Dynamic behavior of dagger-shaped cantilevers for atomic force microscopy

Experimental techniques based on the atomic force microscope (AFM) have been developed for characterizing mechanical properties at the nanoscale and applied to a variety of materials and structures. Atomic force acoustic microscopy (AFAM) is one such technique that uses spectral information of the AFM cantilever as it vibrates in contact with a sample. In this paper, the dynamic behaviour of AFM cantilevers that have a dagger shape is investigated using a power-series method. Dagger-shaped cantilevers have plan-view geometry consisting of a rectangular section at the clamped end and a triangular section at the tip. Their geometry precludes modelling using closed-form expressions. The convergence of the series is demonstrated and the convergence radius is shown to be related to the given geometry. The accuracy and efficiency of the method are investigated by comparison with finite element results for several different cases. AFAM experiments are modelled by including a linear spring at the tip that represents the contact stiffness. The technique developed is shown to be very effective for inversion of experimental frequency information into contact stiffness results for AFAM. In addition, the sensitivities of the frequencies to the contact stiffness are discussed in terms of the various geometric parameters of the problem including the slope, the ratio of the rectangular to triangular lengths and the tip location. Calculations of contact stiffness from experimental data using this model are shown to be very good in comparison with other models. It is anticipated that this approach may be useful for other cantilever geometries as well, such that AFAM accuracy may be improved.