Figure 1

The frictional behavior of graphite at atomic-scale surface steps. (a) Top panel: Color-coded top view of the topography; image size: $1.6\mu \mathrm{m}\times 0.4\mu \mathrm{m}$. Middle panel: Topographical line section along the arrow indicated in the top panel. Individual graphene layers are represented by gray rectangles for illustration purposes, revealing that a double, a fivefold, and a single step are encountered. Bottom panel: Frictional forces recorded during a left-right scan (red) and the corresponding right-left scan (blue) along the same line with an external load of $F_{\mathrm{load}}=6.8\mathrm{nN}$. A higher frictional increase of the frictional forces for upward than for downward scans is obvious (black arrows). (b) Plots of the frictional increase (defined as the difference between the maximum frictional force at the step edge and the friction encountered on the terrace) observed at the three different step edges as a function of the load. For upward scans (top panel), the frictional increase grows linearly with load, while it is constant for downward scans (bottom panel). Parameters: $c_z=0.073\mathrm{N}/\mathrm{m}$, $c_x=20.6\mathrm{N}/\mathrm{m}$, $v_M=6\mu \mathrm{m}/\mathrm{s}$.Reuse & Permissions

Figure 2

The frictional increase observed at a double step on ${\mathrm{MoS}}_2$ (a) and a single step on NaCl (b). On both materials, the additional frictional forces caused by the step edges increase linearly with load for upward scans while they are independent of the actual loading force for downward scans. Note that in agreement with a previous report by Meyer and Amer [22], we observe no change of friction at the edge compared to the on-terrace value for downward scans on NaCl, which leads to a vanishing value for the frictional increase. Parameters: $c_z=0.065\mathrm{N}/\mathrm{m}$, $c_x=18.2\mathrm{N}/\mathrm{m}$, $v_M=2\mu \mathrm{m}/\mathrm{s}$ (a), and $c_z=0.063\mathrm{N}/\mathrm{m}$, $c_x=17.8\mathrm{N}/\mathrm{m}$, $v_M=2\mu \mathrm{m}/\mathrm{s}$ (b).Reuse & Permissions

Figure 3

A schematic of the modified Prandtl-Tomlinson model describing the friction at atomic-scale surface steps (not to scale). $x_t$ represents the position of the tip, which is connected via a spring with spring constant $c_x$ to the body $M$. For sliding, the body $M$ is moved along the $x$ direction while the tip interacts with the tip-sample potential $V(x_t)$ (thick solid line). If $x_t=x_M$, the spring is in its equilibrium position. The tip movement in the interaction potential at the step edge is indicated by arrows. For details, see text.Reuse & Permissions

Figure 4

(a) Density plot of the tip-sample interaction potential assumed in the simulations, featuring a barrier at the top of the step edge and a distinguished potential minimum at its bottom (dark blue). (b) The frictional force acting on the tip (offsets added for clarity) exhibits a distinguished stick-slip movement at the step edge for downward and upward scans. Because of the relative increase of the barrier height and slope with load, the lateral force needed to pull the tip over the step edge increases for upward scans. (c) The tip-sample interaction potential $V(x_t)$ for the same three loads as in (b), illustrating the increase of barrier height and steepness with load. (d) Plot of the frictional increase as calculated in the Prandtl-Tomlinson model for the potential shown in (a). Parameters: $E_0=1.0\mathrm{eV}$, $r_0=0.45\mathrm{nm}$, $m_x=m_z={10}^{-10}\mathrm{kg}$, $c_x=5.0\mathrm{N}/\mathrm{m}$ and ${\gamma }_x={\gamma }_z=2\sqrt{c_x{m}_x}$.Reuse & Permissions