Due to unique electronic properties and chemical stability, tremendous research efforts have been focused on layered transition-metal dichalcogenides (TMDCs) considered as promising potential candidates on electronic, optoelectronic, and photovoltaic applications.^[1–8] As a popular member of TMDCs, MoS₂ monolayer is a two-dimensional (2D) material with bandgap suitable for logic device application,^[9–11] and can be easily synthesized through chemical growth.^[12,13] Unfortunately, when increased the number of layers and applied small strain, MoS₂ undergoes a crossover from direct to indirect band gap, which poses great challenge in making robust optoelectronic devices based on MoS₂.^[14,15] Recently, as a new member of TMDCs, ReS₂ monolayer was experimentally produced through chemical exfoliation process and attracted a lot of attentions.^[16–24] The vanishing interlayer coupling in ReS₂ make its direct band gap (1.55 eV) no dependence on the number of layers. Such insensitivity of band gap on layers enables fabrication of 2D optoelectronic devices without the need for monolayers.^[16]

For prospective applications in low-dimensional spintronic devices, considerable efforts have been devoted to explore the magnetic behavior of TMDCs. It is well known that adding the spin degree of freedom to conventional semiconductor charge-based electronics will add substantially more capability and performance to electronic products, and the operation of spintronic devices require generation and detection of tunable spin currents, which can ideally be done using a ferromagnetic semiconductor.^[25–28] However, pristine (P-) ReS₂ monolayer is intrinsically nonmagnetic. So developing approaches to effectively induce and manipulate the magnetism are crucial for facilitating application of monolayer ReS₂ on low-dimensional spintronic devices. It is well known that vacancy defects in chemically grown 2D materials are noticeable due to the imperfection of the growth process, and can be controllably achieved by chemical vapor deposition, field evaporation, and electron irradiation methods.^[29,30] These structural defects can have a significant influence on the magnetic properties of 2D materials. Recently, Horzum et al. found that some commonly observed vacancies in ReS₂ monolayer, sulfur defects, do not result in any spin polarization, while Re-containing defects induce magnetization with a net magnetic moment of 1–3 μ_B.^[21]

Strain has been known as an effective mechanism for controlling magnetic properties of 2D materials, which can sustain larger strains than bulk crystals. For example, MoS₂ monolayer and graphene have been reported to be strained up to 11% and 15% before rupture.^[31–33] Some theoretical reports have proposed that the appropriate tensile train can induce or manipulate the magnetic properties of vacancy-doped MoS₂ monolayers.^[34–36] The tensile strain appears to be more effective in modulating the magnetic properties of 2D TMDCs than compressive strain, and Re-containing vacancies in ReS₂ monolayer result in a sizable magnetic moment and have higher formation energy than Sulfur vacancies.^[21] Therefore, in this letter we only investigate a possible emergence of magnetism in monosulfur vacancy (V_S) and disulfur vacancy (V_2S) doped ReS₂ monolayers under biaxial tensile strain, and explore the physical mechanism about strain induced magnetism in vacancy-doped ReS₂ monolayer.

Our calculations were performed in the frame work of density functional theory (DFT) as implemented in the Vienna ab-initio simulation package (VASP).^[37,38] The generalized gradient approximation (GGA) within Perdew–Burke–Ernzerhof (PBE) formalism^[39] was employed for exchange-correlation potential. The projector augmented wave (PAW) method^[40] and a plane-wave basis set with an energy cutoff of 500 eV were used in the calculations. A 5×5×1 Monkhorst–Pack k-mesh was used for an integration over the Brillouin zone. The convergence criterion of self-consistency and atomic Hellman–Feynman forces were set to be 10⁻⁵ eV and 0.01 eV/Å, respectively. A large vacuum spacing (at least 15 Å) was used to avoid interaction between ReS₂ layers. The defects were obtained by removing the relevant atoms from a 2×2 supercell. The convergence of defect formation energy was checked by varying the size of supercell.

Calculations with a 3×3 supercell show that the formation energies of defects all are converged on the order of only 0.01 eV. The strain ɛ can be defined as ɛ = (a − a₀)/a₀ × 100%, where a and a₀ is the lattice constants of strain and unstrained system, respectively. Under each biaxial strain, all atoms are fully relaxed with fixed lattice constant.

Due to the distorted 1T structure, there are two and four inequilavent Re and S sites on the ReS₂ monolayer, which are possible for fabricating point defects as shown in Fig. 1(a). We only investigate the relative stability of Sulfur point defects by calculating the formation energy

Fig. 1.

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	Fig. 1. The top view of optimized atomic structures of (a) pristine-, (b) VS-, and (c) V2S-doped ReS2 monolayers. Missing S atoms are shown by empty gray circles. In panel (a), the numbered atoms with black and red colors indicate the possible inequivalent sites for creating Re and S vacancies, respectively.

Due to the distorted lattice structure of ReS₂, the formation of S-vacancies is site dependent. The values presented in Table 1 are the formation energies of those vacancy sites with the lowest formation energy under several strains.

Table 1.

Table 1.

Table 1. Formation energies Ef of VS and V2S doped ReS2 monolayers under several strains for the Re-rich and the S-rich cases. . ɛ VS V2S Re-rich S-rich Re-rich S-rich 0% 1.21 2.08 2.87 4.62 1% 1.36 2.23 3.15 4.90 2% 1.47 2.34 3.36 5.11 3% 1.56 2.43 3.51 5.26 4% 1.62 2.49 3.59 5.34 5% 1.66 2.53 3.61 5.36 6% 1.66 2.53 3.57 5.32 7% 1.64 2.51 3.46 5.21 8% 1.54 2.41 3.06 4.81

Table 1. Formation energies Ef of VS and V2S doped ReS2 monolayers under several strains for the Re-rich and the S-rich cases. .

The unstrained systems are first investigated. Through our calculation results, we find that Vs, shown in Fig. 1(b), is more easily created with the formation energy of 2.08 eV (S-rich condition) in four different monosulfur vacancies. In the plenty of possibility to create disulfur vacancy, V_2S shown in Fig. 1(c) has the lowest formation energy with the value of 4.62 eV (S-rich condition), which is similar with previous calculation result.^[21] Our results show that the formation energy of V_2S is larger than twice that of V_S, suggesting that single sulfur vacancy have no tendency to combine to form the disulfur vacancy and can be introduced more easily in ReS₂ monolayer. The magnetic energy, which represents the energy gained from the spin polarization, is calculated as ΔE_m = E_NM − E_MAG for vacancy-doped ReS₂ monolayer. That the obtained magnetic energy is zero indicates that V_S and V_2S do not induce magnetism in unstrained ReS₂ monolayer. The calculated result is consistent with previous report.^[21]

Next, the strain from 0% to 8% is applied for vacancy-doped ReS₂ monolayer. V_S doped systems remain nonmagnetic when strain is smaller than 8% and then transform to magnetic state. As displayed from the spin resolved charge density in Fig. 2(a), the distribution of spin density is predominantly located at three Re atoms around vacancy, in which anti-ferromagnetic (AFM) couplings are introduced between the spins of Re1 atom (0.263μ_B), Re2 and Re3 atoms with the magnetic moment of −0.065μ_B and −0.188μ_B. Furthermore, the other atoms at vicinity of vacancy also exhibit visible spin polarizations, so that the system exhibits a magnetic behavior with the total magnetic moment of zero. The magnetic energy is only 12 meV, indicating the unstability of the magnetic state at room temperature. The magnetic behavior is different from the formation of a magnetic moment of 2μ_B in V_S-doped MoS₂ monolayer under 9% tensile strain.^[35] In strained V_S-doped MoS₂ monolayer, due to the symmetry of MoS₂ monolayer, three Mo atoms at vicinity of vacancy toward to the vacancy spot form an equilateral triangle, and have to be ferromagnetically (FM) coupled each other under consideration of spin frustration. However, due to distorted 1T structure in ReS₂ monolayer, three Re atoms around S-vacancy (Re1, Re2, and Re3) are inequivalent, and the distances between Re1, Re2, Re3 atoms and missing S atom in P-ReS₂ monolayer are 2.47 Å, 2.48 Å, and 2.46 Å, respectively. This condition improves the possibility of AFM coupling behavior in strained V_S-doped ReS₂ monolayer.

Fig. 2.

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	Fig. 2. The spin resolved charge densities for (a) VS-doped ReS2 monolayer under 8% strain and (b) V2S-doped ReS2 monolayer under 7% strain. The isosurface value is 0.001 e/Å. Red (green) colors represent spin up (down) electrons. The spin polarizations are mainly contributed from the numbered atoms in panels (a) and (b).

For V_2S-doped ReS₂ monolayer, the system remains nonmagnetic when strain is smaller than 6%, and then transforms to a magnetic behavior, in which the spin magnetizations are mostly contributed from Re1 and Re2 atoms around vacancy with local moments of 0.308μ_B and −0.308μ_B, shown in Fig. 2(b). When strain continuously increases to 7%, FM behavior with the total magnetic moment of 1.60μ_B appears in system, in which the local moments are mainly contributed by the 5d orbital of Re1 (0.238μ_B), Re2 (0.238μ_B), Re3 (0.125μ_B), Re4 (0.125μ_B), Re5 (0.093μ_B), and Re6 (0.093μ_B) atoms. We find that the spatial extension of spin density under 6% strain is smaller than that under 7% strain. The magnetic energy (21 meV) under 7% strain is larger than that (12 meV) under 6% strain, indicating stronger magnetic stability with increasing strain.

To get an insight into the strain-induced magnetism, we study the strain effect on electronic structures and stability. Figure 3 shows the strain dependence of band gap and formation energy in vacancy-doped ReS₂ monolayer. All band gaps decrease with increasing tensile strain. For the P-ReS₂ monolayer, the band gap drops steadily with the strain, only 0.78 eV at 8%, which is similar to linearly dropping in P-MoS₂ monolayer.^[34] For V_S- and V_2S-doped ReS₂ monolayers, band gaps are smaller than that of P-ReS₂ monolayer and drop to zero at 8% and 7% strains, respectively. It is interesting that under the strain of 6%, there has been a magnetic state (Re atoms at vicinity of vacancy are AFM coupled with each other) when band gap is 0.19 eV in V_2S-doped ReS₂ monolayer. This is different from the case for vacancy-doped MoS₂ monolayer, where the critical strain value for band gap closing agrees with that for magnetic state emergence.^[34] As shown in Fig. 3(b), the formation energy increase and reach maximum at 5% strain, then drop slightly. Encouragingly, under critical tensile strain of magnetism emergence, the vacancy stability is very close to zero strain, which enables fabrication of magnetic V_S- and V_2S-doped ReS₂ monolayers in experiment.

Fig. 3.

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	Fig. 3. (a) Band gap and (b) formation energy as a function of tensile strain.

To explore the physical mechanism of the emergent magnetic behavior of defective ReS₂ monolayer under tensile strain, the total density of states (DOS) and projected DOS are illustrated in Fig. 4. Comparing the DOS of P-ReS₂ monolayer, we find that the formation of V_S produces a defect level locating close to the valence band maximum (VBM), reducing the band gap from 1.43 eV (band gap of P-ReS₂ monolayer) to 1.19 eV, while the formation of V_2S produces two defect levels locating at close to the VBM and the conduct band minimum (CBM), dramatically reducing the band gap of P-ReS₂ monolayer to 0.85 eV. All these defect states contribute from 5d orbitals of Re atoms at the vicinity of vacancy. With increasing strain, all the conduct band levels shift downwards and start to interact with the valence band levels around the Fermi level. Under the strain of 8%, V_S-doped system exhibits half-metal behavior, with a band gap of 0.08 eV in majority spin channel and a partially occupied band in minority spin channel. Under the strain of 7%, V_2S-doped system is shown to be metallic, i.e., the band gap collapses. The spin majority bands and minority bands split, and magnetism appears.

Fig. 4.

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	Fig. 4. Total DOS (black color) of pristine ReS2 (a), VS-doped ReS2 monolayer under 0% (b) and 8% (c) strains, and V2S-doped ReS2 monolayer under 0% (d) and 7% (e) strains. The corresponding partial DOS (red color) presents the sum of 5d orbitals of nearest Re atoms around the vacancy. Fermi level is denoted by vertical dashed line.

Strain seems indispensable for the magnetism emerging in vacancy-doped ReS₂ monolayer. However, it is important to make sure that magnetism appears before crystal structure breaks. We continue to investigate the V_S- and V_2S-doped ReS₂ monolayers in lager strain condition. As shown in Fig. 5, we find that both the structure break under the same strain of 9%, which is slightly beyond the strain for emergence of magnetism. Although the critical strain that makes material breaking in vacancy-doped ReS₂ monolayer is smaller than that in vacancy-doped MoS₂ monolayer,^[34] it is still plausible to strain-magnetism in vacancy-doped ReS₂ monolayer in experiment.

Fig. 5.

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	Fig. 5. (a) VS- and (b) V2S-doped ReS2 monolayers under 9% strain.

We have systemically explored the strain-induced magnetism in ReS₂ monolayer with V_S and V_2S. When the tensile strain of 8% is applied, V_S-doped ReS₂ monolayer results in magnetic half-metal phase, in which Re atoms around vacancy anti-ferromagnetically coupled each other. For V_2S-doped ReS₂ monolayer, system has a phase transition from magnetic semiconductor under 6% strain to magnetic metal under 7% strain. Furthermore, the partial extension of spin density in ReS₂ monolayer is increasing with increasing strain. Our results show that strain engineering can provide an effective approach to tune magnetism in vacancy doped ReS₂ monolayer for application on low-dimensional spintronic devices.