Configuration-Induced Rich Electronic Properties of Bilayer Graphene

Apr 27, 2015 - The objective of this paper is to investigate the geometric and electronic properties of shift-dependent bilayer graphene along armchai...
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Configuration-Induced Rich Electronic Properties of Bilayer Graphene Ngoc Thanh Thuy Tran,† Shih-Yang Lin,† Olga E. Glukhova,‡ and Ming-Fa Lin*,† †

Department of Physics, National Cheng Kung University, Tainan 701, Taiwan Department of Physics, Saratov State University, Saratov 410012, Russia

‡

S Supporting Information *

ABSTRACT: The objective of this paper is to investigate the geometric and electronic properties of shift-dependent bilayer graphene along armchair and zigzag directions using firstprinciple calculations. The interlayer distance and the total ground state energy gradually decrease and subsequently increase during the stacking configuration sequence: AA → AB → AA′ → AA. Furthermore, there are dramatic changes in which Dirac cones are transformed into parabolic bands or nonvertical Dirac cones, accompanied by a separation of the Dirac cones, creation of an arc-shaped stateless region, distorted energy dispersions, extra low-energy critical points, and splitting of middle-energy states. The density of states (DOS) exhibits many prominent peaks derived from saddle points. All the bilayer systems remain semimetals, with their free carrier densities strongly depending on the stacking configuration. The main features of energy bands and DOS can be used to identify the subangstrom misalignment stackings.

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kish graphite on high-purity SiO2 substrates,32 and on freshly cleaved muscovite mica.35 In addition, the growth of AA stacking was achieved on the (0001) surface of C-terminated 4 H-SiC from a hydrogen-propane mixture,32 and on the (111) surface of diamond using CVD.36 Besides highly symmetric stackings, recent studies have manufactured some asymmetric stackings using specific experimental techniques, such as stacking boundaries of BLG,37,38 and twisted graphene.39−41 Both shift and rotation can drastically change the electronic properties of BLG. Stacking boundaries in which one of the graphene layer is shifted with respect to the other have been grown using the CVD method.37 Moreover, the sliding of graphene flakes on a graphene substrate can also be initiated by the scanning tunneling microscopy tip to overcome the van der Waals interactions.42 In addition, micrometer-size graphite flakes can slide spontaneously after moved by a STM tip.43 Similarly, the second layer of graphene will slide with respect to the first layer by this method with the requirement of an initial activation out of the equilibrium state. FLG exhibits distinct low-lying electronic structures when compared to single layer graphene as a result of interlayer atomic interactions. For the AB stacking, the low-lying energy dispersions consist of two pairs of parabolic bands near the Fermi level (EF), as verified by angle-resolved photoemission spectroscopy (ARPES).44 The tiny overlap of the highest valence band and the lowest conduction band indicates a small number of free carriers. Different from the AB stacking, the AA stacking exhibits two pairs of linear bands (Dirac-cone structure) that intersect each other at EF and result in more

INTRODUCTION Since graphene was first synthesized in 2004,1 it has been extensively studied. Graphene exhibits a lot of remarkable physical properties induced by its hexagonal symmetry, such as an anomalous quantum Hall effect,2 an exceedingly high mobility of charge carriers,3 a high transparency,4 and the largest Young’s modulus of materials ever tested.5 It has potential applications in various areas, especially in electronic devices.4,6 To extend the applications of graphene, numerous methods and techniques have been developed to diversify the electronic structure, including some based on stacking configurations,7−9 doping,10−12 external electric or magnetic fields,13−15 and mechanical strain.16−18 Previous studies have shown that few-layer graphene (FLG) can be manufactured using mechanical exfoliation,19,20 chemical vapor deposition (CVD),21−23 the reduction of graphene oxide,24 an electrochemical method that uses oxalic acid (C2H2O4) as the electrolyte,25 and the flame method.26 This paper focuses on how stacking configurations can enrich the electronic properties of bilayer graphene (BLG). FLG possesses highly symmetric configurations, including AA, AB, and ABC stackings. AB27 and ABC stackings28,29 commonly occur in natural graphite, whereas the AA stacking has been found only in intercalated graphite.30 The geometry of stacked structures can be verified using scanning electron microscopy, transmission electron microscopy, optical spectroscopy, atomic force microscopy, and Raman spectroscopy. AB-stacked graphene is frequently observed and can be produced using the mentioned experimental methods,31−33 with CVD having potential for large-scale production with high quality.33 ABC-stacked graphene has also been synthesized, for instance, from a mixture of hydrogen and methane gases activated by DC discharge,34 from mechanical exfoliation of © XXXX American Chemical Society

Received: November 22, 2014 Revised: April 25, 2015

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The Journal of Physical Chemistry C free carries.45 For trilayer graphene, recent experimental studies have revealed that ABA stacking and ABC stacking, as identified by Raman spectroscopy, have different electronic properties.35,46 The energy band of the former is the superposition of those of monolayer graphene and BLG, whereas the latter consists of two pairs of parabolic bands and one pair of complex bands with a Mexican-hat shape.29 It is noticeable that the lowlying energy dispersions of AB and AA or ABA and ABC stackings are significantly different. A dramatic transformation in electronic properties is expected to occur between these two stackings. The novel Dirac-cone structure in graphene can be tuned by the local defects and the periodic electric potential.47,48 Furthermore, the effect of rotation on stacking configurations has been extensively studied.49,50 The parabolic bands in AB stacking are changed into linear bands when the layers are rotated.49 Recently, intermediate configurations between AA and AB stackings have been studied theoretically with regard to an anomalous optical phonon splitting,51 and electronic transmission.52 The dependence of phonon frequency on the sliding stackings of BLG affects the polarized Raman scattering intensity, which can be used to identify tiny misalignments in BLG.51 However, a full comprehension of the low- and middle-energy electronic properties of various stacking configurations has not been previously achieved. This paper studies the dependence of the interlayer distance, total ground state energy, and electronic structure of BLG on the relative shift between two layers, in which the second layer is shifted along armchair and zigzag directions with respect to the first layer. The effect of the stacking configuration, with different geometric symmetries and interlayer atomic interactions, on the electronic properties is investigated using firstprinciple calculations. The results demonstrate that the interlayer distance, total ground state energy, energy dispersions, band-edge states, density of states (DOS), and free carrier densities are very sensitive to changes in the geometric configuration. Specifically, there is a dramatic transformation between Dirac-cone structures and parabolic bands or nonvertical Dirac-cone structures that can induce a stateless region with special saddle points in the energy-wavevector space. Furthermore, rich energy dispersions are expected to lead to a lot of extra prominent peaks in DOS, including those from the splitting of single-peak structures into twin-peak structures. The free carrier densities at EF strongly depend on the variation of the stateless region. The above-mentioned results can be verified by experimental measurements such as ARPES,44 scanning tunneling spectroscopy (STS),53 and optical spectroscopy.54

employed in the calculations using the semiemprical DFT-D2 correction of Grimme59 to correctly describe the interactions between two layers. In this method, the total energy is given by EDFT+D = EKS‑DFT + Edisp, where EKS‑DFT is the self-consistent Kohn−Sham energy and Edisp is a semiempirical dispersion correction. The k-point mesh is set as 100 × 100 × 1 for the geometry optimization, 200 × 200 × 1 for band structure calculations, and 280 × 280× 1 for the DOS calculations.

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GEOMETRIC STRUCTURES The atomic structures of BLG with an interlayer shift along the armchair (δa) and zigzag (δz) directions are shown in Figure 1.

Figure 1. Geometric structures of shift-dependent BLG along armchair and zigzag directions. First Brillouin zone (BZ) with high symmetric points.

At δa = 0, graphene layers are stacked directly on top of each other (AA stacking). When a shift occurs along the armchair direction in the range of 0 ≤ δa ≤ 12/8 (in units of the C−C bond length b = 1.424 Å) and then along the zigzag direction in the range of 0 ≤ δz ≤ 4/8 (in units of (3)1/2b), stackings of BLG change among the high-symmetry configurations: AA (δa = 0) → AB (δa = 1) → AA′ (δa = 12/8; δz = 0) → AA (δz = 4/ 8). Any C atom in the AA stacking has an equivalent chemical environment with respect to its adjacent layers; a similar phenomenon is also found in the AA′ stacking, although the C atoms have different x-y projections. In AA′ stacking, each A(B) site of the second layer is located at the midpoint between the nearest A(B) sites of the first layer. The interlayer distance, as indicated in Figure 2a, gradually decreases and then increases during the variation of the relative shift between two layers. Furthermore, the interlayer distance of the AB stacking is the shortest (3.26 Å), while that of the AA stacking is the largest (3.52 Å). This result is in good agreement with experimental results (3.35 Å for AB stacking and 3.55 Å for AA stacking) as measured by high-resolution transmission electron microscopy36 and other theoretical results.60,61 This reveals that the total ground state energy is also affected by the relative shift, as shown in Figure 2b. Clearly, the shortest interlayer distance induces the strongest van der Waals interactions among the 2pz orbitals. Therefore, the AB stacking

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COMPUTATIONAL DETAILS The first-principle calculations on configuration-dependent BLG are performed based on density functional theory (DFT) using the VASP code.55,56 The electron−electron interactions are evaluated using the exchange-correlation function under the generalized gradient approximation of Perdew−Burke−Ernzerhof,57 whereas the electron−ion interactions are calculated using the projector augmented wave method.58 A plane-wave basis set with a maximum kinetic energy of 500 eV is used to expand the wave function. To avoid interactions between adjacent unit cells, a vacuum layer with a thickness of 15 Å is added to replicate systems in a direction perpendicular to the interface. All atomic coordinates are relaxed until the Hellmann−Feynman force is less than 0.01 eV/Å. Furthermore, the van der Waals (vdW) force is B

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Figure 2. Variation of (a) total ground states energy (ΔE0) and (b) interlayer distance (Δd) with respect to relative shift, as measured from those of AA-stacked BLG.

is expected to be the most stable configuration of BLG since its ground state energy is the smallest, which is consistent with previous studies.36,62

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ELECTRONIC PROPERTIES The low-energy electronic structures of BLG change dramatically with a variation in the stacking configuration. At δa = 0, the low-lying energy bands of AA stacking possess two pairs of linear bands intersecting at EF = 0, as illustrated in Figure 3a. Superscripts “c” and “v” correspond to the conduction and valence bands, respectively. The band overlap in the two vertical Dirac cones indicates the existence of free electrons and holes. As δa increases from 0, there is a strong hybridization between the conduction band of the lower cone and the valence band of the upper cone. This leads to distorted Dirac-cone structures along kx̂ with an induced arc-shaped stateless region near EF, as shown in Figure 3b for δa = 1/8. The existence of this stateless region reduces the carrier density of free electrons and holes near EF. Obviously, the conduction and valence bands only touch each other at two points on the edge of this arc-shaped region; these points remain Fermimomentum states (kF’s) within the sliding of AA → AB. Moreover, there are two pairs of saddle points on the top and bottom of the arc-shaped stateless region within the range 0 < δa ≤ 4/8. The conservation of the number of electronic states indicates that the low-energy states near the K point are transferred to its neighbor regions, particularly at the induced saddle points. However, the energies of the pairs of saddle points are nearly the same for small δa (energy difference of less than 0.02 eV). The arc-shaped region expands quickly, and energy dispersions are seriously distorted at δa = 4/8 (Figure 3c), where the saddle points energies are completely split (∼0.12 eV splitting energy). It is worth noting that the

Figure 3. Low-lying band structures around K point (circle mark) for various stacking configurations along armchair (δa) and zigzag (δz) directions: (a) δa = 0, (b) δa = 1/8, (c) δa = 4/8, (d) δa = 1, (e) δa = 11/8, (f) δa = 12/8, (g) δz = 1/8, and (h) δz = 3/8.

touching Dirac points start to separate with a small further shift (not shown); this means that a dramatic transformation in band structure occurs there. Thereafter, such pairs of saddle points merge and become a single pair of saddle points for 5/8 ≤ δa < 1 (see Supporting Information in Figure S1 for δa = 6/8). A further increase from δa = 1 to δa = 12/8 also leads to drastic changes in the low-lying band structure. For the AB stacking of δa = 1, the arc-shaped stateless region disappears, and two pairs of parabolic bands with a small overlap appear near EF (Figure 3d). With an increase in δa, the parabolic bands of the first pair become seriously distorted along kŷ and −kŷ simultaneously, as shown in Supporting Information Figure S2 and Figure 3e at δa = 10/8 and δa = 11/8, respectively. The arcshaped stateless region, with two Dirac points at distinct energies, quickly expands. Furthermore, two neighboring conduction or valence bands strongly hybridize with each other. There is also an induced pair of saddle points on the top and bottom of the arc-shaped stateless region situated at around ±0.17 eV for 1 < δa < 12/8. Finally, two pairs of isotropic Dirac cones reform in the AA′ stacking (Figure 3f). In C

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The Journal of Physical Chemistry C contrast to the vertical Dirac-cone structures of AA stacking, the energy spectrum of the AA′ stacking is separated into two Dirac cones, in which the nonvertical Dirac points are located at different wave vectors. A transformation between the nonvertical and vertical Diraccone structures takes place as the configuration gradually moves along the zigzag direction from AA′ stacking back to AA stacking. When δz exceeds 0, two neighboring conduction (valence) bands hybridize with each other, as shown in Figure 3g for δz = 1/8. With a further shift, the conduction band of the lower cone hybridizes with the valence band of the upper cone and the Dirac-cone structures are reformed at δz = 3/8 (Figure 3h). An arc-shaped stateless region along the ky-axis appears near EF. The created stateless region, with two Dirac points at distinct energies, gradually grows, then declines, and finally disappears in the AA stacking (δz = 4/8). One pair of saddle points appears at the top or bottom of the arc-shaped stateless region near δz = 1/8, whereas such points split into two pairs of saddle points in the range of 1/8 < δz < 4/8 and then disappear at δz = 4/8. The saddle point energies gradually decrease from 0.2 to 0.06 eV as the shift occurs. The two-dimensional (2D) band structures along high symmetry points are useful for examining the electronic properties. For graphene, the sp2 orbitals of (2px, 2py, 2 s) possess very strong covalent σ-bonds among the three nearestneighbor carbon atoms, while the 2pz orbitals contribute to the π-bonds, being associated with the highest occupied valence band and the lowest unoccupied conduction band, respectively. The σ and σ* bands form the high-lying electronic structures of BLG within the range of |Ec,v| ≥ 3 eV, in which the extreme energies are at the Γ point. It is noticed that these bands are hardly affected by a variation of the stacking configuration since the effects due to interlayer atomic interactions are negligible. The π and π* bands dominate the low-lying electronic structures within the range of |Ec,v| ≤ 2 eV. The energy dispersions near EF, as indicated in the insets of Figure 4, exhibit a transformation between isotropic and anisotropic characteristics. The linear isotropic dispersions exist near the K point for AA stacking (Figure 4a) and start to transform into anisotropic distorted dispersions at δa = 1/8 (Figure 4b). They change into parabolic isotropic ones at δa = 1 (Figure 4d), and then become anisotropic ones (Figure 4e−h). In addition, the existence of tiny energy spacings are likely along certain specific directions such as K → Γand K→ M (Figure 4c). However, all configuration-dependent BLGs are gapless, mainly due to the spatial inversion and time-reversal symmetries. Except for AA and AB stackings, other stackings exhibit distorted energy dispersions and highly anisotropic 2D band structures. Middleenergy electronic states are closely related to the M and M′ points (saddle points of high anisotropy). These two points still remain saddle points during variations of shift, while the change from 6-fold into 2-fold rotational symmetry results in their energy splitting with a wide range of ∼1 eV. Most π-electronic states accumulate near these two points (see also DOS in Figure 5), which are expected to exhibit very strong absorption peaks with a frequency ω≈ 4−5 eV. They can also be found in carbon-related sp2 systems such as carbon nanotubes,63 graphite,64 and FLG.9 For AA and AB stackings, the low-energy expansion about the K point can be used to obtain the analytic energy dispersions by the tight-binding model.65,66 The phase term hk = ∑j exp(−ik·Rj), where Rj represents the position vector between two different carbon atoms, and plays an important

Figure 4. 2D band structures for various stackings: (a) δa = 0, (b) δa = 1/8, (c) δa = 4/8, (d) δa = 1, (e) δa = 11/8, (f) δa = 12/8, (g) δz = 1/8, and (h) δz = 3/8. Inset shows low-lying bands.

role in low-energy expansions. For AA and AB stackings, the projection of atomic positions of the second layer is just on the top or at the hollow site of the first layer so that the phase term hk is in a simple form. However, for the shift-dependent stacking configurations that do not have the specific position vectors, the extreme energy points will seriously deviate from the K point. As a result, the K point is not a high-symmetric critical point anymore and there exist highly anisotropic and distorted energy bands discussed earlier causing a very complicated phase term hk. In this case, the low-energy expansion still can be done if certain important terms related to the interlayer atomic interactions are neglected to simplify the calculations, leading to inaccurate analytic electronic properties. Therefore, it is very hard to perform the low-energy expansions about the K point to obtain the accurate results compared to the numerical method. Up to now, ARPES has served as a powerful experimental method for investigating the electronic band structure of graphene systems. It has been used to confirm the Dirac-conelike linear energy dispersions for graphene layers on SiC.67 ARPES has been used to verify the effects deriving from stacking configurations,44 doping,68 electrostatic gates,44 and the number of layers. 69,70 The feature-rich bands of configuration-dependent BLG, including the distinct energy spacings along highly symmetric directions, the arc-shaped stateless region, the transformation from Dirac cones into parabolic bands or nonvertical Dirac cones, and the middleD

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Figure 6. Peak energies of DOS at (a) low and (b) middle energy; (c) DOS at EF. Subscript “be” corresponds to band-edge states.

Figure 5. DOS for various stackings: (a) δa = 0, (b) δa = 1/8, (c) δa = 4/8, (d) δa = 1, (e) δa = 11/8, (f) δa = 12/8, (g) δz = 1/8, and (h) δz = 3/8.

maximum four peaks near δa = 4/8 (Figure 5c), which are closely related to the dramatic transformation between the Dirac-cone structure and parabolic band structure. Two-peak structures are observed for the other stacking configurations: 4/ 8 < δa < 12/8 and 1/8 < δz < 4/8. Also note that the low-lying peaks in DOS might merge due to their negligible energy differences. The peak energies of the conduction band decrease near AA and AB stacking configurations, whereas they increase near δa = 4/8 and δz = 1/8. The peak energies in the valence band show the opposite behavior. DOS grows quickly with increasing energy and also exhibits prominent peaks in the middle-energy range. The AA stacking possesses four prominent peaks at E ∼ ±2 eV arising from the M and M′ saddle points, as shown in Figure 5a. Such peaks remain strong during a variation of the stacking configuration. This is a common characteristic of carbon-related sp2 systems since the interatomic interactions due to the sliding process change slightly (≈ 0.1 eV) when compared to middle state energies. As the sliding occurs, there still exist four single peaks

energy band splitting, can furthermore be examined with ARPES. This is one way to identify the tiny misalignment stackings associated with various stacking configurations. The main characteristics of electronic structures are directly reflected in DOS, as shown in Figure 5. The low-energy DOS presents two special structures near EF, namely plateau and dip structures. The former is due to linear energy dispersions corresponding to the AA stacking and the configurations close to AA′ stacking (Figure 5a,e,g), while the latter arises from the band-edge states of parabolic energy bands (Figure 5b−d,f,h). Except for AA, AB, and AA′ stackings, the stackings exhibit symmetric peak structures at low energy (≈ ±0.1 eV), which are caused by the created saddle points at the top and bottom of the arc-shaped stateless region. The variation of the number and energy of peaks with respect to the relative shift is shown in Figure 6a. Two peaks exist in the range of 1/8 ≤ δa < 3/8; these begin to split into three distinct peaks at δa = 3/8. There are a E

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The Journal of Physical Chemistry C for AB stacking (Figure 5d) and configurations closely resembling AA stacking (Figure 5b,h). However, some single peaks might split into twin-peak structures (Figure 5c at δa = 4/ 8 and Figure 5e at δa = 11/8), or six single-peak structures (Figure 5f at δa = 12/8). The splitting and merging of prominent peaks are clearly shown in Figure 6b. Along the armchair direction, four single-peak structures can be observed for δa ≤ 2/8 and δa = 1. In addition, some of these four peaks start to split into two twin-peak structures that result in six peaks near δa = 3/8 and δa = 12/8, and seven peaks near δa = 6/ 8. Furthermore, the energy spacings between saddle points expand quickly, leading to the total splitting into four twin-peak structures for 4/8 ≤ δa ≤ 7/8 (except δa = 6/8) and 9/8 ≤ δa ≤ 11/8. Along the zigzag direction, there are four twin peaks for 1/16 ≤ δz ≤ 4/16. Thereafter, peaks merge and become four single peaks for 5/16 ≤ δz ≤ 8/16. The peak energies also change significantly during a variation of the shift. In addition, merged peaks become sharper because of the higher DOS. The obvious alternation in peak structures is caused by the destruction of the 6-fold rotational symmetry, as mentioned earlier. The value of DOS at EF = 0, which is closely related to the free carrier density, deserves a closer examination, as shown in Figure 6c. All stackings have two kinds of free carrier, namely free electrons in the conduction bands and free holes in the valence bands, mainly owning to the band overlap. Of all the configurations, AA stacking has the highest number of free carriers at EF (with DOS = 0.014 states/eV·atom) due to the largest conduction- and valence-band overlap. It is thus expected to have the highest electrical conductivity among all BLG systems. The DOS at EF during variation gradually declines as a result of the creation of the arc-shaped stateless region, as indicated in Figure 3. It attains its lowest value of 0.002 states/eV·atom at δa = 6/8 since the band overlap is smallest here. Then DOS grows to 0.005 states/eV·atom at δa = 1. With further shifting, there is a tiny decrease at δa = 9/8, followed by an increase. DOS reaches 0.007 states/eV·atom for AA′ stacking, which is about one-half that of the AA stacking. This is because the band overlap in the nonvertical Dirac-cone structure is smaller than that of the vertical one. When the interlayer shift moves along the zigzag direction from AA′ to AA, DOS gradually decreases to 0.005 states/eV·atom at δz = 5/16 and subsequently increases. The DOS at EF decreases with increasing area of the arc-shaped stateless region in the band structure. The above-mentioned DOS peaks can be examined by STS, a method that has been used for carbon nanotubes,71 graphene nanoribbons,72 and FLG.73 The tunneling conductance (dI/ dV) is approximately proportional to DOS and directly reflects the main features in DOS. By means of ARPES, it is difficult to measure the saddle points in the arc-shaped region since many directions need to be taken into account. However, from STS measurements on low-energy peaks in DOS, extra saddle points can be determined. The predicted measured energy range of STS is about ±3 eV; an available range for the verification of the middle-energy splitting prominent peaks, and thus, for the identification of the destruction of the 6-fold rotational symmetry. The free carrier density at EF, shown in Figure 6c, can also be examined using STS. STS measurements on lowand middle-energy peaks are useful in determining the subangstrom misalignment stackings of BLG.

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CONCLUSION

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ASSOCIATED CONTENT

In summary, the geometric structure and electronic properties of configuration-dependent BLGs were studied using firstprinciple calculations. The results demonstrate that the significant changes in electronic structures, especially the lowlying energy bands, essentially depend on the magnitude and direction of the relative shift. Specifically, the Dirac-cone structures transform into parabolic bands or nonvertical Dirac cones, with the creation of an arc-shaped stateless region near EF, distorted energy dispersions, extra low-energy saddle points and the splitting of middle-energy saddle points. The featurerich bands, including the anisotropic low-energy bands and the middle-energy bands splitting, can be examined by ARPES. The M and M′ saddle points are expected to cause strong absorption peaks at about 4−5 eV due to the concentration of most π-electronic states. DOS exhibits many special structures induced by rich energy dispersions. There are plateau or dip structures at EF and symmetric peaks due to the induced saddle points at low energy. Moreover, the middleenergy prominent peaks from the M and M′ points are presented in the single- and twin-peak structures. The height and energy of the peaks in DOS depend on the splitting and merging of peaks as the shift occurs. All the systems are semimetals, in which the free carrier densities show a nonmonotonous dependence on the relative shift. The existence of induced saddle points as well as the splitting of the middle-energy peaks and free carriers at EF could be verified using STS. The low- and middle-energy saddle points and the derived prominent DOS peaks are closely related to stacking configurations; therefore, the experimental measurements on them can be used to resolve the subangstrom misalignment stackings of BLG.

S Supporting Information *

Low-lying band structures around K point for δa = 6/8, and δa = 10/8 are provided. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jp511692e.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +886-6-275-7575. Fax: +886-6-2747995. E-mail: mfl[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Physics Division, National Center for Theoretical Sciences (South), the Nation Science Council of Taiwan (Grant No. NSC 102-2112-M-006-007MY3), and the Ministry of Education of Russia in the framework of the project of the state task (No. 3.1155.2014/ K). We also thank the National Center for High-performance Computing (NCHC) for computer time and facilities.

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REFERENCES

(1) Novoselov, K. S.; Geim, A. K.; Morozov, S.; Jiang, D.; Zhang, Y.; Dubonos, S.; Grigorieva, I.; Firsov, A. Electric field effect in atomically thin carbon films. Science 2004, 306, 666−669.

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DOI: 10.1021/jp511692e J. Phys. Chem. C XXXX, XXX, XXX−XXX