FEM-SIMULATIONS OF VIBRATIONS AND RESONANCES OF STIFF AFM CANTILEVERS

Transcription

1 FEM-SIMULATIONS OF VIBRATIONS AND RESONANCES OF STIFF AFM CANTILEVERS Kai GENG, Ute RABE, Sigrun HIRSEKORN Fraunhofer Institute for Nondestructive Testing (IZFP); Saarbrücken, Germany Phone: , Fax: ; Abstract The quantitative evaluation of atomic force acoustic microscopy images in order to determine local indentation moduli of samples requires the calculation of the vibration behavior and the resonances of the used atomic force microscope cantilevers and of the contact of the sensor tip with the sample surface. Numerical simulations were carried out by finite element methods (FEM) and compared to analytically calculated as well as to measurement results in order to investigate the applicability of the analytical models with effective geometrical data. The numerical simulations take into account the elastic coupling to the chips and the anisotropy of conventional single crystal silicon cantilevers. It was shown that the models of the shape and elasticity of the sensor tip and of its contact to the sample surface are the most critical points rather than the FEM model of the cantilever geometry. Keywords: Finite Element Method (FEM), cantilever beam, Atomic Force Microscopy (AFM), ultrasonic vibrations, resonances 1. Introduction Atomic force microscopy (AFM) [1] is a useful tool for materials characterization with a spatial resolution at a nanoscale level. The AFM sensor is a flexible micro-beam mounted on a chip with a sensor tip at its free end. Commercially available cantilevers inclusive chip are made of silicon single crystals by micromachining technology following some preferential cleavage planes. AFM and its modifications have been intensely used for obtaining high-resolution images of topography and other surface properties such as e.g. adhesion, friction, electric and magnetic forces, ferroelectric polarization, and mechanical stiffness [2] probed by the cantilever tip in contact, intermittent contact, or non-contact with the sample by means of bending or torsion of the beam. High frequency ( 100 khz 3 MHz) dynamic techniques like atomic force acoustic microscopy (AFAM) [3] or ultrasonic atomic force microscopy (UAFM) [4-5] which combine AFM with ultrasound are particularly suitable for stiffness and elastic or viscoelastic measurements. These techniques are based on contact-resonance spectroscopy, i.e. they exploit the resonance spectra of vibrating AFM cantilever beams when the tip is in contact with a sample surface. Contact-resonance vibrations of the cantilever can be excited either by a vibration of the sample surface, which in turn is excited by an ultrasonic transducer coupled to the sample (AFAM) or by a vibration of the cantilever holder (UAFM) or the cantilever itself. Caused by the forces acting between the tip and the sample surface, the contact resonance frequencies of the bending, torsional, and lateral modes of the cantilever shift relative to its free vibration spectra. If an excitation frequency near a flexural contact resonance is applied and, while scanning a sample surface, the amplitude and phase of the resulting cantilever vibration is recorded by lock-in-techniques an elastic stiffness image of the surface is obtained. The spatial resolution depends on the tip-sample contact radius, which is usually in the range from 10 to 100 nm. For quantitative

2 evaluation, complete contact-resonance spectra have to be measured [6]. The contactresonance frequencies of the cantilever provide with the tip-sample contact stiffness [3-6], which depends on the elastic indentation moduli of the tip and the sample and on the shape of contact. The quantitative evaluation procedure requires a convenient analytical or numerical model of the vibration behavior of the cantilever. Analytical models describing the free cantilever-vibration as well as the contact resonances assuming AFM cantilever beams of constant cross-section and rigid clamping at one side are well-known [7]. This approach works well to describe the overall behavior of the vibration modes. However, the resonances of most of the cantilevers show slight systematic deviations from the analytical models, and the theoretical frequencies cannot be fitted to two or more experimental free bending modes simultaneously with an error in the resonance frequencies of less than 10%. This holds especially for stiff cantilevers with spring constants of about 5 to 50 N/m, which are used for contact-resonance applications. Furthermore, the analytic description of the cantilever vibrations used so far in the literature reproduces the contact-resonance frequencies with accuracies of less than 20% [3, 6-9]. Therefore, a more precise modeling shall improve the understanding of the oscillatory behavior of the cantilever in AFAM measurements and help to quantify the tip-sample interactions. Numerical modeling using the finite element method (FEM) seems to be an effective tool for this purpose, since it allows a more detailed description of the cantilever. 2. Demands on the AFM cantilever model Several publications can be found in the literature presenting numerical models and FEM calculations of AFM cantilevers and their vibrations [e.g. 9-15], but they neglect important details in the geometric shape and/or the elastic anisotropy of conventional single crystal silicon cantilevers and/or do not consider the elasticity of the suspension of AFM cantilevers, which influences significantly the resonance frequencies. The objective of this work was to obtain an improved model for AFM cantilever vibration simulations taking into account all important characteristic features and thus allowing the prediction of contact resonance frequency shifts as a function of the tip-sample contact stiffness. This FEM approach should allow a higher precision in the determination of contact stiffness than previously achieved by using analytical models. 3. Discussion of the Results The created FEM model considers the geometrical shape of the cantilevers with a trapezoidal cross-section and a triangular free end, the cubic symmetry of silicon single crystals, and the elastic coupling of the cantilevers to the chips. Figs. 1 and 2 show scanning electron microscope (SEM) micrographs of a conventional stiff single crystal silicon cantilever and the corresponding schematic sketch of the probe geometry. For the description, a Cartesian coordinate system with the x -, y -, and z -axes in the cantilever length, thickness, and width directions, respectively, was used (Fig. 3). These axes coincide with the crystallographic axes [110], [001], and [ 110 ] of the cubic single crystal material, respectively. Fig. 3b shows the FEM model of the cantilever after

3 meshing. It comprises elements with linear dimensions of about 1.5 µm in average for both the beam and the tip. In the regions where higher strain was to be expected, the density of the grid elements was increased. For practical reasons, the length axis of AFM cantilevers inclines by an angle θ (about 11 to 15 ) towards the sample surface. The tip-sample contact forces were modeled as three springs in a coordinate system {X,Y,Z} aligned to the sample surface (Figs. 3a and b). The coordinate system {X,Y,Z} was chosen to coincide with the cantilever system {x,y,z } for θ = 0. The spring constants k N and k S are the tip-sample contact stiffness values in out-of-plane (Y-axis) and in-plane direction (X- and Z-axes), respectively [16]. The FEM model was fitted to the experiments in a two-step iterative procedure. In a first step, measured free resonance frequencies of the lowest bending, torsional, and lateral bending modes of an individual cantilever were used to fit the geometrical cantilever dimensions. Subsequently, tip length and cantilever declination were adapted to the measured contact resonance frequencies of the first and the third bending mode. The FEM model also allows a precise calculation of the spring constant of the cantilever, and consequently a calculation of the force in contact. With the obtained FEM cantilever model, the bending as well as the torsional resonances were predicted and compared to the experimental spectra. For the first three bending modes and for the first free torsional mode errors less than 1% were achieved [16]. In order to investigate the influences of the different parameters several cantilevers of similar shape with only small differences in their geometrical dimensions were calculated. A MATLAB programme was written to render possible flexible variations of individual parameters and thus the adaptation to the real cantilever behaviour. It was shown that the modelling of the shape and elasticity of the sensor tip and of its contact to the sample surface are the most critical points rather than the differences in the analytical and the more realistic FEM model of the cantilever geometry. Figure 1. SEM micrographs showing (a) a side-view of a cantilever, (b) an enlarged side-view of the end of the cantilever carrying the sensor tip, and (c) a view from the bottom side where the tip is mounted.

4 Figure 2. Schematic sketch of the probe geometry used for the FEM simulations. Figure 3. Schematic sketch of the cantilever (a) with its coordinate system {x,y,z } inclined by an angle θ relative to the sample surface coordinate system {X,Y,Z}. The tip-sample forces are modeled by three springs with spring constants k N and k S for vertical and lateral contact stiffness, respectively; (b) FEM model of the cantilever after meshing.

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