NANO LETTERS
Bundling up Carbon Nanotubes through Wigner Defects
2005 Vol. 5, No. 6 1045-1049
Antoˆnio J. R. da Silva,*,† A. Fazzio,† and Alex Antonelli‡ Instituto de Fı´sica, UniVersidade de Sa˜ o Paulo, CP 66318, 05315-970, Sa˜ o Paulo, SP, Brazil, and Instituto de Fı´sica “Gleb Wataghin”, UNICAMP, CP 6165, 13083-970, Campinas, SP, Brazil Received March 9, 2005; Revised Manuscript Received May 5, 2005
ABSTRACT We show, using ab initio total energy density functional theory, that the so-called Wigner defects, an interstitial carbon atom right beside a vacancy, which are present in irradiated graphite, can also exist in bundles of carbon nanotubes. Due to the geometrical structure of a nanotube, however, this defect has a rather low formation energy, lower than the vacancy itself, suggesting that it may be one of the most important defects that are created after electron or ion irradiation. Moreover, they form a strong link between the nanotubes in bundles, increasing their shear modulus by a sizable amount, clearly indicating its importance for the mechanical properties of nanotube bundles.
Single-walled carbon nanotubes (SWNT) have attracted an enormous amount of attention since their discovery1,2 about 10 years ago. No doubt this is related to all their extraordinary materials properties and potential for applications.3 From a mechanical point of view, SWNTs have exceedingly large elastic moduli4,5 around 1 TPa, which would make them perfect for low-density high-modulus fibers. When in bundles, however, a much worse performance is usually obtained, mostly related to the weak interaction between tubes, which makes it relatively easy for them to slide against each other.6 A breakthrough has been recently obtained7 using electron-beam irradiation to generate cross-links between the tubes. This increased the shear modulus by a factor of approximately 30, thus indicating a new way to fabricate stronger fibers, a significant step in the field of materials science. Irradiation by electrons or ions is being used by many groups7-22 as an important tool to modify the structure and the properties of nanotubes. As examples, we can mention the welding of tubes,10,12 and the dramatic increase of shear modulus for a bundle of nanotubes.7 To have a better control of the nanotube engineering, it is very important to have a detailed microscopic understanding of the defects that are created upon irradiation. Properties such as formation energy, geometry, stability and electronic structure are critical in determining what type of defects will be produced and how they will affect the mechanical behavior of the carbon fiber. Few theoretical works have addressed questions such as what will be the effect of electron or ion irradiation on nanotubes * Corresponding author. E-mail:
[email protected]. † Universidade de Sa ˜ o Paulo. ‡ UNICAMP. 10.1021/nl050457c CCC: $30.25 Published on Web 05/14/2005
© 2005 American Chemical Society
and bundles of nanotubes, or what kind of defects will be created during these irradiation procedures.23-27 All of these studies consisted of molecular dynamics simulations with the interaction between the atoms being described by empirical potentials. Even though these works are relevant and indicative of the processes that may occur, they may miss important aspects since empirical potentials cannot describe appropriately features such as re-hybridization, rebonding, or charge redistribution, all of which are quite important when chemical bonds are being broken and made, and specially in stressed configurations that may result from the irradiation procedure. Moreover, recombination barriers and formation energies are usually poorly described by these potentials. Yet another difficulty with these potentials is the important fact that carbon atoms have many possible hybridizations. Therefore, even though crosslinks between tubes in bundles have already been predicted to appear after irradiation,25-27 their detailed structure and properties that will control how they affect the mechanical behavior of bundles have not yet been determined. From all that has been said above, it is clear that a fully quantum mechanical description of the electronic structure of the system is necessary, and that is precisely what we describe below. All our results are based on ab initio total energy density functional theory28,29 calculations. We have used ultrasoft pseudopotentials,30 a plane wave expansion up to 230 eV, and the local density approximation for the exchangecorrelation potential as implemented in the VASP code.31,32 A defect-in-supercell approach was employed, similar to what has been used in the investigation of defects in graphite.33,34 We have in our supercell two (5,5) nanotubes, with 100 atoms each, with lattice primitive vectors appropri-
Figure 1. Final equilibrium geometry for a bundle of nanotubes. The left figure shows both the hexagonal order and the unit cell used in the present work, which consisted of the two tubes marked in black. The top right panel shows a detail of the “line-of-contact” between the tubes in the defect free configuration, corresponding to the region marked by the rectangle in the left panel, as seen from the inside of the top tube. The orange (gray) atoms are in the top (bottom) tube. The atom that will be displaced forming a link between the two tubes is marked in red, whereas its three nearest neighbors are marked in blue. Upon displacement, the red atom will form bonds to the two atoms in the top tube marked green. The lower right panel shows the same atoms after the formation of the defect.
ate for a hexagonal arrangement of tubes in the bundle, as shown in Figure 1. The Brillouin zone sampling had a total of six points, with three of them along the tube axis direction. In all calculations the positions of the unconstrained atoms in the supercell were relaxed until the forces in them were smaller than 0.02 eV/Å. We obtain 1.42 Å for the C-C bonds perpendicular to the tube axis, and 1.41 Å for the others, which gives for the cell length the value of 12.21 Å. For the intertube distance in the bundle we obtain approximately a center-to-center distance of 9.87 Å. From a variety of tests we estimate that the errors in energy differences due to all our approximations are of the order of 0.05 eV. The two tubes in the unit cell are arranged in such a way as if they were obtained by rolling up two neighboring graphene sheets in a graphite structure; in other words, at the “line-of-contact” between them the atomic arrangement resembles that of graphite (see the top right panel in Figure 1). Since the intertube interaction is weak, they have a lot of rotational freedom within the bundle at room temperature,35 which does not make this choice restrictive. However, the curvature will play an important role in the stability of the defect, as discussed below. The structures of a variety of defects in graphite have been recently studied.33,34 In particular, an intimate Frenkel pair defect has been suggested to be responsible for the undesirable energy release in graphite moderators in nuclear reactors, 1046
Figure 2. Final equilibrium geometry for the configuration with the Wigner defect. The interstitial atom (red) can be clearly seen connecting two tubes in the left panel. The top right panel shows a detailed view of the Wigner defect, showing how the interstitial atom ejected from the bottom tube is connected to two atoms (green) in the top tube. It remains connected to the bottom tube via a bond to one of its former nearest neighbors (blue). The bottom right panel shows a detailed view of the defect from a different perspective, where the pentagon formed after the interstitial is ejected can be seen (atoms 7 through 11). The distances between the labeled atoms are presented in Table 1.
the Wigner energy. The procedure we employed in the attempt to produce a Frenkel pair similar to what was found in graphite33,34 was the following: we moved an atom in one of the tubes from the “line-of-contact” and placed it approximately between the two tubes, in the middle of a close by C-C bond in the other tube. The C atoms in the farthest away planes (in both tubes, 20 atoms total) from the defect were held fixed throughout the atomic relaxations. This was done in order to simulate an isolated defect, where the atoms far from the perturbation would not move. During the first few relaxation steps the displaced atom was also kept fixed, otherwise it would tend to return to its original position. All the attempts to create a Frenkel pair defect by displacing one atom from the top tube of Figure 1 (one of the orange atoms) were unsuccessful. Due to the curvature of the tube, the atom always returned to its original position. Eventually, tube rotations may be able to stabilize such configurations, but this may depend on the temperature and the time scales for rotation as compared to the defect relaxation time. On the other hand, when we displaced an atom from the bottom tube of Figure 1 (see right panel at the bottom), a stable Frenkel pair was obtained since the “line-of-contact” and curvature were both favorable. As shown in Figure 1, and in greater detail in Figure 2, the displaced atom (marked in red and labeled 3 in Figure 2) forms two bonds with C atoms of the top tube (marked in green, and labeled 1 and 2 in Figure 2). The bond length between atoms C1 and C2 increases from its equilibrium value of 1.42 Å to 2.16 Å, which clearly indicates that it is broken. This can also be seen in Figure 3 via the charge density plots. Nano Lett., Vol. 5, No. 6, 2005
Figure 3. Charge density plots, in electrons/Å3. The top left panel shows the charge density in a plane that is perpendicular to the tube axis and is farther away from the Wigner defect. Five C-C bonds in each tube can be clearly seen. The top right panel shows a similar plot in a plane that passes through the atoms 1-2-3-4 shown in Figure 2. It can be clearly seen that the C-C bond that existed between atoms 1 and 2 in the upper tube is broken and two new bonds are formed with the interstitial atom. The charge density between the C atoms indicates that the bonds between atoms 2-3 and 3-4 are rather strong, which will have a dramatic influence in the mechanical properties of the nanotubes bundle. The bottom left panel shows the charge density in a plane that approximately passes through the five atoms marked from 7 to 11 in Figure 2. A similar plot is shown in the right bottom panel for the corresponding structure in graphite. It can be seen that the bond between atoms 7 and 8 is much stronger in the case of the nanotubes.
On the other hand, the distances C1-C3 and C2-C3 become 1.50 and 1.42 Å, respectively, clearly showing the formation of two C-C bonds. Curvature effects and the constraints being imposed lead to different bond lengths. The displaced atom remains bonded to its original tube through one of its original neighbors, the atom labeled 4 in Figure 2. The distance C3-C4 decreases from 1.41 to 1.36 Å upon the formation of the defect. This is suggestive of an sp2 hybridization for these two carbon atoms with the formation of a double bond between them, since the C-C distance in the H2CdCH2 molecule is 1.34 Å. These results indicate that a strong bond is being formed between the two tubes, as can be seen from the charge density plots in the two top panels of Figure 3. The atom C4 remains linked to its two other neighbors via the bonds C4-C5 and C4-C6, with bond lengths of 1.42 and 1.50 Å, respectively. As a reference for comparison, we have also studied the Wigner defect in graphite.33,34 We have used a similar approximation as described above; however, the supercell now had 100 C Nano Lett., Vol. 5, No. 6, 2005
atoms, 50 in each of the graphite planes, and 100 k-points were used for the Brillouin zone sampling. The equilibrium C-C bond length was 1.41 Å and the interplanar distance was 3.30 Å. Upon the formation of the Frenkel pair, we obtained for the equivalent distances C1-C2 ) 1.57 Å, C1C3 ) 1.45 Å, C2-C3 ) 1.44 Å, C3-C4 ) 1.32 Å, C4-C5 ) 1.44 Å, and C4-C6 ) 1.44 Å. All these distances are very similar to what has been obtained by other researchers34 and indicate that the links caused by the defects in the nanotube bundles will most likely be stronger than in graphite. An important aspect of this defect in nanotube bundles when compared to graphite is the atomic relaxation around the defect. When the C atom is displaced, it remains bound to one of its three former nearest neighbors (the atoms marked in blue in Figure 1 and Figure 2), as discussed above. The other two C atoms are now in an energetically unfavorable configuration and will tend to approach each other in an attempt to make a chemical bond. This will lead to the formation of a pentagon, which can be clearly seen in Figure 1 and in Figure 2, where the five C atoms are labeled from 7 through 11. In a nanotube, however, due to its curvature allied to a certain degree of flexibility toward compression, these two C atoms can get relatively close to each other, with a final interatomic distance of C7-C8 ) 1.57 Å. This should be contrasted to the similar distance in graphite, C7C8 ) 2.00 Å. Therefore, in the case of nanotubes these two carbon atoms can rebond much more effectively, which should have an important contribution to the stability of the defect. In the two bottom panels of Figure 3 we present charge density plots in planes that pass through the atoms that form this pentagon, for both the nanotube and graphite. It can be clearly seen that there is indeed a much stronger rebonding in the nanotubes when compared to graphite. It should be noted that, due to the curvature of the nanotube, it is very difficult to have a plane passing exactly through all five C atoms (from 7 through 11). That is the reason atom 9 (see Figure 2) appears less clear in the left lower panel of Figure 3. All the relevant interatomic distances are summarized in Table 1. We calculate the defect formation energy as the difference between the total energy for the system with and without the defect, which gives a value of 7.27 eV. However, if we now also allow all the atoms in the bundle to relax, i.e., the 20 C atoms that were held fixed can now relax, we obtain a formation energy of 5.53 eV. This is an extremely low value, showing that this defect may be one of the most important defects in nanotube bundles. The full relaxation allows the tubes to orient themselves in such a way that their interlink bonds are stronger, as indicated by the bond lengths shown in Table 1. Therefore, if the tubes have enough mobility to accommodate the defect, or if the defect concentration is high enough, we expect a formation energy close to 5.5 eV. However, if the defect is isolated and the tubes cannot freely move, the formation energy may be closer to 7.3 eV. To have a feeling about these values, we also calculated the formation energy of two other defects: (i) the Wigner defect in graphite and (ii) a vacancy in an isolated (5,5) 1047
Table 1. Interatomic Distances, in Å, for the Atoms Labeled from 1 to 11 in Figure 2a bond
non-fully relaxed
fully relaxed
graphite
1-2 1-3 2-3 3-4 4-5 4-6 7-8 8-9 9-10 10-11 7-11
2.16 1.50 1.41 1.36 1.42 1.50 1.57 1.41 1.41 1.40 1.42
2.21 1.45 1.43 1.33 1.44 1.45 1.54 1.42 1.40 1.40 1.43
1.57 1.45 1.44 1.32 1.44 1.44 2.00 1.39 1.40 1.40 1.39
a Results are presented for the nanotubes bundle with some of the atoms fixed, as discussed in the text (non-fully relaxed results), and for a situation where all atoms are allowed to relax (fully relaxed results). For graphite all atoms were allowed to relax, and the numbers refer to atoms in similar configurations.
Table 2. Calculated Formation Energies (Ef) for the Defects Considered in This Worka
Ef (eV)
non-fully relaxed
fully relaxed
graphite
vacancy
7.27
5.53
10.67
5.82
a
Results are presented for the Wigner defect in nanotubes bundle with some of the atoms fixed, as discussed in the text (non-fully relaxed), and for a situation where all atoms are allowed to relax (fully relaxed). For the Wigner defect in graphite all atoms were allowed to relax. For the vacancy, an isolated tube was considered, and all atoms were allowed to relax.
nanotube. For the Wigner defect in graphite we obtain a formation energy of 10.67 eV, similarly to other authors.33,34 This means that it is much easier to produce these defects in nanotube bundles than in graphite, indicating that it may be very common in irradiated samples. If we consider a vacancy in an isolated nanotube, we obtain for a fully relaxed structure a formation energy of 5.82 eV. All of these formation energy values are summarized in Table 2. As can be seen, the formation energy for the Wigner defects can be smaller than the formation energy of a vacancy, once more indicating its importance. The strong link that is being formed between the tubes may have a dramatic change in the mechanical properties of nanotube bundles. To estimate the changes in the shear modulus, we perform calculations for two isolated tubes, instead of a full bundle. This should give the correct trend, since the tube-tube interaction is much stronger in the presence of the Wigner defect. We perform relative displacements as illustrated in Figure 4. For the displacements perpendicular to an imaginary line that joins the tubes axes we obtain the dashed curves in Figure 4, whereas for displacements making a 30° angle with respect to the previous displacement direction we have the solid curves in Figure 4. This latter direction would be more representative of a shear displacement of tubes in a bundle. For the dashed curves we obtain36 shear moduli of 3 and 59 GPa for the pure and defective tubes, respectively, whereas for the solid curves we obtain 7 and 97 GPa, respectively. This indicates 1048
Figure 4. Change in energy (per unit length along the nanotubes axes) caused by the relative displacement (perpendicular to the nanotubes axes) of two isolated tubes (not in a bundle), showing the large increase in shear modulus caused by the Wigner defects intertube links.
an increase in the shear modulus by a factor of 20 in the former case and by a factor of 14 in the latter case. Therefore, we show that this defect may be the main reason behind the observed increase in shear modulus of nanotube bundles upon irradiation.7 In the work by Kis and collaborators7 it has been proposed that carbon interstitials or radiation-induced chemical reactions involving, for example, carboxyl groups could form links between the nanotubes in a bundle. These changes would be responsible for the increase in shear modulus; however, no estimates for the formation energies of these structures are provided. We here propose a similar idea but with an alternative local atomic structure, which due to its low formation energy is very likely to be created during the irradiation. Finally, an important question is related to the stability of such a defect. In graphite, it has been proposed1 that this defect is responsible for the Wigner energy release peak at 200 °C, associated with an experimental barrier37 of approximately 1.4 ( 0.4 eV, which agrees nicely with the calculated barrier33,34 of 1.3 eV for the defect recombination. To estimate the barrier, we linearly interpolate the coordinates of the displaced atom between its position in the pure bundle (Figure 1) and in the final defect configuration (Figure 2). We then hold this atom fixed and let all the other atoms relax, except the 20 constrained atoms as discussed above. Ten intermediate configurations were calculated, and the results are plotted in Figure 5. We obtain a recombination barrier of approximately 2.4 eV, indicating that the defect is significantly more stable in nanotube bundles than in graphite. If we assume that the recombination of this defect would also be associated with an energy release, like in graphite, and that to observe such an energy release it would require a recombination rate of the same order as that in graphite, we estimate that in nanotube bundles such a release should be observed around 600 °C. In summary, we have described the structure of a very stable Frenkel pair in nanotube bundles, which provides strong links between nanotubes, thus altering in a dramatic Nano Lett., Vol. 5, No. 6, 2005
Acknowledgment. We acknowledge support from the Brazilian Agencies FAPESP and CNPq. References
Figure 5. Energy as a function of displacement of the interstitial atom from its ideal configuration (zero displacement). The barrier for the Frenkel pair recombination is of the order of 2.4 eV, indicating that this defect will be rather stable.
way the mechanical properties of these bundles. Since these defects have rather low formation energies, they should be prevalent in irradiated samples. They should play an important role in nanotube engineering using irradiation sources, a field that has been gaining a lot of attention lately. In addition to forming the strong links, as already mentioned, the defects could be spots where welding of two tubes will start. Moreover, since they provide a somewhat open local structure in one of the tubes (a small “hole” in the tube wall), they may allow the entrance of atoms or small molecules inside the tube, which may be important for storage purposes.38 Even though we focused on a (5,5) nanotube, we expect that similar results may be found in other types of nanotubes. This expectation is based on two main facts: (i) nanotubes can easily rotate around their axes when inside bundles,35 which will eventually lead to an appropriate alignment between two tubes to form a Wigner defect, and (ii) this alignment does not need to happen at a full line of contact, as in the present case. Even for chiral tubes, rotation will likely produce, as time goes by, regions along the tubes where a favorable orientation between the atoms can result in the formation of a Wigner defect. As the defect is formed, the same rotational freedom will probably allow the tubes to attain the most favorable configuration around the disturbance, which will eventually lead to values for the bond distances C1-C2 and C7-C8 close to what we have obtained here. The bond distance C7-C8, in particular, can become small mostly because of the radial flexibility of the nanotubes, a feature that is also present in chiral tubes. Further studies, however, are necessary in order to confirm these expectations. Finally, it has not escaped our attention that a similar structure will probably exist in multiwalled nanotubes linking, for example, two neighboring walls.39 Some of the ideas discussed here were already foreseen in the study of defects in graphite;33 however, the structure of nanotubes is such that the defect has a combined high stability and low formation energy, which could not be expected from graphite studies. Nano Lett., Vol. 5, No. 6, 2005
(1) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (2) Bethune, D. S.; Klang, C. H.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605. (3) See, for example, Dresselhaus, M. S.; Dresselhaus, G.; Avouris, Ph.; Carbon Nanotubes: Synthesis, Structure, Properties and Applications; Topics in Applied Physics Series; Springer: Heidelberg, 2001. (4) Salvetat, J.-P.; Briggs, G. A. D.; Bonard, J.-M.; Bacsa, R. R.; Kulik, A. J.; Sto¨ckli, T.; Burnham, N. A.; Forro´, L. Phys. ReV. Lett. 1999, 82, 944. (5) Yu, M.-F.; Bradley, S. F.; Arepalli, S.; Ruoff, R. S. Phys. ReV. Lett. 2000, 84, 5552. (6) Ajayan, P. M.; Banhart, F. Nature Mater. 2004, 3, 135. (7) Kis, A.; Csa´nyi, G.; Salvetat, J.-P.; Thien-Nga, L.; Couteau, E.; Kulik, A. J.; Benoit, W.; Brugger, J.; Forro´, L. Nature Mater. 2004, 3, 153. (8) Ajayan, P. M.; Ravikumar, V.; Charlier, J.-C. Phys. ReV. Lett. 1998, 81, 1437. (9) Banhart, F. Rep. Prog. Phys. 1999, 62, 1181. (10) Terrones, M.; Terrones, H.; Banhart, F.; Charlier, J.-C.; Ajayan, P. M. Science 2000, 288, 1226. (11) Ni, B.; Andrews, R.; Jacques, D.; Qian, D.; Wijesundara, M. B. J.; Choi, Y.; Hanley, L.; Sinnott, S. B. J. Phys. Chem. B 2001, 105, 12719. (12) Terrones, M.; Banhart, F.; Grobert, N.; Charlier, J.-C.; Terrones, H.; Ajayan, P. M. Phys. ReV. Lett. 2002, 89, 0755051. (13) Smith, B. W.; Luzzi, D. E. J. Appl. Phys. 2001, 90, 3509. (14) Basiuk, V. A.; Kobayashi, K.; Kaneko, T.; Nagishi, Y.; Basiuk, E. V.; Saniger-Blesa, J.-M. Nano Lett. 2002, 2, 789. (15) Wei, B. Q.; D’Arcy-Gall, J.; Ajayan, P. M.; Ramanath, G. Appl. Phys. Lett. 2003, 83, 3581. (16) Charlier, J.-C.; Terrones, M.; Banhart, F.; Grobert, N.; Terrones, H.; Ajayan, P. M. IEEE Trans. Nanotech. 2003, 2, 349. (17) Khare, B.; Meyyappan, M.; Moore, M. H.; Wilhite, P.; Imanaka, H.; Chen, B. Nano Lett. 2003, 3, 643. (18) Miko´, Cs.; Milas, M.; Seo, J. W.; Couteau, E.; Bari×f0iæ, N.; Gaa´l, R.; Forro´, L. Appl. Phys. Lett. 2003, 83, 4622. (19) Raghuveer, M. S.; Ganesan, P. G.; D’Arcy-Gall, J.; Ramanath, G.; Marshall, M.; Petrov, I. Appl. Phys. Lett. 2004, 84, 4484. (20) Li, J.; Banhart, F. Nano Lett. 2004, 4, 1143. (21) Yoon, M.; Han, S. W.; Kim, G.; Lee, S. B.; Berber, S.; Osawa, E.; Ihm, J.; Terrones, M.; Banhart, F.; Charlier, J.-C.; Grobert, N.; Terrones, H.; Ajayan, P. M.; Tomanek D. Phys. ReV. Lett. 2004, 92, 0755041. (22) Hashimoto, A.; Suenaga, K.; Gloter, A.; Urita, K.; Iijima, S. Nature 2004, 430, 870. (23) Krasheninnikov, A. V.; Nordlund, K.; Sirvio¨, M.; Salonen, E.; Keinonen, J. Phys. ReV. B 2001, 63, 2454051. (24) Krasheninnikov A. V.; Nordlund, K. J. Vac. Sci. Technol. B 2002, 20, 728. (25) Krasheninnikov, A. V.; Nordlund, K.; Keinonen, J. Phys. ReV. B 2002, 65, 1654231. (26) Salonen, E.; Krasheninnikov, A. V.; Nordlund, K. Nucl. Instrum. Methods B 2002, 193, 603. (27) Krasheninnikov, A. V.; Nordlund, K. Nucl. Instrum. Methods B 2004, 216, 355. (28) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (29) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (30) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (31) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. (32) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (33) Telling, R. H.; Ewels, C. P.; El-Barbary, A. A.; Heggie, M. I. Nature Mater. 2003, 2, 333. (34) Ewels, C. P.; Telling, R. H.; El-Barbary, A. A.; Heggie, M. I.; Briddon, P. R. Phys. ReV. Lett. 2003, 91, 0255051-0255054. (35) Kwon, Y.-K.; Toma´nek, D. Phys. ReV. Lett. 2000, 84, 1483. (36) The shear moduli are obtained as the second derivatives of the curves shown in Figure 4 with respect to the displacement. (37) Mitchell, E. W. J.; Taylor M. R. Nature 1965, 208, 638. (38) Meunier, V.; Kephart, J.; Roland, C.; Bernholc, J. Phys. ReV. Lett. 2002, 88, 0755061. (39) Huhtala, M.; Krasheninnikov, A. V.; Aittoniemi, J.; Stuart, S. J.; Nordlund, K.; Kaski, K. Phys. ReV. B 2004, 70, 0454041.
NL050457C 1049
