Degenerate Perturbation in Band-Gap Opening of ... - ACS Publications

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Degenerate Perturbation in Band-Gap Opening of Graphene Superlattice S. L. Xiu,† L. Gong,† V. Wang,§ Y. Y. Liang,*,‡ G. Chen,*,†,‡ and Y. Kawazoe‡ †

Department of Physics, University of Jinan, Jinan, Shandong 250022, P. R. China New Industry Creation Hatchery Center, Tohoku University, Sendai, Miyagi 980-8579, Japan § Department of Applied Physics, Xi’an University of Technology, Xi’an, Shaanxi 710054, P. R. China ‡

S Supporting Information *

ABSTRACT: We study the graphene band-gap engineering by introducing different defects, namely the defects breaking the inversion symmetry and the ones periodically patterning graphene into superlattice such as the regularly arranged antidots, etc. Comparing to the primitive unit cell of graphene, the pseudo-graphene superlattice, referred to the pristine graphene supercell, modulates the boundary condition accordingly. According to the energy band-folding picture, these superlattices can be categorized into two groups on the basis of the Dirac cone position. In some cases, the Dirac points K and K′ in primitive cell are folded to the Γ point of pseudo-superlattices. The coincidence of Dirac points with Γ point results in the fourfold degeneracy. In these systems, a band gap at Γ point can be opened by introducing periodically arranged defects such as the antidots, which could be easily utilized in experiment, for example, by making the graphene nanomesh through lithography technique. In the other cases, the twofold degenerate Dirac points remain nonequivalent with Γ point in pseudosuperlattice, and the method to break the inversion symmetry could open their band gaps. by using block copolymer lithography.6 The graphene nanomesh field-effect transistor with superior on/off ratio is found to support nearly 100 times greater currents than individual nanoribbon devices, which shows great potential to be used in the nano-optoelectronics. From the point of the view of future exploration of its usage, a fundamental understanding on how electronic properties are modulated by the regular patterning is highly desired. In fact, this method could be regarded to pattern graphene into superlattice with carbon vacancies. Superlattices which have been studied extensively and had a huge impact on semiconductor physics would moderate the Born−von Karman boundary conditions.21,22 As to be discussed in this paper, according to the energy band-folding picture, the fourfold degeneracy would be found at the Γ point in some special cases, which could be then degenerated by applying perturbation such as the missing of the pz orbital. A band gap can then be opened with perturbation. In our study, we combine the tight-binding (TB) method and the first-principles method to explore the fundamental physical insights on the band-gap opening in graphene superlattices, which would in turn facilitate corresponding experimental investigations.

1. INTRODUCTION Graphene, the monolayer of carbon, has triggered tremendous research interests as a promising potential candidate to substitute the present silicon-based semiconductor devices.1−5 The crossing of the valence band and the conduction band at the K (K′) point results in the famous Dirac cone. Due to the linear dispersion of energy bands near the Dirac point, the massless fermions provide us many unique and amazing properties. For example, its charge carriers could be tuned continuously between electrons and holes, which has been investigated experimentally to have a group velocity of v0 ≈ 106 m s−1 offering the possibility of high-performance interconnects in a hypothetical carbon-based nano-optoelectronics.4,5 However, since the conductivity due to the semimetal nature of graphene cannot be turned off completely, pristine graphene sheet cannot be used as a transistor in optoelectronics, where high on/off ratios are required. Recently, tremendous efforts have been devoted to tuning electronic properties of the sheet materials.6−20 A methodology known as the quantum confinement by slicing graphene into nanoribbon has been extensively proven efficiently in making a sizable band gap, which, however, in fact cannot carry sufficiently large currents. Substitution doping guest element into graphene could also open the band gap by breaking the equivalence of its AB sublattices, which has been extensively investigated in theory. However, its utilization in experiment may be a challenge due to the difficulties in controlling the precise and uniform doping for tuning the sizable band gap. Interestingly, a most recent experimental study reported an antidot patterned graphene−the graphene nanomesh fabricated © 2014 American Chemical Society

2. COMPUTATIONAL DETAILS For simplicity to facilitate the discussion of the energy bandfolding picture, we would like to first introduce a set of new Received: January 9, 2014 Revised: March 24, 2014 Published: March 26, 2014 8174

dx.doi.org/10.1021/jp5002572 | J. Phys. Chem. C 2014, 118, 8174−8180

The Journal of Physical Chemistry C

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→ ⎯ ⎯→ lattice parameters ( A1, A 2 ) by the matrix M = [m11,m12;m21,m22] on the basis of the primitive unit cell of graphene with lattice ⎯a and → ⎯a . parameters → 1 2 ⎯ ⎧→ → ⎯ → ⎯ ⎪ A1 = m11 a1 + m12 a 2 ⎨ → ⎯ → ⎯ ⎪ ⎯→ ⎩ A 2 = m21 a1 + m22 a 2

(1)

where the m11, m12, m21, and m22 are integers and the coefficient determinant of the matrix M is nonzero (det|M| ≠ 0). The ⎯a and → ⎯a could be then written as [1,0;0,1] primitive cell with → 1 2 following this definition. The TB approximation is adopted, and the Hamiltonian of the primitive cell is H=

∑ Eiai†ai + ∑ tijai†aj i

⟨i , j⟩

(2)

where Ei is the on-site energy of atom i, tij = −2.7 eV is the transfer integral between atom i and j, and a†i (aj) is the creation (annihilation) operator.23,24 Only the first nearest neighbors are taken into account. The interactions with farther neighbors are ignored for simplicity in the qualitative calculation. In order to confirm our conclusions obtained with the TB method, we have also carried out the first-principles calculations within the framework of density functional theory with a plane wave basis set and the projector augmented-wave (PAW) potential as implemented in the Vienna ab initio simulation package (VASP).25,26 The exchange correlation with the generalized gradient approximation by the Perdew, Burke, and Ernzerhof (PBE-GGA)27 is used. The cutoff energy is 400 eV for the plane wave basis set. A repeated-slab geometry with a single carbon layer separated by 10 Å vacuum is adopted to model the graphene sheet. The Monkhorst−Pack 5 × 5 × 1 kmesh is carefully chosen to ensure the accuracy for the calculated hexagonally patterned superlattices whose lattice parameters are >14.23 Å. The atoms are fully relaxed until the forces on each atom are less than 10 meV/Å.

3. RESULTS AND DISCUSSION In our TB studies, we have first treated the all-C system where the on-site energy is zero. The energy band calculated from eq 2 is shown in the upper panel of Figure 1a. The energy gap closes at K and K′ points which are twofold degenerate. A wellknown way to open the band gap in graphene is to break the equivalence of its AB sublattices. For a configuration replacing carbon atoms on A and B sites in primitive unit cell of graphene with boron and nitrogen atoms, namely, the boron nitride sheet, we assume that the transfer integral between the different elements is the same, and the on-site energies of the boron and nitrogen atoms are −1.96 and 1.96 eV, respectively.23,24 The calculated results with TB approximation of the primitive unit cell [1,0;0,1] are shown in the upper panel of Figure 1a by the dashed curve. Due to the breaking of the inversion symmetry, the twofold degeneracy is now removed, resulting in the bandgap opening. Another way to engineer the energy bands of graphene sheet is to introduce the periodically arranged defects. The defects have many different types, such as antidots, Stone−Wales defects, adsorption of functional groups, and so on. We find that the periodical arrangement of the defects in combination with the energy band-folding picture plays a crucial role in determining the properties of the energy bands. The energy

Figure 1. Energy band structures and the band-folding picture illustrated by corresponding h-BZ zones for the primitive unit cell [1,0;0,1] (a) and the pseudo-superlattices [2,0;0,2] (b), [3,0;0,3] (c). The subscripts of k-points are used to aid the band-folding picture illustration. The dashed curves in (a) represent the band structures of BN sheet. The band structures of the superlattices [6,0;0,6], [7,0;0,7], and [8,0;0,8] are shown in (d), (e), and (f), respectively. The upper and the lower panels for each cell are for the pseudo pristine superlattice and the defect patterned superlattice. The structures of the pristine graphene and the D6h C12 flake defect in the red (shaded) circle region are inserted for geometrical illustrations.

bands of [2,0;0,2] and [3,0;0,3] pristine systems are shown in the upper panels of Figure 1b and 1c, respectively. The energy gap of [2,0;0,2] closes at the K2 (K2′) point. However, in the [3,0;0,3] system, it closes at the fourfold degenerate Γ point. The energy band-folding picture can be used to understand these behaviors. For example, as shown in the lower panel of Figure 1b, the first hexagonal Brillouin zone (h-BZ) of [2,0;0,2] pseudo-superlattice is translated within that of primitive graphene, and it is found that the K1′ point in primitive graphene corresponds to the K2 point. Meanwhile, the K1 point is equivalent to the K2′ point. Therefore, K2 and K2′ are twofold 8175



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lower panel of Figure 1d. As a comparison, the twofold degeneracy at K7 (K7′) and K8 (K8′) points of the antidot patterned superlattices [7,0;0,7] and [8,0;0,8], as shown in the lower panels of Figure 1e and 1f, still keeps the closed band gap. To shed light on the influence of antidot, we apply the degenerate perturbation theory to explain the band-gap opening at the Γ point. The antidot is considered as the source of the perturbation. Imagine that the atoms representing the antidot are removed from the primitive graphene gradually, and the corresponding perturbation transfer integral (ξt) between them approaches zero (ξ → 0) smoothly. In the primitive graphene, the Hamiltonian at K (K′) point is HK = νFσ · ℏ k (HK′ = −νFσ · ℏ k.), where νF is the group velocity at Fermi surface and σ is the Pauli matrix.23,24 Because of the fourfold degeneracy of K6 and K6′ at the Γ point, the Hamiltonian of [6,0;0,6] superlattice around Γ point (k⃗ → 0) reads

degenerate. In the lower panel of Figure 1c for [3,0;0,3] pseudo-superlattice, the equivalence of the twofold degenerate Dirac cone points (the K1 and K1′ points referred to the primitive unit cell as shown in Figure 1a) at the Γ point makes it fourfold degenerate. Actually, the fourfold degeneracy is common for all the [3n,0;0,3m] (n and m are nonzero integers) pseudo-superlattices. The energy bands of [6,0;0,6], [7,0;0,7], and [8,0;0,8] pseudo pristine graphene superlattices (PPGSes) sheets are shown in the upper panels of Figure 1d−f, which confirm the above-discussed energy band-folding picture. In the lower panels of Figure 1d−f, we present the antidot regularly patterned superlattices corresponding to the PPGSes shown in the upper panels, respectively. The antidot configuration is schematically illustrated by the inset in the lower panel of Figure 1d. The transfer integrals of the C atoms in the shaded circle are changed from −2.7 to 0 eV in our TB studies to mimic antidot. While the antidot patterned superlattices still keep graphene’s inversion symmetry, a band gap is now opened by removing the fourfold degeneracy by introducing in the antidot for the [6,0;0,6] superlattice, which is shown in the

⎡ ⎤ 1 7 0 0 ℏ(kx − ik y) − ⎢ ⎥ 12 9 ⎢ ⎥ ⎢7 ⎥ 1 0 0 − ⎢ ℏ(kx + ik y) ⎥ 12 9 ⎥ HΓ6(k )⃗ = vF⎢ ⎢ ⎥ 7 1 0 0 − − ℏ(kx + ik y)⎥ ⎢ 9 12 ⎢ ⎥ ⎢ ⎥ 7 1 0 0 − − ℏ(kx − ik y) ⎢⎣ ⎥⎦ 9 12

Similarly, the Hamiltonian of [7,0;0,7] ([8,0;0,8]) at K7 (K8) becomes HK7 = (41/49)νFσ·ℏ k (HK8 = (7/8)νFσ · ℏ k). Therefore, if the perturbation is applied, the band gap, whose value is (|t|/6)(1 − ξ), is opened for the [6,0;0,6] pseudosuperlattice. We also find that the effective velocities of [7,0;0,7] and [8,0;0,8] at the K point are reduced. Inspired by the BN substitution shown in Figure 1a, to open the band gaps in these superlattices, we replaced the C atoms in the shaded circle in the lower panel of Figure 1d (used to illustrate the configuration of the antidot) by D6h (BN)6 flake and performed TB calculations. Due to the breaking of the inversion symmetry, the band-gap opening at K point is realized in our studies. These properties can be generalized to the arbitrary superlattices. On the basis of detailed analysis, we conclude that if the conditions 2m21 + m22 2m11 + m12 m + 2m22 , ∈  or 21 3 3 3 m11 + 2m12 , ∈ 3

(3)

The results discussed above are obtained with the TB approximation in which only the nearest neighbors are taken into consideration. To confirm our conclusions, we have also carried out the first-principles calculations. The calculated results of hexagonal superlattices are shown in Figure 2. The insets in Figure 2 schematically illustrate the studied defects. For the [6,0;0,6], [7,0;0,7], and [8,0;0,8] PPGSes, the studied energy band structures which are shown in the corresponding panels in Figure 2a obviously support the energy band-folding picture. As mentioned above, in the [N,0;0,M] (N and M are nonzero integers) PPGSes (without defects), if both N and M are integer multiple of 3, the Dirac cone would be folded from the K (or K′) point of the h-BZ of primitive cell to the Γ point of the pseudo-superlattices. By elongating the C−C bonds of the D6h C12 flake in the shaded circle as shown in the inset of Figure 2b, which could weaken the transfer integral between nearest C atoms to some sense while keeping the inversion symmetry, we studied the band structures under this perturbation and showed the results in corresponding panels in Figure 2b. The band gap at the Dirac cone opens in the cases of [6,0;0,6] and [6,0;0,9] superlattices. For the other cases, the band structures remain closed at K points of the corresponding superlattices. In order to compare with the results obtained with our TB calculations with zero transfer integral within the C12 flake in the red circle region (see the lower panel of Figure 1d), we have also examined the C12 antidot patterned superlattices (see the structural illustration in the inset of Figure 2c). The corresponding results are shown in Figure 2c. Again, the degenerate perturbation would remove the fourfold degeneracy at Γ points of [6,0;0,6] and [6,0;0,9] superlattices,

(4)

are fulfilled, where  stands for an integer number, the Γ point in the new BZ is equivalent to the Dirac points and the band gap closes at the Γ point with fourfold degeneracy. Otherwise, the closing of the band gap at the Dirac point remains twofold degenerate. It is readily found that [3n,0;0,3m] (n and m are nonzero integers) superlattice is always satisfied under these conditions. These conditions can be applied as the criteria to categorize the superlattices into two groups. 8176



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could be attributed to the local strain induced by the structural distortion due to the C12 antidot. In our TB studies, the structure does not change in order to simplify the discussion, though the transfer integrals in the shaded circle are set to zero. However, in our first-principles studies, the geometrical structures and also the hexagonal lattices are fully optimized after introducing the C12 antidot. The distance between neighboring C12 antidots in [6,0;0,6] superlattice is around 7.8 Å. The stress strain in this case may account for the location of the bottom of conduction band at the M point instead of the Γ point. In order to shed light on the effects of local strain, we have also calculated the energy band structures of the [9,0;0,9] and [12,0;0,12] superlattices, which have the shortest distances between neighboring C12 antidots of 15.1 and 22.5 Å, respectively. The bottoms of the conduction bands are found to move to the Γ point, resulting in the direct band-gap nature of these superlattices. As illustrated by the inset of Figure 2d, we patched the antidot hole with (BN)6 flake which would not only introduce degenerate perturbation but also the AB symmetrical quantum confinement breaking. The energy band structures are then calculated after structural optimization, which are shown in Figure 2d. Now, the band gap opens for all the superlattices. Also, we have noted that our conclusions on the band-gap opening may be applied to the other B−C−Nbased sheet nanostructures. The stabilities and the electronic structures of B−C−N sheet nanostructures were previously studied in detail,29−33 which also shows the relationship between the band-gap opening and the inversion symmetry. Especially, the sheet superlattices of alternatively arranged graphene strips and boron nitride strips were also analyzed with band-folding picture,31 which show similar results to our studies of the graphene superlattices patterned with defect chains. The rectangular superlattice is another type of system, which is also widely studied. The smallest rectangular superlattice is ⎯→ ⎯ → ⎯→ ⎯a a1 + ⎯a 2 and A 2′ = −→ [1,1;-1,1] as defined by the lattices A1′ = → 1 → ⎯ + a 2 in Figure 3a, whose rectangular Brillouin zone (r-BZ) is → ⎯ ⎯→ depicted by reciprocal lattices B1′ and B2′ in Figure 3b. The generalized rectangular superlattice can be [p,p;−q,q] (p and q → ⎯ ⎯a + → ⎯a ) stands for the are nonzero integers), where A1 = p(→ 1 2 ⎯→ ⎯a + → ⎯a ) represents the zigzag armchair edge and A 2 = q(−→ 1 2 → ⎯ edge. Since A1 is always satisfied with eq 4, it is entirely ⎯→ determined by zigzag edge A 2 whether the Γ point of rectangular superlattice would be fourfold degenerate or not. This can only be achieved when q is an integer multiple of 3 to satisfy eq 4. Hereafter, for simplicity, we would like to use the (p,q) instead of the [p,p;−q,q] to define the rectangular ⎯→ ⎯→ supercell p A1′ × q A 2′ . In Figure 3, we used the superlattices (1,1), (1,2), and (1,3) for illustrating the band-folding picture. ⎯→ ⎯ ⎯→ ⎯ The pseudo-superlattice (1,3) with lattices A1* and A 2* could fold the h-BZ’s K and K′ points of the primitive unit cell of graphene into the Γ* point of its r-BZ resulting in the fourfold degeneracy. The calculated energy band structures with firstprinciples method are shown in Figure 3c. The Dirac points are ⎯⎯⎯→ ⎯⎯⎯⎯⎯→ found at the (2/3)Γ′Y′ for (1,1) and the (2/3)Γ″Y″ for (1,2), while it is at the fourfold Γ* point for (1,3). Analogous to the hexagonal superlattices patterned with defects, the C12 antidot acts similarly in affecting the energy band structures of rectangular superlattices. In our first-principles studies, the

Figure 2. Energy band structures of the pseudo pristine superlattices (a) and those of the defect patterned superlattices (b), (c), and (d). The insets schematically illustrate the geometrical configurations of the ideal pristine graphene, the defect formed by elongating bonds of C12 flake in the red (shaded) circle region, the C12 antidot defect, and the (BN)6 substitution patch defect, respectively.

while the band gap still remains closed for the other superlattices due to the preservation of inversion symmetry. It is well-known that the PBE-GGA tends to underestimate the band gap. So, one may wonder whether the band closing in the studied band structures is accurate. In order to clarify this point, we have calculated the energy band structure of the C12 antidot patterned [7,0;0,7] superlattice with the more accurate Heyd− Scuseria−Ernzerhof (HSE) hybrid functional28 which includes a screened short-range Hatree-Fock (HF) exchange term. An empirical parameter α is used to represent the HF exchange fraction, which is 0.25 as in the HSE06 functional for most semiconducting materials. Our calculated energy band structure with the HSE06 functional is almost the same as the one presented in Figure 2c for [7,0;0,7] superlattice, which has the band closing at the K point also. This confirms the validity of PBE-GGA for our studies. As for the energy band structures presented in Figure 2c for [6,0;0,6] superlattices, a discrepancy of the first-principles results as compared to the TB results lies in the band-gap nature. The former shows indirect band-gap character while the latter one owns direct band gap, which 8177



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width of qA′2, which is separated from its nearby images by the defect chains. It needs to be noted that the fabrication of this kind of infinite defect chain should be a great challenge to the experimental studies. Here, we just use this ideal model as a hypothesis as a demonstration for comparison studies. The length of C−C bonds on defect chains equals that on the freestanding qA′2-width armchair nanoribbon. In Figure 4a, we

Figure 3. Schematic structural illustrations of the rectangular superlattices (a) and the corresponding r-BZ zones (b). The primitive unit cell and its corresponding h-BZ zone of graphene are also shown with dotted lines in (a) and (b) for illustration. (c) shows the calculated energy band structures of the pseudo pristine superlattices (1,1), (1,2), and (1,3), supporting the band-folding picture illustrated in (a) and (b). The notions (1,1), (1,2), and (1,3) account for the ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯→ ⎯→ ⎯→ ⎯ supercells A1′ × A 2′ , A1″ × A 2″, and A1* × A 2* shown in (a), which are defined in detail in context.

(8,9) superlattice satisfying eq 4 opens band gap at the Γ point under degenerate perturbation by regularly arranged C12 antidots, while the band gaps of (8,8) and (8,10) remain ⎯→ ⎯ closed at the twofold degenerate points at (2/3)ΓY (see Figure S1 of the Supporting Information). Again, if the (BN)6 flake is used to patch the mesh hole, the breaking of AB sublattice symmetry combined with the degenerate perturbation opens the band gaps of all the regularly patterned (p,q) rectangular superlattices (see Figure S1 in the Supporting Information). It is noted that there are flat bands appearing in these band structures, which are actually also found in our studies of the hexagonal superlattices (it is at 1.95 eV for the [6,0;0,6] superlattice presented in Figure 2c). The charge distribution analysis of the flat bands based on our first-principles calculations indicates that they could be attributed to the dispersive states localized along the edge of the antidot hole. The similar results have also been found in the previous studies.34,35 Now, we consider the special rectangular lattice (1,q). By contracting the 1,2 C−C bond (see Figure 3a) of 3.5% to enhance the transfer integral to form an infinite defect chain ⎯→ along the armchair edge in the A1′ direction, we could get a ribbon structure being analogous to armchair nanoribbon in the

Figure 4. (a) Energy band structures of the pseudo pristine superlattices (1,6), (1,7), and (1,8); (b) band structures of the armchair ribbon separated with its adjacent images by defect chain on the lattice edge of the (1,6), (1,7), and (1,8) superlattices; (c) data of the free-standing armchair nanoribbons of 6A′2, 7A′2, and 8A′2 in width, respectively, which correspond to the ribbons studied in (b).

studied the energy band structures of pseudo-superlattices (1,6), (1,7), and (1,8) of pristine graphene. A 7 × 5 × 1 Monkhorst−Pack k-mesh was adopted in the electronic properties integration in our first-principles calculations. The fourfold degenerate Γ point appears for (1,6). After introducing the defect chain, we also studied the corresponding energy band structures and showed them in Figure 4b. A small band gap opens at the Γ point of (1,6) due to the perturbation in removing the fourfold degeneracy. However, the Dirac cone remains twofold degenerate for both (1,7) and (1,8), which still 8178



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some cases, the K and K′ Dirac points of primitive graphene unit cell could be folded into the Γ point of pseudo pristine superlattice, resulting in the fourfold degenerate Dirac cone at the Γ point. A band gap can be further opened by removing the fourfold degeneracy by introducing structural defects such as antidot, even for the cases that the AB symmetrical quantum confinement preserves. This can be easily utilized in experiment in engineering graphene, for example, by making the graphene nanomesh through lithography technique. In the other cases, the Dirac cone is not folded to the Γ point, remaining twofold degenerate. The band gap could be then opened by breaking the inversion symmetrybreaking the AB symmetry sublattice equivalence.

keep the semimetal nature for the preservation of AB sublattice equivalence. As to the counterparts of the ribbons in defect chain patterned (1,6), (1,7), and (1,8) superlattices, the 6A′2-, 7A′2-, and 8A′2-width free-standing armchair nanoribbons with hydrogen termination on ribbon edges are also studied, which are shown in Figure 4c. Contrary to the ribbons studied in Figure 4b, all the free-standing ribbons are found to have band gaps which decrease from the 6A′2-width ribbon to the 8A′2width one. Following Louie’s method in defining the ribbon width, the studied 6A′2-, 7A′2-, and 8A′2-width ribbons could be sorted into the categories of 3P + 1, 3P, and 3P + 2 (P is an integer) armchair nanoribbons.19 Our calculated results suggest the 3P + 1 and 3P + 2 ribbons to have the biggest and nearly zero gaps, respectively, which agree very well with the conclusions of the armchair nanoribbons.13 Though the ribbons separated by defect chains as studied in Figure 4b have both the same ribbon width and the same C−C bond length on ribbon edges as compared to their free-standing counterparts studied in Figure 4c, the conducting properties are different. This shows that the enhanced transfer integral itself on ribbon edges could not completely account for the band-gap opening of freestanding nanoribbons. The termination of π-electron wave function at the edge of free-standing ribbons plays a crucial role, which in fact act as an infinite high potential wall to block the electron wave from propagating. The effects of the terminated π-electron wave function also play an important role in the freestanding zigzag nanoribbons. We have also examined the ribbon structure separated by the defect chain in (p,1)-type ⎯→ superlattice. The 1,3 C−C bond along the A 2′ direction as marked in Figure 3a is contracted by 3.5% to introduce the infinite defect chain, separating the pA′1-width zigzag ribbon structure with its periodic images. The defect chain patterned (6,1), (7,1), and (8,1) superlattices are studied by using the first-principles method with a 3 × 9 × 1 Monkhorst−Pack kmesh in integrating electronic properties, which are presented in Figure S2 of the Supporting Information. The band crossing ⎯→ ⎯ is found at the at (2/3)ΓY for all the studied (p,1) pseudo pristine superlattices (without defect chain on lattice edge), which could be easily understood by band-folding picture as illustrated in Figure 3. After defect chain is introduced on lattice edge, the defect patterned superlattices still remain semimetallic in character owing to the inversion symmetry preservation. As to the free-standing 6A′1-, 7A′1-, and 8A′1-width zigzag nanoribbons with hydrogen termination on ribbon edges, the calculated ground states are the ones with α-spin and β-spin states on ribbon edges being antiferromagically coupled. The different electron localization of α-spin and β-spin states on both side edges of ribbon results in the band-gap opening, which actually were studied previously in detail by Son et al.19,36 and Jungthawan et al.31

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ASSOCIATED CONTENT

* Supporting Information S

Energy band structures of the supercells (8,8), (8,9), and (8,10) studied with the first-principles method for the corresponding PPGSes, the D6h C12 antidot patterned superlattices, and the D6h (BN)6 substitution flake patterned ones; the calculated band structures (Figure S1) show the band-gap opening mechanism by the degenerate perturbation and the inversion symmetry breaking; the band structures for the corresponding special supercells (6,1), (7,1), and (8,1) for both the pseudo pristine superlattices and the defect chain patterned superlattices, and also for their counterparts free-standing zigzag nanoribbons (Figure S2). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the SR16000 supercomputing resources from the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, and the computing resources from the University of Jinan. This work was jointly supported by the funds from Shandong Province (Grants TSHW20101004 and ZR2010AM027) and the National Natural Science Foundation of China (Grants 11074100 and 11374128).

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REFERENCES

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4. CONCLUSIONS In summary, we have shown the degenerate perturbation in opening band gap of periodically patterned graphene superlattice besides the often adopted method to break the quantum confinement of AB sublattice equivalence. Using the method such as periodically arranged antidots, modification by chemical or physical adsorption, and substitution doping, etc., one can engineer the electronic properties of graphene superlattice. According to the band-folding picture, we proposed a method to sort the patterned superlattices into two categories based on the analysis of the components of basis superlattice vectors. In 8179



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