First-Principles Study of Graphene-6H SiC Surface Interactions
Year 2021,
Volume: 9 Issue: 2, 171 - 177, 30.04.2021
Ahmet Çiçek
,
Bülent Uluğ
Abstract
Interactions of graphene with 6H-SiC {0001} surfaces are numerically investigated from first principles. In order to describe the bulk structure and its 6 bilayer thick surfaces correctly, bare and dipole-corrected atomic relaxations are considered. The obtained lattice parameters and bulk modulus are in good agreement with experimental values. The calculated indirect band gap width of 2.10 eV is smaller than the experimental value due to the nature of the computational method. Geometrical optimization of the surfaces, where dipole correction is applied, reveals that the first two bilayers displace significantly, where the relaxations of the very top bilayer is more pronounced. Band structures of the {0001} surfaces possess two flat bands around the Fermi level due to unsaturated bonds on opposite faces. When one layer of C atoms are introduced on the Si-terminated surface, it behaves as a tightly-bound buffer layer. This is also the case for the C-terminated surface when van der Waals interactions are taken into account. In contrast, disregarding these interactions yields free-standing graphene like behavior for the first C overlayer. On both surfaces, the second C overlayer is free-standing where the corresponding band structures incorporate Dirac-cone like features.
Supporting Institution
Akdeniz University
Project Number
2008.01.0105.010
Thanks
This work is supported by Akdeniz University Scientific Research Projects Coordination Unit under the grant number 2008.01.0105.010.
References
- [1] K. S. Novoselov, D. Jiang, F. Schedin, T. Booth, V. Khotkevich, S. Morozov, and A. K. Geim, "Two-dimensional atomic crystals," Proceedings of the National Academy of Sciences, vol. 102. 30, 2005, pp. 10451-10453.
- [2] Z. Jiang, Y. Zhang, Y.-W. Tan, H. Stormer, and P. Kim, "Quantum Hall effect in graphene," Solid State Communications, vol. 143. 1-2, 2007, pp. 14-19.
- [3] M. Katsnelson, K. Novoselov, and A. Geim, "Chiral tunnelling and the Klein paradox in graphene," Nature Physics, vol. 2. 9, 2006, pp. 620-625.
- [4] Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill, and P. Avouris, "100-GHz transistors from wafer-scale epitaxial graphene," Science, vol. 327. 5966, 2010, pp. 662-662.
- [5] P. Avouris, "Graphene: electronic and photonic properties and devices," Nano Letters, vol. 10. 11, 2010, pp. 4285-4294.
- [6] K. Subrahmanyam, L. Panchakarla, A. Govindaraj, and C. Rao, "Simple method of preparing graphene flakes by an arc-discharge method," The Journal of Physical Chemistry C, vol. 113. 11, 2009, pp. 4257-4259.
- [7] A. Kumar, A. L. M. Reddy, A. Mukherjee, M. Dubey, X. Zhan, N. Singh, L. Ci, W. E. Billups, J. Nagurny, and G. Mital, "Direct synthesis of lithium-intercalated graphene for electrochemical energy storage application," Acs Nano, vol. 5. 6, 2011, pp. 4345-4349.
- [8] A. Mattausch and O. Pankratov, "Ab initio study of graphene on SiC," Physical Review Letters, vol. 99. 7, 2007, p. 076802.
- [9] G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. Van Den Brink, "Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations," Physical Review B, vol. 76. 7, 2007, p. 073103.
- [10] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, "Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition," Nano Letters, vol. 9. 1, 2009, pp. 30-35.
- [11] J. C. Swart, I. M. Ciobîcǎ, R. A. van Santen, and E. van Steen, "Intermediates in the formation of graphitic carbon on a flat FCC-Co (111) surface," The Journal of Physical Chemistry C, vol. 112. 33, 2008, pp. 12899-12904.
- [12] L. Gao, J. R. Guest, and N. P. Guisinger, "Epitaxial graphene on Cu (111)," Nano Letters, vol. 10. 9, 2010, pp. 3512-3516.
- [13] S. Marchini, S. Günther, and J. Wintterlin, "Scanning tunneling microscopy of graphene on Ru (0001)," Physical Review B, vol. 76. 7, 2007, p. 075429.
- [14] P. Sutter, J. T. Sadowski, and E. Sutter, "Graphene on Pt (111): Growth and substrate interaction," Physical Review B, vol. 80. 24, 2009, p. 245411.
- [15] D. Momeni Pakdehi, J. Aprojanz, A. Sinterhauf, K. Pierz, M. Kruskopf, P. Willke, J. Baringhaus, J. Stöckmann, G. Traeger, and F. Hohls, "Minimum resistance anisotropy of epitaxial graphene on SiC," ACS Applied Materials & Interfaces, vol. 10. 6, 2018, pp. 6039-6045.
- [16] Z. Liu, Q. Xu, C. Zhang, Q. Sun, C. Wang, M. Dong, Z. Wang, H. Ohmori, M. Kosinova, and T. Goto, "Laser-induced growth of large-area epitaxial graphene with low sheet resistance on 4H-SiC (0001)," Applied Surface Science, vol. 514. 2020, p. 145938.
- [17] X. Zhang, C. Yan, C. Zeng, T. Sun, Z. Xing, W. Shi, Y. Wang, C. Pang, and B. Zhang, "Epitaxial synthesis of graphene on 4H-SiC by microwave plasma chemical vapor deposition," Materials Research Express, vol. 7. 11, 2020, p. 116410.
- [18] N. Briggs, Z. M. Gebeyehu, A. Vera, T. Zhao, K. Wang, A. D. L. F. Duran, B. Bersch, T. Bowen, K. L. Knappenberger, and J. A. Robinson, "Epitaxial graphene/silicon carbide intercalation: a minireview on graphene modulation and unique 2D materials," Nanoscale, vol. 11. 33, 2019, pp. 15440-15447.
- [19] T. Hu, H. Bao, S. Liu, X. Liu, D. Ma, F. Ma, and K. Xu, "Near-free-standing epitaxial graphene on rough SiC substrate by flash annealing at high temperature," Carbon, vol. 120. 2017, pp. 219-225.
- [20] P. N. First, W. A. de Heer, T. Seyller, C. Berger, J. A. Stroscio, and J.-S. Moon, "Epitaxial graphenes on silicon carbide," MRS Bulletin, vol. 35. 4, 2010, pp. 296-305.
- [21] T. Cavallucci and V. Tozzini, "Intrinsic structural and electronic properties of the Buffer Layer on Silicon Carbide unraveled by Density Functional Theory," Scientific Reports, vol. 8. 1, 2018, pp. 1-10.
- [22] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, and A. N. Marchenkov, "Electronic confinement and coherence in patterned epitaxial graphene," Science, vol. 312. 5777, 2006, pp. 1191-1196.
- [23] A. Mattausch and O. Pankratov, "Density functional study of graphene overlayers on SiC," Physica Status Solidi (b), vol. 245. 7, 2008, pp. 1425-1435.
- [24] F. Varchon, R. Feng, J. Hass, X. Li, B. N. Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, and E. H. Conrad, "Electronic structure of epitaxial graphene layers on SiC: effect of the substrate," Physical Review Letters, vol. 99. 12, 2007, p. 126805.
- [25] L. Magaud, F. Hiebel, F. Varchon, P. Mallet, and J.-Y. Veuillen, "Graphene on the C-terminated SiC (000 1) surface: An ab initio study," Physical Review B, vol. 79. 16, 2009, p. 161405.
- [26] C. P. Huelmo and P. A. Denis, "Unraveling the electromagnetic structure of the epitaxial graphene buffer layer," Journal of Physics: Condensed Matter, vol. 31. 43, 2019, p. 435001.
- [27] K. Hayashi, K. Morita, S. Mizuno, H. Tochihara, and S. Tanaka, "Stable surface termination on vicinal 6H–SiC (0001) surfaces," Surface Science, vol. 603. 3, 2009, pp. 566-570.
- [28] C. Park, B.-H. Cheong, K.-H. Lee, and K.-J. Chang, "Structural and electronic properties of cubic, 2H, 4H, and 6H SiC," Physical Review B, vol. 49. 7, 1994, p. 4485.
- [29] T. Seyller, A. Bostwick, K. Emtsev, K. Horn, L. Ley, J. McChesney, T. Ohta, J. D. Riley, E. Rotenberg, and F. Speck, "Epitaxial graphene: a new material," Physica Status Solidi (b), vol. 245. 7, 2008, pp. 1436-1446.
- [30] K. Emtsev, F. Speck, T. Seyller, L. Ley, and J. D. Riley, "Interaction, growth, and ordering of epitaxial graphene on SiC {0001} surfaces: A comparative photoelectron spectroscopy study," Physical Review B, vol. 77. 15, 2008, p. 155303.
- [31] R. Helbig and F. Engelbrecht, "SiC: Polar properties and their influence on technology and devices," in Advances in Solid State Physics 38: Springer, 1999, pp. 75-86.
- [32] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, and I. Dabo, "QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials," Journal of Physics: Condensed Matter, vol. 21. 39, 2009, p. 395502.
- [33] J. P. Perdew and A. Zunger, "Self-interaction correction to density-functional approximations for many-electron systems," Physical Review B, vol. 23. 10, 1981, p. 5048.
- [34] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, "Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation," Physical Review B, vol. 46. 11, 1992, p. 6671.
- [35] H. J. Monkhorst and J. D. Pack, "Special points for Brillouin-zone integrations," Physical Review B, vol. 13. 12, 1976, p. 5188.
- [36] F. Birch, "Finite elastic strain of cubic crystals," Physical Review, vol. 71. 11, 1947, p. 809.
- [37] G. C. Capitani, S. Di Pierro, and G. Tempesta, "The 6 H-SiC structure model: Further refinement from SCXRD data from a terrestrial moissanite," American Mineralogist, vol. 92. 2-3, 2007, pp. 403-407.
- [38] W. Bassett, M. Weathers, T. C. Wu, and T. Holmquist, "Compressibility of SiC up to 68.4 GPa," Journal of Applied Physics, vol. 74. 6, 1993, pp. 3824-3826.
- [39] M. Yoshida, A. Onodera, M. Ueno, K. Takemura, and O. Shimomura, "Pressure-induced phase transition in SiC," Physical Review B, vol. 48. 14, 1993, p. 10587.
- [40] P. Käckell, B. Wenzien, and F. Bechstedt, "Influence of atomic relaxations on the structural properties of SiC polytypes from ab initio calculations," Physical Review B, vol. 50. 23, 1994, p. 17037.
- [41] J. Wang, L. Zhang, Q. Zeng, G. L. Vignoles, and L. Cheng, "Surface relaxation and oxygen adsorption behavior of different SiC polytypes: a first-principles study," Journal of Physics: Condensed Matter, vol. 22. 26, 2010, p. 265003.
- [42] J. Sołtys, J. Piechota, M. Łopuszyński, and S. Krukowski, "A comparative DFT study of electronic properties of 2H-, 4H-and 6H-SiC (0001) and SiC () clean surfaces: significance of the surface Stark effect," New Journal of Physics, vol. 12. 4, 2010, p. 043024.
- [43] T. Dalibor, G. Pensl, N. Nordell, and A. Schöner, "Electrical properties of the titanium acceptor in silicon carbide," Physical Review B, vol. 55. 20, 1997, p. 13618.
- [44] J. P. Perdew, "Density functional theory and the band gap problem," International Journal of Quantum Chemistry, vol. 28. S19, 1985, pp. 497-523.
- [45] M. Sabisch, P. Krüger, and J. Pollmann, "Ab initio calculations of structural and electronic properties of 6H-SiC (0001) surfaces," Physical Review B, vol. 55. 16, 1997, p. 10561.
- [46] J. Hass, F. Varchon, J.-E. Millan-Otoya, M. Sprinkle, N. Sharma, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, "Why multilayer graphene on 4 H− SiC (000-1) behaves like a single sheet of graphene," Physical Review Letters, vol. 100. 12, 2008, p. 125504.
Year 2021,
Volume: 9 Issue: 2, 171 - 177, 30.04.2021
Ahmet Çiçek
,
Bülent Uluğ
Project Number
2008.01.0105.010
References
- [1] K. S. Novoselov, D. Jiang, F. Schedin, T. Booth, V. Khotkevich, S. Morozov, and A. K. Geim, "Two-dimensional atomic crystals," Proceedings of the National Academy of Sciences, vol. 102. 30, 2005, pp. 10451-10453.
- [2] Z. Jiang, Y. Zhang, Y.-W. Tan, H. Stormer, and P. Kim, "Quantum Hall effect in graphene," Solid State Communications, vol. 143. 1-2, 2007, pp. 14-19.
- [3] M. Katsnelson, K. Novoselov, and A. Geim, "Chiral tunnelling and the Klein paradox in graphene," Nature Physics, vol. 2. 9, 2006, pp. 620-625.
- [4] Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill, and P. Avouris, "100-GHz transistors from wafer-scale epitaxial graphene," Science, vol. 327. 5966, 2010, pp. 662-662.
- [5] P. Avouris, "Graphene: electronic and photonic properties and devices," Nano Letters, vol. 10. 11, 2010, pp. 4285-4294.
- [6] K. Subrahmanyam, L. Panchakarla, A. Govindaraj, and C. Rao, "Simple method of preparing graphene flakes by an arc-discharge method," The Journal of Physical Chemistry C, vol. 113. 11, 2009, pp. 4257-4259.
- [7] A. Kumar, A. L. M. Reddy, A. Mukherjee, M. Dubey, X. Zhan, N. Singh, L. Ci, W. E. Billups, J. Nagurny, and G. Mital, "Direct synthesis of lithium-intercalated graphene for electrochemical energy storage application," Acs Nano, vol. 5. 6, 2011, pp. 4345-4349.
- [8] A. Mattausch and O. Pankratov, "Ab initio study of graphene on SiC," Physical Review Letters, vol. 99. 7, 2007, p. 076802.
- [9] G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. Van Den Brink, "Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations," Physical Review B, vol. 76. 7, 2007, p. 073103.
- [10] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, "Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition," Nano Letters, vol. 9. 1, 2009, pp. 30-35.
- [11] J. C. Swart, I. M. Ciobîcǎ, R. A. van Santen, and E. van Steen, "Intermediates in the formation of graphitic carbon on a flat FCC-Co (111) surface," The Journal of Physical Chemistry C, vol. 112. 33, 2008, pp. 12899-12904.
- [12] L. Gao, J. R. Guest, and N. P. Guisinger, "Epitaxial graphene on Cu (111)," Nano Letters, vol. 10. 9, 2010, pp. 3512-3516.
- [13] S. Marchini, S. Günther, and J. Wintterlin, "Scanning tunneling microscopy of graphene on Ru (0001)," Physical Review B, vol. 76. 7, 2007, p. 075429.
- [14] P. Sutter, J. T. Sadowski, and E. Sutter, "Graphene on Pt (111): Growth and substrate interaction," Physical Review B, vol. 80. 24, 2009, p. 245411.
- [15] D. Momeni Pakdehi, J. Aprojanz, A. Sinterhauf, K. Pierz, M. Kruskopf, P. Willke, J. Baringhaus, J. Stöckmann, G. Traeger, and F. Hohls, "Minimum resistance anisotropy of epitaxial graphene on SiC," ACS Applied Materials & Interfaces, vol. 10. 6, 2018, pp. 6039-6045.
- [16] Z. Liu, Q. Xu, C. Zhang, Q. Sun, C. Wang, M. Dong, Z. Wang, H. Ohmori, M. Kosinova, and T. Goto, "Laser-induced growth of large-area epitaxial graphene with low sheet resistance on 4H-SiC (0001)," Applied Surface Science, vol. 514. 2020, p. 145938.
- [17] X. Zhang, C. Yan, C. Zeng, T. Sun, Z. Xing, W. Shi, Y. Wang, C. Pang, and B. Zhang, "Epitaxial synthesis of graphene on 4H-SiC by microwave plasma chemical vapor deposition," Materials Research Express, vol. 7. 11, 2020, p. 116410.
- [18] N. Briggs, Z. M. Gebeyehu, A. Vera, T. Zhao, K. Wang, A. D. L. F. Duran, B. Bersch, T. Bowen, K. L. Knappenberger, and J. A. Robinson, "Epitaxial graphene/silicon carbide intercalation: a minireview on graphene modulation and unique 2D materials," Nanoscale, vol. 11. 33, 2019, pp. 15440-15447.
- [19] T. Hu, H. Bao, S. Liu, X. Liu, D. Ma, F. Ma, and K. Xu, "Near-free-standing epitaxial graphene on rough SiC substrate by flash annealing at high temperature," Carbon, vol. 120. 2017, pp. 219-225.
- [20] P. N. First, W. A. de Heer, T. Seyller, C. Berger, J. A. Stroscio, and J.-S. Moon, "Epitaxial graphenes on silicon carbide," MRS Bulletin, vol. 35. 4, 2010, pp. 296-305.
- [21] T. Cavallucci and V. Tozzini, "Intrinsic structural and electronic properties of the Buffer Layer on Silicon Carbide unraveled by Density Functional Theory," Scientific Reports, vol. 8. 1, 2018, pp. 1-10.
- [22] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, and A. N. Marchenkov, "Electronic confinement and coherence in patterned epitaxial graphene," Science, vol. 312. 5777, 2006, pp. 1191-1196.
- [23] A. Mattausch and O. Pankratov, "Density functional study of graphene overlayers on SiC," Physica Status Solidi (b), vol. 245. 7, 2008, pp. 1425-1435.
- [24] F. Varchon, R. Feng, J. Hass, X. Li, B. N. Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, and E. H. Conrad, "Electronic structure of epitaxial graphene layers on SiC: effect of the substrate," Physical Review Letters, vol. 99. 12, 2007, p. 126805.
- [25] L. Magaud, F. Hiebel, F. Varchon, P. Mallet, and J.-Y. Veuillen, "Graphene on the C-terminated SiC (000 1) surface: An ab initio study," Physical Review B, vol. 79. 16, 2009, p. 161405.
- [26] C. P. Huelmo and P. A. Denis, "Unraveling the electromagnetic structure of the epitaxial graphene buffer layer," Journal of Physics: Condensed Matter, vol. 31. 43, 2019, p. 435001.
- [27] K. Hayashi, K. Morita, S. Mizuno, H. Tochihara, and S. Tanaka, "Stable surface termination on vicinal 6H–SiC (0001) surfaces," Surface Science, vol. 603. 3, 2009, pp. 566-570.
- [28] C. Park, B.-H. Cheong, K.-H. Lee, and K.-J. Chang, "Structural and electronic properties of cubic, 2H, 4H, and 6H SiC," Physical Review B, vol. 49. 7, 1994, p. 4485.
- [29] T. Seyller, A. Bostwick, K. Emtsev, K. Horn, L. Ley, J. McChesney, T. Ohta, J. D. Riley, E. Rotenberg, and F. Speck, "Epitaxial graphene: a new material," Physica Status Solidi (b), vol. 245. 7, 2008, pp. 1436-1446.
- [30] K. Emtsev, F. Speck, T. Seyller, L. Ley, and J. D. Riley, "Interaction, growth, and ordering of epitaxial graphene on SiC {0001} surfaces: A comparative photoelectron spectroscopy study," Physical Review B, vol. 77. 15, 2008, p. 155303.
- [31] R. Helbig and F. Engelbrecht, "SiC: Polar properties and their influence on technology and devices," in Advances in Solid State Physics 38: Springer, 1999, pp. 75-86.
- [32] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, and I. Dabo, "QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials," Journal of Physics: Condensed Matter, vol. 21. 39, 2009, p. 395502.
- [33] J. P. Perdew and A. Zunger, "Self-interaction correction to density-functional approximations for many-electron systems," Physical Review B, vol. 23. 10, 1981, p. 5048.
- [34] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, "Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation," Physical Review B, vol. 46. 11, 1992, p. 6671.
- [35] H. J. Monkhorst and J. D. Pack, "Special points for Brillouin-zone integrations," Physical Review B, vol. 13. 12, 1976, p. 5188.
- [36] F. Birch, "Finite elastic strain of cubic crystals," Physical Review, vol. 71. 11, 1947, p. 809.
- [37] G. C. Capitani, S. Di Pierro, and G. Tempesta, "The 6 H-SiC structure model: Further refinement from SCXRD data from a terrestrial moissanite," American Mineralogist, vol. 92. 2-3, 2007, pp. 403-407.
- [38] W. Bassett, M. Weathers, T. C. Wu, and T. Holmquist, "Compressibility of SiC up to 68.4 GPa," Journal of Applied Physics, vol. 74. 6, 1993, pp. 3824-3826.
- [39] M. Yoshida, A. Onodera, M. Ueno, K. Takemura, and O. Shimomura, "Pressure-induced phase transition in SiC," Physical Review B, vol. 48. 14, 1993, p. 10587.
- [40] P. Käckell, B. Wenzien, and F. Bechstedt, "Influence of atomic relaxations on the structural properties of SiC polytypes from ab initio calculations," Physical Review B, vol. 50. 23, 1994, p. 17037.
- [41] J. Wang, L. Zhang, Q. Zeng, G. L. Vignoles, and L. Cheng, "Surface relaxation and oxygen adsorption behavior of different SiC polytypes: a first-principles study," Journal of Physics: Condensed Matter, vol. 22. 26, 2010, p. 265003.
- [42] J. Sołtys, J. Piechota, M. Łopuszyński, and S. Krukowski, "A comparative DFT study of electronic properties of 2H-, 4H-and 6H-SiC (0001) and SiC () clean surfaces: significance of the surface Stark effect," New Journal of Physics, vol. 12. 4, 2010, p. 043024.
- [43] T. Dalibor, G. Pensl, N. Nordell, and A. Schöner, "Electrical properties of the titanium acceptor in silicon carbide," Physical Review B, vol. 55. 20, 1997, p. 13618.
- [44] J. P. Perdew, "Density functional theory and the band gap problem," International Journal of Quantum Chemistry, vol. 28. S19, 1985, pp. 497-523.
- [45] M. Sabisch, P. Krüger, and J. Pollmann, "Ab initio calculations of structural and electronic properties of 6H-SiC (0001) surfaces," Physical Review B, vol. 55. 16, 1997, p. 10561.
- [46] J. Hass, F. Varchon, J.-E. Millan-Otoya, M. Sprinkle, N. Sharma, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad, "Why multilayer graphene on 4 H− SiC (000-1) behaves like a single sheet of graphene," Physical Review Letters, vol. 100. 12, 2008, p. 125504.