Theory of higher harmonics imaging in tapping-mode atomic force microscopy

The periodic impact force induced by tip-sample contact in a tapping mode atomic force microscope (AFM) gives rise to the non-harmonic response of a micro-cantilever. These non-harmonic signals contain the full characteristics of tip-sample interaction. A complete theoretical model describing the dynamical behaviour of tip-sample system was developed in this paper. An analytic formula was introduced to describe the relationship between time-varying tip-sample impact force and tip motion. The theoretical analysis and numerical results both show that the time-varying tip-sample impact force can be reconstructed by recording tip motion. This allows for the reconstruction of the characteristics of the tip-sample force, like contact time and maximum contact force. It can also explain the ability of AFM higher harmonics imaging in mapping stiffness and surface energy variations.