Nonlinear modal coupling in a high-stress doubly-clamped nanomechanical resonator

Lulla, Kunal; Cousins, Richard B.; Venkatesan, A.; Patton, M. J.; Armour, A. D.; Mellor, C.J.; Owers‐Bradley, J. R.

doi:10.1088/1367-2630/14/11/113040

Cited by 45 publications

(54 citation statements)

References 33 publications

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“…Based on Ref. [40] and simple expansions of Euler-Bernoulli theory (including non-linear coefficients [38,39,48]) we give the analytic tools enabling the calculation of the "ultimate frequency stability" reached by any doubly-clamped device, depending on stress, dimensions and temperature T [45]. For bottom-up structures like e.g.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear frequency transduction of nanomechanical Brownian motion

et al. 2017

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We report on experiments addressing the non-linear interaction between a nano-mechanical mode and position fluctuations. The Duffing non-linearity transduces the Brownian motion of the mode, and of other non-linearly coupled ones, into frequency noise. This mechanism, ubiquitous to all weaklynonlinear resonators thermalized to a bath, results in a phase diffusion process altering the motion: two limit behaviors appear, analogous to motional narrowing and inhomogeneous broadening in NMR. Their crossover is found to depend non-trivially on the ratio of the frequency noise correlation time to its magnitude. Our measurements obtained over an unprecedented range covering the two limits match the theory of Y. Zhang and M. I. Dykman, Phys. Rev. B 92, 165419 (2015), with no free parameters. We finally discuss the fundamental bound on frequency resolution set by this mechanism, which is not marginal for bottom-up nanostructures.arXiv:1704.06119v2 [cond-mat.mes-hall]

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Section: Introductionmentioning

confidence: 99%

Nonlinear frequency transduction of nanomechanical Brownian motion

et al. 2017

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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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confidence: 99%

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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“…Nonlinear modal interactions have been studied recently in micro-and nanoresonators. [13][14][15][16][17][18] These studies concentrated on mechanical coupling between the modes via the geometric nonlinearity or via the displacement-induced tension, the same mechanism responsible for the Duffing nonlinearity in doubly clamped resonators. By employing a different mode of the same resonator as a phonon cavity, the mechanical mode can be controlled in situ, and its damping characteristics can be modified to a great extent, leading to cooling of the mode and parametric mode splitting.…”

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confidence: 99%

Strong and tunable mode coupling in carbon nanotube resonators

et al. 2012

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The nonlinear interaction between two mechanical resonances of the same freely suspended carbon nanotube resonator is studied. We find that, in the Coulomb-blockade regime, the nonlinear modal interaction is dominated by single-electron-tunneling processes and that the mode-coupling parameter can be tuned with the gate voltage, allowing both mode-softening and mode-stiffening behaviors. This is in striking contrast to tension-induced mode coupling in strings where the coupling parameter is positive and gives rise to a stiffening of the mode. The strength of the mode coupling in carbon nanotubes in the Coulomb-blockade regime is observed to be 6 orders of magnitude larger than the mechanical-mode coupling in micromechanical resonators. Carbon nanotubes present remarkable properties for applications in nanoelectromechanical systems (NEMS), such as low mass density, high Young's modulus, and high crystallinity.1,2 This fact has motivated the use of carbon nanotubes to fabricate high-quality factor (Q) mechanical resonators 3 that can be operated at ultrahigh frequencies 4,5 and can be used as ultrasensitive mass sensors.6-8 Additionally, both the mechanical tension and the electrical properties of carbon nanotubes can be tuned to a large extent by an external electric field, 9 making nanotubes a very versatile component in NEMS devices.Due to the small diameter of carbon nanotubes, they can be easily excited in the nonlinear oscillation regime. 10 Moreover, it has been demonstrated that the nonlinear dynamics of carbon nanotubes can be tuned over a large range 11,12 making nanotube NEMS excellent candidates for the implementation of sensing schemes based on nonlinearity and for the study of fundamental problems on nonlinear dynamics. The nonlinear interaction between mechanical resonance modes is interesting both from a fundamental and from an applied perspective. Nonlinear modal interactions have been studied recently in micro-and nanoresonators. [13][14][15][16][17][18] These studies concentrated on mechanical coupling between the modes via the geometric nonlinearity or via the displacement-induced tension, the same mechanism responsible for the Duffing nonlinearity in doubly clamped resonators. By employing a different mode of the same resonator as a phonon cavity, the mechanical mode can be controlled in situ, and its damping characteristics can be modified to a great extent, leading to cooling of the mode and parametric mode splitting. 13,16 The nonlinear coupling can also be used to detect resonance modes that would otherwise be inaccessible by the experiment 18 to increase the dynamic range of resonators by tuning the nonlinearity constant 18 and for mechanical frequency conversion.17 Additionally, nonlinear coupling has been proposed as a quantum nondemolition scheme to probe mechanical resonators in their quantum ground state 19 and as a way of generating entanglement between different mechanical modes. 20 Furthermore, a recent theoretical paper suggests that the interaction between mechanical resonances c...

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Nonlinear modal coupling in a high-stress doubly-clamped nanomechanical resonator

Cited by 45 publications

References 33 publications

Nonlinear frequency transduction of nanomechanical Brownian motion

Nonlinear frequency transduction of nanomechanical Brownian motion

Tension-induced nonlinearities of flexural modes in nanomechanical resonators

Strong and tunable mode coupling in carbon nanotube resonators

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