A novel scanning probe microscope methodology has been developed that employs an ultrasonic wave launched from the bottom of a sample while the cantilever of an atomic force microscope, driven at a frequency differing from the ultrasonic frequency by the fundamental resonance frequency of the cantilever, engages the sample top surface. The nonlinear mixing of the oscillating cantilever and the ultrasonic wave in the region defined by the cantilever tip-sample surface interaction force generates differencefrequency oscillations at the cantilever fundamental resonance. The resonance-enhanced difference-frequency signals are used to create images of embedded nanoscale features.PACS numbers: 68.37. Tj, 82.35.Np, https://ntrs.nasa.gov/search.jsp?R=20080007508 2018-05-12T16:13:39+00:00Z

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A comprehensive analytical model of the interaction of the cantilever tip of the atomic force microscope (AFM) with the sample surface is developed that accounts for the nonlinearity of the tip-surface interaction force. The interaction is modeled as a nonlinear spring coupled at opposite ends to linear springs representing cantilever and sample surface oscillators. The model leads to a pair of coupled nonlinear differential equations that are solved analytically using a standard iteration procedure.Solutions are obtained for the phase and amplitude signals generated by various acoustic-atomic force microscope (A-AFM) techniques including force modulation microscopy, atomic force acoustic microscopy, ultrasonic force microscopy, heterodyne force microscopy, resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM), and the commonly used intermittent contact mode

Renormalization of the model describing dynamic atomic force microscopy is shown to provide a simple and robust interpretation of cantilever dynamics as a single spring and mass with frequency-dependent cantilever stiffness and damping parameters. Renormalization predicts a bifurcation in the free-space cantilever resonance that leads to the occurrence of multiple stable resonance modes experimentally observed during cantilever-sample “contact.” The bifurcation results from the coupling of the cantilever modes via the nonlinearity of the tip-sample interaction force and the running of the cantilever parameters with frequency. The effective interaction force is represented by a polynomial expansion with coefficients Fij (i,j = 0, 1, 2, …) that account for cantilever-to-sample energy transfer in a single system model. The effective cantilever spring constant obtained from F10 and the interaction force energy transfer factor obtained from F01 are used to show that phase contrast in the linear regime of operation can be expressed in terms of conservative or dissipative force parameters alone when operating in constant amplitude control near the free-space resonance frequency of the cantilever. The model predicts that dissipative force parameters dominate phase contrast at low drive frequencies, while conservative force parameters dominate phase contrast at sufficiently high drive frequencies for appropriate values of F10.

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