Renormalization, resonance bifurcation, and phase contrast in dynamic atomic force microscopy

Renormalization of the model describing dynamic atomic force microscopy is shown to provide a simple and robust interpretation of cantilever dynamics as a single spring and mass with frequency-dependent cantilever stiffness and damping parameters. Renormalization predicts a bifurcation in the free-space cantilever resonance that leads to the occurrence of multiple stable resonance modes experimentally observed during cantilever-sample “contact.” The bifurcation results from the coupling of the cantilever modes via the nonlinearity of the tip-sample interaction force and the running of the cantilever parameters with frequency. The effective interaction force is represented by a polynomial expansion with coefficients Fij (i,j = 0, 1, 2, …) that account for cantilever-to-sample energy transfer in a single system model. The effective cantilever spring constant obtained from F10 and the interaction force energy transfer factor obtained from F01 are used to show that phase contrast in the linear regime of operation can be expressed in terms of conservative or dissipative force parameters alone when operating in constant amplitude control near the free-space resonance frequency of the cantilever. The model predicts that dissipative force parameters dominate phase contrast at low drive frequencies, while conservative force parameters dominate phase contrast at sufficiently high drive frequencies for appropriate values of F10.