Quantitative Formulation of Average Force in Amplitude-Modulation Atomic Force Microscopy

Amplitude-modulation (tapping-mode) atomic force microscopy (AM-AFM) is a technique for obtaining surface topographic images at the atomic or molecular-scale by detecting changes in the cantilever oscillation amplitude. Since it can operate in air or liquid conditions, it has contributed to various material research fields. However, it remains unclear why the tip-sample interaction force estimated from an experimental amplitude value is substantially greater than the actual molecular binding force, despite the successful visualization of molecular dynamics. Here, we performed a theoretical analysis to tackle this question. We show that in general AM-AFM measurements, the cantilever is excited at the resonance slope whereas the conventional equation is only valid for excitation exactly at the resonance frequency. We then derive a force conversion equation for an arbitrary excitation frequency and found that the conventional equation overestimates the actual force by about five times. The theory derived here can be used for diverse AM-AFM applications, and is useful in many fields of material research.