Symmetry Breaking Induced Bandgap in Epitaxial Graphene Layers

NANO LETTERS

Symmetry Breaking Induced Bandgap in Epitaxial Graphene Layers on SiC

2008 Vol. 8, No. 12 4464-4468

Xiangyang Peng* and Rajeev Ahuja Department of Physics, Condensed Matter Theory Group, Uppsala UniVersity, Box 530, SE-751 21 Uppsala, Sweden Received August 7, 2008; Revised Manuscript Received September 30, 2008

ABSTRACT By performing density-functional calculations, we have investigated the electronic bandgap of single epitaxial and multiepitaxial graphene layers on SiC. The calculations show that a defect-free graphene layer above the carbon buffer layer is very flat and no bandgap is found in the Dirac bands. By introducing a finite density of Stone-Wales defects in the graphene layer(s), we find that a bandgap is opened and decreases as the thickness of graphene layers increases, in good agreement with experiments. The band splitting and the charge distribution vary greatly with the number of graphene layers. The bandgap opening is due to the symmetry breaking within the single graphene layer. The narrowing of the bandgap in multiple graphene layers is induced by interlayer interaction.

A flat, single layer of graphite known as graphene has become the focus of studies due to its exceptional properties and fascinating applications. As a novel two-dimensional material ordered like honeycombs, its electron mobility is extremely high, more than 10 times that of silicon wafers.1 It even shows an intriguing quantum Hall effect1 at room temperatures and has the potential to be used as a spintronics2 material. Most of the unique properties of graphene are related to its electronic structure. The carbon atoms in a perfect graphene layer are sp2-bonded, forming three σ bonds with the neighboring atoms in the same layer. Since the σ bonds are very strong, the occupied and unoccupied σ bond related bands are far away from each other. In between are the π and π* bands resulting from the weak pz-like π bonds, which are degenerate only at the corner points (K points) of the hexagonal Brillouin zone. Hence graphene is a zero-gap semimetal. Yet for making electronic devices with graphene, it is crucial to induce a bandgap between π and π* bands to control the transport of electrical charge carriers through the sheet. Generally, there are two approaches to remove the degeneracy. One is by confinement or reducing dimensionality, e.g., the graphene quantum dots (0D)3 and nanoribbons (1D).4-6 The second is to break the equivalence of the A and B sublattices of graphene by interacting with the substrate.7 In the recent experiments, Zhou et al. took the second approach trying to produce a bandgap in graphene layers.8 Epitaxially grown graphene on a SiC substrate is supposed to have strong coupling with the substrate, removing the symmetry of the A and B sublattices and giving rise to an * Corresponding author, [email protected]. 10.1021/nl802409q CCC: $40.75 Published on Web 10/30/2008

 2008 American Chemical Society

energy gap. By angle-resolved photoemission measurements, they found that there is an appreciable bandgap at the Dirac points when graphene is epitaxially grown on the SiC substrate. This gap decreases as the sample thickness increases. Both experiments and calculations show that the graphene layer that is directly bonded to the substrate is devoid of graphene electronic structures and acts as a buffer layer.9,10 The typical graphene electronic structure is recovered only after a subsequent graphene layer (called first layer, consistent with the notations in ref 8) is grown above the buffer layer. However, the calculations9,10 have shown that even at the presence of the substrate, the electronic bands of a single defect-free graphene layer above the buffer layer are gapless and are very similar to the bands of a freestanding graphene layer. The gap opening observed in the experiments8 still remains to be accounted for. As is well-known, graphene layers are not defect-free in reality. Pentagon-heptagon pairs, vacancies, and adatoms are common and stable defects occurring in graphene layers.11-15 In particular, appreciable amounts of defects have been observed in the graphene epilayers on SiC(0001).12,16 The graphene layers on SiC(0001) are grown by thermal desorption at temperatures higher than 1500 K,17,18 which is highly nonequilibrium and can lead to a considerable amount of defects. In addition, defects are often produced during the characterization of the graphene samples by electron beams in the experiments,8,17 such as scanning electron microscopy (SEM) and high-resolution transmission electron microscopy (HRTEM), operating at several hundreds of kiloelectronvolts. It has been shown that using an electron beam of energy 80-140 keV can result in a large yield of defects.11,19 Defects substantially break the symmetry of a

perfect graphene layer and may produce an energy gap. Stone-Wales (SW) defects13 have the lowest formation energy among all the defects commonly appearing in graphene layers,20 indicating that they can exist in appreciable quantity in graphene layer(s). There is a large barrier (4.4 eV) to the motion and annihilation of SW defects20 and therefore they are stable once they are formed. In this paper, we focus our studies on the electronic structure of graphene layer(s) with SW defects occurring in finite density by density functional calculations. Apart from the defects, the thermal fluctuations induce ripples in graphene,21 modulating the morphology and electronic structure of graphene. To see how the height fluctuations might affect the graphene epilayers with SW defects, we displace some atoms from their equilibrium positions and calculate the change of the total energy and bandgap. The calculations are performed by using VASP code22-24 within the generalized gradient approximation25 with spin polarization. The electron-core interaction is described by ultrasoft pseudopotentials. The cutoff for planewave expansion is 286.7 eV. For defect-free graphene layers on the SiC(0001) and SiC(0001j) surface, we use a
3 ×
3 substrate cell to accommodate a 2 × 2 graphene cell. This is an approximation to the realistic situation in which the 6
3 × 6
3 substrate cell is almost commensurate with the 13 × 13 graphene cell. Berger26 found that in experiments the graphene layer on 1 × 1 SiC(0001j) leading to an interface of
3 ×
3 substrate periodicity. In the following, we mainly discuss results of graphene layers on the carbonterminated SiC(0001j) surface. The graphene layers on the Si-terminated SiC(0001) surface are studied in parallel and will be mentioned at the end of this paper. The substrate is simulated by six SiC layers. The bottom layer is saturated by hydrogen atoms and the vacuum space between the slabs is thicker than 12 Å. To address the effect of SW defects in graphene layer, we adopt a 2
3 × 2
3 substrate unit cell. The Brillouin zone of
3 ×
3 (2
3 × 2
3) cell is sampled by a gamma centered 6 × 6 × 1 (3 × 3 × 1) mesh, which includes the important K point. Our calculations find that there is no bandgap for the first defect-free graphene layer, in either AB or AA stacking with respect to the buffer layer, agreeing with the previous calculations.9,10 In the following, we only consider the usual AB stacking. It is supposed that the separation between the first graphene layer and the buffer layer might be as small as 2 Å and the interaction between them is strong, giving rise to bandgap opening.8 By fixing the graphene layer about 2 Å above the buffer layer, we found a very strong repulsion between the two layers. Starting from 2 Å, we increase the separation by 0.05, 0.10, and 0.15 Å and find that the total energy decreases by 0.88, 1.64, and 2.31 eV, respectively, showing that the height of 2 Å is difficult to be accessed by the first graphene layer due to thermal fluctuations. Therefore, the bandgap observed8 is unlikely due to the small separation of the first layer and the buffer layer. Now we address the effect of SW defects on the electronic structure of graphene layers. From now on, unless otherwise stated, we always mean graphene layer as graphene layer Nano Lett., Vol. 8, No. 12, 2008

with SW defects either explicitly or implicitly. The buffer layer, as an interface between the substrate and the graphene layers, is covalently bonded to the substrate and is much buckled (Figure 1a). The buffer layer has a
3 ×
3 substrate periodicity. Two of the three surface carbon atoms in the substrate
3 ×
3 cell are bonded to the buffer layer and go up. The third is left unsaturated and is lower than the other two by 0.45 Å. Accordingly, there are two types of carbon atoms in the buffer layer. The first type goes down due to the bonding to the surface carbon atom. The second type is not bonded to the substrate and is higher than the first type by 0.38 Å. The bonds between the first and second type atoms stretch and those between the second type atoms compress, as shown in Figure 1b. We have also calculated the total energy when the buffer layer is flat and positioned at the height of the first and the second type atoms of the buckled buffer layer, respectively. It is found that the energy increases by 12.14 and 7.64 eV, respectively, showing that the buckling of the buffer layer is not transient and will not disappear just due to thermal fluctuations. The covalent bonding between the buffer layer and the substrate not only leads to buckling but also drastically changes the electronic structure of the buffer layer with respect to that of the free graphene (Figure 2a). The bonding states related bands are submerged in the valence band and the antibonding states related bands are pushed up into the conduction band. The Dirac cones, the typical feature of electronic structure of graphene, are removed from the substrate bandgap. The dangling bond bands with different spin polarities pass through the energy gap of the substrate (Figure 2a). Then a graphene layer with SW defects is put on the buffer layer (Figure 1a). SW defects are topological defects consisting of two five-atom and two seven-atom rings (Figure 1c).13 The buckling of the graphene layer is increased due to the SW defects but it is still smaller than 0.03 Å. The lateral change of the graphene is much more salient. In the SW defect, the C-C bonds are compressed and stretched from 1.38 to 1.65 Å and the C-C-C angle varies from 101.5° to 138.8°, leading the system far away from the hexagonal symmetry (Figure 1c). The distance between the graphene layer and the buffer layer is 3.8 Å, very close to the defectfree situation. As can be seen in Figure 2b, there is an unambiguous bandgap opened between the Dirac bands, in good agreement with the experiment.8 The calculated bandgap is 0.23 eV. The flat bands crossing the upper and the lower Dirac cones are the dangling bond bands of the substrate with opposite spin polarities. To see how the buckling in graphene layer might affect the bandgap, we displaced one C atom in the first graphene layer out of plane by 0.1 Å and found that the energy gap is 0.22 eV, indicating that the buckling in graphene layer will not affect the bandgap much. Since the formation of SW defects is highly nonequilibrium and the barrier for the movement of them is high, the SW defects should distribute randomly within (between) graphene layer(s). We have investigated several different positions of the SW defect within the unit cell and found that the results are insensitive to the position of SW defect. We also calculated the bandgap of an isolated 4465

Figure 1. (a) Graphene layer with SW defects on C-SiC(0001j) and the buffer layer between them. The orange and black spheres stand for substrate Si and C atoms, respectively. The small cyan and blue spheres denote the C atoms in the buffer layer bonded and not bonded to the substrate, respectively. The C atoms in the graphene layer are denoted by blue spheres. (b) The top view of the buffer layer. The dashed diamond shows the periodicity of the buffer layer. (c) The top view of a SW defect in graphene. The bond lengths shown in (b) and (c) are in angstroms.

the first graphene layer within the SiC bandgap (Figure 2b) will be little affected by the buffer layer. The dangling bond states of the unsaturated SiC surface atoms are within the bandgap of SiC, but the surface atoms of the substrate are very far away from the graphene layer above the buffer layer and their interaction is weak.

Figure 2. The band structure of zero (a), one (b), two (c), and three (d) graphene layers with SW defects above the buffer layer and C-SiC(0001j). The Fermi level is indicated by the dashed line.

graphene layer with SW defects. It is 0.25 eV, which is only 0.02 eV larger than that when there is interaction from the buffer layer and substrate. From the band structure, one can see that there are no buffer layer states within the SiC bandgap (Figure 2a), which means that the Dirac bands of 4466

It is interesting to see how the electronic structure changes if the second graphene layer with SW defects is grown above the first one. At the beginning stage of growth, the initial graphene layers are more defect prone.12,16 It is reasonable to suppose that SW defects exist in all of them and distribute randomly between graphene layers. We have checked several relative distributions of SW defects in the first and second graphene layers and found that the bandgap and the geometry of the SW defects are insensitive to different distributions. The SW defects in the first and the second graphene layers have almost identical geometry. In Figure 2c, one can see that there are two sets of Dirac bands of similar dispersion. The bandgap at the K point persists but is reduced to 0.12 eV. The decreasing of the bandgap agrees with the experiments8 and can be ascribed to the interaction between the π (π*) bands of the first and the second graphene layers. Within the tight binding model, these two graphene layers can be viewed as two identical macromolecules with the same electronic states. We use Π1 and Π2 to denote respectively the π states of the first and second graphene layers when Nano Lett., Vol. 8, No. 12, 2008

Figure 3. (a) Schematic diagram of the Dirac band splitting when three graphene layers are grown on the substrate. λ denotes the interaction of the second layer with the first and the third layers. (b) The zoom view of the Dirac bands in Figure 2d near the K point. (c and d) The isocharge density surface plot of bands π2 and π1 at K point, respectively. In these two states, the charge is only distributed in the graphene layers. The substrate and the buffer layer are not shown here.

they are isolated. When they come into interaction, the two states form a bonding state (Π1 + Π2) and an antibonding state (Π1 - Π2), which correspond to the lower band π1 and the upper band π2 in Figure 2c, respectively. The band of a single graphene layer, corresponding to the isolated states Π1 and Π2, is located at the center of bands π1 and π2. Similarly, the π* band of the single layer will split into two bands π1* and π2* with the original band at the center, as indicated in Figure 2c. An interesting conclusion is that the distance between the centers of the two π bands and two π* bands at K point should be equal to the theoretical bandgap (0.23 eV) of the single graphene layer with SW defects. The calculated value of this distance is 0.26 eV, fitting well with the simple tight binding model. Because the π2 band is pushed up and the π1* band is pushed down with respect to the Dirac bands of a single graphene layer, the bandgap is narrowed. After the third layer with SW defects is added, the Dirac band splits into three subbands (Figure 2d). The bandgap is reduced to 0.09 eV, following well with the trend of the experiments.8 The intricate band splitting can still be qualitatively understood within the tight-binding model (Figure 3). The electronic states of the isolated first, second, and third graphene layers are denoted by Π1, Π2, and Π3, respectively, in Figure 3a. Suppose that the first and the third graphene layers already exist before the second layer is inserted between them. They interact and form bonding state (Π1 + Π3) and antibonding state (Π1 - Π3), respectively, as schematically shown in Figure 3a (we first discuss the interaction of the π bands). These two states are energetically Nano Lett., Vol. 8, No. 12, 2008

very close to the π band of a single graphene layer because the interaction between the remote first and third layers is very weak. The charge density of the antibonding state (Π1 - Π3) is zero at the central plane between the first and third layers, and therefore it does not interact with the second layer inserted at this central plane. The π band of the second layer only interacts with (Π1 + Π3) state, whose charge density is not zero at the central plane, and splits into two bands corresponding to bands π1 and π3 in Figure 3b. The band π2 is due to (Π1 - Π3) and is not affected by the second graphene layer. Therefore there are in total three π subbands. The same analysis is applicable to the π* state. The interaction of the three graphene layers results in three π* subbands with π2* band in the middle, which almost coincides with the π* band of a single graphene layer. Our calculations agree well with the physical picture derived above. The distance between band π2 and π2* at K point (Figure 3b) is 0.22 eV, very close to the theoretical bandgap (0.23 eV) of a single graphene layer with SW defects. The charge density of band π2 (derived from the antibonding state (Π1 - Π3)) at K point is zero at the second graphene layer, as depicted in Figure 3c. Band π1 is due to the interaction of all the three graphene layers, and its charge density at K point distributes over all the graphene layers (Figure 2d). The charge distribution of band π3 at K point is similar to that of π1 and is not shown here. For the corresponding π* bands at K point, the charge density is of similar character. It is quite interesting that the charge distribution of a few graphene layers is very different from that of graphite, which contains an infinite number of graphene layers. The interac4467

asymmetry in the buffer layer is not enough to induce a bandgap in the graphene layers above it because the π bonds of the buffer layer are destroyed by the covalent bonding with the substrate. We find that the SW defects in each single graphene layer, which break the symmetry substantially, can open a bandgap in the Dirac bands of the graphene layer. In good agreement with the experiment,8 our calculations show that the bandgap decreases with the increase of the thickness of the graphene layer. The band splitting and the narrowing of the bandgap of Dirac bands are due to the interlayer π (and π*) bond interaction, which can be understood within tight binding model. The charge distribution in the few graphene layers is very different from that in graphite.

Figure 4. The band structure of zero (a), one (b), two (c), and three (d) graphene layers with SW defects above the buffer layer and Si-SiC(0001). The Fermi level is indicated by the dashed line.

tion of the Π2 and (Π1 + Π3) states in the trilayer case is stronger than that of Π2 and Π1 in the bilayer case. Therefore the bands π3 and π1* are further pushed toward each other and the bandgap becomes smaller. With more and more layers grown, the bandgap will decrease until the gap is closed. The bandgap development with the thickness of the graphene layers on Si-terminated SiC(0001) surface is found to be basically the same as discussed above, except that the dangling bond bands due to the surface Si atoms of the substrate are not spin-polarized (Figure 4) since the electrons in Si dangling bonds are more delocalized than those in carbon dangling bonds.10 To address how the bandgap changes when the density of SW defects is lower, we investigated one SW defect in a 6 × 6 and 8 × 8 unit cell of a single graphene layer, respectively. The respective bandgaps are reduced to 0.15 and 0.08 eV due to the decreased perturbation of the symmetry. In reality, however, defects have been found to occur in graphene epilayers on SiC(0001) in an appreciable density,12,16,11 which will lead to an appreciable bandgap. Our results are concerning the graphene with finite density of SW defects. Furthermore, the LDA or GGA Kohn-Sham gap in general considerably underestimates the true bandgap. We expect an overall sizable increase in the value of the calculated bandgap if going beyond LDA/GGA, like using the GW method.27-29 Especially for graphene, the self-energy correction to the bandgap is even larger (0.5-3 eV) due to the weaker screening.29 To simulate the random distribution of the SW defects, we have also calculated the bandgap of two SW defects in different relative positions within an 8 × 8 unit cell of a graphene layer. The bandgap almost does not vary with respect to the change of the distribution of the defects. In summary, the electronic band of a single defect-free graphene layer above the buffer layer is gapless. The

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Acknowledgment. We acknowledge the financial support from STINT (Stiftelsen fo¨r internationalisering av ho¨gre utbildning och forskning). References (1) Geim, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183. (2) Son, Y.-W.; Cohen, M. L.; Louie, S. G. Nature 2006, 444, 347. (3) Trauzettel, B. B.; Bulaev, D. V.; Loss, D.; Burkard, G. Nat. Phys. 2006, 3, 192. ¨ zyilmaz, B.; Zhang, Y.; Kim, P. Phys. ReV. Lett. 2007, (4) Han, M. Y.; O 98, 206805. (5) Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B 1996, 54, 17954. (6) Brey, L.; Fertig, H. A. Phys. ReV. B 2006, 73, 235411. (7) Giovannetti, G.; Khomyakov, P. A.; G, B.; Kelly, P. J.; van den Brink, J. Phys. ReV. B 2007, 76, 73103. (8) Zhou, S. Y.; Gweon, G.-H.; Fedorov, A. V.; First, P. N.; De Heer, W. A.; Lee, D.-H.; Guinea, F.; Neto, A. H. C.; Lanzara, A. Nat. Mater. 2007, 6, 770. (9) Varchon, F.; Feng, R.; Hass, J.; Li, X.; Nguyen, B. N.; Naud, C.; Mallet, P.; Veuillen, J.-Y.; Berger, C.; Conrad, E.; Magaud, L. Phys. ReV. Lett. 2007, 99, 126805. (10) Mattausch, A.; Pankratov, O. Phys. ReV. Lett. 2007, 99, 76802. (11) Hashimoto, A.; Suenaga, K.; Gloter, A.; Urita, K.; Iijima, S. Nature 2004, 430, 870. (12) Rutter, G. M.; Crain, J. N.; Guisinger, N. P.; Li, T.; First, P. N.; Stroscio, J. A. Science 2007, 317, 219. (13) Stones, A. J.; Wales, D. J. Chem. Phys. Lett. 1986, 128, 501. (14) Telling, R. H.; Ewels, C. P.; El-Barbary, A. A.; Heggie, M. I. Nat. Mater. 2003, 2, 333. (15) Nordlund, K.; Keinonen, J.; Mattila, T. Phys. ReV. Lett. 1996, 77, 699. (16) Chen, W.; Xu, H.; Liu, L.; Gao, X.; Qi, D.; Peng, G.; Tan, S. C.; Feng, Y.; Loh, K. P.; Wee, A. T. S. Surf. Sci. 2005, 596, 176. (17) Rollings, E.; Gweon, G.-H.; Zhou, S. Y.; Mun, B. S.; McChesney, J. L.; Hussain, B. S.; Fedorov, A. V.; First, P. N.; De Heer, W. A.; Lanzara, A. J. Phys. Chem. Solids 2006, 67, 2172. (18) Forbeaux, I.; Themlin, J.-M.; Debever, J.-M. Phys. ReV. B 1998, 58, 16396. (19) Urita, K.; Suenaga, K.; Sugai, T.; Shinohara, H.; Iijima, S. Phys. ReV. Lett. 2005, 94, 155502. (20) Li, L.; Reich, S.; Robertson, J. Phys. ReV. B 2005, 72, 184109. (21) Fasolino, A.; Los, J. H.; Katsnelson, M. I. Nat. Mater. 2007, 6, 858. (22) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. (23) Kresse, G.; Futhmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (24) Kresse, G.; Futhmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (25) Perdew, J. P.; Wang, Y. Phys. ReV. B 1986, 33, 8800. (26) Berger, C.; Song, Z.; Li, X.; Wu, X.; Brown, N.; Naud, C.; Mayou, D.; Li, T.; Hass, J.; Marchenkov, A. N.; Conrad, E. H.; First, P. N.; De Heer, W. A. Science 2006, 312, 1191. (27) Hybertsen, M. S.; Louie, S. G. Phys. ReV. B 1986, 34, 5390. (28) Rohlfing, M.; Kru¨ger, P.; Pollmann, J. Phys. ReV. B 1993, 48, 17791. (29) Yang, L.; Park, C. H.; Son, Y. W.; Cohen, M. L.; Louie, S. G. Phys. ReV. B 2007, 99, 186801.

NL802409Q

Nano Lett., Vol. 8, No. 12, 2008