Article pubs.acs.org/JPCC
Why the Band Gap of Graphene Is Tunable on Hexagonal Boron Nitride Erjun Kan,*,† Hao Ren,‡ Fang Wu,§ Zhenyu Li,*,‡ Ruifeng Lu,† Chuanyun Xiao,† Kaiming Deng,*,† and Jinlong Yang‡ †
Department of Applied Physics, Nanjing University of Science and Technology, Nanjing, Jiangsu 210037, People's Republic of China Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China § School of Science, Nanjing Forestry University, Nanjing, Jiangsu 210037, People's Republic of China ‡
ABSTRACT: The electronic properties of a graphene−boron nitride (G/BN) bilayer have been carefully investigated by first-principles calculations. We find that the energy gap of graphene is tunable from 0 to 0.55 eV and sensitive to the stacking order and interlayer distances of the G/BN bilayer. By electronic structure analysis and tight-binding simulations, we conclude that the charge redistribution within graphene and charge transfer between graphene and BN layers determine the energy gap of graphene, through modification of the on-site energy difference of carbon p orbitals at two sublattices. On the basis of the revealed mechanism, we also predict how to engineer the band gap of graphene.
G
raphene, a two-dimensional allotrope of carbon, is a single layer of sp2-bonded carbon atoms densely packed into a honeycomb structure.1 An intense research effort has been devoted to its fundamental physics and possible applications in nanodevices. Accurate experimental measurements have shown that the electronic band structure of graphene has two linear dispersion bands at the Fermi level (the Dirac cone), with gapless excitations. Therefore, carriers in graphene behave like massless Dirac particles.2−4 These unique properties revealed in graphene have inspired more and more research interest, and graphene is expected to be used in the next-generation electronic nanodevices and high-speed switching devices. However, the zero-energy-gap character of graphene has impeded its development in modern electronics. To implant graphene into a practical electronic device, such as a transistor, semiconducting graphene is highly desired. Many methods have been suggested to open an energy gap in graphene, such as molecular adsorption,5 defects,6 hydrogenation,7−9 and graphene−substrate hybrid structures.10,11 Among these methods, graphene−substrate hybrid structures are easier to fabricate. More importantly, if the graphene−substrate interaction is week, many intrinsic properties of graphene can be kept. Therefore, constructing graphene−substrates hybrid structures may be the most promising way to obtain semiconducting graphene. Zhou et al.12 showed that when graphene was epitaxially grown on a SiC substrate, the strong interaction could produce a large energy gap. However, the intrinsic properties of graphene are also destroyed. Inspired by their results, more and more studies have been performed to pursue an energy gap without destroying the intrinsic properties of graphene.13−20 © 2012 American Chemical Society
The energy gap of graphene can be strongly influenced by substrates. For example, the largest energy gap obtained is 44 meV for an oxygen-terminated SiO2 substrate, while it is only 23 meV for a hydrogen-terminated SiO2 substrate.13 Even for the same substrates, such as boron nitride (BN), the energy gap is sensitive to the stacking order of graphene and BN layers.17,18 Although many theoretical investigations have been performed to study the electronic properties of graphene on different substrates, the detailed mechanism of energy-gap opening is still an open question. Previously, it was simply suggested to be a result of symmetry breaking. It is not clear how electrostatics affects the electronic properties of graphene and whether orbital interaction also plays a role. Using a graphene−BN (G/ BN) bilayer as an example, weak physical interlayer interaction suggests no orbital interactions; however, the energy gaps for different stacking orders of the G/BN bilayer are very different. To understand the band gap opening of graphene, two critical problems need to be answered: Why do substrates open an energy gap in graphene? How does the stacking order of hybrid systems affect the electronic properties of graphene?
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RESULTS AND DISCUSSION In this paper, we address the above issues by density functional calculations and a tight-binding model. Since a hexagonal BN layer has a lattice constant similar to that of graphene, and highquality G/BN systems have been fabricated in experiments,19 we take the G/BN bilayers as a model system. Before studying the G/BN bilayer, we explore the electronic properties of freestanding graphene and BN single layers. As plotted in Figure Received: November 7, 2011 Revised: December 21, 2011 Published: January 3, 2012 3142
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Figure 1. (a) Calculated band dispersion of free-standing graphene. Inset pictures are charge densities of the VBM state and CBM state at the K point. (b) Electrostatic potential of graphene. (c) Calculated band dispersion of a free-standing BN layer. Inset pictures are charge densities of the VBM and CBM states at the K point. (d) Electrostatic potential of the BN layer. Pink and azure balls represent boron and nitrogen atoms, respectively.
Figure 2. Geometric structure and band dispersion of the G/BN bilayer with (a) A−B Bernal and (b) A−A stacking. Small pink, small azure, and large blue balls represent boron, nitrogen, and carbon atoms, respectively.
Hamiltonian
1a, the valence band maximum (VBM) state of graphene at the K point is contributed by a π state, while the conduction band minimum (CBM) state is from a π* state. Both these π and π* states come from pz orbitals of carbon. On the other hand, the two sublattices in graphene are equivalent. Therefore, carbon atoms have the same chemical and electrostatic potential environments, as shown in Figure 1c. For a BN single layer, the plotted partial charge density shows that VBM mainly comes from nitrogen atoms, while boron atoms dominate contributions to the CBM state. Different with graphene, the local electrostatic potential distributions around boron and nitrogen atoms are quite different, which is due to their different chemical affinities. To explore the effect of a BN single layer as a substrate, we consider two stacking patterns, which are A−B and A−A stacking as shown in Figure 2. As revealed by previous theoretical and experimental reports, A−B stacking is the ground state of such a hybrid structure, while A−A stacking is the local minimum state with slightly high energy. In the A−B Bernal stacking, carbon atoms of one graphene sublattice are right above N and carbon atoms of the other sublattice are located at the center of hexagons of the BN layer. For the A−A stacking arrangement, carbon atoms are right above B and N atoms. Due to the weak interaction between graphene and the BN single layer, the equilibrium interlayer distance between graphene and the BN layer is about 3.4 Å for both stacking orders. To study the effect of the BN layer on the energy gap of graphene, we calculate the band dispersion of G/BN with different interlayer distances. As shown in Figure 2, the energy gap of graphene is already dozens of millielectronvolts even at equilibrium structures, and it increases as the interlayer distance decreases. Interestingly, the A−A stacking pattern always gives a larger energy gap than the A−B stacking pattern with the same interlayer distance, and this becomes clearer when the interlayer distances become shorter. The largest energy gap obtained in our calculations is 0.55 eV, as shown in Figure 3a. To understand graphene gap opening, we consider the simple nearest neighboring tight-binding (TB) model, with a
H=
∑ εiai†ai + ∑ tijai†aj i
i≠j
where
i = A, B (1)
where εi is the on-site energy of the ith carbon atom, tij is the hopping parameter between two carbon atoms, and a† (a) is the electron creation (annihilation) operator. As reported, tij is taken as 3 eV. Within this model, the energy gap is only determined by the on-site energy difference of carbon atoms at different sublattices, which rigidly moves the energy level of carbon pz orbitals, and the gap opens exactly equal to the difference in on-site energy. The hopping parameter only modifies the band dispersion of graphene and has nothing to do with the opened energy gap. The results obtained by TB simulations are well confirmed by the projected density of states (PDOS) plot of DFT calculations. As shown in Figure 3b,c, the PDOS of two carbon atoms is split at an interlayer distance of 3.4 Å, which reflects the relative movement of the p energy level. As a result, a small energy gap appears in graphene. By reducing the interlayer distances, the split of PDOS becomes more clear, reflecting the larger difference in on-site energy. Since the relative movement of the p energy level is determined by the on-site energy, which can be readily modified by the external electrostatic potential,21 it is interesting to check the electrostatic potential difference (EPD) induced by a BN layer. We define EPDA−A and EPDA−B as the EPDs for different sites of a fictitious honeycomb lattice A−A and A−B stacking with the BN substrate, respectively. As shown in Figure 4a, both EPDA−A and EPDA−B decay very quickly and become zero at distances beyond 4.0 Å. Compared to the open gap, they are very weak, and the difference between them is small. If we put a graphene at a distance of 2.8 Å from the BN substrate, EPDA−A and EPDA−B become quite different (solid lines in Figure 4b). Now, the question is whether the energy gap of graphene is determined by the EPD. To answer this question, we plot the EPD at carbon sites of a G/BN bilayer system with different 3143
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Figure 3. (a) Energy gap of graphene as a function of the interlayer distance in G/BN layers. (b, c) PDOS of graphene in A−B and A−A G/BN bilayers, where CN means the carbon atoms right above nitrogen atoms, CB represents the carbon atoms right above boron atoms, and CH represents the carbon atoms located at the center of hexagons of the BN layer.
Figure 4. (a) EPDA−A (blue dashed line) and EPDA−B (green dashed line) of isolated BN layer. (b) EPDA−A (blue solid line) and EPDA−B (green solid line) of the G/BN bilayer. (c) EPD of carbon atoms at two different sublattices of graphene as a function of the interlayer distance in the G/BN bilayer. (d) Energy gap of graphene as a function of the EPD at its two sublattices in the G/BN bilayer.
Figure 5. (a−c) Differential electron density plots for A−B and A−A stacking G/BN bilayers with interlayer distances of 4.0, 3.4, and 2.8 Å, respectively. The red (blue) color means the electron density is enriched (depleted) with respect to the isolated fragments. Charge densities of states of the valence band (VB) (energy window: from VBM to VBM − 0.05 eV) and conduction band (CB) (energy window: from CBM to CBM + 0.05 eV) states are plotted with interlayer distances of (d) 3.4 and (e) 2.8 Å. The green (yellow) color regions show the VB (CB) states.
interlayer distances in Figure 4c. Interestingly, the EPD is almost the same for the two stacking patterns at a certain interlayer distance. To clarify the relationship between energy gap and EPD, we plot the energy gap as a function of the EPD in Figure 4d. Two notable features are easily observed: First, the energy gap is opened when EPD at different sublattices reaches about 0.03 eV. Second, the energy gaps of the A−A stacking pattern are almost 2 times those of the A−B stacking pattern with similar EPDs, which means the existence of interaction beyond electrostatics. For a G/BN with an interlayer distance of 4.0 Å, there are no interactions other than electrostatics between graphene and BN layers. To explore the effects of the EPD, we plot the differential electron density of the G/BN bilayer, which is defined as the electron density difference between the G/BN bilayer and the separated graphene and BN single layers. The plotted differential electron density clearly shows a redistribution of charge density between two sublattices in the graphene layer (Figure 5a), and the electrons are transferred from CN to CH or CB atoms. Besides, we also found that, in the two different stacking orders, the differential electron densities have
similar shapes, confirming the similar EPDs in the two stacking structures. To reveal the interaction beyond electrostatics, we also plot the differential electron density of the G/BN bilayer with interlayer distances of 2.8 and 3.4 Å in Figure 5b,c. By reducing the interlayer interval of G/BN, the differential electron density is largely increased, reflecting the increasing interlayer electrostatics interactions. Moreover, we also found that electrons are transferred from graphene to the BN layer. To calculate the number of transferred electrons, we adopted a procedure which has been successfully applied in other weak interaction systems.22,23 Because of the weak interaction between graphene and BN, the calculated charge transfer is only 0.011 and 0.005 electron per unit cell for A−A stacking and A−B stacking with an interval of 2.8 Å and 0.002 and 0.001 electron at the equilibrium distances. Although only a small portion of an electron is transferred, we found that the transferred electron comes from CN atoms, which means the split of CN with CH (or 3144
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in the Kresse−Joubert28 implementation is used to describe the electron−ion interaction. The plane-wave basis set cutoff is 450.0 eV. The convergence thresholds for energy and force are 10−5 eV and 0.01 eV/Å, respectively.
CB) atoms is increased. Therefore, the on-site energy difference of CN and CH (or CB) is increased, resulting in a larger energy gap of graphene. Although we have explored the charge transfer between graphene and BN layers, it is interesting to investigate why the charge transfer becomes stronger in the A−A stacking order. In Figure 5d,e, we plotted the charge density of the VBM and CBM states with interlayer distances of 2.8 and 3.4 Å. For both interlayer distances, there is clear evidence of orbital hybridization between BN and the graphene layer, which is surprising considering the commonly accepted weak physical interaction picture. Orbital interaction is stronger for A−A stacking than A−B stacking, leading to relatively stronger charge transfer between graphene and BN layers. As we know, by rotating from A−B stacking to A−A stacking with the same interlayer distances, the distance between carbon and boron atoms is greatly reduced. For example, in the A−B stacking pattern with an interlayer distance of 2.8 Å, the carbon and boron distance is 3.14 Å, while it is only 2.8 Å for A−A stacking. Therefore, VB states of the BN layer overlap much better with the π states of graphene in the A−A stacking. As a result, the orbital interaction becomes stronger in the A−A stacking, as shown in Figure 5d,e. Through differential electron density plots and charge transfer analysis, we found that electrons are transferred from CN to CH (or CB) atoms, which is induced by the EPD. Besides, electrons are also transferred from CN to nitrogen atoms through orbital interactions. Since it directly influences the orbital interactions between graphene and BN layers, the stacking order of th eG/BN bilayer can also determine the energy gap of graphene. On the basis of the explored fundamental physics here, tuning the mutual interaction between graphene and BN or the EPD of carbon atoms through the BN layer becomes a possible and practical way to enlarge the energy gap of graphene. As revealed by our DFT calculations, the EPD of carbon atoms can help the band gap to open. On the other hand, by applying an electric field through the BN layer, the BN substrate will provide a different screening effect and enhance the EPD of carbon atoms. Therefore, the energy gap is greatly enlarged by an external electric field. For example, in A−B stacking without an electric field, the energy gap is roughly 0.03 eV at the equilibrium structure, while it becomes 0.16 eV with an electric field of 1 V/Å, according to our density functional theory calculations. Thus, the revealed mechanism of band gap opening provides the necessary insight for future studies of graphene-based electric devices. In summary, we have performed density functional calculations to study the energy-gap engineering of graphene on a hexagonal BN layer. Our results show that the energy gap of graphene is determined by the on-site energy difference in carbon p orbitals at two different sublattices, which induced orbital interaction and charge transfer between graphene and BN, in addition to electrostatic interaction.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (E.K.);
[email protected] (Z.L.);
[email protected] (K.D.).
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ACKNOWLEDGMENTS The work at the Nanjing University of Science and Technology (NJUST) was supported by NJUST Research funding (Grants 2011ZDJH02, AB41374, and AE88069) and the National Natural Science Foundation of China (Grants 10974096 and 11174150). The work at the University of Science and Technology of China (USTC) was supported by the National Natural Science Foundation of China (Grants 91021004 and 20933006), National Key Basic Research Program under Grant 2011CB921404, USTC-HPC project, and Shanghai Supercomputer Center.
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METHODS Comprehensive electronic structure calculations are performed by means of the density functional theory as implemented in the Vienna Ab Initio Simulation Package (VASP).24,25 The Perdew−Wang functional known as PW9126 is used. Our test calculations using the self-consistent implementation of a nonlocal van der Waals density functional29 give similar electronic structures. The projector augmented wave method27 3145
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