Cantilever dynamics in higher-harmonic atomic force microscopy for enhanced material characterization

Recently a new microcantilever design was introduced to achieve higher-harmonic nonlinear atomic force microscopy (AFM) by utilizing the benefits of intentional nonlinearity in design (Jeong et al., 2016). This microcantilever was designed with an inner Si paddle so that the linearized frequencies of the leading-order bending modes of the base beam and the inner paddle are in 1:3 rational relationship. In this case the strong, non-smooth nonlinearities generated during the AFM tapping operation via the tip-sample interaction trigger an 1:3 internal resonance in the system and a strong 3rd harmonic component in the paddle response. The experimental AFM measurement using this new cantilever system has shown that the 3rd harmonic signal amplified through internal resonance provides stronger sensitivity to material composition compared with the 1st harmonic signal. In this study, we theoretically analyze the dynamics of this new cantilever design to better relate the third harmonic measurements to the material properties of the sample. By tuning the material properties in the numerical model based on the Derjaguin-Muller-Toporov (DMT) contact model, we were able to qualitatively predict the experimental results which validates the theoretical model used for this new cantilever design. The theoretical results, suggest that there may coexist two stable branches in the steady-state dynamics depending on the material properties; the two branches produce dissimilar trends in the amplitude of the third harmonic and phases of the first and third harmonics but very similar trends in the amplitude of the first harmonic. This suggests that in addition to the phase of the first harmonic, the amplitude and phase of the third harmonic may be utilized as channels with which one can qualitatively measure material properties. More importantly, the theoretical study also reveals a strong linear correlation between the amplitude of the third harmonic and the average tip-sample interaction force, which is employed to develop an empirical model relating these two quantities. Since the conservative DMT tip-sample interaction force is directly related to elastic properties of the sample, this indicates the potential for the third harmonic to be used in the construction of enhanced, qualitative compositional maps of elastic properties.