Theory of Q-Controlled dynamic force microscopy in air

Abstract

The theory of dynamic force microscopy in air is developed with respect to the application of the Q-Control technique, which allows to increase or decrease the effective Q factor of the cantilever via an active external feedback. Analytical as well as numerical approaches are applied to solve the equation of motion describing the cantilever dynamics with and without Q-Control in the presence of a model tip-sample interaction force. Based on this analysis, the characteristics of Q-Controlled dynamic force microscopy are compared to conventional dynamic force microscopy carried out in amplitude modulation mode without active Q-Control (“tapping mode”). In the case of negligible tip-sample interaction (i.e., with the tip “far” from the surface), the theory describes how Q-Control alters the shape of the resonance curves of the cantilever by modifying the effective Q factor and shifting the resonance peak. Explicit consideration of tip-sample forces then permits insight into the imaging properties of Q-Controlled dynamic force microscopy. In particular, it is found that an increased Q factor prevents the oscillating cantilever to jump into a repulsive imaging regime during tip-sample approach, which often occurs during conventional tapping mode imaging in air. The analysis reveals in detail which parts of the tip-sample force curve are contributing to the contrast formation for the different imaging conditions if plotted as a function of the tip-sample distance. Based on these findings, we conclude that the restriction of the maximal tip-sample force to specific parts of the attractive regime, which is triggered by the activation of the Q-Control feedback, is the main reason for the enhanced imaging quality reported in several experimental studies if compared to conventional tapping mode imaging without Q-Control.