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Dependence of Electronic and Optical Properties of MoS_{2} Multilayers on the Interlayer Coupling and Van Hove Singularity
Nanoscale Research Lettersvolume 14, Article number: 288 (2019)
Abstract
In this paper, the structural, electronic, and optical properties of MoS_{2} multilayers are investigated by employing the firstprinciples method. Up to sixlayers of MoS_{2} have been comparatively studied. The covalency and ionicity in the MoS_{2} monolayer are shown to be stronger than those in the bulk. As the layer number is increased to two or above two, band splitting is significant due to the interlayer coupling. We found that long plateaus emerged in the imaginary parts of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) and the joint density of states (JDOS) of MoS_{2} multilayers, due to the Van Hove singularities in a twodimensional material. One, two and three small steps appear at the thresholds of both the long plateau of \( {\varepsilon}_2^{xx}\left(\omega \right) \) and JDOS, for monolayer, bilayer, and trilayer, respectively. As the number of layers further increased, the number of small steps increases and the width of the small steps decreases accordingly. Due to interlayer coupling, the longest plateau and shortest plateau of JDOS are from the monolayer and bulk, respectively.
Introduction
Molybdenum disulfide (MoS_{2}) is one of the typical transition metal dichalcogenides and has been widely used as a catalyst [1] and hydrogen storage material [2, 3]. Owing to the strong inplane interactions and weak van der Waals interactions between MoS_{2} atomic layers [4, 5], MoS_{2} crystals have been known as an important solid lubricant for many years [6, 7]. The monolayer MoS_{2}, socalled 1HMoS_{2}, has been exfoliated from bulk MoS_{2} by using micromechanical cleavage [8]. The socalled 2HMoS_{2} (among 1T, 2H, 3R) is the most stable structure of bulk MoS_{2} [9, 10] and is a semiconductor with an indirect bandgap of 1.29 eV [4, 11, 12]. The monolayer MoS_{2} has also drawn great attention due to its twodimensional nature and graphenelike honeycomb structure. It is interesting that monolayer MoS_{2} has a direct bandgap of 1.90 eV [4, 13] which can be used as a conductive channel of fieldeffect transistors [14]. On the other hand, the zero band gap of graphene restricts its applications in optics and transistor application [15,16,17,18]. Moreover, the theoretical and experimental works show that the electronic bandgap decreases as the number of MoS_{2} layers is increased [19,20,21,22]. Interlayer coupling of multilayer MoS_{2} is sensitive to layer thickness [21]. Some investigations on the multilayer MoS_{2} are available [19,20,21,22,23,24,25]; however, the electronic structures and optical properties of multilayer MoS_{2} are still not wellestablished, especially for the layerdependent physical properties related to the interlayer coupling. Van Hove singularity (VHS) plays an important role in optical properties [26, 27] . The only available critical points in twodimensional materials are those of the P_{0} (P_{2}) and P_{1} type, which show as a step and a logarithmic singularity [26, 27]. In this paper, we analyze the electronic and optical properties of MoS_{2} related to Van Hove singularity, layer by layer and up to six atomic layers.
Nowadays, firstprinciples calculations have been successfully performed to study the structural, electronic, and optical properties of a wide variety of materials. In this work, we have systematically studied the electronic and optical properties of monolayer, multilayer and bulk MoS_{2} by using ab initio calculations. Discussions on the optical properties are emphasized. Our results show that, for Ex, the imaginary parts of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) possess long plateaus. At these thresholds of these plateaus, \( {\varepsilon}_2^{xx}\left(\omega \right) \) of the monolayer, bilayer, and trilayer exhibit one, two, and three small steps, respectively. The imaginary part of the dielectric function is also analyzed by the joint density of states and the transition matrix elements. JDOS combined with the band structures and the Van Hove singularities are discussed in detail.
Methods
The present calculations have been performed by using the Vienna ab initio simulation package (VASP) [28, 29], which is based on the density functional theory, the planewave basis and the projector augmented wave (PAW) representation [30]. The exchangecorrelation potential is treated within the generalized gradient approximation (GGA) in the form of PerdewBurkeErnzerhof (PBE) functional [31]. In order to consider the weak interlayer attractions in this layered crystal, PBED2 calculations [32] which include the semiempirical van der Waals correction have been performed. In order to obtain more accurate band gaps, the HeydScuseriaErnzerhof hybrid functional (HSE06) [33,34,35,36] calculations are also performed in this work. The wavefunctions of all the calculated systems are expanded in plane waves, with a kinetic energy cutoff of 500 eV. Brillouin zone (BZ) integrations are calculated by using a special kpoint sampling of the MonkhorstPack scheme [37], with a 45 × 45 × 1 Γcentered grid for the monolayer and multilayer MoS_{2} and 45 × 45 × 11 grid for the bulk MoS_{2} for PBED2 calculations. For HSE06 calculations, a 9 × 9 × 1 Γcentered grid is used for the monolayer and multilayer MoS_{2}. For the monolayer and multilayer MoS_{2}, all the calculations are modeled by a supercell with a vacuum space of 35 Å in the Zdirection to avoid the interactions between adjacent MoS_{2} slabs. All the atomic configurations are fully relaxed until the HellmannFeynman forces on all the atoms are smaller than 0.01 eV/Å. Our spinpolarized calculations show that the band structures of MoS_{2} multilayers are rather insensitive to the spinpolarized effect (see Additional file 1: Figure S1); therefore, all the calculation results presented are based on the nonspinpolarization scheme.
Excitonic effects in monolayer MoS_{2} are found to be significant and have been observed by photoluminescence. We have employed the quasiparticle G_{0}W_{0} method [38], and the BetheSalpeter equation (BSE) [39, 40] to account for the excitonic effects. The band gaps of monolayer MoS_{2} are calculated to be 2.32 and 2.27 eV for the kpoint meshes of 15 × 15 × 1 and 24 × 24 × 1 Γcentered grid, obtained by the G_{0}W_{0} with SOC calculations. The imaginary parts of the dielectric function are shown in Fig. 1, calculated from both the G_{0}W_{0} and the G_{0}W_{0} + BSE methods. Two exciton peaks at 1.84 and 1.99 eV are found, which agrees well with experimental observations [4, 41]. Although the G_{0}W_{0}+BSE scheme could describe the excitonic effects better, in this paper, we present only the results (without excitonic peaks) under the GGAPBE functional.
Results and Discussion
Electronic Structures of MoS_{2} Multilayers
Crystalline MoS_{2} occurs naturally and has three crystalline types: 1T, 2H, and 3R, which corresponds to crystals with trigonal, hexagonal, and rhombohedral primitive unit cells, respectively [9]. 2HMoS_{2} is known as the most stable structure [10]; therefore, we consider only the 2H type of bulk MoS_{2} in this work. Bulk 2HMoS_{2} has a hexagonallayered structure consisting of layers of molybdenum atoms surrounded by six sulfur atoms, with SMoS sheets piled up oppositely (showed in Fig. 2). The neighboring sheets in bulk 2HMoS_{2} are weakly connected with weak van der Waals interactions. A monolayer MoS_{2} can then be easily exfoliated from the bulk. The lattice constants of bulk MoS_{2} are calculated to be a = b = 3.19Å, c = 12.41 Å, which are consistent with the reported values of a = b = 3.18 Å, c = 13.83 Å [18]. The optimized lattice constants for monolayer MoS_{2} are a = b = 3.19 Å, which are in accord with the bulk MoS_{2}. As shown in Table 1, the calculated lattice constants in the a, b directions are the same for different number of layers of MoS_{2}. It was also reported by Kumar et al. [19] that the lattice constants (a, b) of monolayer MoS_{2} are nearly identical to the bulk.
Figure 3 depicts the calculated band structures and electronic density of states (DOS) of different number of layers of MoS_{2}. Results for monolayer, bilayer, trilayer, and fourlayer as well as bulk MoS_{2} are given in Fig. 3, while results for fivelayer and sixlayer MoS_{2} are very similar to those of fourlayer and bulk. For monolayer MoS_{2}, both the valence band maximum (VBM) and the conduction band minimum (CBM) appear at Kpoint of the BZ, exhibiting a direct bandgap of 1.64 eV. For bilayer and trilayer MoS_{2}, both the VBM locates at Γ point while both the CBM lies at K point, causing indirect gaps of 1.17 and 1.08 eV, respectively. However, as the number of MoS_{2} layers increases to four and above four, all the multilayers MoS_{2} show same characters that the VBM locates at Γ point while the CBM lies between Γ and K points, which is the same as in the bulk. Indirect band gaps are 1.03 eV, 1.01 eV, 0.99 eV, 0.93 eV for four, five, sixlayer MoS_{2}, and bulk, respectively. Both the PBED2 and HSE06 calculations (Table 1) show that the fundamental band gap increases monotonically when the number of MoS_{2} layers decreases, which is due to a large confinement of electrons in the slab [4, 5, 19, 42]. Moreover, when the bulk MoS_{2} slab is lessened to a singlelayer, it turns into a direct bandgap semiconductor, as mentioned previously, the bulk MoS_{2} is an indirect gap semiconductor. In Fig. 3a, band structures plot of bulk MoS_{2} show splitting of bands (as compared to those of monolayer MoS_{2}), mainly around the point, owing to interlayer coupling [16]. Band structures for twolayers (2L) and more than 2L MoS_{2} exhibit similar splitting of bands owing again to the interlayer coupling. However, splitting of bands in the bulk is somewhat more significant than those in the multilayers MoS_{2}, indicating a (slightly) stronger interlayer coupling in the bulk than in the multilayers. On the other hand, splitting of bands in the vicinity of point K in BZ is very small. The electronic states at point K for the highest occupied band are mainly composed of d_{xy} and \( {d}_{x^2{y}^2} \) orbitals of Mo atoms, as well as small parts of (p_{x}, p_{y})orbitals of S atoms (shown in Fig. 4b). The Mo atoms are situated in the middle layer of SMoS sheet, which causes a negligible interlayer coupling at K point (since the nearest atoms between MoS_{2} layers are S and S). As shown in Fig. 4, stronger interlayer coupling at point Γ can be found when compared with that at point K, since electronic states at point Γ for the highest occupied band are dominated by \( {d}_{z^2} \) orbitals of Mo atoms and p_{z} orbitals of S atoms. Therefore, SS coupling (interlayer coupling) is clearly stronger at point Γ than that at point K. Our results are consistent with other theoretical work [21]. Generally speaking, the electronic density of states of fewlayer MoS_{2} are similar to those of bulk MoS_{2} (see Fig. 3), since bulk MoS_{2} is actually a layered material with weak interactions between the MoS_{2} layers.
To deeply explore the bonding nature in the monolayer MoS_{2}, the deformation charge density is shown in Fig. 5a. The deformation charge density is given by Δρ_{1}(r) = ρ(r) − ∑_{μ}ρ_{atom}(r − R_{μ}) where ρ(r) is the total charge density and ∑_{μ}ρ_{atom}(r − R_{u}) stands for the superposition of independent atomic charge densities. The results demonstrate that the bonding in the MoS_{2} monolayer is characterized by clear covalent (solid contours lines in between the MoS atoms), as well as strong ionic interactions (represented by alternating areas of dashed and solid contours). To see the bonding strength in the monolayer MoS_{2} as compared to those in the bulk, the charge density differences between monolayer and bulk MoS_{2}, Δρ_{2}(r), is also presented in Fig. 5b. The charge density difference is defined as Δρ_{2}(r) = ρ_{1L}(r) − ρ_{bulk}(r), where ρ_{1L}(r) and ρ_{bulk}(r) are the total charge densities of monolayer and bulk MoS_{2}, respectively. Figure 5b indicates a stronger electronic binding in the monolayer case than those in the bulk, which is reflected by the larger charge accumulation (solid contours lines) in between the MoS atoms in the monolayer, as well as by stronger ionic bonding in the monolayer MoS_{2} since the alternating areas of dashed and solid contours in the Fig. 5b are more significant than those in the bulk. Moreover, the charge differences plot (Fig. 5b) indicates that Mo atom of monolayer lost more electrons than Mo atom in the bulk; therefore, the ionicity of monolayer is stronger than bulk. However, it should be pointed out that the order of magnitude of the charge differences in the Fig. 5b are fairly small (the contour interval in the Fig. 5b is only 2.5 × 10^{−4} e/Å^{3}). Judge from the quantum confinement effect, again, the intralayer interaction of monolayer should be stronger than bulk. Hence, the bandgap of the monolayer (1.64 eV) is expected to be larger than bulk (0.93 eV). Quantum confinement decreases with the increasing layer number [4, 42], which enhances interlayer coupling and reduce intralayer interaction. Thus, the band gap of MoS_{2} decreases with the increase of interlayer coupling. The interlayer charge density redistributions for bilayer MoS_{2}, Δρ_{3}(r), are also presented in Fig. 5c. The Δρ_{3}(r) is given by Δρ_{3}(r) = ρ_{2L}(r) − ρ_{layer1}(r) − ρ_{layer2}(r), where ρ_{2L}(r), ρ_{layer1}(r), ρ_{layer2}(r) are the charge densities of the bilayer MoS_{2}, the first layer of bilayer MoS_{2} and the second layer of bilayer MoS_{2}, respectively. The charge densities of layer1 and layer2 of bilayer MoS_{2} are calculated by using the corresponding structure in bilayer MoS_{2}. Charge transfer from MoS_{2} layers (bilayer) to the intermediate region between the MoS_{2} layers are clearly seen in Fig. 5c, shown as solid contour lines. The ionic interactions between atomic layers in bilayer MoS_{2} are also clear, as seen from the alternating areas of dashed and solid contours. Again, the order of magnitude of interlayer charge densities, Δρ_{3}(r), are very small (the contour interval is only 2.5 × 10^{4} e/Å^{3}). Generally, the interlayer charge density redistributions in 2L, 3L, …, bulk MoS_{2} systems are all very similar.
Optical Properties of MoS_{2} Multilayers
Once the ground state electronic structures of a material are obtained, the optical properties can then be investigated. The imaginary part of the dielectric function \( {\varepsilon}_2^{\alpha \beta}\left(\omega \right) \) is determined by the following equation [43]:
where the indices α and β denote Cartesian directions, c and v refer to conduction and valence bands, E_{ck} and E_{vk} are the energies of conduction bands and valence bands, respectively. The KramersKronig inversion can be applied to acquire the real part of the dielectric function \( {\varepsilon}_1^{\alpha \beta}\left(\omega \right) \) determined by the imaginary part \( {\varepsilon}_2^{\alpha \beta}\left(\omega \right) \):
in which P represents the principal value. Since MoS_{2} has a uniaxial structure, ε^{xx}(ω) is then identical to ε^{yy}(ω). In this work, we need only discuss the electric vector E which is parallel the xy plane, i.e., Ex is parallel to the MoS_{2} xy plane.
For investigating detailed optical spectra of MoS_{2} system, the absorption coefficient α(ω) and the reflectivity R(ω) were calculated by the real part ε_{1}(ω) and the imaginary part of the dielectric function. Equations of parameters mentioned are presented below:
If the matrix element \( \left\langle {u}_{ck+{e}_{\alpha }q}{u}_{vk}\right\rangle \) varies very slowly as kvector, the term \( \left\langle {u}_{ck+{e}_{\alpha }q}{u}_{vk}\right\rangle \left\langle {u}_{ck+{e}_{\beta }q}{u}_{vk}\right\rangle \ast \) in Eq. (1) can be taken outside the summation. In Eq. (1), most of the dispersion in \( {\varepsilon}_2^{\alpha \beta}\left(\omega \right) \) is due to the summation over the delta function δ(E_{ck} − E_{vk} − ℏω). This summation can be transformed into an integration over energy by defining a joint density of states (JDOS) [25, 44],
in which ℏω equals E_{ck} − E_{vk}, S_{k} represents the constant energy surface denoted by E_{ck} − E_{vk} = ℏω = const. The joint density of states J_{cv}(ω) is associated with the transitions from the valence bands to the conduction bands, and the large peaks in J_{cv}(ω) will originate in the spectrum where ∇_{k}(E_{ck} − E_{vk}) ≈ 0. Points in kspace where ∇_{k}(E_{ck} − E_{vk}) = 0 are called critical points or van Hove singularities (VHS), and E_{ck} − E_{vk} are called critical point energies [26, 27]. The critical points ∇_{k}E_{ck} = ∇_{k}E_{vk} = 0 usually occur only at highsymmetry points, while critical points ∇_{k}E_{ck} = ∇_{k}E_{vk} ≠ 0 may occur at any general points in the Brillouin zone [27, 45]. In the twodimensional case, there are three types of critical points, i.e., P_{0} (minimum point), P_{1} (saddle point), and P_{2} (maximum point). At the points P_{0} or P_{2}, a step function singularity occurred in JDOS, while at the saddle point P_{1}, JDOS was described by a logarithmic singularity [27]. In more detail, the E_{c}(k_{x}, k_{y}) − E_{v}(k_{x}, k_{y}) can be expanded in a Taylor series about the critical point. Limiting the expansion to quadratic terms, with the linear term does not occur due to property of the singularity, we then have
Therefore, three types of critical points emerge. For P_{0}, (b_{x} > 0, b_{y} > 0), for P_{1}, (b_{x} > 0, b_{y} < 0) or (b_{x} < 0, b_{y} > 0), and for P_{2}, (b_{x} < 0, b_{y} < 0). In this paper, for the case of MoS_{2} multilayers, only the P_{0} critical point is involved.
Figure 6a gives the imaginary parts of dielectric function, \( {\varepsilon}_2^{xx}\left(\omega \right) \), of MoS_{2} multilayers for Ex. We found an interesting phenomenon that the imaginary parts of dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) possess plateaus, and the plateaus of different layers of MoS_{2} are nearly equal in the range of 1.75 eV~2.19 eV. From the threshold energy up to 1.75 eV, \( {\varepsilon}_2^{xx}\left(\omega \right) \) are quite different for different multilayers of MoS_{2}. The threshold and ending energies of the plateaus in different layers are different, especially, the energy range of \( {\varepsilon}_2^{xx}\left(\omega \right) \) plateau of the monolayer is significantly broader than those of other multilayers. The threshold energy of monolayer MoS_{2} dielectric function is equal to its direct bandgap of 1.64 eV. However, the threshold energy of bilayer dielectric function is not the indirect bandgap of 1.17 eV but the minimum of direct energy gap of 1.62 eV between the valence and conduction bands. This is because that we study only the transitions between valence and conduction bands with the same electron wave vector, which are classified as direct optical transitions [36, 47]. As the number of MoS_{2} layers increased to 4, we found that \( {\varepsilon}_2^{xx}\left(\omega \right) \) of multilayer MoS_{2} systems were almost indistinguishable from bulk. Hence, we discuss here in details only the plateaus of the monolayer, bilayer, and trilayer, as well as bulk MoS_{2}. The \( {\varepsilon}_2^{xx}\left(\omega \right) \) plateaus of monolayer, bilayer, trilayer, and bulk MoS_{2} ended at 2.57 eV, 2.28 eV, 2.21 eV, and 2.19 eV, respectively. To explain this more precisely, JDOS of monolayer, bilayer, trilayer, and bulk MoS_{2} are shown in Fig. 7. From Fig. 7, the plateaus are also shown to be in the JDOS. The plateaus of monolayer, bilayer, and trilayer JDOS ended at 2.57 eV, 2.28 eV, 2.21 eV, respectively, which are exactly the same as those in their \( {\varepsilon}_2^{xx}\left(\omega \right) \). For bulk MoS_{2}, the plateau of JDOS ended at 2.09 eV, which is slightly smaller than 2.19 eV in the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \).
To analyze accurately the electronic transitions and for a detailed analysis of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \), the direct energy gaps, ΔE(NC − NV), between conduction and valence bands of monolayer, bilayer, trilayer, and bulk MoS_{2} are presented in Fig. 8. The notations NC and NV represent the ordinal numbers of conduction and valence bands. Hence, NC = 1, 2, and 3 signify the lowest, the second lowest, and the third lowest unoccupied band of material. On the other hand, NV = 9, 18, and 27 (which is dependent on the number of electrons in the unit cell) signify the highest occupied band of monolayer, bilayer, and trilayer MoS_{2}, respectively. For monolayer, in the region of 0 ~ 2.57 eV, the electronic transitions are found to be contributed only from the highest occupied band NV = 9 to the lowest unoccupied band NC = 1. From Fig. 8a, a minimum appears at high symmetry point K and the threshold of JDOS (Fig. 7a) appears at 1.64 eV which is actually the direct bandgap of the monolayer MoS_{2}. In the vicinity of high symmetry point K, the curve of ΔE(NC = 1 − NV = 9) is similar to a parabola for monolayer MoS_{2}. Therefore, ∇_{k}(E_{ck} − E_{vk}) = 0 at K point, which means a critical point at high symmetry point K. In a twodimensional structure, this critical point belongs to P_{0} type singularity [27], and therefore it leads to a step in the JDOS. Thus, the threshold energy of the JDOS plateau is found at critical point energy 1.64 eV. The ending energy of the JDOS plateau is near 2.57 eV, which is resulted from the appearance of two P_{0} type singularities at point B1 (k = (0.00, 0.16, 0.00)) and point B2 (k = (− 0.10, 0.20, 0.00)). The slopes of the ΔE(NC = 1 − NV = 9) curve near the two critical points B1 and B2 are very small, which give rise to a rapid increase in JDOS (see Eq.(5)). Main critical points for these long plateaus of JDOS are listed in Table 2, including type, location, transition bands, and the direct energy gap ΔE(NC − NV). Furthermore, we found that ∇_{k}E_{ck} = ∇_{k}E_{vk} = 0 happened at high symmetry point K where the slopes of the valence and conduction bands are horizontal. While ∇_{k}E_{ck} = ∇_{k}E_{vk} ≠ 0 happened at points B1 and B2, which means that slopes of two bands are parallel. Simultaneously, analysis on the band structures and direct energy gaps (see Fig. 8a) for the monolayer show that, when the direct energy gap ΔE is below 2.65 eV, only the transitions between NV = 9 and NC = 1 contribute to JDOS; when the ΔE is larger than 2.65 eV, the transitions of NV = 9 to NC = 2 also begin to contribute to JDOS; while when the ΔE reaches above 2.86 eV, the NV = 9 to NC = 3 transitions have effect on JDOS. It should be pointed out that for energy larger than 2.65 eV, many bands in Fig. 8a will contribute to JDOS. JDOS of monolayer MoS_{2} exhibits a plateau in the range of 1.64 ~ 2.57 eV and the variation of the expression M_{vc}^{2}/ω^{2} is found to be small in this range. According to Eqs. (1) and (5), the imaginary part of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) is mainly decided by the JDOS and the transition matrix elements, this gives a similar plateau for the imaginary part of dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) as compared to JDOS.
For bilayer MoS_{2}, in the region of 0 ~ 2.28 eV (the endpoint of JDOS plateau), the electronic transitions are contributed to NV = 17, 18 to NC = 1, 2. The minimum energy in ΔE(NC − NV) is situated at the K point with a gap of 1.62 eV. In Fig. 8b, similar to monolayer MoS_{2}, bilayer MoS_{2} holds two parabolic curves going upward (which come from ΔE(NC = 1 − NV = 18) and ΔE(NC = 2 − NV = 18)) at K point. Therefore, there are two P_{0} type singularities (∇_{k}(E_{ck} − E_{vk}) = 0) at K point, causing a step in the JDOS. The critical point energies are both 1.62 eV, this is because that the conduction bands (NC = 1 and NC = 2) are degenerate at point K (as shown in Fig. 3b), which results in the same direct energy gap between transitions of NV = 18 to NC = 1 and NV = 18 to NC = 2. From Fig. 8b, as the direct energy gap is increased to 1.69 eV, two new parabolas (which come from ΔE(NC = 1 − NV = 17) and ΔE(NC = 2 − NV = 17)) appear and two new singularities emerge again at K point in the direct energy gap graph, leading to a new step in JDOS for bilayer MoS_{2} (see Fig. 7b). As a result, the JDOS of the bilayer MoS_{2} has two steps around the threshold of long plateau (see inset in Fig. 7b). Two parabolas (in Fig. 8b) contribute to the first step and four parabolas contribute to the second step in JDOS. It means that the value of the second step is roughly the double of the first one. As the ΔE reaches to 2.28eV, two new singularities appear at Γ point (where interband transitions come from Γ(NV = 18→NC = 1) and Γ(NV = 18→NC = 2)), which have great contribution to the JDOS and bring the end to the plateau. Our calculations demonstrate that ∇_{k}E_{ck} = ∇_{k}E_{vk} = 0 are satisfied not only at high symmetry point K, but also at high symmetry point Γ. Similar to the case of monolayer, we found that the term of M_{vc}^{2}/ω^{2} is a slowly varying function in the energy range of bilayer JDOS plateau; hence, \( {\varepsilon}_2^{xx}\left(\omega \right) \) of bilayer have a similar plateau in the energy range.
For trilayer MoS_{2}, in the region of 0 ~ 2.21 eV, the JDOS are contributed from electronic transitions of NV = 25, 26, and 27 to NC = 1, 2, and 3. As shown in Fig. 8c, trilayer MoS_{2} have nine singularities at three different energies (ΔE = 1.61 eV, 1.66 eV, and 1.72 eV, respectively) at the K point. Figure 3c depicts that the conduction bands (NC = 1, 2, 3) are threehold degenerate at point K; this means that there are three singularities at each critical point energy. According to our previous analysis, the JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \) of trilayer MoS_{2} will show three steps near the thresholds of the long plateaus, the endpoints of the long plateaus of trilayer JDOS, and \( {\varepsilon}_2^{xx}\left(\omega \right) \) are then owing to the appearance of three singularities at Γ point with ΔE = 2.21 eV (see Fig. 7c), which come from the interband transitions of Γ(NV = 27→NC = 1, 2, 3).
For bulk MoS_{2}, the thresholds of \( {\varepsilon}_2^{xx}\left(\omega \right) \) and JDOS are also located at K point, with the smallest ΔE(NC − NV) equals to 1.59 eV. Nevertheless, there is no obvious step appeared in the thresholds of plateaus for both the \( {\varepsilon}_2^{xx}\left(\omega \right) \) and JDOS (see Fig. 6a and Fig. 7d). Based on the previous analysis, the number of steps in the monolayer, bilayer, and trilayer MoS_{2} are 1, 2, and 3, respectively. As the number of MoS_{2} layers increases, the number of steps also increases in the vicinity of the threshold energy. Thus, in the bulk MoS_{2}, the JDOS curve is composed of numerous small steps around the threshold energy of the long plateau, and finally the small steps disappear near the threshold energy since the width of the small steps decreases. In the region of 0 ~ 2.09 eV, the electron transitions of bulk MoS_{2} are contributed to NV = 17, 18 to NC = 1, 2. The 2.09 eV is the endpoint of JDOS plateau of bulk MoS_{2}, which is attributed to two singularities, i.e., the interband transitions of Γ(NV = 18→NC = 1) as well as Γ(NV = 18→NC = 2), as presented in Fig. 8d. However, the plateau endpoint of the imaginary part of dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) is 2.19 eV, which is greater than the counterpart of JDOS (e.g., 2.09 eV). Checked the transition matrix elements, it verified that some transitions are forbidden by the selection rule in the range of 2.09 eV to 2.19 eV. Therefore, the imaginary part of the dielectric function \( {\varepsilon}_2^{xx}\left(\omega \right) \) is nearly invariable in the range of 2.09 ~ 2.19 eV. As a result, the plateau of \( {\varepsilon}_2^{xx}\left(\omega \right) \) of bulk MoS_{2} is then 1.59 ~ 2.19 eV.
It has been shown that these thresholds of the JDOS plateaus are determined by singularities at the K point for all of the studied materials, see Fig. 8. The endpoint energy of the monolayer JDOS plateau is determined by two critical points at B1 and B2 (Fig. 8a). Nevertheless, the endpoint energies of bilayer, trilayer, and bulk JDOS plateaus are all dependent on the critical points at Γ(Fig. 8b–d). The interlayer coupling near point Γ is significantly larger than the near point K for all the systems of multilayer MoS_{2}. The smallest direct energy gap decreases and the interlayer coupling increases as the number of layers grow. With the layer number increases, a very small decrease of direct energy gap at point K and a more significant decrease of direct energy gap at point Γ can be observed, as a result, a faint red shift in the threshold energy and a bright red shift in the end of both JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \) plateaus can also be found. For monolayer MoS_{2}, the smallest ΔE(NC − NV) at point Γ is 2.75 eV which is larger than that at point B1 (or point B2) with a value around 2.57 eV. When it goes to multilayer and bulk MoS_{2}, the strong interlayer coupling near point Γ makes the smallest ΔE(NC − NV) at Γ less than those at point B1 (or point B2). Hence, monolayer owns the longest plateau of JDOS, which is between 1.64 eV and 2.57 eV. The shortest plateau of JDOS (from 1.59 eV to 2.09 eV) is shown in the bulk.
As the energy is increased to the value larger than the endpoint of long platform of the dielectric function, a peak A can be found at the position around 2.8 eV, for almost all the studied materials (Fig. 6a). The width of peak A for monolayer is narrower compared with those of multilayer MoS_{2}; however, the intensity of peak A for monolayer is found to be a little stronger than multilayers. The differences between the imaginary parts of dielectric function for the monolayer and multilayer MoS_{2} are evident, on the other hand, the differences are small for multilayer MoS_{2}.
In order to explore the detailed optical spectra of MoS_{2} multilayers, the real parts of the dielectric function ε_{1}(ω), the absorption coefficients α(ω), and the reflectivity spectra R(ω) are presented in Fig. 6b–d. Our calculated data of bulk MoS_{2} for the real and imaginary parts of the dielectric function, ε_{1}(ω) and ε_{2}(ω), the absorption coefficient α(ω) and the reflectivity R(ω) agree well with the experimental data, except for the excitonic features near the band edge [48,49,50]. The calculated values of , which is called the static dielectric constant, for MoS_{2} multilayers and bulk can be found in Table 1. From Table 1, the calculated values of \( {\varepsilon}_1^{xx}(0) \) for multilayers and bulk MoS_{2} are all around 15.5, which is very close to the experimental value of 15.0 for bulk MoS_{2} [50]. The values of \( {\varepsilon}_1^{xx}(0) \) increase with the increasing number of MoS_{2} layers. For monolayer MoS_{2}, a clear peak B of \( {\varepsilon}_1^{xx}\left(\omega \right) \) appears about 2.54 eV. Peak B of monolayer is clearly more significant than multilayers, and they are all similar for multilayer MoS_{2}. As the layer number increases, the sharp structures (peak B) also move left slightly. In Fig. 6c, we also observe the emergence of long plateaus in the absorption coefficients, and absorption coefficients are around 1.5 × 10^{5} cm^{−1} at the long plateaus. There are also small steps around the thresholds for the absorption coefficients, which are consistent to those of the imaginary parts of dielectric function. With the layer number increases, the threshold energy of absorption coefficient decreases, while the number of small steps increases at the starting point of the plateau. For monolayer and multilayer MoS_{2}, strong absorption peaks emerge at visible light range (1.65–3.26 eV), and the monolayer MoS_{2} own a highest absorption coefficient of 1.3 × 10^{6} cm^{−1}. The theoretical absorption coefficients for different number of MoS_{2} layers are compared with confocal absorption spectral imaging of MoS_{2} (the inset) [46], as shown in Fig. 6c. For monolayer and multilayer MoS_{2}, a large peak of α(ω) can be found at the position around 2.8 eV for both the calculation and experiment [46, 51]. Furthermore, a smoothly increase of α(ω) can be found between 2.2 and 2.8 eV for both the theoretical and experimental curves. Therefore, from Fig. 6c, the calculated absorption coefficients of MoS_{2} multilayers show fairly good agreement with the experimental data [46], except for the excitonic peaks. The reflectivity spectra are given in Fig. 6d. The reflectivity spectra of MoS_{2} multilayers are all about 0.35–0.36 when energy is zero, which means that MoS_{2} system can reflect about 35 to 36% of the incident light. In the region of visible light, the maximum reflectivity of monolayer MoS_{2} is 64%, while the maxima of multilayer and bulk MoS_{2} are all about 58%. Because of the behaviors discussed, MoS_{2} monolayer and multilayers are being considered for photovoltaic applications.
Conclusions
In this study, by employing ab initio calculations, the electronic and optical properties of MoS_{2} multilayers are investigated. Compared to bulk MoS_{2}, the covalency and ionicity of monolayer MoS_{2} are found to be stronger, which results from larger quantum confinement in the monolayer. With the increase of the layer number, quantum confinement and intralayer interaction both decrease, meanwhile, the interlayer coupling increases, which result in the decrease of the band gap and the minimum direct energy gap. As the layer number becomes larger than two, the optical and electronic properties of MoS_{2} multilayers start to exhibit those of bulk. Band structures of multilayers and bulk show splitting of bands mainly around the Γpoint; however, splitting of bands in the vicinity of K point are tiny, owing to the small interlayer coupling at point K.
For optical properties, Van Hove singularities lead to the occurrence of long plateaus in both JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \). At the beginnings of these long plateaus, monolayer, bilayer, and trilayer structures appear one, two, and three small steps, respectively. With the layer number increases, the number of small steps increases and the width of the small steps decreases, leading to unobvious steps. A small red shift in the threshold energy and a noticeable red shift in the end of both JDOS and \( {\varepsilon}_2^{xx}\left(\omega \right) \) plateaus are observed, since the increased number of layers leads to small changes in the direct energy gap near point K (weak interlayer coupling) and larger changes near point Γ (stronger interlayer coupling). Thus, the longest plateau and shortest plateau of JDOS are from the monolayer and bulk, respectively. Our results demonstrate that the differences between electronic and optical properties for monolayer and multilayer MoS_{2} are significant; however, the differences are not obvious between the multilayer MoS_{2}. The present data can help understand the properties of different layers of MoS_{2}, which should be important for developing optoelectronic devices.
Availability of Data and Materials
The datasets supporting the conclusions of this article are included within the article.
Abbreviations
 ΔE:

The direct energy gap
 1L:

Monolayer MoS_{2}
 2L:

Bilayer MoS_{2}
 3L:

Trilayer MoS_{2}
 4L:

Fourlayer MoS_{2}
 5L:

Fivelayer MoS_{2}
 6L:

Sixlayer MoS_{2}
 BSE:

BetheSalpeter equation
 BZ:

Brillouin zone
 CBM:

Conduction band minimum
 GGA:

Generalized gradient approximation
 GW:

Quasiparticle energy calculation
 JDOS:

Joint density of states
 MoS_{2} :

Molybdenum disulfide
 NC:

The ordinal numbers of conduction band
 NV:

The ordinal numbers of valence band
 PAW:

Projector augmented wave
 PBE:

PerdewBurkeErnzerhof
 VASP:

Vienna ab initio simulation package
 VBM:

Valence band maximum
 VHS:

Van Hove singularity
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Acknowledgements
This work is supported by the National Key R&D Program of China under grant nos. 2016YFA0202601 and 2016YFB0901502.
Funding
This study was funded by the National Key R&D Program of China under grant nos. 2016YFA0202601 and 2016YFB0901502.
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JQH performed the detailed calculations. XHS and SQW discussed on the issues raised during the calculations. KMH and ZZZ designed, discussed, and guided the whole research. Especially, ZZZ cowrote the paper. All authors read and commented on the manuscript.
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Correspondence to ShunQing Wu or ZiZhong Zhu.
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Additional file 1:
Figure S1. The electronic states of MoS_{2} multilayers are insensitive to the spinpolarized effect, due to the overlaps of spinup and spindown band structures for all the cases. (DOC 2814 kb)
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Keywords
 MoS_{2} multilayers
 Electronic properties
 Optical properties
 Van Hove singularities