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Graphene monolayers on GaN(0 0 0 1)
Accepted Manuscript Title: Graphene monolayers on GaN(0001) Author: Miguel Espitia Rico Jairo Arbey Rodr´ıguez Martinez Mar´ıa G. Moreno-Armenta Noboru Takeuchi PII: DOI: Reference:
S0169-4332(14)02522-7 http://dx.doi.org/doi:10.1016/j.apsusc.2014.11.057 APSUSC 29102
To appear in:
APSUSC
Received date: Accepted date:
8-9-2014 10-11-2014
Please cite this article as: M.E. Rico, J.A.R. Martinez, M.G. Moreno-Armenta, N. Takeuchi, Graphene monolayers on GaN(0001), Applied Surface Science (2014), http://dx.doi.org/10.1016/j.apsusc.2014.11.057 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphene monolayers on GaN(0001) Miguel Espitia Rico1,2, Jairo Arbey Rodríguez Martinez2, María G. Moreno-Armenta3 and Noboru Takeuchi3 of Physics, Grupo GEFEM, Universidad Distrital Francisco José de Caldas, Bogotá,
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1Dept.
Colombia.
of Physics, GEMA Grupo de Estudio de Materiales, Universidad Nacional de
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2Dept.
3Centro
us
Colombia, Bogotá, Colombia.
de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de México, Km
an
107 carretera Tijuana-Ensenada, Apdo Postal 14, CP 22800, Ensenada, B.C., México.
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Abstract
The epitaxial growth of graphene on GaN(0001) surfaces is studied by first principles total
d
energy calculations. The idea is to understand how defect-free graphene can be grown on
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substrates. It is found that the most stable structures were a 4×4(0001) GaN/3√3×3√3 graphene
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and a 2√3×2√3 (0001) GaN Northrup bilayer/√21×√21 graphene, grown under N- and Ga-rich conditions, respectively. In these structures, graphene maintains its hexagonal honeycomb structure with the C-C bonds intact. Preservation of the π-network for the graphene layers was demonstrated by the presence of Dirac cones. The band structures for both the N and Ga-rich configurations show metallic characteristics and the Ga-rich configuration is slightly magnetic. This demonstrates that GaN(0001) is an excellent substrate for supporting graphene layers. Keywords: Graphene, Dirac cones, GaN Surface, Interface GaN/Graphene
1. Introduction
Page 1 of 17
Graphene, a single layer of graphite, is a nanomaterial with unique properties, making it one of the most promising materials for applications in many areas including nanoelectronics [1-3]. Graphene’s hexagonal honeycomb lattice has two nonequivalent
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carbon atoms per unit cell, which leads to a unique band structure, characterized by the formation of Dirac cones with a linear dispersion around the points where the valence and
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conduction bands touch. For applications in nanoelectronics, graphene is commonly grown
us
on an insulator or semiconductor substrate [4-8]. Under these conditions, the atomic structure of the graphene layer may become distorted, resulting in subsequent
an
modification of its electronic structure [9]. Therefore, in the resulting graphene/substrate
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system, it is desirable that the graphene layer maintains its honeycomb structure. Using first-principles total energy calculations, we have shown that this condition was satisfied in
d
graphene/GaN(0001) for a wide range of Ga chemical potentials. By itself, GaN is an
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important technological material as it allows the formation of two-dimensional quantum wells and superlattices. GaN is used in a large variety of technological applications such as
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high-power and high-frequency electronic devices. Therefore, graphene/GaN interfaces combine two important fields, extending the possible applications of both. Theoretical studies have suggested that the GaN-graphene interface has a well-defined 1×1 structure with the same lattice constant as bulk GaN. However, the lattice mismatch between graphene and GaN is large (around 30%). More recently, Gohda and Tsuneyuki [10] proposed a GaN-√3×√3/graphene-2×2 superstructure (from now on this will be called Gohda’s model) as the most probable atomic structure for graphene on a GaN(0001) surface. To release the tensile strain generated by a lattice mismatch of 12%, the C-bond network was partially broken, resulting in the formation of C-N bonds.
Page 2 of 17
We present first-principles total energy calculations on the formation of graphene monolayers on GaN(0001) surfaces. Approximately 50 different configurations were calculated and structures with surface formation energies much lower than the 1×1
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GaN(0001)/graphene structure and Gohda’s model were obtained. The calculations were performed under either N- or Ga-rich conditions. Different from Gohda’s model, these
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structures maintained their hexagonal honeycomb lattice structure and the C-C bonds
us
remained intact.
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2. Method
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Calculations were performed using periodic density functional theory (DFT), which were implemented in the plane-wave self-consistent field (PWscf) code in the Quantum
d
ESPRESSO package [11]. The exchange and correlation energies were modeled according to
te
the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) gradient-corrected functional [12]. Long-range interactions were included with the B97-D
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functional based on Beck’s power-series ansatz, and they were explicitly parameterized by including damped atom-pairwise dispersion corrections [13]. Electron-ion interactions were treated with the pseudopotential method [14,15]. The electron wavefunctions were expanded into plane waves with a kinetic-energy cutoff of 30 Ry. For the charge density, a kinetic energy cutoff of 240 Ry was used. A 3×3×1 Monkhorst-Pack mesh [16] was used to generate the k-points in the 2×2 unit cell. The GaN bulk atomic structure was optimized to determine the optimal lattice parameters, a = b = 3.22 Å and c/a = 1.63, and the internal parameter u = 0.38, which were in good agreement with the experimental data [17], within ~1.2%.
Page 3 of 17
To model the GaN(0001) surface, a repeated slab geometry was used. Each slab consisted of four GaN double layers with adatoms on the surface. For the Ga-rich conditions, a bilayer of Ga was added. Dangling bonds on the bottom of the surface were saturated with pseudo-
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H atoms, with a fractional charge of 0.75e. The bottom GaN bilayer and the saturated
pseudo-H atoms were frozen to simulate a bulk-like environment. Two consecutive slabs
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were separated by an empty space ~9.0 Å wide to reduce the slab-slab interactions.
FIG. 1. Relative formation energies (in eV/1×1) of graphene on GaN(0001) as a function of the Ga chemical potential. The zero of energy refers to a clean GaN(0001) surface. The energies of the laterally contracted Ga bilayer (LCGa) with registers A and B proposed by Northrup et al [18] to explain the pseudo (1×1) reconstruction of GaN(0001) are also presented as a reference.
3. Results and Discussion
Page 4 of 17
The surface formation energies of the most relevant configurations, that we have studied, as a function of the Ga chemical potential are described in Figure 1. For N-rich conditions, the graphene layer was on top of an ideally terminated GaN bilayer. For the Ga-rich
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conditions, two possibilities were considered: first, the graphene layer was on top of an uncontracted Ga bilayer, and second, the graphene layer was on top of a laterally contracted
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Ga bilayer (LCB) structure, as proposed by Northrup et al [18-23] to explain the pseudo
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(1×1) GaN (0001) surface grown under Ga-rich conditions. In this model, contraction occurred uniformly within the plane and only the top adlayer was contracted.
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Under N-rich conditions, 4×4 (0001) GaN/3√3×3√3 graphene structures were the most
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stable (labeled I, II, and III in Fig. 1), with lattice mismatches of −0.72%. In structure I, a carbon atom is located at the H3 high-symmetry point (HSP), while the carbon atoms are
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located at T1 and T4 sites in structures IIa and IIb respectively. In structure III graphene's
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H6 HSP is at the H3 HSP of GaN(0001). The energies of I, II, and III were almost degenerated, but configuration I, which has carbon atoms at the H3 sites, had the lowest
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energy and therefore it is the most stable structure under N-rich conditions. These results show that the 4×4 (0001) GaN/3√3×3√3 graphene structures were barely affected by their lateral position. For Ga-rich conditions, 2√3×2√3 (0001) GaN Northrup bilayer/√21×√21 graphene structures were the most stable (curves IV and V in Fig. 1), with lattice mismatches of −1.1%. In both structures, the carbon atoms are on top of the Ga atoms (T1 sites) and they differed from each other in the registries with the underlying contracted Ga layer. The energy separation between IV and V was ~0.2eV/(1×1) cell. Under Ga-rich conditions, the most stable structure was IV, with a graphene layer on a laterally contracted Ga bilayer with Northrup’s B registry [19].
Page 5 of 17
Fig. 2 shows the atomic configuration of the most stable structure under N-rich conditions, a 3√3×3√3 graphene con iguration on a 4×4 (0001) GaN surface with a carbon atom on a H3 site. In Fig. 2a, the values of the lattice constants in the horizontal and vertical directions
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are included. The lattice constant in the horizontal direction is close (maximum
discrepancy ~0.3%) to the value in bulk GaN. The calculated vertical lattice constant (the c
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value of the GaN in the wurtzite phase) is also close to that of the bulk. The interlayer
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distances are within 0.03 Å (~4.6% and 1.5%) of the bulk values of 0.65 Å and 1.97 Å. In the first bilayer, the Ga-N nearest neighbor distances are within 0.04 Å (~2.0%) of the bulk
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value of 1.97 Å. In Fig. 2b, we can see that the values of the lattice constant, the angles, and
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several bond lengths at different sites in the unit cell for the relaxed graphene layer are close to the ideal values for graphene. The graphene layer is 2.39 Å above the top of the Ga
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layer and it is slightly distorted with carbon-carbon bond lengths varying from 1.41 Å to
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1.46 Å (the ideal calculated bond length is 1.42 Å) with a vertical buckling of 0.54 Å. Figs. 2c and 2d reaffirm that the geometry of the GaN unit cell is very close to the bulk geometry.
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Consequently, the most remarkable feature is that the geometry of graphene is maintained and the carbon honeycomb network is preserved. Therefore, the bilayers of GaN and the graphene monolayers in this stable structure are similar to bulk GaN and ideal graphene respectively. Therefore, the graphene layer and the GaN(0001) surface are vertically bound to each other by a weak interactions such as Van der Waals interactions.
Page 6 of 17
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FIG. 2. 3√3×3√3 graphene on a 4×4 (0001) GaN surface where the C atoms are on H3 sites
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(structure I in Fig. 1). (a) The side view and (b) the top view of the graphene layer. (c) The top view of the top most GaN bilayer, (d) a side view of the graphene layer showing a small undulation of ~ 0.5Å. Additionally, a plot of the average separation between the layers is shown. The value of the separation in bulk GaN is presented as a reference and dashed lines are included to guide the eyes. (e) A top view of the first three monolayers: graphene, Ga and N, showing the relative position of a C atom at a H3 site on top of a (0001) GaN substrate.
Figure 3 shows the most stable structure under Ga-rich conditions (structure D in Fig. 1): √21×√21graphene on top of a 2√3×2√3(0001) Northurp’s Ga-bilayer (registry B)GaN(0001), with a carbon on top of a Ga atom. The calculations started with a graphene layer on top of Northup’s Ga bilayer model with a √3×√3 reconstruction, the contracted
Page 7 of 17
layer in registry B. Before the relaxation, the top most Ga layer was formed by 16 atoms, while all other layers had 12 atoms each (our unit cell is formed by 4 √3×√3 cells). After full relaxation, the atoms belonging to the Ga bilayer disorders and they rearrange
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themselves into three layers: the first one (labeled L2 in fig. 3) loses 3 atoms, getting an atomic density closer to the ideal bulk density. These displaced 3 atoms form an
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intermediate layer (L3 in fig. 3), while the number of atoms in the lower Ga layer remains
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unchained (L4 inf fig. 3). Although the Ga bilayer becomes quite disorganized, the atomic positions of the carbon layer are very close to the ideal unsupported graphene monolayer:
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1.42 Å) and the vertical buckling is only 0.24 Å.
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carbon-carbon bondlenghts vary from 1.40 Å to 1.41 Å (the ideal calculated bond length is
Page 8 of 17
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FIG. 3. √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with carbon atoms on top of the Ga atoms (structure IV in Fig. 1). (a) A lateral view of the slab. (b) Top view of the graphene layer (L1). (c) Top view of the top Ga layer (L2). (d) The new Ga layer (L3). (e) The bottom Ga layer that belongs to the initial Ga bilayer, and (f) Side view, top: graphene layer, where a small undulation is observed; bottom: plot of the average vertical separation between the layers, the vertical separations in bulk GaN are plotted as continuous lines.
Page 9 of 17
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FIG. 4. Electronic energy bands for (a) 3√3×3√3 graphene on top of a 4×4 (0001) GaN surface with C atoms on H3, and (b) √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with C atoms on top of the Ga atoms. Both structures exhibit metallic-like behavior, but their most remarkable feature is the presence of Dirac cones, which are highlighted with red circles.
Now, the electronic properties of the carbon monolayers on GaN(0001) surfaces are studied by calculating the band structures of the two stable surfaces. The band structure of the N-rich configuration is shown in Fig. 4a. The existence of Dirac cones is clearly seen at the K point in the surface Brillouin zone, just above the Fermi level. This indicates that
Page 10 of 17
graphene’s π-bond network is preserved, which is a direct consequence of the remarkable feature where the graphene geometry is maintained and the carbon honeycomb network is preserved under N-rich conditions. The band structure of the Ga-rich configuration is
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shown in Fig. 4b, where a clear metallic behavior can be seen, as well as the presence of the Dirac cones. However, the Dirac cones are at the Γ point in the Brillouin zone. This surface
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has a slightly magnetic character with a magnetization of |m| = 0.19 µB/cell, indicating a
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small magnetic moment. The preservation of the Dirac cones in both structures is not difficult to understand because the hexagonal honeycomb lattice is maintained in both
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configurations. 4. Conclusions
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To summarize, first-principles total energy calculations on the formation of graphene monolayers on GaN(0001) surfaces have been presented. Stable structures were obtained
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for a wide range of Ga chemical potentials. The most stable configurations were a 4×4 (0001) GaN/3√3×3√3 graphene under N-rich conditions and a 2√3×2√3 (0001) GaN
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Northrup bilayer/√21×√21 graphene under Ga-rich conditions. These two structures maintain their hexagonal honeycomb lattice with the C-C bonds intact. As a consequence, the Dirac cones remain in both structures. In the first structure, the cones are at the K point in the Brillouin zone, but above the Fermi level, and in the second structure, the cones are at the Γ point at the Fermi level. The last structure exhibits a small magnetic moment. In addition, the formation energy of the 4×4 (0001) GaN/3√3×3√3 graphene structures, which were stable under N-rich conditions, were unaffected by their lateral position. These results demonstrate that GaN(0001) is an excellent substrate for supporting graphene
Page 11 of 17
layers as it maintains most of its properties. For this reason, graphene/GaN interfaces are of potential importance in graphene and GaN applications. Acknowledgements
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N.T. and M.M.A. thank the DGAPA-UNAM projects IN103512-3, IN102714-3 and Conacyt
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project 164485 for partial financial support. The authors are grateful to A. Rodriguez for his technical assistance. Calculations were performed at the DGCTIC-UNAM under projects SC14-
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1-I-48 and SC14-1-I-20. M.E. and J.A. thank Universidad Nacional de Colombia for partial financial support. Some of the calculations were performed at the Grupo de Estudio de
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Materiales (GEMA) cluster. U. N. XcrySDen24 was used to make some of the graphs.
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K. Geim, Proc. Natl. Acad. Sci. USA, PNAS 102 (2005) 10451-10453. [3] A. K. Geim and K. S. Novoselov Nature Materials, 6 (3) (2007) 183-191. [4] S. Gilje, S. Han, M. Wang, K.L. Wang and R.B. Kaner, Nano Letters 7 (11), 2007, pp. 33943398, 788.
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[7] Jinhai Mao, Li Huang, Yi Pan, Min Gao, Junfeng He, Haitao Zhou, Haiming Guo, Yuan Tian, Qiang Zou, Lizhi Zhang, Haigang Zhang, Yeliang Wang, Shixuan Du, Xingjiang Zhou, A. H. Castro Neto and Hong-Jun Gao. Applied Physics Letters. 100 (2012) 093101.
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[8] Ishigami, M., Chen, J.H., Cullen, W.G., Fuhrer, M.S., Williams, E.D. Nano Letters, 7 (6) (2007) 1643-1648.
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[10] Y. Gohda and S. Tsuneyuki, Appl. Phys. Lett. 100 (2012) 053111.
[11] Paolo Giannozzi, Stefano Baroni, Nicola Bonin, et al J. Phys.: Condens. Matter 21 (2009)
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395502; http://www.quantum-espresso.org
[12] J. P. Perdew, K. Burke, and M. Ernzerhof. Phys. Rev. Lett. 77 (1996) 3865.
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[13] Stefan Grimme, J. Comput. Chem. 27(15) (2006) 1787-1799.
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[14] David Vanderbilt, Phys. Rev. B, 41(11) (1990) 7892-7895.
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[15] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt. Phys. Rev. B 47 (1993) 10142.
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[16] H. Monkhorst and J. Pack, Phys. Rev. B 13 (1976) 5188 [17] Leszczynski, M., H. Teisseyre, T. Suski, I. Grzegory, M. Bockowski, J. Jun, S. Porowski, K. Pakula, J.M. Baranowski, C.T. Foxon, T.S. Cheng. Appl. Phys. Lett. 69(1) (1996) 73-75. [18] J. E. Northrup, J. Neugebauer, R. M. Feenstra, and A. R. Smith. Phys. Rev. B 61 (2000) 9932
[19] A. L. Rosa, J. Neugebauer, J. E. Northrup, Chae-Deok Lee, and R. M. Feenstra, Appl. Phys. Lett. 80 (2002) 2008 [20] C. D. Lee, Y. Dong, and R. M. Feenstra, J. E. Northrup and J. Neugebauer, Phys. Rev. B, 68 (2003) 205317
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[21] John E. Northrup and Chris G. Van de Walle, Appl. Phys. Lett. 84 (2004) 4322 [22] John E. Northrup, Appl. Phys. Lett. 86 (2005) 122108. [23] Noboru Takeuchi, Annabella Selloni, T. H. Myers, and A. Doolittle, Phys. Rev. B, 72
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[24] A. Kokalj, Comp. Mater. Sci., 28 (2003) 155–168. Code available from
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http://www.xcrysden.org/http://www.xcrysden.org/
Figure captions
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FIG. 1. Relative formation energies (in eV/1×1) of graphene on GaN(0001) as a function of the Ga chemical potential. The zero of energy refers to a clean GaN(0001) surface. The energies of
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the laterally contracted Ga bilayer (LCGa) with registers A and B proposed by Northrup et al [18] to explain the pseudo (1×1) reconstruction of GaN(0001) are also presented as a reference.
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FIG. 2. 3√3×3√3 graphene on a 4×4 (0001) GaN surface where the C atoms are on H3 sites
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(structure I in Fig. 1). (a) The side view and (b) the top view of the graphene layer. (c) The top view of the top most GaN bilayer, (d) a side view of the graphene layer showing a small
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undulation of ~ 0.5Å. Additionally, a plot of the average separation between the layers is shown. The value of the separation in bulk GaN is presented as a reference and dashed lines are included to guide the eyes. (e) A top view of the first three monolayers: graphene, Ga and N, showing the relative position of a C atom at a H3 site on top of a (0001) GaN substrate. FIG. 3. √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with carbon atoms on top of the Ga atoms (structure IV in Fig. 1). (a) A lateral view of the slab. (b) Top view of the graphene layer (L1). (c) Top view of the top Ga layer (L2). (d) The new Ga layer (L3). (e) The bottom Ga layer that belongs to the initial Ga bilayer, and (f) Side view, top: graphene layer, where a small undulation is observed; bottom: plot of the average vertical separation between the layers, the vertical separations in bulk GaN are plotted as continuous lines.
Page 14 of 17
FIG. 4. Electronic energy bands for (a) 3√3×3√3 graphene on top of a 4×4 (0001) GaN surface with C atoms on H3, and (b) √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with C atoms on top of the Ga atoms. Both structures exhibit metallic-like behavior, but their most remarkable feature is the presence of Dirac cones, which are highlighted
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with red circles.
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*Highlights (for review)
Highlights
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• DFT calculations predict perfect graphene monolayer can be obtained over GaN(0001). • Graphene maintains its hexagonal honeycomb structure with the C-C bonds intact. • Dirac cones indicate preservation of the graphene π-network. • Stable structures were obtained for a wide range of Ga chemical potentials. • GaN(0001) is an excellent substrate for supporting graphene layers
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Graphical Abstract (for review)
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S0169-4332(14)02522-7 http://dx.doi.org/doi:10.1016/j.apsusc.2014.11.057 APSUSC 29102
To appear in:
APSUSC
Received date: Accepted date:
8-9-2014 10-11-2014
Please cite this article as: M.E. Rico, J.A.R. Martinez, M.G. Moreno-Armenta, N. Takeuchi, Graphene monolayers on GaN(0001), Applied Surface Science (2014), http://dx.doi.org/10.1016/j.apsusc.2014.11.057 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphene monolayers on GaN(0001) Miguel Espitia Rico1,2, Jairo Arbey Rodríguez Martinez2, María G. Moreno-Armenta3 and Noboru Takeuchi3 of Physics, Grupo GEFEM, Universidad Distrital Francisco José de Caldas, Bogotá,
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1Dept.
Colombia.
of Physics, GEMA Grupo de Estudio de Materiales, Universidad Nacional de
cr
2Dept.
3Centro
us
Colombia, Bogotá, Colombia.
de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de México, Km
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107 carretera Tijuana-Ensenada, Apdo Postal 14, CP 22800, Ensenada, B.C., México.
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Abstract
The epitaxial growth of graphene on GaN(0001) surfaces is studied by first principles total
d
energy calculations. The idea is to understand how defect-free graphene can be grown on
te
substrates. It is found that the most stable structures were a 4×4(0001) GaN/3√3×3√3 graphene
Ac ce p
and a 2√3×2√3 (0001) GaN Northrup bilayer/√21×√21 graphene, grown under N- and Ga-rich conditions, respectively. In these structures, graphene maintains its hexagonal honeycomb structure with the C-C bonds intact. Preservation of the π-network for the graphene layers was demonstrated by the presence of Dirac cones. The band structures for both the N and Ga-rich configurations show metallic characteristics and the Ga-rich configuration is slightly magnetic. This demonstrates that GaN(0001) is an excellent substrate for supporting graphene layers. Keywords: Graphene, Dirac cones, GaN Surface, Interface GaN/Graphene
1. Introduction
Page 1 of 17
Graphene, a single layer of graphite, is a nanomaterial with unique properties, making it one of the most promising materials for applications in many areas including nanoelectronics [1-3]. Graphene’s hexagonal honeycomb lattice has two nonequivalent
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carbon atoms per unit cell, which leads to a unique band structure, characterized by the formation of Dirac cones with a linear dispersion around the points where the valence and
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conduction bands touch. For applications in nanoelectronics, graphene is commonly grown
us
on an insulator or semiconductor substrate [4-8]. Under these conditions, the atomic structure of the graphene layer may become distorted, resulting in subsequent
an
modification of its electronic structure [9]. Therefore, in the resulting graphene/substrate
M
system, it is desirable that the graphene layer maintains its honeycomb structure. Using first-principles total energy calculations, we have shown that this condition was satisfied in
d
graphene/GaN(0001) for a wide range of Ga chemical potentials. By itself, GaN is an
te
important technological material as it allows the formation of two-dimensional quantum wells and superlattices. GaN is used in a large variety of technological applications such as
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high-power and high-frequency electronic devices. Therefore, graphene/GaN interfaces combine two important fields, extending the possible applications of both. Theoretical studies have suggested that the GaN-graphene interface has a well-defined 1×1 structure with the same lattice constant as bulk GaN. However, the lattice mismatch between graphene and GaN is large (around 30%). More recently, Gohda and Tsuneyuki [10] proposed a GaN-√3×√3/graphene-2×2 superstructure (from now on this will be called Gohda’s model) as the most probable atomic structure for graphene on a GaN(0001) surface. To release the tensile strain generated by a lattice mismatch of 12%, the C-bond network was partially broken, resulting in the formation of C-N bonds.
Page 2 of 17
We present first-principles total energy calculations on the formation of graphene monolayers on GaN(0001) surfaces. Approximately 50 different configurations were calculated and structures with surface formation energies much lower than the 1×1
ip t
GaN(0001)/graphene structure and Gohda’s model were obtained. The calculations were performed under either N- or Ga-rich conditions. Different from Gohda’s model, these
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structures maintained their hexagonal honeycomb lattice structure and the C-C bonds
us
remained intact.
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2. Method
M
Calculations were performed using periodic density functional theory (DFT), which were implemented in the plane-wave self-consistent field (PWscf) code in the Quantum
d
ESPRESSO package [11]. The exchange and correlation energies were modeled according to
te
the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) gradient-corrected functional [12]. Long-range interactions were included with the B97-D
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functional based on Beck’s power-series ansatz, and they were explicitly parameterized by including damped atom-pairwise dispersion corrections [13]. Electron-ion interactions were treated with the pseudopotential method [14,15]. The electron wavefunctions were expanded into plane waves with a kinetic-energy cutoff of 30 Ry. For the charge density, a kinetic energy cutoff of 240 Ry was used. A 3×3×1 Monkhorst-Pack mesh [16] was used to generate the k-points in the 2×2 unit cell. The GaN bulk atomic structure was optimized to determine the optimal lattice parameters, a = b = 3.22 Å and c/a = 1.63, and the internal parameter u = 0.38, which were in good agreement with the experimental data [17], within ~1.2%.
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To model the GaN(0001) surface, a repeated slab geometry was used. Each slab consisted of four GaN double layers with adatoms on the surface. For the Ga-rich conditions, a bilayer of Ga was added. Dangling bonds on the bottom of the surface were saturated with pseudo-
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H atoms, with a fractional charge of 0.75e. The bottom GaN bilayer and the saturated
pseudo-H atoms were frozen to simulate a bulk-like environment. Two consecutive slabs
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were separated by an empty space ~9.0 Å wide to reduce the slab-slab interactions.
FIG. 1. Relative formation energies (in eV/1×1) of graphene on GaN(0001) as a function of the Ga chemical potential. The zero of energy refers to a clean GaN(0001) surface. The energies of the laterally contracted Ga bilayer (LCGa) with registers A and B proposed by Northrup et al [18] to explain the pseudo (1×1) reconstruction of GaN(0001) are also presented as a reference.
3. Results and Discussion
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The surface formation energies of the most relevant configurations, that we have studied, as a function of the Ga chemical potential are described in Figure 1. For N-rich conditions, the graphene layer was on top of an ideally terminated GaN bilayer. For the Ga-rich
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conditions, two possibilities were considered: first, the graphene layer was on top of an uncontracted Ga bilayer, and second, the graphene layer was on top of a laterally contracted
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Ga bilayer (LCB) structure, as proposed by Northrup et al [18-23] to explain the pseudo
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(1×1) GaN (0001) surface grown under Ga-rich conditions. In this model, contraction occurred uniformly within the plane and only the top adlayer was contracted.
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Under N-rich conditions, 4×4 (0001) GaN/3√3×3√3 graphene structures were the most
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stable (labeled I, II, and III in Fig. 1), with lattice mismatches of −0.72%. In structure I, a carbon atom is located at the H3 high-symmetry point (HSP), while the carbon atoms are
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located at T1 and T4 sites in structures IIa and IIb respectively. In structure III graphene's
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H6 HSP is at the H3 HSP of GaN(0001). The energies of I, II, and III were almost degenerated, but configuration I, which has carbon atoms at the H3 sites, had the lowest
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energy and therefore it is the most stable structure under N-rich conditions. These results show that the 4×4 (0001) GaN/3√3×3√3 graphene structures were barely affected by their lateral position. For Ga-rich conditions, 2√3×2√3 (0001) GaN Northrup bilayer/√21×√21 graphene structures were the most stable (curves IV and V in Fig. 1), with lattice mismatches of −1.1%. In both structures, the carbon atoms are on top of the Ga atoms (T1 sites) and they differed from each other in the registries with the underlying contracted Ga layer. The energy separation between IV and V was ~0.2eV/(1×1) cell. Under Ga-rich conditions, the most stable structure was IV, with a graphene layer on a laterally contracted Ga bilayer with Northrup’s B registry [19].
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Fig. 2 shows the atomic configuration of the most stable structure under N-rich conditions, a 3√3×3√3 graphene con iguration on a 4×4 (0001) GaN surface with a carbon atom on a H3 site. In Fig. 2a, the values of the lattice constants in the horizontal and vertical directions
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are included. The lattice constant in the horizontal direction is close (maximum
discrepancy ~0.3%) to the value in bulk GaN. The calculated vertical lattice constant (the c
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value of the GaN in the wurtzite phase) is also close to that of the bulk. The interlayer
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distances are within 0.03 Å (~4.6% and 1.5%) of the bulk values of 0.65 Å and 1.97 Å. In the first bilayer, the Ga-N nearest neighbor distances are within 0.04 Å (~2.0%) of the bulk
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value of 1.97 Å. In Fig. 2b, we can see that the values of the lattice constant, the angles, and
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several bond lengths at different sites in the unit cell for the relaxed graphene layer are close to the ideal values for graphene. The graphene layer is 2.39 Å above the top of the Ga
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layer and it is slightly distorted with carbon-carbon bond lengths varying from 1.41 Å to
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1.46 Å (the ideal calculated bond length is 1.42 Å) with a vertical buckling of 0.54 Å. Figs. 2c and 2d reaffirm that the geometry of the GaN unit cell is very close to the bulk geometry.
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Consequently, the most remarkable feature is that the geometry of graphene is maintained and the carbon honeycomb network is preserved. Therefore, the bilayers of GaN and the graphene monolayers in this stable structure are similar to bulk GaN and ideal graphene respectively. Therefore, the graphene layer and the GaN(0001) surface are vertically bound to each other by a weak interactions such as Van der Waals interactions.
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FIG. 2. 3√3×3√3 graphene on a 4×4 (0001) GaN surface where the C atoms are on H3 sites
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(structure I in Fig. 1). (a) The side view and (b) the top view of the graphene layer. (c) The top view of the top most GaN bilayer, (d) a side view of the graphene layer showing a small undulation of ~ 0.5Å. Additionally, a plot of the average separation between the layers is shown. The value of the separation in bulk GaN is presented as a reference and dashed lines are included to guide the eyes. (e) A top view of the first three monolayers: graphene, Ga and N, showing the relative position of a C atom at a H3 site on top of a (0001) GaN substrate.
Figure 3 shows the most stable structure under Ga-rich conditions (structure D in Fig. 1): √21×√21graphene on top of a 2√3×2√3(0001) Northurp’s Ga-bilayer (registry B)GaN(0001), with a carbon on top of a Ga atom. The calculations started with a graphene layer on top of Northup’s Ga bilayer model with a √3×√3 reconstruction, the contracted
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layer in registry B. Before the relaxation, the top most Ga layer was formed by 16 atoms, while all other layers had 12 atoms each (our unit cell is formed by 4 √3×√3 cells). After full relaxation, the atoms belonging to the Ga bilayer disorders and they rearrange
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themselves into three layers: the first one (labeled L2 in fig. 3) loses 3 atoms, getting an atomic density closer to the ideal bulk density. These displaced 3 atoms form an
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intermediate layer (L3 in fig. 3), while the number of atoms in the lower Ga layer remains
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unchained (L4 inf fig. 3). Although the Ga bilayer becomes quite disorganized, the atomic positions of the carbon layer are very close to the ideal unsupported graphene monolayer:
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1.42 Å) and the vertical buckling is only 0.24 Å.
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carbon-carbon bondlenghts vary from 1.40 Å to 1.41 Å (the ideal calculated bond length is
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FIG. 3. √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with carbon atoms on top of the Ga atoms (structure IV in Fig. 1). (a) A lateral view of the slab. (b) Top view of the graphene layer (L1). (c) Top view of the top Ga layer (L2). (d) The new Ga layer (L3). (e) The bottom Ga layer that belongs to the initial Ga bilayer, and (f) Side view, top: graphene layer, where a small undulation is observed; bottom: plot of the average vertical separation between the layers, the vertical separations in bulk GaN are plotted as continuous lines.
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FIG. 4. Electronic energy bands for (a) 3√3×3√3 graphene on top of a 4×4 (0001) GaN surface with C atoms on H3, and (b) √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with C atoms on top of the Ga atoms. Both structures exhibit metallic-like behavior, but their most remarkable feature is the presence of Dirac cones, which are highlighted with red circles.
Now, the electronic properties of the carbon monolayers on GaN(0001) surfaces are studied by calculating the band structures of the two stable surfaces. The band structure of the N-rich configuration is shown in Fig. 4a. The existence of Dirac cones is clearly seen at the K point in the surface Brillouin zone, just above the Fermi level. This indicates that
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graphene’s π-bond network is preserved, which is a direct consequence of the remarkable feature where the graphene geometry is maintained and the carbon honeycomb network is preserved under N-rich conditions. The band structure of the Ga-rich configuration is
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shown in Fig. 4b, where a clear metallic behavior can be seen, as well as the presence of the Dirac cones. However, the Dirac cones are at the Γ point in the Brillouin zone. This surface
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has a slightly magnetic character with a magnetization of |m| = 0.19 µB/cell, indicating a
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small magnetic moment. The preservation of the Dirac cones in both structures is not difficult to understand because the hexagonal honeycomb lattice is maintained in both
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configurations. 4. Conclusions
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To summarize, first-principles total energy calculations on the formation of graphene monolayers on GaN(0001) surfaces have been presented. Stable structures were obtained
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for a wide range of Ga chemical potentials. The most stable configurations were a 4×4 (0001) GaN/3√3×3√3 graphene under N-rich conditions and a 2√3×2√3 (0001) GaN
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Northrup bilayer/√21×√21 graphene under Ga-rich conditions. These two structures maintain their hexagonal honeycomb lattice with the C-C bonds intact. As a consequence, the Dirac cones remain in both structures. In the first structure, the cones are at the K point in the Brillouin zone, but above the Fermi level, and in the second structure, the cones are at the Γ point at the Fermi level. The last structure exhibits a small magnetic moment. In addition, the formation energy of the 4×4 (0001) GaN/3√3×3√3 graphene structures, which were stable under N-rich conditions, were unaffected by their lateral position. These results demonstrate that GaN(0001) is an excellent substrate for supporting graphene
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layers as it maintains most of its properties. For this reason, graphene/GaN interfaces are of potential importance in graphene and GaN applications. Acknowledgements
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N.T. and M.M.A. thank the DGAPA-UNAM projects IN103512-3, IN102714-3 and Conacyt
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project 164485 for partial financial support. The authors are grateful to A. Rodriguez for his technical assistance. Calculations were performed at the DGCTIC-UNAM under projects SC14-
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1-I-48 and SC14-1-I-20. M.E. and J.A. thank Universidad Nacional de Colombia for partial financial support. Some of the calculations were performed at the Grupo de Estudio de
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Materiales (GEMA) cluster. U. N. XcrySDen24 was used to make some of the graphs.
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References
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Grigorieva1 and A. A. Firsov, Science, 306 (2004) 666-669.
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K. Geim, Proc. Natl. Acad. Sci. USA, PNAS 102 (2005) 10451-10453. [3] A. K. Geim and K. S. Novoselov Nature Materials, 6 (3) (2007) 183-191. [4] S. Gilje, S. Han, M. Wang, K.L. Wang and R.B. Kaner, Nano Letters 7 (11), 2007, pp. 33943398, 788.
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[7] Jinhai Mao, Li Huang, Yi Pan, Min Gao, Junfeng He, Haitao Zhou, Haiming Guo, Yuan Tian, Qiang Zou, Lizhi Zhang, Haigang Zhang, Yeliang Wang, Shixuan Du, Xingjiang Zhou, A. H. Castro Neto and Hong-Jun Gao. Applied Physics Letters. 100 (2012) 093101.
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[8] Ishigami, M., Chen, J.H., Cullen, W.G., Fuhrer, M.S., Williams, E.D. Nano Letters, 7 (6) (2007) 1643-1648.
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[9] Jong-Kwon Lee, Shiro Yamazaki, Hoyeol Yun, ete al, Nano Lett., 13 (8) (2013) 3494-3500.
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[15] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt. Phys. Rev. B 47 (1993) 10142.
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[16] H. Monkhorst and J. Pack, Phys. Rev. B 13 (1976) 5188 [17] Leszczynski, M., H. Teisseyre, T. Suski, I. Grzegory, M. Bockowski, J. Jun, S. Porowski, K. Pakula, J.M. Baranowski, C.T. Foxon, T.S. Cheng. Appl. Phys. Lett. 69(1) (1996) 73-75. [18] J. E. Northrup, J. Neugebauer, R. M. Feenstra, and A. R. Smith. Phys. Rev. B 61 (2000) 9932
[19] A. L. Rosa, J. Neugebauer, J. E. Northrup, Chae-Deok Lee, and R. M. Feenstra, Appl. Phys. Lett. 80 (2002) 2008 [20] C. D. Lee, Y. Dong, and R. M. Feenstra, J. E. Northrup and J. Neugebauer, Phys. Rev. B, 68 (2003) 205317
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[21] John E. Northrup and Chris G. Van de Walle, Appl. Phys. Lett. 84 (2004) 4322 [22] John E. Northrup, Appl. Phys. Lett. 86 (2005) 122108. [23] Noboru Takeuchi, Annabella Selloni, T. H. Myers, and A. Doolittle, Phys. Rev. B, 72
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[24] A. Kokalj, Comp. Mater. Sci., 28 (2003) 155–168. Code available from
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http://www.xcrysden.org/http://www.xcrysden.org/
Figure captions
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FIG. 1. Relative formation energies (in eV/1×1) of graphene on GaN(0001) as a function of the Ga chemical potential. The zero of energy refers to a clean GaN(0001) surface. The energies of
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the laterally contracted Ga bilayer (LCGa) with registers A and B proposed by Northrup et al [18] to explain the pseudo (1×1) reconstruction of GaN(0001) are also presented as a reference.
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FIG. 2. 3√3×3√3 graphene on a 4×4 (0001) GaN surface where the C atoms are on H3 sites
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(structure I in Fig. 1). (a) The side view and (b) the top view of the graphene layer. (c) The top view of the top most GaN bilayer, (d) a side view of the graphene layer showing a small
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undulation of ~ 0.5Å. Additionally, a plot of the average separation between the layers is shown. The value of the separation in bulk GaN is presented as a reference and dashed lines are included to guide the eyes. (e) A top view of the first three monolayers: graphene, Ga and N, showing the relative position of a C atom at a H3 site on top of a (0001) GaN substrate. FIG. 3. √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with carbon atoms on top of the Ga atoms (structure IV in Fig. 1). (a) A lateral view of the slab. (b) Top view of the graphene layer (L1). (c) Top view of the top Ga layer (L2). (d) The new Ga layer (L3). (e) The bottom Ga layer that belongs to the initial Ga bilayer, and (f) Side view, top: graphene layer, where a small undulation is observed; bottom: plot of the average vertical separation between the layers, the vertical separations in bulk GaN are plotted as continuous lines.
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FIG. 4. Electronic energy bands for (a) 3√3×3√3 graphene on top of a 4×4 (0001) GaN surface with C atoms on H3, and (b) √21×√21 graphene on top of a 2√3×2√3 (0001) Ga-bilayer-reg B/GaN surface with C atoms on top of the Ga atoms. Both structures exhibit metallic-like behavior, but their most remarkable feature is the presence of Dirac cones, which are highlighted
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with red circles.
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*Highlights (for review)
Highlights
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• DFT calculations predict perfect graphene monolayer can be obtained over GaN(0001). • Graphene maintains its hexagonal honeycomb structure with the C-C bonds intact. • Dirac cones indicate preservation of the graphene π-network. • Stable structures were obtained for a wide range of Ga chemical potentials. • GaN(0001) is an excellent substrate for supporting graphene layers
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Graphical Abstract (for review)
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