We discuss atomic force acoustic microscopy ͑AFAM͒ methods to determine quantitative values for the elastic properties of thin films. The AFAM approach measures the frequencies of an AFM cantilever's first two flexural resonances while in contact with a material. The indentation modulus M of an unknown or test material can be obtained by comparing the resonant spectrum of the test material to that of a reference material. We examined a niobium film (dϭ280Ϯ30 nm) with AFAM using two separate reference materials and two different cantilever geometries. Data were analyzed by two methods: an analytical model based on conventional beam dynamics, and a finite element method that accommodated variable cantilever cross section and viscous damping. AFAM values of M varied significantly depending on the specific experimental configuration and analysis technique.By averaging values obtained with both reference materials, very good agreement ͑5-10 % difference͒ with values determined by other methods was achieved. These results provide insight into using AFAM methods to attain reliable, accurate measurements of elastic properties on the nanoscale.
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.
The impact of tip-sample dissipation mechanisms in atomic force acoustic microscopy (AFAM) and other techniques is evident in the width (and corresponding Q-factor) of the resonances observed. The Q-factors of resonances while in contact are often several orders of magnitude lower than those for air alone. Dependence on applied load, humidity, and vibration mode have also been observed. Here, two dissipation mechanisms are discussed with application to AFAM. First, frictional losses associated with the tip-sample oscillations are examined. The cyclic energy loss that accompanies tip vibrations is calculated for the combined normal and shear forces present during AFAM measurements using an extension of a Mindlin contact model. The dependence on mode, applied load and friction coefficient, and other parameters is seen explicitly. Second, the dissipation associated with losses from viscoelastic contact is calculated using simple viscoelastic materials models that are coupled with the vibrating-beam boundary value problem. In this case, the dependence of the dissipation on sample properties and tip geometry is determined. The role of each of these mechanisms for specific tip-sample systems is discussed. The results are expected to impact quantitative aspects of nanomechanical characterization techniques. [Work supported by ARL.]
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