Vibrational dynamics of force microscopy: Effect of tip dimensions

The dynamics of a vibrating cantilever with an attached tip in contact with a solid is treated analytically. The tip length is shown to be crucial in determining the resonant response. The finite tip size changes the boundary conditions for the flexural motion, rendering the cantilever-tip-sample combination more rigid and implicating both the normal and lateral stiffnesses of the sample in the analysis. This is confirmed in an experiment with a silica sample, a sapphire tip, and a silicon cantilever. The theory has implications in the field of quantitative analysis with atomic ac force microscopy.