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.

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The operating principle of Pirani pressure sensors is based on the pressure dependence of a suspended strip's electrical conductivity, caused by the thermal conductance of the surrounding gas which changes the Joule heating of the strip. To realize such sensors, not only materials with high temperature dependent electrical conductivity are required, but also minimization of the suspended strip dimensions is essential to maximize the responsivity and minimize the power consumption. Due to this, nanomaterials are especially attractive for this application. Here, we demonstrate the use of a multi-layer suspended graphene strip as a Pirani pressure sensor and compare its behavior with existing models. A clear pressure dependence of the strip's electrical resistance is observed, with a maximum relative change of 2.75% between 1 and 1000 mbar and a power consumption of 8.5 mW. The use of graphene enables miniaturization of the device footprint by 100 times compared to state-of-the-art. Moreover, miniaturization allows for lower power consumption and/or higher responsivity and the sensor's nanogap enables operation near atmospheric pressure that can be used in applications such as barometers for altitude measurement. Furthermore, we demonstrate that the sensor response depends on the type of gas molecules, which opens up the way to selective gas sensing applications. Finally, the graphene synthesis technology is compatible with wafer-scale fabrication, potentially enabling future chip-level integration with readout electronics.@en

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 µm 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 minuscule. 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.

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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.

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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 μm. The large time constants and their scaling with diameter points toward 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 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 an unconventional thermal boundary resistance which occurs solely due to strong deformations within a two-dimensional material.@en

Recent theory has predicted large temperature differences between the in-plane [longitudinal (LA) and transverse (TA)] and out-of-plane [flexural (ZA)] acoustic phonon baths in locally heated suspended graphene. To verify these predictions, and their implications for understanding the nonequilibrium thermodynamics of two-dimensional (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 ΔTLA+TA/ΔTZA between 0.2 and 3.7. These deviations from the generally used local thermal equilibrium assumption (ΔTLA+TA=ΔTZA) can affect the experimental analysis of the thermal properties of 2D materials.@en

Porous, atomically thin graphene membranes have interesting properties for filtration and sieving applications. Here, graphene membranes are used to pump gases through nanopores using optothermal forces, enabling the study of gas flow through nanopores at frequencies above 100 kHz. At these frequencies, the motion of graphene is closely linked to the dynamic gas flow through the nanopore and can thus be used to study gas permeation at the nanoscale. By monitoring the time delay between the actuation force and the membrane mechanical motion, the permeation time-constants of various gases through pores with diameters from 10–400 nm are shown to be significantly different. Thus, a method is presented for differentiating gases based on their molecular mass and for studying gas flow mechanisms. The presented microscopic effusion-based gas sensing methodology provides a nanomechanical alternative for large-scale mass-spectrometry and optical spectrometry based gas characterisation methods.

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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.

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Diamond is a highly desirable material for state-of-the-art micro-electromechanical (MEMS) devices, radio-frequency filters and mass sensors, due to its extreme properties and robustness. However, the fabrication/integration of diamond structures into Si-based components remain costly and complex. In this work, a lithography-free, low-cost method is introduced to fabricate diamond-based micro-resonators: a modified home/office desktop inkjet printer is used to locally deposit nanodiamond ink as ∅50–60 µm spots, which are grown into ≈1 µm thick nanocrystalline diamond film disks by chemical vapor deposition, and suspended by reactive ion etching. The frequency response of the fabricated structures is analyzed by laser interferometry, showing resonance frequencies in the range of ≈9–30 MHz, with Q-factors exceeding 10⁴, and (f₀ × Q) figure of merit up to ≈2.5 × 10¹¹ Hz in vacuum. Analysis in controlled atmospheres shows a clear dependence of the Q-factors on gas pressure up until 1 atm, with Q ∝ 1/P. When applied as mass sensors, the inkjet-printed diamond resonators yield mass responsivities up to 981 Hz fg⁻¹ after Au deposition, and ultrahigh mass resolution up to 278 ± 48 zg, thus outperforming many similar devices produced by traditional top-down, lithography-based techniques. In summary, this work demonstrates the fabrication of functional high-performance diamond-based micro-sensors by direct inkjet printing.

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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 the direct measurement of the mass of the graphene sheet using quartz crystal microbalances (QCMs). 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.

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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 4.1 kHz, which is 100 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 use analytical expressions to investigate the relevant scaling parameters that allow high-frequency, low-temperature stochastic switching to be achieved in mechanical resonators.@en

Worlds first graphene-based Pirani pressure sensor is presented. Due to the decreased area and low thickness, the graphene-based Pirani pressure sensor allows for low power applications down to 0.9 mW. Using an innovative, transfer-free process, suspended graphene beams are realized. This allows for up to 100x miniaturization of the pressure sensor area, while enabling wafer-scale fabrication. The response of the miniaturized pressure sensor is similar to that of the much larger state-of-the-art Si-based Pirani pressure sensors, demonstrating the potential of graphene-based Pirani sensors.@en

We measure the thermal time constants of suspended single-layer molybdenum disulfide drums by their thermomechanical response to a high-frequency modulated laser. From this measurement, the thermal diffusivity of single-layer MoS2 is found to be 1.14×10-5m2/s on average. Using a model for the thermal time constants and a model assuming continuum heat transport, we extract thermal conductivities at room temperature between 10 to 40Wm-1K-1. Significant device-to-device variation in the thermal diffusivity is observed. Based on a statistical analysis we conclude that these variations in thermal diffusivity are caused by microscopic defects that have a large impact on phonon scattering but do not affect the resonance frequency and damping of the membrane's lowest eigenmode. By combining the experimental thermal diffusivity with literature values of the thermal conductivity, a method is presented to determine the specific heat of suspended 2D materials, which is estimated to be 255±104Jkg-1K-1 for single-layer MoS2.

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In the field of nanomechanics, parametric excitations are of interest since they can greatly enhance sensing capabilities and eliminate cross-talk. Above a certain threshold of the parametric pump, the mechanical resonator can be brought into parametric resonance. Here we demonstrate parametric resonance of suspended single-layer graphene membranes by an efficient opto-thermal drive that modulates the intrinsic spring constant. With a large amplitude of the optical drive, a record number of 14 mechanical modes can be brought into parametric resonance by modulating a single parameter: The pre-tension. A detailed analysis of the parametric resonance allows us to study nonlinear dynamics and the loss tangent of graphene resonators. It is found that nonlinear damping, of the van der Pol type, is essential to describe the high amplitude parametric resonance response in atomically thin membranes.

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Micro and nanomechanical sensors are indispensable in modern consumer electronics, automotive and medical industries. Gas pressure sensors are currently the most widespread membrane-based micromechanical sensors. By reducing their size, their unit costs and energy consumption drops, making them more attractive for integration in new applications. Reducing the size requires the membrane to be as thin as possible, but also very strong. Graphene is the perfect material for such a membrane since it is only one atom thick but also the strongest material ever measured. This dissertation investigates the dynamics of suspended graphene membranes for sensing applications. These sensing applications are not restricted to pressure sensors alone, but the dynamics of graphene can also be used as a sensor for other physical properties. Thus, the topic of this thesis goes into the broader subject of the dynamics of interacting graphene membranes. @en

We present suspended graphene pressure sensors fabricated using an innovative surface micro-machining process. The great advantage of this process is that the molybdenum (Mo) catalyst layer for multi-layer graphene chemical vapor deposition (CVD) is also used as a sacrificial layer to suspend the graphene. This method allows for accurate control of the gap size under the beam by simply varying the catalyst thickness. Furthermore, the need for transfer of the graphene layer is eliminated. Using this method, wafer-scale graphene squeeze-film pressure sensors are fabricated and characterized.@en

The thermal response of graphene is expected to be extremely fast due to its low heat capacity and high thermal conductivity. In this work, the thermal response of suspended single-layer graphene membranes is investigated by characterization of their mechanical motion in response to a high-frequency modulated laser. A characteristic delay time τ between the optical intensity and mechanical motion is observed, which is attributed to the time required to raise the temperature of the membrane. We find, however, that the measured time constants are significantly larger than the predicted ones based on values of the specific heat and thermal conductivity. In order to explain the discrepancy between measured and modeled τ, a model is proposed that takes a thermal boundary resistance at the edge of the graphene drum into account. The measurements provide a noninvasive way to characterize thermal properties of suspended atomically thin membranes, providing information that can be hard to obtain by other means.

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Contactless characterization of mechanical resonances using Fabry-Perot interferometry is a powerful tool to study the mechanical and dynamical properties of atomically thin membranes. However, amplitude calibration is often not performed or only possible by making assumptions on the device parameters such as its mass or the temperature. In this work, we demonstrate a calibration technique that directly measures the oscillation amplitude by detecting higher harmonics that arise from nonlinearities in the optical transduction. Employing this technique, we calibrate the resonance amplitude of two-dimensional nanomechanical resonators, without requiring knowledge of their mechanical properties, actuation force, geometric distances, or the laser intensity.

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