## 3 Results and Discussion
### Single Neutral Silicon Vacancy in the H-, O-, or N-Terminated C (001) Surfaces
#### 3.3.1 Formation Energies  
The formation energy \(E_{f}\) of SiV\({}^{0}\), positioned in various C layers of H-, O-, or N-terminated diamond (001) surface and in a 216-atom bulk diamond, calculated using Equation (1), are illustrated in Figure 5. To better distinguish between and use different configurations by name, the formation energies of SiV\({}^{0}\) with their original (prior to geometry optimization) Si or V site (or both) positioned below the dimer rows were illustrated as _configuration 1_; if their original (prior to geometry optimization) Si or V site (or both) was (or were) located between the dimer rows, they were labelled as _configuration 2_. The possible sites for silicon and vacancy were the same as in Figure 2. When its original Si site was located in a specific layer, the V site of this SiV\({}^{0}\) naturally had to be placed in another (upper or lower) C layer, and vice versa. Therefore, the abscissa of the formation energies of SiV\({}^{0}\) were set as \(m-(m+1)\) (where \(m\) refers to the C layer number) for SiV defects.  
Figure 5: Formation energies of SiV\({}^{0}\) in various positions (\(L2\) to \(L7\)) near the diamond (001) surface, under H, O, or N termination and in bulk diamond as a reference. The possible silicon and vacancy sites under different surface states are shown in Figure 2.  
According to Figures 2 and 5, SiV\({}^{0}\) positioned in the second to third layer, with its original silicon and vacancy sites placed between the dimer rows (_configuration 2_) under N termination, led to the smallest formation energy of 4.88 eV; when its silicon and vacancy sites were placed under the dimer rows (_configuration 1_), a close value of 5.16 eV was indicated. Similarly, when SiV\({}^{0}\) was positioned in the second to third layer under O termination, its formation energy was 5.06 eV. The smallest formation energy was 5.39 eV for H termination, when SiV\({}^{0}\) was placed in the third to fourth layer, where its Si and V sites were originally created between the dimer rows (_configuration 2_). Therefore, we concluded that it is easier for SiV\({}^{0}\) to be embedded in the surface layers (_L2_ to _L3_) of diamond (001) surfaces under O or N termination.  
In the case of SiV\({}^{0}\) positioned below the fifth C layer, we found that formation energies fluctuated much less drastically, as confirmed in Table 1, where the situation was similar to that of single Si impurity and V. Furthermore, we can see that these formation energies converged to roughly 6.2 eV in the seventh to eighth layer under H, O, or N termination, and this value was close to the formation energy of SiV\({}^{0}\) in a 216-atom bulk diamond (6.32 eV), as deduced by Equation (1).  
Therefore, as indicated in Figure 5, it is generally easier for SiV\({}^{0}\) to be embedded in an O-terminated diamond (001) surface when it is above the fifth C layer. Their formation energies were almost the same when placed below the fifth C layer among H, O, and N terminations, indicating the effects of the surface-terminating species decline to be negligible. However, it was similarly easy for SiV\({}^{0}\) to form as both _configuration 1_ and 2 at the second to third layers under N termination and as _configuration 2_ at the third to fourth layers under H termination as compared with O termination.