Tip-jump statistics of stick-slip friction

Friction maps with atomic unit cell periodicity on highly oriented graphite in ultra-high vacuum are measured with the friction force microscope. The well-known stick-slip phenomenon is observed where the tip jumps between individual equilibrium positions. We perform a statistical analysis of the frictional forces that are necessary to induce the tip jumps. The corresponding histograms give a direct fingerprint of the distribution function of the tip-jump events. The histograms depend strongly on the atomic structure of the surface, which is related to the two-dimensionality of the surface potential. We compare the experimental to the theoretical distribution function, which is based on the thermally activated Tomlinson model for atomic friction, and quantitative values for the effective energy barriers are extracted.

Figure 1

The potential diagram of the Tomlinson model at zero temperature. In (a) the tip is trapped in a local minimum and is separated from the next minimum to the right by an energy barrier of height $\Delta E$. In (b) the tip base has moved to the right, and the energy barrier has vanished, which causes the tip to “jump” to the next local minimum. However, at nonzero temperature, there is a finite probability that the tip jumps over the energy barrier already in (a) (dashed arrow).

Figure 2

(Color online) The relative probability that a jump occurs as a function of the maximum lateral force $F_m$, which was necessary to induce the jump, after the analytical derivation of the thermally activated Tomlinson model from Sang et al. (Ref. 11). Typical parameters encountered during the experiments were chosen (see text). The four histograms represent the cases for different energy barrier heights $E_0$.

Figure 3

(Color online) (a) A $3\times 3{\mathrm{nm}}^2$ lateral friction force map on HOPG at tip base scan velocity $v=60\mathrm{nm}∕\mathrm{s}$. (b) The schematic representation of the HOPG unit cell. The solid black circles are the carbon atoms in the top layer which are arranged hexagonally around a hollow-site. The big blue circles represent the equilibrium positions where the tip would stick and the blue arrows indicate the jump or slip process.

Figure 4

(Color online) The friction loops from the friction map in Fig. 3a showing the stick-slip behavior of the scanning tip in the trace (blue) and retrace (red) directions at a tip base velocity of $60\mathrm{nm}{\mathrm{s}}^{-1}$.

Figure 5

(Color online) The jump height analysis: The raw data of a single friction trace (blue line) is shown as a function of the tip base position. The differentiated friction signal (black line, bottom) is used to find the position of the slip events, by comparison with an appropriate threshold value (dotted line). The raw data between the slip positions is fitted with a line (black line) and the maximum of the lateral force just before the slip events is then extracted as the jump-height, i.e., the $F_m$ value (red arrows).

Figure 6

The experimental jump-height histogram for a tip base velocity of $120\mathrm{nm}{\mathrm{s}}^{-1}$. The number of jumps is shown as a function of the lateral force necessary to induce the jump. A total of 23 748 jumps were analyzed for this histogram, and a bin width of $0.008\mathrm{nN}$ was chosen. The right-hand scale shows the relative jump probability, which is the number of jumps from the left-hand scale divided by the total number of jumps.

Figure 7

(Color online) The left image shows a zoom of the friction map of image size $6\times 6{\mathrm{nm}}^2$ at a scan speed of $60\mathrm{nm}{\mathrm{s}}^{-1}$. The dissipated energy was calculated from the individual friction loops and is shown in the right part of the graph. For better comparison with other measurements, the dissipated energy is normalized by the number of lattice unit cells, over which the tip was scanned. The black line is a 3-point box average of the raw data (squares) and is meant as a guide to the eye.

Figure 8

(Color online) The friction loops extracted from the friction map in Fig. 7a showing the stick-slip behavior of the scanning tip in the trace and retrace direction on two different positions of the surface: (a) along the center of the hollow-sites [blue lines in Fig. 7a, 7b] between two hollow-site rows [red lines in Fig. 7a]. The dashed lines show the fitted zero friction line which serves as the reference for the jump-height determination.

Figure 9

(Color online) The experimental jump-height distribution functions for a scan speed of $80\mathrm{nm}{\mathrm{s}}^{-1}$ (solid lines, cityscape shape), which were compiled from the jumps at the side (red) and the center (blue) of the hollow-site rows. The dashed lines are the corresponding best fits with the thermally activated Tomlinson model from Sang et al. (Ref. 11) with the energy barrier heights $E_0^{\min }=118\mathrm{meV}$ and $E_0^{\max }=132\mathrm{meV}$, respectively.

Figure 10

(Color online) The experimental jump-height distribution functions for a scan speed of $120\mathrm{nm}{\mathrm{s}}^{-1}$ (solid lines, cityscape shape), which were compiled from the jumps at the side (red) and the center (blue) of the hollow-site rows. The dashed lines are the corresponding best fits with the thermally activated Tomlinson model from Sang et al. (Ref. 11) with the energy barrier heights $E_0^{\min }=118\mathrm{meV}$ and $E_0^{\max }=132\mathrm{meV}$, respectively.