Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators

Westra, H.J.R.; Poot, Menno; Zant, Herre S. J. van der; Venstra, Warner J.

doi:10.1103/physrevlett.105.117205

Cited by 190 publications

(238 citation statements)

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“…Note that the detected high-order modes were electrostatically driven by side gates with a geometry not particularly optimized for the expected mode shapes. Multiple-mode detection has recently shown to be crucial for simultaneous measurements of mass and adsorption position of biomolecules for mass-sensing applications 4 and to quantum non-demolition measurements of the amplitude, making use of the inter-modal coupling 29,30 . In addition, a linear transduction mechanism allows to perform measurements using standard electrical methods, such as network analysers, as well as its incorporation in closed-loop oscillator topologies, both of which would be impaired by quadratic transduction.…”

Section: Discussionmentioning

confidence: 99%

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators

et al. 2014

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Highly sensitive conversion of motion into readable electrical signals is a crucial and challenging issue for nanomechanical resonators. Efficient transduction is particularly difficult to realize in devices of low dimensionality, such as beam resonators based on carbon nanotubes or silicon nanowires, where mechanical vibrations combine very high frequencies with miniscule amplitudes. Here we describe an enhanced piezoresistive transduction mechanism based on the asymmetry of the beam shape at rest. We show that this mechanism enables highly sensitive linear detection of the vibration of low-resistivity silicon beams without the need of exceptionally large piezoresistive coefficients. The general application of this effect is demonstrated by detecting multiple-order modes of silicon nanowire resonators made by either top-down or bottom-up fabrication methods. These results reveal a promising approach for practical applications of the simplest mechanical resonators, facilitating its manufacturability by very large-scale integration technologies.

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Section: Discussionmentioning

confidence: 99%

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators

et al. 2014

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“…In contrast to the results discussed in Refs. [7,8],where it is not taken into account, we explicitly consider the dc deformation of the resonator. This additional aspect creates an asymmetry in our system which leads to a cubic non-linearity.…”

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Tension-induced nonlinearities of flexural modes in nanomechanical resonators

2013

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We consider the tension-induced non-linearities of mechanical resonators, and derive the Hamiltonian of the flexural modes up to the fourth order in the position operators. This tension can be controlled by a nearby gate voltage. We focus on systems which allow large deformations u(x) h compared to the thickness h of the resonator and show that in this case the third-order coupling can become non-zero due to the induced dc deformation and offers the possibility to realize radiationpressure-type equations of motion encountered in optomechanics. The fourth-order coupling is relevant especially for relatively low voltages. It can be detected by accessing the Duffing regime, and by measuring frequency shifts due to mode-mode coupling.PACS numbers: 85.85.+j Recent progress in fabricating nanomechanical resonators has shown how these systems can be used for ultrasensitive measurements of mass, force and charge [1][2][3][4]. Within the couple of past years these systems have also entered the quantum realm [5] as superpositions of vibration states and zero-point vibrations have been measured. Even though such measurements can be performed in a regime where the elastic properties of the resonators could essentially be considered as linear, the extension to non-linear conditions is well within reach of the current experimental techniques.In this paper we consider the generic non-linearities of the resonators, how these show up in measurements, and how they arise when the resonators are manipulated electronically. In general, the effect of non-linearites is twofold: on one hand they modify in an amplitudedependent way the resonant frequency of a given normal mode (Duffing self-non-linearity); on the other, they introduce a coupling between normal modes. Such nonlinearities show up in the presence of strong external driving, which allows to control the coupling of different modes or to detect their occupation numbers.Motivated by the recent advances in fabricating graphene and carbon nanotube resonators [4, 6], we concentrate especially on the regime of thin resonators where the mechanical deformation can be large compared to the resonator thickness. In this case, the major source of non-linearity is the tension induced by the deformation itself. Starting from the mechanical energy of the deformations, we derive the generic Hamiltonian of the flexural modes, including non-linearities up to the fourth order in the vibration amplitudes. In contrast to the results discussed in Refs. [7,8],where it is not taken into account, we explicitly consider the dc deformation of the resonator. This additional aspect creates an asymmetry in our system which leads to a cubic non-linearity. The dc deformation, dictating the strength of the non-linearity, is driven by a nearby gate voltage as in Fig. 1. Concentrating first on the Duffing self-non-linearity of the modes, this then allows us to derive the voltage depen- dence of the Duffing constant and show that it changes sign for a certain value of voltage that depends on mode index and the...

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“…Recently, we also presented a detailed study on the coupling mechanism between the vibrational modes in clamped-clamped resonators. 21 Here, the coupling between the modes is fully described by the displacementinduced tension. A similar analysis can be carried out for the coupling between vibration modes of a cantilever beam.…”

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Q-factor control of a microcantilever by mechanical sideband excitation

Venstra¹,

Westra²,

Zant³

2011

Applied Physics Letters

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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 Â width Â height ¼ 39 lm Â 8 lm Â 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 ...

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Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators

Cited by 190 publications

References 20 publications

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators

Tension-induced nonlinearities of flexural modes in nanomechanical resonators

Q-factor control of a microcantilever by mechanical sideband excitation

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