First-Principles Study of Faceted Single-Crystalline Silicon Carbide

Dec 30, 2008 - It is found that the formation energy (Eform) of the NWs decreases with the increase of wire radius, and that of the NTs decreases with...
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J. Phys. Chem. C 2009, 113, 856–861

First-Principles Study of Faceted Single-Crystalline Silicon Carbide Nanowires and Nanotubes Zhenhai Wang,† Mingwen Zhao,*,†,‡ Tao He,† Xuejuan Zhang,† Zexiao Xi,† Shishen Yan,†,‡ Xiangdong Liu,†,‡ and Yueyuan Xia†,‡ School of Physics, and State Key Laboratory of Crystal Materials, Shandong UniVersity, Jinan 250100, Shandong, China ReceiVed: September 16, 2008; ReVised Manuscript ReceiVed: NoVember 28, 2008

The energetics and atomic and electronic structures of silicon carbide (SiC) nanowires (NWs) and nanotubes (NTs) with radii ranging from 0.45 to 2.9 nm are investigated using density functional theory in conjunction with an atomistic band-order potential. It is found that the formation energy (Eform) of the NWs decreases with the increase of wire radius, and that of the NTs decreases with the increase of wall thickness, irrespective of the tube radius. NTs with faceted single-crystalline walls are energetically more favorable than the cylindrical single- or multiwalled SiC NTs. Due to the surface states, the faceted NWs and NTs possess indirect band gaps, which are narrower than that of bulk SiC crystal. The highest valence band and the lowest conduction band mainly arise from the undercoordinated C and Si atoms on the facets. The surface states can be passivated by surface hydrogenation, and the hydrogenated SiC NWs and NTs become direct-band gap semiconductors with wider band gaps than that of bulk SiC crystal. I. Introduction 1

Since the discovery of carbon nanotubes (CNTs) by Iijima, low-dimensional semiconducting nanomaterials, such as clusters, nanotubes, and nanowires, have attracted considerable attention because of their superior properties resulting from large surfaceto-volume ratios, quantum size-confinement effects, and accordingly, potential applications in various fields. Among these materials, silicon carbide (SiC) has the advantages of high thermal stability, chemical resistivity, excellent mechanical properties, high refractive index, and wide band gap, making it an attractive candidate for building electronic devices for highpower, high-temperature, and high-frequency applications. Onedimensional single-crystalline SiC nanostructures such as nanorods,2 nanobelts,3 nanowires (NWs),4 nanotubes (NTs),5 and nanocables6 have been fabricated via different approaches. SiC NWs and NTs are producible from the reaction between Si and multiwalled CNTs at different substrate temperatures.7 The asproduced SiC NTs can be characterized as rolling up SiC graphene-like sheets,8-10 similar to the formation of multiwalled BN NTs. However, these cylindrical SiC NTs are a metastable phase and likely transform to SiC NWs under certain conditions.7 The SiC NT arrays prepared by using sacrificial alumina membrane as a template have crystalline structures5 which are different from the layered multiwalled NTs.7 Therefore, theoretical models of the crystalline SiC NWs and NTs, rather than cylindrical layered NTs, are highly desirable. The stability, energetics, and electronic structures of singlewalled SiC nanotubes (SWSiCNTs) have been studied extensively using first-principles calculations.8-11 It was revealed that the formation energy (Eform) of SWSiCNTs is approximately proportional to the inverse square of the tube radius, following the classical elastic theory.10 The electronic structures of SWSiCNTs depend on the tube chiralities; e.g., zigzag tubes * Corresponding author. E-mail: [email protected]. † School of Physics, Shandong University. ‡ State Key Laboratory of Crystal Materials, Shandong University.

are direct-band gap semiconductors, whereas armchair and chiral tubes have indirect band gaps.8-10 The nanomechanical response properties of cubic (3C-) SiC NWs have been investigated using molecular dynamics simulations (MDS) with a Tersoff bond-order interatomic potential.12 However, to the best of our knowledge, the relevant Eform and the electronic structures of single-crystalline SiC NTs and NWs have not yet been reported. Theoretical investigation of these issues is important for better understanding the growth mechanisms of these nanomaterials to promote the studies of their syntheses and applications. In this contribution, we report our theoretical study of faceted single-crystalline SiC NWs and NTs with hexagonal cross sections. Both the naked and hydrogenated NWs and NTs were considered. Two different methods, first-principles calculations and classical MDS, were employed to deal with SiC NWs and NTs in different sizes. First-principles calculations using the standard Kohn-Sham self-consistent density functional theory (DFT) were performed to study the surface relaxation, formation energies, and electronic structures of the ultrathin SiC NWs and NTs (0.45 nm < R < 1.1 nm). The Eform of the NWs and NTs as a function of wire radius (R) or wall-thickness (d) obtained from the DFT calculations was then compared with that obtained from the MDS based on the Tersoff potential for the NWs and NTs with a wider range of size (0.45 nm < R < 2.9 nm). Cylindrical layered NTs were also calculated for the purpose of comparison. II. Theoretical Methods and Computational Details In the present DFT calculations, a flexible linear combination of numerical atomic orbital basis sets of double-ζ quality with inclusion of polarization functions (DZP) was adopted for the description of valence electrons, and nonlocal pseudopotentials were employed for the atomic cores. Generalized gradient approximation (GGA) in the form of Perdew, Burke, and Ernzerhof (PBE) was used for the exchange-correlation functional.13 The basis functions were strictly localized within radii

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TABLE 1: Lattice Constants and Bulk Modulus (K) of 3Cand 6H-SiC Crystals Obtained from Different Theoretical Methods, Compared with Experimental Results 3C-SiC

a (Å) c (Å) K (GPa) a

6H-SiC

TP

DFT

exptl

TP

DFT

exptl

4.32 241

4.37 210

4.35 230 ( 4a

3.06 14.97 -

3.09 15.23 -

3.08 15.12 -

Reference 26.

that corresponded to a confinement energy of 0.01 Ry, with the exception of the polarization functions where a fixed radius of 6.0 bohr was specified. An auxiliary basis set with a realspace grid was adopted to expand the electron density for numerical integration. A kinetic energy cutoff of 200 Ry was employed to control the fineness of this mesh. For NWs and NTs, a one-dimensional (1D) periodic boundary condition was applied along the axial direction and a sufficient vacuum space (up to 20 Å) was specified along the radial direction to avoid mirror interactions. The Brillouin zone sampling was carried out by using k-point grids of 1 × 1 × 8 according to the Monkhorst-Pack scheme.14 All of the atomic positions along with the lattice vectors were optimized by using a conjugate gradient (CG) algorithm, until each component of the stress tensors was reduced below 0.02 GPa and the maximum atomic forces was less than 0.01 eV/Å. All of the DFT calculations were performed using the SIESTA code.15-17 The MDS using the Tersoff bond-order interatomic potential (TP)18,19 is more efficient than the DFT calculation. The validity of TP in reproducing the structural and mechanical properties of SiC materials has been proven by several works.20-23 It is suitable for the simulation of large sized SiC nanostructures, which are difficult for DFT calculations. The TP parameters adopted in this work were presented by R. Devanathan et al.,24 and the calculations were performed using the general utility lattice program (GULP) code.25 To check the validity of the above schemes, we calculated the equilibrium lattice constants and the bulk modulus of bulk SiC. As known, natural SiC crystals have more than 200 polymorphous structures. Among them, β- or 3C-SiC is cubic and some of others, such as 2H, 4H, and 6H-SiC, are hexagonal, which are referred to as R-SiC. The optimized lattice constants for 3C- and 6H-SiC crystals obtained from the present DFT and TP calculations are listed in Table 1. Obviously, both the DFT and the TP calculations reproduce the experimental equilibrium lattice constants very well (the maximum difference is less than 1%). The bulk modulus values of 3C-SiC obtained by using the two schemes are 210 GPa (DFT) and 241 GPa (TP), close to the experimental data, 230 ( 4 GPa.26 The present DFT calculation underestimates the bulk modulus of 3C-SiC crystal by about 10%, whereas TP calculation predicts a slightly larger bulk modulus than the experimental result. III. Results and Discussion Among the bulk SiC polymorphs under study (3C, 6H, 4H, and 2H), 2H-SiC crystal is the most unstable form, while 3C-SiC is the most stable one at low temperature. SiC NWs, however, display a different energetic order due to the surface effect. The theoretical work of Ito et al. showed that the hexagonal faceted NWs stabilized a 2H structure when the wire diameter was smaller than 20 nm whereas the NWs in 3C form became energetically more favorable when the diameter was

Figure 1. (color online) Different views of single-crystalline SiC NWs and NTs with hexagonal faceted morphology: (a) a sketch representation of faceted NTs; (b) top view of 2NT; the orange balls represent C atoms, and the red ones are Si atoms; (c) side and (d) top views of NW1. The definitions of the radius (R) and wall thickness (d) are also indicated in (a).

larger than 20 nm.27 In this work, we focused on the NWs and NTs with 2H-SiC single-crystalline structures. The supercells of faceted SiC NWs with hexagonal cross sections were built according to bulk 2H-SiC crystal lattice. Those of faceted SiC NTs were then constructed from the SiC NW supercells by removing different sized hexagonal cores to form thick wall tubes. The morphology of these NWs and NTs was characterized as hexagonal prisms with [0001]-oriented axis enclosed by six facets belonging to the {100} plane group of 2H-SiC crystal, as shown in Figure 1a. The SiC NWs were indexed by the number of SiC pairs (n) in per cell, while the faceted NTs were indexed by the number of (101j0) bilayers of the walls (m). We will hereafter refer to a faceted NT with wall containing m (101j0) bilayers as a mNT. The radius of a NW is defined by the average distance from the center axis to the outmost atoms on the edge of the prism as shown in Figure 1a, while the wall thickness of a NT is represented by the distance between the inner and outer walls, as indicated in Figure 1a. Obviously, the values of n and m are related to the radius of NWs and the wall thickness of NTs, respectively. We first optimized the atomic positions and lattice constants along the axial direction for ultrathin NWs and NTs (R < 1.1 nm) using DFT calculations. The relaxed structures of the 2NT and the NW with n ) 24 are shown in Figure 1b-d, respectively. The surface atoms are found to relax inward, and the relaxation distance of Si atoms is different from that of C atoms. The inward relaxation for the surface Si atoms is much more significant than that for C atoms. We also defined some parameters to characterize the surface relaxation of the NWs and NTs, as shown in Figure 2. These parameters for NWs 1-3 (with n ) 24, 54, and 96, respectively) and for the 3NT obtained from DFT calculations are listed in Table 2. Obviously, both the Si-C bond length and the C-Si-C angle on the facets differ from those of the bulk SiC. The structural relaxation mainly occurs within the first atomic layer. The NWs and the 3NTs (including its inner and outer surfaces) have quite similar extent of surface relaxation. The surface relaxation is responsible for the energetic changes of these nanostructures. To elucidate the energetic evolution of these nanostructures, we defined the formation energies (Eform) by the energy

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Figure 2. (color online) Side view of the (1010) surface. δRβ and dRβ are the distances between adjacent atoms along the directions perpendicular and parallel to the surface, respectively. The subscripts R, β denote to which bilayer the atom belongs. ω is the tilt angle of the Si-C dimer with the surface.

TABLE 2: Structural Relaxation of the (101j0) Facets of NWs and NTs Obtained from the DFT Calculationsa δ11 δ12 δ23 δ33 d11 d12 d22 ω

NW1

NW2

NW3

3NT (outer)

3NT (inner)

0.069a 0.174a 0.615a 0.285a 0.336c 0.155c 0.372c 7.173

0.071a 0.182a 0.607a 0.282a 0.337c 0.154c 0.372c 7.356

0.071a 0.185a 0.606a 0.284a 0.338c 0.153c 0.373c 7.281

0.069a 0.178a 0.608a 0.283a 0.338c 0.154c 0.372c 7.152

0.047a 0.189a 0.597a 0.287a 0.339c 0.337c 0.372c 4.830

a The meanings of the parameters are shown in Figure 2. The distances are relative to the experimental lattice parameters a and c of 2H-SiC crystal, which are 3.08 and 5.04 Å, respectively. The angles (ω) are in degrees.

difference (per SiC pair) between SiC NWs/NTs with optimized configurations and bulk 2H-SiC crystal. The Eform values obtained from the DFT calculations for these thin NWs and NTs are listed in Table 3. The following features can be seen from this table: (1) The Eform of NWs decreases with the increase of wire radius, while that of the faceted NTs decreases with increasing of the wall thickness, irrespective of the tube radius. (2) Both the faceted NWs and NTs are energetically more favorable than the cylindrical layered SiC NTs, which agrees with the experimental findings that the cylindrical NTs are metastable and transform to SiC NWs under certain conditions.7 (3) Faceted NWs are more stable than the faceted NTs with similar diameters. This is due to the larger surface area of NTs in comparison with NWs. (4) The Eform values of the mNTs are very close to that of a NW which has a smaller radius than the mNTs. For example, the NW with a radius of 7.59 Å (n ) 24) has a close value of Eform to that of 3NT (m ) 3) with a radius of 10.67 Å and a wall thickness of 5.15 Å. The energetics of the SiC NWs and NTs in a wide range of size (0.45 nm < R < 2.9 nm) was then studied using TP. The Eform values of ultrathin NWs and NTs (R < 1.1 nm) obtained from TP calculations are also listed in Table 3. It is evident that the main features of the energetic of these NWs and NTs obtained from the TP calculations agree well with those obtained from the DFT calculations, although the Eform values given by the TP calculations are systematically larger than those from the DFT calculations by about 20%. This difference is consistent with the different values of bulk modulus of 3C-SiC crystal given by these two methods. Compared with the DFT and experimental results, the TP calculations overestimate the bulk

modulus. Figure 3 gives the variation of Eform of SiC NWs and NTs as a function of wire/tube radius. It is obvious that the features of the energy evolution mentioned above for the ultrathin NWs and NTs are still kept for the large sized NWs and NTs. The Eform of NWs decreases monotonously with the increase of wire radius. The faceted NTs with the same wall thickness but different radii have almost the same value of Eform which are also close to the Eform of a NW. The Eform of faceted NTs decreases with the increase of wall thickness (represented by m). These features are understandable in terms of a simple model. In view of the similarity of surface relaxation of faceted NWs and NTs which occur mainly within the first layer of the surfaces (see Figure 2 and Table 2), one can naturally suppose that the energy increase of NWs and NTs with respect to bulk crystal arises mainly from the undercoordinated atoms on the facets. In this case, the Eform of a NW can be determined by the expression

Eform ) t × E0

(1)

where t is the ratio of undercoordinated atoms on the facets to the total atoms in NW or NTs and E0 is the energy increase of an undercoordinated atom with respect to the fully coordinated atoms in 2H-SiC crystal. For the NWs of different sizes but identical morphology, the t value is proportional to n-1/2

t ) s × n-1⁄2

(2)

where s depends on the morphology of NWs. For the hexagonal NWs under study, s ) 6. Therefore, Eform is proportional to n-1/2 following the expression

Eform ) E0 × (6/n)1⁄2

(3)

For the cases of the faceted mNTs modeled in this work, a simple relationship between t and m can be proved

t ) 1/m

(4)

Therefore, according to eq 1, the Eform values of mNTs are proportional to 1/m (or to the inverse of wall thickness) following the expression

Eform ) E0/m

(5)

The Eform values of NWs and mNTs as a function of n or m are plotted in Figure 4. The solid lines represent the fitting data according to the eqs 3 and 5, respectively. It is clear that the data of the TP calculations fit the two expressions very well, indicating the rationality of the simple model. The fitting value of E0 is 2.972 eV/SiC for NTs, close to that for NWs, 2.975 eV/SiC. These values agree well with the surface energy of (1010) surface, which is 2.986 eV/SiC. This is also consistent with the results showing similar extent of surface relaxation for NWs and NTs (see Figure 2 and Table 2). The values of n and m are related to the radius of NWs and the wall thickness of NTs, respectively. For large sized NWs and NTs, the radius of NWs is proportional to n-1/2, while the wall thickness of NTs is proportional to m. Therefore, the Eform values of NWs and mNTs are proportional to the inverse of wire radius and wall thickness, respectively. For ultrathin NWs and NTs, however, the variation of Eform as a function of wire radius or wall thickness deviates from the proportional relation due to the size effect. The relationship between the Eform values of NWs and that of mNTs is also understandable. According to the eqs 1 and 5, when the t value of a NW is equal to 1/m, the NW has a Eform value close to that of mNTs. As shown in Table 3 and Figure

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TABLE 3: Structural Parameters, Formation Energies (Eform), and Band Gaps (Egap) of Naked and Hydrogenated SiC NWs and NTs and Single-Walled SiC Nanotubes Obtained from the DFT Calculationsa configurations NWs mNTs SWSiCNTs

NW1 NW2 NW3 2NT-1 2NT-2 3NT-1 (6, 0) (9, 0) (6, 6) (9, 9)

R

d

Eform (DFT)

Eform (TP)

Egapb(bared)

Egapb (hydrogenated)

4.554 7.588 10.64 7.639 10.743 10.67 3.021 4.474 5.145 7.699

2.497 2.512 5.146 -

1.216 0.828 0.632 1.192 1.191 0.828 1.547 1.346 1.304 1.242

1.489 0.993 0.745 1.487 1.487 0.993 -

1.627i 1.577i 1.697i 1.422i 1.339i 1.625i 0.618d 1.485d 2.206i 2.443i

3.848d 3.289d 3.023d 3.791d 3.885d 3.265d -

a

The Eform values given by the TP calculations are also listed in column five for the purpose of comparison. Radius (R) and wall-thickness (d) are in Å. b Eform and Egap are in eV/SiC and eV, respectively. d Direct band gap at the Γ point; iindirect band gap.

Figure 3. (color online) Evolution of formation energy (Eform) of singlecrystalline SiC NWs and NTs as a function of wire/tube radius. The radii of mNTs are the outer radii, as indicated in Figure 1a.

Figure 4. Formation energies (Eform) of NWs and mNTs (from TP simulations) as a function of n-1/2 and 1/m, respectively. The lines represent the fitting data given by the expressions Eform ) 7.287 × n-1/2 (NWs) and Eform ) 2.972/m (mNTs), respectively. The inset shows the results of DFT calculations, where the solid lines are the fitting data of the expressions Eform )5.720 × n-1/2 (NWs) and Eform ) 2.180/m (mNTs).

3, some mNTs really have formation energies close to those of some NWs. The low Eform values for thick-walled mNTs imply the possibility of synthesis of faceted mNTs under proper conditions. It is noteworthy that the formation energies of faceted NTs are quite different from those of SWSiCNTs which can be fitted

by the expression Eform )E + λ/R2, following the classical elastic theory.10 However, those cylindrical NTs are energetically more unfavorable than the faceted NWs and NTs. The metastability of the cylindrical NTs is also proved by the MDS for doublewalled SiC nanotubes which are built analogously to the doublewalled BN nanotubes.28 Our TP MDS indicates that the cylindrical double-walled nanotubes are quite unstable and undergo severe structural distortion even at room temperature. Si-C bonds are formed bridging the inner and outer walls, and eventually, the cylindrical morphology transforms into a faceted configuration. The electronic bands of SiC NWs and NTs are obtained using DFT calculations. It should be stressed that the Kohn-Sham energy gaps differ from the quasi-particle gaps and are usually smaller than the experimental values. Our DFT calculations show that the band gaps (Eg) of 3C-, 2H-, 4H-, and 6H-SiC crystals are 1.37, 2.78, 2.36, and 2.14 eV, respectively, and all of them are indirect band gaps. The experimental values of these gaps for 3C, 4H, and 6H-SiC crystals are 2.3, 3.26, and 3.03 eV, respectively. The present DFT calculations underestimate the Eg values by about 0.9 eV. The energy bands along the Γ (0, 0, 0) π/c-X (0, 0, 1) π/c direction (c is the lattice constant along the axial direction) of naked and hydrogen-passivated NWs and NTs are plotted in Figure 5. The band structures of a (9, 0) SiC SWNT are also presented in this figure for purpose of comparison. It is clear that the naked NW1 and 2NT are indirect gap semiconductors. The valence band maximum (VBM) and the conduction band minimum (CBM) of the 2NT are at the Γ point and X point, respectively. The CBM of NW1 is at (0, 0, 3/5) π/c, while the VBM is at the Γ point. This differs from zigzag SWSiCNTs which have a direct-band gap at the Γ point. The Eg values of the naked NWs and NTs are about 1.42-1.70 eV (see Table 3), narrower than that of 2H-SiC crystal, 2.78 eV. This is due to the surface states of naked NWs and NTs which arise from the dangling bonds on the facets. The surface states appear in the band gap of 2H-SiC crystal and thus narrow the band gaps of the NWs and NTs. This can be further evidenced by the projected electron density of states (PDOS) given by projecting the total density of states onto the atoms on surfaces and those in the interior region (bulk), respectively, as shown in Figure 6. Obviously, the states near the Fermi level mainly arise from the states of the undercoordinated atoms on the facets. The VBM and CBM originate mainly from the C (2p) states and Si (3p) states of the surface atoms, respectively. Such a feature is caused by electron transfer from Si to C atoms due to their different electronegativities.10 The PDOS is also consistent with the spacial distribution of wave functions around the atoms, which

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Figure 7. Isosurfaces of the Kohn-Sham states of the valence band maximum (VBM) and the conduction band minimum (CBM) of NW3 and 3NT at the Γ point. The isovalue is 0.05 Å-3/2.

Figure 5. Energy bands along the Γ (0, 0, 0) π/c-X (0, 0, 1) π/c direction, where c is the lattice constant of the NWs and NTs in the axial direction. (a) (9, 0)-SWSiCNT; (b) naked faceted 2NT; (c) NW1; (d) hydrogenated NW1; (e) hydrogenated 2NT; (f) hydrogenated 3NT. The energy at the Fermi level is set to zero.

see that the CBM wave functions of the NW and NT reside at the six corners of the hexagonal prisms, whereas the VBM wave functions of the NT are on the inner and outer facets. The narrower band gaps of faceted NWs and NTs compared with SiC crystals should make their electronic and optoelectronic properties, such as photoluminescence spectrum, definitely different from those of bulk SiC crystal. Surface states can be passivated by surface hydrogenation. For the hydrogenated (H-) NWs and NTs, the dangling bonds on the facets are saturated by hydrogen atoms. Surface states that appear in the band gaps of naked NWs and NTs are therefore removed in H-NWs and H-NTs, giving rise to wider band gaps than those of corresponding naked NWs and NTs (see Table 2and Figure 5). The Eg values of H-NWs and H-NTs are larger than that of 2H-SiC crystal and decrease with the increase of wire radius or wall thickness, which can be attributed to the quantum confinement effect. Similar results have also been revealed for the hydrogenated SiC NWs in 3C form.29 More interestingly, both H-NWs and H-NTs have direct-band gaps at the Γ point. They may find potential applications in building optoelectronic devices with superior performance that is not accessible for indirect-band gap SiC semiconductors. Conclusion

Figure 6. PDOS of 2NT (a) and of NW1 (b), obtained by projecting the total density of states onto the atoms specified in Figure 1b and d. The energy at the Fermi level is set to zero.

are responsible for the formation of the VBM and CBM. The isosurfaces for the VBM and CBM wave functions are highly located on the facets, as shown in Figure 7. It is interesting to

Our DFT calculations of the faceted single-crystalline SiC NWs and NTs show that the surface relaxation plays an important role in the energetics and electronic properties of these nanostructures. The Eform of NWs decreases with increasing wire radius, whereas that of NTs decreases with the increase of wall thickness, irrespective of the tube radius. The faceted SiC NWs and NTs are energetically more favorable than the cylindrical single-walled SiC NTs built analogously to BN nanotubes. Both the naked faceted SiC NWs and NTs are indirect-band gap semiconductors with narrower band gaps than that of 2H-SiC crystals, due to the surface states. Surface hydrogenation passivates the surface states of the NWs and NTs, making them become direct gap semiconductors. The band gaps of H-NWs and H-NTs are wider than that of 2H-SiC crystal and decrease with the increase of wire radius or tube wall thickness. Acknowledgment. This work is supported by the National Natural Science Foundation of China under grant no. 10675075, the National Basic Research 973 Program of China (grant no. 2005CB623602), and the National Foundation for Fostering Talents of Basic Science (NFFTBS) under grant no. J0730318.

SiC Nanowires and Nanotubes References and Notes (1) Iijima, S. Nature (London) 1991, 354, 56. (2) Zhou, X. T.; Wang, N.; Lai, H. L.; Peng, H. Y.; Bello, I.; Wong, N. B.; Lee, C. S.; Lee, S. T. Appl. Phys. Lett. 1999, 74, 3942. (3) Xi, G. C.; Peng, Y. Y.; Wan, S. M.; Li, T. W.; Yu, W. C.; Qian, Y. T. J. Phys. Chem. B 2004, 108, 20102. (4) Pan, Z. W.; Lai, H. L.; Au, F. C. K.; Duan, X. F.; Zhou, W. Y.; Shi, W. S.; Wang, N.; Lee, C. S.; Wong, N. B.; Lee, S. T.; Xie, S. S. AdV. Mater. 2000, 12, 1186. (5) Wang, H.; Li, X. D.; Kim, T. S.; Kim, D. P. Appl. Phys. Lett. 2005, 86, 173104. (6) Zhang, H. F.; Wang, C. M.; Wang, L. S. Nano Lett. 2002, 2, 941. (7) Sun, X. H.; Li, C. P.; Wong, W. K.; Wong, N. B.; Lee, C. S.; Lee, S. T.; Teo, B. K. J. Am. Chem. Soc. 2002, 124, 48. (8) Mavrandonakis, A.; Froudakis, G. E.; Schnell, M.; Muhlhauser, M. Nano Lett. 2003, 3, 1481. (9) Menon, M.; Richter, E.; Mavrandonakis, A.; Froudakis, G.; Andriotis, A. N. Phys. ReV. B 2004, 69, 115322. (10) Zhao, M. W.; Xia, Y. Y.; Li, F.; Zhang, R. Q.; Lee, S.-T. Phys. ReV. B 2005, 71, 085312. (11) Alam, K. M.; Ray, A. K. Phys. ReV. B 2008, 77, 035436. (12) Makeev, M. A.; Srivastava, D. Phys. ReV. B 2006, 74, 165303. (13) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865; 1997, 78, 1396. (14) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188.

J. Phys. Chem. C, Vol. 113, No. 3, 2009 861 (15) Ordejo´n, P.; Artacho, E.; Soler, J. M. Phys. ReV. B 1996, 53, R10441. (16) Sa´nchez-Portal, D.; Ordejo´n, P.; Artacho, E.; Soler, J. M. Int. J. Quantum Chem. 1997, 65, 453. (17) Soler, J. M.; Artacho, E.; Gale, J; D, Garcı´a A.; Junquera, J.; Ordejo´n, P.; Sa´nchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745, and references therein. (18) Tersoff, J. Phys. ReV. B 1989, 39, 5566. (19) Tersoff, J. Phys. ReV, B 1990, 41, 3248. (20) Gao, F.; Bylaska, E. J.; Weber, W. J.; Corrales, L. R. Phys. ReV. B 2001, 64, 245208. (21) Erhart, P.; Albe, K. Phys. ReV. B 2005, 71, 035211. (22) Baierle, R. J.; Piquini, P. Phys. ReV. B 2006, 74, 155425. (23) Makeev, M. A.; Srivastava, D. Phys. ReV. B 2006, 74, 165303. (24) Devanathan, R.; Rubia, T. D.; Weber, W. J. J. Nucl. Mater. 1998, 253, 47. (25) Gale, J. D. J. Chem. Soc., Faraday Trans. 1997, 93, 629. (26) Bassett, W. A.; Weathers, M. S.; Wu, T. C. J. Appl. Phys. 1993, 74, 3824. (27) Ito, T.; Sano, K.; Akiyama, T.; Nakamura, K. Thin Solid Films 2006, 508, 243. (28) Okada, S.; Saito, S.; Oshiyama, A. Phys. Res. B 2002, 65, 65410. (29) Yan, B. H.; Zhou, G.; Duan, W. H.; Wu, J.; Gu, B. L. Appl. Phys. Lett. 2006, 89, 023104.

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