Cantilever dynamics in heterodyne force microscopy

Experiments in Heterodyne Force Microscopy (HFM) show the possibility to image deeply buried nanoparticles below a surface. However, the contrast mechanism and the motion of the cantilever, which detects the subsurface signal, are not yet understood. We present a numerical study of the cantilever motion in different HFM modes using realistic tip-sample interactions. The results provide information on the sensitivity to the heterodyne signal. The parameters in our calculations are chosen as closely as possible to the situation in real experiments to enable (future) comparisons based on our predictions. In HFM both the tip and the sample are excited at slightly different ultrasonic frequencies such that a difference frequency is generated that can contain subsurface information. We calculate the amplitude and phase of the difference frequency generated by the motion of the cantilever. The amplitude shows a local maximum in the attractive Van-der-Waals regime and an even higher plateau in the repulsive regime. The phase shifts 180° or 90°, depending on the mode of operation. Finally, we observe oscillations in both the amplitude and the phase of the difference frequency, which are caused by a shift of the resonance frequency of the cantilever and an involved transient behavior.