CONTENTS
2.3.2 Graphone-hBN Heterostructures
Owing to its extraordinary electronic, optical, thermal, and mechanical properties and immense potential for nanoelectronic applications, graphene has attracted enormous attention of both academic and industrial research groups [1,2,3,4,5,6,7,8,9]. Graphene is a promising material for several technological applications, such as field effect transistors, photodetectors, electrodes in solar cells and batteries, interconnects, etc. [3,9,10,11,12]. When integrated into functional devices the electronic properties of graphene are heavily influenced by the surrounding materials in these devices. For example, the surrounding dielectrics screen the scattering potential of charged impurities, which limit the conductivity of graphene. As a result, the conductivity of graphene can be modulated over three orders of magnitude by changing its dielectric environment [13]. Likewise, the plasmon excitations in graphene are strongly affected due to the nonlocal screening of electron-electron interactions by the substrate, even in the limit of a weak van der Waals (vdW) interaction [14].
Typically graphene devices are fabricated on SiO2 and SiC substrates; however, carrier mobility in graphene supported on these substrates reduces significantly due to charged surface states, surface roughness, and surface optical phonons [15,16]. Recently, graphene supported on a hexagonal boron nitride (hBN) substrate was found to exhibit much higher mobility than any other substrate [17,18]. High mobility in graphene on hBN is attributed to an extremely flat surface of hBN, which practically eliminates the scattering due to substrate surface roughness and much weaker coupling of electrons in graphene with surface optical phonons of hBN [17,19]. The hBN layer can also be used as a gate dielectric separating the graphene channel and the gate electrode in field effect devices [20]. Moreover, novel field effect tunneling transistors based on the graphene-hBN tunnel junctions have been recently demonstrated [21]. Here we present the electronic structure calculations of a graphene-hBN interface to aid the understanding of experimental devices based on these heterostructures. The sublattice asymmetry induced on the graphene lattice by the highly polar hBN surface is found to modulate the band gap at the Dirac point of graphene [22].
The zero band gap in graphene is a major impediment for its applications in logic devices [3]. A band gap can be opened through quantum confinement by patterning graphene into the so-called graphene nanoribbons (GNRs) [23,24,25]. However, it is difficult to control the band gap in GNRs due to its strong dependence on the width and edge geometry. Alternatively, the band gap can be opened by chemical functionalization of graphene with a variety of species, such as H, F, OH, etc. [26,27]. The band gap opening by H-functionalization/hydrogenation of graphene has especially been a subject of several recent experimental and theoretical studies, which show that the band gap of graphene can be tuned by controlling the degree of hydrogenation [28,29,30,31]. Here we show that the screening of electron-electron interactions by the polarization of surrounding hBN layers can significantly reduce the band gaps in hydrogenated graphene [22]. Such polarization effects are expected to also play an important role in other emerging monolayer materials, such as MoS2, WS2, hBN, etc.
An effective single particle theory such as the density functional theory (DFT) cannot fully capture the effect of dielectric screening, and a more accurate many-body approach that includes the modification of self-energy due to the dielectric screening is required. Here we use many-body perturbation theory based on Green’s function (G) and the screened Coulomb potential (W) approach, i.e., the GW approximation.
Heterostructures of graphene (or graphone) and hBN are modeled using the repeated-slab approach, where the slabs periodic in the xy plane are separated by a large enough vacuum region along the z direction so that their interaction with the periodic images is negligible. The electronic structure calculations are performed in the framework of first-principles DFT at the level of the local density approximation (LDA) as implemented in the ABINIT code [32]. The Troullier-Martins norm-conserving pseudopotentials and the Teter-Pade parametrization for the exchange correlation functional are used [33,34]. A large enough vacuum of 10 Å in the z direction is used to ensure negligible interaction between periodic images. The Brillouin zone is sampled using Monkhorst-Pack meshes [35] of different size, depending on the size of Quasi-Particle Electronic Structure of Pristine and Hydrogenated Graphene the unit cell: 18 × 18 × 1 for Bernal stacked graphene on hBN, 6 × 6 × 1 for misaligned graphene on hBN, and 18 × 18 × 1 for graphone. Wavefunctions are expanded in plane waves with an energy cutoff of 30 Ha. The quasi-particle corrections to the LDA band structure are calculated within the G0W0 approximation, and the screening is calculated using the plasmon-pole model [36]. The Coulomb cutoff technique proposed by Ismail-Beigi et al. is used to minimize the spurious interactions with periodic replicas of the system [37]. The convergence of GW band gap is carefully tested.
Graphene and hBN are bonded by van der Waals interactions. The equilibrium distance between graphene (or graphone) and hBN is determined using the Vienna ab initio simulation package (VASP), which provides a well-tested implementation of vdW interactions [38,39]. The projected augmented wave (PAW) pseudopotentials [40], the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional in the GGA approximation [41], and the DFT-D2 method of Grimme [42] are used. The same values of energy cutoff, vacuum region, and k-point grids as in the ABINIT calculations are used in VASP calculations. The optimized geometries and the electronic structures calculated using VASP and ABINIT without including vdW interactions are found to agree well with each other.
There have been several recent proposals on using hBN as a substrate [17], a dielectric [20], and a tunneling barrier [21] in graphene-based field effect devices. The electronic transport in such devices is controlled by the band offsets at the graphene-hBN interface. The band offsets can be extracted from the band structure of the supercell of a composite system where graphene is adsorbed on a hBN surface. A schematic of graphene adsorbed on a hBN surface is shown in the inset of Figure 2.1(a), and an ideal Bernal stacking arrangement of graphene with respect to the top hBN layer is shown in Figure 2.1(a). The stacking arrangement of Figure 2.1(a), where half of the C atoms in graphene are positioned above the B atoms and the rest of the C atoms are positioned above the center of hBN hexagons, has been found to be a lowest-energy configuration when both graphene and hBN have the same lattice constant [43,44].
The distance between graphene and the hBN surface is determined using the DFT-D2 method of Grimme as implemented in the VASP code [39,42]. The equilibrium distance is found to be 3.14 Å, indicating that graphene is weakly bonded to the hBN surface. A surface-averaged charge density of , a typical wavefunction near the Dirac point of graphene supported on hBN is depicted by grey circles in Figure 2.1(b). The probability density peaks at ≈0.4 Å from the graphene layer and decays rapidly beyond this distance. The wavefunction near the Dirac point can be well approximated by a pz-like orbital given by . The probability density of the parameterized wavefunction with N = 1.87 Å–1 and ζ = 2.39 Å–1 compares well with that obtained from the DFT simulation, as shown in Figure 2.1(b). The same parameterization works well for the freestanding graphene, indicating that the graphene orbitals do not hybridize with hBN orbitals, and the electronic structure of graphene is only weakly influenced by the hBN substrate.
FIGURE 2.1 (a) Atomistic schematic of a Bernal stacked graphene on a hBN substrate. A Bernal stacked graphene lattice is rotated by 30° with respect to the adjacent hBN layer. The solid black lines depict the unit cell. (b) Surface-averaged density for states near the Dirac point of graphene supported on a hBN substrate. (c) DFT and GW band structures of a composite graphene-hBN system shown in the inset of (a). The bands contributed by hBN are identified separately. The inset shows a small band gap opening in graphene at the k-point. (d) DFT and GW band gaps of Bernal stacked graphene on a hBN substrate as a function of the distance between graphene and the hBN surface.
The DFT and GW band structures of the graphene-hBN system are depicted in Figure 2.1(a) and shown in Figure 2.1(c). Three monolayers of the hBN substrate are included in the electronic structure calculation to ensure the convergence of the GW band gap. Due to the weak interaction between graphene and hBN, the linear band structure of graphene near the Dirac point is preserved and a small band gap is opened up at the Dirac point, as shown in the inset of Figure 2.1(c). The band gap opening in graphene is attributed to the sublattice asymmetry induced on the graphene lattice by a weak electrostatic interaction between graphene and Bernal stacked hBN, which unlike graphene has highly polar B-N bonds [45]. As shown in Figure 2.1(d), the graphene band gap can be modulated by changing the distance of the graphene layer from the hBN substrate. The sublattice asymmetry induced by the polar hBN lattice increases as the distance of the graphene layer from the hBN surface decreases, which results in a higher band gap. The GW-corrected band gap of the Bernal graphene on hBN is found to be higher by ≈0.1 eV than the DFT band gap (Figure 2.1(d)).
FIGURE 2.2 Band edges at the graphene-hBN interface obtained from the band structure of Figure 2.1(c). DFT underestimates the hBN band gap and the resulting band offsets.
The bands contributed by hBN are identified in Figure 2.1(c). The band offsets at the graphene-hBN interface extracted from the composite band structure of Figure 2.1(c) are shown in Figure 2.2. There are two important points to be noted from Figure 2.2—first, the asymmetric conduction and valence band offsets. The signatures of such asymmetry have been observed in recent experiments on tunneling across the graphene-hBN interface [21,46]. Second, the DFT band gap of hBN is smaller by ≈1 eV than the GW-corrected band gap, and as a result, DFT underestimates band offsets by ≈0.5 eV. The graphene-hBN band offsets extracted from the recent experimental data on the graphene-hBN interface are in better agreement with the GW band offsets. This clearly indicates the importance of using advanced methods such as the GW approximation in accurate prediction of band offsets at the interfaces of novel materials.
The ideal Bernal stacking considered above is unlikely to be achieved in realistic devices because of the 1.7% lattice mismatch between graphene and hBN. Indeed, recent experiments report the formation of Moiré patterns at the graphene-hBN interface [18,47]. Moreover, contrary to the above calculations, the experimental data do not show any evidence of the band gap opening at the Dirac point of graphene supported on hBN [17]. To address the discrepancy between the above calculations and the experimental data, we compute band gaps in the graphene-hBN system where the stacking arrangement deviates from the ideal Bernal stacking.
The Bernal stacked graphene lattice is rotated by 30° with respect to the underlying hBN layer. This stacking arrangement can be simulated using a small unit cell depicted by the solid black lines in Figure 2.1(a). The unit cell size is, however, much larger when the stacking arrangement deviates from the ideal Bernal stacking. Here we consider three different misalignments where graphene is rotated by 21.8°, 32.2°, and 13.2° with respect to the adjacent hBN monolayer. The supercells for these rotations are shown in Figure 2.3(a–c), and they contain 28 (14 C, 7 B, 7 N), 52 (26 C, 13 B, 13 N), and 76 (38 C, 19 B, 19 N) atoms, respectively. The supercells are constructed using the commensuration conditions presented in [48]. Only one layer of hBN is included to reduce the computational requirement associated with the GW calculation.
FIGURE 2.3 Atomistic schematics of commensurate unit cells of a misaligned graphenehBN system, where graphene is rotated by (a) 21.8°, (b) 32.2°, and (c) 13.2° with respect to the underlying hBN layer. (c) Density of states (DOS) of Bernal stacked (Figure 2.1(a)) and misaligned (Figure 2.3(a)) graphene supported on a hBN substrate illustrating the band gap closing due to misalignment. Due to the high computational requirements, only one hBN layer is included in the DOS calculation.
As pointed out earlier, the band gap opening in the Bernal stacked graphene on hBN results from the sublattice asymmetry induced on the graphene lattice by the underlying highly polar hBN lattice. When the stacking arrangement deviates from the Bernal stacking, the sublattice asymmetry induced by the hBN is significantly reduced. Figure 2.3(d) shows the DFT density of states (DOS) and the DFT and GW band gaps at the Dirac point in the Bernal stacked (Figure 2.1(a)) and misaligned (Figure 2.3(a)) graphene on hBN. As expected, DFT underestimates the band gap of Bernal stacked graphene on hBN compared to the more accurate GW calculation. The DFT band gap of the misaligned graphene on hBN is 0. The gapless nature of misaligned graphene on hBN could be associated with the fact that DFT underestimates the band gaps. To rule out this possibility, we carried out GW calculations on the misaligned supercells shown in Figure 2.3. The GW-corrected band gap of graphene calculated using all three misaligned supercells remains 0. The calculations presented here explain the experimentally observed zero band gap nature of graphene on hBN. As pointed out earlier, only one hBN layer is included in the simulations of misaligned supercells. Including more layers of hBN is not expected to open up a band gap because the induced sublattice asymmetry on the graphene lattice due to hBN layers decreases rapidly as the distance between graphene and hBN layers increases.
2.3.2 GRAPHONE-HBN HETEROSTRUCTURES
Several recent experimental and theoretical studies have shown that the electronic structure of graphene can be significantly altered by chemical functionalization with H, F, Cl, and transition metal atoms [26,31,49,50]. In these graphene-derived materials, the hybridization of the functionalized C atom changes from sp2 to sp3, which results in band gap opening. Hydrogenation of graphene is of particular interest because several experimental and theoretical studies have demonstrated that the band gap of hydrogenated graphene can be tuned by varying the degree of hydrogenation, and this process can even be reversed to recover pristine graphene [31]. Compared to the graphene-hBN system, the hBN substrate has a very different effect on the band gap of hydrogenated graphene [22]. In the following, we investigate the effect of hBN substrate on the band gap of hydrogenated graphene. A single-sided semihydrogenated graphene, also called graphone, is used as an example; however, the results are also applicable to graphene with different hydrogen coverage.
Figure 2.4(a) shows the atomistic schematics of graphone where C atoms belonging to one sublattice are hydrogenated [29]. Graphone can be synthesized by selectively desorbing hydrogen from one side of graphane. The change of hybridization of C atoms from sp2 to sp3 upon hydrogenation results in a nonplanar atomic structure. The optimized atomic structure of graphone in our calculations is virtually identical to the earlier studies [29]. Graphone has a ferromagnetic ground state with a magnetic moment of ≈1 μB located on each nonfunctionalized C atom.
FIGURE 2.4 Atomistic schematics of single-sided semihydrogenated graphene (graphone). The unit cell is depicted by the dotted lines. (b) DFT and GW band structures of graphone. The inset shows the Brillouin zone and high-symmetry directions.
The DFT and GW band structures of graphone are shown in Figure 2.4(b). The DFT (LDA) band structure of graphone shows a large (≈1.5 eV) spin splitting near the Fermi level but no band gap. The GW-corrected band structure shows a large band gap of 2.79 eV and a spin splitting of ≈4 eV throughout the whole Brillouin zone. A large spin splitting makes graphone an attractive material for spintronic applications [51].
When graphone is deposited on the hBN substrate, it binds weakly to the hBN surface via the van der Waals interaction. The equilibrium distance between graphone and hBN obtained from VASP simulations that include the van der Waals interaction using the DFT-D2 method of Grimme is found to be 3.13 Å, as shown in the atomistic schematic of Figure 2.5(a). The surface-averaged density distribution of states at the valence band maximum and the conduction band minimum are shown in Figure 2.5(b). Similar to graphene, decays to an almost negligible value before it reaches the hBN surface, indicating that graphone orbitals do not hybridize with hBN orbitals. Unlike graphene, where the wavefunctions near the band extrema are composed mainly of pz orbitals, in graphone, the wavefunctions near the band extrema are composed of both s and p orbitals. Both spin-split bands near the Fermi level (Figure 2.4(b)) have appreciable s and p contributions, but the p contribution dominates over the s contribution [29].
Now we consider the effect of substrate, hBN in this case, on the band structure of graphone. The DFT and GW band structures of a composite bilayer of graphone and a hBN monolayer are shown in Figure 2.6(a). If the additional bands in Figure 2.6(a) contributed by the hBN monolayer are excluded, the DFT band structure of graphone deposited on the hBN monolayer is virtually identical to that of the freestanding graphone shown in Figure 2.4(b). This is because of the fact that graphone and hBN layers are weakly bound and are essentially electronically isolated systems within the DFT approximation, which does not include the long-range Coulomb interactions. The GW-corrected bands of graphone, however, are significantly altered when it is deposited on the hBN monolayer.
FIGURE 2.5 (a) Atomistic schematic of graphone supported on a hBN substrate. (b) Surface-averaged density for states of graphone at the valence band maximum and the conduction band minimum.
FIGURE 2.6 DFT and GW band structures of graphone-hBN supercell including (a) one, (b) two, and (c) three layers of hBN. The band gap is converged within 0.05 eV for three or more layers of hBN. The bands contributed by hBN are identified separately.
The GW band gap of freestanding graphone (Figure 2.4(b)) decreases from 2.79 eV to 2.16 eV when deposited on the hBN monolayer. The reduction in GW band gap is related to the polarization effects at the graphone-hBN interface. The dielectric properties, and hence the polarization, depend on the thickness of the dielectric material [52]. To accurately estimate the band gap reduction in the adsorbed graphone layer due to the polarization response of the hBN surface, we carry out a convergence study with respect to the thickness of the hBN layer. The DFT and GW band structures of graphone adsorbed on one, two, and three monolayers of hBN are shown in Figure 2.6(a–c). Including more layers of hBN does not affect the DFT bands contributed by the graphone layer. As expected, the polarizability of the hBN layer increases with the number of monolayers, and as a result, the GW band gap reduction in graphone also increases. The comparison of the band structures of graphone deposited on the four-monolayer-thick hBN layer (not shown) and the band structures in Figure 2.6(a–c) indicates that the band gap of graphone is converged within 0.05 eV when three monolayers of hBN are included to mimic the hBN substrate.
The quasi-particle GW band gaps of freestanding graphone and the hBN substrate-supported graphone are 2.79 eV (Figure 2.4(b)) and 1.93 eV (Figure 2.6(c)), respectively, which corresponds to a 0.86 eV band gap reduction. The surrounding dielectric environment strongly influences the electron-electron interactions in nanostructures, which in turn affect their band gaps. Here the band gap reduction in hBN-supported graphone is a result of more effective screening of electron-electron interaction by hBN compared to the screening by the vacuum in freestanding graphone. The screened Coulomb interaction (W) in graphone is reduced by the polarization of the hBN surface, which consequently lowers the band gap. Such band gap reductions have been previously reported in GW calculations on zero-dimensional systems such as molecules adsorbed on metallic or insulating substrates and quantum dots embedded in a dielectric medium, and one-dimensional systems such as GNRs adsorbed on a hBN substrate and CNTs embedded in a dielectric medium [22,53,54,55,56,57,58,59].
FIGURE 2.7 DFT and GW band structures of (a) graphone sandwiched between hBN layers on both sides and (b) graphone supported on a graphite substrate. The bands contributed by hBN and graphite are identified separately. M K
Monolayer materials such as graphene and its derivatives, when used in realistic devices, are often covered with another material such as the gate dielectric or the metal contact. In graphene-based field effect devices, hBN can be used as a substrate as well as a dielectric separating the channel and the metal contact [17,20,21]. The DFT and GW band structures of graphone surrounded by a bottom hBN substrate andaz top hBN dielectric are shown in Figure 2.7(a). Three layers of hBN on each side of graphone are included in the simulation domain to attain convergence of the GW band gap. The band gap of hBN substrate-supported graphone reduces from 1.93 eV (Figure 2.6(c)) to 1.74 eV (Figure 2.7(a)) when covered with a top hBN layer. The reduction in the screened Coulomb potential, W in graphone due to the additional screening from the top hBN dielectric layer, results in further lowering of the graphone band gap. However, the further band gap reduction due to the additional screening from the top hBN dielectric layer is much smaller.
Hydrogenated graphene can be synthesized by hydrogenation of the (0001) graphite surface [60]. Although this process might result in a different configuration of hydrogenated graphene compared to graphone, the polarization of the underlying graphite surface will have a similar effect on the band gap of the hydrogenated graphene on the surface. As an example of such a system, we show the DFT and GW band structures of graphone supported on the graphite substrate. Compared to the band gap reduction of 0.86 eV in hBN-supported graphone, graphite-supported graphone shows a larger band gap reduction of 1.32 eV. This is because of higher polarizability of graphite than of hBN. Thus the band gap reduction is dependent on the electronic structure and dielectric properties of the substrate [55]. The band gap reduction in hydrogenated graphene due to the surrounding dielectric materials can be estimated within a reasonable accuracy using a semiclassical image charge model presented in [22].
FIGURE 2.8 Band offsets extracted from GW band structures in (a) Figure 2.6(c), (b) Figure 2.7(a), and (c) Figure 2.7(b).
The operation of devices employing two-dimensional materials is strongly influenced by the band offsets at their interfaces with the surrounding materials [61]. The conduction and valence band offsets (denoted by ΔEC and ΔEV, respectively) of graphone when integrated into graphone-hBN and graphone-graphite heterostructures are shown in Figure 2.8. The band offsets at the graphone-graphite interface are symmetric, while the band offsets at the graphone-hBN interface are asymmetric. The valence band offset at the interface between the nonhydrogenated side of graphone and hBN is different in Figure 2.8(a) and (b). This is because of the smaller band gap of graphone in the hBN-graphone-hBN heterostructure compared to the graphone-hBN heterostructure. The band offsets at the two graphone-hBN interfaces in the hBN-graphone-hBN heterostructure are also asymmetric. The bands of the hBN layer on the hydrogenated side of graphone are lower in energy than the bands of the hBN layer on the nonhydrogenated side. This asymmetry is a consequence of the electric field depicted by a thick arrow in Figure 2.8(b), which is generated by a dipole layer due to a slight net positive charge on the hydrogenated C atom and a slight net negative charge on the nonhydrogenated C atom [22].
In summary, the electronic structures of atomically thin materials such as graphene and hydrogenated graphene are found to be strongly influenced by the surrounding materials, hBN in this case. The highly polar hBN surface induces a sublattice asymmetry in the perfect Bernal stacked graphene, resulting in a band gap opening on the order of 0.1 eV at the Dirac point of graphene. The induced asymmetry is significantly reduced when the stacking arrangement deviates from the Bernal stacking, which consequently results in closing of the graphene band gap. Hydrogenation of graphene can open up large band gaps; specifically, the band gap of graphone is larger than 2.5 eV. When deposited on the hBN substrate, the band gap of graphone is lowered by ≈1 eV due to the polarization of the hBN substrate surface. Effects such as the band gap modulation by interaction with the surrounding materials can have a profound impact on the devices employing two-dimensional monolayer materials. The band offsets suggest that hBN can also be used as a gate dielectric in the devices based on graphene and hydrogenated graphene.
This work is supported partly by the Interconnect Focus Center funded by the MARCO program of SRC and the State of New York, NSF PetaApps grant number 0749140, and an anonymous gift from Rensselaer. Computing resources of the Computational Center for Nanotechnology Innovations at Rensselaer partly funded by the State of New York and of nanoHUB.org funded by the National Science Foundation have been used for this work. We thank Timothy Boykin, Mathieu Luisier, and Gerhard Klimeck for helpful discussions.
1. ITRS. International Technology Roadmap for Semiconductors. 2009. Available from http://www.itrs.net/Links/2009ITRS/Home2009.htm.
2. A.K. Geim and K.S. Novoselov. The rise of graphene. Nature Materials, 6, 183–191, 2007.
3. F. Schwierz. Graphene transistors. Nature Nanotechnology, 5, 487–496, 2010.
4. 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, 666–669, 2004.
5. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov. Two-dimensional gas of massless Dirac fermions in graphene. Nature, 438, 197–200, 2005.
6. R.R. Nair, P. Blake, A.N. Grigorenko, K.S. Novoselov, T.J. Booth, T. Stauber, N.M.R. Peres, and A.K. Geim. Fine structure constant defines visual transparency of graphene. Science, 320, 1308–1308, 2008.
7. A.A. Balandin, S. Ghosh, W.Z. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C.N. Lau. Superior thermal conductivity of single-layer graphene. Nano Letters, 8, 902–907, 2008.
8. Y.M. Lin, A. Valdes-Garcia, S.J. Han, D.B. Farmer, I. Meric, Y.N. Sun, Y.Q. Wu, C. Dimitrakopoulos, A. Grill, P. Avouris, and K.A. Jenkins. Wafer-scale graphene integrated circuit. Science, 332, 1294–1297, 2011.
9. T. Mueller, F.N.A. Xia, and P. Avouris. Graphene photodetectors for high-speed optical communications. Nature Photonics, 4, 297–301, 2010.
10. H. Park, J.A. Rowehl, K.K. Kim, V. Bulovic, and J. Kong. Doped graphene electrodes for organic solar cells. Nanotechnology, 21, 505204, 2010.
11. M.H. Liang and L.J. Zhi. Graphene-based electrode materials for rechargeable lithium batteries. Journal of Materials Chemistry, 19, 5871–5878, 2009.
12. R. Murali, Y.X. Yang, K. Brenner, T. Beck, and J.D. Meindl. Breakdown current density of graphene nanoribbons. Applied Physics Letters, 94, 243114, 2009.
13. F. Chen, J.L. Xia, D.K. Ferry, and N.J. Tao. Dielectric screening enhanced performance in graphene FET. Nano Letters, 9, 2571–2574, 2009.
14. J. Yan, K.S. Thygesen, and K.W. Jacobsen. Nonlocal screening of plasmons in graphene by semiconducting and metallic substrates: First-principles calculations. Physical Review Letters, 106, 146803, 2011.
15. J.H. Chen, C. Jang, S.D. Xiao, M. Ishigami, and M.S. Fuhrer. Intrinsic and extrinsic performance limits of graphene devices on SiO2. Nature Nanotechnology, 3, 206–209, 2008.
16. L.A. Ponomarenko, R. Yang, T.M. Mohiuddin, M.I. Katsnelson, K.S. Novoselov, S.V. Morozov, A.A. Zhukov, F. Schedin, E.W. Hill, and A.K. Geim. Effect of a high-kappa environment on charge carrier mobility in graphene. Physical Review Letters, 102, 206603, 2009.
17. C.R. Dean, A.F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K.L. Shepard, and J. Hone. Boron nitride substrates for high-quality graphene electronics. Nature Nanotechnology, 5, 722–726, 2010.
18. J.M. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B.J. Leroy. Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nature Materials, 10, 282–285, 2011.
19. J. Schiefele, F. Sols, and F. Guinea. Temperature dependence of the conductivity of graphene on boron nitride. arXiv:1202.2440v1 [cond-mat.mtrl-sci], 2012.
20. I. Meric, C. Dean, A. Young, J. Hone, P. Kim, K.L. Shepard, and IEEE. Graphene field-effect transistors based on boron nitride gate dielectrics. In 2010 International Electron Devices Meeting—Technical Digest, 2010, pp. 556–559.
21. L. Britnell, R.V. Gorbachev, R. Jalil, B.D. Belle, F. Schedin, A. Mishchenko, T. Georgiou, M.I. Katsnelson, L. Eaves, S.V. Morozov, N.M.R. Peres, J. Leist, A.K. Geim, K.S. Novoselov, and L.A. Ponomarenko. Field-effect tunneling transistor based on vertical graphene heterostructures. Science, 335, 947–950, 2012.
22. N. Kharche and S.K. Nayak. Quasiparticle band gap engineering of graphene and graphone on hexagonal boron nitride substrate. Nano Letters, 11, 5274–5278, 2011.
23. A.D. Hernandez-Nieves, B. Partoens, and F.M. Peeters. Electronic and magnetic properties of superlattices of graphene/graphane nanoribbons with different edge hydrogenation. Physical Review B, 82, 165412, 2010.
24. M.Y. Han, B. Ozyilmaz, Y.B. Zhang, and P. Kim. Energy band-gap engineering of graphene nanoribbons. Physical Review Letters, 98, 206805, 2007.
25. Y.W. Son, M.L. Cohen, and S.G. Louie. Energy gaps in graphene nanoribbons. Physical Review Letters, 97, 216803, 2006.
26. L.Z. Li, R. Qin, H. Li, L.L. Yu, Q.H. Liu, G.F. Luo, Z.X. Gao, and J. Lu. Functionalized graphene for high-performance two-dimensional spintronics devices. ACS Nano, 5, 2601–2610, 2011.
27. D.A. Abanin, A.V. Shytov, and L.S. Levitov. Peierls-type instability and tunable band gap in functionalized graphene. Physical Review Letters, 105, 086802, 2010.
28. D. Haberer, D.V. Vyalikh, S. Taioli, B. Dora, M. Farjam, J. Fink, D. Marchenko, T. Pichler, K. Ziegler, S. Simonucci, M.S. Dresselhaus, M. Knupfer, B. Buchner, and A. Gruneis. Tunable band gap in hydrogenated quasi-free-standing graphene. Nano Letters, 10, 3360–3366, 2010.
29. J. Zhou, Q. Wang, Q. Sun, X.S. Chen, Y. Kawazoe, and P. Jena. Ferromagnetism in semihydrogenated graphene sheet. Nano Letters, 9, 3867–3870, 2009.
30. R. Balog, B. Jorgensen, L. Nilsson, M. Andersen, E. Rienks, M. Bianchi, M. Fanetti, E. Laegsgaard, A. Baraldi, S. Lizzit, Z. Sljivancanin, F. Besenbacher, B. Hammer, T.G. Pedersen, P. Hofmann, and L. Hornekaer. Bandgap opening in graphene induced by patterned hydrogen adsorption. Nature Materials, 9, 315–319, 2010.
31. D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, S.V. Morozov, P. Blake, M.P. Halsall, A.C. Ferrari, D.W. Boukhvalov, M.I. Katsnelson, A.K. Geim, and K.S. Novoselov. Control of graphenes properties by reversible hydrogenation: Evidence for graphane. Science, 323, 610–613, 2009.
32. X. Gonze, B. Amadon, P.M. Anglade, J.M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Cote, T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, D.R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M.J.T. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M.J. Verstraete, G. Zerah, and J.W. Zwanziger. ABINIT: First-principles approach to material and nanosystem properties. Computer Physics Communications, 180, 2582–2615, 2009.
33. N. Troullier and J.L. Martins. Efficient pseudopotentials for plane-wave calculations. Physical Review B, 43, 1993–2006, 1991.
34. S. Goedecker, M. Teter, and J. Hutter. Separable dual-space Gaussian pseudopotentials. Physical Review B, 54, 1703–1710, 1996.
35. H.J. Monkhorst and J.D. Pack. Special points for Brillouin-zone integrations. Physical Review B, 13, 5188–5192, 1976.
36. M.S. Hybertsen and S.G. Louie. Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Physical Review B, 34, 5390–5413, 1986.
37. S. Ismail-Beigi. Truncation of periodic image interactions for confined systems. Physical Review B, 73, 233103, 2006.
38. G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational Materials Science, 6, 15–50, 1996.
39. T. Bucko, J. Hafner, S. Lebegue, and J.G. Angyan. Improved description of the structure of molecular and layered crystals: Ab initio DFT calculations with van der Waals corrections. Journal of Physical Chemistry A, 114, 11814–11824, 2010.
40. G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 59, 1758–1775, 1999.
41. J.P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Physical Review Letters, 77, 3865–3868, 1996.
42. S. Grimme. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. Journal of Computational Chemistry, 27, 1787–1799, 2006.
43. B. Sachs, T.O. Wehling, M.I. Katsnelson, and A.I. Lichtenstein. Adhesion and electronic structure of graphene on hexagonal boron nitride substrates. Physical Review B, 84, 195414, 2012.
44. 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, 76, 073103, 2007.
45. E.J. Kan, H. Ren, F. Wu, Z.Y. Li, R.F. Lu, C.Y. Xiao, K.M. Deng, and J.L. Yang. Why the band gap of graphene is tunable on hexagonal boron nitride. Journal of Physical Chemistry C, 116, 3142–3146, 2012.
46. L. Britnell, R.V. Gorbachev, R. Jalil, B.D. Belle, F. Schedin, M.I. Katsnelson, L. Eaves, S.V. Morozov, A.S. Mayorov, N.M.R. Peres, A.H. Castro Neto, J. Leist, A.K. Geim, L.A. Ponomarenko, and K.S. Novoselov. Electron tunneling through ultrathin boron nitride crystalline barriers. Nano Letters, 12, 1707–1710, 2012.
47. R. Decker, Y. Wang, V.W. Brar, W. Regan, H.Z. Tsai, Q. Wu, W. Gannett, A. Zettl, and M.F. Crommie. Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy. Nano Letters, 11, 2291–2295, 2011.
48. S. Shallcross, S. Sharma, and O.A. Pankratov. Quantum interference at the twist boundary in graphene. Physical Review Letters, 101, 056803, 2008.
49. K.J. Jeon, Z. Lee, E. Pollak, L. Moreschini, A. Bostwick, C.M. Park, R. Mendelsberg, V. Radmilovic, R. Kostecki, T.J. Richardson, and E. Rotenberg. Fluorographene: A wide bandgap semiconductor with ultraviolet luminescence. ACS Nano, 5, 1042–1046, 2011.
50. J.T. Robinson, J.S. Burgess, C.E. Junkermeier, S.C. Badescu, T.L. Reinecke, F.K. Perkins, M.K. Zalalutdniov, J.W. Baldwin, J.C. Culbertson, P.E. Sheehan, and E.S. Snow. Properties of fluorinated graphene films. Nano Letters, 10, 3001–3005, 2010.
51. I. Zutic, J. Fabian, and S. Das Sarma. Spintronics: Fundamentals and applications. Reviews of Modern Physics, 76, 323–410, 2004.
52. K. Natori, D. Otani, and N. Sano. Thickness dependence of the effective dielectric constant in a thin film capacitor. Applied Physics Letters, 73, 632–634, 1998.
53. J.B. Neaton, M.S. Hybertsen, and S.G. Louie. Renormalization of molecular electronic levels at metal-molecule interfaces. Physical Review Letters, 97, 216405, 2006.
54. K.S. Thygesen and A. Rubio. Renormalization of molecular quasiparticle levels at metal-molecule interfaces: Trends across binding regimes. Physical Review Letters, 102, 046802, 2009.
55. J.M. Garcia-Lastra, C. Rostgaard, A. Rubio, and K.S. Thygesen. Polarization-induced renormalization of molecular levels at metallic and semiconducting surfaces. Physical Review B, 80, 245427, 2009.
56. J.M. Garcia-Lastra and K.S. Thygesen. Renormalization of optical excitations in molecules near a metal surface. Physical Review Letters, 106, 187402, 2011.
57. A. Franceschetti and A. Zunger. Addition energies and quasiparticle gap of CdSe nano-crystals. Applied Physics Letters, 76, 1731–1733, 2000.
58. X. Jiang, N. Kharche, P. Kohl, T.B. Boykin, G. Klimeck, M. Luisier, P.M. Ajayan, and S.K. Nayak. Giant quasiparticle band gap modulation in graphene nanoribbons supported on weakly interacting surfaces. Submitted for publication, 2012.
59. M. Rohlfing. Redshift of excitons in carbon nanotubes caused by the environment polarizability. Physical Review Letters, 108, 087402, 2012.
60. K.S. Subrahmanyam, P. Kumar, U. Maitra, A. Govindaraj, K. Hembram, U.V. Waghmare, and C.N.R. Rao. Chemical storage of hydrogen in few-layer graphene. Proceedings of the National Academy of Sciences of the United States of America, 108, 2674–2677, 2011.
61. J. Robertson. Band offsets of wide-band-gap oxides and implications for future electronic devices. Journal of Vacuum Science and Technology B, 18, 1785–1791, 2000.