The influence of short-range electrostatic forces on the measured local Contact Potential Difference (CPD) by means of Amplitude Modulation- and Frequency Modulation-Kelvin Probe Force Microscopy (AM- and FM-KPFM) is discussed on the base of numeric and analytic descriptions of both methods. The goal of this work is to help interpreting recent experimental results reporting atomically-resolved CPD images, in particular on bulk insulating samples. The discussion is carried out on the base of spectroscopic curves. The expression of the bias-dependent electrostatic force derives from a previous work and is estimated between a tip with simple geometry and the (001) facet of a perfect alkali halide single crystal. The force, with a short-range character, scales as a second-order polynomial function of the bias voltage. It is stated that the linear term is responsible for the occurrence of the atomic-scale CPD contrast, while the quadratic one, involving the sample polarisation, accounts for the detected signal by the KPFM methods. Nevertheless, analytic and numeric approaches stress the influence of the linear term on the measured CPD which intrinsically hinders the possibility to perform quantitative CPD measurements, but also makes the measured ``pseudo-CPD" strongly deviating from the surface potential. Hence, in the short-range regime, AM- or FM-KPFM measurements neither reflect the CPD nor the local surface potential, but rather an effective value which is convoluted by the geometric parameters of the tip, the so-called local CPD. At last, the influence of long-range, capacitive, electrostatic forces is discussed in conjunction with the short-range ones. This allows us to draw conclusions regarding the distance dependence of the local CPD which then exhibits a resonant behavior as a function of the tip-surface separation. This phenomenon is expected to play a role in the KPFM imaging process.
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