Strain tunable magnetism in SnX2 (X = S, Se) monolayers by hole doping

By first-principles calculations, the magnetism of hole doped tin dichalcogenides SnX2 (X = S, Se) monolayers is systematically studied. It is found that a phase transition from nonmagnetic to ferromagnetic ground state appears once above the critical hole density (~1014 cm−2). The spin magnetic moment can maintain a magnitude of 1.0 μB/hole with excellent stability of ferromagnetic state. Furthermore, we demonstrate that strain is very useful to modulate the DOS near the valence band, resulting in the reduction of the critical hole density to ~1013 cm−2 when the strain reaches 4% (6%) in SnS2 (SnSe2), which can be realized in common field effect transistors. Moreover, the phonon dispersion calculations for the strained SnX2 monolayers indicate that they can keep the dynamical stability under the hole doping. Therefore, the strain tunable magnetic transition in hole doped tin dichalcogenides indicates their potential promising applications in spintronic devices.

Scientific RepoRts | 6:39218 | DOI: 10.1038/srep39218 phenomena. In semiconductor industry, carrier doping is considered as an effective approach to modulate E F , which performs obvious advantages including their completely free of structure disorder and remarkably simple to tune physical properties 26 , compared to previously mentioned approaches. By hole doping in 2D systems, Cao et al. 27 reported the theoretical investigation of magnetism in monolayer GaSe, and Huang et al. 28 demonstrated a robust half-metallic spin-polarized state in silion phosphides (C2/m Si 1 P 3 ). In the previous work, we also explored the hole doping effect on magnetic properties in 2D graphene-like C 2 N 29 .
In recent years, chemically stable and environmentally friendly semiconducting tin dichalcogenides SnX 2 (X = S, Se) have been widely studied for their excellent electrical, optical and magnetic properties, such as lithium ion batteries, photovoltaic devices, as well as field effect transistors [30][31][32][33] . These studies generally indicated the unusual electronic structure with the high DOS and small dispersion near the top of valence band, which would provide a probability for magnetic phase transition. Although several calculations have demonstrated the magnetism in SnX 2 nanostructures, such as manipulating SnSe 2 armchair nanoribbons via edge hydrogenation 22 and doping metal elements (Li, Mg and Al) in single-layer SnS 2 34 , the disadvantage of which would affect their performance in further experiments for their unexpectable complexity and uncontrollability. In this work, by high density hole doping, we mainly study the magnetic characteristic of pristine and strained SnX 2 (X = S, Se) monolayers. Results indicate that ferromagnetic ground state with outstanding stability of spin polarization can be obtained by hole doping. Particularly, by strain engineering, the critical hole density can be reduced to 10 13 cm −2 , which is an order of magnitude smaller than that of pristine structure. Therefore, SnX 2 monolayers can be considered as a viable candidates for spintronic devices.

Computational method
To study the electronic and magnetic properties of the SnX 2 monolayers, density functional theory (DFT) calculations were performed using the Projector-Augmented Wave (PAW) pseudopotential implementation of the Vienna Ab Initio Simulation Package (VASP) [35][36][37] . Electron exchange and correlation effects were described by the generalized gradient approximation (GGA) functional of Perdew-Burke-Ernzerhofer (PBE) formula 38 . The energy cutoff for the plane-wave basis was set as 550 eV on the 11 × 11 × 1 Monkhorst-Pack k-point grid for all simulations. The convergence threshold was 1 × 10 −5 eV for the electronic self-consistent field iterations. The atomic positions were optimized until the maximum Hellma-Feynman force on each atom was less than 10 −2 eV Å −1 . A vacuum spacing of 20 Å was placed to avoid the interactions between the monolayers and its periodic images. Moreover, to examine the dynamical stability of SnX 2 monolayers with doping, the phonon dispersion was calculated by density functional perturbation theory in VASP.

Results and Discussion
SnX 2 (X = S, Se) monolayers with the 1T structure are the hexagonal crystal structure of the CdI 2 -type, as shown in Fig. 1. The SnX 2 monolayers consist of three atomic sublayers with the covalently bonded layer of X-Sn-X. The central Sn atom bonds to six nearest-neighbor X atoms located in the top and bottom sublayers. The structural stability of monolayer SnS 2 has been theoretically predicted by Zhuang et al. 32 For all simulations, the primitive rhombic unit cell was used as the computation unit cell. Before investigating the electronic and magnetic properties, geometric structures of SnX 2 were optimized, and atomic positions were relaxed to zero pressure following the convergence criteria. The optimized lattice constant of SnS 2 and SnSe 2 is 3.700 Å and 3.871 Å, which is in accordance with previous calculations by PBE formula 32,33,39 .
The band structure and DOS of SnX 2 monolayers are displayed in Fig. 2. It is obvious that the conduction band minimum (CBM) is located at the M point and the valance band maximum (VBM) lies between Γ − M point, which means both the SnS 2 and SnSe 2 monolayers are indirect band gap semiconductors. The indirect band gap of SnS 2 (~1.60 eV) is almost double of SnSe 2 (~ 0.83 eV), which is consistent with previous reported value (1.57 eV for SnS 2 and 0.79 eV for SnSe 2 ) 39 by the same method. Due to the underestimation by PBE formula, the band gap is smaller than that of previous reports by simulations with Heyd-Scuseria-Ernzerhof (HSE) hybrid functional calculations and quasiparticle self-consistent GW methods 32 . In order to analyze the contribution of each orbit of atom, the total DOS, together with the 5s, 5p, 4d-orbit partial DOS (PDOS) of Sn atom, and s, p-orbit PDOS of S (Se) atom are depicted in the right diagrams of Fig. 2. It is clear that the high DOS is located around the VBM, which is mainly contributed by S-3p (Se-4p) states, besides, the CBM is co-contributed by Sn-5s and S (Se)-p states.
It is well known that the high DOS around the E F would provide the possibility to develop a spontaneous ferromagnetism. We applied carrier doping to investigate the possible ferromagnetism in SnX 2 monolayers. Figure 3 shows the local magnetic moment (μ B ) per hole with the various hole density. At the nonmagnetic state, the magnetic moment is nearly zero. Once above the critical hole density, the local magnetic moment increases rapidly to a constant value with ~1.0 μ B /hole, implying the transition from nonmagnetic to ferromagnetic ground state. The critical hole density (p c ) of SnS 2 and SnSe 2 is about 2.2 × 10 14 cm −2 and 3.2 × 10 14 cm −2 , respectively. In order to check the stability of ferromagnetic state, the spin polarization energy per hole, Δ E p , is discussed, which is defined by the energy difference between the spin-polarized state (E sp ) and non-spin-polarized state (E nonsp ) for each hole, i.e., Δ E p = (E sp − E nonsp )/number(hole). The hole density dependence of spin polarization energy per hole is also shown in Fig. 3. At the spin polarized states, Δ E p decreases monotonically with the increment of the hole density. At the hole density of 7.0 × 10 14 cm −2 , Δ E p is about − 103 and − 52 meV/hole for SnS 2 and SnSe 2 , respectively, much higher than those in GaSe (− 3 meV/hole) 27 and Si 1 P 3 (− 24 meV/hole) 28 , indicating that the ferromagnetic ground state in hole doped SnX 2 would be much more stable.
We also studied the band structure and DOS of hole doped SnX 2 monolayers. At the first stage, although hole doping leads to the E F shift down to around the VBM, the spin up and spin down bands completely overlap, indicating the nonmagnetic structure. Once the hole density exceeds the critical value, the spin splitting appears around E F , resulting in the magnetic transition. When the hole density continue to increase, the spin up bands shift down, meanwhile, spin down bands shift up gradually. Figure 4 shows the typical electronic structures of hole doped SnX 2 in the spin polarized state with the hole density of 4.2 × 10 14 cm −2 for SnS 2 and 5.4 × 10 14 cm −2 for  To understand the magnetization mechanism of hole doped SnX 2 monolayers, we notice that the magnetic moments are contributed mainly by the p states at the VBM. Like the previous study on d 0 ferromagnetic semiconductors, such as ZnO 12 , GaSe 27 and Si x P y 28 , the band-picture model also can be employed to explain the magnetization mechanism of hole doped SnX 2 monolayers. The ferromagnetic characteristic would be achieved through a p-d hybridization-like p-p interaction between the p states of chalcogens. Figure S1 depicts the evolution of D(E F ) and the energy difference ΔE of the two spin-type bands around the VBM by hole doping for monolayer SnX 2 . A large exchange-splitting of the two spin-type bands appears due to the strong exchange field in the ferromagnetic phase, and the ΔE increases with the increment of hole density when the D(E F ) is large enough, which means a large strength of exchange interaction J in these systems. Consequently, the magnetism of SnX 2 monolayers can be induced by hole doping.
Currently, ultrahigh carrier density accumulation could be achievable using electric-field control in FETs in experiments, including conventional solid state gated voltage in FETs and electric double layer transistors (EDLTs) by using polymer electrolytes or ionic liquid as gated dielectrics. By using EDLTs, an accumulation of extremely high carrier density could up to 10 14 cm −2 in graphene and transitional metal dichalcogenides [40][41][42] , while ~10 13 cm −2 for conventional solid state gated voltage in FETs 43,44 . But doping with high hole density often brings unexpected uncontrollability and difficulty, even by EDLTs. Therefore, it is crucial to reduce the hole density in SnX 2 monolayers for their further practical applications.
The detail DOS of SnSe 2 in the small region from − 1.0 to − 0.4 eV are inset in Fig. 2b. It is find that the high DOS near VBM is located at the deep level, which brings us an insight to analyze the reason why such high critical hole density (~10 14 cm −2 ) must be needed to obtain magnetic transition. Therefore, we devote to modulating the high DOS to the top of valence band. As we known, band structure is highly sensitive to the external conditions such as temperature, pressure or strain, especially in 2D layered structure. They often lead to dramatically changes about electronic, magnetic and optical properties. In particular, the strain engineering is commonly used to tune the band structure, due to the extraordinary break strength and structural stability in a wide range of strain [45][46][47][48][49] . Zhou et al. recently reported the effects of in-plane biaxial strain on the electronic structure of single-layered SnS 2 50 . They found that the tensile strain could result of a larger value of states around the Fermi level. So we believe that the tensile strain should be an effective way to reduce the critical hole density.
In this study, the biaxial tensile strain [2%, 10%], with the increment of 2%, was applied for the SnX 2 monolayers. Figure 5 shows the evolution of DOS near valence band edges for strained SnX 2 monolayers, in addition with unstrained states. It is clear that the DOS near the VBM increase with the increment of strain. When the strain reaches 4% for SnS 2 and 6% for SnSe 2 , the DOS around the VBM are similar, and the VHSs occur, due to the Mexican hat-like band edges around Γ point. Once the E F shift down to the top of valence band, such unusual DOS would lead to the reduction of the hole density, and transitions to magnetism. So our next work studies the local magnetic moment and spin polarization energy dependent on the hole density under various strain strength.
The relations between the spin magnetic moment/hole and the hole density are shown in Fig. 6a and b. At the initial strained SnX 2 monolayers without hole doping, it is nonmagnetic at the ground state. The magnetic moment (~1.0 μ B /hole) appears at the lower hole density in comparison to the unstrained SnX 2 . The inset map plotted relations between the critical hole density and the tensile strain. The critical hole densities reduce sharply in the small range of strain, i.e., 0-4% for SnS 2 and 0-6% for SnSe 2 , and then maintain about 1.5 × 10 13 cm −2 of SnS 2 and 2.0 × 10 13 cm −2 of SnSe 2 , respectively. These values are an order lower than that of unstrained structures, and could be realized in common FETs. Moreover, with the biaxial strain applied, the tendency of critical hole density is consist with that of DOS around the VBM described in Fig. 5. In addition, Fig. S1 also shows the D(E F ) and ΔE for strained SnS 2 (SnSe 2 ) monolayers. It is clear that the SnX 2 monolayers under biaxial strain are more favorable to achieve the spin splitting due to the greatly enhanced D(E F ). Next, we check the stability of spin polarization of the strained structure, as shown in Fig. 6c and d. It is obvious that the spin polarization energy/ hole reduce monotonously at the magnetic state in a large range of hole density. Compared to the unstrained SnX 2 monolayers, the lower spin polarization energy/hole can be obtained with the increment of strain, and finally the spin polarization energy nearly tend to the constant value. For example, at the hole density of 7.0 × 10 14 cm −2  with 6% strain applied, values decrease monotonically to − 180 meV of SnS 2 and − 160 meV of SnSe 2 , which is far more small than that of the unstrained structure. Therefore, by applying tensile strain, the magnetism of SnX 2 monolayers would have promising applications for their achievable hole doping density and stable magnetic phase transition.
In addition, the critical hole density, magnetic moment and spin polarization energy are summarized for SnX 2 in Table 1, in comparison with GaSe 27 and C2/m Si 1 P 3 28 . The magnetism in unstrained structure requires higher hole density doping. But applying tensile strain (≥ 4% for SnS 2 and ≥ 6% for SnSe 2 ), the critical hole density of SnX 2 monolayers is smaller that of Si 1 P 3 and GaSe. Furthermore, SnX 2 monolayers perform better stability of spin polarization under both unstrained and strained state. Therefore, for 2D SnX 2 monolayers, hole doping can induce stable spin polarization in a large range of hole density, and the critical hole density and spin polarization energy can be improved greatly by strain engineering.
The stability of hole doped SnX 2 monolayers is crucial for their applications on magnetism. So, we have also carried out the phonon dispersion calculations to study their dynamic stabilities. The phonon dispersions of SnS 2 (SnSe 2 ) monolayers under the tensile strain of 4% (6%) are shown in Fig. 7. No imaginary frequency is found under the tensile strain, indicating the SnS 2 (SnSe 2 ) monolayers under tensile strain are stable. At 0.1 hole per unit cell doping, large enough to induce magnetism in SnS 2 (SnSe 2 ) monolayer with tensile strain of 4% (6%), there is still no imaginary frequency, except the frequency softening comparing to the SnS 2 (SnSe 2 ) monolayers without doping. Therefore, these results indicate that hole doping in SnX 2 monolayers have negligible effect on their structural stabilities, and will provide feasible theoretical predictions for practical applications in spintronic devices.

Conclusions
In conclusion, electronic and magnetic properties of the SnX 2 (X = S, Se) monolayers are studied by DFT calculations. It is demonstrated that hole doping (~10 14 cm −2 ) can drive a phase transition from nonmagnetic to ferromagnetic ground state. The electron spin magnetic moment can reach a constant magnitude (~1.0 μ B /hole) with excellent stability of ferromagnetic state. Besides, the half-metallic characteristic can be modulated by hole doping. Furthermore, by introducing tensile biaxial strain to SnX 2 monolayers, the critical hole density can reduce to ~10 13 cm −2 , which could be realized by conventional solid high dielectric FETs. The higher spin polarization energy indicates better stability in comparison with the pristine structure. In addition, the hole doped SnX 2 monolayers still remains stable by phonon dispersion calculations. Therefore, investigation of the magnetism for hole doped SnX 2 monolayers under the strain will provide an achievable idea for 2D functional materials in spintronics.