Equivalent point-mass models of continuous atomic force microscope probes

Melcher, John; Hu, Shuiqing; Raman, Arvind

doi:10.1063/1.2767173

Cited by 131 publications

(124 citation statements)

References 24 publications

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“…At first glance this prediction is remarkable since intuition might guide us to expect an infinite equivalent stiffness for the second eigenmode only when the tip mass approaches infinity, as in Ref. 17. The current theory on the other hand suggests this singular condition can be achieved with finite tip mass.…”

Section: A Fundamental and Higher Eigenmodesmentioning

confidence: 92%

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Higher-order eigenmodes of qPlus sensors for high resolution dynamic atomic force microscopy

Tung

Wutscher

Martínez-Martín

et al. 2010

Journal of Applied Physics

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The time response of tuning-fork based sensors can be improved by operating them at higher eigenmodes because a measurement takes at least one oscillation cycle in dynamic force microscopy and the oscillation period of the second eigenmode is only about one sixth of the fundamental mode. Here we study the higher-order eigenmodes of quartz qPlus sensors ͓Bettac et al., Nanotechnology 20, 264009 ͑2009͒; Giessibl and Reichling, Nanotechnology 16, S118 ͑2005͒; Giessibl, Appl. Phys. Lett. 76, 1470 ͑2000͒; and Giessibl, Appl. Phys. Lett. 73, 3956 ͑1998͔͒, their equivalent stiffness, and piezoelectric sensitivity, while paying special attention to the influence of the mass and rotary inertia of the sensing tip which is attached to the end of the qPlus quartz cantilever. A combination of theoretical modeling and scanning laser Doppler vibrometry is used to study the eigenmodes of qPlus sensors with tungsten tips. We find that the geometry of tungsten tips can greatly influence the shape, equivalent stiffness, and piezoelectric sensitivity of the second eigenmode of the quartz cantilever. At a critical tip length it is possible to theoretically achieve infinite equivalent stiffness and infinite piezoelectric sensitivity when the tip becomes a perfect node of vibration and beyond this critical tip length the second eigenmode loses its vibration node and the trajectory of the tip reverses with respect to the beam curvature. The findings have major implications for optimizing tip geometry for high-resolution imaging with qPlus sensors using higher eigenmodes.

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Section: A Fundamental and Higher Eigenmodesmentioning

confidence: 92%

“…The equivalent stiffness of a particular eigenmode is calculated by equating the strain energy of that eigenmode, V = 1 2 ͐ 0 L EI͓q͑t͒ ,xx i ͑x͔͒ 2 dx, to the potential energy of a pointmass oscillator, V = 1 2 k eq q͑t͒ 2 . 17 Using the nondimensional quantity x = x / L, we obtain…”

Section: A Fundamental and Higher Eigenmodesmentioning

confidence: 99%

Higher-order eigenmodes of qPlus sensors for high resolution dynamic atomic force microscopy

Tung

Wutscher

Martínez-Martín

et al. 2010

Journal of Applied Physics

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“…eq /k c =317), while the equivalent mass is independent of eigenmode [56]. The damping is unchanged for higher eigenmodes; C j eq = c. …”

Section: Extension To Multimode Harmonic Oscillatorsmentioning

confidence: 88%

“…In this limit, the equivalent mass M j eq , equivalent stiffness K j eq , and equivalent damping C j eq must be identified. The equivalent stiffness and equivalent mass, are defined as [56] …”

Section: Extension To Multimode Harmonic Oscillatorsmentioning

confidence: 99%

Vibrational spectra of nanowires measured using laser doppler vibrometry and STM studies of epitaxial graphene : an LDRD fellowship report.

Biedermann¹

2009

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A few of the many applications for nanowires are high-aspect ratio conductive atomic force microscope (AFM) cantilever tips, force and mass sensors, and high-frequency resonators. Reliable estimates for the elastic modulus of nanowires and the quality factor of their oscillations are of interest to help enable these applications. Furthermore, a real-time, non-destructive technique to measure the vibrational spectra of nanowires will help enable sensor applications based on nanowires and the use of nanowires as AFM cantilevers (rather than as tips for AFM cantilevers).Laser Doppler vibrometry is used to measure the vibration spectra of individual cantilevered nanowires, specifically multiwalled carbon nanotubes (MWNTs) and silver gallium nanoneedles. Since the entire vibration spectrum is measured with high frequency resolution (100 Hz for a 10 MHz frequency scan), the resonant frequencies and quality factors of the nanowires are accurately determined. Using Euler-Bernoulli beam theory, the elastic modulus and spring constant can be calculated from the resonance frequencies of the oscillation spectrum and the dimensions of the nanowires, which are obtained from parallel SEM studies. Because the diameters of the nanowires 3 studied are smaller than the wavelength of the vibrometer's laser, Mie scattering is used to estimate the lower diameter limit for nanowires whose vibration can be measured in this way. The techniques developed in this thesis can be used to measure the vibrational spectra of any suspended nanowire with high frequency resolution Two different nanowires were measured-MWNTs and Ag 2 Ga nanoneedles. Measurements of the thermal vibration spectra of MWNTs under ambient conditions showed that the elastic modulus, E, of plasma-enhanced chemical vapor deposition (PECVD) MWNTs is 37±26 GPa, well within the range of E previously reported for CVD-grown MWNTs. Since the Ag 2 Ga nanoneedles have a greater optical scattering efficiency than MWNTs, their vibration spectra was more extensively studied. The thermal vibration spectra of Ag 2 Ga nanoneedles was measured under both ambient and low-vacuum conditions. The operational deflection shapes of the vibrating Ag 2 Ga nanoneedles was also measured, allowing confirmation of the eigenmodes of vibration. The modulus of the crystalline nanoneedles was 84.3±1.0 GPa.Gas damping is the dominate mechanism of energy loss for nanowires oscillating under ambient conditions. The measured quality factors, Q, of oscillation are in line with theoretical predictions of air damping in the free molecular gas damping regime. In the free molecular regime, Q gas is linearly proportional to the density and diameter of the nanowire and inversely proportional to the air pressure. Since the density of the Ag 2 Ga nanoneedles is three times that of the MWNTs, the Ag 2 Ga nanoneedles have greater Q at atmospheric pressures. Our initial measurements of Q for Ag 2 Ga nanoneedles in low-vacuum (10 Torr) suggest that the intrinsic Q of these nanoneedles may be on the order of 1000....

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“…where the eigenmode shape was normalized 23 such that ψ 1 (L) = 1. Now, the effective force felt by the tip apex can be described by the thermal force spectrum F T (ω), in units of N 2 /H z, whose expectation value is given by the fluctuation-dissipation theorem 24,25 as…”

Section: Determining the Cantilever Transfer Function From A Thermal mentioning

confidence: 99%

Comparison of photothermal and piezoacoustic excitation methods for frequency and phase modulation atomic force microscopy in liquid environments

et al. 2011

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In attempting to perform frequency modulation atomic force microscopy (FM-AFM) in liquids, a non-flat phase transfer function in the self-excitation system prevents proper tracking of the cantilever natural frequency. This results in frequency-and-phase modulation atomic force microscopy (FPM-AFM) which lies in between phase modulation atomic force microscopy (PM-AFM) and FM-AFM. We derive the theory necessary to recover the conservative force and damping in such a situation, where standard FM-AFM theory no longer applies. Although our recovery procedure applies to all cantilever excitation methods in principle, its practical implementation may be difficult, or even impossible, if the cantilever is driven piezoacoustically. Specifically, we contrast the piezoacoustic excitation method to the photothermal method in the context of force spectroscopy of hydration structures at the mica-water interface. The results clearly demonstrate that photothermal excitation is superior to piezoacoustic excitation, as it allows for accurate quantitative interpretation of the acquired data

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Equivalent point-mass models of continuous atomic force microscope probes

Cited by 131 publications

References 24 publications

Higher-order eigenmodes of qPlus sensors for high resolution dynamic atomic force microscopy

Higher-order eigenmodes of qPlus sensors for high resolution dynamic atomic force microscopy

Vibrational spectra of nanowires measured using laser doppler vibrometry and STM studies of epitaxial graphene : an LDRD fellowship report.

Comparison of photothermal and piezoacoustic excitation methods for frequency and phase modulation atomic force microscopy in liquid environments

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