Lagrangian and impedance spectroscopy treatments of electric force microscopy

TL;DR

This paper develops a rigorous theoretical framework using Lagrangian mechanics to model electrical force microscopy, accounting for scenarios where traditional assumptions about tip charge and sample changes do not hold, enhancing the interpretation of experimental data.

Contribution

It introduces a Lagrangian-based model and perturbation theory for electric force microscopy, relaxing common assumptions and enabling accurate analysis of complex sample interactions.

Findings

Derived coupled equations of motion for cantilever and charge.

Showed limitations of feedback-free time-resolved electric force microscopy.

Related cantilever frequency shift to sample impedance in broadband spectroscopy.

Abstract

Scanning probe microscopy is often extended beyond topographic imaging to study electrical forces and sample properties, with the most widely used experiment being frequency-modulated Kelvin probe force microscopy. The equations commonly used to interpret this experiment, however, rely on two hidden assumptions: (1) the tip charge oscillates in phase with the cantilever motion to keep the tip voltage constant, and (2) any changes in the tip-sample interaction happen slowly. Starting from an electro-mechanical model of the cantilever-sample interaction, we use Lagrangian mechanics to derive coupled equations of motion for the cantilever position and charge. This general approach rigorously describes scanned probe experiments even in the case when the usual assumptions of fast tip charging and slowly changing samples properties are violated. We develop a Magnus-expansion approximation to illustrate how abrupt changes in the tip-sample interaction cause abrupt changes in the cantilever amplitude and phase. We show that feedback-free time-resolved electric force microscopy cannot uniquely determine sub-cycle photocapacitance dynamics. We then use first-order perturbation theory to relate cantilever frequency shift and dissipation to the sample impedance even when the tip charge oscillates out of phase with the cantilever motion. Analogous to the treatment of impedance spectroscopy in electrochemistry, we apply this approximation to determine the cantilever frequency shift and dissipation for an arbitrary sample impedance in (broadband) local dielectric spectroscopy experiments. The general approaches we develop provide a path forward for rigorously modeling the coupled motion of the cantilever position and charge in the wide range of electrical scanned probe microscopy experiments where the hidden assumptions of the conventional equations are violated or inapplicable.