Quantitative formulation of frequency-dependent interaction force for practical applications in amplitude-modulation atomic force microscopy

Amplitude-modulation atomic force microscopy (AM-AFM) measures nanoscale surface structures by detecting changes in the cantilever oscillation amplitude, contributing to materials research. AM-AFM can nondestructively observe fragile molecules, such as biomolecules, even while the probe is in intermittent contact with the sample. However, the interaction force estimated from experimental amplitude values is significantly greater than actual molecular binding forces, despite the successful visualization of molecular dynamics, raising a fundamental question about force quantification. Here, we formulate a quantitative force conversion equation for arbitrary driving frequencies. A comprehensive theoretical analysis reveals that, when the cantilever is excited at the resonance slope, the conventional equation overestimates the actual force by approximately five times since it is valid only for excitation at the resonance frequency. This mathematical framework is validated through simulations and experiments, offering new insights into AM-AFM force estimation. The findings provide a crucial toolset for accurate yet easy-to-use force quantification, enhancing the reliability of practical AM-AFM applications in materials and biomolecular research.