Dynamic behaviour of dagger-shaped cantilevers for atomic force microscopy

Abstract

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