A unified Lagrangian treatment of charge and cantilever dynamics in electric force microscopy

Numerous microcantilever-based electric force microscopy (EFM) protocols have been developed since the invention of scanned probe microscopy. In spite of a common physical basis, a comprehensive formalism for connecting the tip-sample forces to the cantilever dynamics in these experiments is lacking. We present a unified Lagrangian treatment of the EFM, which, significantly, reveals assumptions underlying equations that are widely-used to interpret EFM data. The force is usually stated as F = (1/2)C′ (V - Φ)², with C the tip-sample capacitance, V the tip voltage, and Φ the surface potential, while the associated frequency shift is stated as Δf = -(f₀ / 4k) C′′ (V - Φ)², with
f₀ the cantilever resonance frequency and k the cantilever spring constant. We find that these equations imply assumptions about sample charging that are seldom explicitly stated or experimentally checked. We identify the conditions under which they are valid and analyze the cantilever dissipation that arises from finite equilibration times for sample surface charging.