Electronic and Magnetic Properties of Defected Monolayer WSe2 with Vacancies

By adopting the first-principle methods based on the density functional theory, we studied the structural, electronic, and magnetic properties of defected monolayer WSe2 with vacancies and the influences of external strain on the defected configurations. Our calculations show that the two W atom vacancies (VW2) and one W atom and its nearby three pairs of Se atom vacancies (VWSe6) both induce magnetism into monolayer WSe2 with magnetic moments of 2 and 6 μB, respectively. The magnetic moments are mainly contributed by the atoms around the vacancies. Particularly, monolayer WSe2 with VW2 is half-metallic. Additionally, one Se and one W atom vacancies (VSe, VW), two Se atom vacancies (VSe-Se), and one W atom and the nearby three Se atoms on the same layer vacancy (VWSe3)-doped monolayer WSe2 remain as non-magnetic semiconducting. But the impure electronic states attributed from the W d and Se p orbitals around the vacancies locate around the Fermi level and narrow down the energy gaps. Meanwhile, our calculations indicate that the tensile strain of 0~7% not only manipulates the electronic properties of defected monolayer WSe2 with vacancies by narrowing down their energy gaps, but also controls the magnetic moments of VW-, VW2-, and VWSe6-doped monolayer WSe2.

Indeed, structural engineering methods including irradiation by high energy particles of electron beam [29], ion beam [30] and high energy laser, and chemical etching [31,32] are the effective techniques to induce defects in the 2D materials and have been used to modify the atomic structures. Therefore, it is not only significant but also realistic to study the influence of structural defects such as vacancies on the properties of monolayer WSe 2 , which may offer us the new feature. Additionally, the 2D materials can withstand large strains before rupture and even be stretched beyond the inherent limit of 10% owing to their strong plastic deformation ability as demonstrated on monolayer MoS 2 [33,34]. Thus, strain engineering has been widely used to tune the properties of 2D materials and enhance the relevant performance in the related applications [11,17,[33][34][35][36][37][38][39]. According to Yang et al.'s study, nanoscale local strain modifies the optical band gap and changes the electronic and magnetic properties of monolayer ReSe 2 [38]. Particularly, it was reported that the non-magnetic WS 2 monolayer becomes ferromagnetic under the applied biaxial strain, and the highest magnetic moment reaches 4.85 μ B [39].
In this work, we systematically investigated the effects of vacancy defects and tensile strain on the electronic properties of monolayer WSe 2 . We calculated several vacancy defects of single atom vacancy, double atom vacancy, and big vacancies of four and seven atoms. We found that all the vacancy defects change the electronic properties of monolayer WSe 2 , while only the V W2 and V WSe6 vacancies introduce the magnetism of 2 and 6 μ B , respectively. Additionally, monolayer WSe 2 with V W vacancy converts into magnetic from non-magnetic under the external tensile strain. More importantly, the external biaxial strain effectively modulates not only the energy gaps but also the magnetic moments of V W -, V W2 -, and V WSe6 -doped monolayer WSe 2 . Our calculations suggest defected monolayer WSe 2 with vacancies as potential monolayer magnetic semiconductors.

Computational Methods
All the calculations in the present study were performed by adopting the Vienna Ab initio Simulation Package (VASP) based on density functional theory (DFT) [40,41]. The Perdew-Burke-Ernzerhof (PBE) method was used to calculate the electronic exchange interaction [42]. The ion-electron and electron-electron interactions were calculated by the projector augmented wave (PAW) method and the plane wave basis set [43,44]. The cutoff energy for the plane wave basis set was set to 300 eV, and the first Brillouin zone was sampled by the 3 × 3× 1 k-mesh based on the Monkhorst-Pack method [45]. A vacuum space of 15 Å was added along the vertical direction above the monolayer to remove the interactions between the adjacent images in the periodic slab model. Structure relaxations have been carried out until all the forces on each ion are less than 0.02 eV/Å, and the convergence criteria for the total energy were set as 10 −4 eV. The biaxial tensile strain was imposed on the vacancy defect-doped monolayer WSe 2 , which was calculated by ε = (c − c 0 )/c 0 × 100%, where c and c 0 are the lattice parameters of the strained and free monolayer WSe 2 , respectively.

Results and Discussion
Atomic Structure and Electronic Properties of Monolayer WSe 2 The most stable crystal structure of monolayer WSe 2 , denoted as 1H-WSe 2 , is shown in Fig. 1a, which shows the sandwiched layer of Se-WSe. In 1H-WSe 2 , W atoms and Se atoms occupy the sublattices of hexagonal sheet, and the Se atoms on the lower layer are directly underneath those Se atoms on the upper layer. Our calculated W-W bond length is 3.31 Å and the W-Se bond length is 2.54 Å, agreeing well with previous results [10,11]. As shown in Fig. 1b, the calculated electronic band structure and density of states (DOS) for 1H-WSe 2 indicate that 1H-WSe 2 is non-magnetic semiconducting with a direct band gap of 1.54 eV. Our calculated result agrees well with the previous result of 1.55 eV [12]. To get a more accurate band gap, we adopted the Heyd-Scuseria-Ernzerh (HSE06) [46] method to calculate the electronic band structure. The energy gap of 1H-WSe 2 calculated by HSE06 method is 2.0 eV.

The Magnetic and Electronic Properties of Defected Monolayer WSe 2 with Vacancy
We considered seven vacancy defect configurations for monolayer WSe 2 in the present study. They are the single atom vacancies including one Se atom vacancy (V Se ), one W atom vacancy (V W ), and two atom vacancies of V Se-Se , V Se2 , and V W2 . The two Se atom vacancy V Se-Se means the two Se atoms which are just beneath or above each other are removed, while the V Se2 /V W2 vacancy means that the two adjacent Se/W atoms are removed. We also considered the big vacancies of V WSe3 and V WSe6 . V WSe3 denotes the vacancy of one W atom and the nearby three Se atoms on the same layer, and V WSe6 presents the vacancy of one W atom and the nearby three pairs of Se atoms. The optimized structures of monolayer WSe 2 with vacancies of V Se , V Se-Se , V Se2 , V W , V W2 , V WSe3 , and V WSe6 are shown in the insets of Fig. 2. As we can see, the 5 × 5 × 1 supercell was used for the present study of the defected monolayer WSe 2 . Table 1 summarizes the results for the defected monolayer WSe 2 with vacancies of V Se , V Se-Se , V Se2 , V W , V W2 , V WSe3 , and V WSe6 . We can see that the W-W distances around the vacancies of V Se , V Se-Se , and V Se2 decrease respectively by 0.23, 0.52, and 0.24 Å compared with the original W-W distance in monolayer WSe 2 , which means that the W atoms around the Se atoms vacancies get close to each other. Additionally, the W-W distances around the vacancies of V W , V W2 , and V WSe3 slightly increase by 0.02, 0.01, and 0.06 Å. And those W-W distances around the single atom vacancies (V Se /V W ) are almost equal to the counterpart around the two atoms vacancies (V Se2 /V W2 ). For the bigger vacancy V WSe6 -doped monolayer WSe 2 , the W-W distances between the neighboring W atoms at the corners of the vacancy reduce by 0.58 Å, but the W-W distances at the edges of the vacancy increase by 0.44 Å. The formation energies of the seven vacancy geometries are calculated via: E van-WSe 2 and E WSe 2 are the total energies of the 5 × 5 × 1 supercell of monolayer WSe 2 with and without vacancy defect, and u i and n i (i = Se, W) are the chemical potential and number of the removed i atom. As listed in Table 1, our calculated formation energies for the seven vacancies indicate that V Se , the single Se atom vacancy, should be frequently observed on WSe 2 monolayer, consistent with the previous result of monolayer MoS 2 [17,21]. For the two Se atom vacancies of V Se-Se and V Se2 , the formation energy of V Se2 is a little higher than that of V Se-Se , indicating that V Se-Se is energetically Fig. 2 The optimized atomic structures of monolayer WSe 2 with V Se , V Se-Se , V Se2 , V W , V W2 , V WSe3 , and V WSe6 vacancies. The blue, red, and tangerine balls represent W and Se atoms on the top and bottom layer, respectively preferable than V Se2 . Hence, in the following study, only V Se-Se is studied as the two Se atom vacancies. Additionally, the formation energies for the big size vacancies are higher, which may be generated via certain kind of structural engineering techniques [29][30][31].
We then studied the electronic properties of the defected monolayer WSe 2 with vacancies of V Se , V Se-Se , V W , V W2 , V WSe3 , and V WSe6 . Figure 3 shows the electronic band structures of the six vacancy-doped monolayer WSe 2 . As shown in Fig. 3a, V Se -doped monolayer WSe 2 remains to be semiconducting, but there are obviously extra electronic states generated from the vacancy defect locating in the gap region. Consequently, the energy gap of V Se -doped monolayer WSe 2 reduces to 1.18 eV compared with that of monolayer WSe 2 . The electronic band structure of V Se-Se -doped monolayer WSe 2 is similar with that of V Se -doped monolayer WSe 2 , and their energy gaps are close. V W -and V WSe3 -doped monolayer WSe 2 shown in Fig. 3c and e also maintains the semiconducting feature but with much smaller energy gaps of 0.18 and 0.76 eV, respectively. Different from the above vacancy defects, the majority and the minority spin channels are distributed asymmetrically for the V W2 -and V WSe6 -doped monolayer WSe 2 as shown, in Fig. 3d and f. For the V W2 -doped monolayer WSe 2 , the majority spin channels cross the Fermi level, while the minority spin channels maintain semiconducting with an energy gap of 0.19 eV, and its magnetic moment is 2.0 μ B , while the V WSe6 -doped monolayer WSe 2 is magnetic semiconducting with a magnetic moment of 6.0 μ B .
We also calculated the partial density of states (PDOS) for the six vacancy-doped monolayer WSe 2 to further study their electronic properties. Figure 4 shows that the impure electronic states of V Se -and V Se-Se -doped monolayer WSe 2 are mostly located in conduction band region, and they are mainly derived from the d orbital of W atoms near the vacancy, and little from p orbital of Se atoms around the vacancy. Differently, the impure electronic bands of V W -and V WSe3 -doped monolayer WSe 2 are not only located in the conduction band region, but also being split in the valence band region. For V W vacancy, the conduction bands near the Fermi level mainly come from the d (d xy , d x2 and d z2 ) orbitals of the W atoms around the vacancy, and the valence bands near the Fermi level are mainly from the p orbital of Se atoms around the vacancy. Compared with V W -doped monolayer WSe 2 , the impure electronic states of V WSe3 -doped monolayer WSe 2 are further away from the Fermi level. The conduction bands near the Fermi level are derived from both the Se p z orbital and W d orbitals around the vacancy, while the valence bands near the Fermi level are mainly from the W d orbital around the vacancy. Additionally, W d orbital and the neighboring Se p orbital strongly interact, resulting in the hybridized states around the Fermi level. For the half-metallic V W2 -doped monolayer WSe 2 , the conduction band cross of the Fermi level mainly comes from the Se p x orbital, and the valence bands near the Fermi Table 1 The calculation results for monolayer WSe 2 with V Se , V Se-Se , V Se2 , V W , V W2 , V WSe3, and V WSe6 vacancies The W-W distance between the neighboring W atoms at the corners and at the edges around the V WSe6 , respectively c The energy gap for the half-metal Fig. 3 The electronic band structures of monolayer WSe 2 with a V Se , b V Se-Se , c V W , d V W2 , e V WSe3 , and f V WSe6 vacancies. Blue and red lines represent the majority and minority spin channels, respectively. Fermi level is set as 0 eV level are mainly derived from the W d (d x2 and d z2 ) orbital. As for the magnetic semiconducting V WSe6 -doped monolayer WSe 2 , the conduction bands and the valence bands near the Fermi level are both derived from the W d orbital near the vacancy.

The Electronic and Magnetic Properties of Monolayer WSe 2 with Vacancy Defect Under Tensile Strain
We further studied the electronic and magnetic properties of the vacancy-doped monolayer WSe 2 under the biaxial strain since the strain is an effective way to tune the electronic structures and magnetic moments of the 2D materials. We firstly studied the 1H-WSe 2 monolayer under the biaxial strain. Our calculation result shows that the biaxial strain ranging from 0 to 7% does not induce any magnetism into monolayer WSe 2 , similar with monolayer MoS 2 [34,36]. Additionally, monolayer WSe 2 still keeps the semiconducting nature with the energy gap decreasing to 0.5 eV at 7% strain, and the W-W bond length increases as the applied tensile strain increases. Then, we studied the vacancy-doped monolayer WSe 2 under the tensile strain of 0~7%. Figure 5 shows the electronic band structures for V Se -, V Se-Se -, V W -, V W2 -, V WSe3 -, and V WSe6 -doped monolayer WSe 2 under the biaxial strain of 1%, 4%, and 7%. Similar with the pristine WSe 2 monolayer, V Se -, V Se-Se -, and V WSe3 -doped monolayer WSe 2 all maintain the semiconducting feature under the biaxial strain of 0~7%, and the conduction band minima are getting closer to the Fermi level as the applied tensile strain increases. For the V W -doped monolayer WSe 2 under the biaxial strain larger than 1%, the majority and minority spin channels distribute asymmetrically. Additionally, the V W2 -and V WSe6 -doped monolayer WSe 2 both show magnetic semiconducting feature under the strain of 1~7%. Though the V Se -, V Se-Se -, and V WSe3 -doped monolayer WSe 2 still keep the semiconducting feature under the biaxial strain of 0~7%, the biaxial strain effectively controls their energy gaps as shown in Fig. 6a. The Under the biaxial strain of 0~7%, the V Se -, V Se-Se -, and V WSe3 -doped monolayer WSe 2 remain to be non-magnetic as shown in Fig. 5. In contrast, the non-magnetic V W -doped monolayer WSe 2 become magnetic with the magnetic moment of 4 μ B under the biaxial strain larger than 1%. The spin-resolved charge density shown in Fig. 7a indicates that the magnetic moment mainly arises from the W and Se atoms around the vacancies. As shown in Fig.  7b, the magnetic moment of V W2 -doped monolayer WSe 2 mainly comes from the Se atoms near the vacancy and little from the W atoms around the vacancy. When the applied strain is larger than 1%, more Se atoms are spin-polarized, resulting in the larger magnetic moment of 4 μ B . For V WSe6 vacancy defect, we can see that its magnetic moment remains to be 6 μ B under the strain of 0~6% and then decreases to 4 μ B at the strain of 7% as shown in Fig. 6b. Figure 7c demonstrates that its magnetic moments mainly arise from the six W atoms around V WSe6 . When the applied strain increases to 7%, the nearby Se atoms around the vacancy are more spin-polarized, but the local magnetic moments on the W atoms decrease. Correspondingly, the total magnetic moment of V WSe6 -doped WSe 2 decreases to 4 μ B under 7% strain.

Conclusion
In summary, we studied several vacancy defects for monolayer WSe 2 , including the single Se and W atom vacancies (V Se and V W ), double Se and W atom vacancies (V Se-Se and V W2 ), big vacancy of one W atom and the nearby three Se atoms on the same layer (V WSe3 ), and vacancy of one W atom and the nearby three pairs of Se atoms (V WSe6 ). The V Se -, V Se-Se -, V W -, and V WSe3 -doped monolayer WSe 2 all keep the non-magnetic semiconducting feature as the perfect WSe 2 monolayer, but with smaller energy gaps due to the impure electronic states locating in the energy gap region, which are attributed from the W d and Se p orbital around the vacancies, while V W2 and V WSe6 vacancies induced magnetism into monolayer WSe 2 with magnetic moments of 2 and 6 μ B , respectively. Particularly, monolayer WSe 2 with V W2 vacancy converts into half-metal from semiconducting. More importantly, our calculation results show that the external biaxial strain effectively tunes the magnetism and electronic properties of monolayer WSe 2 .