Imaging Static Charge Distributions: A Comprehensive KPFM Theory

Abstract

We analyze Kelvin probe force microscopy (KPFM) for tip-sample systems that contain static charges by presenting a rigorous derivation for the respective KPFM signal in all common KPFM modes, namely amplitude modulation, frequency modulation, or heterodyne detection in the static, open-loop or closed-loop variant. The electrostatic model employed in the derivation is based on a general electrostatic analysis of an arbitrary tip-sample geometry formed by two metals, and which can include a static charge distribution and dielectric material in-between. The effect of the electrostatic force on the oscillating tip is calculated from this model within the harmonic approximation, and the observables for each of the above KPFM modes are derived from the tip oscillation signal. Our calculation reveals that the KPFM signal can for all modes be written as a weighted sum over all charges, whereby each charge is multiplied with a position-dependent weighting factor depending on the tip-sample geometry, the KPFM mode, and the oscillation amplitude. Interestingly, as the weight function does not depend on the charges itself, the contribution of the void tip-sample system and the charge distribution can be well-separated in the KPFM signal. The weight function for charges allows for a detailed understanding of the KPFM contrast formation, and enables to trace the dependence of the KPFM signal on different parameters such as the tip-sample geometry or the oscillation amplitude.