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2019 | OriginalPaper | Chapter

11. Cantilevers and Detection Methods in Atomic Force Microscopy

Author : Bert Voigtländer

Published in: Atomic Force Microscopy

Publisher: Springer International Publishing

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Abstract

We consider basic requirements for force sensors and introduce a fabrication process for cantilevers. Subsequently, the most common detection method for measuring the cantilever deflection, the beam deflection method, is discussed in detail.

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previous chapter Forces Between Tip and Sample

next chapter Static Atomic Force Microscopy

The bending of the cantilever end by the angle \(\theta \) results in an angle of \(2 \theta \) for the reflected beam. Thus, the linear deflection of the reflected laser beam on the photodiode results (for small angles) as \(\Delta x = 2 \theta L\).

In dynamic AFM the primary bandwidth detecting the oscillatory motion of the cantilever is much larger than 1 kHz used in this quantitative example, however, in this case also the oscillation amplitudes to be detected exceed the pm range by far.

If the cantilever is tilted with respect to the surface by an angle \(\alpha \), the relation between the force perpendicular to the surface and the deflection perpendicular to the surface is modified [9] to \(F = k \Delta z/ \cos ^2 \alpha \). Since \(\alpha \) is usually small (in the range between \(10^{\circ }\) and \(15^{\circ }\)), this correction is small and will be neglected it in the following.

Specific effects occurring close to the kink between the two regions are discussed in Sect. 12.5.

The parameters \(\omega _0\) and Q can be obtained by measuring a resonance curve of the cantilever in response to an external excitation (frequency sweep over the resonance). Alternatively, the thermal noise spectrum can be measured, as described in the next section.

The density and viscosity for the most frequently used fluids (air and water) are: \(\rho _{\mathrm {air}} = 1.2\,\)kg/m\(^3\), \(\eta _{\mathrm {air}} = 1.85\times 10^{-5}\) kg/(m s), and \(\rho _{\mathrm {water}} = 1\times 10^3\) kg/m\(^3\), \(\eta _{\mathrm {water}} = 8.9\times 10^{-4}\) kg/(m s), respectively, under ambient conditions and at sea level [12].

Details of how to extract the noise power spectral density from the time signal without using a spectrum analyzer are given in Appendix B or [15].

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J.E. Sader, J.W.M. Chon, P. Mulvaney, Calibration of rectangular atomic force microscope cantilevers. Rev. Sci. Instrum. 70, 3967 (1999). https://doi.org/10.1063/1.1150021

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Title: Cantilevers and Detection Methods in Atomic Force Microscopy
Author: Bert Voigtländer
Publisher: Springer International Publishing
Book: Atomic Force Microscopy
Print ISBN: 978-3-030-13653-6

Electronic ISBN: 978-3-030-13654-3

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-13654-3_11