Universal Theory of Dynamic Force Microscopy for Exact and Robust Force Reconstruction Using Multiharmonic Signal Analysis

Force reconstruction in dynamic force microscopy (DFM) is a nontrivial problem that requires the deconvolution of integrals. However, conventional reconstruction methods, which recover forces from single-frequency motion of the cantilever at its resonance, exhibit non-negligible error and reconstruction instability in the highly nonlinear force regime when the tip oscillates with its amplitude comparable to the decay length of the interaction. Here, we develop a theoretical platform of DFM based on multiharmonic signal analysis for exact and robust reconstruction of conservative and dissipative forces, valid for all oscillation amplitudes and entire tip-sample distances in both amplitude- and frequency-modulation atomic force microscopy. We achieve accuracy improvement by an order of magnitude for oscillation amplitudes comparable to or larger than the decay length, and by 2 orders of magnitude for smaller amplitudes at the force minimum, even in cases where conventional methods show poor accuracy (≳5%). Moreover, we obtain greater robustness with respect to the oscillation amplitude error, resulting in a fivefold increase in reconstruction precision. Our results demonstrate a fast and versatile reconstruction scheme for nanomechanical force characterization, with higher harmonics measured with sufficient signal-to-noise ratio, which provides unprecedented accuracy and stability beyond conventional methods.