Introduction

Two-dimensional (2D) layered materials and their homo- and hetero-junctions have ignited great enthusiasm in both fundamental research and potential applications1,2,3,4,5,6,7,8. Monolayers of different 2D layered compounds can be stacked on top of each other with no constraints imposed by lattice match or chemistry compatibility9. Among them, the 2D transition-metal dichalcogenides (TMDs) homo and hetero-junctions have tunable band structures and promising in electrical and optical applications. TMDs exhibit direct or indirect band gaps and indirect band gap occupies a major portion of the semiconductors10. Direct band gap semiconductors have advantageous in photon absorption and emission, ultrafast interlayer photoexcited response11 and ultrafast (<100 fs) interlayer charge transfer12 which could be used in the construction of optoelectronic devices, photocatalysis, photoelectric sensor and valleytronic components, etc.13,14,15,16,17,18. Although there have been many studies for band structure modulation of TMDs with different means6,15,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37, to achieve direct energy gaps in general bilayer and few-layer TMDs remains challenging.

The monolayers of group VIB TMDs (MoS2, WS2, MoSe2, and WSe2) show direct band gaps at the K point of Brillouin zone (BZ)38,39,40,41 while most of their homo and heterostructures change to indirect gaps4,42 except for the MoSe2/WSe2 bilayer9,11. The direct–indirect gap transition is correlated to the interlayer electronic hybridizations between orbitals in different layers, dubbed as the quasi-bonding (QB) interaction43, which is mainly at Γ point for valence band (VB@Γ), and, at Λ point in the middle of Γ–K path for conduction band (CB@Λ) for TMDs4. The interlayer QB interaction is a subject of significant interest in many studies8,43,44,45,46 since QB plays a crucial role in shaping 2D materials’ electronic and optical properties42,47. Very recently, efficiently engineering interlayer interaction in energy space has been proposed48 in addition to tuning it in real space (e.g., via decrease interlayer separation by pressure)49.

In the current work, we demonstrate that the TMDs homo and heterobilayers and even few-layers can be tuned to direct gaps by real and/or energy spaces tuning of interlayer interactions, which is based on a classification of the homo and heterobilayers of group VIB TMDs and a basic analysis of the factors that affect band-edge locations in BZ. The real space tuning is achieved by introducing large-angle rotational misalignment between layers, which increases the interlayer separation by ~0.3 Å, decreases the interlayer hopping integral greatly, and make part of the indirect gap bilayers transition to direct gap. All the other bilayers can be further tuned to direct gaps by energy-space tuning, which is applied on top of the real-space tuning by introducing a vertical potential gradient between layers such as by applying external electric field or by dipole-layer adsorption. The energy-space tuning can be introduced efficiently due to the small vertical dielectric constant of 2D semiconducting TMDs50,51. Moreover, for few-layer TMDs, direct gaps may also be achieved with our above method. It is important to note that: (1) in contrast to pressure tuning which strengthens interlayer interaction, here our real-space tuning can weaken interlayer interaction; (2) here the large-angle twisting between layers (e.g., 30° ± 8.2°)52,53 do not induce flat bands, and, it is realized in experiment very recently and reported that large-angle twisted bilayers offer distinctly different platforms from moiré systems53; and (3) the direct gap here needs only the band edges tuning back to the K/K′ valley of the component layers, since robust ultrafast interlayer charge transfer has been generally observed in TMDs homo and heterostructures with random rotation angles11,12,53,54. The current work not only provides a quantitative understanding of band-edges modulation by tuning interlayer interaction in both real and energy spaces but also offers valuable insights for indirect-to-direct gap tuning of 2D semiconductor homo- and hetero-junctions. More importantly, both real and energy spaces tuning (large-angle twisting and dipole-layer adsorption) can be permanently preserved without the need for external factors to maintain, and hence is of great significance for their multiple applications.

Results and discussion

Classification and basic analyses

The ten bilayers composed of MX2 (M = Mo, W; X = S, Se) can be classified into “3 + 1” types as indicated in Fig. 1a, b, including three intrinsic types [types (i) to (ii) to (iii)] and one derived type [type-(iv)]. Type-(i) bilayers are homo ones, and, types (ii)/(iii) are hetero ones with lattice match/mismatch, respectively. For type-(i) [Fig. 1a(i)], there are four homobilayers composed of MX2/MX2 (M = Mo, W; X = S, Se) as indicated by blue lines in Fig. 1b. For type-(ii) [Fig. 1a(ii)], there are two heterobilayers composed of MoX2/WX2 (with different M elements and a same X element) as indicated by black lines in Fig. 1b. Type-(ii) heterobilayers exhibit minimal lattice mismatch (~0.04%) and hence form lattice-matched bilayers. For type-(iii) [Fig. 1a(iii)], there are four heterobilayers of MS2/MSe2 (with Se and S elements, respectively, in the two component layers of bilayer, while the M elements in component layers can be either same or different) as indicated by red lines in Fig. 1b. These four heterobilayers all have ~4% lattice mismatch between component layers. The component layer with Se has larger lattice constants than the component layer with S. This is because, for our studies MX2s, when the X elements changing from S to Se, the lattice constant is enlarged by about ~4%; while the M elements (from Mo to W) almost does not affect lattice constant55.

Fig. 1: The “3 + 1” types of MX2 bilayers (M = Mo, W; X = S, Se): classification, interlayer-interaction-induced band edge change, and (super)cells.
figure 1

a, b Three intrinsic types, (i) to (ii) to (iii), and one derived type-(iv) from types (i) and (ii) with tuning in real and/or energy spaces: type-(i), homobilayers with larger interlayer hopping integral (|t|) and zero energy difference (EDiff = 0) between layers, which prefer indirect gaps; type-(ii), heterobilayers with the same X atoms between layers which have larger |t| while EDiff ≠ 0; type-(iii), heterobilayers with different X atoms with supercells (see text), which have smaller |t| and EDiff ≠ 0, and prefer direct gaps; type-(iv) is the derived type from types (i) and (ii) by decreasing |t| and/or increasing EDiff between layers (see text). c Sketches for the interlayer-interaction-induced band edge change from monolayer to bilayer [ΔE in (b)] using VB@Γ (ΔEΓ) as the example for homobilayers (i) and heterobilayers (ii). d, e Supercells used for type-(iii) heterobilayers and for type-(iv) homo(hetero)-bilayers with 38.2° twist angle between layers as an example. In (d, e) supercells of the two layers are shown with a relative rotation, while in (a)(iii) and (a)(iv) the supercells are aligned and the atomic structure between layers is rotated relatively.

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The lattice mismatch can be largely removed by constructing supercells of MS2-\(\left(\sqrt{13}\times \sqrt{13}\right)R13.9^{\circ }\,\)/MSe2-\(\left(2\sqrt{3}\times 2\sqrt{3}\right)R30^{\circ }\) 55,56, as shown in Fig. 1a(iii), d; with supercells the largest lattice mismatch is then <0.2%. The type-(iv) bilayers [Fig. 1a(iv), b] is a derived type from types (i) and (ii) with real and/or energy spaces tuning (will be discussed later).

For type-(i) [Fig. 1a(i), b], they are homobilayers and hence the component layers match in both real space (lattice and atomic structure) and energy space [EDiff = 0 for band edges between layers, refer to Fig. 1c]. The real-space matching results in a relatively smaller interlayer separation and hence larger interlayer hopping integral (i.e., larger |t|)48. Type-(ii) bilayers are heterobilayers with different M elements and a same X element, the component layers are different in band-edge energies (EDiff ≠ 0, namely, energy-space mismatch) but with essentially the same lattice constant (namely, real-space match) and hence still have larger |t|. For type-(iii) bilayers with different X atoms, the component layers are mismatched in both band-edge energy (EDiff ≠ 0) and real-space lattice. As a result of real-space mismatch, the average interlayer separation is increased by ~0.3 Å relative to that in types (i) and (ii) [see Supplementary Table 1] and hence results in the decrease of interlayer hopping integral (smaller |t|). For bilayers with supercells, the average interlayer separation is the height difference between the average heights of X atoms in adjacent layers across the van der Waals (vdW) gap. For the lattice matched ones in Fig. 1a(i)–(ii), unitcells are used; while for the ones with lattice mismatch in Fig. 1a(iii)–(iv), supercells are used. Note that in Fig. 1a(iv), the real-space matched types (i) and (ii) bilayers change to mismatched in real space due to rotational misalignment (larger-angle twisting) between layers; also, the energy matching (EDiff) can also be tuned by introducing vertical potential gradient between layers (details see below).

The different types of bilayers are summarized in Fig. 1b. In type-(i) bilayers, the energy levels from the component layers are degenerate, and, the interlayer-QB-induced up-shift (ΔE) of valence band edge at Γ point4 is equal to the interlayer hopping integral, namely, ΔE0 = |t|, where the subscript “0” means EDiff = 0 [see Supplementary Note 1]. The ΔE from monolayer to bilayer using VB@Γ (ΔEΓ) as the example for homo and heterobilayers are indicated in Fig. 1c(i) and (ii), respectively. For types (ii) and (iii) bilayers in Fig. 1a, b, they have energy differences (EDiff ≠ 0) between layers as indicated in Fig. 1c(ii). The EDiff and |t| values of some bilayers are listed in Table 1. If |t| << EDiff, the band edge up-shift (ΔE) can be approximated as \(\Delta {{E}}\approx \frac{{t}^{2}}{{E}_{{{{\rm{Diff}}}}}}\). For general cases, including |t| larger than or comparable to EDiff, Fig. 2 shows that \(\Delta {{E}}\) decreases monotonously with the increase of EDiff relative to |t|; also refer to Supplementary Note 1. We use the two formulae in Fig. 1b to simplify thinking, from which qualitatively correct understanding can be given. More discussion can be found in Supplementary Note 1.

Table 1 Energy difference between layers (EDiff) in types (ii) and (iii) and the hopping integral between layers (|t|) in types (i) and (iv)
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Fig. 2: The VB@Γ up-shift in energy (ΔE) from monolayer to bilayer decreases monotonously with the increase of EDiff relative to |t|.
figure 2

In types (ii) and (iii) bilayers EDiff/|2t| < 1, and, EDiff/|2t| can be larger than one with applied potential gradient between layers in type-(iv) bilayers. For CB@Λ, similar conclusion holds.

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The interlayer separation increases (d) increases in types (iii) bilayers by ~0.3 Å, and hence |t| decreases as indicated in Table 1. For type-(ii) bilayers, they are energy mismatched while lattice matched, and hence they are at the intermediate of types (i) and (iii). Based on the above analysis, type-(i) bilayers tend to form indirect band gaps since the stronger interlayer QB interaction may change the band edges from K point to Γ point4, while type-(iii) bilayers may favor direct band gaps due to the weakened interlayer interaction in both real and energy spaces.

To obtain direct band gaps for types (i) and (ii) structures, we also weaken their interlayer interactions by tuning the interlayer electronic coupling in real and/or energy spaces; and this is the derived type-(iv) in Fig. 1a, b. Here, the real-space tuning means increasing the interlayer separation by a large rotation angle between layers (e.g., \({\theta }_{t}\) = 30° ± 8.2°)30,31,32,53, the interlayer separation is then increased (Supplementary Table 2) and the interlayer hopping integral |t| is weakened (refer to Table 1). The possible experimental growth method of larger \({\theta }_{t}\) bilayers31,53 that we studied is also discussed briefly in Supplementary Note 2. Here, the hopping integral of interlayer sulfur p-orbitals (mainly ppσ) is inversely proportional to the square of the interlayer separation \(({t}_{{pp}{\sigma }}\propto \frac{1}{{d}^{2}})\)57. In our calculations, an increase of ~10% in interlayer separation (Supplementary Table 2) reduces interlayer interaction to ~75% (Table 1), which is conducive to the formation of a direct band gap. Supplementary Table 3 provides the summary of bilayers for lattice size, lattice mismatch, energy difference between layers (EDiff), interlayer hopping integral (|t|), interlayer vertical separation (dM–M and dX–X), interlayer binding energy (EB), and twist angle between layers (\({\theta }_{t}\)). The energy-space tuning means increase EDiff, which can be achieved by introducing a vertical potential gradient between layers and hence introduce (increase) EDiff between homo (hetero) layers and thus also decrease the interlayer interaction. It has been demonstrated that a vertical potential gradient can be applied between layers by such as an external electric field34,35,36,37 or a polar/ferroelectric layer vdW adsorption48,58, which may significantly increase EDiff due to the small vertical dielectric constant of 2D semiconducting TMDs50,51.

Verification from detailed band structure analysis

To verify the above basic analysis and to give a deeper understanding on the intra- and interlayer factors that govern the band edges, density-functional theory (DFT) band structure calculations including spin–orbit coupling (SOC) are performed. Beginning from types (i) and (ii) bilayers to provide general insights, then to type-(iii) with band unfolding, and finally about the derived type-(iv) bilayers with tuning in real and energy spaces.

Figure 3 (and Supplementary Fig. 1) shows the band structures of bilayers with larger |t|, namely, types (ii) and (i) in Fig. 1a, b including two heterobilayers with nonzero EDiff and four homobilayers with zero EDiff. As typical examples, Fig. 3b, d shows the band structures of MoSe2/WSe2 heterobilayer and WSe2 homobilayer, and, the component monolayers’ band structures are shown in Fig. 3a, c as references. Figure 3 and Supplementary Fig. 1 show that, except for MoSe2/WSe2, all the other five bilayers have an indirect band gap—just as that expected in Fig. 1b for types (i) and (ii) bilayers with larger |t|. The MoSe2/WSe2 bilayer shows direct band gap, which is consistent with the experimental and theoretical literatures9,11.

Fig. 3: Band edge changes in types (i) and (ii) bilayers, and intra- and interlayer factors that affect the band edges.
figure 3

Band edge changes from monolayer(s) to bilayer in (a–c) for type-(ii) bilayer and in (c, d) for type-(i) bilayer using MoSe2/WSe2 heterobilayer and WSe2 homobilayer as examples. The factors that affect band edges are labeled out in (a–c) and (e–g), see text for details. (c) and (g) are repeated for the convenience of comparing between figures on the same line. h Brillouin zone (BZ) with the relevant k points indicated. To compare the energy changes from monolayers to bilayer, the band energies are corrected by the vacuum level changes from monolayer to bilayer and the interface dipole across vdW gap of heterobilayer (see Supplementary Note 3 for detail).

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The factors that affect band-edge locations [namely, the K, Γ, or Λ point of BZ] are indicated in Fig. 3a–g, which include: (1) ΔEΓ and ΔEΛ as indicated in Fig. 3b, which is the band edges change of VB@Γ and CB@Λ from monolayers to bilayer due to the interlayer QB interaction [comparing Fig. 3e, f, which show the effect of interlayer QB on VB@Γ and CB@Λ]; and, (2) ΔCB and ΔVB in Fig. 3a, c which is the energy difference within monolayers’ VB and CB edges and is related to SOC [comparing Fig. 3f, g which show the SOC effect on VB@K and CB@Λ]. The ΔVB in Fig. 3c is the energy difference between VB@K and VB@Γ, and the ΔCB in Fig. 3a is the energy difference between CB@K and CB@Λ. Larger SOC increases the size of ΔVB but decreases that of ΔCB (or even change the sign of ΔCB in bilayer, as a combination of effects from both SOC and interlayer QB). The interlayer QB effect on VB@K is less apparent than on VB@Γ and CB@Λ [Fig. 3e], and the reason is given below.

In the following, we discuss the factors that affect band-edge locations in BZ from the above-mentioned two main aspects: the SOC effect within monolayer and the interlayer QB interaction.

(i) The SOC effect within monolayer: band structures of component monolayers of bilayer are used as references to see the band structure evolution in different bilayers. For monolayer H-phase TMDs, the VB@Γ is a antibonding state of \({p}_{z}\)–\({d}_{{z}^{2}}\) orbitals43, VB@K is composed of \({p}_{x,y}\) and \({d}_{{xy},{x}^{2}-{y}^{2}}\) orbitals, CB@K mainly has \({p}_{x,y}\) and \({d}_{{z}^{2}}\) orbitals, and CB@Λ is composed of all p, \({d}_{{xy},{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\) orbitals4. For the SOC effect, both the atomic number (Z) and the magnetic quantum number (\({m}_{l}\)) are relevant. Among the four MX2 studied here, WSe2 with heavier elements has the most significant SOC effect; and, the atomic number of W (Z = 74) is much larger than that of Se (Z = 34) and hence the SOC effect mainly from the W d-orbitals. Also, note that \({m}_{l}\) = 0 for \({d}_{{z}^{2}}\) orbital and |\({m}_{l}\)| = 2 for \({d}_{{xy},{x}^{2}-{y}^{2}}\) orbitals. From the above-mentioned band edge orbital-characters (\({d}_{{z}^{2}}\) orbital at VB@Γ and CB@K, and \({d}_{{xy},{x}^{2}-{y}^{2}}\) for VB@K and CB@Λ), the SOC effect is evident mainly at the band edges of VB@K and CB@Λ, which is confirmed by comparing Fig. 3f, g. The SOC effect causes the band splitting of VB@K and CB@Λ within monolayer especially for VB@K, which shifts-up VB@K and favors the valence band maximum (VBM) at the K point. For WSe2, the SOC splitting is about 0.47 eV and it shifts-up VB@K energy by 0.24 eV. Namely, the SOC effect favors the formation of direct band gap at K point. However, the shifts down of CB@Λ within monolayer due to SOC favors the conduction band maximum (CBM) at the Λ point in WSe2, especially for bilayer [Fig. 3c, d].

(ii) The interlayer QB interaction: from monolayer to bilayer, the changes in band edges are due to the interlayer QB interaction at and around VB@Γ and CB@Λ [comparing Fig. 3e and f] due to the interlayer interactions of out-of-plane \({p}_{z}\) orbitals of X atoms across the vdW gap since they are close in real space (details see below). Under interlayer QB interaction, the VBM tends to changes from K point to Γ point and the CBM tends to shifts from K to Λ point. The interlayer orbital couplings is weak for the band edges at K point from symmetry analyses59, and hence the energy changes at K point are minimal for VB@K and CB@K. For the simplicity of discussion, for interlayer QB we focus on the VB@Γ and CB@Λ, and the ΔEs in Figs. 1c and 3b denote the more apparent band-edge energy changes of VB@Γ and CB@Λ from monolayer to bilayer.

To quantify the band-edge changes from monolayer to bilayer for types (i) and (ii) bilayers, the sizes of ΔEΓ vs ΔVB and ΔEΛ vs ΔCB are given in Table 2, which show that: ΔEΓ (ΔVB) are in the range of 0.39–0.51 eV (0.06–0.50 eV), and, ΔEΛ (ΔCB) are in the range of 0.07–0.22 eV (0.06–0.24 eV). The monolayers with larger ΔVB values (~0.5 eV) help to maintain the VB edge at K point from monolayer to bilayer. However, if ΔEΓ > ΔVB, the VB edge shifts from K point to Γ point, as shown in the last column of Table 2. Similarly, the CB band edge shifts from K to Λ point if ΔEΛ > ΔCB.

Table 2 Energy changes of ΔEΓ in VB (ΔEΛ in CB) from monolayer to bilayer vs ΔVB (ΔCB) in monolayer(s) for types (i) and (ii) bilayers, and the resulted band-edge location in BZ
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For type-(ii) bilayer, using MoSe2/WSe2 as the example, the interlayer separation of the heterobilayer almost equal to that of the two corresponding homobilayers of MoSe2 and WSe2 (refer to Supplementary Table 1). However, in Table 2, the ΔEΓ value for MoSe2/WSe2 heterobilayer (of 0.45 eV) is smaller than the average of ΔEΓ of MoSe2 and WSe2 homobilayers (\(\frac{0.51+0.47}{2}=0.49\) eV). This is due to the interlayer energy difference (EDiff ≠ 0) in heterobilayer, which weaken the interlayer QB as indicated in Fig. 2. For the other heterobilayer in Table 2, MoS2/WS2, the same conclusion holds. This conclusion also applies for ΔEΛ of the conductance band (Table 2). These indicate that the QB interaction in type-(ii) bilayers is weaker compared to type-(i), and favor the formation of a direct band gap in heterobilayer compared to homobilayer.

The -pCOHP analysis (details see Supplementary Table 4) helps to understand the orbital-contribution to interlayer QB interactions, namely, which orbital of which atom dominates the interlayer interaction. The magnitude of -pCOHP values in Supplementary Table 4, taking MoSe2/WSe2 as an example, indicate that the interlayer Se-\({p}_{z}\) orbitals dominate the interlayer interaction at VB@Γ; while for CB@Λ both out-of-plane and in-plane p-orbitals of Se are involved. Since Se orbitals dominate interlayer interaction, namely, the X atoms from adjacent layers dominate interlayer orbital-hybridization because they are close in real-space43,60,61, we give dx–x in Supplementary Tables 1–3.

Now move to type-(iii) bilayers. The layer-projected band structures of the type-(iii) bilayers with MS2-\(\left(\sqrt{13}\times \sqrt{13}\right)R13.9^{\circ }\,\)/MSe2-\((2\sqrt{3}\times 2\sqrt{3})R30^{\circ }\) supercells (in short MS2/MSe2 supercell) are shown in Fig. 4b, c and Supplementary Fig. 2, which show that the VB (CB) edge is contributed by the MSe2 (MS2) layer especially for MoS2/WSe2 in Fig. 4b, c, forming type II band alignments. To see clearly the band edge locations in BZ (namely, at K point or not) for the supercell structures, one need to unfold the supercell band structure from supercell BZ to unitcell BZ62. For the \(\sqrt{13}\) supercell, as indicated in Fig. 4a, the K point of unitcell is folded to KS point of supercell. However, for the \(2\sqrt{3}\) supercell, as indicated in Fig. 4d, the K point of unitcell is folded to ΓS point of supercell, which makes the VB edge in Fig. 4c folded to the ΓS point of supercell BZ, which can be unfolded back to the K point of unitcell BZ [see Fig. 4d, g]. The band structures in Fig. 4b, c with supercell BZs are unfolded to Fig. 4f, g with unitcell BZ, respectively, which show that the CB@K (VB@K) of MoS2 (WSe2) contributes to the bilayer CB and VB edges, respectively. Namely, the type-(iii) MoS2/WSe2 bilayer form direct gaps. The same conclusion (that form direct gaps) holds for all of the four type-(iii) bilayers, just as expected in Fig. 1b, and Supplementary Table 5 provides the detailed data (similar to the analysis method of Table 2). In Fig. 4e, h, the band structures of isolated monolayers are also given as references. Note that Fig. 4a, d gives the BZs of unitcell and \(\sqrt{13}\), \(2\sqrt{3}\) supercells, another representation53 is given in Supplementary Fig. 3, which give the same conclusion for BZ folding.

Fig. 4: Direct band gaps in type-(iii) bilayers and band structure unfolding using the MoS2-\(\left(\sqrt{13}\times \sqrt{13}\right)R13.9^{\circ}\)/WSe2-\((2\sqrt{3}\times 2\sqrt{3})R30^{\circ}\) supercell (in short, the MoS2/WSe2 SC) as the example.
figure 4

BZ folding between unitcell (UC) and \(\sqrt{13}\), \(2\sqrt{3}\) supercells in (a, d); and, the layer-projected band structures of the MoS2 and WSe2 layer in the MoS2/WSe2 SC (b, c): the VB (CB) edge of bilayer is from WSe2 (MoS2), forming type II band alignment. In (a, d), larger (smaller) hexagon represents the BZ of unitcell (supercell) and the k-points of supercell has a subscript “S”. For the \(2\sqrt{3}\) supercell in (d), K point of unitcell is folded to ΓS point of supercell, which makes the VB edge from WSe2 folded to the ΓS point in (c) and unfolded back to the K point in (g). The band structures in (b) and (c) are unfolded to (f) and (g), respectively, which show that the MoS2/WSe2 bilayer form direct gap at K point. Comparison of band structures of isolated monolayers with that in SC: e, f for MoS2 and g, h for WSe2. Band structures projected to \({d}_{{x}^{2}-{y}^{2}}+{d}_{{xy}}\) and \({d}_{{z}^{2}}\) orbitals are shown with dots and the dot sizes indicate the relative magnitude of projection.

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In summary, the above detailed band structure analysis confirmed that, just as expected in Fig. 1b, all type-(i) bilayers exhibit indirect band gaps, partial type-(ii) bilayer shows a direct band gap, and all type-(iii) bilayers have direct band gaps. The interlayer separation in type-(iii) bilayers is increased by ~0.3 Å than that in type-(i) bilayers (Supplementary Table 1), and thus type-(iii) bilayers have smaller interlayer hopping integral |t|. In addition, heterobilayers have an energy difference between layers (EDiff ≠ 0). Both smaller |t| and EDiff weaken the effect of interlayer QB interaction (refer to Fig. 2) and facilitate the formation of direct band gaps for type-(iii) bilayers.

Real and energy spaces tuning

For type-(iii) bilayers, two factors (smaller |t| and nonzero EDiff) weaken the effect of interlayer interaction on band edge changes in BZ, and all type-(iii) bilayers show direct band gaps. Inspired by this, for all type-(i) and the one type-(ii) bilayer, MoS2/WS2, that has indirect gaps, we weaken their interlayer interaction by real and/or energy spaces tuning (namely, decreasing |t| and/or increasing EDiff), and this forms the derived type-(iv) bilayers in Fig. 1a(iv). Here, real-space tuning means introducing large twisting angles (\({\theta }_{t}\)) between layers which increase the interlayer separation and decreases the hopping integral |t|, and, energy-space mismatch (EDiff) between layers can be introduced with different means which also weaken the interlayer interaction. The EDiff can be introduced (enhanced) by a vertical potential gradient between layers, such as by applying an external vertical electric field or via dipole-layer vdW adsorption34,35,58.

Figure 5 (and Supplementary Fig. 4, Supplementary Table 6) shows that direct band gaps are obtained with only the real-space tuning or with energy-space tuning combined with real-space tuning. The BZ folding is given in Supplementary Fig. 5a, which shows that the K point of unitcell is folded to the KS point of \(\sqrt{7}\) supercell. So, we can judge direct gap or not directly from the band structure of \(\sqrt{7}\) supercell. For real-space tuning [refer to Fig. 1e], \(\sqrt{7}\) supercells are constructed with interlayer twisting angle \({\theta }_{t}\) = 38.2° 53; and for energy-space tuning a vertical electric field or dipole-layer vdW adsorption (Supplementary Note 5 which includes Supplementary Fig. 6) are used. For most cases, only a small electric field of 0.1–0.2 eV Å−1 is needed to obtain direct gaps (Supplementary Table 6). For MoS2 bilayer, strain is also introduced in addition to the normal real and energy spaces regulation (Supplementary Note 6 and Supplementary Fig. 7) and Supplementary Table 6 provides the detailed data.

Fig. 5: The derived type-(iv) bilayers and trilayers with real and energy spaces tuning: direct gaps with small external electric field (EField).
figure 5

For real-space tuning [refer to Fig. 1e], \(\sqrt{7}\) supercells are constructed with interlayer twisting angle of \({\theta }_{t}\) = 38.2° as the example; for energy-space tuning, vertical EField are applied as examples. a WSe2 homobilayer can change to direct gap under only the real-space tuning. b WS2 homobilayer changes to direct gap under combined real and energy spaces tuning of EField = 0.2 eV Å−1. c The changes of EDiff (ΔEDiff) as a function of EField; for MoS2/WS2 the ΔEDiff values are obtained from CB@K [refer to Supplementary Fig. 4c]. d, e Tuning trilayer to direct gap and the band edge real-space distributions as indicated by partial charge densities together with the percentage of the majority contributions.

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Comparing Figs. 5a and 3d, for the WSe2 homobilayer with and without large-angle rotation between layers, shows that the WSe2 homobilayer undergoes an indirect-to-direct gap transition. Supplementary Fig. 8 shows that with another interlayer rotation angle, \({\theta }_{t}\) = 21.8°, the WSe2 large-angle twisted homobilayer also has a direct gap. Supplementary Table 2 demonstrates that the interlayer separations in type-(iv) bilayers with interlayer rotation have also increased by ~0.3 Å compared to the corresponding unrotated bilayers, which weaken the interlayer interactions by decreasing |t|; and hence favor direct gaps, as expected in Fig. 1a(iv), b.

For the other type-(iv) bilayers and trilayers, to obtain direct gaps, energy-space (EDiff) tuning is also needed in addition to the above real-space tuning of interlayer separation [Fig. 5b–d and Supplementary Fig. 4]. The critical electric field (EField) to obtain direct gap is ~0.2 eV Å−1 for bilayer and ~0.3 eV Å−1 for trilayer. For trilayer, the interlayer QB induced energy up-shift of VB@Γ increases a bit relative to that in bilayer4, and hence the critical EField to obtain a direct gap is a bit larger for trilayer. For even-thicker layers, since the additional up-shift of VB@Γ is not significant4, one expect that the critical EField will not change much with layer number (also see the subsection below).

Figure 5c demonstrates that ΔEDiff increases by 0.10–0.14 eV per 0.1 eV Å−1 EField, implying that the electric field can efficiently increase EDiff and weaken the interlayer interaction. In Supplementary Note 5, we present that the potential gradient can also by applied by dipole-layer vdW adsorption and the dipole-layer-induced equivalent EField is given. Also, direct gap is obtained by dipole-layer vdW adsorption.

From the partial charge density of band edges shown in Fig. 5e, it can be seen that under the influence of vertical electric field, VB@ΓS is primarily contributed by the top two layers, while CB@KS is mainly contributed by the bottom layer. Due to the competition between interlayer QB interaction which mix the band edges of different layers together (tends to equal distribution) and interlayer potential gradient which separate the charge densities of different layers (and the latter increase with distance), for VB@Γ 74% from the top layer while 25% comes from the middle layer. The contribution from the bottom layer to VB@Γ is missing, due to the potential energy difference between top and bottom layers is large (~0.8 eV, which is larger than the QB induced energy-splitting).

In addition to types (i) through (iv), we incorporated two additional groups: type-(v) WSSe/WSSe with different stacking (SSe-SeS, SSe-SSe or SeS-SeS, and SeS-SSe) and type-(vi) W0.5Mo0.5Se2/W0.5Mo0.5Se2. Real and/or energy spaces tuning were applied to them and the transition from indirect-to-direct gap are displayed in detail in Supplementary Note 7 (including Supplementary Figs. 9–11 and Supplementary Table 7).

More discussions

Efficient reducing of interlayer interactions, through increasing real-space interlayer separation and band edge energy-space difference, is of great advantageous for achieving a direct band gap in vdW stacked 2D semiconductors if the 2D monolayer has a direct gap. Importantly, both real and energy spaces tuning (large-angle twisting and dipole-layer vdW adsorption) can be permanently preserved without the need for external factors to maintain. In addition, if change the adsorbed dipole layer to a 2D ferroelectric layer58, the direction of tuning can be inversed by electric field via the polarization reversal of ferroelectric layer, which gives the potential for reversible tuning between direct and indirect gaps. The reversible tuning between direct and indirect gaps is of great significance for 2D materials multiple applications. Note that the electric dipole from the adsorbed dipolar-layer can be either parallel or antiparallel to that from the heterolayers of TMDs, and the net electrical dipole is different for parallel and antiparallel conditions. Finally, the vdW adsorption of 2D metals with very high or low work functions, significant energy difference between layers can also be introduced by degenerate doping to the layer close to 2D metal63.

Interestingly, one may estimate the critical EField for indirect–direct gap transition from some basic parameters but does not need the calculation with external electric field or with dipole-layer vdW adsorption. Namely, based on Fig. 2 above which relate ΔE with EDiff/|2t|, and, the change rate of EDiff with EField is known [Fig. 5c]. Note that t is already known, e.g., refer to Table 1. So, ΔE is now related to EField; and then the energy difference between ΔEΓ and ΔVB, which is the ΔE to overcome for indirect–direct gap transition, can be related to EField. For example, for the WS2 tetralayer with interlayer large-angle twisting, ΔEΓ = |t| = 0.48 eV for WS2 tetralayer and ΔVB = 0.25 eV in WS2 monolayer (Supplementary Fig. 12), and it can be estimated that the critical EField is <0.2 eV Å−1; and DFT calculations with external electric field verified this (not shown). Details about the estimation method can be found in Supplementary Note 6.

The direct gap here for structures consisting of group VIB TMDs needs only the band edges tuning back to the K/K′ valley of the component layers, since robust ultrafast interlayer charge transfer has been generally observed in TMDs homo and heterostructures with random rotation angles11,12,53. There are two possible reasons for this: (1) the equivalence of K/K′ as shown in Supplementary Fig. 5. (2) Due to the small exciton radius of 2D materials54, and hence excitons in 2D materials may only sensitive to the local structure and may not sensitive to the relative rotation between larger supercells.

Conclusions

In conclusion, we demonstrate that, to obtain direct gaps, the interlayer interaction can be reduced efficiently by real and energy spaces tuning. For the real and energy-space matched type-(i) homobilayers (larger |t|, EDiff = 0), the interlayer interaction is the strongest and all have indirect band gaps without further tuning; while for the real energy-space mismatched type-(iii) heterobilayers (smaller |t|, EDiff ≠ 0), the interlayer interaction is the weakest and all show direct band gaps. Motived by this, the derived type-(iv) bilayers and few-layers are proposed with real and/or energy-space tuning to reduce the interlayer interaction and direct gaps are obtained in all systems. These tuning are efficient and hence bilayer and even few-layers can be tuned to direct gaps. The current work not only presents a systematic quantitative analysis of the interlayer interaction by tuning interlayer QB in both real and/or energy spaces but also offers efficient electronic properties tuning method via real and/or energy spaces for homo and heterostructures of TMDs and even other 2D semiconductors, and hence is of great significance for 2D materials’ multiple applications.

Methods

First-principles simulations based on DFT64 calculations were performed using the Vienna Ab initio Simulation Package (VASP)65,66. The projector augmented-wave potential67,68 was adopted to describe the core electrons, and the valence electrons were described by plane-wave basis with an energy cut-off of 500 eV. The exchange-correlation functional adopted the generalized gradient approximation (GGA)69 in the Perdew–Burke–Ernzerhof form (GGA-PBE). The SOC was included for electronic structure calculations, and the many-body dispersion70,71,72 for vdW interaction was included for geometric structure optimization. The convergence criteria were 10−5 eV for the self-consistent energy calculation and 0.01 V/Å for the Hellmann–Feynman force in the geometric optimization. The BZ was sampled using Monkhorst–Pack scheme73 with a k-point density of 2π × 0.03 Å−1. The vacuum separation in the vertical direction was 15 Å to ensure that the interaction between periodic images is negligible. Dipole correction74 is added for heterostructures. Our calculated direct or indirect band gaps are in excellent agreement with experiment, as evidenced in the case of MoSe2/WSe29,11. Additionally, our analyses utilized several packages including Local Orbital Basis Suite Toward Electronic-structure Reconstruction (LOBSTER) package75,76, VASPKIT77, LEEDpat78 and VESTA79. More details on LOBSTER can be found in Supplementary Note 4.