Harmonic balance analysis of tip-sample interaction and coexisting solutions in dynamic atomic force microscopy

This study theoretically and numerically investigates the oscillatory behavior of micro-cantilevers in an atomic force microscope (AFM) operating in dynamic mode, where the oscillations are influenced by interaction forces between the probe tip and the sample surface. We propose a single-degree-of-freedom mass-spring model incorporating the interaction force described by the Morse potential, and derive approximate theoretical solutions for this system. Our findings demonstrate that higher-order approximations of the nonlinear interaction force are crucial for accurately capturing the target amplitude range of probe oscillations, especially under varying quality-factor conditions. Furthermore, we validate the existence of coexisting solutions near the resonance frequency and analyze their basins of attraction as a function of the tip-sample distance. In addition, we confirm that unstable solutions exist together with the stable ones when multiple stable solutions coexist.