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Technical Briefs

# DMCMN: Experimental/Analytical Evaluation of the Effect of Tip Mass on Atomic Force Microscope Cantilever Calibration

[+] Author and Article Information
Matthew S. Allen1

Hartono Sumali

Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185hsumali@sandia.gov

Peter C. Penegor

University of Wisconsin-Madison, 250 Joanne Dr., Brookfield, WI 53005penegor@gmail.com

Note that the Bessel functions in the solution for the hydrodynamic function in Ref. 21 are Bessel functions of the second kind (hence the nomenclature K), not the third kind (Hankel functions) as stated in Sader’s paper.

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(6), 064501 (Oct 30, 2009) (10 pages) doi:10.1115/1.4000160 History: Received May 15, 2008; Revised May 07, 2009; Published October 30, 2009

## Abstract

Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology, and chemistry. Contrary to what the name suggests, the AFM actually measures the displacement of a microscale probe, so one must determine the stiffness of the probe to find the force exerted on a sample. Numerous methods have been proposed for determining the spring constant of AFM cantilever probes, yet most neglect the mass of the probe tip. This work explores the effect of the tip mass on AFM calibration using the method of Sader (1995, “Method for the Calibration of Atomic Force Microscope Cantilevers,” Rev. Sci. Instrum., 66, pp. 3789) and extends that method to account for a massive, rigid tip. One can use this modified method to estimate the spring constant of a cantilever from the measured natural frequency and $Q$-factor for any mode of the probe. This may be helpful when the fundamental mode is difficult to measure or to check for inaccuracies in the calibration obtained with the fundamental mode. The error analysis presented here shows that if the tip is not considered, then the error in the static stiffness is roughly of the same order as the ratio of the tip’s mass to the cantilever beam’s. The area density of the AFM probe is also misestimated if the tip mass is not accounted for, although the trends are different. The model presented here can be used to identify the mass of a probe tip from measurements of the natural frequencies of the probe. These concepts are applied to six low spring-constant, contact-mode AFM cantilevers, and the results suggest that some of the probes are well modeled by an Euler–Bernoulli beam with a constant cross section and a rigid tip, while others are not. One probe is examined in detail, using scanning electron microscopy to quantify the size of the tip and the thickness uniformity of the probe, and laser Doppler vibrometry is used to measure the first four mode shapes. The results suggest that this probe’s thickness is significantly nonuniform, so the models upon which dynamic calibration is based may not be appropriate for this probe.

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## Figures

Figure 7

SEM images of portions of the CSC38-B AFM cantilever shown in Fig. 1, revealing that the thickness of the cantilever is not uniform along its length

Figure 3

Experimentally measured first mode shape for two cantilever AFM probes in vacuum (o) and in air (+) and analytical mode shapes for various tip mass ratios mr. (lines).

Figure 2

Thermal spectrum of the HYDRA cantilever

Figure 1

SEM image of a CSC38-B cantilever, similar to the one discussed in Sec. 3. The shorter cantilever in the background is a CSC38-A cantilever, which is mounted on the same chip.

Figure 6

Experimentally measured fourth mode shape for a CSC38-B AFM probe in vacuum (o) and analytical mode shapes for various tip mass ratios mr (lines)

Figure 5

Experimentally measured third mode shape for a CSC38-B AFM probe in vacuum (o) and analytical mode shapes for various tip mass ratios mr (lines)

Figure 4

Experimentally measured second mode shape for two cantilever AFM probes in vacuum (o) and in air (+) and analytical mode shapes for various tip mass ratios mr (lines)

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