Polarization effects in Non-Contact Atomic Force Microscopy: a key to model the tip-sample interaction above charged adatoms

We discuss the influence of short-range electrostatic forces, so called dipolar forces, between the tip of an atomic force microscope (AFM) and a surface carrying charged adatoms. Dipolar forces are of microscopic character and have their origin in the polarizability of the foremost atoms on tip and surface. In most experiments performed by non-contact AFM, other forces such as binding forces dominate the interaction. However, in the experiments presented by Gross et al. [Science 324, 1428 (2009)], where the charge state of individual gold atoms adsorbed on a thin dielectric layer was determined, binding forces are negligible as the tip-sample distance is relatively large. We develop a model which mimics the experimental tip-sample geometry of the aforementioned experiments. The model includes van der Waals and long-range electrostatic interactions, as well as the short-range electrostatic interaction based on the self-consistent description of electronic polarization effects on neutral and charged adatoms. The model is based on a calculation of the electrostatic energy of the tip-sample geometry. Our calculations of non-contact AFM imaging as well as of bias spectroscopic curves are in good agreement with the experimental ones presented by Gross et al. It is demonstrated that the short-range dipolar force is mainly responsible for the contrast observed in topography imaging above charged species. However, it is the long-range capacitive force which is responsible for the detection of the charge state in bias spectroscopy. We discuss implications of our findings on future experiments which aim to detect single charges by means of Kelvin probe force microscopy. PACS numbers: 07.79.Lh, 41.20.Cv, 34.20.-b