## 3 Results and Discussion
### Formation Energy
#### 3.1.2 Formation Energies of Single PV and PVH Complexes  
According to previous research [20], PV and PVH complexes are proven to be acceptor centers providing hole carriers and reducing carrier concentrations in P-doped diamonds. [PERSON] et al. [5] show that \(\sim\)10\({}^{18}\) cm\({}^{-3}\) hydrogen atoms are detected on (001) surfaces when the background measurement level was \(\sim\)5 \(\times\) 10\({}^{17}\) cm\({}^{-3}\). Considering the layer by layer growth of diamonds via CVD, it is meaningful to analyze the formation of PV and PVH complex in surface layers. The PV complex is modeled by one P atom replacing one C atom with a nearest C atom been removed. To find the possible locations of vacancies around the doped P on the (001) surface, we use the symmetrical approach similar as above. As shown in Figure 3, when doping a P atom at the site 5A on the (001) surface, from the side view, the vacancy has only one symmetric site in the sixth layer, and from the front view, there are two symmetric sites in the fourth layer because of the structure reorganized at the top layer. Three different vacancy locations are calculated around P5A on the (001) surface and these locations are defined as A, B and C, respectively. There are two possible vacancy locations around the doped P atom on (111) surface, one is at the C position where the P-C bond is parallel to the surface normal, and the other is at the C position where the P-C bond is 109\({}^{\circ}\) inclined from the normal, and we defined them as A and B. Figure 4 shows the optimized structures of these three different sites of PV complex (P5 AVA, P5 AVB, and P5 AVC) on the (001) surface. Since the hydrogen atom is small, it is present in the crystal gap (interstitial atom) around the vicinity of the PV complex and has many possible locations. Based on previous research on P-H complexes in surface structures [17] and bulk diamonds [31], there are four H different sites around PV complex: anti-bond (AB), bond-center (BC), cross-line (CL), and on-vacancy (OV) sites, as shown in Figure 5. The AB site is on the opposite extension of the P-C bond, the BC site is inserted between the C and P atoms, the CL site is near the two planes' intersecting line (one plane consists of C\({}_{1}\), C\({}_{3}\), and P and the other consists of C\({}_{2}\), C\({}_{4}\), and P), and the OV site is on the vacancy location. Figure 6 shows the lowest formation energy of PV and PVH complex in each layer of (001) and (111) surface. The lowest formation energies of the PV and PVH complex on the two surfaces show the same trends of increase and decrease. As shown in Figure 6, the lowest formation energies of the PV and PVH complex in each layer of the (111) surface are all higher than those of the (001) surface, which means that the PV and PVH complexes more easily form on the (001) surface, and that is the same for the PH complex [17]. Therefore, the (001) surface contains more P-related complexes than the (111) surface, which may be one of the reasons why the compensation ratios of the (001) surface are approximately one order of magnitude higher than those of the (111) surface [18]. In both (001) and (111) diamonds, the formation energies of the PVH complex in the second and third layer are lower than those of the PV complex, but in the fifth and sixth layer, the formation energies of the PVH complex are same as or higher than those of the PV complex,  
Figure 4: The optimized structures of PV complex when P (yellow) is in the site 5A on the (001) surface. (**a–c**) represent V in the site A, B and C, respectively, the number in the figure represents the length of P–C bonds. Figure 3: The possible vacancy locations around the doped P atom at the site 5A. (**a**) front view of the (001) surface, (**b**) side view of the (001) surface and (**c**) front view of the (111) surface (white: H, gray: C; yellow: P, purple, fuchsia and pink: V in the site A, B, and C, respectively).