Introduction

Two-dimensional (2D) atomic layers that are vertically stacked by van der Waals (vdW) interactions to build heterostructures (2D–vdW–h) might have fascinating properties for potential applications in optoelectronics1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. These hybrids could retain the pristine monolayers’ main features while achieving unusual properties that cannot be obtained otherwise. As a consequence, the interest in identifying the electronic properties of these 2D–vdW–h has increased. Particularly in systems such as graphene (G) and transition metal dichalcogenides (TMDCs), where intensive research studies are conducted in systems like G/G, TMDC/TMDC, and G/TMDC13,14,15,16,17,18,19. Additionally, a new degree of freedom is present when two or more 2D layers are vertically stacked: the interlayer relative angle, θ, made between the crystallographic orientations of both monolayers. For instance, results in twisted G/G exhibit critically dependent electronic and optical properties upon θ13,20,21,22,23. Nevertheless, how much impact the interlayer rotation has on the electronic spectrum still needs to be understood from a fundamental perspective.

Previous studies on G/TMDC have reported that G Dirac cones fall inside the band gap of the TMDC. In addition, they have found direct to indirect electronic transitions, the reduction of the TMDC band gap, and the opening of small gaps close to the Dirac cone, where this latter is now positioned at the Γ point in the reciprocal space24,25,26. Other works on G/TMDC also reported similar changes in the electronic band structure, where once again, the Dirac cone is at Γ after a rotation between monolayers27,28,29. Also, angle-resolved photoemission spectroscopic measurements (ARPES) show interlayer hybridization and mini-gaps with no significant charge transfer between monolayers30,31. Additionally, density functional theory (DFT) calculations have found significant changes in the deeper (higher) valence (conduction) regions assisted by an out-of-plane interaction by relatively weak vdW effects, such as the opening of π-band-energy gaps simultaneously with a splitting around the Γ point at −6.0 eV below the Fermi level32,33,34. However, more detailed studies considering different interlayer relative angles are still awaiting.

This paper discusses the graphene/tungsten disulfide (G-WS2) electronic properties studied with periodic density functional theory (DFT) calculations and analyzed using a general unfolding method that allows us the identification of the band alterations upon interlayer interactions at different angles. First, the physical cause of the G band structure mini-gaps and other electronic deviations is studied when θ = 0.0°. Then, the relative twist-angle impact on the G electronic band structure is unveiled by analyzing G-WS2 with relative θ = 10.9°, 19.1°, and 30.0°. The electronic changes are mainly associated with avoided crossings and moiré replicas, which depend on the interlayer angle. A resume of the unfolding method is in the Supplementary Information, as well as the strain influence on the G-WS2 electronic structure due to the commensurability imposed on the G and WS2 layers with a given θ angle.

Results and discussion

Unfolding bands

The G-WS2 heterostructures supercells with different relative angles, θ = 0.0°, 10.9°, 19.1°, and 30.0°, are constructed using the optimized lattice parameters of the individual unit cell layers, both with hexagonal symmetry but different lattice constants, aG = 2.44 Å, and \({a}_{{{{{\rm{WS}}}}}_{2}}=3.16\) Å. The strain condition to obey the commensurability is applied on the graphene layer. The structural difference induces variations in the band structure and the electronic properties, such as a small p-doping in graphene but not a charge transfer. The commensurability condition imposes up to a strain of 3.6% of the G layer, which can be achieved experimentally35. The four fully-optimized G-WS2 electronic band structures are found through periodic DFT calculations using the SIESTA code36. Then, we employ an unfolding method that allows us to analyze mutual interactions between layers more straightforwardly37,38,39,40,41,42,43. Details of the atomic models and computational calculations are described in the “Methods” section.

To analyze the results, first, we chose the irreducible Brillouin Zone (BZ) of pristine G or WS2 primitive cells, delimited by the high-symmetry points over Γ−K−M−Γ trajectories. Then, we obtain the effective electronic bands by projecting and comparing supercell (SC) electronic states to those of isolated G (WS2) by employing the unfolding method. These are called in this work EEBG (EEB\({}_{{{{{\rm{WS}}}}}_{2}}\)). When comparing, we find that SC electronic states may belong completely (\({W}_{{{{\rm{G}}}}/{{{{\rm{WS}}}}}_{2}}=1\)), partly (\(0 < {W}_{{{{\rm{G}}}}/{{{{\rm{WS}}}}}_{2}} < 1\)), or nothing (\({W}_{{{{\rm{G}}}}/{{{{\rm{WS}}}}}_{2}}=0\)) to isolated G or WS2, where a spectral weight, \({W}_{{{{\rm{G}}}}/{{{{\rm{WS}}}}}_{2}}\), is obtained for each electronic state. Thus, the spectral weight, W, measures how much of the electronic states of isolated monolayers are conserved in 2D–vdW–h upon the presence of other layers. This versatile unfolding method allows to project SC electronic states of G or WS2 onto either primitive cell BZ, labeled BZG or BZ\({}_{{{{{\rm{WS}}}}}_{2}}\). Notice that the unfolding method employed here can project the electronic states distinguishing chemical species, atomic orbitals, and SC oriented at different angles concerning the original ones. For a detailed explanation, the reader can consult ref. 44.

Figure 1 shows the electronic band structure of the four G-WS2 heterostructures along the irreducible BZG. The EEBG of projected out-of-plane Carbon 2pz orbitals or C(2pz) electronic states are shown in red dots where dot size is proportional to WG. Here, tiny gray dots represent SC electronic states with WG = 0, i.e., those states that do not originally belong to the isolated G layer, such as electronic states resulting from the folded G layer or those belonging to WS2. Comparing the four EEBG at different angles with the corresponding band of the isolated G monolayer, we immediately observe that two G properties are conserved. The Dirac point is in the Fermi level (EF) and at the K point of the G primitive cell (PCG), and EEBG shows a linear dispersion around the Dirac point. Contrarily, there are electronic differences from those known for isolated G, like multiple openings of the C(2pz) bands and band splittings seen at Γ for energies smaller than − 7 eV. These band openings are known as mini-gaps. For a quick reference, the band structure of isolated G is shown in the Supplementary Fig. 1(a). We do not include in Fig. 1 the bands belonging to in-plane C orbitals, i.e., C(2s), C(2px), and C(2py), which are ~−3 eV in Γ and disperse down to K and M, since they remain unaffected upon stacking on WS244.

Fig. 1: Effective electronic bands projected onto graphene Brillouin zone for G-WS2 at θ as the twist-angle.
figure 1

θ = a 0.0°, b 10.9°, c 19.1°, and d 30.0°. Red dots show projected SC electronic states onto graphene C(2pz) electronic bands with WG > 0, where dot sizes are proportional to WG values. Tiny gray dots represent SC states with WG = 0, i.e., states that do not originally belong to the isolated G layer in BZG. Here, EF is set to 0 eV.

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We remark that the affected G bands are composed solely of out-of-plane C(2pz) orbitals, evidencing that electronic alterations come mainly from the mutual interaction with WS2 electronic orbitals upon stacking. Looking at the four EEBG, we find that band distortions and mini-gaps are localized at different points in the reciprocal space and occur at different energies, with θ as a parameter. For instance, the θ = 10.9° heterostructure shows the most significant number of mini-gaps and deviations of the C(2pz) bands when going from Γ → K and between − 6.0 and 0.0 eV. In contrast, the case with the smaller number of mini-gaps corresponds with θ = 30.0°. To discuss these deviations, we first analyze the case with θ = 0.0°, followed by the twisted-bilayer cases. The EEBG behaviors are examined in terms of (a) band foldings, (b) electronic repulsion between bands, and (c) the superperiodic potential influence. Band folding occurs because of the SC construction and gives rise to the so-called Bragg diffractions. Electronic repulsion between bands from different layers results in avoided crossings or mini-gaps. Finally, the superperiodic potential due to a second layer allows interactions with the so-called moiré replicas.

Bragg diffractions

Bragg diffractions occur when SC bands cross high-symmetry points or planes of either BZG or BZ\({}_{{{{{\rm{WS}}}}}_{2}}\), corresponding to the individual monolayers. Figure 2 shows BZs and Γ points represented with hexagons and dots, respectively. Here, green, blue, and red hexagons and dots correspond to the reciprocal space of the heterostructure, SC\({}_{{{{\rm{G}}}}-{{{{\rm{WS}}}}}_{2}}\), and the individual layers, PC\({}_{{{{{\rm{WS}}}}}_{2}}\) and PCG. Thus, green, red, and blue triangles show the corresponding irreducible BZs delimited by their high-symmetric points, Γ, K, and M. Notice that the Γ point (black label) at the concentric hexagons is the same for all, supercell and the primitive cells. Here, only K and M points of BZG are identified with red labels. By following the Γ→K direction in BZG (red triangle) when θ = 0.0°, we identify four intersections with green vertices (V), all corresponding to the supercell K − points. The third V intersection also coincides with the K point of the WS2 monolayer. Also, two segments of Γ−K coincide with the borders or perimeters (P) of BZSC, and the fifth intersection is with one ΓSC point or reciprocal lattice point (R). Notice that cone replicas at vertices and supercell K−points are visible in Fig. 1a by looking at the tiny gray dots close to EF. The same happens with the other G-WS2 heterostructures with θ ≠ 0°. A list of the different V, P, and R intersections of the four G-WS2 systems is in Table 1.

Fig. 2: Reciprocal spaces of G-WS2 at θ as the twist-angle.
figure 2

θ = a 0.0°, b 10.9°, c 19.1°, and d 30°. Reciprocal spaces of G-WS2 are shown in green hexagons. Red and blue hexagons depict G and WS2 Brillouin zones, BZG and BZ\({}_{{{{{\rm{WS}}}}}_{2}}\), respectively. Γ points at BZ\({}_{{{{\rm{G}}}}-{{{{\rm{WS}}}}}_{2}}\), BZG, and BZ\({}_{{{{{\rm{WS}}}}}_{2}}\) correspond to green, blue, and red dots. Reciprocal lattice vectors and irreducible BZs are shown with arrows and thick lines using the same color scheme. Γ, K, and M high-symmetry points of irreducible BZG are labeled.

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Table 1 Intersections between the irreducible BZG with either BZsSC and BZ\({}_{{{{{\rm{WS}}}}}_{2}}\) along the Γ→K→M patha.
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In Fig. 3a, the G irreducible BZ is shown where the intersections with supercell and WS2 BZs are marked with dashed lines and labeled from 1 to 10, besides Γ, K, and M points of G. On the left-right side, the ten vertical dashed lines are at k − points of such intersections, also labeled from 1 to 10, where the EEBG of C(2pz) orbitals when θ = 0.0° is displayed. Notice that states with WG = 0 are no longer shown. Here, we indicate some of the most representative alterations of EEBG, where orange and black arrows correspond to Bragg diffractions and avoided crossings (Cn), respectively. At the same time, the distortions indicated by pink arrows (Rn) cannot be associated with any of these electronic features but with moiré replicas that we will discuss later. Recall that these characteristics are always present in the G-WS2 SC band structure; however, they cannot be elucidated before the unfolding process, which also assigns spectral weights and exposes the electronic changes after stacking.

Fig. 3: Irreducible Brillouin zone and effective electronic bands of G-WS2.
figure 3

a k − points intersections between the G irreducible BZ with supercell and WS2 BZs marked with gray dots and labeled from 1 to 10, besides intersections at Γ, K, and M points. b EEBG of C(2pz) bands for G-WS2 2D–vdW–h with θ = 0° and WG > 0, where dot size is proportional to WG. Orange, black, and pink arrows indicate mini-gaps corresponding to Bragg diffractions, avoided crossings labeled as Cn, and moiré replicas labeled as Rn. The ten vertical dashed lines represent the k − points where the intersections occur, marked on a in gray dots.

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In Fig. 3(b), the vertical dashed lines from 1 to 4 in the Γ−K path indicate where Bragg diffractions can be found. Notice that the first intersection takes place at Γ being the same for all BZs, i.e., G, WS2, and G-WS2, while the last one happens at K, where also is located the K − point of the SC. Hence, band gaps due to Bragg diffractions are expected to appear on those intersections, as such as those of WS2. Like the one occurring at vertical line 1, located close to −6.5 eV and pointed out with an orange arrow, where an imperceptible distortion of the C(2pz) band takes place. In the middle between the 4th line and K, there is the MSC high-symmetry point of the supercell, which induces a tiny alteration of the band, also observed in EEBG and pointed by an orange arrow. In general, we find that Bragg diffractions induce minimal band alterations. For instance, a more significant band opening is observed at the intersection with line 4, corresponding to a vertex in the SC. However, at this point, also a band repulsion is observed because of the moiré superpotential, as we will discuss later.

Doing the same analysis for K→M, first, we find an intersection with a green dot (line 5), followed by a green vertex (line 6), and then continue alongside part of the green perimeter reaching an M point of the SC at the same place of the M point of graphene. Again, a minimal band alteration of the C(2pz) band is found in line 5. At the vertex or line 6, a more considerable distortion of C(2pz) is appreciated showing a gap; however, this gap results from a cone replica because of the moiré superpotential, as we discuss in the last part of this section. Finally, the M → Γ trajectory has two intersections with green borders (B) and two with green dots (R), shown with four vertical lines labeled as 7, 8, 9, and 10. Bragg diffractions are not visible along this segment, and the large band deviations suggest another kind of interaction. Because of the absence of vertices along this path (see Fig. 2), there are no cone replicas on this segment of Fig. 1a. Therefore, the origin of such band mini-gaps comes from electronic repulsions, as discussed in the following subsection.

Finally, the analysis of the DOS partial contributions in Fig. 4b–e supports the above observations. Band deviations marked with orange arrows are indistinguishable in DOS compared to the components of isolated G and WS2. For the Bragg diffractions close to EF, slight deviations of DOS are observed but only if such diffractions occur at the same place with other kinds of interactions like moiré replicas, as mentioned before. To corroborate the existence of Bragg diffractions, we construct the same G-WS2 heterostructure with θ = 0.0° but placing an interlayer distance larger than the equilibrium one, for instance, at 5.5 Å. At this layer separation, we assure that mutual interactions between layers significantly decrease. After unfolding, EEBG is identical to pristine G, where all mini-gaps and band alterations in Fig. 3b have disappeared. The reader can see the Supplementary_movie_1.mp4 file in the Supplementary Information, an animation of the mutual interlayer interaction as a function of the interlayer separation.

Fig. 4: Effective electronic bands of G-WS2 and projected density of states.
figure 4

a EEB of G-WS2 with θ = 0.0° projected onto BZG. Red and blue dots represent C(2pz) and WS2 bands. Dot size is proportional to the weights \({W}_{{{{\rm{G}}}}/{{{{\rm{WS}}}}}_{2}}\). b–e Show in solid lines the partial DOS corresponding to C(2pz), W(5d), S(3px), and S(3pz) orbitals. Also, the DOS of isolated monolayers in dotted lines are shown for comparison.

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Avoided crossings

The electronic repulsion between bands breaks them resulting in mini-gaps, also known as avoided crossings gaps, which may take place at any point of the BZ. To visualize them, we project all WS2 (blue dots) and G C(2pz) bands (red dots) on the irreducible BZG, as shown in Fig. 4a for G-WS2 with θ = 0.0°. The WS2 projected states are beyond its irreducible BZ\({}_{{{{{\rm{WS}}}}}_{2}}\), as seen in Fig. 2a. For instance, the K\({}_{{{{{\rm{WS}}}}}_{2}}\) symmetry point of WS2 is at line 4, while M\({}_{{{{{\rm{WS}}}}}_{2}}\) coincides with line 7. Recall that the highest occupied band is in the WS2 monolayer at K\({}_{{{{{\rm{WS}}}}}_{2}}\), reproduced in Fig. 4a at line 4. Here, some mini-gaps are related to the mutual band crossing between G and WS2 bands. We use the same black arrows and Cn as before to label avoided crossings. Because not all WS2 bands interact with those of G, not all the overlappings between G and WS2 bands generate gaps, but only those interacting with C(2pz) and some WS2 orbitals. By projecting the EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) for each orbital of WS2 and looking at those that changed, we gain insights into interlayer interactions. In particular, we find that WS2 bands with S(3pz) orbitals interact with G, while bands without these orbitals do not. For example, the WS2 band at about − 4.2 eV between lines 7 and 8 is constituted by W(5dxz), W(5dyz), S(3px), and S(3py) orbitals, which overlaps C(2pz) band but do not interact.

Mini-gaps marked with black arrows and labeled as Cn in Figs. 3a and 4 come from out-of-plane orbitals interactions. For instance, the avoided crossing C2 located between lines 3 and 4 at ~ −3.2 eV is mainly constituted by S(3pz) orbitals, although W(5dxz), W(5dyz), S(3px), and S(3py) are also present but with a smaller spectral weight. The electronic repulsion in C2 opens a gap of ~ 0.3 eV in both bands belonging to G and WS2, where a kind of resonance is found that shows a mixing between G and WS2 orbitals at the top and bottom of the mini-gap, giving continuity of the upper and bottom bands. These bands suggest that at such k − point and energy, an electron on the G layer will move to WS2 one and vice-versa. In addition, Fig. 4b–e panels show the C(2pz), W(5d), S(3px), and S(3pz) contributions to the total electronic density of states (DOS) of G-WS2, respectively, as solid lines. For comparison, we also include the partial DOS corresponding to both isolated layers in dotted lines. We do not include the partial DOS of in-plane C orbitals because they remain unperturbed upon the presence of WS2, as we mentioned above.

The depletion in the C(2pz) and S(3pz) DOS components ~−3.2 eV, shown in Fig. 4b, e, confirm the presence of the mini-gap, delimited by two sharp peaks at about − 3.0 and − 3.3 eV. Similar behavior occurs for C6 between lines 8 and 9, at ~−5.5 eV, where the C(2pz) band crosses two different WS2 bands, exhibiting two well-defined gaps. Here, both WS2 bands are constituted mainly by S(3pz) orbitals, but there is also the presence of W(5dyz), W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)), W(5dxz), S(3px), and S(3py) orbitals. Again, partial DOS confirms this gap by two depletions of C(2pz) states centered at − 5.2 and − 5.5 eV. In contrast, C1 exhibits a tiny gap between lines 4 and K at ~−1.0 eV. Despite this, the WS2 band is mainly by in-plane W(5dxy) and W(5d\({}_{{{{{\rm{x}}}}}^{2}-{{{{\rm{y}}}}}^{2}}\)) orbitals, while S(3pz) shows a minor contribution according to their spectral weights. Nevertheless, this small amount of S(3pz) orbitals induces a tiny perturbation on the C(2pz) band. Another interesting case occurs in line 7 ~−3.0 eV pointed by a pink arrow. Again, we find a small contribution from S(3pz) orbitals; however, the band distortion is quite significant. In this case, the branch of the S(3pz) band is pushed up to higher energies by ~0.4 eV, simultaneously that the C(2pz) branch is spelled down, evidencing that another electronic interaction concurs additionally to the direct electronic repulsion. It is why we denoted these kinds of mini-gaps, like Rn rather than a Cn. We will discuss this new type of interaction below when we expose the role of interlayer rotations on electronic properties.

At M, about −2.2 eV, many bands converge that are mainly constituted by S(3py) and W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) orbitals, labeled as R3 in Fig. 4a. At the same time, S(3px), W(5dxy), and S(3pz) are also present with a smaller W. Although all these bands cross in a highly symmetric point, a significant band deviation results from the presence of S(3pz) orbitals. The slights deviations on DOS of C(2pz) and S(3pz) orbitals (see Fig. 4b, e) confirm the slight band distortion, such that the peak at −2.2 eV of C(2pz) decreases. Moreover, the large deviation and the small W of the S(3pz) orbitals indicate another kind of interaction. Again, we will go back to this type of band alteration when discussing the role of the interlayer angle in EEB. Between lines 2 and 3, from ~−4.0 to ~−5.5 eV, we find C3 and C4, where the C(2pz) band breaks in multiple small branches because of avoided crossings with the S(3pz) band and the bands made of S(3px), S(3py), and S(3s) orbitals, which are all degenerated. These branches show an extended mixing of C(2pz) and WS2 orbitals, resembling a hybridization. The two sharp peaks in DOS at ~−5.4 and −5.7 eV confirm this, Fig. 4b–e. Moreover, the short branch inside lines 2 and 3, at ~−5.0 eV, bands emerge from a band splitting and the band hybridization, which is inexistent in the isolated WS2 and G layers, as seen in Supplementary Fig. 1a, b. Such band splitting disappears as we increase the distance among layers (see the Supplementary_movie_1.mp4 file in the Supplementary Information).

The two bands at lower energies at Γ dispersing up (down) to line 2 (line 8), marked by H1 (H2) in Fig. 4a, reveal their hybridization nature between C(2pz) and WS2 orbitals. In line 2, both C(2pz) and WS2 bands split into two hybridized bands at ~−5.9 and ~−6.6 eV, dispersing down to reach Γ at ~−6.9 and ~−7.6 eV, respectively. A similar situation is observed inside lines 8 and Γ at around the same energies. From isolated G and WS2 band structures, we observe only one band around this region instead of two. Thus, these hybridized bands emerge as a consequence of mutual interaction. Both hybridized bands have different spectral weights. For instance, the upper band exhibits more G character than WS2, while the bottom one shows a more significant WS2 nature. Confirming this hybridization, DOS shows a decrease of C(2pz) states of about half of its value compared to pristine G at −6.9 eV and extending up to −7.6 eV. Also, the amount of S(3pz) DOS diminishes and extends up to −7.6 eV, where the isolated WS2 layer does not exhibit any occupied states. These hybridized states are mainly constituted by out-of-plane orbitals, while S(3px,y) and W orbitals remain unperturbed for energies below −6.5 eV. Graphite shows the same band splitting at about this energy45, due to the vertical stacking of G layers. Here, WS2 breaks the band degeneracy similarly to that of multiple G layers in graphite.

The remaining mini-gaps indicated with black arrows arise from avoided crossings between C(2pz) and S(3pz) orbitals. However, mini-gaps marked with pink arrows and labeled as Rn neither come strictly from avoided crossings or Bragg diffractions. For example, S(3pz) orbitals are absent around line 4 at ~−2.0 eV and line 7 at ~−2.7 eV, where R1 and R4 indicate small mini-gaps, respectively. The origin of these mini-gaps is attributed to the superperiodic potential due to the moiré pattern. Finally, we observe in Figs. 2a and 3a that lines 4 and 7 coincide with the K and M high-symmetry points of the irreducible BZ of the isolated WS2, suggesting a repulsive interaction with moiré replicas. In the next section, we discuss the origin of Rn mini-gaps.

Moiré replicas

Moiré replicas emerge in 2D–vdW homo and heterostructures because of the superperiodic potential induced by the lattice mismatch between layers, visualized as moiré patterns46. In G-WS2 with θ = 0.0°, the reciprocal lattice vectors, GG and \({{{{\bf{G}}}}}_{{{{{\rm{WS}}}}}_{2}}\), shown in Fig. 2a, are different since G, and WS2 lattice parameters also do. Moiré reciprocal lattice vectors defined as \({{{{\bf{G}}}}}_{{{{\rm{M}}}}}={{{{\bf{G}}}}}_{{{{\rm{G}}}}}-{{{{\bf{G}}}}}_{{{{{\rm{WS}}}}}_{2}}\) describe a superperiodic potential. In particular, the superpotential gives rise to six satellite cone replicas around each K point in BZG46. We show these satellite locations with black circles in Fig. 5a, which delimit a mini BZ also drawn in black lines. In general, any linear combination of GG and \({{{{\bf{G}}}}}_{{{{{\rm{WS}}}}}_{2}}\) represents a scatterer point that might show a complex pattern47. For instance, Pletikosić and collaborators46 reported mini-gaps for G supported on Ir(111) due to moiré replicas localized at the Bragg planes borders delimiting the mini BZ46.

Fig. 5: Effective electronic bands around mini Brillouin zone.
figure 5

a Cone replicas around the K point bordering half of the mini BZ shown in black circles. b EEBG and EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) along the a−b−c−d−e−a path at the equilibrium interlayer distance (with interactions), and c at 5.5 Å, where interlayer interactions are negligible.

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In our case, we found that the Γ−K trajectory intersects the mini BZ at the reciprocal point, corresponding to line 4 in Fig. 3a or point c in Fig. 5a. Also, the K−M path intersects the mini BZ, now corresponding to line 5 or point d, where a small gap is found at an energy of ~ −1.2 eV, as discussed above. To look deeper at these moiré replicas, we examine both EEBG and EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) with and without interactions between the two layers. These are shown in Fig. 5b, c, where EEBG and EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) are displayed along the path delimited by the cone replicas, i.e., a → b→c→d→e→a. In this reciprocal region, the electronic bands predominantly belong to WS2 (blue dots) and only one to C(2pz) (red dots). The main band distortions in G and WS2 upon interlayer interactions (Fig. 5b) appear in the b−c−d region at energies between −1.0 and −4.0 eV. One should notice that these distortions are present even in the absence of avoided crossings between G and WS2 bands, revealing the superperiodic potential effects associated with the moiré pattern. By isolating both layers, i.e., separating them at a distance larger than the equilibrium one, the mutual interactions drop out. This later is shown in Fig. 5c, where both EEBs demonstrate that all the band distortions disappear when the layers are at 5.5 Å.

From this analysis, we found that all mini-gaps labeled as Rn and marked with pink arrows in Fig. 3(a) and Fig. 4 are related to the superperiodic potential or replicas and not to the direct EEBG and EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) bands crossings. Indeed, these mini-gaps appear from the electronic repulsions between bands belonging to G primitive cell and moiré replicas of both G and WS2 unit cell. Because of the size of the G-WS2 supercell with θ = 0.0°, we cannot analyze in detail all Rn whether each mini-gap is associated with G or WS2 replicas. However, a specific examination can be done for the smallest supercell that helps to elucidate the main features, as discussed in the next subsection.

So far, we have found that avoided crossings and moiré replicas are essential to explain the changes in the electronic bands in 2D–vdW heterostructures. These phenomena are related to the reciprocal spaces of SC and each isolated layer. In comparison, Bragg diffractions are responsible for the smaller electronic alterations, which are pretty much insignificant. On the other hand, avoided crossings open well-defined mini-gaps mainly because of the interactions between C(2pz) and S(3pz) out-of-plane orbitals. At deeper energies, these are hybridized, breaking the degeneration of both C(2pz) and S(3pz) bands. Then, moiré replicas arise from the superperiodic potential associated with the lattice mismatch between layers, which are characterized by mini BZs that also produce well-defined mini-gaps. Thus, all these electronic changes strongly depend on the supercell’s symmetry. Therefore, it is expected that the relative angle between the top and bottom layers would play a fundamental role in controlling the electronic properties of moiré G-WS2 heterostructures.

Mini-gaps assisted by the interlayer relative angle

To understand how the interlayer relative angle, θ, influences the electronic properties of G-WS2 2D–vdW heterostructures, let us first discuss the case with θ = 30.0° by doing a similar analysis as done above. From Fig. 2d and following the Γ−K path in BZG (red triangle), we found four crosses with vertices, where the latter is the K point of BZG. Hence, between the Γ and K points, there are 3 cone replicas. We also found that the Γ−K trajectory crosses two green borders, while along K −M there is one reciprocal-lattice point and two green borders, and along M −Γ there are three green vertexes, one reciprocal-lattice point, and one green border. In this case, the BZs of G and WS2 do not coincide except part of their borders, MG−Γ (red line) and Γ−K\({}_{{{{{\rm{WS}}}}}_{2}}\) (blue line). On the other hand, both K-M paths for θ = 0.0° and 30.0° cases are outside the first BZ\({}_{{{{{\rm{WS}}}}}_{2}}\) and do not intersect any of its high-symmetry points. This latter is true for the other two angles, θ = 10.9° and 19.1°, studied here. Notice that although the irreducible BZG is the same for both θ = 0.0° and 30.0° heterostructures, the intersections with the supercell’s high-symmetry points are different (see Table 1). Additionally, we observe that θ = 0.0° and 30.0° exhibit the largest number of intersections with vertexes, reciprocal-lattice points, and borders. In contrast, the θ = 19.1° case has the smallest intersections followed by the θ = 10.9° heterostructure. The above discussion about the similarities and differences in the BZ paths for each θ, together with a comparison of bands’ dispersions with and without interactions, will make it straightforward to determine the mini-gaps nature.

To expose the bands causing avoided crossings and their reliance on the twisted angle, in Fig. 6 we show C(2pz) and S(3pz) bands in red and blue dots, respectively, both projected into the irreducible BZ of G for the four angles studied here. Left-hand side panels, a–d, correspond to isolated or non-interacting G and WS2 monolayers, while right-hand side panels, a′–d′, show the same bands but for the interacting G-WS2 heterostructures. The highest occupied band belonging to WS2 is composed with the combination of S(3pz) and W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) orbitals with the same symmetry, where contributions from W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) orbitals are mainly observed in the highest-occupied band. In both left-handed and right-handed panels in Fig. 6, the labels Cn, Hn, and Rn are displayed to locate the mini-gaps upon interaction.

Fig. 6: Comparison between superposition of electronic bands of isolated G and WS2 layers and effective electronic bands of G-WS2 heterostructures.
figure 6

Left panels, a–d, show C(2pz) (red dots) and S(3pz) (blue dots) bands of isolated G and WS2 layers projected on the G irreducible BZ for the different angles. Right panels, a′–d′, show the C(2pz) and S(3pz) EEBs for the corresponding four interacting G-WS2 cases. The Cn, Hn, and Rn labels indicate mini-gaps associated with avoided crossings, hybridization, and superpotential replicas, respectively. The bands of isolated WS2 are aligned such that the highest valence bands coincide on left-handed and right-handed side panels.

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The band analysis shows that close to −7.0 eV at about the middle of Γ−K and M−Γ trajectories, the band hybridization open gaps, labeled as H1 and H2, observed for all twisted angles θ. These band splittings in Fig. 6a′–d′ originate from the crossing and overlapping of C(2pz) and S(3pz) + W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) bands, as noticed in the corresponding left-hand side panels. The energies at which H1 and H2 occur are slightly different for each θ. However, we should consider that the commensurability condition to treat G-WS2 2D–vdW heterostructures like periodic systems might induce some tensile or compressive strengths on G, slightly altering those energies. As discussed in the Supplementary Methods: graphene under strain, mechanical effect modifies the C(2pz) band dispersion and induces small energy changes on WS2 bands with respect to EF. The tiny energy shift on WS2 bands might be associated with an electrostatic screening that changes as the C to S atoms ratio does. This latter is also influenced by the G strain itself. For more details about the commensurability and its consequences on G-WS2 bands, see the Supplementary Methods: graphene under strain.

Among the four cases, when θ = 0.0° the tensile condition is maximum and the C(2pz) and S(3pz) bands are practically superposed from Γ at ~7.0 eV to halfway K and M, as seen in Fig. 6(a). Here, the energy difference between such bands at Γ before interacting is only δ = 0.173 eV. In contrast, when θ = 30.0° the heterostructure has the smallest compression, and the bands energy difference is about δ = 0.258 eV (see Fig. 6d). Additionally, when θ = 10.9° bands separation is δ = 0.357 eV (see Fig. 6b), while the case with maximum compression is θ = 19.1° also with the largest energy difference, δ = 0.487 eV (see Fig. 6c). Note that G has a tensile condition when θ = 0.0°, while it is compressed for θ = 10.9, 19.1, and 30.0°, exchanging the bands’ energy order at Γ. In the first case a red band is followed by a blue one at deeper energy, while for the other angles, the blue band is followed by a red one.

Comparing the same regions in G-WS2 before and upon interaction in Fig. 6, it is found that the H1 and H2 gaps are originated at the intersection between S(3pz) and C(2pz) bands. Moreover, θ = 0.0° shows a band repulsion with an energy gap of ΔE = 0.665 eV, where S(3pz) and C(2pz) orbitals are combined and split into two bands, where the one pushed down shows more a S(3pz) component, while the one pushed up has more C(2pz) character. In contrast, when θ = 10.9, 19.1, and 30.0° the corresponding split band that is pushed down has more C(2pz) character than the one pushed up with more S(3pz) orbitals, and the energy gap are ΔE = 0.674, 0.800, and 0.838 eV, respectively. By inspecting the spectral weights, Wk, it is found that the predominance of C(2pz) orbitals decrease as the compression strain increases.

All mini-gaps labeled as Cn arise from avoided crossing between C(2pz) and S(3pz)+W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) bands, according to left-hand side panels in Fig. 6 for the isolated G and WS2 layers. Some Cn mini-gaps exhibit a well-defined opening, particularly those bands that show a large spectral weight, Wk. For instance, mini-gaps labeled as C2 in θ = 0.0, 19.1°, and 30.0°, C6 in θ = 0.0°, C9 in θ = 10.9°, or C10 in θ = 19.1°, all of they have predominance of S(3pz) orbitals. In contrast, bands with a small Wk, like the ones of S(3pz)+W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) and energies close to EF, open tiny gaps in the C(2pz) band in the four cases. These gaps are labeled as C1. In particular, Fig. 6a, b show that this band, where W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) dominates, has a very small Wk, being quite imperceptible for the eye. However, our results reveal that Wk increases at this region for the cases with θ = 19.1 and 30.0°, as seen in Fig. 6c, d, where S(3pz) orbitals dominate instead of W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)). In agreement, the right-hand side panels in Fig. 6 show more well-defined C1 gaps as Wk increases. Notice that the band dispersion of S(3pz)+W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)) changes as a function of the twist-angle, shifting the energy where C1 occurs as a function of θ, such that the position of C1 mini-gap decreases in energy from ~−1.0 to ~−2.3 eV as the twist-angle increases from θ = 0.0–30.0° despite the strain effect on G, as we show in the Supplementary Information. In conclusion, the larger weights of S(3pz)+W(5d\({}_{{{{{\rm{z}}}}}^{2}}\)), the C1 mini-gaps are lower energies, indeed, both effects of the interlayer rotation.

Although some avoided crossings nature is directly identified by doing a similar analysis of the electronic bands coming from the projected unit cells, as we did above, there are some mini-gaps whose origin is not discovered in this way. Let us analyze these mini-gaps, which are pointed with pink arrows in Fig. 6 and labeled as Rn, since they are associated with band replicas originated by the moiré superpotential. To gain a deeper insight into their origin, we analyze in Fig. 7 the bands replicas of C(2pz) (green dots) and S(3pz) (orange dots) at each Rn for the cases with θ = 10.9 and 19.1°. Notice that these electronic states, in green and orange dots, do not belong to the projected unit cell but to the replicas. Let us analyze the θ = 10.9° case in detail. From Fig. 6b, b′, the R3, R4, and R5 mini-gaps on EEBG (red dots) clearly lack of avoided crossing with EEB\({}_{{{{{\rm{WS}}}}}_{2}}\) (blue dots). In Fig. 7a, R3 emerges from a crossing with a WS2 band replica with a small S(3pz) character; R4 does from a WS2 band replica with a well-defined S(3pz) character; and the R5 comes from the crossing with a C(2pz) band replica. In contrast, large band deviations observed in R1 and R2 mini-gaps suggest another kind of band interaction together with avoided crossing. For both mini-gaps, the presence of C(2pz) and S(3pz) replicas is observed. We find similar results for the θ = 19.1° case. Here, R1 and R3 emerge from C(2pz) replicas; R4 does from a crossing with a S(3pz) replica; while R2 and R5 show a joint effect composed by a direct avoided crossing and that with C(2pz) replicas.

Fig. 7: Effective electronic bands and orbital analysis around Rn mini-gaps for the G-WS2 at θ as the twist-angle.
figure 7

θ = a 10.9° and b 19.1°. C(2pz) and S(3pz) EEB are shown by red and blue dots. Green and orange dots display C(2pz) and S(3pz) contributions of bands replicas.

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From the above analysis, we infer that all Rn mini-gaps for θ = 0.0 and 30.0°, shown in Fig. 6(a’) and (d’), respectively, emerge from a crossing between C(2pz) and its replica or S(3pz) replicas. Following the band behavior, we find that R5 in θ = 0.0 and R1 for 30.0°, emerge from avoided crossing with S(3pz) and C(2pz) replicas, as it was noticed in R1 for θ = 10.9° and R5 for θ = 19.1°. Although the strain condition imposed in G to obey commensurability, influences on its band dispersion, some k − paths in the θ = 0.0° case are equivalent for θ = 30.0°, such that most of mini-gaps also are. For instance, R2, C4, C5, and C6 along the Γ−M path for the θ = 30.0° case (see Fig. 6d′) are equivalent to R1, C2, C3, and C4 shown in the \(\frac{4}{5}\Gamma -\)K path of θ = 0.0°, as seen in Fig. 6a′. Similarly, R1 and C2 along the \(\frac{5}{6}\Gamma -\)K path of θ = 30.0° coincide with R5 and C6 on M−Γ path when θ = 0.0°. Such equivalence on these paths is confirmed by comparing the bands of the isolated WS2 layer, shown in Fig. 6a, d.

Finally, we show how these mini-gaps relate the cone replicas to the mini Brillouin zones. Figure 5a shows the G-WS2 reciprocal space for θ = 0.0°. Despite that the four G-WS2 systems exhibit two crossings with mini Brillouin zone borders over the Γ−K path of the G Brillouin zone and only one over K−M and M−Γ paths, respectively, the closeness and position of cone replicas over the Γ−K−M−Γ path change respect to the twist angle, θ, suggesting a control of the band gaps. In principle, each crossing border by a mini Brillouin zone, or the closeness to a cone replica, might induce a Rn mini-gap. For instance, R1 and R2 when θ = 0.0° coincide with crossings of mini Brillouin zone borders.

In conclusion, the relative twisted angle in bilayer heterostructure plays a relevant role in the appearance of band gaps and in controlling their energy values and sizes. These minigaps mainly originate from the repulsion between electronic states belonging to both layers, while Bragg diffractions induce only the slightest band deviations that are mostly imperceivable. However, avoided crossings between out-of-plane orbitals belonging to different layers open well-defined band gaps. Using an unfolding methodology to clear the supercell crowed bands and assign a spectral weight, we found that the gap size and band perturbations depend on the interactions revealed with the amount of the weight. It is worth mentioning that a specific control on the mini-gaps energy associated with the twisted angle can be mainly achieved along Γ−K or M−Γ paths. Besides avoided crossings, the superperiodic potential associated with the moiré pattern opens well-defined gaps on graphene or WS2 band replicas. Other van der Waals heterostructures are expected to behave similarly, like WSe2 systems, where minigaps might be localized at similar positions in the BZ but show a different gap size and energy position. However, detailed work must be done. Finally, we remark that the disclosure of mini-gaps and band distortions nature is only possible through an unfolding method to project the supercell wave functions onto each unit cell of its constituents.

Methods

Initial structures

We started by optimizing the structural parameters of the pristine G and WS2 hexagonal unit cells, giving aG = 2.44 Å, and \({a}_{{{{{\rm{WS}}}}}_{2}}=3.16\) Å, for corresponding lattice constants. The average distances between C–C, W–S, and Sup–Sdown atoms are 1.41 Å, 2.41 Å, and 3.16 Å, respectively, also the average interlayer distance is 3.29 Å, which are all in agreement with experimental and theoretical reports48,49. With optimized lattice parameters, we constructed vertically stacked G-WS2 2D–vdW–h composed by one G sheet and one WS2. Although both unit cells have hexagonal symmetry, lattice constants are different. So, G-WS2 2D–vdW–h must be set up by a supercell (SC). The use of SCs in electronic structure calculations has been a common practice due to the ongoing increase in computational resources. This has broadening the field of first-principles calculations towards systems with hundreds or thousands atoms, different composition, perturbations (like impurities), or combined systems such as vertically stacked 2D–vdW–h. In general, to generate a new structure starting from the original unit cell an affine transformation is involved, which consists of two parts: a linear part and a shift of the origin. The linear part implies a change of orientation or length or both of the basis vectors a1, a2, a3, which transform by a matrix P to a new basis vectors set \({{{{\bf{a}}}}}_{{{{\bf{1}}}}}^{{\prime} }\), \({{{{\bf{a}}}}}_{{{{\bf{2}}}}}^{{\prime} }\), \({{{{\bf{a}}}}}_{{{{\bf{3}}}}}^{{\prime} }\), i.e.:

$$\left({{{{\bf{a}}}}}_{{{{\bf{1}}}}}^{{\prime} },{{{{\bf{a}}}}}_{{{{\bf{2}}}}}^{{\prime} },{{{{\bf{a}}}}}_{{{{\bf{3}}}}}^{{\prime} }\right)=\left({{{{\bf{a}}}}}_{{{{\bf{1}}}}},{{{{\bf{a}}}}}_{{{{\bf{2}}}}},{{{{\bf{a}}}}}_{{{{\bf{3}}}}}\right){{{\bf{P}}}}=\left({{{{\bf{a}}}}}_{{{{\bf{1}}}}},{{{{\bf{a}}}}}_{{{{\bf{2}}}}},{{{{\bf{a}}}}}_{{{{\bf{3}}}}}\right)\left[\begin{array}{ccc}{P}_{11}&{P}_{12}&{P}_{13}\\ {P}_{21}&{P}_{22}&{P}_{23}\\ {P}_{31}&{P}_{32}&{P}_{33}\end{array}\right]$$
(1)

In two dimensions, the new basis vectors set \({{{{\bf{a}}}}}_{{{{\bf{1}}}}}^{{\prime} }\), \({{{{\bf{a}}}}}_{{{{\bf{2}}}}}^{{\prime} }\), \({{{{\bf{a}}}}}_{{{{\bf{3}}}}}^{{\prime} }\) can be considered as a three dimensional transformation with invariant a3 axis50. Thus, P33 = 1 and P13 = P23 = P31 = P32 = 0. Hence, given two integer indices (m, n), each layer is constructed by a matrix transformation P acting on its corresponding basis vector, such that P is now:

$${{{\bf{P}}}}=\left[\begin{array}{ccc}m&n&0\\ -n&m+n&0\\ 0&0&1\end{array}\right]$$
(2)

For each G-WS2 2D–vdW–h studied here, it is required a couple of integer indices (m,n) and (p,q) for G and WS2 layer, respectively, such that m ≥ n and p ≥ q. Because of \({a}_{{{{\rm{G}}}}}/{a}_{{{{{\rm{WS}}}}}_{2}}\) is not an integer number, we have imposed the condition that G layers have to be stretched or compressed to assure commensurability, while WS2 remains its optimized lattice parameter. Hence, (m,n) and (p,q) must be chosen in such way that each layer obey commensurability. The stretching/compressing condition used in G layers induces a slight strain, Δ, which is obtained through26:

$$\Delta =\frac{{a}_{{{{\rm{G}}}}}\sqrt{{m}^{2}+{n}^{2}+mn}-{a}_{{{{{\rm{WS}}}}}_{2}}\sqrt{{p}^{2}+{q}^{2}+pq}}{{a}_{{{{{\rm{WS}}}}}_{2}}\sqrt{{p}^{2}+{q}^{2}+pq}}.$$
(3)

The rotation angle (θ) between layers and the total number of atoms (Nt) in the G-WS2 2D–vdW–h are also defined by the (m,n) and (p,q) indices26:

$$\theta =\frac{mp+nq+\left(mq+np/2\right)}{\sqrt{{m}^{2}+{n}^{2}+mn}\times \sqrt{{p}^{2}+{q}^{2}+pq}},$$
(4)
$${N}_{t}=3\left({m}^{2}+{n}^{2}+mn\right)+2\left({p}^{2}+{q}^{2}+pq\right).$$
(5)

Commonly, supercells are expressed as (u1 × u2), a notation obtained in terms of the magnitude of the projected lattice vectors of each SC into the lattice vectors of the pristine unit cell:

$${u}_{1}=\frac{| {{{{{\bf{a}}}}}_{{{{\bf{1}}}}}}_{,{{{\rm{SC}}}}}\cdot {{{{{\bf{a}}}}}_{{{{\bf{1}}}}}}_{,{{{\rm{X}}}}}| }{{a}_{{{{\rm{X}}}}}},\quad \,{{\mbox{and}}}\,\quad {u}_{2}=\frac{| {{{{{\bf{a}}}}}_{{{{\bf{2}}}}}}_{,{{{\rm{SC}}}}}\cdot {{{{{\bf{a}}}}}_{{{{\bf{2}}}}}}_{,{{{\rm{X}}}}}| }{{a}_{{{{\rm{X}}}}}},$$
(6)

where subindex X denotes G or WS2 layer. In this work, we set up four G-WS2 2D–vdW–h with different twisted-angle, θ = 0.0°, 10.9°, 19.1°, and 30.0°. We use θ as the label of each heterostructure. Table 2 resumes the (u1 × u2) notation for G and WS2 layers, (m,n) and (p,q) indices, Nt, θ, and Δ. We remark that induced strain in G layer is smaller than 4%.

Table 2 Structural parameters of the four G-WS2 2D–vdW–h studied here.
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Computational details

Total energy DFT calculations were performed using the computational code SIESTA36. The exchange-correlation energy was described by the local density approximation (LDA) functional51. The electron-ion interaction was treated with norm-conserving pseudopotentials52 in their scalar relativistic and non-local form53. For the expansion of the electronic states, a linear combination of pseudoatomic orbitals (LCAO) was employed, with an optimized double-ζ polarized (DZP) basis set plus diffusive orbitals54. A mesh-cutoff energy of 300 Ry is applied to sample the electronic density in the real space. The electronic relaxation was converged to 10−5. The Monkhorst–Pack scheme55 was used for the Brillouin zone sampling with an optimal 21 × 21 × 1 k-grid. A Fermi-Dirac smearing of 1 K is adopted to get convergence. The structural optimization was carried out with the conjugate gradient algorithm with a maximum value in the interatomic forces of 0.01 meV/Å, together with a value of 0.5 GPa in the stress components. Energy cutoff and k-grid convergence tests were performed on the four systems leading to the values as mentioned earlier as the optimal ones. To avoid spurious interactions, a vacuum space of 55 Å between G and the bottom S layer in the next supercell is applied as well as a slab dipole correction associated with the vertical asymmetry of the heterostructure56. The zigzag directions of each system are taken as reference for the angle between G and WS226. Since both G and WS2 systems have a hexagonal structure, the maximum limit of the rotation angle is 30°. Figure 8a–d shows the four G-WS2 2D–vdW–h studied here. Visualization of atomic models are performed using the VESTA program57.

Fig. 8: Top view of ball-stick models of the four optimized 3 × 3 G-WS2 supercell at θ as the twist-angle.
figure 8

θ = a 0.0°, b 10.9°, c 19.1°, and d 30.0°. Each supercell is marked by blue line.

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