Dependence of the electrostatic patch force evaluation on the lateral resolution of Kelvin probe force microscopy

Kelvin probe force microscopy (KPFM) is widely used to measure the surface potential on samples, from which the electrostatic patch force can be calculated. However, since the KPFM measurements represent a weighted average of local potentials on the sample, the accuracy of the evaluation critically depends on the precision and lateral resolution of the method. In this paper, we investigate the influence of this averaging effect on patch force estimations using both analytic and numerical methods. First, we derive the correlation functions of patch potential and establish the formulas for calculating the electrostatic patch forces in the parallel-plate geometry, with and without consideration of the KPFM measurement effect. Thus, an analytic method is established to determine the accuracy of patch force evaluation when the statistical parameters of the patch potential and the lateral resolution of the KPFM are given. Second, numerical simulations are employed to explore the dependence of estimated patch forces on the lateral resolution of KPFM under more realistic conditions. Both analytic and numerical results show a similar dependence of the patch force estimation on the patch characteristic size, potential fluctuation, and the lateral resolution of KPFM. It is also found that the underestimation of the patch force becomes less sensitive to the resolution of KPFM as the separation between plates increases. The results of this study could provide useful guidance for the accurate evaluation of electrostatic patch forces using KPFM.