Dynamic behaviour of dagger-shaped cantilevers for atomic force microscopy

Abstract

Experimental techniques based on the atomic force microscope (AFM) have been developed for characterizing mechanical properties at the nanoscale and applied to a variety of materials and structures. Atomic force acoustic microscopy (AFAM) is one such technique that uses spectral information of the AFM cantilever as it vibrates in contact with a sample. In this paper, the dynamic behaviour of AFM cantilevers that have a dagger shape is investigated using a power-series method. Dagger-shaped cantilevers have plan-view geometry consisting of a rectangular section at the clamped end and a triangular section at the tip. Their geometry precludes modelling using closed-form expressions. The convergence of the series is demonstrated and the convergence radius is shown to be related to the given geometry. The accuracy and efficiency of the method are investigated by comparison with finite element results for several different cases. AFAM experiments are modelled by including a linear spring at the tip that represents the contact stiffness. The technique developed is shown to be very effective for inversion of experimental frequency information into contact stiffness results for AFAM. In addition, the sensitivities of the frequencies to the contact stiffness are discussed in terms of the various geometric parameters of the problem including the slope, the ratio of the rectangular to triangular lengths and the tip location. Calculations of contact stiffness from experimental data using this model are shown to be very good in comparison with other models. It is anticipated that this approach may be useful for other cantilever geometries as well, such that AFAM accuracy may be improved.

Acoustic Scanning Probe Microscopy: An Overview

“…From the simplest model where the tip-sample contact is modeled as an elastic spring of constant k * and the cantilever mass is assumed to be concentrated in a single point, subsequent improvements have introduced the description of the cantilever as a beam with distributed mass, the tip not placed at the very end of the beam, nonzero tip height, cantilever and tip inclination, normal damping at the contact by a dashpot γ * in parallel with k * [78,79], and the effect of lateral forces by a parallel lateral contact stiffness k lat and lateral dashpot γ lat [76,77]. Finally, a nonuniform cantilever cross section along the axis can be taken into account [80][81][82][83]. Such improvement in the models is fundamental in particular for AFAM and UAFM, which use standard AFM setups, while it is a less pressing requirement for SMM, which uses ad hoc designed probes.…”

“…Such improvement in the models is fundamental in particular for AFAM and UAFM, which use standard AFM setups, while it is a less pressing requirement for SMM, which uses ad hoc designed probes. The simpler models permit analytical solution, while the more comprehensive ones may require approximate solution or finite element methods (FEM) [80][81][82][83][84][85].…”

A new investigation for double tapered atomic force microscope cantilevers by considering the damping effect

“…In the last few years, many researchers have had a growing interest in investigating the dynamic behavior of the AFM cantilever [2][3][4][5][6][7][8][9]. Turner and Wiehn [2] have studied the sensitivity of the vibration modes of the rectangular AFM cantilevers and derived a closed-form expression.…”

“…Wu et al [5] have investigated about the effect of tip length and normal and lateral contact stiffness on the flexural vibration responses of atomic force microscope cantilevers, by ignoring the effects of contact position and angle between cantilever and sample surface. Shen et al [6] have studied dynamic behavior of dagger-shaped AFM cantilever, using a power-series method and ignoring the effect of angle between cantilever and sample surface. Lee et al [7] have studied the flexural sensitivity of a V shaped cantilever of an atomic force microscope, taking into account the angle between the cantilever and the surface without considering the tip dimensions, assuming that the tip located at the end of the cantilever.…”

Sensitivity and resonant frequency of an AFM with sidewall and top-surface probes for both flexural and torsional modes

“…He found that the effect of the cantilever slope is considerable. Shen et al [8] derived the flexural sensitivity of the dagger-shaped cantilever of atomic force microscope using power series and investigated the influence of various design parameters on sensitivity of this kind of AFM. Finally, they compared their analytical results with the results obtained by the finite element method.…”