# Method for Measuring Surface Charge on Insulating Materials Based on the Vibrating Capacitor Method
## 1 Introduction
In KPFM technology, by applying DC and AC biases to the probe, the DC bias is used to compensate for the surface potential of the sample, while the AC bias generates changing electrostatic forces to make the probe vibrate. When the second harmonic amplitude of the probe is zero, the DC bias applied to the probe equals the surface potential of the material. Currently, the vibrating capacitor method can achieve non-contact and non-destructive measurement of surface potential with high spatial resolution, serving as the theoretical basis for many commonly used surface potential measurement devices. However, the vibrating capacitor method was originally designed to detect the contact potential difference or differences in work functions of samples. During the measurement process, the stable current parameters ensure that the applied bias voltage strictly equals the potential difference between the tested metal and the reference voltage. However, unlike conductors, dielectric materials do not have a fixed and uniform surface potential. In addition to being related to surface charge polarity, density, and other conditions, the surface potential of dielectric materials is also influenced by factors such as probe size, probe position, and probe bias voltage [39; 40; 41; 42]. Conversely, due to their extremely low conductivity, the surface charge distribution of dielectric materials often remains stable. Therefore, the surface potential of dielectric materials does not have physical significance; what can be stably measured is their surface charge density. However, so far, the quantitative relationship between the surface potential measurement values obtained by the vibrating capacitor method and the surface charge density remains unclear, affecting the accurate measurement of surface charge density of dielectric materials. This study proposes a method based on Comsol simulations to establish the relationship between surface charge and measured voltage for dielectric materials, while simplifying the simulation process through integration calculation. The aim of this paper is twofold: firstly, to establish an ideal parallel plate capacitor theoretical model and derive the theoretical relationship between surface charge density and measured potential. Additionally, the study aims to create two-dimensional and three-dimensional simulation models of the vibrating capacitor method measurement process. It also investigates the influence of factors such as charged area and probe height on the measured potential using COMSOL Multiphysics 5.6. Lastly, the study seeks to establish an analytical relationship between the measured potential value and the surface charge density distribution. This is based on the potential measurement model of surface point charge distribution. The feasibility of integration calculation as an alternative to simulation methods is then validated using various charge density distributions, including circular trajectory sinusoidal distribution, circular trajectory random distribution, square region random distribution, and others.