Resonance frequencies of AFM cantilevers in contact with a surface

To make the forces in an Atomic Force Microscope that operates in a dynamic mode with one or multiple vibrations applied to the cantilever, quantitative, one needs to relate a change in resonance frequency of the cantilever to a specific tip-sample interaction. Due to the time dependence of the force between the tip and sample caused by the vibrations, this task is not only difficult, but in fact only possible to solve for certain limiting cases, if one follows common theoretical approaches with a Taylor expansion around the deflection point. Here, we present an analytical method for calculating the resonance frequencies of the cantilever that is valid for any tip-sample interaction. Instead of linearizing the tip-sample interaction locally, we calculate an averaged, weighted linearization taking into account all positions of the tip while vibrating. Our method bridges, therefore, the difficult gap between a free oscillating cantilever and a cantilever that is pushed infinitely hard into contact with a surface, which describes a clamped-pinned boundary condition. For a correct description of the cantilever dynamics, we take into account both the tip mass and the tip moment of inertia. Applying our model, we show that it is possible to calculate the modal response of a cantilever as a function of the tip-sample interaction strength. Based on these modal vibration characteristics, we show that the higher resonance frequencies of a cantilever are completely insensitive to the strength of the tip-sample interaction.