## 3 Results and Discussion
### Formation Energy
#### 3.1.1 Formation Energies of the Single P Atom and Single V  
The substitutional location of the P atom and V are shown in Figure 1. These positions are found based on the structural symmetry of the (001) and (111) surface structures. For example, the five outermost atoms in the third C layer of the (001) surface shown in Figure 1a are defined as C31, C32, C33, C34, and C35, respectively. C31, C33, and C35 as well as C32 and C34 are symmetrical, but C31/33/35 and C32/34 are asymmetrical, so the third layer of the (001) surface structure has two doping sites (C31/33/35 and C32/34) with different properties. We define these two sites as 3A and 3B, in which the number represents the doping layer and the capital letter (A and B) refers to the two different doping sites in this layer. Similarly, the doping sites in other layers are defined as 2A, 4A, 4B, 5A, and 6A, respectively. The formation energies of a single P atom and a single V on the (001) and (111) surfaces are shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**Surface** & **Number of Carbon Layer (n)** & \(k\) **Points Set** & **Cutoff Energy (eV)** & **Size (r \(\times\) y \(\times\) z) (Å\({}^{3}\))** \\ \hline (001) & 14 & \(2\times 2\times 1\) & 400 & \(10.06\times 10.06\times 23.27\) \\ (111) & 12 & \(2\times 2\times 1\) & 400 & \(7.55\times 8.71\times 23.06\) \\ \hline \end{tabular}
\end{table} 
Table 1: Calculation and lattice parameters of the two slab models. has a tendency to move upwards towards the vacuum place, which reduces damage to the underlying structure. Conversely, when the P atom is doped at deeper layers, the formation energy significantly increases because it has to destroy the surrounding structure. In addition, regarding the formation energies of the P atom in site A and B in the third and fourth layers of the (001) surface, the energy of the site A is significantly higher than that of the site B. Because of structural reconstruction, the site A is under compressive stress and the site B is under expansive stress. When doping a P atom with a larger volume than a C atom, the expansive stress of site B can be reduced and it eventually results in significant differences in the formation energies at different sites (site A and site B) in the same layer. This is consistent with the computed results of [PERSON] et al. [19] that the formation energy of one single P atom at a site B on the (001) surface is \(\sim\)1.7 eV (in the third layer) and \(\sim\)1.3 eV (in the fourth layer), smaller than their respective counterparts at the site A. Different from doping a P atom, there is no significant increase or decrease in the formation energy of a single V. In general, a single vacancy is more difficult to form on (001) and (111) diamond surfaces than a P atom because of its high formation energy. The energy in the third layer is much lower than that in other layers, which means that vacancies more easily form in the third layer rather than in the second layer. The formation energy of a single V in bulk diamond is 6.609 eV, and we find in the fifth and sixth layer, the formation energy of a vacancy in (001) and (111) surface obviously converge to this number, which means the surface environment has less influence on doping a vacancy in the deeper layer. According to the calculated data, the formation energies of a single P atom on (111) surfaces are generally greater than on (001) surfaces, which means doping P is easier to implement in (001) surfaces, consistent with [PERSON]'s calculated results [19]. According to [PERSON] et al. 's experimental result [30], the controllable range of phosphorus concentrations on (111) surfaces (between \(\sim\)5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\) and \(\sim\)2 \(\times\) 10\({}^{20}\) cm\({}^{-3}\)) is larger than on (001) surfaces (between \(\sim\)5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\) and \(\sim\)8 \(\times\) 10\({}^{18}\) cm\({}^{-3}\)), and the total incorporation efficiency value of (001)-oriented diamonds is two orders of magnitude lower than that of (111)-oriented diamonds.