MoS2 and MoS2

Article pubs.acs.org/JPCC

Electronic Structure of Twisted Bilayers of Graphene/MoS2 and MoS2/ MoS2 Zilu Wang, Qian Chen, and Jinlan Wang* Department of Physics & Key Laboratory of MEMS of Ministry of Education, Southeast University, Nanjing 211189, P.R. China S Supporting Information *

ABSTRACT: Vertically stacked two-dimensional multilayer structures have become a promising prototype for functionalized nanodevices due to their wide range of tunable properties. In this paper we performed first-principles calculations to study the electronic structure of nontwisted and twisted bilayers of hybrid graphene/MoS2 (Gr/MoS2) and MoS2/MoS2. Both twisted bilayers of Gr/MoS2 and MoS2/ MoS2 show significant differences in band structures from the nontwisted ones with the appearance of the crossover between direct and indirect band gap and gap variation. More interestingly, the band structures of twisted Gr/MoS2 with different rotation angles are very different from each other, while those of MoS2/MoS2 are very similar. The variation of band structure with rotation angle in Gr/MoS2 is, indeed, originated from the misorientation-induced lattice strain and the sensitive band-strain dependence of MoS2.

I. INTRODUCTION

Experimentally it is difficult to fabricate layered samples with uniform stacking order from either chemical vapor deposition growth or artificial stacking. Various interlayer orientation angles could emerge as the deposition process differs. Twisted bilayer graphene has been studied by both experiment and theory,23−28 which exhibit critically dependent electronic properties upon the twist angle. For small angles the Fermi velocity is reduced substantially, while for large angles the electronic properties are indistinguishable from that of singlelayer graphene. Nevertheless, how much impact the interlayer rotation has on the electronic properties of graphene/TMDC hybrids and few-layer TMDCs is still unknown. The answer will provide a fundamental understanding of stacking 2D structures and would be of great importance to applications of TMDCbased devices. In this work, by using first-principles calculations we investigate the electronic structures of the bilayers of graphene/MoS2 (Gr/MoS2) hybrids and MoS2/MoS2 and the influence of the interlayer misorientation.

As the superior properties of graphene continue to be revealed, growing interests have been spurred to investigate the graphene analogues and their hybrid nanostructures.1−5 Among the promising graphene-like two-dimensional (2D) material candidates, MoS2, a typical transition metal dichalcogenide (TMDC), has exhibited potential uses in a variety of device applications.6−8 Given the rich electronic properties of TMDCs,9,10 stacked 2D crystals provide great possibilities to achieve desired functionalities with varying thickness and composition.11 Experimentally, graphene/MoS2 heterostructures have been successfully fabricated and shown high-field effect performance by several groups.12−16 Britnell et al.17 built a multilayer heterostructure of graphene and TMDC which yields a large photocurrent with fine-tuned Fermi level and doping effect. Theoretically, Kośmider18 studied the heterojunction of MoS2/WS2 and found a fundamental direct band gap in such bilayer composition. Terrones et al.19 predicted a similar direct band gap in hybrid metal disulfides and diselenides for specific stacking component and position. Bernardi et al.20 reported a high power conversion efficiency of up to ∼1% in graphene/MoS2 solar cells. More recently Kou et al.21 showed that the intrinsic strain and interface polarization have a significant impact on the structural and electronic properties of TMDC heterostructures. Nevertheless, Komsa et al.22 presented a systematic density functional theory (DFT) study on unstrained TMDC heterostructures and revealed that the optical absorption spectrum of the MoS2/WS2 system is merely weakly affected by the interlayer interactions. © 2015 American Chemical Society

II. COMPUTATIONAL MODEL AND METHOD Our theoretical calculations were performed within the framework of the plane-wave pseudopotential DFT method. The ion−electron interactions were described by the projected augmented wave method,29,30 and local density approximation31 was employed to describe the exchange−correlation interactions. We have further performed calculations with the Received: August 1, 2014 Revised: January 13, 2015 Published: January 29, 2015 4752

DOI: 10.1021/jp507751p J. Phys. Chem. C 2015, 119, 4752−4758

Article

The Journal of Physical Chemistry C PBE+DFT-D2 method that includes vdW interactions to justify the LDA results, and the obtained trends are very similar (Figure S1 in the Supporting Information). Therefore, all the results presented are based on LDA calculations. The Brillouin zone was sampled by a k-point mesh of 0.08 Å−1 separation in reciprocal space within Monkhorst−Pack scheme,32 and the kinetic energy cutoff was chosen to be 400 eV. A vacuum region larger than 15 Å was added to avoid spurious interactions between periodic images. All atomic positions were fully optimized until the forces acting on each atom were less than 0.01 eV/Å. All the calculations were performed using the Vienna ab initio simulation package.30,33 The twisted bilayers were modeled using accidental angular commensurations.34 In a hexagonal lattice whose basis vector is a1 and a2, a skewed supercell with basis vector (na1 + ma2) has a corresponding skewed angle, θ = tan−1(√3m/2n + m) (see Figure S2 in the Supporting Information). The graphene/MoS2 hybrid sheets were produced by looking for pairs of commensurate skewed cells of the two lattices with similar periodicity. The rotation angle of graphene relative to MoS2 can be expressed as θGr−θM. We denoted each twisted bilayer heterostructure with a notation of p:q, where p and q refer to the periodicity of two constituent layers. To obtain the energetically most favorable superlattice constant, we inspected the response of the total energy to the applied elastic strain in small bilayer systems (4:3 and √7:2). The optimized supercell constant corresponds to a nearly opposite strain state for the stacking sheets, i.e., −1.5% (−1.6%) and +1.3% (+1.9%), for graphene and MoS2 with a commensurability condition of √7:2 (4:3), respectively, as shown in the inset of Figure 2. Following this trend we rescaled the commensurate lattice constant for each hybrid bilayer so that the compressive and tensile strains suffered by the opposite layers have the same magnitude, and the largest strain is below 3%. The number of atoms inside the unit cell increases as the orientation angle decreases. Therefore, to reduce the computational complexity, we adopted five typical twisted superstructures in this study, namely, √7:2, √37:5, √39:5, √43:5, and √21:√13, respectively. The interlayer displacement is confined to a stacking order where a C atom of graphene and a S atom of MoS2 are superimposed within each unit cell, considering that the interlayer bonding is insensitive to their mutual translation.

Figure 1. (a) Relaxed geometry (top and side view) and (b) electronic band structure and projected density of states of the nontwisted Gr/ MoS2 bilayer. The energy states originated from graphene are denoted by the blue dotted line in the right panel.

graphene, which is attributed to the difference in sublattice onsite energy induced by the substrate.35 Next we focus on the twisted bilayers of the Gr/MoS2 hybrid structure. Figure 2 displays the top view of the optimized configuration for each hybrid bilayer. The largest corrugation of graphene is 0.07 Å, indicating the well-compensated lattice mismatch between graphene and MoS2. The binding energy and interlayer distance for each bilayer system are listed in Table 1. The BE and interlayer distance are in the near range of 20 meV and 3.35 Å, similar to those of nontwisted ones, and they have no clear dependence on the rotation angle. This is reasonable considering the weak interaction between the layers. The interlayer distance is slightly increased when the lateral strain gets larger. Figure 3 depicts the band structure and PDOS for each twisted bilayer, where the Fermi level is set as zero. Similar to the case of nontwisted Gr/MoS2, the band structure of the twisted Gr/MoS2 can be interpreted as a superposition of its constituent states, and there is no obvious hybridization between the orbitals from graphene and MoS2. The opened tiny gap of about 2 meV at the Dirac cone remains irrespective of the rotation angle. The energy gap of graphene at the M point varies from 1.36 to 2.94 eV as a result of the shortened linear dispersion range inside a folded Brillouin zone. For the rotation angles studied here, the direct band gap of monolayer MoS2 is only preserved when θ = 16.1°, which is 1.86 eV at the K point. When θ = 19.1° and 7.6° the valence band minimum (VBM) shifts to the Γ point, and the indirect band gap is 1.28 and 1.38 eV, respectively. When θ = 25.3° the conduction band minimum (CBM) locates at a midpoint between K and Γ, and the indirect band gap becomes 1.85 eV. The transition from direct to indirect band gap of MoS2 is not a result of the band folding effect since the lateral periodicity (2 × 2 and 5 × 5) of the MoS2 supercell preserves the locations of its high-symmetry K points in the shrinked Brillouin zone. Thus, the transition is only attributed to the sensitive band dispersion dependence on the MoS2 lattice strain. Indeed, as previously reported by GGA calculations,36 the direct band gap of monolayer MoS2 is preserved only when the lattice constant is in a narrow range deviating from the optimum value. Within our LDA approximation, the CBM shifts from the K point to a midpoint between Γ−K when the compressive strain is lower than −1.4%, and the VBM changes to Γ when the tensile strain exceeds 0% (Figure S3 in the

III. RESULTS AND DISCUSSION We first study the nontwisted bilayer Gr/MoS2 heterostructure, as shown in Figure 1. At equilibrium position graphene lies 3.32 Å above the topmost S plane of MoS2. The average binding energy (BE) between graphene and MoS2 is about 21 meV per C atom, calculated from BE = (Egr + Emos − Egr@mos)/NCatom. These results are all consistent with the previous study35 and demonstrate the very weak van der Waals interaction between graphene and the MoS2 monolayer. The electronic band structure and projected density of states (PDOS) for this heterobilayer are presented in Figure 1b. To distinguish the contribution of each constituent layer more clearly, the energy states originated primarily from graphene are highlighted by blue dotted lines. Obviously, the electronic structure can be regarded as a simple superposition of the energy states from each constituent. The direct band gap of MoS2 remains the same value as in its monolayer form, 1.83 eV at the K point. A tiny band gap, about 2 meV, is opened in the Dirac cone of 4753


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The Journal of Physical Chemistry C

Figure 2. (a)−(d) Top views of twisted Gr/MoS2 with different interlayer rotation angles, θ = 19.1°, 25.3°, 16.1°, and 7.6°. (e) Charge density difference for twisted Gr/MoS2 in (a) defined as ρdiff = ρGr@mos − ρGr − ρmos; red and blue isosurfaces denote the aggregation and depletion of electrons, respectively. (f) Total energy of twisted Gr/MoS2 in response to the lateral lattice strain.

difference between graphene and its neighboring sulfur plane is listed in Table 1 for each hybrid bilayer. Figure 2e depicts the charge density difference at the Gr/ MoS2 interface (θ = 19.1°), which is defined as ρdiff = ρGr@mos − ρGr − ρmos. There is a weak but clear charge redistribution in the interlayer region for all the bilayer sheets. The charge population analysis from Bader calculation shows that the depletion of charge from graphene is about 5 × 10−4 e per C. To manifest the effect of pure twist deformation on the electronic structure of the stacked bilayer, we further built a pair of twisted Gr/MoS2 with different rotation angles but the same lattice mismatch. The structure is constructed by substituting the bottom MoS2 straight unit cell with a skewed one. The rotation angle between graphene and MoS2 becomes 3° and 24.8° in our simulation (see Figure 4). The band structure and PDOS for the two bilayers are plotted in the same axis with solid and dashed lines, respectively. Interestingly, as clearly seen from the figure, the E−k dispersion relations are almost identical for the two rotation angles, where the only difference between them lies in the 0.02 eV shift of MoS2 energy states. Therefore, we infer that there is only negligible influence of pure twist deformation on the band structure of Gr/MoS2 when the strain impact is identical. We next investigate the twisted MoS2 bilayers, which are produced by stacking a pair of oppositely skewed MoS2 supercells. Since interlayer translation will generate different geometries with the same rotation angle, we have considered two characteristic interlayer registries, where a S atom in the top layer superimposes on a Mo or S atom (S/Mo or S/S) in the bottom layer, respectively, within each periodic supercell. The calculated binding energies and band structures are almost identical between the two stacking orders (Figure S4 in the Supporting Information). Therefore, the following discussions are based on the S/Mo stacking, which is energetically a little more favorable. The top view of the relaxed geometry of MoS2 bilayers is given in Figure 5, where Moiré patterns can be observed for each twisted model. Table 2 lists the binding energy per MoS2 unit and the interlayer distance for each rotation angle. Comparing to the AB stacking the twisted

Table 1. Interlayer Rotation Angle (θ), Lattice Strain (εmos2), Interlayer Distances (deq), Binding Energies (BE), and Electrostatic Potential Difference (Δφ) for Different Twisted Graphene/MoS2 Bilayers system

θ (deg)

εmos2 (%)

BE (meV)

deq (Å)

Δφ

√37:5 √7:2 √39:5 √43:5 √21:√13

25.3 19.1 16.1 7.6 3.0

−2.70 1.40 −1.50 0.90 −0.70

22.6 20.0 20.8 19.9 19.9

3.36 3.34 3.37 3.32 3.32

2.3 5.2 3.2 4.9 3.8

Supporting Information). Therefore, the change of the band structure with respect to the rotation angle in the twisted bilayer is, in fact, a result of the sensitive dependence of band dispersion of MoS2 on lattice strain. An obvious variation of the relative alignment of the graphene Dirac cone to the MoS2 band edge is observed for different twist angles. For θ = 25.3° and 16.1° the Dirac point is far from the band edges of MoS2. On the other hand, the cone gets closer to the MoS2 conduction band and locates 0.1 eV above and 0.03 eV below the CBM for θ = 19.1° and θ = 7.6°, respectively. The relative alignment can also be modulated by the lateral lattice strain. As the compensated lattice shrinks, the CBM of MoS2 rises above the Fermi level. This can be explained by the spontaneous electrical polarization at the interface. Due to the difference in electronegativity between the sulfur and carbon atoms, an electrostatic potential difference between the two constituents will be established upon the adhesion of graphene on MoS2, leading to a spontaneous polarization at the interface. The generated vertical electric field points from MoS2 to graphene. With the increase of lateral strain, the electrostatic potential of both isolated graphene and the MoS2 monolayer increases monotonically. In the twisted Gr/MoS2, as the rotation angle varies, the two constituent layers will undergo opposite strain variations. Thus, the local electrostatic potential of graphene and MoS2 will shift toward the opposite direction, inducing a more pronounced electrostatic potential difference. The corresponding local potential 4754


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Figure 3. Band structure and projected density of states for each twisted Gr/MoS2 in Figure 2. The Fermi level is set as zero.

Figure 4. Arrangement of twisted Gr/MoS2 with the same lattice mismatch but different interlayer rotation angle (a) θ = 3° and (b) θ = 24.8°. (c) Band structure and PDOS for (a) and (b) are plotted by solid and dashed lines, respectively. The effect on electronic structure brought by a pure interlayer twist is negligible.

bilayer is energetically less favorable, as evidenced by the 38% reduction in binding energy and the 14% increase in interlayer distance. As the rotation angle increases, the binding energy is almost invariant, while the interlayer distance has a weak linear decreasing dependence on the rotation angle. Consistently, the corresponding band and PDOS (see Figure 6) for each bilayer

system have a similar dependence on the interlayer misorientation. For AB stacking, the obtained indirect band gap between Γ and I is 1.16 eV, and the direct energy gap at the K point is 1.74 eV. In the case of a twisted bilayer, both the direct gap at the K point and the indirect band gap have very similar values, 1.83 and 1.46 eV, across different rotation angles, 4755


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respectively. The position of the CBM shifts to K point for θ = 38.2° and 27.8°, while it moves to a midpoint between Γ and M for 46.8°. The most favorable AB stacking is the unique conformation of bilayer MoS2 (Figure 6a), which has a spontaneous polarization between the two monolayers, as discussed previously.37 The established electric field points toward the +z direction in our stacking configuration. The intrinsic field breaks the symmetry of bilayer conformation and splits the states from two monolayers. As the electron momentum varies from K to Γ along the high-symmetry path, the band edge states exhibit a gradually enhanced mixing character from the states of constituent monolayers. The split two-edge states at the K point of both the VB and CB are fully localized in the opposite monolayers. Their splitting causes a reduced direct gap at the K point from that of the monolayer. On the other hand, the edge states at Γ originate from a strong mixing of monolayer states, and they are susceptible to interlayer coupling. The distinct behavior between Γ and K states can be understood by their different orbital characters.22 For the AA stacking bilayer, the direct gap at the K point and the indirect band gap between Γ and I are 1.81 and 1.62 eV, respectively. The interlayer interaction is largely reduced due to the absence of intrinsic polarization. All the band edge states present an equal weight of mixing from the states of two monolayers. For the twisted conformations in Figure 5b−d, there is a weak intrinsic polarization at the interface, and their interlayer coupling strength is between that of AB and AA stacking, which

Figure 5. Top view of the optimized structures of bilayer MoS2 with (a) Bernal stacking and (b, c, d) an interlayer twist by a rotation angle of 38.2°, 27.8°, and 46.8°.

Table 2. Interlayer Misorientation (θ), Interlayer Distance (deq), and Binding Energy (BE) per MoS2 Unit for Twisted Bilayer MoS2 Structures from Our Calculations system

θ (deg)

BE (meV)

deq (Å)

AB √7 √13 √19

0 38.2 27.8 46.8

56.7 34.9 35.0 34.9

2.88 3.26 3.27 3.28

Figure 6. Band structure and PDOS for each twisted bilayer MoS2 in Figure 5. Magenta and green denote the contributions from the bottom (1L) and top (2L) layers, respectively. 4756


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Figure 7. Spatial map of charge density difference for the twisted bilayer MoS2 (calculated as ρdiff = ρBL − ρTOP − ρBOT). Regions of electron accumulation and depletion are denoted by red and blue lobes, respectively. Isosurface ρ = 2 × 10−4 e/bohr.3

different rotation angles are very different from each other, which are mainly attributed to the misorientation-induced lattice strain and the sensitive band-strain dependence in MoS2. The influence brought by a pure interlayer rotation is nearly negligible. Furthermore, the position of CBM of MoS2 relative to the Dirac cone can be tuned effectively by strain or twist, which is beneficial for doping MoS2. For twisted bilayer MoS2, the band structure presents a distinct behavior from that of the AB-stacked bilayer, which is a result of reduced interlayer coupling and Brillouin zone folding. The electronic band gap is insensitive to the misorientation angle, which implies that twisting between stacked layers of MoS2 or other TMDCs can be a route for their property engineering.

is judged from their intermediate binding energy, interlayer distance, and the indirect band gap at the Γ point between those values in AB and AA. This can be explained by the fact that under a disordered stacking the interlayer attraction can be enhanced locally where S and Mo atoms in opposite layers get closer occasionally. The states of both the VB and CB at the K point are double degenerate for all three rotation angles, and they are all localized in the opposite monolayers. The band dispersion is also altered by Brillouin zone folding. As the rotation angle decreases, the size of the unit cell increases, which induces a shortening dispersion range and a decreasing energy difference between the VB states at Γ and K. The charge density difference of twisted bilayer MoS2 is plotted in Figure 7, which is defined as ρdiff = ρBL − ρTOP − ρBOT. For all the stacked bilayers there is a tiny but still clear charge accumulation in the middle plane between the two monolayers. The charge depletion occurs at the innermost two neighboring S planes. The spatial redistribution of charge density has a more delocalized character for AB stacking. This is consistent with the relative interlayer bonding strength of AB and twisted stacking order. Excluding the strain effect, the band structures of the twisted Gr/MoS2 and MoS2/MoS2 show insensitive/sensitive dependence on the twisting angle, which come from their different nature of interlayer bonding. For the bilayer systems with little hybridization between the constituent layers, like Gr/MoS2, the combined electronic state is basically a superposition of their respective folded band structure, while for the bilayers such as BL-MoS2 their strong interlayer hybridization is more susceptible to the twisting angle, which imposes different lateral periodicity on the superlattice and modulates the extent of orbital overlap. However, for van der Waals heterostructures, the lattice strain is always an important factor to determine the resultant electronic properties.

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ASSOCIATED CONTENT

S Supporting Information *

Band structures of twisted Gr/MoS2 bilayer calculated using PBE functional with DFT-D2 method, definition of skewed angle in a hexagonal lattice, band gap variation of monolayer MoS2 with the applied strain, and electronic structures of the twisted bilayer MoS2 with S/S stacking order. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-25-52090600-8304. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the NBRP (2011CB302004), NSF (21173040, 21373045, 11404056), and of Jiangsu (BK20130016, BK2012322) and SRFDP (20130092110029) in China. The authors thank the computational resources at the SEU and National Supercomputing Center in Tianjin.

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IV. CONCLUSION In summary, we have explored the electronic properties of twisted bilayers of graphene/MoS2 and MoS2/MoS2 within the framework of density functional theory. Our calculations show that the electronic structures of twisted bilayers of Gr/MoS2 and MoS2/MoS2 are different from those of the nontwisted ones. Moreover, the band structures of twisted Gr/MoS2 with

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