Dynamic properties of AFM cantilevers and the calibration of their spring constants

A hybrid method is introduced for the calibration of the spring constants of atomic force microscopy cantilevers. It is based on the minimization of the difference between the modelled and experimentally determined full-field displacement maps of the surface of the cantilever in motion at several resonant frequencies. The dynamic mechanical response of the cantilever to periodic motion is measured in a vacuum by means of a scanning vibrometer. Given the dimensions of the cantilever, the obtained surface displacements together with analytical or numerical models are used to resolve the physical unknowns of the probe. These are the elastic properties of the cantilever, and the residual stress state built up during the deposition of the reflective coating on the backside of the cantilever. The scanning vibrometry experiment allows the precise determination of the first ten resonant frequencies and the modes associated. After optimization of the elastic properties and the surface stress, the relative agreement between all resonant frequencies is better than 1% with the finite element model and 2% with the Timoshenko beam equation. The agreement between surface displacements is also excellent when the damping constant of the system has been determined, except for the first lateral mode, which exhibits strong coupling to a reflection of the first torsional mode. Because all the displacements at resonance are known, it is possible to decouple these modes, and the result is shown to compare well with the model. The cantilever being fully characterized ( geometry, materials, residual stress state and boundary conditions), it is straightforward to deduce all its spring constants, in the linear and nonlinear elastic regimes.