Multiscale approach for simulations of Kelvin Probe force microscopy with atomic resolution

The distance dependence and atomic-scale contrast recently observed in nominal contact potential difference (CPD) signals simultaneously recorded by the Kelvin probe force microscopy (KPFM) using non-contact atomic force microscopy
is addressed theoretically.
In particular, we consider probing an insulating surface where the applied bias voltage affects electrostatic
forces acting on the atomic scale. Our approach is a multiscale one. First, the electrostatics of the macroscopic
tip-cantilever-sample system is treated, both analytically and numerically.
Then the resulting electric field under the tip apex is inserted into a series of density functional theory calculations for a realistic neutral but reactive silicon nano-scale tip interacting with a NaCl(001) sample.
Theoretical expressions for amplitude modulation (AM) and frequency modulation (FM) KPFM signals and for the corresponding local contact potential differences (LCPD) are obtained and evaluated for several tip oscillation amplitudes A up to 10 nm.
For A = 0.01 nm, the computed LCPD contrast is proportional to the slope of the atomistic force versus bias in the AM mode and to its derivative with respect to the tip-sample separation in the FM mode. Being essentially constant over a few Volts, this slope is the basic quantity which determines variations of the atomic-scale
LCPD contrast.
Already above A = 0.1 nm, the LCPD contrasts in both modes exhibit almost the same spatial dependence as the slope. As the most basic quantity, the slope is shown to be approximately expressed in terms of intrinsic charge distribution and dipole moment and their variation due to the chemical interactions. The slope is also influenced by the macroscopic bodies.
As a second part, we introduce a method to measure the distances between atomic configurations which is useful when seeking the tip-apex structures. The broad application of this method includes conformational search and machine-learning based interatomic potentials.