Classification of dynamic atomic force microscopy control modes based on asymptotic nonlinear mechanics

Control design and improvement of dynamic atomic force microscopy (AFM) modes and development of new dynamic modes are among the central problems of AFM theory and practice. Proper design can speed up the scan, improve quality, and suggest new useful image channels and algorithms for quantitative measurements at the nanometer scale. This paper provides a rigorous modeling based on first principles and mechanical setting of the AFM. A relationship between the empirical ldquoeffectiverdquo parameters of mass-spring models and the properties of cantilever and mechanical characteristics of the AFM was formulated. KBM averaging method was used to derive asymptotic dynamics with amplitude and phase as the state variables. Two equations for steady state of this asymptotic dynamics have four unknowns - amplitude, phase, height, and frequency shift. Keeping two of four unknown variables constant (and re-solving other two by the equations for steady state) determines six dynamic AFM modes, of which four are widely used. They are amplitude modulation (AM) and frequency modulation (FM) with force spectroscopy and imaging operations. The good match between the simulated and experimental results for AM is found.