Article pubs.acs.org/JPCC
Interlayer Electronic Coupling in Arbitrarily Stacked MoS2 Bilayers Controlled by Interlayer S−S Interaction Boxiao Cao and Tianshu Li* Department of Civil and Environmental Engineering, George Washington University, Washington, D.C. 20052, United States S Supporting Information *
ABSTRACT: The vertically heterostructured MoS2 bilayers display a wide range of lattice registry relative to bulk MoS2 through a single or combined in-plane displacement, out-of-plane displacement, and inplane rotation. Here using density functional theory and numerical structural analysis, we examine both the atomic and electronic structures of the arbitrarily stacked MoS2 bilayers that form either commensurate or incommensurate superstructures. Our analysis shows that the interlayer electronic coupling between two MoS2 layers yields an indirect band gap that varies with the lattice registry, thus confirming the previous theoretical findings. The variation of the coupling strength and the indirect band gap with respect to the lattice registry can be attributed to the change in the mean interlayer sulfur− sulfur distance upon displacement. Importantly, our analysis further shows when the twisted MoS2 bilayers form an incommensurate structure, the interlayer sulfur−sulfur distance is uniformly sampled in-plane and yields a distribution independent of the incommensurate twist angle. When a commensurate structure is formed, while the distribution becomes angular dependent, the mean in-plane sulfur−sulfur distance is found nearly independent of the twist angle. Consequently, the magnitude of the indirect band gap in the bilayers exhibits a weak angular dependence, until the twist angle recovers the high symmetry stacking sequence when the gap changes significantly. Our analysis provides the thorough theoretical explanation to the recently measured photoluminescence spectroscopy in twisted MoS2 bilayers and can form the basis for understanding the coupling in vertically heterostructured bilayers composed of other transition-metal dichalcogenide monolayers.
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INTRODUCTION During the past four years, the search for the alternative twodimensional materials has identified the novel properties from monolayer transition-metal dichalcogenides (TMD), e.g., MoS2, that promise many exciting opportunities for the use of next-generation field-effect transistors,1 optoelectronics, flexible electronics,2 valleytronics,3−5 and energy harvesting materials. 6 A more exciting opportunity is to create heterostructures where individual layers of very different characters can be combined to achieve new and novel functionality. Monolayer MoS2 exhibits a direct electronic band gap of ∼1.8 eV, while bulk MoS2 is an indirect semiconductor with a gap of ∼1.2 eV. Upon thinning the bulk MoS2 while keeping the bulk stacking sequence, through, e.g., mechanical exfoliation, the electronic structure of MoS2 undergoes an interesting transformation, featured by an indirect-to-direct transition of the electronic band gap. Theory7,8 predicted that such transition occurs when MoS2 bilayers are thinned to monolayer, and the prediction was subsequently confirmed by the photoluminescence experiments.8,9 Furthermore, the magnitude of the indirect band gap was also predicted to vary with both the number of layers and the interlayer separation distance,7 implying the existence of an interlayer © 2014 American Chemical Society
electronic coupling in MoS2. When MoS2 bilayers are fabricated through the bottom-up approach by stacking one monolayer on top of another, the bilayers may exhibit a range of lattice registry, through a single or combined in-plane displacement, out-of-plane displacement, and in-plane rotation. As the interlayer electronic coupling may vary with these displacements, the optoelectronic behaviors of the MoS2 bilayers are expected to display a range of variations from those of the exfoliated MoS2 bilayers. Indeed, very recent photoluminescence spectroscopy experiments10−12 on the twisted MoS2 bilayers (i.e., one monolayer rotated in-plane with respect to another by a twist angle θt) showed the indirect band gap varies significantly with the twist angle. While such dependence can be potentially employed to tailor the properties of twodimensional heterostructure, the origin of the interlayer electronic coupling, particularly its dependence on lattice registry, needs to be understood. Here we present a thorough computational study on the interlayer electronic coupling in the arbitrarily stacked MoS2 bilayers by combining the ab initio calculations based on the Received: October 8, 2014 Revised: December 11, 2014 Published: December 17, 2014 1247
DOI: 10.1021/jp5101736 J. Phys. Chem. C 2015, 119, 1247−1252
Article
The Journal of Physical Chemistry C
while the bottom of the conduction is composed of the out-ofplane dz2 orbital. Since d states are localized on the Mo atoms which are located inside the S−Mo−S sandwich structure, both energy levels (hence the direct band gap EK−K g ) are relatively insensitive to the change outside the monolayer. Indeed, the direct band gap has been experimentally identified to be nearly invariant with number of layers8,9 and the twist angle in bilayers.10,12 On the other hand, the top of valence band at Γ originates from the combination of both pz orbitals on S atoms and dz2 orbitals on Mo atoms. As the pz orbitals are distributed on the inner S atoms of the bilayers, which overlap the most, the energy level of the valence band top at Γ is expected to vary strongly with the interlayer S−S distance rSS. Specifically, since the pz orbital associated with this state is of the antibonding nature, a decrease in the interlayer S−S distance rSS would raise the energy level, which consequently decreases the indirect band gap EΓ−K g . Because the variations of the lattice registry, which includes the out-of-plane separation, in-plane shift, and in-plane rotation, all may give rise to the change in rSS, the electronic coupling between the two monolayers, particularly the magnitude of the indirect band gap EΓ−K g , is expected to vary accordingly. To demonstrate this, we explicitly examine the variation of the electronic structures of the MoS2 bilayers with respect to all degrees of freedom for the displacement. Translational Displacement. First we consider the influence from the translational displacement, which involves both the in-plane and out-of-plane displacements. While both displacements yield a change in rSS, an in-plane displacement may also alter the stacking sequence. For in-plane displacement, we consider two starting configurations, denoted as A1 and A2, shown in Figure 2. In both A1 and A2, the S atoms in the top
density functional theory (DFT) and the numerical structural analysis. We show that the interlayer coupling in MoS2 bilayers depends on all degrees of freedom for the displacement, i.e., the in-plane shift, the out-of-plane separation, and the in-plane rotation. In particular, we find the complex coupling behavior can be mainly attributed to the interaction arising from the interactions between the S atoms located on the inner sides of both monolayers and can be further well correlated with the mean interlayer S−S distance. This insight allows understanding the twisted MoS2 bilayers with an arbitrary twist angle θt that forms either a commensurate or incommensurate structure.
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THEORETICAL METHODS Our DFT calculations were carried out with the VASP package13−15 using the Perdew−Burke−Ernzerhof functional16 and the projector augmented wave method.17,18 The planewave cutoff was set to be 450 eV, and a Monkhorst−Pack kpoint of 15 × 15 × 1 was used to sample the first Brillouin zone (BZ) of the monolayer MoS2. The sampling of k-points in supercell is achieved to maintain the kpoint density equivalent to that in the first BZ of the monolayer. A vacuum space of 10 Å was left along the direction normal to the basal plane of MoS2 in order to eliminate the interaction between the periodic images. These parameters ensure the errors in the calculated total energy are less than 0.5 meV per MoS2 unit.
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RESULTS AND DISCUSSION As shown in Figure 1, the electronic structure of the bilayer MoS2 with the bulk 2H stacking (also referred as the B2
Figure 1. Band structure of MoS2 bilayers with the bulk 2H stacking sequence (also referred as the B2 in the text) and an interlayer separation distance d = 6.147 Å. The calculated partial charge density is shown as inset for the valence band maximum (VBM) at Γ, VBM at K, and conduction band minimum (CBM) at K. Mo and S atoms are represented by pink and yellow spheres, respectively.
stacking in this paper) is featured by an indirect band gap EΓ−K g between the top of the valence band at Γ and the valley of the conduction band at K (or at the valley between Γ and K, as two valleys are nearly degenerate). The direct band gap EK−K g , which corresponds to the transition at K between the top valence band and bottom conduction band, is nearly identical to that in monolayer MoS2. The calculation of the partial charge density unveils the electronic orbitals responsible for these special states, as shown in Figure 1. At K, the top of valence band is primarily contributed by the in-plane d-orbitals (dxy and dx2−y2),
Figure 2. High symmetry stacking sequences of MoS2 bilayers. Both the side view and top view are shown for each stacking sequence. The five high symmetry stacking sequences can be grouped into two categories on the basis of the dimensionless in-plane interlayer S−S distance t: the A stacking for t = 0 and the B stacking for t = √3/3. The solid arrow indicates that two stacking sequences can be related through an in-plane displacement by √3/3 on the top layer. The dashed arrow represents that the stacking sequences can be transformed via an in-plane rotation of the top layer by 60° with respect to a S−S axis. Note that B1 and B2 correspond to the 3R and 2H stacking sequences in the bulk MoS2, respectively. 1248
DOI: 10.1021/jp5101736 J. Phys. Chem. C 2015, 119, 1247−1252
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The Journal of Physical Chemistry C
while becomes weaker as the interlayer separation distance d increases. This is not unexpected, as when d is sufficiently large, the interlayer coupling becomes negligibly small and the bilayers behave as the superposition of the decoupled individual monolayers. More interestingly, Figure 3 further shows that the variation of E gΓ−K with the in-plane displacement t is virtually independent of either the orientation of the displacement or the starting configuration (i.e., A1 or A2), but only dependent shows on the magnitude of the displacement t. As a result, EΓ−K g a nearly monotonic dependence on t when d is fixed. The nearly isotropic in-plane variation of EΓ−K can be further g illustrated by Figure 3b, which shows the contour of the calculated EΓ−K as a function of the displacement within the x− g y plane. The contour lines are virtually of circular shape and only recover the 6-fold rotational symmetry near the maximum displacement. with the interlayer S−S distance and The variation of EΓ−K g the isotropy of such dependence on the in-plane displacement t suggest that the electronic coupling can be mainly understood by the interlayer S−S interactions. The coupling between two adjacent MoS2 layers involves the interactions from the interlayer S−S, S−Mo, and Mo−Mo couplings, but because the wave functions from the inner S atoms spatially overlap the most, and particularly because the valence band top at Γ is largely contributed by the antibonding pz orbitals on S atoms, the interlayer S−S interaction dominates the electronic coupling between layers. Particularly, the change in the interlayer S−S distance will significantly affect the valence band top and hence the indirect band gap EΓ−K of bilayer MoS2. g This is the main origin for the observed strong dependence of the band gap of MoS2 with the interlayer separation distance and number of layers. Similarly, when the interlayer separation distance is fixed, displacing one layer horizontally with respect to another will also change the interlayer S−S distance, which correspondingly affects the indirect band gap. Because of the symmetry of the pz orbital, the interlayer S−S interaction through pz displays the in-plane isotropy. Here we also note that the less isotropic contour line near the maximum t in Figure 3b can be attributed to the weaker coupling arising from the interlayer Mo−S and Mo−Mo interactions. These interactions are also responsible for the slight scattering of the EΓ−K for a given t. In particular, the high g symmetry stacking sequences can also be reflected by a slight into two at t = 0 (corresponding to A1 and A2) split of EΓ−K g and three at t = √3/3 (corresponding to B1, B2, and B3), respectively. The effect of these interactions becomes more discernible when the interlayer S−S interaction is the weakest, i.e., where the in-plane displacement t reaches its maximum √3/3. However, since the interlayer Mo−S and Mo−Mo interaction are essentially weaker than the interlayer S−S interactions, their effect on the band gap is small, particularly for small t. Therefore, as a good approximation, the electronic coupling between two layers is mainly controlled by the interlayer S−S interaction. In-Plane Rotation. With this understanding, we can now turn to the in-plane rotation where the twsited MoS2 bilayers with a twist angle θt are obtained. Upon rotation, the MoS2 bilayer forms the moire pattern, resulting in either a commensurate or an incommensurate superstructure, depending on the twist angle θt. Only the commensurate superstructures may be modeled directly in ab initio calculation using a supercell combined with the periodic boundary condition.
layer sit directly on top of those in the bottom layer, yielding an in-plane interlayer S−S distance rSS xy = 0. A1 and A2 can be related to each other through rotating the top layer by 60° with respect to a S−S axis. When an in-plane displacement of √3a/ 3, where a is the in-plane lattice constant of MoS2, is applied to the top layer in A2, the bilayers can transform into two high symmetry stacking sequences: B2 and B3. It is noted that B2 also corresponds to the 2H stacking sequence in the bulk MoS2. Similarly, the in-plane displacement of √3a/3 transforms A1 to another two structures which are equivalent to each other through the spatial inversion, thus represented as the B1 stacking (corresponding to the stacking order in the 3R bulk phase). Therefore, all of the B1, B2, and B3 stacking sequences yield an in-plane interlayer S−S distance rSS xy = √3a/3, defining the maximum of rSS xy . Because of the translational symmetry, the in-plane displacement varies between 0 and √3a/3 for any lattice registry. It is then convenient to use the dimensionless t SS ≡ rxy /a ∈ [0, √3/3] to describe the in-plane S−S displacement. Figure 3a shows the calculated indirect band gap EΓ−K of g MoS2 bilayers as a function of the dimensionless in-plane
Figure 3. (a) Variation of the calculated indirect band gap EΓ−K with g the in-plane displacement t for the MoS2 bilayers with an interlayer distance d ranging 6.1−6.8 Å. Note that the A and B stacking sequences yield t = 0 and t = √3/3 = 0.577, respectively. (b) Contour as a function of the in-plane displacement within the x−y of EΓ−K g plane for the MoS2 bilayers with d = 6.1 Å.
displacement t, for different interlayer separation distance d. It varies with both is evident that the indirect band gap EΓ−K g displacements. On one hand, at the fixed in-plane displacement t, the increase in the interlayer separation distance d raises EΓ−K g , consistent with the prediction made in previous study.7 On the other, EΓ−K also increases notably with the in-plane displaceg ment t when d is fixed. The variation of EΓ−K with t is g particularly significant for small d (e.g., ∼ 0.2 eV for d = 6.1 Å), 1249
DOI: 10.1021/jp5101736 J. Phys. Chem. C 2015, 119, 1247−1252
Article
The Journal of Physical Chemistry C The incommensurate superstructure does not possess the translational symmetry, making it difficult to explore the electronic structures theoretically. However, since the coupling of the bilayer MoS2 was found to substantially depend on the interlayer S−S interaction, the electronic properties of the twisted MoS2 bilayers, particularly the coupling at the valence band maximum near Γ, can be understood through examining the variation of the interlayer S−S distance with the twist angle θt. To do this, we consider a MoS2 monolayer overlays another monolayer with the A1 stacking sequence, i.e., rSS xy or t = 0. Choosing a rotational axis to be, for example, the line passing the S atoms on both layers, one rotates the top layer by a twist angle θt, while keeping the interlayer separation distance d fixed. As a consequence, t is no longer a constant throughout the plane but depends on the coordinates of the S atoms on the top layer and t varies between 0 and √3/3. As the interlayer electronic coupling is determined by the interlayer S−S distance, it is of interest to determine the distribution function p(t) in the twisted bilayers, i.e., the probability density of finding a S atom from the top layer that yields an in-plane distance of t with respect to another S atom in the bottom layer. To do this, one chooses a unit cell in the bottom MoS2 layer and folds the xy coordinates of the S atoms in the top MoS2 layer into this unit cell, in order to account for the inplane translational symmetry of MoS2. Interestingly, for the infinitely large and incommensurate MoS2 bilayers, we find that the folded positions are uniformly distributed within the area of the unit cell, and such uniformity is independent of the incommensurate twist angle θit. As a consequence, it can be shown (see Supporting Information) that in a hexagonal lattice the distribution function for the infinitely large and incommensurate bilayers bears the following expression: ⎧4 3 1 0≤t≤ πt , ⎪ ⎪ 3 2 f (t ) = ⎨ ⎪4 3 ⎛ 1 3 −1 1 ⎞ ⎜π − 6 cos ⎟t ,
