Despite the practical and fundamental importance of friction, there are many issues that are still needed to be explored and revealed; one of them is how friction changes across an atomic surface step edge at different temperatures. In this article, a friction force microscope under ultrahigh vacuum conditions has been used to study the temperature dependence of nanoscale friction between a silicon tip and a freshly cleaved HOPG surface with exposed single- and double-layer step edges. For the upward scanning from the lower terrace to the upper terrace, a large resistive force which is linearly dependent on the normal force is observed. A similar resistive force but with a smaller magnitude together with an assistive force is observed for the downward scanning; however, the resistive force is found to be independent of the normal force while the assistive force increases with the normal force. Besides, the resistive force for the double-layer step edge is found to be twice as high as that of the single-step edge, while the assistive force seems to be less influenced by the height of the step. Finally, the experimental results reveal that temperature has a negligible effect on the friction coefficients at the step edges, which is inconsistent with the thermal activated process where friction should decrease with temperature. Based on the theoretical studies, this observation can be explained by a process where the temperature effect is very small compared with the edge Schwoebel–Ehrlich barrier. These findings may help in understanding the temperature effects on macroscopic friction having a lot of step edges at the interface.

Friction at the sliding interface is one of the topmost and oldest technological and scientific topics since it widely determines the efficiency and the lifetime of nearly all moving machinery. The relations between macroscopic friction and normal force were published several hundred years ago,1 showing that dynamic friction is linearly dependent on the normal force and independent of the relative sliding velocity and that the apparent contact area resulted from the collective behavior of many asperities which frequently stick and slip during sliding.2 However, beyond this picture, further details on the energy dissipation and the mechanisms of friction remain elusive. One of the main challenges of getting deep inside into the mechanism is the complex nature of contact interfaces exhibiting surface roughness and constituting a multi-asperity contact. Friction force microscopy (FFM), in which a single-asperity (nanoscale tip) slides on an atomic flat, clean, and low-wear surface, can be viewed as a parsimonious model case.1,3–6 Indeed, nanoscale friction based on the FFM technology has been at the forefront of scientific interest in recent years because it can lead to a fundamental knowledge of how energy is dissipated during the relative sliding process.

For an FFM experiment, the sliding of a tip is usually considered a successive stick-slip motion, where energy is supposed to be dissipated on each slip of the tip together with the injection of phonons into the sample.7 The stick-slip motion of a tip is a thermally activated process that means temperature can help a tip jump out of the potential minimum, thus reducing the average sliding friction. This thermally activated process can be reasonably described by the Prandtl–Tomlinson (PT) model.2,8,9 However, for the case of a surface having atomic step edges, the frictional behavior across its step edges becomes more complicated; especially, whether temperature still has effects on friction remains elusive. Study on the frictional behavior of FFM tip scanning across step edges at different temperatures is very important, which can be considered as a mimic idea model for understanding macroscopic friction behavior. In this paper, we intend to understand the basic phenomena involving a sharp single-asperity sliding across atomic step edges on an atomic flat and homogeneous crystal surface. Here, we chose graphite, the most commonly used solid lubricant, as our sample, which exhibits ultra-low friction and wear, large Young’s modulus, and high load bearing capacity.10 Last but not least, graphite is chemically stable and atomically flat and, more importantly, has a clean surface with single- and double-layer step edges with a well-defined structure, which is ideal for friction study,11 which can be easily carried out by using the scotch tape based cleaving method.

It is commonly believed that there are two types of graphite step edges. The first type is a buried step edge, where the step edge is covered by at least one layer of graphene. On the contrary, the second one is an exposed step edge, where the step edge occurs at the outmost surface distinguished by a much higher friction.11–13 In this article, we focus on the second one, i.e., exposed surface step edges. In previous studies, at exposed step edges, an increase in the friction phenomenon12,14–21 was observed during the upward scanning on graphite, MoS2, and NaCl, which was attributed to the Schwoebel–Ehrlich barrier at the step edges.14–16,22–24 For downward scanning, the situation becomes more complicated. Some studies reported an enhanced friction phenomenon at the step edge11,15–18,25 while others did not.18,20,21 This difference has been partially attributed to tip bluntness, which can affect the interaction energy and the trajectory of a tip as it moves over step edges11,18 and partially attributed to environmental effects.20,26 Recently, the physical (mechanical deformation) and chemical (hydrogen bond formation) contribution to the total friction under ambient conditions at surface step edges have also been discussed.27 

Here, temperature dependence of friction measurement between a silicon tip and a freshly cleaved graphite surface with single- and double-layer step edges has been conducted under UHV conditions. Our results reveal that there is a large resistive force linearly dependent on the applied normal force for the upward scanning from the lower terrace to the upper terrace, while a similar resistive force but independent of the normal force was observed for the downward scanning. This resistive force for the double-layer step edge is twice larger than that of the single-layer step edge. For the downward scanning, there is also an assistive force apart from the resistive force, which slightly linearly increases with normal loads and is independent of the height of step edges. Finally, the experimental results reveal that temperature has a negligible effect on the friction coefficients at the step edges, which is inconsistent with the thermal activated friction where friction should decrease with temperature. Together with the theoretical studies, this observation can be explained by a process where the temperature effect is very small compared with the edge Schwoebel–Ehrlich barrier.

In our experiments, we used a commercial square HOPG sample (NTMDT, ZYA) with a side length of 5 mm and a thickness of 2 mm. As illustrated in Fig. 1(a), in order to control the sample temperature, the HOPG sample was first glued onto a stainless steel substrate by using conductive epoxy glue (Epoxy Technology, USA). Then, the stainless steel substrate was mechanically fixed to a standard low temperature sample holder which was coupled to a flow cryostat using liquid nitrogen as a coolant. By adjusting both the flow rate of liquid nitrogen and the power of resistance heating plate integrated to the temperature control stage, we can vary the temperature of the sample approximately from 90 K to 300 K. In order to prepare a clean sample surface for subsequent frictional and topographic measurements, the HOPG sample was freshly cleaved by scotch tape directly before being transferred into a UHV chamber. Inside the UHV chamber, the HOPG sample was additionally heated up to 500 K for 2.5 h to remove residual adsorption from the HOPG surface. All experiments have been performed within a Scienta Omicron UHV VT-AFM/STM system (Germany) using a PPP-LFMR cantilever obtained from Nanosensors (Switzerland) with a nominal spring constant of k = 0.42 N/m in the normal force direction and a typical initial tip diameter of 10 nm. During the whole set of measurements, an average pressure of about 4 × 10−10 mbar was constantly maintained inside the UHV chamber.

FIG. 1.

Experimental setup and the topographic image of the HOPG surface with step edges used in our measurements. (a) Illustration of the experimental setup. All experiments have been performed using a conventional friction force microscope on a freshly cleaved HOPG sample which was in contact with the temperature control stage under UHV conditions. (b) The typical topographic image of the HOPG surface with a single- and double-layer step edge obtained at T = 297.7 K using the contact mode operation with an applied normal force of 13.1 nN and a scan velocity of 1.25 μm/s. (c) The cross-section height profile across the step edges highlighted in (b). The black arrows in (b) and (c) indicate the scanning direction.

FIG. 1.

Experimental setup and the topographic image of the HOPG surface with step edges used in our measurements. (a) Illustration of the experimental setup. All experiments have been performed using a conventional friction force microscope on a freshly cleaved HOPG sample which was in contact with the temperature control stage under UHV conditions. (b) The typical topographic image of the HOPG surface with a single- and double-layer step edge obtained at T = 297.7 K using the contact mode operation with an applied normal force of 13.1 nN and a scan velocity of 1.25 μm/s. (c) The cross-section height profile across the step edges highlighted in (b). The black arrows in (b) and (c) indicate the scanning direction.

Close modal

All frictional and topographic measurements have been performed in contact mode operation, with normal forces specified in the main text. We used a fixed scanning area of 500 × 125 nm2 for all measurements. If not specified otherwise, the scanning velocity was fixed to v = 1.25 μm/s. Figure 1(b) shows the topographic image of the HOPG surface with two exposed step edges scanned from right to left at T = 297.7 K with the applied normal force FN = 13.1 nN. The cross section profile shown in Fig. 1(b) is then plotted in Fig. 1(c), demonstrating a height of ∼0.34 nm and 0.68 nm for the right and left step edge, respectively, that clearly indicates the single- and double-layer step edges.

To obtain sufficient statistics and reduce experimental errors, each force data point was extracted from a total of 100 loops of the lateral force signal. The error bars in this article are based on the standard deviation of the mean value calculated from the 100 lateral force loops measured for each temperature and normal load. All lateral force values in this article have been calibrated using the approach suggested by Bilas et al.28 In each case, we cooled the sample to a fixed temperature and waited for at least 2 h until a stable state with negligible thermal drifts was reached. In addition, we typically waited for several minutes after bringing the tip into contact with the cooled sample surface. During this period, a thermal equilibrium between the tip and the sample was established. In our experiments, systematic friction measurements have been performed at temperatures of 297.7 K, 187.6 K, and 99.0 K on the same single-layer and double-layer step edges with the applied normal force ranging from 0.3 nN to 42.0 nN in order to study the effects of temperature and normal force.

Figure 2(a) shows the typical measured lateral force curves during scanning from right to left (red line) and the opposite direction (blue line). When the FFM tip scans upward the step edges from the lower terrace to the upper terrace, a resistive friction peak is observed, and this resistive peak for the double-layer step edge (right step edge) is roughly twice as high as the resistive peak for the single-layer step edge (left step edge). Here, the upward resistive force is calculated as the difference between the peak force and the lateral force at the basal plane, as illustrated in Fig. 2(a). In contrast, more complicated lateral force responses are observed when the FFM tip scans downward the step edges from the upper terrace to the lower terrace. For the downward scanning process, not only a similar resistive friction peak with a smaller magnitude but also an opposite assistive friction peak are observed, where the lateral force first increases (resistive force), then decreases (assistive force), which is accomplished by the change in the topographic height, then increases again, and finally decreases to the basal plane value as the FFM tip moves away from the step edge. Similarly, the downward resistive force is calculated as the difference between the peak force and the lateral force at the basal plane, while the downward assistive force is defined as the decrease in the resistive force. When comparing the lateral force curves [Figs. 2(a) and 2(b)] with the topography curve [Fig. 1(c)], an interesting phenomenon which is consistent with the previous results16 was found: maximum of the lateral force appears before the step edge being actually scanned by the tip because the lateral motion induced torque needs to be large enough to overcome the stick of the tip before the step edge. Besides, the x-coordinates of the maximum resistive force for both upward scanning and downward scanning seems to coincide with each other, while the assistive force occurs after the resistive force after the step edge has been scanned (see the insert figures in Fig. 2(a) that illustrates the tip position at which the resistive and assistive force are observed). Figure 2(b) shows the change in friction force, defined as the half value of the lateral force difference. As can be seen in Fig. 2(b), the friction force increases 7 times at the single-layer step edge and 13 times at the double-layer step edge compared with the flat terraces.

FIG. 2.

Typical measured lateral force curves. (a) The measured lateral force while scanning upward and downward the step edges. The vertical black arrows show the resistive force and assistive force at the step edges, while the red and blue horizontal arrows indicate the scanning direction. The inset figures illustrate the tip position at which the resistive and assistive force are observed. (b) The measured friction force defined as the half value of the lateral force difference.

FIG. 2.

Typical measured lateral force curves. (a) The measured lateral force while scanning upward and downward the step edges. The vertical black arrows show the resistive force and assistive force at the step edges, while the red and blue horizontal arrows indicate the scanning direction. The inset figures illustrate the tip position at which the resistive and assistive force are observed. (b) The measured friction force defined as the half value of the lateral force difference.

Close modal

The changes in lateral force as well as the friction force observed at the step edges (Fig. 2) cannot be simply explained by topography induced force variation alone, at which the lateral force for scanning from right to left and the reverse direction should have no hysteresis at the step edge.29,30 In the reported results, the increase in the friction force phenomenon during the upward scanning12,14–21 was attributed to the edge Schwoebel–Ehrlich barrier.22–24 However, for the downward scanning, the results are not consistent. Depending on the tip bluntness and experimental environment, constant, reduced, or enhanced friction has been observed.11,15–18,20,21,25,27 Since our measurements were conducted under UHV conditions on the same single- and double-step with the same tip, we think the resistive force mainly originated from the edge Schwoebel–Ehrlich barrier. However, we also observed an assistive force which happens after the resistive force after the step edge being scanned. This means that the assistive force comes from the mechanical interaction between the side surface of the tip and the step.27 In fact, the assistance force is the lateral component of the total force acting on the tip and therefore should depend on the geometry of the tip and the height of step edges. We did not investigate the influence of the tip geometry, but instead, we show later that the assistive force is actually less influenced by the step height (see Fig. 3 and Table I).

FIG. 3.

Load and temperature dependence of nanoscale friction across HOPG surface step edges. (a) The change in adhesion force at different temperatures. The change in friction between the FFM-tip and the (b) single-layer step edge and (c) double-layer step edge measured for three different temperatures T = 297.7 K, 187.6 K, and 99.0 K as a function of normal loads.

FIG. 3.

Load and temperature dependence of nanoscale friction across HOPG surface step edges. (a) The change in adhesion force at different temperatures. The change in friction between the FFM-tip and the (b) single-layer step edge and (c) double-layer step edge measured for three different temperatures T = 297.7 K, 187.6 K, and 99.0 K as a function of normal loads.

Close modal
TABLE I.

The change in the friction coefficient scanning upward and downward the single- and double-layer step edge at different temperatures.

Friction coefficientUpward resistiveDownward resistiveDownward assistive
Single-layer step edge, T = 297.7 K 0.116 ± 0.008 −0.0005 ± 0.002 −0.045 ± 0.004 
Single-layer step edge, T = 187.6 K 0.084 ± 0.014 −0.009 ± 0.005 −0.038 ± 0.005 
Single-layer step edge, T = 99.0 K 0.143 ± 0.012 0.0009 ± 0.005 −0.073 ± 0.006 
Double-layer step edge, T = 297.7 K 0.214 ± 0.015 −0.045 ± 0.007 −0.080 ± 0.006 
Double-layer step edge, T = 187.6 K 0.207 ± 0.019 −0.055 ± 0.003 −0.066 ± 0.004 
Double-layer step edge, T = 99.0 K 0.152 ± 0.039 −0.026 ± 0.01 −0.047 ± 0.004 
Friction coefficientUpward resistiveDownward resistiveDownward assistive
Single-layer step edge, T = 297.7 K 0.116 ± 0.008 −0.0005 ± 0.002 −0.045 ± 0.004 
Single-layer step edge, T = 187.6 K 0.084 ± 0.014 −0.009 ± 0.005 −0.038 ± 0.005 
Single-layer step edge, T = 99.0 K 0.143 ± 0.012 0.0009 ± 0.005 −0.073 ± 0.006 
Double-layer step edge, T = 297.7 K 0.214 ± 0.015 −0.045 ± 0.007 −0.080 ± 0.006 
Double-layer step edge, T = 187.6 K 0.207 ± 0.019 −0.055 ± 0.003 −0.066 ± 0.004 
Double-layer step edge, T = 99.0 K 0.152 ± 0.039 −0.026 ± 0.01 −0.047 ± 0.004 

We then turn to temperature effects on the resistive and assistive force. To reduce experimental uncertainties, it is best to conduct all measurements at the same step edges, so we need to track the step edges during the temperature change process. To do this, we kept scanning the HOPG surface by using a contact mode and frequently compensating for the thermal drift. Although the tip was slightly worn during this track scanning, we think the shape of the tip did not change too much and only the contact area became larger. In this article, we thus focus on the friction coefficient, which is less influenced by the tip changes and/or adhesion effects. The adhesion force between the silicon tip and the flat surface of HOPG at different temperatures is 8.9 nN, 20.3 nN, and 43.4 nN at temperatures of 297.7 K, 187.6 K, and 99.0 K, respectively, as shown in Fig. 3(a). After having finished all of the measurements, we heated the sample to room temperature and measured the adhesion force again. The measured adhesion force is 29.7 nN, which means that the tip did show some wear during the whole set of measurements. To account for the tip change, the normal load used in this article was defined as the sum of adhesion force and the applied normal force. As shown in Fig. 2(a), the measured lateral force can be divided into two parts, the step edge region and the flat region. Here, we can directly measure the friction coefficient between a silicon tip and the graphite surface by using the friction force at the flat area. The measured friction coefficient at 297.7 K is μ = 0.011 ± 0.001, which is very close to the previous measurement under ambient conditions.

Figures 3(b) and 3(c) show the measured resistive and assistive force as a function of temperature and normal force. In this article, the resistive forces are defined as the positive values; thus, the assistive forces are defined as the negative ones. Again, to account for the tip change, the normal load is the sum of the adhesion force and the applied normal force. For scanning upward surface step edges, the resistive force is linearly dependent on the applied normal force. For downward scanning, the resistive force stays almost constant at different normal forces while the assistive force value increases with the normal load, i.e., a larger normal load leads to a constant resistive and a larger assistive force. However, for all the resistive and assistive forces, the friction coefficient only changed a little bit at different temperatures (see Table I for the results of the linear fit). In addition, the resistive force for the double-layer step edge was observed to be roughly twice larger than that of the single-layer step edge; however, the assistive force seems to be less influenced by the height of the step edge. The downward assistive force results from the mechanical deformation of the tip, meaning that the assistive force depends on the geometry of the tip and the height of the step. In our measurements, all of the assistive forces for the double-layer step were indeed observed to be slightly larger than those for the single-layer step, which is consistent with the fact that a higher step leads to larger deformation of the tip. Considering the assistive friction coefficient, the assistive friction coefficient for a double-layer step should also be larger than a single-layer step, which was observed at a temperature of 297.7 K and 187.6 K. As we further decrease the temperature down to 99.0 K, the wear of the graphene edge becomes very easy, so the violation at 99.0 K may be caused by the different density of defects at two step edges. The resistive force is supposed to be temperature dependent since temperature can help the tip jump out of the edge Schwoebel–Ehrlich barrier, which is not the case in our measurements. In order to get deep inside the mechanism of temperature effects, we turn to theoretical calculation.

In the previous results, the resistive force during the upward and downward scanning across various atomic step edges has been explained in the context of an extended modified thermally activated PT model featuring an additional potential barrier17 at the step edge, called the Ehrlich–Schwoebel barrier, which was adapted from the surface diffusion barrier. To analyze the contribution of temperature to the resistive force, we used the same approach suggested by Hölscher et al.17 and modified this model by introducing temperature effects. A schematic of the theoretical model describing the friction at atomic-scale surface step edges is shown in Fig. 4(a). A ball-like tip with the mass of m = mx = mz = 1010 kg31 representing the FFM tip is driven by a harmonic spring represent the FFM cantilever with a spring constant of kx = 5 N/m and with a sliding velocity of vx = 1 μm/s cross the atomic step edge.17 Here, the substrate was treated as a rigid body. The interaction between the tip and substrate atoms was modeled by the Lenard-Jones (L-J) potential. The L-J potential can be written as

(1)

where ri is the distance between the ball-like tip and the ith atom. The parameters r0 = 0.45 nm and E0 = 1.0 eV represent the equilibrium distance and the binding energy between the tip and substrate carbon atoms, respectively. Figure 4(a) shows the color-coded density plot of the interaction energy between the tip and the substrate with the atomic distance a = 0.3 nm.17 

FIG. 4.

Illustration of the theoretical model and typical potential and lateral force curves. (a) A schematic of the theoretical model describing the friction between an FFM tip and an atomic-scale surface step edge together with the density plot of the tip–sample interaction energy used in the calculation, showing an energy barrier at the high terrace of the step edge and a distinguished potential minimum at the low terrace of the step edge. (b) The tip–sample interaction potential profiles along the sliding path. (c) The frictional force acting on the tip when scanning upward and downward the step edge.

FIG. 4.

Illustration of the theoretical model and typical potential and lateral force curves. (a) A schematic of the theoretical model describing the friction between an FFM tip and an atomic-scale surface step edge together with the density plot of the tip–sample interaction energy used in the calculation, showing an energy barrier at the high terrace of the step edge and a distinguished potential minimum at the low terrace of the step edge. (b) The tip–sample interaction potential profiles along the sliding path. (c) The frictional force acting on the tip when scanning upward and downward the step edge.

Close modal

The mathematical formulation for the dynamics of the tip can be described by the Langevin equation as follows:

(2)
(3)

where FN is the applied normal force, γx and γz are the damping coefficients, and ξ(t) is the thermal activation random force satisfying the fluctuation–dissipation relation,

(4)
(5)
(6)

θ is subjected to a uniform distribution 0,2π.

Because of a lack of a suitable analytical description, we turn to a numerical approach by introducing the fourth-order Runge–Kutta (RK) algorithm developed by Kasdin.32 These equations were solved for variables x and z. Finally, the friction force experienced by the tip can be calculated as

(7)

The density plot of the calculated tip–sample interaction energy is shown in Fig. 4(a) at T = 300 K, FN = 1 nN, and v = 1 μm/s, scanning from left to right. Figure 4(b) shows the plots of the interaction energy profile along the scanning path. Caused by the lack of carbon atoms beyond the step edge, a step-induced Schwoebel–Ehrlich potential barrier can be clearly observed showing an energy barrier at the high terrace of the step edge and a distinguished potential minimum at the low terrace of the step edge.22,23 The corresponding lateral force curve is then plotted, as shown in Fig. 4(c), featuring the resistive forces both scanning up and down the step edge. When the tip scans upward the step edge, the tip first jumps into the local potential minimum before the step edge; then, the tip needs to jump from this minimum over the Schwoebel–Ehrlich barrier which requires a very large lateral force. On the other hand, when the tip scans downward the step edge, the tip first jumps over the Schwoebel–Ehrlich barrier into the minimum at the bottom of the step edge. After this first jump, the tip is temporarily stuck in the minimum at the bottom of the step edge. Then, a slightly large force is needed to help the tip completely jump out of the edge.17 Compared with the downward scan, the lateral force is much higher for the upward scan due to the much higher and steeper energy barrier.

In order to study the effects of temperature, we have performed several calculations for temperatures from 0 K to 300 K and normal forces ranging from 0 nN to 10 nN. Figure 5(a) shows the typical friction increase under different normal forces at T = 0 K (blue symbols) and T = 300 K (red symbols). The resistive force for the upward scanning is linearly dependent on the normal force while the resistive force for the downward scan is nearly normal force independent. Figure 4(b) shows the summary of the temperature effects on the friction increase. One can see that temperature has a negligible effect on the resistive force, consistent with the experimental observation. Actually, the step-induced Schwoebel–Ehrlich potential barrier is around 1 eV, which is two orders of magnitude larger than temperature effects; hence, it is reasonable to have observed the temperature independent resistive force in our measurements.

FIG. 5.

Temperature and normal force dependence of the resistive force across a surface step edge. (a) The resistive force vs normal load at T = 0 K and T = 300 K. (b) Temperature effects on the resistive force while scanning upward and downward a step edge.

FIG. 5.

Temperature and normal force dependence of the resistive force across a surface step edge. (a) The resistive force vs normal load at T = 0 K and T = 300 K. (b) Temperature effects on the resistive force while scanning upward and downward a step edge.

Close modal

To conclude, we present both experimental measurements and theoretical calculations of the temperature effects on atomic friction across HOPG surface step edges. For the upward scanning from the lower terrace to the upper terrace, a large resistive force which is linearly dependent on the normal force is observed. A similar resistive force but with a smaller magnitude together with an assistive force is observed for the downward scanning; however, the resistive force is found to be independent of the normal force compared with the upward scanning while the assistive force increases with the normal force. Furthermore, the resistive force for the double-layer step edge is observed to be twice larger than that of the single-step edge, and at the same time, the assistive force that originated from the horizontal component of the total force acting on the tip is observed to be less influenced by the height of the step edges. Finally, the experimental results reveal that temperature has a negligible effect on the friction coefficients at the step edges, which is inconsistent with the thermal activated friction where friction should decreases with temperature. Together with the theoretical studies, this observation can be explained by a process where temperature effect is very small compared with the edge Schwoebel–Ehrlich barrier.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The author acknowledges the financial support from the NSFC (Grant Nos. 11890672, 61704013, and 11602205) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2682018CX11 and 2682016ZY03).

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