Atomically thin two-dimensional crystals are prospective materials for nanoelectromechanical systems due to their extraordinary mechanical properties [1][2][3][4][5][6][7] (high Young's modulus, elasticity and breaking strength) and low mass. Among this family of 2D materials, graphene is the most studied one so far. Graphene mechanical resonators [8] have been already employed as mass and pressure sensors [9] and provide a platform to study interesting nano-mechanical phenomena such as nonlinear damping [10] .Nevertheless, the lack of a bandgap in graphene may limits its usefulness in certain applications requiring a semiconducting material. Molybdenum disulfide (MoS 2 ), a semiconducting analogue to graphene [11][12][13] , presents excellent mechanical properties similar to graphene [5, 6] in combination to a large intrinsic bandgap [14][15][16][17] . Although multilayered MoS 2 resonators have been recently fabricated by Lee et al. [18] (one device 9 layers thick, the remainder devices 20 to 100 layers thick), single layer MoS 2 mechanical resonators have not been demonstrated so far. Unlike multilayer MoS 2 , which has an indirect bandgap of 1.2 eV, monolayer MoS 2 is a direct bandgap semiconductor (1.8 eV) with potential applications in photodetection [19][20][21][22] , photovoltaics [23] and valleytronics [24][25][26] . Therefore, fabrication of single-layer MoS 2 mechanical resonators is a first and necessary step towards nano-electromechanical systems exploiting the direct bandgap of monolayer MoS 2 . Here, we demonstrate the fabrication of single-layer MoS 2 mechanical resonators. The fabricated resonators have fundamental resonance frequencies in the order of 10 MHz to 30 MHz (depending on their geometry) and their quality factor is about ~55 at room temperature in vacuum. We find that the mechanics of these single-layer resonators lies on the membrane limit (tension dominated) while multilayered MoS 2 resonators can be modeled as circular plates (bending rigidity dominated http://onlinelibrary.wiley.com/doi/10.1002/adma.201303569/abstract have found that direct exfoliation of MoS 2 onto the pre-patterned substrates yield a low density of flakes, and no suspended single-layer MoS 2 devices could be fabricated using this method. We have therefore employed an all-dry transfer technique to deposit the MoS 2 flakes onto the pre-patterned substrates, similar to the one described in [27,28] (see Experimental Section and Supporting Information for a more detailed description of the transfer method). Single-layer devices can be identified at glance by means of optical microscopy [29,30] . Confirmation of the exact layer thickness is performed by combination of atomic force microscopy, Raman spectroscopy [31] and photoluminescence [14][15][16] (see Experimental section and Supporting Information for more details). Figure 1ashows optical images of single layer MoS 2 mechanical resonators. The motion of the drum resonators is detected using an optical interferometer (see the Experimental section and the Supp...
A theoretical and experimental investigation is presented on the intermodal coupling between the flexural vibration modes of a single clamped-clamped beam. Nonlinear coupling allows an arbitrary flexural mode to be used as a self-detector for the amplitude of another mode, presenting a method to measure the energy stored in a specific resonance mode. Experimentally observed complex nonlinear dynamics of the coupled modes are quantitatively captured by a model which couples the modes via the beam extension; the same mechanism is responsible for the well-known Duffing nonlinearity in clamped-clamped beams.PACS numbers: 85.85.+j, 05.45.-a An important topic in nanomechanics is the motion detection of mechanical resonators. Several schemes have been proposed to attain sensitivities near the quantum limit of mechanical motion [1], whereas applicationdriven research is focussed on on-chip detection [2] and readout of resonator arrays [3]. Central in any detection scheme is the coupling of a mechanical resonator to another system, which transduces the motion into a measurable quantity. Examples of sensitive detectors include a single-electron transistor [4], a microwave cavity [5], or an optical interferometer [6]. A second mechanical resonator can also be used to detect the motion of the resonator [7,8]. Such a system of coupled resonators has been proposed as a quantum nondemolition detection scheme, in which one resonator is in a quantum state [9]. Coupling between different mechanical resonators is often present in large-scale integrated arrays due to electrostatic [7] and mechanical interaction [8]. Coupling between individual resonators can also lead to complex behavior, which is theoretically well-documented [10].In this Letter, we study the coupling between vibrational modes in a single beam resonator. We demonstrate that flexural modes are coupled by the displacementinduced tension in the beam. Using this coupling, the displacement of any mode can be detected by measuring the response of another mode, making otherwise undetectable modes visible. We present a general theoretical framework based on the Euler-Bernoulli equation extended with displacement-induced tension. The model quantitatively describes the complex dynamic behavior observed in the regime where two modes are simultaneously driven nonlinear. The coupling mechanism plays an prominent role in the dynamics of carbon nanotube resonators and resonators under high tension, and should be taken into account when describing such systems accurately.Experiments are performed on a single-crystalline silicon beam with dimensions L × w × h = 1000 × 35×6 µm 3 fabricated by patterning a silicon-on-insulator wafer and subsequent wet etching. The resonator is placed in a magnetic field of B = 2.1 T and a magnetomotive technique [3,11] is used to detect the mechanical motion of the beam at room temperature and atmospheric pressure (see Figure 1a). The beam is driven at multiple frequencies by sending alternating currents through a conductive aluminum path, evapo...
Membranes of suspended two-dimensional materials show a large variability in mechanical properties, in part due to static and dynamic wrinkles. As a consequence, experiments typically show a multitude of nanomechanical resonance peaks, which make an unambiguous identification of the vibrational modes difficult. Here, we probe the motion of graphene nanodrum resonators with spatial resolution using a phase-sensitive interferometer. By simultaneously visualizing the local phase and amplitude of the driven motion, we show that unexplained spectral features represent split degenerate modes. When taking these into account, the resonance frequencies up to the eighth vibrational mode agree with theory. The corresponding displacement profiles, however, are remarkably different from theory, as small imperfections increasingly deform the nodal lines for the higher modes. The Brownian motion, which is used to calibrate the local displacement, exhibits a similar mode pattern. The experiments clarify the complicated dynamic behavior of suspended two-dimensional materials, which is crucial for reproducible fabrication and applications.
We have used double clamped beams to implement a mechanical memory. Compressive stress is generated by resistive heating of the beams and beyond the buckling limit the bistable regime is accessed. Bits are written by applying lateral electrostatic forces. The state of the beam is read out by measuring the capacitance between beam and electrodes. Two ways to implement a mechanical memory are discussed: compensation of initial beam imperfections and snap through of the postbuckled beam. Although significant relaxation effects are observed, both methods prove reliable over thousands of write cycles.
We investigate the nonlinear dynamics of microcantilevers. We demonstrate mechanical stiffening of the frequency response at large amplitudes, originating from the geometric nonlinearity. At strong driving the cantilever amplitude is bistable. We map the bistable regime as a function of drive frequency and amplitude, and suggest several applications for the bistable microcantilever, of which a mechanical memory is demonstrated. © 2010 American Institute of Physics. ͓doi:10.1063/1.3511343͔Microcantilevers are widely applied as transducers in sensitive instrumentation, 1,2 with scanning probe microscopy as a clear example. Typically, the cantilever is operated in the linear regime, i.e., it is driven by a harmonic force at moderate strength, and its response is modulated by the parameter to be measured. In clamped-clamped mechanical resonators, additional applications have been proposed based on nonlinear behavior. Nonlinearity in clamped-clamped resonators is due to the extension of the beam, which results in frequency pulling and bistability at strong driving, and can be described by a Duffing equation.3 Applications which employ this bistability are, e.g., elementary mechanical computing functions. 4,5 Since a cantilever beam is clamped only at one side, it can have a nonzero displacement without extending. One would therefore not expect a Duffing-like behavior for a cantilever beam. Nonlinear effects of a different origin have been observed in scanning probe microscopy, due to interactions between the cantilever and its environment. Tipsample interactions either weaken or stiffen the cantilever response, depending on the strength of the softening Van der Waals forces and electrostatic interactions and the hardening short range interactions. 6,7 Weakening also occurs when the cantilever is driven by an electrostatic force.8 Besides nonlinear interactions with the environment, theoretical studies predict intrinsic nonlinear behavior of cantilever beams, [8][9][10][11] of which indications have been reported. 11,12 In this paper, we report a detailed experimental analysis on the nonlinear mechanics of microcantilevers. It is shown that a hardening geometric nonlinearity dominates over softening nonlinear inertia, which effectively leads to a stiffening frequency response for the fundamental mode. At large amplitudes, the mechanical stiffening results in frequency pulling and ultimately in intrinsic bistability of the cantilever. We study the bistability in detail by measuring the cantilever response as a function of the frequency and amplitude, and compare the experimental observations with theory. A good agreement is found. We suggest several applications for the bistable cantilever, and as an example we demonstrate that bit operations can be implemented in the bistable cantilever.Experiments are performed on thin cantilevers with a rectangular cross section, w ϫ h, fabricated from lowpressure chemical vapor deposited silicon nitride using electron beam lithography and an isotropic reactive ion etching release p...
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