Transfer Function Analysis of the Micro Cantilever Used in Atomic Force Microscopy

Current approaches to study the system response of the different modes of dynamic atomic force microscopy (AFM) use model simplification such as single mode approximations or are based on the nontrivial solution of the equation of motion. As an alternative to these approaches, the transfer function analysis gives a more complete description of system dynamics considering the cantilever as an extended vibrating structure. In order to describe the dynamics correctly, the precise experimental configuration has to be taken into account. This includes distinguishing between point loads (mechanical tip sample interaction) and distributed loads induced, e.g., by inertial driving or electrostatic forces. In this work we focus on a transfer function study of two different AFM configurations, the point force and distributed force driven cantilever. Exact analytical expressions of the infinite dimensional transfer function are derived for both possible system outputs: cantilever deflection and slope, which correspond to an interferometric and a light lever deflection detection system, respectively. Frequency response and transfer function infinite product expansion are obtained for the case where system outputs are set at the free end of the cantilever. In the frequency response, it is reflected the full complexity of cantilever dynamics affected by the presence of an infinite number of poles and zeros. An analytical expression for all the zeros and poles of the system is obtained. Using the transfer function, system dynamics are analyzed obtaining modal and antiresonant cantilever shapes along with its step response. From the frequency response, pole-zero investigations, and system dynamics, it is shown that both cantilever actuation and output measurement affect AFM operation. Transfer function analysis of AFM cantilevers improves the possibility of model-based AFM operation to increase imaging and manipulation performance