Congratulations Ryan and Lee. After 8 months of peer review and revisions, our new mathematical description of electric force microscopy is at last in print!
Dwyer, R. P.; Harrell, L. E. & Marohn “Lagrangian and impedance spectroscopy treatments of electric force microscopy”, Phys. Rev. Appl., 2019, 11, 064020, DOI:10.1103/PhysRevApplied.11.064020, arXiv:1807.01219.
In this 32 page manuscript we show that the usual equations used to describe electrical scanned probe microscopy experiments make assumptions which are not valid for a wide variety of semiconductor samples. We derive new equations for cantilever frequency shift and non-contact electrical friction, written in terms of the sample’s electrical impedance, that are valid for semiconductor samples.
Our thanks go out to two anonymous reviewers who read the paper very carefully and made many thoughtful suggestions. As a result of their feedback, we added a multi-page section to the front of the paper highlighting our findings (with many new figures). The paper is considerably stronger as a result of the reviewers’ hard work.
The abstract of the paper reads
Scanning probe microscopy is often extended beyond simple 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 frequency-modulated experiment, however, rely on two hidden assumptions. The first assumption is that the tip charge oscillates in phase with the cantilever motion to keep the tip voltage constant. The second assumption is that any changes in the tip-sample interaction happen slowly. Starting from an electromechanical model of the cantileversample interaction, we use Lagrangian mechanics to derive coupled equations of motion for the cantilever position and charge. We solve these equations analytically using perturbation theory, and, for verification, numerically. 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 subcycle 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 both local dielectric spectroscopy and broadband local dielectric spectroscopy experiments. The general approaches that 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.
This work was funded by the U.S. National Science Foundation.