## I- The stability properties of armchair DWSiCNTs

The electronic structures and stability of armchair DWSiCNTs have been investigated by using density functional theory. To check the stability of armchair DWSiCNTs, we choose *(*5, 5) and (4, 4) armchair single-walled 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 to15). 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.

To consider the interaction energy among the outer and inner tubes the formation energies of the (n, n)@(m, m) DWSiCNTs (with n = 4 and 5 and m = 7 to 15) are calculated from their total energies relative to the corresponding isolated single-walled SiC nanotubes using the following equation [22–24]:

$${\text{E}}_{\text{f}\text{o}\text{r}\text{m}\text{a}\text{t}\text{i}\text{o}\text{n}}=\text{E}\left(\text{n}1,\text{n}2\right)+\text{E}\left(\text{n}3,\text{n}4\right)-\text{E}\left[\left(\text{n}1,\text{n}2\right)@\left(\text{n}3,\text{n}4\right)\right]$$

1

Where \(\text{E}\left(\text{n}1,\text{n}2\right),\text{E}\left(\text{n}3,\text{n}4\right) \text{a}\text{n}\text{d} \text{E}\left[\left(\text{n}1,\text{n}2\right)@\left(\text{n}3,\text{n}4\right)\right]\) indicates the total energy of inner tube, outer tube and DWSiCNTs, respectively.

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 [22–24]:

$${\text{E}}_{\text{b}}=(\text{a}\text{E}\left(\text{S}\text{i}\right)+\text{b}\text{E}(\text{C})-\text{E}(\text{S}\text{i}\text{C}\left)\right)/(\text{b}+\text{a})$$

2

Where E(SiC), E(Si), E(C) represent the total energy of the DWSiCNTs, Silicon, and carbide atoms, and a and b indicate the total number of Si and C atoms in the DWSiCNTs, 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.

Table 1

The armchair DWSiCNTs, number of atoms (N), inter-wall distance (ΔR ), tube diameter (d), the binding energy per atom (Eb), formation energy( EF ), and bandgap (Eg).

Armchair DWSICNTs | N | status | ΔR(Å) | d(Å) | Eb(eV) | EF(eV) | Eg(eV) |

(5,5)@(7,7) | 48 | collapsed | - | - | - | - | - |

(5,5)@(8,8) | 52 | collapsed | - | - | - | - | - |

(5,5)@(9,9) | **56** | **The most stable** | **3.65** | **15.90** | **5.879** | **1.988** | **1.56** |

(5,5)@(10,10) | 60 | stable | 4.18 | 17.18 | 5.876 | 1.697 | 1.84 |

(5,5)@(11,11) | 64 | stable | 5.08 | 18.78 | 5.861 | 1.507 | 1.95 |

(5,5)@(12,12) | 68 | stable | 6.04 | 20.62 | 5.857 | 0.002 | 2.00 |

(5,5)@(13,13) | 72 | unstable | 6.89 | 22.34 | 5.858 | -0.004 | 2.01 |

(5,5)@(14,14) | 76 | unstable | 7.76 | 24.26 | 5.862 | -0.004 | 2.01 |

(5,5)@(15,15) | 80 | unstable | 8.61 | 25.78 | 5.863 | -0.005 | 2.01 |

(4,4)@(6,6) | 40 | collapsed | - | | - | - | - |

(4,4)@(7,7) | 44 | collapsed | - | | - | - | - |

(4,4)@(8,8) | **48** | **The most stable** | 3.65 | **13.97** | 5.863 | 1.875 | 1.22 |

(4,4)@(9,9) | 52 | stable | 4.26 | 15.75 | 5.858 | 1.376 | 1.55 |

(4,4)@(10,10) | 56 | stable | 5.07 | 17.11 | 5.846 | 0.538 | 1.64 |

(4,4)@(11,11) | 60 | stable | 6.11 | 18.79 | 5.843 | 0.0008 | 1.68 |

(4,4)@(12,12) | 64 | unstable | 6.89 | 20.51 | 5.848 | -0.0013 | 1.68 |

(4,4)@(13,13) | 68 | unstable | 7.85 | 22.21 | 5.851 | -0.0018 | 1.69 |

(4,4)@(14,14) | 72 | unstable | 8.71 | 24.16 | 5.854 | -0.0016 | 1.69 |

## II- The electronic properties

In this section, the electronic properties of armchair SWSiCNTs and DWSiCNTs have been investigated. We have studied the effect of the nanotubes' diameter on the value of the bandgap. Figure 2. show 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 are in agreement with previous studies [22, 24]. In addition, the bandgap of double-walled SiC nanotubes is smaller than that of their single-walled constituent nanotubes. 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. Figure 3 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.