Predicting Dislocations and Grain Boundaries in Two-Dimensional

Dec 10, 2012 - extend in third dimension, forming concave dreidel-shaped polyhedra. They include different number of homoelemental bonds and, by react...
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Predicting Dislocations and Grain Boundaries in Two-Dimensional Metal-Disulfides from the First Principles Xiaolong Zou, Yuanyue Liu, and Boris I. Yakobson* Department of Mechanical Engineering and Materials Science, Department of Chemistry, and the Smalley Institute for Nanoscale Science and Technology, Rice University, Houston, Texas 77005, United States. S Supporting Information *

ABSTRACT: Guided by the principles of dislocation theory, we use the first-principles calculations to determine the structure and properties of dislocations and grain boundaries (GB) in single-layer transition metal disulfides MS2 (M = Mo or W). In sharp contrast to other two-dimensional materials (truly planar graphene and h-BN), here the edge dislocations extend in third dimension, forming concave dreidel-shaped polyhedra. They include different number of homoelemental bonds and, by reacting with vacancies, interstitials, and atom substitutions, yield families of the derivative cores for each Burgers vector. The overall structures of GB are controlled by both local-chemical and far-field mechanical energies and display different combinations of dislocation cores. Further, we find two distinct electronic behaviors of GB. Typically, their localized deep-level states act as sinks for carriers but at large 60°-tilt the GB become metallic. The analysis shows how the versatile GB in MS2 (if carefully engineered) should enable new developments for electronic and opto-electronic applications. KEYWORDS: Two-dimensional, transition metal disulfides, dislocation, grain boundary, first principles theory

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While there are considerable difficulties in experimentally obtaining the structural and spectral information for dislocations, the first principles calculations can determine their atomic makeup16 and assess the electronic properties. In this letter, we first derive the unique structures of the elemental dislocation cores and their derivative-families for MS2 (M = Mo or W). The analysis shows how they extend in third dimension, a feature distinctly different from other truly planar 2D materials. Then we use the obtained dislocation cores to build and systematically explore the grain boundaries, their energy variability, and their important contribution into the electronic properties. Unexpectedly, we further discover particular GB types with clearly metallic properties. Our calculations are performed with local density approximation (LDA)17 to the density functional theory (DFT) using projector-augmented wave (PAW)18,19 potentials as implemented in Vienna Ab-initio Simulation Package (VASP).20,21 We adopt the supercell approach and choose the vacuum layer thickness larger than 10 Å. Using the plane-wave-based total energy minimization,22 all structures are fully relaxed until the force on each atom is less than 0.01 eV/Å. The model systems can be constructed either in nanoribbon (NR) configurations or with periodic boundary conditions (PBC) in which the distance between GBs and their images is ∼4 nm in our models, large enough to achieve the convergence of the GB

timulated by the discovery of remarkable physics of one two-dimensional (2D) material, graphene, its inorganic analogs with intrinsic bandgap, such as layered transition metal dichalcogenides, now also receive increasing attention, due to both fundamental interest and great potential for the nextgeneration nanoelectronics and optoelectronics. For example, transistors,1,2 phototransistors,3 and integrated circuits4 have been successfully fabricated based on single-layer MoS2. The performance of these devices relies heavily on the mechanical and electronic properties of the materials. Although several methods, including micromechanical exfoliation,1−5 chemical exfoliation,6−8 liquid exfoliation,9 and thermolysis of single precursor containing Mo and S,10 have been developed to prepare single layer MoS2, the size of the synthesized samples remains limited. Only very recently could large-scale synthesis of monolayer MoS2 be achieved by chemical vapor deposition (CVD) on amorphous SiO2/Si substrate11,12 or graphene.13 However, the carrier mobility (∼0.004−0.04 cm2 V−1 s−1) is significantly reduced compared to mechanically exfoliated MoS2 (0.3−5 cm2 V−1 s−1 14), which can be caused by the inefficient carrier injection through the contacts15 and probably due to the presence of the structural defects, such as the extended grain boundaries (GB) observed in the experiment.11,13 These extended defects can act as undesired sinks of carriers, or increase their scattering, degrading the electrical device performance. Understanding these critical issues requires not only determination of the atomic structure of the dislocation cores but also their influence on the electronic behavior. © 2012 American Chemical Society

Received: October 30, 2012 Revised: December 6, 2012 Published: December 10, 2012 253

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Figure 1. Basic edge dislocations formed by removal of shaded atoms from the lattice (a), with atomic structures of metal-polar shown in (b), sulfur-polar in (d), and a 4|8 in (c). Thin black arrows show the basis vectors, while the Burgers vectors are colored as blue (1, 0), orange (0, 1), and gray (1, 1). Each atomic structure is shown in front, isometric, and side (along the layer) views, with atoms colored blue (metal, M), yellow (front/top layer sulfur, S) and orange (back/bottom layer sulfur, S′).

single-atom sheets of C or h-BN in the triatom thick MS2 each metal atom is 6-fold coordinated by sulfur, arranged in D3h prismatic S3MS′3 units. Accordingly, the dislocation cores are significantly more complex: not the planar polygons but instead the concave three-dimensional (3D) polyhedra of polyelemental composition. For the three cases above, full structural relaxation within the DFT methods reveals the dreidel-like shapes, shown in frontal, isometric, and in-plane projections in Figure 1b−d (the “dreidel” metaphor, used earlier27 for another system, vividly describes our configurations). In addition to the significant long-range elastic strain caused by any dislocation, there are inevitable local distortions in the cores, compared to perfect lattice: change in coordination, the bond lengths, and the angles. The metal atoms within M−M bonds are undercoordinated, only 5-fold. The change of bond angles reaches 50% (from 82 to ∼120°) occurring near the homoelemental bonds. This is largely due to the different orientations of M−S and homoelemental bonds, with S−S in the top and bottom layers, M−M in the middle layer, and M−S connecting different layers. The bond length changes in these dislocation cores do not exceed 0.1 Å compared to perfect lattice. Although the accurate energy of an isolated dislocation is difficult to evaluate28 due to its polarity and well-known logarithmic divergence of the far-field elastic contribution, the typical energy is estimated to be about ∼7 eV, largely due to the extra chemical energy brought about by the homoelemental bonds. This value is half-energy of -and- pair in a large enough unit cell, with the distance between dislocations ∼20 Å, which effectively reduces the dislocation−dislocation interaction. This significant energy can cause variability in the chemical composition of the core and its elemental balance, by means of absorption or emission of point defects, while the vector b for a given dislocation is topological invariant. Efficient interaction of dislocations with the vacancies and interstitials is well-known.28 In the present case, the size of stoichiometric unit MS2 limits the range of possible variability because adding or subtracting a full unit should result in a trivial displacement of the dislocation (climb step). Accordingly, we consider a few examples within the families derived from generic (M-polar) and (S-polar) dislocations. Figure 2 top shows a series of dislocation cores derived from the M-polar by insertion of a single S, or a pair 2S (equivalent

energies (structural model details are shown in the Supporting Information, Figure S1). The GB energies are determined by subtracting the total energies of perfect MS2 lattice of identical number of atoms (and identical perimeter of edges for NR configurations) from the total energies of defected systems. The periodic models give GB energies similar to NR models with negligible differences, and it is easier to study the electronic properties using periodic models that avoid the edges of nanoribbons. A sheet of dichalcogenide MS2 (M = Mo or W) consists of a middle plane of triangularly packed metal atoms, sandwiched between the two layers of S atoms, also packed triangularly in their respective planes. In the most stable 2H phase, these three layers S|M|S′ are stacked so that the frontal view (Figure 1a) displays the alternating M and S|S′ nodes as a honeycomb lattice. It is convenient then, following nanotubes or graphene,23 to choose a basis of two unit vectors along the zigzag node motifs. To construct the dislocations, one removes a half-plane (literally, a half-line in case of 2D-material), reconnects all the bonds seamlessly, and then inspects an arbitrary closed contour in order to determine the resulting topological mismatch, that is the Burgers vector, b. Deleting an armchair half-line of atoms as shown in Figure 1a, left, obviously results in a dislocation with b = (1, 0). Because of the trigonal symmetry, there are three chemically equivalent possibilities, tagged “ ”, with Burgers vectors b = (1, 0), (−1, 1), (0, −1), all M-rich, with distinctive metal−metal homoelemental bond. Deleting an inverted half-line results in an S-rich triplet of inverse dislocations “ ”, with b = (−1, 0), (1, −1), (0, 1), all with distinctive sulfur−sulfur homoelemental bonds. Example of (0, 1) is shown in Figure 1a, right. Similar to the case of h-BN, the energetically unfavorable homoelemental bonds can be avoided in dislocations of √3-times larger Burgers vector b = ±(1, 1), ±(2,-1), ±(−2,1), obtained by removing two parallel zigzag rows, as shown in Figure 1a, middle. The planar views of these three topologically different dislocations are similar to those discussed for nanotubes23 and graphene24,25 or especially for h-BN:26 a pentagon−heptagon 5| 7 with M−M central pair for b = (1, 0) in Figure 1b, a 5|7 with S−S in the center for b = (0, 1) in Figure 1d, or a square− octagon 4|8 for b = (1, 1) shown in Figure 1c, with balanced elemental composition, nonpolar. However, in contrast to 254

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to adding a metal-vacancy, VM in Schottky notations), or a substitution of one M atom with 2S, as in reaction: + VM + 2S. The first two yield configurations whose frontal views are both hexagon−octagon 6|8, while the third one remains a 5|7 but with broken mirror symmetry; see the right top corner in Figure 2. Despite the uncertainty of the absolute energies of the dislocations, the energies of the derived structures relative to the generic can be exactly computed as E = Eder − E⊥ − niμi

where Eder and E⊥ are the total energies of the derivative and the generic form (e.g., a metal-rich 5|7, ) respectively; ni is the number of added atoms and μi is their chemical potential for i = M or S. In thermodynamic equilibrium, μM and μS satisfy the equation μMS2 = μM + 2μS, where μMS2 is the chemical potential of MS2 unit in a pristine sheet. The results are shown in Figure 2 middle panel. They depend explicitly on the chemical potential of sulfur μS, decreasing in the S-rich conditions, with the slopes proportional to the excess of S atoms in the core (1, 2, or 4, as shown by the dotted, dashed, and solid blue lines, respectively). Figure 2 bottom shows the dislocations obtained from the Spolar by deleting a single S, or a pair 2S (equivalent to adding a metal atom), or a substitution of 2S pair with one M, as in reaction: + V2S + M. The first one retains the 5|7 frontal view, only slightly distorted. However, the second one ( + V2S) yields after the reconstruction-relaxation a distinctly different frontal view of a rhomb−hexagon 4|6 dreidel. The third one is + M2S, and after the reconstruction transforms into 4|6+V2S; see the right bottom corner of Figure 2. Again, the relative energies of the derivative cores with respect to the generic are

Figure 2. Top and bottom panels: Derivative dislocation cores formed by the reactions of generic or with the point defects, as indicated by arrows. Middle panel: Energies of derivative cores relative to (blue lines) or (orange lines) as functions of the chemical potential of sulfur in the range −1.4 eV < μS < 0 eV, where MoS2 can remain stable w.r.t. the segregation of the bulk Mo (μS = −1.4 eV) or bulk alpha-S (used as the reference, μS = 0 eV).29,30 The dotted, dashed, and solid lines with slopes 1, 2, and 4, correspond to structures in the top and bottom panels from left to right.

Figure 3. Energies of GBs as functions of tilt angle α, starting from either AC (A-GB, red) or from ZZ (Z-GB, blue), with their structures in the insets near the data points. Shaded lines-areas show the energy ranges due to reconstructions. Red solid circles are for A-GB composed of 5|7, and open circles are for A-GB composed of 4|6 and 6|8 gradually degenerating in all-rhombs GB at 60°. Blue dashes, crosses, and open squares are for ZGB composed of 5|7, 4|6 + 6|8, and 4|8, respectively. 255

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Figure 4. Band structures and LDOS for (a) 21.8° A-GB composed of and , (b) 21.8° A-GB composed of 4|6 of 6|8, (c) 60° Z-GB composed of 4|8, and (d) 60° A-GB composed of 4|6 and 6|8. Green dashed lines indicate the Fermi level. Inset of (d) shows the partial charge density in the range of [−0.1, 0.1] eV. The DOS contribution of the bulk is projected to the Mo and S atoms far from the two parallel GBs.

computed exactly and are shown in Figure 2 middle panel. They all increase with μS with the slopes proportional to the deficit of S atoms in the cores: 1, 2, or 4, as shown by the dotted, dashed, and solid orange lines, respectively. Figure S2 in the Supporting Information shows similar results for WS2. The 4|8 dislocations (Figure 1c) has larger Burgers vector (1, 1), and consequently greater elastic deformation energy, about ∼|b|2 = 3 times greater than the and discussed above. Moreover, DFT-based calculations show that an isolated 4|8 is not stable and undergoes a 0.6 eV exothermic reconstruction splitting into a pair: (1, 1) → (1, 0) + (0, 1); note that in the right-hand side, the first term corresponds to M-rich core while the second one to S-rich, so that the elemental balance is preserved. (The process shown in Figure S3, Supporting Information.) The interplay between the chemical and strain energies should determine the actual structural variety of defects expected in observations of GB: not only generic 5|7, but also 6|8, or 4|6 being most likely encountered at different chemical equilibrium conditions. The discussed dislocation types and reconstructions serve as constituents for the GB. Here, analysis is restricted to the GB lines bisecting the misorientation angle α between the grains, when the adjacent domains can be considered rotated by ±α/2 with respect to their perfect interface. The latter can be either in the AC direction (generating mirror-symmetric armchair AGB) or along the ZZ (asymmetric zigzag Z-GB, invariant to mirror reflection with the element-inversion). Accordingly, Figure 3 presents the GB energies per length versus the tilt angles G(α) with reference alignment along the AC direction (bottom axis, red) or along the ZZ direction (top axis, blue)

(Results for WS2 are shown in Figure S4 in the Supporting Information). Overall, the GB energies increase with the tilt angle, initially linearly, reflecting simply the fact that the dislocation density grows in proportion to α. At larger tilts, the interaction between the dislocations and the cancellation of their stress fields turns these G(α) functions sublinear. It is important to realize that the energy of GB does not play significant role in determining their likelihood of formation or preference, since they are nonequilibrium structures, and the tilt angle is preset by the growth history, not the energy-based equilibrium statistics.31 More relevant is the possibility of the reconstruction of the dislocation cores or their reactions with the point defects, as described above. Because of such possible local reorganizations, the plotted functions are multivalued with the range of energies highlighted by the shaded areas. The “vertical” (at α = const) transitions down in energy are actually possible with annealing and depend on the elemental chemical potentials. The symmetric A-GB are polar (see and in Figure 1), with the excess of one of the elements; therefore, we plot the energies averaged over the M-rich and S-rich GB (at each α) to avoid the dependence on the chemical potentials. The asymmetric Z-GBs are balanced with the equal amount of M and 2S the same as in the perfect lattice. It is interesting to note that at α = 60° tilt graphene would restore its perfect structure with the GB vanished (simply a switch from initial AC interface to ZZ, or vice versa from initial ZZ to AC). In contrast, for MS2, 60°-tilt corresponds to the border between the element-inverted domains, that is, a pronounced GB consisting of a dense row of homoelemental bonds, MM or SS. The monotonous increase of G(α) and its 256

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and of course the topological constraints imposed by the lattice around them. Linear arrays of such dislocations form the grain boundaries between the mutually tilted crystalline domains. We find that typically the GB introduce the midgap electronic states and act as sinks for the carriers. Of particular interest may be the discovery/prediction of the metallic conductivity along the GB bisecting the 60° misoriented domains. These GB can serve as metallic nanowires imbedded in the semiconductor matrix.

maxima at 60° illustrate the dominance of the homoelemental bond energy cost. Reconstruction of dislocation cores is affected by not only the “local chemistry” but also by the overall mechanical stiffness of the sheets. The relatively low flexural rigidity of graphene and h-BN (3.9 and 3.6 eV·Å2/atom, respectively32), permits smaller dislocations fuse into a larger one, since the increase in elastic energy is greatly relieved by the off-plane warping. However, the much higher flexural rigidities of “thick” disulfides (27 and 30 eV·Å2/atom for MoS2 and WS2, comparable with DFTB result for MoS233) prevent such warping. Consequently, the full elastic energy penalty ∼|b|2 makes the dislocations with smaller Burgers vectors preferred, similar to conventional 3D crystals. The extended GB defects bring about important changes to the electronic structures as shown in Figure 4 for MoS2 as example. The computed direct bandgap in perfect MoS2 at K point is 1.8 eV, which is consistent with previous experiments and calculations.5,34−38 The GB adds the localized states with the energies deeply in the middle of the pristine MoS2 gap, seen in the local density of states (LDOS) plots in Figure 4a−c. Computed partial charge density distributions for these localized states show that they mainly originate from the d orbitals of metal atoms in the Mo- and S-rich GBs, or from localized p states of S atoms in 6|8s (Figure S5 in the Supporting Information). By depleting the carriers these deep states may degrade the performance of devices based on polycrystalline MoS2 (A-GB’s of other tilt angles in Figure S6 of the Supporting Information show similar behavior.) For greater tilt α, the cores are placed denser and interact with each other stronger, so that the corresponding local bands become dispersive. Most striking and potentially useful behavior emerges at α = 60° (from AC) GB that show distinct metallic behavior. The states delocalized in one dimension correspond to the bands crossing the Fermi level, rendering perfect metallic stripes embedded in the otherwise semiconducting MoS2. The LDOS contributing to the metallic behavior is visualized through the partial charge density (inset, Figure 4d) and clearly shows the wave functions are indeed associated with the GB and can be detected by scanning tunneling microscopy (STM).39 A 1D metallic system could of course undergo Peierls distortion with a bandgap opening at the Fermi level. Preliminary analysis with greater supercells suggests that it is unlikely, as the periodicity is maintained by the rigid matrix in which the wire is imbedded. Such metallic stripes add new and useful functionality brought about by carefully engineered GB, in contrast to their more often detrimental role. In summary, we predict the structures of dislocations and their assemblies into grain boundaries in the two-dimensional metal disulfides, materials of higher complexity than the monatomic layer graphene or h-BN. For this, we rely equally on topological principles of dislocation theory, as well as the first principles density functional theory calculations, to augment the structural analysis with exact energy evaluation, stability, and electronic structure calculations. We find that the in-plane and through-layer relaxation results in dislocation cores of dreidel-like shapes, complex concave polyhedra with hetero- and homoelemental bonds. Their frontal views in the direction normal to the layer are most likely to be observed and should look as planar polygons: 5|7, 6|8, 4|6, and so forth (all of Burgers vector |b| = 1 lattice parameter), or a less stable 4|8 (|b| = √3). Statistical preference among the core structures depends on the chemical potential of the constituent elements,

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ASSOCIATED CONTENT

S Supporting Information *

Description of the structural models using either PBC or finite NR configurations; the energies of dislocations and GBs in WS2; reconstruction of an isolated 4|8; partial charge density distribution for 21.8° A-GB and 60° Z-GB; and the band structures for 9.4, 32.2, and 46.8° A-GBs composed of 5|7s. 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 This work was supported by the U.S. Army Research Office MURI Grant W911NF-11-1-0362 and partially by the National Science Foundation (CMMI 0708096, NIRT). The computations were performed at the Data Analysis and Visualization Cyberinfrastructure funded by NSF under Grant OCI-0959097.

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REFERENCES

(1) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V.; Kis, A. Nat. Nanotechnol. 2011, 6, 147−150. (2) Zhang, Y.; Ye, J.; Matsuhashi, Y.; Iwasa, Y. Nano Lett. 2012, 12, 1136−1140. (3) Yin, Z. Y.; Li, H.; Li, H.; Jiang, L.; Shi, Y. M.; Sun, Y. H.; Lu, G.; Zhang, Q.; Chen, X. D.; Zhang, H. ACS Nano 2012, 6, 74−80. (4) Radisavljevic, B.; Whitwick, M. B.; Kis, A. ACS Nano 2011, 5, 9934−9938. (5) Splendiani, A.; Sun, L.; Zhang, Y.; Li, T.; Kim, J.; Chim, C.-Y.; Galli, G.; Wang, F. Nano Lett. 2010, 10, 1271−1275. (6) Eda, G.; Yamaguchi, H.; Voiry, D.; Fujita, T.; Chen, M.; Chhowalla, M. Nano Lett. 2011, 11, 5111−5116. (7) Zeng, Z.; Yin, Z.; Huang, X.; Li, H.; He, Q.; Lu, G.; Boey, F.; Zhang, H. Angew. Chem., Int. Ed. 2011, 50, 11093−11097. (8) Ramakrishna Matte, H. S. S.; Gomathi, A.; Manna, A. K.; Late, D. J.; Datta, R.; Pati, S. K.; Rao, C. N. R. Angew. Chem., Int. Ed. 2010, 49, 4059−4062. (9) Coleman, J. N.; Lotya, M.; O’Neill, A.; Bergin, S. D.; King, P. J.; Khan, U.; Young, K.; Gaucher, A.; De, S.; Smith, R. J.; Shvets, I. V.; Arora, S. K.; Stanton, G.; Kim, H. Y.; Lee, K.; Kim, G. T.; Duesberg, G. S.; Hallam, T.; Boland, J. J.; Wang, J. J.; Donegan, J. F.; Grunlan, J. C.; Moriarty, G.; Shmeliov, A.; Nicholls, R. J.; Perkins, J. M.; Grieveson, E. M.; Theuwissen, K.; McComb, D. W.; Nellist, P. D.; Nicolosi, V. Science 2011, 331, 568−571. (10) Liu, K.-K.; Zhang, W.; Lee, Y.-H.; Lin, Y.-C.; Chang, M.-T.; Su, C.-Y.; Chang, C.-S.; Li, H.; Shi, Y.; Zhang, H.; Lai, C.-S.; Li, L.-J. Nano Lett. 2012, 12, 1538−1544. (11) Lee, Y.-H.; Zhang, X.-Q.; Zhang, W.; Chang, M.-T.; Lin, C.-T.; Chang, K.-D.; Yu, Y.-C.; Wang, J. T.-W.; Chang, C.-S.; Li, L.-J.; Lin, T.W. Adv. Mater. 2012, 24, 2320−2325. (12) Zhan, Y. J.; Liu, Z.; Najmaei, S.; Ajayan, P. M.; Lou, J. Small 2012, 8, 966−971. 257

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Letter

(13) Shi, Y.; Zhou, W.; Lu, A. Y.; Fang, W.; Lee, Y. H.; Hsu, A. L.; Kim, S. M.; Kim, K. K.; Yang, H. Y.; Li, L. J.; Idrobo, J. C.; Kong, J. Nano Lett. 2012, 12, 2784−2791. (14) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10451−10453. (15) Popov, I.; Seifert, G.; Tománek, D. Phys. Rev. Lett. 2012, 108, 156802. (16) Yakobson, B. I. NSF-AFOSR Workshop “2D Materials and Devices−Beyond Graphene”; Arlington, VA, USA, May 30-31, 2012. (17) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048−5079. (18) Blöchl, P. E. Phys. Rev. B 1994, 50, 17953−17979. (19) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758−1775. (20) Kresse, G.; Furthmüller, J. Phys. Rev. B 1996, 54, 11169−11186. (21) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15−50. (22) Ihm, J.; Zunger, A.; Cohen, M. L. J. Phys. C: Solid State Phys. 1979, 12, 4409−4422. (23) Yakobson, B. I. Appl. Phys. Lett. 1998, 72, 918−920. (24) Liu, Y.; Yakobson, B. I. Nano Lett. 2010, 10, 2178−2183. (25) Yazyev, O. V.; Louie, S. G. Phys. Rev. B 2010, 81, 194520. (26) Liu, Y.; Zou, X.; Yakobson, B. I. ACS Nano 2012, 6, 7053−7058. (27) Mostoller, M.; Chisholm, M.; Kaplan, T. Phys. Rev. Lett. 1994, 72, 1494−1497. (28) Hirth, J. P.; Lothe, J. Theory of Dislocations, 2nd ed.; Wiley: New York, 1982. (29) Schweiger, H.; Raybaud, P.; Kresse, G.; Toulhoat, H. J. Catal. 2002, 207, 76−87. (30) Bollinger, M.; Jacobsen, K.; Nørskov, J. Phys. Rev. B 2003, 67, 085410. (31) Yakobson, B. I.; Ding, F. ACS Nano 2011, 5, 1569−1574. (32) Kudin, K. N.; Scuseria, G. E.; Yakobson, B. I. Phys. Rev. B 2001, 64, 235406. (33) Lorenz, T.; Teich, D.; Joswig, J. O.; Seifert, G. J. Phys. Chem. C 2012, 116, 11714−11721. (34) Ataca, C.; Ciraci, S. J. Phys. Chem. C 2011, 115, 13303−13311. (35) Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F. Phys. Rev. Lett. 2010, 105, 136805. (36) Li, Y. F.; Zhou, Z.; Zhang, S. B.; Chen, Z. F. J. Am. Chem. Soc. 2008, 130, 16739−16744. (37) Lu, P.; Wu, X.; Guo, W.; Zeng, X. C. Phys. Chem. Chem. Phys. 2012, 14, 13035−13040. (38) Lebègue, S.; Eriksson, O. Phys. Rev. B 2009, 79, 115409. (39) Lahiri, J.; Lin, Y.; Bozkurt, P.; Oleynik, I. I.; Batzill, M. Nat. Nanotechnol. 2010, 5, 326−329.

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