The amplitude phase dynamics and fixed points in tapping-mode atomic force microscopy

The first harmonic of the cantilever deflection in the tapping-mode operation of an atomic force microscope (AFM) is analyzed using asymptotic methods for weakly nonlinear oscillators. The resulting amplitude and phase dynamical equations are obtained which characterize the transient behavior of tapping-mode dynamics. The steady state behavior is analyzed by examining the fixed points of the amplitude phase dynamics and a simple stability criterion is obtained. Further with a simple tip-sample interaction model, the experimentally observed discontinuous jumps in the amplitude versus tip-sample separation plots are explained and the regions of the interaction regime probed by the tip are investigated.