Figure 1G presents Γ-K slices showing the important features within 4 eV of the Fermi level for the 1L, 2L, and bulk WSe 2 regions, along with their second derivatives. All features of the upper bands are well resolved. The spectra are consistent with expectations based on the literature ( 23 ), and density functional theory (DFT, overlaid red dashed lines) reproduces the upper valence band well, with no adjustable parameters other than an energy offset chosen to match the uppermost measured band at Γ. The bands near K are almost unchanged from monolayer to bulk ( 22 , 24 ) because of their in-plane orbital character (W 5d xy and 5 ), and in the monolayer ( 23 , 25 ), the valence band edge is at K. On the other hand, there are strong hybridization effects on the bands near Γ because of their out-of-plane orbital character (Se 4p z and W 5 ). In the bilayer and the bulk, the valence band splits at Γ with a higher-mass band 0.25 eV below that in the monolayer and a lower-mass band that is 0.50 eV higher. In the bilayer, the valence band edge is still at K, whereas in the bulk, it moves to Γ.

Figure 1C shows momentum-integrated spectra taken at points in each region of the WSe 2 flake. The highest intensity peak shifts downward monotonically in energy as the number of layers increases. A SPEM map of the peak energy versus location ( Fig. 1D ) therefore shows contrast between the 1L, 2L, and bulk regions. All spectra were highly consistent within each region, with no spatial variations that would signal fixed charges from contamination or in the substrate, and no drift due to charging resulting from photoemission was detected. From momentum-resolved energy slices, we could determine the orientations of the WSe 2 flake, graphene cap, and graphite support (fig. S2). Figure 1E shows a momentum slice through the graphene K point in the 1L region. The Dirac point energy E D coincides with the Fermi level E F (red dotted line) to within the measurement accuracy of <50 meV, implying minimal charge transfer between WSe 2 and graphene or doping of other origin. This, in turn, implies that there is no significant density of defect states in the gap of the WSe 2 . Figure 1F shows the second derivative of a momentum slice along Γ-K(WSe 2 ) in the 1L region. The valence band of the capping graphene is marked by a white dotted curve. It hybridizes with the WSe 2 bands, producing avoided crossings (white arrows) similar to those seen in graphene on MoS 2 ( 18 ). These features are >3 eV below E F , and the important WSe 2 bands nearer E F ( 22 ) are not affected.

( A ) Optical image and ( B ) schematic cross section of an exfoliated WSe 2 flake with monolayer (1L), bilayer (2L), and bulk regions partially capped with monolayer graphene (G) and supported by a graphite flake on a doped silicon substrate. ( C ) Angle-integrated spectra from each region in (A). ( D ) Map of the energy of peak emission, showing contrast between 1L, 2L, and bulk regions. ( E ) Momentum slice through the graphene K point, showing that E F is at the Dirac point. ( F ) Momentum slice along Γ − K (WSe 2 ) in the 1L region. The intensity is twice-differentiated with respect to energy. Avoided crossings between the graphene valence band (white dotted line) and the monolayer WSe 2 bands are indicated by white arrows. ( G ) Momentum slice of unprocessed (top) and twice-differentiated ARPES (bottom) along Γ − K (WSe 2 ) in the 1L (left), 2L (middle), and bulk (right) regions. Below is the intensity twice-differentiated with respect to energy with overlaid DFT calculation (red dashed lines).

To illustrate our approach and demonstrate its effectiveness, we first studied the effect of hybridization between monolayers of WSe 2 . The optical image ( Fig. 1A ) shows an exfoliated WSe 2 flake that naturally has monolayer (1L), bilayer (2L), and multilayer (bulk) regions; their boundaries are indicated by red dashed lines. Figure 1B is a schematic cross section. The flake is partly capped by a graphene monolayer (G), outlined by a black dashed line, which is essential for the sample to be annealed at 400°C in high vacuum to remove surface contamination without degrading the TMD beneath it. It rests on a thin graphite flake exfoliated directly onto a p-doped silicon chip that serves as an atomically flat conducting substrate (fig. S1). Contamination that is trapped between the layers during transfer collects in blisters, which consolidate upon annealing, leaving the remainder of the interfaces atomically clean ( 21 ). The sample is located by scanning photoemission microscopy (SPEM) using an approximately 1-μm beam spot at 74 eV photon energy (see Materials and Methods).

MoSe 2 /WSe 2 heterostructures

We now turn to the central object of our study, semiconductor heterobilayers. Figure 2A is an optical image of a sample with a MoSe 2 monolayer (green dashed line) partially overlapping a WSe 2 monolayer (red dashed line), forming a heterobilayer region (H) (blue dashed line). The monolayers were aligned during transfer by identifying the crystal axes using polarization-resolved second-harmonic generation (fig. S3) (26–28). As before, we included a protecting graphene cap and a graphite support. Figure 2B shows angle-integrated photoemission spectra from one point in each region. The largest peak is ~200 meV lower in the MoSe 2 monolayer than in the WSe 2 monolayer, whereas in the H region, there are two peaks that are shifted relative to the monolayer peaks. As a result, a map of the energy where the intensity is highest versus position (Fig. 2C) shows contrast between monolayer and H regions. In constant-energy slices, the K points of the two monolayers coincide in momentum space (fig. S4), confirming a twist angle of less than 1° and consistent with lattice constants differing by <1%.

Fig. 2 Bands in a 2D heterostructure. (A) Optical image showing monolayer MoSe 2 and WSe 2 sheets, which overlap, with the MoSe 2 on top, in an aligned heterobilayer region (H). Their boundaries are indicated with color-coded dotted lines. (B) Angle-integrated spectra in each of the three regions. (C) Map of the energy of maximum emission. (D to F) Momentum slices along Γ − K in the three regions, (top) unprocessed and (bottom) twice-differentiated, with cartoons of the structures above. The superposed dashed colored lines are DFT calculations for the MoSe 2 monolayer (green), the WSe 2 monolayer (red), and the commensurate heterobilayer (blue). The graphene valence band is indicated by a white dotted line. The white dashes in the lower panel of (F) indicate the valence band maxima in the MoSe 2 and WSe 2 monolayers and hence the valence band offset. The white dashed lines in the upper panels of (D) to (F) mark the valence band maxima in the isolated MoSe 2 (M) and WSe 2 (W) monolayers and in the aligned heterobilayer (H). (G) A momentum slice near Γ in another heterobilayer intentionally misaligned by about 30°. Here, only two bands are seen, indicating that the third band near Γ in the aligned heterobilayer (F) arises from commensurate domains.

The variation in band structure across the heterojunction is seen in the Γ-K momentum slices in Fig. 2 (D to F) for 1L MoSe 2 , 1L WSe 2 , and the heterobilayer, respectively. The upper valence bands in the monolayer regions are again well matched by DFT (green and red dashed lines). The spin-orbit splitting at K is much smaller in the MoSe 2 than in the WSe 2 , and the valence band edge is substantially lower. In the heterobilayer, the bands near K are very similar to the bands in the monolayers, implying weak interlayer hybridization near K, as was the case for the WSe 2 homobilayer. On the other hand, the bands at Γ are substantially different from those in the monolayers, implying significant hybridization, again as in the WSe 2 homobilayer. Nevertheless, the valence band edge remains at K. This is important for the electrical and optical properties.

Interestingly, we clearly see three bands within 0.5 Å−1 of Γ, not just the two that would be expected from homogeneous hybridization of one band from each monolayer. We note, however, that the third band resembles the upper band in the WSe 2 homobilayer (Fig. 1G), in which the layers are perfectly commensurate, having the bulk 2H stacking. We also recall that when monolayers with mismatched lattice constants are stacked, elastic energy considerations will ensure that any commensurate domains have a finite size. This has been demonstrated for graphene on hBN (29). For zero twist angle, the scale of the domains is , where a is the lattice constant and δa is the difference. Here, this scale is ~100 nm, which is less than the x-ray spot size. The spectrum of the heterobilayer could thus be interpreted as a superposition of spectra from a mixture of incommensurate domains in which hybridization is weak and commensurate domains in which hybridization is similar to that in the homobilayer.

In support of this interpretation, DFT simulations of the commensurate heterobilayer reproduce the uppermost band at Γ (blue lines) (Fig. 2F) and the slightly downward shifted lower band. Adding the hybridized bands of the isolated MoSe 2 and WSe 2 monolayers (green and red lines, respectively) reproduces the three apparent bands in H fairly closely. The remaining small discrepancy can be accounted for by shifts on the order of 100 meV in the incommensurate case, roughly independent of twist angle (30), as predicted by linear-scaling DFT (fig. S5) (31). Additionally, in an intentionally misaligned (by ~30°) MoSe 2 /WSe 2 heterobilayer, where no commensuration is expected, we saw only two bands near Γ, as illustrated in Fig. 2G and fig. S6. The band shifts in the twisted heterobilayer are well matched by DFT predictions for incommensurate layers (fig. S7). Furthermore, in a sample with an aligned bilayer of MoSe 2 on a monolayer of WSe 2 , we observed four bands at Γ rather than three (fig. S8). The combined evidence that aligned heterobilayers are composed of mixtures of incommensurate and commensurate domains is therefore compelling.

The values of key parameters extracted from the μ-ARPES measurements are summarized in Fig. 3. They were consistent across multiple samples and showed no dependence on the orientation of the graphene cap or graphite substrate. The spin-orbit splitting Δ SO at K is 0.49 ± 0.03 eV in WSe 2 and 0.24 ± 0.03 eV in MoSe 2 , in agreement with the literature (23), as are the effective masses of holes at Γ and K. In the WSe 2 monolayer, we find E K − E Γ = 0.50 ± 0.03 eV, consistent with scanning tunneling spectroscopy results (32), and in the MoSe 2 monolayer, we find E K − E Γ = 0.44 ± 0.03 eV. We also record here the valence band width D, which is useful for comparison with band structure calculations (23). As is well known, in both monolayer species, the valence band edge is at K, whereas in the bulk, it is at Γ. In the heterobilayer, we find that the valence band edge is also at K and is higher than the maximum at Γ by 0.14 ± 0.03 eV. We measured a valence band offset (VBO) between the WSe 2 and MoSe 2 monolayers of Δ VBO = 0.30 ± 0.03 eV. Because the bands at Γ in H (Fig. 2F) align well with those in the separate monolayers, we infer that this value is an intrinsic parameter of the heterojunction and that any charge transfer between the layers has negligible effect on the measurement.

Fig. 3 Summary of measured band parameters. Left: Schematic showing the definitions of parameters applicable for monolayers and aligned bilayers. Solid lines signify measured quantities, and dotted lines denote DFT calculations. Main: Graphical illustration of the positions of homologous band edges and hybridization effects. In both 2L WSe 2 and heterobilayer MoSe 2 /WSe 2 , hybridization is almost undetectable at K (red) but much larger at Γ (black). Bottom: Table of quantities determined by fitting the μ-ARPES spectra shown in Figs. 1 and 2. Energies are from Lorentzian fits to the second-derivative curves. The effective masses, which are isotropic within the accuracy of the fits, are obtained from weighted parabolic fits to the above band positions in symmetric windows about K and Γ with widths of 0.08 Å− 1 and 0.15 Å− 1, respectively.

Because we cannot probe the conduction band and the single-particle gaps have not been established incontrovertibly, we show the conduction band edges at K (red dashed line) and Q (blue dashed line) calculated using DFT. Although DFT underestimates these energies, the predictions of variations within the family of materials and across the Brillouin zone are more reliable (23, 24). The conduction band edge in H is predicted to remain at the K point, which, together with our measurements, implies that the band gap in H is direct.