Nonlinear contact resonance spectroscopy in atomic force microscopy

There are several techniques which combine atomic force microscopy with ultrasonics. In atomic force acoustic microscopy the cantilever is forced to ultrasonic vibrations while the tip is in contact with a sample surface. The various physical forces acting between the tip and the surface depend nonlinearly on the distance. Therefore linear approximations are restricted to tip-sample displacements covering small parts of the interaction force curve. This situation is realized for example at high static loads and small amplitudes of the cantilever and the sample surface vibrations. The nonlinearity of the forces becomes noticeable by generation of higher harmonics in the spectrum of the cantilever vibration. A frequency dependent transfer function can be derived from the model of the cantilever as a beam with constant cross-section using the differential equation for flexural vibrations. By multiplying the transfer function with experimental spectra the nonlinear contact and adhesion forces were calculated as a function of time. In situ measurements of the cantilever and the sample vibrations by a calibrated heterodyne Mach-Zehnder interferometer are presented. This allowed us to reconstruct the tip-sample interaction forces as a function of tip-sample distance