Atomic force microscopy cantilever simulation by finite element

Atomic force microscopy cantilever simulation by finite element methods for quantitative atomic force acoustic microscopy measurements F.J. Espinoza Beltrán Centro de Investigación y Estudios Avanzados del IPN. Unidad Querétaro, 76001 Querétaro, Qro., México; and Hamburg University of Technology, Advanced Ceramics Group, 21073 Hamburg, Germany

J. Muñoz-Saldañaa) and D. Torres-Torres Centro de Investigación y Estudios Avanzados del IPN. Unidad Querétaro, 76001 Querétaro, Qro., México

R. Torres-Martínez Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada del IPN. Unidad Querétaro, 76040, Querétaro, Qro., México

G.A. Schneider Hamburg University of Technology, Advanced Ceramics Group, 21073 Hamburg, Germany (Received 14 March 2006; accepted 10 August 2006)

Measurements of vibrational spectra of atomic force microscopy (AFM) microprobes in contact with a sample allow a good correlation between resonance frequencies shifts and the effective elastic modulus of the tip-sample system. In this work we use finite element methods for modeling the AFM microprobe vibration considering actual features of the cantilever geometry. This allowed us to predict the behavior of the cantilevers in contact with any sample for a wide range of effective tip-sample stiffness. Experimental spectra for glass and chromium were well reproduced for the numerical model, and stiffness values were obtained. We present a method to correlate the experimental resonance spectrum to the effective stiffness using realistic geometry of the cantilever to numerically model the vibration of the cantilever in contact with a sample surface. Thus, supported in a reliable finite element method (FEM) model, atomic force acoustic microscopy can be a quantitative technique for elastic-modulus measurements. Considering the possibility of tip-apex wear during atomic force acoustic microscopy measurements, it is necessary to perform a calibration procedure to obtain the tip-sample contact areas before and after each measurement.

I. INTRODUCTION

Atomic force microscopy (AFM) has become in the past years one of the most useful microscopic tools for imaging the surface topography at nanoscale level of several types of materials, whereby it is an essential technique for nanotechnology. AFM is very sensitive for measuring interaction forces between the AFM microprobe and the sample.1–3 The simplest interaction between the AFM microprobe and the sample is the mechanical contact, but it is possible to introduce in a controlled way several additional interaction forces, including electric and magnetic fields. For modulated

a)

Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2006.0379 3072

J. Mater. Res., Vol. 21, No. 12, Dec 2006

forces4 acting on the microprobe–sample contact, it is possible to increase the sensitivity of the measurement, including high-frequency excitations. Heterodyne converter procedures combined with lock-in amplifiers allow amplifying very low signals, due to the interaction between AFM microprobe and the sample, for a frequency range from some kHz to several MHz.5 Atomic force acoustic microscopy (AFAM) is a high-frequency force modulation AFM technique that provides stiffness mapping of surfaces. A piezoelectric transducer attached to the AFM microprobe holder or located in the bottom of the sample generates an ultrasonic mechanical signal when the microprobe is in contact with the sample. Vibrational spectra of the cantilever-sample system provide resonance peaks whose frequencies values are related with the geometry of the microprobe, and mechanical properties (elastic modulus, Poisson ratio) of probe and © 2006 Materials Research Society

F.J. Espinoza Beltrán et al.: AFM cantilever simulation by finite element methods for quantitative AFAM measurements

sample. An AFAM scanning image obtained for a fixed excitation frequency near some resonance shows microscopic details of the surface stiffness of the sample. In the AFAM technique, the following important aspects should be considered before experiments are performed. The cantilever stiffness must be chosen according to the sample stiffness, so that the vibration of the microprobe-sample system represents contributions of both materials. Secondly, sufficiently large static forces acting on the cantilever are necessary to discard water adherence in the contact area (or some small electrostatic charge effects) but short enough to avoid plastic deformation of the sample so that Hertz theory for contact can be used. On the other hand the condition of high applied forces may result in undesirable nanoindentation effects on the sample leading to a much more complex system to be quantified. When working with a high static deflection of the cantilever to get strong output signals an undesirable movement of the cantilever, consisting of a combination of horizontal bending and torsional deformation is present.4 These aspects have seriously limited the analytic description of the experimental cantilever vibration phenomena.5–10 For instance, Rabe et al.5 reported a comparison between experiment and theory for a rectangular cantilever in free vibration and demonstrated that the flexural vibration theory is able to describe AFM cantilever vibrations in the frequency range from 100 kHz to 10 MHz. Calculations showed that the first 4 to 10 modes can be excited with high amplitude concluding that the higher modes are well suited for sensing purposes.5 In another contribution these authors carried out quantitative image analysis of free torsional and flexural vibration modes of rectangular cantilevers (excited in the frequency range of 100 kHz to 3 MHz) using an optical Michelson heterodyne interferometer.11 A comparison of the experimental resonance frequencies of the modes to theoretical models has been undertaken. It showed that irregular geometry of the cantilever strongly influences the vibrational behavior of the cantilever. It has been demonstrated that coupling due to geometrical and mass asymmetries account for a number of resonances besides the standard flexural and torsional modes. Furthermore, these authors concluded that the geometry of the cantilevers plays a dominant role for a successful description of the vibration of the cantilever and suggested that a numerical approach, such as finite element analysis may lead to better agreement with experiment and improve the understanding of the various contributions to the cantilever eigenmodes. Few reports on finite element method (FEM) of AFM cantilever beams under free vibrations12–14 and related to force modulation microscopy12–15 have been reported in the literature. For instance Drobek et al.13 modeled by finite element analysis the V-shaped cantilever as used in force modulation microscopy to get a reliable description

of the mechanical behavior of the system. The method of overtone AFM was applied to Al–Ni–Fe quasi-crystal surfaces to distinguish between different metallurgical phases and different crystallographic orientations of the quasi-crystalline grains in an ingot.13 Further on, Arinero et al.12 used a one-dimensional finite element model for a vibrating rectangular cantilever beam in contact mode, based on a classic finite element scheme. These authors demonstrated that the mode of excitation of the beam strongly influences the cantilever’s frequency response in the contact mode.12 Thus, the AFM cantilever geometry and its elastic properties must be known to discern from the tip-sample contact-mechanics the appropriate elastic modulus of the sample. Kopycinska-Müller et al.16 performed a study with scanning electron microscopy on the changes suffered by the tip apex during AFAM measurements. They found that tip apex, which is usually assumed like a spherical hemisphere, frequently suffers wearing leading eventually to a flat shape with a larger contact area. This paper is an effort to combine in a finite element numerical model with characteristics of the free and coupled tip-sample cantilever vibrations. The irregular geometry of commercial cantilevers, its elastic modulus and density are taken into account to explore the imaging conditions in atomic force acoustic microscopy. Adhesion and friction are not introduced in the finite element model considering only a high static external force as a guarantee of the applicability of Hertz theory.

II. EXPERIMENTAL A. Characterization of the cantilever

FEM simulations were carried out using the reconstruction of the geometry of the beam and the tip of a commercial cantilever model FESP (Veeco Corporation, Santa Barbara, CA) with the nominal dimensions of 225 ␮m length, 28 ␮m width, and 3 ␮m thickness, tip radius of 10–15 nm, spring constant k* of 1–5 N/m, and resonant frequency of 75 kHz. 17 Furthermore, the Young’s modulus of the cantilever silicon material is reported to be 1.69 × 1011 N/m2, with a Poisson ratio of 0.278, and a mass density of 2330 kg/m3. The Young’s modulus Etip and the Poisson ratio vtip of silicon sensor tip are 1.3098 × 1011 N/m2 and 0.181, respectively.5 The two are different because of their different orientation. The chemical composition and real geometrical dimensions of three FESP cantilevers were characterized by an Environmental Scanning Electron Microscopy (Philips Environmental SEM-EDAX). Several micrographs from different views of the cantilever were recorded at different magnifications (400×–200×) using the secondary electron detector, with an acceleration voltage of 5 kV. Image analysis of the micrographs was done


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using the ImageJ software18 based on a previous calibration of the SEM magnifications in a 500 ␮m grid with square elements of 10 ␮m. Measurements of the free vibration of the commercial FESP cantilevers were carried out in the range from 0–4.0 MHz in an atomic force microscope (AFM) (Nanoscope IV, Dimension 3100, Veeco USA, Santa Barbara, CA).

III. RESULTS AND DISCUSSION

B. Simulation by the finite element method

The purpose of the modal analysis is to determine the natural frequencies and the overall vibrational behavior of the cantilever system both for the free vibrations and in contact with a material. The modal analysis problem is solved by means of the classic problem of eigenvalues, which is described in detail in literature. Using analysis of the harmonic response the behavior of the cantilever in a defined range of frequencies was calculated and used to obtain the amplitude and the phase of the cantilevers vibration at a given point on its surface. The maximal and minimal amplitudes are identified in the graph as the resonances and antiresonances of the cantilever, respectively. The harmonic analysis allows one to predict the vibrational spectra of the free cantilever or of the cantilever-sample interaction, and to verify some other effects of its forced vibrations. The differential equation system that represents mathematically the discretized physical model to be solved is: 关M兴

d2关x兴 dt2

+ 关K兴关x兴 = 关F兴sin共␻ t兲 ,

(1)

where [K] and [M] are the stiffness and mass matrixes and [F] is the amplitude of the external force, which is oscillating at the frequency ␻. Finite element software19 (ANSYS-ENGINEERING ANALYSIS SYSTEM, University Introductory, RELEASE 9.0) was used to build a cantilever and tip model applying 15262 elements of the type Solid95. This element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. Further on, modal and harmonic analysis were undertaken. To match the AFM conditions the simulated beam was inclined 15° relative to the sample surface. The results of the numerical simulation of the free vibration of the FESP cantilevers were eventually compared with the experimental measurements. Tip-sample contact interactions were simulated with three COMBIN14 elements. The application of these elements is represented in three dimensions with lines with the origin in the tip end. This element works as both a spring and damping to produce the different vibration modes of the cantilever in contact with a sample surface. The excitation of the cantilever was simulated by applying a harmonic force F sin (␻t) on the fixed end of the 3074

cantilever representing the piezoelectric excitation of the AFM cantilever at the chip where the cantilever is suspended. The magnitude of the force applied to the cantilever (in air) is around 50–1000 nN in the -y direction. Nonlinear behavior like plastic deformation or creep behavior of the contact problem was not simulated.

Representative scanning electron micrographs of the characterized cantilevers are shown in Fig. 1. Image analysis of these micrographs leads to a very good approximation of the dimensions of the used FESP cantilever. For instance, one of the cantilevers is 250 ␮m in length having a trapezoidal cross-section with widths of 34.7 ␮m and 19.4 ␮m, while the thickness varied between 2.9 ␮m and 3.1 ␮m. The free vibration resonance spectrum simulated by the harmonic analysis compared to the experimental measurements is shown in Fig. 2. FEM simulations of the modal analysis for the first six natural eigenmodes for a freely vibrating cantilever are shown in Fig. 3. These eigenmodes were obtained using a cantilevers thickness value of 2.919 ␮m. In Fig. 3 the modal shapes, displayed by each mode the lateral, frontal, and isometric views with a dashed line indicating the undeformed cantilever can be observed. Thus, from the FEM simulation one can unambiguously identify the type of the cantilever oscillations. Flexural oscillations occur in the (a) first, (b) second, (e) fifth, and (f) sixth resonance frequency. Moreover lateral oscillations occur in the (c) third resonance frequency, and finally torsional oscillations are present in the (d) fourth resonance frequency. The simulated resonance frequencies of these eigenmodes are also summarized in Table I. A comparison of the simulated values with the experimental measurements shows a very good matching. For instance, the first torsional mode, the second and the third flexural are all within 1% in respect to the experimental data. The remainder of the vibrational eigenmodes shows differences between 1–7% for the first flexural and lateral modes and the fourth flexural mode. These results indicate that the previous characterization of the cantilevers geometry allowed an excellent simulation of the cantilever behavior under free vibration. We also carried out simulation analysis on the influence of the different geometric parameters on the vibration varying for instance the thickness of the cantilever. It is clear that the variations in cantilever vibration are mostly correlated to the thickness since cantilever stiffness is strongly dependent on this parameter. From the harmonic analysis and an experimental spectrum of resonance of a cantilever-sample interaction, the contact stiffness, k*, can be derived. As mentioned before, to get a realistic simulation of the physical phenomenon of the contact tip-sample, we developed a model



FIG. 1. (a) Cantilever’s length, (b) tip length and cantilever thickness, (c) width of the free part of the cantilever, and (d) cantilever’s top and bottom width.

FIG. 2. Experimental and modeled free vibration modes in flexural vibration spectra. Vertical lines correspond to experimental resonance frequencies.

with normal and lateral spring as well as damping elements (COMBIN14) coupled to the tip apex. These elements interact with the tip apex producing the wellknown deflection and torsion of the cantilever in contact mode. This model was used to reproduce data reported in literature, i.e., a vibrational spectrum for cantilever in contact with glass obtained by Rabe et al.5 The cantilever geometry reported in this reference (230 ␮m length,

28 ␮m width, and 7.4 ␮m thick, as average values) was adapted to our parametric model for considering the more realistic trapezoidal geometry. In Fig. 4 a comparison of experimental data (vertical lines) and simulated flexural vibration spectra for free vibration, and tip-glass and tip-chromium interactions are presented. Simulated vibration resonances are in a good agreement with the experimental results. Free adjustment of geometrical parameters allows reproducing numerically any resonance spectrum in a similar way to analytical modeling. Experimental measurement of the effective stiffness, k* is difficult. We present a method to correlate the experimental resonance spectrum to the effective stiffness using a realistic geometry of the cantilever and model numerically the vibration of the cantilever in contact with a sample surface. For FEM simulation, we used the average parameters and maximum length of cantilever (248 ␮m) given in Ref. 5, and using the geometrical details from the optical microscope image given there. For instance, the parameters for the cantilever were: total length and rectangular lengths of 248 ␮m and 211 ␮m and upper and bottom widths 48 ␮m and 11 ␮m, respectively. The thickness for the best fitting of the free resonance values was 6.4 ␮m. A tip length of 13 ␮m was used. Figure 5 shows the resonance frequencies for the first three flexural modes as a function of effective stiffness k*. This simulation allowed us to reproduce the experimental spectra


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FIG. 3. The finite-element simulations were processed with by using 15262 elements of the type Solid95. (a) First, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth eigenmodes of the cantilever free vibration. TABLE I. Experimental and calculated resonance frequencies of a single crystal silicon cantilever with the average dimensions: 250 ␮m length, 30 ␮m width and 2.919 ␮m. The width is the mean value of the upper and the lower width of the trapezoid. For the calculations a Young’s modulus E of silicon cantilever 1.69 × 1011 N/m2, Poisson ratio of 0.278, and the mass density is 2330 kg/m3, the Young’s modulus and Poisson ratio of silicon sensor tip are 1.3098 × 1011 N/m2 and 0.181, respectively were used.6 Cantilever

First flexural mode

Second flexural mode

First lateral mode

First torsional mode

Third flexural mode

Fourth flexural mode

Experimentally observed frequency (kHz) Simulated (kHz)

65.6 67.0

421.5 422.0

697.7 649.8

1066.4 1066.4

1192.6 1183.5

2343.4 2316.8

reported in Ref. 5, both for free and contact vibration of the AFM microprobe with glass and chromium using the same cantilever geometry. From this figure it is observable that using a suitable geometry (and mechanical parameters) the reproduction of the free vibration allows also to predict with a very good approximation the cantilever vibration in contact with any sample leading to the effective stiffness, k*, for the contact between the AFM tip and glass and chromium. Intersections of horizontal lines, corresponding to resonance frequencies of free vibration (solid), tip-glass contact (dash) and tip-chromium (solid-dot), with the predicted curves allowed obtaining the values showed in Table II. For glass the average value of k* is 3.194 × 103 N/m and a standard deviation of 653 N/m, which corresponds to an error of about 20.5%. In a similar way, for chromium the effective stiffness error is about 40.3%. Please note that in Fig. 5 the resonance frequencies for mode 1 have an inherent high error for the effective stiffness higher than 1000 N/m, due to the fact that these points are in the upper flat region of the curve. Ignoring the stiffness value from mode 1 in the average of stiffness leads to more acceptable values of stiffness. Doing this, the corresponding 3076

k* values are (3.66 ± 0.03) × 103 N/m for glass and (12.47 ± 0.88) × 103 N/m for chromium being the standard deviations 0.8% and 7.0% for glass and chromium, respectively. The combined elastic modulus can be determined from the contact stiffness if the contact area between the AFM tip and sample is known. Contact areas can be obtained from a calibration procedure using a reference material. For instance in nanoindentation procedures it is of common practice to use an indirect method for determining area functions. A series of indentations at varying maximum loads on standard test specimens whose elastic moduli and Poisson ratio are known were undertaken. Since we do not have plastic deformation of the material in the contact area due to the limited stiffness of an AFAM cantilever, such a procedure can not be performed. It has been reported in the literature that for AFAM measurements the contact area changes during the measurement.16 To have a reliable method to obtain the contact area is the main challenge for AFAM measurements. A good approximation can be done by using the Hertz equations approximating the contact between a sphere and a flat surface. The elastic moduli of the samples vary from



30–90 GPa (for glass) and from 132–253 GPa (for chromium).5 Thus, using the Hertz theory,20 and considering the AFM tip as a spherical hemisphere apex with curvature R, the relation between k* and the combined elastic modulus E* is: (2) k* = 公6E*2 FR , where F is the static force applied on the cantilever. For E* the following expression holds: 3

冉

FIG. 4. Flexural vibration spectra showing a comparison between experimental and simulated cantilever and tip coupled to glass.5 The finite-element simulation was processed by using 2131 elements of the type Solid95 and by 3 elements type COMBIN14.

FIG. 5. Resonance frequencies as a function of tip-sample effective stiffness for the first flexural vibrations for the cantilever used for Rabe et al.5 Vertical lines correspond to experimental tip apex-glass and tip apex-chromium contacts. Horizontal lines correspond to: free vibration (solid lines), tip-glass contact (dash lines), and tip-chromium contact (solid-dot lines).

冊冉

冊

1 − ␷Si2 1 − ␷sample2 1 = + . (3) E* ESi Esample The ranges of values for R obtained (5.3–8.9 and 78.0– 131.9 ␮m, for glass and chromium, respectively), by using Eqs. (4) and (5) and Table II, and F ⳱ 500 nN are far away to be real for the case of a rigid AFM tip in contact with a flat rigid surface. Now, if we consider wearing produced by the AFAM measurement the tip apex could be better represented as a flatted apex. Thus, assuming a flattened circular area of the tip apex with radius a, as the contact area between tip and sample, the relation between k* and E* is k* = 2aE* . (4) Here a does not change as a function of the applied force F to the cantilever, but F should be higher than attractive or repulsive forces. The computed range of values obtained for the contact radius a are 33.0–42.8 nm for glass, and 68.5–89.0 nm for chromium. The difference in contact areas is remarkable. This difference can be explained from a lighter wear process in the former material leading to less damage of the tip. An accepted procedure to predict the mechanical properties of materials by AFAM is the use of a material reference having an elastic modulus close to the expected for the analyzed sample.21,22 Hitherto was a constant contact area assumed but it has been proven that it changes during each AFAM measurement.16 Similar results were reported by Passeri et al.23 by supposing that for normal loads higher than a critical value the cantilever tip acts as a flat punch indenting a plain surface. The continuous changing of the contact area does not allow the reproducibility of exactly the same resonance frequencies and experimental resonance spectrum. A much lower wear rate of the tip apex can be obtained for AFM tips covered by a diamond coating.24 Wearing of the contact area with the AFAM measurement should not really be a problem if there are calibration procedures

TABLE II. Effective stiffness for tip-sample (glass and chromium) contact obtained by finite element simulation. Sample

Flexural 1 k* (N/m)

Flexural 2 k* (N/m)

Flexural 3 k* (N/m)

Average k* (N/m)

Averagea k* (N/m)

Glass Chromium

2.27 × 103 2.72 × 104

3.68 × 103 1.34 × 104

3.63 × 104 1.16 × 104

(3.19 ± .65) × 103 (1.74 ± .70) × 104

(3.66 ± .03) × 103 (1.25 ± .09) × 104

a

Average of stiffness values obtained from flexural modes 2 and 3. J. Mater. Res., Vol. 21, No. 12, Dec 2006

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before and after each measurement by means of a wellknown reference material. Additionally, it is not necessary to exploit a reference with elastic modulus value near to one of the samples, but it could be more convenient to use a reference with low friction coefficient to avoid tip apex wearing during calibration. Interaction between tip-apex and sample can be certainly more complex than previously depicted following just the Hertzian contact mechanics model where the tip is assumed to be clamped to the sample. Electrostatic forces and adherence, together with some dissipative processes are also present in the tip and sample interaction, and they could be determinant in the behavior of the cantilever vibration. In these cases the well-known models JKR25 and DMT26 should be considered for describing the tip-sample contact. On the other hand, during AFAM measurements the geometry of the cantilever suffers horizontal deflection and horizontal displacements of the tip-apex even for vertical excitations. Thus, strong elastic and frictional horizontal forces could be present during the AFAM measurements. These strong horizontal forces acting on the tip could also be responsible on the apparent change of contact area between tip and sample. Further work is underway to include stepwise these interactions in the numerical analysis. IV. CONCLUSIONS

Finite element method is a numerical tool that allows vibrational analysis of cantilevers with complex geometry. Reproduction of free vibration spectra of AFM microprobes with measured geometry allowed us to predict the behavior of the cantilevers in contact with any sample for a wide range of effective tip-sample stiffness. Experimental spectra for glass and chromium were well reproduced for the numerical model leading to the determination of stiffness values for each measurement. Assuming the elastic moduli values from the range of values given in literature, it was possible to compute the range of values for the contact area. Furthermore it was proven that the contact area for each measurement was different, which is an evidence of tip apex geometry, which is wearing during measurement acting as a flat punch in contact with the plain surface. Considering a flat contact area, independent of static force, turned out more adequate for these analyses. Thus, supported in a reliable FEM model, atomic force acoustic microscopy can be a quantitative technique for elastic-modulus measurements, being necessary to perform calibration procedure to obtain the tip-sample contact areas before and after each measurement. ACKNOWLEDGMENTS

The authors thank Prof. W. Arnold for helpful discussion. This work was supported by CONACYT. 3078

F.J. Espinoza Beltrán acknowledges gratefully Alexander von Humboldt Foundation, Germany for a Georg Forster-fellowship during the course of this work.

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