Introduction

Two-dimensional (2D) materials have been drawing increasing attention because of their potential applications in next generation electronics. Graphene1, which is currently the most widely investigated 2D material, possesses high mobility2 and displays rich physics3,4,5 driven by the presence of Dirac cone dispersions near the Fermi level at K and K′ points at the Brillouin zone corners6. However, graphene is a gapless system and the spin on the Dirac cone is a pseudospin, limiting its usefulness in semiconductor industry or spintronics device technologies. For these reasons, it is clearly important to find new 2D materials, which contain not only a large band gap but also support spin-polarized states.

A number of 2D materials with spin-polarized states have been synthesized. These include, HgTe quantum wells and related systems7,8, silicene9 and transition-metal dichalcogenide (MX2) family10,11,12. While HgTe quantum well systems and free-standing silicene and related compounds are predicted to be 2D quantum spin Hall (QSH) insulators with small band gaps13,14,15, MX2 family harbors much larger band gaps and a monolayer MoS2 transistor has been reported recently16. Radisavljevic et al. used a hafnium oxide gate dielectric to demonstrate a room-temperature monolayer MoS2 mobility of at least 200 cm2 V−1 s−116. In monolayer MoS2, both the valence and conduction band extremes are located at the corners (K-points) of the 2D hexagonal Brillouin zone17,18,19, resulting in two inequivalent valleys for low energy carriers which are similar to graphene. Bulk MoS2 has been demonstrated with a large indirect gap, while its monolayer thin-film exhibits a direct band gap20,21. Unlike graphene, monolayer MoS2 lacks spatial inversion symmetry and has a strong spin-orbit coupling (SOC) originating from the d orbitals of the heavy transition-metal atoms19. It has been shown that due to the presence of a large direct band gap, inversion symmetry breaking and a strong SOC, optical pumping with circularly polarized light can achieve a valley polarization of 30% in pristine monolayer MoS222,23. In order to achieve continuous and reversible control of valley-contrasting properties, Wu et al. investigated bilayer MoS2 by using polarization-resolved photoluminescence24. In bilayer MoS2 the circularly polarized photoluminescence can be continuously tuned from −15% to 15% as a function of gate voltage, whereas in structurally non-centrosymmetric monolayer MoS2 the photoluminescence polarization is gate independent24. Recently, Cheng et al. have demonstrated that monolayer MoS2 is a 2D diluted magnetic semiconductor25 in which valley polarization can be induced and controlled by magnetic doping26. They have also investigated the interlayer coupling in ultra thin MoS227 and the Rashba effect in MXY (M = Mo, W and X, Y = S, Se, Te) compounds28. A field-dependent unconventional Hall plateau sequence29 is predicted theoretically in trilayer MoS2 and multilayer MoS2 has been shown to be a superconductor at optimal doping30.

Motivated by the intense current interest in 2D transition-metal dichalcogenides, the present work undertakes a comprehensive first-principles investigation of ultra-thin films of these materials and systematically analyzes their spin-polarization characteristics and how these evolve with film thickness and delineates how different chalcogens in Mo and W based films modify their electronic structures. When the SOC is included in the computations, band structures display spin-split valence bands around the K-points for monolayer MX2 thin-films which do not possess inversion symmetry. The spin of the top valence states is found to be aligned along out-of-the-plane direction with the spin pointing up at the K-point and down at the K′-point. The inversion symmetry is restored in bilayer thin films. Here, we find that when an out-of-the-plane electric field Ez is applied, the inversion symmetry is removed and the spin degeneracy at the K-point is lifted. The spin-splitting energy increases with increasing Ez and saturates for Ez larger than a critical field Ec. In the high temperature limit, spin polarization for slabs with even number of layers is zero, while slabs with odd number of layers obey 1/x decay with increasing film thickness x. Finally, we show that the position of the valence band maximum shifts from the K-point to the Γ point as the film thickness increases. In particular, we confirm a transition from an indirect to a direct band gap as the thickness is reduced to a monolayer in MoS2, in agreement with recent experimental findings20,21.

Computational Details

MX2 compounds crystallize in a layered 2H prototype structure with space group P63/mmc (Fig. 1), which contains two inverse MX2 layers. In each layer, intermediate M atom is sandwiched between X atoms, forming strong ionic bonds within a trigonal local structure (Fig. 1 (b)). While the intralayer M-X bonding is strong, the interlayer bonding is weak as it arises from van der Waals forces (Fig. 1 (c)). Unlike the bulk crystal, the space group of monolayer MX2 is not P63/mmc but reduces to due to loss of inversion symmetry. The inversion symmetry is restored in bilayer thin films: In Fig. 1, we used MoS2 as an example where the inversion center is marked by red dot in Fig. 1 (c) lies between two adjacent MoS2 layers.

Figure 1
figure 1

(a) Top view of a monolayer MoS2 thin-film. Gray and yellow dots denote Mo and S atom, respectively. (b) Trigonal local structure of MoS2. (c) Side view of bilayer structure of a MoS2 thin-film. Red dot marks the spatial inversion symmetry point. (d) Band structure of a MoS2 monolayer. Black and red dashed lines give results with and without SOC, respectively. (e) Spin polarization near the K-point around EF. Color bar denotes strength of spin polarization. (f) Spin decay behavior of polarization strength near the K-point.

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The electronic structures of MX2 films were computed using the projector augmented wave method31,32 as implemented in the VASP33,34,35 package within the generalized gradient approximation (GGA)36 scheme. The SOC effects were included in a self-consistent manner. Slabs with vacuum thickness larger than 15 Å were used to model the thin films. A 15 × 15 × 1 Monkhorst-Pack k-point mesh was used in the computations. The experimental lattice constants were used10,37,38. Note that although experimental lattice constants are available for all six compounds studied, the experimental atomic positions are not available for all these compounds. In order to meaningfully unfold trends in electronic structures as a function of the number of layers or strain, it is important of course to use a consistent theoretical framework in computations. Accordingly, relaxed theoretical atomic positions were systematically used in making such comparisons. The atomic positions were relaxed until the residual forces were less than 0.005 eV/Å.

Results and Discussion

Fig. 1 (d) shows the electronic band structure of monolayer MoS2. Without SOC (red dashed line), a direct band gap of ~1.6 eV is obtained between the valence band maximum and conduction band minimum at the K-point and all bands are found to be spin degenerate. When the SOC is turned on (black line), the top valence bands display a significant splitting (~ 150 meV) at the K-point due to the breaking of spatial inversion symmetry (E(k, ↑) = E(−k, ↑)). In sharp contrast, the top valence bands are spin degenerate at the Γ and M points. This is because Γ and M points are time-reversal invariant while K-point is not. Focusing on the split band, we find that it displays nearly 100% out-of-the-plane spin polarization (Fig. 1 (e)), which is consistent with previous results19,39,40. The two bands at K and K′ have opposite spin polarizations, obeying the constraint of time reversal symmetry (E(k, ↑) = E(−k, ↓)). Fig. 1 (f) shows the degree of spin polarization for the top valence band near the K- point. The bands are nearly 100% spin polarized at the K-point and decay very little as the momentum deviates from the K-point within 0.5 Å−1.

We emphasize that the presence of a robust 100% spin-polarized band at the top of valence bands in an MX2 monolayer results from symmetry constraints of mirror symmetry40 and the broken inversion symmetry of the underlying crystal structure and therefore, this polarization cannot be switched off by an external field. In this connection, we have investigated MX2 bilayers in which the inversion symmetry is restored with a center of inversion in the structure lying between the two layers in the bilayer. Fig. 2(a) shows the band structure of a MoS2 bilayer. Since the inversion symmetry is now restored, all bands are spin-degenerate. The energy difference between the top of the valence bands at the K and Γ points, ΔKΓ, becomes smaller. Notably, the splitting at the Γ point is due to the interaction between the two MX2 layers, but the SOC is responsible for the splitting at the K-points [Fig. 2(b)].

Figure 2
figure 2

(a) Band structure of bilayer MoS2. Black and red dashed lines give results at Ez = 0 and Ez = 0.3 (eV/Å), respectively. (b) Evolution of band structure near the K-point (red dashed box in Fig. 2(a)) for varying external electric field, Ez. Color scale gives the degree of spin polarization. Contributions from the first and second layers in MoS2 bilayer are marked with numbers 1 and 2. The unit of electric field is eV/Å. (c) Spin-splitting energy at the K-point as a function of Ez. Blue line gives the intralayer splitting energy, Δintra, While the red line gives the interlayer splitting, Δinter.

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The inversion symmetry of bilayer MX2 can, however, be removed by an out-of-the-plane external field, Ez, which lifts the spin degeneracy of states away from the time-reversal-invariant points in the Brillouin zone. Since K-points are not time-reversal-invariant points, the spin-splitting at the top of the valence bands can be manipulated via Ez. Fig. 2(b) shows how the bilayer MoS2 band structure evolves with increasing Ez. For non-zero Ez, there are four spin-polarized states in a bilayer, two of which come from the first layer and the other two from the second layer in the bilayer system. We define the interlayer splitting energy, Δinter, to be the energy difference between the first-layer spin-up state and the second layer spin-down state. We also define the intralayer splitting energy, Δintra, to be the energy difference between the first-(second-)layer spin-up state and the first-(second-)layer spin-down state. At Ez = 0.1 eV/Å, the states from the second layer are slightly lower than those from the first layer in each split branch, making Δinter less than Δintra, in agreement with previous results41,42,43. [The two topmost valence states are from different layers.] When Ez is larger than a critical value Ec, Δinter becomes larger than Δintra, making the two topmost valence states to come from the same layer. Fig. 2 (c) shows Δinter and Δintra as a function of Ez. We see that Δinter increases linearly as Ez increases, but Δintra remains nearly constant. Δinter is due to the potential difference between two layers caused by Ez. On the other hand, Δintra is the spin-splitting resulting from the effect of SOC within each layer and it therefore depends weakly on Ez. There is a crossover between Δinter and Δintra for 0.1 < Ez < 0.2. This indicates that the maximum of the spin-splitting energy is about 150 meV (solid line in Fig 2(c)), which is comparable to the intrinsic value of the monolayer spin-splitting (Δintra). Δinter under high Ez (Ez > Ec), red dashed line in Fig 2(c), is not relevant to the spin-splitting at the two topmost valence bands, both of which now primarily come from the same layer. Note that the main effect of Ez is to induce a significant band-splitting with little change in the degree of spin polarization. The spin direction reverses when the direction of the Ez reverses, providing an effective pathway for manipulating spin states. These spin-polarized states could be observed via spin sensitive angle-resolved photoemission experiments, important matrix element effects in ARPES and other highly resolved spectroscopies44 notwithstanding and should be accessible to transport experiments in hole-doped samples.

Figure 3(b) shows the band structure of bulk MoS2 at selected kz values. The two well-separated branches can still be seen at the K point because the SOC induced splitting is much larger than the kz dispersion resulting from interlayer coupling. The SOC splitting is about 150 meV, which is comparable to the value of Δintra in the bilayer or the spin-splitting energy in the monolayer, while the interlayer coupling at the K-point is estimated to be only about 25 meV. Note that when the number of layers is finite, the number of states in each branch is the same as the number of MX2 layers.

Figure 3
figure 3

(a) Average spin polarization of the top branch of valence bands as a function of the number of MX2 layers. (b) Band structure of bulk MoS2 at selected kz values (in unit of 2π/c). Solid and dashed lines of various colors refer to different kz values (see legend).

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Concerning the spin-polarization for the top branch of the valence bands at the K-point, it is interesting to consider the degree of spin polarization in the high temperature limit when states in the top branch will become accessible due to thermal fluctuations. Since a multilayer with an even number of layers is inversion symmetric, there will be no net spin polarization in a branch containing an even number of states. Adding one MX2 layer to create a multilayer with an odd number of layers, however, adds a fully spin polarized state into the branch. Despite the mixing of other states through the weak interlayer coupling, the average out-of-the-plane spin polarization for the whole branch becomes nonzero. The degree of spin polarization for the top branch is inversely proportional to the number of layer, x. As shown in Fig. 3(a), the average spin polarization, < Sz >, of the top branch at the K-point as a function of thickness in the high temperature limit can be expressed as

Note that the states at K′ possess opposite spin polarization. Therefore, f(x) will take minus sign for the top branch at K′.

We comment further on the nature of valence bands at the Γ point. Unlike the K-point, since Γ is a time-reversal-invariant point, states at Γ are spin degenerate for any number of layers. With increasing number of layers, as more states interact through interlayer coupling, the bandwidth increases and the top of the valence band at Γ moves to higher energy. In MoS2, the interlayer coupling at Γ is ~0.25 eV, an order of magnitude larger than that for top valence states at the K-point. This drastic difference in the strength of interlayer coupling is due to the orbital character of the relevant states: The top valence states at Γ are primarily associated with out-of-the-plane orbital, while states at the K-points arise from in-plane dxy and orbitals.

Accessibility of spin polarized valence states for spintronics applications will be controlled by the size of SOC spin-splitting energy Δintra and the energy difference between the top valence states at K and Γ, ΔKΓ = EK - EΓ. Accordingly, Fig. 4(a) compares these key quantities in various MX2 materials. Δintra is seen to increase with increasing atomic number as expected since the strength of SOC is larger in heavier atoms. ΔKΓ, on the other hand, does not show any simple trend. Interestingly, the value of ΔKΓ correlates with that of X-M-X bond angle (solid green line). This indicates that the crystal field and M-X bonding, which determine the relative positions of various orbitals, also play an important role in determining ΔKΓ.

Figure 4
figure 4

(a) Spin-splitting energy Δintra (blue line), ΔKΓ (red line) and X-M-X bond angle (green line) for various monolayer MX2 materials. (b) ΔKΓ as a function of in-plane strain in MoS2. (c) ΔKΓ as a function of slab thickness in MoS2.

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In order to help guide search for suitable substrates, we have explored the effect of strain on Δintra and ΔKΓ. Fig. 4(b) shows ΔKΓ as function of strain for monolayer, bilayer and bulk MoS2. ΔKΓ is seen to increase (decrease) as the lattice constant decreases (increases). This trend is consistent with our observation above that ΔKΓ correlates with the X-M-X bond angle in that thin films with larger in-plane lattice constant possess smaller X-M-X bond angles. Interestingly, ΔKΓ is very small for bilayer MoS2 and only a 1% lattice constant increase changes its sign, adding to the possibilities for control of spin-polarized states via an external electric field. Notably, the small value of ΔKΓ in bilayer can change sign if experimental atomic positions are used in the computations instead of self-consistently optimized theoretical positions41,42,43. This discrepancy between experiment and theory is due to overestimation of the bond-length and bond-angle in GGA, which is a typical error of the GGA exchange-correlation functional45. Despite these uncertainties in absolute band positions, the trends shown are fairly robust to whether theoretical or experimental atomic positions are used consistently in the calculations. We suggest that substrates should be chosen with smaller lattice constants so as to have larger ΔKΓ values. Note also that unlike ΔKΓ, which is sensitive to in-plane strain and slab thickness, Δintra is found to be almost constant (not shown) when strain, Ez, or thickness is varied.

Concerning the nature of the band gap and whether it is direct or indirect, we note that all MX2 considered here have valence band maximum at Γ for bulk and at K for monolayer. Take MoS2 as an example. Fig. 4(c) shows that ΔKΓ decreases as slab thickness increases. As shown in Fig. 3(b), the valence band maximum is at Γ and conduction band minimum is away from Γ along the Γ-K direction, so that the bulk band gap is indirect. For monolayer MoS2 shown in Fig. 1(d), on the other hand, both the conduction minimum and valence bands maximum shift to the K-point, making a direct band gap. Our results are consistent with the fact that bulk MoS2 has been demonstrated to support a large indirect-gap, which crosses over from indirect to a direct gap for a monolayer20,21. Such an indirect to direct band gap transition as the thickness is reduced has also been observed directly via ARPES spectroscopy for MoSe246.

Conclusions

We have systematically investigated electronic structures of monolayer and bilayer films of transition-metal dichalcogenides MX2 (M = Mo or W, X = S, Se, or Te) via first-principles calculations and discuss how these electronic structures evolve in the bulk limit. In the monolayer case, the spin-splitting due to SOC occurs at the valence band maximum, resulting in two non-degenerate nearly 100% spin-polarized states with opposite out-of-the-plane polarizations at the K and K' symmetry points in the Brillouin zone. We show that the SOC induced spin splitting can be tuned by an out-of-the-plane external electric field for bilayer MX2 thin films. Our theoretical results concerning the nature of the band gap, whether it is direct or indirect and its evolution with layer thickness are in accord with recent experimental findings in several MX2 materials. Our study provides insight into the degree of spin polarization of the valence bands in ultra-thin transition-metal dichalcogenide films and their viability for spintronics applications.