Simulated measurement of small metal clusters by frequency-modulation non-contact atomic force microscopy

The apparent height and lateral extent of very small metallic clusters and particles adsorbed on flat substrates have been calculated for frequency-modulation non-contact atomic force microscopy (ncAFM). The ncAFM scanning tip was modelled as a Si sphere covered by 1 nm of SiO2. This tip sphere of either 5 or 20 nm total radius (including an SiO2 layer) is attached to a cantilever of spring constant k = 40 N m−1 and oscillated with a 10 nm amplitude. The tip was rastered across the centre of a single cluster of Pd atoms or a single Pd particle located on a flat continuum substrate of alumina or Pd. The clusters were one-atom-thick close-packed arrangements of 19 or 91 atoms (1.4 or 3.0 nm wide); the particles were continuum spheres of diameter 2.0 or 4.0 nm. The tip–substrate and tip–particle interactions were modelled with 6–12 Lennard-Jones potentials. The attractive interaction was taken to be the London–van der Waals dispersion interaction whose magnitude was estimated from Hamaker constants calculated from bulk optical constants of Si, SiO2, Pd, and alumina. The repulsive interaction was determined from estimates of the atomic radii using densities of the bulk materials. These simulations show that the apparent heights of particles imaged by ncAFM range from just 12% of the actual height for the smallest Pd cluster on a Pd substrate to 95% of the actual height for the largest Pd particle on an alumina substrate. The apparent widths of the clusters were similar to those in contact AFM. These results show the most accurate height measurements occur when the lateral extent of the cluster or particle is comparable to or larger than the radius of the tip and when the Hamaker constant for the interaction of the tip with a cluster or particle is larger than that for the tip with the substrate.