Confinement, Transport Gap, and Valley Polarization in Graphene

Letter pubs.acs.org/NanoLett

Confinement, Transport Gap, and Valley Polarization in Graphene from Two Parallel Decorated Line Defects Daniel Gunlycke,*,† Smitha Vasudevan,‡ and Carter T. White† †

Chemistry Division, Naval Research Laboratory, Washington, DC 20375, United States Department of Chemistry, The George Washington University, Washington, DC 20052, United States

‡

S Supporting Information *

ABSTRACT: Quantum transport calculations show that a transport gap approximately Eg = 2ℏvF/W can be engineered in graphene using two parallel transport barriers, separated by W, extended along the zigzag direction. The barriers, modeled by chemically decorated observed line defects, create confinement and resonance bands tracing the bands in zigzag nanoribbons. The resonance bands terminate at the dimensional crossover, where the states become boundary-localized, leaving the transport gap. The structure also allows for nearly perfect valley polarization. KEYWORDS: Graphene, quantum transport, resonant tunneling, transport barrier, transport gap, valley polarization

E

straight and precisely defined. Herein, we consider the 5-5-8 line defect extending along the zigzag direction.15 After its first observation, this line defect has been controllably fabricated.16 As the 5-5-8 line defect is semitransparent with extended states at the Fermi level,17 it is not a good transport barrier by itself. It is, however, a good transport barrier in the presence of a local potential that pulls the conducting states away from the Fermi level. Herein, we show that such a potential can be induced by adsorption of the diatomic gases: hydrogen (H2), oxygen (O2), and fluorine (F2). As we are merely using the adsorbates to generate a local potential, we expect that many other adsorbates will also work well and that the ultimate choice might be one of experimental convenience. First-principles calculations based on density functional theory (DFT) show that all the considered diatomic gases bind to the line defect as paired adatoms. Computational details are provided in Supporting Information. Each adatom pair is roughly located above the central carbon sites, as shown in Figure 1A for the case of hydrogen. The binding energy, adatom separation, and the distance between an adatom and the closest carbon atom beneath it are listed below in Table 1. The binding energies reflect the total energy difference between having the diatomic molecules isolated far above the line defect structure and adsorbed on the surface. That all molecules react exothermically with the line defect, despite the added energy from the dissociation of hydrogen and fluorine, shows that the final structures are very stable. The dissociation of hydrogen

ngineering a gap in graphene without degrading its exceptional transport properties is arguably the main obstacle preventing a breakthrough in graphene-based nanoelectronics. To create such a gap, a lot of effort has been devoted to making graphene nanoribbons isolated either structurally through lithographic cutting,1,2 nanoparticle cutting,3−5 unzipping of nanotubes,6,7 and bottom-up synthesis,8,9 or chemically within functionalized graphene.10−12 The edges in all these nanoribbons are hard boundaries that restrict transport to the dimension extending along the nanoribbon. In contrast, we propose a ribbon structure with soft boundaries formed between two thin parallel transport barriers that in addition to allowing transport along the ribbon is penetrable by electrons in surrounding graphene states. The transport across this railroad track structure is governed by resonant tunneling through quasi-bound states within the railroad track confinement. For highly reflective barriers, the resonances form continuous bands that closely match the band structure of a nanoribbon, except for the boundary-localized states, which cannot carry any transport due to energy and crystal momentum conservation. The resonance bands must therefore terminate at the crossover between extended and boundarylocalized states. As the confined region contains no states near the Fermi level extending across the railroad track structure, electrons approaching this structure experience a sizable transport gap. Thin transport barriers could be achieved through chemical decoration of the graphene surface. Precise control over such decoration on graphene, however, is not an easy task. Therefore, we propose the use of extended line defects,13−15 which are more reactive than graphene itself, to guide the adsorbates. In addition, the extended line defects are both © 2012 American Chemical Society

Received: October 31, 2012 Revised: November 30, 2012 Published: December 5, 2012 259

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and fluorine is indicated by their large adatom separations. Oxygen does not dissociate on the surface, but its unpaired electrons become committed to the carbon−oxygen bonds, eliminating the magnetic moments present in the isolated molecules. To obtain the potential induced by the adsorbates on the line defect and surrounding carbon sites, illustrated in Figure 1B, we have calculated the band structure of periodically repeated arrays of line defects decorated with the considered adsorbates. The band structure for the hydrogen-decorated line defect is shown in Figure 1C. The induced potential has a strong effect on the boundary-localized states due to their large density on the affected sites. To gain better understanding of how the bands are affected by the adsorption, we adopt a nearestneighbor tight-binding model consisting of orthonormal πorbital states at the carbon sites with the parameter γ = −2.6 eV describing the hopping between neighboring sites.18 To capture the effects of the adsorbed molecules, we also introduce two carbon onsite energy parameters, εLD and εZ, defined in Figure 1B. These onsite energies can be determined uniquely from the first-principles energies E± of the boundary-localized states at Γ, shown in Figure 1C. These states are completely localized on the relevant sites with the onsite energies, εLD and εZ. Because the antisymmetric state with respect to reflection about the line defect has a node at the reflection plane,17 we obtain εZ = E−. Making use of the reflection symmetries both across and along the line defect, we find that εLD = E+ + γ − 2γ2/ΔE, where ΔE ≡ E+ − E− is the energy separation between the symmetric and antisymmetric boundary-localized states at Γ. This energy separation arises from interactions across the line defect and has no relation to the transport gap we observe below. The obtained values for εLD and εZ are listed in Table 1. The large negative εLD resulting from hydrogen adsorption is expected, as the electronegativity of hydrogen is only marginally less than that of carbon. Fluorine is considerably more electronegative than carbon but bonds more strongly to the surface, explaining the large positive εLD for fluorine. As can be seen in Figure 1C, the bands from our tight-binding model agree well with the bands from our DFT calculations, not only at Γ, but throughout the first Brillouin zone, indicating that our tightbinding model is capturing the most important interactions. Oxygen and fluorine decoration lead to a similar level of agreement with DFT calculations. The presence of hydrogen, oxygen, or fluorine along the line defect produces an atomically thin but effective barrier to electron transport converting the line defect from semitransparent17 to highly reflective. This is illustrated in Figure 2A,B, where the transmission probability across an undecorated and hydrogen-decorated line defect, respectively, is reported. These results are calculated exactly from the tight-binding model. More details are provided in Supporting Information. The transmission probability is presented as a function of energy and wave vector in the direction of the line defect, two quantities that completely specify the incoming graphene state. The energy dispersion of graphene restricts transmission near the Fermi level to the Dirac cones. From Figure 2A, we find, as previously shown,17 that the valley-averaged transmission probability for the undecorated line defect is approximately 50%. In contrast, the transmission probability across the hydrogen-decorated line defect never exceeds 4% near the Fermi level, as can be seen in Figure 2B. In fact, for a single highly reflective barrier, the maximum transmission is approxiately given by Tmax ≈ (2γ/εLD)2. This shows that

Figure 1. Extended chemically decorated graphene line defect. (A) Top and side view of the atomic structure of the line defect with the gray rectangular box depicting the unit cell in the atomic and electronic structure calculations. (B) The adsorbates are seen as a potential on the carbon sites underneath. This potential is modeled through the onsite energies, εLD and εZ on the center (blue) and offcenter (green) sites, respectively. (C) Band structure of the periodically repeating line defect shown in (A) presented from the zone center Γ to the zone edge along the line defect Y. Dashed (solid) bands are obtained from first-principles (semiempirical) calculations. Green (blue) bands comprise symmetric (antisymmetric) states. The induced potential is uniquely determined from the energies E±.

Table 1. Structural Data and Tight-Binding Parameters for Graphene Line Defects That Have Reacted with Different Diatomic Moleculesa Eb

dCX

dXX

εLD

εZ

molecule

(eV)

(Å)

(Å)

(eV)

(eV)

H2 O2 F2

0.49 0.46 4.36

1.12 1.48 1.39

2.24 1.45 2.39

−27.1 12.8 24.5

−0.30 0.28 0.43

a Eb is the binding energy, dCX is the shortest carbon−adatom bond length, and dCX is the shortest adatom separation. εLD and εZ are onsite energies defined in Figure 1B. The large dXX for hydrogen and fluorine indicate that these molecules have dissociated on the surface.

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fluorine- and oxygen-decorated line defects have also low transmission limits. The highly reflective nature of these chemically decorated line defects make them ideal candidates for the rails in a nanoscale railroad track structure that can be used to produce and control a transport gap in graphene. In particular, two parallel line defects can be used to form a narrow strip in graphene, as shown in Figure 3A. The transport across this structure is driven by resonant tunneling. If the transmission probability through one rail is high, the transmission probability across the entire railroad track structure is generally high too, as can be seen in Figure 3B, for the structure without any chemical decoration. Although some dark resonance bands are emerging, there is no gap due to a symmetric band crossing the Fermi level at the Dirac point.17 For the hydrogen-decorated structure, the transmission probability is shown in Figure 3C. The low-transmission probability through each line defect has begun to isolate the states in the confinement from the outside graphene states, resulting in sharper resonance bands. For a large enough potential, these resonance bands are centered

Figure 2. Transmission across single isolated line defects. (A) Transmission probability across an undecorated line defect as a function of energy and wave vector along the line defect. (B) Transmission probability across a hydrogen-terminated line defect. The transmission probability at the opposite graphene valley is obtained by reflecting the figures around the zone center Γ.

Figure 3. Two parallel decorated line defects. (A) Atomic structure of the line defects showing the confined region in gray. (B) The transmission probability as a function of energy and wave vector along the structure in the absence of chemical decoration. (C) Same figure as in (B), but with hydrogen-decorated line defects. Note that there is a transport gap around the Fermi level at E = 0. (D) The resonance bands in (C) are overlaid with the band structure of a corresponding isolated zigzag nanoribbon. (E) The band structure of the railroad structure in (A) with neighboring railroad tracks separated by 320 zigzag chains. Two sets of bands are drawn, those from Γ to Y and those from X to M. At the resonance bands in (C) and (D), bands associated with the confinement appear broader in the interior of the cones in (E) due to the dispersion across the line defect structure. 261

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around the electronic bands of a corresponding isolated zigzag nanoribbon,19,20 as shown in Figure 3D. An important difference, however, is that the resonance bands closest to the Fermi level terminate abruptly, resulting in a transport gap Eg. To gain further insight into this behavior, we have calculated the band structure in Figure 3E for the railroad track structure periodically repeated with the ratio between the separation of neighboring railroad tracks and the confined width being 20:1. Almost all bands are localized in the direction across the structure. Consistent with Figure 3C, extended states can only be found when the states are allowed to propagate in both graphene and the confined region. As can be seen from Figure 3E, the resonance bands terminate at the dimensional crossover, where the embedded nanoribbon states leave the Dirac cone and become boundary-localized. Because the boundary-localized states do not contribute to the transport across the structure, the transport is not directly affected by any local magnetic moments that could form under some circumstances.21,22 The location of the dimensional crossover can be estimated from an isolated unpolarized zigzag ribbon. The states of the ribbon supports two distinctly different types of states, those that extend both along and across the ribbon and those that only extend along the ribbon edges. Within the Dirac approximation, the former have energies E = ± ℏvF(q2x+q2y )1/2, where qx and qy are real wave vectors that are measured from the center of the Dirac cones and satisfy the transcendental equation qy = ± qx/tan (qxW) with the sign dependent on the considered valley.23 Those states that extend both along and across the ribbon only exist to the critical point qcx = 0, at which qcy = ±1/W.23 Therefore, the transport gap is approximately given by

Eg (W ) =

2ℏvF W

Figure 4. Transport gap and conductance across the confinement. (A) The transport gap Eg as a function of the confinement width W. Circles are numerically determined from the transport calculations. The green curve is given by eq 1. (B) The conductance across the structure per unit length along the structure g, given by eq 2, as a function of energy E.

(1)

where 2ℏvF = √3|γ|a ≈ 1.11 eV nm. This expression agrees well with the numerically obtained values, as can be seen in Figure 4A. The transport gap is also apparent in the conductance G across the structure. This conductance is proportional to the length L of the structure, allowing us to write G = gL, where the conductance across the structure per length unit along the structure g can be expressed as g (E ) =

e 2 2a ah π

∫0

structure. Because there is little overlap between the resonance bands in the two valleys, a carefully designed experiment that fixes the sign of the velocity component will lead to perfect valley polarization. In conclusion, the proposed double barrier structure in graphene could be used to create a transport gap, obtain a ribbon-like confinement, and generate perfect valley polarization. We showed explicitly how these results could be obtained using chemically decorated extended line defects in graphene. We have, however, also obtained similar results without the line defects, that is, by applying large local potentials directly to graphene. See details in Supporting Information. The latter results indicate that the transport gap, confinement, and perfect valley polarization are not sensitive to the nature of the transport barrier, as long as the barriers are highly reflective. Therefore, we also expect that the results presented herein should tolerate a certain level of unevenness in the adsorbate coverage, as long as significant coverage gaps do not exist. Such gaps are unlikely when an excess of adsorbates are present due to the high reactivity at the line defect sites. Finally, we would like to point out that the railroad track structures, presented, herein, could be combined to form more advanced structures on the mesoscopic scale, and ultimately be used in graphene-based scalable electronics.

π /2a

T (E , k y)dk y

(2)

where a is the graphene lattice constant and T(E,ky) is the transmission probability obtained from the quantum transport calculations shown in Figure 3C. As expected, this expression, plotted in Figure 4B, reflects the transport gap. The peaks in this figure are reminiscent of one-dimensional van-Hove singularities, but are actually not singularities at all. The peaks occur at the van Hove singularities because the resonance bands in Figure 3C span a larger portion of the wave vectors at these energies. Only one of the two valleys in graphene are shown in Figures 2 and 3. The transmission probability for the other valley is the reflection of these figures around Γ. Note that this reflection changes the sign of the slope of the two resonance bands forming the transport gap in Figure 3C. Hence, the valley occupied by a transmitted electron with an energy outside the gap and within the window of a single resonance can be determined by the sign of its velocity component along the 262

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(23) Brey, L.; Fertig, H. A. Phys. Rev. B 2006, 73, 235411.

ASSOCIATED CONTENT

S Supporting Information *

Details about the first-principles calculations, the quantum transport calculations, and the transmission probability across the double barrier structure when the barrier is a potential applied directly to graphene. 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: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge support from the U.S. Office of Naval Research, directly and through the U.S. Naval Research Laboratory.

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REFERENCES

(1) 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, 1191−1195. (2) Han, M. Y.; Ö zyilmaz, B.; Zhang, Y.; Kim, P. Phys. Rev. Lett. 2007, 98, 206805. (3) Datta, S. S.; Strachan, D. R.; Khamis, S. M.; Johnson, A. T. C. Nano Lett. 2008, 8, 1912−1915. (4) Ci, L.; Xu, Z.; Wang, L.; Gao, W.; Ding, F.; Kelly, K. F.; Yakobson, B. I.; Ajayan, P. M. Nano Res. 2008, 1, 116−122. (5) Campos, L. C.; Manfrinato, V. R.; Sanchez-Yamagishi, J. D.; Kong, J.; Jarillo-Herrero, P. Nano Lett. 2009, 9, 2600−2604. (6) Kosynkin, D. V.; Higginbotham, A. L.; Sinitskii, A.; Lomeda, J. R.; Dimiev, A.; Price, B. K.; Tour, J. M. Nature 2009, 458, 872−876. (7) Jiao, L.; Zhang, L.; Wang, X.; Diankov, G.; Dai, H. Nature 2009, 458, 877−880. (8) Cai, J.; Ruffieux, P.; Jaafar, R.; Bieri, M.; Braun, T.; Blankenburg, S.; Muoth, M.; Seitsonen, A. P.; Saleh, M.; Feng, X.; Mullen, K.; Fasel, R. Nature 2010, 466, 470−473. (9) Koch, M.; Ample, F.; Joachim, C.; Grill, L. Nat. Nanotechnol. 2012, 7, 713−717. (10) Ruoff, R. Nat. Nanotechnol. 2008, 3, 10−11. (11) Singh, A. K.; Yakobson, B. I. Nano Lett. 2009, 9, 1540−1543. (12) Lee, W.-K.; Robinson, J. T.; Gunlycke, D.; Stine, R. R.; Tamanaha, C. R.; King, W. P.; Sheehan, P. E. Nano Lett. 2011, 11, 5461−5464. (13) Heckl, W. M.; Binnig, G. Ultramicroscopy 1992, 42−44, 1073− 1078. (14) Simonis, P.; Goffaux, C.; Thiry, P. A.; Biro, L. P.; Lambin, P.; Meunier, V. Surf. Sci. 2002, 511, 319−322. (15) Lahiri, J.; Lin, Y.; Bozkurt, P.; Oleynik, I. I.; Batzill, M. Nat. Nanotechnol. 2010, 5, 326−329. (16) Zettl, A.; Chen, J.-H.; Alem, N.; Autes, G.; Gargiulo, F.; Gautam, A.; Linck, M.; Kisielowski, C.; Yazyev, O. V.; Louie, S. G. Growth of a Linear Topological Defect in Graphene as a Gate-tunable Valley Valve, AVS 59th International Exhibition and Symposium, Tampa, FL, October 28−November 2, 2012. (17) Gunlycke, D.; White, C. T. Phys. Rev. Lett. 2011, 106, 136806. (18) White, C. T.; Li, J.; Gunlycke, D.; Mintmire, J. W. Nano Lett. 2007, 7, 825−830. (19) Klein, D. J. Chem. Phys. Lett. 1994, 217, 261−265. (20) Fujita, M.; Wakabayashi, K.; Nakada, K.; Kusakabe, K. J. Phys. Soc. Jpn. 1996, 65, 1920−1923. (21) Okada, S.; Kawai, T.; Nakada, K. J. Phys. Soc. Jpn. 2011, 80, 013709. (22) Kou, L.; Tang, C.; Guo, W.; Chen, C. ACS Nano 2011, 5, 1012−1017. 263

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