Reduction of interfacial friction in commensurate graphene/h-BN heterostructures by surface functionalization

Abstract

The reduction of interfacial friction in commensurately stacked two-dimensional layered materials is important for their application in nanoelectromechanical systems. Our first-principles calculations on the sliding energy corrugation and friction at the interfaces of commensurate fluorinated-graphene/h-BN and oxidized-graphene/h-BN heterostructures show that the sliding energy barriers and shear strengths for these heterostructures are approximately decreased to 50% of those of commensurate graphene/h-BN. The adsorbed F and O atoms significantly suppress the interlayer electrostatic and van der Waals energy corrugations by modifying the geometry and charge redistribution of the graphene layers. Our empirical registry index models further reveal the difference between the roles of the F and O atoms in affecting the sliding energy landscapes, and are also utilized to predict the interlayer superlubricity in a large-scale oxidized-graphene/h-BN system. Surface functionalization is a valid way to control and reduce the interlayer friction in commensurate graphene/h-BN heterostructures.

---

1. Introduction

Understanding interfacial tribology is essential for the actual application of two dimensional (2D) layered materials, that are held together by weak interlayer van der Waals (vdW) forces, in functional devices and nanoelectromechanical systems (NEMS). An important type of solid lubricant is homogenous layered materials such as graphite, hexagonal boron nitride (h-BN), and 2H-molybdenum disulphide.^1–5 The interfacial friction and interlayer potential landscape of homogenous 2D layered materials have been extensively studied using experimental and theoretical methods.^6–20 Recently, wearless friction or superlubricity was observed in a layered graphene system with incommensurate stacking.⁶ However, this ultra-low interlayer friction strongly depends on the degree of commensurability between the lattices of graphene layers, and will disappear when one of the sliding surfaces is rotated by a certain angle or the interface transforms from incommensurate into commensurate. On the other hand, artificial heterogeneous layered materials, which are stacks of different 2D crystals, have attracted great scientific interest as they show novel properties coming from the combination of the unique physical properties of each individual layer and all their different advantages can be properly utilized.^21,22 Due to lattice mismatch between 2D crystals and the presence of moiré patterns, layered heterostructures often exhibit incommensurate interfaces, which remarkably reduce the interlayer friction and sliding resistance^23–25 because of changes in the interatomic distances and interlayer interactions. A recent experimental work²⁶ reported a commensurate–incommensurate transition for graphene on top of h-BN when the rotation angle of graphene reached a critical value, and this led to an alteration in the electronic and optical properties of graphene/h-BN heterostructures. Another theoretical study²⁷ further revealed the important role of the ratio between carbon–boron and carbon–nitrogen interactions in graphene/h-BN heterostructures, determining the strain distribution in the moiré pattern.

For vdW layered materials, commensurate stacking is energetically more favorable than incommensurate stacking.²⁸ Lowering the interlayer friction at commensurate interfaces is an intriguing aspect in the nanofabrication of vdW layered materials and the control of their tribological properties. By using h-BN layers as flat dielectric substrates, graphene/h-BN heterostructures exhibit much higher carrier mobility and current density compared with SiO₂ substrates,^29–31 which has led to the realization of graphene electronic devices with better performance. Experiments have shown that the stacking of graphene on h-BN could adopt the commensurate state because of the very slight lattice mismatch between them.²⁶ Graphene and h-BN have high surface-to-volume ratios and their surfaces are easily decorated by other atoms, molecules and radicals. Moreover, surface functionalization, such as fluorination and oxidation, is an important route to change and monitor the mechanical and physical properties of 2D crystals.^32–36 Further study into the effects of surface functionalization on interlayer interactions and friction in graphene/h-BN systems is necessary for the development of layered heterostructures based on graphene and graphene-like 2D materials.

In this study, we have investigated the sliding energy corrugation and friction at the interfaces of commensurate fluorinated-graphene/h-BN and oxidized-graphene/h-BN heterostructures using first-principles calculations. From the calculated potential energy surfaces (PESs), the sliding energy barriers and shear strengths for these fluorinated and oxidized heterostructures are approximately reduced to 50% of those of commensurate graphene/h-BN. The reduction in the interlayer friction is attributed to fluorination and oxidation induced charge redistribution and a change in vdW interactions. Furthermore, registry index models are developed to describe the role of F and O atoms in the interlayer friction and provide a quantitative way to predict the sliding energy landscape of a large-scale fluorinated or oxidized graphene/h-BN system.

2. Model and method

In our model, two types of heterostructures were established in the rhombus unit cells (a₁ = a₂ = 10.096 Å) in which a 4 × 4 graphene monolayer (32 atoms) was placed on a 4 × 4 h-BN monolayer (32 atoms) with AB stacking and the top surface of the graphene layer was fully decorated with 16 F or O atoms, as shown in Fig. 1 and 2. The initial lengths of the B–N bonds and C–C bonds were 1.45 Å and 1.42 Å, respectively, and there was a vacuum region larger than 2.5 nm in the direction (the z direction) perpendicular to the heterostructure planes. All computations were performed within the framework of density-functional theory (DFT) as implemented with the VASP code by using the projector augmented wave method with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional.^37–39 The influence of vdW interactions was considered by using a modified version of vdW-DF, referred to as “optB86b-vdW,” in which the revPBE exchange functional of the original vdW-DF of Dion et al., is replaced with the optB86b exchange functional to yield more accurate equilibrium interatomic distances and energies for a wide range of systems.^40,41 Those systems were relaxed by using a conjugate-gradient algorithm until the force on each atom was less than 0.1 eV nm⁻¹.

Fig. 1 Initial atomic configurations of fluorinated-graphene/h-BN heterostructures. (a) F–C bonds on the top of B atoms (a-type) and (b) F–C bonds on the hollow sites of the h-BN plane (b-type), and the corresponding PESs when the fluorinated graphene layers slide with respect to the bottom h-BN layers with fixed interlayer distances. The yellow, cyan, blue and pink dots are F, C, N and B atoms, respectively.

Fig. 2 Initial atomic configurations of oxidized-graphene/h-BN heterostructures. (a) O on the C–C bond at 30° to the a₁ axis (a-type) and (b) O on the C–C bond vertical to the a₁ axis (b-type), and the corresponding PESs when the oxidized graphene layers slide with respect to the bottom h-BN layers with fixed interlayer distances. The red, cyan, blue and pink dots are O, C, N and B atoms, respectively.

For fluorinated-graphene/h-BN, there are two stable adsorption sites for the F atoms (Fig. 1): one is to form F–C bonds on the top of B atoms (a-type) with an interlayer distance of 0.332 nm and another is to form F–C bonds on the hollow sites of the h-BN plane (b-type) with an interlayer distance of 0.323 nm. There are also two stable adsorption sites for the O atoms (Fig. 2): one is the O positioning on the C–C bond 30° to the a₁ axis (a-type) with an interlayer distance of 0.327 nm and another is the O positioning on the C–C bond vertical to the a₁ axis (b-type) with an interlayer distance of 0.324 nm. To match with the h-BN layers, the lattice constants of the fluorinated and oxidized graphene layers are compressed to 0.078% and 5.25%, respectively. As a result, the interfaces of the fluorinated-graphene/h-BN and oxidized-graphene/h-BN heterostructures can still be considered to be commensurate. After structural relaxation, the fluorinated and oxidized graphene layers are translated to relatively different positions on the a₁–a₂ plane where the nearest translational positions are separated by 0.036 nm, and the interlayer distances with respect to the bottom h-BN layers are fixed at the initial equilibrium interlayer distances. At different sliding positions, the three coordinates of each atom are fixed, and computations with an energy cutoff of 500 eV and special k points sampled on a 6 × 6 × 1 Monkhorst–Pack mesh⁴² are employed to calculate the total energy. The corresponding PESs for interlayer sliding are constructed and obtained using the deviation between the calculated total energy at different positions and the lowest energy of the system. As the force in the z direction (normal direction) of each atom is hard to modulate to a desired normal force in the DFT calculations, we construct the PES by fixing the interlayer distance to that which has been adopted by previous theoretical studies on interlayer sliding.^{11,13,16,24,25} However, for comparison, we have also studied the case of the normal force approximately being zero, where the z coordinate of each atom is fully relaxed but the x and y coordinates are fixed at different sliding positions.

3. Results and discussion

The fluorination of graphene on one side leads to a corrugated structure, as shown in Fig. 1, where the C atoms bonding with F atoms are lifted up. The formation energies E_f for the a-type and b-type fluorinated heterostructures are 0.96 and 0.79 eV, respectively, which are calculated using E_f = E_t − E_gra/bn − 8E_f2 (E_t is the total energy, E_gra/bn is the energy of the graphene/h-BN heterostructure and E_f2 is the energy of a F₂ molecule). The fluorination of graphene/h-BN is endothermic, and the b-type is more stable than the a-type because of a lower total energy. From the PES of the a-type, the maximum energy barrier (the difference between ΔE_max and ΔE_min) is 7.8 meV per C atom when the fluorinated graphene layer slides relative to the lower h-BN layer, while the maximum energy barrier for the b-type increases to 9.2 meV per atom. According to the maximum static resistance force f_mr acting on the fluorinated graphene layer during the sliding process, the interlayer shear strength τ can be estimated using , where A is the area of the unit cell. The corresponding shear strengths for the a-type and b-type fluorinated graphene layers are 0.488 and 0.625 GPa.

Differently from the fluorinated graphene, the plane of graphene still remains flat after oxidation, as shown in Fig. 2. Using the same calculation method as for fluorination, the formation energies E_f for the a-type and b-type oxidized heterostructures are 0.77 and 0.70 eV, respectively, which indicates an endothermic oxidation process. The maximum energy barriers of the a-type and the b-type are 8.9 and 9.7 meV per atom and the shear strengths are 0.611 and 0.704 GPa, respectively, when the oxidized graphene layers slide relative to the lower h-BN layers. The interlayer energy corrugations and shear strengths of the oxidized graphene/h-BN heterostructures are larger than that of the fluorinated heterostructures. We have also calculated the sliding PES of the AB stacked graphene/h-BN (Fig. 3), and the obtained maximum energy barrier and shear strength are 16.4 meV and 1.121 GPa, respectively. Compared with the graphene/h-BN system, the energy corrugations in the a-type and b-type fluorinated heterostructures are reduced by 52.4% and 44.0%, respectively, and for oxidized heterostructures the energy corrugations in the a-type and b-type are reduced by 46.0% and 41.0%, respectively. As proposed by the Prantl–Tomlinson model, the energy corrugation in the PES and the stiffness of the sliding layer are important factors for predicting actual interlayer friction properties.^43,44 Higher energy corrugation and smaller stiffness will lead to higher interlayer friction. According to a previous study,¹¹ the stiffness K of the sliding layer can be approximately estimated using K = ∂²E/∂s², where E is the potential energy field of the interlayer interaction, which can be deduced using the calculated PES, and s is the relative lateral displacement of the sliding layer. Our calculations show that the stiffnesses of both fluorinated and oxidized graphene are slightly larger than that of pure graphene. These results clearly demonstrate that fluorination and oxidation are valid ways to reduce the interlayer friction in commensurate graphene/h-BN heterostructures.

Fig. 3 PES for a graphene layer sliding on an h-BN layer with a fixed interlayer distance of 0.326 nm.

To understand the roles of the adsorbed F and O atoms, we plot the charge density distributions of the fluorinated, oxidized and pure graphene/h-BN heterostructures in Fig. 4. It can be seen from the magnitudes of charge accumulation that the F, O and N atoms carry negative charges and the B atoms carry positive charges, and fluorination and oxidation slightly influence the charge distribution in the h-BN layer. The charge properties are slightly changed when the graphene layer slides with respect to the bottom h-BN layer. For the b-type heterostructures, the charge density distributions are similar to those of the a-type. The main factors that govern energy corrugation in such layered systems are electrostatic and dispersion (or vdW) interactions. In contrast to graphene/h-BN, the coupling of the repulsive coulomb interaction between the negative F or O atoms and negative N atoms with the attractive coulomb interaction between the negative F or O atoms and positive B atoms weakens the interlayer energy corrugation. Moreover, the interlayer vdW energy corrugations are also weakened by the fluorination and oxidation, as shown by Fig. 5. Due to an increase in the interlayer distance between the C atoms bonding with the F atoms and the h-BN layer, the vdW energy corrugations in the fluorinated systems are lower than those in the oxidized systems. Both the electrostatic and vdW energy corrugations decrease with fluorination and oxidation, which consequently leads to the reduction of interlayer friction.

Fig. 4 2D projections of the charge densities (in units of e Å⁻³) of the (a) a-type fluorinated, (b) a-type oxidized and (c) pure graphene/h-BN heterostructures. The dot denotation is the same as that in Fig. 1 and 2.

Fig. 5 Energy corrugation profiles of the van der Waals energy for the fluorinated, oxidized and pure graphene layers sliding on the h-BN layers with straight paths crossing the maximum and minimum values of ΔE.

Besides the case of fixed interlayer distance, we have also studied the interlayer energy corrugations of pure, a-type fluorinated and a-type oxidized graphene/h-BN heterostructures in which their normal forces are approximately zero, realized by the relaxation of the z coordinates of atoms. The obtained PESs of the three cases are very similar to those of the corresponding heterostructures with fixed interlayer distances, and the maximum energy barriers for the three cases are 11.1, 8.8 and 8.4 meV per atom, respectively. Coinciding with the results shown in Fig. 1–3, fluorination and oxidation significantly reduce the interlayer sliding friction when the normal force is zero. Moreover, the effects of other oxidation behaviour, where the surface of graphene was uniformly decorated with four O–H radicals, on the interlayer sliding friction has been investigated using the same method with the normal force being zero. Here the O–H radicals adsorb on the C atoms and are separated by 0.505 nm. The calculated maximum energy barrier is only 6.6 meV per atom, just 59% of that of the pure graphene/h-BN heterostructure (11.1 meV per atom). This means that the adsorption of O–H radicals can also reduce the interlayer sliding friction.

As the capability and scale of the present DFT calculations are limited, registry index (RI) methods that quantify the registry matching using basic geometric considerations have been recently developed to provide qualitative explanations for the interlayer sliding energy landscapes of layered vdW materials.^13–15 The main contribution to the corrugation energy comes from the electrostatic interactions between the atomic sites, as shown in Fig. 4. Each atom in the unit cell can be ascribed to a circle centered on its position, and then the RI based on the overlap area of the circular projection between the atoms in the top and bottom layers at different sliding positions is used to mimic the sliding energy corrugation.¹³ According to the definition of RI and our DFT results, we establish RI models of the fluorinated and oxidized heterostructures to further elucidate the atomistic mechanisms of surface fluorination and oxidation on the interlayer friction. For the a-type fluorinated-graphene/h-BN, the RI is defined as

Here S^H_C1B, S^H_C1N, S^H_C2B, S^H_C2N, S^H_FB and S^H_FN are the overlap areas of the atomic circle projections between the low C atoms in the top layer and the B atoms in the bottom layer, the low C atoms and the N atoms, the high C atoms and B atoms, the high C atoms and N atoms, the F atoms and B atoms, and the F atoms and N atoms, respectively, when the whole system is at the highest energy state indicated by the DFT results. S^L_C1B, S^L_C1N, S^L_C2B, S^L_C2N, S^L_FB and S^L_FN are the projected circle overlaps at the lowest energy state, as shown in Fig. 6(a). S_C1B, S_C1N, S_C2B, S_C2N, S_FB and S_FN are the projected overlaps of those atoms when the fluorinated graphene layer translates to any position. The atomic radii r defining the circle area, using πr² for the C, B, N and F atoms, are 0.91, 0.86, 0.71 and 0.74 Å, respectively. The mapped RI as a function of the lateral interlayer shifts locates at a range from 0 to 1 and its profile is very consistent with the DFT results, as shown in Fig. 6(a). So the RI model can qualitatively describe the interlayer energy corrugation of the fluorinated heterostructure. For the a-type oxidized-graphene/h-BN, the RI is expressed as

Fig. 6 The profiles of the registry index for (a) a-type fluorinated and (b) a-type oxidized graphene/h-BN, and the corresponding atomic circle configuration and superposition at the highest and lowest total energies. The red, cyan, blue and pink circles are O, C, N and B atoms, respectively. The dark cyan and cyan circles in (a) are the high and low C layers to the bottom h-BN layer, respectively.

Here the definition of each parameter is similar to that for the fluorinated case except that all C atoms are in the same plane, and the atomic radius of the O atoms is 0.82 Å. The obtained RI profile shown in Fig. 6(b) is also consistent with the DFT results. In contrast to the fluorination RI model, there is a coefficient of 0.5 for the O atom related terms in the oxidation RI model, indicating a smaller influence from the O atoms. When this coefficient equals zero, the oxidation RI equation completely becomes the RI of the pure graphene/h-BN system.²⁴ This is because the adsorbed F atoms lead to a corrugated graphene layer, while the C plane still remains flat after oxidation. Therefore, the higher sliding energy barrier and shear strength in the oxidized-graphene/h-BN can be qualitatively explained by the difference between the fluorination and oxidation RI models. Similar trends have also been observed in the RI models of the b-type fluorinated and oxidized graphene/h-BN heterostructures.

The mismatch between the lattice constants of the fluorinated graphene and h-BN is only 0.078%. Such a slight difference means that the interlayer stacking of the fluorinated graphene and h-BN layers in actual situations can be considered as commensurate if no rotation occurs. However, the lattice difference between the oxidized graphene and h-BN is 5.25%, and thus the actual interlayer stacking without any constraints should be incommensurate. Here we have used the oxidation RI model to predict the interlayer friction of oxidized-graphene/h-BN with incommensurate stacking. As shown by the inset of Fig. 7, a graphene flake oxidized on one side with dimensions of 4.796 nm × 8.306 nm is placed on an h-BN layer. The flake is first rotated by an angle and slid along different directions in the xy plane with a fixed interlayer distance of 0.324 nm. Then the corresponding RI at different positions is calculated using our model. The maximum RI for a rotation angle is selected and given in Fig. 7, in which all RIs are lower than 0.42. Smaller maximum RIs indicate lower interlayer energy corrugations and the interlayer sliding in oxidized-graphene/h-BN heterostructures should be superlubricated. This is consistent with other theoretical predictions about interlayer friction in incommensurate systems.^23,24

Fig. 7 The maximum registry index of an oxidized graphene flake sliding on the single-layer h-BN substrate when the graphene flake is rotated by different angles. The inset shows the rotated structure of the oxidized graphene on the h-BN layer.

4. Conclusion

In summary, our first-principles calculations show that surface fluorination and oxidation could remarkably reduce the interfacial friction in commensurate graphene/h-BN heterostructures, as the adsorbed F and O atoms significantly suppress the interlayer electrostatic and vdW energy corrugations. The established empirical registry index models further reveal the difference between the roles of the F and O atoms in affecting the sliding energy landscapes, which are also utilized to predict the interlayer superlubricity in a large-scale oxidized-graphene/h-BN system. These results provide some insights into using surface functionalization to control and reduce interlayer friction in commensurate 2D layered materials.

Acknowledgements

This work is supported by the NSF (11072109, 11472131), the Program for New Century Excellent Talents in University (NCET-13-0855), the Jiangsu NSF (BK20131356), the 973 Program (2013CB932604, 2012CB933403), the Fundamental Research Funds for the Central Universities (NE2012005, NJ20150048, NJ20150048, INMD-2015M02) of China, and the Research Fund of the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (0413G01, MCMS-0414G01), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and sponsored by the Qing Lan Project.

References

1. B. K. Yen, B. E. Schwickert and M. F. Toney, Appl. Phys. Lett., 2004, 84, 4702.
2. L. Rapoport, N. Fleischer and R. Tenne, Adv. Mater., 2003, 15, 651.
3. C. Lee, Q. Li, W. Kalb, X.-Z. Liu, H. Berger, R. W. Carpick and J. Hone, Science, 2010, 328, 76.
4. C. Donnet and A. Erdemir, Surf. Coat. Technol., 2004, 180, 76.
5. S. Brown, J. L. Musfeldt, I. Mihut, J. B. Betts, A. Migliori, A. Zak and R. Tenne, Nano Lett., 2007, 7, 2365.
6. M. Dienwiebel, G. S. Verhoeven, N. Pradeep, J. W. M. Frenken, J. A. Heimberg and H. W. Zandbergen, Phys. Rev. Lett., 2004, 92, 126101.
7. Y. F. Guo, W. L. Guo and C. F. Chen, Phys. Rev. B: Condens. Matter, 2007, 76, 155429.
8. Q. S. Zheng, B. Jiang, S. Liu, Y. Weng, L. Lu, Q. Xue, J. Zhu, Q. Jiang, S. Wang and L. Peng, Phys. Rev. Lett., 2008, 100, 067205.
9. Z. Liu, J. Yang, F. Grey, J. Z. Liu, Y. Liu, Y. Wang, Y. Yang, Y. Cheng and Q. Zheng, Phys. Rev. Lett., 2012, 108, 205503.
10. A. Smolyanitsky, J. P. Killgore and V. K. Tewary, Phys. Rev. B: Condens. Matter, 2012, 85, 035412.
11. L. Xu, T. B. Ma, Y. Z. Hu and H. Wang, Nanotechnology, 2011, 22, 285708.
12. X. Feng, S. Kwon, J. Y. Park and M. Salmeron, ACS Nano, 2013, 7, 1718.
13. N. Marom, J. Bernstein, J. Garel, A. Tkatchenko, E. Joselevich, L. Kronik and O. Hod, Phys. Rev. Lett., 2010, 105, 046801.
14. A. Blumberg, U. Keshet, I. Zaltsman and O. Hod, J. Phys. Chem. Lett., 2012, 3, 1936.
15. O. Hod, Phys. Rev. B: Condens. Matter, 2012, 86, 075444.
16. G. Constantinescu, A. Kuc and T. Heine, Phys. Rev. Lett., 2013, 111, 036104.
17. G. Levita, A. Cavaleiro, E. Molinari, T. Polcar and M. C. Righi, J. Phys. Chem. C, 2014, 118, 13809.
18. J. P. Oviedo, K. C. Santosh, N. Lu, J. G. Wang, K. Cho, R. M. Wallace and M. J. Kim, ACS Nano, 2015, 9, 1543.
19. E. Koren, E. Lörtscher, C. Rawlings, A. W. Knoll and U. Duerig, Science, 2015, 348, 679.
20. W. Gao and A. Tkatchenko, Phys. Rev. Lett., 2015, 114, 096101.
21. A. K. Geim and I. V. Grigorieva, Nature, 2013, 499, 419.
22. E. Gibney, Nature, 2015, 522, 274.
23. L. F. Wang, T. B. Ma, Y. Z. Hu, Q. S. Zheng, H. Wang and J. B. Luo, Nanotechnology, 2014, 25, 385701.
24. I. Leven, D. Krepel, O. Shemesh and O. Hod, J. Phys. Chem. Lett., 2013, 4, 115.
25. X. Y. Zhao, L. Y. Li and M. W. Zhao, J. Phys.: Condens. Matter, 2014, 26, 095002.
26. C. R. Woods, L. Britnell, A. Eckmann, R. S. Ma, J. C. Lu, H. M. Guo, X. Lin, G. L. Yu, Y. Cao, R. V. Gorbachev, A. V. Kretinin, J. Park, L. A. Ponomarenko, M. I. Katsnelson, Yu. N. Gornostyrev, K. Watanabe, T. Taniguchi, C. Casiraghi, H.-J. Gao, A. K. Geim and K. S. Novoselov, Nat. Phys., 2014, 10, 451.
27. M. M. van Wijk, A. Schuring, M. I. Katsnelson and A. Fasolino, Phys. Rev. Lett., 2014, 113, 135504.
28. A. E. Filippov, M. Dienwiebel, J. W. M. Frenken, J. Klafter and M. Urbakh, Phys. Rev. Lett., 2008, 100, 046102.
29. S. D. Sarma and E. H. Hwang, Phys. Rev. B: Condens. Matter, 2011, 83, 121405.
30. J. Xue, J. S. Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. J. Herrero and B. J. LeRoy, Nat. Mater., 2011, 10, 282.
31. E. Kim, T. Yu, E. S. Song and B. Yu, Appl. Phys. Lett., 2011, 98, 262103.
32. Z. H. Zhang, X. C. Zeng and W. L. Guo, J. Am. Chem. Soc., 2011, 133, 14831.
33. Y. Zhao, X. Wu, J. Yang and X. C. Zeng, Phys. Chem. Chem. Phys., 2012, 14, 5545.
34. B. Huang, H. Xiang, J. Yu and S. H. Wei, Phys. Rev. Lett., 2012, 108, 206802.
35. M. Du, X. Li, A. Wang, Y. Wu, X. Hao and M. Zhao, Angew. Chem., Int. Ed., 2014, 53, 3645.
36. Y. F. Guo and W. L. Guo, Nanoscale, 2014, 6, 3731.
37. P. E. Blochl, Phys. Rev. B: Condens. Matter, 1994, 50, 17953.
38. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter, 1999, 59, 1758.
39. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
40. J. Klimes, D. R. Bowler and A. Michelides, J. Phys.: Condens. Matter, 2010, 22, 022201.
41. J. Klimes, D. R. Bowler and A. Michelides, Phys. Rev. B: Condens. Matter, 2011, 83, 195131.
42. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Condens. Matter, 1976, 13, 5188.
43. A. Vanossi, N. Manini, M. Urbakh, S. Zapperi and E. Tosatti, Rev. Mod. Phys., 2013, 85, 529.
44. H. Zhang, Z. Guo, H. Gao and T. Chang, Carbon, 2015, 94, 60.

---