Modeling and experimental vibration analysis of nanomechanical cantilever active probes

Nanomechanical cantilever (NMC) active probes have recently received increased attention in a variety of nanoscale sensing and measurement applications. Current modeling practices call for a uniform cantilever beam without considering the intentional jump discontinuities associated with the piezoelectric layer attachment and the NMC cross-sectional step. This paper presents a comprehensive modeling framework for modal characterization and dynamic response analysis of NMC active probes with geometrical discontinuities. The entire length of the NMC is divided into three segments of uniform beams followed by applying appropriate continuity conditions. The characteristics matrix equation is then used to solve for system natural frequencies and mode shapes. Using an equivalent electromechanical moment of a piezoelectric layer, forced motion analysis of the system is carried out. An experimental setup consisting of a commercial NMC active probe from Veeco and a state-of-the-art microsystem analyzer, the MSA-400 from Polytec, is developed to verify the theoretical developments proposed here. Using a parameter estimation technique based on minimizing the modeling error, optimal values of system parameters are identified. Mode shapes and the modal frequency response of the system for the first three modes determined from the proposed model are compared with those obtained from the experiment and commonly used theory for uniform beams. Results indicate that the uniform beam model fails to accurately predict the actual system response, especially in multiple-mode operation, while the proposed discontinuous beam model demonstrates good agreement with the experimental data. Such detailed and accurate modeling framework can lead to significant enhancement in the sensitivity of piezoelectric-based NMC sensors for use in variety of sensing and imaging applications.