Atomic-Scale Force-Vector Fields

The magnitude and direction of forces acting between individual atoms as a function of their relative position can be described by atomic-scale force-vector fields. We present a noncontact atomic force microscopy based determination of the force fields between an atomically sharp tip and the (001) surface of a KBr crystal in conjunction with atomistic simulations. The direct overlap of experiment and simulation allows identification of the frontmost tip atom and of the surface sublattices. Superposition of vertical and lateral forces reveals the spatial orientation of the interatomic force vectors.

Figure 1

(a) Surface topography of a KBr(001) sample, indicating the position of ${\mathrm{K}}^{+}$ and ${\mathrm{Br}}^{-}$ surface ions according to the force field analysis. The dashed line indicates a representative $y$ position along the sites of topography maxima at which the force field measurements were performed. (b) Measured frequency shift versus relative tip-sample distance (38 curves) as a function of the horizontal tip position along the $x$ axis.

Figure 2

(a) Experimental force curves at positions A–D along a line between the two ionic species. The black dotted curve represents the average over all force curves with a linear extrapolation into the regime of site-specific behavior. Inset: Solid gray curve is a van der Waals fit of the long range part of the force curves. (b) Site-specific part of the short range regime for the same force curves as in (a), where the non-site-specific contributions have been subtracted.

Figure 3

(a) The two-dimensional map of the site-specific vertical tip-sample forces. (b) Simulated map of the site-specific tip-sample forces for a ${\mathrm{K}}^{+}$-terminated KBr-cluster (left) and for a ${\mathrm{Br}}^{-}$-terminated KBr cluster (right). The dotted black lines indicate the minimum of the force curves with respect to the $z$ axis at each $x$ position.

Figure 4

(a) Atomic-scale force-vector field on KBr(001) determined from the AFM experiment. Arrow length and orientation represent magnitude and direction, respectively, of the site-specific force experienced by the tip at the respective position in the $x\mathrm{\text{−}}z$ plane. (b) Simulated force-vector field for the K-terminated tip apex. Length scales for the vectors are indicated in the upper left corners.