Modeling of time-dependent electrostatic effects and AFM-based surface conductivity characterization

Atomic force microscopy (AFM) combined with electrical modes provides a powerful contactless approach to characterize material electrical properties at the nanoscale. However, conventional electrostatic models often overlook dynamic charge effects, which are particularly relevant for two-dimensional (2D) materials deposited on insulating substrates. In this work, we introduce a theoretical framework that extends traditional electrostatic models by incorporating charge dynamics, analyzing two key cases: nonideal conductors and nonideal insulators. Our model establishes a characteristic timescale $\ensuremath{\tau}$, which governs charge redistribution and measurement reliability. Experimental validation using graphene oxide (GO), reduced graphene oxide, and lightly reduced GO demonstrates strong dependence of frequency shift on surface conductivity, confirming our predictions. Temperature-dependent measurements further reveal conductivity variations consistent with disordered electronic materials. These findings offer valuable insight into how finite surface conductivity influences AFM-based techniques. They introduce an approach for analyzing charge dynamics on individual flakes of 2D materials while also presenting a contactless method for estimating surface conductivity.