VOLUME 3, NUMBER 10, OCTOBER 2003 © Copyright 2003 by the American Chemical Society
Ab Initio Modeling of Diamond Nanowire Structures A. S. Barnard,† S. P. Russo,* and I. K. Snook‡ Department of Applied Physics, Royal Melbourne Institute of Technology UniVersity, GPO Box 2476V, Melbourne, 3001 Australia Received March 19, 2003; Revised Manuscript Received April 29, 2003
ABSTRACT Presented are results of our ab initio study of the structural relaxation of diamond nanowires, with dodecahedral and cubododecahedral morphology, showing that the energetic and structural stability of a diamond nanowire is dependent on both the surface morphology and the crystallographic direction of the principal axis. Although all structures under consideration exhibited significant changes in the length and cross-sectional area, nanowires with the principal axis in the [110] direction appear less structurally favorable.
The emergence of molecular nanotechnology has introduced a wide range of potential applications of nanostructured materials for a variety of purposes. One-dimensional (1-D) nanowires have been proposed as important components, playing an integral part in the design and construction of both electronic and optoelectronic nanodevices.1 Significant work has been compiled regarding the structure and properties of semiconductor nanowires including silicon,2,3 silicon carbide,4,5 and carbon.6 The growth of carbon nanowires has been achieved using a number of techniques including laserinduced chemical vapor deposition,7 high-pressure treatment of catalyst-containing thin films,8 annealing of silicon carbide films,9 and annealing of pressed tablets containing graphite.10 Aligned diamond nanowhiskers (a term used to describe particular nanowires) have been formed using air plasma etching of polycrystalline diamond films.11 Dry etching of * To whom correspondence should be addressed. E-mail: salvy.russo@ rmit.edu.au. † E-mail:
[email protected]. ‡ E-mail:
[email protected]. 10.1021/nl034169x CCC: $25.00 Published on Web 07/15/2003
© 2003 American Chemical Society
the diamond films with molybdenum deposits created wellaligned uniformly dispersed nanowhiskers up to 60 nm in diameter with a density of 50/µm2. These diamond nanowhiskers showed well-defined characteristics of diamond.11 Diamond nanocylinders with a diameter of approximately 300 nm have been reported.12 Diamond-based materials have been suggested to be the optimal choice for nanomechanical designs because of their high elastic modulus and strength-to-weight ratio.13,14 This has prompted a number of theoretical studies investigating various aspect of diamond on the nanoscale. Results of these investigations have shown that dehydrogenated C(111) octahedral nanodiamond surfaces are structurally unstable, with their presence inducing phase transitions from the sp3 structure of nanodiamonds to the sp2 structure of carbon onions. However, the presence of cubic surface facets has been found to promote stability.15 For example, whereas cuboctahedral nanodiamond structures have exhibited preferential exfoliation of C(111) surfaces over lower-index surfaces, increasing the C(100) surface area produces a more
stable nanodiamond structure and reduced surface graphitization.16 Attention is now turning to 1-D diamond nanostructures. In a recent paper by Shenderova et al.,17 the stiffness and fracture force of hydrogenated diamond nanorods have been compared with those of single-walled and multiwalled carbon nanotubes. It was determined that the mechanical properties of the nanorods depend on both the diameter of the nanorod and the orientation of the principal axis. The results of their molecular models indicate that diamond nanorods are energetically competitive with nanotubes of a similar diameter and possess desirable mechanical properties, making them a viable target for synthesis. Presented in this letter is an ab initio density functional theory (DFT) study of the structural properties of dehydrogenated diamond nanowires using the Vienna ab initio simulation package (VASP).19 All calculations have been performed in the framework of DFT within the generalized gradient approximation (GGA) with the exchange-correlation functional of Perdew and Wang.21 We used ultrasoft, gradient-corrected Vanderbilt-type pseudopotentials20 and expanded the valence orbitals on a plane wave basis up to a kinetic energy cutoff of 290.00 eV. Preliminary testing determined that a 4 × 4 × 4 Monkhorst-Pack k-point mesh was sufficient in this case and that no advantage could be gained by using a larger k-mesh or plane wave cutoff. Application of the linear tetrahedron method to brillouinzone integration meant that the use of fewer k-points (even in nonperiodic directions) was not recommended. This method has been successfully applied to bulk diamond,22 nanodiamond,23 and fullerenes24 and has been shown to give results that are in excellent agreement with those from experiment and all electron methods. The relaxation technique used here is an efficient matrixdiagonalization routine based on a sequential band-by-band residual minimization method of single-electron energies25,26 with direct inversion in the iterative subspace. Both the ionic positions and super-cell volume have been relaxed. Therefore, because both the symmetry and the lattice parameter are free to change, expansions or contractions over the entire length of the nanowires may occur. This also means that any expansions and contractions may occur independently and that changes in length (for example) are not coupled to changes in width. A detailed description of this technique may be found in ref 27. As previously mentioned, it has been found that nanodiamond C(111) surfaces are unstable and that octahedral nanodiamonds transform into carbon onion. Therefore, three dehydrogenated nanowire morphologies have been considered here and are characterized by pure dodecahedral forms and combinations of cubododecahedral forms. This choice of morphologies succeeds in avoiding the problematic octahedral surfaces while still offering a range of cross sections and surface structures. To assist in clarity, one cubododecahedral group is denoted as “cylindrical” because of its circular cross section, and the other, as “cubic” because of its square or rectangular cross section. Periodic boundary conditions (PBC) have been applied in the x direction, and 1324
Table 1. Details of the Initial 1D Diamond Nanowire Structuresa morphology
atoms
PBC length (nm)
cross section (nm2)
diameter (nm)
dodecahedral dodecahedral dodecahedral cubic cubic cubic cylindrical cylindrical cylindrical
75 144 196 84 132 240 63 128 228
1.0707 1.4275 1.4275 1.5134 1.5134 1.5134 1.0707 1.4275 1.4275
0.254 0.398 0.572 0.202 0.360 0.720 0.223 0.386 0.763
0.505 0.631 0.757 0.451 0.560 0.848 0.472 0.621 0.873
a The diameter refers to the average lateral diameter perpendicular to the nanowire’s principal axis.
sufficient vacuum space has been added in the y and z directions to create infinite 1-D structures. Details of the initial structures are given in Table 1. As indicated in Table 1, three nanowires of each morphology have been considered, with increasing average lateral diameter. The dodecahedral structures are bounded by (110) surfaces in all lateral directions, with a square cross section, and have a principal axis in the [100] direction. The cubic diamond nanowires are bounded by two C(100) surfaces and two C(110) surfaces in the lateral directions, with a rectangular cross section, and have a principal axis in the [110] direction. Finally, the three cylindrical nanowires considered here are bounded by two C(100) surfaces and two C(110) surfaces in the lateral directions, with a circular cross section, and have a principal axis in the [100] direction. The initial structures have been “cleaved” from a bulk diamond lattice, with both C(110) (1 × 1) single dangling bond and C(100) (1×1) double dangling bond surface structures. The results of the structural relaxations of the dodecahedral diamond nanowires are given in Figure 1 (1a-3f). For each nanowire, the initial (top) and final (bottom) configurations are shown from the [100] (right), [110] (center), and principal axis (right) directions. An inspection of Figure 1 (1a-1f) reveals that a reasonable degree of structural relaxation has occurred in the C75 dodecahedral nanowire. The outermost atomic layer has contracted inward, although no surface reconstruction or buckling has occurred. This is not unusual because the diamond C(110) (1 × 1) surface has been found to have no significant reconstructions.28 The outer-layer contraction is, however, more pronounced at the nanowire edges, causing the outer surface of the nanowire to exhibit a slightly convex shape. Similarly, a reasonable degree of structural relaxation occurred in the larger C144 and C196 dodecahedral nanowires shown in Figure 1 (2a-2f, 3a-3f, respectively). Once again, the outer layers contracted, but in both cases, the contraction was less pronounced at the edges. This has resulted in the C(110) (1 × 1) surfaces exhibiting a slightly concave shape with vaguely acute edges. Each of the cubic diamond nanowires underwent a twostage relaxation. In all cases, the initial step in the relaxation involved the reconstruction of the C(100) surfaces to form the C(100) (2 × 1) surface structure. This surface reconstrucNano Lett., Vol. 3, No. 10, 2003
Figure 1. C75 (1), C144 (2), and C196 (3) dodecahedral diamond nanowires showing the initial cleaved structures (from the [100] (a), [110] (b), and principal axis (c) directions) and final relaxed structures (from the [100] (d), [110] (e), and principal axis (f) directions).
Figure 2. C84 (1), C132 (2), and C240 (3) cubic diamond nanowires showing the initial cleaved structures (from the [100] (a), [110] (b), and principal axis (c) directions) and final relaxed structures (from the [100] (d), [110] (e), and principal axis (f) directions).
tion was followed by further relaxation of the entire nanowire, including surface-layer relaxation. Figure 2 (1a-1f) shows the results of the relaxation of the C84 nanowire. Following the reconstruction of the C(100) surfaces, this nanowire was transformed into a nonclassical single-walled nanotube structure. This transition also involved the dissociation of six atoms, as three complete dicarbon molecules, as indicated in Figure 3. The final structure is described as a nonclassical nanotube because of its elliptical shape and the inclusion of the eight-membered and five-membered rings, which are not present in classical nanotubes. This nonclassical ring structure produced some unusual bonding conditions. The majority of the bonds in the structure range in length from 1.40 to 1.58Å; however, in one row of five-membered rings bordering the eight-membered rings, the C-C bond length (of the bonds bordering the five-membered and eight-membered rings) contracted to only ∼1.25 Å. The length of these bonds suggests that they are CdC. This assumption is supported by the electron charge density profile, which shows a large buildup of charge
compared with that of the remaining bonds (Figure 4). The presence of CdC in the structure assists in explaining the metastability of the elliptical shape because the breaking of such bonds represents a considerable potential barrier. The relaxed structures of the C132 and C240 cubic diamond nanowires are shown in Figure 2 (2a-2f and 3a-3f, respectively). As with the C84 cubic nanowire, the first step in the relaxation of each of the larger nanowires involved the reconstruction of the C(100) surfaces to the (2 × 1) surface structure. This was followed by a moderate change in the outer-layer separations. The final relaxed C132 nanowire shows the adoption of a more rounded cross section, and the largest adopted C240 nanowire, a slightly tapered cross section. Overall, a bulk-diamond-like structure was preserved, with C(100) (2 × 1) dimers comparable to bulk-diamond and nanodiamond surfaces.15 Finally, the three cylindrical nanowires have been relaxed. The results of these relaxations are given in Figure 5. In a fashion similar to that of the cubic nanowires, the initial step in the relaxation involved the reconstruction of the C(100)
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Figure 3. C84 cubic diamond nanowire during the relaxation process, showing the expulsion of the outer-layer atoms as C2 dimers.
Figure 5. C63 (1), C128 (2), and C228 (3) cylindrical diamond nanowires showing the initial cleaved structures (from the [100] (a), [110] (b), and principal axis (c) directions) and final relaxed structures (from the [100] (d), [110] (e), and principal axis (f) directions). Figure 4. Electron charge density in the plane of CdC between the eight-membered and five-membered rings along the relaxed C84 cubic diamond nanowire.
surfaces to form the C(100) (2 × 1) surface structure, followed by further relaxation of the entire nanowire. However, each of the final relaxed cylindrical nanowires was found to exhibit a twisted ropelike structure, resulting from the C(100) (2 × 1) surface reconstruction and a significant contraction of the outermost atomic layer. This effect is most pronounced in the C63 cylindrical nanowire where, although a semitubular structure resulted, the nanowire retained the diamond-like structure. The C128 cylindrical nanowire (Figure 5 (2a-2f)) and the largest C228 cylindrical nanowire (Figure 5 (3a-3f)) also exhibited the ropelike structure, which is best appreciated by viewing the electronic charge density in a C(100) plane through the core of the C128 nanowire, as shown in Figure 6. Therefore, with the exception of the smallest cubic nanowire, all of the remaining cubic and dodecahedral 1326
nanowires retained the diamond structure upon relaxation. Although no link between the presence of a C(110) surface and the transformation of nanodiamond particles into carbon onions has been established, the six-membered ring structure of this surface is very similar to the structure of carbon nanotubes. For example, by viewing the diamond nanowires from the [110] direction, the nanowires appear to be very similar to zigzag nanotubes. If the C(110) surface were unstable at the nanoscale, then delamination of the outer surface of the dodecahedral nanowires would have been observed. Because this was not the case, it must be concluded that like diamond nanocrystals23 the C(110) and C(100) surfaces are stable for nanowires with a lateral diameter of less that ∼1 nm. Aside from the retention of the diamond structure, the nanowires did exhibit significant relaxation involving changes in the length and cross-sectional area. Because the nanowires are infinite (due to the periodicity along the principal axis), the change in length (denoted as ∆L) may be considered only in terms of the extension (or contraction) of the length Nano Lett., Vol. 3, No. 10, 2003
Figure 6. Electron charge density in a central (100) place of the relaxed C128 cylindrical diamond nanowire showing the ropelike structure resulting from the relaxation. Table 2. Changes in Energy Per Atom (∆E), Cross-Sectional Area (∆A), and Nanowire Segment Lengths (∆L) Resulting from the Relaxation of Each Nanowire’s Morphologya morphology
atoms
∆E (eV)
∆L (nm)
∆A (nm2)
dodecahedral dodecahedral dodecahedral cubica cubic cubic cylindrical cylindrical cylindrical
75 144 196 84 132 240 63 128 228
-0.2271 -0.2150 -0.2057 -0.9812 -0.4847 -0.4339 -0.7063 -0.5687 -0.2676
+0.0883 +0.1051 +0.0722 -0.0222 -0.0094 -0.0038 +0.0199 +0.0182 +0.0017
-0.0428 -0.0578 -0.0586
a
-0.0265 -0.0615 -0.0336 -0.0421 -0.0448
Nonclassical nanotube.
within the simulation cell and was measured by determining the change in the length of the actual simulation cell itself. However, an expansion or contraction of the cross section (denoted as ∆A) has been measured directly by considering the positions of the ions. It must also be noted that all such changes are not induced by any “end effects” that may be present in finite structures. These results are given in Table 2, along with the change in energy per ion (denoted as ∆E) resulting from the relaxation. The structural changes ∆A and ∆L are also shown in terms of the percentage change in Figure 7. In theory, each plot in Figure 7 should converge to zero as the number of atoms increases (the macroscopic limit), but there are too few data points to deduce anything but the overall trend. The smallest cubic nanowire has been excluded from this comparison because of the conversion to a nonclassical nanotube. However, it is apparent from these Figures (and from Table 2) that the remaining cubic nanowires still exhibit unusual structural changes. The slopes for the cubic nanowires are positive, whereas the cylindrical and dodecahedral nanowire slopes are negative. This is thought to be a product of the cubic nanowire having a Nano Lett., Vol. 3, No. 10, 2003
Figure 7. Percentage change in cross-sectional area ∆A (top) and in periodic segment length ∆L (bottom) for the respective morphologies.
principal axis in the [110] direction, suggesting that this is not an optimal choice for diamond nanowire structures. In conclusion, it has been shown from the ab initio relaxation of diamond nanowires that nanocrystalline diamond may be structurally stable in one dimension. Diamond nanowires with dodecahedral and cubododecahedral morphology retained the diamond structure upon relaxation but did exhibit significant relaxation involving changes in the length and cross-sectional area. The stability, characterized by the variation in these structural properties from that of bulk diamond, has been found to be dependent on both the surface morphology and the crystallographic direction of the principal axis of the nanowire. For example, nanowires having a principal axis in the [110] direction do not represent an optimal choice for diamond nanowire structures. The effects of these structural changes on the electronic properties of these diamond nanowires are still under investigation. As a continuation of this work, octahedral and cuboctahedral diamond nanowire are also being considered. Because the presence of the C(111) surface facets in nanodiamonds promotes instability, it is considered to be important to determine the effects of these surfaces on the structure of diamond nanowires. Acknowledgment. We thank the Victorian Partnership for Advanced Computing and the Australian Partnership for Advanced Computing supercomputer center for their ongoing assistance over the course of this project. 1327
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