A method for calculating capacitances and electrostatic forces in atomic force microscopy

Abstract

A method to calculate electrostatic forces in the context of atomic force microscopy that is useful for the calculation of the electrostatic forces when different length scales are included in the simulation is presented. The versatility of the method allowed for an analysis of the behavior of forces as a function of the tip apex geometry. For example, for flattened, worn out tips, the force at the point of contact with a dielectric sample could be 2.5 times larger than that of a sharp tip. A simple analytical approximation has been also developed for the local characterization of thin films at the nanoscale.

Charge Polarization at Catalytic Metal–Support Junctions. Part B: Theoretical Modeling of Kelvin Probe Force Microscopy

“…While this led to good agreement between experiment and simulation and also to agreement with up to date literature about the direction of charge transfer 26 concerns have been raised about the validity of the approximation assuming a point-like tip and fixed electrical charges, in particular in the 25 presence of conductive samples, [27][28][29][30] and it was suggested to involve also van der Waals forces. 31 Most charges in the oxide support are fixed to defects, and in the metal nanoparticle the charges at the interface are largely immobilized by the attraction to the counter charge in close proximity across the interface, but there are mobile conduction electrons which render the particles polarizable by the applied voltage on the tip.…”

Enhanced dielectric constant resolution of thin insulating films by electrostatic force microscopy

“…However, it is easy to demonstrate that the system fits this condition in the limit 1>2. This condition will be found in most of the systems since the opposite limit (1>2) usually corresponds to the case where the substrate below the thin film is a metal and can be analysed by previous methods 13 . The second and most important problem is that the three series depend on each other and there is not a simple way to express them.…”

“…The tip is placed over a sample at a tip-sample distance D. The tip is characterized by three geometrical parameters: The apex radius Rtip, the half-angle  and the length L. To solve electrostatic problems with this geometry, an algorithm called the Generalized Image Charge Method 13,20 (GICM) has been developed. The GICM replaces the surface charge density by a set of charges inside the metallic tip.…”

“…12 EFM is currently used to determine the static dielectric constants of thin films of a few nanometers thickness directly deposited on a metallic electrode. The dielectric constant can be determined by comparison of the experimental results with simple analytical expressions 13,14,15 . For metallic samples, the tip-sample interaction is very well understood and the force, as well as the force gradient, can be determined by using a simple analytical model.…”

“…For metallic samples, the tip-sample interaction is very well understood and the force, as well as the force gradient, can be determined by using a simple analytical model. 13 The tip-sample interaction due to the thin film dielectric response can be considered as a perturbation of the reference system in absence of the thin film. In contrast with metallic substrates, the electrostatic interactions with dielectric samples depend on the macroscopic geometry of the tip-sample system and, although it is possible to compute it numerically 16 , no simple closed expressions are available.…”