Frequency shifts of cantilever in AFM

The frequency shift and frequency shift image of cantilever in AFM have been studied by numerical integration of the equation of motion of cantilever for silicon tip with rutile TiO2(0 0 1) surface in UHV conditions and by the Hamaker summation method for the tip-surface interaction forces. The effects of the excitation frequency at the cantilever base and the equilibrium position of the tip on the frequency shift have been calculated and the results showed the same phenomena as those measured, e.g., the frequency shift increased dramatically or rapidly before the contact point and was then almost level off after the contact point. The effects of scanning speed and the initial closest distance of tip to the contact point have been calculated at different excitation frequencies at the cantilever base and the results showed that proper frequency shift image could be obtained either by noncontact mode at the excitation frequency slightly less than the resonance frequency of free cantilever, or by tapping mode at the excitation frequency a few times smaller than the resonance frequency of free cantilever.