Relations between interaction force and frequency shift in large-amplitude dynamic force microscopy

Large-amplitude dynamic force microscopy based on measuring shifts of the resonance frequency of the force sensor has proved to be a powerful imaging tool. General expressions relating arbitrary interaction forces to resonance frequency shifts are derived using variational methods and Fourier expansion of the tip motion. This framework can be applied to tapping force microscopy which operates explicitly in the repulsive contact force regime as well as to noncontact force microscopy. In particular, it is shown that for interactions with a range much shorter than the vibration amplitude, the frequency shift can be expressed in terms of a convolution product involving the interaction force and a weakly divergent kernel. The convolution can be inverted, thus enabling one to recover unequivocally interaction potentials and forces from measured frequency shift data. It is shown in an example from noncontact force microscopy that in fact short-range repulsive interactions play a crucial role in atomic resolution imaging.