Contrast inversion in electrostatic force microscopy imaging of trapped charges: tip-sample distance and dielectric constant dependence

We present a numerical and analytical study of the behavior of both electrostatic force and force gradient created by a charge trapped below the surface of a dielectric on an atomic force microscope tip as a function of the dielectric constant and tip-sample distance. As expected, the force decreases monotonously when the dielectric constant increases. However, a maximum in the dielectric constant dependence of the force gradient is found. This maximum occurs in the typical experimental parameters' range and depends on the tip-sample distance and the sample thickness. The analytical study permits us to understand the physical origin of this phenomenon and is in good agreement with the numerical simulation for small tip-sample distances. We also report a study exemplifying a possible contrast inversion in electrostatic force microscopy (EFM) signals while scanning, at different heights, two charges trapped in a sample having heterogeneous dielectric domains. In addition to this particular contrast inversion effect, this study can be considered as a way to gain insight into the mechanisms of EFM image formation as a function of the dielectric constant and tip-sample.