, it is often difficult to use conventional methods to study the phase transition in these systems due to their small size and sensitivity to degradation. Degradation is an important parameter which has been shown to greatly influence the superconductivity in layered systems. Since the CDW state competes with the onset of superconductivity, it is expected that $T_{CDW}$ will also be affected by the degradation. Here, we probe the CDW phase transition by the mechanical resonances of suspended 2H-TaS2 and 2H-TaSe2 membranes and study the effect of disorder on the CDW state. Pristine flakes show the transition near the reported values of 75 K and 122 K respectively. We then study the effect of degradation on 2H-TaS2 which displays an enhancement of $T_{CDW}$ up to 129 K after degradation in ambient air. Finally, we study a sample with local degradation and observe that multiple phase transitions occur at 87 K, 103 K and 118 K with a hysteresis in temperature in the same membrane. The observed spatial variations in the Raman spectra suggest that variations in crystal structure cause domains with different transition temperatures which could result in the hysteresis. This work shows the potential of using nanomechanical resonance to characterize the CDW in suspended 2D materials and demonstrate that degradation can have a large effect on transition temperatures., Comment: The following article has been accepted by Applied Physics Letters. After it is published, it will be found at [https://doi.org/10.1063/5.0051112]

Two-dimensional (2D) magnetic materials with strong magnetostriction are interesting systems for strain-tuning the magnetization, enabling potential for realizing spintronic and nanomagnetic devices. Realizing this potential requires understanding of the magneto-mechanical coupling in the 2D limit. In this work, we suspend thin Cr$_2$Ge$_2$Te$_6$ layers, creating nanomechanical membrane resonators. We probe its mechanical and magnetic properties as a function of temperature and strain. Pronounced signatures of magneto-elastic coupling are observed in the temperature-dependent resonance frequency of these membranes near $T_{\rm C}$. We further utilize Cr$_2$Ge$_2$Te$_6$ in heterostructures with thin layers of WSe$_2$ and FePS$_3$, which have positive thermal expansion coefficients, to compensate the negative thermal expansion coefficient of Cr$_2$Ge$_2$Te$_6$ and quantitatively probe the corresponding $T_{\rm C}$. Finally, we induce a strain of $0.016\%$ in a suspended heterostructure via electrostatic force and demonstrate a resulting enhancement of $T_{\rm C}$ by $2.5 \pm 0.6$ K in the absence of an external magnetic field.

Silicon p-n junction diodes emit low-intensity, broad-spectrum light near 1120 nm in forward bias and between 400-900 nm in reverse bias (avalanche). For the first time, we experimentally achieve optical absorption sensing of pigment in solution with silicon micro LEDs designed in a standard silicon-on-insulator CMOS technology. By driving a single LED in both forward and avalanche modes of operation, we steer its electroluminescent spectrum between visible and near-infrared (NIR). We then characterize the vertical optical transmission of both visible and NIR light from the LED through the same micro-droplet specimen to a vertically mounted discrete silicon photodiode. The effective absorption coefficient of carmine solution in glycerol at varying concentrations were extracted from the color ratio in optical coupling. By computing the LED-specific molar absorption coefficient of carmine, we estimate the concentration (0.040 mo/L) and validate the same with a commercial spectrophotometer (0.030 mol/L ). With a maximum observed sensitivity of 1260 /cm /mol L, the sensor is a significant step forward towards low-cost CMOS-integrated optical sensors with silicon LED as the light source intended for biochemical analyses in food sector and plant/human health., Comment: IEEE Sensors 2020, 4 pages, 5 figures (https://ieeexplore.ieee.org/document/9278700)

Semi-permeable membranes are important elements in water purification and energy generation applications, for which the atomic thickness and strength of graphene can enhance efficiency and permeation rate while maintaining good selectivity. Here, we show that an osmotic pressure difference forms across a suspended graphene membrane as a response to a sucrose concentration difference, providing evidence for its semi-permeability. This osmotic pressure difference is detected via the deflection of the graphene membrane that is measured by atomic force microscopy. Using this technique, the time dependence of this deflection allows us to measure the water permeation rate of a single 3.4 $\mu$m diameter graphene membrane. Its value is close to the expected value of a single nanopore in graphene. The method thus allows one to experimentally study the semi-permeability of graphene membranes at the microscale when the leakage rate is miniscule. It can therefore find use in the development of graphene membranes for filtration, and can enable sensors that measure the concentration and composition of solutions.

The unique properties and atomic thickness of two-dimensional (2D) materials enable smaller and better nanoelectromechanical sensors with novel functionalities. During the last decade, many studies have successfully shown the feasibility of using suspended membranes of 2D materials in pressure sensors, microphones, accelerometers, and mass and gas sensors. In this review, we explain the different sensing concepts and give an overview of the relevant material properties, fabrication routes, and device operation principles. Finally, we discuss sensor readout and integration methods and provide comparisons against the state of the art to show both the challenges and promises of 2D material-based nanoelectromechanical sensing., Comment: Review paper

Resonant sensors determine a sensed parameter by measuring the resonance frequency of a resonator. For fast continuous sensing, it is desirable to operate resonant sensors in a closed-loop configuration, where a feedback loop ensures that the resonator is always actuated near its resonance frequency, so that the precision is maximized even in the presence of drifts or fluctuations of the resonance frequency. However, in a closed-loop configuration, the precision is not only determined by the resonator itself, but also by the feedback loop, even if the feedback circuit is noiseless. Therefore, to characterize the intrinsic precision of resonant sensors, the open-loop configuration is often employed. To link these measurements to the actual closed-loop performance of the resonator, it is desirable to have a relation that determines the closed-loop precision of the resonator from open-loop characterisation data. In this work, we present a methodology to estimate the closed-loop resonant sensor precision by relying only on an open-loop characterization of the resonator. The procedure is beneficial for fast performance estimation and benchmarking of resonant sensors, because it does not require actual closed-loop sensor operation, thus being independent on the particular implementation of the feedback loop. We validate the methodology experimentally by determining the closed-loop precision of a mechanical resonator from an open-loop measurement and comparing this to an actual closed-loop measurement., Comment: Accepted version of publication in IEEE Sensors Journal

Mechanical sources of nonlinear damping play a central role in modern physics, from solid-state physics to thermodynamics. The microscopic theory of mechanical dissipation [M. I . Dykman, M. A. Krivoglaz, Physica Status Solidi (b) 68, 111 (1975)] suggests that nonlinear damping of a resonant mode can be strongly enhanced when it is coupled to a vibration mode that is close to twice its resonance frequency. To date, no experimental evidence of this enhancement has been realized. In this letter, we experimentally show that nanoresonators driven into parametric-direct internal resonance provide supporting evidence for the microscopic theory of nonlinear dissipation. By regulating the drive level, we tune the parametric resonance of a graphene nanodrum over a range of 40-70 MHz to reach successive two-to-one internal resonances, leading to a nearly two-fold increase of the nonlinear damping. Our study opens up an exciting route towards utilizing modal interactions and parametric resonance to realize resonators with engineered nonlinear dissipation over wide frequency range.

Diamagnetic levitation is a promising technique for realizing resonant sensors and energy harvesters, since it offers thermal and mechanical isolation from the environment at zero power. To advance the application of diamagnetically levitating resonators, it is important to characterize their dynamics in the presence of both magnetic and gravitational fields. Here we experimentally actuate and measure rigid body modes of a diamagnetically levitating graphite plate. We numerically calculate the magnetic field and determine the influence of magnetic force on the resonance frequencies of the levitating plate. By analyzing damping mechanisms, we conclude that eddy current damping dominates dissipation in mm-sized plates. We use finite element simulations to model eddy current damping and find close agreement with experimental results. We also study the size-dependent Q-factors (Qs) of diamagnetically levitating plates and show that Qs above 100 million are theoretically attainable by reducing the size of the diamagnetic resonator down to microscale, making these systems of interest for next generation low-noise resonant sensors and oscillators., Comment: 6 pages, 4 figures

The high flexibility, impermeability and strength of graphene membranes are key properties that can enable the next generation of nanomechanical sensors. However, for capacitive pressure sensors the sensitivity offered by a single suspended graphene membrane is too small to compete with commercial sensors. Here, we realize highly sensitive capacitive pressure sensors consisting of arrays of nearly ten thousand small, freestanding double-layer graphene membranes. We fabricate large arrays of small diameter membranes using a procedure that maintains the superior material and mechanical properties of graphene, even after high-temperature anneals. These sensors are readout using a low cost battery-powered circuit board, with a responsivity of up to 47.8 aF Pa$^{-1}$ mm$^{-2}$, thereby outperforming commercial sensors.

Porous, atomically thin graphene membranes have interesting properties for filtration and sieving applications because they can accommodate small pore sizes, while maintaining high permeability. These membranes are therefore receiving much attention for novel gas and water purification applications. Here we show that the atomic thickness and high resonance frequency of porous graphene membranes enables an effusion based gas sensing method that distinguishes gases based on their molecular mass. Graphene membranes are used to pump gases through nanopores using optothermal forces. By monitoring the time delay between the actuation force and the membrane mechanical motion, the permeation time-constants of various gases are shown to be significantly different. The measured linear relation between the effusion time constant and the square root of the molecular mass provides a method for sensing gases based on their molecular mass. The presented microscopic effusion based gas sensor can provide a small, low-power alternative for large, high-power, mass-spectrometry and optical spectrometry based gas sensing methods., Comment: 12 pages

Two-dimensional (2D) materials enable new types of magnetic and electronic phases mediated by their reduced dimensionality like magic-angle induced phase transitions, 2D Ising antiferromagnets and ferromagnetism in 2D atomic layers and heterostructures. However, only a few methods are available to study these phase transitions, which for example is particularly challenging for antiferromagnetic materials. Here, we demonstrate that these phases can be probed by the mechanical motion: the temperature dependent resonance frequency and quality factor of multilayer 2D material membranes show clear anomalies near the phase transition temperature, which are correlated to anomalies in the specific heat of the materials. The observed coupling of mechanical degrees of freedom to magnetic and electronic order is attributed to thermodynamic relations that are not restricted to van der Waals materials. Nanomechanical resonators, therefore, offer the potential to characterize phase transitions in a wide variety of materials, including those that are antiferromagnetic, insulating or so thin that conventional bulk characterization methods become unsuitable.

Recent experiments have shown surprisingly large thermal time constants in suspended graphene ranging from 10 to 100 ns in drums with a diameter ranging from 2 to 7 microns. The large time constants and their scaling with diameter points towards a thermal resistance at the edge of the drum. However, an explanation of the microscopic origin of this resistance is lacking. Here, we show how phonon scattering at a kink in the graphene, e.g. formed by sidewall adhesion at the edge of the suspended membrane, can cause a large thermal time constant. This kink strongly limits the fraction of flexural phonons that cross the suspended graphene edge, which causes a thermal interface resistance at its boundary. Our model predicts thermal time constants that are of the same order of magnitude as experimental data, and shows a similar dependence on the circumference. Furthermore, the model predicts the relative in-plane and out-of-plane phonon contributions to graphene's thermal expansion force, in agreement with experiments. We thus show, that in contrast to conventional thermal (Kapitza) resistance which occurs between two different materials, in 2D materials another type of thermal interface resistance can be geometrically induced in a single material.

Recent theory has predicted large temperature differences between the in-plane (LA and TA) and out-of-plane (ZA) acoustic phonon baths in locally-heated suspended graphene. To verify these predictions, and their implications for understanding the nonequilibrium thermodynamics of 2D materials, experimental techniques are needed. Here, we present a method to determine the acoustic phonon bath temperatures from the frequency-dependent mechanical response of suspended graphene to a power modulated laser. The mechanical motion reveals two counteracting contributions to the thermal expansion force, that are attributed to fast positive thermal expansion by the in-plane phonons and slower negative thermal expansion by the out-of-plane phonons. The magnitude of the two forces reveals that the in-plane and flexural acoustic phonons are at very different temperatures in the steady-state, with typically observed values of the ratio $\Delta T_{\mathrm{LA+TA}}/\Delta T_{\mathrm{ZA}}$ between 0.2 and 3.7. These deviations from the generally used local thermal equilibrium assumption ($\Delta T_{\mathrm{LA+TA}}=\Delta T_{\mathrm{ZA}}$) can affect the experimental analysis of thermal properties of 2D materials.

Complex oxide thin films and heterostructures exhibit a profusion of exotic phenomena, often resulting from the intricate interplay between film and substrate. Recently it has become possible to isolate epitaxially grown single-crystalline layers of these materials, enabling the study of their properties in the absence of interface effects. In this work, we create ultrathin membranes of strongly correlated materials and demonstrate top-down fabrication of nanomechanical resonators made out of \ce{SrTiO3} and \ce{SrRuO3}. Using laser interferometry, we successfully actuate and measure the motion of the nanodrum resonators. By measuring their temperature-dependent mechanical response, we observe signatures of structural phase transitions in \ce{SrTiO3}, which affect the strain and mechanical dissipation in the resonators. This approach can be extended to investigate phase transitions in a wide range of materials. Our study demonstrates the feasibility of integrating ultrathin complex oxide membranes for realizing nanoelectromechanical systems on arbitrary substrates.

Current wafer-scale fabrication methods for graphene-based electronics and sensors involve the transfer of single-layer graphene by a support polymer. This often leaves some polymer residue on the graphene, which can strongly impact its electronic, thermal, and mechanical resonance properties. To assess the cleanliness of graphene fabrication methods, it is thus of considerable interest to quantify the amount of contamination on top of the graphene. Here, we present a methodology for direct measurement of the mass of the graphene sheet using quartz crystal microbalances (QCM). By monitoring the QCM resonance frequency during removal of graphene in an oxygen plasma, the total mass of the graphene and contamination is determined with sub-graphene-monolayer accuracy. Since the etch-rate of the contamination is higher than that of graphene, quantitative measurements of the mass of contaminants below, on top, and between graphene layers are obtained. We find that polymer-based dry transfer methods can increase the mass of a graphene sheet by a factor of 10. The presented mass measurement method is conceptually straightforward to interpret and can be used for standardized testing of graphene transfer procedures in order to improve the quality of graphene devices in future applications.

Stochastic switching between the two bistable states of a strongly driven mechanical resonator enables detection of weak signals based on probability distributions, in a manner that mimics biological systems. However, conventional silicon resonators at the microscale require a large amount of fluctuation power to achieve a switching rate in the order of a few Hertz. Here, we employ graphene membrane resonators of atomic thickness to achieve a stochastic switching rate of 7.8 kHz, which is 200 times faster than current state-of-the-art. The (effective) temperature of the fluctuations is approximately 400 K, which is 3000 times lower than the state-of-the-art. This shows that these membranes are potentially useful to transduce weak signals in the audible frequency domain. Furthermore, we perform numerical simulations to understand the transition dynamics of the resonator and derive simple analytical expressions to investigate the relevant scaling parameters that allow high-frequency, low-temperature stochastic switching to be achieved in mechanical resonators.