MoS₂/MX₂ heterobilayers: bandgap engineering via tensile strain or external electrical field

Abstract

We have performed a comprehensive first-principles study of the electronic and magnetic properties of two-dimensional (2D) transition-metal dichalcogenide (TMD) heterobilayers MX₂/MoS₂ (M = Mo, Cr, W, Fe, V; X = S, Se). For M = Mo, Cr, W; X = S, Se, all heterobilayers show semiconducting characteristics with an indirect bandgap with the exception of the WSe₂/MoS₂ heterobilayer which retains the direct-bandgap character of the constituent monolayer. For M = Fe, V; X = S, Se, the MX₂/MoS₂ heterobilayers exhibit metallic characters. Particular attention of this study has been focused on engineering the bandgap of the TMD heterobilayer materials via application of either a tensile strain or an external electric field. We find that with increasing either the biaxial or uniaxial tensile strain, the MX₂/MoS₂ (M = Mo, Cr, W; X = S, Se) heterobilayers can undergo a semiconductor-to-metal transition. For the WSe₂/MoS₂ heterobilayer, a direct-to-indirect bandgap transition may occur beyond a critical biaxial or uniaxial strain. For M (=Fe, V) and X (=S, Se), the magnetic moments of both metal and chalcogen atoms are enhanced when the MX₂/MoS₂ heterobilayers are under a biaxial tensile strain. Moreover, the bandgap of MX₂/MoS₂ (M = Mo, Cr, W; X = S, Se) heterobilayers can be reduced by the vertical electric field. For two heterobilayers MSe₂/MoS₂ (M = Mo, Cr), PBE calculations suggest that the indirect-to-direct bandgap transition may occur under an external electric field. The transition is attributed to the enhanced spontaneous polarization. The tunable bandgaps in general and possible indirect–direct bandgap transitions due to tensile strain or external electric field make the TMD heterobilayer materials a viable candidate for optoelectronic applications.

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Introduction

Two dimensional transition-metal dichalcogenides (TMDs) have attracted intensive interest recently owing to their novel electronic and catalytic properties that differ from their bulk counterparts.^1–3 For example, as a representative of 2D TMD materials, the 2D molybdenum disulfide (MoS₂) monolayer possesses a direct bandgap of 1.8–1.9 eV while the MoS₂ bilayer possesses an indirect bandgap of ∼1.53 eV; the MoS₂ transistors exhibit a high on/off ratio of 1 × 10⁸ at room temperature. Moreover, the MoS₂-based integrated circuits have been fabricated and reported in the literature.^4–7

Tunable electronic properties of 2D TMD materials are crucial for their applications in optoelectronics. Heterostructures have been widely used in conventional semiconductors for achieving tunable electronic properties. For the development of future 2D materials, the van der Waals heterostructures have been recognized as one of the most promising candidates⁸ and the TMD-based hybrid multilayered structures are prototype van der Waals heterostructures. Recently, the vertical field-effect transistor and memory cell made of TMD/graphene heterostructures have been reported.^9–12 The Moiŕe pattern of the nanometer-scale MoS₂/MoSe₂ heterobilayer has been theoretically studied.¹³ Note however that although many MX₂ (e.g., MoS₂ and MoSe₂) monolayers are direct-gap semiconductors, their bilayers are indirect-gap semiconductors. Recent theoretical studies suggest that the direct-bandgap character can be retained only in several heterobilayer structures^14,15 and the heterobilayers are more desirable for optoelectronic applications.^16,17 To achieve tunable bandgaps for 2D materials, two widely used engineering strategies are the application of either an external electric field or a tensile strain.^18–31 Previous theoretical studies have also shown that the bandgap of the MoS₂ monolayer is insensitive to the external electric field, whereas the indirect bandgap of the MoS₂ bilayer decreases with the increase of the vertical electric field.^18,19 The MoS₂ or MoSe₂ trilayer exhibits similar bandgap behavior as the bilayer counterpart when under the external electric field.²⁰ On the other hand, previous theoretical studies show that the monolayer of TMDs can undergo the direct-to-indirect bandgap transition under the increasing tensile strain, a promising way to tune the bandgap of TMD monolayers.^21,23 Photoluminescence spectroscopy measurements have confirmed that the optical bandgap of the MoS₂ monolayer and bilayer decreases with the uniaxial strain and exhibits a direct-to-indirect transition.²⁵ Moreover, ultra high strain tenability has been demonstrated in trilayer MoS₂ sheets.²⁶ Also, under the tensile strain, the nonmagnetic NbS₂ and NbSe₂ layers can be changed to ferromagnetic.²⁴

To date, most studies of TMD heterostructures are concerned about the Mo and W systems. In view of successful synthesis of nanosheets of V, Nb, Ti, and Cu systems,^32–35 it is timely to examine the electronic properties of TMD heterostructures and the effect of the external electric field or tensile strain on their bandgaps.^36,37 In this study, our focus is placed on numerous MoS₂-based heterobilayers, including CrS₂/MoS₂, CrSe₂/MoS₂, MoSe₂/MoS₂, WS₂/MoS₂, WSe₂/MoS₂, VS₂/MoS₂, and VSe₂/MoS₂. For these heterobilayer systems, the lattice mismatch is typically less than 5%. We find that under a vertical electric field the indirect-to-direct bandgap transition may occur for two heterobilayers. A direct-to-indirect bandgap transition may occur only for the WSe₂/MoS₂ heterobilayer under an increasing tensile strain. In general, either the vertical electric field or the tensile strain can notably affect the bandgap of the TMD heterobilayers.

Computational methods

All calculations are performed within the framework of spin-polarized plane-wave density functional theory (PW-DFT), implemented in the Vienna ab initio simulation package (VASP).^38,39 The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional and projector augmented wave (PAW) potentials are used.^40–42 The effect of van der Waals interaction is accounted for using a dispersion-corrected PBE method.^43,44 More specifically, we adopt a 1 × 1 unit cell for the investigation. The vacuum size is larger than 15 Å between two adjacent images. An energy cutoff of 500 eV is adopted for the plane-wave expansion of the electronic wave function. Geometry structures are relaxed until the force on each atom is less than 0.01 eV Å⁻¹ and the energy convergence criteria of 1 × 10⁻⁵ eV are met. The 2D Brillouin zone integration using the Γ-center scheme is sampled with a 9 × 9 × 1 grid for geometry optimization and a 15 × 15 × 1 grid for static electronic structure calculations. For each heterobilayer system, the unit cell is optimized to obtain the lattice parameters at the lowest total energy.

Biaxial tensile strain is applied to all MX₂/MoS₂ heterobilayers in a symmetric manner while a uniaxial tensile strain is applied in either x- or y-direction (see Fig. 1). The direction of the external electric field is normal to the plane of heterobilayers, and in VASP, the external uniform field is treated by adding an artificial dipole sheet (i.e., dipole correction) in the unit cell.⁴⁵ The geometries are kept fixed when applying the external electric field to neglect the geometric distributions to the electronic structures. The Bader's atom in molecule (AIM) method (based on charge density topological analysis) is used for the charge population calculation.⁴⁶ For a few systems, the hybrid HSE06 functional is also used to confirm the trend of bandgap change.⁴⁷ In particular, the WSe₂/MoS₂ heterobilayer is treated as a special system for which both the HSE06 calculation and the PBE calculation including the spin–orbit (SO) coupling effect⁴⁸ are reported.

Fig. 1 Atomic models of the MX₂/MoS₂ heterobilayer with four different types of layer-on-layer stacking: (a) AA stacking, (b) C7 stacking, (c) C27 stacking and (d) T stacking. For each stacking configuration, the left and right panels display the side and top views, respectively. The d_s−x denotes the interlayer height difference between X (top-layer) and S (lower-layer) atoms. (e) The tensile strain can be applied along x- or/and y-directions.

Results and discussion

1. Heterobilayers of MX₂/MoS₂

It is known that monolayer MX₂ exhibits two possible structures, namely, H or T phase. The H structure is only considered here because it is more stable than T for most of the MX₂ structures considered in this study.³⁶ Moreover, following a previous study,¹⁷ we consider four different types of bilayer stacking, namely, AA, C7, C27, and T stacking, to describe how a H–MX₂ monolayer is superimposed on the 2H–MoS₂ monolayer (see Fig. 1). A testing calculation suggests that the electronic structure is more or less the same for the four different stackings, consistent with a previous study.¹⁷ Therefore, only the C7 stacking that gives rise to the lowest energy in most heterobilayer systems is reported for the electronic structure calculations. The optimized cell parameters and the vertical height differences between interlayer X and S atoms (d_s−x, as shown in Fig. 1(a)) are listed in ESI Table S1.† The d_s−x of X and S atoms in different MX₂/MoS₂ heterobilayers is less than 3.2 Å due to van der Waals interaction between MX₂ and MoS₂ layers.

The computed electronic bandgaps of MX₂ monolayers, bilayers, and MX₂/MoS₂ heterobilayers, as well as the binding energies per unit cell of MX₂/MoS₂ heterobilayers are listed in Table 1. The binding energies are defined as E_b = E(MX₂/MoS₂ heterobilayer) − E(MX₂ monolayer) − E(MoS₂ monolayer), where E(MX₂/MoS₂ heterobilayer) is the total energy of the MX₂/MoS₂ heterobilayer and E(MX₂ monolayer) is the total energy of the MX₂ monolayer. For M = Mo, W, Cr, the MX₂ monolayers are direct semiconductors with the conduction band minimum (CBM) and valence band maximum (VBM) being located at the K point (ESI Fig. S1†). However, their corresponding bilayers become indirect semiconductors. For example, the MoS₂ monolayer is a direct semiconductor with a computed bandgap of 1.67 eV (PBE), while the bilayer is an indirect semiconductor with a bandgap of 1.25 eV (PBE). As shown in Fig. 2, the VBM of the bilayer structures relocates to the Γ point from the K point (for the monolayer). The partial charge density at the Γ point is contributed from both monolayers, and it exhibits a strong upward shift, overtaking the energy at the K point.¹⁴ For MoS₂, WS₂, CrS₂, and CrSe₂ bilayers, their CBM is still located at the K point. For MoSe₂, the CBM moves to the Λ point (Fig. 2), and the energy at the Λ point is 5 meV below that at the K point. The WSe₂ bilayer has a nearly degenerate energy for the two valleys.

Table 1 Computed bandgap E_g1 (in eV) of the MX₂ monolayer, bilayer E_g2, and MX₂/MoS₂ heterobilayer E_g3, as well as the binding energies per unit cell E_b (in eV) of the MX₂/MoS₂ heterobilayers

	E g1 (eV)	E g2	E g3	E b (eV)
MoS2	1.67 Direct	1.25 Indirect	—	—
MoSe2	1.46 Direct	1.20 Indirect	0.74 Indirect	−0.16
WS2	1.81 Direct	1.43 Indirect	1.16 Indirect	−0.22
WSe2	1.55 Direct	1.38 Indirect	0.57 Direct	−0.16
CrS2	0.93 Direct	0.68 Indirect	0.39 Indirect	−0.14
CrSe2	0.74 Direct	0.60 Indirect	0.69 Indirect	−0.22
FeS2	Metal	Metal	Metal	−0.31
VS2	Metal	Metal	Metal	−0.23
VSe2	Metal	Metal	Metal	−0.16

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Fig. 2 Computed band structures (PBE) of the homogeneous bilayer of (a) MoS₂, (b) MoSe₂, (c) WS₂, (d) WSe₂, (e) CrS₂, and (f) CrSe₂. All bilayers show an indirect bandgap.

As shown in Fig. 3, most MX₂/MoS₂ heterobilayers are indirect semiconductors, whereas only the WSe₂/MoS₂ heterobilayer possesses a direct bandgap of 0.57 eV. Different from their own bilayers, the CBM of heterostructures is all located at the K point, while the VBM is located at the Γ point. For the WSe₂/MoS₂ heterobilayer, however, the VBM is still located at the K point, resulting in a direct-bandgap semiconductor (PBE). The VBM of MoSe₂/MoS₂ at the Γ point (V1, Fig. 3(a)) shows a mixing of densities from both monolayers. The CBM (C1, Fig. 3(a)) and the valence band edge (PBE, V2, Fig. 3(a)) at the K point are localized for MoSe₂ and MoS₂, respectively. The CBM and VBM positions of MX₂ monolayers are shown in ESI Fig. S2.† One can see that the band structures of WS₂/MoS₂, WSe₂/MoS₂, and CrS₂/MoS₂ are similar to those of MoSe₂/MoS₂, showing type II alignment of the band edges, which may be of advantageous for the separation of electron–hole pairs.¹⁴

Fig. 3 Computed band structures (PBE) and partial charge density of C1, V1, and V2 states of the heterobilayer: (a) MoSe₂/MoS₂ and (b) CrSe₂/MoS₂. The isosurface value in (a) and (b) is 0.02 e Å⁻³. Computed band structures (PBE) of heterobilayer: (c) WS₂/MoS₂, (d) WSe₂/MoS₂, and (e) CrS₂/MoS₂. Only the WSe₂/MoS₂ heterobilayer exhibits a direct bandgap.

For CrSe₂/MoS₂, the VBM at the Γ point is over that at the K point by 67 meV (Fig. 3(b)), and the VBM at the Γ point is mainly due to the CrSe₂ layer with little contribution from the MoS₂ layer. However, different from other heterobilayers, the CBM and VBM at the K point are both due to CrSe₂, which exhibit the type I alignment. For MX₂ (M = Fe, V), the monolayer, bilayer, and MX₂/MoS₂ heterobilayers all exhibit metallic character, while the ferromagnetism is still kept by the heterobilayer. As shown in Table 1, the binding energies of all the MX₂ and MoS₂ heterobilayers are in the range of −0.31 to −0.14 eV, further supporting the weak van der Waals interaction between the MX₂ and MoS₂ layers.

2. Tunable bandgaps via tensile strain

Strain modulation has been commonly used in low-dimensional systems to tune the electronic structures. For TMD monolayers, the strain-induced bandgap modification has been predicted from recent first-principles calculations.^21,22,24 The photoluminescence spectroscopy measurement has further confirmed the strain effect on the electronic structure of both monolayer and bilayer TMD systems. Hence, it is of both fundamental and practical interest to examine the effect of tensile strains on the electronic properties of MX₂/MoS₂ heterobilayers. As such, first, we have applied in-plane tensile strain by stretching the hexagonal cell biaxially,²⁴ and the biaxial strain is defined as ε = Δa/a₀, where a₀ is unstrained cell parameters and Δa + a₀ is strained cell parameters.

As mentioned above, among the heterobilayers considered in this study, only the WSe₂/MoS₂ heterobilayer exhibits the direct-bandgap character (Fig. 3(d)). Nevertheless, we find that a 1% biaxial strain can turn the heterobilayer into an indirect semiconductor as the VBM is relocated from the K to the Γ point. The latter is 16 meV higher than that of the K point. The CBM is still located at the K point regardless of the strain. As the energy difference between the valence band at the K point and the Γ point is just 100 meV for the unstrained WSe₂/MoS₂ heterobilayer, the mixing feature of the Γ point renders it more easily affected by the tensile strain. Hence, even a relatively small strain (1%) can result in higher Γ point than the K point in the energy diagram, leading to an indirect bandgap. On further increasing the biaxial strain, the energy difference between the valence band edges at these two points becomes greater. And the indirect bandgap decreases with the biaxial tensile strain, as shown in Fig. 4.

Fig. 4 Computed electronic bandgaps (PBE) of MX₂/MoS₂ (M = Mo, W, Cr) heterobilayers versus the biaxial tensile strain, ranging from 0 to 8%.

The computed electronic bandgaps of the semiconducting MX₂/MoS₂ (M = Mo, W, Cr; X = S, Se) heterobilayers as a function of the biaxial tensile strain are shown in Fig. 4. For the unstrained MoSe₂/MoS₂ heterobilayer, it is an indirect semiconductor with a bandgap of 0.74 eV. With the 2% biaxial tensile strain, the bandgap is reduced to 0.39 eV but still indirect. When the tensile strain increases to 4%, the bandgap is further reduced to 0.045 eV. Eventually the MoSe₂/MoS₂ heterobilayer turns into a metal when the biaxial strain reaches 6%. For the WS₂/MoS₂ heterobilayer, it turns into a metal when the biaxial tensile strain reaches 8%.

As shown in Fig. 4, the bandgaps of MX₂/MoS₂ (M = Mo, W, Cr; X = S, Se) generally decrease with the biaxial tensile strain, and undergo a semiconductor-to-metal transition at certain critical strains. To gain more insight into this transition, we have analyzed the band structures and partial density of states (PDOS) of the unstrained and strained MX₂/MoS₂ heterobilayers. Here, we use the PDOS of the WS₂/MoS₂ heterobilayer as an example (see Fig. 5). The unstrained WS₂/MoS₂ heterobilayer is an indirect semiconductor with a bandgap of 1.16 eV. The VBM is mainly contributed by the d orbital of W in the WS₂ layer, while the CBM is mainly contributed by the d orbital of Mo in the MoS₂ layer. With a 4% biaxial tensile strain, the CBM is shifted toward the Fermi level, resulting in a reduced (indirect) bandgap (0.36 eV) for the WS₂/MoS₂ heterobilayer. With an 8% biaxial tensile strain, the shift of the CBM leads to the semiconductor-to-metal transition (see the bottom panel of Fig. 5).

Fig. 5 Computed partial density of states (PDOS) of the WS₂/MoS₂ heterobilayer under 0%, 4%, or 8% biaxial tensile strain. The vertical dashed line represents the Fermi level.

For the semiconducting CrS₂/MoS₂ heterobilayer, the PBE calculation suggests that it becomes a metal with a 4% biaxial tensile strain. Here, a 2 × 2 supercell is used. Under a 2% biaxial tensile strain the CrS₂/MoS₂ heterobilayer undergoes a nonmagnetic-to-antiferromagnetic transition. When the biaxial tensile strain increases to 10%, the CrS₂/MoS₂ heterobilayer turns into a strong antiferromagnetic coupling metal. Bader charge analysis suggests that the charge transfer between CrS₂ and MoS₂ layers is nearly zero under the 0% strain, and increases to 0.1 e under the 10% strain, indicating that the charge transfer between CrS₂ and MoS₂ layers increases with increasing tensile strain, leading to spontaneous polarization between CrS₂ and MoS₂ layers. In stark contrast, the CrS₂ monolayer cannot become magnetic even under a tensile strain as high as 15%. These results indicate that the charge transfer between MoS₂ and CrS₂ layers plays a key role in the nonmagnetic-to-antiferromagnetic transition.

Note also that several metallic heterobilayers MX₂/MoS₂ (M = Fe, V; X = S, Se) still maintain their metallic character under the biaxial tensile strain. Nevertheless, we find that the magnetic moment of M and X atoms increases with the increase of the biaxial tensile strain from 0% to 10% (see Table 2). A close examination of the PDOS of the VS₂/MoS₂ heterobilayer with 0%, 4% or 8% biaxial tensile strain (Fig. 6(a)) reveals that the state corresponding to the Fermi level is mainly contributed by d-states of V, which becomes more localized with increasing strain. As shown in Fig. 6(b), the spin charge density of the VS₂/MoS₂ heterobilayer with a 4% biaxial tensile strain suggests that the magnetism is mainly contributed by the V atom (0.98 μ_B) while the S atoms of VS₂ carry a small magnetic moment of −0.06 μ_B, consistent with the analysis based on PDOS. As a result, nano-mechanical modulation of strain can turn the nonmagnetic CrS₂/MoS₂ heterobilayer into antiferromagnetic. The strain can also enhance the spin polarization of the MX₂/MoS₂ (M = Fe, V; X = S, Se) heterobilayers. This feature may be exploited in spintronic applications such as mechanical nano-switch for spin-polarized transport.

Table 2 Calculated magnetic moment μ (μ_B) of the M and X atoms in MX₂/MoS₂ (M = Fe, V; X = S, Se) heterobilayers. The magnetic moment of X atoms is from MX₂

Strain	FeS2		VS2		VSe2
	Fe	S	V	S	V	Se
0%	1.05	−0.03	0.91	−0.04	1.02	−0.05
2%	1.50	−0.04	0.94	−0.05	1.05	−0.06
4%	1.60	−0.06	0.98	−0.06	1.08	−0.07
6%	1.72	−0.07	1.01	−0.07	1.11	−0.08
8%	1.84	−0.09	1.14	−0.08	1.15	−0.09
10%	1.98	−0.10	1.19	−0.09	1.18	−0.10

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Fig. 6 (a) Computed PDOS of the VS₂/MoS₂ heterobilayer under 0%, 4% or 8% biaxial tensile strain. The vertical dashed line represents the Fermi level. (b) The spin charge density of the VS₂/MoS₂ heterobilayer with a 4% biaxial tensile strain. The isosurface value is 0.01 e Å⁻³. The blue indicates the positive values.

Besides biaxial tensile strains, we also investigate effects of a uniaxial tensile strain in either x- or y-direction (Fig. 1(e)). Our calculations suggest that the bandgaps in both cases are reduced with increasing strain, as shown in Fig. 7. As mentioned above, MoSe₂ (WS₂, CrSe₂, CrS₂)/MoS₂ heterobilayers are indirect semiconductors. Under a uniaxial tensile strain these heterobilayers remain indirect semiconductors, the same behavior as under a biaxial tensile strain. However, the WSe₂/MoS₂ heterobilayer is predicted to be a direct semiconductor based on the PBE calculation. With a 2% uniaxial tensile strain along either x- or y-direction, the heterobilayer still remains a direct semiconductor, which is very different from that under the biaxial tensile strain for which the heterobilayer becomes an indirect semiconductor under only 1% biaxial tensile strain. When the uniaxial tensile strain increases to 4%, the WSe₂/MoS₂ heterobilayer turns into an indirect semiconductor.

Fig. 7 Computed bandgaps of MX₂/MoS₂ (M = Mo, W, Cr) heterobilayers versus the uniaxial tensile strain in the (a) x- or (b) y-direction, ranging from 0 to 8%.

Since the WSe₂/MoS₂ heterobilayer is the only system here showing a direct bandgap (Fig. 3(d)), additional PBE calculations including the spin-orbit coupling effects are presented in ESI Fig. S3–S5.† Under either the biaxial or uniaxial tensile strain, the bandgap is still direct but much smaller. Moreover, the direct-to-indirect transition is not seen with increasing strain. Nevertheless, the bandgap still decreases with increasing strain and exhibits a semiconductor-to-metal transition, consistent with the PBE results. Moreover, HSE06 calculations are also performed for the WSe₂/MoS₂ heterobilayer. Although HSE06 tends to overestimate the bandgap (see ESI Fig. S6† for a test calculation with the bilayer MoS₂), the overall trend in bandgap reduction with increasing strain is the same as that predicted from the PBE calculations (see ESI Fig. S3–S5†). However, the direct-to-indirect transition does not occur until 4% biaxial strain (ESI Fig. S3(l)†) or 6% uniaxial strain (ESI Fig. S4 and S5†).

3. External electric field in the normal direction

MX₂ (M = Mo, W, Cr; X = S, Se) monolayers are direct-bandgap semiconductors, whereas their homogeneous bilayers are indirect-gap semiconductors. Importantly, among the TMD heterobilayers considered, only the WSe₂/MoS₂ heterobilayer is a direct-bandgap semiconductor, while the MoSe₂/MoS₂ heterobilayer possesses a quasi-direct bandgap with only 0.1 eV difference between the direct and indirect bandgap (Fig. 3(a)), consistent with the previous study.¹⁵ Note that the HSE06 calculation suggests that the MoSe₂/MoS₂ heterobilayer is a direct bandgap semiconductor (ESI Fig. S6(b)†). Previous studies also predicted direct-bandgap characters of WS₂/WSe₂ and MoTe₂/MoS₂ heterobilayers.^14,15 We have computed the dipole moments of WSe₂/MoS₂ and MoSe₂/MoS₂ heterobilayers, and found that the dipole moments of both systems are about 0.01 e Å greater than those of the MS₂/MoS₂ (M = Mo, W, Cr) systems, suggesting that the stronger spontaneous polarization in the MSe₂/MoS₂ systems is responsible for the underlying direct-bandgap or quasi-direct-bandgap characters. This large difference in spontaneous polarization may stem from the electronegativity difference between S and Se. Assuming this explanation is valid, one could ask if a vertical electric field is applied to the system to increase the spontaneous polarization in MoSe₂/MoS₂, will the system undergo an indirect-to-direct bandgap transition? Our test calculation shows that the answer to this question is yes. As shown in Fig. 8(b), the applied 0.1 V Å⁻¹ electric field can induce the indirect-to-direct bandgap transition in the MoSe₂/MoS₂ heterobilayer. Indeed, the VBM is moved from the Γ point to the K point, and the direct transition of K–K is 0.03 eV narrower than the indirect transition of Γ–K. Further increasing the external field will reduce the direct bandgap more significantly than the indirect bandgap (Fig. 8(a)). Results of a Bader charge population analysis are presented in ESI Table S2.† One can see that the charge transfer between the MoS₂ and MoSe₂ layers indeed increases with the external electric field. We have also examined the bandgaps of the MoSe₂/MoS₂ heterobilayer with the geometry optimized under different electric fields; the results are nearly the same as those without the geometric optimization under the electric field (see ESI Table S3†).

Fig. 8 (a) Computed bandgaps (PBE) of MX₂/MoS₂ (M = Mo, W, Cr; X = S, Se) heterobilayers versus the applied electric field in the normal direction, whose strength varies from 0 to 0.6 V Å⁻¹. E_D indicates the direct bandgap of K–K transition, E_I indicates the indirect bandgap of Γ–K transition. A crossover of the E_D and E_I curves for the heterobilayer (MoSe₂/MoS₂ and CrSe₂/MoS₂) indicates an indirect-direct bandgap transition. The WSe₂/MoS₂ heterobilayer is always a direct-bandgap semiconductor for field strength <0.6 V Å⁻¹. (b) Computed band structures of the MoSe₂/MoS₂ heterobilayer under an electric field of 0.1 V Å⁻¹ or 0.6 V Å⁻¹.

We have also examined the spontaneous polarization in the CrSe₂/MoS₂ heterobilayer which possesses a dipole moment of 0.005 e Å. Under an external field of 0.5 V Å⁻¹, an indirect-to-direct bandgap transition is predicted. The WSe₂/MoS₂ heterobilayer always retains the direct-bandgap feature under the external electric field (Fig. 8(a)), and its direct bandgap exhibits a steeper reduction with the increase of external electric field. Finally, although the indirect-to-direct bandgap transition is not observed for WS₂/MoS₂ and CrS₂/MoS₂ heterobilayers, their indirect bandgaps exhibit a nearly linear reduction with increase of the electric field. In summary, it appears that the external electric field not only can modify bandgaps of these heterobilayers but also can induce an indirect-to-direct bandgap semiconducting transition beyond a critical field.

Conclusion

We have performed a systematic study of electronic and magnetic properties of MX₂/MoS₂ (M = Mo, W, Cr, Fe, V; X = S, Se) heterobilayers. Our PBE calculations suggest that MX₂/MoS₂ (M = Mo, W, Cr; X = S, Se) heterobilayers are indirect-bandgap semiconductors with the exception of the WSe₂/MoS₂ heterobilayer which can retain the direct-bandgap semiconducting character. Either a vertical electric field or a tensile strain can induce modulation of the bandgaps for these systems. Typically, an increase of the tensile strain decreases the bandgap of heterobilayers. Beyond a critical strain, the semiconductor-to-metal transition may occur. For the WSe₂/MoS₂ heterobilayer, a direct-to-indirect bandgap transition may occur beyond a critical biaxial or uniaxial strain; however, its bandgap is always direct regardless of the strength of the external electric field (<0.6 V Å⁻¹). Moreover, unusual antiferromagnetism is observed in the CrS₂/MoS₂ system with a 2% biaxial tensile strain. The magnetic moment of M and X atoms (M = Fe, V; X = S, Se) increases with increase of the biaxial tensile strain for the MX₂/MoS₂ heterobilayers. The spontaneous polarization in the S/Se interface is more enhanced than the S/S interface. When an electric field is applied in the same direction as the spontaneous polarization, the indirect-to-direct bandgap semiconducting transition can be observed in two heterobilayers (MoSe₂/MoS₂ and CrSe₂/CrS₂). These theoretical predictions suggest that TMD heterobilayer materials are very promising for optoelectronic applications due to their tunable bandgaps by applying tensile strain or vertical electric field, possible direct-to-indirect bandgap transition in the WSe₂/MoS₂ heterobilayer by the strain, and possible indirect-to-direct bandgap transition in MoSe₂/MoS₂ and CrSe₂/CrS₂ by the vertical electric field.

Acknowledgements

XCZ is grateful to valuable discussions with Professors Ali Adibi, Eric Vogel, Joshua Robinson, and Ali Eftekhar. The USTC group is supported by the National Basic Research Programs of China (nos 2011CB921400 and 2012CB 922001), NSFC (grant nos 21121003, 11004180, and 51172223), One Hundred Person Project of CAS, Strategic Priority Research Program of CAS (XDB01020300), Shanghai Supercomputer Center, and Hefei Supercomputer Center. UNL group is supported by ARL (grant no. W911NF1020099), NSF (grant no. DMR-0820521), and a grant from USTC for (1000plan) Qianren-B summer research and a grant from UNL Nebraska Center for Energy Sciences Research.

References

1. Y. Li, Z. Zhou, S. Zhang and Z. Chen, J. Am. Chem. Soc., 2008, 130, 16739–16744.
2. K. F. Mak, C. Lee, J. Hone, J. Shan and T. F. Heinz, Phys. Rev. Lett., 2010, 105, 136805.
3. A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli and F. Wang, Nano Lett., 2010, 10, 1271–1275.
4. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti and A. Kis, Nat. Nanotechnol., 2011, 6, 147–150.
5. H. Wang, L. L. Yu, Y. H. Lee, Y. M. Shi, A. Hsu, M. L. Chin, L. J. Li, M. Dubey, J. Kong and T. Palacios, Nano Lett., 2012, 12, 4674–4680.
6. B. Radisavljevic, M. B. Whitwick and A. Kis, ACS Nano, 2011, 5, 9934–9938.
7. Y. Yoon, K. Ganapathi and S. Salahuddin, Nano Lett., 2011, 11, 3768–3773.
8. A. Geim and I. Grigorieva, Nature, 2013, 499, 419–425.
9. S. Bertolazzi, D. Krasnozhon and A. Kis, ACS Nano, 2013, 7, 3246–3252.
10. L. Britnell, R. Gorbachev, R. Jalil, B. Belle, F. Schedin, A. Mishchenko, T. Georgiou, M. Katsnelson, L. Eaves and S. Morozov, Science, 2012, 335, 947–950.
11. T. Georgiou, R. Jalil, B. D. Belle, L. Britnell, R. V. Gorbachev, S. V. Morozov, Y.-J. Kim, A. Gholinia, S. J. Haigh and O. Makarovsky, Nat. Nanotechnol., 2012, 8, 100–103.
12. W. J. Yu, Z. Li, H. Zhou, Y. Chen, Y. Wang, Y. Huang and X. Duan, Nat. Mater., 2012, 12, 246–252.
13. J. Kang, J. Li, S.-S. Li, J.-B. Xia and L.-W. Wang, Nano Lett., 2013, 13, 5485–5490.
14. H.-P. Komsa and A. V. Krasheninnikov, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 085318.
15. H. Terrones, F. López-Urías and M. Terrones, Sci. Rep., 2013, 3, 1549.
16. Y. Ma, Y. Dai, M. Guo, C. Niu and B. Huang, Nanoscale, 2011, 3, 3883–3887.
17. K. Kośmider and J. Fernández-Rossier, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 075451.
18. A. Ramasubramaniam, D. Naveh and E. Towe, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 205325.
19. Q. Liu, L. Li, Y. Li, Z. Gao, Z. Chen and J. Lu, J. Phys. Chem. C, 2012, 116, 21556–21562.
20. L. Kou, T. Frauenheim and C. Chen, J. Phys. Chem. Lett., 2013, 4, 1730–1746.
21. P. Johari and V. B. Shenoy, ACS Nano, 2012, 6, 5449–5456.
22. Y. Ma, Y. Dai, M. Guo, C. Niu, Y. Zhu and B. Huang, ACS Nano, 2012, 6, 1695–1701.
23. W. S. Yun, S. Han, S. C. Hong, I. G. Kim and J. Lee, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 033305.
24. Y. Zhou, Z. Wang, P. Yang, X. Zu, L. Yang, X. Sun and F. Gao, ACS Nano, 2012, 6, 9727–9736.
25. H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, S. T. Pantelides and K. I. Bolotin, Nano Lett., 2013, 13, 3626–3630.
26. Y. Y. Hui, X. Liu, W. Jie, N. Y. Chan, J. Hao, Y.-T. Hsu, L.-J. Li, W. Guo and S. P. Lau, ACS Nano, 2013, 7, 7126–7131.
27. A. Lu, R. Zhang and S. Lee, Appl. Phys. Lett., 2007, 91, 263107.
28. Z. Zhang and W. Guo, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 075403.
29. C. Zhang, A. De Sarkar and R.-Q. Zhang, J. Phys. Chem. C, 2011, 115, 23682–23687.
30. Z. Zhang, W. Guo and B. I. Yakobson, Nanoscale, 2013, 6381–6387.
31. S. Bhattacharyya and A. K. Singh, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 075454.
32. J. N. Coleman, M. Lotya, A. O’Neill, S. D. Bergin, P. J. King, U. Khan, K. Young, A. Gaucher, S. De and R. J. Smith, Science, 2011, 331, 568–571.
33. J. Feng, X. Sun, C. Wu, L. Peng, C. Lin, S. Hu, J. Yang and Y. Xie, J. Am. Chem. Soc., 2011, 133, 17832–17838.
34. R. J. Smith, P. J. King, M. Lotya, C. Wirtz, U. Khan, S. De, A. O'Neill, G. S. Duesberg, J. C. Grunlan and G. Moriarty, Adv. Mater., 2011, 23, 3944–3948.
35. X. Zhang and Y. Xie, Chem. Soc. Rev., 2013, 42, 8187–8199.
36. C. Ataca, H. Sahin and S. Ciraci, J. Phys. Chem. C, 2012, 116, 8983–8999.
37. H. L. Zhuang and R. G. Hennig, J. Phys. Chem. C, 2013, 117, 20440–20445.
38. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169.
39. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
40. G. Kresse and J. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 558.
41. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758.
42. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
43. J. Klimeš, D. R. Bowler and A. Michaelides, J. Phys.: Condens. Matter, 2010, 22, 022201.
44. J. Klimeš, D. R. Bowler and A. Michaelides, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 195131.
45. J. Neugebauer and M. Scheffler, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 16067–16080.
46. E. Sanville, S. D. Kenny, R. Smith and G. Henkelman, J. Comput. Chem., 2007, 28, 899–908.
47. J. Paier, M. Marsman, K. Hummer, G. Kresse, I. C. Gerber and J. G. Ángyán, J. Chem. Phys., 2006, 125, 249901.
48. Y.-S. Kim, K. Hummer and G. Kresse, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 035203.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c3nr06072a

‡ Both authors contribute equally to this work.

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