# Characterization of Monovacancy Defects in Vanadium Diselenide Monolayer: A DFT Study
## 3 Results  
The top and side views of the unit cell of the optimized T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are shown in Figure 1a,b, respectively. Both the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers belong to the D\({}_{3\text{h}}\) point group symmetry. The optimized lattice constant is 2.90 A for the T-VSe\({}_{2}\) monolayer and 3.33 A for the H-VSe\({}_{2}\) monolayer. The Se\(-\)V bonds in the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are 2.49 A and 2.50 A, respectively, and the Se\(-\)Se bonds in T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are 3.69 A and 3.33 A, respectively. The thickness of the T-VSe\({}_{2}\) monolayer is 3.14 A, while the thickness of the H-VSe\({}_{2}\) monolayer is 3.20 A. The obtained geometry and structural parameters of the considered T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are in good agreement with previous works [32; 77; 78]. The electron localization function (ELF) can be defined as the probability of finding an electron pair in a space region. The value of the ELF ranges from 0 (a free electronic state) to 1 (perfect localization), representing the charge localization in real space [79]. The ELF for the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers is analyzed and plotted in Figure 2a,d, respectively. The isosurface value of 0.7 is adopted in Figure 2. From Figure 2a,d it is seen that the electron localization basin is spherical and completely surrounds the respective atomic cores, suggesting an ionic type of bonding in the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers, as it is expected in 2D TMDs [80]. The local density of states (LDOS) and band structure of  
Figure 1: The optimized unit cell structure of the T-VSe\({}_{2}\) (**a**) and H-VSe\({}_{2}\) (**b**) monolayers. the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are also analyzed and depicted in Figure 2b,c and e,f, respectively. The H (hexagonal symmetry) and T (trigonal symmetry) phases of VSe\({}_{2}\) have substantially different electronic structures due to the difference in the structure symmetry [81]. According to Figure 2c, the T-VSe\({}_{2}\) monolayer has no band gap, as has been reported previously [32]. From Figure 2f, it is seen that the H-VSe\({}_{2}\) monolayer possesses a moderate direct band gap of \(\sim\)0.2 eV, which is also consistent with previous DFT studies [77; 82]. According to the LDOS plots in Figure 2b,e, the states in the vicinity of the Fermi level, in the case of the T-VSe\({}_{2}\) monolayer, and valance bands minimum (VBM) and conduction bands maximum (CBM), in the case of the H-VSe\({}_{2}\) monolayer, are mainly in the V \(d\) states. The stability of the considered monovacancy defects in the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers is evaluated based on their formation energy \(E_{\rm form}\) and the energy needed for the migration of these defects \(E_{\rm migration}\). The calculated \(E_{\rm form}\) for the monovacancy defects in the T-VSe\({}_{2}\) monolayer and the \(E_{\rm form}\) and \(E_{\rm migration}\) for the monovacancy defects in the H-VSe\({}_{2}\) monolayer are collected in Table 1. According to Table 1, the \(E_{\rm form}\) of the VSe\({}_{\rm Se}\) defect in the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are 5.31 eV and 5.60 eV, respectively. In turn, the \(E_{\rm form}\) of the V\({}_{\rm V}\) defect in the T-VSe\({}_{2}\) and H-VSe\({}_{2}\) monolayers are 10.08 eV and 13.03 eV, respectively. It should be noted that while the \(E_{\rm forms}\) of a sulfur vacancy and molybdenum vacancy in a MoS\({}_{2}\) monolayer are 2.12 eV and 6.20 eV [55], respectively, the selenide and palladium vacancies in a PdSe\({}_{2}\) monolayer are lower than 1.82 eV and 1.84 eV [59], respectively. Therefore, the formation of the chalcogen vacancy and transition metal vacancies in VSe\({}_{2}\) monolayers are less favorable than other common 2D TMDs. The migration process of the defects, which can tune the properties of 2D materials such as 2D TMDs, is also important to consider [83; 84].