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Electronic States of Defect Sites of Graphene Model Compounds: A DFT and Direct Molecular Orbital-Molecular Dynamics Study Hiroto Tachikawa* and Hiroshi Kawabata‡ DiVision of Materials Chemistry, Graduate School of Engineering, Hokkaido UniVersity, Sapporo 060-8628, Japan, and Ministry of Education, Culture, Sports, Science and Technology, National Institute of Science and Technology Policy, Kasumigaseki, Chiyoda, 100-0013, Tokyo, Japan ReceiVed: January 13, 2009; ReVised Manuscript ReceiVed: March 4, 2009
Electronic states of normal graphene, the defective graphene (one carbon atom is removed from the normal graphene), the defective graphene anion (defective graphene plus an excess electron), and the defective graphene cation (defective graphene plus one hole) have been investigated by means of density functional theory (DFT) and direct molecular orbital-molecular dynamics (MO-MD) methods in order to elucidate the effect of vacancy defect on the electronic states of graphene. The HOMO and LUMO of normal graphene were widely delocalized as π-conjugated orbitals over the graphene surface in the normal graphene. On the other hand, the excess electron in defective graphene anion was localized in the defect site, indicating that the excess electron on the graphene circuit is efficiently trapped and stabilized by the vacancy defect site of graphene. The direct MO-MD calculations showed that the trapped electron in the defect site is stable at low temperature. Around room temperature (300 K), the structural change of the graphene backbone was found and the vacancy defect was reconstructed by thermal activation. The excess electron escaped from the defect site of the reconstructed graphene, while the spin density delocalized the graphene. 1. Introduction Graphene is one sheet of graphite composed of a layer structure and is a one-atom-thick planar sheet of sp2-bonded carbon atoms. Graphene shows quite different electronic property from most conventional three-dimensional materials.1-3 Namely, intrinsic graphene acts as a semimetal or zero-gap semiconductor. Experimental results from transport measurements show that graphene has a remarkably high electron mobility at room temperature, with reported values in excess of 15 000 cm2/Vs.1-3 Additionally, the symmetry of the experimentally measured conductance indicates that the mobilities for holes and electrons are nearly the same. The mobility is nearly independent of temperature between 10 and 100 K.4-6 Recently, the importance of vacancy in graphene and carbon nanotube has been pointed out by several groups.7-19 Acid treatment is known to create defects in carbon materials.13-15 The C-C bond cleavage is caused by electrophilic attack. Coleman et al. artificially made the defect in graphene by acid treatment.16 They treated the graphene surface by using nonoxidizing acid (HCl) and found the carbon vacancy defect with an in-plane orbital. They also carried out band calculation for two vacancy sites and showed that the band gap in the defect site is significantly small. Theoretical calculations about the defect in the graphene have been carried out by several groups. Yazyev and Helm investigated the magnetism in graphene induced by single carbon atom defects.8,18 They considered two types of defects, i.e., the hydrogen chemisorption defect and the vacancy defect. The magnetism due to the defect-induced extended states has been found. Using first-principles calculation, Duplock et al. investigated the adsorption of atomic hydrogen on the defect of * To whom correspondence should be addressed at Hokkaido University. E-mail:
[email protected]. Fax: +81 11706-7897. ‡ National Institute of Science and Technology Policy.
graphene. They found that a new energy gap was opened as one of the electronic density of states.19 Thus, the electronic states of defect in graphene were well understood theoretically. However, the dynamical feature, i.e., temperature effects on the structures and electronic states of defective graphene, is scarcely known. In the present study, structures and electronic states of normal graphene, the defective graphene (one carbon atom is removed from the normal graphene), the defective graphene anion (defective graphene plus an excess electron), and the defective graphene cation (defective graphene plus one hole) have been investigated by means of both DFT and direct molecular orbital-molecular dynamics (MO-MD) methods20-23 to elucidate the effect of the defect on the electronic states of graphene. In particular, we focus our attention on the recover reaction of the defect vacancy by thermal activation. In previous papers,20-23 we investigated the graphene and graphene-lithium interaction systems. Also, diffusion dynamics of lithium ion and atom (Li+ and Li) on the graphene surface were calculated by using the direct molecular orbital-molecular dynamics (MO-MD) method developed by us. In this method, the potential energy surface and the energy gradient of all atoms of the system are explicitly calculated quantum chemically. Therefore, different characteristics from those of the usual classical MD calculation are obtained. It was found that the Li+ ion diffuses preferentially along the node of HOMO of graphene. Also, it was found that diffusion of lithium atom is much different from that of Li+ ion. The Li atom moves together with spin density distribution generated on the graphite surface. Hence, the diffusion rate for the Li atom is slower than that of the Li+ ion. The diffusion coefficients calculated were in good agreement with the experiments. In this work, the same method was applied to the defective graphene.
10.1021/jp900365h CCC: $40.75 2009 American Chemical Society Published on Web 04/08/2009
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2. Method of Calculation In the present study, normal graphene and defective graphenes were examined to elucidate the effect of the defect on the electronic states of graphene. First, the structure of graphene (normal graphene) with n ) 37 (where n is the number of benzene rings in the graphene) was optimized at the B3LYP/6-31G(d) level. The edges of graphenes were terminated by hydrogen atoms. The defective graphene was made by removing one carbon atom from the graphene. Namely, the defective graphene has a vacancy in the central region of graphene. The radical anion and cation of defective graphenes were examined to elucidate the effect of the defect on the electron and hole transport processes in graphene. The excitation energies of normal graphene, defective graphene, and ion radicals of defective graphene were calculated at the time-dependent (TD)-DFT, PW91PW91/3-21G(d) level. Static density functional theory (DFT) calculations were carried out with the Gaussian 03 program package.24 Temperature effects on the structures and electronic states of defective graphene (defective graphene anion) were investigated by means of the direct molecular orbital-molecular dynamics (MO-MD) method. The total energy and energy gradient on the multidimensional potential energy surface of the system were calculated at each time step at the PM3 level of theory, and then the classical equation of motion is fulldimensionally solved. Therefore, charges and electronic states of all atoms are treated exactly the same within the level of theory. This point is much different from the usual classical molecular dynamics (MD) calculation where the charges of all atoms are constant during the diffusion. We carried out the direct MO-MD calculations under constant temperature condition. The mean temperature of the system is defined by
T)
1 〈Σm V2 〉 3kN i i i
(1)
where N is number of atoms, Vi and mi are velocity and mass of the ith atom, and k is Boltzmann’s constant. We choose temperatures in the range 100-300 K. The velocities of atoms at the starting point were adjusted to the selected temperature. To keep a constant temperature in the system, Berendsen’s method25 was used: i.e., the bath relaxation time (τ) was introduced in the calculation. We have chosen τ ) 0.01 ps in all trajectory calculations. The equations of motion for n atoms in the system are given by
dQj ∂H ) dt j ∂Pj ∂H ∂U ))∂t j ∂Qj
(2)
where j ) 1 - 3N, and H is classical Hamiltonian. Qj, Pj, and U are Cartesian coordinate of jth mode, conjugated momentum, and potential energy of the system, respectively. These equations were numerically solved by the velocity Verlet algorithm method. No symmetry restriction was applied to the calculation of the gradients. The time step size was chosen by 0.50 fs, and a total of 4000 steps were calculated for each dynamics calculation. More details of direct MO-MD and direct ab initio MD methods are described elsewhere.26-29 3. Results A. Electronic States of Normal Graphene. The structure of graphene used in the present study is illustrated in Figure 1.
Figure 1. Optimized structure of graphene (n ) 37) obtained by at the B3LYP/6-31G(d) level. The isosurface represents the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The edge region of graphene is terminated by hydrogen atoms.
The graphene is composed of 37 benzene rings (i.e., n ) 37, where nmeans the number of benzene rings in graphene) and the edge is terminated by hydrogen atoms. The geometry of graphene was optimized at the B3LYP/6-31G(d) level. The excitation energies of the graphene were calculated be 1.51 (first), 1.65 (second), and 2.06 eV (third) at the TD-PW91PW91/3-21G(d) level. The electronic states of the graphene were calculated at the B3LYP/6-31G(d) level. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are widely delocalized over the graphene surface, as shown in Figure 1. The results indicate that both hole and excess electron can move widely on the normal graphene surface. This result supports strongly the mobilities of the excess electron and hole observed experimentally.1-5 B. Electronic States of Graphene with Defect. To elucidate the effects of defect on the electronic states of graphene, a defect was made artificially on the graphene. The defect was made by removing one carbon atom around the center of graphene, as shown in Figure 2. The geometry of the graphene with a defect was optimized at the B3LYP/6-31G(d) level and the structure of the defective graphene was illustrated in Figure 2. The graphene obtained by the geometry optimization is called hereafter “defective graphene”. The defect graphene has a singlet state and a neutral charge at the electronic ground state. The relaxation energy obtained by the geometry optimization was calculated to be 9.3 kcal/mol. The C-C bond length and C-C-C angle were changed from r1 ) 1.424 Å, r2 ) 1.424 Å, and θ ) 120.1° to r1 ) 1.390 Å, r2 ) 1.390 Å, and θ ) 124.3°, respectively. The bond lengths were slightly shortened and the angle (θ) was wider from the structural relaxation. Isosurface representations of HOMO and LUMO are illustrated in Figure 2. HOMO is widely distributed over the graphene, whereas LUMO is localized near the vacancy defect. This feature is much different from those of normal graphene.
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Figure 2. Optimized structure of the defective graphene used in the present study calculated at the B3LYP/6-31G(d) level. One carbon atom was removed from the graphene. HOMO and LUMO of the defective graphene are illustrated by isosurface representation.
The excitation energies of defective graphene are calculated to be 0.04 (1st excitation), 0.08 (2nd), and 0.70 eV (3rd), indicating that the introduction of vacancy defect decreases significantly the band gap of the graphene. C. Electron Capture of Defective Graphene. In the previous section, the electronic states of defective graphene were investigated on the basis of DFT calculation. Here, we consider the anionic state of the defective graphene (i.e., the defective graphene plus an excess electron). The structure of the anion system of the defective graphene (called the “defective graphene anion”) was optimized at the B3LYP/6-31G(d) level, and the optimized structure was given in Figure 3. The adiabatic electron affinity of defective graphene is calculated to be 2.97 eV at the B3LYP/6-31G(d) level. The structure of the defective graphene anion is slightly changed from that of the defective graphene by the electron capture (r1 ) 1.428 Å, r2 ) 1.428 Å, and θ ) 117.8°) after the structural relaxation. The stabilization energy from the vertical electron attachment point of the defective graphene was 16.3 kcal/mol. The special distribution of spin density of the defective graphene anion is illustrated in Figure 3 (lower). As shown in Figure 3, the spin density of the excess electron is localized on the defect site of graphene, and there is no distribution in the other region. This result indicates that an excess electron moving on graphene is easily trapped and is localized in the defect site. D. Hole Capture of Defective Graphene. The structure around the defect site was slightly deformed after the hole capture of defective graphene. The adiabatic ionization potential (IP) of the defect site was calculated to be 5.10 eV. The stabilization energy was 14.7 kcal/mol. The structural parameters were changed to r1 ) 1.388 Å, r2 ) 1.388 Å, and θ ) 124.4° after the structural relaxation. The spin density of the unpaired electron on the defective graphene cation was illustrated in Figure 4. The distribution of spin density in the defect cation is
Figure 3. Optimized structure of the defective graphene anion (defective graphene plus an excess electron) calculated at the B3LYP/6-31G(d) level. The isosurface indicates the spin density distribution of excess electron in the defective graphene anion.
much different from that of the anion. Namely, the excess electron in the anion was localized in the defect site as shown in the previous section. On the other hand, a hole (unpaired electron) in the defective graphene cation is delocalized on the bulk surface of graphene. It can be concluded therefore that the defect site can trap an excess electron efficiently, whereas the defect site traps the hole partially. Thus, the calculations clearly indicate that the defect site on graphene plays an important role in determining the electronic state of graphene. Especially, the conductivity of the excess electron on graphene is strongly affected by the existence of the vacancy defect in the graphene surface. E. Temperature Effects on Spin Density of the Defective Graphene Anion. In previous sections, it was shown that an excess electron is localized in the defect site, whereas the hole is delocalized over the graphene surface. In this section, the temperature effects on the structure and spin density of the defective graphene anion are investigated by means of the direct MO-MD method. Three temperatures (100, 200, and 300 K) were examined as simulation temperatures. The defective graphene anion used in the dynamics calculation is illustrated in Figure 5. The carbon atoms on the defective graphene anion are classified into three regions around the
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Figure 6. Time profiles of distances (R1, R2, and R3) of the defective graphene anion (n ) 19) at 200 K obtained by the direct MO-MD method. The lower panel indicates an extended view of the time profile of distances in the time range 1.0-1.5 ps.
Figure 4. Optimized structure of the defective graphene cation (defective graphene plus a hole) calculated at the B3LYP/6-31G(d) level. The isosurface indicates the spin density distribution of an unpaired electron in the defective graphene cation.
Figure 5. Structure of the defective graphene anion (n ) 19) and geometrical parameters used in the dynamics calculation. Three shells indicate the regions of carbon atoms around the defect site of graphene.
defect. One is the carbon atoms located near the vacancy defect (denoted by “1st shell carbons”). The second one is denoted by “2nd shell carbon atoms” where the carbon atoms are located in the middle region. The last one is “3rd shell carbon atoms” which belong to the others. Thus, the carbon atoms in the first and second shells are located near the vacancy defect. The carbon atoms C1, C2, and C3 belong to the first shell, while nine carbon atoms are located in the second shell. After thermal activation, the carbon backbone of the defective graphene anion vibrates gradually as a function of time. Time profiles of distances (R1, R2, and R3) of the defective graphene anion at 200 K are plotted for example in Figure 6. The distances are R1 ) 2.721 Å, R2 ) 2.722 Å, and R3 ) 2.751 Å at time zero. The distances vibrate periodically in the range 2.62-2.92 Å. A total of 15 geometries of defective graphene anion were randomly sampled from the time period 1.0-1.5 ps, and then the spin densities were calculated at the B3LYP/3-21G(d) level. The time profiles of the distances (R1-R3) in the sampled region are given in Figure 6B. The averaged spin densities of carbon atoms in the shells 1-3 are given in Table 1. At 0.0 K, the spin densities on the first, second, and third shells were calculated to be 1.073, -0.128, and 0.055, respectively, indicating that most spin is localized in the first shell. These values were hardly changed at 100 K. The values at 200 K were 1.068 (1st shell), -0.136 (2nd), and 0.067 (3rd), which are slightly changed by the increase of temperature. These results indicate that the effects of temperature on the spin densities are significantly small in the temperature range 0-200 K. F. Reconstruction Dynamics of the Defective Graphene Anion. Five trajectories were run from different initial conditions at 300 K. Among them, two trajectories gave the large
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TABLE 1: Temperature Dependence of Spin Densities in the Carbon Atoms on Three Regions of Defective Graphene Anion Calculated at the B3LYP/3-21G(d) Level temp/K
1st shell
2nd shell
3rd shell
0.0 100 200 300 (before) recovered defect
1.073 1.073 1.068 1.066 0.985
-0.128 -0.129 -0.136 -0.128 -0.065
0.055 0.057 0.067 0.062 0.080
structural change of the graphene backbone after several femtoseconds. One of the typical trajectories for the structural change was given in Figure 7. The potential energy of the system decreases suddenly at 0.51 ps from 30 to -8 kcal/mol. Time profiles of distances R1, R2, and R3 are plotted in Figure 7B. All distances were suddenly shortened from 2.75 to 2.20 Å at 0.50-0.52 ps, indicating that weak bonds between R1 ()C1-C2), R2 ()C2-C3), and R3 ()C3-C1) are newly formed at time 0.50 ps by structural deformation. This indicates that the defect in the graphene backbone is reconstructed by thermal activation. The spin densities on the defective graphene anion before and after structural reconstruction are illustrated in Figure 8. At time zero, before the structural reconstruction, the excess electron is localized in the defect site. After the structural change, the weak bonds between carbon atoms are formed. Immediately, the molecular orbital of the excess electron is widely delocalized over the graphene surface. These results strongly indicate that the excess electron can escape from the
Figure 7. (A) Potential energy of a trajectory for the defective graphene anion at 300 K plotted as a function of time, obtained by means of the direct MO-MD method. (B) Time profiles of distances (R1, R2, and R3) of the defective graphene anion at 300 K obtained by the direct MO-MD method. Three weak bonds between three carbon atoms were formed in the final state by thermal activation.
Figure 8. Snapshots of the defective graphene anion at 300 K obtained by the direct MO-MD method. Spin densities are also given by the isosurface representation. At time zero, the spin density is localized in the vacancy defect, while the excess electron is widely distributed over the ring after the structural reconstruction at 0.60 ps.
defect site by thermal activation due to the structural reconstruction of the defect site. For one of the trajectories another type of structural deformation occurred as presented in Figure 9. The potential energy of the system decreases suddenly at 1.30-1.50 ps from 4 to -10 kcal/mol. Time profiles of distances R1, R2, and R3 are plotted in Figure 9B. Three C-C bonds vibrated periodically in the range 2.20-2.80 Å at time 0.00-1.20 ps. At 1.30 ps, one of the C-C bonds (R1) was suddenly shortened from 2.40 to 1.45 A at 1.30 ps, indicating that a C-C single bond between C1-C2 atoms is newly formed at time 1.30 ps by structural deformation. The other C-C bonds (R2 and R3) were still longer during the simulation, indicating that only one single bond is reconstructed by the thermal activation in this sample trajectory. This indicates that the defect in the graphene backbone is reconstructed by thermal activation. The spin densities on graphene before and after structural reconstruction are illustrated in Figure 10. At time zero, the excess electron is localized in the defect site. After the structural reconstruction, distribution of the excess electron is widely delocalized over the graphene surface. The excess electron can escape from the defect site by thermal activation and structural reconstruction to from a C-C single bond at 300 K. The averaged spin densities in the three shells 1-3 are given in Table 1. Before the structural reconstruction at 300 K, the spin densities on the first, second, and third shells were calculated to be 1.073, -0.128, and 0.055, respectively. After the structural reconstruction at 300 K, these values were changed to 0.985 (1st), -0.065 (2nd), and 0.080 (3rd). Namely, a portion of the localized excess electron is distributed over the graphene by the structural reconstruction. The partial charge of the carbon atom in the third shell is changed from -0.1 to -0.7 by the structural reconstruction, indicating that the charge is also transferred from the defect site to the bulk surface.
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Figure 9. (A) Potential energy of a trajectory for the defective graphene anion at 300 K, obtained by means of the direct MO-MD method. (B) Time profiles of distances (R1, R2, and R3) of the defective graphene anion at 300 K. One C-C bond was formed in the final state by thermal activation.
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Figure 11. A model of the vacancy defect for excess electron and the thermal reconstruction process derived from the present study.
at the B3/6-31G(d) level. The calculation indicates that the reconstruction reaction is exothermic by 14.7 kcal/mol.
Figure 10. Snapshots of the defective graphene anion at 300 K obtained by the direct MO-MD method. Spin densities are also given by isosurface representation. At time zero, the spin density is localized in the vacancy defect, while the excess electron is widely distributed over the ring after the structural reconstruction at 2.0 ps. One C-C bond was formed in the final state by thermal activation.
To obtain the energetics of the reconstruction before and after structural reconstruction, the energy calculation was carried out
4. Discussion A. Summary. In the present study, electronic states of the defective graphene have been investigated by means of DFT and direct MO-MD methods. The molecular orbitals of neutral graphene (HOMO and LUMO) are widely delocalized over the graphene surface. If a defect is made artificially in the graphene surface, the HOMO is still widely delocalized over the graphene, indicating that the effect of the defect on the HOMO is significantly small. On the other hand, the shape of LUMO is drastically changed by the defect formation: LUMO is localized around the defect site. The spin density of the defective graphene anion is localized in the defect site of graphene. Therefore, an excess electron is easily trapped in the defect site. By thermal activation of the defective graphene anion, reconstruction reactions of carbon backbone occur efficiently. Immediately, a portion of the excess electron escapes from the vacancy and is delocalized over the graphene surface due to the reconstruction of the vacancy defect. B. Model of Electron Transport in Graphene. A model obtained by the present calculation is schematically illustrated in Figure 11. The present calculation showed that excess electron can move widely on the normal graphene surface because the singly occupied molecular orbital (SOMO) of the normal graphene anion is fully delocalized. If a defect exists on the surface of the graphene circuit, the movement of the excess electron is stopped and is stabilized in the defect vacancy. The excess electron in the defect site is significantly stable at low temperature. By thermal activation, it is possible for a
Defective Grapheme portion of the excess electron to escape from the defect because the structural deformation (thermal reconstruction) of the defect site occurs. C. Comparison with Previous Studies. Coleman et al. introduced experimentally a defect vacancy in the graphene sheet by acid treatment (HCl). They measured K-edge X-ray absorption spectra of the defective graphene and suggested that the defect vacancy has metallic conductivity. The present calculation strongly supports this feature: namely, the band gap of graphene is significantly decreased by the formation of the vacancy defect. The first excitation energy is changed from 1.51 to 0.04 eV by the formation of the vacancy defect at the PW91PW91/3-21G(d)//B3LYP/6-31G(d) level. The present calculation showed that a vacancy defect is partially recovered by thermal activation. The recovered defect with a C-C bond was predicted by Yazyev and Helm on the basis of theoretical calculation.8 The present results support their prediction. D. Remarks. In the present study, the dynamics calculations were carried out at the PM3 level because the present system is too large to treat the ab initio MD level and the dynamics calculation for the present system needs a long simulation time (∼2.00 ps). The present direct MO-MD calculations showed that the reconstruction of the carbon backbone was found at 300 K. The temperature would be varied by using the levels of theory used in the dynamics calculation. Therefore, it should be noted that the results obtained in the present calculation are qualitatively effective. More accurate wave functions may provide deeper insight in the dynamics. Despite the several assumptions introduced here, the results enable us to obtain valuable information on the mechanism of the reconstruction reaction of the defective graphene anion. Acknowledgment. The authors are indebted to the Computer Center at the Institute for Molecular Science (IMS) for the use of the computing facilities. H.T. also acknowledges partial support from a Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan. This work is partially supported by the IKETANI foundation. Supporting Information Available: Figures showing the optimized structure of the defective graphene anion and the defective graphene cation radical calculated at the B3LYP/6-31G(d) level and potential energy of a trajectory for the defective graphene anion and time profiles of the temperature of the system, and time profiles of the temperature of the system and kinetic, potential, and total energies of a trajectory for the defective graphene anion at 300 K. This material is available free of charge via the Internet at http://pubs.acs.org.
J. Phys. Chem. C, Vol. 113, No. 18, 2009 7609 References and Notes (1) Geim, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183. (2) Novoselov, K. Nat. Mater. 2007, 6, 720. (3) de Heer, W. A.; Berger, C.; Wu, X.; First, P. N.; Conrad, E. H.; Li, X.; Li, T.; Sprinkle, M.; Hass, J.; Sadowski, M. L.; Marek Potemski, M.; Martinez, G. Solid State Commun. 2007, 143, 92. (4) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonosand, S. V.; Firsov, A. A. Nature (London) 2005, 438, 197. (5) Morozov, S. V.; Novoselov, K. S.; Katsnelson, M. I.; Schedin, F.; Elias, D. C.; Jaszczak, J. A.; Geim, A. K. Phys. ReV. Lett. 2008, 100, 016602. (6) Chen, J.-H.; Jang, C.; Xiao, S.; Ishigami, M.; Fuhrer, M. S. Nat. Nanotechnol. 2008, 3, 206. (7) Khare, R.; Mielke, S. L.; Paci, J. T.; Zhang, S.; Ballarini, R.; Schatz, G. C.; Belytschko, T. Phys. ReV. B 2007, 75, 075412. (8) Yazyev, O. V.; Helm, L. Phys. ReV. B 2007, 75, 125408. (9) Rocha, A. R.; Padilha, J. E.; Fazzio, A.; da Silva, A. J. R. Phys. ReV. B 2008, 77, 153406. (10) Li, T. C.; Lu, S.-P. Phys. ReV. B 2007, 77, 085408. (11) Pimenta, M. A.; Dresselhaus, G.; Dresselhaus, M. S.; Cancado, L. G.; Jorio, A.; Saito, R. Phys. Chem. Chem. Phys. 2007, 9, 1276. (12) Kuemmeth, F.; Ilani, S.; Ralph, D. C.; McEuen, P. L. Nature (London) 2008, 452, 448. (13) Balasubramanian, K.; Burghard, M. Small 2005, 1, 180. (14) Itkis, M. E.; Perea, D. E.; Jung, R.; Niyogi, S.; Haddon, R. C. J. Am. Chem. Soc. 2005, 127, 3439. (15) Strano, M. S. J. Chem. Phys. B 2003, 107, 6970. (16) Coleman, V. A.; Knut, R.; Karis, O.; Grennberg, H.; Jansson, U.; Quinlan, R.; Holloway, B. C.; Sanyal, B.; Eriksson, O. J. Phys. D 2008, 41, 062001. (17) Boukhvalov, D. W.; Katsnelson, M. I. Nano Lett. 2008, 8, 4373. (18) Yazyev, O. V.; Helm, L. J. Phys. Conf. Ser. 2007, 61, 1294. (19) Duplock, E. J.; Scheffler, M.; Lindan, P. J. D. Phys. ReV. Lett. 2004, 92, 225502. (20) Tachikawa, H.; Shimizu, A. J. Phys. Chem. B 2005, 109, 13255. (21) Tachikawa, H.; Shimizu, A. J. Phys. Chem. B 2006, 110, 20445. (22) Tachikawa, H. J. Phys. Chem. C 2007, 111, 13087. (23) Tachikawa, H. J. Phys. Chem. C 2008, 112, 10193. (24) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; MCossi, G.; Scalmani, N. R.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision B.04, an ab initio MO calculation program; Gaussian, Inc., Pittsburgh, PA, 2003. (25) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (26) Tachikawa, H. Phys. Chem. Chem. Phys. 2008, 10, 2200. (27) Tachikawa, H.; Abe, S. J. Chem. Phys. 2007, 126, 194310. (28) Tachikawa, H. J. Chem. Phys. 2006, 125, 144307. (29) Tachikawa, H. J. Chem. Phys. 2006, 125, 133119.
JP900365H
