Simulation of dynamic modes of atomic force microscopy using a 3D finite element model

In dynamic atomic force microscopy (AFM), a cantilever is used to extract surface properties by observing the changes of its dynamic characteristics due to the tip-sample interaction. Analytical/numerical models that can accurately simulate the surface-coupled dynamics of the cantilever are essential for quantitative/qualitative understanding/explanation of measured results. In this paper, a comprehensive finite element model is developed for numerical simulation of free and surface-coupled dynamics of tip-cantilever system in dynamic modes of AFM. Representing the cantilever by three-dimensional beam elements, this model can simulate all the cantilever dynamic modes within a framework of the standard finite element method. Formulations reflecting the exact excitation mechanisms are derived for tapping mode (TM), torsional resonance (TR) mode, and lateral excitation (LE) mode. Under tip-sample interaction, the location/geometry of the tip and the tilting of the cantilever to the sample surface become important and can be easily addressed in this model. With the application of this finite element model, in TM, we show that phase shift of cantilever response and interaction force sensitively depend on in-plane mechanical properties but it is not true for cantilever vibration amplitude. The excitation of higher-order modes of vertical bending is observed in our simulation and its influence on data acquisition and interpretation is discussed. In TR and LE modes, torsion and lateral bending of the cantilever are coupled. We investigate how the torsion and lateral bending contribute to the twist of the cantilever and demonstrate that the lateral bending contribution can be ignored only if the lateral interaction is relatively weak compared to the cantilever stiffness of lateral bending. Tip eccentricity, if any, strongly affects amplitude and phase of cantilever responses and therefore cannot be ignored in TR and LE modes. The differences of TR and LE modes due to their different excitation mechanisms are also discussed. The presented model is capable of accurately simulating the cantilever dynamics under complex tip-sample interaction and might be used for quantitative extraction of surface properties. (c) 2006 Elsevier B.V. All rights reserved.