Frequency-modulation atomic force microscopy

Dynamic atomic force microscopy using the resonance-frequency change of an oscillating cantilever as the interaction-sensitive signal. Small-amplitude oscillations and weak force gradients give the small-amplitude AFM frequency shift; larger amplitudes require an averaged interaction analysis.

At equilibrium separation $z_0$, expanding the tip force gives $m_{\mathrm{eff}}\ddot{x}=-[c-F^{\prime }(z_0)]x$. Thus $\omega /{\omega }_0=\sqrt{1-F^{\prime }(z_0)/c}$, and Taylor expansion gives the displayed result for $|F^{\prime }|\ll c$. The displacement convention takes positive $z$ away from the surface. A positive gradient of an attractive force softens the resonance; a large-amplitude oscillation samples more than the local gradient.