Resonance frequency analysis for surface-coupled atomic force microscopy cantilever in ambient and liquid environments

Shifts in the resonance frequencies of surface-coupled atomic force microscope (AFM) probes are used as the basis for the detection mechanisms in a number of scanning probe microscopy techniques including atomic force acoustic microscopy (AFAM), force modulation microscopy, and resonance enhanced piezoresponse force microscopy (PFM). Here, we analyze resonance characteristics for AFM cantilever coupled to surface in liquid environment, and derive approximate expressions for resonant frequencies as a function of vertical and lateral spring constant of the tip-surface junction. This analysis provides a simplified framework for the interpretation of AFAM and PFM data in ambient, liquid, and vacuum environments.