Nonlinear dynamics of microcantilevers in tapping mode atomic force microscopy: A comparison between theory and experiment

Lee, S. I.; Howell, Stephen W.; Raman, Arvind; Reifenberger, R.

doi:10.1103/physrevb.66.115409

Cited by 226 publications

(174 citation statements)

References 21 publications

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“…23 If nanowires and nanotubes are used in AFM-type tapping mode, nonlinearities in tipsurface interaction become important as well. 24 We have shown that for large aspect ratio resonators, one is forced to work close to the nonlinear regime or even in it, a rather undesirable situation for using nanoresonators as linear sensors. This new nonlinear regime that promises to dominate the nanoscale beyond the conventional dynamic range, however, might offer as yet unexplored opportunities for noise reduction and signal enhancement in nanoresonators.…”

Section: ͑9͒mentioning

confidence: 99%

Dynamic range of nanotube- and nanowire-based electromechanical systems

Postma

Kozinsky

Husain

et al. 2005

Applied Physics Letters

268

223

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Nanomechanical resonators with high aspect ratio, such as nanotubes and nanowires are of interest due to their expected high sensitivity. However, a strongly nonlinear response combined with a high thermomechanical noise level limits the useful linear dynamic range of this type of device. We derive the equations governing this behavior and find a strong dependence ͓ϰd ͱ ͑d / L͒ 5 ͔ of the dynamic range on aspect ratio. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1929098͔The limits of mechanical detection with nanoelectromechanical systems ͑NEMS͒ are being actively pursued for sensing applications, such as the attainment of sub-attonewton force sensing for magnetic resonance force microscopy, 1 sub-attogram mass sensing, 2,3 mechanical single spin detection, 4 or the study of mechanical motion approaching the quantum regime. [5][6][7][8] Applications like these require both high responsivity and ultra-high-frequency operation. 9 Both can be attained simultaneously with small diameter, large aspect ratio doubly clamped resonators. Nanoscale materials such as carbon nanotubes or nanowires are a natural choice for these resonators due to their intrinsic small size. We recently reported a bottom-up nanomechanical resonator, a Pt nanowire, and found that it takes a very low driving power to bring this device into the nonlinear regime. 10 Here, we show how the onset of this nonlinear regime decreases with decreasing diameter, while the thermomechanical noise increases with aspect ratio. We conclude that the useful linear dynamic range of such devices is severely limited, with the result that many applications will involve operation in the nonlinear regime.A typical layout for a doubly clamped nanomechanical resonator is shown in Fig. 1. The resonator can be driven and detected in several ways, e.g., magnetomotively, 11 or optically. 12 The driving force f͑t͒ leads to a time dependent bending profile z͑x , t͒, which can be found by solving the differential equationwith boundary conditions z͑0͒ = z͑L͒ = z x ͑0͒ = z x ͑L͒ = 0. Here, A is the cross-sectional area, E is Young's modulus, is the density, and I is the moment of inertia about the longitudinal axis of the beam. The term in between brackets describes tension in the beam, and is a sum of residual tension T 0 and a bending-induced tension, respectively. Since Eq. ͑1͒ cannot be solved exactly we use the Galerkin discretization procedure, 13 representing the solution to Eq. ͑1͒ in terms of a linearly independent set of basis functions n ͑x͒ where each basis function satisfies the boundary conditions. The error associated with this approximation technique is ͑2͒The Galerkin procedure requires this error to be orthogonal to each basis function, or in other words, the error is a residual that cannot be expressed in terms of the given finite set of basis functions:We are interested in the response of the beam at resonance when the first mode is dominant, so it suffices to consider the case n = 1. For a doubly clamped beam, the simplest function that approximate...

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Section: ͑9͒mentioning

confidence: 99%

Dynamic range of nanotube- and nanowire-based electromechanical systems

Postma

Kozinsky

Husain

et al. 2005

Applied Physics Letters

268

223

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“…This is a challenging problem in the sense that the mode shapes change as the AFM probe passes from one operating phase to another. In this paper we limit the test functions used in the discretization process to the unforced mode shapes of the system, which is classically the case when modeling AFM systems [4,2].…”

Section: Nonlinear Dynamic Analysismentioning

confidence: 99%

“…where d = Z − h is the tip-sample approach and τ = ML 4 EI is a time constant, M = ρA c and c is the nondimensional viscous damping coefficient. In the nondimensional form, the dot and prime denote the derivatives with respect to the nondimensional time t and the nondimensional space x, respectively.…”

Section: Problem Formulationmentioning

confidence: 99%

Nonlinear Dynamical analysis of an AFM tapping mode microcantilever beam

Manoubi

Najar

Choura

et al. 2012

MATEC Web of Conferences

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Abstract. We focus in this paper on the modeling and dynamical analysis of a tapping mode atomic force microscopy (AFM) microcantilever beam. This latter is subjected to a harmonic excitation of its base displacement and to Van der Waals and DMT contact forces at its free end. For AFM design purposes, we derive a mathematical model for accurate description of the AFM microbeam dynamics. We solve the resulting equations of motions and associated boundary conditions using the Galerkin method. We find that using one-mode approximation in tapping mode operating in the neighborhood of the contact region one-mode approximation may lead to erroneous results.

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“…We demonstrate that the bifurcation diagram and phase portraits of the system are very different from those of an impact oscillator. The more general criterion of using the AFM tip displacement [8,12] to tell whether the contact occurs is also adopted in this study.…”

Section: Introductionmentioning

confidence: 99%

“…Since Burham et al [6] first linked the AFM intermittent contact dynamics to an impact oscillator model developed by Pippard for a pin hitting a loudspeaker [7], many theories/models have been developed [3,4,[8][9][10][11][12][13]. In those studies [3,4,[8][9][10][11][12][13], either one degree-of-freedom (DOF) model of a spring-mass system as shown in Fig. 1 or the single mode analysis is applied to the AFM cantilever of a continuous system.…”

Section: Introductionmentioning

confidence: 99%

Multi-modal analysis on the intermittent contact dynamics of atomic force microscope

Zhang

Murphy

2011

Journal of Sound and Vibration

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a b s t r a c tA multi-modal analysis on the intermittent contact between an atomic force microscope (AFM) with a soft sample is presented. The intermittent contact induces the participation of the higher modes into the motion and various subharmonic motions are shown. The AFM tip mass enhances the coupling of different modes. The AFM tip mass is modeled by the Dirac delta function and the coupling effects are analyzed via the Galerkin method. The necessity of applying multi-modal analysis to the intermittent contact problem is demonstrated. Unlike the impact oscillator model which assumes the impact/contact time is infinitesimal, the contact time can be a significant fractional portion in each cycle, especially for the soft sample case and thus results in different dynamic behavior from that of an impact oscillator.

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Nonlinear dynamics of microcantilevers in tapping mode atomic force microscopy: A comparison between theory and experiment

Cited by 226 publications

References 21 publications

Dynamic range of nanotube- and nanowire-based electromechanical systems

Dynamic range of nanotube- and nanowire-based electromechanical systems

Nonlinear Dynamical analysis of an AFM tapping mode microcantilever beam

Multi-modal analysis on the intermittent contact dynamics of atomic force microscope

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