Nanofriction oscillation driven by sublayer indirect contact of silicon tip sliding on few-layer graphene

The main characteristic of friction behavior at the nanoscale is generally distinct from that at the macroscale, resulting in a new research field of nanotribology.^1–4 A fundamental understanding of nanofriction is essential for various technological applications including nanoelectromechanical systems (NEMS).⁵ Owing to its fundamental and practical significance, nanofriction has been paid much attention in the past decades. Numerous studies have illustrated that nanofriction is sensitive to a variety of factors, such as normal load,^6–8 contact area,^2,9 contact edge,^10,11 temperature,¹² environment,^13–15 velocity,¹⁶ surface topography^17–19 and the commensurability.^20,21 Besides, some of the entirely new friction phenomena, like the thickness-dependent friction^22,23 and the negative friction coefficient,²⁴ were reported. Despite the great progress made in recent years, many key aspects of nanofriction still need to be explored. Particularly, with enormous new two-dimensional materials predicted and fabricated, nanofriction with few layers 2D van der Waals materials as lubrications become possible and the theoretical studies become urging and promising.

Graphene is especially promising as an extremely thin but effective solid 2D van der Waals lubricant for MEMS.^25,26 Nanofriction of few layers graphene has been investigated by experiment and molecular dynamics simulation in recent years.^2,22 Previous studies have revealed an interesting and important phenomenon in which sliding friction varies regularly with the thickness of graphene sheets.^22,23 In most circumstances, nanofriction decreases with the increase of graphene thickness, which was initially observed in an AFM experiment of one- and two-layer epitaxial graphene samples grown on SiC, and the observation was explained via an electron-phonon coupling effect.²³ Since then, similar thickness-dependence nanofriction behaviors have been observed both in experiments and atomistic simulations.²² Typically, these observations could be explained by the folding effect associated with the surface deformation of the graphene layer.^17,22 However, further research revealed that the reduction in contact area caused by the folding effect changing with the increasing number of graphene layers is not proportional to that in friction, thus the mechanism behind this phenomenon is still unclear.² 

Furthermore, different trends may also occur when the system environments changes. For example, AFM measurement on suspended graphene shows that friction increases with the increasing number of layers in the case of small loadings, but decreases with the increase of the number of layers under large loadings due to the strong adhesive force of the sliding tip,²⁷ which means that the normal loadings can have great impact on the thickness-dependent friction in the suspended graphene systems. Another recent experiment indicates that the thickness-dependent friction could be altered by scanning AFM tips with different radii against substrates of controlled nanoscale roughness covered with graphene.¹⁴ Moreover, Ye et al. demonstrate that the thickness-dependent nanofriction can be altered in the system with few-layer graphene by changing the surface roughness of the substrate. When the substrate is atomically smooth, the nanofriction is monotonically increased, and the nanofriction is oscillated when the substrate roughness (rms) is 0.1 nm, which shows that the substrate have great effect on the trends of thickness-dependent friction.²⁸ In all, monotonically decreasing nanofriction of few-layer graphene systems only appears in the right combination of substrate surface roughness, tip radius, and the relative adhesion between the tip and the graphene as well as between the graphene and the substrate and so on. Obviously other combinations of these factors could cause other phenomena.²⁸ Therefore, nanofriction with few-layer graphene as lubricant is still mysterious.

In this work, the atomic-scale friction behavior of silicon tip sliding on graphene layers with or without a silicon substrate was investigated by molecular dynamics (MD) simulations. The results interestingly reveal a unique even-odd oscillation friction behavior, which is robust to the normal loadings and the radius of the sliding tip, but can be affected by the substrate. More specifically, the oscillation of nanofriction occurs both in the substrate and suspended graphene systems. The even-odd oscillations in nanofriction of suspended system are attributed to the indirect contact between the top-layer and second-layer graphene, while the direct contact dominates the substrate system due to the change of contact configuration by the substrate. The results not only illustrated a unique even-odd oscillation phenomenon in the systems with few-layer graphene, but also introduced a new way to design the nanofriction system with two-dimensional van der Waals materials as lubrications, which is significantly meaningful for MEMS.

To clearly observe the thickness-dependent nanofriction behavior of few-layer graphene system, we carried out the molecular dynamics simulation of a silicon tip sliding on large graphene nanosheets using the LAMMPS package.²⁹ In our simulations, the covalent bonds of C-C in graphene and Si-Si in smooth substrate and in the crystalline Si tip were described using the Tersoff³⁰ and Stillinger-Weber³¹ potentials, respectively. A typical 6-12 Lennard-Jones potential was employed to describe van der Waals adhesive interaction between graphene and the tip, graphene and the substrate, and between the graphene layers. The Lennard-Jones parameters were chosen such that the work of adhesion (Ead) or the pull-off force (fad) obtained by the molecular dynamics calculation were at the same scale as those from experimental measurements.^27,32–36 Above potential parameter settings have been successfully described the similar system.² The simulations were performed at 300K using a Nose-Hoover thermostat³⁷ and the timestep for all the simulations is 1 fs.

Two systems were considered in our molecular dynamic simulations, as illustrated in Figure 1(a) and (b). Figure 1(a) shows the system in which the silicon tip is placed on the suspended graphene, and Figure 1(b) shows the system in which the silicon tip is placed on the graphene on a smooth silicon (111) substrate. The bottom layer graphene is incommensurately contact with a substrate and the multilayer graphene is AB stack. The sliding tip has a radius of 0.8 nm and its (111) surface is in contact with graphene. The length and width of the graphene are 38.2 and 38.2nm, respectively. The smooth silicon substrate is about 1nm thick (containing three layers of silicon atoms), and its length and width are 39 and 39nm, respectively. To incorporate the compliance of atomic force microscope cantilever in experiments, we coupled harmonic springs on the silicon tip to pull the tip in the x direction and to apply a fixed normal force in the z direction. The stiffness of the normal spring is 0.16Nm-1 and the lateral spring is 30.0Nm-1, comparable to the previous experiments.^2,22 The boundaries are all fixed and non-periodic in all three dimensions.

In our simulations, all the atoms of the silicon substrate are fixed at their initial positions, and the silicon tip is set to be rigid so that the tip slides as a whole during the simulation without deformation. After energy minimum process, a normal loading of 0.8nN is applied to the tip and therefore the tip is displaced towards the graphene. Then, the entire system was relaxed for 1ns after adding the tip and further relaxed after the normal loading was imposed. The friction tests were performed by displacing the lateral spring along the x direction with a constant velocity of 2 ms-1 under a given normal loading, and calculating the lateral force acting on the virtual atoms. To enhance the damping of oscillation (primarily along the lateral direction), we artificially decreased the tip mass by a factor of ten.² The instantaneous resistance force on the tip along the sliding direction is recorded for each timestep, and then the mean friction force is obtained by averaging the sliding resistance force over time. Besides the rigid tip and the fixed atoms at the boundaries, all the other atoms in both graphene and Si substrate were subjected to the Nose-Hoover thermostat.³⁷ The contact atom number of the tip with the top-layer graphene and the top-layer graphene with the sublayer graphene are estimated by setting the cutoffs of 3.0 and 3.4Å, respectively. Ye et al and Li et al. have successfully using the same approach to study the effect of pre-existing wrinkles on the contact interface and the effect of substrate roughness on thickness-dependent friction.^2,28 It should be noted that the qualitative conclusions related to contact area are not affected even when we change the cutoff distance slightly.

To adequately investigate the remarkable effect of graphene thickness upon the nanofriction force, we firstly perform a series of calculations based on a system in which 8nm silicon tips were slid over suspended graphene with different thickness. The normal loadings are 0.8nN, which is comparable to that used in previous experiments and simulation studies. As can be clearly seen from Figure 2, in different thicknesses of the suspended graphene system, the silicon tip is subjected to tangential stick-slip resistance during sliding. And we also see that the period of the stick-slip motion is approximately 0.246nm, which is the same as the lattice constant in the zigzag direction of graphene.

In order to observe the change of friction with the thickness of graphene, we obtained the mean friction force by averaging the instantaneous frictional force of the probe during sliding, as shown in Figure 3(a). It is shown that the mean friction forces of the tip on different graphene layers exhibit the even-odd oscillation behavior, i.e the friction force is about 0.25nN when the silicon tip slides on single layer graphene, while it becomes to 0.15, 0.35 and 0.20nN on two, three and four layers, respectively. In general, the mean friction forces on odd number layers are larger than those on even number layers. Since the results we have observed have not been reported, considering that the results may be caused by statistical errors, we have performed a more accurate comparison simulation of the suspended system by shortening the timestep. And we found that the change of mean friction force is about 0.001nN, which is about 1% of the even odd oscillation amplitude. Therefore, the observed “oscillation” is not statistically significant.

Our results differ from those of similar systems in which friction force is monotonically decreased with thickness. As discussed in the introduction section, many studies have researched the thickness-dependent phenomena in graphene systems. For example, Lee has concluded that friction is monotonically decreased with thickness for atomically thin materials weakly bound to substrates;²² Deng and Spear have discovered that the relationship of thickness-dependent nanofriction can be dramatically influenced by applying different loadings,²⁴ as well as the size of tips.¹⁴ Ye et al. have shown the similar friction oscillation phenomenon caused by substrate roughness.²⁸ Based on the above studies, in order to explore the source of this odd-even oscillation phenomenon, we will further study the effects of normal load, tip size and substrate on this intriguing phenomenon.

Firstly, we apply different normal loadings to check if the even-odd oscillations exist to investigate the effect of the loading on the even-odd oscillation phenomena. Here, the normal loadings were chosen as 0.4 and 1.6 nN, and results are shown in Figure 3(c) and (e), respectively. From Figure 3(a), (c) and (e), we clearly see that, with the normal force increasing from 0.4 to 1.6nN, the mean friction force will become larger. And this is consistent with most of previous studies.^6,27 Moreover, we also found that the friction forces of all three systems have an even-odd oscillating behaviour with increasing graphene thickness, which means that the mean friction force even-odd oscillation phenomenon is independent of the normal loadings. In summary, the above simulation results confirm that the “oscillation” is not caused by the load.

In general, the magnitude of friction force both in macro-scale and micro-scale can be well understood by the classical contact theory. From the classical contact theory, the variation of the contact area between tip and graphene mainly determines the magnitude of sliding friction force, while the contact area is always the contact between tip and top graphene layer. Based on that, we formulate a bold hypothesis that the contact between top-layer and sub-layer graphene is named indirect contact, and we will show below that it can also have a great impact on the thickness-dependent nanofriction. Therefore, to adequately understand the even-odd oscillation phenomenon, we calculated both the direct and indirect contact atom number of three suspended graphene systems under various normal loadings applied, as shown in Figure 3(b), (d) and (f), respectively. In Figure 3(b), (d) and (f), the changes of the direct and the indirect contact atom number of all three systems exhibit the same trend. In this study, as the thickness of graphene increased from one layers to four layers, the number of the direct contact atoms (represented by red line) monotonously increased from 200 to 250, and this is caused by a decrease in the deformation degree of graphene layers downward. However, the number of atoms in indirect contact (represented by blue line) has the similar even-odd oscillation behavior to the friction force. The more indirect contact, the greater the friction. Having done these studies, we conclude that it is the indirect contacts that induce the thickness-dependent even-odd oscillations, and applying different normal loadings cannot change this relationship.

More explicitly, to deeply analyze these thickness-dependent even-odd oscillation phenomena, we also made the contour diagram of atom positions in the upper two graphene layers under the applied loading of 0.8nN, as shown in Figure 4(a)–(f), respectively. Here the left side of each picture shows the top view, and the right side is the side view.

From the top view, we can clearly see a noticeable pattern that three groove marks intersect at the contact point of the silicon tip and the topmost graphene layer. The groove marks are caused by the combined application of the normal loading, the adhesive force of the silicon tip and the flexibility of few-layer graphene. Few previous studies have reported this pattern, especially those systems where friction shows a monotonous decrease. Comparing to similar system researches, we find that the size and adhesive force of the tip can be the main reason for the appearance of groove marks in the suspended system, and the groove marks can be the main cause of even-odd oscillations. From Figure Figure 4(a) to (f), we can see the surface pattern evolves with the thickness of graphene due to the change of graphite quality. The system with odd number graphene layers, i.e. monolayer, tri-layer and five layers clearly shows a relatively larger ripple in the atomic distribution along the x dimension, as shown in Figure 4(a), (c) and (e), respectively. This larger fluctuation along the sliding direction, in turn, will result in a larger atom number of indirect contacts between the top and the second layer graphene. As for the system with even number graphene layers, i.e. bilayer, four layers and six layers, both the top and the second layer graphene show small fluctuations along the sliding direction. And the smaller fluctuation between the top and second graphene layers will have relatively smaller atom numbers of indirect contact, as shown in Figure 4(b), (d) and (f), respectively. Of course, the numbers of different indirect contact atoms will dramatically change the interaction of the top and sublayers and have an effect on friction, resulting in even-odd oscillations of the friction force.

Next, we turned to investigate the effect of tip size on the even-odd oscillations phenomena of friction force. Previous studies have shown that the variation in friction with the number of layers was noticeably dependent upon the dimension of the AFM tip.^14,38 With this in mind, we used the 12nm tip to observe the effect of the tip size on the thickness-dependent even-odd oscillation phenomenon. In the MD simulations, the applied normal loading is 0.8nN for comparison. For different thickness of graphene, the trends in friction force and the number of atoms in direct and indirect contact are shown in Figures 5(a) and (b), respectively. Compared to the 8 nm tip, the 12 nm tip is subject to greater friction due to more interaction with the graphene layer. Interestingly, the mean friction force shows a thickness-dependent even-odd oscillation behaviour, which is similar to the case of 8nm tip. As shown in Figure 5(b), the direct contact atom number is three times larger than that of 8nm, and it still shows a monotonously decreasing trend with the increase of graphene thickness. The indirect contact atom number has the similar even-odd oscillations to the friction force and the groove marks also exist. However, 12nm is still much smaller than that of experiment. If the size is larger enough, the groove marks will be disappearing and will not play an important role.

Finally, we explored the effect of rigid substrates on the thickness-dependent friction. For simplicity, the substrate is also chosen to be a clean Si (111) surface. The simulation results under the normal loading of 0.8nN are presented in Figure 5(c). It can be seen from Figure 5(c) that the mean friction force exhibits a tendency of even-odd oscillation as the thickness of the graphene changes. Compared with the suspended case, the mean friction force on rigid substrate is larger and tends to decreases the thickness of the graphene increases, which is consistent with previous studies. As for the “oscillation”, it can be caused by the adhesive force of the substrate and the sliding tip. And the surface roughness of the substrate may be the reason why the experiment did not observe this phenomenon. In Figure 5(d) we present the number of the direct and indirect contact atoms in different graphene thickness systems. Unlike the suspended system, with the graphene thickness increasing, the number of atoms in direct contact on the Si (111) substrate system shows even-odd oscillation phenomenon that is in accordance with friction force. However, in this case, the indirect contact is reflected on the top layer because the rigid substrate has a supporting effect, preventing the Out-of-plane deformation of the graphene layer. Likewise, the thickness-dependent friction even-odd oscillation behaviour still exists in the substrate system, though it is explained by the change in the number of direct contact. Overall, in our study the even-odd oscillation is a generality characteristic of silicon tip sliding on graphene layers. And this property may also be found in other two-dimensional van der Waals layered materials, such as black phosphorus, MoS2 et al.

In summary, using the MD simulation we have studied a typical model in which a silicon tip slides on graphene layers of different thickness. We found that the mean friction of these systems clearly exhibits even-odd oscillations associated with the thickness of graphene. Furthermore, we have demonstrated that such even-odd oscillation behavior is independent of the silicon tip size, as well as the applied normal loading. We have successfully introduced the concept of indirect contact to understand this intriguing phenomenon, and the even-odd oscillations in nanofriction are attributed to the change of indirect contact between the top and second graphene layers. We also demonstrated that such indirect contact oscillations disappear and are reflected by the direct contact oscillations between the tip and the top-layer graphene when graphene layers are on a rigid silicon substrate. Our findings are particularly beneficial in providing atomic level insight into the friction of tip on graphene systems, which is useful for the design of NEMS devices based on graphene, and providing a new way to understand two-dimensional van der Waals lubrications.

This work is partly supported by NSF of China (Grant No. 11774078) and partly by NSF of China-Henan Joint Fund (Grant No. U1604131, U1604251). The MD simulations were carried out at the high-performance computer center of Zhengzhou University and the high-performance computer center of Henan University.