Stages of melting of graphene model in two-dimensional space

References

• K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004), pp. 666–669.10.1126/science.1102896
   PubMed Web of Science ®Google Scholar
• A.K. Geim, Graphene: Status and prospects, Science 324 (2009), pp. 1530–1534.10.1126/science.1158877
   PubMed Web of Science ®Google Scholar
• Y. Zhu, S. Murali, W. Cai, X. Li, J.W. Suk, J.R. Potts, and R.S. Ruoff, Graphene and graphene oxide: Synthesis, properties, and applications, Adv. Mater. 22 (2010), pp. 3906–3924.10.1002/adma.201001068
   PubMed Web of Science ®Google Scholar
• W. Choi, I. Lahiri, R. Seelaboyina, and Y.S. Kang, Synthesis of graphene and its applications: A review, Crit. Rev. Solid State Mater. Sci. 35 (2010), pp. 52–71.10.1080/10408430903505036
   Web of Science ®Google Scholar
• F. Banhart, J. Kotakoski, and A.V. Krasheninnikov, Structural defects in graphene, ACS Nano 5 (2011), pp. 26–41.10.1021/nn102598m
   PubMed Web of Science ®Google Scholar
• S. Amini, H. Kalaantari, J. Garay, A.A. Balandin, and R. Abbaschian, Growth of graphene and graphite nanocrystals from a molten phase, J. Mater. Sci. 46 (2011), pp. 6255–6263.10.1007/s10853-011-5432-9
   Web of Science ®Google Scholar
• A. Fasolino, J.H. Los, and M.I. Katsnelson, Intrinsic ripples in graphene, Nat. Mater. 6 (2007), pp. 858–861.10.1038/nmat2011
   PubMed Web of Science ®Google Scholar
• P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M.C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon, Phy. Rev. Lett. 108 (2012), 155501 ( 5pp).10.1103/PhysRevLett.108.155501
   PubMed Web of Science ®Google Scholar
• A. Kara, H. Enriquez, A.P. Seitsonen, L.C. Lew Yan Voon, S. Vizzini, B. Aufray, and H. Oughaddou, A review on silicene – new candidate for electronics, Surf. Sci. Rep. 67 (2012), pp. 1–18.10.1016/j.surfrep.2011.10.001
   Web of Science ®Google Scholar
• K.V. Zakharchenko, A. Fasolino, J.H. Los, M.I. Katsnelson, Melting of graphene: From two to one dimension, J. Phys.: Condens. Matter 23 (2011), 202202 ( 4pp).
   PubMed Web of Science ®Google Scholar
• S.K. Singh, M. Neek-Amal, and F.M. Peeters, Melting of graphene clusters, Phys. Rev. B 87 (2013), 134103 (9pp).10.1103/PhysRevB.87.134103
   Web of Science ®Google Scholar
• J.H. Los, K.V. Zakharchenko, M.I. Katsnelson, and A. Fasolino, Melting temperature of graphene, Phys. Rev. B 91 (2015), 045415 ( 7pp).10.1103/PhysRevB.91.045415
   Google Scholar
• L. Berezinsky, Destruction of Long-range order in one dimensional and two dimensional system having a continuous symmmetry group I, Classical Syst. Sov. Phys. JETP 32 (1971), pp. 493–500
   Google Scholar
• L. Berezinsky, Destruction of long-range order in one dimensional and two dimensional system having a continuous symmmetry group II, Quantum Syst. Sov. Phys. JETP 34 (1972), pp. 610–616.
   Google Scholar
• J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C: Solid State Phys. 6 (1973), pp. 1181–1203.10.1088/0022-3719/6/7/010
   Web of Science ®Google Scholar
• B.I. Halperin and D.R. Nelson, Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978), pp. 121–124.10.1103/PhysRevLett.41.121
   Web of Science ®Google Scholar
• A.P. Young, On the theory of the phase transition in the two-dimensional planar spin model, J. Phys. C: Solid State Phys. 11 (1978), pp. L453.10.1088/0022-3719/11/11/003
   Google Scholar
• A.P. Young, Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19 (1979), pp. 1855–1866.10.1103/PhysRevB.19.1855
   Web of Science ®Google Scholar
• D.R. Nelson and B.I. Halperin, Dislocation-mediated melting in two dimensions, Phys. Rev. B 19 (1979), pp. 2457–2484.10.1103/PhysRevB.19.2457
   Web of Science ®Google Scholar
• A.Z. Patashinski, R. Orlik, A.C. Mitus, B.A. Grzybowski, and M.A. Ratner, Melting in 2D Lennard-Jones systems: What type of phase transition? J. Phys. Chem. C 114 (2010), pp. 20749–20755.10.1021/jp1069412
   Web of Science ®Google Scholar
• K.J. Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60 (1988), pp. 161–207.10.1103/RevModPhys.60.161
   Web of Science ®Google Scholar
• M.A. Glaser and N.A. Clark, Melting and liquid structure in two dimensions, Adv. Chem. Phys. 83 (1993), pp. 543–709.
   Google Scholar
• A. Patrykiejew, S. Sokolowski, and K. Binder, Phase transitions in adsorbed layers formed on crystals of square and rectangular surface lattice, Surf. Sci. Rep. 37 (2000), pp. 209–344.
   Web of Science ®Google Scholar
• A.Z. Patashinski and M.A. Ratner, Inherent amorphous structures and statistical mechanics of melting, J. Chem. Phys. 106 (1997), pp. 7249–7256.10.1063/1.473685
   Web of Science ®Google Scholar
• K. Chen, T. Kaplan, and M. Mostoller, Melting in two-dimensional Lennard-Jones systems: Observation of a metastable hexatic phase, Phys. Rev. Lett. 74 (1995), pp. 4019–4022.10.1103/PhysRevLett.74.4019
   PubMed Web of Science ®Google Scholar
• K. Bagchi, H.C. Andersen, and W. Swope, Observation of a two-stage melting transition in two dimensions, Phys. Rev. E 53 (1996), pp. 3794–3803.10.1103/PhysRevE.53.3794
   PubMed Web of Science ®Google Scholar
• K. Bagchi, H.C. Andersen, and W. Swope, Computer simulation study of the melting transition in two dimensions, Phys. Rev. Lett. 76 (1996), pp. 255–258.10.1103/PhysRevLett.76.255
   PubMed Web of Science ®Google Scholar
• S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995), pp. 1–19.10.1006/jcph.1995.1039
   Web of Science ®Google Scholar
• J.H. Los and A. Fasolino, Intrinsic long-range bond-order potential for carbon: Performance in Monte Carlo simulations of graphitization, Phys. Rev. B 68 (2003), p. 024107 (14 pp). 10.1103/PhysRevB.68.024107
   Web of Science ®Google Scholar
• J.H. Los, L.M. Ghiringhelli, E.J. Meijer, and A. Fasolino, Improved long-range reactive bond-order potential for carbon. I. Construction, Phys. Rev. B 72 (2005), p. 214102 (14 pp).10.1103/PhysRevB.72.214102
   Web of Science ®Google Scholar
• S. Le Roux and V. Petkov, ISAACS – interactive structure analysis of amorphous and crystalline systems, J. Appl. Crystallogr. 43 (2010), pp. 181–185.10.1107/S0021889809051929
   Web of Science ®Google Scholar
• W. Humphrey, A. Dalke, and K. Schulten, VMD: Visual molecular dynamics, J. Mol. Graphics 14 (1996), pp. 33–38.10.1016/0263-7855(96)00018-5
   PubMed Web of Science ®Google Scholar
• Y. Kowaki, A. Harada, F. Shomojo, and K. Hoshino, Radius dependence of the melting temperature of single-walled carbon nanotubes: Molecular-dynamics simulations, J. Phys.: Condens. Matter 19 (2007), p. 436224 (9 pp).
   Web of Science ®Google Scholar
• S.G. Kim and D. Tománek, Melting the fullerenes: A molecular dynamics study, Phys. Rev. Lett. 72 (1994), pp. 2418–2421.10.1103/PhysRevLett.72.2418
   PubMed Web of Science ®Google Scholar
• K. Zhang, G.M. Stocks, and J. Zhong, Melting and premelting of carbon nanotubes, Nanotechnology 18 (2007), p. 285703 (5 pp).10.1088/0957-4484/18/28/285703
   Web of Science ®Google Scholar
• V.V. Hoang, ‘Graphenization’ of 2D simple monatomic liquids, J. Phys.: Condens. Matter 26 (2014), p. 205101 (9 pp).
   PubMed Web of Science ®Google Scholar
• N.H. March and M.P. Tosi, Introduction to Liquid State Physics, World Scientific, Singapore, 2002.10.1142/4717
   Google Scholar
• L. Li, S. Reich, and J. Robertson, Defect energies of graphite: Density-functional calculations, Phys. Rev. B 72 (2005), p. 184109 (10 pp).10.1103/PhysRevB.72.184109
   Web of Science ®Google Scholar
• J. Ma, D. Alfè, A. Michaelides, and E. Wang, Stone-wales defects in graphene and other planar sp2-bonded materials, Phys. Rev. B 80 (2009), p. 033407 (4 pp).10.1103/PhysRevB.80.033407
   Web of Science ®Google Scholar
• M.H. Gass, U. Bangert, A.L. Bleloch, P. Wang, R.R. Nair, and A.K. Geim, Free-standing graphene at atomic resolution, Nature Nanotechnol. 3 (2008), pp. 676–681.10.1038/nnano.2008.280
   PubMed Web of Science ®Google Scholar
• J.C. Meyer, C. Kisielowski, R. Erni, M.D. Rossell, M.F. Crommie, and A. Zettl, Direct imaging of lattice atoms and topological defects in graphene membranes, Nano Lett. 8 (2008), pp. 3582–3586.10.1021/nl801386m
   PubMed Web of Science ®Google Scholar
• M.M. Ugeda, I. Brihuega, F. Guinea, and J.M. Gómez-Rodríguez, Missing atom as a source of carbon magnetism, Phys. Rev. Lett. 104 (2010), p. 096804 (4 pp).10.1103/PhysRevLett.104.096804
   PubMed Web of Science ®Google Scholar
• A.V. Krasheninnikov, P.O. Lehtinen, A.S. Foster, and R.M. Nieminen, Bending the rules: Contrasting vacancy energetics and migration in graphite and carbon nanotubes, Chem. Phys. Lett. 418 (2006), pp. 132–136.10.1016/j.cplett.2005.10.106
   Web of Science ®Google Scholar
• A.A. El-Barbary, R.H. Telling, C.P. Ewels, M.I. Heggie, and P.R. Briddon, Structure and energetics of the vacancy in graphite, Phys. Rev. B 68 (2003), p. 144107 (7 pp).10.1103/PhysRevB.68.144107
   Web of Science ®Google Scholar
• G.D. Lee, C.Z. Wang, E. Yoon, N.M. Hwang, D.Y. Kim, and K.M. Ho, Diffusion, coalescence, and reconstruction of vacancy defects in graphene layers, Phys. Rev. Lett. 95 (2005), p. 205501 (4 pp).10.1103/PhysRevLett.95.205501
   PubMed Web of Science ®Google Scholar
• O. Lehtinen, N. Vats, G. Algara-Siller, P. Knyrim, and U. Kaiser, Implantation and atomic-scale investigation of self-interstitials in graphene, Nano Lett. 15 (2014), pp. 235–241.
   PubMed Web of Science ®Google Scholar
• J. Kotakoski, A.V. Krasheninnikov, U. Kaiser, and J.C. Meyer, From point defects in graphene to two-dimensional amorphous carbon, Phys. Rev. Lett. 106 (2011), p. 105505 (4 pp).10.1103/PhysRevLett.106.105505
   PubMed Web of Science ®Google Scholar
• J. Kotakoski, A.V. Krasheninnikov, and K. Nordlund, Energetics, structure, and long-range interaction of vacancy-type defects in carbon nanotubes: Atomistic simulations, Phys. Rev. B 74 (2006), p. 245420 (5 pp).10.1103/PhysRevB.74.245420
   Web of Science ®Google Scholar
• A.C. Mitus, A.Z. Patashinski, A. Patrykiejew, and S. Sokolowski, Local structure, fluctuations, and freezing in two dimensions, Phys. Rev. B 66 (2002), p. 184202 (12 pp).10.1103/PhysRevB.66.184202
   Web of Science ®Google Scholar
• T.P. Duy and V.V. Hoang, Atomic mechanism of homogeneous melting of bcc fe at the limit of superheating, Phys. B 407 (2012), pp. 978–984.
   Web of Science ®Google Scholar
• M.H. Cohen and G.S. Grest, Liquid-glass transition, a free-volume approach, Phys. Rev. B 20 (1979), pp. 1077–1098.10.1103/PhysRevB.20.1077
   Web of Science ®Google Scholar
• A.Z. Patashinski, R. Orlik, A.C. Mitus, B.A. Grzybowski, and M.A. Ratner, Melting in 2D Lennard-Jones systems: What type of phase transition? J. Phys. Chem. C 114 (2010), pp. 20749–20755.10.1021/jp1069412
   Web of Science ®Google Scholar
• S.K. Singh, S. Costamagna, M. Neek-Amal, and F.M. Peeters, Melting of partially fluorinated graphene: From detachment of fluorine atoms to large defects and random coils, J. Phys. Chem. C 118 (2014), pp. 4460–4464.10.1021/jp4109333
   Web of Science ®Google Scholar
• S.K. Singh, M. Neek-Amal, S. Costamagna, and F.M. Peeters, Rippling, buckling, and melting of single- and multilayer MoS2, Phys. Rev. B 91 (2015), p. 014101 (7 pp).10.1103/PhysRevB.91.014101
   Google Scholar
• V.O. Özçelik, H.H. Gurel, and S. Ciraci, Self-healing of vacancy defects in single-layer graphene and silicene, Phys. Rev. B 88 (2013), p. 045440 (11 pp).10.1103/PhysRevB.88.045440
   Web of Science ®Google Scholar
• J. Kotakoski, F.R. Eder, and J.C. Meyer, Atomic structure and energetics of large vacancies in graphene, Phys. Rev. B 89 (2014), p. 201406(R) (5 pp).10.1103/PhysRevB.89.201406
   Web of Science ®Google Scholar