  An analytical model of the electrostatic force between the tip of a
non-contact Atomic Force Microscope (nc-AFM) and the (001) surface of an ionic
crystal is reported. The model is able to account for the atomic contrast of
the local contact potential difference (CPD) observed while nc-AFM-based Kelvin
Probe Force Microscopy (KPFM) experiments. With the goal in mind to put in
evidence this short-range electrostatic force, the Madelung potential arising
at the surface of the ionic crystal is primarily derived. The expression of the
force which is deduced can be split into two major contributions: the first
stands for the coupling between the microscopic structure of the tip apex and
the capacitor formed between the tip, the ionic crystal and the
counter-electrode; the second term depicts the influence of the Madelung
surface potential on the mesoscopic part of the tip, independently from its
microscopic structure. These short-range electrostatic forces are in the range
of ten pico-Newtons. When explicitly considering the crystal polarization, an
analytical expression of the bias voltage to be applied on the tip to
compensate for the local CPD, i.e. to cancel the short-range electrostatic
force, is derived. The compensated CPD has the lateral periodicity of the
Madelung surface potential. However, the strong dependence on the tip geometry,
the applied modulation voltage as well as the tip-sample distance, which can
even lead to an overestimation of the real surface potential, makes
quantitative KPFM measurements of the local CPD extremely difficult.
