Valence Band Splitting on Multilayer MoS2: Mixing of Spin–Orbit

Letter pubs.acs.org/JPCL

Valence Band Splitting on Multilayer MoS2: Mixing of Spin−Orbit Coupling and Interlayer Coupling Xiaofeng Fan,*,† David J. Singh,‡ and Weitao Zheng*,† †

College of Materials Science and Engineering, Jilin University, Changchun 130012, China Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, United States

‡

S Supporting Information *

ABSTRACT: Understanding the origin of valence band splitting is important because it governs the unique spin and valley physics in few-layer MoS2. We explore the effects of spin−orbit coupling and interlayer coupling on few-layer MoS2 using first-principles methods. We find spin−orbit coupling has a major contribution to the valence band splitting at K in multilayer MoS2. In doublelayer MoS2, the interlayer coupling leads to the widening of the gap between the already spin−orbit split states. This is also the case for the bands of the K-point in bulk MoS2. In triple-layer MoS2, the strength of interlayer coupling of the spin-up channel becomes different from that of spin-down at K. This combined with spin−orbit coupling results in the band splitting in two main valence bands at K. With the increase of pressure, this phenomenon becomes more obvious with a decrease of main energy gap in the splitting valence bands at the K valley.

A

while bulk MoS2 is an indirect-gap semiconductor with a band gap of 1.29 eV, thinning MoS2 leads to a transition to a direct gap3 and single-layer MoS2 has a direct band gap of about 1.8 eV.5,29 Therefore, even though the interlayer bonding is relatively weak, it can affect the states near band edges.30 In particular, the states at the top of the valence band at Γ point (VB-Γ) and the bottom of the conduction band along Λ (CBΛ) are quite sensitive to the interlayer coupling. When compared with the VB-Γ and CB-Λ states, the interlayer coupling effects on the top valence band and bottom conduction band at K (VB-K, CB-K) are relatively weak. There remains an open question about the origin of the splitting at K point, which then governs the unique spin and valley physics. In the single-layer limit, the splitting can be attributed entirely to the SOC. In the bulk limit, it is considered to be a result of a combination of SOC and interlayer coupling. However, there is disagreement about the relative strength of these mechanisms.31−38 Here, we explore the effect of SOC in combination with interlayer coupling on few-layer MoS2 using first-principles calculations. The crystal structure is important. In bulk MoS2 there is an inversion center, but not a glide-free mirror plane in between the sheets and a mirror plane, but not an inversion center in the center of the sheets, i.e., in the plane of the Mo. Therefore, multilayer MoS2 is centrosymmetric for even numbers of layers and noncentrosymmetric for odd numbers of layers. We analyze the splitting of states at VB-Γ, VB-K, CBΛ, and CB-K and explore the changes in splitting with layer

new class of two-dimensional (2D) materials, single-layer and/or few-layer sheets of hexagonal transition-metal dichalcogenides (h-TMDs), has attracted broad attention because of the remarkable physical properties of these systems with potential applications in electronic and optoelectronic devices.1−5The single-layer h-TMDs are direct band gap semiconductors with band splitting at valence band maximum due to spin−orbit coupling (SOC), in contrast to graphene.6−9 This offers opportunities to use these materials not only in more standard semiconductor device structures as compared to graphene but also for possibly novel devices that manipulate the spin degrees of freedom and valley polarization.10−12 Singleand few-layer materials have weak screening leading to tightly bound excitons and enhanced electron−electron interactions. In addition, h-TMDs have proven particularly interesting platforms for exploring many novel quantum phenomena,11,13−16 such as spin and valley Hall effects and superconductivity.17−19 Single-layer h-TMDs also exhibit many fascinating optical properties, such as surface sensitive luminescence20,21 and strain-control of the optical band gap,22−25 and other phenomena.26 The purpose of this Letter is to present a detailed study of the role of spin−orbit coupling and its evolution with the number of layers in the prototypical material, MoS2. Bulk MoS2 is a layered van der Waals (vdW) compound.27 A layer of hexagonal MoS2 consists of a sheet containing three atomic layers: a central Mo layer and S layers in both sides. Importantly, the states near the band gap come mainly from the d-orbitals of Mo.28 The vdW type crystal structure and the dorbital character may lead to the expectation that interlayer coupling has a weak effect on the electronic structure and in particular the nature of the states near the band edge. However, © XXXX American Chemical Society

Received: March 27, 2016 Accepted: May 25, 2016

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DOI: 10.1021/acs.jpclett.6b00693 J. Phys. Chem. Lett. 2016, 7, 2175−2181

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The Journal of Physical Chemistry Letters

Figure 1. Band structures of double-layer MoS2 calculated without spin−orbit coupling (a) and with spin−orbit coupling (b); the changes of conduction band splitting (ΔΛ) at Λ point and valence band splitting at Γ point (ΔΓ) and K point (ΔK) with the distance between two layers of double-layer MoS2, calculated without spin−orbit coupling (c) and with spin−orbit coupling (d). Note that the red circles in panels c and d represent the data from the equilibrium (or stable) state (ES) and the dot and dash−dot lines present the conduction band splitting (ΔΛ) at Λ point and valence band splitting (ΔK) at K point of single-layer MoS2, respectively. At the ES, the distance between two layers of MoS2 marked with the blue arrow is 3.083 Å and the distance between two layers of Mo is 6.198 Å, which is similar to half of the lattice constant (c/2 = 6.187 Å) of bulk MoS2.

spin-up state is much different from that of spin-down state at VB-K (note that K and K′ are the time reversal connected pair). The spin-down state at K around Mo is more localized than the spin-up state (Figure S2). For double-layer MoS2, the interaction between two layers becomes important to the states near the Fermi level.39 An obvious effect is that the direct band gap (K−K) of single-layer MoS2 becomes indirect (Γ−K) because of a rise in energy of VB-Γ (Figure 1a,b). This is due to the large band splitting (0.618 eV) at VB-Γ (Figure 1a). The band splitting at the conduction band bottom around Λ is about 0.352 eV (Figure 1a). However, without SOC, the band splitting at VB-K is just 74 meV (Figure 1a). The difference in the effect of the interlayer coupling on different states at VB-Γ, VB-K, and CB-Λ is related to the charge distribution near sulfur atoms, which form the outside of the layers, which reflects the hybridization between d-Mo and p-S. The charge on sulfur atoms for the states at VB-Γ reflects sizable hybridization which is reflected in turn in the effect of interlayer coupling on these states, as discussed previously.30 The weak interlayer coupling at VB-K may make the SOC more important. To explore the rules of SOC and interlayer coupling in double-layer MoS2, we calculated the changes in band structure with interlayer distance with and without SOC. As shown in Figure 1c, the band splittings without SOC at VB-Γ, VB-K, and CB-Λ quickly go to zero with increasing distance, especially at VB-K. With SOC (Figure 1d), the band splittings at VB-K and CB-Λ converge toward constant values reflecting the single-layer values, while the VB-Γ splitting approaches zero. Obviously, there is no spin−orbit effect at VB-Γ. It is well-known that the spin-up and spin-down states at VBK′ are reversed compared with that at VB-K in single-layer MoS2. For double-layer MoS2 (Figure 2a,b), both of the split bands at VB-K have 2-fold degeneracy. As shown in Figure 2c, the upper of these bands is composed of the spin-up state of

separation for double-layer MoS2. It is found that intralayer SOC has a major contribution to the splitting at VB-K, while interlayer coupling is effective in breaking the degeneracy of states at VB-K. The charge and spin density shows the effect of spin−orbit coupling leading to separation of spins for a given valley in real space. For triple-layer MoS2, interlayer coupling combined with intralayer SOC makes the splitting complicated because of the absence of inversion symmetry. The intralayer SOC results in the splitting of the two main bands, while in each main band, the triple-degeneracy is broken mainly because of the difference between interlayer coupling of the spin-up channel and that of spin-down. For double-layer MoS2 in which the double-degeneracy of states in each of the main bands is not broken, the splitting of both main bands is increased because of the strengthening of interlayer coupling. For triple-layer MoS2 under large pressure, the splitting of triple-degeneracy in each main band is very obvious. Thus, the symmetry difference between odd and even layered MoS2 is important in the sense that the associated band splittings are sizable. The structure of single-layer MoS2 has the hexagonal symmetry with space group P6̅m2, which as mentioned is noncentrosymmetric. The six sulfur atoms near each Mo atom form a trigonal prismatic structure with the mirror symmetry in the c direction. An obvious band splitting attributed to the SOC is found at the valence band maximum around K (K′) point (see the Supporting Information, Figure S1). In addition, the SOC also results in an obvious band splitting at the conduction band minimum around Λ point, while sizable splitting at VB-Γ is not found and that of CB-K is about 8 meV. The states of VB-Γ and CB-K come mostly from dz2 orbitals of Mo, and the effect of the absence of inversion symmetry is very weak. On the other hand, the states at VB-K and CB-Λ are mainly from dx2−y2 and dxy orbitals of Mo leading to a sizable splitting. The band splitting at VB-K (149 meV) is larger than that at CB-Λ (about 79 meV). We also note that the charge distribution of 2176

DOI: 10.1021/acs.jpclett.6b00693 J. Phys. Chem. Lett. 2016, 7, 2175−2181

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value (166 meV) of the double layer at VB-K, the increased splitting from interlayer coupling is approximately 17 meV. This is much less than that (74 meV) from interlayer coupling without the consideration of the spin−orbit effect. This phenomenon can be understood by a model Hamiltonian with spin−orbit coupling with the formula ⎞ ⎛ E (k ) M1(k) ⎟ H(k) = ⎜⎜ ⎟ ⎝ M1*(k) E(k) − E0(k)⎠

(1)

where E0(k) is the value of band splitting due to the contribution of spin−orbit coupling and M1(k) represents the coupling parameters of interlayer coupling with the same k point and same energy level, respectively. The results from the model Hamiltonian are consistent with that from density functional theory (DFT; Table S2). For the spin-down channel, the mechanism of band splitting is the same as that of the spinup channel. Therefore, the contribution of intralayer SOC (149 meV) to the band splitting at VB-K is much larger than that of interlayer coupling (17 meV) in double-layer MoS2. The same mechanism for the splitting at VB-K (shown in Figure 2a) can be used for bulk MoS2. The effective contribution of interlayer coupling is increased to 59 meV because the band splitting at VB-K is 0.208 eV. Without the spin−orbit effect, the band splitting due to interlayer coupling would be 146 meV, which is very similar to the value from SOC (149 meV) in the singlelayer material. This may be the origin of the disagreement about the relative strength of the two effects in the bulk limit. Based on the model Hamiltonian mentioned above and analysis (Tables S1 and S2), the effective strength of interlayer coupling (146/2 meV) almost becomes of the same importance as the intralayer SOC effect (149 meV) for the splitting at VB-K in bulk limit. For triple-layer MoS2, the band splitting near the band gap is complicated because there are three states from three layers which are coupling with each other and hybridized with intralayer SOC (Figure 3). As mentioned, the triple-layer case does not have inversion symmetry. For VB-Γ, there is no SOC effect and the three degenerate split due to interlayer coupling. It is found that the two splitting values (ΔΓ1 and ΔΓ2 in Figure 3a) which control the relative energy difference of three states

Figure 2. Schematic of valence band splitting of valence band maximum at K point due to the spin−orbit coupling in each layer (intra-SOC) and interlayer coupling (LC) in band structure of doublelayer MoS2 (a), schematic structure of double-layer MoS2 (b), and the isosurface of band-decomposed charge density of four states at valence band maximum of K point including the states ∼|1↑⟩, ∼|2↓⟩, ∼|1↓⟩, and ∼|2↑⟩ shown in panel a after considering the effects of intra-SOC and interlayer coupling.

first layer and spin-down state of second layer (∼|1↑⟩, ∼|2↓⟩). The lower band is with the spin-down state of the first layer and spin-up state of the second layer (∼|1↓⟩, ∼|2↑⟩). Obviously, the energies of spin-up and spin-down of the second layer at VB-K are reversed compared with that of the first layer (Figures S3 and S4) . Because there is inversion symmetry for the doublelayer system (note that time reversal connects K with K′), the energies of states |1↑⟩ and |2↓⟩ with same energy cannot be split. Therefore, we can understand the splitting at VB-K based on the intralayer SOC and interlayer coupling, as illustrated in Figure 2a. Because of the splitting from intralayer SOC, the energies of |1↑⟩ and |2↑⟩ are very different. This substantially reduces the increase of the splitting of two states from the contribution of interlayer coupling. From the band splitting

Figure 3. Band structure of triple-layer MoS2 calculated without spin−orbit coupling (a) and with spin−orbit coupling (b) and schematic of valence band splitting of valence band maximum at K point due to the spin−orbit (SOC) and interlayer coupling in band structure of triple-layer MoS2 (c). Note that in the inset of panel c, the band structure is plotted with two directions, K → Γ and K → M, and the lengths for K → Γ and K → M are 1/ 10 of total lengths in the two directions, respectively. 2177

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Figure 4. Band structures of double-layer (a) and triple-layer MoS2 (b) under pressure of 15 GPa calculated with spin−orbit coupling and valence bands of triple-layer MoS2 under 15 GPa near the K point and the related schematic about band splitting due to the spin−orbit coupling and the layer’s coupling (c).

difference between the interlayer coupling strength of the spinup channel and that of the spin-down. From the above analysis, there is no difference in the splitting value (149 meV) induced by SOC at VB-K for monolayer, bilayer, and bulk MoS2. This is because the SOC happens mainly intralayer and the SOC between layers can be ignored, though the layer−coupling effect is obvious for multilayer MoS2. It can be easily understood from geometry aspect, because Mo atoms in different layers are far from each other, compared to those in the same layer. While small pressures do not induce the substantial splitting from intra-SOC in singlelayer MoS2 and raise the SOC between layers, it is possible that there could be a strong effect on the splitting from interlayer coupling in triple-layer MoS2, e.g., due to changes in the layer separation under pressure. For double-layer MoS2, the degenerate states in each main band are not split because of the symmetry and the energy gap between both main bands is increased with the strengthening of interlayer coupling under pressure, as in Figure 4a. In the triple-layer case, the strengthening of interlayer coupling under pressure naturally has additional effects, such as the increase of energy gap between ∼|1↑⟩ and ∼|3↑⟩, as in Figure 4b,c. Besides the enhanced interlayer coupling, the energy gap between main bands Δ′SOC has been decreased to approximately 136 meV under 15 GPa in Figure 4b. This may be explained by considering the interlayer couplings of both spin channels independently. The model Hamiltonian of spin-up and spindown channel can be expressed by

after the hybridization are 0.293 and 0.502 eV, respectively. The much different values of both splittings suggest that there is significant coupling between the first layer and third layer because the values of the splitting should be the same with only nearest-neighbor interactions for the three degenerate states in the absence of interactions with other states. Without spin− orbit coupling, the values of the splitting at CB-Λ (ΔΛ1 and ΔΛ2) are 0.241 and 0.225 eV and that at VB-K (ΔK1 and ΔK2) are 49 and 55 meV, respectively. Based on the nearest-neighbor interlayer coupling strength (74/2 meV) at VB-K from the double layer, the interlayer coupling strength between the first layer and third layer of the triple layer is about 2 meV at VB-K, from the model Hamiltonian (Table S1). With intralayer SOC, the states at VB-K (Figures 3b and S5) are mainly separated into two main bands, which is much different from that without SOC. Therefore, a schematic coupling model based on the intra-SOC and near-neighbor interlayer coupling (including the nearest-neighbor and second nearest-neighbor), as shown in Figure 3c, can be applied. With this model, the spin-up and spin-down bands of each layer are split mainly by the intraSOC. Then the interlayer coupling will perturb these states for each spin channel. For example, the spin-up channel is composed of two degenerate upper states (|1↑⟩, |3↑⟩) and one lower state (|2↑⟩) in Figure 3c, and interlayer coupling will result in the splitting of two degenerate upper states (|1↑⟩, | 3↑⟩) with the increase of energy gap between |2↑⟩ and |3↑⟩. If considering both the spin-up and spin-down channels with the same interlayer coupling, there should be two main bands, and each main band is composed of two degenerate states with one single state. The interlayer coupling does not change the energy gap between the main bands. However, it is found that the energy gap Δ′SOC is 148.7 meV, i.e., a little less than the splitting from intra-SOC (ΔSOC = 149 meV). In addition, it is interesting that the two degenerate states in each main band, e.g., the upper states ∼|1↑⟩ and ∼|2↓⟩ and the lower states ∼| 2↑⟩ and ∼|3↓⟩, are split, as shown in the inset of Figure 3c. In addition, it is found that the splitting values are so large (e.g., 11.3 meV between ∼|1↑⟩ and ∼|2↓⟩) that the above proposed model is not enough for the full origin. We propose that the splitting of degenerate states in each main band is due to the

⎛ E(k) − E0 M1(k) M 2 (k ) ⎞ ⎜ ⎟ H(k , ↑ ) = ⎜ M1*(k) E (k ) M1(k) ⎟ ⎜ ⎟ ⎜ ⎟ * * M ( k ) M ( k ) E ( k ) E − ⎝ 2 1 0⎠ ⎛ E (k ) M1(k) M 2(k)⎞ ⎜ ⎟ H(k , ↓ ) = ⎜ M1*(k) E(k) − E0 M1(k) ⎟ ⎜ ⎟ ⎜ * ⎟ * M ( k ) M ( k ) E ( k ) ⎝ 2 ⎠ 1 2178

(2)

DOI: 10.1021/acs.jpclett.6b00693 J. Phys. Chem. Lett. 2016, 7, 2175−2181

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The Journal of Physical Chemistry Letters where M1(k) and M2(k) represent the coupling parameters of the nearest-neighbor and second nearest-neighbor layercoupling. The fitting results of the model Hamiltonian are consistent with that from DFT for 0 and 15 GPa (see Figure S5 and Tables S3 and S4, ). Obviously, the complex splitting behavior of valence bands in the K valley in the triple-layer is due to the intralayer SOC with the different strengths of both the interlayer couplings of spin-up and spin-down channels. In summary, we study in detail the band splitting at valence band maximum of multilayer MoS2 by first-principles methods. We propose a model based on the intralayer spin−orbit coupling to understand the valence band splitting at the K point of multilayer MoS2 and bulk MoS2 with the effects of layer coupling. It is also found that the direct interaction between second nearest-neighbor layers is weak at VB-K. While the two degenerate states in each main band of double-layer MoS2 are not split because of the symmetry, this effect appears in triplelayer MoS2. Especially under pressure, the interlayer coupling strength of the spin-up channel becomes obviously different from that of spin-down channel. This results in the decrease of band gap between two main bands and can be explained with a model Hamiltonian with independent consideration of spin-up and spin-down channels.

MoS2 structures under pressure. The electronic properties can be analyzed with and/or without spin−orbit coupling to explore the band splitting near the band gap. The calculated band gap of single-layer MoS2 without the consideration of spin−orbit interaction is 1.66 eV and less than the experimental report of about 1.8 eV. Obviously, the band gap from PBE is underestimated, as is common in typical density functional calculations. Though the band gap is underestimated by PBE, the band structure near the Fermi level does not have obvious differences compared with that from other many-body methods.

COMPUTATIONAL METHODS The present calculations were performed within density functional theory using accurate frozen-core full-potential projector augmented-wave (PAW) pseudopotentials, as implemented in the VASP code.40−42 We used the generalized gradient approximation (GGA) with the parametrization of Perdew−Burke−Ernzerhof (PBE) with added van der Waals corrections.43 The k-space integrals and the plane-wave basis sets are chosen to ensure that the total energy is converged at the 1 meV/atom level. A kinetic energy cutoff of 500 eV for the plane wave expansion is found to be sufficient. The effect of dispersion interaction is included by the empirical correction scheme of Grimme (DFT+D/PBE).44 This approach has been successful in describing layered structures.45,46 Our calculated lattice constants a and c of bulk MoS2 are 3.191 and 12.374 Å, respectively, similar to the experimental values (3.160 and 12.294 Å). The vdW-DF,47−49 which is a nonlocal correlation functional that approximately accounts for dispersion interactions, is expected to have a better performace than a semiempirical method, such as DFT-D2. We also perform the calculations by the vdW-DF method. In the case of MoS2, by comparing the results from DFT-D2 and vdW-DF for bulk and double-layer,50 the results of both methods are similar (Table S5). We modeled the single and multilayer MoS2 sheets using supercells with vacuum spaces of 24.748 Å along the z direction. The Brillouin zones were sampled with the Γcentered k-point grid of 18 × 18 × 1. Importantly, although only the even layered stacks have inversion, the symmetry does allow a net polarization or electric dipole along z in either case. This is because of the mirror in the center of the central MoS2 sheet for odd numbers and the inversion for even numbers. The pressure dependence was modeled using the method of adding stress to the stress tensor in the VASP code.41,42 Specifically, the structure of bulk MoS2 was optimized under a specified hydrostatic pressure of 15 GPa and then the resulting structural parameters including the in-plane lattice constant (a and b), thickness of each layer of Mo−S−Mo, and distance between layers were used to construct double- and triple-layer

Corresponding Authors

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b00693. Band structures of single-layer and bulk MoS2; charge and spin-charge distributions; and band-splitting parameters of single-layer, double-layer, triple-layer, and bulk MoS2 (PDF)

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AUTHOR INFORMATION

*E-mail: xff[email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The support from the National Natural Science Foundation of China (Grants 11504123 and 51372095) is highly appreciated. Work at the University of Missouri was supported by the Department of Energy, BES through the MAGICS center.

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REFERENCES

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Letter

The Journal of Physical Chemistry Letters

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DOI: 10.1021/acs.jpclett.6b00693 J. Phys. Chem. Lett. 2016, 7, 2175−2181

Letter

The Journal of Physical Chemistry Letters (50) Rydberg, H.; Dion, M.; Jacobson, N.; Schröder, E.; Hyldgaard, P.; Simak, S. I.; Langreth, D. C.; Lundqvist, B. I. Van der Waals Density Functional for Layered Structures. Phys. Rev. Lett. 2003, 91, 126402.

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DOI: 10.1021/acs.jpclett.6b00693 J. Phys. Chem. Lett. 2016, 7, 2175−2181