Study of uncertainties of height measurements of monoatomic steps on Si 5 × 5 using DFT

Abstract

The development of nanotechnology gives rise to new demands on standards for dimensional measurements. Monoatomic steps on, e.g. silicon are a suitable length standard with a very low nominal value. The quantum-mechanical nature of objects consisting of only a few atomic layers in one or more dimensions can no longer be neglected and it is necessary to make a transition from the classical picture to a quantum approach in the field of uncertainty analysis. In this contribution, sources of uncertainty for height measurements using atomic force microscopy (AFM) in contact mode are discussed. Results of density functional theory (DFT) modeling of AFM scans on a monoatomic step on silicon are presented. Van der Waals forces for the interaction of a spherical tip and an infinite step are calculated classically. Height measurements in constant force mode at different forces are simulated. In our approach, we model the tip apex and the monoatomic step as systems of individual atoms. As interatomic forces act on the sample and the tip of the microscope, the atoms of both relax in order to reach equilibrium positions. This leads to changes in those quantities that are finally interpreted as the resultant height of the step. The presence of van der Waals forces induces differences between the forces acting on atoms at different distances of the step. The behavior of different tips is studied along with their impact on the resulting AFM scans. Because the shape of the tip apex is usually unknown in real experiments, this variance in the height result due to different tips is interpreted as a source of uncertainty.