Secondary resonances in tapping mode atomic force microscopy

This work investigates a dual-frequency excitation scheme for tapping-mode atomic force microscopy (AFM) and demonstrates, through numerical simulations, that it can significantly enhance nanoscale compositional mapping of heterogeneous surfaces compared to conventional single-frequency operation. The AFM cantilever is modeled as an Euler–Bernoulli beam with a nonlinear tip–sample interaction and reduced using a two-mode Galerkin approach. Rather than driving the cantilever solely near its first resonance, we apply two distinct excitations whose difference frequency is tuned close to that resonance. Due to nonlinear intermodulation at the intermittent tip–sample contact, this scheme generates a strong secondary (combination) resonance without directly exciting the cantilever at its natural frequency. The amplitude of this secondary resonance is shown to be highly sensitive to nanoscale variations in both long-range attractive forces (characterized by the Hamaker constant) and short-range repulsive contact forces (described by the effective elastic modulus of the sample). By contrast, the conventional single-frequency tapping response exhibits only weak variations under the same parameter changes. These results indicate that tapping can function as an intrinsic nonlinear amplifier of material contrast, enabling the simultaneous acquisition of high-resolution topography and spatially resolved functional contrast (chemical/mechanical). The proposed method therefore provides a pathway toward quantitative nanoscale property mapping—such as adhesion and stiffness—without requiring continuous high-force contact with the surface.