Finite-Element Simulation of Cantilever Vibrations in Atomic Force Acoustic Microscopy

Atomic Force Acoustic Microscopy has been proven to be a powerful technique for materials characterization with nanoscale lateral resolution. This technique allows one to obtain images of elastic properties of materials. By means of spectroscopic measurements of the tip-sample contact-resonance frequencies, it is possible to obtain quantitative values of the mechanical stiffness of the sample surface. For quantitative analysis a reliable relation between the spectroscopic data and the contact stiffness is required based on a correct geometrical model of the cantilever vibrations. This model must be precise enough for predicting the resonance frequencies of the tip-sample interaction when excited over a wide range of frequencies. Analytical models have served as a good reference for understanding the vibrational behavior of the AFM cantilever. They have certain limits, however, for reproducing the tip-sample contact-resonances due to the cantilever geometries used. For obtaining the local elastic modulus of samples, it is necessary to know the tip-sample contact area which is usually obtained by a calibration procedure with a reference sample. In this work we show that finite-element modeling may be used to replace the analytical inversion procedure for AFAM data. First, the three first bending modes of cantilever resonances were used for finding the geometrical dimension of the cantilever employed. Then the normal and in-plane stiffness of the sample were obtained for each measurement on the surface to be measured. A calibration was needed to obtain the tip position of the cantilever by making measurements on a sample with known surface elasticity, here crystalline silicon. The method developed in this work was applied to AFAM measurements on silicon, zerodur, and strontium titanate.