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Aug 10, 2012 - The electronic structures of BN-embedded graphene (BNG) were ..... When WA varies (Figure 6c), the gap falls into three hierarchies as ...
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BN-Embedded Graphene with a Ubiquitous Gap Opening Ruiqi Zhao,†,‡,§ Jinying Wang,†,‡ Mingmei Yang,†,‡ Zhongfan Liu,*,†,‡ and Zhirong Liu*,†,‡ †

College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China Center for Nanochemistry, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, and Beijing National Laboratory for Molecular Sciences (BNLMS), Peking University, Beijing 100871, China § College of Physics and Chemistry, Henan Polytechnic University, Henan 454003, China ‡

S Supporting Information *

ABSTRACT: The electronic structures of BN-embedded graphene (BNG) were theoretically studied. A nonzero gap was found to exist in BNG regardless of the edge structures (zigzag/armchair) and the symmetries of the superlattice and BN quantum dots (QDs). The size of the gap is mainly determined by the width of the carbon wall between neighboring BN QDs. It is insensitive to the size of BN QDs, and thus obeys a universal scaling law. This significant and stable energy gap renders BNG as a promising way to control the electronic properties of graphene. The comparison with graphene antidot lattices and nanoribbons was also provided.

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INTRODUCTION Graphene, a layer of carbon that is one-atom thick, has triggered a great deal of study in both fundamental and applied science since its discovery in 2004.1 Pristine graphene is a zerogap semiconductor, so it is crucial to introduce a nonzero gap in graphene to facilitate its applications in building electronic nanodevices.2,3 Considerable attempts have been made to open a gap in graphene, for example, by cutting graphene into nanoribbons,3−5 by constructing periodic holes on graphene,6−9 by chemical functionalization,10−14 and by utilizing deformations and/or electrostatic gating.15−18 However, there is a drawback that the opened gap may be sensitive to parameters. For example, when the quantum confinement effect was applied in constructing graphene nanoribbons (GNRs), the resulting system with armchair edges is metallic if the ribbon width NA satisfies NA = 3p + 2 (where p is an integer), whereas it is semiconducting if NA = 3p or 3p + 1.3,4 For graphene antidot lattices (GALs), the gap depends in a sensitive way on the wall width and the symmetry of the antidot lattice;7,8 for example, a nonzero gap is opened in hexagonal GALs with an even wall width, whereas the gap remains closed in those with an odd one.7 A similar pattern restriction was observed in chemical functionalization.13 Therefore, the development of novel means to achieve a stable gap opening will be extremely valuable for device applications. Boron nitride (BN) has numerous structures analogous to carbon. Single-layer hexagonal BN (h-BN), the bond length of which is only 1.7% larger than the C−C in graphene,19 has a wide band gap of 5.9 eV.20 Therefore, h-BN is the ideal candidate to hybridize with graphene. Recently, a hybrid BN− graphene material consisting of randomly distributed domains of BN and C phases was realized experimentally.21 Magnetoelectric transport measurements performed on the hybrid system show an anomalous insulator-to-metal transition.22 © 2012 American Chemical Society

Some calculations have been conducted that suggest BN embedded graphene shows great potential for tailoring the band gap of graphene.23−27 However, for such a graphene embedded with periodic BN quantum dots (QDs), the intrinsic factors governing its electronic properties still remain unclear. Is it really determined by the edge structures, geometric shapes, and dimensions of BN QDs (the concentration of BN)? It would be important to explore the key factors controlling the electronic properties of BN-embedded graphene (denoted as BNG hereafter). In this work, systematic studies of the electronic properties of BNG were performed by varying edge structures, symmetries, and dimensions of the superlattice and BN QDs. Interestingly, our calculations suggest that the nonzero gaps existing in BNG are not sensitive to the edge structures, symmetries, and concentration of BN. However, it is mainly determined by the least number of carbon chains between BN QDs, opening an opportunity to engineer electronic properties of graphene controllably.

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COMPUTATIONAL METHODS

We calculated the electronic structures of BNG using density functional theory (DFT) and a single-orbital tight-binding (TB) approach. DFT calculations were performed using the Vienna Ab-initio Simulation Package (VASP)28 with the generalized gradient approximation29 and the Perdew− Burke−Ernzerhof exchange-correlation functional.30 An energy cutoff of 520 eV was used, and the k-points sampling was done by the Monkhorst−Pack scheme31 with different meshes depending on the Brillouin zone dimensions. For instance, we used 21 × 21 × 1 for the lattice shown in Figure 1 with {R, Received: July 5, 2012 Revised: August 3, 2012 Published: August 10, 2012 21098

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Figure 1. Schematic diagram of a BN-embedded graphene with zigzag-edged hexagonal BN QDs and a hexagonal superlattice (ZHH-BNG). The structure of this ZHH-BNG is {R, W} = {3, 4} (this notation will be described below).

W} = {1, 1}. For other superlattices, the meshes that provide a similar sampling density in k-space were used. The vacuum separation was set as 15 Å to avoid any interactions between adjacent layers. The convergence criterion for total energy was 10−5 eV, and the maximum residual forces were smaller than 0.01 eV/Å. The obtained gap values were used to fit the parameters of the TB model. The resulting fitting parameters are listed in Table 1. The data obtained from the TB model

agree well with those from DFT calculations (Figure 2). The TB model was then used to systematically study the properties

of BNG and GALs containing up to thousands of atoms to reveal the scaling behavior of the opening gap. Figures presented in this paper were prepared using TB data unless otherwise specified. It is noted that our TB model only contains the contribution of atomic π states. Similar schemes were adopted in previous works in describing the electronic structures of BN nanoribbons (BNNRs)32 and GNRs confined by BNNRs.33 Strictly speaking, the σ and π states in h-BN are not well separated as those in graphene, and thus the contribution of σ states may not be necessarily negligible. However, as will be shown below, the gap of BNG systems was attributed mainly to the graphene part, while the BN part acts largely as a neutralized matrix. A numerical comparison indicated that the TB model reproduced the band structure of h-BN near the K point very well (Figure S1b in the Supporting Information). Therefore, we expect that the conclusion in this paper is not affected by the simple TB scheme.

Figure 2. Comparison of band gaps obtained from DFT and TB. The list of the calculated systems is provided in Table S1 (Supporting Information).

RESULTS AND DISCUSSION We first consider BNG with zigzag-edged hexagonal BN QDs and a hexagonal superlattice (ZHH-BNG), as shown in Figure 1. These hexagonal-shaped patterns are energetically more stable than other patterns.26 Following a previous convention in GALs,7 we designated BNG by the radius of the BN QDs, R (measured in a unit as the number of hexagonal carbon chains replaced with BN atoms, or equivalently, the number of the outermost B/N atoms on each side of the BN QDs), and the wall width, W (measured as the number of carbon chains between neighboring BN QDs), that is, as {R, W}. The electronic band structures of ZHH-BNG with a fixed R = 1 and different W are shown in Figure 3. The appearing band gaps are all direct because the bands near the Fermi level are mainly attributed to the graphene part. Unlike the case of GALs where

Table 1. Tight-Binding Model Parametersa εC

εB

εN

tCC

tCB

tCN

tBN

0.00

2.76

−1.64

2.65

2.25

1.70

2.40

ε denotes the on-site energy, and t is the nearest-neighboring hopping parameter (both in eV).

a

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where the gap values for various R overlap with each other and can be well described by a universal scaling function of Eg = 3.8 exp(−0.62W0.5) (Figure 4b). Therefore, W is the crucial factor that determines the gap size, while the influence of R and BN content is less important. The important role of W was also pointed out by Shinde and Kumar based on limited data;25 however, it is only true in the case with large BN QDs. In the case with small BN QDs, the influence of BN QDs and GALs can be clearly observed (Figure 4a). It is also noted that the gap of the GALs decreases continuously with increasing R and becomes tiny under large R (Figure 4b), so the gap spans a wide range with persistent fluctuations with W. In contrast, the gap of ZHH-BNG is larger and has a more compact distribution. Therefore, ZHH-BNG is more effective in modifying the gap size of graphene than GALs. One may question the reliability of the simple TB model used to reveal the electronic properties of BNG. To explore the possible influence of the TB model, we also conducted extra calculations by adopting TB parameters taken from ref 33, which contain next-nearest neighboring interactions. The results (Figure S3 in the Supporting Information) confirm that our conclusions are not sensitive to the TB parameters. We also noticed that a similar TB model was used to study electronic properties of BNNRs.32 The novel properties of BNG revealed above are not restricted to hexagonal systems. The results for four systems with various symmetries of superlattices and BN QDs are compared with the corresponding GALs in Figure 5. To reveal possible scaling behaviors for large values of R, only data for BNG with more than 100 atoms in one BN QD (and the corresponding GALs) are shown. The number of atoms in a

Figure 3. Energy band structures for ZHH-BNG with a system notation {R, W} of (a) {1, 1}, (b) {1, 2}, (c) {1, 3}, and (d) {1, 4} obtained from TB calculations. The inset in (a) shows the first Brillouin zone with three high symmetrical points labeled. The gap value is also labeled in each panel. A comparison with DFT calculations is provided in Figure S2 (Supporting Information).

the gap is alternatively closed and opened as W increases, the gap of ZHH-BNG is always open. The gap with W = 1 is 0.97 eV, and it increases to 1.16 eV for W = 2. It then drops dramatically to 0.40 eV for W = 3 and increases slightly to 0.54 eV when W = 4. Therefore, the gap of ZHH-BNG with R = 1 does not change monotonically with respect to W, but varies in an irregular pattern. The irregular gap variation in ZHH-BNG with R = 1 is closely related to the properties of the corresponding GALs. BNG can be regarded as the product of patching the holes in GALs with BN domains, so their band structures may be influenced by the original GALs. The connection between BNG and GALs is clearly demonstrated when their gaps are plotted in the same graph as a function of W (Figure 4). For

Figure 4. Band gaps of ZHH-BNG compared with those of the corresponding GALs as a function of the wall width W with (a) R = 1−4 and (b) R = 5−24. The solid blue line in panel (b) is the fitting exponential curve. The other solid lines are guides. Figure 5. Band gaps of various BNG (blue diamonds) and GALs (red circles) as a function of W: (a) Zigzag-edged hexagonal superlattices and rhombohedral BN QDs (or holes) (denoted as ZHR-), (b) zigzagedged rhombohedral superlattices and rhombohedral BN QDs (or holes) (denoted as ZRR-), (c) zigzag-edged rhombohedral superlattices and hexagonal BN QDs (or holes) (denoted as ZRH-), (d) armchair-edged hexagonal superlattices and hexagonal BN QDs (or holes) (denoted as AHH-). Schematic diagrams of BNG patterns are shown as insets. The solid blue lines in (a, b) are fittings of BNG data points.

small R, the gap of ZHH-BNG wiggles with W, which is obviously in concert with the alternate gap opening/closing behavior of ZHH-GALs (Figure 4a). This effect weakens as R increases. When R = 4, this wiggle becomes unobservable and the gap decreases smoothly with increasing W (Figure 4a). The gap of ZHH-BNG at any fixed W increases with increasing R when R is small (Figure 4a) (i.e., with increasing BN concentration). However, this effect disappears when R ≥ 5, 21100

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supercell is up to 6000 in our calculations. For ZHR- and ZRRGALs (Figure 5a,b, respectively), the gaps are close to zero in most cases (ZHR-GALs with W ≤ R are composed of isolated GNRs, but not interconnected 2D patterns, so their data were not included in Figure 5a). In contrast, both ZHR- and ZRRBNG possess nonzero gaps, which, similar to that of ZHHBNG, are independent of R and obey the exponential scaling law with respect to W (3.8 exp(−0.57W0.5) for ZHR-BNG and 4.0 exp(−0.68W0.5) for ZRR-BNG). For ZRH-GALs (Figure 5c), sometimes the gaps are zero and sometimes they are not. A detailed inspection revealed that the gap of ZRH-GALs is opened only when L ≡ 2R + W = 3p (where p is an integer, and L is the distance between the centers of neighboring holes), and flat bands with E = 0 appear when R > W (data not shown here). Instead, there are ubiquitous nonzero gaps in ZRHBNG, which are always larger than those in ZRH-GALs (Figure 5c). The gap of ZRH-BNG shows a large dispersion with respect to R at small W. This dispersion may originate from the fact that the carbon regions are not in the form of interconnected ribbons as in other BNG systems we have discussed and thus are not well described solely by W (e.g., the carbon-region shape of BNG with R = 2 would be very different from that of R = 10 for the same W). For the armchair-edged cases (Figure 5d), AHH-GALs can be classified into three groups, where 1/3 of the systems have a nonzero gap and 2/3 have a zero gap. This is opposite to zigzag carbon nanotubes34,35 (in which the axis is along the armchair direction) and armchair-edged GNRs3,4 where 2/3 of the systems have a nonzero gap. AHH-BNG can also be classified into three groups according to their gap behavior despite the fact that their gaps are all nonzero. The gap of AHH-BNG fluctuates with W even for large R, although the amplitude is much smaller than that of AHH-GALs. This suggests that armchair edges have a different effect as zigzag ones. Similar fluctuations of the gap were also observed by Seol and Guo in BN-confined armchair GNRs.33 The superlattice designed above possesses only one type of edge, that is, a zigzag or armchair one. What happens if both types of edges exist in the same superlattice? We have studied rectangle BNG as such an example (Figure 6a). Rectangle BNG can be specified by two wall widths (WZ and WA, which is the wall width with zigzag and armchair edges, respectively) and two widths of BN QDs (RZ and RA). To simplify the analysis, RZ is set equal to RA and the gap is calculated as a function of WZ (Figure 6b), WA (Figure 6c), and WZA (Figure 6d). When the wall width WZ varies (Figure 6b), the gap varies in a smooth manner, which is similar to those observed in the ZHH-, ZHR-, and ZRR-BNG systems. When WA varies (Figure 6c), the gap falls into three hierarchies as those in AHH-BNG. When both WZ and WA vary (Figure 6d), the results can be regarded as influenced by the combined effect of varying WZ and WA. Overall, the results presented in Figures 4−6 show that there are ubiquitous nonzero gaps in BNG, and the gaps are larger than (or at least comparable to) the optimized gaps in GALs. More importantly, the significant gap dependence on the carbon wall width indicates a controllable opportunity to engineer the electronic properties of graphene. It is also noted that DFT calculations with GGA or LDA usually underestimate the gap. For example, by using the more reliable GW approximation, the gap of graphane was determined to be 5.4 eV in comparison with the GGA gap value of 3.5 eV;36 that is, there is an underestimation of about 35% in the DFT results. Therefore, we expect that the true gap of BNG would be even

Figure 6. (a) Schematic diagram of rectangle BNG possessing both zigzag and armchair edges. (b−d) The band gaps of rectangle BNG (blue diamonds) and the counterpart GALs (red circles) as a function of WZ (b), WA (c), and WZA (WZA = WZ = WA) (d). To simplify calculations, the number of BN chains (or the removed carbon chains) along the zigzag is set equal to that along the armchair direction; that is, R ≡ RZ ≡ RA = 8, 9, 12, 13, 16, 17, 20.

larger than what we predicted while the scaling behaviors are less affected with only the pre-exponential factor getting larger. At present, the fabrication of GALs with controllable W in graphene has been carried out in experiments.37,38 By adopting techniques, such as the secondary growth method,39 to merge GALs with BN patches, the fabrication of BNG is promising in practice. To better understand the origin of the ubiquitous gap of BNG and the difference between BNG and GALs, we examined the wave functions in Figure 7. For ZHH-GALs, the electrons mainly populate the edges of the hole and the electron density decays exponentially when penetrating into the inner carbon region (indicated by the solid envelope curves in the lower part of Figure 7a). This is similar to the edge state of zigzag GNRs.40 In fact, ZHH-GALs (as well as ZHR- and ZRR-GALs) with large R can be regarded as interconnected zigzag GNRs. This explains why ZHH-GALs and related systems have a very small gap when R is large. For ZHH-BNG (Figure 7b), the electrons spread over the carbon region. At the same time, the electrons have a small distribution in the BN regions, which changes the sharp boundary condition for the carbon regions and thus makes the decay in the carbon regions much slower. This is a kind of buffer effect in which the influence of the sharp boundary condition is neutralized to produce stable gaps. For AHH-GALs, the distribution of electron density is more uniform with no decay behavior (Figure 7c). A detailed inspection indicates that the electron distribution fluctuates with a period of three carbon rows when penetrating from the edges into the inner regions, which is similar to the case of armchair GNRs. When the BN atoms are introduced (Figure 7d), the symmetry between two sublattices is broken and the carbon atoms within a row adopt different values depending on which sublattice they reside in. Such a polarization effect makes the electron distribution different from those of armchair GNRs. Therefore, the fact that the electrons mainly distribute in the graphene region, together with the neutralization effect of the BN region on the boundary condition of graphene region, results in the ubiquitous gaps in BNG and the corresponding scaling behavior. 21101

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the unexpected role of zigzag edges in gap engineering. Because of the existence of edge states, zigzag edges in graphene usually associate with the metallic band structure, and thus their abundant performance is often hidden.41,42 The above results, however, show that zigzag edges in BNG are compatible with the gap opening and the gap is even more controllable than those in armchair-edged BNG.

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CONCLUSIONS In summary, we have systematically investigated the electronic properties of BNG. Our results show that there are ubiquitous gaps opened in BNG and the gap values are mainly governed by the width of the wall between BN QDs. Our results open an opportunity to engineer the gap of graphene more controllably.

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ASSOCIATED CONTENT

* Supporting Information S

Description of the detailed information of systems and the data obtained from TB and DFT. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: zfl[email protected] (Zhongfan Liu), LiuZhiRong@pku. edu.cn (Zhirong Liu). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.Q.Z. was supported by the Doctoral Foundation of Henan Polytechnic University (Grant No. B2009-90), the China Postdoctoral Science Foundation (Grant No. 2012M510254), and the high-performance grid computing platform of Henan Polytechnic University. Z.F.L. was supported by the National Natural Science Foundation of China and the Ministry of Science and Technology of China. Z.R.L. was supported by the Trans-Century Training Program Foundation for the Talents by the State Education Commission of China.

Figure 7. Wave functions (u(x,y)) of (a) ZHH-GALs, (b) ZHH-BNG, (c) AHH-GALs, and (d) AHH-BNG at the conduction band minimum (Γ point for ZHH and K point for AHH systems). In the upper part of each panel, the atomic configuration is shown, where the colors indicate the variation in u(x,y) (for ZHH systems with Im u(x,y) = 0) or |u(x,y)| (for AHH systems with Im u(x,y) ≠ 0) ranging from negative (blue) to positive (red) values. In the lower part, u(x,y) or |u(x,y)| are shown for the selected regions, as indicated by dashed rectangles in the upper part.

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Within the linearized Hamiltonian approximation, the electron wave function of graphene close to the K point obeys the equation for massless Dirac fermions:2 −iνFσ ·∇Ψ(r ) = E Ψ(r )

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(1)

This continuous equation has been used to analyze the properties of GALs.6 However, it seems inapplicable to the systems we have studied that possess delicate boundary conditions. An obvious weakness of eq 1 is that it does not distinguish zigzag and armchair edges, and it cannot account for the existence of edge states. It is also unable to properly describe the alternate gap opening/closing observed for armchair edges. As a result, it is necessary to use a methodology with atomic descriptions, such as the TB model, to study GALs and BNG. The above results suggest that BNG is a promising candidate to open a gap in graphene. The ubiquitous gap in BNG and its stability toward the size of BN QDs makes BNG superior to other methods, such as GALs and GNRs. Our work also reveals 21102

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