Simulation of vibrational resonances of stiff AFM cantilevers by finite element methods

We report on the simulation of vibrational resonances of stiff atomic force microscope cantilevers made of silicon by finite element methods (FEM) for application in contact-resonance spectroscopy. The FEM model considers the cubic symmetry of silicon single crystals and the geometrical shape of the cantilevers with a trapezoidal cross section and a triangular free end. Using a two-step iterative procedure, we fitted our FEM model to the experiment. In a first step, we used the measured resonant frequencies of an individual cantilever to fit the geometrical dimensions of the cantilever beam model. In a second step, we measured the resonant frequencies of the same cantilever in contact with a sample and determined the out-of-plane and in-plane tip-sample contact stiffness values by a fitting procedure. The FEM model also allows precise calculation of the spring constant of the cantilever, and consequently calculation of the force in contact. Finally, we compared the contact stiffness values with those predicted by contact mechanics models