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 Si ₃N ₄ 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. (Figure presented.)

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Ultrasound is widely used in medical imaging, and emerging photo-acoustic imaging is crucial for disease diagnosis. Currently, high-end photo-acoustic imaging systems rely on piezo-electric materials for detecting ultrasound waves, which come with sensitivity, noise, and bandwidth limitations. Advanced applications demand a large matrix of broadband, high-resolution, and scalable ultrasound sensors. Silicon photonic circuits have been introduced to meet these requirements by detecting ultrasound-induced deformation and stress in silicon waveguides. Although higher sensitivities could facilitate the exploration of new applications, the high stiffness of the waveguide materials constrains the intrinsic sensitivity of the silicon photonic circuits to ultrasound signals. Here, we explore the impact of the mechanical properties of a polymer cladding on the sensitivity of silicon photonic ultrasound sensors. Our model and experiments reveal that optimizing the polymer cladding's stiffness enhances the resonance wavelength sensitivity. Experimentally, we show a fourfold increase in the sensitivity compared to the sensors without a cladding polymer and, a twofold sensitivity increase compared to the sensors with a cladding polymer of saturated cross-linking density. Interestingly, comparing experiments with the optomechanical model suggests that the change in Young's Modulus alone cannot explain the sensitivity increase. In conclusion, polymer-coated silicon photonic ultrasound sensors exhibit potential for advanced photo-acoustic imaging applications. It offers the prospect of increasing the ultrasound detection sensitivity of silicon photonic ultrasound sensors while using CMOS-compatible processes. This paves the way to integrate the polymer-coated silicon photonic ultrasound sensors with electronics to utilize the sensors in advanced medical imaging applications.

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The unique properties of two-dimensional (2D) materials bring great promise to improve sensor performance and realise novel sensing principles. However, to enable their high-volume production, wafer-scale processes that allow integration with electronic readout circuits need to be developed. In this perspective, we review recent progress in on-chip 2D material sensors, and compare their performance to the state-of-the-art, with a focus on results achieved in the Graphene Flagship programme. We discuss transfer-based and transfer-free production flows and routes for complementary metal-oxide-semiconductor integration and prototype development. Finally, we give an outlook on the future of 2D material sensors, and sketch a roadmap towards realising their industrial and societal impact.@en

Nanomechanical resonances of two-dimensional (2D) materials are sensitive probes for condensedmatter physics, offering new insights into magnetic and electronic phase transitions. Despite extensive research, the influence of the spin dynamics near a 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 FePS3-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 Néel temperature TN. 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.@en

Acoustic levitation is an attractive and versatile technique that offers several advantages in terms of particle size, range, reconfigurability, and ease of use with respect to alternative levitating techniques. In this paper, we study the use of active damping to improve the response time and positioning precision of an acoustic levitator operating in air. We use a laser Doppler vibrometer to measure the velocity of a levitated particle. Using this information, a control algorithm is designed and implemented to provide active damping. By system identification and modeling, we demonstrate that the active damper mechanism is well-predictable by models and can be electronically reconfigured and controlled.

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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 14× 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−1. 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.@en

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 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 manufacturing process to create ultra-large graphene drums with diameters of up to 1000 μm. We solve the inverse problem of a Föppl–von Kármán 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 found that the reconstructed stress distributions are bi-axial, which likely originates from the transfer process. The introduced methodology 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.

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Multi-material direct ink writing (DIW) of smart materials opens new possibilities for manufacturing complex-shaped structures with embedded sensing and actuation capabilities. In this study, DIW of UV-curable piezoelectric actuators is developed, which do not require high-temperature sintering, allowing direct integration with structural materials. Through particle size and ink rheology optimization, the highest d₃₃^*g₃₃ piezoelectric constant compared to other DIW fabricated piezo composites is achieved, enabling tunable actuation performance. This is used to fabricate ultrasound transducers by printing piezoelectric vibrating membranes along with their support structures made from a structural ink. The impact of transducer design and scaling up transducer dimensions on the resonance behavior to design millimeter-scale ultrasound transducers with desired out-of-plane displacement is explored. A significant increase in output pressure with increasing membrane dimensions is observed. Finally, a practical application is demonstrated by using the printed transducer for accurate proximity sensing using time of flight measurements. The scalability and flexibility of the reported DIW of piezo composites can open up new advancements in biomedical, human-computer interaction, and aerospace fields.

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Most microphones detect sound-pressure-induced motion of a membrane. In contrast, we introduce a microphone that operates by monitoring sound-pressure-induced modulation of the air compressibility. By driving a graphene membrane at resonance, the gas, that is trapped in a squeeze-film beneath it, is compressed at high frequency. Since the gas-film stiffness depends on the air pressure, the resonance frequency of the graphene is modulated by variations in sound pressure. We demonstrate that this squeeze-film microphone principle can be used to detect sound and music by tracking the membrane’s resonance frequency using a phase-locked loop. The squeeze-film microphone potentially offers advantages like increased dynamic range, lower susceptibility to pressure- induced failure and vibration-induced noise over conventional devices. Moreover, microphones might become much smaller, as demonstrated in this work with one that operates using a circular graphene membrane with an area that is more than 1000 times smaller than that of MEMS microphones. @en

As a consequence of their high strength, small thickness, and high flexibility, ultrathin graphene membranes show great potential for pressure and sound sensing applications. This study investigates the performance of multi-layer graphene membranes for microphone applications in the presence of air-loading. Since microphones need a flatband response over the full audible bandwidth, they require a sufficiently high mechanical resonance frequency. Reducing membrane thickness facilitates meeting this bandwidth requirement, and therefore, also allows increasing compliance and sensitivity of the membranes. However, at atmospheric pressure, air-loading effects can increase the effective mass, and thus, reduce the bandwidth of graphene and other 2D material-based microphones. To assess the severity of this performance-limiting effect, we characterize the acoustic response of multi-layer graphene membranes with a thickness of 8 nm in the pressure range from 30 to 1000 mbar, in air and helium environments. A bandwidth reduction by a factor ∼ 2.8 × for membranes with a diameter of 500 μm is observed. These measurements show that air-loading effects, which are usually negligible in conventional microphones, can lead to a substantial bandwidth reduction in ultrathin graphene microphones. With analytical and finite element models, we further analyze the performance limits of graphene microphones in the presence of air-loading effects.

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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.

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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, from both scientific and application points of view. However, few techniques are available for the 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 the values reported in the literature for the same materials. Our work provides an optomechanical method for determining the thermal properties of ultrathin suspended membranes, which are difficult to measure otherwise. It provides a route toward improving our understanding of heat transport in the 2D limit and facilitates engineering of 2D structures with a dedicated thermal performance.

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Although strain engineering and soft-clamping techniques for attaining high Q-factors in nanoresonators have received much attention, their impact on nonlinear dynamics is not fully understood. In this study, we show that nonlinearity of high-Q Si₃N₄ nanomechanical string resonators can be substantially tuned by support design. Through careful engineering of support geometries, we control both stress and mechanical nonlinearities, effectively tuning nonlinear stiffness of two orders of magnitude. Our approach also allows control over the sign of the Duffing constant resulting in nonlinear softening of the mechanical mode that conventionally exhibits hardening behavior. We elucidate the influence of support design on the magnitude and trend of the nonlinearity using both analytical and finite element-based reduced-order models that validate our experimental findings. Our work provides evidence of the role of soft-clamping on the nonlinear dynamic response of nanoresonators, offering an alternative pathway for nullifying or enhancing nonlinearity in a reproducible and passive manner.

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The ultimate isolation offered by levitation provides new opportunities for studying fundamental science and realizing ultra-sensitive floating sensors. Among different levitation schemes, diamagnetic levitation is attractive because it allows stable levitation at room temperature without a continuous power supply. While the dynamics of diamagnetically levitating objects in the linear regime are well studied, their nonlinear dynamics have received little attention. Here, we experimentally and theoretically study the nonlinear dynamic response of graphite resonators that levitate in permanent magnetic traps. By large amplitude actuation, we drive the resonators into nonlinear regime and measure their motion using laser Doppler interferometry. Unlike other magnetic levitation systems, here we observe a resonance frequency reduction with amplitude in a diamagnetic levitation system that we attribute to the softening effect of the magnetic force. We then analyze the asymmetric magnetic potential and construct a model that captures the experimental nonlinear dynamic behavior over a wide range of excitation forces. We also investigate the linearity of the damping forces on the levitating resonator, and show that although eddy current damping remains linear over a large range, gas damping opens a route for tuning nonlinear damping forces via the squeeze-film effect.@en

Correction to: Microsystems & Nanoengineering https://doi.org/10.1038/s41378-024-00656-x published online 21 February 2024 After publication of this article¹, it was brought to our attention that two pressure values were not correctly copied from the submitted original work to the published version. Correction 1 (from PDF, Page 4 of 9): “These membranes show resonance frequencies above the audible range (f₀₁ > 20 kHz) at 1 × 10³ mbar by piezo-shaker actuation”. The described phrase needs to be changed reporting the right pressure value of 1 × 10⁻³ mbar. The new phrase will be: “These membranes show resonance frequencies above the audible range (f₀₁ > 20 kHz) at 1 × 10⁻³ mbar by piezo-shaker actuation”. Correction 2 (from PDF, Page 4 of 9): “Energy losses and dampening are minimized due to the low pressure of 1 × 10³ mbar”. Again, the described phrase needs to be changed reporting the right pressure value of 1 × 10⁻³ mbar. The new phrase will be: “Energy losses and dampening are minimized due to the low pressure of 1 × 10⁻³ mbar”.

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Since the performance of micro-electro-mechanical system (MEMS)-based microphones is approaching fundamental physical, design, and material limits, it has become challenging to improve them. Several works have demonstrated graphene’s suitability as a microphone diaphragm. The potential for achieving smaller, more sensitive, and scalable on-chip MEMS microphones is yet to be determined. To address large graphene sizes, graphene-polymer heterostructures have been proposed, but they compromise performance due to added polymer mass and stiffness. This work demonstrates the first wafer-scale integrated MEMS condenser microphones with diameters of 2R = 220–320 μm, thickness of 7 nm multi-layer graphene, that is suspended over a back-plate with a residual gap of 5 μm. The microphones are manufactured with MEMS compatible wafer-scale technologies without any transfer steps or polymer layers that are more prone to contaminate and wrinkle the graphene. Different designs, all electrically integrated are fabricated and characterized allowing us to study the effects of the introduction of a back-plate for capacitive read-out. The devices show high mechanical compliances C_m = 0.081–1.07 μmPa⁻¹ (10–100 × higher than the silicon reported in the state-of-the-art diaphragms) and pull-in voltages in the range of 2–9.5 V. In addition, to validate the proof of concept, we have electrically characterized the graphene microphone when subjected to sound actuation. An estimated sensitivity of S_1kHz = 24.3–321 mV Pa⁻¹ for a V_bias = 1.5 V was determined, which is 1.9–25.5 × higher than of state-of-the-art microphone devices while having a ~9 × smaller area. (Figure presented.).

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Resonators based on two-dimensional (2D) materials have exceptional properties for application as nanomechanical sensors, which allows them to operate at high frequencies with high sensitivity. However, their performance as nanomechanical sensors is currently limited by their low quality ( Q )-factor. Here, we make use of micro-electromechanical systems (MEMS) to apply pure in-plane mechanical strain, enhancing both their resonance frequency and Q-factor. In contrast to earlier work, the 2D material resonators are fabricated on the MEMS actuators without any wet processing steps using a dry-transfer method. A platinum clamp, which is deposited by electron beam-induced deposition, is shown to be effective in fixing the 2D membrane to the MEMS and preventing slippage. By in-plane straining the membranes in a purely mechanical fashion, we increase the tensile energy, thereby diluting dissipation. This way, we show how dissipation dilution can increase the Q -factor of 2D material resonators by 91%. The presented MEMS actuated dissipation dilution method does not only pave the way toward higher Q -factors in resonators based on 2D materials, but also provides a route toward studies of the intrinsic loss mechanisms of 2D materials in the monolayer limit.

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Motion is all around us, the universe is full of moving matter and this motion is surprisingly predictable. The field of science and engineering that studies time-dependent motion in the presence of forces is called Dynamics. In this book we will introduce the core concepts in dynamics and provide a comprehensive toolset to predict and analyse planar 2D motion of point masses and rigid bodies. The material includes kinematic analysis, Newton’s laws, Euler’s laws, the equations of motion, work, energy, impulse and momentum. Vector-based methods are discussed for systematically solving essentially any problem in 2D dynamics. The book provides a bachelor level introduction for any science and engineering student that can serve as a basis for more advanced courses in dynamics.@en

Semiconducting van der Waals magnets exhibit a rich physical phenomenology with different collective excitations, as magnons or excitons, that can be coupled, thereby offering new opportunities for optoelectronic, spintronic, and magnonic devices. In contrast with the well-studied van der Waals magnets CrI₃ or Fe₃GeTe₂, CrPS₄ is a layered metamagnet with a high optical and magnon transport anisotropy. Here, the structural anisotropy of CrPS₄ above and below the magnetic phase transition is investigated by fabricating nanomechanical resonators. A large anisotropy is observed in the resonance frequency of resonators oriented along the crystalline a- and b-axis, indicative of a lattice expansion along the b-axis, boosted at the magnetic phase transition, and a rather small continuous contraction along the a-axis. This behavior in the mechanical response differs from that previously reported in van der Waals magnets, as FePS₃ or CoPS₃, and can be understood from the quasi-1D nature of CrPS₄. The results pinpoint CrPS₄ as a promising material in the field of low-dimensional magnetism and show the potential of mechanical resonators for unraveling the in-plane structural anisotropy coupled to the magnetic ordering that, in a broader context, can be extended to studying structural modifications in other 2D materials and van der Waals heterostructures.

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This paper outlines the first transfer-free multi-layer (ML-gr) graphene-based MEMS condenser microphone with a high aspect-ratio suitable for sensitive miniaturized portable microphones. This fabrication flow avoids thick polymers needed to realize suspended graphene-polymer heterostructures over air cavities, which generally limit the response by introducing higher mass and stiffness. Several devices are characterized both mechanically and electrically, showing pull-in voltages up to 9.5 V and resonance frequency limited bandwidths of up to 127 kHz. The work paves the way to realize next-generation graphene microphones with a 4.5x area reduction compared to current state-of-the-art MEMS microphones.@en