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...

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...

We demonstrate the coupling between the fundamental and second flexural modes of a microcantilever. A mechanical analogue of cavity-optomechanics is then employed, where the mechanical cavity is formed by the second vibrational mode of the same cantilever, coupled to the fundamental mode via the geometric nonlinearity. By exciting the cantilever at the sum and difference frequencies between fundamental and second flexural modes, the motion of the fundamental mode of the cantilever is damped and amplified. This concept makes it possible to enhance or suppress the Q-factor over a wide range. V C 2011 American Institute of Physics. [doi:10.1063/1.3650714] Cantilevers have numerous scientific and technological applications and are used in various instruments. In sensing applications, the sensitivity is related to the Q-factor, and this has motivated researchers to increase the Q-factor of mechanical resonators, in particular, in dissipative environments. Among the techniques that have been employed are applying residual stress, 1 parametric pumping, 2 and selfoscillation by internal 3 and external 4 feedback mechanisms. When increasing the Q-factor in these ways, energy is pumped into the mechanical mode and the resonator heats up. The opposite effect leads to cooling of the resonator and attenuation of its motion. 5 By pumping energy out of the mechanical resonator into a high quality-factor optical or microwave cavity, several groups have shown reduction of the effective temperature of the vibrational mode from room temperature to millikelvin temperatures. 6-14 Such cooling schemes are now employed to bring down the mode temperature to below an average phonon occupation number of one, providing a promising route to study the quantum behavior of a mechanical resonator. [15][16][17] In analogy to cavity optomechanics, where an optical or a microwave cavity is used to extract energy from the resonator, we employ a mechanical cavity to damp the mechanical mode. Here, the fundamental flexural mode of the cantilever is the mode of interest, and the mechanical cavity is formed by the second flexural mode of the same cantilever, which is geometrically coupled to the fundamental mode. In this paper, we demonstrate the presence of this coupling by strongly driving the cantilever on resonance, while monitoring its broadband frequency spectrum. Sidebands appear in the spectrum, which are located at the sum and difference frequencies of fundamental and second modes of the cantilever. Driving the cantilever at these sidebands results in positive or negative additional damping, which is demonstrated in this paper.Cantilevers are fabricated from low pressure chemical vapor deposited silicon nitride by electron beam lithography and isotropic reactive ion etching in a O 2 /CHF 3 plasma. 18 The dimensions are length  width  height ¼ 39 lm  8 lm  70 nm. An optical deflection technique, similar to the one employed in atomic force microscopy, is used to detect the cantilever motion. Figures 1(a) and 1(b) show the cantilever and ...

The cantilever is a prototype of a highly compliant mechanical system and has an instrumental role in nanotechnology, enabling surface microscopy, and ultrasensitive force and mass measurements. Here we report fluctuation-induced transitions between two stable states of a strongly driven microcantilever. Geometric nonlinearity gives rise to an amplitudedependent resonance frequency and bifurcation occurs beyond a critical point. The cantilever response to a weak parametric modulation is amplified by white noise, resulting in an optimum signal-to-noise ratio at finite noise intensity. This stochastic switching suggests new detection schemes for cantilever-based instrumentation, where the detection of weak signals is mediated by the fluctuating environment. For ultrafloppy, cantilevers with nanometer-scale dimensions operating at room temperature-a new transduction paradigm emerges that is based on probability distributions and mimics nature.

The effective Young's modulus of silicon nitride cantilevers is determined for thicknesses in the range of 20-684 nm by measuring resonance frequencies from thermal noise spectra. A significant deviation from the bulk value is observed for cantilevers thinner than 150 nm. To explain the observations we have compared the thickness dependence of the effective Young's modulus for the first and second flexural resonance mode and measured the static curvature profiles of the cantilevers. We conclude that surface stress cannot explain the observed behavior. A surface elasticity model fits the experimental data consistently. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3152772͔ Micro-and nanoelectromechanical systems are widely studied for their application in sensing and actuation devices. 1 Down-scaling of such devices improves their sensitivity however at the same time mechanical properties may start to deviate from the bulk behavior. The finite-size effects have been the subject of theoretical studies for the past years. [2][3][4][5][6][7][8] In experimental work on single-crystalline Si cantilevers it has been shown that the Young's modulus strongly depends on the thickness. 9 This behavior has also been observed for suspended crystalline silver nanowires. 10 In describing the properties of nanoscale devices, the bulk Young's modulus ͑E͒ is generally replaced by the effective Young's modulus ͑E eff ͒ to account for size-dependent effects, including surface stress. The total surface stress ͑⌺͒ can be written as a sum of a strain-independent part ͑ ͒ and a strain-dependent part ͑strain ⑀͒, which is related to surface elasticity ͑C s ͒ ⌺ = + C s ⑀. [11][12][13][14][15] In this letter, we study the size-dependency of the Young's modulus in silicon nitride cantilevers when one dimension ͑cantilever thickness͒ is scaled down from 684 to 20 nm. As the SiN x is amorphous, it is difficult to distinguish between the two contributions since parameters ͑e.g., C s ͒ are unknown and difficult to calculate. However, by comparing the experimental data for the first and second mode to theory, we show that the straindependent part of the total surface stress is responsible for the size-dependency.Cantilevers are fabricated from low-pressure chemical vapor deposited ͑LPCVD͒ silicon nitride ͑SiN x ͒ on Si͑100͒ substrates and are patterned with an electron-beam pattern generator. After resist development we use reactive ion etching in a CHF 3 / O 2 ͑20:1͒ plasma to transfer the pattern to the SiN x layer. After this step cantilevers are released using a KOH etch process ͑15 min etching time at 85°C; Si etch rate about 1 m / min͒, yielding facetting along the ͑111͒ planes, as shown in Fig. 1͑a͒. This process introduces a negligible undercut, so that length corrections can be disregarded. 16 Cantilevers are fabricated with the following dimensions: lengths ͑L͒ from 8 to 100 m, widths ͑w͒ 8, 12, or 17 m, and thicknesses ͑h͒ ranging from 20 to 684 nm.The thickness was measured using an ellipsometer ͑Leitz SP͒ with an accuracy of ...