Electronic structures and stability of double-walled armchair (n,n)@(m,m) SiC nanotubes

In this work, we have investigated the stability and electronic properties of armchair double-walled SiC nanotubes (DWSiCNTs) based on density functional theory with the SIESTA package. The calculation has been performed on the armchair (4,4)@(n,n) and (5,5)@(n,n) DWSiCNTs with (n = 7–15). The stability calculation of DWSiCNTs shows that the armchair DWSiCNTs with difference chirality of 4, (n,n)@(n + 4,n + 4) and inter-wall distance of 3.65 Å are the most stable structures. Considering the electronic band structure points that all armchair nanotubes are semiconductors with indirect bandgap. Moreover, it is revealed that the value of the bandgap increases by increasing inter-wall distances, and the process of change at higher inter-wall distances be almost constant. In addition, the bandgap of double-walled SiC nanotubes is smaller than that of their single-walled nanotubes. The consequences of this investigation can certainly be helpful in future experimental studies.


Introduction
Nanotubes are one of the most important nanostructures currently being studied [1][2][3][4][5][6][7][8][9][10]. Today, there is a lot of growth in this type of quasi-one-dimensional structure due to its amazing physical properties and its many applications in the electronics industry. Extensive research has been conducted in this field since the discovery of carbon nanotubes in 1991 [7][8][9][10]. Meanwhile, silicon carbide (SiC) due to its interesting chemical and physical properties, has high application potential in the optical and electronics industry, because this material has a wide bandgap, high thermal conductivity, and radiation resistance, for use in work environments with conditions hard to fit [11]. Silicon carbide nanotubes (SiCNTs) have been synthesized from the reaction of SiO-derived silicon with multi-walled carbon nanotubes as mold plates at different temperatures and their structure and stability have been investigated using density functional theory [12]. The results indicate that silicon carbide nanotubes with Si-C replacement bonds are more stable than tubes containing C-C or Si-Si bonds [13]. Depending on the diameter and chirality, carbon nanotubes can be metal or semiconductors [7][8][9][10], while all silicon carbide nanotubes are semiconductors and their energy band gaps depend on their diameters and chiralities so SiC nanotubes have superior advantages over carbon nanotubes. Sun and colleagues first synthesized SiC nanotubes in 2002 and studied their structural properties [12]. SiC nanotubes were then successfully synthesized in research of different groups [14][15][16][17]. In 2004, the first computational studies on the structural properties of SiC nanotubes were performed by Menon et al. using the molecular dynamics approach. They investigated zigzag (12,0) and armchair (6,6) single-walled silicon carbide nanotubes [13]. Adhikari et al. examined the SiC nanotubes of the double-walled armchair (n, n) @ (5,5) using the density functional theory (DFT) by the GAUSSIAN computational code [18]. They showed that the binding energy of each atom of the studied double-walled structures depends not only on the number of atoms, but also on the conjugation of single-walled nanotubes, and the formation energy will be minimized when the distance between the walls is about 3.5 angstroms. Which corresponds to nanotubes (9,9) @ (5,5). In 2009, Moradian et al. examined the nanotubes of (11,11) @ (n, n) (n = 8-5) [19]. The results of their research showed that the inner tube (6,6) is the most ideal nanotube for the outer nanotube (11,11) with an in-wall spacing of 4.3 angstroms.
Despite the fact that many theoretical and experimental studies have been done on the structural properties of single-walled silicon carbide nanotubes, but on the structural properties and stability of double-walled silicon carbide nanotubes, few studies have been done. Therefore, in this study, the structural, electronic, and stability properties of armchair double-walled silicon carbide nanotubes have been studied in the framework of density functional theory.

Computational details
In this work, the electronic and stability properties of DWSiCNTs are studied by density functional theory (DFT), as performed in the SIESTA 4.1-b4 open source code [20]. The interactions between the atomic nuclei and valence electrons are calculated by pseudo-potential approximation. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) [21] is applied to calculate the exchange-correlation function. Using the double zeta polarization basis sets (DZP) [22], the Kohn-Sham orbits in Linear Combinations of Atomic Orbitals (LCAO) are expanded. The valence electron wave function is a plane-wave basis set with the cutoff energy of 500 Ry for armchair DWSiCNTs. The Monkhorst-Pack mesh is used with a gamma-centered k-points grid of 1 × 1 × 39 for all structures. The geometric structures are performed by minimizing the forces on atoms with the criterion that all forces on each atom must be smaller than 0.004 eV/Å. The conjugate-gradient minimization scheme is performed for both electronic structure and geometry optimization calculation. The equilibrium Si-C bond length for SiC nanotubes is considered to be about 1.79 Angstroms, which is consistent with the previous results [19]. The optimized lattice constants of c which is equal to the tube length are obtained at about 3.11 Å for the armchair DWSiCNTs. The atomic positions, lattice constants, k-point vector, and mesh cut-off are optimized using the conjugate gradient (CG) algorithm until a force precision of 0.004 eV/Å is achieved. Since the investigated structures are one-dimensional nanotubes, the boundary conditions are applied in such a way that periodic behavior is considered only along the nanotube length (z-direction), and in the other two directions of x and y a sufficient vacuum of about 15 Å is intended, in order to prevent interatomic interactions between the repeated images. The pseudo-potential standards are built by using Trouiller-Martins schemes that are described as the interaction of valence electrons with atomic central. Pseudopotentials with 2s 2 2p 2 , and 3s 2 3p 2 valence electron configurations are used for C and Si atoms, respectively.

The stability properties of armchair DWSiCNTs
The electronic structures and stability of armchair DWS-iCNTs have been investigated by using density functional theory. In this investigation, the inter-wall distances have increased by increasing the diameter of the outer tubes in accordance with previous research on GeC, SiC, ZnO, and AlN double-walled nanotubes [23][24][25][26][27]. The inner tubes have assumed fixed while outer tubes have been considered variable chirality and diameter. To check the stability of armchair DWSiCNTs, we choose (5,5) and (4,4) armchair singlewalled SiCNTs as the inner nanotubes. Thus the calculation have been performed on the armchair (4,4)@(n,n) and (5,5)@(n,n) DWSiCNTs with (n = 7 to 15). Figure 1 shows the top views of optimized geometries of armchair (5,5)@ (n,n) and (4,4)@(n,n) DWSiCNTs. As can be seen in these figures in the lower outer tube diameter, in other words at small interlayer spacing, the DWSiCNTs collapse and lose their tubular shape due to the interlayer interaction between the outer and inner tubes as having occurred for (4,4)@(6,6), (4,4)@(7,7), (5,5)@(7,7) and (5,5)@ (8,8) DWSiCNTs. The results show the DWSiCNTs with large outer tube diameters are stable by maintaining the cylindrical shape.
The stability of DWNTs is corroborated by calculating the formation energy originating from their inter-wall interactions.
Moreover, we have calculated the binding energies (Eb) per atom for all armchair DWSiCNTs according to the following equation [23][24][25][26][27]: where E(SiC), E(Si), E(C) represent the total energy of the DWSiCNTs, Silicon, and Carbon atoms, and a and b indicate the total number of Silicon and Carbon atoms in the DWS-iCNTs, respectively. The obtained results as the number of atoms, inner and outer tube diameters, inter-wall distances, binding energies per atom, formation energies, and electronic bandgaps for all considered armchair DWSiCNTs, are summarized in Table 1. The DWSiCNTs are stable if their formation energy, which is the inter-wall interaction energy, be a positive value. Thus the DWSiCNTs with the highest positive formation energy are the most stable structure. Therefore, (5, 5)@ (9,9) and (4, 4)@ (8,8) nanotubes have the highest formation energy compared with other armchair-DWNTs. On the other hand, the formation energy of armchair DWSiCNT has a maximum value around the inter-wall distance of about 3.65 Å. Thus, (4,4) and (5,5) are the preferable inner nanotubes for the (8,8) and (9,9) outer  nanotubes, respectively. The highest value of the formation energy occurs at average values of the inter-wall distance (ΔR) about 3.65 Å for armchair DWSiCNTs. The calculated formation energy and binding energies per-atom for the armchair (4,4)@(n,n) and (5,5)@(n,n) DWSiCNTs predicted the highest value for n = 8 and 9 respectively, which indicates the favorable outer tube wall for the inner (4,4) and (5,5) armchair SiCNTs are (8,8) and (9,9) tubes respectively. This indicates the armchair DWSiCNTs with difference chirality of 4; (n,n)@(n + 4,n + 4) and average values of inter-wall distance (ΔR) among the outer and inner tubes about 3.65 Å are the most energetically stable structures. Thus, both (4,4)@ (8,8) and (5,5)@ (9,9)DWSiCNTs are energetically favorable DWSiCNTs. The obtained results revealed that the energetic stability of the DWSiCNTs strongly depends on the inter-wall distances among the outer and inner nanotubes. The binding energy in the higher outer tube diameters and interlayer spaces reaches an almost constant value of 5.85 eV. The binding energy of DWSiCNT's at higher inter-wall distances gradually approaches that of the hexagonal SiC nanosheet. While the diameter of the outer tube increases, the interaction between atoms at the edge of the rings is reduced. One can explain that this is the reason that the value of binding energy is almost constant at higher outer tube diameters.

The electronic structures
In this section, the electronic properties of armchair SWS-iCNTs and DWSiCNTs have been investigated. We have studied the effect of the nanotubes' diameter on the value of the bandgap. Figure 2 shows the electronic band structure of a sample of armchair DWSiCNTs and the band structure of their inner and outer SWSiCNTs. The calculations indicate that all studied SiC nanotubes are semiconductors having indirect band gap. The obtained calculations for electronic properties of SWSiCNTs are in agreement with previous studies [23,25]. In addition, the bandgap of double-walled SiC nanotubes is smaller than that of their single-walled constituent nanotubes which is in agreement with previous studies on multi-walled beryllium oxide nanotubes [28] and double-walled GeC, SiC, ZnO and AlN nanotubes [23][24][25][26][27]. For example, the bandgap of (4, 4) and (8,8) SWSiCNTs are 1.66 and 2.16 eV, respectively. While the bandgap of (4, 4) @ (8,8) DWSiCNT is 1.22 eV. It can be said that the electronic properties of SiC nanotubes can be modified by double-walled. This reduction of the band gap in DWSiCNNTs could be related to the weak van der Waals interaction between the internal and external nanotubes, leading to the charge transfer of about 0.2 electrons from the Si atom to C, in both inner and outer nanotubes. In addition, a charge transfer of about 0.02 electrons from the outer to the inner tubes has also been observed. These charges transfers have been obtained from Mulliken charge analysis. According to Fig. 3 the CBM and the VBM in electronic structure of DWSiCNT are determined by the internal and external tubes, respectively, which provide the opportunity for electron transfer between the two SiCNT walls. Electrons are transferred from the external nanotube to the internal nanotube, which shifts the conduction band minimum (CBM) of the internal tube to lower energy, and finally, the CBM of the DWSiCNT is specified by its internal nanotube. Meantime, holes are transferred from the internal nanotube to the external nanotube, which shifts the maximum of the valence band (VBM) of the external tube to higher energy. These displacements lead to the reduction of the band gap of DWSiCNT and form an obvious coupling between its two walls. These results are consistent as Song et al. [24] observed for double-walled germanium carbide nanotubes and also with earlier studies on double-walled SiC, ZnO and AlN nanotubes [23][24][25][26][27]. Figure 3 shows the total density of states (DOS) of the (4,4)@ (8,8) double-walled nanotube and its constituent single-walled nanotubes. Based on this curve, it can be seen that the electronic gap of SWNTs has increased with the increase in their diameter, and the band gap of their combination in double-walled nanotubes, is smaller than the band gap of each of them, which is in accordance with our band structure calculations and with the previously reported data [23][24][25][26][27].
In Fig. 4, the partial density of states (PDOS) for each of the Si and C atoms in the valence and conduction bands for the double-walled nanotube is plotted. This figure shows that the contribution of 3p orbitals of the Si atom and 2p of the C atom in both conduction and valance bands is significant and also there is a strong hybridization between 3p-Si and 2p-C orbitals around the Fermi level. The role of 3s-Si and 2s-C orbitals is negligible and it can be concluded that these states are distributed in the core region. The band reduction of the double-walled nanotubes compared to their constituent SWSiCNTs is attributed to the interlayer interaction and 3p-Si and 2p-C hybridization. The concluded results for the electronic properties of DWSiCNTs are in agreement with earlier studies on double-walled ZnO and AlN nanotubes [26,27]. Figure 5 shows the variation of bandgap as a function of nanotube diameter and inter-wall spacing. It was found that armchair (5, 5) @ (n, n) DWSiCNTs have higher bandgap than (4, 4) @ (n, n) at the same diameters and interlayer distances. In addition, the bandgap of double-walled SiC nanotubes is smaller than that of their single-walled nanotubes Moreover, it is revealed that the value of the bandgap increases by increasing tube diameters and inter-wall distances, and the process of change at higher inter-wall distances would be almost constant. This can be attributed to weaker interaction between the inner and outer tubes at higher interlayer spaces.

Conclusions
In this study, we have investigated the stability and electronic properties of double-walled SiC nanotubes based on density functional theory. The calculation have been performed on the armchair(4,4)@(n,n) and (5,5)@(n,n) DWSiCNTs with (n = 7 to 15). By calculating the formation energy of nanotubes, the result revealed that armchair (5,5)@ (9,9) and (4,4)@ (8,8) DWSiCNTs have the highest formation energy. Thus they are the most stable nanotubes among other DWSiCNTs with an inter-wall distance of about 3.65 Å. The investigation of the electronic characteristics of nanotubes indicates all armchair nanotubes are semiconductors having indirect bandgap. Their bandgap increases with increasing tube diameter, also the trend of change at higher diameters is slow. The bandgap of double-wall SiCNTs is less than its constituent single-walled nanotubes. This band gap reduction in double-walled nanotubes can be attributed to the weak van der Waals (vdW) interactions between the inner and outer tubes. Modulating the bandgap is essential in optoelectronic systems, for instance, diodes and lasers. The consequences of this investigation offer these nanotubes provide a wide range of applications.
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