Theoretical Study of Carbon Clusters in Silicon Carbide Nanowires

Using first-principles methods we performed a theoretical study of carbon clusters in silicon carbide nanowires. We examined small clusters with carbon interstitials and antisites in hydrogen-passivated SiC nanowires growth along the [100] and [111] directions. The formation energies of these clusters were calculated as a function of the carbon concentration. We verified that the energetic stability of the carbon defects in SiC nanowires depends strongly on the composition of the nanowire surface: the energetically most favorable configuration in carbon-coated [100] SiC nanowire is not expected to occur in silicon-coated [100] SiC nanowire. The binding energies of some aggregates were also obtained, and they indicate that the formation of carbon clusters in SiC nanowires is energetically favored.


I. INTRODUCTION
Silicon carbide (SiC) is a wide-band-gap semiconductor with excellent physical, electronic and mechanical properties 1 such as high thermal conductivity, high breakdown field, low density, high saturation velocity, high mechanical strength, and stability at high temperature. These exceptional features make SiC a promising candidate to replace silicon in electronic devices operating in high-power, high-frequency, and high-temperature regimes. 2 In the last years, SiC nanostructures (like nanospheres, 3 nanosprings, 4 nanowires 5 and nanotubes 6 ) have been successfully synthesized, and several theoretical and experimental works [3][4][5][6][7][8][9][10][11] have been performed to investigate their structural and electronic properties. The unique features of SiC combined with quantum-size effects make the SiC nanostructures interesting materials for nanotechnology applications. For instance, SiC nanowires and nanotubes have been considered as candidates for hydrogen storage nanodevices 8 and for building blocks in molecular electronic applications. 12 In particular, silicon carbide nanowires (SiC NWs) have excellent field emission properties, 13 high mechanical stability and high electrical conductance, 7 and could be used as nanoscale field emitters or nanocontacts in harsh environments.
Some optical and electronic properties of semiconductors may be modified by the presence of defects. The most common defects in SiC are vacancies, interstitials, antisites and clusters. These defects are mainly formed during the growth process and ion implantation of dopants. Vacancies and interstitials of C and Si in 3C-, 4H-and 6H-SiC bulks have been thoroughly investigated in theoretical and experimental works. [14][15][16][17][18][19] These investigations have showed that the C and Si vacancies are electron and hole traps, [16][17][18][19] and that C and Si interstitials have higher mobility than vacancies, although the mobility of point defects in SiC is reduced as compared to another semiconductors (like silicon). 14 The high mobility of carbon interstitials favors the formation of carboninterstitial clusters. Using ab initio methods, Gali et al. 20 systematically investigated small clusters of carbon interstitials and antisites in 3C-and 4H-SiC bulks, and verified that the formation of carbon aggregates is energetically favored.
In spite of some theoretical studies on carbon aggregates in SiC bulk, 20-22 the investigation of carbon clusters in SiC NWs is very scarce. Motivated by the lack of studies on C aggregates in SiC NWs and by their energetically favorable formation in SiC bulk, in this work we performed an ab initio study of small carbon clusters in SiC NWs. We considered hydrogen-passivated SiC NWs grown along the [111] and [100] directions and examined clusters with interstitial and antisites carbon atoms. The formation energies of these clusters were determined as a function of the C concentration. We calculated the binding energies of some aggregates, and our results indicate that the formation of carbon clusters in SiC nanowires is energetically favored. Besides [111] and [100] SiC NWs, the carbon clusters were also investigated in 3C-SiC bulk, in order to compare the effect of C defects in SiC bulk and NW.

II. METHODOLOGY
In this work we present ab initio calculations based on density functional theory 23 (DFT) carried out by using the SIESTA code. 24 We used local spin-density approximation [25][26][27] (LSDA) for the exchange-correlation functional and norm-conserving fully-separable pseudopotentials 28 to treat the electronion interactions. The Kohn-Sham orbitals were expanded using a linear combination of numerical pseudoatomic orbitals 29 and a double-zeta basis set with polarization functions 30 (DZP) was employed to describe the valence electrons.
The 3C-SiC bulk, and the [111] and [100] SiC NWs, were modeled within the supercell approach, with 128, 232 and 279 atoms, respectively. The SiC NWs were constructed from the 3C-SiC structure and the dangling bonds of their surfaces were saturated with hydrogen atoms. Due to the periodic boundary conditions, a vacuum region of about 10Å was used to avoid interactions between a NW and its image. We have considered axial lengths (along the NW growth direction) of about 15.2 and 13.2Å for the [111] and [100] SiC NWs, respectively. The geometries were optimized using the conjugated gradient scheme, within a force convergence criterion of 0.05 eV/Å. The Brillouin zone was sampled by using 2 special k points for 3C-SiC bulk, and 1 special k point for [100] and [111] SiC NWs. We verified the convergence of our total-energy results with respect to the number of special k points using up to 9 k points for 3C-SiC bulk and 4 k points for [  Eight C defects were investigated 33 in the present work (see Fig. 2): C Si (carbon antisite), C -C ( 100 split of C interstitial + C site), C -C Si ( 100 split of C interstitial + C antisite), C -Si ( 100 split of C interstitial + Si site), (C -C) i , 2(C -C) C , [(C -C) Si + (C -C) C ], and 2(C -C) Si .
In order to examine the energetic stability of the C defects in the 3C-SiC bulk and in the [100] and [111] SiC NWs, we used the grand canonical potential at T = 0 K (as suggested in the Refs. 34 and 35) Here E tot is the total energy of the considered structure, µ i is the chemical potential of atomic specie i (i = Si, C or H) and N i is the number of atoms i in the system. In the thermodynamic equilibrium, 36 and where ∆H(SiC) is the formation heat of the SiC bulk. Employing diamond structure 37 for bulk phases of Si and C, and zincblend structure 37,38 for SiC, we found ∆H(SiC) = 0.73 eV, which is in good agreement with the experimental value ∆H(SiC) = 0.68 eV. 39 The upper limits of µ Si and µ C are µ bulk Si and µ bulk C , respectively. Hence, the potential fluctuations ∆µ Si = µ Si − µ bulk This range defines the C-rich (or Si-poor, where ∆µ C = 0 and ∆µ Si = −∆H(SiC)) and Si-rich (or C-poor, where ∆µ C = −∆H(SiC) and ∆µ Si = 0) limits. In Table I we present the formation energies, relative to the pristine structures, of C defects in 3C-SiC bulk, and in hydrogen-passivated [100] and [111] SiC NWs. The two values presented in each column correspond to Sirich (or C-poor) and C-rich limits. For 3C-SiC bulk at stoichiometric condition we found that the formation energies of C Si , C -C, C -C Si , (C -C) i , 2(C -C) Si and [(C -C) Si + (C -C) C ] are 3.49, 7.38, 6.87, 8.79, 9.51 and 11.55 eV, respectively. These results are in good agreement with those obtained by Gali Table I). At Si-rich limit C -Si is the most favorable configuration, followed by the C -C and (C -C) i defects. However, under C-rich conditions 2(C -C) Si is the most stable defect, followed by the C -C Si and [(C -C) Si + (C -C) C ] aggregates. Figure 3 summarizes our calculated formation energy results. The energetically most favorable defect in Si-coated [100] SiC NW is C -Si for ∆µ C ≤ −0.43 eV, C -C Si for −0.43 ≤ ∆µ C ≤ −0.21 eV and 2(C -C) Si for ∆µ C ≥ −0.21 eV. At stoichiometric conditions, C -C Si is the most stable configuration, followed by C -Si and (C -C) i . We find total energy differences, under stoichiometric conditions, of 0.12 eV between C -C Si and C -Si, and of 0.27 eV between C -C Si and (C -C) i .
According to Ref. 20, C -C Si is energetically most favorable than C -C in 3C-SiC bulk. This result can be explained by the larger relaxation required to put two C atoms in a C site than to put two C atoms in a Si site. In the present work we observe a similar behavior [ Fig. 4]: under C-rich conditions C -C Si is the most stable complex with one carbon interstitial in 3C-SiC bulk and in [100] and [111] SiC NWs. However, C -C becomes energetically more stable than C -C Si for low carbon concentration: ∆µ C ≤ −0. Among the defects with two carbon interstitials, 2(C   We next examine the energy gain upon the formation of C clusters through the combination of interstitial C atoms. In this case, the energy gain was determined by comparing the total energies of the separated systems (E i ) and the total of a given (stoichiometrically equivalent) C cluster (E c ). We define the binding energy of the C cluster as E b = E c − E i . Our calculated binding energies are summarized in Table II. The results presented in each column correspond to the stoichiometric condition. In 3C-SiC bulk or SiC NW with a carbon antisite (C Si defect), a carbon interstitial can be captured by C site, Si site or C antisite. The last process is more favorable than the first (second) one by 4.00 eV (4.60 eV) in 3C-SiC bulk, 3.93 eV (4.68 eV) in C-coated [100] SiC NW, 4.95 eV (4.65 eV) in Si-coated [100] SiC NW, and 3.62 eV (3.98 eV) in [111] SiC NW. Considering two carbon interstitials, the formation of carbon clusters is energetically favored in 3C-SiC bulk and in both [100] and [111] SiC NW. In fact, the aggregates (C -C) i and 2(C -C) C are more stable than two isolated C -C defects by 5.51 and 2.88 eV in C-coated [100] SiC NW, and by 6.11 and 3.61 eV in [111] SiC NW. Still, in Si-coated [100] SiC NW the energy gain is 13.6 eV to form 2(C -C) Si from isolated (C -C) and C Si defects, and 3.7 eV to form 2(C -C) Si from two isolated C -C Si . In 3C-SiC bulk these energy gains are 10.2 and 2.2 eV, respectively. Here we can infer that, in general, the formation of C clusters is quite likely in SiC NWs, in particular for thin Si-coated [100] SiC NWs. It is worth to note that, in order to get  a complete picture of the formation of C clusters, the C diffusion mechanism is an important issue, however, it is beyond the scope of the present work. Focusing on the electronic properties, we find that similar to the 3C-SiC bulk phase, 20 C Si is an electrically inactive defect in SiC NWs. C Si does not introduce states within the fundamental band gap. On the other hand, the formation of C clusters gives rise to electronic states lying within the energy bandgap. Figure 5(a) presents the electronic band structure and the projected density of states (PDOS) of the pristine Si-coated [100] SiC NW. The pristine system exhibits an energy bandgap (at the Γ point) of 1.98 eV, calculated within our DFT-LSDA approximation, where the highest occupied (lowest unoccupied) states are mostly composed by C 2p (Si 3p) orbitals. The formation of C clusters give rise to localized electronic states within the bandgap, as depicted in Figs. 5(b)-5(d). C -Si defect gives rise to an occupied (empty) state at 1.2 eV (1.5 eV) above the valence band maximum [Fig 5(b1)]. The dispersionless character of those states, along the ΓX direction (i. e. parallel to the NW growth direction), indicate that those defects states are localized around C -Si. Indeed, the PDOS diagram [ Fig. 5(b2)] indicate that there is a significant contribution from the interstitial C atom to the occupied defect state, while the nearest neighbor Si atom contributes to the formation of the lowest unoccupied defect state. At the C-rich limit, the 2(C -C) Si defect is the most favorable one. Its electronic band structure is depicted in Fig. 5(c1). We find an occupied state at 1 eV above the VBM, and three unoccupied states lying within an energy interval of 2.1 and 2.6 eV above the VBM. Those defect states are mostly localized around the interstitial C atoms [ Fig. 5(c2)]. Finally, for C -C Si (energetically most likely at the stoichiometric condition) we find the formation of spin-unpaired states within the nearby the Fermi level, Figs. 5(d1) and 5(d2). Those defect states a localized around C interstitial atoms. In summary, different from C Si defects, we verify that the formation of C clusters in Si coated [100] SiC NWs gives rise to deep states within the bandgap, which may acts as a trap to the electronic carriers.

IV. CONCLUSIONS
We performed an ab initio investigation of small carbon clusters in SiC bulk and NWs. We examined clusters with carbon interstitials and antisites in 3C-SiC bulk and in hydrogen-passivated [100] and [111] SiC NWs. We observed that the composition of the SiC nanowire surface strongly influences the formation of the carbon clusters. In fact, C Si is the energetically most favorable configuration in 3C-SiC bulk, and in C-coated [100] and [111] SiC NWs, but is not expected to occur in Si-coated [100] SiC NW. The energetically most stable defect in Si-coated [100] SiC NW is C -Si at Si-rich limit, C -C Si at stoichiometric conditions and 2(C -C) Si under C-rich conditions.
Comparing the total energies of the C -C and C -C Si defects (in 3C-SiC bulk and in [100] and [111] SiC NWs), we find that C -C Si is more stable than C -C at stoichiometric and C-rich limits. This finding is in accordance with the larger relaxation required to put two C atoms in a C site than to put two C atoms in a Si site. However, we observed that C -C becomes energetically more stable than C -C Si for low C concentration. Further total energy calculation indicate that the formation of carbon clusters in 3C-SiC bulk and in both [100] and [111] SiC NWs is energetically favored. Finally, our electronic band structure calculations indicate that (i) similar to the 3C SiC bulk phase, C Si defects are electrically inactive, while (ii) in Si coated [100] NWs the C clusters gives rive to deep (localized) levels within the NW bandgap.